LMFDB - Embedded newform 1441.1.u.a.916.4

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			916.4
Root			$0.992115 - 0.125333i$ of defining polynomial
Character	$\chi$	$=$	1441.916
Dual form			1441.1.u.a.785.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.450527 + 1.38658i) q^{3} +(0.309017 - 0.951057i) q^{4} +(-0.866986 - 0.629902i) q^{5} +(0.0388067 - 0.119435i) q^{7} +(-0.910614 + 0.661600i) q^{9} +O(q^{10})$	\(q+(0.450527 + 1.38658i) q^{3} +(0.309017 - 0.951057i) q^{4} +(-0.866986 - 0.629902i) q^{5} +(0.0388067 - 0.119435i) q^{7} +(-0.910614 + 0.661600i) q^{9} +(0.876307 - 0.481754i) q^{11} +1.45794 q^{12} +(0.688925 - 0.500534i) q^{13} +(0.482809 - 1.48593i) q^{15} +(-0.809017 - 0.587785i) q^{16} +(-0.866986 + 0.629902i) q^{20} +0.183089 q^{21} +(0.0458709 + 0.141176i) q^{25} +(-0.148122 - 0.107617i) q^{27} +(-0.101597 - 0.0738147i) q^{28} +(1.06279 + 0.998027i) q^{33} +(-0.108877 + 0.0791038i) q^{35} +(0.347824 + 1.07049i) q^{36} +(1.00441 + 0.729747i) q^{39} +(-0.115808 - 0.356420i) q^{41} +1.93717 q^{43} +(-0.187381 - 0.982287i) q^{44} +1.20623 q^{45} +(0.450527 - 1.38658i) q^{48} +(0.796258 + 0.578516i) q^{49} +(-0.263146 - 0.809880i) q^{52} +(-0.500000 + 0.363271i) q^{53} +(-1.06320 - 0.134314i) q^{55} +(-0.574633 + 1.76854i) q^{59} +(-1.26401 - 0.918358i) q^{60} +(-0.866986 - 0.629902i) q^{61} +(0.0436801 + 0.134433i) q^{63} +(-0.809017 + 0.587785i) q^{64} -0.912576 q^{65} +(-0.175086 + 0.127207i) q^{75} +(-0.0235315 - 0.123357i) q^{77} +(0.331159 + 1.01920i) q^{80} +(-0.265337 + 0.816623i) q^{81} +(0.0565777 - 0.174128i) q^{84} -1.61803 q^{89} +(-0.0330462 - 0.101706i) q^{91} +(-0.479249 + 1.01846i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.800000\pi$
$3$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$3$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$4$	0.309017	−	0.951057i	0.309017	−	0.951057i
$5$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$5$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$6$		0			0
$7$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$7$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$8$		0			0
$9$	−0.910614	+	0.661600i	−0.910614	+	0.661600i
$10$		0			0
$11$	0.876307	−	0.481754i	0.876307	−	0.481754i
$11$	0.876307	−	0.481754i	0.876307	−	0.481754i
$12$	1.45794			1.45794
$13$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$13$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$14$		0			0
$15$	0.482809	−	1.48593i	0.482809	−	1.48593i
$16$	−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	$0.400000\pi$
$20$	−0.866986	+	0.629902i	−0.866986	+	0.629902i
$21$	0.183089			0.183089
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	0.0458709	+	0.141176i	0.0458709	+	0.141176i
$26$		0			0
$27$	−0.148122	−	0.107617i	−0.148122	−	0.107617i
$28$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i
$29$		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
$33$	1.06279	+	0.998027i	1.06279	+	0.998027i
$34$		0			0
$35$	−0.108877	+	0.0791038i	−0.108877	+	0.0791038i
$36$	0.347824	+	1.07049i	0.347824	+	1.07049i
$37$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$37$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$38$		0			0
$39$	1.00441	+	0.729747i	1.00441	+	0.729747i
$40$		0			0
$41$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$41$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$42$		0			0
$43$	1.93717			1.93717			0.968583	−	0.248690i	$-0.0800000\pi$
$43$	1.93717			1.93717			0.968583	+	0.248690i	$0.0800000\pi$
$44$	−0.187381	−	0.982287i	−0.187381	−	0.982287i
$45$	1.20623			1.20623
$46$		0			0
$47$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$47$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$48$	0.450527	−	1.38658i	0.450527	−	1.38658i
$49$	0.796258	+	0.578516i	0.796258	+	0.578516i
$50$		0			0
$51$		0			0
$52$	−0.263146	−	0.809880i	−0.263146	−	0.809880i
$53$	−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
$55$	−1.06320	−	0.134314i	−1.06320	−	0.134314i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$59$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$60$	−1.26401	−	0.918358i	−1.26401	−	0.918358i
$61$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$61$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$62$		0			0
$63$	0.0436801	+	0.134433i	0.0436801	+	0.134433i
$64$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$65$	−0.912576			−0.912576
$66$		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
$75$	−0.175086	+	0.127207i	−0.175086	+	0.127207i
$76$		0			0
$77$	−0.0235315	−	0.123357i	−0.0235315	−	0.123357i
$78$		0			0
$79$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$79$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$80$	0.331159	+	1.01920i	0.331159	+	1.01920i
$81$	−0.265337	+	0.816623i	−0.265337	+	0.816623i
$82$		0			0
$83$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$83$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$84$	0.0565777	−	0.174128i	0.0565777	−	0.174128i
$85$		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
$91$	−0.0330462	−	0.101706i	−0.0330462	−	0.101706i
$92$		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
$99$	−0.479249	+	1.01846i	−0.479249	+	1.01846i
$100$	0.148441			0.148441
$101$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$101$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$102$		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
$105$	−0.158736	−	0.115328i	−0.158736	−	0.115328i
$106$		0			0
$107$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$107$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$108$	−0.148122	+	0.107617i	−0.148122	+	0.107617i
$109$	−0.374763			−0.374763			−0.187381	−	0.982287i	$-0.560000\pi$
$109$	−0.374763			−0.374763			−0.187381	+	0.982287i	$0.560000\pi$
$110$		0			0
$111$		0			0
$112$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i
$113$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$113$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.296192	+	0.911586i	−0.296192	+	0.911586i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.535827	−	0.844328i	0.535827	−	0.844328i
$122$		0			0
$123$	0.442031	−	0.321154i	0.442031	−	0.321154i
$124$		0			0
$125$	−0.282001	+	0.867911i	−0.282001	+	0.867911i
$126$		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
$129$	0.872746	+	2.68604i	0.872746	+	2.68604i
$130$		0			0
$131$	1.00000			1.00000
$131$	1.00000			1.00000
$132$	1.27760	−	0.702367i	1.27760	−	0.702367i
$133$		0			0
$134$		0			0
$135$	0.0606317	+	0.186605i	0.0606317	+	0.186605i
$136$		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	$0.600000\pi$
$140$	0.0415873	+	0.127993i	0.0415873	+	0.127993i
$141$		0			0
$142$		0			0
$143$	0.362576	−	0.770513i	0.362576	−	0.770513i
$144$	1.12558			1.12558
$145$		0			0
$146$		0			0
$147$	−0.443422	+	1.36471i	−0.443422	+	1.36471i
$148$		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
$151$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$151$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.00441	−	0.729747i	1.00441	−	0.729747i
$157$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$157$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$158$		0			0
$159$	−0.728969	−	0.529627i	−0.728969	−	0.529627i
$160$		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	$0.800000\pi$
$164$	−0.374763			−0.374763
$165$	−0.292765	−	1.53473i	−0.292765	−	1.53473i
$166$		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
$169$	−0.0849327	+	0.261396i	−0.0849327	+	0.261396i
$170$		0			0
$171$		0			0
$172$	0.598617	−	1.84235i	0.598617	−	1.84235i
$173$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$173$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$174$		0			0
$175$	0.0186414			0.0186414
$176$	−0.992115	−	0.125333i	−0.992115	−	0.125333i
$177$	−2.71111			−2.71111
$178$		0			0
$179$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$179$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	0.535827	−	0.844328i	$-0.320000\pi$
$180$	0.372746	−	1.14720i	0.372746	−	1.14720i
$181$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$181$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$182$		0			0
$183$	0.482809	−	1.48593i	0.482809	−	1.48593i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.0186014	+	0.0135147i	−0.0186014	+	0.0135147i
$190$		0			0
$191$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$191$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$192$	−1.17950	−	0.856954i	−1.17950	−	0.856954i
$193$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$193$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$194$		0			0
$195$	−0.411140	−	1.26536i	−0.411140	−	1.26536i
$196$	0.796258	−	0.578516i	0.796258	−	0.578516i
$197$		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
$205$	−0.124106	+	0.381959i	−0.124106	+	0.381959i
$206$		0			0
$207$		0			0
$208$	−0.851559			−0.851559
$209$		0			0
$210$		0			0
$211$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$211$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$212$	0.190983	+	0.587785i	0.190983	+	0.587785i
$213$		0			0
$214$		0			0
$215$	−1.67950	−	1.22023i	−1.67950	−	1.22023i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.456288	+	0.969661i	−0.456288	+	0.969661i
$221$		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
$225$	−0.135173	−	0.0982088i	−0.135173	−	0.0982088i
$226$		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
$231$	0.160442	−	0.0882039i	0.160442	−	0.0882039i
$232$		0			0
$233$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$233$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$234$		0			0
$235$		0			0
$236$	1.50441	+	1.09302i	1.50441	+	1.09302i
$237$		0			0
$238$		0			0
$239$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$239$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	0.535827	−	0.844328i	$-0.320000\pi$
$240$	−1.26401	+	0.918358i	−1.26401	+	0.918358i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−1.43494			−1.43494
$244$	−0.866986	+	0.629902i	−0.866986	+	0.629902i
$245$	−0.325937	−	1.00313i	−0.325937	−	1.00313i
$246$		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	$0.800000\pi$
$252$	0.141352			0.141352
$253$		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
$260$	−0.282001	+	0.867911i	−0.282001	+	0.867911i
$261$		0			0
$262$		0			0
$263$	1.45794			1.45794			0.728969	−	0.684547i	$-0.240000\pi$
$263$	1.45794			1.45794			0.728969	+	0.684547i	$0.240000\pi$
$264$		0			0
$265$	0.662318			0.662318
$266$		0			0
$267$	−0.728969	−	2.24353i	−0.728969	−	2.24353i
$268$		0			0
$269$	0.303189	+	0.220280i	0.303189	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$269$	0.303189	+	0.220280i	0.303189	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$270$		0			0
$271$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$271$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$272$		0			0
$273$	0.126135	−	0.0916423i	0.126135	−	0.0916423i
$274$		0			0
$275$	0.108209	+	0.101615i	0.108209	+	0.101615i
$276$		0			0
$277$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$277$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$278$		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
$283$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$283$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.0470631			−0.0470631
$288$		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
$295$	1.61221	−	1.17134i	1.61221	−	1.17134i
$296$		0			0
$297$	−0.181646	−	0.0229472i	−0.181646	−	0.0229472i
$298$		0			0
$299$		0			0
$300$	0.0668769	+	0.205826i	0.0668769	+	0.205826i
$301$	0.0751750	−	0.231365i	0.0751750	−	0.231365i
$302$		0			0
$303$	−2.28488	−	1.66006i	−2.28488	−	1.66006i
$304$		0			0
$305$	0.354888	+	1.09223i	0.354888	+	1.09223i
$306$		0			0
$307$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$307$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$308$	−0.124591	−	0.0157395i	−0.124591	−	0.0157395i
$309$		0			0
$310$		0			0
$311$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$311$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$312$		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
$315$	0.0468099	−	0.144066i	0.0468099	−	0.144066i
$316$		0			0
$317$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$317$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$318$		0			0
$319$		0			0
$320$	1.07165			1.07165
$321$	2.19334	−	1.59355i	2.19334	−	1.59355i
$322$		0			0
$323$		0			0
$324$	0.694661	+	0.504701i	0.694661	+	0.504701i
$325$	0.102265	+	0.0742999i	0.102265	+	0.0742999i
$326$		0			0
$327$	−0.168841	−	0.519639i	−0.168841	−	0.519639i
$328$		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
$336$	−0.148122	−	0.107617i	−0.148122	−	0.107617i
$337$	−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
$339$	−2.06720	+	1.50191i	−2.06720	+	1.50191i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.201592	−	0.146465i	0.201592	−	0.146465i
$344$		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
$351$	−0.155911			−0.155911
$352$		0			0
$353$	−0.374763			−0.374763			−0.187381	−	0.982287i	$-0.560000\pi$
$353$	−0.374763			−0.374763			−0.187381	+	0.982287i	$0.560000\pi$
$354$		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
$363$	1.41213	+	0.362574i	1.41213	+	0.362574i
$364$	−0.106940			−0.106940
$365$		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
$369$	0.341264	+	0.247943i	0.341264	+	0.247943i
$370$		0			0
$371$	0.0239838	+	0.0738147i	0.0239838	+	0.0738147i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−1.33048			−1.33048
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$379$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$383$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$384$		0			0
$385$	−0.0573011	+	0.121771i	−0.0573011	+	0.121771i
$386$		0			0
$387$	−1.76401	+	1.28163i	−1.76401	+	1.28163i
$388$		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
$393$	0.450527	+	1.38658i	0.450527	+	1.38658i
$394$		0			0
$395$		0			0
$396$	0.820513	+	0.770513i	0.820513	+	0.770513i
$397$	−1.85955			−1.85955			−0.929776	−	0.368125i	$-0.880000\pi$
$397$	−1.85955			−1.85955			−0.929776	+	0.368125i	$0.880000\pi$
$398$		0			0
$399$		0			0
$400$	0.0458709	−	0.141176i	0.0458709	−	0.141176i
$401$		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
$404$	0.598617	+	1.84235i	0.598617	+	1.84235i
$405$	0.744436	−	0.540864i	0.744436	−	0.540864i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$409$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.188925	+	0.137262i	0.188925	+	0.137262i
$414$		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			$\pi$
$420$	−0.158736	+	0.115328i	−0.158736	+	0.115328i
$421$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$421$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.108877	+	0.0791038i	−0.108877	+	0.0791038i
$428$	−1.85955			−1.85955
$429$	1.23173	+	0.155604i	1.23173	+	0.155604i
$430$		0			0
$431$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$431$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$432$	0.0565777	+	0.174128i	0.0565777	+	0.174128i
$433$		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
$436$	−0.115808	+	0.356420i	−0.115808	+	0.356420i
$437$		0			0
$438$		0			0
$439$	−1.27485			−1.27485			−0.637424	−	0.770513i	$-0.720000\pi$
$439$	−1.27485			−1.27485			−0.637424	+	0.770513i	$0.720000\pi$
$440$		0			0
$441$	−1.10783			−1.10783
$442$		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
$445$	1.40281	+	1.01920i	1.40281	+	1.01920i
$446$		0			0
$447$		0			0
$448$	0.0388067	+	0.119435i	0.0388067	+	0.119435i
$449$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$449$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$450$		0			0
$451$	−0.273190	−	0.256543i	−0.273190	−	0.256543i
$452$	1.75261			1.75261
$453$	2.34039	−	1.70039i	2.34039	−	1.70039i
$454$		0			0
$455$	−0.0354140	+	0.108993i	−0.0354140	+	0.108993i
$456$		0			0
$457$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$457$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.425779	−	0.904827i	$-0.640000\pi$
$458$		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
$467$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$467$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$468$	0.775441	+	0.563391i	0.775441	+	0.563391i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.69755	−	0.933237i	1.69755	−	0.933237i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.214967	−	0.661600i	0.214967	−	0.661600i
$478$		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
$484$	−0.637424	−	0.770513i	−0.637424	−	0.770513i
$485$		0			0
$486$		0			0
$487$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$487$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$488$		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	$0.600000\pi$
$492$	−0.168841	−	0.519639i	−0.168841	−	0.519639i
$493$		0			0
$494$		0			0
$495$	1.05703	−	0.581107i	1.05703	−	0.581107i
$496$		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	$0.600000\pi$
$500$	0.738289	+	0.536399i	0.738289	+	0.536399i
$501$	−0.728969	−	0.529627i	−0.728969	−	0.529627i
$502$		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
$505$	2.07597			2.07597
$506$		0			0
$507$	−0.400711			−0.400711
$508$		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
$516$	2.82427			2.82427
$517$		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
$525$	0.00839847	+	0.0258478i	0.00839847	+	0.0258478i
$526$		0			0
$527$		0			0
$528$	−0.273190	−	1.43211i	−0.273190	−	1.43211i
$529$	1.00000			1.00000
$530$		0			0
$531$	−0.646797	−	1.99064i	−0.646797	−	1.99064i
$532$		0			0
$533$	−0.258183	−	0.187581i	−0.258183	−	0.187581i
$534$		0			0
$535$	−0.615808	+	1.89526i	−0.615808	+	1.89526i
$536$		0			0
$537$	1.50368	−	1.09249i	1.50368	−	1.09249i
$538$		0			0
$539$	0.976468	+	0.123357i	0.976468	+	0.123357i
$540$	0.196208			0.196208
$541$		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
$545$	0.324914	+	0.236064i	0.324914	+	0.236064i
$546$		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
$549$	1.20623			1.20623
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.0627905	−	0.998027i	$-0.480000\pi$
$557$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$558$		0			0
$559$	1.33456	−	0.969617i	1.33456	−	0.969617i
$560$	0.134579			0.134579
$561$		0			0
$562$		0			0
$563$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$563$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$564$		0			0
$565$	0.580394	−	1.78627i	0.580394	−	1.78627i
$566$		0			0
$567$	0.0872363	+	0.0633809i	0.0872363	+	0.0633809i
$568$		0			0
$569$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$569$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$	−0.620759	−	0.582932i	−0.620759	−	0.582932i
$573$	−1.24152			−1.24152
$574$		0			0
$575$		0			0
$576$	0.347824	−	1.07049i	0.347824	−	1.07049i
$577$	0.303189	+	0.220280i	0.303189	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$577$	0.303189	+	0.220280i	0.303189	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$578$		0			0
$579$	−0.574354	+	1.76768i	−0.574354	+	1.76768i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.263146	+	0.559214i	−0.263146	+	0.559214i
$584$		0			0
$585$	0.831004	−	0.603760i	0.831004	−	0.603760i
$586$		0			0
$587$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$587$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$588$	1.16089	+	0.843439i	1.16089	+	0.843439i
$589$		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
$599$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$599$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$600$		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
$604$	−1.98423			−1.98423
$605$	−0.996398	+	0.394502i	−0.996398	+	0.394502i
$606$		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
$613$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$613$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$614$		0			0
$615$	−0.585531			−0.585531
$616$		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
$623$	−0.0627905	+	0.193249i	−0.0627905	+	0.193249i
$624$	−0.383650	−	1.18075i	−0.383650	−	1.18075i
$625$	0.911282	−	0.662085i	0.911282	−	0.662085i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$631$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$632$		0			0
$633$	−2.06720	−	1.50191i	−2.06720	−	1.50191i
$634$		0			0
$635$		0			0
$636$	−0.728969	+	0.529627i	−0.728969	+	0.529627i
$637$	0.838129			0.838129
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$641$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$642$		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
$645$	0.935282	−	2.87850i	0.935282	−	2.87850i
$646$		0			0
$647$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$647$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$648$		0			0
$649$	0.348445	+	1.82662i	0.348445	+	1.82662i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$653$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$654$		0			0
$655$	−0.866986	−	0.629902i	−0.866986	−	0.629902i
$656$	−0.115808	+	0.356420i	−0.115808	+	0.356420i
$657$		0			0
$658$		0			0
$659$	0.125581			0.125581			0.0627905	−	0.998027i	$-0.480000\pi$
$659$	0.125581			0.125581			0.0627905	+	0.998027i	$0.480000\pi$
$660$	−1.55008	−	0.195821i	−1.55008	−	0.195821i
$661$		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
$671$	−1.06320	−	0.134314i	−1.06320	−	0.134314i
$672$		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
$675$	0.00839847	−	0.0258478i	0.00839847	−	0.0258478i
$676$	0.222357	+	0.161552i	0.222357	+	0.161552i
$677$		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
$688$	−1.56720	−	1.13864i	−1.56720	−	1.13864i
$689$	−0.162633	+	0.500534i	−0.162633	+	0.500534i
$690$		0			0
$691$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$691$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$692$		0			0
$693$	0.103041	+	0.0967619i	0.103041	+	0.0967619i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	1.50368	+	1.09249i	1.50368	+	1.09249i
$700$	0.00576052	−	0.0177291i	0.00576052	−	0.0177291i
$701$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$701$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$702$		0			0
$703$		0			0
$704$	−0.425779	+	0.904827i	−0.425779	+	0.904827i
$705$		0			0
$706$		0			0
$707$	0.0751750	+	0.231365i	0.0751750	+	0.231365i
$708$	−0.837780	+	2.57842i	−0.837780	+	2.57842i
$709$		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
$715$	−0.799696	+	0.439637i	−0.799696	+	0.439637i
$716$	−1.27485			−1.27485
$717$	1.50368	−	1.09249i	1.50368	−	1.09249i
$718$		0			0
$719$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$719$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$720$	−0.975863	−	0.709006i	−0.975863	−	0.709006i
$721$		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
$729$	−0.381145	−	1.17304i	−0.381145	−	1.17304i
$730$		0			0
$731$		0			0
$732$	−1.26401	−	0.918358i	−1.26401	−	0.918358i
$733$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$733$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$734$		0			0
$735$	1.24408	−	0.903875i	1.24408	−	0.903875i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.992115	−	0.125333i	$-0.960000\pi$
$739$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.187381	+	0.982287i	$0.560000\pi$
$740$		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
$749$	−0.233525			−0.233525
$750$		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
$755$	−0.657096	+	2.02233i	−0.657096	+	2.02233i
$756$	0.00710509	+	0.0218672i	0.00710509	+	0.0218672i
$757$	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
$763$	−0.0145433	+	0.0447596i	−0.0145433	+	0.0447596i
$764$	0.688925	+	0.500534i	0.688925	+	0.500534i
$765$		0			0
$766$		0			0
$767$	0.489334	+	1.50602i	0.489334	+	1.50602i
$768$	−1.17950	+	0.856954i	−1.17950	+	0.856954i
$769$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$769$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$770$		0			0
$771$		0			0
$772$	1.03137	−	0.749337i	1.03137	−	0.749337i
$773$		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
$780$	−1.33048			−1.33048
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.304144	−	0.936058i	−0.304144	−	0.936058i
$785$		0			0
$786$		0			0
$787$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$787$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$788$		0			0
$789$	0.656841	+	2.02155i	0.656841	+	2.02155i
$790$		0			0
$791$	0.220095			0.220095
$792$		0			0
$793$	−0.912576			−0.912576
$794$		0			0
$795$	0.298393	+	0.918358i	0.298393	+	0.918358i
$796$		0			0
$797$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$797$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	1.47340	−	1.07049i	1.47340	−	1.07049i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−0.168841	+	0.519639i	−0.168841	+	0.519639i
$808$		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
$811$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$811$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$812$		0			0
$813$	−2.89288			−2.89288
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	0.0973807	+	0.0707512i	0.0973807	+	0.0707512i
$820$	0.324914	+	0.236064i	0.324914	+	0.236064i
$821$	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
$825$	−0.0921464	+	0.195821i	−0.0921464	+	0.195821i
$826$		0			0
$827$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	−0.929776	−	0.368125i	$-0.880000\pi$
$827$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	0.0627905	+	0.998027i	$0.480000\pi$
$828$		0			0
$829$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$829$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$830$		0			0
$831$	2.34039	+	1.70039i	2.34039	+	1.70039i
$832$	−0.263146	+	0.809880i	−0.263146	+	0.809880i
$833$		0			0
$834$		0			0
$835$	0.662318			0.662318
$836$		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
$844$	0.541587	+	1.66683i	0.541587	+	1.66683i
$845$	0.238289	−	0.173127i	0.238289	−	0.173127i
$846$		0			0
$847$	−0.0800484	−	0.0967619i	−0.0800484	−	0.0967619i
$848$	0.618034			0.618034
$849$	−1.71963	+	1.24939i	−1.71963	+	1.24939i
$850$		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			$\pi$
$860$	−1.67950	+	1.22023i	−1.67950	+	1.22023i
$861$	−0.0212032	−	0.0652568i	−0.0212032	−	0.0652568i
$862$		0			0
$863$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$863$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−1.17950	+	0.856954i	−1.17950	+	0.856954i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.0927151	+	0.0673615i	0.0927151	+	0.0673615i
$876$		0			0
$877$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$877$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$878$		0			0
$879$		0			0
$880$	0.781202	+	0.733597i	0.781202	+	0.733597i
$881$		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
$885$	2.35050	+	1.70774i	2.35050	+	1.70774i
$886$		0			0
$887$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	−0.992115	−	0.125333i	$-0.960000\pi$
$887$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	0.876307	+	0.481754i	$0.160000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.160895	+	0.843439i	0.160895	+	0.843439i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−0.422178	+	1.29933i	−0.422178	+	1.29933i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.135173	+	0.0982088i	−0.135173	+	0.0982088i
$901$		0			0
$902$		0			0
$903$	0.354674			0.354674
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$907$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.425779	−	0.904827i	$-0.640000\pi$
$908$		0			0
$909$	0.673792	−	2.07372i	0.673792	−	2.07372i
$910$		0			0
$911$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$911$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−1.35458	+	0.984161i	−1.35458	+	0.984161i
$916$		0			0
$917$	0.0388067	−	0.119435i	0.0388067	−	0.119435i
$918$		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
$921$	0.278441	+	0.856954i	0.278441	+	0.856954i
$922$		0			0
$923$		0			0
$924$	−0.0343075	−	0.179846i	−0.0343075	−	0.179846i
$925$		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
$932$	−0.393950	−	1.21245i	−0.393950	−	1.21245i
$933$	−2.28488	+	1.66006i	−2.28488	+	1.66006i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$937$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$938$		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
$944$	1.50441	−	1.09302i	1.50441	−	1.09302i
$945$	0.0246400			0.0246400
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−2.06720	−	1.50191i	−2.06720	−	1.50191i
$952$		0			0
$953$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$953$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$954$		0			0
$955$	0.738289	−	0.536399i	0.738289	−	0.536399i
$956$	−1.27485			−1.27485
$957$		0			0
$958$		0			0
$959$		0			0
$960$	0.482809	+	1.48593i	0.482809	+	1.48593i
$961$	0.309017	−	0.951057i	0.309017	−	0.951057i
$962$		0			0
$963$	1.69334	+	1.23028i	1.69334	+	1.23028i
$964$		0			0
$965$	−0.422178	−	1.29933i	−0.422178	−	1.29933i
$966$		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	$0.400000\pi$
$972$	−0.443422	+	1.36471i	−0.443422	+	1.36471i
$973$		0			0
$974$		0			0
$975$	−0.0569496	+	0.175273i	−0.0569496	+	0.175273i
$976$	0.331159	+	1.01920i	0.331159	+	1.01920i
$977$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$977$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$978$		0			0
$979$	−1.41789	+	0.779494i	−1.41789	+	0.779494i
$980$	−1.05475			−1.05475
$981$	0.341264	−	0.247943i	0.341264	−	0.247943i
$982$		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
$991$	1.07165			1.07165			0.535827	−	0.844328i	$-0.320000\pi$
$991$	1.07165			1.07165			0.535827	+	0.844328i	$0.320000\pi$
$992$		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.916.4	yes	20
11.4	even	5	inner	1441.1.u.a.785.4	✓	20
131.130	odd	2	CM	1441.1.u.a.916.4	yes	20
1441.785	odd	10	inner	1441.1.u.a.785.4	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1441.1.u.a.785.4	✓	20	11.4	even	5	inner
1441.1.u.a.785.4	✓	20	1441.785	odd	10	inner
1441.1.u.a.916.4	yes	20	1.1	even	1	trivial
1441.1.u.a.916.4	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)

\(f(q)\)	\(=\)	\(q+(0.450527 + 1.38658i) q^{3} +(0.309017 - 0.951057i) q^{4} +(-0.866986 - 0.629902i) q^{5} +(0.0388067 - 0.119435i) q^{7} +(-0.910614 + 0.661600i) q^{9} +O(q^{10})\)	\(q+(0.450527 + 1.38658i) q^{3} +(0.309017 - 0.951057i) q^{4} +(-0.866986 - 0.629902i) q^{5} +(0.0388067 - 0.119435i) q^{7} +(-0.910614 + 0.661600i) q^{9} +(0.876307 - 0.481754i) q^{11} +1.45794 q^{12} +(0.688925 - 0.500534i) q^{13} +(0.482809 - 1.48593i) q^{15} +(-0.809017 - 0.587785i) q^{16} +(-0.866986 + 0.629902i) q^{20} +0.183089 q^{21} +(0.0458709 + 0.141176i) q^{25} +(-0.148122 - 0.107617i) q^{27} +(-0.101597 - 0.0738147i) q^{28} +(1.06279 + 0.998027i) q^{33} +(-0.108877 + 0.0791038i) q^{35} +(0.347824 + 1.07049i) q^{36} +(1.00441 + 0.729747i) q^{39} +(-0.115808 - 0.356420i) q^{41} +1.93717 q^{43} +(-0.187381 - 0.982287i) q^{44} +1.20623 q^{45} +(0.450527 - 1.38658i) q^{48} +(0.796258 + 0.578516i) q^{49} +(-0.263146 - 0.809880i) q^{52} +(-0.500000 + 0.363271i) q^{53} +(-1.06320 - 0.134314i) q^{55} +(-0.574633 + 1.76854i) q^{59} +(-1.26401 - 0.918358i) q^{60} +(-0.866986 - 0.629902i) q^{61} +(0.0436801 + 0.134433i) q^{63} +(-0.809017 + 0.587785i) q^{64} -0.912576 q^{65} +(-0.175086 + 0.127207i) q^{75} +(-0.0235315 - 0.123357i) q^{77} +(0.331159 + 1.01920i) q^{80} +(-0.265337 + 0.816623i) q^{81} +(0.0565777 - 0.174128i) q^{84} -1.61803 q^{89} +(-0.0330462 - 0.101706i) q^{91} +(-0.479249 + 1.01846i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) \) 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

Modular forms

Varieties

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Database

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