LMFDB - Embedded newform 2008.1.c.b.501.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2008,1,Mod(501,2008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2008, 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("2008.501");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2008 = 2^{3} \cdot 251 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2008.c (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:	$1.00212254537$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.8096384512.1
Artin image:	$D_7$
Artin field:	Galois closure of 7.1.8096384512.1

Embedding invariants

Embedding label			501.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	2008.501

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.24698 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.24698 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.445042 q^{11} +1.24698 q^{14} +1.00000 q^{16} -1.80194 q^{17} +1.00000 q^{18} -1.80194 q^{19} -0.445042 q^{22} -0.445042 q^{23} +1.00000 q^{25} +1.24698 q^{28} -1.80194 q^{29} -1.80194 q^{31} +1.00000 q^{32} -1.80194 q^{34} +1.00000 q^{36} +1.24698 q^{37} -1.80194 q^{38} -0.445042 q^{41} +1.24698 q^{43} -0.445042 q^{44} -0.445042 q^{46} +0.554958 q^{49} +1.00000 q^{50} -0.445042 q^{53} +1.24698 q^{56} -1.80194 q^{58} +1.24698 q^{59} -0.445042 q^{61} -1.80194 q^{62} +1.24698 q^{63} +1.00000 q^{64} -1.80194 q^{68} +1.00000 q^{72} +1.24698 q^{73} +1.24698 q^{74} -1.80194 q^{76} -0.554958 q^{77} -1.80194 q^{79} +1.00000 q^{81} -0.445042 q^{82} +1.24698 q^{86} -0.445042 q^{88} +1.24698 q^{89} -0.445042 q^{92} +0.554958 q^{98} -0.445042 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - q^7 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} + 3 q^{4} - q^{7} + 3 q^{8} + 3 q^{9} - q^{11} - q^{14} + 3 q^{16} - q^{17} + 3 q^{18} - q^{19} - q^{22} - q^{23} + 3 q^{25} - q^{28} - q^{29} - q^{31} + 3 q^{32} - q^{34} + 3 q^{36} - q^{37} - q^{38} - q^{41} - q^{43} - q^{44} - q^{46} + 2 q^{49} + 3 q^{50} - q^{53} - q^{56} - q^{58} - q^{59} - q^{61} - q^{62} - q^{63} + 3 q^{64} - q^{68} + 3 q^{72} - q^{73} - q^{74} - q^{76} - 2 q^{77} - q^{79} + 3 q^{81} - q^{82} - q^{86} - q^{88} - q^{89} - q^{92} + 2 q^{98} - q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - q^7 + 3 * q^8 + 3 * q^9 - q^11 - q^14 + 3 * q^16 - q^17 + 3 * q^18 - q^19 - q^22 - q^23 + 3 * q^25 - q^28 - q^29 - q^31 + 3 * q^32 - q^34 + 3 * q^36 - q^37 - q^38 - q^41 - q^43 - q^44 - q^46 + 2 * q^49 + 3 * q^50 - q^53 - q^56 - q^58 - q^59 - q^61 - q^62 - q^63 + 3 * q^64 - q^68 + 3 * q^72 - q^73 - q^74 - q^76 - 2 * q^77 - q^79 + 3 * q^81 - q^82 - q^86 - q^88 - q^89 - q^92 + 2 * q^98 - q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2008\mathbb{Z}\right)^\times$.

$n$	$257$	$503$	$1005$
$\chi(n)$	$-1$	$1$	$-1$

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	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	1.00000	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$7$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$8$	1.00000	1.00000
$9$	1.00000	1.00000
$10$	0	0
$11$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$11$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	1.24698	1.24698
$15$	0	0
$16$	1.00000	1.00000
$17$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$17$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$18$	1.00000	1.00000
$19$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$19$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$20$	0	0
$21$	0	0
$22$	−0.445042	−0.445042
$23$	−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	0
$25$	1.00000	1.00000
$26$	0	0
$27$	0	0
$28$	1.24698	1.24698
$29$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$29$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$30$	0	0
$31$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$31$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$32$	1.00000	1.00000
$33$	0	0
$34$	−1.80194	−1.80194
$35$	0	0
$36$	1.00000	1.00000
$37$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$37$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$38$	−1.80194	−1.80194
$39$	0	0
$40$	0	0
$41$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$41$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$42$	0	0
$43$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$43$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$44$	−0.445042	−0.445042
$45$	0	0
$46$	−0.445042	−0.445042
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	0.554958	0.554958
$50$	1.00000	1.00000
$51$	0	0
$52$	0	0
$53$	−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	0
$55$	0	0
$56$	1.24698	1.24698
$57$	0	0
$58$	−1.80194	−1.80194
$59$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$59$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$60$	0	0
$61$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$61$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$62$	−1.80194	−1.80194
$63$	1.24698	1.24698
$64$	1.00000	1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−1.80194	−1.80194
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.00000	1.00000
$73$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$73$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$74$	1.24698	1.24698
$75$	0	0
$76$	−1.80194	−1.80194
$77$	−0.554958	−0.554958
$78$	0	0
$79$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$79$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$80$	0	0
$81$	1.00000	1.00000
$82$	−0.445042	−0.445042
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	1.24698	1.24698
$87$	0	0
$88$	−0.445042	−0.445042
$89$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$89$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$90$	0	0
$91$	0	0
$92$	−0.445042	−0.445042
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.554958	0.554958
$99$	−0.445042	−0.445042
$100$	1.00000	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$103$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$104$	0	0
$105$	0	0
$106$	−0.445042	−0.445042
$107$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$107$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$108$	0	0
$109$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$109$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$110$	0	0
$111$	0	0
$112$	1.24698	1.24698
$113$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$113$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$114$	0	0
$115$	0	0
$116$	−1.80194	−1.80194
$117$	0	0
$118$	1.24698	1.24698
$119$	−2.24698	−2.24698
$120$	0	0
$121$	−0.801938	−0.801938
$122$	−0.445042	−0.445042
$123$	0	0
$124$	−1.80194	−1.80194
$125$	0	0
$126$	1.24698	1.24698
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000	1.00000
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	−2.24698	−2.24698
$134$	0	0
$135$	0	0
$136$	−1.80194	−1.80194
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$139$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.00000	1.00000
$145$	0	0
$146$	1.24698	1.24698
$147$	0	0
$148$	1.24698	1.24698
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−1.80194	−1.80194
$153$	−1.80194	−1.80194
$154$	−0.554958	−0.554958
$155$	0	0
$156$	0	0
$157$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$157$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$158$	−1.80194	−1.80194
$159$	0	0
$160$	0	0
$161$	−0.554958	−0.554958
$162$	1.00000	1.00000
$163$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$163$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$164$	−0.445042	−0.445042
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	−1.80194	−1.80194
$172$	1.24698	1.24698
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	1.24698	1.24698
$176$	−0.445042	−0.445042
$177$	0	0
$178$	1.24698	1.24698
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−0.445042	−0.445042
$185$	0	0
$186$	0	0
$187$	0.801938	0.801938
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0.554958	0.554958
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−0.445042	−0.445042
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	1.00000	1.00000
$201$	0	0
$202$	0	0
$203$	−2.24698	−2.24698
$204$	0	0
$205$	0	0
$206$	−0.445042	−0.445042
$207$	−0.445042	−0.445042
$208$	0	0
$209$	0.801938	0.801938
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−0.445042	−0.445042
$213$	0	0
$214$	−1.80194	−1.80194
$215$	0	0
$216$	0	0
$217$	−2.24698	−2.24698
$218$	1.24698	1.24698
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	1.24698	1.24698
$225$	1.00000	1.00000
$226$	1.24698	1.24698
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$229$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$230$	0	0
$231$	0	0
$232$	−1.80194	−1.80194
$233$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$233$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$234$	0	0
$235$	0	0
$236$	1.24698	1.24698
$237$	0	0
$238$	−2.24698	−2.24698
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$241$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$242$	−0.801938	−0.801938
$243$	0	0
$244$	−0.445042	−0.445042
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−1.80194	−1.80194
$249$	0	0
$250$	0	0
$251$	1.00000	1.00000
$251$	1.00000	1.00000
$252$	1.24698	1.24698
$253$	0.198062	0.198062
$254$	0	0
$255$	0	0
$256$	1.00000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	1.55496	1.55496
$260$	0	0
$261$	−1.80194	−1.80194
$262$	0	0
$263$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$263$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$264$	0	0
$265$	0	0
$266$	−2.24698	−2.24698
$267$	0	0
$268$	0	0
$269$	2.00000	2.00000	1.00000			$0$
$269$	2.00000	2.00000	1.00000			$0$
$270$	0	0
$271$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$271$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$272$	−1.80194	−1.80194
$273$	0	0
$274$	0	0
$275$	−0.445042	−0.445042
$276$	0	0
$277$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$277$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$278$	1.24698	1.24698
$279$	−1.80194	−1.80194
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	2.00000	2.00000	1.00000			$0$
$283$	2.00000	2.00000	1.00000			$0$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.554958	−0.554958
$288$	1.00000	1.00000
$289$	2.24698	2.24698
$290$	0	0
$291$	0	0
$292$	1.24698	1.24698
$293$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$293$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$294$	0	0
$295$	0	0
$296$	1.24698	1.24698
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.55496	1.55496
$302$	0	0
$303$	0	0
$304$	−1.80194	−1.80194
$305$	0	0
$306$	−1.80194	−1.80194
$307$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$307$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$308$	−0.554958	−0.554958
$309$	0	0
$310$	0	0
$311$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$311$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	−0.445042	−0.445042
$315$	0	0
$316$	−1.80194	−1.80194
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0.801938	0.801938
$320$	0	0
$321$	0	0
$322$	−0.554958	−0.554958
$323$	3.24698	3.24698
$324$	1.00000	1.00000
$325$	0	0
$326$	−1.80194	−1.80194
$327$	0	0
$328$	−0.445042	−0.445042
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	1.24698	1.24698
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$337$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$338$	1.00000	1.00000
$339$	0	0
$340$	0	0
$341$	0.801938	0.801938
$342$	−1.80194	−1.80194
$343$	−0.554958	−0.554958
$344$	1.24698	1.24698
$345$	0	0
$346$	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
$349$	2.00000	2.00000	1.00000			$0$
$349$	2.00000	2.00000	1.00000			$0$
$350$	1.24698	1.24698
$351$	0	0
$352$	−0.445042	−0.445042
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	1.24698	1.24698
$357$	0	0
$358$	0	0
$359$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$359$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$360$	0	0
$361$	2.24698	2.24698
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	−0.445042	−0.445042
$369$	−0.445042	−0.445042
$370$	0	0
$371$	−0.554958	−0.554958
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0.801938	0.801938
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	2.00000	2.00000	1.00000			$0$
$379$	2.00000	2.00000	1.00000			$0$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.24698	1.24698
$388$	0	0
$389$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$389$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$390$	0	0
$391$	0.801938	0.801938
$392$	0.554958	0.554958
$393$	0	0
$394$	0	0
$395$	0	0
$396$	−0.445042	−0.445042
$397$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$397$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$398$	0	0
$399$	0	0
$400$	1.00000	1.00000
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	−2.24698	−2.24698
$407$	−0.554958	−0.554958
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	−0.445042	−0.445042
$413$	1.55496	1.55496
$414$	−0.445042	−0.445042
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0.801938	0.801938
$419$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$419$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$420$	0	0
$421$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$421$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$422$	0	0
$423$	0	0
$424$	−0.445042	−0.445042
$425$	−1.80194	−1.80194
$426$	0	0
$427$	−0.554958	−0.554958
$428$	−1.80194	−1.80194
$429$	0	0
$430$	0	0
$431$	−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	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	−2.24698	−2.24698
$435$	0	0
$436$	1.24698	1.24698
$437$	0.801938	0.801938
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0.554958	0.554958
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	1.24698	1.24698
$449$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$449$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$450$	1.00000	1.00000
$451$	0.198062	0.198062
$452$	1.24698	1.24698
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	1.24698	1.24698
$459$	0	0
$460$	0	0
$461$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$461$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−1.80194	−1.80194
$465$	0	0
$466$	−0.445042	−0.445042
$467$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$467$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	1.24698	1.24698
$473$	−0.554958	−0.554958
$474$	0	0
$475$	−1.80194	−1.80194
$476$	−2.24698	−2.24698
$477$	−0.445042	−0.445042
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	−0.445042	−0.445042
$483$	0	0
$484$	−0.801938	−0.801938
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	−0.445042	−0.445042
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	3.24698	3.24698
$494$	0	0
$495$	0	0
$496$	−1.80194	−1.80194
$497$	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	0
$501$	0	0
$502$	1.00000	1.00000
$503$	2.00000	2.00000	1.00000			$0$
$503$	2.00000	2.00000	1.00000			$0$
$504$	1.24698	1.24698
$505$	0	0
$506$	0.198062	0.198062
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	1.55496	1.55496
$512$	1.00000	1.00000
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	1.55496	1.55496
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−1.80194	−1.80194
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	−1.80194	−1.80194
$527$	3.24698	3.24698
$528$	0	0
$529$	−0.801938	−0.801938
$530$	0	0
$531$	1.24698	1.24698
$532$	−2.24698	−2.24698
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	2.00000	2.00000
$539$	−0.246980	−0.246980
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−0.445042	−0.445042
$543$	0	0
$544$	−1.80194	−1.80194
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	−0.445042	−0.445042
$550$	−0.445042	−0.445042
$551$	3.24698	3.24698
$552$	0	0
$553$	−2.24698	−2.24698
$554$	−1.80194	−1.80194
$555$	0	0
$556$	1.24698	1.24698
$557$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$557$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$558$	−1.80194	−1.80194
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$563$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$564$	0	0
$565$	0	0
$566$	2.00000	2.00000
$567$	1.24698	1.24698
$568$	0	0
$569$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$569$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	−0.554958	−0.554958
$575$	−0.445042	−0.445042
$576$	1.00000	1.00000
$577$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$577$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$578$	2.24698	2.24698
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0.198062	0.198062
$584$	1.24698	1.24698
$585$	0	0
$586$	1.24698	1.24698
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	3.24698	3.24698
$590$	0	0
$591$	0	0
$592$	1.24698	1.24698
$593$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$593$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	1.55496	1.55496
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$607$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$608$	−1.80194	−1.80194
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−1.80194	−1.80194
$613$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$613$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$614$	−0.445042	−0.445042
$615$	0	0
$616$	−0.554958	−0.554958
$617$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$617$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	1.24698	1.24698
$623$	1.55496	1.55496
$624$	0	0
$625$	1.00000	1.00000
$626$	0	0
$627$	0	0
$628$	−0.445042	−0.445042
$629$	−2.24698	−2.24698
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	−1.80194	−1.80194
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0.801938	0.801938
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	2.00000	2.00000	1.00000			$0$
$643$	2.00000	2.00000	1.00000			$0$
$644$	−0.554958	−0.554958
$645$	0	0
$646$	3.24698	3.24698
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	1.00000	1.00000
$649$	−0.554958	−0.554958
$650$	0	0
$651$	0	0
$652$	−1.80194	−1.80194
$653$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$653$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$654$	0	0
$655$	0	0
$656$	−0.445042	−0.445042
$657$	1.24698	1.24698
$658$	0	0
$659$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$659$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$660$	0	0
$661$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$661$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	1.24698	1.24698
$667$	0.801938	0.801938
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.198062	0.198062
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	−1.80194	−1.80194
$675$	0	0
$676$	1.00000	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0.801938	0.801938
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	−1.80194	−1.80194
$685$	0	0
$686$	−0.554958	−0.554958
$687$	0	0
$688$	1.24698	1.24698
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	−0.554958	−0.554958
$694$	1.24698	1.24698
$695$	0	0
$696$	0	0
$697$	0.801938	0.801938
$698$	2.00000	2.00000
$699$	0	0
$700$	1.24698	1.24698
$701$	2.00000	2.00000	1.00000			$0$
$701$	2.00000	2.00000	1.00000			$0$
$702$	0	0
$703$	−2.24698	−2.24698
$704$	−0.445042	−0.445042
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	−1.80194	−1.80194
$712$	1.24698	1.24698
$713$	0.801938	0.801938
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	−0.445042	−0.445042
$719$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$719$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$720$	0	0
$721$	−0.554958	−0.554958
$722$	2.24698	2.24698
$723$	0	0
$724$	0	0
$725$	−1.80194	−1.80194
$726$	0	0
$727$	2.00000	2.00000	1.00000			$0$
$727$	2.00000	2.00000	1.00000			$0$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	−2.24698	−2.24698
$732$	0	0
$733$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$733$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$734$	0	0
$735$	0	0
$736$	−0.445042	−0.445042
$737$	0	0
$738$	−0.445042	−0.445042
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	−0.554958	−0.554958
$743$	−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
$746$	0	0
$747$	0	0
$748$	0.801938	0.801938
$749$	−2.24698	−2.24698
$750$	0	0
$751$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$751$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	2.00000	2.00000
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	1.55496	1.55496
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$769$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	1.24698	1.24698
$775$	−1.80194	−1.80194
$776$	0	0
$777$	0	0
$778$	−1.80194	−1.80194
$779$	0.801938	0.801938
$780$	0	0
$781$	0	0
$782$	0.801938	0.801938
$783$	0	0
$784$	0.554958	0.554958
$785$	0	0
$786$	0	0
$787$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$787$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.55496	1.55496
$792$	−0.445042	−0.445042
$793$	0	0
$794$	−1.80194	−1.80194
$795$	0	0
$796$	0	0
$797$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$797$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	1.00000
$801$	1.24698	1.24698
$802$	0	0
$803$	−0.554958	−0.554958
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	−2.24698	−2.24698
$813$	0	0
$814$	−0.554958	−0.554958
$815$	0	0
$816$	0	0
$817$	−2.24698	−2.24698
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	−0.445042	−0.445042
$825$	0	0
$826$	1.55496	1.55496
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−0.445042	−0.445042
$829$	2.00000	2.00000	1.00000			$0$
$829$	2.00000	2.00000	1.00000			$0$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.00000	−1.00000
$834$	0	0
$835$	0	0
$836$	0.801938	0.801938
$837$	0	0
$838$	−0.445042	−0.445042
$839$	−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
$841$	2.24698	2.24698
$842$	−1.80194	−1.80194
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.00000	−1.00000
$848$	−0.445042	−0.445042
$849$	0	0
$850$	−1.80194	−1.80194
$851$	−0.554958	−0.554958
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	−0.554958	−0.554958
$855$	0	0
$856$	−1.80194	−1.80194
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	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	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	−2.24698	−2.24698
$869$	0.801938	0.801938
$870$	0	0
$871$	0	0
$872$	1.24698	1.24698
$873$	0	0
$874$	0.801938	0.801938
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0.554958	0.554958
$883$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$883$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−0.445042	−0.445042
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	1.24698	1.24698
$897$	0	0
$898$	−1.80194	−1.80194
$899$	3.24698	3.24698
$900$	1.00000	1.00000
$901$	0.801938	0.801938
$902$	0.198062	0.198062
$903$	0	0
$904$	1.24698	1.24698
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	1.24698	1.24698
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	1.24698	1.24698
$923$	0	0
$924$	0	0
$925$	1.24698	1.24698
$926$	0	0
$927$	−0.445042	−0.445042
$928$	−1.80194	−1.80194
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−1.00000	−1.00000
$932$	−0.445042	−0.445042
$933$	0	0
$934$	−1.80194	−1.80194
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$941$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$942$	0	0
$943$	0.198062	0.198062
$944$	1.24698	1.24698
$945$	0	0
$946$	−0.554958	−0.554958
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	−1.80194	−1.80194
$951$	0	0
$952$	−2.24698	−2.24698
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	−0.445042	−0.445042
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.24698	2.24698
$962$	0	0
$963$	−1.80194	−1.80194
$964$	−0.445042	−0.445042
$965$	0	0
$966$	0	0
$967$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$967$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$968$	−0.801938	−0.801938
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	1.55496	1.55496
$974$	0	0
$975$	0	0
$976$	−0.445042	−0.445042
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	−0.554958	−0.554958
$980$	0	0
$981$	1.24698	1.24698
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	3.24698	3.24698
$987$	0	0
$988$	0	0
$989$	−0.554958	−0.554958
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	−1.80194	−1.80194
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$997$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$998$	−0.445042	−0.445042
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.1.c.b.501.3	yes	3
8.5	even	2		2008.1.c.a.501.3	✓	3
251.250	odd	2		2008.1.c.a.501.3	✓	3
2008.501	odd	2	CM	2008.1.c.b.501.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.1.c.a.501.3	✓	3	8.5	even	2
2008.1.c.a.501.3	✓	3	251.250	odd	2
2008.1.c.b.501.3	yes	3	1.1	even	1	trivial
2008.1.c.b.501.3	yes	3	2008.501	odd	2	CM

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:	\( N \)	\(=\)	\( 2008 = 2^{3} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2008.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.24698 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.24698 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.445042 q^{11} +1.24698 q^{14} +1.00000 q^{16} -1.80194 q^{17} +1.00000 q^{18} -1.80194 q^{19} -0.445042 q^{22} -0.445042 q^{23} +1.00000 q^{25} +1.24698 q^{28} -1.80194 q^{29} -1.80194 q^{31} +1.00000 q^{32} -1.80194 q^{34} +1.00000 q^{36} +1.24698 q^{37} -1.80194 q^{38} -0.445042 q^{41} +1.24698 q^{43} -0.445042 q^{44} -0.445042 q^{46} +0.554958 q^{49} +1.00000 q^{50} -0.445042 q^{53} +1.24698 q^{56} -1.80194 q^{58} +1.24698 q^{59} -0.445042 q^{61} -1.80194 q^{62} +1.24698 q^{63} +1.00000 q^{64} -1.80194 q^{68} +1.00000 q^{72} +1.24698 q^{73} +1.24698 q^{74} -1.80194 q^{76} -0.554958 q^{77} -1.80194 q^{79} +1.00000 q^{81} -0.445042 q^{82} +1.24698 q^{86} -0.445042 q^{88} +1.24698 q^{89} -0.445042 q^{92} +0.554958 q^{98} -0.445042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} + 3 q^{4} - q^{7} + 3 q^{8} + 3 q^{9} - q^{11} - q^{14} + 3 q^{16} - q^{17} + 3 q^{18} - q^{19} - q^{22} - q^{23} + 3 q^{25} - q^{28} - q^{29} - q^{31} + 3 q^{32} - q^{34} + 3 q^{36} - q^{37} - q^{38} - q^{41} - q^{43} - q^{44} - q^{46} + 2 q^{49} + 3 q^{50} - q^{53} - q^{56} - q^{58} - q^{59} - q^{61} - q^{62} - q^{63} + 3 q^{64} - q^{68} + 3 q^{72} - q^{73} - q^{74} - q^{76} - 2 q^{77} - q^{79} + 3 q^{81} - q^{82} - q^{86} - q^{88} - q^{89} - q^{92} + 2 q^{98} - q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - q^7 + 3 * q^8 + 3 * q^9 - q^11 - q^14 + 3 * q^16 - q^17 + 3 * q^18 - q^19 - q^22 - q^23 + 3 * q^25 - q^28 - q^29 - q^31 + 3 * q^32 - q^34 + 3 * q^36 - q^37 - q^38 - q^41 - q^43 - q^44 - q^46 + 2 * q^49 + 3 * q^50 - q^53 - q^56 - q^58 - q^59 - q^61 - q^62 - q^63 + 3 * q^64 - q^68 + 3 * q^72 - q^73 - q^74 - q^76 - 2 * q^77 - q^79 + 3 * q^81 - q^82 - q^86 - q^88 - q^89 - q^92 + 2 * q^98 - q^99

Introduction

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