LMFDB - Embedded newform 462.4.a.g.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:	$0$
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)

$f(q)$

$=$

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

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +3.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +6.00000 q^{10} -11.0000 q^{11} -12.0000 q^{12} +41.0000 q^{13} +14.0000 q^{14} -9.00000 q^{15} +16.0000 q^{16} +6.00000 q^{17} +18.0000 q^{18} -43.0000 q^{19} +12.0000 q^{20} -21.0000 q^{21} -22.0000 q^{22} +120.000 q^{23} -24.0000 q^{24} -116.000 q^{25} +82.0000 q^{26} -27.0000 q^{27} +28.0000 q^{28} +111.000 q^{29} -18.0000 q^{30} +266.000 q^{31} +32.0000 q^{32} +33.0000 q^{33} +12.0000 q^{34} +21.0000 q^{35} +36.0000 q^{36} -79.0000 q^{37} -86.0000 q^{38} -123.000 q^{39} +24.0000 q^{40} +216.000 q^{41} -42.0000 q^{42} +284.000 q^{43} -44.0000 q^{44} +27.0000 q^{45} +240.000 q^{46} +213.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} -232.000 q^{50} -18.0000 q^{51} +164.000 q^{52} -216.000 q^{53} -54.0000 q^{54} -33.0000 q^{55} +56.0000 q^{56} +129.000 q^{57} +222.000 q^{58} +393.000 q^{59} -36.0000 q^{60} +350.000 q^{61} +532.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} +123.000 q^{65} +66.0000 q^{66} +821.000 q^{67} +24.0000 q^{68} -360.000 q^{69} +42.0000 q^{70} -264.000 q^{71} +72.0000 q^{72} -865.000 q^{73} -158.000 q^{74} +348.000 q^{75} -172.000 q^{76} -77.0000 q^{77} -246.000 q^{78} -484.000 q^{79} +48.0000 q^{80} +81.0000 q^{81} +432.000 q^{82} +1158.00 q^{83} -84.0000 q^{84} +18.0000 q^{85} +568.000 q^{86} -333.000 q^{87} -88.0000 q^{88} +330.000 q^{89} +54.0000 q^{90} +287.000 q^{91} +480.000 q^{92} -798.000 q^{93} +426.000 q^{94} -129.000 q^{95} -96.0000 q^{96} +980.000 q^{97} +98.0000 q^{98} -99.0000 q^{99} +O(q^{100})\)

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
$5$	3.00000	0.268328	0.134164	−	0.990959i	$-0.457165\pi$
$5$	3.00000	0.268328	0.134164	+	0.990959i	$0.457165\pi$
$6$	−6.00000	−0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	6.00000	0.189737
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−12.0000	−0.288675
$13$	41.0000	0.874720	0.437360	−	0.899287i	$-0.355914\pi$
$13$	41.0000	0.874720	0.437360	+	0.899287i	$0.355914\pi$
$14$	14.0000	0.267261
$15$	−9.00000	−0.154919
$16$	16.0000	0.250000
$17$	6.00000	0.0856008	0.0428004	−	0.999084i	$-0.486372\pi$
$17$	6.00000	0.0856008	0.0428004	+	0.999084i	$0.486372\pi$
$18$	18.0000	0.235702
$19$	−43.0000	−0.519204	−0.259602	−	0.965716i	$-0.583591\pi$
$19$	−43.0000	−0.519204	−0.259602	+	0.965716i	$0.583591\pi$
$20$	12.0000	0.134164
$21$	−21.0000	−0.218218
$22$	−22.0000	−0.213201
$23$	120.000	1.08790	0.543951	−	0.839117i	$-0.316928\pi$
$23$	120.000	1.08790	0.543951	+	0.839117i	$0.316928\pi$
$24$	−24.0000	−0.204124
$25$	−116.000	−0.928000
$26$	82.0000	0.618520
$27$	−27.0000	−0.192450
$28$	28.0000	0.188982
$29$	111.000	0.710765	0.355382	−	0.934721i	$-0.384351\pi$
$29$	111.000	0.710765	0.355382	+	0.934721i	$0.384351\pi$
$30$	−18.0000	−0.109545
$31$	266.000	1.54113	0.770565	−	0.637362i	$-0.219974\pi$
$31$	266.000	1.54113	0.770565	+	0.637362i	$0.219974\pi$
$32$	32.0000	0.176777
$33$	33.0000	0.174078
$34$	12.0000	0.0605289
$35$	21.0000	0.101419
$36$	36.0000	0.166667
$37$	−79.0000	−0.351014	−0.175507	−	0.984478i	$-0.556156\pi$
$37$	−79.0000	−0.351014	−0.175507	+	0.984478i	$0.556156\pi$
$38$	−86.0000	−0.367133
$39$	−123.000	−0.505020
$40$	24.0000	0.0948683
$41$	216.000	0.822769	0.411385	−	0.911462i	$-0.365045\pi$
$41$	216.000	0.822769	0.411385	+	0.911462i	$0.365045\pi$
$42$	−42.0000	−0.154303
$43$	284.000	1.00720	0.503600	−	0.863937i	$-0.332009\pi$
$43$	284.000	1.00720	0.503600	+	0.863937i	$0.332009\pi$
$44$	−44.0000	−0.150756
$45$	27.0000	0.0894427
$46$	240.000	0.769262
$47$	213.000	0.661048	0.330524	−	0.943798i	$-0.392775\pi$
$47$	213.000	0.661048	0.330524	+	0.943798i	$0.392775\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	−232.000	−0.656195
$51$	−18.0000	−0.0494217
$52$	164.000	0.437360
$53$	−216.000	−0.559809	−0.279905	−	0.960028i	$-0.590303\pi$
$53$	−216.000	−0.559809	−0.279905	+	0.960028i	$0.590303\pi$
$54$	−54.0000	−0.136083
$55$	−33.0000	−0.0809040
$56$	56.0000	0.133631
$57$	129.000	0.299763
$58$	222.000	0.502587
$59$	393.000	0.867191	0.433595	−	0.901108i	$-0.357245\pi$
$59$	393.000	0.867191	0.433595	+	0.901108i	$0.357245\pi$
$60$	−36.0000	−0.0774597
$61$	350.000	0.734638	0.367319	−	0.930095i	$-0.380276\pi$
$61$	350.000	0.734638	0.367319	+	0.930095i	$0.380276\pi$
$62$	532.000	1.08974
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	123.000	0.234712
$66$	66.0000	0.123091
$67$	821.000	1.49703	0.748516	−	0.663117i	$-0.230767\pi$
$67$	821.000	1.49703	0.748516	+	0.663117i	$0.230767\pi$
$68$	24.0000	0.0428004
$69$	−360.000	−0.628100
$70$	42.0000	0.0717137
$71$	−264.000	−0.441282	−0.220641	−	0.975355i	$-0.570815\pi$
$71$	−264.000	−0.441282	−0.220641	+	0.975355i	$0.570815\pi$
$72$	72.0000	0.117851
$73$	−865.000	−1.38686	−0.693429	−	0.720525i	$-0.743901\pi$
$73$	−865.000	−1.38686	−0.693429	+	0.720525i	$0.743901\pi$
$74$	−158.000	−0.248204
$75$	348.000	0.535781
$76$	−172.000	−0.259602
$77$	−77.0000	−0.113961
$78$	−246.000	−0.357103
$79$	−484.000	−0.689294	−0.344647	−	0.938732i	$-0.612001\pi$
$79$	−484.000	−0.689294	−0.344647	+	0.938732i	$0.612001\pi$
$80$	48.0000	0.0670820
$81$	81.0000	0.111111
$82$	432.000	0.581786
$83$	1158.00	1.53141	0.765705	−	0.643192i	$-0.222390\pi$
$83$	1158.00	1.53141	0.765705	+	0.643192i	$0.222390\pi$
$84$	−84.0000	−0.109109
$85$	18.0000	0.0229691
$86$	568.000	0.712198
$87$	−333.000	−0.410360
$88$	−88.0000	−0.106600
$89$	330.000	0.393033	0.196516	−	0.980501i	$-0.437037\pi$
$89$	330.000	0.393033	0.196516	+	0.980501i	$0.437037\pi$
$90$	54.0000	0.0632456
$91$	287.000	0.330613
$92$	480.000	0.543951
$93$	−798.000	−0.889771
$94$	426.000	0.467431
$95$	−129.000	−0.139317
$96$	−96.0000	−0.102062
$97$	980.000	1.02581	0.512907	−	0.858444i	$-0.328569\pi$
$97$	980.000	1.02581	0.512907	+	0.858444i	$0.328569\pi$
$98$	98.0000	0.101015
$99$	−99.0000	−0.100504
$100$	−464.000	−0.464000
$101$	210.000	0.206889	0.103444	−	0.994635i	$-0.467014\pi$
$101$	210.000	0.206889	0.103444	+	0.994635i	$0.467014\pi$
$102$	−36.0000	−0.0349464
$103$	−1330.00	−1.27232	−0.636159	−	0.771558i	$-0.719478\pi$
$103$	−1330.00	−1.27232	−0.636159	+	0.771558i	$0.719478\pi$
$104$	328.000	0.309260
$105$	−63.0000	−0.0585540
$106$	−432.000	−0.395845
$107$	−1329.00	−1.20074	−0.600370	−	0.799722i	$-0.704980\pi$
$107$	−1329.00	−1.20074	−0.600370	+	0.799722i	$0.704980\pi$
$108$	−108.000	−0.0962250
$109$	−94.0000	−0.0826015	−0.0413008	−	0.999147i	$-0.513150\pi$
$109$	−94.0000	−0.0826015	−0.0413008	+	0.999147i	$0.513150\pi$
$110$	−66.0000	−0.0572078
$111$	237.000	0.202658
$112$	112.000	0.0944911
$113$	198.000	0.164834	0.0824171	−	0.996598i	$-0.473736\pi$
$113$	198.000	0.164834	0.0824171	+	0.996598i	$0.473736\pi$
$114$	258.000	0.211964
$115$	360.000	0.291915
$116$	444.000	0.355382
$117$	369.000	0.291573
$118$	786.000	0.613196
$119$	42.0000	0.0323541
$120$	−72.0000	−0.0547723
$121$	121.000	0.0909091
$122$	700.000	0.519467
$123$	−648.000	−0.475026
$124$	1064.00	0.770565
$125$	−723.000	−0.517337
$126$	126.000	0.0890871
$127$	−1066.00	−0.744821	−0.372410	−	0.928068i	$-0.621469\pi$
$127$	−1066.00	−0.744821	−0.372410	+	0.928068i	$0.621469\pi$
$128$	128.000	0.0883883
$129$	−852.000	−0.581507
$130$	246.000	0.165966
$131$	−120.000	−0.0800340	−0.0400170	−	0.999199i	$-0.512741\pi$
$131$	−120.000	−0.0800340	−0.0400170	+	0.999199i	$0.512741\pi$
$132$	132.000	0.0870388
$133$	−301.000	−0.196241
$134$	1642.00	1.05856
$135$	−81.0000	−0.0516398
$136$	48.0000	0.0302645
$137$	−1938.00	−1.20857	−0.604287	−	0.796767i	$-0.706542\pi$
$137$	−1938.00	−1.20857	−0.604287	+	0.796767i	$0.706542\pi$
$138$	−720.000	−0.444134
$139$	1712.00	1.04468	0.522338	−	0.852739i	$-0.325060\pi$
$139$	1712.00	1.04468	0.522338	+	0.852739i	$0.325060\pi$
$140$	84.0000	0.0507093
$141$	−639.000	−0.381656
$142$	−528.000	−0.312034
$143$	−451.000	−0.263738
$144$	144.000	0.0833333
$145$	333.000	0.190718
$146$	−1730.00	−0.980656
$147$	−147.000	−0.0824786
$148$	−316.000	−0.175507
$149$	−1917.00	−1.05401	−0.527003	−	0.849864i	$-0.676684\pi$
$149$	−1917.00	−1.05401	−0.527003	+	0.849864i	$0.676684\pi$
$150$	696.000	0.378854
$151$	−3022.00	−1.62865	−0.814327	−	0.580406i	$-0.802894\pi$
$151$	−3022.00	−1.62865	−0.814327	+	0.580406i	$0.802894\pi$
$152$	−344.000	−0.183566
$153$	54.0000	0.0285336
$154$	−154.000	−0.0805823
$155$	798.000	0.413528
$156$	−492.000	−0.252510
$157$	−2200.00	−1.11834	−0.559169	−	0.829054i	$-0.688880\pi$
$157$	−2200.00	−1.11834	−0.559169	+	0.829054i	$0.688880\pi$
$158$	−968.000	−0.487405
$159$	648.000	0.323206
$160$	96.0000	0.0474342
$161$	840.000	0.411188
$162$	162.000	0.0785674
$163$	−3715.00	−1.78516	−0.892581	−	0.450888i	$-0.851107\pi$
$163$	−3715.00	−1.78516	−0.892581	+	0.450888i	$0.851107\pi$
$164$	864.000	0.411385
$165$	99.0000	0.0467099
$166$	2316.00	1.08287
$167$	84.0000	0.0389228	0.0194614	−	0.999811i	$-0.493805\pi$
$167$	84.0000	0.0389228	0.0194614	+	0.999811i	$0.493805\pi$
$168$	−168.000	−0.0771517
$169$	−516.000	−0.234866
$170$	36.0000	0.0162416
$171$	−387.000	−0.173068
$172$	1136.00	0.503600
$173$	1812.00	0.796323	0.398161	−	0.917315i	$-0.369648\pi$
$173$	1812.00	0.796323	0.398161	+	0.917315i	$0.369648\pi$
$174$	−666.000	−0.290169
$175$	−812.000	−0.350751
$176$	−176.000	−0.0753778
$177$	−1179.00	−0.500673
$178$	660.000	0.277916
$179$	192.000	0.0801718	0.0400859	−	0.999196i	$-0.487237\pi$
$179$	192.000	0.0801718	0.0400859	+	0.999196i	$0.487237\pi$
$180$	108.000	0.0447214
$181$	−3148.00	−1.29276	−0.646378	−	0.763017i	$-0.723717\pi$
$181$	−3148.00	−1.29276	−0.646378	+	0.763017i	$0.723717\pi$
$182$	574.000	0.233779
$183$	−1050.00	−0.424143
$184$	960.000	0.384631
$185$	−237.000	−0.0941870
$186$	−1596.00	−0.629163
$187$	−66.0000	−0.0258096
$188$	852.000	0.330524
$189$	−189.000	−0.0727393
$190$	−258.000	−0.0985120
$191$	4554.00	1.72521	0.862607	−	0.505875i	$-0.168830\pi$
$191$	4554.00	1.72521	0.862607	+	0.505875i	$0.168830\pi$
$192$	−192.000	−0.0721688
$193$	−940.000	−0.350584	−0.175292	−	0.984517i	$-0.556087\pi$
$193$	−940.000	−0.350584	−0.175292	+	0.984517i	$0.556087\pi$
$194$	1960.00	0.725360
$195$	−369.000	−0.135511
$196$	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
$199$	3956.00	1.40921	0.704607	−	0.709598i	$-0.251124\pi$
$199$	3956.00	1.40921	0.704607	+	0.709598i	$0.251124\pi$
$200$	−928.000	−0.328098
$201$	−2463.00	−0.864312
$202$	420.000	0.146293
$203$	777.000	0.268644
$204$	−72.0000	−0.0247108
$205$	648.000	0.220772
$206$	−2660.00	−0.899665
$207$	1080.00	0.362634
$208$	656.000	0.218680
$209$	473.000	0.156546
$210$	−126.000	−0.0414039
$211$	122.000	0.0398049	0.0199024	−	0.999802i	$-0.493664\pi$
$211$	122.000	0.0398049	0.0199024	+	0.999802i	$0.493664\pi$
$212$	−864.000	−0.279905
$213$	792.000	0.254774
$214$	−2658.00	−0.849052
$215$	852.000	0.270260
$216$	−216.000	−0.0680414
$217$	1862.00	0.582492
$218$	−188.000	−0.0584081
$219$	2595.00	0.800703
$220$	−132.000	−0.0404520
$221$	246.000	0.0748767
$222$	474.000	0.143301
$223$	602.000	0.180775	0.0903877	−	0.995907i	$-0.471189\pi$
$223$	602.000	0.180775	0.0903877	+	0.995907i	$0.471189\pi$
$224$	224.000	0.0668153
$225$	−1044.00	−0.309333
$226$	396.000	0.116555
$227$	−426.000	−0.124558	−0.0622789	−	0.998059i	$-0.519837\pi$
$227$	−426.000	−0.124558	−0.0622789	+	0.998059i	$0.519837\pi$
$228$	516.000	0.149881
$229$	−4396.00	−1.26854	−0.634270	−	0.773111i	$-0.718699\pi$
$229$	−4396.00	−1.26854	−0.634270	+	0.773111i	$0.718699\pi$
$230$	720.000	0.206415
$231$	231.000	0.0657952
$232$	888.000	0.251293
$233$	3198.00	0.899176	0.449588	−	0.893236i	$-0.351571\pi$
$233$	3198.00	0.899176	0.449588	+	0.893236i	$0.351571\pi$
$234$	738.000	0.206173
$235$	639.000	0.177378
$236$	1572.00	0.433595
$237$	1452.00	0.397964
$238$	84.0000	0.0228778
$239$	2997.00	0.811129	0.405564	−	0.914066i	$-0.367075\pi$
$239$	2997.00	0.811129	0.405564	+	0.914066i	$0.367075\pi$
$240$	−144.000	−0.0387298
$241$	−6235.00	−1.66652	−0.833261	−	0.552880i	$-0.813529\pi$
$241$	−6235.00	−1.66652	−0.833261	+	0.552880i	$0.813529\pi$
$242$	242.000	0.0642824
$243$	−243.000	−0.0641500
$244$	1400.00	0.367319
$245$	147.000	0.0383326
$246$	−1296.00	−0.335894
$247$	−1763.00	−0.454158
$248$	2128.00	0.544872
$249$	−3474.00	−0.884160
$250$	−1446.00	−0.365812
$251$	3735.00	0.939247	0.469624	−	0.882867i	$-0.344390\pi$
$251$	3735.00	0.939247	0.469624	+	0.882867i	$0.344390\pi$
$252$	252.000	0.0629941
$253$	−1320.00	−0.328015
$254$	−2132.00	−0.526668
$255$	−54.0000	−0.0132612
$256$	256.000	0.0625000
$257$	1515.00	0.367716	0.183858	−	0.982953i	$-0.441141\pi$
$257$	1515.00	0.367716	0.183858	+	0.982953i	$0.441141\pi$
$258$	−1704.00	−0.411188
$259$	−553.000	−0.132671
$260$	492.000	0.117356
$261$	999.000	0.236922
$262$	−240.000	−0.0565926
$263$	−4317.00	−1.01216	−0.506079	−	0.862487i	$-0.668906\pi$
$263$	−4317.00	−1.01216	−0.506079	+	0.862487i	$0.668906\pi$
$264$	264.000	0.0615457
$265$	−648.000	−0.150213
$266$	−602.000	−0.138763
$267$	−990.000	−0.226918
$268$	3284.00	0.748516
$269$	−1434.00	−0.325028	−0.162514	−	0.986706i	$-0.551960\pi$
$269$	−1434.00	−0.325028	−0.162514	+	0.986706i	$0.551960\pi$
$270$	−162.000	−0.0365148
$271$	−943.000	−0.211377	−0.105689	−	0.994399i	$-0.533705\pi$
$271$	−943.000	−0.211377	−0.105689	+	0.994399i	$0.533705\pi$
$272$	96.0000	0.0214002
$273$	−861.000	−0.190879
$274$	−3876.00	−0.854590
$275$	1276.00	0.279803
$276$	−1440.00	−0.314050
$277$	−5776.00	−1.25287	−0.626437	−	0.779472i	$-0.715488\pi$
$277$	−5776.00	−1.25287	−0.626437	+	0.779472i	$0.715488\pi$
$278$	3424.00	0.738697
$279$	2394.00	0.513710
$280$	168.000	0.0358569
$281$	−1653.00	−0.350924	−0.175462	−	0.984486i	$-0.556142\pi$
$281$	−1653.00	−0.350924	−0.175462	+	0.984486i	$0.556142\pi$
$282$	−1278.00	−0.269872
$283$	677.000	0.142203	0.0711015	−	0.997469i	$-0.477349\pi$
$283$	677.000	0.142203	0.0711015	+	0.997469i	$0.477349\pi$
$284$	−1056.00	−0.220641
$285$	387.000	0.0804347
$286$	−902.000	−0.186491
$287$	1512.00	0.310977
$288$	288.000	0.0589256
$289$	−4877.00	−0.992673
$290$	666.000	0.134858
$291$	−2940.00	−0.592254
$292$	−3460.00	−0.693429
$293$	4752.00	0.947491	0.473745	−	0.880662i	$-0.342902\pi$
$293$	4752.00	0.947491	0.473745	+	0.880662i	$0.342902\pi$
$294$	−294.000	−0.0583212
$295$	1179.00	0.232692
$296$	−632.000	−0.124102
$297$	297.000	0.0580259
$298$	−3834.00	−0.745294
$299$	4920.00	0.951609
$300$	1392.00	0.267891
$301$	1988.00	0.380686
$302$	−6044.00	−1.15163
$303$	−630.000	−0.119447
$304$	−688.000	−0.129801
$305$	1050.00	0.197124
$306$	108.000	0.0201763
$307$	1676.00	0.311578	0.155789	−	0.987790i	$-0.450208\pi$
$307$	1676.00	0.311578	0.155789	+	0.987790i	$0.450208\pi$
$308$	−308.000	−0.0569803
$309$	3990.00	0.734573
$310$	1596.00	0.292409
$311$	−6288.00	−1.14649	−0.573247	−	0.819382i	$-0.694317\pi$
$311$	−6288.00	−1.14649	−0.573247	+	0.819382i	$0.694317\pi$
$312$	−984.000	−0.178551
$313$	−430.000	−0.0776519	−0.0388259	−	0.999246i	$-0.512362\pi$
$313$	−430.000	−0.0776519	−0.0388259	+	0.999246i	$0.512362\pi$
$314$	−4400.00	−0.790785
$315$	189.000	0.0338062
$316$	−1936.00	−0.344647
$317$	7176.00	1.27143	0.635717	−	0.771923i	$-0.280705\pi$
$317$	7176.00	1.27143	0.635717	+	0.771923i	$0.280705\pi$
$318$	1296.00	0.228541
$319$	−1221.00	−0.214304
$320$	192.000	0.0335410
$321$	3987.00	0.693248
$322$	1680.00	0.290754
$323$	−258.000	−0.0444443
$324$	324.000	0.0555556
$325$	−4756.00	−0.811740
$326$	−7430.00	−1.26230
$327$	282.000	0.0476900
$328$	1728.00	0.290893
$329$	1491.00	0.249853
$330$	198.000	0.0330289
$331$	5360.00	0.890067	0.445034	−	0.895514i	$-0.353192\pi$
$331$	5360.00	0.890067	0.445034	+	0.895514i	$0.353192\pi$
$332$	4632.00	0.765705
$333$	−711.000	−0.117005
$334$	168.000	0.0275226
$335$	2463.00	0.401696
$336$	−336.000	−0.0545545
$337$	−9376.00	−1.51556	−0.757779	−	0.652511i	$-0.773716\pi$
$337$	−9376.00	−1.51556	−0.757779	+	0.652511i	$0.773716\pi$
$338$	−1032.00	−0.166075
$339$	−594.000	−0.0951671
$340$	72.0000	0.0114846
$341$	−2926.00	−0.464668
$342$	−774.000	−0.122378
$343$	343.000	0.0539949
$344$	2272.00	0.356099
$345$	−1080.00	−0.168537
$346$	3624.00	0.563085
$347$	1608.00	0.248766	0.124383	−	0.992234i	$-0.460305\pi$
$347$	1608.00	0.248766	0.124383	+	0.992234i	$0.460305\pi$
$348$	−1332.00	−0.205180
$349$	2477.00	0.379916	0.189958	−	0.981792i	$-0.439165\pi$
$349$	2477.00	0.379916	0.189958	+	0.981792i	$0.439165\pi$
$350$	−1624.00	−0.248018
$351$	−1107.00	−0.168340
$352$	−352.000	−0.0533002
$353$	−975.000	−0.147009	−0.0735043	−	0.997295i	$-0.523418\pi$
$353$	−975.000	−0.147009	−0.0735043	+	0.997295i	$0.523418\pi$
$354$	−2358.00	−0.354029
$355$	−792.000	−0.118408
$356$	1320.00	0.196516
$357$	−126.000	−0.0186796
$358$	384.000	0.0566900
$359$	4116.00	0.605109	0.302555	−	0.953132i	$-0.402161\pi$
$359$	4116.00	0.605109	0.302555	+	0.953132i	$0.402161\pi$
$360$	216.000	0.0316228
$361$	−5010.00	−0.730427
$362$	−6296.00	−0.914117
$363$	−363.000	−0.0524864
$364$	1148.00	0.165306
$365$	−2595.00	−0.372133
$366$	−2100.00	−0.299915
$367$	2066.00	0.293854	0.146927	−	0.989147i	$-0.453062\pi$
$367$	2066.00	0.293854	0.146927	+	0.989147i	$0.453062\pi$
$368$	1920.00	0.271975
$369$	1944.00	0.274256
$370$	−474.000	−0.0666002
$371$	−1512.00	−0.211588
$372$	−3192.00	−0.444886
$373$	1778.00	0.246813	0.123407	−	0.992356i	$-0.460618\pi$
$373$	1778.00	0.246813	0.123407	+	0.992356i	$0.460618\pi$
$374$	−132.000	−0.0182502
$375$	2169.00	0.298684
$376$	1704.00	0.233716
$377$	4551.00	0.621720
$378$	−378.000	−0.0514344
$379$	−10717.0	−1.45249	−0.726247	−	0.687434i	$-0.758737\pi$
$379$	−10717.0	−1.45249	−0.726247	+	0.687434i	$0.758737\pi$
$380$	−516.000	−0.0696585
$381$	3198.00	0.430022
$382$	9108.00	1.21991
$383$	9360.00	1.24876	0.624378	−	0.781122i	$-0.285352\pi$
$383$	9360.00	1.24876	0.624378	+	0.781122i	$0.285352\pi$
$384$	−384.000	−0.0510310
$385$	−231.000	−0.0305788
$386$	−1880.00	−0.247900
$387$	2556.00	0.335733
$388$	3920.00	0.512907
$389$	8364.00	1.09016	0.545079	−	0.838385i	$-0.316500\pi$
$389$	8364.00	1.09016	0.545079	+	0.838385i	$0.316500\pi$
$390$	−738.000	−0.0958207
$391$	720.000	0.0931252
$392$	392.000	0.0505076
$393$	360.000	0.0462076
$394$	−2244.00	−0.286932
$395$	−1452.00	−0.184957
$396$	−396.000	−0.0502519
$397$	−646.000	−0.0816670	−0.0408335	−	0.999166i	$-0.513001\pi$
$397$	−646.000	−0.0816670	−0.0408335	+	0.999166i	$0.513001\pi$
$398$	7912.00	0.996464
$399$	903.000	0.113300
$400$	−1856.00	−0.232000
$401$	2808.00	0.349688	0.174844	−	0.984596i	$-0.444058\pi$
$401$	2808.00	0.349688	0.174844	+	0.984596i	$0.444058\pi$
$402$	−4926.00	−0.611161
$403$	10906.0	1.34806
$404$	840.000	0.103444
$405$	243.000	0.0298142
$406$	1554.00	0.189960
$407$	869.000	0.105835
$408$	−144.000	−0.0174732
$409$	8750.00	1.05785	0.528924	−	0.848669i	$-0.322596\pi$
$409$	8750.00	1.05785	0.528924	+	0.848669i	$0.322596\pi$
$410$	1296.00	0.156109
$411$	5814.00	0.697770
$412$	−5320.00	−0.636159
$413$	2751.00	0.327767
$414$	2160.00	0.256421
$415$	3474.00	0.410920
$416$	1312.00	0.154630
$417$	−5136.00	−0.603144
$418$	946.000	0.110695
$419$	13263.0	1.54640	0.773198	−	0.634165i	$-0.218656\pi$
$419$	13263.0	1.54640	0.773198	+	0.634165i	$0.218656\pi$
$420$	−252.000	−0.0292770
$421$	−5785.00	−0.669700	−0.334850	−	0.942271i	$-0.608686\pi$
$421$	−5785.00	−0.669700	−0.334850	+	0.942271i	$0.608686\pi$
$422$	244.000	0.0281463
$423$	1917.00	0.220349
$424$	−1728.00	−0.197922
$425$	−696.000	−0.0794376
$426$	1584.00	0.180153
$427$	2450.00	0.277667
$428$	−5316.00	−0.600370
$429$	1353.00	0.152269
$430$	1704.00	0.191103
$431$	5019.00	0.560920	0.280460	−	0.959866i	$-0.409513\pi$
$431$	5019.00	0.560920	0.280460	+	0.959866i	$0.409513\pi$
$432$	−432.000	−0.0481125
$433$	14768.0	1.63904	0.819521	−	0.573050i	$-0.194240\pi$
$433$	14768.0	1.63904	0.819521	+	0.573050i	$0.194240\pi$
$434$	3724.00	0.411884
$435$	−999.000	−0.110111
$436$	−376.000	−0.0413008
$437$	−5160.00	−0.564843
$438$	5190.00	0.566182
$439$	−2311.00	−0.251248	−0.125624	−	0.992078i	$-0.540093\pi$
$439$	−2311.00	−0.251248	−0.125624	+	0.992078i	$0.540093\pi$
$440$	−264.000	−0.0286039
$441$	441.000	0.0476190
$442$	492.000	0.0529458
$443$	−252.000	−0.0270268	−0.0135134	−	0.999909i	$-0.504302\pi$
$443$	−252.000	−0.0270268	−0.0135134	+	0.999909i	$0.504302\pi$
$444$	948.000	0.101329
$445$	990.000	0.105462
$446$	1204.00	0.127827
$447$	5751.00	0.608530
$448$	448.000	0.0472456
$449$	1980.00	0.208111	0.104056	−	0.994571i	$-0.466818\pi$
$449$	1980.00	0.208111	0.104056	+	0.994571i	$0.466818\pi$
$450$	−2088.00	−0.218732
$451$	−2376.00	−0.248074
$452$	792.000	0.0824171
$453$	9066.00	0.940304
$454$	−852.000	−0.0880756
$455$	861.000	0.0887128
$456$	1032.00	0.105982
$457$	12164.0	1.24509	0.622547	−	0.782582i	$-0.286098\pi$
$457$	12164.0	1.24509	0.622547	+	0.782582i	$0.286098\pi$
$458$	−8792.00	−0.896994
$459$	−162.000	−0.0164739
$460$	1440.00	0.145957
$461$	−18732.0	−1.89249	−0.946243	−	0.323456i	$-0.895155\pi$
$461$	−18732.0	−1.89249	−0.946243	+	0.323456i	$0.895155\pi$
$462$	462.000	0.0465242
$463$	−13051.0	−1.31000	−0.655002	−	0.755628i	$-0.727332\pi$
$463$	−13051.0	−1.31000	−0.655002	+	0.755628i	$0.727332\pi$
$464$	1776.00	0.177691
$465$	−2394.00	−0.238751
$466$	6396.00	0.635813
$467$	6717.00	0.665580	0.332790	−	0.943001i	$-0.392010\pi$
$467$	6717.00	0.665580	0.332790	+	0.943001i	$0.392010\pi$
$468$	1476.00	0.145787
$469$	5747.00	0.565825
$470$	1278.00	0.125425
$471$	6600.00	0.645673
$472$	3144.00	0.306598
$473$	−3124.00	−0.303682
$474$	2904.00	0.281403
$475$	4988.00	0.481821
$476$	168.000	0.0161770
$477$	−1944.00	−0.186603
$478$	5994.00	0.573555
$479$	−18438.0	−1.75878	−0.879388	−	0.476106i	$-0.842048\pi$
$479$	−18438.0	−1.75878	−0.879388	+	0.476106i	$0.842048\pi$
$480$	−288.000	−0.0273861
$481$	−3239.00	−0.307039
$482$	−12470.0	−1.17841
$483$	−2520.00	−0.237400
$484$	484.000	0.0454545
$485$	2940.00	0.275255
$486$	−486.000	−0.0453609
$487$	−10648.0	−0.990774	−0.495387	−	0.868672i	$-0.664974\pi$
$487$	−10648.0	−0.990774	−0.495387	+	0.868672i	$0.664974\pi$
$488$	2800.00	0.259734
$489$	11145.0	1.03066
$490$	294.000	0.0271052
$491$	16863.0	1.54993	0.774966	−	0.632003i	$-0.217767\pi$
$491$	16863.0	1.54993	0.774966	+	0.632003i	$0.217767\pi$
$492$	−2592.00	−0.237513
$493$	666.000	0.0608421
$494$	−3526.00	−0.321138
$495$	−297.000	−0.0269680
$496$	4256.00	0.385282
$497$	−1848.00	−0.166789
$498$	−6948.00	−0.625195
$499$	12683.0	1.13781	0.568907	−	0.822402i	$-0.307366\pi$
$499$	12683.0	1.13781	0.568907	+	0.822402i	$0.307366\pi$
$500$	−2892.00	−0.258668
$501$	−252.000	−0.0224721
$502$	7470.00	0.664148
$503$	14472.0	1.28285	0.641426	−	0.767185i	$-0.278343\pi$
$503$	14472.0	1.28285	0.641426	+	0.767185i	$0.278343\pi$
$504$	504.000	0.0445435
$505$	630.000	0.0555141
$506$	−2640.00	−0.231941
$507$	1548.00	0.135600
$508$	−4264.00	−0.372410
$509$	−10722.0	−0.933682	−0.466841	−	0.884341i	$-0.654608\pi$
$509$	−10722.0	−0.933682	−0.466841	+	0.884341i	$0.654608\pi$
$510$	−108.000	−0.00937710
$511$	−6055.00	−0.524183
$512$	512.000	0.0441942
$513$	1161.00	0.0999209
$514$	3030.00	0.260015
$515$	−3990.00	−0.341399
$516$	−3408.00	−0.290754
$517$	−2343.00	−0.199313
$518$	−1106.00	−0.0938125
$519$	−5436.00	−0.459757
$520$	984.000	0.0829832
$521$	−10341.0	−0.869573	−0.434786	−	0.900534i	$-0.643176\pi$
$521$	−10341.0	−0.869573	−0.434786	+	0.900534i	$0.643176\pi$
$522$	1998.00	0.167529
$523$	5819.00	0.486515	0.243257	−	0.969962i	$-0.421784\pi$
$523$	5819.00	0.486515	0.243257	+	0.969962i	$0.421784\pi$
$524$	−480.000	−0.0400170
$525$	2436.00	0.202506
$526$	−8634.00	−0.715704
$527$	1596.00	0.131922
$528$	528.000	0.0435194
$529$	2233.00	0.183529
$530$	−1296.00	−0.106216
$531$	3537.00	0.289064
$532$	−1204.00	−0.0981203
$533$	8856.00	0.719692
$534$	−1980.00	−0.160455
$535$	−3987.00	−0.322193
$536$	6568.00	0.529281
$537$	−576.000	−0.0462872
$538$	−2868.00	−0.229829
$539$	−539.000	−0.0430730
$540$	−324.000	−0.0258199
$541$	9668.00	0.768318	0.384159	−	0.923267i	$-0.374492\pi$
$541$	9668.00	0.768318	0.384159	+	0.923267i	$0.374492\pi$
$542$	−1886.00	−0.149466
$543$	9444.00	0.746374
$544$	192.000	0.0151322
$545$	−282.000	−0.0221643
$546$	−1722.00	−0.134972
$547$	13160.0	1.02867	0.514334	−	0.857590i	$-0.328039\pi$
$547$	13160.0	1.02867	0.514334	+	0.857590i	$0.328039\pi$
$548$	−7752.00	−0.604287
$549$	3150.00	0.244879
$550$	2552.00	0.197850
$551$	−4773.00	−0.369032
$552$	−2880.00	−0.222067
$553$	−3388.00	−0.260529
$554$	−11552.0	−0.885916
$555$	711.000	0.0543789
$556$	6848.00	0.522338
$557$	−2253.00	−0.171387	−0.0856936	−	0.996322i	$-0.527311\pi$
$557$	−2253.00	−0.171387	−0.0856936	+	0.996322i	$0.527311\pi$
$558$	4788.00	0.363248
$559$	11644.0	0.881017
$560$	336.000	0.0253546
$561$	198.000	0.0149012
$562$	−3306.00	−0.248141
$563$	−984.000	−0.0736601	−0.0368301	−	0.999322i	$-0.511726\pi$
$563$	−984.000	−0.0736601	−0.0368301	+	0.999322i	$0.511726\pi$
$564$	−2556.00	−0.190828
$565$	594.000	0.0442297
$566$	1354.00	0.100553
$567$	567.000	0.0419961
$568$	−2112.00	−0.156017
$569$	−16878.0	−1.24352	−0.621760	−	0.783208i	$-0.713582\pi$
$569$	−16878.0	−1.24352	−0.621760	+	0.783208i	$0.713582\pi$
$570$	774.000	0.0568760
$571$	4736.00	0.347102	0.173551	−	0.984825i	$-0.444476\pi$
$571$	4736.00	0.347102	0.173551	+	0.984825i	$0.444476\pi$
$572$	−1804.00	−0.131869
$573$	−13662.0	−0.996053
$574$	3024.00	0.219894
$575$	−13920.0	−1.00957
$576$	576.000	0.0416667
$577$	−3694.00	−0.266522	−0.133261	−	0.991081i	$-0.542545\pi$
$577$	−3694.00	−0.266522	−0.133261	+	0.991081i	$0.542545\pi$
$578$	−9754.00	−0.701925
$579$	2820.00	0.202410
$580$	1332.00	0.0953591
$581$	8106.00	0.578818
$582$	−5880.00	−0.418787
$583$	2376.00	0.168789
$584$	−6920.00	−0.490328
$585$	1107.00	0.0782373
$586$	9504.00	0.669977
$587$	−21237.0	−1.49326	−0.746631	−	0.665238i	$-0.768330\pi$
$587$	−21237.0	−1.49326	−0.746631	+	0.665238i	$0.768330\pi$
$588$	−588.000	−0.0412393
$589$	−11438.0	−0.800161
$590$	2358.00	0.164538
$591$	3366.00	0.234279
$592$	−1264.00	−0.0877535
$593$	−7416.00	−0.513556	−0.256778	−	0.966470i	$-0.582661\pi$
$593$	−7416.00	−0.513556	−0.256778	+	0.966470i	$0.582661\pi$
$594$	594.000	0.0410305
$595$	126.000	0.00868151
$596$	−7668.00	−0.527003
$597$	−11868.0	−0.813610
$598$	9840.00	0.672889
$599$	−5484.00	−0.374074	−0.187037	−	0.982353i	$-0.559888\pi$
$599$	−5484.00	−0.374074	−0.187037	+	0.982353i	$0.559888\pi$
$600$	2784.00	0.189427
$601$	−11077.0	−0.751814	−0.375907	−	0.926657i	$-0.622669\pi$
$601$	−11077.0	−0.751814	−0.375907	+	0.926657i	$0.622669\pi$
$602$	3976.00	0.269185
$603$	7389.00	0.499011
$604$	−12088.0	−0.814327
$605$	363.000	0.0243935
$606$	−1260.00	−0.0844620
$607$	−5047.00	−0.337482	−0.168741	−	0.985660i	$-0.553970\pi$
$607$	−5047.00	−0.337482	−0.168741	+	0.985660i	$0.553970\pi$
$608$	−1376.00	−0.0917832
$609$	−2331.00	−0.155102
$610$	2100.00	0.139388
$611$	8733.00	0.578231
$612$	216.000	0.0142668
$613$	20066.0	1.32212	0.661059	−	0.750334i	$-0.270107\pi$
$613$	20066.0	1.32212	0.661059	+	0.750334i	$0.270107\pi$
$614$	3352.00	0.220319
$615$	−1944.00	−0.127463
$616$	−616.000	−0.0402911
$617$	−16626.0	−1.08483	−0.542413	−	0.840112i	$-0.682489\pi$
$617$	−16626.0	−1.08483	−0.542413	+	0.840112i	$0.682489\pi$
$618$	7980.00	0.519422
$619$	−10162.0	−0.659847	−0.329923	−	0.944008i	$-0.607023\pi$
$619$	−10162.0	−0.659847	−0.329923	+	0.944008i	$0.607023\pi$
$620$	3192.00	0.206764
$621$	−3240.00	−0.209367
$622$	−12576.0	−0.810694
$623$	2310.00	0.148552
$624$	−1968.00	−0.126255
$625$	12331.0	0.789184
$626$	−860.000	−0.0549082
$627$	−1419.00	−0.0903818
$628$	−8800.00	−0.559169
$629$	−474.000	−0.0300471
$630$	378.000	0.0239046
$631$	−16108.0	−1.01624	−0.508122	−	0.861285i	$-0.669660\pi$
$631$	−16108.0	−1.01624	−0.508122	+	0.861285i	$0.669660\pi$
$632$	−3872.00	−0.243702
$633$	−366.000	−0.0229813
$634$	14352.0	0.899039
$635$	−3198.00	−0.199856
$636$	2592.00	0.161603
$637$	2009.00	0.124960
$638$	−2442.00	−0.151536
$639$	−2376.00	−0.147094
$640$	384.000	0.0237171
$641$	−19950.0	−1.22929	−0.614647	−	0.788802i	$-0.710702\pi$
$641$	−19950.0	−1.22929	−0.614647	+	0.788802i	$0.710702\pi$
$642$	7974.00	0.490200
$643$	−8260.00	−0.506598	−0.253299	−	0.967388i	$-0.581516\pi$
$643$	−8260.00	−0.506598	−0.253299	+	0.967388i	$0.581516\pi$
$644$	3360.00	0.205594
$645$	−2556.00	−0.156035
$646$	−516.000	−0.0314269
$647$	−28833.0	−1.75200	−0.875999	−	0.482314i	$-0.839797\pi$
$647$	−28833.0	−1.75200	−0.875999	+	0.482314i	$0.839797\pi$
$648$	648.000	0.0392837
$649$	−4323.00	−0.261468
$650$	−9512.00	−0.573987
$651$	−5586.00	−0.336302
$652$	−14860.0	−0.892581
$653$	−13152.0	−0.788174	−0.394087	−	0.919073i	$-0.628939\pi$
$653$	−13152.0	−0.788174	−0.394087	+	0.919073i	$0.628939\pi$
$654$	564.000	0.0337219
$655$	−360.000	−0.0214754
$656$	3456.00	0.205692
$657$	−7785.00	−0.462286
$658$	2982.00	0.176672
$659$	−32229.0	−1.90510	−0.952552	−	0.304376i	$-0.901552\pi$
$659$	−32229.0	−1.90510	−0.952552	+	0.304376i	$0.901552\pi$
$660$	396.000	0.0233550
$661$	−460.000	−0.0270680	−0.0135340	−	0.999908i	$-0.504308\pi$
$661$	−460.000	−0.0270680	−0.0135340	+	0.999908i	$0.504308\pi$
$662$	10720.0	0.629373
$663$	−738.000	−0.0432301
$664$	9264.00	0.541435
$665$	−903.000	−0.0526569
$666$	−1422.00	−0.0827348
$667$	13320.0	0.773242
$668$	336.000	0.0194614
$669$	−1806.00	−0.104371
$670$	4926.00	0.284042
$671$	−3850.00	−0.221502
$672$	−672.000	−0.0385758
$673$	−12784.0	−0.732224	−0.366112	−	0.930571i	$-0.619311\pi$
$673$	−12784.0	−0.732224	−0.366112	+	0.930571i	$0.619311\pi$
$674$	−18752.0	−1.07166
$675$	3132.00	0.178594
$676$	−2064.00	−0.117433
$677$	2190.00	0.124326	0.0621629	−	0.998066i	$-0.480200\pi$
$677$	2190.00	0.124326	0.0621629	+	0.998066i	$0.480200\pi$
$678$	−1188.00	−0.0672933
$679$	6860.00	0.387721
$680$	144.000	0.00812081
$681$	1278.00	0.0719135
$682$	−5852.00	−0.328570
$683$	11514.0	0.645053	0.322526	−	0.946560i	$-0.395468\pi$
$683$	11514.0	0.645053	0.322526	+	0.946560i	$0.395468\pi$
$684$	−1548.00	−0.0865340
$685$	−5814.00	−0.324294
$686$	686.000	0.0381802
$687$	13188.0	0.732392
$688$	4544.00	0.251800
$689$	−8856.00	−0.489676
$690$	−2160.00	−0.119174
$691$	−11554.0	−0.636085	−0.318043	−	0.948076i	$-0.603026\pi$
$691$	−11554.0	−0.636085	−0.318043	+	0.948076i	$0.603026\pi$
$692$	7248.00	0.398161
$693$	−693.000	−0.0379869
$694$	3216.00	0.175904
$695$	5136.00	0.280316
$696$	−2664.00	−0.145084
$697$	1296.00	0.0704297
$698$	4954.00	0.268641
$699$	−9594.00	−0.519139
$700$	−3248.00	−0.175376
$701$	−18450.0	−0.994075	−0.497038	−	0.867729i	$-0.665579\pi$
$701$	−18450.0	−0.994075	−0.497038	+	0.867729i	$0.665579\pi$
$702$	−2214.00	−0.119034
$703$	3397.00	0.182248
$704$	−704.000	−0.0376889
$705$	−1917.00	−0.102409
$706$	−1950.00	−0.103951
$707$	1470.00	0.0781967
$708$	−4716.00	−0.250336
$709$	15203.0	0.805304	0.402652	−	0.915353i	$-0.368088\pi$
$709$	15203.0	0.805304	0.402652	+	0.915353i	$0.368088\pi$
$710$	−1584.00	−0.0837274
$711$	−4356.00	−0.229765
$712$	2640.00	0.138958
$713$	31920.0	1.67660
$714$	−252.000	−0.0132085
$715$	−1353.00	−0.0707683
$716$	768.000	0.0400859
$717$	−8991.00	−0.468306
$718$	8232.00	0.427877
$719$	28515.0	1.47904	0.739520	−	0.673134i	$-0.235052\pi$
$719$	28515.0	1.47904	0.739520	+	0.673134i	$0.235052\pi$
$720$	432.000	0.0223607
$721$	−9310.00	−0.480891
$722$	−10020.0	−0.516490
$723$	18705.0	0.962167
$724$	−12592.0	−0.646378
$725$	−12876.0	−0.659590
$726$	−726.000	−0.0371135
$727$	15932.0	0.812772	0.406386	−	0.913702i	$-0.366789\pi$
$727$	15932.0	0.812772	0.406386	+	0.913702i	$0.366789\pi$
$728$	2296.00	0.116889
$729$	729.000	0.0370370
$730$	−5190.00	−0.263138
$731$	1704.00	0.0862171
$732$	−4200.00	−0.212072
$733$	26630.0	1.34188	0.670942	−	0.741510i	$-0.265890\pi$
$733$	26630.0	1.34188	0.670942	+	0.741510i	$0.265890\pi$
$734$	4132.00	0.207786
$735$	−441.000	−0.0221313
$736$	3840.00	0.192316
$737$	−9031.00	−0.451372
$738$	3888.00	0.193929
$739$	−3940.00	−0.196123	−0.0980617	−	0.995180i	$-0.531264\pi$
$739$	−3940.00	−0.196123	−0.0980617	+	0.995180i	$0.531264\pi$
$740$	−948.000	−0.0470935
$741$	5289.00	0.262208
$742$	−3024.00	−0.149615
$743$	25203.0	1.24443	0.622213	−	0.782848i	$-0.286234\pi$
$743$	25203.0	1.24443	0.622213	+	0.782848i	$0.286234\pi$
$744$	−6384.00	−0.314582
$745$	−5751.00	−0.282819
$746$	3556.00	0.174523
$747$	10422.0	0.510470
$748$	−264.000	−0.0129048
$749$	−9303.00	−0.453837
$750$	4338.00	0.211202
$751$	−17611.0	−0.855705	−0.427853	−	0.903849i	$-0.640730\pi$
$751$	−17611.0	−0.855705	−0.427853	+	0.903849i	$0.640730\pi$
$752$	3408.00	0.165262
$753$	−11205.0	−0.542275
$754$	9102.00	0.439622
$755$	−9066.00	−0.437014
$756$	−756.000	−0.0363696
$757$	−28543.0	−1.37043	−0.685213	−	0.728342i	$-0.740291\pi$
$757$	−28543.0	−1.37043	−0.685213	+	0.728342i	$0.740291\pi$
$758$	−21434.0	−1.02707
$759$	3960.00	0.189379
$760$	−1032.00	−0.0492560
$761$	−16884.0	−0.804263	−0.402132	−	0.915582i	$-0.631731\pi$
$761$	−16884.0	−0.804263	−0.402132	+	0.915582i	$0.631731\pi$
$762$	6396.00	0.304072
$763$	−658.000	−0.0312204
$764$	18216.0	0.862607
$765$	162.000	0.00765637
$766$	18720.0	0.883004
$767$	16113.0	0.758549
$768$	−768.000	−0.0360844
$769$	−22963.0	−1.07681	−0.538405	−	0.842686i	$-0.680973\pi$
$769$	−22963.0	−1.07681	−0.538405	+	0.842686i	$0.680973\pi$
$770$	−462.000	−0.0216225
$771$	−4545.00	−0.212301
$772$	−3760.00	−0.175292
$773$	−32601.0	−1.51692	−0.758458	−	0.651722i	$-0.774047\pi$
$773$	−32601.0	−1.51692	−0.758458	+	0.651722i	$0.774047\pi$
$774$	5112.00	0.237399
$775$	−30856.0	−1.43017
$776$	7840.00	0.362680
$777$	1659.00	0.0765975
$778$	16728.0	0.770858
$779$	−9288.00	−0.427185
$780$	−1476.00	−0.0677555
$781$	2904.00	0.133052
$782$	1440.00	0.0658495
$783$	−2997.00	−0.136787
$784$	784.000	0.0357143
$785$	−6600.00	−0.300082
$786$	720.000	0.0326737
$787$	17867.0	0.809263	0.404631	−	0.914480i	$-0.367400\pi$
$787$	17867.0	0.809263	0.404631	+	0.914480i	$0.367400\pi$
$788$	−4488.00	−0.202891
$789$	12951.0	0.584370
$790$	−2904.00	−0.130784
$791$	1386.00	0.0623015
$792$	−792.000	−0.0355335
$793$	14350.0	0.642602
$794$	−1292.00	−0.0577473
$795$	1944.00	0.0867253
$796$	15824.0	0.704607
$797$	17505.0	0.777991	0.388996	−	0.921240i	$-0.372822\pi$
$797$	17505.0	0.777991	0.388996	+	0.921240i	$0.372822\pi$
$798$	1806.00	0.0801149
$799$	1278.00	0.0565862
$800$	−3712.00	−0.164049
$801$	2970.00	0.131011
$802$	5616.00	0.247267
$803$	9515.00	0.418153
$804$	−9852.00	−0.432156
$805$	2520.00	0.110333
$806$	21812.0	0.953220
$807$	4302.00	0.187655
$808$	1680.00	0.0731463
$809$	11079.0	0.481479	0.240740	−	0.970590i	$-0.422610\pi$
$809$	11079.0	0.481479	0.240740	+	0.970590i	$0.422610\pi$
$810$	486.000	0.0210819
$811$	−28021.0	−1.21326	−0.606628	−	0.794986i	$-0.707478\pi$
$811$	−28021.0	−1.21326	−0.606628	+	0.794986i	$0.707478\pi$
$812$	3108.00	0.134322
$813$	2829.00	0.122039
$814$	1738.00	0.0748364
$815$	−11145.0	−0.479009
$816$	−288.000	−0.0123554
$817$	−12212.0	−0.522942
$818$	17500.0	0.748011
$819$	2583.00	0.110204
$820$	2592.00	0.110386
$821$	12309.0	0.523249	0.261624	−	0.965170i	$-0.415742\pi$
$821$	12309.0	0.523249	0.261624	+	0.965170i	$0.415742\pi$
$822$	11628.0	0.493398
$823$	3893.00	0.164886	0.0824432	−	0.996596i	$-0.473728\pi$
$823$	3893.00	0.164886	0.0824432	+	0.996596i	$0.473728\pi$
$824$	−10640.0	−0.449832
$825$	−3828.00	−0.161544
$826$	5502.00	0.231766
$827$	45063.0	1.89479	0.947397	−	0.320062i	$-0.103704\pi$
$827$	45063.0	1.89479	0.947397	+	0.320062i	$0.103704\pi$
$828$	4320.00	0.181317
$829$	−15136.0	−0.634131	−0.317066	−	0.948404i	$-0.602698\pi$
$829$	−15136.0	−0.634131	−0.317066	+	0.948404i	$0.602698\pi$
$830$	6948.00	0.290565
$831$	17328.0	0.723347
$832$	2624.00	0.109340
$833$	294.000	0.0122287
$834$	−10272.0	−0.426487
$835$	252.000	0.0104441
$836$	1892.00	0.0782730
$837$	−7182.00	−0.296590
$838$	26526.0	1.09347
$839$	−20739.0	−0.853385	−0.426692	−	0.904397i	$-0.640321\pi$
$839$	−20739.0	−0.853385	−0.426692	+	0.904397i	$0.640321\pi$
$840$	−504.000	−0.0207020
$841$	−12068.0	−0.494813
$842$	−11570.0	−0.473549
$843$	4959.00	0.202606
$844$	488.000	0.0199024
$845$	−1548.00	−0.0630211
$846$	3834.00	0.155810
$847$	847.000	0.0343604
$848$	−3456.00	−0.139952
$849$	−2031.00	−0.0821010
$850$	−1392.00	−0.0561708
$851$	−9480.00	−0.381869
$852$	3168.00	0.127387
$853$	37610.0	1.50966	0.754831	−	0.655919i	$-0.227719\pi$
$853$	37610.0	1.50966	0.754831	+	0.655919i	$0.227719\pi$
$854$	4900.00	0.196340
$855$	−1161.00	−0.0464390
$856$	−10632.0	−0.424526
$857$	−37470.0	−1.49352	−0.746762	−	0.665091i	$-0.768393\pi$
$857$	−37470.0	−1.49352	−0.746762	+	0.665091i	$0.768393\pi$
$858$	2706.00	0.107671
$859$	−15604.0	−0.619792	−0.309896	−	0.950770i	$-0.600294\pi$
$859$	−15604.0	−0.619792	−0.309896	+	0.950770i	$0.600294\pi$
$860$	3408.00	0.135130
$861$	−4536.00	−0.179543
$862$	10038.0	0.396631
$863$	20004.0	0.789043	0.394521	−	0.918887i	$-0.370910\pi$
$863$	20004.0	0.789043	0.394521	+	0.918887i	$0.370910\pi$
$864$	−864.000	−0.0340207
$865$	5436.00	0.213676
$866$	29536.0	1.15898
$867$	14631.0	0.573120
$868$	7448.00	0.291246
$869$	5324.00	0.207830
$870$	−1998.00	−0.0778604
$871$	33661.0	1.30948
$872$	−752.000	−0.0292041
$873$	8820.00	0.341938
$874$	−10320.0	−0.399404
$875$	−5061.00	−0.195535
$876$	10380.0	0.400351
$877$	1064.00	0.0409678	0.0204839	−	0.999790i	$-0.493479\pi$
$877$	1064.00	0.0409678	0.0204839	+	0.999790i	$0.493479\pi$
$878$	−4622.00	−0.177659
$879$	−14256.0	−0.547034
$880$	−528.000	−0.0202260
$881$	4179.00	0.159812	0.0799058	−	0.996802i	$-0.474538\pi$
$881$	4179.00	0.159812	0.0799058	+	0.996802i	$0.474538\pi$
$882$	882.000	0.0336718
$883$	−44509.0	−1.69632	−0.848158	−	0.529743i	$-0.822288\pi$
$883$	−44509.0	−1.69632	−0.848158	+	0.529743i	$0.822288\pi$
$884$	984.000	0.0374384
$885$	−3537.00	−0.134345
$886$	−504.000	−0.0191108
$887$	12642.0	0.478553	0.239277	−	0.970951i	$-0.423090\pi$
$887$	12642.0	0.478553	0.239277	+	0.970951i	$0.423090\pi$
$888$	1896.00	0.0716504
$889$	−7462.00	−0.281516
$890$	1980.00	0.0745728
$891$	−891.000	−0.0335013
$892$	2408.00	0.0903877
$893$	−9159.00	−0.343219
$894$	11502.0	0.430296
$895$	576.000	0.0215124
$896$	896.000	0.0334077
$897$	−14760.0	−0.549411
$898$	3960.00	0.147157
$899$	29526.0	1.09538
$900$	−4176.00	−0.154667
$901$	−1296.00	−0.0479201
$902$	−4752.00	−0.175415
$903$	−5964.00	−0.219789
$904$	1584.00	0.0582777
$905$	−9444.00	−0.346883
$906$	18132.0	0.664896
$907$	−28.0000	−0.00102505	−0.000512527	−	1.00000i	$-0.500163\pi$
$907$	−28.0000	−0.00102505	−0.000512527		1.00000i	$0.500163\pi$
$908$	−1704.00	−0.0622789
$909$	1890.00	0.0689630
$910$	1722.00	0.0627294
$911$	−9774.00	−0.355463	−0.177732	−	0.984079i	$-0.556876\pi$
$911$	−9774.00	−0.355463	−0.177732	+	0.984079i	$0.556876\pi$
$912$	2064.00	0.0749406
$913$	−12738.0	−0.461737
$914$	24328.0	0.880414
$915$	−3150.00	−0.113810
$916$	−17584.0	−0.634270
$917$	−840.000	−0.0302500
$918$	−324.000	−0.0116488
$919$	23204.0	0.832894	0.416447	−	0.909160i	$-0.363275\pi$
$919$	23204.0	0.832894	0.416447	+	0.909160i	$0.363275\pi$
$920$	2880.00	0.103207
$921$	−5028.00	−0.179890
$922$	−37464.0	−1.33819
$923$	−10824.0	−0.385998
$924$	924.000	0.0328976
$925$	9164.00	0.325741
$926$	−26102.0	−0.926312
$927$	−11970.0	−0.424106
$928$	3552.00	0.125647
$929$	3633.00	0.128304	0.0641522	−	0.997940i	$-0.479566\pi$
$929$	3633.00	0.128304	0.0641522	+	0.997940i	$0.479566\pi$
$930$	−4788.00	−0.168822
$931$	−2107.00	−0.0741720
$932$	12792.0	0.449588
$933$	18864.0	0.661929
$934$	13434.0	0.470636
$935$	−198.000	−0.00692545
$936$	2952.00	0.103087
$937$	18938.0	0.660275	0.330137	−	0.943933i	$-0.392905\pi$
$937$	18938.0	0.660275	0.330137	+	0.943933i	$0.392905\pi$
$938$	11494.0	0.400099
$939$	1290.00	0.0448323
$940$	2556.00	0.0886889
$941$	14784.0	0.512162	0.256081	−	0.966655i	$-0.417569\pi$
$941$	14784.0	0.512162	0.256081	+	0.966655i	$0.417569\pi$
$942$	13200.0	0.456560
$943$	25920.0	0.895092
$944$	6288.00	0.216798
$945$	−567.000	−0.0195180
$946$	−6248.00	−0.214736
$947$	−31266.0	−1.07287	−0.536435	−	0.843941i	$-0.680229\pi$
$947$	−31266.0	−1.07287	−0.536435	+	0.843941i	$0.680229\pi$
$948$	5808.00	0.198982
$949$	−35465.0	−1.21311
$950$	9976.00	0.340699
$951$	−21528.0	−0.734062
$952$	336.000	0.0114389
$953$	33951.0	1.15402	0.577010	−	0.816737i	$-0.304219\pi$
$953$	33951.0	1.15402	0.577010	+	0.816737i	$0.304219\pi$
$954$	−3888.00	−0.131948
$955$	13662.0	0.462923
$956$	11988.0	0.405564
$957$	3663.00	0.123728
$958$	−36876.0	−1.24364
$959$	−13566.0	−0.456798
$960$	−576.000	−0.0193649
$961$	40965.0	1.37508
$962$	−6478.00	−0.217109
$963$	−11961.0	−0.400247
$964$	−24940.0	−0.833261
$965$	−2820.00	−0.0940715
$966$	−5040.00	−0.167867
$967$	16322.0	0.542792	0.271396	−	0.962468i	$-0.412515\pi$
$967$	16322.0	0.542792	0.271396	+	0.962468i	$0.412515\pi$
$968$	968.000	0.0321412
$969$	774.000	0.0256599
$970$	5880.00	0.194634
$971$	40779.0	1.34774	0.673872	−	0.738848i	$-0.264630\pi$
$971$	40779.0	1.34774	0.673872	+	0.738848i	$0.264630\pi$
$972$	−972.000	−0.0320750
$973$	11984.0	0.394850
$974$	−21296.0	−0.700583
$975$	14268.0	0.468658
$976$	5600.00	0.183659
$977$	21804.0	0.713994	0.356997	−	0.934106i	$-0.383801\pi$
$977$	21804.0	0.713994	0.356997	+	0.934106i	$0.383801\pi$
$978$	22290.0	0.728789
$979$	−3630.00	−0.118504
$980$	588.000	0.0191663
$981$	−846.000	−0.0275338
$982$	33726.0	1.09597
$983$	−35424.0	−1.14939	−0.574695	−	0.818368i	$-0.694879\pi$
$983$	−35424.0	−1.14939	−0.574695	+	0.818368i	$0.694879\pi$
$984$	−5184.00	−0.167947
$985$	−3366.00	−0.108883
$986$	1332.00	0.0430218
$987$	−4473.00	−0.144252
$988$	−7052.00	−0.227079
$989$	34080.0	1.09573
$990$	−594.000	−0.0190693
$991$	−37315.0	−1.19612	−0.598058	−	0.801453i	$-0.704061\pi$
$991$	−37315.0	−1.19612	−0.598058	+	0.801453i	$0.704061\pi$
$992$	8512.00	0.272436
$993$	−16080.0	−0.513881
$994$	−3696.00	−0.117938
$995$	11868.0	0.378132
$996$	−13896.0	−0.442080
$997$	30674.0	0.974378	0.487189	−	0.873296i	$-0.338022\pi$
$997$	30674.0	0.974378	0.487189	+	0.873296i	$0.338022\pi$
$998$	25366.0	0.804556
$999$	2133.00	0.0675527

Twists

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

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