LMFDB - Embedded newform 2040.1.cj.c.1109.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2040,1,Mod(149,2040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2040, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 2, 2, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2040.149"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2040.cj (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,-4,0,0,0,-4,0,0,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(18)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.01809262577$
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 4.2.23582400.4

Embedding invariants

Embedding label			1109.2
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2040.1109
Dual form			2040.1.cj.c.149.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.707107 - 0.707107i) q^{2} +(-0.707107 - 0.707107i) q^{3} -1.00000i q^{4} +(-0.707107 - 0.707107i) q^{5} -1.00000 q^{6} +(-0.707107 - 0.707107i) q^{8} +1.00000i q^{9} -1.00000 q^{10} +(-0.707107 + 0.707107i) q^{12} +1.00000i q^{15} -1.00000 q^{16} +(-0.707107 - 0.707107i) q^{17} +(0.707107 + 0.707107i) q^{18} -2.00000 q^{19} +(-0.707107 + 0.707107i) q^{20} +1.00000i q^{24} +1.00000i q^{25} +(0.707107 - 0.707107i) q^{27} +(0.707107 + 0.707107i) q^{30} +(1.00000 - 1.00000i) q^{31} +(-0.707107 + 0.707107i) q^{32} -1.00000 q^{34} +1.00000 q^{36} +(-1.41421 + 1.41421i) q^{38} +1.00000i q^{40} +(0.707107 - 0.707107i) q^{45} -1.41421 q^{47} +(0.707107 + 0.707107i) q^{48} +1.00000i q^{49} +(0.707107 + 0.707107i) q^{50} +1.00000i q^{51} +1.41421 q^{53} -1.00000i q^{54} +(1.41421 + 1.41421i) q^{57} +1.00000 q^{60} +(1.00000 - 1.00000i) q^{61} -1.41421i q^{62} +1.00000i q^{64} +(-0.707107 + 0.707107i) q^{68} +(0.707107 - 0.707107i) q^{72} +(0.707107 - 0.707107i) q^{75} +2.00000i q^{76} +(-1.00000 - 1.00000i) q^{79} +(0.707107 + 0.707107i) q^{80} -1.00000 q^{81} -1.41421 q^{83} +1.00000i q^{85} -1.00000i q^{90} -1.41421 q^{93} +(-1.00000 + 1.00000i) q^{94} +(1.41421 + 1.41421i) q^{95} +1.00000 q^{96} +(0.707107 + 0.707107i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{6} - 4 q^{10} - 4 q^{16} - 8 q^{19} + 4 q^{31} - 4 q^{34} + 4 q^{36} + 4 q^{60} + 4 q^{61} - 4 q^{79} - 4 q^{81} - 4 q^{94} + 4 q^{96}+O(q^{100}) $ 4 * q - 4 * q^6 - 4 * q^10 - 4 * q^16 - 8 * q^19 + 4 * q^31 - 4 * q^34 + 4 * q^36 + 4 * q^60 + 4 * q^61 - 4 * q^79 - 4 * q^81 - 4 * q^94 + 4 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2040\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1021$	$1361$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$	$-1$	$-1$

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
$3$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$3$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$4$		−	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			−1.00000
$7$		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
$9$			1.00000i			1.00000i
$10$	−1.00000			−1.00000
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$			1.00000i			1.00000i
$16$	−1.00000			−1.00000
$17$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$17$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$18$	0.707107	+	0.707107i	0.707107	+	0.707107i
$19$	−2.00000			−2.00000			−1.00000			$\pi$
$19$	−2.00000			−2.00000			−1.00000			$\pi$
$20$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$21$		0			0
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$			1.00000i			1.00000i
$25$			1.00000i			1.00000i
$26$		0			0
$27$	0.707107	−	0.707107i	0.707107	−	0.707107i
$28$		0			0
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$	0.707107	+	0.707107i	0.707107	+	0.707107i
$31$	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
$34$	−1.00000			−1.00000
$35$		0			0
$36$	1.00000			1.00000
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$39$		0			0
$40$			1.00000i			1.00000i
$41$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$41$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	0.707107	−	0.707107i	0.707107	−	0.707107i
$46$		0			0
$47$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$47$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$48$	0.707107	+	0.707107i	0.707107	+	0.707107i
$49$			1.00000i			1.00000i
$50$	0.707107	+	0.707107i	0.707107	+	0.707107i
$51$			1.00000i			1.00000i
$52$		0			0
$53$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$53$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$54$		−	1.00000i		−	1.00000i
$55$		0			0
$56$		0			0
$57$	1.41421	+	1.41421i	1.41421	+	1.41421i
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$	1.00000			1.00000
$61$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$61$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$62$		−	1.41421i		−	1.41421i
$63$		0			0
$64$			1.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$69$		0			0
$70$		0			0
$71$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$71$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$72$	0.707107	−	0.707107i	0.707107	−	0.707107i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$	0.707107	−	0.707107i	0.707107	−	0.707107i
$76$			2.00000i			2.00000i
$77$		0			0
$78$		0			0
$79$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$79$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$80$	0.707107	+	0.707107i	0.707107	+	0.707107i
$81$	−1.00000			−1.00000
$82$		0			0
$83$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$83$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$			1.00000i			1.00000i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		−	1.00000i		−	1.00000i
$91$		0			0
$92$		0			0
$93$	−1.41421			−1.41421
$94$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$95$	1.41421	+	1.41421i	1.41421	+	1.41421i
$96$	1.00000			1.00000
$97$		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	$0.5\pi$
$102$	0.707107	+	0.707107i	0.707107	+	0.707107i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	1.00000	−	1.00000i	1.00000	−	1.00000i
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$109$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$109$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$113$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$114$	2.00000			2.00000
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	0.707107	−	0.707107i	0.707107	−	0.707107i
$121$		−	1.00000i		−	1.00000i
$122$		−	1.41421i		−	1.41421i
$123$		0			0
$124$	−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
$129$		0			0
$130$		0			0
$131$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$131$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.00000			−1.00000
$136$			1.00000i			1.00000i
$137$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$137$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$139$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$140$		0			0
$141$	1.00000	+	1.00000i	1.00000	+	1.00000i
$142$		0			0
$143$		0			0
$144$		−	1.00000i		−	1.00000i
$145$		0			0
$146$		0			0
$147$	0.707107	−	0.707107i	0.707107	−	0.707107i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		−	1.00000i		−	1.00000i
$151$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$151$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$152$	1.41421	+	1.41421i	1.41421	+	1.41421i
$153$	0.707107	−	0.707107i	0.707107	−	0.707107i
$154$		0			0
$155$	−1.41421			−1.41421
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−1.41421			−1.41421
$159$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$160$	1.00000			1.00000
$161$		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	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$167$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$167$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$	0.707107	+	0.707107i	0.707107	+	0.707107i
$171$		−	2.00000i		−	2.00000i
$172$		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
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$181$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$181$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$182$		0			0
$183$	−1.41421			−1.41421
$184$		0			0
$185$		0			0
$186$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$187$		0			0
$188$			1.41421i			1.41421i
$189$		0			0
$190$	2.00000			2.00000
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	0.707107	−	0.707107i	0.707107	−	0.707107i
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		0			0
$196$	1.00000			1.00000
$197$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$197$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$199$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$200$	0.707107	−	0.707107i	0.707107	−	0.707107i
$201$		0			0
$202$		0			0
$203$		0			0
$204$	1.00000			1.00000
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$211$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$212$		−	1.41421i		−	1.41421i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	−1.00000			−1.00000
$217$		0			0
$218$		−	1.41421i		−	1.41421i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−1.00000			−1.00000
$226$	−2.00000			−2.00000
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$	1.41421	−	1.41421i	1.41421	−	1.41421i
$229$	2.00000			2.00000			1.00000			$0$
$229$	2.00000			2.00000			1.00000			$0$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$	1.00000	+	1.00000i	1.00000	+	1.00000i
$236$		0			0
$237$			1.41421i			1.41421i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		−	1.00000i		−	1.00000i
$241$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$241$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$242$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$243$	0.707107	+	0.707107i	0.707107	+	0.707107i
$244$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$245$	0.707107	−	0.707107i	0.707107	−	0.707107i
$246$		0			0
$247$		0			0
$248$	−1.41421			−1.41421
$249$	1.00000	+	1.00000i	1.00000	+	1.00000i
$250$		−	1.00000i		−	1.00000i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	0.707107	−	0.707107i	0.707107	−	0.707107i
$256$	1.00000			1.00000
$257$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$257$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$263$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$269$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$270$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	0.707107	+	0.707107i	0.707107	+	0.707107i
$273$		0			0
$274$	1.00000	−	1.00000i	1.00000	−	1.00000i
$275$		0			0
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$	−1.41421			−1.41421
$279$	1.00000	+	1.00000i	1.00000	+	1.00000i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	1.41421			1.41421
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		−	2.00000i		−	2.00000i
$286$		0			0
$287$		0			0
$288$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$289$			1.00000i			1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$293$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$294$		−	1.00000i		−	1.00000i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$301$		0			0
$302$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$303$		0			0
$304$	2.00000			2.00000
$305$	−1.41421			−1.41421
$306$		−	1.00000i		−	1.00000i
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$311$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$311$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		0			0
$315$		0			0
$316$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$	−1.41421			−1.41421
$319$		0			0
$320$	0.707107	−	0.707107i	0.707107	−	0.707107i
$321$		0			0
$322$		0			0
$323$	1.41421	+	1.41421i	1.41421	+	1.41421i
$324$			1.00000i			1.00000i
$325$		0			0
$326$		0			0
$327$	−1.41421			−1.41421
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.00000			2.00000			1.00000			$0$
$331$	2.00000			2.00000			1.00000			$0$
$332$			1.41421i			1.41421i
$333$		0			0
$334$		−	2.00000i		−	2.00000i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$339$			2.00000i			2.00000i
$340$	1.00000			1.00000
$341$		0			0
$342$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$343$		0			0
$344$		0			0
$345$		0			0
$346$		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			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$353$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−1.00000			−1.00000
$361$	3.00000			3.00000
$362$			1.41421i			1.41421i
$363$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$364$		0			0
$365$		0			0
$366$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$			1.41421i			1.41421i
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	−1.00000			−1.00000
$376$	1.00000	+	1.00000i	1.00000	+	1.00000i
$377$		0			0
$378$		0			0
$379$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$379$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$380$	1.41421	−	1.41421i	1.41421	−	1.41421i
$381$		0			0
$382$		0			0
$383$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$383$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$384$		−	1.00000i		−	1.00000i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	0.707107	−	0.707107i	0.707107	−	0.707107i
$393$		0			0
$394$			2.00000i			2.00000i
$395$			1.41421i			1.41421i
$396$		0			0
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$		−	1.41421i		−	1.41421i
$399$		0			0
$400$		−	1.00000i		−	1.00000i
$401$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$401$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.707107	+	0.707107i	0.707107	+	0.707107i
$406$		0			0
$407$		0			0
$408$	0.707107	−	0.707107i	0.707107	−	0.707107i
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	1.00000	+	1.00000i	1.00000	+	1.00000i
$416$		0			0
$417$			1.41421i			1.41421i
$418$		0			0
$419$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$419$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$421$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$422$			1.41421i			1.41421i
$423$		−	1.41421i		−	1.41421i
$424$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$425$	0.707107	−	0.707107i	0.707107	−	0.707107i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$431$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$432$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$437$		0			0
$438$		0			0
$439$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$439$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$440$		0			0
$441$	−1.00000			−1.00000
$442$		0			0
$443$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$443$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$449$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$450$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$451$		0			0
$452$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$453$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$454$		0			0
$455$		0			0
$456$		−	2.00000i		−	2.00000i
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$	1.41421	−	1.41421i	1.41421	−	1.41421i
$459$	−1.00000			−1.00000
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$	1.00000	+	1.00000i	1.00000	+	1.00000i
$466$		0			0
$467$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$467$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$		0			0
$470$	1.41421			1.41421
$471$		0			0
$472$		0			0
$473$		0			0
$474$	1.00000	+	1.00000i	1.00000	+	1.00000i
$475$		−	2.00000i		−	2.00000i
$476$		0			0
$477$			1.41421i			1.41421i
$478$		0			0
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$481$		0			0
$482$		−	1.41421i		−	1.41421i
$483$		0			0
$484$	−1.00000			−1.00000
$485$		0			0
$486$	1.00000			1.00000
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$	−1.41421			−1.41421
$489$		0			0
$490$		−	1.00000i		−	1.00000i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$497$		0			0
$498$	1.41421			1.41421
$499$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$499$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$500$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$501$	−2.00000			−2.00000
$502$		0			0
$503$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$503$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.707107	+	0.707107i	0.707107	+	0.707107i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		−	1.00000i		−	1.00000i
$511$		0			0
$512$	0.707107	−	0.707107i	0.707107	−	0.707107i
$513$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$514$	1.00000	+	1.00000i	1.00000	+	1.00000i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$521$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$	1.00000	+	1.00000i	1.00000	+	1.00000i
$527$	−1.41421			−1.41421
$528$		0			0
$529$		−	1.00000i		−	1.00000i
$530$	−1.41421			−1.41421
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$			1.00000i			1.00000i
$541$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$541$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$542$		0			0
$543$	1.41421			1.41421
$544$	1.00000			1.00000
$545$	−1.41421			−1.41421
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		−	1.41421i		−	1.41421i
$549$	1.00000	+	1.00000i	1.00000	+	1.00000i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$557$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$557$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$558$	1.41421			1.41421
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$563$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$564$	1.00000	−	1.00000i	1.00000	−	1.00000i
$565$			2.00000i			2.00000i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$571$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$571$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−1.00000			−1.00000
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	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
$585$		0			0
$586$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$587$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$587$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$588$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$589$	−2.00000	+	2.00000i	−2.00000	+	2.00000i
$590$		0			0
$591$	2.00000			2.00000
$592$		0			0
$593$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$593$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.41421			−1.41421
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	−1.00000			−1.00000
$601$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$601$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	−2.00000			−2.00000
$605$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$	1.41421	−	1.41421i	1.41421	−	1.41421i
$609$		0			0
$610$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$611$		0			0
$612$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$617$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$618$		0			0
$619$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$619$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$620$			1.41421i			1.41421i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.00000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$631$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$632$			1.41421i			1.41421i
$633$	1.41421			1.41421
$634$		0			0
$635$		0			0
$636$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$637$		0			0
$638$		0			0
$639$		0			0
$640$		−	1.00000i		−	1.00000i
$641$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$641$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$	2.00000			2.00000
$647$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$647$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$648$	0.707107	+	0.707107i	0.707107	+	0.707107i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−2.00000			−2.00000			−1.00000			$\pi$
$661$	−2.00000			−2.00000			−1.00000			$\pi$
$662$	1.41421	−	1.41421i	1.41421	−	1.41421i
$663$		0			0
$664$	1.00000	+	1.00000i	1.00000	+	1.00000i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$	0.707107	+	0.707107i	0.707107	+	0.707107i
$676$			1.00000i			1.00000i
$677$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$677$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$678$	1.41421	+	1.41421i	1.41421	+	1.41421i
$679$		0			0
$680$	0.707107	−	0.707107i	0.707107	−	0.707107i
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$	−2.00000			−2.00000
$685$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$686$		0			0
$687$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$691$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			1.41421i			1.41421i
$696$		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
$705$		−	1.41421i		−	1.41421i
$706$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$707$		0			0
$708$		0			0
$709$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$709$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$710$		0			0
$711$	1.00000	−	1.00000i	1.00000	−	1.00000i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$719$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$720$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$721$		0			0
$722$	2.12132	−	2.12132i	2.12132	−	2.12132i
$723$	−1.41421			−1.41421
$724$	1.00000	+	1.00000i	1.00000	+	1.00000i
$725$		0			0
$726$			1.00000i			1.00000i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		−	1.00000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$			1.41421i			1.41421i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	−1.00000			−1.00000
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$743$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$744$	1.00000	+	1.00000i	1.00000	+	1.00000i
$745$		0			0
$746$		0			0
$747$		−	1.41421i		−	1.41421i
$748$		0			0
$749$		0			0
$750$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$751$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$751$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$752$	1.41421			1.41421
$753$		0			0
$754$		0			0
$755$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	1.41421			1.41421
$759$		0			0
$760$		−	2.00000i		−	2.00000i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1.00000			−1.00000
$766$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$767$		0			0
$768$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$769$	−2.00000			−2.00000			−1.00000			$\pi$
$769$	−2.00000			−2.00000			−1.00000			$\pi$
$770$		0			0
$771$	1.00000	−	1.00000i	1.00000	−	1.00000i
$772$		0			0
$773$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$773$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$	1.00000	+	1.00000i	1.00000	+	1.00000i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	1.00000i		−	1.00000i
$785$		0			0
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	1.41421	+	1.41421i	1.41421	+	1.41421i
$789$	1.00000	−	1.00000i	1.00000	−	1.00000i
$790$	1.00000	+	1.00000i	1.00000	+	1.00000i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$			1.41421i			1.41421i
$796$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$797$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$797$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$	1.00000	+	1.00000i	1.00000	+	1.00000i
$800$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$809$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$810$	1.00000			1.00000
$811$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$811$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		−	1.00000i		−	1.00000i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$821$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$822$	−1.41421			−1.41421
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$827$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$	1.41421			1.41421
$831$		0			0
$832$		0			0
$833$	0.707107	−	0.707107i	0.707107	−	0.707107i
$834$	1.00000	+	1.00000i	1.00000	+	1.00000i
$835$	−2.00000			−2.00000
$836$		0			0
$837$		−	1.41421i		−	1.41421i
$838$		0			0
$839$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$839$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$			1.00000i			1.00000i
$842$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$843$		0			0
$844$	1.00000	+	1.00000i	1.00000	+	1.00000i
$845$	0.707107	+	0.707107i	0.707107	+	0.707107i
$846$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$847$		0			0
$848$	−1.41421			−1.41421
$849$		0			0
$850$		−	1.00000i		−	1.00000i
$851$		0			0
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$856$		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
$859$	2.00000			2.00000			1.00000			$0$
$859$	2.00000			2.00000			1.00000			$0$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$863$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$864$			1.00000i			1.00000i
$865$		0			0
$866$		0			0
$867$	0.707107	−	0.707107i	0.707107	−	0.707107i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−1.41421			−1.41421
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$			1.41421i			1.41421i
$879$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$880$		0			0
$881$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$881$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$882$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	1.00000	+	1.00000i	1.00000	+	1.00000i
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.82843			2.82843
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$			1.00000i			1.00000i
$901$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$902$		0			0
$903$		0			0
$904$			2.00000i			2.00000i
$905$	1.41421			1.41421
$906$			2.00000i			2.00000i
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$911$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$912$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$913$		0			0
$914$		0			0
$915$	1.00000	+	1.00000i	1.00000	+	1.00000i
$916$		−	2.00000i		−	2.00000i
$917$		0			0
$918$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$929$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$930$	1.41421			1.41421
$931$		−	2.00000i		−	2.00000i
$932$		0			0
$933$		0			0
$934$	1.00000	−	1.00000i	1.00000	−	1.00000i
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$	1.00000	−	1.00000i	1.00000	−	1.00000i
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$947$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$948$	1.41421			1.41421
$949$		0			0
$950$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$951$		0			0
$952$		0			0
$953$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$953$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$954$	1.00000	+	1.00000i	1.00000	+	1.00000i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−1.00000			−1.00000
$961$		−	1.00000i		−	1.00000i
$962$		0			0
$963$		0			0
$964$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$969$		−	2.00000i		−	2.00000i
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	0.707107	−	0.707107i	0.707107	−	0.707107i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$977$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$977$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$981$	1.00000	+	1.00000i	1.00000	+	1.00000i
$982$		0			0
$983$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$983$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$	2.00000			2.00000
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$991$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$992$			1.41421i			1.41421i
$993$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$994$		0			0
$995$	−1.41421			−1.41421
$996$	1.00000	−	1.00000i	1.00000	−	1.00000i
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		−	1.41421i		−	1.41421i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2040.1.cj.c.1109.2	yes	4
3.2	odd	2	inner	2040.1.cj.c.1109.1	yes	4
5.4	even	2	inner	2040.1.cj.c.1109.1	yes	4
8.5	even	2		2040.1.cj.d.1109.1	yes	4
15.14	odd	2	CM	2040.1.cj.c.1109.2	yes	4
17.13	even	4		2040.1.cj.d.149.1	yes	4
24.5	odd	2		2040.1.cj.d.1109.2	yes	4
40.29	even	2		2040.1.cj.d.1109.2	yes	4
51.47	odd	4		2040.1.cj.d.149.2	yes	4
85.64	even	4		2040.1.cj.d.149.2	yes	4
120.29	odd	2		2040.1.cj.d.1109.1	yes	4
136.13	even	4	inner	2040.1.cj.c.149.2	yes	4
255.149	odd	4		2040.1.cj.d.149.1	yes	4
408.149	odd	4	inner	2040.1.cj.c.149.1	✓	4
680.149	even	4	inner	2040.1.cj.c.149.1	✓	4
2040.149	odd	4	inner	2040.1.cj.c.149.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.1.cj.c.149.1	✓	4	408.149	odd	4	inner
2040.1.cj.c.149.1	✓	4	680.149	even	4	inner
2040.1.cj.c.149.2	yes	4	136.13	even	4	inner
2040.1.cj.c.149.2	yes	4	2040.149	odd	4	inner
2040.1.cj.c.1109.1	yes	4	3.2	odd	2	inner
2040.1.cj.c.1109.1	yes	4	5.4	even	2	inner
2040.1.cj.c.1109.2	yes	4	1.1	even	1	trivial
2040.1.cj.c.1109.2	yes	4	15.14	odd	2	CM
2040.1.cj.d.149.1	yes	4	17.13	even	4
2040.1.cj.d.149.1	yes	4	255.149	odd	4
2040.1.cj.d.149.2	yes	4	51.47	odd	4
2040.1.cj.d.149.2	yes	4	85.64	even	4
2040.1.cj.d.1109.1	yes	4	8.5	even	2
2040.1.cj.d.1109.1	yes	4	120.29	odd	2
2040.1.cj.d.1109.2	yes	4	24.5	odd	2
2040.1.cj.d.1109.2	yes	4	40.29	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2040.cj (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{2} +(-0.707107 - 0.707107i) q^{3} -1.00000i q^{4} +(-0.707107 - 0.707107i) q^{5} -1.00000 q^{6} +(-0.707107 - 0.707107i) q^{8} +1.00000i q^{9} -1.00000 q^{10} +(-0.707107 + 0.707107i) q^{12} +1.00000i q^{15} -1.00000 q^{16} +(-0.707107 - 0.707107i) q^{17} +(0.707107 + 0.707107i) q^{18} -2.00000 q^{19} +(-0.707107 + 0.707107i) q^{20} +1.00000i q^{24} +1.00000i q^{25} +(0.707107 - 0.707107i) q^{27} +(0.707107 + 0.707107i) q^{30} +(1.00000 - 1.00000i) q^{31} +(-0.707107 + 0.707107i) q^{32} -1.00000 q^{34} +1.00000 q^{36} +(-1.41421 + 1.41421i) q^{38} +1.00000i q^{40} +(0.707107 - 0.707107i) q^{45} -1.41421 q^{47} +(0.707107 + 0.707107i) q^{48} +1.00000i q^{49} +(0.707107 + 0.707107i) q^{50} +1.00000i q^{51} +1.41421 q^{53} -1.00000i q^{54} +(1.41421 + 1.41421i) q^{57} +1.00000 q^{60} +(1.00000 - 1.00000i) q^{61} -1.41421i q^{62} +1.00000i q^{64} +(-0.707107 + 0.707107i) q^{68} +(0.707107 - 0.707107i) q^{72} +(0.707107 - 0.707107i) q^{75} +2.00000i q^{76} +(-1.00000 - 1.00000i) q^{79} +(0.707107 + 0.707107i) q^{80} -1.00000 q^{81} -1.41421 q^{83} +1.00000i q^{85} -1.00000i q^{90} -1.41421 q^{93} +(-1.00000 + 1.00000i) q^{94} +(1.41421 + 1.41421i) q^{95} +1.00000 q^{96} +(0.707107 + 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{6} - 4 q^{10} - 4 q^{16} - 8 q^{19} + 4 q^{31} - 4 q^{34} + 4 q^{36} + 4 q^{60} + 4 q^{61} - 4 q^{79} - 4 q^{81} - 4 q^{94} + 4 q^{96}+O(q^{100}) \) 4 * q - 4 * q^6 - 4 * q^10 - 4 * q^16 - 8 * q^19 + 4 * q^31 - 4 * q^34 + 4 * q^36 + 4 * q^60 + 4 * q^61 - 4 * q^79 - 4 * q^81 - 4 * q^94 + 4 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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