LMFDB - Embedded newform 50.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,4,Mod(1,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, 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("50.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$50 = 2 \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	50.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:	$2.95009550029$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	50.1

$q$ -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} -2.00000 q^{3} +4.00000 q^{4} +4.00000 q^{6} -26.0000 q^{7} -8.00000 q^{8} -23.0000 q^{9} -28.0000 q^{11} -8.00000 q^{12} -12.0000 q^{13} +52.0000 q^{14} +16.0000 q^{16} +64.0000 q^{17} +46.0000 q^{18} -60.0000 q^{19} +52.0000 q^{21} +56.0000 q^{22} +58.0000 q^{23} +16.0000 q^{24} +24.0000 q^{26} +100.000 q^{27} -104.000 q^{28} +90.0000 q^{29} -128.000 q^{31} -32.0000 q^{32} +56.0000 q^{33} -128.000 q^{34} -92.0000 q^{36} -236.000 q^{37} +120.000 q^{38} +24.0000 q^{39} +242.000 q^{41} -104.000 q^{42} -362.000 q^{43} -112.000 q^{44} -116.000 q^{46} -226.000 q^{47} -32.0000 q^{48} +333.000 q^{49} -128.000 q^{51} -48.0000 q^{52} +108.000 q^{53} -200.000 q^{54} +208.000 q^{56} +120.000 q^{57} -180.000 q^{58} -20.0000 q^{59} +542.000 q^{61} +256.000 q^{62} +598.000 q^{63} +64.0000 q^{64} -112.000 q^{66} +434.000 q^{67} +256.000 q^{68} -116.000 q^{69} -1128.00 q^{71} +184.000 q^{72} -632.000 q^{73} +472.000 q^{74} -240.000 q^{76} +728.000 q^{77} -48.0000 q^{78} -720.000 q^{79} +421.000 q^{81} -484.000 q^{82} +478.000 q^{83} +208.000 q^{84} +724.000 q^{86} -180.000 q^{87} +224.000 q^{88} -490.000 q^{89} +312.000 q^{91} +232.000 q^{92} +256.000 q^{93} +452.000 q^{94} +64.0000 q^{96} -1456.00 q^{97} -666.000 q^{98} +644.000 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

)

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

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−2.00000	−0.384900	−0.192450	−	0.981307i	$-0.561643\pi$
$3$	−2.00000	−0.384900	−0.192450	+	0.981307i	$0.561643\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	4.00000	0.272166
$7$	−26.0000	−1.40387	−0.701934	−	0.712242i	$-0.747680\pi$
$7$	−26.0000	−1.40387	−0.701934	+	0.712242i	$0.747680\pi$
$8$	−8.00000	−0.353553
$9$	−23.0000	−0.851852
$10$	0	0
$11$	−28.0000	−0.767483	−0.383742	−	0.923440i	$-0.625365\pi$
$11$	−28.0000	−0.767483	−0.383742	+	0.923440i	$0.625365\pi$
$12$	−8.00000	−0.192450
$13$	−12.0000	−0.256015	−0.128008	−	0.991773i	$-0.540858\pi$
$13$	−12.0000	−0.256015	−0.128008	+	0.991773i	$0.540858\pi$
$14$	52.0000	0.992685
$15$	0	0
$16$	16.0000	0.250000
$17$	64.0000	0.913075	0.456538	−	0.889704i	$-0.349089\pi$
$17$	64.0000	0.913075	0.456538	+	0.889704i	$0.349089\pi$
$18$	46.0000	0.602350
$19$	−60.0000	−0.724471	−0.362235	−	0.932087i	$-0.617986\pi$
$19$	−60.0000	−0.724471	−0.362235	+	0.932087i	$0.617986\pi$
$20$	0	0
$21$	52.0000	0.540349
$22$	56.0000	0.542693
$23$	58.0000	0.525819	0.262909	−	0.964821i	$-0.415318\pi$
$23$	58.0000	0.525819	0.262909	+	0.964821i	$0.415318\pi$
$24$	16.0000	0.136083
$25$	0	0
$26$	24.0000	0.181030
$27$	100.000	0.712778
$28$	−104.000	−0.701934
$29$	90.0000	0.576296	0.288148	−	0.957586i	$-0.406961\pi$
$29$	90.0000	0.576296	0.288148	+	0.957586i	$0.406961\pi$
$30$	0	0
$31$	−128.000	−0.741596	−0.370798	−	0.928714i	$-0.620916\pi$
$31$	−128.000	−0.741596	−0.370798	+	0.928714i	$0.620916\pi$
$32$	−32.0000	−0.176777
$33$	56.0000	0.295405
$34$	−128.000	−0.645642
$35$	0	0
$36$	−92.0000	−0.425926
$37$	−236.000	−1.04860	−0.524299	−	0.851534i	$-0.675673\pi$
$37$	−236.000	−1.04860	−0.524299	+	0.851534i	$0.675673\pi$
$38$	120.000	0.512278
$39$	24.0000	0.0985404
$40$	0	0
$41$	242.000	0.921806	0.460903	−	0.887450i	$-0.347526\pi$
$41$	242.000	0.921806	0.460903	+	0.887450i	$0.347526\pi$
$42$	−104.000	−0.382084
$43$	−362.000	−1.28383	−0.641913	−	0.766778i	$-0.721859\pi$
$43$	−362.000	−1.28383	−0.641913	+	0.766778i	$0.721859\pi$
$44$	−112.000	−0.383742
$45$	0	0
$46$	−116.000	−0.371810
$47$	−226.000	−0.701393	−0.350697	−	0.936489i	$-0.614055\pi$
$47$	−226.000	−0.701393	−0.350697	+	0.936489i	$0.614055\pi$
$48$	−32.0000	−0.0962250
$49$	333.000	0.970845
$50$	0	0
$51$	−128.000	−0.351443
$52$	−48.0000	−0.128008
$53$	108.000	0.279905	0.139952	−	0.990158i	$-0.455305\pi$
$53$	108.000	0.279905	0.139952	+	0.990158i	$0.455305\pi$
$54$	−200.000	−0.504010
$55$	0	0
$56$	208.000	0.496342
$57$	120.000	0.278849
$58$	−180.000	−0.407503
$59$	−20.0000	−0.0441318	−0.0220659	−	0.999757i	$-0.507024\pi$
$59$	−20.0000	−0.0441318	−0.0220659	+	0.999757i	$0.507024\pi$
$60$	0	0
$61$	542.000	1.13764	0.568820	−	0.822462i	$-0.307400\pi$
$61$	542.000	1.13764	0.568820	+	0.822462i	$0.307400\pi$
$62$	256.000	0.524388
$63$	598.000	1.19589
$64$	64.0000	0.125000
$65$	0	0
$66$	−112.000	−0.208883
$67$	434.000	0.791366	0.395683	−	0.918387i	$-0.370508\pi$
$67$	434.000	0.791366	0.395683	+	0.918387i	$0.370508\pi$
$68$	256.000	0.456538
$69$	−116.000	−0.202388
$70$	0	0
$71$	−1128.00	−1.88548	−0.942739	−	0.333531i	$-0.891760\pi$
$71$	−1128.00	−1.88548	−0.942739	+	0.333531i	$0.891760\pi$
$72$	184.000	0.301175
$73$	−632.000	−1.01329	−0.506644	−	0.862155i	$-0.669114\pi$
$73$	−632.000	−1.01329	−0.506644	+	0.862155i	$0.669114\pi$
$74$	472.000	0.741471
$75$	0	0
$76$	−240.000	−0.362235
$77$	728.000	1.07745
$78$	−48.0000	−0.0696786
$79$	−720.000	−1.02540	−0.512698	−	0.858569i	$-0.671354\pi$
$79$	−720.000	−1.02540	−0.512698	+	0.858569i	$0.671354\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	−484.000	−0.651815
$83$	478.000	0.632136	0.316068	−	0.948736i	$-0.397637\pi$
$83$	478.000	0.632136	0.316068	+	0.948736i	$0.397637\pi$
$84$	208.000	0.270175
$85$	0	0
$86$	724.000	0.907801
$87$	−180.000	−0.221816
$88$	224.000	0.271346
$89$	−490.000	−0.583594	−0.291797	−	0.956480i	$-0.594253\pi$
$89$	−490.000	−0.583594	−0.291797	+	0.956480i	$0.594253\pi$
$90$	0	0
$91$	312.000	0.359412
$92$	232.000	0.262909
$93$	256.000	0.285440
$94$	452.000	0.495960
$95$	0	0
$96$	64.0000	0.0680414
$97$	−1456.00	−1.52407	−0.762033	−	0.647538i	$-0.775799\pi$
$97$	−1456.00	−1.52407	−0.762033	+	0.647538i	$0.775799\pi$
$98$	−666.000	−0.686491
$99$	644.000	0.653782
$100$	0	0
$101$	−578.000	−0.569437	−0.284719	−	0.958611i	$-0.591900\pi$
$101$	−578.000	−0.569437	−0.284719	+	0.958611i	$0.591900\pi$
$102$	256.000	0.248508
$103$	−1462.00	−1.39859	−0.699297	−	0.714831i	$-0.746503\pi$
$103$	−1462.00	−1.39859	−0.699297	+	0.714831i	$0.746503\pi$
$104$	96.0000	0.0905151
$105$	0	0
$106$	−216.000	−0.197922
$107$	−966.000	−0.872773	−0.436387	−	0.899759i	$-0.643742\pi$
$107$	−966.000	−0.872773	−0.436387	+	0.899759i	$0.643742\pi$
$108$	400.000	0.356389
$109$	370.000	0.325134	0.162567	−	0.986698i	$-0.448023\pi$
$109$	370.000	0.325134	0.162567	+	0.986698i	$0.448023\pi$
$110$	0	0
$111$	472.000	0.403606
$112$	−416.000	−0.350967
$113$	528.000	0.439558	0.219779	−	0.975550i	$-0.429466\pi$
$113$	528.000	0.439558	0.219779	+	0.975550i	$0.429466\pi$
$114$	−240.000	−0.197176
$115$	0	0
$116$	360.000	0.288148
$117$	276.000	0.218087
$118$	40.0000	0.0312059
$119$	−1664.00	−1.28184
$120$	0	0
$121$	−547.000	−0.410969
$122$	−1084.00	−0.804432
$123$	−484.000	−0.354803
$124$	−512.000	−0.370798
$125$	0	0
$126$	−1196.00	−0.845620
$127$	1534.00	1.07181	0.535907	−	0.844277i	$-0.319970\pi$
$127$	1534.00	1.07181	0.535907	+	0.844277i	$0.319970\pi$
$128$	−128.000	−0.0883883
$129$	724.000	0.494145
$130$	0	0
$131$	12.0000	0.00800340	0.00400170	−	0.999992i	$-0.498726\pi$
$131$	12.0000	0.00800340	0.00400170	+	0.999992i	$0.498726\pi$
$132$	224.000	0.147702
$133$	1560.00	1.01706
$134$	−868.000	−0.559580
$135$	0	0
$136$	−512.000	−0.322821
$137$	1224.00	0.763309	0.381655	−	0.924305i	$-0.375354\pi$
$137$	1224.00	0.763309	0.381655	+	0.924305i	$0.375354\pi$
$138$	232.000	0.143110
$139$	3100.00	1.89164	0.945822	−	0.324685i	$-0.105258\pi$
$139$	3100.00	1.89164	0.945822	+	0.324685i	$0.105258\pi$
$140$	0	0
$141$	452.000	0.269966
$142$	2256.00	1.33323
$143$	336.000	0.196488
$144$	−368.000	−0.212963
$145$	0	0
$146$	1264.00	0.716503
$147$	−666.000	−0.373679
$148$	−944.000	−0.524299
$149$	250.000	0.137455	0.0687275	−	0.997635i	$-0.478106\pi$
$149$	250.000	0.137455	0.0687275	+	0.997635i	$0.478106\pi$
$150$	0	0
$151$	2152.00	1.15978	0.579892	−	0.814694i	$-0.303095\pi$
$151$	2152.00	1.15978	0.579892	+	0.814694i	$0.303095\pi$
$152$	480.000	0.256139
$153$	−1472.00	−0.777805
$154$	−1456.00	−0.761869
$155$	0	0
$156$	96.0000	0.0492702
$157$	524.000	0.266368	0.133184	−	0.991091i	$-0.457480\pi$
$157$	524.000	0.266368	0.133184	+	0.991091i	$0.457480\pi$
$158$	1440.00	0.725065
$159$	−216.000	−0.107735
$160$	0	0
$161$	−1508.00	−0.738180
$162$	−842.000	−0.408357
$163$	3518.00	1.69050	0.845249	−	0.534373i	$-0.179452\pi$
$163$	3518.00	1.69050	0.845249	+	0.534373i	$0.179452\pi$
$164$	968.000	0.460903
$165$	0	0
$166$	−956.000	−0.446988
$167$	534.000	0.247438	0.123719	−	0.992317i	$-0.460518\pi$
$167$	534.000	0.247438	0.123719	+	0.992317i	$0.460518\pi$
$168$	−416.000	−0.191042
$169$	−2053.00	−0.934456
$170$	0	0
$171$	1380.00	0.617142
$172$	−1448.00	−0.641913
$173$	−4252.00	−1.86863	−0.934317	−	0.356444i	$-0.883989\pi$
$173$	−4252.00	−1.86863	−0.934317	+	0.356444i	$0.883989\pi$
$174$	360.000	0.156848
$175$	0	0
$176$	−448.000	−0.191871
$177$	40.0000	0.0169864
$178$	980.000	0.412664
$179$	2500.00	1.04390	0.521952	−	0.852975i	$-0.325204\pi$
$179$	2500.00	1.04390	0.521952	+	0.852975i	$0.325204\pi$
$180$	0	0
$181$	−2578.00	−1.05868	−0.529340	−	0.848410i	$-0.677561\pi$
$181$	−2578.00	−1.05868	−0.529340	+	0.848410i	$0.677561\pi$
$182$	−624.000	−0.254143
$183$	−1084.00	−0.437878
$184$	−464.000	−0.185905
$185$	0	0
$186$	−512.000	−0.201837
$187$	−1792.00	−0.700770
$188$	−904.000	−0.350697
$189$	−2600.00	−1.00065
$190$	0	0
$191$	−768.000	−0.290945	−0.145473	−	0.989362i	$-0.546470\pi$
$191$	−768.000	−0.290945	−0.145473	+	0.989362i	$0.546470\pi$
$192$	−128.000	−0.0481125
$193$	2608.00	0.972684	0.486342	−	0.873769i	$-0.338331\pi$
$193$	2608.00	0.972684	0.486342	+	0.873769i	$0.338331\pi$
$194$	2912.00	1.07768
$195$	0	0
$196$	1332.00	0.485423
$197$	−5116.00	−1.85025	−0.925127	−	0.379659i	$-0.876041\pi$
$197$	−5116.00	−1.85025	−0.925127	+	0.379659i	$0.876041\pi$
$198$	−1288.00	−0.462294
$199$	−3480.00	−1.23965	−0.619826	−	0.784739i	$-0.712797\pi$
$199$	−3480.00	−1.23965	−0.619826	+	0.784739i	$0.712797\pi$
$200$	0	0
$201$	−868.000	−0.304597
$202$	1156.00	0.402653
$203$	−2340.00	−0.809043
$204$	−512.000	−0.175721
$205$	0	0
$206$	2924.00	0.988955
$207$	−1334.00	−0.447920
$208$	−192.000	−0.0640039
$209$	1680.00	0.556019
$210$	0	0
$211$	3132.00	1.02188	0.510938	−	0.859618i	$-0.329298\pi$
$211$	3132.00	1.02188	0.510938	+	0.859618i	$0.329298\pi$
$212$	432.000	0.139952
$213$	2256.00	0.725721
$214$	1932.00	0.617144
$215$	0	0
$216$	−800.000	−0.252005
$217$	3328.00	1.04110
$218$	−740.000	−0.229904
$219$	1264.00	0.390015
$220$	0	0
$221$	−768.000	−0.233761
$222$	−944.000	−0.285392
$223$	−62.0000	−0.0186181	−0.00930903	−	0.999957i	$-0.502963\pi$
$223$	−62.0000	−0.0186181	−0.00930903	+	0.999957i	$0.502963\pi$
$224$	832.000	0.248171
$225$	0	0
$226$	−1056.00	−0.310814
$227$	5314.00	1.55376	0.776878	−	0.629651i	$-0.216802\pi$
$227$	5314.00	1.55376	0.776878	+	0.629651i	$0.216802\pi$
$228$	480.000	0.139424
$229$	−190.000	−0.0548277	−0.0274139	−	0.999624i	$-0.508727\pi$
$229$	−190.000	−0.0548277	−0.0274139	+	0.999624i	$0.508727\pi$
$230$	0	0
$231$	−1456.00	−0.414709
$232$	−720.000	−0.203751
$233$	2408.00	0.677053	0.338526	−	0.940957i	$-0.390072\pi$
$233$	2408.00	0.677053	0.338526	+	0.940957i	$0.390072\pi$
$234$	−552.000	−0.154211
$235$	0	0
$236$	−80.0000	−0.0220659
$237$	1440.00	0.394675
$238$	3328.00	0.906396
$239$	−5680.00	−1.53727	−0.768637	−	0.639685i	$-0.779065\pi$
$239$	−5680.00	−1.53727	−0.768637	+	0.639685i	$0.779065\pi$
$240$	0	0
$241$	−278.000	−0.0743052	−0.0371526	−	0.999310i	$-0.511829\pi$
$241$	−278.000	−0.0743052	−0.0371526	+	0.999310i	$0.511829\pi$
$242$	1094.00	0.290599
$243$	−3542.00	−0.935059
$244$	2168.00	0.568820
$245$	0	0
$246$	968.000	0.250884
$247$	720.000	0.185476
$248$	1024.00	0.262194
$249$	−956.000	−0.243309
$250$	0	0
$251$	3252.00	0.817787	0.408893	−	0.912582i	$-0.365915\pi$
$251$	3252.00	0.817787	0.408893	+	0.912582i	$0.365915\pi$
$252$	2392.00	0.597944
$253$	−1624.00	−0.403557
$254$	−3068.00	−0.757888
$255$	0	0
$256$	256.000	0.0625000
$257$	−1536.00	−0.372813	−0.186407	−	0.982473i	$-0.559684\pi$
$257$	−1536.00	−0.372813	−0.186407	+	0.982473i	$0.559684\pi$
$258$	−1448.00	−0.349413
$259$	6136.00	1.47209
$260$	0	0
$261$	−2070.00	−0.490919
$262$	−24.0000	−0.00565926
$263$	4858.00	1.13900	0.569500	−	0.821991i	$-0.307137\pi$
$263$	4858.00	1.13900	0.569500	+	0.821991i	$0.307137\pi$
$264$	−448.000	−0.104441
$265$	0	0
$266$	−3120.00	−0.719171
$267$	980.000	0.224626
$268$	1736.00	0.395683
$269$	2610.00	0.591578	0.295789	−	0.955253i	$-0.404417\pi$
$269$	2610.00	0.591578	0.295789	+	0.955253i	$0.404417\pi$
$270$	0	0
$271$	−5168.00	−1.15843	−0.579213	−	0.815176i	$-0.696640\pi$
$271$	−5168.00	−1.15843	−0.579213	+	0.815176i	$0.696640\pi$
$272$	1024.00	0.228269
$273$	−624.000	−0.138338
$274$	−2448.00	−0.539741
$275$	0	0
$276$	−464.000	−0.101194
$277$	1924.00	0.417336	0.208668	−	0.977987i	$-0.433087\pi$
$277$	1924.00	0.417336	0.208668	+	0.977987i	$0.433087\pi$
$278$	−6200.00	−1.33759
$279$	2944.00	0.631730
$280$	0	0
$281$	3042.00	0.645803	0.322901	−	0.946433i	$-0.395342\pi$
$281$	3042.00	0.645803	0.322901	+	0.946433i	$0.395342\pi$
$282$	−904.000	−0.190895
$283$	1718.00	0.360864	0.180432	−	0.983587i	$-0.442250\pi$
$283$	1718.00	0.360864	0.180432	+	0.983587i	$0.442250\pi$
$284$	−4512.00	−0.942739
$285$	0	0
$286$	−672.000	−0.138938
$287$	−6292.00	−1.29409
$288$	736.000	0.150588
$289$	−817.000	−0.166294
$290$	0	0
$291$	2912.00	0.586613
$292$	−2528.00	−0.506644
$293$	−2292.00	−0.456997	−0.228498	−	0.973544i	$-0.573382\pi$
$293$	−2292.00	−0.456997	−0.228498	+	0.973544i	$0.573382\pi$
$294$	1332.00	0.264231
$295$	0	0
$296$	1888.00	0.370736
$297$	−2800.00	−0.547045
$298$	−500.000	−0.0971954
$299$	−696.000	−0.134618
$300$	0	0
$301$	9412.00	1.80232
$302$	−4304.00	−0.820091
$303$	1156.00	0.219176
$304$	−960.000	−0.181118
$305$	0	0
$306$	2944.00	0.549991
$307$	−5406.00	−1.00501	−0.502503	−	0.864576i	$-0.667587\pi$
$307$	−5406.00	−1.00501	−0.502503	+	0.864576i	$0.667587\pi$
$308$	2912.00	0.538723
$309$	2924.00	0.538319
$310$	0	0
$311$	−5688.00	−1.03710	−0.518548	−	0.855048i	$-0.673527\pi$
$311$	−5688.00	−1.03710	−0.518548	+	0.855048i	$0.673527\pi$
$312$	−192.000	−0.0348393
$313$	−7352.00	−1.32767	−0.663833	−	0.747881i	$-0.731072\pi$
$313$	−7352.00	−1.32767	−0.663833	+	0.747881i	$0.731072\pi$
$314$	−1048.00	−0.188351
$315$	0	0
$316$	−2880.00	−0.512698
$317$	3484.00	0.617290	0.308645	−	0.951177i	$-0.400124\pi$
$317$	3484.00	0.617290	0.308645	+	0.951177i	$0.400124\pi$
$318$	432.000	0.0761804
$319$	−2520.00	−0.442298
$320$	0	0
$321$	1932.00	0.335931
$322$	3016.00	0.521972
$323$	−3840.00	−0.661496
$324$	1684.00	0.288752
$325$	0	0
$326$	−7036.00	−1.19536
$327$	−740.000	−0.125144
$328$	−1936.00	−0.325908
$329$	5876.00	0.984664
$330$	0	0
$331$	−7868.00	−1.30654	−0.653269	−	0.757125i	$-0.726603\pi$
$331$	−7868.00	−1.30654	−0.653269	+	0.757125i	$0.726603\pi$
$332$	1912.00	0.316068
$333$	5428.00	0.893251
$334$	−1068.00	−0.174965
$335$	0	0
$336$	832.000	0.135087
$337$	−656.000	−0.106037	−0.0530187	−	0.998594i	$-0.516884\pi$
$337$	−656.000	−0.106037	−0.0530187	+	0.998594i	$0.516884\pi$
$338$	4106.00	0.660760
$339$	−1056.00	−0.169186
$340$	0	0
$341$	3584.00	0.569163
$342$	−2760.00	−0.436385
$343$	260.000	0.0409291
$344$	2896.00	0.453901
$345$	0	0
$346$	8504.00	1.32132
$347$	5754.00	0.890176	0.445088	−	0.895487i	$-0.353172\pi$
$347$	5754.00	0.890176	0.445088	+	0.895487i	$0.353172\pi$
$348$	−720.000	−0.110908
$349$	−3110.00	−0.477004	−0.238502	−	0.971142i	$-0.576656\pi$
$349$	−3110.00	−0.477004	−0.238502	+	0.971142i	$0.576656\pi$
$350$	0	0
$351$	−1200.00	−0.182482
$352$	896.000	0.135673
$353$	7808.00	1.17727	0.588637	−	0.808397i	$-0.299665\pi$
$353$	7808.00	1.17727	0.588637	+	0.808397i	$0.299665\pi$
$354$	−80.0000	−0.0120112
$355$	0	0
$356$	−1960.00	−0.291797
$357$	3328.00	0.493379
$358$	−5000.00	−0.738151
$359$	−9240.00	−1.35841	−0.679204	−	0.733949i	$-0.737675\pi$
$359$	−9240.00	−1.35841	−0.679204	+	0.733949i	$0.737675\pi$
$360$	0	0
$361$	−3259.00	−0.475142
$362$	5156.00	0.748600
$363$	1094.00	0.158182
$364$	1248.00	0.179706
$365$	0	0
$366$	2168.00	0.309626
$367$	3214.00	0.457137	0.228569	−	0.973528i	$-0.426595\pi$
$367$	3214.00	0.457137	0.228569	+	0.973528i	$0.426595\pi$
$368$	928.000	0.131455
$369$	−5566.00	−0.785242
$370$	0	0
$371$	−2808.00	−0.392949
$372$	1024.00	0.142720
$373$	348.000	0.0483077	0.0241538	−	0.999708i	$-0.492311\pi$
$373$	348.000	0.0483077	0.0241538	+	0.999708i	$0.492311\pi$
$374$	3584.00	0.495519
$375$	0	0
$376$	1808.00	0.247980
$377$	−1080.00	−0.147541
$378$	5200.00	0.707564
$379$	4940.00	0.669527	0.334764	−	0.942302i	$-0.391344\pi$
$379$	4940.00	0.669527	0.334764	+	0.942302i	$0.391344\pi$
$380$	0	0
$381$	−3068.00	−0.412542
$382$	1536.00	0.205729
$383$	−6142.00	−0.819430	−0.409715	−	0.912214i	$-0.634372\pi$
$383$	−6142.00	−0.819430	−0.409715	+	0.912214i	$0.634372\pi$
$384$	256.000	0.0340207
$385$	0	0
$386$	−5216.00	−0.687791
$387$	8326.00	1.09363
$388$	−5824.00	−0.762033
$389$	3050.00	0.397535	0.198768	−	0.980047i	$-0.436306\pi$
$389$	3050.00	0.397535	0.198768	+	0.980047i	$0.436306\pi$
$390$	0	0
$391$	3712.00	0.480112
$392$	−2664.00	−0.343246
$393$	−24.0000	−0.00308051
$394$	10232.0	1.30833
$395$	0	0
$396$	2576.00	0.326891
$397$	−5396.00	−0.682160	−0.341080	−	0.940034i	$-0.610793\pi$
$397$	−5396.00	−0.682160	−0.341080	+	0.940034i	$0.610793\pi$
$398$	6960.00	0.876566
$399$	−3120.00	−0.391467
$400$	0	0
$401$	14482.0	1.80348	0.901741	−	0.432276i	$-0.142289\pi$
$401$	14482.0	1.80348	0.901741	+	0.432276i	$0.142289\pi$
$402$	1736.00	0.215383
$403$	1536.00	0.189860
$404$	−2312.00	−0.284719
$405$	0	0
$406$	4680.00	0.572080
$407$	6608.00	0.804782
$408$	1024.00	0.124254
$409$	−1090.00	−0.131778	−0.0658888	−	0.997827i	$-0.520988\pi$
$409$	−1090.00	−0.131778	−0.0658888	+	0.997827i	$0.520988\pi$
$410$	0	0
$411$	−2448.00	−0.293798
$412$	−5848.00	−0.699297
$413$	520.000	0.0619553
$414$	2668.00	0.316727
$415$	0	0
$416$	384.000	0.0452576
$417$	−6200.00	−0.728094
$418$	−3360.00	−0.393165
$419$	−7180.00	−0.837150	−0.418575	−	0.908182i	$-0.637470\pi$
$419$	−7180.00	−0.837150	−0.418575	+	0.908182i	$0.637470\pi$
$420$	0	0
$421$	−8138.00	−0.942095	−0.471047	−	0.882108i	$-0.656124\pi$
$421$	−8138.00	−0.942095	−0.471047	+	0.882108i	$0.656124\pi$
$422$	−6264.00	−0.722575
$423$	5198.00	0.597483
$424$	−864.000	−0.0989612
$425$	0	0
$426$	−4512.00	−0.513162
$427$	−14092.0	−1.59710
$428$	−3864.00	−0.436387
$429$	−672.000	−0.0756281
$430$	0	0
$431$	−208.000	−0.0232460	−0.0116230	−	0.999932i	$-0.503700\pi$
$431$	−208.000	−0.0232460	−0.0116230	+	0.999932i	$0.503700\pi$
$432$	1600.00	0.178195
$433$	−12992.0	−1.44193	−0.720965	−	0.692971i	$-0.756301\pi$
$433$	−12992.0	−1.44193	−0.720965	+	0.692971i	$0.756301\pi$
$434$	−6656.00	−0.736171
$435$	0	0
$436$	1480.00	0.162567
$437$	−3480.00	−0.380940
$438$	−2528.00	−0.275782
$439$	1080.00	0.117416	0.0587080	−	0.998275i	$-0.481302\pi$
$439$	1080.00	0.117416	0.0587080	+	0.998275i	$0.481302\pi$
$440$	0	0
$441$	−7659.00	−0.827017
$442$	1536.00	0.165294
$443$	9078.00	0.973609	0.486805	−	0.873511i	$-0.338162\pi$
$443$	9078.00	0.973609	0.486805	+	0.873511i	$0.338162\pi$
$444$	1888.00	0.201803
$445$	0	0
$446$	124.000	0.0131650
$447$	−500.000	−0.0529065
$448$	−1664.00	−0.175484
$449$	14310.0	1.50408	0.752039	−	0.659119i	$-0.229071\pi$
$449$	14310.0	1.50408	0.752039	+	0.659119i	$0.229071\pi$
$450$	0	0
$451$	−6776.00	−0.707471
$452$	2112.00	0.219779
$453$	−4304.00	−0.446401
$454$	−10628.0	−1.09867
$455$	0	0
$456$	−960.000	−0.0985880
$457$	2344.00	0.239929	0.119965	−	0.992778i	$-0.461722\pi$
$457$	2344.00	0.239929	0.119965	+	0.992778i	$0.461722\pi$
$458$	380.000	0.0387691
$459$	6400.00	0.650820
$460$	0	0
$461$	11382.0	1.14992	0.574959	−	0.818182i	$-0.305018\pi$
$461$	11382.0	1.14992	0.574959	+	0.818182i	$0.305018\pi$
$462$	2912.00	0.293244
$463$	−16062.0	−1.61223	−0.806117	−	0.591756i	$-0.798435\pi$
$463$	−16062.0	−1.61223	−0.806117	+	0.591756i	$0.798435\pi$
$464$	1440.00	0.144074
$465$	0	0
$466$	−4816.00	−0.478749
$467$	−17166.0	−1.70096	−0.850479	−	0.526008i	$-0.823688\pi$
$467$	−17166.0	−1.70096	−0.850479	+	0.526008i	$0.823688\pi$
$468$	1104.00	0.109044
$469$	−11284.0	−1.11097
$470$	0	0
$471$	−1048.00	−0.102525
$472$	160.000	0.0156030
$473$	10136.0	0.985315
$474$	−2880.00	−0.279078
$475$	0	0
$476$	−6656.00	−0.640919
$477$	−2484.00	−0.238437
$478$	11360.0	1.08702
$479$	7520.00	0.717323	0.358661	−	0.933468i	$-0.383233\pi$
$479$	7520.00	0.717323	0.358661	+	0.933468i	$0.383233\pi$
$480$	0	0
$481$	2832.00	0.268458
$482$	556.000	0.0525417
$483$	3016.00	0.284126
$484$	−2188.00	−0.205485
$485$	0	0
$486$	7084.00	0.661187
$487$	11814.0	1.09927	0.549634	−	0.835406i	$-0.314767\pi$
$487$	11814.0	1.09927	0.549634	+	0.835406i	$0.314767\pi$
$488$	−4336.00	−0.402216
$489$	−7036.00	−0.650673
$490$	0	0
$491$	14052.0	1.29156	0.645782	−	0.763522i	$-0.276532\pi$
$491$	14052.0	1.29156	0.645782	+	0.763522i	$0.276532\pi$
$492$	−1936.00	−0.177402
$493$	5760.00	0.526202
$494$	−1440.00	−0.131151
$495$	0	0
$496$	−2048.00	−0.185399
$497$	29328.0	2.64696
$498$	1912.00	0.172046
$499$	7620.00	0.683603	0.341802	−	0.939772i	$-0.388963\pi$
$499$	7620.00	0.683603	0.341802	+	0.939772i	$0.388963\pi$
$500$	0	0
$501$	−1068.00	−0.0952390
$502$	−6504.00	−0.578262
$503$	1818.00	0.161154	0.0805772	−	0.996748i	$-0.474324\pi$
$503$	1818.00	0.161154	0.0805772	+	0.996748i	$0.474324\pi$
$504$	−4784.00	−0.422810
$505$	0	0
$506$	3248.00	0.285358
$507$	4106.00	0.359672
$508$	6136.00	0.535907
$509$	17850.0	1.55440	0.777198	−	0.629256i	$-0.216640\pi$
$509$	17850.0	1.55440	0.777198	+	0.629256i	$0.216640\pi$
$510$	0	0
$511$	16432.0	1.42252
$512$	−512.000	−0.0441942
$513$	−6000.00	−0.516387
$514$	3072.00	0.263619
$515$	0	0
$516$	2896.00	0.247072
$517$	6328.00	0.538308
$518$	−12272.0	−1.04093
$519$	8504.00	0.719237
$520$	0	0
$521$	−19238.0	−1.61772	−0.808860	−	0.588001i	$-0.799915\pi$
$521$	−19238.0	−1.61772	−0.808860	+	0.588001i	$0.799915\pi$
$522$	4140.00	0.347132
$523$	6278.00	0.524891	0.262445	−	0.964947i	$-0.415471\pi$
$523$	6278.00	0.524891	0.262445	+	0.964947i	$0.415471\pi$
$524$	48.0000	0.00400170
$525$	0	0
$526$	−9716.00	−0.805395
$527$	−8192.00	−0.677133
$528$	896.000	0.0738511
$529$	−8803.00	−0.723514
$530$	0	0
$531$	460.000	0.0375938
$532$	6240.00	0.508531
$533$	−2904.00	−0.235997
$534$	−1960.00	−0.158834
$535$	0	0
$536$	−3472.00	−0.279790
$537$	−5000.00	−0.401799
$538$	−5220.00	−0.418309
$539$	−9324.00	−0.745108
$540$	0	0
$541$	−9818.00	−0.780238	−0.390119	−	0.920764i	$-0.627566\pi$
$541$	−9818.00	−0.780238	−0.390119	+	0.920764i	$0.627566\pi$
$542$	10336.0	0.819131
$543$	5156.00	0.407486
$544$	−2048.00	−0.161410
$545$	0	0
$546$	1248.00	0.0978195
$547$	12514.0	0.978172	0.489086	−	0.872236i	$-0.337330\pi$
$547$	12514.0	0.978172	0.489086	+	0.872236i	$0.337330\pi$
$548$	4896.00	0.381655
$549$	−12466.0	−0.969100
$550$	0	0
$551$	−5400.00	−0.417509
$552$	928.000	0.0715549
$553$	18720.0	1.43952
$554$	−3848.00	−0.295101
$555$	0	0
$556$	12400.0	0.945822
$557$	−10596.0	−0.806045	−0.403022	−	0.915190i	$-0.632040\pi$
$557$	−10596.0	−0.806045	−0.403022	+	0.915190i	$0.632040\pi$
$558$	−5888.00	−0.446701
$559$	4344.00	0.328679
$560$	0	0
$561$	3584.00	0.269727
$562$	−6084.00	−0.456651
$563$	−14002.0	−1.04816	−0.524080	−	0.851669i	$-0.675591\pi$
$563$	−14002.0	−1.04816	−0.524080	+	0.851669i	$0.675591\pi$
$564$	1808.00	0.134983
$565$	0	0
$566$	−3436.00	−0.255169
$567$	−10946.0	−0.810739
$568$	9024.00	0.666617
$569$	−7330.00	−0.540052	−0.270026	−	0.962853i	$-0.587032\pi$
$569$	−7330.00	−0.540052	−0.270026	+	0.962853i	$0.587032\pi$
$570$	0	0
$571$	5812.00	0.425963	0.212981	−	0.977056i	$-0.431683\pi$
$571$	5812.00	0.425963	0.212981	+	0.977056i	$0.431683\pi$
$572$	1344.00	0.0982438
$573$	1536.00	0.111985
$574$	12584.0	0.915063
$575$	0	0
$576$	−1472.00	−0.106481
$577$	−16736.0	−1.20750	−0.603751	−	0.797173i	$-0.706328\pi$
$577$	−16736.0	−1.20750	−0.603751	+	0.797173i	$0.706328\pi$
$578$	1634.00	0.117587
$579$	−5216.00	−0.374386
$580$	0	0
$581$	−12428.0	−0.887436
$582$	−5824.00	−0.414798
$583$	−3024.00	−0.214822
$584$	5056.00	0.358251
$585$	0	0
$586$	4584.00	0.323146
$587$	7434.00	0.522716	0.261358	−	0.965242i	$-0.415830\pi$
$587$	7434.00	0.522716	0.261358	+	0.965242i	$0.415830\pi$
$588$	−2664.00	−0.186839
$589$	7680.00	0.537265
$590$	0	0
$591$	10232.0	0.712163
$592$	−3776.00	−0.262150
$593$	−25872.0	−1.79163	−0.895814	−	0.444429i	$-0.853407\pi$
$593$	−25872.0	−1.79163	−0.895814	+	0.444429i	$0.853407\pi$
$594$	5600.00	0.386820
$595$	0	0
$596$	1000.00	0.0687275
$597$	6960.00	0.477142
$598$	1392.00	0.0951892
$599$	−3720.00	−0.253748	−0.126874	−	0.991919i	$-0.540494\pi$
$599$	−3720.00	−0.253748	−0.126874	+	0.991919i	$0.540494\pi$
$600$	0	0
$601$	−12958.0	−0.879481	−0.439740	−	0.898125i	$-0.644930\pi$
$601$	−12958.0	−0.879481	−0.439740	+	0.898125i	$0.644930\pi$
$602$	−18824.0	−1.27443
$603$	−9982.00	−0.674127
$604$	8608.00	0.579892
$605$	0	0
$606$	−2312.00	−0.154981
$607$	7214.00	0.482384	0.241192	−	0.970477i	$-0.422462\pi$
$607$	7214.00	0.482384	0.241192	+	0.970477i	$0.422462\pi$
$608$	1920.00	0.128070
$609$	4680.00	0.311401
$610$	0	0
$611$	2712.00	0.179568
$612$	−5888.00	−0.388902
$613$	4828.00	0.318109	0.159055	−	0.987270i	$-0.449155\pi$
$613$	4828.00	0.318109	0.159055	+	0.987270i	$0.449155\pi$
$614$	10812.0	0.710646
$615$	0	0
$616$	−5824.00	−0.380934
$617$	−27656.0	−1.80452	−0.902260	−	0.431193i	$-0.858093\pi$
$617$	−27656.0	−1.80452	−0.902260	+	0.431193i	$0.858093\pi$
$618$	−5848.00	−0.380649
$619$	−21220.0	−1.37787	−0.688937	−	0.724821i	$-0.741922\pi$
$619$	−21220.0	−1.37787	−0.688937	+	0.724821i	$0.741922\pi$
$620$	0	0
$621$	5800.00	0.374792
$622$	11376.0	0.733338
$623$	12740.0	0.819289
$624$	384.000	0.0246351
$625$	0	0
$626$	14704.0	0.938802
$627$	−3360.00	−0.214012
$628$	2096.00	0.133184
$629$	−15104.0	−0.957450
$630$	0	0
$631$	17672.0	1.11491	0.557457	−	0.830206i	$-0.311777\pi$
$631$	17672.0	1.11491	0.557457	+	0.830206i	$0.311777\pi$
$632$	5760.00	0.362532
$633$	−6264.00	−0.393320
$634$	−6968.00	−0.436490
$635$	0	0
$636$	−864.000	−0.0538677
$637$	−3996.00	−0.248551
$638$	5040.00	0.312752
$639$	25944.0	1.60615
$640$	0	0
$641$	7322.00	0.451173	0.225586	−	0.974223i	$-0.427570\pi$
$641$	7322.00	0.451173	0.225586	+	0.974223i	$0.427570\pi$
$642$	−3864.00	−0.237539
$643$	8238.00	0.505249	0.252624	−	0.967564i	$-0.418706\pi$
$643$	8238.00	0.505249	0.252624	+	0.967564i	$0.418706\pi$
$644$	−6032.00	−0.369090
$645$	0	0
$646$	7680.00	0.467749
$647$	−6426.00	−0.390467	−0.195233	−	0.980757i	$-0.562546\pi$
$647$	−6426.00	−0.390467	−0.195233	+	0.980757i	$0.562546\pi$
$648$	−3368.00	−0.204178
$649$	560.000	0.0338705
$650$	0	0
$651$	−6656.00	−0.400721
$652$	14072.0	0.845249
$653$	5908.00	0.354055	0.177027	−	0.984206i	$-0.443352\pi$
$653$	5908.00	0.354055	0.177027	+	0.984206i	$0.443352\pi$
$654$	1480.00	0.0884902
$655$	0	0
$656$	3872.00	0.230452
$657$	14536.0	0.863171
$658$	−11752.0	−0.696262
$659$	−26780.0	−1.58301	−0.791503	−	0.611166i	$-0.790701\pi$
$659$	−26780.0	−1.58301	−0.791503	+	0.611166i	$0.790701\pi$
$660$	0	0
$661$	−24538.0	−1.44390	−0.721950	−	0.691945i	$-0.756754\pi$
$661$	−24538.0	−1.44390	−0.721950	+	0.691945i	$0.756754\pi$
$662$	15736.0	0.923863
$663$	1536.00	0.0899748
$664$	−3824.00	−0.223494
$665$	0	0
$666$	−10856.0	−0.631624
$667$	5220.00	0.303027
$668$	2136.00	0.123719
$669$	124.000	0.00716609
$670$	0	0
$671$	−15176.0	−0.873119
$672$	−1664.00	−0.0955211
$673$	28848.0	1.65232	0.826158	−	0.563439i	$-0.190522\pi$
$673$	28848.0	1.65232	0.826158	+	0.563439i	$0.190522\pi$
$674$	1312.00	0.0749798
$675$	0	0
$676$	−8212.00	−0.467228
$677$	26884.0	1.52620	0.763099	−	0.646282i	$-0.223677\pi$
$677$	26884.0	1.52620	0.763099	+	0.646282i	$0.223677\pi$
$678$	2112.00	0.119633
$679$	37856.0	2.13959
$680$	0	0
$681$	−10628.0	−0.598041
$682$	−7168.00	−0.402459
$683$	−14282.0	−0.800125	−0.400063	−	0.916488i	$-0.631012\pi$
$683$	−14282.0	−0.800125	−0.400063	+	0.916488i	$0.631012\pi$
$684$	5520.00	0.308571
$685$	0	0
$686$	−520.000	−0.0289412
$687$	380.000	0.0211032
$688$	−5792.00	−0.320956
$689$	−1296.00	−0.0716599
$690$	0	0
$691$	−3428.00	−0.188723	−0.0943613	−	0.995538i	$-0.530081\pi$
$691$	−3428.00	−0.188723	−0.0943613	+	0.995538i	$0.530081\pi$
$692$	−17008.0	−0.934317
$693$	−16744.0	−0.917824
$694$	−11508.0	−0.629449
$695$	0	0
$696$	1440.00	0.0784239
$697$	15488.0	0.841678
$698$	6220.00	0.337293
$699$	−4816.00	−0.260598
$700$	0	0
$701$	26942.0	1.45162	0.725810	−	0.687895i	$-0.241465\pi$
$701$	26942.0	1.45162	0.725810	+	0.687895i	$0.241465\pi$
$702$	2400.00	0.129034
$703$	14160.0	0.759679
$704$	−1792.00	−0.0959354
$705$	0	0
$706$	−15616.0	−0.832459
$707$	15028.0	0.799415
$708$	160.000	0.00849318
$709$	−1950.00	−0.103292	−0.0516458	−	0.998665i	$-0.516447\pi$
$709$	−1950.00	−0.103292	−0.0516458	+	0.998665i	$0.516447\pi$
$710$	0	0
$711$	16560.0	0.873486
$712$	3920.00	0.206332
$713$	−7424.00	−0.389945
$714$	−6656.00	−0.348872
$715$	0	0
$716$	10000.0	0.521952
$717$	11360.0	0.591697
$718$	18480.0	0.960540
$719$	12080.0	0.626576	0.313288	−	0.949658i	$-0.398570\pi$
$719$	12080.0	0.626576	0.313288	+	0.949658i	$0.398570\pi$
$720$	0	0
$721$	38012.0	1.96344
$722$	6518.00	0.335976
$723$	556.000	0.0286001
$724$	−10312.0	−0.529340
$725$	0	0
$726$	−2188.00	−0.111852
$727$	−17226.0	−0.878785	−0.439393	−	0.898295i	$-0.644806\pi$
$727$	−17226.0	−0.878785	−0.439393	+	0.898295i	$0.644806\pi$
$728$	−2496.00	−0.127071
$729$	−4283.00	−0.217599
$730$	0	0
$731$	−23168.0	−1.17223
$732$	−4336.00	−0.218939
$733$	788.000	0.0397073	0.0198536	−	0.999803i	$-0.493680\pi$
$733$	788.000	0.0397073	0.0198536	+	0.999803i	$0.493680\pi$
$734$	−6428.00	−0.323245
$735$	0	0
$736$	−1856.00	−0.0929525
$737$	−12152.0	−0.607360
$738$	11132.0	0.555250
$739$	−2060.00	−0.102542	−0.0512709	−	0.998685i	$-0.516327\pi$
$739$	−2060.00	−0.102542	−0.0512709	+	0.998685i	$0.516327\pi$
$740$	0	0
$741$	−1440.00	−0.0713896
$742$	5616.00	0.277857
$743$	3258.00	0.160867	0.0804337	−	0.996760i	$-0.474369\pi$
$743$	3258.00	0.160867	0.0804337	+	0.996760i	$0.474369\pi$
$744$	−2048.00	−0.100918
$745$	0	0
$746$	−696.000	−0.0341587
$747$	−10994.0	−0.538487
$748$	−7168.00	−0.350385
$749$	25116.0	1.22526
$750$	0	0
$751$	−4528.00	−0.220012	−0.110006	−	0.993931i	$-0.535087\pi$
$751$	−4528.00	−0.220012	−0.110006	+	0.993931i	$0.535087\pi$
$752$	−3616.00	−0.175348
$753$	−6504.00	−0.314766
$754$	2160.00	0.104327
$755$	0	0
$756$	−10400.0	−0.500323
$757$	−18236.0	−0.875560	−0.437780	−	0.899082i	$-0.644235\pi$
$757$	−18236.0	−0.875560	−0.437780	+	0.899082i	$0.644235\pi$
$758$	−9880.00	−0.473427
$759$	3248.00	0.155329
$760$	0	0
$761$	−18678.0	−0.889720	−0.444860	−	0.895600i	$-0.646747\pi$
$761$	−18678.0	−0.889720	−0.444860	+	0.895600i	$0.646747\pi$
$762$	6136.00	0.291711
$763$	−9620.00	−0.456445
$764$	−3072.00	−0.145473
$765$	0	0
$766$	12284.0	0.579424
$767$	240.000	0.0112984
$768$	−512.000	−0.0240563
$769$	27390.0	1.28441	0.642203	−	0.766534i	$-0.278020\pi$
$769$	27390.0	1.28441	0.642203	+	0.766534i	$0.278020\pi$
$770$	0	0
$771$	3072.00	0.143496
$772$	10432.0	0.486342
$773$	−9252.00	−0.430493	−0.215247	−	0.976560i	$-0.569056\pi$
$773$	−9252.00	−0.430493	−0.215247	+	0.976560i	$0.569056\pi$
$774$	−16652.0	−0.773312
$775$	0	0
$776$	11648.0	0.538839
$777$	−12272.0	−0.566609
$778$	−6100.00	−0.281100
$779$	−14520.0	−0.667822
$780$	0	0
$781$	31584.0	1.44707
$782$	−7424.00	−0.339491
$783$	9000.00	0.410771
$784$	5328.00	0.242711
$785$	0	0
$786$	48.0000	0.00217825
$787$	−5726.00	−0.259352	−0.129676	−	0.991556i	$-0.541394\pi$
$787$	−5726.00	−0.259352	−0.129676	+	0.991556i	$0.541394\pi$
$788$	−20464.0	−0.925127
$789$	−9716.00	−0.438401
$790$	0	0
$791$	−13728.0	−0.617082
$792$	−5152.00	−0.231147
$793$	−6504.00	−0.291253
$794$	10792.0	0.482360
$795$	0	0
$796$	−13920.0	−0.619826
$797$	−27236.0	−1.21048	−0.605238	−	0.796045i	$-0.706922\pi$
$797$	−27236.0	−1.21048	−0.605238	+	0.796045i	$0.706922\pi$
$798$	6240.00	0.276809
$799$	−14464.0	−0.640425
$800$	0	0
$801$	11270.0	0.497136
$802$	−28964.0	−1.27525
$803$	17696.0	0.777682
$804$	−3472.00	−0.152299
$805$	0	0
$806$	−3072.00	−0.134251
$807$	−5220.00	−0.227699
$808$	4624.00	0.201326
$809$	10950.0	0.475873	0.237937	−	0.971281i	$-0.423529\pi$
$809$	10950.0	0.475873	0.237937	+	0.971281i	$0.423529\pi$
$810$	0	0
$811$	−8828.00	−0.382236	−0.191118	−	0.981567i	$-0.561211\pi$
$811$	−8828.00	−0.382236	−0.191118	+	0.981567i	$0.561211\pi$
$812$	−9360.00	−0.404522
$813$	10336.0	0.445879
$814$	−13216.0	−0.569067
$815$	0	0
$816$	−2048.00	−0.0878607
$817$	21720.0	0.930094
$818$	2180.00	0.0931808
$819$	−7176.00	−0.306166
$820$	0	0
$821$	−16058.0	−0.682616	−0.341308	−	0.939951i	$-0.610870\pi$
$821$	−16058.0	−0.682616	−0.341308	+	0.939951i	$0.610870\pi$
$822$	4896.00	0.207746
$823$	−41862.0	−1.77305	−0.886523	−	0.462684i	$-0.846887\pi$
$823$	−41862.0	−1.77305	−0.886523	+	0.462684i	$0.846887\pi$
$824$	11696.0	0.494478
$825$	0	0
$826$	−1040.00	−0.0438090
$827$	12154.0	0.511047	0.255524	−	0.966803i	$-0.417752\pi$
$827$	12154.0	0.511047	0.255524	+	0.966803i	$0.417752\pi$
$828$	−5336.00	−0.223960
$829$	−15390.0	−0.644773	−0.322386	−	0.946608i	$-0.604485\pi$
$829$	−15390.0	−0.644773	−0.322386	+	0.946608i	$0.604485\pi$
$830$	0	0
$831$	−3848.00	−0.160633
$832$	−768.000	−0.0320019
$833$	21312.0	0.886455
$834$	12400.0	0.514840
$835$	0	0
$836$	6720.00	0.278010
$837$	−12800.0	−0.528593
$838$	14360.0	0.591955
$839$	−4280.00	−0.176117	−0.0880584	−	0.996115i	$-0.528066\pi$
$839$	−4280.00	−0.176117	−0.0880584	+	0.996115i	$0.528066\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	16276.0	0.666162
$843$	−6084.00	−0.248570
$844$	12528.0	0.510938
$845$	0	0
$846$	−10396.0	−0.422484
$847$	14222.0	0.576947
$848$	1728.00	0.0699761
$849$	−3436.00	−0.138897
$850$	0	0
$851$	−13688.0	−0.551373
$852$	9024.00	0.362860
$853$	−14452.0	−0.580102	−0.290051	−	0.957011i	$-0.593672\pi$
$853$	−14452.0	−0.580102	−0.290051	+	0.957011i	$0.593672\pi$
$854$	28184.0	1.12932
$855$	0	0
$856$	7728.00	0.308572
$857$	22584.0	0.900181	0.450090	−	0.892983i	$-0.351392\pi$
$857$	22584.0	0.900181	0.450090	+	0.892983i	$0.351392\pi$
$858$	1344.00	0.0534772
$859$	−26740.0	−1.06212	−0.531058	−	0.847336i	$-0.678205\pi$
$859$	−26740.0	−1.06212	−0.531058	+	0.847336i	$0.678205\pi$
$860$	0	0
$861$	12584.0	0.498097
$862$	416.000	0.0164374
$863$	498.000	0.0196432	0.00982162	−	0.999952i	$-0.496874\pi$
$863$	498.000	0.0196432	0.00982162	+	0.999952i	$0.496874\pi$
$864$	−3200.00	−0.126003
$865$	0	0
$866$	25984.0	1.01960
$867$	1634.00	0.0640064
$868$	13312.0	0.520552
$869$	20160.0	0.786975
$870$	0	0
$871$	−5208.00	−0.202602
$872$	−2960.00	−0.114952
$873$	33488.0	1.29828
$874$	6960.00	0.269366
$875$	0	0
$876$	5056.00	0.195007
$877$	13244.0	0.509941	0.254970	−	0.966949i	$-0.417934\pi$
$877$	13244.0	0.509941	0.254970	+	0.966949i	$0.417934\pi$
$878$	−2160.00	−0.0830256
$879$	4584.00	0.175898
$880$	0	0
$881$	40842.0	1.56186	0.780932	−	0.624616i	$-0.214745\pi$
$881$	40842.0	1.56186	0.780932	+	0.624616i	$0.214745\pi$
$882$	15318.0	0.584789
$883$	12078.0	0.460314	0.230157	−	0.973154i	$-0.426076\pi$
$883$	12078.0	0.460314	0.230157	+	0.973154i	$0.426076\pi$
$884$	−3072.00	−0.116881
$885$	0	0
$886$	−18156.0	−0.688446
$887$	18294.0	0.692506	0.346253	−	0.938141i	$-0.387454\pi$
$887$	18294.0	0.692506	0.346253	+	0.938141i	$0.387454\pi$
$888$	−3776.00	−0.142696
$889$	−39884.0	−1.50469
$890$	0	0
$891$	−11788.0	−0.443224
$892$	−248.000	−0.00930903
$893$	13560.0	0.508139
$894$	1000.00	0.0374105
$895$	0	0
$896$	3328.00	0.124086
$897$	1392.00	0.0518144
$898$	−28620.0	−1.06354
$899$	−11520.0	−0.427379
$900$	0	0
$901$	6912.00	0.255574
$902$	13552.0	0.500257
$903$	−18824.0	−0.693714
$904$	−4224.00	−0.155407
$905$	0	0
$906$	8608.00	0.315653
$907$	−22566.0	−0.826121	−0.413060	−	0.910704i	$-0.635540\pi$
$907$	−22566.0	−0.826121	−0.413060	+	0.910704i	$0.635540\pi$
$908$	21256.0	0.776878
$909$	13294.0	0.485076
$910$	0	0
$911$	−6768.00	−0.246140	−0.123070	−	0.992398i	$-0.539274\pi$
$911$	−6768.00	−0.246140	−0.123070	+	0.992398i	$0.539274\pi$
$912$	1920.00	0.0697122
$913$	−13384.0	−0.485154
$914$	−4688.00	−0.169656
$915$	0	0
$916$	−760.000	−0.0274139
$917$	−312.000	−0.0112357
$918$	−12800.0	−0.460199
$919$	22200.0	0.796856	0.398428	−	0.917200i	$-0.369556\pi$
$919$	22200.0	0.796856	0.398428	+	0.917200i	$0.369556\pi$
$920$	0	0
$921$	10812.0	0.386827
$922$	−22764.0	−0.813115
$923$	13536.0	0.482712
$924$	−5824.00	−0.207354
$925$	0	0
$926$	32124.0	1.14002
$927$	33626.0	1.19139
$928$	−2880.00	−0.101876
$929$	−6330.00	−0.223553	−0.111776	−	0.993733i	$-0.535654\pi$
$929$	−6330.00	−0.223553	−0.111776	+	0.993733i	$0.535654\pi$
$930$	0	0
$931$	−19980.0	−0.703349
$932$	9632.00	0.338526
$933$	11376.0	0.399178
$934$	34332.0	1.20276
$935$	0	0
$936$	−2208.00	−0.0771055
$937$	19544.0	0.681403	0.340702	−	0.940172i	$-0.389335\pi$
$937$	19544.0	0.681403	0.340702	+	0.940172i	$0.389335\pi$
$938$	22568.0	0.785577
$939$	14704.0	0.511019
$940$	0	0
$941$	−9898.00	−0.342896	−0.171448	−	0.985193i	$-0.554845\pi$
$941$	−9898.00	−0.342896	−0.171448	+	0.985193i	$0.554845\pi$
$942$	2096.00	0.0724961
$943$	14036.0	0.484703
$944$	−320.000	−0.0110330
$945$	0	0
$946$	−20272.0	−0.696723
$947$	−41406.0	−1.42082	−0.710409	−	0.703789i	$-0.751490\pi$
$947$	−41406.0	−1.42082	−0.710409	+	0.703789i	$0.751490\pi$
$948$	5760.00	0.197338
$949$	7584.00	0.259417
$950$	0	0
$951$	−6968.00	−0.237595
$952$	13312.0	0.453198
$953$	−25432.0	−0.864453	−0.432226	−	0.901765i	$-0.642272\pi$
$953$	−25432.0	−0.864453	−0.432226	+	0.901765i	$0.642272\pi$
$954$	4968.00	0.168601
$955$	0	0
$956$	−22720.0	−0.768637
$957$	5040.00	0.170240
$958$	−15040.0	−0.507224
$959$	−31824.0	−1.07159
$960$	0	0
$961$	−13407.0	−0.450035
$962$	−5664.00	−0.189828
$963$	22218.0	0.743474
$964$	−1112.00	−0.0371526
$965$	0	0
$966$	−6032.00	−0.200907
$967$	−12106.0	−0.402588	−0.201294	−	0.979531i	$-0.564515\pi$
$967$	−12106.0	−0.402588	−0.201294	+	0.979531i	$0.564515\pi$
$968$	4376.00	0.145300
$969$	7680.00	0.254610
$970$	0	0
$971$	7812.00	0.258186	0.129093	−	0.991632i	$-0.458793\pi$
$971$	7812.00	0.258186	0.129093	+	0.991632i	$0.458793\pi$
$972$	−14168.0	−0.467530
$973$	−80600.0	−2.65562
$974$	−23628.0	−0.777300
$975$	0	0
$976$	8672.00	0.284410
$977$	−12576.0	−0.411814	−0.205907	−	0.978572i	$-0.566014\pi$
$977$	−12576.0	−0.411814	−0.205907	+	0.978572i	$0.566014\pi$
$978$	14072.0	0.460095
$979$	13720.0	0.447899
$980$	0	0
$981$	−8510.00	−0.276966
$982$	−28104.0	−0.913274
$983$	−4342.00	−0.140883	−0.0704417	−	0.997516i	$-0.522441\pi$
$983$	−4342.00	−0.140883	−0.0704417	+	0.997516i	$0.522441\pi$
$984$	3872.00	0.125442
$985$	0	0
$986$	−11520.0	−0.372081
$987$	−11752.0	−0.378997
$988$	2880.00	0.0927379
$989$	−20996.0	−0.675060
$990$	0	0
$991$	26272.0	0.842137	0.421068	−	0.907029i	$-0.361655\pi$
$991$	26272.0	0.842137	0.421068	+	0.907029i	$0.361655\pi$
$992$	4096.00	0.131097
$993$	15736.0	0.502887
$994$	−58656.0	−1.87169
$995$	0	0
$996$	−3824.00	−0.121655
$997$	−44796.0	−1.42297	−0.711486	−	0.702700i	$-0.751978\pi$
$997$	−44796.0	−1.42297	−0.711486	+	0.702700i	$0.751978\pi$
$998$	−15240.0	−0.483381
$999$	−23600.0	−0.747418

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	50.4.a.b.1.1		1
3.2	odd	2		450.4.a.k.1.1		1
4.3	odd	2		400.4.a.n.1.1		1
5.2	odd	4		10.4.b.a.9.1	✓	2
5.3	odd	4		10.4.b.a.9.2	yes	2
5.4	even	2		50.4.a.d.1.1		1
7.6	odd	2		2450.4.a.o.1.1		1
8.3	odd	2		1600.4.a.t.1.1		1
8.5	even	2		1600.4.a.bh.1.1		1
15.2	even	4		90.4.c.b.19.2		2
15.8	even	4		90.4.c.b.19.1		2
15.14	odd	2		450.4.a.j.1.1		1
20.3	even	4		80.4.c.a.49.2		2
20.7	even	4		80.4.c.a.49.1		2
20.19	odd	2		400.4.a.h.1.1		1
35.13	even	4		490.4.c.b.99.2		2
35.27	even	4		490.4.c.b.99.1		2
35.34	odd	2		2450.4.a.bb.1.1		1
40.3	even	4		320.4.c.c.129.1		2
40.13	odd	4		320.4.c.d.129.2		2
40.19	odd	2		1600.4.a.bg.1.1		1
40.27	even	4		320.4.c.c.129.2		2
40.29	even	2		1600.4.a.u.1.1		1
40.37	odd	4		320.4.c.d.129.1		2
60.23	odd	4		720.4.f.f.289.2		2
60.47	odd	4		720.4.f.f.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.4.b.a.9.1	✓	2	5.2	odd	4
10.4.b.a.9.2	yes	2	5.3	odd	4
50.4.a.b.1.1		1	1.1	even	1	trivial
50.4.a.d.1.1		1	5.4	even	2
80.4.c.a.49.1		2	20.7	even	4
80.4.c.a.49.2		2	20.3	even	4
90.4.c.b.19.1		2	15.8	even	4
90.4.c.b.19.2		2	15.2	even	4
320.4.c.c.129.1		2	40.3	even	4
320.4.c.c.129.2		2	40.27	even	4
320.4.c.d.129.1		2	40.37	odd	4
320.4.c.d.129.2		2	40.13	odd	4
400.4.a.h.1.1		1	20.19	odd	2
400.4.a.n.1.1		1	4.3	odd	2
450.4.a.j.1.1		1	15.14	odd	2
450.4.a.k.1.1		1	3.2	odd	2
490.4.c.b.99.1		2	35.27	even	4
490.4.c.b.99.2		2	35.13	even	4
720.4.f.f.289.1		2	60.47	odd	4
720.4.f.f.289.2		2	60.23	odd	4
1600.4.a.t.1.1		1	8.3	odd	2
1600.4.a.u.1.1		1	40.29	even	2
1600.4.a.bg.1.1		1	40.19	odd	2
1600.4.a.bh.1.1		1	8.5	even	2
2450.4.a.o.1.1		1	7.6	odd	2
2450.4.a.bb.1.1		1	35.34	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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	50.4.a.b.1.1

Level	$50$
Weight	$4$
Character	50.1
Self dual	yes
Analytic conductor	$2.950$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$