LMFDB - Embedded newform 89.2.a.a.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:	$1$
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	$\chi$	$=$	89.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +3.00000 q^{17} +2.00000 q^{18} -5.00000 q^{19} +1.00000 q^{20} +4.00000 q^{21} +2.00000 q^{22} +7.00000 q^{23} -3.00000 q^{24} -4.00000 q^{25} -2.00000 q^{26} +5.00000 q^{27} +4.00000 q^{28} -1.00000 q^{30} -9.00000 q^{31} -5.00000 q^{32} +2.00000 q^{33} -3.00000 q^{34} +4.00000 q^{35} +2.00000 q^{36} -2.00000 q^{37} +5.00000 q^{38} -2.00000 q^{39} -3.00000 q^{40} -4.00000 q^{42} -7.00000 q^{43} +2.00000 q^{44} +2.00000 q^{45} -7.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +4.00000 q^{50} -3.00000 q^{51} -2.00000 q^{52} -3.00000 q^{53} -5.00000 q^{54} +2.00000 q^{55} -12.0000 q^{56} +5.00000 q^{57} +4.00000 q^{59} -1.00000 q^{60} +6.00000 q^{61} +9.00000 q^{62} +8.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} -2.00000 q^{66} +12.0000 q^{67} -3.00000 q^{68} -7.00000 q^{69} -4.00000 q^{70} -10.0000 q^{71} -6.00000 q^{72} +7.00000 q^{73} +2.00000 q^{74} +4.00000 q^{75} +5.00000 q^{76} +8.00000 q^{77} +2.00000 q^{78} -6.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +12.0000 q^{83} -4.00000 q^{84} -3.00000 q^{85} +7.00000 q^{86} -6.00000 q^{88} -1.00000 q^{89} -2.00000 q^{90} -8.00000 q^{91} -7.00000 q^{92} +9.00000 q^{93} +12.0000 q^{94} +5.00000 q^{95} +5.00000 q^{96} +9.00000 q^{97} -9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−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
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	1.00000	0.408248
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	3.00000	1.06066
$9$	−2.00000	−0.666667
$10$	1.00000	0.316228
$11$	−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
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	4.00000	1.06904
$15$	1.00000	0.258199
$16$	−1.00000	−0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	2.00000	0.471405
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	1.00000	0.223607
$21$	4.00000	0.872872
$22$	2.00000	0.426401
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	−3.00000	−0.612372
$25$	−4.00000	−0.800000
$26$	−2.00000	−0.392232
$27$	5.00000	0.962250
$28$	4.00000	0.755929
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−1.00000	−0.182574
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	−5.00000	−0.883883
$33$	2.00000	0.348155
$34$	−3.00000	−0.514496
$35$	4.00000	0.676123
$36$	2.00000	0.333333
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	5.00000	0.811107
$39$	−2.00000	−0.320256
$40$	−3.00000	−0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−4.00000	−0.617213
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	2.00000	0.301511
$45$	2.00000	0.298142
$46$	−7.00000	−1.03209
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	4.00000	0.565685
$51$	−3.00000	−0.420084
$52$	−2.00000	−0.277350
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	−5.00000	−0.680414
$55$	2.00000	0.269680
$56$	−12.0000	−1.60357
$57$	5.00000	0.662266
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−1.00000	−0.129099
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	9.00000	1.14300
$63$	8.00000	1.00791
$64$	7.00000	0.875000
$65$	−2.00000	−0.248069
$66$	−2.00000	−0.246183
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−3.00000	−0.363803
$69$	−7.00000	−0.842701
$70$	−4.00000	−0.478091
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	−6.00000	−0.707107
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	2.00000	0.232495
$75$	4.00000	0.461880
$76$	5.00000	0.573539
$77$	8.00000	0.911685
$78$	2.00000	0.226455
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	−4.00000	−0.436436
$85$	−3.00000	−0.325396
$86$	7.00000	0.754829
$87$	0	0
$88$	−6.00000	−0.639602
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	−2.00000	−0.210819
$91$	−8.00000	−0.838628
$92$	−7.00000	−0.729800
$93$	9.00000	0.933257
$94$	12.0000	1.23771
$95$	5.00000	0.512989
$96$	5.00000	0.510310
$97$	9.00000	0.913812	0.456906	−	0.889515i	$-0.348958\pi$
$97$	9.00000	0.913812	0.456906	+	0.889515i	$0.348958\pi$
$98$	−9.00000	−0.909137
$99$	4.00000	0.402015
$100$	4.00000	0.400000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	3.00000	0.297044
$103$	−9.00000	−0.886796	−0.443398	−	0.896325i	$-0.646227\pi$
$103$	−9.00000	−0.886796	−0.443398	+	0.896325i	$0.646227\pi$
$104$	6.00000	0.588348
$105$	−4.00000	−0.390360
$106$	3.00000	0.291386
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	−5.00000	−0.481125
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	−2.00000	−0.190693
$111$	2.00000	0.189832
$112$	4.00000	0.377964
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−5.00000	−0.468293
$115$	−7.00000	−0.652753
$116$	0	0
$117$	−4.00000	−0.369800
$118$	−4.00000	−0.368230
$119$	−12.0000	−1.10004
$120$	3.00000	0.273861
$121$	−7.00000	−0.636364
$122$	−6.00000	−0.543214
$123$	0	0
$124$	9.00000	0.808224
$125$	9.00000	0.804984
$126$	−8.00000	−0.712697
$127$	3.00000	0.266207	0.133103	−	0.991102i	$-0.457506\pi$
$127$	3.00000	0.266207	0.133103	+	0.991102i	$0.457506\pi$
$128$	3.00000	0.265165
$129$	7.00000	0.616316
$130$	2.00000	0.175412
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	−2.00000	−0.174078
$133$	20.0000	1.73422
$134$	−12.0000	−1.03664
$135$	−5.00000	−0.430331
$136$	9.00000	0.771744
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	7.00000	0.595880
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	−4.00000	−0.338062
$141$	12.0000	1.01058
$142$	10.0000	0.839181
$143$	−4.00000	−0.334497
$144$	2.00000	0.166667
$145$	0	0
$146$	−7.00000	−0.579324
$147$	−9.00000	−0.742307
$148$	2.00000	0.164399
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	−4.00000	−0.326599
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−15.0000	−1.21666
$153$	−6.00000	−0.485071
$154$	−8.00000	−0.644658
$155$	9.00000	0.722897
$156$	2.00000	0.160128
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	6.00000	0.477334
$159$	3.00000	0.237915
$160$	5.00000	0.395285
$161$	−28.0000	−2.20671
$162$	−1.00000	−0.0785674
$163$	25.0000	1.95815	0.979076	−	0.203497i	$-0.0652307\pi$
$163$	25.0000	1.95815	0.979076	+	0.203497i	$0.0652307\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	−12.0000	−0.931381
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	12.0000	0.925820
$169$	−9.00000	−0.692308
$170$	3.00000	0.230089
$171$	10.0000	0.764719
$172$	7.00000	0.533745
$173$	−1.00000	−0.0760286	−0.0380143	−	0.999277i	$-0.512103\pi$
$173$	−1.00000	−0.0760286	−0.0380143	+	0.999277i	$0.512103\pi$
$174$	0	0
$175$	16.0000	1.20949
$176$	2.00000	0.150756
$177$	−4.00000	−0.300658
$178$	1.00000	0.0749532
$179$	14.0000	1.04641	0.523205	−	0.852207i	$-0.324736\pi$
$179$	14.0000	1.04641	0.523205	+	0.852207i	$0.324736\pi$
$180$	−2.00000	−0.149071
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	8.00000	0.592999
$183$	−6.00000	−0.443533
$184$	21.0000	1.54814
$185$	2.00000	0.147043
$186$	−9.00000	−0.659912
$187$	−6.00000	−0.438763
$188$	12.0000	0.875190
$189$	−20.0000	−1.45479
$190$	−5.00000	−0.362738
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	−7.00000	−0.505181
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−9.00000	−0.646162
$195$	2.00000	0.143223
$196$	−9.00000	−0.642857
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	−4.00000	−0.284268
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	−12.0000	−0.848528
$201$	−12.0000	−0.846415
$202$	0	0
$203$	0	0
$204$	3.00000	0.210042
$205$	0	0
$206$	9.00000	0.627060
$207$	−14.0000	−0.973067
$208$	−2.00000	−0.138675
$209$	10.0000	0.691714
$210$	4.00000	0.276026
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	3.00000	0.206041
$213$	10.0000	0.685189
$214$	10.0000	0.683586
$215$	7.00000	0.477396
$216$	15.0000	1.02062
$217$	36.0000	2.44384
$218$	19.0000	1.28684
$219$	−7.00000	−0.473016
$220$	−2.00000	−0.134840
$221$	6.00000	0.403604
$222$	−2.00000	−0.134231
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	20.0000	1.33631
$225$	8.00000	0.533333
$226$	−2.00000	−0.133038
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	−5.00000	−0.331133
$229$	−28.0000	−1.85029	−0.925146	−	0.379611i	$-0.876058\pi$
$229$	−28.0000	−1.85029	−0.925146	+	0.379611i	$0.876058\pi$
$230$	7.00000	0.461566
$231$	−8.00000	−0.526361
$232$	0	0
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	4.00000	0.261488
$235$	12.0000	0.782794
$236$	−4.00000	−0.260378
$237$	6.00000	0.389742
$238$	12.0000	0.777844
$239$	5.00000	0.323423	0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000	0.323423	0.161712	+	0.986838i	$0.448299\pi$
$240$	−1.00000	−0.0645497
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	7.00000	0.449977
$243$	−16.0000	−1.02640
$244$	−6.00000	−0.384111
$245$	−9.00000	−0.574989
$246$	0	0
$247$	−10.0000	−0.636285
$248$	−27.0000	−1.71450
$249$	−12.0000	−0.760469
$250$	−9.00000	−0.569210
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−8.00000	−0.503953
$253$	−14.0000	−0.880172
$254$	−3.00000	−0.188237
$255$	3.00000	0.187867
$256$	−17.0000	−1.06250
$257$	21.0000	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$258$	−7.00000	−0.435801
$259$	8.00000	0.497096
$260$	2.00000	0.124035
$261$	0	0
$262$	−16.0000	−0.988483
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	6.00000	0.369274
$265$	3.00000	0.184289
$266$	−20.0000	−1.22628
$267$	1.00000	0.0611990
$268$	−12.0000	−0.733017
$269$	−13.0000	−0.792624	−0.396312	−	0.918116i	$-0.629710\pi$
$269$	−13.0000	−0.792624	−0.396312	+	0.918116i	$0.629710\pi$
$270$	5.00000	0.304290
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	−3.00000	−0.181902
$273$	8.00000	0.484182
$274$	22.0000	1.32907
$275$	8.00000	0.482418
$276$	7.00000	0.421350
$277$	−29.0000	−1.74244	−0.871221	−	0.490892i	$-0.836671\pi$
$277$	−29.0000	−1.74244	−0.871221	+	0.490892i	$0.836671\pi$
$278$	8.00000	0.479808
$279$	18.0000	1.07763
$280$	12.0000	0.717137
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	−12.0000	−0.714590
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	10.0000	0.593391
$285$	−5.00000	−0.296174
$286$	4.00000	0.236525
$287$	0	0
$288$	10.0000	0.589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−9.00000	−0.527589
$292$	−7.00000	−0.409644
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	9.00000	0.524891
$295$	−4.00000	−0.232889
$296$	−6.00000	−0.348743
$297$	−10.0000	−0.580259
$298$	−4.00000	−0.231714
$299$	14.0000	0.809641
$300$	−4.00000	−0.230940
$301$	28.0000	1.61389
$302$	8.00000	0.460348
$303$	0	0
$304$	5.00000	0.286770
$305$	−6.00000	−0.343559
$306$	6.00000	0.342997
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	−8.00000	−0.455842
$309$	9.00000	0.511992
$310$	−9.00000	−0.511166
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	−6.00000	−0.339683
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	2.00000	0.112867
$315$	−8.00000	−0.450749
$316$	6.00000	0.337526
$317$	17.0000	0.954815	0.477408	−	0.878682i	$-0.341577\pi$
$317$	17.0000	0.954815	0.477408	+	0.878682i	$0.341577\pi$
$318$	−3.00000	−0.168232
$319$	0	0
$320$	−7.00000	−0.391312
$321$	10.0000	0.558146
$322$	28.0000	1.56038
$323$	−15.0000	−0.834622
$324$	−1.00000	−0.0555556
$325$	−8.00000	−0.443760
$326$	−25.0000	−1.38462
$327$	19.0000	1.05070
$328$	0	0
$329$	48.0000	2.64633
$330$	2.00000	0.110096
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	−12.0000	−0.658586
$333$	4.00000	0.219199
$334$	12.0000	0.656611
$335$	−12.0000	−0.655630
$336$	−4.00000	−0.218218
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	9.00000	0.489535
$339$	−2.00000	−0.108625
$340$	3.00000	0.162698
$341$	18.0000	0.974755
$342$	−10.0000	−0.540738
$343$	−8.00000	−0.431959
$344$	−21.0000	−1.13224
$345$	7.00000	0.376867
$346$	1.00000	0.0537603
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	0	0
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	−16.0000	−0.855236
$351$	10.0000	0.533761
$352$	10.0000	0.533002
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	4.00000	0.212598
$355$	10.0000	0.530745
$356$	1.00000	0.0529999
$357$	12.0000	0.635107
$358$	−14.0000	−0.739923
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	6.00000	0.316228
$361$	6.00000	0.315789
$362$	−22.0000	−1.15629
$363$	7.00000	0.367405
$364$	8.00000	0.419314
$365$	−7.00000	−0.366397
$366$	6.00000	0.313625
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−7.00000	−0.364900
$369$	0	0
$370$	−2.00000	−0.103975
$371$	12.0000	0.623009
$372$	−9.00000	−0.466628
$373$	−37.0000	−1.91579	−0.957894	−	0.287123i	$-0.907301\pi$
$373$	−37.0000	−1.91579	−0.957894	+	0.287123i	$0.907301\pi$
$374$	6.00000	0.310253
$375$	−9.00000	−0.464758
$376$	−36.0000	−1.85656
$377$	0	0
$378$	20.0000	1.02869
$379$	17.0000	0.873231	0.436616	−	0.899648i	$-0.356177\pi$
$379$	17.0000	0.873231	0.436616	+	0.899648i	$0.356177\pi$
$380$	−5.00000	−0.256495
$381$	−3.00000	−0.153695
$382$	−15.0000	−0.767467
$383$	13.0000	0.664269	0.332134	−	0.943232i	$-0.392231\pi$
$383$	13.0000	0.664269	0.332134	+	0.943232i	$0.392231\pi$
$384$	−3.00000	−0.153093
$385$	−8.00000	−0.407718
$386$	14.0000	0.712581
$387$	14.0000	0.711660
$388$	−9.00000	−0.456906
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	−2.00000	−0.101274
$391$	21.0000	1.06202
$392$	27.0000	1.36371
$393$	−16.0000	−0.807093
$394$	−8.00000	−0.403034
$395$	6.00000	0.301893
$396$	−4.00000	−0.201008
$397$	4.00000	0.200754	0.100377	−	0.994949i	$-0.467995\pi$
$397$	4.00000	0.200754	0.100377	+	0.994949i	$0.467995\pi$
$398$	−2.00000	−0.100251
$399$	−20.0000	−1.00125
$400$	4.00000	0.200000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	12.0000	0.598506
$403$	−18.0000	−0.896644
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	4.00000	0.198273
$408$	−9.00000	−0.445566
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	22.0000	1.08518
$412$	9.00000	0.443398
$413$	−16.0000	−0.787309
$414$	14.0000	0.688062
$415$	−12.0000	−0.589057
$416$	−10.0000	−0.490290
$417$	8.00000	0.391762
$418$	−10.0000	−0.489116
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	4.00000	0.195180
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	12.0000	0.584151
$423$	24.0000	1.16692
$424$	−9.00000	−0.437079
$425$	−12.0000	−0.582086
$426$	−10.0000	−0.484502
$427$	−24.0000	−1.16144
$428$	10.0000	0.483368
$429$	4.00000	0.193122
$430$	−7.00000	−0.337570
$431$	27.0000	1.30054	0.650272	−	0.759701i	$-0.274655\pi$
$431$	27.0000	1.30054	0.650272	+	0.759701i	$0.274655\pi$
$432$	−5.00000	−0.240563
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	−36.0000	−1.72806
$435$	0	0
$436$	19.0000	0.909935
$437$	−35.0000	−1.67428
$438$	7.00000	0.334473
$439$	−11.0000	−0.525001	−0.262501	−	0.964932i	$-0.584547\pi$
$439$	−11.0000	−0.525001	−0.262501	+	0.964932i	$0.584547\pi$
$440$	6.00000	0.286039
$441$	−18.0000	−0.857143
$442$	−6.00000	−0.285391
$443$	−14.0000	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$443$	−14.0000	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$444$	−2.00000	−0.0949158
$445$	1.00000	0.0474045
$446$	16.0000	0.757622
$447$	−4.00000	−0.189194
$448$	−28.0000	−1.32288
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	−8.00000	−0.377124
$451$	0	0
$452$	−2.00000	−0.0940721
$453$	8.00000	0.375873
$454$	−24.0000	−1.12638
$455$	8.00000	0.375046
$456$	15.0000	0.702439
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	28.0000	1.30835
$459$	15.0000	0.700140
$460$	7.00000	0.326377
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	8.00000	0.372194
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	0	0
$465$	−9.00000	−0.417365
$466$	26.0000	1.20443
$467$	22.0000	1.01804	0.509019	−	0.860755i	$-0.330008\pi$
$467$	22.0000	1.01804	0.509019	+	0.860755i	$0.330008\pi$
$468$	4.00000	0.184900
$469$	−48.0000	−2.21643
$470$	−12.0000	−0.553519
$471$	2.00000	0.0921551
$472$	12.0000	0.552345
$473$	14.0000	0.643721
$474$	−6.00000	−0.275589
$475$	20.0000	0.917663
$476$	12.0000	0.550019
$477$	6.00000	0.274721
$478$	−5.00000	−0.228695
$479$	−26.0000	−1.18797	−0.593985	−	0.804476i	$-0.702446\pi$
$479$	−26.0000	−1.18797	−0.593985	+	0.804476i	$0.702446\pi$
$480$	−5.00000	−0.228218
$481$	−4.00000	−0.182384
$482$	−10.0000	−0.455488
$483$	28.0000	1.27404
$484$	7.00000	0.318182
$485$	−9.00000	−0.408669
$486$	16.0000	0.725775
$487$	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
$489$	−25.0000	−1.13054
$490$	9.00000	0.406579
$491$	−25.0000	−1.12823	−0.564117	−	0.825695i	$-0.690783\pi$
$491$	−25.0000	−1.12823	−0.564117	+	0.825695i	$0.690783\pi$
$492$	0	0
$493$	0	0
$494$	10.0000	0.449921
$495$	−4.00000	−0.179787
$496$	9.00000	0.404112
$497$	40.0000	1.79425
$498$	12.0000	0.537733
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	−9.00000	−0.402492
$501$	12.0000	0.536120
$502$	12.0000	0.535586
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	24.0000	1.06904
$505$	0	0
$506$	14.0000	0.622376
$507$	9.00000	0.399704
$508$	−3.00000	−0.133103
$509$	3.00000	0.132973	0.0664863	−	0.997787i	$-0.478821\pi$
$509$	3.00000	0.132973	0.0664863	+	0.997787i	$0.478821\pi$
$510$	−3.00000	−0.132842
$511$	−28.0000	−1.23865
$512$	11.0000	0.486136
$513$	−25.0000	−1.10378
$514$	−21.0000	−0.926270
$515$	9.00000	0.396587
$516$	−7.00000	−0.308158
$517$	24.0000	1.05552
$518$	−8.00000	−0.351500
$519$	1.00000	0.0438951
$520$	−6.00000	−0.263117
$521$	40.0000	1.75243	0.876216	−	0.481919i	$-0.160060\pi$
$521$	40.0000	1.75243	0.876216	+	0.481919i	$0.160060\pi$
$522$	0	0
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	−16.0000	−0.698963
$525$	−16.0000	−0.698297
$526$	16.0000	0.697633
$527$	−27.0000	−1.17614
$528$	−2.00000	−0.0870388
$529$	26.0000	1.13043
$530$	−3.00000	−0.130312
$531$	−8.00000	−0.347170
$532$	−20.0000	−0.867110
$533$	0	0
$534$	−1.00000	−0.0432742
$535$	10.0000	0.432338
$536$	36.0000	1.55496
$537$	−14.0000	−0.604145
$538$	13.0000	0.560470
$539$	−18.0000	−0.775315
$540$	5.00000	0.215166
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	2.00000	0.0859074
$543$	−22.0000	−0.944110
$544$	−15.0000	−0.643120
$545$	19.0000	0.813871
$546$	−8.00000	−0.342368
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	22.0000	0.939793
$549$	−12.0000	−0.512148
$550$	−8.00000	−0.341121
$551$	0	0
$552$	−21.0000	−0.893819
$553$	24.0000	1.02058
$554$	29.0000	1.23209
$555$	−2.00000	−0.0848953
$556$	8.00000	0.339276
$557$	32.0000	1.35588	0.677942	−	0.735116i	$-0.262872\pi$
$557$	32.0000	1.35588	0.677942	+	0.735116i	$0.262872\pi$
$558$	−18.0000	−0.762001
$559$	−14.0000	−0.592137
$560$	−4.00000	−0.169031
$561$	6.00000	0.253320
$562$	30.0000	1.26547
$563$	5.00000	0.210725	0.105362	−	0.994434i	$-0.466400\pi$
$563$	5.00000	0.210725	0.105362	+	0.994434i	$0.466400\pi$
$564$	−12.0000	−0.505291
$565$	−2.00000	−0.0841406
$566$	−10.0000	−0.420331
$567$	−4.00000	−0.167984
$568$	−30.0000	−1.25877
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	5.00000	0.209427
$571$	25.0000	1.04622	0.523109	−	0.852266i	$-0.324772\pi$
$571$	25.0000	1.04622	0.523109	+	0.852266i	$0.324772\pi$
$572$	4.00000	0.167248
$573$	−15.0000	−0.626634
$574$	0	0
$575$	−28.0000	−1.16768
$576$	−14.0000	−0.583333
$577$	28.0000	1.16566	0.582828	−	0.812596i	$-0.301946\pi$
$577$	28.0000	1.16566	0.582828	+	0.812596i	$0.301946\pi$
$578$	8.00000	0.332756
$579$	14.0000	0.581820
$580$	0	0
$581$	−48.0000	−1.99138
$582$	9.00000	0.373062
$583$	6.00000	0.248495
$584$	21.0000	0.868986
$585$	4.00000	0.165380
$586$	18.0000	0.743573
$587$	−30.0000	−1.23823	−0.619116	−	0.785299i	$-0.712509\pi$
$587$	−30.0000	−1.23823	−0.619116	+	0.785299i	$0.712509\pi$
$588$	9.00000	0.371154
$589$	45.0000	1.85419
$590$	4.00000	0.164677
$591$	−8.00000	−0.329076
$592$	2.00000	0.0821995
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	10.0000	0.410305
$595$	12.0000	0.491952
$596$	−4.00000	−0.163846
$597$	−2.00000	−0.0818546
$598$	−14.0000	−0.572503
$599$	−1.00000	−0.0408589	−0.0204294	−	0.999791i	$-0.506503\pi$
$599$	−1.00000	−0.0408589	−0.0204294	+	0.999791i	$0.506503\pi$
$600$	12.0000	0.489898
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	−28.0000	−1.14119
$603$	−24.0000	−0.977356
$604$	8.00000	0.325515
$605$	7.00000	0.284590
$606$	0	0
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	25.0000	1.01388
$609$	0	0
$610$	6.00000	0.242933
$611$	−24.0000	−0.970936
$612$	6.00000	0.242536
$613$	−11.0000	−0.444286	−0.222143	−	0.975014i	$-0.571305\pi$
$613$	−11.0000	−0.444286	−0.222143	+	0.975014i	$0.571305\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	24.0000	0.966988
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	−9.00000	−0.362033
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	−9.00000	−0.361449
$621$	35.0000	1.40450
$622$	−28.0000	−1.12270
$623$	4.00000	0.160257
$624$	2.00000	0.0800641
$625$	11.0000	0.440000
$626$	−20.0000	−0.799361
$627$	−10.0000	−0.399362
$628$	2.00000	0.0798087
$629$	−6.00000	−0.239236
$630$	8.00000	0.318728
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	−18.0000	−0.716002
$633$	12.0000	0.476957
$634$	−17.0000	−0.675156
$635$	−3.00000	−0.119051
$636$	−3.00000	−0.118958
$637$	18.0000	0.713186
$638$	0	0
$639$	20.0000	0.791188
$640$	−3.00000	−0.118585
$641$	25.0000	0.987441	0.493720	−	0.869621i	$-0.335637\pi$
$641$	25.0000	0.987441	0.493720	+	0.869621i	$0.335637\pi$
$642$	−10.0000	−0.394669
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	28.0000	1.10335
$645$	−7.00000	−0.275625
$646$	15.0000	0.590167
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	3.00000	0.117851
$649$	−8.00000	−0.314027
$650$	8.00000	0.313786
$651$	−36.0000	−1.41095
$652$	−25.0000	−0.979076
$653$	8.00000	0.313064	0.156532	−	0.987673i	$-0.449969\pi$
$653$	8.00000	0.313064	0.156532	+	0.987673i	$0.449969\pi$
$654$	−19.0000	−0.742959
$655$	−16.0000	−0.625172
$656$	0	0
$657$	−14.0000	−0.546192
$658$	−48.0000	−1.87123
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	2.00000	0.0778499
$661$	34.0000	1.32245	0.661223	−	0.750189i	$-0.270038\pi$
$661$	34.0000	1.32245	0.661223	+	0.750189i	$0.270038\pi$
$662$	−10.0000	−0.388661
$663$	−6.00000	−0.233021
$664$	36.0000	1.39707
$665$	−20.0000	−0.775567
$666$	−4.00000	−0.154997
$667$	0	0
$668$	12.0000	0.464294
$669$	16.0000	0.618596
$670$	12.0000	0.463600
$671$	−12.0000	−0.463255
$672$	−20.0000	−0.771517
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	−8.00000	−0.308148
$675$	−20.0000	−0.769800
$676$	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	$0.387574\pi$
$678$	2.00000	0.0768095
$679$	−36.0000	−1.38155
$680$	−9.00000	−0.345134
$681$	−24.0000	−0.919682
$682$	−18.0000	−0.689256
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	−10.0000	−0.382360
$685$	22.0000	0.840577
$686$	8.00000	0.305441
$687$	28.0000	1.06827
$688$	7.00000	0.266872
$689$	−6.00000	−0.228582
$690$	−7.00000	−0.266485
$691$	18.0000	0.684752	0.342376	−	0.939563i	$-0.388768\pi$
$691$	18.0000	0.684752	0.342376	+	0.939563i	$0.388768\pi$
$692$	1.00000	0.0380143
$693$	−16.0000	−0.607790
$694$	2.00000	0.0759190
$695$	8.00000	0.303457
$696$	0	0
$697$	0	0
$698$	8.00000	0.302804
$699$	26.0000	0.983410
$700$	−16.0000	−0.604743
$701$	13.0000	0.491003	0.245502	−	0.969396i	$-0.421047\pi$
$701$	13.0000	0.491003	0.245502	+	0.969396i	$0.421047\pi$
$702$	−10.0000	−0.377426
$703$	10.0000	0.377157
$704$	−14.0000	−0.527645
$705$	−12.0000	−0.451946
$706$	−14.0000	−0.526897
$707$	0	0
$708$	4.00000	0.150329
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	−10.0000	−0.375293
$711$	12.0000	0.450035
$712$	−3.00000	−0.112430
$713$	−63.0000	−2.35937
$714$	−12.0000	−0.449089
$715$	4.00000	0.149592
$716$	−14.0000	−0.523205
$717$	−5.00000	−0.186728
$718$	0	0
$719$	−9.00000	−0.335643	−0.167822	−	0.985817i	$-0.553673\pi$
$719$	−9.00000	−0.335643	−0.167822	+	0.985817i	$0.553673\pi$
$720$	−2.00000	−0.0745356
$721$	36.0000	1.34071
$722$	−6.00000	−0.223297
$723$	−10.0000	−0.371904
$724$	−22.0000	−0.817624
$725$	0	0
$726$	−7.00000	−0.259794
$727$	13.0000	0.482143	0.241072	−	0.970507i	$-0.422501\pi$
$727$	13.0000	0.482143	0.241072	+	0.970507i	$0.422501\pi$
$728$	−24.0000	−0.889499
$729$	13.0000	0.481481
$730$	7.00000	0.259082
$731$	−21.0000	−0.776713
$732$	6.00000	0.221766
$733$	1.00000	0.0369358	0.0184679	−	0.999829i	$-0.494121\pi$
$733$	1.00000	0.0369358	0.0184679	+	0.999829i	$0.494121\pi$
$734$	8.00000	0.295285
$735$	9.00000	0.331970
$736$	−35.0000	−1.29012
$737$	−24.0000	−0.884051
$738$	0	0
$739$	−33.0000	−1.21392	−0.606962	−	0.794731i	$-0.707612\pi$
$739$	−33.0000	−1.21392	−0.606962	+	0.794731i	$0.707612\pi$
$740$	−2.00000	−0.0735215
$741$	10.0000	0.367359
$742$	−12.0000	−0.440534
$743$	39.0000	1.43077	0.715386	−	0.698730i	$-0.246251\pi$
$743$	39.0000	1.43077	0.715386	+	0.698730i	$0.246251\pi$
$744$	27.0000	0.989868
$745$	−4.00000	−0.146549
$746$	37.0000	1.35467
$747$	−24.0000	−0.878114
$748$	6.00000	0.219382
$749$	40.0000	1.46157
$750$	9.00000	0.328634
$751$	10.0000	0.364905	0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000	0.364905	0.182453	+	0.983215i	$0.441596\pi$
$752$	12.0000	0.437595
$753$	12.0000	0.437304
$754$	0	0
$755$	8.00000	0.291150
$756$	20.0000	0.727393
$757$	11.0000	0.399802	0.199901	−	0.979816i	$-0.435938\pi$
$757$	11.0000	0.399802	0.199901	+	0.979816i	$0.435938\pi$
$758$	−17.0000	−0.617468
$759$	14.0000	0.508168
$760$	15.0000	0.544107
$761$	−25.0000	−0.906249	−0.453125	−	0.891447i	$-0.649691\pi$
$761$	−25.0000	−0.906249	−0.453125	+	0.891447i	$0.649691\pi$
$762$	3.00000	0.108679
$763$	76.0000	2.75138
$764$	−15.0000	−0.542681
$765$	6.00000	0.216930
$766$	−13.0000	−0.469709
$767$	8.00000	0.288863
$768$	17.0000	0.613435
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	8.00000	0.288300
$771$	−21.0000	−0.756297
$772$	14.0000	0.503871
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	−14.0000	−0.503220
$775$	36.0000	1.29316
$776$	27.0000	0.969244
$777$	−8.00000	−0.286998
$778$	−12.0000	−0.430221
$779$	0	0
$780$	−2.00000	−0.0716115
$781$	20.0000	0.715656
$782$	−21.0000	−0.750958
$783$	0	0
$784$	−9.00000	−0.321429
$785$	2.00000	0.0713831
$786$	16.0000	0.570701
$787$	−31.0000	−1.10503	−0.552515	−	0.833503i	$-0.686332\pi$
$787$	−31.0000	−1.10503	−0.552515	+	0.833503i	$0.686332\pi$
$788$	−8.00000	−0.284988
$789$	16.0000	0.569615
$790$	−6.00000	−0.213470
$791$	−8.00000	−0.284447
$792$	12.0000	0.426401
$793$	12.0000	0.426132
$794$	−4.00000	−0.141955
$795$	−3.00000	−0.106399
$796$	−2.00000	−0.0708881
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	20.0000	0.707992
$799$	−36.0000	−1.27359
$800$	20.0000	0.707107
$801$	2.00000	0.0706665
$802$	18.0000	0.635602
$803$	−14.0000	−0.494049
$804$	12.0000	0.423207
$805$	28.0000	0.986870
$806$	18.0000	0.634023
$807$	13.0000	0.457622
$808$	0	0
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	1.00000	0.0351364
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	2.00000	0.0701431
$814$	−4.00000	−0.140200
$815$	−25.0000	−0.875712
$816$	3.00000	0.105021
$817$	35.0000	1.22449
$818$	18.0000	0.629355
$819$	16.0000	0.559085
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	−22.0000	−0.767338
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	−27.0000	−0.940590
$825$	−8.00000	−0.278524
$826$	16.0000	0.556711
$827$	17.0000	0.591148	0.295574	−	0.955320i	$-0.404489\pi$
$827$	17.0000	0.591148	0.295574	+	0.955320i	$0.404489\pi$
$828$	14.0000	0.486534
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	12.0000	0.416526
$831$	29.0000	1.00600
$832$	14.0000	0.485363
$833$	27.0000	0.935495
$834$	−8.00000	−0.277017
$835$	12.0000	0.415277
$836$	−10.0000	−0.345857
$837$	−45.0000	−1.55543
$838$	−15.0000	−0.518166
$839$	−51.0000	−1.76072	−0.880358	−	0.474310i	$-0.842698\pi$
$839$	−51.0000	−1.76072	−0.880358	+	0.474310i	$0.842698\pi$
$840$	−12.0000	−0.414039
$841$	−29.0000	−1.00000
$842$	0	0
$843$	30.0000	1.03325
$844$	12.0000	0.413057
$845$	9.00000	0.309609
$846$	−24.0000	−0.825137
$847$	28.0000	0.962091
$848$	3.00000	0.103020
$849$	−10.0000	−0.343199
$850$	12.0000	0.411597
$851$	−14.0000	−0.479914
$852$	−10.0000	−0.342594
$853$	−8.00000	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$853$	−8.00000	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$854$	24.0000	0.821263
$855$	−10.0000	−0.341993
$856$	−30.0000	−1.02538
$857$	4.00000	0.136637	0.0683187	−	0.997664i	$-0.478237\pi$
$857$	4.00000	0.136637	0.0683187	+	0.997664i	$0.478237\pi$
$858$	−4.00000	−0.136558
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	−7.00000	−0.238698
$861$	0	0
$862$	−27.0000	−0.919624
$863$	43.0000	1.46374	0.731869	−	0.681446i	$-0.238649\pi$
$863$	43.0000	1.46374	0.731869	+	0.681446i	$0.238649\pi$
$864$	−25.0000	−0.850517
$865$	1.00000	0.0340010
$866$	0	0
$867$	8.00000	0.271694
$868$	−36.0000	−1.22192
$869$	12.0000	0.407072
$870$	0	0
$871$	24.0000	0.813209
$872$	−57.0000	−1.93026
$873$	−18.0000	−0.609208
$874$	35.0000	1.18389
$875$	−36.0000	−1.21702
$876$	7.00000	0.236508
$877$	−36.0000	−1.21563	−0.607817	−	0.794077i	$-0.707955\pi$
$877$	−36.0000	−1.21563	−0.607817	+	0.794077i	$0.707955\pi$
$878$	11.0000	0.371232
$879$	18.0000	0.607125
$880$	−2.00000	−0.0674200
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	18.0000	0.606092
$883$	−37.0000	−1.24515	−0.622575	−	0.782560i	$-0.713913\pi$
$883$	−37.0000	−1.24515	−0.622575	+	0.782560i	$0.713913\pi$
$884$	−6.00000	−0.201802
$885$	4.00000	0.134459
$886$	14.0000	0.470339
$887$	−45.0000	−1.51095	−0.755476	−	0.655176i	$-0.772594\pi$
$887$	−45.0000	−1.51095	−0.755476	+	0.655176i	$0.772594\pi$
$888$	6.00000	0.201347
$889$	−12.0000	−0.402467
$890$	−1.00000	−0.0335201
$891$	−2.00000	−0.0670025
$892$	16.0000	0.535720
$893$	60.0000	2.00782
$894$	4.00000	0.133780
$895$	−14.0000	−0.467968
$896$	−12.0000	−0.400892
$897$	−14.0000	−0.467446
$898$	9.00000	0.300334
$899$	0	0
$900$	−8.00000	−0.266667
$901$	−9.00000	−0.299833
$902$	0	0
$903$	−28.0000	−0.931782
$904$	6.00000	0.199557
$905$	−22.0000	−0.731305
$906$	−8.00000	−0.265782
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	−24.0000	−0.796468
$909$	0	0
$910$	−8.00000	−0.265197
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	−5.00000	−0.165567
$913$	−24.0000	−0.794284
$914$	2.00000	0.0661541
$915$	6.00000	0.198354
$916$	28.0000	0.925146
$917$	−64.0000	−2.11347
$918$	−15.0000	−0.495074
$919$	−25.0000	−0.824674	−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000	−0.824674	−0.412337	+	0.911031i	$0.635287\pi$
$920$	−21.0000	−0.692349
$921$	2.00000	0.0659022
$922$	30.0000	0.987997
$923$	−20.0000	−0.658308
$924$	8.00000	0.263181
$925$	8.00000	0.263038
$926$	20.0000	0.657241
$927$	18.0000	0.591198
$928$	0	0
$929$	7.00000	0.229663	0.114831	−	0.993385i	$-0.463367\pi$
$929$	7.00000	0.229663	0.114831	+	0.993385i	$0.463367\pi$
$930$	9.00000	0.295122
$931$	−45.0000	−1.47482
$932$	26.0000	0.851658
$933$	−28.0000	−0.916679
$934$	−22.0000	−0.719862
$935$	6.00000	0.196221
$936$	−12.0000	−0.392232
$937$	57.0000	1.86211	0.931054	−	0.364880i	$-0.118890\pi$
$937$	57.0000	1.86211	0.931054	+	0.364880i	$0.118890\pi$
$938$	48.0000	1.56726
$939$	−20.0000	−0.652675
$940$	−12.0000	−0.391397
$941$	−28.0000	−0.912774	−0.456387	−	0.889781i	$-0.650857\pi$
$941$	−28.0000	−0.912774	−0.456387	+	0.889781i	$0.650857\pi$
$942$	−2.00000	−0.0651635
$943$	0	0
$944$	−4.00000	−0.130189
$945$	20.0000	0.650600
$946$	−14.0000	−0.455179
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	−6.00000	−0.194871
$949$	14.0000	0.454459
$950$	−20.0000	−0.648886
$951$	−17.0000	−0.551263
$952$	−36.0000	−1.16677
$953$	−12.0000	−0.388718	−0.194359	−	0.980930i	$-0.562263\pi$
$953$	−12.0000	−0.388718	−0.194359	+	0.980930i	$0.562263\pi$
$954$	−6.00000	−0.194257
$955$	−15.0000	−0.485389
$956$	−5.00000	−0.161712
$957$	0	0
$958$	26.0000	0.840022
$959$	88.0000	2.84167
$960$	7.00000	0.225924
$961$	50.0000	1.61290
$962$	4.00000	0.128965
$963$	20.0000	0.644491
$964$	−10.0000	−0.322078
$965$	14.0000	0.450676
$966$	−28.0000	−0.900885
$967$	−19.0000	−0.610999	−0.305499	−	0.952192i	$-0.598823\pi$
$967$	−19.0000	−0.610999	−0.305499	+	0.952192i	$0.598823\pi$
$968$	−21.0000	−0.674966
$969$	15.0000	0.481869
$970$	9.00000	0.288973
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	16.0000	0.513200
$973$	32.0000	1.02587
$974$	−20.0000	−0.640841
$975$	8.00000	0.256205
$976$	−6.00000	−0.192055
$977$	−9.00000	−0.287936	−0.143968	−	0.989582i	$-0.545986\pi$
$977$	−9.00000	−0.287936	−0.143968	+	0.989582i	$0.545986\pi$
$978$	25.0000	0.799412
$979$	2.00000	0.0639203
$980$	9.00000	0.287494
$981$	38.0000	1.21325
$982$	25.0000	0.797782
$983$	40.0000	1.27580	0.637901	−	0.770118i	$-0.279803\pi$
$983$	40.0000	1.27580	0.637901	+	0.770118i	$0.279803\pi$
$984$	0	0
$985$	−8.00000	−0.254901
$986$	0	0
$987$	−48.0000	−1.52786
$988$	10.0000	0.318142
$989$	−49.0000	−1.55811
$990$	4.00000	0.127128
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	45.0000	1.42875
$993$	−10.0000	−0.317340
$994$	−40.0000	−1.26872
$995$	−2.00000	−0.0634043
$996$	12.0000	0.380235
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	28.0000	0.886325
$999$	−10.0000	−0.316386

Twists

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

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