LMFDB - Embedded newform 6.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6,4,Mod(1,6)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 6 = 2 \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	6.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:	$0.354011460034$
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	$\chi$	$=$	6.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{5} +6.00000 q^{6} -16.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +6.00000 q^{5} +6.00000 q^{6} -16.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -12.0000 q^{10} +12.0000 q^{11} -12.0000 q^{12} +38.0000 q^{13} +32.0000 q^{14} -18.0000 q^{15} +16.0000 q^{16} -126.000 q^{17} -18.0000 q^{18} +20.0000 q^{19} +24.0000 q^{20} +48.0000 q^{21} -24.0000 q^{22} +168.000 q^{23} +24.0000 q^{24} -89.0000 q^{25} -76.0000 q^{26} -27.0000 q^{27} -64.0000 q^{28} +30.0000 q^{29} +36.0000 q^{30} -88.0000 q^{31} -32.0000 q^{32} -36.0000 q^{33} +252.000 q^{34} -96.0000 q^{35} +36.0000 q^{36} +254.000 q^{37} -40.0000 q^{38} -114.000 q^{39} -48.0000 q^{40} +42.0000 q^{41} -96.0000 q^{42} -52.0000 q^{43} +48.0000 q^{44} +54.0000 q^{45} -336.000 q^{46} -96.0000 q^{47} -48.0000 q^{48} -87.0000 q^{49} +178.000 q^{50} +378.000 q^{51} +152.000 q^{52} +198.000 q^{53} +54.0000 q^{54} +72.0000 q^{55} +128.000 q^{56} -60.0000 q^{57} -60.0000 q^{58} -660.000 q^{59} -72.0000 q^{60} -538.000 q^{61} +176.000 q^{62} -144.000 q^{63} +64.0000 q^{64} +228.000 q^{65} +72.0000 q^{66} +884.000 q^{67} -504.000 q^{68} -504.000 q^{69} +192.000 q^{70} +792.000 q^{71} -72.0000 q^{72} +218.000 q^{73} -508.000 q^{74} +267.000 q^{75} +80.0000 q^{76} -192.000 q^{77} +228.000 q^{78} -520.000 q^{79} +96.0000 q^{80} +81.0000 q^{81} -84.0000 q^{82} -492.000 q^{83} +192.000 q^{84} -756.000 q^{85} +104.000 q^{86} -90.0000 q^{87} -96.0000 q^{88} +810.000 q^{89} -108.000 q^{90} -608.000 q^{91} +672.000 q^{92} +264.000 q^{93} +192.000 q^{94} +120.000 q^{95} +96.0000 q^{96} +1154.00 q^{97} +174.000 q^{98} +108.000 q^{99} +O(q^{100})\)

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
$5$	6.00000	0.536656	0.268328	−	0.963328i	$-0.413529\pi$
$5$	6.00000	0.536656	0.268328	+	0.963328i	$0.413529\pi$
$6$	6.00000	0.408248
$7$	−16.0000	−0.863919	−0.431959	−	0.901893i	$-0.642178\pi$
$7$	−16.0000	−0.863919	−0.431959	+	0.901893i	$0.642178\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−12.0000	−0.379473
$11$	12.0000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
$11$	12.0000	0.328921	0.164461	+	0.986384i	$0.447412\pi$
$12$	−12.0000	−0.288675
$13$	38.0000	0.810716	0.405358	−	0.914158i	$-0.367147\pi$
$13$	38.0000	0.810716	0.405358	+	0.914158i	$0.367147\pi$
$14$	32.0000	0.610883
$15$	−18.0000	−0.309839
$16$	16.0000	0.250000
$17$	−126.000	−1.79762	−0.898808	−	0.438342i	$-0.855566\pi$
$17$	−126.000	−1.79762	−0.898808	+	0.438342i	$0.855566\pi$
$18$	−18.0000	−0.235702
$19$	20.0000	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$20$	24.0000	0.268328
$21$	48.0000	0.498784
$22$	−24.0000	−0.232583
$23$	168.000	1.52306	0.761531	−	0.648129i	$-0.224448\pi$
$23$	168.000	1.52306	0.761531	+	0.648129i	$0.224448\pi$
$24$	24.0000	0.204124
$25$	−89.0000	−0.712000
$26$	−76.0000	−0.573263
$27$	−27.0000	−0.192450
$28$	−64.0000	−0.431959
$29$	30.0000	0.192099	0.0960493	−	0.995377i	$-0.469379\pi$
$29$	30.0000	0.192099	0.0960493	+	0.995377i	$0.469379\pi$
$30$	36.0000	0.219089
$31$	−88.0000	−0.509847	−0.254924	−	0.966961i	$-0.582050\pi$
$31$	−88.0000	−0.509847	−0.254924	+	0.966961i	$0.582050\pi$
$32$	−32.0000	−0.176777
$33$	−36.0000	−0.189903
$34$	252.000	1.27111
$35$	−96.0000	−0.463627
$36$	36.0000	0.166667
$37$	254.000	1.12858	0.564288	−	0.825578i	$-0.309151\pi$
$37$	254.000	1.12858	0.564288	+	0.825578i	$0.309151\pi$
$38$	−40.0000	−0.170759
$39$	−114.000	−0.468067
$40$	−48.0000	−0.189737
$41$	42.0000	0.159983	0.0799914	−	0.996796i	$-0.474511\pi$
$41$	42.0000	0.159983	0.0799914	+	0.996796i	$0.474511\pi$
$42$	−96.0000	−0.352693
$43$	−52.0000	−0.184417	−0.0922084	−	0.995740i	$-0.529393\pi$
$43$	−52.0000	−0.184417	−0.0922084	+	0.995740i	$0.529393\pi$
$44$	48.0000	0.164461
$45$	54.0000	0.178885
$46$	−336.000	−1.07697
$47$	−96.0000	−0.297937	−0.148969	−	0.988842i	$-0.547595\pi$
$47$	−96.0000	−0.297937	−0.148969	+	0.988842i	$0.547595\pi$
$48$	−48.0000	−0.144338
$49$	−87.0000	−0.253644
$50$	178.000	0.503460
$51$	378.000	1.03785
$52$	152.000	0.405358
$53$	198.000	0.513158	0.256579	−	0.966523i	$-0.417405\pi$
$53$	198.000	0.513158	0.256579	+	0.966523i	$0.417405\pi$
$54$	54.0000	0.136083
$55$	72.0000	0.176518
$56$	128.000	0.305441
$57$	−60.0000	−0.139424
$58$	−60.0000	−0.135834
$59$	−660.000	−1.45635	−0.728175	−	0.685391i	$-0.759631\pi$
$59$	−660.000	−1.45635	−0.728175	+	0.685391i	$0.759631\pi$
$60$	−72.0000	−0.154919
$61$	−538.000	−1.12924	−0.564622	−	0.825350i	$-0.690978\pi$
$61$	−538.000	−1.12924	−0.564622	+	0.825350i	$0.690978\pi$
$62$	176.000	0.360516
$63$	−144.000	−0.287973
$64$	64.0000	0.125000
$65$	228.000	0.435076
$66$	72.0000	0.134282
$67$	884.000	1.61191	0.805954	−	0.591979i	$-0.201653\pi$
$67$	884.000	1.61191	0.805954	+	0.591979i	$0.201653\pi$
$68$	−504.000	−0.898808
$69$	−504.000	−0.879340
$70$	192.000	0.327834
$71$	792.000	1.32385	0.661923	−	0.749572i	$-0.269740\pi$
$71$	792.000	1.32385	0.661923	+	0.749572i	$0.269740\pi$
$72$	−72.0000	−0.117851
$73$	218.000	0.349520	0.174760	−	0.984611i	$-0.444085\pi$
$73$	218.000	0.349520	0.174760	+	0.984611i	$0.444085\pi$
$74$	−508.000	−0.798024
$75$	267.000	0.411073
$76$	80.0000	0.120745
$77$	−192.000	−0.284161
$78$	228.000	0.330973
$79$	−520.000	−0.740564	−0.370282	−	0.928919i	$-0.620739\pi$
$79$	−520.000	−0.740564	−0.370282	+	0.928919i	$0.620739\pi$
$80$	96.0000	0.134164
$81$	81.0000	0.111111
$82$	−84.0000	−0.113125
$83$	−492.000	−0.650651	−0.325325	−	0.945602i	$-0.605474\pi$
$83$	−492.000	−0.650651	−0.325325	+	0.945602i	$0.605474\pi$
$84$	192.000	0.249392
$85$	−756.000	−0.964703
$86$	104.000	0.130402
$87$	−90.0000	−0.110908
$88$	−96.0000	−0.116291
$89$	810.000	0.964717	0.482359	−	0.875974i	$-0.339780\pi$
$89$	810.000	0.964717	0.482359	+	0.875974i	$0.339780\pi$
$90$	−108.000	−0.126491
$91$	−608.000	−0.700393
$92$	672.000	0.761531
$93$	264.000	0.294360
$94$	192.000	0.210673
$95$	120.000	0.129597
$96$	96.0000	0.102062
$97$	1154.00	1.20795	0.603974	−	0.797004i	$-0.293583\pi$
$97$	1154.00	1.20795	0.603974	+	0.797004i	$0.293583\pi$
$98$	174.000	0.179354
$99$	108.000	0.109640
$100$	−356.000	−0.356000
$101$	−618.000	−0.608845	−0.304422	−	0.952537i	$-0.598463\pi$
$101$	−618.000	−0.608845	−0.304422	+	0.952537i	$0.598463\pi$
$102$	−756.000	−0.733874
$103$	128.000	0.122449	0.0612243	−	0.998124i	$-0.480499\pi$
$103$	128.000	0.122449	0.0612243	+	0.998124i	$0.480499\pi$
$104$	−304.000	−0.286631
$105$	288.000	0.267675
$106$	−396.000	−0.362858
$107$	−1476.00	−1.33355	−0.666777	−	0.745257i	$-0.732327\pi$
$107$	−1476.00	−1.33355	−0.666777	+	0.745257i	$0.732327\pi$
$108$	−108.000	−0.0962250
$109$	1190.00	1.04570	0.522850	−	0.852425i	$-0.324869\pi$
$109$	1190.00	1.04570	0.522850	+	0.852425i	$0.324869\pi$
$110$	−144.000	−0.124817
$111$	−762.000	−0.651584
$112$	−256.000	−0.215980
$113$	−462.000	−0.384613	−0.192307	−	0.981335i	$-0.561597\pi$
$113$	−462.000	−0.384613	−0.192307	+	0.981335i	$0.561597\pi$
$114$	120.000	0.0985880
$115$	1008.00	0.817361
$116$	120.000	0.0960493
$117$	342.000	0.270239
$118$	1320.00	1.02980
$119$	2016.00	1.55300
$120$	144.000	0.109545
$121$	−1187.00	−0.891811
$122$	1076.00	0.798496
$123$	−126.000	−0.0923662
$124$	−352.000	−0.254924
$125$	−1284.00	−0.918756
$126$	288.000	0.203628
$127$	−2536.00	−1.77192	−0.885959	−	0.463763i	$-0.846499\pi$
$127$	−2536.00	−1.77192	−0.885959	+	0.463763i	$0.846499\pi$
$128$	−128.000	−0.0883883
$129$	156.000	0.106473
$130$	−456.000	−0.307645
$131$	2292.00	1.52865	0.764324	−	0.644832i	$-0.223073\pi$
$131$	2292.00	1.52865	0.764324	+	0.644832i	$0.223073\pi$
$132$	−144.000	−0.0949514
$133$	−320.000	−0.208628
$134$	−1768.00	−1.13979
$135$	−162.000	−0.103280
$136$	1008.00	0.635554
$137$	−726.000	−0.452747	−0.226374	−	0.974041i	$-0.572687\pi$
$137$	−726.000	−0.452747	−0.226374	+	0.974041i	$0.572687\pi$
$138$	1008.00	0.621787
$139$	380.000	0.231879	0.115939	−	0.993256i	$-0.463012\pi$
$139$	380.000	0.231879	0.115939	+	0.993256i	$0.463012\pi$
$140$	−384.000	−0.231814
$141$	288.000	0.172014
$142$	−1584.00	−0.936101
$143$	456.000	0.266662
$144$	144.000	0.0833333
$145$	180.000	0.103091
$146$	−436.000	−0.247148
$147$	261.000	0.146442
$148$	1016.00	0.564288
$149$	1590.00	0.874214	0.437107	−	0.899410i	$-0.356003\pi$
$149$	1590.00	0.874214	0.437107	+	0.899410i	$0.356003\pi$
$150$	−534.000	−0.290673
$151$	2432.00	1.31068	0.655342	−	0.755332i	$-0.272524\pi$
$151$	2432.00	1.31068	0.655342	+	0.755332i	$0.272524\pi$
$152$	−160.000	−0.0853797
$153$	−1134.00	−0.599206
$154$	384.000	0.200932
$155$	−528.000	−0.273613
$156$	−456.000	−0.234033
$157$	614.000	0.312118	0.156059	−	0.987748i	$-0.450121\pi$
$157$	614.000	0.312118	0.156059	+	0.987748i	$0.450121\pi$
$158$	1040.00	0.523658
$159$	−594.000	−0.296272
$160$	−192.000	−0.0948683
$161$	−2688.00	−1.31580
$162$	−162.000	−0.0785674
$163$	−1852.00	−0.889938	−0.444969	−	0.895546i	$-0.646785\pi$
$163$	−1852.00	−0.889938	−0.444969	+	0.895546i	$0.646785\pi$
$164$	168.000	0.0799914
$165$	−216.000	−0.101913
$166$	984.000	0.460080
$167$	−2136.00	−0.989752	−0.494876	−	0.868964i	$-0.664787\pi$
$167$	−2136.00	−0.989752	−0.494876	+	0.868964i	$0.664787\pi$
$168$	−384.000	−0.176347
$169$	−753.000	−0.342740
$170$	1512.00	0.682148
$171$	180.000	0.0804967
$172$	−208.000	−0.0922084
$173$	1758.00	0.772591	0.386296	−	0.922375i	$-0.373754\pi$
$173$	1758.00	0.772591	0.386296	+	0.922375i	$0.373754\pi$
$174$	180.000	0.0784239
$175$	1424.00	0.615110
$176$	192.000	0.0822304
$177$	1980.00	0.840824
$178$	−1620.00	−0.682158
$179$	−540.000	−0.225483	−0.112742	−	0.993624i	$-0.535963\pi$
$179$	−540.000	−0.225483	−0.112742	+	0.993624i	$0.535963\pi$
$180$	216.000	0.0894427
$181$	1982.00	0.813928	0.406964	−	0.913444i	$-0.366588\pi$
$181$	1982.00	0.813928	0.406964	+	0.913444i	$0.366588\pi$
$182$	1216.00	0.495252
$183$	1614.00	0.651969
$184$	−1344.00	−0.538484
$185$	1524.00	0.605658
$186$	−528.000	−0.208144
$187$	−1512.00	−0.591275
$188$	−384.000	−0.148969
$189$	432.000	0.166261
$190$	−240.000	−0.0916391
$191$	−2688.00	−1.01831	−0.509154	−	0.860675i	$-0.670042\pi$
$191$	−2688.00	−1.01831	−0.509154	+	0.860675i	$0.670042\pi$
$192$	−192.000	−0.0721688
$193$	−2302.00	−0.858557	−0.429279	−	0.903172i	$-0.641232\pi$
$193$	−2302.00	−0.858557	−0.429279	+	0.903172i	$0.641232\pi$
$194$	−2308.00	−0.854148
$195$	−684.000	−0.251191
$196$	−348.000	−0.126822
$197$	4374.00	1.58190	0.790951	−	0.611880i	$-0.209586\pi$
$197$	4374.00	1.58190	0.790951	+	0.611880i	$0.209586\pi$
$198$	−216.000	−0.0775275
$199$	−1600.00	−0.569955	−0.284977	−	0.958534i	$-0.591986\pi$
$199$	−1600.00	−0.569955	−0.284977	+	0.958534i	$0.591986\pi$
$200$	712.000	0.251730
$201$	−2652.00	−0.930635
$202$	1236.00	0.430518
$203$	−480.000	−0.165958
$204$	1512.00	0.518927
$205$	252.000	0.0858558
$206$	−256.000	−0.0865843
$207$	1512.00	0.507687
$208$	608.000	0.202679
$209$	240.000	0.0794313
$210$	−576.000	−0.189275
$211$	3332.00	1.08713	0.543565	−	0.839367i	$-0.317074\pi$
$211$	3332.00	1.08713	0.543565	+	0.839367i	$0.317074\pi$
$212$	792.000	0.256579
$213$	−2376.00	−0.764323
$214$	2952.00	0.942965
$215$	−312.000	−0.0989685
$216$	216.000	0.0680414
$217$	1408.00	0.440467
$218$	−2380.00	−0.739422
$219$	−654.000	−0.201796
$220$	288.000	0.0882589
$221$	−4788.00	−1.45736
$222$	1524.00	0.460740
$223$	2648.00	0.795171	0.397586	−	0.917565i	$-0.369848\pi$
$223$	2648.00	0.795171	0.397586	+	0.917565i	$0.369848\pi$
$224$	512.000	0.152721
$225$	−801.000	−0.237333
$226$	924.000	0.271963
$227$	2244.00	0.656121	0.328061	−	0.944657i	$-0.393605\pi$
$227$	2244.00	0.656121	0.328061	+	0.944657i	$0.393605\pi$
$228$	−240.000	−0.0697122
$229$	−5650.00	−1.63040	−0.815202	−	0.579177i	$-0.803374\pi$
$229$	−5650.00	−1.63040	−0.815202	+	0.579177i	$0.803374\pi$
$230$	−2016.00	−0.577961
$231$	576.000	0.164061
$232$	−240.000	−0.0679171
$233$	4698.00	1.32093	0.660464	−	0.750858i	$-0.270360\pi$
$233$	4698.00	1.32093	0.660464	+	0.750858i	$0.270360\pi$
$234$	−684.000	−0.191088
$235$	−576.000	−0.159890
$236$	−2640.00	−0.728175
$237$	1560.00	0.427565
$238$	−4032.00	−1.09813
$239$	−1200.00	−0.324776	−0.162388	−	0.986727i	$-0.551920\pi$
$239$	−1200.00	−0.324776	−0.162388	+	0.986727i	$0.551920\pi$
$240$	−288.000	−0.0774597
$241$	−718.000	−0.191911	−0.0959553	−	0.995386i	$-0.530591\pi$
$241$	−718.000	−0.191911	−0.0959553	+	0.995386i	$0.530591\pi$
$242$	2374.00	0.630605
$243$	−243.000	−0.0641500
$244$	−2152.00	−0.564622
$245$	−522.000	−0.136120
$246$	252.000	0.0653127
$247$	760.000	0.195780
$248$	704.000	0.180258
$249$	1476.00	0.375653
$250$	2568.00	0.649658
$251$	6012.00	1.51185	0.755924	−	0.654659i	$-0.227188\pi$
$251$	6012.00	1.51185	0.755924	+	0.654659i	$0.227188\pi$
$252$	−576.000	−0.143986
$253$	2016.00	0.500968
$254$	5072.00	1.25294
$255$	2268.00	0.556971
$256$	256.000	0.0625000
$257$	−2046.00	−0.496599	−0.248300	−	0.968683i	$-0.579872\pi$
$257$	−2046.00	−0.496599	−0.248300	+	0.968683i	$0.579872\pi$
$258$	−312.000	−0.0752879
$259$	−4064.00	−0.974999
$260$	912.000	0.217538
$261$	270.000	0.0640329
$262$	−4584.00	−1.08092
$263$	−6072.00	−1.42363	−0.711817	−	0.702365i	$-0.752127\pi$
$263$	−6072.00	−1.42363	−0.711817	+	0.702365i	$0.752127\pi$
$264$	288.000	0.0671408
$265$	1188.00	0.275390
$266$	640.000	0.147522
$267$	−2430.00	−0.556980
$268$	3536.00	0.805954
$269$	−6930.00	−1.57074	−0.785371	−	0.619025i	$-0.787528\pi$
$269$	−6930.00	−1.57074	−0.785371	+	0.619025i	$0.787528\pi$
$270$	324.000	0.0730297
$271$	1352.00	0.303056	0.151528	−	0.988453i	$-0.451581\pi$
$271$	1352.00	0.303056	0.151528	+	0.988453i	$0.451581\pi$
$272$	−2016.00	−0.449404
$273$	1824.00	0.404372
$274$	1452.00	0.320141
$275$	−1068.00	−0.234192
$276$	−2016.00	−0.439670
$277$	−1186.00	−0.257256	−0.128628	−	0.991693i	$-0.541057\pi$
$277$	−1186.00	−0.257256	−0.128628	+	0.991693i	$0.541057\pi$
$278$	−760.000	−0.163963
$279$	−792.000	−0.169949
$280$	768.000	0.163917
$281$	2442.00	0.518425	0.259213	−	0.965820i	$-0.416537\pi$
$281$	2442.00	0.518425	0.259213	+	0.965820i	$0.416537\pi$
$282$	−576.000	−0.121632
$283$	2828.00	0.594018	0.297009	−	0.954875i	$-0.404011\pi$
$283$	2828.00	0.594018	0.297009	+	0.954875i	$0.404011\pi$
$284$	3168.00	0.661923
$285$	−360.000	−0.0748230
$286$	−912.000	−0.188558
$287$	−672.000	−0.138212
$288$	−288.000	−0.0589256
$289$	10963.0	2.23143
$290$	−360.000	−0.0728963
$291$	−3462.00	−0.697409
$292$	872.000	0.174760
$293$	4758.00	0.948687	0.474344	−	0.880340i	$-0.342685\pi$
$293$	4758.00	0.948687	0.474344	+	0.880340i	$0.342685\pi$
$294$	−522.000	−0.103550
$295$	−3960.00	−0.781560
$296$	−2032.00	−0.399012
$297$	−324.000	−0.0633010
$298$	−3180.00	−0.618163
$299$	6384.00	1.23477
$300$	1068.00	0.205537
$301$	832.000	0.159321
$302$	−4864.00	−0.926794
$303$	1854.00	0.351517
$304$	320.000	0.0603726
$305$	−3228.00	−0.606016
$306$	2268.00	0.423702
$307$	−8476.00	−1.57574	−0.787868	−	0.615844i	$-0.788815\pi$
$307$	−8476.00	−1.57574	−0.787868	+	0.615844i	$0.788815\pi$
$308$	−768.000	−0.142081
$309$	−384.000	−0.0706958
$310$	1056.00	0.193473
$311$	4632.00	0.844555	0.422278	−	0.906467i	$-0.361231\pi$
$311$	4632.00	0.844555	0.422278	+	0.906467i	$0.361231\pi$
$312$	912.000	0.165487
$313$	−4822.00	−0.870785	−0.435392	−	0.900241i	$-0.643390\pi$
$313$	−4822.00	−0.870785	−0.435392	+	0.900241i	$0.643390\pi$
$314$	−1228.00	−0.220701
$315$	−864.000	−0.154542
$316$	−2080.00	−0.370282
$317$	−3426.00	−0.607014	−0.303507	−	0.952829i	$-0.598158\pi$
$317$	−3426.00	−0.607014	−0.303507	+	0.952829i	$0.598158\pi$
$318$	1188.00	0.209496
$319$	360.000	0.0631854
$320$	384.000	0.0670820
$321$	4428.00	0.769928
$322$	5376.00	0.930412
$323$	−2520.00	−0.434107
$324$	324.000	0.0555556
$325$	−3382.00	−0.577230
$326$	3704.00	0.629281
$327$	−3570.00	−0.603735
$328$	−336.000	−0.0565625
$329$	1536.00	0.257393
$330$	432.000	0.0720631
$331$	−2788.00	−0.462968	−0.231484	−	0.972839i	$-0.574358\pi$
$331$	−2788.00	−0.462968	−0.231484	+	0.972839i	$0.574358\pi$
$332$	−1968.00	−0.325325
$333$	2286.00	0.376192
$334$	4272.00	0.699861
$335$	5304.00	0.865040
$336$	768.000	0.124696
$337$	434.000	0.0701528	0.0350764	−	0.999385i	$-0.488833\pi$
$337$	434.000	0.0701528	0.0350764	+	0.999385i	$0.488833\pi$
$338$	1506.00	0.242354
$339$	1386.00	0.222057
$340$	−3024.00	−0.482351
$341$	−1056.00	−0.167700
$342$	−360.000	−0.0569198
$343$	6880.00	1.08305
$344$	416.000	0.0652012
$345$	−3024.00	−0.471903
$346$	−3516.00	−0.546304
$347$	6684.00	1.03405	0.517026	−	0.855970i	$-0.327039\pi$
$347$	6684.00	1.03405	0.517026	+	0.855970i	$0.327039\pi$
$348$	−360.000	−0.0554541
$349$	2630.00	0.403383	0.201692	−	0.979449i	$-0.435356\pi$
$349$	2630.00	0.403383	0.201692	+	0.979449i	$0.435356\pi$
$350$	−2848.00	−0.434949
$351$	−1026.00	−0.156022
$352$	−384.000	−0.0581456
$353$	−7422.00	−1.11907	−0.559537	−	0.828805i	$-0.689021\pi$
$353$	−7422.00	−1.11907	−0.559537	+	0.828805i	$0.689021\pi$
$354$	−3960.00	−0.594553
$355$	4752.00	0.710451
$356$	3240.00	0.482359
$357$	−6048.00	−0.896622
$358$	1080.00	0.159441
$359$	−10440.0	−1.53482	−0.767412	−	0.641154i	$-0.778456\pi$
$359$	−10440.0	−1.53482	−0.767412	+	0.641154i	$0.778456\pi$
$360$	−432.000	−0.0632456
$361$	−6459.00	−0.941682
$362$	−3964.00	−0.575534
$363$	3561.00	0.514887
$364$	−2432.00	−0.350196
$365$	1308.00	0.187572
$366$	−3228.00	−0.461012
$367$	10424.0	1.48264	0.741319	−	0.671153i	$-0.234200\pi$
$367$	10424.0	1.48264	0.741319	+	0.671153i	$0.234200\pi$
$368$	2688.00	0.380765
$369$	378.000	0.0533276
$370$	−3048.00	−0.428265
$371$	−3168.00	−0.443327
$372$	1056.00	0.147180
$373$	3278.00	0.455036	0.227518	−	0.973774i	$-0.426939\pi$
$373$	3278.00	0.455036	0.227518	+	0.973774i	$0.426939\pi$
$374$	3024.00	0.418094
$375$	3852.00	0.530444
$376$	768.000	0.105337
$377$	1140.00	0.155737
$378$	−864.000	−0.117564
$379$	6140.00	0.832165	0.416083	−	0.909327i	$-0.363403\pi$
$379$	6140.00	0.832165	0.416083	+	0.909327i	$0.363403\pi$
$380$	480.000	0.0647986
$381$	7608.00	1.02302
$382$	5376.00	0.720053
$383$	−3072.00	−0.409848	−0.204924	−	0.978778i	$-0.565695\pi$
$383$	−3072.00	−0.409848	−0.204924	+	0.978778i	$0.565695\pi$
$384$	384.000	0.0510310
$385$	−1152.00	−0.152497
$386$	4604.00	0.607092
$387$	−468.000	−0.0614723
$388$	4616.00	0.603974
$389$	6150.00	0.801587	0.400794	−	0.916168i	$-0.368734\pi$
$389$	6150.00	0.801587	0.400794	+	0.916168i	$0.368734\pi$
$390$	1368.00	0.177619
$391$	−21168.0	−2.73788
$392$	696.000	0.0896768
$393$	−6876.00	−0.882566
$394$	−8748.00	−1.11857
$395$	−3120.00	−0.397428
$396$	432.000	0.0548202
$397$	−106.000	−0.0134005	−0.00670024	−	0.999978i	$-0.502133\pi$
$397$	−106.000	−0.0134005	−0.00670024	+	0.999978i	$0.502133\pi$
$398$	3200.00	0.403019
$399$	960.000	0.120451
$400$	−1424.00	−0.178000
$401$	−1758.00	−0.218929	−0.109464	−	0.993991i	$-0.534914\pi$
$401$	−1758.00	−0.218929	−0.109464	+	0.993991i	$0.534914\pi$
$402$	5304.00	0.658058
$403$	−3344.00	−0.413341
$404$	−2472.00	−0.304422
$405$	486.000	0.0596285
$406$	960.000	0.117350
$407$	3048.00	0.371213
$408$	−3024.00	−0.366937
$409$	−3670.00	−0.443691	−0.221846	−	0.975082i	$-0.571208\pi$
$409$	−3670.00	−0.443691	−0.221846	+	0.975082i	$0.571208\pi$
$410$	−504.000	−0.0607092
$411$	2178.00	0.261394
$412$	512.000	0.0612243
$413$	10560.0	1.25817
$414$	−3024.00	−0.358989
$415$	−2952.00	−0.349176
$416$	−1216.00	−0.143316
$417$	−1140.00	−0.133875
$418$	−480.000	−0.0561664
$419$	−9660.00	−1.12631	−0.563153	−	0.826353i	$-0.690412\pi$
$419$	−9660.00	−1.12631	−0.563153	+	0.826353i	$0.690412\pi$
$420$	1152.00	0.133838
$421$	8462.00	0.979602	0.489801	−	0.871834i	$-0.337069\pi$
$421$	8462.00	0.979602	0.489801	+	0.871834i	$0.337069\pi$
$422$	−6664.00	−0.768717
$423$	−864.000	−0.0993123
$424$	−1584.00	−0.181429
$425$	11214.0	1.27990
$426$	4752.00	0.540458
$427$	8608.00	0.975575
$428$	−5904.00	−0.666777
$429$	−1368.00	−0.153957
$430$	624.000	0.0699813
$431$	9792.00	1.09435	0.547174	−	0.837019i	$-0.315704\pi$
$431$	9792.00	1.09435	0.547174	+	0.837019i	$0.315704\pi$
$432$	−432.000	−0.0481125
$433$	−7342.00	−0.814859	−0.407430	−	0.913237i	$-0.633575\pi$
$433$	−7342.00	−0.814859	−0.407430	+	0.913237i	$0.633575\pi$
$434$	−2816.00	−0.311457
$435$	−540.000	−0.0595196
$436$	4760.00	0.522850
$437$	3360.00	0.367805
$438$	1308.00	0.142691
$439$	10640.0	1.15676	0.578382	−	0.815766i	$-0.303684\pi$
$439$	10640.0	1.15676	0.578382	+	0.815766i	$0.303684\pi$
$440$	−576.000	−0.0624085
$441$	−783.000	−0.0845481
$442$	9576.00	1.03051
$443$	−17412.0	−1.86742	−0.933712	−	0.358024i	$-0.883451\pi$
$443$	−17412.0	−1.86742	−0.933712	+	0.358024i	$0.883451\pi$
$444$	−3048.00	−0.325792
$445$	4860.00	0.517722
$446$	−5296.00	−0.562271
$447$	−4770.00	−0.504728
$448$	−1024.00	−0.107990
$449$	−1710.00	−0.179732	−0.0898662	−	0.995954i	$-0.528644\pi$
$449$	−1710.00	−0.179732	−0.0898662	+	0.995954i	$0.528644\pi$
$450$	1602.00	0.167820
$451$	504.000	0.0526218
$452$	−1848.00	−0.192307
$453$	−7296.00	−0.756724
$454$	−4488.00	−0.463948
$455$	−3648.00	−0.375870
$456$	480.000	0.0492940
$457$	−646.000	−0.0661239	−0.0330619	−	0.999453i	$-0.510526\pi$
$457$	−646.000	−0.0661239	−0.0330619	+	0.999453i	$0.510526\pi$
$458$	11300.0	1.15287
$459$	3402.00	0.345952
$460$	4032.00	0.408680
$461$	−6018.00	−0.607996	−0.303998	−	0.952673i	$-0.598322\pi$
$461$	−6018.00	−0.607996	−0.303998	+	0.952673i	$0.598322\pi$
$462$	−1152.00	−0.116008
$463$	−6712.00	−0.673722	−0.336861	−	0.941554i	$-0.609365\pi$
$463$	−6712.00	−0.673722	−0.336861	+	0.941554i	$0.609365\pi$
$464$	480.000	0.0480247
$465$	1584.00	0.157970
$466$	−9396.00	−0.934037
$467$	5364.00	0.531512	0.265756	−	0.964040i	$-0.414378\pi$
$467$	5364.00	0.531512	0.265756	+	0.964040i	$0.414378\pi$
$468$	1368.00	0.135119
$469$	−14144.0	−1.39256
$470$	1152.00	0.113059
$471$	−1842.00	−0.180201
$472$	5280.00	0.514898
$473$	−624.000	−0.0606587
$474$	−3120.00	−0.302334
$475$	−1780.00	−0.171941
$476$	8064.00	0.776498
$477$	1782.00	0.171053
$478$	2400.00	0.229652
$479$	9840.00	0.938624	0.469312	−	0.883032i	$-0.344502\pi$
$479$	9840.00	0.938624	0.469312	+	0.883032i	$0.344502\pi$
$480$	576.000	0.0547723
$481$	9652.00	0.914955
$482$	1436.00	0.135701
$483$	8064.00	0.759678
$484$	−4748.00	−0.445905
$485$	6924.00	0.648253
$486$	486.000	0.0453609
$487$	1424.00	0.132500	0.0662501	−	0.997803i	$-0.478896\pi$
$487$	1424.00	0.132500	0.0662501	+	0.997803i	$0.478896\pi$
$488$	4304.00	0.399248
$489$	5556.00	0.513806
$490$	1044.00	0.0962513
$491$	−4548.00	−0.418021	−0.209011	−	0.977913i	$-0.567024\pi$
$491$	−4548.00	−0.418021	−0.209011	+	0.977913i	$0.567024\pi$
$492$	−504.000	−0.0461831
$493$	−3780.00	−0.345320
$494$	−1520.00	−0.138437
$495$	648.000	0.0588393
$496$	−1408.00	−0.127462
$497$	−12672.0	−1.14370
$498$	−2952.00	−0.265627
$499$	6500.00	0.583126	0.291563	−	0.956552i	$-0.405825\pi$
$499$	6500.00	0.583126	0.291563	+	0.956552i	$0.405825\pi$
$500$	−5136.00	−0.459378
$501$	6408.00	0.571434
$502$	−12024.0	−1.06904
$503$	12168.0	1.07862	0.539308	−	0.842108i	$-0.318686\pi$
$503$	12168.0	1.07862	0.539308	+	0.842108i	$0.318686\pi$
$504$	1152.00	0.101814
$505$	−3708.00	−0.326740
$506$	−4032.00	−0.354238
$507$	2259.00	0.197881
$508$	−10144.0	−0.885959
$509$	−21090.0	−1.83654	−0.918269	−	0.395957i	$-0.870413\pi$
$509$	−21090.0	−1.83654	−0.918269	+	0.395957i	$0.870413\pi$
$510$	−4536.00	−0.393838
$511$	−3488.00	−0.301957
$512$	−512.000	−0.0441942
$513$	−540.000	−0.0464748
$514$	4092.00	0.351149
$515$	768.000	0.0657129
$516$	624.000	0.0532366
$517$	−1152.00	−0.0979979
$518$	8128.00	0.689428
$519$	−5274.00	−0.446056
$520$	−1824.00	−0.153822
$521$	−5238.00	−0.440462	−0.220231	−	0.975448i	$-0.570681\pi$
$521$	−5238.00	−0.440462	−0.220231	+	0.975448i	$0.570681\pi$
$522$	−540.000	−0.0452781
$523$	8588.00	0.718025	0.359012	−	0.933333i	$-0.383114\pi$
$523$	8588.00	0.718025	0.359012	+	0.933333i	$0.383114\pi$
$524$	9168.00	0.764324
$525$	−4272.00	−0.355134
$526$	12144.0	1.00666
$527$	11088.0	0.916510
$528$	−576.000	−0.0474757
$529$	16057.0	1.31972
$530$	−2376.00	−0.194730
$531$	−5940.00	−0.485450
$532$	−1280.00	−0.104314
$533$	1596.00	0.129701
$534$	4860.00	0.393844
$535$	−8856.00	−0.715660
$536$	−7072.00	−0.569895
$537$	1620.00	0.130183
$538$	13860.0	1.11068
$539$	−1044.00	−0.0834291
$540$	−648.000	−0.0516398
$541$	3062.00	0.243338	0.121669	−	0.992571i	$-0.461175\pi$
$541$	3062.00	0.243338	0.121669	+	0.992571i	$0.461175\pi$
$542$	−2704.00	−0.214293
$543$	−5946.00	−0.469921
$544$	4032.00	0.317777
$545$	7140.00	0.561182
$546$	−3648.00	−0.285934
$547$	−8476.00	−0.662537	−0.331268	−	0.943537i	$-0.607477\pi$
$547$	−8476.00	−0.662537	−0.331268	+	0.943537i	$0.607477\pi$
$548$	−2904.00	−0.226374
$549$	−4842.00	−0.376414
$550$	2136.00	0.165599
$551$	600.000	0.0463899
$552$	4032.00	0.310894
$553$	8320.00	0.639787
$554$	2372.00	0.181907
$555$	−4572.00	−0.349677
$556$	1520.00	0.115939
$557$	−12546.0	−0.954383	−0.477191	−	0.878799i	$-0.658345\pi$
$557$	−12546.0	−0.954383	−0.477191	+	0.878799i	$0.658345\pi$
$558$	1584.00	0.120172
$559$	−1976.00	−0.149510
$560$	−1536.00	−0.115907
$561$	4536.00	0.341373
$562$	−4884.00	−0.366582
$563$	−12.0000	−0.000898294 0	−0.000449147	−	1.00000i	$-0.500143\pi$
$563$	−12.0000	−0.000898294 0	−0.000449147		1.00000i	$0.500143\pi$
$564$	1152.00	0.0860070
$565$	−2772.00	−0.206405
$566$	−5656.00	−0.420034
$567$	−1296.00	−0.0959910
$568$	−6336.00	−0.468050
$569$	19290.0	1.42123	0.710614	−	0.703582i	$-0.248417\pi$
$569$	19290.0	1.42123	0.710614	+	0.703582i	$0.248417\pi$
$570$	720.000	0.0529079
$571$	−12148.0	−0.890329	−0.445165	−	0.895449i	$-0.646855\pi$
$571$	−12148.0	−0.890329	−0.445165	+	0.895449i	$0.646855\pi$
$572$	1824.00	0.133331
$573$	8064.00	0.587920
$574$	1344.00	0.0977308
$575$	−14952.0	−1.08442
$576$	576.000	0.0416667
$577$	−10366.0	−0.747907	−0.373953	−	0.927447i	$-0.621998\pi$
$577$	−10366.0	−0.747907	−0.373953	+	0.927447i	$0.621998\pi$
$578$	−21926.0	−1.57786
$579$	6906.00	0.495688
$580$	720.000	0.0515455
$581$	7872.00	0.562109
$582$	6924.00	0.493143
$583$	2376.00	0.168789
$584$	−1744.00	−0.123574
$585$	2052.00	0.145025
$586$	−9516.00	−0.670823
$587$	7644.00	0.537482	0.268741	−	0.963213i	$-0.413393\pi$
$587$	7644.00	0.537482	0.268741	+	0.963213i	$0.413393\pi$
$588$	1044.00	0.0732208
$589$	−1760.00	−0.123123
$590$	7920.00	0.552646
$591$	−13122.0	−0.913311
$592$	4064.00	0.282144
$593$	8658.00	0.599564	0.299782	−	0.954008i	$-0.403086\pi$
$593$	8658.00	0.599564	0.299782	+	0.954008i	$0.403086\pi$
$594$	648.000	0.0447605
$595$	12096.0	0.833425
$596$	6360.00	0.437107
$597$	4800.00	0.329064
$598$	−12768.0	−0.873114
$599$	25800.0	1.75987	0.879933	−	0.475098i	$-0.157587\pi$
$599$	25800.0	1.75987	0.879933	+	0.475098i	$0.157587\pi$
$600$	−2136.00	−0.145336
$601$	16202.0	1.09966	0.549828	−	0.835278i	$-0.314693\pi$
$601$	16202.0	1.09966	0.549828	+	0.835278i	$0.314693\pi$
$602$	−1664.00	−0.112657
$603$	7956.00	0.537302
$604$	9728.00	0.655342
$605$	−7122.00	−0.478596
$606$	−3708.00	−0.248560
$607$	−24136.0	−1.61392	−0.806960	−	0.590605i	$-0.798889\pi$
$607$	−24136.0	−1.61392	−0.806960	+	0.590605i	$0.798889\pi$
$608$	−640.000	−0.0426898
$609$	1440.00	0.0958157
$610$	6456.00	0.428518
$611$	−3648.00	−0.241542
$612$	−4536.00	−0.299603
$613$	−4642.00	−0.305854	−0.152927	−	0.988237i	$-0.548870\pi$
$613$	−4642.00	−0.305854	−0.152927	+	0.988237i	$0.548870\pi$
$614$	16952.0	1.11421
$615$	−756.000	−0.0495689
$616$	1536.00	0.100466
$617$	−6726.00	−0.438863	−0.219432	−	0.975628i	$-0.570420\pi$
$617$	−6726.00	−0.438863	−0.219432	+	0.975628i	$0.570420\pi$
$618$	768.000	0.0499895
$619$	−21220.0	−1.37787	−0.688937	−	0.724821i	$-0.741922\pi$
$619$	−21220.0	−1.37787	−0.688937	+	0.724821i	$0.741922\pi$
$620$	−2112.00	−0.136806
$621$	−4536.00	−0.293113
$622$	−9264.00	−0.597191
$623$	−12960.0	−0.833437
$624$	−1824.00	−0.117017
$625$	3421.00	0.218944
$626$	9644.00	0.615738
$627$	−720.000	−0.0458597
$628$	2456.00	0.156059
$629$	−32004.0	−2.02875
$630$	1728.00	0.109278
$631$	29792.0	1.87956	0.939779	−	0.341783i	$-0.111031\pi$
$631$	29792.0	1.87956	0.939779	+	0.341783i	$0.111031\pi$
$632$	4160.00	0.261829
$633$	−9996.00	−0.627655
$634$	6852.00	0.429223
$635$	−15216.0	−0.950911
$636$	−2376.00	−0.148136
$637$	−3306.00	−0.205633
$638$	−720.000	−0.0446788
$639$	7128.00	0.441282
$640$	−768.000	−0.0474342
$641$	−10158.0	−0.625923	−0.312962	−	0.949766i	$-0.601321\pi$
$641$	−10158.0	−0.625923	−0.312962	+	0.949766i	$0.601321\pi$
$642$	−8856.00	−0.544421
$643$	29828.0	1.82940	0.914698	−	0.404138i	$-0.132429\pi$
$643$	29828.0	1.82940	0.914698	+	0.404138i	$0.132429\pi$
$644$	−10752.0	−0.657901
$645$	936.000	0.0571395
$646$	5040.00	0.306960
$647$	1944.00	0.118124	0.0590622	−	0.998254i	$-0.481189\pi$
$647$	1944.00	0.118124	0.0590622	+	0.998254i	$0.481189\pi$
$648$	−648.000	−0.0392837
$649$	−7920.00	−0.479025
$650$	6764.00	0.408163
$651$	−4224.00	−0.254304
$652$	−7408.00	−0.444969
$653$	26718.0	1.60116	0.800579	−	0.599227i	$-0.204525\pi$
$653$	26718.0	1.60116	0.800579	+	0.599227i	$0.204525\pi$
$654$	7140.00	0.426905
$655$	13752.0	0.820359
$656$	672.000	0.0399957
$657$	1962.00	0.116507
$658$	−3072.00	−0.182005
$659$	4260.00	0.251815	0.125907	−	0.992042i	$-0.459816\pi$
$659$	4260.00	0.251815	0.125907	+	0.992042i	$0.459816\pi$
$660$	−864.000	−0.0509563
$661$	22862.0	1.34528	0.672639	−	0.739971i	$-0.265161\pi$
$661$	22862.0	1.34528	0.672639	+	0.739971i	$0.265161\pi$
$662$	5576.00	0.327368
$663$	14364.0	0.841405
$664$	3936.00	0.230040
$665$	−1920.00	−0.111962
$666$	−4572.00	−0.266008
$667$	5040.00	0.292578
$668$	−8544.00	−0.494876
$669$	−7944.00	−0.459092
$670$	−10608.0	−0.611676
$671$	−6456.00	−0.371432
$672$	−1536.00	−0.0881733
$673$	−32542.0	−1.86390	−0.931948	−	0.362592i	$-0.881892\pi$
$673$	−32542.0	−1.86390	−0.931948	+	0.362592i	$0.881892\pi$
$674$	−868.000	−0.0496055
$675$	2403.00	0.137024
$676$	−3012.00	−0.171370
$677$	14214.0	0.806925	0.403463	−	0.914996i	$-0.367807\pi$
$677$	14214.0	0.806925	0.403463	+	0.914996i	$0.367807\pi$
$678$	−2772.00	−0.157018
$679$	−18464.0	−1.04357
$680$	6048.00	0.341074
$681$	−6732.00	−0.378812
$682$	2112.00	0.118582
$683$	−7092.00	−0.397317	−0.198659	−	0.980069i	$-0.563659\pi$
$683$	−7092.00	−0.397317	−0.198659	+	0.980069i	$0.563659\pi$
$684$	720.000	0.0402484
$685$	−4356.00	−0.242970
$686$	−13760.0	−0.765830
$687$	16950.0	0.941314
$688$	−832.000	−0.0461042
$689$	7524.00	0.416026
$690$	6048.00	0.333686
$691$	−13228.0	−0.728244	−0.364122	−	0.931351i	$-0.618631\pi$
$691$	−13228.0	−0.728244	−0.364122	+	0.931351i	$0.618631\pi$
$692$	7032.00	0.386296
$693$	−1728.00	−0.0947205
$694$	−13368.0	−0.731185
$695$	2280.00	0.124439
$696$	720.000	0.0392120
$697$	−5292.00	−0.287588
$698$	−5260.00	−0.285235
$699$	−14094.0	−0.762638
$700$	5696.00	0.307555
$701$	28062.0	1.51196	0.755982	−	0.654592i	$-0.227160\pi$
$701$	28062.0	1.51196	0.755982	+	0.654592i	$0.227160\pi$
$702$	2052.00	0.110324
$703$	5080.00	0.272540
$704$	768.000	0.0411152
$705$	1728.00	0.0923124
$706$	14844.0	0.791305
$707$	9888.00	0.525992
$708$	7920.00	0.420412
$709$	−27250.0	−1.44343	−0.721717	−	0.692188i	$-0.756647\pi$
$709$	−27250.0	−1.44343	−0.721717	+	0.692188i	$0.756647\pi$
$710$	−9504.00	−0.502364
$711$	−4680.00	−0.246855
$712$	−6480.00	−0.341079
$713$	−14784.0	−0.776529
$714$	12096.0	0.634008
$715$	2736.00	0.143106
$716$	−2160.00	−0.112742
$717$	3600.00	0.187510
$718$	20880.0	1.08529
$719$	−14400.0	−0.746912	−0.373456	−	0.927648i	$-0.621827\pi$
$719$	−14400.0	−0.746912	−0.373456	+	0.927648i	$0.621827\pi$
$720$	864.000	0.0447214
$721$	−2048.00	−0.105786
$722$	12918.0	0.665870
$723$	2154.00	0.110800
$724$	7928.00	0.406964
$725$	−2670.00	−0.136774
$726$	−7122.00	−0.364080
$727$	17984.0	0.917455	0.458727	−	0.888577i	$-0.348305\pi$
$727$	17984.0	0.917455	0.458727	+	0.888577i	$0.348305\pi$
$728$	4864.00	0.247626
$729$	729.000	0.0370370
$730$	−2616.00	−0.132634
$731$	6552.00	0.331511
$732$	6456.00	0.325984
$733$	16598.0	0.836373	0.418186	−	0.908361i	$-0.362666\pi$
$733$	16598.0	0.836373	0.418186	+	0.908361i	$0.362666\pi$
$734$	−20848.0	−1.04838
$735$	1566.00	0.0785888
$736$	−5376.00	−0.269242
$737$	10608.0	0.530191
$738$	−756.000	−0.0377083
$739$	1460.00	0.0726752	0.0363376	−	0.999340i	$-0.488431\pi$
$739$	1460.00	0.0726752	0.0363376	+	0.999340i	$0.488431\pi$
$740$	6096.00	0.302829
$741$	−2280.00	−0.113034
$742$	6336.00	0.313480
$743$	−30072.0	−1.48484	−0.742419	−	0.669936i	$-0.766322\pi$
$743$	−30072.0	−1.48484	−0.742419	+	0.669936i	$0.766322\pi$
$744$	−2112.00	−0.104072
$745$	9540.00	0.469152
$746$	−6556.00	−0.321759
$747$	−4428.00	−0.216884
$748$	−6048.00	−0.295637
$749$	23616.0	1.15208
$750$	−7704.00	−0.375080
$751$	−18088.0	−0.878882	−0.439441	−	0.898271i	$-0.644823\pi$
$751$	−18088.0	−0.878882	−0.439441	+	0.898271i	$0.644823\pi$
$752$	−1536.00	−0.0744843
$753$	−18036.0	−0.872866
$754$	−2280.00	−0.110123
$755$	14592.0	0.703387
$756$	1728.00	0.0831306
$757$	24734.0	1.18755	0.593773	−	0.804633i	$-0.297638\pi$
$757$	24734.0	1.18755	0.593773	+	0.804633i	$0.297638\pi$
$758$	−12280.0	−0.588430
$759$	−6048.00	−0.289234
$760$	−960.000	−0.0458196
$761$	−22278.0	−1.06120	−0.530602	−	0.847621i	$-0.678034\pi$
$761$	−22278.0	−1.06120	−0.530602	+	0.847621i	$0.678034\pi$
$762$	−15216.0	−0.723383
$763$	−19040.0	−0.903400
$764$	−10752.0	−0.509154
$765$	−6804.00	−0.321568
$766$	6144.00	0.289806
$767$	−25080.0	−1.18069
$768$	−768.000	−0.0360844
$769$	16130.0	0.756388	0.378194	−	0.925726i	$-0.376545\pi$
$769$	16130.0	0.756388	0.378194	+	0.925726i	$0.376545\pi$
$770$	2304.00	0.107832
$771$	6138.00	0.286712
$772$	−9208.00	−0.429279
$773$	29718.0	1.38277	0.691386	−	0.722486i	$-0.257001\pi$
$773$	29718.0	1.38277	0.691386	+	0.722486i	$0.257001\pi$
$774$	936.000	0.0434675
$775$	7832.00	0.363011
$776$	−9232.00	−0.427074
$777$	12192.0	0.562916
$778$	−12300.0	−0.566808
$779$	840.000	0.0386343
$780$	−2736.00	−0.125596
$781$	9504.00	0.435442
$782$	42336.0	1.93597
$783$	−810.000	−0.0369694
$784$	−1392.00	−0.0634111
$785$	3684.00	0.167500
$786$	13752.0	0.624068
$787$	9524.00	0.431377	0.215689	−	0.976462i	$-0.430800\pi$
$787$	9524.00	0.431377	0.215689	+	0.976462i	$0.430800\pi$
$788$	17496.0	0.790951
$789$	18216.0	0.821935
$790$	6240.00	0.281024
$791$	7392.00	0.332275
$792$	−864.000	−0.0387638
$793$	−20444.0	−0.915495
$794$	212.000	0.00947556
$795$	−3564.00	−0.158996
$796$	−6400.00	−0.284977
$797$	−33906.0	−1.50692	−0.753458	−	0.657496i	$-0.771616\pi$
$797$	−33906.0	−1.50692	−0.753458	+	0.657496i	$0.771616\pi$
$798$	−1920.00	−0.0851720
$799$	12096.0	0.535577
$800$	2848.00	0.125865
$801$	7290.00	0.321572
$802$	3516.00	0.154806
$803$	2616.00	0.114965
$804$	−10608.0	−0.465318
$805$	−16128.0	−0.706133
$806$	6688.00	0.292276
$807$	20790.0	0.906868
$808$	4944.00	0.215259
$809$	−630.000	−0.0273790	−0.0136895	−	0.999906i	$-0.504358\pi$
$809$	−630.000	−0.0273790	−0.0136895	+	0.999906i	$0.504358\pi$
$810$	−972.000	−0.0421637
$811$	−20788.0	−0.900081	−0.450040	−	0.893008i	$-0.648590\pi$
$811$	−20788.0	−0.900081	−0.450040	+	0.893008i	$0.648590\pi$
$812$	−1920.00	−0.0829788
$813$	−4056.00	−0.174969
$814$	−6096.00	−0.262487
$815$	−11112.0	−0.477591
$816$	6048.00	0.259464
$817$	−1040.00	−0.0445349
$818$	7340.00	0.313737
$819$	−5472.00	−0.233464
$820$	1008.00	0.0429279
$821$	−43098.0	−1.83207	−0.916036	−	0.401097i	$-0.868629\pi$
$821$	−43098.0	−1.83207	−0.916036	+	0.401097i	$0.868629\pi$
$822$	−4356.00	−0.184833
$823$	−14272.0	−0.604484	−0.302242	−	0.953231i	$-0.597735\pi$
$823$	−14272.0	−0.604484	−0.302242	+	0.953231i	$0.597735\pi$
$824$	−1024.00	−0.0432921
$825$	3204.00	0.135211
$826$	−21120.0	−0.889660
$827$	13644.0	0.573698	0.286849	−	0.957976i	$-0.407392\pi$
$827$	13644.0	0.573698	0.286849	+	0.957976i	$0.407392\pi$
$828$	6048.00	0.253844
$829$	−2410.00	−0.100968	−0.0504842	−	0.998725i	$-0.516076\pi$
$829$	−2410.00	−0.100968	−0.0504842	+	0.998725i	$0.516076\pi$
$830$	5904.00	0.246905
$831$	3558.00	0.148527
$832$	2432.00	0.101339
$833$	10962.0	0.455955
$834$	2280.00	0.0946642
$835$	−12816.0	−0.531157
$836$	960.000	0.0397157
$837$	2376.00	0.0981202
$838$	19320.0	0.796418
$839$	23160.0	0.953006	0.476503	−	0.879173i	$-0.341904\pi$
$839$	23160.0	0.953006	0.476503	+	0.879173i	$0.341904\pi$
$840$	−2304.00	−0.0946376
$841$	−23489.0	−0.963098
$842$	−16924.0	−0.692684
$843$	−7326.00	−0.299313
$844$	13328.0	0.543565
$845$	−4518.00	−0.183934
$846$	1728.00	0.0702244
$847$	18992.0	0.770452
$848$	3168.00	0.128290
$849$	−8484.00	−0.342957
$850$	−22428.0	−0.905028
$851$	42672.0	1.71889
$852$	−9504.00	−0.382162
$853$	32078.0	1.28761	0.643804	−	0.765190i	$-0.277355\pi$
$853$	32078.0	1.28761	0.643804	+	0.765190i	$0.277355\pi$
$854$	−17216.0	−0.689835
$855$	1080.00	0.0431991
$856$	11808.0	0.471483
$857$	−14406.0	−0.574212	−0.287106	−	0.957899i	$-0.592693\pi$
$857$	−14406.0	−0.574212	−0.287106	+	0.957899i	$0.592693\pi$
$858$	2736.00	0.108864
$859$	30620.0	1.21623	0.608115	−	0.793849i	$-0.291926\pi$
$859$	30620.0	1.21623	0.608115	+	0.793849i	$0.291926\pi$
$860$	−1248.00	−0.0494842
$861$	2016.00	0.0797969
$862$	−19584.0	−0.773821
$863$	17568.0	0.692957	0.346478	−	0.938058i	$-0.387377\pi$
$863$	17568.0	0.692957	0.346478	+	0.938058i	$0.387377\pi$
$864$	864.000	0.0340207
$865$	10548.0	0.414616
$866$	14684.0	0.576192
$867$	−32889.0	−1.28831
$868$	5632.00	0.220233
$869$	−6240.00	−0.243587
$870$	1080.00	0.0420867
$871$	33592.0	1.30680
$872$	−9520.00	−0.369711
$873$	10386.0	0.402649
$874$	−6720.00	−0.260077
$875$	20544.0	0.793730
$876$	−2616.00	−0.100898
$877$	−21706.0	−0.835758	−0.417879	−	0.908503i	$-0.637226\pi$
$877$	−21706.0	−0.835758	−0.417879	+	0.908503i	$0.637226\pi$
$878$	−21280.0	−0.817956
$879$	−14274.0	−0.547725
$880$	1152.00	0.0441294
$881$	−14958.0	−0.572018	−0.286009	−	0.958227i	$-0.592329\pi$
$881$	−14958.0	−0.572018	−0.286009	+	0.958227i	$0.592329\pi$
$882$	1566.00	0.0597845
$883$	−32812.0	−1.25052	−0.625261	−	0.780415i	$-0.715008\pi$
$883$	−32812.0	−1.25052	−0.625261	+	0.780415i	$0.715008\pi$
$884$	−19152.0	−0.728678
$885$	11880.0	0.451234
$886$	34824.0	1.32047
$887$	−38856.0	−1.47086	−0.735432	−	0.677598i	$-0.763021\pi$
$887$	−38856.0	−1.47086	−0.735432	+	0.677598i	$0.763021\pi$
$888$	6096.00	0.230370
$889$	40576.0	1.53079
$890$	−9720.00	−0.366084
$891$	972.000	0.0365468
$892$	10592.0	0.397586
$893$	−1920.00	−0.0719489
$894$	9540.00	0.356896
$895$	−3240.00	−0.121007
$896$	2048.00	0.0763604
$897$	−19152.0	−0.712895
$898$	3420.00	0.127090
$899$	−2640.00	−0.0979410
$900$	−3204.00	−0.118667
$901$	−24948.0	−0.922462
$902$	−1008.00	−0.0372092
$903$	−2496.00	−0.0919841
$904$	3696.00	0.135981
$905$	11892.0	0.436799
$906$	14592.0	0.535085
$907$	−28276.0	−1.03516	−0.517579	−	0.855635i	$-0.673167\pi$
$907$	−28276.0	−1.03516	−0.517579	+	0.855635i	$0.673167\pi$
$908$	8976.00	0.328061
$909$	−5562.00	−0.202948
$910$	7296.00	0.265780
$911$	8112.00	0.295019	0.147510	−	0.989061i	$-0.452874\pi$
$911$	8112.00	0.295019	0.147510	+	0.989061i	$0.452874\pi$
$912$	−960.000	−0.0348561
$913$	−5904.00	−0.214013
$914$	1292.00	0.0467566
$915$	9684.00	0.349883
$916$	−22600.0	−0.815202
$917$	−36672.0	−1.32063
$918$	−6804.00	−0.244625
$919$	−26080.0	−0.936126	−0.468063	−	0.883695i	$-0.655048\pi$
$919$	−26080.0	−0.936126	−0.468063	+	0.883695i	$0.655048\pi$
$920$	−8064.00	−0.288981
$921$	25428.0	0.909751
$922$	12036.0	0.429918
$923$	30096.0	1.07326
$924$	2304.00	0.0820303
$925$	−22606.0	−0.803547
$926$	13424.0	0.476393
$927$	1152.00	0.0408162
$928$	−960.000	−0.0339586
$929$	49170.0	1.73651	0.868254	−	0.496120i	$-0.165243\pi$
$929$	49170.0	1.73651	0.868254	+	0.496120i	$0.165243\pi$
$930$	−3168.00	−0.111702
$931$	−1740.00	−0.0612526
$932$	18792.0	0.660464
$933$	−13896.0	−0.487604
$934$	−10728.0	−0.375836
$935$	−9072.00	−0.317311
$936$	−2736.00	−0.0955438
$937$	48314.0	1.68447	0.842236	−	0.539110i	$-0.181239\pi$
$937$	48314.0	1.68447	0.842236	+	0.539110i	$0.181239\pi$
$938$	28288.0	0.984687
$939$	14466.0	0.502748
$940$	−2304.00	−0.0799449
$941$	34782.0	1.20495	0.602477	−	0.798137i	$-0.294181\pi$
$941$	34782.0	1.20495	0.602477	+	0.798137i	$0.294181\pi$
$942$	3684.00	0.127422
$943$	7056.00	0.243664
$944$	−10560.0	−0.364088
$945$	2592.00	0.0892251
$946$	1248.00	0.0428922
$947$	−25116.0	−0.861838	−0.430919	−	0.902391i	$-0.641810\pi$
$947$	−25116.0	−0.861838	−0.430919	+	0.902391i	$0.641810\pi$
$948$	6240.00	0.213782
$949$	8284.00	0.283361
$950$	3560.00	0.121581
$951$	10278.0	0.350460
$952$	−16128.0	−0.549067
$953$	−15462.0	−0.525565	−0.262782	−	0.964855i	$-0.584640\pi$
$953$	−15462.0	−0.525565	−0.262782	+	0.964855i	$0.584640\pi$
$954$	−3564.00	−0.120953
$955$	−16128.0	−0.546481
$956$	−4800.00	−0.162388
$957$	−1080.00	−0.0364801
$958$	−19680.0	−0.663708
$959$	11616.0	0.391137
$960$	−1152.00	−0.0387298
$961$	−22047.0	−0.740056
$962$	−19304.0	−0.646971
$963$	−13284.0	−0.444518
$964$	−2872.00	−0.0959553
$965$	−13812.0	−0.460750
$966$	−16128.0	−0.537174
$967$	−736.000	−0.0244759	−0.0122379	−	0.999925i	$-0.503896\pi$
$967$	−736.000	−0.0244759	−0.0122379	+	0.999925i	$0.503896\pi$
$968$	9496.00	0.315303
$969$	7560.00	0.250632
$970$	−13848.0	−0.458384
$971$	−29268.0	−0.967307	−0.483653	−	0.875260i	$-0.660690\pi$
$971$	−29268.0	−0.967307	−0.483653	+	0.875260i	$0.660690\pi$
$972$	−972.000	−0.0320750
$973$	−6080.00	−0.200325
$974$	−2848.00	−0.0936918
$975$	10146.0	0.333264
$976$	−8608.00	−0.282311
$977$	16674.0	0.546007	0.273003	−	0.962013i	$-0.411983\pi$
$977$	16674.0	0.546007	0.273003	+	0.962013i	$0.411983\pi$
$978$	−11112.0	−0.363316
$979$	9720.00	0.317316
$980$	−2088.00	−0.0680599
$981$	10710.0	0.348567
$982$	9096.00	0.295586
$983$	−31272.0	−1.01467	−0.507336	−	0.861749i	$-0.669370\pi$
$983$	−31272.0	−1.01467	−0.507336	+	0.861749i	$0.669370\pi$
$984$	1008.00	0.0326564
$985$	26244.0	0.848937
$986$	7560.00	0.244178
$987$	−4608.00	−0.148606
$988$	3040.00	0.0978900
$989$	−8736.00	−0.280878
$990$	−1296.00	−0.0416056
$991$	−15928.0	−0.510565	−0.255282	−	0.966867i	$-0.582168\pi$
$991$	−15928.0	−0.510565	−0.255282	+	0.966867i	$0.582168\pi$
$992$	2816.00	0.0901291
$993$	8364.00	0.267295
$994$	25344.0	0.808715
$995$	−9600.00	−0.305870
$996$	5904.00	0.187827
$997$	42014.0	1.33460	0.667300	−	0.744789i	$-0.267450\pi$
$997$	42014.0	1.33460	0.667300	+	0.744789i	$0.267450\pi$
$998$	−13000.0	−0.412332
$999$	−6858.00	−0.217195

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6.4.a.a.1.1	✓	1
3.2	odd	2		18.4.a.a.1.1		1
4.3	odd	2		48.4.a.c.1.1		1
5.2	odd	4		150.4.c.d.49.1		2
5.3	odd	4		150.4.c.d.49.2		2
5.4	even	2		150.4.a.i.1.1		1
7.2	even	3		294.4.e.h.67.1		2
7.3	odd	6		294.4.e.g.79.1		2
7.4	even	3		294.4.e.h.79.1		2
7.5	odd	6		294.4.e.g.67.1		2
7.6	odd	2		294.4.a.e.1.1		1
8.3	odd	2		192.4.a.c.1.1		1
8.5	even	2		192.4.a.i.1.1		1
9.2	odd	6		162.4.c.c.109.1		2
9.4	even	3		162.4.c.f.55.1		2
9.5	odd	6		162.4.c.c.55.1		2
9.7	even	3		162.4.c.f.109.1		2
11.10	odd	2		726.4.a.f.1.1		1
12.11	even	2		144.4.a.c.1.1		1
13.5	odd	4		1014.4.b.d.337.2		2
13.8	odd	4		1014.4.b.d.337.1		2
13.12	even	2		1014.4.a.g.1.1		1
15.2	even	4		450.4.c.e.199.2		2
15.8	even	4		450.4.c.e.199.1		2
15.14	odd	2		450.4.a.h.1.1		1
16.3	odd	4		768.4.d.c.385.2		2
16.5	even	4		768.4.d.n.385.2		2
16.11	odd	4		768.4.d.c.385.1		2
16.13	even	4		768.4.d.n.385.1		2
17.16	even	2		1734.4.a.d.1.1		1
19.18	odd	2		2166.4.a.i.1.1		1
20.3	even	4		1200.4.f.j.49.2		2
20.7	even	4		1200.4.f.j.49.1		2
20.19	odd	2		1200.4.a.b.1.1		1
21.2	odd	6		882.4.g.i.361.1		2
21.5	even	6		882.4.g.f.361.1		2
21.11	odd	6		882.4.g.i.667.1		2
21.17	even	6		882.4.g.f.667.1		2
21.20	even	2		882.4.a.n.1.1		1
24.5	odd	2		576.4.a.q.1.1		1
24.11	even	2		576.4.a.r.1.1		1
28.27	even	2		2352.4.a.e.1.1		1
33.32	even	2		2178.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.4.a.a.1.1	✓	1	1.1	even	1	trivial
18.4.a.a.1.1		1	3.2	odd	2
48.4.a.c.1.1		1	4.3	odd	2
144.4.a.c.1.1		1	12.11	even	2
150.4.a.i.1.1		1	5.4	even	2
150.4.c.d.49.1		2	5.2	odd	4
150.4.c.d.49.2		2	5.3	odd	4
162.4.c.c.55.1		2	9.5	odd	6
162.4.c.c.109.1		2	9.2	odd	6
162.4.c.f.55.1		2	9.4	even	3
162.4.c.f.109.1		2	9.7	even	3
192.4.a.c.1.1		1	8.3	odd	2
192.4.a.i.1.1		1	8.5	even	2
294.4.a.e.1.1		1	7.6	odd	2
294.4.e.g.67.1		2	7.5	odd	6
294.4.e.g.79.1		2	7.3	odd	6
294.4.e.h.67.1		2	7.2	even	3
294.4.e.h.79.1		2	7.4	even	3
450.4.a.h.1.1		1	15.14	odd	2
450.4.c.e.199.1		2	15.8	even	4
450.4.c.e.199.2		2	15.2	even	4
576.4.a.q.1.1		1	24.5	odd	2
576.4.a.r.1.1		1	24.11	even	2
726.4.a.f.1.1		1	11.10	odd	2
768.4.d.c.385.1		2	16.11	odd	4
768.4.d.c.385.2		2	16.3	odd	4
768.4.d.n.385.1		2	16.13	even	4
768.4.d.n.385.2		2	16.5	even	4
882.4.a.n.1.1		1	21.20	even	2
882.4.g.f.361.1		2	21.5	even	6
882.4.g.f.667.1		2	21.17	even	6
882.4.g.i.361.1		2	21.2	odd	6
882.4.g.i.667.1		2	21.11	odd	6
1014.4.a.g.1.1		1	13.12	even	2
1014.4.b.d.337.1		2	13.8	odd	4
1014.4.b.d.337.2		2	13.5	odd	4
1200.4.a.b.1.1		1	20.19	odd	2
1200.4.f.j.49.1		2	20.7	even	4
1200.4.f.j.49.2		2	20.3	even	4
1734.4.a.d.1.1		1	17.16	even	2
2166.4.a.i.1.1		1	19.18	odd	2
2178.4.a.e.1.1		1	33.32	even	2
2352.4.a.e.1.1		1	28.27	even	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:	\( N \)	\(=\)	\( 6 = 2 \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	6.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more