LMFDB - Embedded newform 141.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [141,2,Mod(1,141)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(141, 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("141.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 141 = 3 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	141.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:	$1.12589066850$
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	$\chi$	$=$	141.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -2.00000 q^{20} -4.00000 q^{22} +3.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} -2.00000 q^{30} -4.00000 q^{31} -5.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} -1.00000 q^{36} -10.0000 q^{37} -2.00000 q^{39} +6.00000 q^{40} -2.00000 q^{41} +8.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} -1.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} +2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} +6.00000 q^{58} -4.00000 q^{59} -2.00000 q^{60} +14.0000 q^{61} +4.00000 q^{62} +7.00000 q^{64} -4.00000 q^{65} -4.00000 q^{66} -8.00000 q^{67} -2.00000 q^{68} +16.0000 q^{71} +3.00000 q^{72} +2.00000 q^{73} +10.0000 q^{74} -1.00000 q^{75} +2.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -4.00000 q^{83} +4.00000 q^{85} -8.00000 q^{86} -6.00000 q^{87} +12.0000 q^{88} +18.0000 q^{89} -2.00000 q^{90} -4.00000 q^{93} +1.00000 q^{94} -5.00000 q^{96} -14.0000 q^{97} +7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	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
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−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
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	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
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−5.00000	−0.883883
$33$	4.00000	0.696311
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	6.00000	0.948683
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−4.00000	−0.603023
$45$	2.00000	0.298142
$46$	0	0
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	2.00000	0.280056
$52$	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
$55$	8.00000	1.07872
$56$	0	0
$57$	0	0
$58$	6.00000	0.787839
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	−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	$0.146275\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	−4.00000	−0.492366
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	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	$0.462659\pi$
$74$	10.0000	1.16248
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	2.00000	0.226455
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	−8.00000	−0.862662
$87$	−6.00000	−0.643268
$88$	12.0000	1.27920
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	0	0
$93$	−4.00000	−0.414781
$94$	1.00000	0.103142
$95$	0	0
$96$	−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
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−2.00000	−0.198030
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	2.00000	0.194257
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	−1.00000	−0.0962250
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	−8.00000	−0.762770
$111$	−10.0000	−0.949158
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	−2.00000	−0.184900
$118$	4.00000	0.368230
$119$	0	0
$120$	6.00000	0.547723
$121$	5.00000	0.454545
$122$	−14.0000	−1.26750
$123$	−2.00000	−0.180334
$124$	4.00000	0.359211
$125$	−12.0000	−1.07331
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	3.00000	0.265165
$129$	8.00000	0.704361
$130$	4.00000	0.350823
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	8.00000	0.691095
$135$	2.00000	0.172133
$136$	6.00000	0.514496
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	−1.00000	−0.0842152
$142$	−16.0000	−1.34269
$143$	−8.00000	−0.668994
$144$	−1.00000	−0.0833333
$145$	−12.0000	−0.996546
$146$	−2.00000	−0.165521
$147$	−7.00000	−0.577350
$148$	10.0000	0.821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	1.00000	0.0816497
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−8.00000	−0.642575
$156$	2.00000	0.160128
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−8.00000	−0.636446
$159$	−2.00000	−0.158610
$160$	−10.0000	−0.790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	24.0000	1.87983	0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000	1.87983	0.939913	+	0.341415i	$0.110906\pi$
$164$	2.00000	0.156174
$165$	8.00000	0.622799
$166$	4.00000	0.310460
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−4.00000	−0.306786
$171$	0	0
$172$	−8.00000	−0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−4.00000	−0.301511
$177$	−4.00000	−0.300658
$178$	−18.0000	−1.34916
$179$	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
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	0	0
$185$	−20.0000	−1.47043
$186$	4.00000	0.293294
$187$	8.00000	0.585018
$188$	1.00000	0.0729325
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	7.00000	0.505181
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	14.0000	1.00514
$195$	−4.00000	−0.286446
$196$	7.00000	0.500000
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	−4.00000	−0.284268
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	−3.00000	−0.212132
$201$	−8.00000	−0.564276
$202$	10.0000	0.703598
$203$	0	0
$204$	−2.00000	−0.140028
$205$	−4.00000	−0.279372
$206$	16.0000	1.11477
$207$	0	0
$208$	2.00000	0.138675
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	2.00000	0.137361
$213$	16.0000	1.09630
$214$	4.00000	0.273434
$215$	16.0000	1.09119
$216$	3.00000	0.204124
$217$	0	0
$218$	18.0000	1.21911
$219$	2.00000	0.135147
$220$	−8.00000	−0.539360
$221$	−4.00000	−0.269069
$222$	10.0000	0.671156
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	10.0000	0.665190
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	−18.0000	−1.18176
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	2.00000	0.130744
$235$	−2.00000	−0.130466
$236$	4.00000	0.260378
$237$	8.00000	0.519656
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−2.00000	−0.129099
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	−14.0000	−0.896258
$245$	−14.0000	−0.894427
$246$	2.00000	0.127515
$247$	0	0
$248$	−12.0000	−0.762001
$249$	−4.00000	−0.253490
$250$	12.0000	0.758947
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	−4.00000	−0.250982
$255$	4.00000	0.250490
$256$	−17.0000	−1.06250
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	−8.00000	−0.498058
$259$	0	0
$260$	4.00000	0.248069
$261$	−6.00000	−0.371391
$262$	−12.0000	−0.741362
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	12.0000	0.738549
$265$	−4.00000	−0.245718
$266$	0	0
$267$	18.0000	1.10158
$268$	8.00000	0.488678
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	−2.00000	−0.121716
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−14.0000	−0.845771
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−16.0000	−0.959616
$279$	−4.00000	−0.239474
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	1.00000	0.0595491
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	8.00000	0.473050
$287$	0	0
$288$	−5.00000	−0.294628
$289$	−13.0000	−0.764706
$290$	12.0000	0.704664
$291$	−14.0000	−0.820695
$292$	−2.00000	−0.117041
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	7.00000	0.408248
$295$	−8.00000	−0.465778
$296$	−30.0000	−1.74371
$297$	4.00000	0.232104
$298$	−6.00000	−0.347571
$299$	0	0
$300$	1.00000	0.0577350
$301$	0	0
$302$	−4.00000	−0.230174
$303$	−10.0000	−0.574485
$304$	0	0
$305$	28.0000	1.60328
$306$	−2.00000	−0.114332
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	8.00000	0.454369
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	−6.00000	−0.339683
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	2.00000	0.112154
$319$	−24.0000	−1.34374
$320$	14.0000	0.782624
$321$	−4.00000	−0.223258
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	2.00000	0.110940
$326$	−24.0000	−1.32924
$327$	−18.0000	−0.995402
$328$	−6.00000	−0.331295
$329$	0	0
$330$	−8.00000	−0.440386
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	4.00000	0.219529
$333$	−10.0000	−0.547997
$334$	0	0
$335$	−16.0000	−0.874173
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	9.00000	0.489535
$339$	−10.0000	−0.543125
$340$	−4.00000	−0.216930
$341$	−16.0000	−0.866449
$342$	0	0
$343$	0	0
$344$	24.0000	1.29399
$345$	0	0
$346$	−6.00000	−0.322562
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	6.00000	0.321634
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	−20.0000	−1.06600
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	4.00000	0.212598
$355$	32.0000	1.69838
$356$	−18.0000	−0.953998
$357$	0	0
$358$	−20.0000	−1.05703
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	6.00000	0.316228
$361$	−19.0000	−1.00000
$362$	−6.00000	−0.315353
$363$	5.00000	0.262432
$364$	0	0
$365$	4.00000	0.209370
$366$	−14.0000	−0.731792
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	20.0000	1.03975
$371$	0	0
$372$	4.00000	0.207390
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−8.00000	−0.413670
$375$	−12.0000	−0.619677
$376$	−3.00000	−0.154713
$377$	12.0000	0.618031
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	14.0000	0.712581
$387$	8.00000	0.406663
$388$	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
$391$	0	0
$392$	−21.0000	−1.06066
$393$	12.0000	0.605320
$394$	10.0000	0.503793
$395$	16.0000	0.805047
$396$	−4.00000	−0.201008
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	1.00000	0.0500000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	8.00000	0.399004
$403$	8.00000	0.398508
$404$	10.0000	0.497519
$405$	2.00000	0.0993808
$406$	0	0
$407$	−40.0000	−1.98273
$408$	6.00000	0.297044
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	4.00000	0.197546
$411$	14.0000	0.690569
$412$	16.0000	0.788263
$413$	0	0
$414$	0	0
$415$	−8.00000	−0.392705
$416$	10.0000	0.490290
$417$	16.0000	0.783523
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	8.00000	0.389434
$423$	−1.00000	−0.0486217
$424$	−6.00000	−0.291386
$425$	−2.00000	−0.0970143
$426$	−16.0000	−0.775203
$427$	0	0
$428$	4.00000	0.193347
$429$	−8.00000	−0.386244
$430$	−16.0000	−0.771589
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	−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
$436$	18.0000	0.862044
$437$	0	0
$438$	−2.00000	−0.0955637
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	24.0000	1.14416
$441$	−7.00000	−0.333333
$442$	4.00000	0.190261
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	10.0000	0.474579
$445$	36.0000	1.70656
$446$	12.0000	0.568216
$447$	6.00000	0.283790
$448$	0	0
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	1.00000	0.0471405
$451$	−8.00000	−0.376705
$452$	10.0000	0.470360
$453$	4.00000	0.187936
$454$	−4.00000	−0.187729
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−22.0000	−1.02799
$459$	2.00000	0.0933520
$460$	0	0
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	6.00000	0.278543
$465$	−8.00000	−0.370991
$466$	−14.0000	−0.648537
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	2.00000	0.0924500
$469$	0	0
$470$	2.00000	0.0922531
$471$	−18.0000	−0.829396
$472$	−12.0000	−0.552345
$473$	32.0000	1.47136
$474$	−8.00000	−0.367452
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−10.0000	−0.456435
$481$	20.0000	0.911922
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	−5.00000	−0.227273
$485$	−28.0000	−1.27141
$486$	−1.00000	−0.0453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	42.0000	1.90125
$489$	24.0000	1.08532
$490$	14.0000	0.632456
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	2.00000	0.0901670
$493$	−12.0000	−0.540453
$494$	0	0
$495$	8.00000	0.359573
$496$	4.00000	0.179605
$497$	0	0
$498$	4.00000	0.179244
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	12.0000	0.535586
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	−9.00000	−0.399704
$508$	−4.00000	−0.177471
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	−4.00000	−0.177123
$511$	0	0
$512$	11.0000	0.486136
$513$	0	0
$514$	−14.0000	−0.617514
$515$	−32.0000	−1.41009
$516$	−8.00000	−0.352180
$517$	−4.00000	−0.175920
$518$	0	0
$519$	6.00000	0.263371
$520$	−12.0000	−0.526235
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	6.00000	0.262613
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	−8.00000	−0.348485
$528$	−4.00000	−0.174078
$529$	−23.0000	−1.00000
$530$	4.00000	0.173749
$531$	−4.00000	−0.173585
$532$	0	0
$533$	4.00000	0.173259
$534$	−18.0000	−0.778936
$535$	−8.00000	−0.345870
$536$	−24.0000	−1.03664
$537$	20.0000	0.863064
$538$	−6.00000	−0.258678
$539$	−28.0000	−1.20605
$540$	−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	$0.513689\pi$
$542$	−24.0000	−1.03089
$543$	6.00000	0.257485
$544$	−10.0000	−0.428746
$545$	−36.0000	−1.54207
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−14.0000	−0.598050
$549$	14.0000	0.597505
$550$	4.00000	0.170561
$551$	0	0
$552$	0	0
$553$	0	0
$554$	10.0000	0.424859
$555$	−20.0000	−0.848953
$556$	−16.0000	−0.678551
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	4.00000	0.169334
$559$	−16.0000	−0.676728
$560$	0	0
$561$	8.00000	0.337760
$562$	−30.0000	−1.26547
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	1.00000	0.0421076
$565$	−20.0000	−0.841406
$566$	−12.0000	−0.504398
$567$	0	0
$568$	48.0000	2.01404
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	8.00000	0.334497
$573$	0	0
$574$	0	0
$575$	0	0
$576$	7.00000	0.291667
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	13.0000	0.540729
$579$	−14.0000	−0.581820
$580$	12.0000	0.498273
$581$	0	0
$582$	14.0000	0.580319
$583$	−8.00000	−0.331326
$584$	6.00000	0.248282
$585$	−4.00000	−0.165380
$586$	−18.0000	−0.743573
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	7.00000	0.288675
$589$	0	0
$590$	8.00000	0.329355
$591$	−10.0000	−0.411345
$592$	10.0000	0.410997
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−4.00000	−0.163709
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	−3.00000	−0.122474
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	−4.00000	−0.162758
$605$	10.0000	0.406558
$606$	10.0000	0.406222
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	0	0
$609$	0	0
$610$	−28.0000	−1.13369
$611$	2.00000	0.0809113
$612$	−2.00000	−0.0808452
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−20.0000	−0.807134
$615$	−4.00000	−0.161296
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	16.0000	0.643614
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	32.0000	1.28308
$623$	0	0
$624$	2.00000	0.0800641
$625$	−19.0000	−0.760000
$626$	6.00000	0.239808
$627$	0	0
$628$	18.0000	0.718278
$629$	−20.0000	−0.797452
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	24.0000	0.954669
$633$	−8.00000	−0.317971
$634$	−18.0000	−0.714871
$635$	8.00000	0.317470
$636$	2.00000	0.0793052
$637$	14.0000	0.554700
$638$	24.0000	0.950169
$639$	16.0000	0.632950
$640$	6.00000	0.237171
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	4.00000	0.157867
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	16.0000	0.629999
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	3.00000	0.117851
$649$	−16.0000	−0.628055
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−24.0000	−0.939913
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	18.0000	0.703856
$655$	24.0000	0.937758
$656$	2.00000	0.0780869
$657$	2.00000	0.0780274
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	−8.00000	−0.311400
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−4.00000	−0.155464
$663$	−4.00000	−0.155347
$664$	−12.0000	−0.465690
$665$	0	0
$666$	10.0000	0.387492
$667$	0	0
$668$	0	0
$669$	−12.0000	−0.463947
$670$	16.0000	0.618134
$671$	56.0000	2.16186
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	−2.00000	−0.0770371
$675$	−1.00000	−0.0384900
$676$	9.00000	0.346154
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	10.0000	0.384048
$679$	0	0
$680$	12.0000	0.460179
$681$	4.00000	0.153280
$682$	16.0000	0.612672
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	28.0000	1.06983
$686$	0	0
$687$	22.0000	0.839352
$688$	−8.00000	−0.304997
$689$	4.00000	0.152388
$690$	0	0
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−12.0000	−0.455514
$695$	32.0000	1.21383
$696$	−18.0000	−0.682288
$697$	−4.00000	−0.151511
$698$	−22.0000	−0.832712
$699$	14.0000	0.529529
$700$	0	0
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	2.00000	0.0754851
$703$	0	0
$704$	28.0000	1.05529
$705$	−2.00000	−0.0753244
$706$	6.00000	0.225813
$707$	0	0
$708$	4.00000	0.150329
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	−32.0000	−1.20094
$711$	8.00000	0.300023
$712$	54.0000	2.02374
$713$	0	0
$714$	0	0
$715$	−16.0000	−0.598366
$716$	−20.0000	−0.747435
$717$	0	0
$718$	0	0
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	19.0000	0.707107
$723$	2.00000	0.0743808
$724$	−6.00000	−0.222988
$725$	6.00000	0.222834
$726$	−5.00000	−0.185567
$727$	−20.0000	−0.741759	−0.370879	−	0.928681i	$-0.620944\pi$
$727$	−20.0000	−0.741759	−0.370879	+	0.928681i	$0.620944\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	16.0000	0.591781
$732$	−14.0000	−0.517455
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−28.0000	−1.03350
$735$	−14.0000	−0.516398
$736$	0	0
$737$	−32.0000	−1.17874
$738$	2.00000	0.0736210
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	20.0000	0.735215
$741$	0	0
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	−12.0000	−0.439941
$745$	12.0000	0.439646
$746$	−14.0000	−0.512576
$747$	−4.00000	−0.146352
$748$	−8.00000	−0.292509
$749$	0	0
$750$	12.0000	0.438178
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	1.00000	0.0364662
$753$	−12.0000	−0.437304
$754$	−12.0000	−0.437014
$755$	8.00000	0.291150
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	−4.00000	−0.144905
$763$	0	0
$764$	0	0
$765$	4.00000	0.144620
$766$	24.0000	0.867155
$767$	8.00000	0.288863
$768$	−17.0000	−0.613435
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	14.0000	0.503871
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	−8.00000	−0.287554
$775$	4.00000	0.143684
$776$	−42.0000	−1.50771
$777$	0	0
$778$	6.00000	0.215110
$779$	0	0
$780$	4.00000	0.143223
$781$	64.0000	2.29010
$782$	0	0
$783$	−6.00000	−0.214423
$784$	7.00000	0.250000
$785$	−36.0000	−1.28490
$786$	−12.0000	−0.428026
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	10.0000	0.356235
$789$	−24.0000	−0.854423
$790$	−16.0000	−0.569254
$791$	0	0
$792$	12.0000	0.426401
$793$	−28.0000	−0.994309
$794$	2.00000	0.0709773
$795$	−4.00000	−0.141865
$796$	4.00000	0.141776
$797$	50.0000	1.77109	0.885545	−	0.464553i	$-0.153785\pi$
$797$	50.0000	1.77109	0.885545	+	0.464553i	$0.153785\pi$
$798$	0	0
$799$	−2.00000	−0.0707549
$800$	5.00000	0.176777
$801$	18.0000	0.635999
$802$	−10.0000	−0.353112
$803$	8.00000	0.282314
$804$	8.00000	0.282138
$805$	0	0
$806$	−8.00000	−0.281788
$807$	6.00000	0.211210
$808$	−30.0000	−1.05540
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	−2.00000	−0.0702728
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	0	0
$813$	24.0000	0.841717
$814$	40.0000	1.40200
$815$	48.0000	1.68137
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	−34.0000	−1.18878
$819$	0	0
$820$	4.00000	0.139686
$821$	−54.0000	−1.88461	−0.942306	−	0.334751i	$-0.891348\pi$
$821$	−54.0000	−1.88461	−0.942306	+	0.334751i	$0.891348\pi$
$822$	−14.0000	−0.488306
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	−48.0000	−1.67216
$825$	−4.00000	−0.139262
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	8.00000	0.277684
$831$	−10.0000	−0.346896
$832$	−14.0000	−0.485363
$833$	−14.0000	−0.485071
$834$	−16.0000	−0.554035
$835$	0	0
$836$	0	0
$837$	−4.00000	−0.138260
$838$	−4.00000	−0.138178
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−22.0000	−0.758170
$843$	30.0000	1.03325
$844$	8.00000	0.275371
$845$	−18.0000	−0.619219
$846$	1.00000	0.0343807
$847$	0	0
$848$	2.00000	0.0686803
$849$	12.0000	0.411839
$850$	2.00000	0.0685994
$851$	0	0
$852$	−16.0000	−0.548151
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	30.0000	1.02478	0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000	1.02478	0.512390	+	0.858753i	$0.328760\pi$
$858$	8.00000	0.273115
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	−16.0000	−0.545595
$861$	0	0
$862$	32.0000	1.08992
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	−5.00000	−0.170103
$865$	12.0000	0.408012
$866$	−18.0000	−0.611665
$867$	−13.0000	−0.441503
$868$	0	0
$869$	32.0000	1.08553
$870$	12.0000	0.406838
$871$	16.0000	0.542139
$872$	−54.0000	−1.82867
$873$	−14.0000	−0.473828
$874$	0	0
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	−24.0000	−0.809961
$879$	18.0000	0.607125
$880$	−8.00000	−0.269680
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	7.00000	0.235702
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	4.00000	0.134535
$885$	−8.00000	−0.268917
$886$	36.0000	1.20944
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	−30.0000	−1.00673
$889$	0	0
$890$	−36.0000	−1.20672
$891$	4.00000	0.134005
$892$	12.0000	0.401790
$893$	0	0
$894$	−6.00000	−0.200670
$895$	40.0000	1.33705
$896$	0	0
$897$	0	0
$898$	26.0000	0.867631
$899$	24.0000	0.800445
$900$	1.00000	0.0333333
$901$	−4.00000	−0.133259
$902$	8.00000	0.266371
$903$	0	0
$904$	−30.0000	−0.997785
$905$	12.0000	0.398893
$906$	−4.00000	−0.132891
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	−4.00000	−0.132745
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	22.0000	0.727695
$915$	28.0000	0.925651
$916$	−22.0000	−0.726900
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	−10.0000	−0.329332
$923$	−32.0000	−1.05329
$924$	0	0
$925$	10.0000	0.328798
$926$	4.00000	0.131448
$927$	−16.0000	−0.525509
$928$	30.0000	0.984798
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	8.00000	0.262330
$931$	0	0
$932$	−14.0000	−0.458585
$933$	−32.0000	−1.04763
$934$	−12.0000	−0.392652
$935$	16.0000	0.523256
$936$	−6.00000	−0.196116
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	2.00000	0.0652328
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	18.0000	0.586472
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	−32.0000	−1.04041
$947$	−20.0000	−0.649913	−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000	−0.649913	−0.324956	+	0.945729i	$0.605350\pi$
$948$	−8.00000	−0.259828
$949$	−4.00000	−0.129845
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−50.0000	−1.61966	−0.809829	−	0.586665i	$-0.800440\pi$
$953$	−50.0000	−1.61966	−0.809829	+	0.586665i	$0.800440\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	0	0
$957$	−24.0000	−0.775810
$958$	0	0
$959$	0	0
$960$	14.0000	0.451848
$961$	−15.0000	−0.483871
$962$	−20.0000	−0.644826
$963$	−4.00000	−0.128898
$964$	−2.00000	−0.0644157
$965$	−28.0000	−0.901352
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	15.0000	0.482118
$969$	0	0
$970$	28.0000	0.899026
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	32.0000	1.02535
$975$	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	$0.316955\pi$
$978$	−24.0000	−0.767435
$979$	72.0000	2.30113
$980$	14.0000	0.447214
$981$	−18.0000	−0.574696
$982$	−36.0000	−1.14881
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	−6.00000	−0.191273
$985$	−20.0000	−0.637253
$986$	12.0000	0.382158
$987$	0	0
$988$	0	0
$989$	0	0
$990$	−8.00000	−0.254257
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	20.0000	0.635001
$993$	4.00000	0.126936
$994$	0	0
$995$	−8.00000	−0.253617
$996$	4.00000	0.126745
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	−32.0000	−1.01294
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	141.2.a.c.1.1	✓	1
3.2	odd	2		423.2.a.d.1.1		1
4.3	odd	2		2256.2.a.g.1.1		1
5.4	even	2		3525.2.a.j.1.1		1
7.6	odd	2		6909.2.a.b.1.1		1
8.3	odd	2		9024.2.a.bd.1.1		1
8.5	even	2		9024.2.a.d.1.1		1
12.11	even	2		6768.2.a.f.1.1		1
47.46	odd	2		6627.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
141.2.a.c.1.1	✓	1	1.1	even	1	trivial
423.2.a.d.1.1		1	3.2	odd	2
2256.2.a.g.1.1		1	4.3	odd	2
3525.2.a.j.1.1		1	5.4	even	2
6627.2.a.c.1.1		1	47.46	odd	2
6768.2.a.f.1.1		1	12.11	even	2
6909.2.a.b.1.1		1	7.6	odd	2
9024.2.a.d.1.1		1	8.5	even	2
9024.2.a.bd.1.1		1	8.3	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:	\( N \)	\(=\)	\( 141 = 3 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	141.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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