LMFDB - Embedded newform 2670.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2670,2,Mod(1,2670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2670, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2670 = 2 \cdot 3 \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2670.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:	$21.3200573397$
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	$\chi$	$=$	2670.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -4.00000 q^{21} +4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} -4.00000 q^{42} -4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} -1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} +4.00000 q^{56} -4.00000 q^{57} -6.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} -6.00000 q^{61} -4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} +4.00000 q^{69} +4.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} +2.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} +16.0000 q^{77} -2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} -4.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} +6.00000 q^{87} +4.00000 q^{88} +1.00000 q^{89} +1.00000 q^{90} +8.00000 q^{91} -4.00000 q^{92} +4.00000 q^{93} +4.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	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	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	4.00000	1.06904
$15$	−1.00000	−0.258199
$16$	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
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	1.00000	0.223607
$21$	−4.00000	−0.872872
$22$	4.00000	0.852803
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	4.00000	0.755929
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−1.00000	−0.182574
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	−4.00000	−0.696311
$34$	2.00000	0.342997
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	4.00000	0.648886
$39$	−2.00000	−0.320256
$40$	1.00000	0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−4.00000	−0.617213
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	−4.00000	−0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	9.00000	1.28571
$50$	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
$55$	4.00000	0.539360
$56$	4.00000	0.534522
$57$	−4.00000	−0.529813
$58$	−6.00000	−0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	−1.00000	−0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−4.00000	−0.508001
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−4.00000	−0.492366
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	2.00000	0.242536
$69$	4.00000	0.481543
$70$	4.00000	0.478091
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000	0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	2.00000	0.232495
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	16.0000	1.82337
$78$	−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	$0.351411\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	−4.00000	−0.436436
$85$	2.00000	0.216930
$86$	−4.00000	−0.431331
$87$	6.00000	0.643268
$88$	4.00000	0.426401
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	1.00000	0.105409
$91$	8.00000	0.838628
$92$	−4.00000	−0.417029
$93$	4.00000	0.414781
$94$	0	0
$95$	4.00000	0.410391
$96$	−1.00000	−0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	9.00000	0.909137
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	−2.00000	−0.198030
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	2.00000	0.196116
$105$	−4.00000	−0.390360
$106$	−2.00000	−0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	4.00000	0.381385
$111$	−2.00000	−0.189832
$112$	4.00000	0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−4.00000	−0.374634
$115$	−4.00000	−0.373002
$116$	−6.00000	−0.557086
$117$	2.00000	0.184900
$118$	−12.0000	−1.10469
$119$	8.00000	0.733359
$120$	−1.00000	−0.0912871
$121$	5.00000	0.454545
$122$	−6.00000	−0.543214
$123$	−2.00000	−0.180334
$124$	−4.00000	−0.359211
$125$	1.00000	0.0894427
$126$	4.00000	0.356348
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	2.00000	0.175412
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	−4.00000	−0.348155
$133$	16.0000	1.38738
$134$	−4.00000	−0.345547
$135$	−1.00000	−0.0860663
$136$	2.00000	0.171499
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	4.00000	0.340503
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−8.00000	−0.671345
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	−6.00000	−0.498273
$146$	−6.00000	−0.496564
$147$	−9.00000	−0.742307
$148$	2.00000	0.164399
$149$	−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
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	4.00000	0.324443
$153$	2.00000	0.161690
$154$	16.0000	1.28932
$155$	−4.00000	−0.321288
$156$	−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
$160$	1.00000	0.0790569
$161$	−16.0000	−1.26098
$162$	1.00000	0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	2.00000	0.156174
$165$	−4.00000	−0.311400
$166$	4.00000	0.310460
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	−4.00000	−0.308607
$169$	−9.00000	−0.692308
$170$	2.00000	0.153393
$171$	4.00000	0.305888
$172$	−4.00000	−0.304997
$173$	−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
$175$	4.00000	0.302372
$176$	4.00000	0.301511
$177$	12.0000	0.901975
$178$	1.00000	0.0749532
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	1.00000	0.0745356
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	8.00000	0.592999
$183$	6.00000	0.443533
$184$	−4.00000	−0.294884
$185$	2.00000	0.147043
$186$	4.00000	0.293294
$187$	8.00000	0.585018
$188$	0	0
$189$	−4.00000	−0.290957
$190$	4.00000	0.290191
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	2.00000	0.143592
$195$	−2.00000	−0.143223
$196$	9.00000	0.642857
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	4.00000	0.284268
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	1.00000	0.0707107
$201$	4.00000	0.282138
$202$	2.00000	0.140720
$203$	−24.0000	−1.68447
$204$	−2.00000	−0.140028
$205$	2.00000	0.139686
$206$	−4.00000	−0.278693
$207$	−4.00000	−0.278019
$208$	2.00000	0.138675
$209$	16.0000	1.10674
$210$	−4.00000	−0.276026
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−2.00000	−0.137361
$213$	8.00000	0.548151
$214$	12.0000	0.820303
$215$	−4.00000	−0.272798
$216$	−1.00000	−0.0680414
$217$	−16.0000	−1.08615
$218$	14.0000	0.948200
$219$	6.00000	0.405442
$220$	4.00000	0.269680
$221$	4.00000	0.269069
$222$	−2.00000	−0.134231
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	4.00000	0.267261
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−4.00000	−0.264906
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	−4.00000	−0.263752
$231$	−16.0000	−1.05272
$232$	−6.00000	−0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−12.0000	−0.781133
$237$	−8.00000	−0.519656
$238$	8.00000	0.518563
$239$	28.0000	1.81117	0.905585	−	0.424165i	$-0.139432\pi$
$239$	28.0000	1.81117	0.905585	+	0.424165i	$0.139432\pi$
$240$	−1.00000	−0.0645497
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	9.00000	0.574989
$246$	−2.00000	−0.127515
$247$	8.00000	0.509028
$248$	−4.00000	−0.254000
$249$	−4.00000	−0.253490
$250$	1.00000	0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	4.00000	0.251976
$253$	−16.0000	−1.00591
$254$	4.00000	0.250982
$255$	−2.00000	−0.125245
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	4.00000	0.249029
$259$	8.00000	0.497096
$260$	2.00000	0.124035
$261$	−6.00000	−0.371391
$262$	−4.00000	−0.247121
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−4.00000	−0.246183
$265$	−2.00000	−0.122859
$266$	16.0000	0.981023
$267$	−1.00000	−0.0611990
$268$	−4.00000	−0.244339
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	−1.00000	−0.0608581
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	2.00000	0.121268
$273$	−8.00000	−0.484182
$274$	−6.00000	−0.362473
$275$	4.00000	0.241209
$276$	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	$0.519137\pi$
$278$	−4.00000	−0.239904
$279$	−4.00000	−0.239474
$280$	4.00000	0.239046
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	−8.00000	−0.474713
$285$	−4.00000	−0.236940
$286$	8.00000	0.473050
$287$	8.00000	0.472225
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	−6.00000	−0.352332
$291$	−2.00000	−0.117242
$292$	−6.00000	−0.351123
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−9.00000	−0.524891
$295$	−12.0000	−0.698667
$296$	2.00000	0.116248
$297$	−4.00000	−0.232104
$298$	−6.00000	−0.347571
$299$	−8.00000	−0.462652
$300$	−1.00000	−0.0577350
$301$	−16.0000	−0.922225
$302$	4.00000	0.230174
$303$	−2.00000	−0.114897
$304$	4.00000	0.229416
$305$	−6.00000	−0.343559
$306$	2.00000	0.114332
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	16.0000	0.911685
$309$	4.00000	0.227552
$310$	−4.00000	−0.227185
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	−2.00000	−0.113228
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	14.0000	0.790066
$315$	4.00000	0.225374
$316$	8.00000	0.450035
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	2.00000	0.112154
$319$	−24.0000	−1.34374
$320$	1.00000	0.0559017
$321$	−12.0000	−0.669775
$322$	−16.0000	−0.891645
$323$	8.00000	0.445132
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	20.0000	1.10770
$327$	−14.0000	−0.774202
$328$	2.00000	0.110432
$329$	0	0
$330$	−4.00000	−0.220193
$331$	−36.0000	−1.97874	−0.989369	−	0.145424i	$-0.953545\pi$
$331$	−36.0000	−1.97874	−0.989369	+	0.145424i	$0.953545\pi$
$332$	4.00000	0.219529
$333$	2.00000	0.109599
$334$	8.00000	0.437741
$335$	−4.00000	−0.218543
$336$	−4.00000	−0.218218
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	−9.00000	−0.489535
$339$	6.00000	0.325875
$340$	2.00000	0.108465
$341$	−16.0000	−0.866449
$342$	4.00000	0.216295
$343$	8.00000	0.431959
$344$	−4.00000	−0.215666
$345$	4.00000	0.215353
$346$	−18.0000	−0.967686
$347$	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
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	4.00000	0.213809
$351$	−2.00000	−0.106752
$352$	4.00000	0.213201
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	12.0000	0.637793
$355$	−8.00000	−0.424596
$356$	1.00000	0.0529999
$357$	−8.00000	−0.423405
$358$	4.00000	0.211407
$359$	28.0000	1.47778	0.738892	−	0.673824i	$-0.235349\pi$
$359$	28.0000	1.47778	0.738892	+	0.673824i	$0.235349\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	−6.00000	−0.315353
$363$	−5.00000	−0.262432
$364$	8.00000	0.419314
$365$	−6.00000	−0.314054
$366$	6.00000	0.313625
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−4.00000	−0.208514
$369$	2.00000	0.104116
$370$	2.00000	0.103975
$371$	−8.00000	−0.415339
$372$	4.00000	0.207390
$373$	30.0000	1.55334	0.776671	−	0.629907i	$-0.216907\pi$
$373$	30.0000	1.55334	0.776671	+	0.629907i	$0.216907\pi$
$374$	8.00000	0.413670
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−12.0000	−0.618031
$378$	−4.00000	−0.205738
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	4.00000	0.205196
$381$	−4.00000	−0.204926
$382$	−12.0000	−0.613973
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	−1.00000	−0.0510310
$385$	16.0000	0.815436
$386$	−14.0000	−0.712581
$387$	−4.00000	−0.203331
$388$	2.00000	0.101535
$389$	−38.0000	−1.92668	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−38.0000	−1.92668	−0.963338	+	0.268290i	$0.913542\pi$
$390$	−2.00000	−0.101274
$391$	−8.00000	−0.404577
$392$	9.00000	0.454569
$393$	4.00000	0.201773
$394$	2.00000	0.100759
$395$	8.00000	0.402524
$396$	4.00000	0.201008
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	8.00000	0.401004
$399$	−16.0000	−0.801002
$400$	1.00000	0.0500000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	4.00000	0.199502
$403$	−8.00000	−0.398508
$404$	2.00000	0.0995037
$405$	1.00000	0.0496904
$406$	−24.0000	−1.19110
$407$	8.00000	0.396545
$408$	−2.00000	−0.0990148
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	2.00000	0.0987730
$411$	6.00000	0.295958
$412$	−4.00000	−0.197066
$413$	−48.0000	−2.36193
$414$	−4.00000	−0.196589
$415$	4.00000	0.196352
$416$	2.00000	0.0980581
$417$	4.00000	0.195881
$418$	16.0000	0.782586
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	−4.00000	−0.195180
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	2.00000	0.0970143
$426$	8.00000	0.387601
$427$	−24.0000	−1.16144
$428$	12.0000	0.580042
$429$	−8.00000	−0.386244
$430$	−4.00000	−0.192897
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	−1.00000	−0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	−16.0000	−0.768025
$435$	6.00000	0.287678
$436$	14.0000	0.670478
$437$	−16.0000	−0.765384
$438$	6.00000	0.286691
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	4.00000	0.190693
$441$	9.00000	0.428571
$442$	4.00000	0.190261
$443$	−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
$445$	1.00000	0.0474045
$446$	16.0000	0.757622
$447$	6.00000	0.283790
$448$	4.00000	0.188982
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	1.00000	0.0471405
$451$	8.00000	0.376705
$452$	−6.00000	−0.282216
$453$	−4.00000	−0.187936
$454$	−12.0000	−0.563188
$455$	8.00000	0.375046
$456$	−4.00000	−0.187317
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	10.0000	0.467269
$459$	−2.00000	−0.0933520
$460$	−4.00000	−0.186501
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	−16.0000	−0.744387
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	−6.00000	−0.278543
$465$	4.00000	0.185496
$466$	−6.00000	−0.277945
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	2.00000	0.0924500
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−14.0000	−0.645086
$472$	−12.0000	−0.552345
$473$	−16.0000	−0.735681
$474$	−8.00000	−0.367452
$475$	4.00000	0.183533
$476$	8.00000	0.366679
$477$	−2.00000	−0.0915737
$478$	28.0000	1.28069
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	−1.00000	−0.0456435
$481$	4.00000	0.182384
$482$	−6.00000	−0.273293
$483$	16.0000	0.728025
$484$	5.00000	0.227273
$485$	2.00000	0.0908153
$486$	−1.00000	−0.0453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−6.00000	−0.271607
$489$	−20.0000	−0.904431
$490$	9.00000	0.406579
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−2.00000	−0.0901670
$493$	−12.0000	−0.540453
$494$	8.00000	0.359937
$495$	4.00000	0.179787
$496$	−4.00000	−0.179605
$497$	−32.0000	−1.43540
$498$	−4.00000	−0.179244
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	1.00000	0.0447214
$501$	−8.00000	−0.357414
$502$	12.0000	0.535586
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	4.00000	0.178174
$505$	2.00000	0.0889988
$506$	−16.0000	−0.711287
$507$	9.00000	0.399704
$508$	4.00000	0.177471
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	−2.00000	−0.0885615
$511$	−24.0000	−1.06170
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	−14.0000	−0.617514
$515$	−4.00000	−0.176261
$516$	4.00000	0.176090
$517$	0	0
$518$	8.00000	0.351500
$519$	18.0000	0.790112
$520$	2.00000	0.0877058
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−6.00000	−0.262613
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−4.00000	−0.174741
$525$	−4.00000	−0.174574
$526$	0	0
$527$	−8.00000	−0.348485
$528$	−4.00000	−0.174078
$529$	−7.00000	−0.304348
$530$	−2.00000	−0.0868744
$531$	−12.0000	−0.520756
$532$	16.0000	0.693688
$533$	4.00000	0.173259
$534$	−1.00000	−0.0432742
$535$	12.0000	0.518805
$536$	−4.00000	−0.172774
$537$	−4.00000	−0.172613
$538$	−2.00000	−0.0862261
$539$	36.0000	1.55063
$540$	−1.00000	−0.0430331
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	8.00000	0.343629
$543$	6.00000	0.257485
$544$	2.00000	0.0857493
$545$	14.0000	0.599694
$546$	−8.00000	−0.342368
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−6.00000	−0.256307
$549$	−6.00000	−0.256074
$550$	4.00000	0.170561
$551$	−24.0000	−1.02243
$552$	4.00000	0.170251
$553$	32.0000	1.36078
$554$	−2.00000	−0.0849719
$555$	−2.00000	−0.0848953
$556$	−4.00000	−0.169638
$557$	10.0000	0.423714	0.211857	−	0.977301i	$-0.432049\pi$
$557$	10.0000	0.423714	0.211857	+	0.977301i	$0.432049\pi$
$558$	−4.00000	−0.169334
$559$	−8.00000	−0.338364
$560$	4.00000	0.169031
$561$	−8.00000	−0.337760
$562$	10.0000	0.421825
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	20.0000	0.840663
$567$	4.00000	0.167984
$568$	−8.00000	−0.335673
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	−4.00000	−0.167542
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	8.00000	0.334497
$573$	12.0000	0.501307
$574$	8.00000	0.333914
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	−13.0000	−0.540729
$579$	14.0000	0.581820
$580$	−6.00000	−0.249136
$581$	16.0000	0.663792
$582$	−2.00000	−0.0829027
$583$	−8.00000	−0.331326
$584$	−6.00000	−0.248282
$585$	2.00000	0.0826898
$586$	−6.00000	−0.247858
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−9.00000	−0.371154
$589$	−16.0000	−0.659269
$590$	−12.0000	−0.494032
$591$	−2.00000	−0.0822690
$592$	2.00000	0.0821995
$593$	−38.0000	−1.56047	−0.780236	−	0.625485i	$-0.784901\pi$
$593$	−38.0000	−1.56047	−0.780236	+	0.625485i	$0.784901\pi$
$594$	−4.00000	−0.164122
$595$	8.00000	0.327968
$596$	−6.00000	−0.245770
$597$	−8.00000	−0.327418
$598$	−8.00000	−0.327144
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	−1.00000	−0.0408248
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	−16.0000	−0.652111
$603$	−4.00000	−0.162893
$604$	4.00000	0.162758
$605$	5.00000	0.203279
$606$	−2.00000	−0.0812444
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	4.00000	0.162221
$609$	24.0000	0.972529
$610$	−6.00000	−0.242933
$611$	0	0
$612$	2.00000	0.0808452
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	12.0000	0.484281
$615$	−2.00000	−0.0806478
$616$	16.0000	0.644658
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	4.00000	0.160904
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	−4.00000	−0.160644
$621$	4.00000	0.160514
$622$	−32.0000	−1.28308
$623$	4.00000	0.160257
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	−22.0000	−0.879297
$627$	−16.0000	−0.638978
$628$	14.0000	0.558661
$629$	4.00000	0.159490
$630$	4.00000	0.159364
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	8.00000	0.318223
$633$	12.0000	0.476957
$634$	6.00000	0.238290
$635$	4.00000	0.158735
$636$	2.00000	0.0793052
$637$	18.0000	0.713186
$638$	−24.0000	−0.950169
$639$	−8.00000	−0.316475
$640$	1.00000	0.0395285
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	−12.0000	−0.473602
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	−16.0000	−0.630488
$645$	4.00000	0.157500
$646$	8.00000	0.314756
$647$	−36.0000	−1.41531	−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000	−1.41531	−0.707653	+	0.706560i	$0.750246\pi$
$648$	1.00000	0.0392837
$649$	−48.0000	−1.88416
$650$	2.00000	0.0784465
$651$	16.0000	0.627089
$652$	20.0000	0.783260
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	−14.0000	−0.547443
$655$	−4.00000	−0.156293
$656$	2.00000	0.0780869
$657$	−6.00000	−0.234082
$658$	0	0
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	−4.00000	−0.155700
$661$	−6.00000	−0.233373	−0.116686	−	0.993169i	$-0.537227\pi$
$661$	−6.00000	−0.233373	−0.116686	+	0.993169i	$0.537227\pi$
$662$	−36.0000	−1.39918
$663$	−4.00000	−0.155347
$664$	4.00000	0.155230
$665$	16.0000	0.620453
$666$	2.00000	0.0774984
$667$	24.0000	0.929284
$668$	8.00000	0.309529
$669$	−16.0000	−0.618596
$670$	−4.00000	−0.154533
$671$	−24.0000	−0.926510
$672$	−4.00000	−0.154303
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	10.0000	0.385186
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	6.00000	0.230429
$679$	8.00000	0.307012
$680$	2.00000	0.0766965
$681$	12.0000	0.459841
$682$	−16.0000	−0.612672
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	4.00000	0.152944
$685$	−6.00000	−0.229248
$686$	8.00000	0.305441
$687$	−10.0000	−0.381524
$688$	−4.00000	−0.152499
$689$	−4.00000	−0.152388
$690$	4.00000	0.152277
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−18.0000	−0.684257
$693$	16.0000	0.607790
$694$	12.0000	0.455514
$695$	−4.00000	−0.151729
$696$	6.00000	0.227429
$697$	4.00000	0.151511
$698$	−14.0000	−0.529908
$699$	6.00000	0.226941
$700$	4.00000	0.151186
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	−2.00000	−0.0754851
$703$	8.00000	0.301726
$704$	4.00000	0.150756
$705$	0	0
$706$	−6.00000	−0.225813
$707$	8.00000	0.300871
$708$	12.0000	0.450988
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	−8.00000	−0.300235
$711$	8.00000	0.300023
$712$	1.00000	0.0374766
$713$	16.0000	0.599205
$714$	−8.00000	−0.299392
$715$	8.00000	0.299183
$716$	4.00000	0.149487
$717$	−28.0000	−1.04568
$718$	28.0000	1.04495
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	1.00000	0.0372678
$721$	−16.0000	−0.595871
$722$	−3.00000	−0.111648
$723$	6.00000	0.223142
$724$	−6.00000	−0.222988
$725$	−6.00000	−0.222834
$726$	−5.00000	−0.185567
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	8.00000	0.296500
$729$	1.00000	0.0370370
$730$	−6.00000	−0.222070
$731$	−8.00000	−0.295891
$732$	6.00000	0.221766
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	8.00000	0.295285
$735$	−9.00000	−0.331970
$736$	−4.00000	−0.147442
$737$	−16.0000	−0.589368
$738$	2.00000	0.0736210
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	2.00000	0.0735215
$741$	−8.00000	−0.293887
$742$	−8.00000	−0.293689
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	4.00000	0.146647
$745$	−6.00000	−0.219823
$746$	30.0000	1.09838
$747$	4.00000	0.146352
$748$	8.00000	0.292509
$749$	48.0000	1.75388
$750$	−1.00000	−0.0365148
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	−12.0000	−0.437014
$755$	4.00000	0.145575
$756$	−4.00000	−0.145479
$757$	46.0000	1.67190	0.835949	−	0.548807i	$-0.184918\pi$
$757$	46.0000	1.67190	0.835949	+	0.548807i	$0.184918\pi$
$758$	−20.0000	−0.726433
$759$	16.0000	0.580763
$760$	4.00000	0.145095
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	−4.00000	−0.144905
$763$	56.0000	2.02734
$764$	−12.0000	−0.434145
$765$	2.00000	0.0723102
$766$	12.0000	0.433578
$767$	−24.0000	−0.866590
$768$	−1.00000	−0.0360844
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	16.0000	0.576600
$771$	14.0000	0.504198
$772$	−14.0000	−0.503871
$773$	10.0000	0.359675	0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000	0.359675	0.179838	+	0.983696i	$0.442443\pi$
$774$	−4.00000	−0.143777
$775$	−4.00000	−0.143684
$776$	2.00000	0.0717958
$777$	−8.00000	−0.286998
$778$	−38.0000	−1.36237
$779$	8.00000	0.286630
$780$	−2.00000	−0.0716115
$781$	−32.0000	−1.14505
$782$	−8.00000	−0.286079
$783$	6.00000	0.214423
$784$	9.00000	0.321429
$785$	14.0000	0.499681
$786$	4.00000	0.142675
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	2.00000	0.0712470
$789$	0	0
$790$	8.00000	0.284627
$791$	−24.0000	−0.853342
$792$	4.00000	0.142134
$793$	−12.0000	−0.426132
$794$	34.0000	1.20661
$795$	2.00000	0.0709327
$796$	8.00000	0.283552
$797$	22.0000	0.779280	0.389640	−	0.920967i	$-0.372599\pi$
$797$	22.0000	0.779280	0.389640	+	0.920967i	$0.372599\pi$
$798$	−16.0000	−0.566394
$799$	0	0
$800$	1.00000	0.0353553
$801$	1.00000	0.0353333
$802$	2.00000	0.0706225
$803$	−24.0000	−0.846942
$804$	4.00000	0.141069
$805$	−16.0000	−0.563926
$806$	−8.00000	−0.281788
$807$	2.00000	0.0704033
$808$	2.00000	0.0703598
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	1.00000	0.0351364
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	−24.0000	−0.842235
$813$	−8.00000	−0.280572
$814$	8.00000	0.280400
$815$	20.0000	0.700569
$816$	−2.00000	−0.0700140
$817$	−16.0000	−0.559769
$818$	10.0000	0.349642
$819$	8.00000	0.279543
$820$	2.00000	0.0698430
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	6.00000	0.209274
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	−4.00000	−0.139347
$825$	−4.00000	−0.139262
$826$	−48.0000	−1.67013
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−4.00000	−0.139010
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	4.00000	0.138842
$831$	2.00000	0.0693792
$832$	2.00000	0.0693375
$833$	18.0000	0.623663
$834$	4.00000	0.138509
$835$	8.00000	0.276851
$836$	16.0000	0.553372
$837$	4.00000	0.138260
$838$	−36.0000	−1.24360
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	−4.00000	−0.138013
$841$	7.00000	0.241379
$842$	−6.00000	−0.206774
$843$	−10.0000	−0.344418
$844$	−12.0000	−0.413057
$845$	−9.00000	−0.309609
$846$	0	0
$847$	20.0000	0.687208
$848$	−2.00000	−0.0686803
$849$	−20.0000	−0.686398
$850$	2.00000	0.0685994
$851$	−8.00000	−0.274236
$852$	8.00000	0.274075
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	−24.0000	−0.821263
$855$	4.00000	0.136797
$856$	12.0000	0.410152
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	−8.00000	−0.273115
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	−4.00000	−0.136399
$861$	−8.00000	−0.272639
$862$	−12.0000	−0.408722
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	−1.00000	−0.0340207
$865$	−18.0000	−0.612018
$866$	−14.0000	−0.475739
$867$	13.0000	0.441503
$868$	−16.0000	−0.543075
$869$	32.0000	1.08553
$870$	6.00000	0.203419
$871$	−8.00000	−0.271070
$872$	14.0000	0.474100
$873$	2.00000	0.0676897
$874$	−16.0000	−0.541208
$875$	4.00000	0.135225
$876$	6.00000	0.202721
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	−4.00000	−0.134993
$879$	6.00000	0.202375
$880$	4.00000	0.134840
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	9.00000	0.303046
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	4.00000	0.134535
$885$	12.0000	0.403376
$886$	−12.0000	−0.403148
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	−2.00000	−0.0671156
$889$	16.0000	0.536623
$890$	1.00000	0.0335201
$891$	4.00000	0.134005
$892$	16.0000	0.535720
$893$	0	0
$894$	6.00000	0.200670
$895$	4.00000	0.133705
$896$	4.00000	0.133631
$897$	8.00000	0.267112
$898$	2.00000	0.0667409
$899$	24.0000	0.800445
$900$	1.00000	0.0333333
$901$	−4.00000	−0.133259
$902$	8.00000	0.266371
$903$	16.0000	0.532447
$904$	−6.00000	−0.199557
$905$	−6.00000	−0.199447
$906$	−4.00000	−0.132891
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−12.0000	−0.398234
$909$	2.00000	0.0663358
$910$	8.00000	0.265197
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	−4.00000	−0.132453
$913$	16.0000	0.529523
$914$	10.0000	0.330771
$915$	6.00000	0.198354
$916$	10.0000	0.330409
$917$	−16.0000	−0.528367
$918$	−2.00000	−0.0660098
$919$	28.0000	0.923635	0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000	0.923635	0.461817	+	0.886975i	$0.347198\pi$
$920$	−4.00000	−0.131876
$921$	−12.0000	−0.395413
$922$	−34.0000	−1.11973
$923$	−16.0000	−0.526646
$924$	−16.0000	−0.526361
$925$	2.00000	0.0657596
$926$	40.0000	1.31448
$927$	−4.00000	−0.131377
$928$	−6.00000	−0.196960
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	4.00000	0.131165
$931$	36.0000	1.17985
$932$	−6.00000	−0.196537
$933$	32.0000	1.04763
$934$	4.00000	0.130884
$935$	8.00000	0.261628
$936$	2.00000	0.0653720
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	−16.0000	−0.522419
$939$	22.0000	0.717943
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	−14.0000	−0.456145
$943$	−8.00000	−0.260516
$944$	−12.0000	−0.390567
$945$	−4.00000	−0.130120
$946$	−16.0000	−0.520205
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	−8.00000	−0.259828
$949$	−12.0000	−0.389536
$950$	4.00000	0.129777
$951$	−6.00000	−0.194563
$952$	8.00000	0.259281
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	−2.00000	−0.0647524
$955$	−12.0000	−0.388311
$956$	28.0000	0.905585
$957$	24.0000	0.775810
$958$	−24.0000	−0.775405
$959$	−24.0000	−0.775000
$960$	−1.00000	−0.0322749
$961$	−15.0000	−0.483871
$962$	4.00000	0.128965
$963$	12.0000	0.386695
$964$	−6.00000	−0.193247
$965$	−14.0000	−0.450676
$966$	16.0000	0.514792
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	5.00000	0.160706
$969$	−8.00000	−0.256997
$970$	2.00000	0.0642161
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	−1.00000	−0.0320750
$973$	−16.0000	−0.512936
$974$	−8.00000	−0.256337
$975$	−2.00000	−0.0640513
$976$	−6.00000	−0.192055
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−20.0000	−0.639529
$979$	4.00000	0.127841
$980$	9.00000	0.287494
$981$	14.0000	0.446986
$982$	20.0000	0.638226
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−2.00000	−0.0637577
$985$	2.00000	0.0637253
$986$	−12.0000	−0.382158
$987$	0	0
$988$	8.00000	0.254514
$989$	16.0000	0.508770
$990$	4.00000	0.127128
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	−4.00000	−0.127000
$993$	36.0000	1.14243
$994$	−32.0000	−1.01498
$995$	8.00000	0.253617
$996$	−4.00000	−0.126745
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	44.0000	1.39280
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2670.2.a.d.1.1	✓	1
3.2	odd	2		8010.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2670.2.a.d.1.1	✓	1	1.1	even	1	trivial
8010.2.a.e.1.1		1	3.2	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:	\( N \)	\(=\)	\( 2670 = 2 \cdot 3 \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2670.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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