Properties

Label 624.4.a.u.1.1
Level $624$
Weight $4$
Character 624.1
Self dual yes
Analytic conductor $36.817$
Analytic rank $0$
Dimension $3$
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] = [624,4,Mod(1,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.36248.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 54x - 90 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.84636\) of defining polynomial
Character \(\chi\) \(=\) 624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -10.8037 q^{5} +32.1891 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -10.8037 q^{5} +32.1891 q^{7} +9.00000 q^{9} +30.2219 q^{11} +13.0000 q^{13} -32.4111 q^{15} +34.0000 q^{17} -41.8621 q^{19} +96.5674 q^{21} -45.5417 q^{23} -8.28038 q^{25} +27.0000 q^{27} +2.40702 q^{29} -73.7965 q^{31} +90.6658 q^{33} -347.761 q^{35} +401.383 q^{37} +39.0000 q^{39} -353.619 q^{41} +329.274 q^{43} -97.2332 q^{45} -45.1373 q^{47} +693.139 q^{49} +102.000 q^{51} +449.761 q^{53} -326.508 q^{55} -125.586 q^{57} +351.640 q^{59} +872.044 q^{61} +289.702 q^{63} -140.448 q^{65} +177.463 q^{67} -136.625 q^{69} -32.9352 q^{71} +777.906 q^{73} -24.8411 q^{75} +972.818 q^{77} -350.766 q^{79} +81.0000 q^{81} +421.087 q^{83} -367.325 q^{85} +7.22106 q^{87} -1365.76 q^{89} +418.459 q^{91} -221.389 q^{93} +452.265 q^{95} +468.206 q^{97} +271.997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 9 q^{3} + 16 q^{5} + 22 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 9 q^{3} + 16 q^{5} + 22 q^{7} + 27 q^{9} + 20 q^{11} + 39 q^{13} + 48 q^{15} + 102 q^{17} + 38 q^{19} + 66 q^{21} - 32 q^{23} + 161 q^{25} + 81 q^{27} + 350 q^{29} - 50 q^{31} + 60 q^{33} - 232 q^{35} + 542 q^{37} + 117 q^{39} + 500 q^{41} - 420 q^{43} + 144 q^{45} - 324 q^{47} + 1119 q^{49} + 306 q^{51} + 538 q^{53} - 896 q^{55} + 114 q^{57} - 112 q^{59} + 1206 q^{61} + 198 q^{63} + 208 q^{65} - 950 q^{67} - 96 q^{69} + 1008 q^{71} + 1834 q^{73} + 483 q^{75} - 792 q^{77} + 272 q^{79} + 243 q^{81} + 1640 q^{83} + 544 q^{85} + 1050 q^{87} - 1576 q^{89} + 286 q^{91} - 150 q^{93} + 2760 q^{95} + 402 q^{97} + 180 q^{99}+O(q^{100}) \) 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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −10.8037 −0.966311 −0.483156 0.875535i \(-0.660509\pi\)
−0.483156 + 0.875535i \(0.660509\pi\)
\(6\) 0 0
\(7\) 32.1891 1.73805 0.869025 0.494769i \(-0.164747\pi\)
0.869025 + 0.494769i \(0.164747\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 30.2219 0.828387 0.414194 0.910189i \(-0.364064\pi\)
0.414194 + 0.910189i \(0.364064\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) −32.4111 −0.557900
\(16\) 0 0
\(17\) 34.0000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −41.8621 −0.505465 −0.252732 0.967536i \(-0.581329\pi\)
−0.252732 + 0.967536i \(0.581329\pi\)
\(20\) 0 0
\(21\) 96.5674 1.00346
\(22\) 0 0
\(23\) −45.5417 −0.412874 −0.206437 0.978460i \(-0.566187\pi\)
−0.206437 + 0.978460i \(0.566187\pi\)
\(24\) 0 0
\(25\) −8.28038 −0.0662430
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 2.40702 0.0154128 0.00770642 0.999970i \(-0.497547\pi\)
0.00770642 + 0.999970i \(0.497547\pi\)
\(30\) 0 0
\(31\) −73.7965 −0.427556 −0.213778 0.976882i \(-0.568577\pi\)
−0.213778 + 0.976882i \(0.568577\pi\)
\(32\) 0 0
\(33\) 90.6658 0.478269
\(34\) 0 0
\(35\) −347.761 −1.67950
\(36\) 0 0
\(37\) 401.383 1.78343 0.891715 0.452597i \(-0.149502\pi\)
0.891715 + 0.452597i \(0.149502\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) −353.619 −1.34698 −0.673489 0.739197i \(-0.735205\pi\)
−0.673489 + 0.739197i \(0.735205\pi\)
\(42\) 0 0
\(43\) 329.274 1.16776 0.583882 0.811839i \(-0.301533\pi\)
0.583882 + 0.811839i \(0.301533\pi\)
\(44\) 0 0
\(45\) −97.2332 −0.322104
\(46\) 0 0
\(47\) −45.1373 −0.140084 −0.0700420 0.997544i \(-0.522313\pi\)
−0.0700420 + 0.997544i \(0.522313\pi\)
\(48\) 0 0
\(49\) 693.139 2.02081
\(50\) 0 0
\(51\) 102.000 0.280056
\(52\) 0 0
\(53\) 449.761 1.16565 0.582825 0.812598i \(-0.301947\pi\)
0.582825 + 0.812598i \(0.301947\pi\)
\(54\) 0 0
\(55\) −326.508 −0.800479
\(56\) 0 0
\(57\) −125.586 −0.291830
\(58\) 0 0
\(59\) 351.640 0.775927 0.387963 0.921675i \(-0.373179\pi\)
0.387963 + 0.921675i \(0.373179\pi\)
\(60\) 0 0
\(61\) 872.044 1.83039 0.915195 0.403012i \(-0.132037\pi\)
0.915195 + 0.403012i \(0.132037\pi\)
\(62\) 0 0
\(63\) 289.702 0.579350
\(64\) 0 0
\(65\) −140.448 −0.268006
\(66\) 0 0
\(67\) 177.463 0.323591 0.161795 0.986824i \(-0.448272\pi\)
0.161795 + 0.986824i \(0.448272\pi\)
\(68\) 0 0
\(69\) −136.625 −0.238373
\(70\) 0 0
\(71\) −32.9352 −0.0550520 −0.0275260 0.999621i \(-0.508763\pi\)
−0.0275260 + 0.999621i \(0.508763\pi\)
\(72\) 0 0
\(73\) 777.906 1.24722 0.623609 0.781736i \(-0.285666\pi\)
0.623609 + 0.781736i \(0.285666\pi\)
\(74\) 0 0
\(75\) −24.8411 −0.0382454
\(76\) 0 0
\(77\) 972.818 1.43978
\(78\) 0 0
\(79\) −350.766 −0.499547 −0.249774 0.968304i \(-0.580356\pi\)
−0.249774 + 0.968304i \(0.580356\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 421.087 0.556871 0.278436 0.960455i \(-0.410184\pi\)
0.278436 + 0.960455i \(0.410184\pi\)
\(84\) 0 0
\(85\) −367.325 −0.468730
\(86\) 0 0
\(87\) 7.22106 0.00889861
\(88\) 0 0
\(89\) −1365.76 −1.62663 −0.813315 0.581823i \(-0.802340\pi\)
−0.813315 + 0.581823i \(0.802340\pi\)
\(90\) 0 0
\(91\) 418.459 0.482048
\(92\) 0 0
\(93\) −221.389 −0.246850
\(94\) 0 0
\(95\) 452.265 0.488436
\(96\) 0 0
\(97\) 468.206 0.490094 0.245047 0.969511i \(-0.421197\pi\)
0.245047 + 0.969511i \(0.421197\pi\)
\(98\) 0 0
\(99\) 271.997 0.276129
\(100\) 0 0
\(101\) 548.199 0.540077 0.270039 0.962849i \(-0.412964\pi\)
0.270039 + 0.962849i \(0.412964\pi\)
\(102\) 0 0
\(103\) 917.200 0.877422 0.438711 0.898628i \(-0.355435\pi\)
0.438711 + 0.898628i \(0.355435\pi\)
\(104\) 0 0
\(105\) −1043.28 −0.969657
\(106\) 0 0
\(107\) −1279.54 −1.15605 −0.578025 0.816019i \(-0.696177\pi\)
−0.578025 + 0.816019i \(0.696177\pi\)
\(108\) 0 0
\(109\) −753.302 −0.661957 −0.330978 0.943638i \(-0.607379\pi\)
−0.330978 + 0.943638i \(0.607379\pi\)
\(110\) 0 0
\(111\) 1204.15 1.02966
\(112\) 0 0
\(113\) −1195.29 −0.995074 −0.497537 0.867443i \(-0.665762\pi\)
−0.497537 + 0.867443i \(0.665762\pi\)
\(114\) 0 0
\(115\) 492.019 0.398965
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) 1094.43 0.843078
\(120\) 0 0
\(121\) −417.634 −0.313775
\(122\) 0 0
\(123\) −1060.86 −0.777678
\(124\) 0 0
\(125\) 1439.92 1.03032
\(126\) 0 0
\(127\) −1397.95 −0.976758 −0.488379 0.872632i \(-0.662412\pi\)
−0.488379 + 0.872632i \(0.662412\pi\)
\(128\) 0 0
\(129\) 987.822 0.674208
\(130\) 0 0
\(131\) 1692.92 1.12909 0.564546 0.825401i \(-0.309051\pi\)
0.564546 + 0.825401i \(0.309051\pi\)
\(132\) 0 0
\(133\) −1347.51 −0.878523
\(134\) 0 0
\(135\) −291.700 −0.185967
\(136\) 0 0
\(137\) 137.034 0.0854567 0.0427283 0.999087i \(-0.486395\pi\)
0.0427283 + 0.999087i \(0.486395\pi\)
\(138\) 0 0
\(139\) −2586.53 −1.57832 −0.789161 0.614187i \(-0.789484\pi\)
−0.789161 + 0.614187i \(0.789484\pi\)
\(140\) 0 0
\(141\) −135.412 −0.0808775
\(142\) 0 0
\(143\) 392.885 0.229753
\(144\) 0 0
\(145\) −26.0047 −0.0148936
\(146\) 0 0
\(147\) 2079.42 1.16672
\(148\) 0 0
\(149\) −1911.19 −1.05081 −0.525405 0.850852i \(-0.676086\pi\)
−0.525405 + 0.850852i \(0.676086\pi\)
\(150\) 0 0
\(151\) 2007.47 1.08189 0.540946 0.841057i \(-0.318066\pi\)
0.540946 + 0.841057i \(0.318066\pi\)
\(152\) 0 0
\(153\) 306.000 0.161690
\(154\) 0 0
\(155\) 797.274 0.413152
\(156\) 0 0
\(157\) 3067.26 1.55920 0.779599 0.626280i \(-0.215423\pi\)
0.779599 + 0.626280i \(0.215423\pi\)
\(158\) 0 0
\(159\) 1349.28 0.672988
\(160\) 0 0
\(161\) −1465.95 −0.717596
\(162\) 0 0
\(163\) −1188.86 −0.571281 −0.285640 0.958337i \(-0.592206\pi\)
−0.285640 + 0.958337i \(0.592206\pi\)
\(164\) 0 0
\(165\) −979.525 −0.462157
\(166\) 0 0
\(167\) 3325.23 1.54080 0.770401 0.637560i \(-0.220056\pi\)
0.770401 + 0.637560i \(0.220056\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −376.759 −0.168488
\(172\) 0 0
\(173\) −1452.26 −0.638226 −0.319113 0.947717i \(-0.603385\pi\)
−0.319113 + 0.947717i \(0.603385\pi\)
\(174\) 0 0
\(175\) −266.538 −0.115134
\(176\) 0 0
\(177\) 1054.92 0.447982
\(178\) 0 0
\(179\) −2462.51 −1.02825 −0.514125 0.857715i \(-0.671883\pi\)
−0.514125 + 0.857715i \(0.671883\pi\)
\(180\) 0 0
\(181\) 2816.51 1.15663 0.578313 0.815815i \(-0.303711\pi\)
0.578313 + 0.815815i \(0.303711\pi\)
\(182\) 0 0
\(183\) 2616.13 1.05678
\(184\) 0 0
\(185\) −4336.41 −1.72335
\(186\) 0 0
\(187\) 1027.55 0.401827
\(188\) 0 0
\(189\) 869.106 0.334488
\(190\) 0 0
\(191\) 3897.94 1.47667 0.738337 0.674432i \(-0.235611\pi\)
0.738337 + 0.674432i \(0.235611\pi\)
\(192\) 0 0
\(193\) 3906.14 1.45684 0.728420 0.685131i \(-0.240255\pi\)
0.728420 + 0.685131i \(0.240255\pi\)
\(194\) 0 0
\(195\) −421.344 −0.154734
\(196\) 0 0
\(197\) 2945.95 1.06543 0.532717 0.846294i \(-0.321171\pi\)
0.532717 + 0.846294i \(0.321171\pi\)
\(198\) 0 0
\(199\) 535.794 0.190862 0.0954308 0.995436i \(-0.469577\pi\)
0.0954308 + 0.995436i \(0.469577\pi\)
\(200\) 0 0
\(201\) 532.390 0.186825
\(202\) 0 0
\(203\) 77.4799 0.0267883
\(204\) 0 0
\(205\) 3820.39 1.30160
\(206\) 0 0
\(207\) −409.876 −0.137625
\(208\) 0 0
\(209\) −1265.15 −0.418720
\(210\) 0 0
\(211\) −3688.34 −1.20339 −0.601696 0.798725i \(-0.705508\pi\)
−0.601696 + 0.798725i \(0.705508\pi\)
\(212\) 0 0
\(213\) −98.8057 −0.0317843
\(214\) 0 0
\(215\) −3557.37 −1.12842
\(216\) 0 0
\(217\) −2375.44 −0.743114
\(218\) 0 0
\(219\) 2333.72 0.720082
\(220\) 0 0
\(221\) 442.000 0.134535
\(222\) 0 0
\(223\) −2383.77 −0.715827 −0.357913 0.933755i \(-0.616512\pi\)
−0.357913 + 0.933755i \(0.616512\pi\)
\(224\) 0 0
\(225\) −74.5234 −0.0220810
\(226\) 0 0
\(227\) −5654.49 −1.65331 −0.826656 0.562708i \(-0.809760\pi\)
−0.826656 + 0.562708i \(0.809760\pi\)
\(228\) 0 0
\(229\) −4732.77 −1.36572 −0.682861 0.730548i \(-0.739265\pi\)
−0.682861 + 0.730548i \(0.739265\pi\)
\(230\) 0 0
\(231\) 2918.45 0.831256
\(232\) 0 0
\(233\) −1284.93 −0.361281 −0.180640 0.983549i \(-0.557817\pi\)
−0.180640 + 0.983549i \(0.557817\pi\)
\(234\) 0 0
\(235\) 487.649 0.135365
\(236\) 0 0
\(237\) −1052.30 −0.288414
\(238\) 0 0
\(239\) −4658.76 −1.26088 −0.630440 0.776238i \(-0.717125\pi\)
−0.630440 + 0.776238i \(0.717125\pi\)
\(240\) 0 0
\(241\) 2167.09 0.579230 0.289615 0.957143i \(-0.406473\pi\)
0.289615 + 0.957143i \(0.406473\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −7488.46 −1.95274
\(246\) 0 0
\(247\) −544.208 −0.140191
\(248\) 0 0
\(249\) 1263.26 0.321510
\(250\) 0 0
\(251\) −5219.24 −1.31249 −0.656246 0.754547i \(-0.727857\pi\)
−0.656246 + 0.754547i \(0.727857\pi\)
\(252\) 0 0
\(253\) −1376.36 −0.342020
\(254\) 0 0
\(255\) −1101.98 −0.270621
\(256\) 0 0
\(257\) 399.657 0.0970036 0.0485018 0.998823i \(-0.484555\pi\)
0.0485018 + 0.998823i \(0.484555\pi\)
\(258\) 0 0
\(259\) 12920.2 3.09969
\(260\) 0 0
\(261\) 21.6632 0.00513762
\(262\) 0 0
\(263\) −8045.64 −1.88637 −0.943186 0.332267i \(-0.892187\pi\)
−0.943186 + 0.332267i \(0.892187\pi\)
\(264\) 0 0
\(265\) −4859.08 −1.12638
\(266\) 0 0
\(267\) −4097.28 −0.939136
\(268\) 0 0
\(269\) −2261.41 −0.512568 −0.256284 0.966602i \(-0.582498\pi\)
−0.256284 + 0.966602i \(0.582498\pi\)
\(270\) 0 0
\(271\) −6001.69 −1.34530 −0.672651 0.739960i \(-0.734844\pi\)
−0.672651 + 0.739960i \(0.734844\pi\)
\(272\) 0 0
\(273\) 1255.38 0.278311
\(274\) 0 0
\(275\) −250.249 −0.0548749
\(276\) 0 0
\(277\) −1760.39 −0.381848 −0.190924 0.981605i \(-0.561148\pi\)
−0.190924 + 0.981605i \(0.561148\pi\)
\(278\) 0 0
\(279\) −664.168 −0.142519
\(280\) 0 0
\(281\) −6327.06 −1.34321 −0.671603 0.740911i \(-0.734394\pi\)
−0.671603 + 0.740911i \(0.734394\pi\)
\(282\) 0 0
\(283\) −6197.80 −1.30184 −0.650921 0.759146i \(-0.725617\pi\)
−0.650921 + 0.759146i \(0.725617\pi\)
\(284\) 0 0
\(285\) 1356.80 0.281999
\(286\) 0 0
\(287\) −11382.7 −2.34111
\(288\) 0 0
\(289\) −3757.00 −0.764706
\(290\) 0 0
\(291\) 1404.62 0.282956
\(292\) 0 0
\(293\) 7878.93 1.57096 0.785482 0.618885i \(-0.212415\pi\)
0.785482 + 0.618885i \(0.212415\pi\)
\(294\) 0 0
\(295\) −3799.01 −0.749787
\(296\) 0 0
\(297\) 815.992 0.159423
\(298\) 0 0
\(299\) −592.043 −0.114511
\(300\) 0 0
\(301\) 10599.0 2.02963
\(302\) 0 0
\(303\) 1644.60 0.311814
\(304\) 0 0
\(305\) −9421.29 −1.76873
\(306\) 0 0
\(307\) 3503.65 0.651348 0.325674 0.945482i \(-0.394409\pi\)
0.325674 + 0.945482i \(0.394409\pi\)
\(308\) 0 0
\(309\) 2751.60 0.506580
\(310\) 0 0
\(311\) −9298.24 −1.69535 −0.847676 0.530513i \(-0.821999\pi\)
−0.847676 + 0.530513i \(0.821999\pi\)
\(312\) 0 0
\(313\) 7036.43 1.27068 0.635339 0.772233i \(-0.280860\pi\)
0.635339 + 0.772233i \(0.280860\pi\)
\(314\) 0 0
\(315\) −3129.85 −0.559832
\(316\) 0 0
\(317\) 11047.2 1.95733 0.978666 0.205460i \(-0.0658690\pi\)
0.978666 + 0.205460i \(0.0658690\pi\)
\(318\) 0 0
\(319\) 72.7448 0.0127678
\(320\) 0 0
\(321\) −3838.61 −0.667446
\(322\) 0 0
\(323\) −1423.31 −0.245186
\(324\) 0 0
\(325\) −107.645 −0.0183725
\(326\) 0 0
\(327\) −2259.91 −0.382181
\(328\) 0 0
\(329\) −1452.93 −0.243473
\(330\) 0 0
\(331\) 5075.45 0.842816 0.421408 0.906871i \(-0.361536\pi\)
0.421408 + 0.906871i \(0.361536\pi\)
\(332\) 0 0
\(333\) 3612.45 0.594477
\(334\) 0 0
\(335\) −1917.26 −0.312689
\(336\) 0 0
\(337\) −6777.58 −1.09554 −0.547772 0.836628i \(-0.684524\pi\)
−0.547772 + 0.836628i \(0.684524\pi\)
\(338\) 0 0
\(339\) −3585.87 −0.574506
\(340\) 0 0
\(341\) −2230.27 −0.354182
\(342\) 0 0
\(343\) 11270.7 1.77423
\(344\) 0 0
\(345\) 1476.06 0.230343
\(346\) 0 0
\(347\) 8725.29 1.34985 0.674925 0.737886i \(-0.264176\pi\)
0.674925 + 0.737886i \(0.264176\pi\)
\(348\) 0 0
\(349\) 3551.95 0.544790 0.272395 0.962186i \(-0.412184\pi\)
0.272395 + 0.962186i \(0.412184\pi\)
\(350\) 0 0
\(351\) 351.000 0.0533761
\(352\) 0 0
\(353\) 9440.63 1.42344 0.711719 0.702464i \(-0.247917\pi\)
0.711719 + 0.702464i \(0.247917\pi\)
\(354\) 0 0
\(355\) 355.822 0.0531974
\(356\) 0 0
\(357\) 3283.29 0.486751
\(358\) 0 0
\(359\) −1377.77 −0.202551 −0.101275 0.994858i \(-0.532292\pi\)
−0.101275 + 0.994858i \(0.532292\pi\)
\(360\) 0 0
\(361\) −5106.56 −0.744505
\(362\) 0 0
\(363\) −1252.90 −0.181158
\(364\) 0 0
\(365\) −8404.25 −1.20520
\(366\) 0 0
\(367\) 390.300 0.0555136 0.0277568 0.999615i \(-0.491164\pi\)
0.0277568 + 0.999615i \(0.491164\pi\)
\(368\) 0 0
\(369\) −3182.58 −0.448993
\(370\) 0 0
\(371\) 14477.4 2.02596
\(372\) 0 0
\(373\) 4976.32 0.690788 0.345394 0.938458i \(-0.387745\pi\)
0.345394 + 0.938458i \(0.387745\pi\)
\(374\) 0 0
\(375\) 4319.76 0.594857
\(376\) 0 0
\(377\) 31.2913 0.00427476
\(378\) 0 0
\(379\) −6020.58 −0.815980 −0.407990 0.912986i \(-0.633770\pi\)
−0.407990 + 0.912986i \(0.633770\pi\)
\(380\) 0 0
\(381\) −4193.86 −0.563931
\(382\) 0 0
\(383\) 8536.93 1.13895 0.569474 0.822010i \(-0.307147\pi\)
0.569474 + 0.822010i \(0.307147\pi\)
\(384\) 0 0
\(385\) −10510.0 −1.39127
\(386\) 0 0
\(387\) 2963.47 0.389254
\(388\) 0 0
\(389\) −11385.0 −1.48391 −0.741954 0.670451i \(-0.766101\pi\)
−0.741954 + 0.670451i \(0.766101\pi\)
\(390\) 0 0
\(391\) −1548.42 −0.200273
\(392\) 0 0
\(393\) 5078.76 0.651882
\(394\) 0 0
\(395\) 3789.56 0.482718
\(396\) 0 0
\(397\) −8515.61 −1.07654 −0.538270 0.842773i \(-0.680922\pi\)
−0.538270 + 0.842773i \(0.680922\pi\)
\(398\) 0 0
\(399\) −4042.52 −0.507215
\(400\) 0 0
\(401\) −2276.15 −0.283456 −0.141728 0.989906i \(-0.545266\pi\)
−0.141728 + 0.989906i \(0.545266\pi\)
\(402\) 0 0
\(403\) −959.354 −0.118583
\(404\) 0 0
\(405\) −875.099 −0.107368
\(406\) 0 0
\(407\) 12130.6 1.47737
\(408\) 0 0
\(409\) −797.754 −0.0964460 −0.0482230 0.998837i \(-0.515356\pi\)
−0.0482230 + 0.998837i \(0.515356\pi\)
\(410\) 0 0
\(411\) 411.101 0.0493384
\(412\) 0 0
\(413\) 11319.0 1.34860
\(414\) 0 0
\(415\) −4549.29 −0.538111
\(416\) 0 0
\(417\) −7759.59 −0.911244
\(418\) 0 0
\(419\) −14933.0 −1.74111 −0.870557 0.492067i \(-0.836241\pi\)
−0.870557 + 0.492067i \(0.836241\pi\)
\(420\) 0 0
\(421\) −73.7508 −0.00853775 −0.00426888 0.999991i \(-0.501359\pi\)
−0.00426888 + 0.999991i \(0.501359\pi\)
\(422\) 0 0
\(423\) −406.235 −0.0466947
\(424\) 0 0
\(425\) −281.533 −0.0321326
\(426\) 0 0
\(427\) 28070.3 3.18131
\(428\) 0 0
\(429\) 1178.66 0.132648
\(430\) 0 0
\(431\) −3233.48 −0.361371 −0.180686 0.983541i \(-0.557832\pi\)
−0.180686 + 0.983541i \(0.557832\pi\)
\(432\) 0 0
\(433\) −12812.8 −1.42204 −0.711019 0.703172i \(-0.751766\pi\)
−0.711019 + 0.703172i \(0.751766\pi\)
\(434\) 0 0
\(435\) −78.0141 −0.00859883
\(436\) 0 0
\(437\) 1906.47 0.208693
\(438\) 0 0
\(439\) 6964.75 0.757197 0.378598 0.925561i \(-0.376406\pi\)
0.378598 + 0.925561i \(0.376406\pi\)
\(440\) 0 0
\(441\) 6238.25 0.673605
\(442\) 0 0
\(443\) 1254.97 0.134594 0.0672971 0.997733i \(-0.478562\pi\)
0.0672971 + 0.997733i \(0.478562\pi\)
\(444\) 0 0
\(445\) 14755.2 1.57183
\(446\) 0 0
\(447\) −5733.57 −0.606686
\(448\) 0 0
\(449\) −9871.71 −1.03758 −0.518791 0.854901i \(-0.673618\pi\)
−0.518791 + 0.854901i \(0.673618\pi\)
\(450\) 0 0
\(451\) −10687.1 −1.11582
\(452\) 0 0
\(453\) 6022.41 0.624630
\(454\) 0 0
\(455\) −4520.89 −0.465808
\(456\) 0 0
\(457\) −12859.1 −1.31625 −0.658123 0.752911i \(-0.728649\pi\)
−0.658123 + 0.752911i \(0.728649\pi\)
\(458\) 0 0
\(459\) 918.000 0.0933520
\(460\) 0 0
\(461\) −1632.36 −0.164916 −0.0824582 0.996595i \(-0.526277\pi\)
−0.0824582 + 0.996595i \(0.526277\pi\)
\(462\) 0 0
\(463\) −5566.52 −0.558743 −0.279372 0.960183i \(-0.590126\pi\)
−0.279372 + 0.960183i \(0.590126\pi\)
\(464\) 0 0
\(465\) 2391.82 0.238534
\(466\) 0 0
\(467\) −2849.53 −0.282357 −0.141178 0.989984i \(-0.545089\pi\)
−0.141178 + 0.989984i \(0.545089\pi\)
\(468\) 0 0
\(469\) 5712.38 0.562417
\(470\) 0 0
\(471\) 9201.78 0.900203
\(472\) 0 0
\(473\) 9951.30 0.967360
\(474\) 0 0
\(475\) 346.634 0.0334835
\(476\) 0 0
\(477\) 4047.85 0.388550
\(478\) 0 0
\(479\) 7030.66 0.670645 0.335323 0.942103i \(-0.391155\pi\)
0.335323 + 0.942103i \(0.391155\pi\)
\(480\) 0 0
\(481\) 5217.98 0.494635
\(482\) 0 0
\(483\) −4397.84 −0.414304
\(484\) 0 0
\(485\) −5058.35 −0.473583
\(486\) 0 0
\(487\) 16972.0 1.57921 0.789604 0.613617i \(-0.210286\pi\)
0.789604 + 0.613617i \(0.210286\pi\)
\(488\) 0 0
\(489\) −3566.58 −0.329829
\(490\) 0 0
\(491\) −9969.09 −0.916291 −0.458145 0.888877i \(-0.651486\pi\)
−0.458145 + 0.888877i \(0.651486\pi\)
\(492\) 0 0
\(493\) 81.8387 0.00747633
\(494\) 0 0
\(495\) −2938.57 −0.266826
\(496\) 0 0
\(497\) −1060.16 −0.0956831
\(498\) 0 0
\(499\) 10962.9 0.983500 0.491750 0.870736i \(-0.336357\pi\)
0.491750 + 0.870736i \(0.336357\pi\)
\(500\) 0 0
\(501\) 9975.68 0.889582
\(502\) 0 0
\(503\) −4164.12 −0.369123 −0.184562 0.982821i \(-0.559087\pi\)
−0.184562 + 0.982821i \(0.559087\pi\)
\(504\) 0 0
\(505\) −5922.57 −0.521883
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 0 0
\(509\) 15477.5 1.34779 0.673897 0.738826i \(-0.264619\pi\)
0.673897 + 0.738826i \(0.264619\pi\)
\(510\) 0 0
\(511\) 25040.1 2.16773
\(512\) 0 0
\(513\) −1130.28 −0.0972767
\(514\) 0 0
\(515\) −9909.14 −0.847862
\(516\) 0 0
\(517\) −1364.14 −0.116044
\(518\) 0 0
\(519\) −4356.78 −0.368480
\(520\) 0 0
\(521\) 8442.92 0.709964 0.354982 0.934873i \(-0.384487\pi\)
0.354982 + 0.934873i \(0.384487\pi\)
\(522\) 0 0
\(523\) −9748.73 −0.815071 −0.407535 0.913189i \(-0.633612\pi\)
−0.407535 + 0.913189i \(0.633612\pi\)
\(524\) 0 0
\(525\) −799.614 −0.0664724
\(526\) 0 0
\(527\) −2509.08 −0.207395
\(528\) 0 0
\(529\) −10093.0 −0.829535
\(530\) 0 0
\(531\) 3164.76 0.258642
\(532\) 0 0
\(533\) −4597.05 −0.373584
\(534\) 0 0
\(535\) 13823.7 1.11710
\(536\) 0 0
\(537\) −7387.54 −0.593660
\(538\) 0 0
\(539\) 20948.0 1.67402
\(540\) 0 0
\(541\) −12880.6 −1.02362 −0.511812 0.859097i \(-0.671026\pi\)
−0.511812 + 0.859097i \(0.671026\pi\)
\(542\) 0 0
\(543\) 8449.52 0.667778
\(544\) 0 0
\(545\) 8138.44 0.639656
\(546\) 0 0
\(547\) −4314.68 −0.337262 −0.168631 0.985679i \(-0.553935\pi\)
−0.168631 + 0.985679i \(0.553935\pi\)
\(548\) 0 0
\(549\) 7848.39 0.610130
\(550\) 0 0
\(551\) −100.763 −0.00779065
\(552\) 0 0
\(553\) −11290.8 −0.868238
\(554\) 0 0
\(555\) −13009.2 −0.994976
\(556\) 0 0
\(557\) −4145.64 −0.315361 −0.157681 0.987490i \(-0.550402\pi\)
−0.157681 + 0.987490i \(0.550402\pi\)
\(558\) 0 0
\(559\) 4280.56 0.323879
\(560\) 0 0
\(561\) 3082.64 0.231995
\(562\) 0 0
\(563\) 8390.08 0.628063 0.314032 0.949413i \(-0.398320\pi\)
0.314032 + 0.949413i \(0.398320\pi\)
\(564\) 0 0
\(565\) 12913.5 0.961551
\(566\) 0 0
\(567\) 2607.32 0.193117
\(568\) 0 0
\(569\) 7739.75 0.570241 0.285121 0.958492i \(-0.407966\pi\)
0.285121 + 0.958492i \(0.407966\pi\)
\(570\) 0 0
\(571\) −2853.05 −0.209101 −0.104550 0.994520i \(-0.533340\pi\)
−0.104550 + 0.994520i \(0.533340\pi\)
\(572\) 0 0
\(573\) 11693.8 0.852558
\(574\) 0 0
\(575\) 377.103 0.0273500
\(576\) 0 0
\(577\) 14334.9 1.03426 0.517131 0.855906i \(-0.327000\pi\)
0.517131 + 0.855906i \(0.327000\pi\)
\(578\) 0 0
\(579\) 11718.4 0.841107
\(580\) 0 0
\(581\) 13554.4 0.967870
\(582\) 0 0
\(583\) 13592.7 0.965609
\(584\) 0 0
\(585\) −1264.03 −0.0893355
\(586\) 0 0
\(587\) 20453.1 1.43814 0.719071 0.694937i \(-0.244568\pi\)
0.719071 + 0.694937i \(0.244568\pi\)
\(588\) 0 0
\(589\) 3089.28 0.216115
\(590\) 0 0
\(591\) 8837.85 0.615128
\(592\) 0 0
\(593\) −15117.6 −1.04689 −0.523443 0.852061i \(-0.675353\pi\)
−0.523443 + 0.852061i \(0.675353\pi\)
\(594\) 0 0
\(595\) −11823.9 −0.814675
\(596\) 0 0
\(597\) 1607.38 0.110194
\(598\) 0 0
\(599\) −13786.3 −0.940388 −0.470194 0.882563i \(-0.655816\pi\)
−0.470194 + 0.882563i \(0.655816\pi\)
\(600\) 0 0
\(601\) 28653.1 1.94474 0.972368 0.233454i \(-0.0750028\pi\)
0.972368 + 0.233454i \(0.0750028\pi\)
\(602\) 0 0
\(603\) 1597.17 0.107864
\(604\) 0 0
\(605\) 4511.99 0.303204
\(606\) 0 0
\(607\) −5271.71 −0.352508 −0.176254 0.984345i \(-0.556398\pi\)
−0.176254 + 0.984345i \(0.556398\pi\)
\(608\) 0 0
\(609\) 232.440 0.0154662
\(610\) 0 0
\(611\) −586.785 −0.0388523
\(612\) 0 0
\(613\) 14763.7 0.972760 0.486380 0.873747i \(-0.338317\pi\)
0.486380 + 0.873747i \(0.338317\pi\)
\(614\) 0 0
\(615\) 11461.2 0.751479
\(616\) 0 0
\(617\) 16934.0 1.10492 0.552461 0.833539i \(-0.313689\pi\)
0.552461 + 0.833539i \(0.313689\pi\)
\(618\) 0 0
\(619\) −19432.6 −1.26181 −0.630906 0.775859i \(-0.717317\pi\)
−0.630906 + 0.775859i \(0.717317\pi\)
\(620\) 0 0
\(621\) −1229.63 −0.0794577
\(622\) 0 0
\(623\) −43962.6 −2.82716
\(624\) 0 0
\(625\) −14521.4 −0.929369
\(626\) 0 0
\(627\) −3795.46 −0.241748
\(628\) 0 0
\(629\) 13647.0 0.865091
\(630\) 0 0
\(631\) −11565.8 −0.729680 −0.364840 0.931070i \(-0.618876\pi\)
−0.364840 + 0.931070i \(0.618876\pi\)
\(632\) 0 0
\(633\) −11065.0 −0.694779
\(634\) 0 0
\(635\) 15103.0 0.943852
\(636\) 0 0
\(637\) 9010.81 0.560473
\(638\) 0 0
\(639\) −296.417 −0.0183507
\(640\) 0 0
\(641\) −17372.8 −1.07049 −0.535244 0.844698i \(-0.679780\pi\)
−0.535244 + 0.844698i \(0.679780\pi\)
\(642\) 0 0
\(643\) −26204.3 −1.60715 −0.803574 0.595205i \(-0.797071\pi\)
−0.803574 + 0.595205i \(0.797071\pi\)
\(644\) 0 0
\(645\) −10672.1 −0.651495
\(646\) 0 0
\(647\) −18498.7 −1.12405 −0.562025 0.827120i \(-0.689978\pi\)
−0.562025 + 0.827120i \(0.689978\pi\)
\(648\) 0 0
\(649\) 10627.3 0.642768
\(650\) 0 0
\(651\) −7126.33 −0.429037
\(652\) 0 0
\(653\) −15631.9 −0.936790 −0.468395 0.883519i \(-0.655168\pi\)
−0.468395 + 0.883519i \(0.655168\pi\)
\(654\) 0 0
\(655\) −18289.8 −1.09105
\(656\) 0 0
\(657\) 7001.15 0.415740
\(658\) 0 0
\(659\) −2057.87 −0.121644 −0.0608218 0.998149i \(-0.519372\pi\)
−0.0608218 + 0.998149i \(0.519372\pi\)
\(660\) 0 0
\(661\) −27657.5 −1.62746 −0.813730 0.581244i \(-0.802566\pi\)
−0.813730 + 0.581244i \(0.802566\pi\)
\(662\) 0 0
\(663\) 1326.00 0.0776736
\(664\) 0 0
\(665\) 14558.0 0.848926
\(666\) 0 0
\(667\) −109.620 −0.00636357
\(668\) 0 0
\(669\) −7151.32 −0.413283
\(670\) 0 0
\(671\) 26354.9 1.51627
\(672\) 0 0
\(673\) 18122.1 1.03797 0.518987 0.854782i \(-0.326309\pi\)
0.518987 + 0.854782i \(0.326309\pi\)
\(674\) 0 0
\(675\) −223.570 −0.0127485
\(676\) 0 0
\(677\) 17210.1 0.977012 0.488506 0.872560i \(-0.337542\pi\)
0.488506 + 0.872560i \(0.337542\pi\)
\(678\) 0 0
\(679\) 15071.1 0.851808
\(680\) 0 0
\(681\) −16963.5 −0.954540
\(682\) 0 0
\(683\) 15770.7 0.883525 0.441762 0.897132i \(-0.354353\pi\)
0.441762 + 0.897132i \(0.354353\pi\)
\(684\) 0 0
\(685\) −1480.47 −0.0825777
\(686\) 0 0
\(687\) −14198.3 −0.788500
\(688\) 0 0
\(689\) 5846.89 0.323293
\(690\) 0 0
\(691\) −28419.8 −1.56460 −0.782301 0.622901i \(-0.785954\pi\)
−0.782301 + 0.622901i \(0.785954\pi\)
\(692\) 0 0
\(693\) 8755.36 0.479926
\(694\) 0 0
\(695\) 27944.1 1.52515
\(696\) 0 0
\(697\) −12023.1 −0.653380
\(698\) 0 0
\(699\) −3854.78 −0.208586
\(700\) 0 0
\(701\) −9186.30 −0.494952 −0.247476 0.968894i \(-0.579601\pi\)
−0.247476 + 0.968894i \(0.579601\pi\)
\(702\) 0 0
\(703\) −16802.7 −0.901461
\(704\) 0 0
\(705\) 1462.95 0.0781528
\(706\) 0 0
\(707\) 17646.0 0.938681
\(708\) 0 0
\(709\) −12818.9 −0.679018 −0.339509 0.940603i \(-0.610261\pi\)
−0.339509 + 0.940603i \(0.610261\pi\)
\(710\) 0 0
\(711\) −3156.89 −0.166516
\(712\) 0 0
\(713\) 3360.82 0.176527
\(714\) 0 0
\(715\) −4244.61 −0.222013
\(716\) 0 0
\(717\) −13976.3 −0.727969
\(718\) 0 0
\(719\) 20883.9 1.08323 0.541613 0.840628i \(-0.317814\pi\)
0.541613 + 0.840628i \(0.317814\pi\)
\(720\) 0 0
\(721\) 29523.9 1.52500
\(722\) 0 0
\(723\) 6501.26 0.334418
\(724\) 0 0
\(725\) −19.9311 −0.00102099
\(726\) 0 0
\(727\) −898.193 −0.0458214 −0.0229107 0.999738i \(-0.507293\pi\)
−0.0229107 + 0.999738i \(0.507293\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 11195.3 0.566448
\(732\) 0 0
\(733\) 21008.4 1.05861 0.529306 0.848431i \(-0.322452\pi\)
0.529306 + 0.848431i \(0.322452\pi\)
\(734\) 0 0
\(735\) −22465.4 −1.12741
\(736\) 0 0
\(737\) 5363.28 0.268058
\(738\) 0 0
\(739\) 17326.9 0.862490 0.431245 0.902235i \(-0.358075\pi\)
0.431245 + 0.902235i \(0.358075\pi\)
\(740\) 0 0
\(741\) −1632.62 −0.0809391
\(742\) 0 0
\(743\) −4113.25 −0.203096 −0.101548 0.994831i \(-0.532380\pi\)
−0.101548 + 0.994831i \(0.532380\pi\)
\(744\) 0 0
\(745\) 20647.9 1.01541
\(746\) 0 0
\(747\) 3789.79 0.185624
\(748\) 0 0
\(749\) −41187.1 −2.00927
\(750\) 0 0
\(751\) −15309.1 −0.743857 −0.371928 0.928261i \(-0.621303\pi\)
−0.371928 + 0.928261i \(0.621303\pi\)
\(752\) 0 0
\(753\) −15657.7 −0.757768
\(754\) 0 0
\(755\) −21688.1 −1.04544
\(756\) 0 0
\(757\) −32489.1 −1.55989 −0.779944 0.625850i \(-0.784752\pi\)
−0.779944 + 0.625850i \(0.784752\pi\)
\(758\) 0 0
\(759\) −4129.08 −0.197465
\(760\) 0 0
\(761\) 23935.8 1.14017 0.570087 0.821585i \(-0.306910\pi\)
0.570087 + 0.821585i \(0.306910\pi\)
\(762\) 0 0
\(763\) −24248.1 −1.15051
\(764\) 0 0
\(765\) −3305.93 −0.156243
\(766\) 0 0
\(767\) 4571.33 0.215203
\(768\) 0 0
\(769\) 7627.82 0.357693 0.178847 0.983877i \(-0.442763\pi\)
0.178847 + 0.983877i \(0.442763\pi\)
\(770\) 0 0
\(771\) 1198.97 0.0560051
\(772\) 0 0
\(773\) 11075.3 0.515332 0.257666 0.966234i \(-0.417047\pi\)
0.257666 + 0.966234i \(0.417047\pi\)
\(774\) 0 0
\(775\) 611.063 0.0283226
\(776\) 0 0
\(777\) 38760.5 1.78961
\(778\) 0 0
\(779\) 14803.3 0.680850
\(780\) 0 0
\(781\) −995.367 −0.0456044
\(782\) 0 0
\(783\) 64.9896 0.00296620
\(784\) 0 0
\(785\) −33137.7 −1.50667
\(786\) 0 0
\(787\) 12170.8 0.551260 0.275630 0.961264i \(-0.411113\pi\)
0.275630 + 0.961264i \(0.411113\pi\)
\(788\) 0 0
\(789\) −24136.9 −1.08910
\(790\) 0 0
\(791\) −38475.3 −1.72949
\(792\) 0 0
\(793\) 11336.6 0.507659
\(794\) 0 0
\(795\) −14577.2 −0.650316
\(796\) 0 0
\(797\) 24434.7 1.08597 0.542986 0.839742i \(-0.317294\pi\)
0.542986 + 0.839742i \(0.317294\pi\)
\(798\) 0 0
\(799\) −1534.67 −0.0679507
\(800\) 0 0
\(801\) −12291.8 −0.542210
\(802\) 0 0
\(803\) 23509.8 1.03318
\(804\) 0 0
\(805\) 15837.6 0.693421
\(806\) 0 0
\(807\) −6784.23 −0.295931
\(808\) 0 0
\(809\) 12303.4 0.534690 0.267345 0.963601i \(-0.413854\pi\)
0.267345 + 0.963601i \(0.413854\pi\)
\(810\) 0 0
\(811\) 12491.2 0.540845 0.270422 0.962742i \(-0.412837\pi\)
0.270422 + 0.962742i \(0.412837\pi\)
\(812\) 0 0
\(813\) −18005.1 −0.776710
\(814\) 0 0
\(815\) 12844.1 0.552035
\(816\) 0 0
\(817\) −13784.1 −0.590263
\(818\) 0 0
\(819\) 3766.13 0.160683
\(820\) 0 0
\(821\) 14520.7 0.617268 0.308634 0.951181i \(-0.400128\pi\)
0.308634 + 0.951181i \(0.400128\pi\)
\(822\) 0 0
\(823\) 12198.7 0.516670 0.258335 0.966055i \(-0.416826\pi\)
0.258335 + 0.966055i \(0.416826\pi\)
\(824\) 0 0
\(825\) −750.747 −0.0316820
\(826\) 0 0
\(827\) 33507.8 1.40892 0.704462 0.709742i \(-0.251188\pi\)
0.704462 + 0.709742i \(0.251188\pi\)
\(828\) 0 0
\(829\) 59.8986 0.00250949 0.00125474 0.999999i \(-0.499601\pi\)
0.00125474 + 0.999999i \(0.499601\pi\)
\(830\) 0 0
\(831\) −5281.18 −0.220460
\(832\) 0 0
\(833\) 23566.7 0.980239
\(834\) 0 0
\(835\) −35924.7 −1.48889
\(836\) 0 0
\(837\) −1992.51 −0.0822832
\(838\) 0 0
\(839\) −6541.95 −0.269193 −0.134597 0.990900i \(-0.542974\pi\)
−0.134597 + 0.990900i \(0.542974\pi\)
\(840\) 0 0
\(841\) −24383.2 −0.999762
\(842\) 0 0
\(843\) −18981.2 −0.775500
\(844\) 0 0
\(845\) −1825.82 −0.0743316
\(846\) 0 0
\(847\) −13443.3 −0.545356
\(848\) 0 0
\(849\) −18593.4 −0.751618
\(850\) 0 0
\(851\) −18279.7 −0.736333
\(852\) 0 0
\(853\) −39815.6 −1.59819 −0.799097 0.601202i \(-0.794689\pi\)
−0.799097 + 0.601202i \(0.794689\pi\)
\(854\) 0 0
\(855\) 4070.39 0.162812
\(856\) 0 0
\(857\) −22194.9 −0.884671 −0.442335 0.896850i \(-0.645850\pi\)
−0.442335 + 0.896850i \(0.645850\pi\)
\(858\) 0 0
\(859\) 31640.5 1.25676 0.628381 0.777905i \(-0.283718\pi\)
0.628381 + 0.777905i \(0.283718\pi\)
\(860\) 0 0
\(861\) −34148.1 −1.35164
\(862\) 0 0
\(863\) −33171.9 −1.30844 −0.654220 0.756304i \(-0.727003\pi\)
−0.654220 + 0.756304i \(0.727003\pi\)
\(864\) 0 0
\(865\) 15689.7 0.616725
\(866\) 0 0
\(867\) −11271.0 −0.441503
\(868\) 0 0
\(869\) −10600.8 −0.413818
\(870\) 0 0
\(871\) 2307.02 0.0897479
\(872\) 0 0
\(873\) 4213.86 0.163365
\(874\) 0 0
\(875\) 46349.7 1.79075
\(876\) 0 0
\(877\) 5351.15 0.206038 0.103019 0.994679i \(-0.467150\pi\)
0.103019 + 0.994679i \(0.467150\pi\)
\(878\) 0 0
\(879\) 23636.8 0.906996
\(880\) 0 0
\(881\) −2731.65 −0.104462 −0.0522312 0.998635i \(-0.516633\pi\)
−0.0522312 + 0.998635i \(0.516633\pi\)
\(882\) 0 0
\(883\) −6111.50 −0.232920 −0.116460 0.993195i \(-0.537155\pi\)
−0.116460 + 0.993195i \(0.537155\pi\)
\(884\) 0 0
\(885\) −11397.0 −0.432890
\(886\) 0 0
\(887\) −7214.75 −0.273109 −0.136554 0.990633i \(-0.543603\pi\)
−0.136554 + 0.990633i \(0.543603\pi\)
\(888\) 0 0
\(889\) −44998.9 −1.69765
\(890\) 0 0
\(891\) 2447.98 0.0920430
\(892\) 0 0
\(893\) 1889.54 0.0708075
\(894\) 0 0
\(895\) 26604.2 0.993609
\(896\) 0 0
\(897\) −1776.13 −0.0661128
\(898\) 0 0
\(899\) −177.630 −0.00658986
\(900\) 0 0
\(901\) 15291.9 0.565423
\(902\) 0 0
\(903\) 31797.1 1.17181
\(904\) 0 0
\(905\) −30428.7 −1.11766
\(906\) 0 0
\(907\) −27365.0 −1.00181 −0.500905 0.865502i \(-0.666999\pi\)
−0.500905 + 0.865502i \(0.666999\pi\)
\(908\) 0 0
\(909\) 4933.79 0.180026
\(910\) 0 0
\(911\) 39873.8 1.45014 0.725070 0.688675i \(-0.241807\pi\)
0.725070 + 0.688675i \(0.241807\pi\)
\(912\) 0 0
\(913\) 12726.1 0.461305
\(914\) 0 0
\(915\) −28263.9 −1.02117
\(916\) 0 0
\(917\) 54493.6 1.96242
\(918\) 0 0
\(919\) 37427.3 1.34343 0.671715 0.740810i \(-0.265558\pi\)
0.671715 + 0.740810i \(0.265558\pi\)
\(920\) 0 0
\(921\) 10510.9 0.376056
\(922\) 0 0
\(923\) −428.158 −0.0152687
\(924\) 0 0
\(925\) −3323.60 −0.118140
\(926\) 0 0
\(927\) 8254.80 0.292474
\(928\) 0 0
\(929\) −50172.4 −1.77191 −0.885954 0.463774i \(-0.846495\pi\)
−0.885954 + 0.463774i \(0.846495\pi\)
\(930\) 0 0
\(931\) −29016.3 −1.02145
\(932\) 0 0
\(933\) −27894.7 −0.978813
\(934\) 0 0
\(935\) −11101.3 −0.388290
\(936\) 0 0
\(937\) 23584.4 0.822270 0.411135 0.911574i \(-0.365132\pi\)
0.411135 + 0.911574i \(0.365132\pi\)
\(938\) 0 0
\(939\) 21109.3 0.733627
\(940\) 0 0
\(941\) −38535.5 −1.33499 −0.667493 0.744616i \(-0.732632\pi\)
−0.667493 + 0.744616i \(0.732632\pi\)
\(942\) 0 0
\(943\) 16104.4 0.556132
\(944\) 0 0
\(945\) −9389.55 −0.323219
\(946\) 0 0
\(947\) 35237.3 1.20914 0.604572 0.796550i \(-0.293344\pi\)
0.604572 + 0.796550i \(0.293344\pi\)
\(948\) 0 0
\(949\) 10112.8 0.345916
\(950\) 0 0
\(951\) 33141.7 1.13007
\(952\) 0 0
\(953\) 14173.4 0.481765 0.240883 0.970554i \(-0.422563\pi\)
0.240883 + 0.970554i \(0.422563\pi\)
\(954\) 0 0
\(955\) −42112.1 −1.42693
\(956\) 0 0
\(957\) 218.235 0.00737150
\(958\) 0 0
\(959\) 4410.99 0.148528
\(960\) 0 0
\(961\) −24345.1 −0.817196
\(962\) 0 0
\(963\) −11515.8 −0.385350
\(964\) 0 0
\(965\) −42200.7 −1.40776
\(966\) 0 0
\(967\) −22532.8 −0.749334 −0.374667 0.927159i \(-0.622243\pi\)
−0.374667 + 0.927159i \(0.622243\pi\)
\(968\) 0 0
\(969\) −4269.94 −0.141558
\(970\) 0 0
\(971\) −21753.8 −0.718962 −0.359481 0.933152i \(-0.617046\pi\)
−0.359481 + 0.933152i \(0.617046\pi\)
\(972\) 0 0
\(973\) −83258.1 −2.74320
\(974\) 0 0
\(975\) −322.935 −0.0106074
\(976\) 0 0
\(977\) 19179.7 0.628059 0.314030 0.949413i \(-0.398321\pi\)
0.314030 + 0.949413i \(0.398321\pi\)
\(978\) 0 0
\(979\) −41275.9 −1.34748
\(980\) 0 0
\(981\) −6779.72 −0.220652
\(982\) 0 0
\(983\) −51940.4 −1.68529 −0.842646 0.538468i \(-0.819003\pi\)
−0.842646 + 0.538468i \(0.819003\pi\)
\(984\) 0 0
\(985\) −31827.1 −1.02954
\(986\) 0 0
\(987\) −4358.79 −0.140569
\(988\) 0 0
\(989\) −14995.7 −0.482139
\(990\) 0 0
\(991\) −49237.5 −1.57828 −0.789142 0.614210i \(-0.789475\pi\)
−0.789142 + 0.614210i \(0.789475\pi\)
\(992\) 0 0
\(993\) 15226.4 0.486600
\(994\) 0 0
\(995\) −5788.55 −0.184432
\(996\) 0 0
\(997\) 1110.55 0.0352774 0.0176387 0.999844i \(-0.494385\pi\)
0.0176387 + 0.999844i \(0.494385\pi\)
\(998\) 0 0
\(999\) 10837.3 0.343221
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 624.4.a.u.1.1 3
3.2 odd 2 1872.4.a.bj.1.3 3
4.3 odd 2 312.4.a.g.1.1 3
8.3 odd 2 2496.4.a.bo.1.3 3
8.5 even 2 2496.4.a.bk.1.3 3
12.11 even 2 936.4.a.k.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.g.1.1 3 4.3 odd 2
624.4.a.u.1.1 3 1.1 even 1 trivial
936.4.a.k.1.3 3 12.11 even 2
1872.4.a.bj.1.3 3 3.2 odd 2
2496.4.a.bk.1.3 3 8.5 even 2
2496.4.a.bo.1.3 3 8.3 odd 2