LMFDB - Embedded newform 531.4.a.f.1.8

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 49x^{6} + 89x^{5} + 648x^{4} - 1023x^{3} - 1476x^{2} + 1940x - 384 $ x^8 - 2x^7 - 49x^6 + 89x^5 + 648x^4 - 1023x^3 - 1476x^2 + 1940*x - 384
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-5.19624$ 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)

$f(q)$	$=$	$q+5.19624 q^{2} +19.0009 q^{4} +8.30102 q^{5} +21.5539 q^{7} +57.1634 q^{8} +O(q^{10})$	\(q+5.19624 q^{2} +19.0009 q^{4} +8.30102 q^{5} +21.5539 q^{7} +57.1634 q^{8} +43.1341 q^{10} +28.3746 q^{11} -28.2089 q^{13} +111.999 q^{14} +145.027 q^{16} -21.8166 q^{17} -122.388 q^{19} +157.727 q^{20} +147.441 q^{22} -82.1571 q^{23} -56.0930 q^{25} -146.580 q^{26} +409.544 q^{28} -86.9209 q^{29} +131.178 q^{31} +296.290 q^{32} -113.364 q^{34} +178.919 q^{35} +280.467 q^{37} -635.957 q^{38} +474.515 q^{40} -381.788 q^{41} +452.501 q^{43} +539.144 q^{44} -426.908 q^{46} +158.067 q^{47} +121.570 q^{49} -291.473 q^{50} -535.994 q^{52} +162.922 q^{53} +235.539 q^{55} +1232.09 q^{56} -451.662 q^{58} +59.0000 q^{59} -368.344 q^{61} +681.632 q^{62} +379.374 q^{64} -234.162 q^{65} +177.950 q^{67} -414.536 q^{68} +929.708 q^{70} -58.9220 q^{71} +880.299 q^{73} +1457.37 q^{74} -2325.48 q^{76} +611.584 q^{77} -825.284 q^{79} +1203.88 q^{80} -1983.86 q^{82} -1426.68 q^{83} -181.100 q^{85} +2351.30 q^{86} +1621.99 q^{88} -1556.78 q^{89} -608.011 q^{91} -1561.06 q^{92} +821.356 q^{94} -1015.94 q^{95} +1811.25 q^{97} +631.708 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8	$ 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8} + 29 q^{10} + 27 q^{11} + 89 q^{13} + 37 q^{14} + 362 q^{16} - 79 q^{17} + 288 q^{19} - 457 q^{20} + 596 q^{22} - 202 q^{23} + 264 q^{25} - 270 q^{26} + 702 q^{28} + 114 q^{29} + 538 q^{31} - 316 q^{32} + 498 q^{34} + 196 q^{35} + 395 q^{37} - 397 q^{38} + 918 q^{40} + 39 q^{41} + 527 q^{43} - 64 q^{44} - 539 q^{46} - 860 q^{47} + 347 q^{49} + 591 q^{50} - 644 q^{52} + 812 q^{53} + 536 q^{55} + 2218 q^{56} - 1154 q^{58} + 472 q^{59} - 460 q^{61} + 2014 q^{62} - 451 q^{64} + 986 q^{65} + 1934 q^{67} + 69 q^{68} - 1028 q^{70} + 1687 q^{71} + 1980 q^{73} + 2400 q^{74} - 940 q^{76} + 821 q^{77} + 3319 q^{79} + 2119 q^{80} + 429 q^{82} - 2057 q^{83} + 566 q^{85} + 6690 q^{86} + 1189 q^{88} - 1668 q^{89} + 2427 q^{91} + 980 q^{92} + 332 q^{94} - 2146 q^{95} + 1956 q^{97} + 2026 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8 + 29 * q^10 + 27 * q^11 + 89 * q^13 + 37 * q^14 + 362 * q^16 - 79 * q^17 + 288 * q^19 - 457 * q^20 + 596 * q^22 - 202 * q^23 + 264 * q^25 - 270 * q^26 + 702 * q^28 + 114 * q^29 + 538 * q^31 - 316 * q^32 + 498 * q^34 + 196 * q^35 + 395 * q^37 - 397 * q^38 + 918 * q^40 + 39 * q^41 + 527 * q^43 - 64 * q^44 - 539 * q^46 - 860 * q^47 + 347 * q^49 + 591 * q^50 - 644 * q^52 + 812 * q^53 + 536 * q^55 + 2218 * q^56 - 1154 * q^58 + 472 * q^59 - 460 * q^61 + 2014 * q^62 - 451 * q^64 + 986 * q^65 + 1934 * q^67 + 69 * q^68 - 1028 * q^70 + 1687 * q^71 + 1980 * q^73 + 2400 * q^74 - 940 * q^76 + 821 * q^77 + 3319 * q^79 + 2119 * q^80 + 429 * q^82 - 2057 * q^83 + 566 * q^85 + 6690 * q^86 + 1189 * q^88 - 1668 * q^89 + 2427 * q^91 + 980 * q^92 + 332 * q^94 - 2146 * q^95 + 1956 * q^97 + 2026 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	5.19624	1.83715	0.918574	−	0.395249i	$-0.129342\pi$
$2$	5.19624	1.83715	0.918574	+	0.395249i	$0.129342\pi$
$3$	0	0
$3$	0	0
$4$	19.0009	2.37511
$5$	8.30102	0.742466	0.371233	−	0.928540i	$-0.378935\pi$
$5$	8.30102	0.742466	0.371233	+	0.928540i	$0.378935\pi$
$6$	0	0
$7$	21.5539	1.16380	0.581900	−	0.813260i	$-0.302309\pi$
$7$	21.5539	1.16380	0.581900	+	0.813260i	$0.302309\pi$
$8$	57.1634	2.52629
$9$	0	0
$10$	43.1341	1.36402
$11$	28.3746	0.777752	0.388876	−	0.921290i	$-0.372863\pi$
$11$	28.3746	0.777752	0.388876	+	0.921290i	$0.372863\pi$
$12$	0	0
$13$	−28.2089	−0.601826	−0.300913	−	0.953652i	$-0.597291\pi$
$13$	−28.2089	−0.601826	−0.300913	+	0.953652i	$0.597291\pi$
$14$	111.999	2.13807
$15$	0	0
$16$	145.027	2.26605
$17$	−21.8166	−0.311254	−0.155627	−	0.987816i	$-0.549740\pi$
$17$	−21.8166	−0.311254	−0.155627	+	0.987816i	$0.549740\pi$
$18$	0	0
$19$	−122.388	−1.47777	−0.738887	−	0.673830i	$-0.764648\pi$
$19$	−122.388	−1.47777	−0.738887	+	0.673830i	$0.764648\pi$
$20$	157.727	1.76344
$21$	0	0
$22$	147.441	1.42885
$23$	−82.1571	−0.744823	−0.372412	−	0.928068i	$-0.621469\pi$
$23$	−82.1571	−0.744823	−0.372412	+	0.928068i	$0.621469\pi$
$24$	0	0
$25$	−56.0930	−0.448744
$26$	−146.580	−1.10564
$27$	0	0
$28$	409.544	2.76416
$29$	−86.9209	−0.556580	−0.278290	−	0.960497i	$-0.589768\pi$
$29$	−86.9209	−0.556580	−0.278290	+	0.960497i	$0.589768\pi$
$30$	0	0
$31$	131.178	0.760009	0.380004	−	0.924985i	$-0.375922\pi$
$31$	131.178	0.760009	0.380004	+	0.924985i	$0.375922\pi$
$32$	296.290	1.63678
$33$	0	0
$34$	−113.364	−0.571819
$35$	178.919	0.864083
$36$	0	0
$37$	280.467	1.24617	0.623087	−	0.782152i	$-0.285878\pi$
$37$	280.467	1.24617	0.623087	+	0.782152i	$0.285878\pi$
$38$	−635.957	−2.71489
$39$	0	0
$40$	474.515	1.87568
$41$	−381.788	−1.45427	−0.727137	−	0.686493i	$-0.759149\pi$
$41$	−381.788	−1.45427	−0.727137	+	0.686493i	$0.759149\pi$
$42$	0	0
$43$	452.501	1.60478	0.802392	−	0.596798i	$-0.203560\pi$
$43$	452.501	1.60478	0.802392	+	0.596798i	$0.203560\pi$
$44$	539.144	1.84725
$45$	0	0
$46$	−426.908	−1.36835
$47$	158.067	0.490564	0.245282	−	0.969452i	$-0.421119\pi$
$47$	158.067	0.490564	0.245282	+	0.969452i	$0.421119\pi$
$48$	0	0
$49$	121.570	0.354432
$50$	−291.473	−0.824409
$51$	0	0
$52$	−535.994	−1.42940
$53$	162.922	0.422245	0.211123	−	0.977460i	$-0.432288\pi$
$53$	162.922	0.422245	0.211123	+	0.977460i	$0.432288\pi$
$54$	0	0
$55$	235.539	0.577455
$56$	1232.09	2.94010
$57$	0	0
$58$	−451.662	−1.02252
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	0	0
$61$	−368.344	−0.773141	−0.386570	−	0.922260i	$-0.626340\pi$
$61$	−368.344	−0.773141	−0.386570	+	0.922260i	$0.626340\pi$
$62$	681.632	1.39625
$63$	0	0
$64$	379.374	0.740965
$65$	−234.162	−0.446835
$66$	0	0
$67$	177.950	0.324478	0.162239	−	0.986752i	$-0.448128\pi$
$67$	177.950	0.324478	0.162239	+	0.986752i	$0.448128\pi$
$68$	−414.536	−0.739263
$69$	0	0
$70$	929.708	1.58745
$71$	−58.9220	−0.0984894	−0.0492447	−	0.998787i	$-0.515681\pi$
$71$	−58.9220	−0.0984894	−0.0492447	+	0.998787i	$0.515681\pi$
$72$	0	0
$73$	880.299	1.41139	0.705693	−	0.708518i	$-0.250636\pi$
$73$	880.299	1.41139	0.705693	+	0.708518i	$0.250636\pi$
$74$	1457.37	2.28941
$75$	0	0
$76$	−2325.48	−3.50988
$77$	611.584	0.905148
$78$	0	0
$79$	−825.284	−1.17534	−0.587669	−	0.809101i	$-0.699954\pi$
$79$	−825.284	−1.17534	−0.587669	+	0.809101i	$0.699954\pi$
$80$	1203.88	1.68247
$81$	0	0
$82$	−1983.86	−2.67172
$83$	−1426.68	−1.88673	−0.943367	−	0.331752i	$-0.892360\pi$
$83$	−1426.68	−1.88673	−0.943367	+	0.331752i	$0.892360\pi$
$84$	0	0
$85$	−181.100	−0.231095
$86$	2351.30	2.94823
$87$	0	0
$88$	1621.99	1.96483
$89$	−1556.78	−1.85413	−0.927067	−	0.374896i	$-0.877678\pi$
$89$	−1556.78	−1.85413	−0.927067	+	0.374896i	$0.877678\pi$
$90$	0	0
$91$	−608.011	−0.700405
$92$	−1561.06	−1.76904
$93$	0	0
$94$	821.356	0.901238
$95$	−1015.94	−1.09720
$96$	0	0
$97$	1811.25	1.89593	0.947964	−	0.318377i	$-0.103138\pi$
$97$	1811.25	1.89593	0.947964	+	0.318377i	$0.103138\pi$
$98$	631.708	0.651144
$99$	0	0
$100$	−1065.82	−1.06582
$101$	−1132.95	−1.11617	−0.558085	−	0.829784i	$-0.688464\pi$
$101$	−1132.95	−1.11617	−0.558085	+	0.829784i	$0.688464\pi$
$102$	0	0
$103$	1479.79	1.41561	0.707805	−	0.706408i	$-0.249685\pi$
$103$	1479.79	1.41561	0.707805	+	0.706408i	$0.249685\pi$
$104$	−1612.51	−1.52038
$105$	0	0
$106$	846.580	0.775727
$107$	−293.850	−0.265491	−0.132746	−	0.991150i	$-0.542379\pi$
$107$	−293.850	−0.265491	−0.132746	+	0.991150i	$0.542379\pi$
$108$	0	0
$109$	−801.451	−0.704267	−0.352133	−	0.935950i	$-0.614544\pi$
$109$	−801.451	−0.704267	−0.352133	+	0.935950i	$0.614544\pi$
$110$	1223.91	1.06087
$111$	0	0
$112$	3125.90	2.63723
$113$	1834.04	1.52683	0.763415	−	0.645908i	$-0.223521\pi$
$113$	1834.04	1.52683	0.763415	+	0.645908i	$0.223521\pi$
$114$	0	0
$115$	−681.988	−0.553006
$116$	−1651.58	−1.32194
$117$	0	0
$118$	306.578	0.239176
$119$	−470.233	−0.362237
$120$	0	0
$121$	−525.880	−0.395102
$122$	−1914.00	−1.42037
$123$	0	0
$124$	2492.50	1.80511
$125$	−1503.26	−1.07564
$126$	0	0
$127$	745.001	0.520537	0.260268	−	0.965536i	$-0.416189\pi$
$127$	745.001	0.520537	0.260268	+	0.965536i	$0.416189\pi$
$128$	−398.999	−0.275522
$129$	0	0
$130$	−1216.76	−0.820902
$131$	2205.52	1.47097	0.735485	−	0.677541i	$-0.236954\pi$
$131$	2205.52	1.47097	0.735485	+	0.677541i	$0.236954\pi$
$132$	0	0
$133$	−2637.93	−1.71983
$134$	924.669	0.596114
$135$	0	0
$136$	−1247.11	−0.786316
$137$	−744.233	−0.464118	−0.232059	−	0.972702i	$-0.574546\pi$
$137$	−744.233	−0.464118	−0.232059	+	0.972702i	$0.574546\pi$
$138$	0	0
$139$	267.712	0.163360	0.0816801	−	0.996659i	$-0.473971\pi$
$139$	267.712	0.163360	0.0816801	+	0.996659i	$0.473971\pi$
$140$	3399.63	2.05229
$141$	0	0
$142$	−306.173	−0.180940
$143$	−800.416	−0.468071
$144$	0	0
$145$	−721.533	−0.413242
$146$	4574.24	2.59292
$147$	0	0
$148$	5329.13	2.95981
$149$	2478.30	1.36262	0.681308	−	0.731997i	$-0.261411\pi$
$149$	2478.30	1.36262	0.681308	+	0.731997i	$0.261411\pi$
$150$	0	0
$151$	1902.48	1.02531	0.512655	−	0.858595i	$-0.328662\pi$
$151$	1902.48	1.02531	0.512655	+	0.858595i	$0.328662\pi$
$152$	−6996.10	−3.73328
$153$	0	0
$154$	3177.94	1.66289
$155$	1088.91	0.564281
$156$	0	0
$157$	−3638.32	−1.84949	−0.924745	−	0.380588i	$-0.875722\pi$
$157$	−3638.32	−1.84949	−0.924745	+	0.380588i	$0.875722\pi$
$158$	−4288.37	−2.15927
$159$	0	0
$160$	2459.51	1.21526
$161$	−1770.81	−0.866826
$162$	0	0
$163$	1188.42	0.571068	0.285534	−	0.958369i	$-0.407829\pi$
$163$	1188.42	0.571068	0.285534	+	0.958369i	$0.407829\pi$
$164$	−7254.31	−3.45407
$165$	0	0
$166$	−7413.39	−3.46621
$167$	−1667.69	−0.772752	−0.386376	−	0.922341i	$-0.626273\pi$
$167$	−1667.69	−0.772752	−0.386376	+	0.922341i	$0.626273\pi$
$168$	0	0
$169$	−1401.26	−0.637806
$170$	−941.041	−0.424556
$171$	0	0
$172$	8597.92	3.81154
$173$	−1854.57	−0.815032	−0.407516	−	0.913198i	$-0.633605\pi$
$173$	−1854.57	−0.815032	−0.407516	+	0.913198i	$0.633605\pi$
$174$	0	0
$175$	−1209.02	−0.522249
$176$	4115.10	1.76243
$177$	0	0
$178$	−8089.38	−3.40632
$179$	−3650.51	−1.52431	−0.762155	−	0.647394i	$-0.775859\pi$
$179$	−3650.51	−1.52431	−0.762155	+	0.647394i	$0.775859\pi$
$180$	0	0
$181$	912.195	0.374602	0.187301	−	0.982303i	$-0.440026\pi$
$181$	912.195	0.374602	0.187301	+	0.982303i	$0.440026\pi$
$182$	−3159.37	−1.28675
$183$	0	0
$184$	−4696.38	−1.88164
$185$	2328.16	0.925243
$186$	0	0
$187$	−619.039	−0.242078
$188$	3003.42	1.16514
$189$	0	0
$190$	−5279.09	−2.01571
$191$	−2060.65	−0.780646	−0.390323	−	0.920678i	$-0.627637\pi$
$191$	−2060.65	−0.780646	−0.390323	+	0.920678i	$0.627637\pi$
$192$	0	0
$193$	−1917.94	−0.715318	−0.357659	−	0.933852i	$-0.616425\pi$
$193$	−1917.94	−0.715318	−0.357659	+	0.933852i	$0.616425\pi$
$194$	9411.71	3.48310
$195$	0	0
$196$	2309.94	0.841817
$197$	2490.32	0.900651	0.450325	−	0.892864i	$-0.351308\pi$
$197$	2490.32	0.900651	0.450325	+	0.892864i	$0.351308\pi$
$198$	0	0
$199$	3412.12	1.21547	0.607735	−	0.794140i	$-0.292078\pi$
$199$	3412.12	1.21547	0.607735	+	0.794140i	$0.292078\pi$
$200$	−3206.46	−1.13366
$201$	0	0
$202$	−5887.10	−2.05057
$203$	−1873.48	−0.647748
$204$	0	0
$205$	−3169.23	−1.07975
$206$	7689.34	2.60069
$207$	0	0
$208$	−4091.06	−1.36377
$209$	−3472.71	−1.14934
$210$	0	0
$211$	892.077	0.291057	0.145529	−	0.989354i	$-0.453512\pi$
$211$	892.077	0.291057	0.145529	+	0.989354i	$0.453512\pi$
$212$	3095.66	1.00288
$213$	0	0
$214$	−1526.92	−0.487747
$215$	3756.22	1.19150
$216$	0	0
$217$	2827.40	0.884499
$218$	−4164.53	−1.29384
$219$	0	0
$220$	4475.45	1.37152
$221$	615.423	0.187320
$222$	0	0
$223$	−2891.35	−0.868246	−0.434123	−	0.900854i	$-0.642942\pi$
$223$	−2891.35	−0.868246	−0.434123	+	0.900854i	$0.642942\pi$
$224$	6386.20	1.90489
$225$	0	0
$226$	9530.10	2.80501
$227$	1746.45	0.510643	0.255321	−	0.966856i	$-0.417819\pi$
$227$	1746.45	0.510643	0.255321	+	0.966856i	$0.417819\pi$
$228$	0	0
$229$	−1195.00	−0.344837	−0.172418	−	0.985024i	$-0.555158\pi$
$229$	−1195.00	−0.344837	−0.172418	+	0.985024i	$0.555158\pi$
$230$	−3543.77	−1.01595
$231$	0	0
$232$	−4968.69	−1.40608
$233$	4741.93	1.33328	0.666640	−	0.745380i	$-0.267732\pi$
$233$	4741.93	1.33328	0.666640	+	0.745380i	$0.267732\pi$
$234$	0	0
$235$	1312.12	0.364227
$236$	1121.05	0.309213
$237$	0	0
$238$	−2443.45	−0.665483
$239$	2889.11	0.781928	0.390964	−	0.920406i	$-0.372142\pi$
$239$	2889.11	0.781928	0.390964	+	0.920406i	$0.372142\pi$
$240$	0	0
$241$	2648.11	0.707799	0.353899	−	0.935284i	$-0.384856\pi$
$241$	2648.11	0.707799	0.353899	+	0.935284i	$0.384856\pi$
$242$	−2732.60	−0.725860
$243$	0	0
$244$	−6998.87	−1.83630
$245$	1009.16	0.263154
$246$	0	0
$247$	3452.42	0.889362
$248$	7498.58	1.92000
$249$	0	0
$250$	−7811.29	−1.97612
$251$	−2327.49	−0.585299	−0.292650	−	0.956220i	$-0.594537\pi$
$251$	−2327.49	−0.585299	−0.292650	+	0.956220i	$0.594537\pi$
$252$	0	0
$253$	−2331.18	−0.579288
$254$	3871.21	0.956303
$255$	0	0
$256$	−5108.29	−1.24714
$257$	4224.97	1.02547	0.512737	−	0.858546i	$-0.328632\pi$
$257$	4224.97	1.02547	0.512737	+	0.858546i	$0.328632\pi$
$258$	0	0
$259$	6045.15	1.45030
$260$	−4449.30	−1.06128
$261$	0	0
$262$	11460.4	2.70239
$263$	−3245.27	−0.760881	−0.380441	−	0.924805i	$-0.624228\pi$
$263$	−3245.27	−0.760881	−0.380441	+	0.924805i	$0.624228\pi$
$264$	0	0
$265$	1352.42	0.313503
$266$	−13707.3	−3.15959
$267$	0	0
$268$	3381.21	0.770672
$269$	4695.98	1.06438	0.532191	−	0.846624i	$-0.321369\pi$
$269$	4695.98	1.06438	0.532191	+	0.846624i	$0.321369\pi$
$270$	0	0
$271$	1442.96	0.323444	0.161722	−	0.986836i	$-0.448295\pi$
$271$	1442.96	0.323444	0.161722	+	0.986836i	$0.448295\pi$
$272$	−3164.01	−0.705317
$273$	0	0
$274$	−3867.21	−0.852653
$275$	−1591.62	−0.349012
$276$	0	0
$277$	6781.78	1.47104	0.735519	−	0.677504i	$-0.236938\pi$
$277$	6781.78	1.47104	0.735519	+	0.677504i	$0.236938\pi$
$278$	1391.10	0.300117
$279$	0	0
$280$	10227.6	2.18292
$281$	−916.383	−0.194544	−0.0972719	−	0.995258i	$-0.531012\pi$
$281$	−916.383	−0.194544	−0.0972719	+	0.995258i	$0.531012\pi$
$282$	0	0
$283$	−1693.18	−0.355651	−0.177825	−	0.984062i	$-0.556906\pi$
$283$	−1693.18	−0.355651	−0.177825	+	0.984062i	$0.556906\pi$
$284$	−1119.57	−0.233924
$285$	0	0
$286$	−4159.15	−0.859916
$287$	−8229.01	−1.69248
$288$	0	0
$289$	−4437.03	−0.903121
$290$	−3749.26	−0.759186
$291$	0	0
$292$	16726.5	3.35220
$293$	4712.00	0.939515	0.469757	−	0.882796i	$-0.344341\pi$
$293$	4712.00	0.939515	0.469757	+	0.882796i	$0.344341\pi$
$294$	0	0
$295$	489.760	0.0966609
$296$	16032.4	3.14820
$297$	0	0
$298$	12877.8	2.50333
$299$	2317.56	0.448254
$300$	0	0
$301$	9753.15	1.86765
$302$	9885.75	1.88364
$303$	0	0
$304$	−17749.6	−3.34871
$305$	−3057.63	−0.574031
$306$	0	0
$307$	7720.48	1.43528	0.717640	−	0.696414i	$-0.245222\pi$
$307$	7720.48	1.43528	0.717640	+	0.696414i	$0.245222\pi$
$308$	11620.6	2.14983
$309$	0	0
$310$	5658.25	1.03667
$311$	−5773.53	−1.05269	−0.526345	−	0.850271i	$-0.676438\pi$
$311$	−5773.53	−1.05269	−0.526345	+	0.850271i	$0.676438\pi$
$312$	0	0
$313$	−9875.96	−1.78346	−0.891729	−	0.452570i	$-0.850507\pi$
$313$	−9875.96	−1.78346	−0.891729	+	0.452570i	$0.850507\pi$
$314$	−18905.6	−3.39779
$315$	0	0
$316$	−15681.1	−2.79156
$317$	6005.53	1.06405	0.532025	−	0.846729i	$-0.321431\pi$
$317$	6005.53	1.06405	0.532025	+	0.846729i	$0.321431\pi$
$318$	0	0
$319$	−2466.35	−0.432881
$320$	3149.19	0.550141
$321$	0	0
$322$	−9201.53	−1.59249
$323$	2670.09	0.459962
$324$	0	0
$325$	1582.32	0.270066
$326$	6175.31	1.04914
$327$	0	0
$328$	−21824.3	−3.67391
$329$	3406.97	0.570918
$330$	0	0
$331$	6895.07	1.14498	0.572488	−	0.819913i	$-0.305978\pi$
$331$	6895.07	1.14498	0.572488	+	0.819913i	$0.305978\pi$
$332$	−27108.3	−4.48121
$333$	0	0
$334$	−8665.71	−1.41966
$335$	1477.16	0.240914
$336$	0	0
$337$	6764.00	1.09335	0.546675	−	0.837345i	$-0.315893\pi$
$337$	6764.00	1.09335	0.546675	+	0.837345i	$0.315893\pi$
$338$	−7281.28	−1.17174
$339$	0	0
$340$	−3441.07	−0.548878
$341$	3722.13	0.591098
$342$	0	0
$343$	−4772.67	−0.751312
$344$	25866.5	4.05415
$345$	0	0
$346$	−9636.81	−1.49734
$347$	−5474.73	−0.846971	−0.423485	−	0.905903i	$-0.639193\pi$
$347$	−5474.73	−0.846971	−0.423485	+	0.905903i	$0.639193\pi$
$348$	0	0
$349$	−6375.89	−0.977919	−0.488960	−	0.872306i	$-0.662623\pi$
$349$	−6375.89	−0.977919	−0.488960	+	0.872306i	$0.662623\pi$
$350$	−6282.37	−0.959448
$351$	0	0
$352$	8407.11	1.27301
$353$	−4971.14	−0.749539	−0.374770	−	0.927118i	$-0.622278\pi$
$353$	−4971.14	−0.749539	−0.374770	+	0.927118i	$0.622278\pi$
$354$	0	0
$355$	−489.113	−0.0731251
$356$	−29580.2	−4.40378
$357$	0	0
$358$	−18968.9	−2.80039
$359$	−2073.46	−0.304827	−0.152414	−	0.988317i	$-0.548705\pi$
$359$	−2073.46	−0.304827	−0.152414	+	0.988317i	$0.548705\pi$
$360$	0	0
$361$	8119.79	1.18382
$362$	4739.98	0.688199
$363$	0	0
$364$	−11552.8	−1.66354
$365$	7307.38	1.04791
$366$	0	0
$367$	−6485.59	−0.922466	−0.461233	−	0.887279i	$-0.652593\pi$
$367$	−6485.59	−0.922466	−0.461233	+	0.887279i	$0.652593\pi$
$368$	−11915.0	−1.68781
$369$	0	0
$370$	12097.7	1.69981
$371$	3511.59	0.491409
$372$	0	0
$373$	−2610.25	−0.362342	−0.181171	−	0.983452i	$-0.557989\pi$
$373$	−2610.25	−0.362342	−0.181171	+	0.983452i	$0.557989\pi$
$374$	−3216.67	−0.444733
$375$	0	0
$376$	9035.66	1.23931
$377$	2451.94	0.334964
$378$	0	0
$379$	−5609.34	−0.760244	−0.380122	−	0.924936i	$-0.624118\pi$
$379$	−5609.34	−0.760244	−0.380122	+	0.924936i	$0.624118\pi$
$380$	−19303.9	−2.60597
$381$	0	0
$382$	−10707.6	−1.43416
$383$	−2978.80	−0.397414	−0.198707	−	0.980059i	$-0.563674\pi$
$383$	−2978.80	−0.397414	−0.198707	+	0.980059i	$0.563674\pi$
$384$	0	0
$385$	5076.77	0.672042
$386$	−9966.08	−1.31415
$387$	0	0
$388$	34415.5	4.50305
$389$	−10137.6	−1.32132	−0.660662	−	0.750683i	$-0.729724\pi$
$389$	−10137.6	−1.32132	−0.660662	+	0.750683i	$0.729724\pi$
$390$	0	0
$391$	1792.39	0.231829
$392$	6949.36	0.895397
$393$	0	0
$394$	12940.3	1.65463
$395$	−6850.70	−0.872649
$396$	0	0
$397$	12921.9	1.63359	0.816793	−	0.576930i	$-0.195750\pi$
$397$	12921.9	1.63359	0.816793	+	0.576930i	$0.195750\pi$
$398$	17730.2	2.23300
$399$	0	0
$400$	−8135.02	−1.01688
$401$	9556.68	1.19012	0.595060	−	0.803681i	$-0.297128\pi$
$401$	9556.68	1.19012	0.595060	+	0.803681i	$0.297128\pi$
$402$	0	0
$403$	−3700.38	−0.457393
$404$	−21527.1	−2.65103
$405$	0	0
$406$	−9735.07	−1.19001
$407$	7958.14	0.969215
$408$	0	0
$409$	8381.64	1.01331	0.506657	−	0.862148i	$-0.330881\pi$
$409$	8381.64	1.01331	0.506657	+	0.862148i	$0.330881\pi$
$410$	−16468.1	−1.98366
$411$	0	0
$412$	28117.3	3.36224
$413$	1271.68	0.151514
$414$	0	0
$415$	−11842.9	−1.40084
$416$	−8358.00	−0.985059
$417$	0	0
$418$	−18045.0	−2.11151
$419$	9005.40	1.04998	0.524991	−	0.851108i	$-0.324069\pi$
$419$	9005.40	1.04998	0.524991	+	0.851108i	$0.324069\pi$
$420$	0	0
$421$	754.089	0.0872970	0.0436485	−	0.999047i	$-0.486102\pi$
$421$	754.089	0.0872970	0.0436485	+	0.999047i	$0.486102\pi$
$422$	4635.45	0.534716
$423$	0	0
$424$	9313.15	1.06671
$425$	1223.76	0.139673
$426$	0	0
$427$	−7939.24	−0.899782
$428$	−5583.42	−0.630572
$429$	0	0
$430$	19518.2	2.18896
$431$	2505.79	0.280046	0.140023	−	0.990148i	$-0.455282\pi$
$431$	2505.79	0.280046	0.140023	+	0.990148i	$0.455282\pi$
$432$	0	0
$433$	−176.502	−0.0195893	−0.00979465	−	0.999952i	$-0.503118\pi$
$433$	−176.502	−0.0195893	−0.00979465	+	0.999952i	$0.503118\pi$
$434$	14691.8	1.62496
$435$	0	0
$436$	−15228.3	−1.67271
$437$	10055.0	1.10068
$438$	0	0
$439$	−2223.93	−0.241782	−0.120891	−	0.992666i	$-0.538575\pi$
$439$	−2223.93	−0.241782	−0.120891	+	0.992666i	$0.538575\pi$
$440$	13464.2	1.45882
$441$	0	0
$442$	3197.88	0.344135
$443$	6065.11	0.650479	0.325239	−	0.945632i	$-0.394555\pi$
$443$	6065.11	0.650479	0.325239	+	0.945632i	$0.394555\pi$
$444$	0	0
$445$	−12922.8	−1.37663
$446$	−15024.1	−1.59510
$447$	0	0
$448$	8176.99	0.862336
$449$	127.588	0.0134104	0.00670518	−	0.999978i	$-0.497866\pi$
$449$	127.588	0.0134104	0.00670518	+	0.999978i	$0.497866\pi$
$450$	0	0
$451$	−10833.1	−1.13106
$452$	34848.4	3.62640
$453$	0	0
$454$	9074.97	0.938126
$455$	−5047.11	−0.520027
$456$	0	0
$457$	15246.9	1.56065	0.780327	−	0.625372i	$-0.215053\pi$
$457$	15246.9	1.56065	0.780327	+	0.625372i	$0.215053\pi$
$458$	−6209.49	−0.633517
$459$	0	0
$460$	−12958.4	−1.31345
$461$	−8148.20	−0.823210	−0.411605	−	0.911362i	$-0.635032\pi$
$461$	−8148.20	−0.823210	−0.411605	+	0.911362i	$0.635032\pi$
$462$	0	0
$463$	6518.90	0.654340	0.327170	−	0.944966i	$-0.393905\pi$
$463$	6518.90	0.654340	0.327170	+	0.944966i	$0.393905\pi$
$464$	−12605.9	−1.26124
$465$	0	0
$466$	24640.2	2.44943
$467$	263.751	0.0261348	0.0130674	−	0.999915i	$-0.495840\pi$
$467$	263.751	0.0261348	0.0130674	+	0.999915i	$0.495840\pi$
$468$	0	0
$469$	3835.51	0.377628
$470$	6818.10	0.669139
$471$	0	0
$472$	3372.64	0.328895
$473$	12839.5	1.24812
$474$	0	0
$475$	6865.10	0.663142
$476$	−8934.86	−0.860354
$477$	0	0
$478$	15012.5	1.43652
$479$	−3204.17	−0.305642	−0.152821	−	0.988254i	$-0.548836\pi$
$479$	−3204.17	−0.305642	−0.152821	+	0.988254i	$0.548836\pi$
$480$	0	0
$481$	−7911.65	−0.749980
$482$	13760.2	1.30033
$483$	0	0
$484$	−9992.20	−0.938411
$485$	15035.3	1.40766
$486$	0	0
$487$	5211.90	0.484957	0.242478	−	0.970157i	$-0.422040\pi$
$487$	5211.90	0.484957	0.242478	+	0.970157i	$0.422040\pi$
$488$	−21055.8	−1.95318
$489$	0	0
$490$	5243.82	0.483453
$491$	10189.9	0.936589	0.468295	−	0.883572i	$-0.344869\pi$
$491$	10189.9	0.936589	0.468295	+	0.883572i	$0.344869\pi$
$492$	0	0
$493$	1896.32	0.173237
$494$	17939.6	1.63389
$495$	0	0
$496$	19024.4	1.72222
$497$	−1270.00	−0.114622
$498$	0	0
$499$	−20602.0	−1.84824	−0.924120	−	0.382103i	$-0.875200\pi$
$499$	−20602.0	−1.84824	−0.924120	+	0.382103i	$0.875200\pi$
$500$	−28563.3	−2.55478
$501$	0	0
$502$	−12094.2	−1.07528
$503$	8471.66	0.750959	0.375480	−	0.926831i	$-0.377478\pi$
$503$	8471.66	0.750959	0.375480	+	0.926831i	$0.377478\pi$
$504$	0	0
$505$	−9404.67	−0.828718
$506$	−12113.4	−1.06424
$507$	0	0
$508$	14155.7	1.23633
$509$	−11199.1	−0.975230	−0.487615	−	0.873059i	$-0.662133\pi$
$509$	−11199.1	−0.975230	−0.487615	+	0.873059i	$0.662133\pi$
$510$	0	0
$511$	18973.9	1.64257
$512$	−23351.9	−2.01566
$513$	0	0
$514$	21954.0	1.88395
$515$	12283.8	1.05104
$516$	0	0
$517$	4485.10	0.381537
$518$	31412.1	2.66441
$519$	0	0
$520$	−13385.5	−1.12883
$521$	3200.10	0.269096	0.134548	−	0.990907i	$-0.457042\pi$
$521$	3200.10	0.269096	0.134548	+	0.990907i	$0.457042\pi$
$522$	0	0
$523$	16516.2	1.38089	0.690443	−	0.723386i	$-0.257415\pi$
$523$	16516.2	1.38089	0.690443	+	0.723386i	$0.257415\pi$
$524$	41906.9	3.49372
$525$	0	0
$526$	−16863.2	−1.39785
$527$	−2861.86	−0.236555
$528$	0	0
$529$	−5417.21	−0.445238
$530$	7027.48	0.575951
$531$	0	0
$532$	−50123.2	−4.08480
$533$	10769.8	0.875219
$534$	0	0
$535$	−2439.26	−0.197118
$536$	10172.2	0.819724
$537$	0	0
$538$	24401.4	1.95543
$539$	3449.51	0.275660
$540$	0	0
$541$	20852.2	1.65713	0.828566	−	0.559892i	$-0.189157\pi$
$541$	20852.2	1.65713	0.828566	+	0.559892i	$0.189157\pi$
$542$	7497.95	0.594215
$543$	0	0
$544$	−6464.04	−0.509455
$545$	−6652.86	−0.522894
$546$	0	0
$547$	809.628	0.0632856	0.0316428	−	0.999499i	$-0.489926\pi$
$547$	809.628	0.0632856	0.0316428	+	0.999499i	$0.489926\pi$
$548$	−14141.1	−1.10233
$549$	0	0
$550$	−8270.43	−0.641186
$551$	10638.1	0.822499
$552$	0	0
$553$	−17788.1	−1.36786
$554$	35239.7	2.70251
$555$	0	0
$556$	5086.78	0.387999
$557$	21643.2	1.64641	0.823206	−	0.567743i	$-0.192183\pi$
$557$	21643.2	1.64641	0.823206	+	0.567743i	$0.192183\pi$
$558$	0	0
$559$	−12764.5	−0.965800
$560$	25948.2	1.95806
$561$	0	0
$562$	−4761.74	−0.357406
$563$	−23646.1	−1.77010	−0.885050	−	0.465496i	$-0.845876\pi$
$563$	−23646.1	−1.77010	−0.885050	+	0.465496i	$0.845876\pi$
$564$	0	0
$565$	15224.4	1.13362
$566$	−8798.17	−0.653383
$567$	0	0
$568$	−3368.18	−0.248813
$569$	19407.6	1.42990	0.714948	−	0.699177i	$-0.246450\pi$
$569$	19407.6	1.42990	0.714948	+	0.699177i	$0.246450\pi$
$570$	0	0
$571$	16659.0	1.22094	0.610470	−	0.792039i	$-0.290980\pi$
$571$	16659.0	1.22094	0.610470	+	0.792039i	$0.290980\pi$
$572$	−15208.6	−1.11172
$573$	0	0
$574$	−42759.9	−3.10935
$575$	4608.44	0.334235
$576$	0	0
$577$	−19454.1	−1.40361	−0.701805	−	0.712369i	$-0.747622\pi$
$577$	−19454.1	−1.40361	−0.701805	+	0.712369i	$0.747622\pi$
$578$	−23055.9	−1.65917
$579$	0	0
$580$	−13709.8	−0.981496
$581$	−30750.6	−2.19578
$582$	0	0
$583$	4622.84	0.328402
$584$	50320.8	3.56557
$585$	0	0
$586$	24484.7	1.72603
$587$	2742.52	0.192838	0.0964190	−	0.995341i	$-0.469261\pi$
$587$	2742.52	0.192838	0.0964190	+	0.995341i	$0.469261\pi$
$588$	0	0
$589$	−16054.6	−1.12312
$590$	2544.91	0.177580
$591$	0	0
$592$	40675.4	2.82390
$593$	−9829.13	−0.680664	−0.340332	−	0.940305i	$-0.610540\pi$
$593$	−9829.13	−0.680664	−0.340332	+	0.940305i	$0.610540\pi$
$594$	0	0
$595$	−3903.42	−0.268949
$596$	47089.9	3.23637
$597$	0	0
$598$	12042.6	0.823509
$599$	−25261.4	−1.72312	−0.861562	−	0.507653i	$-0.830513\pi$
$599$	−25261.4	−1.72312	−0.861562	+	0.507653i	$0.830513\pi$
$600$	0	0
$601$	23269.7	1.57935	0.789677	−	0.613523i	$-0.210248\pi$
$601$	23269.7	1.57935	0.789677	+	0.613523i	$0.210248\pi$
$602$	50679.7	3.43115
$603$	0	0
$604$	36148.9	2.43523
$605$	−4365.35	−0.293350
$606$	0	0
$607$	11725.1	0.784029	0.392015	−	0.919959i	$-0.371778\pi$
$607$	11725.1	0.784029	0.392015	+	0.919959i	$0.371778\pi$
$608$	−36262.3	−2.41880
$609$	0	0
$610$	−15888.2	−1.05458
$611$	−4458.90	−0.295234
$612$	0	0
$613$	−4075.52	−0.268530	−0.134265	−	0.990945i	$-0.542867\pi$
$613$	−4075.52	−0.268530	−0.134265	+	0.990945i	$0.542867\pi$
$614$	40117.5	2.63682
$615$	0	0
$616$	34960.2	2.28667
$617$	2326.85	0.151824	0.0759121	−	0.997115i	$-0.475813\pi$
$617$	2326.85	0.151824	0.0759121	+	0.997115i	$0.475813\pi$
$618$	0	0
$619$	27579.3	1.79080	0.895400	−	0.445263i	$-0.146890\pi$
$619$	27579.3	1.79080	0.895400	+	0.445263i	$0.146890\pi$
$620$	20690.3	1.34023
$621$	0	0
$622$	−30000.6	−1.93395
$623$	−33554.6	−2.15784
$624$	0	0
$625$	−5466.95	−0.349885
$626$	−51317.8	−3.27648
$627$	0	0
$628$	−69131.5	−4.39275
$629$	−6118.84	−0.387876
$630$	0	0
$631$	−4818.78	−0.304013	−0.152007	−	0.988379i	$-0.548574\pi$
$631$	−4818.78	−0.304013	−0.152007	+	0.988379i	$0.548574\pi$
$632$	−47176.0	−2.96924
$633$	0	0
$634$	31206.2	1.95482
$635$	6184.27	0.386481
$636$	0	0
$637$	−3429.36	−0.213306
$638$	−12815.7	−0.795266
$639$	0	0
$640$	−3312.10	−0.204566
$641$	16896.8	1.04116	0.520580	−	0.853813i	$-0.325716\pi$
$641$	16896.8	1.04116	0.520580	+	0.853813i	$0.325716\pi$
$642$	0	0
$643$	−30887.2	−1.89436	−0.947180	−	0.320702i	$-0.896081\pi$
$643$	−30887.2	−1.89436	−0.947180	+	0.320702i	$0.896081\pi$
$644$	−33646.9	−2.05881
$645$	0	0
$646$	13874.4	0.845019
$647$	4782.21	0.290585	0.145292	−	0.989389i	$-0.453588\pi$
$647$	4782.21	0.290585	0.145292	+	0.989389i	$0.453588\pi$
$648$	0	0
$649$	1674.10	0.101255
$650$	8222.11	0.496150
$651$	0	0
$652$	22581.0	1.35635
$653$	33078.5	1.98233	0.991164	−	0.132639i	$-0.0423451\pi$
$653$	33078.5	1.98233	0.991164	+	0.132639i	$0.0423451\pi$
$654$	0	0
$655$	18308.1	1.09215
$656$	−55369.6	−3.29546
$657$	0	0
$658$	17703.4	1.04886
$659$	−10359.2	−0.612351	−0.306175	−	0.951975i	$-0.599049\pi$
$659$	−10359.2	−0.612351	−0.306175	+	0.951975i	$0.599049\pi$
$660$	0	0
$661$	−14606.6	−0.859502	−0.429751	−	0.902947i	$-0.641399\pi$
$661$	−14606.6	−0.859502	−0.429751	+	0.902947i	$0.641399\pi$
$662$	35828.4	2.10349
$663$	0	0
$664$	−81554.0	−4.76643
$665$	−21897.6	−1.27692
$666$	0	0
$667$	7141.17	0.414553
$668$	−31687.6	−1.83538
$669$	0	0
$670$	7675.70	0.442594
$671$	−10451.6	−0.601312
$672$	0	0
$673$	4901.10	0.280719	0.140359	−	0.990101i	$-0.455174\pi$
$673$	4901.10	0.280719	0.140359	+	0.990101i	$0.455174\pi$
$674$	35147.4	2.00865
$675$	0	0
$676$	−26625.2	−1.51486
$677$	7783.32	0.441857	0.220929	−	0.975290i	$-0.429091\pi$
$677$	7783.32	0.441857	0.220929	+	0.975290i	$0.429091\pi$
$678$	0	0
$679$	39039.6	2.20648
$680$	−10352.3	−0.583813
$681$	0	0
$682$	19341.1	1.08594
$683$	5065.20	0.283770	0.141885	−	0.989883i	$-0.454684\pi$
$683$	5065.20	0.283770	0.141885	+	0.989883i	$0.454684\pi$
$684$	0	0
$685$	−6177.89	−0.344592
$686$	−24800.0	−1.38027
$687$	0	0
$688$	65624.9	3.63652
$689$	−4595.83	−0.254118
$690$	0	0
$691$	−987.814	−0.0543824	−0.0271912	−	0.999630i	$-0.508656\pi$
$691$	−987.814	−0.0543824	−0.0271912	+	0.999630i	$0.508656\pi$
$692$	−35238.6	−1.93579
$693$	0	0
$694$	−28448.0	−1.55601
$695$	2222.29	0.121289
$696$	0	0
$697$	8329.32	0.452648
$698$	−33130.7	−1.79658
$699$	0	0
$700$	−22972.5	−1.24040
$701$	−24301.9	−1.30937	−0.654685	−	0.755902i	$-0.727199\pi$
$701$	−24301.9	−1.30937	−0.654685	+	0.755902i	$0.727199\pi$
$702$	0	0
$703$	−34325.7	−1.84156
$704$	10764.6	0.576287
$705$	0	0
$706$	−25831.3	−1.37701
$707$	−24419.6	−1.29900
$708$	0	0
$709$	19625.0	1.03954	0.519769	−	0.854307i	$-0.326018\pi$
$709$	19625.0	1.03954	0.519769	+	0.854307i	$0.326018\pi$
$710$	−2541.55	−0.134342
$711$	0	0
$712$	−88990.5	−4.68408
$713$	−10777.2	−0.566072
$714$	0	0
$715$	−6644.27	−0.347527
$716$	−69362.9	−3.62041
$717$	0	0
$718$	−10774.2	−0.560012
$719$	12274.5	0.636663	0.318331	−	0.947979i	$-0.396877\pi$
$719$	12274.5	0.636663	0.318331	+	0.947979i	$0.396877\pi$
$720$	0	0
$721$	31895.2	1.64749
$722$	42192.4	2.17484
$723$	0	0
$724$	17332.5	0.889722
$725$	4875.65	0.249762
$726$	0	0
$727$	9818.35	0.500884	0.250442	−	0.968132i	$-0.419424\pi$
$727$	9818.35	0.500884	0.250442	+	0.968132i	$0.419424\pi$
$728$	−34755.9	−1.76942
$729$	0	0
$730$	37970.9	1.92516
$731$	−9872.04	−0.499495
$732$	0	0
$733$	−36133.7	−1.82078	−0.910388	−	0.413756i	$-0.864217\pi$
$733$	−36133.7	−1.82078	−0.910388	+	0.413756i	$0.864217\pi$
$734$	−33700.7	−1.69471
$735$	0	0
$736$	−24342.3	−1.21912
$737$	5049.26	0.252363
$738$	0	0
$739$	15833.8	0.788166	0.394083	−	0.919075i	$-0.371062\pi$
$739$	15833.8	0.788166	0.394083	+	0.919075i	$0.371062\pi$
$740$	44237.2	2.19756
$741$	0	0
$742$	18247.1	0.902792
$743$	−23888.6	−1.17953	−0.589764	−	0.807576i	$-0.700779\pi$
$743$	−23888.6	−1.17953	−0.589764	+	0.807576i	$0.700779\pi$
$744$	0	0
$745$	20572.4	1.01170
$746$	−13563.5	−0.665676
$747$	0	0
$748$	−11762.3	−0.574963
$749$	−6333.62	−0.308979
$750$	0	0
$751$	−29824.3	−1.44914	−0.724571	−	0.689200i	$-0.757962\pi$
$751$	−29824.3	−1.44914	−0.724571	+	0.689200i	$0.757962\pi$
$752$	22924.1	1.11164
$753$	0	0
$754$	12740.9	0.615378
$755$	15792.5	0.761257
$756$	0	0
$757$	7136.54	0.342645	0.171322	−	0.985215i	$-0.445196\pi$
$757$	7136.54	0.342645	0.171322	+	0.985215i	$0.445196\pi$
$758$	−29147.5	−1.39668
$759$	0	0
$760$	−58074.8	−2.77184
$761$	19820.8	0.944157	0.472079	−	0.881557i	$-0.343504\pi$
$761$	19820.8	0.944157	0.472079	+	0.881557i	$0.343504\pi$
$762$	0	0
$763$	−17274.4	−0.819626
$764$	−39154.2	−1.85412
$765$	0	0
$766$	−15478.5	−0.730108
$767$	−1664.32	−0.0783510
$768$	0	0
$769$	23969.6	1.12401	0.562006	−	0.827133i	$-0.310030\pi$
$769$	23969.6	1.12401	0.562006	+	0.827133i	$0.310030\pi$
$770$	26380.1	1.23464
$771$	0	0
$772$	−36442.6	−1.69896
$773$	−41618.3	−1.93649	−0.968245	−	0.250004i	$-0.919568\pi$
$773$	−41618.3	−1.93649	−0.968245	+	0.250004i	$0.919568\pi$
$774$	0	0
$775$	−7358.17	−0.341049
$776$	103537.	4.78966
$777$	0	0
$778$	−52677.2	−2.42747
$779$	46726.2	2.14909
$780$	0	0
$781$	−1671.89	−0.0766004
$782$	9313.70	0.425904
$783$	0	0
$784$	17631.0	0.803161
$785$	−30201.8	−1.37318
$786$	0	0
$787$	17130.9	0.775920	0.387960	−	0.921676i	$-0.373180\pi$
$787$	17130.9	0.775920	0.387960	+	0.921676i	$0.373180\pi$
$788$	47318.4	2.13915
$789$	0	0
$790$	−35597.9	−1.60319
$791$	39530.7	1.77693
$792$	0	0
$793$	10390.6	0.465296
$794$	67145.5	3.00114
$795$	0	0
$796$	64833.3	2.88688
$797$	3545.01	0.157554	0.0787770	−	0.996892i	$-0.474898\pi$
$797$	3545.01	0.157554	0.0787770	+	0.996892i	$0.474898\pi$
$798$	0	0
$799$	−3448.50	−0.152690
$800$	−16619.8	−0.734497
$801$	0	0
$802$	49658.8	2.18643
$803$	24978.1	1.09771
$804$	0	0
$805$	−14699.5	−0.643589
$806$	−19228.1	−0.840298
$807$	0	0
$808$	−64763.4	−2.81976
$809$	11108.5	0.482763	0.241381	−	0.970430i	$-0.422400\pi$
$809$	11108.5	0.482763	0.241381	+	0.970430i	$0.422400\pi$
$810$	0	0
$811$	−11416.7	−0.494320	−0.247160	−	0.968975i	$-0.579497\pi$
$811$	−11416.7	−0.494320	−0.247160	+	0.968975i	$0.579497\pi$
$812$	−35597.9	−1.53847
$813$	0	0
$814$	41352.4	1.78059
$815$	9865.09	0.423999
$816$	0	0
$817$	−55380.6	−2.37151
$818$	43553.0	1.86161
$819$	0	0
$820$	−60218.2	−2.56453
$821$	−27265.7	−1.15905	−0.579524	−	0.814955i	$-0.696762\pi$
$821$	−27265.7	−1.15905	−0.579524	+	0.814955i	$0.696762\pi$
$822$	0	0
$823$	18672.1	0.790849	0.395424	−	0.918499i	$-0.370598\pi$
$823$	18672.1	0.790849	0.395424	+	0.918499i	$0.370598\pi$
$824$	84589.7	3.57624
$825$	0	0
$826$	6607.95	0.278354
$827$	−33292.9	−1.39989	−0.699945	−	0.714197i	$-0.746792\pi$
$827$	−33292.9	−1.39989	−0.699945	+	0.714197i	$0.746792\pi$
$828$	0	0
$829$	21571.9	0.903768	0.451884	−	0.892077i	$-0.350752\pi$
$829$	21571.9	0.903768	0.451884	+	0.892077i	$0.350752\pi$
$830$	−61538.7	−2.57354
$831$	0	0
$832$	−10701.7	−0.445932
$833$	−2652.25	−0.110318
$834$	0	0
$835$	−13843.5	−0.573743
$836$	−65984.7	−2.72982
$837$	0	0
$838$	46794.2	1.92897
$839$	−24952.4	−1.02676	−0.513381	−	0.858161i	$-0.671607\pi$
$839$	−24952.4	−1.02676	−0.513381	+	0.858161i	$0.671607\pi$
$840$	0	0
$841$	−16833.8	−0.690219
$842$	3918.43	0.160378
$843$	0	0
$844$	16950.3	0.691294
$845$	−11631.9	−0.473549
$846$	0	0
$847$	−11334.8	−0.459820
$848$	23628.1	0.956830
$849$	0	0
$850$	6358.95	0.256600
$851$	−23042.3	−0.928180
$852$	0	0
$853$	22193.3	0.890836	0.445418	−	0.895323i	$-0.353055\pi$
$853$	22193.3	0.890836	0.445418	+	0.895323i	$0.353055\pi$
$854$	−41254.2	−1.65303
$855$	0	0
$856$	−16797.5	−0.670708
$857$	−5155.43	−0.205491	−0.102746	−	0.994708i	$-0.532763\pi$
$857$	−5155.43	−0.205491	−0.102746	+	0.994708i	$0.532763\pi$
$858$	0	0
$859$	−18059.6	−0.717327	−0.358664	−	0.933467i	$-0.616767\pi$
$859$	−18059.6	−0.717327	−0.358664	+	0.933467i	$0.616767\pi$
$860$	71371.6	2.82994
$861$	0	0
$862$	13020.7	0.514486
$863$	3135.68	0.123685	0.0618423	−	0.998086i	$-0.480302\pi$
$863$	3135.68	0.123685	0.0618423	+	0.998086i	$0.480302\pi$
$864$	0	0
$865$	−15394.9	−0.605134
$866$	−917.149	−0.0359884
$867$	0	0
$868$	53723.1	2.10078
$869$	−23417.1	−0.914122
$870$	0	0
$871$	−5019.76	−0.195279
$872$	−45813.6	−1.77918
$873$	0	0
$874$	52248.4	2.02211
$875$	−32401.0	−1.25183
$876$	0	0
$877$	24764.4	0.953516	0.476758	−	0.879035i	$-0.341812\pi$
$877$	24764.4	0.953516	0.476758	+	0.879035i	$0.341812\pi$
$878$	−11556.1	−0.444190
$879$	0	0
$880$	34159.5	1.30854
$881$	24195.2	0.925265	0.462633	−	0.886550i	$-0.346905\pi$
$881$	24195.2	0.925265	0.462633	+	0.886550i	$0.346905\pi$
$882$	0	0
$883$	−26387.8	−1.00569	−0.502843	−	0.864378i	$-0.667713\pi$
$883$	−26387.8	−1.00569	−0.502843	+	0.864378i	$0.667713\pi$
$884$	11693.6	0.444907
$885$	0	0
$886$	31515.8	1.19503
$887$	−45619.8	−1.72690	−0.863452	−	0.504431i	$-0.831702\pi$
$887$	−45619.8	−1.72690	−0.863452	+	0.504431i	$0.831702\pi$
$888$	0	0
$889$	16057.7	0.605801
$890$	−67150.1	−2.52908
$891$	0	0
$892$	−54938.2	−2.06218
$893$	−19345.5	−0.724942
$894$	0	0
$895$	−30302.9	−1.13175
$896$	−8599.98	−0.320653
$897$	0	0
$898$	662.978	0.0246368
$899$	−11402.1	−0.423005
$900$	0	0
$901$	−3554.40	−0.131425
$902$	−56291.3	−2.07793
$903$	0	0
$904$	104840.	3.85721
$905$	7572.15	0.278129
$906$	0	0
$907$	−28604.5	−1.04719	−0.523593	−	0.851968i	$-0.675409\pi$
$907$	−28604.5	−1.04719	−0.523593	+	0.851968i	$0.675409\pi$
$908$	33184.1	1.21283
$909$	0	0
$910$	−26226.0	−0.955367
$911$	−23089.0	−0.839706	−0.419853	−	0.907592i	$-0.637918\pi$
$911$	−23089.0	−0.839706	−0.419853	+	0.907592i	$0.637918\pi$
$912$	0	0
$913$	−40481.6	−1.46741
$914$	79226.4	2.86715
$915$	0	0
$916$	−22706.0	−0.819027
$917$	47537.5	1.71192
$918$	0	0
$919$	−14323.8	−0.514145	−0.257072	−	0.966392i	$-0.582758\pi$
$919$	−14323.8	−0.514145	−0.257072	+	0.966392i	$0.582758\pi$
$920$	−38984.7	−1.39705
$921$	0	0
$922$	−42340.0	−1.51236
$923$	1662.12	0.0592735
$924$	0	0
$925$	−15732.2	−0.559213
$926$	33873.8	1.20212
$927$	0	0
$928$	−25753.8	−0.911001
$929$	34587.5	1.22151	0.610753	−	0.791821i	$-0.290867\pi$
$929$	34587.5	1.22151	0.610753	+	0.791821i	$0.290867\pi$
$930$	0	0
$931$	−14878.7	−0.523770
$932$	90101.0	3.16669
$933$	0	0
$934$	1370.51	0.0480134
$935$	−5138.66	−0.179735
$936$	0	0
$937$	30905.7	1.07753	0.538765	−	0.842456i	$-0.318891\pi$
$937$	30905.7	1.07753	0.538765	+	0.842456i	$0.318891\pi$
$938$	19930.2	0.693758
$939$	0	0
$940$	24931.5	0.865081
$941$	20269.1	0.702184	0.351092	−	0.936341i	$-0.385811\pi$
$941$	20269.1	0.702184	0.351092	+	0.936341i	$0.385811\pi$
$942$	0	0
$943$	31366.6	1.08318
$944$	8556.61	0.295015
$945$	0	0
$946$	66717.3	2.29299
$947$	−3806.38	−0.130613	−0.0653066	−	0.997865i	$-0.520803\pi$
$947$	−3806.38	−0.130613	−0.0653066	+	0.997865i	$0.520803\pi$
$948$	0	0
$949$	−24832.2	−0.849408
$950$	35672.7	1.21829
$951$	0	0
$952$	−26880.1	−0.915115
$953$	58087.5	1.97444	0.987219	−	0.159370i	$-0.0509464\pi$
$953$	58087.5	1.97444	0.987219	+	0.159370i	$0.0509464\pi$
$954$	0	0
$955$	−17105.5	−0.579603
$956$	54895.6	1.85717
$957$	0	0
$958$	−16649.6	−0.561509
$959$	−16041.1	−0.540140
$960$	0	0
$961$	−12583.3	−0.422387
$962$	−41110.8	−1.37782
$963$	0	0
$964$	50316.4	1.68110
$965$	−15920.9	−0.531100
$966$	0	0
$967$	−9873.60	−0.328349	−0.164175	−	0.986431i	$-0.552496\pi$
$967$	−9873.60	−0.328349	−0.164175	+	0.986431i	$0.552496\pi$
$968$	−30061.1	−0.998141
$969$	0	0
$970$	78126.9	2.58609
$971$	−55181.6	−1.82375	−0.911875	−	0.410468i	$-0.865366\pi$
$971$	−55181.6	−1.82375	−0.911875	+	0.410468i	$0.865366\pi$
$972$	0	0
$973$	5770.24	0.190119
$974$	27082.3	0.890937
$975$	0	0
$976$	−53419.9	−1.75198
$977$	−14494.0	−0.474620	−0.237310	−	0.971434i	$-0.576266\pi$
$977$	−14494.0	−0.474620	−0.237310	+	0.971434i	$0.576266\pi$
$978$	0	0
$979$	−44172.9	−1.44206
$980$	19174.9	0.625020
$981$	0	0
$982$	52949.4	1.72065
$983$	38986.8	1.26499	0.632495	−	0.774564i	$-0.282031\pi$
$983$	38986.8	1.26499	0.632495	+	0.774564i	$0.282031\pi$
$984$	0	0
$985$	20672.2	0.668703
$986$	9853.74	0.318263
$987$	0	0
$988$	65599.2	2.11234
$989$	−37176.1	−1.19528
$990$	0	0
$991$	−48817.5	−1.56482	−0.782412	−	0.622762i	$-0.786011\pi$
$991$	−48817.5	−1.56482	−0.782412	+	0.622762i	$0.786011\pi$
$992$	38866.7	1.24397
$993$	0	0
$994$	−6599.21	−0.210578
$995$	28324.1	0.902445
$996$	0	0
$997$	−14978.6	−0.475806	−0.237903	−	0.971289i	$-0.576460\pi$
$997$	−14978.6	−0.475806	−0.237903	+	0.971289i	$0.576460\pi$
$998$	−107053.	−3.39549
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.f.1.8		8
3.2	odd	2		177.4.a.c.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.c.1.1	✓	8	3.2	odd	2
531.4.a.f.1.8		8	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)

\(f(q)\)	\(=\)	\(q+5.19624 q^{2} +19.0009 q^{4} +8.30102 q^{5} +21.5539 q^{7} +57.1634 q^{8} +O(q^{10})\)	\(q+5.19624 q^{2} +19.0009 q^{4} +8.30102 q^{5} +21.5539 q^{7} +57.1634 q^{8} +43.1341 q^{10} +28.3746 q^{11} -28.2089 q^{13} +111.999 q^{14} +145.027 q^{16} -21.8166 q^{17} -122.388 q^{19} +157.727 q^{20} +147.441 q^{22} -82.1571 q^{23} -56.0930 q^{25} -146.580 q^{26} +409.544 q^{28} -86.9209 q^{29} +131.178 q^{31} +296.290 q^{32} -113.364 q^{34} +178.919 q^{35} +280.467 q^{37} -635.957 q^{38} +474.515 q^{40} -381.788 q^{41} +452.501 q^{43} +539.144 q^{44} -426.908 q^{46} +158.067 q^{47} +121.570 q^{49} -291.473 q^{50} -535.994 q^{52} +162.922 q^{53} +235.539 q^{55} +1232.09 q^{56} -451.662 q^{58} +59.0000 q^{59} -368.344 q^{61} +681.632 q^{62} +379.374 q^{64} -234.162 q^{65} +177.950 q^{67} -414.536 q^{68} +929.708 q^{70} -58.9220 q^{71} +880.299 q^{73} +1457.37 q^{74} -2325.48 q^{76} +611.584 q^{77} -825.284 q^{79} +1203.88 q^{80} -1983.86 q^{82} -1426.68 q^{83} -181.100 q^{85} +2351.30 q^{86} +1621.99 q^{88} -1556.78 q^{89} -608.011 q^{91} -1561.06 q^{92} +821.356 q^{94} -1015.94 q^{95} +1811.25 q^{97} +631.708 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8	\( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8} + 29 q^{10} + 27 q^{11} + 89 q^{13} + 37 q^{14} + 362 q^{16} - 79 q^{17} + 288 q^{19} - 457 q^{20} + 596 q^{22} - 202 q^{23} + 264 q^{25} - 270 q^{26} + 702 q^{28} + 114 q^{29} + 538 q^{31} - 316 q^{32} + 498 q^{34} + 196 q^{35} + 395 q^{37} - 397 q^{38} + 918 q^{40} + 39 q^{41} + 527 q^{43} - 64 q^{44} - 539 q^{46} - 860 q^{47} + 347 q^{49} + 591 q^{50} - 644 q^{52} + 812 q^{53} + 536 q^{55} + 2218 q^{56} - 1154 q^{58} + 472 q^{59} - 460 q^{61} + 2014 q^{62} - 451 q^{64} + 986 q^{65} + 1934 q^{67} + 69 q^{68} - 1028 q^{70} + 1687 q^{71} + 1980 q^{73} + 2400 q^{74} - 940 q^{76} + 821 q^{77} + 3319 q^{79} + 2119 q^{80} + 429 q^{82} - 2057 q^{83} + 566 q^{85} + 6690 q^{86} + 1189 q^{88} - 1668 q^{89} + 2427 q^{91} + 980 q^{92} + 332 q^{94} - 2146 q^{95} + 1956 q^{97} + 2026 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8 + 29 * q^10 + 27 * q^11 + 89 * q^13 + 37 * q^14 + 362 * q^16 - 79 * q^17 + 288 * q^19 - 457 * q^20 + 596 * q^22 - 202 * q^23 + 264 * q^25 - 270 * q^26 + 702 * q^28 + 114 * q^29 + 538 * q^31 - 316 * q^32 + 498 * q^34 + 196 * q^35 + 395 * q^37 - 397 * q^38 + 918 * q^40 + 39 * q^41 + 527 * q^43 - 64 * q^44 - 539 * q^46 - 860 * q^47 + 347 * q^49 + 591 * q^50 - 644 * q^52 + 812 * q^53 + 536 * q^55 + 2218 * q^56 - 1154 * q^58 + 472 * q^59 - 460 * q^61 + 2014 * q^62 - 451 * q^64 + 986 * q^65 + 1934 * q^67 + 69 * q^68 - 1028 * q^70 + 1687 * q^71 + 1980 * q^73 + 2400 * q^74 - 940 * q^76 + 821 * q^77 + 3319 * q^79 + 2119 * q^80 + 429 * q^82 - 2057 * q^83 + 566 * q^85 + 6690 * q^86 + 1189 * q^88 - 1668 * q^89 + 2427 * q^91 + 980 * q^92 + 332 * q^94 - 2146 * q^95 + 1956 * q^97 + 2026 * q^98

Introduction

L-functions

Modular forms

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