Properties

Label 1680.4.a.bo.1.1
Level $1680$
Weight $4$
Character 1680.1
Self dual yes
Analytic conductor $99.123$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1680.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(99.1232088096\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1680.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} -48.5685 q^{11} -43.6569 q^{13} +15.0000 q^{15} -67.6569 q^{17} +93.2548 q^{19} +21.0000 q^{21} +104.167 q^{23} +25.0000 q^{25} +27.0000 q^{27} -58.7351 q^{29} +9.08831 q^{31} -145.706 q^{33} +35.0000 q^{35} -252.676 q^{37} -130.971 q^{39} +276.274 q^{41} +92.6375 q^{43} +45.0000 q^{45} +582.794 q^{47} +49.0000 q^{49} -202.971 q^{51} +623.019 q^{53} -242.843 q^{55} +279.765 q^{57} +524.999 q^{59} -352.794 q^{61} +63.0000 q^{63} -218.284 q^{65} +736.520 q^{67} +312.500 q^{69} +492.264 q^{71} +1164.75 q^{73} +75.0000 q^{75} -339.980 q^{77} +872.195 q^{79} +81.0000 q^{81} +529.588 q^{83} -338.284 q^{85} -176.205 q^{87} -385.216 q^{89} -305.598 q^{91} +27.2649 q^{93} +466.274 q^{95} -463.892 q^{97} -437.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 10q^{5} + 14q^{7} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 10q^{5} + 14q^{7} + 18q^{9} + 16q^{11} - 76q^{13} + 30q^{15} - 124q^{17} + 96q^{19} + 42q^{21} + 16q^{23} + 50q^{25} + 54q^{27} + 188q^{29} + 120q^{31} + 48q^{33} + 70q^{35} - 132q^{37} - 228q^{39} + 100q^{41} + 536q^{43} + 90q^{45} + 928q^{47} + 98q^{49} - 372q^{51} + 884q^{53} + 80q^{55} + 288q^{57} - 104q^{59} - 468q^{61} + 126q^{63} - 380q^{65} + 1688q^{67} + 48q^{69} + 136q^{71} + 508q^{73} + 150q^{75} + 112q^{77} + 432q^{79} + 162q^{81} + 584q^{83} - 620q^{85} + 564q^{87} - 1404q^{89} - 532q^{91} + 360q^{93} + 480q^{95} - 1188q^{97} + 144q^{99} + O(q^{100}) \)

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
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −48.5685 −1.33127 −0.665635 0.746278i \(-0.731839\pi\)
−0.665635 + 0.746278i \(0.731839\pi\)
\(12\) 0 0
\(13\) −43.6569 −0.931403 −0.465701 0.884942i \(-0.654198\pi\)
−0.465701 + 0.884942i \(0.654198\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) −67.6569 −0.965247 −0.482623 0.875828i \(-0.660316\pi\)
−0.482623 + 0.875828i \(0.660316\pi\)
\(18\) 0 0
\(19\) 93.2548 1.12601 0.563003 0.826455i \(-0.309646\pi\)
0.563003 + 0.826455i \(0.309646\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) 0 0
\(23\) 104.167 0.944357 0.472179 0.881503i \(-0.343468\pi\)
0.472179 + 0.881503i \(0.343468\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −58.7351 −0.376098 −0.188049 0.982160i \(-0.560216\pi\)
−0.188049 + 0.982160i \(0.560216\pi\)
\(30\) 0 0
\(31\) 9.08831 0.0526551 0.0263276 0.999653i \(-0.491619\pi\)
0.0263276 + 0.999653i \(0.491619\pi\)
\(32\) 0 0
\(33\) −145.706 −0.768609
\(34\) 0 0
\(35\) 35.0000 0.169031
\(36\) 0 0
\(37\) −252.676 −1.12269 −0.561347 0.827580i \(-0.689717\pi\)
−0.561347 + 0.827580i \(0.689717\pi\)
\(38\) 0 0
\(39\) −130.971 −0.537745
\(40\) 0 0
\(41\) 276.274 1.05236 0.526180 0.850373i \(-0.323624\pi\)
0.526180 + 0.850373i \(0.323624\pi\)
\(42\) 0 0
\(43\) 92.6375 0.328537 0.164268 0.986416i \(-0.447474\pi\)
0.164268 + 0.986416i \(0.447474\pi\)
\(44\) 0 0
\(45\) 45.0000 0.149071
\(46\) 0 0
\(47\) 582.794 1.80871 0.904354 0.426784i \(-0.140354\pi\)
0.904354 + 0.426784i \(0.140354\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −202.971 −0.557286
\(52\) 0 0
\(53\) 623.019 1.61468 0.807342 0.590083i \(-0.200905\pi\)
0.807342 + 0.590083i \(0.200905\pi\)
\(54\) 0 0
\(55\) −242.843 −0.595362
\(56\) 0 0
\(57\) 279.765 0.650100
\(58\) 0 0
\(59\) 524.999 1.15846 0.579229 0.815165i \(-0.303354\pi\)
0.579229 + 0.815165i \(0.303354\pi\)
\(60\) 0 0
\(61\) −352.794 −0.740502 −0.370251 0.928932i \(-0.620728\pi\)
−0.370251 + 0.928932i \(0.620728\pi\)
\(62\) 0 0
\(63\) 63.0000 0.125988
\(64\) 0 0
\(65\) −218.284 −0.416536
\(66\) 0 0
\(67\) 736.520 1.34299 0.671494 0.741010i \(-0.265653\pi\)
0.671494 + 0.741010i \(0.265653\pi\)
\(68\) 0 0
\(69\) 312.500 0.545225
\(70\) 0 0
\(71\) 492.264 0.822831 0.411415 0.911448i \(-0.365035\pi\)
0.411415 + 0.911448i \(0.365035\pi\)
\(72\) 0 0
\(73\) 1164.75 1.86745 0.933727 0.357987i \(-0.116537\pi\)
0.933727 + 0.357987i \(0.116537\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) −339.980 −0.503173
\(78\) 0 0
\(79\) 872.195 1.24215 0.621074 0.783752i \(-0.286697\pi\)
0.621074 + 0.783752i \(0.286697\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 529.588 0.700359 0.350180 0.936683i \(-0.386121\pi\)
0.350180 + 0.936683i \(0.386121\pi\)
\(84\) 0 0
\(85\) −338.284 −0.431672
\(86\) 0 0
\(87\) −176.205 −0.217140
\(88\) 0 0
\(89\) −385.216 −0.458796 −0.229398 0.973333i \(-0.573676\pi\)
−0.229398 + 0.973333i \(0.573676\pi\)
\(90\) 0 0
\(91\) −305.598 −0.352037
\(92\) 0 0
\(93\) 27.2649 0.0304005
\(94\) 0 0
\(95\) 466.274 0.503565
\(96\) 0 0
\(97\) −463.892 −0.485579 −0.242789 0.970079i \(-0.578062\pi\)
−0.242789 + 0.970079i \(0.578062\pi\)
\(98\) 0 0
\(99\) −437.117 −0.443757
\(100\) 0 0
\(101\) −432.725 −0.426314 −0.213157 0.977018i \(-0.568375\pi\)
−0.213157 + 0.977018i \(0.568375\pi\)
\(102\) 0 0
\(103\) −512.626 −0.490393 −0.245197 0.969473i \(-0.578853\pi\)
−0.245197 + 0.969473i \(0.578853\pi\)
\(104\) 0 0
\(105\) 105.000 0.0975900
\(106\) 0 0
\(107\) −1963.09 −1.77363 −0.886817 0.462122i \(-0.847088\pi\)
−0.886817 + 0.462122i \(0.847088\pi\)
\(108\) 0 0
\(109\) 545.176 0.479068 0.239534 0.970888i \(-0.423005\pi\)
0.239534 + 0.970888i \(0.423005\pi\)
\(110\) 0 0
\(111\) −758.029 −0.648188
\(112\) 0 0
\(113\) −231.823 −0.192992 −0.0964961 0.995333i \(-0.530764\pi\)
−0.0964961 + 0.995333i \(0.530764\pi\)
\(114\) 0 0
\(115\) 520.833 0.422329
\(116\) 0 0
\(117\) −392.912 −0.310468
\(118\) 0 0
\(119\) −473.598 −0.364829
\(120\) 0 0
\(121\) 1027.90 0.772279
\(122\) 0 0
\(123\) 828.823 0.607581
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −2372.90 −1.65796 −0.828979 0.559280i \(-0.811078\pi\)
−0.828979 + 0.559280i \(0.811078\pi\)
\(128\) 0 0
\(129\) 277.913 0.189681
\(130\) 0 0
\(131\) −1200.04 −0.800364 −0.400182 0.916436i \(-0.631053\pi\)
−0.400182 + 0.916436i \(0.631053\pi\)
\(132\) 0 0
\(133\) 652.784 0.425591
\(134\) 0 0
\(135\) 135.000 0.0860663
\(136\) 0 0
\(137\) 2781.25 1.73444 0.867221 0.497924i \(-0.165904\pi\)
0.867221 + 0.497924i \(0.165904\pi\)
\(138\) 0 0
\(139\) 1245.60 0.760078 0.380039 0.924971i \(-0.375911\pi\)
0.380039 + 0.924971i \(0.375911\pi\)
\(140\) 0 0
\(141\) 1748.38 1.04426
\(142\) 0 0
\(143\) 2120.35 1.23995
\(144\) 0 0
\(145\) −293.675 −0.168196
\(146\) 0 0
\(147\) 147.000 0.0824786
\(148\) 0 0
\(149\) 19.4046 0.0106690 0.00533452 0.999986i \(-0.498302\pi\)
0.00533452 + 0.999986i \(0.498302\pi\)
\(150\) 0 0
\(151\) 2349.80 1.26638 0.633192 0.773995i \(-0.281744\pi\)
0.633192 + 0.773995i \(0.281744\pi\)
\(152\) 0 0
\(153\) −608.912 −0.321749
\(154\) 0 0
\(155\) 45.4416 0.0235481
\(156\) 0 0
\(157\) −3898.46 −1.98172 −0.990862 0.134880i \(-0.956935\pi\)
−0.990862 + 0.134880i \(0.956935\pi\)
\(158\) 0 0
\(159\) 1869.06 0.932239
\(160\) 0 0
\(161\) 729.166 0.356934
\(162\) 0 0
\(163\) −1527.54 −0.734024 −0.367012 0.930216i \(-0.619619\pi\)
−0.367012 + 0.930216i \(0.619619\pi\)
\(164\) 0 0
\(165\) −728.528 −0.343732
\(166\) 0 0
\(167\) 998.518 0.462681 0.231340 0.972873i \(-0.425689\pi\)
0.231340 + 0.972873i \(0.425689\pi\)
\(168\) 0 0
\(169\) −291.079 −0.132489
\(170\) 0 0
\(171\) 839.294 0.375336
\(172\) 0 0
\(173\) 685.253 0.301149 0.150575 0.988599i \(-0.451888\pi\)
0.150575 + 0.988599i \(0.451888\pi\)
\(174\) 0 0
\(175\) 175.000 0.0755929
\(176\) 0 0
\(177\) 1575.00 0.668836
\(178\) 0 0
\(179\) −1025.58 −0.428245 −0.214122 0.976807i \(-0.568689\pi\)
−0.214122 + 0.976807i \(0.568689\pi\)
\(180\) 0 0
\(181\) 2899.40 1.19067 0.595333 0.803479i \(-0.297020\pi\)
0.595333 + 0.803479i \(0.297020\pi\)
\(182\) 0 0
\(183\) −1058.38 −0.427529
\(184\) 0 0
\(185\) −1263.38 −0.502084
\(186\) 0 0
\(187\) 3285.99 1.28500
\(188\) 0 0
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) −1074.18 −0.406939 −0.203469 0.979081i \(-0.565222\pi\)
−0.203469 + 0.979081i \(0.565222\pi\)
\(192\) 0 0
\(193\) 898.999 0.335292 0.167646 0.985847i \(-0.446383\pi\)
0.167646 + 0.985847i \(0.446383\pi\)
\(194\) 0 0
\(195\) −654.853 −0.240487
\(196\) 0 0
\(197\) 3063.63 1.10799 0.553996 0.832519i \(-0.313102\pi\)
0.553996 + 0.832519i \(0.313102\pi\)
\(198\) 0 0
\(199\) 949.522 0.338240 0.169120 0.985595i \(-0.445907\pi\)
0.169120 + 0.985595i \(0.445907\pi\)
\(200\) 0 0
\(201\) 2209.56 0.775375
\(202\) 0 0
\(203\) −411.145 −0.142151
\(204\) 0 0
\(205\) 1381.37 0.470630
\(206\) 0 0
\(207\) 937.499 0.314786
\(208\) 0 0
\(209\) −4529.25 −1.49902
\(210\) 0 0
\(211\) −2306.64 −0.752587 −0.376294 0.926500i \(-0.622802\pi\)
−0.376294 + 0.926500i \(0.622802\pi\)
\(212\) 0 0
\(213\) 1476.79 0.475062
\(214\) 0 0
\(215\) 463.188 0.146926
\(216\) 0 0
\(217\) 63.6182 0.0199018
\(218\) 0 0
\(219\) 3494.26 1.07817
\(220\) 0 0
\(221\) 2953.69 0.899033
\(222\) 0 0
\(223\) 3227.61 0.969222 0.484611 0.874730i \(-0.338961\pi\)
0.484611 + 0.874730i \(0.338961\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) 637.820 0.186492 0.0932458 0.995643i \(-0.470276\pi\)
0.0932458 + 0.995643i \(0.470276\pi\)
\(228\) 0 0
\(229\) 544.774 0.157204 0.0786019 0.996906i \(-0.474954\pi\)
0.0786019 + 0.996906i \(0.474954\pi\)
\(230\) 0 0
\(231\) −1019.94 −0.290507
\(232\) 0 0
\(233\) −5748.54 −1.61631 −0.808154 0.588972i \(-0.799533\pi\)
−0.808154 + 0.588972i \(0.799533\pi\)
\(234\) 0 0
\(235\) 2913.97 0.808878
\(236\) 0 0
\(237\) 2616.59 0.717154
\(238\) 0 0
\(239\) 2678.10 0.724820 0.362410 0.932019i \(-0.381954\pi\)
0.362410 + 0.932019i \(0.381954\pi\)
\(240\) 0 0
\(241\) −2202.16 −0.588604 −0.294302 0.955713i \(-0.595087\pi\)
−0.294302 + 0.955713i \(0.595087\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 245.000 0.0638877
\(246\) 0 0
\(247\) −4071.21 −1.04877
\(248\) 0 0
\(249\) 1588.76 0.404353
\(250\) 0 0
\(251\) 5716.90 1.43764 0.718820 0.695196i \(-0.244683\pi\)
0.718820 + 0.695196i \(0.244683\pi\)
\(252\) 0 0
\(253\) −5059.22 −1.25719
\(254\) 0 0
\(255\) −1014.85 −0.249226
\(256\) 0 0
\(257\) 4724.29 1.14666 0.573332 0.819323i \(-0.305650\pi\)
0.573332 + 0.819323i \(0.305650\pi\)
\(258\) 0 0
\(259\) −1768.73 −0.424339
\(260\) 0 0
\(261\) −528.616 −0.125366
\(262\) 0 0
\(263\) −5975.36 −1.40097 −0.700487 0.713665i \(-0.747034\pi\)
−0.700487 + 0.713665i \(0.747034\pi\)
\(264\) 0 0
\(265\) 3115.10 0.722109
\(266\) 0 0
\(267\) −1155.65 −0.264886
\(268\) 0 0
\(269\) 4486.11 1.01681 0.508407 0.861117i \(-0.330234\pi\)
0.508407 + 0.861117i \(0.330234\pi\)
\(270\) 0 0
\(271\) −3827.68 −0.857989 −0.428994 0.903307i \(-0.641132\pi\)
−0.428994 + 0.903307i \(0.641132\pi\)
\(272\) 0 0
\(273\) −916.794 −0.203249
\(274\) 0 0
\(275\) −1214.21 −0.266254
\(276\) 0 0
\(277\) −3420.54 −0.741950 −0.370975 0.928643i \(-0.620976\pi\)
−0.370975 + 0.928643i \(0.620976\pi\)
\(278\) 0 0
\(279\) 81.7948 0.0175517
\(280\) 0 0
\(281\) 5235.92 1.11156 0.555781 0.831329i \(-0.312419\pi\)
0.555781 + 0.831329i \(0.312419\pi\)
\(282\) 0 0
\(283\) 6985.88 1.46738 0.733688 0.679486i \(-0.237797\pi\)
0.733688 + 0.679486i \(0.237797\pi\)
\(284\) 0 0
\(285\) 1398.82 0.290734
\(286\) 0 0
\(287\) 1933.92 0.397755
\(288\) 0 0
\(289\) −335.550 −0.0682984
\(290\) 0 0
\(291\) −1391.68 −0.280349
\(292\) 0 0
\(293\) 7399.70 1.47541 0.737705 0.675123i \(-0.235910\pi\)
0.737705 + 0.675123i \(0.235910\pi\)
\(294\) 0 0
\(295\) 2625.00 0.518078
\(296\) 0 0
\(297\) −1311.35 −0.256203
\(298\) 0 0
\(299\) −4547.58 −0.879577
\(300\) 0 0
\(301\) 648.463 0.124175
\(302\) 0 0
\(303\) −1298.17 −0.246133
\(304\) 0 0
\(305\) −1763.97 −0.331163
\(306\) 0 0
\(307\) −2668.64 −0.496116 −0.248058 0.968745i \(-0.579792\pi\)
−0.248058 + 0.968745i \(0.579792\pi\)
\(308\) 0 0
\(309\) −1537.88 −0.283129
\(310\) 0 0
\(311\) −6189.25 −1.12849 −0.564244 0.825608i \(-0.690832\pi\)
−0.564244 + 0.825608i \(0.690832\pi\)
\(312\) 0 0
\(313\) −2921.59 −0.527598 −0.263799 0.964578i \(-0.584976\pi\)
−0.263799 + 0.964578i \(0.584976\pi\)
\(314\) 0 0
\(315\) 315.000 0.0563436
\(316\) 0 0
\(317\) 9825.56 1.74088 0.870439 0.492276i \(-0.163835\pi\)
0.870439 + 0.492276i \(0.163835\pi\)
\(318\) 0 0
\(319\) 2852.68 0.500687
\(320\) 0 0
\(321\) −5889.26 −1.02401
\(322\) 0 0
\(323\) −6309.33 −1.08687
\(324\) 0 0
\(325\) −1091.42 −0.186281
\(326\) 0 0
\(327\) 1635.53 0.276590
\(328\) 0 0
\(329\) 4079.56 0.683627
\(330\) 0 0
\(331\) 9258.17 1.53739 0.768693 0.639618i \(-0.220907\pi\)
0.768693 + 0.639618i \(0.220907\pi\)
\(332\) 0 0
\(333\) −2274.09 −0.374232
\(334\) 0 0
\(335\) 3682.60 0.600603
\(336\) 0 0
\(337\) −3693.98 −0.597103 −0.298552 0.954394i \(-0.596503\pi\)
−0.298552 + 0.954394i \(0.596503\pi\)
\(338\) 0 0
\(339\) −695.470 −0.111424
\(340\) 0 0
\(341\) −441.406 −0.0700982
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) 1562.50 0.243832
\(346\) 0 0
\(347\) −3832.83 −0.592960 −0.296480 0.955039i \(-0.595813\pi\)
−0.296480 + 0.955039i \(0.595813\pi\)
\(348\) 0 0
\(349\) 8325.22 1.27690 0.638451 0.769662i \(-0.279575\pi\)
0.638451 + 0.769662i \(0.279575\pi\)
\(350\) 0 0
\(351\) −1178.74 −0.179248
\(352\) 0 0
\(353\) −8991.52 −1.35572 −0.677862 0.735189i \(-0.737093\pi\)
−0.677862 + 0.735189i \(0.737093\pi\)
\(354\) 0 0
\(355\) 2461.32 0.367981
\(356\) 0 0
\(357\) −1420.79 −0.210634
\(358\) 0 0
\(359\) 12893.8 1.89557 0.947783 0.318917i \(-0.103319\pi\)
0.947783 + 0.318917i \(0.103319\pi\)
\(360\) 0 0
\(361\) 1837.46 0.267891
\(362\) 0 0
\(363\) 3083.71 0.445875
\(364\) 0 0
\(365\) 5823.77 0.835150
\(366\) 0 0
\(367\) 7480.17 1.06393 0.531964 0.846767i \(-0.321454\pi\)
0.531964 + 0.846767i \(0.321454\pi\)
\(368\) 0 0
\(369\) 2486.47 0.350787
\(370\) 0 0
\(371\) 4361.14 0.610293
\(372\) 0 0
\(373\) −3523.32 −0.489090 −0.244545 0.969638i \(-0.578639\pi\)
−0.244545 + 0.969638i \(0.578639\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) 0 0
\(377\) 2564.19 0.350298
\(378\) 0 0
\(379\) −13515.4 −1.83177 −0.915886 0.401438i \(-0.868510\pi\)
−0.915886 + 0.401438i \(0.868510\pi\)
\(380\) 0 0
\(381\) −7118.69 −0.957222
\(382\) 0 0
\(383\) 657.182 0.0876774 0.0438387 0.999039i \(-0.486041\pi\)
0.0438387 + 0.999039i \(0.486041\pi\)
\(384\) 0 0
\(385\) −1699.90 −0.225026
\(386\) 0 0
\(387\) 833.738 0.109512
\(388\) 0 0
\(389\) −9741.87 −1.26975 −0.634875 0.772615i \(-0.718948\pi\)
−0.634875 + 0.772615i \(0.718948\pi\)
\(390\) 0 0
\(391\) −7047.58 −0.911538
\(392\) 0 0
\(393\) −3600.11 −0.462091
\(394\) 0 0
\(395\) 4360.98 0.555505
\(396\) 0 0
\(397\) 4407.42 0.557184 0.278592 0.960410i \(-0.410132\pi\)
0.278592 + 0.960410i \(0.410132\pi\)
\(398\) 0 0
\(399\) 1958.35 0.245715
\(400\) 0 0
\(401\) −11569.5 −1.44078 −0.720391 0.693568i \(-0.756037\pi\)
−0.720391 + 0.693568i \(0.756037\pi\)
\(402\) 0 0
\(403\) −396.767 −0.0490431
\(404\) 0 0
\(405\) 405.000 0.0496904
\(406\) 0 0
\(407\) 12272.1 1.49461
\(408\) 0 0
\(409\) −3083.03 −0.372729 −0.186364 0.982481i \(-0.559670\pi\)
−0.186364 + 0.982481i \(0.559670\pi\)
\(410\) 0 0
\(411\) 8343.76 1.00138
\(412\) 0 0
\(413\) 3674.99 0.437856
\(414\) 0 0
\(415\) 2647.94 0.313210
\(416\) 0 0
\(417\) 3736.81 0.438831
\(418\) 0 0
\(419\) 5415.21 0.631385 0.315692 0.948862i \(-0.397763\pi\)
0.315692 + 0.948862i \(0.397763\pi\)
\(420\) 0 0
\(421\) 4188.34 0.484863 0.242432 0.970168i \(-0.422055\pi\)
0.242432 + 0.970168i \(0.422055\pi\)
\(422\) 0 0
\(423\) 5245.15 0.602902
\(424\) 0 0
\(425\) −1691.42 −0.193049
\(426\) 0 0
\(427\) −2469.56 −0.279884
\(428\) 0 0
\(429\) 6361.05 0.715884
\(430\) 0 0
\(431\) 9108.41 1.01795 0.508975 0.860781i \(-0.330024\pi\)
0.508975 + 0.860781i \(0.330024\pi\)
\(432\) 0 0
\(433\) −16847.0 −1.86979 −0.934893 0.354930i \(-0.884505\pi\)
−0.934893 + 0.354930i \(0.884505\pi\)
\(434\) 0 0
\(435\) −881.026 −0.0971080
\(436\) 0 0
\(437\) 9714.03 1.06335
\(438\) 0 0
\(439\) −8434.14 −0.916946 −0.458473 0.888708i \(-0.651603\pi\)
−0.458473 + 0.888708i \(0.651603\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) 4298.49 0.461010 0.230505 0.973071i \(-0.425962\pi\)
0.230505 + 0.973071i \(0.425962\pi\)
\(444\) 0 0
\(445\) −1926.08 −0.205180
\(446\) 0 0
\(447\) 58.2139 0.00615978
\(448\) 0 0
\(449\) 10545.0 1.10835 0.554173 0.832402i \(-0.313035\pi\)
0.554173 + 0.832402i \(0.313035\pi\)
\(450\) 0 0
\(451\) −13418.2 −1.40098
\(452\) 0 0
\(453\) 7049.40 0.731147
\(454\) 0 0
\(455\) −1527.99 −0.157436
\(456\) 0 0
\(457\) 11952.4 1.22344 0.611719 0.791075i \(-0.290478\pi\)
0.611719 + 0.791075i \(0.290478\pi\)
\(458\) 0 0
\(459\) −1826.74 −0.185762
\(460\) 0 0
\(461\) −17200.9 −1.73780 −0.868900 0.494988i \(-0.835173\pi\)
−0.868900 + 0.494988i \(0.835173\pi\)
\(462\) 0 0
\(463\) 10368.7 1.04076 0.520381 0.853934i \(-0.325790\pi\)
0.520381 + 0.853934i \(0.325790\pi\)
\(464\) 0 0
\(465\) 136.325 0.0135955
\(466\) 0 0
\(467\) 16879.5 1.67257 0.836284 0.548296i \(-0.184723\pi\)
0.836284 + 0.548296i \(0.184723\pi\)
\(468\) 0 0
\(469\) 5155.64 0.507602
\(470\) 0 0
\(471\) −11695.4 −1.14415
\(472\) 0 0
\(473\) −4499.27 −0.437371
\(474\) 0 0
\(475\) 2331.37 0.225201
\(476\) 0 0
\(477\) 5607.17 0.538228
\(478\) 0 0
\(479\) −7329.12 −0.699115 −0.349558 0.936915i \(-0.613668\pi\)
−0.349558 + 0.936915i \(0.613668\pi\)
\(480\) 0 0
\(481\) 11031.0 1.04568
\(482\) 0 0
\(483\) 2187.50 0.206076
\(484\) 0 0
\(485\) −2319.46 −0.217157
\(486\) 0 0
\(487\) 17209.5 1.60131 0.800655 0.599125i \(-0.204485\pi\)
0.800655 + 0.599125i \(0.204485\pi\)
\(488\) 0 0
\(489\) −4582.61 −0.423789
\(490\) 0 0
\(491\) −11392.5 −1.04712 −0.523560 0.851989i \(-0.675396\pi\)
−0.523560 + 0.851989i \(0.675396\pi\)
\(492\) 0 0
\(493\) 3973.83 0.363027
\(494\) 0 0
\(495\) −2185.58 −0.198454
\(496\) 0 0
\(497\) 3445.85 0.311001
\(498\) 0 0
\(499\) −19079.4 −1.71164 −0.855822 0.517271i \(-0.826948\pi\)
−0.855822 + 0.517271i \(0.826948\pi\)
\(500\) 0 0
\(501\) 2995.55 0.267129
\(502\) 0 0
\(503\) 13499.4 1.19663 0.598317 0.801259i \(-0.295836\pi\)
0.598317 + 0.801259i \(0.295836\pi\)
\(504\) 0 0
\(505\) −2163.62 −0.190654
\(506\) 0 0
\(507\) −873.237 −0.0764928
\(508\) 0 0
\(509\) −4328.85 −0.376960 −0.188480 0.982077i \(-0.560356\pi\)
−0.188480 + 0.982077i \(0.560356\pi\)
\(510\) 0 0
\(511\) 8153.27 0.705831
\(512\) 0 0
\(513\) 2517.88 0.216700
\(514\) 0 0
\(515\) −2563.13 −0.219311
\(516\) 0 0
\(517\) −28305.5 −2.40788
\(518\) 0 0
\(519\) 2055.76 0.173869
\(520\) 0 0
\(521\) 19395.7 1.63098 0.815490 0.578771i \(-0.196468\pi\)
0.815490 + 0.578771i \(0.196468\pi\)
\(522\) 0 0
\(523\) −20413.8 −1.70675 −0.853377 0.521294i \(-0.825450\pi\)
−0.853377 + 0.521294i \(0.825450\pi\)
\(524\) 0 0
\(525\) 525.000 0.0436436
\(526\) 0 0
\(527\) −614.887 −0.0508252
\(528\) 0 0
\(529\) −1316.34 −0.108189
\(530\) 0 0
\(531\) 4724.99 0.386153
\(532\) 0 0
\(533\) −12061.3 −0.980171
\(534\) 0 0
\(535\) −9815.43 −0.793193
\(536\) 0 0
\(537\) −3076.75 −0.247247
\(538\) 0 0
\(539\) −2379.86 −0.190181
\(540\) 0 0
\(541\) −4919.18 −0.390928 −0.195464 0.980711i \(-0.562621\pi\)
−0.195464 + 0.980711i \(0.562621\pi\)
\(542\) 0 0
\(543\) 8698.19 0.687431
\(544\) 0 0
\(545\) 2725.88 0.214246
\(546\) 0 0
\(547\) −15334.2 −1.19862 −0.599308 0.800518i \(-0.704558\pi\)
−0.599308 + 0.800518i \(0.704558\pi\)
\(548\) 0 0
\(549\) −3175.15 −0.246834
\(550\) 0 0
\(551\) −5477.33 −0.423488
\(552\) 0 0
\(553\) 6105.37 0.469487
\(554\) 0 0
\(555\) −3790.14 −0.289879
\(556\) 0 0
\(557\) −8613.78 −0.655256 −0.327628 0.944807i \(-0.606249\pi\)
−0.327628 + 0.944807i \(0.606249\pi\)
\(558\) 0 0
\(559\) −4044.26 −0.306000
\(560\) 0 0
\(561\) 9857.98 0.741897
\(562\) 0 0
\(563\) 2320.81 0.173731 0.0868654 0.996220i \(-0.472315\pi\)
0.0868654 + 0.996220i \(0.472315\pi\)
\(564\) 0 0
\(565\) −1159.12 −0.0863087
\(566\) 0 0
\(567\) 567.000 0.0419961
\(568\) 0 0
\(569\) −1736.04 −0.127906 −0.0639529 0.997953i \(-0.520371\pi\)
−0.0639529 + 0.997953i \(0.520371\pi\)
\(570\) 0 0
\(571\) −23897.8 −1.75148 −0.875738 0.482786i \(-0.839625\pi\)
−0.875738 + 0.482786i \(0.839625\pi\)
\(572\) 0 0
\(573\) −3222.55 −0.234946
\(574\) 0 0
\(575\) 2604.16 0.188871
\(576\) 0 0
\(577\) 8029.26 0.579311 0.289655 0.957131i \(-0.406459\pi\)
0.289655 + 0.957131i \(0.406459\pi\)
\(578\) 0 0
\(579\) 2697.00 0.193581
\(580\) 0 0
\(581\) 3707.12 0.264711
\(582\) 0 0
\(583\) −30259.1 −2.14958
\(584\) 0 0
\(585\) −1964.56 −0.138845
\(586\) 0 0
\(587\) 8015.14 0.563578 0.281789 0.959476i \(-0.409072\pi\)
0.281789 + 0.959476i \(0.409072\pi\)
\(588\) 0 0
\(589\) 847.529 0.0592900
\(590\) 0 0
\(591\) 9190.88 0.639700
\(592\) 0 0
\(593\) 12820.3 0.887801 0.443901 0.896076i \(-0.353594\pi\)
0.443901 + 0.896076i \(0.353594\pi\)
\(594\) 0 0
\(595\) −2367.99 −0.163157
\(596\) 0 0
\(597\) 2848.57 0.195283
\(598\) 0 0
\(599\) 15330.5 1.04572 0.522860 0.852418i \(-0.324865\pi\)
0.522860 + 0.852418i \(0.324865\pi\)
\(600\) 0 0
\(601\) 107.658 0.00730694 0.00365347 0.999993i \(-0.498837\pi\)
0.00365347 + 0.999993i \(0.498837\pi\)
\(602\) 0 0
\(603\) 6628.68 0.447663
\(604\) 0 0
\(605\) 5139.52 0.345374
\(606\) 0 0
\(607\) 8213.06 0.549189 0.274595 0.961560i \(-0.411456\pi\)
0.274595 + 0.961560i \(0.411456\pi\)
\(608\) 0 0
\(609\) −1233.44 −0.0820712
\(610\) 0 0
\(611\) −25443.0 −1.68463
\(612\) 0 0
\(613\) 1242.60 0.0818732 0.0409366 0.999162i \(-0.486966\pi\)
0.0409366 + 0.999162i \(0.486966\pi\)
\(614\) 0 0
\(615\) 4144.11 0.271718
\(616\) 0 0
\(617\) −13170.6 −0.859367 −0.429683 0.902980i \(-0.641375\pi\)
−0.429683 + 0.902980i \(0.641375\pi\)
\(618\) 0 0
\(619\) −19774.1 −1.28399 −0.641993 0.766711i \(-0.721892\pi\)
−0.641993 + 0.766711i \(0.721892\pi\)
\(620\) 0 0
\(621\) 2812.50 0.181742
\(622\) 0 0
\(623\) −2696.51 −0.173409
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −13587.8 −0.865459
\(628\) 0 0
\(629\) 17095.3 1.08368
\(630\) 0 0
\(631\) 14308.8 0.902735 0.451367 0.892338i \(-0.350936\pi\)
0.451367 + 0.892338i \(0.350936\pi\)
\(632\) 0 0
\(633\) −6919.93 −0.434507
\(634\) 0 0
\(635\) −11864.5 −0.741461
\(636\) 0 0
\(637\) −2139.19 −0.133058
\(638\) 0 0
\(639\) 4430.38 0.274277
\(640\) 0 0
\(641\) 11537.5 0.710925 0.355463 0.934691i \(-0.384323\pi\)
0.355463 + 0.934691i \(0.384323\pi\)
\(642\) 0 0
\(643\) −19603.0 −1.20228 −0.601139 0.799144i \(-0.705286\pi\)
−0.601139 + 0.799144i \(0.705286\pi\)
\(644\) 0 0
\(645\) 1389.56 0.0848279
\(646\) 0 0
\(647\) −21650.0 −1.31553 −0.657765 0.753223i \(-0.728498\pi\)
−0.657765 + 0.753223i \(0.728498\pi\)
\(648\) 0 0
\(649\) −25498.4 −1.54222
\(650\) 0 0
\(651\) 190.855 0.0114903
\(652\) 0 0
\(653\) −2927.33 −0.175429 −0.0877145 0.996146i \(-0.527956\pi\)
−0.0877145 + 0.996146i \(0.527956\pi\)
\(654\) 0 0
\(655\) −6000.18 −0.357934
\(656\) 0 0
\(657\) 10482.8 0.622484
\(658\) 0 0
\(659\) 4778.76 0.282480 0.141240 0.989975i \(-0.454891\pi\)
0.141240 + 0.989975i \(0.454891\pi\)
\(660\) 0 0
\(661\) −31510.3 −1.85417 −0.927086 0.374849i \(-0.877695\pi\)
−0.927086 + 0.374849i \(0.877695\pi\)
\(662\) 0 0
\(663\) 8861.06 0.519057
\(664\) 0 0
\(665\) 3263.92 0.190330
\(666\) 0 0
\(667\) −6118.23 −0.355170
\(668\) 0 0
\(669\) 9682.82 0.559581
\(670\) 0 0
\(671\) 17134.7 0.985808
\(672\) 0 0
\(673\) −8992.15 −0.515040 −0.257520 0.966273i \(-0.582905\pi\)
−0.257520 + 0.966273i \(0.582905\pi\)
\(674\) 0 0
\(675\) 675.000 0.0384900
\(676\) 0 0
\(677\) 19340.8 1.09797 0.548985 0.835832i \(-0.315014\pi\)
0.548985 + 0.835832i \(0.315014\pi\)
\(678\) 0 0
\(679\) −3247.25 −0.183531
\(680\) 0 0
\(681\) 1913.46 0.107671
\(682\) 0 0
\(683\) 4255.14 0.238387 0.119194 0.992871i \(-0.461969\pi\)
0.119194 + 0.992871i \(0.461969\pi\)
\(684\) 0 0
\(685\) 13906.3 0.775666
\(686\) 0 0
\(687\) 1634.32 0.0907616
\(688\) 0 0
\(689\) −27199.1 −1.50392
\(690\) 0 0
\(691\) 17505.5 0.963733 0.481867 0.876245i \(-0.339959\pi\)
0.481867 + 0.876245i \(0.339959\pi\)
\(692\) 0 0
\(693\) −3059.82 −0.167724
\(694\) 0 0
\(695\) 6228.02 0.339917
\(696\) 0 0
\(697\) −18691.8 −1.01579
\(698\) 0 0
\(699\) −17245.6 −0.933175
\(700\) 0 0
\(701\) −3240.77 −0.174611 −0.0873054 0.996182i \(-0.527826\pi\)
−0.0873054 + 0.996182i \(0.527826\pi\)
\(702\) 0 0
\(703\) −23563.3 −1.26416
\(704\) 0 0
\(705\) 8741.91 0.467006
\(706\) 0 0
\(707\) −3029.07 −0.161132
\(708\) 0 0
\(709\) 19949.3 1.05672 0.528358 0.849022i \(-0.322808\pi\)
0.528358 + 0.849022i \(0.322808\pi\)
\(710\) 0 0
\(711\) 7849.76 0.414049
\(712\) 0 0
\(713\) 946.698 0.0497253
\(714\) 0 0
\(715\) 10601.7 0.554522
\(716\) 0 0
\(717\) 8034.30 0.418475
\(718\) 0 0
\(719\) 11259.4 0.584011 0.292006 0.956417i \(-0.405677\pi\)
0.292006 + 0.956417i \(0.405677\pi\)
\(720\) 0 0
\(721\) −3588.38 −0.185351
\(722\) 0 0
\(723\) −6606.47 −0.339830
\(724\) 0 0
\(725\) −1468.38 −0.0752195
\(726\) 0 0
\(727\) 12228.5 0.623840 0.311920 0.950108i \(-0.399028\pi\)
0.311920 + 0.950108i \(0.399028\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −6267.56 −0.317119
\(732\) 0 0
\(733\) −26635.1 −1.34214 −0.671072 0.741392i \(-0.734166\pi\)
−0.671072 + 0.741392i \(0.734166\pi\)
\(734\) 0 0
\(735\) 735.000 0.0368856
\(736\) 0 0
\(737\) −35771.7 −1.78788
\(738\) 0 0
\(739\) 6074.00 0.302349 0.151174 0.988507i \(-0.451694\pi\)
0.151174 + 0.988507i \(0.451694\pi\)
\(740\) 0 0
\(741\) −12213.6 −0.605505
\(742\) 0 0
\(743\) 4016.87 0.198337 0.0991686 0.995071i \(-0.468382\pi\)
0.0991686 + 0.995071i \(0.468382\pi\)
\(744\) 0 0
\(745\) 97.0231 0.00477134
\(746\) 0 0
\(747\) 4766.29 0.233453
\(748\) 0 0
\(749\) −13741.6 −0.670370
\(750\) 0 0
\(751\) 23913.2 1.16192 0.580962 0.813931i \(-0.302676\pi\)
0.580962 + 0.813931i \(0.302676\pi\)
\(752\) 0 0
\(753\) 17150.7 0.830022
\(754\) 0 0
\(755\) 11749.0 0.566344
\(756\) 0 0
\(757\) 31044.9 1.49055 0.745275 0.666758i \(-0.232318\pi\)
0.745275 + 0.666758i \(0.232318\pi\)
\(758\) 0 0
\(759\) −15177.6 −0.725842
\(760\) 0 0
\(761\) 14011.8 0.667446 0.333723 0.942671i \(-0.391695\pi\)
0.333723 + 0.942671i \(0.391695\pi\)
\(762\) 0 0
\(763\) 3816.23 0.181071
\(764\) 0 0
\(765\) −3044.56 −0.143891
\(766\) 0 0
\(767\) −22919.8 −1.07899
\(768\) 0 0
\(769\) 3342.49 0.156740 0.0783701 0.996924i \(-0.475028\pi\)
0.0783701 + 0.996924i \(0.475028\pi\)
\(770\) 0 0
\(771\) 14172.9 0.662027
\(772\) 0 0
\(773\) −21074.6 −0.980594 −0.490297 0.871555i \(-0.663112\pi\)
−0.490297 + 0.871555i \(0.663112\pi\)
\(774\) 0 0
\(775\) 227.208 0.0105310
\(776\) 0 0
\(777\) −5306.20 −0.244992
\(778\) 0 0
\(779\) 25763.9 1.18496
\(780\) 0 0
\(781\) −23908.5 −1.09541
\(782\) 0 0
\(783\) −1585.85 −0.0723800
\(784\) 0 0
\(785\) −19492.3 −0.886254
\(786\) 0 0
\(787\) −21394.8 −0.969048 −0.484524 0.874778i \(-0.661007\pi\)
−0.484524 + 0.874778i \(0.661007\pi\)
\(788\) 0 0
\(789\) −17926.1 −0.808853
\(790\) 0 0
\(791\) −1622.76 −0.0729442
\(792\) 0 0
\(793\) 15401.9 0.689706
\(794\) 0 0
\(795\) 9345.29 0.416910
\(796\) 0 0
\(797\) 20645.0 0.917547 0.458773 0.888553i \(-0.348289\pi\)
0.458773 + 0.888553i \(0.348289\pi\)
\(798\) 0 0
\(799\) −39430.0 −1.74585
\(800\) 0 0
\(801\) −3466.95 −0.152932
\(802\) 0 0
\(803\) −56570.4 −2.48608
\(804\) 0 0
\(805\) 3645.83 0.159626
\(806\) 0 0
\(807\) 13458.3 0.587058
\(808\) 0 0
\(809\) −15939.0 −0.692688 −0.346344 0.938108i \(-0.612577\pi\)
−0.346344 + 0.938108i \(0.612577\pi\)
\(810\) 0 0
\(811\) −22829.2 −0.988460 −0.494230 0.869331i \(-0.664550\pi\)
−0.494230 + 0.869331i \(0.664550\pi\)
\(812\) 0 0
\(813\) −11483.0 −0.495360
\(814\) 0 0
\(815\) −7637.69 −0.328266
\(816\) 0 0
\(817\) 8638.90 0.369935
\(818\) 0 0
\(819\) −2750.38 −0.117346
\(820\) 0 0
\(821\) −5700.22 −0.242313 −0.121157 0.992633i \(-0.538660\pi\)
−0.121157 + 0.992633i \(0.538660\pi\)
\(822\) 0 0
\(823\) 32438.0 1.37390 0.686948 0.726707i \(-0.258950\pi\)
0.686948 + 0.726707i \(0.258950\pi\)
\(824\) 0 0
\(825\) −3642.64 −0.153722
\(826\) 0 0
\(827\) 12762.6 0.536638 0.268319 0.963330i \(-0.413532\pi\)
0.268319 + 0.963330i \(0.413532\pi\)
\(828\) 0 0
\(829\) −30766.7 −1.28899 −0.644494 0.764609i \(-0.722932\pi\)
−0.644494 + 0.764609i \(0.722932\pi\)
\(830\) 0 0
\(831\) −10261.6 −0.428365
\(832\) 0 0
\(833\) −3315.19 −0.137892
\(834\) 0 0
\(835\) 4992.59 0.206917
\(836\) 0 0
\(837\) 245.384 0.0101335
\(838\) 0 0
\(839\) 9779.71 0.402423 0.201212 0.979548i \(-0.435512\pi\)
0.201212 + 0.979548i \(0.435512\pi\)
\(840\) 0 0
\(841\) −20939.2 −0.858551
\(842\) 0 0
\(843\) 15707.8 0.641760
\(844\) 0 0
\(845\) −1455.40 −0.0592510
\(846\) 0 0
\(847\) 7195.32 0.291894
\(848\) 0 0
\(849\) 20957.6 0.847190
\(850\) 0 0
\(851\) −26320.4 −1.06023
\(852\) 0 0
\(853\) 24201.5 0.971445 0.485723 0.874113i \(-0.338556\pi\)
0.485723 + 0.874113i \(0.338556\pi\)
\(854\) 0 0
\(855\) 4196.47 0.167855
\(856\) 0 0
\(857\) 21036.7 0.838507 0.419254 0.907869i \(-0.362292\pi\)
0.419254 + 0.907869i \(0.362292\pi\)
\(858\) 0 0
\(859\) 6179.19 0.245438 0.122719 0.992441i \(-0.460839\pi\)
0.122719 + 0.992441i \(0.460839\pi\)
\(860\) 0 0
\(861\) 5801.76 0.229644
\(862\) 0 0
\(863\) −50256.2 −1.98232 −0.991160 0.132671i \(-0.957644\pi\)
−0.991160 + 0.132671i \(0.957644\pi\)
\(864\) 0 0
\(865\) 3426.27 0.134678
\(866\) 0 0
\(867\) −1006.65 −0.0394321
\(868\) 0 0
\(869\) −42361.2 −1.65363
\(870\) 0 0
\(871\) −32154.1 −1.25086
\(872\) 0 0
\(873\) −4175.03 −0.161860
\(874\) 0 0
\(875\) 875.000 0.0338062
\(876\) 0 0
\(877\) −9175.95 −0.353306 −0.176653 0.984273i \(-0.556527\pi\)
−0.176653 + 0.984273i \(0.556527\pi\)
\(878\) 0 0
\(879\) 22199.1 0.851828
\(880\) 0 0
\(881\) −26172.7 −1.00089 −0.500444 0.865769i \(-0.666830\pi\)
−0.500444 + 0.865769i \(0.666830\pi\)
\(882\) 0 0
\(883\) −18615.5 −0.709471 −0.354736 0.934967i \(-0.615429\pi\)
−0.354736 + 0.934967i \(0.615429\pi\)
\(884\) 0 0
\(885\) 7874.99 0.299113
\(886\) 0 0
\(887\) 12837.8 0.485964 0.242982 0.970031i \(-0.421874\pi\)
0.242982 + 0.970031i \(0.421874\pi\)
\(888\) 0 0
\(889\) −16610.3 −0.626649
\(890\) 0 0
\(891\) −3934.05 −0.147919
\(892\) 0 0
\(893\) 54348.4 2.03662
\(894\) 0 0
\(895\) −5127.92 −0.191517
\(896\) 0 0
\(897\) −13642.7 −0.507824
\(898\) 0 0
\(899\) −533.803 −0.0198035
\(900\) 0 0
\(901\) −42151.5 −1.55857
\(902\) 0 0
\(903\) 1945.39 0.0716926
\(904\) 0 0
\(905\) 14497.0 0.532482
\(906\) 0 0
\(907\) 26766.1 0.979885 0.489942 0.871755i \(-0.337018\pi\)
0.489942 + 0.871755i \(0.337018\pi\)
\(908\) 0 0
\(909\) −3894.52 −0.142105
\(910\) 0 0
\(911\) −5022.67 −0.182666 −0.0913328 0.995820i \(-0.529113\pi\)
−0.0913328 + 0.995820i \(0.529113\pi\)
\(912\) 0 0
\(913\) −25721.3 −0.932367
\(914\) 0 0
\(915\) −5291.91 −0.191197
\(916\) 0 0
\(917\) −8400.26 −0.302509
\(918\) 0 0
\(919\) −9541.79 −0.342497 −0.171248 0.985228i \(-0.554780\pi\)
−0.171248 + 0.985228i \(0.554780\pi\)
\(920\) 0 0
\(921\) −8005.93 −0.286432
\(922\) 0 0
\(923\) −21490.7 −0.766387
\(924\) 0 0
\(925\) −6316.90 −0.224539
\(926\) 0 0
\(927\) −4613.63 −0.163464
\(928\) 0 0
\(929\) −25479.6 −0.899846 −0.449923 0.893067i \(-0.648549\pi\)
−0.449923 + 0.893067i \(0.648549\pi\)
\(930\) 0 0
\(931\) 4569.49 0.160858
\(932\) 0 0
\(933\) −18567.7 −0.651533
\(934\) 0 0
\(935\) 16430.0 0.574671
\(936\) 0 0
\(937\) −33608.3 −1.17176 −0.585878 0.810399i \(-0.699250\pi\)
−0.585878 + 0.810399i \(0.699250\pi\)
\(938\) 0 0
\(939\) −8764.77 −0.304609
\(940\) 0 0
\(941\) −19173.6 −0.664232 −0.332116 0.943239i \(-0.607762\pi\)
−0.332116 + 0.943239i \(0.607762\pi\)
\(942\) 0 0
\(943\) 28778.5 0.993804
\(944\) 0 0
\(945\) 945.000 0.0325300
\(946\) 0 0
\(947\) 979.315 0.0336045 0.0168023 0.999859i \(-0.494651\pi\)
0.0168023 + 0.999859i \(0.494651\pi\)
\(948\) 0 0
\(949\) −50849.5 −1.73935
\(950\) 0 0
\(951\) 29476.7 1.00510
\(952\) 0 0
\(953\) 3048.61 0.103624 0.0518122 0.998657i \(-0.483500\pi\)
0.0518122 + 0.998657i \(0.483500\pi\)
\(954\) 0 0
\(955\) −5370.92 −0.181989
\(956\) 0 0
\(957\) 8558.03 0.289072
\(958\) 0 0
\(959\) 19468.8 0.655557
\(960\) 0 0
\(961\) −29708.4 −0.997227
\(962\) 0 0
\(963\) −17667.8 −0.591211
\(964\) 0 0
\(965\) 4495.00 0.149947
\(966\) 0 0
\(967\) 14467.9 0.481133 0.240567 0.970633i \(-0.422667\pi\)
0.240567 + 0.970633i \(0.422667\pi\)
\(968\) 0 0
\(969\) −18928.0 −0.627507
\(970\) 0 0
\(971\) −12952.8 −0.428090 −0.214045 0.976824i \(-0.568664\pi\)
−0.214045 + 0.976824i \(0.568664\pi\)
\(972\) 0 0
\(973\) 8719.23 0.287282
\(974\) 0 0
\(975\) −3274.26 −0.107549
\(976\) 0 0
\(977\) 47244.2 1.54706 0.773529 0.633760i \(-0.218489\pi\)
0.773529 + 0.633760i \(0.218489\pi\)
\(978\) 0 0
\(979\) 18709.4 0.610781
\(980\) 0 0
\(981\) 4906.58 0.159689
\(982\) 0 0
\(983\) 1536.84 0.0498651 0.0249326 0.999689i \(-0.492063\pi\)
0.0249326 + 0.999689i \(0.492063\pi\)
\(984\) 0 0
\(985\) 15318.1 0.495509
\(986\) 0 0
\(987\) 12238.7 0.394692
\(988\) 0 0
\(989\) 9649.73 0.310256
\(990\) 0 0
\(991\) −3785.22 −0.121334 −0.0606668 0.998158i \(-0.519323\pi\)
−0.0606668 + 0.998158i \(0.519323\pi\)
\(992\) 0 0
\(993\) 27774.5 0.887610
\(994\) 0 0
\(995\) 4747.61 0.151266
\(996\) 0 0
\(997\) −25894.9 −0.822566 −0.411283 0.911508i \(-0.634919\pi\)
−0.411283 + 0.911508i \(0.634919\pi\)
\(998\) 0 0
\(999\) −6822.26 −0.216063
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 1680.4.a.bo.1.1 2
4.3 odd 2 105.4.a.e.1.1 2
12.11 even 2 315.4.a.k.1.2 2
20.3 even 4 525.4.d.l.274.4 4
20.7 even 4 525.4.d.l.274.1 4
20.19 odd 2 525.4.a.l.1.2 2
28.27 even 2 735.4.a.o.1.1 2
60.59 even 2 1575.4.a.q.1.1 2
84.83 odd 2 2205.4.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.e.1.1 2 4.3 odd 2
315.4.a.k.1.2 2 12.11 even 2
525.4.a.l.1.2 2 20.19 odd 2
525.4.d.l.274.1 4 20.7 even 4
525.4.d.l.274.4 4 20.3 even 4
735.4.a.o.1.1 2 28.27 even 2
1575.4.a.q.1.1 2 60.59 even 2
1680.4.a.bo.1.1 2 1.1 even 1 trivial
2205.4.a.bb.1.2 2 84.83 odd 2