LMFDB - Embedded newform 139.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [139,2,Mod(1,139)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(139, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("139.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	139.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:	$1.10992058810$
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	$\chi$	$=$	139.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +3.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +3.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +5.00000 q^{11} -2.00000 q^{12} -7.00000 q^{13} +3.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +6.00000 q^{21} +5.00000 q^{22} +2.00000 q^{23} -6.00000 q^{24} -4.00000 q^{25} -7.00000 q^{26} -4.00000 q^{27} -3.00000 q^{28} +9.00000 q^{29} -2.00000 q^{30} +9.00000 q^{31} +5.00000 q^{32} +10.0000 q^{33} -6.00000 q^{34} -3.00000 q^{35} -1.00000 q^{36} +2.00000 q^{37} -2.00000 q^{38} -14.0000 q^{39} +3.00000 q^{40} -6.00000 q^{41} +6.00000 q^{42} -4.00000 q^{43} -5.00000 q^{44} -1.00000 q^{45} +2.00000 q^{46} +8.00000 q^{47} -2.00000 q^{48} +2.00000 q^{49} -4.00000 q^{50} -12.0000 q^{51} +7.00000 q^{52} -4.00000 q^{54} -5.00000 q^{55} -9.00000 q^{56} -4.00000 q^{57} +9.00000 q^{58} +6.00000 q^{59} +2.00000 q^{60} +4.00000 q^{61} +9.00000 q^{62} +3.00000 q^{63} +7.00000 q^{64} +7.00000 q^{65} +10.0000 q^{66} +5.00000 q^{67} +6.00000 q^{68} +4.00000 q^{69} -3.00000 q^{70} +5.00000 q^{71} -3.00000 q^{72} -6.00000 q^{73} +2.00000 q^{74} -8.00000 q^{75} +2.00000 q^{76} +15.0000 q^{77} -14.0000 q^{78} -5.00000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -6.00000 q^{82} +7.00000 q^{83} -6.00000 q^{84} +6.00000 q^{85} -4.00000 q^{86} +18.0000 q^{87} -15.0000 q^{88} +7.00000 q^{89} -1.00000 q^{90} -21.0000 q^{91} -2.00000 q^{92} +18.0000 q^{93} +8.00000 q^{94} +2.00000 q^{95} +10.0000 q^{96} -12.0000 q^{97} +2.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	−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	$0.571783\pi$
$6$	2.00000	0.816497
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	−2.00000	−0.577350
$13$	−7.00000	−1.94145	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−7.00000	−1.94145	−0.970725	+	0.240192i	$0.922790\pi$
$14$	3.00000	0.801784
$15$	−2.00000	−0.516398
$16$	−1.00000	−0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	1.00000	0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	1.00000	0.223607
$21$	6.00000	1.30931
$22$	5.00000	1.06600
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	−6.00000	−1.22474
$25$	−4.00000	−0.800000
$26$	−7.00000	−1.37281
$27$	−4.00000	−0.769800
$28$	−3.00000	−0.566947
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	−2.00000	−0.365148
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	5.00000	0.883883
$33$	10.0000	1.74078
$34$	−6.00000	−1.02899
$35$	−3.00000	−0.507093
$36$	−1.00000	−0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−2.00000	−0.324443
$39$	−14.0000	−2.24179
$40$	3.00000	0.474342
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	6.00000	0.925820
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−5.00000	−0.753778
$45$	−1.00000	−0.149071
$46$	2.00000	0.294884
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−2.00000	−0.288675
$49$	2.00000	0.285714
$50$	−4.00000	−0.565685
$51$	−12.0000	−1.68034
$52$	7.00000	0.970725
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−4.00000	−0.544331
$55$	−5.00000	−0.674200
$56$	−9.00000	−1.20268
$57$	−4.00000	−0.529813
$58$	9.00000	1.18176
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	2.00000	0.258199
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	9.00000	1.14300
$63$	3.00000	0.377964
$64$	7.00000	0.875000
$65$	7.00000	0.868243
$66$	10.0000	1.23091
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	6.00000	0.727607
$69$	4.00000	0.481543
$70$	−3.00000	−0.358569
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	−3.00000	−0.353553
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	2.00000	0.232495
$75$	−8.00000	−0.923760
$76$	2.00000	0.229416
$77$	15.0000	1.70941
$78$	−14.0000	−1.58519
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	−6.00000	−0.662589
$83$	7.00000	0.768350	0.384175	−	0.923260i	$-0.374486\pi$
$83$	7.00000	0.768350	0.384175	+	0.923260i	$0.374486\pi$
$84$	−6.00000	−0.654654
$85$	6.00000	0.650791
$86$	−4.00000	−0.431331
$87$	18.0000	1.92980
$88$	−15.0000	−1.59901
$89$	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
$91$	−21.0000	−2.20140
$92$	−2.00000	−0.208514
$93$	18.0000	1.86651
$94$	8.00000	0.825137
$95$	2.00000	0.205196
$96$	10.0000	1.02062
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	2.00000	0.202031
$99$	5.00000	0.502519
$100$	4.00000	0.400000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	−12.0000	−1.18818
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	21.0000	2.05922
$105$	−6.00000	−0.585540
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	4.00000	0.384900
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−5.00000	−0.476731
$111$	4.00000	0.379663
$112$	−3.00000	−0.283473
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	−4.00000	−0.374634
$115$	−2.00000	−0.186501
$116$	−9.00000	−0.835629
$117$	−7.00000	−0.647150
$118$	6.00000	0.552345
$119$	−18.0000	−1.65006
$120$	6.00000	0.547723
$121$	14.0000	1.27273
$122$	4.00000	0.362143
$123$	−12.0000	−1.08200
$124$	−9.00000	−0.808224
$125$	9.00000	0.804984
$126$	3.00000	0.267261
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	−3.00000	−0.265165
$129$	−8.00000	−0.704361
$130$	7.00000	0.613941
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	−10.0000	−0.870388
$133$	−6.00000	−0.520266
$134$	5.00000	0.431934
$135$	4.00000	0.344265
$136$	18.0000	1.54349
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	4.00000	0.340503
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	3.00000	0.253546
$141$	16.0000	1.34744
$142$	5.00000	0.419591
$143$	−35.0000	−2.92685
$144$	−1.00000	−0.0833333
$145$	−9.00000	−0.747409
$146$	−6.00000	−0.496564
$147$	4.00000	0.329914
$148$	−2.00000	−0.164399
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	−8.00000	−0.653197
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	6.00000	0.486664
$153$	−6.00000	−0.485071
$154$	15.0000	1.20873
$155$	−9.00000	−0.722897
$156$	14.0000	1.12090
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	−5.00000	−0.397779
$159$	0	0
$160$	−5.00000	−0.395285
$161$	6.00000	0.472866
$162$	−11.0000	−0.864242
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	6.00000	0.468521
$165$	−10.0000	−0.778499
$166$	7.00000	0.543305
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	−18.0000	−1.38873
$169$	36.0000	2.76923
$170$	6.00000	0.460179
$171$	−2.00000	−0.152944
$172$	4.00000	0.304997
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	18.0000	1.36458
$175$	−12.0000	−0.907115
$176$	−5.00000	−0.376889
$177$	12.0000	0.901975
$178$	7.00000	0.524672
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	1.00000	0.0745356
$181$	3.00000	0.222988	0.111494	−	0.993765i	$-0.464436\pi$
$181$	3.00000	0.222988	0.111494	+	0.993765i	$0.464436\pi$
$182$	−21.0000	−1.55662
$183$	8.00000	0.591377
$184$	−6.00000	−0.442326
$185$	−2.00000	−0.147043
$186$	18.0000	1.31982
$187$	−30.0000	−2.19382
$188$	−8.00000	−0.583460
$189$	−12.0000	−0.872872
$190$	2.00000	0.145095
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	14.0000	1.01036
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	−12.0000	−0.861550
$195$	14.0000	1.00256
$196$	−2.00000	−0.142857
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	5.00000	0.355335
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	12.0000	0.848528
$201$	10.0000	0.705346
$202$	2.00000	0.140720
$203$	27.0000	1.89503
$204$	12.0000	0.840168
$205$	6.00000	0.419058
$206$	−16.0000	−1.11477
$207$	2.00000	0.139010
$208$	7.00000	0.485363
$209$	−10.0000	−0.691714
$210$	−6.00000	−0.414039
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	0	0
$213$	10.0000	0.685189
$214$	4.00000	0.273434
$215$	4.00000	0.272798
$216$	12.0000	0.816497
$217$	27.0000	1.83288
$218$	−10.0000	−0.677285
$219$	−12.0000	−0.810885
$220$	5.00000	0.337100
$221$	42.0000	2.82523
$222$	4.00000	0.268462
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	15.0000	1.00223
$225$	−4.00000	−0.266667
$226$	1.00000	0.0665190
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	4.00000	0.264906
$229$	−28.0000	−1.85029	−0.925146	−	0.379611i	$-0.876058\pi$
$229$	−28.0000	−1.85029	−0.925146	+	0.379611i	$0.876058\pi$
$230$	−2.00000	−0.131876
$231$	30.0000	1.97386
$232$	−27.0000	−1.77264
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−7.00000	−0.457604
$235$	−8.00000	−0.521862
$236$	−6.00000	−0.390567
$237$	−10.0000	−0.649570
$238$	−18.0000	−1.16677
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	2.00000	0.129099
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	14.0000	0.899954
$243$	−10.0000	−0.641500
$244$	−4.00000	−0.256074
$245$	−2.00000	−0.127775
$246$	−12.0000	−0.765092
$247$	14.0000	0.890799
$248$	−27.0000	−1.71450
$249$	14.0000	0.887214
$250$	9.00000	0.569210
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	−3.00000	−0.188982
$253$	10.0000	0.628695
$254$	5.00000	0.313728
$255$	12.0000	0.751469
$256$	−17.0000	−1.06250
$257$	−19.0000	−1.18519	−0.592594	−	0.805502i	$-0.701896\pi$
$257$	−19.0000	−1.18519	−0.592594	+	0.805502i	$0.701896\pi$
$258$	−8.00000	−0.498058
$259$	6.00000	0.372822
$260$	−7.00000	−0.434122
$261$	9.00000	0.557086
$262$	−9.00000	−0.556022
$263$	−1.00000	−0.0616626	−0.0308313	−	0.999525i	$-0.509815\pi$
$263$	−1.00000	−0.0616626	−0.0308313	+	0.999525i	$0.509815\pi$
$264$	−30.0000	−1.84637
$265$	0	0
$266$	−6.00000	−0.367884
$267$	14.0000	0.856786
$268$	−5.00000	−0.305424
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	4.00000	0.243432
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	6.00000	0.363803
$273$	−42.0000	−2.54196
$274$	−3.00000	−0.181237
$275$	−20.0000	−1.20605
$276$	−4.00000	−0.240772
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	1.00000	0.0599760
$279$	9.00000	0.538816
$280$	9.00000	0.537853
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	16.0000	0.952786
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	−5.00000	−0.296695
$285$	4.00000	0.236940
$286$	−35.0000	−2.06959
$287$	−18.0000	−1.06251
$288$	5.00000	0.294628
$289$	19.0000	1.11765
$290$	−9.00000	−0.528498
$291$	−24.0000	−1.40690
$292$	6.00000	0.351123
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	4.00000	0.233285
$295$	−6.00000	−0.349334
$296$	−6.00000	−0.348743
$297$	−20.0000	−1.16052
$298$	−2.00000	−0.115857
$299$	−14.0000	−0.809641
$300$	8.00000	0.461880
$301$	−12.0000	−0.691669
$302$	−2.00000	−0.115087
$303$	4.00000	0.229794
$304$	2.00000	0.114708
$305$	−4.00000	−0.229039
$306$	−6.00000	−0.342997
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	−15.0000	−0.854704
$309$	−32.0000	−1.82042
$310$	−9.00000	−0.511166
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	42.0000	2.37778
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−22.0000	−1.24153
$315$	−3.00000	−0.169031
$316$	5.00000	0.281272
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	45.0000	2.51952
$320$	−7.00000	−0.391312
$321$	8.00000	0.446516
$322$	6.00000	0.334367
$323$	12.0000	0.667698
$324$	11.0000	0.611111
$325$	28.0000	1.55316
$326$	11.0000	0.609234
$327$	−20.0000	−1.10600
$328$	18.0000	0.993884
$329$	24.0000	1.32316
$330$	−10.0000	−0.550482
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−7.00000	−0.384175
$333$	2.00000	0.109599
$334$	−8.00000	−0.437741
$335$	−5.00000	−0.273179
$336$	−6.00000	−0.327327
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	36.0000	1.95814
$339$	2.00000	0.108625
$340$	−6.00000	−0.325396
$341$	45.0000	2.43689
$342$	−2.00000	−0.108148
$343$	−15.0000	−0.809924
$344$	12.0000	0.646997
$345$	−4.00000	−0.215353
$346$	9.00000	0.483843
$347$	−13.0000	−0.697877	−0.348938	−	0.937146i	$-0.613458\pi$
$347$	−13.0000	−0.697877	−0.348938	+	0.937146i	$0.613458\pi$
$348$	−18.0000	−0.964901
$349$	27.0000	1.44528	0.722638	−	0.691226i	$-0.242929\pi$
$349$	27.0000	1.44528	0.722638	+	0.691226i	$0.242929\pi$
$350$	−12.0000	−0.641427
$351$	28.0000	1.49453
$352$	25.0000	1.33250
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	12.0000	0.637793
$355$	−5.00000	−0.265372
$356$	−7.00000	−0.370999
$357$	−36.0000	−1.90532
$358$	−18.0000	−0.951330
$359$	27.0000	1.42501	0.712503	−	0.701669i	$-0.247562\pi$
$359$	27.0000	1.42501	0.712503	+	0.701669i	$0.247562\pi$
$360$	3.00000	0.158114
$361$	−15.0000	−0.789474
$362$	3.00000	0.157676
$363$	28.0000	1.46962
$364$	21.0000	1.10070
$365$	6.00000	0.314054
$366$	8.00000	0.418167
$367$	5.00000	0.260998	0.130499	−	0.991448i	$-0.458342\pi$
$367$	5.00000	0.260998	0.130499	+	0.991448i	$0.458342\pi$
$368$	−2.00000	−0.104257
$369$	−6.00000	−0.312348
$370$	−2.00000	−0.103975
$371$	0	0
$372$	−18.0000	−0.933257
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	−30.0000	−1.55126
$375$	18.0000	0.929516
$376$	−24.0000	−1.23771
$377$	−63.0000	−3.24467
$378$	−12.0000	−0.617213
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	−2.00000	−0.102598
$381$	10.0000	0.512316
$382$	4.00000	0.204658
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−6.00000	−0.306186
$385$	−15.0000	−0.764471
$386$	−11.0000	−0.559885
$387$	−4.00000	−0.203331
$388$	12.0000	0.609208
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	14.0000	0.708918
$391$	−12.0000	−0.606866
$392$	−6.00000	−0.303046
$393$	−18.0000	−0.907980
$394$	0	0
$395$	5.00000	0.251577
$396$	−5.00000	−0.251259
$397$	32.0000	1.60603	0.803017	−	0.595956i	$-0.203227\pi$
$397$	32.0000	1.60603	0.803017	+	0.595956i	$0.203227\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	4.00000	0.200000
$401$	−4.00000	−0.199750	−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000	−0.199750	−0.0998752	+	0.995000i	$0.531844\pi$
$402$	10.0000	0.498755
$403$	−63.0000	−3.13825
$404$	−2.00000	−0.0995037
$405$	11.0000	0.546594
$406$	27.0000	1.33999
$407$	10.0000	0.495682
$408$	36.0000	1.78227
$409$	−37.0000	−1.82953	−0.914766	−	0.403984i	$-0.867625\pi$
$409$	−37.0000	−1.82953	−0.914766	+	0.403984i	$0.867625\pi$
$410$	6.00000	0.296319
$411$	−6.00000	−0.295958
$412$	16.0000	0.788263
$413$	18.0000	0.885722
$414$	2.00000	0.0982946
$415$	−7.00000	−0.343616
$416$	−35.0000	−1.71602
$417$	2.00000	0.0979404
$418$	−10.0000	−0.489116
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	6.00000	0.292770
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	10.0000	0.486792
$423$	8.00000	0.388973
$424$	0	0
$425$	24.0000	1.16417
$426$	10.0000	0.484502
$427$	12.0000	0.580721
$428$	−4.00000	−0.193347
$429$	−70.0000	−3.37963
$430$	4.00000	0.192897
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	4.00000	0.192450
$433$	7.00000	0.336399	0.168199	−	0.985753i	$-0.446205\pi$
$433$	7.00000	0.336399	0.168199	+	0.985753i	$0.446205\pi$
$434$	27.0000	1.29604
$435$	−18.0000	−0.863034
$436$	10.0000	0.478913
$437$	−4.00000	−0.191346
$438$	−12.0000	−0.573382
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	15.0000	0.715097
$441$	2.00000	0.0952381
$442$	42.0000	1.99774
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−4.00000	−0.189832
$445$	−7.00000	−0.331832
$446$	20.0000	0.947027
$447$	−4.00000	−0.189194
$448$	21.0000	0.992157
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	−4.00000	−0.188562
$451$	−30.0000	−1.41264
$452$	−1.00000	−0.0470360
$453$	−4.00000	−0.187936
$454$	20.0000	0.938647
$455$	21.0000	0.984495
$456$	12.0000	0.561951
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	−28.0000	−1.30835
$459$	24.0000	1.12022
$460$	2.00000	0.0932505
$461$	−25.0000	−1.16437	−0.582183	−	0.813058i	$-0.697801\pi$
$461$	−25.0000	−1.16437	−0.582183	+	0.813058i	$0.697801\pi$
$462$	30.0000	1.39573
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	−9.00000	−0.417815
$465$	−18.0000	−0.834730
$466$	−6.00000	−0.277945
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	7.00000	0.323575
$469$	15.0000	0.692636
$470$	−8.00000	−0.369012
$471$	−44.0000	−2.02741
$472$	−18.0000	−0.828517
$473$	−20.0000	−0.919601
$474$	−10.0000	−0.459315
$475$	8.00000	0.367065
$476$	18.0000	0.825029
$477$	0	0
$478$	−16.0000	−0.731823
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	−10.0000	−0.456435
$481$	−14.0000	−0.638345
$482$	−6.00000	−0.273293
$483$	12.0000	0.546019
$484$	−14.0000	−0.636364
$485$	12.0000	0.544892
$486$	−10.0000	−0.453609
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	−12.0000	−0.543214
$489$	22.0000	0.994874
$490$	−2.00000	−0.0903508
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	12.0000	0.541002
$493$	−54.0000	−2.43204
$494$	14.0000	0.629890
$495$	−5.00000	−0.224733
$496$	−9.00000	−0.404112
$497$	15.0000	0.672842
$498$	14.0000	0.627355
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−9.00000	−0.402492
$501$	−16.0000	−0.714827
$502$	15.0000	0.669483
$503$	−31.0000	−1.38222	−0.691111	−	0.722749i	$-0.742878\pi$
$503$	−31.0000	−1.38222	−0.691111	+	0.722749i	$0.742878\pi$
$504$	−9.00000	−0.400892
$505$	−2.00000	−0.0889988
$506$	10.0000	0.444554
$507$	72.0000	3.19763
$508$	−5.00000	−0.221839
$509$	8.00000	0.354594	0.177297	−	0.984157i	$-0.443265\pi$
$509$	8.00000	0.354594	0.177297	+	0.984157i	$0.443265\pi$
$510$	12.0000	0.531369
$511$	−18.0000	−0.796273
$512$	−11.0000	−0.486136
$513$	8.00000	0.353209
$514$	−19.0000	−0.838054
$515$	16.0000	0.705044
$516$	8.00000	0.352180
$517$	40.0000	1.75920
$518$	6.00000	0.263625
$519$	18.0000	0.790112
$520$	−21.0000	−0.920911
$521$	28.0000	1.22670	0.613351	−	0.789810i	$-0.289821\pi$
$521$	28.0000	1.22670	0.613351	+	0.789810i	$0.289821\pi$
$522$	9.00000	0.393919
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	9.00000	0.393167
$525$	−24.0000	−1.04745
$526$	−1.00000	−0.0436021
$527$	−54.0000	−2.35228
$528$	−10.0000	−0.435194
$529$	−19.0000	−0.826087
$530$	0	0
$531$	6.00000	0.260378
$532$	6.00000	0.260133
$533$	42.0000	1.81922
$534$	14.0000	0.605839
$535$	−4.00000	−0.172935
$536$	−15.0000	−0.647901
$537$	−36.0000	−1.55351
$538$	−6.00000	−0.258678
$539$	10.0000	0.430730
$540$	−4.00000	−0.172133
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	16.0000	0.687259
$543$	6.00000	0.257485
$544$	−30.0000	−1.28624
$545$	10.0000	0.428353
$546$	−42.0000	−1.79743
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	3.00000	0.128154
$549$	4.00000	0.170716
$550$	−20.0000	−0.852803
$551$	−18.0000	−0.766826
$552$	−12.0000	−0.510754
$553$	−15.0000	−0.637865
$554$	12.0000	0.509831
$555$	−4.00000	−0.169791
$556$	−1.00000	−0.0424094
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	9.00000	0.381000
$559$	28.0000	1.18427
$560$	3.00000	0.126773
$561$	−60.0000	−2.53320
$562$	0	0
$563$	−3.00000	−0.126435	−0.0632175	−	0.998000i	$-0.520136\pi$
$563$	−3.00000	−0.126435	−0.0632175	+	0.998000i	$0.520136\pi$
$564$	−16.0000	−0.673722
$565$	−1.00000	−0.0420703
$566$	16.0000	0.672530
$567$	−33.0000	−1.38587
$568$	−15.0000	−0.629386
$569$	29.0000	1.21574	0.607872	−	0.794035i	$-0.292024\pi$
$569$	29.0000	1.21574	0.607872	+	0.794035i	$0.292024\pi$
$570$	4.00000	0.167542
$571$	2.00000	0.0836974	0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000	0.0836974	0.0418487	+	0.999124i	$0.486675\pi$
$572$	35.0000	1.46342
$573$	8.00000	0.334205
$574$	−18.0000	−0.751305
$575$	−8.00000	−0.333623
$576$	7.00000	0.291667
$577$	−20.0000	−0.832611	−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000	−0.832611	−0.416305	+	0.909225i	$0.636675\pi$
$578$	19.0000	0.790296
$579$	−22.0000	−0.914289
$580$	9.00000	0.373705
$581$	21.0000	0.871227
$582$	−24.0000	−0.994832
$583$	0	0
$584$	18.0000	0.744845
$585$	7.00000	0.289414
$586$	0	0
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	−4.00000	−0.164957
$589$	−18.0000	−0.741677
$590$	−6.00000	−0.247016
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	7.00000	0.287456	0.143728	−	0.989617i	$-0.454091\pi$
$593$	7.00000	0.287456	0.143728	+	0.989617i	$0.454091\pi$
$594$	−20.0000	−0.820610
$595$	18.0000	0.737928
$596$	2.00000	0.0819232
$597$	0	0
$598$	−14.0000	−0.572503
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	24.0000	0.979796
$601$	21.0000	0.856608	0.428304	−	0.903635i	$-0.359111\pi$
$601$	21.0000	0.856608	0.428304	+	0.903635i	$0.359111\pi$
$602$	−12.0000	−0.489083
$603$	5.00000	0.203616
$604$	2.00000	0.0813788
$605$	−14.0000	−0.569181
$606$	4.00000	0.162489
$607$	1.00000	0.0405887	0.0202944	−	0.999794i	$-0.493540\pi$
$607$	1.00000	0.0405887	0.0202944	+	0.999794i	$0.493540\pi$
$608$	−10.0000	−0.405554
$609$	54.0000	2.18819
$610$	−4.00000	−0.161955
$611$	−56.0000	−2.26552
$612$	6.00000	0.242536
$613$	−13.0000	−0.525065	−0.262533	−	0.964923i	$-0.584558\pi$
$613$	−13.0000	−0.525065	−0.262533	+	0.964923i	$0.584558\pi$
$614$	−16.0000	−0.645707
$615$	12.0000	0.483887
$616$	−45.0000	−1.81310
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	−32.0000	−1.28723
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	9.00000	0.361449
$621$	−8.00000	−0.321029
$622$	−6.00000	−0.240578
$623$	21.0000	0.841347
$624$	14.0000	0.560449
$625$	11.0000	0.440000
$626$	−10.0000	−0.399680
$627$	−20.0000	−0.798723
$628$	22.0000	0.877896
$629$	−12.0000	−0.478471
$630$	−3.00000	−0.119523
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	15.0000	0.596668
$633$	20.0000	0.794929
$634$	−18.0000	−0.714871
$635$	−5.00000	−0.198419
$636$	0	0
$637$	−14.0000	−0.554700
$638$	45.0000	1.78157
$639$	5.00000	0.197797
$640$	3.00000	0.118585
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	8.00000	0.315735
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	−6.00000	−0.236433
$645$	8.00000	0.315000
$646$	12.0000	0.472134
$647$	−43.0000	−1.69050	−0.845252	−	0.534368i	$-0.820550\pi$
$647$	−43.0000	−1.69050	−0.845252	+	0.534368i	$0.820550\pi$
$648$	33.0000	1.29636
$649$	30.0000	1.17760
$650$	28.0000	1.09825
$651$	54.0000	2.11643
$652$	−11.0000	−0.430793
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	−20.0000	−0.782062
$655$	9.00000	0.351659
$656$	6.00000	0.234261
$657$	−6.00000	−0.234082
$658$	24.0000	0.935617
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	10.0000	0.389249
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	4.00000	0.155464
$663$	84.0000	3.26229
$664$	−21.0000	−0.814958
$665$	6.00000	0.232670
$666$	2.00000	0.0774984
$667$	18.0000	0.696963
$668$	8.00000	0.309529
$669$	40.0000	1.54649
$670$	−5.00000	−0.193167
$671$	20.0000	0.772091
$672$	30.0000	1.15728
$673$	−41.0000	−1.58043	−0.790217	−	0.612827i	$-0.790032\pi$
$673$	−41.0000	−1.58043	−0.790217	+	0.612827i	$0.790032\pi$
$674$	14.0000	0.539260
$675$	16.0000	0.615840
$676$	−36.0000	−1.38462
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	2.00000	0.0768095
$679$	−36.0000	−1.38155
$680$	−18.0000	−0.690268
$681$	40.0000	1.53280
$682$	45.0000	1.72314
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	2.00000	0.0764719
$685$	3.00000	0.114624
$686$	−15.0000	−0.572703
$687$	−56.0000	−2.13653
$688$	4.00000	0.152499
$689$	0	0
$690$	−4.00000	−0.152277
$691$	−6.00000	−0.228251	−0.114125	−	0.993466i	$-0.536407\pi$
$691$	−6.00000	−0.228251	−0.114125	+	0.993466i	$0.536407\pi$
$692$	−9.00000	−0.342129
$693$	15.0000	0.569803
$694$	−13.0000	−0.493473
$695$	−1.00000	−0.0379322
$696$	−54.0000	−2.04686
$697$	36.0000	1.36360
$698$	27.0000	1.02197
$699$	−12.0000	−0.453882
$700$	12.0000	0.453557
$701$	−5.00000	−0.188847	−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000	−0.188847	−0.0944237	+	0.995532i	$0.530101\pi$
$702$	28.0000	1.05679
$703$	−4.00000	−0.150863
$704$	35.0000	1.31911
$705$	−16.0000	−0.602595
$706$	−18.0000	−0.677439
$707$	6.00000	0.225653
$708$	−12.0000	−0.450988
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	−5.00000	−0.187647
$711$	−5.00000	−0.187515
$712$	−21.0000	−0.787008
$713$	18.0000	0.674105
$714$	−36.0000	−1.34727
$715$	35.0000	1.30893
$716$	18.0000	0.672692
$717$	−32.0000	−1.19506
$718$	27.0000	1.00763
$719$	−33.0000	−1.23069	−0.615346	−	0.788257i	$-0.710984\pi$
$719$	−33.0000	−1.23069	−0.615346	+	0.788257i	$0.710984\pi$
$720$	1.00000	0.0372678
$721$	−48.0000	−1.78761
$722$	−15.0000	−0.558242
$723$	−12.0000	−0.446285
$724$	−3.00000	−0.111494
$725$	−36.0000	−1.33701
$726$	28.0000	1.03918
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	63.0000	2.33494
$729$	13.0000	0.481481
$730$	6.00000	0.222070
$731$	24.0000	0.887672
$732$	−8.00000	−0.295689
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	5.00000	0.184553
$735$	−4.00000	−0.147542
$736$	10.0000	0.368605
$737$	25.0000	0.920887
$738$	−6.00000	−0.220863
$739$	−1.00000	−0.0367856	−0.0183928	−	0.999831i	$-0.505855\pi$
$739$	−1.00000	−0.0367856	−0.0183928	+	0.999831i	$0.505855\pi$
$740$	2.00000	0.0735215
$741$	28.0000	1.02861
$742$	0	0
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	−54.0000	−1.97974
$745$	2.00000	0.0732743
$746$	−4.00000	−0.146450
$747$	7.00000	0.256117
$748$	30.0000	1.09691
$749$	12.0000	0.438470
$750$	18.0000	0.657267
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	−8.00000	−0.291730
$753$	30.0000	1.09326
$754$	−63.0000	−2.29432
$755$	2.00000	0.0727875
$756$	12.0000	0.436436
$757$	44.0000	1.59921	0.799604	−	0.600528i	$-0.205043\pi$
$757$	44.0000	1.59921	0.799604	+	0.600528i	$0.205043\pi$
$758$	16.0000	0.581146
$759$	20.0000	0.725954
$760$	−6.00000	−0.217643
$761$	13.0000	0.471250	0.235625	−	0.971844i	$-0.424286\pi$
$761$	13.0000	0.471250	0.235625	+	0.971844i	$0.424286\pi$
$762$	10.0000	0.362262
$763$	−30.0000	−1.08607
$764$	−4.00000	−0.144715
$765$	6.00000	0.216930
$766$	24.0000	0.867155
$767$	−42.0000	−1.51653
$768$	−34.0000	−1.22687
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	−15.0000	−0.540562
$771$	−38.0000	−1.36854
$772$	11.0000	0.395899
$773$	37.0000	1.33080	0.665399	−	0.746488i	$-0.268262\pi$
$773$	37.0000	1.33080	0.665399	+	0.746488i	$0.268262\pi$
$774$	−4.00000	−0.143777
$775$	−36.0000	−1.29316
$776$	36.0000	1.29232
$777$	12.0000	0.430498
$778$	36.0000	1.29066
$779$	12.0000	0.429945
$780$	−14.0000	−0.501280
$781$	25.0000	0.894570
$782$	−12.0000	−0.429119
$783$	−36.0000	−1.28654
$784$	−2.00000	−0.0714286
$785$	22.0000	0.785214
$786$	−18.0000	−0.642039
$787$	−18.0000	−0.641631	−0.320815	−	0.947142i	$-0.603957\pi$
$787$	−18.0000	−0.641631	−0.320815	+	0.947142i	$0.603957\pi$
$788$	0	0
$789$	−2.00000	−0.0712019
$790$	5.00000	0.177892
$791$	3.00000	0.106668
$792$	−15.0000	−0.533002
$793$	−28.0000	−0.994309
$794$	32.0000	1.13564
$795$	0	0
$796$	0	0
$797$	28.0000	0.991811	0.495905	−	0.868377i	$-0.334836\pi$
$797$	28.0000	0.991811	0.495905	+	0.868377i	$0.334836\pi$
$798$	−12.0000	−0.424795
$799$	−48.0000	−1.69812
$800$	−20.0000	−0.707107
$801$	7.00000	0.247333
$802$	−4.00000	−0.141245
$803$	−30.0000	−1.05868
$804$	−10.0000	−0.352673
$805$	−6.00000	−0.211472
$806$	−63.0000	−2.21908
$807$	−12.0000	−0.422420
$808$	−6.00000	−0.211079
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	11.0000	0.386501
$811$	−1.00000	−0.0351147	−0.0175574	−	0.999846i	$-0.505589\pi$
$811$	−1.00000	−0.0351147	−0.0175574	+	0.999846i	$0.505589\pi$
$812$	−27.0000	−0.947514
$813$	32.0000	1.12229
$814$	10.0000	0.350500
$815$	−11.0000	−0.385313
$816$	12.0000	0.420084
$817$	8.00000	0.279885
$818$	−37.0000	−1.29367
$819$	−21.0000	−0.733799
$820$	−6.00000	−0.209529
$821$	34.0000	1.18661	0.593304	−	0.804978i	$-0.297823\pi$
$821$	34.0000	1.18661	0.593304	+	0.804978i	$0.297823\pi$
$822$	−6.00000	−0.209274
$823$	−26.0000	−0.906303	−0.453152	−	0.891434i	$-0.649700\pi$
$823$	−26.0000	−0.906303	−0.453152	+	0.891434i	$0.649700\pi$
$824$	48.0000	1.67216
$825$	−40.0000	−1.39262
$826$	18.0000	0.626300
$827$	38.0000	1.32139	0.660695	−	0.750655i	$-0.270262\pi$
$827$	38.0000	1.32139	0.660695	+	0.750655i	$0.270262\pi$
$828$	−2.00000	−0.0695048
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	−7.00000	−0.242974
$831$	24.0000	0.832551
$832$	−49.0000	−1.69877
$833$	−12.0000	−0.415775
$834$	2.00000	0.0692543
$835$	8.00000	0.276851
$836$	10.0000	0.345857
$837$	−36.0000	−1.24434
$838$	20.0000	0.690889
$839$	−15.0000	−0.517858	−0.258929	−	0.965896i	$-0.583369\pi$
$839$	−15.0000	−0.517858	−0.258929	+	0.965896i	$0.583369\pi$
$840$	18.0000	0.621059
$841$	52.0000	1.79310
$842$	17.0000	0.585859
$843$	0	0
$844$	−10.0000	−0.344214
$845$	−36.0000	−1.23844
$846$	8.00000	0.275046
$847$	42.0000	1.44314
$848$	0	0
$849$	32.0000	1.09824
$850$	24.0000	0.823193
$851$	4.00000	0.137118
$852$	−10.0000	−0.342594
$853$	48.0000	1.64349	0.821744	−	0.569856i	$-0.193001\pi$
$853$	48.0000	1.64349	0.821744	+	0.569856i	$0.193001\pi$
$854$	12.0000	0.410632
$855$	2.00000	0.0683986
$856$	−12.0000	−0.410152
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	−70.0000	−2.38976
$859$	1.00000	0.0341196	0.0170598	−	0.999854i	$-0.494569\pi$
$859$	1.00000	0.0341196	0.0170598	+	0.999854i	$0.494569\pi$
$860$	−4.00000	−0.136399
$861$	−36.0000	−1.22688
$862$	−18.0000	−0.613082
$863$	43.0000	1.46374	0.731869	−	0.681446i	$-0.238649\pi$
$863$	43.0000	1.46374	0.731869	+	0.681446i	$0.238649\pi$
$864$	−20.0000	−0.680414
$865$	−9.00000	−0.306009
$866$	7.00000	0.237870
$867$	38.0000	1.29055
$868$	−27.0000	−0.916440
$869$	−25.0000	−0.848067
$870$	−18.0000	−0.610257
$871$	−35.0000	−1.18593
$872$	30.0000	1.01593
$873$	−12.0000	−0.406138
$874$	−4.00000	−0.135302
$875$	27.0000	0.912767
$876$	12.0000	0.405442
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	−2.00000	−0.0674967
$879$	0	0
$880$	5.00000	0.168550
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	2.00000	0.0673435
$883$	59.0000	1.98551	0.992754	−	0.120164i	$-0.0383421\pi$
$883$	59.0000	1.98551	0.992754	+	0.120164i	$0.0383421\pi$
$884$	−42.0000	−1.41261
$885$	−12.0000	−0.403376
$886$	12.0000	0.403148
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	−12.0000	−0.402694
$889$	15.0000	0.503084
$890$	−7.00000	−0.234641
$891$	−55.0000	−1.84257
$892$	−20.0000	−0.669650
$893$	−16.0000	−0.535420
$894$	−4.00000	−0.133780
$895$	18.0000	0.601674
$896$	−9.00000	−0.300669
$897$	−28.0000	−0.934893
$898$	−6.00000	−0.200223
$899$	81.0000	2.70150
$900$	4.00000	0.133333
$901$	0	0
$902$	−30.0000	−0.998891
$903$	−24.0000	−0.798670
$904$	−3.00000	−0.0997785
$905$	−3.00000	−0.0997234
$906$	−4.00000	−0.132891
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−20.0000	−0.663723
$909$	2.00000	0.0663358
$910$	21.0000	0.696143
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	4.00000	0.132453
$913$	35.0000	1.15833
$914$	−32.0000	−1.05847
$915$	−8.00000	−0.264472
$916$	28.0000	0.925146
$917$	−27.0000	−0.891619
$918$	24.0000	0.792118
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	6.00000	0.197814
$921$	−32.0000	−1.05444
$922$	−25.0000	−0.823331
$923$	−35.0000	−1.15204
$924$	−30.0000	−0.986928
$925$	−8.00000	−0.263038
$926$	36.0000	1.18303
$927$	−16.0000	−0.525509
$928$	45.0000	1.47720
$929$	−52.0000	−1.70606	−0.853032	−	0.521858i	$-0.825239\pi$
$929$	−52.0000	−1.70606	−0.853032	+	0.521858i	$0.825239\pi$
$930$	−18.0000	−0.590243
$931$	−4.00000	−0.131095
$932$	6.00000	0.196537
$933$	−12.0000	−0.392862
$934$	8.00000	0.261768
$935$	30.0000	0.981105
$936$	21.0000	0.686406
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	15.0000	0.489767
$939$	−20.0000	−0.652675
$940$	8.00000	0.260931
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	−44.0000	−1.43360
$943$	−12.0000	−0.390774
$944$	−6.00000	−0.195283
$945$	12.0000	0.390360
$946$	−20.0000	−0.650256
$947$	−57.0000	−1.85225	−0.926126	−	0.377215i	$-0.876882\pi$
$947$	−57.0000	−1.85225	−0.926126	+	0.377215i	$0.876882\pi$
$948$	10.0000	0.324785
$949$	42.0000	1.36338
$950$	8.00000	0.259554
$951$	−36.0000	−1.16738
$952$	54.0000	1.75015
$953$	−56.0000	−1.81402	−0.907009	−	0.421111i	$-0.861640\pi$
$953$	−56.0000	−1.81402	−0.907009	+	0.421111i	$0.861640\pi$
$954$	0	0
$955$	−4.00000	−0.129437
$956$	16.0000	0.517477
$957$	90.0000	2.90929
$958$	−18.0000	−0.581554
$959$	−9.00000	−0.290625
$960$	−14.0000	−0.451848
$961$	50.0000	1.61290
$962$	−14.0000	−0.451378
$963$	4.00000	0.128898
$964$	6.00000	0.193247
$965$	11.0000	0.354103
$966$	12.0000	0.386094
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	−42.0000	−1.34993
$969$	24.0000	0.770991
$970$	12.0000	0.385297
$971$	45.0000	1.44412	0.722059	−	0.691831i	$-0.243196\pi$
$971$	45.0000	1.44412	0.722059	+	0.691831i	$0.243196\pi$
$972$	10.0000	0.320750
$973$	3.00000	0.0961756
$974$	12.0000	0.384505
$975$	56.0000	1.79344
$976$	−4.00000	−0.128037
$977$	45.0000	1.43968	0.719839	−	0.694141i	$-0.244216\pi$
$977$	45.0000	1.43968	0.719839	+	0.694141i	$0.244216\pi$
$978$	22.0000	0.703482
$979$	35.0000	1.11860
$980$	2.00000	0.0638877
$981$	−10.0000	−0.319275
$982$	22.0000	0.702048
$983$	2.00000	0.0637901	0.0318950	−	0.999491i	$-0.489846\pi$
$983$	2.00000	0.0637901	0.0318950	+	0.999491i	$0.489846\pi$
$984$	36.0000	1.14764
$985$	0	0
$986$	−54.0000	−1.71971
$987$	48.0000	1.52786
$988$	−14.0000	−0.445399
$989$	−8.00000	−0.254385
$990$	−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	$0.510113\pi$
$992$	45.0000	1.42875
$993$	8.00000	0.253872
$994$	15.0000	0.475771
$995$	0	0
$996$	−14.0000	−0.443607
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	4.00000	0.126618
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	139.2.a.a.1.1	✓	1
3.2	odd	2		1251.2.a.b.1.1		1
4.3	odd	2		2224.2.a.a.1.1		1
5.4	even	2		3475.2.a.a.1.1		1
7.6	odd	2		6811.2.a.j.1.1		1
8.3	odd	2		8896.2.a.m.1.1		1
8.5	even	2		8896.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
139.2.a.a.1.1	✓	1	1.1	even	1	trivial
1251.2.a.b.1.1		1	3.2	odd	2
2224.2.a.a.1.1		1	4.3	odd	2
3475.2.a.a.1.1		1	5.4	even	2
6811.2.a.j.1.1		1	7.6	odd	2
8896.2.a.c.1.1		1	8.5	even	2
8896.2.a.m.1.1		1	8.3	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:	\( N \)	\(=\)	\( 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	139.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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