LMFDB - Embedded newform 1169.1.f.c.333.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1169,1,Mod(333,1169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1169, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 3]))

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

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

chi := DirichletCharacter("1169.333");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1169 = 7 \cdot 167 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1169.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:	$0.583406999768$
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:	$\Z[a_1, \ldots, a_{7}]$
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			333.4
Root			$-0.327068 + 0.945001i$ of defining polynomial
Character	$\chi$	$=$	1169.333
Dual form			1169.1.f.c.667.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.415415 - 0.719520i) q^{2} +(0.786053 - 1.36148i) q^{3} +(0.154861 - 0.268227i) q^{4} -1.30615 q^{6} +(-0.888835 - 0.458227i) q^{7} -1.08816 q^{8} +(-0.735759 - 1.27437i) q^{9} +O(q^{10})$	\(q+(-0.415415 - 0.719520i) q^{2} +(0.786053 - 1.36148i) q^{3} +(0.154861 - 0.268227i) q^{4} -1.30615 q^{6} +(-0.888835 - 0.458227i) q^{7} -1.08816 q^{8} +(-0.735759 - 1.27437i) q^{9} +(0.327068 - 0.566498i) q^{11} +(-0.243458 - 0.421681i) q^{12} +(0.0395325 + 0.829889i) q^{14} +(0.297176 + 0.514723i) q^{16} +(-0.611291 + 1.05879i) q^{18} +(0.959493 + 1.66189i) q^{19} +(-1.32254 + 0.849945i) q^{21} -0.543476 q^{22} +(-0.855348 + 1.48151i) q^{24} +(-0.500000 + 0.866025i) q^{25} -0.741276 q^{27} +(-0.260554 + 0.167448i) q^{28} +0.0951638 q^{29} +(0.995472 - 1.72421i) q^{31} +(-0.297176 + 0.514723i) q^{32} +(-0.514186 - 0.890596i) q^{33} -0.455761 q^{36} +(0.797176 - 1.38075i) q^{38} +(1.16096 + 0.598514i) q^{42} +(-0.101300 - 0.175457i) q^{44} +(-0.723734 - 1.25354i) q^{47} +0.934383 q^{48} +(0.580057 + 0.814576i) q^{49} +0.830830 q^{50} +(0.307937 + 0.533363i) q^{54} +(0.967192 + 0.498622i) q^{56} +3.01685 q^{57} +(-0.0395325 - 0.0684723i) q^{58} +(0.142315 + 0.246497i) q^{61} -1.65414 q^{62} +(0.0700176 + 1.46985i) q^{63} +1.08816 q^{64} +(-0.427201 + 0.739934i) q^{66} +(0.800620 + 1.38672i) q^{72} +(0.786053 + 1.36148i) q^{75} +0.594351 q^{76} +(-0.550294 + 0.353653i) q^{77} +(0.153077 - 0.265136i) q^{81} +(0.0231684 + 0.486364i) q^{84} +(0.0748038 - 0.129564i) q^{87} +(-0.355901 + 0.616439i) q^{88} +(-0.928368 - 1.60798i) q^{89} +(-1.56499 - 2.71064i) q^{93} +(-0.601300 + 1.04148i) q^{94} +(0.467192 + 0.809200i) q^{96} -1.30972 q^{97} +(0.345139 - 0.755750i) q^{98} -0.962573 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9}+O(q^{10}) $ 20 * q + 2 * q^2 - q^3 - 8 * q^4 + 4 * q^6 + q^7 - 8 * q^8 - 11 * q^9	$ 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9} - q^{11} - 3 q^{12} + 2 q^{14} - 6 q^{16} + 2 q^{19} + 2 q^{21} + 4 q^{22} - 4 q^{24} - 10 q^{25} - 2 q^{27} - 6 q^{28} + 2 q^{29} - q^{31} + 6 q^{32} + q^{33} + 22 q^{36} + 4 q^{38} + 20 q^{42} + 8 q^{44} - q^{47} - 12 q^{48} + q^{49} - 4 q^{50} - 20 q^{54} + 4 q^{56} + 4 q^{57} - 2 q^{58} + 2 q^{61} - 18 q^{62} + 8 q^{64} + 2 q^{66} + 11 q^{72} - q^{75} - 12 q^{76} + 2 q^{77} - 12 q^{81} + 19 q^{84} + q^{87} - 4 q^{88} - q^{89} + q^{93} - 2 q^{94} - 6 q^{96} - 4 q^{97} + 18 q^{98}+O(q^{100}) $ 20 * q + 2 * q^2 - q^3 - 8 * q^4 + 4 * q^6 + q^7 - 8 * q^8 - 11 * q^9 - q^11 - 3 * q^12 + 2 * q^14 - 6 * q^16 + 2 * q^19 + 2 * q^21 + 4 * q^22 - 4 * q^24 - 10 * q^25 - 2 * q^27 - 6 * q^28 + 2 * q^29 - q^31 + 6 * q^32 + q^33 + 22 * q^36 + 4 * q^38 + 20 * q^42 + 8 * q^44 - q^47 - 12 * q^48 + q^49 - 4 * q^50 - 20 * q^54 + 4 * q^56 + 4 * q^57 - 2 * q^58 + 2 * q^61 - 18 * q^62 + 8 * q^64 + 2 * q^66 + 11 * q^72 - q^75 - 12 * q^76 + 2 * q^77 - 12 * q^81 + 19 * q^84 + q^87 - 4 * q^88 - q^89 + q^93 - 2 * q^94 - 6 * q^96 - 4 * q^97 + 18 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1169\mathbb{Z}\right)^\times$.

$n$	$673$	$836$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$2$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$3$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$3$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$4$	0.154861	−	0.268227i	0.154861	−	0.268227i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	−1.30615			−1.30615
$7$	−0.888835	−	0.458227i	−0.888835	−	0.458227i
$7$	−0.888835	−	0.458227i	−0.888835	−	0.458227i
$8$	−1.08816			−1.08816
$9$	−0.735759	−	1.27437i	−0.735759	−	1.27437i
$10$		0			0
$11$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$11$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$12$	−0.243458	−	0.421681i	−0.243458	−	0.421681i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	0.0395325	+	0.829889i	0.0395325	+	0.829889i
$15$		0			0
$16$	0.297176	+	0.514723i	0.297176	+	0.514723i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$	−0.611291	+	1.05879i	−0.611291	+	1.05879i
$19$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$19$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$20$		0			0
$21$	−1.32254	+	0.849945i	−1.32254	+	0.849945i
$22$	−0.543476			−0.543476
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	−0.855348	+	1.48151i	−0.855348	+	1.48151i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$	−0.741276			−0.741276
$28$	−0.260554	+	0.167448i	−0.260554	+	0.167448i
$29$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$29$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$30$		0			0
$31$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$31$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$32$	−0.297176	+	0.514723i	−0.297176	+	0.514723i
$33$	−0.514186	−	0.890596i	−0.514186	−	0.890596i
$34$		0			0
$35$		0			0
$36$	−0.455761			−0.455761
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	0.797176	−	1.38075i	0.797176	−	1.38075i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	1.16096	+	0.598514i	1.16096	+	0.598514i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	−0.101300	−	0.175457i	−0.101300	−	0.175457i
$45$		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	$-0.424242\pi$
$48$	0.934383			0.934383
$49$	0.580057	+	0.814576i	0.580057	+	0.814576i
$50$	0.830830			0.830830
$51$		0			0
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$	0.307937	+	0.533363i	0.307937	+	0.533363i
$55$		0			0
$56$	0.967192	+	0.498622i	0.967192	+	0.498622i
$57$	3.01685			3.01685
$58$	−0.0395325	−	0.0684723i	−0.0395325	−	0.0684723i
$59$		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	$0.787879\pi$
$62$	−1.65414			−1.65414
$63$	0.0700176	+	1.46985i	0.0700176	+	1.46985i
$64$	1.08816			1.08816
$65$		0			0
$66$	−0.427201	+	0.739934i	−0.427201	+	0.739934i
$67$		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			$\pi$
$72$	0.800620	+	1.38672i	0.800620	+	1.38672i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$	0.786053	+	1.36148i	0.786053	+	1.36148i
$76$	0.594351			0.594351
$77$	−0.550294	+	0.353653i	−0.550294	+	0.353653i
$78$		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
$81$	0.153077	−	0.265136i	0.153077	−	0.265136i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	0.0231684	+	0.486364i	0.0231684	+	0.486364i
$85$		0			0
$86$		0			0
$87$	0.0748038	−	0.129564i	0.0748038	−	0.129564i
$88$	−0.355901	+	0.616439i	−0.355901	+	0.616439i
$89$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$89$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−1.56499	−	2.71064i	−1.56499	−	2.71064i
$94$	−0.601300	+	1.04148i	−0.601300	+	1.04148i
$95$		0			0
$96$	0.467192	+	0.809200i	0.467192	+	0.809200i
$97$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$97$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$98$	0.345139	−	0.755750i	0.345139	−	0.755750i
$99$	−0.962573			−0.962573
$100$	0.154861	+	0.268227i	0.154861	+	0.268227i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$107$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$108$	−0.114795	+	0.198830i	−0.114795	+	0.198830i
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$	−0.0282804	−	0.593678i	−0.0282804	−	0.593678i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	−1.25324	−	2.17068i	−1.25324	−	2.17068i
$115$		0			0
$116$	0.0147371	−	0.0255255i	0.0147371	−	0.0255255i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.286053	+	0.495458i	0.286053	+	0.495458i
$122$	0.118239	−	0.204797i	0.118239	−	0.204797i
$123$		0			0
$124$	−0.308319	−	0.534024i	−0.308319	−	0.534024i
$125$		0			0
$126$	1.02850	−	0.660977i	1.02850	−	0.660977i
$127$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$127$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$128$	−0.154861	−	0.268227i	−0.154861	−	0.268227i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	−0.318509			−0.318509
$133$	−0.0913090	−	1.91681i	−0.0913090	−	1.91681i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$137$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	−2.27557			−2.27557
$142$		0			0
$143$		0			0
$144$	0.437299	−	0.757424i	0.437299	−	0.757424i
$145$		0			0
$146$		0			0
$147$	1.56499	−	0.149438i	1.56499	−	0.149438i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	0.653077	−	1.13116i	0.653077	−	1.13116i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	−1.04408	−	1.80840i	−1.04408	−	1.80840i
$153$		0			0
$154$	0.483061	+	0.249035i	0.483061	+	0.249035i
$155$		0			0
$156$		0			0
$157$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$157$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.254361			−0.254361
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.00000			1.00000
$167$	1.00000			1.00000
$168$	1.43913	−	0.924872i	1.43913	−	0.924872i
$169$	1.00000			1.00000
$170$		0			0
$171$	1.41191	−	2.44550i	1.41191	−	2.44550i
$172$		0			0
$173$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$173$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$174$	−0.124299			−0.124299
$175$	0.841254	−	0.540641i	0.841254	−	0.540641i
$176$	0.388786			0.388786
$177$		0			0
$178$	−0.771316	+	1.33596i	−0.771316	+	1.33596i
$179$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$179$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$180$		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
$183$	0.447468			0.447468
$184$		0			0
$185$		0			0
$186$	−1.30024	+	2.25208i	−1.30024	+	2.25208i
$187$		0			0
$188$	−0.448312			−0.448312
$189$	0.658873	+	0.339672i	0.658873	+	0.339672i
$190$		0			0
$191$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$191$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$192$	0.855348	−	1.48151i	0.855348	−	1.48151i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$	0.544078	+	0.942371i	0.544078	+	0.942371i
$195$		0			0
$196$	0.308319	−	0.0294409i	0.308319	−	0.0294409i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	0.399867	+	0.692590i	0.399867	+	0.692590i
$199$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$199$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$200$	0.544078	−	0.942371i	0.544078	−	0.942371i
$201$		0			0
$202$		0			0
$203$	−0.0845850	−	0.0436066i	−0.0845850	−	0.0436066i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.25528			1.25528
$210$		0			0
$211$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$211$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$212$		0			0
$213$		0			0
$214$	0.738471	−	1.27907i	0.738471	−	1.27907i
$215$		0			0
$216$	0.806624			0.806624
$217$	−1.67489	+	1.07639i	−1.67489	+	1.07639i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$223$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$224$	0.500000	−	0.321330i	0.500000	−	0.321330i
$225$	1.47152			1.47152
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	0.467192	−	0.809200i	0.467192	−	0.809200i
$229$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$229$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$230$		0			0
$231$	0.0489319	+	1.02721i	0.0489319	+	1.02721i
$232$	−0.103553			−0.103553
$233$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$233$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$239$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	0.237662	−	0.411642i	0.237662	−	0.411642i
$243$	−0.611291	−	1.05879i	−0.611291	−	1.05879i
$244$	0.0881559			0.0881559
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−1.08323	+	1.87621i	−1.08323	+	1.87621i
$249$		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	$0.181818\pi$
$252$	0.405096	+	0.208842i	0.405096	+	0.208842i
$253$		0			0
$254$	−0.815816	−	1.41303i	−0.815816	−	1.41303i
$255$		0			0
$256$	0.415415	−	0.719520i	0.415415	−	0.719520i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.0700176	−	0.121274i	−0.0700176	−	0.121274i
$262$		0			0
$263$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$263$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$264$	0.559514	+	0.969107i	0.559514	+	0.969107i
$265$		0			0
$266$	−1.34125	+	0.861971i	−1.34125	+	0.861971i
$267$	−2.91899			−2.91899
$268$		0			0
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$		0			0
$273$		0			0
$274$	1.63163			1.63163
$275$	0.327068	+	0.566498i	0.327068	+	0.566498i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$	−2.92971			−2.92971
$280$		0			0
$281$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$281$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$282$	0.945307	+	1.63732i	0.945307	+	1.63732i
$283$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$283$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.874598			0.874598
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$		0			0
$291$	−1.02951	+	1.78316i	−1.02951	+	1.78316i
$292$		0			0
$293$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$293$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$294$	−0.757643	−	1.06396i	−0.757643	−	1.06396i
$295$		0			0
$296$		0			0
$297$	−0.242448	+	0.419932i	−0.242448	+	0.419932i
$298$		0			0
$299$		0			0
$300$	0.486915			0.486915
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.570276	+	0.987747i	−0.570276	+	0.987747i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	0.00964009	+	0.202370i	0.00964009	+	0.202370i
$309$		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
$319$	0.0311250	−	0.0539102i	0.0311250	−	0.0539102i
$320$		0			0
$321$	2.79469			2.79469
$322$		0			0
$323$		0			0
$324$	−0.0474111	−	0.0821184i	−0.0474111	−	0.0821184i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.0688733	+	1.44583i	0.0688733	+	1.44583i
$330$		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
$334$	−0.415415	−	0.719520i	−0.415415	−	0.719520i
$335$		0			0
$336$	−0.830513	−	0.428159i	−0.830513	−	0.428159i
$337$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$337$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$338$	−0.415415	−	0.719520i	−0.415415	−	0.719520i
$339$		0			0
$340$		0			0
$341$	−0.651174	−	1.12787i	−0.651174	−	1.12787i
$342$	−2.34612			−2.34612
$343$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$344$		0			0
$345$		0			0
$346$	−0.815816	+	1.41303i	−0.815816	+	1.41303i
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$	−0.0231684	−	0.0401288i	−0.0231684	−	0.0401288i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−0.738471	−	0.380708i	−0.738471	−	0.380708i
$351$		0			0
$352$	0.194393	+	0.336699i	0.194393	+	0.336699i
$353$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$353$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$354$		0			0
$355$		0			0
$356$	−0.575071			−0.575071
$357$		0			0
$358$	1.20260			1.20260
$359$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$359$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$360$		0			0
$361$	−1.34125	+	2.32312i	−1.34125	+	2.32312i
$362$	−0.601300	−	1.04148i	−0.601300	−	1.04148i
$363$	0.899412			0.899412
$364$		0			0
$365$		0			0
$366$	−0.185885	−	0.321962i	−0.185885	−	0.321962i
$367$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$367$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.969420			−0.969420
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$	0.787535	+	1.36405i	0.787535	+	1.36405i
$377$		0			0
$378$	−0.0293045	−	0.615177i	−0.0293045	−	0.615177i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	1.54370	−	2.67376i	1.54370	−	2.67376i
$382$	−0.481929	+	0.834725i	−0.481929	+	0.834725i
$383$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$383$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$384$	−0.486915			−0.486915
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.202824	+	0.351302i	−0.202824	+	0.351302i
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$	−0.631192	−	0.886386i	−0.631192	−	0.886386i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.149065	+	0.258188i	−0.149065	+	0.258188i
$397$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$397$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$398$	1.63163			1.63163
$399$	−2.68148	−	1.38240i	−2.68148	−	1.38240i
$400$	−0.594351			−0.594351
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	0.00376206	+	0.0789754i	0.00376206	+	0.0789754i
$407$		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
$411$	1.54370	+	2.67376i	1.54370	+	2.67376i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	−0.521461	−	0.903197i	−0.521461	−	0.903197i
$419$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$419$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$420$		0			0
$421$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$421$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$422$	−0.481929	−	0.834725i	−0.481929	−	0.834725i
$423$	−1.06499	+	1.84461i	−1.06499	+	1.84461i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.0135432	−	0.284307i	−0.0135432	−	0.284307i
$428$	0.550583			0.550583
$429$		0			0
$430$		0			0
$431$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$431$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$432$	−0.220289	−	0.381552i	−0.220289	−	0.381552i
$433$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$433$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$434$	1.47025	+	0.757969i	1.47025	+	0.757969i
$435$		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
$441$	0.611291	−	1.33854i	0.611291	−	1.33854i
$442$		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
$446$	−0.771316	−	1.33596i	−0.771316	−	1.33596i
$447$		0			0
$448$	−0.967192	−	0.498622i	−0.967192	−	0.498622i
$449$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$449$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$450$	−0.611291	−	1.05879i	−0.611291	−	1.05879i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	−3.28280			−3.28280
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$	−0.0395325	+	0.0684723i	−0.0395325	+	0.0684723i
$459$		0			0
$460$		0			0
$461$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$461$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$462$	0.718768	−	0.461924i	0.718768	−	0.461924i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	0.0282804	+	0.0489830i	0.0282804	+	0.0489830i
$465$		0			0
$466$	0.827068	−	1.43252i	0.827068	−	1.43252i
$467$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$467$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.911911	+	1.57948i	0.911911	+	1.57948i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.91899			−1.91899
$476$		0			0
$477$		0			0
$478$	0.653077	+	1.13116i	0.653077	+	1.13116i
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.177194			0.177194
$485$		0			0
$486$	−0.507879	+	0.879672i	−0.507879	+	0.879672i
$487$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$487$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$488$	−0.154861	−	0.268227i	−0.154861	−	0.268227i
$489$		0			0
$490$		0			0
$491$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$491$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.18332			1.18332
$497$		0			0
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$		0			0
$501$	0.786053	−	1.36148i	0.786053	−	1.36148i
$502$	−0.698939	−	1.21060i	−0.698939	−	1.21060i
$503$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$503$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$504$	−0.0761901	−	1.59943i	−0.0761901	−	1.59943i
$505$		0			0
$506$		0			0
$507$	0.786053	−	1.36148i	0.786053	−	1.36148i
$508$	0.304124	−	0.526759i	0.304124	−	0.526759i
$509$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$509$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$	−0.711249	−	1.23192i	−0.711249	−	1.23192i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−0.946841			−0.946841
$518$		0			0
$519$	−3.08739			−3.08739
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$	−0.0581728	+	0.100758i	−0.0581728	+	0.100758i
$523$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$523$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$524$		0			0
$525$	−0.0748038	−	1.57033i	−0.0748038	−	1.57033i
$526$	−0.236479			−0.236479
$527$		0			0
$528$	0.305607	−	0.529326i	0.305607	−	0.529326i
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−0.528280	−	0.272347i	−0.528280	−	0.272347i
$533$		0			0
$534$	1.21259	+	2.10027i	1.21259	+	2.10027i
$535$		0			0
$536$		0			0
$537$	1.13779	+	1.97070i	1.13779	+	1.97070i
$538$		0			0
$539$	0.651174	−	0.0621796i	0.651174	−	0.0621796i
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$		0			0
$543$	1.13779	−	1.97070i	1.13779	−	1.97070i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	0.304124	+	0.526759i	0.304124	+	0.526759i
$549$	0.209419	−	0.362724i	0.209419	−	0.362724i
$550$	0.271738	−	0.470664i	0.271738	−	0.470664i
$551$	0.0913090	+	0.158152i	0.0913090	+	0.158152i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$557$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$558$	1.21705	+	2.10798i	1.21705	+	2.10798i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.118239	+	0.204797i	0.118239	+	0.204797i
$563$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$563$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$564$	−0.352397	+	0.610369i	−0.352397	+	0.610369i
$565$		0			0
$566$	−0.543476			−0.543476
$567$	−0.257552	+	0.165519i	−0.257552	+	0.165519i
$568$		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
$573$	−1.82382			−1.82382
$574$		0			0
$575$		0			0
$576$	−0.800620	−	1.38672i	−0.800620	−	1.38672i
$577$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$577$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$578$	−0.415415	+	0.719520i	−0.415415	+	0.719520i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	1.71070			1.71070
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.827068	+	1.43252i	0.827068	+	1.43252i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	0.202272	−	0.442913i	0.202272	−	0.442913i
$589$	3.82059			3.82059
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$	0.402866			0.402866
$595$		0			0
$596$		0			0
$597$	1.54370	+	2.67376i	1.54370	+	2.67376i
$598$		0			0
$599$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$599$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$600$	−0.855348	−	1.48151i	−0.855348	−	1.48151i
$601$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$601$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	−1.14055			−1.14055
$609$	−0.125858	+	0.0808840i	−0.125858	+	0.0808840i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$613$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$614$		0			0
$615$		0			0
$616$	0.598806	−	0.384829i	0.598806	−	0.384829i
$617$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$617$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$	−0.830830			−0.830830
$623$	0.0883470	+	1.85463i	0.0883470	+	1.85463i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$	0.986715	−	1.70904i	0.986715	−	1.70904i
$628$	0.179656	+	0.311173i	0.179656	+	0.311173i
$629$		0			0
$630$		0			0
$631$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$631$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$632$		0			0
$633$	0.911911	−	1.57948i	0.911911	−	1.57948i
$634$	0.653077	−	1.13116i	0.653077	−	1.13116i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.0517192			−0.0517192
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	−1.16096	−	2.01083i	−1.16096	−	2.01083i
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$	−0.166571	+	0.288510i	−0.166571	+	0.288510i
$649$		0			0
$650$		0			0
$651$	0.148930	+	3.12643i	0.148930	+	3.12643i
$652$		0			0
$653$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$653$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	1.01169	−	0.650175i	1.01169	−	0.650175i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		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
$668$	0.154861	−	0.268227i	0.154861	−	0.268227i
$669$	1.45949	−	2.52792i	1.45949	−	2.52792i
$670$		0			0
$671$	0.186186			0.186186
$672$	−0.0444597	−	0.933325i	−0.0444597	−	0.933325i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	0.797176	+	1.38075i	0.797176	+	1.38075i
$675$	0.370638	−	0.641964i	0.370638	−	0.641964i
$676$	0.154861	−	0.268227i	0.154861	−	0.268227i
$677$	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
$679$	1.16413	+	0.600149i	1.16413	+	0.600149i
$680$		0			0
$681$		0			0
$682$	−0.541015	+	0.937065i	−0.541015	+	0.937065i
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$	−0.437299	−	0.757424i	−0.437299	−	0.757424i
$685$		0			0
$686$	−0.653077	+	0.513585i	−0.653077	+	0.513585i
$687$	−0.149608			−0.149608
$688$		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	$0.333333\pi$
$692$	−0.608249			−0.608249
$693$	0.855569	+	0.441076i	0.855569	+	0.441076i
$694$		0			0
$695$		0			0
$696$	−0.0813982	+	0.140986i	−0.0813982	+	0.140986i
$697$		0			0
$698$		0			0
$699$	3.12998			3.12998
$700$	−0.0147371	−	0.309371i	−0.0147371	−	0.309371i
$701$	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
$704$	0.355901	−	0.616439i	0.355901	−	0.616439i
$705$		0			0
$706$	−1.47694			−1.47694
$707$		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
$712$	1.01021	+	1.74973i	1.01021	+	1.74973i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.224156	+	0.388250i	0.224156	+	0.388250i
$717$	−1.23576	+	2.14040i	−1.23576	+	2.14040i
$718$	0.827068	−	1.43252i	0.827068	−	1.43252i
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$	2.22871			2.22871
$723$		0			0
$724$	0.224156	−	0.388250i	0.224156	−	0.388250i
$725$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i
$726$	−0.373629	−	0.647145i	−0.373629	−	0.647145i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−1.61587			−1.61587
$730$		0			0
$731$		0			0
$732$	0.0692952	−	0.120023i	0.0692952	−	0.120023i
$733$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$733$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$734$	0.391751			0.391751
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$739$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$743$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$744$	1.70295	+	2.94960i	1.70295	+	2.94960i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.0845850	−	1.77566i	−0.0845850	−	1.77566i
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$	0.430152	−	0.745045i	0.430152	−	0.745045i
$753$	1.32254	−	2.29071i	1.32254	−	2.29071i
$754$		0			0
$755$		0			0
$756$	0.193143	−	0.124125i	0.193143	−	0.124125i
$757$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$757$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$761$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$762$	−2.56510			−2.56510
$763$		0			0
$764$	−0.359312			−0.359312
$765$		0			0
$766$	−0.0395325	+	0.0684723i	−0.0395325	+	0.0684723i
$767$		0			0
$768$	−0.653077	−	1.13116i	−0.653077	−	1.13116i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	0.995472	+	1.72421i	0.995472	+	1.72421i
$776$	1.42518			1.42518
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−0.0705427			−0.0705427
$784$	−0.246902	+	0.540641i	−0.246902	+	0.540641i
$785$		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	−	0.387519i	−0.223734	−	0.387519i
$790$		0			0
$791$		0			0
$792$	1.04743			1.04743
$793$		0			0
$794$	−0.771316	+	1.33596i	−0.771316	+	1.33596i
$795$		0			0
$796$	0.304124	+	0.526759i	0.304124	+	0.526759i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$	0.119264	+	2.50365i	0.119264	+	2.50365i
$799$		0			0
$800$	−0.297176	−	0.514723i	−0.297176	−	0.514723i
$801$	−1.36611	+	2.36617i	−1.36611	+	2.36617i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$809$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−0.0247953	+	0.0159350i	−0.0247953	+	0.0159350i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.63163			1.63163
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	1.28255	−	2.22144i	1.28255	−	2.22144i
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$	1.02837			1.02837
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$	0.194393	−	0.336699i	0.194393	−	0.336699i
$837$	−0.737920	+	1.27811i	−0.737920	+	1.27811i
$838$	−0.481929	−	0.834725i	−0.481929	−	0.834725i
$839$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$839$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$840$		0			0
$841$	−0.990944			−0.990944
$842$	0.738471	+	1.27907i	0.738471	+	1.27907i
$843$	−0.223734	+	0.387519i	−0.223734	+	0.387519i
$844$	0.179656	−	0.311173i	0.179656	−	0.311173i
$845$		0			0
$846$	1.76965			1.76965
$847$	−0.0272219	−	0.571458i	−0.0272219	−	0.571458i
$848$		0			0
$849$	−0.514186	−	0.890596i	−0.514186	−	0.890596i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$853$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$854$	−0.198939	+	0.127850i	−0.198939	+	0.127850i
$855$		0			0
$856$	−0.967192	−	1.67522i	−0.967192	−	1.67522i
$857$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$857$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$858$		0			0
$859$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$859$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$860$		0			0
$861$		0			0
$862$	1.54263			1.54263
$863$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$863$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$864$	0.220289	−	0.381552i	0.220289	−	0.381552i
$865$		0			0
$866$	−0.195876	−	0.339266i	−0.195876	−	0.339266i
$867$	−1.57211			−1.57211
$868$	0.0293408	+	0.615940i	0.0293408	+	0.615940i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.963639	+	1.66907i	0.963639	+	1.66907i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$877$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$878$		0			0
$879$	−1.56499	+	2.71064i	−1.56499	+	2.71064i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−1.21705	+	0.116214i	−1.21705	+	0.116214i
$883$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$883$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	−1.74555	−	0.899892i	−1.74555	−	0.899892i
$890$		0			0
$891$	−0.100133	−	0.173435i	−0.100133	−	0.173435i
$892$	0.287535	−	0.498026i	0.287535	−	0.498026i
$893$	1.38884	−	2.40553i	1.38884	−	2.40553i
$894$		0			0
$895$		0			0
$896$	0.0147371	+	0.309371i	0.0147371	+	0.309371i
$897$		0			0
$898$	−0.601300	−	1.04148i	−0.601300	−	1.04148i
$899$	0.0947329	−	0.164082i	0.0947329	−	0.164082i
$900$	0.227880	−	0.394700i	0.227880	−	0.394700i
$901$		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	−0.142315	−	0.989821i	$-0.545455\pi$
$907$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$911$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$912$	0.896534	+	1.55284i	0.896534	+	1.55284i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.0294743			−0.0294743
$917$		0			0
$918$		0			0
$919$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$919$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$920$		0			0
$921$		0			0
$922$	0.797176	+	1.38075i	0.797176	+	1.38075i
$923$		0			0
$924$	0.283102	+	0.145949i	0.283102	+	0.145949i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−0.0282804	+	0.0489830i	−0.0282804	+	0.0489830i
$929$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$929$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$930$		0			0
$931$	−0.797176	+	1.74557i	−0.797176	+	1.74557i
$932$	0.616638			0.616638
$933$	−0.786053	−	1.36148i	−0.786053	−	1.36148i
$934$	0.738471	−	1.27907i	0.738471	−	1.27907i
$935$		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	$0.666667\pi$
$942$	0.757643	−	1.31228i	0.757643	−	1.31228i
$943$		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
$950$	0.797176	+	1.38075i	0.797176	+	1.38075i
$951$	2.47152			2.47152
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−0.243458	+	0.421681i	−0.243458	+	0.421681i
$957$	−0.0489319	−	0.0847525i	−0.0489319	−	0.0847525i
$958$		0			0
$959$	1.65210	−	1.06174i	1.65210	−	1.06174i
$960$		0			0
$961$	−1.48193	−	2.56678i	−1.48193	−	2.56678i
$962$		0			0
$963$	1.30794	−	2.26541i	1.30794	−	2.26541i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$967$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$968$	−0.311270	−	0.539136i	−0.311270	−	0.539136i
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	−0.378660			−0.378660
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.0845850	+	0.146505i	−0.0845850	+	0.146505i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$	−1.21456			−1.21456
$980$		0			0
$981$		0			0
$982$	0.415415	+	0.719520i	0.415415	+	0.719520i
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	2.02261	+	1.04273i	2.02261	+	1.04273i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$	0.591660	+	1.02478i	0.591660	+	1.02478i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$997$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1169.1.f.c.333.4	✓	20
7.2	even	3	inner	1169.1.f.c.667.4	yes	20
167.166	odd	2	CM	1169.1.f.c.333.4	✓	20
1169.667	odd	6	inner	1169.1.f.c.667.4	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1169.1.f.c.333.4	✓	20	1.1	even	1	trivial
1169.1.f.c.333.4	✓	20	167.166	odd	2	CM
1169.1.f.c.667.4	yes	20	7.2	even	3	inner
1169.1.f.c.667.4	yes	20	1169.667	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:	\( N \)	\(=\)	\( 1169 = 7 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1169.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.415415 - 0.719520i) q^{2} +(0.786053 - 1.36148i) q^{3} +(0.154861 - 0.268227i) q^{4} -1.30615 q^{6} +(-0.888835 - 0.458227i) q^{7} -1.08816 q^{8} +(-0.735759 - 1.27437i) q^{9} +O(q^{10})\)	\(q+(-0.415415 - 0.719520i) q^{2} +(0.786053 - 1.36148i) q^{3} +(0.154861 - 0.268227i) q^{4} -1.30615 q^{6} +(-0.888835 - 0.458227i) q^{7} -1.08816 q^{8} +(-0.735759 - 1.27437i) q^{9} +(0.327068 - 0.566498i) q^{11} +(-0.243458 - 0.421681i) q^{12} +(0.0395325 + 0.829889i) q^{14} +(0.297176 + 0.514723i) q^{16} +(-0.611291 + 1.05879i) q^{18} +(0.959493 + 1.66189i) q^{19} +(-1.32254 + 0.849945i) q^{21} -0.543476 q^{22} +(-0.855348 + 1.48151i) q^{24} +(-0.500000 + 0.866025i) q^{25} -0.741276 q^{27} +(-0.260554 + 0.167448i) q^{28} +0.0951638 q^{29} +(0.995472 - 1.72421i) q^{31} +(-0.297176 + 0.514723i) q^{32} +(-0.514186 - 0.890596i) q^{33} -0.455761 q^{36} +(0.797176 - 1.38075i) q^{38} +(1.16096 + 0.598514i) q^{42} +(-0.101300 - 0.175457i) q^{44} +(-0.723734 - 1.25354i) q^{47} +0.934383 q^{48} +(0.580057 + 0.814576i) q^{49} +0.830830 q^{50} +(0.307937 + 0.533363i) q^{54} +(0.967192 + 0.498622i) q^{56} +3.01685 q^{57} +(-0.0395325 - 0.0684723i) q^{58} +(0.142315 + 0.246497i) q^{61} -1.65414 q^{62} +(0.0700176 + 1.46985i) q^{63} +1.08816 q^{64} +(-0.427201 + 0.739934i) q^{66} +(0.800620 + 1.38672i) q^{72} +(0.786053 + 1.36148i) q^{75} +0.594351 q^{76} +(-0.550294 + 0.353653i) q^{77} +(0.153077 - 0.265136i) q^{81} +(0.0231684 + 0.486364i) q^{84} +(0.0748038 - 0.129564i) q^{87} +(-0.355901 + 0.616439i) q^{88} +(-0.928368 - 1.60798i) q^{89} +(-1.56499 - 2.71064i) q^{93} +(-0.601300 + 1.04148i) q^{94} +(0.467192 + 0.809200i) q^{96} -1.30972 q^{97} +(0.345139 - 0.755750i) q^{98} -0.962573 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9}+O(q^{10}) \) 20 * q + 2 * q^2 - q^3 - 8 * q^4 + 4 * q^6 + q^7 - 8 * q^8 - 11 * q^9	\( 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9} - q^{11} - 3 q^{12} + 2 q^{14} - 6 q^{16} + 2 q^{19} + 2 q^{21} + 4 q^{22} - 4 q^{24} - 10 q^{25} - 2 q^{27} - 6 q^{28} + 2 q^{29} - q^{31} + 6 q^{32} + q^{33} + 22 q^{36} + 4 q^{38} + 20 q^{42} + 8 q^{44} - q^{47} - 12 q^{48} + q^{49} - 4 q^{50} - 20 q^{54} + 4 q^{56} + 4 q^{57} - 2 q^{58} + 2 q^{61} - 18 q^{62} + 8 q^{64} + 2 q^{66} + 11 q^{72} - q^{75} - 12 q^{76} + 2 q^{77} - 12 q^{81} + 19 q^{84} + q^{87} - 4 q^{88} - q^{89} + q^{93} - 2 q^{94} - 6 q^{96} - 4 q^{97} + 18 q^{98}+O(q^{100}) \) 20 * q + 2 * q^2 - q^3 - 8 * q^4 + 4 * q^6 + q^7 - 8 * q^8 - 11 * q^9 - q^11 - 3 * q^12 + 2 * q^14 - 6 * q^16 + 2 * q^19 + 2 * q^21 + 4 * q^22 - 4 * q^24 - 10 * q^25 - 2 * q^27 - 6 * q^28 + 2 * q^29 - q^31 + 6 * q^32 + q^33 + 22 * q^36 + 4 * q^38 + 20 * q^42 + 8 * q^44 - q^47 - 12 * q^48 + q^49 - 4 * q^50 - 20 * q^54 + 4 * q^56 + 4 * q^57 - 2 * q^58 + 2 * q^61 - 18 * q^62 + 8 * q^64 + 2 * q^66 + 11 * q^72 - q^75 - 12 * q^76 + 2 * q^77 - 12 * q^81 + 19 * q^84 + q^87 - 4 * q^88 - q^89 + q^93 - 2 * q^94 - 6 * q^96 - 4 * q^97 + 18 * q^98

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