LMFDB - Embedded newform 1503.1.f.b.166.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,1,Mod(166,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1503.166");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1503.f (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:	$0.750094713987$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{33}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{33} + \cdots)$

Embedding invariants

Embedding label			166.4
Root			$-0.888835 + 0.458227i$ of defining polynomial
Character	$\chi$	$=$	1503.166
Dual form			1503.1.f.b.1168.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.580057 - 1.00469i) q^{2} +(0.928368 - 0.371662i) q^{3} +(-0.172932 + 0.299527i) q^{4} +(-0.911911 - 0.717135i) q^{6} +(-0.0475819 - 0.0824143i) q^{7} -0.758872 q^{8} +(0.723734 - 0.690079i) q^{9} +O(q^{10})$	\(q+(-0.580057 - 1.00469i) q^{2} +(0.928368 - 0.371662i) q^{3} +(-0.172932 + 0.299527i) q^{4} +(-0.911911 - 0.717135i) q^{6} +(-0.0475819 - 0.0824143i) q^{7} -0.758872 q^{8} +(0.723734 - 0.690079i) q^{9} +(-0.981929 - 1.70075i) q^{11} +(-0.0492216 + 0.342344i) q^{12} +(-0.0552004 + 0.0956100i) q^{14} +(0.613121 + 1.06196i) q^{16} +(-1.11312 - 0.326842i) q^{18} +0.471518 q^{19} +(-0.0748038 - 0.0588264i) q^{21} +(-1.13915 + 1.97306i) q^{22} +(-0.704513 + 0.282044i) q^{24} +(-0.500000 - 0.866025i) q^{25} +(0.415415 - 0.909632i) q^{27} +0.0329138 q^{28} +(0.888835 + 1.53951i) q^{29} +(-0.580057 + 1.00469i) q^{31} +(0.331854 - 0.574788i) q^{32} +(-1.54370 - 1.21398i) q^{33} +(0.0815405 + 0.336115i) q^{36} +(-0.273507 - 0.473728i) q^{38} +(-0.0157117 + 0.109277i) q^{42} +0.679228 q^{44} +(-0.723734 - 1.25354i) q^{47} +(0.963891 + 0.758013i) q^{48} +(0.495472 - 0.858183i) q^{49} +(-0.580057 + 1.00469i) q^{50} +(-1.15486 + 0.110276i) q^{54} +(0.0361086 + 0.0625419i) q^{56} +(0.437742 - 0.175245i) q^{57} +(1.03115 - 1.78600i) q^{58} +(0.142315 + 0.246497i) q^{61} +1.34586 q^{62} +(-0.0913090 - 0.0268107i) q^{63} +0.456265 q^{64} +(-0.324236 + 2.25511i) q^{66} +(-0.549222 + 0.523682i) q^{72} +(-0.786053 - 0.618159i) q^{75} +(-0.0815405 + 0.141232i) q^{76} +(-0.0934441 + 0.161850i) q^{77} +(0.0475819 - 0.998867i) q^{81} +(0.0305561 - 0.0122328i) q^{84} +(1.39734 + 1.09888i) q^{87} +(0.745158 + 1.29065i) q^{88} +1.85674 q^{89} +(-0.165101 + 1.14831i) q^{93} +(-0.839614 + 1.45425i) q^{94} +(0.0944555 - 0.656953i) q^{96} +(0.654861 + 1.13425i) q^{97} -1.14961 q^{98} +(-1.88431 - 0.553283i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9}+O(q^{10}) $ 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9	$ 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9} - q^{11} + q^{14} - 12 q^{16} + 2 q^{18} + 2 q^{19} - q^{21} + q^{22} + q^{24} - 10 q^{25} - 2 q^{27} - q^{29} - q^{31} - q^{33} + q^{38} - 13 q^{42} - 22 q^{44} - q^{47} + 21 q^{48} - 11 q^{49} - q^{50} - 12 q^{54} - q^{56} - q^{57} + q^{58} + 2 q^{61} + 42 q^{62} + 2 q^{63} + 20 q^{64} - 2 q^{66} - 10 q^{72} + q^{75} + q^{77} + q^{81} + 22 q^{84} - q^{87} - q^{88} + 2 q^{89} + 2 q^{93} + q^{94} + 2 q^{97} - 22 q^{98} + 2 q^{99}+O(q^{100}) $ 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9 - q^11 + q^14 - 12 * q^16 + 2 * q^18 + 2 * q^19 - q^21 + q^22 + q^24 - 10 * q^25 - 2 * q^27 - q^29 - q^31 - q^33 + q^38 - 13 * q^42 - 22 * q^44 - q^47 + 21 * q^48 - 11 * q^49 - q^50 - 12 * q^54 - q^56 - q^57 + q^58 + 2 * q^61 + 42 * q^62 + 2 * q^63 + 20 * q^64 - 2 * q^66 - 10 * q^72 + q^75 + q^77 + q^81 + 22 * q^84 - q^87 - q^88 + 2 * q^89 + 2 * q^93 + q^94 + 2 * q^97 - 22 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1503\mathbb{Z}\right)^\times$.

$n$	$172$	$335$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$2$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$3$	0.928368	−	0.371662i	0.928368	−	0.371662i
$3$	0.928368	−	0.371662i	0.928368	−	0.371662i
$4$	−0.172932	+	0.299527i	−0.172932	+	0.299527i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$	−0.911911	−	0.717135i	−0.911911	−	0.717135i
$7$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$7$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$8$	−0.758872			−0.758872
$9$	0.723734	−	0.690079i	0.723734	−	0.690079i
$10$		0			0
$11$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$11$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$12$	−0.0492216	+	0.342344i	−0.0492216	+	0.342344i
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$	−0.0552004	+	0.0956100i	−0.0552004	+	0.0956100i
$15$		0			0
$16$	0.613121	+	1.06196i	0.613121	+	1.06196i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−1.11312	−	0.326842i	−1.11312	−	0.326842i
$19$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$19$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$20$		0			0
$21$	−0.0748038	−	0.0588264i	−0.0748038	−	0.0588264i
$22$	−1.13915	+	1.97306i	−1.13915	+	1.97306i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	−0.704513	+	0.282044i	−0.704513	+	0.282044i
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$		0			0
$27$	0.415415	−	0.909632i	0.415415	−	0.909632i
$28$	0.0329138			0.0329138
$29$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$29$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$30$		0			0
$31$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$31$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$32$	0.331854	−	0.574788i	0.331854	−	0.574788i
$33$	−1.54370	−	1.21398i	−1.54370	−	1.21398i
$34$		0			0
$35$		0			0
$36$	0.0815405	+	0.336115i	0.0815405	+	0.336115i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−0.273507	−	0.473728i	−0.273507	−	0.473728i
$39$		0			0
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$	−0.0157117	+	0.109277i	−0.0157117	+	0.109277i
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$	0.679228			0.679228
$45$		0			0
$46$		0			0
$47$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$47$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$48$	0.963891	+	0.758013i	0.963891	+	0.758013i
$49$	0.495472	−	0.858183i	0.495472	−	0.858183i
$50$	−0.580057	+	1.00469i	−0.580057	+	1.00469i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−1.15486	+	0.110276i	−1.15486	+	0.110276i
$55$		0			0
$56$	0.0361086	+	0.0625419i	0.0361086	+	0.0625419i
$57$	0.437742	−	0.175245i	0.437742	−	0.175245i
$58$	1.03115	−	1.78600i	1.03115	−	1.78600i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$61$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$62$	1.34586			1.34586
$63$	−0.0913090	−	0.0268107i	−0.0913090	−	0.0268107i
$64$	0.456265			0.456265
$65$		0			0
$66$	−0.324236	+	2.25511i	−0.324236	+	2.25511i
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−0.549222	+	0.523682i	−0.549222	+	0.523682i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−0.786053	−	0.618159i	−0.786053	−	0.618159i
$76$	−0.0815405	+	0.141232i	−0.0815405	+	0.141232i
$77$	−0.0934441	+	0.161850i	−0.0934441	+	0.161850i
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	0.0475819	−	0.998867i	0.0475819	−	0.998867i
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$	0.0305561	−	0.0122328i	0.0305561	−	0.0122328i
$85$		0			0
$86$		0			0
$87$	1.39734	+	1.09888i	1.39734	+	1.09888i
$88$	0.745158	+	1.29065i	0.745158	+	1.29065i
$89$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$89$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−0.165101	+	1.14831i	−0.165101	+	1.14831i
$94$	−0.839614	+	1.45425i	−0.839614	+	1.45425i
$95$		0			0
$96$	0.0944555	−	0.656953i	0.0944555	−	0.656953i
$97$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$97$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$98$	−1.14961			−1.14961
$99$	−1.88431	−	0.553283i	−1.88431	−	0.553283i
$100$	0.345864			0.345864
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$107$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$108$	0.200621	+	0.281733i	0.200621	+	0.281733i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	0.0583469	−	0.101060i	0.0583469	−	0.101060i
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$	−0.429982	−	0.338142i	−0.429982	−	0.338142i
$115$		0			0
$116$	−0.614832			−0.614832
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.42837	+	2.47401i	−1.42837	+	2.47401i
$122$	0.165101	−	0.285964i	0.165101	−	0.285964i
$123$		0			0
$124$	−0.200621	−	0.347485i	−0.200621	−	0.347485i
$125$		0			0
$126$	0.0260280	+	0.107289i	0.0260280	+	0.107289i
$127$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$127$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$128$	−0.596514	−	1.03319i	−0.596514	−	1.03319i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	0.630573	−	0.252443i	0.630573	−	0.252443i
$133$	−0.0224357	−	0.0388598i	−0.0224357	−	0.0388598i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$137$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$138$		0			0
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$	−1.13779	−	0.894765i	−1.13779	−	0.894765i
$142$		0			0
$143$		0			0
$144$	1.17657	+	0.345472i	1.17657	+	0.345472i
$145$		0			0
$146$		0			0
$147$	0.141026	−	0.980857i	0.141026	−	0.980857i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	−0.165101	+	1.14831i	−0.165101	+	1.14831i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$	−0.357822			−0.357822
$153$		0			0
$154$	0.216812			0.216812
$155$		0			0
$156$		0			0
$157$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$157$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−1.03115	+	0.531595i	−1.03115	+	0.531595i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$167$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$168$	0.0567665	+	0.0446417i	0.0567665	+	0.0446417i
$169$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$170$		0			0
$171$	0.341254	−	0.325385i	0.341254	−	0.325385i
$172$		0			0
$173$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$173$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$174$	0.293496	−	2.04131i	0.293496	−	2.04131i
$175$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i
$176$	1.20408	−	2.08553i	1.20408	−	2.08553i
$177$		0			0
$178$	−1.07701	−	1.86544i	−1.07701	−	1.86544i
$179$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$179$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$180$		0			0
$181$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$181$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$182$		0			0
$183$	0.223734	+	0.175946i	0.223734	+	0.175946i
$184$		0			0
$185$		0			0
$186$	1.24946	−	0.500207i	1.24946	−	0.500207i
$187$		0			0
$188$	0.500627			0.500627
$189$	−0.0947329	+	0.00904590i	−0.0947329	+	0.00904590i
$190$		0			0
$191$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$191$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$192$	0.423582	−	0.169577i	0.423582	−	0.169577i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$	0.759713	−	1.31586i	0.759713	−	1.31586i
$195$		0			0
$196$	0.171366	+	0.296815i	0.171366	+	0.296815i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	0.537129	+	2.21408i	0.537129	+	2.21408i
$199$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$199$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$200$	0.379436	+	0.657203i	0.379436	+	0.657203i
$201$		0			0
$202$		0			0
$203$	0.0845850	−	0.146505i	0.0845850	−	0.146505i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.462997	−	0.801934i	−0.462997	−	0.801934i
$210$		0			0
$211$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$211$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$212$		0			0
$213$		0			0
$214$	−0.0552004	−	0.0956100i	−0.0552004	−	0.0956100i
$215$		0			0
$216$	−0.315247	+	0.690294i	−0.315247	+	0.690294i
$217$	0.110401			0.110401
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$223$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$224$	−0.0631610			−0.0631610
$225$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	−0.0232089	+	0.161421i	−0.0232089	+	0.161421i
$229$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$229$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$230$		0			0
$231$	−0.0265970	+	0.184986i	−0.0265970	+	0.184986i
$232$	−0.674512	−	1.16829i	−0.674512	−	1.16829i
$233$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$233$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$239$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	3.31414			3.31414
$243$	−0.327068	−	0.945001i	−0.327068	−	0.945001i
$244$	−0.0984432			−0.0984432
$245$		0			0
$246$		0			0
$247$		0			0
$248$	0.440189	−	0.762430i	0.440189	−	0.762430i
$249$		0			0
$250$		0			0
$251$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$251$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$252$	0.0238208	−	0.0227131i	0.0238208	−	0.0227131i
$253$		0			0
$254$	−1.13915	−	1.97306i	−1.13915	−	1.97306i
$255$		0			0
$256$	−0.463891	+	0.803483i	−0.463891	+	0.803483i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	1.70566	+	0.500828i	1.70566	+	0.500828i
$262$		0			0
$263$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$263$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$264$	1.17147	+	0.921253i	1.17147	+	0.921253i
$265$		0			0
$266$	−0.0260280	+	0.0450818i	−0.0260280	+	0.0450818i
$267$	1.72373	−	0.690079i	1.72373	−	0.690079i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$	0.379436	−	0.657203i	0.379436	−	0.657203i
$275$	−0.981929	+	1.70075i	−0.981929	+	1.70075i
$276$		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
$279$	0.273507	+	1.12741i	0.273507	+	1.12741i
$280$		0			0
$281$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$281$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$282$	−0.238979	+	1.66214i	−0.238979	+	1.66214i
$283$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$283$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.156475	−	0.644999i	−0.156475	−	0.644999i
$289$	1.00000			1.00000
$290$		0			0
$291$	1.02951	+	0.809616i	1.02951	+	0.809616i
$292$		0			0
$293$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$293$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$294$	−1.06726	+	0.427266i	−1.06726	+	0.427266i
$295$		0			0
$296$		0			0
$297$	−1.95496	+	0.186677i	−1.95496	+	0.186677i
$298$		0			0
$299$		0			0
$300$	0.321089	−	0.128545i	0.321089	−	0.128545i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.289098	+	0.500732i	0.289098	+	0.500732i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	−0.0323190	−	0.0559781i	−0.0323190	−	0.0559781i
$309$		0			0
$310$		0			0
$311$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$311$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$	0.963857			0.963857
$315$		0			0
$316$		0			0
$317$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$317$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$318$		0			0
$319$	1.74555	−	3.02337i	1.74555	−	3.02337i
$320$		0			0
$321$	0.0883470	−	0.0353688i	0.0883470	−	0.0353688i
$322$		0			0
$323$		0			0
$324$	0.290959	+	0.186988i	0.290959	+	0.186988i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.0688733	+	0.119292i	−0.0688733	+	0.119292i
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$		0			0
$333$		0			0
$334$	1.16011			1.16011
$335$		0			0
$336$	0.0166073	−	0.115506i	0.0166073	−	0.115506i
$337$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$337$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$338$	−0.580057	+	1.00469i	−0.580057	+	1.00469i
$339$		0			0
$340$		0			0
$341$	2.27830			2.27830
$342$	−0.524856	−	0.154112i	−0.524856	−	0.154112i
$343$	−0.189466			−0.189466
$344$		0			0
$345$		0			0
$346$	0.759713	−	1.31586i	0.759713	−	1.31586i
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$	−0.570791	+	0.228510i	−0.570791	+	0.228510i
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$	0.110401			0.110401
$351$		0			0
$352$	−1.30343			−1.30343
$353$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$353$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$354$		0			0
$355$		0			0
$356$	−0.321089	+	0.556143i	−0.321089	+	0.556143i
$357$		0			0
$358$	−0.273507	−	0.473728i	−0.273507	−	0.473728i
$359$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$359$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$360$		0			0
$361$	−0.777671			−0.777671
$362$	−0.839614	−	1.45425i	−0.839614	−	1.45425i
$363$	−0.406556	+	2.82766i	−0.406556	+	2.82766i
$364$		0			0
$365$		0			0
$366$	0.0469928	−	0.326842i	0.0469928	−	0.326842i
$367$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$367$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.315397	−	0.248031i	−0.315397	−	0.248031i
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$	0.549222	+	0.951280i	0.549222	+	0.951280i
$377$		0			0
$378$	0.0640388	+	0.0899299i	0.0640388	+	0.0899299i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	1.82318	−	0.729892i	1.82318	−	0.729892i
$382$	1.15486	−	2.00028i	1.15486	−	2.00028i
$383$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$383$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$384$	−0.937783	−	0.737481i	−0.937783	−	0.737481i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.452986			−0.452986
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	−0.376000	+	0.651251i	−0.376000	+	0.651251i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	0.491580	−	0.468721i	0.491580	−	0.468721i
$397$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$397$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$398$	0.759713	+	1.31586i	0.759713	+	1.31586i
$399$	−0.0352713	−	0.0277377i	−0.0352713	−	0.0277377i
$400$	0.613121	−	1.06196i	0.613121	−	1.06196i
$401$		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
$404$		0			0
$405$		0			0
$406$	−0.196256			−0.196256
$407$		0			0
$408$		0			0
$409$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$409$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$410$		0			0
$411$	0.514186	+	0.404360i	0.514186	+	0.404360i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	−0.537129	+	0.930335i	−0.537129	+	0.930335i
$419$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$419$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$420$		0			0
$421$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$421$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$422$	0.963857			0.963857
$423$	−1.38884	−	0.407799i	−1.38884	−	0.407799i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.0135432	−	0.0234576i	0.0135432	−	0.0234576i
$428$	−0.0164569	+	0.0285041i	−0.0164569	+	0.0285041i
$429$		0			0
$430$		0			0
$431$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$431$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$432$	1.22069	−	0.116562i	1.22069	−	0.116562i
$433$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$433$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$434$	−0.0640388	−	0.110918i	−0.0640388	−	0.110918i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	−0.233624	−	0.963011i	−0.233624	−	0.963011i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$	−1.07701	+	1.86544i	−1.07701	+	1.86544i
$447$		0			0
$448$	−0.0217100	−	0.0376028i	−0.0217100	−	0.0376028i
$449$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$449$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$450$	0.273507	+	1.12741i	0.273507	+	1.12741i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	−0.332190	+	0.132989i	−0.332190	+	0.132989i
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$	−2.06230			−2.06230
$459$		0			0
$460$		0			0
$461$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$461$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$462$	0.201281	−	0.0805807i	0.201281	−	0.0805807i
$463$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$463$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$464$	−1.08993	+	1.88781i	−1.08993	+	1.88781i
$465$		0			0
$466$	1.15486	+	2.00028i	1.15486	+	2.00028i
$467$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$467$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.118239	+	0.822373i	−0.118239	+	0.822373i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.235759	−	0.408346i	−0.235759	−	0.408346i
$476$		0			0
$477$		0			0
$478$	2.15402			2.15402
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.494021	−	0.855670i	−0.494021	−	0.855670i
$485$		0			0
$486$	−0.759713	+	0.876756i	−0.759713	+	0.876756i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	−0.107999	−	0.187059i	−0.107999	−	0.187059i
$489$		0			0
$490$		0			0
$491$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$491$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.42258			−1.42258
$497$		0			0
$498$		0			0
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$		0			0
$501$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$502$	−0.975950	−	1.69039i	−0.975950	−	1.69039i
$503$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$503$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$504$	0.0692919	+	0.0203459i	0.0692919	+	0.0203459i
$505$		0			0
$506$		0			0
$507$	−0.786053	−	0.618159i	−0.786053	−	0.618159i
$508$	−0.339614	+	0.588228i	−0.339614	+	0.588228i
$509$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$509$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$510$		0			0
$511$		0			0
$512$	−0.116694			−0.116694
$513$	0.195876	−	0.428908i	0.195876	−	0.428908i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−1.42131	+	2.46178i	−1.42131	+	2.46178i
$518$		0			0
$519$	1.02951	+	0.809616i	1.02951	+	0.809616i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−0.486206	−	2.00417i	−0.486206	−	2.00417i
$523$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$523$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$524$		0			0
$525$	−0.0135432	+	0.0941952i	−0.0135432	+	0.0941952i
$526$	0.165101	−	0.285964i	0.165101	−	0.285964i
$527$		0			0
$528$	0.342718	−	2.38365i	0.342718	−	2.38365i
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	0.0155194			0.0155194
$533$		0			0
$534$	−1.69318	−	1.33153i	−1.69318	−	1.33153i
$535$		0			0
$536$		0			0
$537$	0.437742	−	0.175245i	0.437742	−	0.175245i
$538$		0			0
$539$	−1.94607			−1.94607
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	1.34378	−	0.537970i	1.34378	−	0.537970i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$547$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$548$	−0.226242			−0.226242
$549$	0.273100	+	0.0801894i	0.273100	+	0.0801894i
$550$	2.27830			2.27830
$551$	0.419102	+	0.725906i	0.419102	+	0.725906i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$557$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$558$	0.974048	−	0.928753i	0.974048	−	0.928753i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.911911	−	1.57948i	0.911911	−	1.57948i
$563$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$563$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$564$	0.464766	−	0.186064i	0.464766	−	0.186064i
$565$		0			0
$566$	−1.51943			−1.51943
$567$	−0.0845850	+	0.0436066i	−0.0845850	+	0.0436066i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$	1.56499	+	1.23072i	1.56499	+	1.23072i
$574$		0			0
$575$		0			0
$576$	0.330214	−	0.314859i	0.330214	−	0.314859i
$577$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$577$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$578$	−0.580057	−	1.00469i	−0.580057	−	1.00469i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	0.216237	−	1.50396i	0.216237	−	1.50396i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−2.30972			−2.30972
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$	0.269405	+	0.211863i	0.269405	+	0.211863i
$589$	−0.273507	+	0.473728i	−0.273507	+	0.473728i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	1.32154	+	1.85585i	1.32154	+	1.85585i
$595$		0			0
$596$		0			0
$597$	−1.21590	+	0.486774i	−1.21590	+	0.486774i
$598$		0			0
$599$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$599$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$600$	0.596514	+	0.469104i	0.596514	+	0.469104i
$601$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$601$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	0.156475	−	0.271023i	0.156475	−	0.271023i
$609$	0.0240754	−	0.167448i	0.0240754	−	0.167448i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$613$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$614$		0			0
$615$		0			0
$616$	0.0709121	−	0.122823i	0.0709121	−	0.122823i
$617$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$617$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$	−1.16011			−1.16011
$623$	−0.0883470	−	0.153022i	−0.0883470	−	0.153022i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$	−0.727880	−	0.572411i	−0.727880	−	0.572411i
$628$	−0.143677	−	0.248856i	−0.143677	−	0.248856i
$629$		0			0
$630$		0			0
$631$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$631$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$632$		0			0
$633$	−0.118239	+	0.822373i	−0.118239	+	0.822373i
$634$	0.911911	−	1.57948i	0.911911	−	1.57948i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−4.05006			−4.05006
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$	−0.0867810	−	0.0682453i	−0.0867810	−	0.0682453i
$643$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$643$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−0.0361086	+	0.758013i	−0.0361086	+	0.758013i
$649$		0			0
$650$		0			0
$651$	0.102493	−	0.0410319i	0.102493	−	0.0410319i
$652$		0			0
$653$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$653$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	0.159802			0.159802
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.172932	−	0.299527i	−0.172932	−	0.299527i
$669$	−1.45949	−	1.14776i	−1.45949	−	1.14776i
$670$		0			0
$671$	0.279486	−	0.484084i	0.279486	−	0.484084i
$672$	−0.0586367	+	0.0234746i	−0.0586367	+	0.0234746i
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$	1.67923			1.67923
$675$	−0.995472	+	0.0950560i	−0.995472	+	0.0950560i
$676$	0.345864			0.345864
$677$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$677$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	0.0623191	−	0.107940i	0.0623191	−	0.107940i
$680$		0			0
$681$		0			0
$682$	−1.32154	−	2.28898i	−1.32154	−	2.28898i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	0.0384478	+	0.158484i	0.0384478	+	0.158484i
$685$		0			0
$686$	0.109901	+	0.190354i	0.109901	+	0.190354i
$687$	0.252989	−	1.75958i	0.252989	−	1.75958i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$	−0.452986			−0.452986
$693$	0.0440606	+	0.181620i	0.0440606	+	0.181620i
$694$		0			0
$695$		0			0
$696$	−1.06041	−	0.833912i	−1.06041	−	0.833912i
$697$		0			0
$698$		0			0
$699$	−1.84833	+	0.739959i	−1.84833	+	0.739959i
$700$	−0.0164569	−	0.0285041i	−0.0164569	−	0.0285041i
$701$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$701$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$702$		0			0
$703$		0			0
$704$	−0.448020	−	0.775993i	−0.448020	−	0.775993i
$705$		0			0
$706$	−0.975950	+	1.69039i	−0.975950	+	1.69039i
$707$		0			0
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$	−1.40903			−1.40903
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.0815405	+	0.141232i	−0.0815405	+	0.141232i
$717$	−0.264241	+	1.83784i	−0.264241	+	1.83784i
$718$	−0.481929	−	0.834725i	−0.481929	−	0.834725i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	0.451093	+	0.781317i	0.451093	+	0.781317i
$723$		0			0
$724$	−0.250314	+	0.433556i	−0.250314	+	0.433556i
$725$	0.888835	−	1.53951i	0.888835	−	1.53951i
$726$	3.07674	−	1.23174i	3.07674	−	1.23174i
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$730$		0			0
$731$		0			0
$732$	−0.0913915	+	0.0365876i	−0.0913915	+	0.0365876i
$733$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$733$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$734$	−0.839614	+	1.45425i	−0.839614	+	1.45425i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$744$	0.125291	−	0.871417i	0.125291	−	0.871417i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.00452808	−	0.00784286i	−0.00452808	−	0.00784286i
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	0.887473	−	1.53715i	0.887473	−	1.53715i
$753$	1.56199	−	0.625325i	1.56199	−	0.625325i
$754$		0			0
$755$		0			0
$756$	0.0136729	−	0.0299394i	0.0136729	−	0.0299394i
$757$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$757$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$761$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$762$	−1.79086	−	1.40835i	−1.79086	−	1.40835i
$763$		0			0
$764$	−0.688596			−0.688596
$765$		0			0
$766$	1.95190			1.95190
$767$		0			0
$768$	−0.132037	+	0.918339i	−0.132037	+	0.918339i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	1.16011			1.16011
$776$	−0.496956	−	0.860752i	−0.496956	−	0.860752i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	1.76962	−	0.168978i	1.76962	−	0.168978i
$784$	1.21514			1.21514
$785$		0			0
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$		0			0
$789$	0.223734	+	0.175946i	0.223734	+	0.175946i
$790$		0			0
$791$		0			0
$792$	1.42995	+	0.419871i	1.42995	+	0.419871i
$793$		0			0
$794$	0.911911	+	1.57948i	0.911911	+	1.57948i
$795$		0			0
$796$	0.226493	−	0.392297i	0.226493	−	0.392297i
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$	−0.00740834	+	0.0515261i	−0.00740834	+	0.0515261i
$799$		0			0
$800$	−0.663708			−0.663708
$801$	1.34378	−	1.28129i	1.34378	−	1.28129i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$809$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	0.0292549	+	0.0506710i	0.0292549	+	0.0506710i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	2.27830			2.27830
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	0.107999	−	0.751148i	0.107999	−	0.751148i
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$	−0.279486	+	1.94387i	−0.279486	+	1.94387i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$	0.320268			0.320268
$837$	0.672932	+	0.945001i	0.672932	+	0.945001i
$838$	−2.30972			−2.30972
$839$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$839$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$840$		0			0
$841$	−1.08006	+	1.87071i	−1.08006	+	1.87071i
$842$	−0.975950	+	1.69039i	−0.975950	+	1.69039i
$843$	1.23576	+	0.971812i	1.23576	+	0.971812i
$844$	−0.143677	−	0.248856i	−0.143677	−	0.248856i
$845$		0			0
$846$	0.395893	+	1.63189i	0.395893	+	1.63189i
$847$	0.271858			0.271858
$848$		0			0
$849$	0.186393	−	1.29639i	0.186393	−	1.29639i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$853$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$854$	−0.0314234			−0.0314234
$855$		0			0
$856$	−0.0722172			−0.0722172
$857$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$857$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$858$		0			0
$859$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$859$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$860$		0			0
$861$		0			0
$862$	0.911911	+	1.57948i	0.911911	+	1.57948i
$863$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$863$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$864$	−0.384989	−	0.540641i	−0.384989	−	0.540641i
$865$		0			0
$866$	1.11312	+	1.92798i	1.11312	+	1.92798i
$867$	0.928368	−	0.371662i	0.928368	−	0.371662i
$868$	−0.0190918	+	0.0330681i	−0.0190918	+	0.0330681i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.25667	+	0.368991i	1.25667	+	0.368991i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$877$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$878$		0			0
$879$	0.283341	−	1.97068i	0.283341	−	1.97068i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.832010	+	0.793320i	−0.832010	+	0.793320i
$883$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$883$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−0.0934441	−	0.161850i	−0.0934441	−	0.161850i
$890$		0			0
$891$	−1.74555	+	0.899892i	−1.74555	+	0.899892i
$892$	0.642178			0.642178
$893$	−0.341254	−	0.591068i	−0.341254	−	0.591068i
$894$		0			0
$895$		0			0
$896$	−0.0567665	+	0.0983225i	−0.0567665	+	0.0983225i
$897$		0			0
$898$	1.11312	+	1.92798i	1.11312	+	1.92798i
$899$	−2.06230			−2.06230
$900$	0.250314	−	0.238674i	0.250314	−	0.238674i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$907$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$911$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$912$	0.454492	+	0.357416i	0.454492	+	0.357416i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.307416	+	0.532461i	0.307416	+	0.532461i
$917$		0			0
$918$		0			0
$919$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$919$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$920$		0			0
$921$		0			0
$922$	1.11312	−	1.92798i	1.11312	−	1.92798i
$923$		0			0
$924$	−0.0508088	−	0.0399565i	−0.0508088	−	0.0399565i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	1.17985			1.17985
$929$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$929$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$930$		0			0
$931$	0.233624	−	0.404648i	0.233624	−	0.404648i
$932$	0.344298	−	0.596342i	0.344298	−	0.596342i
$933$	0.142315	−	0.989821i	0.142315	−	0.989821i
$934$	1.03115	+	1.78600i	1.03115	+	1.78600i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$	0.894814	−	0.358230i	0.894814	−	0.358230i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$947$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$948$		0			0
$949$		0			0
$950$	−0.273507	+	0.473728i	−0.273507	+	0.473728i
$951$	1.23576	+	0.971812i	1.23576	+	0.971812i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−0.321089	−	0.556143i	−0.321089	−	0.556143i
$957$	0.496834	−	3.45556i	0.496834	−	3.45556i
$958$		0			0
$959$	0.0311250	−	0.0539102i	0.0311250	−	0.0539102i
$960$		0			0
$961$	−0.172932	−	0.299527i	−0.172932	−	0.299527i
$962$		0			0
$963$	0.0688733	−	0.0656706i	0.0688733	−	0.0656706i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$967$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$968$	1.08395	−	1.87745i	1.08395	−	1.87745i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	0.339614	+	0.0654552i	0.339614	+	0.0654552i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.174512	+	0.302264i	−0.174512	+	0.302264i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$	−1.82318	−	3.15784i	−1.82318	−	3.15784i
$980$		0			0
$981$		0			0
$982$	2.32023			2.32023
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−0.0196034	+	0.136345i	−0.0196034	+	0.136345i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	0.384989	+	0.666820i	0.384989	+	0.666820i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$997$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.1.f.b.166.4	✓	20
9.7	even	3	inner	1503.1.f.b.1168.4	yes	20
167.166	odd	2	CM	1503.1.f.b.166.4	✓	20
1503.1168	odd	6	inner	1503.1.f.b.1168.4	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1503.1.f.b.166.4	✓	20	1.1	even	1	trivial
1503.1.f.b.166.4	✓	20	167.166	odd	2	CM
1503.1.f.b.1168.4	yes	20	9.7	even	3	inner
1503.1.f.b.1168.4	yes	20	1503.1168	odd	6	inner

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:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1503.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.580057 - 1.00469i) q^{2} +(0.928368 - 0.371662i) q^{3} +(-0.172932 + 0.299527i) q^{4} +(-0.911911 - 0.717135i) q^{6} +(-0.0475819 - 0.0824143i) q^{7} -0.758872 q^{8} +(0.723734 - 0.690079i) q^{9} +O(q^{10})\)	\(q+(-0.580057 - 1.00469i) q^{2} +(0.928368 - 0.371662i) q^{3} +(-0.172932 + 0.299527i) q^{4} +(-0.911911 - 0.717135i) q^{6} +(-0.0475819 - 0.0824143i) q^{7} -0.758872 q^{8} +(0.723734 - 0.690079i) q^{9} +(-0.981929 - 1.70075i) q^{11} +(-0.0492216 + 0.342344i) q^{12} +(-0.0552004 + 0.0956100i) q^{14} +(0.613121 + 1.06196i) q^{16} +(-1.11312 - 0.326842i) q^{18} +0.471518 q^{19} +(-0.0748038 - 0.0588264i) q^{21} +(-1.13915 + 1.97306i) q^{22} +(-0.704513 + 0.282044i) q^{24} +(-0.500000 - 0.866025i) q^{25} +(0.415415 - 0.909632i) q^{27} +0.0329138 q^{28} +(0.888835 + 1.53951i) q^{29} +(-0.580057 + 1.00469i) q^{31} +(0.331854 - 0.574788i) q^{32} +(-1.54370 - 1.21398i) q^{33} +(0.0815405 + 0.336115i) q^{36} +(-0.273507 - 0.473728i) q^{38} +(-0.0157117 + 0.109277i) q^{42} +0.679228 q^{44} +(-0.723734 - 1.25354i) q^{47} +(0.963891 + 0.758013i) q^{48} +(0.495472 - 0.858183i) q^{49} +(-0.580057 + 1.00469i) q^{50} +(-1.15486 + 0.110276i) q^{54} +(0.0361086 + 0.0625419i) q^{56} +(0.437742 - 0.175245i) q^{57} +(1.03115 - 1.78600i) q^{58} +(0.142315 + 0.246497i) q^{61} +1.34586 q^{62} +(-0.0913090 - 0.0268107i) q^{63} +0.456265 q^{64} +(-0.324236 + 2.25511i) q^{66} +(-0.549222 + 0.523682i) q^{72} +(-0.786053 - 0.618159i) q^{75} +(-0.0815405 + 0.141232i) q^{76} +(-0.0934441 + 0.161850i) q^{77} +(0.0475819 - 0.998867i) q^{81} +(0.0305561 - 0.0122328i) q^{84} +(1.39734 + 1.09888i) q^{87} +(0.745158 + 1.29065i) q^{88} +1.85674 q^{89} +(-0.165101 + 1.14831i) q^{93} +(-0.839614 + 1.45425i) q^{94} +(0.0944555 - 0.656953i) q^{96} +(0.654861 + 1.13425i) q^{97} -1.14961 q^{98} +(-1.88431 - 0.553283i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9	\( 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9} - q^{11} + q^{14} - 12 q^{16} + 2 q^{18} + 2 q^{19} - q^{21} + q^{22} + q^{24} - 10 q^{25} - 2 q^{27} - q^{29} - q^{31} - q^{33} + q^{38} - 13 q^{42} - 22 q^{44} - q^{47} + 21 q^{48} - 11 q^{49} - q^{50} - 12 q^{54} - q^{56} - q^{57} + q^{58} + 2 q^{61} + 42 q^{62} + 2 q^{63} + 20 q^{64} - 2 q^{66} - 10 q^{72} + q^{75} + q^{77} + q^{81} + 22 q^{84} - q^{87} - q^{88} + 2 q^{89} + 2 q^{93} + q^{94} + 2 q^{97} - 22 q^{98} + 2 q^{99}+O(q^{100}) \) 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9 - q^11 + q^14 - 12 * q^16 + 2 * q^18 + 2 * q^19 - q^21 + q^22 + q^24 - 10 * q^25 - 2 * q^27 - q^29 - q^31 - q^33 + q^38 - 13 * q^42 - 22 * q^44 - q^47 + 21 * q^48 - 11 * q^49 - q^50 - 12 * q^54 - q^56 - q^57 + q^58 + 2 * q^61 + 42 * q^62 + 2 * q^63 + 20 * q^64 - 2 * q^66 - 10 * q^72 + q^75 + q^77 + q^81 + 22 * q^84 - q^87 - q^88 + 2 * q^89 + 2 * q^93 + q^94 + 2 * q^97 - 22 * q^98 + 2 * q^99

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