Properties

Label 414.4.a.n.1.3
Level $414$
Weight $4$
Character 414.1
Self dual yes
Analytic conductor $24.427$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [414,4,Mod(1,414)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(414, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("414.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 414.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,8,0,16,0,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(24.4267907424\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 219x^{2} - 468x + 3240 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(16.3784\) of defining polynomial
Character \(\chi\) \(=\) 414.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +13.0601 q^{5} +18.7067 q^{7} +8.00000 q^{8} +26.1202 q^{10} +41.1103 q^{11} +32.6366 q^{13} +37.4133 q^{14} +16.0000 q^{16} -76.8570 q^{17} -164.965 q^{19} +52.2405 q^{20} +82.2206 q^{22} +23.0000 q^{23} +45.5668 q^{25} +65.2732 q^{26} +74.8267 q^{28} -42.5314 q^{29} +246.351 q^{31} +32.0000 q^{32} -153.714 q^{34} +244.311 q^{35} +281.671 q^{37} -329.929 q^{38} +104.481 q^{40} +330.365 q^{41} -160.644 q^{43} +164.441 q^{44} +46.0000 q^{46} -578.490 q^{47} +6.93931 q^{49} +91.1336 q^{50} +130.546 q^{52} -621.131 q^{53} +536.906 q^{55} +149.653 q^{56} -85.0628 q^{58} +347.432 q^{59} +372.119 q^{61} +492.702 q^{62} +64.0000 q^{64} +426.238 q^{65} +884.432 q^{67} -307.428 q^{68} +488.623 q^{70} -777.672 q^{71} -188.099 q^{73} +563.343 q^{74} -659.859 q^{76} +769.037 q^{77} +200.062 q^{79} +208.962 q^{80} +660.730 q^{82} +49.9822 q^{83} -1003.76 q^{85} -321.288 q^{86} +328.882 q^{88} -767.598 q^{89} +610.522 q^{91} +92.0000 q^{92} -1156.98 q^{94} -2154.46 q^{95} +1626.31 q^{97} +13.8786 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 16 q^{4} + 18 q^{7} + 32 q^{8} + 42 q^{11} + 108 q^{13} + 36 q^{14} + 64 q^{16} + 26 q^{17} + 132 q^{19} + 84 q^{22} + 92 q^{23} + 260 q^{25} + 216 q^{26} + 72 q^{28} + 252 q^{29} + 428 q^{31}+ \cdots + 2152 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 13.0601 1.16813 0.584066 0.811706i \(-0.301461\pi\)
0.584066 + 0.811706i \(0.301461\pi\)
\(6\) 0 0
\(7\) 18.7067 1.01006 0.505032 0.863100i \(-0.331481\pi\)
0.505032 + 0.863100i \(0.331481\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 26.1202 0.825995
\(11\) 41.1103 1.12684 0.563419 0.826171i \(-0.309486\pi\)
0.563419 + 0.826171i \(0.309486\pi\)
\(12\) 0 0
\(13\) 32.6366 0.696290 0.348145 0.937441i \(-0.386812\pi\)
0.348145 + 0.937441i \(0.386812\pi\)
\(14\) 37.4133 0.714224
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −76.8570 −1.09650 −0.548252 0.836313i \(-0.684706\pi\)
−0.548252 + 0.836313i \(0.684706\pi\)
\(18\) 0 0
\(19\) −164.965 −1.99187 −0.995934 0.0900884i \(-0.971285\pi\)
−0.995934 + 0.0900884i \(0.971285\pi\)
\(20\) 52.2405 0.584066
\(21\) 0 0
\(22\) 82.2206 0.796795
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 45.5668 0.364534
\(26\) 65.2732 0.492351
\(27\) 0 0
\(28\) 74.8267 0.505032
\(29\) −42.5314 −0.272341 −0.136170 0.990685i \(-0.543479\pi\)
−0.136170 + 0.990685i \(0.543479\pi\)
\(30\) 0 0
\(31\) 246.351 1.42729 0.713644 0.700508i \(-0.247043\pi\)
0.713644 + 0.700508i \(0.247043\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) −153.714 −0.775345
\(35\) 244.311 1.17989
\(36\) 0 0
\(37\) 281.671 1.25153 0.625763 0.780013i \(-0.284788\pi\)
0.625763 + 0.780013i \(0.284788\pi\)
\(38\) −329.929 −1.40846
\(39\) 0 0
\(40\) 104.481 0.412997
\(41\) 330.365 1.25840 0.629200 0.777244i \(-0.283383\pi\)
0.629200 + 0.777244i \(0.283383\pi\)
\(42\) 0 0
\(43\) −160.644 −0.569721 −0.284861 0.958569i \(-0.591947\pi\)
−0.284861 + 0.958569i \(0.591947\pi\)
\(44\) 164.441 0.563419
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) −578.490 −1.79535 −0.897675 0.440658i \(-0.854745\pi\)
−0.897675 + 0.440658i \(0.854745\pi\)
\(48\) 0 0
\(49\) 6.93931 0.0202312
\(50\) 91.1336 0.257765
\(51\) 0 0
\(52\) 130.546 0.348145
\(53\) −621.131 −1.60979 −0.804895 0.593418i \(-0.797778\pi\)
−0.804895 + 0.593418i \(0.797778\pi\)
\(54\) 0 0
\(55\) 536.906 1.31630
\(56\) 149.653 0.357112
\(57\) 0 0
\(58\) −85.0628 −0.192574
\(59\) 347.432 0.766640 0.383320 0.923616i \(-0.374781\pi\)
0.383320 + 0.923616i \(0.374781\pi\)
\(60\) 0 0
\(61\) 372.119 0.781065 0.390532 0.920589i \(-0.372291\pi\)
0.390532 + 0.920589i \(0.372291\pi\)
\(62\) 492.702 1.00925
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 426.238 0.813359
\(66\) 0 0
\(67\) 884.432 1.61270 0.806348 0.591442i \(-0.201441\pi\)
0.806348 + 0.591442i \(0.201441\pi\)
\(68\) −307.428 −0.548252
\(69\) 0 0
\(70\) 488.623 0.834308
\(71\) −777.672 −1.29990 −0.649948 0.759978i \(-0.725209\pi\)
−0.649948 + 0.759978i \(0.725209\pi\)
\(72\) 0 0
\(73\) −188.099 −0.301581 −0.150790 0.988566i \(-0.548182\pi\)
−0.150790 + 0.988566i \(0.548182\pi\)
\(74\) 563.343 0.884963
\(75\) 0 0
\(76\) −659.859 −0.995934
\(77\) 769.037 1.13818
\(78\) 0 0
\(79\) 200.062 0.284921 0.142461 0.989800i \(-0.454499\pi\)
0.142461 + 0.989800i \(0.454499\pi\)
\(80\) 208.962 0.292033
\(81\) 0 0
\(82\) 660.730 0.889823
\(83\) 49.9822 0.0660995 0.0330497 0.999454i \(-0.489478\pi\)
0.0330497 + 0.999454i \(0.489478\pi\)
\(84\) 0 0
\(85\) −1003.76 −1.28086
\(86\) −321.288 −0.402854
\(87\) 0 0
\(88\) 328.882 0.398398
\(89\) −767.598 −0.914216 −0.457108 0.889411i \(-0.651115\pi\)
−0.457108 + 0.889411i \(0.651115\pi\)
\(90\) 0 0
\(91\) 610.522 0.703298
\(92\) 92.0000 0.104257
\(93\) 0 0
\(94\) −1156.98 −1.26950
\(95\) −2154.46 −2.32677
\(96\) 0 0
\(97\) 1626.31 1.70234 0.851171 0.524888i \(-0.175893\pi\)
0.851171 + 0.524888i \(0.175893\pi\)
\(98\) 13.8786 0.0143056
\(99\) 0 0
\(100\) 182.267 0.182267
\(101\) −366.127 −0.360703 −0.180352 0.983602i \(-0.557724\pi\)
−0.180352 + 0.983602i \(0.557724\pi\)
\(102\) 0 0
\(103\) 1828.41 1.74911 0.874554 0.484928i \(-0.161154\pi\)
0.874554 + 0.484928i \(0.161154\pi\)
\(104\) 261.093 0.246176
\(105\) 0 0
\(106\) −1242.26 −1.13829
\(107\) −555.660 −0.502034 −0.251017 0.967983i \(-0.580765\pi\)
−0.251017 + 0.967983i \(0.580765\pi\)
\(108\) 0 0
\(109\) −952.852 −0.837309 −0.418654 0.908146i \(-0.637498\pi\)
−0.418654 + 0.908146i \(0.637498\pi\)
\(110\) 1073.81 0.930763
\(111\) 0 0
\(112\) 299.307 0.252516
\(113\) −255.634 −0.212815 −0.106407 0.994323i \(-0.533935\pi\)
−0.106407 + 0.994323i \(0.533935\pi\)
\(114\) 0 0
\(115\) 300.383 0.243573
\(116\) −170.126 −0.136170
\(117\) 0 0
\(118\) 694.864 0.542096
\(119\) −1437.74 −1.10754
\(120\) 0 0
\(121\) 359.058 0.269765
\(122\) 744.238 0.552296
\(123\) 0 0
\(124\) 985.404 0.713644
\(125\) −1037.41 −0.742308
\(126\) 0 0
\(127\) −1543.83 −1.07868 −0.539341 0.842088i \(-0.681327\pi\)
−0.539341 + 0.842088i \(0.681327\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) 852.476 0.575132
\(131\) −116.906 −0.0779702 −0.0389851 0.999240i \(-0.512412\pi\)
−0.0389851 + 0.999240i \(0.512412\pi\)
\(132\) 0 0
\(133\) −3085.94 −2.01192
\(134\) 1768.86 1.14035
\(135\) 0 0
\(136\) −614.856 −0.387672
\(137\) −2770.00 −1.72742 −0.863712 0.503986i \(-0.831866\pi\)
−0.863712 + 0.503986i \(0.831866\pi\)
\(138\) 0 0
\(139\) −1296.27 −0.790992 −0.395496 0.918468i \(-0.629427\pi\)
−0.395496 + 0.918468i \(0.629427\pi\)
\(140\) 977.245 0.589945
\(141\) 0 0
\(142\) −1555.34 −0.919166
\(143\) 1341.70 0.784606
\(144\) 0 0
\(145\) −555.466 −0.318130
\(146\) −376.199 −0.213250
\(147\) 0 0
\(148\) 1126.69 0.625763
\(149\) −200.699 −0.110348 −0.0551742 0.998477i \(-0.517571\pi\)
−0.0551742 + 0.998477i \(0.517571\pi\)
\(150\) 0 0
\(151\) −109.209 −0.0588564 −0.0294282 0.999567i \(-0.509369\pi\)
−0.0294282 + 0.999567i \(0.509369\pi\)
\(152\) −1319.72 −0.704232
\(153\) 0 0
\(154\) 1538.07 0.804815
\(155\) 3217.38 1.66726
\(156\) 0 0
\(157\) −3299.65 −1.67733 −0.838665 0.544647i \(-0.816664\pi\)
−0.838665 + 0.544647i \(0.816664\pi\)
\(158\) 400.125 0.201470
\(159\) 0 0
\(160\) 417.924 0.206499
\(161\) 430.253 0.210613
\(162\) 0 0
\(163\) 238.673 0.114689 0.0573446 0.998354i \(-0.481737\pi\)
0.0573446 + 0.998354i \(0.481737\pi\)
\(164\) 1321.46 0.629200
\(165\) 0 0
\(166\) 99.9644 0.0467394
\(167\) 3124.43 1.44776 0.723880 0.689926i \(-0.242357\pi\)
0.723880 + 0.689926i \(0.242357\pi\)
\(168\) 0 0
\(169\) −1131.85 −0.515181
\(170\) −2007.52 −0.905706
\(171\) 0 0
\(172\) −642.577 −0.284861
\(173\) −2732.43 −1.20083 −0.600413 0.799690i \(-0.704997\pi\)
−0.600413 + 0.799690i \(0.704997\pi\)
\(174\) 0 0
\(175\) 852.403 0.368203
\(176\) 657.765 0.281710
\(177\) 0 0
\(178\) −1535.20 −0.646449
\(179\) 2350.95 0.981665 0.490833 0.871254i \(-0.336693\pi\)
0.490833 + 0.871254i \(0.336693\pi\)
\(180\) 0 0
\(181\) −2224.83 −0.913650 −0.456825 0.889557i \(-0.651013\pi\)
−0.456825 + 0.889557i \(0.651013\pi\)
\(182\) 1221.04 0.497307
\(183\) 0 0
\(184\) 184.000 0.0737210
\(185\) 3678.66 1.46195
\(186\) 0 0
\(187\) −3159.61 −1.23558
\(188\) −2313.96 −0.897675
\(189\) 0 0
\(190\) −4308.92 −1.64527
\(191\) 3099.04 1.17403 0.587013 0.809577i \(-0.300304\pi\)
0.587013 + 0.809577i \(0.300304\pi\)
\(192\) 0 0
\(193\) −679.935 −0.253590 −0.126795 0.991929i \(-0.540469\pi\)
−0.126795 + 0.991929i \(0.540469\pi\)
\(194\) 3252.63 1.20374
\(195\) 0 0
\(196\) 27.7573 0.0101156
\(197\) −49.9118 −0.0180511 −0.00902555 0.999959i \(-0.502873\pi\)
−0.00902555 + 0.999959i \(0.502873\pi\)
\(198\) 0 0
\(199\) −4052.80 −1.44369 −0.721847 0.692052i \(-0.756707\pi\)
−0.721847 + 0.692052i \(0.756707\pi\)
\(200\) 364.534 0.128882
\(201\) 0 0
\(202\) −732.255 −0.255056
\(203\) −795.621 −0.275082
\(204\) 0 0
\(205\) 4314.61 1.46998
\(206\) 3656.81 1.23681
\(207\) 0 0
\(208\) 522.186 0.174072
\(209\) −6781.75 −2.24451
\(210\) 0 0
\(211\) −864.141 −0.281943 −0.140971 0.990014i \(-0.545023\pi\)
−0.140971 + 0.990014i \(0.545023\pi\)
\(212\) −2484.52 −0.804895
\(213\) 0 0
\(214\) −1111.32 −0.354992
\(215\) −2098.03 −0.665510
\(216\) 0 0
\(217\) 4608.41 1.44165
\(218\) −1905.70 −0.592067
\(219\) 0 0
\(220\) 2147.62 0.658149
\(221\) −2508.35 −0.763484
\(222\) 0 0
\(223\) 5682.45 1.70639 0.853196 0.521591i \(-0.174661\pi\)
0.853196 + 0.521591i \(0.174661\pi\)
\(224\) 598.613 0.178556
\(225\) 0 0
\(226\) −511.269 −0.150483
\(227\) −158.099 −0.0462263 −0.0231132 0.999733i \(-0.507358\pi\)
−0.0231132 + 0.999733i \(0.507358\pi\)
\(228\) 0 0
\(229\) 2744.94 0.792100 0.396050 0.918229i \(-0.370381\pi\)
0.396050 + 0.918229i \(0.370381\pi\)
\(230\) 600.766 0.172232
\(231\) 0 0
\(232\) −340.251 −0.0962871
\(233\) 3647.63 1.02560 0.512798 0.858509i \(-0.328609\pi\)
0.512798 + 0.858509i \(0.328609\pi\)
\(234\) 0 0
\(235\) −7555.15 −2.09721
\(236\) 1389.73 0.383320
\(237\) 0 0
\(238\) −2875.48 −0.783149
\(239\) −2580.14 −0.698306 −0.349153 0.937066i \(-0.613531\pi\)
−0.349153 + 0.937066i \(0.613531\pi\)
\(240\) 0 0
\(241\) 4498.59 1.20240 0.601202 0.799097i \(-0.294689\pi\)
0.601202 + 0.799097i \(0.294689\pi\)
\(242\) 718.115 0.190753
\(243\) 0 0
\(244\) 1488.48 0.390532
\(245\) 90.6283 0.0236328
\(246\) 0 0
\(247\) −5383.89 −1.38692
\(248\) 1970.81 0.504623
\(249\) 0 0
\(250\) −2074.81 −0.524891
\(251\) 1285.33 0.323223 0.161612 0.986854i \(-0.448331\pi\)
0.161612 + 0.986854i \(0.448331\pi\)
\(252\) 0 0
\(253\) 945.537 0.234962
\(254\) −3087.66 −0.762743
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 768.793 0.186599 0.0932996 0.995638i \(-0.470259\pi\)
0.0932996 + 0.995638i \(0.470259\pi\)
\(258\) 0 0
\(259\) 5269.13 1.26412
\(260\) 1704.95 0.406679
\(261\) 0 0
\(262\) −233.811 −0.0551333
\(263\) −3454.27 −0.809883 −0.404942 0.914342i \(-0.632708\pi\)
−0.404942 + 0.914342i \(0.632708\pi\)
\(264\) 0 0
\(265\) −8112.04 −1.88045
\(266\) −6171.88 −1.42264
\(267\) 0 0
\(268\) 3537.73 0.806348
\(269\) −4348.89 −0.985712 −0.492856 0.870111i \(-0.664047\pi\)
−0.492856 + 0.870111i \(0.664047\pi\)
\(270\) 0 0
\(271\) 4876.35 1.09305 0.546527 0.837442i \(-0.315950\pi\)
0.546527 + 0.837442i \(0.315950\pi\)
\(272\) −1229.71 −0.274126
\(273\) 0 0
\(274\) −5540.00 −1.22147
\(275\) 1873.27 0.410771
\(276\) 0 0
\(277\) −757.557 −0.164322 −0.0821610 0.996619i \(-0.526182\pi\)
−0.0821610 + 0.996619i \(0.526182\pi\)
\(278\) −2592.53 −0.559316
\(279\) 0 0
\(280\) 1954.49 0.417154
\(281\) −8407.74 −1.78492 −0.892462 0.451122i \(-0.851024\pi\)
−0.892462 + 0.451122i \(0.851024\pi\)
\(282\) 0 0
\(283\) −2522.06 −0.529757 −0.264878 0.964282i \(-0.585332\pi\)
−0.264878 + 0.964282i \(0.585332\pi\)
\(284\) −3110.69 −0.649948
\(285\) 0 0
\(286\) 2683.40 0.554800
\(287\) 6180.03 1.27106
\(288\) 0 0
\(289\) 993.996 0.202319
\(290\) −1110.93 −0.224952
\(291\) 0 0
\(292\) −752.398 −0.150790
\(293\) 3357.13 0.669372 0.334686 0.942330i \(-0.391370\pi\)
0.334686 + 0.942330i \(0.391370\pi\)
\(294\) 0 0
\(295\) 4537.50 0.895537
\(296\) 2253.37 0.442481
\(297\) 0 0
\(298\) −401.399 −0.0780282
\(299\) 750.642 0.145186
\(300\) 0 0
\(301\) −3005.12 −0.575455
\(302\) −218.418 −0.0416177
\(303\) 0 0
\(304\) −2639.43 −0.497967
\(305\) 4859.92 0.912387
\(306\) 0 0
\(307\) −7660.19 −1.42407 −0.712036 0.702143i \(-0.752227\pi\)
−0.712036 + 0.702143i \(0.752227\pi\)
\(308\) 3076.15 0.569090
\(309\) 0 0
\(310\) 6434.75 1.17893
\(311\) 344.406 0.0627958 0.0313979 0.999507i \(-0.490004\pi\)
0.0313979 + 0.999507i \(0.490004\pi\)
\(312\) 0 0
\(313\) 3403.84 0.614685 0.307343 0.951599i \(-0.400560\pi\)
0.307343 + 0.951599i \(0.400560\pi\)
\(314\) −6599.30 −1.18605
\(315\) 0 0
\(316\) 800.249 0.142461
\(317\) −5801.37 −1.02788 −0.513939 0.857827i \(-0.671814\pi\)
−0.513939 + 0.857827i \(0.671814\pi\)
\(318\) 0 0
\(319\) −1748.48 −0.306884
\(320\) 835.848 0.146017
\(321\) 0 0
\(322\) 860.507 0.148926
\(323\) 12678.7 2.18409
\(324\) 0 0
\(325\) 1487.15 0.253822
\(326\) 477.347 0.0810975
\(327\) 0 0
\(328\) 2642.92 0.444911
\(329\) −10821.6 −1.81342
\(330\) 0 0
\(331\) 8400.66 1.39499 0.697495 0.716589i \(-0.254298\pi\)
0.697495 + 0.716589i \(0.254298\pi\)
\(332\) 199.929 0.0330497
\(333\) 0 0
\(334\) 6248.87 1.02372
\(335\) 11550.8 1.88384
\(336\) 0 0
\(337\) 4144.17 0.669873 0.334937 0.942241i \(-0.391285\pi\)
0.334937 + 0.942241i \(0.391285\pi\)
\(338\) −2263.70 −0.364288
\(339\) 0 0
\(340\) −4015.05 −0.640431
\(341\) 10127.6 1.60832
\(342\) 0 0
\(343\) −6286.57 −0.989630
\(344\) −1285.15 −0.201427
\(345\) 0 0
\(346\) −5464.87 −0.849113
\(347\) 3851.70 0.595880 0.297940 0.954585i \(-0.403701\pi\)
0.297940 + 0.954585i \(0.403701\pi\)
\(348\) 0 0
\(349\) 2581.17 0.395893 0.197947 0.980213i \(-0.436573\pi\)
0.197947 + 0.980213i \(0.436573\pi\)
\(350\) 1704.81 0.260359
\(351\) 0 0
\(352\) 1315.53 0.199199
\(353\) 7748.76 1.16834 0.584171 0.811630i \(-0.301420\pi\)
0.584171 + 0.811630i \(0.301420\pi\)
\(354\) 0 0
\(355\) −10156.5 −1.51845
\(356\) −3070.39 −0.457108
\(357\) 0 0
\(358\) 4701.90 0.694142
\(359\) 1957.02 0.287709 0.143855 0.989599i \(-0.454050\pi\)
0.143855 + 0.989599i \(0.454050\pi\)
\(360\) 0 0
\(361\) 20354.3 2.96754
\(362\) −4449.67 −0.646048
\(363\) 0 0
\(364\) 2442.09 0.351649
\(365\) −2456.60 −0.352286
\(366\) 0 0
\(367\) 5825.76 0.828616 0.414308 0.910137i \(-0.364024\pi\)
0.414308 + 0.910137i \(0.364024\pi\)
\(368\) 368.000 0.0521286
\(369\) 0 0
\(370\) 7357.32 1.03375
\(371\) −11619.3 −1.62599
\(372\) 0 0
\(373\) 2339.75 0.324793 0.162396 0.986726i \(-0.448078\pi\)
0.162396 + 0.986726i \(0.448078\pi\)
\(374\) −6319.23 −0.873689
\(375\) 0 0
\(376\) −4627.92 −0.634752
\(377\) −1388.08 −0.189628
\(378\) 0 0
\(379\) 10381.9 1.40707 0.703537 0.710658i \(-0.251603\pi\)
0.703537 + 0.710658i \(0.251603\pi\)
\(380\) −8617.83 −1.16338
\(381\) 0 0
\(382\) 6198.09 0.830162
\(383\) −4681.97 −0.624641 −0.312320 0.949977i \(-0.601106\pi\)
−0.312320 + 0.949977i \(0.601106\pi\)
\(384\) 0 0
\(385\) 10043.7 1.32955
\(386\) −1359.87 −0.179315
\(387\) 0 0
\(388\) 6505.26 0.851171
\(389\) −2896.26 −0.377497 −0.188748 0.982025i \(-0.560443\pi\)
−0.188748 + 0.982025i \(0.560443\pi\)
\(390\) 0 0
\(391\) −1767.71 −0.228637
\(392\) 55.5145 0.00715282
\(393\) 0 0
\(394\) −99.8235 −0.0127641
\(395\) 2612.84 0.332826
\(396\) 0 0
\(397\) −1599.60 −0.202221 −0.101111 0.994875i \(-0.532240\pi\)
−0.101111 + 0.994875i \(0.532240\pi\)
\(398\) −8105.60 −1.02085
\(399\) 0 0
\(400\) 729.069 0.0911336
\(401\) −2132.80 −0.265603 −0.132802 0.991143i \(-0.542397\pi\)
−0.132802 + 0.991143i \(0.542397\pi\)
\(402\) 0 0
\(403\) 8040.06 0.993807
\(404\) −1464.51 −0.180352
\(405\) 0 0
\(406\) −1591.24 −0.194512
\(407\) 11579.6 1.41027
\(408\) 0 0
\(409\) −4181.61 −0.505544 −0.252772 0.967526i \(-0.581342\pi\)
−0.252772 + 0.967526i \(0.581342\pi\)
\(410\) 8629.22 1.03943
\(411\) 0 0
\(412\) 7313.62 0.874554
\(413\) 6499.29 0.774356
\(414\) 0 0
\(415\) 652.773 0.0772130
\(416\) 1044.37 0.123088
\(417\) 0 0
\(418\) −13563.5 −1.58711
\(419\) 3866.07 0.450763 0.225381 0.974271i \(-0.427637\pi\)
0.225381 + 0.974271i \(0.427637\pi\)
\(420\) 0 0
\(421\) −6888.69 −0.797469 −0.398734 0.917066i \(-0.630550\pi\)
−0.398734 + 0.917066i \(0.630550\pi\)
\(422\) −1728.28 −0.199364
\(423\) 0 0
\(424\) −4969.04 −0.569147
\(425\) −3502.13 −0.399713
\(426\) 0 0
\(427\) 6961.10 0.788926
\(428\) −2222.64 −0.251017
\(429\) 0 0
\(430\) −4196.07 −0.470587
\(431\) −12349.8 −1.38021 −0.690105 0.723709i \(-0.742436\pi\)
−0.690105 + 0.723709i \(0.742436\pi\)
\(432\) 0 0
\(433\) −12531.1 −1.39078 −0.695391 0.718632i \(-0.744769\pi\)
−0.695391 + 0.718632i \(0.744769\pi\)
\(434\) 9216.81 1.01940
\(435\) 0 0
\(436\) −3811.41 −0.418654
\(437\) −3794.19 −0.415333
\(438\) 0 0
\(439\) −11534.2 −1.25398 −0.626988 0.779029i \(-0.715713\pi\)
−0.626988 + 0.779029i \(0.715713\pi\)
\(440\) 4295.25 0.465381
\(441\) 0 0
\(442\) −5016.70 −0.539865
\(443\) 2982.50 0.319871 0.159935 0.987127i \(-0.448871\pi\)
0.159935 + 0.987127i \(0.448871\pi\)
\(444\) 0 0
\(445\) −10024.9 −1.06793
\(446\) 11364.9 1.20660
\(447\) 0 0
\(448\) 1197.23 0.126258
\(449\) 12630.8 1.32759 0.663793 0.747917i \(-0.268946\pi\)
0.663793 + 0.747917i \(0.268946\pi\)
\(450\) 0 0
\(451\) 13581.4 1.41801
\(452\) −1022.54 −0.106407
\(453\) 0 0
\(454\) −316.197 −0.0326869
\(455\) 7973.49 0.821545
\(456\) 0 0
\(457\) −13693.1 −1.40161 −0.700804 0.713354i \(-0.747175\pi\)
−0.700804 + 0.713354i \(0.747175\pi\)
\(458\) 5489.88 0.560099
\(459\) 0 0
\(460\) 1201.53 0.121786
\(461\) 4588.64 0.463588 0.231794 0.972765i \(-0.425540\pi\)
0.231794 + 0.972765i \(0.425540\pi\)
\(462\) 0 0
\(463\) 6110.84 0.613380 0.306690 0.951809i \(-0.400778\pi\)
0.306690 + 0.951809i \(0.400778\pi\)
\(464\) −680.503 −0.0680852
\(465\) 0 0
\(466\) 7295.25 0.725206
\(467\) 2078.48 0.205954 0.102977 0.994684i \(-0.467163\pi\)
0.102977 + 0.994684i \(0.467163\pi\)
\(468\) 0 0
\(469\) 16544.8 1.62893
\(470\) −15110.3 −1.48295
\(471\) 0 0
\(472\) 2779.45 0.271048
\(473\) −6604.13 −0.641984
\(474\) 0 0
\(475\) −7516.91 −0.726104
\(476\) −5750.95 −0.553770
\(477\) 0 0
\(478\) −5160.27 −0.493777
\(479\) 13245.4 1.26346 0.631730 0.775189i \(-0.282345\pi\)
0.631730 + 0.775189i \(0.282345\pi\)
\(480\) 0 0
\(481\) 9192.80 0.871425
\(482\) 8997.17 0.850228
\(483\) 0 0
\(484\) 1436.23 0.134883
\(485\) 21239.9 1.98856
\(486\) 0 0
\(487\) 8327.36 0.774843 0.387421 0.921903i \(-0.373366\pi\)
0.387421 + 0.921903i \(0.373366\pi\)
\(488\) 2976.95 0.276148
\(489\) 0 0
\(490\) 181.257 0.0167109
\(491\) 10585.9 0.972984 0.486492 0.873685i \(-0.338276\pi\)
0.486492 + 0.873685i \(0.338276\pi\)
\(492\) 0 0
\(493\) 3268.84 0.298623
\(494\) −10767.8 −0.980698
\(495\) 0 0
\(496\) 3941.62 0.356822
\(497\) −14547.6 −1.31298
\(498\) 0 0
\(499\) −2630.88 −0.236021 −0.118011 0.993012i \(-0.537652\pi\)
−0.118011 + 0.993012i \(0.537652\pi\)
\(500\) −4149.63 −0.371154
\(501\) 0 0
\(502\) 2570.65 0.228553
\(503\) −8978.08 −0.795851 −0.397925 0.917418i \(-0.630270\pi\)
−0.397925 + 0.917418i \(0.630270\pi\)
\(504\) 0 0
\(505\) −4781.67 −0.421349
\(506\) 1891.07 0.166143
\(507\) 0 0
\(508\) −6175.31 −0.539341
\(509\) −10656.1 −0.927942 −0.463971 0.885850i \(-0.653576\pi\)
−0.463971 + 0.885850i \(0.653576\pi\)
\(510\) 0 0
\(511\) −3518.71 −0.304616
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 1537.59 0.131946
\(515\) 23879.2 2.04319
\(516\) 0 0
\(517\) −23781.9 −2.02307
\(518\) 10538.3 0.893870
\(519\) 0 0
\(520\) 3409.90 0.287566
\(521\) −15732.0 −1.32290 −0.661449 0.749990i \(-0.730058\pi\)
−0.661449 + 0.749990i \(0.730058\pi\)
\(522\) 0 0
\(523\) 3937.97 0.329245 0.164623 0.986357i \(-0.447359\pi\)
0.164623 + 0.986357i \(0.447359\pi\)
\(524\) −467.623 −0.0389851
\(525\) 0 0
\(526\) −6908.54 −0.572674
\(527\) −18933.8 −1.56503
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) −16224.1 −1.32968
\(531\) 0 0
\(532\) −12343.8 −1.00596
\(533\) 10782.0 0.876210
\(534\) 0 0
\(535\) −7256.98 −0.586442
\(536\) 7075.46 0.570174
\(537\) 0 0
\(538\) −8697.78 −0.697003
\(539\) 285.277 0.0227973
\(540\) 0 0
\(541\) −7886.04 −0.626704 −0.313352 0.949637i \(-0.601452\pi\)
−0.313352 + 0.949637i \(0.601452\pi\)
\(542\) 9752.71 0.772905
\(543\) 0 0
\(544\) −2459.42 −0.193836
\(545\) −12444.4 −0.978088
\(546\) 0 0
\(547\) 20947.1 1.63735 0.818677 0.574255i \(-0.194708\pi\)
0.818677 + 0.574255i \(0.194708\pi\)
\(548\) −11080.0 −0.863712
\(549\) 0 0
\(550\) 3746.53 0.290459
\(551\) 7016.18 0.542467
\(552\) 0 0
\(553\) 3742.50 0.287789
\(554\) −1515.11 −0.116193
\(555\) 0 0
\(556\) −5185.07 −0.395496
\(557\) −14249.9 −1.08400 −0.542001 0.840378i \(-0.682333\pi\)
−0.542001 + 0.840378i \(0.682333\pi\)
\(558\) 0 0
\(559\) −5242.88 −0.396691
\(560\) 3908.98 0.294973
\(561\) 0 0
\(562\) −16815.5 −1.26213
\(563\) 20886.4 1.56351 0.781755 0.623586i \(-0.214325\pi\)
0.781755 + 0.623586i \(0.214325\pi\)
\(564\) 0 0
\(565\) −3338.62 −0.248596
\(566\) −5044.13 −0.374594
\(567\) 0 0
\(568\) −6221.37 −0.459583
\(569\) −18602.8 −1.37060 −0.685299 0.728262i \(-0.740328\pi\)
−0.685299 + 0.728262i \(0.740328\pi\)
\(570\) 0 0
\(571\) −13118.5 −0.961456 −0.480728 0.876870i \(-0.659628\pi\)
−0.480728 + 0.876870i \(0.659628\pi\)
\(572\) 5366.80 0.392303
\(573\) 0 0
\(574\) 12360.1 0.898779
\(575\) 1048.04 0.0760107
\(576\) 0 0
\(577\) 15596.4 1.12528 0.562640 0.826702i \(-0.309786\pi\)
0.562640 + 0.826702i \(0.309786\pi\)
\(578\) 1987.99 0.143061
\(579\) 0 0
\(580\) −2221.86 −0.159065
\(581\) 935.000 0.0667648
\(582\) 0 0
\(583\) −25534.9 −1.81397
\(584\) −1504.80 −0.106625
\(585\) 0 0
\(586\) 6714.27 0.473317
\(587\) −15569.7 −1.09477 −0.547385 0.836881i \(-0.684377\pi\)
−0.547385 + 0.836881i \(0.684377\pi\)
\(588\) 0 0
\(589\) −40639.2 −2.84297
\(590\) 9075.00 0.633241
\(591\) 0 0
\(592\) 4506.74 0.312882
\(593\) −1616.91 −0.111971 −0.0559854 0.998432i \(-0.517830\pi\)
−0.0559854 + 0.998432i \(0.517830\pi\)
\(594\) 0 0
\(595\) −18777.0 −1.29375
\(596\) −802.797 −0.0551742
\(597\) 0 0
\(598\) 1501.28 0.102662
\(599\) 10595.7 0.722752 0.361376 0.932420i \(-0.382307\pi\)
0.361376 + 0.932420i \(0.382307\pi\)
\(600\) 0 0
\(601\) −5699.09 −0.386806 −0.193403 0.981119i \(-0.561953\pi\)
−0.193403 + 0.981119i \(0.561953\pi\)
\(602\) −6010.23 −0.406908
\(603\) 0 0
\(604\) −436.836 −0.0294282
\(605\) 4689.34 0.315122
\(606\) 0 0
\(607\) 15513.1 1.03733 0.518664 0.854978i \(-0.326430\pi\)
0.518664 + 0.854978i \(0.326430\pi\)
\(608\) −5278.87 −0.352116
\(609\) 0 0
\(610\) 9719.84 0.645155
\(611\) −18880.0 −1.25008
\(612\) 0 0
\(613\) 5817.04 0.383276 0.191638 0.981466i \(-0.438620\pi\)
0.191638 + 0.981466i \(0.438620\pi\)
\(614\) −15320.4 −1.00697
\(615\) 0 0
\(616\) 6152.29 0.402407
\(617\) −18218.6 −1.18874 −0.594370 0.804191i \(-0.702599\pi\)
−0.594370 + 0.804191i \(0.702599\pi\)
\(618\) 0 0
\(619\) 23581.8 1.53123 0.765617 0.643296i \(-0.222434\pi\)
0.765617 + 0.643296i \(0.222434\pi\)
\(620\) 12869.5 0.833632
\(621\) 0 0
\(622\) 688.812 0.0444033
\(623\) −14359.2 −0.923418
\(624\) 0 0
\(625\) −19244.5 −1.23165
\(626\) 6807.68 0.434648
\(627\) 0 0
\(628\) −13198.6 −0.838665
\(629\) −21648.4 −1.37230
\(630\) 0 0
\(631\) −14989.0 −0.945649 −0.472825 0.881157i \(-0.656766\pi\)
−0.472825 + 0.881157i \(0.656766\pi\)
\(632\) 1600.50 0.100735
\(633\) 0 0
\(634\) −11602.7 −0.726820
\(635\) −20162.6 −1.26004
\(636\) 0 0
\(637\) 226.476 0.0140868
\(638\) −3496.96 −0.217000
\(639\) 0 0
\(640\) 1671.70 0.103249
\(641\) 16769.0 1.03328 0.516642 0.856201i \(-0.327182\pi\)
0.516642 + 0.856201i \(0.327182\pi\)
\(642\) 0 0
\(643\) −15863.5 −0.972933 −0.486467 0.873699i \(-0.661714\pi\)
−0.486467 + 0.873699i \(0.661714\pi\)
\(644\) 1721.01 0.105307
\(645\) 0 0
\(646\) 25357.4 1.54438
\(647\) 25100.0 1.52516 0.762582 0.646891i \(-0.223931\pi\)
0.762582 + 0.646891i \(0.223931\pi\)
\(648\) 0 0
\(649\) 14283.0 0.863880
\(650\) 2974.29 0.179479
\(651\) 0 0
\(652\) 954.693 0.0573446
\(653\) −7127.01 −0.427108 −0.213554 0.976931i \(-0.568504\pi\)
−0.213554 + 0.976931i \(0.568504\pi\)
\(654\) 0 0
\(655\) −1526.80 −0.0910795
\(656\) 5285.84 0.314600
\(657\) 0 0
\(658\) −21643.2 −1.28228
\(659\) 18985.3 1.12225 0.561124 0.827731i \(-0.310369\pi\)
0.561124 + 0.827731i \(0.310369\pi\)
\(660\) 0 0
\(661\) 2780.35 0.163605 0.0818026 0.996649i \(-0.473932\pi\)
0.0818026 + 0.996649i \(0.473932\pi\)
\(662\) 16801.3 0.986407
\(663\) 0 0
\(664\) 399.857 0.0233697
\(665\) −40302.7 −2.35018
\(666\) 0 0
\(667\) −978.223 −0.0567870
\(668\) 12497.7 0.723880
\(669\) 0 0
\(670\) 23101.6 1.33208
\(671\) 15297.9 0.880134
\(672\) 0 0
\(673\) −9328.01 −0.534277 −0.267138 0.963658i \(-0.586078\pi\)
−0.267138 + 0.963658i \(0.586078\pi\)
\(674\) 8288.34 0.473672
\(675\) 0 0
\(676\) −4527.41 −0.257590
\(677\) −33535.6 −1.90381 −0.951905 0.306393i \(-0.900878\pi\)
−0.951905 + 0.306393i \(0.900878\pi\)
\(678\) 0 0
\(679\) 30422.9 1.71948
\(680\) −8030.09 −0.452853
\(681\) 0 0
\(682\) 20255.1 1.13726
\(683\) 25507.4 1.42901 0.714506 0.699629i \(-0.246652\pi\)
0.714506 + 0.699629i \(0.246652\pi\)
\(684\) 0 0
\(685\) −36176.5 −2.01786
\(686\) −12573.1 −0.699774
\(687\) 0 0
\(688\) −2570.31 −0.142430
\(689\) −20271.6 −1.12088
\(690\) 0 0
\(691\) −3801.07 −0.209261 −0.104631 0.994511i \(-0.533366\pi\)
−0.104631 + 0.994511i \(0.533366\pi\)
\(692\) −10929.7 −0.600413
\(693\) 0 0
\(694\) 7703.41 0.421351
\(695\) −16929.4 −0.923984
\(696\) 0 0
\(697\) −25390.9 −1.37984
\(698\) 5162.34 0.279939
\(699\) 0 0
\(700\) 3409.61 0.184102
\(701\) 24101.5 1.29858 0.649289 0.760542i \(-0.275067\pi\)
0.649289 + 0.760542i \(0.275067\pi\)
\(702\) 0 0
\(703\) −46465.8 −2.49287
\(704\) 2631.06 0.140855
\(705\) 0 0
\(706\) 15497.5 0.826143
\(707\) −6849.02 −0.364334
\(708\) 0 0
\(709\) 27015.2 1.43100 0.715498 0.698615i \(-0.246200\pi\)
0.715498 + 0.698615i \(0.246200\pi\)
\(710\) −20313.0 −1.07371
\(711\) 0 0
\(712\) −6140.79 −0.323224
\(713\) 5666.07 0.297610
\(714\) 0 0
\(715\) 17522.8 0.916524
\(716\) 9403.79 0.490833
\(717\) 0 0
\(718\) 3914.04 0.203441
\(719\) −12022.5 −0.623593 −0.311797 0.950149i \(-0.600931\pi\)
−0.311797 + 0.950149i \(0.600931\pi\)
\(720\) 0 0
\(721\) 34203.4 1.76671
\(722\) 40708.7 2.09837
\(723\) 0 0
\(724\) −8899.33 −0.456825
\(725\) −1938.02 −0.0992776
\(726\) 0 0
\(727\) 20108.7 1.02585 0.512923 0.858435i \(-0.328563\pi\)
0.512923 + 0.858435i \(0.328563\pi\)
\(728\) 4884.18 0.248653
\(729\) 0 0
\(730\) −4913.20 −0.249104
\(731\) 12346.6 0.624701
\(732\) 0 0
\(733\) 16026.4 0.807568 0.403784 0.914854i \(-0.367695\pi\)
0.403784 + 0.914854i \(0.367695\pi\)
\(734\) 11651.5 0.585920
\(735\) 0 0
\(736\) 736.000 0.0368605
\(737\) 36359.3 1.81725
\(738\) 0 0
\(739\) −121.045 −0.00602532 −0.00301266 0.999995i \(-0.500959\pi\)
−0.00301266 + 0.999995i \(0.500959\pi\)
\(740\) 14714.6 0.730975
\(741\) 0 0
\(742\) −23238.6 −1.14975
\(743\) 32902.8 1.62461 0.812306 0.583231i \(-0.198212\pi\)
0.812306 + 0.583231i \(0.198212\pi\)
\(744\) 0 0
\(745\) −2621.16 −0.128902
\(746\) 4679.50 0.229663
\(747\) 0 0
\(748\) −12638.5 −0.617791
\(749\) −10394.5 −0.507087
\(750\) 0 0
\(751\) −8822.39 −0.428673 −0.214337 0.976760i \(-0.568759\pi\)
−0.214337 + 0.976760i \(0.568759\pi\)
\(752\) −9255.84 −0.448838
\(753\) 0 0
\(754\) −2776.16 −0.134087
\(755\) −1426.28 −0.0687520
\(756\) 0 0
\(757\) 17256.9 0.828552 0.414276 0.910151i \(-0.364035\pi\)
0.414276 + 0.910151i \(0.364035\pi\)
\(758\) 20763.8 0.994952
\(759\) 0 0
\(760\) −17235.7 −0.822636
\(761\) 28563.8 1.36063 0.680313 0.732921i \(-0.261844\pi\)
0.680313 + 0.732921i \(0.261844\pi\)
\(762\) 0 0
\(763\) −17824.7 −0.845736
\(764\) 12396.2 0.587013
\(765\) 0 0
\(766\) −9363.94 −0.441688
\(767\) 11339.0 0.533804
\(768\) 0 0
\(769\) 36869.9 1.72895 0.864475 0.502676i \(-0.167651\pi\)
0.864475 + 0.502676i \(0.167651\pi\)
\(770\) 20087.4 0.940131
\(771\) 0 0
\(772\) −2719.74 −0.126795
\(773\) −31615.1 −1.47104 −0.735520 0.677503i \(-0.763062\pi\)
−0.735520 + 0.677503i \(0.763062\pi\)
\(774\) 0 0
\(775\) 11225.4 0.520296
\(776\) 13010.5 0.601869
\(777\) 0 0
\(778\) −5792.52 −0.266931
\(779\) −54498.6 −2.50656
\(780\) 0 0
\(781\) −31970.3 −1.46477
\(782\) −3535.42 −0.161671
\(783\) 0 0
\(784\) 111.029 0.00505781
\(785\) −43093.9 −1.95934
\(786\) 0 0
\(787\) −22822.8 −1.03373 −0.516865 0.856067i \(-0.672901\pi\)
−0.516865 + 0.856067i \(0.672901\pi\)
\(788\) −199.647 −0.00902555
\(789\) 0 0
\(790\) 5225.68 0.235343
\(791\) −4782.07 −0.214957
\(792\) 0 0
\(793\) 12144.7 0.543847
\(794\) −3199.21 −0.142992
\(795\) 0 0
\(796\) −16211.2 −0.721847
\(797\) 32080.5 1.42578 0.712892 0.701274i \(-0.247385\pi\)
0.712892 + 0.701274i \(0.247385\pi\)
\(798\) 0 0
\(799\) 44461.0 1.96861
\(800\) 1458.14 0.0644412
\(801\) 0 0
\(802\) −4265.60 −0.187810
\(803\) −7732.83 −0.339833
\(804\) 0 0
\(805\) 5619.16 0.246024
\(806\) 16080.1 0.702727
\(807\) 0 0
\(808\) −2929.02 −0.127528
\(809\) 12622.8 0.548570 0.274285 0.961649i \(-0.411559\pi\)
0.274285 + 0.961649i \(0.411559\pi\)
\(810\) 0 0
\(811\) 14725.4 0.637581 0.318791 0.947825i \(-0.396723\pi\)
0.318791 + 0.947825i \(0.396723\pi\)
\(812\) −3182.48 −0.137541
\(813\) 0 0
\(814\) 23159.2 0.997210
\(815\) 3117.10 0.133972
\(816\) 0 0
\(817\) 26500.6 1.13481
\(818\) −8363.23 −0.357474
\(819\) 0 0
\(820\) 17258.4 0.734989
\(821\) −25304.1 −1.07566 −0.537832 0.843052i \(-0.680756\pi\)
−0.537832 + 0.843052i \(0.680756\pi\)
\(822\) 0 0
\(823\) −26359.2 −1.11643 −0.558217 0.829695i \(-0.688514\pi\)
−0.558217 + 0.829695i \(0.688514\pi\)
\(824\) 14627.2 0.618403
\(825\) 0 0
\(826\) 12998.6 0.547553
\(827\) 23399.7 0.983902 0.491951 0.870623i \(-0.336284\pi\)
0.491951 + 0.870623i \(0.336284\pi\)
\(828\) 0 0
\(829\) 31965.7 1.33922 0.669612 0.742711i \(-0.266461\pi\)
0.669612 + 0.742711i \(0.266461\pi\)
\(830\) 1305.55 0.0545978
\(831\) 0 0
\(832\) 2088.74 0.0870362
\(833\) −533.335 −0.0221836
\(834\) 0 0
\(835\) 40805.5 1.69118
\(836\) −27127.0 −1.12226
\(837\) 0 0
\(838\) 7732.13 0.318738
\(839\) 31922.9 1.31359 0.656793 0.754071i \(-0.271912\pi\)
0.656793 + 0.754071i \(0.271912\pi\)
\(840\) 0 0
\(841\) −22580.1 −0.925830
\(842\) −13777.4 −0.563896
\(843\) 0 0
\(844\) −3456.57 −0.140971
\(845\) −14782.1 −0.601799
\(846\) 0 0
\(847\) 6716.77 0.272481
\(848\) −9938.09 −0.402447
\(849\) 0 0
\(850\) −7004.25 −0.282640
\(851\) 6478.44 0.260961
\(852\) 0 0
\(853\) 23656.8 0.949583 0.474791 0.880098i \(-0.342523\pi\)
0.474791 + 0.880098i \(0.342523\pi\)
\(854\) 13922.2 0.557855
\(855\) 0 0
\(856\) −4445.28 −0.177496
\(857\) −31502.3 −1.25566 −0.627829 0.778351i \(-0.716056\pi\)
−0.627829 + 0.778351i \(0.716056\pi\)
\(858\) 0 0
\(859\) −4487.44 −0.178242 −0.0891208 0.996021i \(-0.528406\pi\)
−0.0891208 + 0.996021i \(0.528406\pi\)
\(860\) −8392.13 −0.332755
\(861\) 0 0
\(862\) −24699.7 −0.975956
\(863\) −21730.6 −0.857147 −0.428574 0.903507i \(-0.640984\pi\)
−0.428574 + 0.903507i \(0.640984\pi\)
\(864\) 0 0
\(865\) −35685.9 −1.40273
\(866\) −25062.3 −0.983431
\(867\) 0 0
\(868\) 18433.6 0.720827
\(869\) 8224.63 0.321060
\(870\) 0 0
\(871\) 28864.9 1.12290
\(872\) −7622.82 −0.296033
\(873\) 0 0
\(874\) −7588.37 −0.293685
\(875\) −19406.4 −0.749780
\(876\) 0 0
\(877\) −9754.59 −0.375586 −0.187793 0.982209i \(-0.560133\pi\)
−0.187793 + 0.982209i \(0.560133\pi\)
\(878\) −23068.3 −0.886696
\(879\) 0 0
\(880\) 8590.49 0.329074
\(881\) 25789.6 0.986235 0.493118 0.869963i \(-0.335857\pi\)
0.493118 + 0.869963i \(0.335857\pi\)
\(882\) 0 0
\(883\) −8851.25 −0.337337 −0.168668 0.985673i \(-0.553947\pi\)
−0.168668 + 0.985673i \(0.553947\pi\)
\(884\) −10033.4 −0.381742
\(885\) 0 0
\(886\) 5964.99 0.226183
\(887\) −13140.9 −0.497439 −0.248720 0.968576i \(-0.580010\pi\)
−0.248720 + 0.968576i \(0.580010\pi\)
\(888\) 0 0
\(889\) −28879.9 −1.08954
\(890\) −20049.9 −0.755138
\(891\) 0 0
\(892\) 22729.8 0.853196
\(893\) 95430.4 3.57610
\(894\) 0 0
\(895\) 30703.7 1.14672
\(896\) 2394.45 0.0892780
\(897\) 0 0
\(898\) 25261.7 0.938744
\(899\) −10477.7 −0.388709
\(900\) 0 0
\(901\) 47738.2 1.76514
\(902\) 27162.8 1.00269
\(903\) 0 0
\(904\) −2045.07 −0.0752413
\(905\) −29056.6 −1.06726
\(906\) 0 0
\(907\) −12195.6 −0.446471 −0.223235 0.974765i \(-0.571662\pi\)
−0.223235 + 0.974765i \(0.571662\pi\)
\(908\) −632.394 −0.0231132
\(909\) 0 0
\(910\) 15947.0 0.580920
\(911\) 23873.1 0.868223 0.434111 0.900859i \(-0.357062\pi\)
0.434111 + 0.900859i \(0.357062\pi\)
\(912\) 0 0
\(913\) 2054.78 0.0744835
\(914\) −27386.1 −0.991087
\(915\) 0 0
\(916\) 10979.8 0.396050
\(917\) −2186.92 −0.0787550
\(918\) 0 0
\(919\) −25048.2 −0.899092 −0.449546 0.893257i \(-0.648414\pi\)
−0.449546 + 0.893257i \(0.648414\pi\)
\(920\) 2403.06 0.0861159
\(921\) 0 0
\(922\) 9177.28 0.327806
\(923\) −25380.6 −0.905105
\(924\) 0 0
\(925\) 12834.9 0.456224
\(926\) 12221.7 0.433725
\(927\) 0 0
\(928\) −1361.01 −0.0481435
\(929\) 40641.7 1.43532 0.717660 0.696394i \(-0.245213\pi\)
0.717660 + 0.696394i \(0.245213\pi\)
\(930\) 0 0
\(931\) −1144.74 −0.0402979
\(932\) 14590.5 0.512798
\(933\) 0 0
\(934\) 4156.96 0.145632
\(935\) −41265.0 −1.44332
\(936\) 0 0
\(937\) 39519.4 1.37785 0.688924 0.724833i \(-0.258083\pi\)
0.688924 + 0.724833i \(0.258083\pi\)
\(938\) 33089.6 1.15183
\(939\) 0 0
\(940\) −30220.6 −1.04860
\(941\) 32189.2 1.11513 0.557566 0.830133i \(-0.311735\pi\)
0.557566 + 0.830133i \(0.311735\pi\)
\(942\) 0 0
\(943\) 7598.40 0.262394
\(944\) 5558.91 0.191660
\(945\) 0 0
\(946\) −13208.3 −0.453951
\(947\) 18703.2 0.641786 0.320893 0.947115i \(-0.396017\pi\)
0.320893 + 0.947115i \(0.396017\pi\)
\(948\) 0 0
\(949\) −6138.93 −0.209987
\(950\) −15033.8 −0.513433
\(951\) 0 0
\(952\) −11501.9 −0.391574
\(953\) −19901.0 −0.676449 −0.338224 0.941066i \(-0.609826\pi\)
−0.338224 + 0.941066i \(0.609826\pi\)
\(954\) 0 0
\(955\) 40473.9 1.37142
\(956\) −10320.5 −0.349153
\(957\) 0 0
\(958\) 26490.8 0.893401
\(959\) −51817.4 −1.74481
\(960\) 0 0
\(961\) 30897.9 1.03715
\(962\) 18385.6 0.616191
\(963\) 0 0
\(964\) 17994.3 0.601202
\(965\) −8880.03 −0.296226
\(966\) 0 0
\(967\) −1654.58 −0.0550236 −0.0275118 0.999621i \(-0.508758\pi\)
−0.0275118 + 0.999621i \(0.508758\pi\)
\(968\) 2872.46 0.0953765
\(969\) 0 0
\(970\) 42479.7 1.40613
\(971\) −21532.9 −0.711661 −0.355831 0.934550i \(-0.615802\pi\)
−0.355831 + 0.934550i \(0.615802\pi\)
\(972\) 0 0
\(973\) −24248.8 −0.798953
\(974\) 16654.7 0.547897
\(975\) 0 0
\(976\) 5953.90 0.195266
\(977\) −28732.0 −0.940856 −0.470428 0.882438i \(-0.655901\pi\)
−0.470428 + 0.882438i \(0.655901\pi\)
\(978\) 0 0
\(979\) −31556.2 −1.03017
\(980\) 362.513 0.0118164
\(981\) 0 0
\(982\) 21171.8 0.688004
\(983\) 7497.77 0.243277 0.121639 0.992574i \(-0.461185\pi\)
0.121639 + 0.992574i \(0.461185\pi\)
\(984\) 0 0
\(985\) −651.854 −0.0210861
\(986\) 6537.67 0.211158
\(987\) 0 0
\(988\) −21535.5 −0.693458
\(989\) −3694.82 −0.118795
\(990\) 0 0
\(991\) 32374.5 1.03775 0.518874 0.854851i \(-0.326351\pi\)
0.518874 + 0.854851i \(0.326351\pi\)
\(992\) 7883.23 0.252311
\(993\) 0 0
\(994\) −29095.3 −0.928417
\(995\) −52930.0 −1.68643
\(996\) 0 0
\(997\) 19368.9 0.615264 0.307632 0.951505i \(-0.400463\pi\)
0.307632 + 0.951505i \(0.400463\pi\)
\(998\) −5261.77 −0.166892
\(999\) 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 414.4.a.n.1.3 yes 4
3.2 odd 2 414.4.a.m.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
414.4.a.m.1.2 4 3.2 odd 2
414.4.a.n.1.3 yes 4 1.1 even 1 trivial