LMFDB - Embedded newform 1610.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1610,2,Mod(1,1610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1610, 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("1610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1610 = 2 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1610.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:	$12.8559147254$
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	$\chi$	$=$	1610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} -2.00000 q^{22} +1.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} -2.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -8.00000 q^{37} -4.00000 q^{38} +8.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} -2.00000 q^{42} +10.0000 q^{43} -2.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} -12.0000 q^{47} -2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} -4.00000 q^{52} -4.00000 q^{53} +4.00000 q^{54} -2.00000 q^{55} +1.00000 q^{56} +8.00000 q^{57} -2.00000 q^{58} +10.0000 q^{59} -2.00000 q^{60} -14.0000 q^{61} -4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +4.00000 q^{66} +10.0000 q^{67} -2.00000 q^{68} -2.00000 q^{69} +1.00000 q^{70} -16.0000 q^{71} +1.00000 q^{72} +6.00000 q^{73} -8.00000 q^{74} -2.00000 q^{75} -4.00000 q^{76} -2.00000 q^{77} +8.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -2.00000 q^{82} +12.0000 q^{83} -2.00000 q^{84} -2.00000 q^{85} +10.0000 q^{86} +4.00000 q^{87} -2.00000 q^{88} -6.00000 q^{89} +1.00000 q^{90} -4.00000 q^{91} +1.00000 q^{92} +8.00000 q^{93} -12.0000 q^{94} -4.00000 q^{95} -2.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.00000	−0.816497
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−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	$0.687167\pi$
$14$	1.00000	0.267261
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	1.00000	0.223607
$21$	−2.00000	−0.436436
$22$	−2.00000	−0.426401
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−2.00000	−0.408248
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	4.00000	0.769800
$28$	1.00000	0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	−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	$0.616954\pi$
$32$	1.00000	0.176777
$33$	4.00000	0.696311
$34$	−2.00000	−0.342997
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−4.00000	−0.648886
$39$	8.00000	1.28103
$40$	1.00000	0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−2.00000	−0.308607
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	−2.00000	−0.301511
$45$	1.00000	0.149071
$46$	1.00000	0.147442
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	−2.00000	−0.288675
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	4.00000	0.560112
$52$	−4.00000	−0.554700
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	4.00000	0.544331
$55$	−2.00000	−0.269680
$56$	1.00000	0.133631
$57$	8.00000	1.05963
$58$	−2.00000	−0.262613
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	−2.00000	−0.258199
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	−4.00000	−0.508001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	4.00000	0.492366
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	−2.00000	−0.242536
$69$	−2.00000	−0.240772
$70$	1.00000	0.119523
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	1.00000	0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−8.00000	−0.929981
$75$	−2.00000	−0.230940
$76$	−4.00000	−0.458831
$77$	−2.00000	−0.227921
$78$	8.00000	0.905822
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	−2.00000	−0.220863
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	−2.00000	−0.218218
$85$	−2.00000	−0.216930
$86$	10.0000	1.07833
$87$	4.00000	0.428845
$88$	−2.00000	−0.213201
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	1.00000	0.105409
$91$	−4.00000	−0.419314
$92$	1.00000	0.104257
$93$	8.00000	0.829561
$94$	−12.0000	−1.23771
$95$	−4.00000	−0.410391
$96$	−2.00000	−0.204124
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	1.00000	0.101015
$99$	−2.00000	−0.201008
$100$	1.00000	0.100000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	4.00000	0.396059
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−4.00000	−0.392232
$105$	−2.00000	−0.195180
$106$	−4.00000	−0.388514
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	4.00000	0.384900
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	−2.00000	−0.190693
$111$	16.0000	1.51865
$112$	1.00000	0.0944911
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	8.00000	0.749269
$115$	1.00000	0.0932505
$116$	−2.00000	−0.185695
$117$	−4.00000	−0.369800
$118$	10.0000	0.920575
$119$	−2.00000	−0.183340
$120$	−2.00000	−0.182574
$121$	−7.00000	−0.636364
$122$	−14.0000	−1.26750
$123$	4.00000	0.360668
$124$	−4.00000	−0.359211
$125$	1.00000	0.0894427
$126$	1.00000	0.0890871
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	1.00000	0.0883883
$129$	−20.0000	−1.76090
$130$	−4.00000	−0.350823
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	4.00000	0.348155
$133$	−4.00000	−0.346844
$134$	10.0000	0.863868
$135$	4.00000	0.344265
$136$	−2.00000	−0.171499
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	−2.00000	−0.170251
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	1.00000	0.0845154
$141$	24.0000	2.02116
$142$	−16.0000	−1.34269
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	6.00000	0.496564
$147$	−2.00000	−0.164957
$148$	−8.00000	−0.657596
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	−2.00000	−0.163299
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−4.00000	−0.324443
$153$	−2.00000	−0.161690
$154$	−2.00000	−0.161165
$155$	−4.00000	−0.321288
$156$	8.00000	0.640513
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	−4.00000	−0.318223
$159$	8.00000	0.634441
$160$	1.00000	0.0790569
$161$	1.00000	0.0788110
$162$	−11.0000	−0.864242
$163$	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
$166$	12.0000	0.931381
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−2.00000	−0.154303
$169$	3.00000	0.230769
$170$	−2.00000	−0.153393
$171$	−4.00000	−0.305888
$172$	10.0000	0.762493
$173$	16.0000	1.21646	0.608229	−	0.793762i	$-0.291880\pi$
$173$	16.0000	1.21646	0.608229	+	0.793762i	$0.291880\pi$
$174$	4.00000	0.303239
$175$	1.00000	0.0755929
$176$	−2.00000	−0.150756
$177$	−20.0000	−1.50329
$178$	−6.00000	−0.449719
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	1.00000	0.0745356
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	−4.00000	−0.296500
$183$	28.0000	2.06982
$184$	1.00000	0.0737210
$185$	−8.00000	−0.588172
$186$	8.00000	0.586588
$187$	4.00000	0.292509
$188$	−12.0000	−0.875190
$189$	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	$0.642949\pi$
$192$	−2.00000	−0.144338
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−14.0000	−1.00514
$195$	8.00000	0.572892
$196$	1.00000	0.0714286
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	−2.00000	−0.142134
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	1.00000	0.0707107
$201$	−20.0000	−1.41069
$202$	−12.0000	−0.844317
$203$	−2.00000	−0.140372
$204$	4.00000	0.280056
$205$	−2.00000	−0.139686
$206$	−16.0000	−1.11477
$207$	1.00000	0.0695048
$208$	−4.00000	−0.277350
$209$	8.00000	0.553372
$210$	−2.00000	−0.138013
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−4.00000	−0.274721
$213$	32.0000	2.19260
$214$	2.00000	0.136717
$215$	10.0000	0.681994
$216$	4.00000	0.272166
$217$	−4.00000	−0.271538
$218$	16.0000	1.08366
$219$	−12.0000	−0.810885
$220$	−2.00000	−0.134840
$221$	8.00000	0.538138
$222$	16.0000	1.07385
$223$	28.0000	1.87502	0.937509	−	0.347960i	$-0.113126\pi$
$223$	28.0000	1.87502	0.937509	+	0.347960i	$0.113126\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	−10.0000	−0.665190
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	8.00000	0.529813
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	1.00000	0.0659380
$231$	4.00000	0.263181
$232$	−2.00000	−0.131306
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	−4.00000	−0.261488
$235$	−12.0000	−0.782794
$236$	10.0000	0.650945
$237$	8.00000	0.519656
$238$	−2.00000	−0.129641
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	−2.00000	−0.129099
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−7.00000	−0.449977
$243$	10.0000	0.641500
$244$	−14.0000	−0.896258
$245$	1.00000	0.0638877
$246$	4.00000	0.255031
$247$	16.0000	1.01806
$248$	−4.00000	−0.254000
$249$	−24.0000	−1.52094
$250$	1.00000	0.0632456
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	1.00000	0.0629941
$253$	−2.00000	−0.125739
$254$	0	0
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	−20.0000	−1.24515
$259$	−8.00000	−0.497096
$260$	−4.00000	−0.248069
$261$	−2.00000	−0.123797
$262$	−6.00000	−0.370681
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	4.00000	0.246183
$265$	−4.00000	−0.245718
$266$	−4.00000	−0.245256
$267$	12.0000	0.734388
$268$	10.0000	0.610847
$269$	−28.0000	−1.70719	−0.853595	−	0.520937i	$-0.825583\pi$
$269$	−28.0000	−1.70719	−0.853595	+	0.520937i	$0.825583\pi$
$270$	4.00000	0.243432
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−2.00000	−0.121268
$273$	8.00000	0.484182
$274$	10.0000	0.604122
$275$	−2.00000	−0.120605
$276$	−2.00000	−0.120386
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	−10.0000	−0.599760
$279$	−4.00000	−0.239474
$280$	1.00000	0.0597614
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	24.0000	1.42918
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	−16.0000	−0.949425
$285$	8.00000	0.473879
$286$	8.00000	0.473050
$287$	−2.00000	−0.118056
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	−2.00000	−0.117444
$291$	28.0000	1.64139
$292$	6.00000	0.351123
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	−2.00000	−0.116642
$295$	10.0000	0.582223
$296$	−8.00000	−0.464991
$297$	−8.00000	−0.464207
$298$	8.00000	0.463428
$299$	−4.00000	−0.231326
$300$	−2.00000	−0.115470
$301$	10.0000	0.576390
$302$	0	0
$303$	24.0000	1.37876
$304$	−4.00000	−0.229416
$305$	−14.0000	−0.801638
$306$	−2.00000	−0.114332
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	−2.00000	−0.113961
$309$	32.0000	1.82042
$310$	−4.00000	−0.227185
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	8.00000	0.452911
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	−22.0000	−1.24153
$315$	1.00000	0.0563436
$316$	−4.00000	−0.225018
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	8.00000	0.448618
$319$	4.00000	0.223957
$320$	1.00000	0.0559017
$321$	−4.00000	−0.223258
$322$	1.00000	0.0557278
$323$	8.00000	0.445132
$324$	−11.0000	−0.611111
$325$	−4.00000	−0.221880
$326$	20.0000	1.10770
$327$	−32.0000	−1.76960
$328$	−2.00000	−0.110432
$329$	−12.0000	−0.661581
$330$	4.00000	0.220193
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	12.0000	0.658586
$333$	−8.00000	−0.438397
$334$	12.0000	0.656611
$335$	10.0000	0.546358
$336$	−2.00000	−0.109109
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	3.00000	0.163178
$339$	20.0000	1.08625
$340$	−2.00000	−0.108465
$341$	8.00000	0.433224
$342$	−4.00000	−0.216295
$343$	1.00000	0.0539949
$344$	10.0000	0.539164
$345$	−2.00000	−0.107676
$346$	16.0000	0.860165
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	4.00000	0.214423
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	1.00000	0.0534522
$351$	−16.0000	−0.854017
$352$	−2.00000	−0.106600
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	−20.0000	−1.06299
$355$	−16.0000	−0.849192
$356$	−6.00000	−0.317999
$357$	4.00000	0.211702
$358$	16.0000	0.845626
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	22.0000	1.15629
$363$	14.0000	0.734809
$364$	−4.00000	−0.209657
$365$	6.00000	0.314054
$366$	28.0000	1.46358
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	1.00000	0.0521286
$369$	−2.00000	−0.104116
$370$	−8.00000	−0.415900
$371$	−4.00000	−0.207670
$372$	8.00000	0.414781
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	4.00000	0.206835
$375$	−2.00000	−0.103280
$376$	−12.0000	−0.618853
$377$	8.00000	0.412021
$378$	4.00000	0.205738
$379$	18.0000	0.924598	0.462299	−	0.886724i	$-0.347025\pi$
$379$	18.0000	0.924598	0.462299	+	0.886724i	$0.347025\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	−12.0000	−0.613973
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−2.00000	−0.102062
$385$	−2.00000	−0.101929
$386$	−2.00000	−0.101797
$387$	10.0000	0.508329
$388$	−14.0000	−0.710742
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	8.00000	0.405096
$391$	−2.00000	−0.101144
$392$	1.00000	0.0505076
$393$	12.0000	0.605320
$394$	10.0000	0.503793
$395$	−4.00000	−0.201262
$396$	−2.00000	−0.100504
$397$	−28.0000	−1.40528	−0.702640	−	0.711546i	$-0.747995\pi$
$397$	−28.0000	−1.40528	−0.702640	+	0.711546i	$0.747995\pi$
$398$	−8.00000	−0.401004
$399$	8.00000	0.400501
$400$	1.00000	0.0500000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	−20.0000	−0.997509
$403$	16.0000	0.797017
$404$	−12.0000	−0.597022
$405$	−11.0000	−0.546594
$406$	−2.00000	−0.0992583
$407$	16.0000	0.793091
$408$	4.00000	0.198030
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	−2.00000	−0.0987730
$411$	−20.0000	−0.986527
$412$	−16.0000	−0.788263
$413$	10.0000	0.492068
$414$	1.00000	0.0491473
$415$	12.0000	0.589057
$416$	−4.00000	−0.196116
$417$	20.0000	0.979404
$418$	8.00000	0.391293
$419$	40.0000	1.95413	0.977064	−	0.212946i	$-0.0683059\pi$
$419$	40.0000	1.95413	0.977064	+	0.212946i	$0.0683059\pi$
$420$	−2.00000	−0.0975900
$421$	−40.0000	−1.94948	−0.974740	−	0.223341i	$-0.928304\pi$
$421$	−40.0000	−1.94948	−0.974740	+	0.223341i	$0.928304\pi$
$422$	4.00000	0.194717
$423$	−12.0000	−0.583460
$424$	−4.00000	−0.194257
$425$	−2.00000	−0.0970143
$426$	32.0000	1.55041
$427$	−14.0000	−0.677507
$428$	2.00000	0.0966736
$429$	−16.0000	−0.772487
$430$	10.0000	0.482243
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	4.00000	0.192450
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	−4.00000	−0.192006
$435$	4.00000	0.191785
$436$	16.0000	0.766261
$437$	−4.00000	−0.191346
$438$	−12.0000	−0.573382
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	−2.00000	−0.0953463
$441$	1.00000	0.0476190
$442$	8.00000	0.380521
$443$	−8.00000	−0.380091	−0.190046	−	0.981775i	$-0.560864\pi$
$443$	−8.00000	−0.380091	−0.190046	+	0.981775i	$0.560864\pi$
$444$	16.0000	0.759326
$445$	−6.00000	−0.284427
$446$	28.0000	1.32584
$447$	−16.0000	−0.756774
$448$	1.00000	0.0472456
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	1.00000	0.0471405
$451$	4.00000	0.188353
$452$	−10.0000	−0.470360
$453$	0	0
$454$	24.0000	1.12638
$455$	−4.00000	−0.187523
$456$	8.00000	0.374634
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−10.0000	−0.467269
$459$	−8.00000	−0.373408
$460$	1.00000	0.0466252
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	4.00000	0.186097
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−2.00000	−0.0928477
$465$	8.00000	0.370991
$466$	18.0000	0.833834
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−4.00000	−0.184900
$469$	10.0000	0.461757
$470$	−12.0000	−0.553519
$471$	44.0000	2.02741
$472$	10.0000	0.460287
$473$	−20.0000	−0.919601
$474$	8.00000	0.367452
$475$	−4.00000	−0.183533
$476$	−2.00000	−0.0916698
$477$	−4.00000	−0.183147
$478$	−8.00000	−0.365911
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	−2.00000	−0.0912871
$481$	32.0000	1.45907
$482$	10.0000	0.455488
$483$	−2.00000	−0.0910032
$484$	−7.00000	−0.318182
$485$	−14.0000	−0.635707
$486$	10.0000	0.453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	−14.0000	−0.633750
$489$	−40.0000	−1.80886
$490$	1.00000	0.0451754
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	4.00000	0.180334
$493$	4.00000	0.180151
$494$	16.0000	0.719874
$495$	−2.00000	−0.0898933
$496$	−4.00000	−0.179605
$497$	−16.0000	−0.717698
$498$	−24.0000	−1.07547
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	1.00000	0.0447214
$501$	−24.0000	−1.07224
$502$	−8.00000	−0.357057
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	1.00000	0.0445435
$505$	−12.0000	−0.533993
$506$	−2.00000	−0.0889108
$507$	−6.00000	−0.266469
$508$	0	0
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	4.00000	0.177123
$511$	6.00000	0.265424
$512$	1.00000	0.0441942
$513$	−16.0000	−0.706417
$514$	6.00000	0.264649
$515$	−16.0000	−0.705044
$516$	−20.0000	−0.880451
$517$	24.0000	1.05552
$518$	−8.00000	−0.351500
$519$	−32.0000	−1.40464
$520$	−4.00000	−0.175412
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	−2.00000	−0.0875376
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	−6.00000	−0.262111
$525$	−2.00000	−0.0872872
$526$	12.0000	0.523225
$527$	8.00000	0.348485
$528$	4.00000	0.174078
$529$	1.00000	0.0434783
$530$	−4.00000	−0.173749
$531$	10.0000	0.433963
$532$	−4.00000	−0.173422
$533$	8.00000	0.346518
$534$	12.0000	0.519291
$535$	2.00000	0.0864675
$536$	10.0000	0.431934
$537$	−32.0000	−1.38090
$538$	−28.0000	−1.20717
$539$	−2.00000	−0.0861461
$540$	4.00000	0.172133
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	20.0000	0.859074
$543$	−44.0000	−1.88822
$544$	−2.00000	−0.0857493
$545$	16.0000	0.685365
$546$	8.00000	0.342368
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	10.0000	0.427179
$549$	−14.0000	−0.597505
$550$	−2.00000	−0.0852803
$551$	8.00000	0.340811
$552$	−2.00000	−0.0851257
$553$	−4.00000	−0.170097
$554$	−14.0000	−0.594803
$555$	16.0000	0.679162
$556$	−10.0000	−0.424094
$557$	−16.0000	−0.677942	−0.338971	−	0.940797i	$-0.610079\pi$
$557$	−16.0000	−0.677942	−0.338971	+	0.940797i	$0.610079\pi$
$558$	−4.00000	−0.169334
$559$	−40.0000	−1.69182
$560$	1.00000	0.0422577
$561$	−8.00000	−0.337760
$562$	10.0000	0.421825
$563$	44.0000	1.85438	0.927189	−	0.374593i	$-0.122217\pi$
$563$	44.0000	1.85438	0.927189	+	0.374593i	$0.122217\pi$
$564$	24.0000	1.01058
$565$	−10.0000	−0.420703
$566$	−8.00000	−0.336265
$567$	−11.0000	−0.461957
$568$	−16.0000	−0.671345
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	8.00000	0.335083
$571$	−30.0000	−1.25546	−0.627730	−	0.778431i	$-0.716016\pi$
$571$	−30.0000	−1.25546	−0.627730	+	0.778431i	$0.716016\pi$
$572$	8.00000	0.334497
$573$	24.0000	1.00261
$574$	−2.00000	−0.0834784
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	−13.0000	−0.540729
$579$	4.00000	0.166234
$580$	−2.00000	−0.0830455
$581$	12.0000	0.497844
$582$	28.0000	1.16064
$583$	8.00000	0.331326
$584$	6.00000	0.248282
$585$	−4.00000	−0.165380
$586$	−2.00000	−0.0826192
$587$	10.0000	0.412744	0.206372	−	0.978474i	$-0.433834\pi$
$587$	10.0000	0.412744	0.206372	+	0.978474i	$0.433834\pi$
$588$	−2.00000	−0.0824786
$589$	16.0000	0.659269
$590$	10.0000	0.411693
$591$	−20.0000	−0.822690
$592$	−8.00000	−0.328798
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	−8.00000	−0.328244
$595$	−2.00000	−0.0819920
$596$	8.00000	0.327693
$597$	16.0000	0.654836
$598$	−4.00000	−0.163572
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	−2.00000	−0.0816497
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	10.0000	0.407570
$603$	10.0000	0.407231
$604$	0	0
$605$	−7.00000	−0.284590
$606$	24.0000	0.974933
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	−4.00000	−0.162221
$609$	4.00000	0.162088
$610$	−14.0000	−0.566843
$611$	48.0000	1.94187
$612$	−2.00000	−0.0808452
$613$	−12.0000	−0.484675	−0.242338	−	0.970192i	$-0.577914\pi$
$613$	−12.0000	−0.484675	−0.242338	+	0.970192i	$0.577914\pi$
$614$	2.00000	0.0807134
$615$	4.00000	0.161296
$616$	−2.00000	−0.0805823
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	32.0000	1.28723
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	−4.00000	−0.160644
$621$	4.00000	0.160514
$622$	−16.0000	−0.641542
$623$	−6.00000	−0.240385
$624$	8.00000	0.320256
$625$	1.00000	0.0400000
$626$	−22.0000	−0.879297
$627$	−16.0000	−0.638978
$628$	−22.0000	−0.877896
$629$	16.0000	0.637962
$630$	1.00000	0.0398410
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−4.00000	−0.159111
$633$	−8.00000	−0.317971
$634$	2.00000	0.0794301
$635$	0	0
$636$	8.00000	0.317221
$637$	−4.00000	−0.158486
$638$	4.00000	0.158362
$639$	−16.0000	−0.632950
$640$	1.00000	0.0395285
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	−4.00000	−0.157867
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	1.00000	0.0394055
$645$	−20.0000	−0.787499
$646$	8.00000	0.314756
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	−11.0000	−0.432121
$649$	−20.0000	−0.785069
$650$	−4.00000	−0.156893
$651$	8.00000	0.313545
$652$	20.0000	0.783260
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	−32.0000	−1.25130
$655$	−6.00000	−0.234439
$656$	−2.00000	−0.0780869
$657$	6.00000	0.234082
$658$	−12.0000	−0.467809
$659$	34.0000	1.32445	0.662226	−	0.749304i	$-0.269612\pi$
$659$	34.0000	1.32445	0.662226	+	0.749304i	$0.269612\pi$
$660$	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	$0.537227\pi$
$662$	−8.00000	−0.310929
$663$	−16.0000	−0.621389
$664$	12.0000	0.465690
$665$	−4.00000	−0.155113
$666$	−8.00000	−0.309994
$667$	−2.00000	−0.0774403
$668$	12.0000	0.464294
$669$	−56.0000	−2.16509
$670$	10.0000	0.386334
$671$	28.0000	1.08093
$672$	−2.00000	−0.0771517
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	14.0000	0.539260
$675$	4.00000	0.153960
$676$	3.00000	0.115385
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	20.0000	0.768095
$679$	−14.0000	−0.537271
$680$	−2.00000	−0.0766965
$681$	−48.0000	−1.83936
$682$	8.00000	0.306336
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	−4.00000	−0.152944
$685$	10.0000	0.382080
$686$	1.00000	0.0381802
$687$	20.0000	0.763048
$688$	10.0000	0.381246
$689$	16.0000	0.609551
$690$	−2.00000	−0.0761387
$691$	18.0000	0.684752	0.342376	−	0.939563i	$-0.388768\pi$
$691$	18.0000	0.684752	0.342376	+	0.939563i	$0.388768\pi$
$692$	16.0000	0.608229
$693$	−2.00000	−0.0759737
$694$	20.0000	0.759190
$695$	−10.0000	−0.379322
$696$	4.00000	0.151620
$697$	4.00000	0.151511
$698$	28.0000	1.05982
$699$	−36.0000	−1.36165
$700$	1.00000	0.0377964
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	−16.0000	−0.603881
$703$	32.0000	1.20690
$704$	−2.00000	−0.0753778
$705$	24.0000	0.903892
$706$	14.0000	0.526897
$707$	−12.0000	−0.451306
$708$	−20.0000	−0.751646
$709$	12.0000	0.450669	0.225335	−	0.974281i	$-0.427652\pi$
$709$	12.0000	0.450669	0.225335	+	0.974281i	$0.427652\pi$
$710$	−16.0000	−0.600469
$711$	−4.00000	−0.150012
$712$	−6.00000	−0.224860
$713$	−4.00000	−0.149801
$714$	4.00000	0.149696
$715$	8.00000	0.299183
$716$	16.0000	0.597948
$717$	16.0000	0.597531
$718$	24.0000	0.895672
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	1.00000	0.0372678
$721$	−16.0000	−0.595871
$722$	−3.00000	−0.111648
$723$	−20.0000	−0.743808
$724$	22.0000	0.817624
$725$	−2.00000	−0.0742781
$726$	14.0000	0.519589
$727$	48.0000	1.78022	0.890111	−	0.455744i	$-0.150627\pi$
$727$	48.0000	1.78022	0.890111	+	0.455744i	$0.150627\pi$
$728$	−4.00000	−0.148250
$729$	13.0000	0.481481
$730$	6.00000	0.222070
$731$	−20.0000	−0.739727
$732$	28.0000	1.03491
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−8.00000	−0.295285
$735$	−2.00000	−0.0737711
$736$	1.00000	0.0368605
$737$	−20.0000	−0.736709
$738$	−2.00000	−0.0736210
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	−8.00000	−0.294086
$741$	−32.0000	−1.17555
$742$	−4.00000	−0.146845
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	8.00000	0.293294
$745$	8.00000	0.293097
$746$	4.00000	0.146450
$747$	12.0000	0.439057
$748$	4.00000	0.146254
$749$	2.00000	0.0730784
$750$	−2.00000	−0.0730297
$751$	36.0000	1.31366	0.656829	−	0.754039i	$-0.271897\pi$
$751$	36.0000	1.31366	0.656829	+	0.754039i	$0.271897\pi$
$752$	−12.0000	−0.437595
$753$	16.0000	0.583072
$754$	8.00000	0.291343
$755$	0	0
$756$	4.00000	0.145479
$757$	20.0000	0.726912	0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000	0.726912	0.363456	+	0.931611i	$0.381597\pi$
$758$	18.0000	0.653789
$759$	4.00000	0.145191
$760$	−4.00000	−0.145095
$761$	−2.00000	−0.0724999	−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000	−0.0724999	−0.0362500	+	0.999343i	$0.511541\pi$
$762$	0	0
$763$	16.0000	0.579239
$764$	−12.0000	−0.434145
$765$	−2.00000	−0.0723102
$766$	0	0
$767$	−40.0000	−1.44432
$768$	−2.00000	−0.0721688
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	−2.00000	−0.0720750
$771$	−12.0000	−0.432169
$772$	−2.00000	−0.0719816
$773$	50.0000	1.79838	0.899188	−	0.437564i	$-0.144158\pi$
$773$	50.0000	1.79838	0.899188	+	0.437564i	$0.144158\pi$
$774$	10.0000	0.359443
$775$	−4.00000	−0.143684
$776$	−14.0000	−0.502571
$777$	16.0000	0.573997
$778$	−24.0000	−0.860442
$779$	8.00000	0.286630
$780$	8.00000	0.286446
$781$	32.0000	1.14505
$782$	−2.00000	−0.0715199
$783$	−8.00000	−0.285897
$784$	1.00000	0.0357143
$785$	−22.0000	−0.785214
$786$	12.0000	0.428026
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	10.0000	0.356235
$789$	−24.0000	−0.854423
$790$	−4.00000	−0.142314
$791$	−10.0000	−0.355559
$792$	−2.00000	−0.0710669
$793$	56.0000	1.98862
$794$	−28.0000	−0.993683
$795$	8.00000	0.283731
$796$	−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	$0.372599\pi$
$798$	8.00000	0.283197
$799$	24.0000	0.849059
$800$	1.00000	0.0353553
$801$	−6.00000	−0.212000
$802$	30.0000	1.05934
$803$	−12.0000	−0.423471
$804$	−20.0000	−0.705346
$805$	1.00000	0.0352454
$806$	16.0000	0.563576
$807$	56.0000	1.97129
$808$	−12.0000	−0.422159
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	−11.0000	−0.386501
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	−2.00000	−0.0701862
$813$	−40.0000	−1.40286
$814$	16.0000	0.560800
$815$	20.0000	0.700569
$816$	4.00000	0.140028
$817$	−40.0000	−1.39942
$818$	−2.00000	−0.0699284
$819$	−4.00000	−0.139771
$820$	−2.00000	−0.0698430
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	−20.0000	−0.697580
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	−16.0000	−0.557386
$825$	4.00000	0.139262
$826$	10.0000	0.347945
$827$	−30.0000	−1.04320	−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000	−1.04320	−0.521601	+	0.853189i	$0.674665\pi$
$828$	1.00000	0.0347524
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	12.0000	0.416526
$831$	28.0000	0.971309
$832$	−4.00000	−0.138675
$833$	−2.00000	−0.0692959
$834$	20.0000	0.692543
$835$	12.0000	0.415277
$836$	8.00000	0.276686
$837$	−16.0000	−0.553041
$838$	40.0000	1.38178
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	−2.00000	−0.0690066
$841$	−25.0000	−0.862069
$842$	−40.0000	−1.37849
$843$	−20.0000	−0.688837
$844$	4.00000	0.137686
$845$	3.00000	0.103203
$846$	−12.0000	−0.412568
$847$	−7.00000	−0.240523
$848$	−4.00000	−0.137361
$849$	16.0000	0.549119
$850$	−2.00000	−0.0685994
$851$	−8.00000	−0.274236
$852$	32.0000	1.09630
$853$	−36.0000	−1.23262	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	−36.0000	−1.23262	−0.616308	+	0.787505i	$0.711372\pi$
$854$	−14.0000	−0.479070
$855$	−4.00000	−0.136797
$856$	2.00000	0.0683586
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	−16.0000	−0.546231
$859$	10.0000	0.341196	0.170598	−	0.985341i	$-0.445430\pi$
$859$	10.0000	0.341196	0.170598	+	0.985341i	$0.445430\pi$
$860$	10.0000	0.340997
$861$	4.00000	0.136320
$862$	−16.0000	−0.544962
$863$	40.0000	1.36162	0.680808	−	0.732462i	$-0.261629\pi$
$863$	40.0000	1.36162	0.680808	+	0.732462i	$0.261629\pi$
$864$	4.00000	0.136083
$865$	16.0000	0.544016
$866$	22.0000	0.747590
$867$	26.0000	0.883006
$868$	−4.00000	−0.135769
$869$	8.00000	0.271381
$870$	4.00000	0.135613
$871$	−40.0000	−1.35535
$872$	16.0000	0.541828
$873$	−14.0000	−0.473828
$874$	−4.00000	−0.135302
$875$	1.00000	0.0338062
$876$	−12.0000	−0.405442
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	−12.0000	−0.404980
$879$	4.00000	0.134917
$880$	−2.00000	−0.0674200
$881$	22.0000	0.741199	0.370599	−	0.928793i	$-0.379152\pi$
$881$	22.0000	0.741199	0.370599	+	0.928793i	$0.379152\pi$
$882$	1.00000	0.0336718
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	8.00000	0.269069
$885$	−20.0000	−0.672293
$886$	−8.00000	−0.268765
$887$	−52.0000	−1.74599	−0.872995	−	0.487730i	$-0.837825\pi$
$887$	−52.0000	−1.74599	−0.872995	+	0.487730i	$0.837825\pi$
$888$	16.0000	0.536925
$889$	0	0
$890$	−6.00000	−0.201120
$891$	22.0000	0.737028
$892$	28.0000	0.937509
$893$	48.0000	1.60626
$894$	−16.0000	−0.535120
$895$	16.0000	0.534821
$896$	1.00000	0.0334077
$897$	8.00000	0.267112
$898$	−34.0000	−1.13459
$899$	8.00000	0.266815
$900$	1.00000	0.0333333
$901$	8.00000	0.266519
$902$	4.00000	0.133185
$903$	−20.0000	−0.665558
$904$	−10.0000	−0.332595
$905$	22.0000	0.731305
$906$	0	0
$907$	22.0000	0.730498	0.365249	−	0.930910i	$-0.380984\pi$
$907$	22.0000	0.730498	0.365249	+	0.930910i	$0.380984\pi$
$908$	24.0000	0.796468
$909$	−12.0000	−0.398015
$910$	−4.00000	−0.132599
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	8.00000	0.264906
$913$	−24.0000	−0.794284
$914$	−22.0000	−0.727695
$915$	28.0000	0.925651
$916$	−10.0000	−0.330409
$917$	−6.00000	−0.198137
$918$	−8.00000	−0.264039
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	1.00000	0.0329690
$921$	−4.00000	−0.131804
$922$	−12.0000	−0.395199
$923$	64.0000	2.10659
$924$	4.00000	0.131590
$925$	−8.00000	−0.263038
$926$	−8.00000	−0.262896
$927$	−16.0000	−0.525509
$928$	−2.00000	−0.0656532
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	8.00000	0.262330
$931$	−4.00000	−0.131095
$932$	18.0000	0.589610
$933$	32.0000	1.04763
$934$	−12.0000	−0.392652
$935$	4.00000	0.130814
$936$	−4.00000	−0.130744
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	10.0000	0.326512
$939$	44.0000	1.43589
$940$	−12.0000	−0.391397
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	44.0000	1.43360
$943$	−2.00000	−0.0651290
$944$	10.0000	0.325472
$945$	4.00000	0.130120
$946$	−20.0000	−0.650256
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	8.00000	0.259828
$949$	−24.0000	−0.779073
$950$	−4.00000	−0.129777
$951$	−4.00000	−0.129709
$952$	−2.00000	−0.0648204
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	−4.00000	−0.129505
$955$	−12.0000	−0.388311
$956$	−8.00000	−0.258738
$957$	−8.00000	−0.258603
$958$	−24.0000	−0.775405
$959$	10.0000	0.322917
$960$	−2.00000	−0.0645497
$961$	−15.0000	−0.483871
$962$	32.0000	1.03172
$963$	2.00000	0.0644491
$964$	10.0000	0.322078
$965$	−2.00000	−0.0643823
$966$	−2.00000	−0.0643489
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	−7.00000	−0.224989
$969$	−16.0000	−0.513994
$970$	−14.0000	−0.449513
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	10.0000	0.320750
$973$	−10.0000	−0.320585
$974$	24.0000	0.769010
$975$	8.00000	0.256205
$976$	−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	$0.316955\pi$
$978$	−40.0000	−1.27906
$979$	12.0000	0.383522
$980$	1.00000	0.0319438
$981$	16.0000	0.510841
$982$	0	0
$983$	40.0000	1.27580	0.637901	−	0.770118i	$-0.279803\pi$
$983$	40.0000	1.27580	0.637901	+	0.770118i	$0.279803\pi$
$984$	4.00000	0.127515
$985$	10.0000	0.318626
$986$	4.00000	0.127386
$987$	24.0000	0.763928
$988$	16.0000	0.509028
$989$	10.0000	0.317982
$990$	−2.00000	−0.0635642
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	−4.00000	−0.127000
$993$	16.0000	0.507745
$994$	−16.0000	−0.507489
$995$	−8.00000	−0.253617
$996$	−24.0000	−0.760469
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	−40.0000	−1.26618
$999$	−32.0000	−1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1610.2.a.d.1.1	✓	1
5.4	even	2		8050.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1610.2.a.d.1.1	✓	1	1.1	even	1	trivial
8050.2.a.k.1.1		1	5.4	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:	\( N \)	\(=\)	\( 1610 = 2 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1610.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more