LMFDB - Embedded newform 2008.1.w.a.267.1

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			267.1
Root			$0.637424 - 0.770513i$ of defining polynomial
Character	$\chi$	$=$	2008.267
Dual form			2008.1.w.a.1459.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.309017 - 0.951057i) q^{2} +(-1.11716 - 0.614163i) q^{3} +(-0.809017 - 0.587785i) q^{4} +(-0.929324 + 0.872693i) q^{6} +(-0.809017 + 0.587785i) q^{8} +(0.335019 + 0.527905i) q^{9} +O(q^{10})$	\(q+(0.309017 - 0.951057i) q^{2} +(-1.11716 - 0.614163i) q^{3} +(-0.809017 - 0.587785i) q^{4} +(-0.929324 + 0.872693i) q^{6} +(-0.809017 + 0.587785i) q^{8} +(0.335019 + 0.527905i) q^{9} +(0.844844 + 1.79538i) q^{11} +(0.542804 + 1.15352i) q^{12} +(0.309017 + 0.951057i) q^{16} +(1.45794 + 1.36909i) q^{17} +(0.605594 - 0.155490i) q^{18} +(-0.574633 + 0.227513i) q^{19} +(1.96858 - 0.248690i) q^{22} +(1.26480 - 0.159781i) q^{24} +(0.309017 - 0.951057i) q^{25} +(0.0299991 + 0.476823i) q^{27} +1.00000 q^{32} +(0.158834 - 2.52460i) q^{33} +(1.75261 - 0.963507i) q^{34} +(0.0392590 - 0.624004i) q^{36} +(0.0388067 + 0.616814i) q^{38} +(-0.124591 - 0.0157395i) q^{41} +(-0.929324 + 1.12336i) q^{43} +(0.371808 - 1.94908i) q^{44} +(0.238883 - 1.25227i) q^{48} +(-0.992115 + 0.125333i) q^{49} +(-0.809017 - 0.587785i) q^{50} +(-0.787899 - 2.42490i) q^{51} +(0.462756 + 0.118815i) q^{54} +(0.781687 + 0.0987500i) q^{57} +(-0.620759 + 0.582932i) q^{59} +(0.309017 - 0.951057i) q^{64} +(-2.35195 - 0.931204i) q^{66} +(1.84489 + 0.233064i) q^{67} +(-0.374763 - 1.96457i) q^{68} +(-0.581331 - 0.230165i) q^{72} +(0.541587 - 0.297740i) q^{73} +(-0.929324 + 0.872693i) q^{75} +(0.598617 + 0.153699i) q^{76} +(0.525546 - 1.11684i) q^{81} +(-0.0534698 + 0.113629i) q^{82} +(-0.328407 - 1.72157i) q^{83} +(0.781202 + 1.23098i) q^{86} +(-1.73879 - 0.955910i) q^{88} +(1.41213 + 1.32608i) q^{89} +(-1.11716 - 0.614163i) q^{96} +(-0.0235315 - 0.123357i) q^{97} +(-0.187381 + 0.982287i) q^{98} +(-0.664754 + 1.04749i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2008\mathbb{Z}\right)^\times$.

$n$	$257$	$503$	$1005$
$\chi(n)$	$e\left(\frac{19}{25}\right)$	$-1$	$-1$

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 $

$2$	0.309017	−	0.951057i	0.309017	−	0.951057i
$2$	0.309017	−	0.951057i	0.309017	−	0.951057i
$3$	−1.11716	−	0.614163i	−1.11716	−	0.614163i	−0.187381	−	0.982287i	$-0.560000\pi$
$3$	−1.11716	−	0.614163i	−1.11716	−	0.614163i	−0.929776	+	0.368125i	$0.880000\pi$
$4$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$5$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$5$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$6$	−0.929324	+	0.872693i	−0.929324	+	0.872693i
$7$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$7$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$8$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$9$	0.335019	+	0.527905i	0.335019	+	0.527905i
$10$		0			0
$11$	0.844844	+	1.79538i	0.844844	+	1.79538i	0.535827	+	0.844328i	$0.320000\pi$
$11$	0.844844	+	1.79538i	0.844844	+	1.79538i	0.309017	+	0.951057i	$0.400000\pi$
$12$	0.542804	+	1.15352i	0.542804	+	1.15352i
$13$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$13$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$14$		0			0
$15$		0			0
$16$	0.309017	+	0.951057i	0.309017	+	0.951057i
$17$	1.45794	+	1.36909i	1.45794	+	1.36909i	0.728969	+	0.684547i	$0.240000\pi$
$17$	1.45794	+	1.36909i	1.45794	+	1.36909i	0.728969	+	0.684547i	$0.240000\pi$
$18$	0.605594	−	0.155490i	0.605594	−	0.155490i
$19$	−0.574633	+	0.227513i	−0.574633	+	0.227513i	−0.637424	−	0.770513i	$-0.720000\pi$
$19$	−0.574633	+	0.227513i	−0.574633	+	0.227513i	0.0627905	+	0.998027i	$0.480000\pi$
$20$		0			0
$21$		0			0
$22$	1.96858	−	0.248690i	1.96858	−	0.248690i
$23$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$23$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$24$	1.26480	−	0.159781i	1.26480	−	0.159781i
$25$	0.309017	−	0.951057i	0.309017	−	0.951057i
$26$		0			0
$27$	0.0299991	+	0.476823i	0.0299991	+	0.476823i
$28$		0			0
$29$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$29$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$30$		0			0
$31$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$31$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$32$	1.00000			1.00000
$33$	0.158834	−	2.52460i	0.158834	−	2.52460i
$34$	1.75261	−	0.963507i	1.75261	−	0.963507i
$35$		0			0
$36$	0.0392590	−	0.624004i	0.0392590	−	0.624004i
$37$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$37$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$38$	0.0388067	+	0.616814i	0.0388067	+	0.616814i
$39$		0			0
$40$		0			0
$41$	−0.124591	−	0.0157395i	−0.124591	−	0.0157395i	0.0627905	−	0.998027i	$-0.480000\pi$
$41$	−0.124591	−	0.0157395i	−0.124591	−	0.0157395i	−0.187381	+	0.982287i	$0.560000\pi$
$42$		0			0
$43$	−0.929324	+	1.12336i	−0.929324	+	1.12336i	0.0627905	+	0.998027i	$0.480000\pi$
$43$	−0.929324	+	1.12336i	−0.929324	+	1.12336i	−0.992115	+	0.125333i	$0.960000\pi$
$44$	0.371808	−	1.94908i	0.371808	−	1.94908i
$45$		0			0
$46$		0			0
$47$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$47$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$48$	0.238883	−	1.25227i	0.238883	−	1.25227i
$49$	−0.992115	+	0.125333i	−0.992115	+	0.125333i
$50$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$51$	−0.787899	−	2.42490i	−0.787899	−	2.42490i
$52$		0			0
$53$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$53$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$54$	0.462756	+	0.118815i	0.462756	+	0.118815i
$55$		0			0
$56$		0			0
$57$	0.781687	+	0.0987500i	0.781687	+	0.0987500i
$58$		0			0
$59$	−0.620759	+	0.582932i	−0.620759	+	0.582932i	−0.929776	−	0.368125i	$-0.880000\pi$
$59$	−0.620759	+	0.582932i	−0.620759	+	0.582932i	0.309017	+	0.951057i	$0.400000\pi$
$60$		0			0
$61$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$61$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$62$		0			0
$63$		0			0
$64$	0.309017	−	0.951057i	0.309017	−	0.951057i
$65$		0			0
$66$	−2.35195	−	0.931204i	−2.35195	−	0.931204i
$67$	1.84489	+	0.233064i	1.84489	+	0.233064i	0.968583	−	0.248690i	$-0.0800000\pi$
$67$	1.84489	+	0.233064i	1.84489	+	0.233064i	0.876307	+	0.481754i	$0.160000\pi$
$68$	−0.374763	−	1.96457i	−0.374763	−	1.96457i
$69$		0			0
$70$		0			0
$71$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$71$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$72$	−0.581331	−	0.230165i	−0.581331	−	0.230165i
$73$	0.541587	−	0.297740i	0.541587	−	0.297740i	−0.187381	−	0.982287i	$-0.560000\pi$
$73$	0.541587	−	0.297740i	0.541587	−	0.297740i	0.728969	+	0.684547i	$0.240000\pi$
$74$		0			0
$75$	−0.929324	+	0.872693i	−0.929324	+	0.872693i
$76$	0.598617	+	0.153699i	0.598617	+	0.153699i
$77$		0			0
$78$		0			0
$79$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$79$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$80$		0			0
$81$	0.525546	−	1.11684i	0.525546	−	1.11684i
$82$	−0.0534698	+	0.113629i	−0.0534698	+	0.113629i
$83$	−0.328407	−	1.72157i	−0.328407	−	1.72157i	−0.637424	−	0.770513i	$-0.720000\pi$
$83$	−0.328407	−	1.72157i	−0.328407	−	1.72157i	0.309017	−	0.951057i	$-0.400000\pi$
$84$		0			0
$85$		0			0
$86$	0.781202	+	1.23098i	0.781202	+	1.23098i
$87$		0			0
$88$	−1.73879	−	0.955910i	−1.73879	−	0.955910i
$89$	1.41213	+	1.32608i	1.41213	+	1.32608i	0.876307	+	0.481754i	$0.160000\pi$
$89$	1.41213	+	1.32608i	1.41213	+	1.32608i	0.535827	+	0.844328i	$0.320000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−1.11716	−	0.614163i	−1.11716	−	0.614163i
$97$	−0.0235315	−	0.123357i	−0.0235315	−	0.123357i	0.968583	−	0.248690i	$-0.0800000\pi$
$97$	−0.0235315	−	0.123357i	−0.0235315	−	0.123357i	−0.992115	+	0.125333i	$0.960000\pi$
$98$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$99$	−0.664754	+	1.04749i	−0.664754	+	1.04749i
$100$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$101$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$101$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$102$	−2.54970			−2.54970
$103$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$103$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.812619	−	0.982287i	0.812619	−	0.982287i	−0.187381	−	0.982287i	$-0.560000\pi$
$107$	0.812619	−	0.982287i	0.812619	−	0.982287i	1.00000			$0$
$108$	0.255999	−	0.403391i	0.255999	−	0.403391i
$109$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$109$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.45794			1.45794			0.728969	−	0.684547i	$-0.240000\pi$
$113$	1.45794			1.45794			0.728969	+	0.684547i	$0.240000\pi$
$114$	0.335471	−	0.712913i	0.335471	−	0.712913i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	0.362576	+	0.770513i	0.362576	+	0.770513i
$119$		0			0
$120$		0			0
$121$	−1.87222	+	2.26313i	−1.87222	+	2.26313i
$122$		0			0
$123$	0.129521	+	0.0941025i	0.129521	+	0.0941025i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$127$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$128$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$129$	1.72813	−	0.684214i	1.72813	−	0.684214i
$130$		0			0
$131$	−1.17950	+	1.10762i	−1.17950	+	1.10762i	−0.187381	+	0.982287i	$0.560000\pi$
$131$	−1.17950	+	1.10762i	−1.17950	+	1.10762i	−0.992115	+	0.125333i	$0.960000\pi$
$132$	−1.61242	+	1.94908i	−1.61242	+	1.94908i
$133$		0			0
$134$	0.791759	−	1.68257i	0.791759	−	1.68257i
$135$		0			0
$136$	−1.98423	−	0.250666i	−1.98423	−	0.250666i
$137$	0.542804	−	1.15352i	0.542804	−	1.15352i	−0.425779	−	0.904827i	$-0.640000\pi$
$137$	0.542804	−	1.15352i	0.542804	−	1.15352i	0.968583	−	0.248690i	$-0.0800000\pi$
$138$		0			0
$139$	−1.62954	+	0.645180i	−1.62954	+	0.645180i	−0.992115	−	0.125333i	$-0.960000\pi$
$139$	−1.62954	+	0.645180i	−1.62954	+	0.645180i	−0.637424	+	0.770513i	$0.720000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.398541	+	0.481754i	−0.398541	+	0.481754i
$145$		0			0
$146$	−0.115808	−	0.607087i	−0.115808	−	0.607087i
$147$	1.18532	+	0.469303i	1.18532	+	0.469303i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	0.542804	+	1.15352i	0.542804	+	1.15352i
$151$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$151$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$152$	0.331159	−	0.521823i	0.331159	−	0.521823i
$153$	−0.234316	+	1.22833i	−0.234316	+	1.22833i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$157$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.899777	−	0.844947i	−0.899777	−	0.844947i
$163$	1.69755	+	0.933237i	1.69755	+	0.933237i	0.968583	+	0.248690i	$0.0800000\pi$
$163$	1.69755	+	0.933237i	1.69755	+	0.933237i	0.728969	+	0.684547i	$0.240000\pi$
$164$	0.0915446	+	0.0859661i	0.0915446	+	0.0859661i
$165$		0			0
$166$	−1.73879	−	0.219661i	−1.73879	−	0.219661i
$167$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$167$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$168$		0			0
$169$	−0.425779	+	0.904827i	−0.425779	+	0.904827i
$170$		0			0
$171$	−0.312619	−	0.227131i	−0.312619	−	0.227131i
$172$	1.41213	−	0.362574i	1.41213	−	0.362574i
$173$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$173$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$174$		0			0
$175$		0			0
$176$	−1.44644	+	1.35830i	−1.44644	+	1.35830i
$177$	1.05150	−	0.269980i	1.05150	−	0.269980i
$178$	1.69755	−	0.933237i	1.69755	−	0.933237i
$179$	−0.996398	−	0.394502i	−0.996398	−	0.394502i	−0.187381	−	0.982287i	$-0.560000\pi$
$179$	−0.996398	−	0.394502i	−0.996398	−	0.394502i	−0.809017	+	0.587785i	$0.800000\pi$
$180$		0			0
$181$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$181$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.22632	+	3.77423i	−1.22632	+	3.77423i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$191$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$192$	−0.929324	+	0.872693i	−0.929324	+	0.872693i
$193$	−0.456288	+	0.718995i	−0.456288	+	0.718995i	−0.992115	−	0.125333i	$-0.960000\pi$
$193$	−0.456288	+	0.718995i	−0.456288	+	0.718995i	0.535827	+	0.844328i	$0.320000\pi$
$194$	−0.124591	−	0.0157395i	−0.124591	−	0.0157395i
$195$		0			0
$196$	0.876307	+	0.481754i	0.876307	+	0.481754i
$197$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$197$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$198$	0.790797	+	0.955910i	0.790797	+	0.955910i
$199$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$199$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$200$	0.309017	+	0.951057i	0.309017	+	0.951057i
$201$	−1.91789	−	1.39343i	−1.91789	−	1.39343i
$202$		0			0
$203$		0			0
$204$	−0.787899	+	2.42490i	−0.787899	+	2.42490i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.893950	−	0.839475i	−0.893950	−	0.839475i
$210$		0			0
$211$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.0627905	−	0.998027i	$-0.480000\pi$
$211$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$212$		0			0
$213$		0			0
$214$	−0.683098	−	1.07639i	−0.683098	−	1.07639i
$215$		0			0
$216$	−0.304539	−	0.368125i	−0.304539	−	0.368125i
$217$		0			0
$218$		0			0
$219$	−0.787899			−0.787899
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$223$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$224$		0			0
$225$	0.605594	−	0.155490i	0.605594	−	0.155490i
$226$	0.450527	−	1.38658i	0.450527	−	1.38658i
$227$	1.60528	−	0.202793i	1.60528	−	0.202793i	0.728969	−	0.684547i	$-0.240000\pi$
$227$	1.60528	−	0.202793i	1.60528	−	0.202793i	0.876307	+	0.481754i	$0.160000\pi$
$228$	−0.574354	−	0.539354i	−0.574354	−	0.539354i
$229$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$229$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.41213	−	0.362574i	1.41213	−	0.362574i	0.535827	−	0.844328i	$-0.320000\pi$
$233$	1.41213	−	0.362574i	1.41213	−	0.362574i	0.876307	+	0.481754i	$0.160000\pi$
$234$		0			0
$235$		0			0
$236$	0.844844	−	0.106729i	0.844844	−	0.106729i
$237$		0			0
$238$		0			0
$239$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$239$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$240$		0			0
$241$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$241$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$242$	1.57381	+	2.47993i	1.57381	+	2.47993i
$243$	−0.886520	+	0.644095i	−0.886520	+	0.644095i
$244$		0			0
$245$		0			0
$246$	0.129521	−	0.0941025i	0.129521	−	0.0941025i
$247$		0			0
$248$		0			0
$249$	−0.690441	+	2.12496i	−0.690441	+	2.12496i
$250$		0			0
$251$	0.0627905	−	0.998027i	0.0627905	−	0.998027i
$251$	0.0627905	−	0.998027i	0.0627905	−	0.998027i
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$257$	−0.620759	+	0.582932i	−0.620759	+	0.582932i	−0.929776	−	0.368125i	$-0.880000\pi$
$257$	−0.620759	+	0.582932i	−0.620759	+	0.582932i	0.309017	+	0.951057i	$0.400000\pi$
$258$	−0.116705	−	1.85498i	−0.116705	−	1.85498i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.688925	+	1.46404i	0.688925	+	1.46404i
$263$		0			0		−0.425779	−	0.904827i	$-0.640000\pi$
$263$		0			0		0.425779	+	0.904827i	$0.360000\pi$
$264$	1.35542	+	2.13580i	1.35542	+	2.13580i
$265$		0			0
$266$		0			0
$267$	−0.763146	−	2.34872i	−0.763146	−	2.34872i
$268$	−1.35556	−	1.27295i	−1.35556	−	1.27295i
$269$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$269$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−0.851559	+	1.80965i	−0.851559	+	1.80965i
$273$		0			0
$274$	−0.929324	−	0.872693i	−0.929324	−	0.872693i
$275$	1.96858	−	0.248690i	1.96858	−	0.248690i
$276$		0			0
$277$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$277$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$278$	0.110048	+	1.74915i	0.110048	+	1.74915i
$279$		0			0
$280$		0			0
$281$	−0.0235315	+	0.123357i	−0.0235315	+	0.123357i	−0.992115	−	0.125333i	$-0.960000\pi$
$281$	−0.0235315	+	0.123357i	−0.0235315	+	0.123357i	0.968583	+	0.248690i	$0.0800000\pi$
$282$		0			0
$283$	1.07165			1.07165			0.535827	−	0.844328i	$-0.320000\pi$
$283$	1.07165			1.07165			0.535827	+	0.844328i	$0.320000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.335019	+	0.527905i	0.335019	+	0.527905i
$289$	0.188372	+	2.99408i	0.188372	+	2.99408i
$290$		0			0
$291$	−0.0494726	+	0.152261i	−0.0494726	+	0.152261i
$292$	−0.613161	−	0.0774602i	−0.613161	−	0.0774602i
$293$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$293$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$294$	0.812619	−	0.982287i	0.812619	−	0.982287i
$295$		0			0
$296$		0			0
$297$	−0.830735	+	0.456701i	−0.830735	+	0.456701i
$298$		0			0
$299$		0			0
$300$	1.26480	−	0.159781i	1.26480	−	0.159781i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.393950	−	0.476203i	−0.393950	−	0.476203i
$305$		0			0
$306$	1.09580	+	0.602421i	1.09580	+	0.602421i
$307$	−0.683098	−	0.825723i	−0.683098	−	0.825723i	0.309017	−	0.951057i	$-0.400000\pi$
$307$	−0.683098	−	0.825723i	−0.683098	−	0.825723i	−0.992115	+	0.125333i	$0.960000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$311$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$312$		0			0
$313$	−1.92189	+	0.242791i	−1.92189	+	0.242791i	−0.992115	−	0.125333i	$-0.960000\pi$
$313$	−1.92189	+	0.242791i	−1.92189	+	0.242791i	−0.929776	+	0.368125i	$0.880000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.929776	−	0.368125i	$-0.880000\pi$
$317$		0			0		0.929776	+	0.368125i	$0.120000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−1.51111	+	0.598290i	−1.51111	+	0.598290i
$322$		0			0
$323$	−1.14927	−	0.455027i	−1.14927	−	0.455027i
$324$	−1.08164	+	0.594636i	−1.08164	+	0.594636i
$325$		0			0
$326$	1.41213	−	1.32608i	1.41213	−	1.32608i
$327$		0			0
$328$	0.110048	−	0.0604991i	0.110048	−	0.0604991i
$329$		0			0
$330$		0			0
$331$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$331$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$332$	−0.746226	+	1.58581i	−0.746226	+	1.58581i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.06320	−	1.67534i	−1.06320	−	1.67534i	−0.637424	−	0.770513i	$-0.720000\pi$
$337$	−1.06320	−	1.67534i	−1.06320	−	1.67534i	−0.425779	−	0.904827i	$-0.640000\pi$
$338$	0.728969	+	0.684547i	0.728969	+	0.684547i
$339$	−1.62875	−	0.895411i	−1.62875	−	0.895411i
$340$		0			0
$341$		0			0
$342$	−0.312619	+	0.227131i	−0.312619	+	0.227131i
$343$		0			0
$344$	0.0915446	−	1.45506i	0.0915446	−	1.45506i
$345$		0			0
$346$		0			0
$347$	−0.746226	−	0.410241i	−0.746226	−	0.410241i	0.0627905	−	0.998027i	$-0.480000\pi$
$347$	−0.746226	−	0.410241i	−0.746226	−	0.410241i	−0.809017	+	0.587785i	$0.800000\pi$
$348$		0			0
$349$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$349$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$350$		0			0
$351$		0			0
$352$	0.844844	+	1.79538i	0.844844	+	1.79538i
$353$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$353$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$354$	0.0681659	−	1.08347i	0.0681659	−	1.08347i
$355$		0			0
$356$	−0.362989	−	1.90285i	−0.362989	−	1.90285i
$357$		0			0
$358$	−0.683098	+	0.825723i	−0.683098	+	0.825723i
$359$		0			0		0.535827	−	0.844328i	$-0.320000\pi$
$359$		0			0		−0.535827	+	0.844328i	$0.680000\pi$
$360$		0			0
$361$	−0.450527	+	0.423073i	−0.450527	+	0.423073i
$362$		0			0
$363$	3.48149	−	1.37842i	3.48149	−	1.37842i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.637424	−	0.770513i	$-0.720000\pi$
$367$		0			0		0.637424	+	0.770513i	$0.280000\pi$
$368$		0			0
$369$	−0.0334313	−	0.0710452i	−0.0334313	−	0.0710452i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$373$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$374$	3.21055	+	2.33260i	3.21055	+	2.33260i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$379$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.637424	−	0.770513i	$-0.280000\pi$
$383$		0			0		−0.637424	+	0.770513i	$0.720000\pi$
$384$	0.542804	+	1.15352i	0.542804	+	1.15352i
$385$		0			0
$386$	0.542804	+	0.656137i	0.542804	+	0.656137i
$387$	−0.904369	−	0.114248i	−0.904369	−	0.114248i
$388$	−0.0534698	+	0.113629i	−0.0534698	+	0.113629i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	0.728969	−	0.684547i	0.728969	−	0.684547i
$393$	1.99794	−	0.512984i	1.99794	−	0.512984i
$394$		0			0
$395$		0			0
$396$	1.15349	−	0.456701i	1.15349	−	0.456701i
$397$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$397$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$398$		0			0
$399$		0			0
$400$	1.00000			1.00000
$401$	0.159566	+	0.339095i	0.159566	+	0.339095i	0.968583	−	0.248690i	$-0.0800000\pi$
$401$	0.159566	+	0.339095i	0.159566	+	0.339095i	−0.809017	+	0.587785i	$0.800000\pi$
$402$	−1.91789	+	1.39343i	−1.91789	+	1.39343i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	2.06275	+	1.49867i	2.06275	+	1.49867i
$409$	0.110048	−	1.74915i	0.110048	−	1.74915i	−0.425779	−	0.904827i	$-0.640000\pi$
$409$	0.110048	−	1.74915i	0.110048	−	1.74915i	0.535827	−	0.844328i	$-0.320000\pi$
$410$		0			0
$411$	−1.31484	+	0.955291i	−1.31484	+	0.955291i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	2.21670	+	0.280034i	2.21670	+	0.280034i
$418$	−1.07463	+	0.590785i	−1.07463	+	0.590785i
$419$	−0.328407	−	1.72157i	−0.328407	−	1.72157i	−0.637424	−	0.770513i	$-0.720000\pi$
$419$	−0.328407	−	1.72157i	−0.328407	−	1.72157i	0.309017	−	0.951057i	$-0.400000\pi$
$420$		0			0
$421$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$421$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$422$	−1.56720	−	1.13864i	−1.56720	−	1.13864i
$423$		0			0
$424$		0			0
$425$	1.75261	−	0.963507i	1.75261	−	0.963507i
$426$		0			0
$427$		0			0
$428$	−1.23480	+	0.317042i	−1.23480	+	0.317042i
$429$		0			0
$430$		0			0
$431$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$431$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$432$	−0.444215	+	0.175877i	−0.444215	+	0.175877i
$433$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$433$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.243474	+	0.749337i	−0.243474	+	0.749337i
$439$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$439$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$440$		0			0
$441$	−0.398541	−	0.481754i	−0.398541	−	0.481754i
$442$		0			0
$443$	1.45794	−	1.36909i	1.45794	−	1.36909i	0.728969	−	0.684547i	$-0.240000\pi$
$443$	1.45794	−	1.36909i	1.45794	−	1.36909i	0.728969	−	0.684547i	$-0.240000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.11716	−	1.35041i	−1.11716	−	1.35041i	−0.929776	−	0.368125i	$-0.880000\pi$
$449$	−1.11716	−	1.35041i	−1.11716	−	1.35041i	−0.187381	−	0.982287i	$-0.560000\pi$
$450$	0.0392590	−	0.624004i	0.0392590	−	0.624004i
$451$	−0.0770013	−	0.236986i	−0.0770013	−	0.236986i
$452$	−1.17950	−	0.856954i	−1.17950	−	0.856954i
$453$		0			0
$454$	0.303189	−	1.58937i	0.303189	−	1.58937i
$455$		0			0
$456$	−0.690441	+	0.379573i	−0.690441	+	0.379573i
$457$	−0.574633	−	0.227513i	−0.574633	−	0.227513i	0.0627905	−	0.998027i	$-0.480000\pi$
$457$	−0.574633	−	0.227513i	−0.574633	−	0.227513i	−0.637424	+	0.770513i	$0.720000\pi$
$458$		0			0
$459$	−0.609078	+	0.736249i	−0.609078	+	0.736249i
$460$		0			0
$461$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$461$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$462$		0			0
$463$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$463$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$464$		0			0
$465$		0			0
$466$	0.0915446	−	1.45506i	0.0915446	−	1.45506i
$467$	0.238883	+	0.288760i	0.238883	+	0.288760i	0.876307	−	0.481754i	$-0.160000\pi$
$467$	0.238883	+	0.288760i	0.238883	+	0.288760i	−0.637424	+	0.770513i	$0.720000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.159566	−	0.836475i	0.159566	−	0.836475i
$473$	−2.80200	−	0.719430i	−2.80200	−	0.719430i
$474$		0			0
$475$	0.0388067	+	0.616814i	0.0388067	+	0.616814i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.728969	−	0.684547i	$-0.760000\pi$
$479$		0			0		0.728969	+	0.684547i	$0.240000\pi$
$480$		0			0
$481$		0			0
$482$	1.07165			1.07165
$483$		0			0
$484$	2.84489	−	0.730444i	2.84489	−	0.730444i
$485$		0			0
$486$	0.338621	+	1.04217i	0.338621	+	1.04217i
$487$		0			0		0.992115	−	0.125333i	$-0.0400000\pi$
$487$		0			0		−0.992115	+	0.125333i	$0.960000\pi$
$488$		0			0
$489$	−1.32327	−	2.08515i	−1.32327	−	2.08515i
$490$		0			0
$491$	−0.0534698	−	0.113629i	−0.0534698	−	0.113629i	0.876307	−	0.481754i	$-0.160000\pi$
$491$	−0.0534698	−	0.113629i	−0.0534698	−	0.113629i	−0.929776	+	0.368125i	$0.880000\pi$
$492$	−0.0494726	−	0.152261i	−0.0494726	−	0.152261i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	1.80760	+	1.31330i	1.80760	+	1.31330i
$499$	−1.62954	−	0.895846i	−1.62954	−	0.895846i	−0.992115	−	0.125333i	$-0.960000\pi$
$499$	−1.62954	−	0.895846i	−1.62954	−	0.895846i	−0.637424	−	0.770513i	$-0.720000\pi$
$500$		0			0
$501$		0			0
$502$	−0.929776	−	0.368125i	−0.929776	−	0.368125i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.03137	−	0.749337i	1.03137	−	0.749337i
$508$		0			0
$509$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$509$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$510$		0			0
$511$		0			0
$512$	0.309017	+	0.951057i	0.309017	+	0.951057i
$513$	−0.125722	−	0.267173i	−0.125722	−	0.267173i
$514$	0.362576	+	0.770513i	0.362576	+	0.770513i
$515$		0			0
$516$	−1.80026	−	0.462227i	−1.80026	−	0.462227i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.996398	+	0.394502i	−0.996398	+	0.394502i	−0.809017	−	0.587785i	$-0.800000\pi$
$521$	−0.996398	+	0.394502i	−0.996398	+	0.394502i	−0.187381	+	0.982287i	$0.560000\pi$
$522$		0			0
$523$	0.159566	−	0.339095i	0.159566	−	0.339095i	−0.809017	−	0.587785i	$-0.800000\pi$
$523$	0.159566	−	0.339095i	0.159566	−	0.339095i	0.968583	+	0.248690i	$0.0800000\pi$
$524$	1.60528	−	0.202793i	1.60528	−	0.202793i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	2.45012	−	0.629084i	2.45012	−	0.629084i
$529$	0.0627905	+	0.998027i	0.0627905	+	0.998027i
$530$		0			0
$531$	−0.515699	−	0.132409i	−0.515699	−	0.132409i
$532$		0			0
$533$		0			0
$534$	−2.46959			−2.46959
$535$		0			0
$536$	−1.62954	+	0.895846i	−1.62954	+	0.895846i
$537$	0.870846	+	1.05267i	0.870846	+	1.05267i
$538$		0			0
$539$	−1.06320	−	1.67534i	−1.06320	−	1.67534i
$540$		0			0
$541$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$541$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$542$		0			0
$543$		0			0
$544$	1.45794	+	1.36909i	1.45794	+	1.36909i
$545$		0			0
$546$		0			0
$547$	1.72897	+	0.684547i	1.72897	+	0.684547i	1.00000			$0$
$547$	1.72897	+	0.684547i	1.72897	+	0.684547i	0.728969	+	0.684547i	$0.240000\pi$
$548$	−1.11716	+	0.614163i	−1.11716	+	0.614163i
$549$		0			0
$550$	0.371808	−	1.94908i	0.371808	−	1.94908i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	1.69755	+	0.435857i	1.69755	+	0.435857i
$557$		0			0		−0.876307	−	0.481754i	$-0.840000\pi$
$557$		0			0		0.876307	+	0.481754i	$0.160000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	3.68798	−	3.46325i	3.68798	−	3.46325i
$562$	0.110048	+	0.0604991i	0.110048	+	0.0604991i
$563$	1.03137	+	1.24672i	1.03137	+	1.24672i	0.968583	+	0.248690i	$0.0800000\pi$
$563$	1.03137	+	1.24672i	1.03137	+	1.24672i	0.0627905	+	0.998027i	$0.480000\pi$
$564$		0			0
$565$		0			0
$566$	0.331159	−	1.01920i	0.331159	−	1.01920i
$567$		0			0
$568$		0			0
$569$	0.844844	+	0.106729i	0.844844	+	0.106729i	0.535827	−	0.844328i	$-0.320000\pi$
$569$	0.844844	+	0.106729i	0.844844	+	0.106729i	0.309017	+	0.951057i	$0.400000\pi$
$570$		0			0
$571$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$571$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.605594	−	0.155490i	0.605594	−	0.155490i
$577$	−1.35556	+	1.27295i	−1.35556	+	1.27295i	−0.425779	+	0.904827i	$0.640000\pi$
$577$	−1.35556	+	1.27295i	−1.35556	+	1.27295i	−0.929776	+	0.368125i	$0.880000\pi$
$578$	2.90575	+	0.746070i	2.90575	+	0.746070i
$579$	0.951325	−	0.522996i	0.951325	−	0.522996i
$580$		0			0
$581$		0			0
$582$	0.129521	+	0.0941025i	0.129521	+	0.0941025i
$583$		0			0
$584$	−0.263146	+	0.559214i	−0.263146	+	0.559214i
$585$		0			0
$586$		0			0
$587$	−1.92189	−	0.242791i	−1.92189	−	0.242791i	−0.929776	−	0.368125i	$-0.880000\pi$
$587$	−1.92189	−	0.242791i	−1.92189	−	0.242791i	−0.992115	+	0.125333i	$0.960000\pi$
$588$	−0.683098	−	1.07639i	−0.683098	−	1.07639i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$593$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$594$	0.177637	+	0.931204i	0.177637	+	0.931204i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$599$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$600$	0.238883	−	1.25227i	0.238883	−	1.25227i
$601$	0.331159	−	0.521823i	0.331159	−	0.521823i	−0.637424	−	0.770513i	$-0.720000\pi$
$601$	0.331159	−	0.521823i	0.331159	−	0.521823i	0.968583	+	0.248690i	$0.0800000\pi$
$602$		0			0
$603$	0.495037	+	1.05201i	0.495037	+	1.05201i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$607$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$608$	−0.574633	+	0.227513i	−0.574633	+	0.227513i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.911557	−	0.856009i	0.911557	−	0.856009i
$613$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$613$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$614$	−0.996398	+	0.394502i	−0.996398	+	0.394502i
$615$		0			0
$616$		0			0
$617$	0.844844	+	0.106729i	0.844844	+	0.106729i	0.535827	−	0.844328i	$-0.320000\pi$
$617$	0.844844	+	0.106729i	0.844844	+	0.106729i	0.309017	+	0.951057i	$0.400000\pi$
$618$		0			0
$619$	−0.620759	+	1.31918i	−0.620759	+	1.31918i	0.309017	+	0.951057i	$0.400000\pi$
$619$	−0.620759	+	1.31918i	−0.620759	+	1.31918i	−0.929776	+	0.368125i	$0.880000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$626$	−0.362989	+	1.90285i	−0.362989	+	1.90285i
$627$	0.483109	+	1.48686i	0.483109	+	1.48686i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$631$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$632$		0			0
$633$	−1.80026	+	1.69055i	−1.80026	+	1.69055i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.50441	−	0.595638i	1.50441	−	0.595638i	0.535827	−	0.844328i	$-0.320000\pi$
$641$	1.50441	−	0.595638i	1.50441	−	0.595638i	0.968583	+	0.248690i	$0.0800000\pi$
$642$	0.102049	+	1.62203i	0.102049	+	1.62203i
$643$	−0.273190	+	0.256543i	−0.273190	+	0.256543i	−0.809017	−	0.587785i	$-0.800000\pi$
$643$	−0.273190	+	0.256543i	−0.273190	+	0.256543i	0.535827	+	0.844328i	$0.320000\pi$
$644$		0			0
$645$		0			0
$646$	−0.787899	+	0.952407i	−0.787899	+	0.952407i
$647$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$647$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$648$	0.231288	+	1.21245i	0.231288	+	1.21245i
$649$	−1.57103	−	0.622015i	−1.57103	−	0.622015i
$650$		0			0
$651$		0			0
$652$	−0.824805	−	1.75280i	−0.824805	−	1.75280i
$653$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$653$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$654$		0			0
$655$		0			0
$656$	−0.0235315	−	0.123357i	−0.0235315	−	0.123357i
$657$	0.338621	+	0.186158i	0.338621	+	0.186158i
$658$		0			0
$659$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$659$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$660$		0			0
$661$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$661$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$662$	0.688925	−	0.500534i	0.688925	−	0.500534i
$663$		0			0
$664$	1.27760	+	1.19975i	1.27760	+	1.19975i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$673$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$674$	−1.92189	+	0.493458i	−1.92189	+	0.493458i
$675$	0.462756	+	0.118815i	0.462756	+	0.118815i
$676$	0.876307	−	0.481754i	0.876307	−	0.481754i
$677$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$677$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$678$	−1.35490	+	1.27233i	−1.35490	+	1.27233i
$679$		0			0
$680$		0			0
$681$	−1.91789	−	0.759348i	−1.91789	−	0.759348i
$682$		0			0
$683$	1.84489	−	0.730444i	1.84489	−	0.730444i	0.876307	−	0.481754i	$-0.160000\pi$
$683$	1.84489	−	0.730444i	1.84489	−	0.730444i	0.968583	−	0.248690i	$-0.0800000\pi$
$684$	0.119410	+	0.367505i	0.119410	+	0.367505i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−1.35556	−	0.536702i	−1.35556	−	0.536702i
$689$		0			0
$690$		0			0
$691$	−0.124591	+	0.0157395i	−0.124591	+	0.0157395i	−0.187381	−	0.982287i	$-0.560000\pi$
$691$	−0.124591	+	0.0157395i	−0.124591	+	0.0157395i	0.0627905	+	0.998027i	$0.480000\pi$
$692$		0			0
$693$		0			0
$694$	−0.620759	+	0.582932i	−0.620759	+	0.582932i
$695$		0			0
$696$		0			0
$697$	−0.160097	−	0.193524i	−0.160097	−	0.193524i
$698$		0			0
$699$	−1.80026	−	0.462227i	−1.80026	−	0.462227i
$700$		0			0
$701$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$701$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$702$		0			0
$703$		0			0
$704$	1.96858	−	0.248690i	1.96858	−	0.248690i
$705$		0			0
$706$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$707$		0			0
$708$	−1.00937	−	0.399639i	−1.00937	−	0.399639i
$709$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$709$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$710$		0			0
$711$		0			0
$712$	−1.92189	−	0.242791i	−1.92189	−	0.242791i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.574221	+	0.904827i	0.574221	+	0.904827i
$717$		0			0
$718$		0			0
$719$		0			0		0.876307	−	0.481754i	$-0.160000\pi$
$719$		0			0		−0.876307	+	0.481754i	$0.840000\pi$
$720$		0			0
$721$		0			0
$722$	0.263146	+	0.559214i	0.263146	+	0.559214i
$723$	0.255999	−	1.34200i	0.255999	−	1.34200i
$724$		0			0
$725$		0			0
$726$	−0.235115	−	3.73705i	−0.235115	−	3.73705i
$727$		0			0		0.968583	−	0.248690i	$-0.0800000\pi$
$727$		0			0		−0.968583	+	0.248690i	$0.920000\pi$
$728$		0			0
$729$	0.161379	−	0.0203870i	0.161379	−	0.0203870i
$730$		0			0
$731$	−2.89288	+	0.365456i	−2.89288	+	0.365456i
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.14020	+	3.50919i	1.14020	+	3.50919i
$738$	−0.0778988	+	0.00984091i	−0.0778988	+	0.00984091i
$739$	−1.92189	−	0.493458i	−1.92189	−	0.493458i	−0.992115	−	0.125333i	$-0.960000\pi$
$739$	−1.92189	−	0.493458i	−1.92189	−	0.493458i	−0.929776	−	0.368125i	$-0.880000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$743$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.798803	−	0.750126i	0.798803	−	0.750126i
$748$	3.21055	−	2.33260i	3.21055	−	2.33260i
$749$		0			0
$750$		0			0
$751$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$751$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$752$		0			0
$753$	−0.683098	+	1.07639i	−0.683098	+	1.07639i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$757$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$758$	1.50441	−	1.09302i	1.50441	−	1.09302i
$759$		0			0
$760$		0			0
$761$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$761$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.26480	−	0.159781i	1.26480	−	0.159781i
$769$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$769$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$770$		0			0
$771$	1.05150	−	0.269980i	1.05150	−	0.269980i
$772$	0.791759	−	0.313480i	0.791759	−	0.313480i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$	−0.388122	+	0.824801i	−0.388122	+	0.824801i
$775$		0			0
$776$	0.0915446	+	0.0859661i	0.0915446	+	0.0859661i
$777$		0			0
$778$		0			0
$779$	0.0751750	−	0.0193017i	0.0751750	−	0.0193017i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$785$		0			0
$786$	0.129521	−	2.05868i	0.129521	−	2.05868i
$787$	1.27760	−	0.702367i	1.27760	−	0.702367i	0.309017	−	0.951057i	$-0.400000\pi$
$787$	1.27760	−	0.702367i	1.27760	−	0.702367i	0.968583	+	0.248690i	$0.0800000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.0778988	−	1.23817i	−0.0778988	−	1.23817i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$797$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$798$		0			0
$799$		0			0
$800$	0.309017	−	0.951057i	0.309017	−	0.951057i
$801$	−0.226954	+	1.18974i	−0.226954	+	1.18974i
$802$	0.371808	−	0.0469702i	0.371808	−	0.0469702i
$803$	0.992115	+	0.720814i	0.992115	+	0.720814i
$804$	0.732570	+	2.25462i	0.732570	+	2.25462i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.26480	+	1.52888i	1.26480	+	1.52888i	0.728969	+	0.684547i	$0.240000\pi$
$809$	1.26480	+	1.52888i	1.26480	+	1.52888i	0.535827	+	0.844328i	$0.320000\pi$
$810$		0			0
$811$	1.03799	−	1.63560i	1.03799	−	1.63560i	0.309017	−	0.951057i	$-0.400000\pi$
$811$	1.03799	−	1.63560i	1.03799	−	1.63560i	0.728969	−	0.684547i	$-0.240000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	2.06275	−	1.49867i	2.06275	−	1.49867i
$817$	0.278441	−	0.856954i	0.278441	−	0.856954i
$818$	−1.62954	−	0.645180i	−1.62954	−	0.645180i
$819$		0			0
$820$		0			0
$821$		0			0		−0.187381	−	0.982287i	$-0.560000\pi$
$821$		0			0		0.187381	+	0.982287i	$0.440000\pi$
$822$	0.502226	+	1.54569i	0.502226	+	1.54569i
$823$		0			0		0.929776	−	0.368125i	$-0.120000\pi$
$823$		0			0		−0.929776	+	0.368125i	$0.880000\pi$
$824$		0			0
$825$	−2.35195	−	0.931204i	−2.35195	−	0.931204i
$826$		0			0
$827$	−0.362989	+	0.0931997i	−0.362989	+	0.0931997i	−0.425779	−	0.904827i	$-0.640000\pi$
$827$	−0.362989	+	0.0931997i	−0.362989	+	0.0931997i	0.0627905	+	0.998027i	$0.480000\pi$
$828$		0			0
$829$		0			0		−0.968583	−	0.248690i	$-0.920000\pi$
$829$		0			0		0.968583	+	0.248690i	$0.0800000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$834$	0.951325	−	2.02167i	0.951325	−	2.02167i
$835$		0			0
$836$	0.229790	+	1.20460i	0.229790	+	1.20460i
$837$		0			0
$838$	−1.73879	−	0.219661i	−1.73879	−	0.219661i
$839$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$839$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$840$		0			0
$841$	0.876307	+	0.481754i	0.876307	+	0.481754i
$842$		0			0
$843$	0.102049	−	0.123357i	0.102049	−	0.123357i
$844$	−1.56720	+	1.13864i	−1.56720	+	1.13864i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.19721	−	0.658170i	−1.19721	−	0.658170i
$850$	−0.374763	−	1.96457i	−0.374763	−	1.96457i
$851$		0			0
$852$		0			0
$853$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$853$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$854$		0			0
$855$		0			0
$856$	−0.0800484	+	1.27233i	−0.0800484	+	1.27233i
$857$	−1.80113	−	0.713118i	−1.80113	−	0.713118i	−0.992115	−	0.125333i	$-0.960000\pi$
$857$	−1.80113	−	0.713118i	−1.80113	−	0.713118i	−0.809017	−	0.587785i	$-0.800000\pi$
$858$		0			0
$859$	−1.80113	+	0.713118i	−1.80113	+	0.713118i	−0.809017	+	0.587785i	$0.800000\pi$
$859$	−1.80113	+	0.713118i	−1.80113	+	0.713118i	−0.992115	+	0.125333i	$0.960000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.728969	−	0.684547i	$-0.240000\pi$
$863$		0			0		−0.728969	+	0.684547i	$0.760000\pi$
$864$	0.0299991	+	0.476823i	0.0299991	+	0.476823i
$865$		0			0
$866$	−1.61803			−1.61803
$867$	1.62841	−	3.46055i	1.62841	−	3.46055i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.0572371	−	0.0537492i	0.0572371	−	0.0537492i
$874$		0			0
$875$		0			0
$876$	0.637424	+	0.463116i	0.637424	+	0.463116i
$877$		0			0		0.187381	−	0.982287i	$-0.440000\pi$
$877$		0			0		−0.187381	+	0.982287i	$0.560000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.876307	+	0.481754i	$0.160000\pi$
$881$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.728969	+	0.684547i	$0.240000\pi$
$882$	−0.581331	+	0.230165i	−0.581331	+	0.230165i
$883$	−0.393950	+	0.476203i	−0.393950	+	0.476203i	−0.929776	−	0.368125i	$-0.880000\pi$
$883$	−0.393950	+	0.476203i	−0.393950	+	0.476203i	0.535827	+	0.844328i	$0.320000\pi$
$884$		0			0
$885$		0			0
$886$	−0.851559	−	1.80965i	−0.851559	−	1.80965i
$887$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$887$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	2.44917			2.44917
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.62954	+	0.645180i	−1.62954	+	0.645180i
$899$		0			0
$900$	−0.581331	−	0.230165i	−0.581331	−	0.230165i
$901$		0			0
$902$	−0.249182			−0.249182
$903$		0			0
$904$	−1.17950	+	0.856954i	−1.17950	+	0.856954i
$905$		0			0
$906$		0			0
$907$	0.0702235	+	0.368125i	0.0702235	+	0.368125i	1.00000			$0$
$907$	0.0702235	+	0.368125i	0.0702235	+	0.368125i	−0.929776	+	0.368125i	$0.880000\pi$
$908$	−1.41789	−	0.779494i	−1.41789	−	0.779494i
$909$		0			0
$910$		0			0
$911$		0			0		0.0627905	−	0.998027i	$-0.480000\pi$
$911$		0			0		−0.0627905	+	0.998027i	$0.520000\pi$
$912$	0.147638	+	0.773944i	0.147638	+	0.773944i
$913$	2.81343	−	2.04407i	2.81343	−	2.04407i
$914$	−0.393950	+	0.476203i	−0.393950	+	0.476203i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	0.511999	+	0.806781i	0.511999	+	0.806781i
$919$		0			0		−0.992115	−	0.125333i	$-0.960000\pi$
$919$		0			0		0.992115	+	0.125333i	$0.0400000\pi$
$920$		0			0
$921$	0.255999	+	1.34200i	0.255999	+	1.34200i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.781202	−	0.733597i	0.781202	−	0.733597i	−0.187381	−	0.982287i	$-0.560000\pi$
$929$	0.781202	−	0.733597i	0.781202	−	0.733597i	0.968583	+	0.248690i	$0.0800000\pi$
$930$		0			0
$931$	0.541587	−	0.297740i	0.541587	−	0.297740i
$932$	−1.35556	−	0.536702i	−1.35556	−	0.536702i
$933$		0			0
$934$	0.348445	−	0.137959i	0.348445	−	0.137959i
$935$		0			0
$936$		0			0
$937$	1.96858	+	0.248690i	1.96858	+	0.248690i	1.00000			$0$
$937$	1.96858	+	0.248690i	1.96858	+	0.248690i	0.968583	+	0.248690i	$0.0800000\pi$
$938$		0			0
$939$	2.29617	+	0.909118i	2.29617	+	0.909118i
$940$		0			0
$941$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$941$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$942$		0			0
$943$		0			0
$944$	−0.746226	−	0.410241i	−0.746226	−	0.410241i
$945$		0			0
$946$	−1.55008	+	2.44254i	−1.55008	+	2.44254i
$947$	−0.613161	−	0.0774602i	−0.613161	−	0.0774602i	−0.187381	−	0.982287i	$-0.560000\pi$
$947$	−0.613161	−	0.0774602i	−0.613161	−	0.0774602i	−0.425779	+	0.904827i	$0.640000\pi$
$948$		0			0
$949$		0			0
$950$	0.598617	+	0.153699i	0.598617	+	0.153699i
$951$		0			0
$952$		0			0
$953$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$953$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.637424	+	0.770513i	−0.637424	+	0.770513i
$962$		0			0
$963$	0.790797	+	0.0999009i	0.790797	+	0.0999009i
$964$	0.331159	−	1.01920i	0.331159	−	1.01920i
$965$		0			0
$966$		0			0
$967$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$967$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$968$	0.184426	−	2.93137i	0.184426	−	2.93137i
$969$	1.00445	+	1.21417i	1.00445	+	1.21417i
$970$		0			0
$971$	0.0388067	−	0.616814i	0.0388067	−	0.616814i	−0.929776	−	0.368125i	$-0.880000\pi$
$971$	0.0388067	−	0.616814i	0.0388067	−	0.616814i	0.968583	−	0.248690i	$-0.0800000\pi$
$972$	1.09580			1.09580
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.110048	+	1.74915i	0.110048	+	1.74915i	0.535827	+	0.844328i	$0.320000\pi$
$977$	0.110048	+	1.74915i	0.110048	+	1.74915i	−0.425779	+	0.904827i	$0.640000\pi$
$978$	−2.39201	+	0.614163i	−2.39201	+	0.614163i
$979$	−1.18779	+	3.65565i	−1.18779	+	3.65565i
$980$		0			0
$981$		0			0
$982$	−0.124591	+	0.0157395i	−0.124591	+	0.0157395i
$983$		0			0		0.425779	−	0.904827i	$-0.360000\pi$
$983$		0			0		−0.425779	+	0.904827i	$0.640000\pi$
$984$	−0.160097			−0.160097
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.535827	−	0.844328i	$-0.680000\pi$
$991$		0			0		0.535827	+	0.844328i	$0.320000\pi$
$992$		0			0
$993$	−0.462229	−	0.982287i	−0.462229	−	0.982287i
$994$		0			0
$995$		0			0
$996$	1.80760	−	1.31330i	1.80760	−	1.31330i
$997$		0			0		−0.0627905	−	0.998027i	$-0.520000\pi$
$997$		0			0		0.0627905	+	0.998027i	$0.480000\pi$
$998$	−1.35556	+	1.27295i	−1.35556	+	1.27295i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.1.w.a.267.1	✓	20
8.3	odd	2	CM	2008.1.w.a.267.1	✓	20
251.204	even	25	inner	2008.1.w.a.1459.1	yes	20
2008.1459	odd	50	inner	2008.1.w.a.1459.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.1.w.a.267.1	✓	20	1.1	even	1	trivial
2008.1.w.a.267.1	✓	20	8.3	odd	2	CM
2008.1.w.a.1459.1	yes	20	251.204	even	25	inner
2008.1.w.a.1459.1	yes	20	2008.1459	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)

\(f(q)\)	\(=\)	\(q+(0.309017 - 0.951057i) q^{2} +(-1.11716 - 0.614163i) q^{3} +(-0.809017 - 0.587785i) q^{4} +(-0.929324 + 0.872693i) q^{6} +(-0.809017 + 0.587785i) q^{8} +(0.335019 + 0.527905i) q^{9} +O(q^{10})\)	\(q+(0.309017 - 0.951057i) q^{2} +(-1.11716 - 0.614163i) q^{3} +(-0.809017 - 0.587785i) q^{4} +(-0.929324 + 0.872693i) q^{6} +(-0.809017 + 0.587785i) q^{8} +(0.335019 + 0.527905i) q^{9} +(0.844844 + 1.79538i) q^{11} +(0.542804 + 1.15352i) q^{12} +(0.309017 + 0.951057i) q^{16} +(1.45794 + 1.36909i) q^{17} +(0.605594 - 0.155490i) q^{18} +(-0.574633 + 0.227513i) q^{19} +(1.96858 - 0.248690i) q^{22} +(1.26480 - 0.159781i) q^{24} +(0.309017 - 0.951057i) q^{25} +(0.0299991 + 0.476823i) q^{27} +1.00000 q^{32} +(0.158834 - 2.52460i) q^{33} +(1.75261 - 0.963507i) q^{34} +(0.0392590 - 0.624004i) q^{36} +(0.0388067 + 0.616814i) q^{38} +(-0.124591 - 0.0157395i) q^{41} +(-0.929324 + 1.12336i) q^{43} +(0.371808 - 1.94908i) q^{44} +(0.238883 - 1.25227i) q^{48} +(-0.992115 + 0.125333i) q^{49} +(-0.809017 - 0.587785i) q^{50} +(-0.787899 - 2.42490i) q^{51} +(0.462756 + 0.118815i) q^{54} +(0.781687 + 0.0987500i) q^{57} +(-0.620759 + 0.582932i) q^{59} +(0.309017 - 0.951057i) q^{64} +(-2.35195 - 0.931204i) q^{66} +(1.84489 + 0.233064i) q^{67} +(-0.374763 - 1.96457i) q^{68} +(-0.581331 - 0.230165i) q^{72} +(0.541587 - 0.297740i) q^{73} +(-0.929324 + 0.872693i) q^{75} +(0.598617 + 0.153699i) q^{76} +(0.525546 - 1.11684i) q^{81} +(-0.0534698 + 0.113629i) q^{82} +(-0.328407 - 1.72157i) q^{83} +(0.781202 + 1.23098i) q^{86} +(-1.73879 - 0.955910i) q^{88} +(1.41213 + 1.32608i) q^{89} +(-1.11716 - 0.614163i) q^{96} +(-0.0235315 - 0.123357i) q^{97} +(-0.187381 + 0.982287i) q^{98} +(-0.664754 + 1.04749i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{2} - 5 q^{4} - 5 q^{8}+O(q^{10}) \) 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

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