LMFDB - Embedded newform 869.1.j.a.236.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [869,1,Mod(157,869)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(869, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([8, 5]))

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

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

chi := DirichletCharacter("869.157");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 869 = 11 \cdot 79 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	869.j (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:	$0.433687495978$
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			236.3
Root			$-0.728969 + 0.684547i$ of defining polynomial
Character	$\chi$	$=$	869.236
Dual form			869.1.j.a.394.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.115808 - 0.356420i) q^{2} +(0.695393 - 0.505233i) q^{4} +(-0.393950 + 1.21245i) q^{5} +(-0.563797 - 0.409622i) q^{8} +(0.309017 + 0.951057i) q^{9} +O(q^{10})$	\(q+(-0.115808 - 0.356420i) q^{2} +(0.695393 - 0.505233i) q^{4} +(-0.393950 + 1.21245i) q^{5} +(-0.563797 - 0.409622i) q^{8} +(0.309017 + 0.951057i) q^{9} +0.477765 q^{10} +(-0.187381 + 0.982287i) q^{11} +(0.450527 + 1.38658i) q^{13} +(0.184911 - 0.569097i) q^{16} +(0.303189 - 0.220280i) q^{18} +(-1.56720 - 1.13864i) q^{19} +(0.338621 + 1.04217i) q^{20} +(0.371808 - 0.0469702i) q^{22} +1.93717 q^{23} +(-0.505828 - 0.367505i) q^{25} +(0.442031 - 0.321154i) q^{26} +(-0.613161 - 1.88711i) q^{31} -0.921143 q^{32} +(0.695393 + 0.505233i) q^{36} +(-0.224339 + 0.690446i) q^{38} +(0.718755 - 0.522206i) q^{40} +(0.365980 + 0.777747i) q^{44} -1.27485 q^{45} +(-0.224339 - 0.690446i) q^{46} +(0.309017 - 0.951057i) q^{49} +(-0.0724075 + 0.222847i) q^{50} +(1.01384 + 0.736597i) q^{52} +(-1.11716 - 0.614163i) q^{55} +(-0.601597 + 0.437086i) q^{62} +(-0.0782352 - 0.240783i) q^{64} -1.85865 q^{65} +0.125581 q^{67} +(0.215351 - 0.662783i) q^{72} +(1.50441 - 1.09302i) q^{73} -1.66510 q^{76} +(0.309017 + 0.951057i) q^{79} +(0.617158 + 0.448391i) q^{80} +(-0.809017 + 0.587785i) q^{81} +(0.190983 - 0.587785i) q^{83} +(0.508012 - 0.477055i) q^{88} -0.851559 q^{89} +(0.147638 + 0.454382i) q^{90} +(1.34709 - 0.978720i) q^{92} +(1.99794 - 1.45159i) q^{95} +(0.331159 + 1.01920i) q^{97} -0.374763 q^{98} +(-0.992115 + 0.125333i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 15 q^{20} - 5 q^{22} - 5 q^{25} + 15 q^{26} - 10 q^{32} - 5 q^{36} - 10 q^{40} - 5 q^{49} - 10 q^{50} - 10 q^{62} - 5 q^{64} - 10 q^{76} - 5 q^{79} - 10 q^{80} - 5 q^{81} + 15 q^{83} - 5 q^{88} + 15 q^{92} + 15 q^{95}+O(q^{100}) $ 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 + 15 * q^20 - 5 * q^22 - 5 * q^25 + 15 * q^26 - 10 * q^32 - 5 * q^36 - 10 * q^40 - 5 * q^49 - 10 * q^50 - 10 * q^62 - 5 * q^64 - 10 * q^76 - 5 * q^79 - 10 * q^80 - 5 * q^81 + 15 * q^83 - 5 * q^88 + 15 * q^92 + 15 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/869\mathbb{Z}\right)^\times$.

$n$	$475$	$793$
$\chi(n)$	$e\left(\frac{2}{5}\right)$	$-1$

Coefficient data

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

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

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

$2$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$2$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$3$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$3$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$4$	0.695393	−	0.505233i	0.695393	−	0.505233i
$5$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	0.535827	+	0.844328i	$0.320000\pi$
$5$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	−0.929776	+	0.368125i	$0.880000\pi$
$6$		0			0
$7$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$7$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$8$	−0.563797	−	0.409622i	−0.563797	−	0.409622i
$9$	0.309017	+	0.951057i	0.309017	+	0.951057i
$10$	0.477765			0.477765
$11$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$11$	−0.187381	+	0.982287i	−0.187381	+	0.982287i
$12$		0			0
$13$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$13$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$14$		0			0
$15$		0			0
$16$	0.184911	−	0.569097i	0.184911	−	0.569097i
$17$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$17$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$18$	0.303189	−	0.220280i	0.303189	−	0.220280i
$19$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$19$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$20$	0.338621	+	1.04217i	0.338621	+	1.04217i
$21$		0			0
$22$	0.371808	−	0.0469702i	0.371808	−	0.0469702i
$23$	1.93717			1.93717			0.968583	−	0.248690i	$-0.0800000\pi$
$23$	1.93717			1.93717			0.968583	+	0.248690i	$0.0800000\pi$
$24$		0			0
$25$	−0.505828	−	0.367505i	−0.505828	−	0.367505i
$26$	0.442031	−	0.321154i	0.442031	−	0.321154i
$27$		0			0
$28$		0			0
$29$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$29$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$30$		0			0
$31$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$31$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$32$	−0.921143			−0.921143
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.695393	+	0.505233i	0.695393	+	0.505233i
$37$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$37$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$38$	−0.224339	+	0.690446i	−0.224339	+	0.690446i
$39$		0			0
$40$	0.718755	−	0.522206i	0.718755	−	0.522206i
$41$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$41$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	0.365980	+	0.777747i	0.365980	+	0.777747i
$45$	−1.27485			−1.27485
$46$	−0.224339	−	0.690446i	−0.224339	−	0.690446i
$47$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$47$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$48$		0			0
$49$	0.309017	−	0.951057i	0.309017	−	0.951057i
$50$	−0.0724075	+	0.222847i	−0.0724075	+	0.222847i
$51$		0			0
$52$	1.01384	+	0.736597i	1.01384	+	0.736597i
$53$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$53$		0			0		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			0		0.809017	−	0.587785i	$-0.200000\pi$
$59$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$60$		0			0
$61$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$61$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$62$	−0.601597	+	0.437086i	−0.601597	+	0.437086i
$63$		0			0
$64$	−0.0782352	−	0.240783i	−0.0782352	−	0.240783i
$65$	−1.85865			−1.85865
$66$		0			0
$67$	0.125581			0.125581			0.0627905	−	0.998027i	$-0.480000\pi$
$67$	0.125581			0.125581			0.0627905	+	0.998027i	$0.480000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$71$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$72$	0.215351	−	0.662783i	0.215351	−	0.662783i
$73$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$73$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$74$		0			0
$75$		0			0
$76$	−1.66510			−1.66510
$77$		0			0
$78$		0			0
$79$	0.309017	+	0.951057i	0.309017	+	0.951057i
$79$	0.309017	+	0.951057i	0.309017	+	0.951057i
$80$	0.617158	+	0.448391i	0.617158	+	0.448391i
$81$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$82$		0			0
$83$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$83$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	0.508012	−	0.477055i	0.508012	−	0.477055i
$89$	−0.851559			−0.851559			−0.425779	−	0.904827i	$-0.640000\pi$
$89$	−0.851559			−0.851559			−0.425779	+	0.904827i	$0.640000\pi$
$90$	0.147638	+	0.454382i	0.147638	+	0.454382i
$91$		0			0
$92$	1.34709	−	0.978720i	1.34709	−	0.978720i
$93$		0			0
$94$		0			0
$95$	1.99794	−	1.45159i	1.99794	−	1.45159i
$96$		0			0
$97$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$97$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$98$	−0.374763			−0.374763
$99$	−0.992115	+	0.125333i	−0.992115	+	0.125333i
$100$	−0.537425			−0.537425
$101$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	−0.992115	−	0.125333i	$-0.960000\pi$
$101$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	0.728969	−	0.684547i	$-0.240000\pi$
$102$		0			0
$103$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$103$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$104$	0.313968	−	0.966296i	0.313968	−	0.966296i
$105$		0			0
$106$		0			0
$107$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$107$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	−0.0895243	+	0.469303i	−0.0895243	+	0.469303i
$111$		0			0
$112$		0			0
$113$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$113$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$114$		0			0
$115$	−0.763146	+	2.34872i	−0.763146	+	2.34872i
$116$		0			0
$117$	−1.17950	+	0.856954i	−1.17950	+	0.856954i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.929776	−	0.368125i	−0.929776	−	0.368125i
$122$		0			0
$123$		0			0
$124$	−1.37982	−	1.00250i	−1.37982	−	1.00250i
$125$	−0.386520	+	0.280823i	−0.386520	+	0.280823i
$126$		0			0
$127$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$127$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$128$	−0.821980	+	0.597204i	−0.821980	+	0.597204i
$129$		0			0
$130$	0.215246	+	0.662460i	0.215246	+	0.662460i
$131$	−1.85955			−1.85955			−0.929776	−	0.368125i	$-0.880000\pi$
$131$	−1.85955			−1.85955			−0.929776	+	0.368125i	$0.880000\pi$
$132$		0			0
$133$		0			0
$134$	−0.0145433	−	0.0447596i	−0.0145433	−	0.0447596i
$135$		0			0
$136$		0			0
$137$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$137$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$138$		0			0
$139$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$139$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.44644	+	0.182728i	−1.44644	+	0.182728i
$144$	0.598384			0.598384
$145$		0			0
$146$	−0.563797	−	0.409622i	−0.563797	−	0.409622i
$147$		0			0
$148$		0			0
$149$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$149$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$150$		0			0
$151$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$151$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$152$	0.417171	+	1.28392i	0.417171	+	1.28392i
$153$		0			0
$154$		0			0
$155$	2.52959			2.52959
$156$		0			0
$157$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$157$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$158$	0.303189	−	0.220280i	0.303189	−	0.220280i
$159$		0			0
$160$	0.362884	−	1.11684i	0.362884	−	1.11684i
$161$		0			0
$162$	0.303189	+	0.220280i	0.303189	+	0.220280i
$163$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$163$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$164$		0			0
$165$		0			0
$166$	−0.231616			−0.231616
$167$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.425779	−	0.904827i	$-0.640000\pi$
$167$	−0.613161	−	1.88711i	−0.613161	−	1.88711i	−0.187381	−	0.982287i	$-0.560000\pi$
$168$		0			0
$169$	−0.910614	+	0.661600i	−0.910614	+	0.661600i
$170$		0			0
$171$	0.598617	−	1.84235i	0.598617	−	1.84235i
$172$		0			0
$173$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$173$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$174$		0			0
$175$		0			0
$176$	0.524368	+	0.288274i	0.524368	+	0.288274i
$177$		0			0
$178$	0.0986173	+	0.303513i	0.0986173	+	0.303513i
$179$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$179$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$180$	−0.886520	+	0.644095i	−0.886520	+	0.644095i
$181$	0.450527	−	1.38658i	0.450527	−	1.38658i	−0.425779	−	0.904827i	$-0.640000\pi$
$181$	0.450527	−	1.38658i	0.450527	−	1.38658i	0.876307	−	0.481754i	$-0.160000\pi$
$182$		0			0
$183$		0			0
$184$	−1.09217	−	0.793506i	−1.09217	−	0.793506i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−0.748754	−	0.544002i	−0.748754	−	0.544002i
$191$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$191$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$192$		0			0
$193$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$193$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$194$	0.324914	−	0.236064i	0.324914	−	0.236064i
$195$		0			0
$196$	−0.265616	−	0.817483i	−0.265616	−	0.817483i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	0.159566	+	0.339095i	0.159566	+	0.339095i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	0.134646	+	0.414397i	0.134646	+	0.414397i
$201$		0			0
$202$	−0.258183	+	0.187581i	−0.258183	+	0.187581i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.598617	+	1.84235i	0.598617	+	1.84235i
$208$	0.872407			0.872407
$209$	1.41213	−	1.32608i	1.41213	−	1.32608i
$210$		0			0
$211$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$211$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−1.08716	+	0.137340i	−1.08716	+	0.137340i
$221$		0			0
$222$		0			0
$223$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$223$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$224$		0			0
$225$	0.193209	−	0.594636i	0.193209	−	0.594636i
$226$		0			0
$227$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$227$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$228$		0			0
$229$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$229$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$230$	0.925511			0.925511
$231$		0			0
$232$		0			0
$233$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$233$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$234$	0.442031	+	0.321154i	0.442031	+	0.321154i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$239$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$240$		0			0
$241$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$241$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$242$	−0.0235315	+	0.374023i	−0.0235315	+	0.374023i
$243$		0			0
$244$		0			0
$245$	1.03137	+	0.749337i	1.03137	+	0.749337i
$246$		0			0
$247$	0.872746	−	2.68604i	0.872746	−	2.68604i
$248$	−0.427306	+	1.31511i	−0.427306	+	1.31511i
$249$		0			0
$250$	0.144853	+	0.105242i	0.144853	+	0.105242i
$251$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$251$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$252$		0			0
$253$	−0.362989	+	1.90285i	−0.362989	+	1.90285i
$254$		0			0
$255$		0			0
$256$	0.103225	+	0.0749974i	0.103225	+	0.0749974i
$257$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.992115	−	0.125333i	$-0.960000\pi$
$257$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.187381	+	0.982287i	$0.560000\pi$
$258$		0			0
$259$		0			0
$260$	−1.29249	+	0.939050i	−1.29249	+	0.939050i
$261$		0			0
$262$	0.215351	+	0.662783i	0.215351	+	0.662783i
$263$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$263$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.0873282	−	0.0634476i	0.0873282	−	0.0634476i
$269$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$269$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$270$		0			0
$271$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$271$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.455778	−	0.428004i	0.455778	−	0.428004i
$276$		0			0
$277$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$277$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	0.535827	−	0.844328i	$-0.320000\pi$
$278$		0			0
$279$	1.60528	−	1.16630i	1.60528	−	1.16630i
$280$		0			0
$281$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	0.535827	+	0.844328i	$0.320000\pi$
$281$	−0.393950	+	1.21245i	−0.393950	+	1.21245i	−0.929776	+	0.368125i	$0.880000\pi$
$282$		0			0
$283$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.876307	+	0.481754i	$0.160000\pi$
$283$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.728969	+	0.684547i	$0.240000\pi$
$284$		0			0
$285$		0			0
$286$	0.232637	+	0.494380i	0.232637	+	0.494380i
$287$		0			0
$288$	−0.284649	−	0.876059i	−0.284649	−	0.876059i
$289$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$290$		0			0
$291$		0			0
$292$	0.493928	−	1.52015i	0.493928	−	1.52015i
$293$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$293$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.872746	+	2.68604i	0.872746	+	2.68604i
$300$		0			0
$301$		0			0
$302$	−0.0145433	+	0.0447596i	−0.0145433	+	0.0447596i
$303$		0			0
$304$	−0.937788	+	0.681343i	−0.937788	+	0.681343i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	−0.292947	−	0.901598i	−0.292947	−	0.901598i
$311$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$311$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$312$		0			0
$313$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$313$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$314$		0			0
$315$		0			0
$316$	0.695393	+	0.505233i	0.695393	+	0.505233i
$317$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$317$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$318$		0			0
$319$		0			0
$320$	0.322759			0.322759
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.265616	+	0.817483i	−0.265616	+	0.817483i
$325$	0.281687	−	0.866942i	0.281687	−	0.866942i
$326$	0.531374	−	0.386066i	0.531374	−	0.386066i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	−0.164160	−	0.505233i	−0.164160	−	0.505233i
$333$		0			0
$334$	−0.601597	+	0.437086i	−0.601597	+	0.437086i
$335$	−0.0494726	+	0.152261i	−0.0494726	+	0.152261i
$336$		0			0
$337$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	−0.929776	−	0.368125i	$-0.880000\pi$
$337$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	0.0627905	+	0.998027i	$0.480000\pi$
$338$	0.341264	+	0.247943i	0.341264	+	0.247943i
$339$		0			0
$340$		0			0
$341$	1.96858	−	0.248690i	1.96858	−	0.248690i
$342$	−0.725978			−0.725978
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$347$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$348$		0			0
$349$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$349$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$350$		0			0
$351$		0			0
$352$	0.172605	−	0.904827i	0.172605	−	0.904827i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$	−0.592168	+	0.430235i	−0.592168	+	0.430235i
$357$		0			0
$358$	−0.0715733	+	0.220280i	−0.0715733	+	0.220280i
$359$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$359$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$360$	0.718755	+	0.522206i	0.718755	+	0.522206i
$361$	0.850604	+	2.61789i	0.850604	+	2.61789i
$362$	−0.546380			−0.546380
$363$		0			0
$364$		0			0
$365$	0.732570	+	2.25462i	0.732570	+	2.25462i
$366$		0			0
$367$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$367$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$368$	0.358203	−	1.10244i	0.358203	−	1.10244i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$379$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$380$	0.655964	−	2.01885i	0.655964	−	2.01885i
$381$		0			0
$382$		0			0
$383$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$383$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.745220	+	0.541434i	0.745220	+	0.541434i
$389$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$389$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$390$		0			0
$391$		0			0
$392$	−0.563797	+	0.409622i	−0.563797	+	0.409622i
$393$		0			0
$394$		0			0
$395$	−1.27485			−1.27485
$396$	−0.626587	+	0.588404i	−0.626587	+	0.588404i
$397$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$397$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$398$		0			0
$399$		0			0
$400$	−0.302679	+	0.219909i	−0.302679	+	0.219909i
$401$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$401$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$402$		0			0
$403$	2.34039	−	1.70039i	2.34039	−	1.70039i
$404$	−0.592168	−	0.430235i	−0.592168	−	0.430235i
$405$	−0.393950	−	1.21245i	−0.393950	−	1.21245i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$409$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.587328	−	0.426719i	0.587328	−	0.426719i
$415$	0.637424	+	0.463116i	0.637424	+	0.463116i
$416$	−0.415000	−	1.27724i	−0.415000	−	1.27724i
$417$		0			0
$418$	−0.636179	−	0.349742i	−0.636179	−	0.349742i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$421$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$431$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$432$		0			0
$433$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$433$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.03593	−	2.20573i	−3.03593	−	2.20573i
$438$		0			0
$439$	1.07165			1.07165			0.535827	−	0.844328i	$-0.320000\pi$
$439$	1.07165			1.07165			0.535827	+	0.844328i	$0.320000\pi$
$440$	0.378275	+	0.803876i	0.378275	+	0.803876i
$441$	1.00000			1.00000
$442$		0			0
$443$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$443$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$444$		0			0
$445$	0.335471	−	1.03247i	0.335471	−	1.03247i
$446$	−0.0715733	+	0.220280i	−0.0715733	+	0.220280i
$447$		0			0
$448$		0			0
$449$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$449$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$450$	−0.234316			−0.234316
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$457$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$458$		0			0
$459$		0			0
$460$	0.655964	+	2.01885i	0.655964	+	2.01885i
$461$		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
$467$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	−0.992115	−	0.125333i	$-0.960000\pi$
$467$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	0.876307	+	0.481754i	$0.160000\pi$
$468$	−0.387252	+	1.19184i	−0.387252	+	1.19184i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.374278	+	1.15191i	0.374278	+	1.15191i
$476$		0			0
$477$		0			0
$478$	0.147638	−	0.454382i	0.147638	−	0.454382i
$479$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$479$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$480$		0			0
$481$		0			0
$482$	−0.202967	−	0.624667i	−0.202967	−	0.624667i
$483$		0			0
$484$	−0.832549	+	0.213762i	−0.832549	+	0.213762i
$485$	−1.36620			−1.36620
$486$		0			0
$487$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$487$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$488$		0			0
$489$		0			0
$490$	0.147638	−	0.454382i	0.147638	−	0.454382i
$491$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$491$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$		0			0
$494$	−1.05843			−1.05843
$495$	0.238883	−	1.25227i	0.238883	−	1.25227i
$496$	−1.18733			−1.18733
$497$		0			0
$498$		0			0
$499$	0.303189	−	0.220280i	0.303189	−	0.220280i	−0.425779	−	0.904827i	$-0.640000\pi$
$499$	0.303189	−	0.220280i	0.303189	−	0.220280i	0.728969	+	0.684547i	$0.240000\pi$
$500$	−0.126902	+	0.390565i	−0.126902	+	0.390565i
$501$		0			0
$502$		0			0
$503$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$503$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$504$		0			0
$505$	1.08561			1.08561
$506$	0.720253	−	0.0909891i	0.720253	−	0.0909891i
$507$		0			0
$508$		0			0
$509$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$509$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$510$		0			0
$511$		0			0
$512$	−0.299192	+	0.920819i	−0.299192	+	0.920819i
$513$		0			0
$514$	0.442031	+	0.321154i	0.442031	+	0.321154i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	1.04790	+	0.761344i	1.04790	+	0.761344i
$521$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$521$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$522$		0			0
$523$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$523$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$524$	−1.29312	+	0.939507i	−1.29312	+	0.939507i
$525$		0			0
$526$	−0.202967	−	0.624667i	−0.202967	−	0.624667i
$527$		0			0
$528$		0			0
$529$	2.75261			2.75261
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.0708022	−	0.0514408i	−0.0708022	−	0.0514408i
$537$		0			0
$538$	−0.0470631			−0.0470631
$539$	0.876307	+	0.481754i	0.876307	+	0.481754i
$540$		0			0
$541$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$541$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$547$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.425779	−	0.904827i	$-0.640000\pi$
$548$		0			0
$549$		0			0
$550$	−0.205332	−	0.112882i	−0.205332	−	0.112882i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.386520	+	0.280823i	−0.386520	+	0.280823i
$555$		0			0
$556$		0			0
$557$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$557$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$558$	−0.601597	−	0.437086i	−0.601597	−	0.437086i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.477765			0.477765
$563$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$563$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$564$		0			0
$565$		0			0
$566$	0.229790	−	0.707220i	0.229790	−	0.707220i
$567$		0			0
$568$		0			0
$569$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$569$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$570$		0			0
$571$	−1.98423			−1.98423			−0.992115	−	0.125333i	$-0.960000\pi$
$571$	−1.98423			−1.98423			−0.992115	+	0.125333i	$0.960000\pi$
$572$	−0.913525	+	0.857857i	−0.913525	+	0.857857i
$573$		0			0
$574$		0			0
$575$	−0.979872	−	0.711919i	−0.979872	−	0.711919i
$576$	0.204822	−	0.148812i	0.204822	−	0.148812i
$577$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$577$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$578$	−0.115808	+	0.356420i	−0.115808	+	0.356420i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−1.29591			−1.29591
$585$	−0.574354	−	1.76768i	−0.574354	−	1.76768i
$586$		0			0
$587$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$587$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$588$		0			0
$589$	−1.18779	+	3.65565i	−1.18779	+	3.65565i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.27485			−1.27485			−0.637424	−	0.770513i	$-0.720000\pi$
$593$	−1.27485			−1.27485			−0.637424	+	0.770513i	$0.720000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	0.856288	−	0.622129i	0.856288	−	0.622129i
$599$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$599$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$600$		0			0
$601$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$601$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$602$		0			0
$603$	0.0388067	+	0.119435i	0.0388067	+	0.119435i
$604$	−0.107944			−0.107944
$605$	0.812619	−	0.982287i	0.812619	−	0.982287i
$606$		0			0
$607$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$607$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$608$	1.44362	+	1.04885i	1.44362	+	1.04885i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$613$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.45794			1.45794			0.728969	−	0.684547i	$-0.240000\pi$
$617$	1.45794			1.45794			0.728969	+	0.684547i	$0.240000\pi$
$618$		0			0
$619$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$619$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$620$	1.75906	−	1.27803i	1.75906	−	1.27803i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.381424	−	1.17390i	−0.381424	−	1.17390i
$626$	0.743615			0.743615
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$631$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$632$	0.215351	−	0.662783i	0.215351	−	0.662783i
$633$		0			0
$634$	−0.490571	+	0.356420i	−0.490571	+	0.356420i
$635$		0			0
$636$		0			0
$637$	1.45794			1.45794
$638$		0			0
$639$		0			0
$640$	−0.400262	−	1.23188i	−0.400262	−	1.23188i
$641$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$641$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$642$		0			0
$643$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$643$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$647$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$648$	0.696891			0.696891
$649$		0			0
$650$	−0.341617			−0.341617
$651$		0			0
$652$	1.21875	+	0.885477i	1.21875	+	0.885477i
$653$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$653$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$654$		0			0
$655$	0.732570	−	2.25462i	0.732570	−	2.25462i
$656$		0			0
$657$	1.50441	+	1.09302i	1.50441	+	1.09302i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$	−0.348445	+	0.253160i	−0.348445	+	0.253160i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1.37982	−	1.00250i	−1.37982	−	1.00250i
$669$		0			0
$670$	0.0599983			0.0599983
$671$		0			0
$672$		0			0
$673$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$673$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$674$	0.324914	+	0.236064i	0.324914	+	0.236064i
$675$		0			0
$676$	−0.298973	+	0.920144i	−0.298973	+	0.920144i
$677$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$677$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	−0.316616	−	0.672843i	−0.316616	−	0.672843i
$683$	−1.85955			−1.85955			−0.929776	−	0.368125i	$-0.880000\pi$
$683$	−1.85955			−1.85955			−0.929776	+	0.368125i	$0.880000\pi$
$684$	−0.514543	−	1.58360i	−0.514543	−	1.58360i
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$691$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$692$		0			0
$693$		0			0
$694$	−0.0470631			−0.0470631
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$701$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$702$		0			0
$703$		0			0
$704$	0.251178	−	0.0317312i	0.251178	−	0.0317312i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$709$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$710$		0			0
$711$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$712$	0.480106	+	0.348817i	0.480106	+	0.348817i
$713$	−1.18779	−	3.65565i	−1.18779	−	3.65565i
$714$		0			0
$715$	0.348276	−	1.82573i	0.348276	−	1.82573i
$716$	−0.531233			−0.531233
$717$		0			0
$718$		0			0
$719$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$719$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$720$	−0.235733	+	0.725513i	−0.235733	+	0.725513i
$721$		0			0
$722$	0.834563	−	0.606346i	0.834563	−	0.606346i
$723$		0			0
$724$	−0.387252	−	1.19184i	−0.387252	−	1.19184i
$725$		0			0
$726$		0			0
$727$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$727$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$728$		0			0
$729$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$730$	0.718755	−	0.522206i	0.718755	−	0.522206i
$731$		0			0
$732$		0			0
$733$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$733$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$734$	−0.258183	−	0.187581i	−0.258183	−	0.187581i
$735$		0			0
$736$	−1.78441			−1.78441
$737$	−0.0235315	+	0.123357i	−0.0235315	+	0.123357i
$738$		0			0
$739$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$739$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.450527	−	1.38658i	0.450527	−	1.38658i	−0.425779	−	0.904827i	$-0.640000\pi$
$743$	0.450527	−	1.38658i	0.450527	−	1.38658i	0.876307	−	0.481754i	$-0.160000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.618034			0.618034
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.303189	+	0.220280i	0.303189	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$751$	0.303189	+	0.220280i	0.303189	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.129521	−	0.0941025i	0.129521	−	0.0941025i
$756$		0			0
$757$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$757$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$758$		0			0
$759$		0			0
$760$	−1.72104			−1.72104
$761$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$761$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.113624	+	0.0825527i	−0.113624	+	0.0825527i
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$773$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$774$		0			0
$775$	−0.383371	+	1.17989i	−0.383371	+	1.17989i
$776$	0.230782	−	0.710273i	0.230782	−	0.710273i
$777$		0			0
$778$	−0.601597	−	0.437086i	−0.601597	−	0.437086i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.484103	−	0.351721i	−0.484103	−	0.351721i
$785$		0			0
$786$		0			0
$787$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$787$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$788$		0			0
$789$		0			0
$790$	0.147638	+	0.454382i	0.147638	+	0.454382i
$791$		0			0
$792$	0.610690	+	0.335730i	0.610690	+	0.335730i
$793$		0			0
$794$	−0.0715733	−	0.220280i	−0.0715733	−	0.220280i
$795$		0			0
$796$		0			0
$797$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$797$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$798$		0			0
$799$		0			0
$800$	0.465940	+	0.338525i	0.465940	+	0.338525i
$801$	−0.263146	−	0.809880i	−0.263146	−	0.809880i
$802$		0			0
$803$	0.791759	+	1.68257i	0.791759	+	1.68257i
$804$		0			0
$805$		0			0
$806$	−0.877091	−	0.637244i	−0.877091	−	0.637244i
$807$		0			0
$808$	−0.183384	+	0.564398i	−0.183384	+	0.564398i
$809$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$809$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$810$	−0.386520	+	0.280823i	−0.386520	+	0.280823i
$811$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$811$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.23432			−2.23432
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$821$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$822$		0			0
$823$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$823$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$827$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$828$	1.34709	+	0.978720i	1.34709	+	0.978720i
$829$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$829$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$830$	0.0912451	−	0.280823i	0.0912451	−	0.280823i
$831$		0			0
$832$	0.298618	−	0.216959i	0.298618	−	0.216959i
$833$		0			0
$834$		0			0
$835$	2.52959			2.52959
$836$	0.312008	−	1.63560i	0.312008	−	1.63560i
$837$		0			0
$838$		0			0
$839$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$839$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$840$		0			0
$841$	0.309017	−	0.951057i	0.309017	−	0.951057i
$842$	−0.124106	+	0.381959i	−0.124106	+	0.381959i
$843$		0			0
$844$		0			0
$845$	−0.443422	−	1.36471i	−0.443422	−	1.36471i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$853$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$854$		0			0
$855$	1.99794	+	1.45159i	1.99794	+	1.45159i
$856$		0			0
$857$	1.93717			1.93717			0.968583	−	0.248690i	$-0.0800000\pi$
$857$	1.93717			1.93717			0.968583	+	0.248690i	$0.0800000\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	−0.563797	+	0.409622i	−0.563797	+	0.409622i
$863$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$863$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$864$		0			0
$865$		0			0
$866$	0.587328	+	0.426719i	0.587328	+	0.426719i
$867$		0			0
$868$		0			0
$869$	−0.992115	+	0.125333i	−0.992115	+	0.125333i
$870$		0			0
$871$	0.0565777	+	0.174128i	0.0565777	+	0.174128i
$872$		0			0
$873$	−0.866986	+	0.629902i	−0.866986	+	0.629902i
$874$	−0.434583	+	1.33751i	−0.434583	+	1.33751i
$875$		0			0
$876$		0			0
$877$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.876307	+	0.481754i	$0.160000\pi$
$877$	1.60528	+	1.16630i	1.60528	+	1.16630i	0.728969	+	0.684547i	$0.240000\pi$
$878$	−0.124106	−	0.381959i	−0.124106	−	0.381959i
$879$		0			0
$880$	−0.556093	+	0.522206i	−0.556093	+	0.522206i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.115808	−	0.356420i	−0.115808	−	0.356420i
$883$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$883$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$887$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$888$		0			0
$889$		0			0
$890$	−0.406845			−0.406845
$891$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$892$	−0.531233			−0.531233
$893$		0			0
$894$		0			0
$895$	0.637424	−	0.463116i	0.637424	−	0.463116i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.166073	−	0.511121i	−0.166073	−	0.511121i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.50368	+	1.09249i	1.50368	+	1.09249i
$906$		0			0
$907$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$907$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$908$		0			0
$909$	0.688925	−	0.500534i	0.688925	−	0.500534i
$910$		0			0
$911$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$911$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$912$		0			0
$913$	0.541587	+	0.297740i	0.541587	+	0.297740i
$914$	0.319132			0.319132
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$919$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$920$	1.39235	−	1.01160i	1.39235	−	1.01160i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$929$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$930$		0			0
$931$	−1.56720	+	1.13864i	−1.56720	+	1.13864i
$932$		0			0
$933$		0			0
$934$	0.140447			0.140447
$935$		0			0
$936$	1.01602			1.01602
$937$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$937$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$941$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	2.19334	+	1.59355i	2.19334	+	1.59355i
$950$	0.367220	−	0.266801i	0.367220	−	0.266801i
$951$		0			0
$952$		0			0
$953$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.425779	+	0.904827i	$0.640000\pi$
$953$	−1.41789	+	1.03016i	−1.41789	+	1.03016i	−0.992115	+	0.125333i	$0.960000\pi$
$954$		0			0
$955$		0			0
$956$	1.09580			1.09580
$957$		0			0
$958$	−0.231616			−0.231616
$959$		0			0
$960$		0			0
$961$	−2.37622	+	1.72642i	−2.37622	+	1.72642i
$962$		0			0
$963$		0			0
$964$	1.21875	−	0.885477i	1.21875	−	0.885477i
$965$		0			0
$966$		0			0
$967$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$967$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$968$	0.373413	+	0.588404i	0.373413	+	0.588404i
$969$		0			0
$970$	0.158216	+	0.486940i	0.158216	+	0.486940i
$971$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$971$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.425779	−	0.904827i	$-0.640000\pi$
$972$		0			0
$973$		0			0
$974$	−0.168841	+	0.519639i	−0.168841	+	0.519639i
$975$		0			0
$976$		0			0
$977$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$977$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$978$		0			0
$979$	0.159566	−	0.836475i	0.159566	−	0.836475i
$980$	1.09580			1.09580
$981$		0			0
$982$		0			0
$983$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$983$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.750172	−	2.30879i	−0.750172	−	2.30879i
$989$		0			0
$990$	−0.473998	+	0.0598799i	−0.473998	+	0.0598799i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	0.564809	+	1.73830i	0.564809	+	1.73830i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$997$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$998$	−0.113624	−	0.0825527i	−0.113624	−	0.0825527i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	869.1.j.a.236.3	✓	20
11.9	even	5	inner	869.1.j.a.394.3	yes	20
79.78	odd	2	CM	869.1.j.a.236.3	✓	20
869.394	odd	10	inner	869.1.j.a.394.3	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
869.1.j.a.236.3	✓	20	1.1	even	1	trivial
869.1.j.a.236.3	✓	20	79.78	odd	2	CM
869.1.j.a.394.3	yes	20	11.9	even	5	inner
869.1.j.a.394.3	yes	20	869.394	odd	10	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 869 = 11 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	869.j (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.115808 - 0.356420i) q^{2} +(0.695393 - 0.505233i) q^{4} +(-0.393950 + 1.21245i) q^{5} +(-0.563797 - 0.409622i) q^{8} +(0.309017 + 0.951057i) q^{9} +O(q^{10})\)	\(q+(-0.115808 - 0.356420i) q^{2} +(0.695393 - 0.505233i) q^{4} +(-0.393950 + 1.21245i) q^{5} +(-0.563797 - 0.409622i) q^{8} +(0.309017 + 0.951057i) q^{9} +0.477765 q^{10} +(-0.187381 + 0.982287i) q^{11} +(0.450527 + 1.38658i) q^{13} +(0.184911 - 0.569097i) q^{16} +(0.303189 - 0.220280i) q^{18} +(-1.56720 - 1.13864i) q^{19} +(0.338621 + 1.04217i) q^{20} +(0.371808 - 0.0469702i) q^{22} +1.93717 q^{23} +(-0.505828 - 0.367505i) q^{25} +(0.442031 - 0.321154i) q^{26} +(-0.613161 - 1.88711i) q^{31} -0.921143 q^{32} +(0.695393 + 0.505233i) q^{36} +(-0.224339 + 0.690446i) q^{38} +(0.718755 - 0.522206i) q^{40} +(0.365980 + 0.777747i) q^{44} -1.27485 q^{45} +(-0.224339 - 0.690446i) q^{46} +(0.309017 - 0.951057i) q^{49} +(-0.0724075 + 0.222847i) q^{50} +(1.01384 + 0.736597i) q^{52} +(-1.11716 - 0.614163i) q^{55} +(-0.601597 + 0.437086i) q^{62} +(-0.0782352 - 0.240783i) q^{64} -1.85865 q^{65} +0.125581 q^{67} +(0.215351 - 0.662783i) q^{72} +(1.50441 - 1.09302i) q^{73} -1.66510 q^{76} +(0.309017 + 0.951057i) q^{79} +(0.617158 + 0.448391i) q^{80} +(-0.809017 + 0.587785i) q^{81} +(0.190983 - 0.587785i) q^{83} +(0.508012 - 0.477055i) q^{88} -0.851559 q^{89} +(0.147638 + 0.454382i) q^{90} +(1.34709 - 0.978720i) q^{92} +(1.99794 - 1.45159i) q^{95} +(0.331159 + 1.01920i) q^{97} -0.374763 q^{98} +(-0.992115 + 0.125333i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) \) 20 * q - 5 * q^4 - 5 * q^9	\( 20 q - 5 q^{4} - 5 q^{9} - 5 q^{16} + 15 q^{20} - 5 q^{22} - 5 q^{25} + 15 q^{26} - 10 q^{32} - 5 q^{36} - 10 q^{40} - 5 q^{49} - 10 q^{50} - 10 q^{62} - 5 q^{64} - 10 q^{76} - 5 q^{79} - 10 q^{80} - 5 q^{81} + 15 q^{83} - 5 q^{88} + 15 q^{92} + 15 q^{95}+O(q^{100}) \) 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 + 15 * q^20 - 5 * q^22 - 5 * q^25 + 15 * q^26 - 10 * q^32 - 5 * q^36 - 10 * q^40 - 5 * q^49 - 10 * q^50 - 10 * q^62 - 5 * q^64 - 10 * q^76 - 5 * q^79 - 10 * q^80 - 5 * q^81 + 15 * q^83 - 5 * q^88 + 15 * q^92 + 15 * q^95

Introduction

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