LMFDB - Embedded newform 6045.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
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	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} -3.00000 q^{7} +1.00000 q^{9} -2.00000 q^{10} -2.00000 q^{11} +2.00000 q^{12} +1.00000 q^{13} +6.00000 q^{14} +1.00000 q^{15} -4.00000 q^{16} +2.00000 q^{17} -2.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} -3.00000 q^{21} +4.00000 q^{22} +6.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -6.00000 q^{28} -3.00000 q^{29} -2.00000 q^{30} +1.00000 q^{31} +8.00000 q^{32} -2.00000 q^{33} -4.00000 q^{34} -3.00000 q^{35} +2.00000 q^{36} -6.00000 q^{37} +8.00000 q^{38} +1.00000 q^{39} +3.00000 q^{41} +6.00000 q^{42} +5.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -12.0000 q^{46} -12.0000 q^{47} -4.00000 q^{48} +2.00000 q^{49} -2.00000 q^{50} +2.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -2.00000 q^{54} -2.00000 q^{55} -4.00000 q^{57} +6.00000 q^{58} -5.00000 q^{59} +2.00000 q^{60} +6.00000 q^{61} -2.00000 q^{62} -3.00000 q^{63} -8.00000 q^{64} +1.00000 q^{65} +4.00000 q^{66} +3.00000 q^{67} +4.00000 q^{68} +6.00000 q^{69} +6.00000 q^{70} +8.00000 q^{71} +10.0000 q^{73} +12.0000 q^{74} +1.00000 q^{75} -8.00000 q^{76} +6.00000 q^{77} -2.00000 q^{78} -2.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -9.00000 q^{83} -6.00000 q^{84} +2.00000 q^{85} -10.0000 q^{86} -3.00000 q^{87} -6.00000 q^{89} -2.00000 q^{90} -3.00000 q^{91} +12.0000 q^{92} +1.00000 q^{93} +24.0000 q^{94} -4.00000 q^{95} +8.00000 q^{96} -17.0000 q^{97} -4.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.00000	1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.00000	−0.816497
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	2.00000	0.577350
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	6.00000	1.60357
$15$	1.00000	0.258199
$16$	−4.00000	−1.00000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−2.00000	−0.471405
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	2.00000	0.447214
$21$	−3.00000	−0.654654
$22$	4.00000	0.852803
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	−6.00000	−1.13389
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	−2.00000	−0.365148
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	8.00000	1.41421
$33$	−2.00000	−0.348155
$34$	−4.00000	−0.685994
$35$	−3.00000	−0.507093
$36$	2.00000	0.333333
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	8.00000	1.29777
$39$	1.00000	0.160128
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	6.00000	0.925820
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	−4.00000	−0.603023
$45$	1.00000	0.149071
$46$	−12.0000	−1.76930
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	−4.00000	−0.577350
$49$	2.00000	0.285714
$50$	−2.00000	−0.282843
$51$	2.00000	0.280056
$52$	2.00000	0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−2.00000	−0.272166
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−4.00000	−0.529813
$58$	6.00000	0.787839
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	2.00000	0.258199
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−2.00000	−0.254000
$63$	−3.00000	−0.377964
$64$	−8.00000	−1.00000
$65$	1.00000	0.124035
$66$	4.00000	0.492366
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	4.00000	0.485071
$69$	6.00000	0.722315
$70$	6.00000	0.717137
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	12.0000	1.39497
$75$	1.00000	0.115470
$76$	−8.00000	−0.917663
$77$	6.00000	0.683763
$78$	−2.00000	−0.226455
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	−6.00000	−0.654654
$85$	2.00000	0.216930
$86$	−10.0000	−1.07833
$87$	−3.00000	−0.321634
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	−2.00000	−0.210819
$91$	−3.00000	−0.314485
$92$	12.0000	1.25109
$93$	1.00000	0.103695
$94$	24.0000	2.47541
$95$	−4.00000	−0.410391
$96$	8.00000	0.816497
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	−4.00000	−0.404061
$99$	−2.00000	−0.201008
$100$	2.00000	0.200000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	−4.00000	−0.396059
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	−12.0000	−1.16554
$107$	−19.0000	−1.83680	−0.918400	−	0.395654i	$-0.870518\pi$
$107$	−19.0000	−1.83680	−0.918400	+	0.395654i	$0.870518\pi$
$108$	2.00000	0.192450
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	4.00000	0.381385
$111$	−6.00000	−0.569495
$112$	12.0000	1.13389
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	8.00000	0.749269
$115$	6.00000	0.559503
$116$	−6.00000	−0.557086
$117$	1.00000	0.0924500
$118$	10.0000	0.920575
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−12.0000	−1.08643
$123$	3.00000	0.270501
$124$	2.00000	0.179605
$125$	1.00000	0.0894427
$126$	6.00000	0.534522
$127$	−5.00000	−0.443678	−0.221839	−	0.975083i	$-0.571206\pi$
$127$	−5.00000	−0.443678	−0.221839	+	0.975083i	$0.571206\pi$
$128$	0	0
$129$	5.00000	0.440225
$130$	−2.00000	−0.175412
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	−4.00000	−0.348155
$133$	12.0000	1.04053
$134$	−6.00000	−0.518321
$135$	1.00000	0.0860663
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−12.0000	−1.02151
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	−6.00000	−0.507093
$141$	−12.0000	−1.01058
$142$	−16.0000	−1.34269
$143$	−2.00000	−0.167248
$144$	−4.00000	−0.333333
$145$	−3.00000	−0.249136
$146$	−20.0000	−1.65521
$147$	2.00000	0.164957
$148$	−12.0000	−0.986394
$149$	11.0000	0.901155	0.450578	−	0.892737i	$-0.351218\pi$
$149$	11.0000	0.901155	0.450578	+	0.892737i	$0.351218\pi$
$150$	−2.00000	−0.163299
$151$	−3.00000	−0.244137	−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000	−0.244137	−0.122068	+	0.992522i	$0.538953\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	−12.0000	−0.966988
$155$	1.00000	0.0803219
$156$	2.00000	0.160128
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	4.00000	0.318223
$159$	6.00000	0.475831
$160$	8.00000	0.632456
$161$	−18.0000	−1.41860
$162$	−2.00000	−0.157135
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	6.00000	0.468521
$165$	−2.00000	−0.155700
$166$	18.0000	1.39707
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−4.00000	−0.306786
$171$	−4.00000	−0.305888
$172$	10.0000	0.762493
$173$	−21.0000	−1.59660	−0.798300	−	0.602260i	$-0.794267\pi$
$173$	−21.0000	−1.59660	−0.798300	+	0.602260i	$0.794267\pi$
$174$	6.00000	0.454859
$175$	−3.00000	−0.226779
$176$	8.00000	0.603023
$177$	−5.00000	−0.375823
$178$	12.0000	0.899438
$179$	5.00000	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$180$	2.00000	0.149071
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	6.00000	0.444750
$183$	6.00000	0.443533
$184$	0	0
$185$	−6.00000	−0.441129
$186$	−2.00000	−0.146647
$187$	−4.00000	−0.292509
$188$	−24.0000	−1.75038
$189$	−3.00000	−0.218218
$190$	8.00000	0.580381
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	−8.00000	−0.577350
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	34.0000	2.44106
$195$	1.00000	0.0716115
$196$	4.00000	0.285714
$197$	−19.0000	−1.35369	−0.676847	−	0.736124i	$-0.736654\pi$
$197$	−19.0000	−1.35369	−0.676847	+	0.736124i	$0.736654\pi$
$198$	4.00000	0.284268
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	3.00000	0.211604
$202$	−4.00000	−0.281439
$203$	9.00000	0.631676
$204$	4.00000	0.280056
$205$	3.00000	0.209529
$206$	12.0000	0.836080
$207$	6.00000	0.417029
$208$	−4.00000	−0.277350
$209$	8.00000	0.553372
$210$	6.00000	0.414039
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	12.0000	0.824163
$213$	8.00000	0.548151
$214$	38.0000	2.59763
$215$	5.00000	0.340997
$216$	0	0
$217$	−3.00000	−0.203653
$218$	−28.0000	−1.89640
$219$	10.0000	0.675737
$220$	−4.00000	−0.269680
$221$	2.00000	0.134535
$222$	12.0000	0.805387
$223$	−18.0000	−1.20537	−0.602685	−	0.797980i	$-0.705902\pi$
$223$	−18.0000	−1.20537	−0.602685	+	0.797980i	$0.705902\pi$
$224$	−24.0000	−1.60357
$225$	1.00000	0.0666667
$226$	−18.0000	−1.19734
$227$	−22.0000	−1.46019	−0.730096	−	0.683345i	$-0.760525\pi$
$227$	−22.0000	−1.46019	−0.730096	+	0.683345i	$0.760525\pi$
$228$	−8.00000	−0.529813
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	−12.0000	−0.791257
$231$	6.00000	0.394771
$232$	0	0
$233$	−25.0000	−1.63780	−0.818902	−	0.573933i	$-0.805417\pi$
$233$	−25.0000	−1.63780	−0.818902	+	0.573933i	$0.805417\pi$
$234$	−2.00000	−0.130744
$235$	−12.0000	−0.782794
$236$	−10.0000	−0.650945
$237$	−2.00000	−0.129914
$238$	12.0000	0.777844
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	−4.00000	−0.258199
$241$	25.0000	1.61039	0.805196	−	0.593009i	$-0.202060\pi$
$241$	25.0000	1.61039	0.805196	+	0.593009i	$0.202060\pi$
$242$	14.0000	0.899954
$243$	1.00000	0.0641500
$244$	12.0000	0.768221
$245$	2.00000	0.127775
$246$	−6.00000	−0.382546
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−9.00000	−0.570352
$250$	−2.00000	−0.126491
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	−6.00000	−0.377964
$253$	−12.0000	−0.754434
$254$	10.0000	0.627456
$255$	2.00000	0.125245
$256$	16.0000	1.00000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	−10.0000	−0.622573
$259$	18.0000	1.11847
$260$	2.00000	0.124035
$261$	−3.00000	−0.185695
$262$	−12.0000	−0.741362
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	−24.0000	−1.47153
$267$	−6.00000	−0.367194
$268$	6.00000	0.366508
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	−2.00000	−0.121716
$271$	−21.0000	−1.27566	−0.637830	−	0.770178i	$-0.720168\pi$
$271$	−21.0000	−1.27566	−0.637830	+	0.770178i	$0.720168\pi$
$272$	−8.00000	−0.485071
$273$	−3.00000	−0.181568
$274$	12.0000	0.724947
$275$	−2.00000	−0.120605
$276$	12.0000	0.722315
$277$	3.00000	0.180253	0.0901263	−	0.995930i	$-0.471273\pi$
$277$	3.00000	0.180253	0.0901263	+	0.995930i	$0.471273\pi$
$278$	−8.00000	−0.479808
$279$	1.00000	0.0598684
$280$	0	0
$281$	1.00000	0.0596550	0.0298275	−	0.999555i	$-0.490504\pi$
$281$	1.00000	0.0596550	0.0298275	+	0.999555i	$0.490504\pi$
$282$	24.0000	1.42918
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	16.0000	0.949425
$285$	−4.00000	−0.236940
$286$	4.00000	0.236525
$287$	−9.00000	−0.531253
$288$	8.00000	0.471405
$289$	−13.0000	−0.764706
$290$	6.00000	0.352332
$291$	−17.0000	−0.996558
$292$	20.0000	1.17041
$293$	−4.00000	−0.233682	−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−4.00000	−0.233682	−0.116841	+	0.993151i	$0.537277\pi$
$294$	−4.00000	−0.233285
$295$	−5.00000	−0.291111
$296$	0	0
$297$	−2.00000	−0.116052
$298$	−22.0000	−1.27443
$299$	6.00000	0.346989
$300$	2.00000	0.115470
$301$	−15.0000	−0.864586
$302$	6.00000	0.345261
$303$	2.00000	0.114897
$304$	16.0000	0.917663
$305$	6.00000	0.343559
$306$	−4.00000	−0.228665
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	12.0000	0.683763
$309$	−6.00000	−0.341328
$310$	−2.00000	−0.113592
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	0	0
$313$	−3.00000	−0.169570	−0.0847850	−	0.996399i	$-0.527020\pi$
$313$	−3.00000	−0.169570	−0.0847850	+	0.996399i	$0.527020\pi$
$314$	12.0000	0.677199
$315$	−3.00000	−0.169031
$316$	−4.00000	−0.225018
$317$	32.0000	1.79730	0.898650	−	0.438667i	$-0.144549\pi$
$317$	32.0000	1.79730	0.898650	+	0.438667i	$0.144549\pi$
$318$	−12.0000	−0.672927
$319$	6.00000	0.335936
$320$	−8.00000	−0.447214
$321$	−19.0000	−1.06048
$322$	36.0000	2.00620
$323$	−8.00000	−0.445132
$324$	2.00000	0.111111
$325$	1.00000	0.0554700
$326$	−40.0000	−2.21540
$327$	14.0000	0.774202
$328$	0	0
$329$	36.0000	1.98474
$330$	4.00000	0.220193
$331$	5.00000	0.274825	0.137412	−	0.990514i	$-0.456121\pi$
$331$	5.00000	0.274825	0.137412	+	0.990514i	$0.456121\pi$
$332$	−18.0000	−0.987878
$333$	−6.00000	−0.328798
$334$	24.0000	1.31322
$335$	3.00000	0.163908
$336$	12.0000	0.654654
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	−2.00000	−0.108786
$339$	9.00000	0.488813
$340$	4.00000	0.216930
$341$	−2.00000	−0.108306
$342$	8.00000	0.432590
$343$	15.0000	0.809924
$344$	0	0
$345$	6.00000	0.323029
$346$	42.0000	2.25793
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	−6.00000	−0.321634
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	6.00000	0.320713
$351$	1.00000	0.0533761
$352$	−16.0000	−0.852803
$353$	−15.0000	−0.798369	−0.399185	−	0.916871i	$-0.630707\pi$
$353$	−15.0000	−0.798369	−0.399185	+	0.916871i	$0.630707\pi$
$354$	10.0000	0.531494
$355$	8.00000	0.424596
$356$	−12.0000	−0.635999
$357$	−6.00000	−0.317554
$358$	−10.0000	−0.528516
$359$	1.00000	0.0527780	0.0263890	−	0.999652i	$-0.491599\pi$
$359$	1.00000	0.0527780	0.0263890	+	0.999652i	$0.491599\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−24.0000	−1.26141
$363$	−7.00000	−0.367405
$364$	−6.00000	−0.314485
$365$	10.0000	0.523424
$366$	−12.0000	−0.627250
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	−24.0000	−1.25109
$369$	3.00000	0.156174
$370$	12.0000	0.623850
$371$	−18.0000	−0.934513
$372$	2.00000	0.103695
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	8.00000	0.413670
$375$	1.00000	0.0516398
$376$	0	0
$377$	−3.00000	−0.154508
$378$	6.00000	0.308607
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	−8.00000	−0.410391
$381$	−5.00000	−0.256158
$382$	36.0000	1.84192
$383$	−9.00000	−0.459879	−0.229939	−	0.973205i	$-0.573853\pi$
$383$	−9.00000	−0.459879	−0.229939	+	0.973205i	$0.573853\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	22.0000	1.11977
$387$	5.00000	0.254164
$388$	−34.0000	−1.72609
$389$	−5.00000	−0.253510	−0.126755	−	0.991934i	$-0.540456\pi$
$389$	−5.00000	−0.253510	−0.126755	+	0.991934i	$0.540456\pi$
$390$	−2.00000	−0.101274
$391$	12.0000	0.606866
$392$	0	0
$393$	6.00000	0.302660
$394$	38.0000	1.91441
$395$	−2.00000	−0.100631
$396$	−4.00000	−0.201008
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	8.00000	0.401004
$399$	12.0000	0.600751
$400$	−4.00000	−0.200000
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	−6.00000	−0.299253
$403$	1.00000	0.0498135
$404$	4.00000	0.199007
$405$	1.00000	0.0496904
$406$	−18.0000	−0.893325
$407$	12.0000	0.594818
$408$	0	0
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	−6.00000	−0.296319
$411$	−6.00000	−0.295958
$412$	−12.0000	−0.591198
$413$	15.0000	0.738102
$414$	−12.0000	−0.589768
$415$	−9.00000	−0.441793
$416$	8.00000	0.392232
$417$	4.00000	0.195881
$418$	−16.0000	−0.782586
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	−6.00000	−0.292770
$421$	−24.0000	−1.16969	−0.584844	−	0.811146i	$-0.698844\pi$
$421$	−24.0000	−1.16969	−0.584844	+	0.811146i	$0.698844\pi$
$422$	−26.0000	−1.26566
$423$	−12.0000	−0.583460
$424$	0	0
$425$	2.00000	0.0970143
$426$	−16.0000	−0.775203
$427$	−18.0000	−0.871081
$428$	−38.0000	−1.83680
$429$	−2.00000	−0.0965609
$430$	−10.0000	−0.482243
$431$	9.00000	0.433515	0.216757	−	0.976226i	$-0.430452\pi$
$431$	9.00000	0.433515	0.216757	+	0.976226i	$0.430452\pi$
$432$	−4.00000	−0.192450
$433$	13.0000	0.624740	0.312370	−	0.949960i	$-0.398877\pi$
$433$	13.0000	0.624740	0.312370	+	0.949960i	$0.398877\pi$
$434$	6.00000	0.288009
$435$	−3.00000	−0.143839
$436$	28.0000	1.34096
$437$	−24.0000	−1.14808
$438$	−20.0000	−0.955637
$439$	33.0000	1.57500	0.787502	−	0.616312i	$-0.211374\pi$
$439$	33.0000	1.57500	0.787502	+	0.616312i	$0.211374\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	−4.00000	−0.190261
$443$	−29.0000	−1.37783	−0.688916	−	0.724841i	$-0.741913\pi$
$443$	−29.0000	−1.37783	−0.688916	+	0.724841i	$0.741913\pi$
$444$	−12.0000	−0.569495
$445$	−6.00000	−0.284427
$446$	36.0000	1.70465
$447$	11.0000	0.520282
$448$	24.0000	1.13389
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	−2.00000	−0.0942809
$451$	−6.00000	−0.282529
$452$	18.0000	0.846649
$453$	−3.00000	−0.140952
$454$	44.0000	2.06502
$455$	−3.00000	−0.140642
$456$	0	0
$457$	20.0000	0.935561	0.467780	−	0.883845i	$-0.345054\pi$
$457$	20.0000	0.935561	0.467780	+	0.883845i	$0.345054\pi$
$458$	44.0000	2.05598
$459$	2.00000	0.0933520
$460$	12.0000	0.559503
$461$	4.00000	0.186299	0.0931493	−	0.995652i	$-0.470307\pi$
$461$	4.00000	0.186299	0.0931493	+	0.995652i	$0.470307\pi$
$462$	−12.0000	−0.558291
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	12.0000	0.557086
$465$	1.00000	0.0463739
$466$	50.0000	2.31621
$467$	17.0000	0.786666	0.393333	−	0.919396i	$-0.371322\pi$
$467$	17.0000	0.786666	0.393333	+	0.919396i	$0.371322\pi$
$468$	2.00000	0.0924500
$469$	−9.00000	−0.415581
$470$	24.0000	1.10704
$471$	−6.00000	−0.276465
$472$	0	0
$473$	−10.0000	−0.459800
$474$	4.00000	0.183726
$475$	−4.00000	−0.183533
$476$	−12.0000	−0.550019
$477$	6.00000	0.274721
$478$	−12.0000	−0.548867
$479$	−37.0000	−1.69057	−0.845287	−	0.534313i	$-0.820570\pi$
$479$	−37.0000	−1.69057	−0.845287	+	0.534313i	$0.820570\pi$
$480$	8.00000	0.365148
$481$	−6.00000	−0.273576
$482$	−50.0000	−2.27744
$483$	−18.0000	−0.819028
$484$	−14.0000	−0.636364
$485$	−17.0000	−0.771930
$486$	−2.00000	−0.0907218
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	−4.00000	−0.180702
$491$	−39.0000	−1.76005	−0.880023	−	0.474932i	$-0.842473\pi$
$491$	−39.0000	−1.76005	−0.880023	+	0.474932i	$0.842473\pi$
$492$	6.00000	0.270501
$493$	−6.00000	−0.270226
$494$	8.00000	0.359937
$495$	−2.00000	−0.0898933
$496$	−4.00000	−0.179605
$497$	−24.0000	−1.07655
$498$	18.0000	0.806599
$499$	−39.0000	−1.74588	−0.872940	−	0.487828i	$-0.837789\pi$
$499$	−39.0000	−1.74588	−0.872940	+	0.487828i	$0.837789\pi$
$500$	2.00000	0.0894427
$501$	−12.0000	−0.536120
$502$	42.0000	1.87455
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	24.0000	1.06693
$507$	1.00000	0.0444116
$508$	−10.0000	−0.443678
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	−4.00000	−0.177123
$511$	−30.0000	−1.32712
$512$	−32.0000	−1.41421
$513$	−4.00000	−0.176604
$514$	12.0000	0.529297
$515$	−6.00000	−0.264392
$516$	10.0000	0.440225
$517$	24.0000	1.05552
$518$	−36.0000	−1.58175
$519$	−21.0000	−0.921798
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	6.00000	0.262613
$523$	−29.0000	−1.26808	−0.634041	−	0.773300i	$-0.718605\pi$
$523$	−29.0000	−1.26808	−0.634041	+	0.773300i	$0.718605\pi$
$524$	12.0000	0.524222
$525$	−3.00000	−0.130931
$526$	32.0000	1.39527
$527$	2.00000	0.0871214
$528$	8.00000	0.348155
$529$	13.0000	0.565217
$530$	−12.0000	−0.521247
$531$	−5.00000	−0.216982
$532$	24.0000	1.04053
$533$	3.00000	0.129944
$534$	12.0000	0.519291
$535$	−19.0000	−0.821442
$536$	0	0
$537$	5.00000	0.215766
$538$	20.0000	0.862261
$539$	−4.00000	−0.172292
$540$	2.00000	0.0860663
$541$	42.0000	1.80572	0.902861	−	0.429934i	$-0.141463\pi$
$541$	42.0000	1.80572	0.902861	+	0.429934i	$0.141463\pi$
$542$	42.0000	1.80405
$543$	12.0000	0.514969
$544$	16.0000	0.685994
$545$	14.0000	0.599694
$546$	6.00000	0.256776
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	−12.0000	−0.512615
$549$	6.00000	0.256074
$550$	4.00000	0.170561
$551$	12.0000	0.511217
$552$	0	0
$553$	6.00000	0.255146
$554$	−6.00000	−0.254916
$555$	−6.00000	−0.254686
$556$	8.00000	0.339276
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	−2.00000	−0.0846668
$559$	5.00000	0.211477
$560$	12.0000	0.507093
$561$	−4.00000	−0.168880
$562$	−2.00000	−0.0843649
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	−24.0000	−1.01058
$565$	9.00000	0.378633
$566$	−44.0000	−1.84946
$567$	−3.00000	−0.125988
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	8.00000	0.335083
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	−4.00000	−0.167248
$573$	−18.0000	−0.751961
$574$	18.0000	0.751305
$575$	6.00000	0.250217
$576$	−8.00000	−0.333333
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	26.0000	1.08146
$579$	−11.0000	−0.457144
$580$	−6.00000	−0.249136
$581$	27.0000	1.12015
$582$	34.0000	1.40935
$583$	−12.0000	−0.496989
$584$	0	0
$585$	1.00000	0.0413449
$586$	8.00000	0.330477
$587$	−31.0000	−1.27951	−0.639753	−	0.768580i	$-0.720964\pi$
$587$	−31.0000	−1.27951	−0.639753	+	0.768580i	$0.720964\pi$
$588$	4.00000	0.164957
$589$	−4.00000	−0.164817
$590$	10.0000	0.411693
$591$	−19.0000	−0.781556
$592$	24.0000	0.986394
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	4.00000	0.164122
$595$	−6.00000	−0.245976
$596$	22.0000	0.901155
$597$	−4.00000	−0.163709
$598$	−12.0000	−0.490716
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	−16.0000	−0.652654	−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000	−0.652654	−0.326327	+	0.945257i	$0.605811\pi$
$602$	30.0000	1.22271
$603$	3.00000	0.122169
$604$	−6.00000	−0.244137
$605$	−7.00000	−0.284590
$606$	−4.00000	−0.162489
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	−32.0000	−1.29777
$609$	9.00000	0.364698
$610$	−12.0000	−0.485866
$611$	−12.0000	−0.485468
$612$	4.00000	0.161690
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	24.0000	0.968561
$615$	3.00000	0.120972
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	12.0000	0.482711
$619$	−9.00000	−0.361741	−0.180870	−	0.983507i	$-0.557891\pi$
$619$	−9.00000	−0.361741	−0.180870	+	0.983507i	$0.557891\pi$
$620$	2.00000	0.0803219
$621$	6.00000	0.240772
$622$	−64.0000	−2.56617
$623$	18.0000	0.721155
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	8.00000	0.319489
$628$	−12.0000	−0.478852
$629$	−12.0000	−0.478471
$630$	6.00000	0.239046
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	13.0000	0.516704
$634$	−64.0000	−2.54176
$635$	−5.00000	−0.198419
$636$	12.0000	0.475831
$637$	2.00000	0.0792429
$638$	−12.0000	−0.475085
$639$	8.00000	0.316475
$640$	0	0
$641$	47.0000	1.85639	0.928194	−	0.372096i	$-0.121361\pi$
$641$	47.0000	1.85639	0.928194	+	0.372096i	$0.121361\pi$
$642$	38.0000	1.49974
$643$	−6.00000	−0.236617	−0.118308	−	0.992977i	$-0.537747\pi$
$643$	−6.00000	−0.236617	−0.118308	+	0.992977i	$0.537747\pi$
$644$	−36.0000	−1.41860
$645$	5.00000	0.196875
$646$	16.0000	0.629512
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	0	0
$649$	10.0000	0.392534
$650$	−2.00000	−0.0784465
$651$	−3.00000	−0.117579
$652$	40.0000	1.56652
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	−28.0000	−1.09489
$655$	6.00000	0.234439
$656$	−12.0000	−0.468521
$657$	10.0000	0.390137
$658$	−72.0000	−2.80685
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	−4.00000	−0.155700
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	−10.0000	−0.388661
$663$	2.00000	0.0776736
$664$	0	0
$665$	12.0000	0.465340
$666$	12.0000	0.464991
$667$	−18.0000	−0.696963
$668$	−24.0000	−0.928588
$669$	−18.0000	−0.695920
$670$	−6.00000	−0.231800
$671$	−12.0000	−0.463255
$672$	−24.0000	−0.925820
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	−52.0000	−2.00297
$675$	1.00000	0.0384900
$676$	2.00000	0.0769231
$677$	−4.00000	−0.153732	−0.0768662	−	0.997041i	$-0.524491\pi$
$677$	−4.00000	−0.153732	−0.0768662	+	0.997041i	$0.524491\pi$
$678$	−18.0000	−0.691286
$679$	51.0000	1.95720
$680$	0	0
$681$	−22.0000	−0.843042
$682$	4.00000	0.153168
$683$	−30.0000	−1.14792	−0.573959	−	0.818884i	$-0.694593\pi$
$683$	−30.0000	−1.14792	−0.573959	+	0.818884i	$0.694593\pi$
$684$	−8.00000	−0.305888
$685$	−6.00000	−0.229248
$686$	−30.0000	−1.14541
$687$	−22.0000	−0.839352
$688$	−20.0000	−0.762493
$689$	6.00000	0.228582
$690$	−12.0000	−0.456832
$691$	−26.0000	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$691$	−26.0000	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$692$	−42.0000	−1.59660
$693$	6.00000	0.227921
$694$	−48.0000	−1.82206
$695$	4.00000	0.151729
$696$	0	0
$697$	6.00000	0.227266
$698$	−4.00000	−0.151402
$699$	−25.0000	−0.945587
$700$	−6.00000	−0.226779
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	−2.00000	−0.0754851
$703$	24.0000	0.905177
$704$	16.0000	0.603023
$705$	−12.0000	−0.451946
$706$	30.0000	1.12906
$707$	−6.00000	−0.225653
$708$	−10.0000	−0.375823
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	−16.0000	−0.600469
$711$	−2.00000	−0.0750059
$712$	0	0
$713$	6.00000	0.224702
$714$	12.0000	0.449089
$715$	−2.00000	−0.0747958
$716$	10.0000	0.373718
$717$	6.00000	0.224074
$718$	−2.00000	−0.0746393
$719$	−13.0000	−0.484818	−0.242409	−	0.970174i	$-0.577938\pi$
$719$	−13.0000	−0.484818	−0.242409	+	0.970174i	$0.577938\pi$
$720$	−4.00000	−0.149071
$721$	18.0000	0.670355
$722$	6.00000	0.223297
$723$	25.0000	0.929760
$724$	24.0000	0.891953
$725$	−3.00000	−0.111417
$726$	14.0000	0.519589
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−20.0000	−0.740233
$731$	10.0000	0.369863
$732$	12.0000	0.443533
$733$	5.00000	0.184679	0.0923396	−	0.995728i	$-0.470565\pi$
$733$	5.00000	0.184679	0.0923396	+	0.995728i	$0.470565\pi$
$734$	64.0000	2.36228
$735$	2.00000	0.0737711
$736$	48.0000	1.76930
$737$	−6.00000	−0.221013
$738$	−6.00000	−0.220863
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	−12.0000	−0.441129
$741$	−4.00000	−0.146944
$742$	36.0000	1.32160
$743$	21.0000	0.770415	0.385208	−	0.922830i	$-0.374130\pi$
$743$	21.0000	0.770415	0.385208	+	0.922830i	$0.374130\pi$
$744$	0	0
$745$	11.0000	0.403009
$746$	−20.0000	−0.732252
$747$	−9.00000	−0.329293
$748$	−8.00000	−0.292509
$749$	57.0000	2.08273
$750$	−2.00000	−0.0730297
$751$	7.00000	0.255434	0.127717	−	0.991811i	$-0.459235\pi$
$751$	7.00000	0.255434	0.127717	+	0.991811i	$0.459235\pi$
$752$	48.0000	1.75038
$753$	−21.0000	−0.765283
$754$	6.00000	0.218507
$755$	−3.00000	−0.109181
$756$	−6.00000	−0.218218
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	4.00000	0.145287
$759$	−12.0000	−0.435572
$760$	0	0
$761$	−38.0000	−1.37750	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000	−1.37750	−0.688749	+	0.724999i	$0.741840\pi$
$762$	10.0000	0.362262
$763$	−42.0000	−1.52050
$764$	−36.0000	−1.30243
$765$	2.00000	0.0723102
$766$	18.0000	0.650366
$767$	−5.00000	−0.180540
$768$	16.0000	0.577350
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	−12.0000	−0.432450
$771$	−6.00000	−0.216085
$772$	−22.0000	−0.791797
$773$	−41.0000	−1.47467	−0.737334	−	0.675529i	$-0.763915\pi$
$773$	−41.0000	−1.47467	−0.737334	+	0.675529i	$0.763915\pi$
$774$	−10.0000	−0.359443
$775$	1.00000	0.0359211
$776$	0	0
$777$	18.0000	0.645746
$778$	10.0000	0.358517
$779$	−12.0000	−0.429945
$780$	2.00000	0.0716115
$781$	−16.0000	−0.572525
$782$	−24.0000	−0.858238
$783$	−3.00000	−0.107211
$784$	−8.00000	−0.285714
$785$	−6.00000	−0.214149
$786$	−12.0000	−0.428026
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	−38.0000	−1.35369
$789$	−16.0000	−0.569615
$790$	4.00000	0.142314
$791$	−27.0000	−0.960009
$792$	0	0
$793$	6.00000	0.213066
$794$	−36.0000	−1.27759
$795$	6.00000	0.212798
$796$	−8.00000	−0.283552
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	−24.0000	−0.849591
$799$	−24.0000	−0.849059
$800$	8.00000	0.282843
$801$	−6.00000	−0.212000
$802$	−48.0000	−1.69494
$803$	−20.0000	−0.705785
$804$	6.00000	0.211604
$805$	−18.0000	−0.634417
$806$	−2.00000	−0.0704470
$807$	−10.0000	−0.352017
$808$	0	0
$809$	9.00000	0.316423	0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000	0.316423	0.158212	+	0.987405i	$0.449427\pi$
$810$	−2.00000	−0.0702728
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	18.0000	0.631676
$813$	−21.0000	−0.736502
$814$	−24.0000	−0.841200
$815$	20.0000	0.700569
$816$	−8.00000	−0.280056
$817$	−20.0000	−0.699711
$818$	38.0000	1.32864
$819$	−3.00000	−0.104828
$820$	6.00000	0.209529
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	12.0000	0.418548
$823$	−41.0000	−1.42917	−0.714585	−	0.699549i	$-0.753384\pi$
$823$	−41.0000	−1.42917	−0.714585	+	0.699549i	$0.753384\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	−30.0000	−1.04383
$827$	15.0000	0.521601	0.260801	−	0.965393i	$-0.416014\pi$
$827$	15.0000	0.521601	0.260801	+	0.965393i	$0.416014\pi$
$828$	12.0000	0.417029
$829$	36.0000	1.25033	0.625166	−	0.780492i	$-0.285031\pi$
$829$	36.0000	1.25033	0.625166	+	0.780492i	$0.285031\pi$
$830$	18.0000	0.624789
$831$	3.00000	0.104069
$832$	−8.00000	−0.277350
$833$	4.00000	0.138592
$834$	−8.00000	−0.277017
$835$	−12.0000	−0.415277
$836$	16.0000	0.553372
$837$	1.00000	0.0345651
$838$	36.0000	1.24360
$839$	9.00000	0.310715	0.155357	−	0.987858i	$-0.450347\pi$
$839$	9.00000	0.310715	0.155357	+	0.987858i	$0.450347\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	48.0000	1.65419
$843$	1.00000	0.0344418
$844$	26.0000	0.894957
$845$	1.00000	0.0344010
$846$	24.0000	0.825137
$847$	21.0000	0.721569
$848$	−24.0000	−0.824163
$849$	22.0000	0.755038
$850$	−4.00000	−0.137199
$851$	−36.0000	−1.23406
$852$	16.0000	0.548151
$853$	19.0000	0.650548	0.325274	−	0.945620i	$-0.394544\pi$
$853$	19.0000	0.650548	0.325274	+	0.945620i	$0.394544\pi$
$854$	36.0000	1.23189
$855$	−4.00000	−0.136797
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	4.00000	0.136558
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	10.0000	0.340997
$861$	−9.00000	−0.306719
$862$	−18.0000	−0.613082
$863$	−21.0000	−0.714848	−0.357424	−	0.933942i	$-0.616345\pi$
$863$	−21.0000	−0.714848	−0.357424	+	0.933942i	$0.616345\pi$
$864$	8.00000	0.272166
$865$	−21.0000	−0.714021
$866$	−26.0000	−0.883516
$867$	−13.0000	−0.441503
$868$	−6.00000	−0.203653
$869$	4.00000	0.135691
$870$	6.00000	0.203419
$871$	3.00000	0.101651
$872$	0	0
$873$	−17.0000	−0.575363
$874$	48.0000	1.62362
$875$	−3.00000	−0.101419
$876$	20.0000	0.675737
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	−66.0000	−2.22739
$879$	−4.00000	−0.134917
$880$	8.00000	0.269680
$881$	−29.0000	−0.977035	−0.488517	−	0.872554i	$-0.662462\pi$
$881$	−29.0000	−0.977035	−0.488517	+	0.872554i	$0.662462\pi$
$882$	−4.00000	−0.134687
$883$	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
$885$	−5.00000	−0.168073
$886$	58.0000	1.94855
$887$	21.0000	0.705111	0.352555	−	0.935791i	$-0.385313\pi$
$887$	21.0000	0.705111	0.352555	+	0.935791i	$0.385313\pi$
$888$	0	0
$889$	15.0000	0.503084
$890$	12.0000	0.402241
$891$	−2.00000	−0.0670025
$892$	−36.0000	−1.20537
$893$	48.0000	1.60626
$894$	−22.0000	−0.735790
$895$	5.00000	0.167132
$896$	0	0
$897$	6.00000	0.200334
$898$	−12.0000	−0.400445
$899$	−3.00000	−0.100056
$900$	2.00000	0.0666667
$901$	12.0000	0.399778
$902$	12.0000	0.399556
$903$	−15.0000	−0.499169
$904$	0	0
$905$	12.0000	0.398893
$906$	6.00000	0.199337
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	−44.0000	−1.46019
$909$	2.00000	0.0663358
$910$	6.00000	0.198898
$911$	43.0000	1.42465	0.712327	−	0.701848i	$-0.247641\pi$
$911$	43.0000	1.42465	0.712327	+	0.701848i	$0.247641\pi$
$912$	16.0000	0.529813
$913$	18.0000	0.595713
$914$	−40.0000	−1.32308
$915$	6.00000	0.198354
$916$	−44.0000	−1.45380
$917$	−18.0000	−0.594412
$918$	−4.00000	−0.132020
$919$	31.0000	1.02260	0.511298	−	0.859404i	$-0.329165\pi$
$919$	31.0000	1.02260	0.511298	+	0.859404i	$0.329165\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	−8.00000	−0.263466
$923$	8.00000	0.263323
$924$	12.0000	0.394771
$925$	−6.00000	−0.197279
$926$	−32.0000	−1.05159
$927$	−6.00000	−0.197066
$928$	−24.0000	−0.787839
$929$	−4.00000	−0.131236	−0.0656179	−	0.997845i	$-0.520902\pi$
$929$	−4.00000	−0.131236	−0.0656179	+	0.997845i	$0.520902\pi$
$930$	−2.00000	−0.0655826
$931$	−8.00000	−0.262189
$932$	−50.0000	−1.63780
$933$	32.0000	1.04763
$934$	−34.0000	−1.11251
$935$	−4.00000	−0.130814
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	18.0000	0.587721
$939$	−3.00000	−0.0979013
$940$	−24.0000	−0.782794
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	12.0000	0.390981
$943$	18.0000	0.586161
$944$	20.0000	0.650945
$945$	−3.00000	−0.0975900
$946$	20.0000	0.650256
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	−4.00000	−0.129914
$949$	10.0000	0.324614
$950$	8.00000	0.259554
$951$	32.0000	1.03767
$952$	0	0
$953$	−28.0000	−0.907009	−0.453504	−	0.891254i	$-0.649826\pi$
$953$	−28.0000	−0.907009	−0.453504	+	0.891254i	$0.649826\pi$
$954$	−12.0000	−0.388514
$955$	−18.0000	−0.582466
$956$	12.0000	0.388108
$957$	6.00000	0.193952
$958$	74.0000	2.39083
$959$	18.0000	0.581250
$960$	−8.00000	−0.258199
$961$	1.00000	0.0322581
$962$	12.0000	0.386896
$963$	−19.0000	−0.612266
$964$	50.0000	1.61039
$965$	−11.0000	−0.354103
$966$	36.0000	1.15828
$967$	−44.0000	−1.41494	−0.707472	−	0.706741i	$-0.750165\pi$
$967$	−44.0000	−1.41494	−0.707472	+	0.706741i	$0.750165\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	34.0000	1.09167
$971$	38.0000	1.21948	0.609739	−	0.792602i	$-0.291274\pi$
$971$	38.0000	1.21948	0.609739	+	0.792602i	$0.291274\pi$
$972$	2.00000	0.0641500
$973$	−12.0000	−0.384702
$974$	32.0000	1.02535
$975$	1.00000	0.0320256
$976$	−24.0000	−0.768221
$977$	−28.0000	−0.895799	−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000	−0.895799	−0.447900	+	0.894084i	$0.647828\pi$
$978$	−40.0000	−1.27906
$979$	12.0000	0.383522
$980$	4.00000	0.127775
$981$	14.0000	0.446986
$982$	78.0000	2.48908
$983$	59.0000	1.88181	0.940904	−	0.338674i	$-0.109978\pi$
$983$	59.0000	1.88181	0.940904	+	0.338674i	$0.109978\pi$
$984$	0	0
$985$	−19.0000	−0.605390
$986$	12.0000	0.382158
$987$	36.0000	1.14589
$988$	−8.00000	−0.254514
$989$	30.0000	0.953945
$990$	4.00000	0.127128
$991$	−18.0000	−0.571789	−0.285894	−	0.958261i	$-0.592291\pi$
$991$	−18.0000	−0.571789	−0.285894	+	0.958261i	$0.592291\pi$
$992$	8.00000	0.254000
$993$	5.00000	0.158670
$994$	48.0000	1.52247
$995$	−4.00000	−0.126809
$996$	−18.0000	−0.570352
$997$	4.00000	0.126681	0.0633406	−	0.997992i	$-0.479825\pi$
$997$	4.00000	0.126681	0.0633406	+	0.997992i	$0.479825\pi$
$998$	78.0000	2.46905
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.c.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.c.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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