LMFDB - Embedded newform 167.1.b.a.166.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [167,1,Mod(166,167)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(167, 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("167.166");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 167 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	167.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:	$0.0833438571097$
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 11.1.129891985607.1

Embedding invariants

Embedding label			166.2
Root			$-1.68251$ of defining polynomial
Character	$\chi$	$=$	167.166

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30972 q^{2} -1.91899 q^{3} +0.715370 q^{4} +2.51334 q^{6} +0.830830 q^{7} +0.372786 q^{8} +2.68251 q^{9} -0.284630 q^{11} -1.37279 q^{12} -1.08816 q^{14} -1.20362 q^{16} -3.51334 q^{18} +1.68251 q^{19} -1.59435 q^{21} +0.372786 q^{22} -0.715370 q^{24} +1.00000 q^{25} -3.22871 q^{27} +0.594351 q^{28} +0.830830 q^{29} -1.30972 q^{31} +1.20362 q^{32} +0.546200 q^{33} +1.91899 q^{36} -2.20362 q^{38} +2.08816 q^{42} -0.203616 q^{44} +1.68251 q^{47} +2.30972 q^{48} -0.309721 q^{49} -1.30972 q^{50} +4.22871 q^{54} +0.309721 q^{56} -3.22871 q^{57} -1.08816 q^{58} -1.91899 q^{61} +1.71537 q^{62} +2.22871 q^{63} -0.372786 q^{64} -0.715370 q^{66} +1.00000 q^{72} -1.91899 q^{75} +1.20362 q^{76} -0.236479 q^{77} +3.51334 q^{81} -1.14055 q^{84} -1.59435 q^{87} -0.106106 q^{88} -1.91899 q^{89} +2.51334 q^{93} -2.20362 q^{94} -2.30972 q^{96} -0.284630 q^{97} +0.405649 q^{98} -0.763521 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} - q^{3} + 4 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 4 q^{9} - q^{11} - 3 q^{12} - 2 q^{14} + 3 q^{16} - 3 q^{18} - q^{19} - 2 q^{21} - 2 q^{22} - 4 q^{24} + 5 q^{25} - 2 q^{27} - 3 q^{28} - q^{29}+ \cdots - 3 q^{99}+O(q^{100}) $ 5 * q - q^2 - q^3 + 4 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 4 * q^9 - q^11 - 3 * q^12 - 2 * q^14 + 3 * q^16 - 3 * q^18 - q^19 - 2 * q^21 - 2 * q^22 - 4 * q^24 + 5 * q^25 - 2 * q^27 - 3 * q^28 - q^29 - q^31 - 3 * q^32 - 2 * q^33 + q^36 - 2 * q^38 + 7 * q^42 + 8 * q^44 - q^47 + 6 * q^48 + 4 * q^49 - q^50 + 7 * q^54 - 4 * q^56 - 2 * q^57 - 2 * q^58 - q^61 + 9 * q^62 - 3 * q^63 + 2 * q^64 - 4 * q^66 + 5 * q^72 - q^75 - 3 * q^76 - 2 * q^77 + 3 * q^81 + 5 * q^84 - 2 * q^87 - 4 * q^88 - q^89 - 2 * q^93 - 2 * q^94 - 6 * q^96 - q^97 + 8 * q^98 - 3 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/167\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-1$

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 $

$2$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$2$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$3$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$3$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$4$	0.715370	0.715370
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	2.51334	2.51334
$7$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$7$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$8$	0.372786	0.372786
$9$	2.68251	2.68251
$10$	0	0
$11$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$11$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$12$	−1.37279	−1.37279
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−1.08816	−1.08816
$15$	0	0
$16$	−1.20362	−1.20362
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−3.51334	−3.51334
$19$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$19$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$20$	0	0
$21$	−1.59435	−1.59435
$22$	0.372786	0.372786
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−0.715370	−0.715370
$25$	1.00000	1.00000
$26$	0	0
$27$	−3.22871	−3.22871
$28$	0.594351	0.594351
$29$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$29$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$30$	0	0
$31$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$31$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$32$	1.20362	1.20362
$33$	0.546200	0.546200
$34$	0	0
$35$	0	0
$36$	1.91899	1.91899
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−2.20362	−2.20362
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	2.08816	2.08816
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−0.203616	−0.203616
$45$	0	0
$46$	0	0
$47$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$47$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$48$	2.30972	2.30972
$49$	−0.309721	−0.309721
$50$	−1.30972	−1.30972
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	4.22871	4.22871
$55$	0	0
$56$	0.309721	0.309721
$57$	−3.22871	−3.22871
$58$	−1.08816	−1.08816
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$61$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$62$	1.71537	1.71537
$63$	2.22871	2.22871
$64$	−0.372786	−0.372786
$65$	0	0
$66$	−0.715370	−0.715370
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.00000	1.00000
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−1.91899	−1.91899
$76$	1.20362	1.20362
$77$	−0.236479	−0.236479
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	3.51334	3.51334
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−1.14055	−1.14055
$85$	0	0
$86$	0	0
$87$	−1.59435	−1.59435
$88$	−0.106106	−0.106106
$89$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$89$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.51334	2.51334
$94$	−2.20362	−2.20362
$95$	0	0
$96$	−2.30972	−2.30972
$97$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$97$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$98$	0.405649	0.405649
$99$	−0.763521	−0.763521
$100$	0.715370	0.715370
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$107$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$108$	−2.30972	−2.30972
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−1.00000
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	4.22871	4.22871
$115$	0	0
$116$	0.594351	0.594351
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−0.918986	−0.918986
$122$	2.51334	2.51334
$123$	0	0
$124$	−0.936936	−0.936936
$125$	0	0
$126$	−2.91899	−2.91899
$127$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$127$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$128$	−0.715370	−0.715370
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0.390736	0.390736
$133$	1.39788	1.39788
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$137$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−3.22871	−3.22871
$142$	0	0
$143$	0	0
$144$	−3.22871	−3.22871
$145$	0	0
$146$	0	0
$147$	0.594351	0.594351
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	2.51334	2.51334
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0.627214	0.627214
$153$	0	0
$154$	0.309721	0.309721
$155$	0	0
$156$	0	0
$157$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$157$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−4.60149	−4.60149
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	1.00000
$167$	1.00000	1.00000
$168$	−0.594351	−0.594351
$169$	1.00000	1.00000
$170$	0	0
$171$	4.51334	4.51334
$172$	0	0
$173$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$173$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$174$	2.08816	2.08816
$175$	0.830830	0.830830
$176$	0.342585	0.342585
$177$	0	0
$178$	2.51334	2.51334
$179$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$179$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$180$	0	0
$181$	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	0
$183$	3.68251	3.68251
$184$	0	0
$185$	0	0
$186$	−3.29177	−3.29177
$187$	0	0
$188$	1.20362	1.20362
$189$	−2.68251	−2.68251
$190$	0	0
$191$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$191$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$192$	0.715370	0.715370
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0.372786	0.372786
$195$	0	0
$196$	−0.221566	−0.221566
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	1.00000	1.00000
$199$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$199$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$200$	0.372786	0.372786
$201$	0	0
$202$	0	0
$203$	0.690279	0.690279
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.478891	−0.478891
$210$	0	0
$211$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$211$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$212$	0	0
$213$	0	0
$214$	−1.08816	−1.08816
$215$	0	0
$216$	−1.20362	−1.20362
$217$	−1.08816	−1.08816
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$223$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$224$	1.00000	1.00000
$225$	2.68251	2.68251
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	−2.30972	−2.30972
$229$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$229$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$230$	0	0
$231$	0.453800	0.453800
$232$	0.309721	0.309721
$233$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$233$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$239$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.20362	1.20362
$243$	−3.51334	−3.51334
$244$	−1.37279	−1.37279
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−0.488245	−0.488245
$249$	0	0
$250$	0	0
$251$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$251$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$252$	1.59435	1.59435
$253$	0	0
$254$	0.372786	0.372786
$255$	0	0
$256$	1.30972	1.30972
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.22871	2.22871
$262$	0	0
$263$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$263$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$264$	0.203616	0.203616
$265$	0	0
$266$	−1.83083	−1.83083
$267$	3.68251	3.68251
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0.372786	0.372786
$275$	−0.284630	−0.284630
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	−3.51334	−3.51334
$280$	0	0
$281$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$281$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$282$	4.22871	4.22871
$283$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$283$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	3.22871	3.22871
$289$	1.00000	1.00000
$290$	0	0
$291$	0.546200	0.546200
$292$	0	0
$293$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$293$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$294$	−0.778434	−0.778434
$295$	0	0
$296$	0	0
$297$	0.918986	0.918986
$298$	0	0
$299$	0	0
$300$	−1.37279	−1.37279
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−2.02509	−2.02509
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	−0.169170	−0.169170
$309$	0	0
$310$	0	0
$311$	2.00000	2.00000	1.00000			$0$
$311$	2.00000	2.00000	1.00000			$0$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	1.71537	1.71537
$315$	0	0
$316$	0	0
$317$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$317$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$318$	0	0
$319$	−0.236479	−0.236479
$320$	0	0
$321$	−1.59435	−1.59435
$322$	0	0
$323$	0	0
$324$	2.51334	2.51334
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.39788	1.39788
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	−1.30972	−1.30972
$335$	0	0
$336$	1.91899	1.91899
$337$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$337$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$338$	−1.30972	−1.30972
$339$	0	0
$340$	0	0
$341$	0.372786	0.372786
$342$	−5.91121	−5.91121
$343$	−1.08816	−1.08816
$344$	0	0
$345$	0	0
$346$	0.372786	0.372786
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	−1.14055	−1.14055
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−1.08816	−1.08816
$351$	0	0
$352$	−0.342585	−0.342585
$353$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$353$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$354$	0	0
$355$	0	0
$356$	−1.37279	−1.37279
$357$	0	0
$358$	−2.20362	−2.20362
$359$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$359$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$360$	0	0
$361$	1.83083	1.83083
$362$	−2.20362	−2.20362
$363$	1.76352	1.76352
$364$	0	0
$365$	0	0
$366$	−4.82306	−4.82306
$367$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$367$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	1.79797	1.79797
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0.627214	0.627214
$377$	0	0
$378$	3.51334	3.51334
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0.546200	0.546200
$382$	1.71537	1.71537
$383$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$383$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$384$	1.37279	1.37279
$385$	0	0
$386$	0	0
$387$	0	0
$388$	−0.203616	−0.203616
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−0.115460	−0.115460
$393$	0	0
$394$	0	0
$395$	0	0
$396$	−0.546200	−0.546200
$397$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$397$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$398$	0.372786	0.372786
$399$	−2.68251	−2.68251
$400$	−1.20362	−1.20362
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	−0.904073	−0.904073
$407$	0	0
$408$	0	0
$409$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$409$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$410$	0	0
$411$	0.546200	0.546200
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0.627214	0.627214
$419$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$419$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$420$	0	0
$421$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$421$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$422$	1.71537	1.71537
$423$	4.51334	4.51334
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.59435	−1.59435
$428$	0.594351	0.594351
$429$	0	0
$430$	0	0
$431$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$431$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$432$	3.88612	3.88612
$433$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$433$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$434$	1.42518	1.42518
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−0.830830	−0.830830
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	2.51334	2.51334
$447$	0	0
$448$	−0.309721	−0.309721
$449$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$449$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$450$	−3.51334	−3.51334
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	−1.20362	−1.20362
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−1.08816	−1.08816
$459$	0	0
$460$	0	0
$461$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$461$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$462$	−0.594351	−0.594351
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−1.00000	−1.00000
$465$	0	0
$466$	1.71537	1.71537
$467$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$467$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2.51334	2.51334
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1.68251	1.68251
$476$	0	0
$477$	0	0
$478$	2.51334	2.51334
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−0.657415	−0.657415
$485$	0	0
$486$	4.60149	4.60149
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	−0.715370	−0.715370
$489$	0	0
$490$	0	0
$491$	2.00000	2.00000	1.00000			$0$
$491$	2.00000	2.00000	1.00000			$0$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	1.57640	1.57640
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−1.91899	−1.91899
$502$	−1.08816	−1.08816
$503$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$503$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$504$	0.830830	0.830830
$505$	0	0
$506$	0	0
$507$	−1.91899	−1.91899
$508$	−0.203616	−0.203616
$509$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$509$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−1.00000
$513$	−5.43232	−5.43232
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.478891	−0.478891
$518$	0	0
$519$	0.546200	0.546200
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−2.91899	−2.91899
$523$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$523$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$524$	0	0
$525$	−1.59435	−1.59435
$526$	2.51334	2.51334
$527$	0	0
$528$	−0.657415	−0.657415
$529$	1.00000	1.00000
$530$	0	0
$531$	0	0
$532$	1.00000	1.00000
$533$	0	0
$534$	−4.82306	−4.82306
$535$	0	0
$536$	0	0
$537$	−3.22871	−3.22871
$538$	0	0
$539$	0.0881559	0.0881559
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	−3.22871	−3.22871
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−0.203616	−0.203616
$549$	−5.14769	−5.14769
$550$	0.372786	0.372786
$551$	1.39788	1.39788
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$557$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$558$	4.60149	4.60149
$559$	0	0
$560$	0	0
$561$	0	0
$562$	2.51334	2.51334
$563$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$563$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$564$	−2.30972	−2.30972
$565$	0	0
$566$	0.372786	0.372786
$567$	2.91899	2.91899
$568$	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
$573$	2.51334	2.51334
$574$	0	0
$575$	0	0
$576$	−1.00000	−1.00000
$577$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$577$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$578$	−1.30972	−1.30972
$579$	0	0
$580$	0	0
$581$	0	0
$582$	−0.715370	−0.715370
$583$	0	0
$584$	0	0
$585$	0	0
$586$	1.71537	1.71537
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0.425181	0.425181
$589$	−2.20362	−2.20362
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	−1.20362	−1.20362
$595$	0	0
$596$	0	0
$597$	0.546200	0.546200
$598$	0	0
$599$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$599$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$600$	−0.715370	−0.715370
$601$	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
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	2.02509	2.02509
$609$	−1.32463	−1.32463
$610$	0	0
$611$	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	$0.727273\pi$
$614$	0	0
$615$	0	0
$616$	−0.0881559	−0.0881559
$617$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$617$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	−2.61944	−2.61944
$623$	−1.59435	−1.59435
$624$	0	0
$625$	1.00000	1.00000
$626$	0	0
$627$	0.918986	0.918986
$628$	−0.936936	−0.936936
$629$	0	0
$630$	0	0
$631$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$631$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$632$	0	0
$633$	2.51334	2.51334
$634$	2.51334	2.51334
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0.309721	0.309721
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	2.08816	2.08816
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	1.30972	1.30972
$649$	0	0
$650$	0	0
$651$	2.08816	2.08816
$652$	0	0
$653$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$653$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	−1.83083	−1.83083
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0.715370	0.715370
$669$	3.68251	3.68251
$670$	0	0
$671$	0.546200	0.546200
$672$	−1.91899	−1.91899
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	−2.20362	−2.20362
$675$	−3.22871	−3.22871
$676$	0.715370	0.715370
$677$	2.00000	2.00000	1.00000			$0$
$677$	2.00000	2.00000	1.00000			$0$
$678$	0	0
$679$	−0.236479	−0.236479
$680$	0	0
$681$	0	0
$682$	−0.488245	−0.488245
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	3.22871	3.22871
$685$	0	0
$686$	1.42518	1.42518
$687$	−1.59435	−1.59435
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−0.203616	−0.203616
$693$	−0.634356	−0.634356
$694$	0	0
$695$	0	0
$696$	−0.594351	−0.594351
$697$	0	0
$698$	0	0
$699$	2.51334	2.51334
$700$	0.594351	0.594351
$701$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$701$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$702$	0	0
$703$	0	0
$704$	0.106106	0.106106
$705$	0	0
$706$	−1.08816	−1.08816
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−0.715370	−0.715370
$713$	0	0
$714$	0	0
$715$	0	0
$716$	1.20362	1.20362
$717$	3.68251	3.68251
$718$	1.71537	1.71537
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	−2.39788	−2.39788
$723$	0	0
$724$	1.20362	1.20362
$725$	0.830830	0.830830
$726$	−2.30972	−2.30972
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	3.22871	3.22871
$730$	0	0
$731$	0	0
$732$	2.63436	2.63436
$733$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$733$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$734$	−2.20362	−2.20362
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.00000	2.00000	1.00000			$0$
$743$	2.00000	2.00000	1.00000			$0$
$744$	0.936936	0.936936
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.690279	0.690279
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−2.02509	−2.02509
$753$	−1.59435	−1.59435
$754$	0	0
$755$	0	0
$756$	−1.91899	−1.91899
$757$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$757$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$761$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$762$	−0.715370	−0.715370
$763$	0	0
$764$	−0.936936	−0.936936
$765$	0	0
$766$	−1.08816	−1.08816
$767$	0	0
$768$	−2.51334	−2.51334
$769$	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			$\pi$
$774$	0	0
$775$	−1.30972	−1.30972
$776$	−0.106106	−0.106106
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−2.68251	−2.68251
$784$	0.372786	0.372786
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	3.68251	3.68251
$790$	0	0
$791$	0	0
$792$	−0.284630	−0.284630
$793$	0	0
$794$	2.51334	2.51334
$795$	0	0
$796$	−0.203616	−0.203616
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	3.51334	3.51334
$799$	0	0
$800$	1.20362	1.20362
$801$	−5.14769	−5.14769
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$809$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0.493805	0.493805
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0.372786	0.372786
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−0.715370	−0.715370
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0.546200	0.546200
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−0.342585	−0.342585
$837$	4.22871	4.22871
$838$	1.71537	1.71537
$839$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$839$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$840$	0	0
$841$	−0.309721	−0.309721
$842$	−1.08816	−1.08816
$843$	3.68251	3.68251
$844$	−0.936936	−0.936936
$845$	0	0
$846$	−5.91121	−5.91121
$847$	−0.763521	−0.763521
$848$	0	0
$849$	0.546200	0.546200
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$853$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$854$	2.08816	2.08816
$855$	0	0
$856$	0.309721	0.309721
$857$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$857$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$858$	0	0
$859$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$859$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$860$	0	0
$861$	0	0
$862$	2.51334	2.51334
$863$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$863$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$864$	−3.88612	−3.88612
$865$	0	0
$866$	−2.20362	−2.20362
$867$	−1.91899	−1.91899
$868$	−0.778434	−0.778434
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−0.763521	−0.763521
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$877$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$878$	0	0
$879$	2.51334	2.51334
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1.08816	1.08816
$883$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$883$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−0.236479	−0.236479
$890$	0	0
$891$	−1.00000	−1.00000
$892$	−1.37279	−1.37279
$893$	2.83083	2.83083
$894$	0	0
$895$	0	0
$896$	−0.594351	−0.594351
$897$	0	0
$898$	−2.20362	−2.20362
$899$	−1.08816	−1.08816
$900$	1.91899	1.91899
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$907$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$911$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$912$	3.88612	3.88612
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0.594351	0.594351
$917$	0	0
$918$	0	0
$919$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$919$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$920$	0	0
$921$	0	0
$922$	−2.20362	−2.20362
$923$	0	0
$924$	0.324635	0.324635
$925$	0	0
$926$	0	0
$927$	0	0
$928$	1.00000	1.00000
$929$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$929$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$930$	0	0
$931$	−0.521109	−0.521109
$932$	−0.936936	−0.936936
$933$	−3.83797	−3.83797
$934$	−1.08816	−1.08816
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	−3.29177	−3.29177
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$947$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$948$	0	0
$949$	0	0
$950$	−2.20362	−2.20362
$951$	3.68251	3.68251
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	−1.37279	−1.37279
$957$	0.453800	0.453800
$958$	0	0
$959$	−0.236479	−0.236479
$960$	0	0
$961$	0.715370	0.715370
$962$	0	0
$963$	2.22871	2.22871
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$967$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$968$	−0.342585	−0.342585
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−2.51334	−2.51334
$973$	0	0
$974$	0	0
$975$	0	0
$976$	2.30972	2.30972
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0.546200	0.546200
$980$	0	0
$981$	0	0
$982$	−2.61944	−2.61944
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.68251	−2.68251
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	−1.57640	−1.57640
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$997$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	167.1.b.a.166.2	✓	5
3.2	odd	2		1503.1.d.a.667.4		5
4.3	odd	2		2672.1.g.a.2337.5		5
167.166	odd	2	CM	167.1.b.a.166.2	✓	5
501.500	even	2		1503.1.d.a.667.4		5
668.667	even	2		2672.1.g.a.2337.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
167.1.b.a.166.2	✓	5	1.1	even	1	trivial
167.1.b.a.166.2	✓	5	167.166	odd	2	CM
1503.1.d.a.667.4		5	3.2	odd	2
1503.1.d.a.667.4		5	501.500	even	2
2672.1.g.a.2337.5		5	4.3	odd	2
2672.1.g.a.2337.5		5	668.667	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 167 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	167.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.30972 q^{2} -1.91899 q^{3} +0.715370 q^{4} +2.51334 q^{6} +0.830830 q^{7} +0.372786 q^{8} +2.68251 q^{9} -0.284630 q^{11} -1.37279 q^{12} -1.08816 q^{14} -1.20362 q^{16} -3.51334 q^{18} +1.68251 q^{19} -1.59435 q^{21} +0.372786 q^{22} -0.715370 q^{24} +1.00000 q^{25} -3.22871 q^{27} +0.594351 q^{28} +0.830830 q^{29} -1.30972 q^{31} +1.20362 q^{32} +0.546200 q^{33} +1.91899 q^{36} -2.20362 q^{38} +2.08816 q^{42} -0.203616 q^{44} +1.68251 q^{47} +2.30972 q^{48} -0.309721 q^{49} -1.30972 q^{50} +4.22871 q^{54} +0.309721 q^{56} -3.22871 q^{57} -1.08816 q^{58} -1.91899 q^{61} +1.71537 q^{62} +2.22871 q^{63} -0.372786 q^{64} -0.715370 q^{66} +1.00000 q^{72} -1.91899 q^{75} +1.20362 q^{76} -0.236479 q^{77} +3.51334 q^{81} -1.14055 q^{84} -1.59435 q^{87} -0.106106 q^{88} -1.91899 q^{89} +2.51334 q^{93} -2.20362 q^{94} -2.30972 q^{96} -0.284630 q^{97} +0.405649 q^{98} -0.763521 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} - q^{3} + 4 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 4 q^{9} - q^{11} - 3 q^{12} - 2 q^{14} + 3 q^{16} - 3 q^{18} - q^{19} - 2 q^{21} - 2 q^{22} - 4 q^{24} + 5 q^{25} - 2 q^{27} - 3 q^{28} - q^{29}+ \cdots - 3 q^{99}+O(q^{100}) \) 5 * q - q^2 - q^3 + 4 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 4 * q^9 - q^11 - 3 * q^12 - 2 * q^14 + 3 * q^16 - 3 * q^18 - q^19 - 2 * q^21 - 2 * q^22 - 4 * q^24 + 5 * q^25 - 2 * q^27 - 3 * q^28 - q^29 - q^31 - 3 * q^32 - 2 * q^33 + q^36 - 2 * q^38 + 7 * q^42 + 8 * q^44 - q^47 + 6 * q^48 + 4 * q^49 - q^50 + 7 * q^54 - 4 * q^56 - 2 * q^57 - 2 * q^58 - q^61 + 9 * q^62 - 3 * q^63 + 2 * q^64 - 4 * q^66 + 5 * q^72 - q^75 - 3 * q^76 - 2 * q^77 + 3 * q^81 + 5 * q^84 - 2 * q^87 - 4 * q^88 - q^89 - 2 * q^93 - 2 * q^94 - 6 * q^96 - q^97 + 8 * q^98 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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