LMFDB - Embedded newform 1503.1.f.b.1168.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,1,Mod(166,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1503.166");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1503.f (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.750094713987$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{33}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{33} + \cdots)$

Embedding invariants

Embedding label			1168.2
Root			$0.981929 + 0.189251i$ of defining polynomial
Character	$\chi$	$=$	1503.1168
Dual form			1503.1.f.b.166.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.928368 + 1.60798i) q^{2} +(-0.888835 + 0.458227i) q^{3} +(-1.22373 - 2.11957i) q^{4} +(0.0883470 - 1.85463i) q^{6} +(0.327068 - 0.566498i) q^{7} +2.68757 q^{8} +(0.580057 - 0.814576i) q^{9} +O(q^{10})$	\(q+(-0.928368 + 1.60798i) q^{2} +(-0.888835 + 0.458227i) q^{3} +(-1.22373 - 2.11957i) q^{4} +(0.0883470 - 1.85463i) q^{6} +(0.327068 - 0.566498i) q^{7} +2.68757 q^{8} +(0.580057 - 0.814576i) q^{9} +(-0.235759 + 0.408346i) q^{11} +(2.05894 + 1.32320i) q^{12} +(0.607279 + 1.05184i) q^{14} +(-1.27132 + 2.20198i) q^{16} +(0.771316 + 1.68895i) q^{18} -1.99094 q^{19} +(-0.0311250 + 0.653395i) q^{21} +(-0.437742 - 0.758192i) q^{22} +(-2.38880 + 1.23151i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(-0.142315 + 0.989821i) q^{27} -1.60098 q^{28} +(-0.981929 + 1.70075i) q^{29} +(-0.928368 - 1.60798i) q^{31} +(-1.01671 - 1.76100i) q^{32} +(0.0224357 - 0.470984i) q^{33} +(-2.43639 - 0.232647i) q^{36} +(1.84833 - 3.20140i) q^{38} +(-1.02175 - 0.656639i) q^{42} +1.15402 q^{44} +(-0.580057 + 1.00469i) q^{47} +(0.120983 - 2.53975i) q^{48} +(0.286053 + 0.495458i) q^{49} +(-0.928368 - 1.60798i) q^{50} +(-1.45949 - 1.14776i) q^{54} +(0.879017 - 1.52250i) q^{56} +(1.76962 - 0.912303i) q^{57} +(-1.82318 - 3.15784i) q^{58} +(-0.841254 + 1.45709i) q^{61} +3.44747 q^{62} +(-0.271738 - 0.595023i) q^{63} +1.23291 q^{64} +(0.736504 + 0.473322i) q^{66} +(1.55894 - 2.18923i) q^{72} +(0.0475819 - 0.998867i) q^{75} +(2.43639 + 4.21994i) q^{76} +(0.154218 + 0.267114i) q^{77} +(-0.327068 - 0.945001i) q^{81} +(1.42300 - 0.733610i) q^{84} +(0.0934441 - 1.96163i) q^{87} +(-0.633618 + 1.09746i) q^{88} -1.77767 q^{89} +(1.56199 + 1.00383i) q^{93} +(-1.07701 - 1.86544i) q^{94} +(1.71063 + 1.09936i) q^{96} +(0.959493 - 1.66189i) q^{97} -1.06225 q^{98} +(0.195876 + 0.428908i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9}+O(q^{10}) $ 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9	$ 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9} - q^{11} + q^{14} - 12 q^{16} + 2 q^{18} + 2 q^{19} - q^{21} + q^{22} + q^{24} - 10 q^{25} - 2 q^{27} - q^{29} - q^{31} - q^{33} + q^{38} - 13 q^{42} - 22 q^{44} - q^{47} + 21 q^{48} - 11 q^{49} - q^{50} - 12 q^{54} - q^{56} - q^{57} + q^{58} + 2 q^{61} + 42 q^{62} + 2 q^{63} + 20 q^{64} - 2 q^{66} - 10 q^{72} + q^{75} + q^{77} + q^{81} + 22 q^{84} - q^{87} - q^{88} + 2 q^{89} + 2 q^{93} + q^{94} + 2 q^{97} - 22 q^{98} + 2 q^{99}+O(q^{100}) $ 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9 - q^11 + q^14 - 12 * q^16 + 2 * q^18 + 2 * q^19 - q^21 + q^22 + q^24 - 10 * q^25 - 2 * q^27 - q^29 - q^31 - q^33 + q^38 - 13 * q^42 - 22 * q^44 - q^47 + 21 * q^48 - 11 * q^49 - q^50 - 12 * q^54 - q^56 - q^57 + q^58 + 2 * q^61 + 42 * q^62 + 2 * q^63 + 20 * q^64 - 2 * q^66 - 10 * q^72 + q^75 + q^77 + q^81 + 22 * q^84 - q^87 - q^88 + 2 * q^89 + 2 * q^93 + q^94 + 2 * q^97 - 22 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1503\mathbb{Z}\right)^\times$.

$n$	$172$	$335$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$2$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$3$	−0.888835	+	0.458227i	−0.888835	+	0.458227i
$3$	−0.888835	+	0.458227i	−0.888835	+	0.458227i
$4$	−1.22373	−	2.11957i	−1.22373	−	2.11957i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	0.0883470	−	1.85463i	0.0883470	−	1.85463i
$7$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$7$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$8$	2.68757			2.68757
$9$	0.580057	−	0.814576i	0.580057	−	0.814576i
$10$		0			0
$11$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$11$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$12$	2.05894	+	1.32320i	2.05894	+	1.32320i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$	0.607279	+	1.05184i	0.607279	+	1.05184i
$15$		0			0
$16$	−1.27132	+	2.20198i	−1.27132	+	2.20198i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	0.771316	+	1.68895i	0.771316	+	1.68895i
$19$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$19$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$20$		0			0
$21$	−0.0311250	+	0.653395i	−0.0311250	+	0.653395i
$22$	−0.437742	−	0.758192i	−0.437742	−	0.758192i
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	−2.38880	+	1.23151i	−2.38880	+	1.23151i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$28$	−1.60098			−1.60098
$29$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$29$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$30$		0			0
$31$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$31$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$32$	−1.01671	−	1.76100i	−1.01671	−	1.76100i
$33$	0.0224357	−	0.470984i	0.0224357	−	0.470984i
$34$		0			0
$35$		0			0
$36$	−2.43639	−	0.232647i	−2.43639	−	0.232647i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	1.84833	−	3.20140i	1.84833	−	3.20140i
$39$		0			0
$40$		0			0
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$	−1.02175	−	0.656639i	−1.02175	−	0.656639i
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$	1.15402			1.15402
$45$		0			0
$46$		0			0
$47$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$47$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$48$	0.120983	−	2.53975i	0.120983	−	2.53975i
$49$	0.286053	+	0.495458i	0.286053	+	0.495458i
$50$	−0.928368	−	1.60798i	−0.928368	−	1.60798i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−1.45949	−	1.14776i	−1.45949	−	1.14776i
$55$		0			0
$56$	0.879017	−	1.52250i	0.879017	−	1.52250i
$57$	1.76962	−	0.912303i	1.76962	−	0.912303i
$58$	−1.82318	−	3.15784i	−1.82318	−	3.15784i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$61$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$62$	3.44747			3.44747
$63$	−0.271738	−	0.595023i	−0.271738	−	0.595023i
$64$	1.23291			1.23291
$65$		0			0
$66$	0.736504	+	0.473322i	0.736504	+	0.473322i
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	1.55894	−	2.18923i	1.55894	−	2.18923i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	0.0475819	−	0.998867i	0.0475819	−	0.998867i
$76$	2.43639	+	4.21994i	2.43639	+	4.21994i
$77$	0.154218	+	0.267114i	0.154218	+	0.267114i
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	−0.327068	−	0.945001i	−0.327068	−	0.945001i
$82$		0			0
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$	1.42300	−	0.733610i	1.42300	−	0.733610i
$85$		0			0
$86$		0			0
$87$	0.0934441	−	1.96163i	0.0934441	−	1.96163i
$88$	−0.633618	+	1.09746i	−0.633618	+	1.09746i
$89$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$89$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	1.56199	+	1.00383i	1.56199	+	1.00383i
$94$	−1.07701	−	1.86544i	−1.07701	−	1.86544i
$95$		0			0
$96$	1.71063	+	1.09936i	1.71063	+	1.09936i
$97$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$97$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$98$	−1.06225			−1.06225
$99$	0.195876	+	0.428908i	0.195876	+	0.428908i
$100$	2.44747			2.44747
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$107$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$108$	2.27215	−	0.909632i	2.27215	−	0.909632i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	0.831613	+	1.44040i	0.831613	+	1.44040i
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$	−0.175894	+	3.69247i	−0.175894	+	3.69247i
$115$		0			0
$116$	4.80648			4.80648
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.388835	+	0.673483i	0.388835	+	0.673483i
$122$	−1.56199	−	2.70544i	−1.56199	−	2.70544i
$123$		0			0
$124$	−2.27215	+	3.93548i	−2.27215	+	3.93548i
$125$		0			0
$126$	1.20906	+	0.115451i	1.20906	+	0.115451i
$127$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$127$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$128$	−0.127880	+	0.221494i	−0.127880	+	0.221494i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	−1.02574	+	0.528805i	−1.02574	+	0.528805i
$133$	−0.651174	+	1.12787i	−0.651174	+	1.12787i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$137$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$	0.0552004	−	1.15880i	0.0552004	−	1.15880i
$142$		0			0
$143$		0			0
$144$	1.05625	+	2.31286i	1.05625	+	2.31286i
$145$		0			0
$146$		0			0
$147$	−0.481286	−	0.309304i	−0.481286	−	0.309304i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	1.56199	+	1.00383i	1.56199	+	1.00383i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	−5.35079			−5.35079
$153$		0			0
$154$	−0.572686			−0.572686
$155$		0			0
$156$		0			0
$157$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$157$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	1.82318	+	0.351390i	1.82318	+	0.351390i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$167$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$168$	−0.0836506	+	1.75604i	−0.0836506	+	1.75604i
$169$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$170$		0			0
$171$	−1.15486	+	1.62177i	−1.15486	+	1.62177i
$172$		0			0
$173$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.235759	−	0.971812i	$-0.424242\pi$
$173$	0.959493	−	1.66189i	0.959493	−	1.66189i	0.723734	−	0.690079i	$-0.242424\pi$
$174$	3.06752	+	1.97137i	3.06752	+	1.97137i
$175$	0.327068	+	0.566498i	0.327068	+	0.566498i
$176$	−0.599448	−	1.03827i	−0.599448	−	1.03827i
$177$		0			0
$178$	1.65033	−	2.85846i	1.65033	−	2.85846i
$179$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$179$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$180$		0			0
$181$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$181$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$182$		0			0
$183$	0.0800569	−	1.68060i	0.0800569	−	1.68060i
$184$		0			0
$185$		0			0
$186$	−3.06423	+	1.57972i	−3.06423	+	1.57972i
$187$		0			0
$188$	2.83934			2.83934
$189$	0.514186	+	0.404360i	0.514186	+	0.404360i
$190$		0			0
$191$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$191$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$192$	−1.09585	+	0.564952i	−1.09585	+	0.564952i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$	1.78153	+	3.08569i	1.78153	+	3.08569i
$195$		0			0
$196$	0.700106	−	1.21262i	0.700106	−	1.21262i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−0.871520	−	0.0832201i	−0.871520	−	0.0832201i
$199$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$199$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$200$	−1.34378	+	2.32750i	−1.34378	+	2.32750i
$201$		0			0
$202$		0			0
$203$	0.642315	+	1.11252i	0.642315	+	1.11252i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.469383	−	0.812995i	0.469383	−	0.812995i
$210$		0			0
$211$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$211$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$212$		0			0
$213$		0			0
$214$	0.607279	−	1.05184i	0.607279	−	1.05184i
$215$		0			0
$216$	−0.382481	+	2.66021i	−0.382481	+	2.66021i
$217$	−1.21456			−1.21456
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$223$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$224$	−1.33014			−1.33014
$225$	0.415415	+	0.909632i	0.415415	+	0.909632i
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	−4.09924	−	2.63442i	−4.09924	−	2.63442i
$229$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$229$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$230$		0			0
$231$	−0.259474	−	0.166754i	−0.259474	−	0.166754i
$232$	−2.63900	+	4.57088i	−2.63900	+	4.57088i
$233$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$233$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$239$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	−1.44393			−1.44393
$243$	0.723734	+	0.690079i	0.723734	+	0.690079i
$244$	4.11788			4.11788
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−2.49505	−	4.32155i	−2.49505	−	4.32155i
$249$		0			0
$250$		0			0
$251$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$251$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$252$	−0.928658	+	1.30412i	−0.928658	+	1.30412i
$253$		0			0
$254$	−0.437742	+	0.758192i	−0.437742	+	0.758192i
$255$		0			0
$256$	0.379017	+	0.656476i	0.379017	+	0.656476i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.815816	+	1.78639i	0.815816	+	1.78639i
$262$		0			0
$263$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$263$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$264$	0.0602975	−	1.26580i	0.0602975	−	1.26580i
$265$		0			0
$266$	−1.20906	−	2.09415i	−1.20906	−	2.09415i
$267$	1.58006	−	0.814576i	1.58006	−	0.814576i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$	−1.34378	−	2.32750i	−1.34378	−	2.32750i
$275$	−0.235759	−	0.408346i	−0.235759	−	0.408346i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$	−1.84833	−	0.176494i	−1.84833	−	0.176494i
$280$		0			0
$281$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$281$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$282$	1.81208	+	1.16455i	1.81208	+	1.16455i
$283$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$283$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−2.02422	−	0.193290i	−2.02422	−	0.193290i
$289$	1.00000			1.00000
$290$		0			0
$291$	−0.0913090	+	1.91681i	−0.0913090	+	1.91681i
$292$		0			0
$293$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$293$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$294$	0.944166	−	0.486751i	0.944166	−	0.486751i
$295$		0			0
$296$		0			0
$297$	−0.370638	−	0.291473i	−0.370638	−	0.291473i
$298$		0			0
$299$		0			0
$300$	−2.17540	+	1.12149i	−2.17540	+	1.12149i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	2.53112	−	4.38403i	2.53112	−	4.38403i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	0.377445	−	0.653753i	0.377445	−	0.653753i
$309$		0			0
$310$		0			0
$311$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$311$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	−0.528482			−0.528482
$315$		0			0
$316$		0			0
$317$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$317$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$318$		0			0
$319$	−0.462997	−	0.801934i	−0.462997	−	0.801934i
$320$		0			0
$321$	0.581419	−	0.299742i	0.581419	−	0.299742i
$322$		0			0
$323$		0			0
$324$	−1.60275	+	1.84967i	−1.60275	+	1.84967i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.379436	+	0.657203i	0.379436	+	0.657203i
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$	1.85674			1.85674
$335$		0			0
$336$	−1.39920	−	0.899208i	−1.39920	−	0.899208i
$337$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$337$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$338$	−0.928368	−	1.60798i	−0.928368	−	1.60798i
$339$		0			0
$340$		0			0
$341$	0.875484			0.875484
$342$	−1.53565	−	3.36260i	−1.53565	−	3.36260i
$343$	1.02837			1.02837
$344$		0			0
$345$		0			0
$346$	1.78153	+	3.08569i	1.78153	+	3.08569i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$	−4.27217	+	2.20246i	−4.27217	+	2.20246i
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$	−1.21456			−1.21456
$351$		0			0
$352$	0.958799			0.958799
$353$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$353$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$354$		0			0
$355$		0			0
$356$	2.17540	+	3.76790i	2.17540	+	3.76790i
$357$		0			0
$358$	1.84833	−	3.20140i	1.84833	−	3.20140i
$359$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$359$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$360$		0			0
$361$	2.96386			2.96386
$362$	−1.07701	+	1.86544i	−1.07701	+	1.86544i
$363$	−0.654218	−	0.420441i	−0.654218	−	0.420441i
$364$		0			0
$365$		0			0
$366$	2.62805	+	1.68895i	2.62805	+	1.68895i
$367$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$367$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.216227	−	4.53915i	0.216227	−	4.53915i
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$	−1.55894	+	2.70017i	−1.55894	+	2.70017i
$377$		0			0
$378$	−1.12756	+	0.451405i	−1.12756	+	0.451405i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−0.419102	+	0.216062i	−0.419102	+	0.216062i
$382$	1.45949	+	2.52792i	1.45949	+	2.52792i
$383$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$383$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$384$	0.0121695	−	0.255469i	0.0121695	−	0.255469i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−4.69666			−4.69666
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$	0.768787	+	1.33158i	0.768787	+	1.33158i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	0.669400	−	0.940041i	0.669400	−	0.940041i
$397$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$397$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$398$	1.78153	−	3.08569i	1.78153	−	3.08569i
$399$	0.0619682	−	1.30087i	0.0619682	−	1.30087i
$400$	−1.27132	−	2.20198i	−1.27132	−	2.20198i
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−2.38522			−2.38522
$407$		0			0
$408$		0			0
$409$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$409$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$410$		0			0
$411$	0.0688733	−	1.44583i	0.0688733	−	1.44583i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	0.871520	+	1.50952i	0.871520	+	1.50952i
$419$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$419$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$420$		0			0
$421$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$421$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$422$	−0.528482			−0.528482
$423$	0.481929	+	1.05528i	0.481929	+	1.05528i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.550294	+	0.953137i	0.550294	+	0.953137i
$428$	0.800488	+	1.38649i	0.800488	+	1.38649i
$429$		0			0
$430$		0			0
$431$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$431$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$432$	−1.99864	−	1.57175i	−1.99864	−	1.57175i
$433$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$433$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$434$	1.12756	−	1.95298i	1.12756	−	1.95298i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	0.569516	+	0.0543822i	0.569516	+	0.0543822i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$	1.65033	+	2.85846i	1.65033	+	2.85846i
$447$		0			0
$448$	0.403246	−	0.698442i	0.403246	−	0.698442i
$449$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$449$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$450$	−1.84833	−	0.176494i	−1.84833	−	0.176494i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	4.75597	−	2.45188i	4.75597	−	2.45188i
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	3.64636			3.64636
$459$		0			0
$460$		0			0
$461$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$461$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$462$	0.509023	−	0.262420i	0.509023	−	0.262420i
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$	−2.49668	−	4.32438i	−2.49668	−	4.32438i
$465$		0			0
$466$	1.45949	−	2.52792i	1.45949	−	2.52792i
$467$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$467$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.995472	−	1.72421i	0.995472	−	1.72421i
$476$		0			0
$477$		0			0
$478$	−3.30067			−3.30067
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.951662	−	1.64833i	0.951662	−	1.64833i
$485$		0			0
$486$	−1.78153	+	0.523103i	−1.78153	+	0.523103i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	−2.26092	+	3.91604i	−2.26092	+	3.91604i
$489$		0			0
$490$		0			0
$491$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$491$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	4.72100			4.72100
$497$		0			0
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$		0			0
$501$	0.841254	+	0.540641i	0.841254	+	0.540641i
$502$	1.21590	−	2.10601i	1.21590	−	2.10601i
$503$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$503$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$504$	−0.730313	−	1.59916i	−0.730313	−	1.59916i
$505$		0			0
$506$		0			0
$507$	0.0475819	−	0.998867i	0.0475819	−	0.998867i
$508$	−0.577012	−	0.999415i	−0.577012	−	0.999415i
$509$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$509$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$510$		0			0
$511$		0			0
$512$	−1.66323			−1.66323
$513$	0.283341	−	1.97068i	0.283341	−	1.97068i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−0.273507	−	0.473728i	−0.273507	−	0.473728i
$518$		0			0
$519$	−0.0913090	+	1.91681i	−0.0913090	+	1.91681i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−3.62985	−	0.346609i	−3.62985	−	0.346609i
$523$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$523$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$524$		0			0
$525$	−0.550294	−	0.353653i	−0.550294	−	0.353653i
$526$	−1.56199	−	2.70544i	−1.56199	−	2.70544i
$527$		0			0
$528$	1.00858	+	0.648172i	1.00858	+	0.648172i
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	3.18745			3.18745
$533$		0			0
$534$	−0.157052	+	3.29693i	−0.157052	+	3.29693i
$535$		0			0
$536$		0			0
$537$	1.76962	−	0.912303i	1.76962	−	0.912303i
$538$		0			0
$539$	−0.269758			−0.269758
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−1.03115	+	0.531595i	−1.03115	+	0.531595i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$	3.54263			3.54263
$549$	0.698939	+	1.53046i	0.698939	+	1.53046i
$550$	0.875484			0.875484
$551$	1.95496	−	3.38610i	1.95496	−	3.38610i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$557$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$558$	1.99973	−	2.80822i	1.99973	−	2.80822i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.0883470	−	0.153022i	−0.0883470	−	0.153022i
$563$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$563$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$564$	−2.52371	+	1.30106i	−2.52371	+	1.30106i
$565$		0			0
$566$	−3.56305			−3.56305
$567$	−0.642315	−	0.123796i	−0.642315	−	0.123796i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$	−0.0748038	+	1.57033i	−0.0748038	+	1.57033i
$574$		0			0
$575$		0			0
$576$	0.715158	−	1.00430i	0.715158	−	1.00430i
$577$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$577$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$578$	−0.928368	+	1.60798i	−0.928368	+	1.60798i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−2.99743	−	1.92633i	−2.99743	−	1.92633i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−2.91899			−2.91899
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$	−0.0666248	+	1.39863i	−0.0666248	+	1.39863i
$589$	1.84833	+	3.20140i	1.84833	+	3.20140i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	0.812771	−	0.325385i	0.812771	−	0.325385i
$595$		0			0
$596$		0			0
$597$	1.70566	−	0.879330i	1.70566	−	0.879330i
$598$		0			0
$599$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$599$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$600$	0.127880	−	2.68452i	0.127880	−	2.68452i
$601$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$601$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	2.02422	+	3.50606i	2.02422	+	3.50606i
$609$	−1.08070	−	0.694523i	−1.08070	−	0.694523i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$613$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$614$		0			0
$615$		0			0
$616$	0.414472	+	0.717887i	0.414472	+	0.717887i
$617$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$617$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$	−1.85674			−1.85674
$623$	−0.581419	+	1.00705i	−0.581419	+	1.00705i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$	−0.0446683	+	0.937702i	−0.0446683	+	0.937702i
$628$	0.348311	−	0.603292i	0.348311	−	0.603292i
$629$		0			0
$630$		0			0
$631$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$631$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$632$		0			0
$633$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$634$	−0.0883470	−	0.153022i	−0.0883470	−	0.153022i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	1.71933			1.71933
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	−0.0577910	+	1.21318i	−0.0577910	+	1.21318i
$643$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$643$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−0.879017	−	2.53975i	−0.879017	−	2.53975i
$649$		0			0
$650$		0			0
$651$	1.07954	−	0.556543i	1.07954	−	0.556543i
$652$		0			0
$653$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$653$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−1.40903			−1.40903
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1.22373	+	2.11957i	−1.22373	+	2.11957i
$669$	−0.0845850	+	1.77566i	−0.0845850	+	1.77566i
$670$		0			0
$671$	−0.396666	−	0.687046i	−0.396666	−	0.687046i
$672$	1.18228	−	0.609505i	1.18228	−	0.609505i
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$	2.15402			2.15402
$675$	−0.786053	−	0.618159i	−0.786053	−	0.618159i
$676$	2.44747			2.44747
$677$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$677$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$678$		0			0
$679$	−0.627639	−	1.08710i	−0.627639	−	1.08710i
$680$		0			0
$681$		0			0
$682$	−0.812771	+	1.40776i	−0.812771	+	1.40776i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	4.85071	+	0.463186i	4.85071	+	0.463186i
$685$		0			0
$686$	−0.954707	+	1.65360i	−0.954707	+	1.65360i
$687$	1.65210	+	1.06174i	1.65210	+	1.06174i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$	−4.69666			−4.69666
$693$	0.307040	+	0.0293188i	0.307040	+	0.0293188i
$694$		0			0
$695$		0			0
$696$	0.251137	−	5.27202i	0.251137	−	5.27202i
$697$		0			0
$698$		0			0
$699$	1.39734	−	0.720381i	1.39734	−	0.720381i
$700$	0.800488	−	1.38649i	0.800488	−	1.38649i
$701$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$701$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$702$		0			0
$703$		0			0
$704$	−0.290670	+	0.503455i	−0.290670	+	0.503455i
$705$		0			0
$706$	1.21590	+	2.10601i	1.21590	+	2.10601i
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$	−4.77761			−4.77761
$713$		0			0
$714$		0			0
$715$		0			0
$716$	2.43639	+	4.21994i	2.43639	+	4.21994i
$717$	−1.49547	−	0.961081i	−1.49547	−	0.961081i
$718$	0.264241	−	0.457679i	0.264241	−	0.457679i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	−2.75155	+	4.76582i	−2.75155	+	4.76582i
$723$		0			0
$724$	−1.41967	−	2.45894i	−1.41967	−	2.45894i
$725$	−0.981929	−	1.70075i	−0.981929	−	1.70075i
$726$	1.28342	−	0.661647i	1.28342	−	0.661647i
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$730$		0			0
$731$		0			0
$732$	−3.66012	+	1.88692i	−3.66012	+	1.88692i
$733$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$733$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$734$	−1.07701	−	1.86544i	−1.07701	−	1.86544i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$743$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$744$	4.19794	+	2.69785i	4.19794	+	2.69785i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.213947	+	0.370567i	−0.213947	+	0.370567i
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$	−1.47487	−	2.55455i	−1.47487	−	2.55455i
$753$	1.16413	−	0.600149i	1.16413	−	0.600149i
$754$		0			0
$755$		0			0
$756$	0.227843	−	1.58468i	0.227843	−	1.58468i
$757$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$757$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$761$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$762$	0.0416572	−	0.874493i	0.0416572	−	0.874493i
$763$		0			0
$764$	−3.84768			−3.84768
$765$		0			0
$766$	−2.43181			−2.43181
$767$		0			0
$768$	−0.637698	−	0.409824i	−0.637698	−	0.409824i
$769$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$769$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	1.85674			1.85674
$776$	2.57870	−	4.46644i	2.57870	−	4.46644i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−1.54370	−	1.21398i	−1.54370	−	1.21398i
$784$	−1.45466			−1.45466
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$		0			0
$789$	0.0800569	−	1.68060i	0.0800569	−	1.68060i
$790$		0			0
$791$		0			0
$792$	0.526429	+	1.15272i	0.526429	+	1.15272i
$793$		0			0
$794$	−0.0883470	+	0.153022i	−0.0883470	+	0.153022i
$795$		0			0
$796$	2.34833	+	4.06742i	2.34833	+	4.06742i
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$	2.03425	+	1.30733i	2.03425	+	1.30733i
$799$		0			0
$800$	2.03343			2.03343
$801$	−1.03115	+	1.44805i	−1.03115	+	1.44805i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$809$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	1.57205	−	2.72286i	1.57205	−	2.72286i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.875484			0.875484
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$	2.26092	+	1.45301i	2.26092	+	1.45301i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$	0.396666	+	0.254922i	0.396666	+	0.254922i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$	−2.29760			−2.29760
$837$	1.72373	−	0.690079i	1.72373	−	0.690079i
$838$	−2.91899			−2.91899
$839$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$839$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$840$		0			0
$841$	−1.42837	−	2.47401i	−1.42837	−	2.47401i
$842$	1.21590	+	2.10601i	1.21590	+	2.10601i
$843$	0.00452808	−	0.0950560i	0.00452808	−	0.0950560i
$844$	0.348311	−	0.603292i	0.348311	−	0.603292i
$845$		0			0
$846$	−2.14427	−	0.204753i	−2.14427	−	0.204753i
$847$	0.508702			0.508702
$848$		0			0
$849$	−1.61435	−	1.03748i	−1.61435	−	1.03748i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$853$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$854$	−2.04350			−2.04350
$855$		0			0
$856$	−1.75803			−1.75803
$857$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$857$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$858$		0			0
$859$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$859$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$860$		0			0
$861$		0			0
$862$	−0.0883470	+	0.153022i	−0.0883470	+	0.153022i
$863$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$863$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$864$	1.88777	−	0.755750i	1.88777	−	0.755750i
$865$		0			0
$866$	−0.771316	+	1.33596i	−0.771316	+	1.33596i
$867$	−0.888835	+	0.458227i	−0.888835	+	0.458227i
$868$	1.48630	+	2.57434i	1.48630	+	2.57434i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.797176	−	1.74557i	−0.797176	−	1.74557i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.580057	+	0.814576i	$0.303030\pi$
$877$	0.995472	+	1.72421i	0.995472	+	1.72421i	0.415415	+	0.909632i	$0.363636\pi$
$878$		0			0
$879$	−1.32254	−	0.849945i	−1.32254	−	0.849945i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.616165	+	0.865283i	−0.616165	+	0.865283i
$883$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$883$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	0.154218	−	0.267114i	0.154218	−	0.267114i
$890$		0			0
$891$	0.462997	+	0.0892353i	0.462997	+	0.0892353i
$892$	−4.35079			−4.35079
$893$	1.15486	−	2.00028i	1.15486	−	2.00028i
$894$		0			0
$895$		0			0
$896$	0.0836506	+	0.144887i	0.0836506	+	0.144887i
$897$		0			0
$898$	−0.771316	+	1.33596i	−0.771316	+	1.33596i
$899$	3.64636			3.64636
$900$	1.41967	−	1.99365i	1.41967	−	1.99365i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$907$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$911$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$912$	−0.240871	+	5.05650i	−0.240871	+	5.05650i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−2.40324	+	4.16253i	−2.40324	+	4.16253i
$917$		0			0
$918$		0			0
$919$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$919$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$920$		0			0
$921$		0			0
$922$	−0.771316	−	1.33596i	−0.771316	−	1.33596i
$923$		0			0
$924$	−0.0359191	+	0.754034i	−0.0359191	+	0.754034i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	3.99337			3.99337
$929$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$929$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$930$		0			0
$931$	−0.569516	−	0.986430i	−0.569516	−	0.986430i
$932$	1.92384	+	3.33219i	1.92384	+	3.33219i
$933$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$934$	−1.82318	+	3.15784i	−1.82318	+	3.15784i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$	0.469734	−	0.242165i	0.469734	−	0.242165i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$947$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$948$		0			0
$949$		0			0
$950$	1.84833	+	3.20140i	1.84833	+	3.20140i
$951$	0.00452808	−	0.0950560i	0.00452808	−	0.0950560i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	2.17540	−	3.76790i	2.17540	−	3.76790i
$957$	0.778996	+	0.500630i	0.778996	+	0.500630i
$958$		0			0
$959$	0.473420	+	0.819988i	0.473420	+	0.819988i
$960$		0			0
$961$	−1.22373	+	2.11957i	−1.22373	+	2.11957i
$962$		0			0
$963$	−0.379436	+	0.532843i	−0.379436	+	0.532843i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.786053	+	1.36148i	0.786053	+	1.36148i	0.928368	+	0.371662i	$0.121212\pi$
$967$	0.786053	+	1.36148i	0.786053	+	1.36148i	−0.142315	+	0.989821i	$0.545455\pi$
$968$	1.04502	+	1.81003i	1.04502	+	1.81003i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	0.577012	−	2.37848i	0.577012	−	2.37848i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−2.13900	−	3.70485i	−2.13900	−	3.70485i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$	0.419102	−	0.725906i	0.419102	−	0.725906i
$980$		0			0
$981$		0			0
$982$	3.71347			3.71347
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−0.638404	−	0.410277i	−0.638404	−	0.410277i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−1.88777	+	3.26972i	−1.88777	+	3.26972i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$997$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.1.f.b.1168.2	yes	20
9.4	even	3	inner	1503.1.f.b.166.2	✓	20
167.166	odd	2	CM	1503.1.f.b.1168.2	yes	20
1503.166	odd	6	inner	1503.1.f.b.166.2	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1503.1.f.b.166.2	✓	20	9.4	even	3	inner
1503.1.f.b.166.2	✓	20	1503.166	odd	6	inner
1503.1.f.b.1168.2	yes	20	1.1	even	1	trivial
1503.1.f.b.1168.2	yes	20	167.166	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1503.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.928368 + 1.60798i) q^{2} +(-0.888835 + 0.458227i) q^{3} +(-1.22373 - 2.11957i) q^{4} +(0.0883470 - 1.85463i) q^{6} +(0.327068 - 0.566498i) q^{7} +2.68757 q^{8} +(0.580057 - 0.814576i) q^{9} +O(q^{10})\)	\(q+(-0.928368 + 1.60798i) q^{2} +(-0.888835 + 0.458227i) q^{3} +(-1.22373 - 2.11957i) q^{4} +(0.0883470 - 1.85463i) q^{6} +(0.327068 - 0.566498i) q^{7} +2.68757 q^{8} +(0.580057 - 0.814576i) q^{9} +(-0.235759 + 0.408346i) q^{11} +(2.05894 + 1.32320i) q^{12} +(0.607279 + 1.05184i) q^{14} +(-1.27132 + 2.20198i) q^{16} +(0.771316 + 1.68895i) q^{18} -1.99094 q^{19} +(-0.0311250 + 0.653395i) q^{21} +(-0.437742 - 0.758192i) q^{22} +(-2.38880 + 1.23151i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(-0.142315 + 0.989821i) q^{27} -1.60098 q^{28} +(-0.981929 + 1.70075i) q^{29} +(-0.928368 - 1.60798i) q^{31} +(-1.01671 - 1.76100i) q^{32} +(0.0224357 - 0.470984i) q^{33} +(-2.43639 - 0.232647i) q^{36} +(1.84833 - 3.20140i) q^{38} +(-1.02175 - 0.656639i) q^{42} +1.15402 q^{44} +(-0.580057 + 1.00469i) q^{47} +(0.120983 - 2.53975i) q^{48} +(0.286053 + 0.495458i) q^{49} +(-0.928368 - 1.60798i) q^{50} +(-1.45949 - 1.14776i) q^{54} +(0.879017 - 1.52250i) q^{56} +(1.76962 - 0.912303i) q^{57} +(-1.82318 - 3.15784i) q^{58} +(-0.841254 + 1.45709i) q^{61} +3.44747 q^{62} +(-0.271738 - 0.595023i) q^{63} +1.23291 q^{64} +(0.736504 + 0.473322i) q^{66} +(1.55894 - 2.18923i) q^{72} +(0.0475819 - 0.998867i) q^{75} +(2.43639 + 4.21994i) q^{76} +(0.154218 + 0.267114i) q^{77} +(-0.327068 - 0.945001i) q^{81} +(1.42300 - 0.733610i) q^{84} +(0.0934441 - 1.96163i) q^{87} +(-0.633618 + 1.09746i) q^{88} -1.77767 q^{89} +(1.56199 + 1.00383i) q^{93} +(-1.07701 - 1.86544i) q^{94} +(1.71063 + 1.09936i) q^{96} +(0.959493 - 1.66189i) q^{97} -1.06225 q^{98} +(0.195876 + 0.428908i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9	\( 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9} - q^{11} + q^{14} - 12 q^{16} + 2 q^{18} + 2 q^{19} - q^{21} + q^{22} + q^{24} - 10 q^{25} - 2 q^{27} - q^{29} - q^{31} - q^{33} + q^{38} - 13 q^{42} - 22 q^{44} - q^{47} + 21 q^{48} - 11 q^{49} - q^{50} - 12 q^{54} - q^{56} - q^{57} + q^{58} + 2 q^{61} + 42 q^{62} + 2 q^{63} + 20 q^{64} - 2 q^{66} - 10 q^{72} + q^{75} + q^{77} + q^{81} + 22 q^{84} - q^{87} - q^{88} + 2 q^{89} + 2 q^{93} + q^{94} + 2 q^{97} - 22 q^{98} + 2 q^{99}+O(q^{100}) \) 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9 - q^11 + q^14 - 12 * q^16 + 2 * q^18 + 2 * q^19 - q^21 + q^22 + q^24 - 10 * q^25 - 2 * q^27 - q^29 - q^31 - q^33 + q^38 - 13 * q^42 - 22 * q^44 - q^47 + 21 * q^48 - 11 * q^49 - q^50 - 12 * q^54 - q^56 - q^57 + q^58 + 2 * q^61 + 42 * q^62 + 2 * q^63 + 20 * q^64 - 2 * q^66 - 10 * q^72 + q^75 + q^77 + q^81 + 22 * q^84 - q^87 - q^88 + 2 * q^89 + 2 * q^93 + q^94 + 2 * q^97 - 22 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more