LMFDB - Embedded newform 1397.1.l.a.1015.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1397,1,Mod(126,1397)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1397, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([4, 5]))

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

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

chi := DirichletCharacter("1397.126");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1397 = 11 \cdot 127 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1397.l (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:	$0.697193822648$
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			1015.5
Root			$0.425779 + 0.904827i$ of defining polynomial
Character	$\chi$	$=$	1397.1015
Dual form			1397.1.l.a.1269.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.60528 - 1.16630i) q^{2} +(0.907634 - 2.79341i) q^{4} +(-1.18779 - 3.65565i) q^{8} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})$	\(q+(1.60528 - 1.16630i) q^{2} +(0.907634 - 2.79341i) q^{4} +(-1.18779 - 3.65565i) q^{8} +(-0.809017 + 0.587785i) q^{9} +(0.535827 - 0.844328i) q^{11} +(-0.866986 + 0.629902i) q^{13} +(-3.79411 - 2.75658i) q^{16} +(0.303189 + 0.220280i) q^{17} +(-0.613161 + 1.88711i) q^{18} +(0.541587 + 1.66683i) q^{19} +(-0.124591 - 1.98031i) q^{22} +(0.309017 + 0.951057i) q^{25} +(-0.657096 + 2.02233i) q^{26} +(1.60528 - 1.16630i) q^{31} -5.46182 q^{32} +0.743615 q^{34} +(0.907634 + 2.79341i) q^{36} +(-0.115808 + 0.356420i) q^{37} +(2.81343 + 2.04407i) q^{38} +(-0.393950 - 1.21245i) q^{41} +(-1.87222 - 2.26313i) q^{44} +(0.331159 + 1.01920i) q^{47} +(-0.809017 - 0.587785i) q^{49} +(1.60528 + 1.16630i) q^{50} +(0.972670 + 2.99357i) q^{52} +(0.688925 + 0.500534i) q^{61} +(1.21665 - 3.74447i) q^{62} +(-4.97361 + 3.61354i) q^{64} +(0.890518 - 0.646999i) q^{68} +(-1.17950 - 0.856954i) q^{71} +(3.10969 + 2.25932i) q^{72} +(0.0388067 - 0.119435i) q^{73} +(0.229790 + 0.707220i) q^{74} +5.14772 q^{76} +(-1.56720 + 1.13864i) q^{79} +(0.309017 - 0.951057i) q^{81} +(-2.04648 - 1.48686i) q^{82} +(-3.72302 - 0.955910i) q^{88} +(1.72030 + 1.24987i) q^{94} -1.98423 q^{98} +(0.0627905 + 0.998027i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 10 q^{32} - 10 q^{34} - 5 q^{36} + 15 q^{38} - 5 q^{44} - 5 q^{49} + 15 q^{52} - 10 q^{62} - 5 q^{64} + 15 q^{74} - 5 q^{81} - 5 q^{88}+O(q^{100}) $ 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 - 5 * q^25 - 10 * q^32 - 10 * q^34 - 5 * q^36 + 15 * q^38 - 5 * q^44 - 5 * q^49 + 15 * q^52 - 10 * q^62 - 5 * q^64 + 15 * q^74 - 5 * q^81 - 5 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1397\mathbb{Z}\right)^\times$.

$n$	$255$	$892$
$\chi(n)$	$e\left(\frac{4}{5}\right)$	$-1$

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 $

$2$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$2$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$3$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$3$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$4$	0.907634	−	2.79341i	0.907634	−	2.79341i
$5$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$5$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$6$		0			0
$7$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$7$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$8$	−1.18779	−	3.65565i	−1.18779	−	3.65565i
$9$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$10$		0			0
$11$	0.535827	−	0.844328i	0.535827	−	0.844328i
$11$	0.535827	−	0.844328i	0.535827	−	0.844328i
$12$		0			0
$13$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	−0.929776	−	0.368125i	$-0.880000\pi$
$13$	−0.866986	+	0.629902i	−0.866986	+	0.629902i	0.0627905	+	0.998027i	$0.480000\pi$
$14$		0			0
$15$		0			0
$16$	−3.79411	−	2.75658i	−3.79411	−	2.75658i
$17$	0.303189	+	0.220280i	0.303189	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$17$	0.303189	+	0.220280i	0.303189	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$18$	−0.613161	+	1.88711i	−0.613161	+	1.88711i
$19$	0.541587	+	1.66683i	0.541587	+	1.66683i	0.728969	+	0.684547i	$0.240000\pi$
$19$	0.541587	+	1.66683i	0.541587	+	1.66683i	−0.187381	+	0.982287i	$0.560000\pi$
$20$		0			0
$21$		0			0
$22$	−0.124591	−	1.98031i	−0.124591	−	1.98031i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	0.309017	+	0.951057i	0.309017	+	0.951057i
$26$	−0.657096	+	2.02233i	−0.657096	+	2.02233i
$27$		0			0
$28$		0			0
$29$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$29$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$30$		0			0
$31$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$31$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$32$	−5.46182			−5.46182
$33$		0			0
$34$	0.743615			0.743615
$35$		0			0
$36$	0.907634	+	2.79341i	0.907634	+	2.79341i
$37$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	−0.992115	−	0.125333i	$-0.960000\pi$
$37$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	0.876307	+	0.481754i	$0.160000\pi$
$38$	2.81343	+	2.04407i	2.81343	+	2.04407i
$39$		0			0
$40$		0			0
$41$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$41$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	0.535827	−	0.844328i	$-0.320000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	−1.87222	−	2.26313i	−1.87222	−	2.26313i
$45$		0			0
$46$		0			0
$47$	0.331159	+	1.01920i	0.331159	+	1.01920i	0.968583	+	0.248690i	$0.0800000\pi$
$47$	0.331159	+	1.01920i	0.331159	+	1.01920i	−0.637424	+	0.770513i	$0.720000\pi$
$48$		0			0
$49$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$50$	1.60528	+	1.16630i	1.60528	+	1.16630i
$51$		0			0
$52$	0.972670	+	2.99357i	0.972670	+	2.99357i
$53$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$53$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$59$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$60$		0			0
$61$	0.688925	+	0.500534i	0.688925	+	0.500534i	0.876307	−	0.481754i	$-0.160000\pi$
$61$	0.688925	+	0.500534i	0.688925	+	0.500534i	−0.187381	+	0.982287i	$0.560000\pi$
$62$	1.21665	−	3.74447i	1.21665	−	3.74447i
$63$		0			0
$64$	−4.97361	+	3.61354i	−4.97361	+	3.61354i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	0.890518	−	0.646999i	0.890518	−	0.646999i
$69$		0			0
$70$		0			0
$71$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$71$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$72$	3.10969	+	2.25932i	3.10969	+	2.25932i
$73$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$73$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$74$	0.229790	+	0.707220i	0.229790	+	0.707220i
$75$		0			0
$76$	5.14772			5.14772
$77$		0			0
$78$		0			0
$79$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.637424	+	0.770513i	$0.720000\pi$
$79$	−1.56720	+	1.13864i	−1.56720	+	1.13864i	−0.929776	+	0.368125i	$0.880000\pi$
$80$		0			0
$81$	0.309017	−	0.951057i	0.309017	−	0.951057i
$82$	−2.04648	−	1.48686i	−2.04648	−	1.48686i
$83$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$83$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−3.72302	−	0.955910i	−3.72302	−	0.955910i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	1.72030	+	1.24987i	1.72030	+	1.24987i
$95$		0			0
$96$		0			0
$97$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$97$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$98$	−1.98423			−1.98423
$99$	0.0627905	+	0.998027i	0.0627905	+	0.998027i
$100$	2.93717			2.93717
$101$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$101$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$102$		0			0
$103$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	0.0627905	+	0.998027i	$0.480000\pi$
$103$	−0.574633	+	1.76854i	−0.574633	+	1.76854i	−0.637424	+	0.770513i	$0.720000\pi$
$104$	3.33251	+	2.42121i	3.33251	+	2.42121i
$105$		0			0
$106$		0			0
$107$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$107$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	0.968583	−	0.248690i	$-0.0800000\pi$
$113$	0.0388067	+	0.119435i	0.0388067	+	0.119435i	−0.929776	+	0.368125i	$0.880000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.331159	−	1.01920i	0.331159	−	1.01920i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.425779	−	0.904827i	−0.425779	−	0.904827i
$122$	1.68969			1.68969
$123$		0			0
$124$	−1.80095	−	5.54277i	−1.80095	−	5.54277i
$125$		0			0
$126$		0			0
$127$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$127$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$128$	−2.08174	+	6.40695i	−2.08174	+	6.40695i
$129$		0			0
$130$		0			0
$131$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$131$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.445141	−	1.37000i	0.445141	−	1.37000i
$137$		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			0
$141$		0			0
$142$	−2.89288			−2.89288
$143$	0.0672897	+	1.06954i	0.0672897	+	1.06954i
$144$	4.68978			4.68978
$145$		0			0
$146$	−0.0770013	−	0.236986i	−0.0770013	−	0.236986i
$147$		0			0
$148$	0.890518	+	0.646999i	0.890518	+	0.646999i
$149$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$149$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$150$		0			0
$151$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$151$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$152$	5.45008	−	3.95971i	5.45008	−	3.95971i
$153$	−0.374763			−0.374763
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	−0.929776	−	0.368125i	$-0.880000\pi$
$157$	−0.393950	−	1.21245i	−0.393950	−	1.21245i	0.535827	−	0.844328i	$-0.320000\pi$
$158$	−1.18779	+	3.65565i	−1.18779	+	3.65565i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.613161	−	1.88711i	−0.613161	−	1.88711i
$163$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$163$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$164$	−3.74444			−3.74444
$165$		0			0
$166$		0			0
$167$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$167$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$168$		0			0
$169$	0.0458709	−	0.141176i	0.0458709	−	0.141176i
$170$		0			0
$171$	−1.41789	−	1.03016i	−1.41789	−	1.03016i
$172$		0			0
$173$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$173$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$174$		0			0
$175$		0			0
$176$	−4.36045	+	1.72642i	−4.36045	+	1.72642i
$177$		0			0
$178$		0			0
$179$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	−0.992115	−	0.125333i	$-0.960000\pi$
$179$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	0.728969	−	0.684547i	$-0.240000\pi$
$180$		0			0
$181$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$181$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.348445	−	0.137959i	0.348445	−	0.137959i
$188$	3.14762			3.14762
$189$		0			0
$190$		0			0
$191$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$191$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$192$		0			0
$193$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$193$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$194$		0			0
$195$		0			0
$196$	−2.37622	+	1.72642i	−2.37622	+	1.72642i
$197$	1.93717			1.93717			0.968583	−	0.248690i	$-0.0800000\pi$
$197$	1.93717			1.93717			0.968583	+	0.248690i	$0.0800000\pi$
$198$	1.26480	+	1.52888i	1.26480	+	1.52888i
$199$	−1.27485			−1.27485			−0.637424	−	0.770513i	$-0.720000\pi$
$199$	−1.27485			−1.27485			−0.637424	+	0.770513i	$0.720000\pi$
$200$	3.10969	−	2.25932i	3.10969	−	2.25932i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	1.14020	+	3.50919i	1.14020	+	3.50919i
$207$		0			0
$208$	5.02582			5.02582
$209$	1.69755	+	0.435857i	1.69755	+	0.435857i
$210$		0			0
$211$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$211$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$212$		0			0
$213$		0			0
$214$	2.34039	+	1.70039i	2.34039	+	1.70039i
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.401616			−0.401616
$222$		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
$225$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$226$	0.201592	+	0.146465i	0.201592	+	0.146465i
$227$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$227$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$228$		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			0
$232$		0			0
$233$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$233$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$234$	−0.657096	−	2.02233i	−0.657096	−	2.02233i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$239$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−1.73879	−	0.955910i	−1.73879	−	0.955910i
$243$		0			0
$244$	2.02349	−	1.47015i	2.02349	−	1.47015i
$245$		0			0
$246$		0			0
$247$	−1.51949	−	1.10397i	−1.51949	−	1.10397i
$248$	−6.17033	−	4.48301i	−6.17033	−	4.48301i
$249$		0			0
$250$		0			0
$251$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	−0.637424	−	0.770513i	$-0.720000\pi$
$251$	−0.101597	+	0.0738147i	−0.101597	+	0.0738147i	0.535827	+	0.844328i	$0.320000\pi$
$252$		0			0
$253$		0			0
$254$	−1.98423			−1.98423
$255$		0			0
$256$	2.23091	+	6.86603i	2.23091	+	6.86603i
$257$		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			0
$261$		0			0
$262$	0.992115	−	0.720814i	0.992115	−	0.720814i
$263$	−0.851559			−0.851559			−0.425779	−	0.904827i	$-0.640000\pi$
$263$	−0.851559			−0.851559			−0.425779	+	0.904827i	$0.640000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$269$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.425779	−	0.904827i	$-0.640000\pi$
$270$		0			0
$271$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$271$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$272$	−0.543114	−	1.67153i	−0.543114	−	1.67153i
$273$		0			0
$274$		0			0
$275$	0.968583	+	0.248690i	0.968583	+	0.248690i
$276$		0			0
$277$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$277$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$278$		0			0
$279$	−0.613161	+	1.88711i	−0.613161	+	1.88711i
$280$		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			0		−0.309017	−	0.951057i	$-0.600000\pi$
$283$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$284$	−3.46438	+	2.51702i	−3.46438	+	2.51702i
$285$		0			0
$286$	1.35542	+	1.63842i	1.35542	+	1.63842i
$287$		0			0
$288$	4.41870	−	3.21038i	4.41870	−	3.21038i
$289$	−0.265616	−	0.817483i	−0.265616	−	0.817483i
$290$		0			0
$291$		0			0
$292$	−0.298408	−	0.216806i	−0.298408	−	0.216806i
$293$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$293$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$294$		0			0
$295$		0			0
$296$	1.44051			1.44051
$297$		0			0
$298$	−2.89288			−2.89288
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	2.53993	−	7.81709i	2.53993	−	7.81709i
$305$		0			0
$306$	−0.601597	+	0.437086i	−0.601597	+	0.437086i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$311$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$312$		0			0
$313$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$313$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$314$	−2.04648	−	1.48686i	−2.04648	−	1.48686i
$315$		0			0
$316$	1.75824	+	5.41130i	1.75824	+	5.41130i
$317$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$317$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−0.202967	+	0.624667i	−0.202967	+	0.624667i
$324$	−2.37622	−	1.72642i	−2.37622	−	1.72642i
$325$	−0.866986	−	0.629902i	−0.866986	−	0.629902i
$326$	0.992115	−	3.05342i	0.992115	−	3.05342i
$327$		0			0
$328$	−3.96438	+	2.88029i	−3.96438	+	2.88029i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−0.115808	−	0.356420i	−0.115808	−	0.356420i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$337$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$338$	−0.0910184	−	0.280126i	−0.0910184	−	0.280126i
$339$		0			0
$340$		0			0
$341$	−0.124591	−	1.98031i	−0.124591	−	1.98031i
$342$	−3.47759			−3.47759
$343$		0			0
$344$		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			0
$352$	−2.92659	+	4.61156i	−2.92659	+	4.61156i
$353$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$353$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−1.36699	−	0.993173i	−1.36699	−	0.993173i
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$		0			0
$361$	−1.67600	+	1.21769i	−1.67600	+	1.21769i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	0.728969	+	0.684547i	$0.240000\pi$
$367$	−0.263146	+	0.809880i	−0.263146	+	0.809880i	−0.992115	+	0.125333i	$0.960000\pi$
$368$		0			0
$369$	1.03137	+	0.749337i	1.03137	+	0.749337i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$	0.398449	−	0.627855i	0.398449	−	0.627855i
$375$		0			0
$376$	3.33251	−	2.42121i	3.33251	−	2.42121i
$377$		0			0
$378$		0			0
$379$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$379$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$380$		0			0
$381$		0			0
$382$	−0.378954	−	1.16630i	−0.378954	−	1.16630i
$383$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$383$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.331159	−	1.01920i	0.331159	−	1.01920i	−0.637424	−	0.770513i	$-0.720000\pi$
$389$	0.331159	−	1.01920i	0.331159	−	1.01920i	0.968583	−	0.248690i	$-0.0800000\pi$
$390$		0			0
$391$		0			0
$392$	−1.18779	+	3.65565i	−1.18779	+	3.65565i
$393$		0			0
$394$	3.10969	−	2.25932i	3.10969	−	2.25932i
$395$		0			0
$396$	2.84489	+	0.730444i	2.84489	+	0.730444i
$397$	−0.851559			−0.851559			−0.425779	−	0.904827i	$-0.640000\pi$
$397$	−0.851559			−0.851559			−0.425779	+	0.904827i	$0.640000\pi$
$398$	−2.04648	+	1.48686i	−2.04648	+	1.48686i
$399$		0			0
$400$	1.44922	−	4.46025i	1.44922	−	4.46025i
$401$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$401$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$402$		0			0
$403$	−0.657096	+	2.02233i	−0.657096	+	2.02233i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.238883	+	0.288760i	0.238883	+	0.288760i
$408$		0			0
$409$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$409$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$410$		0			0
$411$		0			0
$412$	4.41870	+	3.21038i	4.41870	+	3.21038i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	4.73532	−	3.44041i	4.73532	−	3.44041i
$417$		0			0
$418$	3.23338	−	1.28019i	3.23338	−	1.28019i
$419$	1.75261			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$419$	1.75261			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$420$		0			0
$421$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$421$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$422$	−0.0770013	+	0.236986i	−0.0770013	+	0.236986i
$423$	−0.866986	−	0.629902i	−0.866986	−	0.629902i
$424$		0			0
$425$	−0.115808	+	0.356420i	−0.115808	+	0.356420i
$426$		0			0
$427$		0			0
$428$	4.28220			4.28220
$429$		0			0
$430$		0			0
$431$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$431$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$432$		0			0
$433$	0.450527	−	1.38658i	0.450527	−	1.38658i	−0.425779	−	0.904827i	$-0.640000\pi$
$433$	0.450527	−	1.38658i	0.450527	−	1.38658i	0.876307	−	0.481754i	$-0.160000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	1.00000			1.00000
$442$	−0.644704	+	0.468405i	−0.644704	+	0.468405i
$443$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	0.876307	−	0.481754i	$-0.160000\pi$
$443$	−0.115808	−	0.356420i	−0.115808	−	0.356420i	−0.992115	+	0.125333i	$0.960000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.0627905	−	0.998027i	$-0.480000\pi$
$449$	1.03137	−	0.749337i	1.03137	−	0.749337i	0.968583	+	0.248690i	$0.0800000\pi$
$450$	−1.98423			−1.98423
$451$	−1.23480	−	0.317042i	−1.23480	−	0.317042i
$452$	0.368852			0.368852
$453$		0			0
$454$	0.992115	+	3.05342i	0.992115	+	3.05342i
$455$		0			0
$456$		0			0
$457$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.929776	−	0.368125i	$-0.880000\pi$
$457$	−1.56720	−	1.13864i	−1.56720	−	1.13864i	−0.637424	−	0.770513i	$-0.720000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−0.374763			−0.374763			−0.187381	−	0.982287i	$-0.560000\pi$
$463$	−0.374763			−0.374763			−0.187381	+	0.982287i	$0.560000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$467$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$468$	−2.54648	−	1.85013i	−2.54648	−	1.85013i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.41789	+	1.03016i	−1.41789	+	1.03016i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$479$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$480$		0			0
$481$	−0.124106	−	0.381959i	−0.124106	−	0.381959i
$482$		0			0
$483$		0			0
$484$	−2.91401	+	0.368125i	−2.91401	+	0.368125i
$485$		0			0
$486$		0			0
$487$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$487$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$488$	1.01148	−	3.11300i	1.01148	−	3.11300i
$489$		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			0
$493$		0			0
$494$	−3.72677			−3.72677
$495$		0			0
$496$	−9.30560			−9.30560
$497$		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			0
$501$		0			0
$502$	−0.0770013	+	0.236986i	−0.0770013	+	0.236986i
$503$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	−0.637424	−	0.770513i	$-0.720000\pi$
$503$	−0.574633	−	1.76854i	−0.574633	−	1.76854i	0.0627905	−	0.998027i	$-0.480000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−2.37622	+	1.72642i	−2.37622	+	1.72642i
$509$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$509$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$510$		0			0
$511$		0			0
$512$	6.13900	+	4.46025i	6.13900	+	4.46025i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	1.03799	+	0.266509i	1.03799	+	0.266509i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.187381	+	0.982287i	$0.560000\pi$
$521$	−0.613161	+	1.88711i	−0.613161	+	1.88711i	−0.425779	+	0.904827i	$0.640000\pi$
$522$		0			0
$523$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$523$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$524$	0.560949	−	1.72642i	0.560949	−	1.72642i
$525$		0			0
$526$	−1.36699	+	0.993173i	−1.36699	+	0.993173i
$527$	0.743615			0.743615
$528$		0			0
$529$	1.00000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.10528	+	0.803030i	1.10528	+	0.803030i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−3.47759			−3.47759
$539$	−0.929776	+	0.368125i	−0.929776	+	0.368125i
$540$		0			0
$541$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$541$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$542$	0.992115	+	3.05342i	0.992115	+	3.05342i
$543$		0			0
$544$	−1.65596	−	1.20313i	−1.65596	−	1.20313i
$545$		0			0
$546$		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
$549$	−0.851559			−0.851559
$550$	1.84489	−	0.730444i	1.84489	−	0.730444i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$557$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$558$	1.21665	+	3.74447i	1.21665	+	3.74447i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$563$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−1.73173	+	5.32971i	−1.73173	+	5.32971i
$569$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$569$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$	3.04874	+	0.782782i	3.04874	+	0.782782i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	1.89975	−	5.84683i	1.89975	−	5.84683i
$577$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.187381	−	0.982287i	$-0.560000\pi$
$577$	−1.17950	−	0.856954i	−1.17950	−	0.856954i	−0.992115	+	0.125333i	$0.960000\pi$
$578$	−1.37982	−	1.00250i	−1.37982	−	1.00250i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.482706			−0.482706
$585$		0			0
$586$		0			0
$587$	0.541587	−	1.66683i	0.541587	−	1.66683i	−0.187381	−	0.982287i	$-0.560000\pi$
$587$	0.541587	−	1.66683i	0.541587	−	1.66683i	0.728969	−	0.684547i	$-0.240000\pi$
$588$		0			0
$589$	2.81343	+	2.04407i	2.81343	+	2.04407i
$590$		0			0
$591$		0			0
$592$	1.42189	−	1.03306i	1.42189	−	1.03306i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$	−3.46438	+	2.51702i	−3.46438	+	2.51702i
$597$		0			0
$598$		0			0
$599$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$599$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$600$		0			0
$601$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$601$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.992115	−	0.125333i	$-0.960000\pi$
$607$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.187381	+	0.982287i	$0.560000\pi$
$608$	−2.95805	−	9.10394i	−2.95805	−	9.10394i
$609$		0			0
$610$		0			0
$611$	−0.929109	−	0.675037i	−0.929109	−	0.675037i
$612$	−0.340147	+	1.04687i	−0.340147	+	1.04687i
$613$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$613$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$614$		0			0
$615$		0			0
$616$		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			0
$624$		0			0
$625$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$626$		0			0
$627$		0			0
$628$	−3.74444			−3.74444
$629$	−0.113624	+	0.0825527i	−0.113624	+	0.0825527i
$630$		0			0
$631$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$631$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$632$	6.02398	+	4.37668i	6.02398	+	4.37668i
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.07165			1.07165
$638$		0			0
$639$	1.45794			1.45794
$640$		0			0
$641$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$641$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$642$		0			0
$643$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.968583	+	0.248690i	$0.0800000\pi$
$643$	1.50441	+	1.09302i	1.50441	+	1.09302i	0.535827	+	0.844328i	$0.320000\pi$
$644$		0			0
$645$		0			0
$646$	0.402732	+	1.23948i	0.402732	+	1.23948i
$647$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$647$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$648$	−3.84378			−3.84378
$649$		0			0
$650$	−2.12641			−2.12641
$651$		0			0
$652$	−1.46858	−	4.51983i	−1.46858	−	4.51983i
$653$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	−0.929776	−	0.368125i	$-0.880000\pi$
$653$	0.0388067	−	0.119435i	0.0388067	−	0.119435i	0.968583	+	0.248690i	$0.0800000\pi$
$654$		0			0
$655$		0			0
$656$	−1.84754	+	5.68613i	−1.84754	+	5.68613i
$657$	0.0388067	+	0.119435i	0.0388067	+	0.119435i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	0.125581			0.125581			0.0627905	−	0.998027i	$-0.480000\pi$
$661$	0.125581			0.125581			0.0627905	+	0.998027i	$0.480000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.601597	−	0.437086i	−0.601597	−	0.437086i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	0.791759	−	0.313480i	0.791759	−	0.313480i
$672$		0			0
$673$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.992115	−	0.125333i	$-0.960000\pi$
$673$	−1.17950	+	0.856954i	−1.17950	+	0.856954i	−0.187381	+	0.982287i	$0.560000\pi$
$674$		0			0
$675$		0			0
$676$	−0.352729	−	0.256273i	−0.352729	−	0.256273i
$677$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$677$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	−2.50964	−	3.03364i	−2.50964	−	3.03364i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	−4.16459	+	3.02575i	−4.16459	+	3.02575i
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$691$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.147638	−	0.454382i	0.147638	−	0.454382i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$701$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$702$		0			0
$703$	−0.656814			−0.656814
$704$	0.386018	+	6.13559i	0.386018	+	6.13559i
$705$		0			0
$706$	2.81343	−	2.04407i	2.81343	−	2.04407i
$707$		0			0
$708$		0			0
$709$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	0.0627905	−	0.998027i	$-0.480000\pi$
$709$	−0.866986	−	0.629902i	−0.866986	−	0.629902i	−0.929776	+	0.368125i	$0.880000\pi$
$710$		0			0
$711$	0.598617	−	1.84235i	0.598617	−	1.84235i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−2.50117			−2.50117
$717$		0			0
$718$		0			0
$719$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	−0.992115	−	0.125333i	$-0.960000\pi$
$719$	−0.115808	+	0.356420i	−0.115808	+	0.356420i	0.876307	+	0.481754i	$0.160000\pi$
$720$		0			0
$721$		0			0
$722$	−1.27026	+	3.90945i	−1.27026	+	3.90945i
$723$		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
$729$	0.309017	+	0.951057i	0.309017	+	0.951057i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$733$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$734$	0.522142	+	1.60699i	0.522142	+	1.60699i
$735$		0			0
$736$		0			0
$737$		0			0
$738$	2.52959			2.52959
$739$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$739$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$740$		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
$748$	−0.0691160	−	1.09857i	−0.0691160	−	1.09857i
$749$		0			0
$750$		0			0
$751$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$751$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$752$	1.55306	−	4.77984i	1.55306	−	4.77984i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.728969	−	0.684547i	$-0.240000\pi$
$757$	1.60528	−	1.16630i	1.60528	−	1.16630i	0.876307	−	0.481754i	$-0.160000\pi$
$758$		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			0
$764$	−1.46858	−	1.06699i	−1.46858	−	1.06699i
$765$		0			0
$766$	0.992115	−	3.05342i	0.992115	−	3.05342i
$767$		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
$773$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	−0.992115	−	0.125333i	$-0.960000\pi$
$773$	−0.263146	−	0.809880i	−0.263146	−	0.809880i	0.728969	−	0.684547i	$-0.240000\pi$
$774$		0			0
$775$	1.60528	+	1.16630i	1.60528	+	1.16630i
$776$		0			0
$777$		0			0
$778$	−0.657096	−	2.02233i	−0.657096	−	2.02233i
$779$	1.80760	−	1.31330i	1.80760	−	1.31330i
$780$		0			0
$781$	−1.35556	+	0.536702i	−1.35556	+	0.536702i
$782$		0			0
$783$		0			0
$784$	1.44922	+	4.46025i	1.44922	+	4.46025i
$785$		0			0
$786$		0			0
$787$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$787$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$788$	1.75824	−	5.41130i	1.75824	−	5.41130i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	3.57386	−	1.41499i	3.57386	−	1.41499i
$793$	−0.912576			−0.912576
$794$	−1.36699	+	0.993173i	−1.36699	+	0.993173i
$795$		0			0
$796$	−1.15710	+	3.56117i	−1.15710	+	3.56117i
$797$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$797$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$798$		0			0
$799$	−0.124106	+	0.381959i	−0.124106	+	0.381959i
$800$	−1.68779	−	5.19450i	−1.68779	−	5.19450i
$801$		0			0
$802$		0			0
$803$	−0.0800484	−	0.0967619i	−0.0800484	−	0.0967619i
$804$		0			0
$805$		0			0
$806$	1.30383	+	4.01277i	1.30383	+	4.01277i
$807$		0			0
$808$		0			0
$809$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.992115	−	0.125333i	$-0.960000\pi$
$809$	−1.41789	−	1.03016i	−1.41789	−	1.03016i	−0.425779	−	0.904827i	$-0.640000\pi$
$810$		0			0
$811$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$811$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$812$		0			0
$813$		0			0
$814$	0.720253	+	0.184930i	0.720253	+	0.184930i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$821$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$822$		0			0
$823$	0.688925	−	0.500534i	0.688925	−	0.500534i	−0.187381	−	0.982287i	$-0.560000\pi$
$823$	0.688925	−	0.500534i	0.688925	−	0.500534i	0.876307	+	0.481754i	$0.160000\pi$
$824$	7.14772			7.14772
$825$		0			0
$826$		0			0
$827$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$827$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$828$		0			0
$829$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$829$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$830$		0			0
$831$		0			0
$832$	2.03587	−	6.26577i	2.03587	−	6.26577i
$833$	−0.115808	−	0.356420i	−0.115808	−	0.356420i
$834$		0			0
$835$		0			0
$836$	2.75828	−	4.34636i	2.75828	−	4.34636i
$837$		0			0
$838$	2.81343	−	2.04407i	2.81343	−	2.04407i
$839$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$839$		0			0		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
$844$	0.113982	+	0.350799i	0.113982	+	0.350799i
$845$		0			0
$846$	−2.12641			−2.12641
$847$		0			0
$848$		0			0
$849$		0			0
$850$	0.229790	+	0.707220i	0.229790	+	0.707220i
$851$		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
$856$	4.53373	−	3.29394i	4.53373	−	3.29394i
$857$		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			$\pi$
$860$		0			0
$861$		0			0
$862$	0.781687	−	2.40578i	0.781687	−	2.40578i
$863$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$863$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$864$		0			0
$865$		0			0
$866$	−0.893950	−	2.75129i	−0.893950	−	2.75129i
$867$		0			0
$868$		0			0
$869$	0.121636	+	1.93334i	0.121636	+	1.93334i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		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			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	1.60528	−	1.16630i	1.60528	−	1.16630i
$883$	0.450527	+	1.38658i	0.450527	+	1.38658i	0.876307	+	0.481754i	$0.160000\pi$
$883$	0.450527	+	1.38658i	0.450527	+	1.38658i	−0.425779	+	0.904827i	$0.640000\pi$
$884$	−0.364520	+	1.12188i	−0.364520	+	1.12188i
$885$		0			0
$886$	−0.601597	−	0.437086i	−0.601597	−	0.437086i
$887$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$887$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.637424	−	0.770513i	−0.637424	−	0.770513i
$892$		0			0
$893$	−1.51949	+	1.10397i	−1.51949	+	1.10397i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.781687	−	2.40578i	0.781687	−	2.40578i
$899$		0			0
$900$	−2.37622	+	1.72642i	−2.37622	+	1.72642i
$901$		0			0
$902$	−2.35195	+	0.931204i	−2.35195	+	0.931204i
$903$		0			0
$904$	0.390518	−	0.283728i	0.390518	−	0.283728i
$905$		0			0
$906$		0			0
$907$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.968583	−	0.248690i	$-0.0800000\pi$
$907$	1.03137	+	0.749337i	1.03137	+	0.749337i	0.0627905	+	0.998027i	$0.480000\pi$
$908$	3.84480	+	2.79341i	3.84480	+	2.79341i
$909$		0			0
$910$		0			0
$911$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$911$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$912$		0			0
$913$		0			0
$914$	−3.84378			−3.84378
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.303189	+	0.220280i	0.303189	+	0.220280i	0.728969	−	0.684547i	$-0.240000\pi$
$919$	0.303189	+	0.220280i	0.303189	+	0.220280i	−0.425779	+	0.904827i	$0.640000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.56240			1.56240
$924$		0			0
$925$	−0.374763			−0.374763
$926$	−0.601597	+	0.437086i	−0.601597	+	0.437086i
$927$	−0.574633	−	1.76854i	−0.574633	−	1.76854i
$928$		0			0
$929$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$929$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$930$		0			0
$931$	0.541587	−	1.66683i	0.541587	−	1.66683i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−4.11920			−4.11920
$937$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$937$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	0.535827	−	0.844328i	$-0.320000\pi$
$941$	−0.101597	−	0.0738147i	−0.101597	−	0.0738147i	−0.637424	+	0.770513i	$0.720000\pi$
$942$		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
$949$	0.0415873	+	0.127993i	0.0415873	+	0.127993i
$950$	−1.07463	+	3.30738i	−1.07463	+	3.30738i
$951$		0			0
$952$		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			0
$956$		0			0
$957$		0			0
$958$	−0.249182			−0.249182
$959$		0			0
$960$		0			0
$961$	0.907634	−	2.79341i	0.907634	−	2.79341i
$962$	−0.644704	−	0.468405i	−0.644704	−	0.468405i
$963$	−1.17950	−	0.856954i	−1.17950	−	0.856954i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−2.80200	+	2.63125i	−2.80200	+	2.63125i
$969$		0			0
$970$		0			0
$971$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.535827	+	0.844328i	$0.320000\pi$
$971$	0.598617	+	1.84235i	0.598617	+	1.84235i	0.0627905	+	0.998027i	$0.480000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.23410	−	3.79816i	−1.23410	−	3.79816i
$977$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.535827	−	0.844328i	$-0.320000\pi$
$977$	1.50441	−	1.09302i	1.50441	−	1.09302i	0.968583	−	0.248690i	$-0.0800000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.0627905	−	0.998027i	$-0.480000\pi$
$983$	0.598617	−	1.84235i	0.598617	−	1.84235i	0.535827	−	0.844328i	$-0.320000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−4.46300	+	3.24256i	−4.46300	+	3.24256i
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−8.76772	+	6.37012i	−8.76772	+	6.37012i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$997$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1397.1.l.a.1015.5	✓	20
11.4	even	5	inner	1397.1.l.a.1269.5	yes	20
127.126	odd	2	CM	1397.1.l.a.1015.5	✓	20
1397.1269	odd	10	inner	1397.1.l.a.1269.5	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1397.1.l.a.1015.5	✓	20	1.1	even	1	trivial
1397.1.l.a.1015.5	✓	20	127.126	odd	2	CM
1397.1.l.a.1269.5	yes	20	11.4	even	5	inner
1397.1.l.a.1269.5	yes	20	1397.1269	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:	\( N \)	\(=\)	\( 1397 = 11 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1397.l (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(1.60528 - 1.16630i) q^{2} +(0.907634 - 2.79341i) q^{4} +(-1.18779 - 3.65565i) q^{8} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})\)	\(q+(1.60528 - 1.16630i) q^{2} +(0.907634 - 2.79341i) q^{4} +(-1.18779 - 3.65565i) q^{8} +(-0.809017 + 0.587785i) q^{9} +(0.535827 - 0.844328i) q^{11} +(-0.866986 + 0.629902i) q^{13} +(-3.79411 - 2.75658i) q^{16} +(0.303189 + 0.220280i) q^{17} +(-0.613161 + 1.88711i) q^{18} +(0.541587 + 1.66683i) q^{19} +(-0.124591 - 1.98031i) q^{22} +(0.309017 + 0.951057i) q^{25} +(-0.657096 + 2.02233i) q^{26} +(1.60528 - 1.16630i) q^{31} -5.46182 q^{32} +0.743615 q^{34} +(0.907634 + 2.79341i) q^{36} +(-0.115808 + 0.356420i) q^{37} +(2.81343 + 2.04407i) q^{38} +(-0.393950 - 1.21245i) q^{41} +(-1.87222 - 2.26313i) q^{44} +(0.331159 + 1.01920i) q^{47} +(-0.809017 - 0.587785i) q^{49} +(1.60528 + 1.16630i) q^{50} +(0.972670 + 2.99357i) q^{52} +(0.688925 + 0.500534i) q^{61} +(1.21665 - 3.74447i) q^{62} +(-4.97361 + 3.61354i) q^{64} +(0.890518 - 0.646999i) q^{68} +(-1.17950 - 0.856954i) q^{71} +(3.10969 + 2.25932i) q^{72} +(0.0388067 - 0.119435i) q^{73} +(0.229790 + 0.707220i) q^{74} +5.14772 q^{76} +(-1.56720 + 1.13864i) q^{79} +(0.309017 - 0.951057i) q^{81} +(-2.04648 - 1.48686i) q^{82} +(-3.72302 - 0.955910i) q^{88} +(1.72030 + 1.24987i) q^{94} -1.98423 q^{98} +(0.0627905 + 0.998027i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{4} - 5 q^{9}+O(q^{10}) \) 20 * q - 5 * q^4 - 5 * q^9	\( 20 q - 5 q^{4} - 5 q^{9} - 5 q^{16} - 5 q^{25} - 10 q^{32} - 10 q^{34} - 5 q^{36} + 15 q^{38} - 5 q^{44} - 5 q^{49} + 15 q^{52} - 10 q^{62} - 5 q^{64} + 15 q^{74} - 5 q^{81} - 5 q^{88}+O(q^{100}) \) 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 - 5 * q^25 - 10 * q^32 - 10 * q^34 - 5 * q^36 + 15 * q^38 - 5 * q^44 - 5 * q^49 + 15 * q^52 - 10 * q^62 - 5 * q^64 + 15 * q^74 - 5 * q^81 - 5 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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