LMFDB - Embedded newform 61.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [61,2,Mod(1,61)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(61, 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("61.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	61.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:	$0.487087452330$
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	$\chi$	$=$	61.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} -3.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} -5.00000 q^{11} +2.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +6.00000 q^{15} -1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +3.00000 q^{20} -2.00000 q^{21} +5.00000 q^{22} -9.00000 q^{23} -6.00000 q^{24} +4.00000 q^{25} -1.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -6.00000 q^{30} -5.00000 q^{32} +10.0000 q^{33} -4.00000 q^{34} -3.00000 q^{35} -1.00000 q^{36} +8.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} -9.00000 q^{40} +5.00000 q^{41} +2.00000 q^{42} -8.00000 q^{43} +5.00000 q^{44} -3.00000 q^{45} +9.00000 q^{46} +4.00000 q^{47} +2.00000 q^{48} -6.00000 q^{49} -4.00000 q^{50} -8.00000 q^{51} -1.00000 q^{52} +6.00000 q^{53} -4.00000 q^{54} +15.0000 q^{55} +3.00000 q^{56} +8.00000 q^{57} +6.00000 q^{58} +9.00000 q^{59} -6.00000 q^{60} -1.00000 q^{61} +1.00000 q^{63} +7.00000 q^{64} -3.00000 q^{65} -10.0000 q^{66} -7.00000 q^{67} -4.00000 q^{68} +18.0000 q^{69} +3.00000 q^{70} -8.00000 q^{71} +3.00000 q^{72} -11.0000 q^{73} -8.00000 q^{74} -8.00000 q^{75} +4.00000 q^{76} -5.00000 q^{77} +2.00000 q^{78} +3.00000 q^{79} +3.00000 q^{80} -11.0000 q^{81} -5.00000 q^{82} +4.00000 q^{83} +2.00000 q^{84} -12.0000 q^{85} +8.00000 q^{86} +12.0000 q^{87} -15.0000 q^{88} -4.00000 q^{89} +3.00000 q^{90} +1.00000 q^{91} +9.00000 q^{92} -4.00000 q^{94} +12.0000 q^{95} +10.0000 q^{96} -14.0000 q^{97} +6.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	−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	$0.734058\pi$
$6$	2.00000	0.816497
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	3.00000	0.948683
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	2.00000	0.577350
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−1.00000	−0.267261
$15$	6.00000	1.54919
$16$	−1.00000	−0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	3.00000	0.670820
$21$	−2.00000	−0.436436
$22$	5.00000	1.06600
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	−6.00000	−1.22474
$25$	4.00000	0.800000
$26$	−1.00000	−0.196116
$27$	4.00000	0.769800
$28$	−1.00000	−0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−6.00000	−1.09545
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−5.00000	−0.883883
$33$	10.0000	1.74078
$34$	−4.00000	−0.685994
$35$	−3.00000	−0.507093
$36$	−1.00000	−0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	4.00000	0.648886
$39$	−2.00000	−0.320256
$40$	−9.00000	−1.42302
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	2.00000	0.308607
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	5.00000	0.753778
$45$	−3.00000	−0.447214
$46$	9.00000	1.32698
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	2.00000	0.288675
$49$	−6.00000	−0.857143
$50$	−4.00000	−0.565685
$51$	−8.00000	−1.12022
$52$	−1.00000	−0.138675
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−4.00000	−0.544331
$55$	15.0000	2.02260
$56$	3.00000	0.400892
$57$	8.00000	1.05963
$58$	6.00000	0.787839
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	−6.00000	−0.774597
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0	0
$63$	1.00000	0.125988
$64$	7.00000	0.875000
$65$	−3.00000	−0.372104
$66$	−10.0000	−1.23091
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	−4.00000	−0.485071
$69$	18.0000	2.16695
$70$	3.00000	0.358569
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	3.00000	0.353553
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	−8.00000	−0.929981
$75$	−8.00000	−0.923760
$76$	4.00000	0.458831
$77$	−5.00000	−0.569803
$78$	2.00000	0.226455
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	3.00000	0.335410
$81$	−11.0000	−1.22222
$82$	−5.00000	−0.552158
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	2.00000	0.218218
$85$	−12.0000	−1.30158
$86$	8.00000	0.862662
$87$	12.0000	1.28654
$88$	−15.0000	−1.59901
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	3.00000	0.316228
$91$	1.00000	0.104828
$92$	9.00000	0.938315
$93$	0	0
$94$	−4.00000	−0.412568
$95$	12.0000	1.23117
$96$	10.0000	1.02062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	6.00000	0.606092
$99$	−5.00000	−0.502519
$100$	−4.00000	−0.400000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	8.00000	0.792118
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	3.00000	0.294174
$105$	6.00000	0.585540
$106$	−6.00000	−0.582772
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	−4.00000	−0.384900
$109$	−17.0000	−1.62830	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−17.0000	−1.62830	−0.814152	+	0.580651i	$0.802798\pi$
$110$	−15.0000	−1.43019
$111$	−16.0000	−1.51865
$112$	−1.00000	−0.0944911
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	−8.00000	−0.749269
$115$	27.0000	2.51776
$116$	6.00000	0.557086
$117$	1.00000	0.0924500
$118$	−9.00000	−0.828517
$119$	4.00000	0.366679
$120$	18.0000	1.64317
$121$	14.0000	1.27273
$122$	1.00000	0.0905357
$123$	−10.0000	−0.901670
$124$	0	0
$125$	3.00000	0.268328
$126$	−1.00000	−0.0890871
$127$	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
$129$	16.0000	1.40872
$130$	3.00000	0.263117
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	−10.0000	−0.870388
$133$	−4.00000	−0.346844
$134$	7.00000	0.604708
$135$	−12.0000	−1.03280
$136$	12.0000	1.02899
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	−18.0000	−1.53226
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	3.00000	0.253546
$141$	−8.00000	−0.673722
$142$	8.00000	0.671345
$143$	−5.00000	−0.418121
$144$	−1.00000	−0.0833333
$145$	18.0000	1.49482
$146$	11.0000	0.910366
$147$	12.0000	0.989743
$148$	−8.00000	−0.657596
$149$	19.0000	1.55654	0.778270	−	0.627929i	$-0.216097\pi$
$149$	19.0000	1.55654	0.778270	+	0.627929i	$0.216097\pi$
$150$	8.00000	0.653197
$151$	11.0000	0.895167	0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000	0.895167	0.447584	+	0.894242i	$0.352285\pi$
$152$	−12.0000	−0.973329
$153$	4.00000	0.323381
$154$	5.00000	0.402911
$155$	0	0
$156$	2.00000	0.160128
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	−3.00000	−0.238667
$159$	−12.0000	−0.951662
$160$	15.0000	1.18585
$161$	−9.00000	−0.709299
$162$	11.0000	0.864242
$163$	18.0000	1.40987	0.704934	−	0.709273i	$-0.250976\pi$
$163$	18.0000	1.40987	0.704934	+	0.709273i	$0.250976\pi$
$164$	−5.00000	−0.390434
$165$	−30.0000	−2.33550
$166$	−4.00000	−0.310460
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−6.00000	−0.462910
$169$	−12.0000	−0.923077
$170$	12.0000	0.920358
$171$	−4.00000	−0.305888
$172$	8.00000	0.609994
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	−12.0000	−0.909718
$175$	4.00000	0.302372
$176$	5.00000	0.376889
$177$	−18.0000	−1.35296
$178$	4.00000	0.299813
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	3.00000	0.223607
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	−1.00000	−0.0741249
$183$	2.00000	0.147844
$184$	−27.0000	−1.99047
$185$	−24.0000	−1.76452
$186$	0	0
$187$	−20.0000	−1.46254
$188$	−4.00000	−0.291730
$189$	4.00000	0.290957
$190$	−12.0000	−0.870572
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	−14.0000	−1.01036
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	14.0000	1.00514
$195$	6.00000	0.429669
$196$	6.00000	0.428571
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	5.00000	0.355335
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	12.0000	0.848528
$201$	14.0000	0.987484
$202$	0	0
$203$	−6.00000	−0.421117
$204$	8.00000	0.560112
$205$	−15.0000	−1.04765
$206$	−4.00000	−0.278693
$207$	−9.00000	−0.625543
$208$	−1.00000	−0.0693375
$209$	20.0000	1.38343
$210$	−6.00000	−0.414039
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−6.00000	−0.412082
$213$	16.0000	1.09630
$214$	2.00000	0.136717
$215$	24.0000	1.63679
$216$	12.0000	0.816497
$217$	0	0
$218$	17.0000	1.15139
$219$	22.0000	1.48662
$220$	−15.0000	−1.01130
$221$	4.00000	0.269069
$222$	16.0000	1.07385
$223$	23.0000	1.54019	0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000	1.54019	0.770097	+	0.637927i	$0.220208\pi$
$224$	−5.00000	−0.334077
$225$	4.00000	0.266667
$226$	−1.00000	−0.0665190
$227$	21.0000	1.39382	0.696909	−	0.717159i	$-0.254558\pi$
$227$	21.0000	1.39382	0.696909	+	0.717159i	$0.254558\pi$
$228$	−8.00000	−0.529813
$229$	9.00000	0.594737	0.297368	−	0.954763i	$-0.403891\pi$
$229$	9.00000	0.594737	0.297368	+	0.954763i	$0.403891\pi$
$230$	−27.0000	−1.78033
$231$	10.0000	0.657952
$232$	−18.0000	−1.18176
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	−1.00000	−0.0653720
$235$	−12.0000	−0.782794
$236$	−9.00000	−0.585850
$237$	−6.00000	−0.389742
$238$	−4.00000	−0.259281
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	−6.00000	−0.387298
$241$	−3.00000	−0.193247	−0.0966235	−	0.995321i	$-0.530804\pi$
$241$	−3.00000	−0.193247	−0.0966235	+	0.995321i	$0.530804\pi$
$242$	−14.0000	−0.899954
$243$	10.0000	0.641500
$244$	1.00000	0.0640184
$245$	18.0000	1.14998
$246$	10.0000	0.637577
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−8.00000	−0.506979
$250$	−3.00000	−0.189737
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−1.00000	−0.0629941
$253$	45.0000	2.82913
$254$	−6.00000	−0.376473
$255$	24.0000	1.50294
$256$	−17.0000	−1.06250
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	−16.0000	−0.996116
$259$	8.00000	0.497096
$260$	3.00000	0.186052
$261$	−6.00000	−0.371391
$262$	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	$0.664210\pi$
$264$	30.0000	1.84637
$265$	−18.0000	−1.10573
$266$	4.00000	0.245256
$267$	8.00000	0.489592
$268$	7.00000	0.427593
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	12.0000	0.730297
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	−4.00000	−0.242536
$273$	−2.00000	−0.121046
$274$	−9.00000	−0.543710
$275$	−20.0000	−1.20605
$276$	−18.0000	−1.08347
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	11.0000	0.659736
$279$	0	0
$280$	−9.00000	−0.537853
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	8.00000	0.476393
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	8.00000	0.474713
$285$	−24.0000	−1.42164
$286$	5.00000	0.295656
$287$	5.00000	0.295141
$288$	−5.00000	−0.294628
$289$	−1.00000	−0.0588235
$290$	−18.0000	−1.05700
$291$	28.0000	1.64139
$292$	11.0000	0.643726
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	−12.0000	−0.699854
$295$	−27.0000	−1.57200
$296$	24.0000	1.39497
$297$	−20.0000	−1.16052
$298$	−19.0000	−1.10064
$299$	−9.00000	−0.520483
$300$	8.00000	0.461880
$301$	−8.00000	−0.461112
$302$	−11.0000	−0.632979
$303$	0	0
$304$	4.00000	0.229416
$305$	3.00000	0.171780
$306$	−4.00000	−0.228665
$307$	−19.0000	−1.08439	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	−19.0000	−1.08439	−0.542194	+	0.840254i	$0.682406\pi$
$308$	5.00000	0.284901
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	−6.00000	−0.339683
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	4.00000	0.225733
$315$	−3.00000	−0.169031
$316$	−3.00000	−0.168763
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	12.0000	0.672927
$319$	30.0000	1.67968
$320$	−21.0000	−1.17394
$321$	4.00000	0.223258
$322$	9.00000	0.501550
$323$	−16.0000	−0.890264
$324$	11.0000	0.611111
$325$	4.00000	0.221880
$326$	−18.0000	−0.996928
$327$	34.0000	1.88020
$328$	15.0000	0.828236
$329$	4.00000	0.220527
$330$	30.0000	1.65145
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	−4.00000	−0.219529
$333$	8.00000	0.438397
$334$	12.0000	0.656611
$335$	21.0000	1.14735
$336$	2.00000	0.109109
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	12.0000	0.652714
$339$	−2.00000	−0.108625
$340$	12.0000	0.650791
$341$	0	0
$342$	4.00000	0.216295
$343$	−13.0000	−0.701934
$344$	−24.0000	−1.29399
$345$	−54.0000	−2.90726
$346$	−10.0000	−0.537603
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	−12.0000	−0.643268
$349$	−24.0000	−1.28469	−0.642345	−	0.766415i	$-0.722038\pi$
$349$	−24.0000	−1.28469	−0.642345	+	0.766415i	$0.722038\pi$
$350$	−4.00000	−0.213809
$351$	4.00000	0.213504
$352$	25.0000	1.33250
$353$	19.0000	1.01127	0.505634	−	0.862748i	$-0.331259\pi$
$353$	19.0000	1.01127	0.505634	+	0.862748i	$0.331259\pi$
$354$	18.0000	0.956689
$355$	24.0000	1.27379
$356$	4.00000	0.212000
$357$	−8.00000	−0.423405
$358$	18.0000	0.951330
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	−9.00000	−0.474342
$361$	−3.00000	−0.157895
$362$	−8.00000	−0.420471
$363$	−28.0000	−1.46962
$364$	−1.00000	−0.0524142
$365$	33.0000	1.72730
$366$	−2.00000	−0.104542
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	9.00000	0.469157
$369$	5.00000	0.260290
$370$	24.0000	1.24770
$371$	6.00000	0.311504
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	20.0000	1.03418
$375$	−6.00000	−0.309839
$376$	12.0000	0.618853
$377$	−6.00000	−0.309016
$378$	−4.00000	−0.205738
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	−12.0000	−0.615587
$381$	−12.0000	−0.614779
$382$	−3.00000	−0.153493
$383$	5.00000	0.255488	0.127744	−	0.991807i	$-0.459226\pi$
$383$	5.00000	0.255488	0.127744	+	0.991807i	$0.459226\pi$
$384$	−6.00000	−0.306186
$385$	15.0000	0.764471
$386$	8.00000	0.407189
$387$	−8.00000	−0.406663
$388$	14.0000	0.710742
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	−6.00000	−0.303822
$391$	−36.0000	−1.82060
$392$	−18.0000	−0.909137
$393$	32.0000	1.61419
$394$	3.00000	0.151138
$395$	−9.00000	−0.452839
$396$	5.00000	0.251259
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	−6.00000	−0.300753
$399$	8.00000	0.400501
$400$	−4.00000	−0.200000
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	−14.0000	−0.698257
$403$	0	0
$404$	0	0
$405$	33.0000	1.63978
$406$	6.00000	0.297775
$407$	−40.0000	−1.98273
$408$	−24.0000	−1.18818
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	15.0000	0.740797
$411$	−18.0000	−0.887875
$412$	−4.00000	−0.197066
$413$	9.00000	0.442861
$414$	9.00000	0.442326
$415$	−12.0000	−0.589057
$416$	−5.00000	−0.245145
$417$	22.0000	1.07734
$418$	−20.0000	−0.978232
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	−6.00000	−0.292770
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	−4.00000	−0.194717
$423$	4.00000	0.194487
$424$	18.0000	0.874157
$425$	16.0000	0.776114
$426$	−16.0000	−0.775203
$427$	−1.00000	−0.0483934
$428$	2.00000	0.0966736
$429$	10.0000	0.482805
$430$	−24.0000	−1.15738
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	−4.00000	−0.192450
$433$	−8.00000	−0.384455	−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000	−0.384455	−0.192228	+	0.981350i	$0.561571\pi$
$434$	0	0
$435$	−36.0000	−1.72607
$436$	17.0000	0.814152
$437$	36.0000	1.72211
$438$	−22.0000	−1.05120
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	45.0000	2.14529
$441$	−6.00000	−0.285714
$442$	−4.00000	−0.190261
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	16.0000	0.759326
$445$	12.0000	0.568855
$446$	−23.0000	−1.08908
$447$	−38.0000	−1.79734
$448$	7.00000	0.330719
$449$	−35.0000	−1.65175	−0.825876	−	0.563852i	$-0.809319\pi$
$449$	−35.0000	−1.65175	−0.825876	+	0.563852i	$0.809319\pi$
$450$	−4.00000	−0.188562
$451$	−25.0000	−1.17720
$452$	−1.00000	−0.0470360
$453$	−22.0000	−1.03365
$454$	−21.0000	−0.985579
$455$	−3.00000	−0.140642
$456$	24.0000	1.12390
$457$	−20.0000	−0.935561	−0.467780	−	0.883845i	$-0.654946\pi$
$457$	−20.0000	−0.935561	−0.467780	+	0.883845i	$0.654946\pi$
$458$	−9.00000	−0.420542
$459$	16.0000	0.746816
$460$	−27.0000	−1.25888
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	−10.0000	−0.465242
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	14.0000	0.648537
$467$	11.0000	0.509019	0.254510	−	0.967070i	$-0.418086\pi$
$467$	11.0000	0.509019	0.254510	+	0.967070i	$0.418086\pi$
$468$	−1.00000	−0.0462250
$469$	−7.00000	−0.323230
$470$	12.0000	0.553519
$471$	8.00000	0.368621
$472$	27.0000	1.24278
$473$	40.0000	1.83920
$474$	6.00000	0.275589
$475$	−16.0000	−0.734130
$476$	−4.00000	−0.183340
$477$	6.00000	0.274721
$478$	−2.00000	−0.0914779
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	−30.0000	−1.36931
$481$	8.00000	0.364769
$482$	3.00000	0.136646
$483$	18.0000	0.819028
$484$	−14.0000	−0.636364
$485$	42.0000	1.90712
$486$	−10.0000	−0.453609
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−3.00000	−0.135804
$489$	−36.0000	−1.62798
$490$	−18.0000	−0.813157
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	10.0000	0.450835
$493$	−24.0000	−1.08091
$494$	4.00000	0.179969
$495$	15.0000	0.674200
$496$	0	0
$497$	−8.00000	−0.358849
$498$	8.00000	0.358489
$499$	13.0000	0.581960	0.290980	−	0.956729i	$-0.406019\pi$
$499$	13.0000	0.581960	0.290980	+	0.956729i	$0.406019\pi$
$500$	−3.00000	−0.134164
$501$	24.0000	1.07224
$502$	12.0000	0.535586
$503$	10.0000	0.445878	0.222939	−	0.974832i	$-0.428435\pi$
$503$	10.0000	0.445878	0.222939	+	0.974832i	$0.428435\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	−45.0000	−2.00049
$507$	24.0000	1.06588
$508$	−6.00000	−0.266207
$509$	28.0000	1.24108	0.620539	−	0.784176i	$-0.286914\pi$
$509$	28.0000	1.24108	0.620539	+	0.784176i	$0.286914\pi$
$510$	−24.0000	−1.06274
$511$	−11.0000	−0.486611
$512$	11.0000	0.486136
$513$	−16.0000	−0.706417
$514$	22.0000	0.970378
$515$	−12.0000	−0.528783
$516$	−16.0000	−0.704361
$517$	−20.0000	−0.879599
$518$	−8.00000	−0.351500
$519$	−20.0000	−0.877903
$520$	−9.00000	−0.394676
$521$	−20.0000	−0.876216	−0.438108	−	0.898922i	$-0.644351\pi$
$521$	−20.0000	−0.876216	−0.438108	+	0.898922i	$0.644351\pi$
$522$	6.00000	0.262613
$523$	−9.00000	−0.393543	−0.196771	−	0.980449i	$-0.563046\pi$
$523$	−9.00000	−0.393543	−0.196771	+	0.980449i	$0.563046\pi$
$524$	16.0000	0.698963
$525$	−8.00000	−0.349149
$526$	16.0000	0.697633
$527$	0	0
$528$	−10.0000	−0.435194
$529$	58.0000	2.52174
$530$	18.0000	0.781870
$531$	9.00000	0.390567
$532$	4.00000	0.173422
$533$	5.00000	0.216574
$534$	−8.00000	−0.346194
$535$	6.00000	0.259403
$536$	−21.0000	−0.907062
$537$	36.0000	1.55351
$538$	18.0000	0.776035
$539$	30.0000	1.29219
$540$	12.0000	0.516398
$541$	28.0000	1.20381	0.601907	−	0.798566i	$-0.294408\pi$
$541$	28.0000	1.20381	0.601907	+	0.798566i	$0.294408\pi$
$542$	−14.0000	−0.601351
$543$	−16.0000	−0.686626
$544$	−20.0000	−0.857493
$545$	51.0000	2.18460
$546$	2.00000	0.0855921
$547$	−29.0000	−1.23995	−0.619975	−	0.784621i	$-0.712857\pi$
$547$	−29.0000	−1.23995	−0.619975	+	0.784621i	$0.712857\pi$
$548$	−9.00000	−0.384461
$549$	−1.00000	−0.0426790
$550$	20.0000	0.852803
$551$	24.0000	1.02243
$552$	54.0000	2.29839
$553$	3.00000	0.127573
$554$	−10.0000	−0.424859
$555$	48.0000	2.03749
$556$	11.0000	0.466504
$557$	−28.0000	−1.18640	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000	−1.18640	−0.593199	+	0.805056i	$0.702135\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	3.00000	0.126773
$561$	40.0000	1.68880
$562$	0	0
$563$	−10.0000	−0.421450	−0.210725	−	0.977545i	$-0.567582\pi$
$563$	−10.0000	−0.421450	−0.210725	+	0.977545i	$0.567582\pi$
$564$	8.00000	0.336861
$565$	−3.00000	−0.126211
$566$	−6.00000	−0.252199
$567$	−11.0000	−0.461957
$568$	−24.0000	−1.00702
$569$	−15.0000	−0.628833	−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000	−0.628833	−0.314416	+	0.949285i	$0.601809\pi$
$570$	24.0000	1.00525
$571$	38.0000	1.59025	0.795125	−	0.606445i	$-0.207405\pi$
$571$	38.0000	1.59025	0.795125	+	0.606445i	$0.207405\pi$
$572$	5.00000	0.209061
$573$	−6.00000	−0.250654
$574$	−5.00000	−0.208696
$575$	−36.0000	−1.50130
$576$	7.00000	0.291667
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	1.00000	0.0415945
$579$	16.0000	0.664937
$580$	−18.0000	−0.747409
$581$	4.00000	0.165948
$582$	−28.0000	−1.16064
$583$	−30.0000	−1.24247
$584$	−33.0000	−1.36555
$585$	−3.00000	−0.124035
$586$	−18.0000	−0.743573
$587$	32.0000	1.32078	0.660391	−	0.750922i	$-0.270391\pi$
$587$	32.0000	1.32078	0.660391	+	0.750922i	$0.270391\pi$
$588$	−12.0000	−0.494872
$589$	0	0
$590$	27.0000	1.11157
$591$	6.00000	0.246807
$592$	−8.00000	−0.328798
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	20.0000	0.820610
$595$	−12.0000	−0.491952
$596$	−19.0000	−0.778270
$597$	−12.0000	−0.491127
$598$	9.00000	0.368037
$599$	−35.0000	−1.43006	−0.715031	−	0.699093i	$-0.753587\pi$
$599$	−35.0000	−1.43006	−0.715031	+	0.699093i	$0.753587\pi$
$600$	−24.0000	−0.979796
$601$	−43.0000	−1.75401	−0.877003	−	0.480484i	$-0.840461\pi$
$601$	−43.0000	−1.75401	−0.877003	+	0.480484i	$0.840461\pi$
$602$	8.00000	0.326056
$603$	−7.00000	−0.285062
$604$	−11.0000	−0.447584
$605$	−42.0000	−1.70754
$606$	0	0
$607$	−26.0000	−1.05531	−0.527654	−	0.849460i	$-0.676928\pi$
$607$	−26.0000	−1.05531	−0.527654	+	0.849460i	$0.676928\pi$
$608$	20.0000	0.811107
$609$	12.0000	0.486265
$610$	−3.00000	−0.121466
$611$	4.00000	0.161823
$612$	−4.00000	−0.161690
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	19.0000	0.766778
$615$	30.0000	1.20972
$616$	−15.0000	−0.604367
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	8.00000	0.321807
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	−36.0000	−1.44463
$622$	15.0000	0.601445
$623$	−4.00000	−0.160257
$624$	2.00000	0.0800641
$625$	−29.0000	−1.16000
$626$	6.00000	0.239808
$627$	−40.0000	−1.59745
$628$	4.00000	0.159617
$629$	32.0000	1.27592
$630$	3.00000	0.119523
$631$	25.0000	0.995234	0.497617	−	0.867397i	$-0.334208\pi$
$631$	25.0000	0.995234	0.497617	+	0.867397i	$0.334208\pi$
$632$	9.00000	0.358001
$633$	−8.00000	−0.317971
$634$	30.0000	1.19145
$635$	−18.0000	−0.714308
$636$	12.0000	0.475831
$637$	−6.00000	−0.237729
$638$	−30.0000	−1.18771
$639$	−8.00000	−0.316475
$640$	−9.00000	−0.355756
$641$	46.0000	1.81689	0.908445	−	0.418004i	$-0.137270\pi$
$641$	46.0000	1.81689	0.908445	+	0.418004i	$0.137270\pi$
$642$	−4.00000	−0.157867
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	9.00000	0.354650
$645$	−48.0000	−1.89000
$646$	16.0000	0.629512
$647$	39.0000	1.53325	0.766624	−	0.642096i	$-0.221935\pi$
$647$	39.0000	1.53325	0.766624	+	0.642096i	$0.221935\pi$
$648$	−33.0000	−1.29636
$649$	−45.0000	−1.76640
$650$	−4.00000	−0.156893
$651$	0	0
$652$	−18.0000	−0.704934
$653$	46.0000	1.80012	0.900060	−	0.435767i	$-0.143523\pi$
$653$	46.0000	1.80012	0.900060	+	0.435767i	$0.143523\pi$
$654$	−34.0000	−1.32951
$655$	48.0000	1.87552
$656$	−5.00000	−0.195217
$657$	−11.0000	−0.429151
$658$	−4.00000	−0.155936
$659$	−34.0000	−1.32445	−0.662226	−	0.749304i	$-0.730388\pi$
$659$	−34.0000	−1.32445	−0.662226	+	0.749304i	$0.730388\pi$
$660$	30.0000	1.16775
$661$	34.0000	1.32245	0.661223	−	0.750189i	$-0.270038\pi$
$661$	34.0000	1.32245	0.661223	+	0.750189i	$0.270038\pi$
$662$	17.0000	0.660724
$663$	−8.00000	−0.310694
$664$	12.0000	0.465690
$665$	12.0000	0.465340
$666$	−8.00000	−0.309994
$667$	54.0000	2.09089
$668$	12.0000	0.464294
$669$	−46.0000	−1.77846
$670$	−21.0000	−0.811301
$671$	5.00000	0.193023
$672$	10.0000	0.385758
$673$	32.0000	1.23351	0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000	1.23351	0.616755	+	0.787155i	$0.288447\pi$
$674$	2.00000	0.0770371
$675$	16.0000	0.615840
$676$	12.0000	0.461538
$677$	−32.0000	−1.22986	−0.614930	−	0.788582i	$-0.710816\pi$
$677$	−32.0000	−1.22986	−0.614930	+	0.788582i	$0.710816\pi$
$678$	2.00000	0.0768095
$679$	−14.0000	−0.537271
$680$	−36.0000	−1.38054
$681$	−42.0000	−1.60944
$682$	0	0
$683$	26.0000	0.994862	0.497431	−	0.867503i	$-0.334277\pi$
$683$	26.0000	0.994862	0.497431	+	0.867503i	$0.334277\pi$
$684$	4.00000	0.152944
$685$	−27.0000	−1.03162
$686$	13.0000	0.496342
$687$	−18.0000	−0.686743
$688$	8.00000	0.304997
$689$	6.00000	0.228582
$690$	54.0000	2.05574
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	−10.0000	−0.380143
$693$	−5.00000	−0.189934
$694$	−8.00000	−0.303676
$695$	33.0000	1.25176
$696$	36.0000	1.36458
$697$	20.0000	0.757554
$698$	24.0000	0.908413
$699$	28.0000	1.05906
$700$	−4.00000	−0.151186
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	−4.00000	−0.150970
$703$	−32.0000	−1.20690
$704$	−35.0000	−1.31911
$705$	24.0000	0.903892
$706$	−19.0000	−0.715074
$707$	0	0
$708$	18.0000	0.676481
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	−24.0000	−0.900704
$711$	3.00000	0.112509
$712$	−12.0000	−0.449719
$713$	0	0
$714$	8.00000	0.299392
$715$	15.0000	0.560968
$716$	18.0000	0.672692
$717$	−4.00000	−0.149383
$718$	−12.0000	−0.447836
$719$	−46.0000	−1.71551	−0.857755	−	0.514058i	$-0.828142\pi$
$719$	−46.0000	−1.71551	−0.857755	+	0.514058i	$0.828142\pi$
$720$	3.00000	0.111803
$721$	4.00000	0.148968
$722$	3.00000	0.111648
$723$	6.00000	0.223142
$724$	−8.00000	−0.297318
$725$	−24.0000	−0.891338
$726$	28.0000	1.03918
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	3.00000	0.111187
$729$	13.0000	0.481481
$730$	−33.0000	−1.22138
$731$	−32.0000	−1.18356
$732$	−2.00000	−0.0739221
$733$	49.0000	1.80986	0.904928	−	0.425564i	$-0.139924\pi$
$733$	49.0000	1.80986	0.904928	+	0.425564i	$0.139924\pi$
$734$	−14.0000	−0.516749
$735$	−36.0000	−1.32788
$736$	45.0000	1.65872
$737$	35.0000	1.28924
$738$	−5.00000	−0.184053
$739$	45.0000	1.65535	0.827676	−	0.561206i	$-0.189663\pi$
$739$	45.0000	1.65535	0.827676	+	0.561206i	$0.189663\pi$
$740$	24.0000	0.882258
$741$	8.00000	0.293887
$742$	−6.00000	−0.220267
$743$	27.0000	0.990534	0.495267	−	0.868741i	$-0.335070\pi$
$743$	27.0000	0.990534	0.495267	+	0.868741i	$0.335070\pi$
$744$	0	0
$745$	−57.0000	−2.08832
$746$	10.0000	0.366126
$747$	4.00000	0.146352
$748$	20.0000	0.731272
$749$	−2.00000	−0.0730784
$750$	6.00000	0.219089
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	−4.00000	−0.145865
$753$	24.0000	0.874609
$754$	6.00000	0.218507
$755$	−33.0000	−1.20099
$756$	−4.00000	−0.145479
$757$	27.0000	0.981332	0.490666	−	0.871348i	$-0.336754\pi$
$757$	27.0000	0.981332	0.490666	+	0.871348i	$0.336754\pi$
$758$	14.0000	0.508503
$759$	−90.0000	−3.26679
$760$	36.0000	1.30586
$761$	−26.0000	−0.942499	−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000	−0.942499	−0.471250	+	0.882000i	$0.656197\pi$
$762$	12.0000	0.434714
$763$	−17.0000	−0.615441
$764$	−3.00000	−0.108536
$765$	−12.0000	−0.433861
$766$	−5.00000	−0.180657
$767$	9.00000	0.324971
$768$	34.0000	1.22687
$769$	−52.0000	−1.87517	−0.937584	−	0.347759i	$-0.886943\pi$
$769$	−52.0000	−1.87517	−0.937584	+	0.347759i	$0.886943\pi$
$770$	−15.0000	−0.540562
$771$	44.0000	1.58462
$772$	8.00000	0.287926
$773$	−46.0000	−1.65451	−0.827253	−	0.561830i	$-0.810097\pi$
$773$	−46.0000	−1.65451	−0.827253	+	0.561830i	$0.810097\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	−42.0000	−1.50771
$777$	−16.0000	−0.573997
$778$	14.0000	0.501924
$779$	−20.0000	−0.716574
$780$	−6.00000	−0.214834
$781$	40.0000	1.43131
$782$	36.0000	1.28736
$783$	−24.0000	−0.857690
$784$	6.00000	0.214286
$785$	12.0000	0.428298
$786$	−32.0000	−1.14140
$787$	48.0000	1.71102	0.855508	−	0.517790i	$-0.173245\pi$
$787$	48.0000	1.71102	0.855508	+	0.517790i	$0.173245\pi$
$788$	3.00000	0.106871
$789$	32.0000	1.13923
$790$	9.00000	0.320206
$791$	1.00000	0.0355559
$792$	−15.0000	−0.533002
$793$	−1.00000	−0.0355110
$794$	−8.00000	−0.283909
$795$	36.0000	1.27679
$796$	−6.00000	−0.212664
$797$	−23.0000	−0.814702	−0.407351	−	0.913272i	$-0.633547\pi$
$797$	−23.0000	−0.814702	−0.407351	+	0.913272i	$0.633547\pi$
$798$	−8.00000	−0.283197
$799$	16.0000	0.566039
$800$	−20.0000	−0.707107
$801$	−4.00000	−0.141333
$802$	−16.0000	−0.564980
$803$	55.0000	1.94091
$804$	−14.0000	−0.493742
$805$	27.0000	0.951625
$806$	0	0
$807$	36.0000	1.26726
$808$	0	0
$809$	−15.0000	−0.527372	−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000	−0.527372	−0.263686	+	0.964609i	$0.584938\pi$
$810$	−33.0000	−1.15950
$811$	7.00000	0.245803	0.122902	−	0.992419i	$-0.460780\pi$
$811$	7.00000	0.245803	0.122902	+	0.992419i	$0.460780\pi$
$812$	6.00000	0.210559
$813$	−28.0000	−0.982003
$814$	40.0000	1.40200
$815$	−54.0000	−1.89154
$816$	8.00000	0.280056
$817$	32.0000	1.11954
$818$	−30.0000	−1.04893
$819$	1.00000	0.0349428
$820$	15.0000	0.523823
$821$	−36.0000	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$821$	−36.0000	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$822$	18.0000	0.627822
$823$	−12.0000	−0.418294	−0.209147	−	0.977884i	$-0.567069\pi$
$823$	−12.0000	−0.418294	−0.209147	+	0.977884i	$0.567069\pi$
$824$	12.0000	0.418040
$825$	40.0000	1.39262
$826$	−9.00000	−0.313150
$827$	8.00000	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$828$	9.00000	0.312772
$829$	3.00000	0.104194	0.0520972	−	0.998642i	$-0.483409\pi$
$829$	3.00000	0.104194	0.0520972	+	0.998642i	$0.483409\pi$
$830$	12.0000	0.416526
$831$	−20.0000	−0.693792
$832$	7.00000	0.242681
$833$	−24.0000	−0.831551
$834$	−22.0000	−0.761798
$835$	36.0000	1.24583
$836$	−20.0000	−0.691714
$837$	0	0
$838$	−4.00000	−0.138178
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	18.0000	0.621059
$841$	7.00000	0.241379
$842$	−8.00000	−0.275698
$843$	0	0
$844$	−4.00000	−0.137686
$845$	36.0000	1.23844
$846$	−4.00000	−0.137523
$847$	14.0000	0.481046
$848$	−6.00000	−0.206041
$849$	−12.0000	−0.411839
$850$	−16.0000	−0.548795
$851$	−72.0000	−2.46813
$852$	−16.0000	−0.548151
$853$	−43.0000	−1.47229	−0.736146	−	0.676823i	$-0.763356\pi$
$853$	−43.0000	−1.47229	−0.736146	+	0.676823i	$0.763356\pi$
$854$	1.00000	0.0342193
$855$	12.0000	0.410391
$856$	−6.00000	−0.205076
$857$	37.0000	1.26390	0.631948	−	0.775011i	$-0.282256\pi$
$857$	37.0000	1.26390	0.631948	+	0.775011i	$0.282256\pi$
$858$	−10.0000	−0.341394
$859$	6.00000	0.204717	0.102359	−	0.994748i	$-0.467361\pi$
$859$	6.00000	0.204717	0.102359	+	0.994748i	$0.467361\pi$
$860$	−24.0000	−0.818393
$861$	−10.0000	−0.340799
$862$	−6.00000	−0.204361
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	−20.0000	−0.680414
$865$	−30.0000	−1.02003
$866$	8.00000	0.271851
$867$	2.00000	0.0679236
$868$	0	0
$869$	−15.0000	−0.508840
$870$	36.0000	1.22051
$871$	−7.00000	−0.237186
$872$	−51.0000	−1.72708
$873$	−14.0000	−0.473828
$874$	−36.0000	−1.21772
$875$	3.00000	0.101419
$876$	−22.0000	−0.743311
$877$	−12.0000	−0.405211	−0.202606	−	0.979260i	$-0.564941\pi$
$877$	−12.0000	−0.405211	−0.202606	+	0.979260i	$0.564941\pi$
$878$	0	0
$879$	−36.0000	−1.21425
$880$	−15.0000	−0.505650
$881$	−29.0000	−0.977035	−0.488517	−	0.872554i	$-0.662462\pi$
$881$	−29.0000	−0.977035	−0.488517	+	0.872554i	$0.662462\pi$
$882$	6.00000	0.202031
$883$	9.00000	0.302874	0.151437	−	0.988467i	$-0.451610\pi$
$883$	9.00000	0.302874	0.151437	+	0.988467i	$0.451610\pi$
$884$	−4.00000	−0.134535
$885$	54.0000	1.81519
$886$	−12.0000	−0.403148
$887$	16.0000	0.537227	0.268614	−	0.963248i	$-0.413434\pi$
$887$	16.0000	0.537227	0.268614	+	0.963248i	$0.413434\pi$
$888$	−48.0000	−1.61077
$889$	6.00000	0.201234
$890$	−12.0000	−0.402241
$891$	55.0000	1.84257
$892$	−23.0000	−0.770097
$893$	−16.0000	−0.535420
$894$	38.0000	1.27091
$895$	54.0000	1.80502
$896$	3.00000	0.100223
$897$	18.0000	0.601003
$898$	35.0000	1.16797
$899$	0	0
$900$	−4.00000	−0.133333
$901$	24.0000	0.799556
$902$	25.0000	0.832409
$903$	16.0000	0.532447
$904$	3.00000	0.0997785
$905$	−24.0000	−0.797787
$906$	22.0000	0.730901
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−21.0000	−0.696909
$909$	0	0
$910$	3.00000	0.0994490
$911$	2.00000	0.0662630	0.0331315	−	0.999451i	$-0.489452\pi$
$911$	2.00000	0.0662630	0.0331315	+	0.999451i	$0.489452\pi$
$912$	−8.00000	−0.264906
$913$	−20.0000	−0.661903
$914$	20.0000	0.661541
$915$	−6.00000	−0.198354
$916$	−9.00000	−0.297368
$917$	−16.0000	−0.528367
$918$	−16.0000	−0.528079
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	81.0000	2.67049
$921$	38.0000	1.25214
$922$	18.0000	0.592798
$923$	−8.00000	−0.263323
$924$	−10.0000	−0.328976
$925$	32.0000	1.05215
$926$	4.00000	0.131448
$927$	4.00000	0.131377
$928$	30.0000	0.984798
$929$	−3.00000	−0.0984268	−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000	−0.0984268	−0.0492134	+	0.998788i	$0.515671\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	14.0000	0.458585
$933$	30.0000	0.982156
$934$	−11.0000	−0.359931
$935$	60.0000	1.96221
$936$	3.00000	0.0980581
$937$	−21.0000	−0.686040	−0.343020	−	0.939328i	$-0.611450\pi$
$937$	−21.0000	−0.686040	−0.343020	+	0.939328i	$0.611450\pi$
$938$	7.00000	0.228558
$939$	12.0000	0.391605
$940$	12.0000	0.391397
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	−8.00000	−0.260654
$943$	−45.0000	−1.46540
$944$	−9.00000	−0.292925
$945$	−12.0000	−0.390360
$946$	−40.0000	−1.30051
$947$	−41.0000	−1.33232	−0.666160	−	0.745808i	$-0.732063\pi$
$947$	−41.0000	−1.33232	−0.666160	+	0.745808i	$0.732063\pi$
$948$	6.00000	0.194871
$949$	−11.0000	−0.357075
$950$	16.0000	0.519109
$951$	60.0000	1.94563
$952$	12.0000	0.388922
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	−6.00000	−0.194257
$955$	−9.00000	−0.291233
$956$	−2.00000	−0.0646846
$957$	−60.0000	−1.93952
$958$	−14.0000	−0.452319
$959$	9.00000	0.290625
$960$	42.0000	1.35554
$961$	−31.0000	−1.00000
$962$	−8.00000	−0.257930
$963$	−2.00000	−0.0644491
$964$	3.00000	0.0966235
$965$	24.0000	0.772587
$966$	−18.0000	−0.579141
$967$	14.0000	0.450210	0.225105	−	0.974335i	$-0.427728\pi$
$967$	14.0000	0.450210	0.225105	+	0.974335i	$0.427728\pi$
$968$	42.0000	1.34993
$969$	32.0000	1.02799
$970$	−42.0000	−1.34854
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	−10.0000	−0.320750
$973$	−11.0000	−0.352644
$974$	12.0000	0.384505
$975$	−8.00000	−0.256205
$976$	1.00000	0.0320092
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	36.0000	1.15115
$979$	20.0000	0.639203
$980$	−18.0000	−0.574989
$981$	−17.0000	−0.542768
$982$	28.0000	0.893516
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	−30.0000	−0.956365
$985$	9.00000	0.286764
$986$	24.0000	0.764316
$987$	−8.00000	−0.254643
$988$	4.00000	0.127257
$989$	72.0000	2.28947
$990$	−15.0000	−0.476731
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	0	0
$993$	34.0000	1.07896
$994$	8.00000	0.253745
$995$	−18.0000	−0.570638
$996$	8.00000	0.253490
$997$	−56.0000	−1.77354	−0.886769	−	0.462213i	$-0.847056\pi$
$997$	−56.0000	−1.77354	−0.886769	+	0.462213i	$0.847056\pi$
$998$	−13.0000	−0.411508
$999$	32.0000	1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.2.a.a.1.1	✓	1
3.2	odd	2		549.2.a.c.1.1		1
4.3	odd	2		976.2.a.b.1.1		1
5.2	odd	4		1525.2.b.a.1099.1		2
5.3	odd	4		1525.2.b.a.1099.2		2
5.4	even	2		1525.2.a.b.1.1		1
7.6	odd	2		2989.2.a.b.1.1		1
8.3	odd	2		3904.2.a.b.1.1		1
8.5	even	2		3904.2.a.j.1.1		1
11.10	odd	2		7381.2.a.c.1.1		1
12.11	even	2		8784.2.a.w.1.1		1
61.60	even	2		3721.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.2.a.a.1.1	✓	1	1.1	even	1	trivial
549.2.a.c.1.1		1	3.2	odd	2
976.2.a.b.1.1		1	4.3	odd	2
1525.2.a.b.1.1		1	5.4	even	2
1525.2.b.a.1099.1		2	5.2	odd	4
1525.2.b.a.1099.2		2	5.3	odd	4
2989.2.a.b.1.1		1	7.6	odd	2
3721.2.a.a.1.1		1	61.60	even	2
3904.2.a.b.1.1		1	8.3	odd	2
3904.2.a.j.1.1		1	8.5	even	2
7381.2.a.c.1.1		1	11.10	odd	2
8784.2.a.w.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:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	61.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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