Properties

Label 1575.4.a.x.1.1
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 525)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.561553 q^{2} -7.68466 q^{4} +7.00000 q^{7} +8.80776 q^{8} +O(q^{10})\) \(q-0.561553 q^{2} -7.68466 q^{4} +7.00000 q^{7} +8.80776 q^{8} +25.8078 q^{11} +15.5464 q^{13} -3.93087 q^{14} +56.5312 q^{16} -95.1619 q^{17} -143.447 q^{19} -14.4924 q^{22} +77.5227 q^{23} -8.73012 q^{26} -53.7926 q^{28} +204.170 q^{29} -40.6998 q^{31} -102.207 q^{32} +53.4384 q^{34} -95.6695 q^{37} +80.5530 q^{38} -24.7301 q^{41} -282.430 q^{43} -198.324 q^{44} -43.5331 q^{46} +257.417 q^{47} +49.0000 q^{49} -119.469 q^{52} +257.978 q^{53} +61.6543 q^{56} -114.652 q^{58} +651.365 q^{59} +451.304 q^{61} +22.8551 q^{62} -394.855 q^{64} -832.071 q^{67} +731.287 q^{68} +174.965 q^{71} +47.4166 q^{73} +53.7235 q^{74} +1102.34 q^{76} +180.654 q^{77} -1.16006 q^{79} +13.8873 q^{82} +1494.03 q^{83} +158.599 q^{86} +227.309 q^{88} -1309.13 q^{89} +108.825 q^{91} -595.736 q^{92} -144.553 q^{94} +1365.33 q^{97} -27.5161 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - 3 q^{4} + 14 q^{7} - 3 q^{8} + O(q^{10}) \) \( 2 q + 3 q^{2} - 3 q^{4} + 14 q^{7} - 3 q^{8} + 31 q^{11} - 39 q^{13} + 21 q^{14} - 23 q^{16} - 79 q^{17} - 56 q^{19} + 4 q^{22} + 254 q^{23} - 203 q^{26} - 21 q^{28} + 62 q^{29} - 135 q^{31} - 291 q^{32} + 111 q^{34} - 113 q^{37} + 392 q^{38} - 235 q^{41} - 804 q^{43} - 174 q^{44} + 585 q^{46} + 152 q^{47} + 98 q^{49} - 375 q^{52} + 149 q^{53} - 21 q^{56} - 621 q^{58} + 441 q^{59} - 223 q^{61} - 313 q^{62} - 431 q^{64} - 1157 q^{67} + 807 q^{68} - 619 q^{71} - 268 q^{73} - 8 q^{74} + 1512 q^{76} + 217 q^{77} - 427 q^{79} - 735 q^{82} + 1211 q^{83} - 1699 q^{86} + 166 q^{88} - 466 q^{89} - 273 q^{91} + 231 q^{92} - 520 q^{94} - 172 q^{97} + 147 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.561553 −0.198539 −0.0992695 0.995061i \(-0.531651\pi\)
−0.0992695 + 0.995061i \(0.531651\pi\)
\(3\) 0 0
\(4\) −7.68466 −0.960582
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 8.80776 0.389252
\(9\) 0 0
\(10\) 0 0
\(11\) 25.8078 0.707394 0.353697 0.935360i \(-0.384924\pi\)
0.353697 + 0.935360i \(0.384924\pi\)
\(12\) 0 0
\(13\) 15.5464 0.331677 0.165838 0.986153i \(-0.446967\pi\)
0.165838 + 0.986153i \(0.446967\pi\)
\(14\) −3.93087 −0.0750407
\(15\) 0 0
\(16\) 56.5312 0.883301
\(17\) −95.1619 −1.35766 −0.678828 0.734297i \(-0.737512\pi\)
−0.678828 + 0.734297i \(0.737512\pi\)
\(18\) 0 0
\(19\) −143.447 −1.73205 −0.866026 0.499999i \(-0.833334\pi\)
−0.866026 + 0.499999i \(0.833334\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −14.4924 −0.140445
\(23\) 77.5227 0.702809 0.351405 0.936224i \(-0.385704\pi\)
0.351405 + 0.936224i \(0.385704\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −8.73012 −0.0658507
\(27\) 0 0
\(28\) −53.7926 −0.363066
\(29\) 204.170 1.30736 0.653681 0.756770i \(-0.273224\pi\)
0.653681 + 0.756770i \(0.273224\pi\)
\(30\) 0 0
\(31\) −40.6998 −0.235803 −0.117902 0.993025i \(-0.537617\pi\)
−0.117902 + 0.993025i \(0.537617\pi\)
\(32\) −102.207 −0.564621
\(33\) 0 0
\(34\) 53.4384 0.269548
\(35\) 0 0
\(36\) 0 0
\(37\) −95.6695 −0.425080 −0.212540 0.977152i \(-0.568174\pi\)
−0.212540 + 0.977152i \(0.568174\pi\)
\(38\) 80.5530 0.343880
\(39\) 0 0
\(40\) 0 0
\(41\) −24.7301 −0.0941999 −0.0471000 0.998890i \(-0.514998\pi\)
−0.0471000 + 0.998890i \(0.514998\pi\)
\(42\) 0 0
\(43\) −282.430 −1.00163 −0.500816 0.865554i \(-0.666967\pi\)
−0.500816 + 0.865554i \(0.666967\pi\)
\(44\) −198.324 −0.679510
\(45\) 0 0
\(46\) −43.5331 −0.139535
\(47\) 257.417 0.798895 0.399448 0.916756i \(-0.369202\pi\)
0.399448 + 0.916756i \(0.369202\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −119.469 −0.318603
\(53\) 257.978 0.668604 0.334302 0.942466i \(-0.391499\pi\)
0.334302 + 0.942466i \(0.391499\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 61.6543 0.147123
\(57\) 0 0
\(58\) −114.652 −0.259562
\(59\) 651.365 1.43730 0.718648 0.695374i \(-0.244761\pi\)
0.718648 + 0.695374i \(0.244761\pi\)
\(60\) 0 0
\(61\) 451.304 0.947271 0.473636 0.880721i \(-0.342941\pi\)
0.473636 + 0.880721i \(0.342941\pi\)
\(62\) 22.8551 0.0468161
\(63\) 0 0
\(64\) −394.855 −0.771201
\(65\) 0 0
\(66\) 0 0
\(67\) −832.071 −1.51722 −0.758609 0.651546i \(-0.774121\pi\)
−0.758609 + 0.651546i \(0.774121\pi\)
\(68\) 731.287 1.30414
\(69\) 0 0
\(70\) 0 0
\(71\) 174.965 0.292458 0.146229 0.989251i \(-0.453286\pi\)
0.146229 + 0.989251i \(0.453286\pi\)
\(72\) 0 0
\(73\) 47.4166 0.0760233 0.0380116 0.999277i \(-0.487898\pi\)
0.0380116 + 0.999277i \(0.487898\pi\)
\(74\) 53.7235 0.0843950
\(75\) 0 0
\(76\) 1102.34 1.66378
\(77\) 180.654 0.267370
\(78\) 0 0
\(79\) −1.16006 −0.00165211 −0.000826057 1.00000i \(-0.500263\pi\)
−0.000826057 1.00000i \(0.500263\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 13.8873 0.0187023
\(83\) 1494.03 1.97580 0.987898 0.155107i \(-0.0495722\pi\)
0.987898 + 0.155107i \(0.0495722\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 158.599 0.198863
\(87\) 0 0
\(88\) 227.309 0.275354
\(89\) −1309.13 −1.55919 −0.779593 0.626286i \(-0.784574\pi\)
−0.779593 + 0.626286i \(0.784574\pi\)
\(90\) 0 0
\(91\) 108.825 0.125362
\(92\) −595.736 −0.675106
\(93\) 0 0
\(94\) −144.553 −0.158612
\(95\) 0 0
\(96\) 0 0
\(97\) 1365.33 1.42916 0.714580 0.699553i \(-0.246618\pi\)
0.714580 + 0.699553i \(0.246618\pi\)
\(98\) −27.5161 −0.0283627
\(99\) 0 0
\(100\) 0 0
\(101\) −1155.99 −1.13887 −0.569434 0.822037i \(-0.692838\pi\)
−0.569434 + 0.822037i \(0.692838\pi\)
\(102\) 0 0
\(103\) 78.2074 0.0748156 0.0374078 0.999300i \(-0.488090\pi\)
0.0374078 + 0.999300i \(0.488090\pi\)
\(104\) 136.929 0.129106
\(105\) 0 0
\(106\) −144.868 −0.132744
\(107\) −94.0814 −0.0850018 −0.0425009 0.999096i \(-0.513533\pi\)
−0.0425009 + 0.999096i \(0.513533\pi\)
\(108\) 0 0
\(109\) −2128.25 −1.87018 −0.935088 0.354417i \(-0.884679\pi\)
−0.935088 + 0.354417i \(0.884679\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 395.719 0.333856
\(113\) −596.493 −0.496578 −0.248289 0.968686i \(-0.579868\pi\)
−0.248289 + 0.968686i \(0.579868\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1568.98 −1.25583
\(117\) 0 0
\(118\) −365.776 −0.285359
\(119\) −666.133 −0.513146
\(120\) 0 0
\(121\) −664.959 −0.499594
\(122\) −253.431 −0.188070
\(123\) 0 0
\(124\) 312.764 0.226508
\(125\) 0 0
\(126\) 0 0
\(127\) −2088.21 −1.45904 −0.729522 0.683957i \(-0.760257\pi\)
−0.729522 + 0.683957i \(0.760257\pi\)
\(128\) 1039.39 0.717735
\(129\) 0 0
\(130\) 0 0
\(131\) −416.142 −0.277546 −0.138773 0.990324i \(-0.544316\pi\)
−0.138773 + 0.990324i \(0.544316\pi\)
\(132\) 0 0
\(133\) −1004.13 −0.654654
\(134\) 467.252 0.301227
\(135\) 0 0
\(136\) −838.164 −0.528470
\(137\) 619.252 0.386177 0.193089 0.981181i \(-0.438150\pi\)
0.193089 + 0.981181i \(0.438150\pi\)
\(138\) 0 0
\(139\) −266.994 −0.162922 −0.0814610 0.996677i \(-0.525959\pi\)
−0.0814610 + 0.996677i \(0.525959\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −98.2520 −0.0580643
\(143\) 401.218 0.234626
\(144\) 0 0
\(145\) 0 0
\(146\) −26.6270 −0.0150936
\(147\) 0 0
\(148\) 735.187 0.408325
\(149\) −2551.80 −1.40303 −0.701515 0.712655i \(-0.747493\pi\)
−0.701515 + 0.712655i \(0.747493\pi\)
\(150\) 0 0
\(151\) 1117.38 0.602195 0.301097 0.953593i \(-0.402647\pi\)
0.301097 + 0.953593i \(0.402647\pi\)
\(152\) −1263.45 −0.674204
\(153\) 0 0
\(154\) −101.447 −0.0530833
\(155\) 0 0
\(156\) 0 0
\(157\) 456.104 0.231854 0.115927 0.993258i \(-0.463016\pi\)
0.115927 + 0.993258i \(0.463016\pi\)
\(158\) 0.651435 0.000328009 0
\(159\) 0 0
\(160\) 0 0
\(161\) 542.659 0.265637
\(162\) 0 0
\(163\) 105.490 0.0506907 0.0253453 0.999679i \(-0.491931\pi\)
0.0253453 + 0.999679i \(0.491931\pi\)
\(164\) 190.043 0.0904868
\(165\) 0 0
\(166\) −838.976 −0.392272
\(167\) −2482.54 −1.15033 −0.575164 0.818038i \(-0.695062\pi\)
−0.575164 + 0.818038i \(0.695062\pi\)
\(168\) 0 0
\(169\) −1955.31 −0.889991
\(170\) 0 0
\(171\) 0 0
\(172\) 2170.38 0.962150
\(173\) 516.708 0.227079 0.113539 0.993534i \(-0.463781\pi\)
0.113539 + 0.993534i \(0.463781\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1458.94 0.624842
\(177\) 0 0
\(178\) 735.146 0.309559
\(179\) −3847.77 −1.60668 −0.803341 0.595519i \(-0.796946\pi\)
−0.803341 + 0.595519i \(0.796946\pi\)
\(180\) 0 0
\(181\) −3026.41 −1.24282 −0.621411 0.783484i \(-0.713440\pi\)
−0.621411 + 0.783484i \(0.713440\pi\)
\(182\) −61.1109 −0.0248892
\(183\) 0 0
\(184\) 682.802 0.273570
\(185\) 0 0
\(186\) 0 0
\(187\) −2455.92 −0.960398
\(188\) −1978.16 −0.767405
\(189\) 0 0
\(190\) 0 0
\(191\) −1271.09 −0.481532 −0.240766 0.970583i \(-0.577399\pi\)
−0.240766 + 0.970583i \(0.577399\pi\)
\(192\) 0 0
\(193\) 1896.09 0.707170 0.353585 0.935402i \(-0.384963\pi\)
0.353585 + 0.935402i \(0.384963\pi\)
\(194\) −766.707 −0.283744
\(195\) 0 0
\(196\) −376.548 −0.137226
\(197\) 4067.89 1.47119 0.735597 0.677419i \(-0.236902\pi\)
0.735597 + 0.677419i \(0.236902\pi\)
\(198\) 0 0
\(199\) 78.1366 0.0278340 0.0139170 0.999903i \(-0.495570\pi\)
0.0139170 + 0.999903i \(0.495570\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 649.152 0.226110
\(203\) 1429.19 0.494136
\(204\) 0 0
\(205\) 0 0
\(206\) −43.9176 −0.0148538
\(207\) 0 0
\(208\) 878.857 0.292970
\(209\) −3702.05 −1.22524
\(210\) 0 0
\(211\) −1293.02 −0.421872 −0.210936 0.977500i \(-0.567651\pi\)
−0.210936 + 0.977500i \(0.567651\pi\)
\(212\) −1982.47 −0.642250
\(213\) 0 0
\(214\) 52.8317 0.0168762
\(215\) 0 0
\(216\) 0 0
\(217\) −284.899 −0.0891253
\(218\) 1195.12 0.371303
\(219\) 0 0
\(220\) 0 0
\(221\) −1479.43 −0.450303
\(222\) 0 0
\(223\) 1786.65 0.536515 0.268257 0.963347i \(-0.413552\pi\)
0.268257 + 0.963347i \(0.413552\pi\)
\(224\) −715.452 −0.213407
\(225\) 0 0
\(226\) 334.962 0.0985901
\(227\) −4242.02 −1.24032 −0.620161 0.784475i \(-0.712933\pi\)
−0.620161 + 0.784475i \(0.712933\pi\)
\(228\) 0 0
\(229\) 3403.36 0.982096 0.491048 0.871132i \(-0.336614\pi\)
0.491048 + 0.871132i \(0.336614\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1798.29 0.508893
\(233\) −3904.09 −1.09771 −0.548853 0.835919i \(-0.684935\pi\)
−0.548853 + 0.835919i \(0.684935\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −5005.51 −1.38064
\(237\) 0 0
\(238\) 374.069 0.101879
\(239\) −6667.83 −1.80463 −0.902314 0.431079i \(-0.858133\pi\)
−0.902314 + 0.431079i \(0.858133\pi\)
\(240\) 0 0
\(241\) −6003.99 −1.60478 −0.802388 0.596803i \(-0.796437\pi\)
−0.802388 + 0.596803i \(0.796437\pi\)
\(242\) 373.410 0.0991888
\(243\) 0 0
\(244\) −3468.12 −0.909932
\(245\) 0 0
\(246\) 0 0
\(247\) −2230.08 −0.574481
\(248\) −358.474 −0.0917869
\(249\) 0 0
\(250\) 0 0
\(251\) −728.467 −0.183189 −0.0915945 0.995796i \(-0.529196\pi\)
−0.0915945 + 0.995796i \(0.529196\pi\)
\(252\) 0 0
\(253\) 2000.69 0.497163
\(254\) 1172.64 0.289677
\(255\) 0 0
\(256\) 2575.17 0.628703
\(257\) −1479.86 −0.359187 −0.179593 0.983741i \(-0.557478\pi\)
−0.179593 + 0.983741i \(0.557478\pi\)
\(258\) 0 0
\(259\) −669.687 −0.160665
\(260\) 0 0
\(261\) 0 0
\(262\) 233.686 0.0551036
\(263\) −6367.31 −1.49287 −0.746435 0.665458i \(-0.768236\pi\)
−0.746435 + 0.665458i \(0.768236\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 563.871 0.129974
\(267\) 0 0
\(268\) 6394.18 1.45741
\(269\) −1787.18 −0.405079 −0.202539 0.979274i \(-0.564919\pi\)
−0.202539 + 0.979274i \(0.564919\pi\)
\(270\) 0 0
\(271\) 3907.65 0.875915 0.437957 0.898996i \(-0.355702\pi\)
0.437957 + 0.898996i \(0.355702\pi\)
\(272\) −5379.62 −1.19922
\(273\) 0 0
\(274\) −347.743 −0.0766712
\(275\) 0 0
\(276\) 0 0
\(277\) −3978.33 −0.862942 −0.431471 0.902127i \(-0.642005\pi\)
−0.431471 + 0.902127i \(0.642005\pi\)
\(278\) 149.931 0.0323464
\(279\) 0 0
\(280\) 0 0
\(281\) 6488.04 1.37738 0.688690 0.725055i \(-0.258186\pi\)
0.688690 + 0.725055i \(0.258186\pi\)
\(282\) 0 0
\(283\) 164.336 0.0345185 0.0172592 0.999851i \(-0.494506\pi\)
0.0172592 + 0.999851i \(0.494506\pi\)
\(284\) −1344.55 −0.280930
\(285\) 0 0
\(286\) −225.305 −0.0465824
\(287\) −173.111 −0.0356042
\(288\) 0 0
\(289\) 4142.79 0.843231
\(290\) 0 0
\(291\) 0 0
\(292\) −364.381 −0.0730266
\(293\) 5004.57 0.997851 0.498925 0.866645i \(-0.333728\pi\)
0.498925 + 0.866645i \(0.333728\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −842.634 −0.165463
\(297\) 0 0
\(298\) 1432.97 0.278556
\(299\) 1205.20 0.233105
\(300\) 0 0
\(301\) −1977.01 −0.378581
\(302\) −627.470 −0.119559
\(303\) 0 0
\(304\) −8109.23 −1.52992
\(305\) 0 0
\(306\) 0 0
\(307\) 190.167 0.0353531 0.0176765 0.999844i \(-0.494373\pi\)
0.0176765 + 0.999844i \(0.494373\pi\)
\(308\) −1388.27 −0.256831
\(309\) 0 0
\(310\) 0 0
\(311\) 1182.57 0.215619 0.107810 0.994172i \(-0.465616\pi\)
0.107810 + 0.994172i \(0.465616\pi\)
\(312\) 0 0
\(313\) 4659.05 0.841358 0.420679 0.907209i \(-0.361792\pi\)
0.420679 + 0.907209i \(0.361792\pi\)
\(314\) −256.127 −0.0460320
\(315\) 0 0
\(316\) 8.91467 0.00158699
\(317\) −3694.55 −0.654595 −0.327297 0.944921i \(-0.606138\pi\)
−0.327297 + 0.944921i \(0.606138\pi\)
\(318\) 0 0
\(319\) 5269.18 0.924820
\(320\) 0 0
\(321\) 0 0
\(322\) −304.732 −0.0527392
\(323\) 13650.7 2.35153
\(324\) 0 0
\(325\) 0 0
\(326\) −59.2379 −0.0100641
\(327\) 0 0
\(328\) −217.817 −0.0366675
\(329\) 1801.92 0.301954
\(330\) 0 0
\(331\) −8632.79 −1.43354 −0.716769 0.697310i \(-0.754380\pi\)
−0.716769 + 0.697310i \(0.754380\pi\)
\(332\) −11481.1 −1.89791
\(333\) 0 0
\(334\) 1394.08 0.228385
\(335\) 0 0
\(336\) 0 0
\(337\) 8136.61 1.31522 0.657610 0.753358i \(-0.271567\pi\)
0.657610 + 0.753358i \(0.271567\pi\)
\(338\) 1098.01 0.176698
\(339\) 0 0
\(340\) 0 0
\(341\) −1050.37 −0.166806
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) −2487.58 −0.389887
\(345\) 0 0
\(346\) −290.159 −0.0450839
\(347\) 8646.33 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(348\) 0 0
\(349\) −7455.43 −1.14350 −0.571748 0.820429i \(-0.693734\pi\)
−0.571748 + 0.820429i \(0.693734\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2637.74 −0.399410
\(353\) 5450.21 0.821771 0.410886 0.911687i \(-0.365220\pi\)
0.410886 + 0.911687i \(0.365220\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10060.2 1.49773
\(357\) 0 0
\(358\) 2160.73 0.318989
\(359\) −4775.79 −0.702107 −0.351053 0.936355i \(-0.614176\pi\)
−0.351053 + 0.936355i \(0.614176\pi\)
\(360\) 0 0
\(361\) 13718.0 2.00000
\(362\) 1699.49 0.246749
\(363\) 0 0
\(364\) −836.281 −0.120420
\(365\) 0 0
\(366\) 0 0
\(367\) −9636.30 −1.37060 −0.685301 0.728260i \(-0.740329\pi\)
−0.685301 + 0.728260i \(0.740329\pi\)
\(368\) 4382.46 0.620792
\(369\) 0 0
\(370\) 0 0
\(371\) 1805.85 0.252709
\(372\) 0 0
\(373\) −12180.4 −1.69082 −0.845412 0.534115i \(-0.820645\pi\)
−0.845412 + 0.534115i \(0.820645\pi\)
\(374\) 1379.13 0.190676
\(375\) 0 0
\(376\) 2267.27 0.310971
\(377\) 3174.11 0.433621
\(378\) 0 0
\(379\) 1689.39 0.228966 0.114483 0.993425i \(-0.463479\pi\)
0.114483 + 0.993425i \(0.463479\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 713.783 0.0956029
\(383\) 1513.13 0.201872 0.100936 0.994893i \(-0.467816\pi\)
0.100936 + 0.994893i \(0.467816\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1064.76 −0.140401
\(387\) 0 0
\(388\) −10492.1 −1.37283
\(389\) −12165.5 −1.58564 −0.792822 0.609453i \(-0.791389\pi\)
−0.792822 + 0.609453i \(0.791389\pi\)
\(390\) 0 0
\(391\) −7377.21 −0.954173
\(392\) 431.580 0.0556074
\(393\) 0 0
\(394\) −2284.34 −0.292089
\(395\) 0 0
\(396\) 0 0
\(397\) −7353.40 −0.929613 −0.464807 0.885412i \(-0.653876\pi\)
−0.464807 + 0.885412i \(0.653876\pi\)
\(398\) −43.8778 −0.00552612
\(399\) 0 0
\(400\) 0 0
\(401\) −7481.62 −0.931707 −0.465853 0.884862i \(-0.654253\pi\)
−0.465853 + 0.884862i \(0.654253\pi\)
\(402\) 0 0
\(403\) −632.735 −0.0782104
\(404\) 8883.42 1.09398
\(405\) 0 0
\(406\) −802.567 −0.0981053
\(407\) −2469.02 −0.300699
\(408\) 0 0
\(409\) 9248.94 1.11817 0.559084 0.829111i \(-0.311153\pi\)
0.559084 + 0.829111i \(0.311153\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −600.997 −0.0718665
\(413\) 4559.55 0.543247
\(414\) 0 0
\(415\) 0 0
\(416\) −1588.96 −0.187272
\(417\) 0 0
\(418\) 2078.89 0.243258
\(419\) 3363.39 0.392154 0.196077 0.980589i \(-0.437180\pi\)
0.196077 + 0.980589i \(0.437180\pi\)
\(420\) 0 0
\(421\) −3638.86 −0.421252 −0.210626 0.977567i \(-0.567550\pi\)
−0.210626 + 0.977567i \(0.567550\pi\)
\(422\) 726.097 0.0837579
\(423\) 0 0
\(424\) 2272.21 0.260255
\(425\) 0 0
\(426\) 0 0
\(427\) 3159.13 0.358035
\(428\) 722.983 0.0816512
\(429\) 0 0
\(430\) 0 0
\(431\) −11243.2 −1.25653 −0.628264 0.778000i \(-0.716234\pi\)
−0.628264 + 0.778000i \(0.716234\pi\)
\(432\) 0 0
\(433\) −187.332 −0.0207912 −0.0103956 0.999946i \(-0.503309\pi\)
−0.0103956 + 0.999946i \(0.503309\pi\)
\(434\) 159.986 0.0176948
\(435\) 0 0
\(436\) 16354.9 1.79646
\(437\) −11120.4 −1.21730
\(438\) 0 0
\(439\) −11479.4 −1.24802 −0.624011 0.781415i \(-0.714498\pi\)
−0.624011 + 0.781415i \(0.714498\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 830.775 0.0894026
\(443\) −9490.56 −1.01786 −0.508928 0.860809i \(-0.669958\pi\)
−0.508928 + 0.860809i \(0.669958\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −1003.30 −0.106519
\(447\) 0 0
\(448\) −2763.99 −0.291487
\(449\) 5546.58 0.582982 0.291491 0.956574i \(-0.405849\pi\)
0.291491 + 0.956574i \(0.405849\pi\)
\(450\) 0 0
\(451\) −638.229 −0.0666364
\(452\) 4583.85 0.477004
\(453\) 0 0
\(454\) 2382.12 0.246252
\(455\) 0 0
\(456\) 0 0
\(457\) −1776.94 −0.181885 −0.0909426 0.995856i \(-0.528988\pi\)
−0.0909426 + 0.995856i \(0.528988\pi\)
\(458\) −1911.16 −0.194984
\(459\) 0 0
\(460\) 0 0
\(461\) 9059.81 0.915309 0.457655 0.889130i \(-0.348690\pi\)
0.457655 + 0.889130i \(0.348690\pi\)
\(462\) 0 0
\(463\) −14878.5 −1.49344 −0.746719 0.665139i \(-0.768372\pi\)
−0.746719 + 0.665139i \(0.768372\pi\)
\(464\) 11542.0 1.15479
\(465\) 0 0
\(466\) 2192.35 0.217937
\(467\) 14275.8 1.41458 0.707288 0.706926i \(-0.249918\pi\)
0.707288 + 0.706926i \(0.249918\pi\)
\(468\) 0 0
\(469\) −5824.50 −0.573455
\(470\) 0 0
\(471\) 0 0
\(472\) 5737.07 0.559470
\(473\) −7288.89 −0.708548
\(474\) 0 0
\(475\) 0 0
\(476\) 5119.01 0.492919
\(477\) 0 0
\(478\) 3744.34 0.358289
\(479\) 15377.8 1.46687 0.733433 0.679761i \(-0.237917\pi\)
0.733433 + 0.679761i \(0.237917\pi\)
\(480\) 0 0
\(481\) −1487.32 −0.140989
\(482\) 3371.56 0.318610
\(483\) 0 0
\(484\) 5109.99 0.479901
\(485\) 0 0
\(486\) 0 0
\(487\) −19683.3 −1.83149 −0.915747 0.401756i \(-0.868400\pi\)
−0.915747 + 0.401756i \(0.868400\pi\)
\(488\) 3974.98 0.368727
\(489\) 0 0
\(490\) 0 0
\(491\) −19585.1 −1.80013 −0.900065 0.435755i \(-0.856481\pi\)
−0.900065 + 0.435755i \(0.856481\pi\)
\(492\) 0 0
\(493\) −19429.3 −1.77495
\(494\) 1252.31 0.114057
\(495\) 0 0
\(496\) −2300.81 −0.208285
\(497\) 1224.75 0.110539
\(498\) 0 0
\(499\) −6888.47 −0.617977 −0.308988 0.951066i \(-0.599990\pi\)
−0.308988 + 0.951066i \(0.599990\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 409.073 0.0363701
\(503\) 14878.0 1.31885 0.659423 0.751772i \(-0.270801\pi\)
0.659423 + 0.751772i \(0.270801\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1123.49 −0.0987062
\(507\) 0 0
\(508\) 16047.2 1.40153
\(509\) −1935.28 −0.168526 −0.0842629 0.996444i \(-0.526854\pi\)
−0.0842629 + 0.996444i \(0.526854\pi\)
\(510\) 0 0
\(511\) 331.917 0.0287341
\(512\) −9761.22 −0.842557
\(513\) 0 0
\(514\) 831.019 0.0713126
\(515\) 0 0
\(516\) 0 0
\(517\) 6643.35 0.565134
\(518\) 376.064 0.0318983
\(519\) 0 0
\(520\) 0 0
\(521\) 6892.28 0.579570 0.289785 0.957092i \(-0.406416\pi\)
0.289785 + 0.957092i \(0.406416\pi\)
\(522\) 0 0
\(523\) 1074.60 0.0898447 0.0449223 0.998990i \(-0.485696\pi\)
0.0449223 + 0.998990i \(0.485696\pi\)
\(524\) 3197.91 0.266606
\(525\) 0 0
\(526\) 3575.58 0.296393
\(527\) 3873.07 0.320140
\(528\) 0 0
\(529\) −6157.23 −0.506060
\(530\) 0 0
\(531\) 0 0
\(532\) 7716.39 0.628849
\(533\) −384.464 −0.0312439
\(534\) 0 0
\(535\) 0 0
\(536\) −7328.69 −0.590580
\(537\) 0 0
\(538\) 1003.60 0.0804239
\(539\) 1264.58 0.101056
\(540\) 0 0
\(541\) 8660.23 0.688230 0.344115 0.938928i \(-0.388179\pi\)
0.344115 + 0.938928i \(0.388179\pi\)
\(542\) −2194.35 −0.173903
\(543\) 0 0
\(544\) 9726.25 0.766562
\(545\) 0 0
\(546\) 0 0
\(547\) 15346.1 1.19954 0.599771 0.800171i \(-0.295258\pi\)
0.599771 + 0.800171i \(0.295258\pi\)
\(548\) −4758.74 −0.370955
\(549\) 0 0
\(550\) 0 0
\(551\) −29287.6 −2.26442
\(552\) 0 0
\(553\) −8.12042 −0.000624440 0
\(554\) 2234.04 0.171328
\(555\) 0 0
\(556\) 2051.76 0.156500
\(557\) 15544.9 1.18251 0.591257 0.806483i \(-0.298632\pi\)
0.591257 + 0.806483i \(0.298632\pi\)
\(558\) 0 0
\(559\) −4390.77 −0.332218
\(560\) 0 0
\(561\) 0 0
\(562\) −3643.38 −0.273464
\(563\) −18511.9 −1.38576 −0.692881 0.721052i \(-0.743659\pi\)
−0.692881 + 0.721052i \(0.743659\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −92.2831 −0.00685326
\(567\) 0 0
\(568\) 1541.05 0.113840
\(569\) −2157.36 −0.158948 −0.0794738 0.996837i \(-0.525324\pi\)
−0.0794738 + 0.996837i \(0.525324\pi\)
\(570\) 0 0
\(571\) 16010.3 1.17340 0.586700 0.809805i \(-0.300427\pi\)
0.586700 + 0.809805i \(0.300427\pi\)
\(572\) −3083.22 −0.225378
\(573\) 0 0
\(574\) 97.2109 0.00706882
\(575\) 0 0
\(576\) 0 0
\(577\) 2164.01 0.156134 0.0780668 0.996948i \(-0.475125\pi\)
0.0780668 + 0.996948i \(0.475125\pi\)
\(578\) −2326.40 −0.167414
\(579\) 0 0
\(580\) 0 0
\(581\) 10458.2 0.746780
\(582\) 0 0
\(583\) 6657.84 0.472967
\(584\) 417.635 0.0295922
\(585\) 0 0
\(586\) −2810.33 −0.198112
\(587\) 27616.6 1.94184 0.970918 0.239412i \(-0.0769546\pi\)
0.970918 + 0.239412i \(0.0769546\pi\)
\(588\) 0 0
\(589\) 5838.26 0.408424
\(590\) 0 0
\(591\) 0 0
\(592\) −5408.32 −0.375474
\(593\) −10205.4 −0.706718 −0.353359 0.935488i \(-0.614961\pi\)
−0.353359 + 0.935488i \(0.614961\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 19609.7 1.34773
\(597\) 0 0
\(598\) −676.783 −0.0462805
\(599\) −11090.5 −0.756501 −0.378251 0.925703i \(-0.623474\pi\)
−0.378251 + 0.925703i \(0.623474\pi\)
\(600\) 0 0
\(601\) −26537.0 −1.80111 −0.900555 0.434742i \(-0.856839\pi\)
−0.900555 + 0.434742i \(0.856839\pi\)
\(602\) 1110.20 0.0751631
\(603\) 0 0
\(604\) −8586.71 −0.578457
\(605\) 0 0
\(606\) 0 0
\(607\) 11234.5 0.751226 0.375613 0.926777i \(-0.377432\pi\)
0.375613 + 0.926777i \(0.377432\pi\)
\(608\) 14661.3 0.977954
\(609\) 0 0
\(610\) 0 0
\(611\) 4001.90 0.264975
\(612\) 0 0
\(613\) 10622.5 0.699900 0.349950 0.936768i \(-0.386199\pi\)
0.349950 + 0.936768i \(0.386199\pi\)
\(614\) −106.789 −0.00701896
\(615\) 0 0
\(616\) 1591.16 0.104074
\(617\) 20433.3 1.33325 0.666623 0.745395i \(-0.267739\pi\)
0.666623 + 0.745395i \(0.267739\pi\)
\(618\) 0 0
\(619\) −7963.57 −0.517097 −0.258549 0.965998i \(-0.583244\pi\)
−0.258549 + 0.965998i \(0.583244\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −664.077 −0.0428088
\(623\) −9163.91 −0.589317
\(624\) 0 0
\(625\) 0 0
\(626\) −2616.30 −0.167042
\(627\) 0 0
\(628\) −3505.01 −0.222715
\(629\) 9104.09 0.577113
\(630\) 0 0
\(631\) −14703.1 −0.927608 −0.463804 0.885938i \(-0.653516\pi\)
−0.463804 + 0.885938i \(0.653516\pi\)
\(632\) −10.2175 −0.000643088 0
\(633\) 0 0
\(634\) 2074.68 0.129963
\(635\) 0 0
\(636\) 0 0
\(637\) 761.773 0.0473824
\(638\) −2958.92 −0.183613
\(639\) 0 0
\(640\) 0 0
\(641\) −3353.41 −0.206633 −0.103317 0.994649i \(-0.532945\pi\)
−0.103317 + 0.994649i \(0.532945\pi\)
\(642\) 0 0
\(643\) −31862.0 −1.95415 −0.977073 0.212906i \(-0.931707\pi\)
−0.977073 + 0.212906i \(0.931707\pi\)
\(644\) −4170.15 −0.255166
\(645\) 0 0
\(646\) −7665.58 −0.466870
\(647\) −8518.46 −0.517612 −0.258806 0.965929i \(-0.583329\pi\)
−0.258806 + 0.965929i \(0.583329\pi\)
\(648\) 0 0
\(649\) 16810.3 1.01673
\(650\) 0 0
\(651\) 0 0
\(652\) −810.651 −0.0486925
\(653\) −28509.3 −1.70851 −0.854254 0.519856i \(-0.825986\pi\)
−0.854254 + 0.519856i \(0.825986\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1398.02 −0.0832068
\(657\) 0 0
\(658\) −1011.87 −0.0599496
\(659\) 27632.4 1.63339 0.816697 0.577066i \(-0.195803\pi\)
0.816697 + 0.577066i \(0.195803\pi\)
\(660\) 0 0
\(661\) −27052.8 −1.59188 −0.795941 0.605374i \(-0.793023\pi\)
−0.795941 + 0.605374i \(0.793023\pi\)
\(662\) 4847.77 0.284613
\(663\) 0 0
\(664\) 13159.1 0.769082
\(665\) 0 0
\(666\) 0 0
\(667\) 15827.9 0.918826
\(668\) 19077.5 1.10498
\(669\) 0 0
\(670\) 0 0
\(671\) 11647.1 0.670094
\(672\) 0 0
\(673\) 1569.99 0.0899235 0.0449618 0.998989i \(-0.485683\pi\)
0.0449618 + 0.998989i \(0.485683\pi\)
\(674\) −4569.13 −0.261122
\(675\) 0 0
\(676\) 15025.9 0.854909
\(677\) 13853.9 0.786482 0.393241 0.919435i \(-0.371354\pi\)
0.393241 + 0.919435i \(0.371354\pi\)
\(678\) 0 0
\(679\) 9557.33 0.540172
\(680\) 0 0
\(681\) 0 0
\(682\) 589.839 0.0331174
\(683\) −28337.1 −1.58754 −0.793769 0.608219i \(-0.791884\pi\)
−0.793769 + 0.608219i \(0.791884\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −192.613 −0.0107201
\(687\) 0 0
\(688\) −15966.1 −0.884742
\(689\) 4010.63 0.221760
\(690\) 0 0
\(691\) 16936.5 0.932410 0.466205 0.884677i \(-0.345621\pi\)
0.466205 + 0.884677i \(0.345621\pi\)
\(692\) −3970.73 −0.218128
\(693\) 0 0
\(694\) −4855.37 −0.265572
\(695\) 0 0
\(696\) 0 0
\(697\) 2353.37 0.127891
\(698\) 4186.62 0.227028
\(699\) 0 0
\(700\) 0 0
\(701\) 29744.0 1.60259 0.801294 0.598271i \(-0.204145\pi\)
0.801294 + 0.598271i \(0.204145\pi\)
\(702\) 0 0
\(703\) 13723.5 0.736261
\(704\) −10190.3 −0.545543
\(705\) 0 0
\(706\) −3060.58 −0.163154
\(707\) −8091.96 −0.430452
\(708\) 0 0
\(709\) 34264.7 1.81500 0.907502 0.420047i \(-0.137986\pi\)
0.907502 + 0.420047i \(0.137986\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −11530.5 −0.606916
\(713\) −3155.16 −0.165725
\(714\) 0 0
\(715\) 0 0
\(716\) 29568.8 1.54335
\(717\) 0 0
\(718\) 2681.86 0.139396
\(719\) 25469.2 1.32106 0.660529 0.750801i \(-0.270332\pi\)
0.660529 + 0.750801i \(0.270332\pi\)
\(720\) 0 0
\(721\) 547.452 0.0282776
\(722\) −7703.40 −0.397079
\(723\) 0 0
\(724\) 23256.9 1.19383
\(725\) 0 0
\(726\) 0 0
\(727\) 24294.6 1.23939 0.619695 0.784843i \(-0.287256\pi\)
0.619695 + 0.784843i \(0.287256\pi\)
\(728\) 958.503 0.0487974
\(729\) 0 0
\(730\) 0 0
\(731\) 26876.6 1.35987
\(732\) 0 0
\(733\) 34870.8 1.75714 0.878569 0.477615i \(-0.158499\pi\)
0.878569 + 0.477615i \(0.158499\pi\)
\(734\) 5411.29 0.272118
\(735\) 0 0
\(736\) −7923.40 −0.396821
\(737\) −21473.9 −1.07327
\(738\) 0 0
\(739\) 7028.81 0.349877 0.174938 0.984579i \(-0.444027\pi\)
0.174938 + 0.984579i \(0.444027\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1014.08 −0.0501725
\(743\) −1368.74 −0.0675831 −0.0337915 0.999429i \(-0.510758\pi\)
−0.0337915 + 0.999429i \(0.510758\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 6839.94 0.335694
\(747\) 0 0
\(748\) 18872.9 0.922541
\(749\) −658.570 −0.0321276
\(750\) 0 0
\(751\) 37144.0 1.80480 0.902400 0.430900i \(-0.141804\pi\)
0.902400 + 0.430900i \(0.141804\pi\)
\(752\) 14552.1 0.705665
\(753\) 0 0
\(754\) −1782.43 −0.0860907
\(755\) 0 0
\(756\) 0 0
\(757\) 12042.4 0.578189 0.289095 0.957301i \(-0.406646\pi\)
0.289095 + 0.957301i \(0.406646\pi\)
\(758\) −948.683 −0.0454588
\(759\) 0 0
\(760\) 0 0
\(761\) −6112.92 −0.291187 −0.145593 0.989345i \(-0.546509\pi\)
−0.145593 + 0.989345i \(0.546509\pi\)
\(762\) 0 0
\(763\) −14897.7 −0.706860
\(764\) 9767.88 0.462552
\(765\) 0 0
\(766\) −849.700 −0.0400795
\(767\) 10126.4 0.476717
\(768\) 0 0
\(769\) −957.146 −0.0448837 −0.0224418 0.999748i \(-0.507144\pi\)
−0.0224418 + 0.999748i \(0.507144\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −14570.8 −0.679295
\(773\) −32867.9 −1.52934 −0.764668 0.644424i \(-0.777097\pi\)
−0.764668 + 0.644424i \(0.777097\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 12025.5 0.556303
\(777\) 0 0
\(778\) 6831.58 0.314812
\(779\) 3547.46 0.163159
\(780\) 0 0
\(781\) 4515.45 0.206883
\(782\) 4142.69 0.189440
\(783\) 0 0
\(784\) 2770.03 0.126186
\(785\) 0 0
\(786\) 0 0
\(787\) 15320.0 0.693900 0.346950 0.937884i \(-0.387217\pi\)
0.346950 + 0.937884i \(0.387217\pi\)
\(788\) −31260.4 −1.41320
\(789\) 0 0
\(790\) 0 0
\(791\) −4175.45 −0.187689
\(792\) 0 0
\(793\) 7016.15 0.314188
\(794\) 4129.32 0.184564
\(795\) 0 0
\(796\) −600.453 −0.0267368
\(797\) −43629.0 −1.93904 −0.969522 0.245004i \(-0.921211\pi\)
−0.969522 + 0.245004i \(0.921211\pi\)
\(798\) 0 0
\(799\) −24496.3 −1.08463
\(800\) 0 0
\(801\) 0 0
\(802\) 4201.33 0.184980
\(803\) 1223.72 0.0537784
\(804\) 0 0
\(805\) 0 0
\(806\) 355.314 0.0155278
\(807\) 0 0
\(808\) −10181.7 −0.443307
\(809\) −127.735 −0.00555119 −0.00277560 0.999996i \(-0.500884\pi\)
−0.00277560 + 0.999996i \(0.500884\pi\)
\(810\) 0 0
\(811\) 16227.5 0.702618 0.351309 0.936260i \(-0.385737\pi\)
0.351309 + 0.936260i \(0.385737\pi\)
\(812\) −10982.9 −0.474659
\(813\) 0 0
\(814\) 1386.48 0.0597005
\(815\) 0 0
\(816\) 0 0
\(817\) 40513.7 1.73488
\(818\) −5193.77 −0.222000
\(819\) 0 0
\(820\) 0 0
\(821\) −34249.2 −1.45591 −0.727957 0.685623i \(-0.759530\pi\)
−0.727957 + 0.685623i \(0.759530\pi\)
\(822\) 0 0
\(823\) −6624.51 −0.280578 −0.140289 0.990111i \(-0.544803\pi\)
−0.140289 + 0.990111i \(0.544803\pi\)
\(824\) 688.832 0.0291221
\(825\) 0 0
\(826\) −2560.43 −0.107856
\(827\) 33786.8 1.42065 0.710327 0.703872i \(-0.248547\pi\)
0.710327 + 0.703872i \(0.248547\pi\)
\(828\) 0 0
\(829\) −30283.8 −1.26876 −0.634378 0.773023i \(-0.718744\pi\)
−0.634378 + 0.773023i \(0.718744\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −6138.57 −0.255789
\(833\) −4662.93 −0.193951
\(834\) 0 0
\(835\) 0 0
\(836\) 28449.0 1.17695
\(837\) 0 0
\(838\) −1888.72 −0.0778578
\(839\) −16810.9 −0.691750 −0.345875 0.938281i \(-0.612418\pi\)
−0.345875 + 0.938281i \(0.612418\pi\)
\(840\) 0 0
\(841\) 17296.6 0.709195
\(842\) 2043.41 0.0836350
\(843\) 0 0
\(844\) 9936.39 0.405242
\(845\) 0 0
\(846\) 0 0
\(847\) −4654.72 −0.188829
\(848\) 14583.8 0.590579
\(849\) 0 0
\(850\) 0 0
\(851\) −7416.56 −0.298750
\(852\) 0 0
\(853\) 14875.8 0.597112 0.298556 0.954392i \(-0.403495\pi\)
0.298556 + 0.954392i \(0.403495\pi\)
\(854\) −1774.02 −0.0710839
\(855\) 0 0
\(856\) −828.647 −0.0330871
\(857\) 3987.55 0.158941 0.0794703 0.996837i \(-0.474677\pi\)
0.0794703 + 0.996837i \(0.474677\pi\)
\(858\) 0 0
\(859\) 39344.7 1.56278 0.781388 0.624046i \(-0.214512\pi\)
0.781388 + 0.624046i \(0.214512\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 6313.63 0.249470
\(863\) 15627.0 0.616397 0.308198 0.951322i \(-0.400274\pi\)
0.308198 + 0.951322i \(0.400274\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 105.197 0.00412787
\(867\) 0 0
\(868\) 2189.35 0.0856122
\(869\) −29.9386 −0.00116870
\(870\) 0 0
\(871\) −12935.7 −0.503226
\(872\) −18745.1 −0.727969
\(873\) 0 0
\(874\) 6244.69 0.241682
\(875\) 0 0
\(876\) 0 0
\(877\) −14519.0 −0.559034 −0.279517 0.960141i \(-0.590174\pi\)
−0.279517 + 0.960141i \(0.590174\pi\)
\(878\) 6446.29 0.247781
\(879\) 0 0
\(880\) 0 0
\(881\) −24177.7 −0.924592 −0.462296 0.886726i \(-0.652974\pi\)
−0.462296 + 0.886726i \(0.652974\pi\)
\(882\) 0 0
\(883\) 10340.3 0.394089 0.197044 0.980395i \(-0.436866\pi\)
0.197044 + 0.980395i \(0.436866\pi\)
\(884\) 11368.9 0.432553
\(885\) 0 0
\(886\) 5329.45 0.202084
\(887\) 3222.62 0.121990 0.0609949 0.998138i \(-0.480573\pi\)
0.0609949 + 0.998138i \(0.480573\pi\)
\(888\) 0 0
\(889\) −14617.5 −0.551467
\(890\) 0 0
\(891\) 0 0
\(892\) −13729.8 −0.515367
\(893\) −36925.6 −1.38373
\(894\) 0 0
\(895\) 0 0
\(896\) 7275.74 0.271278
\(897\) 0 0
\(898\) −3114.70 −0.115745
\(899\) −8309.70 −0.308280
\(900\) 0 0
\(901\) −24549.7 −0.907735
\(902\) 358.399 0.0132299
\(903\) 0 0
\(904\) −5253.77 −0.193294
\(905\) 0 0
\(906\) 0 0
\(907\) −31692.8 −1.16025 −0.580123 0.814529i \(-0.696995\pi\)
−0.580123 + 0.814529i \(0.696995\pi\)
\(908\) 32598.5 1.19143
\(909\) 0 0
\(910\) 0 0
\(911\) −2403.15 −0.0873985 −0.0436993 0.999045i \(-0.513914\pi\)
−0.0436993 + 0.999045i \(0.513914\pi\)
\(912\) 0 0
\(913\) 38557.6 1.39767
\(914\) 997.843 0.0361113
\(915\) 0 0
\(916\) −26153.6 −0.943385
\(917\) −2912.99 −0.104902
\(918\) 0 0
\(919\) 19622.5 0.704339 0.352170 0.935936i \(-0.385444\pi\)
0.352170 + 0.935936i \(0.385444\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −5087.56 −0.181724
\(923\) 2720.07 0.0970014
\(924\) 0 0
\(925\) 0 0
\(926\) 8355.06 0.296506
\(927\) 0 0
\(928\) −20867.7 −0.738165
\(929\) 12930.5 0.456660 0.228330 0.973584i \(-0.426673\pi\)
0.228330 + 0.973584i \(0.426673\pi\)
\(930\) 0 0
\(931\) −7028.90 −0.247436
\(932\) 30001.6 1.05444
\(933\) 0 0
\(934\) −8016.64 −0.280848
\(935\) 0 0
\(936\) 0 0
\(937\) −18717.1 −0.652573 −0.326287 0.945271i \(-0.605797\pi\)
−0.326287 + 0.945271i \(0.605797\pi\)
\(938\) 3270.76 0.113853
\(939\) 0 0
\(940\) 0 0
\(941\) 10152.9 0.351727 0.175863 0.984415i \(-0.443728\pi\)
0.175863 + 0.984415i \(0.443728\pi\)
\(942\) 0 0
\(943\) −1917.15 −0.0662045
\(944\) 36822.4 1.26956
\(945\) 0 0
\(946\) 4093.09 0.140674
\(947\) 15836.2 0.543408 0.271704 0.962381i \(-0.412413\pi\)
0.271704 + 0.962381i \(0.412413\pi\)
\(948\) 0 0
\(949\) 737.158 0.0252151
\(950\) 0 0
\(951\) 0 0
\(952\) −5867.15 −0.199743
\(953\) 4847.86 0.164782 0.0823911 0.996600i \(-0.473744\pi\)
0.0823911 + 0.996600i \(0.473744\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 51240.0 1.73349
\(957\) 0 0
\(958\) −8635.44 −0.291230
\(959\) 4334.76 0.145961
\(960\) 0 0
\(961\) −28134.5 −0.944397
\(962\) 835.207 0.0279918
\(963\) 0 0
\(964\) 46138.6 1.54152
\(965\) 0 0
\(966\) 0 0
\(967\) −17153.8 −0.570454 −0.285227 0.958460i \(-0.592069\pi\)
−0.285227 + 0.958460i \(0.592069\pi\)
\(968\) −5856.80 −0.194468
\(969\) 0 0
\(970\) 0 0
\(971\) −50352.2 −1.66414 −0.832070 0.554670i \(-0.812844\pi\)
−0.832070 + 0.554670i \(0.812844\pi\)
\(972\) 0 0
\(973\) −1868.96 −0.0615788
\(974\) 11053.2 0.363623
\(975\) 0 0
\(976\) 25512.8 0.836725
\(977\) 1510.03 0.0494474 0.0247237 0.999694i \(-0.492129\pi\)
0.0247237 + 0.999694i \(0.492129\pi\)
\(978\) 0 0
\(979\) −33785.7 −1.10296
\(980\) 0 0
\(981\) 0 0
\(982\) 10998.1 0.357396
\(983\) −5310.89 −0.172321 −0.0861603 0.996281i \(-0.527460\pi\)
−0.0861603 + 0.996281i \(0.527460\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 10910.6 0.352396
\(987\) 0 0
\(988\) 17137.4 0.551836
\(989\) −21894.7 −0.703956
\(990\) 0 0
\(991\) 35845.4 1.14901 0.574505 0.818501i \(-0.305195\pi\)
0.574505 + 0.818501i \(0.305195\pi\)
\(992\) 4159.82 0.133140
\(993\) 0 0
\(994\) −687.764 −0.0219462
\(995\) 0 0
\(996\) 0 0
\(997\) 17857.5 0.567253 0.283627 0.958935i \(-0.408462\pi\)
0.283627 + 0.958935i \(0.408462\pi\)
\(998\) 3868.24 0.122692
\(999\) 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 1575.4.a.x.1.1 2
3.2 odd 2 525.4.a.j.1.2 2
5.4 even 2 1575.4.a.o.1.2 2
15.2 even 4 525.4.d.m.274.3 4
15.8 even 4 525.4.d.m.274.2 4
15.14 odd 2 525.4.a.m.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.j.1.2 2 3.2 odd 2
525.4.a.m.1.1 yes 2 15.14 odd 2
525.4.d.m.274.2 4 15.8 even 4
525.4.d.m.274.3 4 15.2 even 4
1575.4.a.o.1.2 2 5.4 even 2
1575.4.a.x.1.1 2 1.1 even 1 trivial