LMFDB - Embedded newform 8670.2.a.o.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:	yes
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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -6.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +7.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -6.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -4.00000 q^{29} +1.00000 q^{30} +3.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} +1.00000 q^{35} +1.00000 q^{36} -5.00000 q^{37} +7.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} +2.00000 q^{41} +1.00000 q^{42} -5.00000 q^{43} -6.00000 q^{44} -1.00000 q^{45} +4.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} +6.00000 q^{55} -1.00000 q^{56} -7.00000 q^{57} -4.00000 q^{58} -2.00000 q^{59} +1.00000 q^{60} +1.00000 q^{61} +3.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} +6.00000 q^{66} -7.00000 q^{67} -4.00000 q^{69} +1.00000 q^{70} +12.0000 q^{71} +1.00000 q^{72} +14.0000 q^{73} -5.00000 q^{74} -1.00000 q^{75} +7.00000 q^{76} +6.00000 q^{77} -1.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -10.0000 q^{83} +1.00000 q^{84} -5.00000 q^{86} +4.00000 q^{87} -6.00000 q^{88} -4.00000 q^{89} -1.00000 q^{90} -1.00000 q^{91} +4.00000 q^{92} -3.00000 q^{93} +4.00000 q^{94} -7.00000 q^{95} -1.00000 q^{96} -1.00000 q^{97} -6.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	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
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−1.00000	−0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	1.00000	0.235702
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	−1.00000	−0.223607
$21$	1.00000	0.218218
$22$	−6.00000	−1.27920
$23$	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
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	1.00000	0.182574
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	1.00000	0.176777
$33$	6.00000	1.04447
$34$	0	0
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	7.00000	1.13555
$39$	−1.00000	−0.160128
$40$	−1.00000	−0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	1.00000	0.154303
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	−6.00000	−0.904534
$45$	−1.00000	−0.149071
$46$	4.00000	0.589768
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	1.00000	0.141421
$51$	0	0
$52$	1.00000	0.138675
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	−1.00000	−0.136083
$55$	6.00000	0.809040
$56$	−1.00000	−0.133631
$57$	−7.00000	−0.927173
$58$	−4.00000	−0.525226
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	1.00000	0.129099
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	3.00000	0.381000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	6.00000	0.738549
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	1.00000	0.119523
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	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	$0.194369\pi$
$74$	−5.00000	−0.581238
$75$	−1.00000	−0.115470
$76$	7.00000	0.802955
$77$	6.00000	0.683763
$78$	−1.00000	−0.113228
$79$	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
$82$	2.00000	0.220863
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	−5.00000	−0.539164
$87$	4.00000	0.428845
$88$	−6.00000	−0.639602
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	−1.00000	−0.105409
$91$	−1.00000	−0.104828
$92$	4.00000	0.417029
$93$	−3.00000	−0.311086
$94$	4.00000	0.412568
$95$	−7.00000	−0.718185
$96$	−1.00000	−0.102062
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	−6.00000	−0.606092
$99$	−6.00000	−0.603023
$100$	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
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	1.00000	0.0980581
$105$	−1.00000	−0.0975900
$106$	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
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	6.00000	0.572078
$111$	5.00000	0.474579
$112$	−1.00000	−0.0944911
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	−7.00000	−0.655610
$115$	−4.00000	−0.373002
$116$	−4.00000	−0.371391
$117$	1.00000	0.0924500
$118$	−2.00000	−0.184115
$119$	0	0
$120$	1.00000	0.0912871
$121$	25.0000	2.27273
$122$	1.00000	0.0905357
$123$	−2.00000	−0.180334
$124$	3.00000	0.269408
$125$	−1.00000	−0.0894427
$126$	−1.00000	−0.0890871
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	1.00000	0.0883883
$129$	5.00000	0.440225
$130$	−1.00000	−0.0877058
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	6.00000	0.522233
$133$	−7.00000	−0.606977
$134$	−7.00000	−0.604708
$135$	1.00000	0.0860663
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	−4.00000	−0.340503
$139$	15.0000	1.27228	0.636142	−	0.771572i	$-0.280529\pi$
$139$	15.0000	1.27228	0.636142	+	0.771572i	$0.280529\pi$
$140$	1.00000	0.0845154
$141$	−4.00000	−0.336861
$142$	12.0000	1.00702
$143$	−6.00000	−0.501745
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	14.0000	1.15865
$147$	6.00000	0.494872
$148$	−5.00000	−0.410997
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	−1.00000	−0.0816497
$151$	−19.0000	−1.54620	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−19.0000	−1.54620	−0.773099	+	0.634285i	$0.781294\pi$
$152$	7.00000	0.567775
$153$	0	0
$154$	6.00000	0.483494
$155$	−3.00000	−0.240966
$156$	−1.00000	−0.0800641
$157$	−5.00000	−0.399043	−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000	−0.399043	−0.199522	+	0.979893i	$0.563939\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	−1.00000	−0.0790569
$161$	−4.00000	−0.315244
$162$	1.00000	0.0785674
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	2.00000	0.156174
$165$	−6.00000	−0.467099
$166$	−10.0000	−0.776151
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	1.00000	0.0771517
$169$	−12.0000	−0.923077
$170$	0	0
$171$	7.00000	0.535303
$172$	−5.00000	−0.381246
$173$	−20.0000	−1.52057	−0.760286	−	0.649589i	$-0.774941\pi$
$173$	−20.0000	−1.52057	−0.760286	+	0.649589i	$0.774941\pi$
$174$	4.00000	0.303239
$175$	−1.00000	−0.0755929
$176$	−6.00000	−0.452267
$177$	2.00000	0.150329
$178$	−4.00000	−0.299813
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	−1.00000	−0.0745356
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	−1.00000	−0.0741249
$183$	−1.00000	−0.0739221
$184$	4.00000	0.294884
$185$	5.00000	0.367607
$186$	−3.00000	−0.219971
$187$	0	0
$188$	4.00000	0.291730
$189$	1.00000	0.0727393
$190$	−7.00000	−0.507833
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	−1.00000	−0.0721688
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	−1.00000	−0.0717958
$195$	1.00000	0.0716115
$196$	−6.00000	−0.428571
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−6.00000	−0.426401
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	1.00000	0.0707107
$201$	7.00000	0.493742
$202$	−10.0000	−0.703598
$203$	4.00000	0.280745
$204$	0	0
$205$	−2.00000	−0.139686
$206$	11.0000	0.766406
$207$	4.00000	0.278019
$208$	1.00000	0.0693375
$209$	−42.0000	−2.90520
$210$	−1.00000	−0.0690066
$211$	21.0000	1.44570	0.722850	−	0.691005i	$-0.242832\pi$
$211$	21.0000	1.44570	0.722850	+	0.691005i	$0.242832\pi$
$212$	2.00000	0.137361
$213$	−12.0000	−0.822226
$214$	−12.0000	−0.820303
$215$	5.00000	0.340997
$216$	−1.00000	−0.0680414
$217$	−3.00000	−0.203653
$218$	−6.00000	−0.406371
$219$	−14.0000	−0.946032
$220$	6.00000	0.404520
$221$	0	0
$222$	5.00000	0.335578
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−1.00000	−0.0668153
$225$	1.00000	0.0666667
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−7.00000	−0.463586
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	−4.00000	−0.263752
$231$	−6.00000	−0.394771
$232$	−4.00000	−0.262613
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	1.00000	0.0653720
$235$	−4.00000	−0.260931
$236$	−2.00000	−0.130189
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	1.00000	0.0645497
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	25.0000	1.60706
$243$	−1.00000	−0.0641500
$244$	1.00000	0.0640184
$245$	6.00000	0.383326
$246$	−2.00000	−0.127515
$247$	7.00000	0.445399
$248$	3.00000	0.190500
$249$	10.0000	0.633724
$250$	−1.00000	−0.0632456
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	−1.00000	−0.0629941
$253$	−24.0000	−1.50887
$254$	7.00000	0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	28.0000	1.74659	0.873296	−	0.487190i	$-0.161978\pi$
$257$	28.0000	1.74659	0.873296	+	0.487190i	$0.161978\pi$
$258$	5.00000	0.311286
$259$	5.00000	0.310685
$260$	−1.00000	−0.0620174
$261$	−4.00000	−0.247594
$262$	−18.0000	−1.11204
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	6.00000	0.369274
$265$	−2.00000	−0.122859
$266$	−7.00000	−0.429198
$267$	4.00000	0.244796
$268$	−7.00000	−0.427593
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	1.00000	0.0608581
$271$	−29.0000	−1.76162	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−29.0000	−1.76162	−0.880812	+	0.473466i	$0.843003\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	−12.0000	−0.724947
$275$	−6.00000	−0.361814
$276$	−4.00000	−0.240772
$277$	−23.0000	−1.38194	−0.690968	−	0.722885i	$-0.742815\pi$
$277$	−23.0000	−1.38194	−0.690968	+	0.722885i	$0.742815\pi$
$278$	15.0000	0.899640
$279$	3.00000	0.179605
$280$	1.00000	0.0597614
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	−4.00000	−0.238197
$283$	−1.00000	−0.0594438	−0.0297219	−	0.999558i	$-0.509462\pi$
$283$	−1.00000	−0.0594438	−0.0297219	+	0.999558i	$0.509462\pi$
$284$	12.0000	0.712069
$285$	7.00000	0.414644
$286$	−6.00000	−0.354787
$287$	−2.00000	−0.118056
$288$	1.00000	0.0589256
$289$	0	0
$290$	4.00000	0.234888
$291$	1.00000	0.0586210
$292$	14.0000	0.819288
$293$	4.00000	0.233682	0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000	0.233682	0.116841	+	0.993151i	$0.462723\pi$
$294$	6.00000	0.349927
$295$	2.00000	0.116445
$296$	−5.00000	−0.290619
$297$	6.00000	0.348155
$298$	−18.0000	−1.04271
$299$	4.00000	0.231326
$300$	−1.00000	−0.0577350
$301$	5.00000	0.288195
$302$	−19.0000	−1.09333
$303$	10.0000	0.574485
$304$	7.00000	0.401478
$305$	−1.00000	−0.0572598
$306$	0	0
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	6.00000	0.341882
$309$	−11.0000	−0.625768
$310$	−3.00000	−0.170389
$311$	−28.0000	−1.58773	−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000	−1.58773	−0.793867	+	0.608091i	$0.791935\pi$
$312$	−1.00000	−0.0566139
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	−5.00000	−0.282166
$315$	1.00000	0.0563436
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	−2.00000	−0.112154
$319$	24.0000	1.34374
$320$	−1.00000	−0.0559017
$321$	12.0000	0.669775
$322$	−4.00000	−0.222911
$323$	0	0
$324$	1.00000	0.0555556
$325$	1.00000	0.0554700
$326$	−24.0000	−1.32924
$327$	6.00000	0.331801
$328$	2.00000	0.110432
$329$	−4.00000	−0.220527
$330$	−6.00000	−0.330289
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	−10.0000	−0.548821
$333$	−5.00000	−0.273998
$334$	4.00000	0.218870
$335$	7.00000	0.382451
$336$	1.00000	0.0545545
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	0	0
$341$	−18.0000	−0.974755
$342$	7.00000	0.378517
$343$	13.0000	0.701934
$344$	−5.00000	−0.269582
$345$	4.00000	0.215353
$346$	−20.0000	−1.07521
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	4.00000	0.214423
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	−1.00000	−0.0534522
$351$	−1.00000	−0.0533761
$352$	−6.00000	−0.319801
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	2.00000	0.106299
$355$	−12.0000	−0.636894
$356$	−4.00000	−0.212000
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−28.0000	−1.47778	−0.738892	−	0.673824i	$-0.764651\pi$
$359$	−28.0000	−1.47778	−0.738892	+	0.673824i	$0.764651\pi$
$360$	−1.00000	−0.0527046
$361$	30.0000	1.57895
$362$	11.0000	0.578147
$363$	−25.0000	−1.31216
$364$	−1.00000	−0.0524142
$365$	−14.0000	−0.732793
$366$	−1.00000	−0.0522708
$367$	−11.0000	−0.574195	−0.287098	−	0.957901i	$-0.592690\pi$
$367$	−11.0000	−0.574195	−0.287098	+	0.957901i	$0.592690\pi$
$368$	4.00000	0.208514
$369$	2.00000	0.104116
$370$	5.00000	0.259938
$371$	−2.00000	−0.103835
$372$	−3.00000	−0.155543
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	4.00000	0.206284
$377$	−4.00000	−0.206010
$378$	1.00000	0.0514344
$379$	−17.0000	−0.873231	−0.436616	−	0.899648i	$-0.643823\pi$
$379$	−17.0000	−0.873231	−0.436616	+	0.899648i	$0.643823\pi$
$380$	−7.00000	−0.359092
$381$	−7.00000	−0.358621
$382$	−18.0000	−0.920960
$383$	28.0000	1.43073	0.715367	−	0.698749i	$-0.246260\pi$
$383$	28.0000	1.43073	0.715367	+	0.698749i	$0.246260\pi$
$384$	−1.00000	−0.0510310
$385$	−6.00000	−0.305788
$386$	−11.0000	−0.559885
$387$	−5.00000	−0.254164
$388$	−1.00000	−0.0507673
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	1.00000	0.0506370
$391$	0	0
$392$	−6.00000	−0.303046
$393$	18.0000	0.907980
$394$	18.0000	0.906827
$395$	0	0
$396$	−6.00000	−0.301511
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−20.0000	−1.00251
$399$	7.00000	0.350438
$400$	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	$0.547867\pi$
$402$	7.00000	0.349128
$403$	3.00000	0.149441
$404$	−10.0000	−0.497519
$405$	−1.00000	−0.0496904
$406$	4.00000	0.198517
$407$	30.0000	1.48704
$408$	0	0
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	−2.00000	−0.0987730
$411$	12.0000	0.591916
$412$	11.0000	0.541931
$413$	2.00000	0.0984136
$414$	4.00000	0.196589
$415$	10.0000	0.490881
$416$	1.00000	0.0490290
$417$	−15.0000	−0.734553
$418$	−42.0000	−2.05429
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	−1.00000	−0.0487950
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	21.0000	1.02226
$423$	4.00000	0.194487
$424$	2.00000	0.0971286
$425$	0	0
$426$	−12.0000	−0.581402
$427$	−1.00000	−0.0483934
$428$	−12.0000	−0.580042
$429$	6.00000	0.289683
$430$	5.00000	0.241121
$431$	26.0000	1.25238	0.626188	−	0.779672i	$-0.284614\pi$
$431$	26.0000	1.25238	0.626188	+	0.779672i	$0.284614\pi$
$432$	−1.00000	−0.0481125
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	−3.00000	−0.144005
$435$	−4.00000	−0.191785
$436$	−6.00000	−0.287348
$437$	28.0000	1.33942
$438$	−14.0000	−0.668946
$439$	−15.0000	−0.715911	−0.357955	−	0.933739i	$-0.616526\pi$
$439$	−15.0000	−0.715911	−0.357955	+	0.933739i	$0.616526\pi$
$440$	6.00000	0.286039
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−14.0000	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$443$	−14.0000	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$444$	5.00000	0.237289
$445$	4.00000	0.189618
$446$	8.00000	0.378811
$447$	18.0000	0.851371
$448$	−1.00000	−0.0472456
$449$	4.00000	0.188772	0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000	0.188772	0.0943858	+	0.995536i	$0.469911\pi$
$450$	1.00000	0.0471405
$451$	−12.0000	−0.565058
$452$	0	0
$453$	19.0000	0.892698
$454$	−12.0000	−0.563188
$455$	1.00000	0.0468807
$456$	−7.00000	−0.327805
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	22.0000	1.02799
$459$	0	0
$460$	−4.00000	−0.186501
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	−6.00000	−0.279145
$463$	−29.0000	−1.34774	−0.673872	−	0.738848i	$-0.735370\pi$
$463$	−29.0000	−1.34774	−0.673872	+	0.738848i	$0.735370\pi$
$464$	−4.00000	−0.185695
$465$	3.00000	0.139122
$466$	−22.0000	−1.01913
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	1.00000	0.0462250
$469$	7.00000	0.323230
$470$	−4.00000	−0.184506
$471$	5.00000	0.230388
$472$	−2.00000	−0.0920575
$473$	30.0000	1.37940
$474$	0	0
$475$	7.00000	0.321182
$476$	0	0
$477$	2.00000	0.0915737
$478$	−12.0000	−0.548867
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	1.00000	0.0456435
$481$	−5.00000	−0.227980
$482$	18.0000	0.819878
$483$	4.00000	0.182006
$484$	25.0000	1.13636
$485$	1.00000	0.0454077
$486$	−1.00000	−0.0453609
$487$	−1.00000	−0.0453143	−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000	−0.0453143	−0.0226572	+	0.999743i	$0.507213\pi$
$488$	1.00000	0.0452679
$489$	24.0000	1.08532
$490$	6.00000	0.271052
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	−2.00000	−0.0901670
$493$	0	0
$494$	7.00000	0.314945
$495$	6.00000	0.269680
$496$	3.00000	0.134704
$497$	−12.0000	−0.538274
$498$	10.0000	0.448111
$499$	−27.0000	−1.20869	−0.604343	−	0.796724i	$-0.706564\pi$
$499$	−27.0000	−1.20869	−0.604343	+	0.796724i	$0.706564\pi$
$500$	−1.00000	−0.0447214
$501$	−4.00000	−0.178707
$502$	−20.0000	−0.892644
$503$	−14.0000	−0.624229	−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000	−0.624229	−0.312115	+	0.950044i	$0.601037\pi$
$504$	−1.00000	−0.0445435
$505$	10.0000	0.444994
$506$	−24.0000	−1.06693
$507$	12.0000	0.532939
$508$	7.00000	0.310575
$509$	−8.00000	−0.354594	−0.177297	−	0.984157i	$-0.556735\pi$
$509$	−8.00000	−0.354594	−0.177297	+	0.984157i	$0.556735\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	1.00000	0.0441942
$513$	−7.00000	−0.309058
$514$	28.0000	1.23503
$515$	−11.0000	−0.484718
$516$	5.00000	0.220113
$517$	−24.0000	−1.05552
$518$	5.00000	0.219687
$519$	20.0000	0.877903
$520$	−1.00000	−0.0438529
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	−4.00000	−0.175075
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	−18.0000	−0.786334
$525$	1.00000	0.0436436
$526$	24.0000	1.04645
$527$	0	0
$528$	6.00000	0.261116
$529$	−7.00000	−0.304348
$530$	−2.00000	−0.0868744
$531$	−2.00000	−0.0867926
$532$	−7.00000	−0.303488
$533$	2.00000	0.0866296
$534$	4.00000	0.173097
$535$	12.0000	0.518805
$536$	−7.00000	−0.302354
$537$	12.0000	0.517838
$538$	16.0000	0.689809
$539$	36.0000	1.55063
$540$	1.00000	0.0430331
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	−29.0000	−1.24566
$543$	−11.0000	−0.472055
$544$	0	0
$545$	6.00000	0.257012
$546$	1.00000	0.0427960
$547$	27.0000	1.15444	0.577218	−	0.816590i	$-0.304138\pi$
$547$	27.0000	1.15444	0.577218	+	0.816590i	$0.304138\pi$
$548$	−12.0000	−0.512615
$549$	1.00000	0.0426790
$550$	−6.00000	−0.255841
$551$	−28.0000	−1.19284
$552$	−4.00000	−0.170251
$553$	0	0
$554$	−23.0000	−0.977176
$555$	−5.00000	−0.212238
$556$	15.0000	0.636142
$557$	−28.0000	−1.18640	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000	−1.18640	−0.593199	+	0.805056i	$0.702135\pi$
$558$	3.00000	0.127000
$559$	−5.00000	−0.211477
$560$	1.00000	0.0422577
$561$	0	0
$562$	16.0000	0.674919
$563$	22.0000	0.927189	0.463595	−	0.886047i	$-0.346559\pi$
$563$	22.0000	0.927189	0.463595	+	0.886047i	$0.346559\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	−1.00000	−0.0420331
$567$	−1.00000	−0.0419961
$568$	12.0000	0.503509
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	7.00000	0.293198
$571$	−21.0000	−0.878823	−0.439411	−	0.898286i	$-0.644813\pi$
$571$	−21.0000	−0.878823	−0.439411	+	0.898286i	$0.644813\pi$
$572$	−6.00000	−0.250873
$573$	18.0000	0.751961
$574$	−2.00000	−0.0834784
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	13.0000	0.541197	0.270599	−	0.962692i	$-0.412778\pi$
$577$	13.0000	0.541197	0.270599	+	0.962692i	$0.412778\pi$
$578$	0	0
$579$	11.0000	0.457144
$580$	4.00000	0.166091
$581$	10.0000	0.414870
$582$	1.00000	0.0414513
$583$	−12.0000	−0.496989
$584$	14.0000	0.579324
$585$	−1.00000	−0.0413449
$586$	4.00000	0.165238
$587$	−40.0000	−1.65098	−0.825488	−	0.564419i	$-0.809100\pi$
$587$	−40.0000	−1.65098	−0.825488	+	0.564419i	$0.809100\pi$
$588$	6.00000	0.247436
$589$	21.0000	0.865290
$590$	2.00000	0.0823387
$591$	−18.0000	−0.740421
$592$	−5.00000	−0.205499
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	−18.0000	−0.737309
$597$	20.0000	0.818546
$598$	4.00000	0.163572
$599$	−18.0000	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$599$	−18.0000	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$600$	−1.00000	−0.0408248
$601$	−1.00000	−0.0407909	−0.0203954	−	0.999792i	$-0.506493\pi$
$601$	−1.00000	−0.0407909	−0.0203954	+	0.999792i	$0.506493\pi$
$602$	5.00000	0.203785
$603$	−7.00000	−0.285062
$604$	−19.0000	−0.773099
$605$	−25.0000	−1.01639
$606$	10.0000	0.406222
$607$	−41.0000	−1.66414	−0.832069	−	0.554672i	$-0.812844\pi$
$607$	−41.0000	−1.66414	−0.832069	+	0.554672i	$0.812844\pi$
$608$	7.00000	0.283887
$609$	−4.00000	−0.162088
$610$	−1.00000	−0.0404888
$611$	4.00000	0.161823
$612$	0	0
$613$	15.0000	0.605844	0.302922	−	0.953015i	$-0.402038\pi$
$613$	15.0000	0.605844	0.302922	+	0.953015i	$0.402038\pi$
$614$	−32.0000	−1.29141
$615$	2.00000	0.0806478
$616$	6.00000	0.241747
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	−11.0000	−0.442485
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	−3.00000	−0.120483
$621$	−4.00000	−0.160514
$622$	−28.0000	−1.12270
$623$	4.00000	0.160257
$624$	−1.00000	−0.0400320
$625$	1.00000	0.0400000
$626$	−19.0000	−0.759393
$627$	42.0000	1.67732
$628$	−5.00000	−0.199522
$629$	0	0
$630$	1.00000	0.0398410
$631$	−13.0000	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$632$	0	0
$633$	−21.0000	−0.834675
$634$	0	0
$635$	−7.00000	−0.277787
$636$	−2.00000	−0.0793052
$637$	−6.00000	−0.237729
$638$	24.0000	0.950169
$639$	12.0000	0.474713
$640$	−1.00000	−0.0395285
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	12.0000	0.473602
$643$	13.0000	0.512670	0.256335	−	0.966588i	$-0.417485\pi$
$643$	13.0000	0.512670	0.256335	+	0.966588i	$0.417485\pi$
$644$	−4.00000	−0.157622
$645$	−5.00000	−0.196875
$646$	0	0
$647$	−14.0000	−0.550397	−0.275198	−	0.961387i	$-0.588744\pi$
$647$	−14.0000	−0.550397	−0.275198	+	0.961387i	$0.588744\pi$
$648$	1.00000	0.0392837
$649$	12.0000	0.471041
$650$	1.00000	0.0392232
$651$	3.00000	0.117579
$652$	−24.0000	−0.939913
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	6.00000	0.234619
$655$	18.0000	0.703318
$656$	2.00000	0.0780869
$657$	14.0000	0.546192
$658$	−4.00000	−0.155936
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	−6.00000	−0.233550
$661$	−15.0000	−0.583432	−0.291716	−	0.956505i	$-0.594226\pi$
$661$	−15.0000	−0.583432	−0.291716	+	0.956505i	$0.594226\pi$
$662$	−19.0000	−0.738456
$663$	0	0
$664$	−10.0000	−0.388075
$665$	7.00000	0.271448
$666$	−5.00000	−0.193746
$667$	−16.0000	−0.619522
$668$	4.00000	0.154765
$669$	−8.00000	−0.309298
$670$	7.00000	0.270434
$671$	−6.00000	−0.231627
$672$	1.00000	0.0385758
$673$	5.00000	0.192736	0.0963679	−	0.995346i	$-0.469277\pi$
$673$	5.00000	0.192736	0.0963679	+	0.995346i	$0.469277\pi$
$674$	30.0000	1.15556
$675$	−1.00000	−0.0384900
$676$	−12.0000	−0.461538
$677$	−28.0000	−1.07613	−0.538064	−	0.842904i	$-0.680844\pi$
$677$	−28.0000	−1.07613	−0.538064	+	0.842904i	$0.680844\pi$
$678$	0	0
$679$	1.00000	0.0383765
$680$	0	0
$681$	12.0000	0.459841
$682$	−18.0000	−0.689256
$683$	42.0000	1.60709	0.803543	−	0.595247i	$-0.202946\pi$
$683$	42.0000	1.60709	0.803543	+	0.595247i	$0.202946\pi$
$684$	7.00000	0.267652
$685$	12.0000	0.458496
$686$	13.0000	0.496342
$687$	−22.0000	−0.839352
$688$	−5.00000	−0.190623
$689$	2.00000	0.0761939
$690$	4.00000	0.152277
$691$	−33.0000	−1.25538	−0.627690	−	0.778464i	$-0.715999\pi$
$691$	−33.0000	−1.25538	−0.627690	+	0.778464i	$0.715999\pi$
$692$	−20.0000	−0.760286
$693$	6.00000	0.227921
$694$	−20.0000	−0.759190
$695$	−15.0000	−0.568982
$696$	4.00000	0.151620
$697$	0	0
$698$	14.0000	0.529908
$699$	22.0000	0.832116
$700$	−1.00000	−0.0377964
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	−1.00000	−0.0377426
$703$	−35.0000	−1.32005
$704$	−6.00000	−0.226134
$705$	4.00000	0.150649
$706$	−6.00000	−0.225813
$707$	10.0000	0.376089
$708$	2.00000	0.0751646
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	−12.0000	−0.450352
$711$	0	0
$712$	−4.00000	−0.149906
$713$	12.0000	0.449404
$714$	0	0
$715$	6.00000	0.224387
$716$	−12.0000	−0.448461
$717$	12.0000	0.448148
$718$	−28.0000	−1.04495
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	−1.00000	−0.0372678
$721$	−11.0000	−0.409661
$722$	30.0000	1.11648
$723$	−18.0000	−0.669427
$724$	11.0000	0.408812
$725$	−4.00000	−0.148556
$726$	−25.0000	−0.927837
$727$	39.0000	1.44643	0.723215	−	0.690623i	$-0.242664\pi$
$727$	39.0000	1.44643	0.723215	+	0.690623i	$0.242664\pi$
$728$	−1.00000	−0.0370625
$729$	1.00000	0.0370370
$730$	−14.0000	−0.518163
$731$	0	0
$732$	−1.00000	−0.0369611
$733$	41.0000	1.51437	0.757185	−	0.653201i	$-0.226574\pi$
$733$	41.0000	1.51437	0.757185	+	0.653201i	$0.226574\pi$
$734$	−11.0000	−0.406017
$735$	−6.00000	−0.221313
$736$	4.00000	0.147442
$737$	42.0000	1.54709
$738$	2.00000	0.0736210
$739$	29.0000	1.06678	0.533391	−	0.845869i	$-0.320917\pi$
$739$	29.0000	1.06678	0.533391	+	0.845869i	$0.320917\pi$
$740$	5.00000	0.183804
$741$	−7.00000	−0.257151
$742$	−2.00000	−0.0734223
$743$	−42.0000	−1.54083	−0.770415	−	0.637542i	$-0.779951\pi$
$743$	−42.0000	−1.54083	−0.770415	+	0.637542i	$0.779951\pi$
$744$	−3.00000	−0.109985
$745$	18.0000	0.659469
$746$	26.0000	0.951928
$747$	−10.0000	−0.365881
$748$	0	0
$749$	12.0000	0.438470
$750$	1.00000	0.0365148
$751$	−7.00000	−0.255434	−0.127717	−	0.991811i	$-0.540765\pi$
$751$	−7.00000	−0.255434	−0.127717	+	0.991811i	$0.540765\pi$
$752$	4.00000	0.145865
$753$	20.0000	0.728841
$754$	−4.00000	−0.145671
$755$	19.0000	0.691481
$756$	1.00000	0.0363696
$757$	13.0000	0.472493	0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000	0.472493	0.236247	+	0.971693i	$0.424083\pi$
$758$	−17.0000	−0.617468
$759$	24.0000	0.871145
$760$	−7.00000	−0.253917
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	−7.00000	−0.253583
$763$	6.00000	0.217215
$764$	−18.0000	−0.651217
$765$	0	0
$766$	28.0000	1.01168
$767$	−2.00000	−0.0722158
$768$	−1.00000	−0.0360844
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	−6.00000	−0.216225
$771$	−28.0000	−1.00840
$772$	−11.0000	−0.395899
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	−5.00000	−0.179721
$775$	3.00000	0.107763
$776$	−1.00000	−0.0358979
$777$	−5.00000	−0.179374
$778$	30.0000	1.07555
$779$	14.0000	0.501602
$780$	1.00000	0.0358057
$781$	−72.0000	−2.57636
$782$	0	0
$783$	4.00000	0.142948
$784$	−6.00000	−0.214286
$785$	5.00000	0.178458
$786$	18.0000	0.642039
$787$	29.0000	1.03374	0.516869	−	0.856064i	$-0.327097\pi$
$787$	29.0000	1.03374	0.516869	+	0.856064i	$0.327097\pi$
$788$	18.0000	0.641223
$789$	−24.0000	−0.854423
$790$	0	0
$791$	0	0
$792$	−6.00000	−0.213201
$793$	1.00000	0.0355110
$794$	2.00000	0.0709773
$795$	2.00000	0.0709327
$796$	−20.0000	−0.708881
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	7.00000	0.247797
$799$	0	0
$800$	1.00000	0.0353553
$801$	−4.00000	−0.141333
$802$	−6.00000	−0.211867
$803$	−84.0000	−2.96430
$804$	7.00000	0.246871
$805$	4.00000	0.140981
$806$	3.00000	0.105670
$807$	−16.0000	−0.563227
$808$	−10.0000	−0.351799
$809$	−46.0000	−1.61727	−0.808637	−	0.588308i	$-0.799794\pi$
$809$	−46.0000	−1.61727	−0.808637	+	0.588308i	$0.799794\pi$
$810$	−1.00000	−0.0351364
$811$	7.00000	0.245803	0.122902	−	0.992419i	$-0.460780\pi$
$811$	7.00000	0.245803	0.122902	+	0.992419i	$0.460780\pi$
$812$	4.00000	0.140372
$813$	29.0000	1.01707
$814$	30.0000	1.05150
$815$	24.0000	0.840683
$816$	0	0
$817$	−35.0000	−1.22449
$818$	−11.0000	−0.384606
$819$	−1.00000	−0.0349428
$820$	−2.00000	−0.0698430
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	12.0000	0.418548
$823$	−23.0000	−0.801730	−0.400865	−	0.916137i	$-0.631290\pi$
$823$	−23.0000	−0.801730	−0.400865	+	0.916137i	$0.631290\pi$
$824$	11.0000	0.383203
$825$	6.00000	0.208893
$826$	2.00000	0.0695889
$827$	46.0000	1.59958	0.799788	−	0.600282i	$-0.204945\pi$
$827$	46.0000	1.59958	0.799788	+	0.600282i	$0.204945\pi$
$828$	4.00000	0.139010
$829$	−13.0000	−0.451509	−0.225754	−	0.974184i	$-0.572485\pi$
$829$	−13.0000	−0.451509	−0.225754	+	0.974184i	$0.572485\pi$
$830$	10.0000	0.347105
$831$	23.0000	0.797861
$832$	1.00000	0.0346688
$833$	0	0
$834$	−15.0000	−0.519408
$835$	−4.00000	−0.138426
$836$	−42.0000	−1.45260
$837$	−3.00000	−0.103695
$838$	26.0000	0.898155
$839$	−46.0000	−1.58810	−0.794048	−	0.607855i	$-0.792030\pi$
$839$	−46.0000	−1.58810	−0.794048	+	0.607855i	$0.792030\pi$
$840$	−1.00000	−0.0345033
$841$	−13.0000	−0.448276
$842$	−1.00000	−0.0344623
$843$	−16.0000	−0.551069
$844$	21.0000	0.722850
$845$	12.0000	0.412813
$846$	4.00000	0.137523
$847$	−25.0000	−0.859010
$848$	2.00000	0.0686803
$849$	1.00000	0.0343199
$850$	0	0
$851$	−20.0000	−0.685591
$852$	−12.0000	−0.411113
$853$	51.0000	1.74621	0.873103	−	0.487535i	$-0.162104\pi$
$853$	51.0000	1.74621	0.873103	+	0.487535i	$0.162104\pi$
$854$	−1.00000	−0.0342193
$855$	−7.00000	−0.239395
$856$	−12.0000	−0.410152
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	6.00000	0.204837
$859$	15.0000	0.511793	0.255897	−	0.966704i	$-0.417629\pi$
$859$	15.0000	0.511793	0.255897	+	0.966704i	$0.417629\pi$
$860$	5.00000	0.170499
$861$	2.00000	0.0681598
$862$	26.0000	0.885564
$863$	−10.0000	−0.340404	−0.170202	−	0.985409i	$-0.554442\pi$
$863$	−10.0000	−0.340404	−0.170202	+	0.985409i	$0.554442\pi$
$864$	−1.00000	−0.0340207
$865$	20.0000	0.680020
$866$	5.00000	0.169907
$867$	0	0
$868$	−3.00000	−0.101827
$869$	0	0
$870$	−4.00000	−0.135613
$871$	−7.00000	−0.237186
$872$	−6.00000	−0.203186
$873$	−1.00000	−0.0338449
$874$	28.0000	0.947114
$875$	1.00000	0.0338062
$876$	−14.0000	−0.473016
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	−15.0000	−0.506225
$879$	−4.00000	−0.134917
$880$	6.00000	0.202260
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	−6.00000	−0.202031
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	0	0
$885$	−2.00000	−0.0672293
$886$	−14.0000	−0.470339
$887$	2.00000	0.0671534	0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000	0.0671534	0.0335767	+	0.999436i	$0.489310\pi$
$888$	5.00000	0.167789
$889$	−7.00000	−0.234772
$890$	4.00000	0.134080
$891$	−6.00000	−0.201008
$892$	8.00000	0.267860
$893$	28.0000	0.936984
$894$	18.0000	0.602010
$895$	12.0000	0.401116
$896$	−1.00000	−0.0334077
$897$	−4.00000	−0.133556
$898$	4.00000	0.133482
$899$	−12.0000	−0.400222
$900$	1.00000	0.0333333
$901$	0	0
$902$	−12.0000	−0.399556
$903$	−5.00000	−0.166390
$904$	0	0
$905$	−11.0000	−0.365652
$906$	19.0000	0.631233
$907$	−5.00000	−0.166022	−0.0830111	−	0.996549i	$-0.526454\pi$
$907$	−5.00000	−0.166022	−0.0830111	+	0.996549i	$0.526454\pi$
$908$	−12.0000	−0.398234
$909$	−10.0000	−0.331679
$910$	1.00000	0.0331497
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−7.00000	−0.231793
$913$	60.0000	1.98571
$914$	14.0000	0.463079
$915$	1.00000	0.0330590
$916$	22.0000	0.726900
$917$	18.0000	0.594412
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−4.00000	−0.131876
$921$	32.0000	1.05444
$922$	24.0000	0.790398
$923$	12.0000	0.394985
$924$	−6.00000	−0.197386
$925$	−5.00000	−0.164399
$926$	−29.0000	−0.952999
$927$	11.0000	0.361287
$928$	−4.00000	−0.131306
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	3.00000	0.0983739
$931$	−42.0000	−1.37649
$932$	−22.0000	−0.720634
$933$	28.0000	0.916679
$934$	−18.0000	−0.588978
$935$	0	0
$936$	1.00000	0.0326860
$937$	9.00000	0.294017	0.147009	−	0.989135i	$-0.453036\pi$
$937$	9.00000	0.294017	0.147009	+	0.989135i	$0.453036\pi$
$938$	7.00000	0.228558
$939$	19.0000	0.620042
$940$	−4.00000	−0.130466
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	5.00000	0.162909
$943$	8.00000	0.260516
$944$	−2.00000	−0.0650945
$945$	−1.00000	−0.0325300
$946$	30.0000	0.975384
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	0	0
$949$	14.0000	0.454459
$950$	7.00000	0.227110
$951$	0	0
$952$	0	0
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	2.00000	0.0647524
$955$	18.0000	0.582466
$956$	−12.0000	−0.388108
$957$	−24.0000	−0.775810
$958$	−16.0000	−0.516937
$959$	12.0000	0.387500
$960$	1.00000	0.0322749
$961$	−22.0000	−0.709677
$962$	−5.00000	−0.161206
$963$	−12.0000	−0.386695
$964$	18.0000	0.579741
$965$	11.0000	0.354103
$966$	4.00000	0.128698
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	25.0000	0.803530
$969$	0	0
$970$	1.00000	0.0321081
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	−1.00000	−0.0320750
$973$	−15.0000	−0.480878
$974$	−1.00000	−0.0320421
$975$	−1.00000	−0.0320256
$976$	1.00000	0.0320092
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	24.0000	0.767435
$979$	24.0000	0.767043
$980$	6.00000	0.191663
$981$	−6.00000	−0.191565
$982$	−8.00000	−0.255290
$983$	32.0000	1.02064	0.510321	−	0.859984i	$-0.329527\pi$
$983$	32.0000	1.02064	0.510321	+	0.859984i	$0.329527\pi$
$984$	−2.00000	−0.0637577
$985$	−18.0000	−0.573528
$986$	0	0
$987$	4.00000	0.127321
$988$	7.00000	0.222700
$989$	−20.0000	−0.635963
$990$	6.00000	0.190693
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	3.00000	0.0952501
$993$	19.0000	0.602947
$994$	−12.0000	−0.380617
$995$	20.0000	0.634043
$996$	10.0000	0.316862
$997$	−1.00000	−0.0316703	−0.0158352	−	0.999875i	$-0.505041\pi$
$997$	−1.00000	−0.0316703	−0.0158352	+	0.999875i	$0.505041\pi$
$998$	−27.0000	−0.854670
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.o.1.1	✓	1
17.16	even	2		8670.2.a.ba.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8670.2.a.o.1.1	✓	1	1.1	even	1	trivial
8670.2.a.ba.1.1	yes	1	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8670.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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