LMFDB - Embedded newform 8005.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8005,2,Mod(1,8005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8005, 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("8005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8005 = 5 \cdot 1601 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8005.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:	$63.9202468180$
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	$\chi$	$=$	8005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -6.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} +6.00000 q^{22} +2.00000 q^{23} -6.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{27} -2.00000 q^{28} +2.00000 q^{29} +2.00000 q^{30} -8.00000 q^{31} -5.00000 q^{32} +12.0000 q^{33} -6.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} +4.00000 q^{39} +3.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} +8.00000 q^{43} +6.00000 q^{44} +1.00000 q^{45} -2.00000 q^{46} +12.0000 q^{47} +2.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -12.0000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -4.00000 q^{54} -6.00000 q^{55} +6.00000 q^{56} +8.00000 q^{57} -2.00000 q^{58} +10.0000 q^{59} +2.00000 q^{60} -2.00000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} -12.0000 q^{66} -8.00000 q^{67} -6.00000 q^{68} -4.00000 q^{69} -2.00000 q^{70} +10.0000 q^{71} +3.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} -2.00000 q^{75} +4.00000 q^{76} -12.0000 q^{77} -4.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} +6.00000 q^{83} +4.00000 q^{84} +6.00000 q^{85} -8.00000 q^{86} -4.00000 q^{87} -18.0000 q^{88} -2.00000 q^{89} -1.00000 q^{90} -4.00000 q^{91} -2.00000 q^{92} +16.0000 q^{93} -12.0000 q^{94} -4.00000 q^{95} +10.0000 q^{96} +10.0000 q^{97} +3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−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
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	2.00000	0.577350
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−2.00000	−0.534522
$15$	−2.00000	−0.516398
$16$	−1.00000	−0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−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
$21$	−4.00000	−0.872872
$22$	6.00000	1.27920
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	−6.00000	−1.22474
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	4.00000	0.769800
$28$	−2.00000	−0.377964
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	2.00000	0.365148
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−5.00000	−0.883883
$33$	12.0000	2.08893
$34$	−6.00000	−1.02899
$35$	2.00000	0.338062
$36$	−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
$39$	4.00000	0.640513
$40$	3.00000	0.474342
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	4.00000	0.617213
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	6.00000	0.904534
$45$	1.00000	0.149071
$46$	−2.00000	−0.294884
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	2.00000	0.288675
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	−12.0000	−1.68034
$52$	2.00000	0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−4.00000	−0.544331
$55$	−6.00000	−0.809040
$56$	6.00000	0.801784
$57$	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
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	8.00000	1.01600
$63$	2.00000	0.251976
$64$	7.00000	0.875000
$65$	−2.00000	−0.248069
$66$	−12.0000	−1.47710
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−6.00000	−0.727607
$69$	−4.00000	−0.481543
$70$	−2.00000	−0.239046
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	3.00000	0.353553
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−2.00000	−0.232495
$75$	−2.00000	−0.230940
$76$	4.00000	0.458831
$77$	−12.0000	−1.36753
$78$	−4.00000	−0.452911
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	−11.0000	−1.22222
$82$	6.00000	0.662589
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	4.00000	0.436436
$85$	6.00000	0.650791
$86$	−8.00000	−0.862662
$87$	−4.00000	−0.428845
$88$	−18.0000	−1.91881
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	−1.00000	−0.105409
$91$	−4.00000	−0.419314
$92$	−2.00000	−0.208514
$93$	16.0000	1.65912
$94$	−12.0000	−1.23771
$95$	−4.00000	−0.410391
$96$	10.0000	1.02062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	3.00000	0.303046
$99$	−6.00000	−0.603023
$100$	−1.00000	−0.100000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	12.0000	1.18818
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−6.00000	−0.588348
$105$	−4.00000	−0.390360
$106$	6.00000	0.582772
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−4.00000	−0.384900
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	6.00000	0.572078
$111$	−4.00000	−0.379663
$112$	−2.00000	−0.188982
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−8.00000	−0.749269
$115$	2.00000	0.186501
$116$	−2.00000	−0.185695
$117$	−2.00000	−0.184900
$118$	−10.0000	−0.920575
$119$	12.0000	1.10004
$120$	−6.00000	−0.547723
$121$	25.0000	2.27273
$122$	2.00000	0.181071
$123$	12.0000	1.08200
$124$	8.00000	0.718421
$125$	1.00000	0.0894427
$126$	−2.00000	−0.178174
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	3.00000	0.265165
$129$	−16.0000	−1.40872
$130$	2.00000	0.175412
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	−12.0000	−1.04447
$133$	−8.00000	−0.693688
$134$	8.00000	0.691095
$135$	4.00000	0.344265
$136$	18.0000	1.54349
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	4.00000	0.340503
$139$	18.0000	1.52674	0.763370	−	0.645961i	$-0.223543\pi$
$139$	18.0000	1.52674	0.763370	+	0.645961i	$0.223543\pi$
$140$	−2.00000	−0.169031
$141$	−24.0000	−2.02116
$142$	−10.0000	−0.839181
$143$	12.0000	1.00349
$144$	−1.00000	−0.0833333
$145$	2.00000	0.166091
$146$	2.00000	0.165521
$147$	6.00000	0.494872
$148$	−2.00000	−0.164399
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	2.00000	0.163299
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−12.0000	−0.973329
$153$	6.00000	0.485071
$154$	12.0000	0.966988
$155$	−8.00000	−0.642575
$156$	−4.00000	−0.320256
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	8.00000	0.636446
$159$	12.0000	0.951662
$160$	−5.00000	−0.395285
$161$	4.00000	0.315244
$162$	11.0000	0.864242
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	6.00000	0.468521
$165$	12.0000	0.934199
$166$	−6.00000	−0.465690
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−12.0000	−0.925820
$169$	−9.00000	−0.692308
$170$	−6.00000	−0.460179
$171$	−4.00000	−0.305888
$172$	−8.00000	−0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	4.00000	0.303239
$175$	2.00000	0.151186
$176$	6.00000	0.452267
$177$	−20.0000	−1.50329
$178$	2.00000	0.149906
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	−1.00000	−0.0745356
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	4.00000	0.296500
$183$	4.00000	0.295689
$184$	6.00000	0.442326
$185$	2.00000	0.147043
$186$	−16.0000	−1.17318
$187$	−36.0000	−2.63258
$188$	−12.0000	−0.875190
$189$	8.00000	0.581914
$190$	4.00000	0.290191
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	−14.0000	−1.01036
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	−10.0000	−0.717958
$195$	4.00000	0.286446
$196$	3.00000	0.214286
$197$	26.0000	1.85242	0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000	1.85242	0.926212	+	0.377004i	$0.123046\pi$
$198$	6.00000	0.426401
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	3.00000	0.212132
$201$	16.0000	1.12855
$202$	14.0000	0.985037
$203$	4.00000	0.280745
$204$	12.0000	0.840168
$205$	−6.00000	−0.419058
$206$	4.00000	0.278693
$207$	2.00000	0.139010
$208$	2.00000	0.138675
$209$	24.0000	1.66011
$210$	4.00000	0.276026
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	6.00000	0.412082
$213$	−20.0000	−1.37038
$214$	6.00000	0.410152
$215$	8.00000	0.545595
$216$	12.0000	0.816497
$217$	−16.0000	−1.08615
$218$	14.0000	0.948200
$219$	4.00000	0.270295
$220$	6.00000	0.404520
$221$	−12.0000	−0.807207
$222$	4.00000	0.268462
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	−10.0000	−0.668153
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	−8.00000	−0.529813
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	−2.00000	−0.131876
$231$	24.0000	1.57908
$232$	6.00000	0.393919
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	2.00000	0.130744
$235$	12.0000	0.782794
$236$	−10.0000	−0.650945
$237$	16.0000	1.03931
$238$	−12.0000	−0.777844
$239$	22.0000	1.42306	0.711531	−	0.702655i	$-0.248002\pi$
$239$	22.0000	1.42306	0.711531	+	0.702655i	$0.248002\pi$
$240$	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	$0.395619\pi$
$242$	−25.0000	−1.60706
$243$	10.0000	0.641500
$244$	2.00000	0.128037
$245$	−3.00000	−0.191663
$246$	−12.0000	−0.765092
$247$	8.00000	0.509028
$248$	−24.0000	−1.52400
$249$	−12.0000	−0.760469
$250$	−1.00000	−0.0632456
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	−2.00000	−0.125988
$253$	−12.0000	−0.754434
$254$	6.00000	0.376473
$255$	−12.0000	−0.751469
$256$	−17.0000	−1.06250
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	16.0000	0.996116
$259$	4.00000	0.248548
$260$	2.00000	0.124035
$261$	2.00000	0.123797
$262$	10.0000	0.617802
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	36.0000	2.21565
$265$	−6.00000	−0.368577
$266$	8.00000	0.490511
$267$	4.00000	0.244796
$268$	8.00000	0.488678
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	−4.00000	−0.243432
$271$	6.00000	0.364474	0.182237	−	0.983255i	$-0.441666\pi$
$271$	6.00000	0.364474	0.182237	+	0.983255i	$0.441666\pi$
$272$	−6.00000	−0.363803
$273$	8.00000	0.484182
$274$	−14.0000	−0.845771
$275$	−6.00000	−0.361814
$276$	4.00000	0.240772
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−18.0000	−1.07957
$279$	−8.00000	−0.478947
$280$	6.00000	0.358569
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	24.0000	1.42918
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−10.0000	−0.593391
$285$	8.00000	0.473879
$286$	−12.0000	−0.709575
$287$	−12.0000	−0.708338
$288$	−5.00000	−0.294628
$289$	19.0000	1.11765
$290$	−2.00000	−0.117444
$291$	−20.0000	−1.17242
$292$	2.00000	0.117041
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	−6.00000	−0.349927
$295$	10.0000	0.582223
$296$	6.00000	0.348743
$297$	−24.0000	−1.39262
$298$	−10.0000	−0.579284
$299$	−4.00000	−0.231326
$300$	2.00000	0.115470
$301$	16.0000	0.922225
$302$	8.00000	0.460348
$303$	28.0000	1.60856
$304$	4.00000	0.229416
$305$	−2.00000	−0.114520
$306$	−6.00000	−0.342997
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	12.0000	0.683763
$309$	8.00000	0.455104
$310$	8.00000	0.454369
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	12.0000	0.679366
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−18.0000	−1.01580
$315$	2.00000	0.112687
$316$	8.00000	0.450035
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	−12.0000	−0.672927
$319$	−12.0000	−0.671871
$320$	7.00000	0.391312
$321$	12.0000	0.669775
$322$	−4.00000	−0.222911
$323$	−24.0000	−1.33540
$324$	11.0000	0.611111
$325$	−2.00000	−0.110940
$326$	−4.00000	−0.221540
$327$	28.0000	1.54840
$328$	−18.0000	−0.993884
$329$	24.0000	1.32316
$330$	−12.0000	−0.660578
$331$	6.00000	0.329790	0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000	0.329790	0.164895	+	0.986311i	$0.447272\pi$
$332$	−6.00000	−0.329293
$333$	2.00000	0.109599
$334$	0	0
$335$	−8.00000	−0.437087
$336$	4.00000	0.218218
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	9.00000	0.489535
$339$	−12.0000	−0.651751
$340$	−6.00000	−0.325396
$341$	48.0000	2.59935
$342$	4.00000	0.216295
$343$	−20.0000	−1.07990
$344$	24.0000	1.29399
$345$	−4.00000	−0.215353
$346$	−6.00000	−0.322562
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	4.00000	0.214423
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	−2.00000	−0.106904
$351$	−8.00000	−0.427008
$352$	30.0000	1.59901
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	20.0000	1.06299
$355$	10.0000	0.530745
$356$	2.00000	0.106000
$357$	−24.0000	−1.27021
$358$	16.0000	0.845626
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	3.00000	0.158114
$361$	−3.00000	−0.157895
$362$	−10.0000	−0.525588
$363$	−50.0000	−2.62432
$364$	4.00000	0.209657
$365$	−2.00000	−0.104685
$366$	−4.00000	−0.209083
$367$	34.0000	1.77479	0.887393	−	0.461014i	$-0.152514\pi$
$367$	34.0000	1.77479	0.887393	+	0.461014i	$0.152514\pi$
$368$	−2.00000	−0.104257
$369$	−6.00000	−0.312348
$370$	−2.00000	−0.103975
$371$	−12.0000	−0.623009
$372$	−16.0000	−0.829561
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	36.0000	1.86152
$375$	−2.00000	−0.103280
$376$	36.0000	1.85656
$377$	−4.00000	−0.206010
$378$	−8.00000	−0.411476
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	4.00000	0.205196
$381$	12.0000	0.614779
$382$	10.0000	0.511645
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	−6.00000	−0.306186
$385$	−12.0000	−0.611577
$386$	−22.0000	−1.11977
$387$	8.00000	0.406663
$388$	−10.0000	−0.507673
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	−4.00000	−0.202548
$391$	12.0000	0.606866
$392$	−9.00000	−0.454569
$393$	20.0000	1.00887
$394$	−26.0000	−1.30986
$395$	−8.00000	−0.402524
$396$	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	$0.649181\pi$
$398$	−4.00000	−0.200502
$399$	16.0000	0.801002
$400$	−1.00000	−0.0500000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	−16.0000	−0.798007
$403$	16.0000	0.797017
$404$	14.0000	0.696526
$405$	−11.0000	−0.546594
$406$	−4.00000	−0.198517
$407$	−12.0000	−0.594818
$408$	−36.0000	−1.78227
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	6.00000	0.296319
$411$	−28.0000	−1.38114
$412$	4.00000	0.197066
$413$	20.0000	0.984136
$414$	−2.00000	−0.0982946
$415$	6.00000	0.294528
$416$	10.0000	0.490290
$417$	−36.0000	−1.76293
$418$	−24.0000	−1.17388
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	4.00000	0.195180
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	14.0000	0.681509
$423$	12.0000	0.583460
$424$	−18.0000	−0.874157
$425$	6.00000	0.291043
$426$	20.0000	0.969003
$427$	−4.00000	−0.193574
$428$	6.00000	0.290021
$429$	−24.0000	−1.15873
$430$	−8.00000	−0.385794
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	−4.00000	−0.192450
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	16.0000	0.768025
$435$	−4.00000	−0.191785
$436$	14.0000	0.670478
$437$	−8.00000	−0.382692
$438$	−4.00000	−0.191127
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−18.0000	−0.858116
$441$	−3.00000	−0.142857
$442$	12.0000	0.570782
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	4.00000	0.189832
$445$	−2.00000	−0.0948091
$446$	2.00000	0.0947027
$447$	−20.0000	−0.945968
$448$	14.0000	0.661438
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	−1.00000	−0.0471405
$451$	36.0000	1.69517
$452$	−6.00000	−0.282216
$453$	16.0000	0.751746
$454$	24.0000	1.12638
$455$	−4.00000	−0.187523
$456$	24.0000	1.12390
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−10.0000	−0.467269
$459$	24.0000	1.12022
$460$	−2.00000	−0.0932505
$461$	34.0000	1.58354	0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000	1.58354	0.791769	+	0.610821i	$0.209160\pi$
$462$	−24.0000	−1.11658
$463$	18.0000	0.836531	0.418265	−	0.908325i	$-0.362638\pi$
$463$	18.0000	0.836531	0.418265	+	0.908325i	$0.362638\pi$
$464$	−2.00000	−0.0928477
$465$	16.0000	0.741982
$466$	−18.0000	−0.833834
$467$	−30.0000	−1.38823	−0.694117	−	0.719862i	$-0.744205\pi$
$467$	−30.0000	−1.38823	−0.694117	+	0.719862i	$0.744205\pi$
$468$	2.00000	0.0924500
$469$	−16.0000	−0.738811
$470$	−12.0000	−0.553519
$471$	−36.0000	−1.65879
$472$	30.0000	1.38086
$473$	−48.0000	−2.20704
$474$	−16.0000	−0.734904
$475$	−4.00000	−0.183533
$476$	−12.0000	−0.550019
$477$	−6.00000	−0.274721
$478$	−22.0000	−1.00626
$479$	−14.0000	−0.639676	−0.319838	−	0.947472i	$-0.603629\pi$
$479$	−14.0000	−0.639676	−0.319838	+	0.947472i	$0.603629\pi$
$480$	10.0000	0.456435
$481$	−4.00000	−0.182384
$482$	−10.0000	−0.455488
$483$	−8.00000	−0.364013
$484$	−25.0000	−1.13636
$485$	10.0000	0.454077
$486$	−10.0000	−0.453609
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−6.00000	−0.271607
$489$	−8.00000	−0.361773
$490$	3.00000	0.135526
$491$	14.0000	0.631811	0.315906	−	0.948791i	$-0.397692\pi$
$491$	14.0000	0.631811	0.315906	+	0.948791i	$0.397692\pi$
$492$	−12.0000	−0.541002
$493$	12.0000	0.540453
$494$	−8.00000	−0.359937
$495$	−6.00000	−0.269680
$496$	8.00000	0.359211
$497$	20.0000	0.897123
$498$	12.0000	0.537733
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	2.00000	0.0892644
$503$	4.00000	0.178351	0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000	0.178351	0.0891756	+	0.996016i	$0.471577\pi$
$504$	6.00000	0.267261
$505$	−14.0000	−0.622992
$506$	12.0000	0.533465
$507$	18.0000	0.799408
$508$	6.00000	0.266207
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	12.0000	0.531369
$511$	−4.00000	−0.176950
$512$	11.0000	0.486136
$513$	−16.0000	−0.706417
$514$	18.0000	0.793946
$515$	−4.00000	−0.176261
$516$	16.0000	0.704361
$517$	−72.0000	−3.16656
$518$	−4.00000	−0.175750
$519$	−12.0000	−0.526742
$520$	−6.00000	−0.263117
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−2.00000	−0.0875376
$523$	−6.00000	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	10.0000	0.436852
$525$	−4.00000	−0.174574
$526$	12.0000	0.523225
$527$	−48.0000	−2.09091
$528$	−12.0000	−0.522233
$529$	−19.0000	−0.826087
$530$	6.00000	0.260623
$531$	10.0000	0.433963
$532$	8.00000	0.346844
$533$	12.0000	0.519778
$534$	−4.00000	−0.173097
$535$	−6.00000	−0.259403
$536$	−24.0000	−1.03664
$537$	32.0000	1.38090
$538$	−14.0000	−0.603583
$539$	18.0000	0.775315
$540$	−4.00000	−0.172133
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	−6.00000	−0.257722
$543$	−20.0000	−0.858282
$544$	−30.0000	−1.28624
$545$	−14.0000	−0.599694
$546$	−8.00000	−0.342368
$547$	−38.0000	−1.62476	−0.812381	−	0.583127i	$-0.801829\pi$
$547$	−38.0000	−1.62476	−0.812381	+	0.583127i	$0.801829\pi$
$548$	−14.0000	−0.598050
$549$	−2.00000	−0.0853579
$550$	6.00000	0.255841
$551$	−8.00000	−0.340811
$552$	−12.0000	−0.510754
$553$	−16.0000	−0.680389
$554$	10.0000	0.424859
$555$	−4.00000	−0.169791
$556$	−18.0000	−0.763370
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	8.00000	0.338667
$559$	−16.0000	−0.676728
$560$	−2.00000	−0.0845154
$561$	72.0000	3.03984
$562$	−30.0000	−1.26547
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	24.0000	1.01058
$565$	6.00000	0.252422
$566$	−14.0000	−0.588464
$567$	−22.0000	−0.923913
$568$	30.0000	1.25877
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	−8.00000	−0.335083
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	−12.0000	−0.501745
$573$	20.0000	0.835512
$574$	12.0000	0.500870
$575$	2.00000	0.0834058
$576$	7.00000	0.291667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−19.0000	−0.790296
$579$	−44.0000	−1.82858
$580$	−2.00000	−0.0830455
$581$	12.0000	0.497844
$582$	20.0000	0.829027
$583$	36.0000	1.49097
$584$	−6.00000	−0.248282
$585$	−2.00000	−0.0826898
$586$	26.0000	1.07405
$587$	−30.0000	−1.23823	−0.619116	−	0.785299i	$-0.712509\pi$
$587$	−30.0000	−1.23823	−0.619116	+	0.785299i	$0.712509\pi$
$588$	−6.00000	−0.247436
$589$	32.0000	1.31854
$590$	−10.0000	−0.411693
$591$	−52.0000	−2.13899
$592$	−2.00000	−0.0821995
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	24.0000	0.984732
$595$	12.0000	0.491952
$596$	−10.0000	−0.409616
$597$	−8.00000	−0.327418
$598$	4.00000	0.163572
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	−6.00000	−0.244949
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	−16.0000	−0.652111
$603$	−8.00000	−0.325785
$604$	8.00000	0.325515
$605$	25.0000	1.01639
$606$	−28.0000	−1.13742
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	20.0000	0.811107
$609$	−8.00000	−0.324176
$610$	2.00000	0.0809776
$611$	−24.0000	−0.970936
$612$	−6.00000	−0.242536
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	32.0000	1.29141
$615$	12.0000	0.483887
$616$	−36.0000	−1.45048
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−8.00000	−0.321807
$619$	−42.0000	−1.68812	−0.844061	−	0.536247i	$-0.819842\pi$
$619$	−42.0000	−1.68812	−0.844061	+	0.536247i	$0.819842\pi$
$620$	8.00000	0.321288
$621$	8.00000	0.321029
$622$	10.0000	0.400963
$623$	−4.00000	−0.160257
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−6.00000	−0.239808
$627$	−48.0000	−1.91694
$628$	−18.0000	−0.718278
$629$	12.0000	0.478471
$630$	−2.00000	−0.0796819
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	−24.0000	−0.954669
$633$	28.0000	1.11290
$634$	2.00000	0.0794301
$635$	−6.00000	−0.238103
$636$	−12.0000	−0.475831
$637$	6.00000	0.237729
$638$	12.0000	0.475085
$639$	10.0000	0.395594
$640$	3.00000	0.118585
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	−12.0000	−0.473602
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	−4.00000	−0.157622
$645$	−16.0000	−0.629999
$646$	24.0000	0.944267
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	−33.0000	−1.29636
$649$	−60.0000	−2.35521
$650$	2.00000	0.0784465
$651$	32.0000	1.25418
$652$	−4.00000	−0.156652
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	−28.0000	−1.09489
$655$	−10.0000	−0.390732
$656$	6.00000	0.234261
$657$	−2.00000	−0.0780274
$658$	−24.0000	−0.935617
$659$	−14.0000	−0.545363	−0.272681	−	0.962104i	$-0.587910\pi$
$659$	−14.0000	−0.545363	−0.272681	+	0.962104i	$0.587910\pi$
$660$	−12.0000	−0.467099
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	−6.00000	−0.233197
$663$	24.0000	0.932083
$664$	18.0000	0.698535
$665$	−8.00000	−0.310227
$666$	−2.00000	−0.0774984
$667$	4.00000	0.154881
$668$	0	0
$669$	4.00000	0.154649
$670$	8.00000	0.309067
$671$	12.0000	0.463255
$672$	20.0000	0.771517
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	−2.00000	−0.0770371
$675$	4.00000	0.153960
$676$	9.00000	0.346154
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	12.0000	0.460857
$679$	20.0000	0.767530
$680$	18.0000	0.690268
$681$	48.0000	1.83936
$682$	−48.0000	−1.83801
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	4.00000	0.152944
$685$	14.0000	0.534913
$686$	20.0000	0.763604
$687$	−20.0000	−0.763048
$688$	−8.00000	−0.304997
$689$	12.0000	0.457164
$690$	4.00000	0.152277
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	−6.00000	−0.228086
$693$	−12.0000	−0.455842
$694$	8.00000	0.303676
$695$	18.0000	0.682779
$696$	−12.0000	−0.454859
$697$	−36.0000	−1.36360
$698$	−22.0000	−0.832712
$699$	−36.0000	−1.36165
$700$	−2.00000	−0.0755929
$701$	−46.0000	−1.73740	−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−46.0000	−1.73740	−0.868698	+	0.495342i	$0.835043\pi$
$702$	8.00000	0.301941
$703$	−8.00000	−0.301726
$704$	−42.0000	−1.58293
$705$	−24.0000	−0.903892
$706$	−10.0000	−0.376355
$707$	−28.0000	−1.05305
$708$	20.0000	0.751646
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	−10.0000	−0.375293
$711$	−8.00000	−0.300023
$712$	−6.00000	−0.224860
$713$	−16.0000	−0.599205
$714$	24.0000	0.898177
$715$	12.0000	0.448775
$716$	16.0000	0.597948
$717$	−44.0000	−1.64321
$718$	36.0000	1.34351
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	−1.00000	−0.0372678
$721$	−8.00000	−0.297936
$722$	3.00000	0.111648
$723$	−20.0000	−0.743808
$724$	−10.0000	−0.371647
$725$	2.00000	0.0742781
$726$	50.0000	1.85567
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	−12.0000	−0.444750
$729$	13.0000	0.481481
$730$	2.00000	0.0740233
$731$	48.0000	1.77534
$732$	−4.00000	−0.147844
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	−34.0000	−1.25496
$735$	6.00000	0.221313
$736$	−10.0000	−0.368605
$737$	48.0000	1.76810
$738$	6.00000	0.220863
$739$	14.0000	0.514998	0.257499	−	0.966279i	$-0.417102\pi$
$739$	14.0000	0.514998	0.257499	+	0.966279i	$0.417102\pi$
$740$	−2.00000	−0.0735215
$741$	−16.0000	−0.587775
$742$	12.0000	0.440534
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	48.0000	1.75977
$745$	10.0000	0.366372
$746$	−2.00000	−0.0732252
$747$	6.00000	0.219529
$748$	36.0000	1.31629
$749$	−12.0000	−0.438470
$750$	2.00000	0.0730297
$751$	38.0000	1.38664	0.693320	−	0.720630i	$-0.256147\pi$
$751$	38.0000	1.38664	0.693320	+	0.720630i	$0.256147\pi$
$752$	−12.0000	−0.437595
$753$	4.00000	0.145768
$754$	4.00000	0.145671
$755$	−8.00000	−0.291150
$756$	−8.00000	−0.290957
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	34.0000	1.23494
$759$	24.0000	0.871145
$760$	−12.0000	−0.435286
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	−12.0000	−0.434714
$763$	−28.0000	−1.01367
$764$	10.0000	0.361787
$765$	6.00000	0.216930
$766$	16.0000	0.578103
$767$	−20.0000	−0.722158
$768$	34.0000	1.22687
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	12.0000	0.432450
$771$	36.0000	1.29651
$772$	−22.0000	−0.791797
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−8.00000	−0.287554
$775$	−8.00000	−0.287368
$776$	30.0000	1.07694
$777$	−8.00000	−0.286998
$778$	2.00000	0.0717035
$779$	24.0000	0.859889
$780$	−4.00000	−0.143223
$781$	−60.0000	−2.14697
$782$	−12.0000	−0.429119
$783$	8.00000	0.285897
$784$	3.00000	0.107143
$785$	18.0000	0.642448
$786$	−20.0000	−0.713376
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	−26.0000	−0.926212
$789$	24.0000	0.854423
$790$	8.00000	0.284627
$791$	12.0000	0.426671
$792$	−18.0000	−0.639602
$793$	4.00000	0.142044
$794$	18.0000	0.638796
$795$	12.0000	0.425596
$796$	−4.00000	−0.141776
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	−16.0000	−0.566394
$799$	72.0000	2.54718
$800$	−5.00000	−0.176777
$801$	−2.00000	−0.0706665
$802$	30.0000	1.05934
$803$	12.0000	0.423471
$804$	−16.0000	−0.564276
$805$	4.00000	0.140981
$806$	−16.0000	−0.563576
$807$	−28.0000	−0.985647
$808$	−42.0000	−1.47755
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	11.0000	0.386501
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	−4.00000	−0.140372
$813$	−12.0000	−0.420858
$814$	12.0000	0.420600
$815$	4.00000	0.140114
$816$	12.0000	0.420084
$817$	−32.0000	−1.11954
$818$	−2.00000	−0.0699284
$819$	−4.00000	−0.139771
$820$	6.00000	0.209529
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	28.0000	0.976612
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−12.0000	−0.418040
$825$	12.0000	0.417786
$826$	−20.0000	−0.695889
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	−2.00000	−0.0695048
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	−6.00000	−0.208263
$831$	20.0000	0.693792
$832$	−14.0000	−0.485363
$833$	−18.0000	−0.623663
$834$	36.0000	1.24658
$835$	0	0
$836$	−24.0000	−0.830057
$837$	−32.0000	−1.10608
$838$	6.00000	0.207267
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	−12.0000	−0.414039
$841$	−25.0000	−0.862069
$842$	−18.0000	−0.620321
$843$	−60.0000	−2.06651
$844$	14.0000	0.481900
$845$	−9.00000	−0.309609
$846$	−12.0000	−0.412568
$847$	50.0000	1.71802
$848$	6.00000	0.206041
$849$	−28.0000	−0.960958
$850$	−6.00000	−0.205798
$851$	4.00000	0.137118
$852$	20.0000	0.685189
$853$	42.0000	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	4.00000	0.136877
$855$	−4.00000	−0.136797
$856$	−18.0000	−0.615227
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	24.0000	0.819346
$859$	22.0000	0.750630	0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000	0.750630	0.375315	+	0.926897i	$0.377534\pi$
$860$	−8.00000	−0.272798
$861$	24.0000	0.817918
$862$	18.0000	0.613082
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−20.0000	−0.680414
$865$	6.00000	0.204006
$866$	26.0000	0.883516
$867$	−38.0000	−1.29055
$868$	16.0000	0.543075
$869$	48.0000	1.62829
$870$	4.00000	0.135613
$871$	16.0000	0.542139
$872$	−42.0000	−1.42230
$873$	10.0000	0.338449
$874$	8.00000	0.270604
$875$	2.00000	0.0676123
$876$	−4.00000	−0.135147
$877$	−58.0000	−1.95852	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−58.0000	−1.95852	−0.979260	+	0.202606i	$0.935059\pi$
$878$	−8.00000	−0.269987
$879$	52.0000	1.75392
$880$	6.00000	0.202260
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	3.00000	0.101015
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	12.0000	0.403604
$885$	−20.0000	−0.672293
$886$	−16.0000	−0.537531
$887$	58.0000	1.94745	0.973725	−	0.227728i	$-0.0731298\pi$
$887$	58.0000	1.94745	0.973725	+	0.227728i	$0.0731298\pi$
$888$	−12.0000	−0.402694
$889$	−12.0000	−0.402467
$890$	2.00000	0.0670402
$891$	66.0000	2.21108
$892$	2.00000	0.0669650
$893$	−48.0000	−1.60626
$894$	20.0000	0.668900
$895$	−16.0000	−0.534821
$896$	6.00000	0.200446
$897$	8.00000	0.267112
$898$	−22.0000	−0.734150
$899$	−16.0000	−0.533630
$900$	−1.00000	−0.0333333
$901$	−36.0000	−1.19933
$902$	−36.0000	−1.19867
$903$	−32.0000	−1.06489
$904$	18.0000	0.598671
$905$	10.0000	0.332411
$906$	−16.0000	−0.531564
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	24.0000	0.796468
$909$	−14.0000	−0.464351
$910$	4.00000	0.132599
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	−8.00000	−0.264906
$913$	−36.0000	−1.19143
$914$	−22.0000	−0.727695
$915$	4.00000	0.132236
$916$	−10.0000	−0.330409
$917$	−20.0000	−0.660458
$918$	−24.0000	−0.792118
$919$	−46.0000	−1.51740	−0.758700	−	0.651440i	$-0.774165\pi$
$919$	−46.0000	−1.51740	−0.758700	+	0.651440i	$0.774165\pi$
$920$	6.00000	0.197814
$921$	64.0000	2.10887
$922$	−34.0000	−1.11973
$923$	−20.0000	−0.658308
$924$	−24.0000	−0.789542
$925$	2.00000	0.0657596
$926$	−18.0000	−0.591517
$927$	−4.00000	−0.131377
$928$	−10.0000	−0.328266
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	−16.0000	−0.524661
$931$	12.0000	0.393284
$932$	−18.0000	−0.589610
$933$	20.0000	0.654771
$934$	30.0000	0.981630
$935$	−36.0000	−1.17733
$936$	−6.00000	−0.196116
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	16.0000	0.522419
$939$	−12.0000	−0.391605
$940$	−12.0000	−0.391397
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	36.0000	1.17294
$943$	−12.0000	−0.390774
$944$	−10.0000	−0.325472
$945$	8.00000	0.260240
$946$	48.0000	1.56061
$947$	−38.0000	−1.23483	−0.617417	−	0.786636i	$-0.711821\pi$
$947$	−38.0000	−1.23483	−0.617417	+	0.786636i	$0.711821\pi$
$948$	−16.0000	−0.519656
$949$	4.00000	0.129845
$950$	4.00000	0.129777
$951$	4.00000	0.129709
$952$	36.0000	1.16677
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	6.00000	0.194257
$955$	−10.0000	−0.323592
$956$	−22.0000	−0.711531
$957$	24.0000	0.775810
$958$	14.0000	0.452319
$959$	28.0000	0.904167
$960$	−14.0000	−0.451848
$961$	33.0000	1.06452
$962$	4.00000	0.128965
$963$	−6.00000	−0.193347
$964$	−10.0000	−0.322078
$965$	22.0000	0.708205
$966$	8.00000	0.257396
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	75.0000	2.41059
$969$	48.0000	1.54198
$970$	−10.0000	−0.321081
$971$	−38.0000	−1.21948	−0.609739	−	0.792602i	$-0.708726\pi$
$971$	−38.0000	−1.21948	−0.609739	+	0.792602i	$0.708726\pi$
$972$	−10.0000	−0.320750
$973$	36.0000	1.15411
$974$	4.00000	0.128168
$975$	4.00000	0.128103
$976$	2.00000	0.0640184
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	8.00000	0.255812
$979$	12.0000	0.383522
$980$	3.00000	0.0958315
$981$	−14.0000	−0.446986
$982$	−14.0000	−0.446758
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	36.0000	1.14764
$985$	26.0000	0.828429
$986$	−12.0000	−0.382158
$987$	−48.0000	−1.52786
$988$	−8.00000	−0.254514
$989$	16.0000	0.508770
$990$	6.00000	0.190693
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	40.0000	1.27000
$993$	−12.0000	−0.380808
$994$	−20.0000	−0.634361
$995$	4.00000	0.126809
$996$	12.0000	0.380235
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	20.0000	0.633089
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8005.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8005.2.a.a.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8005 = 5 \cdot 1601 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8005.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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