LMFDB - Embedded newform 1441.1.u.a.916.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1441,1,Mod(130,1441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1441, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([6, 5]))

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

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

chi := DirichletCharacter("1441.130");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1441 = 11 \cdot 131 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1441.u (of order $10$, degree $4$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.719152683204$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$5$ over $\Q(\zeta_{10})$
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:	$\Z[a_1, \ldots, a_{11}]$
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			916.1
Root			$-0.876307 - 0.481754i$ of defining polynomial
Character	$\chi$	$=$	1441.916
Dual form			1441.1.u.a.785.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.613161 - 1.88711i) q^{3} +(0.309017 - 0.951057i) q^{4} +(1.03137 + 0.749337i) q^{5} +(0.598617 - 1.84235i) q^{7} +(-2.37622 + 1.72642i) q^{9} +O(q^{10})$	\(q+(-0.613161 - 1.88711i) q^{3} +(0.309017 - 0.951057i) q^{4} +(1.03137 + 0.749337i) q^{5} +(0.598617 - 1.84235i) q^{7} +(-2.37622 + 1.72642i) q^{9} +(-0.425779 + 0.904827i) q^{11} -1.98423 q^{12} +(0.303189 - 0.220280i) q^{13} +(0.781687 - 2.40578i) q^{15} +(-0.809017 - 0.587785i) q^{16} +(1.03137 - 0.749337i) q^{20} -3.84378 q^{21} +(0.193209 + 0.594636i) q^{25} +(3.10969 + 2.25932i) q^{27} +(-1.56720 - 1.13864i) q^{28} +(1.96858 + 0.248690i) q^{33} +(1.99794 - 1.45159i) q^{35} +(0.907634 + 2.79341i) q^{36} +(-0.601597 - 0.437086i) q^{39} +(0.450527 + 1.38658i) q^{41} +1.07165 q^{43} +(0.728969 + 0.684547i) q^{44} -3.74444 q^{45} +(-0.613161 + 1.88711i) q^{48} +(-2.22691 - 1.61795i) q^{49} +(-0.115808 - 0.356420i) q^{52} +(-0.500000 + 0.363271i) q^{53} +(-1.11716 + 0.614163i) q^{55} +(0.0388067 - 0.119435i) q^{59} +(-2.04648 - 1.48686i) q^{60} +(1.03137 + 0.749337i) q^{61} +(1.75824 + 5.41130i) q^{63} +(-0.809017 + 0.587785i) q^{64} +0.477765 q^{65} +(1.00368 - 0.729215i) q^{75} +(1.41213 + 1.32608i) q^{77} +(-0.393950 - 1.21245i) q^{80} +(1.44922 - 4.46025i) q^{81} +(-1.18779 + 3.65565i) q^{84} -1.61803 q^{89} +(-0.224339 - 0.690446i) q^{91} +(-0.550370 - 2.88514i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) $ 20 * q - 5 * q^4 - 5 * q^9	$ 20 q - 5 q^{4} - 5 q^{9} - 5 q^{16} - 5 q^{25} + 20 q^{33} + 15 q^{35} - 5 q^{36} - 10 q^{39} - 10 q^{45} - 5 q^{49} - 10 q^{53} - 10 q^{63} - 5 q^{64} - 10 q^{75} - 5 q^{81} - 10 q^{89}+O(q^{100}) $ 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 - 5 * q^25 + 20 * q^33 + 15 * q^35 - 5 * q^36 - 10 * q^39 - 10 * q^45 - 5 * q^49 - 10 * q^53 - 10 * q^63 - 5 * q^64 - 10 * q^75 - 5 * q^81 - 10 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1441\mathbb{Z}\right)^\times$.

$n$	$133$	$1311$
$\chi(n)$	$-1$	$e\left(\frac{4}{5}\right)$

Coefficient data

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

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

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

$2$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$2$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$3$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$3$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$4$	0.309017	−	0.951057i	0.309017	−	0.951057i
$5$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$5$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$6$		0			0
$7$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.0627905	−	0.998027i	$-0.480000\pi$
$7$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$8$		0			0
$9$	−2.37622	+	1.72642i	−2.37622	+	1.72642i
$10$		0			0
$11$	−0.425779	+	0.904827i	−0.425779	+	0.904827i
$11$	−0.425779	+	0.904827i	−0.425779	+	0.904827i
$12$	−1.98423			−1.98423
$13$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$13$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$14$		0			0
$15$	0.781687	−	2.40578i	0.781687	−	2.40578i
$16$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$17$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$17$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$18$		0			0
$19$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$19$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$20$	1.03137	−	0.749337i	1.03137	−	0.749337i
$21$	−3.84378			−3.84378
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	0.193209	+	0.594636i	0.193209	+	0.594636i
$26$		0			0
$27$	3.10969	+	2.25932i	3.10969	+	2.25932i
$28$	−1.56720	−	1.13864i	−1.56720	−	1.13864i
$29$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$29$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$30$		0			0
$31$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$31$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$32$		0			0
$33$	1.96858	+	0.248690i	1.96858	+	0.248690i
$34$		0			0
$35$	1.99794	−	1.45159i	1.99794	−	1.45159i
$36$	0.907634	+	2.79341i	0.907634	+	2.79341i
$37$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$37$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$38$		0			0
$39$	−0.601597	−	0.437086i	−0.601597	−	0.437086i
$40$		0			0
$41$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$41$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$42$		0			0
$43$	1.07165			1.07165			0.535827	−	0.844328i	$-0.320000\pi$
$43$	1.07165			1.07165			0.535827	+	0.844328i	$0.320000\pi$
$44$	0.728969	+	0.684547i	0.728969	+	0.684547i
$45$	−3.74444			−3.74444
$46$		0			0
$47$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$47$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$48$	−0.613161	+	1.88711i	−0.613161	+	1.88711i
$49$	−2.22691	−	1.61795i	−2.22691	−	1.61795i
$50$		0			0
$51$		0			0
$52$	−0.115808	−	0.356420i	−0.115808	−	0.356420i
$53$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$53$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$54$		0			0
$55$	−1.11716	+	0.614163i	−1.11716	+	0.614163i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$59$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$60$	−2.04648	−	1.48686i	−2.04648	−	1.48686i
$61$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$61$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$62$		0			0
$63$	1.75824	+	5.41130i	1.75824	+	5.41130i
$64$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$65$	0.477765			0.477765
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$71$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$72$		0			0
$73$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$73$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$74$		0			0
$75$	1.00368	−	0.729215i	1.00368	−	0.729215i
$76$		0			0
$77$	1.41213	+	1.32608i	1.41213	+	1.32608i
$78$		0			0
$79$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$79$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$80$	−0.393950	−	1.21245i	−0.393950	−	1.21245i
$81$	1.44922	−	4.46025i	1.44922	−	4.46025i
$82$		0			0
$83$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$83$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$84$	−1.18779	+	3.65565i	−1.18779	+	3.65565i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$89$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$90$		0			0
$91$	−0.224339	−	0.690446i	−0.224339	−	0.690446i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$97$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$98$		0			0
$99$	−0.550370	−	2.88514i	−0.550370	−	2.88514i
$100$	0.625237			0.625237
$101$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	−0.929776	−	0.368125i	$-0.880000\pi$
$101$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	0.0627905	+	0.998027i	$0.480000\pi$
$102$		0			0
$103$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$103$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$104$		0			0
$105$	−3.96438	−	2.88029i	−3.96438	−	2.88029i
$106$		0			0
$107$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$107$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$108$	3.10969	−	2.25932i	3.10969	−	2.25932i
$109$	1.45794			1.45794			0.728969	−	0.684547i	$-0.240000\pi$
$109$	1.45794			1.45794			0.728969	+	0.684547i	$0.240000\pi$
$110$		0			0
$111$		0			0
$112$	−1.56720	+	1.13864i	−1.56720	+	1.13864i
$113$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	−0.992115	−	0.125333i	$-0.960000\pi$
$113$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	0.728969	−	0.684547i	$-0.240000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.340147	+	1.04687i	−0.340147	+	1.04687i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.637424	−	0.770513i	−0.637424	−	0.770513i
$122$		0			0
$123$	2.34039	−	1.70039i	2.34039	−	1.70039i
$124$		0			0
$125$	0.147638	−	0.454382i	0.147638	−	0.454382i
$126$		0			0
$127$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$127$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$128$		0			0
$129$	−0.657096	−	2.02233i	−0.657096	−	2.02233i
$130$		0			0
$131$	1.00000			1.00000
$131$	1.00000			1.00000
$132$	0.844844	−	1.79538i	0.844844	−	1.79538i
$133$		0			0
$134$		0			0
$135$	1.51426	+	4.66040i	1.51426	+	4.66040i
$136$		0			0
$137$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$137$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$138$		0			0
$139$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$139$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$140$	−0.763146	−	2.34872i	−0.763146	−	2.34872i
$141$		0			0
$142$		0			0
$143$	0.0702235	+	0.368125i	0.0702235	+	0.368125i
$144$	2.93717			2.93717
$145$		0			0
$146$		0			0
$147$	−1.68779	+	5.19450i	−1.68779	+	5.19450i
$148$		0			0
$149$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$149$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$150$		0			0
$151$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$151$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.601597	+	0.437086i	−0.601597	+	0.437086i
$157$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$157$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$158$		0			0
$159$	0.992115	+	0.720814i	0.992115	+	0.720814i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$163$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$164$	1.45794			1.45794
$165$	1.84399	+	1.73162i	1.84399	+	1.73162i
$166$		0			0
$167$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$167$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$168$		0			0
$169$	−0.265616	+	0.817483i	−0.265616	+	0.817483i
$170$		0			0
$171$		0			0
$172$	0.331159	−	1.01920i	0.331159	−	1.01920i
$173$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$173$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$174$		0			0
$175$	1.21119			1.21119
$176$	0.876307	−	0.481754i	0.876307	−	0.481754i
$177$	−0.249182			−0.249182
$178$		0			0
$179$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$179$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$180$	−1.15710	+	3.56117i	−1.15710	+	3.56117i
$181$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$181$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$182$		0			0
$183$	0.781687	−	2.40578i	0.781687	−	2.40578i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	6.02398	−	4.37668i	6.02398	−	4.37668i
$190$		0			0
$191$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	−0.992115	−	0.125333i	$-0.960000\pi$
$191$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	0.876307	+	0.481754i	$0.160000\pi$
$192$	1.60528	+	1.16630i	1.60528	+	1.16630i
$193$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$193$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$194$		0			0
$195$	−0.292947	−	0.901598i	−0.292947	−	0.901598i
$196$	−2.22691	+	1.61795i	−2.22691	+	1.61795i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−0.574354	+	1.76768i	−0.574354	+	1.76768i
$206$		0			0
$207$		0			0
$208$	−0.374763			−0.374763
$209$		0			0
$210$		0			0
$211$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$211$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$212$	0.190983	+	0.587785i	0.190983	+	0.587785i
$213$		0			0
$214$		0			0
$215$	1.10528	+	0.803030i	1.10528	+	0.803030i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	0.238883	+	1.25227i	0.238883	+	1.25227i
$221$		0			0
$222$		0			0
$223$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$223$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$224$		0			0
$225$	−1.48570	−	1.07942i	−1.48570	−	1.07942i
$226$		0			0
$227$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$227$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$228$		0			0
$229$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$229$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$230$		0			0
$231$	1.63660	−	3.47796i	1.63660	−	3.47796i
$232$		0			0
$233$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$233$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$234$		0			0
$235$		0			0
$236$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i
$237$		0			0
$238$		0			0
$239$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$239$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$240$	−2.04648	+	1.48686i	−2.04648	+	1.48686i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−5.46182			−5.46182
$244$	1.03137	−	0.749337i	1.03137	−	0.749337i
$245$	−1.08439	−	3.33741i	−1.08439	−	3.33741i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$251$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$252$	5.68978			5.68978
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.309017	+	0.951057i	0.309017	+	0.951057i
$257$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$257$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$258$		0			0
$259$		0			0
$260$	0.147638	−	0.454382i	0.147638	−	0.454382i
$261$		0			0
$262$		0			0
$263$	−1.98423			−1.98423			−0.992115	−	0.125333i	$-0.960000\pi$
$263$	−1.98423			−1.98423			−0.992115	+	0.125333i	$0.960000\pi$
$264$		0			0
$265$	−0.787899			−0.787899
$266$		0			0
$267$	0.992115	+	3.05342i	0.992115	+	3.05342i
$268$		0			0
$269$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$269$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$270$		0			0
$271$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$271$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$272$		0			0
$273$	−1.16539	+	0.846708i	−1.16539	+	0.846708i
$274$		0			0
$275$	−0.620307	−	0.0783630i	−0.620307	−	0.0783630i
$276$		0			0
$277$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$277$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$281$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$282$		0			0
$283$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$283$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.82427			2.82427
$288$		0			0
$289$	0.309017	+	0.951057i	0.309017	+	0.951057i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$293$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$294$		0			0
$295$	0.129521	−	0.0941025i	0.129521	−	0.0941025i
$296$		0			0
$297$	−3.36833	+	1.85176i	−3.36833	+	1.85176i
$298$		0			0
$299$		0			0
$300$	−0.383371	−	1.17989i	−0.383371	−	1.17989i
$301$	0.641510	−	1.97437i	0.641510	−	1.97437i
$302$		0			0
$303$	1.72030	+	1.24987i	1.72030	+	1.24987i
$304$		0			0
$305$	0.502226	+	1.54569i	0.502226	+	1.54569i
$306$		0			0
$307$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$307$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$308$	1.69755	−	0.933237i	1.69755	−	0.933237i
$309$		0			0
$310$		0			0
$311$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$311$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$312$		0			0
$313$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$313$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$314$		0			0
$315$	−2.24149	+	6.89859i	−2.24149	+	6.89859i
$316$		0			0
$317$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$317$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$318$		0			0
$319$		0			0
$320$	−1.27485			−1.27485
$321$	0.201592	−	0.146465i	0.201592	−	0.146465i
$322$		0			0
$323$		0			0
$324$	−3.79411	−	2.75658i	−3.79411	−	2.75658i
$325$	0.189565	+	0.137727i	0.189565	+	0.137727i
$326$		0			0
$327$	−0.893950	−	2.75129i	−0.893950	−	2.75129i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$	3.10969	+	2.25932i	3.10969	+	2.25932i
$337$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$337$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$338$		0			0
$339$	−1.36699	+	0.993173i	−1.36699	+	0.993173i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−2.74670	+	1.99559i	−2.74670	+	1.99559i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$347$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$348$		0			0
$349$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$349$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$350$		0			0
$351$	1.44051			1.44051
$352$		0			0
$353$	1.45794			1.45794			0.728969	−	0.684547i	$-0.240000\pi$
$353$	1.45794			1.45794			0.728969	+	0.684547i	$0.240000\pi$
$354$		0			0
$355$		0			0
$356$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$357$		0			0
$358$		0			0
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$		0			0
$361$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$362$		0			0
$363$	−1.06320	+	1.67534i	−1.06320	+	1.67534i
$364$	−0.725978			−0.725978
$365$		0			0
$366$		0			0
$367$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$367$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$368$		0			0
$369$	−3.46438	−	2.51702i	−3.46438	−	2.51702i
$370$		0			0
$371$	0.369966	+	1.13864i	0.369966	+	1.13864i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−0.947996			−0.947996
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$379$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$383$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$384$		0			0
$385$	0.462756	+	2.42585i	0.462756	+	2.42585i
$386$		0			0
$387$	−2.54648	+	1.85013i	−2.54648	+	1.85013i
$388$		0			0
$389$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$389$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−0.613161	−	1.88711i	−0.613161	−	1.88711i
$394$		0			0
$395$		0			0
$396$	−2.91401	−	0.368125i	−2.91401	−	0.368125i
$397$	0.125581			0.125581			0.0627905	−	0.998027i	$-0.480000\pi$
$397$	0.125581			0.125581			0.0627905	+	0.998027i	$0.480000\pi$
$398$		0			0
$399$		0			0
$400$	0.193209	−	0.594636i	0.193209	−	0.594636i
$401$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$401$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$402$		0			0
$403$		0			0
$404$	0.331159	+	1.01920i	0.331159	+	1.01920i
$405$	4.83692	−	3.51422i	4.83692	−	3.51422i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$409$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.196811	−	0.142991i	−0.196811	−	0.142991i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$	−3.96438	+	2.88029i	−3.96438	+	2.88029i
$421$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$421$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.99794	−	1.45159i	1.99794	−	1.45159i
$428$	0.125581			0.125581
$429$	0.651635	−	0.358239i	0.651635	−	0.358239i
$430$		0			0
$431$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$431$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$432$	−1.18779	−	3.65565i	−1.18779	−	3.65565i
$433$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$433$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$434$		0			0
$435$		0			0
$436$	0.450527	−	1.38658i	0.450527	−	1.38658i
$437$		0			0
$438$		0			0
$439$	−1.85955			−1.85955			−0.929776	−	0.368125i	$-0.880000\pi$
$439$	−1.85955			−1.85955			−0.929776	+	0.368125i	$0.880000\pi$
$440$		0			0
$441$	8.08488			8.08488
$442$		0			0
$443$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$443$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$444$		0			0
$445$	−1.66880	−	1.21245i	−1.66880	−	1.21245i
$446$		0			0
$447$		0			0
$448$	0.598617	+	1.84235i	0.598617	+	1.84235i
$449$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$449$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$450$		0			0
$451$	−1.44644	−	0.182728i	−1.44644	−	0.182728i
$452$	−0.851559			−0.851559
$453$	2.81343	−	2.04407i	2.81343	−	2.04407i
$454$		0			0
$455$	0.285999	−	0.880213i	0.285999	−	0.880213i
$456$		0			0
$457$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$457$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$467$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$468$	0.890518	+	0.646999i	0.890518	+	0.646999i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.456288	+	0.969661i	−0.456288	+	0.969661i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.560949	−	1.72642i	0.560949	−	1.72642i
$478$		0			0
$479$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$479$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$485$		0			0
$486$		0			0
$487$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$487$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$491$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$492$	−0.893950	−	2.75129i	−0.893950	−	2.75129i
$493$		0			0
$494$		0			0
$495$	1.59431	−	3.38807i	1.59431	−	3.38807i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$499$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$500$	−0.386520	−	0.280823i	−0.386520	−	0.280823i
$501$	0.992115	+	0.720814i	0.992115	+	0.720814i
$502$		0			0
$503$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$503$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$504$		0			0
$505$	−1.36620			−1.36620
$506$		0			0
$507$	1.70555			1.70555
$508$		0			0
$509$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$509$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$	−2.12641			−2.12641
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$521$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$522$		0			0
$523$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$523$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$524$	0.309017	−	0.951057i	0.309017	−	0.951057i
$525$	−0.742653	−	2.28565i	−0.742653	−	2.28565i
$526$		0			0
$527$		0			0
$528$	−1.44644	−	1.35830i	−1.44644	−	1.35830i
$529$	1.00000			1.00000
$530$		0			0
$531$	0.113982	+	0.350799i	0.113982	+	0.350799i
$532$		0			0
$533$	0.442031	+	0.321154i	0.442031	+	0.321154i
$534$		0			0
$535$	−0.0494726	+	0.152261i	−0.0494726	+	0.152261i
$536$		0			0
$537$	−2.98509	+	2.16880i	−2.98509	+	2.16880i
$538$		0			0
$539$	2.41213	−	1.32608i	2.41213	−	1.32608i
$540$	4.90024			4.90024
$541$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$541$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.50368	+	1.09249i	1.50368	+	1.09249i
$546$		0			0
$547$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$547$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$548$		0			0
$549$	−3.74444			−3.74444
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$557$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$558$		0			0
$559$	0.324914	−	0.236064i	0.324914	−	0.236064i
$560$	−2.46959			−2.46959
$561$		0			0
$562$		0			0
$563$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$563$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$564$		0			0
$565$	0.335471	−	1.03247i	0.335471	−	1.03247i
$566$		0			0
$567$	−7.34982	−	5.33996i	−7.34982	−	5.33996i
$568$		0			0
$569$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$569$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$	0.371808	+	0.0469702i	0.371808	+	0.0469702i
$573$	0.743615			0.743615
$574$		0			0
$575$		0			0
$576$	0.907634	−	2.79341i	0.907634	−	2.79341i
$577$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$577$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$578$		0			0
$579$	1.14020	−	3.50919i	1.14020	−	3.50919i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.115808	−	0.607087i	−0.115808	−	0.607087i
$584$		0			0
$585$	−1.13527	+	0.824825i	−1.13527	+	0.824825i
$586$		0			0
$587$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	−0.992115	−	0.125333i	$-0.960000\pi$
$587$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	0.876307	+	0.481754i	$0.160000\pi$
$588$	4.41870	+	3.21038i	4.41870	+	3.21038i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$599$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$600$		0			0
$601$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$601$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$602$		0			0
$603$		0			0
$604$	1.75261			1.75261
$605$	−0.0800484	−	1.27233i	−0.0800484	−	1.27233i
$606$		0			0
$607$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$607$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$613$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$614$		0			0
$615$	3.68798			3.68798
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$619$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.968583	+	2.98099i	−0.968583	+	2.98099i
$624$	0.229790	+	0.707220i	0.229790	+	0.707220i
$625$	0.998582	−	0.725513i	0.998582	−	0.725513i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$631$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$632$		0			0
$633$	−1.36699	−	0.993173i	−1.36699	−	0.993173i
$634$		0			0
$635$		0			0
$636$	0.992115	−	0.720814i	0.992115	−	0.720814i
$637$	−1.03158			−1.03158
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$641$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$642$		0			0
$643$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$643$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$644$		0			0
$645$	0.837697	−	2.57817i	0.837697	−	2.57817i
$646$		0			0
$647$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$647$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$648$		0			0
$649$	0.0915446	+	0.0859661i	0.0915446	+	0.0859661i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$653$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$654$		0			0
$655$	1.03137	+	0.749337i	1.03137	+	0.749337i
$656$	0.450527	−	1.38658i	0.450527	−	1.38658i
$657$		0			0
$658$		0			0
$659$	1.93717			1.93717			0.968583	−	0.248690i	$-0.0800000\pi$
$659$	1.93717			1.93717			0.968583	+	0.248690i	$0.0800000\pi$
$660$	2.21670	−	1.21864i	2.21670	−	1.21864i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.190983	+	0.587785i	0.190983	+	0.587785i
$669$		0			0
$670$		0			0
$671$	−1.11716	+	0.614163i	−1.11716	+	0.614163i
$672$		0			0
$673$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$673$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$674$		0			0
$675$	−0.742653	+	2.28565i	−0.742653	+	2.28565i
$676$	0.695393	+	0.505233i	0.695393	+	0.505233i
$677$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$677$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$683$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−0.866986	−	0.629902i	−0.866986	−	0.629902i
$689$	−0.0715733	+	0.220280i	−0.0715733	+	0.220280i
$690$		0			0
$691$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$691$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$692$		0			0
$693$	−5.64491	−	0.713118i	−5.64491	−	0.713118i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−2.98509	−	2.16880i	−2.98509	−	2.16880i
$700$	0.374278	−	1.15191i	0.374278	−	1.15191i
$701$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$701$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$702$		0			0
$703$		0			0
$704$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$705$		0			0
$706$		0			0
$707$	0.641510	+	1.97437i	0.641510	+	1.97437i
$708$	−0.0770013	+	0.236986i	−0.0770013	+	0.236986i
$709$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$709$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−0.203423	+	0.432295i	−0.203423	+	0.432295i
$716$	−1.85955			−1.85955
$717$	−2.98509	+	2.16880i	−2.98509	+	2.16880i
$718$		0			0
$719$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$719$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$720$	3.02932	+	2.20093i	3.02932	+	2.20093i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	1.89975	+	5.84683i	1.89975	+	5.84683i
$730$		0			0
$731$		0			0
$732$	−2.04648	−	1.48686i	−2.04648	−	1.48686i
$733$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$733$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$734$		0			0
$735$	−5.63317	+	4.09274i	−5.63317	+	4.09274i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$739$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$743$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.243271			0.243271
$750$		0			0
$751$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$751$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.690441	+	2.12496i	−0.690441	+	2.12496i
$756$	−2.30095	−	7.08161i	−2.30095	−	7.08161i
$757$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$757$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$761$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$762$		0			0
$763$	0.872746	−	2.68604i	0.872746	−	2.68604i
$764$	0.303189	+	0.220280i	0.303189	+	0.220280i
$765$		0			0
$766$		0			0
$767$	−0.0145433	−	0.0447596i	−0.0145433	−	0.0447596i
$768$	1.60528	−	1.16630i	1.60528	−	1.16630i
$769$	−0.851559			−0.851559			−0.425779	−	0.904827i	$-0.640000\pi$
$769$	−0.851559			−0.851559			−0.425779	+	0.904827i	$0.640000\pi$
$770$		0			0
$771$		0			0
$772$	1.50441	−	1.09302i	1.50441	−	1.09302i
$773$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$773$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−0.947996			−0.947996
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.850604	+	2.61789i	0.850604	+	2.61789i
$785$		0			0
$786$		0			0
$787$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.876307	+	0.481754i	$0.160000\pi$
$787$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.728969	+	0.684547i	$0.240000\pi$
$788$		0			0
$789$	1.21665	+	3.74447i	1.21665	+	3.74447i
$790$		0			0
$791$	−1.64961			−1.64961
$792$		0			0
$793$	0.477765			0.477765
$794$		0			0
$795$	0.483109	+	1.48686i	0.483109	+	1.48686i
$796$		0			0
$797$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$797$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	3.84480	−	2.79341i	3.84480	−	2.79341i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−0.893950	+	2.75129i	−0.893950	+	2.75129i
$808$		0			0
$809$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$809$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$810$		0			0
$811$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$811$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	0.535827	−	0.844328i	$-0.320000\pi$
$812$		0			0
$813$	−3.47759			−3.47759
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	1.72508	+	1.25334i	1.72508	+	1.25334i
$820$	1.50368	+	1.09249i	1.50368	+	1.09249i
$821$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$821$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$822$		0			0
$823$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$823$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$824$		0			0
$825$	0.232468	+	1.21864i	0.232468	+	1.21864i
$826$		0			0
$827$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$827$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$828$		0			0
$829$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	0.535827	+	0.844328i	$0.320000\pi$
$829$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	−0.929776	+	0.368125i	$0.880000\pi$
$830$		0			0
$831$	2.81343	+	2.04407i	2.81343	+	2.04407i
$832$	−0.115808	+	0.356420i	−0.115808	+	0.356420i
$833$		0			0
$834$		0			0
$835$	−0.787899			−0.787899
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$839$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$840$		0			0
$841$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$842$		0			0
$843$		0			0
$844$	−0.263146	−	0.809880i	−0.263146	−	0.809880i
$845$	−0.886520	+	0.644095i	−0.886520	+	0.644095i
$846$		0			0
$847$	−1.80113	+	0.713118i	−1.80113	+	0.713118i
$848$	0.618034			0.618034
$849$	−3.18523	+	2.31421i	−3.18523	+	2.31421i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$853$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$	1.10528	−	0.803030i	1.10528	−	0.803030i
$861$	−1.73173	−	5.32971i	−1.73173	−	5.32971i
$862$		0			0
$863$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$863$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1.60528	−	1.16630i	1.60528	−	1.16630i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.748754	−	0.544002i	−0.748754	−	0.544002i
$876$		0			0
$877$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	−0.992115	−	0.125333i	$-0.960000\pi$
$877$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	0.728969	−	0.684547i	$-0.240000\pi$
$878$		0			0
$879$		0			0
$880$	1.26480	+	0.159781i	1.26480	+	0.159781i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$883$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$884$		0			0
$885$	−0.256999	−	0.186721i	−0.256999	−	0.186721i
$886$		0			0
$887$	0.450527	−	1.38658i	0.450527	−	1.38658i	−0.425779	−	0.904827i	$-0.640000\pi$
$887$	0.450527	−	1.38658i	0.450527	−	1.38658i	0.876307	−	0.481754i	$-0.160000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	3.41870	+	3.21038i	3.41870	+	3.21038i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	0.732570	−	2.25462i	0.732570	−	2.25462i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−1.48570	+	1.07942i	−1.48570	+	1.07942i
$901$		0			0
$902$		0			0
$903$	−4.11920			−4.11920
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$907$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$908$		0			0
$909$	0.972670	−	2.99357i	0.972670	−	2.99357i
$910$		0			0
$911$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.992115	−	0.125333i	$-0.960000\pi$
$911$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.187381	+	0.982287i	$0.560000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	2.60895	−	1.89552i	2.60895	−	1.89552i
$916$		0			0
$917$	0.598617	−	1.84235i	0.598617	−	1.84235i
$918$		0			0
$919$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$919$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$920$		0			0
$921$	−0.378954	−	1.16630i	−0.378954	−	1.16630i
$922$		0			0
$923$		0			0
$924$	−2.80200	−	2.63125i	−2.80200	−	2.63125i
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$929$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$930$		0			0
$931$		0			0
$932$	−0.574633	−	1.76854i	−0.574633	−	1.76854i
$933$	1.72030	−	1.24987i	1.72030	−	1.24987i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$937$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$941$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$942$		0			0
$943$		0			0
$944$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i
$945$	9.49258			9.49258
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−1.36699	−	0.993173i	−1.36699	−	0.993173i
$952$		0			0
$953$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$953$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$954$		0			0
$955$	−0.386520	+	0.280823i	−0.386520	+	0.280823i
$956$	−1.85955			−1.85955
$957$		0			0
$958$		0			0
$959$		0			0
$960$	0.781687	+	2.40578i	0.781687	+	2.40578i
$961$	0.309017	−	0.951057i	0.309017	−	0.951057i
$962$		0			0
$963$	−0.298408	−	0.216806i	−0.298408	−	0.216806i
$964$		0			0
$965$	0.732570	+	2.25462i	0.732570	+	2.25462i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$971$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$972$	−1.68779	+	5.19450i	−1.68779	+	5.19450i
$973$		0			0
$974$		0			0
$975$	0.143673	−	0.442180i	0.143673	−	0.442180i
$976$	−0.393950	−	1.21245i	−0.393950	−	1.21245i
$977$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$977$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$978$		0			0
$979$	0.688925	−	1.46404i	0.688925	−	1.46404i
$980$	−3.50916			−3.50916
$981$	−3.46438	+	2.51702i	−3.46438	+	2.51702i
$982$		0			0
$983$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$983$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.27485			−1.27485			−0.637424	−	0.770513i	$-0.720000\pi$
$991$	−1.27485			−1.27485			−0.637424	+	0.770513i	$0.720000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$997$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1441.1.u.a.916.1	yes	20
11.4	even	5	inner	1441.1.u.a.785.1	✓	20
131.130	odd	2	CM	1441.1.u.a.916.1	yes	20
1441.785	odd	10	inner	1441.1.u.a.785.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1441.1.u.a.785.1	✓	20	11.4	even	5	inner
1441.1.u.a.785.1	✓	20	1441.785	odd	10	inner
1441.1.u.a.916.1	yes	20	1.1	even	1	trivial
1441.1.u.a.916.1	yes	20	131.130	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1441 = 11 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1441.u (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.613161 - 1.88711i) q^{3} +(0.309017 - 0.951057i) q^{4} +(1.03137 + 0.749337i) q^{5} +(0.598617 - 1.84235i) q^{7} +(-2.37622 + 1.72642i) q^{9} +O(q^{10})\)	\(q+(-0.613161 - 1.88711i) q^{3} +(0.309017 - 0.951057i) q^{4} +(1.03137 + 0.749337i) q^{5} +(0.598617 - 1.84235i) q^{7} +(-2.37622 + 1.72642i) q^{9} +(-0.425779 + 0.904827i) q^{11} -1.98423 q^{12} +(0.303189 - 0.220280i) q^{13} +(0.781687 - 2.40578i) q^{15} +(-0.809017 - 0.587785i) q^{16} +(1.03137 - 0.749337i) q^{20} -3.84378 q^{21} +(0.193209 + 0.594636i) q^{25} +(3.10969 + 2.25932i) q^{27} +(-1.56720 - 1.13864i) q^{28} +(1.96858 + 0.248690i) q^{33} +(1.99794 - 1.45159i) q^{35} +(0.907634 + 2.79341i) q^{36} +(-0.601597 - 0.437086i) q^{39} +(0.450527 + 1.38658i) q^{41} +1.07165 q^{43} +(0.728969 + 0.684547i) q^{44} -3.74444 q^{45} +(-0.613161 + 1.88711i) q^{48} +(-2.22691 - 1.61795i) q^{49} +(-0.115808 - 0.356420i) q^{52} +(-0.500000 + 0.363271i) q^{53} +(-1.11716 + 0.614163i) q^{55} +(0.0388067 - 0.119435i) q^{59} +(-2.04648 - 1.48686i) q^{60} +(1.03137 + 0.749337i) q^{61} +(1.75824 + 5.41130i) q^{63} +(-0.809017 + 0.587785i) q^{64} +0.477765 q^{65} +(1.00368 - 0.729215i) q^{75} +(1.41213 + 1.32608i) q^{77} +(-0.393950 - 1.21245i) q^{80} +(1.44922 - 4.46025i) q^{81} +(-1.18779 + 3.65565i) q^{84} -1.61803 q^{89} +(-0.224339 - 0.690446i) q^{91} +(-0.550370 - 2.88514i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) \) 20 * q - 5 * q^4 - 5 * q^9	\( 20 q - 5 q^{4} - 5 q^{9} - 5 q^{16} - 5 q^{25} + 20 q^{33} + 15 q^{35} - 5 q^{36} - 10 q^{39} - 10 q^{45} - 5 q^{49} - 10 q^{53} - 10 q^{63} - 5 q^{64} - 10 q^{75} - 5 q^{81} - 10 q^{89}+O(q^{100}) \) 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 - 5 * q^25 + 20 * q^33 + 15 * q^35 - 5 * q^36 - 10 * q^39 - 10 * q^45 - 5 * q^49 - 10 * q^53 - 10 * q^63 - 5 * q^64 - 10 * q^75 - 5 * q^81 - 10 * q^89

Introduction

L-functions

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