LMFDB - Embedded newform 4021.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4021,2,Mod(1,4021)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4021, 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("4021.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4021.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:	$32.1078466528$
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	$\chi$	$=$	4021.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +3.00000 q^{5} -2.00000 q^{6} +4.00000 q^{7} -2.00000 q^{9} -6.00000 q^{10} -2.00000 q^{11} +2.00000 q^{12} -7.00000 q^{13} -8.00000 q^{14} +3.00000 q^{15} -4.00000 q^{16} +2.00000 q^{17} +4.00000 q^{18} -6.00000 q^{19} +6.00000 q^{20} +4.00000 q^{21} +4.00000 q^{22} +2.00000 q^{23} +4.00000 q^{25} +14.0000 q^{26} -5.00000 q^{27} +8.00000 q^{28} -6.00000 q^{30} -8.00000 q^{31} +8.00000 q^{32} -2.00000 q^{33} -4.00000 q^{34} +12.0000 q^{35} -4.00000 q^{36} +5.00000 q^{37} +12.0000 q^{38} -7.00000 q^{39} +2.00000 q^{41} -8.00000 q^{42} -10.0000 q^{43} -4.00000 q^{44} -6.00000 q^{45} -4.00000 q^{46} +4.00000 q^{47} -4.00000 q^{48} +9.00000 q^{49} -8.00000 q^{50} +2.00000 q^{51} -14.0000 q^{52} -5.00000 q^{53} +10.0000 q^{54} -6.00000 q^{55} -6.00000 q^{57} -7.00000 q^{59} +6.00000 q^{60} +7.00000 q^{61} +16.0000 q^{62} -8.00000 q^{63} -8.00000 q^{64} -21.0000 q^{65} +4.00000 q^{66} -9.00000 q^{67} +4.00000 q^{68} +2.00000 q^{69} -24.0000 q^{70} -3.00000 q^{71} -7.00000 q^{73} -10.0000 q^{74} +4.00000 q^{75} -12.0000 q^{76} -8.00000 q^{77} +14.0000 q^{78} -12.0000 q^{80} +1.00000 q^{81} -4.00000 q^{82} -12.0000 q^{83} +8.00000 q^{84} +6.00000 q^{85} +20.0000 q^{86} +6.00000 q^{89} +12.0000 q^{90} -28.0000 q^{91} +4.00000 q^{92} -8.00000 q^{93} -8.00000 q^{94} -18.0000 q^{95} +8.00000 q^{96} +10.0000 q^{97} -18.0000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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 $

$2$	−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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	2.00000	1.00000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	−2.00000	−0.816497
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	−6.00000	−1.89737
$11$	−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
$13$	−7.00000	−1.94145	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−7.00000	−1.94145	−0.970725	+	0.240192i	$0.922790\pi$
$14$	−8.00000	−2.13809
$15$	3.00000	0.774597
$16$	−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	$0.422021\pi$
$18$	4.00000	0.942809
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	6.00000	1.34164
$21$	4.00000	0.872872
$22$	4.00000	0.852803
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	14.0000	2.74563
$27$	−5.00000	−0.962250
$28$	8.00000	1.51186
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−6.00000	−1.09545
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	8.00000	1.41421
$33$	−2.00000	−0.348155
$34$	−4.00000	−0.685994
$35$	12.0000	2.02837
$36$	−4.00000	−0.666667
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	12.0000	1.94666
$39$	−7.00000	−1.12090
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−8.00000	−1.23443
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	−4.00000	−0.603023
$45$	−6.00000	−0.894427
$46$	−4.00000	−0.589768
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	−4.00000	−0.577350
$49$	9.00000	1.28571
$50$	−8.00000	−1.13137
$51$	2.00000	0.280056
$52$	−14.0000	−1.94145
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	10.0000	1.36083
$55$	−6.00000	−0.809040
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	6.00000	0.774597
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	16.0000	2.03200
$63$	−8.00000	−1.00791
$64$	−8.00000	−1.00000
$65$	−21.0000	−2.60473
$66$	4.00000	0.492366
$67$	−9.00000	−1.09952	−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000	−1.09952	−0.549762	+	0.835321i	$0.685282\pi$
$68$	4.00000	0.485071
$69$	2.00000	0.240772
$70$	−24.0000	−2.86855
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	−10.0000	−1.16248
$75$	4.00000	0.461880
$76$	−12.0000	−1.37649
$77$	−8.00000	−0.911685
$78$	14.0000	1.58519
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−12.0000	−1.34164
$81$	1.00000	0.111111
$82$	−4.00000	−0.441726
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	8.00000	0.872872
$85$	6.00000	0.650791
$86$	20.0000	2.15666
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	12.0000	1.26491
$91$	−28.0000	−2.93520
$92$	4.00000	0.417029
$93$	−8.00000	−0.829561
$94$	−8.00000	−0.825137
$95$	−18.0000	−1.84676
$96$	8.00000	0.816497
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−18.0000	−1.81827
$99$	4.00000	0.402015
$100$	8.00000	0.800000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	−4.00000	−0.396059
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	0	0
$105$	12.0000	1.17108
$106$	10.0000	0.971286
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	−10.0000	−0.962250
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	12.0000	1.14416
$111$	5.00000	0.474579
$112$	−16.0000	−1.51186
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	12.0000	1.12390
$115$	6.00000	0.559503
$116$	0	0
$117$	14.0000	1.29430
$118$	14.0000	1.28880
$119$	8.00000	0.733359
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−14.0000	−1.26750
$123$	2.00000	0.180334
$124$	−16.0000	−1.43684
$125$	−3.00000	−0.268328
$126$	16.0000	1.42539
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	42.0000	3.68364
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	−4.00000	−0.348155
$133$	−24.0000	−2.08106
$134$	18.0000	1.55496
$135$	−15.0000	−1.29099
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	−4.00000	−0.340503
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	24.0000	2.02837
$141$	4.00000	0.336861
$142$	6.00000	0.503509
$143$	14.0000	1.17074
$144$	8.00000	0.666667
$145$	0	0
$146$	14.0000	1.15865
$147$	9.00000	0.742307
$148$	10.0000	0.821995
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	−8.00000	−0.653197
$151$	23.0000	1.87171	0.935857	−	0.352381i	$-0.114628\pi$
$151$	23.0000	1.87171	0.935857	+	0.352381i	$0.114628\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	16.0000	1.28932
$155$	−24.0000	−1.92773
$156$	−14.0000	−1.12090
$157$	−16.0000	−1.27694	−0.638470	−	0.769647i	$-0.720432\pi$
$157$	−16.0000	−1.27694	−0.638470	+	0.769647i	$0.720432\pi$
$158$	0	0
$159$	−5.00000	−0.396526
$160$	24.0000	1.89737
$161$	8.00000	0.630488
$162$	−2.00000	−0.157135
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	4.00000	0.312348
$165$	−6.00000	−0.467099
$166$	24.0000	1.86276
$167$	−22.0000	−1.70241	−0.851206	−	0.524832i	$-0.824128\pi$
$167$	−22.0000	−1.70241	−0.851206	+	0.524832i	$0.824128\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	−12.0000	−0.920358
$171$	12.0000	0.917663
$172$	−20.0000	−1.52499
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	0	0
$175$	16.0000	1.20949
$176$	8.00000	0.603023
$177$	−7.00000	−0.526152
$178$	−12.0000	−0.899438
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−12.0000	−0.894427
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	56.0000	4.15100
$183$	7.00000	0.517455
$184$	0	0
$185$	15.0000	1.10282
$186$	16.0000	1.17318
$187$	−4.00000	−0.292509
$188$	8.00000	0.583460
$189$	−20.0000	−1.45479
$190$	36.0000	2.61171
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	−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	$0.629567\pi$
$194$	−20.0000	−1.43592
$195$	−21.0000	−1.50384
$196$	18.0000	1.28571
$197$	15.0000	1.06871	0.534353	−	0.845262i	$-0.320555\pi$
$197$	15.0000	1.06871	0.534353	+	0.845262i	$0.320555\pi$
$198$	−8.00000	−0.568535
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−9.00000	−0.634811
$202$	36.0000	2.53295
$203$	0	0
$204$	4.00000	0.280056
$205$	6.00000	0.419058
$206$	26.0000	1.81151
$207$	−4.00000	−0.278019
$208$	28.0000	1.94145
$209$	12.0000	0.830057
$210$	−24.0000	−1.65616
$211$	−18.0000	−1.23917	−0.619586	−	0.784929i	$-0.712699\pi$
$211$	−18.0000	−1.23917	−0.619586	+	0.784929i	$0.712699\pi$
$212$	−10.0000	−0.686803
$213$	−3.00000	−0.205557
$214$	8.00000	0.546869
$215$	−30.0000	−2.04598
$216$	0	0
$217$	−32.0000	−2.17230
$218$	−22.0000	−1.49003
$219$	−7.00000	−0.473016
$220$	−12.0000	−0.809040
$221$	−14.0000	−0.941742
$222$	−10.0000	−0.671156
$223$	21.0000	1.40626	0.703132	−	0.711059i	$-0.251784\pi$
$223$	21.0000	1.40626	0.703132	+	0.711059i	$0.251784\pi$
$224$	32.0000	2.13809
$225$	−8.00000	−0.533333
$226$	4.00000	0.266076
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	−12.0000	−0.794719
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−12.0000	−0.791257
$231$	−8.00000	−0.526361
$232$	0	0
$233$	−15.0000	−0.982683	−0.491341	−	0.870967i	$-0.663493\pi$
$233$	−15.0000	−0.982683	−0.491341	+	0.870967i	$0.663493\pi$
$234$	−28.0000	−1.83042
$235$	12.0000	0.782794
$236$	−14.0000	−0.911322
$237$	0	0
$238$	−16.0000	−1.03713
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	−12.0000	−0.774597
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	14.0000	0.899954
$243$	16.0000	1.02640
$244$	14.0000	0.896258
$245$	27.0000	1.72497
$246$	−4.00000	−0.255031
$247$	42.0000	2.67240
$248$	0	0
$249$	−12.0000	−0.760469
$250$	6.00000	0.379473
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	−16.0000	−1.00791
$253$	−4.00000	−0.251478
$254$	−32.0000	−2.00786
$255$	6.00000	0.375735
$256$	16.0000	1.00000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	20.0000	1.24515
$259$	20.0000	1.24274
$260$	−42.0000	−2.60473
$261$	0	0
$262$	−32.0000	−1.97697
$263$	−20.0000	−1.23325	−0.616626	−	0.787256i	$-0.711501\pi$
$263$	−20.0000	−1.23325	−0.616626	+	0.787256i	$0.711501\pi$
$264$	0	0
$265$	−15.0000	−0.921443
$266$	48.0000	2.94307
$267$	6.00000	0.367194
$268$	−18.0000	−1.09952
$269$	1.00000	0.0609711	0.0304855	−	0.999535i	$-0.490295\pi$
$269$	1.00000	0.0609711	0.0304855	+	0.999535i	$0.490295\pi$
$270$	30.0000	1.82574
$271$	6.00000	0.364474	0.182237	−	0.983255i	$-0.441666\pi$
$271$	6.00000	0.364474	0.182237	+	0.983255i	$0.441666\pi$
$272$	−8.00000	−0.485071
$273$	−28.0000	−1.69464
$274$	16.0000	0.966595
$275$	−8.00000	−0.482418
$276$	4.00000	0.240772
$277$	32.0000	1.92269	0.961347	−	0.275340i	$-0.0887905\pi$
$277$	32.0000	1.92269	0.961347	+	0.275340i	$0.0887905\pi$
$278$	−24.0000	−1.43942
$279$	16.0000	0.957895
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	−8.00000	−0.476393
$283$	−7.00000	−0.416107	−0.208053	−	0.978117i	$-0.566713\pi$
$283$	−7.00000	−0.416107	−0.208053	+	0.978117i	$0.566713\pi$
$284$	−6.00000	−0.356034
$285$	−18.0000	−1.06623
$286$	−28.0000	−1.65567
$287$	8.00000	0.472225
$288$	−16.0000	−0.942809
$289$	−13.0000	−0.764706
$290$	0	0
$291$	10.0000	0.586210
$292$	−14.0000	−0.819288
$293$	−11.0000	−0.642627	−0.321313	−	0.946973i	$-0.604124\pi$
$293$	−11.0000	−0.642627	−0.321313	+	0.946973i	$0.604124\pi$
$294$	−18.0000	−1.04978
$295$	−21.0000	−1.22267
$296$	0	0
$297$	10.0000	0.580259
$298$	24.0000	1.39028
$299$	−14.0000	−0.809641
$300$	8.00000	0.461880
$301$	−40.0000	−2.30556
$302$	−46.0000	−2.64700
$303$	−18.0000	−1.03407
$304$	24.0000	1.37649
$305$	21.0000	1.20246
$306$	8.00000	0.457330
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	−16.0000	−0.911685
$309$	−13.0000	−0.739544
$310$	48.0000	2.72622
$311$	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
$313$	−21.0000	−1.18699	−0.593495	−	0.804838i	$-0.702252\pi$
$313$	−21.0000	−1.18699	−0.593495	+	0.804838i	$0.702252\pi$
$314$	32.0000	1.80586
$315$	−24.0000	−1.35225
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	10.0000	0.560772
$319$	0	0
$320$	−24.0000	−1.34164
$321$	−4.00000	−0.223258
$322$	−16.0000	−0.891645
$323$	−12.0000	−0.667698
$324$	2.00000	0.111111
$325$	−28.0000	−1.55316
$326$	−20.0000	−1.10770
$327$	11.0000	0.608301
$328$	0	0
$329$	16.0000	0.882109
$330$	12.0000	0.660578
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−24.0000	−1.31717
$333$	−10.0000	−0.547997
$334$	44.0000	2.40757
$335$	−27.0000	−1.47517
$336$	−16.0000	−0.872872
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	−72.0000	−3.91628
$339$	−2.00000	−0.108625
$340$	12.0000	0.650791
$341$	16.0000	0.866449
$342$	−24.0000	−1.29777
$343$	8.00000	0.431959
$344$	0	0
$345$	6.00000	0.323029
$346$	−8.00000	−0.430083
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	−32.0000	−1.71047
$351$	35.0000	1.86816
$352$	−16.0000	−0.852803
$353$	−16.0000	−0.851594	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	−16.0000	−0.851594	−0.425797	+	0.904819i	$0.640006\pi$
$354$	14.0000	0.744092
$355$	−9.00000	−0.477670
$356$	12.0000	0.635999
$357$	8.00000	0.423405
$358$	0	0
$359$	−9.00000	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−4.00000	−0.210235
$363$	−7.00000	−0.367405
$364$	−56.0000	−2.93520
$365$	−21.0000	−1.09919
$366$	−14.0000	−0.731792
$367$	−2.00000	−0.104399	−0.0521996	−	0.998637i	$-0.516623\pi$
$367$	−2.00000	−0.104399	−0.0521996	+	0.998637i	$0.516623\pi$
$368$	−8.00000	−0.417029
$369$	−4.00000	−0.208232
$370$	−30.0000	−1.55963
$371$	−20.0000	−1.03835
$372$	−16.0000	−0.829561
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	8.00000	0.413670
$375$	−3.00000	−0.154919
$376$	0	0
$377$	0	0
$378$	40.0000	2.05738
$379$	22.0000	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$379$	22.0000	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$380$	−36.0000	−1.84676
$381$	16.0000	0.819705
$382$	6.00000	0.306987
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	22.0000	1.11977
$387$	20.0000	1.01666
$388$	20.0000	1.01535
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	42.0000	2.12675
$391$	4.00000	0.202289
$392$	0	0
$393$	16.0000	0.807093
$394$	−30.0000	−1.51138
$395$	0	0
$396$	8.00000	0.402015
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	32.0000	1.60402
$399$	−24.0000	−1.20150
$400$	−16.0000	−0.800000
$401$	9.00000	0.449439	0.224719	−	0.974424i	$-0.427853\pi$
$401$	9.00000	0.449439	0.224719	+	0.974424i	$0.427853\pi$
$402$	18.0000	0.897758
$403$	56.0000	2.78956
$404$	−36.0000	−1.79107
$405$	3.00000	0.149071
$406$	0	0
$407$	−10.0000	−0.495682
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	−12.0000	−0.592638
$411$	−8.00000	−0.394611
$412$	−26.0000	−1.28093
$413$	−28.0000	−1.37779
$414$	8.00000	0.393179
$415$	−36.0000	−1.76717
$416$	−56.0000	−2.74563
$417$	12.0000	0.587643
$418$	−24.0000	−1.17388
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	24.0000	1.17108
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	36.0000	1.75245
$423$	−8.00000	−0.388973
$424$	0	0
$425$	8.00000	0.388057
$426$	6.00000	0.290701
$427$	28.0000	1.35501
$428$	−8.00000	−0.386695
$429$	14.0000	0.675926
$430$	60.0000	2.89346
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	20.0000	0.962250
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	64.0000	3.07210
$435$	0	0
$436$	22.0000	1.05361
$437$	−12.0000	−0.574038
$438$	14.0000	0.668946
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	28.0000	1.33182
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	10.0000	0.474579
$445$	18.0000	0.853282
$446$	−42.0000	−1.98876
$447$	−12.0000	−0.567581
$448$	−32.0000	−1.51186
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	16.0000	0.754247
$451$	−4.00000	−0.188353
$452$	−4.00000	−0.188144
$453$	23.0000	1.08063
$454$	−56.0000	−2.62821
$455$	−84.0000	−3.93798
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	12.0000	0.560723
$459$	−10.0000	−0.466760
$460$	12.0000	0.559503
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	16.0000	0.744387
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	−24.0000	−1.11297
$466$	30.0000	1.38972
$467$	21.0000	0.971764	0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000	0.971764	0.485882	+	0.874024i	$0.338498\pi$
$468$	28.0000	1.29430
$469$	−36.0000	−1.66233
$470$	−24.0000	−1.10704
$471$	−16.0000	−0.737241
$472$	0	0
$473$	20.0000	0.919601
$474$	0	0
$475$	−24.0000	−1.10120
$476$	16.0000	0.733359
$477$	10.0000	0.457869
$478$	18.0000	0.823301
$479$	−13.0000	−0.593985	−0.296993	−	0.954880i	$-0.595984\pi$
$479$	−13.0000	−0.593985	−0.296993	+	0.954880i	$0.595984\pi$
$480$	24.0000	1.09545
$481$	−35.0000	−1.59586
$482$	0	0
$483$	8.00000	0.364013
$484$	−14.0000	−0.636364
$485$	30.0000	1.36223
$486$	−32.0000	−1.45155
$487$	−18.0000	−0.815658	−0.407829	−	0.913058i	$-0.633714\pi$
$487$	−18.0000	−0.815658	−0.407829	+	0.913058i	$0.633714\pi$
$488$	0	0
$489$	10.0000	0.452216
$490$	−54.0000	−2.43947
$491$	−35.0000	−1.57953	−0.789764	−	0.613411i	$-0.789797\pi$
$491$	−35.0000	−1.57953	−0.789764	+	0.613411i	$0.789797\pi$
$492$	4.00000	0.180334
$493$	0	0
$494$	−84.0000	−3.77934
$495$	12.0000	0.539360
$496$	32.0000	1.43684
$497$	−12.0000	−0.538274
$498$	24.0000	1.07547
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	−6.00000	−0.268328
$501$	−22.0000	−0.982888
$502$	40.0000	1.78529
$503$	−2.00000	−0.0891756	−0.0445878	−	0.999005i	$-0.514197\pi$
$503$	−2.00000	−0.0891756	−0.0445878	+	0.999005i	$0.514197\pi$
$504$	0	0
$505$	−54.0000	−2.40297
$506$	8.00000	0.355643
$507$	36.0000	1.59882
$508$	32.0000	1.41977
$509$	−8.00000	−0.354594	−0.177297	−	0.984157i	$-0.556735\pi$
$509$	−8.00000	−0.354594	−0.177297	+	0.984157i	$0.556735\pi$
$510$	−12.0000	−0.531369
$511$	−28.0000	−1.23865
$512$	−32.0000	−1.41421
$513$	30.0000	1.32453
$514$	4.00000	0.176432
$515$	−39.0000	−1.71855
$516$	−20.0000	−0.880451
$517$	−8.00000	−0.351840
$518$	−40.0000	−1.75750
$519$	4.00000	0.175581
$520$	0	0
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	0	0
$523$	−17.0000	−0.743358	−0.371679	−	0.928361i	$-0.621218\pi$
$523$	−17.0000	−0.743358	−0.371679	+	0.928361i	$0.621218\pi$
$524$	32.0000	1.39793
$525$	16.0000	0.698297
$526$	40.0000	1.74408
$527$	−16.0000	−0.696971
$528$	8.00000	0.348155
$529$	−19.0000	−0.826087
$530$	30.0000	1.30312
$531$	14.0000	0.607548
$532$	−48.0000	−2.08106
$533$	−14.0000	−0.606407
$534$	−12.0000	−0.519291
$535$	−12.0000	−0.518805
$536$	0	0
$537$	0	0
$538$	−2.00000	−0.0862261
$539$	−18.0000	−0.775315
$540$	−30.0000	−1.29099
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	−12.0000	−0.515444
$543$	2.00000	0.0858282
$544$	16.0000	0.685994
$545$	33.0000	1.41356
$546$	56.0000	2.39658
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−16.0000	−0.683486
$549$	−14.0000	−0.597505
$550$	16.0000	0.682242
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−64.0000	−2.71910
$555$	15.0000	0.636715
$556$	24.0000	1.01783
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	−32.0000	−1.35467
$559$	70.0000	2.96068
$560$	−48.0000	−2.02837
$561$	−4.00000	−0.168880
$562$	−52.0000	−2.19349
$563$	42.0000	1.77009	0.885044	−	0.465506i	$-0.154128\pi$
$563$	42.0000	1.77009	0.885044	+	0.465506i	$0.154128\pi$
$564$	8.00000	0.336861
$565$	−6.00000	−0.252422
$566$	14.0000	0.588464
$567$	4.00000	0.167984
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	36.0000	1.50787
$571$	17.0000	0.711428	0.355714	−	0.934595i	$-0.384238\pi$
$571$	17.0000	0.711428	0.355714	+	0.934595i	$0.384238\pi$
$572$	28.0000	1.17074
$573$	−3.00000	−0.125327
$574$	−16.0000	−0.667827
$575$	8.00000	0.333623
$576$	16.0000	0.666667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	26.0000	1.08146
$579$	−11.0000	−0.457144
$580$	0	0
$581$	−48.0000	−1.99138
$582$	−20.0000	−0.829027
$583$	10.0000	0.414158
$584$	0	0
$585$	42.0000	1.73649
$586$	22.0000	0.908812
$587$	−20.0000	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$588$	18.0000	0.742307
$589$	48.0000	1.97781
$590$	42.0000	1.72911
$591$	15.0000	0.617018
$592$	−20.0000	−0.821995
$593$	5.00000	0.205325	0.102663	−	0.994716i	$-0.467264\pi$
$593$	5.00000	0.205325	0.102663	+	0.994716i	$0.467264\pi$
$594$	−20.0000	−0.820610
$595$	24.0000	0.983904
$596$	−24.0000	−0.983078
$597$	−16.0000	−0.654836
$598$	28.0000	1.14501
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	32.0000	1.30531	0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000	1.30531	0.652654	+	0.757656i	$0.273656\pi$
$602$	80.0000	3.26056
$603$	18.0000	0.733017
$604$	46.0000	1.87171
$605$	−21.0000	−0.853771
$606$	36.0000	1.46240
$607$	1.00000	0.0405887	0.0202944	−	0.999794i	$-0.493540\pi$
$607$	1.00000	0.0405887	0.0202944	+	0.999794i	$0.493540\pi$
$608$	−48.0000	−1.94666
$609$	0	0
$610$	−42.0000	−1.70053
$611$	−28.0000	−1.13276
$612$	−8.00000	−0.323381
$613$	−45.0000	−1.81753	−0.908766	−	0.417305i	$-0.862975\pi$
$613$	−45.0000	−1.81753	−0.908766	+	0.417305i	$0.862975\pi$
$614$	8.00000	0.322854
$615$	6.00000	0.241943
$616$	0	0
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	26.0000	1.04587
$619$	−33.0000	−1.32638	−0.663191	−	0.748450i	$-0.730798\pi$
$619$	−33.0000	−1.32638	−0.663191	+	0.748450i	$0.730798\pi$
$620$	−48.0000	−1.92773
$621$	−10.0000	−0.401286
$622$	−64.0000	−2.56617
$623$	24.0000	0.961540
$624$	28.0000	1.12090
$625$	−29.0000	−1.16000
$626$	42.0000	1.67866
$627$	12.0000	0.479234
$628$	−32.0000	−1.27694
$629$	10.0000	0.398726
$630$	48.0000	1.91237
$631$	−37.0000	−1.47295	−0.736473	−	0.676467i	$-0.763510\pi$
$631$	−37.0000	−1.47295	−0.736473	+	0.676467i	$0.763510\pi$
$632$	0	0
$633$	−18.0000	−0.715436
$634$	−12.0000	−0.476581
$635$	48.0000	1.90482
$636$	−10.0000	−0.396526
$637$	−63.0000	−2.49615
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	21.0000	0.829450	0.414725	−	0.909947i	$-0.363878\pi$
$641$	21.0000	0.829450	0.414725	+	0.909947i	$0.363878\pi$
$642$	8.00000	0.315735
$643$	−3.00000	−0.118308	−0.0591542	−	0.998249i	$-0.518840\pi$
$643$	−3.00000	−0.118308	−0.0591542	+	0.998249i	$0.518840\pi$
$644$	16.0000	0.630488
$645$	−30.0000	−1.18125
$646$	24.0000	0.944267
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	14.0000	0.549548
$650$	56.0000	2.19650
$651$	−32.0000	−1.25418
$652$	20.0000	0.783260
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	−22.0000	−0.860268
$655$	48.0000	1.87552
$656$	−8.00000	−0.312348
$657$	14.0000	0.546192
$658$	−32.0000	−1.24749
$659$	−22.0000	−0.856998	−0.428499	−	0.903542i	$-0.640958\pi$
$659$	−22.0000	−0.856998	−0.428499	+	0.903542i	$0.640958\pi$
$660$	−12.0000	−0.467099
$661$	−35.0000	−1.36134	−0.680671	−	0.732589i	$-0.738312\pi$
$661$	−35.0000	−1.36134	−0.680671	+	0.732589i	$0.738312\pi$
$662$	16.0000	0.621858
$663$	−14.0000	−0.543715
$664$	0	0
$665$	−72.0000	−2.79204
$666$	20.0000	0.774984
$667$	0	0
$668$	−44.0000	−1.70241
$669$	21.0000	0.811907
$670$	54.0000	2.08620
$671$	−14.0000	−0.540464
$672$	32.0000	1.23443
$673$	20.0000	0.770943	0.385472	−	0.922720i	$-0.374039\pi$
$673$	20.0000	0.770943	0.385472	+	0.922720i	$0.374039\pi$
$674$	−32.0000	−1.23259
$675$	−20.0000	−0.769800
$676$	72.0000	2.76923
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	4.00000	0.153619
$679$	40.0000	1.53506
$680$	0	0
$681$	28.0000	1.07296
$682$	−32.0000	−1.22534
$683$	−26.0000	−0.994862	−0.497431	−	0.867503i	$-0.665723\pi$
$683$	−26.0000	−0.994862	−0.497431	+	0.867503i	$0.665723\pi$
$684$	24.0000	0.917663
$685$	−24.0000	−0.916993
$686$	−16.0000	−0.610883
$687$	−6.00000	−0.228914
$688$	40.0000	1.52499
$689$	35.0000	1.33339
$690$	−12.0000	−0.456832
$691$	18.0000	0.684752	0.342376	−	0.939563i	$-0.388768\pi$
$691$	18.0000	0.684752	0.342376	+	0.939563i	$0.388768\pi$
$692$	8.00000	0.304114
$693$	16.0000	0.607790
$694$	24.0000	0.911028
$695$	36.0000	1.36556
$696$	0	0
$697$	4.00000	0.151511
$698$	12.0000	0.454207
$699$	−15.0000	−0.567352
$700$	32.0000	1.20949
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	−70.0000	−2.64198
$703$	−30.0000	−1.13147
$704$	16.0000	0.603023
$705$	12.0000	0.451946
$706$	32.0000	1.20434
$707$	−72.0000	−2.70784
$708$	−14.0000	−0.526152
$709$	40.0000	1.50223	0.751116	−	0.660171i	$-0.229516\pi$
$709$	40.0000	1.50223	0.751116	+	0.660171i	$0.229516\pi$
$710$	18.0000	0.675528
$711$	0	0
$712$	0	0
$713$	−16.0000	−0.599205
$714$	−16.0000	−0.598785
$715$	42.0000	1.57071
$716$	0	0
$717$	−9.00000	−0.336111
$718$	18.0000	0.671754
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	24.0000	0.894427
$721$	−52.0000	−1.93658
$722$	−34.0000	−1.26535
$723$	0	0
$724$	4.00000	0.148659
$725$	0	0
$726$	14.0000	0.519589
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	42.0000	1.55449
$731$	−20.0000	−0.739727
$732$	14.0000	0.517455
$733$	41.0000	1.51437	0.757185	−	0.653201i	$-0.226574\pi$
$733$	41.0000	1.51437	0.757185	+	0.653201i	$0.226574\pi$
$734$	4.00000	0.147643
$735$	27.0000	0.995910
$736$	16.0000	0.589768
$737$	18.0000	0.663039
$738$	8.00000	0.294484
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	30.0000	1.10282
$741$	42.0000	1.54291
$742$	40.0000	1.46845
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	−40.0000	−1.46450
$747$	24.0000	0.878114
$748$	−8.00000	−0.292509
$749$	−16.0000	−0.584627
$750$	6.00000	0.219089
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−16.0000	−0.583460
$753$	−20.0000	−0.728841
$754$	0	0
$755$	69.0000	2.51117
$756$	−40.0000	−1.45479
$757$	−43.0000	−1.56286	−0.781431	−	0.623992i	$-0.785510\pi$
$757$	−43.0000	−1.56286	−0.781431	+	0.623992i	$0.785510\pi$
$758$	−44.0000	−1.59815
$759$	−4.00000	−0.145191
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	−32.0000	−1.15924
$763$	44.0000	1.59291
$764$	−6.00000	−0.217072
$765$	−12.0000	−0.433861
$766$	−24.0000	−0.867155
$767$	49.0000	1.76929
$768$	16.0000	0.577350
$769$	−12.0000	−0.432731	−0.216366	−	0.976312i	$-0.569420\pi$
$769$	−12.0000	−0.432731	−0.216366	+	0.976312i	$0.569420\pi$
$770$	48.0000	1.72980
$771$	−2.00000	−0.0720282
$772$	−22.0000	−0.791797
$773$	−3.00000	−0.107903	−0.0539513	−	0.998544i	$-0.517182\pi$
$773$	−3.00000	−0.107903	−0.0539513	+	0.998544i	$0.517182\pi$
$774$	−40.0000	−1.43777
$775$	−32.0000	−1.14947
$776$	0	0
$777$	20.0000	0.717496
$778$	−20.0000	−0.717035
$779$	−12.0000	−0.429945
$780$	−42.0000	−1.50384
$781$	6.00000	0.214697
$782$	−8.00000	−0.286079
$783$	0	0
$784$	−36.0000	−1.28571
$785$	−48.0000	−1.71319
$786$	−32.0000	−1.14140
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	30.0000	1.06871
$789$	−20.0000	−0.712019
$790$	0	0
$791$	−8.00000	−0.284447
$792$	0	0
$793$	−49.0000	−1.74004
$794$	40.0000	1.41955
$795$	−15.0000	−0.531995
$796$	−32.0000	−1.13421
$797$	3.00000	0.106265	0.0531327	−	0.998587i	$-0.483079\pi$
$797$	3.00000	0.106265	0.0531327	+	0.998587i	$0.483079\pi$
$798$	48.0000	1.69918
$799$	8.00000	0.283020
$800$	32.0000	1.13137
$801$	−12.0000	−0.423999
$802$	−18.0000	−0.635602
$803$	14.0000	0.494049
$804$	−18.0000	−0.634811
$805$	24.0000	0.845889
$806$	−112.000	−3.94503
$807$	1.00000	0.0352017
$808$	0	0
$809$	8.00000	0.281265	0.140633	−	0.990062i	$-0.455086\pi$
$809$	8.00000	0.281265	0.140633	+	0.990062i	$0.455086\pi$
$810$	−6.00000	−0.210819
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	6.00000	0.210429
$814$	20.0000	0.701000
$815$	30.0000	1.05085
$816$	−8.00000	−0.280056
$817$	60.0000	2.09913
$818$	−12.0000	−0.419570
$819$	56.0000	1.95680
$820$	12.0000	0.419058
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	16.0000	0.558064
$823$	15.0000	0.522867	0.261434	−	0.965221i	$-0.415805\pi$
$823$	15.0000	0.522867	0.261434	+	0.965221i	$0.415805\pi$
$824$	0	0
$825$	−8.00000	−0.278524
$826$	56.0000	1.94849
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	−8.00000	−0.278019
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	72.0000	2.49916
$831$	32.0000	1.11007
$832$	56.0000	1.94145
$833$	18.0000	0.623663
$834$	−24.0000	−0.831052
$835$	−66.0000	−2.28402
$836$	24.0000	0.830057
$837$	40.0000	1.38260
$838$	−8.00000	−0.276355
$839$	−35.0000	−1.20833	−0.604167	−	0.796858i	$-0.706494\pi$
$839$	−35.0000	−1.20833	−0.604167	+	0.796858i	$0.706494\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	12.0000	0.413547
$843$	26.0000	0.895488
$844$	−36.0000	−1.23917
$845$	108.000	3.71531
$846$	16.0000	0.550091
$847$	−28.0000	−0.962091
$848$	20.0000	0.686803
$849$	−7.00000	−0.240239
$850$	−16.0000	−0.548795
$851$	10.0000	0.342796
$852$	−6.00000	−0.205557
$853$	43.0000	1.47229	0.736146	−	0.676823i	$-0.236644\pi$
$853$	43.0000	1.47229	0.736146	+	0.676823i	$0.236644\pi$
$854$	−56.0000	−1.91628
$855$	36.0000	1.23117
$856$	0	0
$857$	21.0000	0.717346	0.358673	−	0.933463i	$-0.383229\pi$
$857$	21.0000	0.717346	0.358673	+	0.933463i	$0.383229\pi$
$858$	−28.0000	−0.955904
$859$	−39.0000	−1.33066	−0.665331	−	0.746548i	$-0.731710\pi$
$859$	−39.0000	−1.33066	−0.665331	+	0.746548i	$0.731710\pi$
$860$	−60.0000	−2.04598
$861$	8.00000	0.272639
$862$	−36.0000	−1.22616
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	−40.0000	−1.36083
$865$	12.0000	0.408012
$866$	−32.0000	−1.08740
$867$	−13.0000	−0.441503
$868$	−64.0000	−2.17230
$869$	0	0
$870$	0	0
$871$	63.0000	2.13467
$872$	0	0
$873$	−20.0000	−0.676897
$874$	24.0000	0.811812
$875$	−12.0000	−0.405674
$876$	−14.0000	−0.473016
$877$	−52.0000	−1.75592	−0.877958	−	0.478738i	$-0.841094\pi$
$877$	−52.0000	−1.75592	−0.877958	+	0.478738i	$0.841094\pi$
$878$	−70.0000	−2.36239
$879$	−11.0000	−0.371021
$880$	24.0000	0.809040
$881$	46.0000	1.54978	0.774890	−	0.632096i	$-0.217805\pi$
$881$	46.0000	1.54978	0.774890	+	0.632096i	$0.217805\pi$
$882$	36.0000	1.21218
$883$	14.0000	0.471138	0.235569	−	0.971858i	$-0.424305\pi$
$883$	14.0000	0.471138	0.235569	+	0.971858i	$0.424305\pi$
$884$	−28.0000	−0.941742
$885$	−21.0000	−0.705907
$886$	−36.0000	−1.20944
$887$	7.00000	0.235037	0.117518	−	0.993071i	$-0.462506\pi$
$887$	7.00000	0.235037	0.117518	+	0.993071i	$0.462506\pi$
$888$	0	0
$889$	64.0000	2.14649
$890$	−36.0000	−1.20672
$891$	−2.00000	−0.0670025
$892$	42.0000	1.40626
$893$	−24.0000	−0.803129
$894$	24.0000	0.802680
$895$	0	0
$896$	0	0
$897$	−14.0000	−0.467446
$898$	−30.0000	−1.00111
$899$	0	0
$900$	−16.0000	−0.533333
$901$	−10.0000	−0.333148
$902$	8.00000	0.266371
$903$	−40.0000	−1.33112
$904$	0	0
$905$	6.00000	0.199447
$906$	−46.0000	−1.52825
$907$	7.00000	0.232431	0.116216	−	0.993224i	$-0.462924\pi$
$907$	7.00000	0.232431	0.116216	+	0.993224i	$0.462924\pi$
$908$	56.0000	1.85843
$909$	36.0000	1.19404
$910$	168.000	5.56915
$911$	−34.0000	−1.12647	−0.563235	−	0.826297i	$-0.690443\pi$
$911$	−34.0000	−1.12647	−0.563235	+	0.826297i	$0.690443\pi$
$912$	24.0000	0.794719
$913$	24.0000	0.794284
$914$	44.0000	1.45539
$915$	21.0000	0.694239
$916$	−12.0000	−0.396491
$917$	64.0000	2.11347
$918$	20.0000	0.660098
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	68.0000	2.23946
$923$	21.0000	0.691223
$924$	−16.0000	−0.526361
$925$	20.0000	0.657596
$926$	0	0
$927$	26.0000	0.853952
$928$	0	0
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	48.0000	1.57398
$931$	−54.0000	−1.76978
$932$	−30.0000	−0.982683
$933$	32.0000	1.04763
$934$	−42.0000	−1.37428
$935$	−12.0000	−0.392442
$936$	0	0
$937$	52.0000	1.69877	0.849383	−	0.527777i	$-0.176974\pi$
$937$	52.0000	1.69877	0.849383	+	0.527777i	$0.176974\pi$
$938$	72.0000	2.35088
$939$	−21.0000	−0.685309
$940$	24.0000	0.782794
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	32.0000	1.04262
$943$	4.00000	0.130258
$944$	28.0000	0.911322
$945$	−60.0000	−1.95180
$946$	−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	$0.732063\pi$
$948$	0	0
$949$	49.0000	1.59061
$950$	48.0000	1.55733
$951$	6.00000	0.194563
$952$	0	0
$953$	23.0000	0.745043	0.372522	−	0.928024i	$-0.378493\pi$
$953$	23.0000	0.745043	0.372522	+	0.928024i	$0.378493\pi$
$954$	−20.0000	−0.647524
$955$	−9.00000	−0.291233
$956$	−18.0000	−0.582162
$957$	0	0
$958$	26.0000	0.840022
$959$	−32.0000	−1.03333
$960$	−24.0000	−0.774597
$961$	33.0000	1.06452
$962$	70.0000	2.25689
$963$	8.00000	0.257796
$964$	0	0
$965$	−33.0000	−1.06231
$966$	−16.0000	−0.514792
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	−12.0000	−0.385496
$970$	−60.0000	−1.92648
$971$	−22.0000	−0.706014	−0.353007	−	0.935621i	$-0.614841\pi$
$971$	−22.0000	−0.706014	−0.353007	+	0.935621i	$0.614841\pi$
$972$	32.0000	1.02640
$973$	48.0000	1.53881
$974$	36.0000	1.15351
$975$	−28.0000	−0.896718
$976$	−28.0000	−0.896258
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	−20.0000	−0.639529
$979$	−12.0000	−0.383522
$980$	54.0000	1.72497
$981$	−22.0000	−0.702406
$982$	70.0000	2.23379
$983$	−44.0000	−1.40338	−0.701691	−	0.712481i	$-0.747571\pi$
$983$	−44.0000	−1.40338	−0.701691	+	0.712481i	$0.747571\pi$
$984$	0	0
$985$	45.0000	1.43382
$986$	0	0
$987$	16.0000	0.509286
$988$	84.0000	2.67240
$989$	−20.0000	−0.635963
$990$	−24.0000	−0.762770
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	−64.0000	−2.03200
$993$	−8.00000	−0.253872
$994$	24.0000	0.761234
$995$	−48.0000	−1.52170
$996$	−24.0000	−0.760469
$997$	−16.0000	−0.506725	−0.253363	−	0.967371i	$-0.581537\pi$
$997$	−16.0000	−0.506725	−0.253363	+	0.967371i	$0.581537\pi$
$998$	50.0000	1.58272
$999$	−25.0000	−0.790965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4021.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4021.2.a.a.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:	\( N \)	\(=\)	\( 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4021.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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