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Properties

Label 2574.2.a.s.1.1
Level $2574$
Weight $2$
Character 2574.1
Self dual yes
Analytic conductor $20.553$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2574,2,Mod(1,2574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2574, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2574.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2574 = 2 \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2574.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: \(20.5534934803\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2574.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} -2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +2.00000 q^{19} -2.00000 q^{20} +1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{25} -1.00000 q^{26} -2.00000 q^{28} -4.00000 q^{29} -6.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} +4.00000 q^{35} -2.00000 q^{37} +2.00000 q^{38} -2.00000 q^{40} +2.00000 q^{41} -4.00000 q^{43} +1.00000 q^{44} -4.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} +4.00000 q^{53} -2.00000 q^{55} -2.00000 q^{56} -4.00000 q^{58} +4.00000 q^{59} -14.0000 q^{61} -6.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +2.00000 q^{67} +4.00000 q^{68} +4.00000 q^{70} -10.0000 q^{73} -2.00000 q^{74} +2.00000 q^{76} -2.00000 q^{77} -16.0000 q^{79} -2.00000 q^{80} +2.00000 q^{82} -12.0000 q^{83} -8.00000 q^{85} -4.00000 q^{86} +1.00000 q^{88} +10.0000 q^{89} +2.00000 q^{91} -4.00000 q^{92} -8.00000 q^{94} -4.00000 q^{95} +14.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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 \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 − 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −12.0000 −1.19404 −0.597022 − 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 − 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) −4.00000 −0.376288 −0.188144 − 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −14.0000 −1.26750
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 − 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) −10.0000 −0.854358 −0.427179 − 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) 8.00000 0.664364
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 22.0000 1.80231 0.901155 − 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 − 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 2.00000 0.162221
\(153\) 0 0
\(154\) −2.00000 −0.161165
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −16.0000 −1.27289
\(159\) 0 0
\(160\) −2.00000 −0.158114
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 − 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 8.00000 0.619059 0.309529 − 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −8.00000 −0.613572
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 20.0000 1.52057 0.760286 − 0.649589i \(-0.225059\pi\)
0.760286 + 0.649589i \(0.225059\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 − 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −14.0000 −0.997459 −0.498729 − 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −12.0000 −0.844317
\(203\) 8.00000 0.561490
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 4.00000 0.274721
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) −14.0000 −0.948200
\(219\) 0 0
\(220\) −2.00000 −0.134840
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 26.0000 1.74109 0.870544 − 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −4.00000 −0.266076
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 − 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) 16.0000 1.04819 0.524097 − 0.851658i \(-0.324403\pi\)
0.524097 + 0.851658i \(0.324403\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) −8.00000 −0.518563
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −14.0000 −0.896258
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) −6.00000 −0.381000
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 − 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −8.00000 −0.491436
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) 2.00000 0.122169
\(269\) 24.0000 1.46331 0.731653 − 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 − 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 − 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 24.0000 1.42665 0.713326 − 0.700832i \(-0.247188\pi\)
0.713326 + 0.700832i \(0.247188\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −1.00000 −0.0591312
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 8.00000 0.469776
\(291\) 0 0
\(292\) −10.0000 −0.585206
\(293\) −2.00000 −0.116841 −0.0584206 − 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) 22.0000 1.27443
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 28.0000 1.60328
\(306\) 0 0
\(307\) −6.00000 −0.342438 −0.171219 − 0.985233i \(-0.554771\pi\)
−0.171219 + 0.985233i \(0.554771\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) 12.0000 0.681554
\(311\) −20.0000 −1.13410 −0.567048 − 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 − 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 2.00000 0.110770
\(327\) 0 0
\(328\) 2.00000 0.110432
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) −14.0000 −0.769510 −0.384755 − 0.923019i \(-0.625714\pi\)
−0.384755 + 0.923019i \(0.625714\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 − 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) −8.00000 −0.433861
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 20.0000 1.07521
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 22.0000 1.17763 0.588817 − 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −16.0000 −0.844448 −0.422224 − 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 20.0000 1.04685
\(366\) 0 0
\(367\) −24.0000 −1.25279 −0.626395 − 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) −8.00000 −0.415339
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 − 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −2.00000 −0.102733 −0.0513665 − 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) −8.00000 −0.408781 −0.204390 − 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) −6.00000 −0.305392
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) −12.0000 −0.608424 −0.304212 − 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) −14.0000 −0.705310
\(395\) 32.0000 1.61009
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 − 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 6.00000 0.298881
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 − 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) −4.00000 −0.197546
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 2.00000 0.0978232
\(419\) −20.0000 −0.977064 −0.488532 − 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 − 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −12.0000 −0.584151
\(423\) 0 0
\(424\) 4.00000 0.194257
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) 28.0000 1.35501
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) −8.00000 −0.385346 −0.192673 − 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) 30.0000 1.44171 0.720854 − 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) 24.0000 1.14546 0.572729 − 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) −20.0000 −0.948091
\(446\) 26.0000 1.23114
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) −6.00000 −0.283158 −0.141579 − 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) −4.00000 −0.188144
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) 14.0000 0.654892 0.327446 − 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −22.0000 −1.02464 −0.512321 − 0.858794i \(-0.671214\pi\)
−0.512321 + 0.858794i \(0.671214\pi\)
\(462\) 0 0
\(463\) −6.00000 −0.278844 −0.139422 − 0.990233i \(-0.544524\pi\)
−0.139422 + 0.990233i \(0.544524\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 16.0000 0.741186
\(467\) 8.00000 0.370196 0.185098 − 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 16.0000 0.738025
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) −8.00000 −0.366679
\(477\) 0 0
\(478\) 24.0000 1.09773
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −28.0000 −1.27141
\(486\) 0 0
\(487\) 34.0000 1.54069 0.770344 − 0.637629i \(-0.220085\pi\)
0.770344 + 0.637629i \(0.220085\pi\)
\(488\) −14.0000 −0.633750
\(489\) 0 0
\(490\) 6.00000 0.271052
\(491\) −24.0000 −1.08310 −0.541552 − 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 0 0
\(493\) −16.0000 −0.720604
\(494\) −2.00000 −0.0899843
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 0 0
\(498\) 0 0
\(499\) −38.0000 −1.70111 −0.850557 − 0.525883i \(-0.823735\pi\)
−0.850557 + 0.525883i \(0.823735\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 24.0000 1.06799
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) −18.0000 −0.797836 −0.398918 − 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 12.0000 0.529297
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) 4.00000 0.175750
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) 4.00000 0.175243 0.0876216 − 0.996154i \(-0.472073\pi\)
0.0876216 + 0.996154i \(0.472073\pi\)
\(522\) 0 0
\(523\) −32.0000 −1.39926 −0.699631 − 0.714504i \(-0.746652\pi\)
−0.699631 + 0.714504i \(0.746652\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −8.00000 −0.347498
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) −2.00000 −0.0866296
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) 24.0000 1.03471
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 − 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) −2.00000 −0.0859074
\(543\) 0 0
\(544\) 4.00000 0.171499
\(545\) 28.0000 1.19939
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 − 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 32.0000 1.36078
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) 0 0
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 18.0000 0.759284
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) 8.00000 0.336563
\(566\) 24.0000 1.00880
\(567\) 0 0
\(568\) 0 0
\(569\) −16.0000 −0.670755 −0.335377 − 0.942084i \(-0.608864\pi\)
−0.335377 + 0.942084i \(0.608864\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 − 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) −1.00000 −0.0418121
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 − 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) 8.00000 0.332182
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) 4.00000 0.165663
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) −8.00000 −0.329355
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) 22.0000 0.901155
\(597\) 0 0
\(598\) 4.00000 0.163572
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 − 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 8.00000 0.326056
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) 20.0000 0.811775 0.405887 − 0.913923i \(-0.366962\pi\)
0.405887 + 0.913923i \(0.366962\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 28.0000 1.13369
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 22.0000 0.888572 0.444286 − 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) −6.00000 −0.242140
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) 6.00000 0.241551 0.120775 − 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 − 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 12.0000 0.481932
\(621\) 0 0
\(622\) −20.0000 −0.801927
\(623\) −20.0000 −0.801283
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) 6.00000 0.238856 0.119428 − 0.992843i \(-0.461894\pi\)
0.119428 + 0.992843i \(0.461894\pi\)
\(632\) −16.0000 −0.636446
\(633\) 0 0
\(634\) 22.0000 0.873732
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) −4.00000 −0.158362
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) −24.0000 −0.947943 −0.473972 − 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) 2.00000 0.0788723 0.0394362 − 0.999222i \(-0.487444\pi\)
0.0394362 + 0.999222i \(0.487444\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) 2.00000 0.0783260
\(653\) −8.00000 −0.313064 −0.156532 − 0.987673i \(-0.550031\pi\)
−0.156532 + 0.987673i \(0.550031\pi\)
\(654\) 0 0
\(655\) −24.0000 −0.937758
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) 16.0000 0.623745
\(659\) −40.0000 −1.55818 −0.779089 − 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) 46.0000 1.78919 0.894596 − 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) −14.0000 −0.544125
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 16.0000 0.619522
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) −14.0000 −0.540464
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 − 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 24.0000 0.922395 0.461197 − 0.887298i \(-0.347420\pi\)
0.461197 + 0.887298i \(0.347420\pi\)
\(678\) 0 0
\(679\) −28.0000 −1.07454
\(680\) −8.00000 −0.306786
\(681\) 0 0
\(682\) −6.00000 −0.229752
\(683\) −44.0000 −1.68361 −0.841807 − 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 0 0
\(685\) 20.0000 0.764161
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −30.0000 −1.14125 −0.570627 − 0.821209i \(-0.693300\pi\)
−0.570627 + 0.821209i \(0.693300\pi\)
\(692\) 20.0000 0.760286
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) 22.0000 0.832712
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) 16.0000 0.604312 0.302156 − 0.953259i \(-0.402294\pi\)
0.302156 + 0.953259i \(0.402294\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 24.0000 0.902613
\(708\) 0 0
\(709\) −14.0000 −0.525781 −0.262891 − 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) −16.0000 −0.597115
\(719\) 20.0000 0.745874 0.372937 − 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) −15.0000 −0.558242
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 − 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 2.00000 0.0741249
\(729\) 0 0
\(730\) 20.0000 0.740233
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) 50.0000 1.84679 0.923396 − 0.383849i \(-0.125402\pi\)
0.923396 + 0.383849i \(0.125402\pi\)
\(734\) −24.0000 −0.885856
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 2.00000 0.0736709
\(738\) 0 0
\(739\) 10.0000 0.367856 0.183928 − 0.982940i \(-0.441119\pi\)
0.183928 + 0.982940i \(0.441119\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) −8.00000 −0.293689
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −44.0000 −1.61204
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) 4.00000 0.146254
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 − 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) −20.0000 −0.727875
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 − 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) −2.00000 −0.0726433
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) −14.0000 −0.507500 −0.253750 − 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) 0 0
\(763\) 28.0000 1.01367
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) −4.00000 −0.144432
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 − 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 4.00000 0.144150
\(771\) 0 0
\(772\) −6.00000 −0.215945
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 0 0
\(775\) 6.00000 0.215526
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) 0 0
\(782\) −16.0000 −0.572159
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) −6.00000 −0.213877 −0.106938 − 0.994266i \(-0.534105\pi\)
−0.106938 + 0.994266i \(0.534105\pi\)
\(788\) −14.0000 −0.498729
\(789\) 0 0
\(790\) 32.0000 1.13851
\(791\) 8.00000 0.284447
\(792\) 0 0
\(793\) 14.0000 0.497155
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) −44.0000 −1.55856 −0.779280 − 0.626676i \(-0.784415\pi\)
−0.779280 + 0.626676i \(0.784415\pi\)
\(798\) 0 0
\(799\) −32.0000 −1.13208
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) −10.0000 −0.352892
\(804\) 0 0
\(805\) −16.0000 −0.563926
\(806\) 6.00000 0.211341
\(807\) 0 0
\(808\) −12.0000 −0.422159
\(809\) 24.0000 0.843795 0.421898 − 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) 42.0000 1.47482 0.737410 − 0.675446i \(-0.236049\pi\)
0.737410 + 0.675446i \(0.236049\pi\)
\(812\) 8.00000 0.280745
\(813\) 0 0
\(814\) −2.00000 −0.0701000
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) 30.0000 1.04893
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 6.00000 0.209401 0.104701 − 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) −20.0000 −0.697156 −0.348578 − 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) −20.0000 −0.695468 −0.347734 − 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 − 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 24.0000 0.833052
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) −12.0000 −0.415775
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 2.00000 0.0691714
\(837\) 0 0
\(838\) −20.0000 −0.690889
\(839\) −8.00000 −0.276191 −0.138095 − 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) −2.00000 −0.0688021
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 4.00000 0.137361
\(849\) 0 0
\(850\) −4.00000 −0.137199
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 − 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 28.0000 0.958140
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) 56.0000 1.91292 0.956462 − 0.291858i \(-0.0942733\pi\)
0.956462 + 0.291858i \(0.0942733\pi\)
\(858\) 0 0
\(859\) −32.0000 −1.09183 −0.545913 − 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) −8.00000 −0.272481
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −40.0000 −1.36004
\(866\) 30.0000 1.01944
\(867\) 0 0
\(868\) 12.0000 0.407307
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −2.00000 −0.0677674
\(872\) −14.0000 −0.474100
\(873\) 0 0
\(874\) −8.00000 −0.270604
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) −10.0000 −0.337676 −0.168838 − 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 24.0000 0.809961
\(879\) 0 0
\(880\) −2.00000 −0.0674200
\(881\) −36.0000 −1.21287 −0.606435 − 0.795133i \(-0.707401\pi\)
−0.606435 + 0.795133i \(0.707401\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 − 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 12.0000 0.402921 0.201460 − 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) −20.0000 −0.670402
\(891\) 0 0
\(892\) 26.0000 0.870544
\(893\) −16.0000 −0.535420
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 16.0000 0.533037
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) −4.00000 −0.133038
\(905\) 4.00000 0.132964
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 − 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) −4.00000 −0.132599
\(911\) 20.0000 0.662630 0.331315 − 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 14.0000 0.463079
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) −24.0000 −0.792550
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 − 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 8.00000 0.263752
\(921\) 0 0
\(922\) −22.0000 −0.724531
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) −6.00000 −0.197172
\(927\) 0 0
\(928\) −4.00000 −0.131306
\(929\) 6.00000 0.196854 0.0984268 − 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 16.0000 0.524097
\(933\) 0 0
\(934\) 8.00000 0.261768
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) −42.0000 −1.37208 −0.686040 − 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) −4.00000 −0.130605
\(939\) 0 0
\(940\) 16.0000 0.521862
\(941\) 30.0000 0.977972 0.488986 − 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −36.0000 −1.16984 −0.584921 − 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) −2.00000 −0.0648886
\(951\) 0 0
\(952\) −8.00000 −0.259281
\(953\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 24.0000 0.776622
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 0 0
\(959\) 20.0000 0.645834
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 2.00000 0.0644826
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 − 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −28.0000 −0.899026
\(971\) 16.0000 0.513464 0.256732 − 0.966483i \(-0.417354\pi\)
0.256732 + 0.966483i \(0.417354\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 34.0000 1.08943
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) 42.0000 1.34370 0.671850 − 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 10.0000 0.319601
\(980\) 6.00000 0.191663
\(981\) 0 0
\(982\) −24.0000 −0.765871
\(983\) −56.0000 −1.78612 −0.893061 − 0.449935i \(-0.851447\pi\)
−0.893061 + 0.449935i \(0.851447\pi\)
\(984\) 0 0
\(985\) 28.0000 0.892154
\(986\) −16.0000 −0.509544
\(987\) 0 0
\(988\) −2.00000 −0.0636285
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 − 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) −6.00000 −0.190500
\(993\) 0 0
\(994\) 0 0
\(995\) 32.0000 1.01447
\(996\) 0 0
\(997\) −42.0000 −1.33015 −0.665077 − 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) −38.0000 −1.20287
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2574.2.a.s.1.1 yes 1
3.2 odd 2 2574.2.a.l.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2574.2.a.l.1.1 ✓ 1 3.2 odd 2
2574.2.a.s.1.1 yes 1 1.1 even 1 trivial