LMFDB - Embedded newform 168.2.ba.b.5.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,2,Mod(5,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))

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

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

chi := DirichletCharacter("168.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	168.ba (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$1.34148675396$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			5.1
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	168.5
Dual form			168.2.ba.b.101.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 1.22474i) q^{2} +(1.50000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(-3.62132 - 2.09077i) q^{5} +2.44949i q^{6} +(1.62132 - 2.09077i) q^{7} +2.82843 q^{8} +(1.50000 - 2.59808i) q^{9} +O(q^{10})$	\(q+(-0.707107 + 1.22474i) q^{2} +(1.50000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(-3.62132 - 2.09077i) q^{5} +2.44949i q^{6} +(1.62132 - 2.09077i) q^{7} +2.82843 q^{8} +(1.50000 - 2.59808i) q^{9} +(5.12132 - 2.95680i) q^{10} +(0.0857864 + 0.148586i) q^{11} +(-3.00000 - 1.73205i) q^{12} +(1.41421 + 3.46410i) q^{14} -7.24264 q^{15} +(-2.00000 + 3.46410i) q^{16} +(2.12132 + 3.67423i) q^{18} +8.36308i q^{20} +(0.621320 - 4.54026i) q^{21} -0.242641 q^{22} +(4.24264 - 2.44949i) q^{24} +(6.24264 + 10.8126i) q^{25} -5.19615i q^{27} +(-5.24264 - 0.717439i) q^{28} +10.4142 q^{29} +(5.12132 - 8.87039i) q^{30} +(-5.37868 + 3.10538i) q^{31} +(-2.82843 - 4.89898i) q^{32} +(0.257359 + 0.148586i) q^{33} +(-10.2426 + 4.18154i) q^{35} -6.00000 q^{36} +(-10.2426 - 5.91359i) q^{40} +(5.12132 + 3.97141i) q^{42} +(0.171573 - 0.297173i) q^{44} +(-10.8640 + 6.27231i) q^{45} +6.92820i q^{48} +(-1.74264 - 6.77962i) q^{49} -17.6569 q^{50} +(5.03553 + 8.72180i) q^{53} +(6.36396 + 3.67423i) q^{54} -0.717439i q^{55} +(4.58579 - 5.91359i) q^{56} +(-7.36396 + 12.7548i) q^{58} +(3.98528 - 2.30090i) q^{59} +(7.24264 + 12.5446i) q^{60} -8.78335i q^{62} +(-3.00000 - 7.34847i) q^{63} +8.00000 q^{64} +(-0.363961 + 0.210133i) q^{66} +(2.12132 - 15.5014i) q^{70} +(4.24264 - 7.34847i) q^{72} +(8.48528 - 4.89898i) q^{73} +(18.7279 + 10.8126i) q^{75} +(0.449747 + 0.0615465i) q^{77} +(-3.86396 + 6.69258i) q^{79} +(14.4853 - 8.36308i) q^{80} +(-4.50000 - 7.79423i) q^{81} -3.76127i q^{83} +(-8.48528 + 3.46410i) q^{84} +(15.6213 - 9.01897i) q^{87} +(0.242641 + 0.420266i) q^{88} -17.7408i q^{90} +(-5.37868 + 9.31615i) q^{93} +(-8.48528 - 4.89898i) q^{96} +11.5300i q^{97} +(9.53553 + 2.65962i) q^{98} +0.514719 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{3} - 4 q^{4} - 6 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q + 6 * q^3 - 4 * q^4 - 6 * q^5 - 2 * q^7 + 6 * q^9	$ 4 q + 6 q^{3} - 4 q^{4} - 6 q^{5} - 2 q^{7} + 6 q^{9} + 12 q^{10} + 6 q^{11} - 12 q^{12} - 12 q^{15} - 8 q^{16} - 6 q^{21} + 16 q^{22} + 8 q^{25} - 4 q^{28} + 36 q^{29} + 12 q^{30} - 30 q^{31} + 18 q^{33} - 24 q^{35} - 24 q^{36} - 24 q^{40} + 12 q^{42} + 12 q^{44} - 18 q^{45} + 10 q^{49} - 48 q^{50} + 6 q^{53} + 24 q^{56} - 4 q^{58} - 18 q^{59} + 12 q^{60} - 12 q^{63} + 32 q^{64} + 24 q^{66} + 24 q^{75} - 18 q^{77} + 10 q^{79} + 24 q^{80} - 18 q^{81} + 54 q^{87} - 16 q^{88} - 30 q^{93} + 24 q^{98} + 36 q^{99}+O(q^{100}) $ 4 * q + 6 * q^3 - 4 * q^4 - 6 * q^5 - 2 * q^7 + 6 * q^9 + 12 * q^10 + 6 * q^11 - 12 * q^12 - 12 * q^15 - 8 * q^16 - 6 * q^21 + 16 * q^22 + 8 * q^25 - 4 * q^28 + 36 * q^29 + 12 * q^30 - 30 * q^31 + 18 * q^33 - 24 * q^35 - 24 * q^36 - 24 * q^40 + 12 * q^42 + 12 * q^44 - 18 * q^45 + 10 * q^49 - 48 * q^50 + 6 * q^53 + 24 * q^56 - 4 * q^58 - 18 * q^59 + 12 * q^60 - 12 * q^63 + 32 * q^64 + 24 * q^66 + 24 * q^75 - 18 * q^77 + 10 * q^79 + 24 * q^80 - 18 * q^81 + 54 * q^87 - 16 * q^88 - 30 * q^93 + 24 * q^98 + 36 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$-1$	$1$

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 $

$2$	−0.707107	+	1.22474i	−0.500000	+	0.866025i
$2$	−0.707107	+	1.22474i	−0.500000	+	0.866025i
$3$	1.50000	−	0.866025i	0.866025	−	0.500000i
$3$	1.50000	−	0.866025i	0.866025	−	0.500000i
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$	−3.62132	−	2.09077i	−1.61950	−	0.935021i	−0.987048	−	0.160424i	$-0.948714\pi$
$5$	−3.62132	−	2.09077i	−1.61950	−	0.935021i	−0.632456	−	0.774597i	$-0.717953\pi$
$6$			2.44949i			1.00000i
$7$	1.62132	−	2.09077i	0.612801	−	0.790237i
$7$	1.62132	−	2.09077i	0.612801	−	0.790237i
$8$	2.82843			1.00000
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$	5.12132	−	2.95680i	1.61950	−	0.935021i
$11$	0.0857864	+	0.148586i	0.0258656	+	0.0448005i	0.878668	−	0.477432i	$-0.158432\pi$
$11$	0.0857864	+	0.148586i	0.0258656	+	0.0448005i	−0.852803	+	0.522233i	$0.825099\pi$
$12$	−3.00000	−	1.73205i	−0.866025	−	0.500000i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	1.41421	+	3.46410i	0.377964	+	0.925820i
$15$	−7.24264			−1.87004
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$	2.12132	+	3.67423i	0.500000	+	0.866025i
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$			8.36308i			1.87004i
$21$	0.621320	−	4.54026i	0.135583	−	0.990766i
$22$	−0.242641			−0.0517312
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	4.24264	−	2.44949i	0.866025	−	0.500000i
$25$	6.24264	+	10.8126i	1.24853	+	2.16251i
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$	−5.24264	−	0.717439i	−0.990766	−	0.135583i
$29$	10.4142			1.93387			0.966935	−	0.255021i	$-0.0820825\pi$
$29$	10.4142			1.93387			0.966935	+	0.255021i	$0.0820825\pi$
$30$	5.12132	−	8.87039i	0.935021	−	1.61950i
$31$	−5.37868	+	3.10538i	−0.966039	+	0.557743i	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−5.37868	+	3.10538i	−0.966039	+	0.557743i	−0.0680129	+	0.997684i	$0.521666\pi$
$32$	−2.82843	−	4.89898i	−0.500000	−	0.866025i
$33$	0.257359	+	0.148586i	0.0448005	+	0.0258656i
$34$		0			0
$35$	−10.2426	+	4.18154i	−1.73132	+	0.706809i
$36$	−6.00000			−1.00000
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$		0			0
$40$	−10.2426	−	5.91359i	−1.61950	−	0.935021i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	5.12132	+	3.97141i	0.790237	+	0.612801i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	0.171573	−	0.297173i	0.0258656	−	0.0448005i
$45$	−10.8640	+	6.27231i	−1.61950	+	0.935021i
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$			6.92820i			1.00000i
$49$	−1.74264	−	6.77962i	−0.248949	−	0.968517i
$50$	−17.6569			−2.49706
$51$		0			0
$52$		0			0
$53$	5.03553	+	8.72180i	0.691684	+	1.19803i	0.971286	+	0.237915i	$0.0764641\pi$
$53$	5.03553	+	8.72180i	0.691684	+	1.19803i	−0.279602	+	0.960116i	$0.590203\pi$
$54$	6.36396	+	3.67423i	0.866025	+	0.500000i
$55$		−	0.717439i		−	0.0967394i
$56$	4.58579	−	5.91359i	0.612801	−	0.790237i
$57$		0			0
$58$	−7.36396	+	12.7548i	−0.966935	+	1.67478i
$59$	3.98528	−	2.30090i	0.518839	−	0.299552i	−0.217620	−	0.976034i	$-0.569829\pi$
$59$	3.98528	−	2.30090i	0.518839	−	0.299552i	0.736460	+	0.676481i	$0.236496\pi$
$60$	7.24264	+	12.5446i	0.935021	+	1.61950i
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		−	8.78335i		−	1.11549i
$63$	−3.00000	−	7.34847i	−0.377964	−	0.925820i
$64$	8.00000			1.00000
$65$		0			0
$66$	−0.363961	+	0.210133i	−0.0448005	+	0.0258656i
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$	2.12132	−	15.5014i	0.253546	−	1.85277i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	4.24264	−	7.34847i	0.500000	−	0.866025i
$73$	8.48528	−	4.89898i	0.993127	−	0.573382i	0.0869195	−	0.996215i	$-0.472298\pi$
$73$	8.48528	−	4.89898i	0.993127	−	0.573382i	0.906208	+	0.422833i	$0.138964\pi$
$74$		0			0
$75$	18.7279	+	10.8126i	2.16251	+	1.24853i
$76$		0			0
$77$	0.449747	+	0.0615465i	0.0512535	+	0.00701388i
$78$		0			0
$79$	−3.86396	+	6.69258i	−0.434730	+	0.752974i	−0.997274	−	0.0737937i	$-0.976489\pi$
$79$	−3.86396	+	6.69258i	−0.434730	+	0.752974i	0.562544	+	0.826767i	$0.309823\pi$
$80$	14.4853	−	8.36308i	1.61950	−	0.935021i
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$		−	3.76127i		−	0.412854i	−0.978462	−	0.206427i	$-0.933816\pi$
$83$		−	3.76127i		−	0.412854i	0.978462	−	0.206427i	$-0.0661835\pi$
$84$	−8.48528	+	3.46410i	−0.925820	+	0.377964i
$85$		0			0
$86$		0			0
$87$	15.6213	−	9.01897i	1.67478	−	0.966935i
$88$	0.242641	+	0.420266i	0.0258656	+	0.0448005i
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		−	17.7408i		−	1.87004i
$91$		0			0
$92$		0			0
$93$	−5.37868	+	9.31615i	−0.557743	+	0.966039i
$94$		0			0
$95$		0			0
$96$	−8.48528	−	4.89898i	−0.866025	−	0.500000i
$97$			11.5300i			1.17070i	0.810782	+	0.585348i	$0.199042\pi$
$97$			11.5300i			1.17070i	−0.810782	+	0.585348i	$0.800958\pi$
$98$	9.53553	+	2.65962i	0.963234	+	0.268662i
$99$	0.514719			0.0517312
$100$	12.4853	−	21.6251i	1.24853	−	2.16251i
$101$	3.00000	−	1.73205i	0.298511	−	0.172345i	−0.343263	−	0.939239i	$-0.611532\pi$
$101$	3.00000	−	1.73205i	0.298511	−	0.172345i	0.641774	+	0.766894i	$0.278199\pi$
$102$		0			0
$103$	12.7279	+	7.34847i	1.25412	+	0.724066i	0.971925	−	0.235291i	$-0.0756043\pi$
$103$	12.7279	+	7.34847i	1.25412	+	0.724066i	0.282194	+	0.959357i	$0.408938\pi$
$104$		0			0
$105$	−11.7426	+	15.1427i	−1.14596	+	1.47778i
$106$	−14.2426			−1.38337
$107$	−10.3284	+	17.8894i	−0.998487	+	1.72943i	−0.451618	+	0.892211i	$0.649153\pi$
$107$	−10.3284	+	17.8894i	−0.998487	+	1.72943i	−0.546869	+	0.837218i	$0.684180\pi$
$108$	−9.00000	+	5.19615i	−0.866025	+	0.500000i
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$	0.878680	+	0.507306i	0.0837788	+	0.0483697i
$111$		0			0
$112$	4.00000	+	9.79796i	0.377964	+	0.925820i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$	−10.4142	−	18.0379i	−0.966935	−	1.67478i
$117$		0			0
$118$			6.50794i			0.599104i
$119$		0			0
$120$	−20.4853			−1.87004
$121$	5.48528	−	9.50079i	0.498662	−	0.863708i
$122$		0			0
$123$		0			0
$124$	10.7574	+	6.21076i	0.966039	+	0.557743i
$125$		−	31.3000i		−	2.79956i
$126$	11.1213	+	1.52192i	0.990766	+	0.135583i
$127$	6.75736			0.599619			0.299809	−	0.953999i	$-0.403077\pi$
$127$	6.75736			0.599619			0.299809	+	0.953999i	$0.403077\pi$
$128$	−5.65685	+	9.79796i	−0.500000	+	0.866025i
$129$		0			0
$130$		0			0
$131$	−18.4706	−	10.6640i	−1.61378	−	0.931717i	−0.988483	−	0.151330i	$-0.951644\pi$
$131$	−18.4706	−	10.6640i	−1.61378	−	0.931717i	−0.625297	−	0.780387i	$-0.715022\pi$
$132$		−	0.594346i		−	0.0517312i
$133$		0			0
$134$		0			0
$135$	−10.8640	+	18.8169i	−0.935021	+	1.61950i
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$	17.4853	+	13.5592i	1.47778	+	1.14596i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	6.00000	+	10.3923i	0.500000	+	0.866025i
$145$	−37.7132	−	21.7737i	−3.13191	−	1.80821i
$146$			13.8564i			1.14676i
$147$	−8.48528	−	8.66025i	−0.699854	−	0.714286i
$148$		0			0
$149$	−1.41421	+	2.44949i	−0.115857	+	0.200670i	−0.918122	−	0.396298i	$-0.870295\pi$
$149$	−1.41421	+	2.44949i	−0.115857	+	0.200670i	0.802265	+	0.596968i	$0.203628\pi$
$150$	−26.4853	+	15.2913i	−2.16251	+	1.24853i
$151$	−11.1066	−	19.2372i	−0.903842	−	1.56550i	−0.822464	−	0.568818i	$-0.807401\pi$
$151$	−11.1066	−	19.2372i	−0.903842	−	1.56550i	−0.0813788	−	0.996683i	$-0.525932\pi$
$152$		0			0
$153$		0			0
$154$	−0.393398	+	0.507306i	−0.0317009	+	0.0408799i
$155$	25.9706			2.08601
$156$		0			0
$157$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$157$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$158$	−5.46447	−	9.46473i	−0.434730	−	0.752974i
$159$	15.1066	+	8.72180i	1.19803	+	0.691684i
$160$			23.6544i			1.87004i
$161$		0			0
$162$	12.7279			1.00000
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$	−0.621320	−	1.07616i	−0.0483697	−	0.0837788i
$166$	4.60660	+	2.65962i	0.357542	+	0.206427i
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$	1.75736	−	12.8418i	0.135583	−	0.990766i
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	15.0000	+	8.66025i	1.14043	+	0.658427i	0.946537	−	0.322596i	$-0.104555\pi$
$173$	15.0000	+	8.66025i	1.14043	+	0.658427i	0.193892	+	0.981023i	$0.437889\pi$
$174$			25.5095i			1.93387i
$175$	32.7279	+	4.47871i	2.47400	+	0.338559i
$176$	−0.686292			−0.0517312
$177$	3.98528	−	6.90271i	0.299552	−	0.518839i
$178$		0			0
$179$	−5.65685	−	9.79796i	−0.422813	−	0.732334i	0.573400	−	0.819275i	$-0.305624\pi$
$179$	−5.65685	−	9.79796i	−0.422813	−	0.732334i	−0.996213	+	0.0869415i	$0.972291\pi$
$180$	21.7279	+	12.5446i	1.61950	+	0.935021i
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$	−7.60660	−	13.1750i	−0.557743	−	0.966039i
$187$		0			0
$188$		0			0
$189$	−10.8640	−	8.42463i	−0.790237	−	0.612801i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	12.0000	−	6.92820i	0.866025	−	0.500000i
$193$	10.7426	+	18.6068i	0.773272	+	1.33935i	0.935760	+	0.352636i	$0.114715\pi$
$193$	10.7426	+	18.6068i	0.773272	+	1.33935i	−0.162488	+	0.986710i	$0.551952\pi$
$194$	−14.1213	−	8.15295i	−1.01385	−	0.585348i
$195$		0			0
$196$	−10.0000	+	9.79796i	−0.714286	+	0.699854i
$197$	−14.1421			−1.00759			−0.503793	−	0.863825i	$-0.668062\pi$
$197$	−14.1421			−1.00759			−0.503793	+	0.863825i	$0.668062\pi$
$198$	−0.363961	+	0.630399i	−0.0258656	+	0.0448005i
$199$	−21.2132	+	12.2474i	−1.50376	+	0.868199i	−0.503774	+	0.863836i	$0.668055\pi$
$199$	−21.2132	+	12.2474i	−1.50376	+	0.868199i	−0.999990	+	0.00436292i	$0.998611\pi$
$200$	17.6569	+	30.5826i	1.24853	+	2.16251i
$201$		0			0
$202$			4.89898i			0.344691i
$203$	16.8848	−	21.7737i	1.18508	−	1.52822i
$204$		0			0
$205$		0			0
$206$	−18.0000	+	10.3923i	−1.25412	+	0.724066i
$207$		0			0
$208$		0			0
$209$		0			0
$210$	−10.2426	−	25.0892i	−0.706809	−	1.73132i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	10.0711	−	17.4436i	0.691684	−	1.19803i
$213$		0			0
$214$	−14.6066	−	25.2994i	−0.998487	−	1.72943i
$215$		0			0
$216$		−	14.6969i		−	1.00000i
$217$	−2.22792	+	16.2804i	−0.151241	+	1.10519i
$218$		0			0
$219$	8.48528	−	14.6969i	0.573382	−	0.993127i
$220$	−1.24264	+	0.717439i	−0.0837788	+	0.0483697i
$221$		0			0
$222$		0			0
$223$		−	29.8651i		−	1.99992i	−0.00910984	−	0.999959i	$-0.502900\pi$
$223$		−	29.8651i		−	1.99992i	0.00910984	−	0.999959i	$-0.497100\pi$
$224$	−14.8284	−	2.02922i	−0.990766	−	0.135583i
$225$	37.4558			2.49706
$226$		0			0
$227$	−25.7132	+	14.8455i	−1.70665	+	0.985332i	−0.767999	+	0.640451i	$0.778747\pi$
$227$	−25.7132	+	14.8455i	−1.70665	+	0.985332i	−0.938647	+	0.344881i	$0.887919\pi$
$228$		0			0
$229$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$		0			0
$231$	0.727922	−	0.297173i	0.0478938	−	0.0195525i
$232$	29.4558			1.93387
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$	−7.97056	−	4.60181i	−0.518839	−	0.299552i
$237$			13.3852i			0.869459i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	14.4853	−	25.0892i	0.935021	−	1.61950i
$241$	−20.2279	+	11.6786i	−1.30300	+	0.752285i	−0.980917	−	0.194429i	$-0.937715\pi$
$241$	−20.2279	+	11.6786i	−1.30300	+	0.752285i	−0.322078	+	0.946713i	$0.604381\pi$
$242$	7.75736	+	13.4361i	0.498662	+	0.863708i
$243$	−13.5000	−	7.79423i	−0.866025	−	0.500000i
$244$		0			0
$245$	−7.86396	+	28.1946i	−0.502410	+	1.80129i
$246$		0			0
$247$		0			0
$248$	−15.2132	+	8.78335i	−0.966039	+	0.557743i
$249$	−3.25736	−	5.64191i	−0.206427	−	0.357542i
$250$	38.3345	+	22.1324i	2.42449	+	1.39978i
$251$			20.4874i			1.29316i	0.762848	+	0.646578i	$0.223800\pi$
$251$			20.4874i			1.29316i	−0.762848	+	0.646578i	$0.776200\pi$
$252$	−9.72792	+	12.5446i	−0.612801	+	0.790237i
$253$		0			0
$254$	−4.77817	+	8.27604i	−0.299809	+	0.519285i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	15.6213	−	27.0569i	0.966935	−	1.67478i
$262$	26.1213	−	15.0812i	1.61378	−	0.931717i
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$	0.727922	+	0.420266i	0.0448005	+	0.0258656i
$265$		−	42.1126i		−	2.58696i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.8345	−	10.8741i	1.14836	−	0.663007i	0.199874	−	0.979822i	$-0.435947\pi$
$269$	18.8345	−	10.8741i	1.14836	−	0.663007i	0.948487	+	0.316815i	$0.102613\pi$
$270$	−15.3640	−	26.6112i	−0.935021	−	1.61950i
$271$	−5.89340	−	3.40256i	−0.357998	−	0.206691i	0.310204	−	0.950670i	$-0.399603\pi$
$271$	−5.89340	−	3.40256i	−0.357998	−	0.206691i	−0.668202	+	0.743980i	$0.732936\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.07107	+	1.85514i	−0.0645878	+	0.111869i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$			18.6323i			1.11549i
$280$	−28.9706	+	11.8272i	−1.73132	+	0.706809i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$283$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−16.9706			−1.00000
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$	53.3345	−	30.7927i	3.13191	−	1.80821i
$291$	9.98528	+	17.2950i	0.585348	+	1.01385i
$292$	−16.9706	−	9.79796i	−0.993127	−	0.573382i
$293$			3.34101i			0.195184i	0.995227	+	0.0975919i	$0.0311140\pi$
$293$			3.34101i			0.195184i	−0.995227	+	0.0975919i	$0.968886\pi$
$294$	16.6066	−	4.26858i	0.968517	−	0.248949i
$295$	−19.2426			−1.12035
$296$		0			0
$297$	0.772078	−	0.445759i	0.0448005	−	0.0258656i
$298$	−2.00000	−	3.46410i	−0.115857	−	0.200670i
$299$		0			0
$300$		−	43.2503i		−	2.49706i
$301$		0			0
$302$	31.4142			1.80768
$303$	3.00000	−	5.19615i	0.172345	−	0.298511i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−0.343146	−	0.840532i	−0.0195525	−	0.0478938i
$309$	25.4558			1.44813
$310$	−18.3640	+	31.8073i	−1.04300	+	1.80653i
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	21.2574	+	12.2729i	1.20154	+	0.693708i	0.960897	−	0.276907i	$-0.0893093\pi$
$313$	21.2574	+	12.2729i	1.20154	+	0.693708i	0.240640	+	0.970614i	$0.422643\pi$
$314$		0			0
$315$	−4.50000	+	32.8835i	−0.253546	+	1.85277i
$316$	15.4558			0.869459
$317$	−15.2782	+	26.4626i	−0.858108	+	1.48629i	0.0156238	+	0.999878i	$0.495027\pi$
$317$	−15.2782	+	26.4626i	−0.858108	+	1.48629i	−0.873732	+	0.486408i	$0.838307\pi$
$318$	−21.3640	+	12.3345i	−1.19803	+	0.691684i
$319$	0.893398	+	1.54741i	0.0500207	+	0.0866384i
$320$	−28.9706	−	16.7262i	−1.61950	−	0.935021i
$321$			35.7787i			1.99697i
$322$		0			0
$323$		0			0
$324$	−9.00000	+	15.5885i	−0.500000	+	0.866025i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	1.75736			0.0967394
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	−6.51472	+	3.76127i	−0.357542	+	0.206427i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	14.4853	+	11.2328i	0.790237	+	0.612801i
$337$	36.4558			1.98588			0.992938	−	0.118633i	$-0.0378512\pi$
$337$	36.4558			1.98588			0.992938	+	0.118633i	$0.0378512\pi$
$338$	9.19239	−	15.9217i	0.500000	−	0.866025i
$339$		0			0
$340$		0			0
$341$	−0.922836	−	0.532799i	−0.0499743	−	0.0288527i
$342$		0			0
$343$	−17.0000	−	7.34847i	−0.917914	−	0.396780i
$344$		0			0
$345$		0			0
$346$	−21.2132	+	12.2474i	−1.14043	+	0.658427i
$347$	14.1421	+	24.4949i	0.759190	+	1.31495i	0.943264	+	0.332043i	$0.107738\pi$
$347$	14.1421	+	24.4949i	0.759190	+	1.31495i	−0.184075	+	0.982912i	$0.558929\pi$
$348$	−31.2426	−	18.0379i	−1.67478	−	0.966935i
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	−28.6274	+	36.9164i	−1.53020	+	1.97327i
$351$		0			0
$352$	0.485281	−	0.840532i	0.0258656	−	0.0448005i
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$	5.63604	+	9.76191i	0.299552	+	0.518839i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	16.0000			0.845626
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$	−30.7279	+	17.7408i	−1.61950	+	0.935021i
$361$	9.50000	+	16.4545i	0.500000	+	0.866025i
$362$		0			0
$363$		−	19.0016i		−	0.997324i
$364$		0			0
$365$	−40.9706			−2.14450
$366$		0			0
$367$	−26.3787	+	15.2297i	−1.37696	+	0.794986i	−0.991792	−	0.127862i	$-0.959188\pi$
$367$	−26.3787	+	15.2297i	−1.37696	+	0.794986i	−0.385164	+	0.922848i	$0.625855\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	26.3995	+	3.61269i	1.37059	+	0.187561i
$372$	21.5147			1.11549
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$	−27.1066	−	46.9500i	−1.39978	−	2.42449i
$376$		0			0
$377$		0			0
$378$	18.0000	−	7.34847i	0.925820	−	0.377964i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	10.1360	−	5.85204i	0.519285	−	0.299809i
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$			19.5959i			1.00000i
$385$	−1.50000	−	1.16320i	−0.0764471	−	0.0592821i
$386$	−30.3848			−1.54654
$387$		0			0
$388$	19.9706	−	11.5300i	1.01385	−	0.585348i
$389$	−15.5563	−	26.9444i	−0.788738	−	1.36613i	−0.926740	−	0.375703i	$-0.877401\pi$
$389$	−15.5563	−	26.9444i	−0.788738	−	1.36613i	0.138002	−	0.990432i	$-0.455932\pi$
$390$		0			0
$391$		0			0
$392$	−4.92893	−	19.1757i	−0.248949	−	0.968517i
$393$	−36.9411			−1.86343
$394$	10.0000	−	17.3205i	0.503793	−	0.872595i
$395$	27.9853	−	16.1573i	1.40809	−	0.812962i
$396$	−0.514719	−	0.891519i	−0.0258656	−	0.0448005i
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$		−	34.6410i		−	1.73640i
$399$		0			0
$400$	−49.9411			−2.49706
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$		0			0
$404$	−6.00000	−	3.46410i	−0.298511	−	0.172345i
$405$			37.6339i			1.87004i
$406$	14.7279	+	36.0759i	0.730934	+	1.79042i
$407$		0			0
$408$		0			0
$409$	9.47056	−	5.46783i	0.468289	−	0.270367i	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	9.47056	−	5.46783i	0.468289	−	0.270367i	0.715523	+	0.698589i	$0.246188\pi$
$410$		0			0
$411$		0			0
$412$		−	29.3939i		−	1.44813i
$413$	1.65076	−	12.0628i	0.0812285	−	0.593572i
$414$		0			0
$415$	−7.86396	+	13.6208i	−0.386027	+	0.668618i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	10.3923i		−	0.507697i	−0.967244	−	0.253849i	$-0.918303\pi$
$419$		−	10.3923i		−	0.507697i	0.967244	−	0.253849i	$-0.0816965\pi$
$420$	37.9706	+	5.19615i	1.85277	+	0.253546i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	14.2426	+	24.6690i	0.691684	+	1.19803i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	41.3137			1.99697
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	18.0000	+	10.3923i	0.866025	+	0.500000i
$433$			39.1918i			1.88344i	0.336399	+	0.941720i	$0.390791\pi$
$433$			39.1918i			1.88344i	−0.336399	+	0.941720i	$0.609209\pi$
$434$	−18.3640	−	14.2406i	−0.881498	−	0.683572i
$435$	−75.4264			−3.61642
$436$		0			0
$437$		0			0
$438$	12.0000	+	20.7846i	0.573382	+	0.993127i
$439$	36.1066	+	20.8462i	1.72327	+	0.994933i	0.911908	+	0.410394i	$0.134609\pi$
$439$	36.1066	+	20.8462i	1.72327	+	0.994933i	0.811366	+	0.584539i	$0.198725\pi$
$440$		−	2.02922i		−	0.0967394i
$441$	−20.2279	−	5.64191i	−0.963234	−	0.268662i
$442$		0			0
$443$	20.5711	−	35.6301i	0.977361	−	1.69284i	0.305448	−	0.952209i	$-0.401194\pi$
$443$	20.5711	−	35.6301i	0.977361	−	1.69284i	0.671913	−	0.740630i	$-0.265473\pi$
$444$		0			0
$445$		0			0
$446$	36.5772	+	21.1178i	1.73198	+	0.999959i
$447$			4.89898i			0.231714i
$448$	12.9706	−	16.7262i	0.612801	−	0.790237i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−26.4853	+	45.8739i	−1.24853	+	2.16251i
$451$		0			0
$452$		0			0
$453$	−33.3198	−	19.2372i	−1.56550	−	0.903842i
$454$		−	41.9895i		−	1.97066i
$455$		0			0
$456$		0			0
$457$	17.9853	−	31.1514i	0.841316	−	1.45720i	−0.0474665	−	0.998873i	$-0.515115\pi$
$457$	17.9853	−	31.1514i	0.841316	−	1.45720i	0.888783	−	0.458329i	$-0.151552\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			38.1051i			1.77473i	0.461065	+	0.887366i	$0.347467\pi$
$461$			38.1051i			1.77473i	−0.461065	+	0.887366i	$0.652533\pi$
$462$	−0.150758	+	1.10165i	−0.00701388	+	0.0512535i
$463$	−26.0000			−1.20832			−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000			−1.20832			−0.604161	+	0.796862i	$0.706492\pi$
$464$	−20.8284	+	36.0759i	−0.966935	+	1.67478i
$465$	38.9558	−	22.4912i	1.80653	−	1.04300i
$466$		0			0
$467$	15.0000	+	8.66025i	0.694117	+	0.400749i	0.805153	−	0.593068i	$-0.202083\pi$
$467$	15.0000	+	8.66025i	0.694117	+	0.400749i	−0.111035	+	0.993816i	$0.535417\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	11.2721	−	6.50794i	0.518839	−	0.299552i
$473$		0			0
$474$	−16.3934	−	9.46473i	−0.752974	−	0.434730i
$475$		0			0
$476$		0			0
$477$	30.2132			1.38337
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$	20.4853	+	35.4815i	0.935021	+	1.61950i
$481$		0			0
$482$		−	33.0321i		−	1.50457i
$483$		0			0
$484$	−21.9411			−0.997324
$485$	24.1066	−	41.7539i	1.09462	−	1.89595i
$486$	19.0919	−	11.0227i	0.866025	−	0.500000i
$487$	18.5919	+	32.2021i	0.842479	+	1.45922i	0.887793	+	0.460243i	$0.152238\pi$
$487$	18.5919	+	32.2021i	0.842479	+	1.45922i	−0.0453143	+	0.998973i	$0.514429\pi$
$488$		0			0
$489$		0			0
$490$	−28.9706	−	29.5680i	−1.30876	−	1.33574i
$491$	−21.6863			−0.978689			−0.489344	−	0.872091i	$-0.662764\pi$
$491$	−21.6863			−0.978689			−0.489344	+	0.872091i	$0.662764\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−1.86396	−	1.07616i	−0.0837788	−	0.0483697i
$496$		−	24.8431i		−	1.11549i
$497$		0			0
$498$	9.21320			0.412854
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	−54.2132	+	31.3000i	−2.42449	+	1.39978i
$501$		0			0
$502$	−25.0919	−	14.4868i	−1.11991	−	0.646578i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	−8.48528	−	20.7846i	−0.377964	−	0.925820i
$505$	−14.4853			−0.644587
$506$		0			0
$507$	−19.5000	+	11.2583i	−0.866025	+	0.500000i
$508$	−6.75736	−	11.7041i	−0.299809	−	0.519285i
$509$	17.3787	+	10.0336i	0.770296	+	0.444731i	0.832980	−	0.553303i	$-0.186633\pi$
$509$	17.3787	+	10.0336i	0.770296	+	0.444731i	−0.0626839	+	0.998033i	$0.519966\pi$
$510$		0			0
$511$	3.51472	−	25.6836i	0.155482	−	1.13618i
$512$	22.6274			1.00000
$513$		0			0
$514$		0			0
$515$	−30.7279	−	53.2223i	−1.35403	−	2.34526i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	30.0000			1.31685
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$	22.0919	+	38.2643i	0.966935	+	1.67478i
$523$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$523$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$524$			42.6559i			1.86343i
$525$	52.9706	−	21.6251i	2.31182	−	0.943799i
$526$		0			0
$527$		0			0
$528$	−1.02944	+	0.594346i	−0.0448005	+	0.0258656i
$529$	−11.5000	−	19.9186i	−0.500000	−	0.866025i
$530$	51.5772	+	29.7781i	2.24037	+	1.29348i
$531$		−	13.8054i		−	0.599104i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	74.8051	−	43.1887i	3.23411	−	1.86721i
$536$		0			0
$537$	−16.9706	−	9.79796i	−0.732334	−	0.422813i
$538$			30.7566i			1.32601i
$539$	0.857864	−	0.840532i	0.0369508	−	0.0362043i
$540$	43.4558			1.87004
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$	8.33452	−	4.81194i	0.357998	−	0.206691i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$	−1.51472	−	2.62357i	−0.0645878	−	0.111869i
$551$		0			0
$552$		0			0
$553$	7.72792	+	18.9295i	0.328625	+	0.804963i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15.9645	−	27.6513i	−0.676436	−	1.17162i	−0.976047	−	0.217560i	$-0.930190\pi$
$557$	−15.9645	−	27.6513i	−0.676436	−	1.17162i	0.299611	−	0.954062i	$-0.403143\pi$
$558$	−22.8198	−	13.1750i	−0.966039	−	0.557743i
$559$		0			0
$560$	6.00000	−	43.8446i	0.253546	−	1.85277i
$561$		0			0
$562$		0			0
$563$	−4.71320	+	2.72117i	−0.198638	+	0.114684i	−0.596020	−	0.802970i	$-0.703252\pi$
$563$	−4.71320	+	2.72117i	−0.198638	+	0.114684i	0.397382	+	0.917653i	$0.369919\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−23.5919	−	3.22848i	−0.990766	−	0.135583i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	12.0000	−	20.7846i	0.500000	−	0.866025i
$577$	−41.2279	+	23.8030i	−1.71634	+	0.990930i	−0.790980	+	0.611842i	$0.790429\pi$
$577$	−41.2279	+	23.8030i	−1.71634	+	0.990930i	−0.925361	+	0.379088i	$0.876238\pi$
$578$	12.0208	+	20.8207i	0.500000	+	0.866025i
$579$	32.2279	+	18.6068i	1.33935	+	0.773272i
$580$			87.0949i			3.61642i
$581$	−7.86396	−	6.09823i	−0.326252	−	0.252997i
$582$	−28.2426			−1.17070
$583$	−0.863961	+	1.49642i	−0.0357816	+	0.0619756i
$584$	24.0000	−	13.8564i	0.993127	−	0.573382i
$585$		0			0
$586$	−4.09188	−	2.36245i	−0.169034	−	0.0975919i
$587$			30.5316i			1.26017i	0.776525	+	0.630087i	$0.216981\pi$
$587$			30.5316i			1.26017i	−0.776525	+	0.630087i	$0.783019\pi$
$588$	−6.51472	+	23.3572i	−0.268662	+	0.963234i
$589$		0			0
$590$	13.6066	−	23.5673i	0.560175	−	0.970251i
$591$	−21.2132	+	12.2474i	−0.872595	+	0.503793i
$592$		0			0
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$			1.26080i			0.0517312i
$595$		0			0
$596$	5.65685			0.231714
$597$	−21.2132	+	36.7423i	−0.868199	+	1.50376i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	52.9706	+	30.5826i	2.16251	+	1.24853i
$601$		−	22.7628i		−	0.928516i	−0.885700	−	0.464258i	$-0.846321\pi$
$601$		−	22.7628i		−	0.928516i	0.885700	−	0.464258i	$-0.153679\pi$
$602$		0			0
$603$		0			0
$604$	−22.2132	+	38.4744i	−0.903842	+	1.56550i
$605$	−39.7279	+	22.9369i	−1.61517	+	0.932519i
$606$	4.24264	+	7.34847i	0.172345	+	0.298511i
$607$	−35.5919	−	20.5490i	−1.44463	−	0.834058i	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−35.5919	−	20.5490i	−1.44463	−	0.834058i	−0.998154	+	0.0607380i	$0.980655\pi$
$608$		0			0
$609$	6.47056	−	47.2832i	0.262200	−	1.91601i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$	1.27208	+	0.174080i	0.0512535	+	0.00701388i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	−18.0000	+	31.1769i	−0.724066	+	1.25412i
$619$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$619$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$620$	−25.9706	−	44.9823i	−1.04300	−	1.80653i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−34.2279	+	59.2845i	−1.36912	+	2.37138i
$626$	−30.0624	+	17.3566i	−1.20154	+	0.693708i
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−37.0919	−	28.7635i	−1.47778	−	1.14596i
$631$	20.7574			0.826337			0.413169	−	0.910654i	$-0.364422\pi$
$631$	20.7574			0.826337			0.413169	+	0.910654i	$0.364422\pi$
$632$	−10.9289	+	18.9295i	−0.434730	+	0.752974i
$633$		0			0
$634$	−21.6066	−	37.4237i	−0.858108	−	1.48629i
$635$	−24.4706	−	14.1281i	−0.971085	−	0.560656i
$636$		−	34.8872i		−	1.38337i
$637$		0			0
$638$	−2.52691			−0.100041
$639$		0			0
$640$	40.9706	−	23.6544i	1.61950	−	0.935021i
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$	−43.8198	−	25.2994i	−1.72943	−	0.998487i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$	−12.7279	−	22.0454i	−0.500000	−	0.866025i
$649$	0.683766	+	0.394773i	0.0268402	+	0.0154962i
$650$		0			0
$651$	10.7574	+	26.3500i	0.421614	+	1.03274i
$652$		0			0
$653$	4.52082	−	7.83028i	0.176913	−	0.306423i	−0.763909	−	0.645325i	$-0.776722\pi$
$653$	4.52082	−	7.83028i	0.176913	−	0.306423i	0.940822	+	0.338902i	$0.110055\pi$
$654$		0			0
$655$	44.5919	+	77.2354i	1.74235	+	3.01784i
$656$		0			0
$657$		−	29.3939i		−	1.14676i
$658$		0			0
$659$	45.2548			1.76288			0.881439	−	0.472298i	$-0.156575\pi$
$659$	45.2548			1.76288			0.881439	+	0.472298i	$0.156575\pi$
$660$	−1.24264	+	2.15232i	−0.0483697	+	0.0837788i
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$		0			0
$663$		0			0
$664$		−	10.6385i		−	0.412854i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−25.8640	−	44.7977i	−0.999959	−	1.73198i
$670$		0			0
$671$		0			0
$672$	−24.0000	+	9.79796i	−0.925820	+	0.377964i
$673$	−50.9411			−1.96364			−0.981818	−	0.189824i	$-0.939208\pi$
$673$	−50.9411			−1.96364			−0.981818	+	0.189824i	$0.939208\pi$
$674$	−25.7782	+	44.6491i	−0.992938	+	1.71982i
$675$	56.1838	−	32.4377i	2.16251	−	1.24853i
$676$	13.0000	+	22.5167i	0.500000	+	0.866025i
$677$	−24.6213	−	14.2151i	−0.946274	−	0.546332i	−0.0543526	−	0.998522i	$-0.517310\pi$
$677$	−24.6213	−	14.2151i	−0.946274	−	0.546332i	−0.891922	+	0.452190i	$0.850643\pi$
$678$		0			0
$679$	24.1066	+	18.6938i	0.925126	+	0.717404i
$680$		0			0
$681$	−25.7132	+	44.5366i	−0.985332	+	1.70665i
$682$	1.30509	−	0.753492i	0.0499743	−	0.0288527i
$683$	21.0858	+	36.5217i	0.806825	+	1.39746i	0.915052	+	0.403336i	$0.132149\pi$
$683$	21.0858	+	36.5217i	0.806825	+	1.39746i	−0.108227	+	0.994126i	$0.534517\pi$
$684$		0			0
$685$		0			0
$686$	21.0208	−	15.6245i	0.802578	−	0.596547i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$691$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$692$		−	34.6410i		−	1.31685i
$693$	0.834524	−	1.07616i	0.0317009	−	0.0408799i
$694$	−40.0000			−1.51838
$695$		0			0
$696$	44.1838	−	25.5095i	1.67478	−	0.966935i
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−24.9706	−	61.1651i	−0.943799	−	2.31182i
$701$	−51.3848			−1.94078			−0.970388	−	0.241551i	$-0.922344\pi$
$701$	−51.3848			−1.94078			−0.970388	+	0.241551i	$0.922344\pi$
$702$		0			0
$703$		0			0
$704$	0.686292	+	1.18869i	0.0258656	+	0.0448005i
$705$		0			0
$706$		0			0
$707$	1.24264	−	9.08052i	0.0467343	−	0.341508i
$708$	−15.9411			−0.599104
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$	11.5919	+	20.0777i	0.434730	+	0.752974i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−11.3137	+	19.5959i	−0.422813	+	0.732334i
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		−	50.1785i		−	1.87004i
$721$	36.0000	−	14.6969i	1.34071	−	0.547343i
$722$	−26.8701			−1.00000
$723$	−20.2279	+	35.0358i	−0.752285	+	1.30300i
$724$		0			0
$725$	65.0122	+	112.604i	2.41449	+	4.18202i
$726$	23.2721	+	13.4361i	0.863708	+	0.498662i
$727$			28.6764i			1.06355i	0.846886	+	0.531775i	$0.178475\pi$
$727$			28.6764i			1.06355i	−0.846886	+	0.531775i	$0.821525\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$	28.9706	−	50.1785i	1.07225	−	1.85719i
$731$		0			0
$732$		0			0
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$		−	43.0762i		−	1.58997i
$735$	12.6213	+	49.1023i	0.465544	+	1.81117i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$	−23.0919	+	29.7781i	−0.847730	+	1.09319i
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$	−15.2132	+	26.3500i	−0.557743	+	0.966039i
$745$	10.2426	−	5.91359i	0.375261	−	0.216657i
$746$		0			0
$747$	−9.77208	−	5.64191i	−0.357542	−	0.206427i
$748$		0			0
$749$	20.6569	+	50.5988i	0.754785	+	1.84884i
$750$	76.6690			2.79956
$751$	25.8345	−	44.7467i	0.942715	−	1.63283i	0.182453	−	0.983215i	$-0.441596\pi$
$751$	25.8345	−	44.7467i	0.942715	−	1.63283i	0.760263	−	0.649616i	$-0.225070\pi$
$752$		0			0
$753$	17.7426	+	30.7312i	0.646578	+	1.11991i
$754$		0			0
$755$			92.8854i			3.38045i
$756$	−3.72792	+	27.2416i	−0.135583	+	0.990766i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$			16.5521i			0.599619i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−24.0000	−	13.8564i	−0.866025	−	0.500000i
$769$		−	47.0116i		−	1.69528i	−0.530572	−	0.847640i	$-0.678023\pi$
$769$		−	47.0116i		−	1.69528i	0.530572	−	0.847640i	$-0.321977\pi$
$770$	2.48528	−	1.01461i	0.0895633	−	0.0365641i
$771$		0			0
$772$	21.4853	−	37.2136i	0.773272	−	1.33935i
$773$	45.0000	−	25.9808i	1.61854	−	0.934463i	0.631239	−	0.775589i	$-0.282547\pi$
$773$	45.0000	−	25.9808i	1.61854	−	0.934463i	0.987299	−	0.158874i	$-0.0507865\pi$
$774$		0			0
$775$	−67.1543	−	38.7716i	−2.41225	−	1.39272i
$776$			32.6118i			1.17070i
$777$		0			0
$778$	44.0000			1.57748
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	54.1138i		−	1.93387i
$784$	26.9706	+	7.52255i	0.963234	+	0.268662i
$785$		0			0
$786$	26.1213	−	45.2435i	0.931717	−	1.61378i
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$	14.1421	+	24.4949i	0.503793	+	0.872595i
$789$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	168.ba (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 1.22474i) q^{2} +(1.50000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(-3.62132 - 2.09077i) q^{5} +2.44949i q^{6} +(1.62132 - 2.09077i) q^{7} +2.82843 q^{8} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(-0.707107 + 1.22474i) q^{2} +(1.50000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(-3.62132 - 2.09077i) q^{5} +2.44949i q^{6} +(1.62132 - 2.09077i) q^{7} +2.82843 q^{8} +(1.50000 - 2.59808i) q^{9} +(5.12132 - 2.95680i) q^{10} +(0.0857864 + 0.148586i) q^{11} +(-3.00000 - 1.73205i) q^{12} +(1.41421 + 3.46410i) q^{14} -7.24264 q^{15} +(-2.00000 + 3.46410i) q^{16} +(2.12132 + 3.67423i) q^{18} +8.36308i q^{20} +(0.621320 - 4.54026i) q^{21} -0.242641 q^{22} +(4.24264 - 2.44949i) q^{24} +(6.24264 + 10.8126i) q^{25} -5.19615i q^{27} +(-5.24264 - 0.717439i) q^{28} +10.4142 q^{29} +(5.12132 - 8.87039i) q^{30} +(-5.37868 + 3.10538i) q^{31} +(-2.82843 - 4.89898i) q^{32} +(0.257359 + 0.148586i) q^{33} +(-10.2426 + 4.18154i) q^{35} -6.00000 q^{36} +(-10.2426 - 5.91359i) q^{40} +(5.12132 + 3.97141i) q^{42} +(0.171573 - 0.297173i) q^{44} +(-10.8640 + 6.27231i) q^{45} +6.92820i q^{48} +(-1.74264 - 6.77962i) q^{49} -17.6569 q^{50} +(5.03553 + 8.72180i) q^{53} +(6.36396 + 3.67423i) q^{54} -0.717439i q^{55} +(4.58579 - 5.91359i) q^{56} +(-7.36396 + 12.7548i) q^{58} +(3.98528 - 2.30090i) q^{59} +(7.24264 + 12.5446i) q^{60} -8.78335i q^{62} +(-3.00000 - 7.34847i) q^{63} +8.00000 q^{64} +(-0.363961 + 0.210133i) q^{66} +(2.12132 - 15.5014i) q^{70} +(4.24264 - 7.34847i) q^{72} +(8.48528 - 4.89898i) q^{73} +(18.7279 + 10.8126i) q^{75} +(0.449747 + 0.0615465i) q^{77} +(-3.86396 + 6.69258i) q^{79} +(14.4853 - 8.36308i) q^{80} +(-4.50000 - 7.79423i) q^{81} -3.76127i q^{83} +(-8.48528 + 3.46410i) q^{84} +(15.6213 - 9.01897i) q^{87} +(0.242641 + 0.420266i) q^{88} -17.7408i q^{90} +(-5.37868 + 9.31615i) q^{93} +(-8.48528 - 4.89898i) q^{96} +11.5300i q^{97} +(9.53553 + 2.65962i) q^{98} +0.514719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{3} - 4 q^{4} - 6 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q + 6 * q^3 - 4 * q^4 - 6 * q^5 - 2 * q^7 + 6 * q^9	\( 4 q + 6 q^{3} - 4 q^{4} - 6 q^{5} - 2 q^{7} + 6 q^{9} + 12 q^{10} + 6 q^{11} - 12 q^{12} - 12 q^{15} - 8 q^{16} - 6 q^{21} + 16 q^{22} + 8 q^{25} - 4 q^{28} + 36 q^{29} + 12 q^{30} - 30 q^{31} + 18 q^{33} - 24 q^{35} - 24 q^{36} - 24 q^{40} + 12 q^{42} + 12 q^{44} - 18 q^{45} + 10 q^{49} - 48 q^{50} + 6 q^{53} + 24 q^{56} - 4 q^{58} - 18 q^{59} + 12 q^{60} - 12 q^{63} + 32 q^{64} + 24 q^{66} + 24 q^{75} - 18 q^{77} + 10 q^{79} + 24 q^{80} - 18 q^{81} + 54 q^{87} - 16 q^{88} - 30 q^{93} + 24 q^{98} + 36 q^{99}+O(q^{100}) \) 4 * q + 6 * q^3 - 4 * q^4 - 6 * q^5 - 2 * q^7 + 6 * q^9 + 12 * q^10 + 6 * q^11 - 12 * q^12 - 12 * q^15 - 8 * q^16 - 6 * q^21 + 16 * q^22 + 8 * q^25 - 4 * q^28 + 36 * q^29 + 12 * q^30 - 30 * q^31 + 18 * q^33 - 24 * q^35 - 24 * q^36 - 24 * q^40 + 12 * q^42 + 12 * q^44 - 18 * q^45 + 10 * q^49 - 48 * q^50 + 6 * q^53 + 24 * q^56 - 4 * q^58 - 18 * q^59 + 12 * q^60 - 12 * q^63 + 32 * q^64 + 24 * q^66 + 24 * q^75 - 18 * q^77 + 10 * q^79 + 24 * q^80 - 18 * q^81 + 54 * q^87 - 16 * q^88 - 30 * q^93 + 24 * q^98 + 36 * q^99

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