LMFDB - Embedded newform 2001.1.bf.b.206.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2001,1,Mod(68,2001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2001, base_ring=CyclotomicField(28))

chi = DirichletCharacter(H, H._module([14, 14, 23]))

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

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

chi := DirichletCharacter("2001.68");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2001 = 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2001.bf (of order $28$, degree $12$, 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:	$0.998629090279$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{28})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{28}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{28} - \cdots)$

Embedding invariants

Embedding label			206.1
Root			$-0.781831 + 0.623490i$ of defining polynomial
Character	$\chi$	$=$	2001.206
Dual form			2001.1.bf.b.68.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.900969 - 1.43388i) q^{2} +(0.623490 - 0.781831i) q^{3} +(-0.810394 - 1.68280i) q^{4} +(-0.559311 - 1.59842i) q^{6} +(-1.46028 - 0.164534i) q^{8} +(-0.222521 - 0.974928i) q^{9} +O(q^{10})$	\(q+(0.900969 - 1.43388i) q^{2} +(0.623490 - 0.781831i) q^{3} +(-0.810394 - 1.68280i) q^{4} +(-0.559311 - 1.59842i) q^{6} +(-1.46028 - 0.164534i) q^{8} +(-0.222521 - 0.974928i) q^{9} +(-1.82094 - 0.415617i) q^{12} +(0.347948 + 0.277479i) q^{13} +(-0.387055 + 0.485352i) q^{16} +(-1.59842 - 0.559311i) q^{18} +(-0.974928 - 0.222521i) q^{23} +(-1.03911 + 1.03911i) q^{24} +(-0.900969 + 0.433884i) q^{25} +(0.711363 - 0.248917i) q^{26} +(-0.900969 - 0.433884i) q^{27} +(0.433884 + 0.900969i) q^{29} +(1.19745 + 0.752407i) q^{31} +(-0.138138 - 0.394777i) q^{32} +(-1.46028 + 1.16453i) q^{36} +(0.433884 - 0.0990311i) q^{39} +(-0.158342 + 0.158342i) q^{41} +(-1.19745 + 1.19745i) q^{46} +(-0.0739590 - 0.656405i) q^{47} +(0.138138 + 0.605223i) q^{48} +(0.623490 + 0.781831i) q^{49} +(-0.189606 + 1.68280i) q^{50} +(0.184967 - 0.810394i) q^{52} +(-1.43388 + 0.900969i) q^{54} +(1.68280 + 0.189606i) q^{58} +1.80194i q^{59} +(2.15773 - 1.03911i) q^{62} +(-1.29575 - 0.295745i) q^{64} +(-0.781831 + 0.623490i) q^{69} +(1.21572 - 1.52446i) q^{71} +(0.164534 + 1.46028i) q^{72} +(-0.559311 + 0.351438i) q^{73} +(-0.222521 + 0.974928i) q^{75} +(0.248917 - 0.711363i) q^{78} +(-0.900969 + 0.433884i) q^{81} +(0.0843826 + 0.369704i) q^{82} +(0.974928 + 0.222521i) q^{87} +(0.415617 + 1.82094i) q^{92} +(1.33485 - 0.467085i) q^{93} +(-1.00784 - 0.485352i) q^{94} +(-0.394777 - 0.138138i) q^{96} +(1.68280 - 0.189606i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{2} - 2 q^{3} - 14 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 12 * q + 2 * q^2 - 2 * q^3 - 14 * q^4 + 2 * q^6 - 2 * q^9	$ 12 q + 2 q^{2} - 2 q^{3} - 14 q^{4} + 2 q^{6} - 2 q^{9} + 12 q^{16} + 2 q^{18} - 2 q^{25} + 4 q^{26} - 2 q^{27} + 2 q^{31} + 2 q^{32} - 2 q^{41} - 2 q^{46} + 2 q^{47} - 2 q^{48} - 2 q^{49} + 2 q^{50} + 10 q^{52} - 12 q^{54} + 2 q^{58} - 4 q^{62} - 14 q^{64} - 14 q^{72} + 2 q^{73} - 2 q^{75} - 10 q^{78} - 2 q^{81} + 4 q^{82} + 2 q^{92} + 2 q^{93} - 4 q^{94} - 12 q^{96} + 2 q^{98}+O(q^{100}) $ 12 * q + 2 * q^2 - 2 * q^3 - 14 * q^4 + 2 * q^6 - 2 * q^9 + 12 * q^16 + 2 * q^18 - 2 * q^25 + 4 * q^26 - 2 * q^27 + 2 * q^31 + 2 * q^32 - 2 * q^41 - 2 * q^46 + 2 * q^47 - 2 * q^48 - 2 * q^49 + 2 * q^50 + 10 * q^52 - 12 * q^54 + 2 * q^58 - 4 * q^62 - 14 * q^64 - 14 * q^72 + 2 * q^73 - 2 * q^75 - 10 * q^78 - 2 * q^81 + 4 * q^82 + 2 * q^92 + 2 * q^93 - 4 * q^94 - 12 * q^96 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2001\mathbb{Z}\right)^\times$.

$n$	$553$	$668$	$1132$
$\chi(n)$	$e\left(\frac{5}{28}\right)$	$-1$	$-1$

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 $

$2$	0.900969	−	1.43388i	0.900969	−	1.43388i		−	1.00000i	$-0.5\pi$
$2$	0.900969	−	1.43388i	0.900969	−	1.43388i	0.900969	−	0.433884i	$-0.142857\pi$
$3$	0.623490	−	0.781831i	0.623490	−	0.781831i
$3$	0.623490	−	0.781831i	0.623490	−	0.781831i
$4$	−0.810394	−	1.68280i	−0.810394	−	1.68280i
$5$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$5$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$6$	−0.559311	−	1.59842i	−0.559311	−	1.59842i
$7$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$7$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$8$	−1.46028	−	0.164534i	−1.46028	−	0.164534i
$9$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$10$		0			0
$11$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$11$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$12$	−1.82094	−	0.415617i	−1.82094	−	0.415617i
$13$	0.347948	+	0.277479i	0.347948	+	0.277479i	0.781831	−	0.623490i	$-0.214286\pi$
$13$	0.347948	+	0.277479i	0.347948	+	0.277479i	−0.433884	+	0.900969i	$0.642857\pi$
$14$		0			0
$15$		0			0
$16$	−0.387055	+	0.485352i	−0.387055	+	0.485352i
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$	−1.59842	−	0.559311i	−1.59842	−	0.559311i
$19$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$19$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.974928	−	0.222521i	−0.974928	−	0.222521i
$23$	−0.974928	−	0.222521i	−0.974928	−	0.222521i
$24$	−1.03911	+	1.03911i	−1.03911	+	1.03911i
$25$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$26$	0.711363	−	0.248917i	0.711363	−	0.248917i
$27$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$28$		0			0
$29$	0.433884	+	0.900969i	0.433884	+	0.900969i
$29$	0.433884	+	0.900969i	0.433884	+	0.900969i
$30$		0			0
$31$	1.19745	+	0.752407i	1.19745	+	0.752407i	0.974928	−	0.222521i	$-0.0714286\pi$
$31$	1.19745	+	0.752407i	1.19745	+	0.752407i	0.222521	+	0.974928i	$0.428571\pi$
$32$	−0.138138	−	0.394777i	−0.138138	−	0.394777i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−1.46028	+	1.16453i	−1.46028	+	1.16453i
$37$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$37$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$38$		0			0
$39$	0.433884	−	0.0990311i	0.433884	−	0.0990311i
$40$		0			0
$41$	−0.158342	+	0.158342i	−0.158342	+	0.158342i	−0.781831	−	0.623490i	$-0.785714\pi$
$41$	−0.158342	+	0.158342i	−0.158342	+	0.158342i	0.623490	+	0.781831i	$0.285714\pi$
$42$		0			0
$43$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$43$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$44$		0			0
$45$		0			0
$46$	−1.19745	+	1.19745i	−1.19745	+	1.19745i
$47$	−0.0739590	−	0.656405i	−0.0739590	−	0.656405i	−0.974928	−	0.222521i	$-0.928571\pi$
$47$	−0.0739590	−	0.656405i	−0.0739590	−	0.656405i	0.900969	−	0.433884i	$-0.142857\pi$
$48$	0.138138	+	0.605223i	0.138138	+	0.605223i
$49$	0.623490	+	0.781831i	0.623490	+	0.781831i
$50$	−0.189606	+	1.68280i	−0.189606	+	1.68280i
$51$		0			0
$52$	0.184967	−	0.810394i	0.184967	−	0.810394i
$53$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$53$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$54$	−1.43388	+	0.900969i	−1.43388	+	0.900969i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.68280	+	0.189606i	1.68280	+	0.189606i
$59$			1.80194i			1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$59$			1.80194i			1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$60$		0			0
$61$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$61$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$62$	2.15773	−	1.03911i	2.15773	−	1.03911i
$63$		0			0
$64$	−1.29575	−	0.295745i	−1.29575	−	0.295745i
$65$		0			0
$66$		0			0
$67$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$67$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$68$		0			0
$69$	−0.781831	+	0.623490i	−0.781831	+	0.623490i
$70$		0			0
$71$	1.21572	−	1.52446i	1.21572	−	1.52446i	0.433884	−	0.900969i	$-0.357143\pi$
$71$	1.21572	−	1.52446i	1.21572	−	1.52446i	0.781831	−	0.623490i	$-0.214286\pi$
$72$	0.164534	+	1.46028i	0.164534	+	1.46028i
$73$	−0.559311	+	0.351438i	−0.559311	+	0.351438i	−0.781831	−	0.623490i	$-0.785714\pi$
$73$	−0.559311	+	0.351438i	−0.559311	+	0.351438i	0.222521	+	0.974928i	$0.428571\pi$
$74$		0			0
$75$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$76$		0			0
$77$		0			0
$78$	0.248917	−	0.711363i	0.248917	−	0.711363i
$79$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$79$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$80$		0			0
$81$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$82$	0.0843826	+	0.369704i	0.0843826	+	0.369704i
$83$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$83$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	0.974928	+	0.222521i	0.974928	+	0.222521i
$88$		0			0
$89$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$89$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$90$		0			0
$91$		0			0
$92$	0.415617	+	1.82094i	0.415617	+	1.82094i
$93$	1.33485	−	0.467085i	1.33485	−	0.467085i
$94$	−1.00784	−	0.485352i	−1.00784	−	0.485352i
$95$		0			0
$96$	−0.394777	−	0.138138i	−0.394777	−	0.138138i
$97$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$97$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$98$	1.68280	−	0.189606i	1.68280	−	0.189606i
$99$		0			0
$100$	1.46028	+	1.16453i	1.46028	+	1.16453i
$101$	0.559311	−	0.351438i	0.559311	−	0.351438i	−0.222521	−	0.974928i	$-0.571429\pi$
$101$	0.559311	−	0.351438i	0.559311	−	0.351438i	0.781831	+	0.623490i	$0.214286\pi$
$102$		0			0
$103$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$103$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$104$	−0.462446	−	0.462446i	−0.462446	−	0.462446i
$105$		0			0
$106$		0			0
$107$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$107$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$108$			1.86777i			1.86777i
$109$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$109$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$113$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$114$		0			0
$115$		0			0
$116$	1.16453	−	1.46028i	1.16453	−	1.46028i
$117$	0.193096	−	0.400969i	0.193096	−	0.400969i
$118$	2.58377	+	1.62349i	2.58377	+	1.62349i
$119$		0			0
$120$		0			0
$121$	0.974928	−	0.222521i	0.974928	−	0.222521i
$122$		0			0
$123$	0.0250721	+	0.222521i	0.0250721	+	0.222521i
$124$	0.295745	−	2.62481i	0.295745	−	2.62481i
$125$		0			0
$126$		0			0
$127$	−0.0250721	−	0.222521i	−0.0250721	−	0.222521i	0.974928	−	0.222521i	$-0.0714286\pi$
$127$	−0.0250721	−	0.222521i	−0.0250721	−	0.222521i	−1.00000			$\pi$
$128$	−1.29575	+	1.29575i	−1.29575	+	1.29575i
$129$		0			0
$130$		0			0
$131$	0.566116	+	0.900969i	0.566116	+	0.900969i	1.00000			$0$
$131$	0.566116	+	0.900969i	0.566116	+	0.900969i	−0.433884	+	0.900969i	$0.642857\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$137$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$138$	0.189606	+	1.68280i	0.189606	+	1.68280i
$139$	0.433884	−	1.90097i	0.433884	−	1.90097i		−	1.00000i	$-0.5\pi$
$139$	0.433884	−	1.90097i	0.433884	−	1.90097i	0.433884	−	0.900969i	$-0.357143\pi$
$140$		0			0
$141$	−0.559311	−	0.351438i	−0.559311	−	0.351438i
$142$	−1.09057	−	3.11668i	−1.09057	−	3.11668i
$143$		0			0
$144$	0.559311	+	0.269350i	0.559311	+	0.269350i
$145$		0			0
$146$			1.11862i			1.11862i
$147$	1.00000			1.00000
$148$		0			0
$149$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$149$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$150$	1.19745	+	1.19745i	1.19745	+	1.19745i
$151$	−1.94986	−	0.445042i	−1.94986	−	0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$151$	−1.94986	−	0.445042i	−1.94986	−	0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.518266	−	0.649885i	−0.518266	−	0.649885i
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.189606	+	1.68280i	−0.189606	+	1.68280i
$163$	−0.656405	+	0.0739590i	−0.656405	+	0.0739590i	−0.433884	−	0.900969i	$-0.642857\pi$
$163$	−0.656405	+	0.0739590i	−0.656405	+	0.0739590i	−0.222521	+	0.974928i	$0.571429\pi$
$164$	0.394777	+	0.138138i	0.394777	+	0.138138i
$165$		0			0
$166$		0			0
$167$	0.400969	+	0.193096i	0.400969	+	0.193096i	0.623490	−	0.781831i	$-0.285714\pi$
$167$	0.400969	+	0.193096i	0.400969	+	0.193096i	−0.222521	+	0.974928i	$0.571429\pi$
$168$		0			0
$169$	−0.178448	−	0.781831i	−0.178448	−	0.781831i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.24698			−1.24698			−0.623490	−	0.781831i	$-0.714286\pi$
$173$	−1.24698			−1.24698			−0.623490	+	0.781831i	$0.714286\pi$
$174$	1.19745	−	1.19745i	1.19745	−	1.19745i
$175$		0			0
$176$		0			0
$177$	1.40881	+	1.12349i	1.40881	+	1.12349i
$178$		0			0
$179$	0.277479	+	1.21572i	0.277479	+	1.21572i	0.900969	+	0.433884i	$0.142857\pi$
$179$	0.277479	+	1.21572i	0.277479	+	1.21572i	−0.623490	+	0.781831i	$0.714286\pi$
$180$		0			0
$181$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$181$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$182$		0			0
$183$		0			0
$184$	1.38705	+	0.485352i	1.38705	+	0.485352i
$185$		0			0
$186$	0.532915	−	2.33485i	0.532915	−	2.33485i
$187$		0			0
$188$	−1.04466	+	0.656405i	−1.04466	+	0.656405i
$189$		0			0
$190$		0			0
$191$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$191$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$192$	−1.03911	+	0.828660i	−1.03911	+	0.828660i
$193$	−0.623490	+	1.78183i	−0.623490	+	1.78183i			1.00000i	$0.5\pi$
$193$	−0.623490	+	1.78183i	−0.623490	+	1.78183i	−0.623490	+	0.781831i	$0.714286\pi$
$194$		0			0
$195$		0			0
$196$	0.810394	−	1.68280i	0.810394	−	1.68280i
$197$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$197$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$198$		0			0
$199$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$199$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$200$	1.38705	−	0.485352i	1.38705	−	0.485352i
$201$		0			0
$202$		−	1.11862i		−	1.11862i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$			1.00000i			1.00000i
$208$	−0.269350	+	0.0614773i	−0.269350	+	0.0614773i
$209$		0			0
$210$		0			0
$211$	0.158342	−	1.40532i	0.158342	−	1.40532i	−0.623490	−	0.781831i	$-0.714286\pi$
$211$	0.158342	−	1.40532i	0.158342	−	1.40532i	0.781831	−	0.623490i	$-0.214286\pi$
$212$		0			0
$213$	−0.433884	−	1.90097i	−0.433884	−	1.90097i
$214$		0			0
$215$		0			0
$216$	1.24428	+	0.781831i	1.24428	+	0.781831i
$217$		0			0
$218$		0			0
$219$	−0.0739590	+	0.656405i	−0.0739590	+	0.656405i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.777479	−	0.974928i	−0.777479	−	0.974928i	0.222521	−	0.974928i	$-0.428571\pi$
$223$	−0.777479	−	0.974928i	−0.777479	−	0.974928i	−1.00000			$\pi$
$224$		0			0
$225$	0.623490	+	0.781831i	0.623490	+	0.781831i
$226$		0			0
$227$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$227$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$228$		0			0
$229$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$229$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$230$		0			0
$231$		0			0
$232$	−0.485352	−	1.38705i	−0.485352	−	1.38705i
$233$			1.24698i			1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$233$			1.24698i			1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$234$	−0.400969	−	0.638138i	−0.400969	−	0.638138i
$235$		0			0
$236$	3.03230	−	1.46028i	3.03230	−	1.46028i
$237$		0			0
$238$		0			0
$239$	−0.678448	+	1.40881i	−0.678448	+	1.40881i	0.222521	+	0.974928i	$0.428571\pi$
$239$	−0.678448	+	1.40881i	−0.678448	+	1.40881i	−0.900969	+	0.433884i	$0.857143\pi$
$240$		0			0
$241$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$241$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$242$	0.559311	−	1.59842i	0.559311	−	1.59842i
$243$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$244$		0			0
$245$		0			0
$246$	0.341658	+	0.164534i	0.341658	+	0.164534i
$247$		0			0
$248$	−1.62481	−	1.29575i	−1.62481	−	1.29575i
$249$		0			0
$250$		0			0
$251$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$251$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$252$		0			0
$253$		0			0
$254$	−0.341658	−	0.164534i	−0.341658	−	0.164534i
$255$		0			0
$256$	0.394777	+	1.72963i	0.394777	+	1.72963i
$257$	0.376510	+	0.781831i	0.376510	+	0.781831i	1.00000			$0$
$257$	0.376510	+	0.781831i	0.376510	+	0.781831i	−0.623490	+	0.781831i	$0.714286\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.781831	−	0.623490i	0.781831	−	0.623490i
$262$	1.80194			1.80194
$263$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$263$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.222521	+	0.0250721i	0.222521	+	0.0250721i	0.222521	−	0.974928i	$-0.428571\pi$
$269$	0.222521	+	0.0250721i	0.222521	+	0.0250721i			1.00000i	$0.5\pi$
$270$		0			0
$271$	−1.00435	−	0.351438i	−1.00435	−	0.351438i	−0.222521	−	0.974928i	$-0.571429\pi$
$271$	−1.00435	−	0.351438i	−1.00435	−	0.351438i	−0.781831	+	0.623490i	$0.785714\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	1.68280	+	0.810394i	1.68280	+	0.810394i
$277$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$277$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$278$	−2.33485	−	2.33485i	−2.33485	−	2.33485i
$279$	0.467085	−	1.33485i	0.467085	−	1.33485i
$280$		0			0
$281$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$281$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$282$	−1.00784	+	0.485352i	−1.00784	+	0.485352i
$283$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$283$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$284$	−3.55057	−	0.810394i	−3.55057	−	0.810394i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.354140	+	0.222521i	−0.354140	+	0.222521i
$289$			1.00000i			1.00000i
$290$		0			0
$291$		0			0
$292$	1.04466	+	0.656405i	1.04466	+	0.656405i
$293$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$293$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$294$	0.900969	−	1.43388i	0.900969	−	1.43388i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.277479	−	0.347948i	−0.277479	−	0.347948i
$300$	1.82094	−	0.415617i	1.82094	−	0.415617i
$301$		0			0
$302$	−2.39490	+	2.39490i	−2.39490	+	2.39490i
$303$	0.0739590	−	0.656405i	0.0739590	−	0.656405i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.33485	−	1.33485i	1.33485	−	1.33485i	0.433884	−	0.900969i	$-0.357143\pi$
$307$	1.33485	−	1.33485i	1.33485	−	1.33485i	0.900969	−	0.433884i	$-0.142857\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.189606	−	1.68280i	0.189606	−	1.68280i	−0.433884	−	0.900969i	$-0.642857\pi$
$311$	0.189606	−	1.68280i	0.189606	−	1.68280i	0.623490	−	0.781831i	$-0.285714\pi$
$312$	−0.649885	+	0.0732245i	−0.649885	+	0.0732245i
$313$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$313$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.43388	−	0.900969i	−1.43388	−	0.900969i	−0.433884	−	0.900969i	$-0.642857\pi$
$317$	−1.43388	−	0.900969i	−1.43388	−	0.900969i	−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.46028	+	1.16453i	1.46028	+	1.16453i
$325$	−0.433884	−	0.0990311i	−0.433884	−	0.0990311i
$326$	−0.485352	+	1.00784i	−0.485352	+	1.00784i
$327$		0			0
$328$	0.257276	−	0.205171i	0.257276	−	0.205171i
$329$		0			0
$330$		0			0
$331$	1.40532	+	1.40532i	1.40532	+	1.40532i	0.781831	+	0.623490i	$0.214286\pi$
$331$	1.40532	+	1.40532i	1.40532	+	1.40532i	0.623490	+	0.781831i	$0.285714\pi$
$332$		0			0
$333$		0			0
$334$	0.638138	−	0.400969i	0.638138	−	0.400969i
$335$		0			0
$336$		0			0
$337$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$337$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$338$	−1.28183	−	0.448532i	−1.28183	−	0.448532i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−1.12349	+	1.78802i	−1.12349	+	1.78802i
$347$	−1.56366			−1.56366			−0.781831	−	0.623490i	$-0.785714\pi$
$347$	−1.56366			−1.56366			−0.781831	+	0.623490i	$0.785714\pi$
$348$	−0.415617	−	1.82094i	−0.415617	−	1.82094i
$349$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$349$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$350$		0			0
$351$	−0.193096	−	0.400969i	−0.193096	−	0.400969i
$352$		0			0
$353$	0.347948	+	1.52446i	0.347948	+	1.52446i	0.781831	+	0.623490i	$0.214286\pi$
$353$	0.347948	+	1.52446i	0.347948	+	1.52446i	−0.433884	+	0.900969i	$0.642857\pi$
$354$	2.88025	−	1.00784i	2.88025	−	1.00784i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	1.99319	+	0.697449i	1.99319	+	0.697449i
$359$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$359$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$360$		0			0
$361$	−0.781831	−	0.623490i	−0.781831	−	0.623490i
$362$		0			0
$363$	0.433884	−	0.900969i	0.433884	−	0.900969i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$367$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$368$	0.485352	−	0.387055i	0.485352	−	0.387055i
$369$	0.189606	+	0.119137i	0.189606	+	0.119137i
$370$		0			0
$371$		0			0
$372$	−1.86777	−	1.86777i	−1.86777	−	1.86777i
$373$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$373$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$374$		0			0
$375$		0			0
$376$			0.970703i			0.970703i
$377$	−0.0990311	+	0.433884i	−0.0990311	+	0.433884i
$378$		0			0
$379$		0			0		−0.846724	−	0.532032i	$-0.821429\pi$
$379$		0			0		0.846724	+	0.532032i	$0.178571\pi$
$380$		0			0
$381$	−0.189606	−	0.119137i	−0.189606	−	0.119137i
$382$		0			0
$383$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$383$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$384$	0.205171	+	1.82094i	0.205171	+	1.82094i
$385$		0			0
$386$	1.99319	+	2.49939i	1.99319	+	2.49939i
$387$		0			0
$388$		0			0
$389$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$389$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$390$		0			0
$391$		0			0
$392$	−0.781831	−	1.24428i	−0.781831	−	1.24428i
$393$	1.05737	+	0.119137i	1.05737	+	0.119137i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.974928	+	1.22252i	0.974928	+	1.22252i	0.974928	+	0.222521i	$0.0714286\pi$
$397$	0.974928	+	1.22252i	0.974928	+	1.22252i			1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	0.138138	−	0.605223i	0.138138	−	0.605223i
$401$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$401$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$402$		0			0
$403$	0.207872	+	0.594065i	0.207872	+	0.594065i
$404$	−1.04466	−	0.656405i	−1.04466	−	0.656405i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.211363	−	0.0739590i	0.211363	−	0.0739590i	−0.222521	−	0.974928i	$-0.571429\pi$
$409$	0.211363	−	0.0739590i	0.211363	−	0.0739590i	0.433884	+	0.900969i	$0.357143\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	1.43388	+	0.900969i	1.43388	+	0.900969i
$415$		0			0
$416$	0.0614773	−	0.175692i	0.0614773	−	0.175692i
$417$	−1.21572	−	1.52446i	−1.21572	−	1.52446i
$418$		0			0
$419$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$419$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$420$		0			0
$421$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$421$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$422$	−1.87241	−	1.49319i	−1.87241	−	1.49319i
$423$	−0.623490	+	0.218169i	−0.623490	+	0.218169i
$424$		0			0
$425$		0			0
$426$	−3.11668	−	1.09057i	−3.11668	−	1.09057i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$431$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$432$	0.559311	−	0.269350i	0.559311	−	0.269350i
$433$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$433$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	0.874573	+	0.697449i	0.874573	+	0.697449i
$439$	−0.678448	−	1.40881i	−0.678448	−	1.40881i	−0.900969	−	0.433884i	$-0.857143\pi$
$439$	−0.678448	−	1.40881i	−0.678448	−	1.40881i	0.222521	−	0.974928i	$-0.428571\pi$
$440$		0			0
$441$	0.623490	−	0.781831i	0.623490	−	0.781831i
$442$		0			0
$443$	−1.87590	−	0.211363i	−1.87590	−	0.211363i	−0.900969	−	0.433884i	$-0.857143\pi$
$443$	−1.87590	−	0.211363i	−1.87590	−	0.211363i	−0.974928	+	0.222521i	$0.928571\pi$
$444$		0			0
$445$		0			0
$446$	−2.09842	+	0.236435i	−2.09842	+	0.236435i
$447$		0			0
$448$		0			0
$449$	−0.900969	+	0.566116i	−0.900969	+	0.566116i	−0.900969	−	0.433884i	$-0.857143\pi$
$449$	−0.900969	+	0.566116i	−0.900969	+	0.566116i			1.00000i	$0.5\pi$
$450$	1.68280	−	0.189606i	1.68280	−	0.189606i
$451$		0			0
$452$		0			0
$453$	−1.56366	+	1.24698i	−1.56366	+	1.24698i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$457$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.87590	−	0.656405i	1.87590	−	0.656405i	0.900969	−	0.433884i	$-0.142857\pi$
$461$	1.87590	−	0.656405i	1.87590	−	0.656405i	0.974928	−	0.222521i	$-0.0714286\pi$
$462$		0			0
$463$			0.445042i			0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$463$			0.445042i			0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$464$	−0.605223	−	0.138138i	−0.605223	−	0.138138i
$465$		0			0
$466$	1.78802	+	1.12349i	1.78802	+	1.12349i
$467$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$467$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$468$	−0.831235			−0.831235
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.296480	−	2.63133i	0.296480	−	2.63133i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	1.40881	+	2.24211i	1.40881	+	2.24211i
$479$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$479$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.16453	−	1.46028i	−1.16453	−	1.46028i
$485$		0			0
$486$	1.19745	+	1.19745i	1.19745	+	1.19745i
$487$	0.347948	−	1.52446i	0.347948	−	1.52446i	−0.433884	−	0.900969i	$-0.642857\pi$
$487$	0.347948	−	1.52446i	0.347948	−	1.52446i	0.781831	−	0.623490i	$-0.214286\pi$
$488$		0			0
$489$	−0.351438	+	0.559311i	−0.351438	+	0.559311i
$490$		0			0
$491$	1.68280	+	1.05737i	1.68280	+	1.05737i	0.900969	+	0.433884i	$0.142857\pi$
$491$	1.68280	+	1.05737i	1.68280	+	1.05737i	0.781831	+	0.623490i	$0.214286\pi$
$492$	0.354140	−	0.222521i	0.354140	−	0.222521i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.828660	+	0.289961i	−0.828660	+	0.289961i
$497$		0			0
$498$		0			0
$499$	−1.75676	−	0.400969i	−1.75676	−	0.400969i	−0.781831	−	0.623490i	$-0.785714\pi$
$499$	−1.75676	−	0.400969i	−1.75676	−	0.400969i	−0.974928	+	0.222521i	$0.928571\pi$
$500$		0			0
$501$	0.400969	−	0.193096i	0.400969	−	0.193096i
$502$		0			0
$503$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$503$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.722521	−	0.347948i	−0.722521	−	0.347948i
$508$	−0.354140	+	0.222521i	−0.354140	+	0.222521i
$509$	−0.678448	−	0.541044i	−0.678448	−	0.541044i	0.222521	−	0.974928i	$-0.428571\pi$
$509$	−0.678448	−	0.541044i	−0.678448	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$510$		0			0
$511$		0			0
$512$	1.10614	+	0.387055i	1.10614	+	0.387055i
$513$		0			0
$514$	1.46028	+	0.164534i	1.46028	+	0.164534i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−0.777479	+	0.974928i	−0.777479	+	0.974928i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−0.189606	−	1.68280i	−0.189606	−	1.68280i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	1.05737	−	1.68280i	1.05737	−	1.68280i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.900969	+	0.433884i	0.900969	+	0.433884i
$530$		0			0
$531$	1.75676	−	0.400969i	1.75676	−	0.400969i
$532$		0			0
$533$	−0.0990311	+	0.0111581i	−0.0990311	+	0.0111581i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1.12349	+	0.541044i	1.12349	+	0.541044i
$538$	0.236435	−	0.296480i	0.236435	−	0.296480i
$539$		0			0
$540$		0			0
$541$	−0.623490	+	1.78183i	−0.623490	+	1.78183i			1.00000i	$0.5\pi$
$541$	−0.623490	+	1.78183i	−0.623490	+	1.78183i	−0.623490	+	0.781831i	$0.714286\pi$
$542$	−1.40881	+	1.12349i	−1.40881	+	1.12349i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.12349	−	0.541044i	1.12349	−	0.541044i	0.222521	−	0.974928i	$-0.428571\pi$
$547$	1.12349	−	0.541044i	1.12349	−	0.541044i	0.900969	+	0.433884i	$0.142857\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	1.24428	−	0.781831i	1.24428	−	0.781831i
$553$		0			0
$554$	0.248917	+	0.711363i	0.248917	+	0.711363i
$555$		0			0
$556$	−3.55057	+	0.810394i	−3.55057	+	0.810394i
$557$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$557$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$558$	−1.49319	−	1.87241i	−1.49319	−	1.87241i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$	−0.138138	+	1.22601i	−0.138138	+	1.22601i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−2.02611	+	2.02611i	−2.02611	+	2.02611i
$569$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$569$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$570$		0			0
$571$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$571$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.974928	−	0.222521i	0.974928	−	0.222521i
$576$			1.32907i			1.32907i
$577$	0.656405	+	1.87590i	0.656405	+	1.87590i	0.433884	+	0.900969i	$0.357143\pi$
$577$	0.656405	+	1.87590i	0.656405	+	1.87590i	0.222521	+	0.974928i	$0.428571\pi$
$578$	1.43388	+	0.900969i	1.43388	+	0.900969i
$579$	1.00435	+	1.59842i	1.00435	+	1.59842i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.874573	−	0.421172i	0.874573	−	0.421172i
$585$		0			0
$586$		0			0
$587$	0.678448	−	1.40881i	0.678448	−	1.40881i	−0.222521	−	0.974928i	$-0.571429\pi$
$587$	0.678448	−	1.40881i	0.678448	−	1.40881i	0.900969	−	0.433884i	$-0.142857\pi$
$588$	−0.810394	−	1.68280i	−0.810394	−	1.68280i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.974928	−	1.22252i	0.974928	−	1.22252i		−	1.00000i	$-0.5\pi$
$593$	0.974928	−	1.22252i	0.974928	−	1.22252i	0.974928	−	0.222521i	$-0.0714286\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−0.748917	+	0.0843826i	−0.748917	+	0.0843826i
$599$	1.00435	+	0.351438i	1.00435	+	0.351438i	0.781831	−	0.623490i	$-0.214286\pi$
$599$	1.00435	+	0.351438i	1.00435	+	0.351438i	0.222521	+	0.974928i	$0.428571\pi$
$600$	0.485352	−	1.38705i	0.485352	−	1.38705i
$601$	−1.40532	−	0.158342i	−1.40532	−	0.158342i	−0.623490	−	0.781831i	$-0.714286\pi$
$601$	−1.40532	−	0.158342i	−1.40532	−	0.158342i	−0.781831	+	0.623490i	$0.785714\pi$
$602$		0			0
$603$		0			0
$604$	0.831235	+	3.64188i	0.831235	+	3.64188i
$605$		0			0
$606$	−0.874573	−	0.697449i	−0.874573	−	0.697449i
$607$	−1.05737	+	1.68280i	−1.05737	+	1.68280i	−0.433884	+	0.900969i	$0.642857\pi$
$607$	−1.05737	+	1.68280i	−1.05737	+	1.68280i	−0.623490	+	0.781831i	$0.714286\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.156405	−	0.248917i	0.156405	−	0.248917i
$612$		0			0
$613$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$613$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$614$	−0.711363	−	3.11668i	−0.711363	−	3.11668i
$615$		0			0
$616$		0			0
$617$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$617$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$618$		0			0
$619$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$619$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$620$		0			0
$621$	0.781831	+	0.623490i	0.781831	+	0.623490i
$622$	−2.24211	−	1.78802i	−2.24211	−	1.78802i
$623$		0			0
$624$	−0.119872	+	0.248917i	−0.119872	+	0.248917i
$625$	0.623490	−	0.781831i	0.623490	−	0.781831i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$631$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$632$		0			0
$633$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$634$	−2.58377	+	1.24428i	−2.58377	+	1.24428i
$635$		0			0
$636$		0			0
$637$			0.445042i			0.445042i
$638$		0			0
$639$	−1.75676	−	0.846011i	−1.75676	−	0.846011i
$640$		0			0
$641$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$641$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$642$		0			0
$643$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$643$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.541044	+	0.678448i	0.541044	+	0.678448i	0.974928	−	0.222521i	$-0.0714286\pi$
$647$	0.541044	+	0.678448i	0.541044	+	0.678448i	−0.433884	+	0.900969i	$0.642857\pi$
$648$	1.38705	−	0.485352i	1.38705	−	0.485352i
$649$		0			0
$650$	−0.532915	+	0.532915i	−0.532915	+	0.532915i
$651$		0			0
$652$	0.656405	+	1.04466i	0.656405	+	1.04466i
$653$	0.752407	+	1.19745i	0.752407	+	1.19745i	0.974928	+	0.222521i	$0.0714286\pi$
$653$	0.752407	+	1.19745i	0.752407	+	1.19745i	−0.222521	+	0.974928i	$0.571429\pi$
$654$		0			0
$655$		0			0
$656$	−0.0155644	−	0.138138i	−0.0155644	−	0.138138i
$657$	0.467085	+	0.467085i	0.467085	+	0.467085i
$658$		0			0
$659$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$659$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$660$		0			0
$661$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$661$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$662$	3.28122	−	0.748917i	3.28122	−	0.748917i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$668$		−	0.831235i		−	0.831235i
$669$	−1.24698			−1.24698
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.52446	−	0.347948i	−1.52446	−	0.347948i	−0.623490	−	0.781831i	$-0.714286\pi$
$673$	−1.52446	−	0.347948i	−1.52446	−	0.347948i	−0.900969	+	0.433884i	$0.857143\pi$
$674$		0			0
$675$	1.00000			1.00000
$676$	−1.17105	+	0.933884i	−1.17105	+	0.933884i
$677$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$677$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.678448	+	0.541044i	0.678448	+	0.541044i	0.900969	−	0.433884i	$-0.142857\pi$
$683$	0.678448	+	0.541044i	0.678448	+	0.541044i	−0.222521	+	0.974928i	$0.571429\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.433884	−	1.90097i	−0.433884	−	1.90097i	−0.433884	−	0.900969i	$-0.642857\pi$
$691$	−0.433884	−	1.90097i	−0.433884	−	1.90097i		−	1.00000i	$-0.5\pi$
$692$	1.01054	+	2.09842i	1.01054	+	2.09842i
$693$		0			0
$694$	−1.40881	+	2.24211i	−1.40881	+	2.24211i
$695$		0			0
$696$	−1.38705	−	0.485352i	−1.38705	−	0.485352i
$697$		0			0
$698$	−1.62349	+	2.58377i	−1.62349	+	2.58377i
$699$	0.974928	+	0.777479i	0.974928	+	0.777479i
$700$		0			0
$701$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$701$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$702$	−0.748917	−	0.0843826i	−0.748917	−	0.0843826i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	2.49939	+	0.874573i	2.49939	+	0.874573i
$707$		0			0
$708$	0.748917	−	3.28122i	0.748917	−	3.28122i
$709$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$709$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$714$		0			0
$715$		0			0
$716$	1.82094	−	1.45215i	1.82094	−	1.45215i
$717$	0.678448	+	1.40881i	0.678448	+	1.40881i
$718$		0			0
$719$	1.90097	+	0.433884i	1.90097	+	0.433884i	1.00000			$0$
$719$	1.90097	+	0.433884i	1.90097	+	0.433884i	0.900969	+	0.433884i	$0.142857\pi$
$720$		0			0
$721$		0			0
$722$	−1.59842	+	0.559311i	−1.59842	+	0.559311i
$723$		0			0
$724$		0			0
$725$	−0.781831	−	0.623490i	−0.781831	−	0.623490i
$726$	−0.900969	−	1.43388i	−0.900969	−	1.43388i
$727$		0			0		−0.846724	−	0.532032i	$-0.821429\pi$
$727$		0			0		0.846724	+	0.532032i	$0.178571\pi$
$728$		0			0
$729$	0.623490	+	0.781831i	0.623490	+	0.781831i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$733$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$734$		0			0
$735$		0			0
$736$	0.0468288	+	0.415617i	0.0468288	+	0.415617i
$737$		0			0
$738$	0.341658	−	0.164534i	0.341658	−	0.164534i
$739$	0.119137	+	0.189606i	0.119137	+	0.189606i	0.900969	−	0.433884i	$-0.142857\pi$
$739$	0.119137	+	0.189606i	0.119137	+	0.189606i	−0.781831	+	0.623490i	$0.785714\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$743$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$744$	−2.02611	+	0.462446i	−2.02611	+	0.462446i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$751$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$752$	0.347213	+	0.218169i	0.347213	+	0.218169i
$753$		0			0
$754$	0.532915	+	0.532915i	0.532915	+	0.532915i
$755$		0			0
$756$		0			0
$757$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$757$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.193096	+	0.400969i	−0.193096	+	0.400969i	−0.974928	−	0.222521i	$-0.928571\pi$
$761$	−0.193096	+	0.400969i	−0.193096	+	0.400969i	0.781831	+	0.623490i	$0.214286\pi$
$762$	−0.341658	+	0.164534i	−0.341658	+	0.164534i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.500000	+	0.626980i	−0.500000	+	0.626980i
$768$	1.59842	+	0.769757i	1.59842	+	0.769757i
$769$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$769$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$770$		0			0
$771$	0.846011	+	0.193096i	0.846011	+	0.193096i
$772$	3.50374	−	0.394777i	3.50374	−	0.394777i
$773$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$773$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$774$		0			0
$775$	−1.40532	−	0.158342i	−1.40532	−	0.158342i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	1.00000i		−	1.00000i
$784$	−0.620788			−0.620788
$785$		0			0
$786$	1.12349	−	1.40881i	1.12349	−	1.40881i
$787$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$787$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	2.63133	−	0.296480i	2.63133	−	0.296480i
$795$		0			0
$796$		0			0
$797$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$797$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$798$		0			0
$799$		0			0
$800$	0.295745	+	0.295745i	0.295745	+	0.295745i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	1.03911	+	0.237169i	1.03911	+	0.237169i
$807$	0.158342	−	0.158342i	0.158342	−	0.158342i
$808$	−0.874573	+	0.421172i	−0.874573	+	0.421172i
$809$	−1.87590	+	0.656405i	−1.87590	+	0.656405i	−0.900969	+	0.433884i	$0.857143\pi$
$809$	−1.87590	+	0.656405i	−1.87590	+	0.656405i	−0.974928	+	0.222521i	$0.928571\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	−0.900969	+	0.566116i	−0.900969	+	0.566116i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.0843826	−	0.369704i	0.0843826	−	0.369704i
$819$		0			0
$820$		0			0
$821$	−0.541044	−	0.678448i	−0.541044	−	0.678448i	0.433884	−	0.900969i	$-0.357143\pi$
$821$	−0.541044	−	0.678448i	−0.541044	−	0.678448i	−0.974928	+	0.222521i	$0.928571\pi$
$822$		0			0
$823$	−0.222521	−	1.97493i	−0.222521	−	1.97493i	−0.222521	−	0.974928i	$-0.571429\pi$
$823$	−0.222521	−	1.97493i	−0.222521	−	1.97493i		−	1.00000i	$-0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$827$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$828$	1.68280	−	0.810394i	1.68280	−	0.810394i
$829$	−1.19745	+	1.19745i	−1.19745	+	1.19745i	−0.222521	+	0.974928i	$0.571429\pi$
$829$	−1.19745	+	1.19745i	−1.19745	+	1.19745i	−0.974928	+	0.222521i	$0.928571\pi$
$830$		0			0
$831$	0.0990311	+	0.433884i	0.0990311	+	0.433884i
$832$	−0.368789	−	0.462446i	−0.368789	−	0.462446i
$833$		0			0
$834$	−3.28122	+	0.369704i	−3.28122	+	0.369704i
$835$		0			0
$836$		0			0
$837$	−0.752407	−	1.19745i	−0.752407	−	1.19745i
$838$		0			0
$839$		0			0		−0.846724	−	0.532032i	$-0.821429\pi$
$839$		0			0		0.846724	+	0.532032i	$0.178571\pi$
$840$		0			0
$841$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$842$		0			0
$843$		0			0
$844$	−2.49319	+	0.872406i	−2.49319	+	0.872406i
$845$		0			0
$846$	−0.248917	+	1.09057i	−0.248917	+	1.09057i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−2.84733	+	2.27067i	−2.84733	+	2.27067i
$853$	0.467085	+	0.467085i	0.467085	+	0.467085i	0.900969	−	0.433884i	$-0.142857\pi$
$853$	0.467085	+	0.467085i	0.467085	+	0.467085i	−0.433884	+	0.900969i	$0.642857\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.56366	−	1.24698i	−1.56366	−	1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$857$	−1.56366	−	1.24698i	−1.56366	−	1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$858$		0			0
$859$	1.97493	−	0.222521i	1.97493	−	0.222521i	0.974928	−	0.222521i	$-0.0714286\pi$
$859$	1.97493	−	0.222521i	1.97493	−	0.222521i	1.00000			$0$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.80194	−	0.867767i	−1.80194	−	0.867767i	−0.900969	−	0.433884i	$-0.857143\pi$
$863$	−1.80194	−	0.867767i	−1.80194	−	0.867767i	−0.900969	−	0.433884i	$-0.857143\pi$
$864$	−0.0468288	+	0.415617i	−0.0468288	+	0.415617i
$865$		0			0
$866$		0			0
$867$	0.781831	+	0.623490i	0.781831	+	0.623490i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	1.16453	−	0.407488i	1.16453	−	0.407488i
$877$	−0.781831	−	0.376510i	−0.781831	−	0.376510i		−	1.00000i	$-0.5\pi$
$877$	−0.781831	−	0.376510i	−0.781831	−	0.376510i	−0.781831	+	0.623490i	$0.785714\pi$
$878$	−2.63133	−	0.296480i	−2.63133	−	0.296480i
$879$		0			0
$880$		0			0
$881$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$881$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$882$	−0.559311	−	1.59842i	−0.559311	−	1.59842i
$883$	−1.40881	−	1.12349i	−1.40881	−	1.12349i	−0.974928	−	0.222521i	$-0.928571\pi$
$883$	−1.40881	−	1.12349i	−1.40881	−	1.12349i	−0.433884	−	0.900969i	$-0.642857\pi$
$884$		0			0
$885$		0			0
$886$	−1.99319	+	2.49939i	−1.99319	+	2.49939i
$887$	0.158342	+	0.158342i	0.158342	+	0.158342i	0.781831	−	0.623490i	$-0.214286\pi$
$887$	0.158342	+	0.158342i	0.158342	+	0.158342i	−0.623490	+	0.781831i	$0.714286\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−1.01054	+	2.09842i	−1.01054	+	2.09842i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−0.445042			−0.445042
$898$			1.80194i			1.80194i
$899$	−0.158342	+	1.40532i	−0.158342	+	1.40532i
$900$	0.810394	−	1.68280i	0.810394	−	1.68280i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	0.379212	+	3.36560i	0.379212	+	3.36560i
$907$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$907$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$908$		0			0
$909$	−0.467085	−	0.467085i	−0.467085	−	0.467085i
$910$		0			0
$911$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$911$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$919$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$920$		0			0
$921$	−0.211363	−	1.87590i	−0.211363	−	1.87590i
$922$	0.748917	−	3.28122i	0.748917	−	3.28122i
$923$	0.846011	−	0.193096i	0.846011	−	0.193096i
$924$		0			0
$925$		0			0
$926$	0.638138	+	0.400969i	0.638138	+	0.400969i
$927$		0			0
$928$	0.295745	−	0.295745i	0.295745	−	0.295745i
$929$			0.445042i			0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$929$			0.445042i			0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$930$		0			0
$931$		0			0
$932$	2.09842	−	1.01054i	2.09842	−	1.01054i
$933$	−1.19745	−	1.19745i	−1.19745	−	1.19745i
$934$		0			0
$935$		0			0
$936$	−0.347948	+	0.553756i	−0.347948	+	0.553756i
$937$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$937$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$941$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$942$		0			0
$943$	0.189606	−	0.119137i	0.189606	−	0.119137i
$944$	−0.874573	−	0.697449i	−0.874573	−	0.697449i
$945$		0			0
$946$		0			0
$947$	0.211363	+	0.0739590i	0.211363	+	0.0739590i	0.433884	−	0.900969i	$-0.357143\pi$
$947$	0.211363	+	0.0739590i	0.211363	+	0.0739590i	−0.222521	+	0.974928i	$0.571429\pi$
$948$		0			0
$949$	−0.292128	−	0.0329149i	−0.292128	−	0.0329149i
$950$		0			0
$951$	−1.59842	+	0.559311i	−1.59842	+	0.559311i
$952$		0			0
$953$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$953$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$954$		0			0
$955$		0			0
$956$	2.92056			2.92056
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.433884	+	0.900969i	0.433884	+	0.900969i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.78183	−	0.623490i	−1.78183	−	0.623490i	−0.781831	−	0.623490i	$-0.785714\pi$
$967$	−1.78183	−	0.623490i	−1.78183	−	0.623490i	−1.00000			$\pi$
$968$	−1.46028	+	0.164534i	−1.46028	+	0.164534i
$969$		0			0
$970$		0			0
$971$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$971$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$972$	1.82094	−	0.415617i	1.82094	−	0.415617i
$973$		0			0
$974$	−1.87241	−	1.87241i	−1.87241	−	1.87241i
$975$	−0.347948	+	0.277479i	−0.347948	+	0.277479i
$976$		0			0
$977$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$977$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$978$	0.485352	+	1.00784i	0.485352	+	1.00784i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	3.03230	−	1.46028i	3.03230	−	1.46028i
$983$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$983$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$984$		−	0.329068i		−	0.329068i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.52446	+	0.347948i	−1.52446	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$991$	−1.52446	+	0.347948i	−1.52446	+	0.347948i	−0.623490	+	0.781831i	$0.714286\pi$
$992$	0.131619	−	0.576661i	0.131619	−	0.576661i
$993$	1.97493	−	0.222521i	1.97493	−	0.222521i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.119137	−	1.05737i	−0.119137	−	1.05737i	−0.900969	−	0.433884i	$-0.857143\pi$
$997$	−0.119137	−	1.05737i	−0.119137	−	1.05737i	0.781831	−	0.623490i	$-0.214286\pi$
$998$	−2.15773	+	2.15773i	−2.15773	+	2.15773i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2001.1.bf.b.206.1	yes	12
3.2	odd	2		2001.1.bf.a.206.1	yes	12
23.22	odd	2	CM	2001.1.bf.b.206.1	yes	12
29.10	odd	28		2001.1.bf.a.68.1	✓	12
69.68	even	2		2001.1.bf.a.206.1	yes	12
87.68	even	28	inner	2001.1.bf.b.68.1	yes	12
667.68	even	28		2001.1.bf.a.68.1	✓	12
2001.68	odd	28	inner	2001.1.bf.b.68.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.1.bf.a.68.1	✓	12	29.10	odd	28
2001.1.bf.a.68.1	✓	12	667.68	even	28
2001.1.bf.a.206.1	yes	12	3.2	odd	2
2001.1.bf.a.206.1	yes	12	69.68	even	2
2001.1.bf.b.68.1	yes	12	87.68	even	28	inner
2001.1.bf.b.68.1	yes	12	2001.68	odd	28	inner
2001.1.bf.b.206.1	yes	12	1.1	even	1	trivial
2001.1.bf.b.206.1	yes	12	23.22	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:	\( N \)	\(=\)	\( 2001 = 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2001.bf (of order \(28\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(0.900969 - 1.43388i) q^{2} +(0.623490 - 0.781831i) q^{3} +(-0.810394 - 1.68280i) q^{4} +(-0.559311 - 1.59842i) q^{6} +(-1.46028 - 0.164534i) q^{8} +(-0.222521 - 0.974928i) q^{9} +O(q^{10})\)	\(q+(0.900969 - 1.43388i) q^{2} +(0.623490 - 0.781831i) q^{3} +(-0.810394 - 1.68280i) q^{4} +(-0.559311 - 1.59842i) q^{6} +(-1.46028 - 0.164534i) q^{8} +(-0.222521 - 0.974928i) q^{9} +(-1.82094 - 0.415617i) q^{12} +(0.347948 + 0.277479i) q^{13} +(-0.387055 + 0.485352i) q^{16} +(-1.59842 - 0.559311i) q^{18} +(-0.974928 - 0.222521i) q^{23} +(-1.03911 + 1.03911i) q^{24} +(-0.900969 + 0.433884i) q^{25} +(0.711363 - 0.248917i) q^{26} +(-0.900969 - 0.433884i) q^{27} +(0.433884 + 0.900969i) q^{29} +(1.19745 + 0.752407i) q^{31} +(-0.138138 - 0.394777i) q^{32} +(-1.46028 + 1.16453i) q^{36} +(0.433884 - 0.0990311i) q^{39} +(-0.158342 + 0.158342i) q^{41} +(-1.19745 + 1.19745i) q^{46} +(-0.0739590 - 0.656405i) q^{47} +(0.138138 + 0.605223i) q^{48} +(0.623490 + 0.781831i) q^{49} +(-0.189606 + 1.68280i) q^{50} +(0.184967 - 0.810394i) q^{52} +(-1.43388 + 0.900969i) q^{54} +(1.68280 + 0.189606i) q^{58} +1.80194i q^{59} +(2.15773 - 1.03911i) q^{62} +(-1.29575 - 0.295745i) q^{64} +(-0.781831 + 0.623490i) q^{69} +(1.21572 - 1.52446i) q^{71} +(0.164534 + 1.46028i) q^{72} +(-0.559311 + 0.351438i) q^{73} +(-0.222521 + 0.974928i) q^{75} +(0.248917 - 0.711363i) q^{78} +(-0.900969 + 0.433884i) q^{81} +(0.0843826 + 0.369704i) q^{82} +(0.974928 + 0.222521i) q^{87} +(0.415617 + 1.82094i) q^{92} +(1.33485 - 0.467085i) q^{93} +(-1.00784 - 0.485352i) q^{94} +(-0.394777 - 0.138138i) q^{96} +(1.68280 - 0.189606i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{2} - 2 q^{3} - 14 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 12 * q + 2 * q^2 - 2 * q^3 - 14 * q^4 + 2 * q^6 - 2 * q^9	\( 12 q + 2 q^{2} - 2 q^{3} - 14 q^{4} + 2 q^{6} - 2 q^{9} + 12 q^{16} + 2 q^{18} - 2 q^{25} + 4 q^{26} - 2 q^{27} + 2 q^{31} + 2 q^{32} - 2 q^{41} - 2 q^{46} + 2 q^{47} - 2 q^{48} - 2 q^{49} + 2 q^{50} + 10 q^{52} - 12 q^{54} + 2 q^{58} - 4 q^{62} - 14 q^{64} - 14 q^{72} + 2 q^{73} - 2 q^{75} - 10 q^{78} - 2 q^{81} + 4 q^{82} + 2 q^{92} + 2 q^{93} - 4 q^{94} - 12 q^{96} + 2 q^{98}+O(q^{100}) \) 12 * q + 2 * q^2 - 2 * q^3 - 14 * q^4 + 2 * q^6 - 2 * q^9 + 12 * q^16 + 2 * q^18 - 2 * q^25 + 4 * q^26 - 2 * q^27 + 2 * q^31 + 2 * q^32 - 2 * q^41 - 2 * q^46 + 2 * q^47 - 2 * q^48 - 2 * q^49 + 2 * q^50 + 10 * q^52 - 12 * q^54 + 2 * q^58 - 4 * q^62 - 14 * q^64 - 14 * q^72 + 2 * q^73 - 2 * q^75 - 10 * q^78 - 2 * q^81 + 4 * q^82 + 2 * q^92 + 2 * q^93 - 4 * q^94 - 12 * q^96 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more