Properties

Label 8001.2.a.t.1.3
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $16$
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] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.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: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} - 1655 x^{7} - 1150 x^{6} + 1279 x^{5} + 474 x^{4} - 280 x^{3} - 83 x^{2} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 889)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.72082\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.72082 q^{2} +0.961212 q^{4} +4.11468 q^{5} -1.00000 q^{7} +1.78756 q^{8} +O(q^{10})\) \(q-1.72082 q^{2} +0.961212 q^{4} +4.11468 q^{5} -1.00000 q^{7} +1.78756 q^{8} -7.08062 q^{10} -1.90264 q^{11} -0.956847 q^{13} +1.72082 q^{14} -4.99850 q^{16} -0.151705 q^{17} -2.06568 q^{19} +3.95508 q^{20} +3.27410 q^{22} +6.66025 q^{23} +11.9306 q^{25} +1.64656 q^{26} -0.961212 q^{28} -4.22490 q^{29} -2.46514 q^{31} +5.02637 q^{32} +0.261057 q^{34} -4.11468 q^{35} +0.302428 q^{37} +3.55465 q^{38} +7.35526 q^{40} +6.55602 q^{41} +1.79751 q^{43} -1.82884 q^{44} -11.4611 q^{46} +0.645672 q^{47} +1.00000 q^{49} -20.5304 q^{50} -0.919732 q^{52} -0.596441 q^{53} -7.82878 q^{55} -1.78756 q^{56} +7.27028 q^{58} -8.74718 q^{59} -11.3645 q^{61} +4.24205 q^{62} +1.34753 q^{64} -3.93712 q^{65} +11.3637 q^{67} -0.145821 q^{68} +7.08062 q^{70} -6.20333 q^{71} -2.03377 q^{73} -0.520423 q^{74} -1.98555 q^{76} +1.90264 q^{77} +8.04816 q^{79} -20.5672 q^{80} -11.2817 q^{82} +9.02002 q^{83} -0.624220 q^{85} -3.09318 q^{86} -3.40110 q^{88} -2.23627 q^{89} +0.956847 q^{91} +6.40191 q^{92} -1.11108 q^{94} -8.49961 q^{95} +2.48387 q^{97} -1.72082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.72082 −1.21680 −0.608401 0.793630i \(-0.708189\pi\)
−0.608401 + 0.793630i \(0.708189\pi\)
\(3\) 0 0
\(4\) 0.961212 0.480606
\(5\) 4.11468 1.84014 0.920071 0.391750i \(-0.128130\pi\)
0.920071 + 0.391750i \(0.128130\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.78756 0.631999
\(9\) 0 0
\(10\) −7.08062 −2.23909
\(11\) −1.90264 −0.573669 −0.286834 0.957980i \(-0.592603\pi\)
−0.286834 + 0.957980i \(0.592603\pi\)
\(12\) 0 0
\(13\) −0.956847 −0.265381 −0.132691 0.991157i \(-0.542362\pi\)
−0.132691 + 0.991157i \(0.542362\pi\)
\(14\) 1.72082 0.459908
\(15\) 0 0
\(16\) −4.99850 −1.24962
\(17\) −0.151705 −0.0367940 −0.0183970 0.999831i \(-0.505856\pi\)
−0.0183970 + 0.999831i \(0.505856\pi\)
\(18\) 0 0
\(19\) −2.06568 −0.473899 −0.236949 0.971522i \(-0.576148\pi\)
−0.236949 + 0.971522i \(0.576148\pi\)
\(20\) 3.95508 0.884384
\(21\) 0 0
\(22\) 3.27410 0.698041
\(23\) 6.66025 1.38876 0.694379 0.719609i \(-0.255679\pi\)
0.694379 + 0.719609i \(0.255679\pi\)
\(24\) 0 0
\(25\) 11.9306 2.38613
\(26\) 1.64656 0.322917
\(27\) 0 0
\(28\) −0.961212 −0.181652
\(29\) −4.22490 −0.784544 −0.392272 0.919849i \(-0.628311\pi\)
−0.392272 + 0.919849i \(0.628311\pi\)
\(30\) 0 0
\(31\) −2.46514 −0.442751 −0.221376 0.975189i \(-0.571055\pi\)
−0.221376 + 0.975189i \(0.571055\pi\)
\(32\) 5.02637 0.888545
\(33\) 0 0
\(34\) 0.261057 0.0447710
\(35\) −4.11468 −0.695509
\(36\) 0 0
\(37\) 0.302428 0.0497188 0.0248594 0.999691i \(-0.492086\pi\)
0.0248594 + 0.999691i \(0.492086\pi\)
\(38\) 3.55465 0.576641
\(39\) 0 0
\(40\) 7.35526 1.16297
\(41\) 6.55602 1.02388 0.511939 0.859022i \(-0.328927\pi\)
0.511939 + 0.859022i \(0.328927\pi\)
\(42\) 0 0
\(43\) 1.79751 0.274118 0.137059 0.990563i \(-0.456235\pi\)
0.137059 + 0.990563i \(0.456235\pi\)
\(44\) −1.82884 −0.275709
\(45\) 0 0
\(46\) −11.4611 −1.68984
\(47\) 0.645672 0.0941809 0.0470905 0.998891i \(-0.485005\pi\)
0.0470905 + 0.998891i \(0.485005\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −20.5304 −2.90344
\(51\) 0 0
\(52\) −0.919732 −0.127544
\(53\) −0.596441 −0.0819274 −0.0409637 0.999161i \(-0.513043\pi\)
−0.0409637 + 0.999161i \(0.513043\pi\)
\(54\) 0 0
\(55\) −7.82878 −1.05563
\(56\) −1.78756 −0.238873
\(57\) 0 0
\(58\) 7.27028 0.954634
\(59\) −8.74718 −1.13879 −0.569393 0.822065i \(-0.692822\pi\)
−0.569393 + 0.822065i \(0.692822\pi\)
\(60\) 0 0
\(61\) −11.3645 −1.45508 −0.727540 0.686065i \(-0.759337\pi\)
−0.727540 + 0.686065i \(0.759337\pi\)
\(62\) 4.24205 0.538741
\(63\) 0 0
\(64\) 1.34753 0.168441
\(65\) −3.93712 −0.488340
\(66\) 0 0
\(67\) 11.3637 1.38830 0.694151 0.719830i \(-0.255780\pi\)
0.694151 + 0.719830i \(0.255780\pi\)
\(68\) −0.145821 −0.0176834
\(69\) 0 0
\(70\) 7.08062 0.846296
\(71\) −6.20333 −0.736200 −0.368100 0.929786i \(-0.619992\pi\)
−0.368100 + 0.929786i \(0.619992\pi\)
\(72\) 0 0
\(73\) −2.03377 −0.238035 −0.119017 0.992892i \(-0.537974\pi\)
−0.119017 + 0.992892i \(0.537974\pi\)
\(74\) −0.520423 −0.0604979
\(75\) 0 0
\(76\) −1.98555 −0.227759
\(77\) 1.90264 0.216826
\(78\) 0 0
\(79\) 8.04816 0.905489 0.452744 0.891640i \(-0.350445\pi\)
0.452744 + 0.891640i \(0.350445\pi\)
\(80\) −20.5672 −2.29949
\(81\) 0 0
\(82\) −11.2817 −1.24586
\(83\) 9.02002 0.990076 0.495038 0.868871i \(-0.335154\pi\)
0.495038 + 0.868871i \(0.335154\pi\)
\(84\) 0 0
\(85\) −0.624220 −0.0677062
\(86\) −3.09318 −0.333547
\(87\) 0 0
\(88\) −3.40110 −0.362558
\(89\) −2.23627 −0.237044 −0.118522 0.992951i \(-0.537816\pi\)
−0.118522 + 0.992951i \(0.537816\pi\)
\(90\) 0 0
\(91\) 0.956847 0.100305
\(92\) 6.40191 0.667446
\(93\) 0 0
\(94\) −1.11108 −0.114600
\(95\) −8.49961 −0.872041
\(96\) 0 0
\(97\) 2.48387 0.252199 0.126099 0.992018i \(-0.459754\pi\)
0.126099 + 0.992018i \(0.459754\pi\)
\(98\) −1.72082 −0.173829
\(99\) 0 0
\(100\) 11.4679 1.14679
\(101\) 17.1534 1.70682 0.853411 0.521238i \(-0.174530\pi\)
0.853411 + 0.521238i \(0.174530\pi\)
\(102\) 0 0
\(103\) 9.27419 0.913813 0.456907 0.889515i \(-0.348957\pi\)
0.456907 + 0.889515i \(0.348957\pi\)
\(104\) −1.71042 −0.167721
\(105\) 0 0
\(106\) 1.02637 0.0996894
\(107\) 16.1726 1.56346 0.781730 0.623617i \(-0.214337\pi\)
0.781730 + 0.623617i \(0.214337\pi\)
\(108\) 0 0
\(109\) −10.4597 −1.00186 −0.500928 0.865489i \(-0.667008\pi\)
−0.500928 + 0.865489i \(0.667008\pi\)
\(110\) 13.4719 1.28450
\(111\) 0 0
\(112\) 4.99850 0.472313
\(113\) −15.1643 −1.42654 −0.713269 0.700891i \(-0.752786\pi\)
−0.713269 + 0.700891i \(0.752786\pi\)
\(114\) 0 0
\(115\) 27.4048 2.55551
\(116\) −4.06102 −0.377056
\(117\) 0 0
\(118\) 15.0523 1.38568
\(119\) 0.151705 0.0139068
\(120\) 0 0
\(121\) −7.37995 −0.670904
\(122\) 19.5563 1.77054
\(123\) 0 0
\(124\) −2.36952 −0.212789
\(125\) 28.5174 2.55067
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) −12.3716 −1.09350
\(129\) 0 0
\(130\) 6.77507 0.594213
\(131\) 12.5063 1.09268 0.546341 0.837563i \(-0.316020\pi\)
0.546341 + 0.837563i \(0.316020\pi\)
\(132\) 0 0
\(133\) 2.06568 0.179117
\(134\) −19.5549 −1.68929
\(135\) 0 0
\(136\) −0.271183 −0.0232538
\(137\) 12.8382 1.09684 0.548422 0.836202i \(-0.315229\pi\)
0.548422 + 0.836202i \(0.315229\pi\)
\(138\) 0 0
\(139\) 15.6034 1.32346 0.661730 0.749742i \(-0.269823\pi\)
0.661730 + 0.749742i \(0.269823\pi\)
\(140\) −3.95508 −0.334266
\(141\) 0 0
\(142\) 10.6748 0.895809
\(143\) 1.82054 0.152241
\(144\) 0 0
\(145\) −17.3841 −1.44367
\(146\) 3.49974 0.289641
\(147\) 0 0
\(148\) 0.290697 0.0238952
\(149\) 2.15071 0.176193 0.0880966 0.996112i \(-0.471922\pi\)
0.0880966 + 0.996112i \(0.471922\pi\)
\(150\) 0 0
\(151\) −1.26618 −0.103041 −0.0515203 0.998672i \(-0.516407\pi\)
−0.0515203 + 0.998672i \(0.516407\pi\)
\(152\) −3.69253 −0.299504
\(153\) 0 0
\(154\) −3.27410 −0.263835
\(155\) −10.1433 −0.814726
\(156\) 0 0
\(157\) −2.86075 −0.228312 −0.114156 0.993463i \(-0.536416\pi\)
−0.114156 + 0.993463i \(0.536416\pi\)
\(158\) −13.8494 −1.10180
\(159\) 0 0
\(160\) 20.6819 1.63505
\(161\) −6.66025 −0.524901
\(162\) 0 0
\(163\) 12.6597 0.991584 0.495792 0.868441i \(-0.334878\pi\)
0.495792 + 0.868441i \(0.334878\pi\)
\(164\) 6.30173 0.492082
\(165\) 0 0
\(166\) −15.5218 −1.20473
\(167\) 17.0641 1.32046 0.660228 0.751065i \(-0.270460\pi\)
0.660228 + 0.751065i \(0.270460\pi\)
\(168\) 0 0
\(169\) −12.0844 −0.929573
\(170\) 1.07417 0.0823850
\(171\) 0 0
\(172\) 1.72779 0.131743
\(173\) 19.1596 1.45667 0.728337 0.685219i \(-0.240294\pi\)
0.728337 + 0.685219i \(0.240294\pi\)
\(174\) 0 0
\(175\) −11.9306 −0.901871
\(176\) 9.51036 0.716870
\(177\) 0 0
\(178\) 3.84821 0.288436
\(179\) 13.6433 1.01975 0.509873 0.860250i \(-0.329692\pi\)
0.509873 + 0.860250i \(0.329692\pi\)
\(180\) 0 0
\(181\) −14.3725 −1.06830 −0.534151 0.845389i \(-0.679368\pi\)
−0.534151 + 0.845389i \(0.679368\pi\)
\(182\) −1.64656 −0.122051
\(183\) 0 0
\(184\) 11.9056 0.877695
\(185\) 1.24439 0.0914897
\(186\) 0 0
\(187\) 0.288642 0.0211076
\(188\) 0.620628 0.0452639
\(189\) 0 0
\(190\) 14.6263 1.06110
\(191\) 24.2155 1.75218 0.876088 0.482152i \(-0.160145\pi\)
0.876088 + 0.482152i \(0.160145\pi\)
\(192\) 0 0
\(193\) 5.45506 0.392664 0.196332 0.980537i \(-0.437097\pi\)
0.196332 + 0.980537i \(0.437097\pi\)
\(194\) −4.27428 −0.306876
\(195\) 0 0
\(196\) 0.961212 0.0686580
\(197\) 5.48164 0.390551 0.195275 0.980748i \(-0.437440\pi\)
0.195275 + 0.980748i \(0.437440\pi\)
\(198\) 0 0
\(199\) −8.82801 −0.625801 −0.312901 0.949786i \(-0.601301\pi\)
−0.312901 + 0.949786i \(0.601301\pi\)
\(200\) 21.3268 1.50803
\(201\) 0 0
\(202\) −29.5178 −2.07686
\(203\) 4.22490 0.296530
\(204\) 0 0
\(205\) 26.9760 1.88408
\(206\) −15.9592 −1.11193
\(207\) 0 0
\(208\) 4.78279 0.331627
\(209\) 3.93025 0.271861
\(210\) 0 0
\(211\) −4.96059 −0.341501 −0.170751 0.985314i \(-0.554619\pi\)
−0.170751 + 0.985314i \(0.554619\pi\)
\(212\) −0.573306 −0.0393748
\(213\) 0 0
\(214\) −27.8300 −1.90242
\(215\) 7.39618 0.504415
\(216\) 0 0
\(217\) 2.46514 0.167344
\(218\) 17.9992 1.21906
\(219\) 0 0
\(220\) −7.52512 −0.507343
\(221\) 0.145159 0.00976444
\(222\) 0 0
\(223\) −28.2073 −1.88890 −0.944449 0.328659i \(-0.893403\pi\)
−0.944449 + 0.328659i \(0.893403\pi\)
\(224\) −5.02637 −0.335838
\(225\) 0 0
\(226\) 26.0950 1.73581
\(227\) 13.3743 0.887684 0.443842 0.896105i \(-0.353615\pi\)
0.443842 + 0.896105i \(0.353615\pi\)
\(228\) 0 0
\(229\) 25.7395 1.70091 0.850456 0.526046i \(-0.176326\pi\)
0.850456 + 0.526046i \(0.176326\pi\)
\(230\) −47.1587 −3.10955
\(231\) 0 0
\(232\) −7.55227 −0.495831
\(233\) −9.69883 −0.635391 −0.317696 0.948193i \(-0.602909\pi\)
−0.317696 + 0.948193i \(0.602909\pi\)
\(234\) 0 0
\(235\) 2.65674 0.173306
\(236\) −8.40790 −0.547307
\(237\) 0 0
\(238\) −0.261057 −0.0169218
\(239\) 2.10529 0.136180 0.0680899 0.997679i \(-0.478310\pi\)
0.0680899 + 0.997679i \(0.478310\pi\)
\(240\) 0 0
\(241\) −11.2452 −0.724367 −0.362183 0.932107i \(-0.617969\pi\)
−0.362183 + 0.932107i \(0.617969\pi\)
\(242\) 12.6995 0.816357
\(243\) 0 0
\(244\) −10.9237 −0.699321
\(245\) 4.11468 0.262878
\(246\) 0 0
\(247\) 1.97654 0.125764
\(248\) −4.40659 −0.279819
\(249\) 0 0
\(250\) −49.0732 −3.10366
\(251\) 5.06128 0.319465 0.159733 0.987160i \(-0.448937\pi\)
0.159733 + 0.987160i \(0.448937\pi\)
\(252\) 0 0
\(253\) −12.6721 −0.796687
\(254\) 1.72082 0.107974
\(255\) 0 0
\(256\) 18.5942 1.16214
\(257\) −26.7217 −1.66685 −0.833426 0.552631i \(-0.813624\pi\)
−0.833426 + 0.552631i \(0.813624\pi\)
\(258\) 0 0
\(259\) −0.302428 −0.0187919
\(260\) −3.78441 −0.234699
\(261\) 0 0
\(262\) −21.5211 −1.32958
\(263\) 4.86117 0.299753 0.149876 0.988705i \(-0.452112\pi\)
0.149876 + 0.988705i \(0.452112\pi\)
\(264\) 0 0
\(265\) −2.45417 −0.150758
\(266\) −3.55465 −0.217950
\(267\) 0 0
\(268\) 10.9230 0.667226
\(269\) −13.3191 −0.812082 −0.406041 0.913855i \(-0.633091\pi\)
−0.406041 + 0.913855i \(0.633091\pi\)
\(270\) 0 0
\(271\) 32.4393 1.97055 0.985275 0.170978i \(-0.0546926\pi\)
0.985275 + 0.170978i \(0.0546926\pi\)
\(272\) 0.758299 0.0459786
\(273\) 0 0
\(274\) −22.0922 −1.33464
\(275\) −22.6997 −1.36885
\(276\) 0 0
\(277\) 12.9208 0.776337 0.388168 0.921588i \(-0.373108\pi\)
0.388168 + 0.921588i \(0.373108\pi\)
\(278\) −26.8505 −1.61039
\(279\) 0 0
\(280\) −7.35526 −0.439561
\(281\) −17.3588 −1.03554 −0.517769 0.855521i \(-0.673237\pi\)
−0.517769 + 0.855521i \(0.673237\pi\)
\(282\) 0 0
\(283\) 22.3800 1.33035 0.665177 0.746685i \(-0.268356\pi\)
0.665177 + 0.746685i \(0.268356\pi\)
\(284\) −5.96272 −0.353822
\(285\) 0 0
\(286\) −3.13281 −0.185247
\(287\) −6.55602 −0.386990
\(288\) 0 0
\(289\) −16.9770 −0.998646
\(290\) 29.9149 1.75666
\(291\) 0 0
\(292\) −1.95488 −0.114401
\(293\) −10.4775 −0.612102 −0.306051 0.952015i \(-0.599008\pi\)
−0.306051 + 0.952015i \(0.599008\pi\)
\(294\) 0 0
\(295\) −35.9919 −2.09553
\(296\) 0.540609 0.0314222
\(297\) 0 0
\(298\) −3.70098 −0.214392
\(299\) −6.37284 −0.368551
\(300\) 0 0
\(301\) −1.79751 −0.103607
\(302\) 2.17887 0.125380
\(303\) 0 0
\(304\) 10.3253 0.592195
\(305\) −46.7615 −2.67756
\(306\) 0 0
\(307\) 9.32113 0.531985 0.265993 0.963975i \(-0.414300\pi\)
0.265993 + 0.963975i \(0.414300\pi\)
\(308\) 1.82884 0.104208
\(309\) 0 0
\(310\) 17.4547 0.991360
\(311\) 2.77055 0.157104 0.0785518 0.996910i \(-0.474970\pi\)
0.0785518 + 0.996910i \(0.474970\pi\)
\(312\) 0 0
\(313\) 2.44798 0.138368 0.0691840 0.997604i \(-0.477960\pi\)
0.0691840 + 0.997604i \(0.477960\pi\)
\(314\) 4.92282 0.277811
\(315\) 0 0
\(316\) 7.73599 0.435183
\(317\) −13.7948 −0.774794 −0.387397 0.921913i \(-0.626626\pi\)
−0.387397 + 0.921913i \(0.626626\pi\)
\(318\) 0 0
\(319\) 8.03847 0.450068
\(320\) 5.54466 0.309956
\(321\) 0 0
\(322\) 11.4611 0.638701
\(323\) 0.313374 0.0174366
\(324\) 0 0
\(325\) −11.4158 −0.633234
\(326\) −21.7850 −1.20656
\(327\) 0 0
\(328\) 11.7193 0.647091
\(329\) −0.645672 −0.0355971
\(330\) 0 0
\(331\) 12.9957 0.714308 0.357154 0.934046i \(-0.383747\pi\)
0.357154 + 0.934046i \(0.383747\pi\)
\(332\) 8.67015 0.475836
\(333\) 0 0
\(334\) −29.3641 −1.60673
\(335\) 46.7582 2.55467
\(336\) 0 0
\(337\) 10.1574 0.553308 0.276654 0.960970i \(-0.410774\pi\)
0.276654 + 0.960970i \(0.410774\pi\)
\(338\) 20.7951 1.13111
\(339\) 0 0
\(340\) −0.600008 −0.0325400
\(341\) 4.69028 0.253993
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 3.21316 0.173242
\(345\) 0 0
\(346\) −32.9701 −1.77248
\(347\) −7.43989 −0.399394 −0.199697 0.979858i \(-0.563996\pi\)
−0.199697 + 0.979858i \(0.563996\pi\)
\(348\) 0 0
\(349\) −16.7754 −0.897965 −0.448982 0.893541i \(-0.648213\pi\)
−0.448982 + 0.893541i \(0.648213\pi\)
\(350\) 20.5304 1.09740
\(351\) 0 0
\(352\) −9.56339 −0.509730
\(353\) 10.5724 0.562712 0.281356 0.959603i \(-0.409216\pi\)
0.281356 + 0.959603i \(0.409216\pi\)
\(354\) 0 0
\(355\) −25.5248 −1.35471
\(356\) −2.14953 −0.113925
\(357\) 0 0
\(358\) −23.4776 −1.24083
\(359\) 20.3552 1.07431 0.537154 0.843484i \(-0.319499\pi\)
0.537154 + 0.843484i \(0.319499\pi\)
\(360\) 0 0
\(361\) −14.7330 −0.775420
\(362\) 24.7325 1.29991
\(363\) 0 0
\(364\) 0.919732 0.0482071
\(365\) −8.36831 −0.438018
\(366\) 0 0
\(367\) 30.4336 1.58862 0.794311 0.607511i \(-0.207832\pi\)
0.794311 + 0.607511i \(0.207832\pi\)
\(368\) −33.2912 −1.73543
\(369\) 0 0
\(370\) −2.14138 −0.111325
\(371\) 0.596441 0.0309657
\(372\) 0 0
\(373\) 22.5148 1.16577 0.582887 0.812553i \(-0.301923\pi\)
0.582887 + 0.812553i \(0.301923\pi\)
\(374\) −0.496699 −0.0256837
\(375\) 0 0
\(376\) 1.15418 0.0595223
\(377\) 4.04258 0.208203
\(378\) 0 0
\(379\) −32.0316 −1.64535 −0.822676 0.568511i \(-0.807520\pi\)
−0.822676 + 0.568511i \(0.807520\pi\)
\(380\) −8.16993 −0.419108
\(381\) 0 0
\(382\) −41.6705 −2.13205
\(383\) 17.1166 0.874620 0.437310 0.899311i \(-0.355931\pi\)
0.437310 + 0.899311i \(0.355931\pi\)
\(384\) 0 0
\(385\) 7.82878 0.398992
\(386\) −9.38717 −0.477794
\(387\) 0 0
\(388\) 2.38752 0.121208
\(389\) −31.4659 −1.59539 −0.797693 0.603063i \(-0.793947\pi\)
−0.797693 + 0.603063i \(0.793947\pi\)
\(390\) 0 0
\(391\) −1.01040 −0.0510980
\(392\) 1.78756 0.0902856
\(393\) 0 0
\(394\) −9.43290 −0.475222
\(395\) 33.1156 1.66623
\(396\) 0 0
\(397\) −29.3356 −1.47231 −0.736155 0.676812i \(-0.763361\pi\)
−0.736155 + 0.676812i \(0.763361\pi\)
\(398\) 15.1914 0.761476
\(399\) 0 0
\(400\) −59.6352 −2.98176
\(401\) 3.02453 0.151038 0.0755188 0.997144i \(-0.475939\pi\)
0.0755188 + 0.997144i \(0.475939\pi\)
\(402\) 0 0
\(403\) 2.35876 0.117498
\(404\) 16.4880 0.820309
\(405\) 0 0
\(406\) −7.27028 −0.360818
\(407\) −0.575412 −0.0285221
\(408\) 0 0
\(409\) −8.58154 −0.424330 −0.212165 0.977234i \(-0.568051\pi\)
−0.212165 + 0.977234i \(0.568051\pi\)
\(410\) −46.4207 −2.29256
\(411\) 0 0
\(412\) 8.91446 0.439184
\(413\) 8.74718 0.430421
\(414\) 0 0
\(415\) 37.1145 1.82188
\(416\) −4.80946 −0.235803
\(417\) 0 0
\(418\) −6.76324 −0.330801
\(419\) −19.1800 −0.937003 −0.468501 0.883463i \(-0.655206\pi\)
−0.468501 + 0.883463i \(0.655206\pi\)
\(420\) 0 0
\(421\) 32.4549 1.58175 0.790876 0.611976i \(-0.209625\pi\)
0.790876 + 0.611976i \(0.209625\pi\)
\(422\) 8.53628 0.415539
\(423\) 0 0
\(424\) −1.06618 −0.0517781
\(425\) −1.80994 −0.0877951
\(426\) 0 0
\(427\) 11.3645 0.549969
\(428\) 15.5453 0.751408
\(429\) 0 0
\(430\) −12.7275 −0.613773
\(431\) 19.4248 0.935662 0.467831 0.883818i \(-0.345036\pi\)
0.467831 + 0.883818i \(0.345036\pi\)
\(432\) 0 0
\(433\) 16.3673 0.786561 0.393280 0.919419i \(-0.371340\pi\)
0.393280 + 0.919419i \(0.371340\pi\)
\(434\) −4.24205 −0.203625
\(435\) 0 0
\(436\) −10.0540 −0.481498
\(437\) −13.7579 −0.658131
\(438\) 0 0
\(439\) 15.8233 0.755205 0.377602 0.925968i \(-0.376749\pi\)
0.377602 + 0.925968i \(0.376749\pi\)
\(440\) −13.9944 −0.667159
\(441\) 0 0
\(442\) −0.249792 −0.0118814
\(443\) −19.2944 −0.916706 −0.458353 0.888770i \(-0.651561\pi\)
−0.458353 + 0.888770i \(0.651561\pi\)
\(444\) 0 0
\(445\) −9.20155 −0.436195
\(446\) 48.5395 2.29841
\(447\) 0 0
\(448\) −1.34753 −0.0636648
\(449\) −3.84275 −0.181351 −0.0906753 0.995881i \(-0.528903\pi\)
−0.0906753 + 0.995881i \(0.528903\pi\)
\(450\) 0 0
\(451\) −12.4738 −0.587367
\(452\) −14.5761 −0.685602
\(453\) 0 0
\(454\) −23.0147 −1.08014
\(455\) 3.93712 0.184575
\(456\) 0 0
\(457\) 14.7578 0.690342 0.345171 0.938540i \(-0.387821\pi\)
0.345171 + 0.938540i \(0.387821\pi\)
\(458\) −44.2929 −2.06967
\(459\) 0 0
\(460\) 26.3419 1.22820
\(461\) 22.2373 1.03569 0.517847 0.855473i \(-0.326733\pi\)
0.517847 + 0.855473i \(0.326733\pi\)
\(462\) 0 0
\(463\) −23.1151 −1.07425 −0.537124 0.843503i \(-0.680489\pi\)
−0.537124 + 0.843503i \(0.680489\pi\)
\(464\) 21.1181 0.980384
\(465\) 0 0
\(466\) 16.6899 0.773145
\(467\) −27.9366 −1.29275 −0.646376 0.763019i \(-0.723716\pi\)
−0.646376 + 0.763019i \(0.723716\pi\)
\(468\) 0 0
\(469\) −11.3637 −0.524729
\(470\) −4.57176 −0.210880
\(471\) 0 0
\(472\) −15.6362 −0.719712
\(473\) −3.42002 −0.157253
\(474\) 0 0
\(475\) −24.6448 −1.13078
\(476\) 0.145821 0.00668370
\(477\) 0 0
\(478\) −3.62282 −0.165704
\(479\) −0.489744 −0.0223770 −0.0111885 0.999937i \(-0.503561\pi\)
−0.0111885 + 0.999937i \(0.503561\pi\)
\(480\) 0 0
\(481\) −0.289377 −0.0131944
\(482\) 19.3509 0.881410
\(483\) 0 0
\(484\) −7.09369 −0.322441
\(485\) 10.2203 0.464082
\(486\) 0 0
\(487\) −20.6762 −0.936928 −0.468464 0.883483i \(-0.655192\pi\)
−0.468464 + 0.883483i \(0.655192\pi\)
\(488\) −20.3149 −0.919610
\(489\) 0 0
\(490\) −7.08062 −0.319870
\(491\) −34.3912 −1.55205 −0.776026 0.630701i \(-0.782767\pi\)
−0.776026 + 0.630701i \(0.782767\pi\)
\(492\) 0 0
\(493\) 0.640940 0.0288665
\(494\) −3.40126 −0.153030
\(495\) 0 0
\(496\) 12.3220 0.553273
\(497\) 6.20333 0.278257
\(498\) 0 0
\(499\) 6.72669 0.301128 0.150564 0.988600i \(-0.451891\pi\)
0.150564 + 0.988600i \(0.451891\pi\)
\(500\) 27.4112 1.22587
\(501\) 0 0
\(502\) −8.70954 −0.388726
\(503\) 17.0515 0.760290 0.380145 0.924927i \(-0.375874\pi\)
0.380145 + 0.924927i \(0.375874\pi\)
\(504\) 0 0
\(505\) 70.5806 3.14080
\(506\) 21.8064 0.969411
\(507\) 0 0
\(508\) −0.961212 −0.0426469
\(509\) 2.99210 0.132623 0.0663113 0.997799i \(-0.478877\pi\)
0.0663113 + 0.997799i \(0.478877\pi\)
\(510\) 0 0
\(511\) 2.03377 0.0899686
\(512\) −7.25402 −0.320585
\(513\) 0 0
\(514\) 45.9831 2.02823
\(515\) 38.1604 1.68155
\(516\) 0 0
\(517\) −1.22848 −0.0540287
\(518\) 0.520423 0.0228661
\(519\) 0 0
\(520\) −7.03786 −0.308631
\(521\) 25.3004 1.10843 0.554216 0.832373i \(-0.313018\pi\)
0.554216 + 0.832373i \(0.313018\pi\)
\(522\) 0 0
\(523\) −10.6453 −0.465485 −0.232743 0.972538i \(-0.574770\pi\)
−0.232743 + 0.972538i \(0.574770\pi\)
\(524\) 12.0212 0.525150
\(525\) 0 0
\(526\) −8.36519 −0.364740
\(527\) 0.373975 0.0162906
\(528\) 0 0
\(529\) 21.3590 0.928651
\(530\) 4.22317 0.183443
\(531\) 0 0
\(532\) 1.98555 0.0860846
\(533\) −6.27311 −0.271719
\(534\) 0 0
\(535\) 66.5450 2.87699
\(536\) 20.3134 0.877406
\(537\) 0 0
\(538\) 22.9198 0.988142
\(539\) −1.90264 −0.0819527
\(540\) 0 0
\(541\) −41.9713 −1.80449 −0.902244 0.431225i \(-0.858082\pi\)
−0.902244 + 0.431225i \(0.858082\pi\)
\(542\) −55.8222 −2.39777
\(543\) 0 0
\(544\) −0.762528 −0.0326931
\(545\) −43.0383 −1.84356
\(546\) 0 0
\(547\) −3.21658 −0.137531 −0.0687656 0.997633i \(-0.521906\pi\)
−0.0687656 + 0.997633i \(0.521906\pi\)
\(548\) 12.3403 0.527150
\(549\) 0 0
\(550\) 39.0621 1.66561
\(551\) 8.72727 0.371794
\(552\) 0 0
\(553\) −8.04816 −0.342243
\(554\) −22.2344 −0.944648
\(555\) 0 0
\(556\) 14.9981 0.636063
\(557\) 39.5930 1.67761 0.838804 0.544434i \(-0.183255\pi\)
0.838804 + 0.544434i \(0.183255\pi\)
\(558\) 0 0
\(559\) −1.71994 −0.0727457
\(560\) 20.5672 0.869124
\(561\) 0 0
\(562\) 29.8713 1.26004
\(563\) −25.3819 −1.06972 −0.534860 0.844941i \(-0.679636\pi\)
−0.534860 + 0.844941i \(0.679636\pi\)
\(564\) 0 0
\(565\) −62.3963 −2.62503
\(566\) −38.5119 −1.61878
\(567\) 0 0
\(568\) −11.0889 −0.465278
\(569\) 40.5881 1.70154 0.850771 0.525537i \(-0.176136\pi\)
0.850771 + 0.525537i \(0.176136\pi\)
\(570\) 0 0
\(571\) 43.8798 1.83631 0.918156 0.396220i \(-0.129678\pi\)
0.918156 + 0.396220i \(0.129678\pi\)
\(572\) 1.74992 0.0731680
\(573\) 0 0
\(574\) 11.2817 0.470890
\(575\) 79.4610 3.31375
\(576\) 0 0
\(577\) −25.8864 −1.07767 −0.538833 0.842413i \(-0.681135\pi\)
−0.538833 + 0.842413i \(0.681135\pi\)
\(578\) 29.2143 1.21515
\(579\) 0 0
\(580\) −16.7098 −0.693838
\(581\) −9.02002 −0.374213
\(582\) 0 0
\(583\) 1.13481 0.0469992
\(584\) −3.63549 −0.150438
\(585\) 0 0
\(586\) 18.0299 0.744807
\(587\) 24.9754 1.03085 0.515423 0.856936i \(-0.327635\pi\)
0.515423 + 0.856936i \(0.327635\pi\)
\(588\) 0 0
\(589\) 5.09217 0.209819
\(590\) 61.9355 2.54984
\(591\) 0 0
\(592\) −1.51168 −0.0621298
\(593\) 31.7171 1.30246 0.651232 0.758878i \(-0.274252\pi\)
0.651232 + 0.758878i \(0.274252\pi\)
\(594\) 0 0
\(595\) 0.624220 0.0255905
\(596\) 2.06729 0.0846795
\(597\) 0 0
\(598\) 10.9665 0.448453
\(599\) −7.33960 −0.299888 −0.149944 0.988694i \(-0.547909\pi\)
−0.149944 + 0.988694i \(0.547909\pi\)
\(600\) 0 0
\(601\) 41.9979 1.71313 0.856565 0.516039i \(-0.172594\pi\)
0.856565 + 0.516039i \(0.172594\pi\)
\(602\) 3.09318 0.126069
\(603\) 0 0
\(604\) −1.21707 −0.0495219
\(605\) −30.3662 −1.23456
\(606\) 0 0
\(607\) −19.8014 −0.803716 −0.401858 0.915702i \(-0.631635\pi\)
−0.401858 + 0.915702i \(0.631635\pi\)
\(608\) −10.3829 −0.421080
\(609\) 0 0
\(610\) 80.4680 3.25806
\(611\) −0.617809 −0.0249939
\(612\) 0 0
\(613\) 21.9400 0.886150 0.443075 0.896485i \(-0.353888\pi\)
0.443075 + 0.896485i \(0.353888\pi\)
\(614\) −16.0400 −0.647320
\(615\) 0 0
\(616\) 3.40110 0.137034
\(617\) 32.2483 1.29827 0.649133 0.760675i \(-0.275132\pi\)
0.649133 + 0.760675i \(0.275132\pi\)
\(618\) 0 0
\(619\) 28.0029 1.12553 0.562765 0.826617i \(-0.309737\pi\)
0.562765 + 0.826617i \(0.309737\pi\)
\(620\) −9.74982 −0.391562
\(621\) 0 0
\(622\) −4.76761 −0.191164
\(623\) 2.23627 0.0895943
\(624\) 0 0
\(625\) 57.6868 2.30747
\(626\) −4.21253 −0.168366
\(627\) 0 0
\(628\) −2.74978 −0.109728
\(629\) −0.0458799 −0.00182935
\(630\) 0 0
\(631\) −4.65983 −0.185505 −0.0927524 0.995689i \(-0.529567\pi\)
−0.0927524 + 0.995689i \(0.529567\pi\)
\(632\) 14.3866 0.572268
\(633\) 0 0
\(634\) 23.7383 0.942770
\(635\) −4.11468 −0.163286
\(636\) 0 0
\(637\) −0.956847 −0.0379116
\(638\) −13.8327 −0.547644
\(639\) 0 0
\(640\) −50.9052 −2.01220
\(641\) 20.5992 0.813619 0.406809 0.913513i \(-0.366641\pi\)
0.406809 + 0.913513i \(0.366641\pi\)
\(642\) 0 0
\(643\) −40.0660 −1.58005 −0.790025 0.613074i \(-0.789933\pi\)
−0.790025 + 0.613074i \(0.789933\pi\)
\(644\) −6.40191 −0.252271
\(645\) 0 0
\(646\) −0.539260 −0.0212169
\(647\) −16.4147 −0.645328 −0.322664 0.946514i \(-0.604578\pi\)
−0.322664 + 0.946514i \(0.604578\pi\)
\(648\) 0 0
\(649\) 16.6428 0.653286
\(650\) 19.6445 0.770520
\(651\) 0 0
\(652\) 12.1687 0.476561
\(653\) −8.93179 −0.349528 −0.174764 0.984610i \(-0.555916\pi\)
−0.174764 + 0.984610i \(0.555916\pi\)
\(654\) 0 0
\(655\) 51.4596 2.01069
\(656\) −32.7703 −1.27946
\(657\) 0 0
\(658\) 1.11108 0.0433145
\(659\) −26.5807 −1.03544 −0.517718 0.855552i \(-0.673218\pi\)
−0.517718 + 0.855552i \(0.673218\pi\)
\(660\) 0 0
\(661\) 44.3578 1.72532 0.862660 0.505785i \(-0.168797\pi\)
0.862660 + 0.505785i \(0.168797\pi\)
\(662\) −22.3632 −0.869171
\(663\) 0 0
\(664\) 16.1239 0.625727
\(665\) 8.49961 0.329601
\(666\) 0 0
\(667\) −28.1389 −1.08954
\(668\) 16.4022 0.634619
\(669\) 0 0
\(670\) −80.4623 −3.10853
\(671\) 21.6227 0.834734
\(672\) 0 0
\(673\) −37.6652 −1.45189 −0.725943 0.687755i \(-0.758596\pi\)
−0.725943 + 0.687755i \(0.758596\pi\)
\(674\) −17.4790 −0.673266
\(675\) 0 0
\(676\) −11.6157 −0.446758
\(677\) 8.65007 0.332449 0.166225 0.986088i \(-0.446842\pi\)
0.166225 + 0.986088i \(0.446842\pi\)
\(678\) 0 0
\(679\) −2.48387 −0.0953221
\(680\) −1.11583 −0.0427903
\(681\) 0 0
\(682\) −8.07111 −0.309059
\(683\) 10.7169 0.410071 0.205035 0.978755i \(-0.434269\pi\)
0.205035 + 0.978755i \(0.434269\pi\)
\(684\) 0 0
\(685\) 52.8252 2.01835
\(686\) 1.72082 0.0657011
\(687\) 0 0
\(688\) −8.98484 −0.342544
\(689\) 0.570702 0.0217420
\(690\) 0 0
\(691\) −16.7214 −0.636110 −0.318055 0.948072i \(-0.603030\pi\)
−0.318055 + 0.948072i \(0.603030\pi\)
\(692\) 18.4164 0.700086
\(693\) 0 0
\(694\) 12.8027 0.485983
\(695\) 64.2029 2.43535
\(696\) 0 0
\(697\) −0.994585 −0.0376726
\(698\) 28.8673 1.09264
\(699\) 0 0
\(700\) −11.4679 −0.433445
\(701\) 26.9332 1.01725 0.508627 0.860987i \(-0.330153\pi\)
0.508627 + 0.860987i \(0.330153\pi\)
\(702\) 0 0
\(703\) −0.624718 −0.0235617
\(704\) −2.56387 −0.0966294
\(705\) 0 0
\(706\) −18.1932 −0.684709
\(707\) −17.1534 −0.645118
\(708\) 0 0
\(709\) 3.60561 0.135411 0.0677057 0.997705i \(-0.478432\pi\)
0.0677057 + 0.997705i \(0.478432\pi\)
\(710\) 43.9234 1.64842
\(711\) 0 0
\(712\) −3.99748 −0.149812
\(713\) −16.4184 −0.614875
\(714\) 0 0
\(715\) 7.49094 0.280145
\(716\) 13.1141 0.490096
\(717\) 0 0
\(718\) −35.0276 −1.30722
\(719\) −48.7999 −1.81993 −0.909964 0.414686i \(-0.863891\pi\)
−0.909964 + 0.414686i \(0.863891\pi\)
\(720\) 0 0
\(721\) −9.27419 −0.345389
\(722\) 25.3528 0.943532
\(723\) 0 0
\(724\) −13.8150 −0.513432
\(725\) −50.4057 −1.87202
\(726\) 0 0
\(727\) 36.5934 1.35717 0.678587 0.734520i \(-0.262593\pi\)
0.678587 + 0.734520i \(0.262593\pi\)
\(728\) 1.71042 0.0633926
\(729\) 0 0
\(730\) 14.4003 0.532981
\(731\) −0.272692 −0.0100859
\(732\) 0 0
\(733\) −23.0187 −0.850214 −0.425107 0.905143i \(-0.639764\pi\)
−0.425107 + 0.905143i \(0.639764\pi\)
\(734\) −52.3707 −1.93304
\(735\) 0 0
\(736\) 33.4769 1.23397
\(737\) −21.6212 −0.796425
\(738\) 0 0
\(739\) −41.9045 −1.54148 −0.770740 0.637149i \(-0.780113\pi\)
−0.770740 + 0.637149i \(0.780113\pi\)
\(740\) 1.19613 0.0439705
\(741\) 0 0
\(742\) −1.02637 −0.0376791
\(743\) −8.99203 −0.329886 −0.164943 0.986303i \(-0.552744\pi\)
−0.164943 + 0.986303i \(0.552744\pi\)
\(744\) 0 0
\(745\) 8.84950 0.324221
\(746\) −38.7439 −1.41852
\(747\) 0 0
\(748\) 0.277446 0.0101444
\(749\) −16.1726 −0.590932
\(750\) 0 0
\(751\) −36.1068 −1.31756 −0.658778 0.752338i \(-0.728926\pi\)
−0.658778 + 0.752338i \(0.728926\pi\)
\(752\) −3.22739 −0.117691
\(753\) 0 0
\(754\) −6.95654 −0.253342
\(755\) −5.20995 −0.189609
\(756\) 0 0
\(757\) 39.8948 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(758\) 55.1205 2.00207
\(759\) 0 0
\(760\) −15.1936 −0.551130
\(761\) 12.5087 0.453440 0.226720 0.973960i \(-0.427200\pi\)
0.226720 + 0.973960i \(0.427200\pi\)
\(762\) 0 0
\(763\) 10.4597 0.378666
\(764\) 23.2763 0.842106
\(765\) 0 0
\(766\) −29.4546 −1.06424
\(767\) 8.36971 0.302213
\(768\) 0 0
\(769\) 28.6812 1.03427 0.517136 0.855904i \(-0.326998\pi\)
0.517136 + 0.855904i \(0.326998\pi\)
\(770\) −13.4719 −0.485494
\(771\) 0 0
\(772\) 5.24347 0.188717
\(773\) −31.2783 −1.12500 −0.562501 0.826797i \(-0.690161\pi\)
−0.562501 + 0.826797i \(0.690161\pi\)
\(774\) 0 0
\(775\) −29.4106 −1.05646
\(776\) 4.44007 0.159389
\(777\) 0 0
\(778\) 54.1471 1.94127
\(779\) −13.5426 −0.485215
\(780\) 0 0
\(781\) 11.8027 0.422335
\(782\) 1.73871 0.0621761
\(783\) 0 0
\(784\) −4.99850 −0.178518
\(785\) −11.7711 −0.420128
\(786\) 0 0
\(787\) 12.1523 0.433184 0.216592 0.976262i \(-0.430506\pi\)
0.216592 + 0.976262i \(0.430506\pi\)
\(788\) 5.26902 0.187701
\(789\) 0 0
\(790\) −56.9860 −2.02747
\(791\) 15.1643 0.539180
\(792\) 0 0
\(793\) 10.8741 0.386152
\(794\) 50.4812 1.79151
\(795\) 0 0
\(796\) −8.48559 −0.300764
\(797\) −40.0647 −1.41916 −0.709582 0.704623i \(-0.751116\pi\)
−0.709582 + 0.704623i \(0.751116\pi\)
\(798\) 0 0
\(799\) −0.0979520 −0.00346529
\(800\) 59.9677 2.12018
\(801\) 0 0
\(802\) −5.20466 −0.183783
\(803\) 3.86954 0.136553
\(804\) 0 0
\(805\) −27.4048 −0.965894
\(806\) −4.05899 −0.142972
\(807\) 0 0
\(808\) 30.6627 1.07871
\(809\) 17.2428 0.606226 0.303113 0.952955i \(-0.401974\pi\)
0.303113 + 0.952955i \(0.401974\pi\)
\(810\) 0 0
\(811\) −11.7905 −0.414019 −0.207010 0.978339i \(-0.566373\pi\)
−0.207010 + 0.978339i \(0.566373\pi\)
\(812\) 4.06102 0.142514
\(813\) 0 0
\(814\) 0.990179 0.0347058
\(815\) 52.0907 1.82466
\(816\) 0 0
\(817\) −3.71307 −0.129904
\(818\) 14.7673 0.516325
\(819\) 0 0
\(820\) 25.9296 0.905502
\(821\) −22.4832 −0.784670 −0.392335 0.919822i \(-0.628333\pi\)
−0.392335 + 0.919822i \(0.628333\pi\)
\(822\) 0 0
\(823\) 15.2298 0.530878 0.265439 0.964128i \(-0.414483\pi\)
0.265439 + 0.964128i \(0.414483\pi\)
\(824\) 16.5782 0.577529
\(825\) 0 0
\(826\) −15.0523 −0.523737
\(827\) 30.0385 1.04454 0.522271 0.852780i \(-0.325085\pi\)
0.522271 + 0.852780i \(0.325085\pi\)
\(828\) 0 0
\(829\) 16.9736 0.589517 0.294758 0.955572i \(-0.404761\pi\)
0.294758 + 0.955572i \(0.404761\pi\)
\(830\) −63.8673 −2.21687
\(831\) 0 0
\(832\) −1.28938 −0.0447012
\(833\) −0.151705 −0.00525628
\(834\) 0 0
\(835\) 70.2132 2.42983
\(836\) 3.77780 0.130658
\(837\) 0 0
\(838\) 33.0052 1.14015
\(839\) 24.8666 0.858490 0.429245 0.903188i \(-0.358780\pi\)
0.429245 + 0.903188i \(0.358780\pi\)
\(840\) 0 0
\(841\) −11.1502 −0.384491
\(842\) −55.8489 −1.92468
\(843\) 0 0
\(844\) −4.76818 −0.164128
\(845\) −49.7237 −1.71055
\(846\) 0 0
\(847\) 7.37995 0.253578
\(848\) 2.98131 0.102378
\(849\) 0 0
\(850\) 3.11458 0.106829
\(851\) 2.01424 0.0690474
\(852\) 0 0
\(853\) −3.82276 −0.130889 −0.0654443 0.997856i \(-0.520846\pi\)
−0.0654443 + 0.997856i \(0.520846\pi\)
\(854\) −19.5563 −0.669203
\(855\) 0 0
\(856\) 28.9095 0.988106
\(857\) 11.5675 0.395137 0.197569 0.980289i \(-0.436696\pi\)
0.197569 + 0.980289i \(0.436696\pi\)
\(858\) 0 0
\(859\) 21.9356 0.748434 0.374217 0.927341i \(-0.377912\pi\)
0.374217 + 0.927341i \(0.377912\pi\)
\(860\) 7.10930 0.242425
\(861\) 0 0
\(862\) −33.4266 −1.13851
\(863\) 37.0986 1.26285 0.631426 0.775436i \(-0.282470\pi\)
0.631426 + 0.775436i \(0.282470\pi\)
\(864\) 0 0
\(865\) 78.8355 2.68049
\(866\) −28.1651 −0.957089
\(867\) 0 0
\(868\) 2.36952 0.0804267
\(869\) −15.3128 −0.519451
\(870\) 0 0
\(871\) −10.8734 −0.368430
\(872\) −18.6974 −0.633173
\(873\) 0 0
\(874\) 23.6749 0.800815
\(875\) −28.5174 −0.964063
\(876\) 0 0
\(877\) 44.2268 1.49343 0.746716 0.665143i \(-0.231629\pi\)
0.746716 + 0.665143i \(0.231629\pi\)
\(878\) −27.2290 −0.918934
\(879\) 0 0
\(880\) 39.1321 1.31914
\(881\) −42.8685 −1.44427 −0.722137 0.691750i \(-0.756840\pi\)
−0.722137 + 0.691750i \(0.756840\pi\)
\(882\) 0 0
\(883\) −19.5065 −0.656447 −0.328223 0.944600i \(-0.606450\pi\)
−0.328223 + 0.944600i \(0.606450\pi\)
\(884\) 0.139528 0.00469285
\(885\) 0 0
\(886\) 33.2022 1.11545
\(887\) 30.7871 1.03373 0.516864 0.856067i \(-0.327099\pi\)
0.516864 + 0.856067i \(0.327099\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 15.8342 0.530763
\(891\) 0 0
\(892\) −27.1132 −0.907815
\(893\) −1.33375 −0.0446322
\(894\) 0 0
\(895\) 56.1378 1.87648
\(896\) 12.3716 0.413306
\(897\) 0 0
\(898\) 6.61267 0.220668
\(899\) 10.4149 0.347358
\(900\) 0 0
\(901\) 0.0904833 0.00301444
\(902\) 21.4651 0.714710
\(903\) 0 0
\(904\) −27.1072 −0.901571
\(905\) −59.1384 −1.96583
\(906\) 0 0
\(907\) 58.4950 1.94229 0.971147 0.238483i \(-0.0766500\pi\)
0.971147 + 0.238483i \(0.0766500\pi\)
\(908\) 12.8555 0.426626
\(909\) 0 0
\(910\) −6.77507 −0.224591
\(911\) 17.4513 0.578188 0.289094 0.957301i \(-0.406646\pi\)
0.289094 + 0.957301i \(0.406646\pi\)
\(912\) 0 0
\(913\) −17.1619 −0.567975
\(914\) −25.3955 −0.840009
\(915\) 0 0
\(916\) 24.7411 0.817469
\(917\) −12.5063 −0.412995
\(918\) 0 0
\(919\) 11.1696 0.368452 0.184226 0.982884i \(-0.441022\pi\)
0.184226 + 0.982884i \(0.441022\pi\)
\(920\) 48.9879 1.61508
\(921\) 0 0
\(922\) −38.2664 −1.26024
\(923\) 5.93564 0.195374
\(924\) 0 0
\(925\) 3.60815 0.118635
\(926\) 39.7768 1.30715
\(927\) 0 0
\(928\) −21.2359 −0.697102
\(929\) −31.6788 −1.03935 −0.519673 0.854365i \(-0.673946\pi\)
−0.519673 + 0.854365i \(0.673946\pi\)
\(930\) 0 0
\(931\) −2.06568 −0.0676998
\(932\) −9.32263 −0.305373
\(933\) 0 0
\(934\) 48.0738 1.57302
\(935\) 1.18767 0.0388409
\(936\) 0 0
\(937\) 9.99759 0.326607 0.163304 0.986576i \(-0.447785\pi\)
0.163304 + 0.986576i \(0.447785\pi\)
\(938\) 19.5549 0.638491
\(939\) 0 0
\(940\) 2.55369 0.0832921
\(941\) −40.8696 −1.33231 −0.666155 0.745813i \(-0.732061\pi\)
−0.666155 + 0.745813i \(0.732061\pi\)
\(942\) 0 0
\(943\) 43.6648 1.42192
\(944\) 43.7228 1.42305
\(945\) 0 0
\(946\) 5.88523 0.191345
\(947\) 20.8268 0.676781 0.338391 0.941006i \(-0.390117\pi\)
0.338391 + 0.941006i \(0.390117\pi\)
\(948\) 0 0
\(949\) 1.94600 0.0631700
\(950\) 42.4092 1.37594
\(951\) 0 0
\(952\) 0.271183 0.00878910
\(953\) −21.2199 −0.687381 −0.343690 0.939083i \(-0.611677\pi\)
−0.343690 + 0.939083i \(0.611677\pi\)
\(954\) 0 0
\(955\) 99.6393 3.22425
\(956\) 2.02363 0.0654489
\(957\) 0 0
\(958\) 0.842759 0.0272283
\(959\) −12.8382 −0.414568
\(960\) 0 0
\(961\) −24.9231 −0.803971
\(962\) 0.497965 0.0160550
\(963\) 0 0
\(964\) −10.8090 −0.348135
\(965\) 22.4459 0.722558
\(966\) 0 0
\(967\) −30.7468 −0.988751 −0.494375 0.869249i \(-0.664603\pi\)
−0.494375 + 0.869249i \(0.664603\pi\)
\(968\) −13.1921 −0.424011
\(969\) 0 0
\(970\) −17.5873 −0.564695
\(971\) 42.4586 1.36256 0.681280 0.732022i \(-0.261423\pi\)
0.681280 + 0.732022i \(0.261423\pi\)
\(972\) 0 0
\(973\) −15.6034 −0.500221
\(974\) 35.5799 1.14005
\(975\) 0 0
\(976\) 56.8056 1.81830
\(977\) 46.3699 1.48351 0.741753 0.670673i \(-0.233995\pi\)
0.741753 + 0.670673i \(0.233995\pi\)
\(978\) 0 0
\(979\) 4.25483 0.135985
\(980\) 3.95508 0.126341
\(981\) 0 0
\(982\) 59.1809 1.88854
\(983\) −38.3831 −1.22423 −0.612115 0.790769i \(-0.709681\pi\)
−0.612115 + 0.790769i \(0.709681\pi\)
\(984\) 0 0
\(985\) 22.5552 0.718669
\(986\) −1.10294 −0.0351248
\(987\) 0 0
\(988\) 1.89987 0.0604429
\(989\) 11.9719 0.380683
\(990\) 0 0
\(991\) −46.2877 −1.47038 −0.735188 0.677863i \(-0.762906\pi\)
−0.735188 + 0.677863i \(0.762906\pi\)
\(992\) −12.3907 −0.393404
\(993\) 0 0
\(994\) −10.6748 −0.338584
\(995\) −36.3245 −1.15156
\(996\) 0 0
\(997\) 5.68852 0.180157 0.0900786 0.995935i \(-0.471288\pi\)
0.0900786 + 0.995935i \(0.471288\pi\)
\(998\) −11.5754 −0.366413
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8001.2.a.t.1.3 16
3.2 odd 2 889.2.a.c.1.14 16
21.20 even 2 6223.2.a.k.1.14 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
889.2.a.c.1.14 16 3.2 odd 2
6223.2.a.k.1.14 16 21.20 even 2
8001.2.a.t.1.3 16 1.1 even 1 trivial