LMFDB - Embedded newform 8670.2.a.z.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:	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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} -5.00000 q^{22} +7.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +4.00000 q^{29} +1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -5.00000 q^{33} +1.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} +1.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} +1.00000 q^{42} +6.00000 q^{43} -5.00000 q^{44} +1.00000 q^{45} +7.00000 q^{46} -7.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} +13.0000 q^{53} +1.00000 q^{54} -5.00000 q^{55} +1.00000 q^{56} -4.00000 q^{57} +4.00000 q^{58} +9.00000 q^{59} +1.00000 q^{60} +10.0000 q^{61} +8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} -5.00000 q^{66} +4.00000 q^{67} +7.00000 q^{69} +1.00000 q^{70} +10.0000 q^{71} +1.00000 q^{72} +8.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -5.00000 q^{77} +1.00000 q^{78} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -10.0000 q^{83} +1.00000 q^{84} +6.00000 q^{86} +4.00000 q^{87} -5.00000 q^{88} +7.00000 q^{89} +1.00000 q^{90} +1.00000 q^{91} +7.00000 q^{92} +8.00000 q^{93} -7.00000 q^{94} -4.00000 q^{95} +1.00000 q^{96} +12.0000 q^{97} -6.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	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
$19$	−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
$21$	1.00000	0.218218
$22$	−5.00000	−1.06600
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	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	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	1.00000	0.182574
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	−5.00000	−0.870388
$34$	0	0
$35$	1.00000	0.169031
$36$	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
$39$	1.00000	0.160128
$40$	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	$0.549916\pi$
$42$	1.00000	0.154303
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	−5.00000	−0.753778
$45$	1.00000	0.149071
$46$	7.00000	1.03209
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	1.00000	0.141421
$51$	0	0
$52$	1.00000	0.138675
$53$	13.0000	1.78569	0.892844	−	0.450367i	$-0.148707\pi$
$53$	13.0000	1.78569	0.892844	+	0.450367i	$0.148707\pi$
$54$	1.00000	0.136083
$55$	−5.00000	−0.674200
$56$	1.00000	0.133631
$57$	−4.00000	−0.529813
$58$	4.00000	0.525226
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	1.00000	0.129099
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	8.00000	1.01600
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	−5.00000	−0.615457
$67$	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
$69$	7.00000	0.842701
$70$	1.00000	0.119523
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	1.00000	0.117851
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	−6.00000	−0.697486
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	−5.00000	−0.569803
$78$	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
$86$	6.00000	0.646997
$87$	4.00000	0.428845
$88$	−5.00000	−0.533002
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	1.00000	0.105409
$91$	1.00000	0.104828
$92$	7.00000	0.729800
$93$	8.00000	0.829561
$94$	−7.00000	−0.721995
$95$	−4.00000	−0.410391
$96$	1.00000	0.102062
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	−6.00000	−0.606092
$99$	−5.00000	−0.502519
$100$	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	$0.5\pi$
$104$	1.00000	0.0980581
$105$	1.00000	0.0975900
$106$	13.0000	1.26267
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	−5.00000	−0.476731
$111$	−6.00000	−0.569495
$112$	1.00000	0.0944911
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	−4.00000	−0.374634
$115$	7.00000	0.652753
$116$	4.00000	0.371391
$117$	1.00000	0.0924500
$118$	9.00000	0.828517
$119$	0	0
$120$	1.00000	0.0912871
$121$	14.0000	1.27273
$122$	10.0000	0.905357
$123$	−2.00000	−0.180334
$124$	8.00000	0.718421
$125$	1.00000	0.0894427
$126$	1.00000	0.0890871
$127$	−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
$130$	1.00000	0.0877058
$131$	7.00000	0.611593	0.305796	−	0.952097i	$-0.401077\pi$
$131$	7.00000	0.611593	0.305796	+	0.952097i	$0.401077\pi$
$132$	−5.00000	−0.435194
$133$	−4.00000	−0.346844
$134$	4.00000	0.345547
$135$	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	$0.671323\pi$
$138$	7.00000	0.595880
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	1.00000	0.0845154
$141$	−7.00000	−0.589506
$142$	10.0000	0.839181
$143$	−5.00000	−0.418121
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	8.00000	0.662085
$147$	−6.00000	−0.494872
$148$	−6.00000	−0.493197
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	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
$154$	−5.00000	−0.402911
$155$	8.00000	0.642575
$156$	1.00000	0.0800641
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	0	0
$159$	13.0000	1.03097
$160$	1.00000	0.0790569
$161$	7.00000	0.551677
$162$	1.00000	0.0785674
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	−2.00000	−0.156174
$165$	−5.00000	−0.389249
$166$	−10.0000	−0.776151
$167$	−15.0000	−1.16073	−0.580367	−	0.814355i	$-0.697091\pi$
$167$	−15.0000	−1.16073	−0.580367	+	0.814355i	$0.697091\pi$
$168$	1.00000	0.0771517
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−4.00000	−0.305888
$172$	6.00000	0.457496
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	4.00000	0.303239
$175$	1.00000	0.0755929
$176$	−5.00000	−0.376889
$177$	9.00000	0.676481
$178$	7.00000	0.524672
$179$	21.0000	1.56961	0.784807	−	0.619740i	$-0.212762\pi$
$179$	21.0000	1.56961	0.784807	+	0.619740i	$0.212762\pi$
$180$	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	$0.195291\pi$
$182$	1.00000	0.0741249
$183$	10.0000	0.739221
$184$	7.00000	0.516047
$185$	−6.00000	−0.441129
$186$	8.00000	0.586588
$187$	0	0
$188$	−7.00000	−0.510527
$189$	1.00000	0.0727393
$190$	−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
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	12.0000	0.861550
$195$	1.00000	0.0716115
$196$	−6.00000	−0.428571
$197$	−7.00000	−0.498729	−0.249365	−	0.968410i	$-0.580222\pi$
$197$	−7.00000	−0.498729	−0.249365	+	0.968410i	$0.580222\pi$
$198$	−5.00000	−0.355335
$199$	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
$201$	4.00000	0.282138
$202$	−10.0000	−0.703598
$203$	4.00000	0.280745
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	7.00000	0.486534
$208$	1.00000	0.0693375
$209$	20.0000	1.38343
$210$	1.00000	0.0690066
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	13.0000	0.892844
$213$	10.0000	0.685189
$214$	12.0000	0.820303
$215$	6.00000	0.409197
$216$	1.00000	0.0680414
$217$	8.00000	0.543075
$218$	−16.0000	−1.08366
$219$	8.00000	0.540590
$220$	−5.00000	−0.337100
$221$	0	0
$222$	−6.00000	−0.402694
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	0	0
$227$	−10.0000	−0.663723	−0.331862	−	0.943328i	$-0.607677\pi$
$227$	−10.0000	−0.663723	−0.331862	+	0.943328i	$0.607677\pi$
$228$	−4.00000	−0.264906
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	7.00000	0.461566
$231$	−5.00000	−0.328976
$232$	4.00000	0.262613
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	1.00000	0.0653720
$235$	−7.00000	−0.456630
$236$	9.00000	0.585850
$237$	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
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	14.0000	0.899954
$243$	1.00000	0.0641500
$244$	10.0000	0.640184
$245$	−6.00000	−0.383326
$246$	−2.00000	−0.127515
$247$	−4.00000	−0.254514
$248$	8.00000	0.508001
$249$	−10.0000	−0.633724
$250$	1.00000	0.0632456
$251$	−31.0000	−1.95670	−0.978351	−	0.206951i	$-0.933646\pi$
$251$	−31.0000	−1.95670	−0.978351	+	0.206951i	$0.933646\pi$
$252$	1.00000	0.0629941
$253$	−35.0000	−2.20043
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	6.00000	0.373544
$259$	−6.00000	−0.372822
$260$	1.00000	0.0620174
$261$	4.00000	0.247594
$262$	7.00000	0.432461
$263$	−31.0000	−1.91154	−0.955771	−	0.294112i	$-0.904976\pi$
$263$	−31.0000	−1.91154	−0.955771	+	0.294112i	$0.904976\pi$
$264$	−5.00000	−0.307729
$265$	13.0000	0.798584
$266$	−4.00000	−0.245256
$267$	7.00000	0.428393
$268$	4.00000	0.244339
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	1.00000	0.0608581
$271$	−18.0000	−1.09342	−0.546711	−	0.837321i	$-0.684120\pi$
$271$	−18.0000	−1.09342	−0.546711	+	0.837321i	$0.684120\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	−12.0000	−0.724947
$275$	−5.00000	−0.301511
$276$	7.00000	0.421350
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	−4.00000	−0.239904
$279$	8.00000	0.478947
$280$	1.00000	0.0597614
$281$	−17.0000	−1.01413	−0.507067	−	0.861906i	$-0.669271\pi$
$281$	−17.0000	−1.01413	−0.507067	+	0.861906i	$0.669271\pi$
$282$	−7.00000	−0.416844
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	10.0000	0.593391
$285$	−4.00000	−0.236940
$286$	−5.00000	−0.295656
$287$	−2.00000	−0.118056
$288$	1.00000	0.0589256
$289$	0	0
$290$	4.00000	0.234888
$291$	12.0000	0.703452
$292$	8.00000	0.468165
$293$	−29.0000	−1.69420	−0.847099	−	0.531435i	$-0.821653\pi$
$293$	−29.0000	−1.69420	−0.847099	+	0.531435i	$0.821653\pi$
$294$	−6.00000	−0.349927
$295$	9.00000	0.524000
$296$	−6.00000	−0.348743
$297$	−5.00000	−0.290129
$298$	4.00000	0.231714
$299$	7.00000	0.404820
$300$	1.00000	0.0577350
$301$	6.00000	0.345834
$302$	−8.00000	−0.460348
$303$	−10.0000	−0.574485
$304$	−4.00000	−0.229416
$305$	10.0000	0.572598
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	−5.00000	−0.284901
$309$	0	0
$310$	8.00000	0.454369
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	1.00000	0.0566139
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	17.0000	0.959366
$315$	1.00000	0.0563436
$316$	0	0
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	13.0000	0.729004
$319$	−20.0000	−1.11979
$320$	1.00000	0.0559017
$321$	12.0000	0.669775
$322$	7.00000	0.390095
$323$	0	0
$324$	1.00000	0.0555556
$325$	1.00000	0.0554700
$326$	2.00000	0.110770
$327$	−16.0000	−0.884802
$328$	−2.00000	−0.110432
$329$	−7.00000	−0.385922
$330$	−5.00000	−0.275241
$331$	25.0000	1.37412	0.687062	−	0.726599i	$-0.258900\pi$
$331$	25.0000	1.37412	0.687062	+	0.726599i	$0.258900\pi$
$332$	−10.0000	−0.548821
$333$	−6.00000	−0.328798
$334$	−15.0000	−0.820763
$335$	4.00000	0.218543
$336$	1.00000	0.0545545
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	0	0
$341$	−40.0000	−2.16612
$342$	−4.00000	−0.216295
$343$	−13.0000	−0.701934
$344$	6.00000	0.323498
$345$	7.00000	0.376867
$346$	−13.0000	−0.698884
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	4.00000	0.214423
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	1.00000	0.0534522
$351$	1.00000	0.0533761
$352$	−5.00000	−0.266501
$353$	−28.0000	−1.49029	−0.745145	−	0.666903i	$-0.767620\pi$
$353$	−28.0000	−1.49029	−0.745145	+	0.666903i	$0.767620\pi$
$354$	9.00000	0.478345
$355$	10.0000	0.530745
$356$	7.00000	0.370999
$357$	0	0
$358$	21.0000	1.10988
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	22.0000	1.15629
$363$	14.0000	0.734809
$364$	1.00000	0.0524142
$365$	8.00000	0.418739
$366$	10.0000	0.522708
$367$	11.0000	0.574195	0.287098	−	0.957901i	$-0.407310\pi$
$367$	11.0000	0.574195	0.287098	+	0.957901i	$0.407310\pi$
$368$	7.00000	0.364900
$369$	−2.00000	−0.104116
$370$	−6.00000	−0.311925
$371$	13.0000	0.674926
$372$	8.00000	0.414781
$373$	−7.00000	−0.362446	−0.181223	−	0.983442i	$-0.558006\pi$
$373$	−7.00000	−0.362446	−0.181223	+	0.983442i	$0.558006\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−7.00000	−0.360997
$377$	4.00000	0.206010
$378$	1.00000	0.0514344
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	−4.00000	−0.205196
$381$	−4.00000	−0.204926
$382$	4.00000	0.204658
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	1.00000	0.0510310
$385$	−5.00000	−0.254824
$386$	0	0
$387$	6.00000	0.304997
$388$	12.0000	0.609208
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	1.00000	0.0506370
$391$	0	0
$392$	−6.00000	−0.303046
$393$	7.00000	0.353103
$394$	−7.00000	−0.352655
$395$	0	0
$396$	−5.00000	−0.251259
$397$	9.00000	0.451697	0.225849	−	0.974162i	$-0.427485\pi$
$397$	9.00000	0.451697	0.225849	+	0.974162i	$0.427485\pi$
$398$	20.0000	1.00251
$399$	−4.00000	−0.200250
$400$	1.00000	0.0500000
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	4.00000	0.199502
$403$	8.00000	0.398508
$404$	−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
$412$	0	0
$413$	9.00000	0.442861
$414$	7.00000	0.344031
$415$	−10.0000	−0.490881
$416$	1.00000	0.0490290
$417$	−4.00000	−0.195881
$418$	20.0000	0.978232
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	1.00000	0.0487950
$421$	−12.0000	−0.584844	−0.292422	−	0.956289i	$-0.594461\pi$
$421$	−12.0000	−0.584844	−0.292422	+	0.956289i	$0.594461\pi$
$422$	23.0000	1.11962
$423$	−7.00000	−0.340352
$424$	13.0000	0.631336
$425$	0	0
$426$	10.0000	0.484502
$427$	10.0000	0.483934
$428$	12.0000	0.580042
$429$	−5.00000	−0.241402
$430$	6.00000	0.289346
$431$	40.0000	1.92673	0.963366	−	0.268190i	$-0.0864254\pi$
$431$	40.0000	1.92673	0.963366	+	0.268190i	$0.0864254\pi$
$432$	1.00000	0.0481125
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	8.00000	0.384012
$435$	4.00000	0.191785
$436$	−16.0000	−0.766261
$437$	−28.0000	−1.33942
$438$	8.00000	0.382255
$439$	−18.0000	−0.859093	−0.429547	−	0.903045i	$-0.641327\pi$
$439$	−18.0000	−0.859093	−0.429547	+	0.903045i	$0.641327\pi$
$440$	−5.00000	−0.238366
$441$	−6.00000	−0.285714
$442$	0	0
$443$	30.0000	1.42534	0.712672	−	0.701498i	$-0.247485\pi$
$443$	30.0000	1.42534	0.712672	+	0.701498i	$0.247485\pi$
$444$	−6.00000	−0.284747
$445$	7.00000	0.331832
$446$	19.0000	0.899676
$447$	4.00000	0.189194
$448$	1.00000	0.0472456
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	1.00000	0.0471405
$451$	10.0000	0.470882
$452$	0	0
$453$	−8.00000	−0.375873
$454$	−10.0000	−0.469323
$455$	1.00000	0.0468807
$456$	−4.00000	−0.187317
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	0	0
$459$	0	0
$460$	7.00000	0.326377
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	−5.00000	−0.232621
$463$	−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
$465$	8.00000	0.370991
$466$	0	0
$467$	−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
$469$	4.00000	0.184703
$470$	−7.00000	−0.322886
$471$	17.0000	0.783319
$472$	9.00000	0.414259
$473$	−30.0000	−1.37940
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	13.0000	0.595229
$478$	−12.0000	−0.548867
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	1.00000	0.0456435
$481$	−6.00000	−0.273576
$482$	−7.00000	−0.318841
$483$	7.00000	0.318511
$484$	14.0000	0.636364
$485$	12.0000	0.544892
$486$	1.00000	0.0453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	10.0000	0.452679
$489$	2.00000	0.0904431
$490$	−6.00000	−0.271052
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	−2.00000	−0.0901670
$493$	0	0
$494$	−4.00000	−0.179969
$495$	−5.00000	−0.224733
$496$	8.00000	0.359211
$497$	10.0000	0.448561
$498$	−10.0000	−0.448111
$499$	5.00000	0.223831	0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000	0.223831	0.111915	+	0.993718i	$0.464301\pi$
$500$	1.00000	0.0447214
$501$	−15.0000	−0.670151
$502$	−31.0000	−1.38360
$503$	25.0000	1.11469	0.557347	−	0.830279i	$-0.311819\pi$
$503$	25.0000	1.11469	0.557347	+	0.830279i	$0.311819\pi$
$504$	1.00000	0.0445435
$505$	−10.0000	−0.444994
$506$	−35.0000	−1.55594
$507$	−12.0000	−0.532939
$508$	−4.00000	−0.177471
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	6.00000	0.264649
$515$	0	0
$516$	6.00000	0.264135
$517$	35.0000	1.53930
$518$	−6.00000	−0.263625
$519$	−13.0000	−0.570637
$520$	1.00000	0.0438529
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	4.00000	0.175075
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	7.00000	0.305796
$525$	1.00000	0.0436436
$526$	−31.0000	−1.35166
$527$	0	0
$528$	−5.00000	−0.217597
$529$	26.0000	1.13043
$530$	13.0000	0.564684
$531$	9.00000	0.390567
$532$	−4.00000	−0.173422
$533$	−2.00000	−0.0866296
$534$	7.00000	0.302920
$535$	12.0000	0.518805
$536$	4.00000	0.172774
$537$	21.0000	0.906217
$538$	6.00000	0.258678
$539$	30.0000	1.29219
$540$	1.00000	0.0430331
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	−18.0000	−0.773166
$543$	22.0000	0.944110
$544$	0	0
$545$	−16.0000	−0.685365
$546$	1.00000	0.0427960
$547$	−38.0000	−1.62476	−0.812381	−	0.583127i	$-0.801829\pi$
$547$	−38.0000	−1.62476	−0.812381	+	0.583127i	$0.801829\pi$
$548$	−12.0000	−0.512615
$549$	10.0000	0.426790
$550$	−5.00000	−0.213201
$551$	−16.0000	−0.681623
$552$	7.00000	0.297940
$553$	0	0
$554$	1.00000	0.0424859
$555$	−6.00000	−0.254686
$556$	−4.00000	−0.169638
$557$	−17.0000	−0.720313	−0.360157	−	0.932892i	$-0.617277\pi$
$557$	−17.0000	−0.720313	−0.360157	+	0.932892i	$0.617277\pi$
$558$	8.00000	0.338667
$559$	6.00000	0.253773
$560$	1.00000	0.0422577
$561$	0	0
$562$	−17.0000	−0.717102
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	−7.00000	−0.294753
$565$	0	0
$566$	−10.0000	−0.420331
$567$	1.00000	0.0419961
$568$	10.0000	0.419591
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	−4.00000	−0.167542
$571$	−1.00000	−0.0418487	−0.0209243	−	0.999781i	$-0.506661\pi$
$571$	−1.00000	−0.0418487	−0.0209243	+	0.999781i	$0.506661\pi$
$572$	−5.00000	−0.209061
$573$	4.00000	0.167102
$574$	−2.00000	−0.0834784
$575$	7.00000	0.291920
$576$	1.00000	0.0416667
$577$	−42.0000	−1.74848	−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000	−1.74848	−0.874241	+	0.485491i	$0.838641\pi$
$578$	0	0
$579$	0	0
$580$	4.00000	0.166091
$581$	−10.0000	−0.414870
$582$	12.0000	0.497416
$583$	−65.0000	−2.69202
$584$	8.00000	0.331042
$585$	1.00000	0.0413449
$586$	−29.0000	−1.19798
$587$	26.0000	1.07313	0.536567	−	0.843857i	$-0.319721\pi$
$587$	26.0000	1.07313	0.536567	+	0.843857i	$0.319721\pi$
$588$	−6.00000	−0.247436
$589$	−32.0000	−1.31854
$590$	9.00000	0.370524
$591$	−7.00000	−0.287942
$592$	−6.00000	−0.246598
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	−5.00000	−0.205152
$595$	0	0
$596$	4.00000	0.163846
$597$	20.0000	0.818546
$598$	7.00000	0.286251
$599$	−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
$601$	−21.0000	−0.856608	−0.428304	−	0.903635i	$-0.640889\pi$
$601$	−21.0000	−0.856608	−0.428304	+	0.903635i	$0.640889\pi$
$602$	6.00000	0.244542
$603$	4.00000	0.162893
$604$	−8.00000	−0.325515
$605$	14.0000	0.569181
$606$	−10.0000	−0.406222
$607$	−25.0000	−1.01472	−0.507359	−	0.861735i	$-0.669378\pi$
$607$	−25.0000	−1.01472	−0.507359	+	0.861735i	$0.669378\pi$
$608$	−4.00000	−0.162221
$609$	4.00000	0.162088
$610$	10.0000	0.404888
$611$	−7.00000	−0.283190
$612$	0	0
$613$	−29.0000	−1.17130	−0.585649	−	0.810564i	$-0.699160\pi$
$613$	−29.0000	−1.17130	−0.585649	+	0.810564i	$0.699160\pi$
$614$	−10.0000	−0.403567
$615$	−2.00000	−0.0806478
$616$	−5.00000	−0.201456
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	−11.0000	−0.442127	−0.221064	−	0.975259i	$-0.570953\pi$
$619$	−11.0000	−0.442127	−0.221064	+	0.975259i	$0.570953\pi$
$620$	8.00000	0.321288
$621$	7.00000	0.280900
$622$	28.0000	1.12270
$623$	7.00000	0.280449
$624$	1.00000	0.0400320
$625$	1.00000	0.0400000
$626$	8.00000	0.319744
$627$	20.0000	0.798723
$628$	17.0000	0.678374
$629$	0	0
$630$	1.00000	0.0398410
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	23.0000	0.914168
$634$	−22.0000	−0.873732
$635$	−4.00000	−0.158735
$636$	13.0000	0.515484
$637$	−6.00000	−0.237729
$638$	−20.0000	−0.791808
$639$	10.0000	0.395594
$640$	1.00000	0.0395285
$641$	−21.0000	−0.829450	−0.414725	−	0.909947i	$-0.636122\pi$
$641$	−21.0000	−0.829450	−0.414725	+	0.909947i	$0.636122\pi$
$642$	12.0000	0.473602
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	7.00000	0.275839
$645$	6.00000	0.236250
$646$	0	0
$647$	−25.0000	−0.982851	−0.491426	−	0.870919i	$-0.663524\pi$
$647$	−25.0000	−0.982851	−0.491426	+	0.870919i	$0.663524\pi$
$648$	1.00000	0.0392837
$649$	−45.0000	−1.76640
$650$	1.00000	0.0392232
$651$	8.00000	0.313545
$652$	2.00000	0.0783260
$653$	21.0000	0.821794	0.410897	−	0.911682i	$-0.365216\pi$
$653$	21.0000	0.821794	0.410897	+	0.911682i	$0.365216\pi$
$654$	−16.0000	−0.625650
$655$	7.00000	0.273513
$656$	−2.00000	−0.0780869
$657$	8.00000	0.312110
$658$	−7.00000	−0.272888
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	−5.00000	−0.194625
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	25.0000	0.971653
$663$	0	0
$664$	−10.0000	−0.388075
$665$	−4.00000	−0.155113
$666$	−6.00000	−0.232495
$667$	28.0000	1.08416
$668$	−15.0000	−0.580367
$669$	19.0000	0.734582
$670$	4.00000	0.154533
$671$	−50.0000	−1.93023
$672$	1.00000	0.0385758
$673$	50.0000	1.92736	0.963679	−	0.267063i	$-0.0860531\pi$
$673$	50.0000	1.92736	0.963679	+	0.267063i	$0.0860531\pi$
$674$	−8.00000	−0.308148
$675$	1.00000	0.0384900
$676$	−12.0000	−0.461538
$677$	−27.0000	−1.03769	−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000	−1.03769	−0.518847	+	0.854867i	$0.673639\pi$
$678$	0	0
$679$	12.0000	0.460518
$680$	0	0
$681$	−10.0000	−0.383201
$682$	−40.0000	−1.53168
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−4.00000	−0.152944
$685$	−12.0000	−0.458496
$686$	−13.0000	−0.496342
$687$	0	0
$688$	6.00000	0.228748
$689$	13.0000	0.495261
$690$	7.00000	0.266485
$691$	11.0000	0.418460	0.209230	−	0.977866i	$-0.432904\pi$
$691$	11.0000	0.418460	0.209230	+	0.977866i	$0.432904\pi$
$692$	−13.0000	−0.494186
$693$	−5.00000	−0.189934
$694$	−2.00000	−0.0759190
$695$	−4.00000	−0.151729
$696$	4.00000	0.151620
$697$	0	0
$698$	−8.00000	−0.302804
$699$	0	0
$700$	1.00000	0.0377964
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	1.00000	0.0377426
$703$	24.0000	0.905177
$704$	−5.00000	−0.188445
$705$	−7.00000	−0.263635
$706$	−28.0000	−1.05379
$707$	−10.0000	−0.376089
$708$	9.00000	0.338241
$709$	−52.0000	−1.95290	−0.976450	−	0.215742i	$-0.930783\pi$
$709$	−52.0000	−1.95290	−0.976450	+	0.215742i	$0.930783\pi$
$710$	10.0000	0.375293
$711$	0	0
$712$	7.00000	0.262336
$713$	56.0000	2.09722
$714$	0	0
$715$	−5.00000	−0.186989
$716$	21.0000	0.784807
$717$	−12.0000	−0.448148
$718$	−6.00000	−0.223918
$719$	50.0000	1.86469	0.932343	−	0.361576i	$-0.117761\pi$
$719$	50.0000	1.86469	0.932343	+	0.361576i	$0.117761\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	−3.00000	−0.111648
$723$	−7.00000	−0.260333
$724$	22.0000	0.817624
$725$	4.00000	0.148556
$726$	14.0000	0.519589
$727$	17.0000	0.630495	0.315248	−	0.949009i	$-0.397912\pi$
$727$	17.0000	0.630495	0.315248	+	0.949009i	$0.397912\pi$
$728$	1.00000	0.0370625
$729$	1.00000	0.0370370
$730$	8.00000	0.296093
$731$	0	0
$732$	10.0000	0.369611
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	11.0000	0.406017
$735$	−6.00000	−0.221313
$736$	7.00000	0.258023
$737$	−20.0000	−0.736709
$738$	−2.00000	−0.0736210
$739$	51.0000	1.87607	0.938033	−	0.346547i	$-0.112646\pi$
$739$	51.0000	1.87607	0.938033	+	0.346547i	$0.112646\pi$
$740$	−6.00000	−0.220564
$741$	−4.00000	−0.146944
$742$	13.0000	0.477245
$743$	−13.0000	−0.476924	−0.238462	−	0.971152i	$-0.576643\pi$
$743$	−13.0000	−0.476924	−0.238462	+	0.971152i	$0.576643\pi$
$744$	8.00000	0.293294
$745$	4.00000	0.146549
$746$	−7.00000	−0.256288
$747$	−10.0000	−0.365881
$748$	0	0
$749$	12.0000	0.438470
$750$	1.00000	0.0365148
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	−7.00000	−0.255264
$753$	−31.0000	−1.12970
$754$	4.00000	0.145671
$755$	−8.00000	−0.291150
$756$	1.00000	0.0363696
$757$	35.0000	1.27210	0.636048	−	0.771649i	$-0.280568\pi$
$757$	35.0000	1.27210	0.636048	+	0.771649i	$0.280568\pi$
$758$	28.0000	1.01701
$759$	−35.0000	−1.27042
$760$	−4.00000	−0.145095
$761$	−9.00000	−0.326250	−0.163125	−	0.986605i	$-0.552157\pi$
$761$	−9.00000	−0.326250	−0.163125	+	0.986605i	$0.552157\pi$
$762$	−4.00000	−0.144905
$763$	−16.0000	−0.579239
$764$	4.00000	0.144715
$765$	0	0
$766$	−16.0000	−0.578103
$767$	9.00000	0.324971
$768$	1.00000	0.0360844
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	−5.00000	−0.180187
$771$	6.00000	0.216085
$772$	0	0
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	6.00000	0.215666
$775$	8.00000	0.287368
$776$	12.0000	0.430775
$777$	−6.00000	−0.215249
$778$	−36.0000	−1.29066
$779$	8.00000	0.286630
$780$	1.00000	0.0358057
$781$	−50.0000	−1.78914
$782$	0	0
$783$	4.00000	0.142948
$784$	−6.00000	−0.214286
$785$	17.0000	0.606756
$786$	7.00000	0.249682
$787$	48.0000	1.71102	0.855508	−	0.517790i	$-0.173245\pi$
$787$	48.0000	1.71102	0.855508	+	0.517790i	$0.173245\pi$
$788$	−7.00000	−0.249365
$789$	−31.0000	−1.10363
$790$	0	0
$791$	0	0
$792$	−5.00000	−0.177667
$793$	10.0000	0.355110
$794$	9.00000	0.319398
$795$	13.0000	0.461062
$796$	20.0000	0.708881
$797$	37.0000	1.31061	0.655304	−	0.755366i	$-0.272541\pi$
$797$	37.0000	1.31061	0.655304	+	0.755366i	$0.272541\pi$
$798$	−4.00000	−0.141598
$799$	0	0
$800$	1.00000	0.0353553
$801$	7.00000	0.247333
$802$	−27.0000	−0.953403
$803$	−40.0000	−1.41157
$804$	4.00000	0.141069
$805$	7.00000	0.246718
$806$	8.00000	0.281788
$807$	6.00000	0.211210
$808$	−10.0000	−0.351799
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	1.00000	0.0351364
$811$	−29.0000	−1.01833	−0.509164	−	0.860670i	$-0.670045\pi$
$811$	−29.0000	−1.01833	−0.509164	+	0.860670i	$0.670045\pi$
$812$	4.00000	0.140372
$813$	−18.0000	−0.631288
$814$	30.0000	1.05150
$815$	2.00000	0.0700569
$816$	0	0
$817$	−24.0000	−0.839654
$818$	−11.0000	−0.384606
$819$	1.00000	0.0349428
$820$	−2.00000	−0.0698430
$821$	48.0000	1.67521	0.837606	−	0.546275i	$-0.183955\pi$
$821$	48.0000	1.67521	0.837606	+	0.546275i	$0.183955\pi$
$822$	−12.0000	−0.418548
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	−5.00000	−0.174078
$826$	9.00000	0.313150
$827$	−2.00000	−0.0695468	−0.0347734	−	0.999395i	$-0.511071\pi$
$827$	−2.00000	−0.0695468	−0.0347734	+	0.999395i	$0.511071\pi$
$828$	7.00000	0.243267
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	−10.0000	−0.347105
$831$	1.00000	0.0346896
$832$	1.00000	0.0346688
$833$	0	0
$834$	−4.00000	−0.138509
$835$	−15.0000	−0.519096
$836$	20.0000	0.691714
$837$	8.00000	0.276520
$838$	−15.0000	−0.518166
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	1.00000	0.0345033
$841$	−13.0000	−0.448276
$842$	−12.0000	−0.413547
$843$	−17.0000	−0.585511
$844$	23.0000	0.791693
$845$	−12.0000	−0.412813
$846$	−7.00000	−0.240665
$847$	14.0000	0.481046
$848$	13.0000	0.446422
$849$	−10.0000	−0.343199
$850$	0	0
$851$	−42.0000	−1.43974
$852$	10.0000	0.342594
$853$	37.0000	1.26686	0.633428	−	0.773802i	$-0.281647\pi$
$853$	37.0000	1.26686	0.633428	+	0.773802i	$0.281647\pi$
$854$	10.0000	0.342193
$855$	−4.00000	−0.136797
$856$	12.0000	0.410152
$857$	−32.0000	−1.09310	−0.546550	−	0.837427i	$-0.684059\pi$
$857$	−32.0000	−1.09310	−0.546550	+	0.837427i	$0.684059\pi$
$858$	−5.00000	−0.170697
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	6.00000	0.204598
$861$	−2.00000	−0.0681598
$862$	40.0000	1.36241
$863$	−43.0000	−1.46374	−0.731869	−	0.681446i	$-0.761351\pi$
$863$	−43.0000	−1.46374	−0.731869	+	0.681446i	$0.761351\pi$
$864$	1.00000	0.0340207
$865$	−13.0000	−0.442013
$866$	16.0000	0.543702
$867$	0	0
$868$	8.00000	0.271538
$869$	0	0
$870$	4.00000	0.135613
$871$	4.00000	0.135535
$872$	−16.0000	−0.541828
$873$	12.0000	0.406138
$874$	−28.0000	−0.947114
$875$	1.00000	0.0338062
$876$	8.00000	0.270295
$877$	−41.0000	−1.38447	−0.692236	−	0.721671i	$-0.743374\pi$
$877$	−41.0000	−1.38447	−0.692236	+	0.721671i	$0.743374\pi$
$878$	−18.0000	−0.607471
$879$	−29.0000	−0.978146
$880$	−5.00000	−0.168550
$881$	−3.00000	−0.101073	−0.0505363	−	0.998722i	$-0.516093\pi$
$881$	−3.00000	−0.101073	−0.0505363	+	0.998722i	$0.516093\pi$
$882$	−6.00000	−0.202031
$883$	50.0000	1.68263	0.841317	−	0.540542i	$-0.181781\pi$
$883$	50.0000	1.68263	0.841317	+	0.540542i	$0.181781\pi$
$884$	0	0
$885$	9.00000	0.302532
$886$	30.0000	1.00787
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−6.00000	−0.201347
$889$	−4.00000	−0.134156
$890$	7.00000	0.234641
$891$	−5.00000	−0.167506
$892$	19.0000	0.636167
$893$	28.0000	0.936984
$894$	4.00000	0.133780
$895$	21.0000	0.701953
$896$	1.00000	0.0334077
$897$	7.00000	0.233723
$898$	18.0000	0.600668
$899$	32.0000	1.06726
$900$	1.00000	0.0333333
$901$	0	0
$902$	10.0000	0.332964
$903$	6.00000	0.199667
$904$	0	0
$905$	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	$0.653896\pi$
$908$	−10.0000	−0.331862
$909$	−10.0000	−0.331679
$910$	1.00000	0.0331497
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	−4.00000	−0.132453
$913$	50.0000	1.65476
$914$	−8.00000	−0.264616
$915$	10.0000	0.330590
$916$	0	0
$917$	7.00000	0.231160
$918$	0	0
$919$	28.0000	0.923635	0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000	0.923635	0.461817	+	0.886975i	$0.347198\pi$
$920$	7.00000	0.230783
$921$	−10.0000	−0.329511
$922$	2.00000	0.0658665
$923$	10.0000	0.329154
$924$	−5.00000	−0.164488
$925$	−6.00000	−0.197279
$926$	−29.0000	−0.952999
$927$	0	0
$928$	4.00000	0.131306
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	8.00000	0.262330
$931$	24.0000	0.786568
$932$	0	0
$933$	28.0000	0.916679
$934$	−18.0000	−0.588978
$935$	0	0
$936$	1.00000	0.0326860
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	4.00000	0.130605
$939$	8.00000	0.261070
$940$	−7.00000	−0.228315
$941$	−58.0000	−1.89075	−0.945373	−	0.325991i	$-0.894302\pi$
$941$	−58.0000	−1.89075	−0.945373	+	0.325991i	$0.894302\pi$
$942$	17.0000	0.553890
$943$	−14.0000	−0.455903
$944$	9.00000	0.292925
$945$	1.00000	0.0325300
$946$	−30.0000	−0.975384
$947$	22.0000	0.714904	0.357452	−	0.933932i	$-0.383646\pi$
$947$	22.0000	0.714904	0.357452	+	0.933932i	$0.383646\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	−4.00000	−0.129777
$951$	−22.0000	−0.713399
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	13.0000	0.420891
$955$	4.00000	0.129437
$956$	−12.0000	−0.388108
$957$	−20.0000	−0.646508
$958$	16.0000	0.516937
$959$	−12.0000	−0.387500
$960$	1.00000	0.0322749
$961$	33.0000	1.06452
$962$	−6.00000	−0.193448
$963$	12.0000	0.386695
$964$	−7.00000	−0.225455
$965$	0	0
$966$	7.00000	0.225221
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	14.0000	0.449977
$969$	0	0
$970$	12.0000	0.385297
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	1.00000	0.0320750
$973$	−4.00000	−0.128234
$974$	−32.0000	−1.02535
$975$	1.00000	0.0320256
$976$	10.0000	0.320092
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	2.00000	0.0639529
$979$	−35.0000	−1.11860
$980$	−6.00000	−0.191663
$981$	−16.0000	−0.510841
$982$	3.00000	0.0957338
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	−2.00000	−0.0637577
$985$	−7.00000	−0.223039
$986$	0	0
$987$	−7.00000	−0.222812
$988$	−4.00000	−0.127257
$989$	42.0000	1.33552
$990$	−5.00000	−0.158910
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	8.00000	0.254000
$993$	25.0000	0.793351
$994$	10.0000	0.317181
$995$	20.0000	0.634043
$996$	−10.0000	−0.316862
$997$	23.0000	0.728417	0.364209	−	0.931317i	$-0.381339\pi$
$997$	23.0000	0.728417	0.364209	+	0.931317i	$0.381339\pi$
$998$	5.00000	0.158272
$999$	−6.00000	−0.189832

Twists

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

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

Related objects

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