LMFDB - Embedded newform 1169.1.f.c.667.6

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			667.6
Root			$0.235759 - 0.971812i$ of defining polynomial
Character	$\chi$	$=$	1169.667
Dual form			1169.1.f.c.333.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.142315 - 0.246497i) q^{2} +(0.888835 + 1.53951i) q^{3} +(0.459493 + 0.795865i) q^{4} +0.505978 q^{6} +(-0.327068 - 0.945001i) q^{7} +0.546200 q^{8} +(-1.08006 + 1.87071i) q^{9} +O(q^{10})$	\(q+(0.142315 - 0.246497i) q^{2} +(0.888835 + 1.53951i) q^{3} +(0.459493 + 0.795865i) q^{4} +0.505978 q^{6} +(-0.327068 - 0.945001i) q^{7} +0.546200 q^{8} +(-1.08006 + 1.87071i) q^{9} +(-0.235759 - 0.408346i) q^{11} +(-0.816827 + 1.41479i) q^{12} +(-0.279486 - 0.0538665i) q^{14} +(-0.381761 + 0.661229i) q^{16} +(0.307416 + 0.532461i) q^{18} +(-0.415415 + 0.719520i) q^{19} +(1.16413 - 1.34347i) q^{21} -0.134208 q^{22} +(0.485482 + 0.840880i) q^{24} +(-0.500000 - 0.866025i) q^{25} -2.06230 q^{27} +(0.601808 - 0.694523i) q^{28} +1.96386 q^{29} +(-0.928368 - 1.60798i) q^{31} +(0.381761 + 0.661229i) q^{32} +(0.419102 - 0.725906i) q^{33} -1.98511 q^{36} +(0.118239 + 0.204797i) q^{38} +(-0.165489 - 0.478150i) q^{42} +(0.216659 - 0.375265i) q^{44} +(0.995472 - 1.72421i) q^{47} -1.35729 q^{48} +(-0.786053 + 0.618159i) q^{49} -0.284630 q^{50} +(-0.293496 + 0.508350i) q^{54} +(-0.178645 - 0.516160i) q^{56} -1.47694 q^{57} +(0.279486 - 0.484084i) q^{58} +(-0.841254 + 1.45709i) q^{61} -0.528482 q^{62} +(2.12108 + 0.408804i) q^{63} -0.546200 q^{64} +(-0.119289 - 0.206614i) q^{66} +(-0.589927 + 1.02178i) q^{72} +(0.888835 - 1.53951i) q^{75} -0.763521 q^{76} +(-0.308779 + 0.356349i) q^{77} +(-0.752989 - 1.30422i) q^{81} +(1.60413 + 0.309171i) q^{84} +(1.74555 + 3.02337i) q^{87} +(-0.128772 - 0.223039i) q^{88} +(-0.0475819 + 0.0824143i) q^{89} +(1.65033 - 2.85846i) q^{93} +(-0.283341 - 0.490761i) q^{94} +(-0.678645 + 1.17545i) q^{96} -1.91899 q^{97} +(0.0405070 + 0.281733i) q^{98} +1.01853 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1169\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$2$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$3$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$3$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$4$	0.459493	+	0.795865i	0.459493	+	0.795865i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$	0.505978			0.505978
$7$	−0.327068	−	0.945001i	−0.327068	−	0.945001i
$7$	−0.327068	−	0.945001i	−0.327068	−	0.945001i
$8$	0.546200			0.546200
$9$	−1.08006	+	1.87071i	−1.08006	+	1.87071i
$10$		0			0
$11$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$11$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$12$	−0.816827	+	1.41479i	−0.816827	+	1.41479i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−0.279486	−	0.0538665i	−0.279486	−	0.0538665i
$15$		0			0
$16$	−0.381761	+	0.661229i	−0.381761	+	0.661229i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$	0.307416	+	0.532461i	0.307416	+	0.532461i
$19$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$19$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$20$		0			0
$21$	1.16413	−	1.34347i	1.16413	−	1.34347i
$22$	−0.134208			−0.134208
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	0.485482	+	0.840880i	0.485482	+	0.840880i
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$		0			0
$27$	−2.06230			−2.06230
$28$	0.601808	−	0.694523i	0.601808	−	0.694523i
$29$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$29$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$30$		0			0
$31$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$31$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$32$	0.381761	+	0.661229i	0.381761	+	0.661229i
$33$	0.419102	−	0.725906i	0.419102	−	0.725906i
$34$		0			0
$35$		0			0
$36$	−1.98511			−1.98511
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$	0.118239	+	0.204797i	0.118239	+	0.204797i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−0.165489	−	0.478150i	−0.165489	−	0.478150i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	0.216659	−	0.375265i	0.216659	−	0.375265i
$45$		0			0
$46$		0			0
$47$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$47$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$48$	−1.35729			−1.35729
$49$	−0.786053	+	0.618159i	−0.786053	+	0.618159i
$50$	−0.284630			−0.284630
$51$		0			0
$52$		0			0
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$	−0.293496	+	0.508350i	−0.293496	+	0.508350i
$55$		0			0
$56$	−0.178645	−	0.516160i	−0.178645	−	0.516160i
$57$	−1.47694			−1.47694
$58$	0.279486	−	0.484084i	0.279486	−	0.484084i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$61$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$62$	−0.528482			−0.528482
$63$	2.12108	+	0.408804i	2.12108	+	0.408804i
$64$	−0.546200			−0.546200
$65$		0			0
$66$	−0.119289	−	0.206614i	−0.119289	−	0.206614i
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−0.589927	+	1.02178i	−0.589927	+	1.02178i
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$		0			0
$75$	0.888835	−	1.53951i	0.888835	−	1.53951i
$76$	−0.763521			−0.763521
$77$	−0.308779	+	0.356349i	−0.308779	+	0.356349i
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	−0.752989	−	1.30422i	−0.752989	−	1.30422i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	1.60413	+	0.309171i	1.60413	+	0.309171i
$85$		0			0
$86$		0			0
$87$	1.74555	+	3.02337i	1.74555	+	3.02337i
$88$	−0.128772	−	0.223039i	−0.128772	−	0.223039i
$89$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$89$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	1.65033	−	2.85846i	1.65033	−	2.85846i
$94$	−0.283341	−	0.490761i	−0.283341	−	0.490761i
$95$		0			0
$96$	−0.678645	+	1.17545i	−0.678645	+	1.17545i
$97$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$97$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$98$	0.0405070	+	0.281733i	0.0405070	+	0.281733i
$99$	1.01853			1.01853
$100$	0.459493	−	0.795865i	0.459493	−	0.795865i
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$107$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$108$	−0.947613	−	1.64131i	−0.947613	−	1.64131i
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$		0			0
$112$	0.749723	+	0.144497i	0.749723	+	0.144497i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	−0.210191	+	0.364061i	−0.210191	+	0.364061i
$115$		0			0
$116$	0.902379	+	1.56297i	0.902379	+	1.56297i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.388835	−	0.673483i	0.388835	−	0.673483i
$122$	0.239446	+	0.414732i	0.239446	+	0.414732i
$123$		0			0
$124$	0.853157	−	1.47771i	0.853157	−	1.47771i
$125$		0			0
$126$	0.402630	−	0.464659i	0.402630	−	0.464659i
$127$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$127$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$128$	−0.459493	+	0.795865i	−0.459493	+	0.795865i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	0.770297			0.770297
$133$	0.815816	+	0.157236i	0.815816	+	0.157236i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$137$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	3.53924			3.53924
$142$		0			0
$143$		0			0
$144$	−0.824646	−	1.42833i	−0.824646	−	1.42833i
$145$		0			0
$146$		0			0
$147$	−1.65033	−	0.660694i	−1.65033	−	0.660694i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	−0.252989	−	0.438190i	−0.252989	−	0.438190i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$	−0.226900	+	0.393002i	−0.226900	+	0.393002i
$153$		0			0
$154$	0.0438951	+	0.126827i	0.0438951	+	0.126827i
$155$		0			0
$156$		0			0
$157$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$157$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.428646			−0.428646
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.00000			1.00000
$167$	1.00000			1.00000
$168$	0.635847	−	0.733806i	0.635847	−	0.733806i
$169$	1.00000			1.00000
$170$		0			0
$171$	−0.897344	−	1.55424i	−0.897344	−	1.55424i
$172$		0			0
$173$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$173$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$174$	0.993668			0.993668
$175$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$176$	0.360014			0.360014
$177$		0			0
$178$	0.0135432	+	0.0234576i	0.0135432	+	0.0234576i
$179$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$179$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$180$		0			0
$181$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$181$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$182$		0			0
$183$	−2.99094			−2.99094
$184$		0			0
$185$		0			0
$186$	−0.469734	−	0.813603i	−0.469734	−	0.813603i
$187$		0			0
$188$	1.82965			1.82965
$189$	0.674512	+	1.94888i	0.674512	+	1.94888i
$190$		0			0
$191$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$191$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$192$	−0.485482	−	0.840880i	−0.485482	−	0.840880i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$	−0.273100	+	0.473023i	−0.273100	+	0.473023i
$195$		0			0
$196$	−0.853157	−	0.341553i	−0.853157	−	0.341553i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	0.144952	−	0.251065i	0.144952	−	0.251065i
$199$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$199$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$200$	−0.273100	−	0.473023i	−0.273100	−	0.473023i
$201$		0			0
$202$		0			0
$203$	−0.642315	−	1.85585i	−0.642315	−	1.85585i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.391751			0.391751
$210$		0			0
$211$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$211$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$212$		0			0
$213$		0			0
$214$	−0.0930932	−	0.161242i	−0.0930932	−	0.161242i
$215$		0			0
$216$	−1.12643			−1.12643
$217$	−1.21590	+	1.40323i	−1.21590	+	1.40323i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$223$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$224$	0.500000	−	0.577031i	0.500000	−	0.577031i
$225$	2.16011			2.16011
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	−0.678645	−	1.17545i	−0.678645	−	1.17545i
$229$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$229$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$230$		0			0
$231$	−0.823056	−	0.158631i	−0.823056	−	0.158631i
$232$	1.07266			1.07266
$233$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$233$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$239$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	−0.110674	−	0.191693i	−0.110674	−	0.191693i
$243$	0.307416	−	0.532461i	0.307416	−	0.532461i
$244$	−1.54620			−1.54620
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−0.507075	−	0.878279i	−0.507075	−	0.878279i
$249$		0			0
$250$		0			0
$251$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$251$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$252$	0.649267	+	1.87593i	0.649267	+	1.87593i
$253$		0			0
$254$	0.205996	−	0.356796i	0.205996	−	0.356796i
$255$		0			0
$256$	−0.142315	−	0.246497i	−0.142315	−	0.246497i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−2.12108	+	3.67381i	−2.12108	+	3.67381i
$262$		0			0
$263$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$263$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$264$	0.228914	−	0.396490i	0.228914	−	0.396490i
$265$		0			0
$266$	0.154861	−	0.178719i	0.154861	−	0.178719i
$267$	−0.169170			−0.169170
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$		0			0
$274$	−0.411992			−0.411992
$275$	−0.235759	+	0.408346i	−0.235759	+	0.408346i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$	4.01076			4.01076
$280$		0			0
$281$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$281$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$282$	0.503687	−	0.872411i	0.503687	−	0.872411i
$283$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$283$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.64929			−1.64929
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$		0			0
$291$	−1.70566	−	2.95429i	−1.70566	−	2.95429i
$292$		0			0
$293$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$293$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$294$	−0.397725	+	0.312775i	−0.397725	+	0.312775i
$295$		0			0
$296$		0			0
$297$	0.486206	+	0.842133i	0.486206	+	0.842133i
$298$		0			0
$299$		0			0
$300$	1.63365			1.63365
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.317178	−	0.549369i	−0.317178	−	0.549369i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	−0.425488	−	0.0820060i	−0.425488	−	0.0820060i
$309$		0			0
$310$		0			0
$311$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$311$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	0.447468			0.447468
$315$		0			0
$316$		0			0
$317$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$317$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$318$		0			0
$319$	−0.462997	−	0.801934i	−0.462997	−	0.801934i
$320$		0			0
$321$	1.16284			1.16284
$322$		0			0
$323$		0			0
$324$	0.691986	−	1.19856i	0.691986	−	1.19856i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.95496	−	0.376789i	−1.95496	−	0.376789i
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$	0.142315	−	0.246497i	0.142315	−	0.246497i
$335$		0			0
$336$	0.443926	+	1.28264i	0.443926	+	1.28264i
$337$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$337$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$338$	0.142315	−	0.246497i	0.142315	−	0.246497i
$339$		0			0
$340$		0			0
$341$	−0.437742	+	0.758192i	−0.437742	+	0.758192i
$342$	−0.510821			−0.510821
$343$	0.841254	+	0.540641i	0.841254	+	0.540641i
$344$		0			0
$345$		0			0
$346$	0.205996	+	0.356796i	0.205996	+	0.356796i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$	−1.60413	+	2.77844i	−1.60413	+	2.77844i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	0.0930932	+	0.268975i	0.0930932	+	0.268975i
$351$		0			0
$352$	0.180007	−	0.311781i	0.180007	−	0.311781i
$353$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$353$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$354$		0			0
$355$		0			0
$356$	−0.0874542			−0.0874542
$357$		0			0
$358$	0.566682			0.566682
$359$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$359$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$360$		0			0
$361$	0.154861	+	0.268227i	0.154861	+	0.268227i
$362$	−0.283341	+	0.490761i	−0.283341	+	0.490761i
$363$	1.38244			1.38244
$364$		0			0
$365$		0			0
$366$	−0.425656	+	0.737257i	−0.425656	+	0.737257i
$367$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$367$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	3.03327			3.03327
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$	0.543727	−	0.941763i	0.543727	−	0.941763i
$377$		0			0
$378$	0.576384	+	0.111089i	0.576384	+	0.111089i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	1.28656	+	2.22839i	1.28656	+	2.22839i
$382$	−0.223734	−	0.387519i	−0.223734	−	0.387519i
$383$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$383$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$384$	−1.63365			−1.63365
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.881761	−	1.52725i	−0.881761	−	1.52725i
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	−0.429342	+	0.337639i	−0.429342	+	0.337639i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	0.468008	+	0.810614i	0.468008	+	0.810614i
$397$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$397$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$398$	−0.411992			−0.411992
$399$	0.483061	+	1.39571i	0.483061	+	1.39571i
$400$	0.763521			0.763521
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−0.548871	−	0.105786i	−0.548871	−	0.105786i
$407$		0			0
$408$		0			0
$409$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$409$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$410$		0			0
$411$	1.28656	−	2.22839i	1.28656	−	2.22839i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	0.0557520	−	0.0965653i	0.0557520	−	0.0965653i
$419$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$419$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$420$		0			0
$421$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$421$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$422$	−0.223734	+	0.387519i	−0.223734	+	0.387519i
$423$	2.15033	+	3.72449i	2.15033	+	3.72449i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.65210	+	0.318417i	1.65210	+	0.318417i
$428$	0.601142			0.601142
$429$		0			0
$430$		0			0
$431$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$431$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$432$	0.787305	−	1.36365i	0.787305	−	1.36365i
$433$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$433$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$434$	0.172850	+	0.499416i	0.172850	+	0.499416i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−0.307416	−	2.13813i	−0.307416	−	2.13813i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$	0.0135432	−	0.0234576i	0.0135432	−	0.0234576i
$447$		0			0
$448$	0.178645	+	0.516160i	0.178645	+	0.516160i
$449$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$449$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$450$	0.307416	−	0.532461i	0.307416	−	0.532461i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	−0.806706			−0.806706
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	0.279486	+	0.484084i	0.279486	+	0.484084i
$459$		0			0
$460$		0			0
$461$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$461$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$462$	−0.156235	+	0.180305i	−0.156235	+	0.180305i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	−0.749723	+	1.29856i	−0.749723	+	1.29856i
$465$		0			0
$466$	0.264241	+	0.457679i	0.264241	+	0.457679i
$467$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$467$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1.39734	+	2.42027i	−1.39734	+	2.42027i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.830830			0.830830
$476$		0			0
$477$		0			0
$478$	−0.252989	+	0.438190i	−0.252989	+	0.438190i
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.714669			0.714669
$485$		0			0
$486$	−0.0874998	−	0.151554i	−0.0874998	−	0.151554i
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$	−0.459493	+	0.795865i	−0.459493	+	0.795865i
$489$		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
$496$	1.41766			1.41766
$497$		0			0
$498$		0			0
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$		0			0
$501$	0.888835	+	1.53951i	0.888835	+	1.53951i
$502$	−0.186393	+	0.322842i	−0.186393	+	0.322842i
$503$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$503$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$504$	1.15853	+	0.223289i	1.15853	+	0.223289i
$505$		0			0
$506$		0			0
$507$	0.888835	+	1.53951i	0.888835	+	1.53951i
$508$	0.665101	+	1.15199i	0.665101	+	1.15199i
$509$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$509$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$	0.856711	−	1.48387i	0.856711	−	1.48387i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−0.938766			−0.938766
$518$		0			0
$519$	−2.57312			−2.57312
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$	0.603722	+	1.04568i	0.603722	+	1.04568i
$523$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$523$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$524$		0			0
$525$	−1.74555	−	0.336426i	−1.74555	−	0.336426i
$526$	−0.478891			−0.478891
$527$		0			0
$528$	0.319993	+	0.554244i	0.319993	+	0.554244i
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	0.249723	+	0.721528i	0.249723	+	0.721528i
$533$		0			0
$534$	−0.0240754	+	0.0416998i	−0.0240754	+	0.0416998i
$535$		0			0
$536$		0			0
$537$	−1.76962	+	3.06507i	−1.76962	+	3.06507i
$538$		0			0
$539$	0.437742	+	0.175245i	0.437742	+	0.175245i
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$	−1.76962	−	3.06507i	−1.76962	−	3.06507i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	0.665101	−	1.15199i	0.665101	−	1.15199i
$549$	−1.81720	−	3.14749i	−1.81720	−	3.14749i
$550$	0.0671040	+	0.116228i	0.0671040	+	0.116228i
$551$	−0.815816	+	1.41303i	−0.815816	+	1.41303i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$557$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$558$	0.570791	−	0.988639i	0.570791	−	0.988639i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.239446	−	0.414732i	0.239446	−	0.414732i
$563$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$563$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$564$	1.62626	+	2.81676i	1.62626	+	2.81676i
$565$		0			0
$566$	−0.134208			−0.134208
$567$	−0.986206	+	1.13814i	−0.986206	+	1.13814i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$	2.79469			2.79469
$574$		0			0
$575$		0			0
$576$	0.589927	−	1.02178i	0.589927	−	1.02178i
$577$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$577$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$578$	0.142315	+	0.246497i	0.142315	+	0.246497i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−0.970964			−0.970964
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.264241	−	0.457679i	0.264241	−	0.457679i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	−0.232493	−	1.61703i	−0.232493	−	1.61703i
$589$	1.54263			1.54263
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$	0.276777			0.276777
$595$		0			0
$596$		0			0
$597$	1.28656	−	2.22839i	1.28656	−	2.22839i
$598$		0			0
$599$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$599$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$600$	0.485482	−	0.840880i	0.485482	−	0.840880i
$601$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$601$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	−0.634356			−0.634356
$609$	2.28618	−	2.63839i	2.28618	−	2.63839i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$613$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$614$		0			0
$615$		0			0
$616$	−0.168655	+	0.194638i	−0.168655	+	0.194638i
$617$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$617$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$	0.284630			0.284630
$623$	0.0934441	+	0.0180099i	0.0934441	+	0.0180099i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$	0.348202	+	0.603104i	0.348202	+	0.603104i
$628$	−0.722372	+	1.25118i	−0.722372	+	1.25118i
$629$		0			0
$630$		0			0
$631$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$631$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$632$		0			0
$633$	−1.39734	−	2.42027i	−1.39734	−	2.42027i
$634$	−0.252989	−	0.438190i	−0.252989	−	0.438190i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.263565			−0.263565
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$	0.165489	−	0.286636i	0.165489	−	0.286636i
$643$		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	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$	−0.411283	−	0.712363i	−0.411283	−	0.712363i
$649$		0			0
$650$		0			0
$651$	−3.24102	−	0.624655i	−3.24102	−	0.624655i
$652$		0			0
$653$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$653$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−0.371098	+	0.428269i	−0.371098	+	0.428269i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.459493	+	0.795865i	0.459493	+	0.795865i
$669$	0.0845850	+	0.146505i	0.0845850	+	0.146505i
$670$		0			0
$671$	0.793332			0.793332
$672$	1.33276	+	0.256869i	1.33276	+	0.256869i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	0.118239	−	0.204797i	0.118239	−	0.204797i
$675$	1.03115	+	1.78600i	1.03115	+	1.78600i
$676$	0.459493	+	0.795865i	0.459493	+	0.795865i
$677$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$677$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$678$		0			0
$679$	0.627639	+	1.81344i	0.627639	+	1.81344i
$680$		0			0
$681$		0			0
$682$	0.124594	+	0.215804i	0.124594	+	0.215804i
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	0.824646	−	1.42833i	0.824646	−	1.42833i
$685$		0			0
$686$	0.252989	−	0.130425i	0.252989	−	0.130425i
$687$	−3.49109			−3.49109
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$	−1.33020			−1.33020
$693$	−0.333129	−	0.962514i	−0.333129	−	0.962514i
$694$		0			0
$695$		0			0
$696$	0.953418	+	1.65137i	0.953418	+	1.65137i
$697$		0			0
$698$		0			0
$699$	−3.30067			−3.30067
$700$	−0.902379	−	0.173919i	−0.902379	−	0.173919i
$701$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$701$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$702$		0			0
$703$		0			0
$704$	0.128772	+	0.223039i	0.128772	+	0.223039i
$705$		0			0
$706$	0.186186			0.186186
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$	−0.0259893	+	0.0450147i	−0.0259893	+	0.0450147i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.914825	+	1.58452i	−0.914825	+	1.58452i
$717$	−1.58006	−	2.73674i	−1.58006	−	2.73674i
$718$	0.264241	+	0.457679i	0.264241	+	0.457679i
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$		0			0
$722$	0.0881559			0.0881559
$723$		0			0
$724$	−0.914825	−	1.58452i	−0.914825	−	1.58452i
$725$	−0.981929	−	1.70075i	−0.981929	−	1.70075i
$726$	0.196742	−	0.340767i	0.196742	−	0.340767i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−0.413008			−0.413008
$730$		0			0
$731$		0			0
$732$	−1.37432	−	2.38039i	−1.37432	−	2.38039i
$733$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$733$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$734$	−0.330203			−0.330203
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$743$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$744$	0.901412	−	1.56129i	0.901412	−	1.56129i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.642315	−	0.123796i	−0.642315	−	0.123796i
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	0.760064	+	1.31647i	0.760064	+	1.31647i
$753$	−1.16413	−	2.01633i	−1.16413	−	2.01633i
$754$		0			0
$755$		0			0
$756$	−1.24111	+	1.43232i	−1.24111	+	1.43232i
$757$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$757$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$761$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$762$	0.732387			0.732387
$763$		0			0
$764$	1.44474			1.44474
$765$		0			0
$766$	0.279486	+	0.484084i	0.279486	+	0.484084i
$767$		0			0
$768$	0.252989	−	0.438190i	0.252989	−	0.438190i
$769$		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	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$	−0.928368	+	1.60798i	−0.928368	+	1.60798i
$776$	−1.04815			−1.04815
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−4.05006			−4.05006
$784$	−0.108660	−	0.755750i	−0.108660	−	0.755750i
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$		0			0
$789$	1.49547	−	2.59023i	1.49547	−	2.59023i
$790$		0			0
$791$		0			0
$792$	0.556323			0.556323
$793$		0			0
$794$	0.0135432	+	0.0234576i	0.0135432	+	0.0234576i
$795$		0			0
$796$	0.665101	−	1.15199i	0.665101	−	1.15199i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$	0.412785	+	0.0795577i	0.412785	+	0.0795577i
$799$		0			0
$800$	0.381761	−	0.661229i	0.381761	−	0.661229i
$801$	−0.102782	−	0.178024i	−0.102782	−	0.178024i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$809$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	1.18186	−	1.36394i	1.18186	−	1.36394i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.411992			−0.411992
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$	−0.366193	−	0.634266i	−0.366193	−	0.634266i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$	−0.838204			−0.838204
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$	0.180007	+	0.311781i	0.180007	+	0.311781i
$837$	1.91457	+	3.31614i	1.91457	+	3.31614i
$838$	−0.223734	+	0.387519i	−0.223734	+	0.387519i
$839$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$839$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$840$		0			0
$841$	2.85674			2.85674
$842$	−0.0930932	+	0.161242i	−0.0930932	+	0.161242i
$843$	1.49547	+	2.59023i	1.49547	+	2.59023i
$844$	−0.722372	−	1.25118i	−0.722372	−	1.25118i
$845$		0			0
$846$	1.22410			1.22410
$847$	−0.763617	−	0.147175i	−0.763617	−	0.147175i
$848$		0			0
$849$	0.419102	−	0.725906i	0.419102	−	0.725906i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$853$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$854$	0.313607	−	0.361922i	0.313607	−	0.361922i
$855$		0			0
$856$	0.178645	−	0.309422i	0.178645	−	0.309422i
$857$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$857$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$858$		0			0
$859$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$859$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$860$		0			0
$861$		0			0
$862$	−0.0270865			−0.0270865
$863$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$863$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$864$	−0.787305	−	1.36365i	−0.787305	−	1.36365i
$865$		0			0
$866$	0.165101	−	0.285964i	0.165101	−	0.285964i
$867$	−1.77767			−1.77767
$868$	−1.67548	−	0.322922i	−1.67548	−	0.322922i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	2.07261	−	3.58987i	2.07261	−	3.58987i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$877$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$878$		0			0
$879$	1.65033	+	2.85846i	1.65033	+	2.85846i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.570791	−	0.228510i	−0.570791	−	0.228510i
$883$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$883$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−0.473420	−	1.36786i	−0.473420	−	1.36786i
$890$		0			0
$891$	−0.355048	+	0.614961i	−0.355048	+	0.614961i
$892$	0.0437271	+	0.0757376i	0.0437271	+	0.0757376i
$893$	0.827068	+	1.43252i	0.827068	+	1.43252i
$894$		0			0
$895$		0			0
$896$	0.902379	+	0.173919i	0.902379	+	0.173919i
$897$		0			0
$898$	−0.283341	+	0.490761i	−0.283341	+	0.490761i
$899$	−1.82318	−	3.15784i	−1.82318	−	3.15784i
$900$	0.992557	+	1.71916i	0.992557	+	1.71916i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$907$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$911$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$912$	0.563838	−	0.976597i	0.563838	−	0.976597i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.80476			−1.80476
$917$		0			0
$918$		0			0
$919$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$919$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$920$		0			0
$921$		0			0
$922$	0.118239	−	0.204797i	0.118239	−	0.204797i
$923$		0			0
$924$	−0.251940	−	0.727932i	−0.251940	−	0.727932i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	0.749723	+	1.29856i	0.749723	+	1.29856i
$929$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$929$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$930$		0			0
$931$	−0.118239	−	0.822373i	−0.118239	−	0.822373i
$932$	−1.70631			−1.70631
$933$	−0.888835	+	1.53951i	−0.888835	+	1.53951i
$934$	−0.0930932	−	0.161242i	−0.0930932	−	0.161242i
$935$		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	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$	0.397725	+	0.688881i	0.397725	+	0.688881i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$947$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$948$		0			0
$949$		0			0
$950$	0.118239	−	0.204797i	0.118239	−	0.204797i
$951$	3.16011			3.16011
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−0.816827	−	1.41479i	−0.816827	−	1.41479i
$957$	0.823056	−	1.42558i	0.823056	−	1.42558i
$958$		0			0
$959$	−0.947890	+	1.09392i	−0.947890	+	1.09392i
$960$		0			0
$961$	−1.22373	+	2.11957i	−1.22373	+	2.11957i
$962$		0			0
$963$	0.706504	+	1.22370i	0.706504	+	1.22370i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$967$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$968$	0.212382	−	0.367857i	0.212382	−	0.367857i
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	0.565022			0.565022
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.642315	−	1.11252i	−0.642315	−	1.11252i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$	0.0448714			0.0448714
$980$		0			0
$981$		0			0
$982$	−0.142315	+	0.246497i	−0.142315	+	0.246497i
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1.15757	−	3.34459i	−1.15757	−	3.34459i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$	0.708829	−	1.22773i	0.708829	−	1.22773i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$997$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$998$		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.667.6	yes	20
7.4	even	3	inner	1169.1.f.c.333.6	✓	20
167.166	odd	2	CM	1169.1.f.c.667.6	yes	20
1169.333	odd	6	inner	1169.1.f.c.333.6	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1169.1.f.c.333.6	✓	20	7.4	even	3	inner
1169.1.f.c.333.6	✓	20	1169.333	odd	6	inner
1169.1.f.c.667.6	yes	20	1.1	even	1	trivial
1169.1.f.c.667.6	yes	20	167.166	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1169 = 7 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1169.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.142315 - 0.246497i) q^{2} +(0.888835 + 1.53951i) q^{3} +(0.459493 + 0.795865i) q^{4} +0.505978 q^{6} +(-0.327068 - 0.945001i) q^{7} +0.546200 q^{8} +(-1.08006 + 1.87071i) q^{9} +O(q^{10})\)	\(q+(0.142315 - 0.246497i) q^{2} +(0.888835 + 1.53951i) q^{3} +(0.459493 + 0.795865i) q^{4} +0.505978 q^{6} +(-0.327068 - 0.945001i) q^{7} +0.546200 q^{8} +(-1.08006 + 1.87071i) q^{9} +(-0.235759 - 0.408346i) q^{11} +(-0.816827 + 1.41479i) q^{12} +(-0.279486 - 0.0538665i) q^{14} +(-0.381761 + 0.661229i) q^{16} +(0.307416 + 0.532461i) q^{18} +(-0.415415 + 0.719520i) q^{19} +(1.16413 - 1.34347i) q^{21} -0.134208 q^{22} +(0.485482 + 0.840880i) q^{24} +(-0.500000 - 0.866025i) q^{25} -2.06230 q^{27} +(0.601808 - 0.694523i) q^{28} +1.96386 q^{29} +(-0.928368 - 1.60798i) q^{31} +(0.381761 + 0.661229i) q^{32} +(0.419102 - 0.725906i) q^{33} -1.98511 q^{36} +(0.118239 + 0.204797i) q^{38} +(-0.165489 - 0.478150i) q^{42} +(0.216659 - 0.375265i) q^{44} +(0.995472 - 1.72421i) q^{47} -1.35729 q^{48} +(-0.786053 + 0.618159i) q^{49} -0.284630 q^{50} +(-0.293496 + 0.508350i) q^{54} +(-0.178645 - 0.516160i) q^{56} -1.47694 q^{57} +(0.279486 - 0.484084i) q^{58} +(-0.841254 + 1.45709i) q^{61} -0.528482 q^{62} +(2.12108 + 0.408804i) q^{63} -0.546200 q^{64} +(-0.119289 - 0.206614i) q^{66} +(-0.589927 + 1.02178i) q^{72} +(0.888835 - 1.53951i) q^{75} -0.763521 q^{76} +(-0.308779 + 0.356349i) q^{77} +(-0.752989 - 1.30422i) q^{81} +(1.60413 + 0.309171i) q^{84} +(1.74555 + 3.02337i) q^{87} +(-0.128772 - 0.223039i) q^{88} +(-0.0475819 + 0.0824143i) q^{89} +(1.65033 - 2.85846i) q^{93} +(-0.283341 - 0.490761i) q^{94} +(-0.678645 + 1.17545i) q^{96} -1.91899 q^{97} +(0.0405070 + 0.281733i) q^{98} +1.01853 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9}+O(q^{10}) \) 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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