LMFDB - Embedded newform 2151.4.a.a.1.7

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:	$1$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-2.22859 q^{2} -3.03338 q^{4} +7.56659 q^{5} -1.62323 q^{7} +24.5889 q^{8} +O(q^{10})$	\(q-2.22859 q^{2} -3.03338 q^{4} +7.56659 q^{5} -1.62323 q^{7} +24.5889 q^{8} -16.8628 q^{10} +38.9779 q^{11} -66.8951 q^{13} +3.61752 q^{14} -30.5316 q^{16} -47.5417 q^{17} +134.969 q^{19} -22.9523 q^{20} -86.8658 q^{22} +57.2778 q^{23} -67.7468 q^{25} +149.082 q^{26} +4.92387 q^{28} -44.0027 q^{29} -56.6464 q^{31} -128.669 q^{32} +105.951 q^{34} -12.2823 q^{35} +223.857 q^{37} -300.791 q^{38} +186.054 q^{40} +22.2896 q^{41} -232.281 q^{43} -118.235 q^{44} -127.649 q^{46} -349.350 q^{47} -340.365 q^{49} +150.980 q^{50} +202.918 q^{52} +271.409 q^{53} +294.929 q^{55} -39.9135 q^{56} +98.0640 q^{58} +304.493 q^{59} +204.324 q^{61} +126.242 q^{62} +531.003 q^{64} -506.168 q^{65} -747.410 q^{67} +144.212 q^{68} +27.3723 q^{70} -136.286 q^{71} -35.8194 q^{73} -498.886 q^{74} -409.412 q^{76} -63.2701 q^{77} -1065.39 q^{79} -231.020 q^{80} -49.6743 q^{82} -1133.83 q^{83} -359.729 q^{85} +517.661 q^{86} +958.423 q^{88} -700.985 q^{89} +108.586 q^{91} -173.745 q^{92} +778.558 q^{94} +1021.26 q^{95} +289.621 q^{97} +758.535 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * 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 $

$2$	−2.22859	−0.787926	−0.393963	−	0.919126i	$-0.628896\pi$
$2$	−2.22859	−0.787926	−0.393963	+	0.919126i	$0.628896\pi$
$3$	0	0
$3$	0	0
$4$	−3.03338	−0.379172
$5$	7.56659	0.676776	0.338388	−	0.941007i	$-0.390118\pi$
$5$	7.56659	0.676776	0.338388	+	0.941007i	$0.390118\pi$
$6$	0	0
$7$	−1.62323	−0.0876462	−0.0438231	−	0.999039i	$-0.513954\pi$
$7$	−1.62323	−0.0876462	−0.0438231	+	0.999039i	$0.513954\pi$
$8$	24.5889	1.08669
$9$	0	0
$10$	−16.8628	−0.533250
$11$	38.9779	1.06839	0.534194	−	0.845362i	$-0.320615\pi$
$11$	38.9779	1.06839	0.534194	+	0.845362i	$0.320615\pi$
$12$	0	0
$13$	−66.8951	−1.42718	−0.713591	−	0.700562i	$-0.752933\pi$
$13$	−66.8951	−1.42718	−0.713591	+	0.700562i	$0.752933\pi$
$14$	3.61752	0.0690588
$15$	0	0
$16$	−30.5316	−0.477057
$17$	−47.5417	−0.678268	−0.339134	−	0.940738i	$-0.610134\pi$
$17$	−47.5417	−0.678268	−0.339134	+	0.940738i	$0.610134\pi$
$18$	0	0
$19$	134.969	1.62969	0.814843	−	0.579682i	$-0.196823\pi$
$19$	134.969	1.62969	0.814843	+	0.579682i	$0.196823\pi$
$20$	−22.9523	−0.256615
$21$	0	0
$22$	−86.8658	−0.841811
$23$	57.2778	0.519272	0.259636	−	0.965707i	$-0.416397\pi$
$23$	57.2778	0.519272	0.259636	+	0.965707i	$0.416397\pi$
$24$	0	0
$25$	−67.7468	−0.541974
$26$	149.082	1.12451
$27$	0	0
$28$	4.92387	0.0332330
$29$	−44.0027	−0.281762	−0.140881	−	0.990027i	$-0.544993\pi$
$29$	−44.0027	−0.281762	−0.140881	+	0.990027i	$0.544993\pi$
$30$	0	0
$31$	−56.6464	−0.328193	−0.164097	−	0.986444i	$-0.552471\pi$
$31$	−56.6464	−0.328193	−0.164097	+	0.986444i	$0.552471\pi$
$32$	−128.669	−0.710801
$33$	0	0
$34$	105.951	0.534426
$35$	−12.2823	−0.0593169
$36$	0	0
$37$	223.857	0.994644	0.497322	−	0.867566i	$-0.334317\pi$
$37$	223.857	0.994644	0.497322	+	0.867566i	$0.334317\pi$
$38$	−300.791	−1.28407
$39$	0	0
$40$	186.054	0.735443
$41$	22.2896	0.0849035	0.0424518	−	0.999099i	$-0.486483\pi$
$41$	22.2896	0.0849035	0.0424518	+	0.999099i	$0.486483\pi$
$42$	0	0
$43$	−232.281	−0.823781	−0.411891	−	0.911233i	$-0.635131\pi$
$43$	−232.281	−0.823781	−0.411891	+	0.911233i	$0.635131\pi$
$44$	−118.235	−0.405103
$45$	0	0
$46$	−127.649	−0.409148
$47$	−349.350	−1.08421	−0.542105	−	0.840311i	$-0.682373\pi$
$47$	−349.350	−1.08421	−0.542105	+	0.840311i	$0.682373\pi$
$48$	0	0
$49$	−340.365	−0.992318
$50$	150.980	0.427036
$51$	0	0
$52$	202.918	0.541148
$53$	271.409	0.703414	0.351707	−	0.936110i	$-0.385601\pi$
$53$	271.409	0.703414	0.351707	+	0.936110i	$0.385601\pi$
$54$	0	0
$55$	294.929	0.723059
$56$	−39.9135	−0.0952439
$57$	0	0
$58$	98.0640	0.222008
$59$	304.493	0.671891	0.335946	−	0.941881i	$-0.390944\pi$
$59$	304.493	0.671891	0.335946	+	0.941881i	$0.390944\pi$
$60$	0	0
$61$	204.324	0.428870	0.214435	−	0.976738i	$-0.431209\pi$
$61$	204.324	0.428870	0.214435	+	0.976738i	$0.431209\pi$
$62$	126.242	0.258592
$63$	0	0
$64$	531.003	1.03712
$65$	−506.168	−0.965883
$66$	0	0
$67$	−747.410	−1.36285	−0.681423	−	0.731890i	$-0.738639\pi$
$67$	−747.410	−1.36285	−0.681423	+	0.731890i	$0.738639\pi$
$68$	144.212	0.257180
$69$	0	0
$70$	27.3723	0.0467373
$71$	−136.286	−0.227804	−0.113902	−	0.993492i	$-0.536335\pi$
$71$	−136.286	−0.227804	−0.113902	+	0.993492i	$0.536335\pi$
$72$	0	0
$73$	−35.8194	−0.0574294	−0.0287147	−	0.999588i	$-0.509141\pi$
$73$	−35.8194	−0.0574294	−0.0287147	+	0.999588i	$0.509141\pi$
$74$	−498.886	−0.783707
$75$	0	0
$76$	−409.412	−0.617931
$77$	−63.2701	−0.0936402
$78$	0	0
$79$	−1065.39	−1.51729	−0.758644	−	0.651505i	$-0.774138\pi$
$79$	−1065.39	−1.51729	−0.758644	+	0.651505i	$0.774138\pi$
$80$	−231.020	−0.322861
$81$	0	0
$82$	−49.6743	−0.0668977
$83$	−1133.83	−1.49945	−0.749723	−	0.661752i	$-0.769813\pi$
$83$	−1133.83	−1.49945	−0.749723	+	0.661752i	$0.769813\pi$
$84$	0	0
$85$	−359.729	−0.459036
$86$	517.661	0.649079
$87$	0	0
$88$	958.423	1.16100
$89$	−700.985	−0.834880	−0.417440	−	0.908705i	$-0.637073\pi$
$89$	−700.985	−0.834880	−0.417440	+	0.908705i	$0.637073\pi$
$90$	0	0
$91$	108.586	0.125087
$92$	−173.745	−0.196893
$93$	0	0
$94$	778.558	0.854278
$95$	1021.26	1.10293
$96$	0	0
$97$	289.621	0.303160	0.151580	−	0.988445i	$-0.451564\pi$
$97$	289.621	0.303160	0.151580	+	0.988445i	$0.451564\pi$
$98$	758.535	0.781874
$99$	0	0
$100$	205.501	0.205501
$101$	1516.74	1.49427	0.747136	−	0.664671i	$-0.231428\pi$
$101$	1516.74	1.49427	0.747136	+	0.664671i	$0.231428\pi$
$102$	0	0
$103$	1442.48	1.37992	0.689960	−	0.723847i	$-0.257628\pi$
$103$	1442.48	1.37992	0.689960	+	0.723847i	$0.257628\pi$
$104$	−1644.88	−1.55090
$105$	0	0
$106$	−604.861	−0.554239
$107$	2046.85	1.84932	0.924658	−	0.380798i	$-0.124351\pi$
$107$	2046.85	1.84932	0.924658	+	0.380798i	$0.124351\pi$
$108$	0	0
$109$	1368.28	1.20236	0.601180	−	0.799114i	$-0.294698\pi$
$109$	1368.28	1.20236	0.601180	+	0.799114i	$0.294698\pi$
$110$	−657.277	−0.569718
$111$	0	0
$112$	49.5599	0.0418122
$113$	211.999	0.176489	0.0882443	−	0.996099i	$-0.471874\pi$
$113$	211.999	0.176489	0.0882443	+	0.996099i	$0.471874\pi$
$114$	0	0
$115$	433.398	0.351431
$116$	133.477	0.106836
$117$	0	0
$118$	−678.590	−0.529401
$119$	77.1712	0.0594477
$120$	0	0
$121$	188.274	0.141453
$122$	−455.356	−0.337918
$123$	0	0
$124$	171.830	0.124442
$125$	−1458.44	−1.04357
$126$	0	0
$127$	175.224	0.122430	0.0612150	−	0.998125i	$-0.480502\pi$
$127$	175.224	0.122430	0.0612150	+	0.998125i	$0.480502\pi$
$128$	−154.040	−0.106370
$129$	0	0
$130$	1128.04	0.761045
$131$	140.174	0.0934890	0.0467445	−	0.998907i	$-0.485115\pi$
$131$	140.174	0.0934890	0.0467445	+	0.998907i	$0.485115\pi$
$132$	0	0
$133$	−219.086	−0.142836
$134$	1665.67	1.07382
$135$	0	0
$136$	−1169.00	−0.737065
$137$	1067.03	0.665418	0.332709	−	0.943030i	$-0.392037\pi$
$137$	1067.03	0.665418	0.332709	+	0.943030i	$0.392037\pi$
$138$	0	0
$139$	−559.665	−0.341512	−0.170756	−	0.985313i	$-0.554621\pi$
$139$	−559.665	−0.341512	−0.170756	+	0.985313i	$0.554621\pi$
$140$	37.2569	0.0224913
$141$	0	0
$142$	303.725	0.179493
$143$	−2607.43	−1.52479
$144$	0	0
$145$	−332.950	−0.190690
$146$	79.8269	0.0452502
$147$	0	0
$148$	−679.042	−0.377141
$149$	−938.041	−0.515754	−0.257877	−	0.966178i	$-0.583023\pi$
$149$	−938.041	−0.515754	−0.257877	+	0.966178i	$0.583023\pi$
$150$	0	0
$151$	−2207.46	−1.18967	−0.594837	−	0.803847i	$-0.702783\pi$
$151$	−2207.46	−1.18967	−0.594837	+	0.803847i	$0.702783\pi$
$152$	3318.74	1.77096
$153$	0	0
$154$	141.003	0.0737816
$155$	−428.620	−0.222113
$156$	0	0
$157$	1889.51	0.960504	0.480252	−	0.877130i	$-0.340545\pi$
$157$	1889.51	0.960504	0.480252	+	0.877130i	$0.340545\pi$
$158$	2374.32	1.19551
$159$	0	0
$160$	−973.582	−0.481053
$161$	−92.9752	−0.0455122
$162$	0	0
$163$	−818.239	−0.393187	−0.196593	−	0.980485i	$-0.562988\pi$
$163$	−818.239	−0.393187	−0.196593	+	0.980485i	$0.562988\pi$
$164$	−67.6126	−0.0321930
$165$	0	0
$166$	2526.84	1.18145
$167$	3476.52	1.61090	0.805452	−	0.592661i	$-0.201923\pi$
$167$	3476.52	1.61090	0.805452	+	0.592661i	$0.201923\pi$
$168$	0	0
$169$	2277.96	1.03685
$170$	801.689	0.361686
$171$	0	0
$172$	704.597	0.312355
$173$	−3803.22	−1.67141	−0.835705	−	0.549179i	$-0.814940\pi$
$173$	−3803.22	−1.67141	−0.835705	+	0.549179i	$0.814940\pi$
$174$	0	0
$175$	109.969	0.0475020
$176$	−1190.06	−0.509682
$177$	0	0
$178$	1562.21	0.657824
$179$	1212.48	0.506287	0.253143	−	0.967429i	$-0.418536\pi$
$179$	1212.48	0.506287	0.253143	+	0.967429i	$0.418536\pi$
$180$	0	0
$181$	−1668.80	−0.685308	−0.342654	−	0.939462i	$-0.611326\pi$
$181$	−1668.80	−0.685308	−0.342654	+	0.939462i	$0.611326\pi$
$182$	−241.995	−0.0985595
$183$	0	0
$184$	1408.40	0.564286
$185$	1693.83	0.673152
$186$	0	0
$187$	−1853.08	−0.724654
$188$	1059.71	0.411102
$189$	0	0
$190$	−2275.96	−0.869030
$191$	−2007.41	−0.760477	−0.380238	−	0.924889i	$-0.624158\pi$
$191$	−2007.41	−0.760477	−0.380238	+	0.924889i	$0.624158\pi$
$192$	0	0
$193$	−934.206	−0.348423	−0.174211	−	0.984708i	$-0.555738\pi$
$193$	−934.206	−0.348423	−0.174211	+	0.984708i	$0.555738\pi$
$194$	−645.447	−0.238868
$195$	0	0
$196$	1032.46	0.376259
$197$	−441.335	−0.159613	−0.0798067	−	0.996810i	$-0.525430\pi$
$197$	−441.335	−0.159613	−0.0798067	+	0.996810i	$0.525430\pi$
$198$	0	0
$199$	−2860.50	−1.01897	−0.509487	−	0.860479i	$-0.670165\pi$
$199$	−2860.50	−1.01897	−0.509487	+	0.860479i	$0.670165\pi$
$200$	−1665.82	−0.588956
$201$	0	0
$202$	−3380.20	−1.17738
$203$	71.4265	0.0246954
$204$	0	0
$205$	168.656	0.0574607
$206$	−3214.70	−1.08728
$207$	0	0
$208$	2042.42	0.680847
$209$	5260.81	1.74114
$210$	0	0
$211$	−4209.67	−1.37349	−0.686744	−	0.726899i	$-0.740961\pi$
$211$	−4209.67	−1.37349	−0.686744	+	0.726899i	$0.740961\pi$
$212$	−823.287	−0.266715
$213$	0	0
$214$	−4561.60	−1.45713
$215$	−1757.58	−0.557515
$216$	0	0
$217$	91.9502	0.0287649
$218$	−3049.33	−0.947371
$219$	0	0
$220$	−894.632	−0.274164
$221$	3180.31	0.968013
$222$	0	0
$223$	404.102	0.121348	0.0606742	−	0.998158i	$-0.480675\pi$
$223$	404.102	0.121348	0.0606742	+	0.998158i	$0.480675\pi$
$224$	208.859	0.0622990
$225$	0	0
$226$	−472.460	−0.139060
$227$	−42.7969	−0.0125134	−0.00625668	−	0.999980i	$-0.501992\pi$
$227$	−42.7969	−0.0125134	−0.00625668	+	0.999980i	$0.501992\pi$
$228$	0	0
$229$	44.4481	0.0128263	0.00641313	−	0.999979i	$-0.497959\pi$
$229$	44.4481	0.0128263	0.00641313	+	0.999979i	$0.497959\pi$
$230$	−965.867	−0.276902
$231$	0	0
$232$	−1081.98	−0.306187
$233$	6321.45	1.77739	0.888695	−	0.458499i	$-0.151613\pi$
$233$	6321.45	1.77739	0.888695	+	0.458499i	$0.151613\pi$
$234$	0	0
$235$	−2643.38	−0.733767
$236$	−923.641	−0.254762
$237$	0	0
$238$	−171.983	−0.0468404
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−3009.60	−0.804420	−0.402210	−	0.915547i	$-0.631758\pi$
$241$	−3009.60	−0.804420	−0.402210	+	0.915547i	$0.631758\pi$
$242$	−419.586	−0.111455
$243$	0	0
$244$	−619.793	−0.162615
$245$	−2575.40	−0.671577
$246$	0	0
$247$	−9028.78	−2.32586
$248$	−1392.87	−0.356643
$249$	0	0
$250$	3250.26	0.822257
$251$	−4948.72	−1.24447	−0.622233	−	0.782832i	$-0.713774\pi$
$251$	−4948.72	−1.24447	−0.622233	+	0.782832i	$0.713774\pi$
$252$	0	0
$253$	2232.57	0.554784
$254$	−390.503	−0.0964659
$255$	0	0
$256$	−3904.73	−0.953303
$257$	7267.34	1.76391	0.881954	−	0.471336i	$-0.156228\pi$
$257$	7267.34	1.76391	0.881954	+	0.471336i	$0.156228\pi$
$258$	0	0
$259$	−363.371	−0.0871768
$260$	1535.40	0.366236
$261$	0	0
$262$	−312.391	−0.0736625
$263$	4679.22	1.09708	0.548541	−	0.836123i	$-0.315183\pi$
$263$	4679.22	1.09708	0.548541	+	0.836123i	$0.315183\pi$
$264$	0	0
$265$	2053.64	0.476054
$266$	488.254	0.112544
$267$	0	0
$268$	2267.18	0.516753
$269$	−5597.08	−1.26862	−0.634312	−	0.773077i	$-0.718716\pi$
$269$	−5597.08	−1.26862	−0.634312	+	0.773077i	$0.718716\pi$
$270$	0	0
$271$	−2154.99	−0.483049	−0.241525	−	0.970395i	$-0.577647\pi$
$271$	−2154.99	−0.483049	−0.241525	+	0.970395i	$0.577647\pi$
$272$	1451.53	0.323572
$273$	0	0
$274$	−2377.97	−0.524300
$275$	−2640.62	−0.579039
$276$	0	0
$277$	−4870.28	−1.05641	−0.528207	−	0.849116i	$-0.677135\pi$
$277$	−4870.28	−1.05641	−0.528207	+	0.849116i	$0.677135\pi$
$278$	1247.26	0.269086
$279$	0	0
$280$	−302.009	−0.0644588
$281$	−4702.84	−0.998392	−0.499196	−	0.866489i	$-0.666371\pi$
$281$	−4702.84	−0.998392	−0.499196	+	0.866489i	$0.666371\pi$
$282$	0	0
$283$	6748.56	1.41753	0.708764	−	0.705446i	$-0.249253\pi$
$283$	6748.56	1.41753	0.708764	+	0.705446i	$0.249253\pi$
$284$	413.405	0.0863771
$285$	0	0
$286$	5810.90	1.20142
$287$	−36.1811	−0.00744147
$288$	0	0
$289$	−2652.78	−0.539952
$290$	742.010	0.150249
$291$	0	0
$292$	108.654	0.0217756
$293$	3507.99	0.699451	0.349726	−	0.936852i	$-0.386275\pi$
$293$	3507.99	0.699451	0.349726	+	0.936852i	$0.386275\pi$
$294$	0	0
$295$	2303.97	0.454720
$296$	5504.39	1.08087
$297$	0	0
$298$	2090.51	0.406376
$299$	−3831.61	−0.741096
$300$	0	0
$301$	377.047	0.0722013
$302$	4919.53	0.937375
$303$	0	0
$304$	−4120.83	−0.777453
$305$	1546.04	0.290249
$306$	0	0
$307$	−8762.95	−1.62908	−0.814541	−	0.580106i	$-0.803011\pi$
$307$	−8762.95	−1.62908	−0.814541	+	0.580106i	$0.803011\pi$
$308$	191.922	0.0355057
$309$	0	0
$310$	955.219	0.175009
$311$	−7206.97	−1.31405	−0.657026	−	0.753868i	$-0.728186\pi$
$311$	−7206.97	−1.31405	−0.657026	+	0.753868i	$0.728186\pi$
$312$	0	0
$313$	1055.53	0.190613	0.0953064	−	0.995448i	$-0.469617\pi$
$313$	1055.53	0.190613	0.0953064	+	0.995448i	$0.469617\pi$
$314$	−4210.94	−0.756807
$315$	0	0
$316$	3231.73	0.575313
$317$	−1929.88	−0.341934	−0.170967	−	0.985277i	$-0.554689\pi$
$317$	−1929.88	−0.341934	−0.170967	+	0.985277i	$0.554689\pi$
$318$	0	0
$319$	−1715.13	−0.301031
$320$	4017.88	0.701895
$321$	0	0
$322$	207.204	0.0358603
$323$	−6416.67	−1.10536
$324$	0	0
$325$	4531.93	0.773496
$326$	1823.52	0.309802
$327$	0	0
$328$	548.076	0.0922635
$329$	567.075	0.0950269
$330$	0	0
$331$	−6957.14	−1.15528	−0.577642	−	0.816290i	$-0.696027\pi$
$331$	−6957.14	−1.15528	−0.577642	+	0.816290i	$0.696027\pi$
$332$	3439.33	0.568548
$333$	0	0
$334$	−7747.74	−1.26927
$335$	−5655.35	−0.922342
$336$	0	0
$337$	210.468	0.0340206	0.0170103	−	0.999855i	$-0.494585\pi$
$337$	210.468	0.0340206	0.0170103	+	0.999855i	$0.494585\pi$
$338$	−5076.65	−0.816962
$339$	0	0
$340$	1091.19	0.174054
$341$	−2207.96	−0.350638
$342$	0	0
$343$	1109.26	0.174619
$344$	−5711.55	−0.895192
$345$	0	0
$346$	8475.84	1.31695
$347$	−6194.34	−0.958299	−0.479150	−	0.877733i	$-0.659055\pi$
$347$	−6194.34	−0.958299	−0.479150	+	0.877733i	$0.659055\pi$
$348$	0	0
$349$	−5809.17	−0.890996	−0.445498	−	0.895283i	$-0.646973\pi$
$349$	−5809.17	−0.890996	−0.445498	+	0.895283i	$0.646973\pi$
$350$	−245.075	−0.0374281
$351$	0	0
$352$	−5015.23	−0.759411
$353$	3179.78	0.479441	0.239720	−	0.970842i	$-0.422944\pi$
$353$	3179.78	0.479441	0.239720	+	0.970842i	$0.422944\pi$
$354$	0	0
$355$	−1031.22	−0.154173
$356$	2126.35	0.316563
$357$	0	0
$358$	−2702.13	−0.398917
$359$	−12263.4	−1.80289	−0.901446	−	0.432892i	$-0.857493\pi$
$359$	−12263.4	−1.80289	−0.901446	+	0.432892i	$0.857493\pi$
$360$	0	0
$361$	11357.7	1.65588
$362$	3719.07	0.539972
$363$	0	0
$364$	−329.383	−0.0474296
$365$	−271.031	−0.0388669
$366$	0	0
$367$	11659.6	1.65838	0.829191	−	0.558966i	$-0.188802\pi$
$367$	11659.6	1.65838	0.829191	+	0.558966i	$0.188802\pi$
$368$	−1748.79	−0.247722
$369$	0	0
$370$	−3774.86	−0.530394
$371$	−440.560	−0.0616516
$372$	0	0
$373$	−7869.91	−1.09246	−0.546231	−	0.837635i	$-0.683938\pi$
$373$	−7869.91	−1.09246	−0.546231	+	0.837635i	$0.683938\pi$
$374$	4129.75	0.570974
$375$	0	0
$376$	−8590.12	−1.17820
$377$	2943.57	0.402126
$378$	0	0
$379$	12703.3	1.72170	0.860851	−	0.508858i	$-0.169932\pi$
$379$	12703.3	1.72170	0.860851	+	0.508858i	$0.169932\pi$
$380$	−3097.85	−0.418201
$381$	0	0
$382$	4473.70	0.599200
$383$	−7322.97	−0.976987	−0.488494	−	0.872567i	$-0.662453\pi$
$383$	−7322.97	−0.976987	−0.488494	+	0.872567i	$0.662453\pi$
$384$	0	0
$385$	−478.739	−0.0633734
$386$	2081.96	0.274532
$387$	0	0
$388$	−878.530	−0.114950
$389$	437.356	0.0570047	0.0285023	−	0.999594i	$-0.490926\pi$
$389$	437.356	0.0570047	0.0285023	+	0.999594i	$0.490926\pi$
$390$	0	0
$391$	−2723.09	−0.352206
$392$	−8369.20	−1.07834
$393$	0	0
$394$	983.557	0.125764
$395$	−8061.37	−1.02686
$396$	0	0
$397$	−4016.45	−0.507758	−0.253879	−	0.967236i	$-0.581706\pi$
$397$	−4016.45	−0.507758	−0.253879	+	0.967236i	$0.581706\pi$
$398$	6374.89	0.802876
$399$	0	0
$400$	2068.42	0.258552
$401$	−10409.8	−1.29636	−0.648179	−	0.761488i	$-0.724469\pi$
$401$	−10409.8	−1.29636	−0.648179	+	0.761488i	$0.724469\pi$
$402$	0	0
$403$	3789.37	0.468392
$404$	−4600.85	−0.566586
$405$	0	0
$406$	−159.181	−0.0194581
$407$	8725.46	1.06267
$408$	0	0
$409$	7000.25	0.846308	0.423154	−	0.906058i	$-0.360923\pi$
$409$	7000.25	0.846308	0.423154	+	0.906058i	$0.360923\pi$
$410$	−375.865	−0.0452748
$411$	0	0
$412$	−4375.58	−0.523227
$413$	−494.262	−0.0588887
$414$	0	0
$415$	−8579.22	−1.01479
$416$	8607.31	1.01444
$417$	0	0
$418$	−11724.2	−1.37189
$419$	3651.06	0.425695	0.212847	−	0.977085i	$-0.431726\pi$
$419$	3651.06	0.425695	0.212847	+	0.977085i	$0.431726\pi$
$420$	0	0
$421$	−12000.6	−1.38925	−0.694626	−	0.719371i	$-0.744430\pi$
$421$	−12000.6	−1.38925	−0.694626	+	0.719371i	$0.744430\pi$
$422$	9381.65	1.08221
$423$	0	0
$424$	6673.66	0.764390
$425$	3220.80	0.367604
$426$	0	0
$427$	−331.666	−0.0375888
$428$	−6208.88	−0.701209
$429$	0	0
$430$	3916.92	0.439281
$431$	−14009.2	−1.56566	−0.782828	−	0.622239i	$-0.786223\pi$
$431$	−14009.2	−1.56566	−0.782828	+	0.622239i	$0.786223\pi$
$432$	0	0
$433$	9959.36	1.10535	0.552675	−	0.833397i	$-0.313607\pi$
$433$	9959.36	1.10535	0.552675	+	0.833397i	$0.313607\pi$
$434$	−204.920	−0.0226646
$435$	0	0
$436$	−4150.50	−0.455901
$437$	7730.74	0.846250
$438$	0	0
$439$	3158.90	0.343431	0.171715	−	0.985147i	$-0.445069\pi$
$439$	3158.90	0.343431	0.171715	+	0.985147i	$0.445069\pi$
$440$	7251.99	0.785739
$441$	0	0
$442$	−7087.62	−0.762723
$443$	60.5294	0.00649174	0.00324587	−	0.999995i	$-0.498967\pi$
$443$	60.5294	0.00649174	0.00324587	+	0.999995i	$0.498967\pi$
$444$	0	0
$445$	−5304.07	−0.565027
$446$	−900.579	−0.0956136
$447$	0	0
$448$	−861.940	−0.0908992
$449$	−2757.27	−0.289808	−0.144904	−	0.989446i	$-0.546287\pi$
$449$	−2757.27	−0.289808	−0.144904	+	0.989446i	$0.546287\pi$
$450$	0	0
$451$	868.799	0.0907099
$452$	−643.073	−0.0669195
$453$	0	0
$454$	95.3769	0.00985960
$455$	821.628	0.0846560
$456$	0	0
$457$	−10131.3	−1.03703	−0.518515	−	0.855069i	$-0.673515\pi$
$457$	−10131.3	−1.03703	−0.518515	+	0.855069i	$0.673515\pi$
$458$	−99.0566	−0.0101061
$459$	0	0
$460$	−1314.66	−0.133253
$461$	−7291.54	−0.736661	−0.368331	−	0.929695i	$-0.620071\pi$
$461$	−7291.54	−0.736661	−0.368331	+	0.929695i	$0.620071\pi$
$462$	0	0
$463$	−19259.9	−1.93323	−0.966613	−	0.256241i	$-0.917516\pi$
$463$	−19259.9	−1.93323	−0.966613	+	0.256241i	$0.917516\pi$
$464$	1343.47	0.134416
$465$	0	0
$466$	−14087.9	−1.40045
$467$	−16866.3	−1.67127	−0.835633	−	0.549289i	$-0.814899\pi$
$467$	−16866.3	−1.67127	−0.835633	+	0.549289i	$0.814899\pi$
$468$	0	0
$469$	1213.22	0.119448
$470$	5891.03	0.578155
$471$	0	0
$472$	7487.14	0.730135
$473$	−9053.84	−0.880118
$474$	0	0
$475$	−9143.72	−0.883248
$476$	−234.089	−0.0225409
$477$	0	0
$478$	532.634	0.0509667
$479$	−3929.59	−0.374838	−0.187419	−	0.982280i	$-0.560012\pi$
$479$	−3929.59	−0.374838	−0.187419	+	0.982280i	$0.560012\pi$
$480$	0	0
$481$	−14974.9	−1.41954
$482$	6707.17	0.633824
$483$	0	0
$484$	−571.106	−0.0536350
$485$	2191.44	0.205172
$486$	0	0
$487$	−17435.2	−1.62231	−0.811154	−	0.584833i	$-0.801160\pi$
$487$	−17435.2	−1.62231	−0.811154	+	0.584833i	$0.801160\pi$
$488$	5024.11	0.466047
$489$	0	0
$490$	5739.52	0.529153
$491$	−5882.04	−0.540637	−0.270318	−	0.962771i	$-0.587129\pi$
$491$	−5882.04	−0.540637	−0.270318	+	0.962771i	$0.587129\pi$
$492$	0	0
$493$	2091.96	0.191110
$494$	20121.5	1.83261
$495$	0	0
$496$	1729.51	0.156567
$497$	221.223	0.0199662
$498$	0	0
$499$	−16531.2	−1.48305	−0.741523	−	0.670927i	$-0.765896\pi$
$499$	−16531.2	−1.48305	−0.741523	+	0.670927i	$0.765896\pi$
$500$	4423.98	0.395693
$501$	0	0
$502$	11028.7	0.980547
$503$	−392.858	−0.0348244	−0.0174122	−	0.999848i	$-0.505543\pi$
$503$	−392.858	−0.0348244	−0.0174122	+	0.999848i	$0.505543\pi$
$504$	0	0
$505$	11476.6	1.01129
$506$	−4975.48	−0.437129
$507$	0	0
$508$	−531.520	−0.0464221
$509$	−2914.11	−0.253764	−0.126882	−	0.991918i	$-0.540497\pi$
$509$	−2914.11	−0.253764	−0.126882	+	0.991918i	$0.540497\pi$
$510$	0	0
$511$	58.1432	0.00503347
$512$	9934.37	0.857503
$513$	0	0
$514$	−16195.9	−1.38983
$515$	10914.6	0.933897
$516$	0	0
$517$	−13616.9	−1.15836
$518$	809.807	0.0686889
$519$	0	0
$520$	−12446.1	−1.04961
$521$	12981.9	1.09164	0.545822	−	0.837901i	$-0.316218\pi$
$521$	12981.9	1.09164	0.545822	+	0.837901i	$0.316218\pi$
$522$	0	0
$523$	7367.08	0.615946	0.307973	−	0.951395i	$-0.400349\pi$
$523$	7367.08	0.615946	0.307973	+	0.951395i	$0.400349\pi$
$524$	−425.200	−0.0354484
$525$	0	0
$526$	−10428.1	−0.864421
$527$	2693.07	0.222603
$528$	0	0
$529$	−8886.25	−0.730357
$530$	−4576.73	−0.375095
$531$	0	0
$532$	664.570	0.0541594
$533$	−1491.06	−0.121173
$534$	0	0
$535$	15487.7	1.25157
$536$	−18378.0	−1.48099
$537$	0	0
$538$	12473.6	0.999582
$539$	−13266.7	−1.06018
$540$	0	0
$541$	13315.9	1.05821	0.529107	−	0.848555i	$-0.322527\pi$
$541$	13315.9	1.05821	0.529107	+	0.848555i	$0.322527\pi$
$542$	4802.60	0.380607
$543$	0	0
$544$	6117.13	0.482114
$545$	10353.2	0.813728
$546$	0	0
$547$	15337.6	1.19888	0.599442	−	0.800418i	$-0.295389\pi$
$547$	15337.6	1.19888	0.599442	+	0.800418i	$0.295389\pi$
$548$	−3236.69	−0.252308
$549$	0	0
$550$	5884.88	0.456240
$551$	−5939.00	−0.459183
$552$	0	0
$553$	1729.37	0.132985
$554$	10853.9	0.832376
$555$	0	0
$556$	1697.67	0.129492
$557$	96.5578	0.00734522	0.00367261	−	0.999993i	$-0.498831\pi$
$557$	96.5578	0.00734522	0.00367261	+	0.999993i	$0.498831\pi$
$558$	0	0
$559$	15538.5	1.17569
$560$	374.999	0.0282975
$561$	0	0
$562$	10480.7	0.786659
$563$	−9577.93	−0.716984	−0.358492	−	0.933533i	$-0.616709\pi$
$563$	−9577.93	−0.716984	−0.358492	+	0.933533i	$0.616709\pi$
$564$	0	0
$565$	1604.11	0.119443
$566$	−15039.8	−1.11691
$567$	0	0
$568$	−3351.11	−0.247552
$569$	−21002.6	−1.54740	−0.773702	−	0.633549i	$-0.781597\pi$
$569$	−21002.6	−1.54740	−0.773702	+	0.633549i	$0.781597\pi$
$570$	0	0
$571$	13290.0	0.974024	0.487012	−	0.873395i	$-0.338087\pi$
$571$	13290.0	0.974024	0.487012	+	0.873395i	$0.338087\pi$
$572$	7909.32	0.578156
$573$	0	0
$574$	80.6329	0.00586333
$575$	−3880.39	−0.281432
$576$	0	0
$577$	4354.62	0.314186	0.157093	−	0.987584i	$-0.449788\pi$
$577$	4354.62	0.314186	0.157093	+	0.987584i	$0.449788\pi$
$578$	5911.97	0.425442
$579$	0	0
$580$	1009.96	0.0723042
$581$	1840.47	0.131421
$582$	0	0
$583$	10579.0	0.751519
$584$	−880.761	−0.0624078
$585$	0	0
$586$	−7817.89	−0.551116
$587$	−15338.3	−1.07850	−0.539249	−	0.842146i	$-0.681292\pi$
$587$	−15338.3	−1.07850	−0.539249	+	0.842146i	$0.681292\pi$
$588$	0	0
$589$	−7645.52	−0.534852
$590$	−5134.61	−0.358286
$591$	0	0
$592$	−6834.71	−0.474502
$593$	−8570.56	−0.593509	−0.296754	−	0.954954i	$-0.595904\pi$
$593$	−8570.56	−0.593509	−0.296754	+	0.954954i	$0.595904\pi$
$594$	0	0
$595$	583.923	0.0402328
$596$	2845.43	0.195559
$597$	0	0
$598$	8539.09	0.583929
$599$	10518.4	0.717479	0.358739	−	0.933438i	$-0.383207\pi$
$599$	10518.4	0.717479	0.358739	+	0.933438i	$0.383207\pi$
$600$	0	0
$601$	−25037.9	−1.69936	−0.849681	−	0.527297i	$-0.823206\pi$
$601$	−25037.9	−1.69936	−0.849681	+	0.527297i	$0.823206\pi$
$602$	−840.283	−0.0568893
$603$	0	0
$604$	6696.06	0.451091
$605$	1424.59	0.0957320
$606$	0	0
$607$	14311.6	0.956983	0.478491	−	0.878092i	$-0.341184\pi$
$607$	14311.6	0.956983	0.478491	+	0.878092i	$0.341184\pi$
$608$	−17366.3	−1.15838
$609$	0	0
$610$	−3445.49	−0.228695
$611$	23369.8	1.54737
$612$	0	0
$613$	−136.735	−0.00900927	−0.00450464	−	0.999990i	$-0.501434\pi$
$613$	−136.735	−0.00900927	−0.00450464	+	0.999990i	$0.501434\pi$
$614$	19529.1	1.28360
$615$	0	0
$616$	−1555.74	−0.101757
$617$	23886.2	1.55854	0.779272	−	0.626686i	$-0.215589\pi$
$617$	23886.2	1.55854	0.779272	+	0.626686i	$0.215589\pi$
$618$	0	0
$619$	6455.43	0.419169	0.209585	−	0.977791i	$-0.432789\pi$
$619$	6455.43	0.419169	0.209585	+	0.977791i	$0.432789\pi$
$620$	1300.17	0.0842192
$621$	0	0
$622$	16061.4	1.03538
$623$	1137.86	0.0731741
$624$	0	0
$625$	−2567.03	−0.164290
$626$	−2352.34	−0.150189
$627$	0	0
$628$	−5731.59	−0.364196
$629$	−10642.5	−0.674636
$630$	0	0
$631$	22867.0	1.44266	0.721331	−	0.692590i	$-0.243531\pi$
$631$	22867.0	1.44266	0.721331	+	0.692590i	$0.243531\pi$
$632$	−26196.8	−1.64882
$633$	0	0
$634$	4300.93	0.269419
$635$	1325.85	0.0828577
$636$	0	0
$637$	22768.8	1.41622
$638$	3822.33	0.237190
$639$	0	0
$640$	−1165.56	−0.0719886
$641$	6614.53	0.407579	0.203790	−	0.979015i	$-0.434674\pi$
$641$	6614.53	0.407579	0.203790	+	0.979015i	$0.434674\pi$
$642$	0	0
$643$	−25855.8	−1.58577	−0.792887	−	0.609369i	$-0.791423\pi$
$643$	−25855.8	−1.58577	−0.792887	+	0.609369i	$0.791423\pi$
$644$	282.029	0.0172570
$645$	0	0
$646$	14300.1	0.870946
$647$	−12376.2	−0.752024	−0.376012	−	0.926615i	$-0.622705\pi$
$647$	−12376.2	−0.752024	−0.376012	+	0.926615i	$0.622705\pi$
$648$	0	0
$649$	11868.5	0.717840
$650$	−10099.8	−0.609458
$651$	0	0
$652$	2482.03	0.149085
$653$	−16224.2	−0.972285	−0.486142	−	0.873880i	$-0.661596\pi$
$653$	−16224.2	−0.972285	−0.486142	+	0.873880i	$0.661596\pi$
$654$	0	0
$655$	1060.64	0.0632711
$656$	−680.536	−0.0405038
$657$	0	0
$658$	−1263.78	−0.0748742
$659$	−3173.23	−0.187575	−0.0937873	−	0.995592i	$-0.529897\pi$
$659$	−3173.23	−0.187575	−0.0937873	+	0.995592i	$0.529897\pi$
$660$	0	0
$661$	−32265.5	−1.89861	−0.949307	−	0.314351i	$-0.898213\pi$
$661$	−32265.5	−1.89861	−0.949307	+	0.314351i	$0.898213\pi$
$662$	15504.6	0.910279
$663$	0	0
$664$	−27879.6	−1.62943
$665$	−1657.73	−0.0966679
$666$	0	0
$667$	−2520.38	−0.146311
$668$	−10545.6	−0.610809
$669$	0	0
$670$	12603.5	0.726737
$671$	7964.13	0.458199
$672$	0	0
$673$	−10614.9	−0.607986	−0.303993	−	0.952674i	$-0.598320\pi$
$673$	−10614.9	−0.607986	−0.303993	+	0.952674i	$0.598320\pi$
$674$	−469.048	−0.0268057
$675$	0	0
$676$	−6909.91	−0.393145
$677$	4907.12	0.278576	0.139288	−	0.990252i	$-0.455519\pi$
$677$	4907.12	0.278576	0.139288	+	0.990252i	$0.455519\pi$
$678$	0	0
$679$	−470.122	−0.0265709
$680$	−8845.33	−0.498828
$681$	0	0
$682$	4920.63	0.276277
$683$	9533.17	0.534080	0.267040	−	0.963685i	$-0.413954\pi$
$683$	9533.17	0.534080	0.267040	+	0.963685i	$0.413954\pi$
$684$	0	0
$685$	8073.75	0.450339
$686$	−2472.09	−0.137587
$687$	0	0
$688$	7091.93	0.392990
$689$	−18156.0	−1.00390
$690$	0	0
$691$	−28306.4	−1.55836	−0.779181	−	0.626799i	$-0.784365\pi$
$691$	−28306.4	−1.55836	−0.779181	+	0.626799i	$0.784365\pi$
$692$	11536.6	0.633752
$693$	0	0
$694$	13804.7	0.755069
$695$	−4234.75	−0.231127
$696$	0	0
$697$	−1059.68	−0.0575874
$698$	12946.3	0.702039
$699$	0	0
$700$	−333.576	−0.0180114
$701$	13323.0	0.717836	0.358918	−	0.933369i	$-0.383146\pi$
$701$	13323.0	0.717836	0.358918	+	0.933369i	$0.383146\pi$
$702$	0	0
$703$	30213.8	1.62096
$704$	20697.4	1.10804
$705$	0	0
$706$	−7086.43	−0.377764
$707$	−2462.02	−0.130967
$708$	0	0
$709$	12022.2	0.636814	0.318407	−	0.947954i	$-0.396852\pi$
$709$	12022.2	0.636814	0.318407	+	0.947954i	$0.396852\pi$
$710$	2298.16	0.121477
$711$	0	0
$712$	−17236.5	−0.907252
$713$	−3244.58	−0.170422
$714$	0	0
$715$	−19729.3	−1.03194
$716$	−3677.92	−0.191970
$717$	0	0
$718$	27330.1	1.42055
$719$	2623.35	0.136070	0.0680351	−	0.997683i	$-0.478327\pi$
$719$	2623.35	0.136070	0.0680351	+	0.997683i	$0.478327\pi$
$720$	0	0
$721$	−2341.48	−0.120945
$722$	−25311.6	−1.30471
$723$	0	0
$724$	5062.09	0.259850
$725$	2981.04	0.152708
$726$	0	0
$727$	2918.96	0.148911	0.0744556	−	0.997224i	$-0.476278\pi$
$727$	2918.96	0.148911	0.0744556	+	0.997224i	$0.476278\pi$
$728$	2670.02	0.135931
$729$	0	0
$730$	604.017	0.0306242
$731$	11043.1	0.558745
$732$	0	0
$733$	9290.08	0.468127	0.234063	−	0.972221i	$-0.424798\pi$
$733$	9290.08	0.468127	0.234063	+	0.972221i	$0.424798\pi$
$734$	−25984.5	−1.30668
$735$	0	0
$736$	−7369.86	−0.369099
$737$	−29132.5	−1.45605
$738$	0	0
$739$	9544.96	0.475124	0.237562	−	0.971372i	$-0.423652\pi$
$739$	9544.96	0.475124	0.237562	+	0.971372i	$0.423652\pi$
$740$	−5138.03	−0.255240
$741$	0	0
$742$	981.829	0.0485769
$743$	−7040.93	−0.347654	−0.173827	−	0.984776i	$-0.555613\pi$
$743$	−7040.93	−0.347654	−0.173827	+	0.984776i	$0.555613\pi$
$744$	0	0
$745$	−7097.77	−0.349050
$746$	17538.8	0.860780
$747$	0	0
$748$	5621.07	0.274768
$749$	−3322.52	−0.162086
$750$	0	0
$751$	833.380	0.0404933	0.0202466	−	0.999795i	$-0.493555\pi$
$751$	833.380	0.0404933	0.0202466	+	0.999795i	$0.493555\pi$
$752$	10666.2	0.517230
$753$	0	0
$754$	−6560.01	−0.316845
$755$	−16702.9	−0.805142
$756$	0	0
$757$	25237.0	1.21169	0.605847	−	0.795581i	$-0.292834\pi$
$757$	25237.0	1.21169	0.605847	+	0.795581i	$0.292834\pi$
$758$	−28310.5	−1.35657
$759$	0	0
$760$	25111.5	1.19854
$761$	27696.8	1.31933	0.659665	−	0.751560i	$-0.270698\pi$
$761$	27696.8	1.31933	0.659665	+	0.751560i	$0.270698\pi$
$762$	0	0
$763$	−2221.03	−0.105382
$764$	6089.23	0.288351
$765$	0	0
$766$	16319.9	0.769794
$767$	−20369.1	−0.958911
$768$	0	0
$769$	−25983.5	−1.21845	−0.609227	−	0.792996i	$-0.708520\pi$
$769$	−25983.5	−1.21845	−0.609227	+	0.792996i	$0.708520\pi$
$770$	1066.91	0.0499336
$771$	0	0
$772$	2833.80	0.132112
$773$	29918.0	1.39208	0.696038	−	0.718005i	$-0.254945\pi$
$773$	29918.0	1.39208	0.696038	+	0.718005i	$0.254945\pi$
$774$	0	0
$775$	3837.61	0.177872
$776$	7121.46	0.329440
$777$	0	0
$778$	−974.688	−0.0449155
$779$	3008.40	0.138366
$780$	0	0
$781$	−5312.12	−0.243384
$782$	6068.65	0.277512
$783$	0	0
$784$	10391.9	0.473392
$785$	14297.1	0.650046
$786$	0	0
$787$	5026.15	0.227653	0.113826	−	0.993501i	$-0.463689\pi$
$787$	5026.15	0.227653	0.113826	+	0.993501i	$0.463689\pi$
$788$	1338.74	0.0605209
$789$	0	0
$790$	17965.5	0.809093
$791$	−344.124	−0.0154686
$792$	0	0
$793$	−13668.3	−0.612075
$794$	8951.03	0.400076
$795$	0	0
$796$	8676.98	0.386366
$797$	−33500.9	−1.48891	−0.744456	−	0.667671i	$-0.767291\pi$
$797$	−33500.9	−1.48891	−0.744456	+	0.667671i	$0.767291\pi$
$798$	0	0
$799$	16608.7	0.735386
$800$	8716.88	0.385235
$801$	0	0
$802$	23199.1	1.02143
$803$	−1396.17	−0.0613569
$804$	0	0
$805$	−703.505	−0.0308016
$806$	−8444.96	−0.369058
$807$	0	0
$808$	37295.0	1.62380
$809$	31931.7	1.38771	0.693856	−	0.720114i	$-0.255910\pi$
$809$	31931.7	1.38771	0.693856	+	0.720114i	$0.255910\pi$
$810$	0	0
$811$	−28037.1	−1.21395	−0.606977	−	0.794719i	$-0.707618\pi$
$811$	−28037.1	−1.21395	−0.606977	+	0.794719i	$0.707618\pi$
$812$	−216.663	−0.00936379
$813$	0	0
$814$	−19445.5	−0.837303
$815$	−6191.27	−0.266099
$816$	0	0
$817$	−31350.8	−1.34250
$818$	−15600.7	−0.666829
$819$	0	0
$820$	−511.597	−0.0217875
$821$	−3671.55	−0.156076	−0.0780378	−	0.996950i	$-0.524865\pi$
$821$	−3671.55	−0.156076	−0.0780378	+	0.996950i	$0.524865\pi$
$822$	0	0
$823$	20115.5	0.851983	0.425992	−	0.904727i	$-0.359925\pi$
$823$	20115.5	0.851983	0.425992	+	0.904727i	$0.359925\pi$
$824$	35469.0	1.49954
$825$	0	0
$826$	1101.51	0.0464000
$827$	15185.9	0.638531	0.319265	−	0.947665i	$-0.396564\pi$
$827$	15185.9	0.638531	0.319265	+	0.947665i	$0.396564\pi$
$828$	0	0
$829$	−30638.8	−1.28363	−0.641814	−	0.766860i	$-0.721818\pi$
$829$	−30638.8	−1.28363	−0.641814	+	0.766860i	$0.721818\pi$
$830$	19119.6	0.799579
$831$	0	0
$832$	−35521.5	−1.48015
$833$	16181.5	0.673058
$834$	0	0
$835$	26305.4	1.09022
$836$	−15958.0	−0.660190
$837$	0	0
$838$	−8136.73	−0.335416
$839$	24006.4	0.987832	0.493916	−	0.869510i	$-0.335565\pi$
$839$	24006.4	0.987832	0.493916	+	0.869510i	$0.335565\pi$
$840$	0	0
$841$	−22452.8	−0.920610
$842$	26744.5	1.09463
$843$	0	0
$844$	12769.5	0.520788
$845$	17236.4	0.701716
$846$	0	0
$847$	−305.612	−0.0123978
$848$	−8286.57	−0.335568
$849$	0	0
$850$	−7177.85	−0.289645
$851$	12822.0	0.516491
$852$	0	0
$853$	28817.6	1.15674	0.578369	−	0.815775i	$-0.303690\pi$
$853$	28817.6	1.15674	0.578369	+	0.815775i	$0.303690\pi$
$854$	739.147	0.0296172
$855$	0	0
$856$	50329.9	2.00963
$857$	5011.50	0.199755	0.0998773	−	0.995000i	$-0.468155\pi$
$857$	5011.50	0.199755	0.0998773	+	0.995000i	$0.468155\pi$
$858$	0	0
$859$	24647.2	0.978988	0.489494	−	0.872007i	$-0.337181\pi$
$859$	24647.2	0.978988	0.489494	+	0.872007i	$0.337181\pi$
$860$	5331.39	0.211394
$861$	0	0
$862$	31220.7	1.23362
$863$	1217.26	0.0480139	0.0240069	−	0.999712i	$-0.492358\pi$
$863$	1217.26	0.0480139	0.0240069	+	0.999712i	$0.492358\pi$
$864$	0	0
$865$	−28777.4	−1.13117
$866$	−22195.4	−0.870934
$867$	0	0
$868$	−278.920	−0.0109069
$869$	−41526.6	−1.62105
$870$	0	0
$871$	49998.1	1.94503
$872$	33644.4	1.30659
$873$	0	0
$874$	−17228.7	−0.666783
$875$	2367.38	0.0914651
$876$	0	0
$877$	−30332.2	−1.16790	−0.583948	−	0.811791i	$-0.698493\pi$
$877$	−30332.2	−1.16790	−0.583948	+	0.811791i	$0.698493\pi$
$878$	−7039.90	−0.270598
$879$	0	0
$880$	−9004.67	−0.344940
$881$	−14928.0	−0.570869	−0.285435	−	0.958398i	$-0.592138\pi$
$881$	−14928.0	−0.570869	−0.285435	+	0.958398i	$0.592138\pi$
$882$	0	0
$883$	42204.6	1.60849	0.804246	−	0.594296i	$-0.202569\pi$
$883$	42204.6	1.60849	0.804246	+	0.594296i	$0.202569\pi$
$884$	−9647.08	−0.367043
$885$	0	0
$886$	−134.895	−0.00511501
$887$	40539.2	1.53458	0.767291	−	0.641299i	$-0.221604\pi$
$887$	40539.2	1.53458	0.767291	+	0.641299i	$0.221604\pi$
$888$	0	0
$889$	−284.429	−0.0107305
$890$	11820.6	0.445200
$891$	0	0
$892$	−1225.79	−0.0460119
$893$	−47151.4	−1.76692
$894$	0	0
$895$	9174.37	0.342643
$896$	250.042	0.00932292
$897$	0	0
$898$	6144.84	0.228347
$899$	2492.59	0.0924724
$900$	0	0
$901$	−12903.3	−0.477104
$902$	−1936.20	−0.0714727
$903$	0	0
$904$	5212.83	0.191788
$905$	−12627.1	−0.463800
$906$	0	0
$907$	10277.0	0.376231	0.188116	−	0.982147i	$-0.439762\pi$
$907$	10277.0	0.376231	0.188116	+	0.982147i	$0.439762\pi$
$908$	129.819	0.00474471
$909$	0	0
$910$	−1831.07	−0.0667027
$911$	21455.4	0.780296	0.390148	−	0.920752i	$-0.372424\pi$
$911$	21455.4	0.780296	0.390148	+	0.920752i	$0.372424\pi$
$912$	0	0
$913$	−44194.2	−1.60199
$914$	22578.6	0.817103
$915$	0	0
$916$	−134.828	−0.00486336
$917$	−227.535	−0.00819396
$918$	0	0
$919$	16644.2	0.597432	0.298716	−	0.954342i	$-0.403442\pi$
$919$	16644.2	0.597432	0.298716	+	0.954342i	$0.403442\pi$
$920$	10656.8	0.381895
$921$	0	0
$922$	16249.9	0.580435
$923$	9116.84	0.325119
$924$	0	0
$925$	−15165.6	−0.539072
$926$	42922.5	1.52324
$927$	0	0
$928$	5661.77	0.200276
$929$	−16569.9	−0.585189	−0.292595	−	0.956237i	$-0.594519\pi$
$929$	−16569.9	−0.585189	−0.292595	+	0.956237i	$0.594519\pi$
$930$	0	0
$931$	−45938.8	−1.61717
$932$	−19175.3	−0.673936
$933$	0	0
$934$	37588.2	1.31683
$935$	−14021.5	−0.490428
$936$	0	0
$937$	−4858.08	−0.169377	−0.0846887	−	0.996407i	$-0.526990\pi$
$937$	−4858.08	−0.169377	−0.0846887	+	0.996407i	$0.526990\pi$
$938$	−2703.77	−0.0941165
$939$	0	0
$940$	8018.38	0.278224
$941$	35283.6	1.22233	0.611166	−	0.791503i	$-0.290701\pi$
$941$	35283.6	1.22233	0.611166	+	0.791503i	$0.290701\pi$
$942$	0	0
$943$	1276.70	0.0440880
$944$	−9296.66	−0.320530
$945$	0	0
$946$	20177.3	0.693468
$947$	12888.4	0.442255	0.221128	−	0.975245i	$-0.429026\pi$
$947$	12888.4	0.442255	0.221128	+	0.975245i	$0.429026\pi$
$948$	0	0
$949$	2396.15	0.0819623
$950$	20377.6	0.695934
$951$	0	0
$952$	1897.56	0.0646010
$953$	−34115.5	−1.15961	−0.579806	−	0.814754i	$-0.696872\pi$
$953$	−34115.5	−1.15961	−0.579806	+	0.814754i	$0.696872\pi$
$954$	0	0
$955$	−15189.2	−0.514672
$956$	724.977	0.0245266
$957$	0	0
$958$	8757.46	0.295345
$959$	−1732.03	−0.0583214
$960$	0	0
$961$	−26582.2	−0.892289
$962$	33373.0	1.11849
$963$	0	0
$964$	9129.24	0.305013
$965$	−7068.75	−0.235804
$966$	0	0
$967$	−34005.5	−1.13086	−0.565430	−	0.824796i	$-0.691290\pi$
$967$	−34005.5	−1.13086	−0.565430	+	0.824796i	$0.691290\pi$
$968$	4629.45	0.153715
$969$	0	0
$970$	−4883.83	−0.161660
$971$	33676.4	1.11300	0.556502	−	0.830846i	$-0.312143\pi$
$971$	33676.4	1.11300	0.556502	+	0.830846i	$0.312143\pi$
$972$	0	0
$973$	908.465	0.0299322
$974$	38855.9	1.27826
$975$	0	0
$976$	−6238.35	−0.204595
$977$	25287.3	0.828057	0.414029	−	0.910264i	$-0.364121\pi$
$977$	25287.3	0.828057	0.414029	+	0.910264i	$0.364121\pi$
$978$	0	0
$979$	−27322.9	−0.891976
$980$	7812.16	0.254643
$981$	0	0
$982$	13108.7	0.425982
$983$	19135.5	0.620881	0.310441	−	0.950593i	$-0.399523\pi$
$983$	19135.5	0.620881	0.310441	+	0.950593i	$0.399523\pi$
$984$	0	0
$985$	−3339.40	−0.108023
$986$	−4662.13	−0.150581
$987$	0	0
$988$	27387.7	0.881901
$989$	−13304.6	−0.427766
$990$	0	0
$991$	24219.9	0.776358	0.388179	−	0.921584i	$-0.373104\pi$
$991$	24219.9	0.776358	0.388179	+	0.921584i	$0.373104\pi$
$992$	7288.62	0.233280
$993$	0	0
$994$	−493.016	−0.0157319
$995$	−21644.2	−0.689617
$996$	0	0
$997$	−30861.7	−0.980341	−0.490170	−	0.871627i	$-0.663065\pi$
$997$	−30861.7	−0.980341	−0.490170	+	0.871627i	$0.663065\pi$
$998$	36841.4	1.16853
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.a.1.7		22
3.2	odd	2		239.4.a.a.1.16	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.a.1.16	✓	22	3.2	odd	2
2151.4.a.a.1.7		22	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)

\(f(q)\)	\(=\)	\(q-2.22859 q^{2} -3.03338 q^{4} +7.56659 q^{5} -1.62323 q^{7} +24.5889 q^{8} +O(q^{10})\)	\(q-2.22859 q^{2} -3.03338 q^{4} +7.56659 q^{5} -1.62323 q^{7} +24.5889 q^{8} -16.8628 q^{10} +38.9779 q^{11} -66.8951 q^{13} +3.61752 q^{14} -30.5316 q^{16} -47.5417 q^{17} +134.969 q^{19} -22.9523 q^{20} -86.8658 q^{22} +57.2778 q^{23} -67.7468 q^{25} +149.082 q^{26} +4.92387 q^{28} -44.0027 q^{29} -56.6464 q^{31} -128.669 q^{32} +105.951 q^{34} -12.2823 q^{35} +223.857 q^{37} -300.791 q^{38} +186.054 q^{40} +22.2896 q^{41} -232.281 q^{43} -118.235 q^{44} -127.649 q^{46} -349.350 q^{47} -340.365 q^{49} +150.980 q^{50} +202.918 q^{52} +271.409 q^{53} +294.929 q^{55} -39.9135 q^{56} +98.0640 q^{58} +304.493 q^{59} +204.324 q^{61} +126.242 q^{62} +531.003 q^{64} -506.168 q^{65} -747.410 q^{67} +144.212 q^{68} +27.3723 q^{70} -136.286 q^{71} -35.8194 q^{73} -498.886 q^{74} -409.412 q^{76} -63.2701 q^{77} -1065.39 q^{79} -231.020 q^{80} -49.6743 q^{82} -1133.83 q^{83} -359.729 q^{85} +517.661 q^{86} +958.423 q^{88} -700.985 q^{89} +108.586 q^{91} -173.745 q^{92} +778.558 q^{94} +1021.26 q^{95} +289.621 q^{97} +758.535 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more