LMFDB - Embedded newform 1169.1.f.c.333.5

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			333.5
Root			$0.723734 + 0.690079i$ of defining polynomial
Character	$\chi$	$=$	1169.333
Dual form			1169.1.f.c.667.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.142315 + 0.246497i) q^{2} +(-0.0475819 + 0.0824143i) q^{3} +(0.459493 - 0.795865i) q^{4} -0.0270865 q^{6} +(0.981929 + 0.189251i) q^{7} +0.546200 q^{8} +(0.495472 + 0.858183i) q^{9} +O(q^{10})$	\(q+(0.142315 + 0.246497i) q^{2} +(-0.0475819 + 0.0824143i) q^{3} +(0.459493 - 0.795865i) q^{4} -0.0270865 q^{6} +(0.981929 + 0.189251i) q^{7} +0.546200 q^{8} +(0.495472 + 0.858183i) q^{9} +(-0.723734 + 1.25354i) q^{11} +(0.0437271 + 0.0757376i) q^{12} +(0.0930932 + 0.268975i) q^{14} +(-0.381761 - 0.661229i) q^{16} +(-0.141026 + 0.244264i) q^{18} +(-0.415415 - 0.719520i) q^{19} +(-0.0623191 + 0.0719200i) q^{21} -0.411992 q^{22} +(-0.0259893 + 0.0450147i) q^{24} +(-0.500000 + 0.866025i) q^{25} -0.189466 q^{27} +(0.601808 - 0.694523i) q^{28} -0.654136 q^{29} +(0.786053 - 1.36148i) q^{31} +(0.381761 - 0.661229i) q^{32} +(-0.0688733 - 0.119292i) q^{33} +0.910663 q^{36} +(0.118239 - 0.204797i) q^{38} +(-0.0265970 - 0.00512614i) q^{42} +(0.665101 + 1.15199i) q^{44} +(-0.580057 - 1.00469i) q^{47} +0.0726596 q^{48} +(0.928368 + 0.371662i) q^{49} -0.284630 q^{50} +(-0.0269638 - 0.0467027i) q^{54} +(0.536330 + 0.103369i) q^{56} +0.0790650 q^{57} +(-0.0930932 - 0.161242i) q^{58} +(-0.841254 - 1.45709i) q^{61} +0.447468 q^{62} +(0.324106 + 0.936443i) q^{63} -0.546200 q^{64} +(0.0196034 - 0.0339541i) q^{66} +(0.270627 + 0.468740i) q^{72} +(-0.0475819 - 0.0824143i) q^{75} -0.763521 q^{76} +(-0.947890 + 1.09392i) q^{77} +(-0.486457 + 0.842568i) q^{81} +(0.0286035 + 0.0826443i) q^{84} +(0.0311250 - 0.0539102i) q^{87} +(-0.395304 + 0.684686i) q^{88} +(0.888835 + 1.53951i) q^{89} +(0.0748038 + 0.129564i) q^{93} +(0.165101 - 0.285964i) q^{94} +(0.0363298 + 0.0629251i) q^{96} -1.91899 q^{97} +(0.0405070 + 0.281733i) q^{98} -1.43436 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$2$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$3$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$3$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$4$	0.459493	−	0.795865i	0.459493	−	0.795865i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	−0.0270865			−0.0270865
$7$	0.981929	+	0.189251i	0.981929	+	0.189251i
$7$	0.981929	+	0.189251i	0.981929	+	0.189251i
$8$	0.546200			0.546200
$9$	0.495472	+	0.858183i	0.495472	+	0.858183i
$10$		0			0
$11$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$11$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$12$	0.0437271	+	0.0757376i	0.0437271	+	0.0757376i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	0.0930932	+	0.268975i	0.0930932	+	0.268975i
$15$		0			0
$16$	−0.381761	−	0.661229i	−0.381761	−	0.661229i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$	−0.141026	+	0.244264i	−0.141026	+	0.244264i
$19$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$19$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$20$		0			0
$21$	−0.0623191	+	0.0719200i	−0.0623191	+	0.0719200i
$22$	−0.411992			−0.411992
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	−0.0259893	+	0.0450147i	−0.0259893	+	0.0450147i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$	−0.189466			−0.189466
$28$	0.601808	−	0.694523i	0.601808	−	0.694523i
$29$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$29$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$30$		0			0
$31$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$31$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$32$	0.381761	−	0.661229i	0.381761	−	0.661229i
$33$	−0.0688733	−	0.119292i	−0.0688733	−	0.119292i
$34$		0			0
$35$		0			0
$36$	0.910663			0.910663
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	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			$\pi$
$42$	−0.0265970	−	0.00512614i	−0.0265970	−	0.00512614i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	0.665101	+	1.15199i	0.665101	+	1.15199i
$45$		0			0
$46$		0			0
$47$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$47$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$48$	0.0726596			0.0726596
$49$	0.928368	+	0.371662i	0.928368	+	0.371662i
$50$	−0.284630			−0.284630
$51$		0			0
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$	−0.0269638	−	0.0467027i	−0.0269638	−	0.0467027i
$55$		0			0
$56$	0.536330	+	0.103369i	0.536330	+	0.103369i
$57$	0.0790650			0.0790650
$58$	−0.0930932	−	0.161242i	−0.0930932	−	0.161242i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$61$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$62$	0.447468			0.447468
$63$	0.324106	+	0.936443i	0.324106	+	0.936443i
$64$	−0.546200			−0.546200
$65$		0			0
$66$	0.0196034	−	0.0339541i	0.0196034	−	0.0339541i
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.270627	+	0.468740i	0.270627	+	0.468740i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i
$76$	−0.763521			−0.763521
$77$	−0.947890	+	1.09392i	−0.947890	+	1.09392i
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	−0.486457	+	0.842568i	−0.486457	+	0.842568i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	0.0286035	+	0.0826443i	0.0286035	+	0.0826443i
$85$		0			0
$86$		0			0
$87$	0.0311250	−	0.0539102i	0.0311250	−	0.0539102i
$88$	−0.395304	+	0.684686i	−0.395304	+	0.684686i
$89$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$89$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	0.0748038	+	0.129564i	0.0748038	+	0.129564i
$94$	0.165101	−	0.285964i	0.165101	−	0.285964i
$95$		0			0
$96$	0.0363298	+	0.0629251i	0.0363298	+	0.0629251i
$97$	−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
$99$	−1.43436			−1.43436
$100$	0.459493	+	0.795865i	0.459493	+	0.795865i
$101$		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
$107$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$107$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$108$	−0.0870582	+	0.150789i	−0.0870582	+	0.150789i
$109$		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
$112$	−0.249723	−	0.721528i	−0.249723	−	0.721528i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	0.0112521	+	0.0194892i	0.0112521	+	0.0194892i
$115$		0			0
$116$	−0.300571	+	0.520604i	−0.300571	+	0.520604i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.547582	−	0.948440i	−0.547582	−	0.948440i
$122$	0.239446	−	0.414732i	0.239446	−	0.414732i
$123$		0			0
$124$	−0.722372	−	1.25118i	−0.722372	−	1.25118i
$125$		0			0
$126$	−0.184705	+	0.213161i	−0.184705	+	0.213161i
$127$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$127$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$128$	−0.459493	−	0.795865i	−0.459493	−	0.795865i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	−0.126587			−0.126587
$133$	−0.271738	−	0.785135i	−0.271738	−	0.785135i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$137$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	0.110401			0.110401
$142$		0			0
$143$		0			0
$144$	0.378303	−	0.655240i	0.378303	−	0.655240i
$145$		0			0
$146$		0			0
$147$	−0.0748038	+	0.0588264i	−0.0748038	+	0.0588264i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	0.0135432	−	0.0234576i	0.0135432	−	0.0234576i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	−0.226900	−	0.393002i	−0.226900	−	0.393002i
$153$		0			0
$154$	−0.404547	−	0.0779701i	−0.404547	−	0.0779701i
$155$		0			0
$156$		0			0
$157$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$157$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.276920			−0.276920
$163$		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
$168$	−0.0340387	+	0.0392827i	−0.0340387	+	0.0392827i
$169$	1.00000			1.00000
$170$		0			0
$171$	0.411653	−	0.713004i	0.411653	−	0.713004i
$172$		0			0
$173$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$173$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$174$	0.0177182			0.0177182
$175$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$176$	1.10517			1.10517
$177$		0			0
$178$	−0.252989	+	0.438190i	−0.252989	+	0.438190i
$179$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$179$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$180$		0			0
$181$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$181$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$182$		0			0
$183$	0.160114			0.160114
$184$		0			0
$185$		0			0
$186$	−0.0212914	+	0.0368778i	−0.0212914	+	0.0368778i
$187$		0			0
$188$	−1.06613			−1.06613
$189$	−0.186042	−	0.0358566i	−0.186042	−	0.0358566i
$190$		0			0
$191$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$191$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$192$	0.0259893	−	0.0450147i	0.0259893	−	0.0450147i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$	−0.273100	−	0.473023i	−0.273100	−	0.473023i
$195$		0			0
$196$	0.722372	−	0.568079i	0.722372	−	0.568079i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−0.204131	−	0.353565i	−0.204131	−	0.353565i
$199$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$199$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$200$	−0.273100	+	0.473023i	−0.273100	+	0.473023i
$201$		0			0
$202$		0			0
$203$	−0.642315	−	0.123796i	−0.642315	−	0.123796i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.20260			1.20260
$210$		0			0
$211$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$211$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$212$		0			0
$213$		0			0
$214$	0.279486	−	0.484084i	0.279486	−	0.484084i
$215$		0			0
$216$	−0.103486			−0.103486
$217$	1.02951	−	1.18812i	1.02951	−	1.18812i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$223$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$224$	0.500000	−	0.577031i	0.500000	−	0.577031i
$225$	−0.990944			−0.990944
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	0.0363298	−	0.0629251i	0.0363298	−	0.0629251i
$229$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$229$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$230$		0			0
$231$	−0.0450525	−	0.130171i	−0.0450525	−	0.130171i
$232$	−0.357289			−0.357289
$233$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$233$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$239$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	0.155858	−	0.269954i	0.155858	−	0.269954i
$243$	−0.141026	−	0.244264i	−0.141026	−	0.244264i
$244$	−1.54620			−1.54620
$245$		0			0
$246$		0			0
$247$		0			0
$248$	0.429342	−	0.743643i	0.429342	−	0.743643i
$249$		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	$0.727273\pi$
$252$	0.894207	+	0.172344i	0.894207	+	0.172344i
$253$		0			0
$254$	0.0671040	+	0.116228i	0.0671040	+	0.116228i
$255$		0			0
$256$	−0.142315	+	0.246497i	−0.142315	+	0.246497i
$257$		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
$261$	−0.324106	−	0.561368i	−0.324106	−	0.561368i
$262$		0			0
$263$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$263$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$264$	−0.0376186	−	0.0651574i	−0.0376186	−	0.0651574i
$265$		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	$-0.333333\pi$
$269$		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
$274$	−0.134208			−0.134208
$275$	−0.723734	−	1.25354i	−0.723734	−	1.25354i
$276$		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
$279$	1.55787			1.55787
$280$		0			0
$281$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$281$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$282$	0.0157117	+	0.0272134i	0.0157117	+	0.0272134i
$283$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$283$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.756607			0.756607
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$		0			0
$291$	0.0913090	−	0.158152i	0.0913090	−	0.158152i
$292$		0			0
$293$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$293$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$294$	−0.0251462	−	0.0100670i	−0.0251462	−	0.0100670i
$295$		0			0
$296$		0			0
$297$	0.137123	−	0.237504i	0.137123	−	0.237504i
$298$		0			0
$299$		0			0
$300$	−0.0874542			−0.0874542
$301$		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			$\pi$
$308$	0.435067	+	1.25704i	0.435067	+	1.25704i
$309$		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	$0.333333\pi$
$314$	−0.528482			−0.528482
$315$		0			0
$316$		0			0
$317$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$317$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$318$		0			0
$319$	0.473420	−	0.819988i	0.473420	−	0.819988i
$320$		0			0
$321$	0.186888			0.186888
$322$		0			0
$323$		0			0
$324$	0.447047	+	0.774308i	0.447047	+	0.774308i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.379436	−	1.09631i	−0.379436	−	1.09631i
$330$		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
$334$	0.142315	+	0.246497i	0.142315	+	0.246497i
$335$		0			0
$336$	0.0713465	+	0.0137509i	0.0713465	+	0.0137509i
$337$	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
$341$	1.13779	+	1.97070i	1.13779	+	1.97070i
$342$	0.234337			0.234337
$343$	0.841254	+	0.540641i	0.841254	+	0.540641i
$344$		0			0
$345$		0			0
$346$	0.0671040	−	0.116228i	0.0671040	−	0.116228i
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$	−0.0286035	−	0.0495427i	−0.0286035	−	0.0495427i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−0.279486	−	0.0538665i	−0.279486	−	0.0538665i
$351$		0			0
$352$	0.552586	+	0.957107i	0.552586	+	0.957107i
$353$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$353$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$354$		0			0
$355$		0			0
$356$	1.63365			1.63365
$357$		0			0
$358$	−0.330203			−0.330203
$359$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$359$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$360$		0			0
$361$	0.154861	−	0.268227i	0.154861	−	0.268227i
$362$	0.165101	+	0.285964i	0.165101	+	0.285964i
$363$	0.104220			0.104220
$364$		0			0
$365$		0			0
$366$	0.0227866	+	0.0394675i	0.0227866	+	0.0394675i
$367$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$367$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.137487			0.137487
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$	−0.316827	−	0.548761i	−0.316827	−	0.548761i
$377$		0			0
$378$	−0.0176380	−	0.0509616i	−0.0176380	−	0.0509616i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−0.0224357	+	0.0388598i	−0.0224357	+	0.0388598i
$382$	0.264241	−	0.457679i	0.264241	−	0.457679i
$383$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$383$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$384$	0.0874542			0.0874542
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.881761	+	1.52725i	−0.881761	+	1.52725i
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$	0.507075	+	0.203002i	0.507075	+	0.203002i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.659078	+	1.14156i	−0.659078	+	1.14156i
$397$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$397$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$398$	−0.134208			−0.134208
$399$	0.0776362	+	0.0149631i	0.0776362	+	0.0149631i
$400$	0.763521			0.763521
$401$		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
$406$	−0.0608956	−	0.175946i	−0.0608956	−	0.175946i
$407$		0			0
$408$		0			0
$409$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$409$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$410$		0			0
$411$	−0.0224357	−	0.0388598i	−0.0224357	−	0.0388598i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	0.171148	+	0.296437i	0.171148	+	0.296437i
$419$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$419$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$420$		0			0
$421$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$421$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$422$	0.264241	+	0.457679i	0.264241	+	0.457679i
$423$	0.574804	−	0.995589i	0.574804	−	0.995589i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.550294	−	1.58997i	−0.550294	−	1.58997i
$428$	−1.80476			−1.80476
$429$		0			0
$430$		0			0
$431$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$431$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$432$	0.0723306	+	0.125280i	0.0723306	+	0.125280i
$433$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$433$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$434$	0.439382	+	0.0846839i	0.439382	+	0.0846839i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	0.141026	+	0.980857i	0.141026	+	0.980857i
$442$		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
$446$	−0.252989	−	0.438190i	−0.252989	−	0.438190i
$447$		0			0
$448$	−0.536330	−	0.103369i	−0.536330	−	0.103369i
$449$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$449$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$450$	−0.141026	−	0.244264i	−0.141026	−	0.244264i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	0.0431853			0.0431853
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$	−0.0930932	+	0.161242i	−0.0930932	+	0.161242i
$459$		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	$0.363636\pi$
$462$	0.0256750	−	0.0296305i	0.0256750	−	0.0296305i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	0.249723	+	0.432533i	0.249723	+	0.432533i
$465$		0			0
$466$	−0.223734	+	0.387519i	−0.223734	+	0.387519i
$467$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$467$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.0883470	−	0.153022i	−0.0883470	−	0.153022i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.830830			0.830830
$476$		0			0
$477$		0			0
$478$	0.0135432	+	0.0234576i	0.0135432	+	0.0234576i
$479$		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
$484$	−1.00644			−1.00644
$485$		0			0
$486$	0.0401402	−	0.0695248i	0.0401402	−	0.0695248i
$487$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$487$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$488$	−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
$496$	−1.20034			−1.20034
$497$		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
$501$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i
$502$	−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	$0.181818\pi$
$504$	0.177027	+	0.511485i	0.177027	+	0.511485i
$505$		0			0
$506$		0			0
$507$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i
$508$	0.216659	−	0.375265i	0.216659	−	0.375265i
$509$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$509$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$	0.0787070	+	0.136324i	0.0787070	+	0.136324i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	1.67923			1.67923
$518$		0			0
$519$	0.0448714			0.0448714
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$	0.0922502	−	0.159782i	0.0922502	−	0.159782i
$523$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$523$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$524$		0			0
$525$	−0.0311250	−	0.0899299i	−0.0311250	−	0.0899299i
$526$	−0.478891			−0.478891
$527$		0			0
$528$	−0.0525862	+	0.0910820i	−0.0525862	+	0.0910820i
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−0.749723	−	0.144497i	−0.749723	−	0.144497i
$533$		0			0
$534$	−0.0240754	−	0.0416998i	−0.0240754	−	0.0416998i
$535$		0			0
$536$		0			0
$537$	−0.0552004	−	0.0956100i	−0.0552004	−	0.0956100i
$538$		0			0
$539$	−1.13779	+	0.894765i	−1.13779	+	0.894765i
$540$		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
$543$	−0.0552004	+	0.0956100i	−0.0552004	+	0.0956100i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	0.216659	+	0.375265i	0.216659	+	0.375265i
$549$	0.833635	−	1.44390i	0.833635	−	1.44390i
$550$	0.205996	−	0.356796i	0.205996	−	0.356796i
$551$	0.271738	+	0.470664i	0.271738	+	0.470664i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$557$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$558$	0.221708	+	0.384009i	0.221708	+	0.384009i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.239446	+	0.414732i	0.239446	+	0.414732i
$563$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$563$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$564$	0.0507284	−	0.0878642i	0.0507284	−	0.0878642i
$565$		0			0
$566$	−0.411992			−0.411992
$567$	−0.637123	+	0.735279i	−0.637123	+	0.735279i
$568$		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
$573$	0.176694			0.176694
$574$		0			0
$575$		0			0
$576$	−0.270627	−	0.468740i	−0.270627	−	0.468740i
$577$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$577$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$578$	0.142315	−	0.246497i	0.142315	−	0.246497i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	0.0519785			0.0519785
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−0.223734	−	0.387519i	−0.223734	−	0.387519i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	0.0124460	+	0.0865641i	0.0124460	+	0.0865641i
$589$	−1.30615			−1.30615
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$	0.0780585			0.0780585
$595$		0			0
$596$		0			0
$597$	−0.0224357	−	0.0388598i	−0.0224357	−	0.0388598i
$598$		0			0
$599$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$599$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$600$	−0.0259893	−	0.0450147i	−0.0259893	−	0.0450147i
$601$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$601$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$602$		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	$0.333333\pi$
$608$	−0.634356			−0.634356
$609$	0.0407651	−	0.0470455i	0.0407651	−	0.0470455i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$613$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$614$		0			0
$615$		0			0
$616$	−0.517738	+	0.597501i	−0.517738	+	0.597501i
$617$	−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	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$	0.284630			0.284630
$623$	0.581419	+	1.67990i	0.581419	+	1.67990i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$	−0.0572220	+	0.0991114i	−0.0572220	+	0.0991114i
$628$	0.853157	+	1.47771i	0.853157	+	1.47771i
$629$		0			0
$630$		0			0
$631$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$631$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$632$		0			0
$633$	−0.0883470	+	0.153022i	−0.0883470	+	0.153022i
$634$	0.0135432	−	0.0234576i	0.0135432	−	0.0234576i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	0.269499			0.269499
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	0.0265970	+	0.0460673i	0.0265970	+	0.0460673i
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$	−0.265703	+	0.460211i	−0.265703	+	0.460211i
$649$		0			0
$650$		0			0
$651$	0.0489319	+	0.141379i	0.0489319	+	0.141379i
$652$		0			0
$653$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$653$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	0.216237	−	0.249551i	0.216237	−	0.249551i
$659$		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
$668$	0.459493	−	0.795865i	0.459493	−	0.795865i
$669$	0.0845850	−	0.146505i	0.0845850	−	0.146505i
$670$		0			0
$671$	2.43538			2.43538
$672$	0.0237646	+	0.0686634i	0.0237646	+	0.0686634i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	0.118239	+	0.204797i	0.118239	+	0.204797i
$675$	0.0947329	−	0.164082i	0.0947329	−	0.164082i
$676$	0.459493	−	0.795865i	0.459493	−	0.795865i
$677$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$677$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	−1.88431	−	0.363170i	−1.88431	−	0.363170i
$680$		0			0
$681$		0			0
$682$	−0.323848	+	0.560921i	−0.323848	+	0.560921i
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$	−0.378303	−	0.655240i	−0.378303	−	0.655240i
$685$		0			0
$686$	−0.0135432	+	0.284307i	−0.0135432	+	0.284307i
$687$	−0.0622501			−0.0622501
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$	−0.433318			−0.433318
$693$	−1.40844	−	0.271454i	−1.40844	−	0.271454i
$694$		0			0
$695$		0			0
$696$	0.0170005	−	0.0294457i	0.0170005	−	0.0294457i
$697$		0			0
$698$		0			0
$699$	−0.149608			−0.149608
$700$	0.300571	+	0.868442i	0.300571	+	0.868442i
$701$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$701$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$702$		0			0
$703$		0			0
$704$	0.395304	−	0.684686i	0.395304	−	0.684686i
$705$		0			0
$706$	−0.558972			−0.558972
$707$		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
$712$	0.485482	+	0.840880i	0.485482	+	0.840880i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.533064	+	0.923294i	0.533064	+	0.923294i
$717$	−0.00452808	+	0.00784286i	−0.00452808	+	0.00784286i
$718$	−0.223734	+	0.387519i	−0.223734	+	0.387519i
$719$		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
$722$	0.0881559			0.0881559
$723$		0			0
$724$	0.533064	−	0.923294i	0.533064	−	0.923294i
$725$	0.327068	−	0.566498i	0.327068	−	0.566498i
$726$	0.0148321	+	0.0256899i	0.0148321	+	0.0256899i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−0.946072			−0.946072
$730$		0			0
$731$		0			0
$732$	0.0735712	−	0.127429i	0.0735712	−	0.127429i
$733$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$733$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$734$	0.566682			0.566682
$735$		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	$0.666667\pi$
$744$	0.0408579	+	0.0707679i	0.0408579	+	0.0707679i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.642315	−	1.85585i	−0.642315	−	1.85585i
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$	−0.442886	+	0.767101i	−0.442886	+	0.767101i
$753$	0.0623191	−	0.107940i	0.0623191	−	0.107940i
$754$		0			0
$755$		0			0
$756$	−0.114022	+	0.131588i	−0.114022	+	0.131588i
$757$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$757$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$761$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$762$	−0.0127717			−0.0127717
$763$		0			0
$764$	−1.70631			−1.70631
$765$		0			0
$766$	−0.0930932	+	0.161242i	−0.0930932	+	0.161242i
$767$		0			0
$768$	−0.0135432	−	0.0234576i	−0.0135432	−	0.0234576i
$769$		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
$775$	0.786053	+	1.36148i	0.786053	+	1.36148i
$776$	−1.04815			−1.04815
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.123936			0.123936
$784$	−0.108660	−	0.755750i	−0.108660	−	0.755750i
$785$		0			0
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$		0			0
$789$	−0.0800569	−	0.138663i	−0.0800569	−	0.138663i
$790$		0			0
$791$		0			0
$792$	−0.783448			−0.783448
$793$		0			0
$794$	−0.252989	+	0.438190i	−0.252989	+	0.438190i
$795$		0			0
$796$	0.216659	+	0.375265i	0.216659	+	0.375265i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$	0.00736041	+	0.0212665i	0.00736041	+	0.0212665i
$799$		0			0
$800$	0.381761	+	0.661229i	0.381761	+	0.661229i
$801$	−0.880786	+	1.52557i	−0.880786	+	1.52557i
$802$		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	0.235759	−	0.971812i	$-0.424242\pi$
$809$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−0.393664	+	0.454313i	−0.393664	+	0.454313i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.134208			−0.134208
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	0.00638587	−	0.0110607i	0.00638587	−	0.0110607i
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$	0.137747			0.137747
$826$		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
$836$	0.552586	−	0.957107i	0.552586	−	0.957107i
$837$	−0.148930	+	0.257955i	−0.148930	+	0.257955i
$838$	0.264241	+	0.457679i	0.264241	+	0.457679i
$839$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$839$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$840$		0			0
$841$	−0.572106			−0.572106
$842$	0.279486	+	0.484084i	0.279486	+	0.484084i
$843$	−0.0800569	+	0.138663i	−0.0800569	+	0.138663i
$844$	0.853157	−	1.47771i	0.853157	−	1.47771i
$845$		0			0
$846$	0.327212			0.327212
$847$	−0.358193	−	1.03493i	−0.358193	−	1.03493i
$848$		0			0
$849$	−0.0688733	−	0.119292i	−0.0688733	−	0.119292i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$853$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$854$	0.313607	−	0.361922i	0.313607	−	0.361922i
$855$		0			0
$856$	−0.536330	−	0.928950i	−0.536330	−	0.928950i
$857$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$857$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$858$		0			0
$859$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$859$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$860$		0			0
$861$		0			0
$862$	0.505978			0.505978
$863$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$863$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$864$	−0.0723306	+	0.125280i	−0.0723306	+	0.125280i
$865$		0			0
$866$	−0.283341	−	0.490761i	−0.283341	−	0.490761i
$867$	0.0951638			0.0951638
$868$	−0.472529	−	1.36528i	−0.472529	−	1.36528i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.950804	−	1.64684i	−0.950804	−	1.64684i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$877$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$878$		0			0
$879$	0.0748038	−	0.129564i	0.0748038	−	0.129564i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.221708	+	0.174353i	−0.221708	+	0.174353i
$883$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$883$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$884$		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
$889$	0.462997	+	0.0892353i	0.462997	+	0.0892353i
$890$		0			0
$891$	−0.704131	−	1.21959i	−0.704131	−	1.21959i
$892$	−0.816827	+	1.41479i	−0.816827	+	1.41479i
$893$	−0.481929	+	0.834725i	−0.481929	+	0.834725i
$894$		0			0
$895$		0			0
$896$	−0.300571	−	0.868442i	−0.300571	−	0.868442i
$897$		0			0
$898$	0.165101	+	0.285964i	0.165101	+	0.285964i
$899$	−0.514186	+	0.890596i	−0.514186	+	0.890596i
$900$	−0.455332	+	0.788658i	−0.455332	+	0.788658i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$907$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$911$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$912$	−0.0301839	−	0.0522800i	−0.0301839	−	0.0522800i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.601142			0.601142
$917$		0			0
$918$		0			0
$919$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$919$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$920$		0			0
$921$		0			0
$922$	0.118239	+	0.204797i	0.118239	+	0.204797i
$923$		0			0
$924$	−0.124300	−	0.0239568i	−0.124300	−	0.0239568i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−0.249723	+	0.432533i	−0.249723	+	0.432533i
$929$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$929$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$930$		0			0
$931$	−0.118239	−	0.822373i	−0.118239	−	0.822373i
$932$	1.44474			1.44474
$933$	0.0475819	+	0.0824143i	0.0475819	+	0.0824143i
$934$	0.279486	−	0.484084i	0.279486	−	0.484084i
$935$		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	$0.666667\pi$
$942$	0.0251462	−	0.0435545i	0.0251462	−	0.0435545i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$947$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$948$		0			0
$949$		0			0
$950$	0.118239	+	0.204797i	0.118239	+	0.204797i
$951$	0.00905615			0.00905615
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	0.0437271	−	0.0757376i	0.0437271	−	0.0757376i
$957$	0.0450525	+	0.0780332i	0.0450525	+	0.0780332i
$958$		0			0
$959$	−0.308779	+	0.356349i	−0.308779	+	0.356349i
$960$		0			0
$961$	−0.735759	−	1.27437i	−0.735759	−	1.27437i
$962$		0			0
$963$	0.973036	−	1.68535i	0.973036	−	1.68535i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$967$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$968$	−0.299089	−	0.518038i	−0.299089	−	0.518038i
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	−0.259202			−0.259202
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.642315	+	1.11252i	−0.642315	+	1.11252i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$	−2.57312			−2.57312
$980$		0			0
$981$		0			0
$982$	−0.142315	−	0.246497i	−0.142315	−	0.246497i
$983$		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
$987$	0.108406	+	0.0208935i	0.108406	+	0.0208935i
$988$		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	$0.666667\pi$
$992$	−0.600168	−	1.03952i	−0.600168	−	1.03952i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$997$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$998$		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.333.5	✓	20
7.2	even	3	inner	1169.1.f.c.667.5	yes	20
167.166	odd	2	CM	1169.1.f.c.333.5	✓	20
1169.667	odd	6	inner	1169.1.f.c.667.5	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1169.1.f.c.333.5	✓	20	1.1	even	1	trivial
1169.1.f.c.333.5	✓	20	167.166	odd	2	CM
1169.1.f.c.667.5	yes	20	7.2	even	3	inner
1169.1.f.c.667.5	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)

\(f(q)\)	\(=\)	\(q+(0.142315 + 0.246497i) q^{2} +(-0.0475819 + 0.0824143i) q^{3} +(0.459493 - 0.795865i) q^{4} -0.0270865 q^{6} +(0.981929 + 0.189251i) q^{7} +0.546200 q^{8} +(0.495472 + 0.858183i) q^{9} +O(q^{10})\)	\(q+(0.142315 + 0.246497i) q^{2} +(-0.0475819 + 0.0824143i) q^{3} +(0.459493 - 0.795865i) q^{4} -0.0270865 q^{6} +(0.981929 + 0.189251i) q^{7} +0.546200 q^{8} +(0.495472 + 0.858183i) q^{9} +(-0.723734 + 1.25354i) q^{11} +(0.0437271 + 0.0757376i) q^{12} +(0.0930932 + 0.268975i) q^{14} +(-0.381761 - 0.661229i) q^{16} +(-0.141026 + 0.244264i) q^{18} +(-0.415415 - 0.719520i) q^{19} +(-0.0623191 + 0.0719200i) q^{21} -0.411992 q^{22} +(-0.0259893 + 0.0450147i) q^{24} +(-0.500000 + 0.866025i) q^{25} -0.189466 q^{27} +(0.601808 - 0.694523i) q^{28} -0.654136 q^{29} +(0.786053 - 1.36148i) q^{31} +(0.381761 - 0.661229i) q^{32} +(-0.0688733 - 0.119292i) q^{33} +0.910663 q^{36} +(0.118239 - 0.204797i) q^{38} +(-0.0265970 - 0.00512614i) q^{42} +(0.665101 + 1.15199i) q^{44} +(-0.580057 - 1.00469i) q^{47} +0.0726596 q^{48} +(0.928368 + 0.371662i) q^{49} -0.284630 q^{50} +(-0.0269638 - 0.0467027i) q^{54} +(0.536330 + 0.103369i) q^{56} +0.0790650 q^{57} +(-0.0930932 - 0.161242i) q^{58} +(-0.841254 - 1.45709i) q^{61} +0.447468 q^{62} +(0.324106 + 0.936443i) q^{63} -0.546200 q^{64} +(0.0196034 - 0.0339541i) q^{66} +(0.270627 + 0.468740i) q^{72} +(-0.0475819 - 0.0824143i) q^{75} -0.763521 q^{76} +(-0.947890 + 1.09392i) q^{77} +(-0.486457 + 0.842568i) q^{81} +(0.0286035 + 0.0826443i) q^{84} +(0.0311250 - 0.0539102i) q^{87} +(-0.395304 + 0.684686i) q^{88} +(0.888835 + 1.53951i) q^{89} +(0.0748038 + 0.129564i) q^{93} +(0.165101 - 0.285964i) q^{94} +(0.0363298 + 0.0629251i) q^{96} -1.91899 q^{97} +(0.0405070 + 0.281733i) q^{98} -1.43436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - q^{3} - 8 q^{4} + 4 q^{6} + q^{7} - 8 q^{8} - 11 q^{9}+O(q^{10}) \) 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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