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Properties

Label 900.4.a.q.1.1
Level $900$
Weight $4$
Character 900.1
Self dual yes
Analytic conductor $53.102$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [900,4,Mod(1,900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("900.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 900.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,28,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(53.1017190052\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 900.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+28.0000 q^{7} +24.0000 q^{11} +70.0000 q^{13} +102.000 q^{17} +20.0000 q^{19} -72.0000 q^{23} -306.000 q^{29} -136.000 q^{31} +214.000 q^{37} +150.000 q^{41} +292.000 q^{43} -72.0000 q^{47} +441.000 q^{49} -414.000 q^{53} +744.000 q^{59} -418.000 q^{61} -188.000 q^{67} -480.000 q^{71} -434.000 q^{73} +672.000 q^{77} +1352.00 q^{79} -612.000 q^{83} +30.0000 q^{89} +1960.00 q^{91} +286.000 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 28.0000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 24.0000 0.657843 0.328921 − 0.944357i \(-0.393315\pi\)
0.328921 + 0.944357i \(0.393315\pi\)
\(12\) 0 0
\(13\) 70.0000 1.49342 0.746712 − 0.665148i \(-0.231631\pi\)
0.746712 + 0.665148i \(0.231631\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 102.000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 20.0000 0.241490 0.120745 − 0.992684i \(-0.461472\pi\)
0.120745 + 0.992684i \(0.461472\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −72.0000 −0.652741 −0.326370 − 0.945242i \(-0.605826\pi\)
−0.326370 + 0.945242i \(0.605826\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −306.000 −1.95941 −0.979703 − 0.200455i \(-0.935758\pi\)
−0.979703 + 0.200455i \(0.935758\pi\)
\(30\) 0 0
\(31\) −136.000 −0.787946 −0.393973 − 0.919122i \(-0.628900\pi\)
−0.393973 + 0.919122i \(0.628900\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 214.000 0.950848 0.475424 − 0.879757i \(-0.342295\pi\)
0.475424 + 0.879757i \(0.342295\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 150.000 0.571367 0.285684 − 0.958324i \(-0.407779\pi\)
0.285684 + 0.958324i \(0.407779\pi\)
\(42\) 0 0
\(43\) 292.000 1.03557 0.517786 − 0.855510i \(-0.326756\pi\)
0.517786 + 0.855510i \(0.326756\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −72.0000 −0.223453 −0.111726 − 0.993739i \(-0.535638\pi\)
−0.111726 + 0.993739i \(0.535638\pi\)
\(48\) 0 0
\(49\) 441.000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −414.000 −1.07297 −0.536484 − 0.843911i \(-0.680248\pi\)
−0.536484 + 0.843911i \(0.680248\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 744.000 1.64170 0.820852 − 0.571141i \(-0.193499\pi\)
0.820852 + 0.571141i \(0.193499\pi\)
\(60\) 0 0
\(61\) −418.000 −0.877367 −0.438684 − 0.898642i \(-0.644555\pi\)
−0.438684 + 0.898642i \(0.644555\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −188.000 −0.342804 −0.171402 − 0.985201i \(-0.554830\pi\)
−0.171402 + 0.985201i \(0.554830\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −480.000 −0.802331 −0.401166 − 0.916006i \(-0.631395\pi\)
−0.401166 + 0.916006i \(0.631395\pi\)
\(72\) 0 0
\(73\) −434.000 −0.695834 −0.347917 − 0.937525i \(-0.613111\pi\)
−0.347917 + 0.937525i \(0.613111\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 672.000 0.994565
\(78\) 0 0
\(79\) 1352.00 1.92547 0.962733 − 0.270452i \(-0.0871732\pi\)
0.962733 + 0.270452i \(0.0871732\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −612.000 −0.809346 −0.404673 − 0.914461i \(-0.632615\pi\)
−0.404673 + 0.914461i \(0.632615\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 30.0000 0.0357303 0.0178651 − 0.999840i \(-0.494313\pi\)
0.0178651 + 0.999840i \(0.494313\pi\)
\(90\) 0 0
\(91\) 1960.00 2.25784
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 286.000 0.299370 0.149685 − 0.988734i \(-0.452174\pi\)
0.149685 + 0.988734i \(0.452174\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1542.00 1.51916 0.759578 − 0.650416i \(-0.225406\pi\)
0.759578 + 0.650416i \(0.225406\pi\)
\(102\) 0 0
\(103\) −1172.00 −1.12117 −0.560585 − 0.828097i \(-0.689424\pi\)
−0.560585 + 0.828097i \(0.689424\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1956.00 1.76723 0.883615 − 0.468214i \(-0.155102\pi\)
0.883615 + 0.468214i \(0.155102\pi\)
\(108\) 0 0
\(109\) −1858.00 −1.63270 −0.816349 − 0.577559i \(-0.804005\pi\)
−0.816349 + 0.577559i \(0.804005\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 174.000 0.144854 0.0724272 − 0.997374i \(-0.476926\pi\)
0.0724272 + 0.997374i \(0.476926\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2856.00 2.20008
\(120\) 0 0
\(121\) −755.000 −0.567243
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2068.00 1.44492 0.722462 − 0.691411i \(-0.243010\pi\)
0.722462 + 0.691411i \(0.243010\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −312.000 −0.208088 −0.104044 − 0.994573i \(-0.533178\pi\)
−0.104044 + 0.994573i \(0.533178\pi\)
\(132\) 0 0
\(133\) 560.000 0.365099
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2646.00 1.65010 0.825048 − 0.565063i \(-0.191148\pi\)
0.825048 + 0.565063i \(0.191148\pi\)
\(138\) 0 0
\(139\) −1276.00 −0.778625 −0.389313 − 0.921106i \(-0.627287\pi\)
−0.389313 + 0.921106i \(0.627287\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1680.00 0.982438
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3198.00 1.75832 0.879162 − 0.476522i \(-0.158103\pi\)
0.879162 + 0.476522i \(0.158103\pi\)
\(150\) 0 0
\(151\) −760.000 −0.409589 −0.204794 − 0.978805i \(-0.565653\pi\)
−0.204794 + 0.978805i \(0.565653\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 166.000 0.0843837 0.0421919 − 0.999110i \(-0.486566\pi\)
0.0421919 + 0.999110i \(0.486566\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2016.00 −0.986851
\(162\) 0 0
\(163\) −3020.00 −1.45119 −0.725597 − 0.688120i \(-0.758436\pi\)
−0.725597 + 0.688120i \(0.758436\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −984.000 −0.455953 −0.227977 − 0.973667i \(-0.573211\pi\)
−0.227977 + 0.973667i \(0.573211\pi\)
\(168\) 0 0
\(169\) 2703.00 1.23031
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1962.00 0.862243 0.431122 − 0.902294i \(-0.358118\pi\)
0.431122 + 0.902294i \(0.358118\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −576.000 −0.240515 −0.120258 − 0.992743i \(-0.538372\pi\)
−0.120258 + 0.992743i \(0.538372\pi\)
\(180\) 0 0
\(181\) −1210.00 −0.496898 −0.248449 − 0.968645i \(-0.579921\pi\)
−0.248449 + 0.968645i \(0.579921\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2448.00 0.957302
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3384.00 −1.28198 −0.640989 − 0.767550i \(-0.721475\pi\)
−0.640989 + 0.767550i \(0.721475\pi\)
\(192\) 0 0
\(193\) 2038.00 0.760096 0.380048 − 0.924967i \(-0.375908\pi\)
0.380048 + 0.924967i \(0.375908\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4098.00 1.48208 0.741042 − 0.671459i \(-0.234332\pi\)
0.741042 + 0.671459i \(0.234332\pi\)
\(198\) 0 0
\(199\) −2248.00 −0.800786 −0.400393 − 0.916343i \(-0.631126\pi\)
−0.400393 + 0.916343i \(0.631126\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8568.00 −2.96234
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 480.000 0.158863
\(210\) 0 0
\(211\) 3260.00 1.06364 0.531819 − 0.846858i \(-0.321509\pi\)
0.531819 + 0.846858i \(0.321509\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −3808.00 −1.19126
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7140.00 2.17325
\(222\) 0 0
\(223\) 2980.00 0.894868 0.447434 − 0.894317i \(-0.352338\pi\)
0.447434 + 0.894317i \(0.352338\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3180.00 −0.929797 −0.464899 − 0.885364i \(-0.653909\pi\)
−0.464899 + 0.885364i \(0.653909\pi\)
\(228\) 0 0
\(229\) 3374.00 0.973625 0.486813 − 0.873506i \(-0.338159\pi\)
0.486813 + 0.873506i \(0.338159\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1950.00 0.548278 0.274139 − 0.961690i \(-0.411607\pi\)
0.274139 + 0.961690i \(0.411607\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2232.00 −0.604084 −0.302042 − 0.953295i \(-0.597668\pi\)
−0.302042 + 0.953295i \(0.597668\pi\)
\(240\) 0 0
\(241\) −1822.00 −0.486993 −0.243497 − 0.969902i \(-0.578294\pi\)
−0.243497 + 0.969902i \(0.578294\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1400.00 0.360647
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1488.00 −0.374190 −0.187095 − 0.982342i \(-0.559907\pi\)
−0.187095 + 0.982342i \(0.559907\pi\)
\(252\) 0 0
\(253\) −1728.00 −0.429401
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2994.00 −0.726695 −0.363347 − 0.931654i \(-0.618366\pi\)
−0.363347 + 0.931654i \(0.618366\pi\)
\(258\) 0 0
\(259\) 5992.00 1.43755
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2472.00 −0.579582 −0.289791 − 0.957090i \(-0.593586\pi\)
−0.289791 + 0.957090i \(0.593586\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3954.00 −0.896207 −0.448103 − 0.893982i \(-0.647900\pi\)
−0.448103 + 0.893982i \(0.647900\pi\)
\(270\) 0 0
\(271\) −2176.00 −0.487759 −0.243879 − 0.969806i \(-0.578420\pi\)
−0.243879 + 0.969806i \(0.578420\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1034.00 −0.224285 −0.112143 − 0.993692i \(-0.535771\pi\)
−0.112143 + 0.993692i \(0.535771\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6654.00 1.41261 0.706307 − 0.707906i \(-0.250360\pi\)
0.706307 + 0.707906i \(0.250360\pi\)
\(282\) 0 0
\(283\) 1756.00 0.368846 0.184423 − 0.982847i \(-0.440958\pi\)
0.184423 + 0.982847i \(0.440958\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4200.00 0.863826
\(288\) 0 0
\(289\) 5491.00 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3234.00 0.644820 0.322410 − 0.946600i \(-0.395507\pi\)
0.322410 + 0.946600i \(0.395507\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5040.00 −0.974818
\(300\) 0 0
\(301\) 8176.00 1.56564
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2036.00 −0.378504 −0.189252 − 0.981929i \(-0.560606\pi\)
−0.189252 + 0.981929i \(0.560606\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 96.0000 0.0175037 0.00875187 − 0.999962i \(-0.497214\pi\)
0.00875187 + 0.999962i \(0.497214\pi\)
\(312\) 0 0
\(313\) −1202.00 −0.217064 −0.108532 − 0.994093i \(-0.534615\pi\)
−0.108532 + 0.994093i \(0.534615\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3798.00 −0.672924 −0.336462 − 0.941697i \(-0.609230\pi\)
−0.336462 + 0.941697i \(0.609230\pi\)
\(318\) 0 0
\(319\) −7344.00 −1.28898
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2040.00 0.351420
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2016.00 −0.337829
\(330\) 0 0
\(331\) −5668.00 −0.941213 −0.470606 − 0.882343i \(-0.655965\pi\)
−0.470606 + 0.882343i \(0.655965\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 454.000 0.0733856 0.0366928 − 0.999327i \(-0.488318\pi\)
0.0366928 + 0.999327i \(0.488318\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3264.00 −0.518345
\(342\) 0 0
\(343\) 2744.00 0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5604.00 −0.866970 −0.433485 − 0.901161i \(-0.642716\pi\)
−0.433485 + 0.901161i \(0.642716\pi\)
\(348\) 0 0
\(349\) −11266.0 −1.72795 −0.863976 − 0.503533i \(-0.832033\pi\)
−0.863976 + 0.503533i \(0.832033\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6426.00 −0.968899 −0.484450 − 0.874819i \(-0.660980\pi\)
−0.484450 + 0.874819i \(0.660980\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6936.00 −1.01969 −0.509844 − 0.860267i \(-0.670297\pi\)
−0.509844 + 0.860267i \(0.670297\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 388.000 0.0551865 0.0275932 − 0.999619i \(-0.491216\pi\)
0.0275932 + 0.999619i \(0.491216\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −11592.0 −1.62217
\(372\) 0 0
\(373\) 8062.00 1.11913 0.559564 − 0.828787i \(-0.310969\pi\)
0.559564 + 0.828787i \(0.310969\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −21420.0 −2.92622
\(378\) 0 0
\(379\) −3388.00 −0.459182 −0.229591 − 0.973287i \(-0.573739\pi\)
−0.229591 + 0.973287i \(0.573739\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6984.00 0.931764 0.465882 − 0.884847i \(-0.345737\pi\)
0.465882 + 0.884847i \(0.345737\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2526.00 0.329237 0.164619 − 0.986357i \(-0.447361\pi\)
0.164619 + 0.986357i \(0.447361\pi\)
\(390\) 0 0
\(391\) −7344.00 −0.949877
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −6146.00 −0.776975 −0.388487 − 0.921454i \(-0.627002\pi\)
−0.388487 + 0.921454i \(0.627002\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9786.00 −1.21868 −0.609339 − 0.792910i \(-0.708565\pi\)
−0.609339 + 0.792910i \(0.708565\pi\)
\(402\) 0 0
\(403\) −9520.00 −1.17674
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5136.00 0.625509
\(408\) 0 0
\(409\) −886.000 −0.107115 −0.0535573 − 0.998565i \(-0.517056\pi\)
−0.0535573 + 0.998565i \(0.517056\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 20832.0 2.48202
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 11352.0 1.32358 0.661792 − 0.749688i \(-0.269796\pi\)
0.661792 + 0.749688i \(0.269796\pi\)
\(420\) 0 0
\(421\) 10190.0 1.17964 0.589822 − 0.807533i \(-0.299198\pi\)
0.589822 + 0.807533i \(0.299198\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −11704.0 −1.32645
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2448.00 −0.273587 −0.136794 − 0.990600i \(-0.543680\pi\)
−0.136794 + 0.990600i \(0.543680\pi\)
\(432\) 0 0
\(433\) 7078.00 0.785559 0.392779 − 0.919633i \(-0.371514\pi\)
0.392779 + 0.919633i \(0.371514\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1440.00 −0.157631
\(438\) 0 0
\(439\) −18088.0 −1.96650 −0.983250 − 0.182264i \(-0.941657\pi\)
−0.983250 + 0.182264i \(0.941657\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3852.00 0.413124 0.206562 − 0.978433i \(-0.433772\pi\)
0.206562 + 0.978433i \(0.433772\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6522.00 −0.685506 −0.342753 − 0.939426i \(-0.611359\pi\)
−0.342753 + 0.939426i \(0.611359\pi\)
\(450\) 0 0
\(451\) 3600.00 0.375870
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2090.00 −0.213930 −0.106965 − 0.994263i \(-0.534113\pi\)
−0.106965 + 0.994263i \(0.534113\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9894.00 0.999587 0.499793 − 0.866145i \(-0.333409\pi\)
0.499793 + 0.866145i \(0.333409\pi\)
\(462\) 0 0
\(463\) −3044.00 −0.305544 −0.152772 − 0.988261i \(-0.548820\pi\)
−0.152772 + 0.988261i \(0.548820\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10236.0 1.01427 0.507137 − 0.861866i \(-0.330704\pi\)
0.507137 + 0.861866i \(0.330704\pi\)
\(468\) 0 0
\(469\) −5264.00 −0.518271
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7008.00 0.681244
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −11496.0 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 14980.0 1.42002
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 15316.0 1.42512 0.712561 − 0.701610i \(-0.247535\pi\)
0.712561 + 0.701610i \(0.247535\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 11616.0 1.06766 0.533832 − 0.845591i \(-0.320752\pi\)
0.533832 + 0.845591i \(0.320752\pi\)
\(492\) 0 0
\(493\) −31212.0 −2.85135
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −13440.0 −1.21301
\(498\) 0 0
\(499\) 14996.0 1.34532 0.672658 − 0.739953i \(-0.265152\pi\)
0.672658 + 0.739953i \(0.265152\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −21648.0 −1.91896 −0.959480 − 0.281778i \(-0.909076\pi\)
−0.959480 + 0.281778i \(0.909076\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3378.00 −0.294160 −0.147080 − 0.989125i \(-0.546987\pi\)
−0.147080 + 0.989125i \(0.546987\pi\)
\(510\) 0 0
\(511\) −12152.0 −1.05200
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1728.00 −0.146997
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 16158.0 1.35872 0.679362 − 0.733804i \(-0.262257\pi\)
0.679362 + 0.733804i \(0.262257\pi\)
\(522\) 0 0
\(523\) 76.0000 0.00635420 0.00317710 − 0.999995i \(-0.498989\pi\)
0.00317710 + 0.999995i \(0.498989\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −13872.0 −1.14663
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 10500.0 0.853294
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10584.0 0.845798
\(540\) 0 0
\(541\) 9278.00 0.737324 0.368662 − 0.929563i \(-0.379816\pi\)
0.368662 + 0.929563i \(0.379816\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −14564.0 −1.13841 −0.569206 − 0.822195i \(-0.692749\pi\)
−0.569206 + 0.822195i \(0.692749\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6120.00 −0.473177
\(552\) 0 0
\(553\) 37856.0 2.91103
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2154.00 0.163856 0.0819281 − 0.996638i \(-0.473892\pi\)
0.0819281 + 0.996638i \(0.473892\pi\)
\(558\) 0 0
\(559\) 20440.0 1.54655
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8700.00 −0.651263 −0.325632 − 0.945497i \(-0.605577\pi\)
−0.325632 + 0.945497i \(0.605577\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4194.00 −0.309001 −0.154501 − 0.987993i \(-0.549377\pi\)
−0.154501 + 0.987993i \(0.549377\pi\)
\(570\) 0 0
\(571\) −8020.00 −0.587787 −0.293894 − 0.955838i \(-0.594951\pi\)
−0.293894 + 0.955838i \(0.594951\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2686.00 0.193795 0.0968974 − 0.995294i \(-0.469108\pi\)
0.0968974 + 0.995294i \(0.469108\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −17136.0 −1.22362
\(582\) 0 0
\(583\) −9936.00 −0.705844
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3012.00 0.211786 0.105893 − 0.994378i \(-0.466230\pi\)
0.105893 + 0.994378i \(0.466230\pi\)
\(588\) 0 0
\(589\) −2720.00 −0.190281
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −15522.0 −1.07489 −0.537447 − 0.843298i \(-0.680611\pi\)
−0.537447 + 0.843298i \(0.680611\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −19224.0 −1.31130 −0.655652 − 0.755063i \(-0.727606\pi\)
−0.655652 + 0.755063i \(0.727606\pi\)
\(600\) 0 0
\(601\) −6502.00 −0.441301 −0.220651 − 0.975353i \(-0.570818\pi\)
−0.220651 + 0.975353i \(0.570818\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −29396.0 −1.96565 −0.982823 − 0.184552i \(-0.940917\pi\)
−0.982823 + 0.184552i \(0.940917\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5040.00 −0.333710
\(612\) 0 0
\(613\) 10006.0 0.659280 0.329640 − 0.944107i \(-0.393073\pi\)
0.329640 + 0.944107i \(0.393073\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23118.0 1.50842 0.754210 − 0.656633i \(-0.228020\pi\)
0.754210 + 0.656633i \(0.228020\pi\)
\(618\) 0 0
\(619\) 14036.0 0.911397 0.455698 − 0.890134i \(-0.349390\pi\)
0.455698 + 0.890134i \(0.349390\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 840.000 0.0540191
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 21828.0 1.38369
\(630\) 0 0
\(631\) −4288.00 −0.270527 −0.135264 − 0.990810i \(-0.543188\pi\)
−0.135264 + 0.990810i \(0.543188\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 30870.0 1.92012
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1314.00 −0.0809671 −0.0404835 − 0.999180i \(-0.512890\pi\)
−0.0404835 + 0.999180i \(0.512890\pi\)
\(642\) 0 0
\(643\) 628.000 0.0385162 0.0192581 − 0.999815i \(-0.493870\pi\)
0.0192581 + 0.999815i \(0.493870\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −10944.0 −0.664997 −0.332498 − 0.943104i \(-0.607892\pi\)
−0.332498 + 0.943104i \(0.607892\pi\)
\(648\) 0 0
\(649\) 17856.0 1.07998
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1098.00 0.0658010 0.0329005 − 0.999459i \(-0.489526\pi\)
0.0329005 + 0.999459i \(0.489526\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −312.000 −0.0184428 −0.00922139 − 0.999957i \(-0.502935\pi\)
−0.00922139 + 0.999957i \(0.502935\pi\)
\(660\) 0 0
\(661\) 8678.00 0.510643 0.255322 − 0.966856i \(-0.417819\pi\)
0.255322 + 0.966856i \(0.417819\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 22032.0 1.27898
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −10032.0 −0.577170
\(672\) 0 0
\(673\) 14470.0 0.828793 0.414396 − 0.910097i \(-0.363993\pi\)
0.414396 + 0.910097i \(0.363993\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −11838.0 −0.672040 −0.336020 − 0.941855i \(-0.609081\pi\)
−0.336020 + 0.941855i \(0.609081\pi\)
\(678\) 0 0
\(679\) 8008.00 0.452605
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −25548.0 −1.43128 −0.715642 − 0.698467i \(-0.753866\pi\)
−0.715642 + 0.698467i \(0.753866\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −28980.0 −1.60239
\(690\) 0 0
\(691\) −18412.0 −1.01364 −0.506820 − 0.862052i \(-0.669179\pi\)
−0.506820 + 0.862052i \(0.669179\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 15300.0 0.831462
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 8814.00 0.474893 0.237447 − 0.971401i \(-0.423690\pi\)
0.237447 + 0.971401i \(0.423690\pi\)
\(702\) 0 0
\(703\) 4280.00 0.229621
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 43176.0 2.29675
\(708\) 0 0
\(709\) −17314.0 −0.917124 −0.458562 − 0.888662i \(-0.651635\pi\)
−0.458562 + 0.888662i \(0.651635\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 9792.00 0.514324
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 768.000 0.0398353 0.0199176 − 0.999802i \(-0.493660\pi\)
0.0199176 + 0.999802i \(0.493660\pi\)
\(720\) 0 0
\(721\) −32816.0 −1.69505
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 18196.0 0.928270 0.464135 − 0.885764i \(-0.346365\pi\)
0.464135 + 0.885764i \(0.346365\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 29784.0 1.50698
\(732\) 0 0
\(733\) 18142.0 0.914175 0.457087 − 0.889422i \(-0.348893\pi\)
0.457087 + 0.889422i \(0.348893\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4512.00 −0.225511
\(738\) 0 0
\(739\) −13660.0 −0.679961 −0.339981 − 0.940432i \(-0.610420\pi\)
−0.339981 + 0.940432i \(0.610420\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −12768.0 −0.630434 −0.315217 − 0.949020i \(-0.602077\pi\)
−0.315217 + 0.949020i \(0.602077\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 54768.0 2.67180
\(750\) 0 0
\(751\) 22952.0 1.11522 0.557610 − 0.830103i \(-0.311718\pi\)
0.557610 + 0.830103i \(0.311718\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −15818.0 −0.759465 −0.379732 − 0.925096i \(-0.623984\pi\)
−0.379732 + 0.925096i \(0.623984\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18558.0 0.884004 0.442002 − 0.897014i \(-0.354268\pi\)
0.442002 + 0.897014i \(0.354268\pi\)
\(762\) 0 0
\(763\) −52024.0 −2.46841
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 52080.0 2.45176
\(768\) 0 0
\(769\) 14978.0 0.702367 0.351184 − 0.936307i \(-0.385779\pi\)
0.351184 + 0.936307i \(0.385779\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8946.00 0.416255 0.208128 − 0.978102i \(-0.433263\pi\)
0.208128 + 0.978102i \(0.433263\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3000.00 0.137980
\(780\) 0 0
\(781\) −11520.0 −0.527808
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 18436.0 0.835035 0.417517 − 0.908669i \(-0.362900\pi\)
0.417517 + 0.908669i \(0.362900\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4872.00 0.218999
\(792\) 0 0
\(793\) −29260.0 −1.31028
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 16314.0 0.725058 0.362529 − 0.931972i \(-0.381913\pi\)
0.362529 + 0.931972i \(0.381913\pi\)
\(798\) 0 0
\(799\) −7344.00 −0.325172
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −10416.0 −0.457749
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 25446.0 1.10585 0.552926 − 0.833231i \(-0.313511\pi\)
0.552926 + 0.833231i \(0.313511\pi\)
\(810\) 0 0
\(811\) 42740.0 1.85056 0.925280 − 0.379284i \(-0.123830\pi\)
0.925280 + 0.379284i \(0.123830\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 5840.00 0.250080
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −29946.0 −1.27299 −0.636494 − 0.771282i \(-0.719616\pi\)
−0.636494 + 0.771282i \(0.719616\pi\)
\(822\) 0 0
\(823\) 32596.0 1.38059 0.690295 − 0.723528i \(-0.257481\pi\)
0.690295 + 0.723528i \(0.257481\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3804.00 0.159949 0.0799746 − 0.996797i \(-0.474516\pi\)
0.0799746 + 0.996797i \(0.474516\pi\)
\(828\) 0 0
\(829\) 3278.00 0.137334 0.0686669 − 0.997640i \(-0.478125\pi\)
0.0686669 + 0.997640i \(0.478125\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 44982.0 1.87099
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 5784.00 0.238005 0.119002 − 0.992894i \(-0.462030\pi\)
0.119002 + 0.992894i \(0.462030\pi\)
\(840\) 0 0
\(841\) 69247.0 2.83927
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −21140.0 −0.857590
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −15408.0 −0.620657
\(852\) 0 0
\(853\) −17306.0 −0.694661 −0.347331 − 0.937743i \(-0.612912\pi\)
−0.347331 + 0.937743i \(0.612912\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 31134.0 1.24098 0.620488 − 0.784216i \(-0.286934\pi\)
0.620488 + 0.784216i \(0.286934\pi\)
\(858\) 0 0
\(859\) −10780.0 −0.428183 −0.214091 − 0.976814i \(-0.568679\pi\)
−0.214091 + 0.976814i \(0.568679\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3456.00 0.136319 0.0681597 − 0.997674i \(-0.478287\pi\)
0.0681597 + 0.997674i \(0.478287\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 32448.0 1.26665
\(870\) 0 0
\(871\) −13160.0 −0.511951
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2618.00 −0.100802 −0.0504011 − 0.998729i \(-0.516050\pi\)
−0.0504011 + 0.998729i \(0.516050\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 26550.0 1.01531 0.507657 − 0.861559i \(-0.330512\pi\)
0.507657 + 0.861559i \(0.330512\pi\)
\(882\) 0 0
\(883\) −27596.0 −1.05173 −0.525866 − 0.850567i \(-0.676259\pi\)
−0.525866 + 0.850567i \(0.676259\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −37848.0 −1.43271 −0.716354 − 0.697737i \(-0.754190\pi\)
−0.716354 + 0.697737i \(0.754190\pi\)
\(888\) 0 0
\(889\) 57904.0 2.18452
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1440.00 −0.0539617
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 41616.0 1.54391
\(900\) 0 0
\(901\) −42228.0 −1.56140
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4804.00 0.175870 0.0879351 − 0.996126i \(-0.471973\pi\)
0.0879351 + 0.996126i \(0.471973\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −28608.0 −1.04042 −0.520211 − 0.854037i \(-0.674147\pi\)
−0.520211 + 0.854037i \(0.674147\pi\)
\(912\) 0 0
\(913\) −14688.0 −0.532423
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −8736.00 −0.314600
\(918\) 0 0
\(919\) −40768.0 −1.46334 −0.731672 − 0.681657i \(-0.761259\pi\)
−0.731672 + 0.681657i \(0.761259\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −33600.0 −1.19822
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −27642.0 −0.976216 −0.488108 − 0.872783i \(-0.662313\pi\)
−0.488108 + 0.872783i \(0.662313\pi\)
\(930\) 0 0
\(931\) 8820.00 0.310487
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −28106.0 −0.979918 −0.489959 − 0.871746i \(-0.662988\pi\)
−0.489959 + 0.871746i \(0.662988\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −14730.0 −0.510291 −0.255146 − 0.966903i \(-0.582123\pi\)
−0.255146 + 0.966903i \(0.582123\pi\)
\(942\) 0 0
\(943\) −10800.0 −0.372955
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9564.00 −0.328182 −0.164091 − 0.986445i \(-0.552469\pi\)
−0.164091 + 0.986445i \(0.552469\pi\)
\(948\) 0 0
\(949\) −30380.0 −1.03917
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −53898.0 −1.83203 −0.916017 − 0.401141i \(-0.868614\pi\)
−0.916017 + 0.401141i \(0.868614\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 74088.0 2.49471
\(960\) 0 0
\(961\) −11295.0 −0.379141
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −15140.0 −0.503485 −0.251742 − 0.967794i \(-0.581004\pi\)
−0.251742 + 0.967794i \(0.581004\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −23808.0 −0.786854 −0.393427 − 0.919356i \(-0.628711\pi\)
−0.393427 + 0.919356i \(0.628711\pi\)
\(972\) 0 0
\(973\) −35728.0 −1.17717
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 23094.0 0.756236 0.378118 − 0.925757i \(-0.376571\pi\)
0.378118 + 0.925757i \(0.376571\pi\)
\(978\) 0 0
\(979\) 720.000 0.0235049
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 7584.00 0.246075 0.123038 − 0.992402i \(-0.460736\pi\)
0.123038 + 0.992402i \(0.460736\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −21024.0 −0.675960
\(990\) 0 0
\(991\) −26752.0 −0.857523 −0.428761 − 0.903418i \(-0.641050\pi\)
−0.428761 + 0.903418i \(0.641050\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −7778.00 −0.247073 −0.123536 − 0.992340i \(-0.539424\pi\)
−0.123536 + 0.992340i \(0.539424\pi\)
\(998\) 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 900.4.a.q.1.1 1
3.2 odd 2 300.4.a.i.1.1 1
5.2 odd 4 900.4.d.h.649.2 2
5.3 odd 4 900.4.d.h.649.1 2
5.4 even 2 180.4.a.d.1.1 1
12.11 even 2 1200.4.a.a.1.1 1
15.2 even 4 300.4.d.b.49.1 2
15.8 even 4 300.4.d.b.49.2 2
15.14 odd 2 60.4.a.a.1.1 ✓ 1
20.19 odd 2 720.4.a.bb.1.1 1
45.4 even 6 1620.4.i.f.541.1 2
45.14 odd 6 1620.4.i.l.541.1 2
45.29 odd 6 1620.4.i.l.1081.1 2
45.34 even 6 1620.4.i.f.1081.1 2
60.23 odd 4 1200.4.f.n.49.1 2
60.47 odd 4 1200.4.f.n.49.2 2
60.59 even 2 240.4.a.i.1.1 1
120.29 odd 2 960.4.a.bc.1.1 1
120.59 even 2 960.4.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.4.a.a.1.1 ✓ 1 15.14 odd 2
180.4.a.d.1.1 1 5.4 even 2
240.4.a.i.1.1 1 60.59 even 2
300.4.a.i.1.1 1 3.2 odd 2
300.4.d.b.49.1 2 15.2 even 4
300.4.d.b.49.2 2 15.8 even 4
720.4.a.bb.1.1 1 20.19 odd 2
900.4.a.q.1.1 1 1.1 even 1 trivial
900.4.d.h.649.1 2 5.3 odd 4
900.4.d.h.649.2 2 5.2 odd 4
960.4.a.r.1.1 1 120.59 even 2
960.4.a.bc.1.1 1 120.29 odd 2
1200.4.a.a.1.1 1 12.11 even 2
1200.4.f.n.49.1 2 60.23 odd 4
1200.4.f.n.49.2 2 60.47 odd 4
1620.4.i.f.541.1 2 45.4 even 6
1620.4.i.f.1081.1 2 45.34 even 6
1620.4.i.l.541.1 2 45.14 odd 6
1620.4.i.l.1081.1 2 45.29 odd 6