Properties

Label 1904.2.a.q.1.3
Level $1904$
Weight $2$
Character 1904.1
Self dual yes
Analytic conductor $15.204$
Analytic rank $1$
Dimension $4$
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] = [1904,2,Mod(1,1904)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1904, 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("1904.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1904 = 2^{4} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1904.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: \(15.2035165449\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 952)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.26498\) of defining polynomial
Character \(\chi\) \(=\) 1904.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.264977 q^{3} +0.163765 q^{5} +1.00000 q^{7} -2.92979 q^{9} +O(q^{10})\) \(q+0.264977 q^{3} +0.163765 q^{5} +1.00000 q^{7} -2.92979 q^{9} -3.23607 q^{11} +2.47214 q^{13} +0.0433939 q^{15} +1.00000 q^{17} -1.47005 q^{19} +0.264977 q^{21} -4.85748 q^{23} -4.97318 q^{25} -1.57126 q^{27} +0.378584 q^{29} -8.78727 q^{31} -0.857484 q^{33} +0.163765 q^{35} +8.56569 q^{37} +0.655059 q^{39} +5.97318 q^{41} -10.0310 q^{43} -0.479796 q^{45} +8.18710 q^{47} +1.00000 q^{49} +0.264977 q^{51} -8.72945 q^{53} -0.529954 q^{55} -0.389529 q^{57} -2.85748 q^{59} -13.2671 q^{61} -2.92979 q^{63} +0.404849 q^{65} -14.9319 q^{67} -1.28712 q^{69} -14.3317 q^{71} -4.91327 q^{73} -1.31778 q^{75} -3.23607 q^{77} +16.1871 q^{79} +8.37301 q^{81} +5.65715 q^{83} +0.163765 q^{85} +0.100316 q^{87} +13.5167 q^{89} +2.47214 q^{91} -2.32843 q^{93} -0.240742 q^{95} -12.0744 q^{97} +9.48099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{3} - q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{3} - q^{5} + 4 q^{7} + 7 q^{9} - 4 q^{11} - 8 q^{13} - 12 q^{15} + 4 q^{17} - 14 q^{19} - 3 q^{21} - 8 q^{23} + 11 q^{25} - 12 q^{27} + 4 q^{29} - 5 q^{31} + 8 q^{33} - q^{35} - 4 q^{37} - 4 q^{39} - 7 q^{41} - 19 q^{43} - 2 q^{45} - 8 q^{47} + 4 q^{49} - 3 q^{51} + 5 q^{53} + 6 q^{55} + 44 q^{57} - 23 q^{61} + 7 q^{63} - 8 q^{65} - 15 q^{67} - 2 q^{69} - 2 q^{71} - 5 q^{73} - 10 q^{75} - 4 q^{77} + 24 q^{79} - 8 q^{81} - 10 q^{83} - q^{85} - 16 q^{87} - 16 q^{89} - 8 q^{91} - 20 q^{93} - 22 q^{95} - 15 q^{97} - 2 q^{99}+O(q^{100}) \) 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
\(3\) 0.264977 0.152985 0.0764923 0.997070i \(-0.475628\pi\)
0.0764923 + 0.997070i \(0.475628\pi\)
\(4\) 0 0
\(5\) 0.163765 0.0732379 0.0366189 0.999329i \(-0.488341\pi\)
0.0366189 + 0.999329i \(0.488341\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −2.92979 −0.976596
\(10\) 0 0
\(11\) −3.23607 −0.975711 −0.487856 0.872924i \(-0.662221\pi\)
−0.487856 + 0.872924i \(0.662221\pi\)
\(12\) 0 0
\(13\) 2.47214 0.685647 0.342824 0.939400i \(-0.388617\pi\)
0.342824 + 0.939400i \(0.388617\pi\)
\(14\) 0 0
\(15\) 0.0433939 0.0112043
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) −1.47005 −0.337252 −0.168626 0.985680i \(-0.553933\pi\)
−0.168626 + 0.985680i \(0.553933\pi\)
\(20\) 0 0
\(21\) 0.264977 0.0578228
\(22\) 0 0
\(23\) −4.85748 −1.01286 −0.506428 0.862282i \(-0.669034\pi\)
−0.506428 + 0.862282i \(0.669034\pi\)
\(24\) 0 0
\(25\) −4.97318 −0.994636
\(26\) 0 0
\(27\) −1.57126 −0.302389
\(28\) 0 0
\(29\) 0.378584 0.0703013 0.0351506 0.999382i \(-0.488809\pi\)
0.0351506 + 0.999382i \(0.488809\pi\)
\(30\) 0 0
\(31\) −8.78727 −1.57824 −0.789120 0.614239i \(-0.789463\pi\)
−0.789120 + 0.614239i \(0.789463\pi\)
\(32\) 0 0
\(33\) −0.857484 −0.149269
\(34\) 0 0
\(35\) 0.163765 0.0276813
\(36\) 0 0
\(37\) 8.56569 1.40819 0.704095 0.710106i \(-0.251353\pi\)
0.704095 + 0.710106i \(0.251353\pi\)
\(38\) 0 0
\(39\) 0.655059 0.104893
\(40\) 0 0
\(41\) 5.97318 0.932854 0.466427 0.884560i \(-0.345541\pi\)
0.466427 + 0.884560i \(0.345541\pi\)
\(42\) 0 0
\(43\) −10.0310 −1.52971 −0.764857 0.644201i \(-0.777190\pi\)
−0.764857 + 0.644201i \(0.777190\pi\)
\(44\) 0 0
\(45\) −0.479796 −0.0715238
\(46\) 0 0
\(47\) 8.18710 1.19421 0.597106 0.802162i \(-0.296317\pi\)
0.597106 + 0.802162i \(0.296317\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.264977 0.0371042
\(52\) 0 0
\(53\) −8.72945 −1.19908 −0.599541 0.800344i \(-0.704650\pi\)
−0.599541 + 0.800344i \(0.704650\pi\)
\(54\) 0 0
\(55\) −0.529954 −0.0714590
\(56\) 0 0
\(57\) −0.389529 −0.0515943
\(58\) 0 0
\(59\) −2.85748 −0.372013 −0.186006 0.982549i \(-0.559555\pi\)
−0.186006 + 0.982549i \(0.559555\pi\)
\(60\) 0 0
\(61\) −13.2671 −1.69867 −0.849337 0.527851i \(-0.822998\pi\)
−0.849337 + 0.527851i \(0.822998\pi\)
\(62\) 0 0
\(63\) −2.92979 −0.369118
\(64\) 0 0
\(65\) 0.404849 0.0502153
\(66\) 0 0
\(67\) −14.9319 −1.82422 −0.912110 0.409946i \(-0.865547\pi\)
−0.912110 + 0.409946i \(0.865547\pi\)
\(68\) 0 0
\(69\) −1.28712 −0.154951
\(70\) 0 0
\(71\) −14.3317 −1.70086 −0.850431 0.526087i \(-0.823658\pi\)
−0.850431 + 0.526087i \(0.823658\pi\)
\(72\) 0 0
\(73\) −4.91327 −0.575055 −0.287528 0.957772i \(-0.592833\pi\)
−0.287528 + 0.957772i \(0.592833\pi\)
\(74\) 0 0
\(75\) −1.31778 −0.152164
\(76\) 0 0
\(77\) −3.23607 −0.368784
\(78\) 0 0
\(79\) 16.1871 1.82119 0.910596 0.413298i \(-0.135623\pi\)
0.910596 + 0.413298i \(0.135623\pi\)
\(80\) 0 0
\(81\) 8.37301 0.930335
\(82\) 0 0
\(83\) 5.65715 0.620953 0.310476 0.950581i \(-0.399511\pi\)
0.310476 + 0.950581i \(0.399511\pi\)
\(84\) 0 0
\(85\) 0.163765 0.0177628
\(86\) 0 0
\(87\) 0.100316 0.0107550
\(88\) 0 0
\(89\) 13.5167 1.43277 0.716385 0.697705i \(-0.245795\pi\)
0.716385 + 0.697705i \(0.245795\pi\)
\(90\) 0 0
\(91\) 2.47214 0.259150
\(92\) 0 0
\(93\) −2.32843 −0.241447
\(94\) 0 0
\(95\) −0.240742 −0.0246996
\(96\) 0 0
\(97\) −12.0744 −1.22597 −0.612984 0.790095i \(-0.710031\pi\)
−0.612984 + 0.790095i \(0.710031\pi\)
\(98\) 0 0
\(99\) 9.48099 0.952875
\(100\) 0 0
\(101\) 15.3916 1.53152 0.765762 0.643124i \(-0.222362\pi\)
0.765762 + 0.643124i \(0.222362\pi\)
\(102\) 0 0
\(103\) −10.7997 −1.06412 −0.532061 0.846706i \(-0.678582\pi\)
−0.532061 + 0.846706i \(0.678582\pi\)
\(104\) 0 0
\(105\) 0.0433939 0.00423482
\(106\) 0 0
\(107\) −9.27857 −0.896993 −0.448496 0.893785i \(-0.648040\pi\)
−0.448496 + 0.893785i \(0.648040\pi\)
\(108\) 0 0
\(109\) −2.90436 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(110\) 0 0
\(111\) 2.26971 0.215431
\(112\) 0 0
\(113\) 8.67456 0.816034 0.408017 0.912974i \(-0.366220\pi\)
0.408017 + 0.912974i \(0.366220\pi\)
\(114\) 0 0
\(115\) −0.795485 −0.0741794
\(116\) 0 0
\(117\) −7.24283 −0.669600
\(118\) 0 0
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −0.527864 −0.0479876
\(122\) 0 0
\(123\) 1.58276 0.142712
\(124\) 0 0
\(125\) −1.63326 −0.146083
\(126\) 0 0
\(127\) −6.19950 −0.550117 −0.275058 0.961428i \(-0.588697\pi\)
−0.275058 + 0.961428i \(0.588697\pi\)
\(128\) 0 0
\(129\) −2.65799 −0.234023
\(130\) 0 0
\(131\) 8.23816 0.719771 0.359886 0.932996i \(-0.382816\pi\)
0.359886 + 0.932996i \(0.382816\pi\)
\(132\) 0 0
\(133\) −1.47005 −0.127469
\(134\) 0 0
\(135\) −0.257317 −0.0221463
\(136\) 0 0
\(137\) 2.02682 0.173163 0.0865814 0.996245i \(-0.472406\pi\)
0.0865814 + 0.996245i \(0.472406\pi\)
\(138\) 0 0
\(139\) 16.6979 1.41630 0.708149 0.706063i \(-0.249531\pi\)
0.708149 + 0.706063i \(0.249531\pi\)
\(140\) 0 0
\(141\) 2.16940 0.182696
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 0.0619988 0.00514872
\(146\) 0 0
\(147\) 0.264977 0.0218549
\(148\) 0 0
\(149\) −13.1004 −1.07322 −0.536612 0.843829i \(-0.680296\pi\)
−0.536612 + 0.843829i \(0.680296\pi\)
\(150\) 0 0
\(151\) −12.5031 −1.01749 −0.508745 0.860917i \(-0.669891\pi\)
−0.508745 + 0.860917i \(0.669891\pi\)
\(152\) 0 0
\(153\) −2.92979 −0.236859
\(154\) 0 0
\(155\) −1.43905 −0.115587
\(156\) 0 0
\(157\) −16.7171 −1.33417 −0.667083 0.744983i \(-0.732457\pi\)
−0.667083 + 0.744983i \(0.732457\pi\)
\(158\) 0 0
\(159\) −2.31311 −0.183441
\(160\) 0 0
\(161\) −4.85748 −0.382823
\(162\) 0 0
\(163\) 11.0089 0.862280 0.431140 0.902285i \(-0.358112\pi\)
0.431140 + 0.902285i \(0.358112\pi\)
\(164\) 0 0
\(165\) −0.140426 −0.0109321
\(166\) 0 0
\(167\) 4.97109 0.384675 0.192337 0.981329i \(-0.438393\pi\)
0.192337 + 0.981329i \(0.438393\pi\)
\(168\) 0 0
\(169\) −6.88854 −0.529888
\(170\) 0 0
\(171\) 4.30692 0.329358
\(172\) 0 0
\(173\) −17.4542 −1.32702 −0.663508 0.748169i \(-0.730933\pi\)
−0.663508 + 0.748169i \(0.730933\pi\)
\(174\) 0 0
\(175\) −4.97318 −0.375937
\(176\) 0 0
\(177\) −0.757168 −0.0569122
\(178\) 0 0
\(179\) 2.38332 0.178138 0.0890688 0.996025i \(-0.471611\pi\)
0.0890688 + 0.996025i \(0.471611\pi\)
\(180\) 0 0
\(181\) −24.7103 −1.83670 −0.918351 0.395767i \(-0.870479\pi\)
−0.918351 + 0.395767i \(0.870479\pi\)
\(182\) 0 0
\(183\) −3.51547 −0.259871
\(184\) 0 0
\(185\) 1.40276 0.103133
\(186\) 0 0
\(187\) −3.23607 −0.236645
\(188\) 0 0
\(189\) −1.57126 −0.114292
\(190\) 0 0
\(191\) 20.3193 1.47025 0.735127 0.677929i \(-0.237122\pi\)
0.735127 + 0.677929i \(0.237122\pi\)
\(192\) 0 0
\(193\) 16.4568 1.18459 0.592294 0.805722i \(-0.298223\pi\)
0.592294 + 0.805722i \(0.298223\pi\)
\(194\) 0 0
\(195\) 0.107276 0.00768218
\(196\) 0 0
\(197\) −3.38067 −0.240863 −0.120432 0.992722i \(-0.538428\pi\)
−0.120432 + 0.992722i \(0.538428\pi\)
\(198\) 0 0
\(199\) −5.14873 −0.364984 −0.182492 0.983207i \(-0.558416\pi\)
−0.182492 + 0.983207i \(0.558416\pi\)
\(200\) 0 0
\(201\) −3.95661 −0.279077
\(202\) 0 0
\(203\) 0.378584 0.0265714
\(204\) 0 0
\(205\) 0.978197 0.0683203
\(206\) 0 0
\(207\) 14.2314 0.989150
\(208\) 0 0
\(209\) 4.75717 0.329060
\(210\) 0 0
\(211\) 11.6256 0.800339 0.400170 0.916441i \(-0.368951\pi\)
0.400170 + 0.916441i \(0.368951\pi\)
\(212\) 0 0
\(213\) −3.79758 −0.260206
\(214\) 0 0
\(215\) −1.64273 −0.112033
\(216\) 0 0
\(217\) −8.78727 −0.596519
\(218\) 0 0
\(219\) −1.30190 −0.0879746
\(220\) 0 0
\(221\) 2.47214 0.166294
\(222\) 0 0
\(223\) 11.6725 0.781646 0.390823 0.920466i \(-0.372190\pi\)
0.390823 + 0.920466i \(0.372190\pi\)
\(224\) 0 0
\(225\) 14.5704 0.971357
\(226\) 0 0
\(227\) −14.9056 −0.989320 −0.494660 0.869087i \(-0.664707\pi\)
−0.494660 + 0.869087i \(0.664707\pi\)
\(228\) 0 0
\(229\) −18.7997 −1.24232 −0.621158 0.783685i \(-0.713338\pi\)
−0.621158 + 0.783685i \(0.713338\pi\)
\(230\) 0 0
\(231\) −0.857484 −0.0564183
\(232\) 0 0
\(233\) 16.7997 1.10058 0.550291 0.834973i \(-0.314517\pi\)
0.550291 + 0.834973i \(0.314517\pi\)
\(234\) 0 0
\(235\) 1.34076 0.0874615
\(236\) 0 0
\(237\) 4.28921 0.278614
\(238\) 0 0
\(239\) −16.5031 −1.06750 −0.533750 0.845643i \(-0.679217\pi\)
−0.533750 + 0.845643i \(0.679217\pi\)
\(240\) 0 0
\(241\) 19.1768 1.23529 0.617643 0.786458i \(-0.288088\pi\)
0.617643 + 0.786458i \(0.288088\pi\)
\(242\) 0 0
\(243\) 6.93243 0.444716
\(244\) 0 0
\(245\) 0.163765 0.0104626
\(246\) 0 0
\(247\) −3.63415 −0.231236
\(248\) 0 0
\(249\) 1.49902 0.0949962
\(250\) 0 0
\(251\) −21.8307 −1.37794 −0.688972 0.724788i \(-0.741938\pi\)
−0.688972 + 0.724788i \(0.741938\pi\)
\(252\) 0 0
\(253\) 15.7191 0.988254
\(254\) 0 0
\(255\) 0.0433939 0.00271743
\(256\) 0 0
\(257\) −21.5167 −1.34218 −0.671088 0.741378i \(-0.734173\pi\)
−0.671088 + 0.741378i \(0.734173\pi\)
\(258\) 0 0
\(259\) 8.56569 0.532246
\(260\) 0 0
\(261\) −1.10917 −0.0686559
\(262\) 0 0
\(263\) 5.71497 0.352400 0.176200 0.984354i \(-0.443619\pi\)
0.176200 + 0.984354i \(0.443619\pi\)
\(264\) 0 0
\(265\) −1.42958 −0.0878183
\(266\) 0 0
\(267\) 3.58162 0.219192
\(268\) 0 0
\(269\) −23.8375 −1.45340 −0.726699 0.686956i \(-0.758946\pi\)
−0.726699 + 0.686956i \(0.758946\pi\)
\(270\) 0 0
\(271\) 27.9930 1.70046 0.850228 0.526415i \(-0.176464\pi\)
0.850228 + 0.526415i \(0.176464\pi\)
\(272\) 0 0
\(273\) 0.655059 0.0396460
\(274\) 0 0
\(275\) 16.0936 0.970478
\(276\) 0 0
\(277\) 2.03155 0.122064 0.0610321 0.998136i \(-0.480561\pi\)
0.0610321 + 0.998136i \(0.480561\pi\)
\(278\) 0 0
\(279\) 25.7448 1.54130
\(280\) 0 0
\(281\) 30.4444 1.81616 0.908081 0.418795i \(-0.137547\pi\)
0.908081 + 0.418795i \(0.137547\pi\)
\(282\) 0 0
\(283\) 7.19482 0.427688 0.213844 0.976868i \(-0.431402\pi\)
0.213844 + 0.976868i \(0.431402\pi\)
\(284\) 0 0
\(285\) −0.0637911 −0.00377866
\(286\) 0 0
\(287\) 5.97318 0.352586
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −3.19944 −0.187554
\(292\) 0 0
\(293\) 22.6592 1.32377 0.661883 0.749607i \(-0.269757\pi\)
0.661883 + 0.749607i \(0.269757\pi\)
\(294\) 0 0
\(295\) −0.467956 −0.0272454
\(296\) 0 0
\(297\) 5.08470 0.295044
\(298\) 0 0
\(299\) −12.0084 −0.694461
\(300\) 0 0
\(301\) −10.0310 −0.578177
\(302\) 0 0
\(303\) 4.07843 0.234300
\(304\) 0 0
\(305\) −2.17268 −0.124407
\(306\) 0 0
\(307\) −8.44735 −0.482116 −0.241058 0.970511i \(-0.577494\pi\)
−0.241058 + 0.970511i \(0.577494\pi\)
\(308\) 0 0
\(309\) −2.86166 −0.162794
\(310\) 0 0
\(311\) 0.203679 0.0115496 0.00577479 0.999983i \(-0.498162\pi\)
0.00577479 + 0.999983i \(0.498162\pi\)
\(312\) 0 0
\(313\) −2.60964 −0.147505 −0.0737527 0.997277i \(-0.523498\pi\)
−0.0737527 + 0.997277i \(0.523498\pi\)
\(314\) 0 0
\(315\) −0.479796 −0.0270335
\(316\) 0 0
\(317\) 28.4253 1.59652 0.798261 0.602312i \(-0.205754\pi\)
0.798261 + 0.602312i \(0.205754\pi\)
\(318\) 0 0
\(319\) −1.22512 −0.0685937
\(320\) 0 0
\(321\) −2.45861 −0.137226
\(322\) 0 0
\(323\) −1.47005 −0.0817955
\(324\) 0 0
\(325\) −12.2944 −0.681969
\(326\) 0 0
\(327\) −0.769588 −0.0425583
\(328\) 0 0
\(329\) 8.18710 0.451370
\(330\) 0 0
\(331\) 28.8204 1.58411 0.792057 0.610447i \(-0.209010\pi\)
0.792057 + 0.610447i \(0.209010\pi\)
\(332\) 0 0
\(333\) −25.0956 −1.37523
\(334\) 0 0
\(335\) −2.44532 −0.133602
\(336\) 0 0
\(337\) 6.95959 0.379113 0.189557 0.981870i \(-0.439295\pi\)
0.189557 + 0.981870i \(0.439295\pi\)
\(338\) 0 0
\(339\) 2.29856 0.124841
\(340\) 0 0
\(341\) 28.4362 1.53991
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −0.210785 −0.0113483
\(346\) 0 0
\(347\) 4.06876 0.218423 0.109211 0.994019i \(-0.465167\pi\)
0.109211 + 0.994019i \(0.465167\pi\)
\(348\) 0 0
\(349\) −3.58568 −0.191937 −0.0959686 0.995384i \(-0.530595\pi\)
−0.0959686 + 0.995384i \(0.530595\pi\)
\(350\) 0 0
\(351\) −3.88436 −0.207332
\(352\) 0 0
\(353\) 19.1219 1.01776 0.508878 0.860839i \(-0.330060\pi\)
0.508878 + 0.860839i \(0.330060\pi\)
\(354\) 0 0
\(355\) −2.34703 −0.124567
\(356\) 0 0
\(357\) 0.264977 0.0140241
\(358\) 0 0
\(359\) 11.8047 0.623027 0.311514 0.950242i \(-0.399164\pi\)
0.311514 + 0.950242i \(0.399164\pi\)
\(360\) 0 0
\(361\) −16.8390 −0.886261
\(362\) 0 0
\(363\) −0.139872 −0.00734137
\(364\) 0 0
\(365\) −0.804621 −0.0421158
\(366\) 0 0
\(367\) −8.73035 −0.455721 −0.227860 0.973694i \(-0.573173\pi\)
−0.227860 + 0.973694i \(0.573173\pi\)
\(368\) 0 0
\(369\) −17.5001 −0.911021
\(370\) 0 0
\(371\) −8.72945 −0.453211
\(372\) 0 0
\(373\) 20.0930 1.04038 0.520188 0.854052i \(-0.325862\pi\)
0.520188 + 0.854052i \(0.325862\pi\)
\(374\) 0 0
\(375\) −0.432776 −0.0223484
\(376\) 0 0
\(377\) 0.935911 0.0482019
\(378\) 0 0
\(379\) −16.5852 −0.851924 −0.425962 0.904741i \(-0.640064\pi\)
−0.425962 + 0.904741i \(0.640064\pi\)
\(380\) 0 0
\(381\) −1.64273 −0.0841594
\(382\) 0 0
\(383\) −13.5745 −0.693627 −0.346813 0.937934i \(-0.612736\pi\)
−0.346813 + 0.937934i \(0.612736\pi\)
\(384\) 0 0
\(385\) −0.529954 −0.0270090
\(386\) 0 0
\(387\) 29.3887 1.49391
\(388\) 0 0
\(389\) 9.46183 0.479734 0.239867 0.970806i \(-0.422896\pi\)
0.239867 + 0.970806i \(0.422896\pi\)
\(390\) 0 0
\(391\) −4.85748 −0.245654
\(392\) 0 0
\(393\) 2.18292 0.110114
\(394\) 0 0
\(395\) 2.65088 0.133380
\(396\) 0 0
\(397\) −13.7383 −0.689506 −0.344753 0.938693i \(-0.612037\pi\)
−0.344753 + 0.938693i \(0.612037\pi\)
\(398\) 0 0
\(399\) −0.389529 −0.0195008
\(400\) 0 0
\(401\) −1.20451 −0.0601506 −0.0300753 0.999548i \(-0.509575\pi\)
−0.0300753 + 0.999548i \(0.509575\pi\)
\(402\) 0 0
\(403\) −21.7233 −1.08212
\(404\) 0 0
\(405\) 1.37121 0.0681358
\(406\) 0 0
\(407\) −27.7191 −1.37399
\(408\) 0 0
\(409\) 33.1739 1.64034 0.820171 0.572118i \(-0.193878\pi\)
0.820171 + 0.572118i \(0.193878\pi\)
\(410\) 0 0
\(411\) 0.537061 0.0264912
\(412\) 0 0
\(413\) −2.85748 −0.140608
\(414\) 0 0
\(415\) 0.926442 0.0454773
\(416\) 0 0
\(417\) 4.42456 0.216672
\(418\) 0 0
\(419\) 29.0969 1.42148 0.710738 0.703457i \(-0.248361\pi\)
0.710738 + 0.703457i \(0.248361\pi\)
\(420\) 0 0
\(421\) 9.31812 0.454137 0.227069 0.973879i \(-0.427086\pi\)
0.227069 + 0.973879i \(0.427086\pi\)
\(422\) 0 0
\(423\) −23.9865 −1.16626
\(424\) 0 0
\(425\) −4.97318 −0.241235
\(426\) 0 0
\(427\) −13.2671 −0.642038
\(428\) 0 0
\(429\) −2.11982 −0.102346
\(430\) 0 0
\(431\) −8.88227 −0.427844 −0.213922 0.976851i \(-0.568624\pi\)
−0.213922 + 0.976851i \(0.568624\pi\)
\(432\) 0 0
\(433\) 16.7749 0.806149 0.403075 0.915167i \(-0.367942\pi\)
0.403075 + 0.915167i \(0.367942\pi\)
\(434\) 0 0
\(435\) 0.0164283 0.000787674 0
\(436\) 0 0
\(437\) 7.14072 0.341587
\(438\) 0 0
\(439\) 38.0488 1.81597 0.907986 0.419001i \(-0.137620\pi\)
0.907986 + 0.419001i \(0.137620\pi\)
\(440\) 0 0
\(441\) −2.92979 −0.139514
\(442\) 0 0
\(443\) −25.8171 −1.22661 −0.613303 0.789848i \(-0.710160\pi\)
−0.613303 + 0.789848i \(0.710160\pi\)
\(444\) 0 0
\(445\) 2.21356 0.104933
\(446\) 0 0
\(447\) −3.47130 −0.164187
\(448\) 0 0
\(449\) 19.2470 0.908323 0.454161 0.890919i \(-0.349939\pi\)
0.454161 + 0.890919i \(0.349939\pi\)
\(450\) 0 0
\(451\) −19.3296 −0.910196
\(452\) 0 0
\(453\) −3.31304 −0.155660
\(454\) 0 0
\(455\) 0.404849 0.0189796
\(456\) 0 0
\(457\) −3.86250 −0.180680 −0.0903401 0.995911i \(-0.528795\pi\)
−0.0903401 + 0.995911i \(0.528795\pi\)
\(458\) 0 0
\(459\) −1.57126 −0.0733400
\(460\) 0 0
\(461\) −4.55474 −0.212136 −0.106068 0.994359i \(-0.533826\pi\)
−0.106068 + 0.994359i \(0.533826\pi\)
\(462\) 0 0
\(463\) 16.1603 0.751032 0.375516 0.926816i \(-0.377465\pi\)
0.375516 + 0.926816i \(0.377465\pi\)
\(464\) 0 0
\(465\) −0.381314 −0.0176830
\(466\) 0 0
\(467\) 3.29659 0.152548 0.0762740 0.997087i \(-0.475698\pi\)
0.0762740 + 0.997087i \(0.475698\pi\)
\(468\) 0 0
\(469\) −14.9319 −0.689490
\(470\) 0 0
\(471\) −4.42964 −0.204107
\(472\) 0 0
\(473\) 32.4610 1.49256
\(474\) 0 0
\(475\) 7.31080 0.335443
\(476\) 0 0
\(477\) 25.5754 1.17102
\(478\) 0 0
\(479\) 15.7996 0.721902 0.360951 0.932585i \(-0.382452\pi\)
0.360951 + 0.932585i \(0.382452\pi\)
\(480\) 0 0
\(481\) 21.1755 0.965522
\(482\) 0 0
\(483\) −1.28712 −0.0585661
\(484\) 0 0
\(485\) −1.97736 −0.0897874
\(486\) 0 0
\(487\) 22.8369 1.03484 0.517419 0.855732i \(-0.326893\pi\)
0.517419 + 0.855732i \(0.326893\pi\)
\(488\) 0 0
\(489\) 2.91709 0.131916
\(490\) 0 0
\(491\) 4.20368 0.189709 0.0948547 0.995491i \(-0.469761\pi\)
0.0948547 + 0.995491i \(0.469761\pi\)
\(492\) 0 0
\(493\) 0.378584 0.0170506
\(494\) 0 0
\(495\) 1.55265 0.0697866
\(496\) 0 0
\(497\) −14.3317 −0.642865
\(498\) 0 0
\(499\) −8.86425 −0.396818 −0.198409 0.980119i \(-0.563577\pi\)
−0.198409 + 0.980119i \(0.563577\pi\)
\(500\) 0 0
\(501\) 1.31723 0.0588493
\(502\) 0 0
\(503\) 20.6333 0.919994 0.459997 0.887920i \(-0.347850\pi\)
0.459997 + 0.887920i \(0.347850\pi\)
\(504\) 0 0
\(505\) 2.52061 0.112166
\(506\) 0 0
\(507\) −1.82531 −0.0810647
\(508\) 0 0
\(509\) 1.96697 0.0871844 0.0435922 0.999049i \(-0.486120\pi\)
0.0435922 + 0.999049i \(0.486120\pi\)
\(510\) 0 0
\(511\) −4.91327 −0.217350
\(512\) 0 0
\(513\) 2.30982 0.101981
\(514\) 0 0
\(515\) −1.76861 −0.0779341
\(516\) 0 0
\(517\) −26.4940 −1.16521
\(518\) 0 0
\(519\) −4.62496 −0.203013
\(520\) 0 0
\(521\) −40.9435 −1.79377 −0.896884 0.442266i \(-0.854175\pi\)
−0.896884 + 0.442266i \(0.854175\pi\)
\(522\) 0 0
\(523\) −37.4879 −1.63923 −0.819615 0.572914i \(-0.805813\pi\)
−0.819615 + 0.572914i \(0.805813\pi\)
\(524\) 0 0
\(525\) −1.31778 −0.0575126
\(526\) 0 0
\(527\) −8.78727 −0.382780
\(528\) 0 0
\(529\) 0.595151 0.0258761
\(530\) 0 0
\(531\) 8.37182 0.363306
\(532\) 0 0
\(533\) 14.7665 0.639609
\(534\) 0 0
\(535\) −1.51950 −0.0656939
\(536\) 0 0
\(537\) 0.631525 0.0272523
\(538\) 0 0
\(539\) −3.23607 −0.139387
\(540\) 0 0
\(541\) −8.60580 −0.369992 −0.184996 0.982739i \(-0.559227\pi\)
−0.184996 + 0.982739i \(0.559227\pi\)
\(542\) 0 0
\(543\) −6.54766 −0.280987
\(544\) 0 0
\(545\) −0.475632 −0.0203738
\(546\) 0 0
\(547\) 15.6887 0.670800 0.335400 0.942076i \(-0.391128\pi\)
0.335400 + 0.942076i \(0.391128\pi\)
\(548\) 0 0
\(549\) 38.8697 1.65892
\(550\) 0 0
\(551\) −0.556536 −0.0237092
\(552\) 0 0
\(553\) 16.1871 0.688346
\(554\) 0 0
\(555\) 0.371699 0.0157777
\(556\) 0 0
\(557\) 12.0673 0.511307 0.255654 0.966768i \(-0.417709\pi\)
0.255654 + 0.966768i \(0.417709\pi\)
\(558\) 0 0
\(559\) −24.7980 −1.04884
\(560\) 0 0
\(561\) −0.857484 −0.0362030
\(562\) 0 0
\(563\) 5.97700 0.251901 0.125950 0.992037i \(-0.459802\pi\)
0.125950 + 0.992037i \(0.459802\pi\)
\(564\) 0 0
\(565\) 1.42059 0.0597646
\(566\) 0 0
\(567\) 8.37301 0.351634
\(568\) 0 0
\(569\) 26.1373 1.09573 0.547866 0.836566i \(-0.315440\pi\)
0.547866 + 0.836566i \(0.315440\pi\)
\(570\) 0 0
\(571\) −7.33818 −0.307093 −0.153547 0.988141i \(-0.549070\pi\)
−0.153547 + 0.988141i \(0.549070\pi\)
\(572\) 0 0
\(573\) 5.38415 0.224926
\(574\) 0 0
\(575\) 24.1571 1.00742
\(576\) 0 0
\(577\) 0.169395 0.00705202 0.00352601 0.999994i \(-0.498878\pi\)
0.00352601 + 0.999994i \(0.498878\pi\)
\(578\) 0 0
\(579\) 4.36068 0.181224
\(580\) 0 0
\(581\) 5.65715 0.234698
\(582\) 0 0
\(583\) 28.2491 1.16996
\(584\) 0 0
\(585\) −1.18612 −0.0490401
\(586\) 0 0
\(587\) −10.4184 −0.430012 −0.215006 0.976613i \(-0.568977\pi\)
−0.215006 + 0.976613i \(0.568977\pi\)
\(588\) 0 0
\(589\) 12.9177 0.532264
\(590\) 0 0
\(591\) −0.895801 −0.0368483
\(592\) 0 0
\(593\) 21.7811 0.894445 0.447222 0.894423i \(-0.352413\pi\)
0.447222 + 0.894423i \(0.352413\pi\)
\(594\) 0 0
\(595\) 0.163765 0.00671371
\(596\) 0 0
\(597\) −1.36429 −0.0558369
\(598\) 0 0
\(599\) −18.9847 −0.775695 −0.387848 0.921723i \(-0.626781\pi\)
−0.387848 + 0.921723i \(0.626781\pi\)
\(600\) 0 0
\(601\) 17.1797 0.700776 0.350388 0.936605i \(-0.386050\pi\)
0.350388 + 0.936605i \(0.386050\pi\)
\(602\) 0 0
\(603\) 43.7472 1.78152
\(604\) 0 0
\(605\) −0.0864456 −0.00351451
\(606\) 0 0
\(607\) −27.7740 −1.12731 −0.563657 0.826009i \(-0.690606\pi\)
−0.563657 + 0.826009i \(0.690606\pi\)
\(608\) 0 0
\(609\) 0.100316 0.00406501
\(610\) 0 0
\(611\) 20.2396 0.818808
\(612\) 0 0
\(613\) −37.0901 −1.49806 −0.749028 0.662538i \(-0.769479\pi\)
−0.749028 + 0.662538i \(0.769479\pi\)
\(614\) 0 0
\(615\) 0.259200 0.0104519
\(616\) 0 0
\(617\) 39.8484 1.60424 0.802119 0.597165i \(-0.203706\pi\)
0.802119 + 0.597165i \(0.203706\pi\)
\(618\) 0 0
\(619\) −21.4852 −0.863562 −0.431781 0.901978i \(-0.642115\pi\)
−0.431781 + 0.901978i \(0.642115\pi\)
\(620\) 0 0
\(621\) 7.63236 0.306276
\(622\) 0 0
\(623\) 13.5167 0.541536
\(624\) 0 0
\(625\) 24.5984 0.983937
\(626\) 0 0
\(627\) 1.26054 0.0503411
\(628\) 0 0
\(629\) 8.56569 0.341536
\(630\) 0 0
\(631\) −37.2884 −1.48443 −0.742213 0.670164i \(-0.766224\pi\)
−0.742213 + 0.670164i \(0.766224\pi\)
\(632\) 0 0
\(633\) 3.08052 0.122440
\(634\) 0 0
\(635\) −1.01526 −0.0402894
\(636\) 0 0
\(637\) 2.47214 0.0979496
\(638\) 0 0
\(639\) 41.9889 1.66105
\(640\) 0 0
\(641\) −40.5188 −1.60040 −0.800198 0.599735i \(-0.795273\pi\)
−0.800198 + 0.599735i \(0.795273\pi\)
\(642\) 0 0
\(643\) 28.2963 1.11590 0.557949 0.829875i \(-0.311588\pi\)
0.557949 + 0.829875i \(0.311588\pi\)
\(644\) 0 0
\(645\) −0.435285 −0.0171393
\(646\) 0 0
\(647\) −37.6767 −1.48122 −0.740611 0.671934i \(-0.765464\pi\)
−0.740611 + 0.671934i \(0.765464\pi\)
\(648\) 0 0
\(649\) 9.24701 0.362977
\(650\) 0 0
\(651\) −2.32843 −0.0912582
\(652\) 0 0
\(653\) 23.0915 0.903639 0.451819 0.892109i \(-0.350775\pi\)
0.451819 + 0.892109i \(0.350775\pi\)
\(654\) 0 0
\(655\) 1.34912 0.0527145
\(656\) 0 0
\(657\) 14.3948 0.561596
\(658\) 0 0
\(659\) −9.40401 −0.366328 −0.183164 0.983082i \(-0.558634\pi\)
−0.183164 + 0.983082i \(0.558634\pi\)
\(660\) 0 0
\(661\) −48.9621 −1.90441 −0.952203 0.305467i \(-0.901187\pi\)
−0.952203 + 0.305467i \(0.901187\pi\)
\(662\) 0 0
\(663\) 0.655059 0.0254404
\(664\) 0 0
\(665\) −0.240742 −0.00933557
\(666\) 0 0
\(667\) −1.83897 −0.0712050
\(668\) 0 0
\(669\) 3.09294 0.119580
\(670\) 0 0
\(671\) 42.9331 1.65742
\(672\) 0 0
\(673\) −28.8959 −1.11386 −0.556928 0.830561i \(-0.688020\pi\)
−0.556928 + 0.830561i \(0.688020\pi\)
\(674\) 0 0
\(675\) 7.81415 0.300767
\(676\) 0 0
\(677\) −39.7744 −1.52865 −0.764327 0.644829i \(-0.776929\pi\)
−0.764327 + 0.644829i \(0.776929\pi\)
\(678\) 0 0
\(679\) −12.0744 −0.463373
\(680\) 0 0
\(681\) −3.94965 −0.151351
\(682\) 0 0
\(683\) −16.6165 −0.635814 −0.317907 0.948122i \(-0.602980\pi\)
−0.317907 + 0.948122i \(0.602980\pi\)
\(684\) 0 0
\(685\) 0.331922 0.0126821
\(686\) 0 0
\(687\) −4.98148 −0.190055
\(688\) 0 0
\(689\) −21.5804 −0.822148
\(690\) 0 0
\(691\) −46.7768 −1.77947 −0.889737 0.456473i \(-0.849113\pi\)
−0.889737 + 0.456473i \(0.849113\pi\)
\(692\) 0 0
\(693\) 9.48099 0.360153
\(694\) 0 0
\(695\) 2.73453 0.103727
\(696\) 0 0
\(697\) 5.97318 0.226250
\(698\) 0 0
\(699\) 4.45153 0.168372
\(700\) 0 0
\(701\) −6.59195 −0.248975 −0.124487 0.992221i \(-0.539729\pi\)
−0.124487 + 0.992221i \(0.539729\pi\)
\(702\) 0 0
\(703\) −12.5920 −0.474915
\(704\) 0 0
\(705\) 0.355271 0.0133803
\(706\) 0 0
\(707\) 15.3916 0.578861
\(708\) 0 0
\(709\) −43.7396 −1.64267 −0.821337 0.570443i \(-0.806771\pi\)
−0.821337 + 0.570443i \(0.806771\pi\)
\(710\) 0 0
\(711\) −47.4248 −1.77857
\(712\) 0 0
\(713\) 42.6840 1.59853
\(714\) 0 0
\(715\) −1.31012 −0.0489957
\(716\) 0 0
\(717\) −4.37295 −0.163311
\(718\) 0 0
\(719\) 9.08673 0.338878 0.169439 0.985541i \(-0.445804\pi\)
0.169439 + 0.985541i \(0.445804\pi\)
\(720\) 0 0
\(721\) −10.7997 −0.402201
\(722\) 0 0
\(723\) 5.08141 0.188980
\(724\) 0 0
\(725\) −1.88277 −0.0699242
\(726\) 0 0
\(727\) −18.7843 −0.696673 −0.348336 0.937370i \(-0.613253\pi\)
−0.348336 + 0.937370i \(0.613253\pi\)
\(728\) 0 0
\(729\) −23.2821 −0.862300
\(730\) 0 0
\(731\) −10.0310 −0.371010
\(732\) 0 0
\(733\) −25.1602 −0.929314 −0.464657 0.885491i \(-0.653822\pi\)
−0.464657 + 0.885491i \(0.653822\pi\)
\(734\) 0 0
\(735\) 0.0433939 0.00160061
\(736\) 0 0
\(737\) 48.3206 1.77991
\(738\) 0 0
\(739\) 30.5073 1.12223 0.561115 0.827738i \(-0.310373\pi\)
0.561115 + 0.827738i \(0.310373\pi\)
\(740\) 0 0
\(741\) −0.962967 −0.0353755
\(742\) 0 0
\(743\) −24.0773 −0.883311 −0.441656 0.897185i \(-0.645609\pi\)
−0.441656 + 0.897185i \(0.645609\pi\)
\(744\) 0 0
\(745\) −2.14538 −0.0786007
\(746\) 0 0
\(747\) −16.5742 −0.606420
\(748\) 0 0
\(749\) −9.27857 −0.339031
\(750\) 0 0
\(751\) 10.6840 0.389866 0.194933 0.980817i \(-0.437551\pi\)
0.194933 + 0.980817i \(0.437551\pi\)
\(752\) 0 0
\(753\) −5.78464 −0.210804
\(754\) 0 0
\(755\) −2.04757 −0.0745189
\(756\) 0 0
\(757\) −39.0013 −1.41753 −0.708764 0.705446i \(-0.750747\pi\)
−0.708764 + 0.705446i \(0.750747\pi\)
\(758\) 0 0
\(759\) 4.16521 0.151188
\(760\) 0 0
\(761\) −52.6056 −1.90695 −0.953476 0.301470i \(-0.902523\pi\)
−0.953476 + 0.301470i \(0.902523\pi\)
\(762\) 0 0
\(763\) −2.90436 −0.105145
\(764\) 0 0
\(765\) −0.479796 −0.0173471
\(766\) 0 0
\(767\) −7.06409 −0.255069
\(768\) 0 0
\(769\) −21.8596 −0.788276 −0.394138 0.919051i \(-0.628957\pi\)
−0.394138 + 0.919051i \(0.628957\pi\)
\(770\) 0 0
\(771\) −5.70144 −0.205332
\(772\) 0 0
\(773\) 16.5478 0.595182 0.297591 0.954693i \(-0.403817\pi\)
0.297591 + 0.954693i \(0.403817\pi\)
\(774\) 0 0
\(775\) 43.7007 1.56978
\(776\) 0 0
\(777\) 2.26971 0.0814254
\(778\) 0 0
\(779\) −8.78085 −0.314607
\(780\) 0 0
\(781\) 46.3784 1.65955
\(782\) 0 0
\(783\) −0.594853 −0.0212583
\(784\) 0 0
\(785\) −2.73767 −0.0977115
\(786\) 0 0
\(787\) 31.1476 1.11029 0.555146 0.831753i \(-0.312662\pi\)
0.555146 + 0.831753i \(0.312662\pi\)
\(788\) 0 0
\(789\) 1.51434 0.0539118
\(790\) 0 0
\(791\) 8.67456 0.308432
\(792\) 0 0
\(793\) −32.7980 −1.16469
\(794\) 0 0
\(795\) −0.378805 −0.0134348
\(796\) 0 0
\(797\) 7.15198 0.253336 0.126668 0.991945i \(-0.459572\pi\)
0.126668 + 0.991945i \(0.459572\pi\)
\(798\) 0 0
\(799\) 8.18710 0.289639
\(800\) 0 0
\(801\) −39.6011 −1.39924
\(802\) 0 0
\(803\) 15.8997 0.561088
\(804\) 0 0
\(805\) −0.795485 −0.0280372
\(806\) 0 0
\(807\) −6.31639 −0.222347
\(808\) 0 0
\(809\) −34.4048 −1.20961 −0.604805 0.796374i \(-0.706749\pi\)
−0.604805 + 0.796374i \(0.706749\pi\)
\(810\) 0 0
\(811\) 29.0670 1.02068 0.510341 0.859972i \(-0.329519\pi\)
0.510341 + 0.859972i \(0.329519\pi\)
\(812\) 0 0
\(813\) 7.41752 0.260144
\(814\) 0 0
\(815\) 1.80286 0.0631516
\(816\) 0 0
\(817\) 14.7460 0.515898
\(818\) 0 0
\(819\) −7.24283 −0.253085
\(820\) 0 0
\(821\) 19.7064 0.687759 0.343879 0.939014i \(-0.388259\pi\)
0.343879 + 0.939014i \(0.388259\pi\)
\(822\) 0 0
\(823\) −3.35262 −0.116865 −0.0584324 0.998291i \(-0.518610\pi\)
−0.0584324 + 0.998291i \(0.518610\pi\)
\(824\) 0 0
\(825\) 4.26442 0.148468
\(826\) 0 0
\(827\) 32.8201 1.14127 0.570633 0.821205i \(-0.306698\pi\)
0.570633 + 0.821205i \(0.306698\pi\)
\(828\) 0 0
\(829\) −24.4048 −0.847615 −0.423808 0.905752i \(-0.639307\pi\)
−0.423808 + 0.905752i \(0.639307\pi\)
\(830\) 0 0
\(831\) 0.538315 0.0186739
\(832\) 0 0
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) 0.814090 0.0281727
\(836\) 0 0
\(837\) 13.8071 0.477242
\(838\) 0 0
\(839\) 22.7307 0.784751 0.392376 0.919805i \(-0.371653\pi\)
0.392376 + 0.919805i \(0.371653\pi\)
\(840\) 0 0
\(841\) −28.8567 −0.995058
\(842\) 0 0
\(843\) 8.06708 0.277845
\(844\) 0 0
\(845\) −1.12810 −0.0388079
\(846\) 0 0
\(847\) −0.527864 −0.0181376
\(848\) 0 0
\(849\) 1.90646 0.0654297
\(850\) 0 0
\(851\) −41.6077 −1.42629
\(852\) 0 0
\(853\) 14.8218 0.507487 0.253744 0.967272i \(-0.418338\pi\)
0.253744 + 0.967272i \(0.418338\pi\)
\(854\) 0 0
\(855\) 0.705322 0.0241215
\(856\) 0 0
\(857\) 0.355245 0.0121349 0.00606747 0.999982i \(-0.498069\pi\)
0.00606747 + 0.999982i \(0.498069\pi\)
\(858\) 0 0
\(859\) −37.2665 −1.27152 −0.635759 0.771888i \(-0.719313\pi\)
−0.635759 + 0.771888i \(0.719313\pi\)
\(860\) 0 0
\(861\) 1.58276 0.0539402
\(862\) 0 0
\(863\) 0.0548924 0.00186856 0.000934279 1.00000i \(-0.499703\pi\)
0.000934279 1.00000i \(0.499703\pi\)
\(864\) 0 0
\(865\) −2.85838 −0.0971878
\(866\) 0 0
\(867\) 0.264977 0.00899910
\(868\) 0 0
\(869\) −52.3826 −1.77696
\(870\) 0 0
\(871\) −36.9136 −1.25077
\(872\) 0 0
\(873\) 35.3754 1.19728
\(874\) 0 0
\(875\) −1.63326 −0.0552142
\(876\) 0 0
\(877\) −11.7330 −0.396195 −0.198098 0.980182i \(-0.563476\pi\)
−0.198098 + 0.980182i \(0.563476\pi\)
\(878\) 0 0
\(879\) 6.00418 0.202516
\(880\) 0 0
\(881\) 42.6705 1.43761 0.718803 0.695214i \(-0.244690\pi\)
0.718803 + 0.695214i \(0.244690\pi\)
\(882\) 0 0
\(883\) −0.555998 −0.0187108 −0.00935541 0.999956i \(-0.502978\pi\)
−0.00935541 + 0.999956i \(0.502978\pi\)
\(884\) 0 0
\(885\) −0.123998 −0.00416813
\(886\) 0 0
\(887\) 16.2420 0.545353 0.272676 0.962106i \(-0.412091\pi\)
0.272676 + 0.962106i \(0.412091\pi\)
\(888\) 0 0
\(889\) −6.19950 −0.207925
\(890\) 0 0
\(891\) −27.0956 −0.907738
\(892\) 0 0
\(893\) −12.0354 −0.402750
\(894\) 0 0
\(895\) 0.390304 0.0130464
\(896\) 0 0
\(897\) −3.18194 −0.106242
\(898\) 0 0
\(899\) −3.32672 −0.110952
\(900\) 0 0
\(901\) −8.72945 −0.290820
\(902\) 0 0
\(903\) −2.65799 −0.0884522
\(904\) 0 0
\(905\) −4.04668 −0.134516
\(906\) 0 0
\(907\) 13.6050 0.451746 0.225873 0.974157i \(-0.427477\pi\)
0.225873 + 0.974157i \(0.427477\pi\)
\(908\) 0 0
\(909\) −45.0942 −1.49568
\(910\) 0 0
\(911\) 13.4163 0.444501 0.222251 0.974990i \(-0.428660\pi\)
0.222251 + 0.974990i \(0.428660\pi\)
\(912\) 0 0
\(913\) −18.3069 −0.605871
\(914\) 0 0
\(915\) −0.575710 −0.0190324
\(916\) 0 0
\(917\) 8.23816 0.272048
\(918\) 0 0
\(919\) 17.9373 0.591695 0.295848 0.955235i \(-0.404398\pi\)
0.295848 + 0.955235i \(0.404398\pi\)
\(920\) 0 0
\(921\) −2.23835 −0.0737563
\(922\) 0 0
\(923\) −35.4299 −1.16619
\(924\) 0 0
\(925\) −42.5987 −1.40064
\(926\) 0 0
\(927\) 31.6407 1.03922
\(928\) 0 0
\(929\) −19.5156 −0.640286 −0.320143 0.947369i \(-0.603731\pi\)
−0.320143 + 0.947369i \(0.603731\pi\)
\(930\) 0 0
\(931\) −1.47005 −0.0481788
\(932\) 0 0
\(933\) 0.0539703 0.00176691
\(934\) 0 0
\(935\) −0.529954 −0.0173314
\(936\) 0 0
\(937\) −48.4958 −1.58429 −0.792145 0.610333i \(-0.791035\pi\)
−0.792145 + 0.610333i \(0.791035\pi\)
\(938\) 0 0
\(939\) −0.691494 −0.0225660
\(940\) 0 0
\(941\) 12.3406 0.402293 0.201146 0.979561i \(-0.435533\pi\)
0.201146 + 0.979561i \(0.435533\pi\)
\(942\) 0 0
\(943\) −29.0146 −0.944846
\(944\) 0 0
\(945\) −0.257317 −0.00837052
\(946\) 0 0
\(947\) −26.3633 −0.856691 −0.428345 0.903615i \(-0.640903\pi\)
−0.428345 + 0.903615i \(0.640903\pi\)
\(948\) 0 0
\(949\) −12.1463 −0.394285
\(950\) 0 0
\(951\) 7.53204 0.244243
\(952\) 0 0
\(953\) −3.60644 −0.116824 −0.0584120 0.998293i \(-0.518604\pi\)
−0.0584120 + 0.998293i \(0.518604\pi\)
\(954\) 0 0
\(955\) 3.32759 0.107678
\(956\) 0 0
\(957\) −0.324630 −0.0104938
\(958\) 0 0
\(959\) 2.02682 0.0654494
\(960\) 0 0
\(961\) 46.2161 1.49084
\(962\) 0 0
\(963\) 27.1842 0.875999
\(964\) 0 0
\(965\) 2.69505 0.0867567
\(966\) 0 0
\(967\) −28.2139 −0.907299 −0.453649 0.891180i \(-0.649878\pi\)
−0.453649 + 0.891180i \(0.649878\pi\)
\(968\) 0 0
\(969\) −0.389529 −0.0125135
\(970\) 0 0
\(971\) 20.0355 0.642971 0.321486 0.946914i \(-0.395818\pi\)
0.321486 + 0.946914i \(0.395818\pi\)
\(972\) 0 0
\(973\) 16.6979 0.535310
\(974\) 0 0
\(975\) −3.25773 −0.104331
\(976\) 0 0
\(977\) −44.5259 −1.42451 −0.712255 0.701921i \(-0.752326\pi\)
−0.712255 + 0.701921i \(0.752326\pi\)
\(978\) 0 0
\(979\) −43.7410 −1.39797
\(980\) 0 0
\(981\) 8.50915 0.271676
\(982\) 0 0
\(983\) 2.54931 0.0813103 0.0406552 0.999173i \(-0.487055\pi\)
0.0406552 + 0.999173i \(0.487055\pi\)
\(984\) 0 0
\(985\) −0.553636 −0.0176403
\(986\) 0 0
\(987\) 2.16940 0.0690526
\(988\) 0 0
\(989\) 48.7254 1.54938
\(990\) 0 0
\(991\) −24.8794 −0.790319 −0.395160 0.918612i \(-0.629311\pi\)
−0.395160 + 0.918612i \(0.629311\pi\)
\(992\) 0 0
\(993\) 7.63675 0.242345
\(994\) 0 0
\(995\) −0.843181 −0.0267306
\(996\) 0 0
\(997\) 3.68557 0.116723 0.0583615 0.998296i \(-0.481412\pi\)
0.0583615 + 0.998296i \(0.481412\pi\)
\(998\) 0 0
\(999\) −13.4589 −0.425821
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 1904.2.a.q.1.3 4
4.3 odd 2 952.2.a.g.1.2 4
8.3 odd 2 7616.2.a.bj.1.3 4
8.5 even 2 7616.2.a.bp.1.2 4
12.11 even 2 8568.2.a.bj.1.2 4
28.27 even 2 6664.2.a.o.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.g.1.2 4 4.3 odd 2
1904.2.a.q.1.3 4 1.1 even 1 trivial
6664.2.a.o.1.3 4 28.27 even 2
7616.2.a.bj.1.3 4 8.3 odd 2
7616.2.a.bp.1.2 4 8.5 even 2
8568.2.a.bj.1.2 4 12.11 even 2