LMFDB - Embedded newform 462.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [462,4,Mod(1,462)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("462.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 462 = 2 \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	462.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:	$27.2588824227$
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	$\chi$	$=$	462.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -7.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -7.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +14.0000 q^{10} +11.0000 q^{11} +12.0000 q^{12} -67.0000 q^{13} -14.0000 q^{14} -21.0000 q^{15} +16.0000 q^{16} +30.0000 q^{17} -18.0000 q^{18} -7.00000 q^{19} -28.0000 q^{20} +21.0000 q^{21} -22.0000 q^{22} +28.0000 q^{23} -24.0000 q^{24} -76.0000 q^{25} +134.000 q^{26} +27.0000 q^{27} +28.0000 q^{28} +121.000 q^{29} +42.0000 q^{30} -310.000 q^{31} -32.0000 q^{32} +33.0000 q^{33} -60.0000 q^{34} -49.0000 q^{35} +36.0000 q^{36} -71.0000 q^{37} +14.0000 q^{38} -201.000 q^{39} +56.0000 q^{40} -180.000 q^{41} -42.0000 q^{42} -108.000 q^{43} +44.0000 q^{44} -63.0000 q^{45} -56.0000 q^{46} +71.0000 q^{47} +48.0000 q^{48} +49.0000 q^{49} +152.000 q^{50} +90.0000 q^{51} -268.000 q^{52} +128.000 q^{53} -54.0000 q^{54} -77.0000 q^{55} -56.0000 q^{56} -21.0000 q^{57} -242.000 q^{58} -429.000 q^{59} -84.0000 q^{60} +22.0000 q^{61} +620.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} +469.000 q^{65} -66.0000 q^{66} -803.000 q^{67} +120.000 q^{68} +84.0000 q^{69} +98.0000 q^{70} +468.000 q^{71} -72.0000 q^{72} -117.000 q^{73} +142.000 q^{74} -228.000 q^{75} -28.0000 q^{76} +77.0000 q^{77} +402.000 q^{78} -96.0000 q^{79} -112.000 q^{80} +81.0000 q^{81} +360.000 q^{82} -1122.00 q^{83} +84.0000 q^{84} -210.000 q^{85} +216.000 q^{86} +363.000 q^{87} -88.0000 q^{88} -1146.00 q^{89} +126.000 q^{90} -469.000 q^{91} +112.000 q^{92} -930.000 q^{93} -142.000 q^{94} +49.0000 q^{95} -96.0000 q^{96} -92.0000 q^{97} -98.0000 q^{98} +99.0000 q^{99} +O(q^{100})\)

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	−0.707107
$2$	−2.00000	−0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−7.00000	−0.626099	−0.313050	−	0.949737i	$-0.601351\pi$
$5$	−7.00000	−0.626099	−0.313050	+	0.949737i	$0.601351\pi$
$6$	−6.00000	−0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	14.0000	0.442719
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	12.0000	0.288675
$13$	−67.0000	−1.42942	−0.714710	−	0.699421i	$-0.753441\pi$
$13$	−67.0000	−1.42942	−0.714710	+	0.699421i	$0.753441\pi$
$14$	−14.0000	−0.267261
$15$	−21.0000	−0.361478
$16$	16.0000	0.250000
$17$	30.0000	0.428004	0.214002	−	0.976833i	$-0.431350\pi$
$17$	30.0000	0.428004	0.214002	+	0.976833i	$0.431350\pi$
$18$	−18.0000	−0.235702
$19$	−7.00000	−0.0845216	−0.0422608	−	0.999107i	$-0.513456\pi$
$19$	−7.00000	−0.0845216	−0.0422608	+	0.999107i	$0.513456\pi$
$20$	−28.0000	−0.313050
$21$	21.0000	0.218218
$22$	−22.0000	−0.213201
$23$	28.0000	0.253844	0.126922	−	0.991913i	$-0.459490\pi$
$23$	28.0000	0.253844	0.126922	+	0.991913i	$0.459490\pi$
$24$	−24.0000	−0.204124
$25$	−76.0000	−0.608000
$26$	134.000	1.01075
$27$	27.0000	0.192450
$28$	28.0000	0.188982
$29$	121.000	0.774798	0.387399	−	0.921912i	$-0.373374\pi$
$29$	121.000	0.774798	0.387399	+	0.921912i	$0.373374\pi$
$30$	42.0000	0.255604
$31$	−310.000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−310.000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−32.0000	−0.176777
$33$	33.0000	0.174078
$34$	−60.0000	−0.302645
$35$	−49.0000	−0.236643
$36$	36.0000	0.166667
$37$	−71.0000	−0.315468	−0.157734	−	0.987482i	$-0.550419\pi$
$37$	−71.0000	−0.315468	−0.157734	+	0.987482i	$0.550419\pi$
$38$	14.0000	0.0597658
$39$	−201.000	−0.825276
$40$	56.0000	0.221359
$41$	−180.000	−0.685641	−0.342820	−	0.939401i	$-0.611382\pi$
$41$	−180.000	−0.685641	−0.342820	+	0.939401i	$0.611382\pi$
$42$	−42.0000	−0.154303
$43$	−108.000	−0.383020	−0.191510	−	0.981491i	$-0.561338\pi$
$43$	−108.000	−0.383020	−0.191510	+	0.981491i	$0.561338\pi$
$44$	44.0000	0.150756
$45$	−63.0000	−0.208700
$46$	−56.0000	−0.179495
$47$	71.0000	0.220349	0.110175	−	0.993912i	$-0.464859\pi$
$47$	71.0000	0.220349	0.110175	+	0.993912i	$0.464859\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	152.000	0.429921
$51$	90.0000	0.247108
$52$	−268.000	−0.714710
$53$	128.000	0.331739	0.165869	−	0.986148i	$-0.446957\pi$
$53$	128.000	0.331739	0.165869	+	0.986148i	$0.446957\pi$
$54$	−54.0000	−0.136083
$55$	−77.0000	−0.188776
$56$	−56.0000	−0.133631
$57$	−21.0000	−0.0487986
$58$	−242.000	−0.547865
$59$	−429.000	−0.946628	−0.473314	−	0.880894i	$-0.656942\pi$
$59$	−429.000	−0.946628	−0.473314	+	0.880894i	$0.656942\pi$
$60$	−84.0000	−0.180739
$61$	22.0000	0.0461772	0.0230886	−	0.999733i	$-0.492650\pi$
$61$	22.0000	0.0461772	0.0230886	+	0.999733i	$0.492650\pi$
$62$	620.000	1.27000
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	469.000	0.894958
$66$	−66.0000	−0.123091
$67$	−803.000	−1.46421	−0.732105	−	0.681192i	$-0.761462\pi$
$67$	−803.000	−1.46421	−0.732105	+	0.681192i	$0.761462\pi$
$68$	120.000	0.214002
$69$	84.0000	0.146557
$70$	98.0000	0.167332
$71$	468.000	0.782273	0.391136	−	0.920333i	$-0.372082\pi$
$71$	468.000	0.782273	0.391136	+	0.920333i	$0.372082\pi$
$72$	−72.0000	−0.117851
$73$	−117.000	−0.187586	−0.0937932	−	0.995592i	$-0.529899\pi$
$73$	−117.000	−0.187586	−0.0937932	+	0.995592i	$0.529899\pi$
$74$	142.000	0.223070
$75$	−228.000	−0.351029
$76$	−28.0000	−0.0422608
$77$	77.0000	0.113961
$78$	402.000	0.583558
$79$	−96.0000	−0.136720	−0.0683598	−	0.997661i	$-0.521777\pi$
$79$	−96.0000	−0.136720	−0.0683598	+	0.997661i	$0.521777\pi$
$80$	−112.000	−0.156525
$81$	81.0000	0.111111
$82$	360.000	0.484821
$83$	−1122.00	−1.48380	−0.741901	−	0.670510i	$-0.766075\pi$
$83$	−1122.00	−1.48380	−0.741901	+	0.670510i	$0.766075\pi$
$84$	84.0000	0.109109
$85$	−210.000	−0.267973
$86$	216.000	0.270836
$87$	363.000	0.447330
$88$	−88.0000	−0.106600
$89$	−1146.00	−1.36490	−0.682448	−	0.730934i	$-0.739085\pi$
$89$	−1146.00	−1.36490	−0.682448	+	0.730934i	$0.739085\pi$
$90$	126.000	0.147573
$91$	−469.000	−0.540270
$92$	112.000	0.126922
$93$	−930.000	−1.03695
$94$	−142.000	−0.155810
$95$	49.0000	0.0529189
$96$	−96.0000	−0.102062
$97$	−92.0000	−0.0963009	−0.0481504	−	0.998840i	$-0.515333\pi$
$97$	−92.0000	−0.0963009	−0.0481504	+	0.998840i	$0.515333\pi$
$98$	−98.0000	−0.101015
$99$	99.0000	0.100504
$100$	−304.000	−0.304000
$101$	−202.000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−202.000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	−180.000	−0.174732
$103$	−1798.00	−1.72002	−0.860011	−	0.510276i	$-0.829543\pi$
$103$	−1798.00	−1.72002	−0.860011	+	0.510276i	$0.829543\pi$
$104$	536.000	0.505376
$105$	−147.000	−0.136626
$106$	−256.000	−0.234575
$107$	−1431.00	−1.29290	−0.646449	−	0.762958i	$-0.723747\pi$
$107$	−1431.00	−1.29290	−0.646449	+	0.762958i	$0.723747\pi$
$108$	108.000	0.0962250
$109$	−278.000	−0.244290	−0.122145	−	0.992512i	$-0.538977\pi$
$109$	−278.000	−0.244290	−0.122145	+	0.992512i	$0.538977\pi$
$110$	154.000	0.133485
$111$	−213.000	−0.182136
$112$	112.000	0.0944911
$113$	−2322.00	−1.93306	−0.966528	−	0.256560i	$-0.917411\pi$
$113$	−2322.00	−1.93306	−0.966528	+	0.256560i	$0.917411\pi$
$114$	42.0000	0.0345058
$115$	−196.000	−0.158931
$116$	484.000	0.387399
$117$	−603.000	−0.476473
$118$	858.000	0.669367
$119$	210.000	0.161770
$120$	168.000	0.127802
$121$	121.000	0.0909091
$122$	−44.0000	−0.0326522
$123$	−540.000	−0.395855
$124$	−1240.00	−0.898027
$125$	1407.00	1.00677
$126$	−126.000	−0.0890871
$127$	2254.00	1.57488	0.787442	−	0.616389i	$-0.211405\pi$
$127$	2254.00	1.57488	0.787442	+	0.616389i	$0.211405\pi$
$128$	−128.000	−0.0883883
$129$	−324.000	−0.221137
$130$	−938.000	−0.632831
$131$	2196.00	1.46462	0.732311	−	0.680971i	$-0.238442\pi$
$131$	2196.00	1.46462	0.732311	+	0.680971i	$0.238442\pi$
$132$	132.000	0.0870388
$133$	−49.0000	−0.0319462
$134$	1606.00	1.03535
$135$	−189.000	−0.120493
$136$	−240.000	−0.151322
$137$	766.000	0.477692	0.238846	−	0.971057i	$-0.423231\pi$
$137$	766.000	0.477692	0.238846	+	0.971057i	$0.423231\pi$
$138$	−168.000	−0.103631
$139$	−1080.00	−0.659024	−0.329512	−	0.944151i	$-0.606884\pi$
$139$	−1080.00	−0.659024	−0.329512	+	0.944151i	$0.606884\pi$
$140$	−196.000	−0.118322
$141$	213.000	0.127219
$142$	−936.000	−0.553151
$143$	−737.000	−0.430986
$144$	144.000	0.0833333
$145$	−847.000	−0.485100
$146$	234.000	0.132644
$147$	147.000	0.0824786
$148$	−284.000	−0.157734
$149$	1373.00	0.754903	0.377451	−	0.926029i	$-0.376800\pi$
$149$	1373.00	0.754903	0.377451	+	0.926029i	$0.376800\pi$
$150$	456.000	0.248215
$151$	−3210.00	−1.72997	−0.864987	−	0.501794i	$-0.832673\pi$
$151$	−3210.00	−1.72997	−0.864987	+	0.501794i	$0.832673\pi$
$152$	56.0000	0.0298829
$153$	270.000	0.142668
$154$	−154.000	−0.0805823
$155$	2170.00	1.12451
$156$	−804.000	−0.412638
$157$	452.000	0.229768	0.114884	−	0.993379i	$-0.463350\pi$
$157$	452.000	0.229768	0.114884	+	0.993379i	$0.463350\pi$
$158$	192.000	0.0966753
$159$	384.000	0.191529
$160$	224.000	0.110680
$161$	196.000	0.0959439
$162$	−162.000	−0.0785674
$163$	−11.0000	−0.00528581	−0.00264290	−	0.999997i	$-0.500841\pi$
$163$	−11.0000	−0.00528581	−0.00264290	+	0.999997i	$0.500841\pi$
$164$	−720.000	−0.342820
$165$	−231.000	−0.108990
$166$	2244.00	1.04921
$167$	504.000	0.233537	0.116769	−	0.993159i	$-0.462746\pi$
$167$	504.000	0.233537	0.116769	+	0.993159i	$0.462746\pi$
$168$	−168.000	−0.0771517
$169$	2292.00	1.04324
$170$	420.000	0.189485
$171$	−63.0000	−0.0281739
$172$	−432.000	−0.191510
$173$	1688.00	0.741828	0.370914	−	0.928667i	$-0.379044\pi$
$173$	1688.00	0.741828	0.370914	+	0.928667i	$0.379044\pi$
$174$	−726.000	−0.316310
$175$	−532.000	−0.229802
$176$	176.000	0.0753778
$177$	−1287.00	−0.546536
$178$	2292.00	0.965127
$179$	460.000	0.192078	0.0960391	−	0.995378i	$-0.469383\pi$
$179$	460.000	0.192078	0.0960391	+	0.995378i	$0.469383\pi$
$180$	−252.000	−0.104350
$181$	2296.00	0.942875	0.471437	−	0.881900i	$-0.343735\pi$
$181$	2296.00	0.942875	0.471437	+	0.881900i	$0.343735\pi$
$182$	938.000	0.382028
$183$	66.0000	0.0266604
$184$	−224.000	−0.0897473
$185$	497.000	0.197514
$186$	1860.00	0.733236
$187$	330.000	0.129048
$188$	284.000	0.110175
$189$	189.000	0.0727393
$190$	−98.0000	−0.0374193
$191$	878.000	0.332617	0.166309	−	0.986074i	$-0.446815\pi$
$191$	878.000	0.332617	0.166309	+	0.986074i	$0.446815\pi$
$192$	192.000	0.0721688
$193$	−2268.00	−0.845877	−0.422938	−	0.906158i	$-0.639001\pi$
$193$	−2268.00	−0.845877	−0.422938	+	0.906158i	$0.639001\pi$
$194$	184.000	0.0680950
$195$	1407.00	0.516704
$196$	196.000	0.0714286
$197$	5298.00	1.91608	0.958038	−	0.286642i	$-0.0925390\pi$
$197$	5298.00	1.91608	0.958038	+	0.286642i	$0.0925390\pi$
$198$	−198.000	−0.0710669
$199$	1044.00	0.371895	0.185948	−	0.982560i	$-0.440464\pi$
$199$	1044.00	0.371895	0.185948	+	0.982560i	$0.440464\pi$
$200$	608.000	0.214960
$201$	−2409.00	−0.845362
$202$	404.000	0.140720
$203$	847.000	0.292846
$204$	360.000	0.123554
$205$	1260.00	0.429279
$206$	3596.00	1.21624
$207$	252.000	0.0846145
$208$	−1072.00	−0.357355
$209$	−77.0000	−0.0254842
$210$	294.000	0.0966092
$211$	4566.00	1.48975	0.744873	−	0.667206i	$-0.232510\pi$
$211$	4566.00	1.48975	0.744873	+	0.667206i	$0.232510\pi$
$212$	512.000	0.165869
$213$	1404.00	0.451646
$214$	2862.00	0.914216
$215$	756.000	0.239808
$216$	−216.000	−0.0680414
$217$	−2170.00	−0.678844
$218$	556.000	0.172739
$219$	−351.000	−0.108303
$220$	−308.000	−0.0943880
$221$	−2010.00	−0.611797
$222$	426.000	0.128789
$223$	198.000	0.0594577	0.0297288	−	0.999558i	$-0.490536\pi$
$223$	198.000	0.0594577	0.0297288	+	0.999558i	$0.490536\pi$
$224$	−224.000	−0.0668153
$225$	−684.000	−0.202667
$226$	4644.00	1.36688
$227$	3762.00	1.09997	0.549984	−	0.835175i	$-0.314634\pi$
$227$	3762.00	1.09997	0.549984	+	0.835175i	$0.314634\pi$
$228$	−84.0000	−0.0243993
$229$	−704.000	−0.203151	−0.101576	−	0.994828i	$-0.532388\pi$
$229$	−704.000	−0.203151	−0.101576	+	0.994828i	$0.532388\pi$
$230$	392.000	0.112381
$231$	231.000	0.0657952
$232$	−968.000	−0.273932
$233$	2210.00	0.621382	0.310691	−	0.950511i	$-0.399440\pi$
$233$	2210.00	0.621382	0.310691	+	0.950511i	$0.399440\pi$
$234$	1206.00	0.336917
$235$	−497.000	−0.137960
$236$	−1716.00	−0.473314
$237$	−288.000	−0.0789351
$238$	−420.000	−0.114389
$239$	2451.00	0.663356	0.331678	−	0.943393i	$-0.392385\pi$
$239$	2451.00	0.663356	0.331678	+	0.943393i	$0.392385\pi$
$240$	−336.000	−0.0903696
$241$	25.0000	0.00668212	0.00334106	−	0.999994i	$-0.498937\pi$
$241$	25.0000	0.00668212	0.00334106	+	0.999994i	$0.498937\pi$
$242$	−242.000	−0.0642824
$243$	243.000	0.0641500
$244$	88.0000	0.0230886
$245$	−343.000	−0.0894427
$246$	1080.00	0.279912
$247$	469.000	0.120817
$248$	2480.00	0.635001
$249$	−3366.00	−0.856673
$250$	−2814.00	−0.711892
$251$	−5267.00	−1.32450	−0.662251	−	0.749282i	$-0.730399\pi$
$251$	−5267.00	−1.32450	−0.662251	+	0.749282i	$0.730399\pi$
$252$	252.000	0.0629941
$253$	308.000	0.0765367
$254$	−4508.00	−1.11361
$255$	−630.000	−0.154714
$256$	256.000	0.0625000
$257$	−1623.00	−0.393930	−0.196965	−	0.980411i	$-0.563109\pi$
$257$	−1623.00	−0.393930	−0.196965	+	0.980411i	$0.563109\pi$
$258$	648.000	0.156367
$259$	−497.000	−0.119236
$260$	1876.00	0.447479
$261$	1089.00	0.258266
$262$	−4392.00	−1.03564
$263$	−2475.00	−0.580285	−0.290143	−	0.956983i	$-0.593703\pi$
$263$	−2475.00	−0.580285	−0.290143	+	0.956983i	$0.593703\pi$
$264$	−264.000	−0.0615457
$265$	−896.000	−0.207701
$266$	98.0000	0.0225893
$267$	−3438.00	−0.788023
$268$	−3212.00	−0.732105
$269$	1746.00	0.395745	0.197873	−	0.980228i	$-0.436597\pi$
$269$	1746.00	0.395745	0.197873	+	0.980228i	$0.436597\pi$
$270$	378.000	0.0852013
$271$	1653.00	0.370526	0.185263	−	0.982689i	$-0.440686\pi$
$271$	1653.00	0.370526	0.185263	+	0.982689i	$0.440686\pi$
$272$	480.000	0.107001
$273$	−1407.00	−0.311925
$274$	−1532.00	−0.337779
$275$	−836.000	−0.183319
$276$	336.000	0.0732783
$277$	−2056.00	−0.445968	−0.222984	−	0.974822i	$-0.571580\pi$
$277$	−2056.00	−0.445968	−0.222984	+	0.974822i	$0.571580\pi$
$278$	2160.00	0.466001
$279$	−2790.00	−0.598684
$280$	392.000	0.0836660
$281$	837.000	0.177691	0.0888456	−	0.996045i	$-0.471682\pi$
$281$	837.000	0.177691	0.0888456	+	0.996045i	$0.471682\pi$
$282$	−426.000	−0.0899572
$283$	8969.00	1.88393	0.941964	−	0.335713i	$-0.108977\pi$
$283$	8969.00	1.88393	0.941964	+	0.335713i	$0.108977\pi$
$284$	1872.00	0.391136
$285$	147.000	0.0305527
$286$	1474.00	0.304753
$287$	−1260.00	−0.259148
$288$	−288.000	−0.0589256
$289$	−4013.00	−0.816813
$290$	1694.00	0.343018
$291$	−276.000	−0.0555993
$292$	−468.000	−0.0937932
$293$	5120.00	1.02087	0.510433	−	0.859918i	$-0.329485\pi$
$293$	5120.00	1.02087	0.510433	+	0.859918i	$0.329485\pi$
$294$	−294.000	−0.0583212
$295$	3003.00	0.592683
$296$	568.000	0.111535
$297$	297.000	0.0580259
$298$	−2746.00	−0.533797
$299$	−1876.00	−0.362849
$300$	−912.000	−0.175514
$301$	−756.000	−0.144768
$302$	6420.00	1.22328
$303$	−606.000	−0.114897
$304$	−112.000	−0.0211304
$305$	−154.000	−0.0289115
$306$	−540.000	−0.100882
$307$	−2892.00	−0.537639	−0.268819	−	0.963191i	$-0.586634\pi$
$307$	−2892.00	−0.537639	−0.268819	+	0.963191i	$0.586634\pi$
$308$	308.000	0.0569803
$309$	−5394.00	−0.993055
$310$	−4340.00	−0.795147
$311$	3208.00	0.584916	0.292458	−	0.956278i	$-0.405527\pi$
$311$	3208.00	0.584916	0.292458	+	0.956278i	$0.405527\pi$
$312$	1608.00	0.291779
$313$	3550.00	0.641079	0.320540	−	0.947235i	$-0.396136\pi$
$313$	3550.00	0.641079	0.320540	+	0.947235i	$0.396136\pi$
$314$	−904.000	−0.162470
$315$	−441.000	−0.0788811
$316$	−384.000	−0.0683598
$317$	−5524.00	−0.978734	−0.489367	−	0.872078i	$-0.662772\pi$
$317$	−5524.00	−0.978734	−0.489367	+	0.872078i	$0.662772\pi$
$318$	−768.000	−0.135432
$319$	1331.00	0.233610
$320$	−448.000	−0.0782624
$321$	−4293.00	−0.746454
$322$	−392.000	−0.0678426
$323$	−210.000	−0.0361756
$324$	324.000	0.0555556
$325$	5092.00	0.869087
$326$	22.0000	0.00373763
$327$	−834.000	−0.141041
$328$	1440.00	0.242411
$329$	497.000	0.0832842
$330$	462.000	0.0770675
$331$	−2144.00	−0.356027	−0.178013	−	0.984028i	$-0.556967\pi$
$331$	−2144.00	−0.356027	−0.178013	+	0.984028i	$0.556967\pi$
$332$	−4488.00	−0.741901
$333$	−639.000	−0.105156
$334$	−1008.00	−0.165136
$335$	5621.00	0.916740
$336$	336.000	0.0545545
$337$	−884.000	−0.142892	−0.0714459	−	0.997444i	$-0.522761\pi$
$337$	−884.000	−0.142892	−0.0714459	+	0.997444i	$0.522761\pi$
$338$	−4584.00	−0.737683
$339$	−6966.00	−1.11605
$340$	−840.000	−0.133986
$341$	−3410.00	−0.541530
$342$	126.000	0.0199219
$343$	343.000	0.0539949
$344$	864.000	0.135418
$345$	−588.000	−0.0917590
$346$	−3376.00	−0.524552
$347$	−4040.00	−0.625010	−0.312505	−	0.949916i	$-0.601168\pi$
$347$	−4040.00	−0.625010	−0.312505	+	0.949916i	$0.601168\pi$
$348$	1452.00	0.223665
$349$	1225.00	0.187888	0.0939438	−	0.995578i	$-0.470053\pi$
$349$	1225.00	0.187888	0.0939438	+	0.995578i	$0.470053\pi$
$350$	1064.00	0.162495
$351$	−1809.00	−0.275092
$352$	−352.000	−0.0533002
$353$	2483.00	0.374382	0.187191	−	0.982324i	$-0.440062\pi$
$353$	2483.00	0.374382	0.187191	+	0.982324i	$0.440062\pi$
$354$	2574.00	0.386459
$355$	−3276.00	−0.489780
$356$	−4584.00	−0.682448
$357$	630.000	0.0933981
$358$	−920.000	−0.135820
$359$	7804.00	1.14730	0.573648	−	0.819102i	$-0.305528\pi$
$359$	7804.00	1.14730	0.573648	+	0.819102i	$0.305528\pi$
$360$	504.000	0.0737865
$361$	−6810.00	−0.992856
$362$	−4592.00	−0.666713
$363$	363.000	0.0524864
$364$	−1876.00	−0.270135
$365$	819.000	0.117448
$366$	−132.000	−0.0188518
$367$	3074.00	0.437225	0.218612	−	0.975812i	$-0.429847\pi$
$367$	3074.00	0.437225	0.218612	+	0.975812i	$0.429847\pi$
$368$	448.000	0.0634609
$369$	−1620.00	−0.228547
$370$	−994.000	−0.139664
$371$	896.000	0.125385
$372$	−3720.00	−0.518476
$373$	674.000	0.0935614	0.0467807	−	0.998905i	$-0.485104\pi$
$373$	674.000	0.0935614	0.0467807	+	0.998905i	$0.485104\pi$
$374$	−660.000	−0.0912508
$375$	4221.00	0.581257
$376$	−568.000	−0.0779052
$377$	−8107.00	−1.10751
$378$	−378.000	−0.0514344
$379$	12051.0	1.63329	0.816647	−	0.577138i	$-0.195830\pi$
$379$	12051.0	1.63329	0.816647	+	0.577138i	$0.195830\pi$
$380$	196.000	0.0264594
$381$	6762.00	0.909259
$382$	−1756.00	−0.235196
$383$	−12664.0	−1.68956	−0.844778	−	0.535116i	$-0.820268\pi$
$383$	−12664.0	−1.68956	−0.844778	+	0.535116i	$0.820268\pi$
$384$	−384.000	−0.0510310
$385$	−539.000	−0.0713506
$386$	4536.00	0.598125
$387$	−972.000	−0.127673
$388$	−368.000	−0.0481504
$389$	264.000	0.0344096	0.0172048	−	0.999852i	$-0.494523\pi$
$389$	264.000	0.0344096	0.0172048	+	0.999852i	$0.494523\pi$
$390$	−2814.00	−0.365365
$391$	840.000	0.108646
$392$	−392.000	−0.0505076
$393$	6588.00	0.845600
$394$	−10596.0	−1.35487
$395$	672.000	0.0856000
$396$	396.000	0.0502519
$397$	−14394.0	−1.81968	−0.909842	−	0.414956i	$-0.863797\pi$
$397$	−14394.0	−1.81968	−0.909842	+	0.414956i	$0.863797\pi$
$398$	−2088.00	−0.262970
$399$	−147.000	−0.0184441
$400$	−1216.00	−0.152000
$401$	−12304.0	−1.53225	−0.766125	−	0.642691i	$-0.777818\pi$
$401$	−12304.0	−1.53225	−0.766125	+	0.642691i	$0.777818\pi$
$402$	4818.00	0.597761
$403$	20770.0	2.56731
$404$	−808.000	−0.0995037
$405$	−567.000	−0.0695666
$406$	−1694.00	−0.207073
$407$	−781.000	−0.0951173
$408$	−720.000	−0.0873660
$409$	−3154.00	−0.381309	−0.190654	−	0.981657i	$-0.561061\pi$
$409$	−3154.00	−0.381309	−0.190654	+	0.981657i	$0.561061\pi$
$410$	−2520.00	−0.303546
$411$	2298.00	0.275796
$412$	−7192.00	−0.860011
$413$	−3003.00	−0.357792
$414$	−504.000	−0.0598315
$415$	7854.00	0.929006
$416$	2144.00	0.252688
$417$	−3240.00	−0.380488
$418$	154.000	0.0180201
$419$	−4635.00	−0.540417	−0.270208	−	0.962802i	$-0.587093\pi$
$419$	−4635.00	−0.540417	−0.270208	+	0.962802i	$0.587093\pi$
$420$	−588.000	−0.0683130
$421$	−7265.00	−0.841032	−0.420516	−	0.907285i	$-0.638151\pi$
$421$	−7265.00	−0.841032	−0.420516	+	0.907285i	$0.638151\pi$
$422$	−9132.00	−1.05341
$423$	639.000	0.0734497
$424$	−1024.00	−0.117287
$425$	−2280.00	−0.260226
$426$	−2808.00	−0.319362
$427$	154.000	0.0174534
$428$	−5724.00	−0.646449
$429$	−2211.00	−0.248830
$430$	−1512.00	−0.169570
$431$	7589.00	0.848142	0.424071	−	0.905629i	$-0.360601\pi$
$431$	7589.00	0.848142	0.424071	+	0.905629i	$0.360601\pi$
$432$	432.000	0.0481125
$433$	5164.00	0.573132	0.286566	−	0.958061i	$-0.407486\pi$
$433$	5164.00	0.573132	0.286566	+	0.958061i	$0.407486\pi$
$434$	4340.00	0.480015
$435$	−2541.00	−0.280073
$436$	−1112.00	−0.122145
$437$	−196.000	−0.0214553
$438$	702.000	0.0765819
$439$	14221.0	1.54608	0.773042	−	0.634354i	$-0.218734\pi$
$439$	14221.0	1.54608	0.773042	+	0.634354i	$0.218734\pi$
$440$	616.000	0.0667424
$441$	441.000	0.0476190
$442$	4020.00	0.432606
$443$	6924.00	0.742594	0.371297	−	0.928514i	$-0.378913\pi$
$443$	6924.00	0.742594	0.371297	+	0.928514i	$0.378913\pi$
$444$	−852.000	−0.0910679
$445$	8022.00	0.854560
$446$	−396.000	−0.0420429
$447$	4119.00	0.435843
$448$	448.000	0.0472456
$449$	2524.00	0.265289	0.132645	−	0.991164i	$-0.457653\pi$
$449$	2524.00	0.265289	0.132645	+	0.991164i	$0.457653\pi$
$450$	1368.00	0.143307
$451$	−1980.00	−0.206729
$452$	−9288.00	−0.966528
$453$	−9630.00	−0.998801
$454$	−7524.00	−0.777795
$455$	3283.00	0.338262
$456$	168.000	0.0172529
$457$	1928.00	0.197348	0.0986740	−	0.995120i	$-0.468540\pi$
$457$	1928.00	0.197348	0.0986740	+	0.995120i	$0.468540\pi$
$458$	1408.00	0.143650
$459$	810.000	0.0823694
$460$	−784.000	−0.0794656
$461$	−12852.0	−1.29843	−0.649216	−	0.760604i	$-0.724903\pi$
$461$	−12852.0	−1.29843	−0.649216	+	0.760604i	$0.724903\pi$
$462$	−462.000	−0.0465242
$463$	4741.00	0.475881	0.237941	−	0.971280i	$-0.423528\pi$
$463$	4741.00	0.475881	0.237941	+	0.971280i	$0.423528\pi$
$464$	1936.00	0.193699
$465$	6510.00	0.649234
$466$	−4420.00	−0.439383
$467$	−6809.00	−0.674696	−0.337348	−	0.941380i	$-0.609530\pi$
$467$	−6809.00	−0.674696	−0.337348	+	0.941380i	$0.609530\pi$
$468$	−2412.00	−0.238237
$469$	−5621.00	−0.553419
$470$	994.000	0.0975528
$471$	1356.00	0.132656
$472$	3432.00	0.334683
$473$	−1188.00	−0.115485
$474$	576.000	0.0558155
$475$	532.000	0.0513891
$476$	840.000	0.0808852
$477$	1152.00	0.110580
$478$	−4902.00	−0.469063
$479$	1734.00	0.165404	0.0827020	−	0.996574i	$-0.473645\pi$
$479$	1734.00	0.165404	0.0827020	+	0.996574i	$0.473645\pi$
$480$	672.000	0.0639010
$481$	4757.00	0.450937
$482$	−50.0000	−0.00472497
$483$	588.000	0.0553932
$484$	484.000	0.0454545
$485$	644.000	0.0602939
$486$	−486.000	−0.0453609
$487$	−7368.00	−0.685577	−0.342788	−	0.939413i	$-0.611371\pi$
$487$	−7368.00	−0.685577	−0.342788	+	0.939413i	$0.611371\pi$
$488$	−176.000	−0.0163261
$489$	−33.0000	−0.00305176
$490$	686.000	0.0632456
$491$	13201.0	1.21335	0.606673	−	0.794952i	$-0.292504\pi$
$491$	13201.0	1.21335	0.606673	+	0.794952i	$0.292504\pi$
$492$	−2160.00	−0.197927
$493$	3630.00	0.331617
$494$	−938.000	−0.0854304
$495$	−693.000	−0.0629253
$496$	−4960.00	−0.449013
$497$	3276.00	0.295671
$498$	6732.00	0.605759
$499$	16539.0	1.48374	0.741871	−	0.670543i	$-0.233939\pi$
$499$	16539.0	1.48374	0.741871	+	0.670543i	$0.233939\pi$
$500$	5628.00	0.503384
$501$	1512.00	0.134833
$502$	10534.0	0.936565
$503$	−14780.0	−1.31015	−0.655077	−	0.755562i	$-0.727364\pi$
$503$	−14780.0	−1.31015	−0.655077	+	0.755562i	$0.727364\pi$
$504$	−504.000	−0.0445435
$505$	1414.00	0.124598
$506$	−616.000	−0.0541196
$507$	6876.00	0.602315
$508$	9016.00	0.787442
$509$	−11766.0	−1.02459	−0.512297	−	0.858808i	$-0.671205\pi$
$509$	−11766.0	−1.02459	−0.512297	+	0.858808i	$0.671205\pi$
$510$	1260.00	0.109399
$511$	−819.000	−0.0709010
$512$	−512.000	−0.0441942
$513$	−189.000	−0.0162662
$514$	3246.00	0.278550
$515$	12586.0	1.07690
$516$	−1296.00	−0.110568
$517$	781.000	0.0664378
$518$	994.000	0.0843125
$519$	5064.00	0.428295
$520$	−3752.00	−0.316416
$521$	−13503.0	−1.13546	−0.567732	−	0.823213i	$-0.692179\pi$
$521$	−13503.0	−1.13546	−0.567732	+	0.823213i	$0.692179\pi$
$522$	−2178.00	−0.182622
$523$	−18681.0	−1.56188	−0.780940	−	0.624606i	$-0.785259\pi$
$523$	−18681.0	−1.56188	−0.780940	+	0.624606i	$0.785259\pi$
$524$	8784.00	0.732311
$525$	−1596.00	−0.132676
$526$	4950.00	0.410324
$527$	−9300.00	−0.768718
$528$	528.000	0.0435194
$529$	−11383.0	−0.935563
$530$	1792.00	0.146867
$531$	−3861.00	−0.315543
$532$	−196.000	−0.0159731
$533$	12060.0	0.980069
$534$	6876.00	0.557217
$535$	10017.0	0.809482
$536$	6424.00	0.517676
$537$	1380.00	0.110896
$538$	−3492.00	−0.279834
$539$	539.000	0.0430730
$540$	−756.000	−0.0602464
$541$	−2000.00	−0.158940	−0.0794702	−	0.996837i	$-0.525323\pi$
$541$	−2000.00	−0.158940	−0.0794702	+	0.996837i	$0.525323\pi$
$542$	−3306.00	−0.262002
$543$	6888.00	0.544369
$544$	−960.000	−0.0756611
$545$	1946.00	0.152950
$546$	2814.00	0.220564
$547$	19368.0	1.51392	0.756961	−	0.653459i	$-0.226683\pi$
$547$	19368.0	1.51392	0.756961	+	0.653459i	$0.226683\pi$
$548$	3064.00	0.238846
$549$	198.000	0.0153924
$550$	1672.00	0.129626
$551$	−847.000	−0.0654871
$552$	−672.000	−0.0518156
$553$	−672.000	−0.0516751
$554$	4112.00	0.315347
$555$	1491.00	0.114035
$556$	−4320.00	−0.329512
$557$	4669.00	0.355174	0.177587	−	0.984105i	$-0.443171\pi$
$557$	4669.00	0.355174	0.177587	+	0.984105i	$0.443171\pi$
$558$	5580.00	0.423334
$559$	7236.00	0.547496
$560$	−784.000	−0.0591608
$561$	990.000	0.0745059
$562$	−1674.00	−0.125647
$563$	−9324.00	−0.697975	−0.348987	−	0.937127i	$-0.613474\pi$
$563$	−9324.00	−0.697975	−0.348987	+	0.937127i	$0.613474\pi$
$564$	852.000	0.0636093
$565$	16254.0	1.21028
$566$	−17938.0	−1.33214
$567$	567.000	0.0419961
$568$	−3744.00	−0.276575
$569$	25646.0	1.88952	0.944759	−	0.327765i	$-0.106295\pi$
$569$	25646.0	1.88952	0.944759	+	0.327765i	$0.106295\pi$
$570$	−294.000	−0.0216040
$571$	−12928.0	−0.947496	−0.473748	−	0.880661i	$-0.657099\pi$
$571$	−12928.0	−0.947496	−0.473748	+	0.880661i	$0.657099\pi$
$572$	−2948.00	−0.215493
$573$	2634.00	0.192037
$574$	2520.00	0.183245
$575$	−2128.00	−0.154337
$576$	576.000	0.0416667
$577$	−14862.0	−1.07229	−0.536147	−	0.844125i	$-0.680121\pi$
$577$	−14862.0	−1.07229	−0.536147	+	0.844125i	$0.680121\pi$
$578$	8026.00	0.577574
$579$	−6804.00	−0.488367
$580$	−3388.00	−0.242550
$581$	−7854.00	−0.560824
$582$	552.000	0.0393147
$583$	1408.00	0.100023
$584$	936.000	0.0663218
$585$	4221.00	0.298319
$586$	−10240.0	−0.721861
$587$	−13383.0	−0.941015	−0.470507	−	0.882396i	$-0.655929\pi$
$587$	−13383.0	−0.941015	−0.470507	+	0.882396i	$0.655929\pi$
$588$	588.000	0.0412393
$589$	2170.00	0.151805
$590$	−6006.00	−0.419090
$591$	15894.0	1.10625
$592$	−1136.00	−0.0788671
$593$	−10152.0	−0.703023	−0.351512	−	0.936184i	$-0.614332\pi$
$593$	−10152.0	−0.703023	−0.351512	+	0.936184i	$0.614332\pi$
$594$	−594.000	−0.0410305
$595$	−1470.00	−0.101284
$596$	5492.00	0.377451
$597$	3132.00	0.214714
$598$	3752.00	0.256573
$599$	16592.0	1.13177	0.565885	−	0.824484i	$-0.308534\pi$
$599$	16592.0	1.13177	0.565885	+	0.824484i	$0.308534\pi$
$600$	1824.00	0.124107
$601$	−1001.00	−0.0679395	−0.0339698	−	0.999423i	$-0.510815\pi$
$601$	−1001.00	−0.0679395	−0.0339698	+	0.999423i	$0.510815\pi$
$602$	1512.00	0.102366
$603$	−7227.00	−0.488070
$604$	−12840.0	−0.864987
$605$	−847.000	−0.0569181
$606$	1212.00	0.0812444
$607$	−18427.0	−1.23217	−0.616086	−	0.787679i	$-0.711283\pi$
$607$	−18427.0	−1.23217	−0.616086	+	0.787679i	$0.711283\pi$
$608$	224.000	0.0149414
$609$	2541.00	0.169075
$610$	308.000	0.0204435
$611$	−4757.00	−0.314972
$612$	1080.00	0.0713340
$613$	5606.00	0.369371	0.184685	−	0.982798i	$-0.440873\pi$
$613$	5606.00	0.369371	0.184685	+	0.982798i	$0.440873\pi$
$614$	5784.00	0.380168
$615$	3780.00	0.247844
$616$	−616.000	−0.0402911
$617$	6378.00	0.416157	0.208078	−	0.978112i	$-0.433279\pi$
$617$	6378.00	0.416157	0.208078	+	0.978112i	$0.433279\pi$
$618$	10788.0	0.702196
$619$	−4070.00	−0.264276	−0.132138	−	0.991231i	$-0.542184\pi$
$619$	−4070.00	−0.264276	−0.132138	+	0.991231i	$0.542184\pi$
$620$	8680.00	0.562254
$621$	756.000	0.0488522
$622$	−6416.00	−0.413598
$623$	−8022.00	−0.515882
$624$	−3216.00	−0.206319
$625$	−349.000	−0.0223360
$626$	−7100.00	−0.453312
$627$	−231.000	−0.0147133
$628$	1808.00	0.114884
$629$	−2130.00	−0.135022
$630$	882.000	0.0557773
$631$	23252.0	1.46695	0.733477	−	0.679715i	$-0.237896\pi$
$631$	23252.0	1.46695	0.733477	+	0.679715i	$0.237896\pi$
$632$	768.000	0.0483377
$633$	13698.0	0.860105
$634$	11048.0	0.692070
$635$	−15778.0	−0.986033
$636$	1536.00	0.0957647
$637$	−3283.00	−0.204203
$638$	−2662.00	−0.165187
$639$	4212.00	0.260758
$640$	896.000	0.0553399
$641$	3758.00	0.231563	0.115782	−	0.993275i	$-0.463063\pi$
$641$	3758.00	0.231563	0.115782	+	0.993275i	$0.463063\pi$
$642$	8586.00	0.527823
$643$	13068.0	0.801480	0.400740	−	0.916192i	$-0.368753\pi$
$643$	13068.0	0.801480	0.400740	+	0.916192i	$0.368753\pi$
$644$	784.000	0.0479719
$645$	2268.00	0.138453
$646$	420.000	0.0255800
$647$	−31027.0	−1.88531	−0.942656	−	0.333765i	$-0.891681\pi$
$647$	−31027.0	−1.88531	−0.942656	+	0.333765i	$0.891681\pi$
$648$	−648.000	−0.0392837
$649$	−4719.00	−0.285419
$650$	−10184.0	−0.614537
$651$	−6510.00	−0.391931
$652$	−44.0000	−0.00264290
$653$	−18456.0	−1.10603	−0.553016	−	0.833171i	$-0.686523\pi$
$653$	−18456.0	−1.10603	−0.553016	+	0.833171i	$0.686523\pi$
$654$	1668.00	0.0997308
$655$	−15372.0	−0.916998
$656$	−2880.00	−0.171410
$657$	−1053.00	−0.0625288
$658$	−994.000	−0.0588908
$659$	11141.0	0.658561	0.329281	−	0.944232i	$-0.393194\pi$
$659$	11141.0	0.658561	0.329281	+	0.944232i	$0.393194\pi$
$660$	−924.000	−0.0544949
$661$	5172.00	0.304338	0.152169	−	0.988354i	$-0.451374\pi$
$661$	5172.00	0.304338	0.152169	+	0.988354i	$0.451374\pi$
$662$	4288.00	0.251749
$663$	−6030.00	−0.353221
$664$	8976.00	0.524603
$665$	343.000	0.0200015
$666$	1278.00	0.0743566
$667$	3388.00	0.196677
$668$	2016.00	0.116769
$669$	594.000	0.0343279
$670$	−11242.0	−0.648233
$671$	242.000	0.0139230
$672$	−672.000	−0.0385758
$673$	−10460.0	−0.599113	−0.299557	−	0.954078i	$-0.596839\pi$
$673$	−10460.0	−0.599113	−0.299557	+	0.954078i	$0.596839\pi$
$674$	1768.00	0.101040
$675$	−2052.00	−0.117010
$676$	9168.00	0.521620
$677$	−15546.0	−0.882543	−0.441271	−	0.897374i	$-0.645472\pi$
$677$	−15546.0	−0.882543	−0.441271	+	0.897374i	$0.645472\pi$
$678$	13932.0	0.789167
$679$	−644.000	−0.0363983
$680$	1680.00	0.0947427
$681$	11286.0	0.635067
$682$	6820.00	0.382920
$683$	31138.0	1.74445	0.872227	−	0.489101i	$-0.162675\pi$
$683$	31138.0	1.74445	0.872227	+	0.489101i	$0.162675\pi$
$684$	−252.000	−0.0140869
$685$	−5362.00	−0.299082
$686$	−686.000	−0.0381802
$687$	−2112.00	−0.117289
$688$	−1728.00	−0.0957549
$689$	−8576.00	−0.474194
$690$	1176.00	0.0648834
$691$	21574.0	1.18772	0.593859	−	0.804569i	$-0.297604\pi$
$691$	21574.0	1.18772	0.593859	+	0.804569i	$0.297604\pi$
$692$	6752.00	0.370914
$693$	693.000	0.0379869
$694$	8080.00	0.441949
$695$	7560.00	0.412615
$696$	−2904.00	−0.158155
$697$	−5400.00	−0.293457
$698$	−2450.00	−0.132857
$699$	6630.00	0.358755
$700$	−2128.00	−0.114901
$701$	2074.00	0.111746	0.0558730	−	0.998438i	$-0.482206\pi$
$701$	2074.00	0.111746	0.0558730	+	0.998438i	$0.482206\pi$
$702$	3618.00	0.194519
$703$	497.000	0.0266639
$704$	704.000	0.0376889
$705$	−1491.00	−0.0796515
$706$	−4966.00	−0.264728
$707$	−1414.00	−0.0752177
$708$	−5148.00	−0.273268
$709$	−6829.00	−0.361733	−0.180866	−	0.983508i	$-0.557890\pi$
$709$	−6829.00	−0.361733	−0.180866	+	0.983508i	$0.557890\pi$
$710$	6552.00	0.346327
$711$	−864.000	−0.0455732
$712$	9168.00	0.482564
$713$	−8680.00	−0.455917
$714$	−1260.00	−0.0660425
$715$	5159.00	0.269840
$716$	1840.00	0.0960391
$717$	7353.00	0.382989
$718$	−15608.0	−0.811261
$719$	6921.00	0.358984	0.179492	−	0.983759i	$-0.442555\pi$
$719$	6921.00	0.358984	0.179492	+	0.983759i	$0.442555\pi$
$720$	−1008.00	−0.0521749
$721$	−12586.0	−0.650107
$722$	13620.0	0.702055
$723$	75.0000	0.00385793
$724$	9184.00	0.471437
$725$	−9196.00	−0.471077
$726$	−726.000	−0.0371135
$727$	24064.0	1.22763	0.613813	−	0.789451i	$-0.289635\pi$
$727$	24064.0	1.22763	0.613813	+	0.789451i	$0.289635\pi$
$728$	3752.00	0.191014
$729$	729.000	0.0370370
$730$	−1638.00	−0.0830481
$731$	−3240.00	−0.163934
$732$	264.000	0.0133302
$733$	30262.0	1.52490	0.762451	−	0.647047i	$-0.223996\pi$
$733$	30262.0	1.52490	0.762451	+	0.647047i	$0.223996\pi$
$734$	−6148.00	−0.309165
$735$	−1029.00	−0.0516398
$736$	−896.000	−0.0448736
$737$	−8833.00	−0.441476
$738$	3240.00	0.161607
$739$	24024.0	1.19586	0.597928	−	0.801550i	$-0.295991\pi$
$739$	24024.0	1.19586	0.597928	+	0.801550i	$0.295991\pi$
$740$	1988.00	0.0987572
$741$	1407.00	0.0697536
$742$	−1792.00	−0.0886609
$743$	−227.000	−0.0112084	−0.00560419	−	0.999984i	$-0.501784\pi$
$743$	−227.000	−0.0112084	−0.00560419	+	0.999984i	$0.501784\pi$
$744$	7440.00	0.366618
$745$	−9611.00	−0.472644
$746$	−1348.00	−0.0661579
$747$	−10098.0	−0.494600
$748$	1320.00	0.0645240
$749$	−10017.0	−0.488669
$750$	−8442.00	−0.411011
$751$	−23435.0	−1.13869	−0.569344	−	0.822099i	$-0.692803\pi$
$751$	−23435.0	−1.13869	−0.569344	+	0.822099i	$0.692803\pi$
$752$	1136.00	0.0550873
$753$	−15801.0	−0.764702
$754$	16214.0	0.783129
$755$	22470.0	1.08314
$756$	756.000	0.0363696
$757$	1609.00	0.0772524	0.0386262	−	0.999254i	$-0.487702\pi$
$757$	1609.00	0.0772524	0.0386262	+	0.999254i	$0.487702\pi$
$758$	−24102.0	−1.15491
$759$	924.000	0.0441885
$760$	−392.000	−0.0187097
$761$	6976.00	0.332299	0.166150	−	0.986101i	$-0.446867\pi$
$761$	6976.00	0.332299	0.166150	+	0.986101i	$0.446867\pi$
$762$	−13524.0	−0.642943
$763$	−1946.00	−0.0923328
$764$	3512.00	0.166309
$765$	−1890.00	−0.0893243
$766$	25328.0	1.19470
$767$	28743.0	1.35313
$768$	768.000	0.0360844
$769$	−36079.0	−1.69186	−0.845931	−	0.533292i	$-0.820955\pi$
$769$	−36079.0	−1.69186	−0.845931	+	0.533292i	$0.820955\pi$
$770$	1078.00	0.0504525
$771$	−4869.00	−0.227435
$772$	−9072.00	−0.422938
$773$	−23507.0	−1.09377	−0.546887	−	0.837206i	$-0.684187\pi$
$773$	−23507.0	−1.09377	−0.546887	+	0.837206i	$0.684187\pi$
$774$	1944.00	0.0902786
$775$	23560.0	1.09200
$776$	736.000	0.0340475
$777$	−1491.00	−0.0688408
$778$	−528.000	−0.0243313
$779$	1260.00	0.0579515
$780$	5628.00	0.258352
$781$	5148.00	0.235864
$782$	−1680.00	−0.0768244
$783$	3267.00	0.149110
$784$	784.000	0.0357143
$785$	−3164.00	−0.143857
$786$	−13176.0	−0.597929
$787$	−35145.0	−1.59185	−0.795924	−	0.605397i	$-0.793014\pi$
$787$	−35145.0	−1.59185	−0.795924	+	0.605397i	$0.793014\pi$
$788$	21192.0	0.958038
$789$	−7425.00	−0.335028
$790$	−1344.00	−0.0605283
$791$	−16254.0	−0.730627
$792$	−792.000	−0.0355335
$793$	−1474.00	−0.0660067
$794$	28788.0	1.28671
$795$	−2688.00	−0.119916
$796$	4176.00	0.185948
$797$	34155.0	1.51798	0.758991	−	0.651101i	$-0.225692\pi$
$797$	34155.0	1.51798	0.758991	+	0.651101i	$0.225692\pi$
$798$	294.000	0.0130420
$799$	2130.00	0.0943104
$800$	2432.00	0.107480
$801$	−10314.0	−0.454965
$802$	24608.0	1.08346
$803$	−1287.00	−0.0565595
$804$	−9636.00	−0.422681
$805$	−1372.00	−0.0600704
$806$	−41540.0	−1.81536
$807$	5238.00	0.228484
$808$	1616.00	0.0703598
$809$	−3439.00	−0.149455	−0.0747273	−	0.997204i	$-0.523809\pi$
$809$	−3439.00	−0.149455	−0.0747273	+	0.997204i	$0.523809\pi$
$810$	1134.00	0.0491910
$811$	−1625.00	−0.0703594	−0.0351797	−	0.999381i	$-0.511200\pi$
$811$	−1625.00	−0.0703594	−0.0351797	+	0.999381i	$0.511200\pi$
$812$	3388.00	0.146423
$813$	4959.00	0.213923
$814$	1562.00	0.0672581
$815$	77.0000	0.00330944
$816$	1440.00	0.0617771
$817$	756.000	0.0323734
$818$	6308.00	0.269626
$819$	−4221.00	−0.180090
$820$	5040.00	0.214640
$821$	31531.0	1.34036	0.670182	−	0.742196i	$-0.266216\pi$
$821$	31531.0	1.34036	0.670182	+	0.742196i	$0.266216\pi$
$822$	−4596.00	−0.195017
$823$	−31003.0	−1.31312	−0.656559	−	0.754274i	$-0.727989\pi$
$823$	−31003.0	−1.31312	−0.656559	+	0.754274i	$0.727989\pi$
$824$	14384.0	0.608119
$825$	−2508.00	−0.105839
$826$	6006.00	0.252997
$827$	−15015.0	−0.631345	−0.315673	−	0.948868i	$-0.602230\pi$
$827$	−15015.0	−0.631345	−0.315673	+	0.948868i	$0.602230\pi$
$828$	1008.00	0.0423073
$829$	12280.0	0.514478	0.257239	−	0.966348i	$-0.417187\pi$
$829$	12280.0	0.514478	0.257239	+	0.966348i	$0.417187\pi$
$830$	−15708.0	−0.656907
$831$	−6168.00	−0.257480
$832$	−4288.00	−0.178677
$833$	1470.00	0.0611434
$834$	6480.00	0.269046
$835$	−3528.00	−0.146217
$836$	−308.000	−0.0127421
$837$	−8370.00	−0.345651
$838$	9270.00	0.382132
$839$	26599.0	1.09452	0.547258	−	0.836964i	$-0.315672\pi$
$839$	26599.0	1.09452	0.547258	+	0.836964i	$0.315672\pi$
$840$	1176.00	0.0483046
$841$	−9748.00	−0.399688
$842$	14530.0	0.594699
$843$	2511.00	0.102590
$844$	18264.0	0.744873
$845$	−16044.0	−0.653172
$846$	−1278.00	−0.0519368
$847$	847.000	0.0343604
$848$	2048.00	0.0829347
$849$	26907.0	1.08769
$850$	4560.00	0.184008
$851$	−1988.00	−0.0800796
$852$	5616.00	0.225823
$853$	−3798.00	−0.152451	−0.0762257	−	0.997091i	$-0.524287\pi$
$853$	−3798.00	−0.152451	−0.0762257	+	0.997091i	$0.524287\pi$
$854$	−308.000	−0.0123414
$855$	441.000	0.0176396
$856$	11448.0	0.457108
$857$	−38726.0	−1.54359	−0.771794	−	0.635873i	$-0.780640\pi$
$857$	−38726.0	−1.54359	−0.771794	+	0.635873i	$0.780640\pi$
$858$	4422.00	0.175949
$859$	3296.00	0.130917	0.0654587	−	0.997855i	$-0.479149\pi$
$859$	3296.00	0.130917	0.0654587	+	0.997855i	$0.479149\pi$
$860$	3024.00	0.119904
$861$	−3780.00	−0.149619
$862$	−15178.0	−0.599727
$863$	13488.0	0.532024	0.266012	−	0.963970i	$-0.414294\pi$
$863$	13488.0	0.532024	0.266012	+	0.963970i	$0.414294\pi$
$864$	−864.000	−0.0340207
$865$	−11816.0	−0.464458
$866$	−10328.0	−0.405265
$867$	−12039.0	−0.471587
$868$	−8680.00	−0.339422
$869$	−1056.00	−0.0412225
$870$	5082.00	0.198041
$871$	53801.0	2.09297
$872$	2224.00	0.0863694
$873$	−828.000	−0.0321003
$874$	392.000	0.0151712
$875$	9849.00	0.380522
$876$	−1404.00	−0.0541516
$877$	−17144.0	−0.660105	−0.330052	−	0.943963i	$-0.607066\pi$
$877$	−17144.0	−0.660105	−0.330052	+	0.943963i	$0.607066\pi$
$878$	−28442.0	−1.09325
$879$	15360.0	0.589397
$880$	−1232.00	−0.0471940
$881$	11993.0	0.458632	0.229316	−	0.973352i	$-0.426351\pi$
$881$	11993.0	0.458632	0.229316	+	0.973352i	$0.426351\pi$
$882$	−882.000	−0.0336718
$883$	10483.0	0.399526	0.199763	−	0.979844i	$-0.435983\pi$
$883$	10483.0	0.399526	0.199763	+	0.979844i	$0.435983\pi$
$884$	−8040.00	−0.305899
$885$	9009.00	0.342186
$886$	−13848.0	−0.525093
$887$	4562.00	0.172691	0.0863455	−	0.996265i	$-0.472481\pi$
$887$	4562.00	0.172691	0.0863455	+	0.996265i	$0.472481\pi$
$888$	1704.00	0.0643947
$889$	15778.0	0.595250
$890$	−16044.0	−0.604265
$891$	891.000	0.0335013
$892$	792.000	0.0297288
$893$	−497.000	−0.0186243
$894$	−8238.00	−0.308188
$895$	−3220.00	−0.120260
$896$	−896.000	−0.0334077
$897$	−5628.00	−0.209491
$898$	−5048.00	−0.187588
$899$	−37510.0	−1.39158
$900$	−2736.00	−0.101333
$901$	3840.00	0.141986
$902$	3960.00	0.146179
$903$	−2268.00	−0.0835817
$904$	18576.0	0.683439
$905$	−16072.0	−0.590333
$906$	19260.0	0.706259
$907$	18556.0	0.679318	0.339659	−	0.940549i	$-0.389688\pi$
$907$	18556.0	0.679318	0.339659	+	0.940549i	$0.389688\pi$
$908$	15048.0	0.549984
$909$	−1818.00	−0.0663358
$910$	−6566.00	−0.239188
$911$	5826.00	0.211881	0.105941	−	0.994372i	$-0.466215\pi$
$911$	5826.00	0.211881	0.105941	+	0.994372i	$0.466215\pi$
$912$	−336.000	−0.0121996
$913$	−12342.0	−0.447383
$914$	−3856.00	−0.139546
$915$	−462.000	−0.0166921
$916$	−2816.00	−0.101576
$917$	15372.0	0.553575
$918$	−1620.00	−0.0582440
$919$	41492.0	1.48933	0.744665	−	0.667438i	$-0.232609\pi$
$919$	41492.0	1.48933	0.744665	+	0.667438i	$0.232609\pi$
$920$	1568.00	0.0561907
$921$	−8676.00	−0.310406
$922$	25704.0	0.918130
$923$	−31356.0	−1.11820
$924$	924.000	0.0328976
$925$	5396.00	0.191805
$926$	−9482.00	−0.336499
$927$	−16182.0	−0.573340
$928$	−3872.00	−0.136966
$929$	−15381.0	−0.543202	−0.271601	−	0.962410i	$-0.587553\pi$
$929$	−15381.0	−0.543202	−0.271601	+	0.962410i	$0.587553\pi$
$930$	−13020.0	−0.459078
$931$	−343.000	−0.0120745
$932$	8840.00	0.310691
$933$	9624.00	0.337702
$934$	13618.0	0.477082
$935$	−2310.00	−0.0807969
$936$	4824.00	0.168459
$937$	−55054.0	−1.91946	−0.959731	−	0.280921i	$-0.909360\pi$
$937$	−55054.0	−1.91946	−0.959731	+	0.280921i	$0.909360\pi$
$938$	11242.0	0.391327
$939$	10650.0	0.370127
$940$	−1988.00	−0.0689802
$941$	−3892.00	−0.134831	−0.0674153	−	0.997725i	$-0.521475\pi$
$941$	−3892.00	−0.134831	−0.0674153	+	0.997725i	$0.521475\pi$
$942$	−2712.00	−0.0938023
$943$	−5040.00	−0.174046
$944$	−6864.00	−0.236657
$945$	−1323.00	−0.0455420
$946$	2376.00	0.0816601
$947$	−36158.0	−1.24074	−0.620368	−	0.784311i	$-0.713017\pi$
$947$	−36158.0	−1.24074	−0.620368	+	0.784311i	$0.713017\pi$
$948$	−1152.00	−0.0394675
$949$	7839.00	0.268140
$950$	−1064.00	−0.0363376
$951$	−16572.0	−0.565072
$952$	−1680.00	−0.0571944
$953$	−23959.0	−0.814384	−0.407192	−	0.913343i	$-0.633492\pi$
$953$	−23959.0	−0.814384	−0.407192	+	0.913343i	$0.633492\pi$
$954$	−2304.00	−0.0781916
$955$	−6146.00	−0.208251
$956$	9804.00	0.331678
$957$	3993.00	0.134875
$958$	−3468.00	−0.116958
$959$	5362.00	0.180551
$960$	−1344.00	−0.0451848
$961$	66309.0	2.22581
$962$	−9514.00	−0.318860
$963$	−12879.0	−0.430966
$964$	100.000	0.00334106
$965$	15876.0	0.529603
$966$	−1176.00	−0.0391689
$967$	−56106.0	−1.86582	−0.932910	−	0.360110i	$-0.882739\pi$
$967$	−56106.0	−1.86582	−0.932910	+	0.360110i	$0.882739\pi$
$968$	−968.000	−0.0321412
$969$	−630.000	−0.0208860
$970$	−1288.00	−0.0426342
$971$	−35079.0	−1.15936	−0.579680	−	0.814844i	$-0.696822\pi$
$971$	−35079.0	−1.15936	−0.579680	+	0.814844i	$0.696822\pi$
$972$	972.000	0.0320750
$973$	−7560.00	−0.249088
$974$	14736.0	0.484776
$975$	15276.0	0.501768
$976$	352.000	0.0115443
$977$	−26184.0	−0.857421	−0.428711	−	0.903442i	$-0.641032\pi$
$977$	−26184.0	−0.857421	−0.428711	+	0.903442i	$0.641032\pi$
$978$	66.0000	0.00215792
$979$	−12606.0	−0.411532
$980$	−1372.00	−0.0447214
$981$	−2502.00	−0.0814299
$982$	−26402.0	−0.857965
$983$	−8584.00	−0.278522	−0.139261	−	0.990256i	$-0.544473\pi$
$983$	−8584.00	−0.278522	−0.139261	+	0.990256i	$0.544473\pi$
$984$	4320.00	0.139956
$985$	−37086.0	−1.19965
$986$	−7260.00	−0.234488
$987$	1491.00	0.0480841
$988$	1876.00	0.0604084
$989$	−3024.00	−0.0972271
$990$	1386.00	0.0444949
$991$	−51331.0	−1.64539	−0.822696	−	0.568482i	$-0.807531\pi$
$991$	−51331.0	−1.64539	−0.822696	+	0.568482i	$0.807531\pi$
$992$	9920.00	0.317500
$993$	−6432.00	−0.205552
$994$	−6552.00	−0.209071
$995$	−7308.00	−0.232843
$996$	−13464.0	−0.428337
$997$	−38542.0	−1.22431	−0.612155	−	0.790738i	$-0.709697\pi$
$997$	−38542.0	−1.22431	−0.612155	+	0.790738i	$0.709697\pi$
$998$	−33078.0	−1.04916
$999$	−1917.00	−0.0607119

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	462.4.a.d.1.1	✓	1
3.2	odd	2		1386.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.4.a.d.1.1	✓	1	1.1	even	1	trivial
1386.4.a.l.1.1		1	3.2	odd	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:	\( N \)	\(=\)	\( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	462.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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