Properties

Label 8024.2.a.v.1.3
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 1.3
Root \(3.16635\) 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)
 
\(f(q)\) \(=\) \(q-2.16635 q^{3} +3.18635 q^{5} +1.65768 q^{7} +1.69308 q^{9} +O(q^{10})\) \(q-2.16635 q^{3} +3.18635 q^{5} +1.65768 q^{7} +1.69308 q^{9} +1.87664 q^{11} -1.01378 q^{13} -6.90276 q^{15} +1.00000 q^{17} -1.11204 q^{19} -3.59111 q^{21} +1.18438 q^{23} +5.15283 q^{25} +2.83125 q^{27} -7.37401 q^{29} +10.6347 q^{31} -4.06546 q^{33} +5.28194 q^{35} +2.75358 q^{37} +2.19621 q^{39} -7.46162 q^{41} +3.91216 q^{43} +5.39475 q^{45} -10.3554 q^{47} -4.25210 q^{49} -2.16635 q^{51} +2.85631 q^{53} +5.97962 q^{55} +2.40907 q^{57} -1.00000 q^{59} +0.465944 q^{61} +2.80658 q^{63} -3.23027 q^{65} +12.9041 q^{67} -2.56579 q^{69} -2.88814 q^{71} +15.4871 q^{73} -11.1628 q^{75} +3.11086 q^{77} +15.3000 q^{79} -11.2127 q^{81} +13.9680 q^{83} +3.18635 q^{85} +15.9747 q^{87} +7.92874 q^{89} -1.68053 q^{91} -23.0385 q^{93} -3.54336 q^{95} -8.06172 q^{97} +3.17730 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9}+O(q^{10}) \) 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
\(3\) −2.16635 −1.25074 −0.625372 0.780327i \(-0.715053\pi\)
−0.625372 + 0.780327i \(0.715053\pi\)
\(4\) 0 0
\(5\) 3.18635 1.42498 0.712490 0.701683i \(-0.247568\pi\)
0.712490 + 0.701683i \(0.247568\pi\)
\(6\) 0 0
\(7\) 1.65768 0.626544 0.313272 0.949664i \(-0.398575\pi\)
0.313272 + 0.949664i \(0.398575\pi\)
\(8\) 0 0
\(9\) 1.69308 0.564360
\(10\) 0 0
\(11\) 1.87664 0.565827 0.282914 0.959145i \(-0.408699\pi\)
0.282914 + 0.959145i \(0.408699\pi\)
\(12\) 0 0
\(13\) −1.01378 −0.281173 −0.140587 0.990068i \(-0.544899\pi\)
−0.140587 + 0.990068i \(0.544899\pi\)
\(14\) 0 0
\(15\) −6.90276 −1.78228
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) −1.11204 −0.255120 −0.127560 0.991831i \(-0.540715\pi\)
−0.127560 + 0.991831i \(0.540715\pi\)
\(20\) 0 0
\(21\) −3.59111 −0.783645
\(22\) 0 0
\(23\) 1.18438 0.246961 0.123481 0.992347i \(-0.460594\pi\)
0.123481 + 0.992347i \(0.460594\pi\)
\(24\) 0 0
\(25\) 5.15283 1.03057
\(26\) 0 0
\(27\) 2.83125 0.544874
\(28\) 0 0
\(29\) −7.37401 −1.36932 −0.684659 0.728863i \(-0.740049\pi\)
−0.684659 + 0.728863i \(0.740049\pi\)
\(30\) 0 0
\(31\) 10.6347 1.91005 0.955023 0.296533i \(-0.0958305\pi\)
0.955023 + 0.296533i \(0.0958305\pi\)
\(32\) 0 0
\(33\) −4.06546 −0.707705
\(34\) 0 0
\(35\) 5.28194 0.892811
\(36\) 0 0
\(37\) 2.75358 0.452686 0.226343 0.974048i \(-0.427323\pi\)
0.226343 + 0.974048i \(0.427323\pi\)
\(38\) 0 0
\(39\) 2.19621 0.351675
\(40\) 0 0
\(41\) −7.46162 −1.16531 −0.582654 0.812720i \(-0.697986\pi\)
−0.582654 + 0.812720i \(0.697986\pi\)
\(42\) 0 0
\(43\) 3.91216 0.596600 0.298300 0.954472i \(-0.403580\pi\)
0.298300 + 0.954472i \(0.403580\pi\)
\(44\) 0 0
\(45\) 5.39475 0.804201
\(46\) 0 0
\(47\) −10.3554 −1.51049 −0.755245 0.655443i \(-0.772482\pi\)
−0.755245 + 0.655443i \(0.772482\pi\)
\(48\) 0 0
\(49\) −4.25210 −0.607443
\(50\) 0 0
\(51\) −2.16635 −0.303350
\(52\) 0 0
\(53\) 2.85631 0.392345 0.196172 0.980569i \(-0.437149\pi\)
0.196172 + 0.980569i \(0.437149\pi\)
\(54\) 0 0
\(55\) 5.97962 0.806292
\(56\) 0 0
\(57\) 2.40907 0.319090
\(58\) 0 0
\(59\) −1.00000 −0.130189
\(60\) 0 0
\(61\) 0.465944 0.0596580 0.0298290 0.999555i \(-0.490504\pi\)
0.0298290 + 0.999555i \(0.490504\pi\)
\(62\) 0 0
\(63\) 2.80658 0.353596
\(64\) 0 0
\(65\) −3.23027 −0.400666
\(66\) 0 0
\(67\) 12.9041 1.57648 0.788241 0.615367i \(-0.210992\pi\)
0.788241 + 0.615367i \(0.210992\pi\)
\(68\) 0 0
\(69\) −2.56579 −0.308885
\(70\) 0 0
\(71\) −2.88814 −0.342759 −0.171379 0.985205i \(-0.554822\pi\)
−0.171379 + 0.985205i \(0.554822\pi\)
\(72\) 0 0
\(73\) 15.4871 1.81263 0.906313 0.422606i \(-0.138885\pi\)
0.906313 + 0.422606i \(0.138885\pi\)
\(74\) 0 0
\(75\) −11.1628 −1.28897
\(76\) 0 0
\(77\) 3.11086 0.354515
\(78\) 0 0
\(79\) 15.3000 1.72139 0.860693 0.509125i \(-0.170031\pi\)
0.860693 + 0.509125i \(0.170031\pi\)
\(80\) 0 0
\(81\) −11.2127 −1.24586
\(82\) 0 0
\(83\) 13.9680 1.53319 0.766595 0.642131i \(-0.221949\pi\)
0.766595 + 0.642131i \(0.221949\pi\)
\(84\) 0 0
\(85\) 3.18635 0.345608
\(86\) 0 0
\(87\) 15.9747 1.71267
\(88\) 0 0
\(89\) 7.92874 0.840444 0.420222 0.907421i \(-0.361952\pi\)
0.420222 + 0.907421i \(0.361952\pi\)
\(90\) 0 0
\(91\) −1.68053 −0.176167
\(92\) 0 0
\(93\) −23.0385 −2.38898
\(94\) 0 0
\(95\) −3.54336 −0.363541
\(96\) 0 0
\(97\) −8.06172 −0.818544 −0.409272 0.912412i \(-0.634217\pi\)
−0.409272 + 0.912412i \(0.634217\pi\)
\(98\) 0 0
\(99\) 3.17730 0.319330
\(100\) 0 0
\(101\) 0.969119 0.0964309 0.0482154 0.998837i \(-0.484647\pi\)
0.0482154 + 0.998837i \(0.484647\pi\)
\(102\) 0 0
\(103\) 9.07840 0.894522 0.447261 0.894404i \(-0.352400\pi\)
0.447261 + 0.894404i \(0.352400\pi\)
\(104\) 0 0
\(105\) −11.4425 −1.11668
\(106\) 0 0
\(107\) −13.0608 −1.26264 −0.631319 0.775523i \(-0.717486\pi\)
−0.631319 + 0.775523i \(0.717486\pi\)
\(108\) 0 0
\(109\) 4.47445 0.428575 0.214287 0.976771i \(-0.431257\pi\)
0.214287 + 0.976771i \(0.431257\pi\)
\(110\) 0 0
\(111\) −5.96523 −0.566195
\(112\) 0 0
\(113\) −0.268210 −0.0252311 −0.0126156 0.999920i \(-0.504016\pi\)
−0.0126156 + 0.999920i \(0.504016\pi\)
\(114\) 0 0
\(115\) 3.77386 0.351914
\(116\) 0 0
\(117\) −1.71642 −0.158683
\(118\) 0 0
\(119\) 1.65768 0.151959
\(120\) 0 0
\(121\) −7.47823 −0.679839
\(122\) 0 0
\(123\) 16.1645 1.45750
\(124\) 0 0
\(125\) 0.486961 0.0435551
\(126\) 0 0
\(127\) 13.7994 1.22450 0.612250 0.790665i \(-0.290265\pi\)
0.612250 + 0.790665i \(0.290265\pi\)
\(128\) 0 0
\(129\) −8.47513 −0.746193
\(130\) 0 0
\(131\) 8.52266 0.744628 0.372314 0.928107i \(-0.378564\pi\)
0.372314 + 0.928107i \(0.378564\pi\)
\(132\) 0 0
\(133\) −1.84341 −0.159844
\(134\) 0 0
\(135\) 9.02135 0.776434
\(136\) 0 0
\(137\) −0.0419670 −0.00358548 −0.00179274 0.999998i \(-0.500571\pi\)
−0.00179274 + 0.999998i \(0.500571\pi\)
\(138\) 0 0
\(139\) −2.72151 −0.230835 −0.115418 0.993317i \(-0.536821\pi\)
−0.115418 + 0.993317i \(0.536821\pi\)
\(140\) 0 0
\(141\) 22.4334 1.88924
\(142\) 0 0
\(143\) −1.90250 −0.159095
\(144\) 0 0
\(145\) −23.4962 −1.95125
\(146\) 0 0
\(147\) 9.21155 0.759756
\(148\) 0 0
\(149\) 1.17409 0.0961853 0.0480926 0.998843i \(-0.484686\pi\)
0.0480926 + 0.998843i \(0.484686\pi\)
\(150\) 0 0
\(151\) 6.63061 0.539591 0.269796 0.962918i \(-0.413044\pi\)
0.269796 + 0.962918i \(0.413044\pi\)
\(152\) 0 0
\(153\) 1.69308 0.136877
\(154\) 0 0
\(155\) 33.8858 2.72177
\(156\) 0 0
\(157\) 11.0668 0.883231 0.441615 0.897204i \(-0.354406\pi\)
0.441615 + 0.897204i \(0.354406\pi\)
\(158\) 0 0
\(159\) −6.18778 −0.490723
\(160\) 0 0
\(161\) 1.96333 0.154732
\(162\) 0 0
\(163\) −13.3442 −1.04520 −0.522599 0.852579i \(-0.675038\pi\)
−0.522599 + 0.852579i \(0.675038\pi\)
\(164\) 0 0
\(165\) −12.9540 −1.00846
\(166\) 0 0
\(167\) −20.6402 −1.59719 −0.798594 0.601871i \(-0.794422\pi\)
−0.798594 + 0.601871i \(0.794422\pi\)
\(168\) 0 0
\(169\) −11.9722 −0.920942
\(170\) 0 0
\(171\) −1.88278 −0.143980
\(172\) 0 0
\(173\) 3.23972 0.246312 0.123156 0.992387i \(-0.460699\pi\)
0.123156 + 0.992387i \(0.460699\pi\)
\(174\) 0 0
\(175\) 8.54173 0.645694
\(176\) 0 0
\(177\) 2.16635 0.162833
\(178\) 0 0
\(179\) 6.54149 0.488934 0.244467 0.969658i \(-0.421387\pi\)
0.244467 + 0.969658i \(0.421387\pi\)
\(180\) 0 0
\(181\) −6.11871 −0.454800 −0.227400 0.973801i \(-0.573022\pi\)
−0.227400 + 0.973801i \(0.573022\pi\)
\(182\) 0 0
\(183\) −1.00940 −0.0746169
\(184\) 0 0
\(185\) 8.77388 0.645069
\(186\) 0 0
\(187\) 1.87664 0.137233
\(188\) 0 0
\(189\) 4.69330 0.341387
\(190\) 0 0
\(191\) 22.1982 1.60620 0.803101 0.595842i \(-0.203182\pi\)
0.803101 + 0.595842i \(0.203182\pi\)
\(192\) 0 0
\(193\) 15.9317 1.14679 0.573394 0.819280i \(-0.305627\pi\)
0.573394 + 0.819280i \(0.305627\pi\)
\(194\) 0 0
\(195\) 6.99790 0.501130
\(196\) 0 0
\(197\) −15.5410 −1.10725 −0.553627 0.832765i \(-0.686757\pi\)
−0.553627 + 0.832765i \(0.686757\pi\)
\(198\) 0 0
\(199\) −3.59991 −0.255191 −0.127596 0.991826i \(-0.540726\pi\)
−0.127596 + 0.991826i \(0.540726\pi\)
\(200\) 0 0
\(201\) −27.9547 −1.97178
\(202\) 0 0
\(203\) −12.2237 −0.857938
\(204\) 0 0
\(205\) −23.7753 −1.66054
\(206\) 0 0
\(207\) 2.00526 0.139375
\(208\) 0 0
\(209\) −2.08690 −0.144354
\(210\) 0 0
\(211\) −8.31397 −0.572357 −0.286179 0.958176i \(-0.592385\pi\)
−0.286179 + 0.958176i \(0.592385\pi\)
\(212\) 0 0
\(213\) 6.25672 0.428703
\(214\) 0 0
\(215\) 12.4655 0.850142
\(216\) 0 0
\(217\) 17.6289 1.19673
\(218\) 0 0
\(219\) −33.5505 −2.26713
\(220\) 0 0
\(221\) −1.01378 −0.0681945
\(222\) 0 0
\(223\) −10.3942 −0.696049 −0.348025 0.937485i \(-0.613147\pi\)
−0.348025 + 0.937485i \(0.613147\pi\)
\(224\) 0 0
\(225\) 8.72415 0.581610
\(226\) 0 0
\(227\) −5.78914 −0.384239 −0.192119 0.981372i \(-0.561536\pi\)
−0.192119 + 0.981372i \(0.561536\pi\)
\(228\) 0 0
\(229\) 12.4176 0.820577 0.410288 0.911956i \(-0.365428\pi\)
0.410288 + 0.911956i \(0.365428\pi\)
\(230\) 0 0
\(231\) −6.73922 −0.443408
\(232\) 0 0
\(233\) 7.27800 0.476798 0.238399 0.971167i \(-0.423377\pi\)
0.238399 + 0.971167i \(0.423377\pi\)
\(234\) 0 0
\(235\) −32.9959 −2.15242
\(236\) 0 0
\(237\) −33.1452 −2.15301
\(238\) 0 0
\(239\) −16.8762 −1.09163 −0.545816 0.837905i \(-0.683780\pi\)
−0.545816 + 0.837905i \(0.683780\pi\)
\(240\) 0 0
\(241\) −9.98035 −0.642891 −0.321445 0.946928i \(-0.604169\pi\)
−0.321445 + 0.946928i \(0.604169\pi\)
\(242\) 0 0
\(243\) 15.7970 1.01338
\(244\) 0 0
\(245\) −13.5487 −0.865594
\(246\) 0 0
\(247\) 1.12737 0.0717329
\(248\) 0 0
\(249\) −30.2597 −1.91763
\(250\) 0 0
\(251\) 7.39332 0.466662 0.233331 0.972397i \(-0.425037\pi\)
0.233331 + 0.972397i \(0.425037\pi\)
\(252\) 0 0
\(253\) 2.22266 0.139737
\(254\) 0 0
\(255\) −6.90276 −0.432267
\(256\) 0 0
\(257\) −24.7605 −1.54452 −0.772260 0.635307i \(-0.780874\pi\)
−0.772260 + 0.635307i \(0.780874\pi\)
\(258\) 0 0
\(259\) 4.56456 0.283628
\(260\) 0 0
\(261\) −12.4848 −0.772789
\(262\) 0 0
\(263\) 8.12000 0.500701 0.250350 0.968155i \(-0.419454\pi\)
0.250350 + 0.968155i \(0.419454\pi\)
\(264\) 0 0
\(265\) 9.10121 0.559083
\(266\) 0 0
\(267\) −17.1764 −1.05118
\(268\) 0 0
\(269\) 25.7149 1.56787 0.783933 0.620845i \(-0.213210\pi\)
0.783933 + 0.620845i \(0.213210\pi\)
\(270\) 0 0
\(271\) −18.9557 −1.15147 −0.575737 0.817635i \(-0.695285\pi\)
−0.575737 + 0.817635i \(0.695285\pi\)
\(272\) 0 0
\(273\) 3.64061 0.220340
\(274\) 0 0
\(275\) 9.66999 0.583122
\(276\) 0 0
\(277\) 12.5029 0.751227 0.375613 0.926776i \(-0.377432\pi\)
0.375613 + 0.926776i \(0.377432\pi\)
\(278\) 0 0
\(279\) 18.0054 1.07795
\(280\) 0 0
\(281\) 27.7899 1.65781 0.828904 0.559391i \(-0.188965\pi\)
0.828904 + 0.559391i \(0.188965\pi\)
\(282\) 0 0
\(283\) −22.1837 −1.31868 −0.659342 0.751843i \(-0.729165\pi\)
−0.659342 + 0.751843i \(0.729165\pi\)
\(284\) 0 0
\(285\) 7.67616 0.454696
\(286\) 0 0
\(287\) −12.3690 −0.730117
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 17.4645 1.02379
\(292\) 0 0
\(293\) 13.8763 0.810660 0.405330 0.914170i \(-0.367157\pi\)
0.405330 + 0.914170i \(0.367157\pi\)
\(294\) 0 0
\(295\) −3.18635 −0.185516
\(296\) 0 0
\(297\) 5.31322 0.308305
\(298\) 0 0
\(299\) −1.20071 −0.0694388
\(300\) 0 0
\(301\) 6.48511 0.373796
\(302\) 0 0
\(303\) −2.09945 −0.120610
\(304\) 0 0
\(305\) 1.48466 0.0850114
\(306\) 0 0
\(307\) 25.1804 1.43712 0.718559 0.695466i \(-0.244802\pi\)
0.718559 + 0.695466i \(0.244802\pi\)
\(308\) 0 0
\(309\) −19.6670 −1.11882
\(310\) 0 0
\(311\) 25.2728 1.43309 0.716544 0.697542i \(-0.245723\pi\)
0.716544 + 0.697542i \(0.245723\pi\)
\(312\) 0 0
\(313\) −29.6955 −1.67849 −0.839245 0.543753i \(-0.817003\pi\)
−0.839245 + 0.543753i \(0.817003\pi\)
\(314\) 0 0
\(315\) 8.94276 0.503867
\(316\) 0 0
\(317\) −16.8039 −0.943801 −0.471901 0.881652i \(-0.656432\pi\)
−0.471901 + 0.881652i \(0.656432\pi\)
\(318\) 0 0
\(319\) −13.8383 −0.774798
\(320\) 0 0
\(321\) 28.2944 1.57924
\(322\) 0 0
\(323\) −1.11204 −0.0618757
\(324\) 0 0
\(325\) −5.22385 −0.289767
\(326\) 0 0
\(327\) −9.69323 −0.536037
\(328\) 0 0
\(329\) −17.1659 −0.946388
\(330\) 0 0
\(331\) 7.80132 0.428799 0.214400 0.976746i \(-0.431221\pi\)
0.214400 + 0.976746i \(0.431221\pi\)
\(332\) 0 0
\(333\) 4.66204 0.255478
\(334\) 0 0
\(335\) 41.1169 2.24645
\(336\) 0 0
\(337\) 5.82641 0.317385 0.158692 0.987328i \(-0.449272\pi\)
0.158692 + 0.987328i \(0.449272\pi\)
\(338\) 0 0
\(339\) 0.581038 0.0315577
\(340\) 0 0
\(341\) 19.9574 1.08076
\(342\) 0 0
\(343\) −18.6524 −1.00713
\(344\) 0 0
\(345\) −8.17551 −0.440155
\(346\) 0 0
\(347\) 23.6440 1.26928 0.634639 0.772809i \(-0.281149\pi\)
0.634639 + 0.772809i \(0.281149\pi\)
\(348\) 0 0
\(349\) −11.6135 −0.621659 −0.310829 0.950466i \(-0.600607\pi\)
−0.310829 + 0.950466i \(0.600607\pi\)
\(350\) 0 0
\(351\) −2.87027 −0.153204
\(352\) 0 0
\(353\) 7.73008 0.411431 0.205715 0.978612i \(-0.434048\pi\)
0.205715 + 0.978612i \(0.434048\pi\)
\(354\) 0 0
\(355\) −9.20262 −0.488424
\(356\) 0 0
\(357\) −3.59111 −0.190062
\(358\) 0 0
\(359\) −29.7309 −1.56913 −0.784567 0.620043i \(-0.787115\pi\)
−0.784567 + 0.620043i \(0.787115\pi\)
\(360\) 0 0
\(361\) −17.7634 −0.934914
\(362\) 0 0
\(363\) 16.2005 0.850305
\(364\) 0 0
\(365\) 49.3473 2.58296
\(366\) 0 0
\(367\) 16.5954 0.866270 0.433135 0.901329i \(-0.357407\pi\)
0.433135 + 0.901329i \(0.357407\pi\)
\(368\) 0 0
\(369\) −12.6331 −0.657654
\(370\) 0 0
\(371\) 4.73485 0.245821
\(372\) 0 0
\(373\) −12.4901 −0.646712 −0.323356 0.946277i \(-0.604811\pi\)
−0.323356 + 0.946277i \(0.604811\pi\)
\(374\) 0 0
\(375\) −1.05493 −0.0544763
\(376\) 0 0
\(377\) 7.47565 0.385016
\(378\) 0 0
\(379\) 5.41680 0.278242 0.139121 0.990275i \(-0.455572\pi\)
0.139121 + 0.990275i \(0.455572\pi\)
\(380\) 0 0
\(381\) −29.8944 −1.53153
\(382\) 0 0
\(383\) −19.6960 −1.00642 −0.503211 0.864164i \(-0.667848\pi\)
−0.503211 + 0.864164i \(0.667848\pi\)
\(384\) 0 0
\(385\) 9.91229 0.505177
\(386\) 0 0
\(387\) 6.62361 0.336697
\(388\) 0 0
\(389\) −9.45402 −0.479338 −0.239669 0.970855i \(-0.577039\pi\)
−0.239669 + 0.970855i \(0.577039\pi\)
\(390\) 0 0
\(391\) 1.18438 0.0598969
\(392\) 0 0
\(393\) −18.4631 −0.931339
\(394\) 0 0
\(395\) 48.7512 2.45294
\(396\) 0 0
\(397\) 35.9285 1.80320 0.901600 0.432570i \(-0.142393\pi\)
0.901600 + 0.432570i \(0.142393\pi\)
\(398\) 0 0
\(399\) 3.99347 0.199924
\(400\) 0 0
\(401\) −16.0663 −0.802313 −0.401156 0.916010i \(-0.631392\pi\)
−0.401156 + 0.916010i \(0.631392\pi\)
\(402\) 0 0
\(403\) −10.7813 −0.537053
\(404\) 0 0
\(405\) −35.7277 −1.77532
\(406\) 0 0
\(407\) 5.16748 0.256142
\(408\) 0 0
\(409\) 21.1364 1.04513 0.522565 0.852599i \(-0.324975\pi\)
0.522565 + 0.852599i \(0.324975\pi\)
\(410\) 0 0
\(411\) 0.0909153 0.00448452
\(412\) 0 0
\(413\) −1.65768 −0.0815690
\(414\) 0 0
\(415\) 44.5070 2.18476
\(416\) 0 0
\(417\) 5.89574 0.288716
\(418\) 0 0
\(419\) 31.8210 1.55456 0.777278 0.629158i \(-0.216600\pi\)
0.777278 + 0.629158i \(0.216600\pi\)
\(420\) 0 0
\(421\) 0.808973 0.0394269 0.0197135 0.999806i \(-0.493725\pi\)
0.0197135 + 0.999806i \(0.493725\pi\)
\(422\) 0 0
\(423\) −17.5325 −0.852460
\(424\) 0 0
\(425\) 5.15283 0.249949
\(426\) 0 0
\(427\) 0.772385 0.0373783
\(428\) 0 0
\(429\) 4.12149 0.198988
\(430\) 0 0
\(431\) 28.6050 1.37786 0.688928 0.724830i \(-0.258082\pi\)
0.688928 + 0.724830i \(0.258082\pi\)
\(432\) 0 0
\(433\) 17.4313 0.837696 0.418848 0.908056i \(-0.362434\pi\)
0.418848 + 0.908056i \(0.362434\pi\)
\(434\) 0 0
\(435\) 50.9010 2.44052
\(436\) 0 0
\(437\) −1.31708 −0.0630047
\(438\) 0 0
\(439\) 35.9819 1.71732 0.858661 0.512543i \(-0.171297\pi\)
0.858661 + 0.512543i \(0.171297\pi\)
\(440\) 0 0
\(441\) −7.19915 −0.342817
\(442\) 0 0
\(443\) 5.12059 0.243287 0.121643 0.992574i \(-0.461184\pi\)
0.121643 + 0.992574i \(0.461184\pi\)
\(444\) 0 0
\(445\) 25.2637 1.19762
\(446\) 0 0
\(447\) −2.54349 −0.120303
\(448\) 0 0
\(449\) −9.04482 −0.426851 −0.213426 0.976959i \(-0.568462\pi\)
−0.213426 + 0.976959i \(0.568462\pi\)
\(450\) 0 0
\(451\) −14.0027 −0.659363
\(452\) 0 0
\(453\) −14.3642 −0.674891
\(454\) 0 0
\(455\) −5.35475 −0.251035
\(456\) 0 0
\(457\) 7.05890 0.330202 0.165101 0.986277i \(-0.447205\pi\)
0.165101 + 0.986277i \(0.447205\pi\)
\(458\) 0 0
\(459\) 2.83125 0.132151
\(460\) 0 0
\(461\) 11.8936 0.553939 0.276970 0.960879i \(-0.410670\pi\)
0.276970 + 0.960879i \(0.410670\pi\)
\(462\) 0 0
\(463\) 4.55726 0.211794 0.105897 0.994377i \(-0.466229\pi\)
0.105897 + 0.994377i \(0.466229\pi\)
\(464\) 0 0
\(465\) −73.4086 −3.40424
\(466\) 0 0
\(467\) −1.00374 −0.0464474 −0.0232237 0.999730i \(-0.507393\pi\)
−0.0232237 + 0.999730i \(0.507393\pi\)
\(468\) 0 0
\(469\) 21.3908 0.987735
\(470\) 0 0
\(471\) −23.9747 −1.10470
\(472\) 0 0
\(473\) 7.34171 0.337572
\(474\) 0 0
\(475\) −5.73016 −0.262918
\(476\) 0 0
\(477\) 4.83597 0.221424
\(478\) 0 0
\(479\) −15.8181 −0.722749 −0.361375 0.932421i \(-0.617692\pi\)
−0.361375 + 0.932421i \(0.617692\pi\)
\(480\) 0 0
\(481\) −2.79154 −0.127283
\(482\) 0 0
\(483\) −4.25326 −0.193530
\(484\) 0 0
\(485\) −25.6875 −1.16641
\(486\) 0 0
\(487\) 38.6728 1.75243 0.876215 0.481920i \(-0.160060\pi\)
0.876215 + 0.481920i \(0.160060\pi\)
\(488\) 0 0
\(489\) 28.9082 1.30728
\(490\) 0 0
\(491\) 42.4063 1.91377 0.956885 0.290468i \(-0.0938109\pi\)
0.956885 + 0.290468i \(0.0938109\pi\)
\(492\) 0 0
\(493\) −7.37401 −0.332109
\(494\) 0 0
\(495\) 10.1240 0.455039
\(496\) 0 0
\(497\) −4.78760 −0.214753
\(498\) 0 0
\(499\) 28.3674 1.26990 0.634950 0.772553i \(-0.281020\pi\)
0.634950 + 0.772553i \(0.281020\pi\)
\(500\) 0 0
\(501\) 44.7140 1.99767
\(502\) 0 0
\(503\) −22.7740 −1.01544 −0.507720 0.861522i \(-0.669512\pi\)
−0.507720 + 0.861522i \(0.669512\pi\)
\(504\) 0 0
\(505\) 3.08795 0.137412
\(506\) 0 0
\(507\) 25.9361 1.15186
\(508\) 0 0
\(509\) 22.6639 1.00456 0.502280 0.864705i \(-0.332495\pi\)
0.502280 + 0.864705i \(0.332495\pi\)
\(510\) 0 0
\(511\) 25.6726 1.13569
\(512\) 0 0
\(513\) −3.14847 −0.139008
\(514\) 0 0
\(515\) 28.9270 1.27467
\(516\) 0 0
\(517\) −19.4333 −0.854677
\(518\) 0 0
\(519\) −7.01838 −0.308073
\(520\) 0 0
\(521\) 18.2146 0.797995 0.398998 0.916952i \(-0.369358\pi\)
0.398998 + 0.916952i \(0.369358\pi\)
\(522\) 0 0
\(523\) 43.3071 1.89369 0.946843 0.321696i \(-0.104253\pi\)
0.946843 + 0.321696i \(0.104253\pi\)
\(524\) 0 0
\(525\) −18.5044 −0.807598
\(526\) 0 0
\(527\) 10.6347 0.463254
\(528\) 0 0
\(529\) −21.5972 −0.939010
\(530\) 0 0
\(531\) −1.69308 −0.0734734
\(532\) 0 0
\(533\) 7.56447 0.327653
\(534\) 0 0
\(535\) −41.6164 −1.79923
\(536\) 0 0
\(537\) −14.1712 −0.611531
\(538\) 0 0
\(539\) −7.97965 −0.343708
\(540\) 0 0
\(541\) −12.2653 −0.527328 −0.263664 0.964615i \(-0.584931\pi\)
−0.263664 + 0.964615i \(0.584931\pi\)
\(542\) 0 0
\(543\) 13.2553 0.568838
\(544\) 0 0
\(545\) 14.2572 0.610710
\(546\) 0 0
\(547\) −0.637254 −0.0272470 −0.0136235 0.999907i \(-0.504337\pi\)
−0.0136235 + 0.999907i \(0.504337\pi\)
\(548\) 0 0
\(549\) 0.788880 0.0336686
\(550\) 0 0
\(551\) 8.20021 0.349341
\(552\) 0 0
\(553\) 25.3625 1.07852
\(554\) 0 0
\(555\) −19.0073 −0.806816
\(556\) 0 0
\(557\) 16.5516 0.701312 0.350656 0.936504i \(-0.385959\pi\)
0.350656 + 0.936504i \(0.385959\pi\)
\(558\) 0 0
\(559\) −3.96609 −0.167748
\(560\) 0 0
\(561\) −4.06546 −0.171644
\(562\) 0 0
\(563\) −20.6047 −0.868385 −0.434193 0.900820i \(-0.642966\pi\)
−0.434193 + 0.900820i \(0.642966\pi\)
\(564\) 0 0
\(565\) −0.854612 −0.0359538
\(566\) 0 0
\(567\) −18.5871 −0.780584
\(568\) 0 0
\(569\) −12.1979 −0.511363 −0.255682 0.966761i \(-0.582300\pi\)
−0.255682 + 0.966761i \(0.582300\pi\)
\(570\) 0 0
\(571\) 3.85609 0.161372 0.0806862 0.996740i \(-0.474289\pi\)
0.0806862 + 0.996740i \(0.474289\pi\)
\(572\) 0 0
\(573\) −48.0890 −2.00895
\(574\) 0 0
\(575\) 6.10293 0.254510
\(576\) 0 0
\(577\) 32.8143 1.36608 0.683039 0.730382i \(-0.260658\pi\)
0.683039 + 0.730382i \(0.260658\pi\)
\(578\) 0 0
\(579\) −34.5136 −1.43434
\(580\) 0 0
\(581\) 23.1545 0.960610
\(582\) 0 0
\(583\) 5.36026 0.221999
\(584\) 0 0
\(585\) −5.46911 −0.226120
\(586\) 0 0
\(587\) −37.7219 −1.55695 −0.778475 0.627676i \(-0.784006\pi\)
−0.778475 + 0.627676i \(0.784006\pi\)
\(588\) 0 0
\(589\) −11.8262 −0.487291
\(590\) 0 0
\(591\) 33.6674 1.38489
\(592\) 0 0
\(593\) 27.4049 1.12538 0.562692 0.826667i \(-0.309766\pi\)
0.562692 + 0.826667i \(0.309766\pi\)
\(594\) 0 0
\(595\) 5.28194 0.216539
\(596\) 0 0
\(597\) 7.79868 0.319179
\(598\) 0 0
\(599\) 39.7537 1.62429 0.812146 0.583454i \(-0.198299\pi\)
0.812146 + 0.583454i \(0.198299\pi\)
\(600\) 0 0
\(601\) 32.3539 1.31974 0.659872 0.751378i \(-0.270610\pi\)
0.659872 + 0.751378i \(0.270610\pi\)
\(602\) 0 0
\(603\) 21.8476 0.889704
\(604\) 0 0
\(605\) −23.8283 −0.968757
\(606\) 0 0
\(607\) −23.0704 −0.936398 −0.468199 0.883623i \(-0.655097\pi\)
−0.468199 + 0.883623i \(0.655097\pi\)
\(608\) 0 0
\(609\) 26.4809 1.07306
\(610\) 0 0
\(611\) 10.4981 0.424709
\(612\) 0 0
\(613\) −32.1921 −1.30023 −0.650113 0.759838i \(-0.725278\pi\)
−0.650113 + 0.759838i \(0.725278\pi\)
\(614\) 0 0
\(615\) 51.5057 2.07691
\(616\) 0 0
\(617\) 21.1283 0.850594 0.425297 0.905054i \(-0.360170\pi\)
0.425297 + 0.905054i \(0.360170\pi\)
\(618\) 0 0
\(619\) 2.58930 0.104073 0.0520365 0.998645i \(-0.483429\pi\)
0.0520365 + 0.998645i \(0.483429\pi\)
\(620\) 0 0
\(621\) 3.35328 0.134563
\(622\) 0 0
\(623\) 13.1433 0.526575
\(624\) 0 0
\(625\) −24.2125 −0.968500
\(626\) 0 0
\(627\) 4.52096 0.180550
\(628\) 0 0
\(629\) 2.75358 0.109793
\(630\) 0 0
\(631\) 6.52022 0.259566 0.129783 0.991542i \(-0.458572\pi\)
0.129783 + 0.991542i \(0.458572\pi\)
\(632\) 0 0
\(633\) 18.0110 0.715872
\(634\) 0 0
\(635\) 43.9697 1.74489
\(636\) 0 0
\(637\) 4.31071 0.170797
\(638\) 0 0
\(639\) −4.88985 −0.193439
\(640\) 0 0
\(641\) 27.8628 1.10051 0.550257 0.834995i \(-0.314530\pi\)
0.550257 + 0.834995i \(0.314530\pi\)
\(642\) 0 0
\(643\) −21.7812 −0.858968 −0.429484 0.903074i \(-0.641305\pi\)
−0.429484 + 0.903074i \(0.641305\pi\)
\(644\) 0 0
\(645\) −27.0047 −1.06331
\(646\) 0 0
\(647\) −27.3676 −1.07593 −0.537966 0.842967i \(-0.680807\pi\)
−0.537966 + 0.842967i \(0.680807\pi\)
\(648\) 0 0
\(649\) −1.87664 −0.0736644
\(650\) 0 0
\(651\) −38.1904 −1.49680
\(652\) 0 0
\(653\) 18.2041 0.712380 0.356190 0.934414i \(-0.384076\pi\)
0.356190 + 0.934414i \(0.384076\pi\)
\(654\) 0 0
\(655\) 27.1562 1.06108
\(656\) 0 0
\(657\) 26.2209 1.02297
\(658\) 0 0
\(659\) −25.9510 −1.01091 −0.505454 0.862853i \(-0.668675\pi\)
−0.505454 + 0.862853i \(0.668675\pi\)
\(660\) 0 0
\(661\) −25.9721 −1.01020 −0.505099 0.863061i \(-0.668544\pi\)
−0.505099 + 0.863061i \(0.668544\pi\)
\(662\) 0 0
\(663\) 2.19621 0.0852938
\(664\) 0 0
\(665\) −5.87374 −0.227774
\(666\) 0 0
\(667\) −8.73366 −0.338169
\(668\) 0 0
\(669\) 22.5176 0.870580
\(670\) 0 0
\(671\) 0.874408 0.0337561
\(672\) 0 0
\(673\) 33.9691 1.30941 0.654705 0.755884i \(-0.272793\pi\)
0.654705 + 0.755884i \(0.272793\pi\)
\(674\) 0 0
\(675\) 14.5889 0.561528
\(676\) 0 0
\(677\) −18.6022 −0.714940 −0.357470 0.933925i \(-0.616361\pi\)
−0.357470 + 0.933925i \(0.616361\pi\)
\(678\) 0 0
\(679\) −13.3637 −0.512853
\(680\) 0 0
\(681\) 12.5413 0.480584
\(682\) 0 0
\(683\) −32.5792 −1.24661 −0.623305 0.781979i \(-0.714210\pi\)
−0.623305 + 0.781979i \(0.714210\pi\)
\(684\) 0 0
\(685\) −0.133722 −0.00510924
\(686\) 0 0
\(687\) −26.9009 −1.02633
\(688\) 0 0
\(689\) −2.89568 −0.110317
\(690\) 0 0
\(691\) 23.9518 0.911168 0.455584 0.890193i \(-0.349430\pi\)
0.455584 + 0.890193i \(0.349430\pi\)
\(692\) 0 0
\(693\) 5.26694 0.200074
\(694\) 0 0
\(695\) −8.67168 −0.328936
\(696\) 0 0
\(697\) −7.46162 −0.282629
\(698\) 0 0
\(699\) −15.7667 −0.596352
\(700\) 0 0
\(701\) 35.2196 1.33022 0.665112 0.746743i \(-0.268384\pi\)
0.665112 + 0.746743i \(0.268384\pi\)
\(702\) 0 0
\(703\) −3.06210 −0.115489
\(704\) 0 0
\(705\) 71.4808 2.69212
\(706\) 0 0
\(707\) 1.60649 0.0604182
\(708\) 0 0
\(709\) −36.3697 −1.36589 −0.682947 0.730468i \(-0.739302\pi\)
−0.682947 + 0.730468i \(0.739302\pi\)
\(710\) 0 0
\(711\) 25.9042 0.971482
\(712\) 0 0
\(713\) 12.5955 0.471707
\(714\) 0 0
\(715\) −6.06205 −0.226708
\(716\) 0 0
\(717\) 36.5598 1.36535
\(718\) 0 0
\(719\) −7.15496 −0.266835 −0.133417 0.991060i \(-0.542595\pi\)
−0.133417 + 0.991060i \(0.542595\pi\)
\(720\) 0 0
\(721\) 15.0491 0.560457
\(722\) 0 0
\(723\) 21.6209 0.804092
\(724\) 0 0
\(725\) −37.9970 −1.41117
\(726\) 0 0
\(727\) −31.7924 −1.17911 −0.589557 0.807726i \(-0.700698\pi\)
−0.589557 + 0.807726i \(0.700698\pi\)
\(728\) 0 0
\(729\) −0.583601 −0.0216149
\(730\) 0 0
\(731\) 3.91216 0.144697
\(732\) 0 0
\(733\) −1.25692 −0.0464255 −0.0232127 0.999731i \(-0.507390\pi\)
−0.0232127 + 0.999731i \(0.507390\pi\)
\(734\) 0 0
\(735\) 29.3512 1.08264
\(736\) 0 0
\(737\) 24.2162 0.892017
\(738\) 0 0
\(739\) 36.2812 1.33462 0.667312 0.744778i \(-0.267445\pi\)
0.667312 + 0.744778i \(0.267445\pi\)
\(740\) 0 0
\(741\) −2.44228 −0.0897194
\(742\) 0 0
\(743\) −47.4654 −1.74134 −0.870669 0.491869i \(-0.836314\pi\)
−0.870669 + 0.491869i \(0.836314\pi\)
\(744\) 0 0
\(745\) 3.74106 0.137062
\(746\) 0 0
\(747\) 23.6490 0.865271
\(748\) 0 0
\(749\) −21.6507 −0.791098
\(750\) 0 0
\(751\) −1.05538 −0.0385114 −0.0192557 0.999815i \(-0.506130\pi\)
−0.0192557 + 0.999815i \(0.506130\pi\)
\(752\) 0 0
\(753\) −16.0165 −0.583675
\(754\) 0 0
\(755\) 21.1274 0.768907
\(756\) 0 0
\(757\) −24.2820 −0.882544 −0.441272 0.897373i \(-0.645473\pi\)
−0.441272 + 0.897373i \(0.645473\pi\)
\(758\) 0 0
\(759\) −4.81506 −0.174776
\(760\) 0 0
\(761\) −29.2735 −1.06116 −0.530582 0.847633i \(-0.678027\pi\)
−0.530582 + 0.847633i \(0.678027\pi\)
\(762\) 0 0
\(763\) 7.41720 0.268521
\(764\) 0 0
\(765\) 5.39475 0.195047
\(766\) 0 0
\(767\) 1.01378 0.0366056
\(768\) 0 0
\(769\) −10.3600 −0.373592 −0.186796 0.982399i \(-0.559810\pi\)
−0.186796 + 0.982399i \(0.559810\pi\)
\(770\) 0 0
\(771\) 53.6400 1.93180
\(772\) 0 0
\(773\) −24.4009 −0.877638 −0.438819 0.898576i \(-0.644603\pi\)
−0.438819 + 0.898576i \(0.644603\pi\)
\(774\) 0 0
\(775\) 54.7987 1.96843
\(776\) 0 0
\(777\) −9.88843 −0.354746
\(778\) 0 0
\(779\) 8.29763 0.297293
\(780\) 0 0
\(781\) −5.41999 −0.193942
\(782\) 0 0
\(783\) −20.8776 −0.746106
\(784\) 0 0
\(785\) 35.2629 1.25859
\(786\) 0 0
\(787\) −41.0275 −1.46247 −0.731237 0.682124i \(-0.761057\pi\)
−0.731237 + 0.682124i \(0.761057\pi\)
\(788\) 0 0
\(789\) −17.5908 −0.626249
\(790\) 0 0
\(791\) −0.444607 −0.0158084
\(792\) 0 0
\(793\) −0.472366 −0.0167742
\(794\) 0 0
\(795\) −19.7164 −0.699270
\(796\) 0 0
\(797\) 34.4387 1.21988 0.609941 0.792447i \(-0.291193\pi\)
0.609941 + 0.792447i \(0.291193\pi\)
\(798\) 0 0
\(799\) −10.3554 −0.366348
\(800\) 0 0
\(801\) 13.4240 0.474313
\(802\) 0 0
\(803\) 29.0636 1.02563
\(804\) 0 0
\(805\) 6.25585 0.220490
\(806\) 0 0
\(807\) −55.7076 −1.96100
\(808\) 0 0
\(809\) 19.9459 0.701261 0.350631 0.936514i \(-0.385967\pi\)
0.350631 + 0.936514i \(0.385967\pi\)
\(810\) 0 0
\(811\) 43.3658 1.52278 0.761390 0.648294i \(-0.224517\pi\)
0.761390 + 0.648294i \(0.224517\pi\)
\(812\) 0 0
\(813\) 41.0646 1.44020
\(814\) 0 0
\(815\) −42.5193 −1.48939
\(816\) 0 0
\(817\) −4.35049 −0.152204
\(818\) 0 0
\(819\) −2.84527 −0.0994217
\(820\) 0 0
\(821\) −32.5480 −1.13593 −0.567966 0.823052i \(-0.692270\pi\)
−0.567966 + 0.823052i \(0.692270\pi\)
\(822\) 0 0
\(823\) 10.5485 0.367696 0.183848 0.982955i \(-0.441145\pi\)
0.183848 + 0.982955i \(0.441145\pi\)
\(824\) 0 0
\(825\) −20.9486 −0.729336
\(826\) 0 0
\(827\) 2.14650 0.0746411 0.0373206 0.999303i \(-0.488118\pi\)
0.0373206 + 0.999303i \(0.488118\pi\)
\(828\) 0 0
\(829\) −23.4056 −0.812911 −0.406455 0.913671i \(-0.633235\pi\)
−0.406455 + 0.913671i \(0.633235\pi\)
\(830\) 0 0
\(831\) −27.0857 −0.939592
\(832\) 0 0
\(833\) −4.25210 −0.147327
\(834\) 0 0
\(835\) −65.7669 −2.27596
\(836\) 0 0
\(837\) 30.1094 1.04073
\(838\) 0 0
\(839\) −37.6831 −1.30096 −0.650482 0.759521i \(-0.725433\pi\)
−0.650482 + 0.759521i \(0.725433\pi\)
\(840\) 0 0
\(841\) 25.3760 0.875035
\(842\) 0 0
\(843\) −60.2027 −2.07349
\(844\) 0 0
\(845\) −38.1478 −1.31232
\(846\) 0 0
\(847\) −12.3965 −0.425949
\(848\) 0 0
\(849\) 48.0577 1.64934
\(850\) 0 0
\(851\) 3.26130 0.111796
\(852\) 0 0
\(853\) 29.6183 1.01411 0.507055 0.861914i \(-0.330734\pi\)
0.507055 + 0.861914i \(0.330734\pi\)
\(854\) 0 0
\(855\) −5.99919 −0.205168
\(856\) 0 0
\(857\) −54.3763 −1.85746 −0.928730 0.370757i \(-0.879098\pi\)
−0.928730 + 0.370757i \(0.879098\pi\)
\(858\) 0 0
\(859\) −3.71947 −0.126907 −0.0634534 0.997985i \(-0.520211\pi\)
−0.0634534 + 0.997985i \(0.520211\pi\)
\(860\) 0 0
\(861\) 26.7955 0.913189
\(862\) 0 0
\(863\) 4.22578 0.143847 0.0719236 0.997410i \(-0.477086\pi\)
0.0719236 + 0.997410i \(0.477086\pi\)
\(864\) 0 0
\(865\) 10.3229 0.350989
\(866\) 0 0
\(867\) −2.16635 −0.0735732
\(868\) 0 0
\(869\) 28.7126 0.974007
\(870\) 0 0
\(871\) −13.0819 −0.443264
\(872\) 0 0
\(873\) −13.6491 −0.461954
\(874\) 0 0
\(875\) 0.807225 0.0272892
\(876\) 0 0
\(877\) 1.95400 0.0659820 0.0329910 0.999456i \(-0.489497\pi\)
0.0329910 + 0.999456i \(0.489497\pi\)
\(878\) 0 0
\(879\) −30.0609 −1.01393
\(880\) 0 0
\(881\) 21.4670 0.723241 0.361621 0.932325i \(-0.382224\pi\)
0.361621 + 0.932325i \(0.382224\pi\)
\(882\) 0 0
\(883\) 33.8490 1.13911 0.569554 0.821954i \(-0.307116\pi\)
0.569554 + 0.821954i \(0.307116\pi\)
\(884\) 0 0
\(885\) 6.90276 0.232034
\(886\) 0 0
\(887\) −44.6648 −1.49970 −0.749848 0.661610i \(-0.769874\pi\)
−0.749848 + 0.661610i \(0.769874\pi\)
\(888\) 0 0
\(889\) 22.8750 0.767202
\(890\) 0 0
\(891\) −21.0422 −0.704940
\(892\) 0 0
\(893\) 11.5156 0.385356
\(894\) 0 0
\(895\) 20.8435 0.696720
\(896\) 0 0
\(897\) 2.60116 0.0868502
\(898\) 0 0
\(899\) −78.4202 −2.61546
\(900\) 0 0
\(901\) 2.85631 0.0951576
\(902\) 0 0
\(903\) −14.0490 −0.467523
\(904\) 0 0
\(905\) −19.4964 −0.648081
\(906\) 0 0
\(907\) −17.9524 −0.596099 −0.298049 0.954550i \(-0.596336\pi\)
−0.298049 + 0.954550i \(0.596336\pi\)
\(908\) 0 0
\(909\) 1.64080 0.0544218
\(910\) 0 0
\(911\) −32.1198 −1.06418 −0.532089 0.846688i \(-0.678593\pi\)
−0.532089 + 0.846688i \(0.678593\pi\)
\(912\) 0 0
\(913\) 26.2129 0.867521
\(914\) 0 0
\(915\) −3.21630 −0.106327
\(916\) 0 0
\(917\) 14.1278 0.466542
\(918\) 0 0
\(919\) −2.11835 −0.0698778 −0.0349389 0.999389i \(-0.511124\pi\)
−0.0349389 + 0.999389i \(0.511124\pi\)
\(920\) 0 0
\(921\) −54.5495 −1.79747
\(922\) 0 0
\(923\) 2.92795 0.0963745
\(924\) 0 0
\(925\) 14.1887 0.466523
\(926\) 0 0
\(927\) 15.3705 0.504832
\(928\) 0 0
\(929\) −34.7331 −1.13955 −0.569777 0.821799i \(-0.692971\pi\)
−0.569777 + 0.821799i \(0.692971\pi\)
\(930\) 0 0
\(931\) 4.72852 0.154971
\(932\) 0 0
\(933\) −54.7497 −1.79243
\(934\) 0 0
\(935\) 5.97962 0.195555
\(936\) 0 0
\(937\) 1.39058 0.0454283 0.0227142 0.999742i \(-0.492769\pi\)
0.0227142 + 0.999742i \(0.492769\pi\)
\(938\) 0 0
\(939\) 64.3309 2.09936
\(940\) 0 0
\(941\) 32.5529 1.06119 0.530597 0.847624i \(-0.321968\pi\)
0.530597 + 0.847624i \(0.321968\pi\)
\(942\) 0 0
\(943\) −8.83742 −0.287786
\(944\) 0 0
\(945\) 14.9545 0.486470
\(946\) 0 0
\(947\) 47.4737 1.54269 0.771343 0.636419i \(-0.219585\pi\)
0.771343 + 0.636419i \(0.219585\pi\)
\(948\) 0 0
\(949\) −15.7006 −0.509662
\(950\) 0 0
\(951\) 36.4032 1.18045
\(952\) 0 0
\(953\) 20.7290 0.671479 0.335740 0.941955i \(-0.391014\pi\)
0.335740 + 0.941955i \(0.391014\pi\)
\(954\) 0 0
\(955\) 70.7311 2.28881
\(956\) 0 0
\(957\) 29.9787 0.969074
\(958\) 0 0
\(959\) −0.0695678 −0.00224646
\(960\) 0 0
\(961\) 82.0964 2.64827
\(962\) 0 0
\(963\) −22.1130 −0.712583
\(964\) 0 0
\(965\) 50.7639 1.63415
\(966\) 0 0
\(967\) −45.4293 −1.46091 −0.730454 0.682962i \(-0.760691\pi\)
−0.730454 + 0.682962i \(0.760691\pi\)
\(968\) 0 0
\(969\) 2.40907 0.0773906
\(970\) 0 0
\(971\) 44.3642 1.42371 0.711857 0.702324i \(-0.247854\pi\)
0.711857 + 0.702324i \(0.247854\pi\)
\(972\) 0 0
\(973\) −4.51139 −0.144628
\(974\) 0 0
\(975\) 11.3167 0.362425
\(976\) 0 0
\(977\) 15.1563 0.484892 0.242446 0.970165i \(-0.422050\pi\)
0.242446 + 0.970165i \(0.422050\pi\)
\(978\) 0 0
\(979\) 14.8794 0.475546
\(980\) 0 0
\(981\) 7.57560 0.241870
\(982\) 0 0
\(983\) 16.1736 0.515859 0.257929 0.966164i \(-0.416960\pi\)
0.257929 + 0.966164i \(0.416960\pi\)
\(984\) 0 0
\(985\) −49.5192 −1.57781
\(986\) 0 0
\(987\) 37.1874 1.18369
\(988\) 0 0
\(989\) 4.63350 0.147337
\(990\) 0 0
\(991\) 13.8053 0.438539 0.219270 0.975664i \(-0.429633\pi\)
0.219270 + 0.975664i \(0.429633\pi\)
\(992\) 0 0
\(993\) −16.9004 −0.536318
\(994\) 0 0
\(995\) −11.4706 −0.363642
\(996\) 0 0
\(997\) −29.2008 −0.924798 −0.462399 0.886672i \(-0.653011\pi\)
−0.462399 + 0.886672i \(0.653011\pi\)
\(998\) 0 0
\(999\) 7.79608 0.246657
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.1.3 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8024.2.a.v.1.3 18 1.1 even 1 trivial