Properties

Label 525.4.a.m.1.2
Level $525$
Weight $4$
Character 525.1
Self dual yes
Analytic conductor $30.976$
Analytic rank $0$
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: \(0\)
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 1.2
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 \(-0.283219\pi\)
0.629600 + 0.776920i \(0.283219\pi\)
\(3\) −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 \(-0.302324\pi\)
0.581863 + 0.813287i \(0.302324\pi\)
\(14\) −24.9309 −0.475933
\(15\) 0 0
\(16\) −79.5312 −1.24268
\(17\) 16.1619 0.230579 0.115289 0.993332i \(-0.463220\pi\)
0.115289 + 0.993332i \(0.463220\pi\)
\(18\) 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 \(-0.204855\pi\)
0.799958 + 0.600056i \(0.204855\pi\)
\(24\) 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 \(-0.487742\pi\)
0.0385016 + 0.999259i \(0.487742\pi\)
\(38\) 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 \(-0.124175\pi\)
0.924868 + 0.380287i \(0.124175\pi\)
\(44\) −24.3239 −0.0833400
\(45\) 0 0
\(46\) 628.533 2.01461
\(47\) −105.417 −0.327162 −0.163581 0.986530i \(-0.552304\pi\)
−0.163581 + 0.986530i \(0.552304\pi\)
\(48\) 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 \(-0.545102\pi\)
−0.141220 + 0.989978i \(0.545102\pi\)
\(54\) −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 \(-0.404267\pi\)
0.296242 + 0.955113i \(0.404267\pi\)
\(68\) 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 \(-0.418631\pi\)
0.252854 + 0.967504i \(0.418631\pi\)
\(74\) 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 \(-0.559924\pi\)
−0.187148 + 0.982332i \(0.559924\pi\)
\(84\) 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 \(-0.202379\pi\)
0.804601 + 0.593816i \(0.202379\pi\)
\(98\) 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 \(-0.525116\pi\)
−0.0788228 + 0.996889i \(0.525116\pi\)
\(104\) −644.071 −0.607273
\(105\) 0 0
\(106\) −388.132 −0.355648
\(107\) 1184.08 1.06981 0.534904 0.844913i \(-0.320348\pi\)
0.534904 + 0.844913i \(0.320348\pi\)
\(108\) −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 \(-0.213602\pi\)
0.783168 + 0.621810i \(0.213602\pi\)
\(114\) −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 \(-0.305967\pi\)
0.572516 + 0.819893i \(0.305967\pi\)
\(128\) 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 \(-0.601484\pi\)
−0.313447 + 0.949606i \(0.601484\pi\)
\(138\) −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 \(-0.620482\pi\)
−0.369533 + 0.929217i \(0.620482\pi\)
\(158\) −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 \(-0.398847\pi\)
0.312461 + 0.949931i \(0.398847\pi\)
\(164\) 985.043 0.469018
\(165\) 0 0
\(166\) −1008.02 −0.471312
\(167\) −2111.46 −0.978381 −0.489191 0.872177i \(-0.662708\pi\)
−0.489191 + 0.872177i \(0.662708\pi\)
\(168\) −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 \(-0.476527\pi\)
0.0736756 + 0.997282i \(0.476527\pi\)
\(174\) −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 \(-0.739515\pi\)
−0.683435 + 0.730012i \(0.739515\pi\)
\(194\) 5475.29 2.02630
\(195\) 0 0
\(196\) 229.548 0.0836546
\(197\) −5101.89 −1.84515 −0.922576 0.385816i \(-0.873920\pi\)
−0.922576 + 0.385816i \(0.873920\pi\)
\(198\) −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 \(-0.194283\pi\)
0.819442 + 0.573162i \(0.194283\pi\)
\(224\) 1321.55 0.394195
\(225\) 0 0
\(226\) 6701.04 1.97233
\(227\) 281.023 0.0821682 0.0410841 0.999156i \(-0.486919\pi\)
0.0410841 + 0.999156i \(0.486919\pi\)
\(228\) −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 \(-0.197981\pi\)
0.812729 + 0.582642i \(0.197981\pi\)
\(234\) 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 \(-0.325025\pi\)
0.522431 + 0.852681i \(0.325025\pi\)
\(258\) −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 \(-0.564804\pi\)
−0.202184 + 0.979347i \(0.564804\pi\)
\(264\) −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 \(-0.615707\pi\)
−0.355552 + 0.934656i \(0.615707\pi\)
\(278\) −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 \(-0.693315\pi\)
−0.570668 + 0.821181i \(0.693315\pi\)
\(284\) 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 \(-0.173117\pi\)
0.855715 + 0.517447i \(0.173117\pi\)
\(294\) −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 \(-0.662947\pi\)
−0.489846 + 0.871809i \(0.662947\pi\)
\(308\) 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 \(-0.740850\pi\)
−0.686492 + 0.727137i \(0.740850\pi\)
\(314\) −5178.13 −0.930632
\(315\) 0 0
\(316\) −1994.91 −0.355135
\(317\) 8064.55 1.42886 0.714432 0.699704i \(-0.246685\pi\)
0.714432 + 0.699704i \(0.246685\pi\)
\(318\) 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 \(-0.375093\pi\)
0.382414 + 0.923991i \(0.375093\pi\)
\(338\) 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 \(-0.226274\pi\)
0.757800 + 0.652487i \(0.226274\pi\)
\(348\) −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 \(-0.770739\pi\)
−0.751644 + 0.659569i \(0.770739\pi\)
\(354\) −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 \(-0.734321\pi\)
−0.671433 + 0.741065i \(0.734321\pi\)
\(368\) −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 \(-0.571732\pi\)
−0.223451 + 0.974715i \(0.571732\pi\)
\(374\) −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 \(-0.384456\pi\)
0.355073 + 0.934838i \(0.384456\pi\)
\(384\) −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 \(-0.669841\pi\)
−0.508611 + 0.860996i \(0.669841\pi\)
\(398\) 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 \(-0.293608\pi\)
0.603912 + 0.797051i \(0.293608\pi\)
\(434\) 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 \(-0.344217\pi\)
0.470103 + 0.882611i \(0.344217\pi\)
\(444\) −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 \(-0.601458\pi\)
−0.313369 + 0.949632i \(0.601458\pi\)
\(458\) 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 \(-0.691526\pi\)
−0.566043 + 0.824376i \(0.691526\pi\)
\(464\) −11307.0 −1.13128
\(465\) 0 0
\(466\) 20589.6 2.04677
\(467\) 14923.2 1.47872 0.739359 0.673311i \(-0.235128\pi\)
0.739359 + 0.673311i \(0.235128\pi\)
\(468\) 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 \(-0.207625\pi\)
0.794706 + 0.606994i \(0.207625\pi\)
\(488\) 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 \(-0.739171\pi\)
−0.682647 + 0.730748i \(0.739171\pi\)
\(504\) 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 \(-0.593548\pi\)
−0.289677 + 0.957125i \(0.593548\pi\)
\(524\) −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 \(-0.607924\pi\)
−0.332595 + 0.943070i \(0.607924\pi\)
\(548\) −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 \(-0.549155\pi\)
−0.153813 + 0.988100i \(0.549155\pi\)
\(558\) −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 \(-0.297417\pi\)
0.594331 + 0.804221i \(0.297417\pi\)
\(564\) 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 \(-0.635842\pi\)
−0.413924 + 0.910311i \(0.635842\pi\)
\(578\) −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 \(-0.362966\pi\)
0.417330 + 0.908755i \(0.362966\pi\)
\(588\) −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 \(-0.563919\pi\)
−0.199462 + 0.979906i \(0.563919\pi\)
\(594\) 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 \(-0.306711\pi\)
0.570600 + 0.821228i \(0.306711\pi\)
\(608\) −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 \(-0.471570\pi\)
0.0891965 + 0.996014i \(0.471570\pi\)
\(614\) −18768.8 −1.23363
\(615\) 0 0
\(616\) −429.161 −0.0280704
\(617\) −23226.3 −1.51549 −0.757743 0.652553i \(-0.773698\pi\)
−0.757743 + 0.652553i \(0.773698\pi\)
\(618\) 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 \(-0.370498\pi\)
0.395710 + 0.918375i \(0.370498\pi\)
\(644\) −5787.15 −0.354108
\(645\) 0 0
\(646\) 5033.58 0.306569
\(647\) −9425.54 −0.572730 −0.286365 0.958121i \(-0.592447\pi\)
−0.286365 + 0.958121i \(0.592447\pi\)
\(648\) −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 \(-0.853372\pi\)
−0.895765 + 0.444528i \(0.853372\pi\)
\(654\) −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 \(-0.656402\pi\)
−0.471817 + 0.881697i \(0.656402\pi\)
\(674\) 16851.9 0.963071
\(675\) 0 0
\(676\) 3646.11 0.207448
\(677\) −4559.89 −0.258864 −0.129432 0.991588i \(-0.541315\pi\)
−0.129432 + 0.991588i \(0.541315\pi\)
\(678\) −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 \(-0.785500\pi\)
−0.781411 + 0.624017i \(0.785500\pi\)
\(684\) 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 \(-0.837450\pi\)
−0.872419 + 0.488759i \(0.837450\pi\)
\(728\) 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 \(-0.602357\pi\)
−0.316051 + 0.948742i \(0.602357\pi\)
\(734\) −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 \(-0.417592\pi\)
0.256009 + 0.966674i \(0.417592\pi\)
\(744\) −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 \(-0.597146\pi\)
−0.300477 + 0.953789i \(0.597146\pi\)
\(758\) −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 \(-0.744438\pi\)
−0.694644 + 0.719353i \(0.744438\pi\)
\(774\) 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 \(-0.477143\pi\)
0.0717453 + 0.997423i \(0.477143\pi\)
\(788\) −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 \(-0.274148\pi\)
0.651481 + 0.758665i \(0.274148\pi\)
\(798\) 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 \(-0.146202\pi\)
0.896360 + 0.443327i \(0.146202\pi\)
\(824\) 1945.83 0.0822649
\(825\) 0 0
\(826\) −5244.57 −0.220922
\(827\) −31675.8 −1.33189 −0.665946 0.746000i \(-0.731972\pi\)
−0.665946 + 0.746000i \(0.731972\pi\)
\(828\) 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 \(-0.493069\pi\)
0.0217713 + 0.999763i \(0.493069\pi\)
\(854\) 16811.0 0.673607
\(855\) 0 0
\(856\) −13981.4 −0.558263
\(857\) −12661.6 −0.504679 −0.252340 0.967639i \(-0.581200\pi\)
−0.252340 + 0.967639i \(0.581200\pi\)
\(858\) 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 \(-0.806426\pi\)
−0.820717 + 0.571334i \(0.806426\pi\)
\(864\) 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 \(-0.752887\pi\)
−0.713490 + 0.700665i \(0.752887\pi\)
\(878\) 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 \(-0.272221\pi\)
0.656062 + 0.754707i \(0.272221\pi\)
\(884\) 4129.88 0.157130
\(885\) 0 0
\(886\) 31222.6 1.18391
\(887\) 41295.4 1.56321 0.781603 0.623777i \(-0.214403\pi\)
0.781603 + 0.623777i \(0.214403\pi\)
\(888\) 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 \(-0.942225\pi\)
−0.983573 + 0.180509i \(0.942225\pi\)
\(908\) 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 \(-0.782830\pi\)
−0.776151 + 0.630548i \(0.782830\pi\)
\(938\) −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 \(-0.613518\pi\)
−0.349117 + 0.937079i \(0.613518\pi\)
\(948\) 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 \(-0.777266\pi\)
−0.765010 + 0.644018i \(0.777266\pi\)
\(954\) −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 \(-0.733787\pi\)
−0.670190 + 0.742190i \(0.733787\pi\)
\(968\) 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 \(-0.596420\pi\)
−0.298300 + 0.954472i \(0.596420\pi\)
\(978\) −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 \(-0.460253\pi\)
0.124545 + 0.992214i \(0.460253\pi\)
\(984\) 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 \(-0.819065\pi\)
−0.842750 + 0.538305i \(0.819065\pi\)
\(998\) 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.m.1.2 yes 2
3.2 odd 2 1575.4.a.o.1.1 2
5.2 odd 4 525.4.d.m.274.4 4
5.3 odd 4 525.4.d.m.274.1 4
5.4 even 2 525.4.a.j.1.1 2
15.14 odd 2 1575.4.a.x.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.j.1.1 2 5.4 even 2
525.4.a.m.1.2 yes 2 1.1 even 1 trivial
525.4.d.m.274.1 4 5.3 odd 4
525.4.d.m.274.4 4 5.2 odd 4
1575.4.a.o.1.1 2 3.2 odd 2
1575.4.a.x.1.2 2 15.14 odd 2