LMFDB - Embedded newform 539.4.a.o.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,4,Mod(1,539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, 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("539.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	539.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:	$31.8020294931$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 156x^{8} + 5810x^{6} - 52148x^{4} + 20168x^{2} - 1584 $ x^10 - 156x^8 + 5810x^6 - 52148x^4 + 20168x^2 - 1584
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$6.10533$ of defining polynomial
Character	$\chi$	$=$	539.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.15800 q^{2} -5.25424 q^{3} +18.6049 q^{4} -13.3903 q^{5} -27.1014 q^{6} +54.7001 q^{8} +0.607090 q^{9} +O(q^{10})$	\(q+5.15800 q^{2} -5.25424 q^{3} +18.6049 q^{4} -13.3903 q^{5} -27.1014 q^{6} +54.7001 q^{8} +0.607090 q^{9} -69.0669 q^{10} +11.0000 q^{11} -97.7548 q^{12} +11.3828 q^{13} +70.3557 q^{15} +133.304 q^{16} +106.434 q^{17} +3.13137 q^{18} +112.382 q^{19} -249.125 q^{20} +56.7380 q^{22} +190.167 q^{23} -287.408 q^{24} +54.2992 q^{25} +58.7125 q^{26} +138.675 q^{27} -114.376 q^{29} +362.895 q^{30} +212.830 q^{31} +249.979 q^{32} -57.7967 q^{33} +548.987 q^{34} +11.2949 q^{36} +91.9763 q^{37} +579.667 q^{38} -59.8081 q^{39} -732.449 q^{40} -429.485 q^{41} -262.479 q^{43} +204.654 q^{44} -8.12910 q^{45} +980.882 q^{46} -287.593 q^{47} -700.410 q^{48} +280.075 q^{50} -559.231 q^{51} +211.776 q^{52} +217.762 q^{53} +715.284 q^{54} -147.293 q^{55} -590.484 q^{57} -589.951 q^{58} +10.5891 q^{59} +1308.96 q^{60} +712.550 q^{61} +1097.78 q^{62} +222.959 q^{64} -152.419 q^{65} -298.115 q^{66} +1081.11 q^{67} +1980.20 q^{68} -999.185 q^{69} +538.346 q^{71} +33.2079 q^{72} -302.995 q^{73} +474.413 q^{74} -285.301 q^{75} +2090.86 q^{76} -308.490 q^{78} -30.2060 q^{79} -1784.97 q^{80} -745.023 q^{81} -2215.28 q^{82} -149.101 q^{83} -1425.18 q^{85} -1353.87 q^{86} +600.959 q^{87} +601.701 q^{88} +1055.28 q^{89} -41.9299 q^{90} +3538.05 q^{92} -1118.26 q^{93} -1483.40 q^{94} -1504.83 q^{95} -1313.45 q^{96} -758.953 q^{97} +6.67799 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{2} + 44 q^{4} + 36 q^{8} + 170 q^{9}+O(q^{10}) $ 10 * q + 4 * q^2 + 44 * q^4 + 36 * q^8 + 170 * q^9	$ 10 q + 4 q^{2} + 44 q^{4} + 36 q^{8} + 170 q^{9} + 110 q^{11} + 428 q^{15} + 84 q^{16} - 340 q^{18} + 44 q^{22} + 92 q^{23} + 730 q^{25} + 628 q^{29} - 1128 q^{30} + 1252 q^{32} + 2084 q^{36} + 164 q^{37} + 244 q^{39} + 412 q^{43} + 484 q^{44} + 2920 q^{46} - 412 q^{50} - 84 q^{51} + 1968 q^{53} + 720 q^{57} - 2520 q^{58} + 7256 q^{60} + 1044 q^{64} - 3312 q^{65} + 1916 q^{67} + 1324 q^{71} - 5164 q^{72} + 2576 q^{74} + 8512 q^{78} + 604 q^{79} + 6990 q^{81} + 984 q^{85} - 3632 q^{86} + 396 q^{88} + 6024 q^{92} - 3048 q^{93} + 1480 q^{95} + 1870 q^{99}+O(q^{100}) $ 10 * q + 4 * q^2 + 44 * q^4 + 36 * q^8 + 170 * q^9 + 110 * q^11 + 428 * q^15 + 84 * q^16 - 340 * q^18 + 44 * q^22 + 92 * q^23 + 730 * q^25 + 628 * q^29 - 1128 * q^30 + 1252 * q^32 + 2084 * q^36 + 164 * q^37 + 244 * q^39 + 412 * q^43 + 484 * q^44 + 2920 * q^46 - 412 * q^50 - 84 * q^51 + 1968 * q^53 + 720 * q^57 - 2520 * q^58 + 7256 * q^60 + 1044 * q^64 - 3312 * q^65 + 1916 * q^67 + 1324 * q^71 - 5164 * q^72 + 2576 * q^74 + 8512 * q^78 + 604 * q^79 + 6990 * q^81 + 984 * q^85 - 3632 * q^86 + 396 * q^88 + 6024 * q^92 - 3048 * q^93 + 1480 * q^95 + 1870 * q^99

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 $

$2$	5.15800	1.82363	0.911813	−	0.410605i	$-0.134682\pi$
$2$	5.15800	1.82363	0.911813	+	0.410605i	$0.134682\pi$
$3$	−5.25424	−1.01118	−0.505590	−	0.862774i	$-0.668725\pi$
$3$	−5.25424	−1.01118	−0.505590	+	0.862774i	$0.668725\pi$
$4$	18.6049	2.32561
$5$	−13.3903	−1.19766	−0.598831	−	0.800875i	$-0.704368\pi$
$5$	−13.3903	−1.19766	−0.598831	+	0.800875i	$0.704368\pi$
$6$	−27.1014	−1.84401
$7$	0	0
$7$	0	0
$8$	54.7001	2.41743
$9$	0.607090	0.0224848
$10$	−69.0669	−2.18409
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−97.7548	−2.35161
$13$	11.3828	0.242848	0.121424	−	0.992601i	$-0.461254\pi$
$13$	11.3828	0.242848	0.121424	+	0.992601i	$0.461254\pi$
$14$	0	0
$15$	70.3557	1.21105
$16$	133.304	2.08287
$17$	106.434	1.51847	0.759237	−	0.650814i	$-0.225572\pi$
$17$	106.434	1.51847	0.759237	+	0.650814i	$0.225572\pi$
$18$	3.13137	0.0410039
$19$	112.382	1.35696	0.678481	−	0.734618i	$-0.262639\pi$
$19$	112.382	1.35696	0.678481	+	0.734618i	$0.262639\pi$
$20$	−249.125	−2.78530
$21$	0	0
$22$	56.7380	0.549844
$23$	190.167	1.72403	0.862013	−	0.506886i	$-0.169203\pi$
$23$	190.167	1.72403	0.862013	+	0.506886i	$0.169203\pi$
$24$	−287.408	−2.44445
$25$	54.2992	0.434394
$26$	58.7125	0.442864
$27$	138.675	0.988444
$28$	0	0
$29$	−114.376	−0.732382	−0.366191	−	0.930540i	$-0.619338\pi$
$29$	−114.376	−0.732382	−0.366191	+	0.930540i	$0.619338\pi$
$30$	362.895	2.20851
$31$	212.830	1.23308	0.616540	−	0.787324i	$-0.288534\pi$
$31$	212.830	1.23308	0.616540	+	0.787324i	$0.288534\pi$
$32$	249.979	1.38095
$33$	−57.7967	−0.304882
$34$	548.987	2.76913
$35$	0	0
$36$	11.2949	0.0522911
$37$	91.9763	0.408671	0.204335	−	0.978901i	$-0.434497\pi$
$37$	91.9763	0.408671	0.204335	+	0.978901i	$0.434497\pi$
$38$	579.667	2.47459
$39$	−59.8081	−0.245563
$40$	−732.449	−2.89526
$41$	−429.485	−1.63596	−0.817979	−	0.575248i	$-0.804905\pi$
$41$	−429.485	−1.63596	−0.817979	+	0.575248i	$0.804905\pi$
$42$	0	0
$43$	−262.479	−0.930878	−0.465439	−	0.885080i	$-0.654103\pi$
$43$	−262.479	−0.930878	−0.465439	+	0.885080i	$0.654103\pi$
$44$	204.654	0.701199
$45$	−8.12910	−0.0269292
$46$	980.882	3.14398
$47$	−287.593	−0.892547	−0.446274	−	0.894897i	$-0.647249\pi$
$47$	−287.593	−0.892547	−0.446274	+	0.894897i	$0.647249\pi$
$48$	−700.410	−2.10616
$49$	0	0
$50$	280.075	0.792172
$51$	−559.231	−1.53545
$52$	211.776	0.564771
$53$	217.762	0.564377	0.282188	−	0.959359i	$-0.408940\pi$
$53$	217.762	0.564377	0.282188	+	0.959359i	$0.408940\pi$
$54$	715.284	1.80255
$55$	−147.293	−0.361109
$56$	0	0
$57$	−590.484	−1.37213
$58$	−589.951	−1.33559
$59$	10.5891	0.0233659	0.0116830	−	0.999932i	$-0.496281\pi$
$59$	10.5891	0.0233659	0.0116830	+	0.999932i	$0.496281\pi$
$60$	1308.96	2.81644
$61$	712.550	1.49562	0.747809	−	0.663914i	$-0.231106\pi$
$61$	712.550	1.49562	0.747809	+	0.663914i	$0.231106\pi$
$62$	1097.78	2.24868
$63$	0	0
$64$	222.959	0.435468
$65$	−152.419	−0.290850
$66$	−298.115	−0.555991
$67$	1081.11	1.97132	0.985660	−	0.168743i	$-0.0539707\pi$
$67$	1081.11	1.97132	0.985660	+	0.168743i	$0.0539707\pi$
$68$	1980.20	3.53139
$69$	−999.185	−1.74330
$70$	0	0
$71$	538.346	0.899859	0.449929	−	0.893064i	$-0.351449\pi$
$71$	538.346	0.899859	0.449929	+	0.893064i	$0.351449\pi$
$72$	33.2079	0.0543554
$73$	−302.995	−0.485792	−0.242896	−	0.970052i	$-0.578097\pi$
$73$	−302.995	−0.485792	−0.242896	+	0.970052i	$0.578097\pi$
$74$	474.413	0.745263
$75$	−285.301	−0.439250
$76$	2090.86	3.15577
$77$	0	0
$78$	−308.490	−0.447815
$79$	−30.2060	−0.0430183	−0.0215091	−	0.999769i	$-0.506847\pi$
$79$	−30.2060	−0.0430183	−0.0215091	+	0.999769i	$0.506847\pi$
$80$	−1784.97	−2.49457
$81$	−745.023	−1.02198
$82$	−2215.28	−2.98338
$83$	−149.101	−0.197180	−0.0985901	−	0.995128i	$-0.531433\pi$
$83$	−149.101	−0.197180	−0.0985901	+	0.995128i	$0.531433\pi$
$84$	0	0
$85$	−1425.18	−1.81862
$86$	−1353.87	−1.69757
$87$	600.959	0.740570
$88$	601.701	0.728882
$89$	1055.28	1.25685	0.628426	−	0.777870i	$-0.283700\pi$
$89$	1055.28	1.25685	0.628426	+	0.777870i	$0.283700\pi$
$90$	−41.9299	−0.0491089
$91$	0	0
$92$	3538.05	4.00942
$93$	−1118.26	−1.24687
$94$	−1483.40	−1.62767
$95$	−1504.83	−1.62518
$96$	−1313.45	−1.39639
$97$	−758.953	−0.794433	−0.397216	−	0.917725i	$-0.630024\pi$
$97$	−758.953	−0.794433	−0.397216	+	0.917725i	$0.630024\pi$
$98$	0	0
$99$	6.67799	0.00677943
$100$	1010.23	1.01023
$101$	−1395.04	−1.37437	−0.687185	−	0.726483i	$-0.741154\pi$
$101$	−1395.04	−1.37437	−0.687185	+	0.726483i	$0.741154\pi$
$102$	−2884.51	−2.80009
$103$	−121.578	−0.116305	−0.0581524	−	0.998308i	$-0.518521\pi$
$103$	−121.578	−0.116305	−0.0581524	+	0.998308i	$0.518521\pi$
$104$	622.641	0.587067
$105$	0	0
$106$	1123.22	1.02921
$107$	2096.22	1.89392	0.946959	−	0.321354i	$-0.104138\pi$
$107$	2096.22	1.89392	0.946959	+	0.321354i	$0.104138\pi$
$108$	2580.03	2.29874
$109$	−129.337	−0.113654	−0.0568268	−	0.998384i	$-0.518098\pi$
$109$	−129.337	−0.113654	−0.0568268	+	0.998384i	$0.518098\pi$
$110$	−759.736	−0.658527
$111$	−483.266	−0.413240
$112$	0	0
$113$	−68.9077	−0.0573654	−0.0286827	−	0.999589i	$-0.509131\pi$
$113$	−68.9077	−0.0573654	−0.0286827	+	0.999589i	$0.509131\pi$
$114$	−3045.71	−2.50226
$115$	−2546.39	−2.06480
$116$	−2127.96	−1.70324
$117$	6.91039	0.00546039
$118$	54.6188	0.0426107
$119$	0	0
$120$	3848.47	2.92763
$121$	121.000	0.0909091
$122$	3675.33	2.72745
$123$	2256.62	1.65425
$124$	3959.69	2.86767
$125$	946.702	0.677405
$126$	0	0
$127$	−1860.55	−1.29997	−0.649987	−	0.759945i	$-0.725226\pi$
$127$	−1860.55	−1.29997	−0.649987	+	0.759945i	$0.725226\pi$
$128$	−849.805	−0.586819
$129$	1379.13	0.941285
$130$	−786.176	−0.530401
$131$	−2486.26	−1.65821	−0.829104	−	0.559094i	$-0.811149\pi$
$131$	−2486.26	−1.65821	−0.829104	+	0.559094i	$0.811149\pi$
$132$	−1075.30	−0.709039
$133$	0	0
$134$	5576.35	3.59495
$135$	−1856.89	−1.18382
$136$	5821.96	3.67080
$137$	3038.23	1.89470	0.947348	−	0.320207i	$-0.103752\pi$
$137$	3038.23	1.89470	0.947348	+	0.320207i	$0.103752\pi$
$138$	−5153.79	−3.17913
$139$	1452.20	0.886144	0.443072	−	0.896486i	$-0.353889\pi$
$139$	1452.20	0.886144	0.443072	+	0.896486i	$0.353889\pi$
$140$	0	0
$141$	1511.08	0.902526
$142$	2776.79	1.64101
$143$	125.211	0.0732214
$144$	80.9274	0.0468330
$145$	1531.52	0.877146
$146$	−1562.84	−0.885904
$147$	0	0
$148$	1711.21	0.950411
$149$	−2014.62	−1.10768	−0.553839	−	0.832624i	$-0.686838\pi$
$149$	−2014.62	−1.10768	−0.553839	+	0.832624i	$0.686838\pi$
$150$	−1471.58	−0.801029
$151$	−1125.71	−0.606683	−0.303342	−	0.952882i	$-0.598102\pi$
$151$	−1125.71	−0.606683	−0.303342	+	0.952882i	$0.598102\pi$
$152$	6147.33	3.28035
$153$	64.6151	0.0341426
$154$	0	0
$155$	−2849.86	−1.47681
$156$	−1112.72	−0.571085
$157$	−1266.97	−0.644045	−0.322022	−	0.946732i	$-0.604363\pi$
$157$	−1266.97	−0.644045	−0.322022	+	0.946732i	$0.604363\pi$
$158$	−155.803	−0.0784493
$159$	−1144.18	−0.570687
$160$	−3347.28	−1.65391
$161$	0	0
$162$	−3842.82	−1.86371
$163$	−598.556	−0.287623	−0.143811	−	0.989605i	$-0.545936\pi$
$163$	−598.556	−0.287623	−0.143811	+	0.989605i	$0.545936\pi$
$164$	−7990.53	−3.80461
$165$	773.913	0.365146
$166$	−769.062	−0.359583
$167$	268.021	0.124192	0.0620962	−	0.998070i	$-0.480221\pi$
$167$	268.021	0.124192	0.0620962	+	0.998070i	$0.480221\pi$
$168$	0	0
$169$	−2067.43	−0.941025
$170$	−7351.08	−3.31648
$171$	68.2262	0.0305110
$172$	−4883.41	−2.16486
$173$	735.083	0.323048	0.161524	−	0.986869i	$-0.448359\pi$
$173$	735.083	0.323048	0.161524	+	0.986869i	$0.448359\pi$
$174$	3099.75	1.35052
$175$	0	0
$176$	1466.34	0.628009
$177$	−55.6380	−0.0236272
$178$	5443.15	2.29203
$179$	1181.59	0.493389	0.246694	−	0.969093i	$-0.420656\pi$
$179$	1181.59	0.493389	0.246694	+	0.969093i	$0.420656\pi$
$180$	−151.241	−0.0626270
$181$	−300.947	−0.123587	−0.0617933	−	0.998089i	$-0.519682\pi$
$181$	−300.947	−0.123587	−0.0617933	+	0.998089i	$0.519682\pi$
$182$	0	0
$183$	−3743.91	−1.51234
$184$	10402.2	4.16771
$185$	−1231.59	−0.489449
$186$	−5768.00	−2.27382
$187$	1170.78	0.457837
$188$	−5350.64	−2.07572
$189$	0	0
$190$	−7761.90	−2.96372
$191$	−2477.51	−0.938566	−0.469283	−	0.883048i	$-0.655488\pi$
$191$	−2477.51	−0.938566	−0.469283	+	0.883048i	$0.655488\pi$
$192$	−1171.48	−0.440336
$193$	1032.27	0.384997	0.192499	−	0.981297i	$-0.438341\pi$
$193$	1032.27	0.384997	0.192499	+	0.981297i	$0.438341\pi$
$194$	−3914.68	−1.44875
$195$	800.846	0.294101
$196$	0	0
$197$	−536.555	−0.194051	−0.0970253	−	0.995282i	$-0.530933\pi$
$197$	−536.555	−0.194051	−0.0970253	+	0.995282i	$0.530933\pi$
$198$	34.4451	0.0123632
$199$	2652.91	0.945022	0.472511	−	0.881325i	$-0.343348\pi$
$199$	2652.91	0.945022	0.472511	+	0.881325i	$0.343348\pi$
$200$	2970.17	1.05012
$201$	−5680.41	−1.99336
$202$	−7195.59	−2.50634
$203$	0	0
$204$	−10404.4	−3.57087
$205$	5750.92	1.95932
$206$	−627.096	−0.212096
$207$	115.449	0.0387644
$208$	1517.37	0.505821
$209$	1236.21	0.409139
$210$	0	0
$211$	303.624	0.0990633	0.0495316	−	0.998773i	$-0.484227\pi$
$211$	303.624	0.0990633	0.0495316	+	0.998773i	$0.484227\pi$
$212$	4051.45	1.31252
$213$	−2828.60	−0.909919
$214$	10812.3	3.45380
$215$	3514.67	1.11488
$216$	7585.53	2.38949
$217$	0	0
$218$	−667.120	−0.207262
$219$	1592.01	0.491223
$220$	−2740.37	−0.839800
$221$	1211.52	0.368758
$222$	−2492.68	−0.753595
$223$	3428.91	1.02967	0.514836	−	0.857289i	$-0.327853\pi$
$223$	3428.91	1.02967	0.514836	+	0.857289i	$0.327853\pi$
$224$	0	0
$225$	32.9645	0.00976727
$226$	−355.426	−0.104613
$227$	−4069.09	−1.18976	−0.594879	−	0.803815i	$-0.702800\pi$
$227$	−4069.09	−1.18976	−0.594879	+	0.803815i	$0.702800\pi$
$228$	−10985.9	−3.19105
$229$	5233.39	1.51018	0.755092	−	0.655618i	$-0.227592\pi$
$229$	5233.39	1.51018	0.755092	+	0.655618i	$0.227592\pi$
$230$	−13134.3	−3.76543
$231$	0	0
$232$	−6256.38	−1.77048
$233$	−3435.80	−0.966037	−0.483018	−	0.875610i	$-0.660460\pi$
$233$	−3435.80	−0.966037	−0.483018	+	0.875610i	$0.660460\pi$
$234$	35.6438	0.00995772
$235$	3850.94	1.06897
$236$	197.010	0.0543401
$237$	158.710	0.0434992
$238$	0	0
$239$	−983.050	−0.266060	−0.133030	−	0.991112i	$-0.542471\pi$
$239$	−983.050	−0.266060	−0.133030	+	0.991112i	$0.542471\pi$
$240$	9378.68	2.52246
$241$	2090.62	0.558790	0.279395	−	0.960176i	$-0.409866\pi$
$241$	2090.62	0.558790	0.279395	+	0.960176i	$0.409866\pi$
$242$	624.117	0.165784
$243$	170.313	0.0449612
$244$	13256.9	3.47823
$245$	0	0
$246$	11639.6	3.01673
$247$	1279.23	0.329535
$248$	11641.9	2.98088
$249$	783.413	0.199385
$250$	4883.09	1.23533
$251$	3993.10	1.00415	0.502076	−	0.864823i	$-0.332570\pi$
$251$	3993.10	1.00415	0.502076	+	0.864823i	$0.332570\pi$
$252$	0	0
$253$	2091.84	0.519814
$254$	−9596.68	−2.37067
$255$	7488.25	1.83895
$256$	−6166.97	−1.50561
$257$	−6422.85	−1.55894	−0.779468	−	0.626442i	$-0.784510\pi$
$257$	−6422.85	−1.55894	−0.779468	+	0.626442i	$0.784510\pi$
$258$	7113.55	1.71655
$259$	0	0
$260$	−2835.74	−0.676404
$261$	−69.4365	−0.0164675
$262$	−12824.1	−3.02395
$263$	5224.98	1.22504	0.612522	−	0.790454i	$-0.290155\pi$
$263$	5224.98	1.22504	0.612522	+	0.790454i	$0.290155\pi$
$264$	−3161.49	−0.737030
$265$	−2915.90	−0.675933
$266$	0	0
$267$	−5544.72	−1.27090
$268$	20113.9	4.58453
$269$	1426.26	0.323274	0.161637	−	0.986850i	$-0.448323\pi$
$269$	1426.26	0.323274	0.161637	+	0.986850i	$0.448323\pi$
$270$	−9577.84	−2.15885
$271$	−5724.53	−1.28317	−0.641587	−	0.767050i	$-0.721724\pi$
$271$	−5724.53	−1.28317	−0.641587	+	0.767050i	$0.721724\pi$
$272$	14188.1	3.16278
$273$	0	0
$274$	15671.2	3.45522
$275$	597.292	0.130975
$276$	−18589.8	−4.05425
$277$	3943.45	0.855375	0.427688	−	0.903927i	$-0.359328\pi$
$277$	3943.45	0.855375	0.427688	+	0.903927i	$0.359328\pi$
$278$	7490.45	1.61600
$279$	129.207	0.0277256
$280$	0	0
$281$	−4629.29	−0.982777	−0.491389	−	0.870940i	$-0.663511\pi$
$281$	−4629.29	−0.982777	−0.491389	+	0.870940i	$0.663511\pi$
$282$	7794.16	1.64587
$283$	4289.33	0.900970	0.450485	−	0.892784i	$-0.351251\pi$
$283$	4289.33	0.900970	0.450485	+	0.892784i	$0.351251\pi$
$284$	10015.9	2.09272
$285$	7906.74	1.64335
$286$	645.837	0.133529
$287$	0	0
$288$	151.760	0.0310504
$289$	6415.22	1.30576
$290$	7899.60	1.59959
$291$	3987.72	0.803315
$292$	−5637.19	−1.12977
$293$	1008.22	0.201027	0.100513	−	0.994936i	$-0.467951\pi$
$293$	1008.22	0.201027	0.100513	+	0.994936i	$0.467951\pi$
$294$	0	0
$295$	−141.791	−0.0279845
$296$	5031.12	0.987931
$297$	1525.42	0.298027
$298$	−10391.4	−2.01999
$299$	2164.64	0.418676
$300$	−5308.01	−1.02153
$301$	0	0
$302$	−5806.42	−1.10636
$303$	7329.86	1.38973
$304$	14981.0	2.82637
$305$	−9541.24	−1.79124
$306$	333.284	0.0622634
$307$	4523.79	0.840998	0.420499	−	0.907293i	$-0.361855\pi$
$307$	4523.79	0.840998	0.420499	+	0.907293i	$0.361855\pi$
$308$	0	0
$309$	638.798	0.117605
$310$	−14699.5	−2.69316
$311$	−2100.91	−0.383060	−0.191530	−	0.981487i	$-0.561345\pi$
$311$	−2100.91	−0.383060	−0.191530	+	0.981487i	$0.561345\pi$
$312$	−3271.51	−0.593630
$313$	6700.45	1.21001	0.605003	−	0.796223i	$-0.293172\pi$
$313$	6700.45	1.21001	0.605003	+	0.796223i	$0.293172\pi$
$314$	−6535.02	−1.17450
$315$	0	0
$316$	−561.981	−0.100044
$317$	4517.53	0.800410	0.400205	−	0.916426i	$-0.368939\pi$
$317$	4517.53	0.800410	0.400205	+	0.916426i	$0.368939\pi$
$318$	−5901.66	−1.04072
$319$	−1258.14	−0.220822
$320$	−2985.49	−0.521543
$321$	−11014.1	−1.91509
$322$	0	0
$323$	11961.3	2.06051
$324$	−13861.1	−2.37673
$325$	618.078	0.105492
$326$	−3087.35	−0.524517
$327$	679.569	0.114924
$328$	−23492.9	−3.95481
$329$	0	0
$330$	3991.84	0.665890
$331$	−6849.81	−1.13746	−0.568731	−	0.822524i	$-0.692565\pi$
$331$	−6849.81	−1.13746	−0.568731	+	0.822524i	$0.692565\pi$
$332$	−2774.01	−0.458565
$333$	55.8379	0.00918889
$334$	1382.45	0.226480
$335$	−14476.3	−2.36098
$336$	0	0
$337$	−4537.72	−0.733488	−0.366744	−	0.930322i	$-0.619528\pi$
$337$	−4537.72	−0.733488	−0.366744	+	0.930322i	$0.619528\pi$
$338$	−10663.8	−1.71608
$339$	362.058	0.0580068
$340$	−26515.4	−4.22941
$341$	2341.14	0.371788
$342$	351.911	0.0556408
$343$	0	0
$344$	−14357.7	−2.25033
$345$	13379.4	2.08789
$346$	3791.55	0.589119
$347$	−1544.85	−0.238997	−0.119499	−	0.992834i	$-0.538129\pi$
$347$	−1544.85	−0.238997	−0.119499	+	0.992834i	$0.538129\pi$
$348$	11180.8	1.72228
$349$	−7169.69	−1.09967	−0.549835	−	0.835273i	$-0.685309\pi$
$349$	−7169.69	−1.09967	−0.549835	+	0.835273i	$0.685309\pi$
$350$	0	0
$351$	1578.51	0.240042
$352$	2749.76	0.416372
$353$	−3283.48	−0.495076	−0.247538	−	0.968878i	$-0.579622\pi$
$353$	−3283.48	−0.495076	−0.247538	+	0.968878i	$0.579622\pi$
$354$	−286.980	−0.0430871
$355$	−7208.60	−1.07773
$356$	19633.5	2.92295
$357$	0	0
$358$	6094.66	0.899757
$359$	−5660.76	−0.832210	−0.416105	−	0.909317i	$-0.636605\pi$
$359$	−5660.76	−0.832210	−0.416105	+	0.909317i	$0.636605\pi$
$360$	−444.663	−0.0650994
$361$	5770.78	0.841344
$362$	−1552.28	−0.225376
$363$	−635.764	−0.0919254
$364$	0	0
$365$	4057.18	0.581815
$366$	−19311.1	−2.75794
$367$	−889.943	−0.126579	−0.0632897	−	0.997995i	$-0.520159\pi$
$367$	−889.943	−0.126579	−0.0632897	+	0.997995i	$0.520159\pi$
$368$	25350.0	3.59092
$369$	−260.736	−0.0367842
$370$	−6352.52	−0.892573
$371$	0	0
$372$	−20805.2	−2.89973
$373$	2082.25	0.289047	0.144524	−	0.989501i	$-0.453835\pi$
$373$	2082.25	0.289047	0.144524	+	0.989501i	$0.453835\pi$
$374$	6038.85	0.834924
$375$	−4974.21	−0.684978
$376$	−15731.4	−2.15767
$377$	−1301.92	−0.177858
$378$	0	0
$379$	−8061.41	−1.09258	−0.546289	−	0.837597i	$-0.683960\pi$
$379$	−8061.41	−1.09258	−0.546289	+	0.837597i	$0.683960\pi$
$380$	−27997.2	−3.77954
$381$	9775.76	1.31451
$382$	−12779.0	−1.71159
$383$	−6311.33	−0.842021	−0.421010	−	0.907056i	$-0.638324\pi$
$383$	−6311.33	−0.842021	−0.421010	+	0.907056i	$0.638324\pi$
$384$	4465.08	0.593380
$385$	0	0
$386$	5324.45	0.702091
$387$	−159.349	−0.0209306
$388$	−14120.3	−1.84754
$389$	8121.47	1.05855	0.529274	−	0.848451i	$-0.322464\pi$
$389$	8121.47	1.05855	0.529274	+	0.848451i	$0.322464\pi$
$390$	4130.76	0.536331
$391$	20240.3	2.61789
$392$	0	0
$393$	13063.4	1.67675
$394$	−2767.55	−0.353876
$395$	404.467	0.0515214
$396$	124.244	0.0157663
$397$	7826.23	0.989389	0.494694	−	0.869067i	$-0.335280\pi$
$397$	7826.23	0.989389	0.494694	+	0.869067i	$0.335280\pi$
$398$	13683.7	1.72337
$399$	0	0
$400$	7238.28	0.904786
$401$	−4473.24	−0.557065	−0.278533	−	0.960427i	$-0.589848\pi$
$401$	−4473.24	−0.557065	−0.278533	+	0.960427i	$0.589848\pi$
$402$	−29299.5	−3.63514
$403$	2422.61	0.299451
$404$	−25954.5	−3.19625
$405$	9976.05	1.22399
$406$	0	0
$407$	1011.74	0.123219
$408$	−30590.0	−3.71184
$409$	−8293.12	−1.00261	−0.501306	−	0.865270i	$-0.667147\pi$
$409$	−8293.12	−1.00261	−0.501306	+	0.865270i	$0.667147\pi$
$410$	29663.2	3.57308
$411$	−15963.6	−1.91588
$412$	−2261.94	−0.270480
$413$	0	0
$414$	595.484	0.0706919
$415$	1996.50	0.236155
$416$	2845.46	0.335361
$417$	−7630.22	−0.896051
$418$	6376.34	0.746117
$419$	4095.51	0.477515	0.238757	−	0.971079i	$-0.423260\pi$
$419$	4095.51	0.477515	0.238757	+	0.971079i	$0.423260\pi$
$420$	0	0
$421$	−2046.32	−0.236892	−0.118446	−	0.992960i	$-0.537791\pi$
$421$	−2046.32	−0.236892	−0.118446	+	0.992960i	$0.537791\pi$
$422$	1566.09	0.180654
$423$	−174.595	−0.0200688
$424$	11911.6	1.36434
$425$	5779.29	0.659616
$426$	−14589.9	−1.65935
$427$	0	0
$428$	39000.0	4.40452
$429$	−657.889	−0.0740400
$430$	18128.7	2.03312
$431$	462.608	0.0517008	0.0258504	−	0.999666i	$-0.491771\pi$
$431$	462.608	0.0517008	0.0258504	+	0.999666i	$0.491771\pi$
$432$	18485.9	2.05880
$433$	−8601.97	−0.954698	−0.477349	−	0.878714i	$-0.658402\pi$
$433$	−8601.97	−0.954698	−0.477349	+	0.878714i	$0.658402\pi$
$434$	0	0
$435$	−8047.01	−0.886953
$436$	−2406.31	−0.264315
$437$	21371.4	2.33944
$438$	8211.57	0.895808
$439$	−1757.06	−0.191024	−0.0955122	−	0.995428i	$-0.530449\pi$
$439$	−1757.06	−0.191024	−0.0955122	+	0.995428i	$0.530449\pi$
$440$	−8056.94	−0.872954
$441$	0	0
$442$	6249.01	0.672478
$443$	6555.56	0.703079	0.351540	−	0.936173i	$-0.385658\pi$
$443$	6555.56	0.703079	0.351540	+	0.936173i	$0.385658\pi$
$444$	−8991.13	−0.961036
$445$	−14130.5	−1.50528
$446$	17686.3	1.87774
$447$	10585.3	1.12006
$448$	0	0
$449$	6626.53	0.696492	0.348246	−	0.937403i	$-0.386777\pi$
$449$	6626.53	0.696492	0.348246	+	0.937403i	$0.386777\pi$
$450$	170.031	0.0178119
$451$	−4724.33	−0.493260
$452$	−1282.02	−0.133410
$453$	5914.77	0.613466
$454$	−20988.4	−2.16967
$455$	0	0
$456$	−32299.6	−3.31703
$457$	−3605.49	−0.369054	−0.184527	−	0.982827i	$-0.559075\pi$
$457$	−3605.49	−0.369054	−0.184527	+	0.982827i	$0.559075\pi$
$458$	26993.8	2.75401
$459$	14759.7	1.50093
$460$	−47375.4	−4.80193
$461$	−7460.45	−0.753726	−0.376863	−	0.926269i	$-0.622997\pi$
$461$	−7460.45	−0.753726	−0.376863	+	0.926269i	$0.622997\pi$
$462$	0	0
$463$	−4077.03	−0.409235	−0.204617	−	0.978842i	$-0.565595\pi$
$463$	−4077.03	−0.409235	−0.204617	+	0.978842i	$0.565595\pi$
$464$	−15246.7	−1.52546
$465$	14973.8	1.49332
$466$	−17721.8	−1.76169
$467$	−18140.5	−1.79752	−0.898761	−	0.438439i	$-0.855532\pi$
$467$	−18140.5	−1.79752	−0.898761	+	0.438439i	$0.855532\pi$
$468$	128.567	0.0126988
$469$	0	0
$470$	19863.2	1.94940
$471$	6656.96	0.651245
$472$	579.228	0.0564854
$473$	−2887.27	−0.280670
$474$	818.625	0.0793264
$475$	6102.27	0.589456
$476$	0	0
$477$	132.201	0.0126899
$478$	−5070.57	−0.485193
$479$	276.955	0.0264184	0.0132092	−	0.999913i	$-0.495795\pi$
$479$	276.955	0.0264184	0.0132092	+	0.999913i	$0.495795\pi$
$480$	17587.4	1.67240
$481$	1046.95	0.0992448
$482$	10783.4	1.01903
$483$	0	0
$484$	2251.20	0.211420
$485$	10162.6	0.951462
$486$	878.473	0.0819924
$487$	−15977.9	−1.48671	−0.743356	−	0.668896i	$-0.766767\pi$
$487$	−15977.9	−1.48671	−0.743356	+	0.668896i	$0.766767\pi$
$488$	38976.6	3.61555
$489$	3144.96	0.290839
$490$	0	0
$491$	19876.8	1.82694	0.913468	−	0.406911i	$-0.133394\pi$
$491$	19876.8	1.82694	0.913468	+	0.406911i	$0.133394\pi$
$492$	41984.2	3.84714
$493$	−12173.5	−1.11210
$494$	6598.24	0.600949
$495$	−89.4201	−0.00811947
$496$	28371.1	2.56834
$497$	0	0
$498$	4040.84	0.363603
$499$	2050.43	0.183947	0.0919737	−	0.995761i	$-0.470682\pi$
$499$	2050.43	0.183947	0.0919737	+	0.995761i	$0.470682\pi$
$500$	17613.3	1.57538
$501$	−1408.25	−0.125581
$502$	20596.4	1.83120
$503$	−6830.02	−0.605438	−0.302719	−	0.953080i	$-0.597894\pi$
$503$	−6830.02	−0.605438	−0.302719	+	0.953080i	$0.597894\pi$
$504$	0	0
$505$	18679.9	1.64603
$506$	10789.7	0.947946
$507$	10862.8	0.951545
$508$	−34615.3	−3.02324
$509$	−9498.04	−0.827098	−0.413549	−	0.910482i	$-0.635711\pi$
$509$	−9498.04	−0.827098	−0.413549	+	0.910482i	$0.635711\pi$
$510$	38624.4	3.35356
$511$	0	0
$512$	−25010.7	−2.15885
$513$	15584.6	1.34128
$514$	−33129.0	−2.84292
$515$	1627.96	0.139294
$516$	25658.6	2.18907
$517$	−3163.52	−0.269113
$518$	0	0
$519$	−3862.31	−0.326660
$520$	−8337.33	−0.703108
$521$	−21285.2	−1.78987	−0.894934	−	0.446198i	$-0.852778\pi$
$521$	−21285.2	−1.78987	−0.894934	+	0.446198i	$0.852778\pi$
$522$	−358.153	−0.0300306
$523$	15632.9	1.30704	0.653518	−	0.756911i	$-0.273292\pi$
$523$	15632.9	1.30704	0.653518	+	0.756911i	$0.273292\pi$
$524$	−46256.6	−3.85635
$525$	0	0
$526$	26950.4	2.23402
$527$	22652.4	1.87240
$528$	−7704.51	−0.635030
$529$	23996.6	1.97227
$530$	−15040.2	−1.23265
$531$	6.42857	0.000525379 0
$532$	0	0
$533$	−4888.74	−0.397289
$534$	−28599.6	−2.31765
$535$	−28068.9	−2.26827
$536$	59136.8	4.76552
$537$	−6208.39	−0.498905
$538$	7356.65	0.589531
$539$	0	0
$540$	−34547.3	−2.75311
$541$	23634.6	1.87825	0.939124	−	0.343577i	$-0.111639\pi$
$541$	23634.6	1.87825	0.939124	+	0.343577i	$0.111639\pi$
$542$	−29527.1	−2.34003
$543$	1581.25	0.124968
$544$	26606.2	2.09694
$545$	1731.86	0.136119
$546$	0	0
$547$	−2126.08	−0.166188	−0.0830938	−	0.996542i	$-0.526480\pi$
$547$	−2126.08	−0.166188	−0.0830938	+	0.996542i	$0.526480\pi$
$548$	56526.0	4.40633
$549$	432.582	0.0336287
$550$	3080.83	0.238849
$551$	−12853.8	−0.993814
$552$	−54655.6	−4.21430
$553$	0	0
$554$	20340.3	1.55988
$555$	6471.06	0.494921
$556$	27018.1	2.06083
$557$	−16838.8	−1.28094	−0.640469	−	0.767984i	$-0.721260\pi$
$557$	−16838.8	−1.28094	−0.640469	+	0.767984i	$0.721260\pi$
$558$	666.451	0.0505611
$559$	−2987.75	−0.226062
$560$	0	0
$561$	−6151.54	−0.462956
$562$	−23877.9	−1.79222
$563$	−25789.6	−1.93055	−0.965276	−	0.261233i	$-0.915871\pi$
$563$	−25789.6	−1.93055	−0.965276	+	0.261233i	$0.915871\pi$
$564$	28113.6	2.09893
$565$	922.693	0.0687044
$566$	22124.4	1.64303
$567$	0	0
$568$	29447.6	2.17534
$569$	867.265	0.0638975	0.0319487	−	0.999490i	$-0.489829\pi$
$569$	867.265	0.0638975	0.0319487	+	0.999490i	$0.489829\pi$
$570$	40782.9	2.99686
$571$	−17586.1	−1.28889	−0.644444	−	0.764652i	$-0.722911\pi$
$571$	−17586.1	−1.28889	−0.644444	+	0.764652i	$0.722911\pi$
$572$	2329.54	0.170285
$573$	13017.4	0.949059
$574$	0	0
$575$	10325.9	0.748907
$576$	135.357	0.00979142
$577$	12658.5	0.913313	0.456656	−	0.889643i	$-0.349047\pi$
$577$	12658.5	0.913313	0.456656	+	0.889643i	$0.349047\pi$
$578$	33089.7	2.38123
$579$	−5423.80	−0.389301
$580$	28493.9	2.03990
$581$	0	0
$582$	20568.7	1.46495
$583$	2395.39	0.170166
$584$	−16573.8	−1.17437
$585$	−92.5320	−0.00653971
$586$	5200.40	0.366598
$587$	−11370.4	−0.799501	−0.399751	−	0.916624i	$-0.630903\pi$
$587$	−11370.4	−0.799501	−0.399751	+	0.916624i	$0.630903\pi$
$588$	0	0
$589$	23918.4	1.67324
$590$	−731.360	−0.0510332
$591$	2819.19	0.196220
$592$	12260.8	0.851208
$593$	644.585	0.0446373	0.0223187	−	0.999751i	$-0.492895\pi$
$593$	644.585	0.0446373	0.0223187	+	0.999751i	$0.492895\pi$
$594$	7868.12	0.543490
$595$	0	0
$596$	−37481.8	−2.57603
$597$	−13939.0	−0.955588
$598$	11165.2	0.763509
$599$	−2018.10	−0.137659	−0.0688293	−	0.997628i	$-0.521926\pi$
$599$	−2018.10	−0.137659	−0.0688293	+	0.997628i	$0.521926\pi$
$600$	−15606.0	−1.06186
$601$	15753.1	1.06919	0.534595	−	0.845109i	$-0.320464\pi$
$601$	15753.1	1.06919	0.534595	+	0.845109i	$0.320464\pi$
$602$	0	0
$603$	656.331	0.0443248
$604$	−20943.8	−1.41091
$605$	−1620.22	−0.108878
$606$	37807.4	2.53436
$607$	6567.63	0.439163	0.219582	−	0.975594i	$-0.429531\pi$
$607$	6567.63	0.439163	0.219582	+	0.975594i	$0.429531\pi$
$608$	28093.2	1.87390
$609$	0	0
$610$	−49213.7	−3.26656
$611$	−3273.61	−0.216753
$612$	1202.16	0.0794026
$613$	18409.5	1.21297	0.606487	−	0.795093i	$-0.292578\pi$
$613$	18409.5	1.21297	0.606487	+	0.795093i	$0.292578\pi$
$614$	23333.7	1.53367
$615$	−30216.7	−1.98123
$616$	0	0
$617$	2433.45	0.158779	0.0793897	−	0.996844i	$-0.474703\pi$
$617$	2433.45	0.158779	0.0793897	+	0.996844i	$0.474703\pi$
$618$	3294.92	0.214468
$619$	−4405.44	−0.286057	−0.143029	−	0.989719i	$-0.545684\pi$
$619$	−4405.44	−0.286057	−0.143029	+	0.989719i	$0.545684\pi$
$620$	−53021.4	−3.43450
$621$	26371.4	1.70410
$622$	−10836.5	−0.698558
$623$	0	0
$624$	−7972.63	−0.511476
$625$	−19464.0	−1.24570
$626$	34560.9	2.20660
$627$	−6495.33	−0.413713
$628$	−23571.8	−1.49780
$629$	9789.42	0.620556
$630$	0	0
$631$	3055.50	0.192770	0.0963849	−	0.995344i	$-0.469272\pi$
$631$	3055.50	0.192770	0.0963849	+	0.995344i	$0.469272\pi$
$632$	−1652.27	−0.103994
$633$	−1595.32	−0.100171
$634$	23301.4	1.45965
$635$	24913.2	1.55693
$636$	−21287.3	−1.32720
$637$	0	0
$638$	−6489.46	−0.402696
$639$	326.825	0.0202332
$640$	11379.1	0.702811
$641$	24639.8	1.51827	0.759136	−	0.650932i	$-0.225622\pi$
$641$	24639.8	1.51827	0.759136	+	0.650932i	$0.225622\pi$
$642$	−56810.4	−3.49241
$643$	−16478.2	−1.01063	−0.505315	−	0.862935i	$-0.668624\pi$
$643$	−16478.2	−1.01063	−0.505315	+	0.862935i	$0.668624\pi$
$644$	0	0
$645$	−18466.9	−1.12734
$646$	61696.4	3.75760
$647$	−20280.1	−1.23229	−0.616147	−	0.787631i	$-0.711307\pi$
$647$	−20280.1	−1.23229	−0.616147	+	0.787631i	$0.711307\pi$
$648$	−40752.8	−2.47056
$649$	116.481	0.00704509
$650$	3188.04	0.192377
$651$	0	0
$652$	−11136.1	−0.668900
$653$	−20450.9	−1.22558	−0.612790	−	0.790246i	$-0.709953\pi$
$653$	−20450.9	−1.22558	−0.612790	+	0.790246i	$0.709953\pi$
$654$	3505.21	0.209579
$655$	33291.6	1.98597
$656$	−57251.9	−3.40749
$657$	−183.945	−0.0109230
$658$	0	0
$659$	−27596.1	−1.63125	−0.815623	−	0.578583i	$-0.803606\pi$
$659$	−27596.1	−1.63125	−0.815623	+	0.578583i	$0.803606\pi$
$660$	14398.6	0.849188
$661$	−13723.2	−0.807521	−0.403761	−	0.914865i	$-0.632297\pi$
$661$	−13723.2	−0.807521	−0.403761	+	0.914865i	$0.632297\pi$
$662$	−35331.3	−2.07430
$663$	−6365.62	−0.372881
$664$	−8155.84	−0.476668
$665$	0	0
$666$	288.012	0.0167571
$667$	−21750.6	−1.26265
$668$	4986.52	0.288823
$669$	−18016.3	−1.04118
$670$	−74668.9	−4.30554
$671$	7838.05	0.450946
$672$	0	0
$673$	4400.64	0.252054	0.126027	−	0.992027i	$-0.459777\pi$
$673$	4400.64	0.252054	0.126027	+	0.992027i	$0.459777\pi$
$674$	−23405.6	−1.33761
$675$	7529.94	0.429374
$676$	−38464.4	−2.18846
$677$	17129.0	0.972409	0.486204	−	0.873845i	$-0.338381\pi$
$677$	17129.0	0.972409	0.486204	+	0.873845i	$0.338381\pi$
$678$	1867.49	0.105783
$679$	0	0
$680$	−77957.6	−4.39638
$681$	21380.0	1.20306
$682$	12075.6	0.678002
$683$	−13883.5	−0.777802	−0.388901	−	0.921280i	$-0.627145\pi$
$683$	−13883.5	−0.777802	−0.388901	+	0.921280i	$0.627145\pi$
$684$	1269.34	0.0709569
$685$	−40682.7	−2.26920
$686$	0	0
$687$	−27497.5	−1.52707
$688$	−34989.5	−1.93890
$689$	2478.75	0.137058
$690$	69010.7	3.80752
$691$	26189.7	1.44183	0.720914	−	0.693024i	$-0.243722\pi$
$691$	26189.7	1.44183	0.720914	+	0.693024i	$0.243722\pi$
$692$	13676.2	0.751285
$693$	0	0
$694$	−7968.34	−0.435842
$695$	−19445.4	−1.06130
$696$	32872.5	1.79027
$697$	−45711.8	−2.48416
$698$	−36981.2	−2.00539
$699$	18052.5	0.976837
$700$	0	0
$701$	569.596	0.0306895	0.0153448	−	0.999882i	$-0.495115\pi$
$701$	569.596	0.0306895	0.0153448	+	0.999882i	$0.495115\pi$
$702$	8141.94	0.437746
$703$	10336.5	0.554550
$704$	2452.55	0.131298
$705$	−20233.8	−1.08092
$706$	−16936.2	−0.902834
$707$	0	0
$708$	−1035.14	−0.0549477
$709$	12955.9	0.686276	0.343138	−	0.939285i	$-0.388510\pi$
$709$	12955.9	0.686276	0.343138	+	0.939285i	$0.388510\pi$
$710$	−37181.9	−1.96537
$711$	−18.3378	−0.000967259 0
$712$	57724.1	3.03835
$713$	40473.4	2.12586
$714$	0	0
$715$	−1676.61	−0.0876945
$716$	21983.5	1.14743
$717$	5165.19	0.269034
$718$	−29198.2	−1.51764
$719$	34746.3	1.80225	0.901127	−	0.433556i	$-0.142741\pi$
$719$	34746.3	1.80225	0.901127	+	0.433556i	$0.142741\pi$
$720$	−1083.64	−0.0560901
$721$	0	0
$722$	29765.7	1.53430
$723$	−10984.6	−0.565038
$724$	−5599.09	−0.287415
$725$	−6210.53	−0.318142
$726$	−3279.27	−0.167638
$727$	−21267.9	−1.08498	−0.542491	−	0.840061i	$-0.682519\pi$
$727$	−21267.9	−1.08498	−0.542491	+	0.840061i	$0.682519\pi$
$728$	0	0
$729$	19220.8	0.976515
$730$	20926.9	1.06101
$731$	−27936.8	−1.41351
$732$	−69655.2	−3.51712
$733$	−1639.06	−0.0825920	−0.0412960	−	0.999147i	$-0.513149\pi$
$733$	−1639.06	−0.0825920	−0.0412960	+	0.999147i	$0.513149\pi$
$734$	−4590.32	−0.230833
$735$	0	0
$736$	47537.7	2.38079
$737$	11892.2	0.594375
$738$	−1344.88	−0.0670807
$739$	13974.2	0.695600	0.347800	−	0.937569i	$-0.386929\pi$
$739$	13974.2	0.695600	0.347800	+	0.937569i	$0.386929\pi$
$740$	−22913.6	−1.13827
$741$	−6721.37	−0.333219
$742$	0	0
$743$	1163.51	0.0574495	0.0287248	−	0.999587i	$-0.490855\pi$
$743$	1163.51	0.0574495	0.0287248	+	0.999587i	$0.490855\pi$
$744$	−61169.2	−3.01421
$745$	26976.3	1.32662
$746$	10740.2	0.527114
$747$	−90.5177	−0.00443356
$748$	21782.2	1.06475
$749$	0	0
$750$	−25656.9	−1.24914
$751$	−21209.4	−1.03055	−0.515274	−	0.857025i	$-0.672310\pi$
$751$	−21209.4	−1.03055	−0.515274	+	0.857025i	$0.672310\pi$
$752$	−38337.2	−1.85906
$753$	−20980.7	−1.01538
$754$	−6715.30	−0.324346
$755$	15073.6	0.726601
$756$	0	0
$757$	21791.8	1.04628	0.523141	−	0.852246i	$-0.324760\pi$
$757$	21791.8	1.04628	0.523141	+	0.852246i	$0.324760\pi$
$758$	−41580.7	−1.99245
$759$	−10991.0	−0.525625
$760$	−82314.3	−3.92876
$761$	−21169.8	−1.00841	−0.504207	−	0.863583i	$-0.668215\pi$
$761$	−21169.8	−1.00841	−0.504207	+	0.863583i	$0.668215\pi$
$762$	50423.3	2.39717
$763$	0	0
$764$	−46093.8	−2.18274
$765$	−865.214	−0.0408913
$766$	−32553.8	−1.53553
$767$	120.534	0.00567437
$768$	32402.8	1.52244
$769$	17721.6	0.831026	0.415513	−	0.909587i	$-0.363602\pi$
$769$	17721.6	0.831026	0.415513	+	0.909587i	$0.363602\pi$
$770$	0	0
$771$	33747.2	1.57636
$772$	19205.3	0.895355
$773$	40047.5	1.86340	0.931700	−	0.363229i	$-0.118326\pi$
$773$	40047.5	1.86340	0.931700	+	0.363229i	$0.118326\pi$
$774$	−821.920	−0.0381697
$775$	11556.5	0.535642
$776$	−41514.8	−1.92048
$777$	0	0
$778$	41890.5	1.93040
$779$	−48266.5	−2.21993
$780$	14899.7	0.683967
$781$	5921.81	0.271318
$782$	104399.	4.77405
$783$	−15861.1	−0.723919
$784$	0	0
$785$	16965.0	0.771348
$786$	67381.0	3.05776
$787$	13456.1	0.609475	0.304738	−	0.952436i	$-0.401431\pi$
$787$	13456.1	0.609475	0.304738	+	0.952436i	$0.401431\pi$
$788$	−9982.57	−0.451287
$789$	−27453.3	−1.23874
$790$	2086.24	0.0939557
$791$	0	0
$792$	365.287	0.0163888
$793$	8110.82	0.363208
$794$	40367.7	1.80428
$795$	15320.8	0.683490
$796$	49357.1	2.19776
$797$	−28811.7	−1.28051	−0.640253	−	0.768164i	$-0.721170\pi$
$797$	−28811.7	−1.28051	−0.640253	+	0.768164i	$0.721170\pi$
$798$	0	0
$799$	−30609.7	−1.35531
$800$	13573.6	0.599876
$801$	640.652	0.0282601
$802$	−23073.0	−1.01588
$803$	−3332.94	−0.146472
$804$	−105684.	−4.63579
$805$	0	0
$806$	12495.8	0.546087
$807$	−7493.93	−0.326888
$808$	−76308.7	−3.32244
$809$	−36208.8	−1.57359	−0.786794	−	0.617215i	$-0.788261\pi$
$809$	−36208.8	−1.57359	−0.786794	+	0.617215i	$0.788261\pi$
$810$	51456.4	2.23209
$811$	17670.4	0.765095	0.382547	−	0.923936i	$-0.375047\pi$
$811$	17670.4	0.765095	0.382547	+	0.923936i	$0.375047\pi$
$812$	0	0
$813$	30078.1	1.29752
$814$	5218.55	0.224705
$815$	8014.83	0.344475
$816$	−74547.5	−3.19814
$817$	−29498.0	−1.26317
$818$	−42775.9	−1.82839
$819$	0	0
$820$	106995.	4.55663
$821$	14977.2	0.636674	0.318337	−	0.947978i	$-0.396876\pi$
$821$	14977.2	0.636674	0.318337	+	0.947978i	$0.396876\pi$
$822$	−82340.1	−3.49385
$823$	38737.0	1.64069	0.820345	−	0.571869i	$-0.193782\pi$
$823$	38737.0	1.64069	0.820345	+	0.571869i	$0.193782\pi$
$824$	−6650.30	−0.281158
$825$	−3138.32	−0.132439
$826$	0	0
$827$	−14053.9	−0.590935	−0.295467	−	0.955353i	$-0.595475\pi$
$827$	−14053.9	−0.590935	−0.295467	+	0.955353i	$0.595475\pi$
$828$	2147.91	0.0901512
$829$	36550.8	1.53132	0.765659	−	0.643246i	$-0.222413\pi$
$829$	36550.8	1.53132	0.765659	+	0.643246i	$0.222413\pi$
$830$	10297.9	0.430659
$831$	−20719.8	−0.864938
$832$	2537.91	0.105752
$833$	0	0
$834$	−39356.6	−1.63406
$835$	−3588.88	−0.148740
$836$	22999.5	0.951500
$837$	29514.2	1.21883
$838$	21124.6	0.870809
$839$	9030.94	0.371612	0.185806	−	0.982586i	$-0.440510\pi$
$839$	9030.94	0.371612	0.185806	+	0.982586i	$0.440510\pi$
$840$	0	0
$841$	−11307.1	−0.463616
$842$	−10554.9	−0.432003
$843$	24323.4	0.993765
$844$	5648.90	0.230383
$845$	27683.5	1.12703
$846$	−900.559	−0.0365980
$847$	0	0
$848$	29028.5	1.17552
$849$	−22537.2	−0.911042
$850$	29809.5	1.20289
$851$	17490.9	0.704559
$852$	−52626.0	−2.11612
$853$	585.525	0.0235029	0.0117515	−	0.999931i	$-0.496259\pi$
$853$	585.525	0.0235029	0.0117515	+	0.999931i	$0.496259\pi$
$854$	0	0
$855$	−913.567	−0.0365419
$856$	114663.	4.57841
$857$	−27360.3	−1.09056	−0.545281	−	0.838253i	$-0.683577\pi$
$857$	−27360.3	−1.09056	−0.545281	+	0.838253i	$0.683577\pi$
$858$	−3393.39	−0.135021
$859$	5457.12	0.216757	0.108379	−	0.994110i	$-0.465434\pi$
$859$	5457.12	0.216757	0.108379	+	0.994110i	$0.465434\pi$
$860$	65390.1	2.59277
$861$	0	0
$862$	2386.13	0.0942829
$863$	−26038.3	−1.02706	−0.513530	−	0.858072i	$-0.671662\pi$
$863$	−26038.3	−1.02706	−0.513530	+	0.858072i	$0.671662\pi$
$864$	34665.7	1.36499
$865$	−9842.96	−0.386902
$866$	−44368.9	−1.74101
$867$	−33707.1	−1.32036
$868$	0	0
$869$	−332.266	−0.0129705
$870$	−41506.4	−1.61747
$871$	12306.1	0.478731
$872$	−7074.76	−0.274749
$873$	−460.753	−0.0178627
$874$	110234.	4.26626
$875$	0	0
$876$	29619.2	1.14240
$877$	−19237.0	−0.740694	−0.370347	−	0.928894i	$-0.620761\pi$
$877$	−19237.0	−0.740694	−0.370347	+	0.928894i	$0.620761\pi$
$878$	−9062.89	−0.348357
$879$	−5297.44	−0.203274
$880$	−19634.7	−0.752142
$881$	38840.6	1.48533	0.742663	−	0.669665i	$-0.233562\pi$
$881$	38840.6	1.48533	0.742663	+	0.669665i	$0.233562\pi$
$882$	0	0
$883$	−31189.7	−1.18869	−0.594346	−	0.804209i	$-0.702589\pi$
$883$	−31189.7	−1.18869	−0.594346	+	0.804209i	$0.702589\pi$
$884$	22540.2	0.857590
$885$	745.007	0.0282973
$886$	33813.6	1.28215
$887$	−50428.9	−1.90895	−0.954475	−	0.298292i	$-0.903583\pi$
$887$	−50428.9	−1.90895	−0.954475	+	0.298292i	$0.903583\pi$
$888$	−26434.7	−0.998976
$889$	0	0
$890$	−72885.2	−2.74507
$891$	−8195.25	−0.308138
$892$	63794.6	2.39462
$893$	−32320.3	−1.21115
$894$	54598.9	2.04257
$895$	−15821.9	−0.590913
$896$	0	0
$897$	−11373.5	−0.423357
$898$	34179.6	1.27014
$899$	−24342.7	−0.903086
$900$	613.303	0.0227149
$901$	23177.4	0.856992
$902$	−24368.1	−0.899522
$903$	0	0
$904$	−3769.26	−0.138677
$905$	4029.76	0.148015
$906$	30508.4	1.11873
$907$	−33013.0	−1.20858	−0.604288	−	0.796766i	$-0.706542\pi$
$907$	−33013.0	−1.20858	−0.604288	+	0.796766i	$0.706542\pi$
$908$	−75705.1	−2.76692
$909$	−846.913	−0.0309025
$910$	0	0
$911$	−7759.27	−0.282191	−0.141096	−	0.989996i	$-0.545063\pi$
$911$	−7759.27	−0.282191	−0.141096	+	0.989996i	$0.545063\pi$
$912$	−78713.7	−2.85797
$913$	−1640.11	−0.0594520
$914$	−18597.1	−0.673017
$915$	50132.0	1.81127
$916$	97366.8	3.51211
$917$	0	0
$918$	76130.6	2.73713
$919$	−6365.79	−0.228496	−0.114248	−	0.993452i	$-0.536446\pi$
$919$	−6365.79	−0.228496	−0.114248	+	0.993452i	$0.536446\pi$
$920$	−139288.	−4.99150
$921$	−23769.1	−0.850400
$922$	−38481.0	−1.37451
$923$	6127.90	0.218529
$924$	0	0
$925$	4994.24	0.177524
$926$	−21029.3	−0.746292
$927$	−73.8085	−0.00261509
$928$	−28591.5	−1.01138
$929$	−588.023	−0.0207669	−0.0103834	−	0.999946i	$-0.503305\pi$
$929$	−588.023	−0.0207669	−0.0103834	+	0.999946i	$0.503305\pi$
$930$	77235.0	2.72327
$931$	0	0
$932$	−63922.7	−2.24663
$933$	11038.7	0.387342
$934$	−93568.7	−3.27801
$935$	−15677.0	−0.548334
$936$	377.999	0.0132001
$937$	−20264.2	−0.706512	−0.353256	−	0.935527i	$-0.614926\pi$
$937$	−20264.2	−0.706512	−0.353256	+	0.935527i	$0.614926\pi$
$938$	0	0
$939$	−35205.8	−1.22353
$940$	71646.5	2.48601
$941$	37230.2	1.28976	0.644882	−	0.764282i	$-0.276906\pi$
$941$	37230.2	1.28976	0.644882	+	0.764282i	$0.276906\pi$
$942$	34336.6	1.18763
$943$	−81674.0	−2.82044
$944$	1411.57	0.0486682
$945$	0	0
$946$	−14892.5	−0.511838
$947$	−44388.0	−1.52314	−0.761572	−	0.648081i	$-0.775572\pi$
$947$	−44388.0	−1.52314	−0.761572	+	0.648081i	$0.775572\pi$
$948$	2952.79	0.101162
$949$	−3448.93	−0.117974
$950$	31475.5	1.07495
$951$	−23736.2	−0.809358
$952$	0	0
$953$	−35810.3	−1.21722	−0.608610	−	0.793469i	$-0.708273\pi$
$953$	−35810.3	−1.21722	−0.608610	+	0.793469i	$0.708273\pi$
$954$	681.895	0.0231417
$955$	33174.5	1.12409
$956$	−18289.6	−0.618752
$957$	6610.55	0.223290
$958$	1428.53	0.0481772
$959$	0	0
$960$	15686.5	0.527374
$961$	15505.8	0.520487
$962$	5400.16	0.180986
$963$	1272.60	0.0425844
$964$	38895.8	1.29953
$965$	−13822.4	−0.461096
$966$	0	0
$967$	−19844.1	−0.659920	−0.329960	−	0.943995i	$-0.607035\pi$
$967$	−19844.1	−0.659920	−0.329960	+	0.943995i	$0.607035\pi$
$968$	6618.71	0.219766
$969$	−62847.6	−2.08355
$970$	52418.5	1.73511
$971$	22552.7	0.745368	0.372684	−	0.927958i	$-0.378438\pi$
$971$	22552.7	0.745368	0.372684	+	0.927958i	$0.378438\pi$
$972$	3168.66	0.104562
$973$	0	0
$974$	−82414.1	−2.71121
$975$	−3247.53	−0.106671
$976$	94985.5	3.11518
$977$	29267.6	0.958398	0.479199	−	0.877706i	$-0.340927\pi$
$977$	29267.6	0.958398	0.479199	+	0.877706i	$0.340927\pi$
$978$	16221.7	0.530381
$979$	11608.1	0.378955
$980$	0	0
$981$	−78.5193	−0.00255548
$982$	102524.	3.33165
$983$	−13382.6	−0.434221	−0.217110	−	0.976147i	$-0.569663\pi$
$983$	−13382.6	−0.434221	−0.217110	+	0.976147i	$0.569663\pi$
$984$	123437.	3.99902
$985$	7184.62	0.232407
$986$	−62790.9	−2.02806
$987$	0	0
$988$	23799.9	0.766372
$989$	−49915.0	−1.60486
$990$	−461.229	−0.0148069
$991$	27657.2	0.886537	0.443269	−	0.896389i	$-0.353819\pi$
$991$	27657.2	0.886537	0.443269	+	0.896389i	$0.353819\pi$
$992$	53203.1	1.70282
$993$	35990.6	1.15018
$994$	0	0
$995$	−35523.1	−1.13182
$996$	14575.3	0.463692
$997$	−22489.4	−0.714390	−0.357195	−	0.934030i	$-0.616267\pi$
$997$	−22489.4	−0.714390	−0.357195	+	0.934030i	$0.616267\pi$
$998$	10576.1	0.335451
$999$	12754.8	0.403948

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.4.a.o.1.9	✓	10
7.6	odd	2	inner	539.4.a.o.1.10	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.4.a.o.1.9	✓	10	1.1	even	1	trivial
539.4.a.o.1.10	yes	10	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

\(f(q)\)	\(=\)	\(q+5.15800 q^{2} -5.25424 q^{3} +18.6049 q^{4} -13.3903 q^{5} -27.1014 q^{6} +54.7001 q^{8} +0.607090 q^{9} +O(q^{10})\)	\(q+5.15800 q^{2} -5.25424 q^{3} +18.6049 q^{4} -13.3903 q^{5} -27.1014 q^{6} +54.7001 q^{8} +0.607090 q^{9} -69.0669 q^{10} +11.0000 q^{11} -97.7548 q^{12} +11.3828 q^{13} +70.3557 q^{15} +133.304 q^{16} +106.434 q^{17} +3.13137 q^{18} +112.382 q^{19} -249.125 q^{20} +56.7380 q^{22} +190.167 q^{23} -287.408 q^{24} +54.2992 q^{25} +58.7125 q^{26} +138.675 q^{27} -114.376 q^{29} +362.895 q^{30} +212.830 q^{31} +249.979 q^{32} -57.7967 q^{33} +548.987 q^{34} +11.2949 q^{36} +91.9763 q^{37} +579.667 q^{38} -59.8081 q^{39} -732.449 q^{40} -429.485 q^{41} -262.479 q^{43} +204.654 q^{44} -8.12910 q^{45} +980.882 q^{46} -287.593 q^{47} -700.410 q^{48} +280.075 q^{50} -559.231 q^{51} +211.776 q^{52} +217.762 q^{53} +715.284 q^{54} -147.293 q^{55} -590.484 q^{57} -589.951 q^{58} +10.5891 q^{59} +1308.96 q^{60} +712.550 q^{61} +1097.78 q^{62} +222.959 q^{64} -152.419 q^{65} -298.115 q^{66} +1081.11 q^{67} +1980.20 q^{68} -999.185 q^{69} +538.346 q^{71} +33.2079 q^{72} -302.995 q^{73} +474.413 q^{74} -285.301 q^{75} +2090.86 q^{76} -308.490 q^{78} -30.2060 q^{79} -1784.97 q^{80} -745.023 q^{81} -2215.28 q^{82} -149.101 q^{83} -1425.18 q^{85} -1353.87 q^{86} +600.959 q^{87} +601.701 q^{88} +1055.28 q^{89} -41.9299 q^{90} +3538.05 q^{92} -1118.26 q^{93} -1483.40 q^{94} -1504.83 q^{95} -1313.45 q^{96} -758.953 q^{97} +6.67799 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 44 q^{4} + 36 q^{8} + 170 q^{9}+O(q^{10}) \) 10 * q + 4 * q^2 + 44 * q^4 + 36 * q^8 + 170 * q^9	\( 10 q + 4 q^{2} + 44 q^{4} + 36 q^{8} + 170 q^{9} + 110 q^{11} + 428 q^{15} + 84 q^{16} - 340 q^{18} + 44 q^{22} + 92 q^{23} + 730 q^{25} + 628 q^{29} - 1128 q^{30} + 1252 q^{32} + 2084 q^{36} + 164 q^{37} + 244 q^{39} + 412 q^{43} + 484 q^{44} + 2920 q^{46} - 412 q^{50} - 84 q^{51} + 1968 q^{53} + 720 q^{57} - 2520 q^{58} + 7256 q^{60} + 1044 q^{64} - 3312 q^{65} + 1916 q^{67} + 1324 q^{71} - 5164 q^{72} + 2576 q^{74} + 8512 q^{78} + 604 q^{79} + 6990 q^{81} + 984 q^{85} - 3632 q^{86} + 396 q^{88} + 6024 q^{92} - 3048 q^{93} + 1480 q^{95} + 1870 q^{99}+O(q^{100}) \) 10 * q + 4 * q^2 + 44 * q^4 + 36 * q^8 + 170 * q^9 + 110 * q^11 + 428 * q^15 + 84 * q^16 - 340 * q^18 + 44 * q^22 + 92 * q^23 + 730 * q^25 + 628 * q^29 - 1128 * q^30 + 1252 * q^32 + 2084 * q^36 + 164 * q^37 + 244 * q^39 + 412 * q^43 + 484 * q^44 + 2920 * q^46 - 412 * q^50 - 84 * q^51 + 1968 * q^53 + 720 * q^57 - 2520 * q^58 + 7256 * q^60 + 1044 * q^64 - 3312 * q^65 + 1916 * q^67 + 1324 * q^71 - 5164 * q^72 + 2576 * q^74 + 8512 * q^78 + 604 * q^79 + 6990 * q^81 + 984 * q^85 - 3632 * q^86 + 396 * q^88 + 6024 * q^92 - 3048 * q^93 + 1480 * q^95 + 1870 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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