LMFDB - Embedded newform 1169.1.f.c.333.1

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			333.1
Root			$-0.995472 + 0.0950560i$ of defining polynomial
Character	$\chi$	$=$	1169.333
Dual form			1169.1.f.c.667.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.841254 - 1.45709i) q^{2} +(-0.981929 + 1.70075i) q^{3} +(-0.915415 + 1.58555i) q^{4} +3.30420 q^{6} +(0.723734 + 0.690079i) q^{7} +1.39788 q^{8} +(-1.42837 - 2.47401i) q^{9} +O(q^{10})$	\(q+(-0.841254 - 1.45709i) q^{2} +(-0.981929 + 1.70075i) q^{3} +(-0.915415 + 1.58555i) q^{4} +3.30420 q^{6} +(0.723734 + 0.690079i) q^{7} +1.39788 q^{8} +(-1.42837 - 2.47401i) q^{9} +(0.995472 - 1.72421i) q^{11} +(-1.79774 - 3.11378i) q^{12} +(0.396666 - 1.63508i) q^{14} +(-0.260554 - 0.451293i) q^{16} +(-2.40324 + 4.16253i) q^{18} +(0.142315 + 0.246497i) q^{19} +(-1.88431 + 0.553283i) q^{21} -3.34978 q^{22} +(-1.37262 + 2.37744i) q^{24} +(-0.500000 + 0.866025i) q^{25} +3.64636 q^{27} +(-1.75667 + 0.515804i) q^{28} +0.471518 q^{29} +(0.888835 - 1.53951i) q^{31} +(0.260554 - 0.451293i) q^{32} +(1.95496 + 3.38610i) q^{33} +5.23020 q^{36} +(0.239446 - 0.414732i) q^{38} +(2.39136 + 2.28016i) q^{42} +(1.82254 + 3.15673i) q^{44} +(0.786053 + 1.36148i) q^{47} +1.02338 q^{48} +(0.0475819 + 0.998867i) q^{49} +1.68251 q^{50} +(-3.06752 - 5.31310i) q^{54} +(1.01169 + 0.964646i) q^{56} -0.558972 q^{57} +(-0.396666 - 0.687046i) q^{58} +(0.654861 + 1.13425i) q^{61} -2.99094 q^{62} +(0.673501 - 2.77621i) q^{63} -1.39788 q^{64} +(3.28924 - 5.69713i) q^{66} +(-1.99668 - 3.45836i) q^{72} +(-0.981929 - 1.70075i) q^{75} -0.521109 q^{76} +(1.91030 - 0.560914i) q^{77} +(-2.15210 + 3.72755i) q^{81} +(0.847669 - 3.49414i) q^{84} +(-0.462997 + 0.801934i) q^{87} +(1.39155 - 2.41023i) q^{88} +(0.327068 + 0.566498i) q^{89} +(1.74555 + 3.02337i) q^{93} +(1.32254 - 2.29071i) q^{94} +(0.511691 + 0.886276i) q^{96} +0.830830 q^{97} +(1.41542 - 0.909632i) q^{98} -5.68760 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$2$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$3$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$3$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$4$	−0.915415	+	1.58555i	−0.915415	+	1.58555i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	3.30420			3.30420
$7$	0.723734	+	0.690079i	0.723734	+	0.690079i
$7$	0.723734	+	0.690079i	0.723734	+	0.690079i
$8$	1.39788			1.39788
$9$	−1.42837	−	2.47401i	−1.42837	−	2.47401i
$10$		0			0
$11$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$11$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$12$	−1.79774	−	3.11378i	−1.79774	−	3.11378i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	0.396666	−	1.63508i	0.396666	−	1.63508i
$15$		0			0
$16$	−0.260554	−	0.451293i	−0.260554	−	0.451293i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$	−2.40324	+	4.16253i	−2.40324	+	4.16253i
$19$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$19$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$20$		0			0
$21$	−1.88431	+	0.553283i	−1.88431	+	0.553283i
$22$	−3.34978			−3.34978
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	−1.37262	+	2.37744i	−1.37262	+	2.37744i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$	3.64636			3.64636
$28$	−1.75667	+	0.515804i	−1.75667	+	0.515804i
$29$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$29$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$30$		0			0
$31$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$31$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$32$	0.260554	−	0.451293i	0.260554	−	0.451293i
$33$	1.95496	+	3.38610i	1.95496	+	3.38610i
$34$		0			0
$35$		0			0
$36$	5.23020			5.23020
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	0.239446	−	0.414732i	0.239446	−	0.414732i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	2.39136	+	2.28016i	2.39136	+	2.28016i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	1.82254	+	3.15673i	1.82254	+	3.15673i
$45$		0			0
$46$		0			0
$47$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$47$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$48$	1.02338			1.02338
$49$	0.0475819	+	0.998867i	0.0475819	+	0.998867i
$50$	1.68251			1.68251
$51$		0			0
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$	−3.06752	−	5.31310i	−3.06752	−	5.31310i
$55$		0			0
$56$	1.01169	+	0.964646i	1.01169	+	0.964646i
$57$	−0.558972			−0.558972
$58$	−0.396666	−	0.687046i	−0.396666	−	0.687046i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$61$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$62$	−2.99094			−2.99094
$63$	0.673501	−	2.77621i	0.673501	−	2.77621i
$64$	−1.39788			−1.39788
$65$		0			0
$66$	3.28924	−	5.69713i	3.28924	−	5.69713i
$67$		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			$\pi$
$72$	−1.99668	−	3.45836i	−1.99668	−	3.45836i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$	−0.981929	−	1.70075i	−0.981929	−	1.70075i
$76$	−0.521109			−0.521109
$77$	1.91030	−	0.560914i	1.91030	−	0.560914i
$78$		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
$81$	−2.15210	+	3.72755i	−2.15210	+	3.72755i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	0.847669	−	3.49414i	0.847669	−	3.49414i
$85$		0			0
$86$		0			0
$87$	−0.462997	+	0.801934i	−0.462997	+	0.801934i
$88$	1.39155	−	2.41023i	1.39155	−	2.41023i
$89$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$89$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	1.74555	+	3.02337i	1.74555	+	3.02337i
$94$	1.32254	−	2.29071i	1.32254	−	2.29071i
$95$		0			0
$96$	0.511691	+	0.886276i	0.511691	+	0.886276i
$97$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$97$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$98$	1.41542	−	0.909632i	1.41542	−	0.909632i
$99$	−5.68760			−5.68760
$100$	−0.915415	−	1.58555i	−0.915415	−	1.58555i
$101$		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
$107$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$107$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$108$	−3.33794	+	5.78148i	−3.33794	+	5.78148i
$109$		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
$112$	0.122856	−	0.506419i	0.122856	−	0.506419i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	0.470237	+	0.814475i	0.470237	+	0.814475i
$115$		0			0
$116$	−0.431635	+	0.747613i	−0.431635	+	0.747613i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.48193	−	2.56678i	−1.48193	−	2.56678i
$122$	1.10181	−	1.90839i	1.10181	−	1.90839i
$123$		0			0
$124$	1.62731	+	2.81858i	1.62731	+	2.81858i
$125$		0			0
$126$	−4.61178	+	1.35414i	−4.61178	+	1.35414i
$127$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$127$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$128$	0.915415	+	1.58555i	0.915415	+	1.58555i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	−7.15842			−7.15842
$133$	−0.0671040	+	0.276606i	−0.0671040	+	0.276606i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$137$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	−3.08739			−3.08739
$142$		0			0
$143$		0			0
$144$	−0.744335	+	1.28923i	−0.744335	+	1.28923i
$145$		0			0
$146$		0			0
$147$	−1.74555	−	0.899892i	−1.74555	−	0.899892i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	−1.65210	+	2.86152i	−1.65210	+	2.86152i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	0.198939	+	0.344572i	0.198939	+	0.344572i
$153$		0			0
$154$	−2.42435	−	2.31161i	−2.42435	−	2.31161i
$155$		0			0
$156$		0			0
$157$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$157$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	7.24185			7.24185
$163$		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
$168$	−2.63403	+	0.773421i	−2.63403	+	0.773421i
$169$	1.00000			1.00000
$170$		0			0
$171$	0.406556	−	0.704175i	0.406556	−	0.704175i
$172$		0			0
$173$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$173$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$174$	1.55799			1.55799
$175$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$176$	−1.03750			−1.03750
$177$		0			0
$178$	0.550294	−	0.953137i	0.550294	−	0.953137i
$179$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$179$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$180$		0			0
$181$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$181$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$182$		0			0
$183$	−2.57211			−2.57211
$184$		0			0
$185$		0			0
$186$	2.93689	−	5.08685i	2.93689	−	5.08685i
$187$		0			0
$188$	−2.87826			−2.87826
$189$	2.63900	+	2.51628i	2.63900	+	2.51628i
$190$		0			0
$191$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$191$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$192$	1.37262	−	2.37744i	1.37262	−	2.37744i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$	−0.698939	−	1.21060i	−0.698939	−	1.21060i
$195$		0			0
$196$	−1.62731	−	0.838935i	−1.62731	−	0.838935i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	4.78471	+	8.28737i	4.78471	+	8.28737i
$199$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$199$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$200$	−0.698939	+	1.21060i	−0.698939	+	1.21060i
$201$		0			0
$202$		0			0
$203$	0.341254	+	0.325385i	0.341254	+	0.325385i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.566682			0.566682
$210$		0			0
$211$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$211$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$212$		0			0
$213$		0			0
$214$	−1.21769	+	2.10910i	−1.21769	+	2.10910i
$215$		0			0
$216$	5.09717			5.09717
$217$	1.70566	−	0.500828i	1.70566	−	0.500828i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$223$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$224$	0.500000	−	0.146813i	0.500000	−	0.146813i
$225$	2.85674			2.85674
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	0.511691	−	0.886276i	0.511691	−	0.886276i
$229$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$229$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$230$		0			0
$231$	−0.921801	+	3.79972i	−0.921801	+	3.79972i
$232$	0.659124			0.659124
$233$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$233$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$239$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	−2.49336	+	4.31862i	−2.49336	+	4.31862i
$243$	−2.40324	−	4.16253i	−2.40324	−	4.16253i
$244$	−2.39788			−2.39788
$245$		0			0
$246$		0			0
$247$		0			0
$248$	1.24248	−	2.15204i	1.24248	−	2.15204i
$249$		0			0
$250$		0			0
$251$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$251$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$252$	3.78527	+	3.60925i	3.78527	+	3.60925i
$253$		0			0
$254$	−0.975950	−	1.69039i	−0.975950	−	1.69039i
$255$		0			0
$256$	0.841254	−	1.45709i	0.841254	−	1.45709i
$257$		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
$261$	−0.673501	−	1.16654i	−0.673501	−	1.16654i
$262$		0			0
$263$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$263$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$264$	2.73280	+	4.73335i	2.73280	+	4.73335i
$265$		0			0
$266$	0.459493	−	0.134919i	0.459493	−	0.134919i
$267$	−1.28463			−1.28463
$268$		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
$274$	1.95190			1.95190
$275$	0.995472	+	1.72421i	0.995472	+	1.72421i
$276$		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
$279$	−5.07834			−5.07834
$280$		0			0
$281$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$281$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$282$	2.59728	+	4.49862i	2.59728	+	4.49862i
$283$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$283$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.48867			−1.48867
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$		0			0
$291$	−0.815816	+	1.41303i	−0.815816	+	1.41303i
$292$		0			0
$293$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$293$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$294$	0.157220	+	3.30046i	0.157220	+	3.30046i
$295$		0			0
$296$		0			0
$297$	3.62985	−	6.28709i	3.62985	−	6.28709i
$298$		0			0
$299$		0			0
$300$	3.59549			3.59549
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.0741615	−	0.128451i	0.0741615	−	0.128451i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	−0.859360	+	3.54233i	−0.859360	+	3.54233i
$309$		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	$0.333333\pi$
$314$	0.160114			0.160114
$315$		0			0
$316$		0			0
$317$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$317$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$318$		0			0
$319$	0.469383	−	0.812995i	0.469383	−	0.812995i
$320$		0			0
$321$	2.84262			2.84262
$322$		0			0
$323$		0			0
$324$	−3.94013	−	6.82451i	−3.94013	−	6.82451i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.370638	+	1.52779i	−0.370638	+	1.52779i
$330$		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
$334$	−0.841254	−	1.45709i	−0.841254	−	1.45709i
$335$		0			0
$336$	0.740657	+	0.706215i	0.740657	+	0.706215i
$337$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$337$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$338$	−0.841254	−	1.45709i	−0.841254	−	1.45709i
$339$		0			0
$340$		0			0
$341$	−1.76962	−	3.06507i	−1.76962	−	3.06507i
$342$	−1.36807			−1.36807
$343$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$344$		0			0
$345$		0			0
$346$	−0.975950	+	1.69039i	−0.975950	+	1.69039i
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$	−0.847669	−	1.46821i	−0.847669	−	1.46821i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	1.21769	+	1.16106i	1.21769	+	1.16106i
$351$		0			0
$352$	−0.518749	−	0.898500i	−0.518749	−	0.898500i
$353$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$353$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$354$		0			0
$355$		0			0
$356$	−1.19761			−1.19761
$357$		0			0
$358$	−2.64508			−2.64508
$359$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$359$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$360$		0			0
$361$	0.459493	−	0.795865i	0.459493	−	0.795865i
$362$	1.32254	+	2.29071i	1.32254	+	2.29071i
$363$	5.82059			5.82059
$364$		0			0
$365$		0			0
$366$	2.16379	+	3.74780i	2.16379	+	3.74780i
$367$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$367$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−6.39160			−6.39160
$373$		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
$376$	1.09881	+	1.90319i	1.09881	+	1.90319i
$377$		0			0
$378$	1.44639	−	5.96210i	1.44639	−	5.96210i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−1.13915	+	1.97306i	−1.13915	+	1.97306i
$382$	−0.0800569	+	0.138663i	−0.0800569	+	0.138663i
$383$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$383$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$384$	−3.59549			−3.59549
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.760554	+	1.31732i	−0.760554	+	1.31732i
$389$		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
$392$	0.0665137	+	1.39629i	0.0665137	+	1.39629i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	5.20652	−	9.01795i	5.20652	−	9.01795i
$397$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$397$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$398$	1.95190			1.95190
$399$	−0.404547	−	0.385735i	−0.404547	−	0.385735i
$400$	0.521109			0.521109
$401$		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
$406$	0.187035	−	0.770969i	0.187035	−	0.770969i
$407$		0			0
$408$		0			0
$409$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$409$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$410$		0			0
$411$	−1.13915	−	1.97306i	−1.13915	−	1.97306i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	−0.476723	−	0.825708i	−0.476723	−	0.825708i
$419$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$419$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$420$		0			0
$421$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$421$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$422$	−0.0800569	−	0.138663i	−0.0800569	−	0.138663i
$423$	2.24555	−	3.88940i	2.24555	−	3.88940i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.308779	+	1.27280i	−0.308779	+	1.27280i
$428$	2.65007			2.65007
$429$		0			0
$430$		0			0
$431$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$431$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$432$	−0.950076	−	1.64558i	−0.950076	−	1.64558i
$433$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$433$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$434$	−2.16465	−	2.06399i	−2.16465	−	2.06399i
$435$		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
$441$	2.40324	−	1.54447i	2.40324	−	1.54447i
$442$		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
$446$	0.550294	+	0.953137i	0.550294	+	0.953137i
$447$		0			0
$448$	−1.01169	−	0.964646i	−1.01169	−	0.964646i
$449$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$449$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$450$	−2.40324	−	4.16253i	−2.40324	−	4.16253i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	−0.781374			−0.781374
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$	−0.396666	+	0.687046i	−0.396666	+	0.687046i
$459$		0			0
$460$		0			0
$461$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$461$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$462$	6.31201	−	1.85337i	6.31201	−	1.85337i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	−0.122856	−	0.212793i	−0.122856	−	0.212793i
$465$		0			0
$466$	1.49547	−	2.59023i	1.49547	−	2.59023i
$467$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$467$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.0934441	−	0.161850i	−0.0934441	−	0.161850i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.284630			−0.284630
$476$		0			0
$477$		0			0
$478$	−1.65210	−	2.86152i	−1.65210	−	2.86152i
$479$		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
$484$	5.42632			5.42632
$485$		0			0
$486$	−4.04347	+	7.00349i	−4.04347	+	7.00349i
$487$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$487$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$488$	0.915415	+	1.58555i	0.915415	+	1.58555i
$489$		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
$496$	−0.926360			−0.926360
$497$		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
$501$	−0.981929	+	1.70075i	−0.981929	+	1.70075i
$502$	1.61435	+	2.79614i	1.61435	+	2.79614i
$503$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$503$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$504$	0.941472	−	3.88080i	0.941472	−	3.88080i
$505$		0			0
$506$		0			0
$507$	−0.981929	+	1.70075i	−0.981929	+	1.70075i
$508$	−1.06199	+	1.83941i	−1.06199	+	1.83941i
$509$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$509$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$	0.518932	+	0.898816i	0.518932	+	0.898816i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	3.12998			3.12998
$518$		0			0
$519$	2.27830			2.27830
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$	−1.13317	+	1.96271i	−1.13317	+	1.96271i
$523$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$523$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$524$		0			0
$525$	0.462997	−	1.90850i	0.462997	−	1.90850i
$526$	−2.20362			−2.20362
$527$		0			0
$528$	1.01875	−	1.76452i	1.01875	−	1.76452i
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−0.377144	−	0.359606i	−0.377144	−	0.359606i
$533$		0			0
$534$	1.08070	+	1.87183i	1.08070	+	1.87183i
$535$		0			0
$536$		0			0
$537$	1.54370	+	2.67376i	1.54370	+	2.67376i
$538$		0			0
$539$	1.76962	+	0.912303i	1.76962	+	0.912303i
$540$		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
$543$	1.54370	−	2.67376i	1.54370	−	2.67376i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−1.06199	−	1.83941i	−1.06199	−	1.83941i
$549$	1.87076	−	3.24026i	1.87076	−	3.24026i
$550$	1.67489	−	2.90099i	1.67489	−	2.90099i
$551$	0.0671040	+	0.116228i	0.0671040	+	0.116228i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$557$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$558$	4.27217	+	7.39961i	4.27217	+	7.39961i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.10181	+	1.90839i	1.10181	+	1.90839i
$563$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$563$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$564$	2.82625	−	4.89520i	2.82625	−	4.89520i
$565$		0			0
$566$	−3.34978			−3.34978
$567$	−4.12985	+	1.21263i	−4.12985	+	1.21263i
$568$		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
$573$	0.186888			0.186888
$574$		0			0
$575$		0			0
$576$	1.99668	+	3.45836i	1.99668	+	3.45836i
$577$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$577$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$578$	−0.841254	+	1.45709i	−0.841254	+	1.45709i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	2.74523			2.74523
$583$		0			0
$584$		0			0
$585$		0			0
$586$	1.49547	+	2.59023i	1.49547	+	2.59023i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	3.02472	−	1.94387i	3.02472	−	1.94387i
$589$	0.505978			0.505978
$590$		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	$0.333333\pi$
$594$	−12.2145			−12.2145
$595$		0			0
$596$		0			0
$597$	−1.13915	−	1.97306i	−1.13915	−	1.97306i
$598$		0			0
$599$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$599$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$600$	−1.37262	−	2.37744i	−1.37262	−	2.37744i
$601$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$601$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$602$		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	$0.333333\pi$
$608$	0.148323			0.148323
$609$	−0.888485	+	0.260883i	−0.888485	+	0.260883i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$613$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$614$		0			0
$615$		0			0
$616$	2.67036	−	0.784089i	2.67036	−	0.784089i
$617$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$617$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$618$		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
$622$	−1.68251			−1.68251
$623$	−0.154218	+	0.635697i	−0.154218	+	0.635697i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$	−0.556441	+	0.963784i	−0.556441	+	0.963784i
$628$	−0.0871144	−	0.150887i	−0.0871144	−	0.150887i
$629$		0			0
$630$		0			0
$631$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$631$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$632$		0			0
$633$	−0.0934441	+	0.161850i	−0.0934441	+	0.161850i
$634$	−1.65210	+	2.86152i	−1.65210	+	2.86152i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−1.57948			−1.57948
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	−2.39136	−	4.14197i	−2.39136	−	4.14197i
$643$		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	$0.666667\pi$
$648$	−3.00837	+	5.21066i	−3.00837	+	5.21066i
$649$		0			0
$650$		0			0
$651$	−0.823056	+	3.39268i	−0.823056	+	3.39268i
$652$		0			0
$653$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$653$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	2.53794	−	0.745205i	2.53794	−	0.745205i
$659$		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
$668$	−0.915415	+	1.58555i	−0.915415	+	1.58555i
$669$	0.642315	−	1.11252i	0.642315	−	1.11252i
$670$		0			0
$671$	2.60758			2.60758
$672$	−0.241272	+	0.994535i	−0.241272	+	0.994535i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	0.239446	+	0.414732i	0.239446	+	0.414732i
$675$	−1.82318	+	3.15784i	−1.82318	+	3.15784i
$676$	−0.915415	+	1.58555i	−0.915415	+	1.58555i
$677$	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
$679$	0.601300	+	0.573338i	0.601300	+	0.573338i
$680$		0			0
$681$		0			0
$682$	−2.97740	+	5.15701i	−2.97740	+	5.15701i
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$	0.744335	+	1.28923i	0.744335	+	1.28923i
$685$		0			0
$686$	1.65210	+	0.318417i	1.65210	+	0.318417i
$687$	0.925994			0.925994
$688$		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	$0.333333\pi$
$692$	2.12397			2.12397
$693$	−4.11631	−	3.92489i	−4.11631	−	3.92489i
$694$		0			0
$695$		0			0
$696$	−0.647213	+	1.12101i	−0.647213	+	1.12101i
$697$		0			0
$698$		0			0
$699$	−3.49109			−3.49109
$700$	0.431635	−	1.77922i	0.431635	−	1.77922i
$701$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$701$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$702$		0			0
$703$		0			0
$704$	−1.39155	+	2.41023i	−1.39155	+	2.41023i
$705$		0			0
$706$	2.43538			2.43538
$707$		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
$712$	0.457201	+	0.791895i	0.457201	+	0.791895i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.43913	+	2.49265i	1.43913	+	2.49265i
$717$	−1.92837	+	3.34003i	−1.92837	+	3.34003i
$718$	1.49547	−	2.59023i	1.49547	−	2.59023i
$719$		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
$722$	−1.54620			−1.54620
$723$		0			0
$724$	1.43913	−	2.49265i	1.43913	−	2.49265i
$725$	−0.235759	+	0.408346i	−0.235759	+	0.408346i
$726$	−4.89659	−	8.48115i	−4.89659	−	8.48115i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	5.13503			5.13503
$730$		0			0
$731$		0			0
$732$	2.35454	−	4.07819i	2.35454	−	4.07819i
$733$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$733$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$734$	3.12397			3.12397
$735$		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	$0.666667\pi$
$744$	2.44006	+	4.22631i	2.44006	+	4.22631i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.341254	−	1.40667i	0.341254	−	1.40667i
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$	0.409619	−	0.709481i	0.409619	−	0.709481i
$753$	1.88431	−	3.26372i	1.88431	−	3.26372i
$754$		0			0
$755$		0			0
$756$	−6.40545	+	1.88081i	−6.40545	+	1.88081i
$757$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$757$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$761$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$762$	3.83325			3.83325
$763$		0			0
$764$	0.174229			0.174229
$765$		0			0
$766$	−0.396666	+	0.687046i	−0.396666	+	0.687046i
$767$		0			0
$768$	1.65210	+	2.86152i	1.65210	+	2.86152i
$769$		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
$775$	0.888835	+	1.53951i	0.888835	+	1.53951i
$776$	1.16140			1.16140
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	1.71933			1.71933
$784$	0.438384	−	0.281733i	0.438384	−	0.281733i
$785$		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
$789$	1.28605	+	2.22751i	1.28605	+	2.22751i
$790$		0			0
$791$		0			0
$792$	−7.95057			−7.95057
$793$		0			0
$794$	0.550294	−	0.953137i	0.550294	−	0.953137i
$795$		0			0
$796$	−1.06199	−	1.83941i	−1.06199	−	1.83941i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$	−0.221725	+	0.913964i	−0.221725	+	0.913964i
$799$		0			0
$800$	0.260554	+	0.451293i	0.260554	+	0.451293i
$801$	0.934347	−	1.61834i	0.934347	−	1.61834i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$809$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−0.828301	+	0.243211i	−0.828301	+	0.243211i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.95190			1.95190
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	−1.91663	+	3.31969i	−1.91663	+	3.31969i
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$	−3.90993			−3.90993
$826$		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
$836$	−0.518749	+	0.898500i	−0.518749	+	0.898500i
$837$	3.24102	−	5.61361i	3.24102	−	5.61361i
$838$	−0.0800569	−	0.138663i	−0.0800569	−	0.138663i
$839$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$839$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$840$		0			0
$841$	−0.777671			−0.777671
$842$	−1.21769	−	2.10910i	−1.21769	−	2.10910i
$843$	1.28605	−	2.22751i	1.28605	−	2.22751i
$844$	−0.0871144	+	0.150887i	−0.0871144	+	0.150887i
$845$		0			0
$846$	−7.55629			−7.55629
$847$	0.698756	−	2.88031i	0.698756	−	2.88031i
$848$		0			0
$849$	1.95496	+	3.38610i	1.95496	+	3.38610i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$853$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$854$	2.11435	−	0.620830i	2.11435	−	0.620830i
$855$		0			0
$856$	−1.01169	−	1.75230i	−1.01169	−	1.75230i
$857$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$857$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$858$		0			0
$859$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$859$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$860$		0			0
$861$		0			0
$862$	−1.10059			−1.10059
$863$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$863$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$864$	0.950076	−	1.64558i	0.950076	−	1.64558i
$865$		0			0
$866$	−1.56199	−	2.70544i	−1.56199	−	2.70544i
$867$	1.96386			1.96386
$868$	−0.767304	+	3.16287i	−0.767304	+	3.16287i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1.18673	−	2.05548i	−1.18673	−	2.05548i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$877$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$878$		0			0
$879$	1.74555	−	3.02337i	1.74555	−	3.02337i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−4.27217	−	2.20246i	−4.27217	−	2.20246i
$883$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$883$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$884$		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
$889$	0.839614	+	0.800570i	0.839614	+	0.800570i
$890$		0			0
$891$	4.28471	+	7.42134i	4.28471	+	7.42134i
$892$	0.598806	−	1.03716i	0.598806	−	1.03716i
$893$	−0.223734	+	0.387519i	−0.223734	+	0.387519i
$894$		0			0
$895$		0			0
$896$	−0.431635	+	1.77922i	−0.431635	+	1.77922i
$897$		0			0
$898$	1.32254	+	2.29071i	1.32254	+	2.29071i
$899$	0.419102	−	0.725906i	0.419102	−	0.725906i
$900$	−2.61510	+	4.52948i	−2.61510	+	4.52948i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$907$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$911$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$912$	0.145643	+	0.252260i	0.145643	+	0.252260i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.863269			0.863269
$917$		0			0
$918$		0			0
$919$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$919$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$920$		0			0
$921$		0			0
$922$	0.239446	+	0.414732i	0.239446	+	0.414732i
$923$		0			0
$924$	−5.18079	−	4.93987i	−5.18079	−	4.93987i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	0.122856	−	0.212793i	0.122856	−	0.212793i
$929$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$929$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$930$		0			0
$931$	−0.239446	+	0.153882i	−0.239446	+	0.153882i
$932$	−3.25461			−3.25461
$933$	0.981929	+	1.70075i	0.981929	+	1.70075i
$934$	−1.21769	+	2.10910i	−1.21769	+	2.10910i
$935$		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	$0.666667\pi$
$942$	−0.157220	+	0.272314i	−0.157220	+	0.272314i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$947$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$948$		0			0
$949$		0			0
$950$	0.239446	+	0.414732i	0.239446	+	0.414732i
$951$	3.85674			3.85674
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−1.79774	+	3.11378i	−1.79774	+	3.11378i
$957$	0.921801	+	1.59661i	0.921801	+	1.59661i
$958$		0			0
$959$	−1.11312	+	0.326842i	−1.11312	+	0.326842i
$960$		0			0
$961$	−1.08006	−	1.87071i	−1.08006	−	1.87071i
$962$		0			0
$963$	−2.06752	+	3.58104i	−2.06752	+	3.58104i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$967$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$968$	−2.07155	−	3.58804i	−2.07155	−	3.58804i
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	8.79984			8.79984
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.341254	−	0.591068i	0.341254	−	0.591068i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$	1.30235			1.30235
$980$		0			0
$981$		0			0
$982$	0.841254	+	1.45709i	0.841254	+	1.45709i
$983$		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
$987$	−2.23445	−	2.13054i	−2.23445	−	2.13054i
$988$		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	$0.666667\pi$
$992$	−0.463180	−	0.802251i	−0.463180	−	0.802251i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$997$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$998$		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.333.1	✓	20
7.2	even	3	inner	1169.1.f.c.667.1	yes	20
167.166	odd	2	CM	1169.1.f.c.333.1	✓	20
1169.667	odd	6	inner	1169.1.f.c.667.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1169.1.f.c.333.1	✓	20	1.1	even	1	trivial
1169.1.f.c.333.1	✓	20	167.166	odd	2	CM
1169.1.f.c.667.1	yes	20	7.2	even	3	inner
1169.1.f.c.667.1	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)

\(f(q)\)	\(=\)	\(q+(-0.841254 - 1.45709i) q^{2} +(-0.981929 + 1.70075i) q^{3} +(-0.915415 + 1.58555i) q^{4} +3.30420 q^{6} +(0.723734 + 0.690079i) q^{7} +1.39788 q^{8} +(-1.42837 - 2.47401i) q^{9} +O(q^{10})\)	\(q+(-0.841254 - 1.45709i) q^{2} +(-0.981929 + 1.70075i) q^{3} +(-0.915415 + 1.58555i) q^{4} +3.30420 q^{6} +(0.723734 + 0.690079i) q^{7} +1.39788 q^{8} +(-1.42837 - 2.47401i) q^{9} +(0.995472 - 1.72421i) q^{11} +(-1.79774 - 3.11378i) q^{12} +(0.396666 - 1.63508i) q^{14} +(-0.260554 - 0.451293i) q^{16} +(-2.40324 + 4.16253i) q^{18} +(0.142315 + 0.246497i) q^{19} +(-1.88431 + 0.553283i) q^{21} -3.34978 q^{22} +(-1.37262 + 2.37744i) q^{24} +(-0.500000 + 0.866025i) q^{25} +3.64636 q^{27} +(-1.75667 + 0.515804i) q^{28} +0.471518 q^{29} +(0.888835 - 1.53951i) q^{31} +(0.260554 - 0.451293i) q^{32} +(1.95496 + 3.38610i) q^{33} +5.23020 q^{36} +(0.239446 - 0.414732i) q^{38} +(2.39136 + 2.28016i) q^{42} +(1.82254 + 3.15673i) q^{44} +(0.786053 + 1.36148i) q^{47} +1.02338 q^{48} +(0.0475819 + 0.998867i) q^{49} +1.68251 q^{50} +(-3.06752 - 5.31310i) q^{54} +(1.01169 + 0.964646i) q^{56} -0.558972 q^{57} +(-0.396666 - 0.687046i) q^{58} +(0.654861 + 1.13425i) q^{61} -2.99094 q^{62} +(0.673501 - 2.77621i) q^{63} -1.39788 q^{64} +(3.28924 - 5.69713i) q^{66} +(-1.99668 - 3.45836i) q^{72} +(-0.981929 - 1.70075i) q^{75} -0.521109 q^{76} +(1.91030 - 0.560914i) q^{77} +(-2.15210 + 3.72755i) q^{81} +(0.847669 - 3.49414i) q^{84} +(-0.462997 + 0.801934i) q^{87} +(1.39155 - 2.41023i) q^{88} +(0.327068 + 0.566498i) q^{89} +(1.74555 + 3.02337i) q^{93} +(1.32254 - 2.29071i) q^{94} +(0.511691 + 0.886276i) q^{96} +0.830830 q^{97} +(1.41542 - 0.909632i) q^{98} -5.68760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9}+O(q^{10}) \) 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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