LMFDB - Embedded newform 8670.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8670,2,Mod(1,8670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, 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("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8670.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:	$69.2302985525$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8670.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} -4.00000 q^{22} -8.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} -2.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} +4.00000 q^{39} -1.00000 q^{40} -8.00000 q^{41} +2.00000 q^{42} +6.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} +8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +4.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} +2.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} -6.00000 q^{59} +1.00000 q^{60} -14.0000 q^{61} +4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -4.00000 q^{66} -2.00000 q^{67} -8.00000 q^{69} +2.00000 q^{70} -2.00000 q^{71} -1.00000 q^{72} -4.00000 q^{73} +6.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -8.00000 q^{77} -4.00000 q^{78} +1.00000 q^{80} +1.00000 q^{81} +8.00000 q^{82} -16.0000 q^{83} -2.00000 q^{84} -6.00000 q^{86} -2.00000 q^{87} -4.00000 q^{88} -2.00000 q^{89} -1.00000 q^{90} -8.00000 q^{91} -8.00000 q^{92} -4.00000 q^{93} -8.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} +3.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000	0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	2.00000	0.534522
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−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
$22$	−4.00000	−0.852803
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	1.00000	0.192450
$28$	−2.00000	−0.377964
$29$	−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
$31$	−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
$34$	0	0
$35$	−2.00000	−0.338062
$36$	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	$0.664172\pi$
$38$	4.00000	0.648886
$39$	4.00000	0.640513
$40$	−1.00000	−0.158114
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	2.00000	0.308607
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	8.00000	1.17954
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	0	0
$52$	4.00000	0.554700
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−1.00000	−0.136083
$55$	4.00000	0.539360
$56$	2.00000	0.267261
$57$	−4.00000	−0.529813
$58$	2.00000	0.262613
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	1.00000	0.129099
$61$	−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
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	−4.00000	−0.492366
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	2.00000	0.239046
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−1.00000	−0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	6.00000	0.697486
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	−8.00000	−0.911685
$78$	−4.00000	−0.452911
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	8.00000	0.883452
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−6.00000	−0.646997
$87$	−2.00000	−0.214423
$88$	−4.00000	−0.426401
$89$	−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
$91$	−8.00000	−0.838628
$92$	−8.00000	−0.834058
$93$	−4.00000	−0.414781
$94$	−8.00000	−0.825137
$95$	−4.00000	−0.410391
$96$	−1.00000	−0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	3.00000	0.303046
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	−4.00000	−0.392232
$105$	−2.00000	−0.195180
$106$	2.00000	0.194257
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	−4.00000	−0.381385
$111$	−6.00000	−0.569495
$112$	−2.00000	−0.188982
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	4.00000	0.374634
$115$	−8.00000	−0.746004
$116$	−2.00000	−0.185695
$117$	4.00000	0.369800
$118$	6.00000	0.552345
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	5.00000	0.454545
$122$	14.0000	1.26750
$123$	−8.00000	−0.721336
$124$	−4.00000	−0.359211
$125$	1.00000	0.0894427
$126$	2.00000	0.178174
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−1.00000	−0.0883883
$129$	6.00000	0.528271
$130$	−4.00000	−0.350823
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	4.00000	0.348155
$133$	8.00000	0.693688
$134$	2.00000	0.172774
$135$	1.00000	0.0860663
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	8.00000	0.681005
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	−2.00000	−0.169031
$141$	8.00000	0.673722
$142$	2.00000	0.167836
$143$	16.0000	1.33799
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	4.00000	0.331042
$147$	−3.00000	−0.247436
$148$	−6.00000	−0.493197
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	−1.00000	−0.0816497
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	8.00000	0.644658
$155$	−4.00000	−0.321288
$156$	4.00000	0.320256
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	−1.00000	−0.0790569
$161$	16.0000	1.26098
$162$	−1.00000	−0.0785674
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	−8.00000	−0.624695
$165$	4.00000	0.311400
$166$	16.0000	1.24184
$167$	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
$170$	0	0
$171$	−4.00000	−0.305888
$172$	6.00000	0.457496
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	2.00000	0.151620
$175$	−2.00000	−0.151186
$176$	4.00000	0.301511
$177$	−6.00000	−0.450988
$178$	2.00000	0.149906
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	1.00000	0.0745356
$181$	−26.0000	−1.93256	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000	−1.93256	−0.966282	+	0.257485i	$0.917106\pi$
$182$	8.00000	0.592999
$183$	−14.0000	−1.03491
$184$	8.00000	0.589768
$185$	−6.00000	−0.441129
$186$	4.00000	0.293294
$187$	0	0
$188$	8.00000	0.583460
$189$	−2.00000	−0.145479
$190$	4.00000	0.290191
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	1.00000	0.0721688
$193$	−24.0000	−1.72756	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−24.0000	−1.72756	−0.863779	+	0.503871i	$0.831909\pi$
$194$	0	0
$195$	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	$0.123046\pi$
$198$	−4.00000	−0.284268
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−1.00000	−0.0707107
$201$	−2.00000	−0.141069
$202$	−8.00000	−0.562878
$203$	4.00000	0.280745
$204$	0	0
$205$	−8.00000	−0.558744
$206$	12.0000	0.836080
$207$	−8.00000	−0.556038
$208$	4.00000	0.277350
$209$	−16.0000	−1.10674
$210$	2.00000	0.138013
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−2.00000	−0.137361
$213$	−2.00000	−0.137038
$214$	12.0000	0.820303
$215$	6.00000	0.409197
$216$	−1.00000	−0.0680414
$217$	8.00000	0.543075
$218$	−14.0000	−0.948200
$219$	−4.00000	−0.270295
$220$	4.00000	0.269680
$221$	0	0
$222$	6.00000	0.402694
$223$	28.0000	1.87502	0.937509	−	0.347960i	$-0.113126\pi$
$223$	28.0000	1.87502	0.937509	+	0.347960i	$0.113126\pi$
$224$	2.00000	0.133631
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	−28.0000	−1.85843	−0.929213	−	0.369546i	$-0.879513\pi$
$227$	−28.0000	−1.85843	−0.929213	+	0.369546i	$0.879513\pi$
$228$	−4.00000	−0.264906
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	8.00000	0.527504
$231$	−8.00000	−0.526361
$232$	2.00000	0.131306
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−4.00000	−0.261488
$235$	8.00000	0.521862
$236$	−6.00000	−0.390567
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	1.00000	0.0645497
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	−14.0000	−0.896258
$245$	−3.00000	−0.191663
$246$	8.00000	0.510061
$247$	−16.0000	−1.01806
$248$	4.00000	0.254000
$249$	−16.0000	−1.01396
$250$	−1.00000	−0.0632456
$251$	26.0000	1.64111	0.820553	−	0.571571i	$-0.193666\pi$
$251$	26.0000	1.64111	0.820553	+	0.571571i	$0.193666\pi$
$252$	−2.00000	−0.125988
$253$	−32.0000	−2.01182
$254$	4.00000	0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	−6.00000	−0.373544
$259$	12.0000	0.745644
$260$	4.00000	0.248069
$261$	−2.00000	−0.123797
$262$	−16.0000	−0.988483
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−4.00000	−0.246183
$265$	−2.00000	−0.122859
$266$	−8.00000	−0.490511
$267$	−2.00000	−0.122398
$268$	−2.00000	−0.122169
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	−1.00000	−0.0608581
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	6.00000	0.362473
$275$	4.00000	0.241209
$276$	−8.00000	−0.481543
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−20.0000	−1.19952
$279$	−4.00000	−0.239474
$280$	2.00000	0.119523
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	−8.00000	−0.476393
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	−2.00000	−0.118678
$285$	−4.00000	−0.236940
$286$	−16.0000	−0.946100
$287$	16.0000	0.944450
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	2.00000	0.117444
$291$	0	0
$292$	−4.00000	−0.234082
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	3.00000	0.174964
$295$	−6.00000	−0.349334
$296$	6.00000	0.348743
$297$	4.00000	0.232104
$298$	−16.0000	−0.926855
$299$	−32.0000	−1.85061
$300$	1.00000	0.0577350
$301$	−12.0000	−0.691669
$302$	8.00000	0.460348
$303$	8.00000	0.459588
$304$	−4.00000	−0.229416
$305$	−14.0000	−0.801638
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	−8.00000	−0.455842
$309$	−12.0000	−0.682656
$310$	4.00000	0.227185
$311$	−14.0000	−0.793867	−0.396934	−	0.917847i	$-0.629926\pi$
$311$	−14.0000	−0.793867	−0.396934	+	0.917847i	$0.629926\pi$
$312$	−4.00000	−0.226455
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	4.00000	0.225733
$315$	−2.00000	−0.112687
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	2.00000	0.112154
$319$	−8.00000	−0.447914
$320$	1.00000	0.0559017
$321$	−12.0000	−0.669775
$322$	−16.0000	−0.891645
$323$	0	0
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	−8.00000	−0.443079
$327$	14.0000	0.774202
$328$	8.00000	0.441726
$329$	−16.0000	−0.882109
$330$	−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	$0.570560\pi$
$332$	−16.0000	−0.878114
$333$	−6.00000	−0.328798
$334$	−12.0000	−0.656611
$335$	−2.00000	−0.109272
$336$	−2.00000	−0.109109
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	−3.00000	−0.163178
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−16.0000	−0.866449
$342$	4.00000	0.216295
$343$	20.0000	1.07990
$344$	−6.00000	−0.323498
$345$	−8.00000	−0.430706
$346$	22.0000	1.18273
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	−2.00000	−0.107211
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	2.00000	0.106904
$351$	4.00000	0.213504
$352$	−4.00000	−0.213201
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	6.00000	0.318896
$355$	−2.00000	−0.106149
$356$	−2.00000	−0.106000
$357$	0	0
$358$	−6.00000	−0.317110
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	26.0000	1.36653
$363$	5.00000	0.262432
$364$	−8.00000	−0.419314
$365$	−4.00000	−0.209370
$366$	14.0000	0.731792
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	−8.00000	−0.417029
$369$	−8.00000	−0.416463
$370$	6.00000	0.311925
$371$	4.00000	0.207670
$372$	−4.00000	−0.207390
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−8.00000	−0.412568
$377$	−8.00000	−0.412021
$378$	2.00000	0.102869
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−4.00000	−0.205196
$381$	−4.00000	−0.204926
$382$	−4.00000	−0.204658
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	−1.00000	−0.0510310
$385$	−8.00000	−0.407718
$386$	24.0000	1.22157
$387$	6.00000	0.304997
$388$	0	0
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	−4.00000	−0.202548
$391$	0	0
$392$	3.00000	0.151523
$393$	16.0000	0.807093
$394$	−26.0000	−1.30986
$395$	0	0
$396$	4.00000	0.201008
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	−20.0000	−1.00251
$399$	8.00000	0.400501
$400$	1.00000	0.0500000
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	2.00000	0.0997509
$403$	−16.0000	−0.797017
$404$	8.00000	0.398015
$405$	1.00000	0.0496904
$406$	−4.00000	−0.198517
$407$	−24.0000	−1.18964
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	8.00000	0.395092
$411$	−6.00000	−0.295958
$412$	−12.0000	−0.591198
$413$	12.0000	0.590481
$414$	8.00000	0.393179
$415$	−16.0000	−0.785409
$416$	−4.00000	−0.196116
$417$	20.0000	0.979404
$418$	16.0000	0.782586
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	−2.00000	−0.0975900
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	4.00000	0.194717
$423$	8.00000	0.388973
$424$	2.00000	0.0971286
$425$	0	0
$426$	2.00000	0.0969003
$427$	28.0000	1.35501
$428$	−12.0000	−0.580042
$429$	16.0000	0.772487
$430$	−6.00000	−0.289346
$431$	−14.0000	−0.674356	−0.337178	−	0.941441i	$-0.609472\pi$
$431$	−14.0000	−0.674356	−0.337178	+	0.941441i	$0.609472\pi$
$432$	1.00000	0.0481125
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	−8.00000	−0.384012
$435$	−2.00000	−0.0958927
$436$	14.0000	0.670478
$437$	32.0000	1.53077
$438$	4.00000	0.191127
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	−4.00000	−0.190693
$441$	−3.00000	−0.142857
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−6.00000	−0.284747
$445$	−2.00000	−0.0948091
$446$	−28.0000	−1.32584
$447$	16.0000	0.756774
$448$	−2.00000	−0.0944911
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	−1.00000	−0.0471405
$451$	−32.0000	−1.50682
$452$	−6.00000	−0.282216
$453$	−8.00000	−0.375873
$454$	28.0000	1.31411
$455$	−8.00000	−0.375046
$456$	4.00000	0.187317
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	−6.00000	−0.280362
$459$	0	0
$460$	−8.00000	−0.373002
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	8.00000	0.372194
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	−2.00000	−0.0928477
$465$	−4.00000	−0.185496
$466$	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
$469$	4.00000	0.184703
$470$	−8.00000	−0.369012
$471$	−4.00000	−0.184310
$472$	6.00000	0.276172
$473$	24.0000	1.10352
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	22.0000	1.00521	0.502603	−	0.864517i	$-0.332376\pi$
$479$	22.0000	1.00521	0.502603	+	0.864517i	$0.332376\pi$
$480$	−1.00000	−0.0456435
$481$	−24.0000	−1.09431
$482$	−14.0000	−0.637683
$483$	16.0000	0.728025
$484$	5.00000	0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	14.0000	0.633750
$489$	8.00000	0.361773
$490$	3.00000	0.135526
$491$	−18.0000	−0.812329	−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000	−0.812329	−0.406164	+	0.913800i	$0.633134\pi$
$492$	−8.00000	−0.360668
$493$	0	0
$494$	16.0000	0.719874
$495$	4.00000	0.179787
$496$	−4.00000	−0.179605
$497$	4.00000	0.179425
$498$	16.0000	0.716977
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	1.00000	0.0447214
$501$	12.0000	0.536120
$502$	−26.0000	−1.16044
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	2.00000	0.0890871
$505$	8.00000	0.355995
$506$	32.0000	1.42257
$507$	3.00000	0.133235
$508$	−4.00000	−0.177471
$509$	8.00000	0.354594	0.177297	−	0.984157i	$-0.443265\pi$
$509$	8.00000	0.354594	0.177297	+	0.984157i	$0.443265\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	18.0000	0.793946
$515$	−12.0000	−0.528783
$516$	6.00000	0.264135
$517$	32.0000	1.40736
$518$	−12.0000	−0.527250
$519$	−22.0000	−0.965693
$520$	−4.00000	−0.175412
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	2.00000	0.0875376
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	16.0000	0.698963
$525$	−2.00000	−0.0872872
$526$	−8.00000	−0.348817
$527$	0	0
$528$	4.00000	0.174078
$529$	41.0000	1.78261
$530$	2.00000	0.0868744
$531$	−6.00000	−0.260378
$532$	8.00000	0.346844
$533$	−32.0000	−1.38607
$534$	2.00000	0.0865485
$535$	−12.0000	−0.518805
$536$	2.00000	0.0863868
$537$	6.00000	0.258919
$538$	−30.0000	−1.29339
$539$	−12.0000	−0.516877
$540$	1.00000	0.0430331
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	24.0000	1.03089
$543$	−26.0000	−1.11577
$544$	0	0
$545$	14.0000	0.599694
$546$	8.00000	0.342368
$547$	−44.0000	−1.88130	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000	−1.88130	−0.940652	+	0.339372i	$0.889785\pi$
$548$	−6.00000	−0.256307
$549$	−14.0000	−0.597505
$550$	−4.00000	−0.170561
$551$	8.00000	0.340811
$552$	8.00000	0.340503
$553$	0	0
$554$	2.00000	0.0849719
$555$	−6.00000	−0.254686
$556$	20.0000	0.848189
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	4.00000	0.169334
$559$	24.0000	1.01509
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	2.00000	0.0843649
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	8.00000	0.336861
$565$	−6.00000	−0.252422
$566$	−20.0000	−0.840663
$567$	−2.00000	−0.0839921
$568$	2.00000	0.0839181
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	4.00000	0.167542
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	16.0000	0.668994
$573$	4.00000	0.167102
$574$	−16.0000	−0.667827
$575$	−8.00000	−0.333623
$576$	1.00000	0.0416667
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	0	0
$579$	−24.0000	−0.997406
$580$	−2.00000	−0.0830455
$581$	32.0000	1.32758
$582$	0	0
$583$	−8.00000	−0.331326
$584$	4.00000	0.165521
$585$	4.00000	0.165380
$586$	14.0000	0.578335
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	−3.00000	−0.123718
$589$	16.0000	0.659269
$590$	6.00000	0.247016
$591$	26.0000	1.06950
$592$	−6.00000	−0.246598
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	16.0000	0.655386
$597$	20.0000	0.818546
$598$	32.0000	1.30858
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	−1.00000	−0.0408248
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	12.0000	0.489083
$603$	−2.00000	−0.0814463
$604$	−8.00000	−0.325515
$605$	5.00000	0.203279
$606$	−8.00000	−0.324978
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	4.00000	0.162221
$609$	4.00000	0.162088
$610$	14.0000	0.566843
$611$	32.0000	1.29458
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	10.0000	0.403567
$615$	−8.00000	−0.322591
$616$	8.00000	0.322329
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	12.0000	0.482711
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	−4.00000	−0.160644
$621$	−8.00000	−0.321029
$622$	14.0000	0.561349
$623$	4.00000	0.160257
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	16.0000	0.639489
$627$	−16.0000	−0.638978
$628$	−4.00000	−0.159617
$629$	0	0
$630$	2.00000	0.0796819
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	−2.00000	−0.0794301
$635$	−4.00000	−0.158735
$636$	−2.00000	−0.0793052
$637$	−12.0000	−0.475457
$638$	8.00000	0.316723
$639$	−2.00000	−0.0791188
$640$	−1.00000	−0.0395285
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	12.0000	0.473602
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	16.0000	0.630488
$645$	6.00000	0.236250
$646$	0	0
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	−1.00000	−0.0392837
$649$	−24.0000	−0.942082
$650$	−4.00000	−0.156893
$651$	8.00000	0.313545
$652$	8.00000	0.313304
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	−14.0000	−0.547443
$655$	16.0000	0.625172
$656$	−8.00000	−0.312348
$657$	−4.00000	−0.156055
$658$	16.0000	0.623745
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	4.00000	0.155700
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	16.0000	0.620920
$665$	8.00000	0.310227
$666$	6.00000	0.232495
$667$	16.0000	0.619522
$668$	12.0000	0.464294
$669$	28.0000	1.08254
$670$	2.00000	0.0772667
$671$	−56.0000	−2.16186
$672$	2.00000	0.0771517
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	20.0000	0.770371
$675$	1.00000	0.0384900
$676$	3.00000	0.115385
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	0	0
$681$	−28.0000	−1.07296
$682$	16.0000	0.612672
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−4.00000	−0.152944
$685$	−6.00000	−0.229248
$686$	−20.0000	−0.763604
$687$	6.00000	0.228914
$688$	6.00000	0.228748
$689$	−8.00000	−0.304776
$690$	8.00000	0.304555
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−22.0000	−0.836315
$693$	−8.00000	−0.303895
$694$	−28.0000	−1.06287
$695$	20.0000	0.758643
$696$	2.00000	0.0758098
$697$	0	0
$698$	14.0000	0.529908
$699$	−18.0000	−0.680823
$700$	−2.00000	−0.0755929
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	−4.00000	−0.150970
$703$	24.0000	0.905177
$704$	4.00000	0.150756
$705$	8.00000	0.301297
$706$	10.0000	0.376355
$707$	−16.0000	−0.601742
$708$	−6.00000	−0.225494
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	2.00000	0.0750587
$711$	0	0
$712$	2.00000	0.0749532
$713$	32.0000	1.19841
$714$	0	0
$715$	16.0000	0.598366
$716$	6.00000	0.224231
$717$	0	0
$718$	−36.0000	−1.34351
$719$	−10.0000	−0.372937	−0.186469	−	0.982461i	$-0.559704\pi$
$719$	−10.0000	−0.372937	−0.186469	+	0.982461i	$0.559704\pi$
$720$	1.00000	0.0372678
$721$	24.0000	0.893807
$722$	3.00000	0.111648
$723$	14.0000	0.520666
$724$	−26.0000	−0.966282
$725$	−2.00000	−0.0742781
$726$	−5.00000	−0.185567
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	8.00000	0.296500
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	0	0
$732$	−14.0000	−0.517455
$733$	48.0000	1.77292	0.886460	−	0.462805i	$-0.153157\pi$
$733$	48.0000	1.77292	0.886460	+	0.462805i	$0.153157\pi$
$734$	22.0000	0.812035
$735$	−3.00000	−0.110657
$736$	8.00000	0.294884
$737$	−8.00000	−0.294684
$738$	8.00000	0.294484
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	−6.00000	−0.220564
$741$	−16.0000	−0.587775
$742$	−4.00000	−0.146845
$743$	20.0000	0.733729	0.366864	−	0.930274i	$-0.380431\pi$
$743$	20.0000	0.733729	0.366864	+	0.930274i	$0.380431\pi$
$744$	4.00000	0.146647
$745$	16.0000	0.586195
$746$	16.0000	0.585802
$747$	−16.0000	−0.585409
$748$	0	0
$749$	24.0000	0.876941
$750$	−1.00000	−0.0365148
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	8.00000	0.291730
$753$	26.0000	0.947493
$754$	8.00000	0.291343
$755$	−8.00000	−0.291150
$756$	−2.00000	−0.0727393
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	−4.00000	−0.145287
$759$	−32.0000	−1.16153
$760$	4.00000	0.145095
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	4.00000	0.144905
$763$	−28.0000	−1.01367
$764$	4.00000	0.144715
$765$	0	0
$766$	−8.00000	−0.289052
$767$	−24.0000	−0.866590
$768$	1.00000	0.0360844
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	8.00000	0.288300
$771$	−18.0000	−0.648254
$772$	−24.0000	−0.863779
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	−6.00000	−0.215666
$775$	−4.00000	−0.143684
$776$	0	0
$777$	12.0000	0.430498
$778$	−24.0000	−0.860442
$779$	32.0000	1.14652
$780$	4.00000	0.143223
$781$	−8.00000	−0.286263
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	−3.00000	−0.107143
$785$	−4.00000	−0.142766
$786$	−16.0000	−0.570701
$787$	−24.0000	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$787$	−24.0000	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$788$	26.0000	0.926212
$789$	8.00000	0.284808
$790$	0	0
$791$	12.0000	0.426671
$792$	−4.00000	−0.142134
$793$	−56.0000	−1.98862
$794$	6.00000	0.212932
$795$	−2.00000	−0.0709327
$796$	20.0000	0.708881
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	−8.00000	−0.283197
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−2.00000	−0.0706665
$802$	−12.0000	−0.423735
$803$	−16.0000	−0.564628
$804$	−2.00000	−0.0705346
$805$	16.0000	0.563926
$806$	16.0000	0.563576
$807$	30.0000	1.05605
$808$	−8.00000	−0.281439
$809$	48.0000	1.68759	0.843795	−	0.536666i	$-0.180316\pi$
$809$	48.0000	1.68759	0.843795	+	0.536666i	$0.180316\pi$
$810$	−1.00000	−0.0351364
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	4.00000	0.140372
$813$	−24.0000	−0.841717
$814$	24.0000	0.841200
$815$	8.00000	0.280228
$816$	0	0
$817$	−24.0000	−0.839654
$818$	26.0000	0.909069
$819$	−8.00000	−0.279543
$820$	−8.00000	−0.279372
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	6.00000	0.209274
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	12.0000	0.418040
$825$	4.00000	0.139262
$826$	−12.0000	−0.417533
$827$	−44.0000	−1.53003	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000	−1.53003	−0.765015	+	0.644013i	$0.777268\pi$
$828$	−8.00000	−0.278019
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	16.0000	0.555368
$831$	−2.00000	−0.0693792
$832$	4.00000	0.138675
$833$	0	0
$834$	−20.0000	−0.692543
$835$	12.0000	0.415277
$836$	−16.0000	−0.553372
$837$	−4.00000	−0.138260
$838$	0	0
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	2.00000	0.0690066
$841$	−25.0000	−0.862069
$842$	18.0000	0.620321
$843$	−2.00000	−0.0688837
$844$	−4.00000	−0.137686
$845$	3.00000	0.103203
$846$	−8.00000	−0.275046
$847$	−10.0000	−0.343604
$848$	−2.00000	−0.0686803
$849$	20.0000	0.686398
$850$	0	0
$851$	48.0000	1.64542
$852$	−2.00000	−0.0685189
$853$	−2.00000	−0.0684787	−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000	−0.0684787	−0.0342393	+	0.999414i	$0.510901\pi$
$854$	−28.0000	−0.958140
$855$	−4.00000	−0.136797
$856$	12.0000	0.410152
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	−16.0000	−0.546231
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	6.00000	0.204598
$861$	16.0000	0.545279
$862$	14.0000	0.476842
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	−1.00000	−0.0340207
$865$	−22.0000	−0.748022
$866$	26.0000	0.883516
$867$	0	0
$868$	8.00000	0.271538
$869$	0	0
$870$	2.00000	0.0678064
$871$	−8.00000	−0.271070
$872$	−14.0000	−0.474100
$873$	0	0
$874$	−32.0000	−1.08242
$875$	−2.00000	−0.0676123
$876$	−4.00000	−0.135147
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	−24.0000	−0.809961
$879$	−14.0000	−0.472208
$880$	4.00000	0.134840
$881$	−36.0000	−1.21287	−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000	−1.21287	−0.606435	+	0.795133i	$0.707401\pi$
$882$	3.00000	0.101015
$883$	−46.0000	−1.54802	−0.774012	−	0.633171i	$-0.781753\pi$
$883$	−46.0000	−1.54802	−0.774012	+	0.633171i	$0.781753\pi$
$884$	0	0
$885$	−6.00000	−0.201688
$886$	−24.0000	−0.806296
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	6.00000	0.201347
$889$	8.00000	0.268311
$890$	2.00000	0.0670402
$891$	4.00000	0.134005
$892$	28.0000	0.937509
$893$	−32.0000	−1.07084
$894$	−16.0000	−0.535120
$895$	6.00000	0.200558
$896$	2.00000	0.0668153
$897$	−32.0000	−1.06845
$898$	−12.0000	−0.400445
$899$	8.00000	0.266815
$900$	1.00000	0.0333333
$901$	0	0
$902$	32.0000	1.06548
$903$	−12.0000	−0.399335
$904$	6.00000	0.199557
$905$	−26.0000	−0.864269
$906$	8.00000	0.265782
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	−28.0000	−0.929213
$909$	8.00000	0.265343
$910$	8.00000	0.265197
$911$	−50.0000	−1.65657	−0.828287	−	0.560304i	$-0.810684\pi$
$911$	−50.0000	−1.65657	−0.828287	+	0.560304i	$0.810684\pi$
$912$	−4.00000	−0.132453
$913$	−64.0000	−2.11809
$914$	2.00000	0.0661541
$915$	−14.0000	−0.462826
$916$	6.00000	0.198246
$917$	−32.0000	−1.05673
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	8.00000	0.263752
$921$	−10.0000	−0.329511
$922$	28.0000	0.922131
$923$	−8.00000	−0.263323
$924$	−8.00000	−0.263181
$925$	−6.00000	−0.197279
$926$	−4.00000	−0.131448
$927$	−12.0000	−0.394132
$928$	2.00000	0.0656532
$929$	48.0000	1.57483	0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000	1.57483	0.787414	+	0.616424i	$0.211419\pi$
$930$	4.00000	0.131165
$931$	12.0000	0.393284
$932$	−18.0000	−0.589610
$933$	−14.0000	−0.458339
$934$	12.0000	0.392652
$935$	0	0
$936$	−4.00000	−0.130744
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	−4.00000	−0.130605
$939$	−16.0000	−0.522140
$940$	8.00000	0.260931
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	4.00000	0.130327
$943$	64.0000	2.08413
$944$	−6.00000	−0.195283
$945$	−2.00000	−0.0650600
$946$	−24.0000	−0.780307
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	−16.0000	−0.519382
$950$	4.00000	0.129777
$951$	2.00000	0.0648544
$952$	0	0
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	2.00000	0.0647524
$955$	4.00000	0.129437
$956$	0	0
$957$	−8.00000	−0.258603
$958$	−22.0000	−0.710788
$959$	12.0000	0.387500
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	24.0000	0.773791
$963$	−12.0000	−0.386695
$964$	14.0000	0.450910
$965$	−24.0000	−0.772587
$966$	−16.0000	−0.514792
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	0	0
$971$	−50.0000	−1.60458	−0.802288	−	0.596937i	$-0.796384\pi$
$971$	−50.0000	−1.60458	−0.802288	+	0.596937i	$0.796384\pi$
$972$	1.00000	0.0320750
$973$	−40.0000	−1.28234
$974$	−22.0000	−0.704925
$975$	4.00000	0.128103
$976$	−14.0000	−0.448129
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	−8.00000	−0.255812
$979$	−8.00000	−0.255681
$980$	−3.00000	−0.0958315
$981$	14.0000	0.446986
$982$	18.0000	0.574403
$983$	52.0000	1.65854	0.829271	−	0.558846i	$-0.188756\pi$
$983$	52.0000	1.65854	0.829271	+	0.558846i	$0.188756\pi$
$984$	8.00000	0.255031
$985$	26.0000	0.828429
$986$	0	0
$987$	−16.0000	−0.509286
$988$	−16.0000	−0.509028
$989$	−48.0000	−1.52631
$990$	−4.00000	−0.127128
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	4.00000	0.127000
$993$	−8.00000	−0.253872
$994$	−4.00000	−0.126872
$995$	20.0000	0.634043
$996$	−16.0000	−0.506979
$997$	50.0000	1.58352	0.791758	−	0.610835i	$-0.209166\pi$
$997$	50.0000	1.58352	0.791758	+	0.610835i	$0.209166\pi$
$998$	4.00000	0.126618
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.k.1.1		1
17.16	even	2		510.2.a.a.1.1	✓	1
51.50	odd	2		1530.2.a.p.1.1		1
68.67	odd	2		4080.2.a.s.1.1		1
85.33	odd	4		2550.2.d.n.2449.2		2
85.67	odd	4		2550.2.d.n.2449.1		2
85.84	even	2		2550.2.a.bc.1.1		1
255.254	odd	2		7650.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.a.1.1	✓	1	17.16	even	2
1530.2.a.p.1.1		1	51.50	odd	2
2550.2.a.bc.1.1		1	85.84	even	2
2550.2.d.n.2449.1		2	85.67	odd	4
2550.2.d.n.2449.2		2	85.33	odd	4
4080.2.a.s.1.1		1	68.67	odd	2
7650.2.a.k.1.1		1	255.254	odd	2
8670.2.a.k.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:	\( N \)	\(=\)	\( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8670.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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