LMFDB - Embedded newform 2040.1.fb.a.1859.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2040,1,Mod(299,2040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2040, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 8, 8, 8, 3])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2040.299"); 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.fb (of order $16$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(24)] == traces)

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

Self dual:	no
Analytic conductor:	$1.01809262577$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{16})$
Coefficient field:	$\Q(\zeta_{32})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 1 $ x^16 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{16}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{16} - \cdots)$

Embedding invariants

Embedding label			1859.2
Root			$0.195090 - 0.980785i$ of defining polynomial
Character	$\chi$	$=$	2040.1859
Dual form			2040.1.fb.a.1499.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.555570 - 0.831470i) q^{2} +(0.980785 - 0.195090i) q^{3} +(-0.382683 - 0.923880i) q^{4} +(0.831470 + 0.555570i) q^{5} +(0.382683 - 0.923880i) q^{6} +(-0.980785 - 0.195090i) q^{8} +(0.923880 - 0.382683i) q^{9} +(0.923880 - 0.382683i) q^{10} +(-0.555570 - 0.831470i) q^{12} +(0.923880 + 0.382683i) q^{15} +(-0.707107 + 0.707107i) q^{16} +(-0.195090 - 0.980785i) q^{17} +(0.195090 - 0.980785i) q^{18} +(-0.707107 - 0.292893i) q^{19} +(0.195090 - 0.980785i) q^{20} +(-0.275899 + 1.38704i) q^{23} -1.00000 q^{24} +(0.382683 + 0.923880i) q^{25} +(0.831470 - 0.555570i) q^{27} +(0.831470 - 0.555570i) q^{30} +(-1.08979 + 0.216773i) q^{31} +(0.195090 + 0.980785i) q^{32} +(-0.923880 - 0.382683i) q^{34} +(-0.707107 - 0.707107i) q^{36} +(-0.636379 + 0.425215i) q^{38} +(-0.707107 - 0.707107i) q^{40} +(0.980785 + 0.195090i) q^{45} +(1.00000 + 1.00000i) q^{46} +(0.785695 + 0.785695i) q^{47} +(-0.555570 + 0.831470i) q^{48} +(0.382683 - 0.923880i) q^{49} +(0.980785 + 0.195090i) q^{50} +(-0.382683 - 0.923880i) q^{51} +(-0.149316 + 0.360480i) q^{53} -1.00000i q^{54} +(-0.750661 - 0.149316i) q^{57} -1.00000i q^{60} +(-0.923880 - 1.38268i) q^{61} +(-0.425215 + 1.02656i) q^{62} +(0.923880 + 0.382683i) q^{64} +(-0.831470 + 0.555570i) q^{68} +1.41421i q^{69} +(-0.980785 + 0.195090i) q^{72} +(0.555570 + 0.831470i) q^{75} +0.765367i q^{76} +(-1.63099 - 0.324423i) q^{79} +(-0.980785 + 0.195090i) q^{80} +(0.707107 - 0.707107i) q^{81} +(-0.750661 + 1.81225i) q^{83} +(0.382683 - 0.923880i) q^{85} +(0.707107 - 0.707107i) q^{90} +(1.38704 - 0.275899i) q^{92} +(-1.02656 + 0.425215i) q^{93} +(1.08979 - 0.216773i) q^{94} +(-0.425215 - 0.636379i) q^{95} +(0.382683 + 0.923880i) q^{96} +(-0.555570 - 0.831470i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{24} + 16 q^{46}+O(q^{100}) $ 16 * q - 16 * q^24 + 16 * q^46

Character values

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

$n$	$241$	$511$	$817$	$1021$	$1361$
$\chi(n)$	$e\left(\frac{15}{16}\right)$	$-1$	$-1$	$-1$	$-1$

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 $

$2$	0.555570	−	0.831470i	0.555570	−	0.831470i
$2$	0.555570	−	0.831470i	0.555570	−	0.831470i
$3$	0.980785	−	0.195090i	0.980785	−	0.195090i
$3$	0.980785	−	0.195090i	0.980785	−	0.195090i
$4$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$5$	0.831470	+	0.555570i	0.831470	+	0.555570i
$5$	0.831470	+	0.555570i	0.831470	+	0.555570i
$6$	0.382683	−	0.923880i	0.382683	−	0.923880i
$7$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$7$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$8$	−0.980785	−	0.195090i	−0.980785	−	0.195090i
$9$	0.923880	−	0.382683i	0.923880	−	0.382683i
$10$	0.923880	−	0.382683i	0.923880	−	0.382683i
$11$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$11$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$12$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$	0.923880	+	0.382683i	0.923880	+	0.382683i
$16$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$17$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$17$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$18$	0.195090	−	0.980785i	0.195090	−	0.980785i
$19$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$19$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$20$	0.195090	−	0.980785i	0.195090	−	0.980785i
$21$		0			0
$22$		0			0
$23$	−0.275899	+	1.38704i	−0.275899	+	1.38704i	0.555570	+	0.831470i	$0.312500\pi$
$23$	−0.275899	+	1.38704i	−0.275899	+	1.38704i	−0.831470	+	0.555570i	$0.812500\pi$
$24$	−1.00000			−1.00000
$25$	0.382683	+	0.923880i	0.382683	+	0.923880i
$26$		0			0
$27$	0.831470	−	0.555570i	0.831470	−	0.555570i
$28$		0			0
$29$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$29$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$30$	0.831470	−	0.555570i	0.831470	−	0.555570i
$31$	−1.08979	+	0.216773i	−1.08979	+	0.216773i	−0.707107	−	0.707107i	$-0.750000\pi$
$31$	−1.08979	+	0.216773i	−1.08979	+	0.216773i	−0.382683	+	0.923880i	$0.625000\pi$
$32$	0.195090	+	0.980785i	0.195090	+	0.980785i
$33$		0			0
$34$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$35$		0			0
$36$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$37$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$37$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$38$	−0.636379	+	0.425215i	−0.636379	+	0.425215i
$39$		0			0
$40$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$41$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$41$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$42$		0			0
$43$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$43$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$44$		0			0
$45$	0.980785	+	0.195090i	0.980785	+	0.195090i
$46$	1.00000	+	1.00000i	1.00000	+	1.00000i
$47$	0.785695	+	0.785695i	0.785695	+	0.785695i	0.980785	−	0.195090i	$-0.0625000\pi$
$47$	0.785695	+	0.785695i	0.785695	+	0.785695i	−0.195090	+	0.980785i	$0.562500\pi$
$48$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$49$	0.382683	−	0.923880i	0.382683	−	0.923880i
$50$	0.980785	+	0.195090i	0.980785	+	0.195090i
$51$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$52$		0			0
$53$	−0.149316	+	0.360480i	−0.149316	+	0.360480i	−0.980785	−	0.195090i	$-0.937500\pi$
$53$	−0.149316	+	0.360480i	−0.149316	+	0.360480i	0.831470	+	0.555570i	$0.187500\pi$
$54$		−	1.00000i		−	1.00000i
$55$		0			0
$56$		0			0
$57$	−0.750661	−	0.149316i	−0.750661	−	0.149316i
$58$		0			0
$59$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$59$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$60$		−	1.00000i		−	1.00000i
$61$	−0.923880	−	1.38268i	−0.923880	−	1.38268i	−0.923880	−	0.382683i	$-0.875000\pi$
$61$	−0.923880	−	1.38268i	−0.923880	−	1.38268i		−	1.00000i	$-0.5\pi$
$62$	−0.425215	+	1.02656i	−0.425215	+	1.02656i
$63$		0			0
$64$	0.923880	+	0.382683i	0.923880	+	0.382683i
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$69$			1.41421i			1.41421i
$70$		0			0
$71$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$71$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$72$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$73$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$73$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$74$		0			0
$75$	0.555570	+	0.831470i	0.555570	+	0.831470i
$76$			0.765367i			0.765367i
$77$		0			0
$78$		0			0
$79$	−1.63099	−	0.324423i	−1.63099	−	0.324423i	−0.707107	−	0.707107i	$-0.750000\pi$
$79$	−1.63099	−	0.324423i	−1.63099	−	0.324423i	−0.923880	+	0.382683i	$0.875000\pi$
$80$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$81$	0.707107	−	0.707107i	0.707107	−	0.707107i
$82$		0			0
$83$	−0.750661	+	1.81225i	−0.750661	+	1.81225i	−0.195090	+	0.980785i	$0.562500\pi$
$83$	−0.750661	+	1.81225i	−0.750661	+	1.81225i	−0.555570	+	0.831470i	$0.687500\pi$
$84$		0			0
$85$	0.382683	−	0.923880i	0.382683	−	0.923880i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$89$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$90$	0.707107	−	0.707107i	0.707107	−	0.707107i
$91$		0			0
$92$	1.38704	−	0.275899i	1.38704	−	0.275899i
$93$	−1.02656	+	0.425215i	−1.02656	+	0.425215i
$94$	1.08979	−	0.216773i	1.08979	−	0.216773i
$95$	−0.425215	−	0.636379i	−0.425215	−	0.636379i
$96$	0.382683	+	0.923880i	0.382683	+	0.923880i
$97$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$97$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$98$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$99$		0			0
$100$	0.707107	−	0.707107i	0.707107	−	0.707107i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	−0.980785	−	0.195090i	−0.980785	−	0.195090i
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	0.216773	+	0.324423i	0.216773	+	0.324423i
$107$	0.425215	−	0.636379i	0.425215	−	0.636379i	−0.555570	−	0.831470i	$-0.687500\pi$
$107$	0.425215	−	0.636379i	0.425215	−	0.636379i	0.980785	+	0.195090i	$0.0625000\pi$
$108$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$109$	0.324423	−	0.216773i	0.324423	−	0.216773i	−0.382683	−	0.923880i	$-0.625000\pi$
$109$	0.324423	−	0.216773i	0.324423	−	0.216773i	0.707107	+	0.707107i	$0.250000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.360480	+	1.81225i	−0.360480	+	1.81225i	0.195090	+	0.980785i	$0.437500\pi$
$113$	−0.360480	+	1.81225i	−0.360480	+	1.81225i	−0.555570	+	0.831470i	$0.687500\pi$
$114$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$115$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$121$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$122$	−1.66294			−1.66294
$123$		0			0
$124$	0.617317	+	0.923880i	0.617317	+	0.923880i
$125$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$126$		0			0
$127$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$127$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$128$	0.831470	−	0.555570i	0.831470	−	0.555570i
$129$		0			0
$130$		0			0
$131$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$131$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	1.00000			1.00000
$136$			1.00000i			1.00000i
$137$	1.11114			1.11114			0.555570	−	0.831470i	$-0.312500\pi$
$137$	1.11114			1.11114			0.555570	+	0.831470i	$0.312500\pi$
$138$	1.17588	+	0.785695i	1.17588	+	0.785695i
$139$	−1.92388	+	0.382683i	−1.92388	+	0.382683i	−0.923880	+	0.382683i	$0.875000\pi$
$139$	−1.92388	+	0.382683i	−1.92388	+	0.382683i	−1.00000			$\pi$
$140$		0			0
$141$	0.923880	+	0.617317i	0.923880	+	0.617317i
$142$		0			0
$143$		0			0
$144$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$145$		0			0
$146$		0			0
$147$	0.195090	−	0.980785i	0.195090	−	0.980785i
$148$		0			0
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$	1.00000			1.00000
$151$	1.30656	+	0.541196i	1.30656	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$151$	1.30656	+	0.541196i	1.30656	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$152$	0.636379	+	0.425215i	0.636379	+	0.425215i
$153$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$154$		0			0
$155$	−1.02656	−	0.425215i	−1.02656	−	0.425215i
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$	−1.17588	+	1.17588i	−1.17588	+	1.17588i
$159$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i
$160$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$161$		0			0
$162$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$163$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$163$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$164$		0			0
$165$		0			0
$166$	1.08979	+	1.63099i	1.08979	+	1.63099i
$167$	1.81225	−	0.360480i	1.81225	−	0.360480i	0.831470	−	0.555570i	$-0.187500\pi$
$167$	1.81225	−	0.360480i	1.81225	−	0.360480i	0.980785	+	0.195090i	$0.0625000\pi$
$168$		0			0
$169$			1.00000i			1.00000i
$170$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$171$	−0.765367			−0.765367
$172$		0			0
$173$	0.149316	+	0.750661i	0.149316	+	0.750661i	0.980785	+	0.195090i	$0.0625000\pi$
$173$	0.149316	+	0.750661i	0.149316	+	0.750661i	−0.831470	+	0.555570i	$0.812500\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$179$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$180$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$181$	−1.92388	−	0.382683i	−1.92388	−	0.382683i	−0.923880	−	0.382683i	$-0.875000\pi$
$181$	−1.92388	−	0.382683i	−1.92388	−	0.382683i	−1.00000			$\pi$
$182$		0			0
$183$	−1.17588	−	1.17588i	−1.17588	−	1.17588i
$184$	0.541196	−	1.30656i	0.541196	−	1.30656i
$185$		0			0
$186$	−0.216773	+	1.08979i	−0.216773	+	1.08979i
$187$		0			0
$188$	0.425215	−	1.02656i	0.425215	−	1.02656i
$189$		0			0
$190$	−0.765367			−0.765367
$191$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$191$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$192$	0.980785	+	0.195090i	0.980785	+	0.195090i
$193$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$193$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$194$		0			0
$195$		0			0
$196$	−1.00000			−1.00000
$197$	−0.785695	−	1.17588i	−0.785695	−	1.17588i	−0.980785	−	0.195090i	$-0.937500\pi$
$197$	−0.785695	−	1.17588i	−0.785695	−	1.17588i	0.195090	−	0.980785i	$-0.437500\pi$
$198$		0			0
$199$	1.08979	−	1.63099i	1.08979	−	1.63099i	0.382683	−	0.923880i	$-0.375000\pi$
$199$	1.08979	−	1.63099i	1.08979	−	1.63099i	0.707107	−	0.707107i	$-0.250000\pi$
$200$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$201$		0			0
$202$		0			0
$203$		0			0
$204$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$205$		0			0
$206$		0			0
$207$	0.275899	+	1.38704i	0.275899	+	1.38704i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.216773	−	0.324423i	−0.216773	−	0.324423i	0.707107	−	0.707107i	$-0.250000\pi$
$211$	−0.216773	−	0.324423i	−0.216773	−	0.324423i	−0.923880	+	0.382683i	$0.875000\pi$
$212$	0.390181			0.390181
$213$		0			0
$214$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$215$		0			0
$216$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$217$		0			0
$218$		−	0.390181i		−	0.390181i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$223$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$224$		0			0
$225$	0.707107	+	0.707107i	0.707107	+	0.707107i
$226$	1.30656	+	1.30656i	1.30656	+	1.30656i
$227$	−1.81225	−	0.360480i	−1.81225	−	0.360480i	−0.831470	−	0.555570i	$-0.812500\pi$
$227$	−1.81225	−	0.360480i	−1.81225	−	0.360480i	−0.980785	+	0.195090i	$0.937500\pi$
$228$	0.149316	+	0.750661i	0.149316	+	0.750661i
$229$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$229$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$230$	0.275899	+	1.38704i	0.275899	+	1.38704i
$231$		0			0
$232$		0			0
$233$	1.17588	+	0.785695i	1.17588	+	0.785695i	0.980785	−	0.195090i	$-0.0625000\pi$
$233$	1.17588	+	0.785695i	1.17588	+	0.785695i	0.195090	+	0.980785i	$0.437500\pi$
$234$		0			0
$235$	0.216773	+	1.08979i	0.216773	+	1.08979i
$236$		0			0
$237$	−1.66294			−1.66294
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$241$	1.63099	−	0.324423i	1.63099	−	0.324423i	0.707107	−	0.707107i	$-0.250000\pi$
$241$	1.63099	−	0.324423i	1.63099	−	0.324423i	0.923880	+	0.382683i	$0.125000\pi$
$242$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$243$	0.555570	−	0.831470i	0.555570	−	0.831470i
$244$	−0.923880	+	1.38268i	−0.923880	+	1.38268i
$245$	0.831470	−	0.555570i	0.831470	−	0.555570i
$246$		0			0
$247$		0			0
$248$	1.11114			1.11114
$249$	−0.382683	+	1.92388i	−0.382683	+	1.92388i
$250$	0.707107	+	0.707107i	0.707107	+	0.707107i
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	0.195090	−	0.980785i	0.195090	−	0.980785i
$256$		−	1.00000i		−	1.00000i
$257$	1.53636	+	0.636379i	1.53636	+	0.636379i	0.980785	−	0.195090i	$-0.0625000\pi$
$257$	1.53636	+	0.636379i	1.53636	+	0.636379i	0.555570	+	0.831470i	$0.312500\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.360480	+	0.149316i	−0.360480	+	0.149316i	−0.555570	−	0.831470i	$-0.687500\pi$
$263$	−0.360480	+	0.149316i	−0.360480	+	0.149316i	0.195090	+	0.980785i	$0.437500\pi$
$264$		0			0
$265$	−0.324423	+	0.216773i	−0.324423	+	0.216773i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$269$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$270$	0.555570	−	0.831470i	0.555570	−	0.831470i
$271$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$271$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$272$	0.831470	+	0.555570i	0.831470	+	0.555570i
$273$		0			0
$274$	0.617317	−	0.923880i	0.617317	−	0.923880i
$275$		0			0
$276$	1.30656	−	0.541196i	1.30656	−	0.541196i
$277$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$277$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$278$	−0.750661	+	1.81225i	−0.750661	+	1.81225i
$279$	−0.923880	+	0.617317i	−0.923880	+	0.617317i
$280$		0			0
$281$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$281$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$282$	1.02656	−	0.425215i	1.02656	−	0.425215i
$283$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$283$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$284$		0			0
$285$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$286$		0			0
$287$		0			0
$288$	0.555570	+	0.831470i	0.555570	+	0.831470i
$289$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.17588	−	1.17588i	1.17588	−	1.17588i	0.195090	−	0.980785i	$-0.437500\pi$
$293$	1.17588	−	1.17588i	1.17588	−	1.17588i	0.980785	−	0.195090i	$-0.0625000\pi$
$294$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.555570	−	0.831470i	0.555570	−	0.831470i
$301$		0			0
$302$	1.17588	−	0.785695i	1.17588	−	0.785695i
$303$		0			0
$304$	0.707107	−	0.292893i	0.707107	−	0.292893i
$305$		−	1.66294i		−	1.66294i
$306$	−1.00000			−1.00000
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	−0.923880	+	0.617317i	−0.923880	+	0.617317i
$311$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$311$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$312$		0			0
$313$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$313$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$314$		0			0
$315$		0			0
$316$	0.324423	+	1.63099i	0.324423	+	1.63099i
$317$	−1.81225	−	0.360480i	−1.81225	−	0.360480i	−0.831470	−	0.555570i	$-0.812500\pi$
$317$	−1.81225	−	0.360480i	−1.81225	−	0.360480i	−0.980785	+	0.195090i	$0.937500\pi$
$318$	0.275899	+	0.275899i	0.275899	+	0.275899i
$319$		0			0
$320$	0.555570	+	0.831470i	0.555570	+	0.831470i
$321$	0.292893	−	0.707107i	0.292893	−	0.707107i
$322$		0			0
$323$	−0.149316	+	0.750661i	−0.149316	+	0.750661i
$324$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$325$		0			0
$326$		0			0
$327$	0.275899	−	0.275899i	0.275899	−	0.275899i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.707107	−	1.70711i	−0.707107	−	1.70711i	−0.707107	−	0.707107i	$-0.750000\pi$
$331$	−0.707107	−	1.70711i	−0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$332$	1.96157			1.96157
$333$		0			0
$334$	0.707107	−	1.70711i	0.707107	−	1.70711i
$335$		0			0
$336$		0			0
$337$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$337$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$338$	0.831470	+	0.555570i	0.831470	+	0.555570i
$339$			1.84776i			1.84776i
$340$	−1.00000			−1.00000
$341$		0			0
$342$	−0.425215	+	0.636379i	−0.425215	+	0.636379i
$343$		0			0
$344$		0			0
$345$	−0.785695	+	1.17588i	−0.785695	+	1.17588i
$346$	0.707107	+	0.292893i	0.707107	+	0.292893i
$347$	1.02656	+	1.53636i	1.02656	+	1.53636i	0.831470	+	0.555570i	$0.187500\pi$
$347$	1.02656	+	1.53636i	1.02656	+	1.53636i	0.195090	+	0.980785i	$0.437500\pi$
$348$		0			0
$349$	0.765367	+	1.84776i	0.765367	+	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$349$	0.765367	+	1.84776i	0.765367	+	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.38704	−	1.38704i	1.38704	−	1.38704i	0.555570	−	0.831470i	$-0.312500\pi$
$353$	1.38704	−	1.38704i	1.38704	−	1.38704i	0.831470	−	0.555570i	$-0.187500\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$359$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$360$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$361$	−0.292893	−	0.292893i	−0.292893	−	0.292893i
$362$	−1.38704	+	1.38704i	−1.38704	+	1.38704i
$363$	−0.980785	−	0.195090i	−0.980785	−	0.195090i
$364$		0			0
$365$		0			0
$366$	−1.63099	+	0.324423i	−1.63099	+	0.324423i
$367$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$367$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$368$	−0.785695	−	1.17588i	−0.785695	−	1.17588i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.785695	+	0.785695i	0.785695	+	0.785695i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$			1.00000i			1.00000i
$376$	−0.617317	−	0.923880i	−0.617317	−	0.923880i
$377$		0			0
$378$		0			0
$379$	−0.216773	+	0.324423i	−0.216773	+	0.324423i	−0.923880	−	0.382683i	$-0.875000\pi$
$379$	−0.216773	+	0.324423i	−0.216773	+	0.324423i	0.707107	+	0.707107i	$0.250000\pi$
$380$	−0.425215	+	0.636379i	−0.425215	+	0.636379i
$381$		0			0
$382$		0			0
$383$	−0.636379	−	1.53636i	−0.636379	−	1.53636i	−0.831470	−	0.555570i	$-0.812500\pi$
$383$	−0.636379	−	1.53636i	−0.636379	−	1.53636i	0.195090	−	0.980785i	$-0.437500\pi$
$384$	0.707107	−	0.707107i	0.707107	−	0.707107i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$389$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$390$		0			0
$391$	1.41421			1.41421
$392$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$393$		0			0
$394$	−1.41421			−1.41421
$395$	−1.17588	−	1.17588i	−1.17588	−	1.17588i
$396$		0			0
$397$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$397$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$398$	−0.750661	−	1.81225i	−0.750661	−	1.81225i
$399$		0			0
$400$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$401$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$401$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.980785	−	0.195090i	0.980785	−	0.195090i
$406$		0			0
$407$		0			0
$408$	0.195090	+	0.980785i	0.195090	+	0.980785i
$409$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$409$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$410$		0			0
$411$	1.08979	−	0.216773i	1.08979	−	0.216773i
$412$		0			0
$413$		0			0
$414$	1.30656	+	0.541196i	1.30656	+	0.541196i
$415$	−1.63099	+	1.08979i	−1.63099	+	1.08979i
$416$		0			0
$417$	−1.81225	+	0.750661i	−1.81225	+	0.750661i
$418$		0			0
$419$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$419$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$420$		0			0
$421$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$421$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$422$	−0.390181			−0.390181
$423$	1.02656	+	0.425215i	1.02656	+	0.425215i
$424$	0.216773	−	0.324423i	0.216773	−	0.324423i
$425$	0.831470	−	0.555570i	0.831470	−	0.555570i
$426$		0			0
$427$		0			0
$428$	−0.750661	−	0.149316i	−0.750661	−	0.149316i
$429$		0			0
$430$		0			0
$431$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$431$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$432$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$433$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$433$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$434$		0			0
$435$		0			0
$436$	−0.324423	−	0.216773i	−0.324423	−	0.216773i
$437$	0.601345	−	0.899976i	0.601345	−	0.899976i
$438$		0			0
$439$	0.382683	−	0.0761205i	0.382683	−	0.0761205i		−	1.00000i	$-0.5\pi$
$439$	0.382683	−	0.0761205i	0.382683	−	0.0761205i	0.382683	+	0.923880i	$0.375000\pi$
$440$		0			0
$441$		−	1.00000i		−	1.00000i
$442$		0			0
$443$	−1.66294			−1.66294			−0.831470	−	0.555570i	$-0.812500\pi$
$443$	−1.66294			−1.66294			−0.831470	+	0.555570i	$0.812500\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$449$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$450$	0.980785	−	0.195090i	0.980785	−	0.195090i
$451$		0			0
$452$	1.81225	−	0.360480i	1.81225	−	0.360480i
$453$	1.38704	+	0.275899i	1.38704	+	0.275899i
$454$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$455$		0			0
$456$	0.707107	+	0.292893i	0.707107	+	0.292893i
$457$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$457$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$458$	0.360480	−	1.81225i	0.360480	−	1.81225i
$459$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$460$	1.30656	+	0.541196i	1.30656	+	0.541196i
$461$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$461$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$	−1.08979	−	0.216773i	−1.08979	−	0.216773i
$466$	1.30656	−	0.541196i	1.30656	−	0.541196i
$467$	−0.425215	−	1.02656i	−0.425215	−	1.02656i	−0.980785	−	0.195090i	$-0.937500\pi$
$467$	−0.425215	−	1.02656i	−0.425215	−	1.02656i	0.555570	−	0.831470i	$-0.312500\pi$
$468$		0			0
$469$		0			0
$470$	1.02656	+	0.425215i	1.02656	+	0.425215i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−0.923880	+	1.38268i	−0.923880	+	1.38268i
$475$		−	0.765367i		−	0.765367i
$476$		0			0
$477$			0.390181i			0.390181i
$478$		0			0
$479$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$479$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$480$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$481$		0			0
$482$	0.636379	−	1.53636i	0.636379	−	1.53636i
$483$		0			0
$484$			1.00000i			1.00000i
$485$		0			0
$486$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$487$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$487$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$488$	0.636379	+	1.53636i	0.636379	+	1.53636i
$489$		0			0
$490$		−	1.00000i		−	1.00000i
$491$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$491$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.617317	−	0.923880i	0.617317	−	0.923880i
$497$		0			0
$498$	1.38704	+	1.38704i	1.38704	+	1.38704i
$499$	1.92388	+	0.382683i	1.92388	+	0.382683i	1.00000			$0$
$499$	1.92388	+	0.382683i	1.92388	+	0.382683i	0.923880	+	0.382683i	$0.125000\pi$
$500$	0.980785	−	0.195090i	0.980785	−	0.195090i
$501$	1.70711	−	0.707107i	1.70711	−	0.707107i
$502$		0			0
$503$	−1.02656	−	1.53636i	−1.02656	−	1.53636i	−0.831470	−	0.555570i	$-0.812500\pi$
$503$	−1.02656	−	1.53636i	−1.02656	−	1.53636i	−0.195090	−	0.980785i	$-0.562500\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.195090	+	0.980785i	0.195090	+	0.980785i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$511$		0			0
$512$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$513$	−0.750661	+	0.149316i	−0.750661	+	0.149316i
$514$	1.38268	−	0.923880i	1.38268	−	0.923880i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	0.292893	+	0.707107i	0.292893	+	0.707107i
$520$		0			0
$521$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$521$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i
$527$	0.425215	+	1.02656i	0.425215	+	1.02656i
$528$		0			0
$529$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$530$			0.390181i			0.390181i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.707107	−	0.292893i	0.707107	−	0.292893i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$541$	−1.92388	+	0.382683i	−1.92388	+	0.382683i	−0.923880	+	0.382683i	$0.875000\pi$
$541$	−1.92388	+	0.382683i	−1.92388	+	0.382683i	−1.00000			$\pi$
$542$	−0.425215	+	0.636379i	−0.425215	+	0.636379i
$543$	−1.96157			−1.96157
$544$	0.923880	−	0.382683i	0.923880	−	0.382683i
$545$	0.390181			0.390181
$546$		0			0
$547$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$547$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$548$	−0.425215	−	1.02656i	−0.425215	−	1.02656i
$549$	−1.38268	−	0.923880i	−1.38268	−	0.923880i
$550$		0			0
$551$		0			0
$552$	0.275899	−	1.38704i	0.275899	−	1.38704i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	1.08979	+	1.63099i	1.08979	+	1.63099i
$557$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.831470	−	0.555570i	$-0.812500\pi$
$557$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.555570	−	0.831470i	$-0.687500\pi$
$558$			1.11114i			1.11114i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−0.360480	−	0.149316i	−0.360480	−	0.149316i	0.195090	−	0.980785i	$-0.437500\pi$
$563$	−0.360480	−	0.149316i	−0.360480	−	0.149316i	−0.555570	+	0.831470i	$0.687500\pi$
$564$	0.216773	−	1.08979i	0.216773	−	1.08979i
$565$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$569$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$570$	−0.750661	+	0.149316i	−0.750661	+	0.149316i
$571$	1.38268	−	0.923880i	1.38268	−	0.923880i	0.382683	−	0.923880i	$-0.375000\pi$
$571$	1.38268	−	0.923880i	1.38268	−	0.923880i	1.00000			$0$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.38704	+	0.275899i	−1.38704	+	0.275899i
$576$	1.00000			1.00000
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−0.324423	−	1.63099i	−0.324423	−	1.63099i
$587$	−0.360480	+	0.149316i	−0.360480	+	0.149316i	−0.555570	−	0.831470i	$-0.687500\pi$
$587$	−0.360480	+	0.149316i	−0.360480	+	0.149316i	0.195090	+	0.980785i	$0.437500\pi$
$588$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$589$	0.834089	+	0.165911i	0.834089	+	0.165911i
$590$		0			0
$591$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$592$		0			0
$593$	−0.425215	+	1.02656i	−0.425215	+	1.02656i	0.555570	+	0.831470i	$0.312500\pi$
$593$	−0.425215	+	1.02656i	−0.425215	+	1.02656i	−0.980785	+	0.195090i	$0.937500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.750661	−	1.81225i	0.750661	−	1.81225i
$598$		0			0
$599$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$599$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$600$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$601$	−1.08979	−	0.216773i	−1.08979	−	0.216773i	−0.382683	−	0.923880i	$-0.625000\pi$
$601$	−1.08979	−	0.216773i	−1.08979	−	0.216773i	−0.707107	+	0.707107i	$0.750000\pi$
$602$		0			0
$603$		0			0
$604$		−	1.41421i		−	1.41421i
$605$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$606$		0			0
$607$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$607$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$608$	0.149316	−	0.750661i	0.149316	−	0.750661i
$609$		0			0
$610$	−1.38268	−	0.923880i	−1.38268	−	0.923880i
$611$		0			0
$612$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.02656	+	1.53636i	−1.02656	+	1.53636i	−0.195090	+	0.980785i	$0.562500\pi$
$617$	−1.02656	+	1.53636i	−1.02656	+	1.53636i	−0.831470	+	0.555570i	$0.812500\pi$
$618$		0			0
$619$	0.923880	+	1.38268i	0.923880	+	1.38268i	0.923880	+	0.382683i	$0.125000\pi$
$619$	0.923880	+	1.38268i	0.923880	+	1.38268i			1.00000i	$0.5\pi$
$620$			1.11114i			1.11114i
$621$	0.541196	+	1.30656i	0.541196	+	1.30656i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.541196	−	1.30656i	0.541196	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$631$	0.541196	−	1.30656i	0.541196	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$632$	1.53636	+	0.636379i	1.53636	+	0.636379i
$633$	−0.275899	−	0.275899i	−0.275899	−	0.275899i
$634$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$635$		0			0
$636$	0.382683	−	0.0761205i	0.382683	−	0.0761205i
$637$		0			0
$638$		0			0
$639$		0			0
$640$	1.00000			1.00000
$641$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$641$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$642$	−0.425215	−	0.636379i	−0.425215	−	0.636379i
$643$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$643$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$644$		0			0
$645$		0			0
$646$	0.541196	+	0.541196i	0.541196	+	0.541196i
$647$			1.96157i			1.96157i	0.195090	+	0.980785i	$0.437500\pi$
$647$			1.96157i			1.96157i	−0.195090	+	0.980785i	$0.562500\pi$
$648$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.53636	+	1.02656i	−1.53636	+	1.02656i	−0.555570	+	0.831470i	$0.687500\pi$
$653$	−1.53636	+	1.02656i	−1.53636	+	1.02656i	−0.980785	+	0.195090i	$0.937500\pi$
$654$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$659$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$661$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$662$	−1.81225	−	0.360480i	−1.81225	−	0.360480i
$663$		0			0
$664$	1.08979	−	1.63099i	1.08979	−	1.63099i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1.02656	−	1.53636i	−1.02656	−	1.53636i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$673$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$674$		0			0
$675$	0.831470	+	0.555570i	0.831470	+	0.555570i
$676$	0.923880	−	0.382683i	0.923880	−	0.382683i
$677$	1.38704	−	0.275899i	1.38704	−	0.275899i	0.555570	−	0.831470i	$-0.312500\pi$
$677$	1.38704	−	0.275899i	1.38704	−	0.275899i	0.831470	+	0.555570i	$0.187500\pi$
$678$	1.53636	+	1.02656i	1.53636	+	1.02656i
$679$		0			0
$680$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$681$	−1.84776			−1.84776
$682$		0			0
$683$	0.750661	−	0.149316i	0.750661	−	0.149316i	0.195090	−	0.980785i	$-0.437500\pi$
$683$	0.750661	−	0.149316i	0.750661	−	0.149316i	0.555570	+	0.831470i	$0.312500\pi$
$684$	0.292893	+	0.707107i	0.292893	+	0.707107i
$685$	0.923880	+	0.617317i	0.923880	+	0.617317i
$686$		0			0
$687$	1.53636	−	1.02656i	1.53636	−	1.02656i
$688$		0			0
$689$		0			0
$690$	0.541196	+	1.30656i	0.541196	+	1.30656i
$691$	0.216773	−	1.08979i	0.216773	−	1.08979i	−0.707107	−	0.707107i	$-0.750000\pi$
$691$	0.216773	−	1.08979i	0.216773	−	1.08979i	0.923880	−	0.382683i	$-0.125000\pi$
$692$	0.636379	−	0.425215i	0.636379	−	0.425215i
$693$		0			0
$694$	1.84776			1.84776
$695$	−1.81225	−	0.750661i	−1.81225	−	0.750661i
$696$		0			0
$697$		0			0
$698$	1.96157	+	0.390181i	1.96157	+	0.390181i
$699$	1.30656	+	0.541196i	1.30656	+	0.541196i
$700$		0			0
$701$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$701$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	0.425215	+	1.02656i	0.425215	+	1.02656i
$706$	−0.382683	−	1.92388i	−0.382683	−	1.92388i
$707$		0			0
$708$		0			0
$709$	0.216773	−	0.324423i	0.216773	−	0.324423i	−0.707107	−	0.707107i	$-0.750000\pi$
$709$	0.216773	−	0.324423i	0.216773	−	0.324423i	0.923880	+	0.382683i	$0.125000\pi$
$710$		0			0
$711$	−1.63099	+	0.324423i	−1.63099	+	0.324423i
$712$		0			0
$713$		−	1.57139i		−	1.57139i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$719$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$720$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$721$		0			0
$722$	−0.406255	+	0.0808091i	−0.406255	+	0.0808091i
$723$	1.53636	−	0.636379i	1.53636	−	0.636379i
$724$	0.382683	+	1.92388i	0.382683	+	1.92388i
$725$		0			0
$726$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$	0.382683	−	0.923880i	0.382683	−	0.923880i
$730$		0			0
$731$		0			0
$732$	−0.636379	+	1.53636i	−0.636379	+	1.53636i
$733$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$733$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$734$		0			0
$735$	0.707107	−	0.707107i	0.707107	−	0.707107i
$736$	−1.41421			−1.41421
$737$		0			0
$738$		0			0
$739$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$739$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.425215	−	0.636379i	0.425215	−	0.636379i	−0.555570	−	0.831470i	$-0.687500\pi$
$743$	0.425215	−	0.636379i	0.425215	−	0.636379i	0.980785	+	0.195090i	$0.0625000\pi$
$744$	1.08979	−	0.216773i	1.08979	−	0.216773i
$745$		0			0
$746$		0			0
$747$			1.96157i			1.96157i
$748$		0			0
$749$		0			0
$750$	0.831470	+	0.555570i	0.831470	+	0.555570i
$751$	0.382683	+	1.92388i	0.382683	+	1.92388i	0.382683	+	0.923880i	$0.375000\pi$
$751$	0.382683	+	1.92388i	0.382683	+	1.92388i			1.00000i	$0.5\pi$
$752$	−1.11114			−1.11114
$753$		0			0
$754$		0			0
$755$	0.785695	+	1.17588i	0.785695	+	1.17588i
$756$		0			0
$757$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$757$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$758$	0.149316	+	0.360480i	0.149316	+	0.360480i
$759$		0			0
$760$	0.292893	+	0.707107i	0.292893	+	0.707107i
$761$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$761$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		−	1.00000i		−	1.00000i
$766$	−1.63099	−	0.324423i	−1.63099	−	0.324423i
$767$		0			0
$768$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$769$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$769$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$770$		0			0
$771$	1.63099	+	0.324423i	1.63099	+	0.324423i
$772$		0			0
$773$	1.81225	−	0.750661i	1.81225	−	0.750661i	0.831470	−	0.555570i	$-0.187500\pi$
$773$	1.81225	−	0.750661i	1.81225	−	0.750661i	0.980785	−	0.195090i	$-0.0625000\pi$
$774$		0			0
$775$	−0.617317	−	0.923880i	−0.617317	−	0.923880i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	0.785695	−	1.17588i	0.785695	−	1.17588i
$783$		0			0
$784$	0.382683	+	0.923880i	0.382683	+	0.923880i
$785$		0			0
$786$		0			0
$787$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$787$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$788$	−0.785695	+	1.17588i	−0.785695	+	1.17588i
$789$	−0.324423	+	0.216773i	−0.324423	+	0.216773i
$790$	−1.63099	+	0.324423i	−1.63099	+	0.324423i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−0.275899	+	0.275899i	−0.275899	+	0.275899i
$796$	−1.92388	−	0.382683i	−1.92388	−	0.382683i
$797$	−1.02656	−	0.425215i	−1.02656	−	0.425215i	−0.195090	−	0.980785i	$-0.562500\pi$
$797$	−1.02656	−	0.425215i	−1.02656	−	0.425215i	−0.831470	+	0.555570i	$0.812500\pi$
$798$		0			0
$799$	0.617317	−	0.923880i	0.617317	−	0.923880i
$800$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$801$		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		0.831470	−	0.555570i	$-0.187500\pi$
$809$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$810$	0.382683	−	0.923880i	0.382683	−	0.923880i
$811$	0.324423	+	0.216773i	0.324423	+	0.216773i	0.707107	−	0.707107i	$-0.250000\pi$
$811$	0.324423	+	0.216773i	0.324423	+	0.216773i	−0.382683	+	0.923880i	$0.625000\pi$
$812$		0			0
$813$	−0.750661	+	0.149316i	−0.750661	+	0.149316i
$814$		0			0
$815$		0			0
$816$	0.923880	+	0.382683i	0.923880	+	0.382683i
$817$		0			0
$818$	1.02656	−	1.53636i	1.02656	−	1.53636i
$819$		0			0
$820$		0			0
$821$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$821$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$822$	0.425215	−	1.02656i	0.425215	−	1.02656i
$823$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$823$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.275899	−	1.38704i	0.275899	−	1.38704i	−0.555570	−	0.831470i	$-0.687500\pi$
$827$	0.275899	−	1.38704i	0.275899	−	1.38704i	0.831470	−	0.555570i	$-0.187500\pi$
$828$	1.17588	−	0.785695i	1.17588	−	0.785695i
$829$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$829$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$830$			1.96157i			1.96157i
$831$		0			0
$832$		0			0
$833$	−0.980785	−	0.195090i	−0.980785	−	0.195090i
$834$	−0.382683	+	1.92388i	−0.382683	+	1.92388i
$835$	1.70711	+	0.707107i	1.70711	+	0.707107i
$836$		0			0
$837$	−0.785695	+	0.785695i	−0.785695	+	0.785695i
$838$		0			0
$839$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$839$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$840$		0			0
$841$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$842$	0.750661	−	0.149316i	0.750661	−	0.149316i
$843$		0			0
$844$	−0.216773	+	0.324423i	−0.216773	+	0.324423i
$845$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$846$	0.923880	−	0.617317i	0.923880	−	0.617317i
$847$		0			0
$848$	−0.149316	−	0.360480i	−0.149316	−	0.360480i
$849$		0			0
$850$		−	1.00000i		−	1.00000i
$851$		0			0
$852$		0			0
$853$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$853$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$854$		0			0
$855$	−0.636379	−	0.425215i	−0.636379	−	0.425215i
$856$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$857$	−0.785695	−	1.17588i	−0.785695	−	1.17588i	−0.980785	−	0.195090i	$-0.937500\pi$
$857$	−0.785695	−	1.17588i	−0.785695	−	1.17588i	0.195090	−	0.980785i	$-0.437500\pi$
$858$		0			0
$859$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$859$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.275899	−	0.275899i	−0.275899	−	0.275899i	0.555570	−	0.831470i	$-0.312500\pi$
$863$	−0.275899	−	0.275899i	−0.275899	−	0.275899i	−0.831470	+	0.555570i	$0.812500\pi$
$864$	0.707107	+	0.707107i	0.707107	+	0.707107i
$865$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$866$		0			0
$867$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.360480	+	0.149316i	−0.360480	+	0.149316i
$873$		0			0
$874$	−0.414214	−	1.00000i	−0.414214	−	1.00000i
$875$		0			0
$876$		0			0
$877$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$877$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$878$	0.149316	−	0.360480i	0.149316	−	0.360480i
$879$	0.923880	−	1.38268i	0.923880	−	1.38268i
$880$		0			0
$881$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$881$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$882$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	−0.923880	+	1.38268i	−0.923880	+	1.38268i
$887$	−0.390181	−	1.96157i	−0.390181	−	1.96157i	−0.195090	−	0.980785i	$-0.562500\pi$
$887$	−0.390181	−	1.96157i	−0.390181	−	1.96157i	−0.195090	−	0.980785i	$-0.562500\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.325446	−	0.785695i	−0.325446	−	0.785695i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.382683	−	0.923880i	0.382683	−	0.923880i
$901$	0.382683	+	0.0761205i	0.382683	+	0.0761205i
$902$		0			0
$903$		0			0
$904$	0.707107	−	1.70711i	0.707107	−	1.70711i
$905$	−1.38704	−	1.38704i	−1.38704	−	1.38704i
$906$	1.00000	−	1.00000i	1.00000	−	1.00000i
$907$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$907$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$908$	0.360480	+	1.81225i	0.360480	+	1.81225i
$909$		0			0
$910$		0			0
$911$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$911$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$912$	0.636379	−	0.425215i	0.636379	−	0.425215i
$913$		0			0
$914$		0			0
$915$	−0.324423	−	1.63099i	−0.324423	−	1.63099i
$916$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$917$		0			0
$918$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$919$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$919$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$920$	1.17588	−	0.785695i	1.17588	−	0.785695i
$921$		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		0.195090	−	0.980785i	$-0.437500\pi$
$929$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$930$	−0.785695	+	0.785695i	−0.785695	+	0.785695i
$931$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$932$	0.275899	−	1.38704i	0.275899	−	1.38704i
$933$		0			0
$934$	−1.08979	−	0.216773i	−1.08979	−	0.216773i
$935$		0			0
$936$		0			0
$937$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$937$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$938$		0			0
$939$		0			0
$940$	0.923880	−	0.617317i	0.923880	−	0.617317i
$941$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$941$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$947$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$948$	0.636379	+	1.53636i	0.636379	+	1.53636i
$949$		0			0
$950$	−0.636379	−	0.425215i	−0.636379	−	0.425215i
$951$	−1.84776			−1.84776
$952$		0			0
$953$	1.66294			1.66294			0.831470	−	0.555570i	$-0.187500\pi$
$953$	1.66294			1.66294			0.831470	+	0.555570i	$0.187500\pi$
$954$	0.324423	+	0.216773i	0.324423	+	0.216773i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	0.707107	+	0.707107i	0.707107	+	0.707107i
$961$	0.216773	−	0.0897902i	0.216773	−	0.0897902i
$962$		0			0
$963$	0.149316	−	0.750661i	0.149316	−	0.750661i
$964$	−0.923880	−	1.38268i	−0.923880	−	1.38268i
$965$		0			0
$966$		0			0
$967$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$967$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$968$	0.831470	+	0.555570i	0.831470	+	0.555570i
$969$			0.765367i			0.765367i
$970$		0			0
$971$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$971$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$972$	−0.980785	−	0.195090i	−0.980785	−	0.195090i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.63099	+	0.324423i	1.63099	+	0.324423i
$977$	0.149316	+	0.360480i	0.149316	+	0.360480i	0.980785	−	0.195090i	$-0.0625000\pi$
$977$	0.149316	+	0.360480i	0.149316	+	0.360480i	−0.831470	+	0.555570i	$0.812500\pi$
$978$		0			0
$979$		0			0
$980$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$981$	0.216773	−	0.324423i	0.216773	−	0.324423i
$982$		0			0
$983$	0.750661	−	0.149316i	0.750661	−	0.149316i	0.195090	−	0.980785i	$-0.437500\pi$
$983$	0.750661	−	0.149316i	0.750661	−	0.149316i	0.555570	+	0.831470i	$0.312500\pi$
$984$		0			0
$985$		−	1.41421i		−	1.41421i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.324423	−	0.216773i	−0.324423	−	0.216773i	0.382683	−	0.923880i	$-0.375000\pi$
$991$	−0.324423	−	0.216773i	−0.324423	−	0.216773i	−0.707107	+	0.707107i	$0.750000\pi$
$992$	−0.425215	−	1.02656i	−0.425215	−	1.02656i
$993$	−1.02656	−	1.53636i	−1.02656	−	1.53636i
$994$		0			0
$995$	1.81225	−	0.750661i	1.81225	−	0.750661i
$996$	1.92388	−	0.382683i	1.92388	−	0.382683i
$997$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$997$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$998$	1.38704	−	1.38704i	1.38704	−	1.38704i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2040.1.fb.a.1859.2	yes	16
3.2	odd	2	inner	2040.1.fb.a.1859.1	yes	16
5.4	even	2	inner	2040.1.fb.a.1859.1	yes	16
8.3	odd	2		2040.1.fb.b.1859.1	yes	16
15.14	odd	2	CM	2040.1.fb.a.1859.2	yes	16
17.3	odd	16		2040.1.fb.b.1499.1	yes	16
24.11	even	2		2040.1.fb.b.1859.2	yes	16
40.19	odd	2		2040.1.fb.b.1859.2	yes	16
51.20	even	16		2040.1.fb.b.1499.2	yes	16
85.54	odd	16		2040.1.fb.b.1499.2	yes	16
120.59	even	2		2040.1.fb.b.1859.1	yes	16
136.3	even	16	inner	2040.1.fb.a.1499.2	yes	16
255.224	even	16		2040.1.fb.b.1499.1	yes	16
408.275	odd	16	inner	2040.1.fb.a.1499.1	✓	16
680.139	even	16	inner	2040.1.fb.a.1499.1	✓	16
2040.1499	odd	16	inner	2040.1.fb.a.1499.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.1.fb.a.1499.1	✓	16	408.275	odd	16	inner
2040.1.fb.a.1499.1	✓	16	680.139	even	16	inner
2040.1.fb.a.1499.2	yes	16	136.3	even	16	inner
2040.1.fb.a.1499.2	yes	16	2040.1499	odd	16	inner
2040.1.fb.a.1859.1	yes	16	3.2	odd	2	inner
2040.1.fb.a.1859.1	yes	16	5.4	even	2	inner
2040.1.fb.a.1859.2	yes	16	1.1	even	1	trivial
2040.1.fb.a.1859.2	yes	16	15.14	odd	2	CM
2040.1.fb.b.1499.1	yes	16	17.3	odd	16
2040.1.fb.b.1499.1	yes	16	255.224	even	16
2040.1.fb.b.1499.2	yes	16	51.20	even	16
2040.1.fb.b.1499.2	yes	16	85.54	odd	16
2040.1.fb.b.1859.1	yes	16	8.3	odd	2
2040.1.fb.b.1859.1	yes	16	120.59	even	2
2040.1.fb.b.1859.2	yes	16	24.11	even	2
2040.1.fb.b.1859.2	yes	16	40.19	odd	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.fb (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(0.555570 - 0.831470i) q^{2} +(0.980785 - 0.195090i) q^{3} +(-0.382683 - 0.923880i) q^{4} +(0.831470 + 0.555570i) q^{5} +(0.382683 - 0.923880i) q^{6} +(-0.980785 - 0.195090i) q^{8} +(0.923880 - 0.382683i) q^{9} +(0.923880 - 0.382683i) q^{10} +(-0.555570 - 0.831470i) q^{12} +(0.923880 + 0.382683i) q^{15} +(-0.707107 + 0.707107i) q^{16} +(-0.195090 - 0.980785i) q^{17} +(0.195090 - 0.980785i) q^{18} +(-0.707107 - 0.292893i) q^{19} +(0.195090 - 0.980785i) q^{20} +(-0.275899 + 1.38704i) q^{23} -1.00000 q^{24} +(0.382683 + 0.923880i) q^{25} +(0.831470 - 0.555570i) q^{27} +(0.831470 - 0.555570i) q^{30} +(-1.08979 + 0.216773i) q^{31} +(0.195090 + 0.980785i) q^{32} +(-0.923880 - 0.382683i) q^{34} +(-0.707107 - 0.707107i) q^{36} +(-0.636379 + 0.425215i) q^{38} +(-0.707107 - 0.707107i) q^{40} +(0.980785 + 0.195090i) q^{45} +(1.00000 + 1.00000i) q^{46} +(0.785695 + 0.785695i) q^{47} +(-0.555570 + 0.831470i) q^{48} +(0.382683 - 0.923880i) q^{49} +(0.980785 + 0.195090i) q^{50} +(-0.382683 - 0.923880i) q^{51} +(-0.149316 + 0.360480i) q^{53} -1.00000i q^{54} +(-0.750661 - 0.149316i) q^{57} -1.00000i q^{60} +(-0.923880 - 1.38268i) q^{61} +(-0.425215 + 1.02656i) q^{62} +(0.923880 + 0.382683i) q^{64} +(-0.831470 + 0.555570i) q^{68} +1.41421i q^{69} +(-0.980785 + 0.195090i) q^{72} +(0.555570 + 0.831470i) q^{75} +0.765367i q^{76} +(-1.63099 - 0.324423i) q^{79} +(-0.980785 + 0.195090i) q^{80} +(0.707107 - 0.707107i) q^{81} +(-0.750661 + 1.81225i) q^{83} +(0.382683 - 0.923880i) q^{85} +(0.707107 - 0.707107i) q^{90} +(1.38704 - 0.275899i) q^{92} +(-1.02656 + 0.425215i) q^{93} +(1.08979 - 0.216773i) q^{94} +(-0.425215 - 0.636379i) q^{95} +(0.382683 + 0.923880i) q^{96} +(-0.555570 - 0.831470i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{24} + 16 q^{46}+O(q^{100}) \) 16 * q - 16 * q^24 + 16 * q^46

Introduction

L-functions

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