LMFDB - Embedded newform 2008.1.w.a.1067.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2008,1,Mod(51,2008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2008, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([25, 25, 18]))

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

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

chi := DirichletCharacter("2008.51");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2008 = 2^{3} \cdot 251 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2008.w (of order $50$, degree $20$, 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:	$1.00212254537$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{50})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{15} + x^{10} - x^{5} + 1 $ x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{25}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{25} - \cdots)$

Embedding invariants

Embedding label			1067.1
Root			$0.425779 - 0.904827i$ of defining polynomial
Character	$\chi$	$=$	2008.1067
Dual form			2008.1.w.a.1259.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.809017 - 0.587785i) q^{2} +(-0.824805 + 0.211774i) q^{3} +(0.309017 + 0.951057i) q^{4} +(0.791759 + 0.313480i) q^{6} +(0.309017 - 0.951057i) q^{8} +(-0.240851 + 0.132409i) q^{9} +O(q^{10})$	\(q+(-0.809017 - 0.587785i) q^{2} +(-0.824805 + 0.211774i) q^{3} +(0.309017 + 0.951057i) q^{4} +(0.791759 + 0.313480i) q^{6} +(0.309017 - 0.951057i) q^{8} +(-0.240851 + 0.132409i) q^{9} +(0.0672897 + 0.106032i) q^{11} +(-0.456288 - 0.718995i) q^{12} +(-0.809017 + 0.587785i) q^{16} +(-1.85955 + 0.736249i) q^{17} +(0.272681 + 0.0344476i) q^{18} +(0.303189 - 1.58937i) q^{19} +(0.00788530 - 0.125333i) q^{22} +(-0.0534698 + 0.849878i) q^{24} +(-0.809017 - 0.587785i) q^{25} +(0.791374 - 0.743150i) q^{27} +1.00000 q^{32} +(-0.0779556 - 0.0732052i) q^{33} +(1.93717 + 0.497380i) q^{34} +(-0.200356 - 0.188146i) q^{36} +(-1.17950 + 1.10762i) q^{38} +(0.0915446 + 1.45506i) q^{41} +(0.791759 - 1.68257i) q^{43} +(-0.0800484 + 0.0967619i) q^{44} +(0.542804 - 0.656137i) q^{48} +(0.0627905 - 0.998027i) q^{49} +(0.309017 + 0.951057i) q^{50} +(1.37785 - 1.00107i) q^{51} +(-1.07705 + 0.136063i) q^{54} +(0.0865160 + 1.37513i) q^{57} +(-0.996398 - 0.394502i) q^{59} +(-0.809017 - 0.587785i) q^{64} +(0.0200385 + 0.105045i) q^{66} +(-0.0235315 - 0.374023i) q^{67} +(-1.27485 - 1.54103i) q^{68} +(0.0515014 + 0.269980i) q^{72} +(-1.56720 - 0.402389i) q^{73} +(0.791759 + 0.313480i) q^{75} +(1.60528 - 0.202793i) q^{76} +(-0.348079 + 0.548484i) q^{81} +(0.781202 - 1.23098i) q^{82} +(-1.23480 - 1.49261i) q^{83} +(-1.62954 + 0.895846i) q^{86} +(0.121636 - 0.0312307i) q^{88} +(1.84489 - 0.730444i) q^{89} +(-0.824805 + 0.211774i) q^{96} +(-0.929324 - 1.12336i) q^{97} +(-0.637424 + 0.770513i) q^{98} +(-0.0302463 - 0.0166281i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 5 q^{2} - 5 q^{4} - 5 q^{8}+O(q^{10}) $ 20 * q - 5 * q^2 - 5 * q^4 - 5 * q^8	$ 20 q - 5 q^{2} - 5 q^{4} - 5 q^{8} - 5 q^{11} - 5 q^{16} + 20 q^{22} - 5 q^{25} - 5 q^{27} + 20 q^{32} - 5 q^{33} - 5 q^{44} - 5 q^{50} - 5 q^{54} - 5 q^{59} - 5 q^{64} - 5 q^{66} - 5 q^{81} - 5 q^{83} - 5 q^{88} - 5 q^{99}+O(q^{100}) $ 20 * q - 5 * q^2 - 5 * q^4 - 5 * q^8 - 5 * q^11 - 5 * q^16 + 20 * q^22 - 5 * q^25 - 5 * q^27 + 20 * q^32 - 5 * q^33 - 5 * q^44 - 5 * q^50 - 5 * q^54 - 5 * q^59 - 5 * q^64 - 5 * q^66 - 5 * q^81 - 5 * q^83 - 5 * q^88 - 5 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2008\mathbb{Z}\right)^\times$.

$n$	$257$	$503$	$1005$
$\chi(n)$	$e\left(\frac{3}{25}\right)$	$-1$	$-1$

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 $

$2$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$2$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$3$	−0.824805	+	0.211774i	−0.824805	+	0.211774i	−0.637424	−	0.770513i	$-0.720000\pi$
$3$	−0.824805	+	0.211774i	−0.824805	+	0.211774i	−0.187381	+	0.982287i	$0.560000\pi$
$4$	0.309017	+	0.951057i	0.309017	+	0.951057i
$5$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$5$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$6$	0.791759	+	0.313480i	0.791759	+	0.313480i
$7$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$7$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$8$	0.309017	−	0.951057i	0.309017	−	0.951057i
$9$	−0.240851	+	0.132409i	−0.240851	+	0.132409i
$10$		0			0
$11$	0.0672897	+	0.106032i	0.0672897	+	0.106032i	0.876307	−	0.481754i	$-0.160000\pi$
$11$	0.0672897	+	0.106032i	0.0672897	+	0.106032i	−0.809017	+	0.587785i	$0.800000\pi$
$12$	−0.456288	−	0.718995i	−0.456288	−	0.718995i
$13$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$13$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$14$		0			0
$15$		0			0
$16$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$17$	−1.85955	+	0.736249i	−1.85955	+	0.736249i	−0.929776	+	0.368125i	$0.880000\pi$
$17$	−1.85955	+	0.736249i	−1.85955	+	0.736249i	−0.929776	+	0.368125i	$0.880000\pi$
$18$	0.272681	+	0.0344476i	0.272681	+	0.0344476i
$19$	0.303189	−	1.58937i	0.303189	−	1.58937i	−0.425779	−	0.904827i	$-0.640000\pi$
$19$	0.303189	−	1.58937i	0.303189	−	1.58937i	0.728969	−	0.684547i	$-0.240000\pi$
$20$		0			0
$21$		0			0
$22$	0.00788530	−	0.125333i	0.00788530	−	0.125333i
$23$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$23$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$24$	−0.0534698	+	0.849878i	−0.0534698	+	0.849878i
$25$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$26$		0			0
$27$	0.791374	−	0.743150i	0.791374	−	0.743150i
$28$		0			0
$29$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$29$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$30$		0			0
$31$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$31$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$32$	1.00000			1.00000
$33$	−0.0779556	−	0.0732052i	−0.0779556	−	0.0732052i
$34$	1.93717	+	0.497380i	1.93717	+	0.497380i
$35$		0			0
$36$	−0.200356	−	0.188146i	−0.200356	−	0.188146i
$37$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$37$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$38$	−1.17950	+	1.10762i	−1.17950	+	1.10762i
$39$		0			0
$40$		0			0
$41$	0.0915446	+	1.45506i	0.0915446	+	1.45506i	0.728969	+	0.684547i	$0.240000\pi$
$41$	0.0915446	+	1.45506i	0.0915446	+	1.45506i	−0.637424	+	0.770513i	$0.720000\pi$
$42$		0			0
$43$	0.791759	−	1.68257i	0.791759	−	1.68257i	0.0627905	−	0.998027i	$-0.480000\pi$
$43$	0.791759	−	1.68257i	0.791759	−	1.68257i	0.728969	−	0.684547i	$-0.240000\pi$
$44$	−0.0800484	+	0.0967619i	−0.0800484	+	0.0967619i
$45$		0			0
$46$		0			0
$47$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$47$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$48$	0.542804	−	0.656137i	0.542804	−	0.656137i
$49$	0.0627905	−	0.998027i	0.0627905	−	0.998027i
$50$	0.309017	+	0.951057i	0.309017	+	0.951057i
$51$	1.37785	−	1.00107i	1.37785	−	1.00107i
$52$		0			0
$53$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$53$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$54$	−1.07705	+	0.136063i	−1.07705	+	0.136063i
$55$		0			0
$56$		0			0
$57$	0.0865160	+	1.37513i	0.0865160	+	1.37513i
$58$		0			0
$59$	−0.996398	−	0.394502i	−0.996398	−	0.394502i	−0.187381	−	0.982287i	$-0.560000\pi$
$59$	−0.996398	−	0.394502i	−0.996398	−	0.394502i	−0.809017	+	0.587785i	$0.800000\pi$
$60$		0			0
$61$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$61$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$62$		0			0
$63$		0			0
$64$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$65$		0			0
$66$	0.0200385	+	0.105045i	0.0200385	+	0.105045i
$67$	−0.0235315	−	0.374023i	−0.0235315	−	0.374023i	−0.992115	−	0.125333i	$-0.960000\pi$
$67$	−0.0235315	−	0.374023i	−0.0235315	−	0.374023i	0.968583	−	0.248690i	$-0.0800000\pi$
$68$	−1.27485	−	1.54103i	−1.27485	−	1.54103i
$69$		0			0
$70$		0			0
$71$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$71$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$72$	0.0515014	+	0.269980i	0.0515014	+	0.269980i
$73$	−1.56720	−	0.402389i	−1.56720	−	0.402389i	−0.637424	−	0.770513i	$-0.720000\pi$
$73$	−1.56720	−	0.402389i	−1.56720	−	0.402389i	−0.929776	+	0.368125i	$0.880000\pi$
$74$		0			0
$75$	0.791759	+	0.313480i	0.791759	+	0.313480i
$76$	1.60528	−	0.202793i	1.60528	−	0.202793i
$77$		0			0
$78$		0			0
$79$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$79$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$80$		0			0
$81$	−0.348079	+	0.548484i	−0.348079	+	0.548484i
$82$	0.781202	−	1.23098i	0.781202	−	1.23098i
$83$	−1.23480	−	1.49261i	−1.23480	−	1.49261i	−0.809017	−	0.587785i	$-0.800000\pi$
$83$	−1.23480	−	1.49261i	−1.23480	−	1.49261i	−0.425779	−	0.904827i	$-0.640000\pi$
$84$		0			0
$85$		0			0
$86$	−1.62954	+	0.895846i	−1.62954	+	0.895846i
$87$		0			0
$88$	0.121636	−	0.0312307i	0.121636	−	0.0312307i
$89$	1.84489	−	0.730444i	1.84489	−	0.730444i	0.876307	−	0.481754i	$-0.160000\pi$
$89$	1.84489	−	0.730444i	1.84489	−	0.730444i	0.968583	−	0.248690i	$-0.0800000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−0.824805	+	0.211774i	−0.824805	+	0.211774i
$97$	−0.929324	−	1.12336i	−0.929324	−	1.12336i	−0.992115	−	0.125333i	$-0.960000\pi$
$97$	−0.929324	−	1.12336i	−0.929324	−	1.12336i	0.0627905	−	0.998027i	$-0.480000\pi$
$98$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$99$	−0.0302463	−	0.0166281i	−0.0302463	−	0.0166281i
$100$	0.309017	−	0.951057i	0.309017	−	0.951057i
$101$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$101$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$102$	−1.70312			−1.70312
$103$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$103$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.362576	−	0.770513i	0.362576	−	0.770513i	−0.637424	−	0.770513i	$-0.720000\pi$
$107$	0.362576	−	0.770513i	0.362576	−	0.770513i	1.00000			$0$
$108$	0.951325	+	0.522996i	0.951325	+	0.522996i
$109$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$109$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.85955			−1.85955			−0.929776	−	0.368125i	$-0.880000\pi$
$113$	−1.85955			−1.85955			−0.929776	+	0.368125i	$0.880000\pi$
$114$	0.738289	−	1.16336i	0.738289	−	1.16336i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	0.574221	+	0.904827i	0.574221	+	0.904827i
$119$		0			0
$120$		0			0
$121$	0.419064	−	0.890557i	0.419064	−	0.890557i
$122$		0			0
$123$	−0.383650	−	1.18075i	−0.383650	−	1.18075i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$127$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$128$	0.309017	+	0.951057i	0.309017	+	0.951057i
$129$	−0.296722	+	1.55547i	−0.296722	+	1.55547i
$130$		0			0
$131$	−0.574633	−	0.227513i	−0.574633	−	0.227513i	0.0627905	−	0.998027i	$-0.480000\pi$
$131$	−0.574633	−	0.227513i	−0.574633	−	0.227513i	−0.637424	+	0.770513i	$0.720000\pi$
$132$	0.0455327	−	0.0967619i	0.0455327	−	0.0967619i
$133$		0			0
$134$	−0.200808	+	0.316423i	−0.200808	+	0.316423i
$135$		0			0
$136$	0.125581	+	1.99605i	0.125581	+	1.99605i
$137$	−0.456288	+	0.718995i	−0.456288	+	0.718995i	−0.992115	−	0.125333i	$-0.960000\pi$
$137$	−0.456288	+	0.718995i	−0.456288	+	0.718995i	0.535827	+	0.844328i	$0.320000\pi$
$138$		0			0
$139$	−0.362989	+	1.90285i	−0.362989	+	1.90285i	0.0627905	+	0.998027i	$0.480000\pi$
$139$	−0.362989	+	1.90285i	−0.362989	+	1.90285i	−0.425779	+	0.904827i	$0.640000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.117025	−	0.248690i	0.117025	−	0.248690i
$145$		0			0
$146$	1.03137	+	1.24672i	1.03137	+	1.24672i
$147$	0.159566	+	0.836475i	0.159566	+	0.836475i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−0.456288	−	0.718995i	−0.456288	−	0.718995i
$151$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$151$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$152$	−1.41789	−	0.779494i	−1.41789	−	0.779494i
$153$	0.350389	−	0.423548i	0.350389	−	0.423548i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$157$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.603993	−	0.239138i	0.603993	−	0.239138i
$163$	−1.92189	+	0.493458i	−1.92189	+	0.493458i	−0.929776	+	0.368125i	$0.880000\pi$
$163$	−1.92189	+	0.493458i	−1.92189	+	0.493458i	−0.992115	+	0.125333i	$0.960000\pi$
$164$	−1.35556	+	0.536702i	−1.35556	+	0.536702i
$165$		0			0
$166$	0.121636	+	1.93334i	0.121636	+	1.93334i
$167$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$167$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$168$		0			0
$169$	0.535827	−	0.844328i	0.535827	−	0.844328i
$170$		0			0
$171$	0.137424	+	0.422948i	0.137424	+	0.422948i
$172$	1.84489	+	0.233064i	1.84489	+	0.233064i
$173$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$173$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$174$		0			0
$175$		0			0
$176$	−0.116762	−	0.0462295i	−0.116762	−	0.0462295i
$177$	0.905380	+	0.114376i	0.905380	+	0.114376i
$178$	−1.92189	−	0.493458i	−1.92189	−	0.493458i
$179$	−0.328407	−	1.72157i	−0.328407	−	1.72157i	−0.637424	−	0.770513i	$-0.720000\pi$
$179$	−0.328407	−	1.72157i	−0.328407	−	1.72157i	0.309017	−	0.951057i	$-0.400000\pi$
$180$		0			0
$181$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$181$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.203194	−	0.147629i	−0.203194	−	0.147629i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$191$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$192$	0.791759	+	0.313480i	0.791759	+	0.313480i
$193$	0.939097	+	0.516273i	0.939097	+	0.516273i	0.876307	−	0.481754i	$-0.160000\pi$
$193$	0.939097	+	0.516273i	0.939097	+	0.516273i	0.0627905	+	0.998027i	$0.480000\pi$
$194$	0.0915446	+	1.45506i	0.0915446	+	1.45506i
$195$		0			0
$196$	0.968583	−	0.248690i	0.968583	−	0.248690i
$197$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$197$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$198$	0.0146961	+	0.0312307i	0.0146961	+	0.0312307i
$199$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$199$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$200$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$201$	0.0986173	+	0.303513i	0.0986173	+	0.303513i
$202$		0			0
$203$		0			0
$204$	1.37785	+	1.00107i	1.37785	+	1.00107i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.188925	−	0.0748008i	0.188925	−	0.0748008i
$210$		0			0
$211$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.876307	+	0.481754i	$0.160000\pi$
$211$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.728969	+	0.684547i	$0.240000\pi$
$212$		0			0
$213$		0			0
$214$	−0.746226	+	0.410241i	−0.746226	+	0.410241i
$215$		0			0
$216$	−0.462229	−	0.982287i	−0.462229	−	0.982287i
$217$		0			0
$218$		0			0
$219$	1.37785			1.37785
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$223$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$224$		0			0
$225$	0.272681	+	0.0344476i	0.272681	+	0.0344476i
$226$	1.50441	+	1.09302i	1.50441	+	1.09302i
$227$	0.0388067	−	0.616814i	0.0388067	−	0.616814i	−0.929776	−	0.368125i	$-0.880000\pi$
$227$	0.0388067	−	0.616814i	0.0388067	−	0.616814i	0.968583	−	0.248690i	$-0.0800000\pi$
$228$	−1.28109	+	0.507221i	−1.28109	+	0.507221i
$229$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$229$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.84489	+	0.233064i	1.84489	+	0.233064i	0.968583	−	0.248690i	$-0.0800000\pi$
$233$	1.84489	+	0.233064i	1.84489	+	0.233064i	0.876307	+	0.481754i	$0.160000\pi$
$234$		0			0
$235$		0			0
$236$	0.0672897	−	1.06954i	0.0672897	−	1.06954i
$237$		0			0
$238$		0			0
$239$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$239$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$240$		0			0
$241$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$241$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$242$	−0.862487	+	0.474156i	−0.862487	+	0.474156i
$243$	−0.164529	+	0.506367i	−0.164529	+	0.506367i
$244$		0			0
$245$		0			0
$246$	−0.383650	+	1.18075i	−0.383650	+	1.18075i
$247$		0			0
$248$		0			0
$249$	1.33456	+	0.969617i	1.33456	+	0.969617i
$250$		0			0
$251$	0.728969	+	0.684547i	0.728969	+	0.684547i
$251$	0.728969	+	0.684547i	0.728969	+	0.684547i
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.309017	−	0.951057i	0.309017	−	0.951057i
$257$	−0.996398	−	0.394502i	−0.996398	−	0.394502i	−0.187381	−	0.982287i	$-0.560000\pi$
$257$	−0.996398	−	0.394502i	−0.996398	−	0.394502i	−0.809017	+	0.587785i	$0.800000\pi$
$258$	1.15434	−	1.08399i	1.15434	−	1.08399i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.331159	+	0.521823i	0.331159	+	0.521823i
$263$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$263$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$264$	−0.0937119	+	0.0515186i	−0.0937119	+	0.0515186i
$265$		0			0
$266$		0			0
$267$	−1.36699	+	0.993173i	−1.36699	+	0.993173i
$268$	0.348445	−	0.137959i	0.348445	−	0.137959i
$269$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$269$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	1.07165	−	1.68866i	1.07165	−	1.68866i
$273$		0			0
$274$	0.791759	−	0.313480i	0.791759	−	0.313480i
$275$	0.00788530	−	0.125333i	0.00788530	−	0.125333i
$276$		0			0
$277$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$277$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$278$	1.41213	−	1.32608i	1.41213	−	1.32608i
$279$		0			0
$280$		0			0
$281$	−0.929324	+	1.12336i	−0.929324	+	1.12336i	0.0627905	+	0.998027i	$0.480000\pi$
$281$	−0.929324	+	1.12336i	−0.929324	+	1.12336i	−0.992115	+	0.125333i	$0.960000\pi$
$282$		0			0
$283$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$283$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.240851	+	0.132409i	−0.240851	+	0.132409i
$289$	2.18691	−	2.05364i	2.18691	−	2.05364i
$290$		0			0
$291$	1.00441	+	0.729747i	1.00441	+	0.729747i
$292$	−0.101597	−	1.61484i	−0.101597	−	1.61484i
$293$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$293$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$294$	0.362576	−	0.770513i	0.362576	−	0.770513i
$295$		0			0
$296$		0			0
$297$	0.132049	+	0.0339043i	0.132049	+	0.0339043i
$298$		0			0
$299$		0			0
$300$	−0.0534698	+	0.849878i	−0.0534698	+	0.849878i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.688925	+	1.46404i	0.688925	+	1.46404i
$305$		0			0
$306$	−0.532426	+	0.136704i	−0.532426	+	0.136704i
$307$	−0.746226	−	1.58581i	−0.746226	−	1.58581i	−0.809017	−	0.587785i	$-0.800000\pi$
$307$	−0.746226	−	1.58581i	−0.746226	−	1.58581i	0.0627905	−	0.998027i	$-0.480000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$311$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$312$		0			0
$313$	−0.124591	+	1.98031i	−0.124591	+	1.98031i	0.0627905	+	0.998027i	$0.480000\pi$
$313$	−0.124591	+	1.98031i	−0.124591	+	1.98031i	−0.187381	+	0.982287i	$0.560000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$317$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−0.135880	+	0.712308i	−0.135880	+	0.712308i
$322$		0			0
$323$	0.606379	+	3.17875i	0.606379	+	3.17875i
$324$	−0.629202	−	0.161552i	−0.629202	−	0.161552i
$325$		0			0
$326$	1.84489	+	0.730444i	1.84489	+	0.730444i
$327$		0			0
$328$	1.41213	+	0.362574i	1.41213	+	0.362574i
$329$		0			0
$330$		0			0
$331$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$331$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$332$	1.03799	−	1.63560i	1.03799	−	1.63560i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.110048	−	0.0604991i	0.110048	−	0.0604991i	−0.425779	−	0.904827i	$-0.640000\pi$
$337$	0.110048	−	0.0604991i	0.110048	−	0.0604991i	0.535827	+	0.844328i	$0.320000\pi$
$338$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$339$	1.53377	−	0.393805i	1.53377	−	0.393805i
$340$		0			0
$341$		0			0
$342$	0.137424	−	0.422948i	0.137424	−	0.422948i
$343$		0			0
$344$	−1.35556	−	1.27295i	−1.35556	−	1.27295i
$345$		0			0
$346$		0			0
$347$	1.03799	−	0.266509i	1.03799	−	0.266509i	0.309017	−	0.951057i	$-0.400000\pi$
$347$	1.03799	−	0.266509i	1.03799	−	0.266509i	0.728969	+	0.684547i	$0.240000\pi$
$348$		0			0
$349$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$349$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$350$		0			0
$351$		0			0
$352$	0.0672897	+	0.106032i	0.0672897	+	0.106032i
$353$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$353$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$354$	−0.665239	−	0.624701i	−0.665239	−	0.624701i
$355$		0			0
$356$	1.26480	+	1.52888i	1.26480	+	1.52888i
$357$		0			0
$358$	−0.746226	+	1.58581i	−0.746226	+	1.58581i
$359$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$359$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$360$		0			0
$361$	−1.50441	−	0.595638i	−1.50441	−	0.595638i
$362$		0			0
$363$	−0.157050	+	0.823283i	−0.157050	+	0.823283i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$367$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$368$		0			0
$369$	−0.214712	−	0.338332i	−0.214712	−	0.338332i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$373$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$374$	0.0776134	+	0.238869i	0.0776134	+	0.238869i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$379$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$383$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$384$	−0.456288	−	0.718995i	−0.456288	−	0.718995i
$385$		0			0
$386$	−0.456288	−	0.969661i	−0.456288	−	0.969661i
$387$	0.0320919	+	0.510086i	0.0320919	+	0.510086i
$388$	0.781202	−	1.23098i	0.781202	−	1.23098i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	−0.929776	−	0.368125i	−0.929776	−	0.368125i
$393$	0.522142	+	0.0659619i	0.522142	+	0.0659619i
$394$		0			0
$395$		0			0
$396$	0.00646760	−	0.0339043i	0.00646760	−	0.0339043i
$397$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$397$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$398$		0			0
$399$		0			0
$400$	1.00000			1.00000
$401$	−0.683098	−	1.07639i	−0.683098	−	1.07639i	−0.992115	−	0.125333i	$-0.960000\pi$
$401$	−0.683098	−	1.07639i	−0.683098	−	1.07639i	0.309017	−	0.951057i	$-0.400000\pi$
$402$	0.0986173	−	0.303513i	0.0986173	−	0.303513i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.526292	−	1.61976i	−0.526292	−	1.61976i
$409$	1.41213	+	1.32608i	1.41213	+	1.32608i	0.876307	+	0.481754i	$0.160000\pi$
$409$	1.41213	+	1.32608i	1.41213	+	1.32608i	0.535827	+	0.844328i	$0.320000\pi$
$410$		0			0
$411$	0.224084	−	0.689661i	0.224084	−	0.689661i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.103580	−	1.64636i	−0.103580	−	1.64636i
$418$	−0.196811	−	0.0505324i	−0.196811	−	0.0505324i
$419$	−1.23480	−	1.49261i	−1.23480	−	1.49261i	−0.809017	−	0.587785i	$-0.800000\pi$
$419$	−1.23480	−	1.49261i	−1.23480	−	1.49261i	−0.425779	−	0.904827i	$-0.640000\pi$
$420$		0			0
$421$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$421$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$422$	−0.613161	−	1.88711i	−0.613161	−	1.88711i
$423$		0			0
$424$		0			0
$425$	1.93717	+	0.497380i	1.93717	+	0.497380i
$426$		0			0
$427$		0			0
$428$	0.844844	+	0.106729i	0.844844	+	0.106729i
$429$		0			0
$430$		0			0
$431$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$431$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$432$	−0.203423	+	1.06638i	−0.203423	+	1.06638i
$433$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$433$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−1.11470	−	0.809880i	−1.11470	−	0.809880i
$439$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$439$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$440$		0			0
$441$	0.117025	+	0.248690i	0.117025	+	0.248690i
$442$		0			0
$443$	−1.85955	−	0.736249i	−1.85955	−	0.736249i	−0.929776	−	0.368125i	$-0.880000\pi$
$443$	−1.85955	−	0.736249i	−1.85955	−	0.736249i	−0.929776	−	0.368125i	$-0.880000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.824805	−	1.75280i	−0.824805	−	1.75280i	−0.637424	−	0.770513i	$-0.720000\pi$
$449$	−0.824805	−	1.75280i	−0.824805	−	1.75280i	−0.187381	−	0.982287i	$-0.560000\pi$
$450$	−0.200356	−	0.188146i	−0.200356	−	0.188146i
$451$	−0.148122	+	0.107617i	−0.148122	+	0.107617i
$452$	−0.574633	−	1.76854i	−0.574633	−	1.76854i
$453$		0			0
$454$	−0.393950	+	0.476203i	−0.393950	+	0.476203i
$455$		0			0
$456$	1.33456	+	0.342658i	1.33456	+	0.342658i
$457$	0.303189	+	1.58937i	0.303189	+	1.58937i	0.728969	+	0.684547i	$0.240000\pi$
$457$	0.303189	+	1.58937i	0.303189	+	1.58937i	−0.425779	+	0.904827i	$0.640000\pi$
$458$		0			0
$459$	−0.924459	+	1.96457i	−0.924459	+	1.96457i
$460$		0			0
$461$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$461$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$462$		0			0
$463$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$463$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$464$		0			0
$465$		0			0
$466$	−1.35556	−	1.27295i	−1.35556	−	1.27295i
$467$	0.542804	+	1.15352i	0.542804	+	1.15352i	0.968583	+	0.248690i	$0.0800000\pi$
$467$	0.542804	+	1.15352i	0.542804	+	1.15352i	−0.425779	+	0.904827i	$0.640000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−0.683098	+	0.825723i	−0.683098	+	0.825723i
$473$	0.231683	−	0.0292684i	0.231683	−	0.0292684i
$474$		0			0
$475$	−1.17950	+	1.10762i	−1.17950	+	1.10762i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$479$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$480$		0			0
$481$		0			0
$482$	1.75261			1.75261
$483$		0			0
$484$	0.976468	+	0.123357i	0.976468	+	0.123357i
$485$		0			0
$486$	0.430742	−	0.312952i	0.430742	−	0.312952i
$487$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$487$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$488$		0			0
$489$	1.48068	−	0.814013i	1.48068	−	0.814013i
$490$		0			0
$491$	0.781202	+	1.23098i	0.781202	+	1.23098i	0.968583	+	0.248690i	$0.0800000\pi$
$491$	0.781202	+	1.23098i	0.781202	+	1.23098i	−0.187381	+	0.982287i	$0.560000\pi$
$492$	1.00441	−	0.729747i	1.00441	−	0.729747i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−0.509758	−	1.56887i	−0.509758	−	1.56887i
$499$	−0.362989	+	0.0931997i	−0.362989	+	0.0931997i	−0.425779	−	0.904827i	$-0.640000\pi$
$499$	−0.362989	+	0.0931997i	−0.362989	+	0.0931997i	0.0627905	+	0.998027i	$0.480000\pi$
$500$		0			0
$501$		0			0
$502$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.263146	+	0.809880i	−0.263146	+	0.809880i
$508$		0			0
$509$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$509$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$510$		0			0
$511$		0			0
$512$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$513$	−0.941207	−	1.48310i	−0.941207	−	1.48310i
$514$	0.574221	+	0.904827i	0.574221	+	0.904827i
$515$		0			0
$516$	−1.57103	+	0.198467i	−1.57103	+	0.198467i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.328407	+	1.72157i	−0.328407	+	1.72157i	0.309017	+	0.951057i	$0.400000\pi$
$521$	−0.328407	+	1.72157i	−0.328407	+	1.72157i	−0.637424	+	0.770513i	$0.720000\pi$
$522$		0			0
$523$	−0.683098	+	1.07639i	−0.683098	+	1.07639i	0.309017	+	0.951057i	$0.400000\pi$
$523$	−0.683098	+	1.07639i	−0.683098	+	1.07639i	−0.992115	+	0.125333i	$0.960000\pi$
$524$	0.0388067	−	0.616814i	0.0388067	−	0.616814i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.106096	+	0.0134031i	0.106096	+	0.0134031i
$529$	0.728969	−	0.684547i	0.728969	−	0.684547i
$530$		0			0
$531$	0.292219	−	0.0369159i	0.292219	−	0.0369159i
$532$		0			0
$533$		0			0
$534$	1.68969			1.68969
$535$		0			0
$536$	−0.362989	−	0.0931997i	−0.362989	−	0.0931997i
$537$	0.635456	+	1.35041i	0.635456	+	1.35041i
$538$		0			0
$539$	0.110048	−	0.0604991i	0.110048	−	0.0604991i
$540$		0			0
$541$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$541$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$542$		0			0
$543$		0			0
$544$	−1.85955	+	0.736249i	−1.85955	+	0.736249i
$545$		0			0
$546$		0			0
$547$	0.0702235	+	0.368125i	0.0702235	+	0.368125i	1.00000			$0$
$547$	0.0702235	+	0.368125i	0.0702235	+	0.368125i	−0.929776	+	0.368125i	$0.880000\pi$
$548$	−0.824805	−	0.211774i	−0.824805	−	0.211774i
$549$		0			0
$550$	−0.0800484	+	0.0967619i	−0.0800484	+	0.0967619i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−1.92189	+	0.242791i	−1.92189	+	0.242791i
$557$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$557$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	0.198860	+	0.0787342i	0.198860	+	0.0787342i
$562$	1.41213	−	0.362574i	1.41213	−	0.362574i
$563$	−0.263146	−	0.559214i	−0.263146	−	0.559214i	0.728969	−	0.684547i	$-0.240000\pi$
$563$	−0.263146	−	0.559214i	−0.263146	−	0.559214i	−0.992115	+	0.125333i	$0.960000\pi$
$564$		0			0
$565$		0			0
$566$	−1.41789	−	1.03016i	−1.41789	−	1.03016i
$567$		0			0
$568$		0			0
$569$	0.0672897	+	1.06954i	0.0672897	+	1.06954i	0.876307	+	0.481754i	$0.160000\pi$
$569$	0.0672897	+	1.06954i	0.0672897	+	1.06954i	−0.809017	+	0.587785i	$0.800000\pi$
$570$		0			0
$571$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$571$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.272681	+	0.0344476i	0.272681	+	0.0344476i
$577$	0.348445	+	0.137959i	0.348445	+	0.137959i	0.535827	−	0.844328i	$-0.320000\pi$
$577$	0.348445	+	0.137959i	0.348445	+	0.137959i	−0.187381	+	0.982287i	$0.560000\pi$
$578$	−2.97634	+	0.376000i	−2.97634	+	0.376000i
$579$	−0.883906	−	0.226948i	−0.883906	−	0.226948i
$580$		0			0
$581$		0			0
$582$	−0.383650	−	1.18075i	−0.383650	−	1.18075i
$583$		0			0
$584$	−0.866986	+	1.36615i	−0.866986	+	1.36615i
$585$		0			0
$586$		0			0
$587$	−0.124591	−	1.98031i	−0.124591	−	1.98031i	−0.187381	−	0.982287i	$-0.560000\pi$
$587$	−0.124591	−	1.98031i	−0.124591	−	1.98031i	0.0627905	−	0.998027i	$-0.480000\pi$
$588$	−0.746226	+	0.410241i	−0.746226	+	0.410241i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$593$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$594$	−0.0869011	−	0.105045i	−0.0869011	−	0.105045i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$599$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$600$	0.542804	−	0.656137i	0.542804	−	0.656137i
$601$	−1.41789	−	0.779494i	−1.41789	−	0.779494i	−0.425779	−	0.904827i	$-0.640000\pi$
$601$	−1.41789	−	0.779494i	−1.41789	−	0.779494i	−0.992115	+	0.125333i	$0.960000\pi$
$602$		0			0
$603$	0.0551916	+	0.0869681i	0.0551916	+	0.0869681i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$607$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$608$	0.303189	−	1.58937i	0.303189	−	1.58937i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.511094	+	0.202357i	0.511094	+	0.202357i
$613$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$613$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$614$	−0.328407	+	1.72157i	−0.328407	+	1.72157i
$615$		0			0
$616$		0			0
$617$	0.0672897	+	1.06954i	0.0672897	+	1.06954i	0.876307	+	0.481754i	$0.160000\pi$
$617$	0.0672897	+	1.06954i	0.0672897	+	1.06954i	−0.809017	+	0.587785i	$0.800000\pi$
$618$		0			0
$619$	−0.996398	+	1.57007i	−0.996398	+	1.57007i	−0.187381	+	0.982287i	$0.560000\pi$
$619$	−0.996398	+	1.57007i	−0.996398	+	1.57007i	−0.809017	+	0.587785i	$0.800000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.309017	+	0.951057i	0.309017	+	0.951057i
$626$	1.26480	−	1.52888i	1.26480	−	1.52888i
$627$	−0.139986	+	0.101706i	−0.139986	+	0.101706i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$631$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$632$		0			0
$633$	−1.57103	−	0.622015i	−1.57103	−	0.622015i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.115808	+	0.607087i	−0.115808	+	0.607087i	0.876307	+	0.481754i	$0.160000\pi$
$641$	−0.115808	+	0.607087i	−0.115808	+	0.607087i	−0.992115	+	0.125333i	$0.960000\pi$
$642$	0.528613	−	0.496401i	0.528613	−	0.496401i
$643$	1.18532	+	0.469303i	1.18532	+	0.469303i	0.876307	−	0.481754i	$-0.160000\pi$
$643$	1.18532	+	0.469303i	1.18532	+	0.469303i	0.309017	+	0.951057i	$0.400000\pi$
$644$		0			0
$645$		0			0
$646$	1.37785	−	2.92808i	1.37785	−	2.92808i
$647$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$647$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$648$	0.414077	+	0.500534i	0.414077	+	0.500534i
$649$	−0.0252177	−	0.132196i	−0.0252177	−	0.132196i
$650$		0			0
$651$		0			0
$652$	−1.06320	−	1.67534i	−1.06320	−	1.67534i
$653$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$653$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$654$		0			0
$655$		0			0
$656$	−0.929324	−	1.12336i	−0.929324	−	1.12336i
$657$	0.430742	−	0.110596i	0.430742	−	0.110596i
$658$		0			0
$659$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$659$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$660$		0			0
$661$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$661$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$662$	0.331159	−	1.01920i	0.331159	−	1.01920i
$663$		0			0
$664$	−1.80113	+	0.713118i	−1.80113	+	0.713118i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$673$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$674$	−0.124591	−	0.0157395i	−0.124591	−	0.0157395i
$675$	−1.07705	+	0.136063i	−1.07705	+	0.136063i
$676$	0.968583	+	0.248690i	0.968583	+	0.248690i
$677$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$677$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$678$	−1.47232	−	0.582932i	−1.47232	−	0.582932i
$679$		0			0
$680$		0			0
$681$	0.0986173	+	0.516970i	0.0986173	+	0.516970i
$682$		0			0
$683$	−0.0235315	+	0.123357i	−0.0235315	+	0.123357i	−0.992115	−	0.125333i	$-0.960000\pi$
$683$	−0.0235315	+	0.123357i	−0.0235315	+	0.123357i	0.968583	+	0.248690i	$0.0800000\pi$
$684$	−0.359781	+	0.261396i	−0.359781	+	0.261396i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.348445	+	1.82662i	0.348445	+	1.82662i
$689$		0			0
$690$		0			0
$691$	0.0915446	−	1.45506i	0.0915446	−	1.45506i	−0.637424	−	0.770513i	$-0.720000\pi$
$691$	0.0915446	−	1.45506i	0.0915446	−	1.45506i	0.728969	−	0.684547i	$-0.240000\pi$
$692$		0			0
$693$		0			0
$694$	−0.996398	−	0.394502i	−0.996398	−	0.394502i
$695$		0			0
$696$		0			0
$697$	−1.24152	−	2.63836i	−1.24152	−	2.63836i
$698$		0			0
$699$	−1.57103	+	0.198467i	−1.57103	+	0.198467i
$700$		0			0
$701$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$701$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$702$		0			0
$703$		0			0
$704$	0.00788530	−	0.125333i	0.00788530	−	0.125333i
$705$		0			0
$706$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$707$		0			0
$708$	0.171000	+	0.896412i	0.171000	+	0.896412i
$709$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$709$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$710$		0			0
$711$		0			0
$712$	−0.124591	−	1.98031i	−0.124591	−	1.98031i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.53583	−	0.844328i	1.53583	−	0.844328i
$717$		0			0
$718$		0			0
$719$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$719$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$720$		0			0
$721$		0			0
$722$	0.866986	+	1.36615i	0.866986	+	1.36615i
$723$	0.951325	−	1.14995i	0.951325	−	1.14995i
$724$		0			0
$725$		0			0
$726$	0.610970	−	0.573739i	0.610970	−	0.573739i
$727$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$727$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$728$		0			0
$729$	0.0692581	−	1.10083i	0.0692581	−	1.10083i
$730$		0			0
$731$	−0.233525	+	3.71177i	−0.233525	+	3.71177i
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.0380748	−	0.0276630i	0.0380748	−	0.0276630i
$738$	−0.0251609	+	0.399920i	−0.0251609	+	0.399920i
$739$	−0.124591	+	0.0157395i	−0.124591	+	0.0157395i	−0.187381	−	0.982287i	$-0.560000\pi$
$739$	−0.124591	+	0.0157395i	−0.124591	+	0.0157395i	0.0627905	+	0.998027i	$0.480000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$743$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.495037	+	0.195999i	0.495037	+	0.195999i
$748$	0.0776134	−	0.238869i	0.0776134	−	0.238869i
$749$		0			0
$750$		0			0
$751$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$751$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$752$		0			0
$753$	−0.746226	−	0.410241i	−0.746226	−	0.410241i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$757$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$758$	−0.115808	+	0.356420i	−0.115808	+	0.356420i
$759$		0			0
$760$		0			0
$761$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.0627905	−	0.998027i	$-0.480000\pi$
$761$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.0534698	+	0.849878i	−0.0534698	+	0.849878i
$769$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$769$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$770$		0			0
$771$	0.905380	+	0.114376i	0.905380	+	0.114376i
$772$	−0.200808	+	1.05267i	−0.200808	+	1.05267i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$	0.273858	−	0.431531i	0.273858	−	0.431531i
$775$		0			0
$776$	−1.35556	+	0.536702i	−1.35556	+	0.536702i
$777$		0			0
$778$		0			0
$779$	2.34039	+	0.295660i	2.34039	+	0.295660i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.535827	+	0.844328i	0.535827	+	0.844328i
$785$		0			0
$786$	−0.383650	−	0.360272i	−0.383650	−	0.360272i
$787$	−1.80113	−	0.462452i	−1.80113	−	0.462452i	−0.809017	−	0.587785i	$-0.800000\pi$
$787$	−1.80113	−	0.462452i	−1.80113	−	0.462452i	−0.992115	+	0.125333i	$0.960000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.0251609	+	0.0236276i	−0.0251609	+	0.0236276i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$797$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$798$		0			0
$799$		0			0
$800$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$801$	−0.347626	+	0.420208i	−0.347626	+	0.420208i
$802$	−0.0800484	+	1.27233i	−0.0800484	+	1.27233i
$803$	−0.0627905	−	0.193249i	−0.0627905	−	0.193249i
$804$	−0.258183	+	0.187581i	−0.258183	+	0.187581i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.0534698	−	0.113629i	−0.0534698	−	0.113629i	0.876307	−	0.481754i	$-0.160000\pi$
$809$	−0.0534698	−	0.113629i	−0.0534698	−	0.113629i	−0.929776	+	0.368125i	$0.880000\pi$
$810$		0			0
$811$	−1.73879	−	0.955910i	−1.73879	−	0.955910i	−0.929776	−	0.368125i	$-0.880000\pi$
$811$	−1.73879	−	0.955910i	−1.73879	−	0.955910i	−0.809017	−	0.587785i	$-0.800000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−0.526292	+	1.61976i	−0.526292	+	1.61976i
$817$	−2.43419	−	1.76854i	−2.43419	−	1.76854i
$818$	−0.362989	−	1.90285i	−0.362989	−	1.90285i
$819$		0			0
$820$		0			0
$821$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$821$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$822$	−0.586660	+	0.426234i	−0.586660	+	0.426234i
$823$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$823$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$824$		0			0
$825$	0.0200385	+	0.105045i	0.0200385	+	0.105045i
$826$		0			0
$827$	1.26480	+	0.159781i	1.26480	+	0.159781i	0.728969	−	0.684547i	$-0.240000\pi$
$827$	1.26480	+	0.159781i	1.26480	+	0.159781i	0.535827	+	0.844328i	$0.320000\pi$
$828$		0			0
$829$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$829$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.618034	+	1.90211i	0.618034	+	1.90211i
$834$	−0.883906	+	1.39281i	−0.883906	+	1.39281i
$835$		0			0
$836$	0.129521	+	0.156564i	0.129521	+	0.156564i
$837$		0			0
$838$	0.121636	+	1.93334i	0.121636	+	1.93334i
$839$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$839$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$840$		0			0
$841$	0.968583	−	0.248690i	0.968583	−	0.248690i
$842$		0			0
$843$	0.528613	−	1.12336i	0.528613	−	1.12336i
$844$	−0.613161	+	1.88711i	−0.613161	+	1.88711i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.44556	+	0.371158i	−1.44556	+	0.371158i
$850$	−1.27485	−	1.54103i	−1.27485	−	1.54103i
$851$		0			0
$852$		0			0
$853$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$853$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$854$		0			0
$855$		0			0
$856$	−0.620759	−	0.582932i	−0.620759	−	0.582932i
$857$	0.371808	+	1.94908i	0.371808	+	1.94908i	0.309017	+	0.951057i	$0.400000\pi$
$857$	0.371808	+	1.94908i	0.371808	+	1.94908i	0.0627905	+	0.998027i	$0.480000\pi$
$858$		0			0
$859$	0.371808	−	1.94908i	0.371808	−	1.94908i	0.0627905	−	0.998027i	$-0.480000\pi$
$859$	0.371808	−	1.94908i	0.371808	−	1.94908i	0.309017	−	0.951057i	$-0.400000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$863$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$864$	0.791374	−	0.743150i	0.791374	−	0.743150i
$865$		0			0
$866$	0.618034			0.618034
$867$	−1.36886	+	2.15698i	−1.36886	+	2.15698i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.372572	+	0.147512i	0.372572	+	0.147512i
$874$		0			0
$875$		0			0
$876$	0.425779	+	1.31041i	0.425779	+	1.31041i
$877$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$877$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$881$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$882$	0.0515014	−	0.269980i	0.0515014	−	0.269980i
$883$	0.688925	−	1.46404i	0.688925	−	1.46404i	−0.187381	−	0.982287i	$-0.560000\pi$
$883$	0.688925	−	1.46404i	0.688925	−	1.46404i	0.876307	−	0.481754i	$-0.160000\pi$
$884$		0			0
$885$		0			0
$886$	1.07165	+	1.68866i	1.07165	+	1.68866i
$887$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$887$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.0815788			−0.0815788
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.362989	+	1.90285i	−0.362989	+	1.90285i
$899$		0			0
$900$	0.0515014	+	0.269980i	0.0515014	+	0.269980i
$901$		0			0
$902$	0.183089			0.183089
$903$		0			0
$904$	−0.574633	+	1.76854i	−0.574633	+	1.76854i
$905$		0			0
$906$		0			0
$907$	0.812619	+	0.982287i	0.812619	+	0.982287i	1.00000			$0$
$907$	0.812619	+	0.982287i	0.812619	+	0.982287i	−0.187381	+	0.982287i	$0.560000\pi$
$908$	0.598617	−	0.153699i	0.598617	−	0.153699i
$909$		0			0
$910$		0			0
$911$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$911$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$912$	−0.878275	−	1.06165i	−0.878275	−	1.06165i
$913$	0.0751750	−	0.231365i	0.0751750	−	0.231365i
$914$	0.688925	−	1.46404i	0.688925	−	1.46404i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	1.90265	−	1.04599i	1.90265	−	1.04599i
$919$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$919$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$920$		0			0
$921$	0.951325	+	1.14995i	0.951325	+	1.14995i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.62954	−	0.645180i	−1.62954	−	0.645180i	−0.637424	−	0.770513i	$-0.720000\pi$
$929$	−1.62954	−	0.645180i	−1.62954	−	0.645180i	−0.992115	+	0.125333i	$0.960000\pi$
$930$		0			0
$931$	−1.56720	−	0.402389i	−1.56720	−	0.402389i
$932$	0.348445	+	1.82662i	0.348445	+	1.82662i
$933$		0			0
$934$	0.238883	−	1.25227i	0.238883	−	1.25227i
$935$		0			0
$936$		0			0
$937$	0.00788530	+	0.125333i	0.00788530	+	0.125333i	1.00000			$0$
$937$	0.00788530	+	0.125333i	0.00788530	+	0.125333i	−0.992115	+	0.125333i	$0.960000\pi$
$938$		0			0
$939$	−0.316616	−	1.65976i	−0.316616	−	1.65976i
$940$		0			0
$941$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$941$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$942$		0			0
$943$		0			0
$944$	1.03799	−	0.266509i	1.03799	−	0.266509i
$945$		0			0
$946$	−0.204639	−	0.112501i	−0.204639	−	0.112501i
$947$	−0.101597	−	1.61484i	−0.101597	−	1.61484i	−0.637424	−	0.770513i	$-0.720000\pi$
$947$	−0.101597	−	1.61484i	−0.101597	−	1.61484i	0.535827	−	0.844328i	$-0.320000\pi$
$948$		0			0
$949$		0			0
$950$	1.60528	−	0.202793i	1.60528	−	0.202793i
$951$		0			0
$952$		0			0
$953$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$953$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.425779	+	0.904827i	−0.425779	+	0.904827i
$962$		0			0
$963$	0.0146961	+	0.233587i	0.0146961	+	0.233587i
$964$	−1.41789	−	1.03016i	−1.41789	−	1.03016i
$965$		0			0
$966$		0			0
$967$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$967$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$968$	−0.717472	−	0.673751i	−0.717472	−	0.673751i
$969$	−1.17332	−	2.49343i	−1.17332	−	2.49343i
$970$		0			0
$971$	−1.17950	−	1.10762i	−1.17950	−	1.10762i	−0.992115	−	0.125333i	$-0.960000\pi$
$971$	−1.17950	−	1.10762i	−1.17950	−	1.10762i	−0.187381	−	0.982287i	$-0.560000\pi$
$972$	−0.532426			−0.532426
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.41213	−	1.32608i	1.41213	−	1.32608i	0.535827	−	0.844328i	$-0.320000\pi$
$977$	1.41213	−	1.32608i	1.41213	−	1.32608i	0.876307	−	0.481754i	$-0.160000\pi$
$978$	−1.67636	−	0.211774i	−1.67636	−	0.211774i
$979$	0.201592	+	0.146465i	0.201592	+	0.146465i
$980$		0			0
$981$		0			0
$982$	0.0915446	−	1.45506i	0.0915446	−	1.45506i
$983$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$983$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$984$	−1.24152			−1.24152
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$991$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$992$		0			0
$993$	−0.488983	−	0.770513i	−0.488983	−	0.770513i
$994$		0			0
$995$		0			0
$996$	−0.509758	+	1.56887i	−0.509758	+	1.56887i
$997$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$997$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$998$	0.348445	+	0.137959i	0.348445	+	0.137959i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.1.w.a.1067.1	✓	20
8.3	odd	2	CM	2008.1.w.a.1067.1	✓	20
251.4	even	25	inner	2008.1.w.a.1259.1	yes	20
2008.1259	odd	50	inner	2008.1.w.a.1259.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.1.w.a.1067.1	✓	20	1.1	even	1	trivial
2008.1.w.a.1067.1	✓	20	8.3	odd	2	CM
2008.1.w.a.1259.1	yes	20	251.4	even	25	inner
2008.1.w.a.1259.1	yes	20	2008.1259	odd	50	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:	\( N \)	\(=\)	\( 2008 = 2^{3} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2008.w (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.809017 - 0.587785i) q^{2} +(-0.824805 + 0.211774i) q^{3} +(0.309017 + 0.951057i) q^{4} +(0.791759 + 0.313480i) q^{6} +(0.309017 - 0.951057i) q^{8} +(-0.240851 + 0.132409i) q^{9} +O(q^{10})\)	\(q+(-0.809017 - 0.587785i) q^{2} +(-0.824805 + 0.211774i) q^{3} +(0.309017 + 0.951057i) q^{4} +(0.791759 + 0.313480i) q^{6} +(0.309017 - 0.951057i) q^{8} +(-0.240851 + 0.132409i) q^{9} +(0.0672897 + 0.106032i) q^{11} +(-0.456288 - 0.718995i) q^{12} +(-0.809017 + 0.587785i) q^{16} +(-1.85955 + 0.736249i) q^{17} +(0.272681 + 0.0344476i) q^{18} +(0.303189 - 1.58937i) q^{19} +(0.00788530 - 0.125333i) q^{22} +(-0.0534698 + 0.849878i) q^{24} +(-0.809017 - 0.587785i) q^{25} +(0.791374 - 0.743150i) q^{27} +1.00000 q^{32} +(-0.0779556 - 0.0732052i) q^{33} +(1.93717 + 0.497380i) q^{34} +(-0.200356 - 0.188146i) q^{36} +(-1.17950 + 1.10762i) q^{38} +(0.0915446 + 1.45506i) q^{41} +(0.791759 - 1.68257i) q^{43} +(-0.0800484 + 0.0967619i) q^{44} +(0.542804 - 0.656137i) q^{48} +(0.0627905 - 0.998027i) q^{49} +(0.309017 + 0.951057i) q^{50} +(1.37785 - 1.00107i) q^{51} +(-1.07705 + 0.136063i) q^{54} +(0.0865160 + 1.37513i) q^{57} +(-0.996398 - 0.394502i) q^{59} +(-0.809017 - 0.587785i) q^{64} +(0.0200385 + 0.105045i) q^{66} +(-0.0235315 - 0.374023i) q^{67} +(-1.27485 - 1.54103i) q^{68} +(0.0515014 + 0.269980i) q^{72} +(-1.56720 - 0.402389i) q^{73} +(0.791759 + 0.313480i) q^{75} +(1.60528 - 0.202793i) q^{76} +(-0.348079 + 0.548484i) q^{81} +(0.781202 - 1.23098i) q^{82} +(-1.23480 - 1.49261i) q^{83} +(-1.62954 + 0.895846i) q^{86} +(0.121636 - 0.0312307i) q^{88} +(1.84489 - 0.730444i) q^{89} +(-0.824805 + 0.211774i) q^{96} +(-0.929324 - 1.12336i) q^{97} +(-0.637424 + 0.770513i) q^{98} +(-0.0302463 - 0.0166281i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{2} - 5 q^{4} - 5 q^{8}+O(q^{10}) \) 20 * q - 5 * q^2 - 5 * q^4 - 5 * q^8	\( 20 q - 5 q^{2} - 5 q^{4} - 5 q^{8} - 5 q^{11} - 5 q^{16} + 20 q^{22} - 5 q^{25} - 5 q^{27} + 20 q^{32} - 5 q^{33} - 5 q^{44} - 5 q^{50} - 5 q^{54} - 5 q^{59} - 5 q^{64} - 5 q^{66} - 5 q^{81} - 5 q^{83} - 5 q^{88} - 5 q^{99}+O(q^{100}) \) 20 * q - 5 * q^2 - 5 * q^4 - 5 * q^8 - 5 * q^11 - 5 * q^16 + 20 * q^22 - 5 * q^25 - 5 * q^27 + 20 * q^32 - 5 * q^33 - 5 * q^44 - 5 * q^50 - 5 * q^54 - 5 * q^59 - 5 * q^64 - 5 * q^66 - 5 * q^81 - 5 * q^83 - 5 * q^88 - 5 * q^99

Introduction

L-functions

Modular forms

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