LMFDB - Embedded newform 285.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [285,2,Mod(1,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, 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("285.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 285 = 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	285.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:	$2.27573645761$
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	$\chi$	$=$	285.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} +4.00000 q^{22} -4.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -2.00000 q^{29} -1.00000 q^{30} +5.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +4.00000 q^{35} -1.00000 q^{36} -6.00000 q^{37} -1.00000 q^{38} -2.00000 q^{39} -3.00000 q^{40} -6.00000 q^{41} -4.00000 q^{42} +8.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} -14.0000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -12.0000 q^{56} +1.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} +1.00000 q^{60} +14.0000 q^{61} +4.00000 q^{63} +7.00000 q^{64} +2.00000 q^{65} -4.00000 q^{66} -4.00000 q^{67} -2.00000 q^{68} +4.00000 q^{69} +4.00000 q^{70} -3.00000 q^{72} -14.0000 q^{73} -6.00000 q^{74} -1.00000 q^{75} +1.00000 q^{76} +16.0000 q^{77} -2.00000 q^{78} +16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{84} +2.00000 q^{85} +8.00000 q^{86} +2.00000 q^{87} -12.0000 q^{88} -6.00000 q^{89} +1.00000 q^{90} +8.00000 q^{91} +4.00000 q^{92} -12.0000 q^{94} -1.00000 q^{95} -5.00000 q^{96} -10.0000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−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	$0.227186\pi$
$8$	−3.00000	−1.06066
$9$	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
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−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	$0.636929\pi$
$24$	3.00000	0.612372
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	−4.00000	−0.755929
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	−1.00000	−0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	5.00000	0.883883
$33$	−4.00000	−0.696311
$34$	2.00000	0.342997
$35$	4.00000	0.676123
$36$	−1.00000	−0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−1.00000	−0.162221
$39$	−2.00000	−0.320256
$40$	−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	$0.291172\pi$
$44$	−4.00000	−0.603023
$45$	1.00000	0.149071
$46$	−4.00000	−0.589768
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	−2.00000	−0.280056
$52$	−2.00000	−0.277350
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	−1.00000	−0.136083
$55$	4.00000	0.539360
$56$	−12.0000	−1.60357
$57$	1.00000	0.132453
$58$	−2.00000	−0.262613
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	1.00000	0.129099
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	7.00000	0.875000
$65$	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
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−3.00000	−0.353553
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	−6.00000	−0.697486
$75$	−1.00000	−0.115470
$76$	1.00000	0.114708
$77$	16.0000	1.82337
$78$	−2.00000	−0.226455
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	4.00000	0.436436
$85$	2.00000	0.216930
$86$	8.00000	0.862662
$87$	2.00000	0.214423
$88$	−12.0000	−1.27920
$89$	−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
$91$	8.00000	0.838628
$92$	4.00000	0.417029
$93$	0	0
$94$	−12.0000	−1.23771
$95$	−1.00000	−0.102598
$96$	−5.00000	−0.510310
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	9.00000	0.909137
$99$	4.00000	0.402015
$100$	−1.00000	−0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−2.00000	−0.198030
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−6.00000	−0.588348
$105$	−4.00000	−0.390360
$106$	−14.0000	−1.35980
$107$	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
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	4.00000	0.381385
$111$	6.00000	0.569495
$112$	−4.00000	−0.377964
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	1.00000	0.0936586
$115$	−4.00000	−0.373002
$116$	2.00000	0.185695
$117$	2.00000	0.184900
$118$	4.00000	0.368230
$119$	8.00000	0.733359
$120$	3.00000	0.273861
$121$	5.00000	0.454545
$122$	14.0000	1.26750
$123$	6.00000	0.541002
$124$	0	0
$125$	1.00000	0.0894427
$126$	4.00000	0.356348
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−3.00000	−0.265165
$129$	−8.00000	−0.704361
$130$	2.00000	0.175412
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	4.00000	0.348155
$133$	−4.00000	−0.346844
$134$	−4.00000	−0.345547
$135$	−1.00000	−0.0860663
$136$	−6.00000	−0.514496
$137$	−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
$141$	12.0000	1.01058
$142$	0	0
$143$	8.00000	0.668994
$144$	−1.00000	−0.0833333
$145$	−2.00000	−0.166091
$146$	−14.0000	−1.15865
$147$	−9.00000	−0.742307
$148$	6.00000	0.493197
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−1.00000	−0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	3.00000	0.243332
$153$	2.00000	0.161690
$154$	16.0000	1.28932
$155$	0	0
$156$	2.00000	0.160128
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	16.0000	1.27289
$159$	14.0000	1.11027
$160$	5.00000	0.395285
$161$	−16.0000	−1.26098
$162$	1.00000	0.0785674
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	6.00000	0.468521
$165$	−4.00000	−0.311400
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	12.0000	0.925820
$169$	−9.00000	−0.692308
$170$	2.00000	0.153393
$171$	−1.00000	−0.0764719
$172$	−8.00000	−0.609994
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	2.00000	0.151620
$175$	4.00000	0.302372
$176$	−4.00000	−0.301511
$177$	−4.00000	−0.300658
$178$	−6.00000	−0.449719
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	−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	$0.195291\pi$
$182$	8.00000	0.592999
$183$	−14.0000	−1.03491
$184$	12.0000	0.884652
$185$	−6.00000	−0.441129
$186$	0	0
$187$	8.00000	0.585018
$188$	12.0000	0.875190
$189$	−4.00000	−0.290957
$190$	−1.00000	−0.0725476
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−7.00000	−0.505181
$193$	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
$195$	−2.00000	−0.143223
$196$	−9.00000	−0.642857
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	4.00000	0.284268
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−3.00000	−0.212132
$201$	4.00000	0.282138
$202$	−10.0000	−0.703598
$203$	−8.00000	−0.561490
$204$	2.00000	0.140028
$205$	−6.00000	−0.419058
$206$	16.0000	1.11477
$207$	−4.00000	−0.278019
$208$	−2.00000	−0.138675
$209$	−4.00000	−0.276686
$210$	−4.00000	−0.276026
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	14.0000	0.961524
$213$	0	0
$214$	12.0000	0.820303
$215$	8.00000	0.545595
$216$	3.00000	0.204124
$217$	0	0
$218$	−2.00000	−0.135457
$219$	14.0000	0.946032
$220$	−4.00000	−0.269680
$221$	4.00000	0.269069
$222$	6.00000	0.402694
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	20.0000	1.33631
$225$	1.00000	0.0666667
$226$	−10.0000	−0.665190
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−1.00000	−0.0662266
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	−4.00000	−0.263752
$231$	−16.0000	−1.05272
$232$	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
$236$	−4.00000	−0.260378
$237$	−16.0000	−1.03931
$238$	8.00000	0.518563
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	1.00000	0.0645497
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	−14.0000	−0.896258
$245$	9.00000	0.574989
$246$	6.00000	0.382546
$247$	−2.00000	−0.127257
$248$	0	0
$249$	0	0
$250$	1.00000	0.0632456
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	−4.00000	−0.251976
$253$	−16.0000	−1.00591
$254$	−8.00000	−0.501965
$255$	−2.00000	−0.125245
$256$	−17.0000	−1.06250
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	−8.00000	−0.498058
$259$	−24.0000	−1.49129
$260$	−2.00000	−0.124035
$261$	−2.00000	−0.123797
$262$	4.00000	0.247121
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	12.0000	0.738549
$265$	−14.0000	−0.860013
$266$	−4.00000	−0.245256
$267$	6.00000	0.367194
$268$	4.00000	0.244339
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	−1.00000	−0.0608581
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	−2.00000	−0.121268
$273$	−8.00000	−0.484182
$274$	−6.00000	−0.362473
$275$	4.00000	0.241209
$276$	−4.00000	−0.240772
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	−12.0000	−0.717137
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	12.0000	0.714590
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	8.00000	0.473050
$287$	−24.0000	−1.41668
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	−2.00000	−0.117444
$291$	10.0000	0.586210
$292$	14.0000	0.819288
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	−9.00000	−0.524891
$295$	4.00000	0.232889
$296$	18.0000	1.04623
$297$	−4.00000	−0.232104
$298$	6.00000	0.347571
$299$	−8.00000	−0.462652
$300$	1.00000	0.0577350
$301$	32.0000	1.84445
$302$	0	0
$303$	10.0000	0.574485
$304$	1.00000	0.0573539
$305$	14.0000	0.801638
$306$	2.00000	0.114332
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	−16.0000	−0.911685
$309$	−16.0000	−0.910208
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	6.00000	0.339683
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−18.0000	−1.01580
$315$	4.00000	0.225374
$316$	−16.0000	−0.900070
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	14.0000	0.785081
$319$	−8.00000	−0.447914
$320$	7.00000	0.391312
$321$	−12.0000	−0.669775
$322$	−16.0000	−0.891645
$323$	−2.00000	−0.111283
$324$	−1.00000	−0.0555556
$325$	2.00000	0.110940
$326$	0	0
$327$	2.00000	0.110600
$328$	18.0000	0.993884
$329$	−48.0000	−2.64633
$330$	−4.00000	−0.220193
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	−8.00000	−0.437741
$335$	−4.00000	−0.218543
$336$	4.00000	0.218218
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−9.00000	−0.489535
$339$	10.0000	0.543125
$340$	−2.00000	−0.108465
$341$	0	0
$342$	−1.00000	−0.0540738
$343$	8.00000	0.431959
$344$	−24.0000	−1.29399
$345$	4.00000	0.215353
$346$	−6.00000	−0.322562
$347$	32.0000	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$348$	−2.00000	−0.107211
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	4.00000	0.213809
$351$	−2.00000	−0.106752
$352$	20.0000	1.06600
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	6.00000	0.317999
$357$	−8.00000	−0.423405
$358$	12.0000	0.634220
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	−3.00000	−0.158114
$361$	1.00000	0.0526316
$362$	22.0000	1.15629
$363$	−5.00000	−0.262432
$364$	−8.00000	−0.419314
$365$	−14.0000	−0.732793
$366$	−14.0000	−0.731792
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	4.00000	0.208514
$369$	−6.00000	−0.312348
$370$	−6.00000	−0.311925
$371$	−56.0000	−2.90738
$372$	0	0
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	8.00000	0.413670
$375$	−1.00000	−0.0516398
$376$	36.0000	1.85656
$377$	−4.00000	−0.206010
$378$	−4.00000	−0.205738
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	1.00000	0.0512989
$381$	8.00000	0.409852
$382$	0	0
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	3.00000	0.153093
$385$	16.0000	0.815436
$386$	22.0000	1.11977
$387$	8.00000	0.406663
$388$	10.0000	0.507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−2.00000	−0.101274
$391$	−8.00000	−0.404577
$392$	−27.0000	−1.36371
$393$	−4.00000	−0.201773
$394$	22.0000	1.10834
$395$	16.0000	0.805047
$396$	−4.00000	−0.201008
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	−1.00000	−0.0500000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	10.0000	0.497519
$405$	1.00000	0.0496904
$406$	−8.00000	−0.397033
$407$	−24.0000	−1.18964
$408$	6.00000	0.297044
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	−6.00000	−0.296319
$411$	6.00000	0.295958
$412$	−16.0000	−0.788263
$413$	16.0000	0.787309
$414$	−4.00000	−0.196589
$415$	0	0
$416$	10.0000	0.490290
$417$	4.00000	0.195881
$418$	−4.00000	−0.195646
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	4.00000	0.195180
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	12.0000	0.584151
$423$	−12.0000	−0.583460
$424$	42.0000	2.03970
$425$	2.00000	0.0970143
$426$	0	0
$427$	56.0000	2.71003
$428$	−12.0000	−0.580042
$429$	−8.00000	−0.386244
$430$	8.00000	0.385794
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	1.00000	0.0481125
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	2.00000	0.0957826
$437$	4.00000	0.191346
$438$	14.0000	0.668946
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−12.0000	−0.572078
$441$	9.00000	0.428571
$442$	4.00000	0.190261
$443$	40.0000	1.90046	0.950229	−	0.311553i	$-0.100849\pi$
$443$	40.0000	1.90046	0.950229	+	0.311553i	$0.100849\pi$
$444$	−6.00000	−0.284747
$445$	−6.00000	−0.284427
$446$	−24.0000	−1.13643
$447$	−6.00000	−0.283790
$448$	28.0000	1.32288
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	1.00000	0.0471405
$451$	−24.0000	−1.13012
$452$	10.0000	0.470360
$453$	0	0
$454$	−12.0000	−0.563188
$455$	8.00000	0.375046
$456$	−3.00000	−0.140488
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−26.0000	−1.21490
$459$	−2.00000	−0.0933520
$460$	4.00000	0.186501
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	−16.0000	−0.744387
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	18.0000	0.833834
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	−2.00000	−0.0924500
$469$	−16.0000	−0.738811
$470$	−12.0000	−0.553519
$471$	18.0000	0.829396
$472$	−12.0000	−0.552345
$473$	32.0000	1.47136
$474$	−16.0000	−0.734904
$475$	−1.00000	−0.0458831
$476$	−8.00000	−0.366679
$477$	−14.0000	−0.641016
$478$	−24.0000	−1.09773
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	−5.00000	−0.228218
$481$	−12.0000	−0.547153
$482$	−22.0000	−1.00207
$483$	16.0000	0.728025
$484$	−5.00000	−0.227273
$485$	−10.0000	−0.454077
$486$	−1.00000	−0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−42.0000	−1.90125
$489$	0	0
$490$	9.00000	0.406579
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−6.00000	−0.270501
$493$	−4.00000	−0.180151
$494$	−2.00000	−0.0899843
$495$	4.00000	0.179787
$496$	0	0
$497$	0	0
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−1.00000	−0.0447214
$501$	8.00000	0.357414
$502$	28.0000	1.24970
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	−12.0000	−0.534522
$505$	−10.0000	−0.444994
$506$	−16.0000	−0.711287
$507$	9.00000	0.399704
$508$	8.00000	0.354943
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	−2.00000	−0.0885615
$511$	−56.0000	−2.47729
$512$	−11.0000	−0.486136
$513$	1.00000	0.0441511
$514$	−2.00000	−0.0882162
$515$	16.0000	0.705044
$516$	8.00000	0.352180
$517$	−48.0000	−2.11104
$518$	−24.0000	−1.05450
$519$	6.00000	0.263371
$520$	−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
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−4.00000	−0.174741
$525$	−4.00000	−0.174574
$526$	−4.00000	−0.174408
$527$	0	0
$528$	4.00000	0.174078
$529$	−7.00000	−0.304348
$530$	−14.0000	−0.608121
$531$	4.00000	0.173585
$532$	4.00000	0.173422
$533$	−12.0000	−0.519778
$534$	6.00000	0.259645
$535$	12.0000	0.518805
$536$	12.0000	0.518321
$537$	−12.0000	−0.517838
$538$	−26.0000	−1.12094
$539$	36.0000	1.55063
$540$	1.00000	0.0430331
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	24.0000	1.03089
$543$	−22.0000	−0.944110
$544$	10.0000	0.428746
$545$	−2.00000	−0.0856706
$546$	−8.00000	−0.342368
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	6.00000	0.256307
$549$	14.0000	0.597505
$550$	4.00000	0.170561
$551$	2.00000	0.0852029
$552$	−12.0000	−0.510754
$553$	64.0000	2.72156
$554$	22.0000	0.934690
$555$	6.00000	0.254686
$556$	4.00000	0.169638
$557$	−26.0000	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	−4.00000	−0.169031
$561$	−8.00000	−0.337760
$562$	10.0000	0.421825
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	−12.0000	−0.505291
$565$	−10.0000	−0.420703
$566$	−16.0000	−0.672530
$567$	4.00000	0.167984
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	1.00000	0.0418854
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	−8.00000	−0.334497
$573$	0	0
$574$	−24.0000	−1.00174
$575$	−4.00000	−0.166812
$576$	7.00000	0.291667
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	−13.0000	−0.540729
$579$	−22.0000	−0.914289
$580$	2.00000	0.0830455
$581$	0	0
$582$	10.0000	0.414513
$583$	−56.0000	−2.31928
$584$	42.0000	1.73797
$585$	2.00000	0.0826898
$586$	2.00000	0.0826192
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	9.00000	0.371154
$589$	0	0
$590$	4.00000	0.164677
$591$	−22.0000	−0.904959
$592$	6.00000	0.246598
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	−4.00000	−0.164122
$595$	8.00000	0.327968
$596$	−6.00000	−0.245770
$597$	0	0
$598$	−8.00000	−0.327144
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	3.00000	0.122474
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	32.0000	1.30422
$603$	−4.00000	−0.162893
$604$	0	0
$605$	5.00000	0.203279
$606$	10.0000	0.406222
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	−5.00000	−0.202777
$609$	8.00000	0.324176
$610$	14.0000	0.566843
$611$	−24.0000	−0.970936
$612$	−2.00000	−0.0808452
$613$	−42.0000	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$614$	−12.0000	−0.484281
$615$	6.00000	0.241943
$616$	−48.0000	−1.93398
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	−16.0000	−0.643614
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	−24.0000	−0.961540
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	10.0000	0.399680
$627$	4.00000	0.159745
$628$	18.0000	0.718278
$629$	−12.0000	−0.478471
$630$	4.00000	0.159364
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−48.0000	−1.90934
$633$	−12.0000	−0.476957
$634$	2.00000	0.0794301
$635$	−8.00000	−0.317470
$636$	−14.0000	−0.555136
$637$	18.0000	0.713186
$638$	−8.00000	−0.316723
$639$	0	0
$640$	−3.00000	−0.118585
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−12.0000	−0.473602
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	16.0000	0.630488
$645$	−8.00000	−0.315000
$646$	−2.00000	−0.0786889
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	−3.00000	−0.117851
$649$	16.0000	0.628055
$650$	2.00000	0.0784465
$651$	0	0
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	2.00000	0.0782062
$655$	4.00000	0.156293
$656$	6.00000	0.234261
$657$	−14.0000	−0.546192
$658$	−48.0000	−1.87123
$659$	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
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	−28.0000	−1.08825
$663$	−4.00000	−0.155347
$664$	0	0
$665$	−4.00000	−0.155113
$666$	−6.00000	−0.232495
$667$	8.00000	0.309761
$668$	8.00000	0.309529
$669$	24.0000	0.927894
$670$	−4.00000	−0.154533
$671$	56.0000	2.16186
$672$	−20.0000	−0.771517
$673$	30.0000	1.15642	0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000	1.15642	0.578208	+	0.815890i	$0.303752\pi$
$674$	22.0000	0.847408
$675$	−1.00000	−0.0384900
$676$	9.00000	0.346154
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	10.0000	0.384048
$679$	−40.0000	−1.53506
$680$	−6.00000	−0.230089
$681$	12.0000	0.459841
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	1.00000	0.0382360
$685$	−6.00000	−0.229248
$686$	8.00000	0.305441
$687$	26.0000	0.991962
$688$	−8.00000	−0.304997
$689$	−28.0000	−1.06672
$690$	4.00000	0.152277
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	6.00000	0.228086
$693$	16.0000	0.607790
$694$	32.0000	1.21470
$695$	−4.00000	−0.151729
$696$	−6.00000	−0.227429
$697$	−12.0000	−0.454532
$698$	30.0000	1.13552
$699$	−18.0000	−0.680823
$700$	−4.00000	−0.151186
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	−2.00000	−0.0754851
$703$	6.00000	0.226294
$704$	28.0000	1.05529
$705$	12.0000	0.451946
$706$	−14.0000	−0.526897
$707$	−40.0000	−1.50435
$708$	4.00000	0.150329
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	18.0000	0.674579
$713$	0	0
$714$	−8.00000	−0.299392
$715$	8.00000	0.299183
$716$	−12.0000	−0.448461
$717$	24.0000	0.896296
$718$	−8.00000	−0.298557
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	−1.00000	−0.0372678
$721$	64.0000	2.38348
$722$	1.00000	0.0372161
$723$	22.0000	0.818189
$724$	−22.0000	−0.817624
$725$	−2.00000	−0.0742781
$726$	−5.00000	−0.185567
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	−24.0000	−0.889499
$729$	1.00000	0.0370370
$730$	−14.0000	−0.518163
$731$	16.0000	0.591781
$732$	14.0000	0.517455
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	−20.0000	−0.738213
$735$	−9.00000	−0.331970
$736$	−20.0000	−0.737210
$737$	−16.0000	−0.589368
$738$	−6.00000	−0.220863
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	6.00000	0.220564
$741$	2.00000	0.0734718
$742$	−56.0000	−2.05582
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	34.0000	1.24483
$747$	0	0
$748$	−8.00000	−0.292509
$749$	48.0000	1.75388
$750$	−1.00000	−0.0365148
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	12.0000	0.437595
$753$	−28.0000	−1.02038
$754$	−4.00000	−0.145671
$755$	0	0
$756$	4.00000	0.145479
$757$	−50.0000	−1.81728	−0.908640	−	0.417579i	$-0.862879\pi$
$757$	−50.0000	−1.81728	−0.908640	+	0.417579i	$0.862879\pi$
$758$	4.00000	0.145287
$759$	16.0000	0.580763
$760$	3.00000	0.108821
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	8.00000	0.289809
$763$	−8.00000	−0.289619
$764$	0	0
$765$	2.00000	0.0723102
$766$	32.0000	1.15621
$767$	8.00000	0.288863
$768$	17.0000	0.613435
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	16.0000	0.576600
$771$	2.00000	0.0720282
$772$	−22.0000	−0.791797
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	30.0000	1.07694
$777$	24.0000	0.860995
$778$	6.00000	0.215110
$779$	6.00000	0.214972
$780$	2.00000	0.0716115
$781$	0	0
$782$	−8.00000	−0.286079
$783$	2.00000	0.0714742
$784$	−9.00000	−0.321429
$785$	−18.0000	−0.642448
$786$	−4.00000	−0.142675
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	−22.0000	−0.783718
$789$	4.00000	0.142404
$790$	16.0000	0.569254
$791$	−40.0000	−1.42224
$792$	−12.0000	−0.426401
$793$	28.0000	0.994309
$794$	−2.00000	−0.0709773
$795$	14.0000	0.496529
$796$	0	0
$797$	−38.0000	−1.34603	−0.673015	−	0.739629i	$-0.735001\pi$
$797$	−38.0000	−1.34603	−0.673015	+	0.739629i	$0.735001\pi$
$798$	4.00000	0.141598
$799$	−24.0000	−0.849059
$800$	5.00000	0.176777
$801$	−6.00000	−0.212000
$802$	−6.00000	−0.211867
$803$	−56.0000	−1.97620
$804$	−4.00000	−0.141069
$805$	−16.0000	−0.563926
$806$	0	0
$807$	26.0000	0.915243
$808$	30.0000	1.05540
$809$	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
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	8.00000	0.280745
$813$	−24.0000	−0.841717
$814$	−24.0000	−0.841200
$815$	0	0
$816$	2.00000	0.0700140
$817$	−8.00000	−0.279885
$818$	10.0000	0.349642
$819$	8.00000	0.279543
$820$	6.00000	0.209529
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	6.00000	0.209274
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	−48.0000	−1.67216
$825$	−4.00000	−0.139262
$826$	16.0000	0.556711
$827$	−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
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	14.0000	0.485363
$833$	18.0000	0.623663
$834$	4.00000	0.138509
$835$	−8.00000	−0.276851
$836$	4.00000	0.138343
$837$	0	0
$838$	28.0000	0.967244
$839$	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
$842$	14.0000	0.482472
$843$	−10.0000	−0.344418
$844$	−12.0000	−0.413057
$845$	−9.00000	−0.309609
$846$	−12.0000	−0.412568
$847$	20.0000	0.687208
$848$	14.0000	0.480762
$849$	16.0000	0.549119
$850$	2.00000	0.0685994
$851$	24.0000	0.822709
$852$	0	0
$853$	−2.00000	−0.0684787	−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000	−0.0684787	−0.0342393	+	0.999414i	$0.510901\pi$
$854$	56.0000	1.91628
$855$	−1.00000	−0.0341993
$856$	−36.0000	−1.23045
$857$	−26.0000	−0.888143	−0.444072	−	0.895991i	$-0.646466\pi$
$857$	−26.0000	−0.888143	−0.444072	+	0.895991i	$0.646466\pi$
$858$	−8.00000	−0.273115
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	−8.00000	−0.272798
$861$	24.0000	0.817918
$862$	32.0000	1.08992
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	−5.00000	−0.170103
$865$	−6.00000	−0.204006
$866$	6.00000	0.203888
$867$	13.0000	0.441503
$868$	0	0
$869$	64.0000	2.17105
$870$	2.00000	0.0678064
$871$	−8.00000	−0.271070
$872$	6.00000	0.203186
$873$	−10.0000	−0.338449
$874$	4.00000	0.135302
$875$	4.00000	0.135225
$876$	−14.0000	−0.473016
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	8.00000	0.269987
$879$	−2.00000	−0.0674583
$880$	−4.00000	−0.134840
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	9.00000	0.303046
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	−4.00000	−0.134535
$885$	−4.00000	−0.134459
$886$	40.0000	1.34383
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	−18.0000	−0.604040
$889$	−32.0000	−1.07325
$890$	−6.00000	−0.201120
$891$	4.00000	0.134005
$892$	24.0000	0.803579
$893$	12.0000	0.401565
$894$	−6.00000	−0.200670
$895$	12.0000	0.401116
$896$	−12.0000	−0.400892
$897$	8.00000	0.267112
$898$	−6.00000	−0.200223
$899$	0	0
$900$	−1.00000	−0.0333333
$901$	−28.0000	−0.932815
$902$	−24.0000	−0.799113
$903$	−32.0000	−1.06489
$904$	30.0000	0.997785
$905$	22.0000	0.731305
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	12.0000	0.398234
$909$	−10.0000	−0.331679
$910$	8.00000	0.265197
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	−1.00000	−0.0331133
$913$	0	0
$914$	10.0000	0.330771
$915$	−14.0000	−0.462826
$916$	26.0000	0.859064
$917$	16.0000	0.528367
$918$	−2.00000	−0.0660098
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	12.0000	0.395628
$921$	12.0000	0.395413
$922$	30.0000	0.987997
$923$	0	0
$924$	16.0000	0.526361
$925$	−6.00000	−0.197279
$926$	−4.00000	−0.131448
$927$	16.0000	0.525509
$928$	−10.0000	−0.328266
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−9.00000	−0.294963
$932$	−18.0000	−0.589610
$933$	0	0
$934$	−8.00000	−0.261768
$935$	8.00000	0.261628
$936$	−6.00000	−0.196116
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	−16.0000	−0.522419
$939$	−10.0000	−0.326338
$940$	12.0000	0.391397
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	18.0000	0.586472
$943$	24.0000	0.781548
$944$	−4.00000	−0.130189
$945$	−4.00000	−0.130120
$946$	32.0000	1.04041
$947$	−40.0000	−1.29983	−0.649913	−	0.760009i	$-0.725195\pi$
$947$	−40.0000	−1.29983	−0.649913	+	0.760009i	$0.725195\pi$
$948$	16.0000	0.519656
$949$	−28.0000	−0.908918
$950$	−1.00000	−0.0324443
$951$	−2.00000	−0.0648544
$952$	−24.0000	−0.777844
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	−14.0000	−0.453267
$955$	0	0
$956$	24.0000	0.776215
$957$	8.00000	0.258603
$958$	−16.0000	−0.516937
$959$	−24.0000	−0.775000
$960$	−7.00000	−0.225924
$961$	−31.0000	−1.00000
$962$	−12.0000	−0.386896
$963$	12.0000	0.386695
$964$	22.0000	0.708572
$965$	22.0000	0.708205
$966$	16.0000	0.514792
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	−15.0000	−0.482118
$969$	2.00000	0.0642493
$970$	−10.0000	−0.321081
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	1.00000	0.0320750
$973$	−16.0000	−0.512936
$974$	8.00000	0.256337
$975$	−2.00000	−0.0640513
$976$	−14.0000	−0.448129
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	−9.00000	−0.287494
$981$	−2.00000	−0.0638551
$982$	12.0000	0.382935
$983$	−56.0000	−1.78612	−0.893061	−	0.449935i	$-0.851447\pi$
$983$	−56.0000	−1.78612	−0.893061	+	0.449935i	$0.851447\pi$
$984$	−18.0000	−0.573819
$985$	22.0000	0.700978
$986$	−4.00000	−0.127386
$987$	48.0000	1.52786
$988$	2.00000	0.0636285
$989$	−32.0000	−1.01754
$990$	4.00000	0.127128
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	20.0000	0.633089
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	285.2.a.c.1.1	✓	1
3.2	odd	2		855.2.a.a.1.1		1
4.3	odd	2		4560.2.a.w.1.1		1
5.2	odd	4		1425.2.c.f.799.2		2
5.3	odd	4		1425.2.c.f.799.1		2
5.4	even	2		1425.2.a.c.1.1		1
15.14	odd	2		4275.2.a.j.1.1		1
19.18	odd	2		5415.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.c.1.1	✓	1	1.1	even	1	trivial
855.2.a.a.1.1		1	3.2	odd	2
1425.2.a.c.1.1		1	5.4	even	2
1425.2.c.f.799.1		2	5.3	odd	4
1425.2.c.f.799.2		2	5.2	odd	4
4275.2.a.j.1.1		1	15.14	odd	2
4560.2.a.w.1.1		1	4.3	odd	2
5415.2.a.e.1.1		1	19.18	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:	\( N \)	\(=\)	\( 285 = 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	285.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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