LMFDB - Embedded newform 639.1.g.b.283.1

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			283.1
Root			$0.826239 - 0.563320i$ of defining polynomial
Character	$\chi$	$=$	639.283
Dual form			639.1.g.b.70.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.955573 - 1.65510i) q^{2} +(0.365341 + 0.930874i) q^{3} +(-1.32624 + 2.29711i) q^{4} +(0.900969 - 1.56052i) q^{5} +(1.19158 - 1.49419i) q^{6} +3.15813 q^{8} +(-0.733052 + 0.680173i) q^{9} +O(q^{10})$	\(q+(-0.955573 - 1.65510i) q^{2} +(0.365341 + 0.930874i) q^{3} +(-1.32624 + 2.29711i) q^{4} +(0.900969 - 1.56052i) q^{5} +(1.19158 - 1.49419i) q^{6} +3.15813 q^{8} +(-0.733052 + 0.680173i) q^{9} -3.44377 q^{10} +(-2.62285 - 0.395331i) q^{12} +(1.78181 + 0.268565i) q^{15} +(-1.69158 - 2.92990i) q^{16} +(1.82624 + 0.563320i) q^{18} +1.91115 q^{19} +(2.38980 + 4.13925i) q^{20} +(1.15379 + 2.93982i) q^{24} +(-1.12349 - 1.94594i) q^{25} +(-0.900969 - 0.433884i) q^{27} +(-0.365341 - 0.632789i) q^{29} +(-1.25815 - 3.20571i) q^{30} +(-1.65379 + 2.86445i) q^{32} +(-0.590232 - 2.58597i) q^{36} -0.445042 q^{37} +(-1.82624 - 3.16314i) q^{38} +(2.84537 - 4.92833i) q^{40} +(-0.0747301 - 0.129436i) q^{43} +(0.400969 + 1.75676i) q^{45} +(2.10937 - 2.64506i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-2.14715 + 3.71898i) q^{50} +(0.142820 + 1.90580i) q^{54} +(0.698220 + 1.77904i) q^{57} +(-0.698220 + 1.20935i) q^{58} +(-2.98003 + 3.73684i) q^{60} +2.93812 q^{64} +1.00000 q^{71} +(-2.31507 + 2.14807i) q^{72} -1.80194 q^{73} +(0.425270 + 0.736589i) q^{74} +(1.40097 - 1.75676i) q^{75} +(-2.53464 + 4.39012i) q^{76} +(0.900969 + 1.56052i) q^{79} -6.09624 q^{80} +(0.0747301 - 0.997204i) q^{81} +(0.222521 + 0.385418i) q^{83} +(-0.142820 + 0.247372i) q^{86} +(0.455573 - 0.571270i) q^{87} -1.97766 q^{89} +(2.52446 - 2.34236i) q^{90} +(1.72188 - 2.98239i) q^{95} +(-3.27064 - 0.492970i) q^{96} +1.91115 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$2$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$3$	0.365341	+	0.930874i	0.365341	+	0.930874i
$3$	0.365341	+	0.930874i	0.365341	+	0.930874i
$4$	−1.32624	+	2.29711i	−1.32624	+	2.29711i
$5$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$5$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$6$	1.19158	−	1.49419i	1.19158	−	1.49419i
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	3.15813			3.15813
$9$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$10$	−3.44377			−3.44377
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	−2.62285	−	0.395331i	−2.62285	−	0.395331i
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$	1.78181	+	0.268565i	1.78181	+	0.268565i
$16$	−1.69158	−	2.92990i	−1.69158	−	2.92990i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	1.82624	+	0.563320i	1.82624	+	0.563320i
$19$	1.91115			1.91115			0.955573	−	0.294755i	$-0.0952381\pi$
$19$	1.91115			1.91115			0.955573	+	0.294755i	$0.0952381\pi$
$20$	2.38980	+	4.13925i	2.38980	+	4.13925i
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	1.15379	+	2.93982i	1.15379	+	2.93982i
$25$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$26$		0			0
$27$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$28$		0			0
$29$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	0.623490	−	0.781831i	$-0.285714\pi$
$29$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	−0.988831	+	0.149042i	$0.952381\pi$
$30$	−1.25815	−	3.20571i	−1.25815	−	3.20571i
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	−1.65379	+	2.86445i	−1.65379	+	2.86445i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.590232	−	2.58597i	−0.590232	−	2.58597i
$37$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$37$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$38$	−1.82624	−	3.16314i	−1.82624	−	3.16314i
$39$		0			0
$40$	2.84537	−	4.92833i	2.84537	−	4.92833i
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$43$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$44$		0			0
$45$	0.400969	+	1.75676i	0.400969	+	1.75676i
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	2.10937	−	2.64506i	2.10937	−	2.64506i
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	−2.14715	+	3.71898i	−2.14715	+	3.71898i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	0.142820	+	1.90580i	0.142820	+	1.90580i
$55$		0			0
$56$		0			0
$57$	0.698220	+	1.77904i	0.698220	+	1.77904i
$58$	−0.698220	+	1.20935i	−0.698220	+	1.20935i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$	−2.98003	+	3.73684i	−2.98003	+	3.73684i
$61$		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
$64$	2.93812			2.93812
$65$		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
$72$	−2.31507	+	2.14807i	−2.31507	+	2.14807i
$73$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$73$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$74$	0.425270	+	0.736589i	0.425270	+	0.736589i
$75$	1.40097	−	1.75676i	1.40097	−	1.75676i
$76$	−2.53464	+	4.39012i	−2.53464	+	4.39012i
$77$		0			0
$78$		0			0
$79$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$79$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$80$	−6.09624			−6.09624
$81$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$82$		0			0
$83$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$83$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$84$		0			0
$85$		0			0
$86$	−0.142820	+	0.247372i	−0.142820	+	0.247372i
$87$	0.455573	−	0.571270i	0.455573	−	0.571270i
$88$		0			0
$89$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$89$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$90$	2.52446	−	2.34236i	2.52446	−	2.34236i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.72188	−	2.98239i	1.72188	−	2.98239i
$96$	−3.27064	−	0.492970i	−3.27064	−	0.492970i
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$	1.91115			1.91115
$99$		0			0
$100$	5.96006			5.96006
$101$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$101$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$102$		0			0
$103$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$103$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	0.623490	+	0.781831i	$0.285714\pi$
$104$		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	$0.666667\pi$
$108$	2.19158	−	1.49419i	2.19158	−	1.49419i
$109$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$109$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$110$		0			0
$111$	−0.162592	−	0.414278i	−0.162592	−	0.414278i
$112$		0			0
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$	2.27728	−	2.85562i	2.27728	−	2.85562i
$115$		0			0
$116$	1.93812			1.93812
$117$		0			0
$118$		0			0
$119$		0			0
$120$	5.62718	+	0.848162i	5.62718	+	0.848162i
$121$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.24698			−2.24698
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−1.15379	−	1.99843i	−1.15379	−	1.99843i
$129$	0.0931869	−	0.116853i	0.0931869	−	0.116853i
$130$		0			0
$131$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	0.0747301	+	0.997204i	$0.476190\pi$
$131$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	−0.900969	+	0.433884i	$0.857143\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.48883	+	1.01507i	−1.48883	+	1.01507i
$136$		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
$142$	−0.955573	−	1.65510i	−0.955573	−	1.65510i
$143$		0			0
$144$	3.23286	+	0.997204i	3.23286	+	0.997204i
$145$	−1.31664			−1.31664
$146$	1.72188	+	2.98239i	1.72188	+	2.98239i
$147$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$148$	0.590232	−	1.02231i	0.590232	−	1.02231i
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	−4.24634	−	0.640033i	−4.24634	−	0.640033i
$151$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.623490	+	0.781831i	$0.285714\pi$
$151$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.365341	+	0.930874i	$0.380952\pi$
$152$	6.03564			6.03564
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$157$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$158$	1.72188	−	2.98239i	1.72188	−	2.98239i
$159$		0			0
$160$	2.98003	+	5.16157i	2.98003	+	5.16157i
$161$		0			0
$162$	−1.72188	+	0.829215i	−1.72188	+	0.829215i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	0.425270	−	0.736589i	0.425270	−	0.736589i
$167$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	−0.900969	−	0.433884i	$-0.857143\pi$
$167$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	0.826239	+	0.563320i	$0.190476\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$170$		0			0
$171$	−1.40097	+	1.29991i	−1.40097	+	1.29991i
$172$	0.396440			0.396440
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$	−1.38084	−	0.208129i	−1.38084	−	0.208129i
$175$		0			0
$176$		0			0
$177$		0			0
$178$	1.88980	+	3.27323i	1.88980	+	3.27323i
$179$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$179$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$180$	−4.56726	−	1.40881i	−4.56726	−	1.40881i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−6.58154			−6.58154
$191$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$191$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$192$	1.07341	+	2.73502i	1.07341	+	2.73502i
$193$		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
$196$	−1.32624	−	2.29711i	−1.32624	−	2.29711i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$199$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$200$	−3.54812	−	6.14553i	−3.54812	−	6.14553i
$201$		0			0
$202$	1.40097	−	2.42655i	1.40097	−	2.42655i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	1.39644			1.39644
$207$		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
$213$	0.365341	+	0.930874i	0.365341	+	0.930874i
$214$	0.955573	+	1.65510i	0.955573	+	1.65510i
$215$	−0.269318			−0.269318
$216$	−2.84537	−	1.37026i	−2.84537	−	1.37026i
$217$		0			0
$218$	−0.698220	−	1.20935i	−0.698220	−	1.20935i
$219$	−0.658322	−	1.67738i	−0.658322	−	1.67738i
$220$		0			0
$221$		0			0
$222$	−0.530303	+	0.664979i	−0.530303	+	0.664979i
$223$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$223$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$224$		0			0
$225$	2.14715	+	0.662309i	2.14715	+	0.662309i
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	−5.01265	−	0.755536i	−5.01265	−	0.755536i
$229$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	−0.900969	−	0.433884i	$-0.857143\pi$
$229$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	0.826239	+	0.563320i	$0.190476\pi$
$230$		0			0
$231$		0			0
$232$	−1.15379	−	1.99843i	−1.15379	−	1.99843i
$233$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$233$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.12349	+	1.40881i	−1.12349	+	1.40881i
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$	−2.22721	−	5.67483i	−2.22721	−	5.67483i
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	1.91115			1.91115
$243$	0.955573	−	0.294755i	0.955573	−	0.294755i
$244$		0			0
$245$	0.900969	+	1.56052i	0.900969	+	1.56052i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−0.277479	+	0.347948i	−0.277479	+	0.347948i
$250$	2.14715	+	3.71898i	2.14715	+	3.71898i
$251$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$251$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.736007	+	1.27480i	−0.736007	+	1.27480i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$	−0.282450	−	0.0425725i	−0.282450	−	0.0425725i
$259$		0			0
$260$		0			0
$261$	0.698220	+	0.215372i	0.698220	+	0.215372i
$262$	3.15813			3.15813
$263$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	−0.900969	−	0.433884i	$-0.857143\pi$
$263$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	0.0747301	−	0.997204i	$-0.476190\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.722521	−	1.84095i	−0.722521	−	1.84095i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	3.10273	+	1.49419i	3.10273	+	1.49419i
$271$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$271$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$277$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$278$		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	$0.666667\pi$
$284$	−1.32624	+	2.29711i	−1.32624	+	2.29711i
$285$	3.40530	+	0.513267i	3.40530	+	0.513267i
$286$		0			0
$287$		0			0
$288$	−0.736007	−	3.22466i	−0.736007	−	3.22466i
$289$	1.00000			1.00000
$290$	1.25815	+	2.17918i	1.25815	+	2.17918i
$291$		0			0
$292$	2.38980	−	4.13925i	2.38980	−	4.13925i
$293$	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			$0$
$294$	0.698220	+	1.77904i	0.698220	+	1.77904i
$295$		0			0
$296$	−1.40550			−1.40550
$297$		0			0
$298$		0			0
$299$		0			0
$300$	2.17746	+	5.54807i	2.17746	+	5.54807i
$301$		0			0
$302$	1.88980	−	3.27323i	1.88980	−	3.27323i
$303$	−0.914101	+	1.14625i	−0.914101	+	1.14625i
$304$	−3.23286	−	5.59947i	−3.23286	−	5.59947i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	−0.722521	−	0.108903i	−0.722521	−	0.108903i
$310$		0			0
$311$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$311$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$312$		0			0
$313$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$313$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$314$	−3.77960			−3.77960
$315$		0			0
$316$	−4.77960			−4.77960
$317$		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
$320$	2.64715	−	4.58500i	2.64715	−	4.58500i
$321$	−0.365341	−	0.930874i	−0.365341	−	0.930874i
$322$		0			0
$323$		0			0
$324$	2.19158	+	1.49419i	2.19158	+	1.49419i
$325$		0			0
$326$		0			0
$327$	0.266948	+	0.680173i	0.266948	+	0.680173i
$328$		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	$0.333333\pi$
$332$	−1.18046			−1.18046
$333$	0.326239	−	0.302705i	0.326239	−	0.302705i
$334$	0.285640			0.285640
$335$		0			0
$336$		0			0
$337$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$337$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$338$	−0.955573	+	1.65510i	−0.955573	+	1.65510i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	3.49021	+	1.07659i	3.49021	+	1.07659i
$343$		0			0
$344$	−0.236007	−	0.408776i	−0.236007	−	0.408776i
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$	0.708074	+	1.80414i	0.708074	+	1.80414i
$349$		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
$355$	0.900969	−	1.56052i	0.900969	−	1.56052i
$356$	2.62285	−	4.54291i	2.62285	−	4.54291i
$357$		0			0
$358$	−0.142820	−	0.247372i	−0.142820	−	0.247372i
$359$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$359$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$360$	1.26631	+	5.54807i	1.26631	+	5.54807i
$361$	2.65248			2.65248
$362$		0			0
$363$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$364$		0			0
$365$	−1.62349	+	2.81197i	−1.62349	+	2.81197i
$366$		0			0
$367$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	−0.900969	−	0.433884i	$-0.857143\pi$
$367$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	0.0747301	−	0.997204i	$-0.476190\pi$
$368$		0			0
$369$		0			0
$370$	1.53262			1.53262
$371$		0			0
$372$		0			0
$373$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$373$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$374$		0			0
$375$	−0.820914	−	2.09165i	−0.820914	−	2.09165i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$379$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$380$	4.56726	+	7.91072i	4.56726	+	7.91072i
$381$		0			0
$382$	−1.19158	+	2.06388i	−1.19158	+	2.06388i
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$	1.43876	−	1.80414i	1.43876	−	1.80414i
$385$		0			0
$386$		0			0
$387$	0.142820	+	0.0440542i	0.142820	+	0.0440542i
$388$		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
$392$	−1.57906	+	2.73502i	−1.57906	+	2.73502i
$393$	−1.63402	−	0.246289i	−1.63402	−	0.246289i
$394$		0			0
$395$	3.24698			3.24698
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	1.40097	+	2.42655i	1.40097	+	2.42655i
$399$		0			0
$400$	−3.80095	+	6.58343i	−3.80095	+	6.58343i
$401$		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
$404$	−3.88881			−3.88881
$405$	−1.48883	−	1.01507i	−1.48883	−	1.01507i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$409$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$410$		0			0
$411$		0			0
$412$	−0.969059	−	1.67846i	−0.969059	−	1.67846i
$413$		0			0
$414$		0			0
$415$	0.801938			0.801938
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$419$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	0.623490	+	0.781831i	$0.285714\pi$
$420$		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
$426$	1.19158	−	1.49419i	1.19158	−	1.49419i
$427$		0			0
$428$	1.32624	−	2.29711i	1.32624	−	2.29711i
$429$		0			0
$430$	0.257353	+	0.445748i	0.257353	+	0.445748i
$431$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$431$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$432$	0.252824	+	3.37370i	0.252824	+	3.37370i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−0.481024	−	1.22563i	−0.481024	−	1.22563i
$436$	−0.969059	+	1.67846i	−0.969059	+	1.67846i
$437$		0			0
$438$	−2.14715	+	2.69244i	−2.14715	+	2.69244i
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$	1.16728	+	0.175939i	1.16728	+	0.175939i
$445$	−1.78181	+	3.08619i	−1.78181	+	3.08619i
$446$	0.425270	−	0.736589i	0.425270	−	0.736589i
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−0.955573	−	4.18664i	−0.955573	−	4.18664i
$451$		0			0
$452$		0			0
$453$	−1.23305	+	1.54620i	−1.23305	+	1.54620i
$454$		0			0
$455$		0			0
$456$	2.20507	+	5.61842i	2.20507	+	5.61842i
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$	0.285640			0.285640
$459$		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
$463$	0.733052	−	1.26968i	0.733052	−	1.26968i	−0.222521	−	0.974928i	$-0.571429\pi$
$463$	0.733052	−	1.26968i	0.733052	−	1.26968i	0.955573	−	0.294755i	$-0.0952381\pi$
$464$	−1.23601	+	2.14083i	−1.23601	+	2.14083i
$465$		0			0
$466$	1.40097	+	2.42655i	1.40097	+	2.42655i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.95557	+	0.294755i	1.95557	+	0.294755i
$472$		0			0
$473$		0			0
$474$	3.40530	+	0.513267i	3.40530	+	0.513267i
$475$	−2.14715	−	3.71898i	−2.14715	−	3.71898i
$476$		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	$0.333333\pi$
$480$	−3.71604	+	4.65976i	−3.71604	+	4.65976i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.32624	−	2.29711i	−1.32624	−	2.29711i
$485$		0			0
$486$	−1.40097	−	1.29991i	−1.40097	−	1.29991i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$	1.72188	−	2.98239i	1.72188	−	2.98239i
$491$		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
$498$	0.841040	+	0.126766i	0.841040	+	0.126766i
$499$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$499$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$500$	2.98003	−	5.16157i	2.98003	−	5.16157i
$501$	−0.147791	−	0.0222759i	−0.147791	−	0.0222759i
$502$	1.40097	+	2.42655i	1.40097	+	2.42655i
$503$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$503$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$504$		0			0
$505$	2.64183			2.64183
$506$		0			0
$507$	0.623490	−	0.781831i	0.623490	−	0.781831i
$508$		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
$512$	0.505648			0.505648
$513$	−1.72188	−	0.829215i	−1.72188	−	0.829215i
$514$		0			0
$515$	0.658322	+	1.14025i	0.658322	+	1.14025i
$516$	0.144836	+	0.369035i	0.144836	+	0.369035i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$521$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$522$	−0.310737	−	1.36143i	−0.310737	−	1.36143i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	−2.19158	−	3.79593i	−2.19158	−	3.79593i
$525$		0			0
$526$	−1.57906	+	2.73502i	−1.57906	+	2.73502i
$527$		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
$534$	−2.35654	+	2.95501i	−2.35654	+	2.95501i
$535$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$536$		0			0
$537$	0.0546039	+	0.139129i	0.0546039	+	0.139129i
$538$		0			0
$539$		0			0
$540$	−0.357180	−	4.76623i	−0.357180	−	4.76623i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	−1.19158	−	2.06388i	−1.19158	−	2.06388i
$543$		0			0
$544$		0			0
$545$	0.658322	−	1.14025i	0.658322	−	1.14025i
$546$		0			0
$547$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$547$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.698220	−	1.20935i	−0.698220	−	1.20935i
$552$		0			0
$553$		0			0
$554$	1.40097	−	2.42655i	1.40097	−	2.42655i
$555$	−0.792981	−	0.119523i	−0.792981	−	0.119523i
$556$		0			0
$557$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$557$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$558$		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
$568$	3.15813			3.15813
$569$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$569$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$570$	−2.40451	−	6.12658i	−2.40451	−	6.12658i
$571$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$571$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$572$		0			0
$573$	0.777479	−	0.974928i	0.777479	−	0.974928i
$574$		0			0
$575$		0			0
$576$	−2.15379	+	1.99843i	−2.15379	+	1.99843i
$577$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$577$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$578$	−0.955573	−	1.65510i	−0.955573	−	1.65510i
$579$		0			0
$580$	1.74618	−	3.02448i	1.74618	−	3.02448i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−5.69074			−5.69074
$585$		0			0
$586$	−1.91115			−1.91115
$587$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.623490	+	0.781831i	$0.285714\pi$
$587$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.365341	+	0.930874i	$0.380952\pi$
$588$	1.65379	−	2.07379i	1.65379	−	2.07379i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.752824	+	1.30393i	0.752824	+	1.30393i
$593$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$593$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.535628	−	1.36476i	−0.535628	−	1.36476i
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$	4.42444	−	5.54807i	4.42444	−	5.54807i
$601$		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
$604$	−5.24570			−5.24570
$605$	0.900969	+	1.56052i	0.900969	+	1.56052i
$606$	2.77064	+	0.417607i	2.77064	+	0.417607i
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	−3.16064	+	5.47439i	−3.16064	+	5.47439i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$613$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	0.0747301	+	0.997204i	$0.476190\pi$
$617$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	−0.900969	+	0.433884i	$0.857143\pi$
$618$	0.510177	+	1.29991i	0.510177	+	1.29991i
$619$		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
$622$	−0.850540			−0.850540
$623$		0			0
$624$		0			0
$625$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$626$	−1.82624	+	3.16314i	−1.82624	+	3.16314i
$627$		0			0
$628$	2.62285	+	4.54291i	2.62285	+	4.54291i
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	2.84537	+	4.92833i	2.84537	+	4.92833i
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$640$	−4.15813			−4.15813
$641$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$641$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$642$	−1.19158	+	1.49419i	−1.19158	+	1.49419i
$643$	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
$645$	−0.0983929	−	0.250701i	−0.0983929	−	0.250701i
$646$		0			0
$647$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$647$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$648$	0.236007	−	3.14929i	0.236007	−	3.14929i
$649$		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	$0.666667\pi$
$654$	0.870666	−	1.09178i	0.870666	−	1.09178i
$655$	1.48883	+	2.57873i	1.48883	+	2.57873i
$656$		0			0
$657$	1.32091	−	1.22563i	1.32091	−	1.22563i
$658$		0			0
$659$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.623490	+	0.781831i	$0.285714\pi$
$659$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.365341	+	0.930874i	$0.380952\pi$
$660$		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
$664$	0.702749	+	1.21720i	0.702749	+	1.21720i
$665$		0			0
$666$	−0.812753	−	0.250701i	−0.812753	−	0.250701i
$667$		0			0
$668$	−0.198220	−	0.343327i	−0.198220	−	0.343327i
$669$	−0.277479	+	0.347948i	−0.277479	+	0.347948i
$670$		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
$675$	0.167917	+	2.24070i	0.167917	+	2.24070i
$676$	2.65248			2.65248
$677$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.623490	+	0.781831i	$0.285714\pi$
$677$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.365341	+	0.930874i	$0.380952\pi$
$678$		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			$\pi$
$684$	−1.12802	−	4.94217i	−1.12802	−	4.94217i
$685$		0			0
$686$		0			0
$687$	−0.147791	−	0.0222759i	−0.147791	−	0.0222759i
$688$	−0.252824	+	0.437904i	−0.252824	+	0.437904i
$689$		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
$696$	1.43876	−	1.80414i	1.43876	−	1.80414i
$697$		0			0
$698$		0			0
$699$	−0.535628	−	1.36476i	−0.535628	−	1.36476i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−0.850540			−0.850540
$704$		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	$0.333333\pi$
$710$	−3.44377			−3.44377
$711$	−1.72188	−	0.531130i	−1.72188	−	0.531130i
$712$	−6.24570			−6.24570
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.198220	+	0.343327i	−0.198220	+	0.343327i
$717$		0			0
$718$	−1.57906	−	2.73502i	−1.57906	−	2.73502i
$719$	1.91115			1.91115			0.955573	−	0.294755i	$-0.0952381\pi$
$719$	1.91115			1.91115			0.955573	+	0.294755i	$0.0952381\pi$
$720$	4.46886	−	4.14650i	4.46886	−	4.14650i
$721$		0			0
$722$	−2.53464	−	4.39012i	−2.53464	−	4.39012i
$723$		0			0
$724$		0			0
$725$	−0.820914	+	1.42186i	−0.820914	+	1.42186i
$726$	0.698220	+	1.77904i	0.698220	+	1.77904i
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	0.623490	+	0.781831i	0.623490	+	0.781831i
$730$	6.20545			6.20545
$731$		0			0
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$	−1.57906	+	2.73502i	−1.57906	+	2.73502i
$735$	−1.12349	+	1.40881i	−1.12349	+	1.40881i
$736$		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	$0.666667\pi$
$740$	−1.06356	−	1.84214i	−1.06356	−	1.84214i
$741$		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
$746$	−3.44377			−3.44377
$747$	−0.425270	−	0.131178i	−0.425270	−	0.131178i
$748$		0			0
$749$		0			0
$750$	−2.67746	+	3.35742i	−2.67746	+	3.35742i
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$	−0.535628	−	1.36476i	−0.535628	−	1.36476i
$754$		0			0
$755$	3.56362			3.56362
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	1.88980	+	3.27323i	1.88980	+	3.27323i
$759$		0			0
$760$	5.43792	−	9.41876i	5.43792	−	9.41876i
$761$		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
$764$	3.30759			3.30759
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−1.45557	−	0.219392i	−1.45557	−	0.219392i
$769$		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			$\pi$
$774$	−0.0635609	−	0.278479i	−0.0635609	−	0.278479i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.0546039	+	0.728639i	0.0546039	+	0.728639i
$784$	3.38316			3.38316
$785$	−1.78181	−	3.08619i	−1.78181	−	3.08619i
$786$	1.15379	+	2.93982i	1.15379	+	2.93982i
$787$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$787$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$788$		0			0
$789$	1.03030	−	1.29196i	1.03030	−	1.29196i
$790$	−3.10273	−	5.37408i	−3.10273	−	5.37408i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	1.94440	−	3.36781i	1.94440	−	3.36781i
$797$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.222521	+	0.974928i	$0.571429\pi$
$797$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.733052	+	0.680173i	$0.761905\pi$
$798$		0			0
$799$		0			0
$800$	7.43208			7.43208
$801$	1.44973	−	1.34515i	1.44973	−	1.34515i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	2.31507	+	4.00982i	2.31507	+	4.00982i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−0.257353	+	3.43414i	−0.257353	+	3.43414i
$811$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$811$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$812$		0			0
$813$	0.455573	+	1.16078i	0.455573	+	1.16078i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.142820	−	0.247372i	−0.142820	−	0.247372i
$818$	−3.77960			−3.77960
$819$		0			0
$820$		0			0
$821$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$821$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$	−1.15379	+	1.99843i	−1.15379	+	1.99843i
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	1.91115			1.91115			0.955573	−	0.294755i	$-0.0952381\pi$
$829$	1.91115			1.91115			0.955573	+	0.294755i	$0.0952381\pi$
$830$	−0.766310	−	1.32729i	−0.766310	−	1.32729i
$831$	−0.914101	+	1.14625i	−0.914101	+	1.14625i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	0.134659	+	0.233236i	0.134659	+	0.233236i
$836$		0			0
$837$		0			0
$838$	1.39644			1.39644
$839$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$839$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$840$		0			0
$841$	0.233052	−	0.403658i	0.233052	−	0.403658i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.80194			−1.80194
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−2.62285	−	0.395331i	−2.62285	−	0.395331i
$853$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	0.623490	−	0.781831i	$-0.285714\pi$
$853$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	−0.988831	+	0.149042i	$0.952381\pi$
$854$		0			0
$855$	0.766310	+	3.35742i	0.766310	+	3.35742i
$856$	−3.15813			−3.15813
$857$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	0.623490	−	0.781831i	$-0.285714\pi$
$857$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	−0.988831	+	0.149042i	$0.952381\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$	0.357180	−	0.618654i	0.357180	−	0.618654i
$861$		0			0
$862$	0.425270	+	0.736589i	0.425270	+	0.736589i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	2.73286	−	1.86323i	2.73286	−	1.86323i
$865$		0			0
$866$		0			0
$867$	0.365341	+	0.930874i	0.365341	+	0.930874i
$868$		0			0
$869$		0			0
$870$	−1.56889	+	1.96732i	−1.56889	+	1.96732i
$871$		0			0
$872$	2.30759			2.30759
$873$		0			0
$874$		0			0
$875$		0			0
$876$	4.72622	+	0.712362i	4.72622	+	0.712362i
$877$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	0.0747301	+	0.997204i	$0.476190\pi$
$877$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	−0.900969	+	0.433884i	$0.857143\pi$
$878$		0			0
$879$	0.988831	+	0.149042i	0.988831	+	0.149042i
$880$		0			0
$881$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$881$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$882$	−1.40097	+	1.29991i	−1.40097	+	1.29991i
$883$		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	$0.666667\pi$
$888$	−0.513486	−	1.30834i	−0.513486	−	1.30834i
$889$		0			0
$890$	6.81060			6.81060
$891$		0			0
$892$	−1.18046			−1.18046
$893$		0			0
$894$		0			0
$895$	0.134659	−	0.233236i	0.134659	−	0.233236i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−4.36904	+	4.05387i	−4.36904	+	4.05387i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	3.73738	+	0.563320i	3.73738	+	0.563320i
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$		0			0
$909$	−1.40097	−	0.432142i	−1.40097	−	0.432142i
$910$		0			0
$911$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$911$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$912$	4.03130	−	5.05510i	4.03130	−	5.05510i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.198220	−	0.343327i	−0.198220	−	0.343327i
$917$		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
$926$	−2.80194			−2.80194
$927$	−0.162592	−	0.712362i	−0.162592	−	0.712362i
$928$	2.41679			2.41679
$929$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$929$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$930$		0			0
$931$	−0.955573	+	1.65510i	−0.955573	+	1.65510i
$932$	1.94440	−	3.36781i	1.94440	−	3.36781i
$933$	0.440071	+	0.0663300i	0.440071	+	0.0663300i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	1.19158	−	1.49419i	1.19158	−	1.49419i
$940$		0			0
$941$	0.733052	−	1.26968i	0.733052	−	1.26968i	−0.222521	−	0.974928i	$-0.571429\pi$
$941$	0.733052	−	1.26968i	0.733052	−	1.26968i	0.955573	−	0.294755i	$-0.0952381\pi$
$942$	−1.38084	−	3.51833i	−1.38084	−	3.51833i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	−0.900969	−	0.433884i	$-0.857143\pi$
$947$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	0.0747301	−	0.997204i	$-0.476190\pi$
$948$	−1.74618	−	4.44920i	−1.74618	−	4.44920i
$949$		0			0
$950$	−4.10352	+	7.10751i	−4.10352	+	7.10751i
$951$		0			0
$952$		0			0
$953$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$953$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$954$		0			0
$955$	−2.24698			−2.24698
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	5.23517	+	0.789075i	5.23517	+	0.789075i
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$		0			0
$963$	0.733052	−	0.680173i	0.733052	−	0.680173i
$964$		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	$0.666667\pi$
$968$	−1.57906	+	2.73502i	−1.57906	+	2.73502i
$969$		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	$0.666667\pi$
$972$	−0.590232	+	2.58597i	−0.590232	+	2.58597i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$977$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	0.623490	+	0.781831i	$0.285714\pi$
$978$		0			0
$979$		0			0
$980$	−4.77960			−4.77960
$981$	−0.535628	+	0.496990i	−0.535628	+	0.496990i
$982$		0			0
$983$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$983$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$984$		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
$995$	−1.32091	+	2.28789i	−1.32091	+	2.28789i
$996$	−0.431272	−	1.09886i	−0.431272	−	1.09886i
$997$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	−0.900969	−	0.433884i	$-0.857143\pi$
$997$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	0.0747301	−	0.997204i	$-0.476190\pi$
$998$	2.38316			2.38316
$999$	0.400969	+	0.193096i	0.400969	+	0.193096i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	639.1.g.b.283.1	yes	12
3.2	odd	2		1917.1.g.b.496.6		12
9.2	odd	6		1917.1.g.b.1774.6		12
9.7	even	3	inner	639.1.g.b.70.1	✓	12
71.70	odd	2	CM	639.1.g.b.283.1	yes	12
213.212	even	2		1917.1.g.b.496.6		12
639.70	odd	6	inner	639.1.g.b.70.1	✓	12
639.425	even	6		1917.1.g.b.1774.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
639.1.g.b.70.1	✓	12	9.7	even	3	inner
639.1.g.b.70.1	✓	12	639.70	odd	6	inner
639.1.g.b.283.1	yes	12	1.1	even	1	trivial
639.1.g.b.283.1	yes	12	71.70	odd	2	CM
1917.1.g.b.496.6		12	3.2	odd	2
1917.1.g.b.496.6		12	213.212	even	2
1917.1.g.b.1774.6		12	9.2	odd	6
1917.1.g.b.1774.6		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)

\(f(q)\)	\(=\)	\(q+(-0.955573 - 1.65510i) q^{2} +(0.365341 + 0.930874i) q^{3} +(-1.32624 + 2.29711i) q^{4} +(0.900969 - 1.56052i) q^{5} +(1.19158 - 1.49419i) q^{6} +3.15813 q^{8} +(-0.733052 + 0.680173i) q^{9} +O(q^{10})\)	\(q+(-0.955573 - 1.65510i) q^{2} +(0.365341 + 0.930874i) q^{3} +(-1.32624 + 2.29711i) q^{4} +(0.900969 - 1.56052i) q^{5} +(1.19158 - 1.49419i) q^{6} +3.15813 q^{8} +(-0.733052 + 0.680173i) q^{9} -3.44377 q^{10} +(-2.62285 - 0.395331i) q^{12} +(1.78181 + 0.268565i) q^{15} +(-1.69158 - 2.92990i) q^{16} +(1.82624 + 0.563320i) q^{18} +1.91115 q^{19} +(2.38980 + 4.13925i) q^{20} +(1.15379 + 2.93982i) q^{24} +(-1.12349 - 1.94594i) q^{25} +(-0.900969 - 0.433884i) q^{27} +(-0.365341 - 0.632789i) q^{29} +(-1.25815 - 3.20571i) q^{30} +(-1.65379 + 2.86445i) q^{32} +(-0.590232 - 2.58597i) q^{36} -0.445042 q^{37} +(-1.82624 - 3.16314i) q^{38} +(2.84537 - 4.92833i) q^{40} +(-0.0747301 - 0.129436i) q^{43} +(0.400969 + 1.75676i) q^{45} +(2.10937 - 2.64506i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-2.14715 + 3.71898i) q^{50} +(0.142820 + 1.90580i) q^{54} +(0.698220 + 1.77904i) q^{57} +(-0.698220 + 1.20935i) q^{58} +(-2.98003 + 3.73684i) q^{60} +2.93812 q^{64} +1.00000 q^{71} +(-2.31507 + 2.14807i) q^{72} -1.80194 q^{73} +(0.425270 + 0.736589i) q^{74} +(1.40097 - 1.75676i) q^{75} +(-2.53464 + 4.39012i) q^{76} +(0.900969 + 1.56052i) q^{79} -6.09624 q^{80} +(0.0747301 - 0.997204i) q^{81} +(0.222521 + 0.385418i) q^{83} +(-0.142820 + 0.247372i) q^{86} +(0.455573 - 0.571270i) q^{87} -1.97766 q^{89} +(2.52446 - 2.34236i) q^{90} +(1.72188 - 2.98239i) q^{95} +(-3.27064 - 0.492970i) q^{96} +1.91115 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + q^{3} - 7 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + q^{9}+O(q^{10}) \) 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