LMFDB - Embedded newform 2151.4.a.d.1.20

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:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+0.752561 q^{2} -7.43365 q^{4} -3.00130 q^{5} -33.6762 q^{7} -11.6148 q^{8} +O(q^{10})$	\(q+0.752561 q^{2} -7.43365 q^{4} -3.00130 q^{5} -33.6762 q^{7} -11.6148 q^{8} -2.25866 q^{10} +23.1982 q^{11} -32.9020 q^{13} -25.3434 q^{14} +50.7284 q^{16} +69.6147 q^{17} +46.3891 q^{19} +22.3106 q^{20} +17.4580 q^{22} -13.1075 q^{23} -115.992 q^{25} -24.7608 q^{26} +250.337 q^{28} -43.9956 q^{29} +170.395 q^{31} +131.094 q^{32} +52.3893 q^{34} +101.072 q^{35} +144.835 q^{37} +34.9106 q^{38} +34.8594 q^{40} +216.809 q^{41} -121.796 q^{43} -172.447 q^{44} -9.86422 q^{46} -263.442 q^{47} +791.084 q^{49} -87.2912 q^{50} +244.582 q^{52} +21.3080 q^{53} -69.6246 q^{55} +391.141 q^{56} -33.1094 q^{58} -392.596 q^{59} +777.684 q^{61} +128.233 q^{62} -307.171 q^{64} +98.7487 q^{65} +703.025 q^{67} -517.491 q^{68} +76.0630 q^{70} +1079.54 q^{71} -265.162 q^{73} +108.997 q^{74} -344.840 q^{76} -781.225 q^{77} -564.600 q^{79} -152.251 q^{80} +163.162 q^{82} +179.482 q^{83} -208.934 q^{85} -91.6592 q^{86} -269.441 q^{88} -1355.59 q^{89} +1108.01 q^{91} +97.4368 q^{92} -198.256 q^{94} -139.227 q^{95} +1450.08 q^{97} +595.339 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * 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 $

$2$	0.752561	0.266070	0.133035	−	0.991111i	$-0.457528\pi$
$2$	0.752561	0.266070	0.133035	+	0.991111i	$0.457528\pi$
$3$	0	0
$3$	0	0
$4$	−7.43365	−0.929206
$5$	−3.00130	−0.268444	−0.134222	−	0.990951i	$-0.542854\pi$
$5$	−3.00130	−0.268444	−0.134222	+	0.990951i	$0.542854\pi$
$6$	0	0
$7$	−33.6762	−1.81834	−0.909171	−	0.416423i	$-0.863284\pi$
$7$	−33.6762	−1.81834	−0.909171	+	0.416423i	$0.863284\pi$
$8$	−11.6148	−0.513305
$9$	0	0
$10$	−2.25866	−0.0714251
$11$	23.1982	0.635864	0.317932	−	0.948113i	$-0.397012\pi$
$11$	23.1982	0.635864	0.317932	+	0.948113i	$0.397012\pi$
$12$	0	0
$13$	−32.9020	−0.701952	−0.350976	−	0.936384i	$-0.614150\pi$
$13$	−32.9020	−0.701952	−0.350976	+	0.936384i	$0.614150\pi$
$14$	−25.3434	−0.483807
$15$	0	0
$16$	50.7284	0.792631
$17$	69.6147	0.993178	0.496589	−	0.867986i	$-0.334586\pi$
$17$	69.6147	0.993178	0.496589	+	0.867986i	$0.334586\pi$
$18$	0	0
$19$	46.3891	0.560125	0.280063	−	0.959982i	$-0.409645\pi$
$19$	46.3891	0.560125	0.280063	+	0.959982i	$0.409645\pi$
$20$	22.3106	0.249440
$21$	0	0
$22$	17.4580	0.169185
$23$	−13.1075	−0.118831	−0.0594154	−	0.998233i	$-0.518924\pi$
$23$	−13.1075	−0.118831	−0.0594154	+	0.998233i	$0.518924\pi$
$24$	0	0
$25$	−115.992	−0.927938
$26$	−24.7608	−0.186769
$27$	0	0
$28$	250.337	1.68962
$29$	−43.9956	−0.281716	−0.140858	−	0.990030i	$-0.544986\pi$
$29$	−43.9956	−0.281716	−0.140858	+	0.990030i	$0.544986\pi$
$30$	0	0
$31$	170.395	0.987221	0.493611	−	0.869683i	$-0.335677\pi$
$31$	170.395	0.987221	0.493611	+	0.869683i	$0.335677\pi$
$32$	131.094	0.724201
$33$	0	0
$34$	52.3893	0.264255
$35$	101.072	0.488124
$36$	0	0
$37$	144.835	0.643531	0.321766	−	0.946819i	$-0.395724\pi$
$37$	144.835	0.643531	0.321766	+	0.946819i	$0.395724\pi$
$38$	34.9106	0.149033
$39$	0	0
$40$	34.8594	0.137794
$41$	216.809	0.825851	0.412925	−	0.910765i	$-0.364507\pi$
$41$	216.809	0.825851	0.412925	+	0.910765i	$0.364507\pi$
$42$	0	0
$43$	−121.796	−0.431948	−0.215974	−	0.976399i	$-0.569293\pi$
$43$	−121.796	−0.431948	−0.215974	+	0.976399i	$0.569293\pi$
$44$	−172.447	−0.590849
$45$	0	0
$46$	−9.86422	−0.0316174
$47$	−263.442	−0.817594	−0.408797	−	0.912625i	$-0.634052\pi$
$47$	−263.442	−0.817594	−0.408797	+	0.912625i	$0.634052\pi$
$48$	0	0
$49$	791.084	2.30637
$50$	−87.2912	−0.246897
$51$	0	0
$52$	244.582	0.652258
$53$	21.3080	0.0552240	0.0276120	−	0.999619i	$-0.491210\pi$
$53$	21.3080	0.0552240	0.0276120	+	0.999619i	$0.491210\pi$
$54$	0	0
$55$	−69.6246	−0.170694
$56$	391.141	0.933364
$57$	0	0
$58$	−33.1094	−0.0749564
$59$	−392.596	−0.866299	−0.433149	−	0.901322i	$-0.642598\pi$
$59$	−392.596	−0.866299	−0.433149	+	0.901322i	$0.642598\pi$
$60$	0	0
$61$	777.684	1.63233	0.816166	−	0.577817i	$-0.196095\pi$
$61$	777.684	1.63233	0.816166	+	0.577817i	$0.196095\pi$
$62$	128.233	0.262670
$63$	0	0
$64$	−307.171	−0.599943
$65$	98.7487	0.188435
$66$	0	0
$67$	703.025	1.28191	0.640956	−	0.767577i	$-0.278538\pi$
$67$	703.025	1.28191	0.640956	+	0.767577i	$0.278538\pi$
$68$	−517.491	−0.922868
$69$	0	0
$70$	76.0630	0.129875
$71$	1079.54	1.80448	0.902241	−	0.431233i	$-0.141921\pi$
$71$	1079.54	1.80448	0.902241	+	0.431233i	$0.141921\pi$
$72$	0	0
$73$	−265.162	−0.425135	−0.212568	−	0.977146i	$-0.568183\pi$
$73$	−265.162	−0.425135	−0.212568	+	0.977146i	$0.568183\pi$
$74$	108.997	0.171225
$75$	0	0
$76$	−344.840	−0.520472
$77$	−781.225	−1.15622
$78$	0	0
$79$	−564.600	−0.804082	−0.402041	−	0.915622i	$-0.631699\pi$
$79$	−564.600	−0.804082	−0.402041	+	0.915622i	$0.631699\pi$
$80$	−152.251	−0.212777
$81$	0	0
$82$	163.162	0.219735
$83$	179.482	0.237358	0.118679	−	0.992933i	$-0.462134\pi$
$83$	179.482	0.237358	0.118679	+	0.992933i	$0.462134\pi$
$84$	0	0
$85$	−208.934	−0.266613
$86$	−91.6592	−0.114929
$87$	0	0
$88$	−269.441	−0.326392
$89$	−1355.59	−1.61452	−0.807262	−	0.590194i	$-0.799051\pi$
$89$	−1355.59	−1.61452	−0.807262	+	0.590194i	$0.799051\pi$
$90$	0	0
$91$	1108.01	1.27639
$92$	97.4368	0.110418
$93$	0	0
$94$	−198.256	−0.217538
$95$	−139.227	−0.150362
$96$	0	0
$97$	1450.08	1.51787	0.758936	−	0.651165i	$-0.225720\pi$
$97$	1450.08	1.51787	0.758936	+	0.651165i	$0.225720\pi$
$98$	595.339	0.613656
$99$	0	0
$100$	862.246	0.862246
$101$	−811.220	−0.799202	−0.399601	−	0.916689i	$-0.630851\pi$
$101$	−811.220	−0.799202	−0.399601	+	0.916689i	$0.630851\pi$
$102$	0	0
$103$	267.198	0.255610	0.127805	−	0.991799i	$-0.459207\pi$
$103$	267.198	0.255610	0.127805	+	0.991799i	$0.459207\pi$
$104$	382.149	0.360315
$105$	0	0
$106$	16.0355	0.0146935
$107$	833.616	0.753165	0.376583	−	0.926383i	$-0.377099\pi$
$107$	833.616	0.753165	0.376583	+	0.926383i	$0.377099\pi$
$108$	0	0
$109$	−1349.31	−1.18569	−0.592845	−	0.805316i	$-0.701995\pi$
$109$	−1349.31	−1.18569	−0.592845	+	0.805316i	$0.701995\pi$
$110$	−52.3968	−0.0454167
$111$	0	0
$112$	−1708.34	−1.44127
$113$	−820.078	−0.682712	−0.341356	−	0.939934i	$-0.610886\pi$
$113$	−820.078	−0.682712	−0.341356	+	0.939934i	$0.610886\pi$
$114$	0	0
$115$	39.3396	0.0318995
$116$	327.048	0.261773
$117$	0	0
$118$	−295.452	−0.230496
$119$	−2344.35	−1.80594
$120$	0	0
$121$	−792.845	−0.595677
$122$	585.255	0.434315
$123$	0	0
$124$	−1266.66	−0.917332
$125$	723.289	0.517544
$126$	0	0
$127$	1231.93	0.860758	0.430379	−	0.902648i	$-0.358380\pi$
$127$	1231.93	0.860758	0.430379	+	0.902648i	$0.358380\pi$
$128$	−1279.92	−0.883828
$129$	0	0
$130$	74.3145	0.0501370
$131$	−2054.91	−1.37052	−0.685259	−	0.728299i	$-0.740311\pi$
$131$	−2054.91	−1.37052	−0.685259	+	0.728299i	$0.740311\pi$
$132$	0	0
$133$	−1562.21	−1.01850
$134$	529.069	0.341079
$135$	0	0
$136$	−808.558	−0.509803
$137$	−2948.28	−1.83860	−0.919300	−	0.393558i	$-0.871244\pi$
$137$	−2948.28	−1.83860	−0.919300	+	0.393558i	$0.871244\pi$
$138$	0	0
$139$	−2612.85	−1.59438	−0.797189	−	0.603729i	$-0.793681\pi$
$139$	−2612.85	−1.59438	−0.797189	+	0.603729i	$0.793681\pi$
$140$	−751.336	−0.453568
$141$	0	0
$142$	812.422	0.480119
$143$	−763.266	−0.446346
$144$	0	0
$145$	132.044	0.0756252
$146$	−199.551	−0.113116
$147$	0	0
$148$	−1076.65	−0.597973
$149$	3594.68	1.97643	0.988214	−	0.153082i	$-0.0489198\pi$
$149$	3594.68	1.97643	0.988214	+	0.153082i	$0.0489198\pi$
$150$	0	0
$151$	810.048	0.436562	0.218281	−	0.975886i	$-0.429955\pi$
$151$	810.048	0.436562	0.218281	+	0.975886i	$0.429955\pi$
$152$	−538.798	−0.287515
$153$	0	0
$154$	−587.920	−0.307636
$155$	−511.406	−0.265014
$156$	0	0
$157$	3301.67	1.67836	0.839178	−	0.543856i	$-0.183036\pi$
$157$	3301.67	1.67836	0.839178	+	0.543856i	$0.183036\pi$
$158$	−424.896	−0.213943
$159$	0	0
$160$	−393.453	−0.194408
$161$	441.411	0.216075
$162$	0	0
$163$	−695.732	−0.334319	−0.167159	−	0.985930i	$-0.553459\pi$
$163$	−695.732	−0.334319	−0.167159	+	0.985930i	$0.553459\pi$
$164$	−1611.68	−0.767386
$165$	0	0
$166$	135.071	0.0631539
$167$	−1177.40	−0.545570	−0.272785	−	0.962075i	$-0.587945\pi$
$167$	−1177.40	−0.545570	−0.272785	+	0.962075i	$0.587945\pi$
$168$	0	0
$169$	−1114.46	−0.507263
$170$	−157.236	−0.0709379
$171$	0	0
$172$	905.392	0.401369
$173$	−4089.37	−1.79716	−0.898581	−	0.438808i	$-0.855401\pi$
$173$	−4089.37	−1.79716	−0.898581	+	0.438808i	$0.855401\pi$
$174$	0	0
$175$	3906.17	1.68731
$176$	1176.81	0.504006
$177$	0	0
$178$	−1020.17	−0.429577
$179$	−1112.13	−0.464381	−0.232190	−	0.972670i	$-0.574589\pi$
$179$	−1112.13	−0.464381	−0.232190	+	0.972670i	$0.574589\pi$
$180$	0	0
$181$	−1922.54	−0.789510	−0.394755	−	0.918786i	$-0.629171\pi$
$181$	−1922.54	−0.789510	−0.394755	+	0.918786i	$0.629171\pi$
$182$	833.848	0.339609
$183$	0	0
$184$	152.241	0.0609965
$185$	−434.692	−0.172752
$186$	0	0
$187$	1614.93	0.631527
$188$	1958.33	0.759713
$189$	0	0
$190$	−104.777	−0.0400070
$191$	−979.384	−0.371025	−0.185512	−	0.982642i	$-0.559394\pi$
$191$	−979.384	−0.371025	−0.185512	+	0.982642i	$0.559394\pi$
$192$	0	0
$193$	378.245	0.141071	0.0705354	−	0.997509i	$-0.477529\pi$
$193$	378.245	0.141071	0.0705354	+	0.997509i	$0.477529\pi$
$194$	1091.28	0.403861
$195$	0	0
$196$	−5880.64	−2.14309
$197$	−1369.16	−0.495170	−0.247585	−	0.968866i	$-0.579637\pi$
$197$	−1369.16	−0.495170	−0.247585	+	0.968866i	$0.579637\pi$
$198$	0	0
$199$	727.155	0.259028	0.129514	−	0.991578i	$-0.458658\pi$
$199$	727.155	0.259028	0.129514	+	0.991578i	$0.458658\pi$
$200$	1347.22	0.476315
$201$	0	0
$202$	−610.492	−0.212644
$203$	1481.60	0.512257
$204$	0	0
$205$	−650.709	−0.221695
$206$	201.083	0.0680102
$207$	0	0
$208$	−1669.07	−0.556389
$209$	1076.14	0.356164
$210$	0	0
$211$	918.060	0.299535	0.149767	−	0.988721i	$-0.452148\pi$
$211$	918.060	0.299535	0.149767	+	0.988721i	$0.452148\pi$
$212$	−158.396	−0.0513145
$213$	0	0
$214$	627.347	0.200395
$215$	365.547	0.115954
$216$	0	0
$217$	−5738.25	−1.79511
$218$	−1015.44	−0.315477
$219$	0	0
$220$	517.565	0.158610
$221$	−2290.46	−0.697164
$222$	0	0
$223$	1865.63	0.560233	0.280116	−	0.959966i	$-0.409627\pi$
$223$	1865.63	0.560233	0.280116	+	0.959966i	$0.409627\pi$
$224$	−4414.75	−1.31684
$225$	0	0
$226$	−617.159	−0.181650
$227$	−2009.44	−0.587538	−0.293769	−	0.955876i	$-0.594910\pi$
$227$	−2009.44	−0.587538	−0.293769	+	0.955876i	$0.594910\pi$
$228$	0	0
$229$	1607.07	0.463747	0.231873	−	0.972746i	$-0.425515\pi$
$229$	1607.07	0.463747	0.231873	+	0.972746i	$0.425515\pi$
$230$	29.6055	0.00848751
$231$	0	0
$232$	510.998	0.144606
$233$	5852.65	1.64558	0.822790	−	0.568346i	$-0.192416\pi$
$233$	5852.65	1.64558	0.822790	+	0.568346i	$0.192416\pi$
$234$	0	0
$235$	790.667	0.219478
$236$	2918.42	0.804970
$237$	0	0
$238$	−1764.27	−0.480507
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−4346.82	−1.16184	−0.580919	−	0.813961i	$-0.697307\pi$
$241$	−4346.82	−1.16184	−0.580919	+	0.813961i	$0.697307\pi$
$242$	−596.665	−0.158492
$243$	0	0
$244$	−5781.03	−1.51677
$245$	−2374.28	−0.619131
$246$	0	0
$247$	−1526.29	−0.393181
$248$	−1979.10	−0.506746
$249$	0	0
$250$	544.319	0.137703
$251$	−1748.31	−0.439651	−0.219825	−	0.975539i	$-0.570549\pi$
$251$	−1748.31	−0.439651	−0.219825	+	0.975539i	$0.570549\pi$
$252$	0	0
$253$	−304.071	−0.0755603
$254$	927.104	0.229022
$255$	0	0
$256$	1494.15	0.364782
$257$	3617.27	0.877973	0.438987	−	0.898494i	$-0.355338\pi$
$257$	3617.27	0.877973	0.438987	+	0.898494i	$0.355338\pi$
$258$	0	0
$259$	−4877.47	−1.17016
$260$	−734.064	−0.175095
$261$	0	0
$262$	−1546.44	−0.364655
$263$	−2805.30	−0.657727	−0.328864	−	0.944377i	$-0.606666\pi$
$263$	−2805.30	−0.657727	−0.328864	+	0.944377i	$0.606666\pi$
$264$	0	0
$265$	−63.9515	−0.0148246
$266$	−1175.66	−0.270993
$267$	0	0
$268$	−5226.04	−1.19116
$269$	2889.35	0.654895	0.327448	−	0.944869i	$-0.393812\pi$
$269$	2889.35	0.654895	0.327448	+	0.944869i	$0.393812\pi$
$270$	0	0
$271$	−5034.14	−1.12842	−0.564211	−	0.825631i	$-0.690820\pi$
$271$	−5034.14	−1.12842	−0.564211	+	0.825631i	$0.690820\pi$
$272$	3531.44	0.787224
$273$	0	0
$274$	−2218.76	−0.489197
$275$	−2690.81	−0.590043
$276$	0	0
$277$	−1885.50	−0.408985	−0.204492	−	0.978868i	$-0.565554\pi$
$277$	−1885.50	−0.408985	−0.204492	+	0.978868i	$0.565554\pi$
$278$	−1966.33	−0.424217
$279$	0	0
$280$	−1173.93	−0.250556
$281$	2902.21	0.616125	0.308063	−	0.951366i	$-0.400319\pi$
$281$	2902.21	0.616125	0.308063	+	0.951366i	$0.400319\pi$
$282$	0	0
$283$	6276.62	1.31840	0.659198	−	0.751969i	$-0.270896\pi$
$283$	6276.62	1.31840	0.659198	+	0.751969i	$0.270896\pi$
$284$	−8024.95	−1.67674
$285$	0	0
$286$	−574.404	−0.118760
$287$	−7301.30	−1.50168
$288$	0	0
$289$	−66.7997	−0.0135965
$290$	99.3711	0.0201216
$291$	0	0
$292$	1971.12	0.395038
$293$	7376.15	1.47071	0.735357	−	0.677680i	$-0.237014\pi$
$293$	7376.15	1.47071	0.735357	+	0.677680i	$0.237014\pi$
$294$	0	0
$295$	1178.30	0.232553
$296$	−1682.22	−0.330328
$297$	0	0
$298$	2705.22	0.525869
$299$	431.264	0.0834136
$300$	0	0
$301$	4101.64	0.785430
$302$	609.611	0.116156
$303$	0	0
$304$	2353.24	0.443973
$305$	−2334.06	−0.438190
$306$	0	0
$307$	4405.55	0.819017	0.409508	−	0.912306i	$-0.365700\pi$
$307$	4405.55	0.819017	0.409508	+	0.912306i	$0.365700\pi$
$308$	5807.36	1.07437
$309$	0	0
$310$	−384.865	−0.0705124
$311$	2819.23	0.514032	0.257016	−	0.966407i	$-0.417261\pi$
$311$	2819.23	0.514032	0.257016	+	0.966407i	$0.417261\pi$
$312$	0	0
$313$	1948.14	0.351806	0.175903	−	0.984407i	$-0.443715\pi$
$313$	1948.14	0.351806	0.175903	+	0.984407i	$0.443715\pi$
$314$	2484.71	0.446561
$315$	0	0
$316$	4197.04	0.747159
$317$	925.739	0.164021	0.0820106	−	0.996631i	$-0.473866\pi$
$317$	925.739	0.164021	0.0820106	+	0.996631i	$0.473866\pi$
$318$	0	0
$319$	−1020.62	−0.179133
$320$	921.911	0.161051
$321$	0	0
$322$	332.189	0.0574912
$323$	3229.36	0.556305
$324$	0	0
$325$	3816.38	0.651368
$326$	−523.581	−0.0889524
$327$	0	0
$328$	−2518.19	−0.423913
$329$	8871.70	1.48666
$330$	0	0
$331$	−9467.89	−1.57221	−0.786106	−	0.618092i	$-0.787906\pi$
$331$	−9467.89	−1.57221	−0.786106	+	0.618092i	$0.787906\pi$
$332$	−1334.21	−0.220554
$333$	0	0
$334$	−886.068	−0.145160
$335$	−2109.99	−0.344122
$336$	0	0
$337$	−2069.34	−0.334494	−0.167247	−	0.985915i	$-0.553488\pi$
$337$	−2069.34	−0.334494	−0.167247	+	0.985915i	$0.553488\pi$
$338$	−838.697	−0.134968
$339$	0	0
$340$	1553.15	0.247739
$341$	3952.85	0.627739
$342$	0	0
$343$	−15089.8	−2.37542
$344$	1414.64	0.221721
$345$	0	0
$346$	−3077.50	−0.478172
$347$	−7345.63	−1.13641	−0.568205	−	0.822887i	$-0.692362\pi$
$347$	−7345.63	−1.13641	−0.568205	+	0.822887i	$0.692362\pi$
$348$	0	0
$349$	−2222.32	−0.340853	−0.170427	−	0.985370i	$-0.554515\pi$
$349$	−2222.32	−0.340853	−0.170427	+	0.985370i	$0.554515\pi$
$350$	2939.63	0.448943
$351$	0	0
$352$	3041.15	0.460493
$353$	27.0994	0.00408600	0.00204300	−	0.999998i	$-0.499350\pi$
$353$	27.0994	0.00408600	0.00204300	+	0.999998i	$0.499350\pi$
$354$	0	0
$355$	−3240.03	−0.484403
$356$	10077.0	1.50023
$357$	0	0
$358$	−836.943	−0.123558
$359$	4298.97	0.632008	0.316004	−	0.948758i	$-0.397659\pi$
$359$	4298.97	0.632008	0.316004	+	0.948758i	$0.397659\pi$
$360$	0	0
$361$	−4707.05	−0.686259
$362$	−1446.83	−0.210065
$363$	0	0
$364$	−8236.59	−1.18603
$365$	795.830	0.114125
$366$	0	0
$367$	4846.82	0.689379	0.344690	−	0.938717i	$-0.387984\pi$
$367$	4846.82	0.689379	0.344690	+	0.938717i	$0.387984\pi$
$368$	−664.924	−0.0941890
$369$	0	0
$370$	−327.132	−0.0459643
$371$	−717.570	−0.100416
$372$	0	0
$373$	−1304.18	−0.181040	−0.0905199	−	0.995895i	$-0.528853\pi$
$373$	−1304.18	−0.181040	−0.0905199	+	0.995895i	$0.528853\pi$
$374$	1215.33	0.168031
$375$	0	0
$376$	3059.81	0.419675
$377$	1447.54	0.197751
$378$	0	0
$379$	−5756.62	−0.780205	−0.390103	−	0.920771i	$-0.627560\pi$
$379$	−5756.62	−0.780205	−0.390103	+	0.920771i	$0.627560\pi$
$380$	1034.97	0.139718
$381$	0	0
$382$	−737.046	−0.0987187
$383$	3526.97	0.470547	0.235274	−	0.971929i	$-0.424401\pi$
$383$	3526.97	0.470547	0.235274	+	0.971929i	$0.424401\pi$
$384$	0	0
$385$	2344.69	0.310380
$386$	284.652	0.0375348
$387$	0	0
$388$	−10779.4	−1.41042
$389$	8824.91	1.15023	0.575116	−	0.818072i	$-0.304957\pi$
$389$	8824.91	1.15023	0.575116	+	0.818072i	$0.304957\pi$
$390$	0	0
$391$	−912.476	−0.118020
$392$	−9188.26	−1.18387
$393$	0	0
$394$	−1030.38	−0.131750
$395$	1694.53	0.215851
$396$	0	0
$397$	73.8641	0.00933786	0.00466893	−	0.999989i	$-0.498514\pi$
$397$	73.8641	0.00933786	0.00466893	+	0.999989i	$0.498514\pi$
$398$	547.228	0.0689198
$399$	0	0
$400$	−5884.10	−0.735512
$401$	−1043.42	−0.129940	−0.0649701	−	0.997887i	$-0.520695\pi$
$401$	−1043.42	−0.129940	−0.0649701	+	0.997887i	$0.520695\pi$
$402$	0	0
$403$	−5606.34	−0.692982
$404$	6030.33	0.742624
$405$	0	0
$406$	1115.00	0.136296
$407$	3359.90	0.409199
$408$	0	0
$409$	−2326.15	−0.281224	−0.140612	−	0.990065i	$-0.544907\pi$
$409$	−2326.15	−0.281224	−0.140612	+	0.990065i	$0.544907\pi$
$410$	−489.698	−0.0589865
$411$	0	0
$412$	−1986.26	−0.237514
$413$	13221.1	1.57523
$414$	0	0
$415$	−538.678	−0.0637173
$416$	−4313.27	−0.508354
$417$	0	0
$418$	809.862	0.0947647
$419$	−9008.49	−1.05034	−0.525172	−	0.850996i	$-0.675999\pi$
$419$	−9008.49	−1.05034	−0.525172	+	0.850996i	$0.675999\pi$
$420$	0	0
$421$	−6967.06	−0.806541	−0.403270	−	0.915081i	$-0.632127\pi$
$421$	−6967.06	−0.806541	−0.403270	+	0.915081i	$0.632127\pi$
$422$	690.896	0.0796974
$423$	0	0
$424$	−247.487	−0.0283468
$425$	−8074.76	−0.921608
$426$	0	0
$427$	−26189.4	−2.96814
$428$	−6196.81	−0.699846
$429$	0	0
$430$	275.097	0.0308520
$431$	2738.84	0.306091	0.153045	−	0.988219i	$-0.451092\pi$
$431$	2738.84	0.306091	0.153045	+	0.988219i	$0.451092\pi$
$432$	0	0
$433$	−13846.9	−1.53681	−0.768406	−	0.639962i	$-0.778950\pi$
$433$	−13846.9	−1.53681	−0.768406	+	0.639962i	$0.778950\pi$
$434$	−4318.39	−0.477625
$435$	0	0
$436$	10030.3	1.10175
$437$	−608.046	−0.0665602
$438$	0	0
$439$	12224.5	1.32903	0.664516	−	0.747274i	$-0.268638\pi$
$439$	12224.5	1.32903	0.664516	+	0.747274i	$0.268638\pi$
$440$	808.673	0.0876181
$441$	0	0
$442$	−1723.71	−0.185495
$443$	−11737.7	−1.25886	−0.629431	−	0.777056i	$-0.716712\pi$
$443$	−11737.7	−1.25886	−0.629431	+	0.777056i	$0.716712\pi$
$444$	0	0
$445$	4068.54	0.433410
$446$	1404.00	0.149061
$447$	0	0
$448$	10344.3	1.09090
$449$	−14329.4	−1.50612	−0.753060	−	0.657952i	$-0.771423\pi$
$449$	−14329.4	−1.50612	−0.753060	+	0.657952i	$0.771423\pi$
$450$	0	0
$451$	5029.57	0.525129
$452$	6096.18	0.634381
$453$	0	0
$454$	−1512.23	−0.156327
$455$	−3325.48	−0.342639
$456$	0	0
$457$	12707.3	1.30071	0.650355	−	0.759630i	$-0.274620\pi$
$457$	12707.3	1.30071	0.650355	+	0.759630i	$0.274620\pi$
$458$	1209.42	0.123389
$459$	0	0
$460$	−292.437	−0.0296412
$461$	−15387.8	−1.55463	−0.777313	−	0.629115i	$-0.783418\pi$
$461$	−15387.8	−1.55463	−0.777313	+	0.629115i	$0.783418\pi$
$462$	0	0
$463$	−6482.04	−0.650640	−0.325320	−	0.945604i	$-0.605472\pi$
$463$	−6482.04	−0.650640	−0.325320	+	0.945604i	$0.605472\pi$
$464$	−2231.83	−0.223297
$465$	0	0
$466$	4404.48	0.437840
$467$	11841.8	1.17339	0.586696	−	0.809807i	$-0.300428\pi$
$467$	11841.8	1.17339	0.586696	+	0.809807i	$0.300428\pi$
$468$	0	0
$469$	−23675.2	−2.33096
$470$	595.025	0.0583967
$471$	0	0
$472$	4559.91	0.444675
$473$	−2825.45	−0.274661
$474$	0	0
$475$	−5380.77	−0.519761
$476$	17427.1	1.67809
$477$	0	0
$478$	−179.862	−0.0172107
$479$	10448.0	0.996617	0.498309	−	0.867000i	$-0.333955\pi$
$479$	10448.0	0.996617	0.498309	+	0.867000i	$0.333955\pi$
$480$	0	0
$481$	−4765.35	−0.451728
$482$	−3271.24	−0.309131
$483$	0	0
$484$	5893.74	0.553506
$485$	−4352.13	−0.407464
$486$	0	0
$487$	18864.7	1.75532	0.877662	−	0.479280i	$-0.159102\pi$
$487$	18864.7	1.75532	0.877662	+	0.479280i	$0.159102\pi$
$488$	−9032.62	−0.837884
$489$	0	0
$490$	−1786.79	−0.164733
$491$	−11773.3	−1.08212	−0.541062	−	0.840982i	$-0.681978\pi$
$491$	−11773.3	−1.08212	−0.541062	+	0.840982i	$0.681978\pi$
$492$	0	0
$493$	−3062.74	−0.279795
$494$	−1148.63	−0.104614
$495$	0	0
$496$	8643.87	0.782502
$497$	−36354.9	−3.28116
$498$	0	0
$499$	20448.0	1.83442	0.917212	−	0.398400i	$-0.130434\pi$
$499$	20448.0	1.83442	0.917212	+	0.398400i	$0.130434\pi$
$500$	−5376.68	−0.480905
$501$	0	0
$502$	−1315.71	−0.116978
$503$	−19590.9	−1.73661	−0.868307	−	0.496028i	$-0.834791\pi$
$503$	−19590.9	−1.73661	−0.868307	+	0.496028i	$0.834791\pi$
$504$	0	0
$505$	2434.71	0.214541
$506$	−228.832	−0.0201044
$507$	0	0
$508$	−9157.75	−0.799822
$509$	295.210	0.0257072	0.0128536	−	0.999917i	$-0.495908\pi$
$509$	295.210	0.0257072	0.0128536	+	0.999917i	$0.495908\pi$
$510$	0	0
$511$	8929.64	0.773041
$512$	11363.8	0.980886
$513$	0	0
$514$	2722.22	0.233603
$515$	−801.941	−0.0686170
$516$	0	0
$517$	−6111.36	−0.519879
$518$	−3670.60	−0.311345
$519$	0	0
$520$	−1146.94	−0.0967246
$521$	−20799.5	−1.74903	−0.874513	−	0.485002i	$-0.838819\pi$
$521$	−20799.5	−1.74903	−0.874513	+	0.485002i	$0.838819\pi$
$522$	0	0
$523$	2673.88	0.223557	0.111779	−	0.993733i	$-0.464345\pi$
$523$	2673.88	0.223557	0.111779	+	0.993733i	$0.464345\pi$
$524$	15275.5	1.27350
$525$	0	0
$526$	−2111.16	−0.175002
$527$	11862.0	0.980487
$528$	0	0
$529$	−11995.2	−0.985879
$530$	−48.1274	−0.00394438
$531$	0	0
$532$	11612.9	0.946397
$533$	−7133.45	−0.579708
$534$	0	0
$535$	−2501.93	−0.202183
$536$	−8165.47	−0.658012
$537$	0	0
$538$	2174.41	0.174248
$539$	18351.7	1.46654
$540$	0	0
$541$	−1961.73	−0.155899	−0.0779496	−	0.996957i	$-0.524837\pi$
$541$	−1961.73	−0.155899	−0.0779496	+	0.996957i	$0.524837\pi$
$542$	−3788.50	−0.300240
$543$	0	0
$544$	9126.09	0.719261
$545$	4049.68	0.318292
$546$	0	0
$547$	14181.2	1.10849	0.554245	−	0.832353i	$-0.313007\pi$
$547$	14181.2	1.10849	0.554245	+	0.832353i	$0.313007\pi$
$548$	21916.5	1.70844
$549$	0	0
$550$	−2025.00	−0.156993
$551$	−2040.91	−0.157797
$552$	0	0
$553$	19013.6	1.46210
$554$	−1418.95	−0.108819
$555$	0	0
$556$	19423.0	1.48151
$557$	16561.9	1.25988	0.629939	−	0.776645i	$-0.283080\pi$
$557$	16561.9	1.25988	0.629939	+	0.776645i	$0.283080\pi$
$558$	0	0
$559$	4007.35	0.303207
$560$	5127.23	0.386902
$561$	0	0
$562$	2184.09	0.163933
$563$	14432.0	1.08035	0.540173	−	0.841554i	$-0.318359\pi$
$563$	14432.0	1.08035	0.540173	+	0.841554i	$0.318359\pi$
$564$	0	0
$565$	2461.30	0.183270
$566$	4723.54	0.350787
$567$	0	0
$568$	−12538.6	−0.926249
$569$	−21978.4	−1.61930	−0.809650	−	0.586912i	$-0.800343\pi$
$569$	−21978.4	−1.61930	−0.809650	+	0.586912i	$0.800343\pi$
$570$	0	0
$571$	16447.5	1.20544	0.602718	−	0.797954i	$-0.294084\pi$
$571$	16447.5	1.20544	0.602718	+	0.797954i	$0.294084\pi$
$572$	5673.85	0.414748
$573$	0	0
$574$	−5494.67	−0.399553
$575$	1520.37	0.110268
$576$	0	0
$577$	15459.5	1.11540	0.557701	−	0.830042i	$-0.311684\pi$
$577$	15459.5	1.11540	0.557701	+	0.830042i	$0.311684\pi$
$578$	−50.2708	−0.00361763
$579$	0	0
$580$	−981.568	−0.0702714
$581$	−6044.26	−0.431598
$582$	0	0
$583$	494.305	0.0351150
$584$	3079.79	0.218224
$585$	0	0
$586$	5551.00	0.391314
$587$	−1418.64	−0.0997504	−0.0498752	−	0.998755i	$-0.515882\pi$
$587$	−1418.64	−0.0997504	−0.0498752	+	0.998755i	$0.515882\pi$
$588$	0	0
$589$	7904.47	0.552968
$590$	886.740	0.0618755
$591$	0	0
$592$	7347.23	0.510083
$593$	−1666.29	−0.115390	−0.0576952	−	0.998334i	$-0.518375\pi$
$593$	−1666.29	−0.115390	−0.0576952	+	0.998334i	$0.518375\pi$
$594$	0	0
$595$	7036.11	0.484794
$596$	−26721.6	−1.83651
$597$	0	0
$598$	324.553	0.0221939
$599$	−10349.1	−0.705930	−0.352965	−	0.935637i	$-0.614827\pi$
$599$	−10349.1	−0.705930	−0.352965	+	0.935637i	$0.614827\pi$
$600$	0	0
$601$	10876.6	0.738213	0.369107	−	0.929387i	$-0.379664\pi$
$601$	10876.6	0.738213	0.369107	+	0.929387i	$0.379664\pi$
$602$	3086.73	0.208980
$603$	0	0
$604$	−6021.62	−0.405656
$605$	2379.57	0.159906
$606$	0	0
$607$	5183.10	0.346583	0.173291	−	0.984871i	$-0.444560\pi$
$607$	5183.10	0.346583	0.173291	+	0.984871i	$0.444560\pi$
$608$	6081.34	0.405643
$609$	0	0
$610$	−1756.52	−0.116589
$611$	8667.76	0.573912
$612$	0	0
$613$	9171.98	0.604328	0.302164	−	0.953256i	$-0.402291\pi$
$613$	9171.98	0.604328	0.302164	+	0.953256i	$0.402291\pi$
$614$	3315.45	0.217916
$615$	0	0
$616$	9073.75	0.593493
$617$	−18599.4	−1.21359	−0.606793	−	0.794860i	$-0.707544\pi$
$617$	−18599.4	−1.21359	−0.606793	+	0.794860i	$0.707544\pi$
$618$	0	0
$619$	−8784.45	−0.570399	−0.285199	−	0.958468i	$-0.592060\pi$
$619$	−8784.45	−0.570399	−0.285199	+	0.958468i	$0.592060\pi$
$620$	3801.62	0.246253
$621$	0	0
$622$	2121.64	0.136769
$623$	45651.2	2.93576
$624$	0	0
$625$	12328.2	0.789006
$626$	1466.09	0.0936053
$627$	0	0
$628$	−24543.5	−1.55954
$629$	10082.6	0.639141
$630$	0	0
$631$	−3210.54	−0.202551	−0.101276	−	0.994858i	$-0.532292\pi$
$631$	−3210.54	−0.202551	−0.101276	+	0.994858i	$0.532292\pi$
$632$	6557.70	0.412739
$633$	0	0
$634$	696.675	0.0436412
$635$	−3697.39	−0.231066
$636$	0	0
$637$	−26028.3	−1.61896
$638$	−768.076	−0.0476621
$639$	0	0
$640$	3841.42	0.237259
$641$	−14413.1	−0.888120	−0.444060	−	0.895997i	$-0.646462\pi$
$641$	−14413.1	−0.888120	−0.444060	+	0.895997i	$0.646462\pi$
$642$	0	0
$643$	10078.4	0.618123	0.309062	−	0.951042i	$-0.399985\pi$
$643$	10078.4	0.618123	0.309062	+	0.951042i	$0.399985\pi$
$644$	−3281.30	−0.200778
$645$	0	0
$646$	2430.29	0.148016
$647$	−12853.4	−0.781020	−0.390510	−	0.920599i	$-0.627701\pi$
$647$	−12853.4	−0.781020	−0.390510	+	0.920599i	$0.627701\pi$
$648$	0	0
$649$	−9107.50	−0.550848
$650$	2872.06	0.173310
$651$	0	0
$652$	5171.83	0.310651
$653$	6796.22	0.407284	0.203642	−	0.979045i	$-0.434722\pi$
$653$	6796.22	0.407284	0.203642	+	0.979045i	$0.434722\pi$
$654$	0	0
$655$	6167.39	0.367908
$656$	10998.4	0.654595
$657$	0	0
$658$	6676.50	0.395558
$659$	−3891.94	−0.230058	−0.115029	−	0.993362i	$-0.536696\pi$
$659$	−3891.94	−0.230058	−0.115029	+	0.993362i	$0.536696\pi$
$660$	0	0
$661$	−20779.0	−1.22271	−0.611353	−	0.791358i	$-0.709375\pi$
$661$	−20779.0	−1.22271	−0.611353	+	0.791358i	$0.709375\pi$
$662$	−7125.16	−0.418319
$663$	0	0
$664$	−2084.64	−0.121837
$665$	4688.65	0.273410
$666$	0	0
$667$	576.674	0.0334766
$668$	8752.41	0.506948
$669$	0	0
$670$	−1587.89	−0.0915607
$671$	18040.8	1.03794
$672$	0	0
$673$	−5402.59	−0.309442	−0.154721	−	0.987958i	$-0.549448\pi$
$673$	−5402.59	−0.309442	−0.154721	+	0.987958i	$0.549448\pi$
$674$	−1557.31	−0.0889989
$675$	0	0
$676$	8284.49	0.471352
$677$	10792.8	0.612703	0.306351	−	0.951919i	$-0.400892\pi$
$677$	10792.8	0.612703	0.306351	+	0.951919i	$0.400892\pi$
$678$	0	0
$679$	−48833.2	−2.76001
$680$	2426.72	0.136854
$681$	0	0
$682$	2974.76	0.167023
$683$	−25933.5	−1.45288	−0.726441	−	0.687229i	$-0.758827\pi$
$683$	−25933.5	−1.45288	−0.726441	+	0.687229i	$0.758827\pi$
$684$	0	0
$685$	8848.65	0.493562
$686$	−11356.0	−0.632030
$687$	0	0
$688$	−6178.54	−0.342376
$689$	−701.074	−0.0387646
$690$	0	0
$691$	25389.2	1.39776	0.698878	−	0.715241i	$-0.253683\pi$
$691$	25389.2	1.39776	0.698878	+	0.715241i	$0.253683\pi$
$692$	30398.9	1.66993
$693$	0	0
$694$	−5528.04	−0.302365
$695$	7841.93	0.428002
$696$	0	0
$697$	15093.1	0.820217
$698$	−1672.43	−0.0906910
$699$	0	0
$700$	−29037.1	−1.56786
$701$	207.631	0.0111870	0.00559352	−	0.999984i	$-0.498220\pi$
$701$	207.631	0.0111870	0.00559352	+	0.999984i	$0.498220\pi$
$702$	0	0
$703$	6718.74	0.360458
$704$	−7125.79	−0.381482
$705$	0	0
$706$	20.3940	0.00108716
$707$	27318.8	1.45322
$708$	0	0
$709$	−20461.6	−1.08385	−0.541925	−	0.840427i	$-0.682304\pi$
$709$	−20461.6	−1.08385	−0.541925	+	0.840427i	$0.682304\pi$
$710$	−2438.32	−0.128885
$711$	0	0
$712$	15744.9	0.828743
$713$	−2233.46	−0.117312
$714$	0	0
$715$	2290.79	0.119819
$716$	8267.16	0.431506
$717$	0	0
$718$	3235.24	0.168159
$719$	−35996.1	−1.86708	−0.933539	−	0.358476i	$-0.883297\pi$
$719$	−35996.1	−1.86708	−0.933539	+	0.358476i	$0.883297\pi$
$720$	0	0
$721$	−8998.21	−0.464786
$722$	−3542.35	−0.182593
$723$	0	0
$724$	14291.5	0.733618
$725$	5103.15	0.261415
$726$	0	0
$727$	27928.7	1.42478	0.712391	−	0.701783i	$-0.247612\pi$
$727$	27928.7	1.42478	0.712391	+	0.701783i	$0.247612\pi$
$728$	−12869.3	−0.655177
$729$	0	0
$730$	598.911	0.0303653
$731$	−8478.82	−0.429002
$732$	0	0
$733$	10861.6	0.547315	0.273657	−	0.961827i	$-0.411767\pi$
$733$	10861.6	0.547315	0.273657	+	0.961827i	$0.411767\pi$
$734$	3647.53	0.183423
$735$	0	0
$736$	−1718.32	−0.0860574
$737$	16308.9	0.815123
$738$	0	0
$739$	−3441.72	−0.171320	−0.0856601	−	0.996324i	$-0.527300\pi$
$739$	−3441.72	−0.171320	−0.0856601	+	0.996324i	$0.527300\pi$
$740$	3231.35	0.160523
$741$	0	0
$742$	−540.015	−0.0267178
$743$	−27258.9	−1.34594	−0.672969	−	0.739670i	$-0.734981\pi$
$743$	−27258.9	−1.34594	−0.672969	+	0.739670i	$0.734981\pi$
$744$	0	0
$745$	−10788.7	−0.530561
$746$	−981.475	−0.0481694
$747$	0	0
$748$	−12004.8	−0.586819
$749$	−28073.0	−1.36951
$750$	0	0
$751$	−745.193	−0.0362084	−0.0181042	−	0.999836i	$-0.505763\pi$
$751$	−745.193	−0.0362084	−0.0181042	+	0.999836i	$0.505763\pi$
$752$	−13364.0	−0.648050
$753$	0	0
$754$	1089.36	0.0526158
$755$	−2431.20	−0.117192
$756$	0	0
$757$	−12744.5	−0.611898	−0.305949	−	0.952048i	$-0.598974\pi$
$757$	−12744.5	−0.611898	−0.305949	+	0.952048i	$0.598974\pi$
$758$	−4332.21	−0.207590
$759$	0	0
$760$	1617.09	0.0771818
$761$	−29403.4	−1.40062	−0.700311	−	0.713838i	$-0.746955\pi$
$761$	−29403.4	−1.40062	−0.700311	+	0.713838i	$0.746955\pi$
$762$	0	0
$763$	45439.5	2.15599
$764$	7280.40	0.344759
$765$	0	0
$766$	2654.26	0.125199
$767$	12917.2	0.608100
$768$	0	0
$769$	25855.3	1.21244	0.606220	−	0.795297i	$-0.292685\pi$
$769$	25855.3	1.21244	0.606220	+	0.795297i	$0.292685\pi$
$770$	1764.52	0.0825831
$771$	0	0
$772$	−2811.74	−0.131084
$773$	−20413.7	−0.949842	−0.474921	−	0.880028i	$-0.657523\pi$
$773$	−20413.7	−0.949842	−0.474921	+	0.880028i	$0.657523\pi$
$774$	0	0
$775$	−19764.5	−0.916080
$776$	−16842.4	−0.779131
$777$	0	0
$778$	6641.28	0.306043
$779$	10057.6	0.462580
$780$	0	0
$781$	25043.4	1.14741
$782$	−686.694	−0.0314017
$783$	0	0
$784$	40130.4	1.82810
$785$	−9909.30	−0.450545
$786$	0	0
$787$	5138.39	0.232737	0.116368	−	0.993206i	$-0.462875\pi$
$787$	5138.39	0.232737	0.116368	+	0.993206i	$0.462875\pi$
$788$	10177.9	0.460116
$789$	0	0
$790$	1275.24	0.0574317
$791$	27617.1	1.24140
$792$	0	0
$793$	−25587.4	−1.14582
$794$	55.5872	0.00248453
$795$	0	0
$796$	−5405.41	−0.240691
$797$	1505.72	0.0669202	0.0334601	−	0.999440i	$-0.489347\pi$
$797$	1505.72	0.0669202	0.0334601	+	0.999440i	$0.489347\pi$
$798$	0	0
$799$	−18339.4	−0.812016
$800$	−15205.9	−0.672013
$801$	0	0
$802$	−785.239	−0.0345733
$803$	−6151.27	−0.270328
$804$	0	0
$805$	−1324.81	−0.0580041
$806$	−4219.11	−0.184382
$807$	0	0
$808$	9422.13	0.410234
$809$	−18907.7	−0.821706	−0.410853	−	0.911702i	$-0.634769\pi$
$809$	−18907.7	−0.821706	−0.410853	+	0.911702i	$0.634769\pi$
$810$	0	0
$811$	−31211.7	−1.35141	−0.675703	−	0.737174i	$-0.736160\pi$
$811$	−31211.7	−1.35141	−0.675703	+	0.737174i	$0.736160\pi$
$812$	−11013.7	−0.475992
$813$	0	0
$814$	2528.53	0.108876
$815$	2088.10	0.0897460
$816$	0	0
$817$	−5650.02	−0.241945
$818$	−1750.57	−0.0748255
$819$	0	0
$820$	4837.14	0.206000
$821$	−2614.52	−0.111142	−0.0555710	−	0.998455i	$-0.517698\pi$
$821$	−2614.52	−0.111142	−0.0555710	+	0.998455i	$0.517698\pi$
$822$	0	0
$823$	−41791.5	−1.77006	−0.885030	−	0.465535i	$-0.845862\pi$
$823$	−41791.5	−1.77006	−0.885030	+	0.465535i	$0.845862\pi$
$824$	−3103.44	−0.131206
$825$	0	0
$826$	9949.70	0.419121
$827$	−5166.38	−0.217234	−0.108617	−	0.994084i	$-0.534642\pi$
$827$	−5166.38	−0.217234	−0.108617	+	0.994084i	$0.534642\pi$
$828$	0	0
$829$	−36677.0	−1.53660	−0.768302	−	0.640087i	$-0.778898\pi$
$829$	−36677.0	−1.53660	−0.768302	+	0.640087i	$0.778898\pi$
$830$	−405.388	−0.0169533
$831$	0	0
$832$	10106.5	0.421131
$833$	55071.1	2.29064
$834$	0	0
$835$	3533.74	0.146455
$836$	−7999.66	−0.330950
$837$	0	0
$838$	−6779.44	−0.279465
$839$	−19345.0	−0.796024	−0.398012	−	0.917380i	$-0.630300\pi$
$839$	−19345.0	−0.796024	−0.398012	+	0.917380i	$0.630300\pi$
$840$	0	0
$841$	−22453.4	−0.920636
$842$	−5243.14	−0.214597
$843$	0	0
$844$	−6824.54	−0.278330
$845$	3344.82	0.136172
$846$	0	0
$847$	26700.0	1.08314
$848$	1080.92	0.0437723
$849$	0	0
$850$	−6076.75	−0.245213
$851$	−1898.42	−0.0764714
$852$	0	0
$853$	11302.8	0.453694	0.226847	−	0.973930i	$-0.427158\pi$
$853$	11302.8	0.453694	0.226847	+	0.973930i	$0.427158\pi$
$854$	−19709.1	−0.789734
$855$	0	0
$856$	−9682.25	−0.386604
$857$	40349.8	1.60831	0.804156	−	0.594419i	$-0.202618\pi$
$857$	40349.8	1.60831	0.804156	+	0.594419i	$0.202618\pi$
$858$	0	0
$859$	−15370.1	−0.610501	−0.305251	−	0.952272i	$-0.598740\pi$
$859$	−15370.1	−0.610501	−0.305251	+	0.952272i	$0.598740\pi$
$860$	−2717.35	−0.107745
$861$	0	0
$862$	2061.14	0.0814417
$863$	22607.1	0.891719	0.445860	−	0.895103i	$-0.352898\pi$
$863$	22607.1	0.891719	0.445860	+	0.895103i	$0.352898\pi$
$864$	0	0
$865$	12273.4	0.482438
$866$	−10420.6	−0.408901
$867$	0	0
$868$	42656.2	1.66802
$869$	−13097.7	−0.511287
$870$	0	0
$871$	−23130.9	−0.899841
$872$	15671.9	0.608621
$873$	0	0
$874$	−457.592	−0.0177097
$875$	−24357.6	−0.941072
$876$	0	0
$877$	−42006.4	−1.61739	−0.808697	−	0.588226i	$-0.799827\pi$
$877$	−42006.4	−1.61739	−0.808697	+	0.588226i	$0.799827\pi$
$878$	9199.71	0.353616
$879$	0	0
$880$	−3531.94	−0.135298
$881$	11739.4	0.448933	0.224467	−	0.974482i	$-0.427936\pi$
$881$	11739.4	0.448933	0.224467	+	0.974482i	$0.427936\pi$
$882$	0	0
$883$	2127.39	0.0810787	0.0405393	−	0.999178i	$-0.487092\pi$
$883$	2127.39	0.0810787	0.0405393	+	0.999178i	$0.487092\pi$
$884$	17026.5	0.647809
$885$	0	0
$886$	−8833.35	−0.334946
$887$	−4677.87	−0.177077	−0.0885386	−	0.996073i	$-0.528220\pi$
$887$	−4677.87	−0.177077	−0.0885386	+	0.996073i	$0.528220\pi$
$888$	0	0
$889$	−41486.7	−1.56515
$890$	3061.82	0.115317
$891$	0	0
$892$	−13868.4	−0.520572
$893$	−12220.8	−0.457955
$894$	0	0
$895$	3337.82	0.124660
$896$	43102.8	1.60710
$897$	0	0
$898$	−10783.8	−0.400734
$899$	−7496.63	−0.278116
$900$	0	0
$901$	1483.35	0.0548473
$902$	3785.06	0.139721
$903$	0	0
$904$	9525.02	0.350440
$905$	5770.12	0.211939
$906$	0	0
$907$	16691.6	0.611065	0.305532	−	0.952182i	$-0.401166\pi$
$907$	16691.6	0.611065	0.305532	+	0.952182i	$0.401166\pi$
$908$	14937.5	0.545944
$909$	0	0
$910$	−2502.63	−0.0911662
$911$	−21965.6	−0.798850	−0.399425	−	0.916766i	$-0.630790\pi$
$911$	−21965.6	−0.798850	−0.399425	+	0.916766i	$0.630790\pi$
$912$	0	0
$913$	4163.65	0.150927
$914$	9563.06	0.346081
$915$	0	0
$916$	−11946.4	−0.430916
$917$	69201.4	2.49207
$918$	0	0
$919$	33759.8	1.21179	0.605895	−	0.795545i	$-0.292815\pi$
$919$	33759.8	1.21179	0.605895	+	0.795545i	$0.292815\pi$
$920$	−456.920	−0.0163742
$921$	0	0
$922$	−11580.3	−0.413640
$923$	−35519.1	−1.26666
$924$	0	0
$925$	−16799.7	−0.597157
$926$	−4878.13	−0.173116
$927$	0	0
$928$	−5767.57	−0.204019
$929$	−29364.7	−1.03706	−0.518528	−	0.855061i	$-0.673520\pi$
$929$	−29364.7	−1.03706	−0.518528	+	0.855061i	$0.673520\pi$
$930$	0	0
$931$	36697.7	1.29186
$932$	−43506.6	−1.52908
$933$	0	0
$934$	8911.70	0.312205
$935$	−4846.89	−0.169530
$936$	0	0
$937$	7721.05	0.269195	0.134598	−	0.990900i	$-0.457026\pi$
$937$	7721.05	0.269195	0.134598	+	0.990900i	$0.457026\pi$
$938$	−17817.0	−0.620199
$939$	0	0
$940$	−5877.54	−0.203941
$941$	−26601.6	−0.921560	−0.460780	−	0.887515i	$-0.652430\pi$
$941$	−26601.6	−0.921560	−0.460780	+	0.887515i	$0.652430\pi$
$942$	0	0
$943$	−2841.83	−0.0981366
$944$	−19915.8	−0.686655
$945$	0	0
$946$	−2126.33	−0.0730791
$947$	−20687.2	−0.709868	−0.354934	−	0.934891i	$-0.615497\pi$
$947$	−20687.2	−0.709868	−0.354934	+	0.934891i	$0.615497\pi$
$948$	0	0
$949$	8724.36	0.298424
$950$	−4049.36	−0.138293
$951$	0	0
$952$	27229.1	0.926997
$953$	32236.1	1.09573	0.547865	−	0.836567i	$-0.315441\pi$
$953$	32236.1	1.09573	0.547865	+	0.836567i	$0.315441\pi$
$954$	0	0
$955$	2939.42	0.0995995
$956$	1776.64	0.0601054
$957$	0	0
$958$	7862.73	0.265170
$959$	99286.6	3.34320
$960$	0	0
$961$	−756.514	−0.0253940
$962$	−3586.22	−0.120192
$963$	0	0
$964$	32312.7	1.07959
$965$	−1135.23	−0.0378696
$966$	0	0
$967$	26137.8	0.869218	0.434609	−	0.900619i	$-0.356887\pi$
$967$	26137.8	0.869218	0.434609	+	0.900619i	$0.356887\pi$
$968$	9208.71	0.305764
$969$	0	0
$970$	−3275.24	−0.108414
$971$	13176.7	0.435488	0.217744	−	0.976006i	$-0.430130\pi$
$971$	13176.7	0.435488	0.217744	+	0.976006i	$0.430130\pi$
$972$	0	0
$973$	87990.6	2.89913
$974$	14196.9	0.467040
$975$	0	0
$976$	39450.7	1.29384
$977$	32438.3	1.06222	0.531112	−	0.847302i	$-0.321774\pi$
$977$	32438.3	1.06222	0.531112	+	0.847302i	$0.321774\pi$
$978$	0	0
$979$	−31447.3	−1.02662
$980$	17649.6	0.575301
$981$	0	0
$982$	−8860.15	−0.287921
$983$	39927.1	1.29550	0.647751	−	0.761853i	$-0.275710\pi$
$983$	39927.1	1.29550	0.647751	+	0.761853i	$0.275710\pi$
$984$	0	0
$985$	4109.26	0.132926
$986$	−2304.90	−0.0744451
$987$	0	0
$988$	11345.9	0.365347
$989$	1596.45	0.0513288
$990$	0	0
$991$	−21881.0	−0.701385	−0.350692	−	0.936491i	$-0.614054\pi$
$991$	−21881.0	−0.701385	−0.350692	+	0.936491i	$0.614054\pi$
$992$	22337.8	0.714946
$993$	0	0
$994$	−27359.3	−0.873021
$995$	−2182.41	−0.0695347
$996$	0	0
$997$	14951.7	0.474950	0.237475	−	0.971394i	$-0.423680\pi$
$997$	14951.7	0.474950	0.237475	+	0.971394i	$0.423680\pi$
$998$	15388.4	0.488086
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.d.1.20		32
3.2	odd	2		717.4.a.d.1.13	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.d.1.13	✓	32	3.2	odd	2
2151.4.a.d.1.20		32	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)

\(f(q)\)	\(=\)	\(q+0.752561 q^{2} -7.43365 q^{4} -3.00130 q^{5} -33.6762 q^{7} -11.6148 q^{8} +O(q^{10})\)	\(q+0.752561 q^{2} -7.43365 q^{4} -3.00130 q^{5} -33.6762 q^{7} -11.6148 q^{8} -2.25866 q^{10} +23.1982 q^{11} -32.9020 q^{13} -25.3434 q^{14} +50.7284 q^{16} +69.6147 q^{17} +46.3891 q^{19} +22.3106 q^{20} +17.4580 q^{22} -13.1075 q^{23} -115.992 q^{25} -24.7608 q^{26} +250.337 q^{28} -43.9956 q^{29} +170.395 q^{31} +131.094 q^{32} +52.3893 q^{34} +101.072 q^{35} +144.835 q^{37} +34.9106 q^{38} +34.8594 q^{40} +216.809 q^{41} -121.796 q^{43} -172.447 q^{44} -9.86422 q^{46} -263.442 q^{47} +791.084 q^{49} -87.2912 q^{50} +244.582 q^{52} +21.3080 q^{53} -69.6246 q^{55} +391.141 q^{56} -33.1094 q^{58} -392.596 q^{59} +777.684 q^{61} +128.233 q^{62} -307.171 q^{64} +98.7487 q^{65} +703.025 q^{67} -517.491 q^{68} +76.0630 q^{70} +1079.54 q^{71} -265.162 q^{73} +108.997 q^{74} -344.840 q^{76} -781.225 q^{77} -564.600 q^{79} -152.251 q^{80} +163.162 q^{82} +179.482 q^{83} -208.934 q^{85} -91.6592 q^{86} -269.441 q^{88} -1355.59 q^{89} +1108.01 q^{91} +97.4368 q^{92} -198.256 q^{94} -139.227 q^{95} +1450.08 q^{97} +595.339 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

Introduction

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