LMFDB - Embedded newform 129.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [129,2,Mod(1,129)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(129, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("129.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 129 = 3 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	129.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.03007018607$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	129.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} -4.00000 q^{23} -3.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} +5.00000 q^{32} -6.00000 q^{34} -1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} -6.00000 q^{40} +2.00000 q^{41} -1.00000 q^{43} +2.00000 q^{45} -4.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +4.00000 q^{57} -6.00000 q^{58} -2.00000 q^{60} +14.0000 q^{61} +8.00000 q^{62} +7.00000 q^{64} -4.00000 q^{65} +12.0000 q^{67} +6.00000 q^{68} -4.00000 q^{69} +8.00000 q^{71} -3.00000 q^{72} +2.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} -8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -12.0000 q^{85} -1.00000 q^{86} -6.00000 q^{87} +14.0000 q^{89} +2.00000 q^{90} +4.00000 q^{92} +8.00000 q^{93} +4.00000 q^{94} +8.00000 q^{95} +5.00000 q^{96} -14.0000 q^{97} -7.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	1.00000	0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	−1.00000	−0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−3.00000	−0.612372
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	2.00000	0.365148
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	−1.00000	−0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	4.00000	0.648886
$39$	−2.00000	−0.320256
$40$	−6.00000	−0.948683
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	2.00000	0.298142
$46$	−4.00000	−0.589768
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	−1.00000	−0.141421
$51$	−6.00000	−0.840168
$52$	2.00000	0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	4.00000	0.529813
$58$	−6.00000	−0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−2.00000	−0.258199
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	6.00000	0.727607
$69$	−4.00000	−0.481543
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−3.00000	−0.353553
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	6.00000	0.697486
$75$	−1.00000	−0.115470
$76$	−4.00000	−0.458831
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	−1.00000	−0.107833
$87$	−6.00000	−0.643268
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	4.00000	0.417029
$93$	8.00000	0.829561
$94$	4.00000	0.412568
$95$	8.00000	0.820783
$96$	5.00000	0.510310
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−7.00000	−0.707107
$99$	0	0
$100$	1.00000	0.100000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	−6.00000	−0.594089
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−1.00000	−0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	4.00000	0.374634
$115$	−8.00000	−0.746004
$116$	6.00000	0.557086
$117$	−2.00000	−0.184900
$118$	0	0
$119$	0	0
$120$	−6.00000	−0.547723
$121$	−11.0000	−1.00000
$122$	14.0000	1.26750
$123$	2.00000	0.180334
$124$	−8.00000	−0.718421
$125$	−12.0000	−1.07331
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−3.00000	−0.265165
$129$	−1.00000	−0.0880451
$130$	−4.00000	−0.350823
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	0	0
$134$	12.0000	1.03664
$135$	2.00000	0.172133
$136$	18.0000	1.54349
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−4.00000	−0.340503
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	8.00000	0.671345
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	−12.0000	−0.996546
$146$	2.00000	0.165521
$147$	−7.00000	−0.577350
$148$	−6.00000	−0.493197
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−1.00000	−0.0816497
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−12.0000	−0.973329
$153$	−6.00000	−0.485071
$154$	0	0
$155$	16.0000	1.28515
$156$	2.00000	0.160128
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−8.00000	−0.636446
$159$	−2.00000	−0.158610
$160$	10.0000	0.790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−12.0000	−0.920358
$171$	4.00000	0.305888
$172$	1.00000	0.0762493
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	0	0
$177$	0	0
$178$	14.0000	1.04934
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−2.00000	−0.149071
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	12.0000	0.884652
$185$	12.0000	0.882258
$186$	8.00000	0.586588
$187$	0	0
$188$	−4.00000	−0.291730
$189$	0	0
$190$	8.00000	0.580381
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	7.00000	0.505181
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	−14.0000	−1.00514
$195$	−4.00000	−0.286446
$196$	7.00000	0.500000
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	3.00000	0.212132
$201$	12.0000	0.846415
$202$	−18.0000	−1.26648
$203$	0	0
$204$	6.00000	0.420084
$205$	4.00000	0.279372
$206$	−8.00000	−0.557386
$207$	−4.00000	−0.278019
$208$	2.00000	0.138675
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	2.00000	0.137361
$213$	8.00000	0.548151
$214$	0	0
$215$	−2.00000	−0.136399
$216$	−3.00000	−0.204124
$217$	0	0
$218$	−2.00000	−0.135457
$219$	2.00000	0.135147
$220$	0	0
$221$	12.0000	0.807207
$222$	6.00000	0.402694
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−2.00000	−0.133038
$227$	−28.0000	−1.85843	−0.929213	−	0.369546i	$-0.879513\pi$
$227$	−28.0000	−1.85843	−0.929213	+	0.369546i	$0.879513\pi$
$228$	−4.00000	−0.264906
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	18.0000	1.18176
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−2.00000	−0.130744
$235$	8.00000	0.521862
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	28.0000	1.81117	0.905585	−	0.424165i	$-0.139432\pi$
$239$	28.0000	1.81117	0.905585	+	0.424165i	$0.139432\pi$
$240$	−2.00000	−0.129099
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	−11.0000	−0.707107
$243$	1.00000	0.0641500
$244$	−14.0000	−0.896258
$245$	−14.0000	−0.894427
$246$	2.00000	0.127515
$247$	−8.00000	−0.509028
$248$	−24.0000	−1.52400
$249$	0	0
$250$	−12.0000	−0.758947
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	0	0
$254$	16.0000	1.00393
$255$	−12.0000	−0.751469
$256$	−17.0000	−1.06250
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	−1.00000	−0.0622573
$259$	0	0
$260$	4.00000	0.248069
$261$	−6.00000	−0.371391
$262$	−4.00000	−0.247121
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	14.0000	0.856786
$268$	−12.0000	−0.733017
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	2.00000	0.121716
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	4.00000	0.240772
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−20.0000	−1.19952
$279$	8.00000	0.478947
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	4.00000	0.238197
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	−8.00000	−0.474713
$285$	8.00000	0.473879
$286$	0	0
$287$	0	0
$288$	5.00000	0.294628
$289$	19.0000	1.11765
$290$	−12.0000	−0.704664
$291$	−14.0000	−0.820695
$292$	−2.00000	−0.117041
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	−18.0000	−1.04623
$297$	0	0
$298$	−6.00000	−0.347571
$299$	8.00000	0.462652
$300$	1.00000	0.0577350
$301$	0	0
$302$	16.0000	0.920697
$303$	−18.0000	−1.03407
$304$	−4.00000	−0.229416
$305$	28.0000	1.60328
$306$	−6.00000	−0.342997
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	16.0000	0.908739
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	6.00000	0.339683
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	8.00000	0.450035
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	−2.00000	−0.112154
$319$	0	0
$320$	14.0000	0.782624
$321$	0	0
$322$	0	0
$323$	−24.0000	−1.33540
$324$	−1.00000	−0.0555556
$325$	2.00000	0.110940
$326$	−4.00000	−0.221540
$327$	−2.00000	−0.110600
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	−12.0000	−0.656611
$335$	24.0000	1.31126
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	−9.00000	−0.489535
$339$	−2.00000	−0.108625
$340$	12.0000	0.650791
$341$	0	0
$342$	4.00000	0.216295
$343$	0	0
$344$	3.00000	0.161749
$345$	−8.00000	−0.430706
$346$	−18.0000	−0.967686
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	6.00000	0.321634
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	−14.0000	−0.741999
$357$	0	0
$358$	20.0000	1.05703
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	−6.00000	−0.316228
$361$	−3.00000	−0.157895
$362$	22.0000	1.15629
$363$	−11.0000	−0.577350
$364$	0	0
$365$	4.00000	0.209370
$366$	14.0000	0.731792
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	4.00000	0.208514
$369$	2.00000	0.104116
$370$	12.0000	0.623850
$371$	0	0
$372$	−8.00000	−0.414781
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	−12.0000	−0.618853
$377$	12.0000	0.618031
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	−8.00000	−0.410391
$381$	16.0000	0.819705
$382$	8.00000	0.409316
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	18.0000	0.916176
$387$	−1.00000	−0.0508329
$388$	14.0000	0.710742
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−4.00000	−0.202548
$391$	24.0000	1.21373
$392$	21.0000	1.06066
$393$	−4.00000	−0.201773
$394$	14.0000	0.705310
$395$	−16.0000	−0.805047
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	1.00000	0.0500000
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	12.0000	0.598506
$403$	−16.0000	−0.797017
$404$	18.0000	0.895533
$405$	2.00000	0.0993808
$406$	0	0
$407$	0	0
$408$	18.0000	0.891133
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	4.00000	0.197546
$411$	−18.0000	−0.887875
$412$	8.00000	0.394132
$413$	0	0
$414$	−4.00000	−0.196589
$415$	0	0
$416$	−10.0000	−0.490290
$417$	−20.0000	−0.979404
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	−4.00000	−0.194717
$423$	4.00000	0.194487
$424$	6.00000	0.291386
$425$	6.00000	0.291043
$426$	8.00000	0.387601
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	−1.00000	−0.0481125
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	−12.0000	−0.575356
$436$	2.00000	0.0957826
$437$	−16.0000	−0.765384
$438$	2.00000	0.0955637
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	12.0000	0.570782
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	−6.00000	−0.284747
$445$	28.0000	1.32733
$446$	−16.0000	−0.757622
$447$	−6.00000	−0.283790
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	−1.00000	−0.0471405
$451$	0	0
$452$	2.00000	0.0940721
$453$	16.0000	0.751746
$454$	−28.0000	−1.31411
$455$	0	0
$456$	−12.0000	−0.561951
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	6.00000	0.280362
$459$	−6.00000	−0.280056
$460$	8.00000	0.373002
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	6.00000	0.278543
$465$	16.0000	0.741982
$466$	−18.0000	−0.833834
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	2.00000	0.0924500
$469$	0	0
$470$	8.00000	0.369012
$471$	14.0000	0.645086
$472$	0	0
$473$	0	0
$474$	−8.00000	−0.367452
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	28.0000	1.28069
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	10.0000	0.456435
$481$	−12.0000	−0.547153
$482$	−6.00000	−0.273293
$483$	0	0
$484$	11.0000	0.500000
$485$	−28.0000	−1.27141
$486$	1.00000	0.0453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	−42.0000	−1.90125
$489$	−4.00000	−0.180886
$490$	−14.0000	−0.632456
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−2.00000	−0.0901670
$493$	36.0000	1.62136
$494$	−8.00000	−0.359937
$495$	0	0
$496$	−8.00000	−0.359211
$497$	0	0
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	12.0000	0.536656
$501$	−12.0000	−0.536120
$502$	16.0000	0.714115
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−36.0000	−1.60198
$506$	0	0
$507$	−9.00000	−0.399704
$508$	−16.0000	−0.709885
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	−12.0000	−0.531369
$511$	0	0
$512$	−11.0000	−0.486136
$513$	4.00000	0.176604
$514$	−18.0000	−0.793946
$515$	−16.0000	−0.705044
$516$	1.00000	0.0440225
$517$	0	0
$518$	0	0
$519$	−18.0000	−0.790112
$520$	12.0000	0.526235
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	−6.00000	−0.262613
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	24.0000	1.04645
$527$	−48.0000	−2.09091
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−4.00000	−0.173749
$531$	0	0
$532$	0	0
$533$	−4.00000	−0.173259
$534$	14.0000	0.605839
$535$	0	0
$536$	−36.0000	−1.55496
$537$	20.0000	0.863064
$538$	−10.0000	−0.431131
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	8.00000	0.343629
$543$	22.0000	0.944110
$544$	−30.0000	−1.28624
$545$	−4.00000	−0.171341
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	18.0000	0.768922
$549$	14.0000	0.597505
$550$	0	0
$551$	−24.0000	−1.02243
$552$	12.0000	0.510754
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	12.0000	0.509372
$556$	20.0000	0.848189
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	8.00000	0.338667
$559$	2.00000	0.0845910
$560$	0	0
$561$	0	0
$562$	10.0000	0.421825
$563$	16.0000	0.674320	0.337160	−	0.941447i	$-0.390534\pi$
$563$	16.0000	0.674320	0.337160	+	0.941447i	$0.390534\pi$
$564$	−4.00000	−0.168430
$565$	−4.00000	−0.168281
$566$	−12.0000	−0.504398
$567$	0	0
$568$	−24.0000	−1.00702
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	8.00000	0.335083
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	4.00000	0.166812
$576$	7.00000	0.291667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	19.0000	0.790296
$579$	18.0000	0.748054
$580$	12.0000	0.498273
$581$	0	0
$582$	−14.0000	−0.580319
$583$	0	0
$584$	−6.00000	−0.248282
$585$	−4.00000	−0.165380
$586$	−2.00000	−0.0826192
$587$	20.0000	0.825488	0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000	0.825488	0.412744	+	0.910847i	$0.364570\pi$
$588$	7.00000	0.288675
$589$	32.0000	1.31854
$590$	0	0
$591$	14.0000	0.575883
$592$	−6.00000	−0.246598
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	8.00000	0.327418
$598$	8.00000	0.327144
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	3.00000	0.122474
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	−16.0000	−0.651031
$605$	−22.0000	−0.894427
$606$	−18.0000	−0.731200
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	20.0000	0.811107
$609$	0	0
$610$	28.0000	1.13369
$611$	−8.00000	−0.323645
$612$	6.00000	0.242536
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−28.0000	−1.12999
$615$	4.00000	0.161296
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−8.00000	−0.321807
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	−16.0000	−0.642575
$621$	−4.00000	−0.160514
$622$	12.0000	0.481156
$623$	0	0
$624$	2.00000	0.0800641
$625$	−19.0000	−0.760000
$626$	10.0000	0.399680
$627$	0	0
$628$	−14.0000	−0.558661
$629$	−36.0000	−1.43541
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	24.0000	0.954669
$633$	−4.00000	−0.158986
$634$	30.0000	1.19145
$635$	32.0000	1.26988
$636$	2.00000	0.0793052
$637$	14.0000	0.554700
$638$	0	0
$639$	8.00000	0.316475
$640$	−6.00000	−0.237171
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	0	0
$645$	−2.00000	−0.0787499
$646$	−24.0000	−0.944267
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	−3.00000	−0.117851
$649$	0	0
$650$	2.00000	0.0784465
$651$	0	0
$652$	4.00000	0.156652
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	−2.00000	−0.0782062
$655$	−8.00000	−0.312586
$656$	−2.00000	−0.0780869
$657$	2.00000	0.0780274
$658$	0	0
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	−20.0000	−0.777322
$663$	12.0000	0.466041
$664$	0	0
$665$	0	0
$666$	6.00000	0.232495
$667$	24.0000	0.929284
$668$	12.0000	0.464294
$669$	−16.0000	−0.618596
$670$	24.0000	0.927201
$671$	0	0
$672$	0	0
$673$	50.0000	1.92736	0.963679	−	0.267063i	$-0.0860531\pi$
$673$	50.0000	1.92736	0.963679	+	0.267063i	$0.0860531\pi$
$674$	18.0000	0.693334
$675$	−1.00000	−0.0384900
$676$	9.00000	0.346154
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	−2.00000	−0.0768095
$679$	0	0
$680$	36.0000	1.38054
$681$	−28.0000	−1.07296
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−4.00000	−0.152944
$685$	−36.0000	−1.37549
$686$	0	0
$687$	6.00000	0.228914
$688$	1.00000	0.0381246
$689$	4.00000	0.152388
$690$	−8.00000	−0.304555
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−20.0000	−0.759190
$695$	−40.0000	−1.51729
$696$	18.0000	0.682288
$697$	−12.0000	−0.454532
$698$	14.0000	0.529908
$699$	−18.0000	−0.680823
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	−2.00000	−0.0754851
$703$	24.0000	0.905177
$704$	0	0
$705$	8.00000	0.301297
$706$	26.0000	0.978523
$707$	0	0
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	16.0000	0.600469
$711$	−8.00000	−0.300023
$712$	−42.0000	−1.57402
$713$	−32.0000	−1.19841
$714$	0	0
$715$	0	0
$716$	−20.0000	−0.747435
$717$	28.0000	1.04568
$718$	4.00000	0.149279
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	−3.00000	−0.111648
$723$	−6.00000	−0.223142
$724$	−22.0000	−0.817624
$725$	6.00000	0.222834
$726$	−11.0000	−0.408248
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	6.00000	0.221918
$732$	−14.0000	−0.517455
$733$	−10.0000	−0.369358	−0.184679	−	0.982799i	$-0.559125\pi$
$733$	−10.0000	−0.369358	−0.184679	+	0.982799i	$0.559125\pi$
$734$	−8.00000	−0.295285
$735$	−14.0000	−0.516398
$736$	−20.0000	−0.737210
$737$	0	0
$738$	2.00000	0.0736210
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	−12.0000	−0.441129
$741$	−8.00000	−0.293887
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	−24.0000	−0.879883
$745$	−12.0000	−0.439646
$746$	−10.0000	−0.366126
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−12.0000	−0.438178
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	−4.00000	−0.145865
$753$	16.0000	0.583072
$754$	12.0000	0.437014
$755$	32.0000	1.16460
$756$	0	0
$757$	−50.0000	−1.81728	−0.908640	−	0.417579i	$-0.862879\pi$
$757$	−50.0000	−1.81728	−0.908640	+	0.417579i	$0.862879\pi$
$758$	−28.0000	−1.01701
$759$	0	0
$760$	−24.0000	−0.870572
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	16.0000	0.579619
$763$	0	0
$764$	−8.00000	−0.289430
$765$	−12.0000	−0.433861
$766$	−16.0000	−0.578103
$767$	0	0
$768$	−17.0000	−0.613435
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	−18.0000	−0.647834
$773$	−46.0000	−1.65451	−0.827253	−	0.561830i	$-0.810097\pi$
$773$	−46.0000	−1.65451	−0.827253	+	0.561830i	$0.810097\pi$
$774$	−1.00000	−0.0359443
$775$	−8.00000	−0.287368
$776$	42.0000	1.50771
$777$	0	0
$778$	−6.00000	−0.215110
$779$	8.00000	0.286630
$780$	4.00000	0.143223
$781$	0	0
$782$	24.0000	0.858238
$783$	−6.00000	−0.214423
$784$	7.00000	0.250000
$785$	28.0000	0.999363
$786$	−4.00000	−0.142675
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−14.0000	−0.498729
$789$	24.0000	0.854423
$790$	−16.0000	−0.569254
$791$	0	0
$792$	0	0
$793$	−28.0000	−0.994309
$794$	−18.0000	−0.638796
$795$	−4.00000	−0.141865
$796$	−8.00000	−0.283552
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	−5.00000	−0.176777
$801$	14.0000	0.494666
$802$	−22.0000	−0.776847
$803$	0	0
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−16.0000	−0.563576
$807$	−10.0000	−0.352017
$808$	54.0000	1.89971
$809$	50.0000	1.75791	0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000	1.75791	0.878953	+	0.476908i	$0.158243\pi$
$810$	2.00000	0.0702728
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	0	0
$815$	−8.00000	−0.280228
$816$	6.00000	0.210042
$817$	−4.00000	−0.139942
$818$	18.0000	0.629355
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−26.0000	−0.907406	−0.453703	−	0.891153i	$-0.649897\pi$
$821$	−26.0000	−0.907406	−0.453703	+	0.891153i	$0.649897\pi$
$822$	−18.0000	−0.627822
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	0	0
$827$	24.0000	0.834562	0.417281	−	0.908778i	$-0.362983\pi$
$827$	24.0000	0.834562	0.417281	+	0.908778i	$0.362983\pi$
$828$	4.00000	0.139010
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	−14.0000	−0.485363
$833$	42.0000	1.45521
$834$	−20.0000	−0.692543
$835$	−24.0000	−0.830554
$836$	0	0
$837$	8.00000	0.276520
$838$	−4.00000	−0.138178
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−34.0000	−1.17172
$843$	10.0000	0.344418
$844$	4.00000	0.137686
$845$	−18.0000	−0.619219
$846$	4.00000	0.137523
$847$	0	0
$848$	2.00000	0.0686803
$849$	−12.0000	−0.411839
$850$	6.00000	0.205798
$851$	−24.0000	−0.822709
$852$	−8.00000	−0.274075
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	0	0
$857$	26.0000	0.888143	0.444072	−	0.895991i	$-0.353534\pi$
$857$	26.0000	0.888143	0.444072	+	0.895991i	$0.353534\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	2.00000	0.0681994
$861$	0	0
$862$	36.0000	1.22616
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	5.00000	0.170103
$865$	−36.0000	−1.22404
$866$	18.0000	0.611665
$867$	19.0000	0.645274
$868$	0	0
$869$	0	0
$870$	−12.0000	−0.406838
$871$	−24.0000	−0.813209
$872$	6.00000	0.203186
$873$	−14.0000	−0.473828
$874$	−16.0000	−0.541208
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	−40.0000	−1.34993
$879$	−2.00000	−0.0674583
$880$	0	0
$881$	−38.0000	−1.28025	−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000	−1.28025	−0.640126	+	0.768270i	$0.721118\pi$
$882$	−7.00000	−0.235702
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−16.0000	−0.537531
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	−18.0000	−0.604040
$889$	0	0
$890$	28.0000	0.938562
$891$	0	0
$892$	16.0000	0.535720
$893$	16.0000	0.535420
$894$	−6.00000	−0.200670
$895$	40.0000	1.33705
$896$	0	0
$897$	8.00000	0.267112
$898$	30.0000	1.00111
$899$	−48.0000	−1.60089
$900$	1.00000	0.0333333
$901$	12.0000	0.399778
$902$	0	0
$903$	0	0
$904$	6.00000	0.199557
$905$	44.0000	1.46261
$906$	16.0000	0.531564
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	28.0000	0.929213
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	18.0000	0.595387
$915$	28.0000	0.925651
$916$	−6.00000	−0.198246
$917$	0	0
$918$	−6.00000	−0.198030
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	24.0000	0.791257
$921$	−28.0000	−0.922631
$922$	−18.0000	−0.592798
$923$	−16.0000	−0.526646
$924$	0	0
$925$	−6.00000	−0.197279
$926$	16.0000	0.525793
$927$	−8.00000	−0.262754
$928$	−30.0000	−0.984798
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	16.0000	0.524661
$931$	−28.0000	−0.917663
$932$	18.0000	0.589610
$933$	12.0000	0.392862
$934$	−12.0000	−0.392652
$935$	0	0
$936$	6.00000	0.196116
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	−8.00000	−0.260931
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	14.0000	0.456145
$943$	−8.00000	−0.260516
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	8.00000	0.259828
$949$	−4.00000	−0.129845
$950$	−4.00000	−0.129777
$951$	30.0000	0.972817
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	−2.00000	−0.0647524
$955$	16.0000	0.517748
$956$	−28.0000	−0.905585
$957$	0	0
$958$	36.0000	1.16311
$959$	0	0
$960$	14.0000	0.451848
$961$	33.0000	1.06452
$962$	−12.0000	−0.386896
$963$	0	0
$964$	6.00000	0.193247
$965$	36.0000	1.15888
$966$	0	0
$967$	−56.0000	−1.80084	−0.900419	−	0.435023i	$-0.856740\pi$
$967$	−56.0000	−1.80084	−0.900419	+	0.435023i	$0.856740\pi$
$968$	33.0000	1.06066
$969$	−24.0000	−0.770991
$970$	−28.0000	−0.899026
$971$	32.0000	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	24.0000	0.769010
$975$	2.00000	0.0640513
$976$	−14.0000	−0.448129
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	−4.00000	−0.127906
$979$	0	0
$980$	14.0000	0.447214
$981$	−2.00000	−0.0638551
$982$	−12.0000	−0.382935
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−6.00000	−0.191273
$985$	28.0000	0.892154
$986$	36.0000	1.14647
$987$	0	0
$988$	8.00000	0.254514
$989$	4.00000	0.127193
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	40.0000	1.27000
$993$	−20.0000	−0.634681
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	4.00000	0.126618
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.2.a.b.1.1	✓	1
3.2	odd	2		387.2.a.a.1.1		1
4.3	odd	2		2064.2.a.e.1.1		1
5.4	even	2		3225.2.a.b.1.1		1
7.6	odd	2		6321.2.a.f.1.1		1
8.3	odd	2		8256.2.a.ba.1.1		1
8.5	even	2		8256.2.a.f.1.1		1
12.11	even	2		6192.2.a.g.1.1		1
15.14	odd	2		9675.2.a.s.1.1		1
43.42	odd	2		5547.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.a.b.1.1	✓	1	1.1	even	1	trivial
387.2.a.a.1.1		1	3.2	odd	2
2064.2.a.e.1.1		1	4.3	odd	2
3225.2.a.b.1.1		1	5.4	even	2
5547.2.a.b.1.1		1	43.42	odd	2
6192.2.a.g.1.1		1	12.11	even	2
6321.2.a.f.1.1		1	7.6	odd	2
8256.2.a.f.1.1		1	8.5	even	2
8256.2.a.ba.1.1		1	8.3	odd	2
9675.2.a.s.1.1		1	15.14	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:	\( N \)	\(=\)	\( 129 = 3 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	129.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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