LMFDB - Embedded newform 1591.1.d.b.1590.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1591,1,Mod(1590,1591)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1591, 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("1591.1590");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1591 = 37 \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1591.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:	$0.794012435099$
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 11.1.10194147150028951.1

Embedding invariants

Embedding label			1590.5
Root			$-1.68251$ of defining polynomial
Character	$\chi$	$=$	1591.1590

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.91899 q^{2} +2.68251 q^{4} -1.68251 q^{5} +3.22871 q^{8} +1.00000 q^{9} +O(q^{10})$	$q+1.91899 q^{2} +2.68251 q^{4} -1.68251 q^{5} +3.22871 q^{8} +1.00000 q^{9} -3.22871 q^{10} -0.284630 q^{11} +3.51334 q^{16} +1.91899 q^{18} +0.284630 q^{19} -4.51334 q^{20} -0.546200 q^{22} +1.83083 q^{25} -0.830830 q^{29} +3.51334 q^{32} +2.68251 q^{36} -1.00000 q^{37} +0.546200 q^{38} -5.43232 q^{40} -1.30972 q^{41} -1.00000 q^{43} -0.763521 q^{44} -1.68251 q^{45} -1.91899 q^{47} +1.00000 q^{49} +3.51334 q^{50} -1.91899 q^{53} +0.478891 q^{55} -1.59435 q^{58} +1.30972 q^{61} +3.22871 q^{64} -1.30972 q^{67} +3.22871 q^{72} -1.91899 q^{74} +0.763521 q^{76} -5.91121 q^{80} +1.00000 q^{81} -2.51334 q^{82} +0.830830 q^{83} -1.91899 q^{86} -0.918986 q^{88} +0.284630 q^{89} -3.22871 q^{90} -3.68251 q^{94} -0.478891 q^{95} +1.91899 q^{98} -0.284630 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 4 q^{4} + q^{5} + 2 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q + q^2 + 4 * q^4 + q^5 + 2 * q^8 + 5 * q^9	$ 5 q + q^{2} + 4 q^{4} + q^{5} + 2 q^{8} + 5 q^{9} - 2 q^{10} - q^{11} + 3 q^{16} + q^{18} + q^{19} - 8 q^{20} + 2 q^{22} + 4 q^{25} + q^{29} + 3 q^{32} + 4 q^{36} - 5 q^{37} - 2 q^{38} - 4 q^{40} - q^{41} - 5 q^{43} - 3 q^{44} + q^{45} - q^{47} + 5 q^{49} + 3 q^{50} - q^{53} + 2 q^{55} - 2 q^{58} + q^{61} + 2 q^{64} - q^{67} + 2 q^{72} - q^{74} + 3 q^{76} - 6 q^{80} + 5 q^{81} + 2 q^{82} - q^{83} - q^{86} + 4 q^{88} + q^{89} - 2 q^{90} - 9 q^{94} - 2 q^{95} + q^{98} - q^{99}+O(q^{100}) $ 5 * q + q^2 + 4 * q^4 + q^5 + 2 * q^8 + 5 * q^9 - 2 * q^10 - q^11 + 3 * q^16 + q^18 + q^19 - 8 * q^20 + 2 * q^22 + 4 * q^25 + q^29 + 3 * q^32 + 4 * q^36 - 5 * q^37 - 2 * q^38 - 4 * q^40 - q^41 - 5 * q^43 - 3 * q^44 + q^45 - q^47 + 5 * q^49 + 3 * q^50 - q^53 + 2 * q^55 - 2 * q^58 + q^61 + 2 * q^64 - q^67 + 2 * q^72 - q^74 + 3 * q^76 - 6 * q^80 + 5 * q^81 + 2 * q^82 - q^83 - q^86 + 4 * q^88 + q^89 - 2 * q^90 - 9 * q^94 - 2 * q^95 + q^98 - q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1591\mathbb{Z}\right)^\times$.

$n$	$519$	$1334$
$\chi(n)$	$-1$	$-1$

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	$-0.0909091\pi$
$2$	1.91899	1.91899	0.959493	+	0.281733i	$0.0909091\pi$
$3$	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	$-0.818182\pi$
$5$	−1.68251	−1.68251	−0.841254	+	0.540641i	$0.818182\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	3.22871	3.22871
$9$	1.00000	1.00000
$10$	−3.22871	−3.22871
$11$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$11$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	3.51334	3.51334
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.91899	1.91899
$19$	0.284630	0.284630	0.142315	−	0.989821i	$-0.454545\pi$
$19$	0.284630	0.284630	0.142315	+	0.989821i	$0.454545\pi$
$20$	−4.51334	−4.51334
$21$	0	0
$22$	−0.546200	−0.546200
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	1.83083	1.83083
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.830830	−0.830830	−0.415415	−	0.909632i	$-0.636364\pi$
$29$	−0.830830	−0.830830	−0.415415	+	0.909632i	$0.636364\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	3.51334	3.51334
$33$	0	0
$34$	0	0
$35$	0	0
$36$	2.68251	2.68251
$37$	−1.00000	−1.00000
$37$	−1.00000	−1.00000
$38$	0.546200	0.546200
$39$	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
$43$	−1.00000	−1.00000
$43$	−1.00000	−1.00000
$44$	−0.763521	−0.763521
$45$	−1.68251	−1.68251
$46$	0	0
$47$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$47$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$48$	0	0
$49$	1.00000	1.00000
$50$	3.51334	3.51334
$51$	0	0
$52$	0	0
$53$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$53$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$54$	0	0
$55$	0.478891	0.478891
$56$	0	0
$57$	0	0
$58$	−1.59435	−1.59435
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	1.30972	1.30972	0.654861	−	0.755750i	$-0.272727\pi$
$61$	1.30972	1.30972	0.654861	+	0.755750i	$0.272727\pi$
$62$	0	0
$63$	0	0
$64$	3.22871	3.22871
$65$	0	0
$66$	0	0
$67$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$67$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	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			$\pi$
$74$	−1.91899	−1.91899
$75$	0	0
$76$	0.763521	0.763521
$77$	0	0
$78$	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
$83$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$83$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$84$	0	0
$85$	0	0
$86$	−1.91899	−1.91899
$87$	0	0
$88$	−0.918986	−0.918986
$89$	0.284630	0.284630	0.142315	−	0.989821i	$-0.454545\pi$
$89$	0.284630	0.284630	0.142315	+	0.989821i	$0.454545\pi$
$90$	−3.22871	−3.22871
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−3.68251	−3.68251
$95$	−0.478891	−0.478891
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.91899	1.91899
$99$	−0.284630	−0.284630
$100$	4.91121	4.91121
$101$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$101$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	−3.68251	−3.68251
$107$	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
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0.918986	0.918986
$111$	0	0
$112$	0	0
$113$	1.30972	1.30972	0.654861	−	0.755750i	$-0.272727\pi$
$113$	1.30972	1.30972	0.654861	+	0.755750i	$0.272727\pi$
$114$	0	0
$115$	0	0
$116$	−2.22871	−2.22871
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−0.918986	−0.918986
$122$	2.51334	2.51334
$123$	0	0
$124$	0	0
$125$	−1.39788	−1.39788
$126$	0	0
$127$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$127$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$128$	2.68251	2.68251
$129$	0	0
$130$	0	0
$131$	1.30972	1.30972	0.654861	−	0.755750i	$-0.272727\pi$
$131$	1.30972	1.30972	0.654861	+	0.755750i	$0.272727\pi$
$132$	0	0
$133$	0	0
$134$	−2.51334	−2.51334
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$139$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	3.51334	3.51334
$145$	1.39788	1.39788
$146$	0	0
$147$	0	0
$148$	−2.68251	−2.68251
$149$	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			$\pi$
$152$	0.918986	0.918986
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	−5.91121	−5.91121
$161$	0	0
$162$	1.91899	1.91899
$163$	−0.830830	−0.830830	−0.415415	−	0.909632i	$-0.636364\pi$
$163$	−0.830830	−0.830830	−0.415415	+	0.909632i	$0.636364\pi$
$164$	−3.51334	−3.51334
$165$	0	0
$166$	1.59435	1.59435
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	0.284630	0.284630
$172$	−2.68251	−2.68251
$173$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$173$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−1.00000
$177$	0	0
$178$	0.546200	0.546200
$179$	−0.830830	−0.830830	−0.415415	−	0.909632i	$-0.636364\pi$
$179$	−0.830830	−0.830830	−0.415415	+	0.909632i	$0.636364\pi$
$180$	−4.51334	−4.51334
$181$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$181$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.68251	1.68251
$186$	0	0
$187$	0	0
$188$	−5.14769	−5.14769
$189$	0	0
$190$	−0.918986	−0.918986
$191$	−2.00000	−2.00000	−1.00000			$\pi$
$191$	−2.00000	−2.00000	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	2.68251	2.68251
$197$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$197$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$198$	−0.546200	−0.546200
$199$	1.91899	1.91899	0.959493	−	0.281733i	$-0.0909091\pi$
$199$	1.91899	1.91899	0.959493	+	0.281733i	$0.0909091\pi$
$200$	5.91121	5.91121
$201$	0	0
$202$	3.22871	3.22871
$203$	0	0
$204$	0	0
$205$	2.20362	2.20362
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.0810141	−0.0810141
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−5.14769	−5.14769
$213$	0	0
$214$	3.22871	3.22871
$215$	1.68251	1.68251
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	1.28463	1.28463
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	1.83083	1.83083
$226$	2.51334	2.51334
$227$	1.91899	1.91899	0.959493	−	0.281733i	$-0.0909091\pi$
$227$	1.91899	1.91899	0.959493	+	0.281733i	$0.0909091\pi$
$228$	0	0
$229$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$229$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$230$	0	0
$231$	0	0
$232$	−2.68251	−2.68251
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	3.22871	3.22871
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−0.830830	−0.830830	−0.415415	−	0.909632i	$-0.636364\pi$
$241$	−0.830830	−0.830830	−0.415415	+	0.909632i	$0.636364\pi$
$242$	−1.76352	−1.76352
$243$	0	0
$244$	3.51334	3.51334
$245$	−1.68251	−1.68251
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−2.68251	−2.68251
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	3.22871	3.22871
$255$	0	0
$256$	1.91899	1.91899
$257$	−0.830830	−0.830830	−0.415415	−	0.909632i	$-0.636364\pi$
$257$	−0.830830	−0.830830	−0.415415	+	0.909632i	$0.636364\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−0.830830	−0.830830
$262$	2.51334	2.51334
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	3.22871	3.22871
$266$	0	0
$267$	0	0
$268$	−3.51334	−3.51334
$269$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$269$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$270$	0	0
$271$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$271$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.521109	−0.521109
$276$	0	0
$277$	1.91899	1.91899	0.959493	−	0.281733i	$-0.0909091\pi$
$277$	1.91899	1.91899	0.959493	+	0.281733i	$0.0909091\pi$
$278$	−3.68251	−3.68251
$279$	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			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	3.51334	3.51334
$289$	1.00000	1.00000
$290$	2.68251	2.68251
$291$	0	0
$292$	0	0
$293$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$293$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$294$	0	0
$295$	0	0
$296$	−3.22871	−3.22871
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	1.00000	1.00000
$305$	−2.20362	−2.20362
$306$	0	0
$307$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$307$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0.284630	0.284630	0.142315	−	0.989821i	$-0.454545\pi$
$313$	0.284630	0.284630	0.142315	+	0.989821i	$0.454545\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$317$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$318$	0	0
$319$	0.236479	0.236479
$320$	−5.43232	−5.43232
$321$	0	0
$322$	0	0
$323$	0	0
$324$	2.68251	2.68251
$325$	0	0
$326$	−1.59435	−1.59435
$327$	0	0
$328$	−4.22871	−4.22871
$329$	0	0
$330$	0	0
$331$	−1.68251	−1.68251	−0.841254	−	0.540641i	$-0.818182\pi$
$331$	−1.68251	−1.68251	−0.841254	+	0.540641i	$0.818182\pi$
$332$	2.22871	2.22871
$333$	−1.00000	−1.00000
$334$	0	0
$335$	2.20362	2.20362
$336$	0	0
$337$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$337$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$338$	1.91899	1.91899
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0.546200	0.546200
$343$	0	0
$344$	−3.22871	−3.22871
$345$	0	0
$346$	−2.51334	−2.51334
$347$	−1.68251	−1.68251	−0.841254	−	0.540641i	$-0.818182\pi$
$347$	−1.68251	−1.68251	−0.841254	+	0.540641i	$0.818182\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−1.00000
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0.763521	0.763521
$357$	0	0
$358$	−1.59435	−1.59435
$359$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$359$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$360$	−5.43232	−5.43232
$361$	−0.918986	−0.918986
$362$	−0.546200	−0.546200
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$367$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$368$	0	0
$369$	−1.30972	−1.30972
$370$	3.22871	3.22871
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	−6.19584	−6.19584
$377$	0	0
$378$	0	0
$379$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$379$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$380$	−1.28463	−1.28463
$381$	0	0
$382$	−3.83797	−3.83797
$383$	−1.68251	−1.68251	−0.841254	−	0.540641i	$-0.818182\pi$
$383$	−1.68251	−1.68251	−0.841254	+	0.540641i	$0.818182\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.00000	−1.00000
$388$	0	0
$389$	1.91899	1.91899	0.959493	−	0.281733i	$-0.0909091\pi$
$389$	1.91899	1.91899	0.959493	+	0.281733i	$0.0909091\pi$
$390$	0	0
$391$	0	0
$392$	3.22871	3.22871
$393$	0	0
$394$	−0.546200	−0.546200
$395$	0	0
$396$	−0.763521	−0.763521
$397$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$397$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$398$	3.68251	3.68251
$399$	0	0
$400$	6.43232	6.43232
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	4.51334	4.51334
$405$	−1.68251	−1.68251
$406$	0	0
$407$	0.284630	0.284630
$408$	0	0
$409$	1.91899	1.91899	0.959493	−	0.281733i	$-0.0909091\pi$
$409$	1.91899	1.91899	0.959493	+	0.281733i	$0.0909091\pi$
$410$	4.22871	4.22871
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.39788	−1.39788
$416$	0	0
$417$	0	0
$418$	−0.155465	−0.155465
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	1.30972	1.30972	0.654861	−	0.755750i	$-0.272727\pi$
$421$	1.30972	1.30972	0.654861	+	0.755750i	$0.272727\pi$
$422$	0	0
$423$	−1.91899	−1.91899
$424$	−6.19584	−6.19584
$425$	0	0
$426$	0	0
$427$	0	0
$428$	4.51334	4.51334
$429$	0	0
$430$	3.22871	3.22871
$431$	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	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	1.54620	1.54620
$441$	1.00000	1.00000
$442$	0	0
$443$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$443$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$444$	0	0
$445$	−0.478891	−0.478891
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.30972	1.30972	0.654861	−	0.755750i	$-0.272727\pi$
$449$	1.30972	1.30972	0.654861	+	0.755750i	$0.272727\pi$
$450$	3.51334	3.51334
$451$	0.372786	0.372786
$452$	3.51334	3.51334
$453$	0	0
$454$	3.68251	3.68251
$455$	0	0
$456$	0	0
$457$	−0.830830	−0.830830	−0.415415	−	0.909632i	$-0.636364\pi$
$457$	−0.830830	−0.830830	−0.415415	+	0.909632i	$0.636364\pi$
$458$	−2.51334	−2.51334
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	1.30972	1.30972	0.654861	−	0.755750i	$-0.272727\pi$
$463$	1.30972	1.30972	0.654861	+	0.755750i	$0.272727\pi$
$464$	−2.91899	−2.91899
$465$	0	0
$466$	0	0
$467$	1.30972	1.30972	0.654861	−	0.755750i	$-0.272727\pi$
$467$	1.30972	1.30972	0.654861	+	0.755750i	$0.272727\pi$
$468$	0	0
$469$	0	0
$470$	6.19584	6.19584
$471$	0	0
$472$	0	0
$473$	0.284630	0.284630
$474$	0	0
$475$	0.521109	0.521109
$476$	0	0
$477$	−1.91899	−1.91899
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	−1.59435	−1.59435
$483$	0	0
$484$	−2.46519	−2.46519
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	4.22871	4.22871
$489$	0	0
$490$	−3.22871	−3.22871
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0.478891	0.478891
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.91899	1.91899	0.959493	−	0.281733i	$-0.0909091\pi$
$499$	1.91899	1.91899	0.959493	+	0.281733i	$0.0909091\pi$
$500$	−3.74982	−3.74982
$501$	0	0
$502$	0	0
$503$	−0.830830	−0.830830	−0.415415	−	0.909632i	$-0.636364\pi$
$503$	−0.830830	−0.830830	−0.415415	+	0.909632i	$0.636364\pi$
$504$	0	0
$505$	−2.83083	−2.83083
$506$	0	0
$507$	0	0
$508$	4.51334	4.51334
$509$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$509$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	1.00000
$513$	0	0
$514$	−1.59435	−1.59435
$515$	0	0
$516$	0	0
$517$	0.546200	0.546200
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−1.59435	−1.59435
$523$	−1.68251	−1.68251	−0.841254	−	0.540641i	$-0.818182\pi$
$523$	−1.68251	−1.68251	−0.841254	+	0.540641i	$0.818182\pi$
$524$	3.51334	3.51334
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	6.19584	6.19584
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−2.83083	−2.83083
$536$	−4.22871	−4.22871
$537$	0	0
$538$	1.59435	1.59435
$539$	−0.284630	−0.284630
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	1.59435	1.59435
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	1.30972	1.30972
$550$	−1.00000	−1.00000
$551$	−0.236479	−0.236479
$552$	0	0
$553$	0	0
$554$	3.68251	3.68251
$555$	0	0
$556$	−5.14769	−5.14769
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	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
$565$	−2.20362	−2.20362
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	3.22871	3.22871
$577$	1.91899	1.91899	0.959493	−	0.281733i	$-0.0909091\pi$
$577$	1.91899	1.91899	0.959493	+	0.281733i	$0.0909091\pi$
$578$	1.91899	1.91899
$579$	0	0
$580$	3.74982	3.74982
$581$	0	0
$582$	0	0
$583$	0.546200	0.546200
$584$	0	0
$585$	0	0
$586$	1.59435	1.59435
$587$	−0.830830	−0.830830	−0.415415	−	0.909632i	$-0.636364\pi$
$587$	−0.830830	−0.830830	−0.415415	+	0.909632i	$0.636364\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−3.51334	−3.51334
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$599$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	−1.30972	−1.30972
$604$	0	0
$605$	1.54620	1.54620
$606$	0	0
$607$	−1.68251	−1.68251	−0.841254	−	0.540641i	$-0.818182\pi$
$607$	−1.68251	−1.68251	−0.841254	+	0.540641i	$0.818182\pi$
$608$	1.00000	1.00000
$609$	0	0
$610$	−4.22871	−4.22871
$611$	0	0
$612$	0	0
$613$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$613$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$614$	1.59435	1.59435
$615$	0	0
$616$	0	0
$617$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$617$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$618$	0	0
$619$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$619$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0.521109	0.521109
$626$	0.546200	0.546200
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0.284630	0.284630	0.142315	−	0.989821i	$-0.454545\pi$
$631$	0.284630	0.284630	0.142315	+	0.989821i	$0.454545\pi$
$632$	0	0
$633$	0	0
$634$	3.22871	3.22871
$635$	−2.83083	−2.83083
$636$	0	0
$637$	0	0
$638$	0.453800	0.453800
$639$	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			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.284630	0.284630	0.142315	−	0.989821i	$-0.454545\pi$
$647$	0.284630	0.284630	0.142315	+	0.989821i	$0.454545\pi$
$648$	3.22871	3.22871
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−2.22871	−2.22871
$653$	0.284630	0.284630	0.142315	−	0.989821i	$-0.454545\pi$
$653$	0.284630	0.284630	0.142315	+	0.989821i	$0.454545\pi$
$654$	0	0
$655$	−2.20362	−2.20362
$656$	−4.60149	−4.60149
$657$	0	0
$658$	0	0
$659$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$659$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−3.22871	−3.22871
$663$	0	0
$664$	2.68251	2.68251
$665$	0	0
$666$	−1.91899	−1.91899
$667$	0	0
$668$	0	0
$669$	0	0
$670$	4.22871	4.22871
$671$	−0.372786	−0.372786
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	−0.546200	−0.546200
$675$	0	0
$676$	2.68251	2.68251
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0.763521	0.763521
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−3.51334	−3.51334
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−3.51334	−3.51334
$693$	0	0
$694$	−3.22871	−3.22871
$695$	3.22871	3.22871
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	−0.284630	−0.284630
$704$	−0.918986	−0.918986
$705$	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
$711$	0	0
$712$	0.918986	0.918986
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−2.22871	−2.22871
$717$	0	0
$718$	−2.51334	−2.51334
$719$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$719$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$720$	−5.91121	−5.91121
$721$	0	0
$722$	−1.76352	−1.76352
$723$	0	0
$724$	−0.763521	−0.763521
$725$	−1.52111	−1.52111
$726$	0	0
$727$	1.91899	1.91899	0.959493	−	0.281733i	$-0.0909091\pi$
$727$	1.91899	1.91899	0.959493	+	0.281733i	$0.0909091\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−2.51334	−2.51334
$735$	0	0
$736$	0	0
$737$	0.372786	0.372786
$738$	−2.51334	−2.51334
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	4.51334	4.51334
$741$	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
$747$	0.830830	0.830830
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−6.74204	−6.74204
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0.284630	0.284630	0.142315	−	0.989821i	$-0.454545\pi$
$757$	0.284630	0.284630	0.142315	+	0.989821i	$0.454545\pi$
$758$	−3.68251	−3.68251
$759$	0	0
$760$	−1.54620	−1.54620
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	−5.36501	−5.36501
$765$	0	0
$766$	−3.22871	−3.22871
$767$	0	0
$768$	0	0
$769$	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			$\pi$
$774$	−1.91899	−1.91899
$775$	0	0
$776$	0	0
$777$	0	0
$778$	3.68251	3.68251
$779$	−0.372786	−0.372786
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	3.51334	3.51334
$785$	0	0
$786$	0	0
$787$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$787$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$788$	−0.763521	−0.763521
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−0.918986	−0.918986
$793$	0	0
$794$	3.22871	3.22871
$795$	0	0
$796$	5.14769	5.14769
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	6.43232	6.43232
$801$	0.284630	0.284630
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	5.43232	5.43232
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−3.22871	−3.22871
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0.546200	0.546200
$815$	1.39788	1.39788
$816$	0	0
$817$	−0.284630	−0.284630
$818$	3.68251	3.68251
$819$	0	0
$820$	5.91121	5.91121
$821$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$821$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$822$	0	0
$823$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$823$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	1.91899	1.91899	0.959493	−	0.281733i	$-0.0909091\pi$
$829$	1.91899	1.91899	0.959493	+	0.281733i	$0.0909091\pi$
$830$	−2.68251	−2.68251
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−0.217321	−0.217321
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−0.309721	−0.309721
$842$	2.51334	2.51334
$843$	0	0
$844$	0	0
$845$	−1.68251	−1.68251
$846$	−3.68251	−3.68251
$847$	0	0
$848$	−6.74204	−6.74204
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	−0.478891	−0.478891
$856$	5.43232	5.43232
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−1.68251	−1.68251	−0.841254	−	0.540641i	$-0.818182\pi$
$859$	−1.68251	−1.68251	−0.841254	+	0.540641i	$0.818182\pi$
$860$	4.51334	4.51334
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	2.20362	2.20362
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$877$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$878$	0	0
$879$	0	0
$880$	1.68251	1.68251
$881$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$881$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$882$	1.91899	1.91899
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−2.51334	−2.51334
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	−0.918986	−0.918986
$891$	−0.284630	−0.284630
$892$	0	0
$893$	−0.546200	−0.546200
$894$	0	0
$895$	1.39788	1.39788
$896$	0	0
$897$	0	0
$898$	2.51334	2.51334
$899$	0	0
$900$	4.91121	4.91121
$901$	0	0
$902$	0.715370	0.715370
$903$	0	0
$904$	4.22871	4.22871
$905$	0.478891	0.478891
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	5.14769	5.14769
$909$	1.68251	1.68251
$910$	0	0
$911$	−0.830830	−0.830830	−0.415415	−	0.909632i	$-0.636364\pi$
$911$	−0.830830	−0.830830	−0.415415	+	0.909632i	$0.636364\pi$
$912$	0	0
$913$	−0.236479	−0.236479
$914$	−1.59435	−1.59435
$915$	0	0
$916$	−3.51334	−3.51334
$917$	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
$925$	−1.83083	−1.83083
$926$	2.51334	2.51334
$927$	0	0
$928$	−2.91899	−2.91899
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0.284630	0.284630
$932$	0	0
$933$	0	0
$934$	2.51334	2.51334
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	8.66103	8.66103
$941$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$941$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0.546200	0.546200
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	1.00000	1.00000
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	−3.68251	−3.68251
$955$	3.36501	3.36501
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	1.68251	1.68251
$964$	−2.22871	−2.22871
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−2.96714	−2.96714
$969$	0	0
$970$	0	0
$971$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$971$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	4.60149	4.60149
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	−0.0810141	−0.0810141
$980$	−4.51334	−4.51334
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0.478891	0.478891
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0.918986	0.918986
$991$	−1.68251	−1.68251	−0.841254	−	0.540641i	$-0.818182\pi$
$991$	−1.68251	−1.68251	−0.841254	+	0.540641i	$0.818182\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.22871	−3.22871
$996$	0	0
$997$	0.284630	0.284630	0.142315	−	0.989821i	$-0.454545\pi$
$997$	0.284630	0.284630	0.142315	+	0.989821i	$0.454545\pi$
$998$	3.68251	3.68251
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1591.1.d.b.1590.5	yes	5
37.36	even	2		1591.1.d.a.1590.1	✓	5
43.42	odd	2		1591.1.d.a.1590.1	✓	5
1591.1590	odd	2	CM	1591.1.d.b.1590.5	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1591.1.d.a.1590.1	✓	5	37.36	even	2
1591.1.d.a.1590.1	✓	5	43.42	odd	2
1591.1.d.b.1590.5	yes	5	1.1	even	1	trivial
1591.1.d.b.1590.5	yes	5	1591.1590	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:	\( N \)	\(=\)	\( 1591 = 37 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1591.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.91899 q^{2} +2.68251 q^{4} -1.68251 q^{5} +3.22871 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.91899 q^{2} +2.68251 q^{4} -1.68251 q^{5} +3.22871 q^{8} +1.00000 q^{9} -3.22871 q^{10} -0.284630 q^{11} +3.51334 q^{16} +1.91899 q^{18} +0.284630 q^{19} -4.51334 q^{20} -0.546200 q^{22} +1.83083 q^{25} -0.830830 q^{29} +3.51334 q^{32} +2.68251 q^{36} -1.00000 q^{37} +0.546200 q^{38} -5.43232 q^{40} -1.30972 q^{41} -1.00000 q^{43} -0.763521 q^{44} -1.68251 q^{45} -1.91899 q^{47} +1.00000 q^{49} +3.51334 q^{50} -1.91899 q^{53} +0.478891 q^{55} -1.59435 q^{58} +1.30972 q^{61} +3.22871 q^{64} -1.30972 q^{67} +3.22871 q^{72} -1.91899 q^{74} +0.763521 q^{76} -5.91121 q^{80} +1.00000 q^{81} -2.51334 q^{82} +0.830830 q^{83} -1.91899 q^{86} -0.918986 q^{88} +0.284630 q^{89} -3.22871 q^{90} -3.68251 q^{94} -0.478891 q^{95} +1.91899 q^{98} -0.284630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 4 q^{4} + q^{5} + 2 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q + q^2 + 4 * q^4 + q^5 + 2 * q^8 + 5 * q^9	\( 5 q + q^{2} + 4 q^{4} + q^{5} + 2 q^{8} + 5 q^{9} - 2 q^{10} - q^{11} + 3 q^{16} + q^{18} + q^{19} - 8 q^{20} + 2 q^{22} + 4 q^{25} + q^{29} + 3 q^{32} + 4 q^{36} - 5 q^{37} - 2 q^{38} - 4 q^{40} - q^{41} - 5 q^{43} - 3 q^{44} + q^{45} - q^{47} + 5 q^{49} + 3 q^{50} - q^{53} + 2 q^{55} - 2 q^{58} + q^{61} + 2 q^{64} - q^{67} + 2 q^{72} - q^{74} + 3 q^{76} - 6 q^{80} + 5 q^{81} + 2 q^{82} - q^{83} - q^{86} + 4 q^{88} + q^{89} - 2 q^{90} - 9 q^{94} - 2 q^{95} + q^{98} - q^{99}+O(q^{100}) \) 5 * q + q^2 + 4 * q^4 + q^5 + 2 * q^8 + 5 * q^9 - 2 * q^10 - q^11 + 3 * q^16 + q^18 + q^19 - 8 * q^20 + 2 * q^22 + 4 * q^25 + q^29 + 3 * q^32 + 4 * q^36 - 5 * q^37 - 2 * q^38 - 4 * q^40 - q^41 - 5 * q^43 - 3 * q^44 + q^45 - q^47 + 5 * q^49 + 3 * q^50 - q^53 + 2 * q^55 - 2 * q^58 + q^61 + 2 * q^64 - q^67 + 2 * q^72 - q^74 + 3 * q^76 - 6 * q^80 + 5 * q^81 + 2 * q^82 - q^83 - q^86 + 4 * q^88 + q^89 - 2 * q^90 - 9 * q^94 - 2 * q^95 + q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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