LMFDB - Embedded newform 3640.1.cl.e.1357.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,1,Mod(853,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 3, 2, 1]))

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

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

chi := DirichletCharacter("3640.853");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3640.cl (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:	$1.81659664598$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.123032000.2

Embedding invariants

Embedding label			1357.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3640.1357
Dual form			3640.1.cl.e.853.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +(-1.41421 + 1.41421i) q^{3} +1.00000 q^{4} +(0.707107 - 0.707107i) q^{5} +(1.41421 - 1.41421i) q^{6} -1.00000i q^{7} -1.00000 q^{8} -3.00000i q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +(-1.41421 + 1.41421i) q^{3} +1.00000 q^{4} +(0.707107 - 0.707107i) q^{5} +(1.41421 - 1.41421i) q^{6} -1.00000i q^{7} -1.00000 q^{8} -3.00000i q^{9} +(-0.707107 + 0.707107i) q^{10} +(-1.41421 + 1.41421i) q^{12} +(-0.707107 - 0.707107i) q^{13} +1.00000i q^{14} +2.00000i q^{15} +1.00000 q^{16} +3.00000i q^{18} +(0.707107 - 0.707107i) q^{20} +(1.41421 + 1.41421i) q^{21} +(-1.00000 - 1.00000i) q^{23} +(1.41421 - 1.41421i) q^{24} -1.00000i q^{25} +(0.707107 + 0.707107i) q^{26} +(2.82843 + 2.82843i) q^{27} -1.00000i q^{28} -2.00000i q^{30} -1.00000 q^{32} +(-0.707107 - 0.707107i) q^{35} -3.00000i q^{36} +2.00000 q^{39} +(-0.707107 + 0.707107i) q^{40} +(-1.41421 - 1.41421i) q^{42} +(-2.12132 - 2.12132i) q^{45} +(1.00000 + 1.00000i) q^{46} +(-1.41421 + 1.41421i) q^{48} -1.00000 q^{49} +1.00000i q^{50} +(-0.707107 - 0.707107i) q^{52} +(-2.82843 - 2.82843i) q^{54} +1.00000i q^{56} +(-1.41421 + 1.41421i) q^{59} +2.00000i q^{60} -1.41421 q^{61} -3.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} +2.82843 q^{69} +(0.707107 + 0.707107i) q^{70} +(-1.00000 + 1.00000i) q^{71} +3.00000i q^{72} +(1.41421 + 1.41421i) q^{75} -2.00000 q^{78} +(0.707107 - 0.707107i) q^{80} -5.00000 q^{81} +1.41421i q^{83} +(1.41421 + 1.41421i) q^{84} +(2.12132 + 2.12132i) q^{90} +(-0.707107 + 0.707107i) q^{91} +(-1.00000 - 1.00000i) q^{92} +(1.41421 - 1.41421i) q^{96} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{16} - 4 q^{23} - 4 q^{32} + 8 q^{39} + 4 q^{46} - 4 q^{49} - 12 q^{63} + 4 q^{64} - 4 q^{65} - 4 q^{71} - 8 q^{78} - 20 q^{81} - 4 q^{92} + 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^16 - 4 * q^23 - 4 * q^32 + 8 * q^39 + 4 * q^46 - 4 * q^49 - 12 * q^63 + 4 * q^64 - 4 * q^65 - 4 * q^71 - 8 * q^78 - 20 * q^81 - 4 * q^92 + 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3640\mathbb{Z}\right)^\times$.

$n$	$521$	$561$	$911$	$1457$	$1821$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$	$e\left(\frac{1}{4}\right)$	$-1$

Coefficient data

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

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

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

$2$	−1.00000			−1.00000
$2$	−1.00000			−1.00000
$3$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$3$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$4$	1.00000			1.00000
$5$	0.707107	−	0.707107i	0.707107	−	0.707107i
$5$	0.707107	−	0.707107i	0.707107	−	0.707107i
$6$	1.41421	−	1.41421i	1.41421	−	1.41421i
$7$		−	1.00000i		−	1.00000i
$7$		−	1.00000i		−	1.00000i
$8$	−1.00000			−1.00000
$9$		−	3.00000i		−	3.00000i
$10$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$11$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$11$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$12$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$13$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$13$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$14$			1.00000i			1.00000i
$15$			2.00000i			2.00000i
$16$	1.00000			1.00000
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$			3.00000i			3.00000i
$19$		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
$21$	1.41421	+	1.41421i	1.41421	+	1.41421i
$22$		0			0
$23$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$23$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$24$	1.41421	−	1.41421i	1.41421	−	1.41421i
$25$		−	1.00000i		−	1.00000i
$26$	0.707107	+	0.707107i	0.707107	+	0.707107i
$27$	2.82843	+	2.82843i	2.82843	+	2.82843i
$28$		−	1.00000i		−	1.00000i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		−	2.00000i		−	2.00000i
$31$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$31$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$32$	−1.00000			−1.00000
$33$		0			0
$34$		0			0
$35$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$36$		−	3.00000i		−	3.00000i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$	2.00000			2.00000
$40$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$41$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$41$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$42$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	−2.12132	−	2.12132i	−2.12132	−	2.12132i
$46$	1.00000	+	1.00000i	1.00000	+	1.00000i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$49$	−1.00000			−1.00000
$50$			1.00000i			1.00000i
$51$		0			0
$52$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$	−2.82843	−	2.82843i	−2.82843	−	2.82843i
$55$		0			0
$56$			1.00000i			1.00000i
$57$		0			0
$58$		0			0
$59$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$59$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$60$			2.00000i			2.00000i
$61$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$61$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$62$		0			0
$63$	−3.00000			−3.00000
$64$	1.00000			1.00000
$65$	−1.00000			−1.00000
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	2.82843			2.82843
$70$	0.707107	+	0.707107i	0.707107	+	0.707107i
$71$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$71$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$72$			3.00000i			3.00000i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	1.41421	+	1.41421i	1.41421	+	1.41421i
$76$		0			0
$77$		0			0
$78$	−2.00000			−2.00000
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	0.707107	−	0.707107i	0.707107	−	0.707107i
$81$	−5.00000			−5.00000
$82$		0			0
$83$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$83$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$84$	1.41421	+	1.41421i	1.41421	+	1.41421i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$89$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$90$	2.12132	+	2.12132i	2.12132	+	2.12132i
$91$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$92$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	1.41421	−	1.41421i	1.41421	−	1.41421i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	1.00000			1.00000
$99$		0			0
$100$		−	1.00000i		−	1.00000i
$101$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$101$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$	0.707107	+	0.707107i	0.707107	+	0.707107i
$105$	2.00000			2.00000
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$	2.82843	+	2.82843i	2.82843	+	2.82843i
$109$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$109$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$110$		0			0
$111$		0			0
$112$		−	1.00000i		−	1.00000i
$113$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$113$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$114$		0			0
$115$	−1.41421			−1.41421
$116$		0			0
$117$	−2.12132	+	2.12132i	−2.12132	+	2.12132i
$118$	1.41421	−	1.41421i	1.41421	−	1.41421i
$119$		0			0
$120$		−	2.00000i		−	2.00000i
$121$			1.00000i			1.00000i
$122$	1.41421			1.41421
$123$		0			0
$124$		0			0
$125$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$126$	3.00000			3.00000
$127$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$127$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$128$	−1.00000			−1.00000
$129$		0			0
$130$	1.00000			1.00000
$131$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$131$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	4.00000			4.00000
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	−2.82843			−2.82843
$139$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$139$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$140$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$141$		0			0
$142$	1.00000	−	1.00000i	1.00000	−	1.00000i
$143$		0			0
$144$		−	3.00000i		−	3.00000i
$145$		0			0
$146$		0			0
$147$	1.41421	−	1.41421i	1.41421	−	1.41421i
$148$		0			0
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$151$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$151$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	2.00000			2.00000
$157$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$157$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$161$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$162$	5.00000			5.00000
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		−	1.41421i		−	1.41421i
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$169$			1.00000i			1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$	−1.00000			−1.00000
$176$		0			0
$177$		−	4.00000i		−	4.00000i
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	−2.12132	−	2.12132i	−2.12132	−	2.12132i
$181$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$181$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$182$	0.707107	−	0.707107i	0.707107	−	0.707107i
$183$	2.00000	−	2.00000i	2.00000	−	2.00000i
$184$	1.00000	+	1.00000i	1.00000	+	1.00000i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	2.82843	−	2.82843i	2.82843	−	2.82843i
$190$		0			0
$191$	2.00000			2.00000			1.00000			$0$
$191$	2.00000			2.00000			1.00000			$0$
$192$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$	1.41421	−	1.41421i	1.41421	−	1.41421i
$196$	−1.00000			−1.00000
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			1.00000i			1.00000i
$201$		0			0
$202$			1.41421i			1.41421i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.00000	+	3.00000i	−3.00000	+	3.00000i
$208$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$209$		0			0
$210$	−2.00000			−2.00000
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		−	2.82843i		−	2.82843i
$214$		0			0
$215$		0			0
$216$	−2.82843	−	2.82843i	−2.82843	−	2.82843i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$			1.00000i			1.00000i
$225$	−3.00000			−3.00000
$226$	1.00000	−	1.00000i	1.00000	−	1.00000i
$227$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$227$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$229$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$230$	1.41421			1.41421
$231$		0			0
$232$		0			0
$233$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$233$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$234$	2.12132	−	2.12132i	2.12132	−	2.12132i
$235$		0			0
$236$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$237$		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			$\pi$
$240$			2.00000i			2.00000i
$241$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$241$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$242$		−	1.00000i		−	1.00000i
$243$	4.24264	−	4.24264i	4.24264	−	4.24264i
$244$	−1.41421			−1.41421
$245$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$250$	0.707107	+	0.707107i	0.707107	+	0.707107i
$251$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$251$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$252$	−3.00000			−3.00000
$253$		0			0
$254$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		0			0
$260$	−1.00000			−1.00000
$261$		0			0
$262$	1.41421			1.41421
$263$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$263$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$269$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$270$	−4.00000			−4.00000
$271$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$271$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$272$		0			0
$273$		−	2.00000i		−	2.00000i
$274$		0			0
$275$		0			0
$276$	2.82843			2.82843
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$			1.41421i			1.41421i
$279$		0			0
$280$	0.707107	+	0.707107i	0.707107	+	0.707107i
$281$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$281$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−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
$288$			3.00000i			3.00000i
$289$		−	1.00000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$293$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$294$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$295$			2.00000i			2.00000i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			1.41421i			1.41421i
$300$	1.41421	+	1.41421i	1.41421	+	1.41421i
$301$		0			0
$302$	1.00000	+	1.00000i	1.00000	+	1.00000i
$303$	2.00000	+	2.00000i	2.00000	+	2.00000i
$304$		0			0
$305$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$306$		0			0
$307$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$307$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$	−2.00000			−2.00000
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$315$	−2.12132	+	2.12132i	−2.12132	+	2.12132i
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$	0.707107	−	0.707107i	0.707107	−	0.707107i
$321$		0			0
$322$	1.00000	−	1.00000i	1.00000	−	1.00000i
$323$		0			0
$324$	−5.00000			−5.00000
$325$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$331$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$332$			1.41421i			1.41421i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	1.41421	+	1.41421i	1.41421	+	1.41421i
$337$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$337$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$338$		−	1.00000i		−	1.00000i
$339$		−	2.82843i		−	2.82843i
$340$		0			0
$341$		0			0
$342$		0			0
$343$			1.00000i			1.00000i
$344$		0			0
$345$	2.00000	−	2.00000i	2.00000	−	2.00000i
$346$		0			0
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		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	$0.250000\pi$
$350$	1.00000			1.00000
$351$		−	4.00000i		−	4.00000i
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$			4.00000i			4.00000i
$355$			1.41421i			1.41421i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$359$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$360$	2.12132	+	2.12132i	2.12132	+	2.12132i
$361$		−	1.00000i		−	1.00000i
$362$			1.41421i			1.41421i
$363$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$364$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$365$		0			0
$366$	−2.00000	+	2.00000i	−2.00000	+	2.00000i
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$	2.00000			2.00000
$376$		0			0
$377$		0			0
$378$	−2.82843	+	2.82843i	−2.82843	+	2.82843i
$379$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$379$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$380$		0			0
$381$			2.82843i			2.82843i
$382$	−2.00000			−2.00000
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	1.41421	−	1.41421i	1.41421	−	1.41421i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$391$		0			0
$392$	1.00000			1.00000
$393$	2.00000	−	2.00000i	2.00000	−	2.00000i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$397$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$398$		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	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		−	1.41421i		−	1.41421i
$405$	−3.53553	+	3.53553i	−3.53553	+	3.53553i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$409$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.41421	+	1.41421i	1.41421	+	1.41421i
$414$	3.00000	−	3.00000i	3.00000	−	3.00000i
$415$	1.00000	+	1.00000i	1.00000	+	1.00000i
$416$	0.707107	+	0.707107i	0.707107	+	0.707107i
$417$	2.00000	+	2.00000i	2.00000	+	2.00000i
$418$		0			0
$419$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$419$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$420$	2.00000			2.00000
$421$		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
$424$		0			0
$425$		0			0
$426$			2.82843i			2.82843i
$427$			1.41421i			1.41421i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$431$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$432$	2.82843	+	2.82843i	2.82843	+	2.82843i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$			3.00000i			3.00000i
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		−	1.00000i		−	1.00000i
$449$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$449$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$450$	3.00000			3.00000
$451$		0			0
$452$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$453$	2.82843			2.82843
$454$	1.41421			1.41421
$455$			1.00000i			1.00000i
$456$		0			0
$457$	2.00000			2.00000			1.00000			$0$
$457$	2.00000			2.00000			1.00000			$0$
$458$		0			0
$459$		0			0
$460$	−1.41421			−1.41421
$461$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$461$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$	−2.00000			−2.00000			−1.00000			$\pi$
$463$	−2.00000			−2.00000			−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$	−2.12132	+	2.12132i	−2.12132	+	2.12132i
$469$		0			0
$470$		0			0
$471$	−4.00000			−4.00000
$472$	1.41421	−	1.41421i	1.41421	−	1.41421i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	1.00000	+	1.00000i	1.00000	+	1.00000i
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$		−	2.00000i		−	2.00000i
$481$		0			0
$482$		0			0
$483$		−	2.82843i		−	2.82843i
$484$			1.00000i			1.00000i
$485$		0			0
$486$	−4.24264	+	4.24264i	−4.24264	+	4.24264i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	1.41421			1.41421
$489$		0			0
$490$	0.707107	−	0.707107i	0.707107	−	0.707107i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.00000	+	1.00000i	1.00000	+	1.00000i
$498$	2.00000	+	2.00000i	2.00000	+	2.00000i
$499$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$499$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$500$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$501$		0			0
$502$			1.41421i			1.41421i
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$	3.00000			3.00000
$505$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$506$		0			0
$507$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$508$	1.00000	−	1.00000i	1.00000	−	1.00000i
$509$		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
$512$	−1.00000			−1.00000
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	1.00000			1.00000
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$	−1.41421			−1.41421
$525$	1.41421	−	1.41421i	1.41421	−	1.41421i
$526$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$527$		0			0
$528$		0			0
$529$			1.00000i			1.00000i
$530$		0			0
$531$	4.24264	+	4.24264i	4.24264	+	4.24264i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		−	1.41421i		−	1.41421i
$539$		0			0
$540$	4.00000			4.00000
$541$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$541$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$542$		0			0
$543$	2.00000	+	2.00000i	2.00000	+	2.00000i
$544$		0			0
$545$		0			0
$546$			2.00000i			2.00000i
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$			4.24264i			4.24264i
$550$		0			0
$551$		0			0
$552$	−2.82843			−2.82843
$553$		0			0
$554$		0			0
$555$		0			0
$556$		−	1.41421i		−	1.41421i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$561$		0			0
$562$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$563$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$563$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$564$		0			0
$565$			1.41421i			1.41421i
$566$		0			0
$567$			5.00000i			5.00000i
$568$	1.00000	−	1.00000i	1.00000	−	1.00000i
$569$	−2.00000			−2.00000			−1.00000			$\pi$
$569$	−2.00000			−2.00000			−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	+	2.82843i	−2.82843	+	2.82843i
$574$		0			0
$575$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$576$		−	3.00000i		−	3.00000i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$			1.00000i			1.00000i
$579$		0			0
$580$		0			0
$581$	1.41421			1.41421
$582$		0			0
$583$		0			0
$584$		0			0
$585$			3.00000i			3.00000i
$586$	1.41421			1.41421
$587$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$587$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$588$	1.41421	−	1.41421i	1.41421	−	1.41421i
$589$		0			0
$590$		−	2.00000i		−	2.00000i
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		−	1.41421i		−	1.41421i
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$605$	0.707107	+	0.707107i	0.707107	+	0.707107i
$606$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$	1.00000	−	1.00000i	1.00000	−	1.00000i
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$			1.41421i			1.41421i
$615$		0			0
$616$		0			0
$617$	−2.00000			−2.00000			−1.00000			$\pi$
$617$	−2.00000			−2.00000			−1.00000			$\pi$
$618$		0			0
$619$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$619$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$620$		0			0
$621$		−	5.65685i		−	5.65685i
$622$		0			0
$623$		0			0
$624$	2.00000			2.00000
$625$	−1.00000			−1.00000
$626$		0			0
$627$		0			0
$628$	1.41421	+	1.41421i	1.41421	+	1.41421i
$629$		0			0
$630$	2.12132	−	2.12132i	2.12132	−	2.12132i
$631$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$631$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	1.41421i		−	1.41421i
$636$		0			0
$637$	0.707107	+	0.707107i	0.707107	+	0.707107i
$638$		0			0
$639$	3.00000	+	3.00000i	3.00000	+	3.00000i
$640$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$643$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$644$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	5.00000			5.00000
$649$		0			0
$650$	0.707107	−	0.707107i	0.707107	−	0.707107i
$651$		0			0
$652$		0			0
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$661$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$662$		0			0
$663$		0			0
$664$		−	1.41421i		−	1.41421i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$673$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$673$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$674$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$675$	2.82843	−	2.82843i	2.82843	−	2.82843i
$676$			1.00000i			1.00000i
$677$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$677$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$678$			2.82843i			2.82843i
$679$		0			0
$680$		0			0
$681$	2.00000	−	2.00000i	2.00000	−	2.00000i
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		−	1.00000i		−	1.00000i
$687$		0			0
$688$		0			0
$689$		0			0
$690$	−2.00000	+	2.00000i	−2.00000	+	2.00000i
$691$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$691$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−2.82843			−2.82843
$700$	−1.00000			−1.00000
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$			4.00000i			4.00000i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.41421			−1.41421
$708$		−	4.00000i		−	4.00000i
$709$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$709$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$710$		−	1.41421i		−	1.41421i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	2.82843			2.82843
$718$	1.00000	−	1.00000i	1.00000	−	1.00000i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$	−2.12132	−	2.12132i	−2.12132	−	2.12132i
$721$		0			0
$722$			1.00000i			1.00000i
$723$		0			0
$724$		−	1.41421i		−	1.41421i
$725$		0			0
$726$	1.41421	+	1.41421i	1.41421	+	1.41421i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$	0.707107	−	0.707107i	0.707107	−	0.707107i
$729$			7.00000i			7.00000i
$730$		0			0
$731$		0			0
$732$	2.00000	−	2.00000i	2.00000	−	2.00000i
$733$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$733$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$		−	2.00000i		−	2.00000i
$736$	1.00000	+	1.00000i	1.00000	+	1.00000i
$737$		0			0
$738$		0			0
$739$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$739$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$743$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	4.24264			4.24264
$748$		0			0
$749$		0			0
$750$	−2.00000			−2.00000
$751$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$751$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$752$		0			0
$753$	2.00000	+	2.00000i	2.00000	+	2.00000i
$754$		0			0
$755$	−1.41421			−1.41421
$756$	2.82843	−	2.82843i	2.82843	−	2.82843i
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$761$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$762$		−	2.82843i		−	2.82843i
$763$		0			0
$764$	2.00000			2.00000
$765$		0			0
$766$		0			0
$767$	2.00000			2.00000
$768$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$769$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$769$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$773$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	1.41421	−	1.41421i	1.41421	−	1.41421i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.00000			−1.00000
$785$	2.00000			2.00000
$786$	−2.00000	+	2.00000i	−2.00000	+	2.00000i
$787$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$787$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$788$		0			0
$789$			2.82843i			2.82843i
$790$		0			0
$791$	1.00000	+	1.00000i	1.00000	+	1.00000i
$792$		0			0
$793$	1.00000	+	1.00000i	1.00000	+	1.00000i
$794$			1.41421i			1.41421i
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$			1.00000i			1.00000i
$801$		0			0
$802$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$803$		0			0
$804$		0			0
$805$			1.41421i			1.41421i
$806$		0			0
$807$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$808$			1.41421i			1.41421i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	3.53553	−	3.53553i	3.53553	−	3.53553i
$811$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$811$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	2.12132	+	2.12132i	2.12132	+	2.12132i
$820$		0			0
$821$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$821$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$822$		0			0
$823$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$823$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−3.00000	+	3.00000i	−3.00000	+	3.00000i
$829$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$829$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$830$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$831$		0			0
$832$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$833$		0			0
$834$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$835$		0			0
$836$		0			0
$837$		0			0
$838$			1.41421i			1.41421i
$839$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$839$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$840$	−2.00000			−2.00000
$841$	−1.00000			−1.00000
$842$		0			0
$843$	−2.82843			−2.82843
$844$		0			0
$845$	0.707107	+	0.707107i	0.707107	+	0.707107i
$846$		0			0
$847$	1.00000			1.00000
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		−	2.82843i		−	2.82843i
$853$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$853$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$854$		−	1.41421i		−	1.41421i
$855$		0			0
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$859$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$863$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$863$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$864$	−2.82843	−	2.82843i	−2.82843	−	2.82843i
$865$		0			0
$866$		0			0
$867$	1.41421	+	1.41421i	1.41421	+	1.41421i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$	2.00000	−	2.00000i	2.00000	−	2.00000i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		−	3.00000i		−	3.00000i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$	−2.82843	−	2.82843i	−2.82843	−	2.82843i
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$			1.00000i			1.00000i
$897$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$898$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$899$		0			0
$900$	−3.00000			−3.00000
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.00000	−	1.00000i	1.00000	−	1.00000i
$905$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$906$	−2.82843			−2.82843
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$	−1.41421			−1.41421
$909$	−4.24264			−4.24264
$910$		−	1.00000i		−	1.00000i
$911$	2.00000			2.00000			1.00000			$0$
$911$	2.00000			2.00000			1.00000			$0$
$912$		0			0
$913$		0			0
$914$	−2.00000			−2.00000
$915$		−	2.82843i		−	2.82843i
$916$		0			0
$917$			1.41421i			1.41421i
$918$		0			0
$919$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$919$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$920$	1.41421			1.41421
$921$	2.00000	+	2.00000i	2.00000	+	2.00000i
$922$		0			0
$923$	1.41421			1.41421
$924$		0			0
$925$		0			0
$926$	2.00000			2.00000
$927$		0			0
$928$		0			0
$929$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$929$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$930$		0			0
$931$		0			0
$932$	1.00000	+	1.00000i	1.00000	+	1.00000i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	2.12132	−	2.12132i	2.12132	−	2.12132i
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$941$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$942$	4.00000			4.00000
$943$		0			0
$944$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$945$		−	4.00000i		−	4.00000i
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$953$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$954$		0			0
$955$	1.41421	−	1.41421i	1.41421	−	1.41421i
$956$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$957$		0			0
$958$		0			0
$959$		0			0
$960$			2.00000i			2.00000i
$961$		−	1.00000i		−	1.00000i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$			2.82843i			2.82843i
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		−	1.00000i		−	1.00000i
$969$		0			0
$970$		0			0
$971$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$971$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$972$	4.24264	−	4.24264i	4.24264	−	4.24264i
$973$	−1.41421			−1.41421
$974$		0			0
$975$		−	2.00000i		−	2.00000i
$976$	−1.41421			−1.41421
$977$	−2.00000			−2.00000			−1.00000			$\pi$
$977$	−2.00000			−2.00000			−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		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	$0.5\pi$
$992$		0			0
$993$		0			0
$994$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$995$		0			0
$996$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−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	3640.1.cl.e.1357.1	yes	4
5.3	odd	4		3640.1.cm.e.2813.1	yes	4
7.6	odd	2	inner	3640.1.cl.e.1357.2	yes	4
8.5	even	2	inner	3640.1.cl.e.1357.2	yes	4
13.8	odd	4		3640.1.cm.e.3037.1	yes	4
35.13	even	4		3640.1.cm.e.2813.2	yes	4
40.13	odd	4		3640.1.cm.e.2813.2	yes	4
56.13	odd	2	CM	3640.1.cl.e.1357.1	yes	4
65.8	even	4	inner	3640.1.cl.e.853.1	✓	4
91.34	even	4		3640.1.cm.e.3037.2	yes	4
104.21	odd	4		3640.1.cm.e.3037.2	yes	4
280.13	even	4		3640.1.cm.e.2813.1	yes	4
455.398	odd	4	inner	3640.1.cl.e.853.2	yes	4
520.333	even	4	inner	3640.1.cl.e.853.2	yes	4
728.125	even	4		3640.1.cm.e.3037.1	yes	4
3640.853	odd	4	inner	3640.1.cl.e.853.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.1.cl.e.853.1	✓	4	65.8	even	4	inner
3640.1.cl.e.853.1	✓	4	3640.853	odd	4	inner
3640.1.cl.e.853.2	yes	4	455.398	odd	4	inner
3640.1.cl.e.853.2	yes	4	520.333	even	4	inner
3640.1.cl.e.1357.1	yes	4	1.1	even	1	trivial
3640.1.cl.e.1357.1	yes	4	56.13	odd	2	CM
3640.1.cl.e.1357.2	yes	4	7.6	odd	2	inner
3640.1.cl.e.1357.2	yes	4	8.5	even	2	inner
3640.1.cm.e.2813.1	yes	4	5.3	odd	4
3640.1.cm.e.2813.1	yes	4	280.13	even	4
3640.1.cm.e.2813.2	yes	4	35.13	even	4
3640.1.cm.e.2813.2	yes	4	40.13	odd	4
3640.1.cm.e.3037.1	yes	4	13.8	odd	4
3640.1.cm.e.3037.1	yes	4	728.125	even	4
3640.1.cm.e.3037.2	yes	4	91.34	even	4
3640.1.cm.e.3037.2	yes	4	104.21	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:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3640.cl (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +(-1.41421 + 1.41421i) q^{3} +1.00000 q^{4} +(0.707107 - 0.707107i) q^{5} +(1.41421 - 1.41421i) q^{6} -1.00000i q^{7} -1.00000 q^{8} -3.00000i q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +(-1.41421 + 1.41421i) q^{3} +1.00000 q^{4} +(0.707107 - 0.707107i) q^{5} +(1.41421 - 1.41421i) q^{6} -1.00000i q^{7} -1.00000 q^{8} -3.00000i q^{9} +(-0.707107 + 0.707107i) q^{10} +(-1.41421 + 1.41421i) q^{12} +(-0.707107 - 0.707107i) q^{13} +1.00000i q^{14} +2.00000i q^{15} +1.00000 q^{16} +3.00000i q^{18} +(0.707107 - 0.707107i) q^{20} +(1.41421 + 1.41421i) q^{21} +(-1.00000 - 1.00000i) q^{23} +(1.41421 - 1.41421i) q^{24} -1.00000i q^{25} +(0.707107 + 0.707107i) q^{26} +(2.82843 + 2.82843i) q^{27} -1.00000i q^{28} -2.00000i q^{30} -1.00000 q^{32} +(-0.707107 - 0.707107i) q^{35} -3.00000i q^{36} +2.00000 q^{39} +(-0.707107 + 0.707107i) q^{40} +(-1.41421 - 1.41421i) q^{42} +(-2.12132 - 2.12132i) q^{45} +(1.00000 + 1.00000i) q^{46} +(-1.41421 + 1.41421i) q^{48} -1.00000 q^{49} +1.00000i q^{50} +(-0.707107 - 0.707107i) q^{52} +(-2.82843 - 2.82843i) q^{54} +1.00000i q^{56} +(-1.41421 + 1.41421i) q^{59} +2.00000i q^{60} -1.41421 q^{61} -3.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} +2.82843 q^{69} +(0.707107 + 0.707107i) q^{70} +(-1.00000 + 1.00000i) q^{71} +3.00000i q^{72} +(1.41421 + 1.41421i) q^{75} -2.00000 q^{78} +(0.707107 - 0.707107i) q^{80} -5.00000 q^{81} +1.41421i q^{83} +(1.41421 + 1.41421i) q^{84} +(2.12132 + 2.12132i) q^{90} +(-0.707107 + 0.707107i) q^{91} +(-1.00000 - 1.00000i) q^{92} +(1.41421 - 1.41421i) q^{96} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{16} - 4 q^{23} - 4 q^{32} + 8 q^{39} + 4 q^{46} - 4 q^{49} - 12 q^{63} + 4 q^{64} - 4 q^{65} - 4 q^{71} - 8 q^{78} - 20 q^{81} - 4 q^{92} + 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^16 - 4 * q^23 - 4 * q^32 + 8 * q^39 + 4 * q^46 - 4 * q^49 - 12 * q^63 + 4 * q^64 - 4 * q^65 - 4 * q^71 - 8 * q^78 - 20 * q^81 - 4 * q^92 + 4 * q^98

Introduction

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