LMFDB - Embedded newform 871.1.d.a.870.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [871,1,Mod(870,871)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(871, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("871.870");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 871 = 13 \cdot 67 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	871.d (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:	$0.434685626003$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{11}$
Projective field:	Galois closure of 11.1.501292001353351.1

Embedding invariants

Embedding label			870.1
Root			$1.30972$ of defining polynomial
Character	$\chi$	$=$	871.870

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.91899 q^{2} +2.68251 q^{4} +1.68251 q^{5} -0.284630 q^{7} -3.22871 q^{8} +1.00000 q^{9} -3.22871 q^{10} +0.830830 q^{11} +1.00000 q^{13} +0.546200 q^{14} +3.51334 q^{16} -0.284630 q^{17} -1.91899 q^{18} +4.51334 q^{20} -1.59435 q^{22} -1.91899 q^{23} +1.83083 q^{25} -1.91899 q^{26} -0.763521 q^{28} -1.30972 q^{29} -1.30972 q^{31} -3.51334 q^{32} +0.546200 q^{34} -0.478891 q^{35} +2.68251 q^{36} -5.43232 q^{40} -1.30972 q^{41} +2.22871 q^{44} +1.68251 q^{45} +3.68251 q^{46} -0.918986 q^{49} -3.51334 q^{50} +2.68251 q^{52} +1.39788 q^{55} +0.918986 q^{56} +2.51334 q^{58} +2.51334 q^{62} -0.284630 q^{63} +3.22871 q^{64} +1.68251 q^{65} +1.00000 q^{67} -0.763521 q^{68} +0.918986 q^{70} -3.22871 q^{72} -0.236479 q^{77} +5.91121 q^{80} +1.00000 q^{81} +2.51334 q^{82} -0.478891 q^{85} -2.68251 q^{88} -3.22871 q^{90} -0.284630 q^{91} -5.14769 q^{92} +0.830830 q^{97} +1.76352 q^{98} +0.830830 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 4 q^{4} - q^{5} - q^{7} - 2 q^{8} + 5 q^{9} - 2 q^{10} - q^{11} + 5 q^{13} - 2 q^{14} + 3 q^{16} - q^{17} - q^{18} + 8 q^{20} - 2 q^{22} - q^{23} + 4 q^{25} - q^{26} - 3 q^{28} - q^{29}+ \cdots - q^{99}+O(q^{100}) $ 5 * q - q^2 + 4 * q^4 - q^5 - q^7 - 2 * q^8 + 5 * q^9 - 2 * q^10 - q^11 + 5 * q^13 - 2 * q^14 + 3 * q^16 - q^17 - q^18 + 8 * q^20 - 2 * q^22 - q^23 + 4 * q^25 - q^26 - 3 * q^28 - q^29 - q^31 - 3 * q^32 - 2 * q^34 - 2 * q^35 + 4 * q^36 - 4 * q^40 - q^41 - 3 * q^44 - q^45 + 9 * q^46 + 4 * q^49 - 3 * q^50 + 4 * q^52 - 2 * q^55 - 4 * q^56 - 2 * q^58 - 2 * q^62 - q^63 + 2 * q^64 - q^65 + 5 * q^67 - 3 * q^68 - 4 * q^70 - 2 * q^72 - 2 * q^77 + 6 * q^80 + 5 * q^81 - 2 * q^82 - 2 * q^85 - 4 * q^88 - 2 * q^90 - q^91 - 3 * q^92 - q^97 + 8 * q^98 - q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/871\mathbb{Z}\right)^\times$.

$n$	$404$	$470$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$2$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	2.68251	2.68251
$5$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$5$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$6$	0	0
$7$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$7$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$8$	−3.22871	−3.22871
$9$	1.00000	1.00000
$10$	−3.22871	−3.22871
$11$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$11$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$12$	0	0
$13$	1.00000	1.00000
$13$	1.00000	1.00000
$14$	0.546200	0.546200
$15$	0	0
$16$	3.51334	3.51334
$17$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$17$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$18$	−1.91899	−1.91899
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	4.51334	4.51334
$21$	0	0
$22$	−1.59435	−1.59435
$23$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$23$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$24$	0	0
$25$	1.83083	1.83083
$26$	−1.91899	−1.91899
$27$	0	0
$28$	−0.763521	−0.763521
$29$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$29$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$30$	0	0
$31$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$31$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$32$	−3.51334	−3.51334
$33$	0	0
$34$	0.546200	0.546200
$35$	−0.478891	−0.478891
$36$	2.68251	2.68251
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	−5.43232	−5.43232
$41$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$41$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	2.22871	2.22871
$45$	1.68251	1.68251
$46$	3.68251	3.68251
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−0.918986	−0.918986
$50$	−3.51334	−3.51334
$51$	0	0
$52$	2.68251	2.68251
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	1.39788	1.39788
$56$	0.918986	0.918986
$57$	0	0
$58$	2.51334	2.51334
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	2.51334	2.51334
$63$	−0.284630	−0.284630
$64$	3.22871	3.22871
$65$	1.68251	1.68251
$66$	0	0
$67$	1.00000	1.00000
$67$	1.00000	1.00000
$68$	−0.763521	−0.763521
$69$	0	0
$70$	0.918986	0.918986
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−3.22871	−3.22871
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.236479	−0.236479
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	5.91121	5.91121
$81$	1.00000	1.00000
$82$	2.51334	2.51334
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	−0.478891	−0.478891
$86$	0	0
$87$	0	0
$88$	−2.68251	−2.68251
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−3.22871	−3.22871
$91$	−0.284630	−0.284630
$92$	−5.14769	−5.14769
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$97$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$98$	1.76352	1.76352
$99$	0.830830	0.830830
$100$	4.91121	4.91121
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$103$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$104$	−3.22871	−3.22871
$105$	0	0
$106$	0	0
$107$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$107$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$108$	0	0
$109$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$109$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$110$	−2.68251	−2.68251
$111$	0	0
$112$	−1.00000	−1.00000
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	−3.22871	−3.22871
$116$	−3.51334	−3.51334
$117$	1.00000	1.00000
$118$	0	0
$119$	0.0810141	0.0810141
$120$	0	0
$121$	−0.309721	−0.309721
$122$	0	0
$123$	0	0
$124$	−3.51334	−3.51334
$125$	1.39788	1.39788
$126$	0.546200	0.546200
$127$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$127$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$128$	−2.68251	−2.68251
$129$	0	0
$130$	−3.22871	−3.22871
$131$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$131$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$132$	0	0
$133$	0	0
$134$	−1.91899	−1.91899
$135$	0	0
$136$	0.918986	0.918986
$137$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$137$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−1.28463	−1.28463
$141$	0	0
$142$	0	0
$143$	0.830830	0.830830
$144$	3.51334	3.51334
$145$	−2.20362	−2.20362
$146$	0	0
$147$	0	0
$148$	0	0
$149$	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			$\pi$
$152$	0	0
$153$	−0.284630	−0.284630
$154$	0.453800	0.453800
$155$	−2.20362	−2.20362
$156$	0	0
$157$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$157$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$158$	0	0
$159$	0	0
$160$	−5.91121	−5.91121
$161$	0.546200	0.546200
$162$	−1.91899	−1.91899
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−3.51334	−3.51334
$165$	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
$170$	0.918986	0.918986
$171$	0	0
$172$	0	0
$173$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$173$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$174$	0	0
$175$	−0.521109	−0.521109
$176$	2.91899	2.91899
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	4.51334	4.51334
$181$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$181$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$182$	0.546200	0.546200
$183$	0	0
$184$	6.19584	6.19584
$185$	0	0
$186$	0	0
$187$	−0.236479	−0.236479
$188$	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			$\pi$
$194$	−1.59435	−1.59435
$195$	0	0
$196$	−2.46519	−2.46519
$197$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$197$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$198$	−1.59435	−1.59435
$199$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$199$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$200$	−5.91121	−5.91121
$201$	0	0
$202$	0	0
$203$	0.372786	0.372786
$204$	0	0
$205$	−2.20362	−2.20362
$206$	2.51334	2.51334
$207$	−1.91899	−1.91899
$208$	3.51334	3.51334
$209$	0	0
$210$	0	0
$211$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$211$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$212$	0	0
$213$	0	0
$214$	−3.22871	−3.22871
$215$	0	0
$216$	0	0
$217$	0.372786	0.372786
$218$	3.68251	3.68251
$219$	0	0
$220$	3.74982	3.74982
$221$	−0.284630	−0.284630
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	1.00000	1.00000
$225$	1.83083	1.83083
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$229$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$230$	6.19584	6.19584
$231$	0	0
$232$	4.22871	4.22871
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	−1.91899	−1.91899
$235$	0	0
$236$	0	0
$237$	0	0
$238$	−0.155465	−0.155465
$239$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$239$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0.594351	0.594351
$243$	0	0
$244$	0	0
$245$	−1.54620	−1.54620
$246$	0	0
$247$	0	0
$248$	4.22871	4.22871
$249$	0	0
$250$	−2.68251	−2.68251
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−0.763521	−0.763521
$253$	−1.59435	−1.59435
$254$	−1.59435	−1.59435
$255$	0	0
$256$	1.91899	1.91899
$257$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$257$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$258$	0	0
$259$	0	0
$260$	4.51334	4.51334
$261$	−1.30972	−1.30972
$262$	−3.22871	−3.22871
$263$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$263$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	2.68251	2.68251
$269$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$269$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$270$	0	0
$271$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$271$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$272$	−1.00000	−1.00000
$273$	0	0
$274$	3.68251	3.68251
$275$	1.52111	1.52111
$276$	0	0
$277$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$277$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$278$	0	0
$279$	−1.30972	−1.30972
$280$	1.54620	1.54620
$281$	2.00000	2.00000	1.00000			$0$
$281$	2.00000	2.00000	1.00000			$0$
$282$	0	0
$283$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$283$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$284$	0	0
$285$	0	0
$286$	−1.59435	−1.59435
$287$	0.372786	0.372786
$288$	−3.51334	−3.51334
$289$	−0.918986	−0.918986
$290$	4.22871	4.22871
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.91899	−1.91899
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0.546200	0.546200
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	−0.634356	−0.634356
$309$	0	0
$310$	4.22871	4.22871
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	−1.59435	−1.59435
$315$	−0.478891	−0.478891
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	−1.08816	−1.08816
$320$	5.43232	5.43232
$321$	0	0
$322$	−1.04815	−1.04815
$323$	0	0
$324$	2.68251	2.68251
$325$	1.83083	1.83083
$326$	0	0
$327$	0	0
$328$	4.22871	4.22871
$329$	0	0
$330$	0	0
$331$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$331$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.68251	1.68251
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1.91899	−1.91899
$339$	0	0
$340$	−1.28463	−1.28463
$341$	−1.08816	−1.08816
$342$	0	0
$343$	0.546200	0.546200
$344$	0	0
$345$	0	0
$346$	−1.59435	−1.59435
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	1.00000	1.00000
$351$	0	0
$352$	−2.91899	−2.91899
$353$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$353$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−5.43232	−5.43232
$361$	1.00000	1.00000
$362$	−3.22871	−3.22871
$363$	0	0
$364$	−0.763521	−0.763521
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	−6.74204	−6.74204
$369$	−1.30972	−1.30972
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0.453800	0.453800
$375$	0	0
$376$	0	0
$377$	−1.30972	−1.30972
$378$	0	0
$379$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$379$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$383$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$384$	0	0
$385$	−0.397877	−0.397877
$386$	0	0
$387$	0	0
$388$	2.22871	2.22871
$389$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$389$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$390$	0	0
$391$	0.546200	0.546200
$392$	2.96714	2.96714
$393$	0	0
$394$	−1.59435	−1.59435
$395$	0	0
$396$	2.22871	2.22871
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	2.51334	2.51334
$399$	0	0
$400$	6.43232	6.43232
$401$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$401$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$402$	0	0
$403$	−1.30972	−1.30972
$404$	0	0
$405$	1.68251	1.68251
$406$	−0.715370	−0.715370
$407$	0	0
$408$	0	0
$409$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$409$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$410$	4.22871	4.22871
$411$	0	0
$412$	−3.51334	−3.51334
$413$	0	0
$414$	3.68251	3.68251
$415$	0	0
$416$	−3.51334	−3.51334
$417$	0	0
$418$	0	0
$419$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$419$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	3.68251	3.68251
$423$	0	0
$424$	0	0
$425$	−0.521109	−0.521109
$426$	0	0
$427$	0	0
$428$	4.51334	4.51334
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	−0.715370	−0.715370
$435$	0	0
$436$	−5.14769	−5.14769
$437$	0	0
$438$	0	0
$439$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$439$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$440$	−4.51334	−4.51334
$441$	−0.918986	−0.918986
$442$	0.546200	0.546200
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−0.918986	−0.918986
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−3.51334	−3.51334
$451$	−1.08816	−1.08816
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.478891	−0.478891
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	3.68251	3.68251
$459$	0	0
$460$	−8.66103	−8.66103
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$463$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$464$	−4.60149	−4.60149
$465$	0	0
$466$	0	0
$467$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$467$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$468$	2.68251	2.68251
$469$	−0.284630	−0.284630
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0.217321	0.217321
$477$	0	0
$478$	−3.22871	−3.22871
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−0.830830	−0.830830
$485$	1.39788	1.39788
$486$	0	0
$487$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$487$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$488$	0	0
$489$	0	0
$490$	2.96714	2.96714
$491$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$491$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$492$	0	0
$493$	0.372786	0.372786
$494$	0	0
$495$	1.39788	1.39788
$496$	−4.60149	−4.60149
$497$	0	0
$498$	0	0
$499$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$499$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$500$	3.74982	3.74982
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0.918986	0.918986
$505$	0	0
$506$	3.05954	3.05954
$507$	0	0
$508$	2.22871	2.22871
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−1.00000
$513$	0	0
$514$	2.51334	2.51334
$515$	−2.20362	−2.20362
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−5.43232	−5.43232
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	2.51334	2.51334
$523$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$523$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$524$	4.51334	4.51334
$525$	0	0
$526$	0.546200	0.546200
$527$	0.372786	0.372786
$528$	0	0
$529$	2.68251	2.68251
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.30972	−1.30972
$534$	0	0
$535$	2.83083	2.83083
$536$	−3.22871	−3.22871
$537$	0	0
$538$	0.546200	0.546200
$539$	−0.763521	−0.763521
$540$	0	0
$541$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$541$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$542$	−3.22871	−3.22871
$543$	0	0
$544$	1.00000	1.00000
$545$	−3.22871	−3.22871
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−5.14769	−5.14769
$549$	0	0
$550$	−2.91899	−2.91899
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0.546200	0.546200
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	2.51334	2.51334
$559$	0	0
$560$	−1.68251	−1.68251
$561$	0	0
$562$	−3.83797	−3.83797
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	3.68251	3.68251
$567$	−0.284630	−0.284630
$568$	0	0
$569$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$569$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$570$	0	0
$571$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$571$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$572$	2.22871	2.22871
$573$	0	0
$574$	−0.715370	−0.715370
$575$	−3.51334	−3.51334
$576$	3.22871	3.22871
$577$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$577$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$578$	1.76352	1.76352
$579$	0	0
$580$	−5.91121	−5.91121
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	1.68251	1.68251
$586$	0	0
$587$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$587$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.00000	2.00000	1.00000			$0$
$593$	2.00000	2.00000	1.00000			$0$
$594$	0	0
$595$	0.136307	0.136307
$596$	0	0
$597$	0	0
$598$	3.68251	3.68251
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$601$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$602$	0	0
$603$	1.00000	1.00000
$604$	0	0
$605$	−0.521109	−0.521109
$606$	0	0
$607$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$607$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−0.763521	−0.763521
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0.763521	0.763521
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	−5.91121	−5.91121
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0.521109	0.521109
$626$	0	0
$627$	0	0
$628$	2.22871	2.22871
$629$	0	0
$630$	0.918986	0.918986
$631$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$631$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.39788	1.39788
$636$	0	0
$637$	−0.918986	−0.918986
$638$	2.08816	2.08816
$639$	0	0
$640$	−4.51334	−4.51334
$641$	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			$\pi$
$644$	1.46519	1.46519
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−3.22871	−3.22871
$649$	0	0
$650$	−3.51334	−3.51334
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	2.83083	2.83083
$656$	−4.60149	−4.60149
$657$	0	0
$658$	0	0
$659$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$659$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$660$	0	0
$661$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$661$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$662$	−3.22871	−3.22871
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.51334	2.51334
$668$	0	0
$669$	0	0
$670$	−3.22871	−3.22871
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	2.68251	2.68251
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	−0.236479	−0.236479
$680$	1.54620	1.54620
$681$	0	0
$682$	2.08816	2.08816
$683$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$683$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$684$	0	0
$685$	−3.22871	−3.22871
$686$	−1.04815	−1.04815
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	2.22871	2.22871
$693$	−0.236479	−0.236479
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0.372786	0.372786
$698$	0	0
$699$	0	0
$700$	−1.39788	−1.39788
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	2.68251	2.68251
$705$	0	0
$706$	0.546200	0.546200
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.51334	2.51334
$714$	0	0
$715$	1.39788	1.39788
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$719$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$720$	5.91121	5.91121
$721$	0.372786	0.372786
$722$	−1.91899	−1.91899
$723$	0	0
$724$	4.51334	4.51334
$725$	−2.39788	−2.39788
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0.918986	0.918986
$729$	1.00000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$733$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$734$	0	0
$735$	0	0
$736$	6.74204	6.74204
$737$	0.830830	0.830830
$738$	2.51334	2.51334
$739$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$739$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	−0.634356	−0.634356
$749$	−0.478891	−0.478891
$750$	0	0
$751$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$751$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$752$	0	0
$753$	0	0
$754$	2.51334	2.51334
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−3.22871	−3.22871
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0.546200	0.546200
$764$	0	0
$765$	−0.478891	−0.478891
$766$	−1.59435	−1.59435
$767$	0	0
$768$	0	0
$769$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$769$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$770$	0.763521	0.763521
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−2.39788	−2.39788
$776$	−2.68251	−2.68251
$777$	0	0
$778$	−1.59435	−1.59435
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−1.04815	−1.04815
$783$	0	0
$784$	−3.22871	−3.22871
$785$	1.39788	1.39788
$786$	0	0
$787$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$787$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$788$	2.22871	2.22871
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−2.68251	−2.68251
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−3.51334	−3.51334
$797$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$797$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$798$	0	0
$799$	0	0
$800$	−6.43232	−6.43232
$801$	0	0
$802$	0.546200	0.546200
$803$	0	0
$804$	0	0
$805$	0.918986	0.918986
$806$	2.51334	2.51334
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−3.22871	−3.22871
$811$	2.00000	2.00000	1.00000			$0$
$811$	2.00000	2.00000	1.00000			$0$
$812$	1.00000	1.00000
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	3.68251	3.68251
$819$	−0.284630	−0.284630
$820$	−5.91121	−5.91121
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$823$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$824$	4.22871	4.22871
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−5.14769	−5.14769
$829$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$829$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$830$	0	0
$831$	0	0
$832$	3.22871	3.22871
$833$	0.261571	0.261571
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−1.59435	−1.59435
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	0.715370	0.715370
$842$	0	0
$843$	0	0
$844$	−5.14769	−5.14769
$845$	1.68251	1.68251
$846$	0	0
$847$	0.0881559	0.0881559
$848$	0	0
$849$	0	0
$850$	1.00000	1.00000
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−5.43232	−5.43232
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$859$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	1.39788	1.39788
$866$	0	0
$867$	0	0
$868$	1.00000	1.00000
$869$	0	0
$870$	0	0
$871$	1.00000	1.00000
$872$	6.19584	6.19584
$873$	0.830830	0.830830
$874$	0	0
$875$	−0.397877	−0.397877
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−1.59435	−1.59435
$879$	0	0
$880$	4.91121	4.91121
$881$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$881$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$882$	1.76352	1.76352
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	−0.763521	−0.763521
$885$	0	0
$886$	0	0
$887$	2.00000	2.00000	1.00000			$0$
$887$	2.00000	2.00000	1.00000			$0$
$888$	0	0
$889$	−0.236479	−0.236479
$890$	0	0
$891$	0.830830	0.830830
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0.763521	0.763521
$897$	0	0
$898$	0	0
$899$	1.71537	1.71537
$900$	4.91121	4.91121
$901$	0	0
$902$	2.08816	2.08816
$903$	0	0
$904$	0	0
$905$	2.83083	2.83083
$906$	0	0
$907$	2.00000	2.00000	1.00000			$0$
$907$	2.00000	2.00000	1.00000			$0$
$908$	0	0
$909$	0	0
$910$	0.918986	0.918986
$911$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$911$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−5.14769	−5.14769
$917$	−0.478891	−0.478891
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	10.4246	10.4246
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−3.22871	−3.22871
$927$	−1.30972	−1.30972
$928$	4.60149	4.60149
$929$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$929$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0.546200	0.546200
$935$	−0.397877	−0.397877
$936$	−3.22871	−3.22871
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0.546200	0.546200
$939$	0	0
$940$	0	0
$941$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$941$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$942$	0	0
$943$	2.51334	2.51334
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	−0.261571	−0.261571
$953$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$953$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$954$	0	0
$955$	0	0
$956$	4.51334	4.51334
$957$	0	0
$958$	0	0
$959$	0.546200	0.546200
$960$	0	0
$961$	0.715370	0.715370
$962$	0	0
$963$	1.68251	1.68251
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	1.00000	1.00000
$969$	0	0
$970$	−2.68251	−2.68251
$971$	2.00000	2.00000	1.00000			$0$
$971$	2.00000	2.00000	1.00000			$0$
$972$	0	0
$973$	0	0
$974$	3.68251	3.68251
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	−4.14769	−4.14769
$981$	−1.91899	−1.91899
$982$	2.51334	2.51334
$983$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$983$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$984$	0	0
$985$	1.39788	1.39788
$986$	−0.715370	−0.715370
$987$	0	0
$988$	0	0
$989$	0	0
$990$	−2.68251	−2.68251
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	4.60149	4.60149
$993$	0	0
$994$	0	0
$995$	−2.20362	−2.20362
$996$	0	0
$997$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$997$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$998$	0.546200	0.546200
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	871.1.d.a.870.1	✓	5
13.12	even	2		871.1.d.b.870.5	yes	5
67.66	odd	2		871.1.d.b.870.5	yes	5
871.870	odd	2	CM	871.1.d.a.870.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
871.1.d.a.870.1	✓	5	1.1	even	1	trivial
871.1.d.a.870.1	✓	5	871.870	odd	2	CM
871.1.d.b.870.5	yes	5	13.12	even	2
871.1.d.b.870.5	yes	5	67.66	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 871 = 13 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	871.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.91899 q^{2} +2.68251 q^{4} +1.68251 q^{5} -0.284630 q^{7} -3.22871 q^{8} +1.00000 q^{9} -3.22871 q^{10} +0.830830 q^{11} +1.00000 q^{13} +0.546200 q^{14} +3.51334 q^{16} -0.284630 q^{17} -1.91899 q^{18} +4.51334 q^{20} -1.59435 q^{22} -1.91899 q^{23} +1.83083 q^{25} -1.91899 q^{26} -0.763521 q^{28} -1.30972 q^{29} -1.30972 q^{31} -3.51334 q^{32} +0.546200 q^{34} -0.478891 q^{35} +2.68251 q^{36} -5.43232 q^{40} -1.30972 q^{41} +2.22871 q^{44} +1.68251 q^{45} +3.68251 q^{46} -0.918986 q^{49} -3.51334 q^{50} +2.68251 q^{52} +1.39788 q^{55} +0.918986 q^{56} +2.51334 q^{58} +2.51334 q^{62} -0.284630 q^{63} +3.22871 q^{64} +1.68251 q^{65} +1.00000 q^{67} -0.763521 q^{68} +0.918986 q^{70} -3.22871 q^{72} -0.236479 q^{77} +5.91121 q^{80} +1.00000 q^{81} +2.51334 q^{82} -0.478891 q^{85} -2.68251 q^{88} -3.22871 q^{90} -0.284630 q^{91} -5.14769 q^{92} +0.830830 q^{97} +1.76352 q^{98} +0.830830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 4 q^{4} - q^{5} - q^{7} - 2 q^{8} + 5 q^{9} - 2 q^{10} - q^{11} + 5 q^{13} - 2 q^{14} + 3 q^{16} - q^{17} - q^{18} + 8 q^{20} - 2 q^{22} - q^{23} + 4 q^{25} - q^{26} - 3 q^{28} - q^{29}+ \cdots - q^{99}+O(q^{100}) \) 5 * q - q^2 + 4 * q^4 - q^5 - q^7 - 2 * q^8 + 5 * q^9 - 2 * q^10 - q^11 + 5 * q^13 - 2 * q^14 + 3 * q^16 - q^17 - q^18 + 8 * q^20 - 2 * q^22 - q^23 + 4 * q^25 - q^26 - 3 * q^28 - q^29 - q^31 - 3 * q^32 - 2 * q^34 - 2 * q^35 + 4 * q^36 - 4 * q^40 - q^41 - 3 * q^44 - q^45 + 9 * q^46 + 4 * q^49 - 3 * q^50 + 4 * q^52 - 2 * q^55 - 4 * q^56 - 2 * q^58 - 2 * q^62 - q^63 + 2 * q^64 - q^65 + 5 * q^67 - 3 * q^68 - 4 * q^70 - 2 * q^72 - 2 * q^77 + 6 * q^80 + 5 * q^81 - 2 * q^82 - 2 * q^85 - 4 * q^88 - 2 * q^90 - q^91 - 3 * q^92 - q^97 + 8 * q^98 - q^99

Introduction

L-functions

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