Properties

Label 2352.4.a.f.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $1$
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -6.00000 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -6.00000 q^{5} +9.00000 q^{9} +30.0000 q^{11} +53.0000 q^{13} +18.0000 q^{15} -84.0000 q^{17} +97.0000 q^{19} -84.0000 q^{23} -89.0000 q^{25} -27.0000 q^{27} -180.000 q^{29} -179.000 q^{31} -90.0000 q^{33} -145.000 q^{37} -159.000 q^{39} +126.000 q^{41} +325.000 q^{43} -54.0000 q^{45} +366.000 q^{47} +252.000 q^{51} -768.000 q^{53} -180.000 q^{55} -291.000 q^{57} +264.000 q^{59} +818.000 q^{61} -318.000 q^{65} +523.000 q^{67} +252.000 q^{69} +342.000 q^{71} -43.0000 q^{73} +267.000 q^{75} +1171.00 q^{79} +81.0000 q^{81} +810.000 q^{83} +504.000 q^{85} +540.000 q^{87} -600.000 q^{89} +537.000 q^{93} -582.000 q^{95} +386.000 q^{97} +270.000 q^{99} +O(q^{100})\)

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
\(5\) −6.00000 −0.536656 −0.268328 0.963328i \(-0.586471\pi\)
−0.268328 + 0.963328i \(0.586471\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 30.0000 0.822304 0.411152 0.911567i \(-0.365127\pi\)
0.411152 + 0.911567i \(0.365127\pi\)
\(12\) 0 0
\(13\) 53.0000 1.13074 0.565368 0.824839i \(-0.308734\pi\)
0.565368 + 0.824839i \(0.308734\pi\)
\(14\) 0 0
\(15\) 18.0000 0.309839
\(16\) 0 0
\(17\) −84.0000 −1.19841 −0.599206 0.800595i \(-0.704517\pi\)
−0.599206 + 0.800595i \(0.704517\pi\)
\(18\) 0 0
\(19\) 97.0000 1.17123 0.585614 0.810590i \(-0.300854\pi\)
0.585614 + 0.810590i \(0.300854\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −84.0000 −0.761531 −0.380765 0.924672i \(-0.624339\pi\)
−0.380765 + 0.924672i \(0.624339\pi\)
\(24\) 0 0
\(25\) −89.0000 −0.712000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −180.000 −1.15259 −0.576296 0.817241i \(-0.695502\pi\)
−0.576296 + 0.817241i \(0.695502\pi\)
\(30\) 0 0
\(31\) −179.000 −1.03708 −0.518538 0.855055i \(-0.673523\pi\)
−0.518538 + 0.855055i \(0.673523\pi\)
\(32\) 0 0
\(33\) −90.0000 −0.474757
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −145.000 −0.644266 −0.322133 0.946694i \(-0.604400\pi\)
−0.322133 + 0.946694i \(0.604400\pi\)
\(38\) 0 0
\(39\) −159.000 −0.652830
\(40\) 0 0
\(41\) 126.000 0.479949 0.239974 0.970779i \(-0.422861\pi\)
0.239974 + 0.970779i \(0.422861\pi\)
\(42\) 0 0
\(43\) 325.000 1.15261 0.576303 0.817236i \(-0.304495\pi\)
0.576303 + 0.817236i \(0.304495\pi\)
\(44\) 0 0
\(45\) −54.0000 −0.178885
\(46\) 0 0
\(47\) 366.000 1.13588 0.567942 0.823068i \(-0.307740\pi\)
0.567942 + 0.823068i \(0.307740\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 252.000 0.691903
\(52\) 0 0
\(53\) −768.000 −1.99043 −0.995216 0.0976975i \(-0.968852\pi\)
−0.995216 + 0.0976975i \(0.968852\pi\)
\(54\) 0 0
\(55\) −180.000 −0.441294
\(56\) 0 0
\(57\) −291.000 −0.676209
\(58\) 0 0
\(59\) 264.000 0.582540 0.291270 0.956641i \(-0.405922\pi\)
0.291270 + 0.956641i \(0.405922\pi\)
\(60\) 0 0
\(61\) 818.000 1.71695 0.858477 0.512852i \(-0.171411\pi\)
0.858477 + 0.512852i \(0.171411\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −318.000 −0.606816
\(66\) 0 0
\(67\) 523.000 0.953651 0.476826 0.878998i \(-0.341787\pi\)
0.476826 + 0.878998i \(0.341787\pi\)
\(68\) 0 0
\(69\) 252.000 0.439670
\(70\) 0 0
\(71\) 342.000 0.571661 0.285831 0.958280i \(-0.407731\pi\)
0.285831 + 0.958280i \(0.407731\pi\)
\(72\) 0 0
\(73\) −43.0000 −0.0689420 −0.0344710 0.999406i \(-0.510975\pi\)
−0.0344710 + 0.999406i \(0.510975\pi\)
\(74\) 0 0
\(75\) 267.000 0.411073
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1171.00 1.66769 0.833847 0.551996i \(-0.186134\pi\)
0.833847 + 0.551996i \(0.186134\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 810.000 1.07119 0.535597 0.844474i \(-0.320087\pi\)
0.535597 + 0.844474i \(0.320087\pi\)
\(84\) 0 0
\(85\) 504.000 0.643135
\(86\) 0 0
\(87\) 540.000 0.665449
\(88\) 0 0
\(89\) −600.000 −0.714605 −0.357303 0.933989i \(-0.616304\pi\)
−0.357303 + 0.933989i \(0.616304\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 537.000 0.598756
\(94\) 0 0
\(95\) −582.000 −0.628547
\(96\) 0 0
\(97\) 386.000 0.404045 0.202022 0.979381i \(-0.435249\pi\)
0.202022 + 0.979381i \(0.435249\pi\)
\(98\) 0 0
\(99\) 270.000 0.274101
\(100\) 0 0
\(101\) 618.000 0.608845 0.304422 0.952537i \(-0.401537\pi\)
0.304422 + 0.952537i \(0.401537\pi\)
\(102\) 0 0
\(103\) −1475.00 −1.41103 −0.705515 0.708695i \(-0.749284\pi\)
−0.705515 + 0.708695i \(0.749284\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1884.00 −1.70218 −0.851090 0.525021i \(-0.824058\pi\)
−0.851090 + 0.525021i \(0.824058\pi\)
\(108\) 0 0
\(109\) 413.000 0.362920 0.181460 0.983398i \(-0.441918\pi\)
0.181460 + 0.983398i \(0.441918\pi\)
\(110\) 0 0
\(111\) 435.000 0.371967
\(112\) 0 0
\(113\) −882.000 −0.734262 −0.367131 0.930169i \(-0.619660\pi\)
−0.367131 + 0.930169i \(0.619660\pi\)
\(114\) 0 0
\(115\) 504.000 0.408680
\(116\) 0 0
\(117\) 477.000 0.376912
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −431.000 −0.323817
\(122\) 0 0
\(123\) −378.000 −0.277098
\(124\) 0 0
\(125\) 1284.00 0.918756
\(126\) 0 0
\(127\) −2483.00 −1.73489 −0.867443 0.497536i \(-0.834238\pi\)
−0.867443 + 0.497536i \(0.834238\pi\)
\(128\) 0 0
\(129\) −975.000 −0.665457
\(130\) 0 0
\(131\) −2118.00 −1.41260 −0.706300 0.707913i \(-0.749637\pi\)
−0.706300 + 0.707913i \(0.749637\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 162.000 0.103280
\(136\) 0 0
\(137\) 3012.00 1.87834 0.939170 0.343453i \(-0.111597\pi\)
0.939170 + 0.343453i \(0.111597\pi\)
\(138\) 0 0
\(139\) 37.0000 0.0225777 0.0112888 0.999936i \(-0.496407\pi\)
0.0112888 + 0.999936i \(0.496407\pi\)
\(140\) 0 0
\(141\) −1098.00 −0.655803
\(142\) 0 0
\(143\) 1590.00 0.929808
\(144\) 0 0
\(145\) 1080.00 0.618546
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1644.00 −0.903904 −0.451952 0.892042i \(-0.649272\pi\)
−0.451952 + 0.892042i \(0.649272\pi\)
\(150\) 0 0
\(151\) −1088.00 −0.586359 −0.293179 0.956057i \(-0.594713\pi\)
−0.293179 + 0.956057i \(0.594713\pi\)
\(152\) 0 0
\(153\) −756.000 −0.399470
\(154\) 0 0
\(155\) 1074.00 0.556553
\(156\) 0 0
\(157\) 506.000 0.257218 0.128609 0.991695i \(-0.458949\pi\)
0.128609 + 0.991695i \(0.458949\pi\)
\(158\) 0 0
\(159\) 2304.00 1.14918
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1844.00 −0.886093 −0.443047 0.896499i \(-0.646102\pi\)
−0.443047 + 0.896499i \(0.646102\pi\)
\(164\) 0 0
\(165\) 540.000 0.254781
\(166\) 0 0
\(167\) −162.000 −0.0750655 −0.0375327 0.999295i \(-0.511950\pi\)
−0.0375327 + 0.999295i \(0.511950\pi\)
\(168\) 0 0
\(169\) 612.000 0.278562
\(170\) 0 0
\(171\) 873.000 0.390409
\(172\) 0 0
\(173\) −2724.00 −1.19712 −0.598560 0.801078i \(-0.704260\pi\)
−0.598560 + 0.801078i \(0.704260\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −792.000 −0.336330
\(178\) 0 0
\(179\) 1254.00 0.523622 0.261811 0.965119i \(-0.415680\pi\)
0.261811 + 0.965119i \(0.415680\pi\)
\(180\) 0 0
\(181\) −1807.00 −0.742062 −0.371031 0.928620i \(-0.620996\pi\)
−0.371031 + 0.928620i \(0.620996\pi\)
\(182\) 0 0
\(183\) −2454.00 −0.991284
\(184\) 0 0
\(185\) 870.000 0.345750
\(186\) 0 0
\(187\) −2520.00 −0.985458
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −714.000 −0.270488 −0.135244 0.990812i \(-0.543182\pi\)
−0.135244 + 0.990812i \(0.543182\pi\)
\(192\) 0 0
\(193\) −3709.00 −1.38331 −0.691657 0.722226i \(-0.743119\pi\)
−0.691657 + 0.722226i \(0.743119\pi\)
\(194\) 0 0
\(195\) 954.000 0.350345
\(196\) 0 0
\(197\) −1044.00 −0.377573 −0.188787 0.982018i \(-0.560455\pi\)
−0.188787 + 0.982018i \(0.560455\pi\)
\(198\) 0 0
\(199\) 136.000 0.0484462 0.0242231 0.999707i \(-0.492289\pi\)
0.0242231 + 0.999707i \(0.492289\pi\)
\(200\) 0 0
\(201\) −1569.00 −0.550591
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −756.000 −0.257567
\(206\) 0 0
\(207\) −756.000 −0.253844
\(208\) 0 0
\(209\) 2910.00 0.963105
\(210\) 0 0
\(211\) −1484.00 −0.484184 −0.242092 0.970253i \(-0.577834\pi\)
−0.242092 + 0.970253i \(0.577834\pi\)
\(212\) 0 0
\(213\) −1026.00 −0.330049
\(214\) 0 0
\(215\) −1950.00 −0.618553
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 129.000 0.0398037
\(220\) 0 0
\(221\) −4452.00 −1.35509
\(222\) 0 0
\(223\) 2032.00 0.610192 0.305096 0.952322i \(-0.401311\pi\)
0.305096 + 0.952322i \(0.401311\pi\)
\(224\) 0 0
\(225\) −801.000 −0.237333
\(226\) 0 0
\(227\) −6198.00 −1.81223 −0.906114 0.423034i \(-0.860965\pi\)
−0.906114 + 0.423034i \(0.860965\pi\)
\(228\) 0 0
\(229\) −4591.00 −1.32481 −0.662406 0.749145i \(-0.730464\pi\)
−0.662406 + 0.749145i \(0.730464\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4530.00 1.27369 0.636846 0.770991i \(-0.280239\pi\)
0.636846 + 0.770991i \(0.280239\pi\)
\(234\) 0 0
\(235\) −2196.00 −0.609580
\(236\) 0 0
\(237\) −3513.00 −0.962843
\(238\) 0 0
\(239\) −1530.00 −0.414090 −0.207045 0.978331i \(-0.566385\pi\)
−0.207045 + 0.978331i \(0.566385\pi\)
\(240\) 0 0
\(241\) 5534.00 1.47915 0.739577 0.673072i \(-0.235025\pi\)
0.739577 + 0.673072i \(0.235025\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5141.00 1.32435
\(248\) 0 0
\(249\) −2430.00 −0.618454
\(250\) 0 0
\(251\) 468.000 0.117689 0.0588444 0.998267i \(-0.481258\pi\)
0.0588444 + 0.998267i \(0.481258\pi\)
\(252\) 0 0
\(253\) −2520.00 −0.626210
\(254\) 0 0
\(255\) −1512.00 −0.371314
\(256\) 0 0
\(257\) −2490.00 −0.604365 −0.302183 0.953250i \(-0.597715\pi\)
−0.302183 + 0.953250i \(0.597715\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1620.00 −0.384197
\(262\) 0 0
\(263\) −1572.00 −0.368569 −0.184285 0.982873i \(-0.558997\pi\)
−0.184285 + 0.982873i \(0.558997\pi\)
\(264\) 0 0
\(265\) 4608.00 1.06818
\(266\) 0 0
\(267\) 1800.00 0.412578
\(268\) 0 0
\(269\) 1806.00 0.409345 0.204672 0.978831i \(-0.434387\pi\)
0.204672 + 0.978831i \(0.434387\pi\)
\(270\) 0 0
\(271\) 6112.00 1.37003 0.685014 0.728530i \(-0.259796\pi\)
0.685014 + 0.728530i \(0.259796\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2670.00 −0.585480
\(276\) 0 0
\(277\) −4231.00 −0.917748 −0.458874 0.888501i \(-0.651747\pi\)
−0.458874 + 0.888501i \(0.651747\pi\)
\(278\) 0 0
\(279\) −1611.00 −0.345692
\(280\) 0 0
\(281\) −3816.00 −0.810119 −0.405060 0.914290i \(-0.632749\pi\)
−0.405060 + 0.914290i \(0.632749\pi\)
\(282\) 0 0
\(283\) 3997.00 0.839565 0.419783 0.907625i \(-0.362106\pi\)
0.419783 + 0.907625i \(0.362106\pi\)
\(284\) 0 0
\(285\) 1746.00 0.362892
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2143.00 0.436190
\(290\) 0 0
\(291\) −1158.00 −0.233275
\(292\) 0 0
\(293\) 4608.00 0.918779 0.459389 0.888235i \(-0.348068\pi\)
0.459389 + 0.888235i \(0.348068\pi\)
\(294\) 0 0
\(295\) −1584.00 −0.312624
\(296\) 0 0
\(297\) −810.000 −0.158252
\(298\) 0 0
\(299\) −4452.00 −0.861090
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1854.00 −0.351517
\(304\) 0 0
\(305\) −4908.00 −0.921414
\(306\) 0 0
\(307\) 631.000 0.117306 0.0586532 0.998278i \(-0.481319\pi\)
0.0586532 + 0.998278i \(0.481319\pi\)
\(308\) 0 0
\(309\) 4425.00 0.814658
\(310\) 0 0
\(311\) −3894.00 −0.709995 −0.354998 0.934867i \(-0.615518\pi\)
−0.354998 + 0.934867i \(0.615518\pi\)
\(312\) 0 0
\(313\) −2185.00 −0.394580 −0.197290 0.980345i \(-0.563214\pi\)
−0.197290 + 0.980345i \(0.563214\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3504.00 0.620834 0.310417 0.950601i \(-0.399531\pi\)
0.310417 + 0.950601i \(0.399531\pi\)
\(318\) 0 0
\(319\) −5400.00 −0.947780
\(320\) 0 0
\(321\) 5652.00 0.982754
\(322\) 0 0
\(323\) −8148.00 −1.40361
\(324\) 0 0
\(325\) −4717.00 −0.805083
\(326\) 0 0
\(327\) −1239.00 −0.209532
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2945.00 −0.489039 −0.244519 0.969644i \(-0.578630\pi\)
−0.244519 + 0.969644i \(0.578630\pi\)
\(332\) 0 0
\(333\) −1305.00 −0.214755
\(334\) 0 0
\(335\) −3138.00 −0.511783
\(336\) 0 0
\(337\) 4277.00 0.691344 0.345672 0.938355i \(-0.387651\pi\)
0.345672 + 0.938355i \(0.387651\pi\)
\(338\) 0 0
\(339\) 2646.00 0.423926
\(340\) 0 0
\(341\) −5370.00 −0.852791
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1512.00 −0.235952
\(346\) 0 0
\(347\) −7188.00 −1.11202 −0.556012 0.831175i \(-0.687669\pi\)
−0.556012 + 0.831175i \(0.687669\pi\)
\(348\) 0 0
\(349\) −9406.00 −1.44267 −0.721335 0.692587i \(-0.756471\pi\)
−0.721335 + 0.692587i \(0.756471\pi\)
\(350\) 0 0
\(351\) −1431.00 −0.217610
\(352\) 0 0
\(353\) 3390.00 0.511137 0.255569 0.966791i \(-0.417737\pi\)
0.255569 + 0.966791i \(0.417737\pi\)
\(354\) 0 0
\(355\) −2052.00 −0.306785
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4812.00 0.707431 0.353715 0.935353i \(-0.384918\pi\)
0.353715 + 0.935353i \(0.384918\pi\)
\(360\) 0 0
\(361\) 2550.00 0.371774
\(362\) 0 0
\(363\) 1293.00 0.186956
\(364\) 0 0
\(365\) 258.000 0.0369982
\(366\) 0 0
\(367\) 7099.00 1.00971 0.504857 0.863203i \(-0.331545\pi\)
0.504857 + 0.863203i \(0.331545\pi\)
\(368\) 0 0
\(369\) 1134.00 0.159983
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 2963.00 0.411309 0.205655 0.978625i \(-0.434068\pi\)
0.205655 + 0.978625i \(0.434068\pi\)
\(374\) 0 0
\(375\) −3852.00 −0.530444
\(376\) 0 0
\(377\) −9540.00 −1.30328
\(378\) 0 0
\(379\) 11899.0 1.61269 0.806346 0.591444i \(-0.201442\pi\)
0.806346 + 0.591444i \(0.201442\pi\)
\(380\) 0 0
\(381\) 7449.00 1.00164
\(382\) 0 0
\(383\) −2568.00 −0.342607 −0.171304 0.985218i \(-0.554798\pi\)
−0.171304 + 0.985218i \(0.554798\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2925.00 0.384202
\(388\) 0 0
\(389\) −10146.0 −1.32242 −0.661212 0.750199i \(-0.729957\pi\)
−0.661212 + 0.750199i \(0.729957\pi\)
\(390\) 0 0
\(391\) 7056.00 0.912627
\(392\) 0 0
\(393\) 6354.00 0.815565
\(394\) 0 0
\(395\) −7026.00 −0.894978
\(396\) 0 0
\(397\) −6229.00 −0.787467 −0.393734 0.919225i \(-0.628817\pi\)
−0.393734 + 0.919225i \(0.628817\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2472.00 −0.307845 −0.153922 0.988083i \(-0.549191\pi\)
−0.153922 + 0.988083i \(0.549191\pi\)
\(402\) 0 0
\(403\) −9487.00 −1.17266
\(404\) 0 0
\(405\) −486.000 −0.0596285
\(406\) 0 0
\(407\) −4350.00 −0.529783
\(408\) 0 0
\(409\) −7075.00 −0.855345 −0.427673 0.903934i \(-0.640666\pi\)
−0.427673 + 0.903934i \(0.640666\pi\)
\(410\) 0 0
\(411\) −9036.00 −1.08446
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4860.00 −0.574863
\(416\) 0 0
\(417\) −111.000 −0.0130352
\(418\) 0 0
\(419\) 4158.00 0.484801 0.242400 0.970176i \(-0.422065\pi\)
0.242400 + 0.970176i \(0.422065\pi\)
\(420\) 0 0
\(421\) −6595.00 −0.763469 −0.381735 0.924272i \(-0.624673\pi\)
−0.381735 + 0.924272i \(0.624673\pi\)
\(422\) 0 0
\(423\) 3294.00 0.378628
\(424\) 0 0
\(425\) 7476.00 0.853269
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −4770.00 −0.536825
\(430\) 0 0
\(431\) −1518.00 −0.169651 −0.0848254 0.996396i \(-0.527033\pi\)
−0.0848254 + 0.996396i \(0.527033\pi\)
\(432\) 0 0
\(433\) 8567.00 0.950817 0.475408 0.879765i \(-0.342300\pi\)
0.475408 + 0.879765i \(0.342300\pi\)
\(434\) 0 0
\(435\) −3240.00 −0.357117
\(436\) 0 0
\(437\) −8148.00 −0.891926
\(438\) 0 0
\(439\) −10640.0 −1.15676 −0.578382 0.815766i \(-0.696316\pi\)
−0.578382 + 0.815766i \(0.696316\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7032.00 −0.754177 −0.377088 0.926177i \(-0.623075\pi\)
−0.377088 + 0.926177i \(0.623075\pi\)
\(444\) 0 0
\(445\) 3600.00 0.383497
\(446\) 0 0
\(447\) 4932.00 0.521869
\(448\) 0 0
\(449\) −14814.0 −1.55705 −0.778525 0.627613i \(-0.784032\pi\)
−0.778525 + 0.627613i \(0.784032\pi\)
\(450\) 0 0
\(451\) 3780.00 0.394664
\(452\) 0 0
\(453\) 3264.00 0.338534
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −11251.0 −1.15164 −0.575820 0.817576i \(-0.695317\pi\)
−0.575820 + 0.817576i \(0.695317\pi\)
\(458\) 0 0
\(459\) 2268.00 0.230634
\(460\) 0 0
\(461\) −3852.00 −0.389166 −0.194583 0.980886i \(-0.562335\pi\)
−0.194583 + 0.980886i \(0.562335\pi\)
\(462\) 0 0
\(463\) 475.000 0.0476784 0.0238392 0.999716i \(-0.492411\pi\)
0.0238392 + 0.999716i \(0.492411\pi\)
\(464\) 0 0
\(465\) −3222.00 −0.321326
\(466\) 0 0
\(467\) −5934.00 −0.587993 −0.293997 0.955806i \(-0.594985\pi\)
−0.293997 + 0.955806i \(0.594985\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −1518.00 −0.148505
\(472\) 0 0
\(473\) 9750.00 0.947792
\(474\) 0 0
\(475\) −8633.00 −0.833914
\(476\) 0 0
\(477\) −6912.00 −0.663477
\(478\) 0 0
\(479\) 13368.0 1.27516 0.637578 0.770386i \(-0.279936\pi\)
0.637578 + 0.770386i \(0.279936\pi\)
\(480\) 0 0
\(481\) −7685.00 −0.728494
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2316.00 −0.216833
\(486\) 0 0
\(487\) −6653.00 −0.619048 −0.309524 0.950892i \(-0.600170\pi\)
−0.309524 + 0.950892i \(0.600170\pi\)
\(488\) 0 0
\(489\) 5532.00 0.511586
\(490\) 0 0
\(491\) −15444.0 −1.41951 −0.709754 0.704450i \(-0.751194\pi\)
−0.709754 + 0.704450i \(0.751194\pi\)
\(492\) 0 0
\(493\) 15120.0 1.38128
\(494\) 0 0
\(495\) −1620.00 −0.147098
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −683.000 −0.0612731 −0.0306366 0.999531i \(-0.509753\pi\)
−0.0306366 + 0.999531i \(0.509753\pi\)
\(500\) 0 0
\(501\) 486.000 0.0433391
\(502\) 0 0
\(503\) −9882.00 −0.875977 −0.437989 0.898980i \(-0.644309\pi\)
−0.437989 + 0.898980i \(0.644309\pi\)
\(504\) 0 0
\(505\) −3708.00 −0.326740
\(506\) 0 0
\(507\) −1836.00 −0.160828
\(508\) 0 0
\(509\) 4206.00 0.366263 0.183131 0.983088i \(-0.441377\pi\)
0.183131 + 0.983088i \(0.441377\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2619.00 −0.225403
\(514\) 0 0
\(515\) 8850.00 0.757238
\(516\) 0 0
\(517\) 10980.0 0.934042
\(518\) 0 0
\(519\) 8172.00 0.691158
\(520\) 0 0
\(521\) 9060.00 0.761854 0.380927 0.924605i \(-0.375605\pi\)
0.380927 + 0.924605i \(0.375605\pi\)
\(522\) 0 0
\(523\) 15679.0 1.31089 0.655444 0.755243i \(-0.272481\pi\)
0.655444 + 0.755243i \(0.272481\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 15036.0 1.24284
\(528\) 0 0
\(529\) −5111.00 −0.420071
\(530\) 0 0
\(531\) 2376.00 0.194180
\(532\) 0 0
\(533\) 6678.00 0.542695
\(534\) 0 0
\(535\) 11304.0 0.913485
\(536\) 0 0
\(537\) −3762.00 −0.302313
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7711.00 −0.612794 −0.306397 0.951904i \(-0.599124\pi\)
−0.306397 + 0.951904i \(0.599124\pi\)
\(542\) 0 0
\(543\) 5421.00 0.428430
\(544\) 0 0
\(545\) −2478.00 −0.194763
\(546\) 0 0
\(547\) −4292.00 −0.335489 −0.167745 0.985830i \(-0.553648\pi\)
−0.167745 + 0.985830i \(0.553648\pi\)
\(548\) 0 0
\(549\) 7362.00 0.572318
\(550\) 0 0
\(551\) −17460.0 −1.34995
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −2610.00 −0.199619
\(556\) 0 0
\(557\) −9858.00 −0.749905 −0.374952 0.927044i \(-0.622341\pi\)
−0.374952 + 0.927044i \(0.622341\pi\)
\(558\) 0 0
\(559\) 17225.0 1.30329
\(560\) 0 0
\(561\) 7560.00 0.568954
\(562\) 0 0
\(563\) 13890.0 1.03978 0.519888 0.854235i \(-0.325974\pi\)
0.519888 + 0.854235i \(0.325974\pi\)
\(564\) 0 0
\(565\) 5292.00 0.394046
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 19038.0 1.40266 0.701331 0.712836i \(-0.252590\pi\)
0.701331 + 0.712836i \(0.252590\pi\)
\(570\) 0 0
\(571\) 8053.00 0.590206 0.295103 0.955465i \(-0.404646\pi\)
0.295103 + 0.955465i \(0.404646\pi\)
\(572\) 0 0
\(573\) 2142.00 0.156166
\(574\) 0 0
\(575\) 7476.00 0.542210
\(576\) 0 0
\(577\) −17137.0 −1.23643 −0.618217 0.786007i \(-0.712145\pi\)
−0.618217 + 0.786007i \(0.712145\pi\)
\(578\) 0 0
\(579\) 11127.0 0.798657
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −23040.0 −1.63674
\(584\) 0 0
\(585\) −2862.00 −0.202272
\(586\) 0 0
\(587\) −18144.0 −1.27578 −0.637890 0.770127i \(-0.720193\pi\)
−0.637890 + 0.770127i \(0.720193\pi\)
\(588\) 0 0
\(589\) −17363.0 −1.21465
\(590\) 0 0
\(591\) 3132.00 0.217992
\(592\) 0 0
\(593\) −24702.0 −1.71061 −0.855303 0.518128i \(-0.826629\pi\)
−0.855303 + 0.518128i \(0.826629\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −408.000 −0.0279704
\(598\) 0 0
\(599\) 2172.00 0.148156 0.0740781 0.997252i \(-0.476399\pi\)
0.0740781 + 0.997252i \(0.476399\pi\)
\(600\) 0 0
\(601\) 4175.00 0.283364 0.141682 0.989912i \(-0.454749\pi\)
0.141682 + 0.989912i \(0.454749\pi\)
\(602\) 0 0
\(603\) 4707.00 0.317884
\(604\) 0 0
\(605\) 2586.00 0.173778
\(606\) 0 0
\(607\) −2261.00 −0.151188 −0.0755940 0.997139i \(-0.524085\pi\)
−0.0755940 + 0.997139i \(0.524085\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 19398.0 1.28438
\(612\) 0 0
\(613\) −16318.0 −1.07517 −0.537584 0.843210i \(-0.680663\pi\)
−0.537584 + 0.843210i \(0.680663\pi\)
\(614\) 0 0
\(615\) 2268.00 0.148707
\(616\) 0 0
\(617\) −26550.0 −1.73235 −0.866177 0.499737i \(-0.833430\pi\)
−0.866177 + 0.499737i \(0.833430\pi\)
\(618\) 0 0
\(619\) −19925.0 −1.29379 −0.646893 0.762581i \(-0.723932\pi\)
−0.646893 + 0.762581i \(0.723932\pi\)
\(620\) 0 0
\(621\) 2268.00 0.146557
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) 0 0
\(627\) −8730.00 −0.556049
\(628\) 0 0
\(629\) 12180.0 0.772096
\(630\) 0 0
\(631\) 6832.00 0.431026 0.215513 0.976501i \(-0.430858\pi\)
0.215513 + 0.976501i \(0.430858\pi\)
\(632\) 0 0
\(633\) 4452.00 0.279544
\(634\) 0 0
\(635\) 14898.0 0.931038
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3078.00 0.190554
\(640\) 0 0
\(641\) 10212.0 0.629251 0.314625 0.949216i \(-0.398121\pi\)
0.314625 + 0.949216i \(0.398121\pi\)
\(642\) 0 0
\(643\) −3779.00 −0.231772 −0.115886 0.993263i \(-0.536971\pi\)
−0.115886 + 0.993263i \(0.536971\pi\)
\(644\) 0 0
\(645\) 5850.00 0.357122
\(646\) 0 0
\(647\) −16998.0 −1.03286 −0.516430 0.856329i \(-0.672739\pi\)
−0.516430 + 0.856329i \(0.672739\pi\)
\(648\) 0 0
\(649\) 7920.00 0.479025
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −21750.0 −1.30344 −0.651718 0.758462i \(-0.725951\pi\)
−0.651718 + 0.758462i \(0.725951\pi\)
\(654\) 0 0
\(655\) 12708.0 0.758080
\(656\) 0 0
\(657\) −387.000 −0.0229807
\(658\) 0 0
\(659\) 10944.0 0.646916 0.323458 0.946243i \(-0.395155\pi\)
0.323458 + 0.946243i \(0.395155\pi\)
\(660\) 0 0
\(661\) 10955.0 0.644630 0.322315 0.946633i \(-0.395539\pi\)
0.322315 + 0.946633i \(0.395539\pi\)
\(662\) 0 0
\(663\) 13356.0 0.782359
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 15120.0 0.877734
\(668\) 0 0
\(669\) −6096.00 −0.352294
\(670\) 0 0
\(671\) 24540.0 1.41186
\(672\) 0 0
\(673\) 25103.0 1.43782 0.718908 0.695106i \(-0.244642\pi\)
0.718908 + 0.695106i \(0.244642\pi\)
\(674\) 0 0
\(675\) 2403.00 0.137024
\(676\) 0 0
\(677\) −5604.00 −0.318138 −0.159069 0.987267i \(-0.550849\pi\)
−0.159069 + 0.987267i \(0.550849\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 18594.0 1.04629
\(682\) 0 0
\(683\) −10968.0 −0.614464 −0.307232 0.951635i \(-0.599403\pi\)
−0.307232 + 0.951635i \(0.599403\pi\)
\(684\) 0 0
\(685\) −18072.0 −1.00802
\(686\) 0 0
\(687\) 13773.0 0.764880
\(688\) 0 0
\(689\) −40704.0 −2.25065
\(690\) 0 0
\(691\) −8405.00 −0.462723 −0.231361 0.972868i \(-0.574318\pi\)
−0.231361 + 0.972868i \(0.574318\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −222.000 −0.0121165
\(696\) 0 0
\(697\) −10584.0 −0.575176
\(698\) 0 0
\(699\) −13590.0 −0.735366
\(700\) 0 0
\(701\) 468.000 0.0252156 0.0126078 0.999921i \(-0.495987\pi\)
0.0126078 + 0.999921i \(0.495987\pi\)
\(702\) 0 0
\(703\) −14065.0 −0.754583
\(704\) 0 0
\(705\) 6588.00 0.351941
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −25066.0 −1.32775 −0.663874 0.747844i \(-0.731089\pi\)
−0.663874 + 0.747844i \(0.731089\pi\)
\(710\) 0 0
\(711\) 10539.0 0.555898
\(712\) 0 0
\(713\) 15036.0 0.789765
\(714\) 0 0
\(715\) −9540.00 −0.498987
\(716\) 0 0
\(717\) 4590.00 0.239075
\(718\) 0 0
\(719\) −11082.0 −0.574811 −0.287405 0.957809i \(-0.592793\pi\)
−0.287405 + 0.957809i \(0.592793\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −16602.0 −0.853990
\(724\) 0 0
\(725\) 16020.0 0.820645
\(726\) 0 0
\(727\) −13481.0 −0.687734 −0.343867 0.939018i \(-0.611737\pi\)
−0.343867 + 0.939018i \(0.611737\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −27300.0 −1.38130
\(732\) 0 0
\(733\) 24317.0 1.22533 0.612666 0.790342i \(-0.290097\pi\)
0.612666 + 0.790342i \(0.290097\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 15690.0 0.784191
\(738\) 0 0
\(739\) 18217.0 0.906797 0.453399 0.891308i \(-0.350211\pi\)
0.453399 + 0.891308i \(0.350211\pi\)
\(740\) 0 0
\(741\) −15423.0 −0.764613
\(742\) 0 0
\(743\) −19782.0 −0.976758 −0.488379 0.872632i \(-0.662412\pi\)
−0.488379 + 0.872632i \(0.662412\pi\)
\(744\) 0 0
\(745\) 9864.00 0.485086
\(746\) 0 0
\(747\) 7290.00 0.357064
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4921.00 0.239108 0.119554 0.992828i \(-0.461854\pi\)
0.119554 + 0.992828i \(0.461854\pi\)
\(752\) 0 0
\(753\) −1404.00 −0.0679477
\(754\) 0 0
\(755\) 6528.00 0.314673
\(756\) 0 0
\(757\) 18098.0 0.868934 0.434467 0.900688i \(-0.356937\pi\)
0.434467 + 0.900688i \(0.356937\pi\)
\(758\) 0 0
\(759\) 7560.00 0.361542
\(760\) 0 0
\(761\) −24468.0 −1.16552 −0.582762 0.812643i \(-0.698028\pi\)
−0.582762 + 0.812643i \(0.698028\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 4536.00 0.214378
\(766\) 0 0
\(767\) 13992.0 0.658699
\(768\) 0 0
\(769\) 21719.0 1.01847 0.509237 0.860626i \(-0.329928\pi\)
0.509237 + 0.860626i \(0.329928\pi\)
\(770\) 0 0
\(771\) 7470.00 0.348931
\(772\) 0 0
\(773\) −30306.0 −1.41013 −0.705065 0.709142i \(-0.749082\pi\)
−0.705065 + 0.709142i \(0.749082\pi\)
\(774\) 0 0
\(775\) 15931.0 0.738398
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12222.0 0.562129
\(780\) 0 0
\(781\) 10260.0 0.470079
\(782\) 0 0
\(783\) 4860.00 0.221816
\(784\) 0 0
\(785\) −3036.00 −0.138038
\(786\) 0 0
\(787\) −27296.0 −1.23634 −0.618169 0.786046i \(-0.712125\pi\)
−0.618169 + 0.786046i \(0.712125\pi\)
\(788\) 0 0
\(789\) 4716.00 0.212793
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 43354.0 1.94142
\(794\) 0 0
\(795\) −13824.0 −0.616713
\(796\) 0 0
\(797\) −35100.0 −1.55998 −0.779991 0.625791i \(-0.784776\pi\)
−0.779991 + 0.625791i \(0.784776\pi\)
\(798\) 0 0
\(799\) −30744.0 −1.36126
\(800\) 0 0
\(801\) −5400.00 −0.238202
\(802\) 0 0
\(803\) −1290.00 −0.0566913
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −5418.00 −0.236335
\(808\) 0 0
\(809\) 44394.0 1.92931 0.964654 0.263520i \(-0.0848836\pi\)
0.964654 + 0.263520i \(0.0848836\pi\)
\(810\) 0 0
\(811\) 8584.00 0.371671 0.185835 0.982581i \(-0.440501\pi\)
0.185835 + 0.982581i \(0.440501\pi\)
\(812\) 0 0
\(813\) −18336.0 −0.790986
\(814\) 0 0
\(815\) 11064.0 0.475528
\(816\) 0 0
\(817\) 31525.0 1.34996
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9834.00 −0.418038 −0.209019 0.977912i \(-0.567027\pi\)
−0.209019 + 0.977912i \(0.567027\pi\)
\(822\) 0 0
\(823\) −43856.0 −1.85750 −0.928751 0.370704i \(-0.879116\pi\)
−0.928751 + 0.370704i \(0.879116\pi\)
\(824\) 0 0
\(825\) 8010.00 0.338027
\(826\) 0 0
\(827\) −13266.0 −0.557804 −0.278902 0.960320i \(-0.589970\pi\)
−0.278902 + 0.960320i \(0.589970\pi\)
\(828\) 0 0
\(829\) 17453.0 0.731204 0.365602 0.930771i \(-0.380863\pi\)
0.365602 + 0.930771i \(0.380863\pi\)
\(830\) 0 0
\(831\) 12693.0 0.529862
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 972.000 0.0402844
\(836\) 0 0
\(837\) 4833.00 0.199585
\(838\) 0 0
\(839\) 35172.0 1.44729 0.723643 0.690175i \(-0.242466\pi\)
0.723643 + 0.690175i \(0.242466\pi\)
\(840\) 0 0
\(841\) 8011.00 0.328468
\(842\) 0 0
\(843\) 11448.0 0.467722
\(844\) 0 0
\(845\) −3672.00 −0.149492
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −11991.0 −0.484723
\(850\) 0 0
\(851\) 12180.0 0.490629
\(852\) 0 0
\(853\) 3503.00 0.140610 0.0703051 0.997526i \(-0.477603\pi\)
0.0703051 + 0.997526i \(0.477603\pi\)
\(854\) 0 0
\(855\) −5238.00 −0.209516
\(856\) 0 0
\(857\) 22848.0 0.910703 0.455352 0.890312i \(-0.349514\pi\)
0.455352 + 0.890312i \(0.349514\pi\)
\(858\) 0 0
\(859\) 13456.0 0.534474 0.267237 0.963631i \(-0.413889\pi\)
0.267237 + 0.963631i \(0.413889\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −40710.0 −1.60578 −0.802888 0.596130i \(-0.796704\pi\)
−0.802888 + 0.596130i \(0.796704\pi\)
\(864\) 0 0
\(865\) 16344.0 0.642442
\(866\) 0 0
\(867\) −6429.00 −0.251834
\(868\) 0 0
\(869\) 35130.0 1.37135
\(870\) 0 0
\(871\) 27719.0 1.07833
\(872\) 0 0
\(873\) 3474.00 0.134682
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2906.00 0.111891 0.0559456 0.998434i \(-0.482183\pi\)
0.0559456 + 0.998434i \(0.482183\pi\)
\(878\) 0 0
\(879\) −13824.0 −0.530457
\(880\) 0 0
\(881\) −19188.0 −0.733780 −0.366890 0.930264i \(-0.619577\pi\)
−0.366890 + 0.930264i \(0.619577\pi\)
\(882\) 0 0
\(883\) 17251.0 0.657466 0.328733 0.944423i \(-0.393378\pi\)
0.328733 + 0.944423i \(0.393378\pi\)
\(884\) 0 0
\(885\) 4752.00 0.180493
\(886\) 0 0
\(887\) 2094.00 0.0792668 0.0396334 0.999214i \(-0.487381\pi\)
0.0396334 + 0.999214i \(0.487381\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2430.00 0.0913671
\(892\) 0 0
\(893\) 35502.0 1.33038
\(894\) 0 0
\(895\) −7524.00 −0.281005
\(896\) 0 0
\(897\) 13356.0 0.497150
\(898\) 0 0
\(899\) 32220.0 1.19532
\(900\) 0 0
\(901\) 64512.0 2.38536
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10842.0 0.398232
\(906\) 0 0
\(907\) 40267.0 1.47414 0.737069 0.675817i \(-0.236209\pi\)
0.737069 + 0.675817i \(0.236209\pi\)
\(908\) 0 0
\(909\) 5562.00 0.202948
\(910\) 0 0
\(911\) −17604.0 −0.640227 −0.320113 0.947379i \(-0.603721\pi\)
−0.320113 + 0.947379i \(0.603721\pi\)
\(912\) 0 0
\(913\) 24300.0 0.880846
\(914\) 0 0
\(915\) 14724.0 0.531979
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −3509.00 −0.125953 −0.0629767 0.998015i \(-0.520059\pi\)
−0.0629767 + 0.998015i \(0.520059\pi\)
\(920\) 0 0
\(921\) −1893.00 −0.0677269
\(922\) 0 0
\(923\) 18126.0 0.646397
\(924\) 0 0
\(925\) 12905.0 0.458718
\(926\) 0 0
\(927\) −13275.0 −0.470343
\(928\) 0 0
\(929\) −34638.0 −1.22329 −0.611645 0.791133i \(-0.709492\pi\)
−0.611645 + 0.791133i \(0.709492\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 11682.0 0.409916
\(934\) 0 0
\(935\) 15120.0 0.528852
\(936\) 0 0
\(937\) −17353.0 −0.605014 −0.302507 0.953147i \(-0.597824\pi\)
−0.302507 + 0.953147i \(0.597824\pi\)
\(938\) 0 0
\(939\) 6555.00 0.227811
\(940\) 0 0
\(941\) −46920.0 −1.62545 −0.812725 0.582648i \(-0.802017\pi\)
−0.812725 + 0.582648i \(0.802017\pi\)
\(942\) 0 0
\(943\) −10584.0 −0.365496
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −18354.0 −0.629804 −0.314902 0.949124i \(-0.601972\pi\)
−0.314902 + 0.949124i \(0.601972\pi\)
\(948\) 0 0
\(949\) −2279.00 −0.0779552
\(950\) 0 0
\(951\) −10512.0 −0.358438
\(952\) 0 0
\(953\) 35568.0 1.20898 0.604491 0.796612i \(-0.293376\pi\)
0.604491 + 0.796612i \(0.293376\pi\)
\(954\) 0 0
\(955\) 4284.00 0.145159
\(956\) 0 0
\(957\) 16200.0 0.547201
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2250.00 0.0755262
\(962\) 0 0
\(963\) −16956.0 −0.567393
\(964\) 0 0
\(965\) 22254.0 0.742364
\(966\) 0 0
\(967\) 27343.0 0.909298 0.454649 0.890671i \(-0.349765\pi\)
0.454649 + 0.890671i \(0.349765\pi\)
\(968\) 0 0
\(969\) 24444.0 0.810376
\(970\) 0 0
\(971\) −51024.0 −1.68634 −0.843171 0.537645i \(-0.819314\pi\)
−0.843171 + 0.537645i \(0.819314\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 14151.0 0.464815
\(976\) 0 0
\(977\) −2226.00 −0.0728926 −0.0364463 0.999336i \(-0.511604\pi\)
−0.0364463 + 0.999336i \(0.511604\pi\)
\(978\) 0 0
\(979\) −18000.0 −0.587623
\(980\) 0 0
\(981\) 3717.00 0.120973
\(982\) 0 0
\(983\) −35304.0 −1.14550 −0.572748 0.819731i \(-0.694123\pi\)
−0.572748 + 0.819731i \(0.694123\pi\)
\(984\) 0 0
\(985\) 6264.00 0.202627
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −27300.0 −0.877745
\(990\) 0 0
\(991\) 2341.00 0.0750397 0.0375198 0.999296i \(-0.488054\pi\)
0.0375198 + 0.999296i \(0.488054\pi\)
\(992\) 0 0
\(993\) 8835.00 0.282347
\(994\) 0 0
\(995\) −816.000 −0.0259989
\(996\) 0 0
\(997\) 29015.0 0.921679 0.460840 0.887483i \(-0.347548\pi\)
0.460840 + 0.887483i \(0.347548\pi\)
\(998\) 0 0
\(999\) 3915.00 0.123989
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.f.1.1 1
4.3 odd 2 294.4.a.d.1.1 1
7.2 even 3 336.4.q.f.193.1 2
7.4 even 3 336.4.q.f.289.1 2
7.6 odd 2 2352.4.a.bf.1.1 1
12.11 even 2 882.4.a.o.1.1 1
28.3 even 6 294.4.e.i.79.1 2
28.11 odd 6 42.4.e.a.37.1 yes 2
28.19 even 6 294.4.e.i.67.1 2
28.23 odd 6 42.4.e.a.25.1 2
28.27 even 2 294.4.a.c.1.1 1
84.11 even 6 126.4.g.b.37.1 2
84.23 even 6 126.4.g.b.109.1 2
84.47 odd 6 882.4.g.g.361.1 2
84.59 odd 6 882.4.g.g.667.1 2
84.83 odd 2 882.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.4.e.a.25.1 2 28.23 odd 6
42.4.e.a.37.1 yes 2 28.11 odd 6
126.4.g.b.37.1 2 84.11 even 6
126.4.g.b.109.1 2 84.23 even 6
294.4.a.c.1.1 1 28.27 even 2
294.4.a.d.1.1 1 4.3 odd 2
294.4.e.i.67.1 2 28.19 even 6
294.4.e.i.79.1 2 28.3 even 6
336.4.q.f.193.1 2 7.2 even 3
336.4.q.f.289.1 2 7.4 even 3
882.4.a.l.1.1 1 84.83 odd 2
882.4.a.o.1.1 1 12.11 even 2
882.4.g.g.361.1 2 84.47 odd 6
882.4.g.g.667.1 2 84.59 odd 6
2352.4.a.f.1.1 1 1.1 even 1 trivial
2352.4.a.bf.1.1 1 7.6 odd 2