LMFDB - Embedded newform 5.4.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5,4,Mod(1,5)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	5.a (trivial)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$0.295009550029$
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	$\chi$	$=$	5.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

f(q)

=

q-4.00000 q^{2} +2.00000 q^{3} +8.00000 q^{4} -5.00000 q^{5} -8.00000 q^{6} +6.00000 q^{7} -23.0000 q^{9} +20.0000 q^{10} +32.0000 q^{11} +16.0000 q^{12} -38.0000 q^{13} -24.0000 q^{14} -10.0000 q^{15} -64.0000 q^{16} +26.0000 q^{17} +92.0000 q^{18} +100.000 q^{19} -40.0000 q^{20} +12.0000 q^{21} -128.000 q^{22} -78.0000 q^{23} +25.0000 q^{25} +152.000 q^{26} -100.000 q^{27} +48.0000 q^{28} -50.0000 q^{29} +40.0000 q^{30} -108.000 q^{31} +256.000 q^{32} +64.0000 q^{33} -104.000 q^{34} -30.0000 q^{35} -184.000 q^{36} +266.000 q^{37} -400.000 q^{38} -76.0000 q^{39} +22.0000 q^{41} -48.0000 q^{42} +442.000 q^{43} +256.000 q^{44} +115.000 q^{45} +312.000 q^{46} -514.000 q^{47} -128.000 q^{48} -307.000 q^{49} -100.000 q^{50} +52.0000 q^{51} -304.000 q^{52} +2.00000 q^{53} +400.000 q^{54} -160.000 q^{55} +200.000 q^{57} +200.000 q^{58} +500.000 q^{59} -80.0000 q^{60} -518.000 q^{61} +432.000 q^{62} -138.000 q^{63} -512.000 q^{64} +190.000 q^{65} -256.000 q^{66} +126.000 q^{67} +208.000 q^{68} -156.000 q^{69} +120.000 q^{70} +412.000 q^{71} -878.000 q^{73} -1064.00 q^{74} +50.0000 q^{75} +800.000 q^{76} +192.000 q^{77} +304.000 q^{78} +600.000 q^{79} +320.000 q^{80} +421.000 q^{81} -88.0000 q^{82} +282.000 q^{83} +96.0000 q^{84} -130.000 q^{85} -1768.00 q^{86} -100.000 q^{87} -150.000 q^{89} -460.000 q^{90} -228.000 q^{91} -624.000 q^{92} -216.000 q^{93} +2056.00 q^{94} -500.000 q^{95} +512.000 q^{96} +386.000 q^{97} +1228.00 q^{98} -736.000 q^{99} +O(q^{100})

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−4.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−4.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	8.00000	1.00000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−8.00000	−0.544331
$7$	6.00000	0.323970	0.161985	−	0.986793i	$-0.448210\pi$
$7$	6.00000	0.323970	0.161985	+	0.986793i	$0.448210\pi$
$8$	0	0
$9$	−23.0000	−0.851852
$10$	20.0000	0.632456
$11$	32.0000	0.877124	0.438562	−	0.898701i	$-0.355488\pi$
$11$	32.0000	0.877124	0.438562	+	0.898701i	$0.355488\pi$
$12$	16.0000	0.384900
$13$	−38.0000	−0.810716	−0.405358	−	0.914158i	$-0.632853\pi$
$13$	−38.0000	−0.810716	−0.405358	+	0.914158i	$0.632853\pi$
$14$	−24.0000	−0.458162
$15$	−10.0000	−0.172133
$16$	−64.0000	−1.00000
$17$	26.0000	0.370937	0.185468	−	0.982650i	$-0.440620\pi$
$17$	26.0000	0.370937	0.185468	+	0.982650i	$0.440620\pi$
$18$	92.0000	1.20470
$19$	100.000	1.20745	0.603726	−	0.797192i	$-0.293682\pi$
$19$	100.000	1.20745	0.603726	+	0.797192i	$0.293682\pi$
$20$	−40.0000	−0.447214
$21$	12.0000	0.124696
$22$	−128.000	−1.24044
$23$	−78.0000	−0.707136	−0.353568	−	0.935409i	$-0.615032\pi$
$23$	−78.0000	−0.707136	−0.353568	+	0.935409i	$0.615032\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	152.000	1.14653
$27$	−100.000	−0.712778
$28$	48.0000	0.323970
$29$	−50.0000	−0.320164	−0.160082	−	0.987104i	$-0.551176\pi$
$29$	−50.0000	−0.320164	−0.160082	+	0.987104i	$0.551176\pi$
$30$	40.0000	0.243432
$31$	−108.000	−0.625722	−0.312861	−	0.949799i	$-0.601287\pi$
$31$	−108.000	−0.625722	−0.312861	+	0.949799i	$0.601287\pi$
$32$	256.000	1.41421
$33$	64.0000	0.337605
$34$	−104.000	−0.524584
$35$	−30.0000	−0.144884
$36$	−184.000	−0.851852
$37$	266.000	1.18190	0.590948	−	0.806710i	$-0.298754\pi$
$37$	266.000	1.18190	0.590948	+	0.806710i	$0.298754\pi$
$38$	−400.000	−1.70759
$39$	−76.0000	−0.312045
$40$	0	0
$41$	22.0000	0.0838006	0.0419003	−	0.999122i	$-0.486659\pi$
$41$	22.0000	0.0838006	0.0419003	+	0.999122i	$0.486659\pi$
$42$	−48.0000	−0.176347
$43$	442.000	1.56754	0.783772	−	0.621049i	$-0.213293\pi$
$43$	442.000	1.56754	0.783772	+	0.621049i	$0.213293\pi$
$44$	256.000	0.877124
$45$	115.000	0.380960
$46$	312.000	1.00004
$47$	−514.000	−1.59520	−0.797602	−	0.603184i	$-0.793899\pi$
$47$	−514.000	−1.59520	−0.797602	+	0.603184i	$0.793899\pi$
$48$	−128.000	−0.384900
$49$	−307.000	−0.895044
$50$	−100.000	−0.282843
$51$	52.0000	0.142774
$52$	−304.000	−0.810716
$53$	2.00000	0.00518342	0.00259171	−	0.999997i	$-0.499175\pi$
$53$	2.00000	0.00518342	0.00259171	+	0.999997i	$0.499175\pi$
$54$	400.000	1.00802
$55$	−160.000	−0.392262
$56$	0	0
$57$	200.000	0.464748
$58$	200.000	0.452781
$59$	500.000	1.10330	0.551648	−	0.834077i	$-0.313999\pi$
$59$	500.000	1.10330	0.551648	+	0.834077i	$0.313999\pi$
$60$	−80.0000	−0.172133
$61$	−518.000	−1.08726	−0.543632	−	0.839324i	$-0.682951\pi$
$61$	−518.000	−1.08726	−0.543632	+	0.839324i	$0.682951\pi$
$62$	432.000	0.884904
$63$	−138.000	−0.275974
$64$	−512.000	−1.00000
$65$	190.000	0.362563
$66$	−256.000	−0.477446
$67$	126.000	0.229751	0.114876	−	0.993380i	$-0.463353\pi$
$67$	126.000	0.229751	0.114876	+	0.993380i	$0.463353\pi$
$68$	208.000	0.370937
$69$	−156.000	−0.272177
$70$	120.000	0.204896
$71$	412.000	0.688668	0.344334	−	0.938847i	$-0.388105\pi$
$71$	412.000	0.688668	0.344334	+	0.938847i	$0.388105\pi$
$72$	0	0
$73$	−878.000	−1.40770	−0.703850	−	0.710348i	$-0.748537\pi$
$73$	−878.000	−1.40770	−0.703850	+	0.710348i	$0.748537\pi$
$74$	−1064.00	−1.67145
$75$	50.0000	0.0769800
$76$	800.000	1.20745
$77$	192.000	0.284161
$78$	304.000	0.441298
$79$	600.000	0.854497	0.427249	−	0.904134i	$-0.359483\pi$
$79$	600.000	0.854497	0.427249	+	0.904134i	$0.359483\pi$
$80$	320.000	0.447214
$81$	421.000	0.577503
$82$	−88.0000	−0.118512
$83$	282.000	0.372934	0.186467	−	0.982461i	$-0.440296\pi$
$83$	282.000	0.372934	0.186467	+	0.982461i	$0.440296\pi$
$84$	96.0000	0.124696
$85$	−130.000	−0.165888
$86$	−1768.00	−2.21684
$87$	−100.000	−0.123231
$88$	0	0
$89$	−150.000	−0.178651	−0.0893257	−	0.996002i	$-0.528471\pi$
$89$	−150.000	−0.178651	−0.0893257	+	0.996002i	$0.528471\pi$
$90$	−460.000	−0.538758
$91$	−228.000	−0.262647
$92$	−624.000	−0.707136
$93$	−216.000	−0.240840
$94$	2056.00	2.25596
$95$	−500.000	−0.539989
$96$	512.000	0.544331
$97$	386.000	0.404045	0.202022	−	0.979381i	$-0.435249\pi$
$97$	386.000	0.404045	0.202022	+	0.979381i	$0.435249\pi$
$98$	1228.00	1.26578
$99$	−736.000	−0.747180
$100$	200.000	0.200000
$101$	702.000	0.691600	0.345800	−	0.938308i	$-0.387608\pi$
$101$	702.000	0.691600	0.345800	+	0.938308i	$0.387608\pi$
$102$	−208.000	−0.201912
$103$	−598.000	−0.572065	−0.286032	−	0.958220i	$-0.592337\pi$
$103$	−598.000	−0.572065	−0.286032	+	0.958220i	$0.592337\pi$
$104$	0	0
$105$	−60.0000	−0.0557657
$106$	−8.00000	−0.00733046
$107$	−1194.00	−1.07877	−0.539385	−	0.842059i	$-0.681343\pi$
$107$	−1194.00	−1.07877	−0.539385	+	0.842059i	$0.681343\pi$
$108$	−800.000	−0.712778
$109$	−550.000	−0.483307	−0.241653	−	0.970363i	$-0.577690\pi$
$109$	−550.000	−0.483307	−0.241653	+	0.970363i	$0.577690\pi$
$110$	640.000	0.554742
$111$	532.000	0.454912
$112$	−384.000	−0.323970
$113$	1562.00	1.30036	0.650180	−	0.759781i	$-0.274694\pi$
$113$	1562.00	1.30036	0.650180	+	0.759781i	$0.274694\pi$
$114$	−800.000	−0.657253
$115$	390.000	0.316241
$116$	−400.000	−0.320164
$117$	874.000	0.690610
$118$	−2000.00	−1.56030
$119$	156.000	0.120172
$120$	0	0
$121$	−307.000	−0.230654
$122$	2072.00	1.53762
$123$	44.0000	0.0322548
$124$	−864.000	−0.625722
$125$	−125.000	−0.0894427
$126$	552.000	0.390286
$127$	1846.00	1.28981	0.644906	−	0.764262i	$-0.276897\pi$
$127$	1846.00	1.28981	0.644906	+	0.764262i	$0.276897\pi$
$128$	0	0
$129$	884.000	0.603348
$130$	−760.000	−0.512742
$131$	−2208.00	−1.47262	−0.736312	−	0.676642i	$-0.763435\pi$
$131$	−2208.00	−1.47262	−0.736312	+	0.676642i	$0.763435\pi$
$132$	512.000	0.337605
$133$	600.000	0.391177
$134$	−504.000	−0.324918
$135$	500.000	0.318764
$136$	0	0
$137$	−2334.00	−1.45553	−0.727763	−	0.685829i	$-0.759440\pi$
$137$	−2334.00	−1.45553	−0.727763	+	0.685829i	$0.759440\pi$
$138$	624.000	0.384916
$139$	−700.000	−0.427146	−0.213573	−	0.976927i	$-0.568510\pi$
$139$	−700.000	−0.427146	−0.213573	+	0.976927i	$0.568510\pi$
$140$	−240.000	−0.144884
$141$	−1028.00	−0.613994
$142$	−1648.00	−0.973923
$143$	−1216.00	−0.711098
$144$	1472.00	0.851852
$145$	250.000	0.143182
$146$	3512.00	1.99079
$147$	−614.000	−0.344502
$148$	2128.00	1.18190
$149$	2050.00	1.12713	0.563566	−	0.826071i	$-0.309429\pi$
$149$	2050.00	1.12713	0.563566	+	0.826071i	$0.309429\pi$
$150$	−200.000	−0.108866
$151$	1852.00	0.998103	0.499052	−	0.866572i	$-0.333682\pi$
$151$	1852.00	0.998103	0.499052	+	0.866572i	$0.333682\pi$
$152$	0	0
$153$	−598.000	−0.315983
$154$	−768.000	−0.401865
$155$	540.000	0.279831
$156$	−608.000	−0.312045
$157$	−2494.00	−1.26779	−0.633894	−	0.773420i	$-0.718545\pi$
$157$	−2494.00	−1.26779	−0.633894	+	0.773420i	$0.718545\pi$
$158$	−2400.00	−1.20844
$159$	4.00000	0.00199510
$160$	−1280.00	−0.632456
$161$	−468.000	−0.229090
$162$	−1684.00	−0.816713
$163$	2762.00	1.32722	0.663609	−	0.748080i	$-0.269024\pi$
$163$	2762.00	1.32722	0.663609	+	0.748080i	$0.269024\pi$
$164$	176.000	0.0838006
$165$	−320.000	−0.150982
$166$	−1128.00	−0.527408
$167$	3126.00	1.44849	0.724243	−	0.689545i	$-0.242189\pi$
$167$	3126.00	1.44849	0.724243	+	0.689545i	$0.242189\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	520.000	0.234601
$171$	−2300.00	−1.02857
$172$	3536.00	1.56754
$173$	−78.0000	−0.0342788	−0.0171394	−	0.999853i	$-0.505456\pi$
$173$	−78.0000	−0.0342788	−0.0171394	+	0.999853i	$0.505456\pi$
$174$	400.000	0.174275
$175$	150.000	0.0647939
$176$	−2048.00	−0.877124
$177$	1000.00	0.424659
$178$	600.000	0.252651
$179$	−1300.00	−0.542830	−0.271415	−	0.962462i	$-0.587492\pi$
$179$	−1300.00	−0.542830	−0.271415	+	0.962462i	$0.587492\pi$
$180$	920.000	0.380960
$181$	1742.00	0.715369	0.357685	−	0.933842i	$-0.383566\pi$
$181$	1742.00	0.715369	0.357685	+	0.933842i	$0.383566\pi$
$182$	912.000	0.371439
$183$	−1036.00	−0.418488
$184$	0	0
$185$	−1330.00	−0.528560
$186$	864.000	0.340600
$187$	832.000	0.325358
$188$	−4112.00	−1.59520
$189$	−600.000	−0.230918
$190$	2000.00	0.763659
$191$	3772.00	1.42897	0.714483	−	0.699653i	$-0.246662\pi$
$191$	3772.00	1.42897	0.714483	+	0.699653i	$0.246662\pi$
$192$	−1024.00	−0.384900
$193$	−358.000	−0.133520	−0.0667601	−	0.997769i	$-0.521266\pi$
$193$	−358.000	−0.133520	−0.0667601	+	0.997769i	$0.521266\pi$
$194$	−1544.00	−0.571406
$195$	380.000	0.139551
$196$	−2456.00	−0.895044
$197$	−2214.00	−0.800716	−0.400358	−	0.916359i	$-0.631114\pi$
$197$	−2214.00	−0.800716	−0.400358	+	0.916359i	$0.631114\pi$
$198$	2944.00	1.05667
$199$	−2600.00	−0.926176	−0.463088	−	0.886312i	$-0.653259\pi$
$199$	−2600.00	−0.926176	−0.463088	+	0.886312i	$0.653259\pi$
$200$	0	0
$201$	252.000	0.0884314
$202$	−2808.00	−0.978070
$203$	−300.000	−0.103724
$204$	416.000	0.142774
$205$	−110.000	−0.0374767
$206$	2392.00	0.809022
$207$	1794.00	0.602375
$208$	2432.00	0.810716
$209$	3200.00	1.05908
$210$	240.000	0.0788646
$211$	−1168.00	−0.381083	−0.190541	−	0.981679i	$-0.561024\pi$
$211$	−1168.00	−0.381083	−0.190541	+	0.981679i	$0.561024\pi$
$212$	16.0000	0.00518342
$213$	824.000	0.265068
$214$	4776.00	1.52561
$215$	−2210.00	−0.701027
$216$	0	0
$217$	−648.000	−0.202715
$218$	2200.00	0.683499
$219$	−1756.00	−0.541824
$220$	−1280.00	−0.392262
$221$	−988.000	−0.300724
$222$	−2128.00	−0.643342
$223$	−6478.00	−1.94529	−0.972643	−	0.232303i	$-0.925374\pi$
$223$	−6478.00	−1.94529	−0.972643	+	0.232303i	$0.925374\pi$
$224$	1536.00	0.458162
$225$	−575.000	−0.170370
$226$	−6248.00	−1.83899
$227$	646.000	0.188883	0.0944417	−	0.995530i	$-0.469893\pi$
$227$	646.000	0.188883	0.0944417	+	0.995530i	$0.469893\pi$
$228$	1600.00	0.464748
$229$	3750.00	1.08213	0.541063	−	0.840982i	$-0.318022\pi$
$229$	3750.00	1.08213	0.541063	+	0.840982i	$0.318022\pi$
$230$	−1560.00	−0.447232
$231$	384.000	0.109374
$232$	0	0
$233$	1482.00	0.416691	0.208346	−	0.978055i	$-0.433192\pi$
$233$	1482.00	0.416691	0.208346	+	0.978055i	$0.433192\pi$
$234$	−3496.00	−0.976670
$235$	2570.00	0.713397
$236$	4000.00	1.10330
$237$	1200.00	0.328896
$238$	−624.000	−0.169949
$239$	1400.00	0.378906	0.189453	−	0.981890i	$-0.439329\pi$
$239$	1400.00	0.378906	0.189453	+	0.981890i	$0.439329\pi$
$240$	640.000	0.172133
$241$	3022.00	0.807735	0.403867	−	0.914817i	$-0.367666\pi$
$241$	3022.00	0.807735	0.403867	+	0.914817i	$0.367666\pi$
$242$	1228.00	0.326194
$243$	3542.00	0.935059
$244$	−4144.00	−1.08726
$245$	1535.00	0.400276
$246$	−176.000	−0.0456152
$247$	−3800.00	−0.978900
$248$	0	0
$249$	564.000	0.143542
$250$	500.000	0.126491
$251$	−1248.00	−0.313837	−0.156918	−	0.987612i	$-0.550156\pi$
$251$	−1248.00	−0.313837	−0.156918	+	0.987612i	$0.550156\pi$
$252$	−1104.00	−0.275974
$253$	−2496.00	−0.620246
$254$	−7384.00	−1.82407
$255$	−260.000	−0.0638503
$256$	4096.00	1.00000
$257$	2106.00	0.511162	0.255581	−	0.966788i	$-0.417733\pi$
$257$	2106.00	0.511162	0.255581	+	0.966788i	$0.417733\pi$
$258$	−3536.00	−0.853263
$259$	1596.00	0.382898
$260$	1520.00	0.362563
$261$	1150.00	0.272733
$262$	8832.00	2.08261
$263$	−3638.00	−0.852961	−0.426480	−	0.904497i	$-0.640247\pi$
$263$	−3638.00	−0.852961	−0.426480	+	0.904497i	$0.640247\pi$
$264$	0	0
$265$	−10.0000	−0.00231809
$266$	−2400.00	−0.553208
$267$	−300.000	−0.0687629
$268$	1008.00	0.229751
$269$	−6550.00	−1.48461	−0.742306	−	0.670061i	$-0.766268\pi$
$269$	−6550.00	−1.48461	−0.742306	+	0.670061i	$0.766268\pi$
$270$	−2000.00	−0.450800
$271$	−4388.00	−0.983587	−0.491793	−	0.870712i	$-0.663658\pi$
$271$	−4388.00	−0.983587	−0.491793	+	0.870712i	$0.663658\pi$
$272$	−1664.00	−0.370937
$273$	−456.000	−0.101093
$274$	9336.00	2.05842
$275$	800.000	0.175425
$276$	−1248.00	−0.272177
$277$	546.000	0.118433	0.0592165	−	0.998245i	$-0.481140\pi$
$277$	546.000	0.118433	0.0592165	+	0.998245i	$0.481140\pi$
$278$	2800.00	0.604075
$279$	2484.00	0.533022
$280$	0	0
$281$	−6858.00	−1.45592	−0.727961	−	0.685619i	$-0.759532\pi$
$281$	−6858.00	−1.45592	−0.727961	+	0.685619i	$0.759532\pi$
$282$	4112.00	0.868319
$283$	9282.00	1.94967	0.974837	−	0.222920i	$-0.0715588\pi$
$283$	9282.00	1.94967	0.974837	+	0.222920i	$0.0715588\pi$
$284$	3296.00	0.688668
$285$	−1000.00	−0.207842
$286$	4864.00	1.00564
$287$	132.000	0.0271488
$288$	−5888.00	−1.20470
$289$	−4237.00	−0.862406
$290$	−1000.00	−0.202490
$291$	772.000	0.155517
$292$	−7024.00	−1.40770
$293$	4842.00	0.965436	0.482718	−	0.875776i	$-0.339650\pi$
$293$	4842.00	0.965436	0.482718	+	0.875776i	$0.339650\pi$
$294$	2456.00	0.487200
$295$	−2500.00	−0.493409
$296$	0	0
$297$	−3200.00	−0.625195
$298$	−8200.00	−1.59400
$299$	2964.00	0.573286
$300$	400.000	0.0769800
$301$	2652.00	0.507836
$302$	−7408.00	−1.41153
$303$	1404.00	0.266197
$304$	−6400.00	−1.20745
$305$	2590.00	0.486239
$306$	2392.00	0.446868
$307$	−2594.00	−0.482239	−0.241120	−	0.970495i	$-0.577515\pi$
$307$	−2594.00	−0.482239	−0.241120	+	0.970495i	$0.577515\pi$
$308$	1536.00	0.284161
$309$	−1196.00	−0.220188
$310$	−2160.00	−0.395741
$311$	7332.00	1.33685	0.668424	−	0.743781i	$-0.266969\pi$
$311$	7332.00	1.33685	0.668424	+	0.743781i	$0.266969\pi$
$312$	0	0
$313$	1562.00	0.282075	0.141037	−	0.990004i	$-0.454956\pi$
$313$	1562.00	0.282075	0.141037	+	0.990004i	$0.454956\pi$
$314$	9976.00	1.79292
$315$	690.000	0.123419
$316$	4800.00	0.854497
$317$	1426.00	0.252657	0.126328	−	0.991988i	$-0.459681\pi$
$317$	1426.00	0.252657	0.126328	+	0.991988i	$0.459681\pi$
$318$	−16.0000	−0.00282150
$319$	−1600.00	−0.280824
$320$	2560.00	0.447214
$321$	−2388.00	−0.415219
$322$	1872.00	0.323983
$323$	2600.00	0.447888
$324$	3368.00	0.577503
$325$	−950.000	−0.162143
$326$	−11048.0	−1.87697
$327$	−1100.00	−0.186025
$328$	0	0
$329$	−3084.00	−0.516798
$330$	1280.00	0.213520
$331$	−4008.00	−0.665558	−0.332779	−	0.943005i	$-0.607986\pi$
$331$	−4008.00	−0.665558	−0.332779	+	0.943005i	$0.607986\pi$
$332$	2256.00	0.372934
$333$	−6118.00	−1.00680
$334$	−12504.0	−2.04847
$335$	−630.000	−0.102748
$336$	−768.000	−0.124696
$337$	8866.00	1.43312	0.716561	−	0.697525i	$-0.245715\pi$
$337$	8866.00	1.43312	0.716561	+	0.697525i	$0.245715\pi$
$338$	3012.00	0.484708
$339$	3124.00	0.500509
$340$	−1040.00	−0.165888
$341$	−3456.00	−0.548835
$342$	9200.00	1.45462
$343$	−3900.00	−0.613936
$344$	0	0
$345$	780.000	0.121721
$346$	312.000	0.0484775
$347$	−1714.00	−0.265165	−0.132583	−	0.991172i	$-0.542327\pi$
$347$	−1714.00	−0.265165	−0.132583	+	0.991172i	$0.542327\pi$
$348$	−800.000	−0.123231
$349$	1150.00	0.176384	0.0881921	−	0.996103i	$-0.471891\pi$
$349$	1150.00	0.176384	0.0881921	+	0.996103i	$0.471891\pi$
$350$	−600.000	−0.0916324
$351$	3800.00	0.577860
$352$	8192.00	1.24044
$353$	−4398.00	−0.663122	−0.331561	−	0.943434i	$-0.607575\pi$
$353$	−4398.00	−0.663122	−0.331561	+	0.943434i	$0.607575\pi$
$354$	−4000.00	−0.600558
$355$	−2060.00	−0.307982
$356$	−1200.00	−0.178651
$357$	312.000	0.0462543
$358$	5200.00	0.767677
$359$	1800.00	0.264625	0.132312	−	0.991208i	$-0.457760\pi$
$359$	1800.00	0.264625	0.132312	+	0.991208i	$0.457760\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	−6968.00	−1.01168
$363$	−614.000	−0.0887786
$364$	−1824.00	−0.262647
$365$	4390.00	0.629543
$366$	4144.00	0.591832
$367$	−5874.00	−0.835478	−0.417739	−	0.908567i	$-0.637177\pi$
$367$	−5874.00	−0.835478	−0.417739	+	0.908567i	$0.637177\pi$
$368$	4992.00	0.707136
$369$	−506.000	−0.0713857
$370$	5320.00	0.747496
$371$	12.0000	0.00167927
$372$	−1728.00	−0.240840
$373$	−2078.00	−0.288458	−0.144229	−	0.989544i	$-0.546070\pi$
$373$	−2078.00	−0.288458	−0.144229	+	0.989544i	$0.546070\pi$
$374$	−3328.00	−0.460125
$375$	−250.000	−0.0344265
$376$	0	0
$377$	1900.00	0.259562
$378$	2400.00	0.326568
$379$	7900.00	1.07070	0.535351	−	0.844630i	$-0.320179\pi$
$379$	7900.00	1.07070	0.535351	+	0.844630i	$0.320179\pi$
$380$	−4000.00	−0.539989
$381$	3692.00	0.496449
$382$	−15088.0	−2.02086
$383$	−7518.00	−1.00301	−0.501504	−	0.865155i	$-0.667220\pi$
$383$	−7518.00	−1.00301	−0.501504	+	0.865155i	$0.667220\pi$
$384$	0	0
$385$	−960.000	−0.127081
$386$	1432.00	0.188826
$387$	−10166.0	−1.33531
$388$	3088.00	0.404045
$389$	−1950.00	−0.254162	−0.127081	−	0.991892i	$-0.540561\pi$
$389$	−1950.00	−0.254162	−0.127081	+	0.991892i	$0.540561\pi$
$390$	−1520.00	−0.197354
$391$	−2028.00	−0.262303
$392$	0	0
$393$	−4416.00	−0.566814
$394$	8856.00	1.13238
$395$	−3000.00	−0.382143
$396$	−5888.00	−0.747180
$397$	13786.0	1.74282	0.871410	−	0.490555i	$-0.163206\pi$
$397$	13786.0	1.74282	0.871410	+	0.490555i	$0.163206\pi$
$398$	10400.0	1.30981
$399$	1200.00	0.150564
$400$	−1600.00	−0.200000
$401$	6402.00	0.797258	0.398629	−	0.917112i	$-0.369486\pi$
$401$	6402.00	0.797258	0.398629	+	0.917112i	$0.369486\pi$
$402$	−1008.00	−0.125061
$403$	4104.00	0.507282
$404$	5616.00	0.691600
$405$	−2105.00	−0.258267
$406$	1200.00	0.146687
$407$	8512.00	1.03667
$408$	0	0
$409$	11150.0	1.34800	0.674000	−	0.738731i	$-0.264575\pi$
$409$	11150.0	1.34800	0.674000	+	0.738731i	$0.264575\pi$
$410$	440.000	0.0530001
$411$	−4668.00	−0.560232
$412$	−4784.00	−0.572065
$413$	3000.00	0.357434
$414$	−7176.00	−0.851887
$415$	−1410.00	−0.166781
$416$	−9728.00	−1.14653
$417$	−1400.00	−0.164408
$418$	−12800.0	−1.49777
$419$	−13700.0	−1.59735	−0.798674	−	0.601764i	$-0.794465\pi$
$419$	−13700.0	−1.59735	−0.798674	+	0.601764i	$0.794465\pi$
$420$	−480.000	−0.0557657
$421$	−5438.00	−0.629529	−0.314765	−	0.949170i	$-0.601926\pi$
$421$	−5438.00	−0.629529	−0.314765	+	0.949170i	$0.601926\pi$
$422$	4672.00	0.538932
$423$	11822.0	1.35888
$424$	0	0
$425$	650.000	0.0741874
$426$	−3296.00	−0.374863
$427$	−3108.00	−0.352240
$428$	−9552.00	−1.07877
$429$	−2432.00	−0.273702
$430$	8840.00	0.991402
$431$	7692.00	0.859653	0.429827	−	0.902911i	$-0.358575\pi$
$431$	7692.00	0.859653	0.429827	+	0.902911i	$0.358575\pi$
$432$	6400.00	0.712778
$433$	−1118.00	−0.124082	−0.0620412	−	0.998074i	$-0.519761\pi$
$433$	−1118.00	−0.124082	−0.0620412	+	0.998074i	$0.519761\pi$
$434$	2592.00	0.286682
$435$	500.000	0.0551107
$436$	−4400.00	−0.483307
$437$	−7800.00	−0.853832
$438$	7024.00	0.766255
$439$	−2600.00	−0.282668	−0.141334	−	0.989962i	$-0.545139\pi$
$439$	−2600.00	−0.282668	−0.141334	+	0.989962i	$0.545139\pi$
$440$	0	0
$441$	7061.00	0.762445
$442$	3952.00	0.425288
$443$	−11958.0	−1.28249	−0.641243	−	0.767337i	$-0.721581\pi$
$443$	−11958.0	−1.28249	−0.641243	+	0.767337i	$0.721581\pi$
$444$	4256.00	0.454912
$445$	750.000	0.0798953
$446$	25912.0	2.75105
$447$	4100.00	0.433833
$448$	−3072.00	−0.323970
$449$	−17050.0	−1.79207	−0.896035	−	0.443984i	$-0.853565\pi$
$449$	−17050.0	−1.79207	−0.896035	+	0.443984i	$0.853565\pi$
$450$	2300.00	0.240940
$451$	704.000	0.0735035
$452$	12496.0	1.30036
$453$	3704.00	0.384170
$454$	−2584.00	−0.267121
$455$	1140.00	0.117459
$456$	0	0
$457$	−9494.00	−0.971796	−0.485898	−	0.874016i	$-0.661507\pi$
$457$	−9494.00	−0.971796	−0.485898	+	0.874016i	$0.661507\pi$
$458$	−15000.0	−1.53036
$459$	−2600.00	−0.264396
$460$	3120.00	0.316241
$461$	−11418.0	−1.15356	−0.576778	−	0.816901i	$-0.695690\pi$
$461$	−11418.0	−1.15356	−0.576778	+	0.816901i	$0.695690\pi$
$462$	−1536.00	−0.154678
$463$	7962.00	0.799191	0.399596	−	0.916692i	$-0.369151\pi$
$463$	7962.00	0.799191	0.399596	+	0.916692i	$0.369151\pi$
$464$	3200.00	0.320164
$465$	1080.00	0.107707
$466$	−5928.00	−0.589290
$467$	6526.00	0.646654	0.323327	−	0.946287i	$-0.395199\pi$
$467$	6526.00	0.646654	0.323327	+	0.946287i	$0.395199\pi$
$468$	6992.00	0.690610
$469$	756.000	0.0744325
$470$	−10280.0	−1.00890
$471$	−4988.00	−0.487972
$472$	0	0
$473$	14144.0	1.37493
$474$	−4800.00	−0.465129
$475$	2500.00	0.241490
$476$	1248.00	0.120172
$477$	−46.0000	−0.00441550
$478$	−5600.00	−0.535854
$479$	17400.0	1.65976	0.829881	−	0.557940i	$-0.188408\pi$
$479$	17400.0	1.65976	0.829881	+	0.557940i	$0.188408\pi$
$480$	−2560.00	−0.243432
$481$	−10108.0	−0.958181
$482$	−12088.0	−1.14231
$483$	−936.000	−0.0881770
$484$	−2456.00	−0.230654
$485$	−1930.00	−0.180694
$486$	−14168.0	−1.32237
$487$	1166.00	0.108494	0.0542469	−	0.998528i	$-0.482724\pi$
$487$	1166.00	0.108494	0.0542469	+	0.998528i	$0.482724\pi$
$488$	0	0
$489$	5524.00	0.510846
$490$	−6140.00	−0.566075
$491$	7072.00	0.650010	0.325005	−	0.945712i	$-0.394634\pi$
$491$	7072.00	0.650010	0.325005	+	0.945712i	$0.394634\pi$
$492$	352.000	0.0322548
$493$	−1300.00	−0.118761
$494$	15200.0	1.38437
$495$	3680.00	0.334149
$496$	6912.00	0.625722
$497$	2472.00	0.223107
$498$	−2256.00	−0.203000
$499$	100.000	0.00897117	0.00448559	−	0.999990i	$-0.498572\pi$
$499$	100.000	0.00897117	0.00448559	+	0.999990i	$0.498572\pi$
$500$	−1000.00	−0.0894427
$501$	6252.00	0.557522
$502$	4992.00	0.443832
$503$	2602.00	0.230651	0.115325	−	0.993328i	$-0.463209\pi$
$503$	2602.00	0.230651	0.115325	+	0.993328i	$0.463209\pi$
$504$	0	0
$505$	−3510.00	−0.309293
$506$	9984.00	0.877160
$507$	−1506.00	−0.131921
$508$	14768.0	1.28981
$509$	11150.0	0.970953	0.485476	−	0.874250i	$-0.338646\pi$
$509$	11150.0	0.970953	0.485476	+	0.874250i	$0.338646\pi$
$510$	1040.00	0.0902980
$511$	−5268.00	−0.456052
$512$	−16384.0	−1.41421
$513$	−10000.0	−0.860645
$514$	−8424.00	−0.722892
$515$	2990.00	0.255835
$516$	7072.00	0.603348
$517$	−16448.0	−1.39919
$518$	−6384.00	−0.541500
$519$	−156.000	−0.0131939
$520$	0	0
$521$	−3638.00	−0.305919	−0.152959	−	0.988232i	$-0.548880\pi$
$521$	−3638.00	−0.305919	−0.152959	+	0.988232i	$0.548880\pi$
$522$	−4600.00	−0.385702
$523$	−2078.00	−0.173737	−0.0868686	−	0.996220i	$-0.527686\pi$
$523$	−2078.00	−0.173737	−0.0868686	+	0.996220i	$0.527686\pi$
$524$	−17664.0	−1.47262
$525$	300.000	0.0249392
$526$	14552.0	1.20627
$527$	−2808.00	−0.232103
$528$	−4096.00	−0.337605
$529$	−6083.00	−0.499959
$530$	40.0000	0.00327828
$531$	−11500.0	−0.939845
$532$	4800.00	0.391177
$533$	−836.000	−0.0679384
$534$	1200.00	0.0972455
$535$	5970.00	0.482440
$536$	0	0
$537$	−2600.00	−0.208935
$538$	26200.0	2.09956
$539$	−9824.00	−0.785064
$540$	4000.00	0.318764
$541$	5622.00	0.446781	0.223391	−	0.974729i	$-0.428287\pi$
$541$	5622.00	0.446781	0.223391	+	0.974729i	$0.428287\pi$
$542$	17552.0	1.39100
$543$	3484.00	0.275346
$544$	6656.00	0.524584
$545$	2750.00	0.216141
$546$	1824.00	0.142967
$547$	16486.0	1.28865	0.644324	−	0.764753i	$-0.277139\pi$
$547$	16486.0	1.28865	0.644324	+	0.764753i	$0.277139\pi$
$548$	−18672.0	−1.45553
$549$	11914.0	0.926188
$550$	−3200.00	−0.248088
$551$	−5000.00	−0.386583
$552$	0	0
$553$	3600.00	0.276831
$554$	−2184.00	−0.167490
$555$	−2660.00	−0.203443
$556$	−5600.00	−0.427146
$557$	11706.0	0.890483	0.445242	−	0.895410i	$-0.353118\pi$
$557$	11706.0	0.890483	0.445242	+	0.895410i	$0.353118\pi$
$558$	−9936.00	−0.753807
$559$	−16796.0	−1.27083
$560$	1920.00	0.144884
$561$	1664.00	0.125230
$562$	27432.0	2.05898
$563$	−25038.0	−1.87429	−0.937146	−	0.348939i	$-0.886542\pi$
$563$	−25038.0	−1.87429	−0.937146	+	0.348939i	$0.886542\pi$
$564$	−8224.00	−0.613994
$565$	−7810.00	−0.581538
$566$	−37128.0	−2.75725
$567$	2526.00	0.187094
$568$	0	0
$569$	17550.0	1.29303	0.646515	−	0.762901i	$-0.276226\pi$
$569$	17550.0	1.29303	0.646515	+	0.762901i	$0.276226\pi$
$570$	4000.00	0.293933
$571$	10712.0	0.785084	0.392542	−	0.919734i	$-0.371596\pi$
$571$	10712.0	0.785084	0.392542	+	0.919734i	$0.371596\pi$
$572$	−9728.00	−0.711098
$573$	7544.00	0.550009
$574$	−528.000	−0.0383942
$575$	−1950.00	−0.141427
$576$	11776.0	0.851852
$577$	−13654.0	−0.985136	−0.492568	−	0.870274i	$-0.663942\pi$
$577$	−13654.0	−0.985136	−0.492568	+	0.870274i	$0.663942\pi$
$578$	16948.0	1.21963
$579$	−716.000	−0.0513920
$580$	2000.00	0.143182
$581$	1692.00	0.120819
$582$	−3088.00	−0.219934
$583$	64.0000	0.00454650
$584$	0	0
$585$	−4370.00	−0.308850
$586$	−19368.0	−1.36533
$587$	14166.0	0.996071	0.498035	−	0.867157i	$-0.334055\pi$
$587$	14166.0	0.996071	0.498035	+	0.867157i	$0.334055\pi$
$588$	−4912.00	−0.344502
$589$	−10800.0	−0.755528
$590$	10000.0	0.697786
$591$	−4428.00	−0.308196
$592$	−17024.0	−1.18190
$593$	17842.0	1.23555	0.617777	−	0.786354i	$-0.288034\pi$
$593$	17842.0	1.23555	0.617777	+	0.786354i	$0.288034\pi$
$594$	12800.0	0.884159
$595$	−780.000	−0.0537427
$596$	16400.0	1.12713
$597$	−5200.00	−0.356485
$598$	−11856.0	−0.810749
$599$	−17600.0	−1.20053	−0.600264	−	0.799802i	$-0.704938\pi$
$599$	−17600.0	−1.20053	−0.600264	+	0.799802i	$0.704938\pi$
$600$	0	0
$601$	27302.0	1.85303	0.926516	−	0.376256i	$-0.122789\pi$
$601$	27302.0	1.85303	0.926516	+	0.376256i	$0.122789\pi$
$602$	−10608.0	−0.718189
$603$	−2898.00	−0.195714
$604$	14816.0	0.998103
$605$	1535.00	0.103151
$606$	−5616.00	−0.376459
$607$	−3794.00	−0.253696	−0.126848	−	0.991922i	$-0.540486\pi$
$607$	−3794.00	−0.253696	−0.126848	+	0.991922i	$0.540486\pi$
$608$	25600.0	1.70759
$609$	−600.000	−0.0399232
$610$	−10360.0	−0.687646
$611$	19532.0	1.29326
$612$	−4784.00	−0.315983
$613$	−13238.0	−0.872231	−0.436116	−	0.899891i	$-0.643646\pi$
$613$	−13238.0	−0.872231	−0.436116	+	0.899891i	$0.643646\pi$
$614$	10376.0	0.681989
$615$	−220.000	−0.0144248
$616$	0	0
$617$	−11574.0	−0.755189	−0.377595	−	0.925971i	$-0.623249\pi$
$617$	−11574.0	−0.755189	−0.377595	+	0.925971i	$0.623249\pi$
$618$	4784.00	0.311393
$619$	8300.00	0.538942	0.269471	−	0.963008i	$-0.413151\pi$
$619$	8300.00	0.538942	0.269471	+	0.963008i	$0.413151\pi$
$620$	4320.00	0.279831
$621$	7800.00	0.504031
$622$	−29328.0	−1.89059
$623$	−900.000	−0.0578776
$624$	4864.00	0.312045
$625$	625.000	0.0400000
$626$	−6248.00	−0.398914
$627$	6400.00	0.407642
$628$	−19952.0	−1.26779
$629$	6916.00	0.438409
$630$	−2760.00	−0.174541
$631$	−7508.00	−0.473675	−0.236837	−	0.971549i	$-0.576111\pi$
$631$	−7508.00	−0.473675	−0.236837	+	0.971549i	$0.576111\pi$
$632$	0	0
$633$	−2336.00	−0.146679
$634$	−5704.00	−0.357310
$635$	−9230.00	−0.576821
$636$	32.0000	0.00199510
$637$	11666.0	0.725626
$638$	6400.00	0.397145
$639$	−9476.00	−0.586643
$640$	0	0
$641$	−27378.0	−1.68700	−0.843499	−	0.537130i	$-0.819508\pi$
$641$	−27378.0	−1.68700	−0.843499	+	0.537130i	$0.819508\pi$
$642$	9552.00	0.587208
$643$	1842.00	0.112973	0.0564863	−	0.998403i	$-0.482010\pi$
$643$	1842.00	0.112973	0.0564863	+	0.998403i	$0.482010\pi$
$644$	−3744.00	−0.229090
$645$	−4420.00	−0.269825
$646$	−10400.0	−0.633409
$647$	−10114.0	−0.614563	−0.307282	−	0.951619i	$-0.599419\pi$
$647$	−10114.0	−0.614563	−0.307282	+	0.951619i	$0.599419\pi$
$648$	0	0
$649$	16000.0	0.967727
$650$	3800.00	0.229305
$651$	−1296.00	−0.0780250
$652$	22096.0	1.32722
$653$	10402.0	0.623372	0.311686	−	0.950185i	$-0.399106\pi$
$653$	10402.0	0.623372	0.311686	+	0.950185i	$0.399106\pi$
$654$	4400.00	0.263079
$655$	11040.0	0.658578
$656$	−1408.00	−0.0838006
$657$	20194.0	1.19915
$658$	12336.0	0.730862
$659$	7100.00	0.419692	0.209846	−	0.977734i	$-0.432704\pi$
$659$	7100.00	0.419692	0.209846	+	0.977734i	$0.432704\pi$
$660$	−2560.00	−0.150982
$661$	−7118.00	−0.418847	−0.209424	−	0.977825i	$-0.567159\pi$
$661$	−7118.00	−0.418847	−0.209424	+	0.977825i	$0.567159\pi$
$662$	16032.0	0.941241
$663$	−1976.00	−0.115749
$664$	0	0
$665$	−3000.00	−0.174940
$666$	24472.0	1.42383
$667$	3900.00	0.226400
$668$	25008.0	1.44849
$669$	−12956.0	−0.748741
$670$	2520.00	0.145308
$671$	−16576.0	−0.953665
$672$	3072.00	0.176347
$673$	−31278.0	−1.79150	−0.895749	−	0.444560i	$-0.853360\pi$
$673$	−31278.0	−1.79150	−0.895749	+	0.444560i	$0.853360\pi$
$674$	−35464.0	−2.02674
$675$	−2500.00	−0.142556
$676$	−6024.00	−0.342740
$677$	−30054.0	−1.70616	−0.853079	−	0.521782i	$-0.825268\pi$
$677$	−30054.0	−1.70616	−0.853079	+	0.521782i	$0.825268\pi$
$678$	−12496.0	−0.707826
$679$	2316.00	0.130898
$680$	0	0
$681$	1292.00	0.0727012
$682$	13824.0	0.776171
$683$	−4518.00	−0.253113	−0.126557	−	0.991959i	$-0.540393\pi$
$683$	−4518.00	−0.253113	−0.126557	+	0.991959i	$0.540393\pi$
$684$	−18400.0	−1.02857
$685$	11670.0	0.650931
$686$	15600.0	0.868237
$687$	7500.00	0.416511
$688$	−28288.0	−1.56754
$689$	−76.0000	−0.00420228
$690$	−3120.00	−0.172140
$691$	29272.0	1.61152	0.805759	−	0.592243i	$-0.201758\pi$
$691$	29272.0	1.61152	0.805759	+	0.592243i	$0.201758\pi$
$692$	−624.000	−0.0342788
$693$	−4416.00	−0.242063
$694$	6856.00	0.375000
$695$	3500.00	0.191025
$696$	0	0
$697$	572.000	0.0310847
$698$	−4600.00	−0.249445
$699$	2964.00	0.160385
$700$	1200.00	0.0647939
$701$	−5798.00	−0.312393	−0.156196	−	0.987726i	$-0.549923\pi$
$701$	−5798.00	−0.312393	−0.156196	+	0.987726i	$0.549923\pi$
$702$	−15200.0	−0.817218
$703$	26600.0	1.42708
$704$	−16384.0	−0.877124
$705$	5140.00	0.274587
$706$	17592.0	0.937796
$707$	4212.00	0.224057
$708$	8000.00	0.424659
$709$	8950.00	0.474082	0.237041	−	0.971500i	$-0.423822\pi$
$709$	8950.00	0.474082	0.237041	+	0.971500i	$0.423822\pi$
$710$	8240.00	0.435552
$711$	−13800.0	−0.727905
$712$	0	0
$713$	8424.00	0.442470
$714$	−1248.00	−0.0654135
$715$	6080.00	0.318013
$716$	−10400.0	−0.542830
$717$	2800.00	0.145841
$718$	−7200.00	−0.374236
$719$	7800.00	0.404577	0.202289	−	0.979326i	$-0.435162\pi$
$719$	7800.00	0.404577	0.202289	+	0.979326i	$0.435162\pi$
$720$	−7360.00	−0.380960
$721$	−3588.00	−0.185332
$722$	−12564.0	−0.647623
$723$	6044.00	0.310897
$724$	13936.0	0.715369
$725$	−1250.00	−0.0640329
$726$	2456.00	0.125552
$727$	−8554.00	−0.436383	−0.218191	−	0.975906i	$-0.570016\pi$
$727$	−8554.00	−0.436383	−0.218191	+	0.975906i	$0.570016\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	−17560.0	−0.890308
$731$	11492.0	0.581460
$732$	−8288.00	−0.418488
$733$	2882.00	0.145224	0.0726119	−	0.997360i	$-0.476867\pi$
$733$	2882.00	0.145224	0.0726119	+	0.997360i	$0.476867\pi$
$734$	23496.0	1.18154
$735$	3070.00	0.154066
$736$	−19968.0	−1.00004
$737$	4032.00	0.201521
$738$	2024.00	0.100955
$739$	18700.0	0.930840	0.465420	−	0.885090i	$-0.345903\pi$
$739$	18700.0	0.930840	0.465420	+	0.885090i	$0.345903\pi$
$740$	−10640.0	−0.528560
$741$	−7600.00	−0.376779
$742$	−48.0000	−0.00237485
$743$	12242.0	0.604462	0.302231	−	0.953235i	$-0.402269\pi$
$743$	12242.0	0.604462	0.302231	+	0.953235i	$0.402269\pi$
$744$	0	0
$745$	−10250.0	−0.504068
$746$	8312.00	0.407941
$747$	−6486.00	−0.317685
$748$	6656.00	0.325358
$749$	−7164.00	−0.349488
$750$	1000.00	0.0486864
$751$	−31148.0	−1.51346	−0.756729	−	0.653729i	$-0.773204\pi$
$751$	−31148.0	−1.51346	−0.756729	+	0.653729i	$0.773204\pi$
$752$	32896.0	1.59520
$753$	−2496.00	−0.120796
$754$	−7600.00	−0.367076
$755$	−9260.00	−0.446365
$756$	−4800.00	−0.230918
$757$	−7694.00	−0.369410	−0.184705	−	0.982794i	$-0.559133\pi$
$757$	−7694.00	−0.369410	−0.184705	+	0.982794i	$0.559133\pi$
$758$	−31600.0	−1.51420
$759$	−4992.00	−0.238733
$760$	0	0
$761$	−4518.00	−0.215213	−0.107607	−	0.994194i	$-0.534319\pi$
$761$	−4518.00	−0.215213	−0.107607	+	0.994194i	$0.534319\pi$
$762$	−14768.0	−0.702084
$763$	−3300.00	−0.156577
$764$	30176.0	1.42897
$765$	2990.00	0.141312
$766$	30072.0	1.41847
$767$	−19000.0	−0.894459
$768$	8192.00	0.384900
$769$	−39550.0	−1.85463	−0.927314	−	0.374283i	$-0.877889\pi$
$769$	−39550.0	−1.85463	−0.927314	+	0.374283i	$0.877889\pi$
$770$	3840.00	0.179719
$771$	4212.00	0.196746
$772$	−2864.00	−0.133520
$773$	22122.0	1.02933	0.514666	−	0.857391i	$-0.327916\pi$
$773$	22122.0	1.02933	0.514666	+	0.857391i	$0.327916\pi$
$774$	40664.0	1.88842
$775$	−2700.00	−0.125144
$776$	0	0
$777$	3192.00	0.147378
$778$	7800.00	0.359439
$779$	2200.00	0.101185
$780$	3040.00	0.139551
$781$	13184.0	0.604047
$782$	8112.00	0.370952
$783$	5000.00	0.228206
$784$	19648.0	0.895044
$785$	12470.0	0.566972
$786$	17664.0	0.801595
$787$	−16634.0	−0.753416	−0.376708	−	0.926332i	$-0.622944\pi$
$787$	−16634.0	−0.753416	−0.376708	+	0.926332i	$0.622944\pi$
$788$	−17712.0	−0.800716
$789$	−7276.00	−0.328305
$790$	12000.0	0.540431
$791$	9372.00	0.421277
$792$	0	0
$793$	19684.0	0.881462
$794$	−55144.0	−2.46472
$795$	−20.0000	−0.000892235 0
$796$	−20800.0	−0.926176
$797$	27586.0	1.22603	0.613015	−	0.790071i	$-0.289956\pi$
$797$	27586.0	1.22603	0.613015	+	0.790071i	$0.289956\pi$
$798$	−4800.00	−0.212930
$799$	−13364.0	−0.591720
$800$	6400.00	0.282843
$801$	3450.00	0.152184
$802$	−25608.0	−1.12749
$803$	−28096.0	−1.23473
$804$	2016.00	0.0884314
$805$	2340.00	0.102452
$806$	−16416.0	−0.717406
$807$	−13100.0	−0.571427
$808$	0	0
$809$	3850.00	0.167316	0.0836581	−	0.996495i	$-0.473340\pi$
$809$	3850.00	0.167316	0.0836581	+	0.996495i	$0.473340\pi$
$810$	8420.00	0.365245
$811$	10032.0	0.434366	0.217183	−	0.976131i	$-0.430313\pi$
$811$	10032.0	0.434366	0.217183	+	0.976131i	$0.430313\pi$
$812$	−2400.00	−0.103724
$813$	−8776.00	−0.378583
$814$	−34048.0	−1.46607
$815$	−13810.0	−0.593550
$816$	−3328.00	−0.142774
$817$	44200.0	1.89273
$818$	−44600.0	−1.90636
$819$	5244.00	0.223736
$820$	−880.000	−0.0374767
$821$	20562.0	0.874079	0.437039	−	0.899442i	$-0.356027\pi$
$821$	20562.0	0.874079	0.437039	+	0.899442i	$0.356027\pi$
$822$	18672.0	0.792288
$823$	10322.0	0.437184	0.218592	−	0.975816i	$-0.429854\pi$
$823$	10322.0	0.437184	0.218592	+	0.975816i	$0.429854\pi$
$824$	0	0
$825$	1600.00	0.0675210
$826$	−12000.0	−0.505488
$827$	8846.00	0.371954	0.185977	−	0.982554i	$-0.440455\pi$
$827$	8846.00	0.371954	0.185977	+	0.982554i	$0.440455\pi$
$828$	14352.0	0.602375
$829$	−25350.0	−1.06205	−0.531026	−	0.847355i	$-0.678194\pi$
$829$	−25350.0	−1.06205	−0.531026	+	0.847355i	$0.678194\pi$
$830$	5640.00	0.235864
$831$	1092.00	0.0455849
$832$	19456.0	0.810716
$833$	−7982.00	−0.332005
$834$	5600.00	0.232509
$835$	−15630.0	−0.647783
$836$	25600.0	1.05908
$837$	10800.0	0.446001
$838$	54800.0	2.25899
$839$	46000.0	1.89284	0.946422	−	0.322932i	$-0.104669\pi$
$839$	46000.0	1.89284	0.946422	+	0.322932i	$0.104669\pi$
$840$	0	0
$841$	−21889.0	−0.897495
$842$	21752.0	0.890289
$843$	−13716.0	−0.560385
$844$	−9344.00	−0.381083
$845$	3765.00	0.153278
$846$	−47288.0	−1.92174
$847$	−1842.00	−0.0747248
$848$	−128.000	−0.00518342
$849$	18564.0	0.750430
$850$	−2600.00	−0.104917
$851$	−20748.0	−0.835761
$852$	6592.00	0.265068
$853$	−16998.0	−0.682298	−0.341149	−	0.940009i	$-0.610816\pi$
$853$	−16998.0	−0.682298	−0.341149	+	0.940009i	$0.610816\pi$
$854$	12432.0	0.498143
$855$	11500.0	0.459990
$856$	0	0
$857$	−26494.0	−1.05603	−0.528015	−	0.849235i	$-0.677064\pi$
$857$	−26494.0	−1.05603	−0.528015	+	0.849235i	$0.677064\pi$
$858$	9728.00	0.387073
$859$	−21500.0	−0.853982	−0.426991	−	0.904256i	$-0.640426\pi$
$859$	−21500.0	−0.853982	−0.426991	+	0.904256i	$0.640426\pi$
$860$	−17680.0	−0.701027
$861$	264.000	0.0104496
$862$	−30768.0	−1.21573
$863$	25762.0	1.01616	0.508082	−	0.861309i	$-0.330355\pi$
$863$	25762.0	1.01616	0.508082	+	0.861309i	$0.330355\pi$
$864$	−25600.0	−1.00802
$865$	390.000	0.0153299
$866$	4472.00	0.175479
$867$	−8474.00	−0.331940
$868$	−5184.00	−0.202715
$869$	19200.0	0.749500
$870$	−2000.00	−0.0779383
$871$	−4788.00	−0.186263
$872$	0	0
$873$	−8878.00	−0.344186
$874$	31200.0	1.20750
$875$	−750.000	−0.0289767
$876$	−14048.0	−0.541824
$877$	30546.0	1.17613	0.588064	−	0.808814i	$-0.299890\pi$
$877$	30546.0	1.17613	0.588064	+	0.808814i	$0.299890\pi$
$878$	10400.0	0.399753
$879$	9684.00	0.371596
$880$	10240.0	0.392262
$881$	32942.0	1.25976	0.629878	−	0.776694i	$-0.283105\pi$
$881$	32942.0	1.25976	0.629878	+	0.776694i	$0.283105\pi$
$882$	−28244.0	−1.07826
$883$	−27118.0	−1.03351	−0.516757	−	0.856132i	$-0.672861\pi$
$883$	−27118.0	−1.03351	−0.516757	+	0.856132i	$0.672861\pi$
$884$	−7904.00	−0.300724
$885$	−5000.00	−0.189913
$886$	47832.0	1.81371
$887$	−38634.0	−1.46246	−0.731230	−	0.682131i	$-0.761054\pi$
$887$	−38634.0	−1.46246	−0.731230	+	0.682131i	$0.761054\pi$
$888$	0	0
$889$	11076.0	0.417860
$890$	−3000.00	−0.112989
$891$	13472.0	0.506542
$892$	−51824.0	−1.94529
$893$	−51400.0	−1.92613
$894$	−16400.0	−0.613532
$895$	6500.00	0.242761
$896$	0	0
$897$	5928.00	0.220658
$898$	68200.0	2.53437
$899$	5400.00	0.200334
$900$	−4600.00	−0.170370
$901$	52.0000	0.00192272
$902$	−2816.00	−0.103950
$903$	5304.00	0.195466
$904$	0	0
$905$	−8710.00	−0.319923
$906$	−14816.0	−0.543299
$907$	−1794.00	−0.0656767	−0.0328384	−	0.999461i	$-0.510455\pi$
$907$	−1794.00	−0.0656767	−0.0328384	+	0.999461i	$0.510455\pi$
$908$	5168.00	0.188883
$909$	−16146.0	−0.589141
$910$	−4560.00	−0.166113
$911$	41732.0	1.51772	0.758860	−	0.651254i	$-0.225757\pi$
$911$	41732.0	1.51772	0.758860	+	0.651254i	$0.225757\pi$
$912$	−12800.0	−0.464748
$913$	9024.00	0.327109
$914$	37976.0	1.37433
$915$	5180.00	0.187154
$916$	30000.0	1.08213
$917$	−13248.0	−0.477086
$918$	10400.0	0.373912
$919$	29200.0	1.04812	0.524058	−	0.851682i	$-0.324417\pi$
$919$	29200.0	1.04812	0.524058	+	0.851682i	$0.324417\pi$
$920$	0	0
$921$	−5188.00	−0.185614
$922$	45672.0	1.63137
$923$	−15656.0	−0.558314
$924$	3072.00	0.109374
$925$	6650.00	0.236379
$926$	−31848.0	−1.13023
$927$	13754.0	0.487315
$928$	−12800.0	−0.452781
$929$	−48650.0	−1.71814	−0.859071	−	0.511856i	$-0.828958\pi$
$929$	−48650.0	−1.71814	−0.859071	+	0.511856i	$0.828958\pi$
$930$	−4320.00	−0.152321
$931$	−30700.0	−1.08072
$932$	11856.0	0.416691
$933$	14664.0	0.514553
$934$	−26104.0	−0.914506
$935$	−4160.00	−0.145504
$936$	0	0
$937$	−11334.0	−0.395161	−0.197580	−	0.980287i	$-0.563308\pi$
$937$	−11334.0	−0.395161	−0.197580	+	0.980287i	$0.563308\pi$
$938$	−3024.00	−0.105263
$939$	3124.00	0.108571
$940$	20560.0	0.713397
$941$	−31178.0	−1.08010	−0.540050	−	0.841633i	$-0.681595\pi$
$941$	−31178.0	−1.08010	−0.540050	+	0.841633i	$0.681595\pi$
$942$	19952.0	0.690097
$943$	−1716.00	−0.0592584
$944$	−32000.0	−1.10330
$945$	3000.00	0.103270
$946$	−56576.0	−1.94444
$947$	4686.00	0.160797	0.0803984	−	0.996763i	$-0.474381\pi$
$947$	4686.00	0.160797	0.0803984	+	0.996763i	$0.474381\pi$
$948$	9600.00	0.328896
$949$	33364.0	1.14124
$950$	−10000.0	−0.341519
$951$	2852.00	0.0972476
$952$	0	0
$953$	−598.000	−0.0203265	−0.0101632	−	0.999948i	$-0.503235\pi$
$953$	−598.000	−0.0203265	−0.0101632	+	0.999948i	$0.503235\pi$
$954$	184.000	0.00624447
$955$	−18860.0	−0.639053
$956$	11200.0	0.378906
$957$	−3200.00	−0.108089
$958$	−69600.0	−2.34726
$959$	−14004.0	−0.471546
$960$	5120.00	0.172133
$961$	−18127.0	−0.608472
$962$	40432.0	1.35507
$963$	27462.0	0.918952
$964$	24176.0	0.807735
$965$	1790.00	0.0597121
$966$	3744.00	0.124701
$967$	41726.0	1.38761	0.693804	−	0.720163i	$-0.255933\pi$
$967$	41726.0	1.38761	0.693804	+	0.720163i	$0.255933\pi$
$968$	0	0
$969$	5200.00	0.172392
$970$	7720.00	0.255540
$971$	24312.0	0.803511	0.401756	−	0.915747i	$-0.368400\pi$
$971$	24312.0	0.803511	0.401756	+	0.915747i	$0.368400\pi$
$972$	28336.0	0.935059
$973$	−4200.00	−0.138382
$974$	−4664.00	−0.153433
$975$	−1900.00	−0.0624089
$976$	33152.0	1.08726
$977$	40946.0	1.34082	0.670409	−	0.741992i	$-0.266119\pi$
$977$	40946.0	1.34082	0.670409	+	0.741992i	$0.266119\pi$
$978$	−22096.0	−0.722446
$979$	−4800.00	−0.156699
$980$	12280.0	0.400276
$981$	12650.0	0.411706
$982$	−28288.0	−0.919253
$983$	42282.0	1.37191	0.685954	−	0.727645i	$-0.259385\pi$
$983$	42282.0	1.37191	0.685954	+	0.727645i	$0.259385\pi$
$984$	0	0
$985$	11070.0	0.358091
$986$	5200.00	0.167953
$987$	−6168.00	−0.198916
$988$	−30400.0	−0.978900
$989$	−34476.0	−1.10847
$990$	−14720.0	−0.472558
$991$	1172.00	0.0375679	0.0187840	−	0.999824i	$-0.494021\pi$
$991$	1172.00	0.0375679	0.0187840	+	0.999824i	$0.494021\pi$
$992$	−27648.0	−0.884904
$993$	−8016.00	−0.256173
$994$	−9888.00	−0.315521
$995$	13000.0	0.414199
$996$	4512.00	0.143542
$997$	−31614.0	−1.00424	−0.502119	−	0.864798i	$-0.667446\pi$
$997$	−31614.0	−1.00424	−0.502119	+	0.864798i	$0.667446\pi$
$998$	−400.000	−0.0126872
$999$	−26600.0	−0.842429

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.4.a.a.1.1	✓	1
3.2	odd	2		45.4.a.d.1.1		1
4.3	odd	2		80.4.a.d.1.1		1
5.2	odd	4		25.4.b.a.24.1		2
5.3	odd	4		25.4.b.a.24.2		2
5.4	even	2		25.4.a.c.1.1		1
7.2	even	3		245.4.e.f.116.1		2
7.3	odd	6		245.4.e.g.226.1		2
7.4	even	3		245.4.e.f.226.1		2
7.5	odd	6		245.4.e.g.116.1		2
7.6	odd	2		245.4.a.a.1.1		1
8.3	odd	2		320.4.a.h.1.1		1
8.5	even	2		320.4.a.g.1.1		1
9.2	odd	6		405.4.e.c.271.1		2
9.4	even	3		405.4.e.l.136.1		2
9.5	odd	6		405.4.e.c.136.1		2
9.7	even	3		405.4.e.l.271.1		2
11.10	odd	2		605.4.a.d.1.1		1
12.11	even	2		720.4.a.u.1.1		1
13.12	even	2		845.4.a.b.1.1		1
15.2	even	4		225.4.b.c.199.2		2
15.8	even	4		225.4.b.c.199.1		2
15.14	odd	2		225.4.a.b.1.1		1
16.3	odd	4		1280.4.d.l.641.1		2
16.5	even	4		1280.4.d.e.641.1		2
16.11	odd	4		1280.4.d.l.641.2		2
16.13	even	4		1280.4.d.e.641.2		2
17.16	even	2		1445.4.a.a.1.1		1
19.18	odd	2		1805.4.a.h.1.1		1
20.3	even	4		400.4.c.k.49.1		2
20.7	even	4		400.4.c.k.49.2		2
20.19	odd	2		400.4.a.m.1.1		1
21.20	even	2		2205.4.a.q.1.1		1
35.34	odd	2		1225.4.a.k.1.1		1
40.19	odd	2		1600.4.a.s.1.1		1
40.29	even	2		1600.4.a.bi.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.4.a.a.1.1	✓	1	1.1	even	1	trivial
25.4.a.c.1.1		1	5.4	even	2
25.4.b.a.24.1		2	5.2	odd	4
25.4.b.a.24.2		2	5.3	odd	4
45.4.a.d.1.1		1	3.2	odd	2
80.4.a.d.1.1		1	4.3	odd	2
225.4.a.b.1.1		1	15.14	odd	2
225.4.b.c.199.1		2	15.8	even	4
225.4.b.c.199.2		2	15.2	even	4
245.4.a.a.1.1		1	7.6	odd	2
245.4.e.f.116.1		2	7.2	even	3
245.4.e.f.226.1		2	7.4	even	3
245.4.e.g.116.1		2	7.5	odd	6
245.4.e.g.226.1		2	7.3	odd	6
320.4.a.g.1.1		1	8.5	even	2
320.4.a.h.1.1		1	8.3	odd	2
400.4.a.m.1.1		1	20.19	odd	2
400.4.c.k.49.1		2	20.3	even	4
400.4.c.k.49.2		2	20.7	even	4
405.4.e.c.136.1		2	9.5	odd	6
405.4.e.c.271.1		2	9.2	odd	6
405.4.e.l.136.1		2	9.4	even	3
405.4.e.l.271.1		2	9.7	even	3
605.4.a.d.1.1		1	11.10	odd	2
720.4.a.u.1.1		1	12.11	even	2
845.4.a.b.1.1		1	13.12	even	2
1225.4.a.k.1.1		1	35.34	odd	2
1280.4.d.e.641.1		2	16.5	even	4
1280.4.d.e.641.2		2	16.13	even	4
1280.4.d.l.641.1		2	16.3	odd	4
1280.4.d.l.641.2		2	16.11	odd	4
1445.4.a.a.1.1		1	17.16	even	2
1600.4.a.s.1.1		1	40.19	odd	2
1600.4.a.bi.1.1		1	40.29	even	2
1805.4.a.h.1.1		1	19.18	odd	2
2205.4.a.q.1.1		1	21.20	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	5.4.a.a.1.1

Level	$5$
Weight	$4$
Character	5.1
Self dual	yes
Analytic conductor	$0.295$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$