LMFDB - Embedded newform 8670.2.a.bb.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:	$0$
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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} +4.00000 q^{21} +4.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} -2.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +4.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} +4.00000 q^{42} -12.0000 q^{43} +4.00000 q^{44} +1.00000 q^{45} +4.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} +4.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} -2.00000 q^{61} -4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} +4.00000 q^{69} +4.00000 q^{70} +4.00000 q^{71} +1.00000 q^{72} +14.0000 q^{73} +6.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +16.0000 q^{77} -2.00000 q^{78} +12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -4.00000 q^{83} +4.00000 q^{84} -12.0000 q^{86} -2.00000 q^{87} +4.00000 q^{88} +10.0000 q^{89} +1.00000 q^{90} -8.00000 q^{91} +4.00000 q^{92} -4.00000 q^{93} +8.00000 q^{94} -4.00000 q^{95} +1.00000 q^{96} -18.0000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	4.00000	1.06904
$15$	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
$21$	4.00000	0.872872
$22$	4.00000	0.852803
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	−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
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.00000	−0.648886
$39$	−2.00000	−0.320256
$40$	1.00000	0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	4.00000	0.617213
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	4.00000	0.589768
$47$	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
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	4.00000	0.539360
$56$	4.00000	0.534522
$57$	−4.00000	−0.529813
$58$	−2.00000	−0.262613
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	1.00000	0.129099
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−4.00000	−0.508001
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	4.00000	0.492366
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	4.00000	0.478091
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	1.00000	0.117851
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	6.00000	0.697486
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	16.0000	1.82337
$78$	−2.00000	−0.226455
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	−12.0000	−1.29399
$87$	−2.00000	−0.214423
$88$	4.00000	0.426401
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	1.00000	0.105409
$91$	−8.00000	−0.838628
$92$	4.00000	0.417029
$93$	−4.00000	−0.414781
$94$	8.00000	0.825137
$95$	−4.00000	−0.410391
$96$	1.00000	0.102062
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	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	$0.665755\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−2.00000	−0.196116
$105$	4.00000	0.390360
$106$	−2.00000	−0.194257
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	4.00000	0.381385
$111$	6.00000	0.569495
$112$	4.00000	0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−4.00000	−0.374634
$115$	4.00000	0.373002
$116$	−2.00000	−0.185695
$117$	−2.00000	−0.184900
$118$	12.0000	1.10469
$119$	0	0
$120$	1.00000	0.0912871
$121$	5.00000	0.454545
$122$	−2.00000	−0.181071
$123$	−2.00000	−0.180334
$124$	−4.00000	−0.359211
$125$	1.00000	0.0894427
$126$	4.00000	0.356348
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	−12.0000	−1.05654
$130$	−2.00000	−0.175412
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	4.00000	0.348155
$133$	−16.0000	−1.38738
$134$	4.00000	0.345547
$135$	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	$0.582506\pi$
$138$	4.00000	0.340503
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	4.00000	0.338062
$141$	8.00000	0.673722
$142$	4.00000	0.335673
$143$	−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
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	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
$154$	16.0000	1.28932
$155$	−4.00000	−0.321288
$156$	−2.00000	−0.160128
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	12.0000	0.954669
$159$	−2.00000	−0.158610
$160$	1.00000	0.0790569
$161$	16.0000	1.26098
$162$	1.00000	0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−2.00000	−0.156174
$165$	4.00000	0.311400
$166$	−4.00000	−0.310460
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	4.00000	0.308607
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−12.0000	−0.914991
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	−2.00000	−0.151620
$175$	4.00000	0.302372
$176$	4.00000	0.301511
$177$	12.0000	0.901975
$178$	10.0000	0.749532
$179$	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
$183$	−2.00000	−0.147844
$184$	4.00000	0.294884
$185$	6.00000	0.441129
$186$	−4.00000	−0.293294
$187$	0	0
$188$	8.00000	0.583460
$189$	4.00000	0.290957
$190$	−4.00000	−0.290191
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	1.00000	0.0721688
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	−18.0000	−1.29232
$195$	−2.00000	−0.143223
$196$	9.00000	0.642857
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	4.00000	0.284268
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	1.00000	0.0707107
$201$	4.00000	0.282138
$202$	−10.0000	−0.703598
$203$	−8.00000	−0.561490
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	4.00000	0.278019
$208$	−2.00000	−0.138675
$209$	−16.0000	−1.10674
$210$	4.00000	0.276026
$211$	−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
$213$	4.00000	0.274075
$214$	−12.0000	−0.820303
$215$	−12.0000	−0.818393
$216$	1.00000	0.0680414
$217$	−16.0000	−1.08615
$218$	−10.0000	−0.677285
$219$	14.0000	0.946032
$220$	4.00000	0.269680
$221$	0	0
$222$	6.00000	0.402694
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	4.00000	0.267261
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	−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	$0.436476\pi$
$230$	4.00000	0.263752
$231$	16.0000	1.05272
$232$	−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	$0.700718\pi$
$234$	−2.00000	−0.130744
$235$	8.00000	0.521862
$236$	12.0000	0.781133
$237$	12.0000	0.779484
$238$	0	0
$239$	−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
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	9.00000	0.574989
$246$	−2.00000	−0.127515
$247$	8.00000	0.509028
$248$	−4.00000	−0.254000
$249$	−4.00000	−0.253490
$250$	1.00000	0.0632456
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	4.00000	0.251976
$253$	16.0000	1.00591
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−12.0000	−0.747087
$259$	24.0000	1.49129
$260$	−2.00000	−0.124035
$261$	−2.00000	−0.123797
$262$	−20.0000	−1.23560
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	4.00000	0.246183
$265$	−2.00000	−0.122859
$266$	−16.0000	−0.981023
$267$	10.0000	0.611990
$268$	4.00000	0.244339
$269$	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
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	−6.00000	−0.362473
$275$	4.00000	0.241209
$276$	4.00000	0.240772
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	20.0000	1.19952
$279$	−4.00000	−0.239474
$280$	4.00000	0.239046
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	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	$0.297374\pi$
$284$	4.00000	0.237356
$285$	−4.00000	−0.236940
$286$	−8.00000	−0.473050
$287$	−8.00000	−0.472225
$288$	1.00000	0.0589256
$289$	0	0
$290$	−2.00000	−0.117444
$291$	−18.0000	−1.05518
$292$	14.0000	0.819288
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	9.00000	0.524891
$295$	12.0000	0.698667
$296$	6.00000	0.348743
$297$	4.00000	0.232104
$298$	−2.00000	−0.115857
$299$	−8.00000	−0.462652
$300$	1.00000	0.0577350
$301$	−48.0000	−2.76667
$302$	−8.00000	−0.460348
$303$	−10.0000	−0.574485
$304$	−4.00000	−0.229416
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	16.0000	0.911685
$309$	0	0
$310$	−4.00000	−0.227185
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	−2.00000	−0.113228
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−10.0000	−0.564333
$315$	4.00000	0.225374
$316$	12.0000	0.675053
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	−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
$325$	−2.00000	−0.110940
$326$	20.0000	1.10770
$327$	−10.0000	−0.553001
$328$	−2.00000	−0.110432
$329$	32.0000	1.76422
$330$	4.00000	0.220193
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	−4.00000	−0.219529
$333$	6.00000	0.328798
$334$	−12.0000	−0.656611
$335$	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
$339$	6.00000	0.325875
$340$	0	0
$341$	−16.0000	−0.866449
$342$	−4.00000	−0.216295
$343$	8.00000	0.431959
$344$	−12.0000	−0.646997
$345$	4.00000	0.215353
$346$	14.0000	0.752645
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	−2.00000	−0.107211
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	4.00000	0.213809
$351$	−2.00000	−0.106752
$352$	4.00000	0.213201
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	12.0000	0.637793
$355$	4.00000	0.212298
$356$	10.0000	0.529999
$357$	0	0
$358$	12.0000	0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	22.0000	1.15629
$363$	5.00000	0.262432
$364$	−8.00000	−0.419314
$365$	14.0000	0.732793
$366$	−2.00000	−0.104542
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	4.00000	0.208514
$369$	−2.00000	−0.104116
$370$	6.00000	0.311925
$371$	−8.00000	−0.415339
$372$	−4.00000	−0.207390
$373$	38.0000	1.96757	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	38.0000	1.96757	0.983783	+	0.179364i	$0.0574041\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	8.00000	0.412568
$377$	4.00000	0.206010
$378$	4.00000	0.205738
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−4.00000	−0.205196
$381$	−16.0000	−0.819705
$382$	−8.00000	−0.409316
$383$	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
$385$	16.0000	0.815436
$386$	−18.0000	−0.916176
$387$	−12.0000	−0.609994
$388$	−18.0000	−0.913812
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	−2.00000	−0.101274
$391$	0	0
$392$	9.00000	0.454569
$393$	−20.0000	−1.00887
$394$	14.0000	0.705310
$395$	12.0000	0.603786
$396$	4.00000	0.201008
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	−4.00000	−0.200502
$399$	−16.0000	−0.801002
$400$	1.00000	0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	4.00000	0.199502
$403$	8.00000	0.398508
$404$	−10.0000	−0.497519
$405$	1.00000	0.0496904
$406$	−8.00000	−0.397033
$407$	24.0000	1.18964
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	−2.00000	−0.0987730
$411$	−6.00000	−0.295958
$412$	0	0
$413$	48.0000	2.36193
$414$	4.00000	0.196589
$415$	−4.00000	−0.196352
$416$	−2.00000	−0.0980581
$417$	20.0000	0.979404
$418$	−16.0000	−0.782586
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	4.00000	0.195180
$421$	−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
$426$	4.00000	0.193801
$427$	−8.00000	−0.387147
$428$	−12.0000	−0.580042
$429$	−8.00000	−0.386244
$430$	−12.0000	−0.578691
$431$	4.00000	0.192673	0.0963366	−	0.995349i	$-0.469287\pi$
$431$	4.00000	0.192673	0.0963366	+	0.995349i	$0.469287\pi$
$432$	1.00000	0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	−16.0000	−0.768025
$435$	−2.00000	−0.0958927
$436$	−10.0000	−0.478913
$437$	−16.0000	−0.765384
$438$	14.0000	0.668946
$439$	12.0000	0.572729	0.286364	−	0.958121i	$-0.407553\pi$
$439$	12.0000	0.572729	0.286364	+	0.958121i	$0.407553\pi$
$440$	4.00000	0.190693
$441$	9.00000	0.428571
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	6.00000	0.284747
$445$	10.0000	0.474045
$446$	−8.00000	−0.378811
$447$	−2.00000	−0.0945968
$448$	4.00000	0.188982
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	1.00000	0.0471405
$451$	−8.00000	−0.376705
$452$	6.00000	0.282216
$453$	−8.00000	−0.375873
$454$	20.0000	0.938647
$455$	−8.00000	−0.375046
$456$	−4.00000	−0.187317
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	4.00000	0.186501
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	16.0000	0.744387
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	−2.00000	−0.0928477
$465$	−4.00000	−0.185496
$466$	−18.0000	−0.833834
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	−2.00000	−0.0924500
$469$	16.0000	0.738811
$470$	8.00000	0.369012
$471$	−10.0000	−0.460776
$472$	12.0000	0.552345
$473$	−48.0000	−2.20704
$474$	12.0000	0.551178
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	−24.0000	−1.09773
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	1.00000	0.0456435
$481$	−12.0000	−0.547153
$482$	−10.0000	−0.455488
$483$	16.0000	0.728025
$484$	5.00000	0.227273
$485$	−18.0000	−0.817338
$486$	1.00000	0.0453609
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	−2.00000	−0.0905357
$489$	20.0000	0.904431
$490$	9.00000	0.406579
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−2.00000	−0.0901670
$493$	0	0
$494$	8.00000	0.359937
$495$	4.00000	0.179787
$496$	−4.00000	−0.179605
$497$	16.0000	0.717698
$498$	−4.00000	−0.179244
$499$	−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
$502$	−28.0000	−1.24970
$503$	−44.0000	−1.96186	−0.980932	−	0.194354i	$-0.937739\pi$
$503$	−44.0000	−1.96186	−0.980932	+	0.194354i	$0.937739\pi$
$504$	4.00000	0.178174
$505$	−10.0000	−0.444994
$506$	16.0000	0.711287
$507$	−9.00000	−0.399704
$508$	−16.0000	−0.709885
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	0	0
$511$	56.0000	2.47729
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	18.0000	0.793946
$515$	0	0
$516$	−12.0000	−0.528271
$517$	32.0000	1.40736
$518$	24.0000	1.05450
$519$	14.0000	0.614532
$520$	−2.00000	−0.0877058
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−2.00000	−0.0875376
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	−20.0000	−0.873704
$525$	4.00000	0.174574
$526$	−16.0000	−0.697633
$527$	0	0
$528$	4.00000	0.174078
$529$	−7.00000	−0.304348
$530$	−2.00000	−0.0868744
$531$	12.0000	0.520756
$532$	−16.0000	−0.693688
$533$	4.00000	0.173259
$534$	10.0000	0.432742
$535$	−12.0000	−0.518805
$536$	4.00000	0.172774
$537$	12.0000	0.517838
$538$	30.0000	1.29339
$539$	36.0000	1.55063
$540$	1.00000	0.0430331
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	0	0
$545$	−10.0000	−0.428353
$546$	−8.00000	−0.342368
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−6.00000	−0.256307
$549$	−2.00000	−0.0853579
$550$	4.00000	0.170561
$551$	8.00000	0.340811
$552$	4.00000	0.170251
$553$	48.0000	2.04117
$554$	−2.00000	−0.0849719
$555$	6.00000	0.254686
$556$	20.0000	0.848189
$557$	−26.0000	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$558$	−4.00000	−0.169334
$559$	24.0000	1.01509
$560$	4.00000	0.169031
$561$	0	0
$562$	10.0000	0.421825
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	8.00000	0.336861
$565$	6.00000	0.252422
$566$	20.0000	0.840663
$567$	4.00000	0.167984
$568$	4.00000	0.167836
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−4.00000	−0.167542
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	−8.00000	−0.334497
$573$	−8.00000	−0.334205
$574$	−8.00000	−0.333914
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	−18.0000	−0.748054
$580$	−2.00000	−0.0830455
$581$	−16.0000	−0.663792
$582$	−18.0000	−0.746124
$583$	−8.00000	−0.331326
$584$	14.0000	0.579324
$585$	−2.00000	−0.0826898
$586$	−26.0000	−1.07405
$587$	44.0000	1.81607	0.908037	−	0.418890i	$-0.137581\pi$
$587$	44.0000	1.81607	0.908037	+	0.418890i	$0.137581\pi$
$588$	9.00000	0.371154
$589$	16.0000	0.659269
$590$	12.0000	0.494032
$591$	14.0000	0.575883
$592$	6.00000	0.246598
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−2.00000	−0.0819232
$597$	−4.00000	−0.163709
$598$	−8.00000	−0.327144
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	1.00000	0.0408248
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	−48.0000	−1.95633
$603$	4.00000	0.162893
$604$	−8.00000	−0.325515
$605$	5.00000	0.203279
$606$	−10.0000	−0.406222
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	−4.00000	−0.162221
$609$	−8.00000	−0.324176
$610$	−2.00000	−0.0809776
$611$	−16.0000	−0.647291
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−4.00000	−0.161427
$615$	−2.00000	−0.0806478
$616$	16.0000	0.644658
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	−4.00000	−0.160644
$621$	4.00000	0.160514
$622$	−20.0000	−0.801927
$623$	40.0000	1.60257
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	−10.0000	−0.399680
$627$	−16.0000	−0.638978
$628$	−10.0000	−0.399043
$629$	0	0
$630$	4.00000	0.159364
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	12.0000	0.477334
$633$	−4.00000	−0.158986
$634$	−10.0000	−0.397151
$635$	−16.0000	−0.634941
$636$	−2.00000	−0.0793052
$637$	−18.0000	−0.713186
$638$	−8.00000	−0.316723
$639$	4.00000	0.158238
$640$	1.00000	0.0395285
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	−12.0000	−0.473602
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	16.0000	0.630488
$645$	−12.0000	−0.472500
$646$	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
$649$	48.0000	1.88416
$650$	−2.00000	−0.0784465
$651$	−16.0000	−0.627089
$652$	20.0000	0.783260
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	−10.0000	−0.391031
$655$	−20.0000	−0.781465
$656$	−2.00000	−0.0780869
$657$	14.0000	0.546192
$658$	32.0000	1.24749
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	4.00000	0.155700
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	−4.00000	−0.155230
$665$	−16.0000	−0.620453
$666$	6.00000	0.232495
$667$	−8.00000	−0.309761
$668$	−12.0000	−0.464294
$669$	−8.00000	−0.309298
$670$	4.00000	0.154533
$671$	−8.00000	−0.308837
$672$	4.00000	0.154303
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	22.0000	0.847408
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	6.00000	0.230429
$679$	−72.0000	−2.76311
$680$	0	0
$681$	20.0000	0.766402
$682$	−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
$686$	8.00000	0.305441
$687$	6.00000	0.228914
$688$	−12.0000	−0.457496
$689$	4.00000	0.152388
$690$	4.00000	0.152277
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	14.0000	0.532200
$693$	16.0000	0.607790
$694$	−20.0000	−0.759190
$695$	20.0000	0.758643
$696$	−2.00000	−0.0758098
$697$	0	0
$698$	22.0000	0.832712
$699$	−18.0000	−0.680823
$700$	4.00000	0.151186
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	−2.00000	−0.0754851
$703$	−24.0000	−0.905177
$704$	4.00000	0.150756
$705$	8.00000	0.301297
$706$	2.00000	0.0752710
$707$	−40.0000	−1.50435
$708$	12.0000	0.450988
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	4.00000	0.150117
$711$	12.0000	0.450035
$712$	10.0000	0.374766
$713$	−16.0000	−0.599205
$714$	0	0
$715$	−8.00000	−0.299183
$716$	12.0000	0.448461
$717$	−24.0000	−0.896296
$718$	−24.0000	−0.895672
$719$	44.0000	1.64092	0.820462	−	0.571702i	$-0.193717\pi$
$719$	44.0000	1.64092	0.820462	+	0.571702i	$0.193717\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	−3.00000	−0.111648
$723$	−10.0000	−0.371904
$724$	22.0000	0.817624
$725$	−2.00000	−0.0742781
$726$	5.00000	0.185567
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	14.0000	0.518163
$731$	0	0
$732$	−2.00000	−0.0739221
$733$	−18.0000	−0.664845	−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000	−0.664845	−0.332423	+	0.943131i	$0.607866\pi$
$734$	−28.0000	−1.03350
$735$	9.00000	0.331970
$736$	4.00000	0.147442
$737$	16.0000	0.589368
$738$	−2.00000	−0.0736210
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	6.00000	0.220564
$741$	8.00000	0.293887
$742$	−8.00000	−0.293689
$743$	−28.0000	−1.02722	−0.513610	−	0.858024i	$-0.671692\pi$
$743$	−28.0000	−1.02722	−0.513610	+	0.858024i	$0.671692\pi$
$744$	−4.00000	−0.146647
$745$	−2.00000	−0.0732743
$746$	38.0000	1.39128
$747$	−4.00000	−0.146352
$748$	0	0
$749$	−48.0000	−1.75388
$750$	1.00000	0.0365148
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	8.00000	0.291730
$753$	−28.0000	−1.02038
$754$	4.00000	0.145671
$755$	−8.00000	−0.291150
$756$	4.00000	0.145479
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	−20.0000	−0.726433
$759$	16.0000	0.580763
$760$	−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	$0.534685\pi$
$762$	−16.0000	−0.579619
$763$	−40.0000	−1.44810
$764$	−8.00000	−0.289430
$765$	0	0
$766$	8.00000	0.289052
$767$	−24.0000	−0.866590
$768$	1.00000	0.0360844
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	16.0000	0.576600
$771$	18.0000	0.648254
$772$	−18.0000	−0.647834
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−12.0000	−0.431331
$775$	−4.00000	−0.143684
$776$	−18.0000	−0.646162
$777$	24.0000	0.860995
$778$	−18.0000	−0.645331
$779$	8.00000	0.286630
$780$	−2.00000	−0.0716115
$781$	16.0000	0.572525
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	9.00000	0.321429
$785$	−10.0000	−0.356915
$786$	−20.0000	−0.713376
$787$	−36.0000	−1.28326	−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000	−1.28326	−0.641631	+	0.767014i	$0.721742\pi$
$788$	14.0000	0.498729
$789$	−16.0000	−0.569615
$790$	12.0000	0.426941
$791$	24.0000	0.853342
$792$	4.00000	0.142134
$793$	4.00000	0.142044
$794$	30.0000	1.06466
$795$	−2.00000	−0.0709327
$796$	−4.00000	−0.141776
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	−16.0000	−0.566394
$799$	0	0
$800$	1.00000	0.0353553
$801$	10.0000	0.353333
$802$	6.00000	0.211867
$803$	56.0000	1.97620
$804$	4.00000	0.141069
$805$	16.0000	0.563926
$806$	8.00000	0.281788
$807$	30.0000	1.05605
$808$	−10.0000	−0.351799
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	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
$813$	0	0
$814$	24.0000	0.841200
$815$	20.0000	0.700569
$816$	0	0
$817$	48.0000	1.67931
$818$	10.0000	0.349642
$819$	−8.00000	−0.279543
$820$	−2.00000	−0.0698430
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	−6.00000	−0.209274
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	48.0000	1.67013
$827$	−44.0000	−1.53003	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000	−1.53003	−0.765015	+	0.644013i	$0.777268\pi$
$828$	4.00000	0.139010
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	−4.00000	−0.138842
$831$	−2.00000	−0.0693792
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	20.0000	0.692543
$835$	−12.0000	−0.415277
$836$	−16.0000	−0.553372
$837$	−4.00000	−0.138260
$838$	−12.0000	−0.414533
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	4.00000	0.138013
$841$	−25.0000	−0.862069
$842$	−18.0000	−0.620321
$843$	10.0000	0.344418
$844$	−4.00000	−0.137686
$845$	−9.00000	−0.309609
$846$	8.00000	0.275046
$847$	20.0000	0.687208
$848$	−2.00000	−0.0686803
$849$	20.0000	0.686398
$850$	0	0
$851$	24.0000	0.822709
$852$	4.00000	0.137038
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	−8.00000	−0.273754
$855$	−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	$0.377384\pi$
$858$	−8.00000	−0.273115
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	−12.0000	−0.409197
$861$	−8.00000	−0.272639
$862$	4.00000	0.136241
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	1.00000	0.0340207
$865$	14.0000	0.476014
$866$	−14.0000	−0.475739
$867$	0	0
$868$	−16.0000	−0.543075
$869$	48.0000	1.62829
$870$	−2.00000	−0.0678064
$871$	−8.00000	−0.271070
$872$	−10.0000	−0.338643
$873$	−18.0000	−0.609208
$874$	−16.0000	−0.541208
$875$	4.00000	0.135225
$876$	14.0000	0.473016
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	12.0000	0.404980
$879$	−26.0000	−0.876958
$880$	4.00000	0.134840
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	9.00000	0.303046
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	−36.0000	−1.20944
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	6.00000	0.201347
$889$	−64.0000	−2.14649
$890$	10.0000	0.335201
$891$	4.00000	0.134005
$892$	−8.00000	−0.267860
$893$	−32.0000	−1.07084
$894$	−2.00000	−0.0668900
$895$	12.0000	0.401116
$896$	4.00000	0.133631
$897$	−8.00000	−0.267112
$898$	6.00000	0.200223
$899$	8.00000	0.266815
$900$	1.00000	0.0333333
$901$	0	0
$902$	−8.00000	−0.266371
$903$	−48.0000	−1.59734
$904$	6.00000	0.199557
$905$	22.0000	0.731305
$906$	−8.00000	−0.265782
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	20.0000	0.663723
$909$	−10.0000	−0.331679
$910$	−8.00000	−0.265197
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	−4.00000	−0.132453
$913$	−16.0000	−0.529523
$914$	10.0000	0.330771
$915$	−2.00000	−0.0661180
$916$	6.00000	0.198246
$917$	−80.0000	−2.64183
$918$	0	0
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	4.00000	0.131876
$921$	−4.00000	−0.131804
$922$	14.0000	0.461065
$923$	−8.00000	−0.263323
$924$	16.0000	0.526361
$925$	6.00000	0.197279
$926$	40.0000	1.31448
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	−4.00000	−0.131165
$931$	−36.0000	−1.17985
$932$	−18.0000	−0.589610
$933$	−20.0000	−0.654771
$934$	36.0000	1.17796
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	16.0000	0.522419
$939$	−10.0000	−0.326338
$940$	8.00000	0.260931
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	−10.0000	−0.325818
$943$	−8.00000	−0.260516
$944$	12.0000	0.390567
$945$	4.00000	0.130120
$946$	−48.0000	−1.56061
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	12.0000	0.389742
$949$	−28.0000	−0.908918
$950$	−4.00000	−0.129777
$951$	−10.0000	−0.324272
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	−2.00000	−0.0647524
$955$	−8.00000	−0.258874
$956$	−24.0000	−0.776215
$957$	−8.00000	−0.258603
$958$	−20.0000	−0.646171
$959$	−24.0000	−0.775000
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	−12.0000	−0.386896
$963$	−12.0000	−0.386695
$964$	−10.0000	−0.322078
$965$	−18.0000	−0.579441
$966$	16.0000	0.514792
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	−18.0000	−0.577945
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	1.00000	0.0320750
$973$	80.0000	2.56468
$974$	−20.0000	−0.640841
$975$	−2.00000	−0.0640513
$976$	−2.00000	−0.0640184
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	20.0000	0.639529
$979$	40.0000	1.27841
$980$	9.00000	0.287494
$981$	−10.0000	−0.319275
$982$	−12.0000	−0.382935
$983$	28.0000	0.893061	0.446531	−	0.894768i	$-0.352659\pi$
$983$	28.0000	0.893061	0.446531	+	0.894768i	$0.352659\pi$
$984$	−2.00000	−0.0637577
$985$	14.0000	0.446077
$986$	0	0
$987$	32.0000	1.01857
$988$	8.00000	0.254514
$989$	−48.0000	−1.52631
$990$	4.00000	0.127128
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	−4.00000	−0.127000
$993$	28.0000	0.888553
$994$	16.0000	0.507489
$995$	−4.00000	−0.126809
$996$	−4.00000	−0.126745
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	−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.bb.1.1		1
17.16	even	2		510.2.a.c.1.1	✓	1
51.50	odd	2		1530.2.a.d.1.1		1
68.67	odd	2		4080.2.a.x.1.1		1
85.33	odd	4		2550.2.d.b.2449.1		2
85.67	odd	4		2550.2.d.b.2449.2		2
85.84	even	2		2550.2.a.n.1.1		1
255.254	odd	2		7650.2.a.cn.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.c.1.1	✓	1	17.16	even	2
1530.2.a.d.1.1		1	51.50	odd	2
2550.2.a.n.1.1		1	85.84	even	2
2550.2.d.b.2449.1		2	85.33	odd	4
2550.2.d.b.2449.2		2	85.67	odd	4
4080.2.a.x.1.1		1	68.67	odd	2
7650.2.a.cn.1.1		1	255.254	odd	2
8670.2.a.bb.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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