LMFDB - Embedded newform 1002.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1002,2,Mod(1,1002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1002, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1002 = 2 \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1002.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:	$8.00101028253$
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	$\chi$	$=$	1002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} -2.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +1.00000 q^{30} -5.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +3.00000 q^{37} +4.00000 q^{38} -6.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} -8.00000 q^{43} +2.00000 q^{44} -1.00000 q^{45} -8.00000 q^{46} +3.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +4.00000 q^{50} -4.00000 q^{51} -6.00000 q^{52} -7.00000 q^{53} -1.00000 q^{54} -2.00000 q^{55} +1.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} +11.0000 q^{59} -1.00000 q^{60} +4.00000 q^{61} +5.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -2.00000 q^{66} -7.00000 q^{67} -4.00000 q^{68} +8.00000 q^{69} -1.00000 q^{70} -4.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} -3.00000 q^{74} -4.00000 q^{75} -4.00000 q^{76} -2.00000 q^{77} +6.00000 q^{78} +4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -15.0000 q^{83} -1.00000 q^{84} +4.00000 q^{85} +8.00000 q^{86} -2.00000 q^{87} -2.00000 q^{88} +9.00000 q^{89} +1.00000 q^{90} +6.00000 q^{91} +8.00000 q^{92} -5.00000 q^{93} -3.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} -13.0000 q^{97} +6.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	−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
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.00000	0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−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
$22$	−2.00000	−0.426401
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	6.00000	1.17670
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−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
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	4.00000	0.685994
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	4.00000	0.648886
$39$	−6.00000	−0.960769
$40$	1.00000	0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	1.00000	0.154303
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	2.00000	0.301511
$45$	−1.00000	−0.149071
$46$	−8.00000	−1.17954
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	4.00000	0.565685
$51$	−4.00000	−0.560112
$52$	−6.00000	−0.832050
$53$	−7.00000	−0.961524	−0.480762	−	0.876851i	$-0.659640\pi$
$53$	−7.00000	−0.961524	−0.480762	+	0.876851i	$0.659640\pi$
$54$	−1.00000	−0.136083
$55$	−2.00000	−0.269680
$56$	1.00000	0.133631
$57$	−4.00000	−0.529813
$58$	2.00000	0.262613
$59$	11.0000	1.43208	0.716039	−	0.698060i	$-0.245953\pi$
$59$	11.0000	1.43208	0.716039	+	0.698060i	$0.245953\pi$
$60$	−1.00000	−0.129099
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	5.00000	0.635001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	−2.00000	−0.246183
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	−4.00000	−0.485071
$69$	8.00000	0.963087
$70$	−1.00000	−0.119523
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−1.00000	−0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−3.00000	−0.348743
$75$	−4.00000	−0.461880
$76$	−4.00000	−0.458831
$77$	−2.00000	−0.227921
$78$	6.00000	0.679366
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−15.0000	−1.64646	−0.823232	−	0.567705i	$-0.807831\pi$
$83$	−15.0000	−1.64646	−0.823232	+	0.567705i	$0.807831\pi$
$84$	−1.00000	−0.109109
$85$	4.00000	0.433861
$86$	8.00000	0.862662
$87$	−2.00000	−0.214423
$88$	−2.00000	−0.213201
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	1.00000	0.105409
$91$	6.00000	0.628971
$92$	8.00000	0.834058
$93$	−5.00000	−0.518476
$94$	−3.00000	−0.309426
$95$	4.00000	0.410391
$96$	−1.00000	−0.102062
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	6.00000	0.606092
$99$	2.00000	0.201008
$100$	−4.00000	−0.400000
$101$	1.00000	0.0995037	0.0497519	−	0.998762i	$-0.484157\pi$
$101$	1.00000	0.0995037	0.0497519	+	0.998762i	$0.484157\pi$
$102$	4.00000	0.396059
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	6.00000	0.588348
$105$	1.00000	0.0975900
$106$	7.00000	0.679900
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	1.00000	0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	2.00000	0.190693
$111$	3.00000	0.284747
$112$	−1.00000	−0.0944911
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	4.00000	0.374634
$115$	−8.00000	−0.746004
$116$	−2.00000	−0.185695
$117$	−6.00000	−0.554700
$118$	−11.0000	−1.01263
$119$	4.00000	0.366679
$120$	1.00000	0.0912871
$121$	−7.00000	−0.636364
$122$	−4.00000	−0.362143
$123$	−6.00000	−0.541002
$124$	−5.00000	−0.449013
$125$	9.00000	0.804984
$126$	1.00000	0.0890871
$127$	−17.0000	−1.50851	−0.754253	−	0.656584i	$-0.772001\pi$
$127$	−17.0000	−1.50851	−0.754253	+	0.656584i	$0.772001\pi$
$128$	−1.00000	−0.0883883
$129$	−8.00000	−0.704361
$130$	−6.00000	−0.526235
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	2.00000	0.174078
$133$	4.00000	0.346844
$134$	7.00000	0.604708
$135$	−1.00000	−0.0860663
$136$	4.00000	0.342997
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	−8.00000	−0.681005
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	1.00000	0.0845154
$141$	3.00000	0.252646
$142$	4.00000	0.335673
$143$	−12.0000	−1.00349
$144$	1.00000	0.0833333
$145$	2.00000	0.166091
$146$	−6.00000	−0.496564
$147$	−6.00000	−0.494872
$148$	3.00000	0.246598
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	4.00000	0.326599
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	4.00000	0.324443
$153$	−4.00000	−0.323381
$154$	2.00000	0.161165
$155$	5.00000	0.401610
$156$	−6.00000	−0.480384
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−4.00000	−0.318223
$159$	−7.00000	−0.555136
$160$	1.00000	0.0790569
$161$	−8.00000	−0.630488
$162$	−1.00000	−0.0785674
$163$	9.00000	0.704934	0.352467	−	0.935824i	$-0.385343\pi$
$163$	9.00000	0.704934	0.352467	+	0.935824i	$0.385343\pi$
$164$	−6.00000	−0.468521
$165$	−2.00000	−0.155700
$166$	15.0000	1.16423
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	1.00000	0.0771517
$169$	23.0000	1.76923
$170$	−4.00000	−0.306786
$171$	−4.00000	−0.305888
$172$	−8.00000	−0.609994
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	2.00000	0.151620
$175$	4.00000	0.302372
$176$	2.00000	0.150756
$177$	11.0000	0.826811
$178$	−9.00000	−0.674579
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	−1.00000	−0.0745356
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	−6.00000	−0.444750
$183$	4.00000	0.295689
$184$	−8.00000	−0.589768
$185$	−3.00000	−0.220564
$186$	5.00000	0.366618
$187$	−8.00000	−0.585018
$188$	3.00000	0.218797
$189$	−1.00000	−0.0727393
$190$	−4.00000	−0.290191
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	1.00000	0.0721688
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	13.0000	0.933346
$195$	6.00000	0.429669
$196$	−6.00000	−0.428571
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−2.00000	−0.142134
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	4.00000	0.282843
$201$	−7.00000	−0.493742
$202$	−1.00000	−0.0703598
$203$	2.00000	0.140372
$204$	−4.00000	−0.280056
$205$	6.00000	0.419058
$206$	−6.00000	−0.418040
$207$	8.00000	0.556038
$208$	−6.00000	−0.416025
$209$	−8.00000	−0.553372
$210$	−1.00000	−0.0690066
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	−7.00000	−0.480762
$213$	−4.00000	−0.274075
$214$	10.0000	0.683586
$215$	8.00000	0.545595
$216$	−1.00000	−0.0680414
$217$	5.00000	0.339422
$218$	−2.00000	−0.135457
$219$	6.00000	0.405442
$220$	−2.00000	−0.134840
$221$	24.0000	1.61441
$222$	−3.00000	−0.201347
$223$	29.0000	1.94198	0.970992	−	0.239113i	$-0.0768565\pi$
$223$	29.0000	1.94198	0.970992	+	0.239113i	$0.0768565\pi$
$224$	1.00000	0.0668153
$225$	−4.00000	−0.266667
$226$	4.00000	0.266076
$227$	1.00000	0.0663723	0.0331862	−	0.999449i	$-0.489435\pi$
$227$	1.00000	0.0663723	0.0331862	+	0.999449i	$0.489435\pi$
$228$	−4.00000	−0.264906
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	8.00000	0.527504
$231$	−2.00000	−0.131590
$232$	2.00000	0.131306
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	6.00000	0.392232
$235$	−3.00000	−0.195698
$236$	11.0000	0.716039
$237$	4.00000	0.259828
$238$	−4.00000	−0.259281
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	−1.00000	−0.0645497
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	4.00000	0.256074
$245$	6.00000	0.383326
$246$	6.00000	0.382546
$247$	24.0000	1.52708
$248$	5.00000	0.317500
$249$	−15.0000	−0.950586
$250$	−9.00000	−0.569210
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	−1.00000	−0.0629941
$253$	16.0000	1.00591
$254$	17.0000	1.06667
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	8.00000	0.498058
$259$	−3.00000	−0.186411
$260$	6.00000	0.372104
$261$	−2.00000	−0.123797
$262$	12.0000	0.741362
$263$	15.0000	0.924940	0.462470	−	0.886635i	$-0.346963\pi$
$263$	15.0000	0.924940	0.462470	+	0.886635i	$0.346963\pi$
$264$	−2.00000	−0.123091
$265$	7.00000	0.430007
$266$	−4.00000	−0.245256
$267$	9.00000	0.550791
$268$	−7.00000	−0.427593
$269$	27.0000	1.64622	0.823110	−	0.567883i	$-0.192237\pi$
$269$	27.0000	1.64622	0.823110	+	0.567883i	$0.192237\pi$
$270$	1.00000	0.0608581
$271$	−30.0000	−1.82237	−0.911185	−	0.411997i	$-0.864831\pi$
$271$	−30.0000	−1.82237	−0.911185	+	0.411997i	$0.864831\pi$
$272$	−4.00000	−0.242536
$273$	6.00000	0.363137
$274$	−9.00000	−0.543710
$275$	−8.00000	−0.482418
$276$	8.00000	0.481543
$277$	−9.00000	−0.540758	−0.270379	−	0.962754i	$-0.587149\pi$
$277$	−9.00000	−0.540758	−0.270379	+	0.962754i	$0.587149\pi$
$278$	7.00000	0.419832
$279$	−5.00000	−0.299342
$280$	−1.00000	−0.0597614
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	−3.00000	−0.178647
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	−4.00000	−0.237356
$285$	4.00000	0.236940
$286$	12.0000	0.709575
$287$	6.00000	0.354169
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	−2.00000	−0.117444
$291$	−13.0000	−0.762073
$292$	6.00000	0.351123
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	6.00000	0.349927
$295$	−11.0000	−0.640445
$296$	−3.00000	−0.174371
$297$	2.00000	0.116052
$298$	−9.00000	−0.521356
$299$	−48.0000	−2.77591
$300$	−4.00000	−0.230940
$301$	8.00000	0.461112
$302$	−20.0000	−1.15087
$303$	1.00000	0.0574485
$304$	−4.00000	−0.229416
$305$	−4.00000	−0.229039
$306$	4.00000	0.228665
$307$	−27.0000	−1.54097	−0.770486	−	0.637457i	$-0.779986\pi$
$307$	−27.0000	−1.54097	−0.770486	+	0.637457i	$0.779986\pi$
$308$	−2.00000	−0.113961
$309$	6.00000	0.341328
$310$	−5.00000	−0.283981
$311$	−27.0000	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$312$	6.00000	0.339683
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	18.0000	1.01580
$315$	1.00000	0.0563436
$316$	4.00000	0.225018
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	7.00000	0.392541
$319$	−4.00000	−0.223957
$320$	−1.00000	−0.0559017
$321$	−10.0000	−0.558146
$322$	8.00000	0.445823
$323$	16.0000	0.890264
$324$	1.00000	0.0555556
$325$	24.0000	1.33128
$326$	−9.00000	−0.498464
$327$	2.00000	0.110600
$328$	6.00000	0.331295
$329$	−3.00000	−0.165395
$330$	2.00000	0.110096
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−15.0000	−0.823232
$333$	3.00000	0.164399
$334$	−1.00000	−0.0547176
$335$	7.00000	0.382451
$336$	−1.00000	−0.0545545
$337$	29.0000	1.57973	0.789865	−	0.613280i	$-0.210150\pi$
$337$	29.0000	1.57973	0.789865	+	0.613280i	$0.210150\pi$
$338$	−23.0000	−1.25104
$339$	−4.00000	−0.217250
$340$	4.00000	0.216930
$341$	−10.0000	−0.541530
$342$	4.00000	0.216295
$343$	13.0000	0.701934
$344$	8.00000	0.431331
$345$	−8.00000	−0.430706
$346$	4.00000	0.215041
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	−2.00000	−0.107211
$349$	29.0000	1.55233	0.776167	−	0.630527i	$-0.217161\pi$
$349$	29.0000	1.55233	0.776167	+	0.630527i	$0.217161\pi$
$350$	−4.00000	−0.213809
$351$	−6.00000	−0.320256
$352$	−2.00000	−0.106600
$353$	−21.0000	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$354$	−11.0000	−0.584643
$355$	4.00000	0.212298
$356$	9.00000	0.476999
$357$	4.00000	0.211702
$358$	−2.00000	−0.105703
$359$	−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
$362$	−20.0000	−1.05118
$363$	−7.00000	−0.367405
$364$	6.00000	0.314485
$365$	−6.00000	−0.314054
$366$	−4.00000	−0.209083
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	8.00000	0.417029
$369$	−6.00000	−0.312348
$370$	3.00000	0.155963
$371$	7.00000	0.363422
$372$	−5.00000	−0.259238
$373$	13.0000	0.673114	0.336557	−	0.941663i	$-0.390737\pi$
$373$	13.0000	0.673114	0.336557	+	0.941663i	$0.390737\pi$
$374$	8.00000	0.413670
$375$	9.00000	0.464758
$376$	−3.00000	−0.154713
$377$	12.0000	0.618031
$378$	1.00000	0.0514344
$379$	−11.0000	−0.565032	−0.282516	−	0.959263i	$-0.591169\pi$
$379$	−11.0000	−0.565032	−0.282516	+	0.959263i	$0.591169\pi$
$380$	4.00000	0.205196
$381$	−17.0000	−0.870936
$382$	−15.0000	−0.767467
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	−1.00000	−0.0510310
$385$	2.00000	0.101929
$386$	−16.0000	−0.814379
$387$	−8.00000	−0.406663
$388$	−13.0000	−0.659975
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	−6.00000	−0.303822
$391$	−32.0000	−1.61831
$392$	6.00000	0.303046
$393$	−12.0000	−0.605320
$394$	−6.00000	−0.302276
$395$	−4.00000	−0.201262
$396$	2.00000	0.100504
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	8.00000	0.401004
$399$	4.00000	0.200250
$400$	−4.00000	−0.200000
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	7.00000	0.349128
$403$	30.0000	1.49441
$404$	1.00000	0.0497519
$405$	−1.00000	−0.0496904
$406$	−2.00000	−0.0992583
$407$	6.00000	0.297409
$408$	4.00000	0.198030
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	−6.00000	−0.296319
$411$	9.00000	0.443937
$412$	6.00000	0.295599
$413$	−11.0000	−0.541275
$414$	−8.00000	−0.393179
$415$	15.0000	0.736321
$416$	6.00000	0.294174
$417$	−7.00000	−0.342791
$418$	8.00000	0.391293
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	1.00000	0.0487950
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	2.00000	0.0973585
$423$	3.00000	0.145865
$424$	7.00000	0.339950
$425$	16.0000	0.776114
$426$	4.00000	0.193801
$427$	−4.00000	−0.193574
$428$	−10.0000	−0.483368
$429$	−12.0000	−0.579365
$430$	−8.00000	−0.385794
$431$	−39.0000	−1.87856	−0.939282	−	0.343146i	$-0.888507\pi$
$431$	−39.0000	−1.87856	−0.939282	+	0.343146i	$0.888507\pi$
$432$	1.00000	0.0481125
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	−5.00000	−0.240008
$435$	2.00000	0.0958927
$436$	2.00000	0.0957826
$437$	−32.0000	−1.53077
$438$	−6.00000	−0.286691
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	2.00000	0.0953463
$441$	−6.00000	−0.285714
$442$	−24.0000	−1.14156
$443$	9.00000	0.427603	0.213801	−	0.976877i	$-0.431415\pi$
$443$	9.00000	0.427603	0.213801	+	0.976877i	$0.431415\pi$
$444$	3.00000	0.142374
$445$	−9.00000	−0.426641
$446$	−29.0000	−1.37319
$447$	9.00000	0.425685
$448$	−1.00000	−0.0472456
$449$	5.00000	0.235965	0.117982	−	0.993016i	$-0.462357\pi$
$449$	5.00000	0.235965	0.117982	+	0.993016i	$0.462357\pi$
$450$	4.00000	0.188562
$451$	−12.0000	−0.565058
$452$	−4.00000	−0.188144
$453$	20.0000	0.939682
$454$	−1.00000	−0.0469323
$455$	−6.00000	−0.281284
$456$	4.00000	0.187317
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	20.0000	0.934539
$459$	−4.00000	−0.186704
$460$	−8.00000	−0.373002
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	2.00000	0.0930484
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	−2.00000	−0.0928477
$465$	5.00000	0.231869
$466$	9.00000	0.416917
$467$	−38.0000	−1.75843	−0.879215	−	0.476425i	$-0.841932\pi$
$467$	−38.0000	−1.75843	−0.879215	+	0.476425i	$0.841932\pi$
$468$	−6.00000	−0.277350
$469$	7.00000	0.323230
$470$	3.00000	0.138380
$471$	−18.0000	−0.829396
$472$	−11.0000	−0.506316
$473$	−16.0000	−0.735681
$474$	−4.00000	−0.183726
$475$	16.0000	0.734130
$476$	4.00000	0.183340
$477$	−7.00000	−0.320508
$478$	−24.0000	−1.09773
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	1.00000	0.0456435
$481$	−18.0000	−0.820729
$482$	−12.0000	−0.546585
$483$	−8.00000	−0.364013
$484$	−7.00000	−0.318182
$485$	13.0000	0.590300
$486$	−1.00000	−0.0453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−4.00000	−0.181071
$489$	9.00000	0.406994
$490$	−6.00000	−0.271052
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	−6.00000	−0.270501
$493$	8.00000	0.360302
$494$	−24.0000	−1.07981
$495$	−2.00000	−0.0898933
$496$	−5.00000	−0.224507
$497$	4.00000	0.179425
$498$	15.0000	0.672166
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	9.00000	0.402492
$501$	1.00000	0.0446767
$502$	−20.0000	−0.892644
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	1.00000	0.0445435
$505$	−1.00000	−0.0444994
$506$	−16.0000	−0.711287
$507$	23.0000	1.02147
$508$	−17.0000	−0.754253
$509$	−16.0000	−0.709188	−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000	−0.709188	−0.354594	+	0.935020i	$0.615381\pi$
$510$	−4.00000	−0.177123
$511$	−6.00000	−0.265424
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	22.0000	0.970378
$515$	−6.00000	−0.264392
$516$	−8.00000	−0.352180
$517$	6.00000	0.263880
$518$	3.00000	0.131812
$519$	−4.00000	−0.175581
$520$	−6.00000	−0.263117
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	2.00000	0.0875376
$523$	−10.0000	−0.437269	−0.218635	−	0.975807i	$-0.570160\pi$
$523$	−10.0000	−0.437269	−0.218635	+	0.975807i	$0.570160\pi$
$524$	−12.0000	−0.524222
$525$	4.00000	0.174574
$526$	−15.0000	−0.654031
$527$	20.0000	0.871214
$528$	2.00000	0.0870388
$529$	41.0000	1.78261
$530$	−7.00000	−0.304061
$531$	11.0000	0.477359
$532$	4.00000	0.173422
$533$	36.0000	1.55933
$534$	−9.00000	−0.389468
$535$	10.0000	0.432338
$536$	7.00000	0.302354
$537$	2.00000	0.0863064
$538$	−27.0000	−1.16405
$539$	−12.0000	−0.516877
$540$	−1.00000	−0.0430331
$541$	−3.00000	−0.128980	−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000	−0.128980	−0.0644900	+	0.997918i	$0.520542\pi$
$542$	30.0000	1.28861
$543$	20.0000	0.858282
$544$	4.00000	0.171499
$545$	−2.00000	−0.0856706
$546$	−6.00000	−0.256776
$547$	−17.0000	−0.726868	−0.363434	−	0.931620i	$-0.618396\pi$
$547$	−17.0000	−0.726868	−0.363434	+	0.931620i	$0.618396\pi$
$548$	9.00000	0.384461
$549$	4.00000	0.170716
$550$	8.00000	0.341121
$551$	8.00000	0.340811
$552$	−8.00000	−0.340503
$553$	−4.00000	−0.170097
$554$	9.00000	0.382373
$555$	−3.00000	−0.127343
$556$	−7.00000	−0.296866
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	5.00000	0.211667
$559$	48.0000	2.03018
$560$	1.00000	0.0422577
$561$	−8.00000	−0.337760
$562$	−15.0000	−0.632737
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	3.00000	0.126323
$565$	4.00000	0.168281
$566$	8.00000	0.336265
$567$	−1.00000	−0.0419961
$568$	4.00000	0.167836
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	−4.00000	−0.167542
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	−12.0000	−0.501745
$573$	15.0000	0.626634
$574$	−6.00000	−0.250435
$575$	−32.0000	−1.33449
$576$	1.00000	0.0416667
$577$	31.0000	1.29055	0.645273	−	0.763952i	$-0.276743\pi$
$577$	31.0000	1.29055	0.645273	+	0.763952i	$0.276743\pi$
$578$	1.00000	0.0415945
$579$	16.0000	0.664937
$580$	2.00000	0.0830455
$581$	15.0000	0.622305
$582$	13.0000	0.538867
$583$	−14.0000	−0.579821
$584$	−6.00000	−0.248282
$585$	6.00000	0.248069
$586$	−12.0000	−0.495715
$587$	−21.0000	−0.866763	−0.433381	−	0.901211i	$-0.642680\pi$
$587$	−21.0000	−0.866763	−0.433381	+	0.901211i	$0.642680\pi$
$588$	−6.00000	−0.247436
$589$	20.0000	0.824086
$590$	11.0000	0.452863
$591$	6.00000	0.246807
$592$	3.00000	0.123299
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	−2.00000	−0.0820610
$595$	−4.00000	−0.163984
$596$	9.00000	0.368654
$597$	−8.00000	−0.327418
$598$	48.0000	1.96287
$599$	19.0000	0.776319	0.388159	−	0.921592i	$-0.373111\pi$
$599$	19.0000	0.776319	0.388159	+	0.921592i	$0.373111\pi$
$600$	4.00000	0.163299
$601$	27.0000	1.10135	0.550676	−	0.834719i	$-0.314370\pi$
$601$	27.0000	1.10135	0.550676	+	0.834719i	$0.314370\pi$
$602$	−8.00000	−0.326056
$603$	−7.00000	−0.285062
$604$	20.0000	0.813788
$605$	7.00000	0.284590
$606$	−1.00000	−0.0406222
$607$	−38.0000	−1.54237	−0.771186	−	0.636610i	$-0.780336\pi$
$607$	−38.0000	−1.54237	−0.771186	+	0.636610i	$0.780336\pi$
$608$	4.00000	0.162221
$609$	2.00000	0.0810441
$610$	4.00000	0.161955
$611$	−18.0000	−0.728202
$612$	−4.00000	−0.161690
$613$	−4.00000	−0.161558	−0.0807792	−	0.996732i	$-0.525741\pi$
$613$	−4.00000	−0.161558	−0.0807792	+	0.996732i	$0.525741\pi$
$614$	27.0000	1.08963
$615$	6.00000	0.241943
$616$	2.00000	0.0805823
$617$	−26.0000	−1.04672	−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000	−1.04672	−0.523360	+	0.852111i	$0.675322\pi$
$618$	−6.00000	−0.241355
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	5.00000	0.200805
$621$	8.00000	0.321029
$622$	27.0000	1.08260
$623$	−9.00000	−0.360577
$624$	−6.00000	−0.240192
$625$	11.0000	0.440000
$626$	−10.0000	−0.399680
$627$	−8.00000	−0.319489
$628$	−18.0000	−0.718278
$629$	−12.0000	−0.478471
$630$	−1.00000	−0.0398410
$631$	−35.0000	−1.39333	−0.696664	−	0.717398i	$-0.745333\pi$
$631$	−35.0000	−1.39333	−0.696664	+	0.717398i	$0.745333\pi$
$632$	−4.00000	−0.159111
$633$	−2.00000	−0.0794929
$634$	−6.00000	−0.238290
$635$	17.0000	0.674624
$636$	−7.00000	−0.277568
$637$	36.0000	1.42637
$638$	4.00000	0.158362
$639$	−4.00000	−0.158238
$640$	1.00000	0.0395285
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	10.0000	0.394669
$643$	40.0000	1.57745	0.788723	−	0.614749i	$-0.210743\pi$
$643$	40.0000	1.57745	0.788723	+	0.614749i	$0.210743\pi$
$644$	−8.00000	−0.315244
$645$	8.00000	0.315000
$646$	−16.0000	−0.629512
$647$	−44.0000	−1.72982	−0.864909	−	0.501928i	$-0.832624\pi$
$647$	−44.0000	−1.72982	−0.864909	+	0.501928i	$0.832624\pi$
$648$	−1.00000	−0.0392837
$649$	22.0000	0.863576
$650$	−24.0000	−0.941357
$651$	5.00000	0.195965
$652$	9.00000	0.352467
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	−2.00000	−0.0782062
$655$	12.0000	0.468879
$656$	−6.00000	−0.234261
$657$	6.00000	0.234082
$658$	3.00000	0.116952
$659$	9.00000	0.350590	0.175295	−	0.984516i	$-0.443912\pi$
$659$	9.00000	0.350590	0.175295	+	0.984516i	$0.443912\pi$
$660$	−2.00000	−0.0778499
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	−20.0000	−0.777322
$663$	24.0000	0.932083
$664$	15.0000	0.582113
$665$	−4.00000	−0.155113
$666$	−3.00000	−0.116248
$667$	−16.0000	−0.619522
$668$	1.00000	0.0386912
$669$	29.0000	1.12120
$670$	−7.00000	−0.270434
$671$	8.00000	0.308837
$672$	1.00000	0.0385758
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	−29.0000	−1.11704
$675$	−4.00000	−0.153960
$676$	23.0000	0.884615
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	4.00000	0.153619
$679$	13.0000	0.498894
$680$	−4.00000	−0.153393
$681$	1.00000	0.0383201
$682$	10.0000	0.382920
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	−4.00000	−0.152944
$685$	−9.00000	−0.343872
$686$	−13.0000	−0.496342
$687$	−20.0000	−0.763048
$688$	−8.00000	−0.304997
$689$	42.0000	1.60007
$690$	8.00000	0.304555
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−4.00000	−0.152057
$693$	−2.00000	−0.0759737
$694$	−3.00000	−0.113878
$695$	7.00000	0.265525
$696$	2.00000	0.0758098
$697$	24.0000	0.909065
$698$	−29.0000	−1.09767
$699$	−9.00000	−0.340411
$700$	4.00000	0.151186
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	6.00000	0.226455
$703$	−12.0000	−0.452589
$704$	2.00000	0.0753778
$705$	−3.00000	−0.112987
$706$	21.0000	0.790345
$707$	−1.00000	−0.0376089
$708$	11.0000	0.413405
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	−4.00000	−0.150117
$711$	4.00000	0.150012
$712$	−9.00000	−0.337289
$713$	−40.0000	−1.49801
$714$	−4.00000	−0.149696
$715$	12.0000	0.448775
$716$	2.00000	0.0747435
$717$	24.0000	0.896296
$718$	24.0000	0.895672
$719$	4.00000	0.149175	0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000	0.149175	0.0745874	+	0.997214i	$0.476236\pi$
$720$	−1.00000	−0.0372678
$721$	−6.00000	−0.223452
$722$	3.00000	0.111648
$723$	12.0000	0.446285
$724$	20.0000	0.743294
$725$	8.00000	0.297113
$726$	7.00000	0.259794
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	6.00000	0.222070
$731$	32.0000	1.18356
$732$	4.00000	0.147844
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	−8.00000	−0.294884
$737$	−14.0000	−0.515697
$738$	6.00000	0.220863
$739$	11.0000	0.404642	0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000	0.404642	0.202321	+	0.979319i	$0.435152\pi$
$740$	−3.00000	−0.110282
$741$	24.0000	0.881662
$742$	−7.00000	−0.256978
$743$	21.0000	0.770415	0.385208	−	0.922830i	$-0.374130\pi$
$743$	21.0000	0.770415	0.385208	+	0.922830i	$0.374130\pi$
$744$	5.00000	0.183309
$745$	−9.00000	−0.329734
$746$	−13.0000	−0.475964
$747$	−15.0000	−0.548821
$748$	−8.00000	−0.292509
$749$	10.0000	0.365392
$750$	−9.00000	−0.328634
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	3.00000	0.109399
$753$	20.0000	0.728841
$754$	−12.0000	−0.437014
$755$	−20.0000	−0.727875
$756$	−1.00000	−0.0363696
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	11.0000	0.399538
$759$	16.0000	0.580763
$760$	−4.00000	−0.145095
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	17.0000	0.615845
$763$	−2.00000	−0.0724049
$764$	15.0000	0.542681
$765$	4.00000	0.144620
$766$	−16.0000	−0.578103
$767$	−66.0000	−2.38312
$768$	1.00000	0.0360844
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	−2.00000	−0.0720750
$771$	−22.0000	−0.792311
$772$	16.0000	0.575853
$773$	−45.0000	−1.61854	−0.809269	−	0.587439i	$-0.800136\pi$
$773$	−45.0000	−1.61854	−0.809269	+	0.587439i	$0.800136\pi$
$774$	8.00000	0.287554
$775$	20.0000	0.718421
$776$	13.0000	0.466673
$777$	−3.00000	−0.107624
$778$	−34.0000	−1.21896
$779$	24.0000	0.859889
$780$	6.00000	0.214834
$781$	−8.00000	−0.286263
$782$	32.0000	1.14432
$783$	−2.00000	−0.0714742
$784$	−6.00000	−0.214286
$785$	18.0000	0.642448
$786$	12.0000	0.428026
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	6.00000	0.213741
$789$	15.0000	0.534014
$790$	4.00000	0.142314
$791$	4.00000	0.142224
$792$	−2.00000	−0.0710669
$793$	−24.0000	−0.852265
$794$	26.0000	0.922705
$795$	7.00000	0.248264
$796$	−8.00000	−0.283552
$797$	54.0000	1.91278	0.956389	−	0.292096i	$-0.0943526\pi$
$797$	54.0000	1.91278	0.956389	+	0.292096i	$0.0943526\pi$
$798$	−4.00000	−0.141598
$799$	−12.0000	−0.424529
$800$	4.00000	0.141421
$801$	9.00000	0.317999
$802$	24.0000	0.847469
$803$	12.0000	0.423471
$804$	−7.00000	−0.246871
$805$	8.00000	0.281963
$806$	−30.0000	−1.05670
$807$	27.0000	0.950445
$808$	−1.00000	−0.0351799
$809$	49.0000	1.72275	0.861374	−	0.507971i	$-0.169604\pi$
$809$	49.0000	1.72275	0.861374	+	0.507971i	$0.169604\pi$
$810$	1.00000	0.0351364
$811$	23.0000	0.807639	0.403820	−	0.914839i	$-0.367682\pi$
$811$	23.0000	0.807639	0.403820	+	0.914839i	$0.367682\pi$
$812$	2.00000	0.0701862
$813$	−30.0000	−1.05215
$814$	−6.00000	−0.210300
$815$	−9.00000	−0.315256
$816$	−4.00000	−0.140028
$817$	32.0000	1.11954
$818$	−26.0000	−0.909069
$819$	6.00000	0.209657
$820$	6.00000	0.209529
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	−9.00000	−0.313911
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	−6.00000	−0.209020
$825$	−8.00000	−0.278524
$826$	11.0000	0.382739
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	8.00000	0.278019
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	−15.0000	−0.520658
$831$	−9.00000	−0.312207
$832$	−6.00000	−0.208013
$833$	24.0000	0.831551
$834$	7.00000	0.242390
$835$	−1.00000	−0.0346064
$836$	−8.00000	−0.276686
$837$	−5.00000	−0.172825
$838$	16.0000	0.552711
$839$	−25.0000	−0.863096	−0.431548	−	0.902090i	$-0.642032\pi$
$839$	−25.0000	−0.863096	−0.431548	+	0.902090i	$0.642032\pi$
$840$	−1.00000	−0.0345033
$841$	−25.0000	−0.862069
$842$	−2.00000	−0.0689246
$843$	15.0000	0.516627
$844$	−2.00000	−0.0688428
$845$	−23.0000	−0.791224
$846$	−3.00000	−0.103142
$847$	7.00000	0.240523
$848$	−7.00000	−0.240381
$849$	−8.00000	−0.274559
$850$	−16.0000	−0.548795
$851$	24.0000	0.822709
$852$	−4.00000	−0.137038
$853$	−52.0000	−1.78045	−0.890223	−	0.455525i	$-0.849452\pi$
$853$	−52.0000	−1.78045	−0.890223	+	0.455525i	$0.849452\pi$
$854$	4.00000	0.136877
$855$	4.00000	0.136797
$856$	10.0000	0.341793
$857$	27.0000	0.922302	0.461151	−	0.887322i	$-0.347437\pi$
$857$	27.0000	0.922302	0.461151	+	0.887322i	$0.347437\pi$
$858$	12.0000	0.409673
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	8.00000	0.272798
$861$	6.00000	0.204479
$862$	39.0000	1.32835
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	−1.00000	−0.0340207
$865$	4.00000	0.136004
$866$	11.0000	0.373795
$867$	−1.00000	−0.0339618
$868$	5.00000	0.169711
$869$	8.00000	0.271381
$870$	−2.00000	−0.0678064
$871$	42.0000	1.42312
$872$	−2.00000	−0.0677285
$873$	−13.0000	−0.439983
$874$	32.0000	1.08242
$875$	−9.00000	−0.304256
$876$	6.00000	0.202721
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	8.00000	0.269987
$879$	12.0000	0.404750
$880$	−2.00000	−0.0674200
$881$	−28.0000	−0.943344	−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000	−0.943344	−0.471672	+	0.881774i	$0.656349\pi$
$882$	6.00000	0.202031
$883$	6.00000	0.201916	0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000	0.201916	0.100958	+	0.994891i	$0.467809\pi$
$884$	24.0000	0.807207
$885$	−11.0000	−0.369761
$886$	−9.00000	−0.302361
$887$	46.0000	1.54453	0.772264	−	0.635301i	$-0.219124\pi$
$887$	46.0000	1.54453	0.772264	+	0.635301i	$0.219124\pi$
$888$	−3.00000	−0.100673
$889$	17.0000	0.570162
$890$	9.00000	0.301681
$891$	2.00000	0.0670025
$892$	29.0000	0.970992
$893$	−12.0000	−0.401565
$894$	−9.00000	−0.301005
$895$	−2.00000	−0.0668526
$896$	1.00000	0.0334077
$897$	−48.0000	−1.60267
$898$	−5.00000	−0.166852
$899$	10.0000	0.333519
$900$	−4.00000	−0.133333
$901$	28.0000	0.932815
$902$	12.0000	0.399556
$903$	8.00000	0.266223
$904$	4.00000	0.133038
$905$	−20.0000	−0.664822
$906$	−20.0000	−0.664455
$907$	−34.0000	−1.12895	−0.564476	−	0.825450i	$-0.690922\pi$
$907$	−34.0000	−1.12895	−0.564476	+	0.825450i	$0.690922\pi$
$908$	1.00000	0.0331862
$909$	1.00000	0.0331679
$910$	6.00000	0.198898
$911$	−1.00000	−0.0331315	−0.0165657	−	0.999863i	$-0.505273\pi$
$911$	−1.00000	−0.0331315	−0.0165657	+	0.999863i	$0.505273\pi$
$912$	−4.00000	−0.132453
$913$	−30.0000	−0.992855
$914$	2.00000	0.0661541
$915$	−4.00000	−0.132236
$916$	−20.0000	−0.660819
$917$	12.0000	0.396275
$918$	4.00000	0.132020
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	8.00000	0.263752
$921$	−27.0000	−0.889680
$922$	6.00000	0.197599
$923$	24.0000	0.789970
$924$	−2.00000	−0.0657952
$925$	−12.0000	−0.394558
$926$	22.0000	0.722965
$927$	6.00000	0.197066
$928$	2.00000	0.0656532
$929$	−33.0000	−1.08269	−0.541347	−	0.840799i	$-0.682086\pi$
$929$	−33.0000	−1.08269	−0.541347	+	0.840799i	$0.682086\pi$
$930$	−5.00000	−0.163956
$931$	24.0000	0.786568
$932$	−9.00000	−0.294805
$933$	−27.0000	−0.883940
$934$	38.0000	1.24340
$935$	8.00000	0.261628
$936$	6.00000	0.196116
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	−7.00000	−0.228558
$939$	10.0000	0.326338
$940$	−3.00000	−0.0978492
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	18.0000	0.586472
$943$	−48.0000	−1.56310
$944$	11.0000	0.358020
$945$	1.00000	0.0325300
$946$	16.0000	0.520205
$947$	58.0000	1.88475	0.942373	−	0.334563i	$-0.108589\pi$
$947$	58.0000	1.88475	0.942373	+	0.334563i	$0.108589\pi$
$948$	4.00000	0.129914
$949$	−36.0000	−1.16861
$950$	−16.0000	−0.519109
$951$	6.00000	0.194563
$952$	−4.00000	−0.129641
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	7.00000	0.226633
$955$	−15.0000	−0.485389
$956$	24.0000	0.776215
$957$	−4.00000	−0.129302
$958$	−24.0000	−0.775405
$959$	−9.00000	−0.290625
$960$	−1.00000	−0.0322749
$961$	−6.00000	−0.193548
$962$	18.0000	0.580343
$963$	−10.0000	−0.322245
$964$	12.0000	0.386494
$965$	−16.0000	−0.515058
$966$	8.00000	0.257396
$967$	−24.0000	−0.771788	−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000	−0.771788	−0.385894	+	0.922543i	$0.626107\pi$
$968$	7.00000	0.224989
$969$	16.0000	0.513994
$970$	−13.0000	−0.417405
$971$	−47.0000	−1.50830	−0.754151	−	0.656701i	$-0.771951\pi$
$971$	−47.0000	−1.50830	−0.754151	+	0.656701i	$0.771951\pi$
$972$	1.00000	0.0320750
$973$	7.00000	0.224410
$974$	16.0000	0.512673
$975$	24.0000	0.768615
$976$	4.00000	0.128037
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	−9.00000	−0.287788
$979$	18.0000	0.575282
$980$	6.00000	0.191663
$981$	2.00000	0.0638551
$982$	−8.00000	−0.255290
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	6.00000	0.191273
$985$	−6.00000	−0.191176
$986$	−8.00000	−0.254772
$987$	−3.00000	−0.0954911
$988$	24.0000	0.763542
$989$	−64.0000	−2.03508
$990$	2.00000	0.0635642
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	5.00000	0.158750
$993$	20.0000	0.634681
$994$	−4.00000	−0.126872
$995$	8.00000	0.253617
$996$	−15.0000	−0.475293
$997$	−24.0000	−0.760088	−0.380044	−	0.924968i	$-0.624091\pi$
$997$	−24.0000	−0.760088	−0.380044	+	0.924968i	$0.624091\pi$
$998$	24.0000	0.759707
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1002.2.a.c.1.1	✓	1
3.2	odd	2		3006.2.a.f.1.1		1
4.3	odd	2		8016.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.c.1.1	✓	1	1.1	even	1	trivial
3006.2.a.f.1.1		1	3.2	odd	2
8016.2.a.b.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1002 = 2 \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1002.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more