Properties

Label 693.4.a.s.1.2
Level $693$
Weight $4$
Character 693.1
Self dual yes
Analytic conductor $40.888$
Analytic rank $1$
Dimension $8$
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] = [693,4,Mod(1,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("693.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 693.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: \(40.8883236340\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 45x^{6} + 77x^{5} + 540x^{4} - 915x^{3} - 1452x^{2} + 2660x - 672 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.50363\) of defining polynomial
Character \(\chi\) \(=\) 693.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.50363 q^{2} +4.27544 q^{4} +15.3058 q^{5} -7.00000 q^{7} +13.0495 q^{8} -53.6258 q^{10} +11.0000 q^{11} +5.90915 q^{13} +24.5254 q^{14} -79.9241 q^{16} -73.2134 q^{17} +80.3384 q^{19} +65.4390 q^{20} -38.5400 q^{22} -190.324 q^{23} +109.267 q^{25} -20.7035 q^{26} -29.9281 q^{28} -285.553 q^{29} +85.3709 q^{31} +175.629 q^{32} +256.513 q^{34} -107.140 q^{35} -149.061 q^{37} -281.476 q^{38} +199.732 q^{40} +13.9250 q^{41} -148.052 q^{43} +47.0299 q^{44} +666.824 q^{46} +94.7888 q^{47} +49.0000 q^{49} -382.831 q^{50} +25.2642 q^{52} +105.750 q^{53} +168.364 q^{55} -91.3463 q^{56} +1000.47 q^{58} +184.187 q^{59} -189.128 q^{61} -299.108 q^{62} +24.0535 q^{64} +90.4441 q^{65} -193.756 q^{67} -313.020 q^{68} +375.381 q^{70} +356.265 q^{71} +655.740 q^{73} +522.254 q^{74} +343.482 q^{76} -77.0000 q^{77} +110.189 q^{79} -1223.30 q^{80} -48.7881 q^{82} -1278.27 q^{83} -1120.59 q^{85} +518.721 q^{86} +143.544 q^{88} +255.120 q^{89} -41.3640 q^{91} -813.718 q^{92} -332.105 q^{94} +1229.64 q^{95} +372.537 q^{97} -171.678 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8} - 13 q^{10} + 88 q^{11} - 148 q^{13} + 14 q^{14} + 266 q^{16} - 114 q^{17} + 58 q^{19} - 291 q^{20} - 22 q^{22} - 246 q^{23} + 244 q^{25} - 305 q^{26}+ \cdots - 98 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.50363 −1.23872 −0.619361 0.785107i \(-0.712608\pi\)
−0.619361 + 0.785107i \(0.712608\pi\)
\(3\) 0 0
\(4\) 4.27544 0.534430
\(5\) 15.3058 1.36899 0.684495 0.729017i \(-0.260023\pi\)
0.684495 + 0.729017i \(0.260023\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 13.0495 0.576711
\(9\) 0 0
\(10\) −53.6258 −1.69580
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 5.90915 0.126069 0.0630347 0.998011i \(-0.479922\pi\)
0.0630347 + 0.998011i \(0.479922\pi\)
\(14\) 24.5254 0.468193
\(15\) 0 0
\(16\) −79.9241 −1.24881
\(17\) −73.2134 −1.04452 −0.522260 0.852786i \(-0.674911\pi\)
−0.522260 + 0.852786i \(0.674911\pi\)
\(18\) 0 0
\(19\) 80.3384 0.970046 0.485023 0.874501i \(-0.338811\pi\)
0.485023 + 0.874501i \(0.338811\pi\)
\(20\) 65.4390 0.731630
\(21\) 0 0
\(22\) −38.5400 −0.373489
\(23\) −190.324 −1.72544 −0.862722 0.505678i \(-0.831242\pi\)
−0.862722 + 0.505678i \(0.831242\pi\)
\(24\) 0 0
\(25\) 109.267 0.874134
\(26\) −20.7035 −0.156165
\(27\) 0 0
\(28\) −29.9281 −0.201996
\(29\) −285.553 −1.82848 −0.914238 0.405177i \(-0.867210\pi\)
−0.914238 + 0.405177i \(0.867210\pi\)
\(30\) 0 0
\(31\) 85.3709 0.494615 0.247307 0.968937i \(-0.420454\pi\)
0.247307 + 0.968937i \(0.420454\pi\)
\(32\) 175.629 0.970222
\(33\) 0 0
\(34\) 256.513 1.29387
\(35\) −107.140 −0.517430
\(36\) 0 0
\(37\) −149.061 −0.662309 −0.331154 0.943577i \(-0.607438\pi\)
−0.331154 + 0.943577i \(0.607438\pi\)
\(38\) −281.476 −1.20162
\(39\) 0 0
\(40\) 199.732 0.789512
\(41\) 13.9250 0.0530420 0.0265210 0.999648i \(-0.491557\pi\)
0.0265210 + 0.999648i \(0.491557\pi\)
\(42\) 0 0
\(43\) −148.052 −0.525065 −0.262532 0.964923i \(-0.584558\pi\)
−0.262532 + 0.964923i \(0.584558\pi\)
\(44\) 47.0299 0.161137
\(45\) 0 0
\(46\) 666.824 2.13735
\(47\) 94.7888 0.294178 0.147089 0.989123i \(-0.453010\pi\)
0.147089 + 0.989123i \(0.453010\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) −382.831 −1.08281
\(51\) 0 0
\(52\) 25.2642 0.0673754
\(53\) 105.750 0.274074 0.137037 0.990566i \(-0.456242\pi\)
0.137037 + 0.990566i \(0.456242\pi\)
\(54\) 0 0
\(55\) 168.364 0.412766
\(56\) −91.3463 −0.217976
\(57\) 0 0
\(58\) 1000.47 2.26497
\(59\) 184.187 0.406424 0.203212 0.979135i \(-0.434862\pi\)
0.203212 + 0.979135i \(0.434862\pi\)
\(60\) 0 0
\(61\) −189.128 −0.396972 −0.198486 0.980104i \(-0.563602\pi\)
−0.198486 + 0.980104i \(0.563602\pi\)
\(62\) −299.108 −0.612690
\(63\) 0 0
\(64\) 24.0535 0.0469795
\(65\) 90.4441 0.172588
\(66\) 0 0
\(67\) −193.756 −0.353299 −0.176649 0.984274i \(-0.556526\pi\)
−0.176649 + 0.984274i \(0.556526\pi\)
\(68\) −313.020 −0.558224
\(69\) 0 0
\(70\) 375.381 0.640951
\(71\) 356.265 0.595505 0.297752 0.954643i \(-0.403763\pi\)
0.297752 + 0.954643i \(0.403763\pi\)
\(72\) 0 0
\(73\) 655.740 1.05135 0.525675 0.850686i \(-0.323813\pi\)
0.525675 + 0.850686i \(0.323813\pi\)
\(74\) 522.254 0.820416
\(75\) 0 0
\(76\) 343.482 0.518422
\(77\) −77.0000 −0.113961
\(78\) 0 0
\(79\) 110.189 0.156927 0.0784634 0.996917i \(-0.474999\pi\)
0.0784634 + 0.996917i \(0.474999\pi\)
\(80\) −1223.30 −1.70961
\(81\) 0 0
\(82\) −48.7881 −0.0657042
\(83\) −1278.27 −1.69047 −0.845233 0.534399i \(-0.820538\pi\)
−0.845233 + 0.534399i \(0.820538\pi\)
\(84\) 0 0
\(85\) −1120.59 −1.42994
\(86\) 518.721 0.650409
\(87\) 0 0
\(88\) 143.544 0.173885
\(89\) 255.120 0.303850 0.151925 0.988392i \(-0.451453\pi\)
0.151925 + 0.988392i \(0.451453\pi\)
\(90\) 0 0
\(91\) −41.3640 −0.0476498
\(92\) −813.718 −0.922130
\(93\) 0 0
\(94\) −332.105 −0.364405
\(95\) 1229.64 1.32798
\(96\) 0 0
\(97\) 372.537 0.389953 0.194976 0.980808i \(-0.437537\pi\)
0.194976 + 0.980808i \(0.437537\pi\)
\(98\) −171.678 −0.176960
\(99\) 0 0
\(100\) 467.164 0.467164
\(101\) −1143.78 −1.12684 −0.563419 0.826171i \(-0.690514\pi\)
−0.563419 + 0.826171i \(0.690514\pi\)
\(102\) 0 0
\(103\) 471.721 0.451263 0.225631 0.974213i \(-0.427556\pi\)
0.225631 + 0.974213i \(0.427556\pi\)
\(104\) 77.1113 0.0727056
\(105\) 0 0
\(106\) −370.510 −0.339501
\(107\) −1562.49 −1.41170 −0.705849 0.708363i \(-0.749434\pi\)
−0.705849 + 0.708363i \(0.749434\pi\)
\(108\) 0 0
\(109\) −895.894 −0.787258 −0.393629 0.919269i \(-0.628781\pi\)
−0.393629 + 0.919269i \(0.628781\pi\)
\(110\) −589.884 −0.511302
\(111\) 0 0
\(112\) 559.469 0.472008
\(113\) −341.558 −0.284345 −0.142173 0.989842i \(-0.545409\pi\)
−0.142173 + 0.989842i \(0.545409\pi\)
\(114\) 0 0
\(115\) −2913.05 −2.36212
\(116\) −1220.87 −0.977194
\(117\) 0 0
\(118\) −645.322 −0.503447
\(119\) 512.493 0.394792
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 662.634 0.491738
\(123\) 0 0
\(124\) 364.998 0.264337
\(125\) −240.809 −0.172309
\(126\) 0 0
\(127\) −2446.85 −1.70963 −0.854813 0.518935i \(-0.826329\pi\)
−0.854813 + 0.518935i \(0.826329\pi\)
\(128\) −1489.31 −1.02842
\(129\) 0 0
\(130\) −316.883 −0.213788
\(131\) −2075.09 −1.38398 −0.691989 0.721908i \(-0.743265\pi\)
−0.691989 + 0.721908i \(0.743265\pi\)
\(132\) 0 0
\(133\) −562.368 −0.366643
\(134\) 678.849 0.437639
\(135\) 0 0
\(136\) −955.396 −0.602386
\(137\) −199.066 −0.124141 −0.0620707 0.998072i \(-0.519770\pi\)
−0.0620707 + 0.998072i \(0.519770\pi\)
\(138\) 0 0
\(139\) 2358.20 1.43899 0.719495 0.694497i \(-0.244373\pi\)
0.719495 + 0.694497i \(0.244373\pi\)
\(140\) −458.073 −0.276530
\(141\) 0 0
\(142\) −1248.22 −0.737664
\(143\) 65.0006 0.0380114
\(144\) 0 0
\(145\) −4370.61 −2.50317
\(146\) −2297.47 −1.30233
\(147\) 0 0
\(148\) −637.300 −0.353958
\(149\) 616.754 0.339104 0.169552 0.985521i \(-0.445768\pi\)
0.169552 + 0.985521i \(0.445768\pi\)
\(150\) 0 0
\(151\) −997.331 −0.537494 −0.268747 0.963211i \(-0.586610\pi\)
−0.268747 + 0.963211i \(0.586610\pi\)
\(152\) 1048.37 0.559436
\(153\) 0 0
\(154\) 269.780 0.141165
\(155\) 1306.67 0.677123
\(156\) 0 0
\(157\) 3181.76 1.61740 0.808701 0.588220i \(-0.200171\pi\)
0.808701 + 0.588220i \(0.200171\pi\)
\(158\) −386.061 −0.194388
\(159\) 0 0
\(160\) 2688.14 1.32822
\(161\) 1332.27 0.652157
\(162\) 0 0
\(163\) −2715.81 −1.30502 −0.652511 0.757779i \(-0.726284\pi\)
−0.652511 + 0.757779i \(0.726284\pi\)
\(164\) 59.5356 0.0283473
\(165\) 0 0
\(166\) 4478.60 2.09402
\(167\) 193.315 0.0895759 0.0447880 0.998997i \(-0.485739\pi\)
0.0447880 + 0.998997i \(0.485739\pi\)
\(168\) 0 0
\(169\) −2162.08 −0.984106
\(170\) 3926.13 1.77129
\(171\) 0 0
\(172\) −632.990 −0.280611
\(173\) −2560.26 −1.12516 −0.562580 0.826743i \(-0.690191\pi\)
−0.562580 + 0.826743i \(0.690191\pi\)
\(174\) 0 0
\(175\) −764.868 −0.330392
\(176\) −879.165 −0.376532
\(177\) 0 0
\(178\) −893.847 −0.376386
\(179\) −2783.34 −1.16221 −0.581107 0.813827i \(-0.697380\pi\)
−0.581107 + 0.813827i \(0.697380\pi\)
\(180\) 0 0
\(181\) 3055.72 1.25486 0.627430 0.778673i \(-0.284107\pi\)
0.627430 + 0.778673i \(0.284107\pi\)
\(182\) 144.924 0.0590248
\(183\) 0 0
\(184\) −2483.62 −0.995083
\(185\) −2281.49 −0.906694
\(186\) 0 0
\(187\) −805.347 −0.314935
\(188\) 405.264 0.157218
\(189\) 0 0
\(190\) −4308.21 −1.64500
\(191\) −3141.28 −1.19003 −0.595014 0.803715i \(-0.702854\pi\)
−0.595014 + 0.803715i \(0.702854\pi\)
\(192\) 0 0
\(193\) −1695.74 −0.632446 −0.316223 0.948685i \(-0.602415\pi\)
−0.316223 + 0.948685i \(0.602415\pi\)
\(194\) −1305.23 −0.483043
\(195\) 0 0
\(196\) 209.497 0.0763472
\(197\) 3061.85 1.10735 0.553675 0.832733i \(-0.313225\pi\)
0.553675 + 0.832733i \(0.313225\pi\)
\(198\) 0 0
\(199\) 66.9619 0.0238533 0.0119266 0.999929i \(-0.496204\pi\)
0.0119266 + 0.999929i \(0.496204\pi\)
\(200\) 1425.87 0.504123
\(201\) 0 0
\(202\) 4007.40 1.39584
\(203\) 1998.87 0.691099
\(204\) 0 0
\(205\) 213.133 0.0726140
\(206\) −1652.74 −0.558989
\(207\) 0 0
\(208\) −472.284 −0.157437
\(209\) 883.722 0.292480
\(210\) 0 0
\(211\) −913.636 −0.298092 −0.149046 0.988830i \(-0.547620\pi\)
−0.149046 + 0.988830i \(0.547620\pi\)
\(212\) 452.130 0.146474
\(213\) 0 0
\(214\) 5474.39 1.74870
\(215\) −2266.06 −0.718809
\(216\) 0 0
\(217\) −597.596 −0.186947
\(218\) 3138.89 0.975193
\(219\) 0 0
\(220\) 719.829 0.220595
\(221\) −432.629 −0.131682
\(222\) 0 0
\(223\) 153.648 0.0461391 0.0230695 0.999734i \(-0.492656\pi\)
0.0230695 + 0.999734i \(0.492656\pi\)
\(224\) −1229.40 −0.366710
\(225\) 0 0
\(226\) 1196.69 0.352225
\(227\) −497.228 −0.145384 −0.0726921 0.997354i \(-0.523159\pi\)
−0.0726921 + 0.997354i \(0.523159\pi\)
\(228\) 0 0
\(229\) 3374.79 0.973854 0.486927 0.873443i \(-0.338118\pi\)
0.486927 + 0.873443i \(0.338118\pi\)
\(230\) 10206.3 2.92600
\(231\) 0 0
\(232\) −3726.32 −1.05450
\(233\) −4849.94 −1.36365 −0.681825 0.731516i \(-0.738813\pi\)
−0.681825 + 0.731516i \(0.738813\pi\)
\(234\) 0 0
\(235\) 1450.82 0.402727
\(236\) 787.479 0.217206
\(237\) 0 0
\(238\) −1795.59 −0.489037
\(239\) −7350.07 −1.98927 −0.994637 0.103430i \(-0.967018\pi\)
−0.994637 + 0.103430i \(0.967018\pi\)
\(240\) 0 0
\(241\) 919.171 0.245680 0.122840 0.992426i \(-0.460800\pi\)
0.122840 + 0.992426i \(0.460800\pi\)
\(242\) −423.940 −0.112611
\(243\) 0 0
\(244\) −808.604 −0.212154
\(245\) 749.983 0.195570
\(246\) 0 0
\(247\) 474.731 0.122293
\(248\) 1114.05 0.285250
\(249\) 0 0
\(250\) 843.706 0.213443
\(251\) 4350.71 1.09408 0.547041 0.837106i \(-0.315754\pi\)
0.547041 + 0.837106i \(0.315754\pi\)
\(252\) 0 0
\(253\) −2093.56 −0.520241
\(254\) 8572.86 2.11775
\(255\) 0 0
\(256\) 5025.56 1.22694
\(257\) 4128.98 1.00217 0.501087 0.865397i \(-0.332934\pi\)
0.501087 + 0.865397i \(0.332934\pi\)
\(258\) 0 0
\(259\) 1043.42 0.250329
\(260\) 386.689 0.0922362
\(261\) 0 0
\(262\) 7270.34 1.71436
\(263\) 6853.69 1.60691 0.803454 0.595367i \(-0.202993\pi\)
0.803454 + 0.595367i \(0.202993\pi\)
\(264\) 0 0
\(265\) 1618.59 0.375205
\(266\) 1970.33 0.454169
\(267\) 0 0
\(268\) −828.392 −0.188814
\(269\) 7519.13 1.70427 0.852136 0.523320i \(-0.175307\pi\)
0.852136 + 0.523320i \(0.175307\pi\)
\(270\) 0 0
\(271\) −5354.31 −1.20019 −0.600095 0.799929i \(-0.704870\pi\)
−0.600095 + 0.799929i \(0.704870\pi\)
\(272\) 5851.51 1.30441
\(273\) 0 0
\(274\) 697.455 0.153777
\(275\) 1201.93 0.263561
\(276\) 0 0
\(277\) 6529.96 1.41642 0.708208 0.706004i \(-0.249504\pi\)
0.708208 + 0.706004i \(0.249504\pi\)
\(278\) −8262.26 −1.78251
\(279\) 0 0
\(280\) −1398.13 −0.298407
\(281\) −1530.30 −0.324876 −0.162438 0.986719i \(-0.551936\pi\)
−0.162438 + 0.986719i \(0.551936\pi\)
\(282\) 0 0
\(283\) −5563.44 −1.16860 −0.584298 0.811540i \(-0.698630\pi\)
−0.584298 + 0.811540i \(0.698630\pi\)
\(284\) 1523.19 0.318256
\(285\) 0 0
\(286\) −227.738 −0.0470855
\(287\) −97.4751 −0.0200480
\(288\) 0 0
\(289\) 447.195 0.0910228
\(290\) 15313.0 3.10073
\(291\) 0 0
\(292\) 2803.58 0.561873
\(293\) −4599.78 −0.917140 −0.458570 0.888658i \(-0.651638\pi\)
−0.458570 + 0.888658i \(0.651638\pi\)
\(294\) 0 0
\(295\) 2819.12 0.556391
\(296\) −1945.16 −0.381961
\(297\) 0 0
\(298\) −2160.88 −0.420055
\(299\) −1124.65 −0.217526
\(300\) 0 0
\(301\) 1036.37 0.198456
\(302\) 3494.28 0.665806
\(303\) 0 0
\(304\) −6420.97 −1.21141
\(305\) −2894.74 −0.543451
\(306\) 0 0
\(307\) 5902.91 1.09738 0.548692 0.836025i \(-0.315126\pi\)
0.548692 + 0.836025i \(0.315126\pi\)
\(308\) −329.209 −0.0609040
\(309\) 0 0
\(310\) −4578.08 −0.838766
\(311\) −1759.91 −0.320886 −0.160443 0.987045i \(-0.551292\pi\)
−0.160443 + 0.987045i \(0.551292\pi\)
\(312\) 0 0
\(313\) −2376.13 −0.429095 −0.214548 0.976714i \(-0.568828\pi\)
−0.214548 + 0.976714i \(0.568828\pi\)
\(314\) −11147.7 −2.00351
\(315\) 0 0
\(316\) 471.106 0.0838664
\(317\) −1891.58 −0.335147 −0.167573 0.985860i \(-0.553593\pi\)
−0.167573 + 0.985860i \(0.553593\pi\)
\(318\) 0 0
\(319\) −3141.08 −0.551307
\(320\) 368.158 0.0643145
\(321\) 0 0
\(322\) −4667.77 −0.807841
\(323\) −5881.84 −1.01323
\(324\) 0 0
\(325\) 645.674 0.110202
\(326\) 9515.20 1.61656
\(327\) 0 0
\(328\) 181.714 0.0305899
\(329\) −663.522 −0.111189
\(330\) 0 0
\(331\) −8905.72 −1.47886 −0.739430 0.673234i \(-0.764905\pi\)
−0.739430 + 0.673234i \(0.764905\pi\)
\(332\) −5465.18 −0.903436
\(333\) 0 0
\(334\) −677.306 −0.110960
\(335\) −2965.58 −0.483663
\(336\) 0 0
\(337\) 10164.2 1.64297 0.821484 0.570232i \(-0.193147\pi\)
0.821484 + 0.570232i \(0.193147\pi\)
\(338\) 7575.14 1.21903
\(339\) 0 0
\(340\) −4791.01 −0.764203
\(341\) 939.079 0.149132
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) −1932.01 −0.302811
\(345\) 0 0
\(346\) 8970.20 1.39376
\(347\) −7678.94 −1.18797 −0.593987 0.804475i \(-0.702447\pi\)
−0.593987 + 0.804475i \(0.702447\pi\)
\(348\) 0 0
\(349\) 8494.69 1.30289 0.651447 0.758694i \(-0.274162\pi\)
0.651447 + 0.758694i \(0.274162\pi\)
\(350\) 2679.82 0.409263
\(351\) 0 0
\(352\) 1931.92 0.292533
\(353\) −1477.92 −0.222838 −0.111419 0.993774i \(-0.535540\pi\)
−0.111419 + 0.993774i \(0.535540\pi\)
\(354\) 0 0
\(355\) 5452.91 0.815240
\(356\) 1090.75 0.162387
\(357\) 0 0
\(358\) 9751.78 1.43966
\(359\) −1354.69 −0.199158 −0.0995791 0.995030i \(-0.531750\pi\)
−0.0995791 + 0.995030i \(0.531750\pi\)
\(360\) 0 0
\(361\) −404.749 −0.0590100
\(362\) −10706.1 −1.55442
\(363\) 0 0
\(364\) −176.850 −0.0254655
\(365\) 10036.6 1.43929
\(366\) 0 0
\(367\) −2054.98 −0.292286 −0.146143 0.989263i \(-0.546686\pi\)
−0.146143 + 0.989263i \(0.546686\pi\)
\(368\) 15211.5 2.15476
\(369\) 0 0
\(370\) 7993.50 1.12314
\(371\) −740.253 −0.103590
\(372\) 0 0
\(373\) 6632.28 0.920661 0.460330 0.887748i \(-0.347731\pi\)
0.460330 + 0.887748i \(0.347731\pi\)
\(374\) 2821.64 0.390116
\(375\) 0 0
\(376\) 1236.94 0.169656
\(377\) −1687.37 −0.230515
\(378\) 0 0
\(379\) −578.193 −0.0783636 −0.0391818 0.999232i \(-0.512475\pi\)
−0.0391818 + 0.999232i \(0.512475\pi\)
\(380\) 5257.26 0.709715
\(381\) 0 0
\(382\) 11005.9 1.47411
\(383\) 5643.47 0.752918 0.376459 0.926433i \(-0.377142\pi\)
0.376459 + 0.926433i \(0.377142\pi\)
\(384\) 0 0
\(385\) −1178.54 −0.156011
\(386\) 5941.25 0.783424
\(387\) 0 0
\(388\) 1592.76 0.208403
\(389\) 13487.4 1.75794 0.878970 0.476877i \(-0.158231\pi\)
0.878970 + 0.476877i \(0.158231\pi\)
\(390\) 0 0
\(391\) 13934.2 1.80226
\(392\) 639.424 0.0823873
\(393\) 0 0
\(394\) −10727.6 −1.37170
\(395\) 1686.53 0.214831
\(396\) 0 0
\(397\) −15505.0 −1.96014 −0.980068 0.198664i \(-0.936340\pi\)
−0.980068 + 0.198664i \(0.936340\pi\)
\(398\) −234.610 −0.0295476
\(399\) 0 0
\(400\) −8733.05 −1.09163
\(401\) 11389.7 1.41839 0.709196 0.705011i \(-0.249058\pi\)
0.709196 + 0.705011i \(0.249058\pi\)
\(402\) 0 0
\(403\) 504.469 0.0623558
\(404\) −4890.18 −0.602217
\(405\) 0 0
\(406\) −7003.31 −0.856079
\(407\) −1639.67 −0.199694
\(408\) 0 0
\(409\) 13512.3 1.63360 0.816798 0.576924i \(-0.195747\pi\)
0.816798 + 0.576924i \(0.195747\pi\)
\(410\) −746.740 −0.0899485
\(411\) 0 0
\(412\) 2016.82 0.241169
\(413\) −1289.31 −0.153614
\(414\) 0 0
\(415\) −19564.9 −2.31423
\(416\) 1037.82 0.122315
\(417\) 0 0
\(418\) −3096.24 −0.362301
\(419\) −8247.54 −0.961620 −0.480810 0.876825i \(-0.659657\pi\)
−0.480810 + 0.876825i \(0.659657\pi\)
\(420\) 0 0
\(421\) 2630.75 0.304549 0.152275 0.988338i \(-0.451340\pi\)
0.152275 + 0.988338i \(0.451340\pi\)
\(422\) 3201.05 0.369252
\(423\) 0 0
\(424\) 1379.99 0.158062
\(425\) −7999.79 −0.913051
\(426\) 0 0
\(427\) 1323.89 0.150041
\(428\) −6680.34 −0.754454
\(429\) 0 0
\(430\) 7939.43 0.890404
\(431\) 10428.7 1.16551 0.582753 0.812649i \(-0.301976\pi\)
0.582753 + 0.812649i \(0.301976\pi\)
\(432\) 0 0
\(433\) −8822.44 −0.979167 −0.489583 0.871956i \(-0.662851\pi\)
−0.489583 + 0.871956i \(0.662851\pi\)
\(434\) 2093.76 0.231575
\(435\) 0 0
\(436\) −3830.35 −0.420735
\(437\) −15290.3 −1.67376
\(438\) 0 0
\(439\) −16899.6 −1.83730 −0.918652 0.395068i \(-0.870721\pi\)
−0.918652 + 0.395068i \(0.870721\pi\)
\(440\) 2197.06 0.238047
\(441\) 0 0
\(442\) 1515.77 0.163117
\(443\) −9130.18 −0.979206 −0.489603 0.871946i \(-0.662858\pi\)
−0.489603 + 0.871946i \(0.662858\pi\)
\(444\) 0 0
\(445\) 3904.81 0.415968
\(446\) −538.325 −0.0571534
\(447\) 0 0
\(448\) −168.375 −0.0177566
\(449\) −6461.47 −0.679143 −0.339572 0.940580i \(-0.610282\pi\)
−0.339572 + 0.940580i \(0.610282\pi\)
\(450\) 0 0
\(451\) 153.175 0.0159928
\(452\) −1460.31 −0.151963
\(453\) 0 0
\(454\) 1742.11 0.180090
\(455\) −633.109 −0.0652321
\(456\) 0 0
\(457\) −9239.47 −0.945743 −0.472871 0.881132i \(-0.656782\pi\)
−0.472871 + 0.881132i \(0.656782\pi\)
\(458\) −11824.0 −1.20633
\(459\) 0 0
\(460\) −12454.6 −1.26239
\(461\) −7393.95 −0.747007 −0.373504 0.927629i \(-0.621844\pi\)
−0.373504 + 0.927629i \(0.621844\pi\)
\(462\) 0 0
\(463\) −3360.31 −0.337293 −0.168646 0.985677i \(-0.553940\pi\)
−0.168646 + 0.985677i \(0.553940\pi\)
\(464\) 22822.6 2.28343
\(465\) 0 0
\(466\) 16992.4 1.68918
\(467\) 10685.1 1.05878 0.529389 0.848379i \(-0.322421\pi\)
0.529389 + 0.848379i \(0.322421\pi\)
\(468\) 0 0
\(469\) 1356.29 0.133534
\(470\) −5083.13 −0.498866
\(471\) 0 0
\(472\) 2403.54 0.234389
\(473\) −1628.58 −0.158313
\(474\) 0 0
\(475\) 8778.31 0.847951
\(476\) 2191.14 0.210989
\(477\) 0 0
\(478\) 25751.9 2.46416
\(479\) 18204.4 1.73649 0.868247 0.496133i \(-0.165247\pi\)
0.868247 + 0.496133i \(0.165247\pi\)
\(480\) 0 0
\(481\) −880.821 −0.0834969
\(482\) −3220.44 −0.304330
\(483\) 0 0
\(484\) 517.329 0.0485846
\(485\) 5701.97 0.533841
\(486\) 0 0
\(487\) −6213.98 −0.578197 −0.289099 0.957299i \(-0.593356\pi\)
−0.289099 + 0.957299i \(0.593356\pi\)
\(488\) −2468.02 −0.228938
\(489\) 0 0
\(490\) −2627.67 −0.242257
\(491\) −17022.7 −1.56461 −0.782306 0.622894i \(-0.785957\pi\)
−0.782306 + 0.622894i \(0.785957\pi\)
\(492\) 0 0
\(493\) 20906.3 1.90988
\(494\) −1663.28 −0.151487
\(495\) 0 0
\(496\) −6823.19 −0.617682
\(497\) −2493.85 −0.225080
\(498\) 0 0
\(499\) −20595.2 −1.84763 −0.923814 0.382843i \(-0.874945\pi\)
−0.923814 + 0.382843i \(0.874945\pi\)
\(500\) −1029.56 −0.0920871
\(501\) 0 0
\(502\) −15243.3 −1.35526
\(503\) −19173.9 −1.69965 −0.849824 0.527067i \(-0.823292\pi\)
−0.849824 + 0.527067i \(0.823292\pi\)
\(504\) 0 0
\(505\) −17506.5 −1.54263
\(506\) 7335.07 0.644434
\(507\) 0 0
\(508\) −10461.4 −0.913677
\(509\) −3120.30 −0.271719 −0.135859 0.990728i \(-0.543380\pi\)
−0.135859 + 0.990728i \(0.543380\pi\)
\(510\) 0 0
\(511\) −4590.18 −0.397373
\(512\) −5693.25 −0.491423
\(513\) 0 0
\(514\) −14466.4 −1.24141
\(515\) 7220.06 0.617774
\(516\) 0 0
\(517\) 1042.68 0.0886980
\(518\) −3655.78 −0.310088
\(519\) 0 0
\(520\) 1180.25 0.0995333
\(521\) 7932.70 0.667059 0.333530 0.942740i \(-0.391760\pi\)
0.333530 + 0.942740i \(0.391760\pi\)
\(522\) 0 0
\(523\) −4156.12 −0.347484 −0.173742 0.984791i \(-0.555586\pi\)
−0.173742 + 0.984791i \(0.555586\pi\)
\(524\) −8871.91 −0.739640
\(525\) 0 0
\(526\) −24012.8 −1.99051
\(527\) −6250.29 −0.516635
\(528\) 0 0
\(529\) 24056.1 1.97716
\(530\) −5670.95 −0.464774
\(531\) 0 0
\(532\) −2404.37 −0.195945
\(533\) 82.2850 0.00668698
\(534\) 0 0
\(535\) −23915.1 −1.93260
\(536\) −2528.41 −0.203751
\(537\) 0 0
\(538\) −26344.3 −2.11112
\(539\) 539.000 0.0430730
\(540\) 0 0
\(541\) −21965.6 −1.74561 −0.872804 0.488070i \(-0.837701\pi\)
−0.872804 + 0.488070i \(0.837701\pi\)
\(542\) 18759.5 1.48670
\(543\) 0 0
\(544\) −12858.4 −1.01342
\(545\) −13712.4 −1.07775
\(546\) 0 0
\(547\) 11875.0 0.928224 0.464112 0.885776i \(-0.346373\pi\)
0.464112 + 0.885776i \(0.346373\pi\)
\(548\) −851.096 −0.0663449
\(549\) 0 0
\(550\) −4211.14 −0.326479
\(551\) −22940.8 −1.77371
\(552\) 0 0
\(553\) −771.322 −0.0593127
\(554\) −22878.6 −1.75455
\(555\) 0 0
\(556\) 10082.3 0.769040
\(557\) −2292.04 −0.174357 −0.0871786 0.996193i \(-0.527785\pi\)
−0.0871786 + 0.996193i \(0.527785\pi\)
\(558\) 0 0
\(559\) −874.864 −0.0661946
\(560\) 8563.11 0.646174
\(561\) 0 0
\(562\) 5361.62 0.402431
\(563\) 7686.51 0.575395 0.287698 0.957721i \(-0.407110\pi\)
0.287698 + 0.957721i \(0.407110\pi\)
\(564\) 0 0
\(565\) −5227.80 −0.389266
\(566\) 19492.3 1.44756
\(567\) 0 0
\(568\) 4649.07 0.343434
\(569\) 1044.55 0.0769592 0.0384796 0.999259i \(-0.487749\pi\)
0.0384796 + 0.999259i \(0.487749\pi\)
\(570\) 0 0
\(571\) −1085.27 −0.0795399 −0.0397700 0.999209i \(-0.512663\pi\)
−0.0397700 + 0.999209i \(0.512663\pi\)
\(572\) 277.907 0.0203144
\(573\) 0 0
\(574\) 341.517 0.0248339
\(575\) −20796.1 −1.50827
\(576\) 0 0
\(577\) −12880.9 −0.929354 −0.464677 0.885480i \(-0.653830\pi\)
−0.464677 + 0.885480i \(0.653830\pi\)
\(578\) −1566.81 −0.112752
\(579\) 0 0
\(580\) −18686.3 −1.33777
\(581\) 8947.91 0.638936
\(582\) 0 0
\(583\) 1163.25 0.0826365
\(584\) 8557.06 0.606325
\(585\) 0 0
\(586\) 16115.9 1.13608
\(587\) 4693.01 0.329985 0.164993 0.986295i \(-0.447240\pi\)
0.164993 + 0.986295i \(0.447240\pi\)
\(588\) 0 0
\(589\) 6858.55 0.479799
\(590\) −9877.15 −0.689213
\(591\) 0 0
\(592\) 11913.5 0.827101
\(593\) −13825.6 −0.957418 −0.478709 0.877974i \(-0.658895\pi\)
−0.478709 + 0.877974i \(0.658895\pi\)
\(594\) 0 0
\(595\) 7844.11 0.540466
\(596\) 2636.90 0.181227
\(597\) 0 0
\(598\) 3940.36 0.269454
\(599\) −13616.7 −0.928823 −0.464411 0.885620i \(-0.653734\pi\)
−0.464411 + 0.885620i \(0.653734\pi\)
\(600\) 0 0
\(601\) 25768.9 1.74898 0.874490 0.485044i \(-0.161196\pi\)
0.874490 + 0.485044i \(0.161196\pi\)
\(602\) −3631.05 −0.245831
\(603\) 0 0
\(604\) −4264.03 −0.287253
\(605\) 1852.00 0.124454
\(606\) 0 0
\(607\) 19103.1 1.27738 0.638691 0.769463i \(-0.279476\pi\)
0.638691 + 0.769463i \(0.279476\pi\)
\(608\) 14109.7 0.941161
\(609\) 0 0
\(610\) 10142.1 0.673184
\(611\) 560.121 0.0370869
\(612\) 0 0
\(613\) −6915.62 −0.455659 −0.227830 0.973701i \(-0.573163\pi\)
−0.227830 + 0.973701i \(0.573163\pi\)
\(614\) −20681.6 −1.35935
\(615\) 0 0
\(616\) −1004.81 −0.0657223
\(617\) −16084.3 −1.04948 −0.524742 0.851261i \(-0.675838\pi\)
−0.524742 + 0.851261i \(0.675838\pi\)
\(618\) 0 0
\(619\) 12041.5 0.781891 0.390945 0.920414i \(-0.372148\pi\)
0.390945 + 0.920414i \(0.372148\pi\)
\(620\) 5586.58 0.361875
\(621\) 0 0
\(622\) 6166.09 0.397488
\(623\) −1785.84 −0.114845
\(624\) 0 0
\(625\) −17344.1 −1.11002
\(626\) 8325.09 0.531530
\(627\) 0 0
\(628\) 13603.4 0.864389
\(629\) 10913.2 0.691795
\(630\) 0 0
\(631\) 19654.5 1.23999 0.619994 0.784607i \(-0.287135\pi\)
0.619994 + 0.784607i \(0.287135\pi\)
\(632\) 1437.91 0.0905014
\(633\) 0 0
\(634\) 6627.39 0.415153
\(635\) −37450.9 −2.34046
\(636\) 0 0
\(637\) 289.548 0.0180099
\(638\) 11005.2 0.682915
\(639\) 0 0
\(640\) −22795.0 −1.40789
\(641\) −9941.56 −0.612586 −0.306293 0.951937i \(-0.599089\pi\)
−0.306293 + 0.951937i \(0.599089\pi\)
\(642\) 0 0
\(643\) 15162.4 0.929930 0.464965 0.885329i \(-0.346067\pi\)
0.464965 + 0.885329i \(0.346067\pi\)
\(644\) 5696.03 0.348532
\(645\) 0 0
\(646\) 20607.8 1.25511
\(647\) −7769.45 −0.472100 −0.236050 0.971741i \(-0.575853\pi\)
−0.236050 + 0.971741i \(0.575853\pi\)
\(648\) 0 0
\(649\) 2026.05 0.122542
\(650\) −2262.20 −0.136509
\(651\) 0 0
\(652\) −11611.3 −0.697443
\(653\) 17240.5 1.03319 0.516594 0.856230i \(-0.327200\pi\)
0.516594 + 0.856230i \(0.327200\pi\)
\(654\) 0 0
\(655\) −31760.8 −1.89465
\(656\) −1112.94 −0.0662396
\(657\) 0 0
\(658\) 2324.74 0.137732
\(659\) 26820.8 1.58542 0.792709 0.609600i \(-0.208670\pi\)
0.792709 + 0.609600i \(0.208670\pi\)
\(660\) 0 0
\(661\) 33944.1 1.99738 0.998692 0.0511251i \(-0.0162807\pi\)
0.998692 + 0.0511251i \(0.0162807\pi\)
\(662\) 31202.4 1.83189
\(663\) 0 0
\(664\) −16680.8 −0.974910
\(665\) −8607.49 −0.501931
\(666\) 0 0
\(667\) 54347.5 3.15494
\(668\) 826.508 0.0478721
\(669\) 0 0
\(670\) 10390.3 0.599123
\(671\) −2080.40 −0.119692
\(672\) 0 0
\(673\) −9225.59 −0.528411 −0.264205 0.964466i \(-0.585110\pi\)
−0.264205 + 0.964466i \(0.585110\pi\)
\(674\) −35611.7 −2.03518
\(675\) 0 0
\(676\) −9243.86 −0.525936
\(677\) −23388.8 −1.32778 −0.663889 0.747831i \(-0.731095\pi\)
−0.663889 + 0.747831i \(0.731095\pi\)
\(678\) 0 0
\(679\) −2607.76 −0.147388
\(680\) −14623.1 −0.824661
\(681\) 0 0
\(682\) −3290.19 −0.184733
\(683\) −8866.19 −0.496713 −0.248357 0.968669i \(-0.579891\pi\)
−0.248357 + 0.968669i \(0.579891\pi\)
\(684\) 0 0
\(685\) −3046.86 −0.169948
\(686\) 1201.75 0.0668847
\(687\) 0 0
\(688\) 11833.0 0.655709
\(689\) 624.895 0.0345524
\(690\) 0 0
\(691\) 26611.8 1.46507 0.732534 0.680731i \(-0.238338\pi\)
0.732534 + 0.680731i \(0.238338\pi\)
\(692\) −10946.2 −0.601320
\(693\) 0 0
\(694\) 26904.2 1.47157
\(695\) 36094.0 1.96996
\(696\) 0 0
\(697\) −1019.50 −0.0554034
\(698\) −29762.3 −1.61392
\(699\) 0 0
\(700\) −3270.15 −0.176571
\(701\) −19326.3 −1.04129 −0.520644 0.853774i \(-0.674308\pi\)
−0.520644 + 0.853774i \(0.674308\pi\)
\(702\) 0 0
\(703\) −11975.3 −0.642470
\(704\) 264.589 0.0141649
\(705\) 0 0
\(706\) 5178.09 0.276034
\(707\) 8006.48 0.425905
\(708\) 0 0
\(709\) 24749.2 1.31097 0.655484 0.755209i \(-0.272465\pi\)
0.655484 + 0.755209i \(0.272465\pi\)
\(710\) −19105.0 −1.00986
\(711\) 0 0
\(712\) 3329.18 0.175234
\(713\) −16248.1 −0.853430
\(714\) 0 0
\(715\) 994.885 0.0520372
\(716\) −11900.0 −0.621122
\(717\) 0 0
\(718\) 4746.34 0.246702
\(719\) 3517.02 0.182424 0.0912119 0.995832i \(-0.470926\pi\)
0.0912119 + 0.995832i \(0.470926\pi\)
\(720\) 0 0
\(721\) −3302.05 −0.170561
\(722\) 1418.09 0.0730969
\(723\) 0 0
\(724\) 13064.5 0.670635
\(725\) −31201.4 −1.59833
\(726\) 0 0
\(727\) 27443.7 1.40004 0.700021 0.714123i \(-0.253174\pi\)
0.700021 + 0.714123i \(0.253174\pi\)
\(728\) −539.779 −0.0274801
\(729\) 0 0
\(730\) −35164.6 −1.78288
\(731\) 10839.4 0.548441
\(732\) 0 0
\(733\) −24925.8 −1.25601 −0.628005 0.778210i \(-0.716128\pi\)
−0.628005 + 0.778210i \(0.716128\pi\)
\(734\) 7199.88 0.362061
\(735\) 0 0
\(736\) −33426.4 −1.67406
\(737\) −2131.31 −0.106524
\(738\) 0 0
\(739\) 37245.9 1.85401 0.927004 0.375053i \(-0.122375\pi\)
0.927004 + 0.375053i \(0.122375\pi\)
\(740\) −9754.38 −0.484565
\(741\) 0 0
\(742\) 2593.57 0.128319
\(743\) −13860.1 −0.684357 −0.342178 0.939635i \(-0.611165\pi\)
−0.342178 + 0.939635i \(0.611165\pi\)
\(744\) 0 0
\(745\) 9439.90 0.464230
\(746\) −23237.1 −1.14044
\(747\) 0 0
\(748\) −3443.22 −0.168311
\(749\) 10937.4 0.533572
\(750\) 0 0
\(751\) 20219.8 0.982467 0.491234 0.871028i \(-0.336546\pi\)
0.491234 + 0.871028i \(0.336546\pi\)
\(752\) −7575.91 −0.367374
\(753\) 0 0
\(754\) 5911.94 0.285544
\(755\) −15264.9 −0.735825
\(756\) 0 0
\(757\) 4381.34 0.210360 0.105180 0.994453i \(-0.466458\pi\)
0.105180 + 0.994453i \(0.466458\pi\)
\(758\) 2025.78 0.0970706
\(759\) 0 0
\(760\) 16046.2 0.765863
\(761\) 14022.2 0.667942 0.333971 0.942583i \(-0.391611\pi\)
0.333971 + 0.942583i \(0.391611\pi\)
\(762\) 0 0
\(763\) 6271.26 0.297556
\(764\) −13430.4 −0.635987
\(765\) 0 0
\(766\) −19772.6 −0.932656
\(767\) 1088.39 0.0512377
\(768\) 0 0
\(769\) 10002.3 0.469042 0.234521 0.972111i \(-0.424648\pi\)
0.234521 + 0.972111i \(0.424648\pi\)
\(770\) 4129.19 0.193254
\(771\) 0 0
\(772\) −7250.04 −0.337998
\(773\) 42649.3 1.98446 0.992232 0.124405i \(-0.0397021\pi\)
0.992232 + 0.124405i \(0.0397021\pi\)
\(774\) 0 0
\(775\) 9328.20 0.432360
\(776\) 4861.41 0.224890
\(777\) 0 0
\(778\) −47254.9 −2.17760
\(779\) 1118.71 0.0514532
\(780\) 0 0
\(781\) 3918.91 0.179551
\(782\) −48820.4 −2.23250
\(783\) 0 0
\(784\) −3916.28 −0.178402
\(785\) 48699.3 2.21421
\(786\) 0 0
\(787\) 9129.62 0.413514 0.206757 0.978392i \(-0.433709\pi\)
0.206757 + 0.978392i \(0.433709\pi\)
\(788\) 13090.8 0.591801
\(789\) 0 0
\(790\) −5908.97 −0.266116
\(791\) 2390.90 0.107472
\(792\) 0 0
\(793\) −1117.58 −0.0500461
\(794\) 54323.8 2.42806
\(795\) 0 0
\(796\) 286.292 0.0127479
\(797\) 31429.2 1.39684 0.698419 0.715690i \(-0.253887\pi\)
0.698419 + 0.715690i \(0.253887\pi\)
\(798\) 0 0
\(799\) −6939.81 −0.307275
\(800\) 19190.4 0.848105
\(801\) 0 0
\(802\) −39905.4 −1.75699
\(803\) 7213.14 0.316994
\(804\) 0 0
\(805\) 20391.4 0.892796
\(806\) −1767.47 −0.0772415
\(807\) 0 0
\(808\) −14925.8 −0.649860
\(809\) 18839.2 0.818727 0.409363 0.912371i \(-0.365751\pi\)
0.409363 + 0.912371i \(0.365751\pi\)
\(810\) 0 0
\(811\) 10325.1 0.447059 0.223529 0.974697i \(-0.428242\pi\)
0.223529 + 0.974697i \(0.428242\pi\)
\(812\) 8546.06 0.369345
\(813\) 0 0
\(814\) 5744.79 0.247365
\(815\) −41567.6 −1.78656
\(816\) 0 0
\(817\) −11894.3 −0.509337
\(818\) −47342.2 −2.02357
\(819\) 0 0
\(820\) 911.239 0.0388071
\(821\) 31707.7 1.34788 0.673938 0.738788i \(-0.264601\pi\)
0.673938 + 0.738788i \(0.264601\pi\)
\(822\) 0 0
\(823\) −27144.9 −1.14971 −0.574856 0.818254i \(-0.694942\pi\)
−0.574856 + 0.818254i \(0.694942\pi\)
\(824\) 6155.72 0.260248
\(825\) 0 0
\(826\) 4517.25 0.190285
\(827\) −10010.3 −0.420909 −0.210455 0.977604i \(-0.567494\pi\)
−0.210455 + 0.977604i \(0.567494\pi\)
\(828\) 0 0
\(829\) −36122.6 −1.51338 −0.756689 0.653776i \(-0.773184\pi\)
−0.756689 + 0.653776i \(0.773184\pi\)
\(830\) 68548.4 2.86669
\(831\) 0 0
\(832\) 142.136 0.00592269
\(833\) −3587.45 −0.149217
\(834\) 0 0
\(835\) 2958.84 0.122629
\(836\) 3778.30 0.156310
\(837\) 0 0
\(838\) 28896.4 1.19118
\(839\) 6552.24 0.269617 0.134808 0.990872i \(-0.456958\pi\)
0.134808 + 0.990872i \(0.456958\pi\)
\(840\) 0 0
\(841\) 57151.4 2.34333
\(842\) −9217.20 −0.377251
\(843\) 0 0
\(844\) −3906.20 −0.159309
\(845\) −33092.3 −1.34723
\(846\) 0 0
\(847\) −847.000 −0.0343604
\(848\) −8452.01 −0.342268
\(849\) 0 0
\(850\) 28028.3 1.13102
\(851\) 28369.8 1.14278
\(852\) 0 0
\(853\) −29048.8 −1.16601 −0.583007 0.812467i \(-0.698124\pi\)
−0.583007 + 0.812467i \(0.698124\pi\)
\(854\) −4638.43 −0.185859
\(855\) 0 0
\(856\) −20389.7 −0.814141
\(857\) 4964.30 0.197873 0.0989365 0.995094i \(-0.468456\pi\)
0.0989365 + 0.995094i \(0.468456\pi\)
\(858\) 0 0
\(859\) 21708.3 0.862255 0.431127 0.902291i \(-0.358116\pi\)
0.431127 + 0.902291i \(0.358116\pi\)
\(860\) −9688.40 −0.384153
\(861\) 0 0
\(862\) −36538.3 −1.44374
\(863\) −11197.9 −0.441692 −0.220846 0.975309i \(-0.570882\pi\)
−0.220846 + 0.975309i \(0.570882\pi\)
\(864\) 0 0
\(865\) −39186.7 −1.54033
\(866\) 30910.6 1.21291
\(867\) 0 0
\(868\) −2554.99 −0.0999101
\(869\) 1212.08 0.0473152
\(870\) 0 0
\(871\) −1144.93 −0.0445402
\(872\) −11691.0 −0.454020
\(873\) 0 0
\(874\) 53571.6 2.07332
\(875\) 1685.66 0.0651266
\(876\) 0 0
\(877\) −20949.4 −0.806626 −0.403313 0.915062i \(-0.632141\pi\)
−0.403313 + 0.915062i \(0.632141\pi\)
\(878\) 59210.2 2.27591
\(879\) 0 0
\(880\) −13456.3 −0.515468
\(881\) 35886.4 1.37236 0.686178 0.727434i \(-0.259287\pi\)
0.686178 + 0.727434i \(0.259287\pi\)
\(882\) 0 0
\(883\) −7181.51 −0.273700 −0.136850 0.990592i \(-0.543698\pi\)
−0.136850 + 0.990592i \(0.543698\pi\)
\(884\) −1849.68 −0.0703749
\(885\) 0 0
\(886\) 31988.8 1.21296
\(887\) 25132.3 0.951365 0.475682 0.879617i \(-0.342201\pi\)
0.475682 + 0.879617i \(0.342201\pi\)
\(888\) 0 0
\(889\) 17127.9 0.646178
\(890\) −13681.0 −0.515268
\(891\) 0 0
\(892\) 656.912 0.0246581
\(893\) 7615.18 0.285366
\(894\) 0 0
\(895\) −42601.1 −1.59106
\(896\) 10425.1 0.388705
\(897\) 0 0
\(898\) 22638.6 0.841269
\(899\) −24377.9 −0.904392
\(900\) 0 0
\(901\) −7742.34 −0.286276
\(902\) −536.669 −0.0198106
\(903\) 0 0
\(904\) −4457.15 −0.163985
\(905\) 46770.1 1.71789
\(906\) 0 0
\(907\) −9475.36 −0.346884 −0.173442 0.984844i \(-0.555489\pi\)
−0.173442 + 0.984844i \(0.555489\pi\)
\(908\) −2125.87 −0.0776977
\(909\) 0 0
\(910\) 2218.18 0.0808044
\(911\) 43008.3 1.56414 0.782068 0.623193i \(-0.214165\pi\)
0.782068 + 0.623193i \(0.214165\pi\)
\(912\) 0 0
\(913\) −14061.0 −0.509694
\(914\) 32371.7 1.17151
\(915\) 0 0
\(916\) 14428.7 0.520457
\(917\) 14525.6 0.523094
\(918\) 0 0
\(919\) 23979.4 0.860727 0.430363 0.902656i \(-0.358385\pi\)
0.430363 + 0.902656i \(0.358385\pi\)
\(920\) −38013.8 −1.36226
\(921\) 0 0
\(922\) 25905.7 0.925334
\(923\) 2105.22 0.0750750
\(924\) 0 0
\(925\) −16287.4 −0.578947
\(926\) 11773.3 0.417812
\(927\) 0 0
\(928\) −50151.4 −1.77403
\(929\) 27635.2 0.975975 0.487988 0.872851i \(-0.337731\pi\)
0.487988 + 0.872851i \(0.337731\pi\)
\(930\) 0 0
\(931\) 3936.58 0.138578
\(932\) −20735.7 −0.728776
\(933\) 0 0
\(934\) −37436.8 −1.31153
\(935\) −12326.5 −0.431143
\(936\) 0 0
\(937\) −5950.96 −0.207481 −0.103740 0.994604i \(-0.533081\pi\)
−0.103740 + 0.994604i \(0.533081\pi\)
\(938\) −4751.94 −0.165412
\(939\) 0 0
\(940\) 6202.88 0.215230
\(941\) −21970.4 −0.761122 −0.380561 0.924756i \(-0.624269\pi\)
−0.380561 + 0.924756i \(0.624269\pi\)
\(942\) 0 0
\(943\) −2650.26 −0.0915210
\(944\) −14720.9 −0.507549
\(945\) 0 0
\(946\) 5705.94 0.196106
\(947\) −18563.0 −0.636976 −0.318488 0.947927i \(-0.603175\pi\)
−0.318488 + 0.947927i \(0.603175\pi\)
\(948\) 0 0
\(949\) 3874.86 0.132543
\(950\) −30756.0 −1.05037
\(951\) 0 0
\(952\) 6687.77 0.227681
\(953\) 642.345 0.0218338 0.0109169 0.999940i \(-0.496525\pi\)
0.0109169 + 0.999940i \(0.496525\pi\)
\(954\) 0 0
\(955\) −48079.8 −1.62914
\(956\) −31424.8 −1.06313
\(957\) 0 0
\(958\) −63781.5 −2.15103
\(959\) 1393.46 0.0469210
\(960\) 0 0
\(961\) −22502.8 −0.755356
\(962\) 3086.07 0.103429
\(963\) 0 0
\(964\) 3929.86 0.131299
\(965\) −25954.6 −0.865812
\(966\) 0 0
\(967\) 27334.8 0.909026 0.454513 0.890740i \(-0.349813\pi\)
0.454513 + 0.890740i \(0.349813\pi\)
\(968\) 1578.99 0.0524283
\(969\) 0 0
\(970\) −19977.6 −0.661281
\(971\) −53121.2 −1.75565 −0.877827 0.478978i \(-0.841007\pi\)
−0.877827 + 0.478978i \(0.841007\pi\)
\(972\) 0 0
\(973\) −16507.4 −0.543887
\(974\) 21771.5 0.716225
\(975\) 0 0
\(976\) 15115.9 0.495745
\(977\) 11393.0 0.373075 0.186537 0.982448i \(-0.440273\pi\)
0.186537 + 0.982448i \(0.440273\pi\)
\(978\) 0 0
\(979\) 2806.32 0.0916143
\(980\) 3206.51 0.104519
\(981\) 0 0
\(982\) 59641.4 1.93812
\(983\) −50295.2 −1.63191 −0.815955 0.578115i \(-0.803788\pi\)
−0.815955 + 0.578115i \(0.803788\pi\)
\(984\) 0 0
\(985\) 46864.0 1.51595
\(986\) −73247.9 −2.36581
\(987\) 0 0
\(988\) 2029.69 0.0653572
\(989\) 28177.9 0.905970
\(990\) 0 0
\(991\) −4361.21 −0.139797 −0.0698983 0.997554i \(-0.522267\pi\)
−0.0698983 + 0.997554i \(0.522267\pi\)
\(992\) 14993.6 0.479886
\(993\) 0 0
\(994\) 8737.54 0.278811
\(995\) 1024.90 0.0326549
\(996\) 0 0
\(997\) −25946.3 −0.824199 −0.412100 0.911139i \(-0.635204\pi\)
−0.412100 + 0.911139i \(0.635204\pi\)
\(998\) 72157.9 2.28870
\(999\) 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 693.4.a.s.1.2 8
3.2 odd 2 693.4.a.t.1.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.4.a.s.1.2 8 1.1 even 1 trivial
693.4.a.t.1.7 yes 8 3.2 odd 2