Properties

Label 5220.2.a.x.1.3
Level $5220$
Weight $2$
Character 5220.1
Self dual yes
Analytic conductor $41.682$
Analytic rank $1$
Dimension $3$
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] = [5220,2,Mod(1,5220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5220, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5220.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5220.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: \(41.6819098551\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 580)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.51414\) of defining polynomial
Character \(\chi\) \(=\) 5220.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +1.32088 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +1.32088 q^{7} -5.32088 q^{11} -5.02827 q^{13} +6.34916 q^{17} -4.34916 q^{19} +1.70739 q^{23} +1.00000 q^{25} +1.00000 q^{29} +8.34916 q^{31} +1.32088 q^{35} -6.93438 q^{37} +1.02827 q^{41} +10.7357 q^{43} +0.679116 q^{47} -5.25526 q^{49} -2.38650 q^{53} -5.32088 q^{55} -10.4431 q^{59} -6.38650 q^{61} -5.02827 q^{65} -5.70739 q^{67} +3.61350 q^{71} -6.73566 q^{73} -7.02827 q^{77} -11.3774 q^{79} +3.96265 q^{83} +6.34916 q^{85} -2.58522 q^{89} -6.64177 q^{91} -4.34916 q^{95} -15.3209 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} - 4 q^{7} - 8 q^{11} - 2 q^{13} - 2 q^{17} + 8 q^{19} + 3 q^{25} + 3 q^{29} + 4 q^{31} - 4 q^{35} - 10 q^{37} - 10 q^{41} + 14 q^{43} + 10 q^{47} + 3 q^{49} - 10 q^{53} - 8 q^{55} - 8 q^{59} - 22 q^{61} - 2 q^{65} - 12 q^{67} + 8 q^{71} - 2 q^{73} - 8 q^{77} - 12 q^{83} - 2 q^{85} - 18 q^{89} - 4 q^{91} + 8 q^{95} - 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.32088 0.499247 0.249624 0.968343i \(-0.419693\pi\)
0.249624 + 0.968343i \(0.419693\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.32088 −1.60431 −0.802154 0.597118i \(-0.796312\pi\)
−0.802154 + 0.597118i \(0.796312\pi\)
\(12\) 0 0
\(13\) −5.02827 −1.39459 −0.697296 0.716783i \(-0.745614\pi\)
−0.697296 + 0.716783i \(0.745614\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.34916 1.53990 0.769949 0.638106i \(-0.220282\pi\)
0.769949 + 0.638106i \(0.220282\pi\)
\(18\) 0 0
\(19\) −4.34916 −0.997765 −0.498883 0.866670i \(-0.666256\pi\)
−0.498883 + 0.866670i \(0.666256\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.70739 0.356015 0.178008 0.984029i \(-0.443035\pi\)
0.178008 + 0.984029i \(0.443035\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 8.34916 1.49955 0.749777 0.661691i \(-0.230161\pi\)
0.749777 + 0.661691i \(0.230161\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.32088 0.223270
\(36\) 0 0
\(37\) −6.93438 −1.14000 −0.570002 0.821643i \(-0.693058\pi\)
−0.570002 + 0.821643i \(0.693058\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.02827 0.160589 0.0802947 0.996771i \(-0.474414\pi\)
0.0802947 + 0.996771i \(0.474414\pi\)
\(42\) 0 0
\(43\) 10.7357 1.63717 0.818587 0.574383i \(-0.194758\pi\)
0.818587 + 0.574383i \(0.194758\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.679116 0.0990592 0.0495296 0.998773i \(-0.484228\pi\)
0.0495296 + 0.998773i \(0.484228\pi\)
\(48\) 0 0
\(49\) −5.25526 −0.750752
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.38650 −0.327812 −0.163906 0.986476i \(-0.552409\pi\)
−0.163906 + 0.986476i \(0.552409\pi\)
\(54\) 0 0
\(55\) −5.32088 −0.717468
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.4431 −1.35957 −0.679785 0.733412i \(-0.737927\pi\)
−0.679785 + 0.733412i \(0.737927\pi\)
\(60\) 0 0
\(61\) −6.38650 −0.817708 −0.408854 0.912600i \(-0.634072\pi\)
−0.408854 + 0.912600i \(0.634072\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.02827 −0.623681
\(66\) 0 0
\(67\) −5.70739 −0.697269 −0.348634 0.937259i \(-0.613354\pi\)
−0.348634 + 0.937259i \(0.613354\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.61350 0.428843 0.214421 0.976741i \(-0.431213\pi\)
0.214421 + 0.976741i \(0.431213\pi\)
\(72\) 0 0
\(73\) −6.73566 −0.788350 −0.394175 0.919035i \(-0.628970\pi\)
−0.394175 + 0.919035i \(0.628970\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.02827 −0.800946
\(78\) 0 0
\(79\) −11.3774 −1.28006 −0.640031 0.768349i \(-0.721078\pi\)
−0.640031 + 0.768349i \(0.721078\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.96265 0.434958 0.217479 0.976065i \(-0.430217\pi\)
0.217479 + 0.976065i \(0.430217\pi\)
\(84\) 0 0
\(85\) 6.34916 0.688663
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.58522 −0.274033 −0.137016 0.990569i \(-0.543751\pi\)
−0.137016 + 0.990569i \(0.543751\pi\)
\(90\) 0 0
\(91\) −6.64177 −0.696247
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.34916 −0.446214
\(96\) 0 0
\(97\) −15.3209 −1.55560 −0.777800 0.628512i \(-0.783664\pi\)
−0.777800 + 0.628512i \(0.783664\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −11.8688 −1.18099 −0.590493 0.807043i \(-0.701067\pi\)
−0.590493 + 0.807043i \(0.701067\pi\)
\(102\) 0 0
\(103\) −3.65084 −0.359728 −0.179864 0.983691i \(-0.557566\pi\)
−0.179864 + 0.983691i \(0.557566\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.32088 0.514389 0.257195 0.966360i \(-0.417202\pi\)
0.257195 + 0.966360i \(0.417202\pi\)
\(108\) 0 0
\(109\) −14.3118 −1.37082 −0.685411 0.728156i \(-0.740378\pi\)
−0.685411 + 0.728156i \(0.740378\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.03735 −0.191657 −0.0958287 0.995398i \(-0.530550\pi\)
−0.0958287 + 0.995398i \(0.530550\pi\)
\(114\) 0 0
\(115\) 1.70739 0.159215
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 8.38650 0.768790
\(120\) 0 0
\(121\) 17.3118 1.57380
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −1.26434 −0.112192 −0.0560959 0.998425i \(-0.517865\pi\)
−0.0560959 + 0.998425i \(0.517865\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.70739 0.149175 0.0745877 0.997214i \(-0.476236\pi\)
0.0745877 + 0.997214i \(0.476236\pi\)
\(132\) 0 0
\(133\) −5.74474 −0.498132
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −21.7639 −1.85942 −0.929709 0.368294i \(-0.879942\pi\)
−0.929709 + 0.368294i \(0.879942\pi\)
\(138\) 0 0
\(139\) −16.6983 −1.41633 −0.708166 0.706046i \(-0.750477\pi\)
−0.708166 + 0.706046i \(0.750477\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 26.7549 2.23735
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 6.44305 0.524328 0.262164 0.965023i \(-0.415564\pi\)
0.262164 + 0.965023i \(0.415564\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.34916 0.670621
\(156\) 0 0
\(157\) 10.7357 0.856799 0.428400 0.903589i \(-0.359078\pi\)
0.428400 + 0.903589i \(0.359078\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.25526 0.177740
\(162\) 0 0
\(163\) 3.90611 0.305950 0.152975 0.988230i \(-0.451115\pi\)
0.152975 + 0.988230i \(0.451115\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.51960 0.427120 0.213560 0.976930i \(-0.431494\pi\)
0.213560 + 0.976930i \(0.431494\pi\)
\(168\) 0 0
\(169\) 12.2835 0.944888
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.80128 −0.441063 −0.220532 0.975380i \(-0.570779\pi\)
−0.220532 + 0.975380i \(0.570779\pi\)
\(174\) 0 0
\(175\) 1.32088 0.0998495
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −14.0565 −1.05064 −0.525318 0.850906i \(-0.676054\pi\)
−0.525318 + 0.850906i \(0.676054\pi\)
\(180\) 0 0
\(181\) −7.08482 −0.526611 −0.263305 0.964713i \(-0.584813\pi\)
−0.263305 + 0.964713i \(0.584813\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.93438 −0.509826
\(186\) 0 0
\(187\) −33.7831 −2.47047
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.547875 0.0396428 0.0198214 0.999804i \(-0.493690\pi\)
0.0198214 + 0.999804i \(0.493690\pi\)
\(192\) 0 0
\(193\) 4.40571 0.317130 0.158565 0.987349i \(-0.449313\pi\)
0.158565 + 0.987349i \(0.449313\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.64177 0.330712 0.165356 0.986234i \(-0.447123\pi\)
0.165356 + 0.986234i \(0.447123\pi\)
\(198\) 0 0
\(199\) 18.4431 1.30739 0.653697 0.756757i \(-0.273217\pi\)
0.653697 + 0.756757i \(0.273217\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.32088 0.0927079
\(204\) 0 0
\(205\) 1.02827 0.0718178
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 23.1414 1.60072
\(210\) 0 0
\(211\) −4.54787 −0.313089 −0.156544 0.987671i \(-0.550035\pi\)
−0.156544 + 0.987671i \(0.550035\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.7357 0.732166
\(216\) 0 0
\(217\) 11.0283 0.748648
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −31.9253 −2.14753
\(222\) 0 0
\(223\) 1.12217 0.0751459 0.0375730 0.999294i \(-0.488037\pi\)
0.0375730 + 0.999294i \(0.488037\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.63270 −0.639345 −0.319672 0.947528i \(-0.603573\pi\)
−0.319672 + 0.947528i \(0.603573\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.05655 0.527802 0.263901 0.964550i \(-0.414991\pi\)
0.263901 + 0.964550i \(0.414991\pi\)
\(234\) 0 0
\(235\) 0.679116 0.0443006
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −15.6135 −1.00995 −0.504977 0.863133i \(-0.668499\pi\)
−0.504977 + 0.863133i \(0.668499\pi\)
\(240\) 0 0
\(241\) 18.5105 1.19237 0.596184 0.802848i \(-0.296683\pi\)
0.596184 + 0.802848i \(0.296683\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.25526 −0.335747
\(246\) 0 0
\(247\) 21.8688 1.39148
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.67912 0.421582 0.210791 0.977531i \(-0.432396\pi\)
0.210791 + 0.977531i \(0.432396\pi\)
\(252\) 0 0
\(253\) −9.08482 −0.571158
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 26.4249 1.64834 0.824170 0.566342i \(-0.191642\pi\)
0.824170 + 0.566342i \(0.191642\pi\)
\(258\) 0 0
\(259\) −9.15951 −0.569145
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −27.4340 −1.69165 −0.845826 0.533459i \(-0.820892\pi\)
−0.845826 + 0.533459i \(0.820892\pi\)
\(264\) 0 0
\(265\) −2.38650 −0.146602
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.9717 0.668958 0.334479 0.942403i \(-0.391440\pi\)
0.334479 + 0.942403i \(0.391440\pi\)
\(270\) 0 0
\(271\) 20.1504 1.22405 0.612026 0.790838i \(-0.290355\pi\)
0.612026 + 0.790838i \(0.290355\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.32088 −0.320861
\(276\) 0 0
\(277\) 3.15951 0.189837 0.0949184 0.995485i \(-0.469741\pi\)
0.0949184 + 0.995485i \(0.469741\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.3118 0.615151 0.307576 0.951524i \(-0.400482\pi\)
0.307576 + 0.951524i \(0.400482\pi\)
\(282\) 0 0
\(283\) 0.423851 0.0251953 0.0125977 0.999921i \(-0.495990\pi\)
0.0125977 + 0.999921i \(0.495990\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.35823 0.0801738
\(288\) 0 0
\(289\) 23.3118 1.37128
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −19.2462 −1.12437 −0.562187 0.827010i \(-0.690040\pi\)
−0.562187 + 0.827010i \(0.690040\pi\)
\(294\) 0 0
\(295\) −10.4431 −0.608018
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −8.58522 −0.496496
\(300\) 0 0
\(301\) 14.1806 0.817355
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6.38650 −0.365690
\(306\) 0 0
\(307\) 22.7357 1.29759 0.648796 0.760962i \(-0.275273\pi\)
0.648796 + 0.760962i \(0.275273\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −14.8031 −0.839409 −0.419704 0.907661i \(-0.637866\pi\)
−0.419704 + 0.907661i \(0.637866\pi\)
\(312\) 0 0
\(313\) −14.9717 −0.846252 −0.423126 0.906071i \(-0.639067\pi\)
−0.423126 + 0.906071i \(0.639067\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −23.1222 −1.29867 −0.649335 0.760502i \(-0.724953\pi\)
−0.649335 + 0.760502i \(0.724953\pi\)
\(318\) 0 0
\(319\) −5.32088 −0.297912
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −27.6135 −1.53646
\(324\) 0 0
\(325\) −5.02827 −0.278918
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.897033 0.0494550
\(330\) 0 0
\(331\) −9.82048 −0.539783 −0.269891 0.962891i \(-0.586988\pi\)
−0.269891 + 0.962891i \(0.586988\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −5.70739 −0.311828
\(336\) 0 0
\(337\) −8.09389 −0.440903 −0.220451 0.975398i \(-0.570753\pi\)
−0.220451 + 0.975398i \(0.570753\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −44.4249 −2.40574
\(342\) 0 0
\(343\) −16.1878 −0.874058
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.37743 0.396041 0.198021 0.980198i \(-0.436549\pi\)
0.198021 + 0.980198i \(0.436549\pi\)
\(348\) 0 0
\(349\) 8.82956 0.472635 0.236318 0.971676i \(-0.424059\pi\)
0.236318 + 0.971676i \(0.424059\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.7266 −0.730593 −0.365296 0.930891i \(-0.619032\pi\)
−0.365296 + 0.930891i \(0.619032\pi\)
\(354\) 0 0
\(355\) 3.61350 0.191784
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.70739 0.0901126 0.0450563 0.998984i \(-0.485653\pi\)
0.0450563 + 0.998984i \(0.485653\pi\)
\(360\) 0 0
\(361\) −0.0848216 −0.00446429
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.73566 −0.352561
\(366\) 0 0
\(367\) −25.9627 −1.35524 −0.677620 0.735412i \(-0.736988\pi\)
−0.677620 + 0.735412i \(0.736988\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.15230 −0.163659
\(372\) 0 0
\(373\) 9.92531 0.513913 0.256956 0.966423i \(-0.417280\pi\)
0.256956 + 0.966423i \(0.417280\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.02827 −0.258969
\(378\) 0 0
\(379\) 34.2070 1.75710 0.878548 0.477655i \(-0.158513\pi\)
0.878548 + 0.477655i \(0.158513\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 21.4340 1.09523 0.547613 0.836732i \(-0.315537\pi\)
0.547613 + 0.836732i \(0.315537\pi\)
\(384\) 0 0
\(385\) −7.02827 −0.358194
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 36.3684 1.84395 0.921975 0.387251i \(-0.126575\pi\)
0.921975 + 0.387251i \(0.126575\pi\)
\(390\) 0 0
\(391\) 10.8405 0.548227
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −11.3774 −0.572461
\(396\) 0 0
\(397\) −27.4713 −1.37875 −0.689373 0.724406i \(-0.742114\pi\)
−0.689373 + 0.724406i \(0.742114\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.3118 −1.11420 −0.557099 0.830446i \(-0.688086\pi\)
−0.557099 + 0.830446i \(0.688086\pi\)
\(402\) 0 0
\(403\) −41.9819 −2.09127
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 36.8970 1.82892
\(408\) 0 0
\(409\) 4.95358 0.244939 0.122469 0.992472i \(-0.460919\pi\)
0.122469 + 0.992472i \(0.460919\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −13.7941 −0.678762
\(414\) 0 0
\(415\) 3.96265 0.194519
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.198716 0.00970793 0.00485397 0.999988i \(-0.498455\pi\)
0.00485397 + 0.999988i \(0.498455\pi\)
\(420\) 0 0
\(421\) 26.4996 1.29151 0.645756 0.763544i \(-0.276542\pi\)
0.645756 + 0.763544i \(0.276542\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.34916 0.307979
\(426\) 0 0
\(427\) −8.43584 −0.408239
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 21.2835 1.02519 0.512596 0.858630i \(-0.328684\pi\)
0.512596 + 0.858630i \(0.328684\pi\)
\(432\) 0 0
\(433\) −24.8031 −1.19196 −0.595981 0.802998i \(-0.703237\pi\)
−0.595981 + 0.802998i \(0.703237\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.42571 −0.355220
\(438\) 0 0
\(439\) −32.3118 −1.54216 −0.771079 0.636739i \(-0.780283\pi\)
−0.771079 + 0.636739i \(0.780283\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −25.3774 −1.20572 −0.602859 0.797848i \(-0.705972\pi\)
−0.602859 + 0.797848i \(0.705972\pi\)
\(444\) 0 0
\(445\) −2.58522 −0.122551
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −13.7266 −0.647798 −0.323899 0.946092i \(-0.604994\pi\)
−0.323899 + 0.946092i \(0.604994\pi\)
\(450\) 0 0
\(451\) −5.47133 −0.257635
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.64177 −0.311371
\(456\) 0 0
\(457\) −15.0101 −0.702144 −0.351072 0.936348i \(-0.614183\pi\)
−0.351072 + 0.936348i \(0.614183\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −32.7549 −1.52555 −0.762773 0.646666i \(-0.776163\pi\)
−0.762773 + 0.646666i \(0.776163\pi\)
\(462\) 0 0
\(463\) −24.5369 −1.14033 −0.570164 0.821531i \(-0.693120\pi\)
−0.570164 + 0.821531i \(0.693120\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.60442 0.398165 0.199083 0.979983i \(-0.436204\pi\)
0.199083 + 0.979983i \(0.436204\pi\)
\(468\) 0 0
\(469\) −7.53880 −0.348110
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −57.1232 −2.62653
\(474\) 0 0
\(475\) −4.34916 −0.199553
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.26434 −0.149151 −0.0745757 0.997215i \(-0.523760\pi\)
−0.0745757 + 0.997215i \(0.523760\pi\)
\(480\) 0 0
\(481\) 34.8680 1.58984
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −15.3209 −0.695686
\(486\) 0 0
\(487\) 34.2070 1.55007 0.775033 0.631920i \(-0.217733\pi\)
0.775033 + 0.631920i \(0.217733\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8.93438 0.403203 0.201601 0.979468i \(-0.435385\pi\)
0.201601 + 0.979468i \(0.435385\pi\)
\(492\) 0 0
\(493\) 6.34916 0.285952
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.77301 0.214099
\(498\) 0 0
\(499\) 28.6236 1.28137 0.640685 0.767804i \(-0.278651\pi\)
0.640685 + 0.767804i \(0.278651\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0.0192012 0.000856140 0 0.000428070 1.00000i \(-0.499864\pi\)
0.000428070 1.00000i \(0.499864\pi\)
\(504\) 0 0
\(505\) −11.8688 −0.528153
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8.25526 0.365908 0.182954 0.983121i \(-0.441434\pi\)
0.182954 + 0.983121i \(0.441434\pi\)
\(510\) 0 0
\(511\) −8.89703 −0.393582
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.65084 −0.160875
\(516\) 0 0
\(517\) −3.61350 −0.158921
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 20.2553 0.887399 0.443700 0.896176i \(-0.353666\pi\)
0.443700 + 0.896176i \(0.353666\pi\)
\(522\) 0 0
\(523\) 1.82048 0.0796042 0.0398021 0.999208i \(-0.487327\pi\)
0.0398021 + 0.999208i \(0.487327\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 53.0101 2.30916
\(528\) 0 0
\(529\) −20.0848 −0.873253
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −5.17044 −0.223957
\(534\) 0 0
\(535\) 5.32088 0.230042
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 27.9627 1.20444
\(540\) 0 0
\(541\) 1.03920 0.0446788 0.0223394 0.999750i \(-0.492889\pi\)
0.0223394 + 0.999750i \(0.492889\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −14.3118 −0.613051
\(546\) 0 0
\(547\) 37.7567 1.61436 0.807180 0.590305i \(-0.200992\pi\)
0.807180 + 0.590305i \(0.200992\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −4.34916 −0.185280
\(552\) 0 0
\(553\) −15.0283 −0.639067
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −45.8506 −1.94275 −0.971376 0.237545i \(-0.923657\pi\)
−0.971376 + 0.237545i \(0.923657\pi\)
\(558\) 0 0
\(559\) −53.9819 −2.28319
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −33.3027 −1.40354 −0.701772 0.712402i \(-0.747607\pi\)
−0.701772 + 0.712402i \(0.747607\pi\)
\(564\) 0 0
\(565\) −2.03735 −0.0857118
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23.6700 0.992300 0.496150 0.868237i \(-0.334747\pi\)
0.496150 + 0.868237i \(0.334747\pi\)
\(570\) 0 0
\(571\) 14.5671 0.609613 0.304807 0.952414i \(-0.401408\pi\)
0.304807 + 0.952414i \(0.401408\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.70739 0.0712031
\(576\) 0 0
\(577\) −13.7639 −0.573000 −0.286500 0.958080i \(-0.592492\pi\)
−0.286500 + 0.958080i \(0.592492\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.23421 0.217152
\(582\) 0 0
\(583\) 12.6983 0.525911
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.04748 0.0432339 0.0216170 0.999766i \(-0.493119\pi\)
0.0216170 + 0.999766i \(0.493119\pi\)
\(588\) 0 0
\(589\) −36.3118 −1.49620
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −14.3865 −0.590783 −0.295391 0.955376i \(-0.595450\pi\)
−0.295391 + 0.955376i \(0.595450\pi\)
\(594\) 0 0
\(595\) 8.38650 0.343813
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 5.98080 0.244369 0.122184 0.992507i \(-0.461010\pi\)
0.122184 + 0.992507i \(0.461010\pi\)
\(600\) 0 0
\(601\) −7.08482 −0.288996 −0.144498 0.989505i \(-0.546157\pi\)
−0.144498 + 0.989505i \(0.546157\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 17.3118 0.703825
\(606\) 0 0
\(607\) 32.7175 1.32796 0.663982 0.747749i \(-0.268865\pi\)
0.663982 + 0.747749i \(0.268865\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.41478 −0.138147
\(612\) 0 0
\(613\) 9.22699 0.372675 0.186337 0.982486i \(-0.440338\pi\)
0.186337 + 0.982486i \(0.440338\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.281683 −0.0113401 −0.00567006 0.999984i \(-0.501805\pi\)
−0.00567006 + 0.999984i \(0.501805\pi\)
\(618\) 0 0
\(619\) 21.2462 0.853957 0.426978 0.904262i \(-0.359578\pi\)
0.426978 + 0.904262i \(0.359578\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.41478 −0.136810
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −44.0275 −1.75549
\(630\) 0 0
\(631\) 37.8688 1.50753 0.753766 0.657143i \(-0.228235\pi\)
0.753766 + 0.657143i \(0.228235\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.26434 −0.0501737
\(636\) 0 0
\(637\) 26.4249 1.04699
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 36.7658 1.45216 0.726081 0.687609i \(-0.241340\pi\)
0.726081 + 0.687609i \(0.241340\pi\)
\(642\) 0 0
\(643\) −24.9728 −0.984830 −0.492415 0.870360i \(-0.663886\pi\)
−0.492415 + 0.870360i \(0.663886\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.56522 0.140163 0.0700816 0.997541i \(-0.477674\pi\)
0.0700816 + 0.997541i \(0.477674\pi\)
\(648\) 0 0
\(649\) 55.5663 2.18117
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 21.5652 0.843912 0.421956 0.906616i \(-0.361344\pi\)
0.421956 + 0.906616i \(0.361344\pi\)
\(654\) 0 0
\(655\) 1.70739 0.0667132
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 40.4623 1.57619 0.788093 0.615556i \(-0.211069\pi\)
0.788093 + 0.615556i \(0.211069\pi\)
\(660\) 0 0
\(661\) 26.1987 1.01901 0.509506 0.860467i \(-0.329828\pi\)
0.509506 + 0.860467i \(0.329828\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.74474 −0.222771
\(666\) 0 0
\(667\) 1.70739 0.0661104
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 33.9819 1.31185
\(672\) 0 0
\(673\) 3.35823 0.129450 0.0647251 0.997903i \(-0.479383\pi\)
0.0647251 + 0.997903i \(0.479383\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15.6965 0.603264 0.301632 0.953424i \(-0.402469\pi\)
0.301632 + 0.953424i \(0.402469\pi\)
\(678\) 0 0
\(679\) −20.2371 −0.776629
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 40.1998 1.53820 0.769101 0.639128i \(-0.220704\pi\)
0.769101 + 0.639128i \(0.220704\pi\)
\(684\) 0 0
\(685\) −21.7639 −0.831557
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 25.7084 0.977995 0.488998 0.872285i \(-0.337363\pi\)
0.488998 + 0.872285i \(0.337363\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.6983 −0.633403
\(696\) 0 0
\(697\) 6.52867 0.247291
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 8.51775 0.321711 0.160855 0.986978i \(-0.448575\pi\)
0.160855 + 0.986978i \(0.448575\pi\)
\(702\) 0 0
\(703\) 30.1587 1.13746
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −15.6773 −0.589604
\(708\) 0 0
\(709\) 35.0848 1.31764 0.658819 0.752301i \(-0.271056\pi\)
0.658819 + 0.752301i \(0.271056\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 14.2553 0.533864
\(714\) 0 0
\(715\) 26.7549 1.00058
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −31.3292 −1.16838 −0.584190 0.811617i \(-0.698588\pi\)
−0.584190 + 0.811617i \(0.698588\pi\)
\(720\) 0 0
\(721\) −4.82234 −0.179593
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) −13.9627 −0.517846 −0.258923 0.965898i \(-0.583368\pi\)
−0.258923 + 0.965898i \(0.583368\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 68.1624 2.52108
\(732\) 0 0
\(733\) −29.5761 −1.09242 −0.546210 0.837648i \(-0.683930\pi\)
−0.546210 + 0.837648i \(0.683930\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 30.3684 1.11863
\(738\) 0 0
\(739\) −6.66819 −0.245293 −0.122647 0.992450i \(-0.539138\pi\)
−0.122647 + 0.992450i \(0.539138\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 33.7002 1.23634 0.618170 0.786045i \(-0.287874\pi\)
0.618170 + 0.786045i \(0.287874\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7.02827 0.256808
\(750\) 0 0
\(751\) −48.3129 −1.76296 −0.881481 0.472220i \(-0.843453\pi\)
−0.881481 + 0.472220i \(0.843453\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 6.44305 0.234487
\(756\) 0 0
\(757\) 42.3985 1.54100 0.770500 0.637440i \(-0.220007\pi\)
0.770500 + 0.637440i \(0.220007\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 44.7933 1.62375 0.811877 0.583828i \(-0.198446\pi\)
0.811877 + 0.583828i \(0.198446\pi\)
\(762\) 0 0
\(763\) −18.9043 −0.684380
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 52.5105 1.89605
\(768\) 0 0
\(769\) −52.2080 −1.88267 −0.941335 0.337473i \(-0.890428\pi\)
−0.941335 + 0.337473i \(0.890428\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 24.2179 0.871058 0.435529 0.900175i \(-0.356561\pi\)
0.435529 + 0.900175i \(0.356561\pi\)
\(774\) 0 0
\(775\) 8.34916 0.299911
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.47213 −0.160231
\(780\) 0 0
\(781\) −19.2270 −0.687996
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 10.7357 0.383172
\(786\) 0 0
\(787\) −1.43398 −0.0511159 −0.0255579 0.999673i \(-0.508136\pi\)
−0.0255579 + 0.999673i \(0.508136\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.69110 −0.0956845
\(792\) 0 0
\(793\) 32.1131 1.14037
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 14.4732 0.512666 0.256333 0.966588i \(-0.417486\pi\)
0.256333 + 0.966588i \(0.417486\pi\)
\(798\) 0 0
\(799\) 4.31181 0.152541
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 35.8397 1.26476
\(804\) 0 0
\(805\) 2.25526 0.0794876
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −49.2654 −1.73208 −0.866039 0.499976i \(-0.833342\pi\)
−0.866039 + 0.499976i \(0.833342\pi\)
\(810\) 0 0
\(811\) −4.69832 −0.164980 −0.0824901 0.996592i \(-0.526287\pi\)
−0.0824901 + 0.996592i \(0.526287\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.90611 0.136825
\(816\) 0 0
\(817\) −46.6911 −1.63351
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −34.7730 −1.21359 −0.606793 0.794860i \(-0.707544\pi\)
−0.606793 + 0.794860i \(0.707544\pi\)
\(822\) 0 0
\(823\) −44.1323 −1.53836 −0.769178 0.639035i \(-0.779334\pi\)
−0.769178 + 0.639035i \(0.779334\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −22.6610 −0.787999 −0.394000 0.919111i \(-0.628909\pi\)
−0.394000 + 0.919111i \(0.628909\pi\)
\(828\) 0 0
\(829\) −37.9144 −1.31682 −0.658410 0.752659i \(-0.728771\pi\)
−0.658410 + 0.752659i \(0.728771\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −33.3665 −1.15608
\(834\) 0 0
\(835\) 5.51960 0.191014
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −30.7175 −1.06049 −0.530243 0.847846i \(-0.677899\pi\)
−0.530243 + 0.847846i \(0.677899\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.2835 0.422567
\(846\) 0 0
\(847\) 22.8669 0.785716
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −11.8397 −0.405859
\(852\) 0 0
\(853\) 1.25341 0.0429159 0.0214579 0.999770i \(-0.493169\pi\)
0.0214579 + 0.999770i \(0.493169\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.85783 0.268418 0.134209 0.990953i \(-0.457151\pi\)
0.134209 + 0.990953i \(0.457151\pi\)
\(858\) 0 0
\(859\) −14.4913 −0.494438 −0.247219 0.968960i \(-0.579517\pi\)
−0.247219 + 0.968960i \(0.579517\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9.69646 0.330071 0.165036 0.986288i \(-0.447226\pi\)
0.165036 + 0.986288i \(0.447226\pi\)
\(864\) 0 0
\(865\) −5.80128 −0.197250
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 60.5380 2.05361
\(870\) 0 0
\(871\) 28.6983 0.972405
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.32088 0.0446540
\(876\) 0 0
\(877\) 6.27341 0.211838 0.105919 0.994375i \(-0.466222\pi\)
0.105919 + 0.994375i \(0.466222\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −19.5953 −0.660184 −0.330092 0.943949i \(-0.607080\pi\)
−0.330092 + 0.943949i \(0.607080\pi\)
\(882\) 0 0
\(883\) −5.24619 −0.176548 −0.0882742 0.996096i \(-0.528135\pi\)
−0.0882742 + 0.996096i \(0.528135\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32.6044 1.09475 0.547375 0.836888i \(-0.315627\pi\)
0.547375 + 0.836888i \(0.315627\pi\)
\(888\) 0 0
\(889\) −1.67004 −0.0560114
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.95358 −0.0988378
\(894\) 0 0
\(895\) −14.0565 −0.469859
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 8.34916 0.278460
\(900\) 0 0
\(901\) −15.1523 −0.504796
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.08482 −0.235507
\(906\) 0 0
\(907\) −1.96265 −0.0651688 −0.0325844 0.999469i \(-0.510374\pi\)
−0.0325844 + 0.999469i \(0.510374\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −50.7066 −1.67998 −0.839992 0.542599i \(-0.817440\pi\)
−0.839992 + 0.542599i \(0.817440\pi\)
\(912\) 0 0
\(913\) −21.0848 −0.697806
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.25526 0.0744754
\(918\) 0 0
\(919\) 56.2371 1.85509 0.927546 0.373710i \(-0.121914\pi\)
0.927546 + 0.373710i \(0.121914\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −18.1696 −0.598061
\(924\) 0 0
\(925\) −6.93438 −0.228001
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 5.34009 0.175203 0.0876013 0.996156i \(-0.472080\pi\)
0.0876013 + 0.996156i \(0.472080\pi\)
\(930\) 0 0
\(931\) 22.8560 0.749074
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −33.7831 −1.10483
\(936\) 0 0
\(937\) −3.01013 −0.0983366 −0.0491683 0.998791i \(-0.515657\pi\)
−0.0491683 + 0.998791i \(0.515657\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.829557 0.0270428 0.0135214 0.999909i \(-0.495696\pi\)
0.0135214 + 0.999909i \(0.495696\pi\)
\(942\) 0 0
\(943\) 1.75566 0.0571723
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −21.3027 −0.692246 −0.346123 0.938189i \(-0.612502\pi\)
−0.346123 + 0.938189i \(0.612502\pi\)
\(948\) 0 0
\(949\) 33.8688 1.09943
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 40.0203 1.29638 0.648192 0.761477i \(-0.275526\pi\)
0.648192 + 0.761477i \(0.275526\pi\)
\(954\) 0 0
\(955\) 0.547875 0.0177288
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −28.7476 −0.928310
\(960\) 0 0
\(961\) 38.7084 1.24866
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.40571 0.141825
\(966\) 0 0
\(967\) −31.0365 −0.998068 −0.499034 0.866582i \(-0.666312\pi\)
−0.499034 + 0.866582i \(0.666312\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2.66819 0.0856262 0.0428131 0.999083i \(-0.486368\pi\)
0.0428131 + 0.999083i \(0.486368\pi\)
\(972\) 0 0
\(973\) −22.0565 −0.707100
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −45.6519 −1.46053 −0.730267 0.683162i \(-0.760604\pi\)
−0.730267 + 0.683162i \(0.760604\pi\)
\(978\) 0 0
\(979\) 13.7557 0.439633
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 17.4521 0.556636 0.278318 0.960489i \(-0.410223\pi\)
0.278318 + 0.960489i \(0.410223\pi\)
\(984\) 0 0
\(985\) 4.64177 0.147899
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 18.3300 0.582859
\(990\) 0 0
\(991\) −4.49960 −0.142935 −0.0714673 0.997443i \(-0.522768\pi\)
−0.0714673 + 0.997443i \(0.522768\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 18.4431 0.584684
\(996\) 0 0
\(997\) −21.1896 −0.671083 −0.335541 0.942025i \(-0.608919\pi\)
−0.335541 + 0.942025i \(0.608919\pi\)
\(998\) 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 5220.2.a.x.1.3 3
3.2 odd 2 580.2.a.c.1.3 3
12.11 even 2 2320.2.a.m.1.1 3
15.2 even 4 2900.2.c.f.349.1 6
15.8 even 4 2900.2.c.f.349.6 6
15.14 odd 2 2900.2.a.g.1.1 3
24.5 odd 2 9280.2.a.bk.1.1 3
24.11 even 2 9280.2.a.bw.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
580.2.a.c.1.3 3 3.2 odd 2
2320.2.a.m.1.1 3 12.11 even 2
2900.2.a.g.1.1 3 15.14 odd 2
2900.2.c.f.349.1 6 15.2 even 4
2900.2.c.f.349.6 6 15.8 even 4
5220.2.a.x.1.3 3 1.1 even 1 trivial
9280.2.a.bk.1.1 3 24.5 odd 2
9280.2.a.bw.1.3 3 24.11 even 2