Properties

Label 1521.4.a.v.1.3
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 2x^{3} - 25x^{2} + 24x + 78 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.46610\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.46610 q^{2} -5.85055 q^{4} -9.85055 q^{5} +29.9396 q^{7} -20.3063 q^{8} +O(q^{10})\) \(q+1.46610 q^{2} -5.85055 q^{4} -9.85055 q^{5} +29.9396 q^{7} -20.3063 q^{8} -14.4419 q^{10} -46.9257 q^{11} +43.8945 q^{14} +17.0333 q^{16} +48.2616 q^{17} +120.300 q^{19} +57.6311 q^{20} -68.7979 q^{22} -130.697 q^{23} -27.9667 q^{25} -175.163 q^{28} +194.946 q^{29} -32.0123 q^{31} +187.423 q^{32} +70.7565 q^{34} -294.921 q^{35} -32.4250 q^{37} +176.372 q^{38} +200.028 q^{40} +241.825 q^{41} +96.4087 q^{43} +274.541 q^{44} -191.615 q^{46} -539.015 q^{47} +553.380 q^{49} -41.0021 q^{50} +152.277 q^{53} +462.244 q^{55} -607.963 q^{56} +285.810 q^{58} -327.792 q^{59} -98.4180 q^{61} -46.9332 q^{62} +138.515 q^{64} -441.151 q^{67} -282.357 q^{68} -432.385 q^{70} -345.049 q^{71} +773.839 q^{73} -47.5383 q^{74} -703.822 q^{76} -1404.94 q^{77} -150.332 q^{79} -167.787 q^{80} +354.540 q^{82} -337.966 q^{83} -475.403 q^{85} +141.345 q^{86} +952.889 q^{88} -169.913 q^{89} +764.649 q^{92} -790.250 q^{94} -1185.02 q^{95} +214.201 q^{97} +811.311 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8} - 62 q^{10} - 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} + 124 q^{19} + 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} - 144 q^{28} - 194 q^{29} + 26 q^{31} - 654 q^{32} + 1062 q^{34} - 88 q^{35} + 102 q^{37} - 332 q^{38} - 998 q^{40} + 1054 q^{41} + 450 q^{43} + 44 q^{44} - 172 q^{46} + 96 q^{47} + 1070 q^{49} - 996 q^{50} - 262 q^{53} + 204 q^{55} - 2164 q^{56} + 722 q^{58} - 308 q^{59} - 928 q^{61} - 2780 q^{62} + 1026 q^{64} - 1134 q^{67} - 1786 q^{68} - 2324 q^{70} - 1064 q^{71} + 952 q^{73} - 1158 q^{74} - 1708 q^{76} - 2508 q^{77} - 746 q^{79} + 2922 q^{80} + 1734 q^{82} + 404 q^{83} - 1394 q^{85} - 3168 q^{86} + 3060 q^{88} - 1620 q^{89} - 332 q^{92} - 772 q^{94} - 2204 q^{95} + 2166 q^{97} + 1906 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.46610 0.518345 0.259173 0.965831i \(-0.416550\pi\)
0.259173 + 0.965831i \(0.416550\pi\)
\(3\) 0 0
\(4\) −5.85055 −0.731318
\(5\) −9.85055 −0.881060 −0.440530 0.897738i \(-0.645209\pi\)
−0.440530 + 0.897738i \(0.645209\pi\)
\(6\) 0 0
\(7\) 29.9396 1.61659 0.808293 0.588780i \(-0.200392\pi\)
0.808293 + 0.588780i \(0.200392\pi\)
\(8\) −20.3063 −0.897421
\(9\) 0 0
\(10\) −14.4419 −0.456693
\(11\) −46.9257 −1.28624 −0.643120 0.765765i \(-0.722360\pi\)
−0.643120 + 0.765765i \(0.722360\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 43.8945 0.837950
\(15\) 0 0
\(16\) 17.0333 0.266145
\(17\) 48.2616 0.688539 0.344270 0.938871i \(-0.388127\pi\)
0.344270 + 0.938871i \(0.388127\pi\)
\(18\) 0 0
\(19\) 120.300 1.45257 0.726283 0.687396i \(-0.241246\pi\)
0.726283 + 0.687396i \(0.241246\pi\)
\(20\) 57.6311 0.644335
\(21\) 0 0
\(22\) −68.7979 −0.666717
\(23\) −130.697 −1.18488 −0.592440 0.805615i \(-0.701835\pi\)
−0.592440 + 0.805615i \(0.701835\pi\)
\(24\) 0 0
\(25\) −27.9667 −0.223734
\(26\) 0 0
\(27\) 0 0
\(28\) −175.163 −1.18224
\(29\) 194.946 1.24829 0.624147 0.781307i \(-0.285447\pi\)
0.624147 + 0.781307i \(0.285447\pi\)
\(30\) 0 0
\(31\) −32.0123 −0.185470 −0.0927351 0.995691i \(-0.529561\pi\)
−0.0927351 + 0.995691i \(0.529561\pi\)
\(32\) 187.423 1.03538
\(33\) 0 0
\(34\) 70.7565 0.356901
\(35\) −294.921 −1.42431
\(36\) 0 0
\(37\) −32.4250 −0.144071 −0.0720355 0.997402i \(-0.522949\pi\)
−0.0720355 + 0.997402i \(0.522949\pi\)
\(38\) 176.372 0.752931
\(39\) 0 0
\(40\) 200.028 0.790681
\(41\) 241.825 0.921140 0.460570 0.887623i \(-0.347645\pi\)
0.460570 + 0.887623i \(0.347645\pi\)
\(42\) 0 0
\(43\) 96.4087 0.341911 0.170956 0.985279i \(-0.445314\pi\)
0.170956 + 0.985279i \(0.445314\pi\)
\(44\) 274.541 0.940651
\(45\) 0 0
\(46\) −191.615 −0.614177
\(47\) −539.015 −1.67284 −0.836419 0.548090i \(-0.815355\pi\)
−0.836419 + 0.548090i \(0.815355\pi\)
\(48\) 0 0
\(49\) 553.380 1.61335
\(50\) −41.0021 −0.115971
\(51\) 0 0
\(52\) 0 0
\(53\) 152.277 0.394657 0.197328 0.980337i \(-0.436774\pi\)
0.197328 + 0.980337i \(0.436774\pi\)
\(54\) 0 0
\(55\) 462.244 1.13325
\(56\) −607.963 −1.45076
\(57\) 0 0
\(58\) 285.810 0.647047
\(59\) −327.792 −0.723304 −0.361652 0.932313i \(-0.617787\pi\)
−0.361652 + 0.932313i \(0.617787\pi\)
\(60\) 0 0
\(61\) −98.4180 −0.206576 −0.103288 0.994651i \(-0.532936\pi\)
−0.103288 + 0.994651i \(0.532936\pi\)
\(62\) −46.9332 −0.0961376
\(63\) 0 0
\(64\) 138.515 0.270537
\(65\) 0 0
\(66\) 0 0
\(67\) −441.151 −0.804405 −0.402202 0.915551i \(-0.631755\pi\)
−0.402202 + 0.915551i \(0.631755\pi\)
\(68\) −282.357 −0.503541
\(69\) 0 0
\(70\) −432.385 −0.738284
\(71\) −345.049 −0.576757 −0.288379 0.957516i \(-0.593116\pi\)
−0.288379 + 0.957516i \(0.593116\pi\)
\(72\) 0 0
\(73\) 773.839 1.24070 0.620349 0.784326i \(-0.286991\pi\)
0.620349 + 0.784326i \(0.286991\pi\)
\(74\) −47.5383 −0.0746785
\(75\) 0 0
\(76\) −703.822 −1.06229
\(77\) −1404.94 −2.07932
\(78\) 0 0
\(79\) −150.332 −0.214097 −0.107049 0.994254i \(-0.534140\pi\)
−0.107049 + 0.994254i \(0.534140\pi\)
\(80\) −167.787 −0.234489
\(81\) 0 0
\(82\) 354.540 0.477469
\(83\) −337.966 −0.446947 −0.223473 0.974710i \(-0.571740\pi\)
−0.223473 + 0.974710i \(0.571740\pi\)
\(84\) 0 0
\(85\) −475.403 −0.606644
\(86\) 141.345 0.177228
\(87\) 0 0
\(88\) 952.889 1.15430
\(89\) −169.913 −0.202368 −0.101184 0.994868i \(-0.532263\pi\)
−0.101184 + 0.994868i \(0.532263\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 764.649 0.866524
\(93\) 0 0
\(94\) −790.250 −0.867108
\(95\) −1185.02 −1.27980
\(96\) 0 0
\(97\) 214.201 0.224215 0.112107 0.993696i \(-0.464240\pi\)
0.112107 + 0.993696i \(0.464240\pi\)
\(98\) 811.311 0.836273
\(99\) 0 0
\(100\) 163.621 0.163621
\(101\) −1595.11 −1.57148 −0.785741 0.618556i \(-0.787718\pi\)
−0.785741 + 0.618556i \(0.787718\pi\)
\(102\) 0 0
\(103\) −1570.30 −1.50219 −0.751096 0.660193i \(-0.770475\pi\)
−0.751096 + 0.660193i \(0.770475\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 223.253 0.204568
\(107\) 6.73302 0.00608323 0.00304161 0.999995i \(-0.499032\pi\)
0.00304161 + 0.999995i \(0.499032\pi\)
\(108\) 0 0
\(109\) −542.422 −0.476648 −0.238324 0.971186i \(-0.576598\pi\)
−0.238324 + 0.971186i \(0.576598\pi\)
\(110\) 677.697 0.587417
\(111\) 0 0
\(112\) 509.969 0.430246
\(113\) 1442.82 1.20114 0.600572 0.799571i \(-0.294940\pi\)
0.600572 + 0.799571i \(0.294940\pi\)
\(114\) 0 0
\(115\) 1287.44 1.04395
\(116\) −1140.54 −0.912900
\(117\) 0 0
\(118\) −480.577 −0.374921
\(119\) 1444.93 1.11308
\(120\) 0 0
\(121\) 871.025 0.654414
\(122\) −144.291 −0.107078
\(123\) 0 0
\(124\) 187.289 0.135638
\(125\) 1506.81 1.07818
\(126\) 0 0
\(127\) −2492.84 −1.74176 −0.870881 0.491494i \(-0.836451\pi\)
−0.870881 + 0.491494i \(0.836451\pi\)
\(128\) −1296.31 −0.895144
\(129\) 0 0
\(130\) 0 0
\(131\) −744.561 −0.496585 −0.248292 0.968685i \(-0.579869\pi\)
−0.248292 + 0.968685i \(0.579869\pi\)
\(132\) 0 0
\(133\) 3601.74 2.34820
\(134\) −646.772 −0.416959
\(135\) 0 0
\(136\) −980.016 −0.617909
\(137\) 222.167 0.138547 0.0692736 0.997598i \(-0.477932\pi\)
0.0692736 + 0.997598i \(0.477932\pi\)
\(138\) 0 0
\(139\) −777.145 −0.474220 −0.237110 0.971483i \(-0.576200\pi\)
−0.237110 + 0.971483i \(0.576200\pi\)
\(140\) 1725.45 1.04162
\(141\) 0 0
\(142\) −505.876 −0.298959
\(143\) 0 0
\(144\) 0 0
\(145\) −1920.32 −1.09982
\(146\) 1134.53 0.643110
\(147\) 0 0
\(148\) 189.704 0.105362
\(149\) −1778.29 −0.977742 −0.488871 0.872356i \(-0.662591\pi\)
−0.488871 + 0.872356i \(0.662591\pi\)
\(150\) 0 0
\(151\) 1166.00 0.628394 0.314197 0.949358i \(-0.398265\pi\)
0.314197 + 0.949358i \(0.398265\pi\)
\(152\) −2442.85 −1.30356
\(153\) 0 0
\(154\) −2059.78 −1.07780
\(155\) 315.338 0.163410
\(156\) 0 0
\(157\) 517.628 0.263129 0.131564 0.991308i \(-0.458000\pi\)
0.131564 + 0.991308i \(0.458000\pi\)
\(158\) −220.402 −0.110976
\(159\) 0 0
\(160\) −1846.22 −0.912227
\(161\) −3913.02 −1.91546
\(162\) 0 0
\(163\) 610.188 0.293212 0.146606 0.989195i \(-0.453165\pi\)
0.146606 + 0.989195i \(0.453165\pi\)
\(164\) −1414.81 −0.673647
\(165\) 0 0
\(166\) −495.493 −0.231673
\(167\) −2983.01 −1.38223 −0.691115 0.722745i \(-0.742880\pi\)
−0.691115 + 0.722745i \(0.742880\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −696.990 −0.314451
\(171\) 0 0
\(172\) −564.044 −0.250046
\(173\) −978.212 −0.429896 −0.214948 0.976625i \(-0.568958\pi\)
−0.214948 + 0.976625i \(0.568958\pi\)
\(174\) 0 0
\(175\) −837.313 −0.361685
\(176\) −799.298 −0.342326
\(177\) 0 0
\(178\) −249.110 −0.104897
\(179\) −1852.94 −0.773717 −0.386858 0.922139i \(-0.626440\pi\)
−0.386858 + 0.922139i \(0.626440\pi\)
\(180\) 0 0
\(181\) 852.777 0.350201 0.175101 0.984551i \(-0.443975\pi\)
0.175101 + 0.984551i \(0.443975\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2653.98 1.06334
\(185\) 319.403 0.126935
\(186\) 0 0
\(187\) −2264.71 −0.885627
\(188\) 3153.53 1.22338
\(189\) 0 0
\(190\) −1737.36 −0.663377
\(191\) −4441.29 −1.68252 −0.841258 0.540633i \(-0.818185\pi\)
−0.841258 + 0.540633i \(0.818185\pi\)
\(192\) 0 0
\(193\) −2482.43 −0.925851 −0.462925 0.886397i \(-0.653200\pi\)
−0.462925 + 0.886397i \(0.653200\pi\)
\(194\) 314.041 0.116221
\(195\) 0 0
\(196\) −3237.57 −1.17987
\(197\) −1260.23 −0.455775 −0.227888 0.973687i \(-0.573182\pi\)
−0.227888 + 0.973687i \(0.573182\pi\)
\(198\) 0 0
\(199\) −5520.96 −1.96669 −0.983344 0.181756i \(-0.941822\pi\)
−0.983344 + 0.181756i \(0.941822\pi\)
\(200\) 567.901 0.200783
\(201\) 0 0
\(202\) −2338.60 −0.814570
\(203\) 5836.60 2.01798
\(204\) 0 0
\(205\) −2382.11 −0.811580
\(206\) −2302.21 −0.778654
\(207\) 0 0
\(208\) 0 0
\(209\) −5645.18 −1.86835
\(210\) 0 0
\(211\) −4527.79 −1.47728 −0.738640 0.674100i \(-0.764532\pi\)
−0.738640 + 0.674100i \(0.764532\pi\)
\(212\) −890.901 −0.288620
\(213\) 0 0
\(214\) 9.87129 0.00315321
\(215\) −949.679 −0.301244
\(216\) 0 0
\(217\) −958.435 −0.299829
\(218\) −795.245 −0.247068
\(219\) 0 0
\(220\) −2704.38 −0.828770
\(221\) 0 0
\(222\) 0 0
\(223\) 4481.17 1.34566 0.672829 0.739798i \(-0.265079\pi\)
0.672829 + 0.739798i \(0.265079\pi\)
\(224\) 5611.37 1.67377
\(225\) 0 0
\(226\) 2115.32 0.622607
\(227\) 5759.75 1.68409 0.842044 0.539408i \(-0.181352\pi\)
0.842044 + 0.539408i \(0.181352\pi\)
\(228\) 0 0
\(229\) −4635.08 −1.33753 −0.668766 0.743473i \(-0.733177\pi\)
−0.668766 + 0.743473i \(0.733177\pi\)
\(230\) 1887.51 0.541126
\(231\) 0 0
\(232\) −3958.63 −1.12024
\(233\) −5886.33 −1.65505 −0.827524 0.561431i \(-0.810251\pi\)
−0.827524 + 0.561431i \(0.810251\pi\)
\(234\) 0 0
\(235\) 5309.59 1.47387
\(236\) 1917.76 0.528965
\(237\) 0 0
\(238\) 2118.42 0.576961
\(239\) −2135.84 −0.578060 −0.289030 0.957320i \(-0.593333\pi\)
−0.289030 + 0.957320i \(0.593333\pi\)
\(240\) 0 0
\(241\) 4692.92 1.25435 0.627173 0.778880i \(-0.284212\pi\)
0.627173 + 0.778880i \(0.284212\pi\)
\(242\) 1277.01 0.339212
\(243\) 0 0
\(244\) 575.799 0.151073
\(245\) −5451.09 −1.42146
\(246\) 0 0
\(247\) 0 0
\(248\) 650.051 0.166445
\(249\) 0 0
\(250\) 2209.13 0.558871
\(251\) 3902.88 0.981464 0.490732 0.871310i \(-0.336729\pi\)
0.490732 + 0.871310i \(0.336729\pi\)
\(252\) 0 0
\(253\) 6133.06 1.52404
\(254\) −3654.76 −0.902834
\(255\) 0 0
\(256\) −3008.64 −0.734531
\(257\) 4130.83 1.00262 0.501312 0.865267i \(-0.332851\pi\)
0.501312 + 0.865267i \(0.332851\pi\)
\(258\) 0 0
\(259\) −970.790 −0.232903
\(260\) 0 0
\(261\) 0 0
\(262\) −1091.60 −0.257402
\(263\) −6352.17 −1.48932 −0.744661 0.667443i \(-0.767389\pi\)
−0.744661 + 0.667443i \(0.767389\pi\)
\(264\) 0 0
\(265\) −1500.01 −0.347716
\(266\) 5280.52 1.21718
\(267\) 0 0
\(268\) 2580.97 0.588276
\(269\) −181.524 −0.0411439 −0.0205719 0.999788i \(-0.506549\pi\)
−0.0205719 + 0.999788i \(0.506549\pi\)
\(270\) 0 0
\(271\) 3460.35 0.775651 0.387825 0.921733i \(-0.373226\pi\)
0.387825 + 0.921733i \(0.373226\pi\)
\(272\) 822.053 0.183251
\(273\) 0 0
\(274\) 325.719 0.0718153
\(275\) 1312.36 0.287776
\(276\) 0 0
\(277\) −6437.94 −1.39646 −0.698228 0.715876i \(-0.746028\pi\)
−0.698228 + 0.715876i \(0.746028\pi\)
\(278\) −1139.37 −0.245810
\(279\) 0 0
\(280\) 5988.77 1.27820
\(281\) 2974.26 0.631421 0.315711 0.948855i \(-0.397757\pi\)
0.315711 + 0.948855i \(0.397757\pi\)
\(282\) 0 0
\(283\) 3035.72 0.637649 0.318825 0.947814i \(-0.396712\pi\)
0.318825 + 0.947814i \(0.396712\pi\)
\(284\) 2018.72 0.421793
\(285\) 0 0
\(286\) 0 0
\(287\) 7240.15 1.48910
\(288\) 0 0
\(289\) −2583.81 −0.525914
\(290\) −2815.39 −0.570087
\(291\) 0 0
\(292\) −4527.38 −0.907346
\(293\) 1955.74 0.389952 0.194976 0.980808i \(-0.437537\pi\)
0.194976 + 0.980808i \(0.437537\pi\)
\(294\) 0 0
\(295\) 3228.93 0.637274
\(296\) 658.431 0.129292
\(297\) 0 0
\(298\) −2607.16 −0.506808
\(299\) 0 0
\(300\) 0 0
\(301\) 2886.44 0.552729
\(302\) 1709.47 0.325725
\(303\) 0 0
\(304\) 2049.10 0.386593
\(305\) 969.471 0.182006
\(306\) 0 0
\(307\) −1027.56 −0.191029 −0.0955147 0.995428i \(-0.530450\pi\)
−0.0955147 + 0.995428i \(0.530450\pi\)
\(308\) 8219.66 1.52064
\(309\) 0 0
\(310\) 462.318 0.0847029
\(311\) −3405.61 −0.620947 −0.310474 0.950582i \(-0.600488\pi\)
−0.310474 + 0.950582i \(0.600488\pi\)
\(312\) 0 0
\(313\) 4813.20 0.869196 0.434598 0.900625i \(-0.356890\pi\)
0.434598 + 0.900625i \(0.356890\pi\)
\(314\) 758.895 0.136391
\(315\) 0 0
\(316\) 879.525 0.156573
\(317\) −1141.33 −0.202219 −0.101110 0.994875i \(-0.532239\pi\)
−0.101110 + 0.994875i \(0.532239\pi\)
\(318\) 0 0
\(319\) −9147.98 −1.60561
\(320\) −1364.45 −0.238359
\(321\) 0 0
\(322\) −5736.88 −0.992870
\(323\) 5805.88 1.00015
\(324\) 0 0
\(325\) 0 0
\(326\) 894.597 0.151985
\(327\) 0 0
\(328\) −4910.58 −0.826650
\(329\) −16137.9 −2.70429
\(330\) 0 0
\(331\) 7652.57 1.27076 0.635382 0.772198i \(-0.280843\pi\)
0.635382 + 0.772198i \(0.280843\pi\)
\(332\) 1977.29 0.326860
\(333\) 0 0
\(334\) −4373.40 −0.716472
\(335\) 4345.57 0.708729
\(336\) 0 0
\(337\) −2503.69 −0.404702 −0.202351 0.979313i \(-0.564858\pi\)
−0.202351 + 0.979313i \(0.564858\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 2781.37 0.443650
\(341\) 1502.20 0.238559
\(342\) 0 0
\(343\) 6298.69 0.991537
\(344\) −1957.71 −0.306838
\(345\) 0 0
\(346\) −1434.16 −0.222835
\(347\) −4496.12 −0.695574 −0.347787 0.937574i \(-0.613067\pi\)
−0.347787 + 0.937574i \(0.613067\pi\)
\(348\) 0 0
\(349\) 3577.61 0.548726 0.274363 0.961626i \(-0.411533\pi\)
0.274363 + 0.961626i \(0.411533\pi\)
\(350\) −1227.59 −0.187478
\(351\) 0 0
\(352\) −8794.96 −1.33174
\(353\) 8045.17 1.21303 0.606517 0.795070i \(-0.292566\pi\)
0.606517 + 0.795070i \(0.292566\pi\)
\(354\) 0 0
\(355\) 3398.92 0.508157
\(356\) 994.086 0.147996
\(357\) 0 0
\(358\) −2716.60 −0.401052
\(359\) 2172.90 0.319447 0.159724 0.987162i \(-0.448940\pi\)
0.159724 + 0.987162i \(0.448940\pi\)
\(360\) 0 0
\(361\) 7613.14 1.10995
\(362\) 1250.26 0.181525
\(363\) 0 0
\(364\) 0 0
\(365\) −7622.74 −1.09313
\(366\) 0 0
\(367\) 7662.76 1.08990 0.544949 0.838469i \(-0.316549\pi\)
0.544949 + 0.838469i \(0.316549\pi\)
\(368\) −2226.20 −0.315349
\(369\) 0 0
\(370\) 468.278 0.0657962
\(371\) 4559.10 0.637996
\(372\) 0 0
\(373\) 10542.5 1.46346 0.731732 0.681593i \(-0.238712\pi\)
0.731732 + 0.681593i \(0.238712\pi\)
\(374\) −3320.30 −0.459060
\(375\) 0 0
\(376\) 10945.4 1.50124
\(377\) 0 0
\(378\) 0 0
\(379\) 5475.54 0.742110 0.371055 0.928611i \(-0.378996\pi\)
0.371055 + 0.928611i \(0.378996\pi\)
\(380\) 6933.03 0.935939
\(381\) 0 0
\(382\) −6511.39 −0.872124
\(383\) −808.085 −0.107810 −0.0539050 0.998546i \(-0.517167\pi\)
−0.0539050 + 0.998546i \(0.517167\pi\)
\(384\) 0 0
\(385\) 13839.4 1.83200
\(386\) −3639.49 −0.479910
\(387\) 0 0
\(388\) −1253.19 −0.163972
\(389\) 7060.26 0.920230 0.460115 0.887859i \(-0.347808\pi\)
0.460115 + 0.887859i \(0.347808\pi\)
\(390\) 0 0
\(391\) −6307.65 −0.815836
\(392\) −11237.1 −1.44786
\(393\) 0 0
\(394\) −1847.63 −0.236249
\(395\) 1480.85 0.188633
\(396\) 0 0
\(397\) −1419.89 −0.179502 −0.0897511 0.995964i \(-0.528607\pi\)
−0.0897511 + 0.995964i \(0.528607\pi\)
\(398\) −8094.29 −1.01942
\(399\) 0 0
\(400\) −476.365 −0.0595456
\(401\) −10670.7 −1.32886 −0.664428 0.747353i \(-0.731325\pi\)
−0.664428 + 0.747353i \(0.731325\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 9332.28 1.14925
\(405\) 0 0
\(406\) 8557.05 1.04601
\(407\) 1521.56 0.185310
\(408\) 0 0
\(409\) 6351.56 0.767883 0.383942 0.923357i \(-0.374566\pi\)
0.383942 + 0.923357i \(0.374566\pi\)
\(410\) −3492.42 −0.420678
\(411\) 0 0
\(412\) 9187.09 1.09858
\(413\) −9813.97 −1.16928
\(414\) 0 0
\(415\) 3329.15 0.393787
\(416\) 0 0
\(417\) 0 0
\(418\) −8276.40 −0.968450
\(419\) −5617.87 −0.655015 −0.327507 0.944849i \(-0.606209\pi\)
−0.327507 + 0.944849i \(0.606209\pi\)
\(420\) 0 0
\(421\) −1518.29 −0.175765 −0.0878825 0.996131i \(-0.528010\pi\)
−0.0878825 + 0.996131i \(0.528010\pi\)
\(422\) −6638.20 −0.765741
\(423\) 0 0
\(424\) −3092.17 −0.354173
\(425\) −1349.72 −0.154050
\(426\) 0 0
\(427\) −2946.60 −0.333948
\(428\) −39.3918 −0.00444878
\(429\) 0 0
\(430\) −1392.33 −0.156149
\(431\) 4970.13 0.555459 0.277730 0.960659i \(-0.410418\pi\)
0.277730 + 0.960659i \(0.410418\pi\)
\(432\) 0 0
\(433\) −3298.71 −0.366111 −0.183055 0.983103i \(-0.558599\pi\)
−0.183055 + 0.983103i \(0.558599\pi\)
\(434\) −1405.16 −0.155415
\(435\) 0 0
\(436\) 3173.46 0.348581
\(437\) −15722.9 −1.72112
\(438\) 0 0
\(439\) −6048.47 −0.657581 −0.328790 0.944403i \(-0.606641\pi\)
−0.328790 + 0.944403i \(0.606641\pi\)
\(440\) −9386.47 −1.01701
\(441\) 0 0
\(442\) 0 0
\(443\) −6822.62 −0.731722 −0.365861 0.930670i \(-0.619225\pi\)
−0.365861 + 0.930670i \(0.619225\pi\)
\(444\) 0 0
\(445\) 1673.74 0.178299
\(446\) 6569.86 0.697515
\(447\) 0 0
\(448\) 4147.09 0.437347
\(449\) −410.410 −0.0431369 −0.0215684 0.999767i \(-0.506866\pi\)
−0.0215684 + 0.999767i \(0.506866\pi\)
\(450\) 0 0
\(451\) −11347.8 −1.18481
\(452\) −8441.30 −0.878419
\(453\) 0 0
\(454\) 8444.38 0.872939
\(455\) 0 0
\(456\) 0 0
\(457\) −16642.8 −1.70354 −0.851771 0.523915i \(-0.824471\pi\)
−0.851771 + 0.523915i \(0.824471\pi\)
\(458\) −6795.50 −0.693303
\(459\) 0 0
\(460\) −7532.21 −0.763459
\(461\) 11729.7 1.18505 0.592523 0.805553i \(-0.298132\pi\)
0.592523 + 0.805553i \(0.298132\pi\)
\(462\) 0 0
\(463\) −3564.93 −0.357832 −0.178916 0.983864i \(-0.557259\pi\)
−0.178916 + 0.983864i \(0.557259\pi\)
\(464\) 3320.56 0.332227
\(465\) 0 0
\(466\) −8629.95 −0.857886
\(467\) −1134.81 −0.112447 −0.0562233 0.998418i \(-0.517906\pi\)
−0.0562233 + 0.998418i \(0.517906\pi\)
\(468\) 0 0
\(469\) −13207.9 −1.30039
\(470\) 7784.40 0.763974
\(471\) 0 0
\(472\) 6656.25 0.649107
\(473\) −4524.05 −0.439780
\(474\) 0 0
\(475\) −3364.40 −0.324988
\(476\) −8453.65 −0.814018
\(477\) 0 0
\(478\) −3131.36 −0.299635
\(479\) 19372.6 1.84793 0.923963 0.382481i \(-0.124930\pi\)
0.923963 + 0.382481i \(0.124930\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 6880.29 0.650184
\(483\) 0 0
\(484\) −5095.97 −0.478585
\(485\) −2110.00 −0.197547
\(486\) 0 0
\(487\) −9045.95 −0.841707 −0.420853 0.907129i \(-0.638269\pi\)
−0.420853 + 0.907129i \(0.638269\pi\)
\(488\) 1998.51 0.185385
\(489\) 0 0
\(490\) −7991.86 −0.736807
\(491\) −14403.3 −1.32385 −0.661927 0.749568i \(-0.730261\pi\)
−0.661927 + 0.749568i \(0.730261\pi\)
\(492\) 0 0
\(493\) 9408.40 0.859499
\(494\) 0 0
\(495\) 0 0
\(496\) −545.273 −0.0493619
\(497\) −10330.6 −0.932378
\(498\) 0 0
\(499\) −9319.75 −0.836091 −0.418045 0.908426i \(-0.637285\pi\)
−0.418045 + 0.908426i \(0.637285\pi\)
\(500\) −8815.64 −0.788495
\(501\) 0 0
\(502\) 5722.02 0.508737
\(503\) −2745.98 −0.243414 −0.121707 0.992566i \(-0.538837\pi\)
−0.121707 + 0.992566i \(0.538837\pi\)
\(504\) 0 0
\(505\) 15712.7 1.38457
\(506\) 8991.69 0.789979
\(507\) 0 0
\(508\) 14584.5 1.27378
\(509\) 1182.11 0.102939 0.0514697 0.998675i \(-0.483609\pi\)
0.0514697 + 0.998675i \(0.483609\pi\)
\(510\) 0 0
\(511\) 23168.4 2.00570
\(512\) 5959.48 0.514403
\(513\) 0 0
\(514\) 6056.22 0.519705
\(515\) 15468.3 1.32352
\(516\) 0 0
\(517\) 25293.7 2.15167
\(518\) −1423.28 −0.120724
\(519\) 0 0
\(520\) 0 0
\(521\) −10858.8 −0.913115 −0.456558 0.889694i \(-0.650918\pi\)
−0.456558 + 0.889694i \(0.650918\pi\)
\(522\) 0 0
\(523\) −10161.7 −0.849602 −0.424801 0.905287i \(-0.639656\pi\)
−0.424801 + 0.905287i \(0.639656\pi\)
\(524\) 4356.09 0.363162
\(525\) 0 0
\(526\) −9312.93 −0.771983
\(527\) −1544.96 −0.127703
\(528\) 0 0
\(529\) 4914.73 0.403939
\(530\) −2199.16 −0.180237
\(531\) 0 0
\(532\) −21072.1 −1.71728
\(533\) 0 0
\(534\) 0 0
\(535\) −66.3239 −0.00535969
\(536\) 8958.14 0.721889
\(537\) 0 0
\(538\) −266.132 −0.0213267
\(539\) −25967.8 −2.07516
\(540\) 0 0
\(541\) −9573.04 −0.760771 −0.380386 0.924828i \(-0.624209\pi\)
−0.380386 + 0.924828i \(0.624209\pi\)
\(542\) 5073.23 0.402055
\(543\) 0 0
\(544\) 9045.34 0.712896
\(545\) 5343.15 0.419955
\(546\) 0 0
\(547\) 15958.9 1.24745 0.623724 0.781645i \(-0.285619\pi\)
0.623724 + 0.781645i \(0.285619\pi\)
\(548\) −1299.80 −0.101322
\(549\) 0 0
\(550\) 1924.05 0.149167
\(551\) 23452.0 1.81323
\(552\) 0 0
\(553\) −4500.89 −0.346107
\(554\) −9438.67 −0.723846
\(555\) 0 0
\(556\) 4546.72 0.346806
\(557\) 2145.09 0.163178 0.0815892 0.996666i \(-0.474000\pi\)
0.0815892 + 0.996666i \(0.474000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −5023.47 −0.379072
\(561\) 0 0
\(562\) 4360.57 0.327294
\(563\) 22318.6 1.67072 0.835362 0.549701i \(-0.185258\pi\)
0.835362 + 0.549701i \(0.185258\pi\)
\(564\) 0 0
\(565\) −14212.6 −1.05828
\(566\) 4450.67 0.330523
\(567\) 0 0
\(568\) 7006.67 0.517594
\(569\) 19753.0 1.45534 0.727671 0.685927i \(-0.240603\pi\)
0.727671 + 0.685927i \(0.240603\pi\)
\(570\) 0 0
\(571\) −10640.6 −0.779850 −0.389925 0.920847i \(-0.627499\pi\)
−0.389925 + 0.920847i \(0.627499\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 10614.8 0.771870
\(575\) 3655.17 0.265098
\(576\) 0 0
\(577\) 6547.89 0.472430 0.236215 0.971701i \(-0.424093\pi\)
0.236215 + 0.971701i \(0.424093\pi\)
\(578\) −3788.13 −0.272605
\(579\) 0 0
\(580\) 11234.9 0.804319
\(581\) −10118.6 −0.722529
\(582\) 0 0
\(583\) −7145.69 −0.507623
\(584\) −15713.8 −1.11343
\(585\) 0 0
\(586\) 2867.32 0.202130
\(587\) −5900.33 −0.414877 −0.207439 0.978248i \(-0.566513\pi\)
−0.207439 + 0.978248i \(0.566513\pi\)
\(588\) 0 0
\(589\) −3851.08 −0.269408
\(590\) 4733.94 0.330328
\(591\) 0 0
\(592\) −552.303 −0.0383437
\(593\) 15261.5 1.05686 0.528428 0.848978i \(-0.322782\pi\)
0.528428 + 0.848978i \(0.322782\pi\)
\(594\) 0 0
\(595\) −14233.4 −0.980693
\(596\) 10404.0 0.715041
\(597\) 0 0
\(598\) 0 0
\(599\) 18900.6 1.28925 0.644623 0.764501i \(-0.277014\pi\)
0.644623 + 0.764501i \(0.277014\pi\)
\(600\) 0 0
\(601\) −18507.0 −1.25610 −0.628049 0.778174i \(-0.716146\pi\)
−0.628049 + 0.778174i \(0.716146\pi\)
\(602\) 4231.81 0.286505
\(603\) 0 0
\(604\) −6821.72 −0.459556
\(605\) −8580.08 −0.576578
\(606\) 0 0
\(607\) −6408.50 −0.428522 −0.214261 0.976776i \(-0.568734\pi\)
−0.214261 + 0.976776i \(0.568734\pi\)
\(608\) 22547.0 1.50395
\(609\) 0 0
\(610\) 1421.34 0.0943418
\(611\) 0 0
\(612\) 0 0
\(613\) −3507.26 −0.231088 −0.115544 0.993302i \(-0.536861\pi\)
−0.115544 + 0.993302i \(0.536861\pi\)
\(614\) −1506.51 −0.0990192
\(615\) 0 0
\(616\) 28529.1 1.86602
\(617\) −14507.8 −0.946614 −0.473307 0.880897i \(-0.656940\pi\)
−0.473307 + 0.880897i \(0.656940\pi\)
\(618\) 0 0
\(619\) 4750.85 0.308486 0.154243 0.988033i \(-0.450706\pi\)
0.154243 + 0.988033i \(0.450706\pi\)
\(620\) −1844.90 −0.119505
\(621\) 0 0
\(622\) −4992.98 −0.321865
\(623\) −5087.14 −0.327146
\(624\) 0 0
\(625\) −11347.0 −0.726209
\(626\) 7056.65 0.450544
\(627\) 0 0
\(628\) −3028.40 −0.192431
\(629\) −1564.88 −0.0991986
\(630\) 0 0
\(631\) 4825.19 0.304418 0.152209 0.988348i \(-0.451361\pi\)
0.152209 + 0.988348i \(0.451361\pi\)
\(632\) 3052.69 0.192135
\(633\) 0 0
\(634\) −1673.30 −0.104819
\(635\) 24555.8 1.53460
\(636\) 0 0
\(637\) 0 0
\(638\) −13411.9 −0.832258
\(639\) 0 0
\(640\) 12769.3 0.788675
\(641\) 5911.89 0.364284 0.182142 0.983272i \(-0.441697\pi\)
0.182142 + 0.983272i \(0.441697\pi\)
\(642\) 0 0
\(643\) 23408.3 1.43567 0.717833 0.696216i \(-0.245134\pi\)
0.717833 + 0.696216i \(0.245134\pi\)
\(644\) 22893.3 1.40081
\(645\) 0 0
\(646\) 8512.02 0.518422
\(647\) 16398.7 0.996442 0.498221 0.867050i \(-0.333987\pi\)
0.498221 + 0.867050i \(0.333987\pi\)
\(648\) 0 0
\(649\) 15381.9 0.930342
\(650\) 0 0
\(651\) 0 0
\(652\) −3569.93 −0.214431
\(653\) −27529.6 −1.64979 −0.824897 0.565283i \(-0.808767\pi\)
−0.824897 + 0.565283i \(0.808767\pi\)
\(654\) 0 0
\(655\) 7334.34 0.437521
\(656\) 4119.07 0.245157
\(657\) 0 0
\(658\) −23659.8 −1.40175
\(659\) 24179.7 1.42930 0.714650 0.699482i \(-0.246586\pi\)
0.714650 + 0.699482i \(0.246586\pi\)
\(660\) 0 0
\(661\) 4525.04 0.266269 0.133134 0.991098i \(-0.457496\pi\)
0.133134 + 0.991098i \(0.457496\pi\)
\(662\) 11219.4 0.658695
\(663\) 0 0
\(664\) 6862.84 0.401099
\(665\) −35479.1 −2.06890
\(666\) 0 0
\(667\) −25478.8 −1.47908
\(668\) 17452.2 1.01085
\(669\) 0 0
\(670\) 6371.05 0.367366
\(671\) 4618.34 0.265706
\(672\) 0 0
\(673\) 3287.18 0.188279 0.0941393 0.995559i \(-0.469990\pi\)
0.0941393 + 0.995559i \(0.469990\pi\)
\(674\) −3670.66 −0.209775
\(675\) 0 0
\(676\) 0 0
\(677\) 9724.21 0.552041 0.276020 0.961152i \(-0.410984\pi\)
0.276020 + 0.961152i \(0.410984\pi\)
\(678\) 0 0
\(679\) 6413.10 0.362463
\(680\) 9653.69 0.544415
\(681\) 0 0
\(682\) 2202.38 0.123656
\(683\) −14548.7 −0.815065 −0.407532 0.913191i \(-0.633611\pi\)
−0.407532 + 0.913191i \(0.633611\pi\)
\(684\) 0 0
\(685\) −2188.46 −0.122068
\(686\) 9234.52 0.513959
\(687\) 0 0
\(688\) 1642.15 0.0909979
\(689\) 0 0
\(690\) 0 0
\(691\) 6728.96 0.370451 0.185226 0.982696i \(-0.440698\pi\)
0.185226 + 0.982696i \(0.440698\pi\)
\(692\) 5723.07 0.314391
\(693\) 0 0
\(694\) −6591.76 −0.360548
\(695\) 7655.31 0.417816
\(696\) 0 0
\(697\) 11670.9 0.634241
\(698\) 5245.15 0.284429
\(699\) 0 0
\(700\) 4898.74 0.264507
\(701\) 29159.8 1.57111 0.785557 0.618789i \(-0.212376\pi\)
0.785557 + 0.618789i \(0.212376\pi\)
\(702\) 0 0
\(703\) −3900.73 −0.209273
\(704\) −6499.92 −0.347976
\(705\) 0 0
\(706\) 11795.0 0.628770
\(707\) −47757.0 −2.54044
\(708\) 0 0
\(709\) −20489.0 −1.08531 −0.542653 0.839957i \(-0.682580\pi\)
−0.542653 + 0.839957i \(0.682580\pi\)
\(710\) 4983.16 0.263401
\(711\) 0 0
\(712\) 3450.31 0.181609
\(713\) 4183.91 0.219760
\(714\) 0 0
\(715\) 0 0
\(716\) 10840.7 0.565833
\(717\) 0 0
\(718\) 3185.70 0.165584
\(719\) 17990.1 0.933127 0.466563 0.884488i \(-0.345492\pi\)
0.466563 + 0.884488i \(0.345492\pi\)
\(720\) 0 0
\(721\) −47014.0 −2.42842
\(722\) 11161.6 0.575337
\(723\) 0 0
\(724\) −4989.21 −0.256108
\(725\) −5452.00 −0.279286
\(726\) 0 0
\(727\) −37652.7 −1.92086 −0.960428 0.278528i \(-0.910153\pi\)
−0.960428 + 0.278528i \(0.910153\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −11175.7 −0.566618
\(731\) 4652.84 0.235419
\(732\) 0 0
\(733\) 4524.26 0.227977 0.113989 0.993482i \(-0.463637\pi\)
0.113989 + 0.993482i \(0.463637\pi\)
\(734\) 11234.4 0.564944
\(735\) 0 0
\(736\) −24495.6 −1.22679
\(737\) 20701.3 1.03466
\(738\) 0 0
\(739\) 818.302 0.0407331 0.0203665 0.999793i \(-0.493517\pi\)
0.0203665 + 0.999793i \(0.493517\pi\)
\(740\) −1868.68 −0.0928300
\(741\) 0 0
\(742\) 6684.10 0.330702
\(743\) −39002.2 −1.92578 −0.962888 0.269901i \(-0.913009\pi\)
−0.962888 + 0.269901i \(0.913009\pi\)
\(744\) 0 0
\(745\) 17517.2 0.861449
\(746\) 15456.4 0.758579
\(747\) 0 0
\(748\) 13249.8 0.647675
\(749\) 201.584 0.00983407
\(750\) 0 0
\(751\) −22377.7 −1.08732 −0.543658 0.839307i \(-0.682961\pi\)
−0.543658 + 0.839307i \(0.682961\pi\)
\(752\) −9181.18 −0.445217
\(753\) 0 0
\(754\) 0 0
\(755\) −11485.7 −0.553653
\(756\) 0 0
\(757\) −34512.6 −1.65704 −0.828521 0.559958i \(-0.810817\pi\)
−0.828521 + 0.559958i \(0.810817\pi\)
\(758\) 8027.70 0.384669
\(759\) 0 0
\(760\) 24063.4 1.14852
\(761\) −19975.6 −0.951529 −0.475764 0.879573i \(-0.657828\pi\)
−0.475764 + 0.879573i \(0.657828\pi\)
\(762\) 0 0
\(763\) −16239.9 −0.770542
\(764\) 25984.0 1.23046
\(765\) 0 0
\(766\) −1184.73 −0.0558827
\(767\) 0 0
\(768\) 0 0
\(769\) −33064.9 −1.55052 −0.775260 0.631642i \(-0.782381\pi\)
−0.775260 + 0.631642i \(0.782381\pi\)
\(770\) 20290.0 0.949610
\(771\) 0 0
\(772\) 14523.6 0.677091
\(773\) 18564.0 0.863779 0.431890 0.901927i \(-0.357847\pi\)
0.431890 + 0.901927i \(0.357847\pi\)
\(774\) 0 0
\(775\) 895.279 0.0414960
\(776\) −4349.64 −0.201215
\(777\) 0 0
\(778\) 10351.1 0.476997
\(779\) 29091.6 1.33802
\(780\) 0 0
\(781\) 16191.7 0.741848
\(782\) −9247.66 −0.422885
\(783\) 0 0
\(784\) 9425.86 0.429385
\(785\) −5098.91 −0.231832
\(786\) 0 0
\(787\) 36436.8 1.65036 0.825179 0.564871i \(-0.191074\pi\)
0.825179 + 0.564871i \(0.191074\pi\)
\(788\) 7373.04 0.333317
\(789\) 0 0
\(790\) 2171.08 0.0977768
\(791\) 43197.5 1.94175
\(792\) 0 0
\(793\) 0 0
\(794\) −2081.71 −0.0930441
\(795\) 0 0
\(796\) 32300.7 1.43827
\(797\) −12365.6 −0.549576 −0.274788 0.961505i \(-0.588608\pi\)
−0.274788 + 0.961505i \(0.588608\pi\)
\(798\) 0 0
\(799\) −26013.7 −1.15181
\(800\) −5241.61 −0.231649
\(801\) 0 0
\(802\) −15644.4 −0.688806
\(803\) −36313.0 −1.59584
\(804\) 0 0
\(805\) 38545.4 1.68763
\(806\) 0 0
\(807\) 0 0
\(808\) 32390.9 1.41028
\(809\) 8093.75 0.351744 0.175872 0.984413i \(-0.443726\pi\)
0.175872 + 0.984413i \(0.443726\pi\)
\(810\) 0 0
\(811\) 15984.7 0.692105 0.346052 0.938215i \(-0.387522\pi\)
0.346052 + 0.938215i \(0.387522\pi\)
\(812\) −34147.3 −1.47578
\(813\) 0 0
\(814\) 2230.77 0.0960546
\(815\) −6010.68 −0.258337
\(816\) 0 0
\(817\) 11598.0 0.496649
\(818\) 9312.03 0.398029
\(819\) 0 0
\(820\) 13936.6 0.593523
\(821\) 26861.9 1.14188 0.570942 0.820990i \(-0.306578\pi\)
0.570942 + 0.820990i \(0.306578\pi\)
\(822\) 0 0
\(823\) −5205.94 −0.220495 −0.110248 0.993904i \(-0.535164\pi\)
−0.110248 + 0.993904i \(0.535164\pi\)
\(824\) 31886.9 1.34810
\(825\) 0 0
\(826\) −14388.3 −0.606092
\(827\) −46621.5 −1.96032 −0.980162 0.198200i \(-0.936491\pi\)
−0.980162 + 0.198200i \(0.936491\pi\)
\(828\) 0 0
\(829\) −41829.7 −1.75248 −0.876241 0.481874i \(-0.839956\pi\)
−0.876241 + 0.481874i \(0.839956\pi\)
\(830\) 4880.87 0.204118
\(831\) 0 0
\(832\) 0 0
\(833\) 26707.0 1.11086
\(834\) 0 0
\(835\) 29384.3 1.21783
\(836\) 33027.4 1.36636
\(837\) 0 0
\(838\) −8236.37 −0.339524
\(839\) −12685.1 −0.521975 −0.260988 0.965342i \(-0.584048\pi\)
−0.260988 + 0.965342i \(0.584048\pi\)
\(840\) 0 0
\(841\) 13614.9 0.558238
\(842\) −2225.97 −0.0911069
\(843\) 0 0
\(844\) 26490.1 1.08036
\(845\) 0 0
\(846\) 0 0
\(847\) 26078.2 1.05792
\(848\) 2593.77 0.105036
\(849\) 0 0
\(850\) −1978.83 −0.0798509
\(851\) 4237.85 0.170707
\(852\) 0 0
\(853\) −37493.3 −1.50498 −0.752488 0.658606i \(-0.771146\pi\)
−0.752488 + 0.658606i \(0.771146\pi\)
\(854\) −4320.01 −0.173100
\(855\) 0 0
\(856\) −136.723 −0.00545921
\(857\) 11826.3 0.471386 0.235693 0.971828i \(-0.424264\pi\)
0.235693 + 0.971828i \(0.424264\pi\)
\(858\) 0 0
\(859\) −36498.7 −1.44973 −0.724866 0.688890i \(-0.758099\pi\)
−0.724866 + 0.688890i \(0.758099\pi\)
\(860\) 5556.14 0.220306
\(861\) 0 0
\(862\) 7286.72 0.287920
\(863\) −2292.79 −0.0904372 −0.0452186 0.998977i \(-0.514398\pi\)
−0.0452186 + 0.998977i \(0.514398\pi\)
\(864\) 0 0
\(865\) 9635.92 0.378764
\(866\) −4836.24 −0.189772
\(867\) 0 0
\(868\) 5607.37 0.219270
\(869\) 7054.45 0.275381
\(870\) 0 0
\(871\) 0 0
\(872\) 11014.6 0.427753
\(873\) 0 0
\(874\) −23051.3 −0.892132
\(875\) 45113.2 1.74298
\(876\) 0 0
\(877\) −20800.3 −0.800885 −0.400442 0.916322i \(-0.631144\pi\)
−0.400442 + 0.916322i \(0.631144\pi\)
\(878\) −8867.68 −0.340854
\(879\) 0 0
\(880\) 7873.52 0.301610
\(881\) 19255.9 0.736377 0.368188 0.929751i \(-0.379978\pi\)
0.368188 + 0.929751i \(0.379978\pi\)
\(882\) 0 0
\(883\) 1744.49 0.0664857 0.0332429 0.999447i \(-0.489417\pi\)
0.0332429 + 0.999447i \(0.489417\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −10002.7 −0.379284
\(887\) −2970.70 −0.112454 −0.0562268 0.998418i \(-0.517907\pi\)
−0.0562268 + 0.998418i \(0.517907\pi\)
\(888\) 0 0
\(889\) −74634.6 −2.81571
\(890\) 2453.87 0.0924202
\(891\) 0 0
\(892\) −26217.3 −0.984104
\(893\) −64843.6 −2.42991
\(894\) 0 0
\(895\) 18252.5 0.681691
\(896\) −38810.9 −1.44708
\(897\) 0 0
\(898\) −601.703 −0.0223598
\(899\) −6240.66 −0.231521
\(900\) 0 0
\(901\) 7349.12 0.271736
\(902\) −16637.1 −0.614140
\(903\) 0 0
\(904\) −29298.4 −1.07793
\(905\) −8400.32 −0.308548
\(906\) 0 0
\(907\) 33003.3 1.20822 0.604111 0.796900i \(-0.293528\pi\)
0.604111 + 0.796900i \(0.293528\pi\)
\(908\) −33697.7 −1.23160
\(909\) 0 0
\(910\) 0 0
\(911\) −14977.0 −0.544686 −0.272343 0.962200i \(-0.587799\pi\)
−0.272343 + 0.962200i \(0.587799\pi\)
\(912\) 0 0
\(913\) 15859.3 0.574881
\(914\) −24400.1 −0.883022
\(915\) 0 0
\(916\) 27117.7 0.978161
\(917\) −22291.9 −0.802773
\(918\) 0 0
\(919\) 11425.4 0.410107 0.205054 0.978751i \(-0.434263\pi\)
0.205054 + 0.978751i \(0.434263\pi\)
\(920\) −26143.1 −0.936862
\(921\) 0 0
\(922\) 17196.9 0.614263
\(923\) 0 0
\(924\) 0 0
\(925\) 906.820 0.0322336
\(926\) −5226.55 −0.185481
\(927\) 0 0
\(928\) 36537.3 1.29245
\(929\) −11954.2 −0.422179 −0.211090 0.977467i \(-0.567701\pi\)
−0.211090 + 0.977467i \(0.567701\pi\)
\(930\) 0 0
\(931\) 66571.7 2.34350
\(932\) 34438.2 1.21037
\(933\) 0 0
\(934\) −1663.74 −0.0582861
\(935\) 22308.7 0.780290
\(936\) 0 0
\(937\) 42546.4 1.48338 0.741692 0.670740i \(-0.234024\pi\)
0.741692 + 0.670740i \(0.234024\pi\)
\(938\) −19364.1 −0.674051
\(939\) 0 0
\(940\) −31064.0 −1.07787
\(941\) −20665.1 −0.715903 −0.357951 0.933740i \(-0.616525\pi\)
−0.357951 + 0.933740i \(0.616525\pi\)
\(942\) 0 0
\(943\) −31605.9 −1.09144
\(944\) −5583.37 −0.192503
\(945\) 0 0
\(946\) −6632.72 −0.227958
\(947\) 9493.73 0.325771 0.162885 0.986645i \(-0.447920\pi\)
0.162885 + 0.986645i \(0.447920\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −4932.56 −0.168456
\(951\) 0 0
\(952\) −29341.3 −0.998904
\(953\) −53334.1 −1.81287 −0.906433 0.422349i \(-0.861206\pi\)
−0.906433 + 0.422349i \(0.861206\pi\)
\(954\) 0 0
\(955\) 43749.2 1.48240
\(956\) 12495.9 0.422746
\(957\) 0 0
\(958\) 28402.2 0.957864
\(959\) 6651.58 0.223974
\(960\) 0 0
\(961\) −28766.2 −0.965601
\(962\) 0 0
\(963\) 0 0
\(964\) −27456.1 −0.917326
\(965\) 24453.3 0.815730
\(966\) 0 0
\(967\) −42110.1 −1.40038 −0.700191 0.713956i \(-0.746902\pi\)
−0.700191 + 0.713956i \(0.746902\pi\)
\(968\) −17687.3 −0.587285
\(969\) 0 0
\(970\) −3093.47 −0.102397
\(971\) 13827.7 0.457006 0.228503 0.973543i \(-0.426617\pi\)
0.228503 + 0.973543i \(0.426617\pi\)
\(972\) 0 0
\(973\) −23267.4 −0.766618
\(974\) −13262.3 −0.436295
\(975\) 0 0
\(976\) −1676.38 −0.0549791
\(977\) −1133.89 −0.0371302 −0.0185651 0.999828i \(-0.505910\pi\)
−0.0185651 + 0.999828i \(0.505910\pi\)
\(978\) 0 0
\(979\) 7973.31 0.260294
\(980\) 31891.9 1.03954
\(981\) 0 0
\(982\) −21116.7 −0.686214
\(983\) −26250.2 −0.851729 −0.425865 0.904787i \(-0.640030\pi\)
−0.425865 + 0.904787i \(0.640030\pi\)
\(984\) 0 0
\(985\) 12414.0 0.401565
\(986\) 13793.7 0.445517
\(987\) 0 0
\(988\) 0 0
\(989\) −12600.3 −0.405124
\(990\) 0 0
\(991\) −29360.4 −0.941133 −0.470566 0.882365i \(-0.655950\pi\)
−0.470566 + 0.882365i \(0.655950\pi\)
\(992\) −5999.84 −0.192031
\(993\) 0 0
\(994\) −15145.7 −0.483293
\(995\) 54384.5 1.73277
\(996\) 0 0
\(997\) 16835.9 0.534803 0.267401 0.963585i \(-0.413835\pi\)
0.267401 + 0.963585i \(0.413835\pi\)
\(998\) −13663.7 −0.433384
\(999\) 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 1521.4.a.v.1.3 4
3.2 odd 2 507.4.a.m.1.2 4
13.3 even 3 117.4.g.e.100.2 8
13.9 even 3 117.4.g.e.55.2 8
13.12 even 2 1521.4.a.bb.1.2 4
39.5 even 4 507.4.b.h.337.5 8
39.8 even 4 507.4.b.h.337.4 8
39.29 odd 6 39.4.e.c.22.3 yes 8
39.35 odd 6 39.4.e.c.16.3 8
39.38 odd 2 507.4.a.i.1.3 4
156.35 even 6 624.4.q.i.289.4 8
156.107 even 6 624.4.q.i.529.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.c.16.3 8 39.35 odd 6
39.4.e.c.22.3 yes 8 39.29 odd 6
117.4.g.e.55.2 8 13.9 even 3
117.4.g.e.100.2 8 13.3 even 3
507.4.a.i.1.3 4 39.38 odd 2
507.4.a.m.1.2 4 3.2 odd 2
507.4.b.h.337.4 8 39.8 even 4
507.4.b.h.337.5 8 39.5 even 4
624.4.q.i.289.4 8 156.35 even 6
624.4.q.i.529.4 8 156.107 even 6
1521.4.a.v.1.3 4 1.1 even 1 trivial
1521.4.a.bb.1.2 4 13.12 even 2