Properties

Label 1575.4.a.r.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: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 315)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.56155 q^{2} -1.43845 q^{4} +7.00000 q^{7} +24.1771 q^{8} +O(q^{10})\) \(q-2.56155 q^{2} -1.43845 q^{4} +7.00000 q^{7} +24.1771 q^{8} +6.24621 q^{11} +56.3542 q^{13} -17.9309 q^{14} -50.4233 q^{16} -24.6004 q^{17} -90.7083 q^{19} -16.0000 q^{22} +69.8617 q^{23} -144.354 q^{26} -10.0691 q^{28} -228.847 q^{29} -67.8920 q^{31} -64.2547 q^{32} +63.0152 q^{34} +58.8466 q^{37} +232.354 q^{38} +19.2007 q^{41} -365.218 q^{43} -8.98485 q^{44} -178.955 q^{46} +195.153 q^{47} +49.0000 q^{49} -81.0625 q^{52} +511.201 q^{53} +169.240 q^{56} +586.203 q^{58} -284.000 q^{59} -123.460 q^{61} +173.909 q^{62} +567.978 q^{64} -144.968 q^{67} +35.3863 q^{68} -73.0284 q^{71} -638.850 q^{73} -150.739 q^{74} +130.479 q^{76} +43.7235 q^{77} +976.189 q^{79} -49.1837 q^{82} +484.466 q^{83} +935.525 q^{86} +151.015 q^{88} +1017.30 q^{89} +394.479 q^{91} -100.492 q^{92} -499.896 q^{94} -1806.67 q^{97} -125.516 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 7 q^{4} + 14 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 7 q^{4} + 14 q^{7} + 3 q^{8} - 4 q^{11} + 22 q^{13} - 7 q^{14} - 39 q^{16} + 58 q^{17} - 32 q^{22} + 82 q^{23} - 198 q^{26} - 49 q^{28} - 334 q^{29} - 210 q^{31} + 123 q^{32} + 192 q^{34} - 6 q^{37} + 374 q^{38} - 176 q^{41} - 46 q^{43} + 48 q^{44} - 160 q^{46} + 514 q^{47} + 98 q^{49} + 110 q^{52} + 808 q^{53} + 21 q^{56} + 422 q^{58} - 568 q^{59} - 618 q^{61} - 48 q^{62} + 769 q^{64} - 694 q^{67} - 424 q^{68} - 814 q^{71} - 82 q^{73} - 252 q^{74} - 374 q^{76} - 28 q^{77} + 600 q^{79} - 354 q^{82} - 268 q^{83} + 1434 q^{86} + 368 q^{88} + 72 q^{89} + 154 q^{91} - 168 q^{92} - 2 q^{94} - 1626 q^{97} - 49 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.56155 −0.905646 −0.452823 0.891601i \(-0.649583\pi\)
−0.452823 + 0.891601i \(0.649583\pi\)
\(3\) 0 0
\(4\) −1.43845 −0.179806
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 24.1771 1.06849
\(9\) 0 0
\(10\) 0 0
\(11\) 6.24621 0.171209 0.0856047 0.996329i \(-0.472718\pi\)
0.0856047 + 0.996329i \(0.472718\pi\)
\(12\) 0 0
\(13\) 56.3542 1.20229 0.601147 0.799138i \(-0.294710\pi\)
0.601147 + 0.799138i \(0.294710\pi\)
\(14\) −17.9309 −0.342302
\(15\) 0 0
\(16\) −50.4233 −0.787864
\(17\) −24.6004 −0.350969 −0.175484 0.984482i \(-0.556149\pi\)
−0.175484 + 0.984482i \(0.556149\pi\)
\(18\) 0 0
\(19\) −90.7083 −1.09526 −0.547629 0.836721i \(-0.684470\pi\)
−0.547629 + 0.836721i \(0.684470\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −16.0000 −0.155055
\(23\) 69.8617 0.633356 0.316678 0.948533i \(-0.397433\pi\)
0.316678 + 0.948533i \(0.397433\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −144.354 −1.08885
\(27\) 0 0
\(28\) −10.0691 −0.0679602
\(29\) −228.847 −1.46537 −0.732685 0.680568i \(-0.761733\pi\)
−0.732685 + 0.680568i \(0.761733\pi\)
\(30\) 0 0
\(31\) −67.8920 −0.393347 −0.196674 0.980469i \(-0.563014\pi\)
−0.196674 + 0.980469i \(0.563014\pi\)
\(32\) −64.2547 −0.354961
\(33\) 0 0
\(34\) 63.0152 0.317853
\(35\) 0 0
\(36\) 0 0
\(37\) 58.8466 0.261468 0.130734 0.991417i \(-0.458267\pi\)
0.130734 + 0.991417i \(0.458267\pi\)
\(38\) 232.354 0.991916
\(39\) 0 0
\(40\) 0 0
\(41\) 19.2007 0.0731379 0.0365689 0.999331i \(-0.488357\pi\)
0.0365689 + 0.999331i \(0.488357\pi\)
\(42\) 0 0
\(43\) −365.218 −1.29524 −0.647618 0.761965i \(-0.724235\pi\)
−0.647618 + 0.761965i \(0.724235\pi\)
\(44\) −8.98485 −0.0307845
\(45\) 0 0
\(46\) −178.955 −0.573596
\(47\) 195.153 0.605661 0.302830 0.953044i \(-0.402068\pi\)
0.302830 + 0.953044i \(0.402068\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −81.0625 −0.216180
\(53\) 511.201 1.32488 0.662442 0.749113i \(-0.269520\pi\)
0.662442 + 0.749113i \(0.269520\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 169.240 0.403850
\(57\) 0 0
\(58\) 586.203 1.32711
\(59\) −284.000 −0.626672 −0.313336 0.949642i \(-0.601447\pi\)
−0.313336 + 0.949642i \(0.601447\pi\)
\(60\) 0 0
\(61\) −123.460 −0.259139 −0.129569 0.991570i \(-0.541359\pi\)
−0.129569 + 0.991570i \(0.541359\pi\)
\(62\) 173.909 0.356233
\(63\) 0 0
\(64\) 567.978 1.10933
\(65\) 0 0
\(66\) 0 0
\(67\) −144.968 −0.264338 −0.132169 0.991227i \(-0.542194\pi\)
−0.132169 + 0.991227i \(0.542194\pi\)
\(68\) 35.3863 0.0631062
\(69\) 0 0
\(70\) 0 0
\(71\) −73.0284 −0.122069 −0.0610344 0.998136i \(-0.519440\pi\)
−0.0610344 + 0.998136i \(0.519440\pi\)
\(72\) 0 0
\(73\) −638.850 −1.02427 −0.512135 0.858905i \(-0.671145\pi\)
−0.512135 + 0.858905i \(0.671145\pi\)
\(74\) −150.739 −0.236797
\(75\) 0 0
\(76\) 130.479 0.196934
\(77\) 43.7235 0.0647111
\(78\) 0 0
\(79\) 976.189 1.39025 0.695126 0.718888i \(-0.255349\pi\)
0.695126 + 0.718888i \(0.255349\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −49.1837 −0.0662370
\(83\) 484.466 0.640687 0.320344 0.947301i \(-0.396202\pi\)
0.320344 + 0.947301i \(0.396202\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 935.525 1.17303
\(87\) 0 0
\(88\) 151.015 0.182935
\(89\) 1017.30 1.21161 0.605806 0.795612i \(-0.292851\pi\)
0.605806 + 0.795612i \(0.292851\pi\)
\(90\) 0 0
\(91\) 394.479 0.454425
\(92\) −100.492 −0.113881
\(93\) 0 0
\(94\) −499.896 −0.548514
\(95\) 0 0
\(96\) 0 0
\(97\) −1806.67 −1.89113 −0.945564 0.325437i \(-0.894489\pi\)
−0.945564 + 0.325437i \(0.894489\pi\)
\(98\) −125.516 −0.129378
\(99\) 0 0
\(100\) 0 0
\(101\) 483.053 0.475897 0.237948 0.971278i \(-0.423525\pi\)
0.237948 + 0.971278i \(0.423525\pi\)
\(102\) 0 0
\(103\) 339.049 0.324345 0.162172 0.986762i \(-0.448150\pi\)
0.162172 + 0.986762i \(0.448150\pi\)
\(104\) 1362.48 1.28464
\(105\) 0 0
\(106\) −1309.47 −1.19987
\(107\) −450.847 −0.407336 −0.203668 0.979040i \(-0.565286\pi\)
−0.203668 + 0.979040i \(0.565286\pi\)
\(108\) 0 0
\(109\) −1841.70 −1.61838 −0.809189 0.587548i \(-0.800093\pi\)
−0.809189 + 0.587548i \(0.800093\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −352.963 −0.297785
\(113\) 1874.72 1.56069 0.780347 0.625347i \(-0.215042\pi\)
0.780347 + 0.625347i \(0.215042\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 329.184 0.263482
\(117\) 0 0
\(118\) 727.481 0.567543
\(119\) −172.203 −0.132654
\(120\) 0 0
\(121\) −1291.98 −0.970687
\(122\) 316.250 0.234688
\(123\) 0 0
\(124\) 97.6591 0.0707262
\(125\) 0 0
\(126\) 0 0
\(127\) 38.0984 0.0266196 0.0133098 0.999911i \(-0.495763\pi\)
0.0133098 + 0.999911i \(0.495763\pi\)
\(128\) −940.868 −0.649702
\(129\) 0 0
\(130\) 0 0
\(131\) 1551.51 1.03478 0.517389 0.855750i \(-0.326904\pi\)
0.517389 + 0.855750i \(0.326904\pi\)
\(132\) 0 0
\(133\) −634.958 −0.413969
\(134\) 371.343 0.239396
\(135\) 0 0
\(136\) −594.765 −0.375005
\(137\) −1203.24 −0.750364 −0.375182 0.926951i \(-0.622420\pi\)
−0.375182 + 0.926951i \(0.622420\pi\)
\(138\) 0 0
\(139\) 1897.00 1.15756 0.578781 0.815483i \(-0.303529\pi\)
0.578781 + 0.815483i \(0.303529\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 187.066 0.110551
\(143\) 352.000 0.205844
\(144\) 0 0
\(145\) 0 0
\(146\) 1636.45 0.927626
\(147\) 0 0
\(148\) −84.6477 −0.0470135
\(149\) −704.888 −0.387562 −0.193781 0.981045i \(-0.562075\pi\)
−0.193781 + 0.981045i \(0.562075\pi\)
\(150\) 0 0
\(151\) −3035.21 −1.63578 −0.817888 0.575378i \(-0.804855\pi\)
−0.817888 + 0.575378i \(0.804855\pi\)
\(152\) −2193.06 −1.17027
\(153\) 0 0
\(154\) −112.000 −0.0586053
\(155\) 0 0
\(156\) 0 0
\(157\) −2713.65 −1.37944 −0.689722 0.724074i \(-0.742267\pi\)
−0.689722 + 0.724074i \(0.742267\pi\)
\(158\) −2500.56 −1.25908
\(159\) 0 0
\(160\) 0 0
\(161\) 489.032 0.239386
\(162\) 0 0
\(163\) 465.259 0.223570 0.111785 0.993732i \(-0.464343\pi\)
0.111785 + 0.993732i \(0.464343\pi\)
\(164\) −27.6193 −0.0131506
\(165\) 0 0
\(166\) −1240.98 −0.580236
\(167\) −4156.06 −1.92578 −0.962891 0.269892i \(-0.913012\pi\)
−0.962891 + 0.269892i \(0.913012\pi\)
\(168\) 0 0
\(169\) 978.792 0.445513
\(170\) 0 0
\(171\) 0 0
\(172\) 525.346 0.232891
\(173\) 4241.17 1.86387 0.931936 0.362622i \(-0.118119\pi\)
0.931936 + 0.362622i \(0.118119\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −314.955 −0.134890
\(177\) 0 0
\(178\) −2605.87 −1.09729
\(179\) −2940.35 −1.22778 −0.613889 0.789392i \(-0.710396\pi\)
−0.613889 + 0.789392i \(0.710396\pi\)
\(180\) 0 0
\(181\) −1986.35 −0.815716 −0.407858 0.913045i \(-0.633724\pi\)
−0.407858 + 0.913045i \(0.633724\pi\)
\(182\) −1010.48 −0.411548
\(183\) 0 0
\(184\) 1689.05 0.676732
\(185\) 0 0
\(186\) 0 0
\(187\) −153.659 −0.0600891
\(188\) −280.718 −0.108901
\(189\) 0 0
\(190\) 0 0
\(191\) 1615.88 0.612150 0.306075 0.952007i \(-0.400984\pi\)
0.306075 + 0.952007i \(0.400984\pi\)
\(192\) 0 0
\(193\) 2052.44 0.765482 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(194\) 4627.88 1.71269
\(195\) 0 0
\(196\) −70.4839 −0.0256866
\(197\) 4468.58 1.61611 0.808054 0.589108i \(-0.200521\pi\)
0.808054 + 0.589108i \(0.200521\pi\)
\(198\) 0 0
\(199\) 1543.07 0.549675 0.274838 0.961491i \(-0.411376\pi\)
0.274838 + 0.961491i \(0.411376\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −1237.37 −0.430994
\(203\) −1601.93 −0.553858
\(204\) 0 0
\(205\) 0 0
\(206\) −868.492 −0.293741
\(207\) 0 0
\(208\) −2841.56 −0.947245
\(209\) −566.583 −0.187519
\(210\) 0 0
\(211\) 1284.02 0.418937 0.209469 0.977815i \(-0.432827\pi\)
0.209469 + 0.977815i \(0.432827\pi\)
\(212\) −735.335 −0.238222
\(213\) 0 0
\(214\) 1154.87 0.368902
\(215\) 0 0
\(216\) 0 0
\(217\) −475.244 −0.148671
\(218\) 4717.62 1.46568
\(219\) 0 0
\(220\) 0 0
\(221\) −1386.33 −0.421968
\(222\) 0 0
\(223\) −3815.31 −1.14571 −0.572853 0.819658i \(-0.694163\pi\)
−0.572853 + 0.819658i \(0.694163\pi\)
\(224\) −449.783 −0.134162
\(225\) 0 0
\(226\) −4802.18 −1.41344
\(227\) 2271.53 0.664172 0.332086 0.943249i \(-0.392248\pi\)
0.332086 + 0.943249i \(0.392248\pi\)
\(228\) 0 0
\(229\) −2367.54 −0.683195 −0.341598 0.939846i \(-0.610968\pi\)
−0.341598 + 0.939846i \(0.610968\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −5532.84 −1.56573
\(233\) −1617.71 −0.454849 −0.227425 0.973796i \(-0.573030\pi\)
−0.227425 + 0.973796i \(0.573030\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 408.519 0.112679
\(237\) 0 0
\(238\) 441.106 0.120137
\(239\) −1935.29 −0.523781 −0.261891 0.965098i \(-0.584346\pi\)
−0.261891 + 0.965098i \(0.584346\pi\)
\(240\) 0 0
\(241\) 477.901 0.127736 0.0638679 0.997958i \(-0.479656\pi\)
0.0638679 + 0.997958i \(0.479656\pi\)
\(242\) 3309.49 0.879099
\(243\) 0 0
\(244\) 177.591 0.0465947
\(245\) 0 0
\(246\) 0 0
\(247\) −5111.79 −1.31682
\(248\) −1641.43 −0.420286
\(249\) 0 0
\(250\) 0 0
\(251\) −4769.70 −1.19944 −0.599722 0.800208i \(-0.704722\pi\)
−0.599722 + 0.800208i \(0.704722\pi\)
\(252\) 0 0
\(253\) 436.371 0.108436
\(254\) −97.5910 −0.0241079
\(255\) 0 0
\(256\) −2133.74 −0.520933
\(257\) 682.524 0.165660 0.0828302 0.996564i \(-0.473604\pi\)
0.0828302 + 0.996564i \(0.473604\pi\)
\(258\) 0 0
\(259\) 411.926 0.0988256
\(260\) 0 0
\(261\) 0 0
\(262\) −3974.27 −0.937142
\(263\) 3029.11 0.710202 0.355101 0.934828i \(-0.384447\pi\)
0.355101 + 0.934828i \(0.384447\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1626.48 0.374909
\(267\) 0 0
\(268\) 208.529 0.0475295
\(269\) −6187.33 −1.40241 −0.701205 0.712960i \(-0.747354\pi\)
−0.701205 + 0.712960i \(0.747354\pi\)
\(270\) 0 0
\(271\) −7558.90 −1.69436 −0.847178 0.531309i \(-0.821700\pi\)
−0.847178 + 0.531309i \(0.821700\pi\)
\(272\) 1240.43 0.276516
\(273\) 0 0
\(274\) 3082.17 0.679564
\(275\) 0 0
\(276\) 0 0
\(277\) −3685.36 −0.799393 −0.399697 0.916647i \(-0.630885\pi\)
−0.399697 + 0.916647i \(0.630885\pi\)
\(278\) −4859.26 −1.04834
\(279\) 0 0
\(280\) 0 0
\(281\) −7969.11 −1.69180 −0.845902 0.533338i \(-0.820937\pi\)
−0.845902 + 0.533338i \(0.820937\pi\)
\(282\) 0 0
\(283\) 2479.73 0.520864 0.260432 0.965492i \(-0.416135\pi\)
0.260432 + 0.965492i \(0.416135\pi\)
\(284\) 105.048 0.0219487
\(285\) 0 0
\(286\) −901.667 −0.186422
\(287\) 134.405 0.0276435
\(288\) 0 0
\(289\) −4307.82 −0.876821
\(290\) 0 0
\(291\) 0 0
\(292\) 918.952 0.184170
\(293\) −5950.02 −1.18636 −0.593181 0.805069i \(-0.702128\pi\)
−0.593181 + 0.805069i \(0.702128\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1422.74 0.279375
\(297\) 0 0
\(298\) 1805.61 0.350994
\(299\) 3937.00 0.761480
\(300\) 0 0
\(301\) −2556.52 −0.489554
\(302\) 7774.86 1.48143
\(303\) 0 0
\(304\) 4573.81 0.862915
\(305\) 0 0
\(306\) 0 0
\(307\) 3129.90 0.581865 0.290933 0.956744i \(-0.406034\pi\)
0.290933 + 0.956744i \(0.406034\pi\)
\(308\) −62.8939 −0.0116354
\(309\) 0 0
\(310\) 0 0
\(311\) −7261.25 −1.32395 −0.661973 0.749527i \(-0.730281\pi\)
−0.661973 + 0.749527i \(0.730281\pi\)
\(312\) 0 0
\(313\) 2310.83 0.417303 0.208652 0.977990i \(-0.433093\pi\)
0.208652 + 0.977990i \(0.433093\pi\)
\(314\) 6951.16 1.24929
\(315\) 0 0
\(316\) −1404.20 −0.249975
\(317\) 4701.40 0.832987 0.416494 0.909139i \(-0.363259\pi\)
0.416494 + 0.909139i \(0.363259\pi\)
\(318\) 0 0
\(319\) −1429.42 −0.250885
\(320\) 0 0
\(321\) 0 0
\(322\) −1252.68 −0.216799
\(323\) 2231.46 0.384401
\(324\) 0 0
\(325\) 0 0
\(326\) −1191.79 −0.202475
\(327\) 0 0
\(328\) 464.218 0.0781468
\(329\) 1366.07 0.228918
\(330\) 0 0
\(331\) 1366.18 0.226864 0.113432 0.993546i \(-0.463816\pi\)
0.113432 + 0.993546i \(0.463816\pi\)
\(332\) −696.879 −0.115199
\(333\) 0 0
\(334\) 10646.0 1.74408
\(335\) 0 0
\(336\) 0 0
\(337\) 740.632 0.119718 0.0598588 0.998207i \(-0.480935\pi\)
0.0598588 + 0.998207i \(0.480935\pi\)
\(338\) −2507.23 −0.403477
\(339\) 0 0
\(340\) 0 0
\(341\) −424.068 −0.0673448
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) −8829.90 −1.38394
\(345\) 0 0
\(346\) −10864.0 −1.68801
\(347\) −2605.56 −0.403094 −0.201547 0.979479i \(-0.564597\pi\)
−0.201547 + 0.979479i \(0.564597\pi\)
\(348\) 0 0
\(349\) 4665.07 0.715517 0.357758 0.933814i \(-0.383541\pi\)
0.357758 + 0.933814i \(0.383541\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −401.349 −0.0607726
\(353\) 2964.75 0.447019 0.223509 0.974702i \(-0.428249\pi\)
0.223509 + 0.974702i \(0.428249\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1463.33 −0.217855
\(357\) 0 0
\(358\) 7531.87 1.11193
\(359\) −4267.55 −0.627389 −0.313695 0.949524i \(-0.601567\pi\)
−0.313695 + 0.949524i \(0.601567\pi\)
\(360\) 0 0
\(361\) 1369.00 0.199592
\(362\) 5088.15 0.738749
\(363\) 0 0
\(364\) −567.437 −0.0817082
\(365\) 0 0
\(366\) 0 0
\(367\) 9280.33 1.31997 0.659985 0.751279i \(-0.270562\pi\)
0.659985 + 0.751279i \(0.270562\pi\)
\(368\) −3522.66 −0.498998
\(369\) 0 0
\(370\) 0 0
\(371\) 3578.41 0.500759
\(372\) 0 0
\(373\) −10781.1 −1.49657 −0.748286 0.663376i \(-0.769123\pi\)
−0.748286 + 0.663376i \(0.769123\pi\)
\(374\) 393.606 0.0544195
\(375\) 0 0
\(376\) 4718.24 0.647140
\(377\) −12896.5 −1.76181
\(378\) 0 0
\(379\) −5914.16 −0.801557 −0.400779 0.916175i \(-0.631260\pi\)
−0.400779 + 0.916175i \(0.631260\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4139.15 −0.554391
\(383\) −11513.9 −1.53612 −0.768059 0.640379i \(-0.778777\pi\)
−0.768059 + 0.640379i \(0.778777\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5257.44 −0.693256
\(387\) 0 0
\(388\) 2598.80 0.340036
\(389\) −5399.73 −0.703797 −0.351898 0.936038i \(-0.614464\pi\)
−0.351898 + 0.936038i \(0.614464\pi\)
\(390\) 0 0
\(391\) −1718.62 −0.222288
\(392\) 1184.68 0.152641
\(393\) 0 0
\(394\) −11446.5 −1.46362
\(395\) 0 0
\(396\) 0 0
\(397\) −2622.13 −0.331488 −0.165744 0.986169i \(-0.553003\pi\)
−0.165744 + 0.986169i \(0.553003\pi\)
\(398\) −3952.66 −0.497811
\(399\) 0 0
\(400\) 0 0
\(401\) 11119.1 1.38469 0.692344 0.721568i \(-0.256578\pi\)
0.692344 + 0.721568i \(0.256578\pi\)
\(402\) 0 0
\(403\) −3826.00 −0.472920
\(404\) −694.846 −0.0855690
\(405\) 0 0
\(406\) 4103.42 0.501599
\(407\) 367.568 0.0447658
\(408\) 0 0
\(409\) −6589.18 −0.796611 −0.398305 0.917253i \(-0.630402\pi\)
−0.398305 + 0.917253i \(0.630402\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −487.704 −0.0583191
\(413\) −1988.00 −0.236860
\(414\) 0 0
\(415\) 0 0
\(416\) −3621.02 −0.426767
\(417\) 0 0
\(418\) 1451.33 0.169825
\(419\) 11871.6 1.38416 0.692081 0.721820i \(-0.256694\pi\)
0.692081 + 0.721820i \(0.256694\pi\)
\(420\) 0 0
\(421\) −1731.57 −0.200455 −0.100227 0.994965i \(-0.531957\pi\)
−0.100227 + 0.994965i \(0.531957\pi\)
\(422\) −3289.09 −0.379409
\(423\) 0 0
\(424\) 12359.3 1.41562
\(425\) 0 0
\(426\) 0 0
\(427\) −864.222 −0.0979452
\(428\) 648.519 0.0732415
\(429\) 0 0
\(430\) 0 0
\(431\) −10653.8 −1.19066 −0.595330 0.803481i \(-0.702979\pi\)
−0.595330 + 0.803481i \(0.702979\pi\)
\(432\) 0 0
\(433\) −2642.01 −0.293226 −0.146613 0.989194i \(-0.546837\pi\)
−0.146613 + 0.989194i \(0.546837\pi\)
\(434\) 1217.36 0.134644
\(435\) 0 0
\(436\) 2649.19 0.290994
\(437\) −6337.04 −0.693688
\(438\) 0 0
\(439\) 8858.21 0.963051 0.481525 0.876432i \(-0.340083\pi\)
0.481525 + 0.876432i \(0.340083\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3551.17 0.382153
\(443\) −6621.73 −0.710176 −0.355088 0.934833i \(-0.615549\pi\)
−0.355088 + 0.934833i \(0.615549\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 9773.13 1.03760
\(447\) 0 0
\(448\) 3975.85 0.419288
\(449\) −13081.7 −1.37497 −0.687487 0.726196i \(-0.741286\pi\)
−0.687487 + 0.726196i \(0.741286\pi\)
\(450\) 0 0
\(451\) 119.932 0.0125219
\(452\) −2696.68 −0.280622
\(453\) 0 0
\(454\) −5818.65 −0.601504
\(455\) 0 0
\(456\) 0 0
\(457\) −12167.0 −1.24540 −0.622699 0.782461i \(-0.713964\pi\)
−0.622699 + 0.782461i \(0.713964\pi\)
\(458\) 6064.59 0.618733
\(459\) 0 0
\(460\) 0 0
\(461\) −11283.8 −1.14000 −0.570000 0.821645i \(-0.693057\pi\)
−0.570000 + 0.821645i \(0.693057\pi\)
\(462\) 0 0
\(463\) −16542.9 −1.66051 −0.830254 0.557385i \(-0.811805\pi\)
−0.830254 + 0.557385i \(0.811805\pi\)
\(464\) 11539.2 1.15451
\(465\) 0 0
\(466\) 4143.85 0.411932
\(467\) −10266.9 −1.01734 −0.508668 0.860963i \(-0.669862\pi\)
−0.508668 + 0.860963i \(0.669862\pi\)
\(468\) 0 0
\(469\) −1014.77 −0.0999103
\(470\) 0 0
\(471\) 0 0
\(472\) −6866.29 −0.669590
\(473\) −2281.23 −0.221757
\(474\) 0 0
\(475\) 0 0
\(476\) 247.704 0.0238519
\(477\) 0 0
\(478\) 4957.36 0.474360
\(479\) 7967.98 0.760055 0.380027 0.924975i \(-0.375915\pi\)
0.380027 + 0.924975i \(0.375915\pi\)
\(480\) 0 0
\(481\) 3316.25 0.314362
\(482\) −1224.17 −0.115683
\(483\) 0 0
\(484\) 1858.45 0.174535
\(485\) 0 0
\(486\) 0 0
\(487\) −9956.62 −0.926443 −0.463221 0.886243i \(-0.653306\pi\)
−0.463221 + 0.886243i \(0.653306\pi\)
\(488\) −2984.91 −0.276886
\(489\) 0 0
\(490\) 0 0
\(491\) 18660.8 1.71518 0.857589 0.514336i \(-0.171962\pi\)
0.857589 + 0.514336i \(0.171962\pi\)
\(492\) 0 0
\(493\) 5629.71 0.514299
\(494\) 13094.1 1.19258
\(495\) 0 0
\(496\) 3423.34 0.309904
\(497\) −511.199 −0.0461377
\(498\) 0 0
\(499\) 4074.21 0.365504 0.182752 0.983159i \(-0.441499\pi\)
0.182752 + 0.983159i \(0.441499\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 12217.8 1.08627
\(503\) −4255.51 −0.377224 −0.188612 0.982052i \(-0.560399\pi\)
−0.188612 + 0.982052i \(0.560399\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1117.79 −0.0982050
\(507\) 0 0
\(508\) −54.8025 −0.00478636
\(509\) −10171.8 −0.885771 −0.442885 0.896578i \(-0.646045\pi\)
−0.442885 + 0.896578i \(0.646045\pi\)
\(510\) 0 0
\(511\) −4471.95 −0.387138
\(512\) 12992.6 1.12148
\(513\) 0 0
\(514\) −1748.32 −0.150030
\(515\) 0 0
\(516\) 0 0
\(517\) 1218.97 0.103695
\(518\) −1055.17 −0.0895010
\(519\) 0 0
\(520\) 0 0
\(521\) −1680.39 −0.141303 −0.0706517 0.997501i \(-0.522508\pi\)
−0.0706517 + 0.997501i \(0.522508\pi\)
\(522\) 0 0
\(523\) −13211.8 −1.10461 −0.552305 0.833642i \(-0.686251\pi\)
−0.552305 + 0.833642i \(0.686251\pi\)
\(524\) −2231.76 −0.186059
\(525\) 0 0
\(526\) −7759.23 −0.643191
\(527\) 1670.17 0.138053
\(528\) 0 0
\(529\) −7286.34 −0.598861
\(530\) 0 0
\(531\) 0 0
\(532\) 913.354 0.0744341
\(533\) 1082.04 0.0879333
\(534\) 0 0
\(535\) 0 0
\(536\) −3504.90 −0.282441
\(537\) 0 0
\(538\) 15849.2 1.27009
\(539\) 306.064 0.0244585
\(540\) 0 0
\(541\) 9650.84 0.766954 0.383477 0.923551i \(-0.374727\pi\)
0.383477 + 0.923551i \(0.374727\pi\)
\(542\) 19362.5 1.53449
\(543\) 0 0
\(544\) 1580.69 0.124580
\(545\) 0 0
\(546\) 0 0
\(547\) 23864.3 1.86538 0.932689 0.360682i \(-0.117456\pi\)
0.932689 + 0.360682i \(0.117456\pi\)
\(548\) 1730.80 0.134920
\(549\) 0 0
\(550\) 0 0
\(551\) 20758.3 1.60496
\(552\) 0 0
\(553\) 6833.33 0.525466
\(554\) 9440.25 0.723967
\(555\) 0 0
\(556\) −2728.73 −0.208136
\(557\) −2314.22 −0.176044 −0.0880221 0.996119i \(-0.528055\pi\)
−0.0880221 + 0.996119i \(0.528055\pi\)
\(558\) 0 0
\(559\) −20581.5 −1.55726
\(560\) 0 0
\(561\) 0 0
\(562\) 20413.3 1.53218
\(563\) −7017.86 −0.525342 −0.262671 0.964885i \(-0.584603\pi\)
−0.262671 + 0.964885i \(0.584603\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −6351.95 −0.471718
\(567\) 0 0
\(568\) −1765.61 −0.130429
\(569\) −5302.37 −0.390662 −0.195331 0.980737i \(-0.562578\pi\)
−0.195331 + 0.980737i \(0.562578\pi\)
\(570\) 0 0
\(571\) 17767.2 1.30216 0.651082 0.759008i \(-0.274315\pi\)
0.651082 + 0.759008i \(0.274315\pi\)
\(572\) −506.333 −0.0370120
\(573\) 0 0
\(574\) −344.286 −0.0250352
\(575\) 0 0
\(576\) 0 0
\(577\) 6089.57 0.439363 0.219681 0.975572i \(-0.429498\pi\)
0.219681 + 0.975572i \(0.429498\pi\)
\(578\) 11034.7 0.794089
\(579\) 0 0
\(580\) 0 0
\(581\) 3391.26 0.242157
\(582\) 0 0
\(583\) 3193.07 0.226833
\(584\) −15445.5 −1.09442
\(585\) 0 0
\(586\) 15241.3 1.07442
\(587\) −26543.9 −1.86641 −0.933205 0.359344i \(-0.883000\pi\)
−0.933205 + 0.359344i \(0.883000\pi\)
\(588\) 0 0
\(589\) 6158.37 0.430817
\(590\) 0 0
\(591\) 0 0
\(592\) −2967.24 −0.206001
\(593\) 16365.0 1.13327 0.566635 0.823969i \(-0.308245\pi\)
0.566635 + 0.823969i \(0.308245\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1013.94 0.0696859
\(597\) 0 0
\(598\) −10084.8 −0.689631
\(599\) 17516.3 1.19482 0.597411 0.801935i \(-0.296196\pi\)
0.597411 + 0.801935i \(0.296196\pi\)
\(600\) 0 0
\(601\) 8693.80 0.590062 0.295031 0.955488i \(-0.404670\pi\)
0.295031 + 0.955488i \(0.404670\pi\)
\(602\) 6548.67 0.443362
\(603\) 0 0
\(604\) 4365.99 0.294122
\(605\) 0 0
\(606\) 0 0
\(607\) 18096.9 1.21010 0.605051 0.796186i \(-0.293153\pi\)
0.605051 + 0.796186i \(0.293153\pi\)
\(608\) 5828.44 0.388774
\(609\) 0 0
\(610\) 0 0
\(611\) 10997.7 0.728183
\(612\) 0 0
\(613\) −4641.61 −0.305828 −0.152914 0.988239i \(-0.548866\pi\)
−0.152914 + 0.988239i \(0.548866\pi\)
\(614\) −8017.40 −0.526964
\(615\) 0 0
\(616\) 1057.11 0.0691429
\(617\) 14676.1 0.957600 0.478800 0.877924i \(-0.341072\pi\)
0.478800 + 0.877924i \(0.341072\pi\)
\(618\) 0 0
\(619\) −19645.3 −1.27563 −0.637813 0.770192i \(-0.720161\pi\)
−0.637813 + 0.770192i \(0.720161\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 18600.1 1.19903
\(623\) 7121.09 0.457946
\(624\) 0 0
\(625\) 0 0
\(626\) −5919.32 −0.377929
\(627\) 0 0
\(628\) 3903.44 0.248032
\(629\) −1447.65 −0.0917671
\(630\) 0 0
\(631\) 26231.2 1.65491 0.827456 0.561531i \(-0.189788\pi\)
0.827456 + 0.561531i \(0.189788\pi\)
\(632\) 23601.4 1.48546
\(633\) 0 0
\(634\) −12042.9 −0.754391
\(635\) 0 0
\(636\) 0 0
\(637\) 2761.35 0.171756
\(638\) 3661.55 0.227213
\(639\) 0 0
\(640\) 0 0
\(641\) −30882.4 −1.90293 −0.951466 0.307754i \(-0.900423\pi\)
−0.951466 + 0.307754i \(0.900423\pi\)
\(642\) 0 0
\(643\) 6216.88 0.381290 0.190645 0.981659i \(-0.438942\pi\)
0.190645 + 0.981659i \(0.438942\pi\)
\(644\) −703.447 −0.0430430
\(645\) 0 0
\(646\) −5716.00 −0.348132
\(647\) −21210.4 −1.28882 −0.644410 0.764680i \(-0.722897\pi\)
−0.644410 + 0.764680i \(0.722897\pi\)
\(648\) 0 0
\(649\) −1773.92 −0.107292
\(650\) 0 0
\(651\) 0 0
\(652\) −669.251 −0.0401992
\(653\) 32938.7 1.97395 0.986977 0.160864i \(-0.0514281\pi\)
0.986977 + 0.160864i \(0.0514281\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −968.165 −0.0576227
\(657\) 0 0
\(658\) −3499.27 −0.207319
\(659\) −9543.51 −0.564131 −0.282066 0.959395i \(-0.591020\pi\)
−0.282066 + 0.959395i \(0.591020\pi\)
\(660\) 0 0
\(661\) 13274.5 0.781116 0.390558 0.920578i \(-0.372282\pi\)
0.390558 + 0.920578i \(0.372282\pi\)
\(662\) −3499.55 −0.205459
\(663\) 0 0
\(664\) 11713.0 0.684565
\(665\) 0 0
\(666\) 0 0
\(667\) −15987.6 −0.928101
\(668\) 5978.27 0.346267
\(669\) 0 0
\(670\) 0 0
\(671\) −771.159 −0.0443670
\(672\) 0 0
\(673\) 13575.3 0.777545 0.388772 0.921334i \(-0.372899\pi\)
0.388772 + 0.921334i \(0.372899\pi\)
\(674\) −1897.17 −0.108422
\(675\) 0 0
\(676\) −1407.94 −0.0801058
\(677\) 12020.7 0.682414 0.341207 0.939988i \(-0.389164\pi\)
0.341207 + 0.939988i \(0.389164\pi\)
\(678\) 0 0
\(679\) −12646.7 −0.714779
\(680\) 0 0
\(681\) 0 0
\(682\) 1086.27 0.0609905
\(683\) 18391.9 1.03038 0.515188 0.857077i \(-0.327722\pi\)
0.515188 + 0.857077i \(0.327722\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −878.613 −0.0489003
\(687\) 0 0
\(688\) 18415.5 1.02047
\(689\) 28808.3 1.59290
\(690\) 0 0
\(691\) 15594.1 0.858508 0.429254 0.903184i \(-0.358777\pi\)
0.429254 + 0.903184i \(0.358777\pi\)
\(692\) −6100.69 −0.335135
\(693\) 0 0
\(694\) 6674.28 0.365061
\(695\) 0 0
\(696\) 0 0
\(697\) −472.346 −0.0256691
\(698\) −11949.8 −0.648004
\(699\) 0 0
\(700\) 0 0
\(701\) 31093.7 1.67531 0.837656 0.546198i \(-0.183925\pi\)
0.837656 + 0.546198i \(0.183925\pi\)
\(702\) 0 0
\(703\) −5337.88 −0.286375
\(704\) 3547.71 0.189928
\(705\) 0 0
\(706\) −7594.36 −0.404841
\(707\) 3381.37 0.179872
\(708\) 0 0
\(709\) −2494.67 −0.132143 −0.0660714 0.997815i \(-0.521047\pi\)
−0.0660714 + 0.997815i \(0.521047\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 24595.3 1.29459
\(713\) −4743.06 −0.249129
\(714\) 0 0
\(715\) 0 0
\(716\) 4229.54 0.220762
\(717\) 0 0
\(718\) 10931.6 0.568192
\(719\) −34467.1 −1.78777 −0.893885 0.448295i \(-0.852031\pi\)
−0.893885 + 0.448295i \(0.852031\pi\)
\(720\) 0 0
\(721\) 2373.34 0.122591
\(722\) −3506.77 −0.180759
\(723\) 0 0
\(724\) 2857.27 0.146670
\(725\) 0 0
\(726\) 0 0
\(727\) −9314.97 −0.475204 −0.237602 0.971363i \(-0.576361\pi\)
−0.237602 + 0.971363i \(0.576361\pi\)
\(728\) 9537.35 0.485547
\(729\) 0 0
\(730\) 0 0
\(731\) 8984.49 0.454588
\(732\) 0 0
\(733\) −16146.3 −0.813611 −0.406805 0.913515i \(-0.633357\pi\)
−0.406805 + 0.913515i \(0.633357\pi\)
\(734\) −23772.1 −1.19543
\(735\) 0 0
\(736\) −4488.95 −0.224816
\(737\) −905.500 −0.0452571
\(738\) 0 0
\(739\) −36749.1 −1.82928 −0.914640 0.404268i \(-0.867526\pi\)
−0.914640 + 0.404268i \(0.867526\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −9166.27 −0.453510
\(743\) −2527.09 −0.124778 −0.0623890 0.998052i \(-0.519872\pi\)
−0.0623890 + 0.998052i \(0.519872\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 27616.2 1.35536
\(747\) 0 0
\(748\) 221.031 0.0108044
\(749\) −3155.93 −0.153959
\(750\) 0 0
\(751\) 15828.0 0.769070 0.384535 0.923111i \(-0.374362\pi\)
0.384535 + 0.923111i \(0.374362\pi\)
\(752\) −9840.28 −0.477178
\(753\) 0 0
\(754\) 33035.0 1.59557
\(755\) 0 0
\(756\) 0 0
\(757\) 20845.9 1.00087 0.500435 0.865774i \(-0.333174\pi\)
0.500435 + 0.865774i \(0.333174\pi\)
\(758\) 15149.4 0.725927
\(759\) 0 0
\(760\) 0 0
\(761\) −2420.55 −0.115302 −0.0576511 0.998337i \(-0.518361\pi\)
−0.0576511 + 0.998337i \(0.518361\pi\)
\(762\) 0 0
\(763\) −12891.9 −0.611690
\(764\) −2324.35 −0.110068
\(765\) 0 0
\(766\) 29493.5 1.39118
\(767\) −16004.6 −0.753445
\(768\) 0 0
\(769\) 22646.3 1.06196 0.530980 0.847384i \(-0.321824\pi\)
0.530980 + 0.847384i \(0.321824\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2952.33 −0.137638
\(773\) 27620.2 1.28516 0.642580 0.766219i \(-0.277864\pi\)
0.642580 + 0.766219i \(0.277864\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −43680.0 −2.02064
\(777\) 0 0
\(778\) 13831.7 0.637391
\(779\) −1741.67 −0.0801049
\(780\) 0 0
\(781\) −456.151 −0.0208993
\(782\) 4402.35 0.201314
\(783\) 0 0
\(784\) −2470.74 −0.112552
\(785\) 0 0
\(786\) 0 0
\(787\) −14767.1 −0.668859 −0.334429 0.942421i \(-0.608544\pi\)
−0.334429 + 0.942421i \(0.608544\pi\)
\(788\) −6427.82 −0.290586
\(789\) 0 0
\(790\) 0 0
\(791\) 13123.0 0.589887
\(792\) 0 0
\(793\) −6957.50 −0.311561
\(794\) 6716.71 0.300211
\(795\) 0 0
\(796\) −2219.62 −0.0988348
\(797\) −7549.97 −0.335551 −0.167775 0.985825i \(-0.553658\pi\)
−0.167775 + 0.985825i \(0.553658\pi\)
\(798\) 0 0
\(799\) −4800.85 −0.212568
\(800\) 0 0
\(801\) 0 0
\(802\) −28482.1 −1.25404
\(803\) −3990.39 −0.175365
\(804\) 0 0
\(805\) 0 0
\(806\) 9800.50 0.428298
\(807\) 0 0
\(808\) 11678.8 0.508489
\(809\) −17920.0 −0.778783 −0.389391 0.921072i \(-0.627315\pi\)
−0.389391 + 0.921072i \(0.627315\pi\)
\(810\) 0 0
\(811\) 25536.6 1.10569 0.552843 0.833285i \(-0.313543\pi\)
0.552843 + 0.833285i \(0.313543\pi\)
\(812\) 2304.29 0.0995869
\(813\) 0 0
\(814\) −941.545 −0.0405420
\(815\) 0 0
\(816\) 0 0
\(817\) 33128.3 1.41862
\(818\) 16878.5 0.721447
\(819\) 0 0
\(820\) 0 0
\(821\) 13688.8 0.581904 0.290952 0.956738i \(-0.406028\pi\)
0.290952 + 0.956738i \(0.406028\pi\)
\(822\) 0 0
\(823\) 15102.9 0.639678 0.319839 0.947472i \(-0.396371\pi\)
0.319839 + 0.947472i \(0.396371\pi\)
\(824\) 8197.22 0.346558
\(825\) 0 0
\(826\) 5092.37 0.214511
\(827\) −36290.3 −1.52592 −0.762961 0.646445i \(-0.776255\pi\)
−0.762961 + 0.646445i \(0.776255\pi\)
\(828\) 0 0
\(829\) −39405.1 −1.65090 −0.825449 0.564477i \(-0.809078\pi\)
−0.825449 + 0.564477i \(0.809078\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 32007.9 1.33374
\(833\) −1205.42 −0.0501384
\(834\) 0 0
\(835\) 0 0
\(836\) 815.000 0.0337170
\(837\) 0 0
\(838\) −30409.6 −1.25356
\(839\) 33093.9 1.36177 0.680886 0.732389i \(-0.261595\pi\)
0.680886 + 0.732389i \(0.261595\pi\)
\(840\) 0 0
\(841\) 27981.8 1.14731
\(842\) 4435.50 0.181541
\(843\) 0 0
\(844\) −1847.00 −0.0753274
\(845\) 0 0
\(846\) 0 0
\(847\) −9043.89 −0.366885
\(848\) −25776.4 −1.04383
\(849\) 0 0
\(850\) 0 0
\(851\) 4111.12 0.165602
\(852\) 0 0
\(853\) −29441.6 −1.18178 −0.590892 0.806750i \(-0.701224\pi\)
−0.590892 + 0.806750i \(0.701224\pi\)
\(854\) 2213.75 0.0887037
\(855\) 0 0
\(856\) −10900.2 −0.435233
\(857\) −16012.9 −0.638260 −0.319130 0.947711i \(-0.603391\pi\)
−0.319130 + 0.947711i \(0.603391\pi\)
\(858\) 0 0
\(859\) −13404.3 −0.532421 −0.266211 0.963915i \(-0.585772\pi\)
−0.266211 + 0.963915i \(0.585772\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 27290.2 1.07832
\(863\) 2058.73 0.0812050 0.0406025 0.999175i \(-0.487072\pi\)
0.0406025 + 0.999175i \(0.487072\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 6767.66 0.265559
\(867\) 0 0
\(868\) 683.614 0.0267320
\(869\) 6097.48 0.238024
\(870\) 0 0
\(871\) −8169.54 −0.317812
\(872\) −44527.0 −1.72922
\(873\) 0 0
\(874\) 16232.7 0.628236
\(875\) 0 0
\(876\) 0 0
\(877\) 16477.0 0.634422 0.317211 0.948355i \(-0.397254\pi\)
0.317211 + 0.948355i \(0.397254\pi\)
\(878\) −22690.8 −0.872183
\(879\) 0 0
\(880\) 0 0
\(881\) −43307.7 −1.65616 −0.828079 0.560612i \(-0.810566\pi\)
−0.828079 + 0.560612i \(0.810566\pi\)
\(882\) 0 0
\(883\) −15197.9 −0.579217 −0.289608 0.957145i \(-0.593525\pi\)
−0.289608 + 0.957145i \(0.593525\pi\)
\(884\) 1994.17 0.0758723
\(885\) 0 0
\(886\) 16961.9 0.643168
\(887\) 44953.6 1.70168 0.850842 0.525422i \(-0.176093\pi\)
0.850842 + 0.525422i \(0.176093\pi\)
\(888\) 0 0
\(889\) 266.689 0.0100613
\(890\) 0 0
\(891\) 0 0
\(892\) 5488.13 0.206005
\(893\) −17702.0 −0.663355
\(894\) 0 0
\(895\) 0 0
\(896\) −6586.08 −0.245564
\(897\) 0 0
\(898\) 33509.5 1.24524
\(899\) 15536.9 0.576400
\(900\) 0 0
\(901\) −12575.7 −0.464993
\(902\) −307.212 −0.0113404
\(903\) 0 0
\(904\) 45325.2 1.66758
\(905\) 0 0
\(906\) 0 0
\(907\) 38388.7 1.40537 0.702687 0.711499i \(-0.251983\pi\)
0.702687 + 0.711499i \(0.251983\pi\)
\(908\) −3267.48 −0.119422
\(909\) 0 0
\(910\) 0 0
\(911\) 46222.1 1.68102 0.840508 0.541799i \(-0.182257\pi\)
0.840508 + 0.541799i \(0.182257\pi\)
\(912\) 0 0
\(913\) 3026.08 0.109692
\(914\) 31166.3 1.12789
\(915\) 0 0
\(916\) 3405.59 0.122842
\(917\) 10860.6 0.391109
\(918\) 0 0
\(919\) −30946.4 −1.11080 −0.555402 0.831582i \(-0.687436\pi\)
−0.555402 + 0.831582i \(0.687436\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 28904.1 1.03244
\(923\) −4115.46 −0.146763
\(924\) 0 0
\(925\) 0 0
\(926\) 42375.6 1.50383
\(927\) 0 0
\(928\) 14704.5 0.520149
\(929\) 1907.48 0.0673652 0.0336826 0.999433i \(-0.489276\pi\)
0.0336826 + 0.999433i \(0.489276\pi\)
\(930\) 0 0
\(931\) −4444.71 −0.156466
\(932\) 2326.99 0.0817845
\(933\) 0 0
\(934\) 26299.2 0.921347
\(935\) 0 0
\(936\) 0 0
\(937\) 3334.99 0.116275 0.0581374 0.998309i \(-0.481484\pi\)
0.0581374 + 0.998309i \(0.481484\pi\)
\(938\) 2599.40 0.0904834
\(939\) 0 0
\(940\) 0 0
\(941\) −9632.00 −0.333681 −0.166841 0.985984i \(-0.553357\pi\)
−0.166841 + 0.985984i \(0.553357\pi\)
\(942\) 0 0
\(943\) 1341.40 0.0463223
\(944\) 14320.2 0.493732
\(945\) 0 0
\(946\) 5843.48 0.200833
\(947\) 53606.0 1.83945 0.919727 0.392559i \(-0.128410\pi\)
0.919727 + 0.392559i \(0.128410\pi\)
\(948\) 0 0
\(949\) −36001.9 −1.23148
\(950\) 0 0
\(951\) 0 0
\(952\) −4163.36 −0.141739
\(953\) −28433.6 −0.966481 −0.483240 0.875488i \(-0.660540\pi\)
−0.483240 + 0.875488i \(0.660540\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 2783.82 0.0941790
\(957\) 0 0
\(958\) −20410.4 −0.688341
\(959\) −8422.70 −0.283611
\(960\) 0 0
\(961\) −25181.7 −0.845278
\(962\) −8494.75 −0.284700
\(963\) 0 0
\(964\) −687.436 −0.0229676
\(965\) 0 0
\(966\) 0 0
\(967\) −33877.6 −1.12661 −0.563304 0.826250i \(-0.690470\pi\)
−0.563304 + 0.826250i \(0.690470\pi\)
\(968\) −31236.4 −1.03717
\(969\) 0 0
\(970\) 0 0
\(971\) −10208.6 −0.337394 −0.168697 0.985668i \(-0.553956\pi\)
−0.168697 + 0.985668i \(0.553956\pi\)
\(972\) 0 0
\(973\) 13279.0 0.437517
\(974\) 25504.4 0.839029
\(975\) 0 0
\(976\) 6225.27 0.204166
\(977\) 35478.1 1.16177 0.580883 0.813987i \(-0.302707\pi\)
0.580883 + 0.813987i \(0.302707\pi\)
\(978\) 0 0
\(979\) 6354.27 0.207439
\(980\) 0 0
\(981\) 0 0
\(982\) −47800.7 −1.55334
\(983\) −54435.8 −1.76626 −0.883130 0.469128i \(-0.844568\pi\)
−0.883130 + 0.469128i \(0.844568\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −14420.8 −0.465773
\(987\) 0 0
\(988\) 7353.04 0.236773
\(989\) −25514.7 −0.820346
\(990\) 0 0
\(991\) −47319.6 −1.51681 −0.758404 0.651784i \(-0.774021\pi\)
−0.758404 + 0.651784i \(0.774021\pi\)
\(992\) 4362.38 0.139623
\(993\) 0 0
\(994\) 1309.46 0.0417844
\(995\) 0 0
\(996\) 0 0
\(997\) −26273.1 −0.834580 −0.417290 0.908773i \(-0.637020\pi\)
−0.417290 + 0.908773i \(0.637020\pi\)
\(998\) −10436.3 −0.331017
\(999\) 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.r.1.1 2
3.2 odd 2 1575.4.a.u.1.2 2
5.4 even 2 315.4.a.j.1.2 yes 2
15.14 odd 2 315.4.a.h.1.1 2
35.34 odd 2 2205.4.a.ba.1.2 2
105.104 even 2 2205.4.a.y.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
315.4.a.h.1.1 2 15.14 odd 2
315.4.a.j.1.2 yes 2 5.4 even 2
1575.4.a.r.1.1 2 1.1 even 1 trivial
1575.4.a.u.1.2 2 3.2 odd 2
2205.4.a.y.1.1 2 105.104 even 2
2205.4.a.ba.1.2 2 35.34 odd 2