LMFDB - Embedded newform 2151.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$0$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.24697 q^{2} +19.5307 q^{4} +10.0958 q^{5} +0.804440 q^{7} -60.5014 q^{8} +O(q^{10})$	\(q-5.24697 q^{2} +19.5307 q^{4} +10.0958 q^{5} +0.804440 q^{7} -60.5014 q^{8} -52.9726 q^{10} +0.829353 q^{11} -86.7323 q^{13} -4.22087 q^{14} +161.203 q^{16} +50.2442 q^{17} +121.120 q^{19} +197.179 q^{20} -4.35159 q^{22} -76.0475 q^{23} -23.0740 q^{25} +455.082 q^{26} +15.7113 q^{28} +40.2936 q^{29} -81.0073 q^{31} -361.819 q^{32} -263.630 q^{34} +8.12150 q^{35} -321.670 q^{37} -635.511 q^{38} -610.813 q^{40} -147.774 q^{41} +32.4253 q^{43} +16.1979 q^{44} +399.019 q^{46} +271.172 q^{47} -342.353 q^{49} +121.068 q^{50} -1693.95 q^{52} -110.872 q^{53} +8.37302 q^{55} -48.6697 q^{56} -211.419 q^{58} +817.455 q^{59} -492.812 q^{61} +425.043 q^{62} +608.826 q^{64} -875.636 q^{65} +862.135 q^{67} +981.306 q^{68} -42.6133 q^{70} +395.164 q^{71} -1025.61 q^{73} +1687.79 q^{74} +2365.55 q^{76} +0.667165 q^{77} +210.661 q^{79} +1627.48 q^{80} +775.365 q^{82} +879.930 q^{83} +507.258 q^{85} -170.135 q^{86} -50.1770 q^{88} -461.975 q^{89} -69.7709 q^{91} -1485.26 q^{92} -1422.83 q^{94} +1222.80 q^{95} -372.791 q^{97} +1796.32 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8}+O(q^{10}) $ 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8	$ 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8} - 36 q^{10} + 132 q^{11} + 104 q^{13} + 280 q^{14} + 822 q^{16} + 408 q^{17} + 20 q^{19} + 800 q^{20} - 2 q^{22} + 276 q^{23} + 1477 q^{25} + 780 q^{26} + 224 q^{28} + 696 q^{29} - 380 q^{31} + 896 q^{32} - 72 q^{34} + 700 q^{35} + 224 q^{37} + 988 q^{38} - 258 q^{40} + 2706 q^{41} - 156 q^{43} + 1584 q^{44} + 428 q^{46} + 1316 q^{47} + 2135 q^{49} + 1400 q^{50} + 1092 q^{52} + 1484 q^{53} - 992 q^{55} + 3360 q^{56} - 120 q^{58} + 3186 q^{59} - 254 q^{61} + 1240 q^{62} + 3054 q^{64} + 5120 q^{65} + 288 q^{67} + 9420 q^{68} + 1108 q^{70} + 4468 q^{71} - 1770 q^{73} + 6214 q^{74} + 720 q^{76} + 6352 q^{77} - 746 q^{79} + 7040 q^{80} + 276 q^{82} + 5484 q^{83} + 588 q^{85} + 10152 q^{86} + 1186 q^{88} + 11570 q^{89} + 1768 q^{91} + 15366 q^{92} - 2142 q^{94} + 5736 q^{95} + 2390 q^{97} + 6912 q^{98}+O(q^{100}) $ 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8 - 36 * q^10 + 132 * q^11 + 104 * q^13 + 280 * q^14 + 822 * q^16 + 408 * q^17 + 20 * q^19 + 800 * q^20 - 2 * q^22 + 276 * q^23 + 1477 * q^25 + 780 * q^26 + 224 * q^28 + 696 * q^29 - 380 * q^31 + 896 * q^32 - 72 * q^34 + 700 * q^35 + 224 * q^37 + 988 * q^38 - 258 * q^40 + 2706 * q^41 - 156 * q^43 + 1584 * q^44 + 428 * q^46 + 1316 * q^47 + 2135 * q^49 + 1400 * q^50 + 1092 * q^52 + 1484 * q^53 - 992 * q^55 + 3360 * q^56 - 120 * q^58 + 3186 * q^59 - 254 * q^61 + 1240 * q^62 + 3054 * q^64 + 5120 * q^65 + 288 * q^67 + 9420 * q^68 + 1108 * q^70 + 4468 * q^71 - 1770 * q^73 + 6214 * q^74 + 720 * q^76 + 6352 * q^77 - 746 * q^79 + 7040 * q^80 + 276 * q^82 + 5484 * q^83 + 588 * q^85 + 10152 * q^86 + 1186 * q^88 + 11570 * q^89 + 1768 * q^91 + 15366 * q^92 - 2142 * q^94 + 5736 * q^95 + 2390 * q^97 + 6912 * q^98

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 $

$2$	−5.24697	−1.85509	−0.927543	−	0.373718i	$-0.878083\pi$
$2$	−5.24697	−1.85509	−0.927543	+	0.373718i	$0.878083\pi$
$3$	0	0
$3$	0	0
$4$	19.5307	2.44134
$5$	10.0958	0.903000	0.451500	−	0.892271i	$-0.350889\pi$
$5$	10.0958	0.903000	0.451500	+	0.892271i	$0.350889\pi$
$6$	0	0
$7$	0.804440	0.0434357	0.0217178	−	0.999764i	$-0.493086\pi$
$7$	0.804440	0.0434357	0.0217178	+	0.999764i	$0.493086\pi$
$8$	−60.5014	−2.67381
$9$	0	0
$10$	−52.9726	−1.67514
$11$	0.829353	0.0227327	0.0113663	−	0.999935i	$-0.496382\pi$
$11$	0.829353	0.0227327	0.0113663	+	0.999935i	$0.496382\pi$
$12$	0	0
$13$	−86.7323	−1.85040	−0.925201	−	0.379478i	$-0.876104\pi$
$13$	−86.7323	−1.85040	−0.925201	+	0.379478i	$0.876104\pi$
$14$	−4.22087	−0.0805768
$15$	0	0
$16$	161.203	2.51880
$17$	50.2442	0.716824	0.358412	−	0.933563i	$-0.383318\pi$
$17$	50.2442	0.716824	0.358412	+	0.933563i	$0.383318\pi$
$18$	0	0
$19$	121.120	1.46246	0.731230	−	0.682131i	$-0.238947\pi$
$19$	121.120	1.46246	0.731230	+	0.682131i	$0.238947\pi$
$20$	197.179	2.20453
$21$	0	0
$22$	−4.35159	−0.0421711
$23$	−76.0475	−0.689435	−0.344717	−	0.938707i	$-0.612025\pi$
$23$	−76.0475	−0.689435	−0.344717	+	0.938707i	$0.612025\pi$
$24$	0	0
$25$	−23.0740	−0.184592
$26$	455.082	3.43265
$27$	0	0
$28$	15.7113	0.106041
$29$	40.2936	0.258011	0.129006	−	0.991644i	$-0.458821\pi$
$29$	40.2936	0.258011	0.129006	+	0.991644i	$0.458821\pi$
$30$	0	0
$31$	−81.0073	−0.469334	−0.234667	−	0.972076i	$-0.575400\pi$
$31$	−81.0073	−0.469334	−0.234667	+	0.972076i	$0.575400\pi$
$32$	−361.819	−1.99879
$33$	0	0
$34$	−263.630	−1.32977
$35$	8.12150	0.0392224
$36$	0	0
$37$	−321.670	−1.42925	−0.714624	−	0.699509i	$-0.753402\pi$
$37$	−321.670	−1.42925	−0.714624	+	0.699509i	$0.753402\pi$
$38$	−635.511	−2.71299
$39$	0	0
$40$	−610.813	−2.41445
$41$	−147.774	−0.562887	−0.281444	−	0.959578i	$-0.590813\pi$
$41$	−147.774	−0.562887	−0.281444	+	0.959578i	$0.590813\pi$
$42$	0	0
$43$	32.4253	0.114996	0.0574978	−	0.998346i	$-0.481688\pi$
$43$	32.4253	0.114996	0.0574978	+	0.998346i	$0.481688\pi$
$44$	16.1979	0.0554982
$45$	0	0
$46$	399.019	1.27896
$47$	271.172	0.841585	0.420792	−	0.907157i	$-0.361752\pi$
$47$	271.172	0.841585	0.420792	+	0.907157i	$0.361752\pi$
$48$	0	0
$49$	−342.353	−0.998113
$50$	121.068	0.342433
$51$	0	0
$52$	−1693.95	−4.51746
$53$	−110.872	−0.287347	−0.143674	−	0.989625i	$-0.545892\pi$
$53$	−110.872	−0.287347	−0.143674	+	0.989625i	$0.545892\pi$
$54$	0	0
$55$	8.37302	0.0205276
$56$	−48.6697	−0.116139
$57$	0	0
$58$	−211.419	−0.478633
$59$	817.455	1.80379	0.901894	−	0.431957i	$-0.142177\pi$
$59$	817.455	1.80379	0.901894	+	0.431957i	$0.142177\pi$
$60$	0	0
$61$	−492.812	−1.03440	−0.517198	−	0.855866i	$-0.673025\pi$
$61$	−492.812	−1.03440	−0.517198	+	0.855866i	$0.673025\pi$
$62$	425.043	0.870654
$63$	0	0
$64$	608.826	1.18911
$65$	−875.636	−1.67091
$66$	0	0
$67$	862.135	1.57204	0.786019	−	0.618202i	$-0.212139\pi$
$67$	862.135	1.57204	0.786019	+	0.618202i	$0.212139\pi$
$68$	981.306	1.75001
$69$	0	0
$70$	−42.6133	−0.0727609
$71$	395.164	0.660526	0.330263	−	0.943889i	$-0.392863\pi$
$71$	395.164	0.660526	0.330263	+	0.943889i	$0.392863\pi$
$72$	0	0
$73$	−1025.61	−1.64436	−0.822181	−	0.569227i	$-0.807243\pi$
$73$	−1025.61	−1.64436	−0.822181	+	0.569227i	$0.807243\pi$
$74$	1687.79	2.65138
$75$	0	0
$76$	2365.55	3.57036
$77$	0.667165	0.000987409 0
$78$	0	0
$79$	210.661	0.300015	0.150007	−	0.988685i	$-0.452070\pi$
$79$	210.661	0.300015	0.150007	+	0.988685i	$0.452070\pi$
$80$	1627.48	2.27448
$81$	0	0
$82$	775.365	1.04420
$83$	879.930	1.16367	0.581836	−	0.813306i	$-0.302334\pi$
$83$	879.930	1.16367	0.581836	+	0.813306i	$0.302334\pi$
$84$	0	0
$85$	507.258	0.647292
$86$	−170.135	−0.213327
$87$	0	0
$88$	−50.1770	−0.0607828
$89$	−461.975	−0.550216	−0.275108	−	0.961413i	$-0.588714\pi$
$89$	−461.975	−0.550216	−0.275108	+	0.961413i	$0.588714\pi$
$90$	0	0
$91$	−69.7709	−0.0803734
$92$	−1485.26	−1.68314
$93$	0	0
$94$	−1422.83	−1.56121
$95$	1222.80	1.32060
$96$	0	0
$97$	−372.791	−0.390218	−0.195109	−	0.980782i	$-0.562506\pi$
$97$	−372.791	−0.390218	−0.195109	+	0.980782i	$0.562506\pi$
$98$	1796.32	1.85159
$99$	0	0
$100$	−450.651	−0.450651
$101$	982.265	0.967713	0.483856	−	0.875147i	$-0.339236\pi$
$101$	982.265	0.967713	0.483856	+	0.875147i	$0.339236\pi$
$102$	0	0
$103$	1237.81	1.18413	0.592065	−	0.805890i	$-0.298313\pi$
$103$	1237.81	1.18413	0.592065	+	0.805890i	$0.298313\pi$
$104$	5247.43	4.94762
$105$	0	0
$106$	581.741	0.533054
$107$	1287.57	1.16331	0.581657	−	0.813434i	$-0.302405\pi$
$107$	1287.57	1.16331	0.581657	+	0.813434i	$0.302405\pi$
$108$	0	0
$109$	2123.88	1.86634	0.933169	−	0.359438i	$-0.117032\pi$
$109$	2123.88	1.86634	0.933169	+	0.359438i	$0.117032\pi$
$110$	−43.9330	−0.0380804
$111$	0	0
$112$	129.678	0.109406
$113$	−633.826	−0.527658	−0.263829	−	0.964569i	$-0.584985\pi$
$113$	−633.826	−0.527658	−0.263829	+	0.964569i	$0.584985\pi$
$114$	0	0
$115$	−767.763	−0.622559
$116$	786.962	0.629893
$117$	0	0
$118$	−4289.16	−3.34618
$119$	40.4184	0.0311357
$120$	0	0
$121$	−1330.31	−0.999483
$122$	2585.77	1.91889
$123$	0	0
$124$	−1582.13	−1.14580
$125$	−1494.93	−1.06969
$126$	0	0
$127$	−567.851	−0.396761	−0.198380	−	0.980125i	$-0.563568\pi$
$127$	−567.851	−0.396761	−0.198380	+	0.980125i	$0.563568\pi$
$128$	−299.943	−0.207121
$129$	0	0
$130$	4594.44	3.09968
$131$	1255.97	0.837671	0.418836	−	0.908062i	$-0.362438\pi$
$131$	1255.97	0.837671	0.418836	+	0.908062i	$0.362438\pi$
$132$	0	0
$133$	97.4334	0.0635229
$134$	−4523.60	−2.91627
$135$	0	0
$136$	−3039.85	−1.91665
$137$	−1960.41	−1.22255	−0.611273	−	0.791420i	$-0.709342\pi$
$137$	−1960.41	−1.22255	−0.611273	+	0.791420i	$0.709342\pi$
$138$	0	0
$139$	36.7466	0.0224231	0.0112115	−	0.999937i	$-0.496431\pi$
$139$	36.7466	0.0224231	0.0112115	+	0.999937i	$0.496431\pi$
$140$	158.619	0.0957552
$141$	0	0
$142$	−2073.41	−1.22533
$143$	−71.9318	−0.0420646
$144$	0	0
$145$	406.797	0.232984
$146$	5381.34	3.05043
$147$	0	0
$148$	−6282.44	−3.48928
$149$	1648.67	0.906470	0.453235	−	0.891391i	$-0.350270\pi$
$149$	1648.67	0.906470	0.453235	+	0.891391i	$0.350270\pi$
$150$	0	0
$151$	418.856	0.225735	0.112868	−	0.993610i	$-0.463996\pi$
$151$	418.856	0.225735	0.112868	+	0.993610i	$0.463996\pi$
$152$	−7327.91	−3.91034
$153$	0	0
$154$	−3.50060	−0.00183173
$155$	−817.837	−0.423808
$156$	0	0
$157$	1258.72	0.639854	0.319927	−	0.947442i	$-0.396342\pi$
$157$	1258.72	0.639854	0.319927	+	0.947442i	$0.396342\pi$
$158$	−1105.33	−0.556553
$159$	0	0
$160$	−3652.86	−1.80490
$161$	−61.1756	−0.0299460
$162$	0	0
$163$	1096.41	0.526855	0.263427	−	0.964679i	$-0.415147\pi$
$163$	1096.41	0.526855	0.263427	+	0.964679i	$0.415147\pi$
$164$	−2886.13	−1.37420
$165$	0	0
$166$	−4616.97	−2.15871
$167$	−2947.26	−1.36567	−0.682833	−	0.730575i	$-0.739252\pi$
$167$	−2947.26	−1.36567	−0.682833	+	0.730575i	$0.739252\pi$
$168$	0	0
$169$	5325.50	2.42399
$170$	−2661.57	−1.20078
$171$	0	0
$172$	633.290	0.280744
$173$	3254.19	1.43012	0.715062	−	0.699061i	$-0.246398\pi$
$173$	3254.19	1.43012	0.715062	+	0.699061i	$0.246398\pi$
$174$	0	0
$175$	−18.5616	−0.00801786
$176$	133.695	0.0572591
$177$	0	0
$178$	2423.97	1.02070
$179$	690.970	0.288523	0.144261	−	0.989540i	$-0.453919\pi$
$179$	690.970	0.288523	0.144261	+	0.989540i	$0.453919\pi$
$180$	0	0
$181$	4480.00	1.83975	0.919877	−	0.392207i	$-0.128288\pi$
$181$	4480.00	1.83975	0.919877	+	0.392207i	$0.128288\pi$
$182$	366.086	0.149100
$183$	0	0
$184$	4600.98	1.84342
$185$	−3247.53	−1.29061
$186$	0	0
$187$	41.6702	0.0162953
$188$	5296.18	2.05459
$189$	0	0
$190$	−6416.02	−2.44983
$191$	3071.51	1.16359	0.581797	−	0.813334i	$-0.302350\pi$
$191$	3071.51	1.16359	0.581797	+	0.813334i	$0.302350\pi$
$192$	0	0
$193$	−1090.71	−0.406793	−0.203397	−	0.979096i	$-0.565198\pi$
$193$	−1090.71	−0.406793	−0.203397	+	0.979096i	$0.565198\pi$
$194$	1956.02	0.723888
$195$	0	0
$196$	−6686.40	−2.43673
$197$	−4977.93	−1.80032	−0.900160	−	0.435560i	$-0.856550\pi$
$197$	−4977.93	−1.80032	−0.900160	+	0.435560i	$0.856550\pi$
$198$	0	0
$199$	−1050.73	−0.374294	−0.187147	−	0.982332i	$-0.559924\pi$
$199$	−1050.73	−0.374294	−0.187147	+	0.982332i	$0.559924\pi$
$200$	1396.01	0.493563
$201$	0	0
$202$	−5153.92	−1.79519
$203$	32.4137	0.0112069
$204$	0	0
$205$	−1491.90	−0.508287
$206$	−6494.77	−2.19666
$207$	0	0
$208$	−13981.5	−4.66080
$209$	100.451	0.0332456
$210$	0	0
$211$	−467.437	−0.152510	−0.0762551	−	0.997088i	$-0.524296\pi$
$211$	−467.437	−0.152510	−0.0762551	+	0.997088i	$0.524296\pi$
$212$	−2165.41	−0.701513
$213$	0	0
$214$	−6755.87	−2.15804
$215$	327.361	0.103841
$216$	0	0
$217$	−65.1655	−0.0203858
$218$	−11143.9	−3.46222
$219$	0	0
$220$	163.531	0.0501149
$221$	−4357.80	−1.32641
$222$	0	0
$223$	−1929.50	−0.579411	−0.289706	−	0.957116i	$-0.593557\pi$
$223$	−1929.50	−0.579411	−0.289706	+	0.957116i	$0.593557\pi$
$224$	−291.061	−0.0868185
$225$	0	0
$226$	3325.67	0.978851
$227$	−3767.18	−1.10148	−0.550741	−	0.834676i	$-0.685655\pi$
$227$	−3767.18	−1.10148	−0.550741	+	0.834676i	$0.685655\pi$
$228$	0	0
$229$	3912.29	1.12896	0.564480	−	0.825447i	$-0.309077\pi$
$229$	3912.29	1.12896	0.564480	+	0.825447i	$0.309077\pi$
$230$	4028.43	1.15490
$231$	0	0
$232$	−2437.82	−0.689873
$233$	2768.47	0.778406	0.389203	−	0.921152i	$-0.372750\pi$
$233$	2768.47	0.778406	0.389203	+	0.921152i	$0.372750\pi$
$234$	0	0
$235$	2737.71	0.759950
$236$	15965.5	4.40366
$237$	0	0
$238$	−212.074	−0.0577594
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	649.271	0.173540	0.0867701	−	0.996228i	$-0.472345\pi$
$241$	649.271	0.173540	0.0867701	+	0.996228i	$0.472345\pi$
$242$	6980.11	1.85413
$243$	0	0
$244$	−9624.98	−2.52531
$245$	−3456.34	−0.901296
$246$	0	0
$247$	−10505.0	−2.70614
$248$	4901.05	1.25491
$249$	0	0
$250$	7843.86	1.98436
$251$	−2455.96	−0.617604	−0.308802	−	0.951126i	$-0.599928\pi$
$251$	−2455.96	−0.617604	−0.308802	+	0.951126i	$0.599928\pi$
$252$	0	0
$253$	−63.0702	−0.0156727
$254$	2979.50	0.736025
$255$	0	0
$256$	−3296.82	−0.804887
$257$	−382.561	−0.0928541	−0.0464271	−	0.998922i	$-0.514784\pi$
$257$	−382.561	−0.0928541	−0.0464271	+	0.998922i	$0.514784\pi$
$258$	0	0
$259$	−258.764	−0.0620803
$260$	−17101.8	−4.07927
$261$	0	0
$262$	−6590.06	−1.55395
$263$	2500.94	0.586367	0.293183	−	0.956056i	$-0.405285\pi$
$263$	2500.94	0.586367	0.293183	+	0.956056i	$0.405285\pi$
$264$	0	0
$265$	−1119.34	−0.259474
$266$	−511.231	−0.117840
$267$	0	0
$268$	16838.1	3.83788
$269$	7654.24	1.73490	0.867448	−	0.497527i	$-0.165759\pi$
$269$	7654.24	1.73490	0.867448	+	0.497527i	$0.165759\pi$
$270$	0	0
$271$	−7244.71	−1.62393	−0.811964	−	0.583707i	$-0.801602\pi$
$271$	−7244.71	−1.62393	−0.811964	+	0.583707i	$0.801602\pi$
$272$	8099.54	1.80554
$273$	0	0
$274$	10286.2	2.26793
$275$	−19.1365	−0.00419626
$276$	0	0
$277$	3761.47	0.815901	0.407951	−	0.913004i	$-0.366244\pi$
$277$	3761.47	0.815901	0.407951	+	0.913004i	$0.366244\pi$
$278$	−192.809	−0.0415967
$279$	0	0
$280$	−491.362	−0.104873
$281$	1308.50	0.277789	0.138894	−	0.990307i	$-0.455645\pi$
$281$	1308.50	0.277789	0.138894	+	0.990307i	$0.455645\pi$
$282$	0	0
$283$	−6594.73	−1.38521	−0.692607	−	0.721315i	$-0.743538\pi$
$283$	−6594.73	−1.38521	−0.692607	+	0.721315i	$0.743538\pi$
$284$	7717.84	1.61257
$285$	0	0
$286$	377.424	0.0780334
$287$	−118.875	−0.0244494
$288$	0	0
$289$	−2388.52	−0.486163
$290$	−2134.45	−0.432205
$291$	0	0
$292$	−20030.9	−4.01445
$293$	54.5617	0.0108789	0.00543947	−	0.999985i	$-0.498269\pi$
$293$	54.5617	0.0108789	0.00543947	+	0.999985i	$0.498269\pi$
$294$	0	0
$295$	8252.89	1.62882
$296$	19461.5	3.82154
$297$	0	0
$298$	−8650.51	−1.68158
$299$	6595.78	1.27573
$300$	0	0
$301$	26.0842	0.00499491
$302$	−2197.73	−0.418758
$303$	0	0
$304$	19524.9	3.68365
$305$	−4975.36	−0.934059
$306$	0	0
$307$	401.434	0.0746288	0.0373144	−	0.999304i	$-0.488120\pi$
$307$	401.434	0.0746288	0.0373144	+	0.999304i	$0.488120\pi$
$308$	13.0302	0.00241060
$309$	0	0
$310$	4291.17	0.786200
$311$	−181.977	−0.0331800	−0.0165900	−	0.999862i	$-0.505281\pi$
$311$	−181.977	−0.0331800	−0.0165900	+	0.999862i	$0.505281\pi$
$312$	0	0
$313$	3794.05	0.685151	0.342575	−	0.939490i	$-0.388701\pi$
$313$	3794.05	0.685151	0.342575	+	0.939490i	$0.388701\pi$
$314$	−6604.49	−1.18698
$315$	0	0
$316$	4114.35	0.732438
$317$	5855.74	1.03751	0.518756	−	0.854923i	$-0.326395\pi$
$317$	5855.74	1.03751	0.518756	+	0.854923i	$0.326395\pi$
$318$	0	0
$319$	33.4176	0.00586529
$320$	6146.61	1.07377
$321$	0	0
$322$	320.987	0.0555525
$323$	6085.56	1.04833
$324$	0	0
$325$	2001.26	0.341569
$326$	−5752.82	−0.977360
$327$	0	0
$328$	8940.52	1.50505
$329$	218.141	0.0365548
$330$	0	0
$331$	5803.68	0.963744	0.481872	−	0.876242i	$-0.339957\pi$
$331$	5803.68	0.963744	0.481872	+	0.876242i	$0.339957\pi$
$332$	17185.7	2.84092
$333$	0	0
$334$	15464.2	2.53343
$335$	8703.98	1.41955
$336$	0	0
$337$	−9871.07	−1.59558	−0.797792	−	0.602933i	$-0.793999\pi$
$337$	−9871.07	−1.59558	−0.797792	+	0.602933i	$0.793999\pi$
$338$	−27942.8	−4.49670
$339$	0	0
$340$	9907.11	1.58026
$341$	−67.1837	−0.0106692
$342$	0	0
$343$	−551.325	−0.0867894
$344$	−1961.78	−0.307477
$345$	0	0
$346$	−17074.7	−2.65300
$347$	−1360.44	−0.210468	−0.105234	−	0.994447i	$-0.533559\pi$
$347$	−1360.44	−0.210468	−0.105234	+	0.994447i	$0.533559\pi$
$348$	0	0
$349$	7884.72	1.20934	0.604670	−	0.796476i	$-0.293305\pi$
$349$	7884.72	1.20934	0.604670	+	0.796476i	$0.293305\pi$
$350$	97.3923	0.0148738
$351$	0	0
$352$	−300.076	−0.0454377
$353$	−3655.13	−0.551113	−0.275557	−	0.961285i	$-0.588862\pi$
$353$	−3655.13	−0.551113	−0.275557	+	0.961285i	$0.588862\pi$
$354$	0	0
$355$	3989.51	0.596454
$356$	−9022.70	−1.34326
$357$	0	0
$358$	−3625.50	−0.535234
$359$	−10645.2	−1.56499	−0.782497	−	0.622654i	$-0.786054\pi$
$359$	−10645.2	−1.56499	−0.782497	+	0.622654i	$0.786054\pi$
$360$	0	0
$361$	7810.96	1.13879
$362$	−23506.4	−3.41290
$363$	0	0
$364$	−1362.68	−0.196219
$365$	−10354.4	−1.48486
$366$	0	0
$367$	5002.35	0.711500	0.355750	−	0.934581i	$-0.384225\pi$
$367$	5002.35	0.711500	0.355750	+	0.934581i	$0.384225\pi$
$368$	−12259.1	−1.73655
$369$	0	0
$370$	17039.7	2.39419
$371$	−89.1896	−0.0124811
$372$	0	0
$373$	9773.39	1.35669	0.678347	−	0.734741i	$-0.262697\pi$
$373$	9773.39	1.35669	0.678347	+	0.734741i	$0.262697\pi$
$374$	−218.642	−0.0302292
$375$	0	0
$376$	−16406.3	−2.25024
$377$	−3494.75	−0.477424
$378$	0	0
$379$	565.733	0.0766748	0.0383374	−	0.999265i	$-0.487794\pi$
$379$	565.733	0.0766748	0.0383374	+	0.999265i	$0.487794\pi$
$380$	23882.3	3.22404
$381$	0	0
$382$	−16116.1	−2.15857
$383$	7907.81	1.05501	0.527507	−	0.849551i	$-0.323127\pi$
$383$	7907.81	1.05501	0.527507	+	0.849551i	$0.323127\pi$
$384$	0	0
$385$	6.73559	0.000891630 0
$386$	5722.93	0.754636
$387$	0	0
$388$	−7280.88	−0.952656
$389$	−2665.09	−0.347367	−0.173683	−	0.984802i	$-0.555567\pi$
$389$	−2665.09	−0.347367	−0.173683	+	0.984802i	$0.555567\pi$
$390$	0	0
$391$	−3820.95	−0.494204
$392$	20712.8	2.66876
$393$	0	0
$394$	26119.1	3.33975
$395$	2126.80	0.270913
$396$	0	0
$397$	1845.78	0.233343	0.116671	−	0.993171i	$-0.462778\pi$
$397$	1845.78	0.233343	0.116671	+	0.993171i	$0.462778\pi$
$398$	5513.17	0.694347
$399$	0	0
$400$	−3719.60	−0.464950
$401$	−8399.16	−1.04597	−0.522985	−	0.852342i	$-0.675182\pi$
$401$	−8399.16	−1.04597	−0.522985	+	0.852342i	$0.675182\pi$
$402$	0	0
$403$	7025.95	0.868456
$404$	19184.3	2.36252
$405$	0	0
$406$	−170.074	−0.0207897
$407$	−266.778	−0.0324906
$408$	0	0
$409$	103.382	0.0124986	0.00624929	−	0.999980i	$-0.498011\pi$
$409$	103.382	0.0124986	0.00624929	+	0.999980i	$0.498011\pi$
$410$	7827.96	0.942916
$411$	0	0
$412$	24175.4	2.89086
$413$	657.593	0.0783487
$414$	0	0
$415$	8883.63	1.05080
$416$	31381.4	3.69856
$417$	0	0
$418$	−527.063	−0.0616735
$419$	10313.4	1.20249	0.601247	−	0.799063i	$-0.294671\pi$
$419$	10313.4	1.20249	0.601247	+	0.799063i	$0.294671\pi$
$420$	0	0
$421$	4651.67	0.538500	0.269250	−	0.963070i	$-0.413224\pi$
$421$	4651.67	0.538500	0.269250	+	0.963070i	$0.413224\pi$
$422$	2452.63	0.282919
$423$	0	0
$424$	6707.90	0.768312
$425$	−1159.33	−0.132320
$426$	0	0
$427$	−396.438	−0.0449297
$428$	25147.3	2.84004
$429$	0	0
$430$	−1717.65	−0.192634
$431$	−17304.9	−1.93398	−0.966991	−	0.254811i	$-0.917987\pi$
$431$	−17304.9	−1.93398	−0.966991	+	0.254811i	$0.917987\pi$
$432$	0	0
$433$	4053.60	0.449893	0.224947	−	0.974371i	$-0.427779\pi$
$433$	4053.60	0.449893	0.224947	+	0.974371i	$0.427779\pi$
$434$	341.922	0.0378174
$435$	0	0
$436$	41480.9	4.55637
$437$	−9210.84	−1.00827
$438$	0	0
$439$	11949.7	1.29916	0.649578	−	0.760295i	$-0.274946\pi$
$439$	11949.7	1.29916	0.649578	+	0.760295i	$0.274946\pi$
$440$	−506.579	−0.0548869
$441$	0	0
$442$	22865.3	2.46061
$443$	9196.35	0.986302	0.493151	−	0.869944i	$-0.335845\pi$
$443$	9196.35	0.986302	0.493151	+	0.869944i	$0.335845\pi$
$444$	0	0
$445$	−4664.02	−0.496845
$446$	10124.0	1.07486
$447$	0	0
$448$	489.764	0.0516499
$449$	14223.8	1.49501	0.747506	−	0.664255i	$-0.231251\pi$
$449$	14223.8	1.49501	0.747506	+	0.664255i	$0.231251\pi$
$450$	0	0
$451$	−122.557	−0.0127959
$452$	−12379.1	−1.28819
$453$	0	0
$454$	19766.3	2.04334
$455$	−704.396	−0.0725772
$456$	0	0
$457$	8717.68	0.892333	0.446166	−	0.894950i	$-0.352789\pi$
$457$	8717.68	0.892333	0.446166	+	0.894950i	$0.352789\pi$
$458$	−20527.7	−2.09432
$459$	0	0
$460$	−14995.0	−1.51988
$461$	627.326	0.0633785	0.0316893	−	0.999498i	$-0.489911\pi$
$461$	627.326	0.0633785	0.0316893	+	0.999498i	$0.489911\pi$
$462$	0	0
$463$	−2601.88	−0.261165	−0.130583	−	0.991437i	$-0.541685\pi$
$463$	−2601.88	−0.261165	−0.130583	+	0.991437i	$0.541685\pi$
$464$	6495.46	0.649880
$465$	0	0
$466$	−14526.1	−1.44401
$467$	12141.3	1.20307	0.601533	−	0.798848i	$-0.294557\pi$
$467$	12141.3	1.20307	0.601533	+	0.798848i	$0.294557\pi$
$468$	0	0
$469$	693.536	0.0682825
$470$	−14364.7	−1.40977
$471$	0	0
$472$	−49457.1	−4.82299
$473$	26.8920	0.00261416
$474$	0	0
$475$	−2794.71	−0.269958
$476$	789.402	0.0760129
$477$	0	0
$478$	1254.03	0.119995
$479$	7262.80	0.692789	0.346394	−	0.938089i	$-0.387406\pi$
$479$	7262.80	0.692789	0.346394	+	0.938089i	$0.387406\pi$
$480$	0	0
$481$	27899.2	2.64468
$482$	−3406.71	−0.321932
$483$	0	0
$484$	−25982.0	−2.44008
$485$	−3763.64	−0.352367
$486$	0	0
$487$	6346.92	0.590568	0.295284	−	0.955410i	$-0.404586\pi$
$487$	6346.92	0.590568	0.295284	+	0.955410i	$0.404586\pi$
$488$	29815.8	2.76578
$489$	0	0
$490$	18135.3	1.67198
$491$	4676.73	0.429853	0.214926	−	0.976630i	$-0.431049\pi$
$491$	4676.73	0.429853	0.214926	+	0.976630i	$0.431049\pi$
$492$	0	0
$493$	2024.52	0.184949
$494$	55119.4	5.02012
$495$	0	0
$496$	−13058.7	−1.18216
$497$	317.885	0.0286904
$498$	0	0
$499$	−4655.75	−0.417676	−0.208838	−	0.977950i	$-0.566968\pi$
$499$	−4655.75	−0.417676	−0.208838	+	0.977950i	$0.566968\pi$
$500$	−29197.1	−2.61147
$501$	0	0
$502$	12886.3	1.14571
$503$	−2338.40	−0.207285	−0.103642	−	0.994615i	$-0.533050\pi$
$503$	−2338.40	−0.207285	−0.103642	+	0.994615i	$0.533050\pi$
$504$	0	0
$505$	9916.79	0.873844
$506$	330.928	0.0290742
$507$	0	0
$508$	−11090.5	−0.968628
$509$	−7899.22	−0.687872	−0.343936	−	0.938993i	$-0.611760\pi$
$509$	−7899.22	−0.687872	−0.343936	+	0.938993i	$0.611760\pi$
$510$	0	0
$511$	−825.040	−0.0714239
$512$	19697.8	1.70025
$513$	0	0
$514$	2007.29	0.172252
$515$	12496.8	1.06927
$516$	0	0
$517$	224.897	0.0191315
$518$	1357.73	0.115164
$519$	0	0
$520$	52977.2	4.46770
$521$	−7815.67	−0.657219	−0.328609	−	0.944466i	$-0.606580\pi$
$521$	−7815.67	−0.657219	−0.328609	+	0.944466i	$0.606580\pi$
$522$	0	0
$523$	19469.3	1.62779	0.813895	−	0.581012i	$-0.197343\pi$
$523$	19469.3	1.62779	0.813895	+	0.581012i	$0.197343\pi$
$524$	24530.1	2.04504
$525$	0	0
$526$	−13122.4	−1.08776
$527$	−4070.15	−0.336430
$528$	0	0
$529$	−6383.78	−0.524680
$530$	5873.17	0.481347
$531$	0	0
$532$	1902.95	0.155081
$533$	12816.8	1.04157
$534$	0	0
$535$	12999.1	1.05047
$536$	−52160.4	−4.20333
$537$	0	0
$538$	−40161.6	−3.21838
$539$	−283.932	−0.0226898
$540$	0	0
$541$	−8167.14	−0.649044	−0.324522	−	0.945878i	$-0.605203\pi$
$541$	−8167.14	−0.649044	−0.324522	+	0.945878i	$0.605203\pi$
$542$	38012.8	3.01253
$543$	0	0
$544$	−18179.3	−1.43278
$545$	21442.4	1.68530
$546$	0	0
$547$	−300.206	−0.0234660	−0.0117330	−	0.999931i	$-0.503735\pi$
$547$	−300.206	−0.0234660	−0.0117330	+	0.999931i	$0.503735\pi$
$548$	−38288.2	−2.98465
$549$	0	0
$550$	100.409	0.00778443
$551$	4880.34	0.377331
$552$	0	0
$553$	169.464	0.0130313
$554$	−19736.3	−1.51357
$555$	0	0
$556$	717.688	0.0547424
$557$	−21047.1	−1.60106	−0.800532	−	0.599289i	$-0.795450\pi$
$557$	−21047.1	−1.60106	−0.800532	+	0.599289i	$0.795450\pi$
$558$	0	0
$559$	−2812.32	−0.212788
$560$	1309.21	0.0987935
$561$	0	0
$562$	−6865.67	−0.515322
$563$	337.232	0.0252445	0.0126222	−	0.999920i	$-0.495982\pi$
$563$	337.232	0.0252445	0.0126222	+	0.999920i	$0.495982\pi$
$564$	0	0
$565$	−6399.01	−0.476475
$566$	34602.3	2.56969
$567$	0	0
$568$	−23908.0	−1.76612
$569$	16539.0	1.21854	0.609270	−	0.792963i	$-0.291462\pi$
$569$	16539.0	1.21854	0.609270	+	0.792963i	$0.291462\pi$
$570$	0	0
$571$	−9813.91	−0.719263	−0.359632	−	0.933094i	$-0.617098\pi$
$571$	−9813.91	−0.719263	−0.359632	+	0.933094i	$0.617098\pi$
$572$	−1404.88	−0.102694
$573$	0	0
$574$	623.734	0.0453557
$575$	1754.72	0.127264
$576$	0	0
$577$	9201.66	0.663900	0.331950	−	0.943297i	$-0.392293\pi$
$577$	9201.66	0.663900	0.331950	+	0.943297i	$0.392293\pi$
$578$	12532.5	0.901873
$579$	0	0
$580$	7945.05	0.568793
$581$	707.851	0.0505449
$582$	0	0
$583$	−91.9519	−0.00653217
$584$	62050.7	4.39671
$585$	0	0
$586$	−286.284	−0.0201813
$587$	22231.6	1.56320	0.781599	−	0.623782i	$-0.214405\pi$
$587$	22231.6	1.56320	0.781599	+	0.623782i	$0.214405\pi$
$588$	0	0
$589$	−9811.57	−0.686382
$590$	−43302.7	−3.02160
$591$	0	0
$592$	−51854.3	−3.60000
$593$	−11178.1	−0.774081	−0.387040	−	0.922063i	$-0.626503\pi$
$593$	−11178.1	−0.774081	−0.387040	+	0.922063i	$0.626503\pi$
$594$	0	0
$595$	408.058	0.0281156
$596$	32199.6	2.21300
$597$	0	0
$598$	−34607.9	−2.36659
$599$	−3692.27	−0.251856	−0.125928	−	0.992039i	$-0.540191\pi$
$599$	−3692.27	−0.251856	−0.125928	+	0.992039i	$0.540191\pi$
$600$	0	0
$601$	−14403.8	−0.977608	−0.488804	−	0.872394i	$-0.662567\pi$
$601$	−14403.8	−0.977608	−0.488804	+	0.872394i	$0.662567\pi$
$602$	−136.863	−0.00926599
$603$	0	0
$604$	8180.57	0.551097
$605$	−13430.6	−0.902533
$606$	0	0
$607$	17874.2	1.19521	0.597604	−	0.801792i	$-0.296120\pi$
$607$	17874.2	1.19521	0.597604	+	0.801792i	$0.296120\pi$
$608$	−43823.3	−2.92314
$609$	0	0
$610$	26105.6	1.73276
$611$	−23519.4	−1.55727
$612$	0	0
$613$	315.861	0.0208116	0.0104058	−	0.999946i	$-0.496688\pi$
$613$	315.861	0.0208116	0.0104058	+	0.999946i	$0.496688\pi$
$614$	−2106.31	−0.138443
$615$	0	0
$616$	−40.3644	−0.00264014
$617$	7184.59	0.468786	0.234393	−	0.972142i	$-0.424690\pi$
$617$	7184.59	0.468786	0.234393	+	0.972142i	$0.424690\pi$
$618$	0	0
$619$	−2840.58	−0.184447	−0.0922235	−	0.995738i	$-0.529397\pi$
$619$	−2840.58	−0.184447	−0.0922235	+	0.995738i	$0.529397\pi$
$620$	−15972.9	−1.03466
$621$	0	0
$622$	954.829	0.0615517
$623$	−371.631	−0.0238990
$624$	0	0
$625$	−12208.3	−0.781334
$626$	−19907.3	−1.27101
$627$	0	0
$628$	24583.8	1.56210
$629$	−16162.0	−1.02452
$630$	0	0
$631$	836.274	0.0527599	0.0263800	−	0.999652i	$-0.491602\pi$
$631$	836.274	0.0527599	0.0263800	+	0.999652i	$0.491602\pi$
$632$	−12745.3	−0.802183
$633$	0	0
$634$	−30724.9	−1.92467
$635$	−5732.93	−0.358275
$636$	0	0
$637$	29693.1	1.84691
$638$	−175.341	−0.0108806
$639$	0	0
$640$	−3028.17	−0.187030
$641$	31860.5	1.96320	0.981601	−	0.190944i	$-0.0611548\pi$
$641$	31860.5	1.96320	0.981601	+	0.190944i	$0.0611548\pi$
$642$	0	0
$643$	26267.6	1.61103	0.805517	−	0.592573i	$-0.201888\pi$
$643$	26267.6	1.61103	0.805517	+	0.592573i	$0.201888\pi$
$644$	−1194.80	−0.0731085
$645$	0	0
$646$	−31930.8	−1.94474
$647$	−20151.8	−1.22450	−0.612248	−	0.790666i	$-0.709735\pi$
$647$	−20151.8	−1.22450	−0.612248	+	0.790666i	$0.709735\pi$
$648$	0	0
$649$	677.959	0.0410049
$650$	−10500.6	−0.633639
$651$	0	0
$652$	21413.6	1.28623
$653$	−26410.1	−1.58270	−0.791352	−	0.611361i	$-0.790622\pi$
$653$	−26410.1	−1.58270	−0.791352	+	0.611361i	$0.790622\pi$
$654$	0	0
$655$	12680.1	0.756417
$656$	−23821.6	−1.41780
$657$	0	0
$658$	−1144.58	−0.0678122
$659$	−720.199	−0.0425721	−0.0212860	−	0.999773i	$-0.506776\pi$
$659$	−720.199	−0.0425721	−0.0212860	+	0.999773i	$0.506776\pi$
$660$	0	0
$661$	−20377.6	−1.19909	−0.599544	−	0.800342i	$-0.704651\pi$
$661$	−20377.6	−1.19909	−0.599544	+	0.800342i	$0.704651\pi$
$662$	−30451.8	−1.78783
$663$	0	0
$664$	−53237.0	−3.11144
$665$	983.672	0.0573612
$666$	0	0
$667$	−3064.22	−0.177882
$668$	−57562.2	−3.33406
$669$	0	0
$670$	−45669.6	−2.63339
$671$	−408.716	−0.0235146
$672$	0	0
$673$	−7590.31	−0.434747	−0.217374	−	0.976088i	$-0.569749\pi$
$673$	−7590.31	−0.434747	−0.217374	+	0.976088i	$0.569749\pi$
$674$	51793.3	2.95994
$675$	0	0
$676$	104011.	5.91778
$677$	−27177.2	−1.54284	−0.771421	−	0.636325i	$-0.780454\pi$
$677$	−27177.2	−1.54284	−0.771421	+	0.636325i	$0.780454\pi$
$678$	0	0
$679$	−299.888	−0.0169494
$680$	−30689.8	−1.73074
$681$	0	0
$682$	352.511	0.0197923
$683$	23544.6	1.31905	0.659524	−	0.751684i	$-0.270758\pi$
$683$	23544.6	1.31905	0.659524	+	0.751684i	$0.270758\pi$
$684$	0	0
$685$	−19792.0	−1.10396
$686$	2892.79	0.161002
$687$	0	0
$688$	5227.07	0.289652
$689$	9616.17	0.531708
$690$	0	0
$691$	−5593.57	−0.307944	−0.153972	−	0.988075i	$-0.549207\pi$
$691$	−5593.57	−0.307944	−0.153972	+	0.988075i	$0.549207\pi$
$692$	63556.7	3.49142
$693$	0	0
$694$	7138.22	0.390437
$695$	370.988	0.0202480
$696$	0	0
$697$	−7424.78	−0.403491
$698$	−41370.9	−2.24343
$699$	0	0
$700$	−362.522	−0.0195743
$701$	29596.7	1.59465	0.797327	−	0.603548i	$-0.206247\pi$
$701$	29596.7	1.59465	0.797327	+	0.603548i	$0.206247\pi$
$702$	0	0
$703$	−38960.5	−2.09022
$704$	504.932	0.0270317
$705$	0	0
$706$	19178.4	1.02236
$707$	790.173	0.0420333
$708$	0	0
$709$	11431.8	0.605542	0.302771	−	0.953063i	$-0.402088\pi$
$709$	11431.8	0.605542	0.302771	+	0.953063i	$0.402088\pi$
$710$	−20932.9	−1.10647
$711$	0	0
$712$	27950.1	1.47117
$713$	6160.40	0.323575
$714$	0	0
$715$	−726.212	−0.0379843
$716$	13495.1	0.704382
$717$	0	0
$718$	55855.2	2.90320
$719$	−14447.3	−0.749364	−0.374682	−	0.927153i	$-0.622248\pi$
$719$	−14447.3	−0.749364	−0.374682	+	0.927153i	$0.622248\pi$
$720$	0	0
$721$	995.746	0.0514335
$722$	−40983.9	−2.11255
$723$	0	0
$724$	87497.6	4.49147
$725$	−929.732	−0.0476267
$726$	0	0
$727$	−30498.2	−1.55587	−0.777935	−	0.628345i	$-0.783733\pi$
$727$	−30498.2	−1.55587	−0.777935	+	0.628345i	$0.783733\pi$
$728$	4221.24	0.214903
$729$	0	0
$730$	54329.2	2.75454
$731$	1629.18	0.0824317
$732$	0	0
$733$	−3580.60	−0.180426	−0.0902132	−	0.995922i	$-0.528755\pi$
$733$	−3580.60	−0.180426	−0.0902132	+	0.995922i	$0.528755\pi$
$734$	−26247.2	−1.31989
$735$	0	0
$736$	27515.4	1.37803
$737$	715.015	0.0357366
$738$	0	0
$739$	25098.0	1.24932	0.624658	−	0.780899i	$-0.285238\pi$
$739$	25098.0	1.24932	0.624658	+	0.780899i	$0.285238\pi$
$740$	−63426.6	−3.15082
$741$	0	0
$742$	467.976	0.0231535
$743$	15887.1	0.784443	0.392221	−	0.919871i	$-0.371707\pi$
$743$	15887.1	0.784443	0.392221	+	0.919871i	$0.371707\pi$
$744$	0	0
$745$	16644.7	0.818542
$746$	−51280.7	−2.51678
$747$	0	0
$748$	813.850	0.0397825
$749$	1035.78	0.0505293
$750$	0	0
$751$	−5066.16	−0.246161	−0.123080	−	0.992397i	$-0.539277\pi$
$751$	−5066.16	−0.246161	−0.123080	+	0.992397i	$0.539277\pi$
$752$	43713.8	2.11979
$753$	0	0
$754$	18336.9	0.885663
$755$	4228.71	0.203839
$756$	0	0
$757$	29118.9	1.39808	0.699039	−	0.715084i	$-0.253612\pi$
$757$	29118.9	1.39808	0.699039	+	0.715084i	$0.253612\pi$
$758$	−2968.39	−0.142238
$759$	0	0
$760$	−73981.4	−3.53104
$761$	39582.7	1.88551	0.942755	−	0.333487i	$-0.108225\pi$
$761$	39582.7	1.88551	0.942755	+	0.333487i	$0.108225\pi$
$762$	0	0
$763$	1708.53	0.0810656
$764$	59988.7	2.84073
$765$	0	0
$766$	−41492.1	−1.95714
$767$	−70899.8	−3.33773
$768$	0	0
$769$	3581.25	0.167936	0.0839682	−	0.996468i	$-0.473241\pi$
$769$	3581.25	0.167936	0.0839682	+	0.996468i	$0.473241\pi$
$770$	−35.3415	−0.00165405
$771$	0	0
$772$	−21302.4	−0.993121
$773$	1820.20	0.0846935	0.0423468	−	0.999103i	$-0.486517\pi$
$773$	1820.20	0.0846935	0.0423468	+	0.999103i	$0.486517\pi$
$774$	0	0
$775$	1869.16	0.0866351
$776$	22554.4	1.04337
$777$	0	0
$778$	13983.7	0.644395
$779$	−17898.3	−0.823200
$780$	0	0
$781$	327.730	0.0150155
$782$	20048.4	0.916790
$783$	0	0
$784$	−55188.5	−2.51405
$785$	12707.9	0.577788
$786$	0	0
$787$	−38042.7	−1.72310	−0.861548	−	0.507676i	$-0.830505\pi$
$787$	−38042.7	−1.72310	−0.861548	+	0.507676i	$0.830505\pi$
$788$	−97222.6	−4.39519
$789$	0	0
$790$	−11159.2	−0.502567
$791$	−509.875	−0.0229192
$792$	0	0
$793$	42742.8	1.91405
$794$	−9684.76	−0.432871
$795$	0	0
$796$	−20521.6	−0.913779
$797$	34264.4	1.52285	0.761423	−	0.648255i	$-0.224501\pi$
$797$	34264.4	1.52285	0.761423	+	0.648255i	$0.224501\pi$
$798$	0	0
$799$	13624.8	0.603268
$800$	8348.59	0.368959
$801$	0	0
$802$	44070.1	1.94036
$803$	−850.592	−0.0373807
$804$	0	0
$805$	−617.619	−0.0270413
$806$	−36865.0	−1.61106
$807$	0	0
$808$	−59428.4	−2.58748
$809$	8842.40	0.384280	0.192140	−	0.981368i	$-0.438457\pi$
$809$	8842.40	0.384280	0.192140	+	0.981368i	$0.438457\pi$
$810$	0	0
$811$	16775.4	0.726341	0.363170	−	0.931723i	$-0.381694\pi$
$811$	16775.4	0.726341	0.363170	+	0.931723i	$0.381694\pi$
$812$	633.064	0.0273598
$813$	0	0
$814$	1399.78	0.0602729
$815$	11069.2	0.475749
$816$	0	0
$817$	3927.34	0.168177
$818$	−542.444	−0.0231859
$819$	0	0
$820$	−29137.9	−1.24090
$821$	−16529.4	−0.702657	−0.351328	−	0.936252i	$-0.614270\pi$
$821$	−16529.4	−0.702657	−0.351328	+	0.936252i	$0.614270\pi$
$822$	0	0
$823$	17811.2	0.754386	0.377193	−	0.926135i	$-0.376889\pi$
$823$	17811.2	0.754386	0.377193	+	0.926135i	$0.376889\pi$
$824$	−74889.4	−3.16614
$825$	0	0
$826$	−3450.37	−0.145344
$827$	−18973.0	−0.797772	−0.398886	−	0.917001i	$-0.630603\pi$
$827$	−18973.0	−0.797772	−0.398886	+	0.917001i	$0.630603\pi$
$828$	0	0
$829$	−29108.6	−1.21952	−0.609762	−	0.792585i	$-0.708735\pi$
$829$	−29108.6	−1.21952	−0.609762	+	0.792585i	$0.708735\pi$
$830$	−46612.2	−1.94932
$831$	0	0
$832$	−52804.9	−2.20034
$833$	−17201.3	−0.715472
$834$	0	0
$835$	−29755.1	−1.23320
$836$	1961.88	0.0811639
$837$	0	0
$838$	−54114.4	−2.23073
$839$	32777.9	1.34877	0.674386	−	0.738379i	$-0.264408\pi$
$839$	32777.9	1.34877	0.674386	+	0.738379i	$0.264408\pi$
$840$	0	0
$841$	−22765.4	−0.933430
$842$	−24407.2	−0.998964
$843$	0	0
$844$	−9129.37	−0.372329
$845$	53765.4	2.18886
$846$	0	0
$847$	−1070.16	−0.0434132
$848$	−17872.9	−0.723771
$849$	0	0
$850$	6082.99	0.245465
$851$	24462.2	0.985373
$852$	0	0
$853$	11519.2	0.462380	0.231190	−	0.972909i	$-0.425738\pi$
$853$	11519.2	0.462380	0.231190	+	0.972909i	$0.425738\pi$
$854$	2080.10	0.0833484
$855$	0	0
$856$	−77900.0	−3.11048
$857$	21343.7	0.850743	0.425372	−	0.905019i	$-0.360143\pi$
$857$	21343.7	0.850743	0.425372	+	0.905019i	$0.360143\pi$
$858$	0	0
$859$	21764.2	0.864475	0.432238	−	0.901760i	$-0.357724\pi$
$859$	21764.2	0.864475	0.432238	+	0.901760i	$0.357724\pi$
$860$	6393.60	0.253511
$861$	0	0
$862$	90798.2	3.58770
$863$	22120.1	0.872510	0.436255	−	0.899823i	$-0.356305\pi$
$863$	22120.1	0.872510	0.436255	+	0.899823i	$0.356305\pi$
$864$	0	0
$865$	32853.8	1.29140
$866$	−21269.1	−0.834590
$867$	0	0
$868$	−1272.73	−0.0497687
$869$	174.712	0.00682014
$870$	0	0
$871$	−74775.0	−2.90890
$872$	−128498.	−4.99023
$873$	0	0
$874$	48329.0	1.87043
$875$	−1202.58	−0.0464625
$876$	0	0
$877$	45552.7	1.75394	0.876970	−	0.480545i	$-0.159561\pi$
$877$	45552.7	1.75394	0.876970	+	0.480545i	$0.159561\pi$
$878$	−62699.9	−2.41004
$879$	0	0
$880$	1349.76	0.0517050
$881$	−16750.7	−0.640576	−0.320288	−	0.947320i	$-0.603780\pi$
$881$	−16750.7	−0.640576	−0.320288	+	0.947320i	$0.603780\pi$
$882$	0	0
$883$	−7820.30	−0.298045	−0.149023	−	0.988834i	$-0.547613\pi$
$883$	−7820.30	−0.298045	−0.149023	+	0.988834i	$0.547613\pi$
$884$	−85111.0	−3.23823
$885$	0	0
$886$	−48253.0	−1.82967
$887$	−3307.54	−0.125205	−0.0626023	−	0.998039i	$-0.519940\pi$
$887$	−3307.54	−0.125205	−0.0626023	+	0.998039i	$0.519940\pi$
$888$	0	0
$889$	−456.802	−0.0172336
$890$	24472.0	0.921690
$891$	0	0
$892$	−37684.5	−1.41454
$893$	32844.2	1.23078
$894$	0	0
$895$	6975.93	0.260536
$896$	−241.286	−0.00899642
$897$	0	0
$898$	−74631.7	−2.77337
$899$	−3264.07	−0.121093
$900$	0	0
$901$	−5570.66	−0.205978
$902$	643.052	0.0237376
$903$	0	0
$904$	38347.4	1.41086
$905$	45229.3	1.66130
$906$	0	0
$907$	22776.0	0.833807	0.416904	−	0.908951i	$-0.363115\pi$
$907$	22776.0	0.833807	0.416904	+	0.908951i	$0.363115\pi$
$908$	−73575.7	−2.68909
$909$	0	0
$910$	3695.95	0.134637
$911$	19433.9	0.706777	0.353388	−	0.935477i	$-0.385029\pi$
$911$	19433.9	0.706777	0.353388	+	0.935477i	$0.385029\pi$
$912$	0	0
$913$	729.773	0.0264534
$914$	−45741.5	−1.65535
$915$	0	0
$916$	76409.9	2.75617
$917$	1010.36	0.0363848
$918$	0	0
$919$	3518.51	0.126295	0.0631473	−	0.998004i	$-0.479886\pi$
$919$	3518.51	0.126295	0.0631473	+	0.998004i	$0.479886\pi$
$920$	46450.8	1.66460
$921$	0	0
$922$	−3291.56	−0.117573
$923$	−34273.5	−1.22224
$924$	0	0
$925$	7422.20	0.263827
$926$	13652.0	0.484484
$927$	0	0
$928$	−14579.0	−0.515709
$929$	29305.0	1.03495	0.517474	−	0.855699i	$-0.326872\pi$
$929$	29305.0	1.03495	0.517474	+	0.855699i	$0.326872\pi$
$930$	0	0
$931$	−41465.6	−1.45970
$932$	54070.3	1.90035
$933$	0	0
$934$	−63705.0	−2.23179
$935$	420.696	0.0147147
$936$	0	0
$937$	3989.30	0.139087	0.0695436	−	0.997579i	$-0.477846\pi$
$937$	3989.30	0.139087	0.0695436	+	0.997579i	$0.477846\pi$
$938$	−3638.96	−0.126670
$939$	0	0
$940$	53469.4	1.85530
$941$	−5735.38	−0.198691	−0.0993453	−	0.995053i	$-0.531675\pi$
$941$	−5735.38	−0.198691	−0.0993453	+	0.995053i	$0.531675\pi$
$942$	0	0
$943$	11237.8	0.388074
$944$	131776.	4.54339
$945$	0	0
$946$	−141.102	−0.00484949
$947$	9209.74	0.316026	0.158013	−	0.987437i	$-0.449491\pi$
$947$	9209.74	0.316026	0.158013	+	0.987437i	$0.449491\pi$
$948$	0	0
$949$	88953.4	3.04273
$950$	14663.8	0.500795
$951$	0	0
$952$	−2445.37	−0.0832510
$953$	−37747.8	−1.28307	−0.641537	−	0.767092i	$-0.721703\pi$
$953$	−37747.8	−1.28307	−0.641537	+	0.767092i	$0.721703\pi$
$954$	0	0
$955$	31009.4	1.05072
$956$	−4667.84	−0.157917
$957$	0	0
$958$	−38107.7	−1.28518
$959$	−1577.03	−0.0531021
$960$	0	0
$961$	−23228.8	−0.779726
$962$	−146386.	−4.90611
$963$	0	0
$964$	12680.7	0.423671
$965$	−11011.6	−0.367334
$966$	0	0
$967$	57837.6	1.92340	0.961702	−	0.274099i	$-0.0883795\pi$
$967$	57837.6	1.92340	0.961702	+	0.274099i	$0.0883795\pi$
$968$	80485.8	2.67243
$969$	0	0
$970$	19747.7	0.653671
$971$	47828.7	1.58074	0.790369	−	0.612631i	$-0.209889\pi$
$971$	47828.7	1.58074	0.790369	+	0.612631i	$0.209889\pi$
$972$	0	0
$973$	29.5604	0.000973961 0
$974$	−33302.1	−1.09555
$975$	0	0
$976$	−79443.0	−2.60544
$977$	39670.3	1.29904	0.649522	−	0.760343i	$-0.274969\pi$
$977$	39670.3	1.29904	0.649522	+	0.760343i	$0.274969\pi$
$978$	0	0
$979$	−383.140	−0.0125079
$980$	−67504.8	−2.20037
$981$	0	0
$982$	−24538.7	−0.797413
$983$	51418.1	1.66834	0.834172	−	0.551505i	$-0.185946\pi$
$983$	51418.1	1.66834	0.834172	+	0.551505i	$0.185946\pi$
$984$	0	0
$985$	−50256.4	−1.62569
$986$	−10622.6	−0.343096
$987$	0	0
$988$	−205170.	−6.60661
$989$	−2465.86	−0.0792820
$990$	0	0
$991$	28309.2	0.907438	0.453719	−	0.891145i	$-0.350097\pi$
$991$	28309.2	0.907438	0.453719	+	0.891145i	$0.350097\pi$
$992$	29310.0	0.938097
$993$	0	0
$994$	−1667.94	−0.0532231
$995$	−10608.0	−0.337987
$996$	0	0
$997$	−25372.2	−0.805962	−0.402981	−	0.915208i	$-0.632026\pi$
$997$	−25372.2	−0.805962	−0.402981	+	0.915208i	$0.632026\pi$
$998$	24428.6	0.774824
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.h.1.2	yes	59
3.2	odd	2		2151.4.a.g.1.58	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.58	✓	59	3.2	odd	2
2151.4.a.h.1.2	yes	59	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-5.24697 q^{2} +19.5307 q^{4} +10.0958 q^{5} +0.804440 q^{7} -60.5014 q^{8} +O(q^{10})\)	\(q-5.24697 q^{2} +19.5307 q^{4} +10.0958 q^{5} +0.804440 q^{7} -60.5014 q^{8} -52.9726 q^{10} +0.829353 q^{11} -86.7323 q^{13} -4.22087 q^{14} +161.203 q^{16} +50.2442 q^{17} +121.120 q^{19} +197.179 q^{20} -4.35159 q^{22} -76.0475 q^{23} -23.0740 q^{25} +455.082 q^{26} +15.7113 q^{28} +40.2936 q^{29} -81.0073 q^{31} -361.819 q^{32} -263.630 q^{34} +8.12150 q^{35} -321.670 q^{37} -635.511 q^{38} -610.813 q^{40} -147.774 q^{41} +32.4253 q^{43} +16.1979 q^{44} +399.019 q^{46} +271.172 q^{47} -342.353 q^{49} +121.068 q^{50} -1693.95 q^{52} -110.872 q^{53} +8.37302 q^{55} -48.6697 q^{56} -211.419 q^{58} +817.455 q^{59} -492.812 q^{61} +425.043 q^{62} +608.826 q^{64} -875.636 q^{65} +862.135 q^{67} +981.306 q^{68} -42.6133 q^{70} +395.164 q^{71} -1025.61 q^{73} +1687.79 q^{74} +2365.55 q^{76} +0.667165 q^{77} +210.661 q^{79} +1627.48 q^{80} +775.365 q^{82} +879.930 q^{83} +507.258 q^{85} -170.135 q^{86} -50.1770 q^{88} -461.975 q^{89} -69.7709 q^{91} -1485.26 q^{92} -1422.83 q^{94} +1222.80 q^{95} -372.791 q^{97} +1796.32 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8}+O(q^{10}) \) 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8	\( 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8} - 36 q^{10} + 132 q^{11} + 104 q^{13} + 280 q^{14} + 822 q^{16} + 408 q^{17} + 20 q^{19} + 800 q^{20} - 2 q^{22} + 276 q^{23} + 1477 q^{25} + 780 q^{26} + 224 q^{28} + 696 q^{29} - 380 q^{31} + 896 q^{32} - 72 q^{34} + 700 q^{35} + 224 q^{37} + 988 q^{38} - 258 q^{40} + 2706 q^{41} - 156 q^{43} + 1584 q^{44} + 428 q^{46} + 1316 q^{47} + 2135 q^{49} + 1400 q^{50} + 1092 q^{52} + 1484 q^{53} - 992 q^{55} + 3360 q^{56} - 120 q^{58} + 3186 q^{59} - 254 q^{61} + 1240 q^{62} + 3054 q^{64} + 5120 q^{65} + 288 q^{67} + 9420 q^{68} + 1108 q^{70} + 4468 q^{71} - 1770 q^{73} + 6214 q^{74} + 720 q^{76} + 6352 q^{77} - 746 q^{79} + 7040 q^{80} + 276 q^{82} + 5484 q^{83} + 588 q^{85} + 10152 q^{86} + 1186 q^{88} + 11570 q^{89} + 1768 q^{91} + 15366 q^{92} - 2142 q^{94} + 5736 q^{95} + 2390 q^{97} + 6912 q^{98}+O(q^{100}) \) 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8 - 36 * q^10 + 132 * q^11 + 104 * q^13 + 280 * q^14 + 822 * q^16 + 408 * q^17 + 20 * q^19 + 800 * q^20 - 2 * q^22 + 276 * q^23 + 1477 * q^25 + 780 * q^26 + 224 * q^28 + 696 * q^29 - 380 * q^31 + 896 * q^32 - 72 * q^34 + 700 * q^35 + 224 * q^37 + 988 * q^38 - 258 * q^40 + 2706 * q^41 - 156 * q^43 + 1584 * q^44 + 428 * q^46 + 1316 * q^47 + 2135 * q^49 + 1400 * q^50 + 1092 * q^52 + 1484 * q^53 - 992 * q^55 + 3360 * q^56 - 120 * q^58 + 3186 * q^59 - 254 * q^61 + 1240 * q^62 + 3054 * q^64 + 5120 * q^65 + 288 * q^67 + 9420 * q^68 + 1108 * q^70 + 4468 * q^71 - 1770 * q^73 + 6214 * q^74 + 720 * q^76 + 6352 * q^77 - 746 * q^79 + 7040 * q^80 + 276 * q^82 + 5484 * q^83 + 588 * q^85 + 10152 * q^86 + 1186 * q^88 + 11570 * q^89 + 1768 * q^91 + 15366 * q^92 - 2142 * q^94 + 5736 * q^95 + 2390 * q^97 + 6912 * q^98

Introduction

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