LMFDB - Embedded newform 2001.1.bf.b.620.1

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			620.1
Root			$0.974928 - 0.222521i$ of defining polynomial
Character	$\chi$	$=$	2001.620
Dual form			2001.1.bf.b.965.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.623490 - 1.78183i) q^{2} +(-0.222521 + 0.974928i) q^{3} +(-2.00435 + 1.59842i) q^{4} +(1.87590 - 0.211363i) q^{6} +(2.49939 + 1.57047i) q^{8} +(-0.900969 - 0.433884i) q^{9} +O(q^{10})$	\(q+(-0.623490 - 1.78183i) q^{2} +(-0.222521 + 0.974928i) q^{3} +(-2.00435 + 1.59842i) q^{4} +(1.87590 - 0.211363i) q^{6} +(2.49939 + 1.57047i) q^{8} +(-0.900969 - 0.433884i) q^{9} +(-1.11233 - 2.30978i) q^{12} +(-1.75676 - 0.400969i) q^{13} +(0.669500 - 2.93327i) q^{16} +(-0.211363 + 1.87590i) q^{18} +(-0.433884 - 0.900969i) q^{23} +(-2.08726 + 2.08726i) q^{24} +(0.623490 + 0.781831i) q^{25} +(0.380863 + 3.38025i) q^{26} +(0.623490 - 0.781831i) q^{27} +(0.781831 - 0.623490i) q^{29} +(1.33485 - 0.467085i) q^{31} +(-2.71075 + 0.305428i) q^{32} +(2.49939 - 0.570469i) q^{36} +(0.781831 - 1.62349i) q^{39} +(0.752407 - 0.752407i) q^{41} +(-1.33485 + 1.33485i) q^{46} +(-1.05737 - 1.68280i) q^{47} +(2.71075 + 1.30543i) q^{48} +(-0.222521 - 0.974928i) q^{49} +(1.00435 - 1.59842i) q^{50} +(4.16208 - 2.00435i) q^{52} +(-1.78183 - 0.623490i) q^{54} +(-1.59842 - 1.00435i) q^{58} -1.24698i q^{59} +(-1.66453 - 2.08726i) q^{62} +(0.928918 + 1.92892i) q^{64} +(0.974928 - 0.222521i) q^{69} +(-0.193096 + 0.846011i) q^{71} +(-1.57047 - 2.49939i) q^{72} +(1.87590 + 0.656405i) q^{73} +(-0.900969 + 0.433884i) q^{75} +(-3.38025 - 0.380863i) q^{78} +(0.623490 + 0.781831i) q^{81} +(-1.80978 - 0.871544i) q^{82} +(0.433884 + 0.900969i) q^{87} +(2.30978 + 1.11233i) q^{92} +(0.158342 + 1.40532i) q^{93} +(-2.33921 + 2.93327i) q^{94} +(0.305428 - 2.71075i) q^{96} +(-1.59842 + 1.00435i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2001\mathbb{Z}\right)^\times$.

$n$	$553$	$668$	$1132$
$\chi(n)$	$e\left(\frac{25}{28}\right)$	$-1$	$-1$

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 $

$2$	−0.623490	−	1.78183i	−0.623490	−	1.78183i	−0.623490	−	0.781831i	$-0.714286\pi$
$2$	−0.623490	−	1.78183i	−0.623490	−	1.78183i		−	1.00000i	$-0.5\pi$
$3$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$3$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$4$	−2.00435	+	1.59842i	−2.00435	+	1.59842i
$5$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$5$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$6$	1.87590	−	0.211363i	1.87590	−	0.211363i
$7$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$7$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$8$	2.49939	+	1.57047i	2.49939	+	1.57047i
$9$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$10$		0			0
$11$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$11$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$12$	−1.11233	−	2.30978i	−1.11233	−	2.30978i
$13$	−1.75676	−	0.400969i	−1.75676	−	0.400969i	−0.781831	−	0.623490i	$-0.785714\pi$
$13$	−1.75676	−	0.400969i	−1.75676	−	0.400969i	−0.974928	+	0.222521i	$0.928571\pi$
$14$		0			0
$15$		0			0
$16$	0.669500	−	2.93327i	0.669500	−	2.93327i
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$	−0.211363	+	1.87590i	−0.211363	+	1.87590i
$19$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$19$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.433884	−	0.900969i	−0.433884	−	0.900969i
$23$	−0.433884	−	0.900969i	−0.433884	−	0.900969i
$24$	−2.08726	+	2.08726i	−2.08726	+	2.08726i
$25$	0.623490	+	0.781831i	0.623490	+	0.781831i
$26$	0.380863	+	3.38025i	0.380863	+	3.38025i
$27$	0.623490	−	0.781831i	0.623490	−	0.781831i
$28$		0			0
$29$	0.781831	−	0.623490i	0.781831	−	0.623490i
$29$	0.781831	−	0.623490i	0.781831	−	0.623490i
$30$		0			0
$31$	1.33485	−	0.467085i	1.33485	−	0.467085i	0.433884	−	0.900969i	$-0.357143\pi$
$31$	1.33485	−	0.467085i	1.33485	−	0.467085i	0.900969	+	0.433884i	$0.142857\pi$
$32$	−2.71075	+	0.305428i	−2.71075	+	0.305428i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	2.49939	−	0.570469i	2.49939	−	0.570469i
$37$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$37$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$38$		0			0
$39$	0.781831	−	1.62349i	0.781831	−	1.62349i
$40$		0			0
$41$	0.752407	−	0.752407i	0.752407	−	0.752407i	−0.222521	−	0.974928i	$-0.571429\pi$
$41$	0.752407	−	0.752407i	0.752407	−	0.752407i	0.974928	+	0.222521i	$0.0714286\pi$
$42$		0			0
$43$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$43$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$44$		0			0
$45$		0			0
$46$	−1.33485	+	1.33485i	−1.33485	+	1.33485i
$47$	−1.05737	−	1.68280i	−1.05737	−	1.68280i	−0.623490	−	0.781831i	$-0.714286\pi$
$47$	−1.05737	−	1.68280i	−1.05737	−	1.68280i	−0.433884	−	0.900969i	$-0.642857\pi$
$48$	2.71075	+	1.30543i	2.71075	+	1.30543i
$49$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$50$	1.00435	−	1.59842i	1.00435	−	1.59842i
$51$		0			0
$52$	4.16208	−	2.00435i	4.16208	−	2.00435i
$53$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$53$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$54$	−1.78183	−	0.623490i	−1.78183	−	0.623490i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−1.59842	−	1.00435i	−1.59842	−	1.00435i
$59$		−	1.24698i		−	1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$59$		−	1.24698i		−	1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$60$		0			0
$61$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$61$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$62$	−1.66453	−	2.08726i	−1.66453	−	2.08726i
$63$		0			0
$64$	0.928918	+	1.92892i	0.928918	+	1.92892i
$65$		0			0
$66$		0			0
$67$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$67$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$68$		0			0
$69$	0.974928	−	0.222521i	0.974928	−	0.222521i
$70$		0			0
$71$	−0.193096	+	0.846011i	−0.193096	+	0.846011i	0.781831	+	0.623490i	$0.214286\pi$
$71$	−0.193096	+	0.846011i	−0.193096	+	0.846011i	−0.974928	+	0.222521i	$0.928571\pi$
$72$	−1.57047	−	2.49939i	−1.57047	−	2.49939i
$73$	1.87590	+	0.656405i	1.87590	+	0.656405i	0.974928	+	0.222521i	$0.0714286\pi$
$73$	1.87590	+	0.656405i	1.87590	+	0.656405i	0.900969	+	0.433884i	$0.142857\pi$
$74$		0			0
$75$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$76$		0			0
$77$		0			0
$78$	−3.38025	−	0.380863i	−3.38025	−	0.380863i
$79$		0			0		−0.846724	−	0.532032i	$-0.821429\pi$
$79$		0			0		0.846724	+	0.532032i	$0.178571\pi$
$80$		0			0
$81$	0.623490	+	0.781831i	0.623490	+	0.781831i
$82$	−1.80978	−	0.871544i	−1.80978	−	0.871544i
$83$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$83$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	0.433884	+	0.900969i	0.433884	+	0.900969i
$88$		0			0
$89$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$89$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$90$		0			0
$91$		0			0
$92$	2.30978	+	1.11233i	2.30978	+	1.11233i
$93$	0.158342	+	1.40532i	0.158342	+	1.40532i
$94$	−2.33921	+	2.93327i	−2.33921	+	2.93327i
$95$		0			0
$96$	0.305428	−	2.71075i	0.305428	−	2.71075i
$97$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$97$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$98$	−1.59842	+	1.00435i	−1.59842	+	1.00435i
$99$		0			0
$100$	−2.49939	−	0.570469i	−2.49939	−	0.570469i
$101$	−1.87590	−	0.656405i	−1.87590	−	0.656405i	−0.974928	−	0.222521i	$-0.928571\pi$
$101$	−1.87590	−	0.656405i	−1.87590	−	0.656405i	−0.900969	−	0.433884i	$-0.857143\pi$
$102$		0			0
$103$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$103$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$104$	−3.76111	−	3.76111i	−3.76111	−	3.76111i
$105$		0			0
$106$		0			0
$107$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$107$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$108$			2.56366i			2.56366i
$109$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$109$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$113$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$114$		0			0
$115$		0			0
$116$	−0.570469	+	2.49939i	−0.570469	+	2.49939i
$117$	1.40881	+	1.12349i	1.40881	+	1.12349i
$118$	−2.22191	+	0.777479i	−2.22191	+	0.777479i
$119$		0			0
$120$		0			0
$121$	0.433884	−	0.900969i	0.433884	−	0.900969i
$122$		0			0
$123$	0.566116	+	0.900969i	0.566116	+	0.900969i
$124$	−1.92892	+	3.06986i	−1.92892	+	3.06986i
$125$		0			0
$126$		0			0
$127$	−0.566116	−	0.900969i	−0.566116	−	0.900969i	0.433884	−	0.900969i	$-0.357143\pi$
$127$	−0.566116	−	0.900969i	−0.566116	−	0.900969i	−1.00000			$\pi$
$128$	0.928918	−	0.928918i	0.928918	−	0.928918i
$129$		0			0
$130$		0			0
$131$	0.218169	−	0.623490i	0.218169	−	0.623490i	−0.781831	−	0.623490i	$-0.785714\pi$
$131$	0.218169	−	0.623490i	0.218169	−	0.623490i	1.00000			$0$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$137$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$138$	−1.00435	−	1.59842i	−1.00435	−	1.59842i
$139$	0.781831	−	0.376510i	0.781831	−	0.376510i		−	1.00000i	$-0.5\pi$
$139$	0.781831	−	0.376510i	0.781831	−	0.376510i	0.781831	+	0.623490i	$0.214286\pi$
$140$		0			0
$141$	1.87590	−	0.656405i	1.87590	−	0.656405i
$142$	1.62784	−	0.183414i	1.62784	−	0.183414i
$143$		0			0
$144$	−1.87590	+	2.35230i	−1.87590	+	2.35230i
$145$		0			0
$146$		−	3.75179i		−	3.75179i
$147$	1.00000			1.00000
$148$		0			0
$149$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$149$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$150$	1.33485	+	1.33485i	1.33485	+	1.33485i
$151$	−0.867767	−	1.80194i	−0.867767	−	1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$151$	−0.867767	−	1.80194i	−0.867767	−	1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.02795	+	4.50374i	1.02795	+	4.50374i
$157$		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
$162$	1.00435	−	1.59842i	1.00435	−	1.59842i
$163$	−1.68280	+	1.05737i	−1.68280	+	1.05737i	−0.781831	+	0.623490i	$0.785714\pi$
$163$	−1.68280	+	1.05737i	−1.68280	+	1.05737i	−0.900969	+	0.433884i	$0.857143\pi$
$164$	−0.305428	+	2.71075i	−0.305428	+	2.71075i
$165$		0			0
$166$		0			0
$167$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.222521	+	0.974928i	$0.571429\pi$
$167$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.900969	+	0.433884i	$0.857143\pi$
$168$		0			0
$169$	2.02446	+	0.974928i	2.02446	+	0.974928i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.445042			0.445042			0.222521	−	0.974928i	$-0.428571\pi$
$173$	0.445042			0.445042			0.222521	+	0.974928i	$0.428571\pi$
$174$	1.33485	−	1.33485i	1.33485	−	1.33485i
$175$		0			0
$176$		0			0
$177$	1.21572	+	0.277479i	1.21572	+	0.277479i
$178$		0			0
$179$	−0.400969	−	0.193096i	−0.400969	−	0.193096i	0.222521	−	0.974928i	$-0.428571\pi$
$179$	−0.400969	−	0.193096i	−0.400969	−	0.193096i	−0.623490	+	0.781831i	$0.714286\pi$
$180$		0			0
$181$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$181$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$182$		0			0
$183$		0			0
$184$	0.330500	−	2.93327i	0.330500	−	2.93327i
$185$		0			0
$186$	2.40532	−	1.15834i	2.40532	−	1.15834i
$187$		0			0
$188$	4.80917	+	1.68280i	4.80917	+	1.68280i
$189$		0			0
$190$		0			0
$191$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$191$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$192$	−2.08726	+	0.476403i	−2.08726	+	0.476403i
$193$	0.222521	+	0.0250721i	0.222521	+	0.0250721i	0.222521	−	0.974928i	$-0.428571\pi$
$193$	0.222521	+	0.0250721i	0.222521	+	0.0250721i			1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	2.00435	+	1.59842i	2.00435	+	1.59842i
$197$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$197$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$198$		0			0
$199$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$199$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$200$	0.330500	+	2.93327i	0.330500	+	2.93327i
$201$		0			0
$202$			3.75179i			3.75179i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$			1.00000i			1.00000i
$208$	−2.35230	+	4.88460i	−2.35230	+	4.88460i
$209$		0			0
$210$		0			0
$211$	−0.752407	+	1.19745i	−0.752407	+	1.19745i	0.222521	+	0.974928i	$0.428571\pi$
$211$	−0.752407	+	1.19745i	−0.752407	+	1.19745i	−0.974928	+	0.222521i	$0.928571\pi$
$212$		0			0
$213$	−0.781831	−	0.376510i	−0.781831	−	0.376510i
$214$		0			0
$215$		0			0
$216$	2.78618	−	0.974928i	2.78618	−	0.974928i
$217$		0			0
$218$		0			0
$219$	−1.05737	+	1.68280i	−1.05737	+	1.68280i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.0990311	−	0.433884i	−0.0990311	−	0.433884i	0.900969	−	0.433884i	$-0.142857\pi$
$223$	−0.0990311	−	0.433884i	−0.0990311	−	0.433884i	−1.00000			$\pi$
$224$		0			0
$225$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$226$		0			0
$227$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$227$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$228$		0			0
$229$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$229$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$230$		0			0
$231$		0			0
$232$	2.93327	−	0.330500i	2.93327	−	0.330500i
$233$		−	0.445042i		−	0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$233$		−	0.445042i		−	0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$234$	1.12349	−	3.21075i	1.12349	−	3.21075i
$235$		0			0
$236$	1.99319	+	2.49939i	1.99319	+	2.49939i
$237$		0			0
$238$		0			0
$239$	1.52446	+	1.21572i	1.52446	+	1.21572i	0.900969	+	0.433884i	$0.142857\pi$
$239$	1.52446	+	1.21572i	1.52446	+	1.21572i	0.623490	+	0.781831i	$0.285714\pi$
$240$		0			0
$241$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$241$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$242$	−1.87590	−	0.211363i	−1.87590	−	0.211363i
$243$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$244$		0			0
$245$		0			0
$246$	1.25241	−	1.57047i	1.25241	−	1.57047i
$247$		0			0
$248$	4.06986	+	0.928918i	4.06986	+	0.928918i
$249$		0			0
$250$		0			0
$251$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$251$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$252$		0			0
$253$		0			0
$254$	−1.25241	+	1.57047i	−1.25241	+	1.57047i
$255$		0			0
$256$	−0.305428	−	0.147086i	−0.305428	−	0.147086i
$257$	1.22252	−	0.974928i	1.22252	−	0.974928i	0.222521	−	0.974928i	$-0.428571\pi$
$257$	1.22252	−	0.974928i	1.22252	−	0.974928i	1.00000			$0$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.974928	+	0.222521i	−0.974928	+	0.222521i
$262$	−1.24698			−1.24698
$263$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$263$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.900969	+	0.566116i	0.900969	+	0.566116i	0.900969	−	0.433884i	$-0.142857\pi$
$269$	0.900969	+	0.566116i	0.900969	+	0.566116i			1.00000i	$0.5\pi$
$270$		0			0
$271$	0.0739590	−	0.656405i	0.0739590	−	0.656405i	−0.900969	−	0.433884i	$-0.857143\pi$
$271$	0.0739590	−	0.656405i	0.0739590	−	0.656405i	0.974928	−	0.222521i	$-0.0714286\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−1.59842	+	2.00435i	−1.59842	+	2.00435i
$277$	0.400969	−	1.75676i	0.400969	−	1.75676i	−0.222521	−	0.974928i	$-0.571429\pi$
$277$	0.400969	−	1.75676i	0.400969	−	1.75676i	0.623490	−	0.781831i	$-0.285714\pi$
$278$	−1.15834	−	1.15834i	−1.15834	−	1.15834i
$279$	−1.40532	−	0.158342i	−1.40532	−	0.158342i
$280$		0			0
$281$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$281$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$282$	−2.33921	−	2.93327i	−2.33921	−	2.93327i
$283$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$283$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$284$	−0.965245	−	2.00435i	−0.965245	−	2.00435i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	2.57482	+	0.900969i	2.57482	+	0.900969i
$289$			1.00000i			1.00000i
$290$		0			0
$291$		0			0
$292$	−4.80917	+	1.68280i	−4.80917	+	1.68280i
$293$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$293$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$294$	−0.623490	−	1.78183i	−0.623490	−	1.78183i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.400969	+	1.75676i	0.400969	+	1.75676i
$300$	1.11233	−	2.30978i	1.11233	−	2.30978i
$301$		0			0
$302$	−2.66971	+	2.66971i	−2.66971	+	2.66971i
$303$	1.05737	−	1.68280i	1.05737	−	1.68280i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	0.158342	−	0.158342i	0.158342	−	0.158342i	−0.623490	−	0.781831i	$-0.714286\pi$
$307$	0.158342	−	0.158342i	0.158342	−	0.158342i	0.781831	+	0.623490i	$0.214286\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.00435	+	1.59842i	−1.00435	+	1.59842i	−0.222521	+	0.974928i	$0.571429\pi$
$311$	−1.00435	+	1.59842i	−1.00435	+	1.59842i	−0.781831	+	0.623490i	$0.785714\pi$
$312$	4.50374	−	2.82989i	4.50374	−	2.82989i
$313$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$313$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.78183	+	0.623490i	−1.78183	+	0.623490i	−0.781831	+	0.623490i	$0.785714\pi$
$317$	−1.78183	+	0.623490i	−1.78183	+	0.623490i	−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−2.49939	−	0.570469i	−2.49939	−	0.570469i
$325$	−0.781831	−	1.62349i	−0.781831	−	1.62349i
$326$	2.93327	+	2.33921i	2.93327	+	2.33921i
$327$		0			0
$328$	3.06219	−	0.698924i	3.06219	−	0.698924i
$329$		0			0
$330$		0			0
$331$	−1.19745	−	1.19745i	−1.19745	−	1.19745i	−0.974928	−	0.222521i	$-0.928571\pi$
$331$	−1.19745	−	1.19745i	−1.19745	−	1.19745i	−0.222521	−	0.974928i	$-0.571429\pi$
$332$		0			0
$333$		0			0
$334$	3.21075	+	1.12349i	3.21075	+	1.12349i
$335$		0			0
$336$		0			0
$337$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$337$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$338$	0.474928	−	4.21510i	0.474928	−	4.21510i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.277479	−	0.792990i	−0.277479	−	0.792990i
$347$	1.94986			1.94986			0.974928	−	0.222521i	$-0.0714286\pi$
$347$	1.94986			1.94986			0.974928	+	0.222521i	$0.0714286\pi$
$348$	−2.30978	−	1.11233i	−2.30978	−	1.11233i
$349$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$349$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$350$		0			0
$351$	−1.40881	+	1.12349i	−1.40881	+	1.12349i
$352$		0			0
$353$	−1.75676	−	0.846011i	−1.75676	−	0.846011i	−0.974928	−	0.222521i	$-0.928571\pi$
$353$	−1.75676	−	0.846011i	−1.75676	−	0.846011i	−0.781831	−	0.623490i	$-0.785714\pi$
$354$	−0.263565	−	2.33921i	−0.263565	−	2.33921i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.0940653	+	0.834853i	−0.0940653	+	0.834853i
$359$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$359$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$360$		0			0
$361$	0.974928	+	0.222521i	0.974928	+	0.222521i
$362$		0			0
$363$	0.781831	+	0.623490i	0.781831	+	0.623490i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$367$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$368$	−2.93327	+	0.669500i	−2.93327	+	0.669500i
$369$	−1.00435	+	0.351438i	−1.00435	+	0.351438i
$370$		0			0
$371$		0			0
$372$	−2.56366	−	2.56366i	−2.56366	−	2.56366i
$373$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$373$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$374$		0			0
$375$		0			0
$376$		−	5.86654i		−	5.86654i
$377$	−1.62349	+	0.781831i	−1.62349	+	0.781831i
$378$		0			0
$379$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$379$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$380$		0			0
$381$	1.00435	−	0.351438i	1.00435	−	0.351438i
$382$		0			0
$383$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$383$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$384$	0.698924	+	1.11233i	0.698924	+	1.11233i
$385$		0			0
$386$	−0.0940653	−	0.412127i	−0.0940653	−	0.412127i
$387$		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
$392$	0.974928	−	2.78618i	0.974928	−	2.78618i
$393$	0.559311	+	0.351438i	0.559311	+	0.351438i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.433884	+	1.90097i	0.433884	+	1.90097i	0.433884	+	0.900969i	$0.357143\pi$
$397$	0.433884	+	1.90097i	0.433884	+	1.90097i			1.00000i	$0.500000\pi$
$398$		0			0
$399$		0			0
$400$	2.71075	−	1.30543i	2.71075	−	1.30543i
$401$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$401$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$402$		0			0
$403$	−2.53230	+	0.285322i	−2.53230	+	0.285322i
$404$	4.80917	−	1.68280i	4.80917	−	1.68280i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.119137	−	1.05737i	−0.119137	−	1.05737i	−0.900969	−	0.433884i	$-0.857143\pi$
$409$	−0.119137	−	1.05737i	−0.119137	−	1.05737i	0.781831	−	0.623490i	$-0.214286\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	1.78183	−	0.623490i	1.78183	−	0.623490i
$415$		0			0
$416$	4.88460	+	0.550362i	4.88460	+	0.550362i
$417$	0.193096	+	0.846011i	0.193096	+	0.846011i
$418$		0			0
$419$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$419$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$420$		0			0
$421$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$421$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$422$	2.60277	+	0.594065i	2.60277	+	0.594065i
$423$	0.222521	+	1.97493i	0.222521	+	1.97493i
$424$		0			0
$425$		0			0
$426$	−0.183414	+	1.62784i	−0.183414	+	1.62784i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$431$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$432$	−1.87590	−	2.35230i	−1.87590	−	2.35230i
$433$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$433$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	3.65773	+	0.834853i	3.65773	+	0.834853i
$439$	1.52446	−	1.21572i	1.52446	−	1.21572i	0.623490	−	0.781831i	$-0.285714\pi$
$439$	1.52446	−	1.21572i	1.52446	−	1.21572i	0.900969	−	0.433884i	$-0.142857\pi$
$440$		0			0
$441$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$442$		0			0
$443$	0.189606	+	0.119137i	0.189606	+	0.119137i	0.623490	−	0.781831i	$-0.285714\pi$
$443$	0.189606	+	0.119137i	0.189606	+	0.119137i	−0.433884	+	0.900969i	$0.642857\pi$
$444$		0			0
$445$		0			0
$446$	−0.711363	+	0.446979i	−0.711363	+	0.446979i
$447$		0			0
$448$		0			0
$449$	0.623490	+	0.218169i	0.623490	+	0.218169i	0.623490	−	0.781831i	$-0.285714\pi$
$449$	0.623490	+	0.218169i	0.623490	+	0.218169i			1.00000i	$0.5\pi$
$450$	−1.59842	+	1.00435i	−1.59842	+	1.00435i
$451$		0			0
$452$		0			0
$453$	1.94986	−	0.445042i	1.94986	−	0.445042i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$457$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−0.189606	−	1.68280i	−0.189606	−	1.68280i	−0.623490	−	0.781831i	$-0.714286\pi$
$461$	−0.189606	−	1.68280i	−0.189606	−	1.68280i	0.433884	−	0.900969i	$-0.357143\pi$
$462$		0			0
$463$			1.80194i			1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$463$			1.80194i			1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$464$	−1.30543	−	2.71075i	−1.30543	−	2.71075i
$465$		0			0
$466$	−0.792990	+	0.277479i	−0.792990	+	0.277479i
$467$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$467$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$468$	−4.61956			−4.61956
$469$		0			0
$470$		0			0
$471$		0			0
$472$	1.95834	−	3.11668i	1.95834	−	3.11668i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	1.21572	−	3.47431i	1.21572	−	3.47431i
$479$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$479$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.570469	+	2.49939i	0.570469	+	2.49939i
$485$		0			0
$486$	1.33485	+	1.33485i	1.33485	+	1.33485i
$487$	−1.75676	+	0.846011i	−1.75676	+	0.846011i	−0.781831	+	0.623490i	$0.785714\pi$
$487$	−1.75676	+	0.846011i	−1.75676	+	0.846011i	−0.974928	+	0.222521i	$0.928571\pi$
$488$		0			0
$489$	−0.656405	−	1.87590i	−0.656405	−	1.87590i
$490$		0			0
$491$	−1.59842	+	0.559311i	−1.59842	+	0.559311i	−0.974928	−	0.222521i	$-0.928571\pi$
$491$	−1.59842	+	0.559311i	−1.59842	+	0.559311i	−0.623490	+	0.781831i	$0.714286\pi$
$492$	−2.57482	−	0.900969i	−2.57482	−	0.900969i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.476403	−	4.22820i	−0.476403	−	4.22820i
$497$		0			0
$498$		0			0
$499$	0.541044	+	1.12349i	0.541044	+	1.12349i	0.974928	+	0.222521i	$0.0714286\pi$
$499$	0.541044	+	1.12349i	0.541044	+	1.12349i	−0.433884	+	0.900969i	$0.642857\pi$
$500$		0			0
$501$	−1.12349	−	1.40881i	−1.12349	−	1.40881i
$502$		0			0
$503$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$503$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.40097	+	1.75676i	−1.40097	+	1.75676i
$508$	2.57482	+	0.900969i	2.57482	+	0.900969i
$509$	1.52446	+	0.347948i	1.52446	+	0.347948i	0.900969	−	0.433884i	$-0.142857\pi$
$509$	1.52446	+	0.347948i	1.52446	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$510$		0			0
$511$		0			0
$512$	0.0754345	−	0.669500i	0.0754345	−	0.669500i
$513$		0			0
$514$	−2.49939	−	1.57047i	−2.49939	−	1.57047i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−0.0990311	+	0.433884i	−0.0990311	+	0.433884i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	1.00435	+	1.59842i	1.00435	+	1.59842i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	0.559311	+	1.59842i	0.559311	+	1.59842i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$530$		0			0
$531$	−0.541044	+	1.12349i	−0.541044	+	1.12349i
$532$		0			0
$533$	−1.62349	+	1.02011i	−1.62349	+	1.02011i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.277479	−	0.347948i	0.277479	−	0.347948i
$538$	0.446979	−	1.95834i	0.446979	−	1.95834i
$539$		0			0
$540$		0			0
$541$	0.222521	+	0.0250721i	0.222521	+	0.0250721i	0.222521	−	0.974928i	$-0.428571\pi$
$541$	0.222521	+	0.0250721i	0.222521	+	0.0250721i			1.00000i	$0.5\pi$
$542$	−1.21572	+	0.277479i	−1.21572	+	0.277479i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.277479	+	0.347948i	0.277479	+	0.347948i	0.900969	−	0.433884i	$-0.142857\pi$
$547$	0.277479	+	0.347948i	0.277479	+	0.347948i	−0.623490	+	0.781831i	$0.714286\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	2.78618	+	0.974928i	2.78618	+	0.974928i
$553$		0			0
$554$	−3.38025	+	0.380863i	−3.38025	+	0.380863i
$555$		0			0
$556$	−0.965245	+	2.00435i	−0.965245	+	2.00435i
$557$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$557$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$558$	0.594065	+	2.60277i	0.594065	+	2.60277i
$559$		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	$0.750000\pi$
$564$	−2.71075	+	4.31413i	−2.71075	+	4.31413i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−1.81126	+	1.81126i	−1.81126	+	1.81126i
$569$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$569$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$570$		0			0
$571$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$571$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.433884	−	0.900969i	0.433884	−	0.900969i
$576$		−	2.14094i		−	2.14094i
$577$	1.68280	−	0.189606i	1.68280	−	0.189606i	0.781831	−	0.623490i	$-0.214286\pi$
$577$	1.68280	−	0.189606i	1.68280	−	0.189606i	0.900969	+	0.433884i	$0.142857\pi$
$578$	1.78183	−	0.623490i	1.78183	−	0.623490i
$579$	−0.0739590	+	0.211363i	−0.0739590	+	0.211363i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	3.65773	+	4.58665i	3.65773	+	4.58665i
$585$		0			0
$586$		0			0
$587$	−1.52446	−	1.21572i	−1.52446	−	1.21572i	−0.900969	−	0.433884i	$-0.857143\pi$
$587$	−1.52446	−	1.21572i	−1.52446	−	1.21572i	−0.623490	−	0.781831i	$-0.714286\pi$
$588$	−2.00435	+	1.59842i	−2.00435	+	1.59842i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.433884	−	1.90097i	0.433884	−	1.90097i		−	1.00000i	$-0.5\pi$
$593$	0.433884	−	1.90097i	0.433884	−	1.90097i	0.433884	−	0.900969i	$-0.357143\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	2.88025	−	1.80978i	2.88025	−	1.80978i
$599$	−0.0739590	+	0.656405i	−0.0739590	+	0.656405i	0.900969	+	0.433884i	$0.142857\pi$
$599$	−0.0739590	+	0.656405i	−0.0739590	+	0.656405i	−0.974928	+	0.222521i	$0.928571\pi$
$600$	−2.93327	−	0.330500i	−2.93327	−	0.330500i
$601$	1.19745	+	0.752407i	1.19745	+	0.752407i	0.974928	−	0.222521i	$-0.0714286\pi$
$601$	1.19745	+	0.752407i	1.19745	+	0.752407i	0.222521	+	0.974928i	$0.428571\pi$
$602$		0			0
$603$		0			0
$604$	4.61956	+	2.22466i	4.61956	+	2.22466i
$605$		0			0
$606$	−3.65773	−	0.834853i	−3.65773	−	0.834853i
$607$	−0.559311	−	1.59842i	−0.559311	−	1.59842i	−0.781831	−	0.623490i	$-0.785714\pi$
$607$	−0.559311	−	1.59842i	−0.559311	−	1.59842i	0.222521	−	0.974928i	$-0.428571\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.18280	+	3.38025i	1.18280	+	3.38025i
$612$		0			0
$613$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$613$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$614$	−0.380863	−	0.183414i	−0.380863	−	0.183414i
$615$		0			0
$616$		0			0
$617$		0			0		−0.846724	−	0.532032i	$-0.821429\pi$
$617$		0			0		0.846724	+	0.532032i	$0.178571\pi$
$618$		0			0
$619$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$619$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$620$		0			0
$621$	−0.974928	−	0.222521i	−0.974928	−	0.222521i
$622$	3.47431	+	0.792990i	3.47431	+	0.792990i
$623$		0			0
$624$	−4.23870	−	3.38025i	−4.23870	−	3.38025i
$625$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$631$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$632$		0			0
$633$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$634$	2.22191	+	2.78618i	2.22191	+	2.78618i
$635$		0			0
$636$		0			0
$637$			1.80194i			1.80194i
$638$		0			0
$639$	0.541044	−	0.678448i	0.541044	−	0.678448i
$640$		0			0
$641$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$641$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$642$		0			0
$643$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$643$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.347948	−	1.52446i	−0.347948	−	1.52446i	−0.781831	−	0.623490i	$-0.785714\pi$
$647$	−0.347948	−	1.52446i	−0.347948	−	1.52446i	0.433884	−	0.900969i	$-0.357143\pi$
$648$	0.330500	+	2.93327i	0.330500	+	2.93327i
$649$		0			0
$650$	−2.40532	+	2.40532i	−2.40532	+	2.40532i
$651$		0			0
$652$	1.68280	−	4.80917i	1.68280	−	4.80917i
$653$	−0.467085	+	1.33485i	−0.467085	+	1.33485i	0.433884	+	0.900969i	$0.357143\pi$
$653$	−0.467085	+	1.33485i	−0.467085	+	1.33485i	−0.900969	+	0.433884i	$0.857143\pi$
$654$		0			0
$655$		0			0
$656$	−1.70328	−	2.71075i	−1.70328	−	2.71075i
$657$	−1.40532	−	1.40532i	−1.40532	−	1.40532i
$658$		0			0
$659$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$659$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$660$		0			0
$661$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$661$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$662$	−1.38705	+	2.88025i	−1.38705	+	2.88025i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$668$		−	4.61956i		−	4.61956i
$669$	0.445042			0.445042
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.846011	+	1.75676i	0.846011	+	1.75676i	0.623490	+	0.781831i	$0.285714\pi$
$673$	0.846011	+	1.75676i	0.846011	+	1.75676i	0.222521	+	0.974928i	$0.428571\pi$
$674$		0			0
$675$	1.00000			1.00000
$676$	−5.61607	+	1.28183i	−5.61607	+	1.28183i
$677$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$677$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.52446	−	0.347948i	−1.52446	−	0.347948i	−0.623490	−	0.781831i	$-0.714286\pi$
$683$	−1.52446	−	0.347948i	−1.52446	−	0.347948i	−0.900969	+	0.433884i	$0.857143\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.781831	−	0.376510i	−0.781831	−	0.376510i		−	1.00000i	$-0.5\pi$
$691$	−0.781831	−	0.376510i	−0.781831	−	0.376510i	−0.781831	+	0.623490i	$0.785714\pi$
$692$	−0.892021	+	0.711363i	−0.892021	+	0.711363i
$693$		0			0
$694$	−1.21572	−	3.47431i	−1.21572	−	3.47431i
$695$		0			0
$696$	−0.330500	+	2.93327i	−0.330500	+	2.93327i
$697$		0			0
$698$	−0.777479	−	2.22191i	−0.777479	−	2.22191i
$699$	0.433884	+	0.0990311i	0.433884	+	0.0990311i
$700$		0			0
$701$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$701$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$702$	2.88025	+	1.80978i	2.88025	+	1.80978i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.412127	+	3.65773i	−0.412127	+	3.65773i
$707$		0			0
$708$	−2.88025	+	1.38705i	−2.88025	+	1.38705i
$709$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$709$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$714$		0			0
$715$		0			0
$716$	1.11233	−	0.253882i	1.11233	−	0.253882i
$717$	−1.52446	+	1.21572i	−1.52446	+	1.21572i
$718$		0			0
$719$	0.376510	+	0.781831i	0.376510	+	0.781831i	1.00000			$0$
$719$	0.376510	+	0.781831i	0.376510	+	0.781831i	−0.623490	+	0.781831i	$0.714286\pi$
$720$		0			0
$721$		0			0
$722$	−0.211363	−	1.87590i	−0.211363	−	1.87590i
$723$		0			0
$724$		0			0
$725$	0.974928	+	0.222521i	0.974928	+	0.222521i
$726$	0.623490	−	1.78183i	0.623490	−	1.78183i
$727$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$727$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$728$		0			0
$729$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$733$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$734$		0			0
$735$		0			0
$736$	1.45133	+	2.30978i	1.45133	+	2.30978i
$737$		0			0
$738$	1.25241	+	1.57047i	1.25241	+	1.57047i
$739$	0.351438	−	1.00435i	0.351438	−	1.00435i	−0.623490	−	0.781831i	$-0.714286\pi$
$739$	0.351438	−	1.00435i	0.351438	−	1.00435i	0.974928	−	0.222521i	$-0.0714286\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$743$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$744$	−1.81126	+	3.76111i	−1.81126	+	3.76111i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$751$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$752$	−5.64402	+	1.97493i	−5.64402	+	1.97493i
$753$		0			0
$754$	2.40532	+	2.40532i	2.40532	+	2.40532i
$755$		0			0
$756$		0			0
$757$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$757$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.40881	−	1.12349i	−1.40881	−	1.12349i	−0.974928	−	0.222521i	$-0.928571\pi$
$761$	−1.40881	−	1.12349i	−1.40881	−	1.12349i	−0.433884	−	0.900969i	$-0.642857\pi$
$762$	−1.25241	−	1.57047i	−1.25241	−	1.57047i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.500000	+	2.19064i	−0.500000	+	2.19064i
$768$	0.211363	−	0.265041i	0.211363	−	0.265041i
$769$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$769$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$770$		0			0
$771$	0.678448	+	1.40881i	0.678448	+	1.40881i
$772$	−0.486086	+	0.305428i	−0.486086	+	0.305428i
$773$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$773$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$774$		0			0
$775$	1.19745	+	0.752407i	1.19745	+	0.752407i
$776$		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
$784$	−3.00870			−3.00870
$785$		0			0
$786$	0.277479	−	1.21572i	0.277479	−	1.21572i
$787$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$787$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	3.11668	−	1.95834i	3.11668	−	1.95834i
$795$		0			0
$796$		0			0
$797$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$797$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$798$		0			0
$799$		0			0
$800$	−1.92892	−	1.92892i	−1.92892	−	1.92892i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	2.08726	+	4.33424i	2.08726	+	4.33424i
$807$	−0.752407	+	0.752407i	−0.752407	+	0.752407i
$808$	−3.65773	−	4.58665i	−3.65773	−	4.58665i
$809$	0.189606	+	1.68280i	0.189606	+	1.68280i	0.623490	+	0.781831i	$0.285714\pi$
$809$	0.189606	+	1.68280i	0.189606	+	1.68280i	−0.433884	+	0.900969i	$0.642857\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	0.623490	+	0.218169i	0.623490	+	0.218169i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.80978	+	0.871544i	−1.80978	+	0.871544i
$819$		0			0
$820$		0			0
$821$	0.347948	+	1.52446i	0.347948	+	1.52446i	0.781831	+	0.623490i	$0.214286\pi$
$821$	0.347948	+	1.52446i	0.347948	+	1.52446i	−0.433884	+	0.900969i	$0.642857\pi$
$822$		0			0
$823$	−0.900969	−	1.43388i	−0.900969	−	1.43388i	−0.900969	−	0.433884i	$-0.857143\pi$
$823$	−0.900969	−	1.43388i	−0.900969	−	1.43388i		−	1.00000i	$-0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$827$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$828$	−1.59842	−	2.00435i	−1.59842	−	2.00435i
$829$	−1.33485	+	1.33485i	−1.33485	+	1.33485i	−0.433884	+	0.900969i	$0.642857\pi$
$829$	−1.33485	+	1.33485i	−1.33485	+	1.33485i	−0.900969	+	0.433884i	$0.857143\pi$
$830$		0			0
$831$	1.62349	+	0.781831i	1.62349	+	0.781831i
$832$	−0.858449	−	3.76111i	−0.858449	−	3.76111i
$833$		0			0
$834$	1.38705	−	0.871544i	1.38705	−	0.871544i
$835$		0			0
$836$		0			0
$837$	0.467085	−	1.33485i	0.467085	−	1.33485i
$838$		0			0
$839$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$839$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$840$		0			0
$841$	0.222521	−	0.974928i	0.222521	−	0.974928i
$842$		0			0
$843$		0			0
$844$	−0.405935	−	3.60277i	−0.405935	−	3.60277i
$845$		0			0
$846$	3.38025	−	1.62784i	3.38025	−	1.62784i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	2.16889	−	0.495034i	2.16889	−	0.495034i
$853$	−1.40532	−	1.40532i	−1.40532	−	1.40532i	−0.781831	−	0.623490i	$-0.785714\pi$
$853$	−1.40532	−	1.40532i	−1.40532	−	1.40532i	−0.623490	−	0.781831i	$-0.714286\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.94986	+	0.445042i	1.94986	+	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$857$	1.94986	+	0.445042i	1.94986	+	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$858$		0			0
$859$	1.43388	−	0.900969i	1.43388	−	0.900969i	0.433884	−	0.900969i	$-0.357143\pi$
$859$	1.43388	−	0.900969i	1.43388	−	0.900969i	1.00000			$0$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.24698	−	1.56366i	1.24698	−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$863$	1.24698	−	1.56366i	1.24698	−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$864$	−1.45133	+	2.30978i	−1.45133	+	2.30978i
$865$		0			0
$866$		0			0
$867$	−0.974928	−	0.222521i	−0.974928	−	0.222521i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−0.570469	−	5.06305i	−0.570469	−	5.06305i
$877$	0.974928	−	1.22252i	0.974928	−	1.22252i		−	1.00000i	$-0.5\pi$
$877$	0.974928	−	1.22252i	0.974928	−	1.22252i	0.974928	−	0.222521i	$-0.0714286\pi$
$878$	−3.11668	−	1.95834i	−3.11668	−	1.95834i
$879$		0			0
$880$		0			0
$881$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$881$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$882$	1.87590	−	0.211363i	1.87590	−	0.211363i
$883$	−1.21572	−	0.277479i	−1.21572	−	0.277479i	−0.433884	−	0.900969i	$-0.642857\pi$
$883$	−1.21572	−	0.277479i	−1.21572	−	0.277479i	−0.781831	+	0.623490i	$0.785714\pi$
$884$		0			0
$885$		0			0
$886$	0.0940653	−	0.412127i	0.0940653	−	0.412127i
$887$	−0.752407	−	0.752407i	−0.752407	−	0.752407i	0.222521	−	0.974928i	$-0.428571\pi$
$887$	−0.752407	−	0.752407i	−0.752407	−	0.752407i	−0.974928	+	0.222521i	$0.928571\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	0.892021	+	0.711363i	0.892021	+	0.711363i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1.80194			−1.80194
$898$		−	1.24698i		−	1.24698i
$899$	0.752407	−	1.19745i	0.752407	−	1.19745i
$900$	2.00435	+	1.59842i	2.00435	+	1.59842i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−2.00870	−	3.19684i	−2.00870	−	3.19684i
$907$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$907$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$908$		0			0
$909$	1.40532	+	1.40532i	1.40532	+	1.40532i
$910$		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		−0.222521	−	0.974928i	$-0.571429\pi$
$919$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$920$		0			0
$921$	0.119137	+	0.189606i	0.119137	+	0.189606i
$922$	−2.88025	+	1.38705i	−2.88025	+	1.38705i
$923$	0.678448	−	1.40881i	0.678448	−	1.40881i
$924$		0			0
$925$		0			0
$926$	3.21075	−	1.12349i	3.21075	−	1.12349i
$927$		0			0
$928$	−1.92892	+	1.92892i	−1.92892	+	1.92892i
$929$			1.80194i			1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$929$			1.80194i			1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$930$		0			0
$931$		0			0
$932$	0.711363	+	0.892021i	0.711363	+	0.892021i
$933$	−1.33485	−	1.33485i	−1.33485	−	1.33485i
$934$		0			0
$935$		0			0
$936$	1.75676	+	5.02053i	1.75676	+	5.02053i
$937$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$937$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$941$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$942$		0			0
$943$	−1.00435	−	0.351438i	−1.00435	−	0.351438i
$944$	−3.65773	−	0.834853i	−3.65773	−	0.834853i
$945$		0			0
$946$		0			0
$947$	−0.119137	+	1.05737i	−0.119137	+	1.05737i	0.781831	+	0.623490i	$0.214286\pi$
$947$	−0.119137	+	1.05737i	−0.119137	+	1.05737i	−0.900969	+	0.433884i	$0.857143\pi$
$948$		0			0
$949$	−3.03230	−	1.90532i	−3.03230	−	1.90532i
$950$		0			0
$951$	−0.211363	−	1.87590i	−0.211363	−	1.87590i
$952$		0			0
$953$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$953$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$954$		0			0
$955$		0			0
$956$	−4.99877			−4.99877
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.781831	−	0.623490i	0.781831	−	0.623490i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.0250721	+	0.222521i	−0.0250721	+	0.222521i	0.974928	+	0.222521i	$0.0714286\pi$
$967$	−0.0250721	+	0.222521i	−0.0250721	+	0.222521i	−1.00000			$\pi$
$968$	2.49939	−	1.57047i	2.49939	−	1.57047i
$969$		0			0
$970$		0			0
$971$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$971$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$972$	1.11233	−	2.30978i	1.11233	−	2.30978i
$973$		0			0
$974$	2.60277	+	2.60277i	2.60277	+	2.60277i
$975$	1.75676	−	0.400969i	1.75676	−	0.400969i
$976$		0			0
$977$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$977$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$978$	−2.93327	+	2.33921i	−2.93327	+	2.33921i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	1.99319	+	2.49939i	1.99319	+	2.49939i
$983$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$983$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$984$			3.14094i			3.14094i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.846011	−	1.75676i	0.846011	−	1.75676i	0.222521	−	0.974928i	$-0.428571\pi$
$991$	0.846011	−	1.75676i	0.846011	−	1.75676i	0.623490	−	0.781831i	$-0.285714\pi$
$992$	−3.47579	+	1.67385i	−3.47579	+	1.67385i
$993$	1.43388	−	0.900969i	1.43388	−	0.900969i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.351438	−	0.559311i	−0.351438	−	0.559311i	0.623490	−	0.781831i	$-0.285714\pi$
$997$	−0.351438	−	0.559311i	−0.351438	−	0.559311i	−0.974928	+	0.222521i	$0.928571\pi$
$998$	1.66453	−	1.66453i	1.66453	−	1.66453i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2001.1.bf.b.620.1	yes	12
3.2	odd	2		2001.1.bf.a.620.1	✓	12
23.22	odd	2	CM	2001.1.bf.b.620.1	yes	12
29.8	odd	28		2001.1.bf.a.965.1	yes	12
69.68	even	2		2001.1.bf.a.620.1	✓	12
87.8	even	28	inner	2001.1.bf.b.965.1	yes	12
667.298	even	28		2001.1.bf.a.965.1	yes	12
2001.965	odd	28	inner	2001.1.bf.b.965.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.1.bf.a.620.1	✓	12	3.2	odd	2
2001.1.bf.a.620.1	✓	12	69.68	even	2
2001.1.bf.a.965.1	yes	12	29.8	odd	28
2001.1.bf.a.965.1	yes	12	667.298	even	28
2001.1.bf.b.620.1	yes	12	1.1	even	1	trivial
2001.1.bf.b.620.1	yes	12	23.22	odd	2	CM
2001.1.bf.b.965.1	yes	12	87.8	even	28	inner
2001.1.bf.b.965.1	yes	12	2001.965	odd	28	inner

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)

\(f(q)\)	\(=\)	\(q+(-0.623490 - 1.78183i) q^{2} +(-0.222521 + 0.974928i) q^{3} +(-2.00435 + 1.59842i) q^{4} +(1.87590 - 0.211363i) q^{6} +(2.49939 + 1.57047i) q^{8} +(-0.900969 - 0.433884i) q^{9} +O(q^{10})\)	\(q+(-0.623490 - 1.78183i) q^{2} +(-0.222521 + 0.974928i) q^{3} +(-2.00435 + 1.59842i) q^{4} +(1.87590 - 0.211363i) q^{6} +(2.49939 + 1.57047i) q^{8} +(-0.900969 - 0.433884i) q^{9} +(-1.11233 - 2.30978i) q^{12} +(-1.75676 - 0.400969i) q^{13} +(0.669500 - 2.93327i) q^{16} +(-0.211363 + 1.87590i) q^{18} +(-0.433884 - 0.900969i) q^{23} +(-2.08726 + 2.08726i) q^{24} +(0.623490 + 0.781831i) q^{25} +(0.380863 + 3.38025i) q^{26} +(0.623490 - 0.781831i) q^{27} +(0.781831 - 0.623490i) q^{29} +(1.33485 - 0.467085i) q^{31} +(-2.71075 + 0.305428i) q^{32} +(2.49939 - 0.570469i) q^{36} +(0.781831 - 1.62349i) q^{39} +(0.752407 - 0.752407i) q^{41} +(-1.33485 + 1.33485i) q^{46} +(-1.05737 - 1.68280i) q^{47} +(2.71075 + 1.30543i) q^{48} +(-0.222521 - 0.974928i) q^{49} +(1.00435 - 1.59842i) q^{50} +(4.16208 - 2.00435i) q^{52} +(-1.78183 - 0.623490i) q^{54} +(-1.59842 - 1.00435i) q^{58} -1.24698i q^{59} +(-1.66453 - 2.08726i) q^{62} +(0.928918 + 1.92892i) q^{64} +(0.974928 - 0.222521i) q^{69} +(-0.193096 + 0.846011i) q^{71} +(-1.57047 - 2.49939i) q^{72} +(1.87590 + 0.656405i) q^{73} +(-0.900969 + 0.433884i) q^{75} +(-3.38025 - 0.380863i) q^{78} +(0.623490 + 0.781831i) q^{81} +(-1.80978 - 0.871544i) q^{82} +(0.433884 + 0.900969i) q^{87} +(2.30978 + 1.11233i) q^{92} +(0.158342 + 1.40532i) q^{93} +(-2.33921 + 2.93327i) q^{94} +(0.305428 - 2.71075i) q^{96} +(-1.59842 + 1.00435i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{2} - 2 q^{3} - 14 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 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