LMFDB - Embedded newform 142.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [142,2,Mod(1,142)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(142, 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("142.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 142 = 2 \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	142.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:	$1.13387570870$
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	$\chi$	$=$	142.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +2.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} -3.00000 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +2.00000 q^{18} +5.00000 q^{19} -2.00000 q^{20} +1.00000 q^{21} +2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +3.00000 q^{26} +5.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -2.00000 q^{30} +1.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +6.00000 q^{34} +2.00000 q^{35} -2.00000 q^{36} +6.00000 q^{37} -5.00000 q^{38} +3.00000 q^{39} +2.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} +5.00000 q^{43} -2.00000 q^{44} +4.00000 q^{45} +1.00000 q^{46} -3.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} -3.00000 q^{52} -6.00000 q^{53} -5.00000 q^{54} +4.00000 q^{55} +1.00000 q^{56} -5.00000 q^{57} -6.00000 q^{58} +2.00000 q^{59} +2.00000 q^{60} -6.00000 q^{61} -1.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -2.00000 q^{66} -14.0000 q^{67} -6.00000 q^{68} +1.00000 q^{69} -2.00000 q^{70} -1.00000 q^{71} +2.00000 q^{72} -17.0000 q^{73} -6.00000 q^{74} +1.00000 q^{75} +5.00000 q^{76} +2.00000 q^{77} -3.00000 q^{78} +10.0000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} +1.00000 q^{84} +12.0000 q^{85} -5.00000 q^{86} -6.00000 q^{87} +2.00000 q^{88} +9.00000 q^{89} -4.00000 q^{90} +3.00000 q^{91} -1.00000 q^{92} -1.00000 q^{93} +3.00000 q^{94} -10.0000 q^{95} +1.00000 q^{96} -6.00000 q^{97} +6.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	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
$9$	−2.00000	−0.666667
$10$	2.00000	0.632456
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.00000	−0.288675
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	1.00000	0.267261
$15$	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	$0.759367\pi$
$18$	2.00000	0.471405
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	−2.00000	−0.447214
$21$	1.00000	0.218218
$22$	2.00000	0.426401
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	3.00000	0.588348
$27$	5.00000	0.962250
$28$	−1.00000	−0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−2.00000	−0.365148
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	6.00000	1.02899
$35$	2.00000	0.338062
$36$	−2.00000	−0.333333
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−5.00000	−0.811107
$39$	3.00000	0.480384
$40$	2.00000	0.316228
$41$	−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
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	−2.00000	−0.301511
$45$	4.00000	0.596285
$46$	1.00000	0.147442
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	1.00000	0.141421
$51$	6.00000	0.840168
$52$	−3.00000	−0.416025
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−5.00000	−0.680414
$55$	4.00000	0.539360
$56$	1.00000	0.133631
$57$	−5.00000	−0.662266
$58$	−6.00000	−0.787839
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	2.00000	0.258199
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−1.00000	−0.127000
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	−2.00000	−0.246183
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	−6.00000	−0.727607
$69$	1.00000	0.120386
$70$	−2.00000	−0.239046
$71$	−1.00000	−0.118678
$71$	−1.00000	−0.118678
$72$	2.00000	0.235702
$73$	−17.0000	−1.98970	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−17.0000	−1.98970	−0.994850	+	0.101361i	$0.967680\pi$
$74$	−6.00000	−0.697486
$75$	1.00000	0.115470
$76$	5.00000	0.573539
$77$	2.00000	0.227921
$78$	−3.00000	−0.339683
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	1.00000	0.109109
$85$	12.0000	1.30158
$86$	−5.00000	−0.539164
$87$	−6.00000	−0.643268
$88$	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	$0.341725\pi$
$90$	−4.00000	−0.421637
$91$	3.00000	0.314485
$92$	−1.00000	−0.104257
$93$	−1.00000	−0.103695
$94$	3.00000	0.309426
$95$	−10.0000	−1.02598
$96$	1.00000	0.102062
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	6.00000	0.606092
$99$	4.00000	0.402015
$100$	−1.00000	−0.100000
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	−6.00000	−0.594089
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	3.00000	0.294174
$105$	−2.00000	−0.195180
$106$	6.00000	0.582772
$107$	−17.0000	−1.64345	−0.821726	−	0.569883i	$-0.806989\pi$
$107$	−17.0000	−1.64345	−0.821726	+	0.569883i	$0.806989\pi$
$108$	5.00000	0.481125
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	−4.00000	−0.381385
$111$	−6.00000	−0.569495
$112$	−1.00000	−0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	5.00000	0.468293
$115$	2.00000	0.186501
$116$	6.00000	0.557086
$117$	6.00000	0.554700
$118$	−2.00000	−0.184115
$119$	6.00000	0.550019
$120$	−2.00000	−0.182574
$121$	−7.00000	−0.636364
$122$	6.00000	0.543214
$123$	6.00000	0.541002
$124$	1.00000	0.0898027
$125$	12.0000	1.07331
$126$	−2.00000	−0.178174
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	−5.00000	−0.440225
$130$	−6.00000	−0.526235
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	2.00000	0.174078
$133$	−5.00000	−0.433555
$134$	14.0000	1.20942
$135$	−10.0000	−0.860663
$136$	6.00000	0.514496
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	−1.00000	−0.0851257
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	2.00000	0.169031
$141$	3.00000	0.252646
$142$	1.00000	0.0839181
$143$	6.00000	0.501745
$144$	−2.00000	−0.166667
$145$	−12.0000	−0.996546
$146$	17.0000	1.40693
$147$	6.00000	0.494872
$148$	6.00000	0.493197
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	−1.00000	−0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−5.00000	−0.405554
$153$	12.0000	0.970143
$154$	−2.00000	−0.161165
$155$	−2.00000	−0.160644
$156$	3.00000	0.240192
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−10.0000	−0.795557
$159$	6.00000	0.475831
$160$	2.00000	0.158114
$161$	1.00000	0.0788110
$162$	−1.00000	−0.0785674
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	−6.00000	−0.468521
$165$	−4.00000	−0.311400
$166$	−4.00000	−0.310460
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	−1.00000	−0.0771517
$169$	−4.00000	−0.307692
$170$	−12.0000	−0.920358
$171$	−10.0000	−0.764719
$172$	5.00000	0.381246
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	6.00000	0.454859
$175$	1.00000	0.0755929
$176$	−2.00000	−0.150756
$177$	−2.00000	−0.150329
$178$	−9.00000	−0.674579
$179$	21.0000	1.56961	0.784807	−	0.619740i	$-0.212762\pi$
$179$	21.0000	1.56961	0.784807	+	0.619740i	$0.212762\pi$
$180$	4.00000	0.298142
$181$	−19.0000	−1.41226	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	−19.0000	−1.41226	−0.706129	+	0.708083i	$0.749560\pi$
$182$	−3.00000	−0.222375
$183$	6.00000	0.443533
$184$	1.00000	0.0737210
$185$	−12.0000	−0.882258
$186$	1.00000	0.0733236
$187$	12.0000	0.877527
$188$	−3.00000	−0.218797
$189$	−5.00000	−0.363696
$190$	10.0000	0.725476
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	−1.00000	−0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	6.00000	0.430775
$195$	−6.00000	−0.429669
$196$	−6.00000	−0.428571
$197$	27.0000	1.92367	0.961835	−	0.273629i	$-0.0882242\pi$
$197$	27.0000	1.92367	0.961835	+	0.273629i	$0.0882242\pi$
$198$	−4.00000	−0.284268
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	1.00000	0.0707107
$201$	14.0000	0.987484
$202$	8.00000	0.562878
$203$	−6.00000	−0.421117
$204$	6.00000	0.420084
$205$	12.0000	0.838116
$206$	4.00000	0.278693
$207$	2.00000	0.139010
$208$	−3.00000	−0.208013
$209$	−10.0000	−0.691714
$210$	2.00000	0.138013
$211$	−26.0000	−1.78991	−0.894957	−	0.446153i	$-0.852794\pi$
$211$	−26.0000	−1.78991	−0.894957	+	0.446153i	$0.852794\pi$
$212$	−6.00000	−0.412082
$213$	1.00000	0.0685189
$214$	17.0000	1.16210
$215$	−10.0000	−0.681994
$216$	−5.00000	−0.340207
$217$	−1.00000	−0.0678844
$218$	−12.0000	−0.812743
$219$	17.0000	1.14875
$220$	4.00000	0.269680
$221$	18.0000	1.21081
$222$	6.00000	0.402694
$223$	22.0000	1.47323	0.736614	−	0.676313i	$-0.236423\pi$
$223$	22.0000	1.47323	0.736614	+	0.676313i	$0.236423\pi$
$224$	1.00000	0.0668153
$225$	2.00000	0.133333
$226$	−6.00000	−0.399114
$227$	22.0000	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$228$	−5.00000	−0.331133
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	−2.00000	−0.131876
$231$	−2.00000	−0.131590
$232$	−6.00000	−0.393919
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	−6.00000	−0.392232
$235$	6.00000	0.391397
$236$	2.00000	0.130189
$237$	−10.0000	−0.649570
$238$	−6.00000	−0.388922
$239$	25.0000	1.61712	0.808558	−	0.588417i	$-0.200249\pi$
$239$	25.0000	1.61712	0.808558	+	0.588417i	$0.200249\pi$
$240$	2.00000	0.129099
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	7.00000	0.449977
$243$	−16.0000	−1.02640
$244$	−6.00000	−0.384111
$245$	12.0000	0.766652
$246$	−6.00000	−0.382546
$247$	−15.0000	−0.954427
$248$	−1.00000	−0.0635001
$249$	−4.00000	−0.253490
$250$	−12.0000	−0.758947
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	2.00000	0.125988
$253$	2.00000	0.125739
$254$	−16.0000	−1.00393
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	−4.00000	−0.249513	−0.124757	−	0.992187i	$-0.539815\pi$
$257$	−4.00000	−0.249513	−0.124757	+	0.992187i	$0.539815\pi$
$258$	5.00000	0.311286
$259$	−6.00000	−0.372822
$260$	6.00000	0.372104
$261$	−12.0000	−0.742781
$262$	−3.00000	−0.185341
$263$	26.0000	1.60323	0.801614	−	0.597841i	$-0.203975\pi$
$263$	26.0000	1.60323	0.801614	+	0.597841i	$0.203975\pi$
$264$	−2.00000	−0.123091
$265$	12.0000	0.737154
$266$	5.00000	0.306570
$267$	−9.00000	−0.550791
$268$	−14.0000	−0.855186
$269$	7.00000	0.426798	0.213399	−	0.976965i	$-0.431547\pi$
$269$	7.00000	0.426798	0.213399	+	0.976965i	$0.431547\pi$
$270$	10.0000	0.608581
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−6.00000	−0.363803
$273$	−3.00000	−0.181568
$274$	−8.00000	−0.483298
$275$	2.00000	0.120605
$276$	1.00000	0.0601929
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	8.00000	0.479808
$279$	−2.00000	−0.119737
$280$	−2.00000	−0.119523
$281$	28.0000	1.67034	0.835170	−	0.549992i	$-0.185369\pi$
$281$	28.0000	1.67034	0.835170	+	0.549992i	$0.185369\pi$
$282$	−3.00000	−0.178647
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	−1.00000	−0.0593391
$285$	10.0000	0.592349
$286$	−6.00000	−0.354787
$287$	6.00000	0.354169
$288$	2.00000	0.117851
$289$	19.0000	1.11765
$290$	12.0000	0.704664
$291$	6.00000	0.351726
$292$	−17.0000	−0.994850
$293$	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
$295$	−4.00000	−0.232889
$296$	−6.00000	−0.348743
$297$	−10.0000	−0.580259
$298$	−3.00000	−0.173785
$299$	3.00000	0.173494
$300$	1.00000	0.0577350
$301$	−5.00000	−0.288195
$302$	0	0
$303$	8.00000	0.459588
$304$	5.00000	0.286770
$305$	12.0000	0.687118
$306$	−12.0000	−0.685994
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	2.00000	0.113961
$309$	4.00000	0.227552
$310$	2.00000	0.113592
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	−3.00000	−0.169842
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	−2.00000	−0.112867
$315$	−4.00000	−0.225374
$316$	10.0000	0.562544
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	−6.00000	−0.336463
$319$	−12.0000	−0.671871
$320$	−2.00000	−0.111803
$321$	17.0000	0.948847
$322$	−1.00000	−0.0557278
$323$	−30.0000	−1.66924
$324$	1.00000	0.0555556
$325$	3.00000	0.166410
$326$	6.00000	0.332309
$327$	−12.0000	−0.663602
$328$	6.00000	0.331295
$329$	3.00000	0.165395
$330$	4.00000	0.220193
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	4.00000	0.219529
$333$	−12.0000	−0.657596
$334$	24.0000	1.31322
$335$	28.0000	1.52980
$336$	1.00000	0.0545545
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	4.00000	0.217571
$339$	−6.00000	−0.325875
$340$	12.0000	0.650791
$341$	−2.00000	−0.108306
$342$	10.0000	0.540738
$343$	13.0000	0.701934
$344$	−5.00000	−0.269582
$345$	−2.00000	−0.107676
$346$	9.00000	0.483843
$347$	−22.0000	−1.18102	−0.590511	−	0.807030i	$-0.701074\pi$
$347$	−22.0000	−1.18102	−0.590511	+	0.807030i	$0.701074\pi$
$348$	−6.00000	−0.321634
$349$	29.0000	1.55233	0.776167	−	0.630527i	$-0.217161\pi$
$349$	29.0000	1.55233	0.776167	+	0.630527i	$0.217161\pi$
$350$	−1.00000	−0.0534522
$351$	−15.0000	−0.800641
$352$	2.00000	0.106600
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	2.00000	0.106299
$355$	2.00000	0.106149
$356$	9.00000	0.476999
$357$	−6.00000	−0.317554
$358$	−21.0000	−1.10988
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	−4.00000	−0.210819
$361$	6.00000	0.315789
$362$	19.0000	0.998618
$363$	7.00000	0.367405
$364$	3.00000	0.157243
$365$	34.0000	1.77964
$366$	−6.00000	−0.313625
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	−1.00000	−0.0521286
$369$	12.0000	0.624695
$370$	12.0000	0.623850
$371$	6.00000	0.311504
$372$	−1.00000	−0.0518476
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	−12.0000	−0.620505
$375$	−12.0000	−0.619677
$376$	3.00000	0.154713
$377$	−18.0000	−0.927047
$378$	5.00000	0.257172
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	−10.0000	−0.512989
$381$	−16.0000	−0.819705
$382$	20.0000	1.02329
$383$	−19.0000	−0.970855	−0.485427	−	0.874277i	$-0.661336\pi$
$383$	−19.0000	−0.970855	−0.485427	+	0.874277i	$0.661336\pi$
$384$	1.00000	0.0510310
$385$	−4.00000	−0.203859
$386$	2.00000	0.101797
$387$	−10.0000	−0.508329
$388$	−6.00000	−0.304604
$389$	−29.0000	−1.47036	−0.735179	−	0.677873i	$-0.762902\pi$
$389$	−29.0000	−1.47036	−0.735179	+	0.677873i	$0.762902\pi$
$390$	6.00000	0.303822
$391$	6.00000	0.303433
$392$	6.00000	0.303046
$393$	−3.00000	−0.151330
$394$	−27.0000	−1.36024
$395$	−20.0000	−1.00631
$396$	4.00000	0.201008
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	26.0000	1.30326
$399$	5.00000	0.250313
$400$	−1.00000	−0.0500000
$401$	−20.0000	−0.998752	−0.499376	−	0.866385i	$-0.666437\pi$
$401$	−20.0000	−0.998752	−0.499376	+	0.866385i	$0.666437\pi$
$402$	−14.0000	−0.698257
$403$	−3.00000	−0.149441
$404$	−8.00000	−0.398015
$405$	−2.00000	−0.0993808
$406$	6.00000	0.297775
$407$	−12.0000	−0.594818
$408$	−6.00000	−0.297044
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	−12.0000	−0.592638
$411$	−8.00000	−0.394611
$412$	−4.00000	−0.197066
$413$	−2.00000	−0.0984136
$414$	−2.00000	−0.0982946
$415$	−8.00000	−0.392705
$416$	3.00000	0.147087
$417$	8.00000	0.391762
$418$	10.0000	0.489116
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	−2.00000	−0.0975900
$421$	−15.0000	−0.731055	−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000	−0.731055	−0.365528	+	0.930800i	$0.619111\pi$
$422$	26.0000	1.26566
$423$	6.00000	0.291730
$424$	6.00000	0.291386
$425$	6.00000	0.291043
$426$	−1.00000	−0.0484502
$427$	6.00000	0.290360
$428$	−17.0000	−0.821726
$429$	−6.00000	−0.289683
$430$	10.0000	0.482243
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	5.00000	0.240563
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	1.00000	0.0480015
$435$	12.0000	0.575356
$436$	12.0000	0.574696
$437$	−5.00000	−0.239182
$438$	−17.0000	−0.812291
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	−4.00000	−0.190693
$441$	12.0000	0.571429
$442$	−18.0000	−0.856173
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−6.00000	−0.284747
$445$	−18.0000	−0.853282
$446$	−22.0000	−1.04173
$447$	−3.00000	−0.141895
$448$	−1.00000	−0.0472456
$449$	−42.0000	−1.98210	−0.991051	−	0.133482i	$-0.957384\pi$
$449$	−42.0000	−1.98210	−0.991051	+	0.133482i	$0.957384\pi$
$450$	−2.00000	−0.0942809
$451$	12.0000	0.565058
$452$	6.00000	0.282216
$453$	0	0
$454$	−22.0000	−1.03251
$455$	−6.00000	−0.281284
$456$	5.00000	0.234146
$457$	−20.0000	−0.935561	−0.467780	−	0.883845i	$-0.654946\pi$
$457$	−20.0000	−0.935561	−0.467780	+	0.883845i	$0.654946\pi$
$458$	−2.00000	−0.0934539
$459$	−30.0000	−1.40028
$460$	2.00000	0.0932505
$461$	5.00000	0.232873	0.116437	−	0.993198i	$-0.462853\pi$
$461$	5.00000	0.232873	0.116437	+	0.993198i	$0.462853\pi$
$462$	2.00000	0.0930484
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	6.00000	0.278543
$465$	2.00000	0.0927478
$466$	1.00000	0.0463241
$467$	16.0000	0.740392	0.370196	−	0.928954i	$-0.379291\pi$
$467$	16.0000	0.740392	0.370196	+	0.928954i	$0.379291\pi$
$468$	6.00000	0.277350
$469$	14.0000	0.646460
$470$	−6.00000	−0.276759
$471$	−2.00000	−0.0921551
$472$	−2.00000	−0.0920575
$473$	−10.0000	−0.459800
$474$	10.0000	0.459315
$475$	−5.00000	−0.229416
$476$	6.00000	0.275010
$477$	12.0000	0.549442
$478$	−25.0000	−1.14347
$479$	7.00000	0.319838	0.159919	−	0.987130i	$-0.448877\pi$
$479$	7.00000	0.319838	0.159919	+	0.987130i	$0.448877\pi$
$480$	−2.00000	−0.0912871
$481$	−18.0000	−0.820729
$482$	26.0000	1.18427
$483$	−1.00000	−0.0455016
$484$	−7.00000	−0.318182
$485$	12.0000	0.544892
$486$	16.0000	0.725775
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	6.00000	0.271607
$489$	6.00000	0.271329
$490$	−12.0000	−0.542105
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	6.00000	0.270501
$493$	−36.0000	−1.62136
$494$	15.0000	0.674882
$495$	−8.00000	−0.359573
$496$	1.00000	0.0449013
$497$	1.00000	0.0448561
$498$	4.00000	0.179244
$499$	7.00000	0.313363	0.156682	−	0.987649i	$-0.449920\pi$
$499$	7.00000	0.313363	0.156682	+	0.987649i	$0.449920\pi$
$500$	12.0000	0.536656
$501$	24.0000	1.07224
$502$	21.0000	0.937276
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	−2.00000	−0.0890871
$505$	16.0000	0.711991
$506$	−2.00000	−0.0889108
$507$	4.00000	0.177646
$508$	16.0000	0.709885
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	12.0000	0.531369
$511$	17.0000	0.752036
$512$	−1.00000	−0.0441942
$513$	25.0000	1.10378
$514$	4.00000	0.176432
$515$	8.00000	0.352522
$516$	−5.00000	−0.220113
$517$	6.00000	0.263880
$518$	6.00000	0.263625
$519$	9.00000	0.395056
$520$	−6.00000	−0.263117
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	12.0000	0.525226
$523$	32.0000	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$524$	3.00000	0.131056
$525$	−1.00000	−0.0436436
$526$	−26.0000	−1.13365
$527$	−6.00000	−0.261364
$528$	2.00000	0.0870388
$529$	−22.0000	−0.956522
$530$	−12.0000	−0.521247
$531$	−4.00000	−0.173585
$532$	−5.00000	−0.216777
$533$	18.0000	0.779667
$534$	9.00000	0.389468
$535$	34.0000	1.46995
$536$	14.0000	0.604708
$537$	−21.0000	−0.906217
$538$	−7.00000	−0.301791
$539$	12.0000	0.516877
$540$	−10.0000	−0.430331
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	−20.0000	−0.859074
$543$	19.0000	0.815368
$544$	6.00000	0.257248
$545$	−24.0000	−1.02805
$546$	3.00000	0.128388
$547$	19.0000	0.812381	0.406191	−	0.913788i	$-0.366857\pi$
$547$	19.0000	0.812381	0.406191	+	0.913788i	$0.366857\pi$
$548$	8.00000	0.341743
$549$	12.0000	0.512148
$550$	−2.00000	−0.0852803
$551$	30.0000	1.27804
$552$	−1.00000	−0.0425628
$553$	−10.0000	−0.425243
$554$	2.00000	0.0849719
$555$	12.0000	0.509372
$556$	−8.00000	−0.339276
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	2.00000	0.0846668
$559$	−15.0000	−0.634432
$560$	2.00000	0.0845154
$561$	−12.0000	−0.506640
$562$	−28.0000	−1.18111
$563$	34.0000	1.43293	0.716465	−	0.697623i	$-0.245759\pi$
$563$	34.0000	1.43293	0.716465	+	0.697623i	$0.245759\pi$
$564$	3.00000	0.126323
$565$	−12.0000	−0.504844
$566$	6.00000	0.252199
$567$	−1.00000	−0.0419961
$568$	1.00000	0.0419591
$569$	−5.00000	−0.209611	−0.104805	−	0.994493i	$-0.533422\pi$
$569$	−5.00000	−0.209611	−0.104805	+	0.994493i	$0.533422\pi$
$570$	−10.0000	−0.418854
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	6.00000	0.250873
$573$	20.0000	0.835512
$574$	−6.00000	−0.250435
$575$	1.00000	0.0417029
$576$	−2.00000	−0.0833333
$577$	46.0000	1.91501	0.957503	−	0.288425i	$-0.0931316\pi$
$577$	46.0000	1.91501	0.957503	+	0.288425i	$0.0931316\pi$
$578$	−19.0000	−0.790296
$579$	2.00000	0.0831172
$580$	−12.0000	−0.498273
$581$	−4.00000	−0.165948
$582$	−6.00000	−0.248708
$583$	12.0000	0.496989
$584$	17.0000	0.703465
$585$	−12.0000	−0.496139
$586$	−12.0000	−0.495715
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	6.00000	0.247436
$589$	5.00000	0.206021
$590$	4.00000	0.164677
$591$	−27.0000	−1.11063
$592$	6.00000	0.246598
$593$	25.0000	1.02663	0.513313	−	0.858201i	$-0.328418\pi$
$593$	25.0000	1.02663	0.513313	+	0.858201i	$0.328418\pi$
$594$	10.0000	0.410305
$595$	−12.0000	−0.491952
$596$	3.00000	0.122885
$597$	26.0000	1.06411
$598$	−3.00000	−0.122679
$599$	−7.00000	−0.286012	−0.143006	−	0.989722i	$-0.545677\pi$
$599$	−7.00000	−0.286012	−0.143006	+	0.989722i	$0.545677\pi$
$600$	−1.00000	−0.0408248
$601$	16.0000	0.652654	0.326327	−	0.945257i	$-0.394189\pi$
$601$	16.0000	0.652654	0.326327	+	0.945257i	$0.394189\pi$
$602$	5.00000	0.203785
$603$	28.0000	1.14025
$604$	0	0
$605$	14.0000	0.569181
$606$	−8.00000	−0.324978
$607$	−44.0000	−1.78590	−0.892952	−	0.450151i	$-0.851370\pi$
$607$	−44.0000	−1.78590	−0.892952	+	0.450151i	$0.851370\pi$
$608$	−5.00000	−0.202777
$609$	6.00000	0.243132
$610$	−12.0000	−0.485866
$611$	9.00000	0.364101
$612$	12.0000	0.485071
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	−22.0000	−0.887848
$615$	−12.0000	−0.483887
$616$	−2.00000	−0.0805823
$617$	−13.0000	−0.523360	−0.261680	−	0.965155i	$-0.584277\pi$
$617$	−13.0000	−0.523360	−0.261680	+	0.965155i	$0.584277\pi$
$618$	−4.00000	−0.160904
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	−2.00000	−0.0803219
$621$	−5.00000	−0.200643
$622$	4.00000	0.160385
$623$	−9.00000	−0.360577
$624$	3.00000	0.120096
$625$	−19.0000	−0.760000
$626$	−19.0000	−0.759393
$627$	10.0000	0.399362
$628$	2.00000	0.0798087
$629$	−36.0000	−1.43541
$630$	4.00000	0.159364
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	−10.0000	−0.397779
$633$	26.0000	1.03341
$634$	−14.0000	−0.556011
$635$	−32.0000	−1.26988
$636$	6.00000	0.237915
$637$	18.0000	0.713186
$638$	12.0000	0.475085
$639$	2.00000	0.0791188
$640$	2.00000	0.0790569
$641$	5.00000	0.197488	0.0987441	−	0.995113i	$-0.468517\pi$
$641$	5.00000	0.197488	0.0987441	+	0.995113i	$0.468517\pi$
$642$	−17.0000	−0.670936
$643$	−9.00000	−0.354925	−0.177463	−	0.984128i	$-0.556789\pi$
$643$	−9.00000	−0.354925	−0.177463	+	0.984128i	$0.556789\pi$
$644$	1.00000	0.0394055
$645$	10.0000	0.393750
$646$	30.0000	1.18033
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−1.00000	−0.0392837
$649$	−4.00000	−0.157014
$650$	−3.00000	−0.117670
$651$	1.00000	0.0391931
$652$	−6.00000	−0.234978
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	12.0000	0.469237
$655$	−6.00000	−0.234439
$656$	−6.00000	−0.234261
$657$	34.0000	1.32647
$658$	−3.00000	−0.116952
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	−4.00000	−0.155700
$661$	−37.0000	−1.43913	−0.719567	−	0.694423i	$-0.755660\pi$
$661$	−37.0000	−1.43913	−0.719567	+	0.694423i	$0.755660\pi$
$662$	12.0000	0.466393
$663$	−18.0000	−0.699062
$664$	−4.00000	−0.155230
$665$	10.0000	0.387783
$666$	12.0000	0.464991
$667$	−6.00000	−0.232321
$668$	−24.0000	−0.928588
$669$	−22.0000	−0.850569
$670$	−28.0000	−1.08173
$671$	12.0000	0.463255
$672$	−1.00000	−0.0385758
$673$	−28.0000	−1.07932	−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000	−1.07932	−0.539660	+	0.841883i	$0.681447\pi$
$674$	18.0000	0.693334
$675$	−5.00000	−0.192450
$676$	−4.00000	−0.153846
$677$	12.0000	0.461197	0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000	0.461197	0.230599	+	0.973049i	$0.425932\pi$
$678$	6.00000	0.230429
$679$	6.00000	0.230259
$680$	−12.0000	−0.460179
$681$	−22.0000	−0.843042
$682$	2.00000	0.0765840
$683$	−28.0000	−1.07139	−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000	−1.07139	−0.535695	+	0.844411i	$0.679950\pi$
$684$	−10.0000	−0.382360
$685$	−16.0000	−0.611329
$686$	−13.0000	−0.496342
$687$	−2.00000	−0.0763048
$688$	5.00000	0.190623
$689$	18.0000	0.685745
$690$	2.00000	0.0761387
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−9.00000	−0.342129
$693$	−4.00000	−0.151947
$694$	22.0000	0.835109
$695$	16.0000	0.606915
$696$	6.00000	0.227429
$697$	36.0000	1.36360
$698$	−29.0000	−1.09767
$699$	1.00000	0.0378235
$700$	1.00000	0.0377964
$701$	−5.00000	−0.188847	−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000	−0.188847	−0.0944237	+	0.995532i	$0.530101\pi$
$702$	15.0000	0.566139
$703$	30.0000	1.13147
$704$	−2.00000	−0.0753778
$705$	−6.00000	−0.225973
$706$	14.0000	0.526897
$707$	8.00000	0.300871
$708$	−2.00000	−0.0751646
$709$	−7.00000	−0.262891	−0.131445	−	0.991323i	$-0.541962\pi$
$709$	−7.00000	−0.262891	−0.131445	+	0.991323i	$0.541962\pi$
$710$	−2.00000	−0.0750587
$711$	−20.0000	−0.750059
$712$	−9.00000	−0.337289
$713$	−1.00000	−0.0374503
$714$	6.00000	0.224544
$715$	−12.0000	−0.448775
$716$	21.0000	0.784807
$717$	−25.0000	−0.933642
$718$	4.00000	0.149279
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	4.00000	0.149071
$721$	4.00000	0.148968
$722$	−6.00000	−0.223297
$723$	26.0000	0.966950
$724$	−19.0000	−0.706129
$725$	−6.00000	−0.222834
$726$	−7.00000	−0.259794
$727$	−21.0000	−0.778847	−0.389423	−	0.921059i	$-0.627326\pi$
$727$	−21.0000	−0.778847	−0.389423	+	0.921059i	$0.627326\pi$
$728$	−3.00000	−0.111187
$729$	13.0000	0.481481
$730$	−34.0000	−1.25840
$731$	−30.0000	−1.10959
$732$	6.00000	0.221766
$733$	−11.0000	−0.406294	−0.203147	−	0.979148i	$-0.565117\pi$
$733$	−11.0000	−0.406294	−0.203147	+	0.979148i	$0.565117\pi$
$734$	−10.0000	−0.369107
$735$	−12.0000	−0.442627
$736$	1.00000	0.0368605
$737$	28.0000	1.03139
$738$	−12.0000	−0.441726
$739$	13.0000	0.478213	0.239106	−	0.970993i	$-0.423146\pi$
$739$	13.0000	0.478213	0.239106	+	0.970993i	$0.423146\pi$
$740$	−12.0000	−0.441129
$741$	15.0000	0.551039
$742$	−6.00000	−0.220267
$743$	−3.00000	−0.110059	−0.0550297	−	0.998485i	$-0.517525\pi$
$743$	−3.00000	−0.110059	−0.0550297	+	0.998485i	$0.517525\pi$
$744$	1.00000	0.0366618
$745$	−6.00000	−0.219823
$746$	0	0
$747$	−8.00000	−0.292705
$748$	12.0000	0.438763
$749$	17.0000	0.621166
$750$	12.0000	0.438178
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	−3.00000	−0.109399
$753$	21.0000	0.765283
$754$	18.0000	0.655521
$755$	0	0
$756$	−5.00000	−0.181848
$757$	−37.0000	−1.34479	−0.672394	−	0.740193i	$-0.734734\pi$
$757$	−37.0000	−1.34479	−0.672394	+	0.740193i	$0.734734\pi$
$758$	0	0
$759$	−2.00000	−0.0725954
$760$	10.0000	0.362738
$761$	4.00000	0.145000	0.0724999	−	0.997368i	$-0.476902\pi$
$761$	4.00000	0.145000	0.0724999	+	0.997368i	$0.476902\pi$
$762$	16.0000	0.579619
$763$	−12.0000	−0.434429
$764$	−20.0000	−0.723575
$765$	−24.0000	−0.867722
$766$	19.0000	0.686498
$767$	−6.00000	−0.216647
$768$	−1.00000	−0.0360844
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	4.00000	0.144150
$771$	4.00000	0.144056
$772$	−2.00000	−0.0719816
$773$	−39.0000	−1.40273	−0.701366	−	0.712801i	$-0.747426\pi$
$773$	−39.0000	−1.40273	−0.701366	+	0.712801i	$0.747426\pi$
$774$	10.0000	0.359443
$775$	−1.00000	−0.0359211
$776$	6.00000	0.215387
$777$	6.00000	0.215249
$778$	29.0000	1.03970
$779$	−30.0000	−1.07486
$780$	−6.00000	−0.214834
$781$	2.00000	0.0715656
$782$	−6.00000	−0.214560
$783$	30.0000	1.07211
$784$	−6.00000	−0.214286
$785$	−4.00000	−0.142766
$786$	3.00000	0.107006
$787$	47.0000	1.67537	0.837685	−	0.546154i	$-0.183909\pi$
$787$	47.0000	1.67537	0.837685	+	0.546154i	$0.183909\pi$
$788$	27.0000	0.961835
$789$	−26.0000	−0.925625
$790$	20.0000	0.711568
$791$	−6.00000	−0.213335
$792$	−4.00000	−0.142134
$793$	18.0000	0.639199
$794$	−18.0000	−0.638796
$795$	−12.0000	−0.425596
$796$	−26.0000	−0.921546
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	−5.00000	−0.176998
$799$	18.0000	0.636794
$800$	1.00000	0.0353553
$801$	−18.0000	−0.635999
$802$	20.0000	0.706225
$803$	34.0000	1.19983
$804$	14.0000	0.493742
$805$	−2.00000	−0.0704907
$806$	3.00000	0.105670
$807$	−7.00000	−0.246412
$808$	8.00000	0.281439
$809$	48.0000	1.68759	0.843795	−	0.536666i	$-0.180316\pi$
$809$	48.0000	1.68759	0.843795	+	0.536666i	$0.180316\pi$
$810$	2.00000	0.0702728
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	−6.00000	−0.210559
$813$	−20.0000	−0.701431
$814$	12.0000	0.420600
$815$	12.0000	0.420342
$816$	6.00000	0.210042
$817$	25.0000	0.874639
$818$	19.0000	0.664319
$819$	−6.00000	−0.209657
$820$	12.0000	0.419058
$821$	4.00000	0.139601	0.0698005	−	0.997561i	$-0.477764\pi$
$821$	4.00000	0.139601	0.0698005	+	0.997561i	$0.477764\pi$
$822$	8.00000	0.279032
$823$	−55.0000	−1.91718	−0.958590	−	0.284791i	$-0.908076\pi$
$823$	−55.0000	−1.91718	−0.958590	+	0.284791i	$0.908076\pi$
$824$	4.00000	0.139347
$825$	−2.00000	−0.0696311
$826$	2.00000	0.0695889
$827$	−54.0000	−1.87776	−0.938882	−	0.344239i	$-0.888137\pi$
$827$	−54.0000	−1.87776	−0.938882	+	0.344239i	$0.888137\pi$
$828$	2.00000	0.0695048
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	8.00000	0.277684
$831$	2.00000	0.0693792
$832$	−3.00000	−0.104006
$833$	36.0000	1.24733
$834$	−8.00000	−0.277017
$835$	48.0000	1.66111
$836$	−10.0000	−0.345857
$837$	5.00000	0.172825
$838$	20.0000	0.690889
$839$	−28.0000	−0.966667	−0.483334	−	0.875436i	$-0.660574\pi$
$839$	−28.0000	−0.966667	−0.483334	+	0.875436i	$0.660574\pi$
$840$	2.00000	0.0690066
$841$	7.00000	0.241379
$842$	15.0000	0.516934
$843$	−28.0000	−0.964371
$844$	−26.0000	−0.894957
$845$	8.00000	0.275208
$846$	−6.00000	−0.206284
$847$	7.00000	0.240523
$848$	−6.00000	−0.206041
$849$	6.00000	0.205919
$850$	−6.00000	−0.205798
$851$	−6.00000	−0.205677
$852$	1.00000	0.0342594
$853$	−40.0000	−1.36957	−0.684787	−	0.728743i	$-0.740105\pi$
$853$	−40.0000	−1.36957	−0.684787	+	0.728743i	$0.740105\pi$
$854$	−6.00000	−0.205316
$855$	20.0000	0.683986
$856$	17.0000	0.581048
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	6.00000	0.204837
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−10.0000	−0.340997
$861$	−6.00000	−0.204479
$862$	−16.0000	−0.544962
$863$	29.0000	0.987171	0.493586	−	0.869697i	$-0.335686\pi$
$863$	29.0000	0.987171	0.493586	+	0.869697i	$0.335686\pi$
$864$	−5.00000	−0.170103
$865$	18.0000	0.612018
$866$	2.00000	0.0679628
$867$	−19.0000	−0.645274
$868$	−1.00000	−0.0339422
$869$	−20.0000	−0.678454
$870$	−12.0000	−0.406838
$871$	42.0000	1.42312
$872$	−12.0000	−0.406371
$873$	12.0000	0.406138
$874$	5.00000	0.169128
$875$	−12.0000	−0.405674
$876$	17.0000	0.574377
$877$	8.00000	0.270141	0.135070	−	0.990836i	$-0.456874\pi$
$877$	8.00000	0.270141	0.135070	+	0.990836i	$0.456874\pi$
$878$	20.0000	0.674967
$879$	−12.0000	−0.404750
$880$	4.00000	0.134840
$881$	33.0000	1.11180	0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000	1.11180	0.555899	+	0.831250i	$0.312374\pi$
$882$	−12.0000	−0.404061
$883$	−14.0000	−0.471138	−0.235569	−	0.971858i	$-0.575695\pi$
$883$	−14.0000	−0.471138	−0.235569	+	0.971858i	$0.575695\pi$
$884$	18.0000	0.605406
$885$	4.00000	0.134459
$886$	−12.0000	−0.403148
$887$	−52.0000	−1.74599	−0.872995	−	0.487730i	$-0.837825\pi$
$887$	−52.0000	−1.74599	−0.872995	+	0.487730i	$0.837825\pi$
$888$	6.00000	0.201347
$889$	−16.0000	−0.536623
$890$	18.0000	0.603361
$891$	−2.00000	−0.0670025
$892$	22.0000	0.736614
$893$	−15.0000	−0.501956
$894$	3.00000	0.100335
$895$	−42.0000	−1.40391
$896$	1.00000	0.0334077
$897$	−3.00000	−0.100167
$898$	42.0000	1.40156
$899$	6.00000	0.200111
$900$	2.00000	0.0666667
$901$	36.0000	1.19933
$902$	−12.0000	−0.399556
$903$	5.00000	0.166390
$904$	−6.00000	−0.199557
$905$	38.0000	1.26316
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	22.0000	0.730096
$909$	16.0000	0.530687
$910$	6.00000	0.198898
$911$	1.00000	0.0331315	0.0165657	−	0.999863i	$-0.494727\pi$
$911$	1.00000	0.0331315	0.0165657	+	0.999863i	$0.494727\pi$
$912$	−5.00000	−0.165567
$913$	−8.00000	−0.264761
$914$	20.0000	0.661541
$915$	−12.0000	−0.396708
$916$	2.00000	0.0660819
$917$	−3.00000	−0.0990687
$918$	30.0000	0.990148
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	−2.00000	−0.0659380
$921$	−22.0000	−0.724925
$922$	−5.00000	−0.164666
$923$	3.00000	0.0987462
$924$	−2.00000	−0.0657952
$925$	−6.00000	−0.197279
$926$	−14.0000	−0.460069
$927$	8.00000	0.262754
$928$	−6.00000	−0.196960
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	−2.00000	−0.0655826
$931$	−30.0000	−0.983210
$932$	−1.00000	−0.0327561
$933$	4.00000	0.130954
$934$	−16.0000	−0.523536
$935$	−24.0000	−0.784884
$936$	−6.00000	−0.196116
$937$	32.0000	1.04539	0.522697	−	0.852518i	$-0.324926\pi$
$937$	32.0000	1.04539	0.522697	+	0.852518i	$0.324926\pi$
$938$	−14.0000	−0.457116
$939$	−19.0000	−0.620042
$940$	6.00000	0.195698
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	2.00000	0.0651635
$943$	6.00000	0.195387
$944$	2.00000	0.0650945
$945$	10.0000	0.325300
$946$	10.0000	0.325128
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	−10.0000	−0.324785
$949$	51.0000	1.65553
$950$	5.00000	0.162221
$951$	−14.0000	−0.453981
$952$	−6.00000	−0.194461
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	−12.0000	−0.388514
$955$	40.0000	1.29437
$956$	25.0000	0.808558
$957$	12.0000	0.387905
$958$	−7.00000	−0.226160
$959$	−8.00000	−0.258333
$960$	2.00000	0.0645497
$961$	−30.0000	−0.967742
$962$	18.0000	0.580343
$963$	34.0000	1.09563
$964$	−26.0000	−0.837404
$965$	4.00000	0.128765
$966$	1.00000	0.0321745
$967$	−29.0000	−0.932577	−0.466289	−	0.884633i	$-0.654409\pi$
$967$	−29.0000	−0.932577	−0.466289	+	0.884633i	$0.654409\pi$
$968$	7.00000	0.224989
$969$	30.0000	0.963739
$970$	−12.0000	−0.385297
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	−16.0000	−0.513200
$973$	8.00000	0.256468
$974$	−32.0000	−1.02535
$975$	−3.00000	−0.0960769
$976$	−6.00000	−0.192055
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−6.00000	−0.191859
$979$	−18.0000	−0.575282
$980$	12.0000	0.383326
$981$	−24.0000	−0.766261
$982$	−36.0000	−1.14881
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	−6.00000	−0.191273
$985$	−54.0000	−1.72058
$986$	36.0000	1.14647
$987$	−3.00000	−0.0954911
$988$	−15.0000	−0.477214
$989$	−5.00000	−0.158991
$990$	8.00000	0.254257
$991$	−35.0000	−1.11181	−0.555906	−	0.831245i	$-0.687628\pi$
$991$	−35.0000	−1.11181	−0.555906	+	0.831245i	$0.687628\pi$
$992$	−1.00000	−0.0317500
$993$	12.0000	0.380808
$994$	−1.00000	−0.0317181
$995$	52.0000	1.64851
$996$	−4.00000	−0.126745
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	−7.00000	−0.221581
$999$	30.0000	0.949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	142.2.a.a.1.1	✓	1
3.2	odd	2		1278.2.a.k.1.1		1
4.3	odd	2		1136.2.a.d.1.1		1
5.4	even	2		3550.2.a.n.1.1		1
7.6	odd	2		6958.2.a.g.1.1		1
8.3	odd	2		4544.2.a.g.1.1		1
8.5	even	2		4544.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
142.2.a.a.1.1	✓	1	1.1	even	1	trivial
1136.2.a.d.1.1		1	4.3	odd	2
1278.2.a.k.1.1		1	3.2	odd	2
3550.2.a.n.1.1		1	5.4	even	2
4544.2.a.g.1.1		1	8.3	odd	2
4544.2.a.m.1.1		1	8.5	even	2
6958.2.a.g.1.1		1	7.6	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:	\( N \)	\(=\)	\( 142 = 2 \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	142.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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