LMFDB - Embedded newform 520.1.bo.a.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [520,1,Mod(109,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 2, 3]))

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

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

chi := DirichletCharacter("520.109");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 520 = 2^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	520.bo (of order $4$, degree $2$, 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:	$0.259513806569$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.0.439400.1

Embedding invariants

Embedding label			109.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	520.109
Dual form			520.1.bo.a.229.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{2} +1.41421 q^{3} -1.00000i q^{4} +(-0.707107 + 0.707107i) q^{5} +(-1.00000 + 1.00000i) q^{6} +(0.707107 + 0.707107i) q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+(-0.707107 + 0.707107i) q^{2} +1.41421 q^{3} -1.00000i q^{4} +(-0.707107 + 0.707107i) q^{5} +(-1.00000 + 1.00000i) q^{6} +(0.707107 + 0.707107i) q^{8} +1.00000 q^{9} -1.00000i q^{10} -1.41421i q^{12} +(0.707107 + 0.707107i) q^{13} +(-1.00000 + 1.00000i) q^{15} -1.00000 q^{16} +(-0.707107 + 0.707107i) q^{18} +(0.707107 + 0.707107i) q^{20} +(1.00000 + 1.00000i) q^{24} -1.00000i q^{25} -1.00000 q^{26} -1.41421i q^{30} +(1.00000 - 1.00000i) q^{31} +(0.707107 - 0.707107i) q^{32} -1.00000i q^{36} +(-1.41421 - 1.41421i) q^{37} +(1.00000 + 1.00000i) q^{39} -1.00000 q^{40} +(-1.00000 + 1.00000i) q^{41} -1.41421i q^{43} +(-0.707107 + 0.707107i) q^{45} -1.41421 q^{48} +1.00000i q^{49} +(0.707107 + 0.707107i) q^{50} +(0.707107 - 0.707107i) q^{52} -1.41421 q^{53} +(1.00000 + 1.00000i) q^{60} +1.41421i q^{62} +1.00000i q^{64} -1.00000 q^{65} +(-1.41421 + 1.41421i) q^{67} +(1.00000 - 1.00000i) q^{71} +(0.707107 + 0.707107i) q^{72} +2.00000 q^{74} -1.41421i q^{75} -1.41421 q^{78} +(0.707107 - 0.707107i) q^{80} -1.00000 q^{81} -1.41421i q^{82} +(1.41421 - 1.41421i) q^{83} +(1.00000 + 1.00000i) q^{86} +(-1.00000 - 1.00000i) q^{89} -1.00000i q^{90} +(1.41421 - 1.41421i) q^{93} +(1.00000 - 1.00000i) q^{96} +(-0.707107 - 0.707107i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{6} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^6 + 4 * q^9	$ 4 q - 4 q^{6} + 4 q^{9} - 4 q^{15} - 4 q^{16} + 4 q^{24} - 4 q^{26} + 4 q^{31} + 4 q^{39} - 4 q^{40} - 4 q^{41} + 4 q^{60} - 4 q^{65} + 4 q^{71} + 8 q^{74} - 4 q^{81} + 4 q^{86} - 4 q^{89} + 4 q^{96}+O(q^{100}) $ 4 * q - 4 * q^6 + 4 * q^9 - 4 * q^15 - 4 * q^16 + 4 * q^24 - 4 * q^26 + 4 * q^31 + 4 * q^39 - 4 * q^40 - 4 * q^41 + 4 * q^60 - 4 * q^65 + 4 * q^71 + 8 * q^74 - 4 * q^81 + 4 * q^86 - 4 * q^89 + 4 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/520\mathbb{Z}\right)^\times$.

$n$	$41$	$261$	$391$	$417$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$2$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$3$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$3$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$4$		−	1.00000i		−	1.00000i
$5$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$5$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$6$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	0.707107	+	0.707107i	0.707107	+	0.707107i
$9$	1.00000			1.00000
$10$		−	1.00000i		−	1.00000i
$11$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$11$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$12$		−	1.41421i		−	1.41421i
$13$	0.707107	+	0.707107i	0.707107	+	0.707107i
$13$	0.707107	+	0.707107i	0.707107	+	0.707107i
$14$		0			0
$15$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$16$	−1.00000			−1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$	0.707107	+	0.707107i	0.707107	+	0.707107i
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	1.00000	+	1.00000i	1.00000	+	1.00000i
$25$		−	1.00000i		−	1.00000i
$26$	−1.00000			−1.00000
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		−	1.41421i		−	1.41421i
$31$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$31$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$32$	0.707107	−	0.707107i	0.707107	−	0.707107i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		−	1.00000i		−	1.00000i
$37$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$37$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$38$		0			0
$39$	1.00000	+	1.00000i	1.00000	+	1.00000i
$40$	−1.00000			−1.00000
$41$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$41$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$42$		0			0
$43$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$43$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$44$		0			0
$45$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$	−1.41421			−1.41421
$49$			1.00000i			1.00000i
$50$	0.707107	+	0.707107i	0.707107	+	0.707107i
$51$		0			0
$52$	0.707107	−	0.707107i	0.707107	−	0.707107i
$53$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$53$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$	1.00000	+	1.00000i	1.00000	+	1.00000i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$			1.41421i			1.41421i
$63$		0			0
$64$			1.00000i			1.00000i
$65$	−1.00000			−1.00000
$66$		0			0
$67$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$67$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$71$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$72$	0.707107	+	0.707107i	0.707107	+	0.707107i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$	2.00000			2.00000
$75$		−	1.41421i		−	1.41421i
$76$		0			0
$77$		0			0
$78$	−1.41421			−1.41421
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	0.707107	−	0.707107i	0.707107	−	0.707107i
$81$	−1.00000			−1.00000
$82$		−	1.41421i		−	1.41421i
$83$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$83$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$84$		0			0
$85$		0			0
$86$	1.00000	+	1.00000i	1.00000	+	1.00000i
$87$		0			0
$88$		0			0
$89$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$89$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$90$		−	1.00000i		−	1.00000i
$91$		0			0
$92$		0			0
$93$	1.41421	−	1.41421i	1.41421	−	1.41421i
$94$		0			0
$95$		0			0
$96$	1.00000	−	1.00000i	1.00000	−	1.00000i
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$99$		0			0
$100$	−1.00000			−1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$			1.00000i			1.00000i
$105$		0			0
$106$	1.00000	−	1.00000i	1.00000	−	1.00000i
$107$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$107$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$		0			0
$111$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.707107	+	0.707107i	0.707107	+	0.707107i
$118$		0			0
$119$		0			0
$120$	−1.41421			−1.41421
$121$			1.00000i			1.00000i
$122$		0			0
$123$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$124$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$125$	0.707107	+	0.707107i	0.707107	+	0.707107i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$129$		−	2.00000i		−	2.00000i
$130$	0.707107	−	0.707107i	0.707107	−	0.707107i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		−	2.00000i		−	2.00000i
$135$		0			0
$136$		0			0
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$			1.41421i			1.41421i
$143$		0			0
$144$	−1.00000			−1.00000
$145$		0			0
$146$		0			0
$147$			1.41421i			1.41421i
$148$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$149$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$149$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$150$	1.00000	+	1.00000i	1.00000	+	1.00000i
$151$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$151$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			1.41421i			1.41421i
$156$	1.00000	−	1.00000i	1.00000	−	1.00000i
$157$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$157$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$	−2.00000			−2.00000
$160$			1.00000i			1.00000i
$161$		0			0
$162$	0.707107	−	0.707107i	0.707107	−	0.707107i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$	1.00000	+	1.00000i	1.00000	+	1.00000i
$165$		0			0
$166$			2.00000i			2.00000i
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$			1.00000i			1.00000i
$170$		0			0
$171$		0			0
$172$	−1.41421			−1.41421
$173$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$173$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	1.41421			1.41421
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	0.707107	+	0.707107i	0.707107	+	0.707107i
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.00000			2.00000
$186$			2.00000i			2.00000i
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.00000			−2.00000			−1.00000			$\pi$
$191$	−2.00000			−2.00000			−1.00000			$\pi$
$192$			1.41421i			1.41421i
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$	−1.41421			−1.41421
$196$	1.00000			1.00000
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	0.707107	−	0.707107i	0.707107	−	0.707107i
$201$	−2.00000	+	2.00000i	−2.00000	+	2.00000i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		−	1.41421i		−	1.41421i
$206$		0			0
$207$		0			0
$208$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			1.41421i			1.41421i
$213$	1.41421	−	1.41421i	1.41421	−	1.41421i
$214$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$215$	1.00000	+	1.00000i	1.00000	+	1.00000i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	2.82843			2.82843
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$		−	1.00000i		−	1.00000i
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$229$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	−1.00000			−1.00000
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$239$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$240$	1.00000	−	1.00000i	1.00000	−	1.00000i
$241$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$241$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$242$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$243$	−1.41421			−1.41421
$244$		0			0
$245$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$246$		−	2.00000i		−	2.00000i
$247$		0			0
$248$	1.41421			1.41421
$249$	2.00000	−	2.00000i	2.00000	−	2.00000i
$250$	−1.00000			−1.00000
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$	1.41421	+	1.41421i	1.41421	+	1.41421i
$259$		0			0
$260$			1.00000i			1.00000i
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	1.00000	−	1.00000i	1.00000	−	1.00000i
$266$		0			0
$267$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$268$	1.41421	+	1.41421i	1.41421	+	1.41421i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$271$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$277$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$278$		0			0
$279$	1.00000	−	1.00000i	1.00000	−	1.00000i
$280$		0			0
$281$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$281$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$282$		0			0
$283$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$283$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$284$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.707107	−	0.707107i	0.707107	−	0.707107i
$289$	−1.00000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$295$		0			0
$296$		−	2.00000i		−	2.00000i
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−1.41421			−1.41421
$301$		0			0
$302$	−1.41421			−1.41421
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$307$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$308$		0			0
$309$		0			0
$310$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$			1.41421i			1.41421i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$315$		0			0
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$	1.41421	−	1.41421i	1.41421	−	1.41421i
$319$		0			0
$320$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$321$	2.00000			2.00000
$322$		0			0
$323$		0			0
$324$			1.00000i			1.00000i
$325$	0.707107	−	0.707107i	0.707107	−	0.707107i
$326$		0			0
$327$		0			0
$328$	−1.41421			−1.41421
$329$		0			0
$330$		0			0
$331$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$331$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$332$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$333$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$334$		0			0
$335$		−	2.00000i		−	2.00000i
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	1.00000	−	1.00000i	1.00000	−	1.00000i
$345$		0			0
$346$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$347$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$347$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$349$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$			1.41421i			1.41421i
$356$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$357$		0			0
$358$		0			0
$359$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$359$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$360$	−1.00000			−1.00000
$361$		−	1.00000i		−	1.00000i
$362$		0			0
$363$			1.41421i			1.41421i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$370$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$371$		0			0
$372$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$373$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$373$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$	1.00000	+	1.00000i	1.00000	+	1.00000i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$379$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$380$		0			0
$381$		0			0
$382$	1.41421	−	1.41421i	1.41421	−	1.41421i
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$385$		0			0
$386$		0			0
$387$		−	1.41421i		−	1.41421i
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$	1.00000	−	1.00000i	1.00000	−	1.00000i
$391$		0			0
$392$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$		0			0
$400$			1.00000i			1.00000i
$401$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$401$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$402$		−	2.82843i		−	2.82843i
$403$	1.41421			1.41421
$404$		0			0
$405$	0.707107	−	0.707107i	0.707107	−	0.707107i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$409$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$410$	1.00000	+	1.00000i	1.00000	+	1.00000i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$			2.00000i			2.00000i
$416$	1.00000			1.00000
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$		0			0
$423$		0			0
$424$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$425$		0			0
$426$			2.00000i			2.00000i
$427$		0			0
$428$		−	1.41421i		−	1.41421i
$429$		0			0
$430$	−1.41421			−1.41421
$431$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$431$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$439$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$440$		0			0
$441$			1.00000i			1.00000i
$442$		0			0
$443$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$443$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$444$	−2.00000	+	2.00000i	−2.00000	+	2.00000i
$445$	1.41421			1.41421
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$449$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$450$	0.707107	+	0.707107i	0.707107	+	0.707107i
$451$		0			0
$452$		0			0
$453$	1.41421	+	1.41421i	1.41421	+	1.41421i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$461$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$			2.00000i			2.00000i
$466$		0			0
$467$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$467$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$468$	0.707107	−	0.707107i	0.707107	−	0.707107i
$469$		0			0
$470$		0			0
$471$	2.00000			2.00000
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.41421			−1.41421
$478$		−	1.41421i		−	1.41421i
$479$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$479$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$480$			1.41421i			1.41421i
$481$		−	2.00000i		−	2.00000i
$482$	−1.41421			−1.41421
$483$		0			0
$484$	1.00000			1.00000
$485$		0			0
$486$	1.00000	−	1.00000i	1.00000	−	1.00000i
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$		0			0
$490$	1.00000			1.00000
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$	1.41421	+	1.41421i	1.41421	+	1.41421i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$497$		0			0
$498$			2.82843i			2.82843i
$499$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$499$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$500$	0.707107	−	0.707107i	0.707107	−	0.707107i
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			1.41421i			1.41421i
$508$		0			0
$509$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$509$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$510$		0			0
$511$		0			0
$512$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$513$		0			0
$514$		0			0
$515$		0			0
$516$	−2.00000			−2.00000
$517$		0			0
$518$		0			0
$519$			2.00000i			2.00000i
$520$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$523$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1.00000			−1.00000
$530$			1.41421i			1.41421i
$531$		0			0
$532$		0			0
$533$	−1.41421			−1.41421
$534$	2.00000			2.00000
$535$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$536$	−2.00000			−2.00000
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$541$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$542$	−1.41421			−1.41421
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$547$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$555$	2.82843			2.82843
$556$		0			0
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$			1.41421i			1.41421i
$559$	1.00000	−	1.00000i	1.00000	−	1.00000i
$560$		0			0
$561$		0			0
$562$	1.41421			1.41421
$563$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$563$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$564$		0			0
$565$		0			0
$566$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$567$		0			0
$568$	1.41421			1.41421
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$	−2.82843			−2.82843
$574$		0			0
$575$		0			0
$576$			1.00000i			1.00000i
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	0.707107	−	0.707107i	0.707107	−	0.707107i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−1.00000			−1.00000
$586$		0			0
$587$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$587$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$588$	1.41421			1.41421
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.41421	+	1.41421i	1.41421	+	1.41421i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2.00000			−2.00000			−1.00000			$\pi$
$599$	−2.00000			−2.00000			−1.00000			$\pi$
$600$	1.00000	−	1.00000i	1.00000	−	1.00000i
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$604$	1.00000	−	1.00000i	1.00000	−	1.00000i
$605$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$613$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$614$	2.00000			2.00000
$615$		−	2.00000i		−	2.00000i
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$619$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$620$	1.41421			1.41421
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$625$	−1.00000			−1.00000
$626$		0			0
$627$		0			0
$628$		−	1.41421i		−	1.41421i
$629$		0			0
$630$		0			0
$631$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$631$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$			2.00000i			2.00000i
$637$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$638$		0			0
$639$	1.00000	−	1.00000i	1.00000	−	1.00000i
$640$	1.00000			1.00000
$641$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$641$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$642$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$	1.41421	+	1.41421i	1.41421	+	1.41421i
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$649$		0			0
$650$			1.00000i			1.00000i
$651$		0			0
$652$		0			0
$653$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$653$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$	1.00000	−	1.00000i	1.00000	−	1.00000i
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$661$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$662$		0			0
$663$		0			0
$664$	2.00000			2.00000
$665$		0			0
$666$	2.00000			2.00000
$667$		0			0
$668$		0			0
$669$		0			0
$670$	1.41421	+	1.41421i	1.41421	+	1.41421i
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	1.00000			1.00000
$677$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$677$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$			1.41421i			1.41421i
$689$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$690$		0			0
$691$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$691$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$692$	1.41421			1.41421
$693$		0			0
$694$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$709$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$710$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$711$		0			0
$712$		−	1.41421i		−	1.41421i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$718$	−1.41421			−1.41421
$719$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$719$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$720$	0.707107	−	0.707107i	0.707107	−	0.707107i
$721$		0			0
$722$	0.707107	+	0.707107i	0.707107	+	0.707107i
$723$	1.41421	+	1.41421i	1.41421	+	1.41421i
$724$		0			0
$725$		0			0
$726$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−1.00000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$736$		0			0
$737$		0			0
$738$		−	1.41421i		−	1.41421i
$739$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$739$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$740$		−	2.00000i		−	2.00000i
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$	2.00000			2.00000
$745$		0			0
$746$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$747$	1.41421	−	1.41421i	1.41421	−	1.41421i
$748$		0			0
$749$		0			0
$750$	−1.41421			−1.41421
$751$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$751$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.41421			−1.41421
$756$		0			0
$757$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$757$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$761$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$			2.00000i			2.00000i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.41421			1.41421
$769$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$769$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$	1.00000	+	1.00000i	1.00000	+	1.00000i
$775$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$			1.41421i			1.41421i
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	1.00000i		−	1.00000i
$785$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$786$		0			0
$787$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$787$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	1.41421	−	1.41421i	1.41421	−	1.41421i
$796$		0			0
$797$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$797$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$798$		0			0
$799$		0			0
$800$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$801$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$802$	−1.41421			−1.41421
$803$		0			0
$804$	2.00000	+	2.00000i	2.00000	+	2.00000i
$805$		0			0
$806$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$807$		0			0
$808$		0			0
$809$	2.00000			2.00000			1.00000			$0$
$809$	2.00000			2.00000			1.00000			$0$
$810$			1.00000i			1.00000i
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$	1.41421	+	1.41421i	1.41421	+	1.41421i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		−	1.41421i		−	1.41421i
$819$		0			0
$820$	−1.41421			−1.41421
$821$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$821$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$831$			2.00000i			2.00000i
$832$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$839$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$		0			0
$843$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$844$		0			0
$845$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$846$		0			0
$847$		0			0
$848$	1.41421			1.41421
$849$			2.00000i			2.00000i
$850$		0			0
$851$		0			0
$852$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$853$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$853$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$854$		0			0
$855$		0			0
$856$	1.00000	+	1.00000i	1.00000	+	1.00000i
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$	1.00000	−	1.00000i	1.00000	−	1.00000i
$861$		0			0
$862$			1.41421i			1.41421i
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$866$		0			0
$867$	−1.41421			−1.41421
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−2.00000			−2.00000
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$	1.41421	+	1.41421i	1.41421	+	1.41421i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$883$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$883$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$884$		0			0
$885$		0			0
$886$	1.00000	−	1.00000i	1.00000	−	1.00000i
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		−	2.82843i		−	2.82843i
$889$		0			0
$890$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.41421			−1.41421
$899$		0			0
$900$	−1.00000			−1.00000
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−2.00000			−2.00000
$907$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$907$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−2.00000			−2.00000			−1.00000			$\pi$
$911$	−2.00000			−2.00000			−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−2.00000			−2.00000			−1.00000			$\pi$
$919$	−2.00000			−2.00000			−1.00000			$\pi$
$920$		0			0
$921$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$922$		0			0
$923$	1.41421			1.41421
$924$		0			0
$925$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$929$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$930$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$931$		0			0
$932$		0			0
$933$		0			0
$934$	1.00000	+	1.00000i	1.00000	+	1.00000i
$935$		0			0
$936$			1.00000i			1.00000i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$	1.00000	−	1.00000i	1.00000	−	1.00000i
$955$	1.41421	−	1.41421i	1.41421	−	1.41421i
$956$	1.00000	+	1.00000i	1.00000	+	1.00000i
$957$		0			0
$958$	1.41421			1.41421
$959$		0			0
$960$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$961$		−	1.00000i		−	1.00000i
$962$	1.41421	+	1.41421i	1.41421	+	1.41421i
$963$	1.41421			1.41421
$964$	1.00000	−	1.00000i	1.00000	−	1.00000i
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$			1.41421i			1.41421i
$973$		0			0
$974$		0			0
$975$	1.00000	−	1.00000i	1.00000	−	1.00000i
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$981$		0			0
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$	−2.00000			−2.00000
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		−	1.41421i		−	1.41421i
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$997$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$997$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	520.1.bo.a.109.1	✓	4
4.3	odd	2		2080.1.cs.a.369.1		4
5.2	odd	4		2600.1.r.a.2501.1		4
5.3	odd	4		2600.1.r.a.2501.2		4
5.4	even	2	inner	520.1.bo.a.109.2	yes	4
8.3	odd	2		2080.1.cs.a.369.2		4
8.5	even	2	inner	520.1.bo.a.109.2	yes	4
13.8	odd	4	inner	520.1.bo.a.229.1	yes	4
20.19	odd	2		2080.1.cs.a.369.2		4
40.13	odd	4		2600.1.r.a.2501.1		4
40.19	odd	2		2080.1.cs.a.369.1		4
40.29	even	2	RM	520.1.bo.a.109.1	✓	4
40.37	odd	4		2600.1.r.a.2501.2		4
52.47	even	4		2080.1.cs.a.1009.1		4
65.8	even	4		2600.1.r.a.2101.1		4
65.34	odd	4	inner	520.1.bo.a.229.2	yes	4
65.47	even	4		2600.1.r.a.2101.2		4
104.21	odd	4	inner	520.1.bo.a.229.2	yes	4
104.99	even	4		2080.1.cs.a.1009.2		4
260.99	even	4		2080.1.cs.a.1009.2		4
520.99	even	4		2080.1.cs.a.1009.1		4
520.229	odd	4	inner	520.1.bo.a.229.1	yes	4
520.333	even	4		2600.1.r.a.2101.2		4
520.437	even	4		2600.1.r.a.2101.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.1.bo.a.109.1	✓	4	1.1	even	1	trivial
520.1.bo.a.109.1	✓	4	40.29	even	2	RM
520.1.bo.a.109.2	yes	4	5.4	even	2	inner
520.1.bo.a.109.2	yes	4	8.5	even	2	inner
520.1.bo.a.229.1	yes	4	13.8	odd	4	inner
520.1.bo.a.229.1	yes	4	520.229	odd	4	inner
520.1.bo.a.229.2	yes	4	65.34	odd	4	inner
520.1.bo.a.229.2	yes	4	104.21	odd	4	inner
2080.1.cs.a.369.1		4	4.3	odd	2
2080.1.cs.a.369.1		4	40.19	odd	2
2080.1.cs.a.369.2		4	8.3	odd	2
2080.1.cs.a.369.2		4	20.19	odd	2
2080.1.cs.a.1009.1		4	52.47	even	4
2080.1.cs.a.1009.1		4	520.99	even	4
2080.1.cs.a.1009.2		4	104.99	even	4
2080.1.cs.a.1009.2		4	260.99	even	4
2600.1.r.a.2101.1		4	65.8	even	4
2600.1.r.a.2101.1		4	520.437	even	4
2600.1.r.a.2101.2		4	65.47	even	4
2600.1.r.a.2101.2		4	520.333	even	4
2600.1.r.a.2501.1		4	5.2	odd	4
2600.1.r.a.2501.1		4	40.13	odd	4
2600.1.r.a.2501.2		4	5.3	odd	4
2600.1.r.a.2501.2		4	40.37	odd	4

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:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	520.bo (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{2} +1.41421 q^{3} -1.00000i q^{4} +(-0.707107 + 0.707107i) q^{5} +(-1.00000 + 1.00000i) q^{6} +(0.707107 + 0.707107i) q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{2} +1.41421 q^{3} -1.00000i q^{4} +(-0.707107 + 0.707107i) q^{5} +(-1.00000 + 1.00000i) q^{6} +(0.707107 + 0.707107i) q^{8} +1.00000 q^{9} -1.00000i q^{10} -1.41421i q^{12} +(0.707107 + 0.707107i) q^{13} +(-1.00000 + 1.00000i) q^{15} -1.00000 q^{16} +(-0.707107 + 0.707107i) q^{18} +(0.707107 + 0.707107i) q^{20} +(1.00000 + 1.00000i) q^{24} -1.00000i q^{25} -1.00000 q^{26} -1.41421i q^{30} +(1.00000 - 1.00000i) q^{31} +(0.707107 - 0.707107i) q^{32} -1.00000i q^{36} +(-1.41421 - 1.41421i) q^{37} +(1.00000 + 1.00000i) q^{39} -1.00000 q^{40} +(-1.00000 + 1.00000i) q^{41} -1.41421i q^{43} +(-0.707107 + 0.707107i) q^{45} -1.41421 q^{48} +1.00000i q^{49} +(0.707107 + 0.707107i) q^{50} +(0.707107 - 0.707107i) q^{52} -1.41421 q^{53} +(1.00000 + 1.00000i) q^{60} +1.41421i q^{62} +1.00000i q^{64} -1.00000 q^{65} +(-1.41421 + 1.41421i) q^{67} +(1.00000 - 1.00000i) q^{71} +(0.707107 + 0.707107i) q^{72} +2.00000 q^{74} -1.41421i q^{75} -1.41421 q^{78} +(0.707107 - 0.707107i) q^{80} -1.00000 q^{81} -1.41421i q^{82} +(1.41421 - 1.41421i) q^{83} +(1.00000 + 1.00000i) q^{86} +(-1.00000 - 1.00000i) q^{89} -1.00000i q^{90} +(1.41421 - 1.41421i) q^{93} +(1.00000 - 1.00000i) q^{96} +(-0.707107 - 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{6} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^6 + 4 * q^9	\( 4 q - 4 q^{6} + 4 q^{9} - 4 q^{15} - 4 q^{16} + 4 q^{24} - 4 q^{26} + 4 q^{31} + 4 q^{39} - 4 q^{40} - 4 q^{41} + 4 q^{60} - 4 q^{65} + 4 q^{71} + 8 q^{74} - 4 q^{81} + 4 q^{86} - 4 q^{89} + 4 q^{96}+O(q^{100}) \) 4 * q - 4 * q^6 + 4 * q^9 - 4 * q^15 - 4 * q^16 + 4 * q^24 - 4 * q^26 + 4 * q^31 + 4 * q^39 - 4 * q^40 - 4 * q^41 + 4 * q^60 - 4 * q^65 + 4 * q^71 + 8 * q^74 - 4 * q^81 + 4 * q^86 - 4 * q^89 + 4 * q^96

Introduction

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