LMFDB - Embedded newform 531.4.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,4,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	531.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:	$31.3300142130$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 34x^{5} + 25x^{4} + 315x^{3} - 146x^{2} - 736x + 512 $ x^7 - x^6 - 34x^5 + 25x^4 + 315x^3 - 146x^2 - 736*x + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.29817$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.29817 q^{2} -6.31476 q^{4} -7.05496 q^{5} -33.4497 q^{7} +18.5830 q^{8} +O(q^{10})$	\(q-1.29817 q^{2} -6.31476 q^{4} -7.05496 q^{5} -33.4497 q^{7} +18.5830 q^{8} +9.15853 q^{10} +27.5842 q^{11} -55.6756 q^{13} +43.4234 q^{14} +26.3942 q^{16} -108.437 q^{17} -141.505 q^{19} +44.5504 q^{20} -35.8089 q^{22} -142.351 q^{23} -75.2276 q^{25} +72.2763 q^{26} +211.227 q^{28} +97.2883 q^{29} -221.538 q^{31} -182.928 q^{32} +140.770 q^{34} +235.987 q^{35} +339.976 q^{37} +183.698 q^{38} -131.102 q^{40} +266.877 q^{41} -67.2505 q^{43} -174.187 q^{44} +184.795 q^{46} +262.017 q^{47} +775.886 q^{49} +97.6581 q^{50} +351.578 q^{52} -380.876 q^{53} -194.605 q^{55} -621.596 q^{56} -126.297 q^{58} +59.0000 q^{59} -15.4226 q^{61} +287.593 q^{62} +26.3175 q^{64} +392.789 q^{65} -172.954 q^{67} +684.753 q^{68} -306.350 q^{70} +616.313 q^{71} -210.965 q^{73} -441.346 q^{74} +893.572 q^{76} -922.684 q^{77} +543.689 q^{79} -186.210 q^{80} -346.452 q^{82} -350.751 q^{83} +765.019 q^{85} +87.3025 q^{86} +512.596 q^{88} -1447.29 q^{89} +1862.34 q^{91} +898.911 q^{92} -340.143 q^{94} +998.315 q^{95} -625.674 q^{97} -1007.23 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8}+O(q^{10}) $ 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8	$ 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8} - 79 q^{10} + 131 q^{11} - 123 q^{13} + 117 q^{14} + 202 q^{16} + 235 q^{17} - 80 q^{19} - 61 q^{20} + 688 q^{22} + 274 q^{23} + 193 q^{25} + 180 q^{26} - 118 q^{28} + 406 q^{29} - 346 q^{31} + 854 q^{32} + 178 q^{34} + 424 q^{35} - 157 q^{37} + 129 q^{38} - 590 q^{40} + 825 q^{41} - 815 q^{43} + 1690 q^{44} + 1457 q^{46} + 1196 q^{47} + 914 q^{49} - 713 q^{50} + 1030 q^{52} + 900 q^{53} - 1044 q^{55} - 2172 q^{56} + 1242 q^{58} + 413 q^{59} + 420 q^{61} - 646 q^{62} + 3541 q^{64} - 190 q^{65} + 1316 q^{67} + 611 q^{68} + 4658 q^{70} + 173 q^{71} - 418 q^{73} - 660 q^{74} + 1540 q^{76} + 753 q^{77} + 2635 q^{79} - 6155 q^{80} - 125 q^{82} - 457 q^{83} + 1270 q^{85} - 3482 q^{86} + 7685 q^{88} - 592 q^{89} + 3179 q^{91} + 3500 q^{92} + 2064 q^{94} + 2250 q^{95} - 1906 q^{97} - 2994 q^{98}+O(q^{100}) $ 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8 - 79 * q^10 + 131 * q^11 - 123 * q^13 + 117 * q^14 + 202 * q^16 + 235 * q^17 - 80 * q^19 - 61 * q^20 + 688 * q^22 + 274 * q^23 + 193 * q^25 + 180 * q^26 - 118 * q^28 + 406 * q^29 - 346 * q^31 + 854 * q^32 + 178 * q^34 + 424 * q^35 - 157 * q^37 + 129 * q^38 - 590 * q^40 + 825 * q^41 - 815 * q^43 + 1690 * q^44 + 1457 * q^46 + 1196 * q^47 + 914 * q^49 - 713 * q^50 + 1030 * q^52 + 900 * q^53 - 1044 * q^55 - 2172 * q^56 + 1242 * q^58 + 413 * q^59 + 420 * q^61 - 646 * q^62 + 3541 * q^64 - 190 * q^65 + 1316 * q^67 + 611 * q^68 + 4658 * q^70 + 173 * q^71 - 418 * q^73 - 660 * q^74 + 1540 * q^76 + 753 * q^77 + 2635 * q^79 - 6155 * q^80 - 125 * q^82 - 457 * q^83 + 1270 * q^85 - 3482 * q^86 + 7685 * q^88 - 592 * q^89 + 3179 * q^91 + 3500 * q^92 + 2064 * q^94 + 2250 * q^95 - 1906 * q^97 - 2994 * 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 $

$2$	−1.29817	−0.458972	−0.229486	−	0.973312i	$-0.573705\pi$
$2$	−1.29817	−0.458972	−0.229486	+	0.973312i	$0.573705\pi$
$3$	0	0
$3$	0	0
$4$	−6.31476	−0.789345
$5$	−7.05496	−0.631015	−0.315507	−	0.948923i	$-0.602175\pi$
$5$	−7.05496	−0.631015	−0.315507	+	0.948923i	$0.602175\pi$
$6$	0	0
$7$	−33.4497	−1.80612	−0.903058	−	0.429518i	$-0.858684\pi$
$7$	−33.4497	−1.80612	−0.903058	+	0.429518i	$0.858684\pi$
$8$	18.5830	0.821259
$9$	0	0
$10$	9.15853	0.289618
$11$	27.5842	0.756086	0.378043	−	0.925788i	$-0.376597\pi$
$11$	27.5842	0.756086	0.378043	+	0.925788i	$0.376597\pi$
$12$	0	0
$13$	−55.6756	−1.18782	−0.593909	−	0.804532i	$-0.702416\pi$
$13$	−55.6756	−1.18782	−0.593909	+	0.804532i	$0.702416\pi$
$14$	43.4234	0.828957
$15$	0	0
$16$	26.3942	0.412410
$17$	−108.437	−1.54705	−0.773525	−	0.633766i	$-0.781508\pi$
$17$	−108.437	−1.54705	−0.773525	+	0.633766i	$0.781508\pi$
$18$	0	0
$19$	−141.505	−1.70861	−0.854304	−	0.519773i	$-0.826017\pi$
$19$	−141.505	−1.70861	−0.854304	+	0.519773i	$0.826017\pi$
$20$	44.5504	0.498088
$21$	0	0
$22$	−35.8089	−0.347022
$23$	−142.351	−1.29053	−0.645265	−	0.763959i	$-0.723253\pi$
$23$	−142.351	−1.29053	−0.645265	+	0.763959i	$0.723253\pi$
$24$	0	0
$25$	−75.2276	−0.601820
$26$	72.2763	0.545175
$27$	0	0
$28$	211.227	1.42565
$29$	97.2883	0.622965	0.311482	−	0.950252i	$-0.399175\pi$
$29$	97.2883	0.622965	0.311482	+	0.950252i	$0.399175\pi$
$30$	0	0
$31$	−221.538	−1.28353	−0.641763	−	0.766903i	$-0.721797\pi$
$31$	−221.538	−1.28353	−0.641763	+	0.766903i	$0.721797\pi$
$32$	−182.928	−1.01054
$33$	0	0
$34$	140.770	0.710052
$35$	235.987	1.13969
$36$	0	0
$37$	339.976	1.51059	0.755293	−	0.655387i	$-0.227494\pi$
$37$	339.976	1.51059	0.755293	+	0.655387i	$0.227494\pi$
$38$	183.698	0.784204
$39$	0	0
$40$	−131.102	−0.518227
$41$	266.877	1.01657	0.508283	−	0.861190i	$-0.330280\pi$
$41$	266.877	1.01657	0.508283	+	0.861190i	$0.330280\pi$
$42$	0	0
$43$	−67.2505	−0.238503	−0.119251	−	0.992864i	$-0.538049\pi$
$43$	−67.2505	−0.238503	−0.119251	+	0.992864i	$0.538049\pi$
$44$	−174.187	−0.596812
$45$	0	0
$46$	184.795	0.592317
$47$	262.017	0.813174	0.406587	−	0.913612i	$-0.366719\pi$
$47$	262.017	0.813174	0.406587	+	0.913612i	$0.366719\pi$
$48$	0	0
$49$	775.886	2.26206
$50$	97.6581	0.276219
$51$	0	0
$52$	351.578	0.937598
$53$	−380.876	−0.987121	−0.493560	−	0.869712i	$-0.664305\pi$
$53$	−380.876	−0.987121	−0.493560	+	0.869712i	$0.664305\pi$
$54$	0	0
$55$	−194.605	−0.477101
$56$	−621.596	−1.48329
$57$	0	0
$58$	−126.297	−0.285923
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	0	0
$61$	−15.4226	−0.0323715	−0.0161857	−	0.999869i	$-0.505152\pi$
$61$	−15.4226	−0.0323715	−0.0161857	+	0.999869i	$0.505152\pi$
$62$	287.593	0.589103
$63$	0	0
$64$	26.3175	0.0514014
$65$	392.789	0.749531
$66$	0	0
$67$	−172.954	−0.315369	−0.157685	−	0.987490i	$-0.550403\pi$
$67$	−172.954	−0.315369	−0.157685	+	0.987490i	$0.550403\pi$
$68$	684.753	1.22116
$69$	0	0
$70$	−306.350	−0.523084
$71$	616.313	1.03018	0.515091	−	0.857136i	$-0.327758\pi$
$71$	616.313	1.03018	0.515091	+	0.857136i	$0.327758\pi$
$72$	0	0
$73$	−210.965	−0.338242	−0.169121	−	0.985595i	$-0.554093\pi$
$73$	−210.965	−0.338242	−0.169121	+	0.985595i	$0.554093\pi$
$74$	−441.346	−0.693317
$75$	0	0
$76$	893.572	1.34868
$77$	−922.684	−1.36558
$78$	0	0
$79$	543.689	0.774302	0.387151	−	0.922016i	$-0.373459\pi$
$79$	543.689	0.774302	0.387151	+	0.922016i	$0.373459\pi$
$80$	−186.210	−0.260237
$81$	0	0
$82$	−346.452	−0.466575
$83$	−350.751	−0.463855	−0.231927	−	0.972733i	$-0.574503\pi$
$83$	−350.751	−0.463855	−0.231927	+	0.972733i	$0.574503\pi$
$84$	0	0
$85$	765.019	0.976211
$86$	87.3025	0.109466
$87$	0	0
$88$	512.596	0.620942
$89$	−1447.29	−1.72373	−0.861867	−	0.507135i	$-0.830705\pi$
$89$	−1447.29	−1.72373	−0.861867	+	0.507135i	$0.830705\pi$
$90$	0	0
$91$	1862.34	2.14534
$92$	898.911	1.01867
$93$	0	0
$94$	−340.143	−0.373224
$95$	998.315	1.07816
$96$	0	0
$97$	−625.674	−0.654924	−0.327462	−	0.944864i	$-0.606193\pi$
$97$	−625.674	−0.654924	−0.327462	+	0.944864i	$0.606193\pi$
$98$	−1007.23	−1.03822
$99$	0	0
$100$	475.044	0.475044
$101$	902.589	0.889218	0.444609	−	0.895725i	$-0.353343\pi$
$101$	902.589	0.889218	0.444609	+	0.895725i	$0.353343\pi$
$102$	0	0
$103$	700.033	0.669673	0.334836	−	0.942276i	$-0.391319\pi$
$103$	700.033	0.669673	0.334836	+	0.942276i	$0.391319\pi$
$104$	−1034.62	−0.975507
$105$	0	0
$106$	494.442	0.453061
$107$	−1645.87	−1.48703	−0.743513	−	0.668721i	$-0.766842\pi$
$107$	−1645.87	−1.48703	−0.743513	+	0.668721i	$0.766842\pi$
$108$	0	0
$109$	−253.521	−0.222779	−0.111390	−	0.993777i	$-0.535530\pi$
$109$	−253.521	−0.222779	−0.111390	+	0.993777i	$0.535530\pi$
$110$	252.630	0.218976
$111$	0	0
$112$	−882.880	−0.744860
$113$	947.504	0.788793	0.394397	−	0.918940i	$-0.370954\pi$
$113$	947.504	0.788793	0.394397	+	0.918940i	$0.370954\pi$
$114$	0	0
$115$	1004.28	0.814343
$116$	−614.352	−0.491734
$117$	0	0
$118$	−76.5920	−0.0597531
$119$	3627.19	2.79415
$120$	0	0
$121$	−570.113	−0.428335
$122$	20.0211	0.0148576
$123$	0	0
$124$	1398.96	1.01314
$125$	1412.60	1.01077
$126$	0	0
$127$	−2655.70	−1.85555	−0.927775	−	0.373139i	$-0.878281\pi$
$127$	−2655.70	−1.85555	−0.927775	+	0.373139i	$0.878281\pi$
$128$	1429.26	0.986952
$129$	0	0
$130$	−509.907	−0.344014
$131$	1677.06	1.11851	0.559256	−	0.828995i	$-0.311087\pi$
$131$	1677.06	1.11851	0.559256	+	0.828995i	$0.311087\pi$
$132$	0	0
$133$	4733.32	3.08595
$134$	224.524	0.144746
$135$	0	0
$136$	−2015.08	−1.27053
$137$	−687.255	−0.428585	−0.214293	−	0.976769i	$-0.568745\pi$
$137$	−687.255	−0.428585	−0.214293	+	0.976769i	$0.568745\pi$
$138$	0	0
$139$	−149.695	−0.0913451	−0.0456726	−	0.998956i	$-0.514543\pi$
$139$	−149.695	−0.0913451	−0.0456726	+	0.998956i	$0.514543\pi$
$140$	−1490.20	−0.899605
$141$	0	0
$142$	−800.078	−0.472825
$143$	−1535.77	−0.898092
$144$	0	0
$145$	−686.365	−0.393100
$146$	273.869	0.155243
$147$	0	0
$148$	−2146.87	−1.19237
$149$	−221.901	−0.122005	−0.0610027	−	0.998138i	$-0.519430\pi$
$149$	−221.901	−0.122005	−0.0610027	+	0.998138i	$0.519430\pi$
$150$	0	0
$151$	−1262.38	−0.680340	−0.340170	−	0.940364i	$-0.610485\pi$
$151$	−1262.38	−0.680340	−0.340170	+	0.940364i	$0.610485\pi$
$152$	−2629.59	−1.40321
$153$	0	0
$154$	1197.80	0.626762
$155$	1562.94	0.809924
$156$	0	0
$157$	−3446.33	−1.75189	−0.875947	−	0.482407i	$-0.839763\pi$
$157$	−3446.33	−1.75189	−0.875947	+	0.482407i	$0.839763\pi$
$158$	−705.801	−0.355383
$159$	0	0
$160$	1290.55	0.637668
$161$	4761.60	2.33085
$162$	0	0
$163$	−1915.37	−0.920388	−0.460194	−	0.887818i	$-0.652220\pi$
$163$	−1915.37	−0.920388	−0.460194	+	0.887818i	$0.652220\pi$
$164$	−1685.26	−0.802421
$165$	0	0
$166$	455.334	0.212896
$167$	3013.60	1.39640	0.698201	−	0.715902i	$-0.253984\pi$
$167$	3013.60	1.39640	0.698201	+	0.715902i	$0.253984\pi$
$168$	0	0
$169$	902.774	0.410912
$170$	−993.123	−0.448053
$171$	0	0
$172$	424.671	0.188261
$173$	1913.98	0.841138	0.420569	−	0.907261i	$-0.361830\pi$
$173$	1913.98	0.841138	0.420569	+	0.907261i	$0.361830\pi$
$174$	0	0
$175$	2516.34	1.08696
$176$	728.063	0.311817
$177$	0	0
$178$	1878.83	0.791146
$179$	−1507.53	−0.629485	−0.314742	−	0.949177i	$-0.601918\pi$
$179$	−1507.53	−0.629485	−0.314742	+	0.949177i	$0.601918\pi$
$180$	0	0
$181$	−1970.30	−0.809123	−0.404562	−	0.914511i	$-0.632576\pi$
$181$	−1970.30	−0.809123	−0.404562	+	0.914511i	$0.632576\pi$
$182$	−2417.63	−0.984650
$183$	0	0
$184$	−2645.30	−1.05986
$185$	−2398.52	−0.953202
$186$	0	0
$187$	−2991.15	−1.16970
$188$	−1654.58	−0.641874
$189$	0	0
$190$	−1295.98	−0.494844
$191$	2132.62	0.807911	0.403955	−	0.914779i	$-0.367635\pi$
$191$	2132.62	0.807911	0.403955	+	0.914779i	$0.367635\pi$
$192$	0	0
$193$	1319.47	0.492112	0.246056	−	0.969256i	$-0.420865\pi$
$193$	1319.47	0.492112	0.246056	+	0.969256i	$0.420865\pi$
$194$	812.231	0.300592
$195$	0	0
$196$	−4899.53	−1.78554
$197$	−187.057	−0.0676511	−0.0338255	−	0.999428i	$-0.510769\pi$
$197$	−187.057	−0.0676511	−0.0338255	+	0.999428i	$0.510769\pi$
$198$	0	0
$199$	3324.89	1.18440	0.592200	−	0.805791i	$-0.298260\pi$
$199$	3324.89	1.18440	0.592200	+	0.805791i	$0.298260\pi$
$200$	−1397.95	−0.494251
$201$	0	0
$202$	−1171.71	−0.408126
$203$	−3254.27	−1.12515
$204$	0	0
$205$	−1882.81	−0.641468
$206$	−908.761	−0.307361
$207$	0	0
$208$	−1469.51	−0.489868
$209$	−3903.31	−1.29185
$210$	0	0
$211$	4844.60	1.58065	0.790323	−	0.612691i	$-0.209913\pi$
$211$	4844.60	1.58065	0.790323	+	0.612691i	$0.209913\pi$
$212$	2405.14	0.779179
$213$	0	0
$214$	2136.61	0.682504
$215$	474.450	0.150499
$216$	0	0
$217$	7410.38	2.31820
$218$	329.114	0.102249
$219$	0	0
$220$	1228.88	0.376597
$221$	6037.30	1.83761
$222$	0	0
$223$	−4223.02	−1.26814	−0.634068	−	0.773277i	$-0.718616\pi$
$223$	−4223.02	−1.26814	−0.634068	+	0.773277i	$0.718616\pi$
$224$	6118.89	1.82516
$225$	0	0
$226$	−1230.02	−0.362034
$227$	−6199.87	−1.81277	−0.906387	−	0.422448i	$-0.861171\pi$
$227$	−6199.87	−1.81277	−0.906387	+	0.422448i	$0.861171\pi$
$228$	0	0
$229$	−2502.28	−0.722075	−0.361037	−	0.932551i	$-0.617577\pi$
$229$	−2502.28	−0.722075	−0.361037	+	0.932551i	$0.617577\pi$
$230$	−1303.72	−0.373761
$231$	0	0
$232$	1807.91	0.511615
$233$	5317.19	1.49503	0.747513	−	0.664247i	$-0.231248\pi$
$233$	5317.19	1.49503	0.747513	+	0.664247i	$0.231248\pi$
$234$	0	0
$235$	−1848.52	−0.513125
$236$	−372.571	−0.102764
$237$	0	0
$238$	−4708.71	−1.28244
$239$	−6431.88	−1.74077	−0.870385	−	0.492372i	$-0.836130\pi$
$239$	−6431.88	−1.74077	−0.870385	+	0.492372i	$0.836130\pi$
$240$	0	0
$241$	2651.24	0.708638	0.354319	−	0.935125i	$-0.384713\pi$
$241$	2651.24	0.708638	0.354319	+	0.935125i	$0.384713\pi$
$242$	740.103	0.196594
$243$	0	0
$244$	97.3899	0.0255522
$245$	−5473.84	−1.42739
$246$	0	0
$247$	7878.40	2.02952
$248$	−4116.83	−1.05411
$249$	0	0
$250$	−1833.79	−0.463916
$251$	−3189.73	−0.802126	−0.401063	−	0.916050i	$-0.631359\pi$
$251$	−3189.73	−0.802126	−0.401063	+	0.916050i	$0.631359\pi$
$252$	0	0
$253$	−3926.63	−0.975751
$254$	3447.54	0.851646
$255$	0	0
$256$	−2065.96	−0.504385
$257$	−6997.92	−1.69851	−0.849257	−	0.527980i	$-0.822950\pi$
$257$	−6997.92	−1.69851	−0.849257	+	0.527980i	$0.822950\pi$
$258$	0	0
$259$	−11372.1	−2.72830
$260$	−2480.37	−0.591638
$261$	0	0
$262$	−2177.10	−0.513366
$263$	4833.10	1.13316	0.566582	−	0.824006i	$-0.308266\pi$
$263$	4833.10	1.13316	0.566582	+	0.824006i	$0.308266\pi$
$264$	0	0
$265$	2687.07	0.622888
$266$	−6144.65	−1.41636
$267$	0	0
$268$	1092.16	0.248935
$269$	−4142.13	−0.938849	−0.469424	−	0.882973i	$-0.655539\pi$
$269$	−4142.13	−0.938849	−0.469424	+	0.882973i	$0.655539\pi$
$270$	0	0
$271$	−3931.65	−0.881295	−0.440648	−	0.897680i	$-0.645251\pi$
$271$	−3931.65	−0.881295	−0.440648	+	0.897680i	$0.645251\pi$
$272$	−2862.11	−0.638018
$273$	0	0
$274$	892.173	0.196709
$275$	−2075.09	−0.455028
$276$	0	0
$277$	−2635.02	−0.571564	−0.285782	−	0.958295i	$-0.592253\pi$
$277$	−2635.02	−0.571564	−0.285782	+	0.958295i	$0.592253\pi$
$278$	194.329	0.0419248
$279$	0	0
$280$	4385.33	0.935978
$281$	2791.96	0.592721	0.296360	−	0.955076i	$-0.404227\pi$
$281$	2791.96	0.592721	0.296360	+	0.955076i	$0.404227\pi$
$282$	0	0
$283$	−915.281	−0.192254	−0.0961268	−	0.995369i	$-0.530645\pi$
$283$	−915.281	−0.192254	−0.0961268	+	0.995369i	$0.530645\pi$
$284$	−3891.87	−0.813169
$285$	0	0
$286$	1993.68	0.412199
$287$	−8926.97	−1.83604
$288$	0	0
$289$	6845.59	1.39336
$290$	891.017	0.180422
$291$	0	0
$292$	1332.20	0.266989
$293$	−2631.82	−0.524752	−0.262376	−	0.964966i	$-0.584506\pi$
$293$	−2631.82	−0.524752	−0.262376	+	0.964966i	$0.584506\pi$
$294$	0	0
$295$	−416.243	−0.0821511
$296$	6317.76	1.24058
$297$	0	0
$298$	288.065	0.0559971
$299$	7925.47	1.53291
$300$	0	0
$301$	2249.51	0.430763
$302$	1638.79	0.312257
$303$	0	0
$304$	−3734.93	−0.704647
$305$	108.806	0.0204269
$306$	0	0
$307$	−6778.17	−1.26010	−0.630050	−	0.776555i	$-0.716965\pi$
$307$	−6778.17	−1.26010	−0.630050	+	0.776555i	$0.716965\pi$
$308$	5826.52	1.07791
$309$	0	0
$310$	−2028.96	−0.371732
$311$	−5915.03	−1.07849	−0.539245	−	0.842149i	$-0.681290\pi$
$311$	−5915.03	−1.07849	−0.539245	+	0.842149i	$0.681290\pi$
$312$	0	0
$313$	6150.93	1.11077	0.555385	−	0.831593i	$-0.312571\pi$
$313$	6150.93	1.11077	0.555385	+	0.831593i	$0.312571\pi$
$314$	4473.92	0.804070
$315$	0	0
$316$	−3433.27	−0.611191
$317$	3992.23	0.707337	0.353669	−	0.935371i	$-0.384934\pi$
$317$	3992.23	0.707337	0.353669	+	0.935371i	$0.384934\pi$
$318$	0	0
$319$	2683.62	0.471015
$320$	−185.669	−0.0324350
$321$	0	0
$322$	−6181.36	−1.06979
$323$	15344.4	2.64330
$324$	0	0
$325$	4188.34	0.714853
$326$	2486.47	0.422432
$327$	0	0
$328$	4959.37	0.834864
$329$	−8764.42	−1.46869
$330$	0	0
$331$	−4224.03	−0.701431	−0.350715	−	0.936482i	$-0.614062\pi$
$331$	−4224.03	−0.701431	−0.350715	+	0.936482i	$0.614062\pi$
$332$	2214.91	0.366141
$333$	0	0
$334$	−3912.16	−0.640909
$335$	1220.19	0.199003
$336$	0	0
$337$	2266.93	0.366431	0.183216	−	0.983073i	$-0.441349\pi$
$337$	2266.93	0.366431	0.183216	+	0.983073i	$0.441349\pi$
$338$	−1171.95	−0.188597
$339$	0	0
$340$	−4830.91	−0.770567
$341$	−6110.93	−0.970456
$342$	0	0
$343$	−14479.9	−2.27942
$344$	−1249.71	−0.195872
$345$	0	0
$346$	−2484.66	−0.386059
$347$	270.565	0.0418579	0.0209290	−	0.999781i	$-0.493338\pi$
$347$	270.565	0.0418579	0.0209290	+	0.999781i	$0.493338\pi$
$348$	0	0
$349$	2623.43	0.402375	0.201188	−	0.979553i	$-0.435520\pi$
$349$	2623.43	0.402375	0.201188	+	0.979553i	$0.435520\pi$
$350$	−3266.64	−0.498883
$351$	0	0
$352$	−5045.92	−0.764058
$353$	4779.31	0.720615	0.360307	−	0.932834i	$-0.382672\pi$
$353$	4779.31	0.720615	0.360307	+	0.932834i	$0.382672\pi$
$354$	0	0
$355$	−4348.06	−0.650060
$356$	9139.28	1.36062
$357$	0	0
$358$	1957.02	0.288916
$359$	5676.39	0.834509	0.417254	−	0.908790i	$-0.362992\pi$
$359$	5676.39	0.834509	0.417254	+	0.908790i	$0.362992\pi$
$360$	0	0
$361$	13164.8	1.91934
$362$	2557.78	0.371365
$363$	0	0
$364$	−11760.2	−1.69341
$365$	1488.35	0.213435
$366$	0	0
$367$	4096.80	0.582701	0.291350	−	0.956616i	$-0.405895\pi$
$367$	4096.80	0.582701	0.291350	+	0.956616i	$0.405895\pi$
$368$	−3757.24	−0.532227
$369$	0	0
$370$	3113.68	0.437493
$371$	12740.2	1.78286
$372$	0	0
$373$	11745.0	1.63038	0.815190	−	0.579194i	$-0.196633\pi$
$373$	11745.0	1.63038	0.815190	+	0.579194i	$0.196633\pi$
$374$	3883.01	0.536860
$375$	0	0
$376$	4869.06	0.667826
$377$	−5416.58	−0.739969
$378$	0	0
$379$	−973.770	−0.131977	−0.0659884	−	0.997820i	$-0.521020\pi$
$379$	−973.770	−0.131977	−0.0659884	+	0.997820i	$0.521020\pi$
$380$	−6304.12	−0.851038
$381$	0	0
$382$	−2768.50	−0.370808
$383$	3020.77	0.403013	0.201506	−	0.979487i	$-0.435416\pi$
$383$	3020.77	0.403013	0.201506	+	0.979487i	$0.435416\pi$
$384$	0	0
$385$	6509.50	0.861700
$386$	−1712.90	−0.225865
$387$	0	0
$388$	3950.98	0.516961
$389$	327.769	0.0427213	0.0213606	−	0.999772i	$-0.493200\pi$
$389$	327.769	0.0427213	0.0213606	+	0.999772i	$0.493200\pi$
$390$	0	0
$391$	15436.1	1.99651
$392$	14418.3	1.85774
$393$	0	0
$394$	242.832	0.0310499
$395$	−3835.71	−0.488596
$396$	0	0
$397$	5076.62	0.641785	0.320892	−	0.947116i	$-0.396017\pi$
$397$	5076.62	0.641785	0.320892	+	0.947116i	$0.396017\pi$
$398$	−4316.27	−0.543606
$399$	0	0
$400$	−1985.57	−0.248197
$401$	−6277.21	−0.781718	−0.390859	−	0.920450i	$-0.627822\pi$
$401$	−6277.21	−0.781718	−0.390859	+	0.920450i	$0.627822\pi$
$402$	0	0
$403$	12334.2	1.52460
$404$	−5699.63	−0.701899
$405$	0	0
$406$	4224.59	0.516411
$407$	9377.96	1.14213
$408$	0	0
$409$	−6562.26	−0.793357	−0.396679	−	0.917958i	$-0.629837\pi$
$409$	−6562.26	−0.793357	−0.396679	+	0.917958i	$0.629837\pi$
$410$	2444.20	0.294416
$411$	0	0
$412$	−4420.54	−0.528603
$413$	−1973.54	−0.235136
$414$	0	0
$415$	2474.54	0.292699
$416$	10184.6	1.20034
$417$	0	0
$418$	5067.16	0.592925
$419$	−10923.8	−1.27365	−0.636827	−	0.771006i	$-0.719754\pi$
$419$	−10923.8	−1.27365	−0.636827	+	0.771006i	$0.719754\pi$
$420$	0	0
$421$	1049.00	0.121437	0.0607187	−	0.998155i	$-0.480661\pi$
$421$	1049.00	0.121437	0.0607187	+	0.998155i	$0.480661\pi$
$422$	−6289.11	−0.725472
$423$	0	0
$424$	−7077.82	−0.810682
$425$	8157.45	0.931046
$426$	0	0
$427$	515.882	0.0584667
$428$	10393.2	1.17378
$429$	0	0
$430$	−615.916	−0.0690746
$431$	8997.43	1.00555	0.502774	−	0.864418i	$-0.332313\pi$
$431$	8997.43	1.00555	0.502774	+	0.864418i	$0.332313\pi$
$432$	0	0
$433$	−7942.35	−0.881490	−0.440745	−	0.897632i	$-0.645286\pi$
$433$	−7942.35	−0.881490	−0.440745	+	0.897632i	$0.645286\pi$
$434$	−9619.92	−1.06399
$435$	0	0
$436$	1600.93	0.175850
$437$	20143.4	2.20501
$438$	0	0
$439$	−3365.40	−0.365881	−0.182941	−	0.983124i	$-0.558562\pi$
$439$	−3365.40	−0.365881	−0.182941	+	0.983124i	$0.558562\pi$
$440$	−3616.34	−0.391824
$441$	0	0
$442$	−7837.43	−0.843413
$443$	4056.81	0.435090	0.217545	−	0.976050i	$-0.430195\pi$
$443$	4056.81	0.435090	0.217545	+	0.976050i	$0.430195\pi$
$444$	0	0
$445$	10210.6	1.08770
$446$	5482.19	0.582039
$447$	0	0
$448$	−880.315	−0.0928370
$449$	−10892.7	−1.14490	−0.572449	−	0.819940i	$-0.694007\pi$
$449$	−10892.7	−1.14490	−0.572449	+	0.819940i	$0.694007\pi$
$450$	0	0
$451$	7361.58	0.768611
$452$	−5983.26	−0.622630
$453$	0	0
$454$	8048.48	0.832013
$455$	−13138.7	−1.35374
$456$	0	0
$457$	9685.62	0.991409	0.495705	−	0.868491i	$-0.334910\pi$
$457$	9685.62	0.991409	0.495705	+	0.868491i	$0.334910\pi$
$458$	3248.38	0.331412
$459$	0	0
$460$	−6341.78	−0.642798
$461$	−145.044	−0.0146538	−0.00732689	−	0.999973i	$-0.502332\pi$
$461$	−145.044	−0.0146538	−0.00732689	+	0.999973i	$0.502332\pi$
$462$	0	0
$463$	−12516.4	−1.25634	−0.628169	−	0.778077i	$-0.716195\pi$
$463$	−12516.4	−1.25634	−0.628169	+	0.778077i	$0.716195\pi$
$464$	2567.85	0.256917
$465$	0	0
$466$	−6902.62	−0.686175
$467$	−2625.49	−0.260156	−0.130078	−	0.991504i	$-0.541523\pi$
$467$	−2625.49	−0.260156	−0.130078	+	0.991504i	$0.541523\pi$
$468$	0	0
$469$	5785.28	0.569593
$470$	2399.69	0.235510
$471$	0	0
$472$	1096.40	0.106919
$473$	−1855.05	−0.180328
$474$	0	0
$475$	10645.1	1.02828
$476$	−22904.8	−2.20555
$477$	0	0
$478$	8349.67	0.798964
$479$	16253.8	1.55043	0.775216	−	0.631697i	$-0.217641\pi$
$479$	16253.8	1.55043	0.775216	+	0.631697i	$0.217641\pi$
$480$	0	0
$481$	−18928.4	−1.79430
$482$	−3441.76	−0.325245
$483$	0	0
$484$	3600.13	0.338104
$485$	4414.11	0.413266
$486$	0	0
$487$	−1328.98	−0.123659	−0.0618296	−	0.998087i	$-0.519694\pi$
$487$	−1328.98	−0.123659	−0.0618296	+	0.998087i	$0.519694\pi$
$488$	−286.597	−0.0265854
$489$	0	0
$490$	7105.97	0.655133
$491$	−14197.4	−1.30493	−0.652463	−	0.757820i	$-0.726264\pi$
$491$	−14197.4	−1.30493	−0.652463	+	0.757820i	$0.726264\pi$
$492$	0	0
$493$	−10549.6	−0.963757
$494$	−10227.5	−0.931491
$495$	0	0
$496$	−5847.31	−0.529339
$497$	−20615.5	−1.86063
$498$	0	0
$499$	−7543.06	−0.676701	−0.338350	−	0.941020i	$-0.609869\pi$
$499$	−7543.06	−0.676701	−0.338350	+	0.941020i	$0.609869\pi$
$500$	−8920.21	−0.797848
$501$	0	0
$502$	4140.80	0.368154
$503$	8035.31	0.712280	0.356140	−	0.934433i	$-0.384093\pi$
$503$	8035.31	0.712280	0.356140	+	0.934433i	$0.384093\pi$
$504$	0	0
$505$	−6367.73	−0.561109
$506$	5097.43	0.447842
$507$	0	0
$508$	16770.1	1.46467
$509$	2243.76	0.195389	0.0976946	−	0.995216i	$-0.468853\pi$
$509$	2243.76	0.195389	0.0976946	+	0.995216i	$0.468853\pi$
$510$	0	0
$511$	7056.74	0.610904
$512$	−8752.11	−0.755453
$513$	0	0
$514$	9084.48	0.779570
$515$	−4938.70	−0.422573
$516$	0	0
$517$	7227.53	0.614829
$518$	14762.9	1.25221
$519$	0	0
$520$	7299.19	0.615559
$521$	−5397.02	−0.453834	−0.226917	−	0.973914i	$-0.572865\pi$
$521$	−5397.02	−0.453834	−0.226917	+	0.973914i	$0.572865\pi$
$522$	0	0
$523$	−229.273	−0.0191690	−0.00958451	−	0.999954i	$-0.503051\pi$
$523$	−229.273	−0.0191690	−0.00958451	+	0.999954i	$0.503051\pi$
$524$	−10590.2	−0.882892
$525$	0	0
$526$	−6274.19	−0.520090
$527$	24022.9	1.98568
$528$	0	0
$529$	8096.74	0.665467
$530$	−3488.27	−0.285888
$531$	0	0
$532$	−29889.8	−2.43588
$533$	−14858.5	−1.20750
$534$	0	0
$535$	11611.5	0.938336
$536$	−3214.01	−0.259000
$537$	0	0
$538$	5377.19	0.430905
$539$	21402.2	1.71031
$540$	0	0
$541$	21118.4	1.67829	0.839143	−	0.543911i	$-0.183057\pi$
$541$	21118.4	1.67829	0.839143	+	0.543911i	$0.183057\pi$
$542$	5103.95	0.404490
$543$	0	0
$544$	19836.2	1.56336
$545$	1788.58	0.140577
$546$	0	0
$547$	700.599	0.0547632	0.0273816	−	0.999625i	$-0.491283\pi$
$547$	700.599	0.0547632	0.0273816	+	0.999625i	$0.491283\pi$
$548$	4339.85	0.338302
$549$	0	0
$550$	2693.82	0.208845
$551$	−13766.8	−1.06440
$552$	0	0
$553$	−18186.3	−1.39848
$554$	3420.70	0.262332
$555$	0	0
$556$	945.288	0.0721028
$557$	−13.7365	−0.00104495	−0.000522474	−	1.00000i	$-0.500166\pi$
$557$	−13.7365	−0.00104495	−0.000522474		1.00000i	$0.500166\pi$
$558$	0	0
$559$	3744.21	0.283298
$560$	6228.68	0.470018
$561$	0	0
$562$	−3624.44	−0.272042
$563$	20483.4	1.53334	0.766672	−	0.642039i	$-0.221911\pi$
$563$	20483.4	1.53334	0.766672	+	0.642039i	$0.221911\pi$
$564$	0	0
$565$	−6684.60	−0.497740
$566$	1188.19	0.0882390
$567$	0	0
$568$	11452.9	0.846046
$569$	16699.3	1.23035	0.615175	−	0.788390i	$-0.289085\pi$
$569$	16699.3	1.23035	0.615175	+	0.788390i	$0.289085\pi$
$570$	0	0
$571$	−2261.27	−0.165729	−0.0828643	−	0.996561i	$-0.526407\pi$
$571$	−2261.27	−0.165729	−0.0828643	+	0.996561i	$0.526407\pi$
$572$	9697.99	0.708904
$573$	0	0
$574$	11588.7	0.842689
$575$	10708.7	0.776667
$576$	0	0
$577$	−12365.4	−0.892160	−0.446080	−	0.894993i	$-0.647180\pi$
$577$	−12365.4	−0.892160	−0.446080	+	0.894993i	$0.647180\pi$
$578$	−8886.73	−0.639514
$579$	0	0
$580$	4334.23	0.310291
$581$	11732.5	0.837776
$582$	0	0
$583$	−10506.2	−0.746348
$584$	−3920.36	−0.277784
$585$	0	0
$586$	3416.54	0.240847
$587$	−17739.0	−1.24730	−0.623651	−	0.781703i	$-0.714351\pi$
$587$	−17739.0	−1.24730	−0.623651	+	0.781703i	$0.714351\pi$
$588$	0	0
$589$	31348.8	2.19304
$590$	540.353	0.0377051
$591$	0	0
$592$	8973.40	0.622981
$593$	−24070.9	−1.66690	−0.833451	−	0.552594i	$-0.813638\pi$
$593$	−24070.9	−1.66690	−0.833451	+	0.552594i	$0.813638\pi$
$594$	0	0
$595$	−25589.7	−1.76315
$596$	1401.25	0.0963044
$597$	0	0
$598$	−10288.6	−0.703565
$599$	−28779.3	−1.96309	−0.981544	−	0.191238i	$-0.938750\pi$
$599$	−28779.3	−1.96309	−0.981544	+	0.191238i	$0.938750\pi$
$600$	0	0
$601$	−11612.0	−0.788128	−0.394064	−	0.919083i	$-0.628931\pi$
$601$	−11612.0	−0.788128	−0.394064	+	0.919083i	$0.628931\pi$
$602$	−2920.25	−0.197708
$603$	0	0
$604$	7971.65	0.537023
$605$	4022.13	0.270285
$606$	0	0
$607$	−821.129	−0.0549071	−0.0274535	−	0.999623i	$-0.508740\pi$
$607$	−821.129	−0.0549071	−0.0274535	+	0.999623i	$0.508740\pi$
$608$	25885.3	1.72662
$609$	0	0
$610$	−141.248	−0.00937536
$611$	−14588.0	−0.965903
$612$	0	0
$613$	7705.99	0.507736	0.253868	−	0.967239i	$-0.418297\pi$
$613$	7705.99	0.507736	0.253868	+	0.967239i	$0.418297\pi$
$614$	8799.21	0.578350
$615$	0	0
$616$	−17146.2	−1.12149
$617$	447.613	0.0292062	0.0146031	−	0.999893i	$-0.495352\pi$
$617$	447.613	0.0292062	0.0146031	+	0.999893i	$0.495352\pi$
$618$	0	0
$619$	−5073.90	−0.329462	−0.164731	−	0.986339i	$-0.552676\pi$
$619$	−5073.90	−0.329462	−0.164731	+	0.986339i	$0.552676\pi$
$620$	−9869.58	−0.639309
$621$	0	0
$622$	7678.70	0.494997
$623$	48411.4	3.11326
$624$	0	0
$625$	−562.371	−0.0359917
$626$	−7984.94	−0.509812
$627$	0	0
$628$	21762.8	1.38285
$629$	−36866.0	−2.33695
$630$	0	0
$631$	20851.5	1.31551	0.657753	−	0.753234i	$-0.271507\pi$
$631$	20851.5	1.31551	0.657753	+	0.753234i	$0.271507\pi$
$632$	10103.4	0.635902
$633$	0	0
$634$	−5182.59	−0.324648
$635$	18735.8	1.17088
$636$	0	0
$637$	−43197.9	−2.68691
$638$	−3483.79	−0.216183
$639$	0	0
$640$	−10083.4	−0.622781
$641$	5938.37	0.365915	0.182957	−	0.983121i	$-0.441433\pi$
$641$	5938.37	0.365915	0.182957	+	0.983121i	$0.441433\pi$
$642$	0	0
$643$	943.102	0.0578418	0.0289209	−	0.999582i	$-0.490793\pi$
$643$	943.102	0.0578418	0.0289209	+	0.999582i	$0.490793\pi$
$644$	−30068.3	−1.83984
$645$	0	0
$646$	−19919.7	−1.21320
$647$	−14236.8	−0.865082	−0.432541	−	0.901614i	$-0.642383\pi$
$647$	−14236.8	−0.865082	−0.432541	+	0.901614i	$0.642383\pi$
$648$	0	0
$649$	1627.47	0.0984340
$650$	−5437.17	−0.328098
$651$	0	0
$652$	12095.1	0.726503
$653$	−8903.59	−0.533575	−0.266788	−	0.963755i	$-0.585962\pi$
$653$	−8903.59	−0.533575	−0.266788	+	0.963755i	$0.585962\pi$
$654$	0	0
$655$	−11831.6	−0.705798
$656$	7044.01	0.419242
$657$	0	0
$658$	11377.7	0.674086
$659$	4548.25	0.268854	0.134427	−	0.990924i	$-0.457081\pi$
$659$	4548.25	0.268854	0.134427	+	0.990924i	$0.457081\pi$
$660$	0	0
$661$	−3357.62	−0.197574	−0.0987870	−	0.995109i	$-0.531496\pi$
$661$	−3357.62	−0.197574	−0.0987870	+	0.995109i	$0.531496\pi$
$662$	5483.50	0.321937
$663$	0	0
$664$	−6518.00	−0.380945
$665$	−33393.4	−1.94728
$666$	0	0
$667$	−13849.1	−0.803955
$668$	−19030.1	−1.10224
$669$	0	0
$670$	−1584.01	−0.0913366
$671$	−425.419	−0.0244756
$672$	0	0
$673$	−14707.2	−0.842382	−0.421191	−	0.906972i	$-0.638388\pi$
$673$	−14707.2	−0.842382	−0.421191	+	0.906972i	$0.638388\pi$
$674$	−2942.85	−0.168182
$675$	0	0
$676$	−5700.80	−0.324351
$677$	7070.52	0.401392	0.200696	−	0.979654i	$-0.435680\pi$
$677$	7070.52	0.401392	0.200696	+	0.979654i	$0.435680\pi$
$678$	0	0
$679$	20928.6	1.18287
$680$	14216.3	0.801722
$681$	0	0
$682$	7933.02	0.445412
$683$	−16587.9	−0.929308	−0.464654	−	0.885492i	$-0.653821\pi$
$683$	−16587.9	−0.929308	−0.464654	+	0.885492i	$0.653821\pi$
$684$	0	0
$685$	4848.56	0.270444
$686$	18797.4	1.04619
$687$	0	0
$688$	−1775.03	−0.0983608
$689$	21205.5	1.17252
$690$	0	0
$691$	−32722.2	−1.80147	−0.900733	−	0.434373i	$-0.856970\pi$
$691$	−32722.2	−1.80147	−0.900733	+	0.434373i	$0.856970\pi$
$692$	−12086.3	−0.663948
$693$	0	0
$694$	−351.239	−0.0192116
$695$	1056.09	0.0576401
$696$	0	0
$697$	−28939.4	−1.57268
$698$	−3405.65	−0.184679
$699$	0	0
$700$	−15890.1	−0.857984
$701$	24968.5	1.34529	0.672645	−	0.739965i	$-0.265158\pi$
$701$	24968.5	1.34529	0.672645	+	0.739965i	$0.265158\pi$
$702$	0	0
$703$	−48108.4	−2.58100
$704$	725.947	0.0388639
$705$	0	0
$706$	−6204.35	−0.330742
$707$	−30191.4	−1.60603
$708$	0	0
$709$	33965.7	1.79916	0.899582	−	0.436752i	$-0.143871\pi$
$709$	33965.7	1.79916	0.899582	+	0.436752i	$0.143871\pi$
$710$	5644.52	0.298359
$711$	0	0
$712$	−26894.9	−1.41563
$713$	31536.0	1.65643
$714$	0	0
$715$	10834.8	0.566709
$716$	9519.66	0.496880
$717$	0	0
$718$	−7368.92	−0.383016
$719$	−14477.7	−0.750941	−0.375470	−	0.926834i	$-0.622519\pi$
$719$	−14477.7	−0.750941	−0.375470	+	0.926834i	$0.622519\pi$
$720$	0	0
$721$	−23415.9	−1.20951
$722$	−17090.1	−0.880925
$723$	0	0
$724$	12442.0	0.638677
$725$	−7318.76	−0.374913
$726$	0	0
$727$	18345.6	0.935900	0.467950	−	0.883755i	$-0.344993\pi$
$727$	18345.6	0.935900	0.467950	+	0.883755i	$0.344993\pi$
$728$	34607.7	1.76188
$729$	0	0
$730$	−1932.13	−0.0979609
$731$	7292.45	0.368975
$732$	0	0
$733$	−24879.5	−1.25368	−0.626839	−	0.779149i	$-0.715652\pi$
$733$	−24879.5	−1.25368	−0.626839	+	0.779149i	$0.715652\pi$
$734$	−5318.34	−0.267443
$735$	0	0
$736$	26039.9	1.30414
$737$	−4770.80	−0.238446
$738$	0	0
$739$	522.214	0.0259945	0.0129973	−	0.999916i	$-0.495863\pi$
$739$	522.214	0.0259945	0.0129973	+	0.999916i	$0.495863\pi$
$740$	15146.1	0.752405
$741$	0	0
$742$	−16539.0	−0.818281
$743$	1698.03	0.0838422	0.0419211	−	0.999121i	$-0.486652\pi$
$743$	1698.03	0.0838422	0.0419211	+	0.999121i	$0.486652\pi$
$744$	0	0
$745$	1565.50	0.0769872
$746$	−15247.0	−0.748299
$747$	0	0
$748$	18888.4	0.923298
$749$	55053.8	2.68574
$750$	0	0
$751$	−21329.8	−1.03640	−0.518199	−	0.855260i	$-0.673397\pi$
$751$	−21329.8	−1.03640	−0.518199	+	0.855260i	$0.673397\pi$
$752$	6915.75	0.335361
$753$	0	0
$754$	7031.64	0.339625
$755$	8906.06	0.429304
$756$	0	0
$757$	−2570.95	−0.123438	−0.0617192	−	0.998094i	$-0.519658\pi$
$757$	−2570.95	−0.123438	−0.0617192	+	0.998094i	$0.519658\pi$
$758$	1264.12	0.0605736
$759$	0	0
$760$	18551.7	0.885447
$761$	16345.4	0.778606	0.389303	−	0.921110i	$-0.372716\pi$
$761$	16345.4	0.778606	0.389303	+	0.921110i	$0.372716\pi$
$762$	0	0
$763$	8480.23	0.402365
$764$	−13467.0	−0.637720
$765$	0	0
$766$	−3921.47	−0.184972
$767$	−3284.86	−0.154641
$768$	0	0
$769$	35378.7	1.65902	0.829512	−	0.558489i	$-0.188619\pi$
$769$	35378.7	1.65902	0.829512	+	0.558489i	$0.188619\pi$
$770$	−8450.42	−0.395496
$771$	0	0
$772$	−8332.13	−0.388446
$773$	8622.56	0.401206	0.200603	−	0.979673i	$-0.435710\pi$
$773$	8622.56	0.401206	0.200603	+	0.979673i	$0.435710\pi$
$774$	0	0
$775$	16665.7	0.772452
$776$	−11626.9	−0.537862
$777$	0	0
$778$	−425.500	−0.0196079
$779$	−37764.6	−1.73691
$780$	0	0
$781$	17000.5	0.778906
$782$	−20038.7	−0.916344
$783$	0	0
$784$	20478.9	0.932895
$785$	24313.7	1.10547
$786$	0	0
$787$	18664.1	0.845366	0.422683	−	0.906278i	$-0.361088\pi$
$787$	18664.1	0.845366	0.422683	+	0.906278i	$0.361088\pi$
$788$	1181.22	0.0534000
$789$	0	0
$790$	4979.39	0.224252
$791$	−31693.8	−1.42465
$792$	0	0
$793$	858.662	0.0384514
$794$	−6590.32	−0.294561
$795$	0	0
$796$	−20995.9	−0.934900
$797$	41335.9	1.83713	0.918564	−	0.395272i	$-0.129350\pi$
$797$	41335.9	1.83713	0.918564	+	0.395272i	$0.129350\pi$
$798$	0	0
$799$	−28412.4	−1.25802
$800$	13761.2	0.608166
$801$	0	0
$802$	8148.88	0.358787
$803$	−5819.31	−0.255740
$804$	0	0
$805$	−33592.9	−1.47080
$806$	−16011.9	−0.699747
$807$	0	0
$808$	16772.8	0.730278
$809$	−19310.4	−0.839206	−0.419603	−	0.907708i	$-0.637831\pi$
$809$	−19310.4	−0.839206	−0.419603	+	0.907708i	$0.637831\pi$
$810$	0	0
$811$	6287.79	0.272249	0.136125	−	0.990692i	$-0.456535\pi$
$811$	6287.79	0.272249	0.136125	+	0.990692i	$0.456535\pi$
$812$	20549.9	0.888129
$813$	0	0
$814$	−12174.2	−0.524207
$815$	13512.8	0.580778
$816$	0	0
$817$	9516.31	0.407508
$818$	8518.93	0.364129
$819$	0	0
$820$	11889.5	0.506339
$821$	10235.6	0.435109	0.217554	−	0.976048i	$-0.430192\pi$
$821$	10235.6	0.435109	0.217554	+	0.976048i	$0.430192\pi$
$822$	0	0
$823$	−7825.03	−0.331426	−0.165713	−	0.986174i	$-0.552992\pi$
$823$	−7825.03	−0.331426	−0.165713	+	0.986174i	$0.552992\pi$
$824$	13008.7	0.549975
$825$	0	0
$826$	2561.98	0.107921
$827$	−15916.1	−0.669233	−0.334617	−	0.942354i	$-0.608607\pi$
$827$	−15916.1	−0.669233	−0.334617	+	0.942354i	$0.608607\pi$
$828$	0	0
$829$	−26623.7	−1.11541	−0.557707	−	0.830038i	$-0.688319\pi$
$829$	−26623.7	−1.11541	−0.557707	+	0.830038i	$0.688319\pi$
$830$	−3212.36	−0.134341
$831$	0	0
$832$	−1465.24	−0.0610555
$833$	−84134.7	−3.49951
$834$	0	0
$835$	−21260.8	−0.881150
$836$	24648.5	1.01972
$837$	0	0
$838$	14180.9	0.584572
$839$	19230.4	0.791307	0.395653	−	0.918400i	$-0.370518\pi$
$839$	19230.4	0.791307	0.395653	+	0.918400i	$0.370518\pi$
$840$	0	0
$841$	−14924.0	−0.611915
$842$	−1361.78	−0.0557363
$843$	0	0
$844$	−30592.5	−1.24767
$845$	−6369.04	−0.259292
$846$	0	0
$847$	19070.1	0.773622
$848$	−10052.9	−0.407098
$849$	0	0
$850$	−10589.7	−0.427324
$851$	−48395.8	−1.94946
$852$	0	0
$853$	21069.1	0.845710	0.422855	−	0.906197i	$-0.361028\pi$
$853$	21069.1	0.845710	0.422855	+	0.906197i	$0.361028\pi$
$854$	−669.701	−0.0268346
$855$	0	0
$856$	−30585.1	−1.22123
$857$	8066.10	0.321508	0.160754	−	0.986994i	$-0.448607\pi$
$857$	8066.10	0.321508	0.160754	+	0.986994i	$0.448607\pi$
$858$	0	0
$859$	−17890.8	−0.710625	−0.355313	−	0.934748i	$-0.615626\pi$
$859$	−17890.8	−0.710625	−0.355313	+	0.934748i	$0.615626\pi$
$860$	−2996.03	−0.118795
$861$	0	0
$862$	−11680.2	−0.461518
$863$	17900.4	0.706069	0.353035	−	0.935610i	$-0.385150\pi$
$863$	17900.4	0.706069	0.353035	+	0.935610i	$0.385150\pi$
$864$	0	0
$865$	−13503.0	−0.530770
$866$	10310.5	0.404579
$867$	0	0
$868$	−46794.7	−1.82986
$869$	14997.2	0.585438
$870$	0	0
$871$	9629.34	0.374601
$872$	−4711.18	−0.182960
$873$	0	0
$874$	−26149.5	−1.01204
$875$	−47251.0	−1.82557
$876$	0	0
$877$	45119.0	1.73724	0.868620	−	0.495478i	$-0.165007\pi$
$877$	45119.0	1.73724	0.868620	+	0.495478i	$0.165007\pi$
$878$	4368.86	0.167929
$879$	0	0
$880$	−5136.45	−0.196761
$881$	43192.5	1.65175	0.825876	−	0.563852i	$-0.190681\pi$
$881$	43192.5	1.65175	0.825876	+	0.563852i	$0.190681\pi$
$882$	0	0
$883$	36353.7	1.38550	0.692751	−	0.721177i	$-0.256399\pi$
$883$	36353.7	1.38550	0.692751	+	0.721177i	$0.256399\pi$
$884$	−38124.1	−1.45051
$885$	0	0
$886$	−5266.43	−0.199694
$887$	4579.23	0.173343	0.0866717	−	0.996237i	$-0.472377\pi$
$887$	4579.23	0.173343	0.0866717	+	0.996237i	$0.472377\pi$
$888$	0	0
$889$	88832.4	3.35134
$890$	−13255.0	−0.499224
$891$	0	0
$892$	26667.3	1.00100
$893$	−37076.9	−1.38940
$894$	0	0
$895$	10635.5	0.397214
$896$	−47808.4	−1.78255
$897$	0	0
$898$	14140.6	0.525476
$899$	−21553.0	−0.799592
$900$	0	0
$901$	41301.1	1.52712
$902$	−9556.58	−0.352771
$903$	0	0
$904$	17607.4	0.647804
$905$	13900.4	0.510569
$906$	0	0
$907$	−47448.3	−1.73704	−0.868520	−	0.495654i	$-0.834929\pi$
$907$	−47448.3	−1.73704	−0.868520	+	0.495654i	$0.834929\pi$
$908$	39150.7	1.43090
$909$	0	0
$910$	17056.3	0.621329
$911$	25809.9	0.938660	0.469330	−	0.883023i	$-0.344495\pi$
$911$	25809.9	0.938660	0.469330	+	0.883023i	$0.344495\pi$
$912$	0	0
$913$	−9675.18	−0.350714
$914$	−12573.6	−0.455029
$915$	0	0
$916$	15801.3	0.569966
$917$	−56097.2	−2.02016
$918$	0	0
$919$	27429.5	0.984565	0.492283	−	0.870435i	$-0.336163\pi$
$919$	27429.5	0.984565	0.492283	+	0.870435i	$0.336163\pi$
$920$	18662.5	0.668787
$921$	0	0
$922$	188.292	0.00672567
$923$	−34313.6	−1.22367
$924$	0	0
$925$	−25575.6	−0.909102
$926$	16248.3	0.576624
$927$	0	0
$928$	−17796.7	−0.629533
$929$	15216.0	0.537375	0.268688	−	0.963227i	$-0.413410\pi$
$929$	15216.0	0.537375	0.268688	+	0.963227i	$0.413410\pi$
$930$	0	0
$931$	−109792.	−3.86497
$932$	−33576.8	−1.18009
$933$	0	0
$934$	3408.32	0.119404
$935$	21102.4	0.738099
$936$	0	0
$937$	44431.6	1.54911	0.774556	−	0.632506i	$-0.217973\pi$
$937$	44431.6	1.54911	0.774556	+	0.632506i	$0.217973\pi$
$938$	−7510.27	−0.261427
$939$	0	0
$940$	11673.0	0.405032
$941$	5816.82	0.201512	0.100756	−	0.994911i	$-0.467874\pi$
$941$	5816.82	0.201512	0.100756	+	0.994911i	$0.467874\pi$
$942$	0	0
$943$	−37990.2	−1.31191
$944$	1557.26	0.0536912
$945$	0	0
$946$	2408.17	0.0827657
$947$	−19125.2	−0.656267	−0.328134	−	0.944631i	$-0.606420\pi$
$947$	−19125.2	−0.656267	−0.328134	+	0.944631i	$0.606420\pi$
$948$	0	0
$949$	11745.6	0.401769
$950$	−13819.1	−0.471950
$951$	0	0
$952$	67404.0	2.29472
$953$	−20408.0	−0.693682	−0.346841	−	0.937924i	$-0.612746\pi$
$953$	−20408.0	−0.693682	−0.346841	+	0.937924i	$0.612746\pi$
$954$	0	0
$955$	−15045.5	−0.509804
$956$	40615.8	1.37407
$957$	0	0
$958$	−21100.2	−0.711605
$959$	22988.5	0.774075
$960$	0	0
$961$	19287.9	0.647440
$962$	24572.2	0.823535
$963$	0	0
$964$	−16742.0	−0.559359
$965$	−9308.81	−0.310530
$966$	0	0
$967$	−16417.9	−0.545981	−0.272991	−	0.962017i	$-0.588013\pi$
$967$	−16417.9	−0.545981	−0.272991	+	0.962017i	$0.588013\pi$
$968$	−10594.4	−0.351774
$969$	0	0
$970$	−5730.25	−0.189678
$971$	567.686	0.0187620	0.00938101	−	0.999956i	$-0.497014\pi$
$971$	567.686	0.0187620	0.00938101	+	0.999956i	$0.497014\pi$
$972$	0	0
$973$	5007.26	0.164980
$974$	1725.25	0.0567561
$975$	0	0
$976$	−407.067	−0.0133503
$977$	−906.220	−0.0296751	−0.0148375	−	0.999890i	$-0.504723\pi$
$977$	−906.220	−0.0296751	−0.0148375	+	0.999890i	$0.504723\pi$
$978$	0	0
$979$	−39922.3	−1.30329
$980$	34566.0	1.12670
$981$	0	0
$982$	18430.6	0.598925
$983$	−54807.1	−1.77831	−0.889153	−	0.457609i	$-0.848706\pi$
$983$	−54807.1	−1.77831	−0.889153	+	0.457609i	$0.848706\pi$
$984$	0	0
$985$	1319.68	0.0426888
$986$	13695.2	0.442338
$987$	0	0
$988$	−49750.2	−1.60199
$989$	9573.16	0.307795
$990$	0	0
$991$	−22970.2	−0.736298	−0.368149	−	0.929767i	$-0.620008\pi$
$991$	−22970.2	−0.736298	−0.368149	+	0.929767i	$0.620008\pi$
$992$	40525.4	1.29706
$993$	0	0
$994$	26762.4	0.853976
$995$	−23457.0	−0.747374
$996$	0	0
$997$	−41803.1	−1.32790	−0.663951	−	0.747776i	$-0.731122\pi$
$997$	−41803.1	−1.32790	−0.663951	+	0.747776i	$0.731122\pi$
$998$	9792.16	0.310587
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.d.1.3		7
3.2	odd	2		177.4.a.a.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.a.1.5	✓	7	3.2	odd	2
531.4.a.d.1.3		7	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-1.29817 q^{2} -6.31476 q^{4} -7.05496 q^{5} -33.4497 q^{7} +18.5830 q^{8} +O(q^{10})\)	\(q-1.29817 q^{2} -6.31476 q^{4} -7.05496 q^{5} -33.4497 q^{7} +18.5830 q^{8} +9.15853 q^{10} +27.5842 q^{11} -55.6756 q^{13} +43.4234 q^{14} +26.3942 q^{16} -108.437 q^{17} -141.505 q^{19} +44.5504 q^{20} -35.8089 q^{22} -142.351 q^{23} -75.2276 q^{25} +72.2763 q^{26} +211.227 q^{28} +97.2883 q^{29} -221.538 q^{31} -182.928 q^{32} +140.770 q^{34} +235.987 q^{35} +339.976 q^{37} +183.698 q^{38} -131.102 q^{40} +266.877 q^{41} -67.2505 q^{43} -174.187 q^{44} +184.795 q^{46} +262.017 q^{47} +775.886 q^{49} +97.6581 q^{50} +351.578 q^{52} -380.876 q^{53} -194.605 q^{55} -621.596 q^{56} -126.297 q^{58} +59.0000 q^{59} -15.4226 q^{61} +287.593 q^{62} +26.3175 q^{64} +392.789 q^{65} -172.954 q^{67} +684.753 q^{68} -306.350 q^{70} +616.313 q^{71} -210.965 q^{73} -441.346 q^{74} +893.572 q^{76} -922.684 q^{77} +543.689 q^{79} -186.210 q^{80} -346.452 q^{82} -350.751 q^{83} +765.019 q^{85} +87.3025 q^{86} +512.596 q^{88} -1447.29 q^{89} +1862.34 q^{91} +898.911 q^{92} -340.143 q^{94} +998.315 q^{95} -625.674 q^{97} -1007.23 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8}+O(q^{10}) \) 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8	\( 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8} - 79 q^{10} + 131 q^{11} - 123 q^{13} + 117 q^{14} + 202 q^{16} + 235 q^{17} - 80 q^{19} - 61 q^{20} + 688 q^{22} + 274 q^{23} + 193 q^{25} + 180 q^{26} - 118 q^{28} + 406 q^{29} - 346 q^{31} + 854 q^{32} + 178 q^{34} + 424 q^{35} - 157 q^{37} + 129 q^{38} - 590 q^{40} + 825 q^{41} - 815 q^{43} + 1690 q^{44} + 1457 q^{46} + 1196 q^{47} + 914 q^{49} - 713 q^{50} + 1030 q^{52} + 900 q^{53} - 1044 q^{55} - 2172 q^{56} + 1242 q^{58} + 413 q^{59} + 420 q^{61} - 646 q^{62} + 3541 q^{64} - 190 q^{65} + 1316 q^{67} + 611 q^{68} + 4658 q^{70} + 173 q^{71} - 418 q^{73} - 660 q^{74} + 1540 q^{76} + 753 q^{77} + 2635 q^{79} - 6155 q^{80} - 125 q^{82} - 457 q^{83} + 1270 q^{85} - 3482 q^{86} + 7685 q^{88} - 592 q^{89} + 3179 q^{91} + 3500 q^{92} + 2064 q^{94} + 2250 q^{95} - 1906 q^{97} - 2994 q^{98}+O(q^{100}) \) 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8 - 79 * q^10 + 131 * q^11 - 123 * q^13 + 117 * q^14 + 202 * q^16 + 235 * q^17 - 80 * q^19 - 61 * q^20 + 688 * q^22 + 274 * q^23 + 193 * q^25 + 180 * q^26 - 118 * q^28 + 406 * q^29 - 346 * q^31 + 854 * q^32 + 178 * q^34 + 424 * q^35 - 157 * q^37 + 129 * q^38 - 590 * q^40 + 825 * q^41 - 815 * q^43 + 1690 * q^44 + 1457 * q^46 + 1196 * q^47 + 914 * q^49 - 713 * q^50 + 1030 * q^52 + 900 * q^53 - 1044 * q^55 - 2172 * q^56 + 1242 * q^58 + 413 * q^59 + 420 * q^61 - 646 * q^62 + 3541 * q^64 - 190 * q^65 + 1316 * q^67 + 611 * q^68 + 4658 * q^70 + 173 * q^71 - 418 * q^73 - 660 * q^74 + 1540 * q^76 + 753 * q^77 + 2635 * q^79 - 6155 * q^80 - 125 * q^82 - 457 * q^83 + 1270 * q^85 - 3482 * q^86 + 7685 * q^88 - 592 * q^89 + 3179 * q^91 + 3500 * q^92 + 2064 * q^94 + 2250 * q^95 - 1906 * q^97 - 2994 * q^98

Introduction

L-functions

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