LMFDB - Embedded newform 536.1.h.b.133.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [536,1,Mod(133,536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(536, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("536.133");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 536 = 2^{3} \cdot 67 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	536.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$0.267498846771$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.153990656.1
Artin image:	$D_7$
Artin field:	Galois closure of 7.1.153990656.1

Embedding invariants

Embedding label			133.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	536.133

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} +1.24698 q^{5} -0.445042 q^{6} +1.00000 q^{8} -0.801938 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} +1.24698 q^{5} -0.445042 q^{6} +1.00000 q^{8} -0.801938 q^{9} +1.24698 q^{10} -1.80194 q^{11} -0.445042 q^{12} -1.80194 q^{13} -0.554958 q^{15} +1.00000 q^{16} +1.24698 q^{17} -0.801938 q^{18} +1.24698 q^{20} -1.80194 q^{22} -0.445042 q^{23} -0.445042 q^{24} +0.554958 q^{25} -1.80194 q^{26} +0.801938 q^{27} -0.554958 q^{30} +1.00000 q^{32} +0.801938 q^{33} +1.24698 q^{34} -0.801938 q^{36} +0.801938 q^{39} +1.24698 q^{40} +1.24698 q^{43} -1.80194 q^{44} -1.00000 q^{45} -0.445042 q^{46} -1.80194 q^{47} -0.445042 q^{48} +1.00000 q^{49} +0.554958 q^{50} -0.554958 q^{51} -1.80194 q^{52} -0.445042 q^{53} +0.801938 q^{54} -2.24698 q^{55} -0.554958 q^{60} -0.445042 q^{61} +1.00000 q^{64} -2.24698 q^{65} +0.801938 q^{66} +1.00000 q^{67} +1.24698 q^{68} +0.198062 q^{69} +1.24698 q^{71} -0.801938 q^{72} -1.80194 q^{73} -0.246980 q^{75} +0.801938 q^{78} +1.24698 q^{80} +0.445042 q^{81} +1.55496 q^{85} +1.24698 q^{86} -1.80194 q^{88} -0.445042 q^{89} -1.00000 q^{90} -0.445042 q^{92} -1.80194 q^{94} -0.445042 q^{96} +1.00000 q^{98} +1.44504 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} + 3 q^{8} + 2 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 + 3 * q^8 + 2 * q^9	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} + 3 q^{8} + 2 q^{9} - q^{10} - q^{11} - q^{12} - q^{13} - 2 q^{15} + 3 q^{16} - q^{17} + 2 q^{18} - q^{20} - q^{22} - q^{23} - q^{24} + 2 q^{25} - q^{26} - 2 q^{27} - 2 q^{30} + 3 q^{32} - 2 q^{33} - q^{34} + 2 q^{36} - 2 q^{39} - q^{40} - q^{43} - q^{44} - 3 q^{45} - q^{46} - q^{47} - q^{48} + 3 q^{49} + 2 q^{50} - 2 q^{51} - q^{52} - q^{53} - 2 q^{54} - 2 q^{55} - 2 q^{60} - q^{61} + 3 q^{64} - 2 q^{65} - 2 q^{66} + 3 q^{67} - q^{68} + 5 q^{69} - q^{71} + 2 q^{72} - q^{73} + 4 q^{75} - 2 q^{78} - q^{80} + q^{81} + 5 q^{85} - q^{86} - q^{88} - q^{89} - 3 q^{90} - q^{92} - q^{94} - q^{96} + 3 q^{98} + 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 + 3 * q^8 + 2 * q^9 - q^10 - q^11 - q^12 - q^13 - 2 * q^15 + 3 * q^16 - q^17 + 2 * q^18 - q^20 - q^22 - q^23 - q^24 + 2 * q^25 - q^26 - 2 * q^27 - 2 * q^30 + 3 * q^32 - 2 * q^33 - q^34 + 2 * q^36 - 2 * q^39 - q^40 - q^43 - q^44 - 3 * q^45 - q^46 - q^47 - q^48 + 3 * q^49 + 2 * q^50 - 2 * q^51 - q^52 - q^53 - 2 * q^54 - 2 * q^55 - 2 * q^60 - q^61 + 3 * q^64 - 2 * q^65 - 2 * q^66 + 3 * q^67 - q^68 + 5 * q^69 - q^71 + 2 * q^72 - q^73 + 4 * q^75 - 2 * q^78 - q^80 + q^81 + 5 * q^85 - q^86 - q^88 - q^89 - 3 * q^90 - q^92 - q^94 - q^96 + 3 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/536\mathbb{Z}\right)^\times$.

$n$	$135$	$269$	$337$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	1.00000	1.00000
$2$	1.00000	1.00000
$3$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$3$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$4$	1.00000	1.00000
$5$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$5$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$6$	−0.445042	−0.445042
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000	1.00000
$9$	−0.801938	−0.801938
$10$	1.24698	1.24698
$11$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$11$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$12$	−0.445042	−0.445042
$13$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$13$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$14$	0	0
$15$	−0.554958	−0.554958
$16$	1.00000	1.00000
$17$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$17$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$18$	−0.801938	−0.801938
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	1.24698	1.24698
$21$	0	0
$22$	−1.80194	−1.80194
$23$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$23$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$24$	−0.445042	−0.445042
$25$	0.554958	0.554958
$26$	−1.80194	−1.80194
$27$	0.801938	0.801938
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−0.554958	−0.554958
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	1.00000	1.00000
$33$	0.801938	0.801938
$34$	1.24698	1.24698
$35$	0	0
$36$	−0.801938	−0.801938
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0.801938	0.801938
$40$	1.24698	1.24698
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$43$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$44$	−1.80194	−1.80194
$45$	−1.00000	−1.00000
$46$	−0.445042	−0.445042
$47$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$47$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$48$	−0.445042	−0.445042
$49$	1.00000	1.00000
$50$	0.554958	0.554958
$51$	−0.554958	−0.554958
$52$	−1.80194	−1.80194
$53$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$53$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$54$	0.801938	0.801938
$55$	−2.24698	−2.24698
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−0.554958	−0.554958
$61$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$61$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	1.00000
$65$	−2.24698	−2.24698
$66$	0.801938	0.801938
$67$	1.00000	1.00000
$67$	1.00000	1.00000
$68$	1.24698	1.24698
$69$	0.198062	0.198062
$70$	0	0
$71$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$71$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$72$	−0.801938	−0.801938
$73$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$73$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$74$	0	0
$75$	−0.246980	−0.246980
$76$	0	0
$77$	0	0
$78$	0.801938	0.801938
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	1.24698	1.24698
$81$	0.445042	0.445042
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	1.55496	1.55496
$86$	1.24698	1.24698
$87$	0	0
$88$	−1.80194	−1.80194
$89$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$89$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$90$	−1.00000	−1.00000
$91$	0	0
$92$	−0.445042	−0.445042
$93$	0	0
$94$	−1.80194	−1.80194
$95$	0	0
$96$	−0.445042	−0.445042
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.00000	1.00000
$99$	1.44504	1.44504
$100$	0.554958	0.554958
$101$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$101$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$102$	−0.554958	−0.554958
$103$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$103$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$104$	−1.80194	−1.80194
$105$	0	0
$106$	−0.445042	−0.445042
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0.801938	0.801938
$109$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$109$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$110$	−2.24698	−2.24698
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	−0.554958	−0.554958
$116$	0	0
$117$	1.44504	1.44504
$118$	0	0
$119$	0	0
$120$	−0.554958	−0.554958
$121$	2.24698	2.24698
$122$	−0.445042	−0.445042
$123$	0	0
$124$	0	0
$125$	−0.554958	−0.554958
$126$	0	0
$127$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$127$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$128$	1.00000	1.00000
$129$	−0.554958	−0.554958
$130$	−2.24698	−2.24698
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0.801938	0.801938
$133$	0	0
$134$	1.00000	1.00000
$135$	1.00000	1.00000
$136$	1.24698	1.24698
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0.198062	0.198062
$139$	2.00000	2.00000	1.00000			$0$
$139$	2.00000	2.00000	1.00000			$0$
$140$	0	0
$141$	0.801938	0.801938
$142$	1.24698	1.24698
$143$	3.24698	3.24698
$144$	−0.801938	−0.801938
$145$	0	0
$146$	−1.80194	−1.80194
$147$	−0.445042	−0.445042
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−0.246980	−0.246980
$151$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$151$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$152$	0	0
$153$	−1.00000	−1.00000
$154$	0	0
$155$	0	0
$156$	0.801938	0.801938
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0.198062	0.198062
$160$	1.24698	1.24698
$161$	0	0
$162$	0.445042	0.445042
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	1.00000	1.00000
$166$	0	0
$167$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$167$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$168$	0	0
$169$	2.24698	2.24698
$170$	1.55496	1.55496
$171$	0	0
$172$	1.24698	1.24698
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	−1.80194	−1.80194
$177$	0	0
$178$	−0.445042	−0.445042
$179$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$179$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$180$	−1.00000	−1.00000
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0.198062	0.198062
$184$	−0.445042	−0.445042
$185$	0	0
$186$	0	0
$187$	−2.24698	−2.24698
$188$	−1.80194	−1.80194
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−0.445042	−0.445042
$193$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$193$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$194$	0	0
$195$	1.00000	1.00000
$196$	1.00000	1.00000
$197$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$197$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$198$	1.44504	1.44504
$199$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$199$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$200$	0.554958	0.554958
$201$	−0.445042	−0.445042
$202$	−0.445042	−0.445042
$203$	0	0
$204$	−0.554958	−0.554958
$205$	0	0
$206$	−0.445042	−0.445042
$207$	0.356896	0.356896
$208$	−1.80194	−1.80194
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−0.445042	−0.445042
$213$	−0.554958	−0.554958
$214$	0	0
$215$	1.55496	1.55496
$216$	0.801938	0.801938
$217$	0	0
$218$	1.24698	1.24698
$219$	0.801938	0.801938
$220$	−2.24698	−2.24698
$221$	−2.24698	−2.24698
$222$	0	0
$223$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$223$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$224$	0	0
$225$	−0.445042	−0.445042
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	2.00000	2.00000	1.00000			$0$
$229$	2.00000	2.00000	1.00000			$0$
$230$	−0.554958	−0.554958
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	1.44504	1.44504
$235$	−2.24698	−2.24698
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−0.554958	−0.554958
$241$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$241$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$242$	2.24698	2.24698
$243$	−1.00000	−1.00000
$244$	−0.445042	−0.445042
$245$	1.24698	1.24698
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−0.554958	−0.554958
$251$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$251$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$252$	0	0
$253$	0.801938	0.801938
$254$	1.24698	1.24698
$255$	−0.692021	−0.692021
$256$	1.00000	1.00000
$257$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$257$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$258$	−0.554958	−0.554958
$259$	0	0
$260$	−2.24698	−2.24698
$261$	0	0
$262$	0	0
$263$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$263$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$264$	0.801938	0.801938
$265$	−0.554958	−0.554958
$266$	0	0
$267$	0.198062	0.198062
$268$	1.00000	1.00000
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	1.00000	1.00000
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	1.24698	1.24698
$273$	0	0
$274$	0	0
$275$	−1.00000	−1.00000
$276$	0.198062	0.198062
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	2.00000	2.00000
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0.801938	0.801938
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	1.24698	1.24698
$285$	0	0
$286$	3.24698	3.24698
$287$	0	0
$288$	−0.801938	−0.801938
$289$	0.554958	0.554958
$290$	0	0
$291$	0	0
$292$	−1.80194	−1.80194
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−0.445042	−0.445042
$295$	0	0
$296$	0	0
$297$	−1.44504	−1.44504
$298$	0	0
$299$	0.801938	0.801938
$300$	−0.246980	−0.246980
$301$	0	0
$302$	−1.80194	−1.80194
$303$	0.198062	0.198062
$304$	0	0
$305$	−0.554958	−0.554958
$306$	−1.00000	−1.00000
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0.198062	0.198062
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0.801938	0.801938
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0.198062	0.198062
$319$	0	0
$320$	1.24698	1.24698
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0.445042	0.445042
$325$	−1.00000	−1.00000
$326$	0	0
$327$	−0.554958	−0.554958
$328$	0	0
$329$	0	0
$330$	1.00000	1.00000
$331$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$331$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$332$	0	0
$333$	0	0
$334$	−1.80194	−1.80194
$335$	1.24698	1.24698
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	2.24698	2.24698
$339$	0	0
$340$	1.55496	1.55496
$341$	0	0
$342$	0	0
$343$	0	0
$344$	1.24698	1.24698
$345$	0.246980	0.246980
$346$	0	0
$347$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$347$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	−1.44504	−1.44504
$352$	−1.80194	−1.80194
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	1.55496	1.55496
$356$	−0.445042	−0.445042
$357$	0	0
$358$	−0.445042	−0.445042
$359$	2.00000	2.00000	1.00000			$0$
$359$	2.00000	2.00000	1.00000			$0$
$360$	−1.00000	−1.00000
$361$	1.00000	1.00000
$362$	0	0
$363$	−1.00000	−1.00000
$364$	0	0
$365$	−2.24698	−2.24698
$366$	0.198062	0.198062
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	−0.445042	−0.445042
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$373$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$374$	−2.24698	−2.24698
$375$	0.246980	0.246980
$376$	−1.80194	−1.80194
$377$	0	0
$378$	0	0
$379$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$379$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$380$	0	0
$381$	−0.554958	−0.554958
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−0.445042	−0.445042
$385$	0	0
$386$	−0.445042	−0.445042
$387$	−1.00000	−1.00000
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	1.00000	1.00000
$391$	−0.554958	−0.554958
$392$	1.00000	1.00000
$393$	0	0
$394$	−1.80194	−1.80194
$395$	0	0
$396$	1.44504	1.44504
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	1.24698	1.24698
$399$	0	0
$400$	0.554958	0.554958
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−0.445042	−0.445042
$403$	0	0
$404$	−0.445042	−0.445042
$405$	0.554958	0.554958
$406$	0	0
$407$	0	0
$408$	−0.554958	−0.554958
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	−0.445042	−0.445042
$413$	0	0
$414$	0.356896	0.356896
$415$	0	0
$416$	−1.80194	−1.80194
$417$	−0.890084	−0.890084
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	1.44504	1.44504
$424$	−0.445042	−0.445042
$425$	0.692021	0.692021
$426$	−0.554958	−0.554958
$427$	0	0
$428$	0	0
$429$	−1.44504	−1.44504
$430$	1.55496	1.55496
$431$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$431$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$432$	0.801938	0.801938
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	1.24698	1.24698
$437$	0	0
$438$	0.801938	0.801938
$439$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$439$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$440$	−2.24698	−2.24698
$441$	−0.801938	−0.801938
$442$	−2.24698	−2.24698
$443$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$443$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$444$	0	0
$445$	−0.554958	−0.554958
$446$	1.24698	1.24698
$447$	0	0
$448$	0	0
$449$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$449$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$450$	−0.445042	−0.445042
$451$	0	0
$452$	0	0
$453$	0.801938	0.801938
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$457$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$458$	2.00000	2.00000
$459$	1.00000	1.00000
$460$	−0.554958	−0.554958
$461$	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
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	1.44504	1.44504
$469$	0	0
$470$	−2.24698	−2.24698
$471$	0	0
$472$	0	0
$473$	−2.24698	−2.24698
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.356896	0.356896
$478$	0	0
$479$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$479$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$480$	−0.554958	−0.554958
$481$	0	0
$482$	−1.80194	−1.80194
$483$	0	0
$484$	2.24698	2.24698
$485$	0	0
$486$	−1.00000	−1.00000
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	−0.445042	−0.445042
$489$	0	0
$490$	1.24698	1.24698
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	1.80194	1.80194
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$499$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$500$	−0.554958	−0.554958
$501$	0.801938	0.801938
$502$	1.24698	1.24698
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−0.554958	−0.554958
$506$	0.801938	0.801938
$507$	−1.00000	−1.00000
$508$	1.24698	1.24698
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	−0.692021	−0.692021
$511$	0	0
$512$	1.00000	1.00000
$513$	0	0
$514$	−1.80194	−1.80194
$515$	−0.554958	−0.554958
$516$	−0.554958	−0.554958
$517$	3.24698	3.24698
$518$	0	0
$519$	0	0
$520$	−2.24698	−2.24698
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	−0.445042	−0.445042
$527$	0	0
$528$	0.801938	0.801938
$529$	−0.801938	−0.801938
$530$	−0.554958	−0.554958
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0.198062	0.198062
$535$	0	0
$536$	1.00000	1.00000
$537$	0.198062	0.198062
$538$	0	0
$539$	−1.80194	−1.80194
$540$	1.00000	1.00000
$541$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$541$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$542$	0	0
$543$	0	0
$544$	1.24698	1.24698
$545$	1.55496	1.55496
$546$	0	0
$547$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$547$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$548$	0	0
$549$	0.356896	0.356896
$550$	−1.00000	−1.00000
$551$	0	0
$552$	0.198062	0.198062
$553$	0	0
$554$	0	0
$555$	0	0
$556$	2.00000	2.00000
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	−2.24698	−2.24698
$560$	0	0
$561$	1.00000	1.00000
$562$	0	0
$563$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$563$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$564$	0.801938	0.801938
$565$	0	0
$566$	0	0
$567$	0	0
$568$	1.24698	1.24698
$569$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$569$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	3.24698	3.24698
$573$	0	0
$574$	0	0
$575$	−0.246980	−0.246980
$576$	−0.801938	−0.801938
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0.554958	0.554958
$579$	0.198062	0.198062
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0.801938	0.801938
$584$	−1.80194	−1.80194
$585$	1.80194	1.80194
$586$	0	0
$587$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$587$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$588$	−0.445042	−0.445042
$589$	0	0
$590$	0	0
$591$	0.801938	0.801938
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	−1.44504	−1.44504
$595$	0	0
$596$	0	0
$597$	−0.554958	−0.554958
$598$	0.801938	0.801938
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−0.246980	−0.246980
$601$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$601$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$602$	0	0
$603$	−0.801938	−0.801938
$604$	−1.80194	−1.80194
$605$	2.80194	2.80194
$606$	0.198062	0.198062
$607$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$607$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$608$	0	0
$609$	0	0
$610$	−0.554958	−0.554958
$611$	3.24698	3.24698
$612$	−1.00000	−1.00000
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.00000	2.00000	1.00000			$0$
$617$	2.00000	2.00000	1.00000			$0$
$618$	0.198062	0.198062
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−0.356896	−0.356896
$622$	0	0
$623$	0	0
$624$	0.801938	0.801938
$625$	−1.24698	−1.24698
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.55496	1.55496
$636$	0.198062	0.198062
$637$	−1.80194	−1.80194
$638$	0	0
$639$	−1.00000	−1.00000
$640$	1.24698	1.24698
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	−0.692021	−0.692021
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0.445042	0.445042
$649$	0	0
$650$	−1.00000	−1.00000
$651$	0	0
$652$	0	0
$653$	2.00000	2.00000	1.00000			$0$
$653$	2.00000	2.00000	1.00000			$0$
$654$	−0.554958	−0.554958
$655$	0	0
$656$	0	0
$657$	1.44504	1.44504
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	1.00000	1.00000
$661$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$661$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$662$	1.24698	1.24698
$663$	1.00000	1.00000
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−1.80194	−1.80194
$669$	−0.554958	−0.554958
$670$	1.24698	1.24698
$671$	0.801938	0.801938
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0.445042	0.445042
$676$	2.24698	2.24698
$677$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$677$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$678$	0	0
$679$	0	0
$680$	1.55496	1.55496
$681$	0	0
$682$	0	0
$683$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$683$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−0.890084	−0.890084
$688$	1.24698	1.24698
$689$	0.801938	0.801938
$690$	0.246980	0.246980
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	1.24698	1.24698
$695$	2.49396	2.49396
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.00000	2.00000	1.00000			$0$
$701$	2.00000	2.00000	1.00000			$0$
$702$	−1.44504	−1.44504
$703$	0	0
$704$	−1.80194	−1.80194
$705$	1.00000	1.00000
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	1.55496	1.55496
$711$	0	0
$712$	−0.445042	−0.445042
$713$	0	0
$714$	0	0
$715$	4.04892	4.04892
$716$	−0.445042	−0.445042
$717$	0	0
$718$	2.00000	2.00000
$719$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$719$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$720$	−1.00000	−1.00000
$721$	0	0
$722$	1.00000	1.00000
$723$	0.801938	0.801938
$724$	0	0
$725$	0	0
$726$	−1.00000	−1.00000
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	−2.24698	−2.24698
$731$	1.55496	1.55496
$732$	0.198062	0.198062
$733$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$733$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$734$	0	0
$735$	−0.554958	−0.554958
$736$	−0.445042	−0.445042
$737$	−1.80194	−1.80194
$738$	0	0
$739$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$739$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$743$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$744$	0	0
$745$	0	0
$746$	−1.80194	−1.80194
$747$	0	0
$748$	−2.24698	−2.24698
$749$	0	0
$750$	0.246980	0.246980
$751$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$751$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$752$	−1.80194	−1.80194
$753$	−0.554958	−0.554958
$754$	0	0
$755$	−2.24698	−2.24698
$756$	0	0
$757$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$757$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$758$	−1.80194	−1.80194
$759$	−0.356896	−0.356896
$760$	0	0
$761$	2.00000	2.00000	1.00000			$0$
$761$	2.00000	2.00000	1.00000			$0$
$762$	−0.554958	−0.554958
$763$	0	0
$764$	0	0
$765$	−1.24698	−1.24698
$766$	0	0
$767$	0	0
$768$	−0.445042	−0.445042
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0.801938	0.801938
$772$	−0.445042	−0.445042
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−1.00000	−1.00000
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	1.00000	1.00000
$781$	−2.24698	−2.24698
$782$	−0.554958	−0.554958
$783$	0	0
$784$	1.00000	1.00000
$785$	0	0
$786$	0	0
$787$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$787$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$788$	−1.80194	−1.80194
$789$	0.198062	0.198062
$790$	0	0
$791$	0	0
$792$	1.44504	1.44504
$793$	0.801938	0.801938
$794$	0	0
$795$	0.246980	0.246980
$796$	1.24698	1.24698
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	−2.24698	−2.24698
$800$	0.554958	0.554958
$801$	0.356896	0.356896
$802$	0	0
$803$	3.24698	3.24698
$804$	−0.445042	−0.445042
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−0.445042	−0.445042
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0.554958	0.554958
$811$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$811$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−0.554958	−0.554958
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$823$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$824$	−0.445042	−0.445042
$825$	0.445042	0.445042
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0.356896	0.356896
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	−1.80194	−1.80194
$833$	1.24698	1.24698
$834$	−0.890084	−0.890084
$835$	−2.24698	−2.24698
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$839$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.80194	2.80194
$846$	1.44504	1.44504
$847$	0	0
$848$	−0.445042	−0.445042
$849$	0	0
$850$	0.692021	0.692021
$851$	0	0
$852$	−0.554958	−0.554958
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	−1.44504	−1.44504
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	1.55496	1.55496
$861$	0	0
$862$	−1.80194	−1.80194
$863$	2.00000	2.00000	1.00000			$0$
$863$	2.00000	2.00000	1.00000			$0$
$864$	0.801938	0.801938
$865$	0	0
$866$	0	0
$867$	−0.246980	−0.246980
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−1.80194	−1.80194
$872$	1.24698	1.24698
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0.801938	0.801938
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−0.445042	−0.445042
$879$	0	0
$880$	−2.24698	−2.24698
$881$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$881$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$882$	−0.801938	−0.801938
$883$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$883$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$884$	−2.24698	−2.24698
$885$	0	0
$886$	−1.80194	−1.80194
$887$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$887$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$888$	0	0
$889$	0	0
$890$	−0.554958	−0.554958
$891$	−0.801938	−0.801938
$892$	1.24698	1.24698
$893$	0	0
$894$	0	0
$895$	−0.554958	−0.554958
$896$	0	0
$897$	−0.356896	−0.356896
$898$	1.24698	1.24698
$899$	0	0
$900$	−0.445042	−0.445042
$901$	−0.554958	−0.554958
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0.801938	0.801938
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0.356896	0.356896
$910$	0	0
$911$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$911$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$912$	0	0
$913$	0	0
$914$	1.24698	1.24698
$915$	0.246980	0.246980
$916$	2.00000	2.00000
$917$	0	0
$918$	1.00000	1.00000
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	−0.554958	−0.554958
$921$	0	0
$922$	0	0
$923$	−2.24698	−2.24698
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.356896	0.356896
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.80194	−2.80194
$936$	1.44504	1.44504
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	−2.24698	−2.24698
$941$	2.00000	2.00000	1.00000			$0$
$941$	2.00000	2.00000	1.00000			$0$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	−2.24698	−2.24698
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	3.24698	3.24698
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$953$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$954$	0.356896	0.356896
$955$	0	0
$956$	0	0
$957$	0	0
$958$	−1.80194	−1.80194
$959$	0	0
$960$	−0.554958	−0.554958
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	−1.80194	−1.80194
$965$	−0.554958	−0.554958
$966$	0	0
$967$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$967$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$968$	2.24698	2.24698
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−1.00000	−1.00000
$973$	0	0
$974$	0	0
$975$	0.445042	0.445042
$976$	−0.445042	−0.445042
$977$	2.00000	2.00000	1.00000			$0$
$977$	2.00000	2.00000	1.00000			$0$
$978$	0	0
$979$	0.801938	0.801938
$980$	1.24698	1.24698
$981$	−1.00000	−1.00000
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−2.24698	−2.24698
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.554958	−0.554958
$990$	1.80194	1.80194
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	−0.554958	−0.554958
$994$	0	0
$995$	1.55496	1.55496
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−0.445042	−0.445042
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	536.1.h.b.133.2	yes	3
4.3	odd	2		2144.1.h.b.401.2		3
8.3	odd	2		2144.1.h.a.401.2		3
8.5	even	2		536.1.h.a.133.2	✓	3
67.66	odd	2		536.1.h.a.133.2	✓	3
268.267	even	2		2144.1.h.a.401.2		3
536.133	odd	2	CM	536.1.h.b.133.2	yes	3
536.267	even	2		2144.1.h.b.401.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
536.1.h.a.133.2	✓	3	8.5	even	2
536.1.h.a.133.2	✓	3	67.66	odd	2
536.1.h.b.133.2	yes	3	1.1	even	1	trivial
536.1.h.b.133.2	yes	3	536.133	odd	2	CM
2144.1.h.a.401.2		3	8.3	odd	2
2144.1.h.a.401.2		3	268.267	even	2
2144.1.h.b.401.2		3	4.3	odd	2
2144.1.h.b.401.2		3	536.267	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 536 = 2^{3} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	536.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} +1.24698 q^{5} -0.445042 q^{6} +1.00000 q^{8} -0.801938 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} +1.24698 q^{5} -0.445042 q^{6} +1.00000 q^{8} -0.801938 q^{9} +1.24698 q^{10} -1.80194 q^{11} -0.445042 q^{12} -1.80194 q^{13} -0.554958 q^{15} +1.00000 q^{16} +1.24698 q^{17} -0.801938 q^{18} +1.24698 q^{20} -1.80194 q^{22} -0.445042 q^{23} -0.445042 q^{24} +0.554958 q^{25} -1.80194 q^{26} +0.801938 q^{27} -0.554958 q^{30} +1.00000 q^{32} +0.801938 q^{33} +1.24698 q^{34} -0.801938 q^{36} +0.801938 q^{39} +1.24698 q^{40} +1.24698 q^{43} -1.80194 q^{44} -1.00000 q^{45} -0.445042 q^{46} -1.80194 q^{47} -0.445042 q^{48} +1.00000 q^{49} +0.554958 q^{50} -0.554958 q^{51} -1.80194 q^{52} -0.445042 q^{53} +0.801938 q^{54} -2.24698 q^{55} -0.554958 q^{60} -0.445042 q^{61} +1.00000 q^{64} -2.24698 q^{65} +0.801938 q^{66} +1.00000 q^{67} +1.24698 q^{68} +0.198062 q^{69} +1.24698 q^{71} -0.801938 q^{72} -1.80194 q^{73} -0.246980 q^{75} +0.801938 q^{78} +1.24698 q^{80} +0.445042 q^{81} +1.55496 q^{85} +1.24698 q^{86} -1.80194 q^{88} -0.445042 q^{89} -1.00000 q^{90} -0.445042 q^{92} -1.80194 q^{94} -0.445042 q^{96} +1.00000 q^{98} +1.44504 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} + 3 q^{8} + 2 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 + 3 * q^8 + 2 * q^9	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} + 3 q^{8} + 2 q^{9} - q^{10} - q^{11} - q^{12} - q^{13} - 2 q^{15} + 3 q^{16} - q^{17} + 2 q^{18} - q^{20} - q^{22} - q^{23} - q^{24} + 2 q^{25} - q^{26} - 2 q^{27} - 2 q^{30} + 3 q^{32} - 2 q^{33} - q^{34} + 2 q^{36} - 2 q^{39} - q^{40} - q^{43} - q^{44} - 3 q^{45} - q^{46} - q^{47} - q^{48} + 3 q^{49} + 2 q^{50} - 2 q^{51} - q^{52} - q^{53} - 2 q^{54} - 2 q^{55} - 2 q^{60} - q^{61} + 3 q^{64} - 2 q^{65} - 2 q^{66} + 3 q^{67} - q^{68} + 5 q^{69} - q^{71} + 2 q^{72} - q^{73} + 4 q^{75} - 2 q^{78} - q^{80} + q^{81} + 5 q^{85} - q^{86} - q^{88} - q^{89} - 3 q^{90} - q^{92} - q^{94} - q^{96} + 3 q^{98} + 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 + 3 * q^8 + 2 * q^9 - q^10 - q^11 - q^12 - q^13 - 2 * q^15 + 3 * q^16 - q^17 + 2 * q^18 - q^20 - q^22 - q^23 - q^24 + 2 * q^25 - q^26 - 2 * q^27 - 2 * q^30 + 3 * q^32 - 2 * q^33 - q^34 + 2 * q^36 - 2 * q^39 - q^40 - q^43 - q^44 - 3 * q^45 - q^46 - q^47 - q^48 + 3 * q^49 + 2 * q^50 - 2 * q^51 - q^52 - q^53 - 2 * q^54 - 2 * q^55 - 2 * q^60 - q^61 + 3 * q^64 - 2 * q^65 - 2 * q^66 + 3 * q^67 - q^68 + 5 * q^69 - q^71 + 2 * q^72 - q^73 + 4 * q^75 - 2 * q^78 - q^80 + q^81 + 5 * q^85 - q^86 - q^88 - q^89 - 3 * q^90 - q^92 - q^94 - q^96 + 3 * q^98 + 4 * q^99

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