LMFDB - Embedded newform 583.1.n.a.428.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [583,1,Mod(43,583)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(583, base_ring=CyclotomicField(26))

chi = DirichletCharacter(H, H._module([13, 11]))

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

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

chi := DirichletCharacter("583.43");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 583 = 11 \cdot 53 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	583.n (of order $26$, degree $12$, 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:	$0.290954902365$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{26})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{26}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{26} - \cdots)$

Embedding invariants

Embedding label			428.1
Root			$0.970942 + 0.239316i$ of defining polynomial
Character	$\chi$	$=$	583.428
Dual form			583.1.n.a.252.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.764919 - 1.45743i) q^{3} +(0.970942 + 0.239316i) q^{4} +(-0.764919 + 0.527986i) q^{5} +(-0.970942 - 1.40665i) q^{9} +O(q^{10})$	\(q+(0.764919 - 1.45743i) q^{3} +(0.970942 + 0.239316i) q^{4} +(-0.764919 + 0.527986i) q^{5} +(-0.970942 - 1.40665i) q^{9} +(-0.748511 - 0.663123i) q^{11} +(1.09148 - 1.23202i) q^{12} +(0.184402 + 1.51868i) q^{15} +(0.885456 + 0.464723i) q^{16} +(-0.869047 + 0.329586i) q^{20} +1.32625i q^{23} +(-0.0482722 + 0.127283i) q^{25} +(-1.15883 + 0.140707i) q^{27} +(0.317391 + 0.358261i) q^{31} +(-1.53901 + 0.583668i) q^{33} +(-0.606094 - 1.59814i) q^{36} +(-1.00599 - 0.527986i) q^{37} +(-0.568065 - 0.822984i) q^{44} +(1.48538 + 0.563332i) q^{45} +(-1.00599 + 1.45743i) q^{47} +(1.35460 - 0.935016i) q^{48} +(-0.970942 - 0.239316i) q^{49} +(-0.354605 - 0.935016i) q^{53} +(0.922670 + 0.112032i) q^{55} +(1.10312 - 1.59814i) q^{59} +(-0.184402 + 1.51868i) q^{60} +(0.748511 + 0.663123i) q^{64} +(-0.475142 + 1.92773i) q^{67} +(1.93291 + 1.01447i) q^{69} +(0.869047 + 1.65583i) q^{71} +(0.148582 + 0.167715i) q^{75} +(-0.922670 + 0.112032i) q^{80} +(-0.0752430 + 0.198399i) q^{81} +(-0.530851 - 1.39974i) q^{89} +(-0.317391 + 1.28771i) q^{92} +(0.764919 - 0.188536i) q^{93} +(-0.850405 - 1.23202i) q^{97} +(-0.206022 + 1.69675i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{4} - q^{9}+O(q^{10}) $ 12 * q + q^4 - q^9	$ 12 q + q^{4} - q^{9} - q^{11} - 13 q^{15} - q^{16} - q^{25} - 12 q^{36} + 2 q^{37} + q^{44} + 2 q^{47} + 13 q^{48} - q^{49} - q^{53} + 2 q^{59} + 13 q^{60} + q^{64} - 14 q^{81} + 2 q^{89} - 2 q^{97} - q^{99}+O(q^{100}) $ 12 * q + q^4 - q^9 - q^11 - 13 * q^15 - q^16 - q^25 - 12 * q^36 + 2 * q^37 + q^44 + 2 * q^47 + 13 * q^48 - q^49 - q^53 + 2 * q^59 + 13 * q^60 + q^64 - 14 * q^81 + 2 * q^89 - 2 * q^97 - q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/583\mathbb{Z}\right)^\times$.

$n$	$266$	$320$
$\chi(n)$	$-1$	$e\left(\frac{1}{26}\right)$

Coefficient data

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

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

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

$2$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$2$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$3$	0.764919	−	1.45743i	0.764919	−	1.45743i	−0.120537	−	0.992709i	$-0.538462\pi$
$3$	0.764919	−	1.45743i	0.764919	−	1.45743i	0.885456	−	0.464723i	$-0.153846\pi$
$4$	0.970942	+	0.239316i	0.970942	+	0.239316i
$5$	−0.764919	+	0.527986i	−0.764919	+	0.527986i	−0.885456	−	0.464723i	$-0.846154\pi$
$5$	−0.764919	+	0.527986i	−0.764919	+	0.527986i	0.120537	+	0.992709i	$0.461538\pi$
$6$		0			0
$7$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$7$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$8$		0			0
$9$	−0.970942	−	1.40665i	−0.970942	−	1.40665i
$10$		0			0
$11$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$11$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$12$	1.09148	−	1.23202i	1.09148	−	1.23202i
$13$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$13$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$14$		0			0
$15$	0.184402	+	1.51868i	0.184402	+	1.51868i
$16$	0.885456	+	0.464723i	0.885456	+	0.464723i
$17$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$17$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$18$		0			0
$19$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$19$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$20$	−0.869047	+	0.329586i	−0.869047	+	0.329586i
$21$		0			0
$22$		0			0
$23$			1.32625i			1.32625i	0.748511	+	0.663123i	$0.230769\pi$
$23$			1.32625i			1.32625i	−0.748511	+	0.663123i	$0.769231\pi$
$24$		0			0
$25$	−0.0482722	+	0.127283i	−0.0482722	+	0.127283i
$26$		0			0
$27$	−1.15883	+	0.140707i	−1.15883	+	0.140707i
$28$		0			0
$29$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$29$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$30$		0			0
$31$	0.317391	+	0.358261i	0.317391	+	0.358261i	0.885456	−	0.464723i	$-0.153846\pi$
$31$	0.317391	+	0.358261i	0.317391	+	0.358261i	−0.568065	+	0.822984i	$0.692308\pi$
$32$		0			0
$33$	−1.53901	+	0.583668i	−1.53901	+	0.583668i
$34$		0			0
$35$		0			0
$36$	−0.606094	−	1.59814i	−0.606094	−	1.59814i
$37$	−1.00599	−	0.527986i	−1.00599	−	0.527986i	−0.120537	−	0.992709i	$-0.538462\pi$
$37$	−1.00599	−	0.527986i	−1.00599	−	0.527986i	−0.885456	+	0.464723i	$0.846154\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$41$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$42$		0			0
$43$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$43$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$44$	−0.568065	−	0.822984i	−0.568065	−	0.822984i
$45$	1.48538	+	0.563332i	1.48538	+	0.563332i
$46$		0			0
$47$	−1.00599	+	1.45743i	−1.00599	+	1.45743i	−0.120537	+	0.992709i	$0.538462\pi$
$47$	−1.00599	+	1.45743i	−1.00599	+	1.45743i	−0.885456	+	0.464723i	$0.846154\pi$
$48$	1.35460	−	0.935016i	1.35460	−	0.935016i
$49$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$53$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$54$		0			0
$55$	0.922670	+	0.112032i	0.922670	+	0.112032i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.10312	−	1.59814i	1.10312	−	1.59814i	0.354605	−	0.935016i	$-0.384615\pi$
$59$	1.10312	−	1.59814i	1.10312	−	1.59814i	0.748511	−	0.663123i	$-0.230769\pi$
$60$	−0.184402	+	1.51868i	−0.184402	+	1.51868i
$61$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$61$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$62$		0			0
$63$		0			0
$64$	0.748511	+	0.663123i	0.748511	+	0.663123i
$65$		0			0
$66$		0			0
$67$	−0.475142	+	1.92773i	−0.475142	+	1.92773i	−0.120537	+	0.992709i	$0.538462\pi$
$67$	−0.475142	+	1.92773i	−0.475142	+	1.92773i	−0.354605	+	0.935016i	$0.615385\pi$
$68$		0			0
$69$	1.93291	+	1.01447i	1.93291	+	1.01447i
$70$		0			0
$71$	0.869047	+	1.65583i	0.869047	+	1.65583i	0.748511	+	0.663123i	$0.230769\pi$
$71$	0.869047	+	1.65583i	0.869047	+	1.65583i	0.120537	+	0.992709i	$0.461538\pi$
$72$		0			0
$73$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$73$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$74$		0			0
$75$	0.148582	+	0.167715i	0.148582	+	0.167715i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$79$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$80$	−0.922670	+	0.112032i	−0.922670	+	0.112032i
$81$	−0.0752430	+	0.198399i	−0.0752430	+	0.198399i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.530851	−	1.39974i	−0.530851	−	1.39974i	−0.885456	−	0.464723i	$-0.846154\pi$
$89$	−0.530851	−	1.39974i	−0.530851	−	1.39974i	0.354605	−	0.935016i	$-0.384615\pi$
$90$		0			0
$91$		0			0
$92$	−0.317391	+	1.28771i	−0.317391	+	1.28771i
$93$	0.764919	−	0.188536i	0.764919	−	0.188536i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.850405	−	1.23202i	−0.850405	−	1.23202i	−0.970942	−	0.239316i	$-0.923077\pi$
$97$	−0.850405	−	1.23202i	−0.850405	−	1.23202i	0.120537	−	0.992709i	$-0.461538\pi$
$98$		0			0
$99$	−0.206022	+	1.69675i	−0.206022	+	1.69675i
$100$	−0.0773304	+	0.112032i	−0.0773304	+	0.112032i
$101$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$101$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$102$		0			0
$103$	0.869047	−	1.65583i	0.869047	−	1.65583i	0.120537	−	0.992709i	$-0.461538\pi$
$103$	0.869047	−	1.65583i	0.869047	−	1.65583i	0.748511	−	0.663123i	$-0.230769\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	−1.15883	−	0.140707i	−1.15883	−	0.140707i
$109$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$109$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$110$		0			0
$111$	−1.53901	+	1.06230i	−1.53901	+	1.06230i
$112$		0			0
$113$	−0.0290582	+	0.239316i	−0.0290582	+	0.239316i	0.970942	+	0.239316i	$0.0769231\pi$
$113$	−0.0290582	+	0.239316i	−0.0290582	+	0.239316i	−1.00000			$\pi$
$114$		0			0
$115$	−0.700239	−	1.01447i	−0.700239	−	1.01447i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.120537	+	0.992709i	0.120537	+	0.992709i
$122$		0			0
$123$		0			0
$124$	0.222431	+	0.423807i	0.222431	+	0.423807i
$125$	−0.252710	−	1.02529i	−0.252710	−	1.02529i
$126$		0			0
$127$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$127$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$131$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$132$	−1.63397	+	0.198399i	−1.63397	+	0.198399i
$133$		0			0
$134$		0			0
$135$	0.812117	−	0.719473i	0.812117	−	0.719473i
$136$		0			0
$137$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$137$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$138$		0			0
$139$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$139$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$140$		0			0
$141$	1.35460	+	2.58098i	1.35460	+	2.58098i
$142$		0			0
$143$		0			0
$144$	−0.206022	−	1.69675i	−0.206022	−	1.69675i
$145$		0			0
$146$		0			0
$147$	−1.09148	+	1.23202i	−1.09148	+	1.23202i
$148$	−0.850405	−	0.753393i	−0.850405	−	0.753393i
$149$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$149$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$150$		0			0
$151$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$151$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.431935	−	0.106462i	−0.431935	−	0.106462i
$156$		0			0
$157$	1.85640	+	0.225408i	1.85640	+	0.225408i	0.970942	−	0.239316i	$-0.0769231\pi$
$157$	1.85640	+	0.225408i	1.85640	+	0.225408i	0.885456	+	0.464723i	$0.153846\pi$
$158$		0			0
$159$	−1.63397	−	0.198399i	−1.63397	−	0.198399i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.88546	−	0.464723i	−1.88546	−	0.464723i	−0.885456	−	0.464723i	$-0.846154\pi$
$163$	−1.88546	−	0.464723i	−1.88546	−	0.464723i	−1.00000			$\pi$
$164$		0			0
$165$	0.869047	−	1.25903i	0.869047	−	1.25903i
$166$		0			0
$167$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$167$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$168$		0			0
$169$	0.885456	−	0.464723i	0.885456	−	0.464723i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$173$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$174$		0			0
$175$		0			0
$176$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$177$	−1.48538	−	2.83016i	−1.48538	−	2.83016i
$178$		0			0
$179$	1.74851	−	0.663123i	1.74851	−	0.663123i	0.748511	−	0.663123i	$-0.230769\pi$
$179$	1.74851	−	0.663123i	1.74851	−	0.663123i	1.00000			$0$
$180$	1.30741	+	0.902438i	1.30741	+	0.902438i
$181$	−1.31658	−	1.48611i	−1.31658	−	1.48611i	−0.748511	−	0.663123i	$-0.769231\pi$
$181$	−1.31658	−	1.48611i	−1.31658	−	1.48611i	−0.568065	−	0.822984i	$-0.692308\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.04827	−	0.127283i	1.04827	−	0.127283i
$186$		0			0
$187$		0			0
$188$	−1.32555	+	1.17433i	−1.32555	+	1.17433i
$189$		0			0
$190$		0			0
$191$	1.09148	+	0.753393i	1.09148	+	0.753393i	0.970942	−	0.239316i	$-0.0769231\pi$
$191$	1.09148	+	0.753393i	1.09148	+	0.753393i	0.120537	+	0.992709i	$0.461538\pi$
$192$	1.53901	−	0.583668i	1.53901	−	0.583668i
$193$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$193$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$194$		0			0
$195$		0			0
$196$	−0.885456	−	0.464723i	−0.885456	−	0.464723i
$197$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$197$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$198$		0			0
$199$	0.234068	−	0.0576926i	0.234068	−	0.0576926i	−0.120537	−	0.992709i	$-0.538462\pi$
$199$	0.234068	−	0.0576926i	0.234068	−	0.0576926i	0.354605	+	0.935016i	$0.384615\pi$
$200$		0			0
$201$	2.44608	+	2.16704i	2.44608	+	2.16704i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.86557	−	1.28771i	1.86557	−	1.28771i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$213$	3.07801			3.07801
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	0.869047	+	0.329586i	0.869047	+	0.329586i
$221$		0			0
$222$		0			0
$223$	−1.32555	−	1.17433i	−1.32555	−	1.17433i	−0.970942	−	0.239316i	$-0.923077\pi$
$223$	−1.32555	−	1.17433i	−1.32555	−	1.17433i	−0.354605	−	0.935016i	$-0.615385\pi$
$224$		0			0
$225$	0.225913	−	0.0556825i	0.225913	−	0.0556825i
$226$		0			0
$227$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$227$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$228$		0			0
$229$	0.530851	+	1.39974i	0.530851	+	1.39974i	0.885456	+	0.464723i	$0.153846\pi$
$229$	0.530851	+	1.39974i	0.530851	+	1.39974i	−0.354605	+	0.935016i	$0.615385\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$233$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$234$		0			0
$235$		−	1.64597i		−	1.64597i
$236$	1.45352	−	1.28771i	1.45352	−	1.28771i
$237$		0			0
$238$		0			0
$239$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$239$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$240$	−0.542488	+	1.43042i	−0.542488	+	1.43042i
$241$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$241$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$242$		0			0
$243$	−0.542488	−	0.612343i	−0.542488	−	0.612343i
$244$		0			0
$245$	0.869047	−	0.329586i	0.869047	−	0.329586i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.114544	+	0.464723i	−0.114544	+	0.464723i	0.885456	+	0.464723i	$0.153846\pi$
$251$	−0.114544	+	0.464723i	−0.114544	+	0.464723i	−1.00000			$1.00000\pi$
$252$		0			0
$253$	0.879463	−	0.992709i	0.879463	−	0.992709i
$254$		0			0
$255$		0			0
$256$	0.568065	+	0.822984i	0.568065	+	0.822984i
$257$	−1.53901	−	0.583668i	−1.53901	−	0.583668i	−0.568065	−	0.822984i	$-0.692308\pi$
$257$	−1.53901	−	0.583668i	−1.53901	−	0.583668i	−0.970942	+	0.239316i	$0.923077\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$263$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$264$		0			0
$265$	0.764919	+	0.527986i	0.764919	+	0.527986i
$266$		0			0
$267$	−2.44608	−	0.297008i	−2.44608	−	0.297008i
$268$	−0.922670	+	1.75800i	−0.922670	+	1.75800i
$269$	−0.688601	−	0.169725i	−0.688601	−	0.169725i	−0.120537	−	0.992709i	$-0.538462\pi$
$269$	−0.688601	−	0.169725i	−0.688601	−	0.169725i	−0.568065	+	0.822984i	$0.692308\pi$
$270$		0			0
$271$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$271$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.120537	−	0.0632625i	0.120537	−	0.0632625i
$276$	1.63397	+	1.44757i	1.63397	+	1.44757i
$277$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$277$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$278$		0			0
$279$	0.195780	−	0.794309i	0.195780	−	0.794309i
$280$		0			0
$281$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$281$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$282$		0			0
$283$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$283$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$284$	0.447528	+	1.81569i	0.447528	+	1.81569i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$290$		0			0
$291$	−2.44608	+	0.297008i	−2.44608	+	0.297008i
$292$		0			0
$293$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$293$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$294$		0			0
$295$			1.80488i			1.80488i
$296$		0			0
$297$	0.960699	+	0.663123i	0.960699	+	0.663123i
$298$		0			0
$299$		0			0
$300$	0.104128	+	0.198399i	0.104128	+	0.198399i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$307$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$308$		0			0
$309$	−1.74851	−	2.53316i	−1.74851	−	2.53316i
$310$		0			0
$311$	0.136945	−	1.12785i	0.136945	−	1.12785i	−0.748511	−	0.663123i	$-0.769231\pi$
$311$	0.136945	−	1.12785i	0.136945	−	1.12785i	0.885456	−	0.464723i	$-0.153846\pi$
$312$		0			0
$313$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$313$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.709210			0.709210			0.354605	−	0.935016i	$-0.384615\pi$
$317$	0.709210			0.709210			0.354605	+	0.935016i	$0.384615\pi$
$318$		0			0
$319$		0			0
$320$	−0.922670	−	0.112032i	−0.922670	−	0.112032i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.120537	+	0.174628i	−0.120537	+	0.174628i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.71945	−	0.423807i	1.71945	−	0.423807i	0.748511	−	0.663123i	$-0.230769\pi$
$331$	1.71945	−	0.423807i	1.71945	−	0.423807i	0.970942	+	0.239316i	$0.0769231\pi$
$332$		0			0
$333$	0.234068	+	1.92773i	0.234068	+	1.92773i
$334$		0			0
$335$	−0.654366	−	1.72542i	−0.654366	−	1.72542i
$336$		0			0
$337$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$337$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$338$		0			0
$339$	0.326559	+	0.225408i	0.326559	+	0.225408i
$340$		0			0
$341$		−	0.478631i		−	0.478631i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−2.01415	+	0.244562i	−2.01415	+	0.244562i
$346$		0			0
$347$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$347$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$348$		0			0
$349$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$349$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.222431	+	0.423807i	0.222431	+	0.423807i	0.970942	−	0.239316i	$-0.0769231\pi$
$353$	0.222431	+	0.423807i	0.222431	+	0.423807i	−0.748511	+	0.663123i	$0.769231\pi$
$354$		0			0
$355$	−1.53901	−	0.807733i	−1.53901	−	0.807733i
$356$	−0.180446	−	1.48611i	−0.180446	−	1.48611i
$357$		0			0
$358$		0			0
$359$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$359$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$360$		0			0
$361$	−0.885456	+	0.464723i	−0.885456	+	0.464723i
$362$		0			0
$363$	1.53901	+	0.583668i	1.53901	+	0.583668i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−1.45352	−	0.358261i	−1.45352	−	0.358261i	−0.568065	−	0.822984i	$-0.692308\pi$
$367$	−1.45352	−	0.358261i	−1.45352	−	0.358261i	−0.885456	+	0.464723i	$0.846154\pi$
$368$	−0.616337	+	1.17433i	−0.616337	+	1.17433i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.787812			0.787812
$373$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$373$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$374$		0			0
$375$	−1.68759	−	0.415953i	−1.68759	−	0.415953i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.24006	−	0.470293i	−1.24006	−	0.470293i	−0.354605	−	0.935016i	$-0.615385\pi$
$379$	−1.24006	−	0.470293i	−1.24006	−	0.470293i	−0.885456	+	0.464723i	$0.846154\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.616337	+	0.695701i	−0.616337	+	0.695701i	−0.970942	−	0.239316i	$-0.923077\pi$
$383$	−0.616337	+	0.695701i	−0.616337	+	0.695701i	0.354605	+	0.935016i	$0.384615\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.530851	−	1.39974i	−0.530851	−	1.39974i
$389$	−0.764919	−	1.45743i	−0.764919	−	1.45743i	−0.885456	−	0.464723i	$-0.846154\pi$
$389$	−0.764919	−	1.45743i	−0.764919	−	1.45743i	0.120537	−	0.992709i	$-0.461538\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.606094	+	1.59814i	−0.606094	+	1.59814i
$397$	1.31658	−	0.159861i	1.31658	−	0.159861i	0.568065	−	0.822984i	$-0.307692\pi$
$397$	1.31658	−	0.159861i	1.31658	−	0.159861i	0.748511	+	0.663123i	$0.230769\pi$
$398$		0			0
$399$		0			0
$400$	−0.101894	+	0.0902706i	−0.101894	+	0.0902706i
$401$			1.87003i			1.87003i	0.354605	+	0.935016i	$0.384615\pi$
$401$			1.87003i			1.87003i	−0.354605	+	0.935016i	$0.615385\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.0471973	−	0.191487i	−0.0471973	−	0.191487i
$406$		0			0
$407$	0.402877	+	1.06230i	0.402877	+	1.06230i
$408$		0			0
$409$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$409$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$410$		0			0
$411$		0			0
$412$	1.24006	−	1.39974i	1.24006	−	1.39974i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.09148	+	0.753393i	−1.09148	+	0.753393i	−0.970942	−	0.239316i	$-0.923077\pi$
$419$	−1.09148	+	0.753393i	−1.09148	+	0.753393i	−0.120537	+	0.992709i	$0.538462\pi$
$420$		0			0
$421$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$421$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$422$		0			0
$423$	3.02686			3.02686
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$431$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$432$	−1.09148	−	0.413943i	−1.09148	−	0.413943i
$433$	1.10312	+	1.59814i	1.10312	+	1.59814i	0.748511	+	0.663123i	$0.230769\pi$
$433$	1.10312	+	1.59814i	1.10312	+	1.59814i	0.354605	+	0.935016i	$0.384615\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$439$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$440$		0			0
$441$	0.606094	+	1.59814i	0.606094	+	1.59814i
$442$		0			0
$443$	0.475142	+	1.92773i	0.475142	+	1.92773i	0.354605	+	0.935016i	$0.384615\pi$
$443$	0.475142	+	1.92773i	0.475142	+	1.92773i	0.120537	+	0.992709i	$0.461538\pi$
$444$	−1.74851	+	0.663123i	−1.74851	+	0.663123i
$445$	1.14510	+	0.790406i	1.14510	+	0.790406i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.688601	−	1.81569i	0.688601	−	1.81569i	0.120537	−	0.992709i	$-0.461538\pi$
$449$	0.688601	−	1.81569i	0.688601	−	1.81569i	0.568065	−	0.822984i	$-0.307692\pi$
$450$		0			0
$451$		0			0
$452$	−0.0854858	+	0.225408i	−0.0854858	+	0.225408i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$457$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$458$		0			0
$459$		0			0
$460$	−0.437112	−	1.15257i	−0.437112	−	1.15257i
$461$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$461$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$462$		0			0
$463$	−0.475142	+	1.92773i	−0.475142	+	1.92773i	−0.120537	+	0.992709i	$0.538462\pi$
$463$	−0.475142	+	1.92773i	−0.475142	+	1.92773i	−0.354605	+	0.935016i	$0.615385\pi$
$464$		0			0
$465$	−0.485557	+	0.548081i	−0.485557	+	0.548081i
$466$		0			0
$467$	1.32555	−	0.695701i	1.32555	−	0.695701i	0.354605	−	0.935016i	$-0.384615\pi$
$467$	1.32555	−	0.695701i	1.32555	−	0.695701i	0.970942	+	0.239316i	$0.0769231\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.74851	−	2.53316i	1.74851	−	2.53316i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.970942	+	1.40665i	−0.970942	+	1.40665i
$478$		0			0
$479$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$479$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$485$	1.30098	+	0.493398i	1.30098	+	0.493398i
$486$		0			0
$487$	0.213460	−	0.112032i	0.213460	−	0.112032i	−0.354605	−	0.935016i	$-0.615385\pi$
$487$	0.213460	−	0.112032i	0.213460	−	0.112032i	0.568065	+	0.822984i	$0.307692\pi$
$488$		0			0
$489$	−2.11952	+	2.39245i	−2.11952	+	2.39245i
$490$		0			0
$491$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$491$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−0.738268	−	1.40665i	−0.738268	−	1.40665i
$496$	0.114544	+	0.464723i	0.114544	+	0.464723i
$497$		0			0
$498$		0			0
$499$	1.09148	+	1.23202i	1.09148	+	1.23202i	0.970942	+	0.239316i	$0.0769231\pi$
$499$	1.09148	+	1.23202i	1.09148	+	1.23202i	0.120537	+	0.992709i	$0.461538\pi$
$500$		−	1.05597i		−	1.05597i
$501$		0			0
$502$		0			0
$503$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$503$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	1.64597i		−	1.64597i
$508$		0			0
$509$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$509$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.209504	+	1.72542i	0.209504	+	1.72542i
$516$		0			0
$517$	1.71945	−	0.423807i	1.71945	−	0.423807i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.00599	+	1.45743i	1.00599	+	1.45743i	0.885456	+	0.464723i	$0.153846\pi$
$521$	1.00599	+	1.45743i	1.00599	+	1.45743i	0.120537	+	0.992709i	$0.461538\pi$
$522$		0			0
$523$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$523$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−1.63397	−	0.198399i	−1.63397	−	0.198399i
$529$	−0.758927			−0.758927
$530$		0			0
$531$	−3.31909			−3.31909
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.371013	−	3.05557i	0.371013	−	3.05557i
$538$		0			0
$539$	0.568065	+	0.822984i	0.568065	+	0.822984i
$540$	0.960699	−	0.504214i	0.960699	−	0.504214i
$541$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$541$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$542$		0			0
$543$	−3.17297	+	0.782068i	−3.17297	+	0.782068i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$547$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	0.616337	−	1.62515i	0.616337	−	1.62515i
$556$		0			0
$557$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$557$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$563$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$564$	0.697573	+	2.83016i	0.697573	+	2.83016i
$565$	−0.104128	−	0.198399i	−0.104128	−	0.198399i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$569$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$570$		0			0
$571$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$571$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$572$		0			0
$573$	1.93291	−	1.01447i	1.93291	−	1.01447i
$574$		0			0
$575$	−0.168809	−	0.0640208i	−0.168809	−	0.0640208i
$576$	0.206022	−	1.69675i	0.206022	−	1.69675i
$577$	−0.136945	+	0.198399i	−0.136945	+	0.198399i	−0.885456	−	0.464723i	$-0.846154\pi$
$577$	−0.136945	+	0.198399i	−0.136945	+	0.198399i	0.748511	+	0.663123i	$0.230769\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.354605	+	0.935016i	−0.354605	+	0.935016i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.88546	+	0.464723i	1.88546	+	0.464723i	1.00000			$0$
$587$	1.88546	+	0.464723i	1.88546	+	0.464723i	0.885456	+	0.464723i	$0.153846\pi$
$588$	−1.35460	+	0.935016i	−1.35460	+	0.935016i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.645395	−	0.935016i	−0.645395	−	0.935016i
$593$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$593$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.0949602	−	0.385269i	0.0949602	−	0.385269i
$598$		0			0
$599$	1.56806	+	0.822984i	1.56806	+	0.822984i	1.00000			$0$
$599$	1.56806	+	0.822984i	1.56806	+	0.822984i	0.568065	+	0.822984i	$0.307692\pi$
$600$		0			0
$601$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$601$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$602$		0			0
$603$	3.17297	−	1.20335i	3.17297	−	1.20335i
$604$		0			0
$605$	−0.616337	−	0.695701i	−0.616337	−	0.695701i
$606$		0			0
$607$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$607$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$617$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$618$		0			0
$619$	−0.627974	−	1.65583i	−0.627974	−	1.65583i	−0.748511	−	0.663123i	$-0.769231\pi$
$619$	−0.627974	−	1.65583i	−0.627974	−	1.65583i	0.120537	−	0.992709i	$-0.461538\pi$
$620$	−0.393906	−	0.206738i	−0.393906	−	0.206738i
$621$	−0.186612	−	1.53689i	−0.186612	−	1.53689i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.632746	+	0.560564i	0.632746	+	0.560564i
$626$		0			0
$627$		0			0
$628$	1.74851	+	0.663123i	1.74851	+	0.663123i
$629$		0			0
$630$		0			0
$631$	−0.393906	+	0.271894i	−0.393906	+	0.271894i	−0.748511	−	0.663123i	$-0.769231\pi$
$631$	−0.393906	+	0.271894i	−0.393906	+	0.271894i	0.354605	+	0.935016i	$0.384615\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−1.53901	−	0.583668i	−1.53901	−	0.583668i
$637$		0			0
$638$		0			0
$639$	1.48538	−	2.83016i	1.48538	−	2.83016i
$640$		0			0
$641$	0.764919	−	0.527986i	0.764919	−	0.527986i	−0.120537	−	0.992709i	$-0.538462\pi$
$641$	0.764919	−	0.527986i	0.764919	−	0.527986i	0.885456	+	0.464723i	$0.153846\pi$
$642$		0			0
$643$	0.0854858	−	0.704039i	0.0854858	−	0.704039i	−0.885456	−	0.464723i	$-0.846154\pi$
$643$	0.0854858	−	0.704039i	0.0854858	−	0.704039i	0.970942	−	0.239316i	$-0.0769231\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.180446	−	0.159861i	−0.180446	−	0.159861i	0.568065	−	0.822984i	$-0.307692\pi$
$647$	−0.180446	−	0.159861i	−0.180446	−	0.159861i	−0.748511	+	0.663123i	$0.769231\pi$
$648$		0			0
$649$	−1.88546	+	0.464723i	−1.88546	+	0.464723i
$650$		0			0
$651$		0			0
$652$	−1.71945	−	0.902438i	−1.71945	−	0.902438i
$653$	−0.0854858	−	0.225408i	−0.0854858	−	0.225408i	0.885456	−	0.464723i	$-0.153846\pi$
$653$	−0.0854858	−	0.225408i	−0.0854858	−	0.225408i	−0.970942	+	0.239316i	$0.923077\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$	1.14510	−	1.01447i	1.14510	−	1.01447i
$661$	−0.688601	+	1.81569i	−0.688601	+	1.81569i	−0.120537	+	0.992709i	$0.538462\pi$
$661$	−0.688601	+	1.81569i	−0.688601	+	1.81569i	−0.568065	+	0.822984i	$0.692308\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−2.72545	+	1.03363i	−2.72545	+	1.03363i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$673$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$674$		0			0
$675$	0.0380294	−	0.154291i	0.0380294	−	0.154291i
$676$	0.970942	−	0.239316i	0.970942	−	0.239316i
$677$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$677$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.136945	−	0.198399i	0.136945	−	0.198399i	−0.748511	−	0.663123i	$-0.769231\pi$
$683$	0.136945	−	0.198399i	0.136945	−	0.198399i	0.885456	+	0.464723i	$0.153846\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	2.44608	+	0.297008i	2.44608	+	0.297008i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.31658	+	0.159861i	1.31658	+	0.159861i	0.748511	−	0.663123i	$-0.230769\pi$
$691$	1.31658	+	0.159861i	1.31658	+	0.159861i	0.568065	+	0.822984i	$0.307692\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$701$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$702$		0			0
$703$		0			0
$704$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$705$	−2.39889	−	1.25903i	−2.39889	−	1.25903i
$706$		0			0
$707$		0			0
$708$	−0.764919	−	3.10340i	−0.764919	−	3.10340i
$709$	−1.85640	+	0.704039i	−1.85640	+	0.704039i	−0.885456	+	0.464723i	$0.846154\pi$
$709$	−1.85640	+	0.704039i	−1.85640	+	0.704039i	−0.970942	+	0.239316i	$0.923077\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.475142	+	0.420939i	−0.475142	+	0.420939i
$714$		0			0
$715$		0			0
$716$	1.85640	−	0.225408i	1.85640	−	0.225408i
$717$		0			0
$718$		0			0
$719$			1.64597i			1.64597i	0.568065	+	0.822984i	$0.307692\pi$
$719$			1.64597i			1.64597i	−0.568065	+	0.822984i	$0.692308\pi$
$720$	1.05345	+	1.18910i	1.05345	+	1.18910i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−0.922670	−	1.75800i	−0.922670	−	1.75800i
$725$		0			0
$726$		0			0
$727$	0.241073	+	1.98542i	0.241073	+	1.98542i	0.120537	+	0.992709i	$0.461538\pi$
$727$	0.241073	+	1.98542i	0.241073	+	1.98542i	0.120537	+	0.992709i	$0.461538\pi$
$728$		0			0
$729$	−1.51343	+	0.373027i	−1.51343	+	0.373027i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$733$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$734$		0			0
$735$	0.184402	−	1.51868i	0.184402	−	1.51868i
$736$		0			0
$737$	1.63397	−	1.12785i	1.63397	−	1.12785i
$738$		0			0
$739$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$739$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$740$	1.04827	+	0.127283i	1.04827	+	0.127283i
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.136945	+	0.198399i	0.136945	+	0.198399i	0.885456	−	0.464723i	$-0.153846\pi$
$751$	0.136945	+	0.198399i	0.136945	+	0.198399i	−0.748511	+	0.663123i	$0.769231\pi$
$752$	−1.56806	+	0.822984i	−1.56806	+	0.822984i
$753$	0.589686	+	0.522416i	0.589686	+	0.522416i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.180446	−	1.48611i	−0.180446	−	1.48611i	−0.748511	−	0.663123i	$-0.769231\pi$
$757$	−0.180446	−	1.48611i	−0.180446	−	1.48611i	0.568065	−	0.822984i	$-0.307692\pi$
$758$		0			0
$759$	−0.774087	−	2.04110i	−0.774087	−	2.04110i
$760$		0			0
$761$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$761$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$762$		0			0
$763$		0			0
$764$	0.879463	+	0.992709i	0.879463	+	0.992709i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.63397	−	0.198399i	1.63397	−	0.198399i
$769$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$769$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$770$		0			0
$771$	−2.02787	+	1.79654i	−2.02787	+	1.79654i
$772$		0			0
$773$	−0.879463	−	0.992709i	−0.879463	−	0.992709i	0.120537	−	0.992709i	$-0.461538\pi$
$773$	−0.879463	−	0.992709i	−0.879463	−	0.992709i	−1.00000			$\pi$
$774$		0			0
$775$	−0.0609218	+	0.0231046i	−0.0609218	+	0.0231046i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	0.447528	−	1.81569i	0.447528	−	1.81569i
$782$		0			0
$783$		0			0
$784$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$785$	−1.53901	+	0.807733i	−1.53901	+	0.807733i
$786$		0			0
$787$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$787$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	1.35460	−	0.710951i	1.35460	−	0.710951i
$796$	0.241073			0.241073
$797$	1.97094	+	0.239316i	1.97094	+	0.239316i	1.00000			$0$
$797$	1.97094	+	0.239316i	1.97094	+	0.239316i	0.970942	+	0.239316i	$0.0769231\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−1.45352	+	2.10579i	−1.45352	+	2.10579i
$802$		0			0
$803$		0			0
$804$	1.85640	+	2.68946i	1.85640	+	2.68946i
$805$		0			0
$806$		0			0
$807$	−0.774087	+	0.873764i	−0.774087	+	0.873764i
$808$		0			0
$809$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$809$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$810$		0			0
$811$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$811$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.68759	−	0.640018i	1.68759	−	0.640018i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$821$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$822$		0			0
$823$	−0.688601	+	1.81569i	−0.688601	+	1.81569i	−0.120537	+	0.992709i	$0.538462\pi$
$823$	−0.688601	+	1.81569i	−0.688601	+	1.81569i	−0.568065	+	0.822984i	$0.692308\pi$
$824$		0			0
$825$		−	0.224065i		−	0.224065i
$826$		0			0
$827$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$827$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$828$	2.11952	−	0.803830i	2.11952	−	0.803830i
$829$	−0.317391	−	1.28771i	−0.317391	−	1.28771i	−0.885456	−	0.464723i	$-0.846154\pi$
$829$	−0.317391	−	1.28771i	−0.317391	−	1.28771i	0.568065	−	0.822984i	$-0.307692\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.418211	−	0.370502i	−0.418211	−	0.370502i
$838$		0			0
$839$	0.645395	+	0.935016i	0.645395	+	0.935016i	1.00000			$0$
$839$	0.645395	+	0.935016i	0.645395	+	0.935016i	−0.354605	+	0.935016i	$0.615385\pi$
$840$		0			0
$841$	0.120537	−	0.992709i	0.120537	−	0.992709i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.431935	+	0.822984i	−0.431935	+	0.822984i
$846$		0			0
$847$		0			0
$848$	0.120537	−	0.992709i	0.120537	−	0.992709i
$849$		0			0
$850$		0			0
$851$	0.700239	−	1.33419i	0.700239	−	1.33419i
$852$	2.98857	+	0.736617i	2.98857	+	0.736617i
$853$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$853$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$857$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$858$		0			0
$859$	0.530851	+	0.470293i	0.530851	+	0.470293i	0.885456	−	0.464723i	$-0.153846\pi$
$859$	0.530851	+	0.470293i	0.530851	+	0.470293i	−0.354605	+	0.935016i	$0.615385\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.0854858	+	0.704039i	0.0854858	+	0.704039i	0.970942	+	0.239316i	$0.0769231\pi$
$863$	0.0854858	+	0.704039i	0.0854858	+	0.704039i	−0.885456	+	0.464723i	$0.846154\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.393906	+	1.59814i	0.393906	+	1.59814i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.907336	+	2.39245i	−0.907336	+	2.39245i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$877$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$878$		0			0
$879$		0			0
$880$	0.764919	+	0.527986i	0.764919	+	0.527986i
$881$	−1.85640	+	0.704039i	−1.85640	+	0.704039i	−0.885456	+	0.464723i	$0.846154\pi$
$881$	−1.85640	+	0.704039i	−1.85640	+	0.704039i	−0.970942	+	0.239316i	$0.923077\pi$
$882$		0			0
$883$	−0.922670	−	1.75800i	−0.922670	−	1.75800i	−0.568065	−	0.822984i	$-0.692308\pi$
$883$	−0.922670	−	1.75800i	−0.922670	−	1.75800i	−0.354605	−	0.935016i	$-0.615385\pi$
$884$		0			0
$885$	2.63049	+	1.38059i	2.63049	+	1.38059i
$886$		0			0
$887$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$887$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.187883	−	0.0986088i	0.187883	−	0.0986088i
$892$	−1.00599	−	1.45743i	−1.00599	−	1.45743i
$893$		0			0
$894$		0			0
$895$	−0.987350	+	1.43042i	−0.987350	+	1.43042i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.232674			0.232674
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.79172	+	0.441619i	1.79172	+	0.441619i
$906$		0			0
$907$	−1.10312	+	1.59814i	−1.10312	+	1.59814i	−0.354605	+	0.935016i	$0.615385\pi$
$907$	−1.10312	+	1.59814i	−1.10312	+	1.59814i	−0.748511	+	0.663123i	$0.769231\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.32555	−	0.695701i	1.32555	−	0.695701i	0.354605	−	0.935016i	$-0.384615\pi$
$911$	1.32555	−	0.695701i	1.32555	−	0.695701i	0.970942	+	0.239316i	$0.0769231\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.180446	+	1.48611i	0.180446	+	1.48611i
$917$		0			0
$918$		0			0
$919$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$919$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.115765	−	0.102559i	0.115765	−	0.102559i
$926$		0			0
$927$	−3.17297	+	0.385269i	−3.17297	+	0.385269i
$928$		0			0
$929$	0.402877	−	1.06230i	0.402877	−	1.06230i	−0.568065	−	0.822984i	$-0.692308\pi$
$929$	0.402877	−	1.06230i	0.402877	−	1.06230i	0.970942	−	0.239316i	$-0.0769231\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−1.53901	−	1.06230i	−1.53901	−	1.06230i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$937$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$938$		0			0
$939$		0			0
$940$	0.393906	−	1.59814i	0.393906	−	1.59814i
$941$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$941$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$942$		0			0
$943$		0			0
$944$	1.71945	−	0.902438i	1.71945	−	0.902438i
$945$		0			0
$946$		0			0
$947$	0.213460	−	1.75800i	0.213460	−	1.75800i	−0.354605	−	0.935016i	$-0.615385\pi$
$947$	0.213460	−	1.75800i	0.213460	−	1.75800i	0.568065	−	0.822984i	$-0.307692\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	0.542488	−	1.03363i	0.542488	−	1.03363i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$	−1.23267			−1.23267
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−0.869047	+	1.25903i	−0.869047	+	1.25903i
$961$	0.0929232	−	0.765291i	0.0929232	−	0.765291i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$967$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.0854858	+	0.225408i	0.0854858	+	0.225408i	0.970942	−	0.239316i	$-0.0769231\pi$
$971$	0.0854858	+	0.225408i	0.0854858	+	0.225408i	−0.885456	+	0.464723i	$0.846154\pi$
$972$	−0.380181	−	0.724375i	−0.380181	−	0.724375i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	0.929446i		−	0.929446i	−0.885456	−	0.464723i	$-0.846154\pi$
$977$		−	0.929446i		−	0.929446i	0.885456	−	0.464723i	$-0.153846\pi$
$978$		0			0
$979$	−0.530851	+	1.39974i	−0.530851	+	1.39974i
$980$	0.922670	−	0.112032i	0.922670	−	0.112032i
$981$		0			0
$982$		0			0
$983$	0.530851	−	0.470293i	0.530851	−	0.470293i	−0.354605	−	0.935016i	$-0.615385\pi$
$983$	0.530851	−	0.470293i	0.530851	−	0.470293i	0.885456	+	0.464723i	$0.153846\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.56806	−	0.822984i	−1.56806	−	0.822984i	−0.568065	−	0.822984i	$-0.692308\pi$
$991$	−1.56806	−	0.822984i	−1.56806	−	0.822984i	−1.00000			$\pi$
$992$		0			0
$993$	0.697573	−	2.83016i	0.697573	−	2.83016i
$994$		0			0
$995$	−0.148582	+	0.167715i	−0.148582	+	0.167715i
$996$		0			0
$997$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$997$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$998$		0			0
$999$	1.24006	+	0.470293i	1.24006	+	0.470293i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	583.1.n.a.428.1	yes	12
11.10	odd	2	CM	583.1.n.a.428.1	yes	12
53.40	even	26	inner	583.1.n.a.252.1	✓	12
583.252	odd	26	inner	583.1.n.a.252.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
583.1.n.a.252.1	✓	12	53.40	even	26	inner
583.1.n.a.252.1	✓	12	583.252	odd	26	inner
583.1.n.a.428.1	yes	12	1.1	even	1	trivial
583.1.n.a.428.1	yes	12	11.10	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 583 = 11 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	583.n (of order \(26\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(0.764919 - 1.45743i) q^{3} +(0.970942 + 0.239316i) q^{4} +(-0.764919 + 0.527986i) q^{5} +(-0.970942 - 1.40665i) q^{9} +O(q^{10})\)	\(q+(0.764919 - 1.45743i) q^{3} +(0.970942 + 0.239316i) q^{4} +(-0.764919 + 0.527986i) q^{5} +(-0.970942 - 1.40665i) q^{9} +(-0.748511 - 0.663123i) q^{11} +(1.09148 - 1.23202i) q^{12} +(0.184402 + 1.51868i) q^{15} +(0.885456 + 0.464723i) q^{16} +(-0.869047 + 0.329586i) q^{20} +1.32625i q^{23} +(-0.0482722 + 0.127283i) q^{25} +(-1.15883 + 0.140707i) q^{27} +(0.317391 + 0.358261i) q^{31} +(-1.53901 + 0.583668i) q^{33} +(-0.606094 - 1.59814i) q^{36} +(-1.00599 - 0.527986i) q^{37} +(-0.568065 - 0.822984i) q^{44} +(1.48538 + 0.563332i) q^{45} +(-1.00599 + 1.45743i) q^{47} +(1.35460 - 0.935016i) q^{48} +(-0.970942 - 0.239316i) q^{49} +(-0.354605 - 0.935016i) q^{53} +(0.922670 + 0.112032i) q^{55} +(1.10312 - 1.59814i) q^{59} +(-0.184402 + 1.51868i) q^{60} +(0.748511 + 0.663123i) q^{64} +(-0.475142 + 1.92773i) q^{67} +(1.93291 + 1.01447i) q^{69} +(0.869047 + 1.65583i) q^{71} +(0.148582 + 0.167715i) q^{75} +(-0.922670 + 0.112032i) q^{80} +(-0.0752430 + 0.198399i) q^{81} +(-0.530851 - 1.39974i) q^{89} +(-0.317391 + 1.28771i) q^{92} +(0.764919 - 0.188536i) q^{93} +(-0.850405 - 1.23202i) q^{97} +(-0.206022 + 1.69675i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{4} - q^{9}+O(q^{10}) \) 12 * q + q^4 - q^9	\( 12 q + q^{4} - q^{9} - q^{11} - 13 q^{15} - q^{16} - q^{25} - 12 q^{36} + 2 q^{37} + q^{44} + 2 q^{47} + 13 q^{48} - q^{49} - q^{53} + 2 q^{59} + 13 q^{60} + q^{64} - 14 q^{81} + 2 q^{89} - 2 q^{97} - q^{99}+O(q^{100}) \) 12 * q + q^4 - q^9 - q^11 - 13 * q^15 - q^16 - q^25 - 12 * q^36 + 2 * q^37 + q^44 + 2 * q^47 + 13 * q^48 - q^49 - q^53 + 2 * q^59 + 13 * q^60 + q^64 - 14 * q^81 + 2 * q^89 - 2 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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