Properties

Label 1521.4.a.bk.1.4
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.04224\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.04224 q^{2} -3.82924 q^{4} -12.0825 q^{5} +29.7373 q^{7} +24.1582 q^{8} +O(q^{10})\) \(q-2.04224 q^{2} -3.82924 q^{4} -12.0825 q^{5} +29.7373 q^{7} +24.1582 q^{8} +24.6753 q^{10} -28.0636 q^{11} -60.7308 q^{14} -18.7029 q^{16} +50.6556 q^{17} +105.148 q^{19} +46.2667 q^{20} +57.3126 q^{22} +160.592 q^{23} +20.9857 q^{25} -113.871 q^{28} -140.105 q^{29} +223.593 q^{31} -155.070 q^{32} -103.451 q^{34} -359.300 q^{35} -228.352 q^{37} -214.739 q^{38} -291.890 q^{40} +295.902 q^{41} +192.103 q^{43} +107.462 q^{44} -327.968 q^{46} -36.9300 q^{47} +541.307 q^{49} -42.8579 q^{50} -149.102 q^{53} +339.077 q^{55} +718.399 q^{56} +286.128 q^{58} -438.867 q^{59} +286.146 q^{61} -456.631 q^{62} +466.313 q^{64} -537.128 q^{67} -193.973 q^{68} +733.777 q^{70} -102.729 q^{71} -75.5209 q^{73} +466.350 q^{74} -402.639 q^{76} -834.535 q^{77} +17.5526 q^{79} +225.977 q^{80} -604.304 q^{82} -1463.08 q^{83} -612.044 q^{85} -392.322 q^{86} -677.965 q^{88} +334.905 q^{89} -614.946 q^{92} +75.4200 q^{94} -1270.45 q^{95} -748.756 q^{97} -1105.48 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 60 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} + 3140 q^{64} - 1200 q^{68} + 4380 q^{74} - 4320 q^{77} + 1100 q^{79} - 4780 q^{82} + 6340 q^{88} + 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.04224 −0.722042 −0.361021 0.932558i \(-0.617572\pi\)
−0.361021 + 0.932558i \(0.617572\pi\)
\(3\) 0 0
\(4\) −3.82924 −0.478656
\(5\) −12.0825 −1.08069 −0.540344 0.841444i \(-0.681706\pi\)
−0.540344 + 0.841444i \(0.681706\pi\)
\(6\) 0 0
\(7\) 29.7373 1.60566 0.802832 0.596206i \(-0.203326\pi\)
0.802832 + 0.596206i \(0.203326\pi\)
\(8\) 24.1582 1.06765
\(9\) 0 0
\(10\) 24.6753 0.780302
\(11\) −28.0636 −0.769226 −0.384613 0.923078i \(-0.625665\pi\)
−0.384613 + 0.923078i \(0.625665\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −60.7308 −1.15936
\(15\) 0 0
\(16\) −18.7029 −0.292233
\(17\) 50.6556 0.722693 0.361347 0.932432i \(-0.382317\pi\)
0.361347 + 0.932432i \(0.382317\pi\)
\(18\) 0 0
\(19\) 105.148 1.26962 0.634808 0.772670i \(-0.281079\pi\)
0.634808 + 0.772670i \(0.281079\pi\)
\(20\) 46.2667 0.517277
\(21\) 0 0
\(22\) 57.3126 0.555413
\(23\) 160.592 1.45590 0.727951 0.685629i \(-0.240473\pi\)
0.727951 + 0.685629i \(0.240473\pi\)
\(24\) 0 0
\(25\) 20.9857 0.167886
\(26\) 0 0
\(27\) 0 0
\(28\) −113.871 −0.768560
\(29\) −140.105 −0.897132 −0.448566 0.893750i \(-0.648065\pi\)
−0.448566 + 0.893750i \(0.648065\pi\)
\(30\) 0 0
\(31\) 223.593 1.29544 0.647718 0.761880i \(-0.275724\pi\)
0.647718 + 0.761880i \(0.275724\pi\)
\(32\) −155.070 −0.856647
\(33\) 0 0
\(34\) −103.451 −0.521815
\(35\) −359.300 −1.73522
\(36\) 0 0
\(37\) −228.352 −1.01462 −0.507308 0.861765i \(-0.669359\pi\)
−0.507308 + 0.861765i \(0.669359\pi\)
\(38\) −214.739 −0.916716
\(39\) 0 0
\(40\) −291.890 −1.15380
\(41\) 295.902 1.12713 0.563563 0.826073i \(-0.309430\pi\)
0.563563 + 0.826073i \(0.309430\pi\)
\(42\) 0 0
\(43\) 192.103 0.681291 0.340645 0.940192i \(-0.389354\pi\)
0.340645 + 0.940192i \(0.389354\pi\)
\(44\) 107.462 0.368194
\(45\) 0 0
\(46\) −327.968 −1.05122
\(47\) −36.9300 −0.114613 −0.0573063 0.998357i \(-0.518251\pi\)
−0.0573063 + 0.998357i \(0.518251\pi\)
\(48\) 0 0
\(49\) 541.307 1.57815
\(50\) −42.8579 −0.121220
\(51\) 0 0
\(52\) 0 0
\(53\) −149.102 −0.386429 −0.193214 0.981157i \(-0.561891\pi\)
−0.193214 + 0.981157i \(0.561891\pi\)
\(54\) 0 0
\(55\) 339.077 0.831293
\(56\) 718.399 1.71429
\(57\) 0 0
\(58\) 286.128 0.647767
\(59\) −438.867 −0.968400 −0.484200 0.874957i \(-0.660889\pi\)
−0.484200 + 0.874957i \(0.660889\pi\)
\(60\) 0 0
\(61\) 286.146 0.600610 0.300305 0.953843i \(-0.402912\pi\)
0.300305 + 0.953843i \(0.402912\pi\)
\(62\) −456.631 −0.935358
\(63\) 0 0
\(64\) 466.313 0.910768
\(65\) 0 0
\(66\) 0 0
\(67\) −537.128 −0.979412 −0.489706 0.871888i \(-0.662896\pi\)
−0.489706 + 0.871888i \(0.662896\pi\)
\(68\) −193.973 −0.345921
\(69\) 0 0
\(70\) 733.777 1.25290
\(71\) −102.729 −0.171713 −0.0858567 0.996307i \(-0.527363\pi\)
−0.0858567 + 0.996307i \(0.527363\pi\)
\(72\) 0 0
\(73\) −75.5209 −0.121083 −0.0605414 0.998166i \(-0.519283\pi\)
−0.0605414 + 0.998166i \(0.519283\pi\)
\(74\) 466.350 0.732596
\(75\) 0 0
\(76\) −402.639 −0.607709
\(77\) −834.535 −1.23512
\(78\) 0 0
\(79\) 17.5526 0.0249978 0.0124989 0.999922i \(-0.496021\pi\)
0.0124989 + 0.999922i \(0.496021\pi\)
\(80\) 225.977 0.315813
\(81\) 0 0
\(82\) −604.304 −0.813831
\(83\) −1463.08 −1.93487 −0.967434 0.253122i \(-0.918543\pi\)
−0.967434 + 0.253122i \(0.918543\pi\)
\(84\) 0 0
\(85\) −612.044 −0.781005
\(86\) −392.322 −0.491920
\(87\) 0 0
\(88\) −677.965 −0.821265
\(89\) 334.905 0.398875 0.199438 0.979911i \(-0.436089\pi\)
0.199438 + 0.979911i \(0.436089\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −614.946 −0.696876
\(93\) 0 0
\(94\) 75.4200 0.0827551
\(95\) −1270.45 −1.37206
\(96\) 0 0
\(97\) −748.756 −0.783760 −0.391880 0.920016i \(-0.628175\pi\)
−0.391880 + 0.920016i \(0.628175\pi\)
\(98\) −1105.48 −1.13949
\(99\) 0 0
\(100\) −80.3594 −0.0803594
\(101\) 784.002 0.772387 0.386194 0.922418i \(-0.373790\pi\)
0.386194 + 0.922418i \(0.373790\pi\)
\(102\) 0 0
\(103\) 396.040 0.378864 0.189432 0.981894i \(-0.439335\pi\)
0.189432 + 0.981894i \(0.439335\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 304.502 0.279018
\(107\) 1436.59 1.29795 0.648974 0.760810i \(-0.275198\pi\)
0.648974 + 0.760810i \(0.275198\pi\)
\(108\) 0 0
\(109\) −1977.92 −1.73807 −0.869037 0.494746i \(-0.835261\pi\)
−0.869037 + 0.494746i \(0.835261\pi\)
\(110\) −692.477 −0.600228
\(111\) 0 0
\(112\) −556.175 −0.469228
\(113\) −122.405 −0.101902 −0.0509509 0.998701i \(-0.516225\pi\)
−0.0509509 + 0.998701i \(0.516225\pi\)
\(114\) 0 0
\(115\) −1940.35 −1.57338
\(116\) 536.496 0.429417
\(117\) 0 0
\(118\) 896.273 0.699225
\(119\) 1506.36 1.16040
\(120\) 0 0
\(121\) −543.436 −0.408291
\(122\) −584.379 −0.433665
\(123\) 0 0
\(124\) −856.192 −0.620067
\(125\) 1256.75 0.899256
\(126\) 0 0
\(127\) 2309.61 1.61374 0.806868 0.590731i \(-0.201161\pi\)
0.806868 + 0.590731i \(0.201161\pi\)
\(128\) 288.232 0.199034
\(129\) 0 0
\(130\) 0 0
\(131\) 1444.26 0.963250 0.481625 0.876377i \(-0.340047\pi\)
0.481625 + 0.876377i \(0.340047\pi\)
\(132\) 0 0
\(133\) 3126.83 2.03858
\(134\) 1096.94 0.707176
\(135\) 0 0
\(136\) 1223.75 0.771584
\(137\) 735.918 0.458932 0.229466 0.973317i \(-0.426302\pi\)
0.229466 + 0.973317i \(0.426302\pi\)
\(138\) 0 0
\(139\) −1505.14 −0.918449 −0.459225 0.888320i \(-0.651873\pi\)
−0.459225 + 0.888320i \(0.651873\pi\)
\(140\) 1375.85 0.830573
\(141\) 0 0
\(142\) 209.797 0.123984
\(143\) 0 0
\(144\) 0 0
\(145\) 1692.81 0.969520
\(146\) 154.232 0.0874269
\(147\) 0 0
\(148\) 874.415 0.485652
\(149\) −427.843 −0.235237 −0.117618 0.993059i \(-0.537526\pi\)
−0.117618 + 0.993059i \(0.537526\pi\)
\(150\) 0 0
\(151\) −1601.83 −0.863278 −0.431639 0.902046i \(-0.642065\pi\)
−0.431639 + 0.902046i \(0.642065\pi\)
\(152\) 2540.20 1.35551
\(153\) 0 0
\(154\) 1704.32 0.891807
\(155\) −2701.55 −1.39996
\(156\) 0 0
\(157\) −730.346 −0.371261 −0.185631 0.982620i \(-0.559433\pi\)
−0.185631 + 0.982620i \(0.559433\pi\)
\(158\) −35.8467 −0.0180495
\(159\) 0 0
\(160\) 1873.62 0.925767
\(161\) 4775.58 2.33769
\(162\) 0 0
\(163\) 1898.36 0.912215 0.456107 0.889925i \(-0.349243\pi\)
0.456107 + 0.889925i \(0.349243\pi\)
\(164\) −1133.08 −0.539505
\(165\) 0 0
\(166\) 2987.97 1.39706
\(167\) −1427.50 −0.661457 −0.330729 0.943726i \(-0.607294\pi\)
−0.330729 + 0.943726i \(0.607294\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 1249.94 0.563919
\(171\) 0 0
\(172\) −735.611 −0.326104
\(173\) 2044.40 0.898454 0.449227 0.893418i \(-0.351699\pi\)
0.449227 + 0.893418i \(0.351699\pi\)
\(174\) 0 0
\(175\) 624.058 0.269568
\(176\) 524.871 0.224793
\(177\) 0 0
\(178\) −683.958 −0.288004
\(179\) 3889.72 1.62420 0.812098 0.583521i \(-0.198325\pi\)
0.812098 + 0.583521i \(0.198325\pi\)
\(180\) 0 0
\(181\) −2477.02 −1.01721 −0.508606 0.861000i \(-0.669839\pi\)
−0.508606 + 0.861000i \(0.669839\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 3879.61 1.55440
\(185\) 2759.05 1.09648
\(186\) 0 0
\(187\) −1421.58 −0.555914
\(188\) 141.414 0.0548600
\(189\) 0 0
\(190\) 2594.57 0.990683
\(191\) −2276.81 −0.862535 −0.431267 0.902224i \(-0.641933\pi\)
−0.431267 + 0.902224i \(0.641933\pi\)
\(192\) 0 0
\(193\) 3922.42 1.46291 0.731456 0.681888i \(-0.238841\pi\)
0.731456 + 0.681888i \(0.238841\pi\)
\(194\) 1529.14 0.565907
\(195\) 0 0
\(196\) −2072.80 −0.755392
\(197\) 5063.23 1.83117 0.915584 0.402128i \(-0.131729\pi\)
0.915584 + 0.402128i \(0.131729\pi\)
\(198\) 0 0
\(199\) 3270.06 1.16487 0.582433 0.812879i \(-0.302101\pi\)
0.582433 + 0.812879i \(0.302101\pi\)
\(200\) 506.977 0.179243
\(201\) 0 0
\(202\) −1601.12 −0.557696
\(203\) −4166.34 −1.44049
\(204\) 0 0
\(205\) −3575.22 −1.21807
\(206\) −808.810 −0.273555
\(207\) 0 0
\(208\) 0 0
\(209\) −2950.84 −0.976622
\(210\) 0 0
\(211\) −2812.18 −0.917527 −0.458764 0.888558i \(-0.651708\pi\)
−0.458764 + 0.888558i \(0.651708\pi\)
\(212\) 570.948 0.184966
\(213\) 0 0
\(214\) −2933.87 −0.937173
\(215\) −2321.08 −0.736262
\(216\) 0 0
\(217\) 6649.05 2.08003
\(218\) 4039.39 1.25496
\(219\) 0 0
\(220\) −1298.41 −0.397903
\(221\) 0 0
\(222\) 0 0
\(223\) −917.736 −0.275588 −0.137794 0.990461i \(-0.544001\pi\)
−0.137794 + 0.990461i \(0.544001\pi\)
\(224\) −4611.35 −1.37549
\(225\) 0 0
\(226\) 249.981 0.0735774
\(227\) 1336.39 0.390746 0.195373 0.980729i \(-0.437408\pi\)
0.195373 + 0.980729i \(0.437408\pi\)
\(228\) 0 0
\(229\) 164.820 0.0475617 0.0237808 0.999717i \(-0.492430\pi\)
0.0237808 + 0.999717i \(0.492430\pi\)
\(230\) 3962.66 1.13604
\(231\) 0 0
\(232\) −3384.68 −0.957824
\(233\) 4243.42 1.19312 0.596558 0.802570i \(-0.296535\pi\)
0.596558 + 0.802570i \(0.296535\pi\)
\(234\) 0 0
\(235\) 446.205 0.123860
\(236\) 1680.53 0.463530
\(237\) 0 0
\(238\) −3076.35 −0.837859
\(239\) 2491.07 0.674200 0.337100 0.941469i \(-0.390554\pi\)
0.337100 + 0.941469i \(0.390554\pi\)
\(240\) 0 0
\(241\) 2917.40 0.779776 0.389888 0.920862i \(-0.372514\pi\)
0.389888 + 0.920862i \(0.372514\pi\)
\(242\) 1109.83 0.294803
\(243\) 0 0
\(244\) −1095.72 −0.287485
\(245\) −6540.32 −1.70549
\(246\) 0 0
\(247\) 0 0
\(248\) 5401.60 1.38307
\(249\) 0 0
\(250\) −2566.58 −0.649300
\(251\) 1313.88 0.330403 0.165202 0.986260i \(-0.447173\pi\)
0.165202 + 0.986260i \(0.447173\pi\)
\(252\) 0 0
\(253\) −4506.79 −1.11992
\(254\) −4716.78 −1.16519
\(255\) 0 0
\(256\) −4319.15 −1.05448
\(257\) 987.582 0.239703 0.119851 0.992792i \(-0.461758\pi\)
0.119851 + 0.992792i \(0.461758\pi\)
\(258\) 0 0
\(259\) −6790.57 −1.62913
\(260\) 0 0
\(261\) 0 0
\(262\) −2949.53 −0.695507
\(263\) 6986.45 1.63803 0.819017 0.573769i \(-0.194519\pi\)
0.819017 + 0.573769i \(0.194519\pi\)
\(264\) 0 0
\(265\) 1801.52 0.417609
\(266\) −6385.74 −1.47194
\(267\) 0 0
\(268\) 2056.79 0.468801
\(269\) 5904.34 1.33827 0.669134 0.743142i \(-0.266665\pi\)
0.669134 + 0.743142i \(0.266665\pi\)
\(270\) 0 0
\(271\) −2131.54 −0.477793 −0.238897 0.971045i \(-0.576786\pi\)
−0.238897 + 0.971045i \(0.576786\pi\)
\(272\) −947.408 −0.211195
\(273\) 0 0
\(274\) −1502.92 −0.331368
\(275\) −588.934 −0.129142
\(276\) 0 0
\(277\) 4032.41 0.874673 0.437336 0.899298i \(-0.355922\pi\)
0.437336 + 0.899298i \(0.355922\pi\)
\(278\) 3073.86 0.663159
\(279\) 0 0
\(280\) −8680.03 −1.85261
\(281\) 2298.29 0.487916 0.243958 0.969786i \(-0.421554\pi\)
0.243958 + 0.969786i \(0.421554\pi\)
\(282\) 0 0
\(283\) 6656.80 1.39825 0.699127 0.714998i \(-0.253572\pi\)
0.699127 + 0.714998i \(0.253572\pi\)
\(284\) 393.373 0.0821916
\(285\) 0 0
\(286\) 0 0
\(287\) 8799.33 1.80978
\(288\) 0 0
\(289\) −2347.01 −0.477715
\(290\) −3457.13 −0.700034
\(291\) 0 0
\(292\) 289.188 0.0579570
\(293\) −7466.99 −1.48883 −0.744413 0.667719i \(-0.767271\pi\)
−0.744413 + 0.667719i \(0.767271\pi\)
\(294\) 0 0
\(295\) 5302.59 1.04654
\(296\) −5516.57 −1.08326
\(297\) 0 0
\(298\) 873.759 0.169851
\(299\) 0 0
\(300\) 0 0
\(301\) 5712.64 1.09392
\(302\) 3271.32 0.623323
\(303\) 0 0
\(304\) −1966.58 −0.371024
\(305\) −3457.34 −0.649071
\(306\) 0 0
\(307\) −3965.99 −0.737299 −0.368650 0.929568i \(-0.620180\pi\)
−0.368650 + 0.929568i \(0.620180\pi\)
\(308\) 3195.64 0.591196
\(309\) 0 0
\(310\) 5517.23 1.01083
\(311\) −7372.29 −1.34419 −0.672097 0.740463i \(-0.734606\pi\)
−0.672097 + 0.740463i \(0.734606\pi\)
\(312\) 0 0
\(313\) 8249.55 1.48975 0.744875 0.667204i \(-0.232509\pi\)
0.744875 + 0.667204i \(0.232509\pi\)
\(314\) 1491.54 0.268066
\(315\) 0 0
\(316\) −67.2133 −0.0119653
\(317\) 5575.26 0.987817 0.493909 0.869514i \(-0.335568\pi\)
0.493909 + 0.869514i \(0.335568\pi\)
\(318\) 0 0
\(319\) 3931.85 0.690098
\(320\) −5634.21 −0.984256
\(321\) 0 0
\(322\) −9752.88 −1.68791
\(323\) 5326.35 0.917543
\(324\) 0 0
\(325\) 0 0
\(326\) −3876.91 −0.658657
\(327\) 0 0
\(328\) 7148.46 1.20338
\(329\) −1098.20 −0.184029
\(330\) 0 0
\(331\) −4157.36 −0.690361 −0.345180 0.938536i \(-0.612182\pi\)
−0.345180 + 0.938536i \(0.612182\pi\)
\(332\) 5602.50 0.926136
\(333\) 0 0
\(334\) 2915.30 0.477600
\(335\) 6489.82 1.05844
\(336\) 0 0
\(337\) 3225.18 0.521326 0.260663 0.965430i \(-0.416059\pi\)
0.260663 + 0.965430i \(0.416059\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 2343.67 0.373833
\(341\) −6274.82 −0.996483
\(342\) 0 0
\(343\) 5897.11 0.928321
\(344\) 4640.87 0.727381
\(345\) 0 0
\(346\) −4175.15 −0.648721
\(347\) −3290.49 −0.509057 −0.254529 0.967065i \(-0.581920\pi\)
−0.254529 + 0.967065i \(0.581920\pi\)
\(348\) 0 0
\(349\) −4491.52 −0.688899 −0.344449 0.938805i \(-0.611934\pi\)
−0.344449 + 0.938805i \(0.611934\pi\)
\(350\) −1274.48 −0.194639
\(351\) 0 0
\(352\) 4351.81 0.658955
\(353\) −5897.88 −0.889270 −0.444635 0.895712i \(-0.646667\pi\)
−0.444635 + 0.895712i \(0.646667\pi\)
\(354\) 0 0
\(355\) 1241.21 0.185569
\(356\) −1282.43 −0.190924
\(357\) 0 0
\(358\) −7943.75 −1.17274
\(359\) 9277.20 1.36388 0.681938 0.731410i \(-0.261137\pi\)
0.681938 + 0.731410i \(0.261137\pi\)
\(360\) 0 0
\(361\) 4197.19 0.611924
\(362\) 5058.67 0.734469
\(363\) 0 0
\(364\) 0 0
\(365\) 912.477 0.130853
\(366\) 0 0
\(367\) −6574.36 −0.935092 −0.467546 0.883969i \(-0.654862\pi\)
−0.467546 + 0.883969i \(0.654862\pi\)
\(368\) −3003.54 −0.425463
\(369\) 0 0
\(370\) −5634.65 −0.791707
\(371\) −4433.89 −0.620474
\(372\) 0 0
\(373\) −5345.55 −0.742043 −0.371021 0.928624i \(-0.620992\pi\)
−0.371021 + 0.928624i \(0.620992\pi\)
\(374\) 2903.20 0.401393
\(375\) 0 0
\(376\) −892.162 −0.122366
\(377\) 0 0
\(378\) 0 0
\(379\) −1038.51 −0.140751 −0.0703757 0.997521i \(-0.522420\pi\)
−0.0703757 + 0.997521i \(0.522420\pi\)
\(380\) 4864.87 0.656743
\(381\) 0 0
\(382\) 4649.80 0.622786
\(383\) −6749.19 −0.900437 −0.450219 0.892918i \(-0.648654\pi\)
−0.450219 + 0.892918i \(0.648654\pi\)
\(384\) 0 0
\(385\) 10083.2 1.33478
\(386\) −8010.54 −1.05628
\(387\) 0 0
\(388\) 2867.17 0.375151
\(389\) 1246.11 0.162417 0.0812083 0.996697i \(-0.474122\pi\)
0.0812083 + 0.996697i \(0.474122\pi\)
\(390\) 0 0
\(391\) 8134.89 1.05217
\(392\) 13077.0 1.68492
\(393\) 0 0
\(394\) −10340.3 −1.32218
\(395\) −212.079 −0.0270148
\(396\) 0 0
\(397\) 8355.69 1.05632 0.528161 0.849144i \(-0.322882\pi\)
0.528161 + 0.849144i \(0.322882\pi\)
\(398\) −6678.25 −0.841082
\(399\) 0 0
\(400\) −392.494 −0.0490618
\(401\) −3283.66 −0.408923 −0.204461 0.978875i \(-0.565544\pi\)
−0.204461 + 0.978875i \(0.565544\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −3002.14 −0.369708
\(405\) 0 0
\(406\) 8508.68 1.04010
\(407\) 6408.37 0.780470
\(408\) 0 0
\(409\) 10928.9 1.32127 0.660636 0.750706i \(-0.270287\pi\)
0.660636 + 0.750706i \(0.270287\pi\)
\(410\) 7301.47 0.879497
\(411\) 0 0
\(412\) −1516.53 −0.181345
\(413\) −13050.7 −1.55492
\(414\) 0 0
\(415\) 17677.6 2.09099
\(416\) 0 0
\(417\) 0 0
\(418\) 6026.33 0.705162
\(419\) −7302.94 −0.851485 −0.425742 0.904844i \(-0.639987\pi\)
−0.425742 + 0.904844i \(0.639987\pi\)
\(420\) 0 0
\(421\) 7580.99 0.877612 0.438806 0.898582i \(-0.355401\pi\)
0.438806 + 0.898582i \(0.355401\pi\)
\(422\) 5743.15 0.662493
\(423\) 0 0
\(424\) −3602.03 −0.412571
\(425\) 1063.04 0.121330
\(426\) 0 0
\(427\) 8509.19 0.964377
\(428\) −5501.06 −0.621270
\(429\) 0 0
\(430\) 4740.21 0.531612
\(431\) 10056.7 1.12394 0.561968 0.827159i \(-0.310044\pi\)
0.561968 + 0.827159i \(0.310044\pi\)
\(432\) 0 0
\(433\) 2733.38 0.303367 0.151683 0.988429i \(-0.451531\pi\)
0.151683 + 0.988429i \(0.451531\pi\)
\(434\) −13579.0 −1.50187
\(435\) 0 0
\(436\) 7573.93 0.831939
\(437\) 16886.0 1.84844
\(438\) 0 0
\(439\) 6744.23 0.733223 0.366611 0.930374i \(-0.380518\pi\)
0.366611 + 0.930374i \(0.380518\pi\)
\(440\) 8191.48 0.887531
\(441\) 0 0
\(442\) 0 0
\(443\) −8655.69 −0.928317 −0.464158 0.885752i \(-0.653643\pi\)
−0.464158 + 0.885752i \(0.653643\pi\)
\(444\) 0 0
\(445\) −4046.48 −0.431059
\(446\) 1874.24 0.198986
\(447\) 0 0
\(448\) 13866.9 1.46239
\(449\) 6522.46 0.685555 0.342777 0.939417i \(-0.388632\pi\)
0.342777 + 0.939417i \(0.388632\pi\)
\(450\) 0 0
\(451\) −8304.07 −0.867014
\(452\) 468.719 0.0487759
\(453\) 0 0
\(454\) −2729.23 −0.282135
\(455\) 0 0
\(456\) 0 0
\(457\) 1551.23 0.158782 0.0793909 0.996844i \(-0.474702\pi\)
0.0793909 + 0.996844i \(0.474702\pi\)
\(458\) −336.603 −0.0343415
\(459\) 0 0
\(460\) 7430.06 0.753105
\(461\) 7766.25 0.784621 0.392310 0.919833i \(-0.371676\pi\)
0.392310 + 0.919833i \(0.371676\pi\)
\(462\) 0 0
\(463\) 2004.52 0.201205 0.100603 0.994927i \(-0.467923\pi\)
0.100603 + 0.994927i \(0.467923\pi\)
\(464\) 2620.37 0.262172
\(465\) 0 0
\(466\) −8666.10 −0.861479
\(467\) −18674.3 −1.85042 −0.925209 0.379458i \(-0.876111\pi\)
−0.925209 + 0.379458i \(0.876111\pi\)
\(468\) 0 0
\(469\) −15972.7 −1.57261
\(470\) −911.259 −0.0894324
\(471\) 0 0
\(472\) −10602.2 −1.03391
\(473\) −5391.11 −0.524067
\(474\) 0 0
\(475\) 2206.61 0.213150
\(476\) −5768.22 −0.555433
\(477\) 0 0
\(478\) −5087.36 −0.486800
\(479\) −9313.02 −0.888357 −0.444178 0.895938i \(-0.646504\pi\)
−0.444178 + 0.895938i \(0.646504\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −5958.03 −0.563031
\(483\) 0 0
\(484\) 2080.95 0.195431
\(485\) 9046.81 0.846999
\(486\) 0 0
\(487\) 3536.80 0.329092 0.164546 0.986369i \(-0.447384\pi\)
0.164546 + 0.986369i \(0.447384\pi\)
\(488\) 6912.76 0.641241
\(489\) 0 0
\(490\) 13356.9 1.23144
\(491\) 3361.78 0.308992 0.154496 0.987993i \(-0.450625\pi\)
0.154496 + 0.987993i \(0.450625\pi\)
\(492\) 0 0
\(493\) −7097.10 −0.648351
\(494\) 0 0
\(495\) 0 0
\(496\) −4181.84 −0.378569
\(497\) −3054.87 −0.275714
\(498\) 0 0
\(499\) −4027.43 −0.361308 −0.180654 0.983547i \(-0.557821\pi\)
−0.180654 + 0.983547i \(0.557821\pi\)
\(500\) −4812.40 −0.430434
\(501\) 0 0
\(502\) −2683.26 −0.238565
\(503\) −1766.67 −0.156604 −0.0783022 0.996930i \(-0.524950\pi\)
−0.0783022 + 0.996930i \(0.524950\pi\)
\(504\) 0 0
\(505\) −9472.67 −0.834709
\(506\) 9203.96 0.808628
\(507\) 0 0
\(508\) −8844.05 −0.772424
\(509\) −6816.27 −0.593567 −0.296784 0.954945i \(-0.595914\pi\)
−0.296784 + 0.954945i \(0.595914\pi\)
\(510\) 0 0
\(511\) −2245.79 −0.194418
\(512\) 6514.89 0.562344
\(513\) 0 0
\(514\) −2016.88 −0.173076
\(515\) −4785.13 −0.409433
\(516\) 0 0
\(517\) 1036.39 0.0881630
\(518\) 13868.0 1.17630
\(519\) 0 0
\(520\) 0 0
\(521\) 5442.27 0.457640 0.228820 0.973469i \(-0.426513\pi\)
0.228820 + 0.973469i \(0.426513\pi\)
\(522\) 0 0
\(523\) 20728.5 1.73307 0.866535 0.499117i \(-0.166342\pi\)
0.866535 + 0.499117i \(0.166342\pi\)
\(524\) −5530.43 −0.461065
\(525\) 0 0
\(526\) −14268.0 −1.18273
\(527\) 11326.2 0.936202
\(528\) 0 0
\(529\) 13622.8 1.11965
\(530\) −3679.13 −0.301531
\(531\) 0 0
\(532\) −11973.4 −0.975775
\(533\) 0 0
\(534\) 0 0
\(535\) −17357.5 −1.40268
\(536\) −12976.0 −1.04567
\(537\) 0 0
\(538\) −12058.1 −0.966285
\(539\) −15191.0 −1.21396
\(540\) 0 0
\(541\) 8577.44 0.681651 0.340825 0.940127i \(-0.389294\pi\)
0.340825 + 0.940127i \(0.389294\pi\)
\(542\) 4353.13 0.344987
\(543\) 0 0
\(544\) −7855.14 −0.619093
\(545\) 23898.1 1.87832
\(546\) 0 0
\(547\) 8723.99 0.681921 0.340961 0.940078i \(-0.389248\pi\)
0.340961 + 0.940078i \(0.389248\pi\)
\(548\) −2818.01 −0.219671
\(549\) 0 0
\(550\) 1202.75 0.0932459
\(551\) −14731.8 −1.13901
\(552\) 0 0
\(553\) 521.968 0.0401380
\(554\) −8235.17 −0.631550
\(555\) 0 0
\(556\) 5763.56 0.439621
\(557\) 965.006 0.0734087 0.0367043 0.999326i \(-0.488314\pi\)
0.0367043 + 0.999326i \(0.488314\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 6719.95 0.507089
\(561\) 0 0
\(562\) −4693.66 −0.352296
\(563\) 14605.2 1.09331 0.546657 0.837357i \(-0.315900\pi\)
0.546657 + 0.837357i \(0.315900\pi\)
\(564\) 0 0
\(565\) 1478.95 0.110124
\(566\) −13594.8 −1.00960
\(567\) 0 0
\(568\) −2481.74 −0.183330
\(569\) −7802.48 −0.574863 −0.287432 0.957801i \(-0.592801\pi\)
−0.287432 + 0.957801i \(0.592801\pi\)
\(570\) 0 0
\(571\) 11988.2 0.878618 0.439309 0.898336i \(-0.355223\pi\)
0.439309 + 0.898336i \(0.355223\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −17970.4 −1.30674
\(575\) 3370.14 0.244425
\(576\) 0 0
\(577\) 5576.90 0.402374 0.201187 0.979553i \(-0.435520\pi\)
0.201187 + 0.979553i \(0.435520\pi\)
\(578\) 4793.17 0.344930
\(579\) 0 0
\(580\) −6482.19 −0.464066
\(581\) −43508.1 −3.10675
\(582\) 0 0
\(583\) 4184.33 0.297251
\(584\) −1824.45 −0.129274
\(585\) 0 0
\(586\) 15249.4 1.07499
\(587\) −26754.0 −1.88119 −0.940593 0.339535i \(-0.889730\pi\)
−0.940593 + 0.339535i \(0.889730\pi\)
\(588\) 0 0
\(589\) 23510.5 1.64471
\(590\) −10829.2 −0.755644
\(591\) 0 0
\(592\) 4270.85 0.296505
\(593\) 3589.40 0.248565 0.124283 0.992247i \(-0.460337\pi\)
0.124283 + 0.992247i \(0.460337\pi\)
\(594\) 0 0
\(595\) −18200.5 −1.25403
\(596\) 1638.32 0.112597
\(597\) 0 0
\(598\) 0 0
\(599\) 7462.78 0.509050 0.254525 0.967066i \(-0.418081\pi\)
0.254525 + 0.967066i \(0.418081\pi\)
\(600\) 0 0
\(601\) −16511.0 −1.12063 −0.560316 0.828279i \(-0.689320\pi\)
−0.560316 + 0.828279i \(0.689320\pi\)
\(602\) −11666.6 −0.789858
\(603\) 0 0
\(604\) 6133.80 0.413213
\(605\) 6566.04 0.441235
\(606\) 0 0
\(607\) −11953.6 −0.799309 −0.399654 0.916666i \(-0.630870\pi\)
−0.399654 + 0.916666i \(0.630870\pi\)
\(608\) −16305.3 −1.08761
\(609\) 0 0
\(610\) 7060.73 0.468657
\(611\) 0 0
\(612\) 0 0
\(613\) −4575.65 −0.301482 −0.150741 0.988573i \(-0.548166\pi\)
−0.150741 + 0.988573i \(0.548166\pi\)
\(614\) 8099.51 0.532361
\(615\) 0 0
\(616\) −20160.9 −1.31868
\(617\) 19231.0 1.25480 0.627400 0.778697i \(-0.284119\pi\)
0.627400 + 0.778697i \(0.284119\pi\)
\(618\) 0 0
\(619\) −11715.6 −0.760727 −0.380363 0.924837i \(-0.624201\pi\)
−0.380363 + 0.924837i \(0.624201\pi\)
\(620\) 10344.9 0.670099
\(621\) 0 0
\(622\) 15056.0 0.970564
\(623\) 9959.17 0.640459
\(624\) 0 0
\(625\) −17807.8 −1.13970
\(626\) −16847.6 −1.07566
\(627\) 0 0
\(628\) 2796.68 0.177706
\(629\) −11567.3 −0.733256
\(630\) 0 0
\(631\) −8780.09 −0.553930 −0.276965 0.960880i \(-0.589329\pi\)
−0.276965 + 0.960880i \(0.589329\pi\)
\(632\) 424.040 0.0266889
\(633\) 0 0
\(634\) −11386.0 −0.713245
\(635\) −27905.7 −1.74395
\(636\) 0 0
\(637\) 0 0
\(638\) −8029.78 −0.498279
\(639\) 0 0
\(640\) −3482.55 −0.215094
\(641\) −24991.7 −1.53996 −0.769980 0.638068i \(-0.779734\pi\)
−0.769980 + 0.638068i \(0.779734\pi\)
\(642\) 0 0
\(643\) −2353.86 −0.144365 −0.0721827 0.997391i \(-0.522996\pi\)
−0.0721827 + 0.997391i \(0.522996\pi\)
\(644\) −18286.8 −1.11895
\(645\) 0 0
\(646\) −10877.7 −0.662504
\(647\) 5910.80 0.359162 0.179581 0.983743i \(-0.442526\pi\)
0.179581 + 0.983743i \(0.442526\pi\)
\(648\) 0 0
\(649\) 12316.2 0.744918
\(650\) 0 0
\(651\) 0 0
\(652\) −7269.28 −0.436637
\(653\) −5924.34 −0.355034 −0.177517 0.984118i \(-0.556806\pi\)
−0.177517 + 0.984118i \(0.556806\pi\)
\(654\) 0 0
\(655\) −17450.2 −1.04097
\(656\) −5534.23 −0.329383
\(657\) 0 0
\(658\) 2242.79 0.132877
\(659\) 12839.5 0.758964 0.379482 0.925199i \(-0.376102\pi\)
0.379482 + 0.925199i \(0.376102\pi\)
\(660\) 0 0
\(661\) 10265.5 0.604057 0.302028 0.953299i \(-0.402336\pi\)
0.302028 + 0.953299i \(0.402336\pi\)
\(662\) 8490.35 0.498469
\(663\) 0 0
\(664\) −35345.4 −2.06577
\(665\) −37779.8 −2.20306
\(666\) 0 0
\(667\) −22499.7 −1.30614
\(668\) 5466.25 0.316610
\(669\) 0 0
\(670\) −13253.8 −0.764236
\(671\) −8030.27 −0.462004
\(672\) 0 0
\(673\) 9862.82 0.564909 0.282454 0.959281i \(-0.408851\pi\)
0.282454 + 0.959281i \(0.408851\pi\)
\(674\) −6586.61 −0.376419
\(675\) 0 0
\(676\) 0 0
\(677\) −32615.5 −1.85158 −0.925788 0.378043i \(-0.876597\pi\)
−0.925788 + 0.378043i \(0.876597\pi\)
\(678\) 0 0
\(679\) −22266.0 −1.25845
\(680\) −14785.9 −0.833841
\(681\) 0 0
\(682\) 12814.7 0.719502
\(683\) 21627.4 1.21164 0.605820 0.795602i \(-0.292845\pi\)
0.605820 + 0.795602i \(0.292845\pi\)
\(684\) 0 0
\(685\) −8891.70 −0.495962
\(686\) −12043.3 −0.670286
\(687\) 0 0
\(688\) −3592.90 −0.199096
\(689\) 0 0
\(690\) 0 0
\(691\) 14233.1 0.783580 0.391790 0.920055i \(-0.371856\pi\)
0.391790 + 0.920055i \(0.371856\pi\)
\(692\) −7828.49 −0.430050
\(693\) 0 0
\(694\) 6719.98 0.367561
\(695\) 18185.8 0.992557
\(696\) 0 0
\(697\) 14989.1 0.814566
\(698\) 9172.78 0.497414
\(699\) 0 0
\(700\) −2389.67 −0.129030
\(701\) 28747.0 1.54887 0.774437 0.632651i \(-0.218033\pi\)
0.774437 + 0.632651i \(0.218033\pi\)
\(702\) 0 0
\(703\) −24010.8 −1.28817
\(704\) −13086.4 −0.700586
\(705\) 0 0
\(706\) 12044.9 0.642090
\(707\) 23314.1 1.24019
\(708\) 0 0
\(709\) −1818.65 −0.0963339 −0.0481670 0.998839i \(-0.515338\pi\)
−0.0481670 + 0.998839i \(0.515338\pi\)
\(710\) −2534.86 −0.133988
\(711\) 0 0
\(712\) 8090.70 0.425859
\(713\) 35907.3 1.88603
\(714\) 0 0
\(715\) 0 0
\(716\) −14894.7 −0.777431
\(717\) 0 0
\(718\) −18946.3 −0.984776
\(719\) −26141.8 −1.35595 −0.677973 0.735087i \(-0.737141\pi\)
−0.677973 + 0.735087i \(0.737141\pi\)
\(720\) 0 0
\(721\) 11777.2 0.608328
\(722\) −8571.68 −0.441835
\(723\) 0 0
\(724\) 9485.10 0.486894
\(725\) −2940.20 −0.150616
\(726\) 0 0
\(727\) −1340.10 −0.0683652 −0.0341826 0.999416i \(-0.510883\pi\)
−0.0341826 + 0.999416i \(0.510883\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1863.50 −0.0944811
\(731\) 9731.11 0.492364
\(732\) 0 0
\(733\) −32517.1 −1.63854 −0.819269 0.573409i \(-0.805620\pi\)
−0.819269 + 0.573409i \(0.805620\pi\)
\(734\) 13426.4 0.675176
\(735\) 0 0
\(736\) −24903.0 −1.24719
\(737\) 15073.7 0.753389
\(738\) 0 0
\(739\) −22170.5 −1.10359 −0.551797 0.833978i \(-0.686058\pi\)
−0.551797 + 0.833978i \(0.686058\pi\)
\(740\) −10565.1 −0.524838
\(741\) 0 0
\(742\) 9055.07 0.448008
\(743\) 30387.8 1.50043 0.750216 0.661193i \(-0.229949\pi\)
0.750216 + 0.661193i \(0.229949\pi\)
\(744\) 0 0
\(745\) 5169.39 0.254217
\(746\) 10916.9 0.535786
\(747\) 0 0
\(748\) 5443.57 0.266092
\(749\) 42720.3 2.08407
\(750\) 0 0
\(751\) −16898.3 −0.821077 −0.410539 0.911843i \(-0.634659\pi\)
−0.410539 + 0.911843i \(0.634659\pi\)
\(752\) 690.699 0.0334936
\(753\) 0 0
\(754\) 0 0
\(755\) 19354.0 0.932934
\(756\) 0 0
\(757\) 32925.8 1.58086 0.790428 0.612555i \(-0.209858\pi\)
0.790428 + 0.612555i \(0.209858\pi\)
\(758\) 2120.89 0.101628
\(759\) 0 0
\(760\) −30691.8 −1.46488
\(761\) 16792.0 0.799882 0.399941 0.916541i \(-0.369031\pi\)
0.399941 + 0.916541i \(0.369031\pi\)
\(762\) 0 0
\(763\) −58817.9 −2.79076
\(764\) 8718.46 0.412857
\(765\) 0 0
\(766\) 13783.5 0.650153
\(767\) 0 0
\(768\) 0 0
\(769\) −1541.57 −0.0722891 −0.0361445 0.999347i \(-0.511508\pi\)
−0.0361445 + 0.999347i \(0.511508\pi\)
\(770\) −20592.4 −0.963765
\(771\) 0 0
\(772\) −15019.9 −0.700231
\(773\) 38071.2 1.77144 0.885721 0.464218i \(-0.153665\pi\)
0.885721 + 0.464218i \(0.153665\pi\)
\(774\) 0 0
\(775\) 4692.26 0.217485
\(776\) −18088.6 −0.836782
\(777\) 0 0
\(778\) −2544.85 −0.117272
\(779\) 31113.6 1.43102
\(780\) 0 0
\(781\) 2882.93 0.132086
\(782\) −16613.4 −0.759712
\(783\) 0 0
\(784\) −10124.0 −0.461189
\(785\) 8824.38 0.401217
\(786\) 0 0
\(787\) 20049.8 0.908129 0.454065 0.890969i \(-0.349973\pi\)
0.454065 + 0.890969i \(0.349973\pi\)
\(788\) −19388.3 −0.876498
\(789\) 0 0
\(790\) 433.117 0.0195058
\(791\) −3640.00 −0.163620
\(792\) 0 0
\(793\) 0 0
\(794\) −17064.4 −0.762709
\(795\) 0 0
\(796\) −12521.9 −0.557570
\(797\) 22401.9 0.995627 0.497813 0.867284i \(-0.334136\pi\)
0.497813 + 0.867284i \(0.334136\pi\)
\(798\) 0 0
\(799\) −1870.71 −0.0828298
\(800\) −3254.24 −0.143819
\(801\) 0 0
\(802\) 6706.02 0.295259
\(803\) 2119.39 0.0931401
\(804\) 0 0
\(805\) −57700.7 −2.52631
\(806\) 0 0
\(807\) 0 0
\(808\) 18940.1 0.824640
\(809\) 41966.4 1.82381 0.911903 0.410406i \(-0.134613\pi\)
0.911903 + 0.410406i \(0.134613\pi\)
\(810\) 0 0
\(811\) −13029.1 −0.564133 −0.282067 0.959395i \(-0.591020\pi\)
−0.282067 + 0.959395i \(0.591020\pi\)
\(812\) 15953.9 0.689500
\(813\) 0 0
\(814\) −13087.4 −0.563532
\(815\) −22936.8 −0.985819
\(816\) 0 0
\(817\) 20199.4 0.864977
\(818\) −22319.5 −0.954014
\(819\) 0 0
\(820\) 13690.4 0.583036
\(821\) −8898.14 −0.378255 −0.189127 0.981953i \(-0.560566\pi\)
−0.189127 + 0.981953i \(0.560566\pi\)
\(822\) 0 0
\(823\) 22723.6 0.962448 0.481224 0.876598i \(-0.340192\pi\)
0.481224 + 0.876598i \(0.340192\pi\)
\(824\) 9567.61 0.404494
\(825\) 0 0
\(826\) 26652.7 1.12272
\(827\) −19073.3 −0.801989 −0.400994 0.916081i \(-0.631335\pi\)
−0.400994 + 0.916081i \(0.631335\pi\)
\(828\) 0 0
\(829\) 42503.8 1.78072 0.890361 0.455255i \(-0.150452\pi\)
0.890361 + 0.455255i \(0.150452\pi\)
\(830\) −36102.0 −1.50978
\(831\) 0 0
\(832\) 0 0
\(833\) 27420.2 1.14052
\(834\) 0 0
\(835\) 17247.7 0.714829
\(836\) 11299.5 0.467465
\(837\) 0 0
\(838\) 14914.4 0.614808
\(839\) 19427.2 0.799408 0.399704 0.916644i \(-0.369113\pi\)
0.399704 + 0.916644i \(0.369113\pi\)
\(840\) 0 0
\(841\) −4759.60 −0.195154
\(842\) −15482.2 −0.633673
\(843\) 0 0
\(844\) 10768.5 0.439180
\(845\) 0 0
\(846\) 0 0
\(847\) −16160.3 −0.655578
\(848\) 2788.64 0.112927
\(849\) 0 0
\(850\) −2170.99 −0.0876052
\(851\) −36671.5 −1.47718
\(852\) 0 0
\(853\) 26851.8 1.07783 0.538914 0.842361i \(-0.318835\pi\)
0.538914 + 0.842361i \(0.318835\pi\)
\(854\) −17377.8 −0.696320
\(855\) 0 0
\(856\) 34705.4 1.38576
\(857\) −41539.4 −1.65573 −0.827864 0.560929i \(-0.810444\pi\)
−0.827864 + 0.560929i \(0.810444\pi\)
\(858\) 0 0
\(859\) −11936.2 −0.474107 −0.237054 0.971497i \(-0.576182\pi\)
−0.237054 + 0.971497i \(0.576182\pi\)
\(860\) 8887.99 0.352416
\(861\) 0 0
\(862\) −20538.3 −0.811529
\(863\) −41128.6 −1.62229 −0.811143 0.584848i \(-0.801154\pi\)
−0.811143 + 0.584848i \(0.801154\pi\)
\(864\) 0 0
\(865\) −24701.3 −0.970948
\(866\) −5582.22 −0.219043
\(867\) 0 0
\(868\) −25460.8 −0.995619
\(869\) −492.590 −0.0192290
\(870\) 0 0
\(871\) 0 0
\(872\) −47782.9 −1.85566
\(873\) 0 0
\(874\) −34485.3 −1.33465
\(875\) 37372.3 1.44390
\(876\) 0 0
\(877\) −6406.86 −0.246687 −0.123343 0.992364i \(-0.539362\pi\)
−0.123343 + 0.992364i \(0.539362\pi\)
\(878\) −13773.4 −0.529417
\(879\) 0 0
\(880\) −6341.73 −0.242931
\(881\) 2938.07 0.112357 0.0561783 0.998421i \(-0.482108\pi\)
0.0561783 + 0.998421i \(0.482108\pi\)
\(882\) 0 0
\(883\) 3022.06 0.115176 0.0575881 0.998340i \(-0.481659\pi\)
0.0575881 + 0.998340i \(0.481659\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 17677.0 0.670284
\(887\) 10060.4 0.380830 0.190415 0.981704i \(-0.439017\pi\)
0.190415 + 0.981704i \(0.439017\pi\)
\(888\) 0 0
\(889\) 68681.5 2.59112
\(890\) 8263.89 0.311243
\(891\) 0 0
\(892\) 3514.24 0.131912
\(893\) −3883.13 −0.145514
\(894\) 0 0
\(895\) −46997.3 −1.75525
\(896\) 8571.24 0.319582
\(897\) 0 0
\(898\) −13320.5 −0.494999
\(899\) −31326.5 −1.16218
\(900\) 0 0
\(901\) −7552.84 −0.279269
\(902\) 16958.9 0.626020
\(903\) 0 0
\(904\) −2957.09 −0.108796
\(905\) 29928.4 1.09929
\(906\) 0 0
\(907\) −43158.6 −1.58000 −0.789999 0.613108i \(-0.789919\pi\)
−0.789999 + 0.613108i \(0.789919\pi\)
\(908\) −5117.36 −0.187033
\(909\) 0 0
\(910\) 0 0
\(911\) 32665.9 1.18800 0.594001 0.804464i \(-0.297547\pi\)
0.594001 + 0.804464i \(0.297547\pi\)
\(912\) 0 0
\(913\) 41059.3 1.48835
\(914\) −3167.98 −0.114647
\(915\) 0 0
\(916\) −631.137 −0.0227657
\(917\) 42948.4 1.54665
\(918\) 0 0
\(919\) 18989.9 0.681633 0.340816 0.940130i \(-0.389297\pi\)
0.340816 + 0.940130i \(0.389297\pi\)
\(920\) −46875.3 −1.67982
\(921\) 0 0
\(922\) −15860.6 −0.566529
\(923\) 0 0
\(924\) 0 0
\(925\) −4792.12 −0.170340
\(926\) −4093.72 −0.145279
\(927\) 0 0
\(928\) 21726.0 0.768525
\(929\) 5596.81 0.197659 0.0988295 0.995104i \(-0.468490\pi\)
0.0988295 + 0.995104i \(0.468490\pi\)
\(930\) 0 0
\(931\) 56917.6 2.00365
\(932\) −16249.1 −0.571091
\(933\) 0 0
\(934\) 38137.5 1.33608
\(935\) 17176.1 0.600770
\(936\) 0 0
\(937\) 40294.4 1.40487 0.702433 0.711750i \(-0.252097\pi\)
0.702433 + 0.711750i \(0.252097\pi\)
\(938\) 32620.2 1.13549
\(939\) 0 0
\(940\) −1708.63 −0.0592865
\(941\) −43648.8 −1.51213 −0.756063 0.654499i \(-0.772880\pi\)
−0.756063 + 0.654499i \(0.772880\pi\)
\(942\) 0 0
\(943\) 47519.5 1.64098
\(944\) 8208.10 0.282999
\(945\) 0 0
\(946\) 11010.0 0.378398
\(947\) 10486.6 0.359839 0.179919 0.983681i \(-0.442416\pi\)
0.179919 + 0.983681i \(0.442416\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −4506.44 −0.153903
\(951\) 0 0
\(952\) 36390.9 1.23890
\(953\) −33058.8 −1.12369 −0.561846 0.827242i \(-0.689909\pi\)
−0.561846 + 0.827242i \(0.689909\pi\)
\(954\) 0 0
\(955\) 27509.5 0.932131
\(956\) −9538.90 −0.322709
\(957\) 0 0
\(958\) 19019.5 0.641431
\(959\) 21884.2 0.736891
\(960\) 0 0
\(961\) 20202.8 0.678152
\(962\) 0 0
\(963\) 0 0
\(964\) −11171.4 −0.373244
\(965\) −47392.5 −1.58095
\(966\) 0 0
\(967\) −53634.9 −1.78364 −0.891821 0.452389i \(-0.850572\pi\)
−0.891821 + 0.452389i \(0.850572\pi\)
\(968\) −13128.4 −0.435913
\(969\) 0 0
\(970\) −18475.8 −0.611569
\(971\) 4086.80 0.135069 0.0675344 0.997717i \(-0.478487\pi\)
0.0675344 + 0.997717i \(0.478487\pi\)
\(972\) 0 0
\(973\) −44758.8 −1.47472
\(974\) −7223.00 −0.237618
\(975\) 0 0
\(976\) −5351.76 −0.175518
\(977\) −14381.0 −0.470919 −0.235459 0.971884i \(-0.575659\pi\)
−0.235459 + 0.971884i \(0.575659\pi\)
\(978\) 0 0
\(979\) −9398.64 −0.306825
\(980\) 25044.5 0.816343
\(981\) 0 0
\(982\) −6865.57 −0.223105
\(983\) 12916.5 0.419099 0.209549 0.977798i \(-0.432800\pi\)
0.209549 + 0.977798i \(0.432800\pi\)
\(984\) 0 0
\(985\) −61176.2 −1.97892
\(986\) 14494.0 0.468137
\(987\) 0 0
\(988\) 0 0
\(989\) 30850.3 0.991893
\(990\) 0 0
\(991\) −5838.98 −0.187166 −0.0935829 0.995611i \(-0.529832\pi\)
−0.0935829 + 0.995611i \(0.529832\pi\)
\(992\) −34672.5 −1.10973
\(993\) 0 0
\(994\) 6238.79 0.199077
\(995\) −39510.3 −1.25886
\(996\) 0 0
\(997\) 44290.1 1.40690 0.703452 0.710743i \(-0.251641\pi\)
0.703452 + 0.710743i \(0.251641\pi\)
\(998\) 8224.99 0.260879
\(999\) 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.bk.1.4 10
3.2 odd 2 507.4.a.r.1.7 10
13.6 odd 12 117.4.q.e.10.4 10
13.11 odd 12 117.4.q.e.82.4 10
13.12 even 2 inner 1521.4.a.bk.1.7 10
39.5 even 4 507.4.b.i.337.4 10
39.8 even 4 507.4.b.i.337.7 10
39.11 even 12 39.4.j.c.4.2 10
39.32 even 12 39.4.j.c.10.2 yes 10
39.38 odd 2 507.4.a.r.1.4 10
156.11 odd 12 624.4.bv.h.433.4 10
156.71 odd 12 624.4.bv.h.49.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.2 10 39.11 even 12
39.4.j.c.10.2 yes 10 39.32 even 12
117.4.q.e.10.4 10 13.6 odd 12
117.4.q.e.82.4 10 13.11 odd 12
507.4.a.r.1.4 10 39.38 odd 2
507.4.a.r.1.7 10 3.2 odd 2
507.4.b.i.337.4 10 39.5 even 4
507.4.b.i.337.7 10 39.8 even 4
624.4.bv.h.49.2 10 156.71 odd 12
624.4.bv.h.433.4 10 156.11 odd 12
1521.4.a.bk.1.4 10 1.1 even 1 trivial
1521.4.a.bk.1.7 10 13.12 even 2 inner