LMFDB - Embedded newform 639.1.g.b.70.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [639,1,Mod(70,639)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(639, 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("639.70");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 639 = 3^{2} \cdot 71 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	639.g (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:	$0.318902543072$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{21}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{21} - \cdots)$

Embedding invariants

Embedding label			70.3
Root			$-0.733052 + 0.680173i$ of defining polynomial
Character	$\chi$	$=$	639.70
Dual form			639.1.g.b.283.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.365341 + 0.632789i) q^{2} +(0.0747301 + 0.997204i) q^{3} +(0.233052 + 0.403658i) q^{4} +(0.222521 + 0.385418i) q^{5} +(-0.658322 - 0.317031i) q^{6} -1.07126 q^{8} +(-0.988831 + 0.149042i) q^{9} +O(q^{10})$	\(q+(-0.365341 + 0.632789i) q^{2} +(0.0747301 + 0.997204i) q^{3} +(0.233052 + 0.403658i) q^{4} +(0.222521 + 0.385418i) q^{5} +(-0.658322 - 0.317031i) q^{6} -1.07126 q^{8} +(-0.988831 + 0.149042i) q^{9} -0.325184 q^{10} +(-0.385113 + 0.262566i) q^{12} +(-0.367711 + 0.250701i) q^{15} +(0.158322 - 0.274221i) q^{16} +(0.266948 - 0.680173i) q^{18} +0.730682 q^{19} +(-0.103718 + 0.179645i) q^{20} +(-0.0800550 - 1.06826i) q^{24} +(0.400969 - 0.694498i) q^{25} +(-0.222521 - 0.974928i) q^{27} +(-0.0747301 + 0.129436i) q^{29} +(-0.0243010 - 0.324275i) q^{30} +(-0.419945 - 0.727366i) q^{32} +(-0.290611 - 0.364415i) q^{36} +1.24698 q^{37} +(-0.266948 + 0.462368i) q^{38} +(-0.238377 - 0.412881i) q^{40} +(-0.955573 + 1.65510i) q^{43} +(-0.277479 - 0.347948i) q^{45} +(0.285286 + 0.137386i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(0.292981 + 0.507458i) q^{50} +(0.698220 + 0.215372i) q^{54} +(0.0546039 + 0.728639i) q^{57} +(-0.0546039 - 0.0945768i) q^{58} +(-0.186893 - 0.0900030i) q^{60} +0.930336 q^{64} +1.00000 q^{71} +(1.05929 - 0.159662i) q^{72} -0.445042 q^{73} +(-0.455573 + 0.789075i) q^{74} +(0.722521 + 0.347948i) q^{75} +(0.170287 + 0.294945i) q^{76} +(0.222521 - 0.385418i) q^{79} +0.140920 q^{80} +(0.955573 - 0.294755i) q^{81} +(-0.623490 + 1.07992i) q^{83} +(-0.698220 - 1.20935i) q^{86} +(-0.134659 - 0.0648483i) q^{87} +1.65248 q^{89} +(0.321552 - 0.0484662i) q^{90} +(0.162592 + 0.281618i) q^{95} +(0.693950 - 0.473127i) q^{96} +0.730682 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{2} + q^{3} - 7 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + q^{9}+O(q^{10}) $ 12 * q - q^2 + q^3 - 7 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + q^9	$ 12 q - q^{2} + q^{3} - 7 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + q^{9} + 4 q^{10} + 2 q^{15} - 8 q^{16} + 13 q^{18} + 2 q^{19} + 7 q^{20} - 6 q^{24} - 4 q^{25} - 2 q^{27} - q^{29} - 9 q^{30} - 4 q^{37} - 13 q^{38} + 2 q^{40} - q^{43} - 4 q^{45} - 5 q^{48} - 6 q^{49} - 3 q^{50} - q^{54} - q^{57} + q^{58} - 7 q^{60} + 12 q^{64} + 12 q^{71} - 6 q^{72} - 4 q^{73} + 5 q^{74} + 8 q^{75} + 2 q^{79} - 10 q^{80} + q^{81} + 2 q^{83} + q^{86} - 5 q^{87} + 2 q^{89} + 12 q^{90} - 2 q^{95} - 7 q^{96} + 2 q^{98}+O(q^{100}) $ 12 * q - q^2 + q^3 - 7 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + q^9 + 4 * q^10 + 2 * q^15 - 8 * q^16 + 13 * q^18 + 2 * q^19 + 7 * q^20 - 6 * q^24 - 4 * q^25 - 2 * q^27 - q^29 - 9 * q^30 - 4 * q^37 - 13 * q^38 + 2 * q^40 - q^43 - 4 * q^45 - 5 * q^48 - 6 * q^49 - 3 * q^50 - q^54 - q^57 + q^58 - 7 * q^60 + 12 * q^64 + 12 * q^71 - 6 * q^72 - 4 * q^73 + 5 * q^74 + 8 * q^75 + 2 * q^79 - 10 * q^80 + q^81 + 2 * q^83 + q^86 - 5 * q^87 + 2 * q^89 + 12 * q^90 - 2 * q^95 - 7 * q^96 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/639\mathbb{Z}\right)^\times$.

$n$	$433$	$569$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$2$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	0.623490	+	0.781831i	$0.285714\pi$
$3$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$3$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$4$	0.233052	+	0.403658i	0.233052	+	0.403658i
$5$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$5$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$6$	−0.658322	−	0.317031i	−0.658322	−	0.317031i
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$	−1.07126			−1.07126
$9$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$10$	−0.325184			−0.325184
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	−0.385113	+	0.262566i	−0.385113	+	0.262566i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$	−0.367711	+	0.250701i	−0.367711	+	0.250701i
$16$	0.158322	−	0.274221i	0.158322	−	0.274221i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	0.266948	−	0.680173i	0.266948	−	0.680173i
$19$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$19$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$20$	−0.103718	+	0.179645i	−0.103718	+	0.179645i
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	−0.0800550	−	1.06826i	−0.0800550	−	1.06826i
$25$	0.400969	−	0.694498i	0.400969	−	0.694498i
$26$		0			0
$27$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$28$		0			0
$29$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	−0.900969	−	0.433884i	$-0.857143\pi$
$29$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	0.826239	+	0.563320i	$0.190476\pi$
$30$	−0.0243010	−	0.324275i	−0.0243010	−	0.324275i
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	−0.419945	−	0.727366i	−0.419945	−	0.727366i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.290611	−	0.364415i	−0.290611	−	0.364415i
$37$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$37$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$38$	−0.266948	+	0.462368i	−0.266948	+	0.462368i
$39$		0			0
$40$	−0.238377	−	0.412881i	−0.238377	−	0.412881i
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.222521	+	0.974928i	$0.571429\pi$
$43$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.733052	+	0.680173i	$0.761905\pi$
$44$		0			0
$45$	−0.277479	−	0.347948i	−0.277479	−	0.347948i
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	0.285286	+	0.137386i	0.285286	+	0.137386i
$49$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$50$	0.292981	+	0.507458i	0.292981	+	0.507458i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	0.698220	+	0.215372i	0.698220	+	0.215372i
$55$		0			0
$56$		0			0
$57$	0.0546039	+	0.728639i	0.0546039	+	0.728639i
$58$	−0.0546039	−	0.0945768i	−0.0546039	−	0.0945768i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$	−0.186893	−	0.0900030i	−0.186893	−	0.0900030i
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	0.930336			0.930336
$65$		0			0
$66$		0			0
$67$		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
$71$	1.00000			1.00000
$71$	1.00000			1.00000
$72$	1.05929	−	0.159662i	1.05929	−	0.159662i
$73$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$73$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$74$	−0.455573	+	0.789075i	−0.455573	+	0.789075i
$75$	0.722521	+	0.347948i	0.722521	+	0.347948i
$76$	0.170287	+	0.294945i	0.170287	+	0.294945i
$77$		0			0
$78$		0			0
$79$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$79$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$80$	0.140920			0.140920
$81$	0.955573	−	0.294755i	0.955573	−	0.294755i
$82$		0			0
$83$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$83$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$84$		0			0
$85$		0			0
$86$	−0.698220	−	1.20935i	−0.698220	−	1.20935i
$87$	−0.134659	−	0.0648483i	−0.134659	−	0.0648483i
$88$		0			0
$89$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$89$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$90$	0.321552	−	0.0484662i	0.321552	−	0.0484662i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.162592	+	0.281618i	0.162592	+	0.281618i
$96$	0.693950	−	0.473127i	0.693950	−	0.473127i
$97$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$97$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$98$	0.730682			0.730682
$99$		0			0
$100$	0.373786			0.373786
$101$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$101$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$102$		0			0
$103$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$103$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$107$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$108$	0.341678	−	0.317031i	0.341678	−	0.317031i
$109$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$109$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$110$		0			0
$111$	0.0931869	+	1.24349i	0.0931869	+	1.24349i
$112$		0			0
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$	−0.481024	−	0.231649i	−0.481024	−	0.231649i
$115$		0			0
$116$	−0.0696640			−0.0696640
$117$		0			0
$118$		0			0
$119$		0			0
$120$	0.393912	−	0.268565i	0.393912	−	0.268565i
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.801938			0.801938
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	0.0800550	−	0.138659i	0.0800550	−	0.138659i
$129$	−1.72188	−	0.829215i	−1.72188	−	0.829215i
$130$		0			0
$131$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$131$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	0.326239	−	0.302705i	0.326239	−	0.302705i
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$		0			0
$142$	−0.365341	+	0.632789i	−0.365341	+	0.632789i
$143$		0			0
$144$	−0.115683	+	0.294755i	−0.115683	+	0.294755i
$145$	−0.0665160			−0.0665160
$146$	0.162592	−	0.281618i	0.162592	−	0.281618i
$147$	0.826239	−	0.563320i	0.826239	−	0.563320i
$148$	0.290611	+	0.503353i	0.290611	+	0.503353i
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	−0.484144	+	0.330084i	−0.484144	+	0.330084i
$151$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	0.0747301	+	0.997204i	$0.476190\pi$
$151$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	−0.900969	+	0.433884i	$0.857143\pi$
$152$	−0.782747			−0.782747
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	−0.900969	−	0.433884i	$-0.857143\pi$
$157$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	0.0747301	−	0.997204i	$-0.476190\pi$
$158$	0.162592	+	0.281618i	0.162592	+	0.281618i
$159$		0			0
$160$	0.186893	−	0.323708i	0.186893	−	0.323708i
$161$		0			0
$162$	−0.162592	+	0.712362i	−0.162592	+	0.712362i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	−0.455573	−	0.789075i	−0.455573	−	0.789075i
$167$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$167$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$170$		0			0
$171$	−0.722521	+	0.108903i	−0.722521	+	0.108903i
$172$	−0.890792			−0.890792
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$	0.0902318	−	0.0615190i	0.0902318	−	0.0615190i
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.603718	+	1.04567i	−0.603718	+	1.04567i
$179$	1.91115			1.91115			0.955573	−	0.294755i	$-0.0952381\pi$
$179$	1.91115			1.91115			0.955573	+	0.294755i	$0.0952381\pi$
$180$	0.0757848	−	0.193096i	0.0757848	−	0.193096i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.277479	+	0.480608i	0.277479	+	0.480608i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−0.237606			−0.237606
$191$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$191$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$192$	0.0695241	+	0.927735i	0.0695241	+	0.927735i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$	0.233052	−	0.403658i	0.233052	−	0.403658i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$199$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$200$	−0.429540	+	0.743985i	−0.429540	+	0.743985i
$201$		0			0
$202$	0.722521	+	1.25144i	0.722521	+	1.25144i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	0.109208			0.109208
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$		0			0
$213$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$214$	0.365341	−	0.632789i	0.365341	−	0.632789i
$215$	−0.850540			−0.850540
$216$	0.238377	+	1.04440i	0.238377	+	1.04440i
$217$		0			0
$218$	−0.0546039	+	0.0945768i	−0.0546039	+	0.0945768i
$219$	−0.0332580	−	0.443797i	−0.0332580	−	0.443797i
$220$		0			0
$221$		0			0
$222$	−0.820914	−	0.395331i	−0.820914	−	0.395331i
$223$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$223$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$224$		0			0
$225$	−0.292981	+	0.746503i	−0.292981	+	0.746503i
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	−0.281395	+	0.191852i	−0.281395	+	0.191852i
$229$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$229$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$230$		0			0
$231$		0			0
$232$	0.0800550	−	0.138659i	0.0800550	−	0.138659i
$233$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$233$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.400969	+	0.193096i	0.400969	+	0.193096i
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$	0.0105309	+	0.140526i	0.0105309	+	0.140526i
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	0.730682			0.730682
$243$	0.365341	+	0.930874i	0.365341	+	0.930874i
$244$		0			0
$245$	0.222521	−	0.385418i	0.222521	−	0.385418i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−1.12349	−	0.541044i	−1.12349	−	0.541044i
$250$	−0.292981	+	0.507458i	−0.292981	+	0.507458i
$251$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$251$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.523663	+	0.907011i	0.523663	+	0.907011i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$	1.15379	−	0.786643i	1.15379	−	0.786643i
$259$		0			0
$260$		0			0
$261$	0.0546039	−	0.139129i	0.0546039	−	0.139129i
$262$	−1.07126			−1.07126
$263$	0.733052	−	1.26968i	0.733052	−	1.26968i	−0.222521	−	0.974928i	$-0.571429\pi$
$263$	0.733052	−	1.26968i	0.733052	−	1.26968i	0.955573	−	0.294755i	$-0.0952381\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.123490	+	1.64786i	0.123490	+	1.64786i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	0.0723603	+	0.317031i	0.0723603	+	0.317031i
$271$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$271$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$277$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$	0.233052	+	0.403658i	0.233052	+	0.403658i
$285$	−0.268680	+	0.183183i	−0.268680	+	0.183183i
$286$		0			0
$287$		0			0
$288$	0.523663	+	0.656652i	0.523663	+	0.656652i
$289$	1.00000			1.00000
$290$	0.0243010	−	0.0420906i	0.0243010	−	0.0420906i
$291$		0			0
$292$	−0.103718	−	0.179645i	−0.103718	−	0.179645i
$293$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$293$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$294$	0.0546039	+	0.728639i	0.0546039	+	0.728639i
$295$		0			0
$296$	−1.33583			−1.33583
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.0279331	+	0.372741i	0.0279331	+	0.372741i
$301$		0			0
$302$	−0.603718	−	1.04567i	−0.603718	−	1.04567i
$303$	1.78181	+	0.858075i	1.78181	+	0.858075i
$304$	0.115683	−	0.200369i	0.115683	−	0.200369i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	0.123490	−	0.0841939i	0.123490	−	0.0841939i
$310$		0			0
$311$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$311$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$312$		0			0
$313$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$313$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	0.623490	+	0.781831i	$0.285714\pi$
$314$	1.20744			1.20744
$315$		0			0
$316$	0.207436			0.207436
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$	0.207019	+	0.358568i	0.207019	+	0.358568i
$321$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i
$322$		0			0
$323$		0			0
$324$	0.341678	+	0.317031i	0.341678	+	0.317031i
$325$		0			0
$326$		0			0
$327$	0.0111692	+	0.149042i	0.0111692	+	0.149042i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	−0.581222			−0.581222
$333$	−1.23305	+	0.185853i	−1.23305	+	0.185853i
$334$	1.39644			1.39644
$335$		0			0
$336$		0			0
$337$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$337$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$338$	−0.365341	−	0.632789i	−0.365341	−	0.632789i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	0.195054	−	0.496990i	0.195054	−	0.496990i
$343$		0			0
$344$	1.02366	−	1.77304i	1.02366	−	1.77304i
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$	−0.00520599	−	0.0694692i	−0.00520599	−	0.0694692i
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$	0.222521	+	0.385418i	0.222521	+	0.385418i
$356$	0.385113	+	0.667035i	0.385113	+	0.667035i
$357$		0			0
$358$	−0.698220	+	1.20935i	−0.698220	+	1.20935i
$359$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$359$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$360$	0.297251	+	0.372741i	0.297251	+	0.372741i
$361$	−0.466104			−0.466104
$362$		0			0
$363$	0.826239	−	0.563320i	0.826239	−	0.563320i
$364$		0			0
$365$	−0.0990311	−	0.171527i	−0.0990311	−	0.171527i
$366$		0			0
$367$	0.733052	−	1.26968i	0.733052	−	1.26968i	−0.222521	−	0.974928i	$-0.571429\pi$
$367$	0.733052	−	1.26968i	0.733052	−	1.26968i	0.955573	−	0.294755i	$-0.0952381\pi$
$368$		0			0
$369$		0			0
$370$	−0.405498			−0.405498
$371$		0			0
$372$		0			0
$373$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$373$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$374$		0			0
$375$	0.0599289	+	0.799695i	0.0599289	+	0.799695i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$379$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$380$	−0.0757848	+	0.131263i	−0.0757848	+	0.131263i
$381$		0			0
$382$	0.658322	+	1.14025i	0.658322	+	1.14025i
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$	0.144254	+	0.0694692i	0.144254	+	0.0694692i
$385$		0			0
$386$		0			0
$387$	0.698220	−	1.77904i	0.698220	−	1.77904i
$388$		0			0
$389$		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
$392$	0.535628	+	0.927735i	0.535628	+	0.927735i
$393$	−1.21135	+	0.825886i	−1.21135	+	0.825886i
$394$		0			0
$395$	0.198062			0.198062
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	0.722521	−	1.25144i	0.722521	−	1.25144i
$399$		0			0
$400$	−0.126964	−	0.219908i	−0.126964	−	0.219908i
$401$		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
$404$	0.921795			0.921795
$405$	0.326239	+	0.302705i	0.326239	+	0.302705i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	−0.900969	−	0.433884i	$-0.857143\pi$
$409$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	0.0747301	−	0.997204i	$-0.476190\pi$
$410$		0			0
$411$		0			0
$412$	0.0348320	−	0.0603308i	0.0348320	−	0.0603308i
$413$		0			0
$414$		0			0
$415$	−0.554958			−0.554958
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$419$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	−0.658322	−	0.317031i	−0.658322	−	0.317031i
$427$		0			0
$428$	−0.233052	−	0.403658i	−0.233052	−	0.403658i
$429$		0			0
$430$	0.310737	−	0.538212i	0.310737	−	0.538212i
$431$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$431$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$432$	−0.302576	−	0.0933323i	−0.302576	−	0.0933323i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−0.00497075	−	0.0663300i	−0.00497075	−	0.0663300i
$436$	0.0348320	+	0.0603308i	0.0348320	+	0.0603308i
$437$		0			0
$438$	0.292981	+	0.141092i	0.292981	+	0.141092i
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	0.623490	+	0.781831i	0.623490	+	0.781831i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	−0.480228	+	0.327414i	−0.480228	+	0.327414i
$445$	0.367711	+	0.636894i	0.367711	+	0.636894i
$446$	−0.455573	−	0.789075i	−0.455573	−	0.789075i
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−0.365341	−	0.458123i	−0.365341	−	0.458123i
$451$		0			0
$452$		0			0
$453$	−1.48883	−	0.716983i	−1.48883	−	0.716983i
$454$		0			0
$455$		0			0
$456$	−0.0584948	−	0.780559i	−0.0584948	−	0.780559i
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	1.39644			1.39644
$459$		0			0
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.623490	+	0.781831i	$0.285714\pi$
$463$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.365341	+	0.930874i	$0.380952\pi$
$464$	0.0236628	+	0.0409852i	0.0236628	+	0.0409852i
$465$		0			0
$466$	0.722521	−	1.25144i	0.722521	−	1.25144i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.36534	−	0.930874i	1.36534	−	0.930874i
$472$		0			0
$473$		0			0
$474$	−0.268680	+	0.183183i	−0.268680	+	0.183183i
$475$	0.292981	−	0.507458i	0.292981	−	0.507458i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$	0.336770	+	0.162180i	0.336770	+	0.162180i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.233052	−	0.403658i	0.233052	−	0.403658i
$485$		0			0
$486$	−0.722521	−	0.108903i	−0.722521	−	0.108903i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$	0.162592	+	0.281618i	0.162592	+	0.281618i
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	0.752824	−	0.513267i	0.752824	−	0.513267i
$499$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$499$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$500$	0.186893	+	0.323708i	0.186893	+	0.323708i
$501$	1.57906	−	1.07659i	1.57906	−	1.07659i
$502$	0.722521	−	1.25144i	0.722521	−	1.25144i
$503$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$503$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$504$		0			0
$505$	0.880142			0.880142
$506$		0			0
$507$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$508$		0			0
$509$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$509$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$	−0.605152			−0.605152
$513$	−0.162592	−	0.712362i	−0.162592	−	0.712362i
$514$		0			0
$515$	0.0332580	−	0.0576046i	0.0332580	−	0.0576046i
$516$	−0.0665690	−	0.888301i	−0.0665690	−	0.888301i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$521$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$522$	0.0680900	+	0.0853822i	0.0680900	+	0.0853822i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	−0.341678	+	0.591804i	−0.341678	+	0.591804i
$525$		0			0
$526$	0.535628	+	0.927735i	0.535628	+	0.927735i
$527$		0			0
$528$		0			0
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	−1.08786	−	0.523887i	−1.08786	−	0.523887i
$535$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$536$		0			0
$537$	0.142820	+	1.90580i	0.142820	+	1.90580i
$538$		0			0
$539$		0			0
$540$	0.198220	+	0.0611427i	0.198220	+	0.0611427i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	0.658322	−	1.14025i	0.658322	−	1.14025i
$543$		0			0
$544$		0			0
$545$	0.0332580	+	0.0576046i	0.0332580	+	0.0576046i
$546$		0			0
$547$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$547$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.0546039	+	0.0945768i	−0.0546039	+	0.0945768i
$552$		0			0
$553$		0			0
$554$	0.722521	+	1.25144i	0.722521	+	1.25144i
$555$	−0.458528	+	0.312619i	−0.458528	+	0.312619i
$556$		0			0
$557$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$557$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−1.07126			−1.07126
$569$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.222521	+	0.974928i	$0.571429\pi$
$569$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.733052	+	0.680173i	$0.761905\pi$
$570$	−0.0177563	−	0.236942i	−0.0177563	−	0.236942i
$571$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$571$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$572$		0			0
$573$	1.62349	+	0.781831i	1.62349	+	0.781831i
$574$		0			0
$575$		0			0
$576$	−0.919945	+	0.138659i	−0.919945	+	0.138659i
$577$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$577$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$578$	−0.365341	+	0.632789i	−0.365341	+	0.632789i
$579$		0			0
$580$	−0.0155017	−	0.0268497i	−0.0155017	−	0.0268497i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.476754			0.476754
$585$		0			0
$586$	−0.730682			−0.730682
$587$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	0.0747301	+	0.997204i	$0.476190\pi$
$587$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	−0.900969	+	0.433884i	$0.857143\pi$
$588$	0.419945	+	0.202235i	0.419945	+	0.202235i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.197424	−	0.341948i	0.197424	−	0.341948i
$593$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$593$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.147791	−	1.97213i	−0.147791	−	1.97213i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	−0.774005	−	0.372741i	−0.774005	−	0.372741i
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$		0			0
$604$	−0.770226			−0.770226
$605$	0.222521	−	0.385418i	0.222521	−	0.385418i
$606$	−1.19395	+	0.814021i	−1.19395	+	0.814021i
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	−0.306846	−	0.531473i	−0.306846	−	0.531473i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$613$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$617$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$618$	0.00816111	+	0.108903i	0.00816111	+	0.108903i
$619$		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
$622$	0.911146			0.911146
$623$		0			0
$624$		0			0
$625$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$626$	−0.266948	−	0.462368i	−0.266948	−	0.462368i
$627$		0			0
$628$	0.385113	−	0.667035i	0.385113	−	0.667035i
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	−0.238377	+	0.412881i	−0.238377	+	0.412881i
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$640$	0.0712557			0.0712557
$641$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$641$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$642$	0.658322	+	0.317031i	0.658322	+	0.317031i
$643$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$643$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$644$		0			0
$645$	−0.0635609	−	0.848162i	−0.0635609	−	0.848162i
$646$		0			0
$647$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$647$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$648$	−1.02366	+	0.315758i	−1.02366	+	0.315758i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$	−0.0983929	−	0.0473835i	−0.0983929	−	0.0473835i
$655$	−0.326239	+	0.565062i	−0.326239	+	0.565062i
$656$		0			0
$657$	0.440071	−	0.0663300i	0.440071	−	0.0663300i
$658$		0			0
$659$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	0.0747301	+	0.997204i	$0.476190\pi$
$659$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	−0.900969	+	0.433884i	$0.857143\pi$
$660$		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
$664$	0.667917	−	1.15687i	0.667917	−	1.15687i
$665$		0			0
$666$	0.332879	−	0.848162i	0.332879	−	0.848162i
$667$		0			0
$668$	0.445396	−	0.771449i	0.445396	−	0.771449i
$669$	−1.12349	−	0.541044i	−1.12349	−	0.541044i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$		0			0
$675$	−0.766310	−	0.236375i	−0.766310	−	0.236375i
$676$	−0.466104			−0.466104
$677$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	0.0747301	+	0.997204i	$0.476190\pi$
$677$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	−0.900969	+	0.433884i	$0.857143\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	−0.212344	−	0.266271i	−0.212344	−	0.266271i
$685$		0			0
$686$		0			0
$687$	1.57906	−	1.07659i	1.57906	−	1.07659i
$688$	0.302576	+	0.524077i	0.302576	+	0.524077i
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$	0.144254	+	0.0694692i	0.144254	+	0.0694692i
$697$		0			0
$698$		0			0
$699$	−0.147791	−	1.97213i	−0.147791	−	1.97213i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	0.911146			0.911146
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$	−0.325184			−0.325184
$711$	−0.162592	+	0.414278i	−0.162592	+	0.414278i
$712$	−1.77023			−1.77023
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.445396	+	0.771449i	0.445396	+	0.771449i
$717$		0			0
$718$	0.535628	−	0.927735i	0.535628	−	0.927735i
$719$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$719$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$720$	−0.139346	+	0.0210030i	−0.139346	+	0.0210030i
$721$		0			0
$722$	0.170287	−	0.294945i	0.170287	−	0.294945i
$723$		0			0
$724$		0			0
$725$	0.0599289	+	0.103800i	0.0599289	+	0.103800i
$726$	0.0546039	+	0.728639i	0.0546039	+	0.728639i
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$730$	0.144721			0.144721
$731$		0			0
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$	0.535628	+	0.927735i	0.535628	+	0.927735i
$735$	0.400969	+	0.193096i	0.400969	+	0.193096i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$739$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$740$	−0.129334	+	0.224013i	−0.129334	+	0.224013i
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$		0			0
$746$	−0.325184			−0.325184
$747$	0.455573	−	1.16078i	0.455573	−	1.16078i
$748$		0			0
$749$		0			0
$750$	−0.527933	−	0.254239i	−0.527933	−	0.254239i
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$	−0.147791	−	1.97213i	−0.147791	−	1.97213i
$754$		0			0
$755$	−0.735422			−0.735422
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−0.603718	+	1.04567i	−0.603718	+	1.04567i
$759$		0			0
$760$	−0.174178	−	0.301685i	−0.174178	−	0.301685i
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$		0			0
$764$	0.839890			0.839890
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.865341	+	0.589980i	−0.865341	+	0.589980i
$769$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$769$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$	0.870666	+	1.09178i	0.870666	+	1.09178i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.142820	+	0.0440542i	0.142820	+	0.0440542i
$784$	−0.316644			−0.316644
$785$	0.367711	−	0.636894i	0.367711	−	0.636894i
$786$	−0.0800550	−	1.06826i	−0.0800550	−	1.06826i
$787$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$787$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$788$		0			0
$789$	1.32091	+	0.636119i	1.32091	+	0.636119i
$790$	−0.0723603	+	0.125332i	−0.0723603	+	0.125332i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−0.460898	−	0.798298i	−0.460898	−	0.798298i
$797$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	0.623490	−	0.781831i	$-0.285714\pi$
$797$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	−0.988831	+	0.149042i	$0.952381\pi$
$798$		0			0
$799$		0			0
$800$	−0.673539			−0.673539
$801$	−1.63402	+	0.246289i	−1.63402	+	0.246289i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−1.05929	+	1.83475i	−1.05929	+	1.83475i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−0.310737	+	0.0958497i	−0.310737	+	0.0958497i
$811$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$811$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$812$		0			0
$813$	−0.134659	−	1.79690i	−0.134659	−	1.79690i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.698220	+	1.20935i	−0.698220	+	1.20935i
$818$	1.20744			1.20744
$819$		0			0
$820$		0			0
$821$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$821$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$	0.0800550	+	0.138659i	0.0800550	+	0.138659i
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$829$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$830$	0.202749	−	0.351172i	0.202749	−	0.351172i
$831$	1.78181	+	0.858075i	1.78181	+	0.858075i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	0.425270	−	0.736589i	0.425270	−	0.736589i
$836$		0			0
$837$		0			0
$838$	0.109208			0.109208
$839$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$839$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	0.623490	+	0.781831i	$0.285714\pi$
$840$		0			0
$841$	0.488831	+	0.846680i	0.488831	+	0.846680i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.445042			−0.445042
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−0.385113	+	0.262566i	−0.385113	+	0.262566i
$853$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	−0.900969	−	0.433884i	$-0.857143\pi$
$853$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	0.826239	+	0.563320i	$0.190476\pi$
$854$		0			0
$855$	−0.202749	−	0.254239i	−0.202749	−	0.254239i
$856$	1.07126			1.07126
$857$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	−0.900969	−	0.433884i	$-0.857143\pi$
$857$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	0.826239	+	0.563320i	$0.190476\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$	−0.198220	−	0.343327i	−0.198220	−	0.343327i
$861$		0			0
$862$	−0.455573	+	0.789075i	−0.455573	+	0.789075i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−0.615683	+	0.571270i	−0.615683	+	0.571270i
$865$		0			0
$866$		0			0
$867$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$868$		0			0
$869$		0			0
$870$	0.0437890	+	0.0210877i	0.0437890	+	0.0210877i
$871$		0			0
$872$	−0.160110			−0.160110
$873$		0			0
$874$		0			0
$875$		0			0
$876$	0.171391	−	0.116853i	0.171391	−	0.116853i
$877$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$877$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$878$		0			0
$879$	−0.826239	+	0.563320i	−0.826239	+	0.563320i
$880$		0			0
$881$	1.91115			1.91115			0.955573	−	0.294755i	$-0.0952381\pi$
$881$	1.91115			1.91115			0.955573	+	0.294755i	$0.0952381\pi$
$882$	−0.722521	+	0.108903i	−0.722521	+	0.108903i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		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	$0.333333\pi$
$888$	−0.0998270	−	1.33210i	−0.0998270	−	1.33210i
$889$		0			0
$890$	−0.537359			−0.537359
$891$		0			0
$892$	−0.581222			−0.581222
$893$		0			0
$894$		0			0
$895$	0.425270	+	0.736589i	0.425270	+	0.736589i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.369611	+	0.0557099i	−0.369611	+	0.0557099i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	0.997630	−	0.680173i	0.997630	−	0.680173i
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$	−0.722521	+	1.84095i	−0.722521	+	1.84095i
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$	0.208453	+	0.100386i	0.208453	+	0.100386i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.445396	−	0.771449i	0.445396	−	0.771449i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.500000	−	0.866025i	0.500000	−	0.866025i
$926$	−1.44504			−1.44504
$927$	0.0931869	+	0.116853i	0.0931869	+	0.116853i
$928$	0.125530			0.125530
$929$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$929$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	0.623490	+	0.781831i	$0.285714\pi$
$930$		0			0
$931$	−0.365341	−	0.632789i	−0.365341	−	0.632789i
$932$	−0.460898	−	0.798298i	−0.460898	−	0.798298i
$933$	1.03030	−	0.702449i	1.03030	−	0.702449i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	−0.658322	−	0.317031i	−0.658322	−	0.317031i
$940$		0			0
$941$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.623490	+	0.781831i	$0.285714\pi$
$941$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.365341	+	0.930874i	$0.380952\pi$
$942$	0.0902318	+	1.20406i	0.0902318	+	1.20406i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.733052	−	1.26968i	0.733052	−	1.26968i	−0.222521	−	0.974928i	$-0.571429\pi$
$947$	0.733052	−	1.26968i	0.733052	−	1.26968i	0.955573	−	0.294755i	$-0.0952381\pi$
$948$	0.0155017	+	0.206856i	0.0155017	+	0.206856i
$949$		0			0
$950$	0.214076	+	0.370790i	0.214076	+	0.370790i
$951$		0			0
$952$		0			0
$953$	1.91115			1.91115			0.955573	−	0.294755i	$-0.0952381\pi$
$953$	1.91115			1.91115			0.955573	+	0.294755i	$0.0952381\pi$
$954$		0			0
$955$	0.801938			0.801938
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−0.342095	+	0.233236i	−0.342095	+	0.233236i
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$		0			0
$963$	0.988831	−	0.149042i	0.988831	−	0.149042i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$	0.535628	+	0.927735i	0.535628	+	0.927735i
$969$		0			0
$970$		0			0
$971$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$971$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$972$	−0.290611	+	0.364415i	−0.290611	+	0.364415i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$977$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$978$		0			0
$979$		0			0
$980$	0.207436			0.207436
$981$	−0.147791	+	0.0222759i	−0.147791	+	0.0222759i
$982$		0			0
$983$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$983$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.440071	−	0.762226i	−0.440071	−	0.762226i
$996$	−0.0434348	−	0.579597i	−0.0434348	−	0.579597i
$997$	0.733052	−	1.26968i	0.733052	−	1.26968i	−0.222521	−	0.974928i	$-0.571429\pi$
$997$	0.733052	−	1.26968i	0.733052	−	1.26968i	0.955573	−	0.294755i	$-0.0952381\pi$
$998$	−1.31664			−1.31664
$999$	−0.277479	−	1.21572i	−0.277479	−	1.21572i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	639.1.g.b.70.3	✓	12
3.2	odd	2		1917.1.g.b.1774.4		12
9.4	even	3	inner	639.1.g.b.283.3	yes	12
9.5	odd	6		1917.1.g.b.496.4		12
71.70	odd	2	CM	639.1.g.b.70.3	✓	12
213.212	even	2		1917.1.g.b.1774.4		12
639.212	even	6		1917.1.g.b.496.4		12
639.283	odd	6	inner	639.1.g.b.283.3	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
639.1.g.b.70.3	✓	12	1.1	even	1	trivial
639.1.g.b.70.3	✓	12	71.70	odd	2	CM
639.1.g.b.283.3	yes	12	9.4	even	3	inner
639.1.g.b.283.3	yes	12	639.283	odd	6	inner
1917.1.g.b.496.4		12	9.5	odd	6
1917.1.g.b.496.4		12	639.212	even	6
1917.1.g.b.1774.4		12	3.2	odd	2
1917.1.g.b.1774.4		12	213.212	even	2

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:	\( N \)	\(=\)	\( 639 = 3^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	639.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.365341 + 0.632789i) q^{2} +(0.0747301 + 0.997204i) q^{3} +(0.233052 + 0.403658i) q^{4} +(0.222521 + 0.385418i) q^{5} +(-0.658322 - 0.317031i) q^{6} -1.07126 q^{8} +(-0.988831 + 0.149042i) q^{9} +O(q^{10})\)	\(q+(-0.365341 + 0.632789i) q^{2} +(0.0747301 + 0.997204i) q^{3} +(0.233052 + 0.403658i) q^{4} +(0.222521 + 0.385418i) q^{5} +(-0.658322 - 0.317031i) q^{6} -1.07126 q^{8} +(-0.988831 + 0.149042i) q^{9} -0.325184 q^{10} +(-0.385113 + 0.262566i) q^{12} +(-0.367711 + 0.250701i) q^{15} +(0.158322 - 0.274221i) q^{16} +(0.266948 - 0.680173i) q^{18} +0.730682 q^{19} +(-0.103718 + 0.179645i) q^{20} +(-0.0800550 - 1.06826i) q^{24} +(0.400969 - 0.694498i) q^{25} +(-0.222521 - 0.974928i) q^{27} +(-0.0747301 + 0.129436i) q^{29} +(-0.0243010 - 0.324275i) q^{30} +(-0.419945 - 0.727366i) q^{32} +(-0.290611 - 0.364415i) q^{36} +1.24698 q^{37} +(-0.266948 + 0.462368i) q^{38} +(-0.238377 - 0.412881i) q^{40} +(-0.955573 + 1.65510i) q^{43} +(-0.277479 - 0.347948i) q^{45} +(0.285286 + 0.137386i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(0.292981 + 0.507458i) q^{50} +(0.698220 + 0.215372i) q^{54} +(0.0546039 + 0.728639i) q^{57} +(-0.0546039 - 0.0945768i) q^{58} +(-0.186893 - 0.0900030i) q^{60} +0.930336 q^{64} +1.00000 q^{71} +(1.05929 - 0.159662i) q^{72} -0.445042 q^{73} +(-0.455573 + 0.789075i) q^{74} +(0.722521 + 0.347948i) q^{75} +(0.170287 + 0.294945i) q^{76} +(0.222521 - 0.385418i) q^{79} +0.140920 q^{80} +(0.955573 - 0.294755i) q^{81} +(-0.623490 + 1.07992i) q^{83} +(-0.698220 - 1.20935i) q^{86} +(-0.134659 - 0.0648483i) q^{87} +1.65248 q^{89} +(0.321552 - 0.0484662i) q^{90} +(0.162592 + 0.281618i) q^{95} +(0.693950 - 0.473127i) q^{96} +0.730682 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + q^{3} - 7 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + q^{9}+O(q^{10}) \) 12 * q - q^2 + q^3 - 7 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + q^9	\( 12 q - q^{2} + q^{3} - 7 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + q^{9} + 4 q^{10} + 2 q^{15} - 8 q^{16} + 13 q^{18} + 2 q^{19} + 7 q^{20} - 6 q^{24} - 4 q^{25} - 2 q^{27} - q^{29} - 9 q^{30} - 4 q^{37} - 13 q^{38} + 2 q^{40} - q^{43} - 4 q^{45} - 5 q^{48} - 6 q^{49} - 3 q^{50} - q^{54} - q^{57} + q^{58} - 7 q^{60} + 12 q^{64} + 12 q^{71} - 6 q^{72} - 4 q^{73} + 5 q^{74} + 8 q^{75} + 2 q^{79} - 10 q^{80} + q^{81} + 2 q^{83} + q^{86} - 5 q^{87} + 2 q^{89} + 12 q^{90} - 2 q^{95} - 7 q^{96} + 2 q^{98}+O(q^{100}) \) 12 * q - q^2 + q^3 - 7 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + q^9 + 4 * q^10 + 2 * q^15 - 8 * q^16 + 13 * q^18 + 2 * q^19 + 7 * q^20 - 6 * q^24 - 4 * q^25 - 2 * q^27 - q^29 - 9 * q^30 - 4 * q^37 - 13 * q^38 + 2 * q^40 - q^43 - 4 * q^45 - 5 * q^48 - 6 * q^49 - 3 * q^50 - q^54 - q^57 + q^58 - 7 * q^60 + 12 * q^64 + 12 * q^71 - 6 * q^72 - 4 * q^73 + 5 * q^74 + 8 * q^75 + 2 * q^79 - 10 * q^80 + q^81 + 2 * q^83 + q^86 - 5 * q^87 + 2 * q^89 + 12 * q^90 - 2 * q^95 - 7 * q^96 + 2 * q^98

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