Properties

Label 2352.4.a.cc.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{505}) \)
Defining polynomial: \(x^{2} - x - 126\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(11.7361\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -6.73610 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -6.73610 q^{5} +9.00000 q^{9} +53.6805 q^{11} +1.73610 q^{13} -20.2083 q^{15} -46.9444 q^{17} +20.1527 q^{19} -118.944 q^{23} -79.6249 q^{25} +27.0000 q^{27} -103.681 q^{29} +157.528 q^{31} +161.042 q^{33} -37.7361 q^{37} +5.20831 q^{39} -287.250 q^{41} -504.875 q^{43} -60.6249 q^{45} +220.500 q^{47} -140.833 q^{51} +292.319 q^{53} -361.597 q^{55} +60.4582 q^{57} +595.875 q^{59} -265.389 q^{61} -11.6946 q^{65} -936.735 q^{67} -356.833 q^{69} +545.944 q^{71} +299.374 q^{73} -238.875 q^{75} +940.333 q^{79} +81.0000 q^{81} -611.319 q^{83} +316.222 q^{85} -311.042 q^{87} -1176.19 q^{89} +472.583 q^{93} -135.751 q^{95} +1482.49 q^{97} +483.125 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 9q^{5} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 9q^{5} + 18q^{9} - 5q^{11} - 19q^{13} + 27q^{15} - 4q^{17} - 117q^{19} - 148q^{23} + 43q^{25} + 54q^{27} - 95q^{29} + 360q^{31} - 15q^{33} - 53q^{37} - 57q^{39} - 170q^{41} - 403q^{43} + 81q^{45} - 368q^{47} - 12q^{51} + 697q^{53} - 1285q^{55} - 351q^{57} + 585q^{59} - 1160q^{61} - 338q^{65} - 233q^{67} - 444q^{69} - 616q^{71} - 817q^{73} + 129q^{75} + 802q^{79} + 162q^{81} + 283q^{83} + 992q^{85} - 285q^{87} - 1858q^{89} + 1080q^{93} - 2294q^{95} + 1729q^{97} - 45q^{99} + O(q^{100}) \)

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
\(5\) −6.73610 −0.602495 −0.301248 0.953546i \(-0.597403\pi\)
−0.301248 + 0.953546i \(0.597403\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 53.6805 1.47139 0.735695 0.677313i \(-0.236856\pi\)
0.735695 + 0.677313i \(0.236856\pi\)
\(12\) 0 0
\(13\) 1.73610 0.0370391 0.0185195 0.999828i \(-0.494105\pi\)
0.0185195 + 0.999828i \(0.494105\pi\)
\(14\) 0 0
\(15\) −20.2083 −0.347851
\(16\) 0 0
\(17\) −46.9444 −0.669747 −0.334873 0.942263i \(-0.608694\pi\)
−0.334873 + 0.942263i \(0.608694\pi\)
\(18\) 0 0
\(19\) 20.1527 0.243334 0.121667 0.992571i \(-0.461176\pi\)
0.121667 + 0.992571i \(0.461176\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −118.944 −1.07833 −0.539166 0.842200i \(-0.681260\pi\)
−0.539166 + 0.842200i \(0.681260\pi\)
\(24\) 0 0
\(25\) −79.6249 −0.636999
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −103.681 −0.663896 −0.331948 0.943298i \(-0.607706\pi\)
−0.331948 + 0.943298i \(0.607706\pi\)
\(30\) 0 0
\(31\) 157.528 0.912672 0.456336 0.889808i \(-0.349162\pi\)
0.456336 + 0.889808i \(0.349162\pi\)
\(32\) 0 0
\(33\) 161.042 0.849507
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −37.7361 −0.167670 −0.0838348 0.996480i \(-0.526717\pi\)
−0.0838348 + 0.996480i \(0.526717\pi\)
\(38\) 0 0
\(39\) 5.20831 0.0213845
\(40\) 0 0
\(41\) −287.250 −1.09417 −0.547084 0.837078i \(-0.684262\pi\)
−0.547084 + 0.837078i \(0.684262\pi\)
\(42\) 0 0
\(43\) −504.875 −1.79053 −0.895264 0.445537i \(-0.853013\pi\)
−0.895264 + 0.445537i \(0.853013\pi\)
\(44\) 0 0
\(45\) −60.6249 −0.200832
\(46\) 0 0
\(47\) 220.500 0.684323 0.342162 0.939641i \(-0.388841\pi\)
0.342162 + 0.939641i \(0.388841\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −140.833 −0.386678
\(52\) 0 0
\(53\) 292.319 0.757607 0.378803 0.925477i \(-0.376336\pi\)
0.378803 + 0.925477i \(0.376336\pi\)
\(54\) 0 0
\(55\) −361.597 −0.886505
\(56\) 0 0
\(57\) 60.4582 0.140489
\(58\) 0 0
\(59\) 595.875 1.31485 0.657426 0.753519i \(-0.271645\pi\)
0.657426 + 0.753519i \(0.271645\pi\)
\(60\) 0 0
\(61\) −265.389 −0.557043 −0.278521 0.960430i \(-0.589844\pi\)
−0.278521 + 0.960430i \(0.589844\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −11.6946 −0.0223159
\(66\) 0 0
\(67\) −936.735 −1.70807 −0.854033 0.520218i \(-0.825851\pi\)
−0.854033 + 0.520218i \(0.825851\pi\)
\(68\) 0 0
\(69\) −356.833 −0.622575
\(70\) 0 0
\(71\) 545.944 0.912558 0.456279 0.889837i \(-0.349182\pi\)
0.456279 + 0.889837i \(0.349182\pi\)
\(72\) 0 0
\(73\) 299.374 0.479988 0.239994 0.970774i \(-0.422855\pi\)
0.239994 + 0.970774i \(0.422855\pi\)
\(74\) 0 0
\(75\) −238.875 −0.367772
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 940.333 1.33919 0.669593 0.742728i \(-0.266468\pi\)
0.669593 + 0.742728i \(0.266468\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −611.319 −0.808445 −0.404223 0.914661i \(-0.632458\pi\)
−0.404223 + 0.914661i \(0.632458\pi\)
\(84\) 0 0
\(85\) 316.222 0.403519
\(86\) 0 0
\(87\) −311.042 −0.383301
\(88\) 0 0
\(89\) −1176.19 −1.40086 −0.700429 0.713722i \(-0.747008\pi\)
−0.700429 + 0.713722i \(0.747008\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 472.583 0.526931
\(94\) 0 0
\(95\) −135.751 −0.146608
\(96\) 0 0
\(97\) 1482.49 1.55179 0.775895 0.630862i \(-0.217299\pi\)
0.775895 + 0.630862i \(0.217299\pi\)
\(98\) 0 0
\(99\) 483.125 0.490463
\(100\) 0 0
\(101\) 617.944 0.608789 0.304395 0.952546i \(-0.401546\pi\)
0.304395 + 0.952546i \(0.401546\pi\)
\(102\) 0 0
\(103\) 816.764 0.781341 0.390670 0.920531i \(-0.372243\pi\)
0.390670 + 0.920531i \(0.372243\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1685.38 −1.52272 −0.761361 0.648328i \(-0.775469\pi\)
−0.761361 + 0.648328i \(0.775469\pi\)
\(108\) 0 0
\(109\) 91.5968 0.0804898 0.0402449 0.999190i \(-0.487186\pi\)
0.0402449 + 0.999190i \(0.487186\pi\)
\(110\) 0 0
\(111\) −113.208 −0.0968041
\(112\) 0 0
\(113\) 614.194 0.511314 0.255657 0.966767i \(-0.417708\pi\)
0.255657 + 0.966767i \(0.417708\pi\)
\(114\) 0 0
\(115\) 801.222 0.649690
\(116\) 0 0
\(117\) 15.6249 0.0123464
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1550.60 1.16499
\(122\) 0 0
\(123\) −861.750 −0.631718
\(124\) 0 0
\(125\) 1378.37 0.986284
\(126\) 0 0
\(127\) −234.278 −0.163691 −0.0818457 0.996645i \(-0.526081\pi\)
−0.0818457 + 0.996645i \(0.526081\pi\)
\(128\) 0 0
\(129\) −1514.62 −1.03376
\(130\) 0 0
\(131\) −2664.76 −1.77726 −0.888631 0.458622i \(-0.848343\pi\)
−0.888631 + 0.458622i \(0.848343\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −181.875 −0.115950
\(136\) 0 0
\(137\) −1046.31 −0.652496 −0.326248 0.945284i \(-0.605784\pi\)
−0.326248 + 0.945284i \(0.605784\pi\)
\(138\) 0 0
\(139\) −647.265 −0.394966 −0.197483 0.980306i \(-0.563277\pi\)
−0.197483 + 0.980306i \(0.563277\pi\)
\(140\) 0 0
\(141\) 661.499 0.395094
\(142\) 0 0
\(143\) 93.1949 0.0544989
\(144\) 0 0
\(145\) 698.403 0.399994
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1326.81 0.729505 0.364752 0.931105i \(-0.381154\pi\)
0.364752 + 0.931105i \(0.381154\pi\)
\(150\) 0 0
\(151\) −3683.24 −1.98502 −0.992508 0.122178i \(-0.961012\pi\)
−0.992508 + 0.122178i \(0.961012\pi\)
\(152\) 0 0
\(153\) −422.500 −0.223249
\(154\) 0 0
\(155\) −1061.12 −0.549881
\(156\) 0 0
\(157\) −2818.25 −1.43262 −0.716308 0.697784i \(-0.754169\pi\)
−0.716308 + 0.697784i \(0.754169\pi\)
\(158\) 0 0
\(159\) 876.958 0.437405
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4106.61 −1.97334 −0.986670 0.162733i \(-0.947969\pi\)
−0.986670 + 0.162733i \(0.947969\pi\)
\(164\) 0 0
\(165\) −1084.79 −0.511824
\(166\) 0 0
\(167\) −1045.56 −0.484476 −0.242238 0.970217i \(-0.577881\pi\)
−0.242238 + 0.970217i \(0.577881\pi\)
\(168\) 0 0
\(169\) −2193.99 −0.998628
\(170\) 0 0
\(171\) 181.374 0.0811114
\(172\) 0 0
\(173\) −1910.31 −0.839525 −0.419763 0.907634i \(-0.637887\pi\)
−0.419763 + 0.907634i \(0.637887\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1787.62 0.759130
\(178\) 0 0
\(179\) −231.140 −0.0965151 −0.0482576 0.998835i \(-0.515367\pi\)
−0.0482576 + 0.998835i \(0.515367\pi\)
\(180\) 0 0
\(181\) 3308.40 1.35863 0.679314 0.733848i \(-0.262278\pi\)
0.679314 + 0.733848i \(0.262278\pi\)
\(182\) 0 0
\(183\) −796.167 −0.321609
\(184\) 0 0
\(185\) 254.194 0.101020
\(186\) 0 0
\(187\) −2520.00 −0.985458
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1800.75 −0.682187 −0.341093 0.940029i \(-0.610797\pi\)
−0.341093 + 0.940029i \(0.610797\pi\)
\(192\) 0 0
\(193\) −1283.36 −0.478644 −0.239322 0.970940i \(-0.576925\pi\)
−0.239322 + 0.970940i \(0.576925\pi\)
\(194\) 0 0
\(195\) −35.0837 −0.0128841
\(196\) 0 0
\(197\) 60.7514 0.0219714 0.0109857 0.999940i \(-0.496503\pi\)
0.0109857 + 0.999940i \(0.496503\pi\)
\(198\) 0 0
\(199\) −3860.39 −1.37515 −0.687577 0.726112i \(-0.741325\pi\)
−0.687577 + 0.726112i \(0.741325\pi\)
\(200\) 0 0
\(201\) −2810.21 −0.986153
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1934.94 0.659231
\(206\) 0 0
\(207\) −1070.50 −0.359444
\(208\) 0 0
\(209\) 1081.81 0.358039
\(210\) 0 0
\(211\) −491.637 −0.160406 −0.0802031 0.996779i \(-0.525557\pi\)
−0.0802031 + 0.996779i \(0.525557\pi\)
\(212\) 0 0
\(213\) 1637.83 0.526866
\(214\) 0 0
\(215\) 3400.89 1.07878
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 898.123 0.277121
\(220\) 0 0
\(221\) −81.5003 −0.0248068
\(222\) 0 0
\(223\) −823.376 −0.247253 −0.123626 0.992329i \(-0.539452\pi\)
−0.123626 + 0.992329i \(0.539452\pi\)
\(224\) 0 0
\(225\) −716.624 −0.212333
\(226\) 0 0
\(227\) 3758.87 1.09905 0.549527 0.835476i \(-0.314808\pi\)
0.549527 + 0.835476i \(0.314808\pi\)
\(228\) 0 0
\(229\) −2440.10 −0.704132 −0.352066 0.935975i \(-0.614521\pi\)
−0.352066 + 0.935975i \(0.614521\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1949.86 −0.548239 −0.274120 0.961696i \(-0.588386\pi\)
−0.274120 + 0.961696i \(0.588386\pi\)
\(234\) 0 0
\(235\) −1485.31 −0.412301
\(236\) 0 0
\(237\) 2821.00 0.773180
\(238\) 0 0
\(239\) −2705.64 −0.732273 −0.366137 0.930561i \(-0.619320\pi\)
−0.366137 + 0.930561i \(0.619320\pi\)
\(240\) 0 0
\(241\) 604.211 0.161496 0.0807482 0.996735i \(-0.474269\pi\)
0.0807482 + 0.996735i \(0.474269\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 34.9872 0.00901288
\(248\) 0 0
\(249\) −1833.96 −0.466756
\(250\) 0 0
\(251\) 1631.82 0.410356 0.205178 0.978725i \(-0.434223\pi\)
0.205178 + 0.978725i \(0.434223\pi\)
\(252\) 0 0
\(253\) −6385.00 −1.58665
\(254\) 0 0
\(255\) 948.667 0.232972
\(256\) 0 0
\(257\) −7106.55 −1.72488 −0.862441 0.506158i \(-0.831065\pi\)
−0.862441 + 0.506158i \(0.831065\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −933.125 −0.221299
\(262\) 0 0
\(263\) −5570.25 −1.30599 −0.652996 0.757361i \(-0.726488\pi\)
−0.652996 + 0.757361i \(0.726488\pi\)
\(264\) 0 0
\(265\) −1969.09 −0.456455
\(266\) 0 0
\(267\) −3528.58 −0.808786
\(268\) 0 0
\(269\) 3913.18 0.886955 0.443477 0.896286i \(-0.353745\pi\)
0.443477 + 0.896286i \(0.353745\pi\)
\(270\) 0 0
\(271\) −6072.51 −1.36118 −0.680588 0.732666i \(-0.738276\pi\)
−0.680588 + 0.732666i \(0.738276\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4274.31 −0.937274
\(276\) 0 0
\(277\) 6140.62 1.33196 0.665982 0.745968i \(-0.268013\pi\)
0.665982 + 0.745968i \(0.268013\pi\)
\(278\) 0 0
\(279\) 1417.75 0.304224
\(280\) 0 0
\(281\) −365.751 −0.0776472 −0.0388236 0.999246i \(-0.512361\pi\)
−0.0388236 + 0.999246i \(0.512361\pi\)
\(282\) 0 0
\(283\) 710.348 0.149208 0.0746039 0.997213i \(-0.476231\pi\)
0.0746039 + 0.997213i \(0.476231\pi\)
\(284\) 0 0
\(285\) −407.252 −0.0846440
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2709.22 −0.551440
\(290\) 0 0
\(291\) 4447.46 0.895926
\(292\) 0 0
\(293\) 296.373 0.0590932 0.0295466 0.999563i \(-0.490594\pi\)
0.0295466 + 0.999563i \(0.490594\pi\)
\(294\) 0 0
\(295\) −4013.87 −0.792192
\(296\) 0 0
\(297\) 1449.37 0.283169
\(298\) 0 0
\(299\) −206.500 −0.0399404
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 1853.83 0.351485
\(304\) 0 0
\(305\) 1787.69 0.335616
\(306\) 0 0
\(307\) −8596.87 −1.59821 −0.799103 0.601194i \(-0.794692\pi\)
−0.799103 + 0.601194i \(0.794692\pi\)
\(308\) 0 0
\(309\) 2450.29 0.451107
\(310\) 0 0
\(311\) 10475.3 1.90997 0.954984 0.296658i \(-0.0958723\pi\)
0.954984 + 0.296658i \(0.0958723\pi\)
\(312\) 0 0
\(313\) 5496.52 0.992594 0.496297 0.868153i \(-0.334693\pi\)
0.496297 + 0.868153i \(0.334693\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6556.37 1.16165 0.580824 0.814029i \(-0.302730\pi\)
0.580824 + 0.814029i \(0.302730\pi\)
\(318\) 0 0
\(319\) −5565.62 −0.976850
\(320\) 0 0
\(321\) −5056.13 −0.879145
\(322\) 0 0
\(323\) −946.057 −0.162972
\(324\) 0 0
\(325\) −138.237 −0.0235939
\(326\) 0 0
\(327\) 274.790 0.0464708
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −102.376 −0.0170003 −0.00850017 0.999964i \(-0.502706\pi\)
−0.00850017 + 0.999964i \(0.502706\pi\)
\(332\) 0 0
\(333\) −339.625 −0.0558899
\(334\) 0 0
\(335\) 6309.95 1.02910
\(336\) 0 0
\(337\) 1333.50 0.215549 0.107775 0.994175i \(-0.465627\pi\)
0.107775 + 0.994175i \(0.465627\pi\)
\(338\) 0 0
\(339\) 1842.58 0.295208
\(340\) 0 0
\(341\) 8456.17 1.34290
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2403.67 0.375099
\(346\) 0 0
\(347\) 1101.00 0.170331 0.0851655 0.996367i \(-0.472858\pi\)
0.0851655 + 0.996367i \(0.472858\pi\)
\(348\) 0 0
\(349\) 9467.89 1.45216 0.726081 0.687609i \(-0.241340\pi\)
0.726081 + 0.687609i \(0.241340\pi\)
\(350\) 0 0
\(351\) 46.8748 0.00712818
\(352\) 0 0
\(353\) −1684.69 −0.254015 −0.127007 0.991902i \(-0.540537\pi\)
−0.127007 + 0.991902i \(0.540537\pi\)
\(354\) 0 0
\(355\) −3677.53 −0.549812
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6635.25 0.975474 0.487737 0.872991i \(-0.337823\pi\)
0.487737 + 0.872991i \(0.337823\pi\)
\(360\) 0 0
\(361\) −6452.87 −0.940788
\(362\) 0 0
\(363\) 4651.79 0.672605
\(364\) 0 0
\(365\) −2016.62 −0.289191
\(366\) 0 0
\(367\) 7121.97 1.01298 0.506490 0.862246i \(-0.330943\pi\)
0.506490 + 0.862246i \(0.330943\pi\)
\(368\) 0 0
\(369\) −2585.25 −0.364723
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −13621.3 −1.89085 −0.945424 0.325843i \(-0.894352\pi\)
−0.945424 + 0.325843i \(0.894352\pi\)
\(374\) 0 0
\(375\) 4135.12 0.569432
\(376\) 0 0
\(377\) −180.000 −0.0245901
\(378\) 0 0
\(379\) 3331.51 0.451525 0.225763 0.974182i \(-0.427513\pi\)
0.225763 + 0.974182i \(0.427513\pi\)
\(380\) 0 0
\(381\) −702.834 −0.0945073
\(382\) 0 0
\(383\) −3275.75 −0.437031 −0.218515 0.975833i \(-0.570121\pi\)
−0.218515 + 0.975833i \(0.570121\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4543.87 −0.596842
\(388\) 0 0
\(389\) −14975.4 −1.95188 −0.975941 0.218035i \(-0.930035\pi\)
−0.975941 + 0.218035i \(0.930035\pi\)
\(390\) 0 0
\(391\) 5583.78 0.722209
\(392\) 0 0
\(393\) −7994.29 −1.02610
\(394\) 0 0
\(395\) −6334.18 −0.806853
\(396\) 0 0
\(397\) −10758.6 −1.36010 −0.680050 0.733166i \(-0.738042\pi\)
−0.680050 + 0.733166i \(0.738042\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2607.00 −0.324657 −0.162328 0.986737i \(-0.551900\pi\)
−0.162328 + 0.986737i \(0.551900\pi\)
\(402\) 0 0
\(403\) 273.484 0.0338045
\(404\) 0 0
\(405\) −545.624 −0.0669439
\(406\) 0 0
\(407\) −2025.69 −0.246707
\(408\) 0 0
\(409\) 6214.64 0.751330 0.375665 0.926756i \(-0.377414\pi\)
0.375665 + 0.926756i \(0.377414\pi\)
\(410\) 0 0
\(411\) −3138.92 −0.376719
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4117.91 0.487085
\(416\) 0 0
\(417\) −1941.79 −0.228034
\(418\) 0 0
\(419\) 13111.4 1.52872 0.764358 0.644792i \(-0.223056\pi\)
0.764358 + 0.644792i \(0.223056\pi\)
\(420\) 0 0
\(421\) −8410.12 −0.973596 −0.486798 0.873514i \(-0.661835\pi\)
−0.486798 + 0.873514i \(0.661835\pi\)
\(422\) 0 0
\(423\) 1984.50 0.228108
\(424\) 0 0
\(425\) 3737.95 0.426628
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 279.585 0.0314650
\(430\) 0 0
\(431\) −7783.31 −0.869858 −0.434929 0.900465i \(-0.643226\pi\)
−0.434929 + 0.900465i \(0.643226\pi\)
\(432\) 0 0
\(433\) 13870.1 1.53939 0.769695 0.638412i \(-0.220409\pi\)
0.769695 + 0.638412i \(0.220409\pi\)
\(434\) 0 0
\(435\) 2095.21 0.230937
\(436\) 0 0
\(437\) −2397.05 −0.262395
\(438\) 0 0
\(439\) −5679.63 −0.617480 −0.308740 0.951146i \(-0.599907\pi\)
−0.308740 + 0.951146i \(0.599907\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2316.93 0.248489 0.124245 0.992252i \(-0.460349\pi\)
0.124245 + 0.992252i \(0.460349\pi\)
\(444\) 0 0
\(445\) 7922.97 0.844010
\(446\) 0 0
\(447\) 3980.42 0.421180
\(448\) 0 0
\(449\) 9526.86 1.00134 0.500669 0.865639i \(-0.333088\pi\)
0.500669 + 0.865639i \(0.333088\pi\)
\(450\) 0 0
\(451\) −15419.7 −1.60995
\(452\) 0 0
\(453\) −11049.7 −1.14605
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16627.3 −1.70195 −0.850974 0.525208i \(-0.823987\pi\)
−0.850974 + 0.525208i \(0.823987\pi\)
\(458\) 0 0
\(459\) −1267.50 −0.128893
\(460\) 0 0
\(461\) −11338.3 −1.14550 −0.572749 0.819730i \(-0.694123\pi\)
−0.572749 + 0.819730i \(0.694123\pi\)
\(462\) 0 0
\(463\) −9207.87 −0.924246 −0.462123 0.886816i \(-0.652912\pi\)
−0.462123 + 0.886816i \(0.652912\pi\)
\(464\) 0 0
\(465\) −3183.37 −0.317474
\(466\) 0 0
\(467\) 17215.7 1.70588 0.852941 0.522007i \(-0.174816\pi\)
0.852941 + 0.522007i \(0.174816\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −8454.74 −0.827121
\(472\) 0 0
\(473\) −27101.9 −2.63456
\(474\) 0 0
\(475\) −1604.66 −0.155004
\(476\) 0 0
\(477\) 2630.88 0.252536
\(478\) 0 0
\(479\) −10293.8 −0.981911 −0.490956 0.871185i \(-0.663352\pi\)
−0.490956 + 0.871185i \(0.663352\pi\)
\(480\) 0 0
\(481\) −65.5137 −0.00621033
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −9986.18 −0.934946
\(486\) 0 0
\(487\) −11109.0 −1.03367 −0.516833 0.856086i \(-0.672889\pi\)
−0.516833 + 0.856086i \(0.672889\pi\)
\(488\) 0 0
\(489\) −12319.8 −1.13931
\(490\) 0 0
\(491\) 6573.88 0.604226 0.302113 0.953272i \(-0.402308\pi\)
0.302113 + 0.953272i \(0.402308\pi\)
\(492\) 0 0
\(493\) 4867.22 0.444642
\(494\) 0 0
\(495\) −3254.38 −0.295502
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 16841.4 1.51087 0.755435 0.655223i \(-0.227425\pi\)
0.755435 + 0.655223i \(0.227425\pi\)
\(500\) 0 0
\(501\) −3136.67 −0.279712
\(502\) 0 0
\(503\) 18436.6 1.63429 0.817146 0.576431i \(-0.195555\pi\)
0.817146 + 0.576431i \(0.195555\pi\)
\(504\) 0 0
\(505\) −4162.53 −0.366793
\(506\) 0 0
\(507\) −6581.96 −0.576558
\(508\) 0 0
\(509\) 20825.2 1.81348 0.906738 0.421694i \(-0.138564\pi\)
0.906738 + 0.421694i \(0.138564\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 544.123 0.0468297
\(514\) 0 0
\(515\) −5501.80 −0.470754
\(516\) 0 0
\(517\) 11836.5 1.00691
\(518\) 0 0
\(519\) −5730.92 −0.484700
\(520\) 0 0
\(521\) 14384.7 1.20961 0.604805 0.796373i \(-0.293251\pi\)
0.604805 + 0.796373i \(0.293251\pi\)
\(522\) 0 0
\(523\) −6487.49 −0.542405 −0.271203 0.962522i \(-0.587421\pi\)
−0.271203 + 0.962522i \(0.587421\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7395.05 −0.611259
\(528\) 0 0
\(529\) 1980.77 0.162799
\(530\) 0 0
\(531\) 5362.87 0.438284
\(532\) 0 0
\(533\) −498.695 −0.0405270
\(534\) 0 0
\(535\) 11352.9 0.917433
\(536\) 0 0
\(537\) −693.420 −0.0557230
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 14416.6 1.14569 0.572844 0.819664i \(-0.305840\pi\)
0.572844 + 0.819664i \(0.305840\pi\)
\(542\) 0 0
\(543\) 9925.21 0.784404
\(544\) 0 0
\(545\) −617.006 −0.0484947
\(546\) 0 0
\(547\) 4881.38 0.381559 0.190780 0.981633i \(-0.438898\pi\)
0.190780 + 0.981633i \(0.438898\pi\)
\(548\) 0 0
\(549\) −2388.50 −0.185681
\(550\) 0 0
\(551\) −2089.44 −0.161549
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 762.583 0.0583240
\(556\) 0 0
\(557\) −3153.13 −0.239860 −0.119930 0.992782i \(-0.538267\pi\)
−0.119930 + 0.992782i \(0.538267\pi\)
\(558\) 0 0
\(559\) −876.514 −0.0663195
\(560\) 0 0
\(561\) −7560.00 −0.568954
\(562\) 0 0
\(563\) −6795.87 −0.508724 −0.254362 0.967109i \(-0.581866\pi\)
−0.254362 + 0.967109i \(0.581866\pi\)
\(564\) 0 0
\(565\) −4137.28 −0.308065
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10708.3 −0.788955 −0.394477 0.918906i \(-0.629074\pi\)
−0.394477 + 0.918906i \(0.629074\pi\)
\(570\) 0 0
\(571\) −6531.66 −0.478706 −0.239353 0.970933i \(-0.576935\pi\)
−0.239353 + 0.970933i \(0.576935\pi\)
\(572\) 0 0
\(573\) −5402.25 −0.393861
\(574\) 0 0
\(575\) 9470.94 0.686896
\(576\) 0 0
\(577\) 17461.9 1.25987 0.629937 0.776646i \(-0.283081\pi\)
0.629937 + 0.776646i \(0.283081\pi\)
\(578\) 0 0
\(579\) −3850.08 −0.276345
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 15691.9 1.11473
\(584\) 0 0
\(585\) −105.251 −0.00743863
\(586\) 0 0
\(587\) −9254.42 −0.650717 −0.325358 0.945591i \(-0.605485\pi\)
−0.325358 + 0.945591i \(0.605485\pi\)
\(588\) 0 0
\(589\) 3174.61 0.222084
\(590\) 0 0
\(591\) 182.254 0.0126852
\(592\) 0 0
\(593\) −21019.2 −1.45557 −0.727786 0.685804i \(-0.759451\pi\)
−0.727786 + 0.685804i \(0.759451\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −11581.2 −0.793945
\(598\) 0 0
\(599\) 26971.7 1.83979 0.919894 0.392167i \(-0.128275\pi\)
0.919894 + 0.392167i \(0.128275\pi\)
\(600\) 0 0
\(601\) −26510.3 −1.79930 −0.899648 0.436616i \(-0.856177\pi\)
−0.899648 + 0.436616i \(0.856177\pi\)
\(602\) 0 0
\(603\) −8430.62 −0.569355
\(604\) 0 0
\(605\) −10445.0 −0.701899
\(606\) 0 0
\(607\) −29412.0 −1.96671 −0.983356 0.181687i \(-0.941844\pi\)
−0.983356 + 0.181687i \(0.941844\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 382.810 0.0253467
\(612\) 0 0
\(613\) −6068.64 −0.399853 −0.199926 0.979811i \(-0.564070\pi\)
−0.199926 + 0.979811i \(0.564070\pi\)
\(614\) 0 0
\(615\) 5804.83 0.380607
\(616\) 0 0
\(617\) 1904.56 0.124270 0.0621352 0.998068i \(-0.480209\pi\)
0.0621352 + 0.998068i \(0.480209\pi\)
\(618\) 0 0
\(619\) 406.478 0.0263937 0.0131969 0.999913i \(-0.495799\pi\)
0.0131969 + 0.999913i \(0.495799\pi\)
\(620\) 0 0
\(621\) −3211.50 −0.207525
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 668.244 0.0427676
\(626\) 0 0
\(627\) 3245.42 0.206714
\(628\) 0 0
\(629\) 1771.50 0.112296
\(630\) 0 0
\(631\) 15294.2 0.964903 0.482452 0.875923i \(-0.339746\pi\)
0.482452 + 0.875923i \(0.339746\pi\)
\(632\) 0 0
\(633\) −1474.91 −0.0926105
\(634\) 0 0
\(635\) 1578.12 0.0986233
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 4913.49 0.304186
\(640\) 0 0
\(641\) −15722.2 −0.968782 −0.484391 0.874852i \(-0.660959\pi\)
−0.484391 + 0.874852i \(0.660959\pi\)
\(642\) 0 0
\(643\) −18529.9 −1.13646 −0.568232 0.822868i \(-0.692372\pi\)
−0.568232 + 0.822868i \(0.692372\pi\)
\(644\) 0 0
\(645\) 10202.7 0.622836
\(646\) 0 0
\(647\) 8638.95 0.524934 0.262467 0.964941i \(-0.415464\pi\)
0.262467 + 0.964941i \(0.415464\pi\)
\(648\) 0 0
\(649\) 31986.9 1.93466
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27884.4 1.67106 0.835528 0.549448i \(-0.185162\pi\)
0.835528 + 0.549448i \(0.185162\pi\)
\(654\) 0 0
\(655\) 17950.1 1.07079
\(656\) 0 0
\(657\) 2694.37 0.159996
\(658\) 0 0
\(659\) 5767.55 0.340928 0.170464 0.985364i \(-0.445473\pi\)
0.170464 + 0.985364i \(0.445473\pi\)
\(660\) 0 0
\(661\) 4090.12 0.240677 0.120338 0.992733i \(-0.461602\pi\)
0.120338 + 0.992733i \(0.461602\pi\)
\(662\) 0 0
\(663\) −244.501 −0.0143222
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12332.2 0.715900
\(668\) 0 0
\(669\) −2470.13 −0.142751
\(670\) 0 0
\(671\) −14246.2 −0.819627
\(672\) 0 0
\(673\) −21667.7 −1.24106 −0.620528 0.784185i \(-0.713082\pi\)
−0.620528 + 0.784185i \(0.713082\pi\)
\(674\) 0 0
\(675\) −2149.87 −0.122591
\(676\) 0 0
\(677\) −15318.9 −0.869648 −0.434824 0.900515i \(-0.643189\pi\)
−0.434824 + 0.900515i \(0.643189\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 11276.6 0.634539
\(682\) 0 0
\(683\) −9981.36 −0.559189 −0.279595 0.960118i \(-0.590200\pi\)
−0.279595 + 0.960118i \(0.590200\pi\)
\(684\) 0 0
\(685\) 7048.02 0.393126
\(686\) 0 0
\(687\) −7320.29 −0.406531
\(688\) 0 0
\(689\) 507.497 0.0280611
\(690\) 0 0
\(691\) −23243.1 −1.27961 −0.639805 0.768537i \(-0.720985\pi\)
−0.639805 + 0.768537i \(0.720985\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4360.04 0.237965
\(696\) 0 0
\(697\) 13484.8 0.732815
\(698\) 0 0
\(699\) −5849.59 −0.316526
\(700\) 0 0
\(701\) −11295.6 −0.608598 −0.304299 0.952577i \(-0.598422\pi\)
−0.304299 + 0.952577i \(0.598422\pi\)
\(702\) 0 0
\(703\) −760.485 −0.0407998
\(704\) 0 0
\(705\) −4455.93 −0.238042
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6868.11 0.363804 0.181902 0.983317i \(-0.441775\pi\)
0.181902 + 0.983317i \(0.441775\pi\)
\(710\) 0 0
\(711\) 8463.00 0.446395
\(712\) 0 0
\(713\) −18737.1 −0.984163
\(714\) 0 0
\(715\) −627.770 −0.0328353
\(716\) 0 0
\(717\) −8116.92 −0.422778
\(718\) 0 0
\(719\) 31479.5 1.63281 0.816404 0.577481i \(-0.195964\pi\)
0.816404 + 0.577481i \(0.195964\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1812.63 0.0932400
\(724\) 0 0
\(725\) 8255.55 0.422901
\(726\) 0 0
\(727\) −26045.8 −1.32873 −0.664363 0.747410i \(-0.731297\pi\)
−0.664363 + 0.747410i \(0.731297\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 23701.0 1.19920
\(732\) 0 0
\(733\) −3538.40 −0.178300 −0.0891499 0.996018i \(-0.528415\pi\)
−0.0891499 + 0.996018i \(0.528415\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −50284.4 −2.51323
\(738\) 0 0
\(739\) −10426.9 −0.519024 −0.259512 0.965740i \(-0.583562\pi\)
−0.259512 + 0.965740i \(0.583562\pi\)
\(740\) 0 0
\(741\) 104.962 0.00520359
\(742\) 0 0
\(743\) −2518.88 −0.124372 −0.0621862 0.998065i \(-0.519807\pi\)
−0.0621862 + 0.998065i \(0.519807\pi\)
\(744\) 0 0
\(745\) −8937.50 −0.439523
\(746\) 0 0
\(747\) −5501.87 −0.269482
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −14820.3 −0.720105 −0.360053 0.932932i \(-0.617241\pi\)
−0.360053 + 0.932932i \(0.617241\pi\)
\(752\) 0 0
\(753\) 4895.46 0.236919
\(754\) 0 0
\(755\) 24810.7 1.19596
\(756\) 0 0
\(757\) 8892.84 0.426969 0.213485 0.976946i \(-0.431519\pi\)
0.213485 + 0.976946i \(0.431519\pi\)
\(758\) 0 0
\(759\) −19155.0 −0.916050
\(760\) 0 0
\(761\) 22795.0 1.08583 0.542916 0.839787i \(-0.317320\pi\)
0.542916 + 0.839787i \(0.317320\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2846.00 0.134506
\(766\) 0 0
\(767\) 1034.50 0.0487009
\(768\) 0 0
\(769\) −13025.5 −0.610809 −0.305405 0.952223i \(-0.598792\pi\)
−0.305405 + 0.952223i \(0.598792\pi\)
\(770\) 0 0
\(771\) −21319.7 −0.995861
\(772\) 0 0
\(773\) 26467.0 1.23150 0.615751 0.787941i \(-0.288853\pi\)
0.615751 + 0.787941i \(0.288853\pi\)
\(774\) 0 0
\(775\) −12543.1 −0.581371
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5788.87 −0.266249
\(780\) 0 0
\(781\) 29306.5 1.34273
\(782\) 0 0
\(783\) −2799.37 −0.127767
\(784\) 0 0
\(785\) 18984.0 0.863144
\(786\) 0 0
\(787\) −33216.6 −1.50450 −0.752252 0.658876i \(-0.771032\pi\)
−0.752252 + 0.658876i \(0.771032\pi\)
\(788\) 0 0
\(789\) −16710.7 −0.754015
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −460.743 −0.0206324
\(794\) 0 0
\(795\) −5907.28 −0.263534
\(796\) 0 0
\(797\) −15363.9 −0.682830 −0.341415 0.939913i \(-0.610906\pi\)
−0.341415 + 0.939913i \(0.610906\pi\)
\(798\) 0 0
\(799\) −10351.2 −0.458323
\(800\) 0 0
\(801\) −10585.7 −0.466953
\(802\) 0 0
\(803\) 16070.6 0.706249
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 11739.5 0.512083
\(808\) 0 0
\(809\) 26742.7 1.16220 0.581102 0.813831i \(-0.302622\pi\)
0.581102 + 0.813831i \(0.302622\pi\)
\(810\) 0 0
\(811\) 23651.0 1.02404 0.512022 0.858973i \(-0.328897\pi\)
0.512022 + 0.858973i \(0.328897\pi\)
\(812\) 0 0
\(813\) −18217.5 −0.785876
\(814\) 0 0
\(815\) 27662.5 1.18893
\(816\) 0 0
\(817\) −10174.6 −0.435697
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 16974.1 0.721558 0.360779 0.932651i \(-0.382511\pi\)
0.360779 + 0.932651i \(0.382511\pi\)
\(822\) 0 0
\(823\) −22328.9 −0.945730 −0.472865 0.881135i \(-0.656780\pi\)
−0.472865 + 0.881135i \(0.656780\pi\)
\(824\) 0 0
\(825\) −12822.9 −0.541135
\(826\) 0 0
\(827\) −15731.8 −0.661485 −0.330743 0.943721i \(-0.607299\pi\)
−0.330743 + 0.943721i \(0.607299\pi\)
\(828\) 0 0
\(829\) −38025.4 −1.59309 −0.796547 0.604576i \(-0.793342\pi\)
−0.796547 + 0.604576i \(0.793342\pi\)
\(830\) 0 0
\(831\) 18421.9 0.769010
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 7042.97 0.291895
\(836\) 0 0
\(837\) 4253.25 0.175644
\(838\) 0 0
\(839\) −16546.5 −0.680868 −0.340434 0.940268i \(-0.610574\pi\)
−0.340434 + 0.940268i \(0.610574\pi\)
\(840\) 0 0
\(841\) −13639.4 −0.559242
\(842\) 0 0
\(843\) −1097.25 −0.0448296
\(844\) 0 0
\(845\) 14778.9 0.601669
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 2131.04 0.0861452
\(850\) 0 0
\(851\) 4488.50 0.180803
\(852\) 0 0
\(853\) 13511.9 0.542365 0.271182 0.962528i \(-0.412585\pi\)
0.271182 + 0.962528i \(0.412585\pi\)
\(854\) 0 0
\(855\) −1221.76 −0.0488692
\(856\) 0 0
\(857\) 17178.2 0.684711 0.342355 0.939570i \(-0.388775\pi\)
0.342355 + 0.939570i \(0.388775\pi\)
\(858\) 0 0
\(859\) 35914.3 1.42652 0.713260 0.700899i \(-0.247218\pi\)
0.713260 + 0.700899i \(0.247218\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 882.508 0.0348099 0.0174049 0.999849i \(-0.494460\pi\)
0.0174049 + 0.999849i \(0.494460\pi\)
\(864\) 0 0
\(865\) 12868.0 0.505810
\(866\) 0 0
\(867\) −8127.67 −0.318374
\(868\) 0 0
\(869\) 50477.6 1.97046
\(870\) 0 0
\(871\) −1626.27 −0.0632652
\(872\) 0 0
\(873\) 13342.4 0.517263
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 17420.1 0.670737 0.335368 0.942087i \(-0.391139\pi\)
0.335368 + 0.942087i \(0.391139\pi\)
\(878\) 0 0
\(879\) 889.120 0.0341175
\(880\) 0 0
\(881\) 38097.8 1.45692 0.728460 0.685088i \(-0.240236\pi\)
0.728460 + 0.685088i \(0.240236\pi\)
\(882\) 0 0
\(883\) 13870.6 0.528633 0.264317 0.964436i \(-0.414854\pi\)
0.264317 + 0.964436i \(0.414854\pi\)
\(884\) 0 0
\(885\) −12041.6 −0.457372
\(886\) 0 0
\(887\) 17467.8 0.661230 0.330615 0.943766i \(-0.392744\pi\)
0.330615 + 0.943766i \(0.392744\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4348.12 0.163488
\(892\) 0 0
\(893\) 4443.67 0.166519
\(894\) 0 0
\(895\) 1556.98 0.0581499
\(896\) 0 0
\(897\) −619.499 −0.0230596
\(898\) 0 0
\(899\) −16332.6 −0.605919
\(900\) 0 0
\(901\) −13722.8 −0.507405
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −22285.7 −0.818567
\(906\) 0 0
\(907\) 28536.3 1.04469 0.522344 0.852735i \(-0.325058\pi\)
0.522344 + 0.852735i \(0.325058\pi\)
\(908\) 0 0
\(909\) 5561.49 0.202930
\(910\) 0 0
\(911\) 50235.7 1.82698 0.913492 0.406857i \(-0.133375\pi\)
0.913492 + 0.406857i \(0.133375\pi\)
\(912\) 0 0
\(913\) −32815.9 −1.18954
\(914\) 0 0
\(915\) 5363.07 0.193768
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −5317.08 −0.190854 −0.0954268 0.995436i \(-0.530422\pi\)
−0.0954268 + 0.995436i \(0.530422\pi\)
\(920\) 0 0
\(921\) −25790.6 −0.922725
\(922\) 0 0
\(923\) 947.814 0.0338003
\(924\) 0 0
\(925\) 3004.73 0.106805
\(926\) 0 0
\(927\) 7350.87 0.260447
\(928\) 0 0
\(929\) 25493.1 0.900325 0.450162 0.892947i \(-0.351366\pi\)
0.450162 + 0.892947i \(0.351366\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 31425.9 1.10272
\(934\) 0 0
\(935\) 16975.0 0.593734
\(936\) 0 0
\(937\) 12691.4 0.442486 0.221243 0.975219i \(-0.428989\pi\)
0.221243 + 0.975219i \(0.428989\pi\)
\(938\) 0 0
\(939\) 16489.6 0.573074
\(940\) 0 0
\(941\) −15378.4 −0.532753 −0.266377 0.963869i \(-0.585826\pi\)
−0.266377 + 0.963869i \(0.585826\pi\)
\(942\) 0 0
\(943\) 34166.8 1.17988
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −50442.9 −1.73091 −0.865455 0.500986i \(-0.832971\pi\)
−0.865455 + 0.500986i \(0.832971\pi\)
\(948\) 0 0
\(949\) 519.745 0.0177783
\(950\) 0 0
\(951\) 19669.1 0.670678
\(952\) 0 0
\(953\) 31787.3 1.08047 0.540237 0.841513i \(-0.318335\pi\)
0.540237 + 0.841513i \(0.318335\pi\)
\(954\) 0 0
\(955\) 12130.0 0.411014
\(956\) 0 0
\(957\) −16696.9 −0.563984
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −4975.99 −0.167030
\(962\) 0 0
\(963\) −15168.4 −0.507574
\(964\) 0 0
\(965\) 8644.84 0.288381
\(966\) 0 0
\(967\) −1005.61 −0.0334417 −0.0167209 0.999860i \(-0.505323\pi\)
−0.0167209 + 0.999860i \(0.505323\pi\)
\(968\) 0 0
\(969\) −2838.17 −0.0940921
\(970\) 0 0
\(971\) −7070.91 −0.233693 −0.116847 0.993150i \(-0.537279\pi\)
−0.116847 + 0.993150i \(0.537279\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −414.711 −0.0136219
\(976\) 0 0
\(977\) −11097.1 −0.363384 −0.181692 0.983355i \(-0.558157\pi\)
−0.181692 + 0.983355i \(0.558157\pi\)
\(978\) 0 0
\(979\) −63138.7 −2.06121
\(980\) 0 0
\(981\) 824.371 0.0268299
\(982\) 0 0
\(983\) 17775.5 0.576755 0.288377 0.957517i \(-0.406884\pi\)
0.288377 + 0.957517i \(0.406884\pi\)
\(984\) 0 0
\(985\) −409.228 −0.0132376
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 60052.0 1.93078
\(990\) 0 0
\(991\) −4691.86 −0.150395 −0.0751977 0.997169i \(-0.523959\pi\)
−0.0751977 + 0.997169i \(0.523959\pi\)
\(992\) 0 0
\(993\) −307.129 −0.00981515
\(994\) 0 0
\(995\) 26004.0 0.828523
\(996\) 0 0
\(997\) −8471.39 −0.269099 −0.134549 0.990907i \(-0.542959\pi\)
−0.134549 + 0.990907i \(0.542959\pi\)
\(998\) 0 0
\(999\) −1018.87 −0.0322680
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 2352.4.a.cc.1.1 2
4.3 odd 2 1176.4.a.r.1.1 2
7.2 even 3 336.4.q.g.193.2 4
7.4 even 3 336.4.q.g.289.2 4
7.6 odd 2 2352.4.a.bo.1.2 2
28.11 odd 6 168.4.q.d.121.2 yes 4
28.23 odd 6 168.4.q.d.25.2 4
28.27 even 2 1176.4.a.u.1.2 2
84.11 even 6 504.4.s.f.289.1 4
84.23 even 6 504.4.s.f.361.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.d.25.2 4 28.23 odd 6
168.4.q.d.121.2 yes 4 28.11 odd 6
336.4.q.g.193.2 4 7.2 even 3
336.4.q.g.289.2 4 7.4 even 3
504.4.s.f.289.1 4 84.11 even 6
504.4.s.f.361.1 4 84.23 even 6
1176.4.a.r.1.1 2 4.3 odd 2
1176.4.a.u.1.2 2 28.27 even 2
2352.4.a.bo.1.2 2 7.6 odd 2
2352.4.a.cc.1.1 2 1.1 even 1 trivial