LMFDB - Embedded newform 507.1.o.a.53.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,1,Mod(14,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(26))

chi = DirichletCharacter(H, H._module([13, 18]))

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

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

chi := DirichletCharacter("507.14");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	507.o (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:	$0.253025961405$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{13}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{13} - \cdots)$

Embedding invariants

Embedding label			53.1
Root			$0.748511 - 0.663123i$ of defining polynomial
Character	$\chi$	$=$	507.53
Dual form			507.1.o.a.287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.354605 - 0.935016i) q^{3} +(0.120537 - 0.992709i) q^{4} +(-0.402877 - 0.583668i) q^{7} +(-0.748511 + 0.663123i) q^{9} +O(q^{10})$	\(q+(-0.354605 - 0.935016i) q^{3} +(0.120537 - 0.992709i) q^{4} +(-0.402877 - 0.583668i) q^{7} +(-0.748511 + 0.663123i) q^{9} +(-0.970942 + 0.239316i) q^{12} +1.00000 q^{13} +(-0.970942 - 0.239316i) q^{16} -1.49702 q^{19} +(-0.402877 + 0.583668i) q^{21} +(0.885456 - 0.464723i) q^{25} +(0.885456 + 0.464723i) q^{27} +(-0.627974 + 0.329586i) q^{28} +(0.213460 + 0.112032i) q^{31} +(0.568065 + 0.822984i) q^{36} +(1.00599 + 0.527986i) q^{37} +(-0.354605 - 0.935016i) q^{39} +(1.00599 - 0.527986i) q^{43} +(0.120537 + 0.992709i) q^{48} +(0.176246 - 0.464723i) q^{49} +(0.120537 - 0.992709i) q^{52} +(0.530851 + 1.39974i) q^{57} +(0.136945 - 0.198399i) q^{61} +(0.688601 + 0.169725i) q^{63} +(-0.354605 + 0.935016i) q^{64} +(0.213460 + 1.75800i) q^{67} +(1.45352 + 1.28771i) q^{73} +(-0.748511 - 0.663123i) q^{75} +(-0.180446 + 1.48611i) q^{76} +(-0.0854858 - 0.704039i) q^{79} +(0.120537 - 0.992709i) q^{81} +(0.530851 + 0.470293i) q^{84} +(-0.402877 - 0.583668i) q^{91} +(0.0290582 - 0.239316i) q^{93} +(-1.94188 - 0.478631i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{3} - q^{4} - 2 q^{7} - q^{9}+O(q^{10}) $ 12 * q - q^3 - q^4 - 2 * q^7 - q^9	$ 12 q - q^{3} - q^{4} - 2 q^{7} - q^{9} - q^{12} + 12 q^{13} - q^{16} - 2 q^{19} - 2 q^{21} - q^{25} - q^{27} - 2 q^{28} - 2 q^{31} - q^{36} - 2 q^{37} - q^{39} - 2 q^{43} - q^{48} - 3 q^{49} - q^{52} - 2 q^{57} - 2 q^{61} - 2 q^{63} - q^{64} - 2 q^{67} - 2 q^{73} - q^{75} - 2 q^{76} - 2 q^{79} - q^{81} - 2 q^{84} - 2 q^{91} + 11 q^{93} - 2 q^{97}+O(q^{100}) $ 12 * q - q^3 - q^4 - 2 * q^7 - q^9 - q^12 + 12 * q^13 - q^16 - 2 * q^19 - 2 * q^21 - q^25 - q^27 - 2 * q^28 - 2 * q^31 - q^36 - 2 * q^37 - q^39 - 2 * q^43 - q^48 - 3 * q^49 - q^52 - 2 * q^57 - 2 * q^61 - 2 * q^63 - q^64 - 2 * q^67 - 2 * q^73 - q^75 - 2 * q^76 - 2 * q^79 - q^81 - 2 * q^84 - 2 * q^91 + 11 * q^93 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{10}{13}\right)$

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		0.748511	−	0.663123i	$-0.230769\pi$
$2$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$3$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$3$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$4$	0.120537	−	0.992709i	0.120537	−	0.992709i
$5$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$5$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$6$		0			0
$7$	−0.402877	−	0.583668i	−0.402877	−	0.583668i	0.568065	−	0.822984i	$-0.307692\pi$
$7$	−0.402877	−	0.583668i	−0.402877	−	0.583668i	−0.970942	+	0.239316i	$0.923077\pi$
$8$		0			0
$9$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$10$		0			0
$11$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$11$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$12$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$13$	1.00000			1.00000
$13$	1.00000			1.00000
$14$		0			0
$15$		0			0
$16$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$17$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$17$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$18$		0			0
$19$	−1.49702			−1.49702			−0.748511	−	0.663123i	$-0.769231\pi$
$19$	−1.49702			−1.49702			−0.748511	+	0.663123i	$0.769231\pi$
$20$		0			0
$21$	−0.402877	+	0.583668i	−0.402877	+	0.583668i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	0.885456	−	0.464723i	0.885456	−	0.464723i
$26$		0			0
$27$	0.885456	+	0.464723i	0.885456	+	0.464723i
$28$	−0.627974	+	0.329586i	−0.627974	+	0.329586i
$29$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$29$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$30$		0			0
$31$	0.213460	+	0.112032i	0.213460	+	0.112032i	0.568065	−	0.822984i	$-0.307692\pi$
$31$	0.213460	+	0.112032i	0.213460	+	0.112032i	−0.354605	+	0.935016i	$0.615385\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.568065	+	0.822984i	0.568065	+	0.822984i
$37$	1.00599	+	0.527986i	1.00599	+	0.527986i	0.885456	−	0.464723i	$-0.153846\pi$
$37$	1.00599	+	0.527986i	1.00599	+	0.527986i	0.120537	+	0.992709i	$0.461538\pi$
$38$		0			0
$39$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$40$		0			0
$41$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$41$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$42$		0			0
$43$	1.00599	−	0.527986i	1.00599	−	0.527986i	0.120537	−	0.992709i	$-0.461538\pi$
$43$	1.00599	−	0.527986i	1.00599	−	0.527986i	0.885456	+	0.464723i	$0.153846\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$47$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$48$	0.120537	+	0.992709i	0.120537	+	0.992709i
$49$	0.176246	−	0.464723i	0.176246	−	0.464723i
$50$		0			0
$51$		0			0
$52$	0.120537	−	0.992709i	0.120537	−	0.992709i
$53$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$53$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.530851	+	1.39974i	0.530851	+	1.39974i
$58$		0			0
$59$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$59$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$60$		0			0
$61$	0.136945	−	0.198399i	0.136945	−	0.198399i	−0.748511	−	0.663123i	$-0.769231\pi$
$61$	0.136945	−	0.198399i	0.136945	−	0.198399i	0.885456	+	0.464723i	$0.153846\pi$
$62$		0			0
$63$	0.688601	+	0.169725i	0.688601	+	0.169725i
$64$	−0.354605	+	0.935016i	−0.354605	+	0.935016i
$65$		0			0
$66$		0			0
$67$	0.213460	+	1.75800i	0.213460	+	1.75800i	0.568065	+	0.822984i	$0.307692\pi$
$67$	0.213460	+	1.75800i	0.213460	+	1.75800i	−0.354605	+	0.935016i	$0.615385\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$71$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$72$		0			0
$73$	1.45352	+	1.28771i	1.45352	+	1.28771i	0.885456	+	0.464723i	$0.153846\pi$
$73$	1.45352	+	1.28771i	1.45352	+	1.28771i	0.568065	+	0.822984i	$0.307692\pi$
$74$		0			0
$75$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$76$	−0.180446	+	1.48611i	−0.180446	+	1.48611i
$77$		0			0
$78$		0			0
$79$	−0.0854858	−	0.704039i	−0.0854858	−	0.704039i	−0.970942	−	0.239316i	$-0.923077\pi$
$79$	−0.0854858	−	0.704039i	−0.0854858	−	0.704039i	0.885456	−	0.464723i	$-0.153846\pi$
$80$		0			0
$81$	0.120537	−	0.992709i	0.120537	−	0.992709i
$82$		0			0
$83$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$83$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$84$	0.530851	+	0.470293i	0.530851	+	0.470293i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−0.402877	−	0.583668i	−0.402877	−	0.583668i
$92$		0			0
$93$	0.0290582	−	0.239316i	0.0290582	−	0.239316i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.94188	−	0.478631i	−1.94188	−	0.478631i	−0.970942	−	0.239316i	$-0.923077\pi$
$97$	−1.94188	−	0.478631i	−1.94188	−	0.478631i	−0.970942	−	0.239316i	$-0.923077\pi$
$98$		0			0
$99$		0			0
$100$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$101$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$101$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$102$		0			0
$103$	0.688601	+	1.81569i	0.688601	+	1.81569i	0.568065	+	0.822984i	$0.307692\pi$
$103$	0.688601	+	1.81569i	0.688601	+	1.81569i	0.120537	+	0.992709i	$0.461538\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$107$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$108$	0.568065	−	0.822984i	0.568065	−	0.822984i
$109$	−1.71945	+	0.902438i	−1.71945	+	0.902438i	−0.748511	+	0.663123i	$0.769231\pi$
$109$	−1.71945	+	0.902438i	−1.71945	+	0.902438i	−0.970942	+	0.239316i	$0.923077\pi$
$110$		0			0
$111$	0.136945	−	1.12785i	0.136945	−	1.12785i
$112$	0.251489	+	0.663123i	0.251489	+	0.663123i
$113$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$113$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.120537	+	0.992709i	0.120537	+	0.992709i
$122$		0			0
$123$		0			0
$124$	0.136945	−	0.198399i	0.136945	−	0.198399i
$125$		0			0
$126$		0			0
$127$	−0.234068	−	1.92773i	−0.234068	−	1.92773i	−0.354605	−	0.935016i	$-0.615385\pi$
$127$	−0.234068	−	1.92773i	−0.234068	−	1.92773i	0.120537	−	0.992709i	$-0.461538\pi$
$128$		0			0
$129$	−0.850405	−	0.753393i	−0.850405	−	0.753393i
$130$		0			0
$131$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$131$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$132$		0			0
$133$	0.603116	+	0.873764i	0.603116	+	0.873764i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$137$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$138$		0			0
$139$	−1.71945	−	0.423807i	−1.71945	−	0.423807i	−0.748511	−	0.663123i	$-0.769231\pi$
$139$	−1.71945	−	0.423807i	−1.71945	−	0.423807i	−0.970942	+	0.239316i	$0.923077\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.885456	−	0.464723i	0.885456	−	0.464723i
$145$		0			0
$146$		0			0
$147$	−0.497021			−0.497021
$148$	0.645395	−	0.935016i	0.645395	−	0.935016i
$149$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$149$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$150$		0			0
$151$	0.213460	−	1.75800i	0.213460	−	1.75800i	−0.354605	−	0.935016i	$-0.615385\pi$
$151$	0.213460	−	1.75800i	0.213460	−	1.75800i	0.568065	−	0.822984i	$-0.307692\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$157$	−0.234068	+	0.0576926i	−0.234068	+	0.0576926i	−0.354605	−	0.935016i	$-0.615385\pi$
$157$	−0.234068	+	0.0576926i	−0.234068	+	0.0576926i	0.120537	+	0.992709i	$0.461538\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.56806	+	0.822984i	1.56806	+	0.822984i	1.00000			$0$
$163$	1.56806	+	0.822984i	1.56806	+	0.822984i	0.568065	+	0.822984i	$0.307692\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$167$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$168$		0			0
$169$	1.00000			1.00000
$170$		0			0
$171$	1.12054	−	0.992709i	1.12054	−	0.992709i
$172$	−0.402877	−	1.06230i	−0.402877	−	1.06230i
$173$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$173$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$174$		0			0
$175$	−0.627974	−	0.329586i	−0.627974	−	0.329586i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$179$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$180$		0			0
$181$	−1.71945	+	0.423807i	−1.71945	+	0.423807i	−0.970942	−	0.239316i	$-0.923077\pi$
$181$	−1.71945	+	0.423807i	−1.71945	+	0.423807i	−0.748511	+	0.663123i	$0.769231\pi$
$182$		0			0
$183$	−0.234068	−	0.0576926i	−0.234068	−	0.0576926i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.0854858	−	0.704039i	−0.0854858	−	0.704039i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	1.00000			1.00000
$193$	0.645395	−	0.935016i	0.645395	−	0.935016i	−0.354605	−	0.935016i	$-0.615385\pi$
$193$	0.645395	−	0.935016i	0.645395	−	0.935016i	1.00000			$0$
$194$		0			0
$195$		0			0
$196$	−0.440091	−	0.230977i	−0.440091	−	0.230977i
$197$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$197$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$198$		0			0
$199$	1.45352	+	0.358261i	1.45352	+	0.358261i	0.885456	−	0.464723i	$-0.153846\pi$
$199$	1.45352	+	0.358261i	1.45352	+	0.358261i	0.568065	+	0.822984i	$0.307692\pi$
$200$		0			0
$201$	1.56806	−	0.822984i	1.56806	−	0.822984i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$209$		0			0
$210$		0			0
$211$	0.213460	+	1.75800i	0.213460	+	1.75800i	0.568065	+	0.822984i	$0.307692\pi$
$211$	0.213460	+	1.75800i	0.213460	+	1.75800i	−0.354605	+	0.935016i	$0.615385\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.0206083	−	0.169725i	−0.0206083	−	0.169725i
$218$		0			0
$219$	0.688601	−	1.81569i	0.688601	−	1.81569i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.94188	+	0.478631i	−1.94188	+	0.478631i	−0.970942	+	0.239316i	$0.923077\pi$
$223$	−1.94188	+	0.478631i	−1.94188	+	0.478631i	−0.970942	+	0.239316i	$0.923077\pi$
$224$		0			0
$225$	−0.354605	+	0.935016i	−0.354605	+	0.935016i
$226$		0			0
$227$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$227$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$228$	1.45352	−	0.358261i	1.45352	−	0.358261i
$229$	−1.71945	+	0.902438i	−1.71945	+	0.902438i	−0.748511	+	0.663123i	$0.769231\pi$
$229$	−1.71945	+	0.902438i	−1.71945	+	0.902438i	−0.970942	+	0.239316i	$0.923077\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$233$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.627974	+	0.329586i	−0.627974	+	0.329586i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−0.234068	−	0.0576926i	−0.234068	−	0.0576926i	0.120537	−	0.992709i	$-0.461538\pi$
$241$	−0.234068	−	0.0576926i	−0.234068	−	0.0576926i	−0.354605	+	0.935016i	$0.615385\pi$
$242$		0			0
$243$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$244$	−0.180446	−	0.159861i	−0.180446	−	0.159861i
$245$		0			0
$246$		0			0
$247$	−1.49702			−1.49702
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$251$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$252$	0.251489	−	0.663123i	0.251489	−	0.663123i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.885456	+	0.464723i	0.885456	+	0.464723i
$257$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$257$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$258$		0			0
$259$	−0.0971229	−	0.799879i	−0.0971229	−	0.799879i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$263$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.77091			1.77091
$269$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$269$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$270$		0			0
$271$	0.0290582	+	0.239316i	0.0290582	+	0.239316i	1.00000			$0$
$271$	0.0290582	+	0.239316i	0.0290582	+	0.239316i	−0.970942	+	0.239316i	$0.923077\pi$
$272$		0			0
$273$	−0.402877	+	0.583668i	−0.402877	+	0.583668i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.402877	+	0.583668i	−0.402877	+	0.583668i	−0.970942	−	0.239316i	$-0.923077\pi$
$277$	−0.402877	+	0.583668i	−0.402877	+	0.583668i	0.568065	+	0.822984i	$0.307692\pi$
$278$		0			0
$279$	−0.234068	+	0.0576926i	−0.234068	+	0.0576926i
$280$		0			0
$281$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$281$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$282$		0			0
$283$	−1.71945	−	0.902438i	−1.71945	−	0.902438i	−0.970942	−	0.239316i	$-0.923077\pi$
$283$	−1.71945	−	0.902438i	−1.71945	−	0.902438i	−0.748511	−	0.663123i	$-0.769231\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.354605	+	0.935016i	−0.354605	+	0.935016i
$290$		0			0
$291$	0.241073	+	1.98542i	0.241073	+	1.98542i
$292$	1.45352	−	1.28771i	1.45352	−	1.28771i
$293$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$293$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$301$	−0.713460	−	0.374453i	−0.713460	−	0.374453i
$302$		0			0
$303$		0			0
$304$	1.45352	+	0.358261i	1.45352	+	0.358261i
$305$		0			0
$306$		0			0
$307$	1.00599	+	0.527986i	1.00599	+	0.527986i	0.885456	−	0.464723i	$-0.153846\pi$
$307$	1.00599	+	0.527986i	1.00599	+	0.527986i	0.120537	+	0.992709i	$0.461538\pi$
$308$		0			0
$309$	1.45352	−	1.28771i	1.45352	−	1.28771i
$310$		0			0
$311$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$311$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$312$		0			0
$313$	−0.627974	+	0.329586i	−0.627974	+	0.329586i	−0.748511	−	0.663123i	$-0.769231\pi$
$313$	−0.627974	+	0.329586i	−0.627974	+	0.329586i	0.120537	+	0.992709i	$0.461538\pi$
$314$		0			0
$315$		0			0
$316$	−0.709210			−0.709210
$317$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$317$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$325$	0.885456	−	0.464723i	0.885456	−	0.464723i
$326$		0			0
$327$	1.45352	+	1.28771i	1.45352	+	1.28771i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.00599	+	1.45743i	1.00599	+	1.45743i	0.885456	+	0.464723i	$0.153846\pi$
$331$	1.00599	+	1.45743i	1.00599	+	1.45743i	0.120537	+	0.992709i	$0.461538\pi$
$332$		0			0
$333$	−1.10312	+	0.271894i	−1.10312	+	0.271894i
$334$		0			0
$335$		0			0
$336$	0.530851	−	0.470293i	0.530851	−	0.470293i
$337$	1.13613			1.13613			0.568065	−	0.822984i	$-0.307692\pi$
$337$	1.13613			1.13613			0.568065	+	0.822984i	$0.307692\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.03085	+	0.254082i	−1.03085	+	0.254082i
$344$		0			0
$345$		0			0
$346$		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
$349$	−1.49702	−	1.32625i	−1.49702	−	1.32625i	−0.748511	−	0.663123i	$-0.769231\pi$
$349$	−1.49702	−	1.32625i	−1.49702	−	1.32625i	−0.748511	−	0.663123i	$-0.769231\pi$
$350$		0			0
$351$	0.885456	+	0.464723i	0.885456	+	0.464723i
$352$		0			0
$353$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$353$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$359$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$360$		0			0
$361$	1.24107			1.24107
$362$		0			0
$363$	0.885456	−	0.464723i	0.885456	−	0.464723i
$364$	−0.627974	+	0.329586i	−0.627974	+	0.329586i
$365$		0			0
$366$		0			0
$367$	0.530851	−	0.470293i	0.530851	−	0.470293i	−0.354605	−	0.935016i	$-0.615385\pi$
$367$	0.530851	−	0.470293i	0.530851	−	0.470293i	0.885456	+	0.464723i	$0.153846\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.234068	−	0.0576926i	−0.234068	−	0.0576926i
$373$	−0.180446	−	0.159861i	−0.180446	−	0.159861i	0.568065	−	0.822984i	$-0.307692\pi$
$373$	−0.180446	−	0.159861i	−0.180446	−	0.159861i	−0.748511	+	0.663123i	$0.769231\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.530851	+	1.39974i	0.530851	+	1.39974i	0.885456	+	0.464723i	$0.153846\pi$
$379$	0.530851	+	1.39974i	0.530851	+	1.39974i	−0.354605	+	0.935016i	$0.615385\pi$
$380$		0			0
$381$	−1.71945	+	0.902438i	−1.71945	+	0.902438i
$382$		0			0
$383$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$383$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.402877	+	1.06230i	−0.402877	+	1.06230i
$388$	−0.709210	+	1.87003i	−0.709210	+	1.87003i
$389$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$389$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.234068	+	0.0576926i	−0.234068	+	0.0576926i	−0.354605	−	0.935016i	$-0.615385\pi$
$397$	−0.234068	+	0.0576926i	−0.234068	+	0.0576926i	0.120537	+	0.992709i	$0.461538\pi$
$398$		0			0
$399$	0.603116	−	0.873764i	0.603116	−	0.873764i
$400$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$401$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$401$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$402$		0			0
$403$	0.213460	+	0.112032i	0.213460	+	0.112032i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.530851	+	1.39974i	0.530851	+	1.39974i	0.885456	+	0.464723i	$0.153846\pi$
$409$	0.530851	+	1.39974i	0.530851	+	1.39974i	−0.354605	+	0.935016i	$0.615385\pi$
$410$		0			0
$411$		0			0
$412$	1.88546	−	0.464723i	1.88546	−	0.464723i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.213460	+	1.75800i	0.213460	+	1.75800i
$418$		0			0
$419$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$419$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$420$		0			0
$421$	0.688601	−	1.81569i	0.688601	−	1.81569i	0.120537	−	0.992709i	$-0.461538\pi$
$421$	0.688601	−	1.81569i	0.688601	−	1.81569i	0.568065	−	0.822984i	$-0.307692\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.170972			−0.170972
$428$		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	$0.538462\pi$
$432$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$433$	1.45352	−	0.358261i	1.45352	−	0.358261i	0.568065	−	0.822984i	$-0.307692\pi$
$433$	1.45352	−	0.358261i	1.45352	−	0.358261i	0.885456	+	0.464723i	$0.153846\pi$
$434$		0			0
$435$		0			0
$436$	0.688601	+	1.81569i	0.688601	+	1.81569i
$437$		0			0
$438$		0			0
$439$	−1.32555	+	0.695701i	−1.32555	+	0.695701i	−0.970942	−	0.239316i	$-0.923077\pi$
$439$	−1.32555	+	0.695701i	−1.32555	+	0.695701i	−0.354605	+	0.935016i	$0.615385\pi$
$440$		0			0
$441$	0.176246	+	0.464723i	0.176246	+	0.464723i
$442$		0			0
$443$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$443$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$444$	−1.10312	−	0.271894i	−1.10312	−	0.271894i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.688601	−	0.169725i	0.688601	−	0.169725i
$449$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$449$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.71945	+	0.423807i	−1.71945	+	0.423807i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.688601	−	1.81569i	0.688601	−	1.81569i	0.120537	−	0.992709i	$-0.461538\pi$
$457$	0.688601	−	1.81569i	0.688601	−	1.81569i	0.568065	−	0.822984i	$-0.307692\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$461$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$462$		0			0
$463$	1.12054	−	0.992709i	1.12054	−	0.992709i	0.120537	−	0.992709i	$-0.461538\pi$
$463$	1.12054	−	0.992709i	1.12054	−	0.992709i	1.00000			$0$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$467$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$468$	0.568065	+	0.822984i	0.568065	+	0.822984i
$469$	0.940091	−	0.832848i	0.940091	−	0.832848i
$470$		0			0
$471$	0.136945	+	0.198399i	0.136945	+	0.198399i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.32555	+	0.695701i	−1.32555	+	0.695701i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$479$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$480$		0			0
$481$	1.00599	+	0.527986i	1.00599	+	0.527986i
$482$		0			0
$483$		0			0
$484$	1.00000			1.00000
$485$		0			0
$486$		0			0
$487$	0.0290582	+	0.239316i	0.0290582	+	0.239316i	1.00000			$0$
$487$	0.0290582	+	0.239316i	0.0290582	+	0.239316i	−0.970942	+	0.239316i	$0.923077\pi$
$488$		0			0
$489$	0.213460	−	1.75800i	0.213460	−	1.75800i
$490$		0			0
$491$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$491$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.180446	−	0.159861i	−0.180446	−	0.159861i
$497$		0			0
$498$		0			0
$499$	−1.10312	−	1.59814i	−1.10312	−	1.59814i	−0.748511	−	0.663123i	$-0.769231\pi$
$499$	−1.10312	−	1.59814i	−1.10312	−	1.59814i	−0.354605	−	0.935016i	$-0.615385\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$503$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$508$	−1.94188			−1.94188
$509$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$509$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$510$		0			0
$511$	0.166003	−	1.36716i	0.166003	−	1.36716i
$512$		0			0
$513$	−1.32555	−	0.695701i	−1.32555	−	0.695701i
$514$		0			0
$515$		0			0
$516$	−0.850405	+	0.753393i	−0.850405	+	0.753393i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$521$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$522$		0			0
$523$	1.45352	+	0.358261i	1.45352	+	0.358261i	0.885456	−	0.464723i	$-0.153846\pi$
$523$	1.45352	+	0.358261i	1.45352	+	0.358261i	0.568065	+	0.822984i	$0.307692\pi$
$524$		0			0
$525$	−0.0854858	+	0.704039i	−0.0854858	+	0.704039i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	1.00000			1.00000
$530$		0			0
$531$		0			0
$532$	0.940091	−	0.493398i	0.940091	−	0.493398i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.71945	−	0.423807i	−1.71945	−	0.423807i	−0.748511	−	0.663123i	$-0.769231\pi$
$541$	−1.71945	−	0.423807i	−1.71945	−	0.423807i	−0.970942	+	0.239316i	$0.923077\pi$
$542$		0			0
$543$	1.00599	+	1.45743i	1.00599	+	1.45743i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.180446	−	0.159861i	−0.180446	−	0.159861i	0.568065	−	0.822984i	$-0.307692\pi$
$547$	−0.180446	−	0.159861i	−0.180446	−	0.159861i	−0.748511	+	0.663123i	$0.769231\pi$
$548$		0			0
$549$	0.0290582	+	0.239316i	0.0290582	+	0.239316i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.376485	+	0.333536i	−0.376485	+	0.333536i
$554$		0			0
$555$		0			0
$556$	−0.627974	+	1.65583i	−0.627974	+	1.65583i
$557$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$557$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$558$		0			0
$559$	1.00599	−	0.527986i	1.00599	−	0.527986i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$563$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.627974	+	0.329586i	−0.627974	+	0.329586i
$568$		0			0
$569$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$569$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$570$		0			0
$571$	0.251489	−	0.663123i	0.251489	−	0.663123i	−0.748511	−	0.663123i	$-0.769231\pi$
$571$	0.251489	−	0.663123i	0.251489	−	0.663123i	1.00000			$0$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$577$	1.77091			1.77091			0.885456	−	0.464723i	$-0.153846\pi$
$577$	1.77091			1.77091			0.885456	+	0.464723i	$0.153846\pi$
$578$		0			0
$579$	−1.10312	−	0.271894i	−1.10312	−	0.271894i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	−0.0599093	+	0.493398i	−0.0599093	+	0.493398i
$589$	−0.319554	−	0.167715i	−0.319554	−	0.167715i
$590$		0			0
$591$		0			0
$592$	−0.850405	−	0.753393i	−0.850405	−	0.753393i
$593$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$593$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.180446	−	1.48611i	−0.180446	−	1.48611i
$598$		0			0
$599$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$599$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$600$		0			0
$601$	−0.850405	−	0.753393i	−0.850405	−	0.753393i	0.120537	−	0.992709i	$-0.461538\pi$
$601$	−0.850405	−	0.753393i	−0.850405	−	0.753393i	−0.970942	+	0.239316i	$0.923077\pi$
$602$		0			0
$603$	−1.32555	−	1.17433i	−1.32555	−	1.17433i
$604$	−1.71945	−	0.423807i	−1.71945	−	0.423807i
$605$		0			0
$606$		0			0
$607$	−0.402877	−	1.06230i	−0.402877	−	1.06230i	−0.970942	−	0.239316i	$-0.923077\pi$
$607$	−0.402877	−	1.06230i	−0.402877	−	1.06230i	0.568065	−	0.822984i	$-0.307692\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.10312	−	0.271894i	−1.10312	−	0.271894i	−0.354605	−	0.935016i	$-0.615385\pi$
$613$	−1.10312	−	0.271894i	−1.10312	−	0.271894i	−0.748511	+	0.663123i	$0.769231\pi$
$614$		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
$619$	−0.402877	−	1.06230i	−0.402877	−	1.06230i	−0.970942	−	0.239316i	$-0.923077\pi$
$619$	−0.402877	−	1.06230i	−0.402877	−	1.06230i	0.568065	−	0.822984i	$-0.307692\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.120537	+	0.992709i	0.120537	+	0.992709i
$625$	0.568065	−	0.822984i	0.568065	−	0.822984i
$626$		0			0
$627$		0			0
$628$	0.0290582	+	0.239316i	0.0290582	+	0.239316i
$629$		0			0
$630$		0			0
$631$	0.645395	−	0.935016i	0.645395	−	0.935016i	−0.354605	−	0.935016i	$-0.615385\pi$
$631$	0.645395	−	0.935016i	0.645395	−	0.935016i	1.00000			$0$
$632$		0			0
$633$	1.56806	−	0.822984i	1.56806	−	0.822984i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.176246	−	0.464723i	0.176246	−	0.464723i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$641$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$642$		0			0
$643$	−0.402877	+	1.06230i	−0.402877	+	1.06230i	0.568065	+	0.822984i	$0.307692\pi$
$643$	−0.402877	+	1.06230i	−0.402877	+	1.06230i	−0.970942	+	0.239316i	$0.923077\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$647$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.151388	+	0.0794545i	−0.151388	+	0.0794545i
$652$	1.00599	−	1.45743i	1.00599	−	1.45743i
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.94188			−1.94188
$658$		0			0
$659$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$659$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$660$		0			0
$661$	−0.850405	−	1.23202i	−0.850405	−	1.23202i	−0.970942	−	0.239316i	$-0.923077\pi$
$661$	−0.850405	−	1.23202i	−0.850405	−	1.23202i	0.120537	−	0.992709i	$-0.461538\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.13613	+	1.64597i	1.13613	+	1.64597i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.530851	+	1.39974i	0.530851	+	1.39974i	0.885456	+	0.464723i	$0.153846\pi$
$673$	0.530851	+	1.39974i	0.530851	+	1.39974i	−0.354605	+	0.935016i	$0.615385\pi$
$674$		0			0
$675$	1.00000			1.00000
$676$	0.120537	−	0.992709i	0.120537	−	0.992709i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	0.502979	+	1.32625i	0.502979	+	1.32625i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$683$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$684$	−0.850405	−	1.23202i	−0.850405	−	1.23202i
$685$		0			0
$686$		0			0
$687$	1.45352	+	1.28771i	1.45352	+	1.28771i
$688$	−1.10312	+	0.271894i	−1.10312	+	0.271894i
$689$		0			0
$690$		0			0
$691$	0.136945	+	0.198399i	0.136945	+	0.198399i	0.885456	−	0.464723i	$-0.153846\pi$
$691$	0.136945	+	0.198399i	0.136945	+	0.198399i	−0.748511	+	0.663123i	$0.769231\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.402877	+	0.583668i	−0.402877	+	0.583668i
$701$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$701$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$702$		0			0
$703$	−1.50599	−	0.790406i	−1.50599	−	0.790406i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.627974	+	1.65583i	−0.627974	+	1.65583i	0.120537	+	0.992709i	$0.461538\pi$
$709$	−0.627974	+	1.65583i	−0.627974	+	1.65583i	−0.748511	+	0.663123i	$0.769231\pi$
$710$		0			0
$711$	0.530851	+	0.470293i	0.530851	+	0.470293i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$719$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$720$		0			0
$721$	0.782340	−	1.13342i	0.782340	−	1.13342i
$722$		0			0
$723$	0.0290582	+	0.239316i	0.0290582	+	0.239316i
$724$	0.213460	+	1.75800i	0.213460	+	1.75800i
$725$		0			0
$726$		0			0
$727$	0.645395	−	0.935016i	0.645395	−	0.935016i	−0.354605	−	0.935016i	$-0.615385\pi$
$727$	0.645395	−	0.935016i	0.645395	−	0.935016i	1.00000			$0$
$728$		0			0
$729$	0.568065	+	0.822984i	0.568065	+	0.822984i
$730$		0			0
$731$		0			0
$732$	−0.0854858	+	0.225408i	−0.0854858	+	0.225408i
$733$	−0.402877	−	1.06230i	−0.402877	−	1.06230i	−0.970942	−	0.239316i	$-0.923077\pi$
$733$	−0.402877	−	1.06230i	−0.402877	−	1.06230i	0.568065	−	0.822984i	$-0.307692\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.234068	−	0.0576926i	−0.234068	−	0.0576926i	0.120537	−	0.992709i	$-0.461538\pi$
$739$	−0.234068	−	0.0576926i	−0.234068	−	0.0576926i	−0.354605	+	0.935016i	$0.615385\pi$
$740$		0			0
$741$	0.530851	+	1.39974i	0.530851	+	1.39974i
$742$		0			0
$743$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$743$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.45352	+	1.28771i	1.45352	+	1.28771i	0.885456	+	0.464723i	$0.153846\pi$
$751$	1.45352	+	1.28771i	1.45352	+	1.28771i	0.568065	+	0.822984i	$0.307692\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.709210			−0.709210
$757$	0.136945	−	1.12785i	0.136945	−	1.12785i	−0.748511	−	0.663123i	$-0.769231\pi$
$757$	0.136945	−	1.12785i	0.136945	−	1.12785i	0.885456	−	0.464723i	$-0.153846\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$761$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$762$		0			0
$763$	1.21945	+	0.640018i	1.21945	+	0.640018i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.120537	−	0.992709i	0.120537	−	0.992709i
$769$	0.213460	−	1.75800i	0.213460	−	1.75800i	−0.354605	−	0.935016i	$-0.615385\pi$
$769$	0.213460	−	1.75800i	0.213460	−	1.75800i	0.568065	−	0.822984i	$-0.307692\pi$
$770$		0			0
$771$		0			0
$772$	−0.850405	−	0.753393i	−0.850405	−	0.753393i
$773$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$773$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$774$		0			0
$775$	0.241073			0.241073
$776$		0			0
$777$	−0.713460	+	0.374453i	−0.713460	+	0.374453i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.282340	+	0.409041i	−0.282340	+	0.409041i
$785$		0			0
$786$		0			0
$787$	−0.234068	+	1.92773i	−0.234068	+	1.92773i	0.120537	+	0.992709i	$0.461538\pi$
$787$	−0.234068	+	1.92773i	−0.234068	+	1.92773i	−0.354605	+	0.935016i	$0.615385\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.136945	−	0.198399i	0.136945	−	0.198399i
$794$		0			0
$795$		0			0
$796$	0.530851	−	1.39974i	0.530851	−	1.39974i
$797$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$797$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−0.627974	−	1.65583i	−0.627974	−	1.65583i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$809$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$810$		0			0
$811$	1.45352	+	0.358261i	1.45352	+	0.358261i	0.885456	−	0.464723i	$-0.153846\pi$
$811$	1.45352	+	0.358261i	1.45352	+	0.358261i	0.568065	+	0.822984i	$0.307692\pi$
$812$		0			0
$813$	0.213460	−	0.112032i	0.213460	−	0.112032i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.50599	+	0.790406i	−1.50599	+	0.790406i
$818$		0			0
$819$	0.688601	+	0.169725i	0.688601	+	0.169725i
$820$		0			0
$821$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$821$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$822$		0			0
$823$	−1.49702			−1.49702			−0.748511	−	0.663123i	$-0.769231\pi$
$823$	−1.49702			−1.49702			−0.748511	+	0.663123i	$0.769231\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$827$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$828$		0			0
$829$	1.88546	+	0.464723i	1.88546	+	0.464723i	1.00000			$0$
$829$	1.88546	+	0.464723i	1.88546	+	0.464723i	0.885456	+	0.464723i	$0.153846\pi$
$830$		0			0
$831$	0.688601	+	0.169725i	0.688601	+	0.169725i
$832$	−0.354605	+	0.935016i	−0.354605	+	0.935016i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.136945	+	0.198399i	0.136945	+	0.198399i
$838$		0			0
$839$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$839$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$840$		0			0
$841$	0.120537	−	0.992709i	0.120537	−	0.992709i
$842$		0			0
$843$		0			0
$844$	1.77091			1.77091
$845$		0			0
$846$		0			0
$847$	0.530851	−	0.470293i	0.530851	−	0.470293i
$848$		0			0
$849$	−0.234068	+	1.92773i	−0.234068	+	1.92773i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.10312	−	1.59814i	−1.10312	−	1.59814i	−0.748511	−	0.663123i	$-0.769231\pi$
$853$	−1.10312	−	1.59814i	−1.10312	−	1.59814i	−0.354605	−	0.935016i	$-0.615385\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$857$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$858$		0			0
$859$	0.688601	+	0.169725i	0.688601	+	0.169725i	0.568065	−	0.822984i	$-0.307692\pi$
$859$	0.688601	+	0.169725i	0.688601	+	0.169725i	0.120537	+	0.992709i	$0.461538\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$863$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1.00000			1.00000
$868$	−0.170972			−0.170972
$869$		0			0
$870$		0			0
$871$	0.213460	+	1.75800i	0.213460	+	1.75800i
$872$		0			0
$873$	1.77091	−	0.929446i	1.77091	−	0.929446i
$874$		0			0
$875$		0			0
$876$	−1.71945	−	0.902438i	−1.71945	−	0.902438i
$877$	1.00599	−	0.527986i	1.00599	−	0.527986i	0.120537	−	0.992709i	$-0.461538\pi$
$877$	1.00599	−	0.527986i	1.00599	−	0.527986i	0.885456	+	0.464723i	$0.153846\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$881$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$882$		0			0
$883$	−0.850405	+	0.753393i	−0.850405	+	0.753393i	−0.970942	−	0.239316i	$-0.923077\pi$
$883$	−0.850405	+	0.753393i	−0.850405	+	0.753393i	0.120537	+	0.992709i	$0.461538\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$887$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$888$		0			0
$889$	−1.03085	+	0.913254i	−1.03085	+	0.913254i
$890$		0			0
$891$		0			0
$892$	0.241073	+	1.98542i	0.241073	+	1.98542i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.885456	+	0.464723i	0.885456	+	0.464723i
$901$		0			0
$902$		0			0
$903$	−0.0971229	+	0.799879i	−0.0971229	+	0.799879i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.71945	+	0.423807i	−1.71945	+	0.423807i	−0.970942	−	0.239316i	$-0.923077\pi$
$907$	−1.71945	+	0.423807i	−1.71945	+	0.423807i	−0.748511	+	0.663123i	$0.769231\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$911$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$912$	−0.180446	−	1.48611i	−0.180446	−	1.48611i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.688601	+	1.81569i	0.688601	+	1.81569i
$917$		0			0
$918$		0			0
$919$	−1.71945	+	0.423807i	−1.71945	+	0.423807i	−0.970942	−	0.239316i	$-0.923077\pi$
$919$	−1.71945	+	0.423807i	−1.71945	+	0.423807i	−0.748511	+	0.663123i	$0.769231\pi$
$920$		0			0
$921$	0.136945	−	1.12785i	0.136945	−	1.12785i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.13613			1.13613
$926$		0			0
$927$	−1.71945	−	0.902438i	−1.71945	−	0.902438i
$928$		0			0
$929$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$929$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$930$		0			0
$931$	−0.263844	+	0.695701i	−0.263844	+	0.695701i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.180446	+	1.48611i	−0.180446	+	1.48611i	0.568065	+	0.822984i	$0.307692\pi$
$937$	−0.180446	+	1.48611i	−0.180446	+	1.48611i	−0.748511	+	0.663123i	$0.769231\pi$
$938$		0			0
$939$	0.530851	+	0.470293i	0.530851	+	0.470293i
$940$		0			0
$941$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$941$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$947$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$948$	0.251489	+	0.663123i	0.251489	+	0.663123i
$949$	1.45352	+	1.28771i	1.45352	+	1.28771i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$953$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.535051	−	0.775155i	−0.535051	−	0.775155i
$962$		0			0
$963$		0			0
$964$	−0.0854858	+	0.225408i	−0.0854858	+	0.225408i
$965$		0			0
$966$		0			0
$967$	0.0290582	+	0.239316i	0.0290582	+	0.239316i	1.00000			$0$
$967$	0.0290582	+	0.239316i	0.0290582	+	0.239316i	−0.970942	+	0.239316i	$0.923077\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$971$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$972$	0.120537	+	0.992709i	0.120537	+	0.992709i
$973$	0.445365	+	1.17433i	0.445365	+	1.17433i
$974$		0			0
$975$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$976$	−0.180446	+	0.159861i	−0.180446	+	0.159861i
$977$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$977$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.688601	−	1.81569i	0.688601	−	1.81569i
$982$		0			0
$983$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$983$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.180446	+	1.48611i	−0.180446	+	1.48611i
$989$		0			0
$990$		0			0
$991$	−1.49702			−1.49702			−0.748511	−	0.663123i	$-0.769231\pi$
$991$	−1.49702			−1.49702			−0.748511	+	0.663123i	$0.769231\pi$
$992$		0			0
$993$	1.00599	−	1.45743i	1.00599	−	1.45743i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.00599	+	1.45743i	1.00599	+	1.45743i	0.885456	+	0.464723i	$0.153846\pi$
$997$	1.00599	+	1.45743i	1.00599	+	1.45743i	0.120537	+	0.992709i	$0.461538\pi$
$998$		0			0
$999$	0.645395	+	0.935016i	0.645395	+	0.935016i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.1.o.a.53.1	✓	12
3.2	odd	2	CM	507.1.o.a.53.1	✓	12
169.118	even	13	inner	507.1.o.a.287.1	yes	12
507.287	odd	26	inner	507.1.o.a.287.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.1.o.a.53.1	✓	12	1.1	even	1	trivial
507.1.o.a.53.1	✓	12	3.2	odd	2	CM
507.1.o.a.287.1	yes	12	169.118	even	13	inner
507.1.o.a.287.1	yes	12	507.287	odd	26	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	507.o (of order \(26\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.354605 - 0.935016i) q^{3} +(0.120537 - 0.992709i) q^{4} +(-0.402877 - 0.583668i) q^{7} +(-0.748511 + 0.663123i) q^{9} +O(q^{10})\)	\(q+(-0.354605 - 0.935016i) q^{3} +(0.120537 - 0.992709i) q^{4} +(-0.402877 - 0.583668i) q^{7} +(-0.748511 + 0.663123i) q^{9} +(-0.970942 + 0.239316i) q^{12} +1.00000 q^{13} +(-0.970942 - 0.239316i) q^{16} -1.49702 q^{19} +(-0.402877 + 0.583668i) q^{21} +(0.885456 - 0.464723i) q^{25} +(0.885456 + 0.464723i) q^{27} +(-0.627974 + 0.329586i) q^{28} +(0.213460 + 0.112032i) q^{31} +(0.568065 + 0.822984i) q^{36} +(1.00599 + 0.527986i) q^{37} +(-0.354605 - 0.935016i) q^{39} +(1.00599 - 0.527986i) q^{43} +(0.120537 + 0.992709i) q^{48} +(0.176246 - 0.464723i) q^{49} +(0.120537 - 0.992709i) q^{52} +(0.530851 + 1.39974i) q^{57} +(0.136945 - 0.198399i) q^{61} +(0.688601 + 0.169725i) q^{63} +(-0.354605 + 0.935016i) q^{64} +(0.213460 + 1.75800i) q^{67} +(1.45352 + 1.28771i) q^{73} +(-0.748511 - 0.663123i) q^{75} +(-0.180446 + 1.48611i) q^{76} +(-0.0854858 - 0.704039i) q^{79} +(0.120537 - 0.992709i) q^{81} +(0.530851 + 0.470293i) q^{84} +(-0.402877 - 0.583668i) q^{91} +(0.0290582 - 0.239316i) q^{93} +(-1.94188 - 0.478631i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} - q^{4} - 2 q^{7} - q^{9}+O(q^{10}) \) 12 * q - q^3 - q^4 - 2 * q^7 - q^9	\( 12 q - q^{3} - q^{4} - 2 q^{7} - q^{9} - q^{12} + 12 q^{13} - q^{16} - 2 q^{19} - 2 q^{21} - q^{25} - q^{27} - 2 q^{28} - 2 q^{31} - q^{36} - 2 q^{37} - q^{39} - 2 q^{43} - q^{48} - 3 q^{49} - q^{52} - 2 q^{57} - 2 q^{61} - 2 q^{63} - q^{64} - 2 q^{67} - 2 q^{73} - q^{75} - 2 q^{76} - 2 q^{79} - q^{81} - 2 q^{84} - 2 q^{91} + 11 q^{93} - 2 q^{97}+O(q^{100}) \) 12 * q - q^3 - q^4 - 2 * q^7 - q^9 - q^12 + 12 * q^13 - q^16 - 2 * q^19 - 2 * q^21 - q^25 - q^27 - 2 * q^28 - 2 * q^31 - q^36 - 2 * q^37 - q^39 - 2 * q^43 - q^48 - 3 * q^49 - q^52 - 2 * q^57 - 2 * q^61 - 2 * q^63 - q^64 - 2 * q^67 - 2 * q^73 - q^75 - 2 * q^76 - 2 * q^79 - q^81 - 2 * q^84 - 2 * q^91 + 11 * q^93 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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