LMFDB - Embedded newform 71.1.b.a.70.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [71,1,Mod(70,71)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(71, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("71.70");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 71 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	71.b (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:	$0.0354336158969$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.357911.1
Artin image:	$D_7$
Artin field:	Galois closure of 7.1.357911.1

Embedding invariants

Embedding label			70.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	71.70

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.80194 q^{2} -0.445042 q^{3} +2.24698 q^{4} +1.24698 q^{5} +0.801938 q^{6} -2.24698 q^{8} -0.801938 q^{9} +O(q^{10})$	$q-1.80194 q^{2} -0.445042 q^{3} +2.24698 q^{4} +1.24698 q^{5} +0.801938 q^{6} -2.24698 q^{8} -0.801938 q^{9} -2.24698 q^{10} -1.00000 q^{12} -0.554958 q^{15} +1.80194 q^{16} +1.44504 q^{18} -1.80194 q^{19} +2.80194 q^{20} +1.00000 q^{24} +0.554958 q^{25} +0.801938 q^{27} -0.445042 q^{29} +1.00000 q^{30} -1.00000 q^{32} -1.80194 q^{36} -1.80194 q^{37} +3.24698 q^{38} -2.80194 q^{40} +1.24698 q^{43} -1.00000 q^{45} -0.801938 q^{48} +1.00000 q^{49} -1.00000 q^{50} -1.44504 q^{54} +0.801938 q^{57} +0.801938 q^{58} -1.24698 q^{60} +1.00000 q^{71} +1.80194 q^{72} +1.24698 q^{73} +3.24698 q^{74} -0.246980 q^{75} -4.04892 q^{76} +1.24698 q^{79} +2.24698 q^{80} +0.445042 q^{81} -1.80194 q^{83} -2.24698 q^{86} +0.198062 q^{87} -0.445042 q^{89} +1.80194 q^{90} -2.24698 q^{95} +0.445042 q^{96} -1.80194 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - q^{3} + 2 q^{4} - q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 3 * q - q^2 - q^3 + 2 * q^4 - q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9	$ 3 q - q^{2} - q^{3} + 2 q^{4} - q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 3 q^{12} - 2 q^{15} + q^{16} + 4 q^{18} - q^{19} + 4 q^{20} + 3 q^{24} + 2 q^{25} - 2 q^{27} - q^{29} + 3 q^{30} - 3 q^{32} - q^{36} - q^{37} + 5 q^{38} - 4 q^{40} - q^{43} - 3 q^{45} + 2 q^{48} + 3 q^{49} - 3 q^{50} - 4 q^{54} - 2 q^{57} - 2 q^{58} + q^{60} + 3 q^{71} + q^{72} - q^{73} + 5 q^{74} + 4 q^{75} - 3 q^{76} - q^{79} + 2 q^{80} + q^{81} - q^{83} - 2 q^{86} + 5 q^{87} - q^{89} + q^{90} - 2 q^{95} + q^{96} - q^{98}+O(q^{100}) $ 3 * q - q^2 - q^3 + 2 * q^4 - q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 3 * q^12 - 2 * q^15 + q^16 + 4 * q^18 - q^19 + 4 * q^20 + 3 * q^24 + 2 * q^25 - 2 * q^27 - q^29 + 3 * q^30 - 3 * q^32 - q^36 - q^37 + 5 * q^38 - 4 * q^40 - q^43 - 3 * q^45 + 2 * q^48 + 3 * q^49 - 3 * q^50 - 4 * q^54 - 2 * q^57 - 2 * q^58 + q^60 + 3 * q^71 + q^72 - q^73 + 5 * q^74 + 4 * q^75 - 3 * q^76 - q^79 + 2 * q^80 + q^81 - q^83 - 2 * q^86 + 5 * q^87 - q^89 + q^90 - 2 * q^95 + q^96 - q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/71\mathbb{Z}\right)^\times$.

$n$	$7$
$\chi(n)$	$-1$

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 $

$2$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$2$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$3$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$3$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$4$	2.24698	2.24698
$5$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$5$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$6$	0.801938	0.801938
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−2.24698	−2.24698
$9$	−0.801938	−0.801938
$10$	−2.24698	−2.24698
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−1.00000	−1.00000
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−0.554958	−0.554958
$16$	1.80194	1.80194
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.44504	1.44504
$19$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$19$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$20$	2.80194	2.80194
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.00000	1.00000
$25$	0.554958	0.554958
$26$	0	0
$27$	0.801938	0.801938
$28$	0	0
$29$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$29$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$30$	1.00000	1.00000
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.00000	−1.00000
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−1.80194	−1.80194
$37$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$37$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$38$	3.24698	3.24698
$39$	0	0
$40$	−2.80194	−2.80194
$41$	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	$0.285714\pi$
$44$	0	0
$45$	−1.00000	−1.00000
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−0.801938	−0.801938
$49$	1.00000	1.00000
$50$	−1.00000	−1.00000
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−1.44504	−1.44504
$55$	0	0
$56$	0	0
$57$	0.801938	0.801938
$58$	0.801938	0.801938
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−1.24698	−1.24698
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.00000	1.00000
$71$	1.00000	1.00000
$72$	1.80194	1.80194
$73$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$73$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$74$	3.24698	3.24698
$75$	−0.246980	−0.246980
$76$	−4.04892	−4.04892
$77$	0	0
$78$	0	0
$79$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$79$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$80$	2.24698	2.24698
$81$	0.445042	0.445042
$82$	0	0
$83$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$83$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$84$	0	0
$85$	0	0
$86$	−2.24698	−2.24698
$87$	0.198062	0.198062
$88$	0	0
$89$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$89$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$90$	1.80194	1.80194
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.24698	−2.24698
$96$	0.445042	0.445042
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.80194	−1.80194
$99$	0	0
$100$	1.24698	1.24698
$101$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$101$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$102$	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	0
$107$	2.00000	2.00000	1.00000			$0$
$107$	2.00000	2.00000	1.00000			$0$
$108$	1.80194	1.80194
$109$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$109$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$110$	0	0
$111$	0.801938	0.801938
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	−1.44504	−1.44504
$115$	0	0
$116$	−1.00000	−1.00000
$117$	0	0
$118$	0	0
$119$	0	0
$120$	1.24698	1.24698
$121$	1.00000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−0.554958	−0.554958
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000	1.00000
$129$	−0.554958	−0.554958
$130$	0	0
$131$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$131$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00000	1.00000
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	−1.80194	−1.80194
$143$	0	0
$144$	−1.44504	−1.44504
$145$	−0.554958	−0.554958
$146$	−2.24698	−2.24698
$147$	−0.445042	−0.445042
$148$	−4.04892	−4.04892
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0.445042	0.445042
$151$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$151$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$152$	4.04892	4.04892
$153$	0	0
$154$	0	0
$155$	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	$0.571429\pi$
$158$	−2.24698	−2.24698
$159$	0	0
$160$	−1.24698	−1.24698
$161$	0	0
$162$	−0.801938	−0.801938
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	3.24698	3.24698
$167$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$167$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	1.44504	1.44504
$172$	2.80194	2.80194
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	−0.356896	−0.356896
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0.801938	0.801938
$179$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$179$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$180$	−2.24698	−2.24698
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.24698	−2.24698
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	4.04892	4.04892
$191$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$191$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	2.24698	2.24698
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$199$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$200$	−1.24698	−1.24698
$201$	0	0
$202$	3.24698	3.24698
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0.801938	0.801938
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	−0.445042	−0.445042
$214$	−3.60388	−3.60388
$215$	1.55496	1.55496
$216$	−1.80194	−1.80194
$217$	0	0
$218$	0.801938	0.801938
$219$	−0.554958	−0.554958
$220$	0	0
$221$	0	0
$222$	−1.44504	−1.44504
$223$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$223$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$224$	0	0
$225$	−0.445042	−0.445042
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	1.80194	1.80194
$229$	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
$232$	1.00000	1.00000
$233$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$233$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−0.554958	−0.554958
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−1.00000	−1.00000
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−1.80194	−1.80194
$243$	−1.00000	−1.00000
$244$	0	0
$245$	1.24698	1.24698
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.801938	0.801938
$250$	1.00000	1.00000
$251$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$251$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−1.80194	−1.80194
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	1.00000	1.00000
$259$	0	0
$260$	0	0
$261$	0.356896	0.356896
$262$	−2.24698	−2.24698
$263$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$263$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.198062	0.198062
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−1.80194	−1.80194
$271$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$271$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$277$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	2.24698	2.24698
$285$	1.00000	1.00000
$286$	0	0
$287$	0	0
$288$	0.801938	0.801938
$289$	1.00000	1.00000
$290$	1.00000	1.00000
$291$	0	0
$292$	2.80194	2.80194
$293$	2.00000	2.00000	1.00000			$0$
$293$	2.00000	2.00000	1.00000			$0$
$294$	0.801938	0.801938
$295$	0	0
$296$	4.04892	4.04892
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−0.554958	−0.554958
$301$	0	0
$302$	0.801938	0.801938
$303$	0.801938	0.801938
$304$	−3.24698	−3.24698
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0.198062	0.198062
$310$	0	0
$311$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$311$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$312$	0	0
$313$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$313$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$314$	0.801938	0.801938
$315$	0	0
$316$	2.80194	2.80194
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−0.890084	−0.890084
$322$	0	0
$323$	0	0
$324$	1.00000	1.00000
$325$	0	0
$326$	0	0
$327$	0.198062	0.198062
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−4.04892	−4.04892
$333$	1.44504	1.44504
$334$	−2.24698	−2.24698
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1.80194	−1.80194
$339$	0	0
$340$	0	0
$341$	0	0
$342$	−2.60388	−2.60388
$343$	0	0
$344$	−2.80194	−2.80194
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0.445042	0.445042
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	1.24698	1.24698
$356$	−1.00000	−1.00000
$357$	0	0
$358$	−2.24698	−2.24698
$359$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$359$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$360$	2.24698	2.24698
$361$	2.24698	2.24698
$362$	0	0
$363$	−0.445042	−0.445042
$364$	0	0
$365$	1.55496	1.55496
$366$	0	0
$367$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$367$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$368$	0	0
$369$	0	0
$370$	4.04892	4.04892
$371$	0	0
$372$	0	0
$373$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$373$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$374$	0	0
$375$	0.246980	0.246980
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$379$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$380$	−5.04892	−5.04892
$381$	0	0
$382$	0.801938	0.801938
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−0.445042	−0.445042
$385$	0	0
$386$	0	0
$387$	−1.00000	−1.00000
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−2.24698	−2.24698
$393$	−0.554958	−0.554958
$394$	0	0
$395$	1.55496	1.55496
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	3.24698	3.24698
$399$	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
$404$	−4.04892	−4.04892
$405$	0.554958	0.554958
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$409$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$410$	0	0
$411$	0	0
$412$	−1.00000	−1.00000
$413$	0	0
$414$	0	0
$415$	−2.24698	−2.24698
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−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
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0.801938	0.801938
$427$	0	0
$428$	4.49396	4.49396
$429$	0	0
$430$	−2.80194	−2.80194
$431$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$431$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$432$	1.44504	1.44504
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0.246980	0.246980
$436$	−1.00000	−1.00000
$437$	0	0
$438$	1.00000	1.00000
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−0.801938	−0.801938
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	1.80194	1.80194
$445$	−0.554958	−0.554958
$446$	3.24698	3.24698
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0.801938	0.801938
$451$	0	0
$452$	0	0
$453$	0.198062	0.198062
$454$	0	0
$455$	0	0
$456$	−1.80194	−1.80194
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−2.24698	−2.24698
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$463$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$464$	−0.801938	−0.801938
$465$	0	0
$466$	3.24698	3.24698
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0.198062	0.198062
$472$	0	0
$473$	0	0
$474$	1.00000	1.00000
$475$	−1.00000	−1.00000
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0.554958	0.554958
$481$	0	0
$482$	0	0
$483$	0	0
$484$	2.24698	2.24698
$485$	0	0
$486$	1.80194	1.80194
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−2.24698	−2.24698
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−1.44504	−1.44504
$499$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$499$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$500$	−1.24698	−1.24698
$501$	−0.554958	−0.554958
$502$	3.24698	3.24698
$503$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$503$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$504$	0	0
$505$	−2.24698	−2.24698
$506$	0	0
$507$	−0.445042	−0.445042
$508$	0	0
$509$	2.00000	2.00000	1.00000			$0$
$509$	2.00000	2.00000	1.00000			$0$
$510$	0	0
$511$	0	0
$512$	2.24698	2.24698
$513$	−1.44504	−1.44504
$514$	0	0
$515$	−0.554958	−0.554958
$516$	−1.24698	−1.24698
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$521$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$522$	−0.643104	−0.643104
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	2.80194	2.80194
$525$	0	0
$526$	−2.24698	−2.24698
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	−0.356896	−0.356896
$535$	2.49396	2.49396
$536$	0	0
$537$	−0.554958	−0.554958
$538$	0	0
$539$	0	0
$540$	2.24698	2.24698
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0.801938	0.801938
$543$	0	0
$544$	0	0
$545$	−0.554958	−0.554958
$546$	0	0
$547$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$547$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.801938	0.801938
$552$	0	0
$553$	0	0
$554$	3.24698	3.24698
$555$	1.00000	1.00000
$556$	0	0
$557$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$557$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−2.24698	−2.24698
$569$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$569$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$570$	−1.80194	−1.80194
$571$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$571$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$572$	0	0
$573$	0.198062	0.198062
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$577$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$578$	−1.80194	−1.80194
$579$	0	0
$580$	−1.24698	−1.24698
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−2.80194	−2.80194
$585$	0	0
$586$	−3.60388	−3.60388
$587$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$587$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$588$	−1.00000	−1.00000
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−3.24698	−3.24698
$593$	−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
$597$	0.801938	0.801938
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0.554958	0.554958
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	−1.00000	−1.00000
$605$	1.24698	1.24698
$606$	−1.44504	−1.44504
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	1.80194	1.80194
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−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	0
$615$	0	0
$616$	0	0
$617$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$617$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$618$	−0.356896	−0.356896
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	3.24698	3.24698
$623$	0	0
$624$	0	0
$625$	−1.24698	−1.24698
$626$	3.24698	3.24698
$627$	0	0
$628$	−1.00000	−1.00000
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	−2.80194	−2.80194
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−0.801938	−0.801938
$640$	1.24698	1.24698
$641$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$641$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$642$	1.60388	1.60388
$643$	2.00000	2.00000	1.00000			$0$
$643$	2.00000	2.00000	1.00000			$0$
$644$	0	0
$645$	−0.692021	−0.692021
$646$	0	0
$647$	2.00000	2.00000	1.00000			$0$
$647$	2.00000	2.00000	1.00000			$0$
$648$	−1.00000	−1.00000
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−0.356896	−0.356896
$655$	1.55496	1.55496
$656$	0	0
$657$	−1.00000	−1.00000
$658$	0	0
$659$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$659$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	4.04892	4.04892
$665$	0	0
$666$	−2.60388	−2.60388
$667$	0	0
$668$	2.80194	2.80194
$669$	0.801938	0.801938
$670$	0	0
$671$	0	0
$672$	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
$677$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$677$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	3.24698	3.24698
$685$	0	0
$686$	0	0
$687$	−0.554958	−0.554958
$688$	2.24698	2.24698
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	−0.445042	−0.445042
$697$	0	0
$698$	0	0
$699$	0.801938	0.801938
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	3.24698	3.24698
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	−2.24698	−2.24698
$711$	−1.00000	−1.00000
$712$	1.00000	1.00000
$713$	0	0
$714$	0	0
$715$	0	0
$716$	2.80194	2.80194
$717$	0	0
$718$	−2.24698	−2.24698
$719$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$719$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$720$	−1.80194	−1.80194
$721$	0	0
$722$	−4.04892	−4.04892
$723$	0	0
$724$	0	0
$725$	−0.246980	−0.246980
$726$	0.801938	0.801938
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	−2.80194	−2.80194
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−2.24698	−2.24698
$735$	−0.554958	−0.554958
$736$	0	0
$737$	0	0
$738$	0	0
$739$	2.00000	2.00000	1.00000			$0$
$739$	2.00000	2.00000	1.00000			$0$
$740$	−5.04892	−5.04892
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	−2.24698	−2.24698
$747$	1.44504	1.44504
$748$	0	0
$749$	0	0
$750$	−0.445042	−0.445042
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0.801938	0.801938
$754$	0	0
$755$	−0.554958	−0.554958
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0.801938	0.801938
$759$	0	0
$760$	5.04892	5.04892
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	−1.00000	−1.00000
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0.801938	0.801938
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	1.80194	1.80194
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−0.356896	−0.356896
$784$	1.80194	1.80194
$785$	−0.554958	−0.554958
$786$	1.00000	1.00000
$787$	−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
$789$	−0.554958	−0.554958
$790$	−2.80194	−2.80194
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−4.04892	−4.04892
$797$	−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
$800$	−0.554958	−0.554958
$801$	0.356896	0.356896
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	4.04892	4.04892
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−1.00000	−1.00000
$811$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$811$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$812$	0	0
$813$	0.198062	0.198062
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.24698	−2.24698
$818$	0.801938	0.801938
$819$	0	0
$820$	0	0
$821$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$821$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	1.00000	1.00000
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$829$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$830$	4.04892	4.04892
$831$	0.801938	0.801938
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.55496	1.55496
$836$	0	0
$837$	0	0
$838$	0.801938	0.801938
$839$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$839$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$840$	0	0
$841$	−0.801938	−0.801938
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.24698	1.24698
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	−1.00000	−1.00000
$853$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$853$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$854$	0	0
$855$	1.80194	1.80194
$856$	−4.49396	−4.49396
$857$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$857$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	3.49396	3.49396
$861$	0	0
$862$	3.24698	3.24698
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−0.801938	−0.801938
$865$	0	0
$866$	0	0
$867$	−0.445042	−0.445042
$868$	0	0
$869$	0	0
$870$	−0.445042	−0.445042
$871$	0	0
$872$	1.00000	1.00000
$873$	0	0
$874$	0	0
$875$	0	0
$876$	−1.24698	−1.24698
$877$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$877$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$878$	0	0
$879$	−0.890084	−0.890084
$880$	0	0
$881$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$881$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$882$	1.44504	1.44504
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−1.80194	−1.80194
$889$	0	0
$890$	1.00000	1.00000
$891$	0	0
$892$	−4.04892	−4.04892
$893$	0	0
$894$	0	0
$895$	1.55496	1.55496
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−1.00000	−1.00000
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−0.356896	−0.356896
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	1.44504	1.44504
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	1.44504	1.44504
$913$	0	0
$914$	0	0
$915$	0	0
$916$	2.80194	2.80194
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−1.00000	−1.00000
$926$	3.24698	3.24698
$927$	0.356896	0.356896
$928$	0.445042	0.445042
$929$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$929$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$930$	0	0
$931$	−1.80194	−1.80194
$932$	−4.04892	−4.04892
$933$	0.801938	0.801938
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0.801938	0.801938
$940$	0	0
$941$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$941$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$942$	−0.356896	−0.356896
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$947$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$948$	−1.24698	−1.24698
$949$	0	0
$950$	1.80194	1.80194
$951$	0	0
$952$	0	0
$953$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$953$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$954$	0	0
$955$	−0.554958	−0.554958
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	−1.60388	−1.60388
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−2.24698	−2.24698
$969$	0	0
$970$	0	0
$971$	2.00000	2.00000	1.00000			$0$
$971$	2.00000	2.00000	1.00000			$0$
$972$	−2.24698	−2.24698
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$977$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$978$	0	0
$979$	0	0
$980$	2.80194	2.80194
$981$	0.356896	0.356896
$982$	0	0
$983$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$983$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.24698	−2.24698
$996$	1.80194	1.80194
$997$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$997$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$998$	0.801938	0.801938
$999$	−1.44504	−1.44504

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	71.1.b.a.70.1	✓	3
3.2	odd	2		639.1.d.a.496.3		3
4.3	odd	2		1136.1.h.a.993.2		3
5.2	odd	4		1775.1.c.a.1774.1		6
5.3	odd	4		1775.1.c.a.1774.6		6
5.4	even	2		1775.1.d.b.851.3		3
7.2	even	3		3479.1.g.e.851.3		6
7.3	odd	6		3479.1.g.d.1206.3		6
7.4	even	3		3479.1.g.e.1206.3		6
7.5	odd	6		3479.1.g.d.851.3		6
7.6	odd	2		3479.1.d.e.638.1		3
71.70	odd	2	CM	71.1.b.a.70.1	✓	3
213.212	even	2		639.1.d.a.496.3		3
284.283	even	2		1136.1.h.a.993.2		3
355.212	even	4		1775.1.c.a.1774.1		6
355.283	even	4		1775.1.c.a.1774.6		6
355.354	odd	2		1775.1.d.b.851.3		3
497.212	odd	6		3479.1.g.e.851.3		6
497.283	even	6		3479.1.g.d.1206.3		6
497.354	odd	6		3479.1.g.e.1206.3		6
497.425	even	6		3479.1.g.d.851.3		6
497.496	even	2		3479.1.d.e.638.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.1.b.a.70.1	✓	3	1.1	even	1	trivial
71.1.b.a.70.1	✓	3	71.70	odd	2	CM
639.1.d.a.496.3		3	3.2	odd	2
639.1.d.a.496.3		3	213.212	even	2
1136.1.h.a.993.2		3	4.3	odd	2
1136.1.h.a.993.2		3	284.283	even	2
1775.1.c.a.1774.1		6	5.2	odd	4
1775.1.c.a.1774.1		6	355.212	even	4
1775.1.c.a.1774.6		6	5.3	odd	4
1775.1.c.a.1774.6		6	355.283	even	4
1775.1.d.b.851.3		3	5.4	even	2
1775.1.d.b.851.3		3	355.354	odd	2
3479.1.d.e.638.1		3	7.6	odd	2
3479.1.d.e.638.1		3	497.496	even	2
3479.1.g.d.851.3		6	7.5	odd	6
3479.1.g.d.851.3		6	497.425	even	6
3479.1.g.d.1206.3		6	7.3	odd	6
3479.1.g.d.1206.3		6	497.283	even	6
3479.1.g.e.851.3		6	7.2	even	3
3479.1.g.e.851.3		6	497.212	odd	6
3479.1.g.e.1206.3		6	7.4	even	3
3479.1.g.e.1206.3		6	497.354	odd	6

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:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	71.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.80194 q^{2} -0.445042 q^{3} +2.24698 q^{4} +1.24698 q^{5} +0.801938 q^{6} -2.24698 q^{8} -0.801938 q^{9} +O(q^{10})\)	\(q-1.80194 q^{2} -0.445042 q^{3} +2.24698 q^{4} +1.24698 q^{5} +0.801938 q^{6} -2.24698 q^{8} -0.801938 q^{9} -2.24698 q^{10} -1.00000 q^{12} -0.554958 q^{15} +1.80194 q^{16} +1.44504 q^{18} -1.80194 q^{19} +2.80194 q^{20} +1.00000 q^{24} +0.554958 q^{25} +0.801938 q^{27} -0.445042 q^{29} +1.00000 q^{30} -1.00000 q^{32} -1.80194 q^{36} -1.80194 q^{37} +3.24698 q^{38} -2.80194 q^{40} +1.24698 q^{43} -1.00000 q^{45} -0.801938 q^{48} +1.00000 q^{49} -1.00000 q^{50} -1.44504 q^{54} +0.801938 q^{57} +0.801938 q^{58} -1.24698 q^{60} +1.00000 q^{71} +1.80194 q^{72} +1.24698 q^{73} +3.24698 q^{74} -0.246980 q^{75} -4.04892 q^{76} +1.24698 q^{79} +2.24698 q^{80} +0.445042 q^{81} -1.80194 q^{83} -2.24698 q^{86} +0.198062 q^{87} -0.445042 q^{89} +1.80194 q^{90} -2.24698 q^{95} +0.445042 q^{96} -1.80194 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - q^{3} + 2 q^{4} - q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 3 * q - q^2 - q^3 + 2 * q^4 - q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9	\( 3 q - q^{2} - q^{3} + 2 q^{4} - q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 3 q^{12} - 2 q^{15} + q^{16} + 4 q^{18} - q^{19} + 4 q^{20} + 3 q^{24} + 2 q^{25} - 2 q^{27} - q^{29} + 3 q^{30} - 3 q^{32} - q^{36} - q^{37} + 5 q^{38} - 4 q^{40} - q^{43} - 3 q^{45} + 2 q^{48} + 3 q^{49} - 3 q^{50} - 4 q^{54} - 2 q^{57} - 2 q^{58} + q^{60} + 3 q^{71} + q^{72} - q^{73} + 5 q^{74} + 4 q^{75} - 3 q^{76} - q^{79} + 2 q^{80} + q^{81} - q^{83} - 2 q^{86} + 5 q^{87} - q^{89} + q^{90} - 2 q^{95} + q^{96} - q^{98}+O(q^{100}) \) 3 * q - q^2 - q^3 + 2 * q^4 - q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 3 * q^12 - 2 * q^15 + q^16 + 4 * q^18 - q^19 + 4 * q^20 + 3 * q^24 + 2 * q^25 - 2 * q^27 - q^29 + 3 * q^30 - 3 * q^32 - q^36 - q^37 + 5 * q^38 - 4 * q^40 - q^43 - 3 * q^45 + 2 * q^48 + 3 * q^49 - 3 * q^50 - 4 * q^54 - 2 * q^57 - 2 * q^58 + q^60 + 3 * q^71 + q^72 - q^73 + 5 * q^74 + 4 * q^75 - 3 * q^76 - q^79 + 2 * q^80 + q^81 - q^83 - 2 * q^86 + 5 * q^87 - q^89 + q^90 - 2 * q^95 + q^96 - q^98

Introduction

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