LMFDB - Embedded newform 89.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [89,2,Mod(1,89)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(89, 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("89.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	89.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:	$0.710668577989$
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	$\chi$	$=$	89.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} +2.00000 q^{14} -4.00000 q^{15} -1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} +2.00000 q^{20} +4.00000 q^{21} -4.00000 q^{22} +2.00000 q^{23} -6.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -4.00000 q^{27} -2.00000 q^{28} -6.00000 q^{29} -4.00000 q^{30} +6.00000 q^{31} +5.00000 q^{32} -8.00000 q^{33} +6.00000 q^{34} -4.00000 q^{35} -1.00000 q^{36} +10.0000 q^{37} -2.00000 q^{38} +4.00000 q^{39} +6.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} +2.00000 q^{43} +4.00000 q^{44} -2.00000 q^{45} +2.00000 q^{46} +12.0000 q^{47} -2.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +12.0000 q^{51} -2.00000 q^{52} -6.00000 q^{53} -4.00000 q^{54} +8.00000 q^{55} -6.00000 q^{56} -4.00000 q^{57} -6.00000 q^{58} -10.0000 q^{59} +4.00000 q^{60} -6.00000 q^{61} +6.00000 q^{62} +2.00000 q^{63} +7.00000 q^{64} -4.00000 q^{65} -8.00000 q^{66} +12.0000 q^{67} -6.00000 q^{68} +4.00000 q^{69} -4.00000 q^{70} +4.00000 q^{71} -3.00000 q^{72} +10.0000 q^{73} +10.0000 q^{74} -2.00000 q^{75} +2.00000 q^{76} -8.00000 q^{77} +4.00000 q^{78} -12.0000 q^{79} +2.00000 q^{80} -11.0000 q^{81} -6.00000 q^{82} -6.00000 q^{83} -4.00000 q^{84} -12.0000 q^{85} +2.00000 q^{86} -12.0000 q^{87} +12.0000 q^{88} +1.00000 q^{89} -2.00000 q^{90} +4.00000 q^{91} -2.00000 q^{92} +12.0000 q^{93} +12.0000 q^{94} +4.00000 q^{95} +10.0000 q^{96} -18.0000 q^{97} -3.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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	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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	2.00000	0.816497
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−2.00000	−0.577350
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	2.00000	0.534522
$15$	−4.00000	−1.03280
$16$	−1.00000	−0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	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	$0.573682\pi$
$20$	2.00000	0.447214
$21$	4.00000	0.872872
$22$	−4.00000	−0.852803
$23$	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
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	−4.00000	−0.769800
$28$	−2.00000	−0.377964
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−4.00000	−0.730297
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	5.00000	0.883883
$33$	−8.00000	−1.39262
$34$	6.00000	1.02899
$35$	−4.00000	−0.676123
$36$	−1.00000	−0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	−2.00000	−0.324443
$39$	4.00000	0.640513
$40$	6.00000	0.948683
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	4.00000	0.617213
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	4.00000	0.603023
$45$	−2.00000	−0.298142
$46$	2.00000	0.294884
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−2.00000	−0.288675
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	12.0000	1.68034
$52$	−2.00000	−0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−4.00000	−0.544331
$55$	8.00000	1.07872
$56$	−6.00000	−0.801784
$57$	−4.00000	−0.529813
$58$	−6.00000	−0.787839
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	4.00000	0.516398
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	6.00000	0.762001
$63$	2.00000	0.251976
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	−8.00000	−0.984732
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−6.00000	−0.727607
$69$	4.00000	0.481543
$70$	−4.00000	−0.478091
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	−3.00000	−0.353553
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	10.0000	1.16248
$75$	−2.00000	−0.230940
$76$	2.00000	0.229416
$77$	−8.00000	−0.911685
$78$	4.00000	0.452911
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	2.00000	0.223607
$81$	−11.0000	−1.22222
$82$	−6.00000	−0.662589
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	−4.00000	−0.436436
$85$	−12.0000	−1.30158
$86$	2.00000	0.215666
$87$	−12.0000	−1.28654
$88$	12.0000	1.27920
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	−2.00000	−0.210819
$91$	4.00000	0.419314
$92$	−2.00000	−0.208514
$93$	12.0000	1.24434
$94$	12.0000	1.23771
$95$	4.00000	0.410391
$96$	10.0000	1.02062
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	−3.00000	−0.303046
$99$	−4.00000	−0.402015
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	12.0000	1.18818
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	−6.00000	−0.588348
$105$	−8.00000	−0.780720
$106$	−6.00000	−0.582772
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	4.00000	0.384900
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	8.00000	0.762770
$111$	20.0000	1.89832
$112$	−2.00000	−0.188982
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−4.00000	−0.374634
$115$	−4.00000	−0.373002
$116$	6.00000	0.557086
$117$	2.00000	0.184900
$118$	−10.0000	−0.920575
$119$	12.0000	1.10004
$120$	12.0000	1.09545
$121$	5.00000	0.454545
$122$	−6.00000	−0.543214
$123$	−12.0000	−1.08200
$124$	−6.00000	−0.538816
$125$	12.0000	1.07331
$126$	2.00000	0.178174
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	−3.00000	−0.265165
$129$	4.00000	0.352180
$130$	−4.00000	−0.350823
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	8.00000	0.696311
$133$	−4.00000	−0.346844
$134$	12.0000	1.03664
$135$	8.00000	0.688530
$136$	−18.0000	−1.54349
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	4.00000	0.340503
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	4.00000	0.338062
$141$	24.0000	2.02116
$142$	4.00000	0.335673
$143$	−8.00000	−0.668994
$144$	−1.00000	−0.0833333
$145$	12.0000	0.996546
$146$	10.0000	0.827606
$147$	−6.00000	−0.494872
$148$	−10.0000	−0.821995
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	−2.00000	−0.163299
$151$	22.0000	1.79033	0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000	1.79033	0.895167	+	0.445730i	$0.147056\pi$
$152$	6.00000	0.486664
$153$	6.00000	0.485071
$154$	−8.00000	−0.644658
$155$	−12.0000	−0.963863
$156$	−4.00000	−0.320256
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	−12.0000	−0.954669
$159$	−12.0000	−0.951662
$160$	−10.0000	−0.790569
$161$	4.00000	0.315244
$162$	−11.0000	−0.864242
$163$	22.0000	1.72317	0.861586	−	0.507611i	$-0.169471\pi$
$163$	22.0000	1.72317	0.861586	+	0.507611i	$0.169471\pi$
$164$	6.00000	0.468521
$165$	16.0000	1.24560
$166$	−6.00000	−0.465690
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−12.0000	−0.925820
$169$	−9.00000	−0.692308
$170$	−12.0000	−0.920358
$171$	−2.00000	−0.152944
$172$	−2.00000	−0.152499
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	−12.0000	−0.909718
$175$	−2.00000	−0.151186
$176$	4.00000	0.301511
$177$	−20.0000	−1.50329
$178$	1.00000	0.0749532
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	2.00000	0.149071
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	4.00000	0.296500
$183$	−12.0000	−0.887066
$184$	−6.00000	−0.442326
$185$	−20.0000	−1.47043
$186$	12.0000	0.879883
$187$	−24.0000	−1.75505
$188$	−12.0000	−0.875190
$189$	−8.00000	−0.581914
$190$	4.00000	0.290191
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	14.0000	1.01036
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−18.0000	−1.29232
$195$	−8.00000	−0.572892
$196$	3.00000	0.214286
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	−4.00000	−0.284268
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	3.00000	0.212132
$201$	24.0000	1.69283
$202$	−6.00000	−0.422159
$203$	−12.0000	−0.842235
$204$	−12.0000	−0.840168
$205$	12.0000	0.838116
$206$	−6.00000	−0.418040
$207$	2.00000	0.139010
$208$	−2.00000	−0.138675
$209$	8.00000	0.553372
$210$	−8.00000	−0.552052
$211$	18.0000	1.23917	0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000	1.23917	0.619586	+	0.784929i	$0.287301\pi$
$212$	6.00000	0.412082
$213$	8.00000	0.548151
$214$	−8.00000	−0.546869
$215$	−4.00000	−0.272798
$216$	12.0000	0.816497
$217$	12.0000	0.814613
$218$	2.00000	0.135457
$219$	20.0000	1.35147
$220$	−8.00000	−0.539360
$221$	12.0000	0.807207
$222$	20.0000	1.34231
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	10.0000	0.668153
$225$	−1.00000	−0.0666667
$226$	−14.0000	−0.931266
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	4.00000	0.264906
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	−4.00000	−0.263752
$231$	−16.0000	−1.05272
$232$	18.0000	1.18176
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	2.00000	0.130744
$235$	−24.0000	−1.56559
$236$	10.0000	0.650945
$237$	−24.0000	−1.55897
$238$	12.0000	0.777844
$239$	10.0000	0.646846	0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000	0.646846	0.323423	+	0.946254i	$0.395166\pi$
$240$	4.00000	0.258199
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	5.00000	0.321412
$243$	−10.0000	−0.641500
$244$	6.00000	0.384111
$245$	6.00000	0.383326
$246$	−12.0000	−0.765092
$247$	−4.00000	−0.254514
$248$	−18.0000	−1.14300
$249$	−12.0000	−0.760469
$250$	12.0000	0.758947
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−2.00000	−0.125988
$253$	−8.00000	−0.502956
$254$	6.00000	0.376473
$255$	−24.0000	−1.50294
$256$	−17.0000	−1.06250
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	4.00000	0.249029
$259$	20.0000	1.24274
$260$	4.00000	0.248069
$261$	−6.00000	−0.371391
$262$	8.00000	0.494242
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	24.0000	1.47710
$265$	12.0000	0.737154
$266$	−4.00000	−0.245256
$267$	2.00000	0.122398
$268$	−12.0000	−0.733017
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	8.00000	0.486864
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−6.00000	−0.363803
$273$	8.00000	0.484182
$274$	10.0000	0.604122
$275$	4.00000	0.241209
$276$	−4.00000	−0.240772
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	4.00000	0.239904
$279$	6.00000	0.359211
$280$	12.0000	0.717137
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	24.0000	1.42918
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−4.00000	−0.237356
$285$	8.00000	0.473879
$286$	−8.00000	−0.473050
$287$	−12.0000	−0.708338
$288$	5.00000	0.294628
$289$	19.0000	1.11765
$290$	12.0000	0.704664
$291$	−36.0000	−2.11036
$292$	−10.0000	−0.585206
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−6.00000	−0.349927
$295$	20.0000	1.16445
$296$	−30.0000	−1.74371
$297$	16.0000	0.928414
$298$	2.00000	0.115857
$299$	4.00000	0.231326
$300$	2.00000	0.115470
$301$	4.00000	0.230556
$302$	22.0000	1.26596
$303$	−12.0000	−0.689382
$304$	2.00000	0.114708
$305$	12.0000	0.687118
$306$	6.00000	0.342997
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	8.00000	0.455842
$309$	−12.0000	−0.682656
$310$	−12.0000	−0.681554
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	−12.0000	−0.679366
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	−14.0000	−0.790066
$315$	−4.00000	−0.225374
$316$	12.0000	0.675053
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	−12.0000	−0.672927
$319$	24.0000	1.34374
$320$	−14.0000	−0.782624
$321$	−16.0000	−0.893033
$322$	4.00000	0.222911
$323$	−12.0000	−0.667698
$324$	11.0000	0.611111
$325$	−2.00000	−0.110940
$326$	22.0000	1.21847
$327$	4.00000	0.221201
$328$	18.0000	0.993884
$329$	24.0000	1.32316
$330$	16.0000	0.880771
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	6.00000	0.329293
$333$	10.0000	0.547997
$334$	0	0
$335$	−24.0000	−1.31126
$336$	−4.00000	−0.218218
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−9.00000	−0.489535
$339$	−28.0000	−1.52075
$340$	12.0000	0.650791
$341$	−24.0000	−1.29967
$342$	−2.00000	−0.108148
$343$	−20.0000	−1.07990
$344$	−6.00000	−0.323498
$345$	−8.00000	−0.430706
$346$	22.0000	1.18273
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	12.0000	0.643268
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	−2.00000	−0.106904
$351$	−8.00000	−0.427008
$352$	−20.0000	−1.06600
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	−20.0000	−1.06299
$355$	−8.00000	−0.424596
$356$	−1.00000	−0.0529999
$357$	24.0000	1.27021
$358$	4.00000	0.211407
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	6.00000	0.316228
$361$	−15.0000	−0.789474
$362$	−14.0000	−0.735824
$363$	10.0000	0.524864
$364$	−4.00000	−0.209657
$365$	−20.0000	−1.04685
$366$	−12.0000	−0.627250
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	−2.00000	−0.104257
$369$	−6.00000	−0.312348
$370$	−20.0000	−1.03975
$371$	−12.0000	−0.623009
$372$	−12.0000	−0.622171
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−24.0000	−1.24101
$375$	24.0000	1.23935
$376$	−36.0000	−1.85656
$377$	−12.0000	−0.618031
$378$	−8.00000	−0.411476
$379$	−22.0000	−1.13006	−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000	−1.13006	−0.565032	+	0.825069i	$0.691136\pi$
$380$	−4.00000	−0.205196
$381$	12.0000	0.614779
$382$	6.00000	0.306987
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	−6.00000	−0.306186
$385$	16.0000	0.815436
$386$	−14.0000	−0.712581
$387$	2.00000	0.101666
$388$	18.0000	0.913812
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	−8.00000	−0.405096
$391$	12.0000	0.606866
$392$	9.00000	0.454569
$393$	16.0000	0.807093
$394$	10.0000	0.503793
$395$	24.0000	1.20757
$396$	4.00000	0.201008
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	20.0000	1.00251
$399$	−8.00000	−0.400501
$400$	1.00000	0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	24.0000	1.19701
$403$	12.0000	0.597763
$404$	6.00000	0.298511
$405$	22.0000	1.09319
$406$	−12.0000	−0.595550
$407$	−40.0000	−1.98273
$408$	−36.0000	−1.78227
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	12.0000	0.592638
$411$	20.0000	0.986527
$412$	6.00000	0.295599
$413$	−20.0000	−0.984136
$414$	2.00000	0.0982946
$415$	12.0000	0.589057
$416$	10.0000	0.490290
$417$	8.00000	0.391762
$418$	8.00000	0.391293
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	8.00000	0.390360
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	18.0000	0.876226
$423$	12.0000	0.583460
$424$	18.0000	0.874157
$425$	−6.00000	−0.291043
$426$	8.00000	0.387601
$427$	−12.0000	−0.580721
$428$	8.00000	0.386695
$429$	−16.0000	−0.772487
$430$	−4.00000	−0.192897
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	4.00000	0.192450
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	12.0000	0.576018
$435$	24.0000	1.15071
$436$	−2.00000	−0.0957826
$437$	−4.00000	−0.191346
$438$	20.0000	0.955637
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	−24.0000	−1.14416
$441$	−3.00000	−0.142857
$442$	12.0000	0.570782
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−20.0000	−0.949158
$445$	−2.00000	−0.0948091
$446$	−16.0000	−0.757622
$447$	4.00000	0.189194
$448$	14.0000	0.661438
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	−1.00000	−0.0471405
$451$	24.0000	1.13012
$452$	14.0000	0.658505
$453$	44.0000	2.06730
$454$	0	0
$455$	−8.00000	−0.375046
$456$	12.0000	0.561951
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	2.00000	0.0934539
$459$	−24.0000	−1.12022
$460$	4.00000	0.186501
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	−16.0000	−0.744387
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	6.00000	0.278543
$465$	−24.0000	−1.11297
$466$	26.0000	1.20443
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	−2.00000	−0.0924500
$469$	24.0000	1.10822
$470$	−24.0000	−1.10704
$471$	−28.0000	−1.29017
$472$	30.0000	1.38086
$473$	−8.00000	−0.367840
$474$	−24.0000	−1.10236
$475$	2.00000	0.0917663
$476$	−12.0000	−0.550019
$477$	−6.00000	−0.274721
$478$	10.0000	0.457389
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	−20.0000	−0.912871
$481$	20.0000	0.911922
$482$	−14.0000	−0.637683
$483$	8.00000	0.364013
$484$	−5.00000	−0.227273
$485$	36.0000	1.63468
$486$	−10.0000	−0.453609
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	18.0000	0.814822
$489$	44.0000	1.98975
$490$	6.00000	0.271052
$491$	−14.0000	−0.631811	−0.315906	−	0.948791i	$-0.602308\pi$
$491$	−14.0000	−0.631811	−0.315906	+	0.948791i	$0.602308\pi$
$492$	12.0000	0.541002
$493$	−36.0000	−1.62136
$494$	−4.00000	−0.179969
$495$	8.00000	0.359573
$496$	−6.00000	−0.269408
$497$	8.00000	0.358849
$498$	−12.0000	−0.537733
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	−12.0000	−0.535586
$503$	−2.00000	−0.0891756	−0.0445878	−	0.999005i	$-0.514197\pi$
$503$	−2.00000	−0.0891756	−0.0445878	+	0.999005i	$0.514197\pi$
$504$	−6.00000	−0.267261
$505$	12.0000	0.533993
$506$	−8.00000	−0.355643
$507$	−18.0000	−0.799408
$508$	−6.00000	−0.266207
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	−24.0000	−1.06274
$511$	20.0000	0.884748
$512$	−11.0000	−0.486136
$513$	8.00000	0.353209
$514$	18.0000	0.793946
$515$	12.0000	0.528783
$516$	−4.00000	−0.176090
$517$	−48.0000	−2.11104
$518$	20.0000	0.878750
$519$	44.0000	1.93139
$520$	12.0000	0.526235
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	−6.00000	−0.262613
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−8.00000	−0.349482
$525$	−4.00000	−0.174574
$526$	16.0000	0.697633
$527$	36.0000	1.56818
$528$	8.00000	0.348155
$529$	−19.0000	−0.826087
$530$	12.0000	0.521247
$531$	−10.0000	−0.433963
$532$	4.00000	0.173422
$533$	−12.0000	−0.519778
$534$	2.00000	0.0865485
$535$	16.0000	0.691740
$536$	−36.0000	−1.55496
$537$	8.00000	0.345225
$538$	−14.0000	−0.603583
$539$	12.0000	0.516877
$540$	−8.00000	−0.344265
$541$	−6.00000	−0.257960	−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000	−0.257960	−0.128980	+	0.991647i	$0.541170\pi$
$542$	−8.00000	−0.343629
$543$	−28.0000	−1.20160
$544$	30.0000	1.28624
$545$	−4.00000	−0.171341
$546$	8.00000	0.342368
$547$	6.00000	0.256541	0.128271	−	0.991739i	$-0.459057\pi$
$547$	6.00000	0.256541	0.128271	+	0.991739i	$0.459057\pi$
$548$	−10.0000	−0.427179
$549$	−6.00000	−0.256074
$550$	4.00000	0.170561
$551$	12.0000	0.511217
$552$	−12.0000	−0.510754
$553$	−24.0000	−1.02058
$554$	−2.00000	−0.0849719
$555$	−40.0000	−1.69791
$556$	−4.00000	−0.169638
$557$	34.0000	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$558$	6.00000	0.254000
$559$	4.00000	0.169182
$560$	4.00000	0.169031
$561$	−48.0000	−2.02656
$562$	−6.00000	−0.253095
$563$	22.0000	0.927189	0.463595	−	0.886047i	$-0.346559\pi$
$563$	22.0000	0.927189	0.463595	+	0.886047i	$0.346559\pi$
$564$	−24.0000	−1.01058
$565$	28.0000	1.17797
$566$	4.00000	0.168133
$567$	−22.0000	−0.923913
$568$	−12.0000	−0.503509
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	8.00000	0.335083
$571$	−2.00000	−0.0836974	−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000	−0.0836974	−0.0418487	+	0.999124i	$0.513325\pi$
$572$	8.00000	0.334497
$573$	12.0000	0.501307
$574$	−12.0000	−0.500870
$575$	−2.00000	−0.0834058
$576$	7.00000	0.291667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	19.0000	0.790296
$579$	−28.0000	−1.16364
$580$	−12.0000	−0.498273
$581$	−12.0000	−0.497844
$582$	−36.0000	−1.49225
$583$	24.0000	0.993978
$584$	−30.0000	−1.24141
$585$	−4.00000	−0.165380
$586$	−6.00000	−0.247858
$587$	24.0000	0.990586	0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000	0.990586	0.495293	+	0.868726i	$0.335061\pi$
$588$	6.00000	0.247436
$589$	−12.0000	−0.494451
$590$	20.0000	0.823387
$591$	20.0000	0.822690
$592$	−10.0000	−0.410997
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	16.0000	0.656488
$595$	−24.0000	−0.983904
$596$	−2.00000	−0.0819232
$597$	40.0000	1.63709
$598$	4.00000	0.163572
$599$	10.0000	0.408589	0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000	0.408589	0.204294	+	0.978909i	$0.434510\pi$
$600$	6.00000	0.244949
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	4.00000	0.163028
$603$	12.0000	0.488678
$604$	−22.0000	−0.895167
$605$	−10.0000	−0.406558
$606$	−12.0000	−0.487467
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	−10.0000	−0.405554
$609$	−24.0000	−0.972529
$610$	12.0000	0.485866
$611$	24.0000	0.970936
$612$	−6.00000	−0.242536
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	−32.0000	−1.29141
$615$	24.0000	0.967773
$616$	24.0000	0.966988
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−12.0000	−0.482711
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	12.0000	0.481932
$621$	−8.00000	−0.321029
$622$	−16.0000	−0.641542
$623$	2.00000	0.0801283
$624$	−4.00000	−0.160128
$625$	−19.0000	−0.760000
$626$	2.00000	0.0799361
$627$	16.0000	0.638978
$628$	14.0000	0.558661
$629$	60.0000	2.39236
$630$	−4.00000	−0.159364
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	36.0000	1.43200
$633$	36.0000	1.43087
$634$	−2.00000	−0.0794301
$635$	−12.0000	−0.476205
$636$	12.0000	0.475831
$637$	−6.00000	−0.237729
$638$	24.0000	0.950169
$639$	4.00000	0.158238
$640$	6.00000	0.237171
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	−16.0000	−0.631470
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	−4.00000	−0.157622
$645$	−8.00000	−0.315000
$646$	−12.0000	−0.472134
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	33.0000	1.29636
$649$	40.0000	1.57014
$650$	−2.00000	−0.0784465
$651$	24.0000	0.940634
$652$	−22.0000	−0.861586
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	4.00000	0.156412
$655$	−16.0000	−0.625172
$656$	6.00000	0.234261
$657$	10.0000	0.390137
$658$	24.0000	0.935617
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	−16.0000	−0.622799
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	28.0000	1.08825
$663$	24.0000	0.932083
$664$	18.0000	0.698535
$665$	8.00000	0.310227
$666$	10.0000	0.387492
$667$	−12.0000	−0.464642
$668$	0	0
$669$	−32.0000	−1.23719
$670$	−24.0000	−0.927201
$671$	24.0000	0.926510
$672$	20.0000	0.771517
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	2.00000	0.0770371
$675$	4.00000	0.153960
$676$	9.00000	0.346154
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	−28.0000	−1.07533
$679$	−36.0000	−1.38155
$680$	36.0000	1.38054
$681$	0	0
$682$	−24.0000	−0.919007
$683$	6.00000	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$684$	2.00000	0.0764719
$685$	−20.0000	−0.764161
$686$	−20.0000	−0.763604
$687$	4.00000	0.152610
$688$	−2.00000	−0.0762493
$689$	−12.0000	−0.457164
$690$	−8.00000	−0.304555
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	−22.0000	−0.836315
$693$	−8.00000	−0.303895
$694$	−16.0000	−0.607352
$695$	−8.00000	−0.303457
$696$	36.0000	1.36458
$697$	−36.0000	−1.36360
$698$	−14.0000	−0.529908
$699$	52.0000	1.96682
$700$	2.00000	0.0755929
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	−8.00000	−0.301941
$703$	−20.0000	−0.754314
$704$	−28.0000	−1.05529
$705$	−48.0000	−1.80778
$706$	34.0000	1.27961
$707$	−12.0000	−0.451306
$708$	20.0000	0.751646
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	−8.00000	−0.300235
$711$	−12.0000	−0.450035
$712$	−3.00000	−0.112430
$713$	12.0000	0.449404
$714$	24.0000	0.898177
$715$	16.0000	0.598366
$716$	−4.00000	−0.149487
$717$	20.0000	0.746914
$718$	18.0000	0.671754
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	2.00000	0.0745356
$721$	−12.0000	−0.446903
$722$	−15.0000	−0.558242
$723$	−28.0000	−1.04133
$724$	14.0000	0.520306
$725$	6.00000	0.222834
$726$	10.0000	0.371135
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	−12.0000	−0.444750
$729$	13.0000	0.481481
$730$	−20.0000	−0.740233
$731$	12.0000	0.443836
$732$	12.0000	0.443533
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	16.0000	0.590571
$735$	12.0000	0.442627
$736$	10.0000	0.368605
$737$	−48.0000	−1.76810
$738$	−6.00000	−0.220863
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	20.0000	0.735215
$741$	−8.00000	−0.293887
$742$	−12.0000	−0.440534
$743$	−42.0000	−1.54083	−0.770415	−	0.637542i	$-0.779951\pi$
$743$	−42.0000	−1.54083	−0.770415	+	0.637542i	$0.779951\pi$
$744$	−36.0000	−1.31982
$745$	−4.00000	−0.146549
$746$	14.0000	0.512576
$747$	−6.00000	−0.219529
$748$	24.0000	0.877527
$749$	−16.0000	−0.584627
$750$	24.0000	0.876356
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−12.0000	−0.437595
$753$	−24.0000	−0.874609
$754$	−12.0000	−0.437014
$755$	−44.0000	−1.60132
$756$	8.00000	0.290957
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	−22.0000	−0.799076
$759$	−16.0000	−0.580763
$760$	−12.0000	−0.435286
$761$	−2.00000	−0.0724999	−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000	−0.0724999	−0.0362500	+	0.999343i	$0.511541\pi$
$762$	12.0000	0.434714
$763$	4.00000	0.144810
$764$	−6.00000	−0.217072
$765$	−12.0000	−0.433861
$766$	14.0000	0.505841
$767$	−20.0000	−0.722158
$768$	−34.0000	−1.22687
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	16.0000	0.576600
$771$	36.0000	1.29651
$772$	14.0000	0.503871
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	2.00000	0.0718885
$775$	−6.00000	−0.215526
$776$	54.0000	1.93849
$777$	40.0000	1.43499
$778$	−30.0000	−1.07555
$779$	12.0000	0.429945
$780$	8.00000	0.286446
$781$	−16.0000	−0.572525
$782$	12.0000	0.429119
$783$	24.0000	0.857690
$784$	3.00000	0.107143
$785$	28.0000	0.999363
$786$	16.0000	0.570701
$787$	2.00000	0.0712923	0.0356462	−	0.999364i	$-0.488651\pi$
$787$	2.00000	0.0712923	0.0356462	+	0.999364i	$0.488651\pi$
$788$	−10.0000	−0.356235
$789$	32.0000	1.13923
$790$	24.0000	0.853882
$791$	−28.0000	−0.995565
$792$	12.0000	0.426401
$793$	−12.0000	−0.426132
$794$	−14.0000	−0.496841
$795$	24.0000	0.851192
$796$	−20.0000	−0.708881
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	−8.00000	−0.283197
$799$	72.0000	2.54718
$800$	−5.00000	−0.176777
$801$	1.00000	0.0353333
$802$	−18.0000	−0.635602
$803$	−40.0000	−1.41157
$804$	−24.0000	−0.846415
$805$	−8.00000	−0.281963
$806$	12.0000	0.422682
$807$	−28.0000	−0.985647
$808$	18.0000	0.633238
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	22.0000	0.773001
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	12.0000	0.421117
$813$	−16.0000	−0.561144
$814$	−40.0000	−1.40200
$815$	−44.0000	−1.54125
$816$	−12.0000	−0.420084
$817$	−4.00000	−0.139942
$818$	30.0000	1.04893
$819$	4.00000	0.139771
$820$	−12.0000	−0.419058
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	20.0000	0.697580
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	18.0000	0.627060
$825$	8.00000	0.278524
$826$	−20.0000	−0.695889
$827$	−2.00000	−0.0695468	−0.0347734	−	0.999395i	$-0.511071\pi$
$827$	−2.00000	−0.0695468	−0.0347734	+	0.999395i	$0.511071\pi$
$828$	−2.00000	−0.0695048
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	12.0000	0.416526
$831$	−4.00000	−0.138758
$832$	14.0000	0.485363
$833$	−18.0000	−0.623663
$834$	8.00000	0.277017
$835$	0	0
$836$	−8.00000	−0.276686
$837$	−24.0000	−0.829561
$838$	6.00000	0.207267
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	24.0000	0.828079
$841$	7.00000	0.241379
$842$	−30.0000	−1.03387
$843$	−12.0000	−0.413302
$844$	−18.0000	−0.619586
$845$	18.0000	0.619219
$846$	12.0000	0.412568
$847$	10.0000	0.343604
$848$	6.00000	0.206041
$849$	8.00000	0.274559
$850$	−6.00000	−0.205798
$851$	20.0000	0.685591
$852$	−8.00000	−0.274075
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	−12.0000	−0.410632
$855$	4.00000	0.136797
$856$	24.0000	0.820303
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	−16.0000	−0.546231
$859$	−34.0000	−1.16007	−0.580033	−	0.814593i	$-0.696960\pi$
$859$	−34.0000	−1.16007	−0.580033	+	0.814593i	$0.696960\pi$
$860$	4.00000	0.136399
$861$	−24.0000	−0.817918
$862$	18.0000	0.613082
$863$	38.0000	1.29354	0.646768	−	0.762687i	$-0.276120\pi$
$863$	38.0000	1.29354	0.646768	+	0.762687i	$0.276120\pi$
$864$	−20.0000	−0.680414
$865$	−44.0000	−1.49604
$866$	−6.00000	−0.203888
$867$	38.0000	1.29055
$868$	−12.0000	−0.407307
$869$	48.0000	1.62829
$870$	24.0000	0.813676
$871$	24.0000	0.813209
$872$	−6.00000	−0.203186
$873$	−18.0000	−0.609208
$874$	−4.00000	−0.135302
$875$	24.0000	0.811348
$876$	−20.0000	−0.675737
$877$	−30.0000	−1.01303	−0.506514	−	0.862232i	$-0.669066\pi$
$877$	−30.0000	−1.01303	−0.506514	+	0.862232i	$0.669066\pi$
$878$	10.0000	0.337484
$879$	−12.0000	−0.404750
$880$	−8.00000	−0.269680
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	−3.00000	−0.101015
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	−12.0000	−0.403604
$885$	40.0000	1.34459
$886$	−4.00000	−0.134383
$887$	−30.0000	−1.00730	−0.503651	−	0.863907i	$-0.668010\pi$
$887$	−30.0000	−1.00730	−0.503651	+	0.863907i	$0.668010\pi$
$888$	−60.0000	−2.01347
$889$	12.0000	0.402467
$890$	−2.00000	−0.0670402
$891$	44.0000	1.47406
$892$	16.0000	0.535720
$893$	−24.0000	−0.803129
$894$	4.00000	0.133780
$895$	−8.00000	−0.267411
$896$	−6.00000	−0.200446
$897$	8.00000	0.267112
$898$	18.0000	0.600668
$899$	−36.0000	−1.20067
$900$	1.00000	0.0333333
$901$	−36.0000	−1.19933
$902$	24.0000	0.799113
$903$	8.00000	0.266223
$904$	42.0000	1.39690
$905$	28.0000	0.930751
$906$	44.0000	1.46180
$907$	−16.0000	−0.531271	−0.265636	−	0.964073i	$-0.585582\pi$
$907$	−16.0000	−0.531271	−0.265636	+	0.964073i	$0.585582\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	−8.00000	−0.265197
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	4.00000	0.132453
$913$	24.0000	0.794284
$914$	−38.0000	−1.25693
$915$	24.0000	0.793416
$916$	−2.00000	−0.0660819
$917$	16.0000	0.528367
$918$	−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	$0.284384\pi$
$920$	12.0000	0.395628
$921$	−64.0000	−2.10887
$922$	18.0000	0.592798
$923$	8.00000	0.263323
$924$	16.0000	0.526361
$925$	−10.0000	−0.328798
$926$	4.00000	0.131448
$927$	−6.00000	−0.197066
$928$	−30.0000	−0.984798
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	−24.0000	−0.786991
$931$	6.00000	0.196642
$932$	−26.0000	−0.851658
$933$	−32.0000	−1.04763
$934$	−4.00000	−0.130884
$935$	48.0000	1.56977
$936$	−6.00000	−0.196116
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	24.0000	0.783628
$939$	4.00000	0.130535
$940$	24.0000	0.782794
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	−28.0000	−0.912289
$943$	−12.0000	−0.390774
$944$	10.0000	0.325472
$945$	16.0000	0.520480
$946$	−8.00000	−0.260102
$947$	20.0000	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$947$	20.0000	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$948$	24.0000	0.779484
$949$	20.0000	0.649227
$950$	2.00000	0.0648886
$951$	−4.00000	−0.129709
$952$	−36.0000	−1.16677
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	−6.00000	−0.194257
$955$	−12.0000	−0.388311
$956$	−10.0000	−0.323423
$957$	48.0000	1.55162
$958$	−28.0000	−0.904639
$959$	20.0000	0.645834
$960$	−28.0000	−0.903696
$961$	5.00000	0.161290
$962$	20.0000	0.644826
$963$	−8.00000	−0.257796
$964$	14.0000	0.450910
$965$	28.0000	0.901352
$966$	8.00000	0.257396
$967$	50.0000	1.60789	0.803946	−	0.594703i	$-0.202730\pi$
$967$	50.0000	1.60789	0.803946	+	0.594703i	$0.202730\pi$
$968$	−15.0000	−0.482118
$969$	−24.0000	−0.770991
$970$	36.0000	1.15589
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	10.0000	0.320750
$973$	8.00000	0.256468
$974$	20.0000	0.640841
$975$	−4.00000	−0.128103
$976$	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	$0.407031\pi$
$978$	44.0000	1.40696
$979$	−4.00000	−0.127841
$980$	−6.00000	−0.191663
$981$	2.00000	0.0638551
$982$	−14.0000	−0.446758
$983$	32.0000	1.02064	0.510321	−	0.859984i	$-0.329527\pi$
$983$	32.0000	1.02064	0.510321	+	0.859984i	$0.329527\pi$
$984$	36.0000	1.14764
$985$	−20.0000	−0.637253
$986$	−36.0000	−1.14647
$987$	48.0000	1.52786
$988$	4.00000	0.127257
$989$	4.00000	0.127193
$990$	8.00000	0.254257
$991$	50.0000	1.58830	0.794151	−	0.607720i	$-0.207916\pi$
$991$	50.0000	1.58830	0.794151	+	0.607720i	$0.207916\pi$
$992$	30.0000	0.952501
$993$	56.0000	1.77711
$994$	8.00000	0.253745
$995$	−40.0000	−1.26809
$996$	12.0000	0.380235
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	−34.0000	−1.07625
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	89.2.a.b.1.1	✓	1
3.2	odd	2		801.2.a.a.1.1		1
4.3	odd	2		1424.2.a.a.1.1		1
5.4	even	2		2225.2.a.a.1.1		1
7.6	odd	2		4361.2.a.b.1.1		1
8.3	odd	2		5696.2.a.o.1.1		1
8.5	even	2		5696.2.a.d.1.1		1
89.88	even	2		7921.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
89.2.a.b.1.1	✓	1	1.1	even	1	trivial
801.2.a.a.1.1		1	3.2	odd	2
1424.2.a.a.1.1		1	4.3	odd	2
2225.2.a.a.1.1		1	5.4	even	2
4361.2.a.b.1.1		1	7.6	odd	2
5696.2.a.d.1.1		1	8.5	even	2
5696.2.a.o.1.1		1	8.3	odd	2
7921.2.a.b.1.1		1	89.88	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	89.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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