LMFDB - Embedded newform 118.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [118,2,Mod(1,118)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(118, 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("118.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 118 = 2 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	118.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:	$0.942234743851$
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	$\chi$	$=$	118.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +2.00000 q^{18} +3.00000 q^{19} -3.00000 q^{20} +1.00000 q^{21} +2.00000 q^{22} +1.00000 q^{24} +4.00000 q^{25} +2.00000 q^{26} +5.00000 q^{27} -1.00000 q^{28} -1.00000 q^{29} -3.00000 q^{30} +10.0000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +2.00000 q^{34} +3.00000 q^{35} -2.00000 q^{36} -12.0000 q^{37} -3.00000 q^{38} +2.00000 q^{39} +3.00000 q^{40} +7.00000 q^{41} -1.00000 q^{42} -6.00000 q^{43} -2.00000 q^{44} +6.00000 q^{45} -6.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -4.00000 q^{50} +2.00000 q^{51} -2.00000 q^{52} -11.0000 q^{53} -5.00000 q^{54} +6.00000 q^{55} +1.00000 q^{56} -3.00000 q^{57} +1.00000 q^{58} -1.00000 q^{59} +3.00000 q^{60} -12.0000 q^{61} -10.0000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -2.00000 q^{66} +10.0000 q^{67} -2.00000 q^{68} -3.00000 q^{70} +4.00000 q^{71} +2.00000 q^{72} +12.0000 q^{73} +12.0000 q^{74} -4.00000 q^{75} +3.00000 q^{76} +2.00000 q^{77} -2.00000 q^{78} -15.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -7.00000 q^{82} -14.0000 q^{83} +1.00000 q^{84} +6.00000 q^{85} +6.00000 q^{86} +1.00000 q^{87} +2.00000 q^{88} +4.00000 q^{89} -6.00000 q^{90} +2.00000 q^{91} -10.0000 q^{93} +6.00000 q^{94} -9.00000 q^{95} +1.00000 q^{96} +6.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	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
$10$	3.00000	0.948683
$11$	−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	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	1.00000	0.267261
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	2.00000	0.471405
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	−3.00000	−0.670820
$21$	1.00000	0.218218
$22$	2.00000	0.426401
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	2.00000	0.392232
$27$	5.00000	0.962250
$28$	−1.00000	−0.188982
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	−3.00000	−0.547723
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	2.00000	0.342997
$35$	3.00000	0.507093
$36$	−2.00000	−0.333333
$37$	−12.0000	−1.97279	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	−12.0000	−1.97279	−0.986394	+	0.164399i	$0.947432\pi$
$38$	−3.00000	−0.486664
$39$	2.00000	0.320256
$40$	3.00000	0.474342
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	−1.00000	−0.154303
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	−2.00000	−0.301511
$45$	6.00000	0.894427
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	−4.00000	−0.565685
$51$	2.00000	0.280056
$52$	−2.00000	−0.277350
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	−5.00000	−0.680414
$55$	6.00000	0.809040
$56$	1.00000	0.133631
$57$	−3.00000	−0.397360
$58$	1.00000	0.131306
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	3.00000	0.387298
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	−10.0000	−1.27000
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	−2.00000	−0.246183
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	−3.00000	−0.358569
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	2.00000	0.235702
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	12.0000	1.39497
$75$	−4.00000	−0.461880
$76$	3.00000	0.344124
$77$	2.00000	0.227921
$78$	−2.00000	−0.226455
$79$	−15.0000	−1.68763	−0.843816	−	0.536633i	$-0.819696\pi$
$79$	−15.0000	−1.68763	−0.843816	+	0.536633i	$0.819696\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	−7.00000	−0.773021
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	1.00000	0.109109
$85$	6.00000	0.650791
$86$	6.00000	0.646997
$87$	1.00000	0.107211
$88$	2.00000	0.213201
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	−6.00000	−0.632456
$91$	2.00000	0.209657
$92$	0	0
$93$	−10.0000	−1.03695
$94$	6.00000	0.618853
$95$	−9.00000	−0.923381
$96$	1.00000	0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	6.00000	0.606092
$99$	4.00000	0.402015
$100$	4.00000	0.400000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	−2.00000	−0.198030
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	2.00000	0.196116
$105$	−3.00000	−0.292770
$106$	11.0000	1.06841
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	5.00000	0.481125
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	−6.00000	−0.572078
$111$	12.0000	1.13899
$112$	−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	$0.408930\pi$
$114$	3.00000	0.280976
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	4.00000	0.369800
$118$	1.00000	0.0920575
$119$	2.00000	0.183340
$120$	−3.00000	−0.273861
$121$	−7.00000	−0.636364
$122$	12.0000	1.08643
$123$	−7.00000	−0.631169
$124$	10.0000	0.898027
$125$	3.00000	0.268328
$126$	−2.00000	−0.178174
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	−1.00000	−0.0883883
$129$	6.00000	0.528271
$130$	−6.00000	−0.526235
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	2.00000	0.174078
$133$	−3.00000	−0.260133
$134$	−10.0000	−0.863868
$135$	−15.0000	−1.29099
$136$	2.00000	0.171499
$137$	−5.00000	−0.427179	−0.213589	−	0.976924i	$-0.568515\pi$
$137$	−5.00000	−0.427179	−0.213589	+	0.976924i	$0.568515\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	3.00000	0.253546
$141$	6.00000	0.505291
$142$	−4.00000	−0.335673
$143$	4.00000	0.334497
$144$	−2.00000	−0.166667
$145$	3.00000	0.249136
$146$	−12.0000	−0.993127
$147$	6.00000	0.494872
$148$	−12.0000	−0.986394
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	4.00000	0.326599
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−3.00000	−0.243332
$153$	4.00000	0.323381
$154$	−2.00000	−0.161165
$155$	−30.0000	−2.40966
$156$	2.00000	0.160128
$157$	−12.0000	−0.957704	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−12.0000	−0.957704	−0.478852	+	0.877896i	$0.658947\pi$
$158$	15.0000	1.19334
$159$	11.0000	0.872357
$160$	3.00000	0.237171
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	7.00000	0.546608
$165$	−6.00000	−0.467099
$166$	14.0000	1.08661
$167$	−9.00000	−0.696441	−0.348220	−	0.937413i	$-0.613214\pi$
$167$	−9.00000	−0.696441	−0.348220	+	0.937413i	$0.613214\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	−6.00000	−0.460179
$171$	−6.00000	−0.458831
$172$	−6.00000	−0.457496
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	−1.00000	−0.0758098
$175$	−4.00000	−0.302372
$176$	−2.00000	−0.150756
$177$	1.00000	0.0751646
$178$	−4.00000	−0.299813
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	6.00000	0.447214
$181$	−1.00000	−0.0743294	−0.0371647	−	0.999309i	$-0.511833\pi$
$181$	−1.00000	−0.0743294	−0.0371647	+	0.999309i	$0.511833\pi$
$182$	−2.00000	−0.148250
$183$	12.0000	0.887066
$184$	0	0
$185$	36.0000	2.64677
$186$	10.0000	0.733236
$187$	4.00000	0.292509
$188$	−6.00000	−0.437595
$189$	−5.00000	−0.363696
$190$	9.00000	0.652929
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	−1.00000	−0.0721688
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	0	0
$195$	−6.00000	−0.429669
$196$	−6.00000	−0.428571
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−4.00000	−0.284268
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	−4.00000	−0.282843
$201$	−10.0000	−0.705346
$202$	−14.0000	−0.985037
$203$	1.00000	0.0701862
$204$	2.00000	0.140028
$205$	−21.0000	−1.46670
$206$	−4.00000	−0.278693
$207$	0	0
$208$	−2.00000	−0.138675
$209$	−6.00000	−0.415029
$210$	3.00000	0.207020
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−11.0000	−0.755483
$213$	−4.00000	−0.274075
$214$	−3.00000	−0.205076
$215$	18.0000	1.22759
$216$	−5.00000	−0.340207
$217$	−10.0000	−0.678844
$218$	4.00000	0.270914
$219$	−12.0000	−0.810885
$220$	6.00000	0.404520
$221$	4.00000	0.269069
$222$	−12.0000	−0.805387
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	1.00000	0.0668153
$225$	−8.00000	−0.533333
$226$	−6.00000	−0.399114
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	−3.00000	−0.198680
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	1.00000	0.0656532
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	−4.00000	−0.261488
$235$	18.0000	1.17419
$236$	−1.00000	−0.0650945
$237$	15.0000	0.974355
$238$	−2.00000	−0.129641
$239$	11.0000	0.711531	0.355765	−	0.934575i	$-0.384220\pi$
$239$	11.0000	0.711531	0.355765	+	0.934575i	$0.384220\pi$
$240$	3.00000	0.193649
$241$	9.00000	0.579741	0.289870	−	0.957066i	$-0.406388\pi$
$241$	9.00000	0.579741	0.289870	+	0.957066i	$0.406388\pi$
$242$	7.00000	0.449977
$243$	−16.0000	−1.02640
$244$	−12.0000	−0.768221
$245$	18.0000	1.14998
$246$	7.00000	0.446304
$247$	−6.00000	−0.381771
$248$	−10.0000	−0.635001
$249$	14.0000	0.887214
$250$	−3.00000	−0.189737
$251$	1.00000	0.0631194	0.0315597	−	0.999502i	$-0.489953\pi$
$251$	1.00000	0.0631194	0.0315597	+	0.999502i	$0.489953\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	−1.00000	−0.0627456
$255$	−6.00000	−0.375735
$256$	1.00000	0.0625000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	−6.00000	−0.373544
$259$	12.0000	0.745644
$260$	6.00000	0.372104
$261$	2.00000	0.123797
$262$	−10.0000	−0.617802
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	−2.00000	−0.123091
$265$	33.0000	2.02717
$266$	3.00000	0.183942
$267$	−4.00000	−0.244796
$268$	10.0000	0.610847
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	15.0000	0.912871
$271$	−3.00000	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$272$	−2.00000	−0.121268
$273$	−2.00000	−0.121046
$274$	5.00000	0.302061
$275$	−8.00000	−0.482418
$276$	0	0
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	20.0000	1.19952
$279$	−20.0000	−1.19737
$280$	−3.00000	−0.179284
$281$	11.0000	0.656205	0.328102	−	0.944642i	$-0.393591\pi$
$281$	11.0000	0.656205	0.328102	+	0.944642i	$0.393591\pi$
$282$	−6.00000	−0.357295
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	4.00000	0.237356
$285$	9.00000	0.533114
$286$	−4.00000	−0.236525
$287$	−7.00000	−0.413197
$288$	2.00000	0.117851
$289$	−13.0000	−0.764706
$290$	−3.00000	−0.176166
$291$	0	0
$292$	12.0000	0.702247
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	−6.00000	−0.349927
$295$	3.00000	0.174667
$296$	12.0000	0.697486
$297$	−10.0000	−0.580259
$298$	18.0000	1.04271
$299$	0	0
$300$	−4.00000	−0.230940
$301$	6.00000	0.345834
$302$	−2.00000	−0.115087
$303$	−14.0000	−0.804279
$304$	3.00000	0.172062
$305$	36.0000	2.06135
$306$	−4.00000	−0.228665
$307$	5.00000	0.285365	0.142683	−	0.989769i	$-0.454427\pi$
$307$	5.00000	0.285365	0.142683	+	0.989769i	$0.454427\pi$
$308$	2.00000	0.113961
$309$	−4.00000	−0.227552
$310$	30.0000	1.70389
$311$	−5.00000	−0.283524	−0.141762	−	0.989901i	$-0.545277\pi$
$311$	−5.00000	−0.283524	−0.141762	+	0.989901i	$0.545277\pi$
$312$	−2.00000	−0.113228
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	12.0000	0.677199
$315$	−6.00000	−0.338062
$316$	−15.0000	−0.843816
$317$	−34.0000	−1.90963	−0.954815	−	0.297200i	$-0.903947\pi$
$317$	−34.0000	−1.90963	−0.954815	+	0.297200i	$0.903947\pi$
$318$	−11.0000	−0.616849
$319$	2.00000	0.111979
$320$	−3.00000	−0.167705
$321$	−3.00000	−0.167444
$322$	0	0
$323$	−6.00000	−0.333849
$324$	1.00000	0.0555556
$325$	−8.00000	−0.443760
$326$	−20.0000	−1.10770
$327$	4.00000	0.221201
$328$	−7.00000	−0.386510
$329$	6.00000	0.330791
$330$	6.00000	0.330289
$331$	−29.0000	−1.59398	−0.796992	−	0.603990i	$-0.793577\pi$
$331$	−29.0000	−1.59398	−0.796992	+	0.603990i	$0.793577\pi$
$332$	−14.0000	−0.768350
$333$	24.0000	1.31519
$334$	9.00000	0.492458
$335$	−30.0000	−1.63908
$336$	1.00000	0.0545545
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	9.00000	0.489535
$339$	−6.00000	−0.325875
$340$	6.00000	0.325396
$341$	−20.0000	−1.08306
$342$	6.00000	0.324443
$343$	13.0000	0.701934
$344$	6.00000	0.323498
$345$	0	0
$346$	−12.0000	−0.645124
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	1.00000	0.0536056
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	4.00000	0.213809
$351$	−10.0000	−0.533761
$352$	2.00000	0.106600
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	−1.00000	−0.0531494
$355$	−12.0000	−0.636894
$356$	4.00000	0.212000
$357$	−2.00000	−0.105851
$358$	4.00000	0.211407
$359$	29.0000	1.53056	0.765281	−	0.643697i	$-0.222600\pi$
$359$	29.0000	1.53056	0.765281	+	0.643697i	$0.222600\pi$
$360$	−6.00000	−0.316228
$361$	−10.0000	−0.526316
$362$	1.00000	0.0525588
$363$	7.00000	0.367405
$364$	2.00000	0.104828
$365$	−36.0000	−1.88433
$366$	−12.0000	−0.627250
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	0	0
$369$	−14.0000	−0.728811
$370$	−36.0000	−1.87155
$371$	11.0000	0.571092
$372$	−10.0000	−0.518476
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	−4.00000	−0.206835
$375$	−3.00000	−0.154919
$376$	6.00000	0.309426
$377$	2.00000	0.103005
$378$	5.00000	0.257172
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	−9.00000	−0.461690
$381$	−1.00000	−0.0512316
$382$	−6.00000	−0.306987
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	1.00000	0.0510310
$385$	−6.00000	−0.305788
$386$	5.00000	0.254493
$387$	12.0000	0.609994
$388$	0	0
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	6.00000	0.303822
$391$	0	0
$392$	6.00000	0.303046
$393$	−10.0000	−0.504433
$394$	−18.0000	−0.906827
$395$	45.0000	2.26420
$396$	4.00000	0.201008
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	25.0000	1.25314
$399$	3.00000	0.150188
$400$	4.00000	0.200000
$401$	−32.0000	−1.59800	−0.799002	−	0.601329i	$-0.794638\pi$
$401$	−32.0000	−1.59800	−0.799002	+	0.601329i	$0.794638\pi$
$402$	10.0000	0.498755
$403$	−20.0000	−0.996271
$404$	14.0000	0.696526
$405$	−3.00000	−0.149071
$406$	−1.00000	−0.0496292
$407$	24.0000	1.18964
$408$	−2.00000	−0.0990148
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	21.0000	1.03712
$411$	5.00000	0.246632
$412$	4.00000	0.197066
$413$	1.00000	0.0492068
$414$	0	0
$415$	42.0000	2.06170
$416$	2.00000	0.0980581
$417$	20.0000	0.979404
$418$	6.00000	0.293470
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	−3.00000	−0.146385
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	−12.0000	−0.584151
$423$	12.0000	0.583460
$424$	11.0000	0.534207
$425$	−8.00000	−0.388057
$426$	4.00000	0.193801
$427$	12.0000	0.580721
$428$	3.00000	0.145010
$429$	−4.00000	−0.193122
$430$	−18.0000	−0.868037
$431$	−34.0000	−1.63772	−0.818861	−	0.573992i	$-0.805394\pi$
$431$	−34.0000	−1.63772	−0.818861	+	0.573992i	$0.805394\pi$
$432$	5.00000	0.240563
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	10.0000	0.480015
$435$	−3.00000	−0.143839
$436$	−4.00000	−0.191565
$437$	0	0
$438$	12.0000	0.573382
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	−6.00000	−0.286039
$441$	12.0000	0.571429
$442$	−4.00000	−0.190261
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	12.0000	0.569495
$445$	−12.0000	−0.568855
$446$	24.0000	1.13643
$447$	18.0000	0.851371
$448$	−1.00000	−0.0472456
$449$	5.00000	0.235965	0.117982	−	0.993016i	$-0.462357\pi$
$449$	5.00000	0.235965	0.117982	+	0.993016i	$0.462357\pi$
$450$	8.00000	0.377124
$451$	−14.0000	−0.659234
$452$	6.00000	0.282216
$453$	−2.00000	−0.0939682
$454$	−28.0000	−1.31411
$455$	−6.00000	−0.281284
$456$	3.00000	0.140488
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−26.0000	−1.21490
$459$	−10.0000	−0.466760
$460$	0	0
$461$	−26.0000	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$462$	2.00000	0.0930484
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	−1.00000	−0.0464238
$465$	30.0000	1.39122
$466$	−14.0000	−0.648537
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	4.00000	0.184900
$469$	−10.0000	−0.461757
$470$	−18.0000	−0.830278
$471$	12.0000	0.552931
$472$	1.00000	0.0460287
$473$	12.0000	0.551761
$474$	−15.0000	−0.688973
$475$	12.0000	0.550598
$476$	2.00000	0.0916698
$477$	22.0000	1.00731
$478$	−11.0000	−0.503128
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	−3.00000	−0.136931
$481$	24.0000	1.09431
$482$	−9.00000	−0.409939
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	16.0000	0.725775
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	12.0000	0.543214
$489$	−20.0000	−0.904431
$490$	−18.0000	−0.813157
$491$	−27.0000	−1.21849	−0.609246	−	0.792981i	$-0.708528\pi$
$491$	−27.0000	−1.21849	−0.609246	+	0.792981i	$0.708528\pi$
$492$	−7.00000	−0.315584
$493$	2.00000	0.0900755
$494$	6.00000	0.269953
$495$	−12.0000	−0.539360
$496$	10.0000	0.449013
$497$	−4.00000	−0.179425
$498$	−14.0000	−0.627355
$499$	39.0000	1.74588	0.872940	−	0.487828i	$-0.162211\pi$
$499$	39.0000	1.74588	0.872940	+	0.487828i	$0.162211\pi$
$500$	3.00000	0.134164
$501$	9.00000	0.402090
$502$	−1.00000	−0.0446322
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	−2.00000	−0.0890871
$505$	−42.0000	−1.86898
$506$	0	0
$507$	9.00000	0.399704
$508$	1.00000	0.0443678
$509$	8.00000	0.354594	0.177297	−	0.984157i	$-0.443265\pi$
$509$	8.00000	0.354594	0.177297	+	0.984157i	$0.443265\pi$
$510$	6.00000	0.265684
$511$	−12.0000	−0.530849
$512$	−1.00000	−0.0441942
$513$	15.0000	0.662266
$514$	3.00000	0.132324
$515$	−12.0000	−0.528783
$516$	6.00000	0.264135
$517$	12.0000	0.527759
$518$	−12.0000	−0.527250
$519$	−12.0000	−0.526742
$520$	−6.00000	−0.263117
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	−2.00000	−0.0875376
$523$	−11.0000	−0.480996	−0.240498	−	0.970650i	$-0.577311\pi$
$523$	−11.0000	−0.480996	−0.240498	+	0.970650i	$0.577311\pi$
$524$	10.0000	0.436852
$525$	4.00000	0.174574
$526$	9.00000	0.392419
$527$	−20.0000	−0.871214
$528$	2.00000	0.0870388
$529$	−23.0000	−1.00000
$530$	−33.0000	−1.43343
$531$	2.00000	0.0867926
$532$	−3.00000	−0.130066
$533$	−14.0000	−0.606407
$534$	4.00000	0.173097
$535$	−9.00000	−0.389104
$536$	−10.0000	−0.431934
$537$	4.00000	0.172613
$538$	18.0000	0.776035
$539$	12.0000	0.516877
$540$	−15.0000	−0.645497
$541$	−20.0000	−0.859867	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000	−0.859867	−0.429934	+	0.902861i	$0.641463\pi$
$542$	3.00000	0.128861
$543$	1.00000	0.0429141
$544$	2.00000	0.0857493
$545$	12.0000	0.514024
$546$	2.00000	0.0855921
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−5.00000	−0.213589
$549$	24.0000	1.02430
$550$	8.00000	0.341121
$551$	−3.00000	−0.127804
$552$	0	0
$553$	15.0000	0.637865
$554$	−1.00000	−0.0424859
$555$	−36.0000	−1.52811
$556$	−20.0000	−0.848189
$557$	45.0000	1.90671	0.953356	−	0.301849i	$-0.0976040\pi$
$557$	45.0000	1.90671	0.953356	+	0.301849i	$0.0976040\pi$
$558$	20.0000	0.846668
$559$	12.0000	0.507546
$560$	3.00000	0.126773
$561$	−4.00000	−0.168880
$562$	−11.0000	−0.464007
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	6.00000	0.252646
$565$	−18.0000	−0.757266
$566$	−16.0000	−0.672530
$567$	−1.00000	−0.0419961
$568$	−4.00000	−0.167836
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	−9.00000	−0.376969
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	4.00000	0.167248
$573$	−6.00000	−0.250654
$574$	7.00000	0.292174
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	−23.0000	−0.957503	−0.478751	−	0.877951i	$-0.658910\pi$
$577$	−23.0000	−0.957503	−0.478751	+	0.877951i	$0.658910\pi$
$578$	13.0000	0.540729
$579$	5.00000	0.207793
$580$	3.00000	0.124568
$581$	14.0000	0.580818
$582$	0	0
$583$	22.0000	0.911147
$584$	−12.0000	−0.496564
$585$	−12.0000	−0.496139
$586$	9.00000	0.371787
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	6.00000	0.247436
$589$	30.0000	1.23613
$590$	−3.00000	−0.123508
$591$	−18.0000	−0.740421
$592$	−12.0000	−0.493197
$593$	−35.0000	−1.43728	−0.718639	−	0.695383i	$-0.755235\pi$
$593$	−35.0000	−1.43728	−0.718639	+	0.695383i	$0.755235\pi$
$594$	10.0000	0.410305
$595$	−6.00000	−0.245976
$596$	−18.0000	−0.737309
$597$	25.0000	1.02318
$598$	0	0
$599$	−27.0000	−1.10319	−0.551595	−	0.834112i	$-0.685981\pi$
$599$	−27.0000	−1.10319	−0.551595	+	0.834112i	$0.685981\pi$
$600$	4.00000	0.163299
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	−6.00000	−0.244542
$603$	−20.0000	−0.814463
$604$	2.00000	0.0813788
$605$	21.0000	0.853771
$606$	14.0000	0.568711
$607$	7.00000	0.284121	0.142061	−	0.989858i	$-0.454627\pi$
$607$	7.00000	0.284121	0.142061	+	0.989858i	$0.454627\pi$
$608$	−3.00000	−0.121666
$609$	−1.00000	−0.0405220
$610$	−36.0000	−1.45760
$611$	12.0000	0.485468
$612$	4.00000	0.161690
$613$	4.00000	0.161558	0.0807792	−	0.996732i	$-0.474259\pi$
$613$	4.00000	0.161558	0.0807792	+	0.996732i	$0.474259\pi$
$614$	−5.00000	−0.201784
$615$	21.0000	0.846802
$616$	−2.00000	−0.0805823
$617$	−39.0000	−1.57008	−0.785040	−	0.619445i	$-0.787358\pi$
$617$	−39.0000	−1.57008	−0.785040	+	0.619445i	$0.787358\pi$
$618$	4.00000	0.160904
$619$	−3.00000	−0.120580	−0.0602901	−	0.998181i	$-0.519203\pi$
$619$	−3.00000	−0.120580	−0.0602901	+	0.998181i	$0.519203\pi$
$620$	−30.0000	−1.20483
$621$	0	0
$622$	5.00000	0.200482
$623$	−4.00000	−0.160257
$624$	2.00000	0.0800641
$625$	−29.0000	−1.16000
$626$	4.00000	0.159872
$627$	6.00000	0.239617
$628$	−12.0000	−0.478852
$629$	24.0000	0.956943
$630$	6.00000	0.239046
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	15.0000	0.596668
$633$	−12.0000	−0.476957
$634$	34.0000	1.35031
$635$	−3.00000	−0.119051
$636$	11.0000	0.436178
$637$	12.0000	0.475457
$638$	−2.00000	−0.0791808
$639$	−8.00000	−0.316475
$640$	3.00000	0.118585
$641$	−38.0000	−1.50091	−0.750455	−	0.660922i	$-0.770166\pi$
$641$	−38.0000	−1.50091	−0.750455	+	0.660922i	$0.770166\pi$
$642$	3.00000	0.118401
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	0	0
$645$	−18.0000	−0.708749
$646$	6.00000	0.236067
$647$	27.0000	1.06148	0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000	1.06148	0.530740	+	0.847535i	$0.321914\pi$
$648$	−1.00000	−0.0392837
$649$	2.00000	0.0785069
$650$	8.00000	0.313786
$651$	10.0000	0.391931
$652$	20.0000	0.783260
$653$	21.0000	0.821794	0.410897	−	0.911682i	$-0.365216\pi$
$653$	21.0000	0.821794	0.410897	+	0.911682i	$0.365216\pi$
$654$	−4.00000	−0.156412
$655$	−30.0000	−1.17220
$656$	7.00000	0.273304
$657$	−24.0000	−0.936329
$658$	−6.00000	−0.233904
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	−6.00000	−0.233550
$661$	−19.0000	−0.739014	−0.369507	−	0.929228i	$-0.620473\pi$
$661$	−19.0000	−0.739014	−0.369507	+	0.929228i	$0.620473\pi$
$662$	29.0000	1.12712
$663$	−4.00000	−0.155347
$664$	14.0000	0.543305
$665$	9.00000	0.349005
$666$	−24.0000	−0.929981
$667$	0	0
$668$	−9.00000	−0.348220
$669$	24.0000	0.927894
$670$	30.0000	1.15900
$671$	24.0000	0.926510
$672$	−1.00000	−0.0385758
$673$	40.0000	1.54189	0.770943	−	0.636904i	$-0.219785\pi$
$673$	40.0000	1.54189	0.770943	+	0.636904i	$0.219785\pi$
$674$	−20.0000	−0.770371
$675$	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	$0.387574\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	−6.00000	−0.230089
$681$	−28.0000	−1.07296
$682$	20.0000	0.765840
$683$	−42.0000	−1.60709	−0.803543	−	0.595247i	$-0.797054\pi$
$683$	−42.0000	−1.60709	−0.803543	+	0.595247i	$0.797054\pi$
$684$	−6.00000	−0.229416
$685$	15.0000	0.573121
$686$	−13.0000	−0.496342
$687$	−26.0000	−0.991962
$688$	−6.00000	−0.228748
$689$	22.0000	0.838133
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	12.0000	0.456172
$693$	−4.00000	−0.151947
$694$	18.0000	0.683271
$695$	60.0000	2.27593
$696$	−1.00000	−0.0379049
$697$	−14.0000	−0.530288
$698$	−2.00000	−0.0757011
$699$	−14.0000	−0.529529
$700$	−4.00000	−0.151186
$701$	4.00000	0.151078	0.0755390	−	0.997143i	$-0.475932\pi$
$701$	4.00000	0.151078	0.0755390	+	0.997143i	$0.475932\pi$
$702$	10.0000	0.377426
$703$	−36.0000	−1.35777
$704$	−2.00000	−0.0753778
$705$	−18.0000	−0.677919
$706$	24.0000	0.903252
$707$	−14.0000	−0.526524
$708$	1.00000	0.0375823
$709$	−17.0000	−0.638448	−0.319224	−	0.947679i	$-0.603422\pi$
$709$	−17.0000	−0.638448	−0.319224	+	0.947679i	$0.603422\pi$
$710$	12.0000	0.450352
$711$	30.0000	1.12509
$712$	−4.00000	−0.149906
$713$	0	0
$714$	2.00000	0.0748481
$715$	−12.0000	−0.448775
$716$	−4.00000	−0.149487
$717$	−11.0000	−0.410803
$718$	−29.0000	−1.08227
$719$	−52.0000	−1.93927	−0.969636	−	0.244551i	$-0.921359\pi$
$719$	−52.0000	−1.93927	−0.969636	+	0.244551i	$0.921359\pi$
$720$	6.00000	0.223607
$721$	−4.00000	−0.148968
$722$	10.0000	0.372161
$723$	−9.00000	−0.334714
$724$	−1.00000	−0.0371647
$725$	−4.00000	−0.148556
$726$	−7.00000	−0.259794
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	−2.00000	−0.0741249
$729$	13.0000	0.481481
$730$	36.0000	1.33242
$731$	12.0000	0.443836
$732$	12.0000	0.443533
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	−10.0000	−0.369107
$735$	−18.0000	−0.663940
$736$	0	0
$737$	−20.0000	−0.736709
$738$	14.0000	0.515347
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	36.0000	1.32339
$741$	6.00000	0.220416
$742$	−11.0000	−0.403823
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	10.0000	0.366618
$745$	54.0000	1.97841
$746$	−6.00000	−0.219676
$747$	28.0000	1.02447
$748$	4.00000	0.146254
$749$	−3.00000	−0.109618
$750$	3.00000	0.109545
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	−6.00000	−0.218797
$753$	−1.00000	−0.0364420
$754$	−2.00000	−0.0728357
$755$	−6.00000	−0.218362
$756$	−5.00000	−0.181848
$757$	−13.0000	−0.472493	−0.236247	−	0.971693i	$-0.575917\pi$
$757$	−13.0000	−0.472493	−0.236247	+	0.971693i	$0.575917\pi$
$758$	−5.00000	−0.181608
$759$	0	0
$760$	9.00000	0.326464
$761$	−35.0000	−1.26875	−0.634375	−	0.773026i	$-0.718742\pi$
$761$	−35.0000	−1.26875	−0.634375	+	0.773026i	$0.718742\pi$
$762$	1.00000	0.0362262
$763$	4.00000	0.144810
$764$	6.00000	0.217072
$765$	−12.0000	−0.433861
$766$	24.0000	0.867155
$767$	2.00000	0.0722158
$768$	−1.00000	−0.0360844
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	6.00000	0.216225
$771$	3.00000	0.108042
$772$	−5.00000	−0.179954
$773$	4.00000	0.143870	0.0719350	−	0.997409i	$-0.477083\pi$
$773$	4.00000	0.143870	0.0719350	+	0.997409i	$0.477083\pi$
$774$	−12.0000	−0.431331
$775$	40.0000	1.43684
$776$	0	0
$777$	−12.0000	−0.430498
$778$	10.0000	0.358517
$779$	21.0000	0.752403
$780$	−6.00000	−0.214834
$781$	−8.00000	−0.286263
$782$	0	0
$783$	−5.00000	−0.178685
$784$	−6.00000	−0.214286
$785$	36.0000	1.28490
$786$	10.0000	0.356688
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	18.0000	0.641223
$789$	9.00000	0.320408
$790$	−45.0000	−1.60103
$791$	−6.00000	−0.213335
$792$	−4.00000	−0.142134
$793$	24.0000	0.852265
$794$	22.0000	0.780751
$795$	−33.0000	−1.17039
$796$	−25.0000	−0.886102
$797$	48.0000	1.70025	0.850124	−	0.526583i	$-0.176527\pi$
$797$	48.0000	1.70025	0.850124	+	0.526583i	$0.176527\pi$
$798$	−3.00000	−0.106199
$799$	12.0000	0.424529
$800$	−4.00000	−0.141421
$801$	−8.00000	−0.282666
$802$	32.0000	1.12996
$803$	−24.0000	−0.846942
$804$	−10.0000	−0.352673
$805$	0	0
$806$	20.0000	0.704470
$807$	18.0000	0.633630
$808$	−14.0000	−0.492518
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	3.00000	0.105409
$811$	−18.0000	−0.632065	−0.316033	−	0.948748i	$-0.602351\pi$
$811$	−18.0000	−0.632065	−0.316033	+	0.948748i	$0.602351\pi$
$812$	1.00000	0.0350931
$813$	3.00000	0.105215
$814$	−24.0000	−0.841200
$815$	−60.0000	−2.10171
$816$	2.00000	0.0700140
$817$	−18.0000	−0.629740
$818$	−8.00000	−0.279713
$819$	−4.00000	−0.139771
$820$	−21.0000	−0.733352
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	−5.00000	−0.174395
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	−4.00000	−0.139347
$825$	8.00000	0.278524
$826$	−1.00000	−0.0347945
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	−49.0000	−1.70184	−0.850920	−	0.525295i	$-0.823955\pi$
$829$	−49.0000	−1.70184	−0.850920	+	0.525295i	$0.823955\pi$
$830$	−42.0000	−1.45784
$831$	−1.00000	−0.0346896
$832$	−2.00000	−0.0693375
$833$	12.0000	0.415775
$834$	−20.0000	−0.692543
$835$	27.0000	0.934374
$836$	−6.00000	−0.207514
$837$	50.0000	1.72825
$838$	−28.0000	−0.967244
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	3.00000	0.103510
$841$	−28.0000	−0.965517
$842$	8.00000	0.275698
$843$	−11.0000	−0.378860
$844$	12.0000	0.413057
$845$	27.0000	0.928828
$846$	−12.0000	−0.412568
$847$	7.00000	0.240523
$848$	−11.0000	−0.377742
$849$	−16.0000	−0.549119
$850$	8.00000	0.274398
$851$	0	0
$852$	−4.00000	−0.137038
$853$	−1.00000	−0.0342393	−0.0171197	−	0.999853i	$-0.505450\pi$
$853$	−1.00000	−0.0342393	−0.0171197	+	0.999853i	$0.505450\pi$
$854$	−12.0000	−0.410632
$855$	18.0000	0.615587
$856$	−3.00000	−0.102538
$857$	20.0000	0.683187	0.341593	−	0.939848i	$-0.389033\pi$
$857$	20.0000	0.683187	0.341593	+	0.939848i	$0.389033\pi$
$858$	4.00000	0.136558
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	18.0000	0.613795
$861$	7.00000	0.238559
$862$	34.0000	1.15804
$863$	−28.0000	−0.953131	−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000	−0.953131	−0.476566	+	0.879139i	$0.658119\pi$
$864$	−5.00000	−0.170103
$865$	−36.0000	−1.22404
$866$	−11.0000	−0.373795
$867$	13.0000	0.441503
$868$	−10.0000	−0.339422
$869$	30.0000	1.01768
$870$	3.00000	0.101710
$871$	−20.0000	−0.677674
$872$	4.00000	0.135457
$873$	0	0
$874$	0	0
$875$	−3.00000	−0.101419
$876$	−12.0000	−0.405442
$877$	31.0000	1.04680	0.523398	−	0.852088i	$-0.324664\pi$
$877$	31.0000	1.04680	0.523398	+	0.852088i	$0.324664\pi$
$878$	−28.0000	−0.944954
$879$	9.00000	0.303562
$880$	6.00000	0.202260
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	−12.0000	−0.404061
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	4.00000	0.134535
$885$	−3.00000	−0.100844
$886$	−8.00000	−0.268765
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	−12.0000	−0.402694
$889$	−1.00000	−0.0335389
$890$	12.0000	0.402241
$891$	−2.00000	−0.0670025
$892$	−24.0000	−0.803579
$893$	−18.0000	−0.602347
$894$	−18.0000	−0.602010
$895$	12.0000	0.401116
$896$	1.00000	0.0334077
$897$	0	0
$898$	−5.00000	−0.166852
$899$	−10.0000	−0.333519
$900$	−8.00000	−0.266667
$901$	22.0000	0.732926
$902$	14.0000	0.466149
$903$	−6.00000	−0.199667
$904$	−6.00000	−0.199557
$905$	3.00000	0.0997234
$906$	2.00000	0.0664455
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	28.0000	0.929213
$909$	−28.0000	−0.928701
$910$	6.00000	0.198898
$911$	31.0000	1.02708	0.513538	−	0.858067i	$-0.328335\pi$
$911$	31.0000	1.02708	0.513538	+	0.858067i	$0.328335\pi$
$912$	−3.00000	−0.0993399
$913$	28.0000	0.926665
$914$	10.0000	0.330771
$915$	−36.0000	−1.19012
$916$	26.0000	0.859064
$917$	−10.0000	−0.330229
$918$	10.0000	0.330049
$919$	−22.0000	−0.725713	−0.362857	−	0.931845i	$-0.618198\pi$
$919$	−22.0000	−0.725713	−0.362857	+	0.931845i	$0.618198\pi$
$920$	0	0
$921$	−5.00000	−0.164756
$922$	26.0000	0.856264
$923$	−8.00000	−0.263323
$924$	−2.00000	−0.0657952
$925$	−48.0000	−1.57823
$926$	−32.0000	−1.05159
$927$	−8.00000	−0.262754
$928$	1.00000	0.0328266
$929$	46.0000	1.50921	0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000	1.50921	0.754606	+	0.656179i	$0.227828\pi$
$930$	−30.0000	−0.983739
$931$	−18.0000	−0.589926
$932$	14.0000	0.458585
$933$	5.00000	0.163693
$934$	−4.00000	−0.130884
$935$	−12.0000	−0.392442
$936$	−4.00000	−0.130744
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	10.0000	0.326512
$939$	4.00000	0.130535
$940$	18.0000	0.587095
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	−12.0000	−0.390981
$943$	0	0
$944$	−1.00000	−0.0325472
$945$	15.0000	0.487950
$946$	−12.0000	−0.390154
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	15.0000	0.487177
$949$	−24.0000	−0.779073
$950$	−12.0000	−0.389331
$951$	34.0000	1.10253
$952$	−2.00000	−0.0648204
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	−22.0000	−0.712276
$955$	−18.0000	−0.582466
$956$	11.0000	0.355765
$957$	−2.00000	−0.0646508
$958$	−8.00000	−0.258468
$959$	5.00000	0.161458
$960$	3.00000	0.0968246
$961$	69.0000	2.22581
$962$	−24.0000	−0.773791
$963$	−6.00000	−0.193347
$964$	9.00000	0.289870
$965$	15.0000	0.482867
$966$	0	0
$967$	−26.0000	−0.836104	−0.418052	−	0.908423i	$-0.637287\pi$
$967$	−26.0000	−0.836104	−0.418052	+	0.908423i	$0.637287\pi$
$968$	7.00000	0.224989
$969$	6.00000	0.192748
$970$	0	0
$971$	−27.0000	−0.866471	−0.433236	−	0.901281i	$-0.642628\pi$
$971$	−27.0000	−0.866471	−0.433236	+	0.901281i	$0.642628\pi$
$972$	−16.0000	−0.513200
$973$	20.0000	0.641171
$974$	−11.0000	−0.352463
$975$	8.00000	0.256205
$976$	−12.0000	−0.384111
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	20.0000	0.639529
$979$	−8.00000	−0.255681
$980$	18.0000	0.574989
$981$	8.00000	0.255420
$982$	27.0000	0.861605
$983$	10.0000	0.318950	0.159475	−	0.987202i	$-0.449020\pi$
$983$	10.0000	0.318950	0.159475	+	0.987202i	$0.449020\pi$
$984$	7.00000	0.223152
$985$	−54.0000	−1.72058
$986$	−2.00000	−0.0636930
$987$	−6.00000	−0.190982
$988$	−6.00000	−0.190885
$989$	0	0
$990$	12.0000	0.381385
$991$	−26.0000	−0.825917	−0.412959	−	0.910750i	$-0.635505\pi$
$991$	−26.0000	−0.825917	−0.412959	+	0.910750i	$0.635505\pi$
$992$	−10.0000	−0.317500
$993$	29.0000	0.920287
$994$	4.00000	0.126872
$995$	75.0000	2.37766
$996$	14.0000	0.443607
$997$	−29.0000	−0.918439	−0.459220	−	0.888323i	$-0.651871\pi$
$997$	−29.0000	−0.918439	−0.459220	+	0.888323i	$0.651871\pi$
$998$	−39.0000	−1.23452
$999$	−60.0000	−1.89832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	118.2.a.a.1.1	✓	1
3.2	odd	2		1062.2.a.l.1.1		1
4.3	odd	2		944.2.a.f.1.1		1
5.2	odd	4		2950.2.c.j.1299.1		2
5.3	odd	4		2950.2.c.j.1299.2		2
5.4	even	2		2950.2.a.s.1.1		1
7.6	odd	2		5782.2.a.d.1.1		1
8.3	odd	2		3776.2.a.l.1.1		1
8.5	even	2		3776.2.a.t.1.1		1
12.11	even	2		8496.2.a.v.1.1		1
59.58	odd	2		6962.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
118.2.a.a.1.1	✓	1	1.1	even	1	trivial
944.2.a.f.1.1		1	4.3	odd	2
1062.2.a.l.1.1		1	3.2	odd	2
2950.2.a.s.1.1		1	5.4	even	2
2950.2.c.j.1299.1		2	5.2	odd	4
2950.2.c.j.1299.2		2	5.3	odd	4
3776.2.a.l.1.1		1	8.3	odd	2
3776.2.a.t.1.1		1	8.5	even	2
5782.2.a.d.1.1		1	7.6	odd	2
6962.2.a.i.1.1		1	59.58	odd	2
8496.2.a.v.1.1		1	12.11	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:	\( N \)	\(=\)	\( 118 = 2 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	118.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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