↑

Properties

Label 180.4.a.d.1.1
Level $180$
Weight $4$
Character 180.1
Self dual yes
Analytic conductor $10.620$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [180,4,Mod(1,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 180.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: \(10.6203438010\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 180.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.00000 q^{5} -28.0000 q^{7} +24.0000 q^{11} -70.0000 q^{13} -102.000 q^{17} +20.0000 q^{19} +72.0000 q^{23} +25.0000 q^{25} -306.000 q^{29} -136.000 q^{31} -140.000 q^{35} -214.000 q^{37} +150.000 q^{41} -292.000 q^{43} +72.0000 q^{47} +441.000 q^{49} +414.000 q^{53} +120.000 q^{55} +744.000 q^{59} -418.000 q^{61} -350.000 q^{65} +188.000 q^{67} -480.000 q^{71} +434.000 q^{73} -672.000 q^{77} +1352.00 q^{79} +612.000 q^{83} -510.000 q^{85} +30.0000 q^{89} +1960.00 q^{91} +100.000 q^{95} -286.000 q^{97} +O(q^{100})\)

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
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −28.0000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 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 \(-0.768369\pi\)
−0.746712 + 0.665148i \(0.768369\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −102.000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 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 \(-0.394174\pi\)
0.326370 + 0.945242i \(0.394174\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 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
\(35\) −140.000 −0.676123
\(36\) 0 0
\(37\) −214.000 −0.950848 −0.475424 − 0.879757i \(-0.657705\pi\)
−0.475424 + 0.879757i \(0.657705\pi\)
\(38\) 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 \(-0.673244\pi\)
−0.517786 + 0.855510i \(0.673244\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 72.0000 0.223453 0.111726 − 0.993739i \(-0.464362\pi\)
0.111726 + 0.993739i \(0.464362\pi\)
\(48\) 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 \(-0.319752\pi\)
0.536484 + 0.843911i \(0.319752\pi\)
\(54\) 0 0
\(55\) 120.000 0.294196
\(56\) 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
\(65\) −350.000 −0.667879
\(66\) 0 0
\(67\) 188.000 0.342804 0.171402 − 0.985201i \(-0.445170\pi\)
0.171402 + 0.985201i \(0.445170\pi\)
\(68\) 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 \(-0.386889\pi\)
0.347917 + 0.937525i \(0.386889\pi\)
\(74\) 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 \(-0.367385\pi\)
0.404673 + 0.914461i \(0.367385\pi\)
\(84\) 0 0
\(85\) −510.000 −0.650791
\(86\) 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
\(95\) 100.000 0.107998
\(96\) 0 0
\(97\) −286.000 −0.299370 −0.149685 − 0.988734i \(-0.547826\pi\)
−0.149685 + 0.988734i \(0.547826\pi\)
\(98\) 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 \(-0.310576\pi\)
0.560585 + 0.828097i \(0.310576\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1956.00 −1.76723 −0.883615 − 0.468214i \(-0.844898\pi\)
−0.883615 + 0.468214i \(0.844898\pi\)
\(108\) 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 \(-0.523074\pi\)
−0.0724272 + 0.997374i \(0.523074\pi\)
\(114\) 0 0
\(115\) 360.000 0.291915
\(116\) 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
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −2068.00 −1.44492 −0.722462 − 0.691411i \(-0.756990\pi\)
−0.722462 + 0.691411i \(0.756990\pi\)
\(128\) 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 \(-0.808852\pi\)
−0.825048 + 0.565063i \(0.808852\pi\)
\(138\) 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
\(145\) −1530.00 −0.876273
\(146\) 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
\(155\) −680.000 −0.352380
\(156\) 0 0
\(157\) −166.000 −0.0843837 −0.0421919 − 0.999110i \(-0.513434\pi\)
−0.0421919 + 0.999110i \(0.513434\pi\)
\(158\) 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 \(-0.241564\pi\)
0.725597 + 0.688120i \(0.241564\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 984.000 0.455953 0.227977 − 0.973667i \(-0.426789\pi\)
0.227977 + 0.973667i \(0.426789\pi\)
\(168\) 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 \(-0.641882\pi\)
−0.431122 + 0.902294i \(0.641882\pi\)
\(174\) 0 0
\(175\) −700.000 −0.302372
\(176\) 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
\(185\) −1070.00 −0.425232
\(186\) 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 \(-0.624092\pi\)
−0.380048 + 0.924967i \(0.624092\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4098.00 −1.48208 −0.741042 − 0.671459i \(-0.765668\pi\)
−0.741042 + 0.671459i \(0.765668\pi\)
\(198\) 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
\(205\) 750.000 0.255523
\(206\) 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
\(215\) −1460.00 −0.463122
\(216\) 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 \(-0.647662\pi\)
−0.447434 + 0.894317i \(0.647662\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3180.00 0.929797 0.464899 − 0.885364i \(-0.346091\pi\)
0.464899 + 0.885364i \(0.346091\pi\)
\(228\) 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 \(-0.588393\pi\)
−0.274139 + 0.961690i \(0.588393\pi\)
\(234\) 0 0
\(235\) 360.000 0.0999311
\(236\) 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
\(245\) 2205.00 0.574989
\(246\) 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 \(-0.381634\pi\)
0.363347 + 0.931654i \(0.381634\pi\)
\(258\) 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 \(-0.406414\pi\)
0.289791 + 0.957090i \(0.406414\pi\)
\(264\) 0 0
\(265\) 2070.00 0.479846
\(266\) 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
\(275\) 600.000 0.131569
\(276\) 0 0
\(277\) 1034.00 0.224285 0.112143 − 0.993692i \(-0.464229\pi\)
0.112143 + 0.993692i \(0.464229\pi\)
\(278\) 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 \(-0.559042\pi\)
−0.184423 + 0.982847i \(0.559042\pi\)
\(284\) 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 \(-0.604493\pi\)
−0.322410 + 0.946600i \(0.604493\pi\)
\(294\) 0 0
\(295\) 3720.00 0.734192
\(296\) 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
\(305\) −2090.00 −0.392371
\(306\) 0 0
\(307\) 2036.00 0.378504 0.189252 − 0.981929i \(-0.439394\pi\)
0.189252 + 0.981929i \(0.439394\pi\)
\(308\) 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 \(-0.465385\pi\)
0.108532 + 0.994093i \(0.465385\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3798.00 0.672924 0.336462 − 0.941697i \(-0.390770\pi\)
0.336462 + 0.941697i \(0.390770\pi\)
\(318\) 0 0
\(319\) −7344.00 −1.28898
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2040.00 −0.351420
\(324\) 0 0
\(325\) −1750.00 −0.298685
\(326\) 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
\(335\) 940.000 0.153307
\(336\) 0 0
\(337\) −454.000 −0.0733856 −0.0366928 − 0.999327i \(-0.511682\pi\)
−0.0366928 + 0.999327i \(0.511682\pi\)
\(338\) 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 \(-0.357284\pi\)
0.433485 + 0.901161i \(0.357284\pi\)
\(348\) 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 \(-0.339020\pi\)
0.484450 + 0.874819i \(0.339020\pi\)
\(354\) 0 0
\(355\) −2400.00 −0.358813
\(356\) 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
\(365\) 2170.00 0.311186
\(366\) 0 0
\(367\) −388.000 −0.0551865 −0.0275932 − 0.999619i \(-0.508784\pi\)
−0.0275932 + 0.999619i \(0.508784\pi\)
\(368\) 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 \(-0.689031\pi\)
−0.559564 + 0.828787i \(0.689031\pi\)
\(374\) 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 \(-0.654263\pi\)
−0.465882 + 0.884847i \(0.654263\pi\)
\(384\) 0 0
\(385\) −3360.00 −0.444783
\(386\) 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
\(395\) 6760.00 0.861095
\(396\) 0 0
\(397\) 6146.00 0.776975 0.388487 − 0.921454i \(-0.372998\pi\)
0.388487 + 0.921454i \(0.372998\pi\)
\(398\) 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
\(415\) 3060.00 0.361951
\(416\) 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
\(425\) −2550.00 −0.291043
\(426\) 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 \(-0.628486\pi\)
−0.392779 + 0.919633i \(0.628486\pi\)
\(434\) 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 \(-0.566228\pi\)
−0.206562 + 0.978433i \(0.566228\pi\)
\(444\) 0 0
\(445\) 150.000 0.0159791
\(446\) 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
\(455\) 9800.00 1.00974
\(456\) 0 0
\(457\) 2090.00 0.213930 0.106965 − 0.994263i \(-0.465887\pi\)
0.106965 + 0.994263i \(0.465887\pi\)
\(458\) 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 \(-0.451180\pi\)
0.152772 + 0.988261i \(0.451180\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10236.0 −1.01427 −0.507137 − 0.861866i \(-0.669296\pi\)
−0.507137 + 0.861866i \(0.669296\pi\)
\(468\) 0 0
\(469\) −5264.00 −0.518271
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −7008.00 −0.681244
\(474\) 0 0
\(475\) 500.000 0.0482980
\(476\) 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
\(485\) −1430.00 −0.133882
\(486\) 0 0
\(487\) −15316.0 −1.42512 −0.712561 − 0.701610i \(-0.752465\pi\)
−0.712561 + 0.701610i \(0.752465\pi\)
\(488\) 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 \(-0.0909240\pi\)
0.959480 + 0.281778i \(0.0909240\pi\)
\(504\) 0 0
\(505\) 7710.00 0.679387
\(506\) 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
\(515\) 5860.00 0.501403
\(516\) 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 \(-0.501011\pi\)
−0.00317710 + 0.999995i \(0.501011\pi\)
\(524\) 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
\(535\) −9780.00 −0.790329
\(536\) 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
\(545\) −9290.00 −0.730165
\(546\) 0 0
\(547\) 14564.0 1.13841 0.569206 − 0.822195i \(-0.307251\pi\)
0.569206 + 0.822195i \(0.307251\pi\)
\(548\) 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 \(-0.526108\pi\)
−0.0819281 + 0.996638i \(0.526108\pi\)
\(558\) 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 \(-0.394423\pi\)
0.325632 + 0.945497i \(0.394423\pi\)
\(564\) 0 0
\(565\) −870.000 −0.0647808
\(566\) 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
\(575\) 1800.00 0.130548
\(576\) 0 0
\(577\) −2686.00 −0.193795 −0.0968974 − 0.995294i \(-0.530892\pi\)
−0.0968974 + 0.995294i \(0.530892\pi\)
\(578\) 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 \(-0.533770\pi\)
−0.105893 + 0.994378i \(0.533770\pi\)
\(588\) 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 \(-0.319389\pi\)
0.537447 + 0.843298i \(0.319389\pi\)
\(594\) 0 0
\(595\) 14280.0 0.983904
\(596\) 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
\(605\) −3775.00 −0.253679
\(606\) 0 0
\(607\) 29396.0 1.96565 0.982823 − 0.184552i \(-0.0590834\pi\)
0.982823 + 0.184552i \(0.0590834\pi\)
\(608\) 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 \(-0.606927\pi\)
−0.329640 + 0.944107i \(0.606927\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −23118.0 −1.50842 −0.754210 − 0.656633i \(-0.771980\pi\)
−0.754210 + 0.656633i \(0.771980\pi\)
\(618\) 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
\(625\) 625.000 0.0400000
\(626\) 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
\(635\) −10340.0 −0.646190
\(636\) 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 \(-0.506130\pi\)
−0.0192581 + 0.999815i \(0.506130\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10944.0 0.664997 0.332498 − 0.943104i \(-0.392108\pi\)
0.332498 + 0.943104i \(0.392108\pi\)
\(648\) 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 \(-0.510474\pi\)
−0.0329005 + 0.999459i \(0.510474\pi\)
\(654\) 0 0
\(655\) −1560.00 −0.0930599
\(656\) 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
\(665\) −2800.00 −0.163277
\(666\) 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 \(-0.636007\pi\)
−0.414396 + 0.910097i \(0.636007\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11838.0 0.672040 0.336020 − 0.941855i \(-0.390919\pi\)
0.336020 + 0.941855i \(0.390919\pi\)
\(678\) 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 \(-0.246134\pi\)
0.715642 + 0.698467i \(0.246134\pi\)
\(684\) 0 0
\(685\) −13230.0 −0.737945
\(686\) 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
\(695\) −6380.00 −0.348212
\(696\) 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
\(715\) −8400.00 −0.439360
\(716\) 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
\(725\) −7650.00 −0.391881
\(726\) 0 0
\(727\) −18196.0 −0.928270 −0.464135 − 0.885764i \(-0.653635\pi\)
−0.464135 + 0.885764i \(0.653635\pi\)
\(728\) 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 \(-0.651107\pi\)
−0.457087 + 0.889422i \(0.651107\pi\)
\(734\) 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 \(-0.397923\pi\)
0.315217 + 0.949020i \(0.397923\pi\)
\(744\) 0 0
\(745\) 15990.0 0.786347
\(746\) 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
\(755\) −3800.00 −0.183174
\(756\) 0 0
\(757\) 15818.0 0.759465 0.379732 − 0.925096i \(-0.376016\pi\)
0.379732 + 0.925096i \(0.376016\pi\)
\(758\) 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 \(-0.566737\pi\)
−0.208128 + 0.978102i \(0.566737\pi\)
\(774\) 0 0
\(775\) −3400.00 −0.157589
\(776\) 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
\(785\) −830.000 −0.0377375
\(786\) 0 0
\(787\) −18436.0 −0.835035 −0.417517 − 0.908669i \(-0.637100\pi\)
−0.417517 + 0.908669i \(0.637100\pi\)
\(788\) 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 \(-0.618087\pi\)
−0.362529 + 0.931972i \(0.618087\pi\)
\(798\) 0 0
\(799\) −7344.00 −0.325172
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10416.0 0.457749
\(804\) 0 0
\(805\) −10080.0 −0.441333
\(806\) 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
\(815\) 15100.0 0.648994
\(816\) 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 \(-0.742519\pi\)
−0.690295 + 0.723528i \(0.742519\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3804.00 −0.159949 −0.0799746 − 0.996797i \(-0.525484\pi\)
−0.0799746 + 0.996797i \(0.525484\pi\)
\(828\) 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
\(835\) 4920.00 0.203909
\(836\) 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
\(845\) 13515.0 0.550213
\(846\) 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 \(-0.387088\pi\)
0.347331 + 0.937743i \(0.387088\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −31134.0 −1.24098 −0.620488 − 0.784216i \(-0.713066\pi\)
−0.620488 + 0.784216i \(0.713066\pi\)
\(858\) 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 \(-0.521713\pi\)
−0.0681597 + 0.997674i \(0.521713\pi\)
\(864\) 0 0
\(865\) −9810.00 −0.385607
\(866\) 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
\(875\) −3500.00 −0.135225
\(876\) 0 0
\(877\) 2618.00 0.100802 0.0504011 − 0.998729i \(-0.483950\pi\)
0.0504011 + 0.998729i \(0.483950\pi\)
\(878\) 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 \(-0.323741\pi\)
0.525866 + 0.850567i \(0.323741\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 37848.0 1.43271 0.716354 − 0.697737i \(-0.245810\pi\)
0.716354 + 0.697737i \(0.245810\pi\)
\(888\) 0 0
\(889\) 57904.0 2.18452
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1440.00 0.0539617
\(894\) 0 0
\(895\) −2880.00 −0.107562
\(896\) 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
\(905\) −6050.00 −0.222220
\(906\) 0 0
\(907\) −4804.00 −0.175870 −0.0879351 − 0.996126i \(-0.528027\pi\)
−0.0879351 + 0.996126i \(0.528027\pi\)
\(908\) 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
\(925\) −5350.00 −0.190170
\(926\) 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
\(935\) −12240.0 −0.428119
\(936\) 0 0
\(937\) 28106.0 0.979918 0.489959 − 0.871746i \(-0.337012\pi\)
0.489959 + 0.871746i \(0.337012\pi\)
\(938\) 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 \(-0.447531\pi\)
0.164091 + 0.986445i \(0.447531\pi\)
\(948\) 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 \(-0.131386\pi\)
0.916017 + 0.401141i \(0.131386\pi\)
\(954\) 0 0
\(955\) −16920.0 −0.573318
\(956\) 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
\(965\) −10190.0 −0.339925
\(966\) 0 0
\(967\) 15140.0 0.503485 0.251742 − 0.967794i \(-0.418996\pi\)
0.251742 + 0.967794i \(0.418996\pi\)
\(968\) 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 \(-0.623429\pi\)
−0.378118 + 0.925757i \(0.623429\pi\)
\(978\) 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 \(-0.539264\pi\)
−0.123038 + 0.992402i \(0.539264\pi\)
\(984\) 0 0
\(985\) −20490.0 −0.662808
\(986\) 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
\(995\) −11240.0 −0.358123
\(996\) 0 0
\(997\) 7778.00 0.247073 0.123536 − 0.992340i \(-0.460576\pi\)
0.123536 + 0.992340i \(0.460576\pi\)
\(998\) 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 180.4.a.d.1.1 1
3.2 odd 2 60.4.a.a.1.1 ✓ 1
4.3 odd 2 720.4.a.bb.1.1 1
5.2 odd 4 900.4.d.h.649.1 2
5.3 odd 4 900.4.d.h.649.2 2
5.4 even 2 900.4.a.q.1.1 1
9.2 odd 6 1620.4.i.l.1081.1 2
9.4 even 3 1620.4.i.f.541.1 2
9.5 odd 6 1620.4.i.l.541.1 2
9.7 even 3 1620.4.i.f.1081.1 2
12.11 even 2 240.4.a.i.1.1 1
15.2 even 4 300.4.d.b.49.2 2
15.8 even 4 300.4.d.b.49.1 2
15.14 odd 2 300.4.a.i.1.1 1
24.5 odd 2 960.4.a.bc.1.1 1
24.11 even 2 960.4.a.r.1.1 1
60.23 odd 4 1200.4.f.n.49.2 2
60.47 odd 4 1200.4.f.n.49.1 2
60.59 even 2 1200.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.4.a.a.1.1 ✓ 1 3.2 odd 2
180.4.a.d.1.1 1 1.1 even 1 trivial
240.4.a.i.1.1 1 12.11 even 2
300.4.a.i.1.1 1 15.14 odd 2
300.4.d.b.49.1 2 15.8 even 4
300.4.d.b.49.2 2 15.2 even 4
720.4.a.bb.1.1 1 4.3 odd 2
900.4.a.q.1.1 1 5.4 even 2
900.4.d.h.649.1 2 5.2 odd 4
900.4.d.h.649.2 2 5.3 odd 4
960.4.a.r.1.1 1 24.11 even 2
960.4.a.bc.1.1 1 24.5 odd 2
1200.4.a.a.1.1 1 60.59 even 2
1200.4.f.n.49.1 2 60.47 odd 4
1200.4.f.n.49.2 2 60.23 odd 4
1620.4.i.f.541.1 2 9.4 even 3
1620.4.i.f.1081.1 2 9.7 even 3
1620.4.i.l.541.1 2 9.5 odd 6
1620.4.i.l.1081.1 2 9.2 odd 6