LMFDB - Embedded newform 462.4.a.e.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)

$f(q)$

$=$

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

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -4.00000 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -8.00000 q^{10} -11.0000 q^{11} -12.0000 q^{12} +62.0000 q^{13} -14.0000 q^{14} +12.0000 q^{15} +16.0000 q^{16} -120.000 q^{17} +18.0000 q^{18} +118.000 q^{19} -16.0000 q^{20} +21.0000 q^{21} -22.0000 q^{22} -188.000 q^{23} -24.0000 q^{24} -109.000 q^{25} +124.000 q^{26} -27.0000 q^{27} -28.0000 q^{28} +62.0000 q^{29} +24.0000 q^{30} -322.000 q^{31} +32.0000 q^{32} +33.0000 q^{33} -240.000 q^{34} +28.0000 q^{35} +36.0000 q^{36} -198.000 q^{37} +236.000 q^{38} -186.000 q^{39} -32.0000 q^{40} +48.0000 q^{41} +42.0000 q^{42} +32.0000 q^{43} -44.0000 q^{44} -36.0000 q^{45} -376.000 q^{46} -326.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} -218.000 q^{50} +360.000 q^{51} +248.000 q^{52} -482.000 q^{53} -54.0000 q^{54} +44.0000 q^{55} -56.0000 q^{56} -354.000 q^{57} +124.000 q^{58} +400.000 q^{59} +48.0000 q^{60} +70.0000 q^{61} -644.000 q^{62} -63.0000 q^{63} +64.0000 q^{64} -248.000 q^{65} +66.0000 q^{66} -124.000 q^{67} -480.000 q^{68} +564.000 q^{69} +56.0000 q^{70} -712.000 q^{71} +72.0000 q^{72} +304.000 q^{73} -396.000 q^{74} +327.000 q^{75} +472.000 q^{76} +77.0000 q^{77} -372.000 q^{78} -1016.00 q^{79} -64.0000 q^{80} +81.0000 q^{81} +96.0000 q^{82} +430.000 q^{83} +84.0000 q^{84} +480.000 q^{85} +64.0000 q^{86} -186.000 q^{87} -88.0000 q^{88} +442.000 q^{89} -72.0000 q^{90} -434.000 q^{91} -752.000 q^{92} +966.000 q^{93} -652.000 q^{94} -472.000 q^{95} -96.0000 q^{96} -966.000 q^{97} +98.0000 q^{98} -99.0000 q^{99} +O(q^{100})\)

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
$5$	−4.00000	−0.357771	−0.178885	−	0.983870i	$-0.557249\pi$
$5$	−4.00000	−0.357771	−0.178885	+	0.983870i	$0.557249\pi$
$6$	−6.00000	−0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−8.00000	−0.252982
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−12.0000	−0.288675
$13$	62.0000	1.32275	0.661373	−	0.750057i	$-0.269974\pi$
$13$	62.0000	1.32275	0.661373	+	0.750057i	$0.269974\pi$
$14$	−14.0000	−0.267261
$15$	12.0000	0.206559
$16$	16.0000	0.250000
$17$	−120.000	−1.71202	−0.856008	−	0.516962i	$-0.827063\pi$
$17$	−120.000	−1.71202	−0.856008	+	0.516962i	$0.827063\pi$
$18$	18.0000	0.235702
$19$	118.000	1.42479	0.712396	−	0.701777i	$-0.247610\pi$
$19$	118.000	1.42479	0.712396	+	0.701777i	$0.247610\pi$
$20$	−16.0000	−0.178885
$21$	21.0000	0.218218
$22$	−22.0000	−0.213201
$23$	−188.000	−1.70438	−0.852189	−	0.523234i	$-0.824726\pi$
$23$	−188.000	−1.70438	−0.852189	+	0.523234i	$0.824726\pi$
$24$	−24.0000	−0.204124
$25$	−109.000	−0.872000
$26$	124.000	0.935323
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	62.0000	0.397004	0.198502	−	0.980101i	$-0.436392\pi$
$29$	62.0000	0.397004	0.198502	+	0.980101i	$0.436392\pi$
$30$	24.0000	0.146059
$31$	−322.000	−1.86558	−0.932789	−	0.360423i	$-0.882632\pi$
$31$	−322.000	−1.86558	−0.932789	+	0.360423i	$0.882632\pi$
$32$	32.0000	0.176777
$33$	33.0000	0.174078
$34$	−240.000	−1.21058
$35$	28.0000	0.135225
$36$	36.0000	0.166667
$37$	−198.000	−0.879757	−0.439878	−	0.898057i	$-0.644978\pi$
$37$	−198.000	−0.879757	−0.439878	+	0.898057i	$0.644978\pi$
$38$	236.000	1.00748
$39$	−186.000	−0.763688
$40$	−32.0000	−0.126491
$41$	48.0000	0.182838	0.0914188	−	0.995813i	$-0.470860\pi$
$41$	48.0000	0.182838	0.0914188	+	0.995813i	$0.470860\pi$
$42$	42.0000	0.154303
$43$	32.0000	0.113487	0.0567437	−	0.998389i	$-0.481928\pi$
$43$	32.0000	0.113487	0.0567437	+	0.998389i	$0.481928\pi$
$44$	−44.0000	−0.150756
$45$	−36.0000	−0.119257
$46$	−376.000	−1.20518
$47$	−326.000	−1.01174	−0.505872	−	0.862608i	$-0.668829\pi$
$47$	−326.000	−1.01174	−0.505872	+	0.862608i	$0.668829\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	−218.000	−0.616597
$51$	360.000	0.988433
$52$	248.000	0.661373
$53$	−482.000	−1.24920	−0.624602	−	0.780943i	$-0.714739\pi$
$53$	−482.000	−1.24920	−0.624602	+	0.780943i	$0.714739\pi$
$54$	−54.0000	−0.136083
$55$	44.0000	0.107872
$56$	−56.0000	−0.133631
$57$	−354.000	−0.822604
$58$	124.000	0.280724
$59$	400.000	0.882637	0.441318	−	0.897351i	$-0.354511\pi$
$59$	400.000	0.882637	0.441318	+	0.897351i	$0.354511\pi$
$60$	48.0000	0.103280
$61$	70.0000	0.146928	0.0734638	−	0.997298i	$-0.476595\pi$
$61$	70.0000	0.146928	0.0734638	+	0.997298i	$0.476595\pi$
$62$	−644.000	−1.31916
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	−248.000	−0.473240
$66$	66.0000	0.123091
$67$	−124.000	−0.226105	−0.113052	−	0.993589i	$-0.536063\pi$
$67$	−124.000	−0.226105	−0.113052	+	0.993589i	$0.536063\pi$
$68$	−480.000	−0.856008
$69$	564.000	0.984023
$70$	56.0000	0.0956183
$71$	−712.000	−1.19012	−0.595062	−	0.803680i	$-0.702873\pi$
$71$	−712.000	−1.19012	−0.595062	+	0.803680i	$0.702873\pi$
$72$	72.0000	0.117851
$73$	304.000	0.487404	0.243702	−	0.969850i	$-0.421638\pi$
$73$	304.000	0.487404	0.243702	+	0.969850i	$0.421638\pi$
$74$	−396.000	−0.622082
$75$	327.000	0.503449
$76$	472.000	0.712396
$77$	77.0000	0.113961
$78$	−372.000	−0.540009
$79$	−1016.00	−1.44695	−0.723474	−	0.690351i	$-0.757456\pi$
$79$	−1016.00	−1.44695	−0.723474	+	0.690351i	$0.757456\pi$
$80$	−64.0000	−0.0894427
$81$	81.0000	0.111111
$82$	96.0000	0.129286
$83$	430.000	0.568658	0.284329	−	0.958727i	$-0.408229\pi$
$83$	430.000	0.568658	0.284329	+	0.958727i	$0.408229\pi$
$84$	84.0000	0.109109
$85$	480.000	0.612510
$86$	64.0000	0.0802476
$87$	−186.000	−0.229210
$88$	−88.0000	−0.106600
$89$	442.000	0.526426	0.263213	−	0.964738i	$-0.415218\pi$
$89$	442.000	0.526426	0.263213	+	0.964738i	$0.415218\pi$
$90$	−72.0000	−0.0843274
$91$	−434.000	−0.499951
$92$	−752.000	−0.852189
$93$	966.000	1.07709
$94$	−652.000	−0.715411
$95$	−472.000	−0.509749
$96$	−96.0000	−0.102062
$97$	−966.000	−1.01116	−0.505580	−	0.862780i	$-0.668721\pi$
$97$	−966.000	−1.01116	−0.505580	+	0.862780i	$0.668721\pi$
$98$	98.0000	0.101015
$99$	−99.0000	−0.100504
$100$	−436.000	−0.436000
$101$	546.000	0.537911	0.268956	−	0.963153i	$-0.413322\pi$
$101$	546.000	0.537911	0.268956	+	0.963153i	$0.413322\pi$
$102$	720.000	0.698928
$103$	98.0000	0.0937498	0.0468749	−	0.998901i	$-0.485074\pi$
$103$	98.0000	0.0937498	0.0468749	+	0.998901i	$0.485074\pi$
$104$	496.000	0.467662
$105$	−84.0000	−0.0780720
$106$	−964.000	−0.883320
$107$	−2064.00	−1.86481	−0.932404	−	0.361418i	$-0.882293\pi$
$107$	−2064.00	−1.86481	−0.932404	+	0.361418i	$0.882293\pi$
$108$	−108.000	−0.0962250
$109$	1306.00	1.14763	0.573817	−	0.818984i	$-0.305462\pi$
$109$	1306.00	1.14763	0.573817	+	0.818984i	$0.305462\pi$
$110$	88.0000	0.0762770
$111$	594.000	0.507928
$112$	−112.000	−0.0944911
$113$	478.000	0.397933	0.198967	−	0.980006i	$-0.436241\pi$
$113$	478.000	0.397933	0.198967	+	0.980006i	$0.436241\pi$
$114$	−708.000	−0.581669
$115$	752.000	0.609777
$116$	248.000	0.198502
$117$	558.000	0.440916
$118$	800.000	0.624118
$119$	840.000	0.647081
$120$	96.0000	0.0730297
$121$	121.000	0.0909091
$122$	140.000	0.103893
$123$	−144.000	−0.105561
$124$	−1288.00	−0.932789
$125$	936.000	0.669747
$126$	−126.000	−0.0890871
$127$	824.000	0.575734	0.287867	−	0.957670i	$-0.407054\pi$
$127$	824.000	0.575734	0.287867	+	0.957670i	$0.407054\pi$
$128$	128.000	0.0883883
$129$	−96.0000	−0.0655219
$130$	−496.000	−0.334631
$131$	−22.0000	−0.0146729	−0.00733645	−	0.999973i	$-0.502335\pi$
$131$	−22.0000	−0.0146729	−0.00733645	+	0.999973i	$0.502335\pi$
$132$	132.000	0.0870388
$133$	−826.000	−0.538521
$134$	−248.000	−0.159880
$135$	108.000	0.0688530
$136$	−960.000	−0.605289
$137$	−2106.00	−1.31334	−0.656671	−	0.754178i	$-0.728036\pi$
$137$	−2106.00	−1.31334	−0.656671	+	0.754178i	$0.728036\pi$
$138$	1128.00	0.695810
$139$	578.000	0.352700	0.176350	−	0.984328i	$-0.443571\pi$
$139$	578.000	0.352700	0.176350	+	0.984328i	$0.443571\pi$
$140$	112.000	0.0676123
$141$	978.000	0.584131
$142$	−1424.00	−0.841545
$143$	−682.000	−0.398823
$144$	144.000	0.0833333
$145$	−248.000	−0.142036
$146$	608.000	0.344647
$147$	−147.000	−0.0824786
$148$	−792.000	−0.439878
$149$	946.000	0.520130	0.260065	−	0.965591i	$-0.416256\pi$
$149$	946.000	0.520130	0.260065	+	0.965591i	$0.416256\pi$
$150$	654.000	0.355993
$151$	2368.00	1.27619	0.638096	−	0.769956i	$-0.279722\pi$
$151$	2368.00	1.27619	0.638096	+	0.769956i	$0.279722\pi$
$152$	944.000	0.503740
$153$	−1080.00	−0.570672
$154$	154.000	0.0805823
$155$	1288.00	0.667449
$156$	−744.000	−0.381844
$157$	208.000	0.105734	0.0528669	−	0.998602i	$-0.483164\pi$
$157$	208.000	0.105734	0.0528669	+	0.998602i	$0.483164\pi$
$158$	−2032.00	−1.02315
$159$	1446.00	0.721228
$160$	−128.000	−0.0632456
$161$	1316.00	0.644195
$162$	162.000	0.0785674
$163$	3852.00	1.85099	0.925497	−	0.378756i	$-0.123648\pi$
$163$	3852.00	1.85099	0.925497	+	0.378756i	$0.123648\pi$
$164$	192.000	0.0914188
$165$	−132.000	−0.0622799
$166$	860.000	0.402102
$167$	28.0000	0.0129743	0.00648714	−	0.999979i	$-0.497935\pi$
$167$	28.0000	0.0129743	0.00648714	+	0.999979i	$0.497935\pi$
$168$	168.000	0.0771517
$169$	1647.00	0.749659
$170$	960.000	0.433110
$171$	1062.00	0.474931
$172$	128.000	0.0567437
$173$	3646.00	1.60231	0.801157	−	0.598455i	$-0.204218\pi$
$173$	3646.00	1.60231	0.801157	+	0.598455i	$0.204218\pi$
$174$	−372.000	−0.162076
$175$	763.000	0.329585
$176$	−176.000	−0.0753778
$177$	−1200.00	−0.509591
$178$	884.000	0.372239
$179$	948.000	0.395848	0.197924	−	0.980217i	$-0.436580\pi$
$179$	948.000	0.395848	0.197924	+	0.980217i	$0.436580\pi$
$180$	−144.000	−0.0596285
$181$	2228.00	0.914950	0.457475	−	0.889223i	$-0.348754\pi$
$181$	2228.00	0.914950	0.457475	+	0.889223i	$0.348754\pi$
$182$	−868.000	−0.353519
$183$	−210.000	−0.0848287
$184$	−1504.00	−0.602589
$185$	792.000	0.314751
$186$	1932.00	0.761619
$187$	1320.00	0.516192
$188$	−1304.00	−0.505872
$189$	189.000	0.0727393
$190$	−944.000	−0.360447
$191$	−3412.00	−1.29258	−0.646292	−	0.763090i	$-0.723681\pi$
$191$	−3412.00	−1.29258	−0.646292	+	0.763090i	$0.723681\pi$
$192$	−192.000	−0.0721688
$193$	−3894.00	−1.45231	−0.726156	−	0.687530i	$-0.758695\pi$
$193$	−3894.00	−1.45231	−0.726156	+	0.687530i	$0.758695\pi$
$194$	−1932.00	−0.714998
$195$	744.000	0.273225
$196$	196.000	0.0714286
$197$	−1122.00	−0.405783	−0.202891	−	0.979201i	$-0.565034\pi$
$197$	−1122.00	−0.405783	−0.202891	+	0.979201i	$0.565034\pi$
$198$	−198.000	−0.0710669
$199$	−1406.00	−0.500848	−0.250424	−	0.968136i	$-0.580570\pi$
$199$	−1406.00	−0.500848	−0.250424	+	0.968136i	$0.580570\pi$
$200$	−872.000	−0.308299
$201$	372.000	0.130542
$202$	1092.00	0.380361
$203$	−434.000	−0.150053
$204$	1440.00	0.494217
$205$	−192.000	−0.0654140
$206$	196.000	0.0662911
$207$	−1692.00	−0.568126
$208$	992.000	0.330687
$209$	−1298.00	−0.429591
$210$	−168.000	−0.0552052
$211$	−4288.00	−1.39904	−0.699522	−	0.714612i	$-0.746604\pi$
$211$	−4288.00	−1.39904	−0.699522	+	0.714612i	$0.746604\pi$
$212$	−1928.00	−0.624602
$213$	2136.00	0.687119
$214$	−4128.00	−1.31862
$215$	−128.000	−0.0406025
$216$	−216.000	−0.0680414
$217$	2254.00	0.705122
$218$	2612.00	0.811500
$219$	−912.000	−0.281403
$220$	176.000	0.0539360
$221$	−7440.00	−2.26456
$222$	1188.00	0.359159
$223$	4102.00	1.23179	0.615897	−	0.787826i	$-0.288794\pi$
$223$	4102.00	1.23179	0.615897	+	0.787826i	$0.288794\pi$
$224$	−224.000	−0.0668153
$225$	−981.000	−0.290667
$226$	956.000	0.281381
$227$	−6670.00	−1.95024	−0.975118	−	0.221688i	$-0.928843\pi$
$227$	−6670.00	−1.95024	−0.975118	+	0.221688i	$0.928843\pi$
$228$	−1416.00	−0.411302
$229$	−1708.00	−0.492873	−0.246436	−	0.969159i	$-0.579260\pi$
$229$	−1708.00	−0.492873	−0.246436	+	0.969159i	$0.579260\pi$
$230$	1504.00	0.431177
$231$	−231.000	−0.0657952
$232$	496.000	0.140362
$233$	3198.00	0.899176	0.449588	−	0.893236i	$-0.351571\pi$
$233$	3198.00	0.899176	0.449588	+	0.893236i	$0.351571\pi$
$234$	1116.00	0.311774
$235$	1304.00	0.361973
$236$	1600.00	0.441318
$237$	3048.00	0.835396
$238$	1680.00	0.457556
$239$	3760.00	1.01763	0.508816	−	0.860875i	$-0.330083\pi$
$239$	3760.00	1.01763	0.508816	+	0.860875i	$0.330083\pi$
$240$	192.000	0.0516398
$241$	6848.00	1.83037	0.915184	−	0.403037i	$-0.132046\pi$
$241$	6848.00	1.83037	0.915184	+	0.403037i	$0.132046\pi$
$242$	242.000	0.0642824
$243$	−243.000	−0.0641500
$244$	280.000	0.0734638
$245$	−196.000	−0.0511101
$246$	−288.000	−0.0746431
$247$	7316.00	1.88464
$248$	−2576.00	−0.659581
$249$	−1290.00	−0.328315
$250$	1872.00	0.473583
$251$	−6800.00	−1.71001	−0.855004	−	0.518621i	$-0.826446\pi$
$251$	−6800.00	−1.71001	−0.855004	+	0.518621i	$0.826446\pi$
$252$	−252.000	−0.0629941
$253$	2068.00	0.513890
$254$	1648.00	0.407105
$255$	−1440.00	−0.353633
$256$	256.000	0.0625000
$257$	1466.00	0.355823	0.177912	−	0.984046i	$-0.443066\pi$
$257$	1466.00	0.355823	0.177912	+	0.984046i	$0.443066\pi$
$258$	−192.000	−0.0463310
$259$	1386.00	0.332517
$260$	−992.000	−0.236620
$261$	558.000	0.132335
$262$	−44.0000	−0.0103753
$263$	576.000	0.135048	0.0675241	−	0.997718i	$-0.478490\pi$
$263$	576.000	0.135048	0.0675241	+	0.997718i	$0.478490\pi$
$264$	264.000	0.0615457
$265$	1928.00	0.446929
$266$	−1652.00	−0.380792
$267$	−1326.00	−0.303932
$268$	−496.000	−0.113052
$269$	−1420.00	−0.321855	−0.160927	−	0.986966i	$-0.551448\pi$
$269$	−1420.00	−0.321855	−0.160927	+	0.986966i	$0.551448\pi$
$270$	216.000	0.0486864
$271$	−2644.00	−0.592663	−0.296331	−	0.955085i	$-0.595763\pi$
$271$	−2644.00	−0.592663	−0.296331	+	0.955085i	$0.595763\pi$
$272$	−1920.00	−0.428004
$273$	1302.00	0.288647
$274$	−4212.00	−0.928672
$275$	1199.00	0.262918
$276$	2256.00	0.492012
$277$	−78.0000	−0.0169190	−0.00845951	−	0.999964i	$-0.502693\pi$
$277$	−78.0000	−0.0169190	−0.00845951	+	0.999964i	$0.502693\pi$
$278$	1156.00	0.249397
$279$	−2898.00	−0.621859
$280$	224.000	0.0478091
$281$	−330.000	−0.0700575	−0.0350287	−	0.999386i	$-0.511152\pi$
$281$	−330.000	−0.0700575	−0.0350287	+	0.999386i	$0.511152\pi$
$282$	1956.00	0.413043
$283$	−6078.00	−1.27668	−0.638339	−	0.769756i	$-0.720378\pi$
$283$	−6078.00	−1.27668	−0.638339	+	0.769756i	$0.720378\pi$
$284$	−2848.00	−0.595062
$285$	1416.00	0.294304
$286$	−1364.00	−0.282011
$287$	−336.000	−0.0691061
$288$	288.000	0.0589256
$289$	9487.00	1.93100
$290$	−496.000	−0.100435
$291$	2898.00	0.583793
$292$	1216.00	0.243702
$293$	5382.00	1.07311	0.536553	−	0.843867i	$-0.319726\pi$
$293$	5382.00	1.07311	0.536553	+	0.843867i	$0.319726\pi$
$294$	−294.000	−0.0583212
$295$	−1600.00	−0.315782
$296$	−1584.00	−0.311041
$297$	297.000	0.0580259
$298$	1892.00	0.367787
$299$	−11656.0	−2.25446
$300$	1308.00	0.251725
$301$	−224.000	−0.0428942
$302$	4736.00	0.902405
$303$	−1638.00	−0.310563
$304$	1888.00	0.356198
$305$	−280.000	−0.0525664
$306$	−2160.00	−0.403526
$307$	10594.0	1.96948	0.984742	−	0.174021i	$-0.0556762\pi$
$307$	10594.0	1.96948	0.984742	+	0.174021i	$0.0556762\pi$
$308$	308.000	0.0569803
$309$	−294.000	−0.0541265
$310$	2576.00	0.471958
$311$	−1458.00	−0.265838	−0.132919	−	0.991127i	$-0.542435\pi$
$311$	−1458.00	−0.265838	−0.132919	+	0.991127i	$0.542435\pi$
$312$	−1488.00	−0.270005
$313$	7662.00	1.38365	0.691824	−	0.722066i	$-0.256807\pi$
$313$	7662.00	1.38365	0.691824	+	0.722066i	$0.256807\pi$
$314$	416.000	0.0747651
$315$	252.000	0.0450749
$316$	−4064.00	−0.723474
$317$	2262.00	0.400778	0.200389	−	0.979716i	$-0.435779\pi$
$317$	2262.00	0.400778	0.200389	+	0.979716i	$0.435779\pi$
$318$	2892.00	0.509985
$319$	−682.000	−0.119701
$320$	−256.000	−0.0447214
$321$	6192.00	1.07665
$322$	2632.00	0.455514
$323$	−14160.0	−2.43927
$324$	324.000	0.0555556
$325$	−6758.00	−1.15344
$326$	7704.00	1.30885
$327$	−3918.00	−0.662587
$328$	384.000	0.0646428
$329$	2282.00	0.382403
$330$	−264.000	−0.0440386
$331$	−8780.00	−1.45798	−0.728992	−	0.684523i	$-0.760011\pi$
$331$	−8780.00	−1.45798	−0.728992	+	0.684523i	$0.760011\pi$
$332$	1720.00	0.284329
$333$	−1782.00	−0.293252
$334$	56.0000	0.00917420
$335$	496.000	0.0808937
$336$	336.000	0.0545545
$337$	2174.00	0.351410	0.175705	−	0.984443i	$-0.443779\pi$
$337$	2174.00	0.351410	0.175705	+	0.984443i	$0.443779\pi$
$338$	3294.00	0.530089
$339$	−1434.00	−0.229747
$340$	1920.00	0.306255
$341$	3542.00	0.562493
$342$	2124.00	0.335827
$343$	−343.000	−0.0539949
$344$	256.000	0.0401238
$345$	−2256.00	−0.352055
$346$	7292.00	1.13301
$347$	7656.00	1.18443	0.592213	−	0.805782i	$-0.298255\pi$
$347$	7656.00	1.18443	0.592213	+	0.805782i	$0.298255\pi$
$348$	−744.000	−0.114605
$349$	−3046.00	−0.467188	−0.233594	−	0.972334i	$-0.575049\pi$
$349$	−3046.00	−0.467188	−0.233594	+	0.972334i	$0.575049\pi$
$350$	1526.00	0.233052
$351$	−1674.00	−0.254563
$352$	−352.000	−0.0533002
$353$	−10390.0	−1.56658	−0.783292	−	0.621654i	$-0.786461\pi$
$353$	−10390.0	−1.56658	−0.783292	+	0.621654i	$0.786461\pi$
$354$	−2400.00	−0.360335
$355$	2848.00	0.425792
$356$	1768.00	0.263213
$357$	−2520.00	−0.373593
$358$	1896.00	0.279907
$359$	−1848.00	−0.271682	−0.135841	−	0.990731i	$-0.543374\pi$
$359$	−1848.00	−0.271682	−0.135841	+	0.990731i	$0.543374\pi$
$360$	−288.000	−0.0421637
$361$	7065.00	1.03003
$362$	4456.00	0.646967
$363$	−363.000	−0.0524864
$364$	−1736.00	−0.249976
$365$	−1216.00	−0.174379
$366$	−420.000	−0.0599829
$367$	−7314.00	−1.04029	−0.520147	−	0.854077i	$-0.674123\pi$
$367$	−7314.00	−1.04029	−0.520147	+	0.854077i	$0.674123\pi$
$368$	−3008.00	−0.426095
$369$	432.000	0.0609459
$370$	1584.00	0.222563
$371$	3374.00	0.472155
$372$	3864.00	0.538546
$373$	−6034.00	−0.837610	−0.418805	−	0.908076i	$-0.637551\pi$
$373$	−6034.00	−0.837610	−0.418805	+	0.908076i	$0.637551\pi$
$374$	2640.00	0.365003
$375$	−2808.00	−0.386679
$376$	−2608.00	−0.357706
$377$	3844.00	0.525135
$378$	378.000	0.0514344
$379$	−3220.00	−0.436412	−0.218206	−	0.975903i	$-0.570021\pi$
$379$	−3220.00	−0.436412	−0.218206	+	0.975903i	$0.570021\pi$
$380$	−1888.00	−0.254875
$381$	−2472.00	−0.332400
$382$	−6824.00	−0.913995
$383$	−5074.00	−0.676943	−0.338472	−	0.940977i	$-0.609910\pi$
$383$	−5074.00	−0.676943	−0.338472	+	0.940977i	$0.609910\pi$
$384$	−384.000	−0.0510310
$385$	−308.000	−0.0407718
$386$	−7788.00	−1.02694
$387$	288.000	0.0378291
$388$	−3864.00	−0.505580
$389$	−11166.0	−1.45537	−0.727685	−	0.685912i	$-0.759404\pi$
$389$	−11166.0	−1.45537	−0.727685	+	0.685912i	$0.759404\pi$
$390$	1488.00	0.193200
$391$	22560.0	2.91792
$392$	392.000	0.0505076
$393$	66.0000	0.00847140
$394$	−2244.00	−0.286932
$395$	4064.00	0.517676
$396$	−396.000	−0.0502519
$397$	−4608.00	−0.582541	−0.291271	−	0.956641i	$-0.594078\pi$
$397$	−4608.00	−0.582541	−0.291271	+	0.956641i	$0.594078\pi$
$398$	−2812.00	−0.354153
$399$	2478.00	0.310915
$400$	−1744.00	−0.218000
$401$	−2498.00	−0.311083	−0.155541	−	0.987829i	$-0.549712\pi$
$401$	−2498.00	−0.311083	−0.155541	+	0.987829i	$0.549712\pi$
$402$	744.000	0.0923068
$403$	−19964.0	−2.46769
$404$	2184.00	0.268956
$405$	−324.000	−0.0397523
$406$	−868.000	−0.106104
$407$	2178.00	0.265257
$408$	2880.00	0.349464
$409$	8876.00	1.07308	0.536540	−	0.843875i	$-0.319731\pi$
$409$	8876.00	1.07308	0.536540	+	0.843875i	$0.319731\pi$
$410$	−384.000	−0.0462547
$411$	6318.00	0.758258
$412$	392.000	0.0468749
$413$	−2800.00	−0.333605
$414$	−3384.00	−0.401726
$415$	−1720.00	−0.203449
$416$	1984.00	0.233831
$417$	−1734.00	−0.203632
$418$	−2596.00	−0.303767
$419$	9252.00	1.07873	0.539367	−	0.842071i	$-0.318663\pi$
$419$	9252.00	1.07873	0.539367	+	0.842071i	$0.318663\pi$
$420$	−336.000	−0.0390360
$421$	4750.00	0.549883	0.274942	−	0.961461i	$-0.411341\pi$
$421$	4750.00	0.549883	0.274942	+	0.961461i	$0.411341\pi$
$422$	−8576.00	−0.989273
$423$	−2934.00	−0.337248
$424$	−3856.00	−0.441660
$425$	13080.0	1.49288
$426$	4272.00	0.485866
$427$	−490.000	−0.0555334
$428$	−8256.00	−0.932404
$429$	2046.00	0.230261
$430$	−256.000	−0.0287103
$431$	−8008.00	−0.894969	−0.447485	−	0.894292i	$-0.647680\pi$
$431$	−8008.00	−0.894969	−0.447485	+	0.894292i	$0.647680\pi$
$432$	−432.000	−0.0481125
$433$	−11734.0	−1.30231	−0.651155	−	0.758945i	$-0.725715\pi$
$433$	−11734.0	−1.30231	−0.651155	+	0.758945i	$0.725715\pi$
$434$	4508.00	0.498597
$435$	744.000	0.0820048
$436$	5224.00	0.573817
$437$	−22184.0	−2.42839
$438$	−1824.00	−0.198982
$439$	−8212.00	−0.892796	−0.446398	−	0.894835i	$-0.647293\pi$
$439$	−8212.00	−0.892796	−0.446398	+	0.894835i	$0.647293\pi$
$440$	352.000	0.0381385
$441$	441.000	0.0476190
$442$	−14880.0	−1.60129
$443$	−5012.00	−0.537533	−0.268767	−	0.963205i	$-0.586616\pi$
$443$	−5012.00	−0.537533	−0.268767	+	0.963205i	$0.586616\pi$
$444$	2376.00	0.253964
$445$	−1768.00	−0.188340
$446$	8204.00	0.871010
$447$	−2838.00	−0.300297
$448$	−448.000	−0.0472456
$449$	−13602.0	−1.42966	−0.714831	−	0.699297i	$-0.753496\pi$
$449$	−13602.0	−1.42966	−0.714831	+	0.699297i	$0.753496\pi$
$450$	−1962.00	−0.205532
$451$	−528.000	−0.0551276
$452$	1912.00	0.198967
$453$	−7104.00	−0.736810
$454$	−13340.0	−1.37902
$455$	1736.00	0.178868
$456$	−2832.00	−0.290835
$457$	5962.00	0.610264	0.305132	−	0.952310i	$-0.401299\pi$
$457$	5962.00	0.610264	0.305132	+	0.952310i	$0.401299\pi$
$458$	−3416.00	−0.348514
$459$	3240.00	0.329478
$460$	3008.00	0.304889
$461$	9506.00	0.960387	0.480194	−	0.877162i	$-0.340566\pi$
$461$	9506.00	0.960387	0.480194	+	0.877162i	$0.340566\pi$
$462$	−462.000	−0.0465242
$463$	−9768.00	−0.980470	−0.490235	−	0.871590i	$-0.663089\pi$
$463$	−9768.00	−0.980470	−0.490235	+	0.871590i	$0.663089\pi$
$464$	992.000	0.0992510
$465$	−3864.00	−0.385352
$466$	6396.00	0.635813
$467$	−2068.00	−0.204916	−0.102458	−	0.994737i	$-0.532671\pi$
$467$	−2068.00	−0.204916	−0.102458	+	0.994737i	$0.532671\pi$
$468$	2232.00	0.220458
$469$	868.000	0.0854595
$470$	2608.00	0.255953
$471$	−624.000	−0.0610454
$472$	3200.00	0.312059
$473$	−352.000	−0.0342177
$474$	6096.00	0.590714
$475$	−12862.0	−1.24242
$476$	3360.00	0.323541
$477$	−4338.00	−0.416401
$478$	7520.00	0.719575
$479$	−10920.0	−1.04164	−0.520822	−	0.853665i	$-0.674374\pi$
$479$	−10920.0	−1.04164	−0.520822	+	0.853665i	$0.674374\pi$
$480$	384.000	0.0365148
$481$	−12276.0	−1.16370
$482$	13696.0	1.29426
$483$	−3948.00	−0.371926
$484$	484.000	0.0454545
$485$	3864.00	0.361763
$486$	−486.000	−0.0453609
$487$	−2724.00	−0.253462	−0.126731	−	0.991937i	$-0.540449\pi$
$487$	−2724.00	−0.253462	−0.126731	+	0.991937i	$0.540449\pi$
$488$	560.000	0.0519467
$489$	−11556.0	−1.06867
$490$	−392.000	−0.0361403
$491$	−12460.0	−1.14524	−0.572619	−	0.819822i	$-0.694073\pi$
$491$	−12460.0	−1.14524	−0.572619	+	0.819822i	$0.694073\pi$
$492$	−576.000	−0.0527807
$493$	−7440.00	−0.679677
$494$	14632.0	1.33264
$495$	396.000	0.0359573
$496$	−5152.00	−0.466394
$497$	4984.00	0.449825
$498$	−2580.00	−0.232154
$499$	22028.0	1.97617	0.988085	−	0.153910i	$-0.0491866\pi$
$499$	22028.0	1.97617	0.988085	+	0.153910i	$0.0491866\pi$
$500$	3744.00	0.334874
$501$	−84.0000	−0.00749071
$502$	−13600.0	−1.20916
$503$	−20724.0	−1.83705	−0.918526	−	0.395360i	$-0.870620\pi$
$503$	−20724.0	−1.83705	−0.918526	+	0.395360i	$0.870620\pi$
$504$	−504.000	−0.0445435
$505$	−2184.00	−0.192449
$506$	4136.00	0.363375
$507$	−4941.00	−0.432816
$508$	3296.00	0.287867
$509$	3740.00	0.325683	0.162841	−	0.986652i	$-0.447934\pi$
$509$	3740.00	0.325683	0.162841	+	0.986652i	$0.447934\pi$
$510$	−2880.00	−0.250056
$511$	−2128.00	−0.184221
$512$	512.000	0.0441942
$513$	−3186.00	−0.274201
$514$	2932.00	0.251605
$515$	−392.000	−0.0335409
$516$	−384.000	−0.0327610
$517$	3586.00	0.305052
$518$	2772.00	0.235125
$519$	−10938.0	−0.925096
$520$	−1984.00	−0.167316
$521$	15090.0	1.26892	0.634458	−	0.772958i	$-0.281223\pi$
$521$	15090.0	1.26892	0.634458	+	0.772958i	$0.281223\pi$
$522$	1116.00	0.0935747
$523$	−10694.0	−0.894103	−0.447052	−	0.894508i	$-0.647526\pi$
$523$	−10694.0	−0.894103	−0.447052	+	0.894508i	$0.647526\pi$
$524$	−88.0000	−0.00733645
$525$	−2289.00	−0.190286
$526$	1152.00	0.0954935
$527$	38640.0	3.19390
$528$	528.000	0.0435194
$529$	23177.0	1.90491
$530$	3856.00	0.316026
$531$	3600.00	0.294212
$532$	−3304.00	−0.269260
$533$	2976.00	0.241848
$534$	−2652.00	−0.214912
$535$	8256.00	0.667174
$536$	−992.000	−0.0799401
$537$	−2844.00	−0.228543
$538$	−2840.00	−0.227586
$539$	−539.000	−0.0430730
$540$	432.000	0.0344265
$541$	10438.0	0.829510	0.414755	−	0.909933i	$-0.363867\pi$
$541$	10438.0	0.829510	0.414755	+	0.909933i	$0.363867\pi$
$542$	−5288.00	−0.419076
$543$	−6684.00	−0.528247
$544$	−3840.00	−0.302645
$545$	−5224.00	−0.410590
$546$	2604.00	0.204104
$547$	−5852.00	−0.457429	−0.228714	−	0.973494i	$-0.573452\pi$
$547$	−5852.00	−0.457429	−0.228714	+	0.973494i	$0.573452\pi$
$548$	−8424.00	−0.656671
$549$	630.000	0.0489759
$550$	2398.00	0.185911
$551$	7316.00	0.565648
$552$	4512.00	0.347905
$553$	7112.00	0.546895
$554$	−156.000	−0.0119635
$555$	−2376.00	−0.181722
$556$	2312.00	0.176350
$557$	9990.00	0.759946	0.379973	−	0.924998i	$-0.375933\pi$
$557$	9990.00	0.759946	0.379973	+	0.924998i	$0.375933\pi$
$558$	−5796.00	−0.439721
$559$	1984.00	0.150115
$560$	448.000	0.0338062
$561$	−3960.00	−0.298024
$562$	−660.000	−0.0495381
$563$	3258.00	0.243887	0.121943	−	0.992537i	$-0.461087\pi$
$563$	3258.00	0.243887	0.121943	+	0.992537i	$0.461087\pi$
$564$	3912.00	0.292065
$565$	−1912.00	−0.142369
$566$	−12156.0	−0.902747
$567$	−567.000	−0.0419961
$568$	−5696.00	−0.420773
$569$	10478.0	0.771987	0.385994	−	0.922501i	$-0.373859\pi$
$569$	10478.0	0.771987	0.385994	+	0.922501i	$0.373859\pi$
$570$	2832.00	0.208104
$571$	−8116.00	−0.594823	−0.297412	−	0.954749i	$-0.596123\pi$
$571$	−8116.00	−0.594823	−0.297412	+	0.954749i	$0.596123\pi$
$572$	−2728.00	−0.199412
$573$	10236.0	0.746274
$574$	−672.000	−0.0488654
$575$	20492.0	1.48622
$576$	576.000	0.0416667
$577$	5910.00	0.426406	0.213203	−	0.977008i	$-0.431610\pi$
$577$	5910.00	0.426406	0.213203	+	0.977008i	$0.431610\pi$
$578$	18974.0	1.36542
$579$	11682.0	0.838493
$580$	−992.000	−0.0710182
$581$	−3010.00	−0.214933
$582$	5796.00	0.412804
$583$	5302.00	0.376649
$584$	2432.00	0.172323
$585$	−2232.00	−0.157747
$586$	10764.0	0.758800
$587$	12888.0	0.906209	0.453105	−	0.891457i	$-0.350316\pi$
$587$	12888.0	0.906209	0.453105	+	0.891457i	$0.350316\pi$
$588$	−588.000	−0.0412393
$589$	−37996.0	−2.65806
$590$	−3200.00	−0.223291
$591$	3366.00	0.234279
$592$	−3168.00	−0.219939
$593$	16440.0	1.13847	0.569233	−	0.822177i	$-0.307240\pi$
$593$	16440.0	1.13847	0.569233	+	0.822177i	$0.307240\pi$
$594$	594.000	0.0410305
$595$	−3360.00	−0.231507
$596$	3784.00	0.260065
$597$	4218.00	0.289165
$598$	−23312.0	−1.59414
$599$	−18840.0	−1.28511	−0.642556	−	0.766239i	$-0.722126\pi$
$599$	−18840.0	−1.28511	−0.642556	+	0.766239i	$0.722126\pi$
$600$	2616.00	0.177996
$601$	−16068.0	−1.09056	−0.545281	−	0.838254i	$-0.683577\pi$
$601$	−16068.0	−1.09056	−0.545281	+	0.838254i	$0.683577\pi$
$602$	−448.000	−0.0303308
$603$	−1116.00	−0.0753682
$604$	9472.00	0.638096
$605$	−484.000	−0.0325246
$606$	−3276.00	−0.219601
$607$	13272.0	0.887469	0.443735	−	0.896158i	$-0.353653\pi$
$607$	13272.0	0.887469	0.443735	+	0.896158i	$0.353653\pi$
$608$	3776.00	0.251870
$609$	1302.00	0.0866333
$610$	−560.000	−0.0371701
$611$	−20212.0	−1.33828
$612$	−4320.00	−0.285336
$613$	25022.0	1.64866	0.824330	−	0.566109i	$-0.191552\pi$
$613$	25022.0	1.64866	0.824330	+	0.566109i	$0.191552\pi$
$614$	21188.0	1.39264
$615$	576.000	0.0377668
$616$	616.000	0.0402911
$617$	7202.00	0.469922	0.234961	−	0.972005i	$-0.424504\pi$
$617$	7202.00	0.469922	0.234961	+	0.972005i	$0.424504\pi$
$618$	−588.000	−0.0382732
$619$	16228.0	1.05373	0.526865	−	0.849949i	$-0.323367\pi$
$619$	16228.0	1.05373	0.526865	+	0.849949i	$0.323367\pi$
$620$	5152.00	0.333725
$621$	5076.00	0.328008
$622$	−2916.00	−0.187976
$623$	−3094.00	−0.198970
$624$	−2976.00	−0.190922
$625$	9881.00	0.632384
$626$	15324.0	0.978387
$627$	3894.00	0.248025
$628$	832.000	0.0528669
$629$	23760.0	1.50616
$630$	504.000	0.0318728
$631$	−25348.0	−1.59919	−0.799594	−	0.600541i	$-0.794952\pi$
$631$	−25348.0	−1.59919	−0.799594	+	0.600541i	$0.794952\pi$
$632$	−8128.00	−0.511574
$633$	12864.0	0.807738
$634$	4524.00	0.283393
$635$	−3296.00	−0.205981
$636$	5784.00	0.360614
$637$	3038.00	0.188964
$638$	−1364.00	−0.0846415
$639$	−6408.00	−0.396708
$640$	−512.000	−0.0316228
$641$	28938.0	1.78312	0.891562	−	0.452899i	$-0.149610\pi$
$641$	28938.0	1.78312	0.891562	+	0.452899i	$0.149610\pi$
$642$	12384.0	0.761305
$643$	20776.0	1.27422	0.637112	−	0.770772i	$-0.280129\pi$
$643$	20776.0	1.27422	0.637112	+	0.770772i	$0.280129\pi$
$644$	5264.00	0.322097
$645$	384.000	0.0234418
$646$	−28320.0	−1.72482
$647$	−30114.0	−1.82984	−0.914918	−	0.403640i	$-0.867745\pi$
$647$	−30114.0	−1.82984	−0.914918	+	0.403640i	$0.867745\pi$
$648$	648.000	0.0392837
$649$	−4400.00	−0.266125
$650$	−13516.0	−0.815602
$651$	−6762.00	−0.407102
$652$	15408.0	0.925497
$653$	9850.00	0.590291	0.295146	−	0.955452i	$-0.404632\pi$
$653$	9850.00	0.590291	0.295146	+	0.955452i	$0.404632\pi$
$654$	−7836.00	−0.468520
$655$	88.0000	0.00524953
$656$	768.000	0.0457094
$657$	2736.00	0.162468
$658$	4564.00	0.270400
$659$	32360.0	1.91285	0.956424	−	0.291982i	$-0.0943147\pi$
$659$	32360.0	1.91285	0.956424	+	0.291982i	$0.0943147\pi$
$660$	−528.000	−0.0311400
$661$	29836.0	1.75565	0.877826	−	0.478980i	$-0.158993\pi$
$661$	29836.0	1.75565	0.877826	+	0.478980i	$0.158993\pi$
$662$	−17560.0	−1.03095
$663$	22320.0	1.30745
$664$	3440.00	0.201051
$665$	3304.00	0.192667
$666$	−3564.00	−0.207361
$667$	−11656.0	−0.676645
$668$	112.000	0.00648714
$669$	−12306.0	−0.711177
$670$	992.000	0.0572005
$671$	−770.000	−0.0443003
$672$	672.000	0.0385758
$673$	−24082.0	−1.37934	−0.689668	−	0.724126i	$-0.742243\pi$
$673$	−24082.0	−1.37934	−0.689668	+	0.724126i	$0.742243\pi$
$674$	4348.00	0.248485
$675$	2943.00	0.167816
$676$	6588.00	0.374829
$677$	−3690.00	−0.209480	−0.104740	−	0.994500i	$-0.533401\pi$
$677$	−3690.00	−0.209480	−0.104740	+	0.994500i	$0.533401\pi$
$678$	−2868.00	−0.162456
$679$	6762.00	0.382182
$680$	3840.00	0.216555
$681$	20010.0	1.12597
$682$	7084.00	0.397742
$683$	−7596.00	−0.425553	−0.212777	−	0.977101i	$-0.568251\pi$
$683$	−7596.00	−0.425553	−0.212777	+	0.977101i	$0.568251\pi$
$684$	4248.00	0.237465
$685$	8424.00	0.469875
$686$	−686.000	−0.0381802
$687$	5124.00	0.284560
$688$	512.000	0.0283718
$689$	−29884.0	−1.65238
$690$	−4512.00	−0.248940
$691$	12960.0	0.713490	0.356745	−	0.934202i	$-0.383886\pi$
$691$	12960.0	0.713490	0.356745	+	0.934202i	$0.383886\pi$
$692$	14584.0	0.801157
$693$	693.000	0.0379869
$694$	15312.0	0.837515
$695$	−2312.00	−0.126186
$696$	−1488.00	−0.0810381
$697$	−5760.00	−0.313021
$698$	−6092.00	−0.330352
$699$	−9594.00	−0.519139
$700$	3052.00	0.164793
$701$	11034.0	0.594506	0.297253	−	0.954799i	$-0.403930\pi$
$701$	11034.0	0.594506	0.297253	+	0.954799i	$0.403930\pi$
$702$	−3348.00	−0.180003
$703$	−23364.0	−1.25347
$704$	−704.000	−0.0376889
$705$	−3912.00	−0.208985
$706$	−20780.0	−1.10774
$707$	−3822.00	−0.203311
$708$	−4800.00	−0.254795
$709$	−694.000	−0.0367612	−0.0183806	−	0.999831i	$-0.505851\pi$
$709$	−694.000	−0.0367612	−0.0183806	+	0.999831i	$0.505851\pi$
$710$	5696.00	0.301080
$711$	−9144.00	−0.482316
$712$	3536.00	0.186120
$713$	60536.0	3.17965
$714$	−5040.00	−0.264170
$715$	2728.00	0.142687
$716$	3792.00	0.197924
$717$	−11280.0	−0.587530
$718$	−3696.00	−0.192108
$719$	−3762.00	−0.195131	−0.0975653	−	0.995229i	$-0.531105\pi$
$719$	−3762.00	−0.195131	−0.0975653	+	0.995229i	$0.531105\pi$
$720$	−576.000	−0.0298142
$721$	−686.000	−0.0354341
$722$	14130.0	0.728344
$723$	−20544.0	−1.05676
$724$	8912.00	0.457475
$725$	−6758.00	−0.346187
$726$	−726.000	−0.0371135
$727$	−1274.00	−0.0649932	−0.0324966	−	0.999472i	$-0.510346\pi$
$727$	−1274.00	−0.0649932	−0.0324966	+	0.999472i	$0.510346\pi$
$728$	−3472.00	−0.176759
$729$	729.000	0.0370370
$730$	−2432.00	−0.123305
$731$	−3840.00	−0.194292
$732$	−840.000	−0.0424143
$733$	−21026.0	−1.05950	−0.529750	−	0.848154i	$-0.677714\pi$
$733$	−21026.0	−1.05950	−0.529750	+	0.848154i	$0.677714\pi$
$734$	−14628.0	−0.735599
$735$	588.000	0.0295084
$736$	−6016.00	−0.301294
$737$	1364.00	0.0681731
$738$	864.000	0.0430952
$739$	12692.0	0.631776	0.315888	−	0.948796i	$-0.397698\pi$
$739$	12692.0	0.631776	0.315888	+	0.948796i	$0.397698\pi$
$740$	3168.00	0.157376
$741$	−21948.0	−1.08810
$742$	6748.00	0.333864
$743$	2152.00	0.106257	0.0531287	−	0.998588i	$-0.483081\pi$
$743$	2152.00	0.106257	0.0531287	+	0.998588i	$0.483081\pi$
$744$	7728.00	0.380809
$745$	−3784.00	−0.186087
$746$	−12068.0	−0.592280
$747$	3870.00	0.189553
$748$	5280.00	0.258096
$749$	14448.0	0.704831
$750$	−5616.00	−0.273423
$751$	18740.0	0.910562	0.455281	−	0.890348i	$-0.349539\pi$
$751$	18740.0	0.910562	0.455281	+	0.890348i	$0.349539\pi$
$752$	−5216.00	−0.252936
$753$	20400.0	0.987274
$754$	7688.00	0.371327
$755$	−9472.00	−0.456585
$756$	756.000	0.0363696
$757$	−36894.0	−1.77138	−0.885690	−	0.464276i	$-0.846314\pi$
$757$	−36894.0	−1.77138	−0.885690	+	0.464276i	$0.846314\pi$
$758$	−6440.00	−0.308590
$759$	−6204.00	−0.296694
$760$	−3776.00	−0.180224
$761$	−18732.0	−0.892292	−0.446146	−	0.894960i	$-0.647204\pi$
$761$	−18732.0	−0.892292	−0.446146	+	0.894960i	$0.647204\pi$
$762$	−4944.00	−0.235042
$763$	−9142.00	−0.433765
$764$	−13648.0	−0.646292
$765$	4320.00	0.204170
$766$	−10148.0	−0.478671
$767$	24800.0	1.16750
$768$	−768.000	−0.0360844
$769$	−16292.0	−0.763985	−0.381993	−	0.924165i	$-0.624762\pi$
$769$	−16292.0	−0.763985	−0.381993	+	0.924165i	$0.624762\pi$
$770$	−616.000	−0.0288300
$771$	−4398.00	−0.205435
$772$	−15576.0	−0.726156
$773$	23336.0	1.08582	0.542909	−	0.839791i	$-0.317323\pi$
$773$	23336.0	1.08582	0.542909	+	0.839791i	$0.317323\pi$
$774$	576.000	0.0267492
$775$	35098.0	1.62678
$776$	−7728.00	−0.357499
$777$	−4158.00	−0.191979
$778$	−22332.0	−1.02910
$779$	5664.00	0.260506
$780$	2976.00	0.136613
$781$	7832.00	0.358836
$782$	45120.0	2.06328
$783$	−1674.00	−0.0764034
$784$	784.000	0.0357143
$785$	−832.000	−0.0378285
$786$	132.000	0.00599018
$787$	9110.00	0.412626	0.206313	−	0.978486i	$-0.433854\pi$
$787$	9110.00	0.412626	0.206313	+	0.978486i	$0.433854\pi$
$788$	−4488.00	−0.202891
$789$	−1728.00	−0.0779701
$790$	8128.00	0.366052
$791$	−3346.00	−0.150405
$792$	−792.000	−0.0355335
$793$	4340.00	0.194348
$794$	−9216.00	−0.411919
$795$	−5784.00	−0.258034
$796$	−5624.00	−0.250424
$797$	20144.0	0.895279	0.447639	−	0.894214i	$-0.352265\pi$
$797$	20144.0	0.895279	0.447639	+	0.894214i	$0.352265\pi$
$798$	4956.00	0.219850
$799$	39120.0	1.73212
$800$	−3488.00	−0.154149
$801$	3978.00	0.175475
$802$	−4996.00	−0.219969
$803$	−3344.00	−0.146958
$804$	1488.00	0.0652708
$805$	−5264.00	−0.230474
$806$	−39928.0	−1.74492
$807$	4260.00	0.185823
$808$	4368.00	0.190180
$809$	10946.0	0.475699	0.237850	−	0.971302i	$-0.423557\pi$
$809$	10946.0	0.475699	0.237850	+	0.971302i	$0.423557\pi$
$810$	−648.000	−0.0281091
$811$	13874.0	0.600718	0.300359	−	0.953826i	$-0.402894\pi$
$811$	13874.0	0.600718	0.300359	+	0.953826i	$0.402894\pi$
$812$	−1736.00	−0.0750267
$813$	7932.00	0.342174
$814$	4356.00	0.187565
$815$	−15408.0	−0.662232
$816$	5760.00	0.247108
$817$	3776.00	0.161696
$818$	17752.0	0.758783
$819$	−3906.00	−0.166650
$820$	−768.000	−0.0327070
$821$	7570.00	0.321796	0.160898	−	0.986971i	$-0.448561\pi$
$821$	7570.00	0.321796	0.160898	+	0.986971i	$0.448561\pi$
$822$	12636.0	0.536169
$823$	14372.0	0.608720	0.304360	−	0.952557i	$-0.401557\pi$
$823$	14372.0	0.608720	0.304360	+	0.952557i	$0.401557\pi$
$824$	784.000	0.0331456
$825$	−3597.00	−0.151796
$826$	−5600.00	−0.235895
$827$	25036.0	1.05270	0.526352	−	0.850266i	$-0.323559\pi$
$827$	25036.0	1.05270	0.526352	+	0.850266i	$0.323559\pi$
$828$	−6768.00	−0.284063
$829$	−38684.0	−1.62069	−0.810344	−	0.585954i	$-0.800720\pi$
$829$	−38684.0	−1.62069	−0.810344	+	0.585954i	$0.800720\pi$
$830$	−3440.00	−0.143860
$831$	234.000	0.00976820
$832$	3968.00	0.165343
$833$	−5880.00	−0.244574
$834$	−3468.00	−0.143989
$835$	−112.000	−0.00464182
$836$	−5192.00	−0.214796
$837$	8694.00	0.359031
$838$	18504.0	0.762781
$839$	31110.0	1.28014	0.640069	−	0.768317i	$-0.278906\pi$
$839$	31110.0	1.28014	0.640069	+	0.768317i	$0.278906\pi$
$840$	−672.000	−0.0276026
$841$	−20545.0	−0.842388
$842$	9500.00	0.388826
$843$	990.000	0.0404477
$844$	−17152.0	−0.699522
$845$	−6588.00	−0.268206
$846$	−5868.00	−0.238470
$847$	−847.000	−0.0343604
$848$	−7712.00	−0.312301
$849$	18234.0	0.737090
$850$	26160.0	1.05562
$851$	37224.0	1.49944
$852$	8544.00	0.343559
$853$	−25922.0	−1.04051	−0.520253	−	0.854012i	$-0.674163\pi$
$853$	−25922.0	−1.04051	−0.520253	+	0.854012i	$0.674163\pi$
$854$	−980.000	−0.0392680
$855$	−4248.00	−0.169916
$856$	−16512.0	−0.659309
$857$	−14076.0	−0.561058	−0.280529	−	0.959845i	$-0.590510\pi$
$857$	−14076.0	−0.561058	−0.280529	+	0.959845i	$0.590510\pi$
$858$	4092.00	0.162819
$859$	−23920.0	−0.950105	−0.475052	−	0.879958i	$-0.657571\pi$
$859$	−23920.0	−0.950105	−0.475052	+	0.879958i	$0.657571\pi$
$860$	−512.000	−0.0203012
$861$	1008.00	0.0398984
$862$	−16016.0	−0.632839
$863$	−34316.0	−1.35357	−0.676785	−	0.736181i	$-0.736627\pi$
$863$	−34316.0	−1.35357	−0.676785	+	0.736181i	$0.736627\pi$
$864$	−864.000	−0.0340207
$865$	−14584.0	−0.573261
$866$	−23468.0	−0.920872
$867$	−28461.0	−1.11486
$868$	9016.00	0.352561
$869$	11176.0	0.436271
$870$	1488.00	0.0579861
$871$	−7688.00	−0.299079
$872$	10448.0	0.405750
$873$	−8694.00	−0.337053
$874$	−44368.0	−1.71713
$875$	−6552.00	−0.253141
$876$	−3648.00	−0.140701
$877$	2954.00	0.113739	0.0568697	−	0.998382i	$-0.481888\pi$
$877$	2954.00	0.113739	0.0568697	+	0.998382i	$0.481888\pi$
$878$	−16424.0	−0.631302
$879$	−16146.0	−0.619558
$880$	704.000	0.0269680
$881$	−35994.0	−1.37647	−0.688234	−	0.725489i	$-0.741614\pi$
$881$	−35994.0	−1.37647	−0.688234	+	0.725489i	$0.741614\pi$
$882$	882.000	0.0336718
$883$	−36844.0	−1.40419	−0.702095	−	0.712084i	$-0.747752\pi$
$883$	−36844.0	−1.40419	−0.702095	+	0.712084i	$0.747752\pi$
$884$	−29760.0	−1.13228
$885$	4800.00	0.182317
$886$	−10024.0	−0.380094
$887$	35000.0	1.32490	0.662449	−	0.749107i	$-0.269517\pi$
$887$	35000.0	1.32490	0.662449	+	0.749107i	$0.269517\pi$
$888$	4752.00	0.179580
$889$	−5768.00	−0.217607
$890$	−3536.00	−0.133176
$891$	−891.000	−0.0335013
$892$	16408.0	0.615897
$893$	−38468.0	−1.44153
$894$	−5676.00	−0.212342
$895$	−3792.00	−0.141623
$896$	−896.000	−0.0334077
$897$	34968.0	1.30161
$898$	−27204.0	−1.01092
$899$	−19964.0	−0.740641
$900$	−3924.00	−0.145333
$901$	57840.0	2.13866
$902$	−1056.00	−0.0389811
$903$	672.000	0.0247650
$904$	3824.00	0.140691
$905$	−8912.00	−0.327342
$906$	−14208.0	−0.521004
$907$	−5460.00	−0.199886	−0.0999428	−	0.994993i	$-0.531866\pi$
$907$	−5460.00	−0.199886	−0.0999428	+	0.994993i	$0.531866\pi$
$908$	−26680.0	−0.975118
$909$	4914.00	0.179304
$910$	3472.00	0.126479
$911$	27116.0	0.986162	0.493081	−	0.869984i	$-0.335871\pi$
$911$	27116.0	0.986162	0.493081	+	0.869984i	$0.335871\pi$
$912$	−5664.00	−0.205651
$913$	−4730.00	−0.171457
$914$	11924.0	0.431522
$915$	840.000	0.0303492
$916$	−6832.00	−0.246436
$917$	154.000	0.00554583
$918$	6480.00	0.232976
$919$	50056.0	1.79673	0.898365	−	0.439249i	$-0.144756\pi$
$919$	50056.0	1.79673	0.898365	+	0.439249i	$0.144756\pi$
$920$	6016.00	0.215589
$921$	−31782.0	−1.13708
$922$	19012.0	0.679096
$923$	−44144.0	−1.57423
$924$	−924.000	−0.0328976
$925$	21582.0	0.767148
$926$	−19536.0	−0.693297
$927$	882.000	0.0312499
$928$	1984.00	0.0701810
$929$	−28294.0	−0.999242	−0.499621	−	0.866244i	$-0.666527\pi$
$929$	−28294.0	−0.999242	−0.499621	+	0.866244i	$0.666527\pi$
$930$	−7728.00	−0.272485
$931$	5782.00	0.203542
$932$	12792.0	0.449588
$933$	4374.00	0.153482
$934$	−4136.00	−0.144897
$935$	−5280.00	−0.184679
$936$	4464.00	0.155887
$937$	15368.0	0.535806	0.267903	−	0.963446i	$-0.413669\pi$
$937$	15368.0	0.535806	0.267903	+	0.963446i	$0.413669\pi$
$938$	1736.00	0.0604290
$939$	−22986.0	−0.798850
$940$	5216.00	0.180986
$941$	−2394.00	−0.0829354	−0.0414677	−	0.999140i	$-0.513203\pi$
$941$	−2394.00	−0.0829354	−0.0414677	+	0.999140i	$0.513203\pi$
$942$	−1248.00	−0.0431656
$943$	−9024.00	−0.311624
$944$	6400.00	0.220659
$945$	−756.000	−0.0260240
$946$	−704.000	−0.0241956
$947$	20772.0	0.712776	0.356388	−	0.934338i	$-0.384008\pi$
$947$	20772.0	0.712776	0.356388	+	0.934338i	$0.384008\pi$
$948$	12192.0	0.417698
$949$	18848.0	0.644712
$950$	−25724.0	−0.878523
$951$	−6786.00	−0.231389
$952$	6720.00	0.228778
$953$	−49790.0	−1.69240	−0.846200	−	0.532866i	$-0.821115\pi$
$953$	−49790.0	−1.69240	−0.846200	+	0.532866i	$0.821115\pi$
$954$	−8676.00	−0.294440
$955$	13648.0	0.462449
$956$	15040.0	0.508816
$957$	2046.00	0.0691095
$958$	−21840.0	−0.736554
$959$	14742.0	0.496396
$960$	768.000	0.0258199
$961$	73893.0	2.48038
$962$	−24552.0	−0.822857
$963$	−18576.0	−0.621603
$964$	27392.0	0.915184
$965$	15576.0	0.519595
$966$	−7896.00	−0.262991
$967$	24456.0	0.813291	0.406645	−	0.913586i	$-0.366699\pi$
$967$	24456.0	0.813291	0.406645	+	0.913586i	$0.366699\pi$
$968$	968.000	0.0321412
$969$	42480.0	1.40831
$970$	7728.00	0.255805
$971$	32820.0	1.08470	0.542350	−	0.840153i	$-0.317535\pi$
$971$	32820.0	1.08470	0.542350	+	0.840153i	$0.317535\pi$
$972$	−972.000	−0.0320750
$973$	−4046.00	−0.133308
$974$	−5448.00	−0.179225
$975$	20274.0	0.665936
$976$	1120.00	0.0367319
$977$	21650.0	0.708951	0.354475	−	0.935065i	$-0.384659\pi$
$977$	21650.0	0.708951	0.354475	+	0.935065i	$0.384659\pi$
$978$	−23112.0	−0.755665
$979$	−4862.00	−0.158723
$980$	−784.000	−0.0255551
$981$	11754.0	0.382545
$982$	−24920.0	−0.809806
$983$	−36502.0	−1.18437	−0.592184	−	0.805803i	$-0.701734\pi$
$983$	−36502.0	−1.18437	−0.592184	+	0.805803i	$0.701734\pi$
$984$	−1152.00	−0.0373216
$985$	4488.00	0.145177
$986$	−14880.0	−0.480604
$987$	−6846.00	−0.220781
$988$	29264.0	0.942320
$989$	−6016.00	−0.193425
$990$	792.000	0.0254257
$991$	−39016.0	−1.25064	−0.625320	−	0.780368i	$-0.715032\pi$
$991$	−39016.0	−1.25064	−0.625320	+	0.780368i	$0.715032\pi$
$992$	−10304.0	−0.329791
$993$	26340.0	0.841767
$994$	9968.00	0.318074
$995$	5624.00	0.179189
$996$	−5160.00	−0.164157
$997$	−12558.0	−0.398913	−0.199456	−	0.979907i	$-0.563918\pi$
$997$	−12558.0	−0.398913	−0.199456	+	0.979907i	$0.563918\pi$
$998$	44056.0	1.39736
$999$	5346.00	0.169309

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.4.a.e.1.1	✓	1	1.1	even	1	trivial
1386.4.a.e.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

Downloads

Learn more