Properties

Label 525.4.a.j.1.1
Level $525$
Weight $4$
Character 525.1
Self dual yes
Analytic conductor $30.976$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.9760027530\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 525.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.56155 q^{2} +3.00000 q^{3} +4.68466 q^{4} -10.6847 q^{6} +7.00000 q^{7} +11.8078 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.56155 q^{2} +3.00000 q^{3} +4.68466 q^{4} -10.6847 q^{6} +7.00000 q^{7} +11.8078 q^{8} +9.00000 q^{9} -5.19224 q^{11} +14.0540 q^{12} -54.5464 q^{13} -24.9309 q^{14} -79.5312 q^{16} -16.1619 q^{17} -32.0540 q^{18} +87.4470 q^{19} +21.0000 q^{21} +18.4924 q^{22} -176.477 q^{23} +35.4233 q^{24} +194.270 q^{26} +27.0000 q^{27} +32.7926 q^{28} +142.170 q^{29} -94.3002 q^{31} +188.793 q^{32} -15.5767 q^{33} +57.5616 q^{34} +42.1619 q^{36} -17.3305 q^{37} -311.447 q^{38} -163.639 q^{39} +210.270 q^{41} -74.7926 q^{42} -521.570 q^{43} -24.3239 q^{44} +628.533 q^{46} +105.417 q^{47} -238.594 q^{48} +49.0000 q^{49} -48.4858 q^{51} -255.531 q^{52} +108.978 q^{53} -96.1619 q^{54} +82.6543 q^{56} +262.341 q^{57} -506.348 q^{58} +210.365 q^{59} -674.304 q^{61} +335.855 q^{62} +63.0000 q^{63} -36.1449 q^{64} +55.4773 q^{66} -324.929 q^{67} -75.7131 q^{68} -529.432 q^{69} +793.965 q^{71} +106.270 q^{72} -315.417 q^{73} +61.7235 q^{74} +409.659 q^{76} -36.3457 q^{77} +582.810 q^{78} -425.840 q^{79} +81.0000 q^{81} -748.887 q^{82} +283.029 q^{83} +98.3778 q^{84} +1857.60 q^{86} +426.511 q^{87} -61.3087 q^{88} -843.131 q^{89} -381.825 q^{91} -826.736 q^{92} -282.901 q^{93} -375.447 q^{94} +566.378 q^{96} -1537.33 q^{97} -174.516 q^{98} -46.7301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9} - 31 q^{11} - 9 q^{12} - 39 q^{13} - 21 q^{14} - 23 q^{16} + 79 q^{17} - 27 q^{18} - 56 q^{19} + 42 q^{21} + 4 q^{22} - 254 q^{23} + 9 q^{24} + 203 q^{26} + 54 q^{27} - 21 q^{28} - 62 q^{29} - 135 q^{31} + 291 q^{32} - 93 q^{33} + 111 q^{34} - 27 q^{36} - 113 q^{37} - 392 q^{38} - 117 q^{39} + 235 q^{41} - 63 q^{42} - 804 q^{43} + 174 q^{44} + 585 q^{46} - 152 q^{47} - 69 q^{48} + 98 q^{49} + 237 q^{51} - 375 q^{52} - 149 q^{53} - 81 q^{54} + 21 q^{56} - 168 q^{57} - 621 q^{58} - 441 q^{59} - 223 q^{61} + 313 q^{62} + 126 q^{63} - 431 q^{64} + 12 q^{66} - 1157 q^{67} - 807 q^{68} - 762 q^{69} + 619 q^{71} + 27 q^{72} - 268 q^{73} + 8 q^{74} + 1512 q^{76} - 217 q^{77} + 609 q^{78} - 427 q^{79} + 162 q^{81} - 735 q^{82} - 1211 q^{83} - 63 q^{84} + 1699 q^{86} - 186 q^{87} + 166 q^{88} + 466 q^{89} - 273 q^{91} - 231 q^{92} - 405 q^{93} - 520 q^{94} + 873 q^{96} - 172 q^{97} - 147 q^{98} - 279 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.56155 −1.25920 −0.629600 0.776920i \(-0.716781\pi\)
−0.629600 + 0.776920i \(0.716781\pi\)
\(3\) 3.00000 0.577350
\(4\) 4.68466 0.585582
\(5\) 0 0
\(6\) −10.6847 −0.726999
\(7\) 7.00000 0.377964
\(8\) 11.8078 0.521834
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −5.19224 −0.142320 −0.0711599 0.997465i \(-0.522670\pi\)
−0.0711599 + 0.997465i \(0.522670\pi\)
\(12\) 14.0540 0.338086
\(13\) −54.5464 −1.16373 −0.581863 0.813287i \(-0.697676\pi\)
−0.581863 + 0.813287i \(0.697676\pi\)
\(14\) −24.9309 −0.475933
\(15\) 0 0
\(16\) −79.5312 −1.24268
\(17\) −16.1619 −0.230579 −0.115289 0.993332i \(-0.536780\pi\)
−0.115289 + 0.993332i \(0.536780\pi\)
\(18\) −32.0540 −0.419733
\(19\) 87.4470 1.05588 0.527940 0.849282i \(-0.322965\pi\)
0.527940 + 0.849282i \(0.322965\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) 18.4924 0.179209
\(23\) −176.477 −1.59992 −0.799958 0.600056i \(-0.795145\pi\)
−0.799958 + 0.600056i \(0.795145\pi\)
\(24\) 35.4233 0.301281
\(25\) 0 0
\(26\) 194.270 1.46536
\(27\) 27.0000 0.192450
\(28\) 32.7926 0.221329
\(29\) 142.170 0.910358 0.455179 0.890400i \(-0.349575\pi\)
0.455179 + 0.890400i \(0.349575\pi\)
\(30\) 0 0
\(31\) −94.3002 −0.546349 −0.273174 0.961965i \(-0.588074\pi\)
−0.273174 + 0.961965i \(0.588074\pi\)
\(32\) 188.793 1.04294
\(33\) −15.5767 −0.0821684
\(34\) 57.5616 0.290345
\(35\) 0 0
\(36\) 42.1619 0.195194
\(37\) −17.3305 −0.0770031 −0.0385016 0.999259i \(-0.512258\pi\)
−0.0385016 + 0.999259i \(0.512258\pi\)
\(38\) −311.447 −1.32956
\(39\) −163.639 −0.671878
\(40\) 0 0
\(41\) 210.270 0.800942 0.400471 0.916309i \(-0.368846\pi\)
0.400471 + 0.916309i \(0.368846\pi\)
\(42\) −74.7926 −0.274780
\(43\) −521.570 −1.84974 −0.924868 0.380287i \(-0.875825\pi\)
−0.924868 + 0.380287i \(0.875825\pi\)
\(44\) −24.3239 −0.0833400
\(45\) 0 0
\(46\) 628.533 2.01461
\(47\) 105.417 0.327162 0.163581 0.986530i \(-0.447696\pi\)
0.163581 + 0.986530i \(0.447696\pi\)
\(48\) −238.594 −0.717459
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −48.4858 −0.133125
\(52\) −255.531 −0.681458
\(53\) 108.978 0.282440 0.141220 0.989978i \(-0.454898\pi\)
0.141220 + 0.989978i \(0.454898\pi\)
\(54\) −96.1619 −0.242333
\(55\) 0 0
\(56\) 82.6543 0.197235
\(57\) 262.341 0.609612
\(58\) −506.348 −1.14632
\(59\) 210.365 0.464189 0.232094 0.972693i \(-0.425442\pi\)
0.232094 + 0.972693i \(0.425442\pi\)
\(60\) 0 0
\(61\) −674.304 −1.41534 −0.707670 0.706543i \(-0.750254\pi\)
−0.707670 + 0.706543i \(0.750254\pi\)
\(62\) 335.855 0.687962
\(63\) 63.0000 0.125988
\(64\) −36.1449 −0.0705955
\(65\) 0 0
\(66\) 55.4773 0.103466
\(67\) −324.929 −0.592484 −0.296242 0.955113i \(-0.595733\pi\)
−0.296242 + 0.955113i \(0.595733\pi\)
\(68\) −75.7131 −0.135023
\(69\) −529.432 −0.923712
\(70\) 0 0
\(71\) 793.965 1.32713 0.663565 0.748118i \(-0.269042\pi\)
0.663565 + 0.748118i \(0.269042\pi\)
\(72\) 106.270 0.173945
\(73\) −315.417 −0.505709 −0.252854 0.967504i \(-0.581369\pi\)
−0.252854 + 0.967504i \(0.581369\pi\)
\(74\) 61.7235 0.0969623
\(75\) 0 0
\(76\) 409.659 0.618304
\(77\) −36.3457 −0.0537918
\(78\) 582.810 0.846028
\(79\) −425.840 −0.606465 −0.303233 0.952917i \(-0.598066\pi\)
−0.303233 + 0.952917i \(0.598066\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −748.887 −1.00855
\(83\) 283.029 0.374295 0.187148 0.982332i \(-0.440076\pi\)
0.187148 + 0.982332i \(0.440076\pi\)
\(84\) 98.3778 0.127785
\(85\) 0 0
\(86\) 1857.60 2.32919
\(87\) 426.511 0.525596
\(88\) −61.3087 −0.0742674
\(89\) −843.131 −1.00418 −0.502088 0.864817i \(-0.667435\pi\)
−0.502088 + 0.864817i \(0.667435\pi\)
\(90\) 0 0
\(91\) −381.825 −0.439847
\(92\) −826.736 −0.936882
\(93\) −282.901 −0.315435
\(94\) −375.447 −0.411962
\(95\) 0 0
\(96\) 566.378 0.602143
\(97\) −1537.33 −1.60920 −0.804601 0.593816i \(-0.797621\pi\)
−0.804601 + 0.593816i \(0.797621\pi\)
\(98\) −174.516 −0.179886
\(99\) −46.7301 −0.0474399
\(100\) 0 0
\(101\) −1589.99 −1.56644 −0.783219 0.621745i \(-0.786424\pi\)
−0.783219 + 0.621745i \(0.786424\pi\)
\(102\) 172.685 0.167631
\(103\) 164.793 0.157646 0.0788228 0.996889i \(-0.474884\pi\)
0.0788228 + 0.996889i \(0.474884\pi\)
\(104\) −644.071 −0.607273
\(105\) 0 0
\(106\) −388.132 −0.355648
\(107\) −1184.08 −1.06981 −0.534904 0.844913i \(-0.679652\pi\)
−0.534904 + 0.844913i \(0.679652\pi\)
\(108\) 126.486 0.112695
\(109\) 333.247 0.292837 0.146419 0.989223i \(-0.453225\pi\)
0.146419 + 0.989223i \(0.453225\pi\)
\(110\) 0 0
\(111\) −51.9915 −0.0444578
\(112\) −556.719 −0.469687
\(113\) −1881.49 −1.56634 −0.783168 0.621810i \(-0.786398\pi\)
−0.783168 + 0.621810i \(0.786398\pi\)
\(114\) −934.341 −0.767623
\(115\) 0 0
\(116\) 666.020 0.533090
\(117\) −490.918 −0.387909
\(118\) −749.224 −0.584506
\(119\) −113.133 −0.0871507
\(120\) 0 0
\(121\) −1304.04 −0.979745
\(122\) 2401.57 1.78220
\(123\) 630.810 0.462424
\(124\) −441.764 −0.319932
\(125\) 0 0
\(126\) −224.378 −0.158644
\(127\) −1638.79 −1.14503 −0.572516 0.819893i \(-0.694033\pi\)
−0.572516 + 0.819893i \(0.694033\pi\)
\(128\) −1381.61 −0.954048
\(129\) −1564.71 −1.06795
\(130\) 0 0
\(131\) −598.142 −0.398931 −0.199465 0.979905i \(-0.563921\pi\)
−0.199465 + 0.979905i \(0.563921\pi\)
\(132\) −72.9716 −0.0481164
\(133\) 612.129 0.399085
\(134\) 1157.25 0.746055
\(135\) 0 0
\(136\) −190.836 −0.120324
\(137\) 1005.25 0.626894 0.313447 0.949606i \(-0.398516\pi\)
0.313447 + 0.949606i \(0.398516\pi\)
\(138\) 1885.60 1.16314
\(139\) −1875.01 −1.14414 −0.572072 0.820204i \(-0.693860\pi\)
−0.572072 + 0.820204i \(0.693860\pi\)
\(140\) 0 0
\(141\) 316.250 0.188887
\(142\) −2827.75 −1.67112
\(143\) 283.218 0.165621
\(144\) −715.781 −0.414225
\(145\) 0 0
\(146\) 1123.37 0.636788
\(147\) 147.000 0.0824786
\(148\) −81.1875 −0.0450917
\(149\) −1051.80 −0.578299 −0.289150 0.957284i \(-0.593373\pi\)
−0.289150 + 0.957284i \(0.593373\pi\)
\(150\) 0 0
\(151\) −750.383 −0.404406 −0.202203 0.979344i \(-0.564810\pi\)
−0.202203 + 0.979344i \(0.564810\pi\)
\(152\) 1032.55 0.550994
\(153\) −145.457 −0.0768597
\(154\) 129.447 0.0677346
\(155\) 0 0
\(156\) −766.594 −0.393440
\(157\) 1453.90 0.739067 0.369533 0.929217i \(-0.379518\pi\)
0.369533 + 0.929217i \(0.379518\pi\)
\(158\) 1516.65 0.763660
\(159\) 326.935 0.163067
\(160\) 0 0
\(161\) −1235.34 −0.604711
\(162\) −288.486 −0.139911
\(163\) −1300.49 −0.624921 −0.312461 0.949931i \(-0.601153\pi\)
−0.312461 + 0.949931i \(0.601153\pi\)
\(164\) 985.043 0.469018
\(165\) 0 0
\(166\) −1008.02 −0.471312
\(167\) 2111.46 0.978381 0.489191 0.872177i \(-0.337292\pi\)
0.489191 + 0.872177i \(0.337292\pi\)
\(168\) 247.963 0.113874
\(169\) 778.310 0.354260
\(170\) 0 0
\(171\) 787.023 0.351960
\(172\) −2443.38 −1.08317
\(173\) −335.292 −0.147351 −0.0736756 0.997282i \(-0.523473\pi\)
−0.0736756 + 0.997282i \(0.523473\pi\)
\(174\) −1519.04 −0.661829
\(175\) 0 0
\(176\) 412.945 0.176857
\(177\) 631.094 0.267999
\(178\) 3002.85 1.26446
\(179\) 2322.23 0.969672 0.484836 0.874605i \(-0.338879\pi\)
0.484836 + 0.874605i \(0.338879\pi\)
\(180\) 0 0
\(181\) −1525.59 −0.626500 −0.313250 0.949671i \(-0.601418\pi\)
−0.313250 + 0.949671i \(0.601418\pi\)
\(182\) 1359.89 0.553855
\(183\) −2022.91 −0.817147
\(184\) −2083.80 −0.834891
\(185\) 0 0
\(186\) 1007.57 0.397195
\(187\) 83.9165 0.0328160
\(188\) 493.841 0.191580
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) 293.912 0.111344 0.0556721 0.998449i \(-0.482270\pi\)
0.0556721 + 0.998449i \(0.482270\pi\)
\(192\) −108.435 −0.0407583
\(193\) 3664.91 1.36687 0.683435 0.730012i \(-0.260485\pi\)
0.683435 + 0.730012i \(0.260485\pi\)
\(194\) 5475.29 2.02630
\(195\) 0 0
\(196\) 229.548 0.0836546
\(197\) 5101.89 1.84515 0.922576 0.385816i \(-0.126080\pi\)
0.922576 + 0.385816i \(0.126080\pi\)
\(198\) 166.432 0.0597363
\(199\) 5025.86 1.79032 0.895161 0.445743i \(-0.147061\pi\)
0.895161 + 0.445743i \(0.147061\pi\)
\(200\) 0 0
\(201\) −974.787 −0.342071
\(202\) 5662.85 1.97246
\(203\) 995.193 0.344083
\(204\) −227.139 −0.0779556
\(205\) 0 0
\(206\) −586.918 −0.198507
\(207\) −1588.30 −0.533305
\(208\) 4338.14 1.44614
\(209\) −454.045 −0.150273
\(210\) 0 0
\(211\) −3267.98 −1.06624 −0.533122 0.846039i \(-0.678981\pi\)
−0.533122 + 0.846039i \(0.678981\pi\)
\(212\) 510.526 0.165392
\(213\) 2381.89 0.766219
\(214\) 4217.17 1.34710
\(215\) 0 0
\(216\) 318.810 0.100427
\(217\) −660.101 −0.206500
\(218\) −1186.88 −0.368741
\(219\) −946.250 −0.291971
\(220\) 0 0
\(221\) 881.575 0.268331
\(222\) 185.170 0.0559812
\(223\) −5457.65 −1.63888 −0.819442 0.573162i \(-0.805717\pi\)
−0.819442 + 0.573162i \(0.805717\pi\)
\(224\) 1321.55 0.394195
\(225\) 0 0
\(226\) 6701.04 1.97233
\(227\) −281.023 −0.0821682 −0.0410841 0.999156i \(-0.513081\pi\)
−0.0410841 + 0.999156i \(0.513081\pi\)
\(228\) 1228.98 0.356978
\(229\) 2776.64 0.801248 0.400624 0.916243i \(-0.368793\pi\)
0.400624 + 0.916243i \(0.368793\pi\)
\(230\) 0 0
\(231\) −109.037 −0.0310567
\(232\) 1678.71 0.475056
\(233\) −5781.09 −1.62546 −0.812729 0.582642i \(-0.802019\pi\)
−0.812729 + 0.582642i \(0.802019\pi\)
\(234\) 1748.43 0.488455
\(235\) 0 0
\(236\) 985.486 0.271821
\(237\) −1277.52 −0.350143
\(238\) 402.931 0.109740
\(239\) 1588.17 0.429833 0.214916 0.976632i \(-0.431052\pi\)
0.214916 + 0.976632i \(0.431052\pi\)
\(240\) 0 0
\(241\) −4330.01 −1.15735 −0.578673 0.815560i \(-0.696429\pi\)
−0.578673 + 0.815560i \(0.696429\pi\)
\(242\) 4644.41 1.23369
\(243\) 243.000 0.0641500
\(244\) −3158.88 −0.828798
\(245\) 0 0
\(246\) −2246.66 −0.582284
\(247\) −4769.92 −1.22876
\(248\) −1113.47 −0.285104
\(249\) 849.088 0.216099
\(250\) 0 0
\(251\) 1400.53 0.352195 0.176097 0.984373i \(-0.443653\pi\)
0.176097 + 0.984373i \(0.443653\pi\)
\(252\) 295.133 0.0737764
\(253\) 916.312 0.227700
\(254\) 5836.64 1.44182
\(255\) 0 0
\(256\) 5209.83 1.27193
\(257\) −4304.86 −1.04486 −0.522431 0.852681i \(-0.674975\pi\)
−0.522431 + 0.852681i \(0.674975\pi\)
\(258\) 5572.80 1.34476
\(259\) −121.313 −0.0291045
\(260\) 0 0
\(261\) 1279.53 0.303453
\(262\) 2130.31 0.502333
\(263\) 1724.69 0.404369 0.202184 0.979347i \(-0.435196\pi\)
0.202184 + 0.979347i \(0.435196\pi\)
\(264\) −183.926 −0.0428783
\(265\) 0 0
\(266\) −2180.13 −0.502527
\(267\) −2529.39 −0.579761
\(268\) −1522.18 −0.346948
\(269\) 8004.82 1.81436 0.907180 0.420744i \(-0.138231\pi\)
0.907180 + 0.420744i \(0.138231\pi\)
\(270\) 0 0
\(271\) −1963.65 −0.440160 −0.220080 0.975482i \(-0.570632\pi\)
−0.220080 + 0.975482i \(0.570632\pi\)
\(272\) 1285.38 0.286535
\(273\) −1145.47 −0.253946
\(274\) −3580.26 −0.789384
\(275\) 0 0
\(276\) −2480.21 −0.540909
\(277\) 3278.33 0.711104 0.355552 0.934656i \(-0.384293\pi\)
0.355552 + 0.934656i \(0.384293\pi\)
\(278\) 6677.93 1.44070
\(279\) −848.702 −0.182116
\(280\) 0 0
\(281\) 2859.04 0.606961 0.303480 0.952838i \(-0.401851\pi\)
0.303480 + 0.952838i \(0.401851\pi\)
\(282\) −1126.34 −0.237846
\(283\) 5433.66 1.14134 0.570668 0.821181i \(-0.306685\pi\)
0.570668 + 0.821181i \(0.306685\pi\)
\(284\) 3719.45 0.777144
\(285\) 0 0
\(286\) −1008.70 −0.208550
\(287\) 1471.89 0.302728
\(288\) 1699.13 0.347647
\(289\) −4651.79 −0.946833
\(290\) 0 0
\(291\) −4612.00 −0.929073
\(292\) −1477.62 −0.296134
\(293\) −8583.43 −1.71143 −0.855715 0.517447i \(-0.826883\pi\)
−0.855715 + 0.517447i \(0.826883\pi\)
\(294\) −523.548 −0.103857
\(295\) 0 0
\(296\) −204.634 −0.0401829
\(297\) −140.190 −0.0273895
\(298\) 3746.03 0.728194
\(299\) 9626.20 1.86186
\(300\) 0 0
\(301\) −3650.99 −0.699135
\(302\) 2672.53 0.509228
\(303\) −4769.98 −0.904384
\(304\) −6954.77 −1.31212
\(305\) 0 0
\(306\) 518.054 0.0967816
\(307\) 5269.83 0.979691 0.489846 0.871809i \(-0.337053\pi\)
0.489846 + 0.871809i \(0.337053\pi\)
\(308\) −170.267 −0.0314995
\(309\) 494.378 0.0910167
\(310\) 0 0
\(311\) −4761.43 −0.868154 −0.434077 0.900876i \(-0.642925\pi\)
−0.434077 + 0.900876i \(0.642925\pi\)
\(312\) −1932.21 −0.350609
\(313\) 7602.95 1.37298 0.686492 0.727137i \(-0.259150\pi\)
0.686492 + 0.727137i \(0.259150\pi\)
\(314\) −5178.13 −0.930632
\(315\) 0 0
\(316\) −1994.91 −0.355135
\(317\) −8064.55 −1.42886 −0.714432 0.699704i \(-0.753315\pi\)
−0.714432 + 0.699704i \(0.753315\pi\)
\(318\) −1164.39 −0.205333
\(319\) −738.182 −0.129562
\(320\) 0 0
\(321\) −3552.24 −0.617654
\(322\) 4399.73 0.761452
\(323\) −1413.31 −0.243464
\(324\) 379.457 0.0650647
\(325\) 0 0
\(326\) 4631.76 0.786901
\(327\) 999.741 0.169070
\(328\) 2482.82 0.417959
\(329\) 737.917 0.123655
\(330\) 0 0
\(331\) 6960.79 1.15589 0.577945 0.816076i \(-0.303855\pi\)
0.577945 + 0.816076i \(0.303855\pi\)
\(332\) 1325.90 0.219181
\(333\) −155.974 −0.0256677
\(334\) −7520.08 −1.23198
\(335\) 0 0
\(336\) −1670.16 −0.271174
\(337\) −4731.61 −0.764828 −0.382414 0.923991i \(-0.624907\pi\)
−0.382414 + 0.923991i \(0.624907\pi\)
\(338\) −2771.99 −0.446084
\(339\) −5644.48 −0.904325
\(340\) 0 0
\(341\) 489.629 0.0777563
\(342\) −2803.02 −0.443187
\(343\) 343.000 0.0539949
\(344\) −6158.58 −0.965256
\(345\) 0 0
\(346\) 1194.16 0.185544
\(347\) −9796.67 −1.51560 −0.757800 0.652487i \(-0.773726\pi\)
−0.757800 + 0.652487i \(0.773726\pi\)
\(348\) 1998.06 0.307779
\(349\) 12702.4 1.94827 0.974134 0.225971i \(-0.0725554\pi\)
0.974134 + 0.225971i \(0.0725554\pi\)
\(350\) 0 0
\(351\) −1472.75 −0.223959
\(352\) −980.256 −0.148431
\(353\) 9970.21 1.50329 0.751644 0.659569i \(-0.229261\pi\)
0.751644 + 0.659569i \(0.229261\pi\)
\(354\) −2247.67 −0.337465
\(355\) 0 0
\(356\) −3949.78 −0.588028
\(357\) −339.400 −0.0503165
\(358\) −8270.73 −1.22101
\(359\) 4388.21 0.645128 0.322564 0.946548i \(-0.395455\pi\)
0.322564 + 0.946548i \(0.395455\pi\)
\(360\) 0 0
\(361\) 787.970 0.114881
\(362\) 5433.49 0.788889
\(363\) −3912.12 −0.565656
\(364\) −1788.72 −0.257567
\(365\) 0 0
\(366\) 7204.71 1.02895
\(367\) 9441.30 1.34287 0.671433 0.741065i \(-0.265679\pi\)
0.671433 + 0.741065i \(0.265679\pi\)
\(368\) 14035.5 1.98818
\(369\) 1892.43 0.266981
\(370\) 0 0
\(371\) 762.847 0.106752
\(372\) −1325.29 −0.184713
\(373\) 3219.40 0.446901 0.223451 0.974715i \(-0.428268\pi\)
0.223451 + 0.974715i \(0.428268\pi\)
\(374\) −298.873 −0.0413218
\(375\) 0 0
\(376\) 1244.73 0.170724
\(377\) −7754.89 −1.05941
\(378\) −673.133 −0.0915933
\(379\) −14011.4 −1.89899 −0.949495 0.313783i \(-0.898403\pi\)
−0.949495 + 0.313783i \(0.898403\pi\)
\(380\) 0 0
\(381\) −4916.37 −0.661085
\(382\) −1046.78 −0.140204
\(383\) −5322.87 −0.710147 −0.355073 0.934838i \(-0.615544\pi\)
−0.355073 + 0.934838i \(0.615544\pi\)
\(384\) −4144.83 −0.550820
\(385\) 0 0
\(386\) −13052.8 −1.72116
\(387\) −4694.13 −0.616579
\(388\) −7201.88 −0.942320
\(389\) −3844.51 −0.501091 −0.250545 0.968105i \(-0.580610\pi\)
−0.250545 + 0.968105i \(0.580610\pi\)
\(390\) 0 0
\(391\) 2852.21 0.368907
\(392\) 578.580 0.0745478
\(393\) −1794.43 −0.230323
\(394\) −18170.7 −2.32341
\(395\) 0 0
\(396\) −218.915 −0.0277800
\(397\) 8046.40 1.01722 0.508611 0.860996i \(-0.330159\pi\)
0.508611 + 0.860996i \(0.330159\pi\)
\(398\) −17899.9 −2.25437
\(399\) 1836.39 0.230412
\(400\) 0 0
\(401\) 7741.38 0.964055 0.482027 0.876156i \(-0.339901\pi\)
0.482027 + 0.876156i \(0.339901\pi\)
\(402\) 3471.76 0.430735
\(403\) 5143.74 0.635801
\(404\) −7448.58 −0.917279
\(405\) 0 0
\(406\) −3544.43 −0.433269
\(407\) 89.9840 0.0109591
\(408\) −572.509 −0.0694691
\(409\) −8966.94 −1.08407 −0.542037 0.840354i \(-0.682347\pi\)
−0.542037 + 0.840354i \(0.682347\pi\)
\(410\) 0 0
\(411\) 3015.76 0.361937
\(412\) 771.997 0.0923145
\(413\) 1472.55 0.175447
\(414\) 5656.80 0.671537
\(415\) 0 0
\(416\) −10298.0 −1.21370
\(417\) −5625.02 −0.660571
\(418\) 1617.11 0.189223
\(419\) −12413.6 −1.44736 −0.723681 0.690135i \(-0.757551\pi\)
−0.723681 + 0.690135i \(0.757551\pi\)
\(420\) 0 0
\(421\) −1672.14 −0.193575 −0.0967875 0.995305i \(-0.530857\pi\)
−0.0967875 + 0.995305i \(0.530857\pi\)
\(422\) 11639.1 1.34261
\(423\) 948.750 0.109054
\(424\) 1286.79 0.147387
\(425\) 0 0
\(426\) −8483.24 −0.964823
\(427\) −4720.13 −0.534948
\(428\) −5547.02 −0.626461
\(429\) 849.653 0.0956216
\(430\) 0 0
\(431\) 16021.8 1.79059 0.895296 0.445472i \(-0.146964\pi\)
0.895296 + 0.445472i \(0.146964\pi\)
\(432\) −2147.34 −0.239153
\(433\) −10882.7 −1.20782 −0.603912 0.797051i \(-0.706392\pi\)
−0.603912 + 0.797051i \(0.706392\pi\)
\(434\) 2350.99 0.260025
\(435\) 0 0
\(436\) 1561.15 0.171480
\(437\) −15432.4 −1.68932
\(438\) 3370.12 0.367650
\(439\) 7738.40 0.841307 0.420653 0.907221i \(-0.361801\pi\)
0.420653 + 0.907221i \(0.361801\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) −3139.78 −0.337882
\(443\) −8766.56 −0.940207 −0.470103 0.882611i \(-0.655783\pi\)
−0.470103 + 0.882611i \(0.655783\pi\)
\(444\) −243.562 −0.0260337
\(445\) 0 0
\(446\) 19437.7 2.06368
\(447\) −3155.39 −0.333881
\(448\) −253.014 −0.0266826
\(449\) 3099.58 0.325786 0.162893 0.986644i \(-0.447917\pi\)
0.162893 + 0.986644i \(0.447917\pi\)
\(450\) 0 0
\(451\) −1091.77 −0.113990
\(452\) −8814.15 −0.917219
\(453\) −2251.15 −0.233484
\(454\) 1000.88 0.103466
\(455\) 0 0
\(456\) 3097.66 0.318117
\(457\) 6122.94 0.626737 0.313369 0.949632i \(-0.398542\pi\)
0.313369 + 0.949632i \(0.398542\pi\)
\(458\) −9889.16 −1.00893
\(459\) −436.372 −0.0443749
\(460\) 0 0
\(461\) −10412.2 −1.05194 −0.525970 0.850503i \(-0.676297\pi\)
−0.525970 + 0.850503i \(0.676297\pi\)
\(462\) 388.341 0.0391066
\(463\) 11278.5 1.13209 0.566043 0.824376i \(-0.308474\pi\)
0.566043 + 0.824376i \(0.308474\pi\)
\(464\) −11307.0 −1.13128
\(465\) 0 0
\(466\) 20589.6 2.04677
\(467\) −14923.2 −1.47872 −0.739359 0.673311i \(-0.764872\pi\)
−0.739359 + 0.673311i \(0.764872\pi\)
\(468\) −2299.78 −0.227153
\(469\) −2274.50 −0.223938
\(470\) 0 0
\(471\) 4361.69 0.426701
\(472\) 2483.93 0.242230
\(473\) 2708.11 0.263254
\(474\) 4549.95 0.440899
\(475\) 0 0
\(476\) −529.992 −0.0510339
\(477\) 980.804 0.0941466
\(478\) −5656.34 −0.541245
\(479\) −4674.21 −0.445867 −0.222933 0.974834i \(-0.571563\pi\)
−0.222933 + 0.974834i \(0.571563\pi\)
\(480\) 0 0
\(481\) 945.316 0.0896106
\(482\) 15421.6 1.45733
\(483\) −3706.02 −0.349130
\(484\) −6108.99 −0.573721
\(485\) 0 0
\(486\) −865.457 −0.0807777
\(487\) −17081.7 −1.58941 −0.794706 0.606994i \(-0.792375\pi\)
−0.794706 + 0.606994i \(0.792375\pi\)
\(488\) −7962.02 −0.738573
\(489\) −3901.47 −0.360799
\(490\) 0 0
\(491\) 18203.9 1.67318 0.836588 0.547832i \(-0.184547\pi\)
0.836588 + 0.547832i \(0.184547\pi\)
\(492\) 2955.13 0.270787
\(493\) −2297.75 −0.209909
\(494\) 16988.3 1.54725
\(495\) 0 0
\(496\) 7499.81 0.678934
\(497\) 5557.75 0.501608
\(498\) −3024.07 −0.272112
\(499\) 7109.47 0.637803 0.318901 0.947788i \(-0.396686\pi\)
0.318901 + 0.947788i \(0.396686\pi\)
\(500\) 0 0
\(501\) 6334.38 0.564869
\(502\) −4988.07 −0.443483
\(503\) 15402.0 1.36529 0.682647 0.730748i \(-0.260829\pi\)
0.682647 + 0.730748i \(0.260829\pi\)
\(504\) 743.889 0.0657450
\(505\) 0 0
\(506\) −3263.49 −0.286719
\(507\) 2334.93 0.204532
\(508\) −7677.17 −0.670511
\(509\) 6404.72 0.557730 0.278865 0.960330i \(-0.410042\pi\)
0.278865 + 0.960330i \(0.410042\pi\)
\(510\) 0 0
\(511\) −2207.92 −0.191140
\(512\) −7502.22 −0.647567
\(513\) 2361.07 0.203204
\(514\) 15332.0 1.31569
\(515\) 0 0
\(516\) −7330.13 −0.625370
\(517\) −547.348 −0.0465616
\(518\) 432.064 0.0366483
\(519\) −1005.88 −0.0850732
\(520\) 0 0
\(521\) −8916.72 −0.749806 −0.374903 0.927064i \(-0.622324\pi\)
−0.374903 + 0.927064i \(0.622324\pi\)
\(522\) −4557.13 −0.382107
\(523\) 6929.40 0.579353 0.289677 0.957125i \(-0.406452\pi\)
0.289677 + 0.957125i \(0.406452\pi\)
\(524\) −2802.09 −0.233607
\(525\) 0 0
\(526\) −6142.58 −0.509181
\(527\) 1524.07 0.125977
\(528\) 1238.83 0.102109
\(529\) 18977.2 1.55973
\(530\) 0 0
\(531\) 1893.28 0.154730
\(532\) 2867.61 0.233697
\(533\) −11469.5 −0.932078
\(534\) 9008.56 0.730035
\(535\) 0 0
\(536\) −3836.69 −0.309178
\(537\) 6966.68 0.559840
\(538\) −28509.6 −2.28464
\(539\) −254.420 −0.0203314
\(540\) 0 0
\(541\) −6929.23 −0.550667 −0.275334 0.961349i \(-0.588788\pi\)
−0.275334 + 0.961349i \(0.588788\pi\)
\(542\) 6993.65 0.554249
\(543\) −4576.78 −0.361710
\(544\) −3051.25 −0.240480
\(545\) 0 0
\(546\) 4079.67 0.319769
\(547\) 8509.95 0.665190 0.332595 0.943070i \(-0.392076\pi\)
0.332595 + 0.943070i \(0.392076\pi\)
\(548\) 4709.26 0.367098
\(549\) −6068.74 −0.471780
\(550\) 0 0
\(551\) 12432.4 0.961228
\(552\) −6251.41 −0.482024
\(553\) −2980.88 −0.229222
\(554\) −11676.0 −0.895422
\(555\) 0 0
\(556\) −8783.76 −0.669990
\(557\) 4043.94 0.307625 0.153813 0.988100i \(-0.450845\pi\)
0.153813 + 0.988100i \(0.450845\pi\)
\(558\) 3022.70 0.229321
\(559\) 28449.8 2.15259
\(560\) 0 0
\(561\) 251.750 0.0189463
\(562\) −10182.6 −0.764285
\(563\) −15878.9 −1.18866 −0.594331 0.804221i \(-0.702583\pi\)
−0.594331 + 0.804221i \(0.702583\pi\)
\(564\) 1481.52 0.110609
\(565\) 0 0
\(566\) −19352.3 −1.43717
\(567\) 567.000 0.0419961
\(568\) 9374.95 0.692543
\(569\) 11611.6 0.855510 0.427755 0.903895i \(-0.359305\pi\)
0.427755 + 0.903895i \(0.359305\pi\)
\(570\) 0 0
\(571\) 17395.7 1.27493 0.637466 0.770478i \(-0.279982\pi\)
0.637466 + 0.770478i \(0.279982\pi\)
\(572\) 1326.78 0.0969850
\(573\) 881.736 0.0642846
\(574\) −5242.21 −0.381195
\(575\) 0 0
\(576\) −325.304 −0.0235318
\(577\) 11474.0 0.827848 0.413924 0.910311i \(-0.364158\pi\)
0.413924 + 0.910311i \(0.364158\pi\)
\(578\) 16567.6 1.19225
\(579\) 10994.7 0.789162
\(580\) 0 0
\(581\) 1981.20 0.141470
\(582\) 16425.9 1.16989
\(583\) −565.841 −0.0401968
\(584\) −3724.37 −0.263896
\(585\) 0 0
\(586\) 30570.3 2.15503
\(587\) −11870.4 −0.834659 −0.417330 0.908755i \(-0.637034\pi\)
−0.417330 + 0.908755i \(0.637034\pi\)
\(588\) 688.645 0.0482980
\(589\) −8246.26 −0.576878
\(590\) 0 0
\(591\) 15305.7 1.06530
\(592\) 1378.32 0.0956899
\(593\) 5760.65 0.398923 0.199462 0.979906i \(-0.436081\pi\)
0.199462 + 0.979906i \(0.436081\pi\)
\(594\) 499.295 0.0344888
\(595\) 0 0
\(596\) −4927.31 −0.338642
\(597\) 15077.6 1.03364
\(598\) −34284.2 −2.34446
\(599\) −21696.5 −1.47996 −0.739978 0.672631i \(-0.765164\pi\)
−0.739978 + 0.672631i \(0.765164\pi\)
\(600\) 0 0
\(601\) −12403.0 −0.841812 −0.420906 0.907104i \(-0.638288\pi\)
−0.420906 + 0.907104i \(0.638288\pi\)
\(602\) 13003.2 0.880350
\(603\) −2924.36 −0.197495
\(604\) −3515.29 −0.236813
\(605\) 0 0
\(606\) 16988.5 1.13880
\(607\) −17066.5 −1.14120 −0.570600 0.821228i \(-0.693289\pi\)
−0.570600 + 0.821228i \(0.693289\pi\)
\(608\) 16509.3 1.10122
\(609\) 2985.58 0.198656
\(610\) 0 0
\(611\) −5750.10 −0.380727
\(612\) −681.418 −0.0450077
\(613\) −2707.50 −0.178393 −0.0891965 0.996014i \(-0.528430\pi\)
−0.0891965 + 0.996014i \(0.528430\pi\)
\(614\) −18768.8 −1.23363
\(615\) 0 0
\(616\) −429.161 −0.0280704
\(617\) 23226.3 1.51549 0.757743 0.652553i \(-0.226302\pi\)
0.757743 + 0.652553i \(0.226302\pi\)
\(618\) −1760.75 −0.114608
\(619\) −2298.43 −0.149243 −0.0746216 0.997212i \(-0.523775\pi\)
−0.0746216 + 0.997212i \(0.523775\pi\)
\(620\) 0 0
\(621\) −4764.89 −0.307904
\(622\) 16958.1 1.09318
\(623\) −5901.91 −0.379543
\(624\) 13014.4 0.834926
\(625\) 0 0
\(626\) −27078.3 −1.72886
\(627\) −1362.14 −0.0867599
\(628\) 6811.01 0.432785
\(629\) 280.094 0.0177553
\(630\) 0 0
\(631\) −663.913 −0.0418858 −0.0209429 0.999781i \(-0.506667\pi\)
−0.0209429 + 0.999781i \(0.506667\pi\)
\(632\) −5028.22 −0.316474
\(633\) −9803.95 −0.615596
\(634\) 28722.3 1.79923
\(635\) 0 0
\(636\) 1531.58 0.0954890
\(637\) −2672.77 −0.166247
\(638\) 2629.08 0.163144
\(639\) 7145.68 0.442377
\(640\) 0 0
\(641\) 15215.6 0.937566 0.468783 0.883313i \(-0.344693\pi\)
0.468783 + 0.883313i \(0.344693\pi\)
\(642\) 12651.5 0.777749
\(643\) −12904.0 −0.791420 −0.395710 0.918375i \(-0.629502\pi\)
−0.395710 + 0.918375i \(0.629502\pi\)
\(644\) −5787.15 −0.354108
\(645\) 0 0
\(646\) 5033.58 0.306569
\(647\) 9425.54 0.572730 0.286365 0.958121i \(-0.407553\pi\)
0.286365 + 0.958121i \(0.407553\pi\)
\(648\) 956.429 0.0579816
\(649\) −1092.26 −0.0660632
\(650\) 0 0
\(651\) −1980.30 −0.119223
\(652\) −6092.35 −0.365943
\(653\) 29894.7 1.79153 0.895765 0.444528i \(-0.146628\pi\)
0.895765 + 0.444528i \(0.146628\pi\)
\(654\) −3560.63 −0.212892
\(655\) 0 0
\(656\) −16723.0 −0.995312
\(657\) −2838.75 −0.168570
\(658\) −2628.13 −0.155707
\(659\) −11593.6 −0.685313 −0.342656 0.939461i \(-0.611327\pi\)
−0.342656 + 0.939461i \(0.611327\pi\)
\(660\) 0 0
\(661\) −17149.2 −1.00911 −0.504557 0.863378i \(-0.668344\pi\)
−0.504557 + 0.863378i \(0.668344\pi\)
\(662\) −24791.2 −1.45550
\(663\) 2644.72 0.154921
\(664\) 3341.94 0.195320
\(665\) 0 0
\(666\) 555.511 0.0323208
\(667\) −25089.9 −1.45650
\(668\) 9891.47 0.572923
\(669\) −16372.9 −0.946210
\(670\) 0 0
\(671\) 3501.15 0.201431
\(672\) 3964.64 0.227589
\(673\) 16475.0 0.943633 0.471817 0.881697i \(-0.343598\pi\)
0.471817 + 0.881697i \(0.343598\pi\)
\(674\) 16851.9 0.963071
\(675\) 0 0
\(676\) 3646.11 0.207448
\(677\) 4559.89 0.258864 0.129432 0.991588i \(-0.458685\pi\)
0.129432 + 0.991588i \(0.458685\pi\)
\(678\) 20103.1 1.13872
\(679\) −10761.3 −0.608221
\(680\) 0 0
\(681\) −843.070 −0.0474398
\(682\) −1743.84 −0.0979106
\(683\) 27895.9 1.56282 0.781411 0.624017i \(-0.214500\pi\)
0.781411 + 0.624017i \(0.214500\pi\)
\(684\) 3686.93 0.206101
\(685\) 0 0
\(686\) −1221.61 −0.0679904
\(687\) 8329.93 0.462601
\(688\) 41481.1 2.29862
\(689\) −5944.37 −0.328683
\(690\) 0 0
\(691\) −28178.5 −1.55132 −0.775659 0.631152i \(-0.782582\pi\)
−0.775659 + 0.631152i \(0.782582\pi\)
\(692\) −1570.73 −0.0862862
\(693\) −327.111 −0.0179306
\(694\) 34891.4 1.90844
\(695\) 0 0
\(696\) 5036.14 0.274274
\(697\) −3398.37 −0.184680
\(698\) −45240.4 −2.45326
\(699\) −17343.3 −0.938458
\(700\) 0 0
\(701\) 3912.96 0.210828 0.105414 0.994428i \(-0.466383\pi\)
0.105414 + 0.994428i \(0.466383\pi\)
\(702\) 5245.29 0.282009
\(703\) −1515.50 −0.0813060
\(704\) 187.673 0.0100471
\(705\) 0 0
\(706\) −35509.4 −1.89294
\(707\) −11130.0 −0.592058
\(708\) 2956.46 0.156936
\(709\) −7782.72 −0.412251 −0.206126 0.978526i \(-0.566086\pi\)
−0.206126 + 0.978526i \(0.566086\pi\)
\(710\) 0 0
\(711\) −3832.56 −0.202155
\(712\) −9955.49 −0.524014
\(713\) 16641.8 0.874112
\(714\) 1208.79 0.0633584
\(715\) 0 0
\(716\) 10878.8 0.567823
\(717\) 4764.50 0.248164
\(718\) −15628.9 −0.812345
\(719\) −27868.8 −1.44552 −0.722762 0.691097i \(-0.757128\pi\)
−0.722762 + 0.691097i \(0.757128\pi\)
\(720\) 0 0
\(721\) 1153.55 0.0595844
\(722\) −2806.40 −0.144658
\(723\) −12990.0 −0.668194
\(724\) −7146.89 −0.366868
\(725\) 0 0
\(726\) 13933.2 0.712274
\(727\) 34202.4 1.74484 0.872419 0.488759i \(-0.162550\pi\)
0.872419 + 0.488759i \(0.162550\pi\)
\(728\) −4508.50 −0.229527
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 8429.58 0.426510
\(732\) −9476.65 −0.478507
\(733\) 12544.2 0.632101 0.316051 0.948742i \(-0.397643\pi\)
0.316051 + 0.948742i \(0.397643\pi\)
\(734\) −33625.7 −1.69094
\(735\) 0 0
\(736\) −33317.6 −1.66862
\(737\) 1687.11 0.0843221
\(738\) −6739.99 −0.336182
\(739\) 4563.19 0.227144 0.113572 0.993530i \(-0.463771\pi\)
0.113572 + 0.993530i \(0.463771\pi\)
\(740\) 0 0
\(741\) −14309.7 −0.709422
\(742\) −2716.92 −0.134422
\(743\) −10369.7 −0.512017 −0.256009 0.966674i \(-0.582408\pi\)
−0.256009 + 0.966674i \(0.582408\pi\)
\(744\) −3340.42 −0.164605
\(745\) 0 0
\(746\) −11466.1 −0.562738
\(747\) 2547.26 0.124765
\(748\) 393.120 0.0192164
\(749\) −8288.57 −0.404349
\(750\) 0 0
\(751\) −36808.0 −1.78847 −0.894237 0.447595i \(-0.852281\pi\)
−0.894237 + 0.447595i \(0.852281\pi\)
\(752\) −8383.92 −0.406556
\(753\) 4201.60 0.203340
\(754\) 27619.4 1.33401
\(755\) 0 0
\(756\) 885.400 0.0425948
\(757\) 12516.6 0.600955 0.300477 0.953789i \(-0.402854\pi\)
0.300477 + 0.953789i \(0.402854\pi\)
\(758\) 49902.3 2.39121
\(759\) 2748.93 0.131462
\(760\) 0 0
\(761\) 11745.1 0.559473 0.279736 0.960077i \(-0.409753\pi\)
0.279736 + 0.960077i \(0.409753\pi\)
\(762\) 17509.9 0.832438
\(763\) 2332.73 0.110682
\(764\) 1376.88 0.0652011
\(765\) 0 0
\(766\) 18957.7 0.894216
\(767\) −11474.6 −0.540189
\(768\) 15629.5 0.734350
\(769\) 36497.1 1.71147 0.855735 0.517414i \(-0.173105\pi\)
0.855735 + 0.517414i \(0.173105\pi\)
\(770\) 0 0
\(771\) −12914.6 −0.603252
\(772\) 17168.8 0.800414
\(773\) 29858.1 1.38929 0.694644 0.719353i \(-0.255562\pi\)
0.694644 + 0.719353i \(0.255562\pi\)
\(774\) 16718.4 0.776396
\(775\) 0 0
\(776\) −18152.5 −0.839737
\(777\) −363.940 −0.0168035
\(778\) 13692.4 0.630973
\(779\) 18387.5 0.845699
\(780\) 0 0
\(781\) −4122.45 −0.188877
\(782\) −10158.3 −0.464527
\(783\) 3838.60 0.175199
\(784\) −3897.03 −0.177525
\(785\) 0 0
\(786\) 6390.94 0.290022
\(787\) −3168.00 −0.143491 −0.0717453 0.997423i \(-0.522857\pi\)
−0.0717453 + 0.997423i \(0.522857\pi\)
\(788\) 23900.6 1.08049
\(789\) 5174.07 0.233463
\(790\) 0 0
\(791\) −13170.5 −0.592019
\(792\) −551.778 −0.0247558
\(793\) 36780.8 1.64707
\(794\) −28657.7 −1.28089
\(795\) 0 0
\(796\) 23544.5 1.04838
\(797\) −29317.0 −1.30296 −0.651481 0.758665i \(-0.725852\pi\)
−0.651481 + 0.758665i \(0.725852\pi\)
\(798\) −6540.39 −0.290134
\(799\) −1703.74 −0.0754366
\(800\) 0 0
\(801\) −7588.18 −0.334725
\(802\) −27571.3 −1.21394
\(803\) 1637.72 0.0719724
\(804\) −4566.54 −0.200310
\(805\) 0 0
\(806\) −18319.7 −0.800600
\(807\) 24014.5 1.04752
\(808\) −18774.3 −0.817422
\(809\) 16657.3 0.723904 0.361952 0.932197i \(-0.382110\pi\)
0.361952 + 0.932197i \(0.382110\pi\)
\(810\) 0 0
\(811\) 5144.55 0.222749 0.111375 0.993779i \(-0.464475\pi\)
0.111375 + 0.993779i \(0.464475\pi\)
\(812\) 4662.14 0.201489
\(813\) −5890.95 −0.254126
\(814\) −320.483 −0.0137997
\(815\) 0 0
\(816\) 3856.13 0.165431
\(817\) −45609.7 −1.95310
\(818\) 31936.2 1.36507
\(819\) −3436.42 −0.146616
\(820\) 0 0
\(821\) −5217.18 −0.221779 −0.110890 0.993833i \(-0.535370\pi\)
−0.110890 + 0.993833i \(0.535370\pi\)
\(822\) −10740.8 −0.455751
\(823\) −42326.5 −1.79272 −0.896360 0.443327i \(-0.853798\pi\)
−0.896360 + 0.443327i \(0.853798\pi\)
\(824\) 1945.83 0.0822649
\(825\) 0 0
\(826\) −5244.57 −0.220922
\(827\) 31675.8 1.33189 0.665946 0.746000i \(-0.268028\pi\)
0.665946 + 0.746000i \(0.268028\pi\)
\(828\) −7440.62 −0.312294
\(829\) −3471.22 −0.145429 −0.0727144 0.997353i \(-0.523166\pi\)
−0.0727144 + 0.997353i \(0.523166\pi\)
\(830\) 0 0
\(831\) 9835.00 0.410556
\(832\) 1971.57 0.0821539
\(833\) −791.934 −0.0329399
\(834\) 20033.8 0.831791
\(835\) 0 0
\(836\) −2127.05 −0.0879970
\(837\) −2546.11 −0.105145
\(838\) 44211.7 1.82252
\(839\) −20964.9 −0.862682 −0.431341 0.902189i \(-0.641959\pi\)
−0.431341 + 0.902189i \(0.641959\pi\)
\(840\) 0 0
\(841\) −4176.57 −0.171248
\(842\) 5955.41 0.243749
\(843\) 8577.12 0.350429
\(844\) −15309.4 −0.624373
\(845\) 0 0
\(846\) −3379.02 −0.137321
\(847\) −9128.28 −0.370309
\(848\) −8667.17 −0.350981
\(849\) 16301.0 0.658950
\(850\) 0 0
\(851\) 3058.44 0.123199
\(852\) 11158.4 0.448685
\(853\) −1084.77 −0.0435426 −0.0217713 0.999763i \(-0.506931\pi\)
−0.0217713 + 0.999763i \(0.506931\pi\)
\(854\) 16811.0 0.673607
\(855\) 0 0
\(856\) −13981.4 −0.558263
\(857\) 12661.6 0.504679 0.252340 0.967639i \(-0.418800\pi\)
0.252340 + 0.967639i \(0.418800\pi\)
\(858\) −3026.09 −0.120407
\(859\) −39678.7 −1.57604 −0.788021 0.615648i \(-0.788894\pi\)
−0.788021 + 0.615648i \(0.788894\pi\)
\(860\) 0 0
\(861\) 4415.67 0.174780
\(862\) −57062.6 −2.25471
\(863\) 41614.0 1.64143 0.820717 0.571334i \(-0.193574\pi\)
0.820717 + 0.571334i \(0.193574\pi\)
\(864\) 5097.40 0.200714
\(865\) 0 0
\(866\) 38759.2 1.52089
\(867\) −13955.4 −0.546654
\(868\) −3092.35 −0.120923
\(869\) 2211.06 0.0863120
\(870\) 0 0
\(871\) 17723.7 0.689489
\(872\) 3934.90 0.152813
\(873\) −13836.0 −0.536400
\(874\) 54963.3 2.12719
\(875\) 0 0
\(876\) −4432.86 −0.170973
\(877\) 37061.0 1.42698 0.713490 0.700665i \(-0.247113\pi\)
0.713490 + 0.700665i \(0.247113\pi\)
\(878\) −27560.7 −1.05937
\(879\) −25750.3 −0.988095
\(880\) 0 0
\(881\) −25468.7 −0.973962 −0.486981 0.873412i \(-0.661902\pi\)
−0.486981 + 0.873412i \(0.661902\pi\)
\(882\) −1570.64 −0.0599619
\(883\) −34428.3 −1.31212 −0.656062 0.754707i \(-0.727779\pi\)
−0.656062 + 0.754707i \(0.727779\pi\)
\(884\) 4129.88 0.157130
\(885\) 0 0
\(886\) 31222.6 1.18391
\(887\) −41295.4 −1.56321 −0.781603 0.623777i \(-0.785597\pi\)
−0.781603 + 0.623777i \(0.785597\pi\)
\(888\) −613.903 −0.0231996
\(889\) −11471.5 −0.432782
\(890\) 0 0
\(891\) −420.571 −0.0158133
\(892\) −25567.2 −0.959702
\(893\) 9218.37 0.345443
\(894\) 11238.1 0.420423
\(895\) 0 0
\(896\) −9671.26 −0.360596
\(897\) 28878.6 1.07495
\(898\) −11039.3 −0.410230
\(899\) −13406.7 −0.497373
\(900\) 0 0
\(901\) −1761.30 −0.0651247
\(902\) 3888.40 0.143536
\(903\) −10953.0 −0.403646
\(904\) −22216.2 −0.817368
\(905\) 0 0
\(906\) 8017.59 0.294003
\(907\) 53733.8 1.96715 0.983573 0.180509i \(-0.0577746\pi\)
0.983573 + 0.180509i \(0.0577746\pi\)
\(908\) −1316.50 −0.0481162
\(909\) −14309.9 −0.522146
\(910\) 0 0
\(911\) 24296.8 0.883634 0.441817 0.897105i \(-0.354334\pi\)
0.441817 + 0.897105i \(0.354334\pi\)
\(912\) −20864.3 −0.757550
\(913\) −1469.55 −0.0532696
\(914\) −21807.2 −0.789187
\(915\) 0 0
\(916\) 13007.6 0.469197
\(917\) −4186.99 −0.150782
\(918\) 1554.16 0.0558769
\(919\) 4280.46 0.153645 0.0768223 0.997045i \(-0.475523\pi\)
0.0768223 + 0.997045i \(0.475523\pi\)
\(920\) 0 0
\(921\) 15809.5 0.565625
\(922\) 37083.6 1.32460
\(923\) −43307.9 −1.54442
\(924\) −510.801 −0.0181863
\(925\) 0 0
\(926\) −40168.9 −1.42552
\(927\) 1483.13 0.0525485
\(928\) 26840.7 0.949450
\(929\) −31884.5 −1.12604 −0.563022 0.826442i \(-0.690361\pi\)
−0.563022 + 0.826442i \(0.690361\pi\)
\(930\) 0 0
\(931\) 4284.90 0.150840
\(932\) −27082.4 −0.951839
\(933\) −14284.3 −0.501229
\(934\) 53149.6 1.86200
\(935\) 0 0
\(936\) −5796.64 −0.202424
\(937\) 44523.1 1.55230 0.776151 0.630548i \(-0.217170\pi\)
0.776151 + 0.630548i \(0.217170\pi\)
\(938\) 8100.76 0.281982
\(939\) 22808.8 0.792693
\(940\) 0 0
\(941\) 46374.9 1.60657 0.803283 0.595598i \(-0.203085\pi\)
0.803283 + 0.595598i \(0.203085\pi\)
\(942\) −15534.4 −0.537301
\(943\) −37107.9 −1.28144
\(944\) −16730.6 −0.576836
\(945\) 0 0
\(946\) −9645.09 −0.331489
\(947\) 20348.2 0.698234 0.349117 0.937079i \(-0.386482\pi\)
0.349117 + 0.937079i \(0.386482\pi\)
\(948\) −5984.74 −0.205037
\(949\) 17204.8 0.588507
\(950\) 0 0
\(951\) −24193.6 −0.824956
\(952\) −1335.85 −0.0454782
\(953\) 45012.9 1.53002 0.765010 0.644018i \(-0.222734\pi\)
0.765010 + 0.644018i \(0.222734\pi\)
\(954\) −3493.18 −0.118549
\(955\) 0 0
\(956\) 7440.02 0.251702
\(957\) −2214.55 −0.0748027
\(958\) 16647.4 0.561435
\(959\) 7036.76 0.236944
\(960\) 0 0
\(961\) −20898.5 −0.701503
\(962\) −3366.79 −0.112838
\(963\) −10656.7 −0.356603
\(964\) −20284.6 −0.677721
\(965\) 0 0
\(966\) 13199.2 0.439624
\(967\) 40305.8 1.34038 0.670190 0.742190i \(-0.266213\pi\)
0.670190 + 0.742190i \(0.266213\pi\)
\(968\) −15397.8 −0.511265
\(969\) −4239.93 −0.140564
\(970\) 0 0
\(971\) 33991.8 1.12343 0.561713 0.827332i \(-0.310142\pi\)
0.561713 + 0.827332i \(0.310142\pi\)
\(972\) 1138.37 0.0375651
\(973\) −13125.0 −0.432445
\(974\) 60837.2 2.00139
\(975\) 0 0
\(976\) 53628.2 1.75881
\(977\) 18219.0 0.596600 0.298300 0.954472i \(-0.403580\pi\)
0.298300 + 0.954472i \(0.403580\pi\)
\(978\) 13895.3 0.454317
\(979\) 4377.73 0.142914
\(980\) 0 0
\(981\) 2999.22 0.0976125
\(982\) −64834.1 −2.10686
\(983\) −7676.89 −0.249089 −0.124545 0.992214i \(-0.539747\pi\)
−0.124545 + 0.992214i \(0.539747\pi\)
\(984\) 7448.45 0.241309
\(985\) 0 0
\(986\) 8183.55 0.264318
\(987\) 2213.75 0.0713925
\(988\) −22345.4 −0.719537
\(989\) 92045.3 2.95942
\(990\) 0 0
\(991\) 50585.6 1.62150 0.810748 0.585395i \(-0.199061\pi\)
0.810748 + 0.585395i \(0.199061\pi\)
\(992\) −17803.2 −0.569810
\(993\) 20882.4 0.667354
\(994\) −19794.2 −0.631625
\(995\) 0 0
\(996\) 3977.69 0.126544
\(997\) 53060.5 1.68550 0.842750 0.538305i \(-0.180935\pi\)
0.842750 + 0.538305i \(0.180935\pi\)
\(998\) −25320.8 −0.803121
\(999\) −467.923 −0.0148193
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 525.4.a.j.1.1 2
3.2 odd 2 1575.4.a.x.1.2 2
5.2 odd 4 525.4.d.m.274.1 4
5.3 odd 4 525.4.d.m.274.4 4
5.4 even 2 525.4.a.m.1.2 yes 2
15.14 odd 2 1575.4.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.j.1.1 2 1.1 even 1 trivial
525.4.a.m.1.2 yes 2 5.4 even 2
525.4.d.m.274.1 4 5.2 odd 4
525.4.d.m.274.4 4 5.3 odd 4
1575.4.a.o.1.1 2 15.14 odd 2
1575.4.a.x.1.2 2 3.2 odd 2