LMFDB - Embedded newform 141.2.a.d.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:	$1$
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)

$f(q)$

$=$

$q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +2.00000 q^{12} -4.00000 q^{13} +1.00000 q^{15} +4.00000 q^{16} +8.00000 q^{17} -6.00000 q^{19} +2.00000 q^{20} +3.00000 q^{21} +3.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} +6.00000 q^{28} -1.00000 q^{29} +4.00000 q^{31} +3.00000 q^{33} +3.00000 q^{35} -2.00000 q^{36} +1.00000 q^{37} +4.00000 q^{39} -10.0000 q^{41} -8.00000 q^{43} +6.00000 q^{44} -1.00000 q^{45} -1.00000 q^{47} -4.00000 q^{48} +2.00000 q^{49} -8.00000 q^{51} +8.00000 q^{52} +10.0000 q^{53} +3.00000 q^{55} +6.00000 q^{57} -10.0000 q^{59} -2.00000 q^{60} +2.00000 q^{61} -3.00000 q^{63} -8.00000 q^{64} +4.00000 q^{65} +4.00000 q^{67} -16.0000 q^{68} -3.00000 q^{69} -6.00000 q^{71} -8.00000 q^{73} +4.00000 q^{75} +12.0000 q^{76} +9.00000 q^{77} -3.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -18.0000 q^{83} -6.00000 q^{84} -8.00000 q^{85} +1.00000 q^{87} -2.00000 q^{89} +12.0000 q^{91} -6.00000 q^{92} -4.00000 q^{93} +6.00000 q^{95} +5.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−2.00000	−1.00000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	2.00000	0.577350
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	4.00000	1.00000
$17$	8.00000	1.94029	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000	1.94029	0.970143	+	0.242536i	$0.0779791\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	2.00000	0.447214
$21$	3.00000	0.654654
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	6.00000	1.13389
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	3.00000	0.507093
$36$	−2.00000	−0.333333
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	6.00000	0.904534
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	−4.00000	−0.577350
$49$	2.00000	0.285714
$50$	0	0
$51$	−8.00000	−1.12022
$52$	8.00000	1.10940
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	−2.00000	−0.258199
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	−8.00000	−1.00000
$65$	4.00000	0.496139
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−16.0000	−1.94029
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	12.0000	1.37649
$77$	9.00000	1.02565
$78$	0	0
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	−18.0000	−1.97576	−0.987878	−	0.155230i	$-0.950388\pi$
$83$	−18.0000	−1.97576	−0.987878	+	0.155230i	$0.950388\pi$
$84$	−6.00000	−0.654654
$85$	−8.00000	−0.867722
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	−6.00000	−0.625543
$93$	−4.00000	−0.414781
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	8.00000	0.800000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	2.00000	0.192450
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	−12.0000	−1.13389
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	2.00000	0.185695
$117$	−4.00000	−0.369800
$118$	0	0
$119$	−24.0000	−2.20008
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	10.0000	0.901670
$124$	−8.00000	−0.718421
$125$	9.00000	0.804984
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	−6.00000	−0.522233
$133$	18.0000	1.56080
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	−6.00000	−0.507093
$141$	1.00000	0.0842152
$142$	0	0
$143$	12.0000	1.00349
$144$	4.00000	0.333333
$145$	1.00000	0.0830455
$146$	0	0
$147$	−2.00000	−0.164957
$148$	−2.00000	−0.164399
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	8.00000	0.646762
$154$	0	0
$155$	−4.00000	−0.321288
$156$	−8.00000	−0.640513
$157$	9.00000	0.718278	0.359139	−	0.933284i	$-0.383070\pi$
$157$	9.00000	0.718278	0.359139	+	0.933284i	$0.383070\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	−9.00000	−0.709299
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	20.0000	1.56174
$165$	−3.00000	−0.233550
$166$	0	0
$167$	−19.0000	−1.47026	−0.735132	−	0.677924i	$-0.762880\pi$
$167$	−19.0000	−1.47026	−0.735132	+	0.677924i	$0.762880\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−6.00000	−0.458831
$172$	16.0000	1.21999
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	12.0000	0.907115
$176$	−12.0000	−0.904534
$177$	10.0000	0.751646
$178$	0	0
$179$	5.00000	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$180$	2.00000	0.149071
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−24.0000	−1.75505
$188$	2.00000	0.145865
$189$	3.00000	0.218218
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	8.00000	0.577350
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	−4.00000	−0.285714
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	−28.0000	−1.98487	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−28.0000	−1.98487	−0.992434	+	0.122782i	$0.960818\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	3.00000	0.210559
$204$	16.0000	1.12022
$205$	10.0000	0.698430
$206$	0	0
$207$	3.00000	0.208514
$208$	−16.0000	−1.10940
$209$	18.0000	1.24509
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−20.0000	−1.37361
$213$	6.00000	0.411113
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	8.00000	0.540590
$220$	−6.00000	−0.404520
$221$	−32.0000	−2.15255
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	11.0000	0.730096	0.365048	−	0.930989i	$-0.381053\pi$
$227$	11.0000	0.730096	0.365048	+	0.930989i	$0.381053\pi$
$228$	−12.0000	−0.794719
$229$	−24.0000	−1.58596	−0.792982	−	0.609245i	$-0.791473\pi$
$229$	−24.0000	−1.58596	−0.792982	+	0.609245i	$0.791473\pi$
$230$	0	0
$231$	−9.00000	−0.592157
$232$	0	0
$233$	11.0000	0.720634	0.360317	−	0.932830i	$-0.382669\pi$
$233$	11.0000	0.720634	0.360317	+	0.932830i	$0.382669\pi$
$234$	0	0
$235$	1.00000	0.0652328
$236$	20.0000	1.30189
$237$	3.00000	0.194871
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	4.00000	0.258199
$241$	−17.0000	−1.09507	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000	−1.09507	−0.547533	+	0.836784i	$0.684433\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−4.00000	−0.256074
$245$	−2.00000	−0.127775
$246$	0	0
$247$	24.0000	1.52708
$248$	0	0
$249$	18.0000	1.14070
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	6.00000	0.377964
$253$	−9.00000	−0.565825
$254$	0	0
$255$	8.00000	0.500979
$256$	16.0000	1.00000
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	−8.00000	−0.496139
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	0	0
$267$	2.00000	0.122398
$268$	−8.00000	−0.488678
$269$	−28.0000	−1.70719	−0.853595	−	0.520937i	$-0.825583\pi$
$269$	−28.0000	−1.70719	−0.853595	+	0.520937i	$0.825583\pi$
$270$	0	0
$271$	−3.00000	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$272$	32.0000	1.94029
$273$	−12.0000	−0.726273
$274$	0	0
$275$	12.0000	0.723627
$276$	6.00000	0.361158
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	3.00000	0.178331	0.0891657	−	0.996017i	$-0.471580\pi$
$283$	3.00000	0.178331	0.0891657	+	0.996017i	$0.471580\pi$
$284$	12.0000	0.712069
$285$	−6.00000	−0.355409
$286$	0	0
$287$	30.0000	1.77084
$288$	0	0
$289$	47.0000	2.76471
$290$	0	0
$291$	−5.00000	−0.293105
$292$	16.0000	0.936329
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	10.0000	0.582223
$296$	0	0
$297$	3.00000	0.174078
$298$	0	0
$299$	−12.0000	−0.693978
$300$	−8.00000	−0.461880
$301$	24.0000	1.38334
$302$	0	0
$303$	−12.0000	−0.689382
$304$	−24.0000	−1.37649
$305$	−2.00000	−0.114520
$306$	0	0
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	−18.0000	−1.02565
$309$	−11.0000	−0.625768
$310$	0	0
$311$	−7.00000	−0.396934	−0.198467	−	0.980108i	$-0.563596\pi$
$311$	−7.00000	−0.396934	−0.198467	+	0.980108i	$0.563596\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	6.00000	0.337526
$317$	−23.0000	−1.29181	−0.645904	−	0.763418i	$-0.723520\pi$
$317$	−23.0000	−1.29181	−0.645904	+	0.763418i	$0.723520\pi$
$318$	0	0
$319$	3.00000	0.167968
$320$	8.00000	0.447214
$321$	−9.00000	−0.502331
$322$	0	0
$323$	−48.0000	−2.67079
$324$	−2.00000	−0.111111
$325$	16.0000	0.887520
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	36.0000	1.97576
$333$	1.00000	0.0547997
$334$	0	0
$335$	−4.00000	−0.218543
$336$	12.0000	0.654654
$337$	−25.0000	−1.36184	−0.680918	−	0.732359i	$-0.738419\pi$
$337$	−25.0000	−1.36184	−0.680918	+	0.732359i	$0.738419\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	16.0000	0.867722
$341$	−12.0000	−0.649836
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	−2.00000	−0.107211
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	12.0000	0.638696	0.319348	−	0.947638i	$-0.396536\pi$
$353$	12.0000	0.638696	0.319348	+	0.947638i	$0.396536\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	4.00000	0.212000
$357$	24.0000	1.27021
$358$	0	0
$359$	9.00000	0.475002	0.237501	−	0.971387i	$-0.423672\pi$
$359$	9.00000	0.475002	0.237501	+	0.971387i	$0.423672\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	2.00000	0.104973
$364$	−24.0000	−1.25794
$365$	8.00000	0.418739
$366$	0	0
$367$	−30.0000	−1.56599	−0.782994	−	0.622030i	$-0.786308\pi$
$367$	−30.0000	−1.56599	−0.782994	+	0.622030i	$0.786308\pi$
$368$	12.0000	0.625543
$369$	−10.0000	−0.520579
$370$	0	0
$371$	−30.0000	−1.55752
$372$	8.00000	0.414781
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	−13.0000	−0.667765	−0.333883	−	0.942615i	$-0.608359\pi$
$379$	−13.0000	−0.667765	−0.333883	+	0.942615i	$0.608359\pi$
$380$	−12.0000	−0.615587
$381$	−8.00000	−0.409852
$382$	0	0
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	0	0
$385$	−9.00000	−0.458682
$386$	0	0
$387$	−8.00000	−0.406663
$388$	−10.0000	−0.507673
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	0	0
$393$	−14.0000	−0.706207
$394$	0	0
$395$	3.00000	0.150946
$396$	6.00000	0.301511
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	−18.0000	−0.901127
$400$	−16.0000	−0.800000
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	−24.0000	−1.19404
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−3.00000	−0.148704
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	−22.0000	−1.08386
$413$	30.0000	1.47620
$414$	0	0
$415$	18.0000	0.883585
$416$	0	0
$417$	10.0000	0.489702
$418$	0	0
$419$	−7.00000	−0.341972	−0.170986	−	0.985273i	$-0.554695\pi$
$419$	−7.00000	−0.341972	−0.170986	+	0.985273i	$0.554695\pi$
$420$	6.00000	0.292770
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	0	0
$423$	−1.00000	−0.0486217
$424$	0	0
$425$	−32.0000	−1.55223
$426$	0	0
$427$	−6.00000	−0.290360
$428$	−18.0000	−0.870063
$429$	−12.0000	−0.579365
$430$	0	0
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	−4.00000	−0.192450
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	0	0
$435$	−1.00000	−0.0479463
$436$	0	0
$437$	−18.0000	−0.861057
$438$	0	0
$439$	37.0000	1.76591	0.882957	−	0.469454i	$-0.155549\pi$
$439$	37.0000	1.76591	0.882957	+	0.469454i	$0.155549\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	2.00000	0.0949158
$445$	2.00000	0.0948091
$446$	0	0
$447$	0	0
$448$	24.0000	1.13389
$449$	27.0000	1.27421	0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000	1.27421	0.637104	+	0.770778i	$0.280132\pi$
$450$	0	0
$451$	30.0000	1.41264
$452$	−12.0000	−0.564433
$453$	−10.0000	−0.469841
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	−29.0000	−1.35656	−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000	−1.35656	−0.678281	+	0.734802i	$0.737275\pi$
$458$	0	0
$459$	−8.00000	−0.373408
$460$	6.00000	0.279751
$461$	−37.0000	−1.72326	−0.861631	−	0.507535i	$-0.830557\pi$
$461$	−37.0000	−1.72326	−0.861631	+	0.507535i	$0.830557\pi$
$462$	0	0
$463$	10.0000	0.464739	0.232370	−	0.972628i	$-0.425352\pi$
$463$	10.0000	0.464739	0.232370	+	0.972628i	$0.425352\pi$
$464$	−4.00000	−0.185695
$465$	4.00000	0.185496
$466$	0	0
$467$	−17.0000	−0.786666	−0.393333	−	0.919396i	$-0.628678\pi$
$467$	−17.0000	−0.786666	−0.393333	+	0.919396i	$0.628678\pi$
$468$	8.00000	0.369800
$469$	−12.0000	−0.554109
$470$	0	0
$471$	−9.00000	−0.414698
$472$	0	0
$473$	24.0000	1.10352
$474$	0	0
$475$	24.0000	1.10120
$476$	48.0000	2.20008
$477$	10.0000	0.457869
$478$	0	0
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	9.00000	0.409514
$484$	4.00000	0.181818
$485$	−5.00000	−0.227038
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	38.0000	1.71492	0.857458	−	0.514554i	$-0.172042\pi$
$491$	38.0000	1.71492	0.857458	+	0.514554i	$0.172042\pi$
$492$	−20.0000	−0.901670
$493$	−8.00000	−0.360302
$494$	0	0
$495$	3.00000	0.134840
$496$	16.0000	0.718421
$497$	18.0000	0.807410
$498$	0	0
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	−18.0000	−0.804984
$501$	19.0000	0.848857
$502$	0	0
$503$	−33.0000	−1.47140	−0.735699	−	0.677309i	$-0.763146\pi$
$503$	−33.0000	−1.47140	−0.735699	+	0.677309i	$0.763146\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	−3.00000	−0.133235
$508$	−16.0000	−0.709885
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	0	0
$513$	6.00000	0.264906
$514$	0	0
$515$	−11.0000	−0.484718
$516$	−16.0000	−0.704361
$517$	3.00000	0.131940
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	−17.0000	−0.743358	−0.371679	−	0.928361i	$-0.621218\pi$
$523$	−17.0000	−0.743358	−0.371679	+	0.928361i	$0.621218\pi$
$524$	−28.0000	−1.22319
$525$	−12.0000	−0.523723
$526$	0	0
$527$	32.0000	1.39394
$528$	12.0000	0.522233
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−10.0000	−0.433963
$532$	−36.0000	−1.56080
$533$	40.0000	1.73259
$534$	0	0
$535$	−9.00000	−0.389104
$536$	0	0
$537$	−5.00000	−0.215766
$538$	0	0
$539$	−6.00000	−0.258438
$540$	−2.00000	−0.0860663
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	18.0000	0.772454
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	−20.0000	−0.854358
$549$	2.00000	0.0853579
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	9.00000	0.382719
$554$	0	0
$555$	1.00000	0.0424476
$556$	20.0000	0.848189
$557$	−13.0000	−0.550828	−0.275414	−	0.961326i	$-0.588815\pi$
$557$	−13.0000	−0.550828	−0.275414	+	0.961326i	$0.588815\pi$
$558$	0	0
$559$	32.0000	1.35346
$560$	12.0000	0.507093
$561$	24.0000	1.01328
$562$	0	0
$563$	32.0000	1.34864	0.674320	−	0.738440i	$-0.264437\pi$
$563$	32.0000	1.34864	0.674320	+	0.738440i	$0.264437\pi$
$564$	−2.00000	−0.0842152
$565$	−6.00000	−0.252422
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	−45.0000	−1.88650	−0.943249	−	0.332086i	$-0.892248\pi$
$569$	−45.0000	−1.88650	−0.943249	+	0.332086i	$0.892248\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−24.0000	−1.00349
$573$	4.00000	0.167102
$574$	0	0
$575$	−12.0000	−0.500435
$576$	−8.00000	−0.333333
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	14.0000	0.581820
$580$	−2.00000	−0.0830455
$581$	54.0000	2.24030
$582$	0	0
$583$	−30.0000	−1.24247
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	4.00000	0.164957
$589$	−24.0000	−0.988903
$590$	0	0
$591$	−22.0000	−0.904959
$592$	4.00000	0.164399
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	24.0000	0.983904
$596$	0	0
$597$	28.0000	1.14596
$598$	0	0
$599$	25.0000	1.02147	0.510736	−	0.859738i	$-0.329373\pi$
$599$	25.0000	1.02147	0.510736	+	0.859738i	$0.329373\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	−20.0000	−0.813788
$605$	2.00000	0.0813116
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	−3.00000	−0.121566
$610$	0	0
$611$	4.00000	0.161823
$612$	−16.0000	−0.646762
$613$	15.0000	0.605844	0.302922	−	0.953015i	$-0.402038\pi$
$613$	15.0000	0.605844	0.302922	+	0.953015i	$0.402038\pi$
$614$	0	0
$615$	−10.0000	−0.403239
$616$	0	0
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	1.00000	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$620$	8.00000	0.321288
$621$	−3.00000	−0.120386
$622$	0	0
$623$	6.00000	0.240385
$624$	16.0000	0.640513
$625$	11.0000	0.440000
$626$	0	0
$627$	−18.0000	−0.718851
$628$	−18.0000	−0.718278
$629$	8.00000	0.318981
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	0	0
$635$	−8.00000	−0.317470
$636$	20.0000	0.793052
$637$	−8.00000	−0.316972
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	25.0000	0.987441	0.493720	−	0.869621i	$-0.335637\pi$
$641$	25.0000	0.987441	0.493720	+	0.869621i	$0.335637\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	18.0000	0.709299
$645$	−8.00000	−0.315000
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	30.0000	1.17760
$650$	0	0
$651$	12.0000	0.470317
$652$	4.00000	0.156652
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	0	0
$655$	−14.0000	−0.547025
$656$	−40.0000	−1.56174
$657$	−8.00000	−0.312110
$658$	0	0
$659$	26.0000	1.01282	0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000	1.01282	0.506408	+	0.862294i	$0.330973\pi$
$660$	6.00000	0.233550
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	32.0000	1.24278
$664$	0	0
$665$	−18.0000	−0.698010
$666$	0	0
$667$	−3.00000	−0.116160
$668$	38.0000	1.47026
$669$	−24.0000	−0.927894
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	−6.00000	−0.230769
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	0	0
$679$	−15.0000	−0.575647
$680$	0	0
$681$	−11.0000	−0.421521
$682$	0	0
$683$	−28.0000	−1.07139	−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000	−1.07139	−0.535695	+	0.844411i	$0.679950\pi$
$684$	12.0000	0.458831
$685$	−10.0000	−0.382080
$686$	0	0
$687$	24.0000	0.915657
$688$	−32.0000	−1.21999
$689$	−40.0000	−1.52388
$690$	0	0
$691$	−2.00000	−0.0760836	−0.0380418	−	0.999276i	$-0.512112\pi$
$691$	−2.00000	−0.0760836	−0.0380418	+	0.999276i	$0.512112\pi$
$692$	36.0000	1.36851
$693$	9.00000	0.341882
$694$	0	0
$695$	10.0000	0.379322
$696$	0	0
$697$	−80.0000	−3.03022
$698$	0	0
$699$	−11.0000	−0.416058
$700$	−24.0000	−0.907115
$701$	−45.0000	−1.69963	−0.849813	−	0.527084i	$-0.823285\pi$
$701$	−45.0000	−1.69963	−0.849813	+	0.527084i	$0.823285\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	24.0000	0.904534
$705$	−1.00000	−0.0376622
$706$	0	0
$707$	−36.0000	−1.35392
$708$	−20.0000	−0.751646
$709$	−7.00000	−0.262891	−0.131445	−	0.991323i	$-0.541962\pi$
$709$	−7.00000	−0.262891	−0.131445	+	0.991323i	$0.541962\pi$
$710$	0	0
$711$	−3.00000	−0.112509
$712$	0	0
$713$	12.0000	0.449404
$714$	0	0
$715$	−12.0000	−0.448775
$716$	−10.0000	−0.373718
$717$	−4.00000	−0.149383
$718$	0	0
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	−4.00000	−0.149071
$721$	−33.0000	−1.22898
$722$	0	0
$723$	17.0000	0.632237
$724$	36.0000	1.33793
$725$	4.00000	0.148556
$726$	0	0
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−64.0000	−2.36713
$732$	4.00000	0.147844
$733$	37.0000	1.36663	0.683313	−	0.730125i	$-0.260538\pi$
$733$	37.0000	1.36663	0.683313	+	0.730125i	$0.260538\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	0	0
$737$	−12.0000	−0.442026
$738$	0	0
$739$	−23.0000	−0.846069	−0.423034	−	0.906114i	$-0.639035\pi$
$739$	−23.0000	−0.846069	−0.423034	+	0.906114i	$0.639035\pi$
$740$	2.00000	0.0735215
$741$	−24.0000	−0.881662
$742$	0	0
$743$	28.0000	1.02722	0.513610	−	0.858024i	$-0.328308\pi$
$743$	28.0000	1.02722	0.513610	+	0.858024i	$0.328308\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−18.0000	−0.658586
$748$	48.0000	1.75505
$749$	−27.0000	−0.986559
$750$	0	0
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	−4.00000	−0.145865
$753$	8.00000	0.291536
$754$	0	0
$755$	−10.0000	−0.363937
$756$	−6.00000	−0.218218
$757$	24.0000	0.872295	0.436147	−	0.899875i	$-0.356343\pi$
$757$	24.0000	0.872295	0.436147	+	0.899875i	$0.356343\pi$
$758$	0	0
$759$	9.00000	0.326679
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	0	0
$764$	8.00000	0.289430
$765$	−8.00000	−0.289241
$766$	0	0
$767$	40.0000	1.44432
$768$	−16.0000	−0.577350
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	15.0000	0.540212
$772$	28.0000	1.00774
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	3.00000	0.107624
$778$	0	0
$779$	60.0000	2.14972
$780$	8.00000	0.286446
$781$	18.0000	0.644091
$782$	0	0
$783$	1.00000	0.0357371
$784$	8.00000	0.285714
$785$	−9.00000	−0.321224
$786$	0	0
$787$	10.0000	0.356462	0.178231	−	0.983989i	$-0.442963\pi$
$787$	10.0000	0.356462	0.178231	+	0.983989i	$0.442963\pi$
$788$	−44.0000	−1.56744
$789$	−18.0000	−0.640817
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	−8.00000	−0.284088
$794$	0	0
$795$	10.0000	0.354663
$796$	56.0000	1.98487
$797$	−23.0000	−0.814702	−0.407351	−	0.913272i	$-0.633547\pi$
$797$	−23.0000	−0.814702	−0.407351	+	0.913272i	$0.633547\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	24.0000	0.846942
$804$	8.00000	0.282138
$805$	9.00000	0.317208
$806$	0	0
$807$	28.0000	0.985647
$808$	0	0
$809$	−27.0000	−0.949269	−0.474635	−	0.880183i	$-0.657420\pi$
$809$	−27.0000	−0.949269	−0.474635	+	0.880183i	$0.657420\pi$
$810$	0	0
$811$	−13.0000	−0.456492	−0.228246	−	0.973604i	$-0.573299\pi$
$811$	−13.0000	−0.456492	−0.228246	+	0.973604i	$0.573299\pi$
$812$	−6.00000	−0.210559
$813$	3.00000	0.105215
$814$	0	0
$815$	2.00000	0.0700569
$816$	−32.0000	−1.12022
$817$	48.0000	1.67931
$818$	0	0
$819$	12.0000	0.419314
$820$	−20.0000	−0.698430
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	0	0
$823$	21.0000	0.732014	0.366007	−	0.930612i	$-0.380725\pi$
$823$	21.0000	0.732014	0.366007	+	0.930612i	$0.380725\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	−44.0000	−1.53003	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000	−1.53003	−0.765015	+	0.644013i	$0.777268\pi$
$828$	−6.00000	−0.208514
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	22.0000	0.763172
$832$	32.0000	1.10940
$833$	16.0000	0.554367
$834$	0	0
$835$	19.0000	0.657522
$836$	−36.0000	−1.24509
$837$	−4.00000	−0.138260
$838$	0	0
$839$	−7.00000	−0.241667	−0.120833	−	0.992673i	$-0.538557\pi$
$839$	−7.00000	−0.241667	−0.120833	+	0.992673i	$0.538557\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	−15.0000	−0.516627
$844$	40.0000	1.37686
$845$	−3.00000	−0.103203
$846$	0	0
$847$	6.00000	0.206162
$848$	40.0000	1.37361
$849$	−3.00000	−0.102960
$850$	0	0
$851$	3.00000	0.102839
$852$	−12.0000	−0.411113
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	0	0
$857$	23.0000	0.785665	0.392833	−	0.919610i	$-0.371495\pi$
$857$	23.0000	0.785665	0.392833	+	0.919610i	$0.371495\pi$
$858$	0	0
$859$	6.00000	0.204717	0.102359	−	0.994748i	$-0.467361\pi$
$859$	6.00000	0.204717	0.102359	+	0.994748i	$0.467361\pi$
$860$	−16.0000	−0.545595
$861$	−30.0000	−1.02240
$862$	0	0
$863$	14.0000	0.476566	0.238283	−	0.971196i	$-0.423415\pi$
$863$	14.0000	0.476566	0.238283	+	0.971196i	$0.423415\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	0	0
$867$	−47.0000	−1.59620
$868$	24.0000	0.814613
$869$	9.00000	0.305304
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	5.00000	0.169224
$874$	0	0
$875$	−27.0000	−0.912767
$876$	−16.0000	−0.540590
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	0	0
$879$	9.00000	0.303562
$880$	12.0000	0.404520
$881$	35.0000	1.17918	0.589590	−	0.807703i	$-0.299289\pi$
$881$	35.0000	1.17918	0.589590	+	0.807703i	$0.299289\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	64.0000	2.15255
$885$	−10.0000	−0.336146
$886$	0	0
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	0	0
$891$	−3.00000	−0.100504
$892$	−48.0000	−1.60716
$893$	6.00000	0.200782
$894$	0	0
$895$	−5.00000	−0.167132
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	−4.00000	−0.133407
$900$	8.00000	0.266667
$901$	80.0000	2.66519
$902$	0	0
$903$	−24.0000	−0.798670
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	−19.0000	−0.630885	−0.315442	−	0.948945i	$-0.602153\pi$
$907$	−19.0000	−0.630885	−0.315442	+	0.948945i	$0.602153\pi$
$908$	−22.0000	−0.730096
$909$	12.0000	0.398015
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	24.0000	0.794719
$913$	54.0000	1.78714
$914$	0	0
$915$	2.00000	0.0661180
$916$	48.0000	1.58596
$917$	−42.0000	−1.38696
$918$	0	0
$919$	−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
$921$	−7.00000	−0.230658
$922$	0	0
$923$	24.0000	0.789970
$924$	18.0000	0.592157
$925$	−4.00000	−0.131519
$926$	0	0
$927$	11.0000	0.361287
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	−22.0000	−0.720634
$933$	7.00000	0.229170
$934$	0	0
$935$	24.0000	0.784884
$936$	0	0
$937$	−58.0000	−1.89478	−0.947389	−	0.320085i	$-0.896288\pi$
$937$	−58.0000	−1.89478	−0.947389	+	0.320085i	$0.896288\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	−2.00000	−0.0652328
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	0	0
$943$	−30.0000	−0.976934
$944$	−40.0000	−1.30189
$945$	−3.00000	−0.0975900
$946$	0	0
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	−6.00000	−0.194871
$949$	32.0000	1.03876
$950$	0	0
$951$	23.0000	0.745826
$952$	0	0
$953$	−21.0000	−0.680257	−0.340128	−	0.940379i	$-0.610471\pi$
$953$	−21.0000	−0.680257	−0.340128	+	0.940379i	$0.610471\pi$
$954$	0	0
$955$	4.00000	0.129437
$956$	−8.00000	−0.258738
$957$	−3.00000	−0.0969762
$958$	0	0
$959$	−30.0000	−0.968751
$960$	−8.00000	−0.258199
$961$	−15.0000	−0.483871
$962$	0	0
$963$	9.00000	0.290021
$964$	34.0000	1.09507
$965$	14.0000	0.450676
$966$	0	0
$967$	−37.0000	−1.18984	−0.594920	−	0.803785i	$-0.702816\pi$
$967$	−37.0000	−1.18984	−0.594920	+	0.803785i	$0.702816\pi$
$968$	0	0
$969$	48.0000	1.54198
$970$	0	0
$971$	9.00000	0.288824	0.144412	−	0.989518i	$-0.453871\pi$
$971$	9.00000	0.288824	0.144412	+	0.989518i	$0.453871\pi$
$972$	2.00000	0.0641500
$973$	30.0000	0.961756
$974$	0	0
$975$	−16.0000	−0.512410
$976$	8.00000	0.256074
$977$	−10.0000	−0.319928	−0.159964	−	0.987123i	$-0.551138\pi$
$977$	−10.0000	−0.319928	−0.159964	+	0.987123i	$0.551138\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	4.00000	0.127775
$981$	0	0
$982$	0	0
$983$	−12.0000	−0.382741	−0.191370	−	0.981518i	$-0.561293\pi$
$983$	−12.0000	−0.382741	−0.191370	+	0.981518i	$0.561293\pi$
$984$	0	0
$985$	−22.0000	−0.700978
$986$	0	0
$987$	−3.00000	−0.0954911
$988$	−48.0000	−1.52708
$989$	−24.0000	−0.763156
$990$	0	0
$991$	15.0000	0.476491	0.238245	−	0.971205i	$-0.423428\pi$
$991$	15.0000	0.476491	0.238245	+	0.971205i	$0.423428\pi$
$992$	0	0
$993$	−20.0000	−0.634681
$994$	0	0
$995$	28.0000	0.887660
$996$	−36.0000	−1.14070
$997$	16.0000	0.506725	0.253363	−	0.967371i	$-0.418463\pi$
$997$	16.0000	0.506725	0.253363	+	0.967371i	$0.418463\pi$
$998$	0	0
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	141.2.a.d.1.1	✓	1
3.2	odd	2		423.2.a.c.1.1		1
4.3	odd	2		2256.2.a.k.1.1		1
5.4	even	2		3525.2.a.h.1.1		1
7.6	odd	2		6909.2.a.h.1.1		1
8.3	odd	2		9024.2.a.r.1.1		1
8.5	even	2		9024.2.a.bl.1.1		1
12.11	even	2		6768.2.a.m.1.1		1
47.46	odd	2		6627.2.a.e.1.1		1

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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