LMFDB - Embedded newform 639.1.g.b.283.4

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			283.4
Root			$-0.988831 + 0.149042i$ of defining polynomial
Character	$\chi$	$=$	639.283
Dual form			639.1.g.b.70.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0747301 - 0.129436i) q^{2} +(0.955573 + 0.294755i) q^{3} +(0.488831 - 0.846680i) q^{4} +(-0.623490 + 1.07992i) q^{5} +(-0.0332580 - 0.145713i) q^{6} -0.295582 q^{8} +(0.826239 + 0.563320i) q^{9} +O(q^{10})$	\(q+(-0.0747301 - 0.129436i) q^{2} +(0.955573 + 0.294755i) q^{3} +(0.488831 - 0.846680i) q^{4} +(-0.623490 + 1.07992i) q^{5} +(-0.0332580 - 0.145713i) q^{6} -0.295582 q^{8} +(0.826239 + 0.563320i) q^{9} +0.186374 q^{10} +(0.716677 - 0.664979i) q^{12} +(-0.914101 + 0.848162i) q^{15} +(-0.466742 - 0.808421i) q^{16} +(0.0111692 - 0.149042i) q^{18} +0.149460 q^{19} +(0.609562 + 1.05579i) q^{20} +(-0.282450 - 0.0871242i) q^{24} +(-0.277479 - 0.480608i) q^{25} +(0.623490 + 0.781831i) q^{27} +(-0.955573 - 1.65510i) q^{29} +(0.178094 + 0.0549346i) q^{30} +(-0.217550 + 0.376808i) q^{32} +(0.880843 - 0.424191i) q^{36} -1.80194 q^{37} +(-0.0111692 - 0.0193456i) q^{38} +(0.184292 - 0.319203i) q^{40} +(-0.365341 - 0.632789i) q^{43} +(-1.12349 + 0.541044i) q^{45} +(-0.207720 - 0.910080i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.0414721 + 0.0718317i) q^{50} +(0.0546039 - 0.139129i) q^{54} +(0.142820 + 0.0440542i) q^{57} +(-0.142820 + 0.247372i) q^{58} +(0.271281 + 1.18856i) q^{60} -0.868454 q^{64} +1.00000 q^{71} +(-0.244221 - 0.166507i) q^{72} +1.24698 q^{73} +(0.134659 + 0.233236i) q^{74} +(-0.123490 - 0.541044i) q^{75} +(0.0730607 - 0.126545i) q^{76} +(-0.623490 - 1.07992i) q^{79} +1.16404 q^{80} +(0.365341 + 0.930874i) q^{81} +(0.900969 + 1.56052i) q^{83} +(-0.0546039 + 0.0945768i) q^{86} +(-0.425270 - 1.86323i) q^{87} -1.46610 q^{89} +(0.153989 + 0.104988i) q^{90} +(-0.0931869 + 0.161404i) q^{95} +(-0.318951 + 0.295943i) q^{96} +0.149460 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/639\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$2$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$3$	0.955573	+	0.294755i	0.955573	+	0.294755i
$3$	0.955573	+	0.294755i	0.955573	+	0.294755i
$4$	0.488831	−	0.846680i	0.488831	−	0.846680i
$5$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$5$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$6$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	−0.295582			−0.295582
$9$	0.826239	+	0.563320i	0.826239	+	0.563320i
$10$	0.186374			0.186374
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	0.716677	−	0.664979i	0.716677	−	0.664979i
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$	−0.914101	+	0.848162i	−0.914101	+	0.848162i
$16$	−0.466742	−	0.808421i	−0.466742	−	0.808421i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	0.0111692	−	0.149042i	0.0111692	−	0.149042i
$19$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$19$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$20$	0.609562	+	1.05579i	0.609562	+	1.05579i
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	−0.282450	−	0.0871242i	−0.282450	−	0.0871242i
$25$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$26$		0			0
$27$	0.623490	+	0.781831i	0.623490	+	0.781831i
$28$		0			0
$29$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$29$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$30$	0.178094	+	0.0549346i	0.178094	+	0.0549346i
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	−0.217550	+	0.376808i	−0.217550	+	0.376808i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.880843	−	0.424191i	0.880843	−	0.424191i
$37$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$37$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$38$	−0.0111692	−	0.0193456i	−0.0111692	−	0.0193456i
$39$		0			0
$40$	0.184292	−	0.319203i	0.184292	−	0.319203i
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	0.623490	−	0.781831i	$-0.285714\pi$
$43$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	−0.988831	+	0.149042i	$0.952381\pi$
$44$		0			0
$45$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	−0.207720	−	0.910080i	−0.207720	−	0.910080i
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	−0.0414721	+	0.0718317i	−0.0414721	+	0.0718317i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	0.0546039	−	0.139129i	0.0546039	−	0.139129i
$55$		0			0
$56$		0			0
$57$	0.142820	+	0.0440542i	0.142820	+	0.0440542i
$58$	−0.142820	+	0.247372i	−0.142820	+	0.247372i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$	0.271281	+	1.18856i	0.271281	+	1.18856i
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	−0.868454			−0.868454
$65$		0			0
$66$		0			0
$67$		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
$71$	1.00000			1.00000
$71$	1.00000			1.00000
$72$	−0.244221	−	0.166507i	−0.244221	−	0.166507i
$73$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$73$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$74$	0.134659	+	0.233236i	0.134659	+	0.233236i
$75$	−0.123490	−	0.541044i	−0.123490	−	0.541044i
$76$	0.0730607	−	0.126545i	0.0730607	−	0.126545i
$77$		0			0
$78$		0			0
$79$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$79$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$80$	1.16404			1.16404
$81$	0.365341	+	0.930874i	0.365341	+	0.930874i
$82$		0			0
$83$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$83$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$84$		0			0
$85$		0			0
$86$	−0.0546039	+	0.0945768i	−0.0546039	+	0.0945768i
$87$	−0.425270	−	1.86323i	−0.425270	−	1.86323i
$88$		0			0
$89$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$89$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$90$	0.153989	+	0.104988i	0.153989	+	0.104988i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.0931869	+	0.161404i	−0.0931869	+	0.161404i
$96$	−0.318951	+	0.295943i	−0.318951	+	0.295943i
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$	0.149460			0.149460
$99$		0			0
$100$	−0.542561			−0.542561
$101$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	−0.900969	−	0.433884i	$-0.857143\pi$
$101$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	0.0747301	−	0.997204i	$-0.476190\pi$
$102$		0			0
$103$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.222521	+	0.974928i	$0.571429\pi$
$103$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.733052	+	0.680173i	$0.761905\pi$
$104$		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	$0.666667\pi$
$108$	0.966742	−	0.145713i	0.966742	−	0.145713i
$109$	1.91115			1.91115			0.955573	−	0.294755i	$-0.0952381\pi$
$109$	1.91115			1.91115			0.955573	+	0.294755i	$0.0952381\pi$
$110$		0			0
$111$	−1.72188	−	0.531130i	−1.72188	−	0.531130i
$112$		0			0
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$	−0.00497075	−	0.0217783i	−0.00497075	−	0.0217783i
$115$		0			0
$116$	−1.86845			−1.86845
$117$		0			0
$118$		0			0
$119$		0			0
$120$	0.270191	−	0.250701i	0.270191	−	0.250701i
$121$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.554958			−0.554958
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	0.282450	+	0.489217i	0.282450	+	0.489217i
$129$	−0.162592	−	0.712362i	−0.162592	−	0.712362i
$130$		0			0
$131$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$131$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.23305	+	0.185853i	−1.23305	+	0.185853i
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$		0			0
$142$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i
$143$		0			0
$144$	0.0697593	−	0.930874i	0.0697593	−	0.930874i
$145$	2.38316			2.38316
$146$	−0.0931869	−	0.161404i	−0.0931869	−	0.161404i
$147$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$148$	−0.880843	+	1.52566i	−0.880843	+	1.52566i
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	−0.0608024	+	0.0564163i	−0.0608024	+	0.0564163i
$151$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$151$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$152$	−0.0441777			−0.0441777
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.733052	−	1.26968i	0.733052	−	1.26968i	−0.222521	−	0.974928i	$-0.571429\pi$
$157$	0.733052	−	1.26968i	0.733052	−	1.26968i	0.955573	−	0.294755i	$-0.0952381\pi$
$158$	−0.0931869	+	0.161404i	−0.0931869	+	0.161404i
$159$		0			0
$160$	−0.271281	−	0.469872i	−0.271281	−	0.469872i
$161$		0			0
$162$	0.0931869	−	0.116853i	0.0931869	−	0.116853i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	0.134659	−	0.233236i	0.134659	−	0.233236i
$167$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$167$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	0.623490	+	0.781831i	$0.285714\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$170$		0			0
$171$	0.123490	+	0.0841939i	0.123490	+	0.0841939i
$172$	−0.714360			−0.714360
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$	−0.209389	+	0.194285i	−0.209389	+	0.194285i
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.109562	+	0.189767i	0.109562	+	0.189767i
$179$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$179$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$180$	−0.0911053	+	1.21572i	−0.0911053	+	1.21572i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.12349	−	1.94594i	1.12349	−	1.94594i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	0.0278555			0.0278555
$191$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$191$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$192$	−0.829871	−	0.255981i	−0.829871	−	0.255981i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$	0.488831	+	0.846680i	0.488831	+	0.846680i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$199$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$200$	0.0820177	+	0.142059i	0.0820177	+	0.142059i
$201$		0			0
$202$	−0.123490	+	0.213891i	−0.123490	+	0.213891i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	0.285640			0.285640
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$		0			0
$213$	0.955573	+	0.294755i	0.955573	+	0.294755i
$214$	0.0747301	+	0.129436i	0.0747301	+	0.129436i
$215$	0.911146			0.911146
$216$	−0.184292	−	0.231095i	−0.184292	−	0.231095i
$217$		0			0
$218$	−0.142820	−	0.247372i	−0.142820	−	0.247372i
$219$	1.19158	+	0.367554i	1.19158	+	0.367554i
$220$		0			0
$221$		0			0
$222$	0.0599289	+	0.262566i	0.0599289	+	0.262566i
$223$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$223$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$224$		0			0
$225$	0.0414721	−	0.553406i	0.0414721	−	0.553406i
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	0.107115	−	0.0993879i	0.107115	−	0.0993879i
$229$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$229$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	0.623490	+	0.781831i	$0.285714\pi$
$230$		0			0
$231$		0			0
$232$	0.282450	+	0.489217i	0.282450	+	0.489217i
$233$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$233$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.277479	−	1.21572i	−0.277479	−	1.21572i
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$	1.11232	+	0.343105i	1.11232	+	0.343105i
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	0.149460			0.149460
$243$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$244$		0			0
$245$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	0.400969	+	1.75676i	0.400969	+	1.75676i
$250$	0.0414721	+	0.0718317i	0.0414721	+	0.0718317i
$251$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$251$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.392012	+	0.678985i	−0.392012	+	0.678985i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$	−0.0800550	+	0.0742802i	−0.0800550	+	0.0742802i
$259$		0			0
$260$		0			0
$261$	0.142820	−	1.90580i	0.142820	−	1.90580i
$262$	−0.295582			−0.295582
$263$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.623490	+	0.781831i	$0.285714\pi$
$263$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.365341	+	0.930874i	$0.380952\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1.40097	−	0.432142i	−1.40097	−	0.432142i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	0.116202	+	0.145713i	0.116202	+	0.145713i
$271$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$271$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	−0.900969	−	0.433884i	$-0.857143\pi$
$277$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	0.0747301	−	0.997204i	$-0.476190\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$	0.488831	−	0.846680i	0.488831	−	0.846680i
$285$	−0.136622	+	0.126766i	−0.136622	+	0.126766i
$286$		0			0
$287$		0			0
$288$	−0.392012	+	0.188783i	−0.392012	+	0.188783i
$289$	1.00000			1.00000
$290$	−0.178094	−	0.308467i	−0.178094	−	0.308467i
$291$		0			0
$292$	0.609562	−	1.05579i	0.609562	−	1.05579i
$293$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$293$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$294$	0.142820	+	0.0440542i	0.142820	+	0.0440542i
$295$		0			0
$296$	0.532620			0.532620
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.518457	−	0.159923i	−0.518457	−	0.159923i
$301$		0			0
$302$	0.109562	−	0.189767i	0.109562	−	0.189767i
$303$	−0.367711	−	1.61105i	−0.367711	−	1.61105i
$304$	−0.0697593	−	0.120827i	−0.0697593	−	0.120827i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	−1.40097	+	1.29991i	−1.40097	+	1.29991i
$310$		0			0
$311$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$311$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$312$		0			0
$313$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$313$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$314$	−0.219124			−0.219124
$315$		0			0
$316$	−1.21912			−1.21912
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$	0.541472	−	0.937857i	0.541472	−	0.937857i
$321$	−0.955573	−	0.294755i	−0.955573	−	0.294755i
$322$		0			0
$323$		0			0
$324$	0.966742	+	0.145713i	0.966742	+	0.145713i
$325$		0			0
$326$		0			0
$327$	1.82624	+	0.563320i	1.82624	+	0.563320i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	1.76169			1.76169
$333$	−1.48883	−	1.01507i	−1.48883	−	1.01507i
$334$	0.109208			0.109208
$335$		0			0
$336$		0			0
$337$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$337$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$338$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	0.00166935	−	0.0222759i	0.00166935	−	0.0222759i
$343$		0			0
$344$	0.107988	+	0.187041i	0.107988	+	0.187041i
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$	−1.78544	−	0.550736i	−1.78544	−	0.550736i
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$356$	−0.716677	+	1.24132i	−0.716677	+	1.24132i
$357$		0			0
$358$	−0.0546039	−	0.0945768i	−0.0546039	−	0.0945768i
$359$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$359$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$360$	0.332083	−	0.159923i	0.332083	−	0.159923i
$361$	−0.977662			−0.977662
$362$		0			0
$363$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$364$		0			0
$365$	−0.777479	+	1.34663i	−0.777479	+	1.34663i
$366$		0			0
$367$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.623490	+	0.781831i	$0.285714\pi$
$367$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.365341	+	0.930874i	$0.380952\pi$
$368$		0			0
$369$		0			0
$370$	−0.335834			−0.335834
$371$		0			0
$372$		0			0
$373$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$373$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$374$		0			0
$375$	−0.530303	−	0.163577i	−0.530303	−	0.163577i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$379$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$380$	0.0911053	+	0.157799i	0.0911053	+	0.157799i
$381$		0			0
$382$	0.0332580	−	0.0576046i	0.0332580	−	0.0576046i
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$	0.125702	+	0.550736i	0.125702	+	0.550736i
$385$		0			0
$386$		0			0
$387$	0.0546039	−	0.728639i	0.0546039	−	0.728639i
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	0.147791	−	0.255981i	0.147791	−	0.255981i
$393$	1.44973	−	1.34515i	1.44973	−	1.34515i
$394$		0			0
$395$	1.55496			1.55496
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−0.123490	−	0.213891i	−0.123490	−	0.213891i
$399$		0			0
$400$	−0.259022	+	0.448640i	−0.259022	+	0.448640i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$		0			0
$404$	−1.61556			−1.61556
$405$	−1.23305	−	0.185853i	−1.23305	−	0.185853i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.733052	−	1.26968i	0.733052	−	1.26968i	−0.222521	−	0.974928i	$-0.571429\pi$
$409$	0.733052	−	1.26968i	0.733052	−	1.26968i	0.955573	−	0.294755i	$-0.0952381\pi$
$410$		0			0
$411$		0			0
$412$	0.934227	+	1.61813i	0.934227	+	1.61813i
$413$		0			0
$414$		0			0
$415$	−2.24698			−2.24698
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.222521	+	0.974928i	$0.571429\pi$
$419$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.733052	+	0.680173i	$0.761905\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i
$427$		0			0
$428$	−0.488831	+	0.846680i	−0.488831	+	0.846680i
$429$		0			0
$430$	−0.0680900	−	0.117935i	−0.0680900	−	0.117935i
$431$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$431$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$432$	0.341040	−	0.868956i	0.341040	−	0.868956i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	2.27728	+	0.702449i	2.27728	+	0.702449i
$436$	0.934227	−	1.61813i	0.934227	−	1.61813i
$437$		0			0
$438$	−0.0414721	−	0.181701i	−0.0414721	−	0.181701i
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$	−1.29141	+	1.19825i	−1.29141	+	1.19825i
$445$	0.914101	−	1.58327i	0.914101	−	1.58327i
$446$	0.134659	−	0.233236i	0.134659	−	0.233236i
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−0.0747301	+	0.0359881i	−0.0747301	+	0.0359881i
$451$		0			0
$452$		0			0
$453$	0.326239	+	1.42935i	0.326239	+	1.42935i
$454$		0			0
$455$		0			0
$456$	−0.0422150	−	0.0130216i	−0.0422150	−	0.0130216i
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$	0.109208			0.109208
$459$		0			0
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	0.0747301	+	0.997204i	$0.476190\pi$
$463$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	−0.900969	+	0.433884i	$0.857143\pi$
$464$	−0.892012	+	1.54501i	−0.892012	+	1.54501i
$465$		0			0
$466$	−0.123490	−	0.213891i	−0.123490	−	0.213891i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.07473	−	0.997204i	1.07473	−	0.997204i
$472$		0			0
$473$		0			0
$474$	−0.136622	+	0.126766i	−0.136622	+	0.126766i
$475$	−0.0414721	−	0.0718317i	−0.0414721	−	0.0718317i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$	−0.120731	−	0.528958i	−0.120731	−	0.528958i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.488831	+	0.846680i	0.488831	+	0.846680i
$485$		0			0
$486$	0.123490	−	0.0841939i	0.123490	−	0.0841939i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$	−0.0931869	+	0.161404i	−0.0931869	+	0.161404i
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	0.197424	−	0.183183i	0.197424	−	0.183183i
$499$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$499$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$500$	−0.271281	+	0.469872i	−0.271281	+	0.469872i
$501$	−0.535628	+	0.496990i	−0.535628	+	0.496990i
$502$	−0.123490	−	0.213891i	−0.123490	−	0.213891i
$503$	1.91115			1.91115			0.955573	−	0.294755i	$-0.0952381\pi$
$503$	1.91115			1.91115			0.955573	+	0.294755i	$0.0952381\pi$
$504$		0			0
$505$	2.06061			2.06061
$506$		0			0
$507$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$508$		0			0
$509$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$509$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$510$		0			0
$511$		0			0
$512$	0.682080			0.682080
$513$	0.0931869	+	0.116853i	0.0931869	+	0.116853i
$514$		0			0
$515$	−1.19158	−	2.06388i	−1.19158	−	2.06388i
$516$	−0.682623	−	0.210561i	−0.682623	−	0.210561i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$521$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$522$	−0.257353	+	0.123935i	−0.257353	+	0.123935i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	−0.966742	−	1.67445i	−0.966742	−	1.67445i
$525$		0			0
$526$	0.147791	−	0.255981i	0.147791	−	0.255981i
$527$		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
$534$	0.0487597	+	0.213630i	0.0487597	+	0.213630i
$535$	0.623490	−	1.07992i	0.623490	−	1.07992i
$536$		0			0
$537$	0.698220	+	0.215372i	0.698220	+	0.215372i
$538$		0			0
$539$		0			0
$540$	−0.445396	+	1.13485i	−0.445396	+	1.13485i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	0.0332580	+	0.0576046i	0.0332580	+	0.0576046i
$543$		0			0
$544$		0			0
$545$	−1.19158	+	2.06388i	−1.19158	+	2.06388i
$546$		0			0
$547$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	−0.900969	−	0.433884i	$-0.857143\pi$
$547$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	0.0747301	−	0.997204i	$-0.476190\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.142820	−	0.247372i	−0.142820	−	0.247372i
$552$		0			0
$553$		0			0
$554$	−0.123490	+	0.213891i	−0.123490	+	0.213891i
$555$	1.64715	−	1.52833i	1.64715	−	1.52833i
$556$		0			0
$557$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$557$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−0.295582			−0.295582
$569$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	0.623490	−	0.781831i	$-0.285714\pi$
$569$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	−0.988831	+	0.149042i	$0.952381\pi$
$570$	0.0266179	+	0.00821054i	0.0266179	+	0.00821054i
$571$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$571$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$572$		0			0
$573$	0.0990311	+	0.433884i	0.0990311	+	0.433884i
$574$		0			0
$575$		0			0
$576$	−0.717550	−	0.489217i	−0.717550	−	0.489217i
$577$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$577$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$578$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i
$579$		0			0
$580$	1.16496	−	2.01777i	1.16496	−	2.01777i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.368584			−0.368584
$585$		0			0
$586$	−0.149460			−0.149460
$587$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$587$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$588$	0.217550	+	0.953150i	0.217550	+	0.953150i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.841040	+	1.45672i	0.841040	+	1.45672i
$593$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$593$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.57906	+	0.487076i	1.57906	+	0.487076i
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$	0.0365013	+	0.159923i	0.0365013	+	0.159923i
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$		0			0
$604$	1.43335			1.43335
$605$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$606$	−0.181049	+	0.167989i	−0.181049	+	0.167989i
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	−0.0325151	+	0.0563178i	−0.0325151	+	0.0563178i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$613$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$617$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$618$	0.272950	+	0.0841939i	0.272950	+	0.0841939i
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$	−0.269318			−0.269318
$623$		0			0
$624$		0			0
$625$	0.623490	−	1.07992i	0.623490	−	1.07992i
$626$	−0.0111692	+	0.0193456i	−0.0111692	+	0.0193456i
$627$		0			0
$628$	−0.716677	−	1.24132i	−0.716677	−	1.24132i
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	0.184292	+	0.319203i	0.184292	+	0.319203i
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	0.826239	+	0.563320i	0.826239	+	0.563320i
$640$	−0.704418			−0.704418
$641$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$641$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$642$	0.0332580	+	0.145713i	0.0332580	+	0.145713i
$643$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$643$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$644$		0			0
$645$	0.870666	+	0.268565i	0.870666	+	0.268565i
$646$		0			0
$647$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$647$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$648$	−0.107988	−	0.275149i	−0.107988	−	0.275149i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$	−0.0635609	−	0.278479i	−0.0635609	−	0.278479i
$655$	1.23305	+	2.13571i	1.23305	+	2.13571i
$656$		0			0
$657$	1.03030	+	0.702449i	1.03030	+	0.702449i
$658$		0			0
$659$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$659$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$660$		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
$664$	−0.266310	−	0.461262i	−0.266310	−	0.461262i
$665$		0			0
$666$	−0.0201262	+	0.268565i	−0.0201262	+	0.268565i
$667$		0			0
$668$	0.357180	+	0.618654i	0.357180	+	0.618654i
$669$	0.400969	+	1.75676i	0.400969	+	1.75676i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$		0			0
$675$	0.202749	−	0.516596i	0.202749	−	0.516596i
$676$	−0.977662			−0.977662
$677$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$677$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$678$		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			$\pi$
$684$	0.131651	−	0.0633997i	0.131651	−	0.0633997i
$685$		0			0
$686$		0			0
$687$	−0.535628	+	0.496990i	−0.535628	+	0.496990i
$688$	−0.341040	+	0.590699i	−0.341040	+	0.590699i
$689$		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			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$	0.125702	+	0.550736i	0.125702	+	0.550736i
$697$		0			0
$698$		0			0
$699$	1.57906	+	0.487076i	1.57906	+	0.487076i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−0.269318			−0.269318
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$	0.186374			0.186374
$711$	0.0931869	−	1.24349i	0.0931869	−	1.24349i
$712$	0.433353			0.433353
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.357180	−	0.618654i	0.357180	−	0.618654i
$717$		0			0
$718$	0.147791	+	0.255981i	0.147791	+	0.255981i
$719$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$719$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$720$	0.961771	+	0.655725i	0.961771	+	0.655725i
$721$		0			0
$722$	0.0730607	+	0.126545i	0.0730607	+	0.126545i
$723$		0			0
$724$		0			0
$725$	−0.530303	+	0.918512i	−0.530303	+	0.918512i
$726$	0.142820	+	0.0440542i	0.142820	+	0.0440542i
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$730$	0.232404			0.232404
$731$		0			0
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$	0.147791	−	0.255981i	0.147791	−	0.255981i
$735$	−0.277479	−	1.21572i	−0.277479	−	1.21572i
$736$		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	$0.666667\pi$
$740$	−1.09839	−	1.90247i	−1.09839	−	1.90247i
$741$		0			0
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$	0.186374			0.186374
$747$	−0.134659	+	1.79690i	−0.134659	+	1.79690i
$748$		0			0
$749$		0			0
$750$	0.0184568	+	0.0808646i	0.0184568	+	0.0808646i
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$	1.57906	+	0.487076i	1.57906	+	0.487076i
$754$		0			0
$755$	−1.82820			−1.82820
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	0.109562	+	0.189767i	0.109562	+	0.189767i
$759$		0			0
$760$	0.0275443	−	0.0477082i	0.0275443	−	0.0477082i
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$		0			0
$764$	0.435100			0.435100
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.574730	+	0.533272i	−0.574730	+	0.533272i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$	−0.0983929	+	0.0473835i	−0.0983929	+	0.0473835i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.698220	−	1.77904i	0.698220	−	1.77904i
$784$	0.933484			0.933484
$785$	0.914101	+	1.58327i	0.914101	+	1.58327i
$786$	−0.282450	−	0.0871242i	−0.282450	−	0.0871242i
$787$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$787$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$788$		0			0
$789$	0.440071	+	1.92808i	0.440071	+	1.92808i
$790$	−0.116202	−	0.201268i	−0.116202	−	0.201268i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.807782	−	1.39912i	0.807782	−	1.39912i
$797$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	−0.900969	−	0.433884i	$-0.857143\pi$
$797$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	0.826239	+	0.563320i	$0.190476\pi$
$798$		0			0
$799$		0			0
$800$	0.241462			0.241462
$801$	−1.21135	−	0.825886i	−1.21135	−	0.825886i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.244221	+	0.423003i	0.244221	+	0.423003i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	0.0680900	+	0.173490i	0.0680900	+	0.173490i
$811$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$811$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$812$		0			0
$813$	−0.425270	−	0.131178i	−0.425270	−	0.131178i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.0546039	−	0.0945768i	−0.0546039	−	0.0945768i
$818$	−0.219124			−0.219124
$819$		0			0
$820$		0			0
$821$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$821$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$	0.282450	−	0.489217i	0.282450	−	0.489217i
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$829$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$830$	0.167917	+	0.290841i	0.167917	+	0.290841i
$831$	−0.367711	−	1.61105i	−0.367711	−	1.61105i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−0.455573	−	0.789075i	−0.455573	−	0.789075i
$836$		0			0
$837$		0			0
$838$	0.285640			0.285640
$839$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$839$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$840$		0			0
$841$	−1.32624	+	2.29711i	−1.32624	+	2.29711i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.24698			1.24698
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	0.716677	−	0.664979i	0.716677	−	0.664979i
$853$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$853$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$854$		0			0
$855$	−0.167917	+	0.0808646i	−0.167917	+	0.0808646i
$856$	0.295582			0.295582
$857$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$857$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$	0.445396	−	0.771449i	0.445396	−	0.771449i
$861$		0			0
$862$	0.134659	+	0.233236i	0.134659	+	0.233236i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−0.430241	+	0.0648483i	−0.430241	+	0.0648483i
$865$		0			0
$866$		0			0
$867$	0.955573	+	0.294755i	0.955573	+	0.294755i
$868$		0			0
$869$		0			0
$870$	−0.0792592	−	0.347257i	−0.0792592	−	0.347257i
$871$		0			0
$872$	−0.564900			−0.564900
$873$		0			0
$874$		0			0
$875$		0			0
$876$	0.893681	−	0.829215i	0.893681	−	0.829215i
$877$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$877$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$878$		0			0
$879$	0.733052	−	0.680173i	0.733052	−	0.680173i
$880$		0			0
$881$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$881$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$882$	0.123490	+	0.0841939i	0.123490	+	0.0841939i
$883$		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	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$	0.508957	+	0.156992i	0.508957	+	0.156992i
$889$		0			0
$890$	−0.273243			−0.273243
$891$		0			0
$892$	1.76169			1.76169
$893$		0			0
$894$		0			0
$895$	−0.455573	+	0.789075i	−0.455573	+	0.789075i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.448285	−	0.305636i	−0.448285	−	0.305636i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	0.160629	−	0.149042i	0.160629	−	0.149042i
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$		0			0
$909$	0.123490	−	1.64786i	0.123490	−	1.64786i
$910$		0			0
$911$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$911$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$912$	−0.0310458	−	0.136021i	−0.0310458	−	0.136021i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.357180	+	0.618654i	0.357180	+	0.618654i
$917$		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
$926$	0.246980			0.246980
$927$	−1.72188	+	0.829215i	−1.72188	+	0.829215i
$928$	0.831540			0.831540
$929$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$929$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$930$		0			0
$931$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i
$932$	0.807782	−	1.39912i	0.807782	−	1.39912i
$933$	1.32091	−	1.22563i	1.32091	−	1.22563i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i
$940$		0			0
$941$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	0.0747301	+	0.997204i	$0.476190\pi$
$941$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	−0.900969	+	0.433884i	$0.857143\pi$
$942$	−0.209389	−	0.0645880i	−0.209389	−	0.0645880i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.623490	+	0.781831i	$0.285714\pi$
$947$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.365341	+	0.930874i	$0.380952\pi$
$948$	−1.16496	−	0.359343i	−1.16496	−	0.359343i
$949$		0			0
$950$	−0.00619842	+	0.0107360i	−0.00619842	+	0.0107360i
$951$		0			0
$952$		0			0
$953$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$953$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$954$		0			0
$955$	−0.554958			−0.554958
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	0.793854	−	0.736589i	0.793854	−	0.736589i
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$		0			0
$963$	−0.826239	−	0.563320i	−0.826239	−	0.563320i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$	0.147791	−	0.255981i	0.147791	−	0.255981i
$969$		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	$0.666667\pi$
$972$	0.880843	+	0.424191i	0.880843	+	0.424191i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.222521	+	0.974928i	$0.571429\pi$
$977$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.733052	+	0.680173i	$0.761905\pi$
$978$		0			0
$979$		0			0
$980$	−1.21912			−1.21912
$981$	1.57906	+	1.07659i	1.57906	+	1.07659i
$982$		0			0
$983$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	−0.900969	−	0.433884i	$-0.857143\pi$
$983$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	0.0747301	−	0.997204i	$-0.476190\pi$
$984$		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
$995$	−1.03030	+	1.78454i	−1.03030	+	1.78454i
$996$	1.68342	+	0.519266i	1.68342	+	0.519266i
$997$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.623490	+	0.781831i	$0.285714\pi$
$997$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.365341	+	0.930874i	$0.380952\pi$
$998$	−0.0665160			−0.0665160
$999$	−1.12349	−	1.40881i	−1.12349	−	1.40881i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	639.1.g.b.283.4	yes	12
3.2	odd	2		1917.1.g.b.496.3		12
9.2	odd	6		1917.1.g.b.1774.3		12
9.7	even	3	inner	639.1.g.b.70.4	✓	12
71.70	odd	2	CM	639.1.g.b.283.4	yes	12
213.212	even	2		1917.1.g.b.496.3		12
639.70	odd	6	inner	639.1.g.b.70.4	✓	12
639.425	even	6		1917.1.g.b.1774.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
639.1.g.b.70.4	✓	12	9.7	even	3	inner
639.1.g.b.70.4	✓	12	639.70	odd	6	inner
639.1.g.b.283.4	yes	12	1.1	even	1	trivial
639.1.g.b.283.4	yes	12	71.70	odd	2	CM
1917.1.g.b.496.3		12	3.2	odd	2
1917.1.g.b.496.3		12	213.212	even	2
1917.1.g.b.1774.3		12	9.2	odd	6
1917.1.g.b.1774.3		12	639.425	even	6

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)

\(f(q)\)	\(=\)	\(q+(-0.0747301 - 0.129436i) q^{2} +(0.955573 + 0.294755i) q^{3} +(0.488831 - 0.846680i) q^{4} +(-0.623490 + 1.07992i) q^{5} +(-0.0332580 - 0.145713i) q^{6} -0.295582 q^{8} +(0.826239 + 0.563320i) q^{9} +O(q^{10})\)	\(q+(-0.0747301 - 0.129436i) q^{2} +(0.955573 + 0.294755i) q^{3} +(0.488831 - 0.846680i) q^{4} +(-0.623490 + 1.07992i) q^{5} +(-0.0332580 - 0.145713i) q^{6} -0.295582 q^{8} +(0.826239 + 0.563320i) q^{9} +0.186374 q^{10} +(0.716677 - 0.664979i) q^{12} +(-0.914101 + 0.848162i) q^{15} +(-0.466742 - 0.808421i) q^{16} +(0.0111692 - 0.149042i) q^{18} +0.149460 q^{19} +(0.609562 + 1.05579i) q^{20} +(-0.282450 - 0.0871242i) q^{24} +(-0.277479 - 0.480608i) q^{25} +(0.623490 + 0.781831i) q^{27} +(-0.955573 - 1.65510i) q^{29} +(0.178094 + 0.0549346i) q^{30} +(-0.217550 + 0.376808i) q^{32} +(0.880843 - 0.424191i) q^{36} -1.80194 q^{37} +(-0.0111692 - 0.0193456i) q^{38} +(0.184292 - 0.319203i) q^{40} +(-0.365341 - 0.632789i) q^{43} +(-1.12349 + 0.541044i) q^{45} +(-0.207720 - 0.910080i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.0414721 + 0.0718317i) q^{50} +(0.0546039 - 0.139129i) q^{54} +(0.142820 + 0.0440542i) q^{57} +(-0.142820 + 0.247372i) q^{58} +(0.271281 + 1.18856i) q^{60} -0.868454 q^{64} +1.00000 q^{71} +(-0.244221 - 0.166507i) q^{72} +1.24698 q^{73} +(0.134659 + 0.233236i) q^{74} +(-0.123490 - 0.541044i) q^{75} +(0.0730607 - 0.126545i) q^{76} +(-0.623490 - 1.07992i) q^{79} +1.16404 q^{80} +(0.365341 + 0.930874i) q^{81} +(0.900969 + 1.56052i) q^{83} +(-0.0546039 + 0.0945768i) q^{86} +(-0.425270 - 1.86323i) q^{87} -1.46610 q^{89} +(0.153989 + 0.104988i) q^{90} +(-0.0931869 + 0.161404i) q^{95} +(-0.318951 + 0.295943i) q^{96} +0.149460 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + q^{3} - 7 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more