Properties

Label 8016.2.a.s.1.4
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $5$
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] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, 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("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.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: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.284897.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 5x^{2} + 10x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.435916\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.07208 q^{5} -1.37406 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.07208 q^{5} -1.37406 q^{7} +1.00000 q^{9} +5.27463 q^{11} -6.01023 q^{13} -2.07208 q^{15} +0.202551 q^{17} +0.336870 q^{19} +1.37406 q^{21} +2.04334 q^{23} -0.706472 q^{25} -1.00000 q^{27} -6.05765 q^{29} +5.31798 q^{31} -5.27463 q^{33} -2.84717 q^{35} +1.31836 q^{37} +6.01023 q^{39} -11.1438 q^{41} +0.965489 q^{43} +2.07208 q^{45} -4.19640 q^{47} -5.11196 q^{49} -0.202551 q^{51} +8.35963 q^{53} +10.9295 q^{55} -0.336870 q^{57} -0.730755 q^{59} -10.2419 q^{61} -1.37406 q^{63} -12.4537 q^{65} -0.951843 q^{67} -2.04334 q^{69} -12.3009 q^{71} +12.2704 q^{73} +0.706472 q^{75} -7.24767 q^{77} +15.3159 q^{79} +1.00000 q^{81} -0.663677 q^{83} +0.419704 q^{85} +6.05765 q^{87} -10.6676 q^{89} +8.25842 q^{91} -5.31798 q^{93} +0.698022 q^{95} +2.07616 q^{97} +5.27463 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9} + 3 q^{11} - 14 q^{13} - q^{15} - 13 q^{17} + 2 q^{19} - 4 q^{21} + 5 q^{23} + 2 q^{25} - 5 q^{27} + 13 q^{29} - 2 q^{31} - 3 q^{33} + 12 q^{35} - 5 q^{37} + 14 q^{39} - 20 q^{41} + 20 q^{43} + q^{45} - q^{47} - 9 q^{49} + 13 q^{51} - 3 q^{53} + 3 q^{55} - 2 q^{57} + q^{59} - 34 q^{61} + 4 q^{63} - 22 q^{65} + 16 q^{67} - 5 q^{69} + 5 q^{71} - 12 q^{73} - 2 q^{75} - 8 q^{77} + 20 q^{79} + 5 q^{81} + 15 q^{83} - 27 q^{85} - 13 q^{87} - 48 q^{89} - 7 q^{91} + 2 q^{93} + 5 q^{95} - 21 q^{97} + 3 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.07208 0.926664 0.463332 0.886185i \(-0.346654\pi\)
0.463332 + 0.886185i \(0.346654\pi\)
\(6\) 0 0
\(7\) −1.37406 −0.519346 −0.259673 0.965697i \(-0.583615\pi\)
−0.259673 + 0.965697i \(0.583615\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.27463 1.59036 0.795181 0.606372i \(-0.207376\pi\)
0.795181 + 0.606372i \(0.207376\pi\)
\(12\) 0 0
\(13\) −6.01023 −1.66694 −0.833469 0.552567i \(-0.813648\pi\)
−0.833469 + 0.552567i \(0.813648\pi\)
\(14\) 0 0
\(15\) −2.07208 −0.535010
\(16\) 0 0
\(17\) 0.202551 0.0491260 0.0245630 0.999698i \(-0.492181\pi\)
0.0245630 + 0.999698i \(0.492181\pi\)
\(18\) 0 0
\(19\) 0.336870 0.0772832 0.0386416 0.999253i \(-0.487697\pi\)
0.0386416 + 0.999253i \(0.487697\pi\)
\(20\) 0 0
\(21\) 1.37406 0.299845
\(22\) 0 0
\(23\) 2.04334 0.426066 0.213033 0.977045i \(-0.431666\pi\)
0.213033 + 0.977045i \(0.431666\pi\)
\(24\) 0 0
\(25\) −0.706472 −0.141294
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.05765 −1.12488 −0.562439 0.826839i \(-0.690137\pi\)
−0.562439 + 0.826839i \(0.690137\pi\)
\(30\) 0 0
\(31\) 5.31798 0.955137 0.477568 0.878595i \(-0.341518\pi\)
0.477568 + 0.878595i \(0.341518\pi\)
\(32\) 0 0
\(33\) −5.27463 −0.918196
\(34\) 0 0
\(35\) −2.84717 −0.481259
\(36\) 0 0
\(37\) 1.31836 0.216736 0.108368 0.994111i \(-0.465437\pi\)
0.108368 + 0.994111i \(0.465437\pi\)
\(38\) 0 0
\(39\) 6.01023 0.962407
\(40\) 0 0
\(41\) −11.1438 −1.74037 −0.870183 0.492728i \(-0.836000\pi\)
−0.870183 + 0.492728i \(0.836000\pi\)
\(42\) 0 0
\(43\) 0.965489 0.147236 0.0736179 0.997287i \(-0.476545\pi\)
0.0736179 + 0.997287i \(0.476545\pi\)
\(44\) 0 0
\(45\) 2.07208 0.308888
\(46\) 0 0
\(47\) −4.19640 −0.612108 −0.306054 0.952014i \(-0.599009\pi\)
−0.306054 + 0.952014i \(0.599009\pi\)
\(48\) 0 0
\(49\) −5.11196 −0.730279
\(50\) 0 0
\(51\) −0.202551 −0.0283629
\(52\) 0 0
\(53\) 8.35963 1.14828 0.574142 0.818756i \(-0.305336\pi\)
0.574142 + 0.818756i \(0.305336\pi\)
\(54\) 0 0
\(55\) 10.9295 1.47373
\(56\) 0 0
\(57\) −0.336870 −0.0446195
\(58\) 0 0
\(59\) −0.730755 −0.0951362 −0.0475681 0.998868i \(-0.515147\pi\)
−0.0475681 + 0.998868i \(0.515147\pi\)
\(60\) 0 0
\(61\) −10.2419 −1.31134 −0.655671 0.755047i \(-0.727614\pi\)
−0.655671 + 0.755047i \(0.727614\pi\)
\(62\) 0 0
\(63\) −1.37406 −0.173115
\(64\) 0 0
\(65\) −12.4537 −1.54469
\(66\) 0 0
\(67\) −0.951843 −0.116286 −0.0581430 0.998308i \(-0.518518\pi\)
−0.0581430 + 0.998308i \(0.518518\pi\)
\(68\) 0 0
\(69\) −2.04334 −0.245989
\(70\) 0 0
\(71\) −12.3009 −1.45984 −0.729922 0.683531i \(-0.760444\pi\)
−0.729922 + 0.683531i \(0.760444\pi\)
\(72\) 0 0
\(73\) 12.2704 1.43614 0.718072 0.695968i \(-0.245025\pi\)
0.718072 + 0.695968i \(0.245025\pi\)
\(74\) 0 0
\(75\) 0.706472 0.0815764
\(76\) 0 0
\(77\) −7.24767 −0.825949
\(78\) 0 0
\(79\) 15.3159 1.72317 0.861587 0.507610i \(-0.169471\pi\)
0.861587 + 0.507610i \(0.169471\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.663677 −0.0728480 −0.0364240 0.999336i \(-0.511597\pi\)
−0.0364240 + 0.999336i \(0.511597\pi\)
\(84\) 0 0
\(85\) 0.419704 0.0455232
\(86\) 0 0
\(87\) 6.05765 0.649448
\(88\) 0 0
\(89\) −10.6676 −1.13076 −0.565381 0.824830i \(-0.691271\pi\)
−0.565381 + 0.824830i \(0.691271\pi\)
\(90\) 0 0
\(91\) 8.25842 0.865718
\(92\) 0 0
\(93\) −5.31798 −0.551448
\(94\) 0 0
\(95\) 0.698022 0.0716155
\(96\) 0 0
\(97\) 2.07616 0.210802 0.105401 0.994430i \(-0.466387\pi\)
0.105401 + 0.994430i \(0.466387\pi\)
\(98\) 0 0
\(99\) 5.27463 0.530121
\(100\) 0 0
\(101\) −17.3465 −1.72604 −0.863021 0.505168i \(-0.831430\pi\)
−0.863021 + 0.505168i \(0.831430\pi\)
\(102\) 0 0
\(103\) −16.8119 −1.65653 −0.828264 0.560338i \(-0.810671\pi\)
−0.828264 + 0.560338i \(0.810671\pi\)
\(104\) 0 0
\(105\) 2.84717 0.277855
\(106\) 0 0
\(107\) 3.15743 0.305240 0.152620 0.988285i \(-0.451229\pi\)
0.152620 + 0.988285i \(0.451229\pi\)
\(108\) 0 0
\(109\) −5.11105 −0.489550 −0.244775 0.969580i \(-0.578714\pi\)
−0.244775 + 0.969580i \(0.578714\pi\)
\(110\) 0 0
\(111\) −1.31836 −0.125133
\(112\) 0 0
\(113\) 8.36231 0.786660 0.393330 0.919397i \(-0.371323\pi\)
0.393330 + 0.919397i \(0.371323\pi\)
\(114\) 0 0
\(115\) 4.23397 0.394820
\(116\) 0 0
\(117\) −6.01023 −0.555646
\(118\) 0 0
\(119\) −0.278318 −0.0255134
\(120\) 0 0
\(121\) 16.8218 1.52925
\(122\) 0 0
\(123\) 11.1438 1.00480
\(124\) 0 0
\(125\) −11.8243 −1.05760
\(126\) 0 0
\(127\) −7.98111 −0.708209 −0.354104 0.935206i \(-0.615214\pi\)
−0.354104 + 0.935206i \(0.615214\pi\)
\(128\) 0 0
\(129\) −0.965489 −0.0850066
\(130\) 0 0
\(131\) −0.126390 −0.0110428 −0.00552138 0.999985i \(-0.501758\pi\)
−0.00552138 + 0.999985i \(0.501758\pi\)
\(132\) 0 0
\(133\) −0.462879 −0.0401367
\(134\) 0 0
\(135\) −2.07208 −0.178337
\(136\) 0 0
\(137\) 8.31975 0.710805 0.355402 0.934713i \(-0.384344\pi\)
0.355402 + 0.934713i \(0.384344\pi\)
\(138\) 0 0
\(139\) −2.22951 −0.189105 −0.0945525 0.995520i \(-0.530142\pi\)
−0.0945525 + 0.995520i \(0.530142\pi\)
\(140\) 0 0
\(141\) 4.19640 0.353401
\(142\) 0 0
\(143\) −31.7018 −2.65103
\(144\) 0 0
\(145\) −12.5519 −1.04238
\(146\) 0 0
\(147\) 5.11196 0.421627
\(148\) 0 0
\(149\) 9.87451 0.808952 0.404476 0.914549i \(-0.367454\pi\)
0.404476 + 0.914549i \(0.367454\pi\)
\(150\) 0 0
\(151\) 9.00556 0.732862 0.366431 0.930445i \(-0.380580\pi\)
0.366431 + 0.930445i \(0.380580\pi\)
\(152\) 0 0
\(153\) 0.202551 0.0163753
\(154\) 0 0
\(155\) 11.0193 0.885091
\(156\) 0 0
\(157\) −6.54956 −0.522712 −0.261356 0.965242i \(-0.584170\pi\)
−0.261356 + 0.965242i \(0.584170\pi\)
\(158\) 0 0
\(159\) −8.35963 −0.662962
\(160\) 0 0
\(161\) −2.80768 −0.221276
\(162\) 0 0
\(163\) 2.22508 0.174282 0.0871409 0.996196i \(-0.472227\pi\)
0.0871409 + 0.996196i \(0.472227\pi\)
\(164\) 0 0
\(165\) −10.9295 −0.850859
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 23.1228 1.77868
\(170\) 0 0
\(171\) 0.336870 0.0257611
\(172\) 0 0
\(173\) 5.46198 0.415266 0.207633 0.978207i \(-0.433424\pi\)
0.207633 + 0.978207i \(0.433424\pi\)
\(174\) 0 0
\(175\) 0.970736 0.0733807
\(176\) 0 0
\(177\) 0.730755 0.0549269
\(178\) 0 0
\(179\) −25.3449 −1.89437 −0.947183 0.320693i \(-0.896084\pi\)
−0.947183 + 0.320693i \(0.896084\pi\)
\(180\) 0 0
\(181\) −4.47384 −0.332538 −0.166269 0.986080i \(-0.553172\pi\)
−0.166269 + 0.986080i \(0.553172\pi\)
\(182\) 0 0
\(183\) 10.2419 0.757103
\(184\) 0 0
\(185\) 2.73174 0.200842
\(186\) 0 0
\(187\) 1.06839 0.0781281
\(188\) 0 0
\(189\) 1.37406 0.0999483
\(190\) 0 0
\(191\) −23.2392 −1.68153 −0.840764 0.541402i \(-0.817894\pi\)
−0.840764 + 0.541402i \(0.817894\pi\)
\(192\) 0 0
\(193\) 7.47719 0.538220 0.269110 0.963110i \(-0.413271\pi\)
0.269110 + 0.963110i \(0.413271\pi\)
\(194\) 0 0
\(195\) 12.4537 0.891827
\(196\) 0 0
\(197\) 17.5011 1.24690 0.623451 0.781862i \(-0.285730\pi\)
0.623451 + 0.781862i \(0.285730\pi\)
\(198\) 0 0
\(199\) −22.2943 −1.58040 −0.790199 0.612850i \(-0.790023\pi\)
−0.790199 + 0.612850i \(0.790023\pi\)
\(200\) 0 0
\(201\) 0.951843 0.0671378
\(202\) 0 0
\(203\) 8.32358 0.584201
\(204\) 0 0
\(205\) −23.0908 −1.61273
\(206\) 0 0
\(207\) 2.04334 0.142022
\(208\) 0 0
\(209\) 1.77686 0.122908
\(210\) 0 0
\(211\) 12.0997 0.832975 0.416487 0.909142i \(-0.363261\pi\)
0.416487 + 0.909142i \(0.363261\pi\)
\(212\) 0 0
\(213\) 12.3009 0.842841
\(214\) 0 0
\(215\) 2.00057 0.136438
\(216\) 0 0
\(217\) −7.30723 −0.496047
\(218\) 0 0
\(219\) −12.2704 −0.829159
\(220\) 0 0
\(221\) −1.21738 −0.0818899
\(222\) 0 0
\(223\) −4.28101 −0.286678 −0.143339 0.989674i \(-0.545784\pi\)
−0.143339 + 0.989674i \(0.545784\pi\)
\(224\) 0 0
\(225\) −0.706472 −0.0470981
\(226\) 0 0
\(227\) −6.47975 −0.430076 −0.215038 0.976606i \(-0.568988\pi\)
−0.215038 + 0.976606i \(0.568988\pi\)
\(228\) 0 0
\(229\) 15.3238 1.01263 0.506314 0.862349i \(-0.331008\pi\)
0.506314 + 0.862349i \(0.331008\pi\)
\(230\) 0 0
\(231\) 7.24767 0.476862
\(232\) 0 0
\(233\) −3.54194 −0.232040 −0.116020 0.993247i \(-0.537014\pi\)
−0.116020 + 0.993247i \(0.537014\pi\)
\(234\) 0 0
\(235\) −8.69529 −0.567218
\(236\) 0 0
\(237\) −15.3159 −0.994875
\(238\) 0 0
\(239\) 13.3809 0.865537 0.432768 0.901505i \(-0.357537\pi\)
0.432768 + 0.901505i \(0.357537\pi\)
\(240\) 0 0
\(241\) −28.4045 −1.82969 −0.914846 0.403803i \(-0.867688\pi\)
−0.914846 + 0.403803i \(0.867688\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −10.5924 −0.676723
\(246\) 0 0
\(247\) −2.02466 −0.128826
\(248\) 0 0
\(249\) 0.663677 0.0420588
\(250\) 0 0
\(251\) 3.32320 0.209758 0.104879 0.994485i \(-0.466554\pi\)
0.104879 + 0.994485i \(0.466554\pi\)
\(252\) 0 0
\(253\) 10.7779 0.677600
\(254\) 0 0
\(255\) −0.419704 −0.0262829
\(256\) 0 0
\(257\) 5.17474 0.322791 0.161396 0.986890i \(-0.448400\pi\)
0.161396 + 0.986890i \(0.448400\pi\)
\(258\) 0 0
\(259\) −1.81150 −0.112561
\(260\) 0 0
\(261\) −6.05765 −0.374959
\(262\) 0 0
\(263\) 4.72275 0.291217 0.145609 0.989342i \(-0.453486\pi\)
0.145609 + 0.989342i \(0.453486\pi\)
\(264\) 0 0
\(265\) 17.3218 1.06407
\(266\) 0 0
\(267\) 10.6676 0.652846
\(268\) 0 0
\(269\) −5.74547 −0.350307 −0.175154 0.984541i \(-0.556042\pi\)
−0.175154 + 0.984541i \(0.556042\pi\)
\(270\) 0 0
\(271\) 29.7842 1.80926 0.904632 0.426194i \(-0.140146\pi\)
0.904632 + 0.426194i \(0.140146\pi\)
\(272\) 0 0
\(273\) −8.25842 −0.499822
\(274\) 0 0
\(275\) −3.72638 −0.224709
\(276\) 0 0
\(277\) −28.1486 −1.69128 −0.845642 0.533751i \(-0.820782\pi\)
−0.845642 + 0.533751i \(0.820782\pi\)
\(278\) 0 0
\(279\) 5.31798 0.318379
\(280\) 0 0
\(281\) 1.00907 0.0601962 0.0300981 0.999547i \(-0.490418\pi\)
0.0300981 + 0.999547i \(0.490418\pi\)
\(282\) 0 0
\(283\) −4.30214 −0.255736 −0.127868 0.991791i \(-0.540813\pi\)
−0.127868 + 0.991791i \(0.540813\pi\)
\(284\) 0 0
\(285\) −0.698022 −0.0413472
\(286\) 0 0
\(287\) 15.3122 0.903853
\(288\) 0 0
\(289\) −16.9590 −0.997587
\(290\) 0 0
\(291\) −2.07616 −0.121707
\(292\) 0 0
\(293\) −3.15536 −0.184338 −0.0921691 0.995743i \(-0.529380\pi\)
−0.0921691 + 0.995743i \(0.529380\pi\)
\(294\) 0 0
\(295\) −1.51418 −0.0881592
\(296\) 0 0
\(297\) −5.27463 −0.306065
\(298\) 0 0
\(299\) −12.2810 −0.710226
\(300\) 0 0
\(301\) −1.32664 −0.0764663
\(302\) 0 0
\(303\) 17.3465 0.996531
\(304\) 0 0
\(305\) −21.2221 −1.21517
\(306\) 0 0
\(307\) 6.89668 0.393615 0.196807 0.980442i \(-0.436943\pi\)
0.196807 + 0.980442i \(0.436943\pi\)
\(308\) 0 0
\(309\) 16.8119 0.956397
\(310\) 0 0
\(311\) 1.13302 0.0642477 0.0321239 0.999484i \(-0.489773\pi\)
0.0321239 + 0.999484i \(0.489773\pi\)
\(312\) 0 0
\(313\) −10.1280 −0.572466 −0.286233 0.958160i \(-0.592403\pi\)
−0.286233 + 0.958160i \(0.592403\pi\)
\(314\) 0 0
\(315\) −2.84717 −0.160420
\(316\) 0 0
\(317\) −15.8272 −0.888945 −0.444473 0.895792i \(-0.646609\pi\)
−0.444473 + 0.895792i \(0.646609\pi\)
\(318\) 0 0
\(319\) −31.9519 −1.78896
\(320\) 0 0
\(321\) −3.15743 −0.176231
\(322\) 0 0
\(323\) 0.0682334 0.00379661
\(324\) 0 0
\(325\) 4.24606 0.235529
\(326\) 0 0
\(327\) 5.11105 0.282642
\(328\) 0 0
\(329\) 5.76611 0.317896
\(330\) 0 0
\(331\) 9.06680 0.498356 0.249178 0.968458i \(-0.419840\pi\)
0.249178 + 0.968458i \(0.419840\pi\)
\(332\) 0 0
\(333\) 1.31836 0.0722455
\(334\) 0 0
\(335\) −1.97230 −0.107758
\(336\) 0 0
\(337\) 26.3569 1.43575 0.717876 0.696171i \(-0.245114\pi\)
0.717876 + 0.696171i \(0.245114\pi\)
\(338\) 0 0
\(339\) −8.36231 −0.454178
\(340\) 0 0
\(341\) 28.0504 1.51901
\(342\) 0 0
\(343\) 16.6426 0.898614
\(344\) 0 0
\(345\) −4.23397 −0.227949
\(346\) 0 0
\(347\) 30.4435 1.63429 0.817146 0.576430i \(-0.195555\pi\)
0.817146 + 0.576430i \(0.195555\pi\)
\(348\) 0 0
\(349\) −21.7753 −1.16560 −0.582801 0.812615i \(-0.698043\pi\)
−0.582801 + 0.812615i \(0.698043\pi\)
\(350\) 0 0
\(351\) 6.01023 0.320802
\(352\) 0 0
\(353\) −13.4066 −0.713560 −0.356780 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(354\) 0 0
\(355\) −25.4884 −1.35278
\(356\) 0 0
\(357\) 0.278318 0.0147302
\(358\) 0 0
\(359\) −5.78786 −0.305471 −0.152736 0.988267i \(-0.548808\pi\)
−0.152736 + 0.988267i \(0.548808\pi\)
\(360\) 0 0
\(361\) −18.8865 −0.994027
\(362\) 0 0
\(363\) −16.8218 −0.882914
\(364\) 0 0
\(365\) 25.4253 1.33082
\(366\) 0 0
\(367\) 11.1654 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(368\) 0 0
\(369\) −11.1438 −0.580122
\(370\) 0 0
\(371\) −11.4866 −0.596357
\(372\) 0 0
\(373\) −11.7084 −0.606238 −0.303119 0.952953i \(-0.598028\pi\)
−0.303119 + 0.952953i \(0.598028\pi\)
\(374\) 0 0
\(375\) 11.8243 0.610603
\(376\) 0 0
\(377\) 36.4078 1.87510
\(378\) 0 0
\(379\) 8.73464 0.448668 0.224334 0.974512i \(-0.427979\pi\)
0.224334 + 0.974512i \(0.427979\pi\)
\(380\) 0 0
\(381\) 7.98111 0.408884
\(382\) 0 0
\(383\) 30.7953 1.57356 0.786782 0.617230i \(-0.211745\pi\)
0.786782 + 0.617230i \(0.211745\pi\)
\(384\) 0 0
\(385\) −15.0178 −0.765377
\(386\) 0 0
\(387\) 0.965489 0.0490786
\(388\) 0 0
\(389\) −9.79229 −0.496489 −0.248244 0.968697i \(-0.579854\pi\)
−0.248244 + 0.968697i \(0.579854\pi\)
\(390\) 0 0
\(391\) 0.413882 0.0209309
\(392\) 0 0
\(393\) 0.126390 0.00637555
\(394\) 0 0
\(395\) 31.7358 1.59680
\(396\) 0 0
\(397\) −28.2801 −1.41934 −0.709669 0.704535i \(-0.751156\pi\)
−0.709669 + 0.704535i \(0.751156\pi\)
\(398\) 0 0
\(399\) 0.462879 0.0231730
\(400\) 0 0
\(401\) −21.4063 −1.06898 −0.534490 0.845175i \(-0.679496\pi\)
−0.534490 + 0.845175i \(0.679496\pi\)
\(402\) 0 0
\(403\) −31.9623 −1.59215
\(404\) 0 0
\(405\) 2.07208 0.102963
\(406\) 0 0
\(407\) 6.95385 0.344690
\(408\) 0 0
\(409\) −15.5676 −0.769769 −0.384885 0.922965i \(-0.625759\pi\)
−0.384885 + 0.922965i \(0.625759\pi\)
\(410\) 0 0
\(411\) −8.31975 −0.410383
\(412\) 0 0
\(413\) 1.00410 0.0494086
\(414\) 0 0
\(415\) −1.37519 −0.0675056
\(416\) 0 0
\(417\) 2.22951 0.109180
\(418\) 0 0
\(419\) 6.33403 0.309438 0.154719 0.987959i \(-0.450553\pi\)
0.154719 + 0.987959i \(0.450553\pi\)
\(420\) 0 0
\(421\) 6.61588 0.322438 0.161219 0.986919i \(-0.448457\pi\)
0.161219 + 0.986919i \(0.448457\pi\)
\(422\) 0 0
\(423\) −4.19640 −0.204036
\(424\) 0 0
\(425\) −0.143097 −0.00694122
\(426\) 0 0
\(427\) 14.0730 0.681040
\(428\) 0 0
\(429\) 31.7018 1.53058
\(430\) 0 0
\(431\) −12.5186 −0.602999 −0.301499 0.953466i \(-0.597487\pi\)
−0.301499 + 0.953466i \(0.597487\pi\)
\(432\) 0 0
\(433\) −23.6289 −1.13553 −0.567767 0.823189i \(-0.692192\pi\)
−0.567767 + 0.823189i \(0.692192\pi\)
\(434\) 0 0
\(435\) 12.5519 0.601820
\(436\) 0 0
\(437\) 0.688340 0.0329278
\(438\) 0 0
\(439\) −28.0247 −1.33755 −0.668774 0.743466i \(-0.733181\pi\)
−0.668774 + 0.743466i \(0.733181\pi\)
\(440\) 0 0
\(441\) −5.11196 −0.243426
\(442\) 0 0
\(443\) −33.1995 −1.57735 −0.788677 0.614808i \(-0.789234\pi\)
−0.788677 + 0.614808i \(0.789234\pi\)
\(444\) 0 0
\(445\) −22.1041 −1.04784
\(446\) 0 0
\(447\) −9.87451 −0.467048
\(448\) 0 0
\(449\) 20.3396 0.959885 0.479942 0.877300i \(-0.340658\pi\)
0.479942 + 0.877300i \(0.340658\pi\)
\(450\) 0 0
\(451\) −58.7794 −2.76781
\(452\) 0 0
\(453\) −9.00556 −0.423118
\(454\) 0 0
\(455\) 17.1121 0.802229
\(456\) 0 0
\(457\) −4.58262 −0.214366 −0.107183 0.994239i \(-0.534183\pi\)
−0.107183 + 0.994239i \(0.534183\pi\)
\(458\) 0 0
\(459\) −0.202551 −0.00945429
\(460\) 0 0
\(461\) 37.0853 1.72723 0.863616 0.504150i \(-0.168194\pi\)
0.863616 + 0.504150i \(0.168194\pi\)
\(462\) 0 0
\(463\) −28.7461 −1.33594 −0.667972 0.744187i \(-0.732837\pi\)
−0.667972 + 0.744187i \(0.732837\pi\)
\(464\) 0 0
\(465\) −11.0193 −0.511007
\(466\) 0 0
\(467\) −7.43392 −0.344001 −0.172000 0.985097i \(-0.555023\pi\)
−0.172000 + 0.985097i \(0.555023\pi\)
\(468\) 0 0
\(469\) 1.30789 0.0603928
\(470\) 0 0
\(471\) 6.54956 0.301788
\(472\) 0 0
\(473\) 5.09260 0.234158
\(474\) 0 0
\(475\) −0.237989 −0.0109197
\(476\) 0 0
\(477\) 8.35963 0.382761
\(478\) 0 0
\(479\) −37.3938 −1.70857 −0.854284 0.519806i \(-0.826004\pi\)
−0.854284 + 0.519806i \(0.826004\pi\)
\(480\) 0 0
\(481\) −7.92362 −0.361286
\(482\) 0 0
\(483\) 2.80768 0.127754
\(484\) 0 0
\(485\) 4.30198 0.195343
\(486\) 0 0
\(487\) −17.5735 −0.796333 −0.398167 0.917313i \(-0.630353\pi\)
−0.398167 + 0.917313i \(0.630353\pi\)
\(488\) 0 0
\(489\) −2.22508 −0.100622
\(490\) 0 0
\(491\) −3.26494 −0.147345 −0.0736723 0.997283i \(-0.523472\pi\)
−0.0736723 + 0.997283i \(0.523472\pi\)
\(492\) 0 0
\(493\) −1.22699 −0.0552607
\(494\) 0 0
\(495\) 10.9295 0.491244
\(496\) 0 0
\(497\) 16.9021 0.758164
\(498\) 0 0
\(499\) 26.5936 1.19049 0.595246 0.803543i \(-0.297055\pi\)
0.595246 + 0.803543i \(0.297055\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) −22.9368 −1.02270 −0.511351 0.859372i \(-0.670855\pi\)
−0.511351 + 0.859372i \(0.670855\pi\)
\(504\) 0 0
\(505\) −35.9434 −1.59946
\(506\) 0 0
\(507\) −23.1228 −1.02692
\(508\) 0 0
\(509\) −33.2388 −1.47328 −0.736642 0.676283i \(-0.763590\pi\)
−0.736642 + 0.676283i \(0.763590\pi\)
\(510\) 0 0
\(511\) −16.8603 −0.745857
\(512\) 0 0
\(513\) −0.336870 −0.0148732
\(514\) 0 0
\(515\) −34.8357 −1.53504
\(516\) 0 0
\(517\) −22.1345 −0.973473
\(518\) 0 0
\(519\) −5.46198 −0.239754
\(520\) 0 0
\(521\) −4.46558 −0.195641 −0.0978204 0.995204i \(-0.531187\pi\)
−0.0978204 + 0.995204i \(0.531187\pi\)
\(522\) 0 0
\(523\) −1.08095 −0.0472668 −0.0236334 0.999721i \(-0.507523\pi\)
−0.0236334 + 0.999721i \(0.507523\pi\)
\(524\) 0 0
\(525\) −0.970736 −0.0423664
\(526\) 0 0
\(527\) 1.07716 0.0469220
\(528\) 0 0
\(529\) −18.8248 −0.818468
\(530\) 0 0
\(531\) −0.730755 −0.0317121
\(532\) 0 0
\(533\) 66.9767 2.90108
\(534\) 0 0
\(535\) 6.54246 0.282855
\(536\) 0 0
\(537\) 25.3449 1.09371
\(538\) 0 0
\(539\) −26.9637 −1.16141
\(540\) 0 0
\(541\) −16.5141 −0.709998 −0.354999 0.934867i \(-0.615519\pi\)
−0.354999 + 0.934867i \(0.615519\pi\)
\(542\) 0 0
\(543\) 4.47384 0.191991
\(544\) 0 0
\(545\) −10.5905 −0.453648
\(546\) 0 0
\(547\) −0.00160491 −6.86211e−5 0 −3.43105e−5 1.00000i \(-0.500011\pi\)
−3.43105e−5 1.00000i \(0.500011\pi\)
\(548\) 0 0
\(549\) −10.2419 −0.437114
\(550\) 0 0
\(551\) −2.04064 −0.0869341
\(552\) 0 0
\(553\) −21.0450 −0.894924
\(554\) 0 0
\(555\) −2.73174 −0.115956
\(556\) 0 0
\(557\) 19.6063 0.830747 0.415374 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(558\) 0 0
\(559\) −5.80281 −0.245433
\(560\) 0 0
\(561\) −1.06839 −0.0451073
\(562\) 0 0
\(563\) 11.3368 0.477790 0.238895 0.971045i \(-0.423215\pi\)
0.238895 + 0.971045i \(0.423215\pi\)
\(564\) 0 0
\(565\) 17.3274 0.728969
\(566\) 0 0
\(567\) −1.37406 −0.0577051
\(568\) 0 0
\(569\) −35.0972 −1.47135 −0.735676 0.677333i \(-0.763136\pi\)
−0.735676 + 0.677333i \(0.763136\pi\)
\(570\) 0 0
\(571\) 14.7322 0.616525 0.308262 0.951301i \(-0.400252\pi\)
0.308262 + 0.951301i \(0.400252\pi\)
\(572\) 0 0
\(573\) 23.2392 0.970830
\(574\) 0 0
\(575\) −1.44356 −0.0602008
\(576\) 0 0
\(577\) 12.0436 0.501383 0.250691 0.968067i \(-0.419342\pi\)
0.250691 + 0.968067i \(0.419342\pi\)
\(578\) 0 0
\(579\) −7.47719 −0.310741
\(580\) 0 0
\(581\) 0.911932 0.0378333
\(582\) 0 0
\(583\) 44.0940 1.82619
\(584\) 0 0
\(585\) −12.4537 −0.514897
\(586\) 0 0
\(587\) 39.3028 1.62220 0.811101 0.584906i \(-0.198869\pi\)
0.811101 + 0.584906i \(0.198869\pi\)
\(588\) 0 0
\(589\) 1.79146 0.0738160
\(590\) 0 0
\(591\) −17.5011 −0.719900
\(592\) 0 0
\(593\) −15.1187 −0.620849 −0.310424 0.950598i \(-0.600471\pi\)
−0.310424 + 0.950598i \(0.600471\pi\)
\(594\) 0 0
\(595\) −0.576698 −0.0236423
\(596\) 0 0
\(597\) 22.2943 0.912444
\(598\) 0 0
\(599\) −12.0427 −0.492053 −0.246027 0.969263i \(-0.579125\pi\)
−0.246027 + 0.969263i \(0.579125\pi\)
\(600\) 0 0
\(601\) 31.6066 1.28926 0.644630 0.764495i \(-0.277012\pi\)
0.644630 + 0.764495i \(0.277012\pi\)
\(602\) 0 0
\(603\) −0.951843 −0.0387620
\(604\) 0 0
\(605\) 34.8561 1.41710
\(606\) 0 0
\(607\) −37.1053 −1.50606 −0.753028 0.657988i \(-0.771408\pi\)
−0.753028 + 0.657988i \(0.771408\pi\)
\(608\) 0 0
\(609\) −8.32358 −0.337288
\(610\) 0 0
\(611\) 25.2213 1.02035
\(612\) 0 0
\(613\) 10.6070 0.428414 0.214207 0.976788i \(-0.431283\pi\)
0.214207 + 0.976788i \(0.431283\pi\)
\(614\) 0 0
\(615\) 23.0908 0.931113
\(616\) 0 0
\(617\) −16.2374 −0.653693 −0.326847 0.945077i \(-0.605986\pi\)
−0.326847 + 0.945077i \(0.605986\pi\)
\(618\) 0 0
\(619\) 36.8237 1.48007 0.740035 0.672569i \(-0.234809\pi\)
0.740035 + 0.672569i \(0.234809\pi\)
\(620\) 0 0
\(621\) −2.04334 −0.0819965
\(622\) 0 0
\(623\) 14.6579 0.587257
\(624\) 0 0
\(625\) −20.9685 −0.838741
\(626\) 0 0
\(627\) −1.77686 −0.0709611
\(628\) 0 0
\(629\) 0.267035 0.0106474
\(630\) 0 0
\(631\) −12.8100 −0.509958 −0.254979 0.966947i \(-0.582069\pi\)
−0.254979 + 0.966947i \(0.582069\pi\)
\(632\) 0 0
\(633\) −12.0997 −0.480918
\(634\) 0 0
\(635\) −16.5375 −0.656271
\(636\) 0 0
\(637\) 30.7240 1.21733
\(638\) 0 0
\(639\) −12.3009 −0.486615
\(640\) 0 0
\(641\) 12.6972 0.501509 0.250754 0.968051i \(-0.419321\pi\)
0.250754 + 0.968051i \(0.419321\pi\)
\(642\) 0 0
\(643\) −11.3118 −0.446096 −0.223048 0.974807i \(-0.571601\pi\)
−0.223048 + 0.974807i \(0.571601\pi\)
\(644\) 0 0
\(645\) −2.00057 −0.0787725
\(646\) 0 0
\(647\) 16.1608 0.635348 0.317674 0.948200i \(-0.397098\pi\)
0.317674 + 0.948200i \(0.397098\pi\)
\(648\) 0 0
\(649\) −3.85446 −0.151301
\(650\) 0 0
\(651\) 7.30723 0.286393
\(652\) 0 0
\(653\) −28.4137 −1.11191 −0.555957 0.831211i \(-0.687648\pi\)
−0.555957 + 0.831211i \(0.687648\pi\)
\(654\) 0 0
\(655\) −0.261891 −0.0102329
\(656\) 0 0
\(657\) 12.2704 0.478715
\(658\) 0 0
\(659\) −43.1667 −1.68154 −0.840768 0.541396i \(-0.817896\pi\)
−0.840768 + 0.541396i \(0.817896\pi\)
\(660\) 0 0
\(661\) −40.3262 −1.56851 −0.784254 0.620440i \(-0.786954\pi\)
−0.784254 + 0.620440i \(0.786954\pi\)
\(662\) 0 0
\(663\) 1.21738 0.0472792
\(664\) 0 0
\(665\) −0.959125 −0.0371933
\(666\) 0 0
\(667\) −12.3778 −0.479272
\(668\) 0 0
\(669\) 4.28101 0.165514
\(670\) 0 0
\(671\) −54.0223 −2.08551
\(672\) 0 0
\(673\) −16.7667 −0.646310 −0.323155 0.946346i \(-0.604744\pi\)
−0.323155 + 0.946346i \(0.604744\pi\)
\(674\) 0 0
\(675\) 0.706472 0.0271921
\(676\) 0 0
\(677\) 37.1987 1.42966 0.714832 0.699297i \(-0.246503\pi\)
0.714832 + 0.699297i \(0.246503\pi\)
\(678\) 0 0
\(679\) −2.85277 −0.109479
\(680\) 0 0
\(681\) 6.47975 0.248305
\(682\) 0 0
\(683\) −21.4255 −0.819825 −0.409913 0.912125i \(-0.634441\pi\)
−0.409913 + 0.912125i \(0.634441\pi\)
\(684\) 0 0
\(685\) 17.2392 0.658677
\(686\) 0 0
\(687\) −15.3238 −0.584641
\(688\) 0 0
\(689\) −50.2433 −1.91412
\(690\) 0 0
\(691\) 17.1907 0.653966 0.326983 0.945030i \(-0.393968\pi\)
0.326983 + 0.945030i \(0.393968\pi\)
\(692\) 0 0
\(693\) −7.24767 −0.275316
\(694\) 0 0
\(695\) −4.61974 −0.175237
\(696\) 0 0
\(697\) −2.25719 −0.0854972
\(698\) 0 0
\(699\) 3.54194 0.133968
\(700\) 0 0
\(701\) 18.6527 0.704504 0.352252 0.935905i \(-0.385416\pi\)
0.352252 + 0.935905i \(0.385416\pi\)
\(702\) 0 0
\(703\) 0.444114 0.0167501
\(704\) 0 0
\(705\) 8.69529 0.327484
\(706\) 0 0
\(707\) 23.8352 0.896413
\(708\) 0 0
\(709\) −33.3235 −1.25149 −0.625744 0.780028i \(-0.715205\pi\)
−0.625744 + 0.780028i \(0.715205\pi\)
\(710\) 0 0
\(711\) 15.3159 0.574391
\(712\) 0 0
\(713\) 10.8664 0.406951
\(714\) 0 0
\(715\) −65.6887 −2.45662
\(716\) 0 0
\(717\) −13.3809 −0.499718
\(718\) 0 0
\(719\) −33.9117 −1.26469 −0.632346 0.774686i \(-0.717908\pi\)
−0.632346 + 0.774686i \(0.717908\pi\)
\(720\) 0 0
\(721\) 23.1006 0.860312
\(722\) 0 0
\(723\) 28.4045 1.05637
\(724\) 0 0
\(725\) 4.27956 0.158939
\(726\) 0 0
\(727\) 32.3637 1.20030 0.600152 0.799886i \(-0.295107\pi\)
0.600152 + 0.799886i \(0.295107\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.195561 0.00723310
\(732\) 0 0
\(733\) 4.01849 0.148426 0.0742131 0.997242i \(-0.476355\pi\)
0.0742131 + 0.997242i \(0.476355\pi\)
\(734\) 0 0
\(735\) 10.5924 0.390706
\(736\) 0 0
\(737\) −5.02062 −0.184937
\(738\) 0 0
\(739\) 3.69570 0.135948 0.0679742 0.997687i \(-0.478346\pi\)
0.0679742 + 0.997687i \(0.478346\pi\)
\(740\) 0 0
\(741\) 2.02466 0.0743779
\(742\) 0 0
\(743\) −32.3327 −1.18617 −0.593086 0.805139i \(-0.702091\pi\)
−0.593086 + 0.805139i \(0.702091\pi\)
\(744\) 0 0
\(745\) 20.4608 0.749626
\(746\) 0 0
\(747\) −0.663677 −0.0242827
\(748\) 0 0
\(749\) −4.33851 −0.158526
\(750\) 0 0
\(751\) −2.13842 −0.0780322 −0.0390161 0.999239i \(-0.512422\pi\)
−0.0390161 + 0.999239i \(0.512422\pi\)
\(752\) 0 0
\(753\) −3.32320 −0.121104
\(754\) 0 0
\(755\) 18.6603 0.679116
\(756\) 0 0
\(757\) −27.1166 −0.985570 −0.492785 0.870151i \(-0.664021\pi\)
−0.492785 + 0.870151i \(0.664021\pi\)
\(758\) 0 0
\(759\) −10.7779 −0.391212
\(760\) 0 0
\(761\) 17.7022 0.641705 0.320852 0.947129i \(-0.396031\pi\)
0.320852 + 0.947129i \(0.396031\pi\)
\(762\) 0 0
\(763\) 7.02290 0.254246
\(764\) 0 0
\(765\) 0.419704 0.0151744
\(766\) 0 0
\(767\) 4.39200 0.158586
\(768\) 0 0
\(769\) −21.7313 −0.783651 −0.391825 0.920040i \(-0.628156\pi\)
−0.391825 + 0.920040i \(0.628156\pi\)
\(770\) 0 0
\(771\) −5.17474 −0.186364
\(772\) 0 0
\(773\) 5.91086 0.212599 0.106299 0.994334i \(-0.466100\pi\)
0.106299 + 0.994334i \(0.466100\pi\)
\(774\) 0 0
\(775\) −3.75700 −0.134956
\(776\) 0 0
\(777\) 1.81150 0.0649873
\(778\) 0 0
\(779\) −3.75400 −0.134501
\(780\) 0 0
\(781\) −64.8825 −2.32168
\(782\) 0 0
\(783\) 6.05765 0.216483
\(784\) 0 0
\(785\) −13.5712 −0.484378
\(786\) 0 0
\(787\) 39.2576 1.39938 0.699692 0.714445i \(-0.253321\pi\)
0.699692 + 0.714445i \(0.253321\pi\)
\(788\) 0 0
\(789\) −4.72275 −0.168134
\(790\) 0 0
\(791\) −11.4903 −0.408549
\(792\) 0 0
\(793\) 61.5562 2.18592
\(794\) 0 0
\(795\) −17.3218 −0.614342
\(796\) 0 0
\(797\) 48.4430 1.71594 0.857970 0.513700i \(-0.171725\pi\)
0.857970 + 0.513700i \(0.171725\pi\)
\(798\) 0 0
\(799\) −0.849987 −0.0300704
\(800\) 0 0
\(801\) −10.6676 −0.376921
\(802\) 0 0
\(803\) 64.7220 2.28399
\(804\) 0 0
\(805\) −5.81774 −0.205048
\(806\) 0 0
\(807\) 5.74547 0.202250
\(808\) 0 0
\(809\) −26.7053 −0.938910 −0.469455 0.882956i \(-0.655550\pi\)
−0.469455 + 0.882956i \(0.655550\pi\)
\(810\) 0 0
\(811\) 51.0389 1.79222 0.896108 0.443836i \(-0.146383\pi\)
0.896108 + 0.443836i \(0.146383\pi\)
\(812\) 0 0
\(813\) −29.7842 −1.04458
\(814\) 0 0
\(815\) 4.61055 0.161501
\(816\) 0 0
\(817\) 0.325244 0.0113788
\(818\) 0 0
\(819\) 8.25842 0.288573
\(820\) 0 0
\(821\) 50.2885 1.75508 0.877540 0.479504i \(-0.159183\pi\)
0.877540 + 0.479504i \(0.159183\pi\)
\(822\) 0 0
\(823\) 21.0372 0.733310 0.366655 0.930357i \(-0.380503\pi\)
0.366655 + 0.930357i \(0.380503\pi\)
\(824\) 0 0
\(825\) 3.72638 0.129736
\(826\) 0 0
\(827\) −9.90656 −0.344485 −0.172242 0.985055i \(-0.555101\pi\)
−0.172242 + 0.985055i \(0.555101\pi\)
\(828\) 0 0
\(829\) −27.9584 −0.971035 −0.485518 0.874227i \(-0.661369\pi\)
−0.485518 + 0.874227i \(0.661369\pi\)
\(830\) 0 0
\(831\) 28.1486 0.976463
\(832\) 0 0
\(833\) −1.03543 −0.0358757
\(834\) 0 0
\(835\) −2.07208 −0.0717074
\(836\) 0 0
\(837\) −5.31798 −0.183816
\(838\) 0 0
\(839\) −23.6314 −0.815848 −0.407924 0.913016i \(-0.633747\pi\)
−0.407924 + 0.913016i \(0.633747\pi\)
\(840\) 0 0
\(841\) 7.69510 0.265348
\(842\) 0 0
\(843\) −1.00907 −0.0347543
\(844\) 0 0
\(845\) 47.9125 1.64824
\(846\) 0 0
\(847\) −23.1141 −0.794211
\(848\) 0 0
\(849\) 4.30214 0.147649
\(850\) 0 0
\(851\) 2.69385 0.0923441
\(852\) 0 0
\(853\) 24.1256 0.826045 0.413023 0.910721i \(-0.364473\pi\)
0.413023 + 0.910721i \(0.364473\pi\)
\(854\) 0 0
\(855\) 0.698022 0.0238718
\(856\) 0 0
\(857\) −31.9576 −1.09165 −0.545825 0.837899i \(-0.683784\pi\)
−0.545825 + 0.837899i \(0.683784\pi\)
\(858\) 0 0
\(859\) −56.0883 −1.91371 −0.956854 0.290569i \(-0.906156\pi\)
−0.956854 + 0.290569i \(0.906156\pi\)
\(860\) 0 0
\(861\) −15.3122 −0.521840
\(862\) 0 0
\(863\) −49.9022 −1.69869 −0.849344 0.527839i \(-0.823002\pi\)
−0.849344 + 0.527839i \(0.823002\pi\)
\(864\) 0 0
\(865\) 11.3177 0.384812
\(866\) 0 0
\(867\) 16.9590 0.575957
\(868\) 0 0
\(869\) 80.7858 2.74047
\(870\) 0 0
\(871\) 5.72079 0.193842
\(872\) 0 0
\(873\) 2.07616 0.0702674
\(874\) 0 0
\(875\) 16.2473 0.549259
\(876\) 0 0
\(877\) 28.1153 0.949388 0.474694 0.880151i \(-0.342559\pi\)
0.474694 + 0.880151i \(0.342559\pi\)
\(878\) 0 0
\(879\) 3.15536 0.106428
\(880\) 0 0
\(881\) 48.3928 1.63039 0.815197 0.579184i \(-0.196629\pi\)
0.815197 + 0.579184i \(0.196629\pi\)
\(882\) 0 0
\(883\) 35.0305 1.17887 0.589435 0.807816i \(-0.299351\pi\)
0.589435 + 0.807816i \(0.299351\pi\)
\(884\) 0 0
\(885\) 1.51418 0.0508988
\(886\) 0 0
\(887\) 18.6796 0.627200 0.313600 0.949555i \(-0.398465\pi\)
0.313600 + 0.949555i \(0.398465\pi\)
\(888\) 0 0
\(889\) 10.9665 0.367806
\(890\) 0 0
\(891\) 5.27463 0.176707
\(892\) 0 0
\(893\) −1.41364 −0.0473057
\(894\) 0 0
\(895\) −52.5167 −1.75544
\(896\) 0 0
\(897\) 12.2810 0.410049
\(898\) 0 0
\(899\) −32.2144 −1.07441
\(900\) 0 0
\(901\) 1.69325 0.0564105
\(902\) 0 0
\(903\) 1.32664 0.0441479
\(904\) 0 0
\(905\) −9.27018 −0.308151
\(906\) 0 0
\(907\) 40.5486 1.34639 0.673197 0.739463i \(-0.264920\pi\)
0.673197 + 0.739463i \(0.264920\pi\)
\(908\) 0 0
\(909\) −17.3465 −0.575347
\(910\) 0 0
\(911\) −21.2864 −0.705252 −0.352626 0.935764i \(-0.614711\pi\)
−0.352626 + 0.935764i \(0.614711\pi\)
\(912\) 0 0
\(913\) −3.50065 −0.115855
\(914\) 0 0
\(915\) 21.2221 0.701580
\(916\) 0 0
\(917\) 0.173668 0.00573502
\(918\) 0 0
\(919\) −16.0390 −0.529077 −0.264538 0.964375i \(-0.585220\pi\)
−0.264538 + 0.964375i \(0.585220\pi\)
\(920\) 0 0
\(921\) −6.89668 −0.227253
\(922\) 0 0
\(923\) 73.9310 2.43347
\(924\) 0 0
\(925\) −0.931382 −0.0306237
\(926\) 0 0
\(927\) −16.8119 −0.552176
\(928\) 0 0
\(929\) 12.7224 0.417409 0.208704 0.977979i \(-0.433075\pi\)
0.208704 + 0.977979i \(0.433075\pi\)
\(930\) 0 0
\(931\) −1.72206 −0.0564383
\(932\) 0 0
\(933\) −1.13302 −0.0370935
\(934\) 0 0
\(935\) 2.21378 0.0723984
\(936\) 0 0
\(937\) 17.0752 0.557822 0.278911 0.960317i \(-0.410026\pi\)
0.278911 + 0.960317i \(0.410026\pi\)
\(938\) 0 0
\(939\) 10.1280 0.330513
\(940\) 0 0
\(941\) 29.7023 0.968267 0.484134 0.874994i \(-0.339135\pi\)
0.484134 + 0.874994i \(0.339135\pi\)
\(942\) 0 0
\(943\) −22.7706 −0.741512
\(944\) 0 0
\(945\) 2.84717 0.0926184
\(946\) 0 0
\(947\) 1.64205 0.0533594 0.0266797 0.999644i \(-0.491507\pi\)
0.0266797 + 0.999644i \(0.491507\pi\)
\(948\) 0 0
\(949\) −73.7481 −2.39396
\(950\) 0 0
\(951\) 15.8272 0.513233
\(952\) 0 0
\(953\) −4.67237 −0.151353 −0.0756764 0.997132i \(-0.524112\pi\)
−0.0756764 + 0.997132i \(0.524112\pi\)
\(954\) 0 0
\(955\) −48.1535 −1.55821
\(956\) 0 0
\(957\) 31.9519 1.03286
\(958\) 0 0
\(959\) −11.4319 −0.369154
\(960\) 0 0
\(961\) −2.71913 −0.0877138
\(962\) 0 0
\(963\) 3.15743 0.101747
\(964\) 0 0
\(965\) 15.4933 0.498749
\(966\) 0 0
\(967\) 2.49406 0.0802037 0.0401018 0.999196i \(-0.487232\pi\)
0.0401018 + 0.999196i \(0.487232\pi\)
\(968\) 0 0
\(969\) −0.0682334 −0.00219197
\(970\) 0 0
\(971\) −43.9326 −1.40986 −0.704931 0.709275i \(-0.749022\pi\)
−0.704931 + 0.709275i \(0.749022\pi\)
\(972\) 0 0
\(973\) 3.06349 0.0982110
\(974\) 0 0
\(975\) −4.24606 −0.135983
\(976\) 0 0
\(977\) 44.8137 1.43372 0.716859 0.697218i \(-0.245579\pi\)
0.716859 + 0.697218i \(0.245579\pi\)
\(978\) 0 0
\(979\) −56.2676 −1.79832
\(980\) 0 0
\(981\) −5.11105 −0.163183
\(982\) 0 0
\(983\) −34.9940 −1.11613 −0.558067 0.829796i \(-0.688457\pi\)
−0.558067 + 0.829796i \(0.688457\pi\)
\(984\) 0 0
\(985\) 36.2638 1.15546
\(986\) 0 0
\(987\) −5.76611 −0.183537
\(988\) 0 0
\(989\) 1.97282 0.0627322
\(990\) 0 0
\(991\) 26.5450 0.843230 0.421615 0.906775i \(-0.361463\pi\)
0.421615 + 0.906775i \(0.361463\pi\)
\(992\) 0 0
\(993\) −9.06680 −0.287726
\(994\) 0 0
\(995\) −46.1956 −1.46450
\(996\) 0 0
\(997\) 51.9221 1.64439 0.822195 0.569206i \(-0.192749\pi\)
0.822195 + 0.569206i \(0.192749\pi\)
\(998\) 0 0
\(999\) −1.31836 −0.0417110
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 8016.2.a.s.1.4 5
4.3 odd 2 4008.2.a.f.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.f.1.4 5 4.3 odd 2
8016.2.a.s.1.4 5 1.1 even 1 trivial