Properties

Label 8024.2.a.v.1.16
Level $8024$
Weight $2$
Character 8024.1
Self dual yes
Analytic conductor $64.072$
Analytic rank $0$
Dimension $18$
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] = [8024,2,Mod(1,8024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8024, 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("8024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8024.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: \(64.0719625819\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 9 x^{17} - 2 x^{16} + 212 x^{15} - 289 x^{14} - 2094 x^{13} + 3933 x^{12} + 11326 x^{11} - 23166 x^{10} - 36429 x^{9} + 72042 x^{8} + 69272 x^{7} - 119982 x^{6} + \cdots + 1136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(-2.05229\) of defining polynomial
Character \(\chi\) \(=\) 8024.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.05229 q^{3} -3.77561 q^{5} -1.23812 q^{7} +6.31647 q^{9} +O(q^{10})\) \(q+3.05229 q^{3} -3.77561 q^{5} -1.23812 q^{7} +6.31647 q^{9} -6.24004 q^{11} +1.40461 q^{13} -11.5242 q^{15} +1.00000 q^{17} -5.85333 q^{19} -3.77911 q^{21} -4.79049 q^{23} +9.25521 q^{25} +10.1228 q^{27} +3.61882 q^{29} +7.20076 q^{31} -19.0464 q^{33} +4.67466 q^{35} +4.25299 q^{37} +4.28726 q^{39} -1.60416 q^{41} +4.57197 q^{43} -23.8485 q^{45} +10.4542 q^{47} -5.46705 q^{49} +3.05229 q^{51} +3.40906 q^{53} +23.5599 q^{55} -17.8661 q^{57} -1.00000 q^{59} +10.8165 q^{61} -7.82056 q^{63} -5.30324 q^{65} -3.16292 q^{67} -14.6220 q^{69} +3.51156 q^{71} -15.4110 q^{73} +28.2496 q^{75} +7.72594 q^{77} +4.98347 q^{79} +11.9483 q^{81} -1.57429 q^{83} -3.77561 q^{85} +11.0457 q^{87} -10.4632 q^{89} -1.73907 q^{91} +21.9788 q^{93} +22.0999 q^{95} -1.93468 q^{97} -39.4150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9} - q^{11} + 15 q^{13} + 3 q^{15} + 18 q^{17} + 26 q^{19} - 4 q^{21} + 22 q^{23} + 42 q^{25} + 45 q^{27} + 6 q^{29} + 13 q^{31} - 5 q^{33} + 4 q^{35} + 4 q^{37} + 36 q^{39} - 15 q^{41} + 12 q^{43} + 14 q^{45} + 8 q^{47} - 13 q^{49} + 9 q^{51} - 11 q^{53} + 55 q^{55} - 20 q^{57} - 18 q^{59} + 53 q^{61} + 29 q^{63} - 26 q^{65} + 2 q^{67} + 32 q^{69} + 8 q^{71} - 42 q^{73} + 72 q^{75} + 6 q^{77} - 9 q^{79} + 42 q^{81} - 4 q^{83} + 2 q^{85} + 36 q^{87} + 13 q^{89} + 68 q^{91} + q^{93} + 3 q^{95} - 56 q^{97} - 13 q^{99}+O(q^{100}) \) 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
\(3\) 3.05229 1.76224 0.881120 0.472893i \(-0.156790\pi\)
0.881120 + 0.472893i \(0.156790\pi\)
\(4\) 0 0
\(5\) −3.77561 −1.68850 −0.844251 0.535947i \(-0.819955\pi\)
−0.844251 + 0.535947i \(0.819955\pi\)
\(6\) 0 0
\(7\) −1.23812 −0.467966 −0.233983 0.972241i \(-0.575176\pi\)
−0.233983 + 0.972241i \(0.575176\pi\)
\(8\) 0 0
\(9\) 6.31647 2.10549
\(10\) 0 0
\(11\) −6.24004 −1.88144 −0.940722 0.339180i \(-0.889850\pi\)
−0.940722 + 0.339180i \(0.889850\pi\)
\(12\) 0 0
\(13\) 1.40461 0.389568 0.194784 0.980846i \(-0.437599\pi\)
0.194784 + 0.980846i \(0.437599\pi\)
\(14\) 0 0
\(15\) −11.5242 −2.97555
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) −5.85333 −1.34285 −0.671423 0.741074i \(-0.734317\pi\)
−0.671423 + 0.741074i \(0.734317\pi\)
\(20\) 0 0
\(21\) −3.77911 −0.824669
\(22\) 0 0
\(23\) −4.79049 −0.998887 −0.499443 0.866346i \(-0.666462\pi\)
−0.499443 + 0.866346i \(0.666462\pi\)
\(24\) 0 0
\(25\) 9.25521 1.85104
\(26\) 0 0
\(27\) 10.1228 1.94814
\(28\) 0 0
\(29\) 3.61882 0.671999 0.335999 0.941862i \(-0.390926\pi\)
0.335999 + 0.941862i \(0.390926\pi\)
\(30\) 0 0
\(31\) 7.20076 1.29329 0.646647 0.762789i \(-0.276171\pi\)
0.646647 + 0.762789i \(0.276171\pi\)
\(32\) 0 0
\(33\) −19.0464 −3.31555
\(34\) 0 0
\(35\) 4.67466 0.790163
\(36\) 0 0
\(37\) 4.25299 0.699187 0.349593 0.936902i \(-0.386320\pi\)
0.349593 + 0.936902i \(0.386320\pi\)
\(38\) 0 0
\(39\) 4.28726 0.686511
\(40\) 0 0
\(41\) −1.60416 −0.250528 −0.125264 0.992123i \(-0.539978\pi\)
−0.125264 + 0.992123i \(0.539978\pi\)
\(42\) 0 0
\(43\) 4.57197 0.697218 0.348609 0.937268i \(-0.386654\pi\)
0.348609 + 0.937268i \(0.386654\pi\)
\(44\) 0 0
\(45\) −23.8485 −3.55512
\(46\) 0 0
\(47\) 10.4542 1.52491 0.762454 0.647042i \(-0.223994\pi\)
0.762454 + 0.647042i \(0.223994\pi\)
\(48\) 0 0
\(49\) −5.46705 −0.781007
\(50\) 0 0
\(51\) 3.05229 0.427406
\(52\) 0 0
\(53\) 3.40906 0.468270 0.234135 0.972204i \(-0.424774\pi\)
0.234135 + 0.972204i \(0.424774\pi\)
\(54\) 0 0
\(55\) 23.5599 3.17682
\(56\) 0 0
\(57\) −17.8661 −2.36642
\(58\) 0 0
\(59\) −1.00000 −0.130189
\(60\) 0 0
\(61\) 10.8165 1.38490 0.692452 0.721463i \(-0.256530\pi\)
0.692452 + 0.721463i \(0.256530\pi\)
\(62\) 0 0
\(63\) −7.82056 −0.985298
\(64\) 0 0
\(65\) −5.30324 −0.657786
\(66\) 0 0
\(67\) −3.16292 −0.386412 −0.193206 0.981158i \(-0.561889\pi\)
−0.193206 + 0.981158i \(0.561889\pi\)
\(68\) 0 0
\(69\) −14.6220 −1.76028
\(70\) 0 0
\(71\) 3.51156 0.416745 0.208373 0.978049i \(-0.433183\pi\)
0.208373 + 0.978049i \(0.433183\pi\)
\(72\) 0 0
\(73\) −15.4110 −1.80372 −0.901860 0.432029i \(-0.857798\pi\)
−0.901860 + 0.432029i \(0.857798\pi\)
\(74\) 0 0
\(75\) 28.2496 3.26198
\(76\) 0 0
\(77\) 7.72594 0.880452
\(78\) 0 0
\(79\) 4.98347 0.560684 0.280342 0.959900i \(-0.409552\pi\)
0.280342 + 0.959900i \(0.409552\pi\)
\(80\) 0 0
\(81\) 11.9483 1.32759
\(82\) 0 0
\(83\) −1.57429 −0.172801 −0.0864003 0.996261i \(-0.527536\pi\)
−0.0864003 + 0.996261i \(0.527536\pi\)
\(84\) 0 0
\(85\) −3.77561 −0.409522
\(86\) 0 0
\(87\) 11.0457 1.18422
\(88\) 0 0
\(89\) −10.4632 −1.10910 −0.554549 0.832151i \(-0.687109\pi\)
−0.554549 + 0.832151i \(0.687109\pi\)
\(90\) 0 0
\(91\) −1.73907 −0.182305
\(92\) 0 0
\(93\) 21.9788 2.27910
\(94\) 0 0
\(95\) 22.0999 2.26740
\(96\) 0 0
\(97\) −1.93468 −0.196437 −0.0982187 0.995165i \(-0.531314\pi\)
−0.0982187 + 0.995165i \(0.531314\pi\)
\(98\) 0 0
\(99\) −39.4150 −3.96136
\(100\) 0 0
\(101\) −13.4503 −1.33836 −0.669179 0.743102i \(-0.733354\pi\)
−0.669179 + 0.743102i \(0.733354\pi\)
\(102\) 0 0
\(103\) 4.57036 0.450331 0.225166 0.974321i \(-0.427708\pi\)
0.225166 + 0.974321i \(0.427708\pi\)
\(104\) 0 0
\(105\) 14.2684 1.39246
\(106\) 0 0
\(107\) 18.8533 1.82262 0.911311 0.411719i \(-0.135071\pi\)
0.911311 + 0.411719i \(0.135071\pi\)
\(108\) 0 0
\(109\) 13.2595 1.27003 0.635014 0.772501i \(-0.280994\pi\)
0.635014 + 0.772501i \(0.280994\pi\)
\(110\) 0 0
\(111\) 12.9813 1.23213
\(112\) 0 0
\(113\) 7.54665 0.709929 0.354964 0.934880i \(-0.384493\pi\)
0.354964 + 0.934880i \(0.384493\pi\)
\(114\) 0 0
\(115\) 18.0870 1.68662
\(116\) 0 0
\(117\) 8.87215 0.820230
\(118\) 0 0
\(119\) −1.23812 −0.113499
\(120\) 0 0
\(121\) 27.9381 2.53983
\(122\) 0 0
\(123\) −4.89637 −0.441490
\(124\) 0 0
\(125\) −16.0660 −1.43699
\(126\) 0 0
\(127\) −14.1187 −1.25284 −0.626418 0.779488i \(-0.715480\pi\)
−0.626418 + 0.779488i \(0.715480\pi\)
\(128\) 0 0
\(129\) 13.9550 1.22867
\(130\) 0 0
\(131\) 12.4425 1.08711 0.543555 0.839374i \(-0.317078\pi\)
0.543555 + 0.839374i \(0.317078\pi\)
\(132\) 0 0
\(133\) 7.24714 0.628407
\(134\) 0 0
\(135\) −38.2197 −3.28943
\(136\) 0 0
\(137\) 17.7615 1.51746 0.758732 0.651403i \(-0.225819\pi\)
0.758732 + 0.651403i \(0.225819\pi\)
\(138\) 0 0
\(139\) 17.4733 1.48206 0.741031 0.671470i \(-0.234337\pi\)
0.741031 + 0.671470i \(0.234337\pi\)
\(140\) 0 0
\(141\) 31.9094 2.68725
\(142\) 0 0
\(143\) −8.76480 −0.732949
\(144\) 0 0
\(145\) −13.6633 −1.13467
\(146\) 0 0
\(147\) −16.6870 −1.37632
\(148\) 0 0
\(149\) −21.2271 −1.73900 −0.869498 0.493937i \(-0.835557\pi\)
−0.869498 + 0.493937i \(0.835557\pi\)
\(150\) 0 0
\(151\) 21.6764 1.76400 0.881999 0.471250i \(-0.156197\pi\)
0.881999 + 0.471250i \(0.156197\pi\)
\(152\) 0 0
\(153\) 6.31647 0.510656
\(154\) 0 0
\(155\) −27.1872 −2.18373
\(156\) 0 0
\(157\) 20.1659 1.60941 0.804706 0.593674i \(-0.202323\pi\)
0.804706 + 0.593674i \(0.202323\pi\)
\(158\) 0 0
\(159\) 10.4054 0.825204
\(160\) 0 0
\(161\) 5.93122 0.467446
\(162\) 0 0
\(163\) 11.0314 0.864045 0.432022 0.901863i \(-0.357800\pi\)
0.432022 + 0.901863i \(0.357800\pi\)
\(164\) 0 0
\(165\) 71.9117 5.59832
\(166\) 0 0
\(167\) 14.5916 1.12914 0.564568 0.825387i \(-0.309043\pi\)
0.564568 + 0.825387i \(0.309043\pi\)
\(168\) 0 0
\(169\) −11.0271 −0.848237
\(170\) 0 0
\(171\) −36.9724 −2.82735
\(172\) 0 0
\(173\) 15.8605 1.20585 0.602924 0.797798i \(-0.294002\pi\)
0.602924 + 0.797798i \(0.294002\pi\)
\(174\) 0 0
\(175\) −11.4591 −0.866225
\(176\) 0 0
\(177\) −3.05229 −0.229424
\(178\) 0 0
\(179\) 13.6715 1.02186 0.510928 0.859623i \(-0.329302\pi\)
0.510928 + 0.859623i \(0.329302\pi\)
\(180\) 0 0
\(181\) 9.82274 0.730119 0.365059 0.930984i \(-0.381049\pi\)
0.365059 + 0.930984i \(0.381049\pi\)
\(182\) 0 0
\(183\) 33.0149 2.44053
\(184\) 0 0
\(185\) −16.0576 −1.18058
\(186\) 0 0
\(187\) −6.24004 −0.456317
\(188\) 0 0
\(189\) −12.5333 −0.911662
\(190\) 0 0
\(191\) −23.0627 −1.66876 −0.834379 0.551192i \(-0.814173\pi\)
−0.834379 + 0.551192i \(0.814173\pi\)
\(192\) 0 0
\(193\) −11.8172 −0.850624 −0.425312 0.905047i \(-0.639836\pi\)
−0.425312 + 0.905047i \(0.639836\pi\)
\(194\) 0 0
\(195\) −16.1870 −1.15918
\(196\) 0 0
\(197\) −9.34408 −0.665738 −0.332869 0.942973i \(-0.608017\pi\)
−0.332869 + 0.942973i \(0.608017\pi\)
\(198\) 0 0
\(199\) −7.08700 −0.502384 −0.251192 0.967937i \(-0.580823\pi\)
−0.251192 + 0.967937i \(0.580823\pi\)
\(200\) 0 0
\(201\) −9.65414 −0.680950
\(202\) 0 0
\(203\) −4.48055 −0.314473
\(204\) 0 0
\(205\) 6.05669 0.423017
\(206\) 0 0
\(207\) −30.2590 −2.10314
\(208\) 0 0
\(209\) 36.5250 2.52649
\(210\) 0 0
\(211\) 23.3465 1.60724 0.803619 0.595144i \(-0.202905\pi\)
0.803619 + 0.595144i \(0.202905\pi\)
\(212\) 0 0
\(213\) 10.7183 0.734405
\(214\) 0 0
\(215\) −17.2619 −1.17725
\(216\) 0 0
\(217\) −8.91542 −0.605218
\(218\) 0 0
\(219\) −47.0388 −3.17859
\(220\) 0 0
\(221\) 1.40461 0.0944840
\(222\) 0 0
\(223\) 19.3835 1.29802 0.649008 0.760782i \(-0.275184\pi\)
0.649008 + 0.760782i \(0.275184\pi\)
\(224\) 0 0
\(225\) 58.4602 3.89735
\(226\) 0 0
\(227\) 15.9845 1.06093 0.530464 0.847708i \(-0.322018\pi\)
0.530464 + 0.847708i \(0.322018\pi\)
\(228\) 0 0
\(229\) 14.7512 0.974788 0.487394 0.873182i \(-0.337948\pi\)
0.487394 + 0.873182i \(0.337948\pi\)
\(230\) 0 0
\(231\) 23.5818 1.55157
\(232\) 0 0
\(233\) −27.5607 −1.80556 −0.902782 0.430099i \(-0.858479\pi\)
−0.902782 + 0.430099i \(0.858479\pi\)
\(234\) 0 0
\(235\) −39.4711 −2.57481
\(236\) 0 0
\(237\) 15.2110 0.988060
\(238\) 0 0
\(239\) −0.765675 −0.0495274 −0.0247637 0.999693i \(-0.507883\pi\)
−0.0247637 + 0.999693i \(0.507883\pi\)
\(240\) 0 0
\(241\) −4.89629 −0.315398 −0.157699 0.987487i \(-0.550408\pi\)
−0.157699 + 0.987487i \(0.550408\pi\)
\(242\) 0 0
\(243\) 6.10135 0.391402
\(244\) 0 0
\(245\) 20.6414 1.31873
\(246\) 0 0
\(247\) −8.22162 −0.523129
\(248\) 0 0
\(249\) −4.80518 −0.304516
\(250\) 0 0
\(251\) −26.0888 −1.64671 −0.823354 0.567529i \(-0.807900\pi\)
−0.823354 + 0.567529i \(0.807900\pi\)
\(252\) 0 0
\(253\) 29.8929 1.87935
\(254\) 0 0
\(255\) −11.5242 −0.721676
\(256\) 0 0
\(257\) −12.2271 −0.762703 −0.381351 0.924430i \(-0.624541\pi\)
−0.381351 + 0.924430i \(0.624541\pi\)
\(258\) 0 0
\(259\) −5.26572 −0.327196
\(260\) 0 0
\(261\) 22.8582 1.41489
\(262\) 0 0
\(263\) 9.91567 0.611426 0.305713 0.952124i \(-0.401105\pi\)
0.305713 + 0.952124i \(0.401105\pi\)
\(264\) 0 0
\(265\) −12.8713 −0.790675
\(266\) 0 0
\(267\) −31.9368 −1.95450
\(268\) 0 0
\(269\) −12.1519 −0.740913 −0.370456 0.928850i \(-0.620799\pi\)
−0.370456 + 0.928850i \(0.620799\pi\)
\(270\) 0 0
\(271\) 1.35358 0.0822241 0.0411121 0.999155i \(-0.486910\pi\)
0.0411121 + 0.999155i \(0.486910\pi\)
\(272\) 0 0
\(273\) −5.30816 −0.321264
\(274\) 0 0
\(275\) −57.7529 −3.48263
\(276\) 0 0
\(277\) 0.757934 0.0455399 0.0227699 0.999741i \(-0.492751\pi\)
0.0227699 + 0.999741i \(0.492751\pi\)
\(278\) 0 0
\(279\) 45.4834 2.72302
\(280\) 0 0
\(281\) 8.06201 0.480939 0.240470 0.970657i \(-0.422699\pi\)
0.240470 + 0.970657i \(0.422699\pi\)
\(282\) 0 0
\(283\) 6.20592 0.368903 0.184452 0.982842i \(-0.440949\pi\)
0.184452 + 0.982842i \(0.440949\pi\)
\(284\) 0 0
\(285\) 67.4552 3.99570
\(286\) 0 0
\(287\) 1.98615 0.117239
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −5.90522 −0.346170
\(292\) 0 0
\(293\) −17.1148 −0.999857 −0.499928 0.866067i \(-0.666640\pi\)
−0.499928 + 0.866067i \(0.666640\pi\)
\(294\) 0 0
\(295\) 3.77561 0.219824
\(296\) 0 0
\(297\) −63.1667 −3.66531
\(298\) 0 0
\(299\) −6.72876 −0.389134
\(300\) 0 0
\(301\) −5.66065 −0.326275
\(302\) 0 0
\(303\) −41.0543 −2.35851
\(304\) 0 0
\(305\) −40.8387 −2.33842
\(306\) 0 0
\(307\) 3.64310 0.207923 0.103961 0.994581i \(-0.466848\pi\)
0.103961 + 0.994581i \(0.466848\pi\)
\(308\) 0 0
\(309\) 13.9501 0.793591
\(310\) 0 0
\(311\) −1.19420 −0.0677171 −0.0338586 0.999427i \(-0.510780\pi\)
−0.0338586 + 0.999427i \(0.510780\pi\)
\(312\) 0 0
\(313\) −32.6743 −1.84686 −0.923431 0.383764i \(-0.874628\pi\)
−0.923431 + 0.383764i \(0.874628\pi\)
\(314\) 0 0
\(315\) 29.5274 1.66368
\(316\) 0 0
\(317\) 6.97251 0.391615 0.195808 0.980642i \(-0.437267\pi\)
0.195808 + 0.980642i \(0.437267\pi\)
\(318\) 0 0
\(319\) −22.5816 −1.26433
\(320\) 0 0
\(321\) 57.5459 3.21190
\(322\) 0 0
\(323\) −5.85333 −0.325688
\(324\) 0 0
\(325\) 12.9999 0.721106
\(326\) 0 0
\(327\) 40.4718 2.23809
\(328\) 0 0
\(329\) −12.9436 −0.713606
\(330\) 0 0
\(331\) −15.0038 −0.824686 −0.412343 0.911029i \(-0.635289\pi\)
−0.412343 + 0.911029i \(0.635289\pi\)
\(332\) 0 0
\(333\) 26.8638 1.47213
\(334\) 0 0
\(335\) 11.9419 0.652458
\(336\) 0 0
\(337\) −25.9217 −1.41204 −0.706021 0.708190i \(-0.749512\pi\)
−0.706021 + 0.708190i \(0.749512\pi\)
\(338\) 0 0
\(339\) 23.0345 1.25106
\(340\) 0 0
\(341\) −44.9330 −2.43326
\(342\) 0 0
\(343\) 15.4357 0.833452
\(344\) 0 0
\(345\) 55.2068 2.97223
\(346\) 0 0
\(347\) 15.3870 0.826020 0.413010 0.910727i \(-0.364478\pi\)
0.413010 + 0.910727i \(0.364478\pi\)
\(348\) 0 0
\(349\) 24.5559 1.31444 0.657222 0.753697i \(-0.271731\pi\)
0.657222 + 0.753697i \(0.271731\pi\)
\(350\) 0 0
\(351\) 14.2186 0.758930
\(352\) 0 0
\(353\) −10.9374 −0.582139 −0.291070 0.956702i \(-0.594011\pi\)
−0.291070 + 0.956702i \(0.594011\pi\)
\(354\) 0 0
\(355\) −13.2583 −0.703676
\(356\) 0 0
\(357\) −3.77911 −0.200012
\(358\) 0 0
\(359\) −22.8515 −1.20606 −0.603028 0.797720i \(-0.706039\pi\)
−0.603028 + 0.797720i \(0.706039\pi\)
\(360\) 0 0
\(361\) 15.2615 0.803235
\(362\) 0 0
\(363\) 85.2752 4.47579
\(364\) 0 0
\(365\) 58.1858 3.04558
\(366\) 0 0
\(367\) 4.00596 0.209109 0.104555 0.994519i \(-0.466658\pi\)
0.104555 + 0.994519i \(0.466658\pi\)
\(368\) 0 0
\(369\) −10.1326 −0.527484
\(370\) 0 0
\(371\) −4.22083 −0.219135
\(372\) 0 0
\(373\) 2.89796 0.150051 0.0750254 0.997182i \(-0.476096\pi\)
0.0750254 + 0.997182i \(0.476096\pi\)
\(374\) 0 0
\(375\) −49.0381 −2.53231
\(376\) 0 0
\(377\) 5.08302 0.261789
\(378\) 0 0
\(379\) 5.02615 0.258176 0.129088 0.991633i \(-0.458795\pi\)
0.129088 + 0.991633i \(0.458795\pi\)
\(380\) 0 0
\(381\) −43.0945 −2.20780
\(382\) 0 0
\(383\) −12.7587 −0.651941 −0.325971 0.945380i \(-0.605691\pi\)
−0.325971 + 0.945380i \(0.605691\pi\)
\(384\) 0 0
\(385\) −29.1701 −1.48665
\(386\) 0 0
\(387\) 28.8787 1.46798
\(388\) 0 0
\(389\) 1.18517 0.0600907 0.0300453 0.999549i \(-0.490435\pi\)
0.0300453 + 0.999549i \(0.490435\pi\)
\(390\) 0 0
\(391\) −4.79049 −0.242266
\(392\) 0 0
\(393\) 37.9782 1.91575
\(394\) 0 0
\(395\) −18.8156 −0.946717
\(396\) 0 0
\(397\) −13.3590 −0.670471 −0.335235 0.942134i \(-0.608816\pi\)
−0.335235 + 0.942134i \(0.608816\pi\)
\(398\) 0 0
\(399\) 22.1204 1.10740
\(400\) 0 0
\(401\) −26.2766 −1.31219 −0.656094 0.754679i \(-0.727793\pi\)
−0.656094 + 0.754679i \(0.727793\pi\)
\(402\) 0 0
\(403\) 10.1142 0.503826
\(404\) 0 0
\(405\) −45.1122 −2.24164
\(406\) 0 0
\(407\) −26.5388 −1.31548
\(408\) 0 0
\(409\) 30.8658 1.52622 0.763109 0.646270i \(-0.223672\pi\)
0.763109 + 0.646270i \(0.223672\pi\)
\(410\) 0 0
\(411\) 54.2131 2.67414
\(412\) 0 0
\(413\) 1.23812 0.0609240
\(414\) 0 0
\(415\) 5.94390 0.291774
\(416\) 0 0
\(417\) 53.3334 2.61175
\(418\) 0 0
\(419\) 28.5466 1.39459 0.697295 0.716784i \(-0.254387\pi\)
0.697295 + 0.716784i \(0.254387\pi\)
\(420\) 0 0
\(421\) −13.9895 −0.681806 −0.340903 0.940099i \(-0.610733\pi\)
−0.340903 + 0.940099i \(0.610733\pi\)
\(422\) 0 0
\(423\) 66.0339 3.21068
\(424\) 0 0
\(425\) 9.25521 0.448944
\(426\) 0 0
\(427\) −13.3921 −0.648089
\(428\) 0 0
\(429\) −26.7527 −1.29163
\(430\) 0 0
\(431\) −4.67821 −0.225341 −0.112671 0.993632i \(-0.535941\pi\)
−0.112671 + 0.993632i \(0.535941\pi\)
\(432\) 0 0
\(433\) 22.3370 1.07345 0.536723 0.843758i \(-0.319662\pi\)
0.536723 + 0.843758i \(0.319662\pi\)
\(434\) 0 0
\(435\) −41.7042 −1.99956
\(436\) 0 0
\(437\) 28.0403 1.34135
\(438\) 0 0
\(439\) −11.2740 −0.538079 −0.269039 0.963129i \(-0.586706\pi\)
−0.269039 + 0.963129i \(0.586706\pi\)
\(440\) 0 0
\(441\) −34.5324 −1.64440
\(442\) 0 0
\(443\) 16.8230 0.799283 0.399642 0.916671i \(-0.369135\pi\)
0.399642 + 0.916671i \(0.369135\pi\)
\(444\) 0 0
\(445\) 39.5050 1.87272
\(446\) 0 0
\(447\) −64.7914 −3.06453
\(448\) 0 0
\(449\) 7.86623 0.371230 0.185615 0.982623i \(-0.440572\pi\)
0.185615 + 0.982623i \(0.440572\pi\)
\(450\) 0 0
\(451\) 10.0100 0.471354
\(452\) 0 0
\(453\) 66.1626 3.10859
\(454\) 0 0
\(455\) 6.56606 0.307822
\(456\) 0 0
\(457\) −41.6990 −1.95060 −0.975300 0.220886i \(-0.929105\pi\)
−0.975300 + 0.220886i \(0.929105\pi\)
\(458\) 0 0
\(459\) 10.1228 0.472492
\(460\) 0 0
\(461\) 10.0371 0.467472 0.233736 0.972300i \(-0.424905\pi\)
0.233736 + 0.972300i \(0.424905\pi\)
\(462\) 0 0
\(463\) −3.58615 −0.166662 −0.0833312 0.996522i \(-0.526556\pi\)
−0.0833312 + 0.996522i \(0.526556\pi\)
\(464\) 0 0
\(465\) −82.9833 −3.84826
\(466\) 0 0
\(467\) 15.9734 0.739163 0.369581 0.929198i \(-0.379501\pi\)
0.369581 + 0.929198i \(0.379501\pi\)
\(468\) 0 0
\(469\) 3.91608 0.180828
\(470\) 0 0
\(471\) 61.5520 2.83617
\(472\) 0 0
\(473\) −28.5293 −1.31178
\(474\) 0 0
\(475\) −54.1738 −2.48566
\(476\) 0 0
\(477\) 21.5332 0.985937
\(478\) 0 0
\(479\) −22.8787 −1.04536 −0.522678 0.852530i \(-0.675067\pi\)
−0.522678 + 0.852530i \(0.675067\pi\)
\(480\) 0 0
\(481\) 5.97377 0.272381
\(482\) 0 0
\(483\) 18.1038 0.823751
\(484\) 0 0
\(485\) 7.30461 0.331685
\(486\) 0 0
\(487\) 29.9081 1.35527 0.677633 0.735400i \(-0.263006\pi\)
0.677633 + 0.735400i \(0.263006\pi\)
\(488\) 0 0
\(489\) 33.6710 1.52265
\(490\) 0 0
\(491\) 29.4620 1.32960 0.664800 0.747022i \(-0.268517\pi\)
0.664800 + 0.747022i \(0.268517\pi\)
\(492\) 0 0
\(493\) 3.61882 0.162984
\(494\) 0 0
\(495\) 148.816 6.68876
\(496\) 0 0
\(497\) −4.34774 −0.195023
\(498\) 0 0
\(499\) 39.9948 1.79041 0.895207 0.445650i \(-0.147028\pi\)
0.895207 + 0.445650i \(0.147028\pi\)
\(500\) 0 0
\(501\) 44.5379 1.98981
\(502\) 0 0
\(503\) 31.2775 1.39460 0.697298 0.716782i \(-0.254386\pi\)
0.697298 + 0.716782i \(0.254386\pi\)
\(504\) 0 0
\(505\) 50.7831 2.25982
\(506\) 0 0
\(507\) −33.6578 −1.49480
\(508\) 0 0
\(509\) 24.5604 1.08862 0.544311 0.838884i \(-0.316791\pi\)
0.544311 + 0.838884i \(0.316791\pi\)
\(510\) 0 0
\(511\) 19.0807 0.844080
\(512\) 0 0
\(513\) −59.2521 −2.61605
\(514\) 0 0
\(515\) −17.2559 −0.760385
\(516\) 0 0
\(517\) −65.2349 −2.86903
\(518\) 0 0
\(519\) 48.4107 2.12499
\(520\) 0 0
\(521\) −20.1909 −0.884581 −0.442291 0.896872i \(-0.645834\pi\)
−0.442291 + 0.896872i \(0.645834\pi\)
\(522\) 0 0
\(523\) 18.4985 0.808882 0.404441 0.914564i \(-0.367466\pi\)
0.404441 + 0.914564i \(0.367466\pi\)
\(524\) 0 0
\(525\) −34.9764 −1.52650
\(526\) 0 0
\(527\) 7.20076 0.313670
\(528\) 0 0
\(529\) −0.0511709 −0.00222482
\(530\) 0 0
\(531\) −6.31647 −0.274111
\(532\) 0 0
\(533\) −2.25322 −0.0975976
\(534\) 0 0
\(535\) −71.1828 −3.07750
\(536\) 0 0
\(537\) 41.7294 1.80076
\(538\) 0 0
\(539\) 34.1146 1.46942
\(540\) 0 0
\(541\) 11.3867 0.489554 0.244777 0.969579i \(-0.421285\pi\)
0.244777 + 0.969579i \(0.421285\pi\)
\(542\) 0 0
\(543\) 29.9818 1.28664
\(544\) 0 0
\(545\) −50.0626 −2.14445
\(546\) 0 0
\(547\) −9.22381 −0.394382 −0.197191 0.980365i \(-0.563182\pi\)
−0.197191 + 0.980365i \(0.563182\pi\)
\(548\) 0 0
\(549\) 68.3218 2.91590
\(550\) 0 0
\(551\) −21.1822 −0.902391
\(552\) 0 0
\(553\) −6.17015 −0.262381
\(554\) 0 0
\(555\) −49.0125 −2.08046
\(556\) 0 0
\(557\) −0.696560 −0.0295142 −0.0147571 0.999891i \(-0.504698\pi\)
−0.0147571 + 0.999891i \(0.504698\pi\)
\(558\) 0 0
\(559\) 6.42181 0.271614
\(560\) 0 0
\(561\) −19.0464 −0.804140
\(562\) 0 0
\(563\) 6.27115 0.264297 0.132149 0.991230i \(-0.457812\pi\)
0.132149 + 0.991230i \(0.457812\pi\)
\(564\) 0 0
\(565\) −28.4932 −1.19872
\(566\) 0 0
\(567\) −14.7935 −0.621269
\(568\) 0 0
\(569\) 15.8454 0.664275 0.332138 0.943231i \(-0.392230\pi\)
0.332138 + 0.943231i \(0.392230\pi\)
\(570\) 0 0
\(571\) −33.0789 −1.38431 −0.692154 0.721750i \(-0.743338\pi\)
−0.692154 + 0.721750i \(0.743338\pi\)
\(572\) 0 0
\(573\) −70.3940 −2.94075
\(574\) 0 0
\(575\) −44.3370 −1.84898
\(576\) 0 0
\(577\) 10.6859 0.444860 0.222430 0.974949i \(-0.428601\pi\)
0.222430 + 0.974949i \(0.428601\pi\)
\(578\) 0 0
\(579\) −36.0696 −1.49900
\(580\) 0 0
\(581\) 1.94916 0.0808649
\(582\) 0 0
\(583\) −21.2727 −0.881023
\(584\) 0 0
\(585\) −33.4977 −1.38496
\(586\) 0 0
\(587\) −9.22005 −0.380552 −0.190276 0.981731i \(-0.560938\pi\)
−0.190276 + 0.981731i \(0.560938\pi\)
\(588\) 0 0
\(589\) −42.1484 −1.73670
\(590\) 0 0
\(591\) −28.5208 −1.17319
\(592\) 0 0
\(593\) −28.7429 −1.18033 −0.590165 0.807283i \(-0.700937\pi\)
−0.590165 + 0.807283i \(0.700937\pi\)
\(594\) 0 0
\(595\) 4.67466 0.191643
\(596\) 0 0
\(597\) −21.6316 −0.885321
\(598\) 0 0
\(599\) −41.8242 −1.70889 −0.854446 0.519540i \(-0.826103\pi\)
−0.854446 + 0.519540i \(0.826103\pi\)
\(600\) 0 0
\(601\) 30.2501 1.23393 0.616963 0.786992i \(-0.288363\pi\)
0.616963 + 0.786992i \(0.288363\pi\)
\(602\) 0 0
\(603\) −19.9785 −0.813586
\(604\) 0 0
\(605\) −105.483 −4.28851
\(606\) 0 0
\(607\) 3.32219 0.134844 0.0674218 0.997725i \(-0.478523\pi\)
0.0674218 + 0.997725i \(0.478523\pi\)
\(608\) 0 0
\(609\) −13.6759 −0.554177
\(610\) 0 0
\(611\) 14.6841 0.594055
\(612\) 0 0
\(613\) 12.5542 0.507061 0.253531 0.967327i \(-0.418408\pi\)
0.253531 + 0.967327i \(0.418408\pi\)
\(614\) 0 0
\(615\) 18.4868 0.745458
\(616\) 0 0
\(617\) −13.0224 −0.524261 −0.262131 0.965032i \(-0.584425\pi\)
−0.262131 + 0.965032i \(0.584425\pi\)
\(618\) 0 0
\(619\) −8.48932 −0.341215 −0.170607 0.985339i \(-0.554573\pi\)
−0.170607 + 0.985339i \(0.554573\pi\)
\(620\) 0 0
\(621\) −48.4933 −1.94597
\(622\) 0 0
\(623\) 12.9547 0.519021
\(624\) 0 0
\(625\) 14.3828 0.575314
\(626\) 0 0
\(627\) 111.485 4.45228
\(628\) 0 0
\(629\) 4.25299 0.169578
\(630\) 0 0
\(631\) −17.7118 −0.705096 −0.352548 0.935794i \(-0.614685\pi\)
−0.352548 + 0.935794i \(0.614685\pi\)
\(632\) 0 0
\(633\) 71.2602 2.83234
\(634\) 0 0
\(635\) 53.3068 2.11542
\(636\) 0 0
\(637\) −7.67905 −0.304255
\(638\) 0 0
\(639\) 22.1806 0.877453
\(640\) 0 0
\(641\) 13.3703 0.528096 0.264048 0.964510i \(-0.414942\pi\)
0.264048 + 0.964510i \(0.414942\pi\)
\(642\) 0 0
\(643\) 14.2822 0.563237 0.281618 0.959526i \(-0.409129\pi\)
0.281618 + 0.959526i \(0.409129\pi\)
\(644\) 0 0
\(645\) −52.6884 −2.07461
\(646\) 0 0
\(647\) 22.4578 0.882908 0.441454 0.897284i \(-0.354463\pi\)
0.441454 + 0.897284i \(0.354463\pi\)
\(648\) 0 0
\(649\) 6.24004 0.244943
\(650\) 0 0
\(651\) −27.2124 −1.06654
\(652\) 0 0
\(653\) −10.6424 −0.416471 −0.208236 0.978079i \(-0.566772\pi\)
−0.208236 + 0.978079i \(0.566772\pi\)
\(654\) 0 0
\(655\) −46.9781 −1.83559
\(656\) 0 0
\(657\) −97.3429 −3.79771
\(658\) 0 0
\(659\) −43.8459 −1.70799 −0.853996 0.520279i \(-0.825828\pi\)
−0.853996 + 0.520279i \(0.825828\pi\)
\(660\) 0 0
\(661\) −21.7057 −0.844255 −0.422128 0.906536i \(-0.638717\pi\)
−0.422128 + 0.906536i \(0.638717\pi\)
\(662\) 0 0
\(663\) 4.28726 0.166503
\(664\) 0 0
\(665\) −27.3624 −1.06107
\(666\) 0 0
\(667\) −17.3360 −0.671251
\(668\) 0 0
\(669\) 59.1640 2.28741
\(670\) 0 0
\(671\) −67.4951 −2.60562
\(672\) 0 0
\(673\) 31.1139 1.19935 0.599676 0.800243i \(-0.295296\pi\)
0.599676 + 0.800243i \(0.295296\pi\)
\(674\) 0 0
\(675\) 93.6887 3.60608
\(676\) 0 0
\(677\) 13.5726 0.521638 0.260819 0.965388i \(-0.416007\pi\)
0.260819 + 0.965388i \(0.416007\pi\)
\(678\) 0 0
\(679\) 2.39538 0.0919261
\(680\) 0 0
\(681\) 48.7892 1.86961
\(682\) 0 0
\(683\) 19.5991 0.749939 0.374970 0.927037i \(-0.377653\pi\)
0.374970 + 0.927037i \(0.377653\pi\)
\(684\) 0 0
\(685\) −67.0603 −2.56224
\(686\) 0 0
\(687\) 45.0250 1.71781
\(688\) 0 0
\(689\) 4.78838 0.182423
\(690\) 0 0
\(691\) −27.3409 −1.04010 −0.520049 0.854137i \(-0.674086\pi\)
−0.520049 + 0.854137i \(0.674086\pi\)
\(692\) 0 0
\(693\) 48.8006 1.85378
\(694\) 0 0
\(695\) −65.9722 −2.50247
\(696\) 0 0
\(697\) −1.60416 −0.0607620
\(698\) 0 0
\(699\) −84.1233 −3.18184
\(700\) 0 0
\(701\) −45.1033 −1.70353 −0.851764 0.523926i \(-0.824467\pi\)
−0.851764 + 0.523926i \(0.824467\pi\)
\(702\) 0 0
\(703\) −24.8941 −0.938900
\(704\) 0 0
\(705\) −120.477 −4.53744
\(706\) 0 0
\(707\) 16.6531 0.626306
\(708\) 0 0
\(709\) 18.7392 0.703764 0.351882 0.936044i \(-0.385542\pi\)
0.351882 + 0.936044i \(0.385542\pi\)
\(710\) 0 0
\(711\) 31.4779 1.18051
\(712\) 0 0
\(713\) −34.4952 −1.29186
\(714\) 0 0
\(715\) 33.0924 1.23759
\(716\) 0 0
\(717\) −2.33706 −0.0872791
\(718\) 0 0
\(719\) 24.6072 0.917694 0.458847 0.888515i \(-0.348263\pi\)
0.458847 + 0.888515i \(0.348263\pi\)
\(720\) 0 0
\(721\) −5.65867 −0.210740
\(722\) 0 0
\(723\) −14.9449 −0.555807
\(724\) 0 0
\(725\) 33.4930 1.24390
\(726\) 0 0
\(727\) −8.73767 −0.324062 −0.162031 0.986786i \(-0.551805\pi\)
−0.162031 + 0.986786i \(0.551805\pi\)
\(728\) 0 0
\(729\) −17.2219 −0.637850
\(730\) 0 0
\(731\) 4.57197 0.169100
\(732\) 0 0
\(733\) 51.2644 1.89349 0.946746 0.321980i \(-0.104348\pi\)
0.946746 + 0.321980i \(0.104348\pi\)
\(734\) 0 0
\(735\) 63.0036 2.32392
\(736\) 0 0
\(737\) 19.7367 0.727012
\(738\) 0 0
\(739\) 11.2275 0.413012 0.206506 0.978445i \(-0.433791\pi\)
0.206506 + 0.978445i \(0.433791\pi\)
\(740\) 0 0
\(741\) −25.0948 −0.921879
\(742\) 0 0
\(743\) 30.2024 1.10802 0.554009 0.832511i \(-0.313097\pi\)
0.554009 + 0.832511i \(0.313097\pi\)
\(744\) 0 0
\(745\) 80.1453 2.93630
\(746\) 0 0
\(747\) −9.94394 −0.363830
\(748\) 0 0
\(749\) −23.3428 −0.852926
\(750\) 0 0
\(751\) 8.36633 0.305292 0.152646 0.988281i \(-0.451221\pi\)
0.152646 + 0.988281i \(0.451221\pi\)
\(752\) 0 0
\(753\) −79.6304 −2.90189
\(754\) 0 0
\(755\) −81.8415 −2.97852
\(756\) 0 0
\(757\) 10.9573 0.398249 0.199125 0.979974i \(-0.436190\pi\)
0.199125 + 0.979974i \(0.436190\pi\)
\(758\) 0 0
\(759\) 91.2417 3.31186
\(760\) 0 0
\(761\) −10.9348 −0.396385 −0.198192 0.980163i \(-0.563507\pi\)
−0.198192 + 0.980163i \(0.563507\pi\)
\(762\) 0 0
\(763\) −16.4169 −0.594331
\(764\) 0 0
\(765\) −23.8485 −0.862244
\(766\) 0 0
\(767\) −1.40461 −0.0507174
\(768\) 0 0
\(769\) −1.86900 −0.0673979 −0.0336989 0.999432i \(-0.510729\pi\)
−0.0336989 + 0.999432i \(0.510729\pi\)
\(770\) 0 0
\(771\) −37.3205 −1.34407
\(772\) 0 0
\(773\) −12.9374 −0.465324 −0.232662 0.972558i \(-0.574744\pi\)
−0.232662 + 0.972558i \(0.574744\pi\)
\(774\) 0 0
\(775\) 66.6445 2.39394
\(776\) 0 0
\(777\) −16.0725 −0.576598
\(778\) 0 0
\(779\) 9.38969 0.336421
\(780\) 0 0
\(781\) −21.9123 −0.784083
\(782\) 0 0
\(783\) 36.6327 1.30914
\(784\) 0 0
\(785\) −76.1384 −2.71750
\(786\) 0 0
\(787\) 43.7898 1.56094 0.780469 0.625195i \(-0.214980\pi\)
0.780469 + 0.625195i \(0.214980\pi\)
\(788\) 0 0
\(789\) 30.2655 1.07748
\(790\) 0 0
\(791\) −9.34367 −0.332223
\(792\) 0 0
\(793\) 15.1929 0.539514
\(794\) 0 0
\(795\) −39.2868 −1.39336
\(796\) 0 0
\(797\) −0.281773 −0.00998092 −0.00499046 0.999988i \(-0.501589\pi\)
−0.00499046 + 0.999988i \(0.501589\pi\)
\(798\) 0 0
\(799\) 10.4542 0.369845
\(800\) 0 0
\(801\) −66.0906 −2.33519
\(802\) 0 0
\(803\) 96.1652 3.39359
\(804\) 0 0
\(805\) −22.3940 −0.789283
\(806\) 0 0
\(807\) −37.0910 −1.30567
\(808\) 0 0
\(809\) 0.884554 0.0310993 0.0155496 0.999879i \(-0.495050\pi\)
0.0155496 + 0.999879i \(0.495050\pi\)
\(810\) 0 0
\(811\) 29.5777 1.03862 0.519308 0.854587i \(-0.326190\pi\)
0.519308 + 0.854587i \(0.326190\pi\)
\(812\) 0 0
\(813\) 4.13152 0.144899
\(814\) 0 0
\(815\) −41.6502 −1.45894
\(816\) 0 0
\(817\) −26.7612 −0.936257
\(818\) 0 0
\(819\) −10.9848 −0.383840
\(820\) 0 0
\(821\) −27.3064 −0.953002 −0.476501 0.879174i \(-0.658095\pi\)
−0.476501 + 0.879174i \(0.658095\pi\)
\(822\) 0 0
\(823\) −31.1330 −1.08523 −0.542614 0.839982i \(-0.682565\pi\)
−0.542614 + 0.839982i \(0.682565\pi\)
\(824\) 0 0
\(825\) −176.278 −6.13723
\(826\) 0 0
\(827\) 5.37772 0.187002 0.0935008 0.995619i \(-0.470194\pi\)
0.0935008 + 0.995619i \(0.470194\pi\)
\(828\) 0 0
\(829\) 45.5651 1.58254 0.791271 0.611466i \(-0.209420\pi\)
0.791271 + 0.611466i \(0.209420\pi\)
\(830\) 0 0
\(831\) 2.31343 0.0802521
\(832\) 0 0
\(833\) −5.46705 −0.189422
\(834\) 0 0
\(835\) −55.0923 −1.90655
\(836\) 0 0
\(837\) 72.8919 2.51951
\(838\) 0 0
\(839\) −17.6725 −0.610123 −0.305062 0.952333i \(-0.598677\pi\)
−0.305062 + 0.952333i \(0.598677\pi\)
\(840\) 0 0
\(841\) −15.9041 −0.548417
\(842\) 0 0
\(843\) 24.6076 0.847530
\(844\) 0 0
\(845\) 41.6339 1.43225
\(846\) 0 0
\(847\) −34.5908 −1.18855
\(848\) 0 0
\(849\) 18.9422 0.650096
\(850\) 0 0
\(851\) −20.3739 −0.698409
\(852\) 0 0
\(853\) 18.2776 0.625814 0.312907 0.949784i \(-0.398697\pi\)
0.312907 + 0.949784i \(0.398697\pi\)
\(854\) 0 0
\(855\) 139.593 4.77398
\(856\) 0 0
\(857\) −28.2976 −0.966627 −0.483313 0.875447i \(-0.660567\pi\)
−0.483313 + 0.875447i \(0.660567\pi\)
\(858\) 0 0
\(859\) −8.97217 −0.306126 −0.153063 0.988216i \(-0.548914\pi\)
−0.153063 + 0.988216i \(0.548914\pi\)
\(860\) 0 0
\(861\) 6.06230 0.206603
\(862\) 0 0
\(863\) 5.79476 0.197256 0.0986280 0.995124i \(-0.468555\pi\)
0.0986280 + 0.995124i \(0.468555\pi\)
\(864\) 0 0
\(865\) −59.8829 −2.03608
\(866\) 0 0
\(867\) 3.05229 0.103661
\(868\) 0 0
\(869\) −31.0971 −1.05490
\(870\) 0 0
\(871\) −4.44265 −0.150534
\(872\) 0 0
\(873\) −12.2204 −0.413597
\(874\) 0 0
\(875\) 19.8917 0.672461
\(876\) 0 0
\(877\) 53.2033 1.79655 0.898274 0.439437i \(-0.144822\pi\)
0.898274 + 0.439437i \(0.144822\pi\)
\(878\) 0 0
\(879\) −52.2393 −1.76199
\(880\) 0 0
\(881\) 49.5949 1.67090 0.835448 0.549570i \(-0.185208\pi\)
0.835448 + 0.549570i \(0.185208\pi\)
\(882\) 0 0
\(883\) 3.78067 0.127230 0.0636149 0.997975i \(-0.479737\pi\)
0.0636149 + 0.997975i \(0.479737\pi\)
\(884\) 0 0
\(885\) 11.5242 0.387383
\(886\) 0 0
\(887\) −1.03510 −0.0347551 −0.0173776 0.999849i \(-0.505532\pi\)
−0.0173776 + 0.999849i \(0.505532\pi\)
\(888\) 0 0
\(889\) 17.4807 0.586285
\(890\) 0 0
\(891\) −74.5581 −2.49779
\(892\) 0 0
\(893\) −61.1921 −2.04772
\(894\) 0 0
\(895\) −51.6183 −1.72541
\(896\) 0 0
\(897\) −20.5381 −0.685747
\(898\) 0 0
\(899\) 26.0583 0.869093
\(900\) 0 0
\(901\) 3.40906 0.113572
\(902\) 0 0
\(903\) −17.2780 −0.574974
\(904\) 0 0
\(905\) −37.0868 −1.23281
\(906\) 0 0
\(907\) −1.34934 −0.0448040 −0.0224020 0.999749i \(-0.507131\pi\)
−0.0224020 + 0.999749i \(0.507131\pi\)
\(908\) 0 0
\(909\) −84.9585 −2.81790
\(910\) 0 0
\(911\) 45.1573 1.49613 0.748064 0.663626i \(-0.230984\pi\)
0.748064 + 0.663626i \(0.230984\pi\)
\(912\) 0 0
\(913\) 9.82363 0.325115
\(914\) 0 0
\(915\) −124.651 −4.12085
\(916\) 0 0
\(917\) −15.4054 −0.508731
\(918\) 0 0
\(919\) −16.8588 −0.556119 −0.278060 0.960564i \(-0.589691\pi\)
−0.278060 + 0.960564i \(0.589691\pi\)
\(920\) 0 0
\(921\) 11.1198 0.366409
\(922\) 0 0
\(923\) 4.93236 0.162350
\(924\) 0 0
\(925\) 39.3623 1.29422
\(926\) 0 0
\(927\) 28.8685 0.948167
\(928\) 0 0
\(929\) −22.9557 −0.753152 −0.376576 0.926386i \(-0.622899\pi\)
−0.376576 + 0.926386i \(0.622899\pi\)
\(930\) 0 0
\(931\) 32.0005 1.04877
\(932\) 0 0
\(933\) −3.64506 −0.119334
\(934\) 0 0
\(935\) 23.5599 0.770493
\(936\) 0 0
\(937\) −40.3147 −1.31703 −0.658513 0.752570i \(-0.728814\pi\)
−0.658513 + 0.752570i \(0.728814\pi\)
\(938\) 0 0
\(939\) −99.7315 −3.25461
\(940\) 0 0
\(941\) −40.2320 −1.31153 −0.655763 0.754966i \(-0.727653\pi\)
−0.655763 + 0.754966i \(0.727653\pi\)
\(942\) 0 0
\(943\) 7.68473 0.250249
\(944\) 0 0
\(945\) 47.3207 1.53934
\(946\) 0 0
\(947\) 49.0097 1.59260 0.796300 0.604902i \(-0.206788\pi\)
0.796300 + 0.604902i \(0.206788\pi\)
\(948\) 0 0
\(949\) −21.6464 −0.702671
\(950\) 0 0
\(951\) 21.2821 0.690120
\(952\) 0 0
\(953\) 18.1255 0.587142 0.293571 0.955937i \(-0.405156\pi\)
0.293571 + 0.955937i \(0.405156\pi\)
\(954\) 0 0
\(955\) 87.0756 2.81770
\(956\) 0 0
\(957\) −68.9256 −2.22805
\(958\) 0 0
\(959\) −21.9909 −0.710122
\(960\) 0 0
\(961\) 20.8509 0.672611
\(962\) 0 0
\(963\) 119.087 3.83751
\(964\) 0 0
\(965\) 44.6173 1.43628
\(966\) 0 0
\(967\) 54.7068 1.75925 0.879625 0.475667i \(-0.157793\pi\)
0.879625 + 0.475667i \(0.157793\pi\)
\(968\) 0 0
\(969\) −17.8661 −0.573940
\(970\) 0 0
\(971\) −35.9133 −1.15251 −0.576256 0.817269i \(-0.695487\pi\)
−0.576256 + 0.817269i \(0.695487\pi\)
\(972\) 0 0
\(973\) −21.6340 −0.693555
\(974\) 0 0
\(975\) 39.6795 1.27076
\(976\) 0 0
\(977\) 12.7512 0.407948 0.203974 0.978976i \(-0.434614\pi\)
0.203974 + 0.978976i \(0.434614\pi\)
\(978\) 0 0
\(979\) 65.2909 2.08671
\(980\) 0 0
\(981\) 83.7531 2.67403
\(982\) 0 0
\(983\) −32.4439 −1.03480 −0.517400 0.855744i \(-0.673100\pi\)
−0.517400 + 0.855744i \(0.673100\pi\)
\(984\) 0 0
\(985\) 35.2796 1.12410
\(986\) 0 0
\(987\) −39.5077 −1.25754
\(988\) 0 0
\(989\) −21.9020 −0.696442
\(990\) 0 0
\(991\) −4.35920 −0.138474 −0.0692372 0.997600i \(-0.522057\pi\)
−0.0692372 + 0.997600i \(0.522057\pi\)
\(992\) 0 0
\(993\) −45.7961 −1.45329
\(994\) 0 0
\(995\) 26.7577 0.848277
\(996\) 0 0
\(997\) −25.8139 −0.817533 −0.408767 0.912639i \(-0.634041\pi\)
−0.408767 + 0.912639i \(0.634041\pi\)
\(998\) 0 0
\(999\) 43.0522 1.36211
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 8024.2.a.v.1.16 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8024.2.a.v.1.16 18 1.1 even 1 trivial