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Properties

Label 4012.2.a.b.1.1
Level $4012$
Weight $2$
Character 4012.1
Self dual yes
Analytic conductor $32.036$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4012,2,Mod(1,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, 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("4012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.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: \(32.0359812909\)
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 \(\chi\) \(=\) 4012.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -1.00000 q^{5} -5.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -1.00000 q^{5} -5.00000 q^{7} +6.00000 q^{9} -2.00000 q^{11} +4.00000 q^{13} +3.00000 q^{15} +1.00000 q^{17} -3.00000 q^{19} +15.0000 q^{21} -4.00000 q^{23} -4.00000 q^{25} -9.00000 q^{27} -3.00000 q^{29} +8.00000 q^{31} +6.00000 q^{33} +5.00000 q^{35} -12.0000 q^{39} +3.00000 q^{41} +4.00000 q^{43} -6.00000 q^{45} +8.00000 q^{47} +18.0000 q^{49} -3.00000 q^{51} +7.00000 q^{53} +2.00000 q^{55} +9.00000 q^{57} -1.00000 q^{59} +10.0000 q^{61} -30.0000 q^{63} -4.00000 q^{65} +10.0000 q^{67} +12.0000 q^{69} -8.00000 q^{71} -10.0000 q^{73} +12.0000 q^{75} +10.0000 q^{77} +13.0000 q^{79} +9.00000 q^{81} -4.00000 q^{83} -1.00000 q^{85} +9.00000 q^{87} +14.0000 q^{89} -20.0000 q^{91} -24.0000 q^{93} +3.00000 q^{95} +2.00000 q^{97} -12.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 −1.73205 −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214 −0.223607 − 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −5.00000 −1.88982 −0.944911 − 0.327327i \(-0.893852\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 − 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) 15.0000 3.27327
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) 5.00000 0.845154
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) −12.0000 −1.92154
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −6.00000 −0.894427
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 18.0000 2.57143
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 7.00000 0.961524 0.480762 − 0.876851i \(-0.340360\pi\)
0.480762 + 0.876851i \(0.340360\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 9.00000 1.19208
\(58\) 0 0
\(59\) −1.00000 −0.130189
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) −30.0000 −3.77964
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 − 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 12.0000 1.38564
\(76\) 0 0
\(77\) 10.0000 1.13961
\(78\) 0 0
\(79\) 13.0000 1.46261 0.731307 − 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −1.00000 −0.108465
\(86\) 0 0
\(87\) 9.00000 0.964901
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −20.0000 −2.09657
\(92\) 0 0
\(93\) −24.0000 −2.48868
\(94\) 0 0
\(95\) 3.00000 0.307794
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) −12.0000 −1.20605
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 − 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) −15.0000 −1.46385
\(106\) 0 0
\(107\) −3.00000 −0.290021 −0.145010 − 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 − 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 24.0000 2.21880
\(118\) 0 0
\(119\) −5.00000 −0.458349
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −9.00000 −0.811503
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 − 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 15.0000 1.30066
\(134\) 0 0
\(135\) 9.00000 0.774597
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 − 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 3.00000 0.249136
\(146\) 0 0
\(147\) −54.0000 −4.45384
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 − 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 − 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −20.0000 −1.59617 −0.798087 − 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) 0 0
\(159\) −21.0000 −1.66541
\(160\) 0 0
\(161\) 20.0000 1.57622
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 0 0
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) −17.0000 −1.31550 −0.657750 − 0.753237i \(-0.728492\pi\)
−0.657750 + 0.753237i \(0.728492\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −18.0000 −1.37649
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 20.0000 1.51186
\(176\) 0 0
\(177\) 3.00000 0.225494
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 − 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 1.00000 0.0743294 0.0371647 − 0.999309i \(-0.488167\pi\)
0.0371647 + 0.999309i \(0.488167\pi\)
\(182\) 0 0
\(183\) −30.0000 −2.21766
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 0 0
\(189\) 45.0000 3.27327
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 − 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) 3.00000 0.215945 0.107972 − 0.994154i \(-0.465564\pi\)
0.107972 + 0.994154i \(0.465564\pi\)
\(194\) 0 0
\(195\) 12.0000 0.859338
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −5.00000 −0.354441 −0.177220 − 0.984171i \(-0.556711\pi\)
−0.177220 + 0.984171i \(0.556711\pi\)
\(200\) 0 0
\(201\) −30.0000 −2.11604
\(202\) 0 0
\(203\) 15.0000 1.05279
\(204\) 0 0
\(205\) −3.00000 −0.209529
\(206\) 0 0
\(207\) −24.0000 −1.66812
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 6.00000 0.413057 0.206529 − 0.978441i \(-0.433783\pi\)
0.206529 + 0.978441i \(0.433783\pi\)
\(212\) 0 0
\(213\) 24.0000 1.64445
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −40.0000 −2.71538
\(218\) 0 0
\(219\) 30.0000 2.02721
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −24.0000 −1.60000
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 − 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) −30.0000 −1.97386
\(232\) 0 0
\(233\) 4.00000 0.262049 0.131024 − 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 0 0
\(237\) −39.0000 −2.53332
\(238\) 0 0
\(239\) 11.0000 0.711531 0.355765 − 0.934575i \(-0.384220\pi\)
0.355765 + 0.934575i \(0.384220\pi\)
\(240\) 0 0
\(241\) −11.0000 −0.708572 −0.354286 − 0.935137i \(-0.615276\pi\)
−0.354286 + 0.935137i \(0.615276\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −18.0000 −1.14998
\(246\) 0 0
\(247\) −12.0000 −0.763542
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 − 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) 3.00000 0.187867
\(256\) 0 0
\(257\) −27.0000 −1.68421 −0.842107 − 0.539311i \(-0.818685\pi\)
−0.842107 + 0.539311i \(0.818685\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) 0 0
\(263\) −21.0000 −1.29492 −0.647458 − 0.762101i \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) 0 0
\(265\) −7.00000 −0.430007
\(266\) 0 0
\(267\) −42.0000 −2.57036
\(268\) 0 0
\(269\) 12.0000 0.731653 0.365826 − 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) 29.0000 1.76162 0.880812 − 0.473466i \(-0.156997\pi\)
0.880812 + 0.473466i \(0.156997\pi\)
\(272\) 0 0
\(273\) 60.0000 3.63137
\(274\) 0 0
\(275\) 8.00000 0.482418
\(276\) 0 0
\(277\) 11.0000 0.660926 0.330463 − 0.943819i \(-0.392795\pi\)
0.330463 + 0.943819i \(0.392795\pi\)
\(278\) 0 0
\(279\) 48.0000 2.87368
\(280\) 0 0
\(281\) 27.0000 1.61068 0.805342 − 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) −9.00000 −0.533114
\(286\) 0 0
\(287\) −15.0000 −0.885422
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) 0 0
\(293\) −11.0000 −0.642627 −0.321313 − 0.946973i \(-0.604124\pi\)
−0.321313 + 0.946973i \(0.604124\pi\)
\(294\) 0 0
\(295\) 1.00000 0.0582223
\(296\) 0 0
\(297\) 18.0000 1.04447
\(298\) 0 0
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) −25.0000 −1.42683 −0.713413 − 0.700744i \(-0.752851\pi\)
−0.713413 + 0.700744i \(0.752851\pi\)
\(308\) 0 0
\(309\) 48.0000 2.73062
\(310\) 0 0
\(311\) −13.0000 −0.737162 −0.368581 − 0.929596i \(-0.620156\pi\)
−0.368581 + 0.929596i \(0.620156\pi\)
\(312\) 0 0
\(313\) −2.00000 −0.113047 −0.0565233 − 0.998401i \(-0.518002\pi\)
−0.0565233 + 0.998401i \(0.518002\pi\)
\(314\) 0 0
\(315\) 30.0000 1.69031
\(316\) 0 0
\(317\) 6.00000 0.336994 0.168497 − 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 0 0
\(325\) −16.0000 −0.887520
\(326\) 0 0
\(327\) −6.00000 −0.331801
\(328\) 0 0
\(329\) −40.0000 −2.20527
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 − 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −10.0000 −0.546358
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 − 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) 36.0000 1.95525
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) −55.0000 −2.96972
\(344\) 0 0
\(345\) −12.0000 −0.646058
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −24.0000 −1.28469 −0.642345 − 0.766415i \(-0.722038\pi\)
−0.642345 + 0.766415i \(0.722038\pi\)
\(350\) 0 0
\(351\) −36.0000 −1.92154
\(352\) 0 0
\(353\) −12.0000 −0.638696 −0.319348 − 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 0 0
\(357\) 15.0000 0.793884
\(358\) 0 0
\(359\) 17.0000 0.897226 0.448613 − 0.893726i \(-0.351918\pi\)
0.448613 + 0.893726i \(0.351918\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 0 0
\(363\) 21.0000 1.10221
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 − 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) −35.0000 −1.81711
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 − 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) −27.0000 −1.39427
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 − 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) 21.0000 1.07586
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 − 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) −10.0000 −0.509647
\(386\) 0 0
\(387\) 24.0000 1.21999
\(388\) 0 0
\(389\) 10.0000 0.507020 0.253510 − 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) −13.0000 −0.654101
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 − 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) −45.0000 −2.25282
\(400\) 0 0
\(401\) −32.0000 −1.59800 −0.799002 − 0.601329i \(-0.794638\pi\)
−0.799002 + 0.601329i \(0.794638\pi\)
\(402\) 0 0
\(403\) 32.0000 1.59403
\(404\) 0 0
\(405\) −9.00000 −0.447214
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 16.0000 0.791149 0.395575 − 0.918434i \(-0.370545\pi\)
0.395575 + 0.918434i \(0.370545\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) 0 0
\(413\) 5.00000 0.246034
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 − 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) 0 0
\(423\) 48.0000 2.33384
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) −50.0000 −2.41967
\(428\) 0 0
\(429\) 24.0000 1.15873
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −33.0000 −1.58588 −0.792939 − 0.609301i \(-0.791450\pi\)
−0.792939 + 0.609301i \(0.791450\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) 0 0
\(437\) 12.0000 0.574038
\(438\) 0 0
\(439\) 12.0000 0.572729 0.286364 − 0.958121i \(-0.407553\pi\)
0.286364 + 0.958121i \(0.407553\pi\)
\(440\) 0 0
\(441\) 108.000 5.14286
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −14.0000 −0.663664
\(446\) 0 0
\(447\) 54.0000 2.55411
\(448\) 0 0
\(449\) 33.0000 1.55737 0.778683 − 0.627417i \(-0.215888\pi\)
0.778683 + 0.627417i \(0.215888\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 0 0
\(453\) 42.0000 1.97333
\(454\) 0 0
\(455\) 20.0000 0.937614
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) −9.00000 −0.420084
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 − 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 0 0
\(465\) 24.0000 1.11297
\(466\) 0 0
\(467\) −6.00000 −0.277647 −0.138823 − 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) −50.0000 −2.30879
\(470\) 0 0
\(471\) 60.0000 2.76465
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) 12.0000 0.550598
\(476\) 0 0
\(477\) 42.0000 1.92305
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 − 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −60.0000 −2.73009
\(484\) 0 0
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −13.0000 −0.589086 −0.294543 − 0.955638i \(-0.595167\pi\)
−0.294543 + 0.955638i \(0.595167\pi\)
\(488\) 0 0
\(489\) −48.0000 −2.17064
\(490\) 0 0
\(491\) 27.0000 1.21849 0.609246 − 0.792981i \(-0.291472\pi\)
0.609246 + 0.792981i \(0.291472\pi\)
\(492\) 0 0
\(493\) −3.00000 −0.135113
\(494\) 0 0
\(495\) 12.0000 0.539360
\(496\) 0 0
\(497\) 40.0000 1.79425
\(498\) 0 0
\(499\) −19.0000 −0.850557 −0.425278 − 0.905063i \(-0.639824\pi\)
−0.425278 + 0.905063i \(0.639824\pi\)
\(500\) 0 0
\(501\) 51.0000 2.27851
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 − 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −8.00000 −0.354594 −0.177297 − 0.984157i \(-0.556735\pi\)
−0.177297 + 0.984157i \(0.556735\pi\)
\(510\) 0 0
\(511\) 50.0000 2.21187
\(512\) 0 0
\(513\) 27.0000 1.19208
\(514\) 0 0
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −26.0000 −1.13908 −0.569540 − 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) 0 0
\(523\) 11.0000 0.480996 0.240498 − 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(524\) 0 0
\(525\) −60.0000 −2.61861
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 3.00000 0.129701
\(536\) 0 0
\(537\) −60.0000 −2.58919
\(538\) 0 0
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 − 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) 0 0
\(543\) −3.00000 −0.128742
\(544\) 0 0
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 − 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) 60.0000 2.56074
\(550\) 0 0
\(551\) 9.00000 0.383413
\(552\) 0 0
\(553\) −65.0000 −2.76408
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.00000 −0.0423714 −0.0211857 − 0.999776i \(-0.506744\pi\)
−0.0211857 + 0.999776i \(0.506744\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 0 0
\(563\) −44.0000 −1.85438 −0.927189 − 0.374593i \(-0.877783\pi\)
−0.927189 + 0.374593i \(0.877783\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 0 0
\(567\) −45.0000 −1.88982
\(568\) 0 0
\(569\) 28.0000 1.17382 0.586911 − 0.809652i \(-0.300344\pi\)
0.586911 + 0.809652i \(0.300344\pi\)
\(570\) 0 0
\(571\) −22.0000 −0.920671 −0.460336 − 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) −54.0000 −2.25588
\(574\) 0 0
\(575\) 16.0000 0.667246
\(576\) 0 0
\(577\) −23.0000 −0.957503 −0.478751 − 0.877951i \(-0.658910\pi\)
−0.478751 + 0.877951i \(0.658910\pi\)
\(578\) 0 0
\(579\) −9.00000 −0.374027
\(580\) 0 0
\(581\) 20.0000 0.829740
\(582\) 0 0
\(583\) −14.0000 −0.579821
\(584\) 0 0
\(585\) −24.0000 −0.992278
\(586\) 0 0
\(587\) −38.0000 −1.56843 −0.784214 − 0.620491i \(-0.786934\pi\)
−0.784214 + 0.620491i \(0.786934\pi\)
\(588\) 0 0
\(589\) −24.0000 −0.988903
\(590\) 0 0
\(591\) 54.0000 2.22126
\(592\) 0 0
\(593\) 1.00000 0.0410651 0.0205325 − 0.999789i \(-0.493464\pi\)
0.0205325 + 0.999789i \(0.493464\pi\)
\(594\) 0 0
\(595\) 5.00000 0.204980
\(596\) 0 0
\(597\) 15.0000 0.613909
\(598\) 0 0
\(599\) 33.0000 1.34834 0.674172 − 0.738575i \(-0.264501\pi\)
0.674172 + 0.738575i \(0.264501\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 − 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 0 0
\(603\) 60.0000 2.44339
\(604\) 0 0
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) 31.0000 1.25825 0.629126 − 0.777304i \(-0.283413\pi\)
0.629126 + 0.777304i \(0.283413\pi\)
\(608\) 0 0
\(609\) −45.0000 −1.82349
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) 0 0
\(613\) 26.0000 1.05013 0.525065 − 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 0 0
\(615\) 9.00000 0.362915
\(616\) 0 0
\(617\) −7.00000 −0.281809 −0.140905 − 0.990023i \(-0.545001\pi\)
−0.140905 + 0.990023i \(0.545001\pi\)
\(618\) 0 0
\(619\) 11.0000 0.442127 0.221064 − 0.975259i \(-0.429047\pi\)
0.221064 + 0.975259i \(0.429047\pi\)
\(620\) 0 0
\(621\) 36.0000 1.44463
\(622\) 0 0
\(623\) −70.0000 −2.80449
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −18.0000 −0.718851
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 − 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 0 0
\(633\) −18.0000 −0.715436
\(634\) 0 0
\(635\) 7.00000 0.277787
\(636\) 0 0
\(637\) 72.0000 2.85274
\(638\) 0 0
\(639\) −48.0000 −1.89885
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −23.0000 −0.907031 −0.453516 − 0.891248i \(-0.649830\pi\)
−0.453516 + 0.891248i \(0.649830\pi\)
\(644\) 0 0
\(645\) 12.0000 0.472500
\(646\) 0 0
\(647\) −9.00000 −0.353827 −0.176913 − 0.984226i \(-0.556611\pi\)
−0.176913 + 0.984226i \(0.556611\pi\)
\(648\) 0 0
\(649\) 2.00000 0.0785069
\(650\) 0 0
\(651\) 120.000 4.70317
\(652\) 0 0
\(653\) 11.0000 0.430463 0.215232 − 0.976563i \(-0.430949\pi\)
0.215232 + 0.976563i \(0.430949\pi\)
\(654\) 0 0
\(655\) −4.00000 −0.156293
\(656\) 0 0
\(657\) −60.0000 −2.34082
\(658\) 0 0
\(659\) −14.0000 −0.545363 −0.272681 − 0.962104i \(-0.587910\pi\)
−0.272681 + 0.962104i \(0.587910\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 − 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 0 0
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) −15.0000 −0.581675
\(666\) 0 0
\(667\) 12.0000 0.464642
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −20.0000 −0.772091
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 − 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 0 0
\(675\) 36.0000 1.38564
\(676\) 0 0
\(677\) 22.0000 0.845529 0.422764 − 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) −54.0000 −2.06928
\(682\) 0 0
\(683\) −6.00000 −0.229584 −0.114792 − 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(684\) 0 0
\(685\) −3.00000 −0.114624
\(686\) 0 0
\(687\) 42.0000 1.60240
\(688\) 0 0
\(689\) 28.0000 1.06672
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 − 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 0 0
\(693\) 60.0000 2.27921
\(694\) 0 0
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) 3.00000 0.113633
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 − 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 24.0000 0.903892
\(706\) 0 0
\(707\) −10.0000 −0.376089
\(708\) 0 0
\(709\) 25.0000 0.938895 0.469447 − 0.882960i \(-0.344453\pi\)
0.469447 + 0.882960i \(0.344453\pi\)
\(710\) 0 0
\(711\) 78.0000 2.92523
\(712\) 0 0
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 0 0
\(717\) −33.0000 −1.23241
\(718\) 0 0
\(719\) −26.0000 −0.969636 −0.484818 − 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(720\) 0 0
\(721\) 80.0000 2.97936
\(722\) 0 0
\(723\) 33.0000 1.22728
\(724\) 0 0
\(725\) 12.0000 0.445669
\(726\) 0 0
\(727\) 48.0000 1.78022 0.890111 − 0.455744i \(-0.150627\pi\)
0.890111 + 0.455744i \(0.150627\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) 42.0000 1.55131 0.775653 − 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 0 0
\(735\) 54.0000 1.99182
\(736\) 0 0
\(737\) −20.0000 −0.736709
\(738\) 0 0
\(739\) −8.00000 −0.294285 −0.147142 − 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 0 0
\(741\) 36.0000 1.32249
\(742\) 0 0
\(743\) −40.0000 −1.46746 −0.733729 − 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 0 0
\(747\) −24.0000 −0.878114
\(748\) 0 0
\(749\) 15.0000 0.548088
\(750\) 0 0
\(751\) −38.0000 −1.38664 −0.693320 − 0.720630i \(-0.743853\pi\)
−0.693320 + 0.720630i \(0.743853\pi\)
\(752\) 0 0
\(753\) 63.0000 2.29585
\(754\) 0 0
\(755\) 14.0000 0.509512
\(756\) 0 0
\(757\) 9.00000 0.327111 0.163555 − 0.986534i \(-0.447704\pi\)
0.163555 + 0.986534i \(0.447704\pi\)
\(758\) 0 0
\(759\) −24.0000 −0.871145
\(760\) 0 0
\(761\) −15.0000 −0.543750 −0.271875 − 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) 0 0
\(763\) −10.0000 −0.362024
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) −4.00000 −0.144432
\(768\) 0 0
\(769\) −42.0000 −1.51456 −0.757279 − 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(770\) 0 0
\(771\) 81.0000 2.91714
\(772\) 0 0
\(773\) −14.0000 −0.503545 −0.251773 − 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 0 0
\(775\) −32.0000 −1.14947
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −9.00000 −0.322458
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) 0 0
\(783\) 27.0000 0.964901
\(784\) 0 0
\(785\) 20.0000 0.713831
\(786\) 0 0
\(787\) 44.0000 1.56843 0.784215 − 0.620489i \(-0.213066\pi\)
0.784215 + 0.620489i \(0.213066\pi\)
\(788\) 0 0
\(789\) 63.0000 2.24286
\(790\) 0 0
\(791\) 60.0000 2.13335
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) 0 0
\(795\) 21.0000 0.744793
\(796\) 0 0
\(797\) 54.0000 1.91278 0.956389 − 0.292096i \(-0.0943526\pi\)
0.956389 + 0.292096i \(0.0943526\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 84.0000 2.96799
\(802\) 0 0
\(803\) 20.0000 0.705785
\(804\) 0 0
\(805\) −20.0000 −0.704907
\(806\) 0 0
\(807\) −36.0000 −1.26726
\(808\) 0 0
\(809\) 16.0000 0.562530 0.281265 − 0.959630i \(-0.409246\pi\)
0.281265 + 0.959630i \(0.409246\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) −87.0000 −3.05122
\(814\) 0 0
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) 0 0
\(819\) −120.000 −4.19314
\(820\) 0 0
\(821\) 46.0000 1.60541 0.802706 − 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 0 0
\(823\) −46.0000 −1.60346 −0.801730 − 0.597687i \(-0.796087\pi\)
−0.801730 + 0.597687i \(0.796087\pi\)
\(824\) 0 0
\(825\) −24.0000 −0.835573
\(826\) 0 0
\(827\) 20.0000 0.695468 0.347734 − 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 0 0
\(829\) 45.0000 1.56291 0.781457 − 0.623959i \(-0.214477\pi\)
0.781457 + 0.623959i \(0.214477\pi\)
\(830\) 0 0
\(831\) −33.0000 −1.14476
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 17.0000 0.588309
\(836\) 0 0
\(837\) −72.0000 −2.48868
\(838\) 0 0
\(839\) 30.0000 1.03572 0.517858 − 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −81.0000 −2.78979
\(844\) 0 0
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 35.0000 1.20261
\(848\) 0 0
\(849\) −60.0000 −2.05919
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 25.0000 0.855984 0.427992 − 0.903783i \(-0.359221\pi\)
0.427992 + 0.903783i \(0.359221\pi\)
\(854\) 0 0
\(855\) 18.0000 0.615587
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 48.0000 1.63774 0.818869 − 0.573980i \(-0.194601\pi\)
0.818869 + 0.573980i \(0.194601\pi\)
\(860\) 0 0
\(861\) 45.0000 1.53360
\(862\) 0 0
\(863\) −6.00000 −0.204242 −0.102121 − 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 0 0
\(867\) −3.00000 −0.101885
\(868\) 0 0
\(869\) −26.0000 −0.881990
\(870\) 0 0
\(871\) 40.0000 1.35535
\(872\) 0 0
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) −45.0000 −1.52128
\(876\) 0 0
\(877\) −35.0000 −1.18187 −0.590933 − 0.806721i \(-0.701240\pi\)
−0.590933 + 0.806721i \(0.701240\pi\)
\(878\) 0 0
\(879\) 33.0000 1.11306
\(880\) 0 0
\(881\) 24.0000 0.808581 0.404290 − 0.914631i \(-0.367519\pi\)
0.404290 + 0.914631i \(0.367519\pi\)
\(882\) 0 0
\(883\) 51.0000 1.71629 0.858143 − 0.513410i \(-0.171618\pi\)
0.858143 + 0.513410i \(0.171618\pi\)
\(884\) 0 0
\(885\) −3.00000 −0.100844
\(886\) 0 0
\(887\) −18.0000 −0.604381 −0.302190 − 0.953248i \(-0.597718\pi\)
−0.302190 + 0.953248i \(0.597718\pi\)
\(888\) 0 0
\(889\) 35.0000 1.17386
\(890\) 0 0
\(891\) −18.0000 −0.603023
\(892\) 0 0
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) −20.0000 −0.668526
\(896\) 0 0
\(897\) 48.0000 1.60267
\(898\) 0 0
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 7.00000 0.233204
\(902\) 0 0
\(903\) 60.0000 1.99667
\(904\) 0 0
\(905\) −1.00000 −0.0332411
\(906\) 0 0
\(907\) 17.0000 0.564476 0.282238 − 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(908\) 0 0
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −37.0000 −1.22586 −0.612932 − 0.790135i \(-0.710010\pi\)
−0.612932 + 0.790135i \(0.710010\pi\)
\(912\) 0 0
\(913\) 8.00000 0.264761
\(914\) 0 0
\(915\) 30.0000 0.991769
\(916\) 0 0
\(917\) −20.0000 −0.660458
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 − 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 75.0000 2.47133
\(922\) 0 0
\(923\) −32.0000 −1.05329
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −96.0000 −3.15305
\(928\) 0 0
\(929\) −38.0000 −1.24674 −0.623370 − 0.781927i \(-0.714237\pi\)
−0.623370 + 0.781927i \(0.714237\pi\)
\(930\) 0 0
\(931\) −54.0000 −1.76978
\(932\) 0 0
\(933\) 39.0000 1.27680
\(934\) 0 0
\(935\) 2.00000 0.0654070
\(936\) 0 0
\(937\) 12.0000 0.392023 0.196011 − 0.980602i \(-0.437201\pi\)
0.196011 + 0.980602i \(0.437201\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 8.00000 0.260793 0.130396 − 0.991462i \(-0.458375\pi\)
0.130396 + 0.991462i \(0.458375\pi\)
\(942\) 0 0
\(943\) −12.0000 −0.390774
\(944\) 0 0
\(945\) −45.0000 −1.46385
\(946\) 0 0
\(947\) 45.0000 1.46230 0.731152 − 0.682215i \(-0.238983\pi\)
0.731152 + 0.682215i \(0.238983\pi\)
\(948\) 0 0
\(949\) −40.0000 −1.29845
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) −54.0000 −1.74923 −0.874616 − 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 0 0
\(955\) −18.0000 −0.582466
\(956\) 0 0
\(957\) −18.0000 −0.581857
\(958\) 0 0
\(959\) −15.0000 −0.484375
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −18.0000 −0.580042
\(964\) 0 0
\(965\) −3.00000 −0.0965734
\(966\) 0 0
\(967\) −12.0000 −0.385894 −0.192947 − 0.981209i \(-0.561805\pi\)
−0.192947 + 0.981209i \(0.561805\pi\)
\(968\) 0 0
\(969\) 9.00000 0.289122
\(970\) 0 0
\(971\) −57.0000 −1.82922 −0.914609 − 0.404341i \(-0.867501\pi\)
−0.914609 + 0.404341i \(0.867501\pi\)
\(972\) 0 0
\(973\) 20.0000 0.641171
\(974\) 0 0
\(975\) 48.0000 1.53723
\(976\) 0 0
\(977\) −38.0000 −1.21573 −0.607864 − 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) 0 0
\(979\) −28.0000 −0.894884
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) 0 0
\(983\) −50.0000 −1.59475 −0.797376 − 0.603483i \(-0.793779\pi\)
−0.797376 + 0.603483i \(0.793779\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 0 0
\(987\) 120.000 3.81964
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 − 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) 0 0
\(993\) −51.0000 −1.61844
\(994\) 0 0
\(995\) 5.00000 0.158511
\(996\) 0 0
\(997\) 1.00000 0.0316703 0.0158352 − 0.999875i \(-0.494959\pi\)
0.0158352 + 0.999875i \(0.494959\pi\)
\(998\) 0 0
\(999\) 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 4012.2.a.b.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4012.2.a.b.1.1 ✓ 1 1.1 even 1 trivial