Properties

Label 525.4.d.m.274.4
Level $525$
Weight $4$
Character 525.274
Analytic conductor $30.976$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.m.274.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.56155i q^{2} +3.00000i q^{3} -4.68466 q^{4} -10.6847 q^{6} -7.00000i q^{7} +11.8078i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+3.56155i q^{2} +3.00000i q^{3} -4.68466 q^{4} -10.6847 q^{6} -7.00000i q^{7} +11.8078i q^{8} -9.00000 q^{9} -5.19224 q^{11} -14.0540i q^{12} -54.5464i q^{13} +24.9309 q^{14} -79.5312 q^{16} +16.1619i q^{17} -32.0540i q^{18} -87.4470 q^{19} +21.0000 q^{21} -18.4924i q^{22} -176.477i q^{23} -35.4233 q^{24} +194.270 q^{26} -27.0000i q^{27} +32.7926i q^{28} -142.170 q^{29} -94.3002 q^{31} -188.793i q^{32} -15.5767i q^{33} -57.5616 q^{34} +42.1619 q^{36} +17.3305i q^{37} -311.447i q^{38} +163.639 q^{39} +210.270 q^{41} +74.7926i q^{42} -521.570i q^{43} +24.3239 q^{44} +628.533 q^{46} -105.417i q^{47} -238.594i q^{48} -49.0000 q^{49} -48.4858 q^{51} +255.531i q^{52} +108.978i q^{53} +96.1619 q^{54} +82.6543 q^{56} -262.341i q^{57} -506.348i q^{58} -210.365 q^{59} -674.304 q^{61} -335.855i q^{62} +63.0000i q^{63} +36.1449 q^{64} +55.4773 q^{66} +324.929i q^{67} -75.7131i q^{68} +529.432 q^{69} +793.965 q^{71} -106.270i q^{72} -315.417i q^{73} -61.7235 q^{74} +409.659 q^{76} +36.3457i q^{77} +582.810i q^{78} +425.840 q^{79} +81.0000 q^{81} +748.887i q^{82} +283.029i q^{83} -98.3778 q^{84} +1857.60 q^{86} -426.511i q^{87} -61.3087i q^{88} +843.131 q^{89} -381.825 q^{91} +826.736i q^{92} -282.901i q^{93} +375.447 q^{94} +566.378 q^{96} +1537.33i q^{97} -174.516i q^{98} +46.7301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{4} - 18 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{4} - 18 q^{6} - 36 q^{9} - 62 q^{11} + 42 q^{14} - 46 q^{16} + 112 q^{19} + 84 q^{21} - 18 q^{24} + 406 q^{26} + 124 q^{29} - 270 q^{31} - 222 q^{34} - 54 q^{36} + 234 q^{39} + 470 q^{41} - 348 q^{44} + 1170 q^{46} - 196 q^{49} + 474 q^{51} + 162 q^{54} + 42 q^{56} + 882 q^{59} - 446 q^{61} + 862 q^{64} + 24 q^{66} + 1524 q^{69} + 1238 q^{71} - 16 q^{74} + 3024 q^{76} + 854 q^{79} + 324 q^{81} + 126 q^{84} + 3398 q^{86} - 932 q^{89} - 546 q^{91} + 1040 q^{94} + 1746 q^{96} + 558 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.56155i 1.25920i 0.776920 + 0.629600i \(0.216781\pi\)
−0.776920 + 0.629600i \(0.783219\pi\)
\(3\) 3.00000i 0.577350i
\(4\) −4.68466 −0.585582
\(5\) 0 0
\(6\) −10.6847 −0.726999
\(7\) − 7.00000i − 0.377964i
\(8\) 11.8078i 0.521834i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −5.19224 −0.142320 −0.0711599 0.997465i \(-0.522670\pi\)
−0.0711599 + 0.997465i \(0.522670\pi\)
\(12\) − 14.0540i − 0.338086i
\(13\) − 54.5464i − 1.16373i −0.813287 0.581863i \(-0.802324\pi\)
0.813287 0.581863i \(-0.197676\pi\)
\(14\) 24.9309 0.475933
\(15\) 0 0
\(16\) −79.5312 −1.24268
\(17\) 16.1619i 0.230579i 0.993332 + 0.115289i \(0.0367796\pi\)
−0.993332 + 0.115289i \(0.963220\pi\)
\(18\) − 32.0540i − 0.419733i
\(19\) −87.4470 −1.05588 −0.527940 0.849282i \(-0.677035\pi\)
−0.527940 + 0.849282i \(0.677035\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) − 18.4924i − 0.179209i
\(23\) − 176.477i − 1.59992i −0.600056 0.799958i \(-0.704855\pi\)
0.600056 0.799958i \(-0.295145\pi\)
\(24\) −35.4233 −0.301281
\(25\) 0 0
\(26\) 194.270 1.46536
\(27\) − 27.0000i − 0.192450i
\(28\) 32.7926i 0.221329i
\(29\) −142.170 −0.910358 −0.455179 0.890400i \(-0.650425\pi\)
−0.455179 + 0.890400i \(0.650425\pi\)
\(30\) 0 0
\(31\) −94.3002 −0.546349 −0.273174 0.961965i \(-0.588074\pi\)
−0.273174 + 0.961965i \(0.588074\pi\)
\(32\) − 188.793i − 1.04294i
\(33\) − 15.5767i − 0.0821684i
\(34\) −57.5616 −0.290345
\(35\) 0 0
\(36\) 42.1619 0.195194
\(37\) 17.3305i 0.0770031i 0.999259 + 0.0385016i \(0.0122585\pi\)
−0.999259 + 0.0385016i \(0.987742\pi\)
\(38\) − 311.447i − 1.32956i
\(39\) 163.639 0.671878
\(40\) 0 0
\(41\) 210.270 0.800942 0.400471 0.916309i \(-0.368846\pi\)
0.400471 + 0.916309i \(0.368846\pi\)
\(42\) 74.7926i 0.274780i
\(43\) − 521.570i − 1.84974i −0.380287 0.924868i \(-0.624175\pi\)
0.380287 0.924868i \(-0.375825\pi\)
\(44\) 24.3239 0.0833400
\(45\) 0 0
\(46\) 628.533 2.01461
\(47\) − 105.417i − 0.327162i −0.986530 0.163581i \(-0.947696\pi\)
0.986530 0.163581i \(-0.0523045\pi\)
\(48\) − 238.594i − 0.717459i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −48.4858 −0.133125
\(52\) 255.531i 0.681458i
\(53\) 108.978i 0.282440i 0.989978 + 0.141220i \(0.0451025\pi\)
−0.989978 + 0.141220i \(0.954898\pi\)
\(54\) 96.1619 0.242333
\(55\) 0 0
\(56\) 82.6543 0.197235
\(57\) − 262.341i − 0.609612i
\(58\) − 506.348i − 1.14632i
\(59\) −210.365 −0.464189 −0.232094 0.972693i \(-0.574558\pi\)
−0.232094 + 0.972693i \(0.574558\pi\)
\(60\) 0 0
\(61\) −674.304 −1.41534 −0.707670 0.706543i \(-0.750254\pi\)
−0.707670 + 0.706543i \(0.750254\pi\)
\(62\) − 335.855i − 0.687962i
\(63\) 63.0000i 0.125988i
\(64\) 36.1449 0.0705955
\(65\) 0 0
\(66\) 55.4773 0.103466
\(67\) 324.929i 0.592484i 0.955113 + 0.296242i \(0.0957334\pi\)
−0.955113 + 0.296242i \(0.904267\pi\)
\(68\) − 75.7131i − 0.135023i
\(69\) 529.432 0.923712
\(70\) 0 0
\(71\) 793.965 1.32713 0.663565 0.748118i \(-0.269042\pi\)
0.663565 + 0.748118i \(0.269042\pi\)
\(72\) − 106.270i − 0.173945i
\(73\) − 315.417i − 0.505709i −0.967504 0.252854i \(-0.918631\pi\)
0.967504 0.252854i \(-0.0813693\pi\)
\(74\) −61.7235 −0.0969623
\(75\) 0 0
\(76\) 409.659 0.618304
\(77\) 36.3457i 0.0537918i
\(78\) 582.810i 0.846028i
\(79\) 425.840 0.606465 0.303233 0.952917i \(-0.401934\pi\)
0.303233 + 0.952917i \(0.401934\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 748.887i 1.00855i
\(83\) 283.029i 0.374295i 0.982332 + 0.187148i \(0.0599243\pi\)
−0.982332 + 0.187148i \(0.940076\pi\)
\(84\) −98.3778 −0.127785
\(85\) 0 0
\(86\) 1857.60 2.32919
\(87\) − 426.511i − 0.525596i
\(88\) − 61.3087i − 0.0742674i
\(89\) 843.131 1.00418 0.502088 0.864817i \(-0.332565\pi\)
0.502088 + 0.864817i \(0.332565\pi\)
\(90\) 0 0
\(91\) −381.825 −0.439847
\(92\) 826.736i 0.936882i
\(93\) − 282.901i − 0.315435i
\(94\) 375.447 0.411962
\(95\) 0 0
\(96\) 566.378 0.602143
\(97\) 1537.33i 1.60920i 0.593816 + 0.804601i \(0.297621\pi\)
−0.593816 + 0.804601i \(0.702379\pi\)
\(98\) − 174.516i − 0.179886i
\(99\) 46.7301 0.0474399
\(100\) 0 0
\(101\) −1589.99 −1.56644 −0.783219 0.621745i \(-0.786424\pi\)
−0.783219 + 0.621745i \(0.786424\pi\)
\(102\) − 172.685i − 0.167631i
\(103\) 164.793i 0.157646i 0.996889 + 0.0788228i \(0.0251161\pi\)
−0.996889 + 0.0788228i \(0.974884\pi\)
\(104\) 644.071 0.607273
\(105\) 0 0
\(106\) −388.132 −0.355648
\(107\) 1184.08i 1.06981i 0.844913 + 0.534904i \(0.179652\pi\)
−0.844913 + 0.534904i \(0.820348\pi\)
\(108\) 126.486i 0.112695i
\(109\) −333.247 −0.292837 −0.146419 0.989223i \(-0.546775\pi\)
−0.146419 + 0.989223i \(0.546775\pi\)
\(110\) 0 0
\(111\) −51.9915 −0.0444578
\(112\) 556.719i 0.469687i
\(113\) − 1881.49i − 1.56634i −0.621810 0.783168i \(-0.713602\pi\)
0.621810 0.783168i \(-0.286398\pi\)
\(114\) 934.341 0.767623
\(115\) 0 0
\(116\) 666.020 0.533090
\(117\) 490.918i 0.387909i
\(118\) − 749.224i − 0.584506i
\(119\) 113.133 0.0871507
\(120\) 0 0
\(121\) −1304.04 −0.979745
\(122\) − 2401.57i − 1.78220i
\(123\) 630.810i 0.462424i
\(124\) 441.764 0.319932
\(125\) 0 0
\(126\) −224.378 −0.158644
\(127\) 1638.79i 1.14503i 0.819893 + 0.572516i \(0.194033\pi\)
−0.819893 + 0.572516i \(0.805967\pi\)
\(128\) − 1381.61i − 0.954048i
\(129\) 1564.71 1.06795
\(130\) 0 0
\(131\) −598.142 −0.398931 −0.199465 0.979905i \(-0.563921\pi\)
−0.199465 + 0.979905i \(0.563921\pi\)
\(132\) 72.9716i 0.0481164i
\(133\) 612.129i 0.399085i
\(134\) −1157.25 −0.746055
\(135\) 0 0
\(136\) −190.836 −0.120324
\(137\) − 1005.25i − 0.626894i −0.949606 0.313447i \(-0.898516\pi\)
0.949606 0.313447i \(-0.101484\pi\)
\(138\) 1885.60i 1.16314i
\(139\) 1875.01 1.14414 0.572072 0.820204i \(-0.306140\pi\)
0.572072 + 0.820204i \(0.306140\pi\)
\(140\) 0 0
\(141\) 316.250 0.188887
\(142\) 2827.75i 1.67112i
\(143\) 283.218i 0.165621i
\(144\) 715.781 0.414225
\(145\) 0 0
\(146\) 1123.37 0.636788
\(147\) − 147.000i − 0.0824786i
\(148\) − 81.1875i − 0.0450917i
\(149\) 1051.80 0.578299 0.289150 0.957284i \(-0.406627\pi\)
0.289150 + 0.957284i \(0.406627\pi\)
\(150\) 0 0
\(151\) −750.383 −0.404406 −0.202203 0.979344i \(-0.564810\pi\)
−0.202203 + 0.979344i \(0.564810\pi\)
\(152\) − 1032.55i − 0.550994i
\(153\) − 145.457i − 0.0768597i
\(154\) −129.447 −0.0677346
\(155\) 0 0
\(156\) −766.594 −0.393440
\(157\) − 1453.90i − 0.739067i −0.929217 0.369533i \(-0.879518\pi\)
0.929217 0.369533i \(-0.120482\pi\)
\(158\) 1516.65i 0.763660i
\(159\) −326.935 −0.163067
\(160\) 0 0
\(161\) −1235.34 −0.604711
\(162\) 288.486i 0.139911i
\(163\) − 1300.49i − 0.624921i −0.949931 0.312461i \(-0.898847\pi\)
0.949931 0.312461i \(-0.101153\pi\)
\(164\) −985.043 −0.469018
\(165\) 0 0
\(166\) −1008.02 −0.471312
\(167\) − 2111.46i − 0.978381i −0.872177 0.489191i \(-0.837292\pi\)
0.872177 0.489191i \(-0.162708\pi\)
\(168\) 247.963i 0.113874i
\(169\) −778.310 −0.354260
\(170\) 0 0
\(171\) 787.023 0.351960
\(172\) 2443.38i 1.08317i
\(173\) − 335.292i − 0.147351i −0.997282 0.0736756i \(-0.976527\pi\)
0.997282 0.0736756i \(-0.0234729\pi\)
\(174\) 1519.04 0.661829
\(175\) 0 0
\(176\) 412.945 0.176857
\(177\) − 631.094i − 0.267999i
\(178\) 3002.85i 1.26446i
\(179\) −2322.23 −0.969672 −0.484836 0.874605i \(-0.661121\pi\)
−0.484836 + 0.874605i \(0.661121\pi\)
\(180\) 0 0
\(181\) −1525.59 −0.626500 −0.313250 0.949671i \(-0.601418\pi\)
−0.313250 + 0.949671i \(0.601418\pi\)
\(182\) − 1359.89i − 0.553855i
\(183\) − 2022.91i − 0.817147i
\(184\) 2083.80 0.834891
\(185\) 0 0
\(186\) 1007.57 0.397195
\(187\) − 83.9165i − 0.0328160i
\(188\) 493.841i 0.191580i
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) 293.912 0.111344 0.0556721 0.998449i \(-0.482270\pi\)
0.0556721 + 0.998449i \(0.482270\pi\)
\(192\) 108.435i 0.0407583i
\(193\) 3664.91i 1.36687i 0.730012 + 0.683435i \(0.239515\pi\)
−0.730012 + 0.683435i \(0.760485\pi\)
\(194\) −5475.29 −2.02630
\(195\) 0 0
\(196\) 229.548 0.0836546
\(197\) − 5101.89i − 1.84515i −0.385816 0.922576i \(-0.626080\pi\)
0.385816 0.922576i \(-0.373920\pi\)
\(198\) 166.432i 0.0597363i
\(199\) −5025.86 −1.79032 −0.895161 0.445743i \(-0.852939\pi\)
−0.895161 + 0.445743i \(0.852939\pi\)
\(200\) 0 0
\(201\) −974.787 −0.342071
\(202\) − 5662.85i − 1.97246i
\(203\) 995.193i 0.344083i
\(204\) 227.139 0.0779556
\(205\) 0 0
\(206\) −586.918 −0.198507
\(207\) 1588.30i 0.533305i
\(208\) 4338.14i 1.44614i
\(209\) 454.045 0.150273
\(210\) 0 0
\(211\) −3267.98 −1.06624 −0.533122 0.846039i \(-0.678981\pi\)
−0.533122 + 0.846039i \(0.678981\pi\)
\(212\) − 510.526i − 0.165392i
\(213\) 2381.89i 0.766219i
\(214\) −4217.17 −1.34710
\(215\) 0 0
\(216\) 318.810 0.100427
\(217\) 660.101i 0.206500i
\(218\) − 1186.88i − 0.368741i
\(219\) 946.250 0.291971
\(220\) 0 0
\(221\) 881.575 0.268331
\(222\) − 185.170i − 0.0559812i
\(223\) − 5457.65i − 1.63888i −0.573162 0.819442i \(-0.694283\pi\)
0.573162 0.819442i \(-0.305717\pi\)
\(224\) −1321.55 −0.394195
\(225\) 0 0
\(226\) 6701.04 1.97233
\(227\) 281.023i 0.0821682i 0.999156 + 0.0410841i \(0.0130812\pi\)
−0.999156 + 0.0410841i \(0.986919\pi\)
\(228\) 1228.98i 0.356978i
\(229\) −2776.64 −0.801248 −0.400624 0.916243i \(-0.631207\pi\)
−0.400624 + 0.916243i \(0.631207\pi\)
\(230\) 0 0
\(231\) −109.037 −0.0310567
\(232\) − 1678.71i − 0.475056i
\(233\) − 5781.09i − 1.62546i −0.582642 0.812729i \(-0.697981\pi\)
0.582642 0.812729i \(-0.302019\pi\)
\(234\) −1748.43 −0.488455
\(235\) 0 0
\(236\) 985.486 0.271821
\(237\) 1277.52i 0.350143i
\(238\) 402.931i 0.109740i
\(239\) −1588.17 −0.429833 −0.214916 0.976632i \(-0.568948\pi\)
−0.214916 + 0.976632i \(0.568948\pi\)
\(240\) 0 0
\(241\) −4330.01 −1.15735 −0.578673 0.815560i \(-0.696429\pi\)
−0.578673 + 0.815560i \(0.696429\pi\)
\(242\) − 4644.41i − 1.23369i
\(243\) 243.000i 0.0641500i
\(244\) 3158.88 0.828798
\(245\) 0 0
\(246\) −2246.66 −0.582284
\(247\) 4769.92i 1.22876i
\(248\) − 1113.47i − 0.285104i
\(249\) −849.088 −0.216099
\(250\) 0 0
\(251\) 1400.53 0.352195 0.176097 0.984373i \(-0.443653\pi\)
0.176097 + 0.984373i \(0.443653\pi\)
\(252\) − 295.133i − 0.0737764i
\(253\) 916.312i 0.227700i
\(254\) −5836.64 −1.44182
\(255\) 0 0
\(256\) 5209.83 1.27193
\(257\) 4304.86i 1.04486i 0.852681 + 0.522431i \(0.174975\pi\)
−0.852681 + 0.522431i \(0.825025\pi\)
\(258\) 5572.80i 1.34476i
\(259\) 121.313 0.0291045
\(260\) 0 0
\(261\) 1279.53 0.303453
\(262\) − 2130.31i − 0.502333i
\(263\) 1724.69i 0.404369i 0.979347 + 0.202184i \(0.0648041\pi\)
−0.979347 + 0.202184i \(0.935196\pi\)
\(264\) 183.926 0.0428783
\(265\) 0 0
\(266\) −2180.13 −0.502527
\(267\) 2529.39i 0.579761i
\(268\) − 1522.18i − 0.346948i
\(269\) −8004.82 −1.81436 −0.907180 0.420744i \(-0.861769\pi\)
−0.907180 + 0.420744i \(0.861769\pi\)
\(270\) 0 0
\(271\) −1963.65 −0.440160 −0.220080 0.975482i \(-0.570632\pi\)
−0.220080 + 0.975482i \(0.570632\pi\)
\(272\) − 1285.38i − 0.286535i
\(273\) − 1145.47i − 0.253946i
\(274\) 3580.26 0.789384
\(275\) 0 0
\(276\) −2480.21 −0.540909
\(277\) − 3278.33i − 0.711104i −0.934656 0.355552i \(-0.884293\pi\)
0.934656 0.355552i \(-0.115707\pi\)
\(278\) 6677.93i 1.44070i
\(279\) 848.702 0.182116
\(280\) 0 0
\(281\) 2859.04 0.606961 0.303480 0.952838i \(-0.401851\pi\)
0.303480 + 0.952838i \(0.401851\pi\)
\(282\) 1126.34i 0.237846i
\(283\) 5433.66i 1.14134i 0.821181 + 0.570668i \(0.193315\pi\)
−0.821181 + 0.570668i \(0.806685\pi\)
\(284\) −3719.45 −0.777144
\(285\) 0 0
\(286\) −1008.70 −0.208550
\(287\) − 1471.89i − 0.302728i
\(288\) 1699.13i 0.347647i
\(289\) 4651.79 0.946833
\(290\) 0 0
\(291\) −4612.00 −0.929073
\(292\) 1477.62i 0.296134i
\(293\) − 8583.43i − 1.71143i −0.517447 0.855715i \(-0.673117\pi\)
0.517447 0.855715i \(-0.326883\pi\)
\(294\) 523.548 0.103857
\(295\) 0 0
\(296\) −204.634 −0.0401829
\(297\) 140.190i 0.0273895i
\(298\) 3746.03i 0.728194i
\(299\) −9626.20 −1.86186
\(300\) 0 0
\(301\) −3650.99 −0.699135
\(302\) − 2672.53i − 0.509228i
\(303\) − 4769.98i − 0.904384i
\(304\) 6954.77 1.31212
\(305\) 0 0
\(306\) 518.054 0.0967816
\(307\) − 5269.83i − 0.979691i −0.871809 0.489846i \(-0.837053\pi\)
0.871809 0.489846i \(-0.162947\pi\)
\(308\) − 170.267i − 0.0314995i
\(309\) −494.378 −0.0910167
\(310\) 0 0
\(311\) −4761.43 −0.868154 −0.434077 0.900876i \(-0.642925\pi\)
−0.434077 + 0.900876i \(0.642925\pi\)
\(312\) 1932.21i 0.350609i
\(313\) 7602.95i 1.37298i 0.727137 + 0.686492i \(0.240850\pi\)
−0.727137 + 0.686492i \(0.759150\pi\)
\(314\) 5178.13 0.930632
\(315\) 0 0
\(316\) −1994.91 −0.355135
\(317\) 8064.55i 1.42886i 0.699704 + 0.714432i \(0.253315\pi\)
−0.699704 + 0.714432i \(0.746685\pi\)
\(318\) − 1164.39i − 0.205333i
\(319\) 738.182 0.129562
\(320\) 0 0
\(321\) −3552.24 −0.617654
\(322\) − 4399.73i − 0.761452i
\(323\) − 1413.31i − 0.243464i
\(324\) −379.457 −0.0650647
\(325\) 0 0
\(326\) 4631.76 0.786901
\(327\) − 999.741i − 0.169070i
\(328\) 2482.82i 0.417959i
\(329\) −737.917 −0.123655
\(330\) 0 0
\(331\) 6960.79 1.15589 0.577945 0.816076i \(-0.303855\pi\)
0.577945 + 0.816076i \(0.303855\pi\)
\(332\) − 1325.90i − 0.219181i
\(333\) − 155.974i − 0.0256677i
\(334\) 7520.08 1.23198
\(335\) 0 0
\(336\) −1670.16 −0.271174
\(337\) 4731.61i 0.764828i 0.923991 + 0.382414i \(0.124907\pi\)
−0.923991 + 0.382414i \(0.875093\pi\)
\(338\) − 2771.99i − 0.446084i
\(339\) 5644.48 0.904325
\(340\) 0 0
\(341\) 489.629 0.0777563
\(342\) 2803.02i 0.443187i
\(343\) 343.000i 0.0539949i
\(344\) 6158.58 0.965256
\(345\) 0 0
\(346\) 1194.16 0.185544
\(347\) 9796.67i 1.51560i 0.652487 + 0.757800i \(0.273726\pi\)
−0.652487 + 0.757800i \(0.726274\pi\)
\(348\) 1998.06i 0.307779i
\(349\) −12702.4 −1.94827 −0.974134 0.225971i \(-0.927445\pi\)
−0.974134 + 0.225971i \(0.927445\pi\)
\(350\) 0 0
\(351\) −1472.75 −0.223959
\(352\) 980.256i 0.148431i
\(353\) 9970.21i 1.50329i 0.659569 + 0.751644i \(0.270739\pi\)
−0.659569 + 0.751644i \(0.729261\pi\)
\(354\) 2247.67 0.337465
\(355\) 0 0
\(356\) −3949.78 −0.588028
\(357\) 339.400i 0.0503165i
\(358\) − 8270.73i − 1.22101i
\(359\) −4388.21 −0.645128 −0.322564 0.946548i \(-0.604545\pi\)
−0.322564 + 0.946548i \(0.604545\pi\)
\(360\) 0 0
\(361\) 787.970 0.114881
\(362\) − 5433.49i − 0.788889i
\(363\) − 3912.12i − 0.565656i
\(364\) 1788.72 0.257567
\(365\) 0 0
\(366\) 7204.71 1.02895
\(367\) − 9441.30i − 1.34287i −0.741065 0.671433i \(-0.765679\pi\)
0.741065 0.671433i \(-0.234321\pi\)
\(368\) 14035.5i 1.98818i
\(369\) −1892.43 −0.266981
\(370\) 0 0
\(371\) 762.847 0.106752
\(372\) 1325.29i 0.184713i
\(373\) 3219.40i 0.446901i 0.974715 + 0.223451i \(0.0717322\pi\)
−0.974715 + 0.223451i \(0.928268\pi\)
\(374\) 298.873 0.0413218
\(375\) 0 0
\(376\) 1244.73 0.170724
\(377\) 7754.89i 1.05941i
\(378\) − 673.133i − 0.0915933i
\(379\) 14011.4 1.89899 0.949495 0.313783i \(-0.101597\pi\)
0.949495 + 0.313783i \(0.101597\pi\)
\(380\) 0 0
\(381\) −4916.37 −0.661085
\(382\) 1046.78i 0.140204i
\(383\) − 5322.87i − 0.710147i −0.934838 0.355073i \(-0.884456\pi\)
0.934838 0.355073i \(-0.115544\pi\)
\(384\) 4144.83 0.550820
\(385\) 0 0
\(386\) −13052.8 −1.72116
\(387\) 4694.13i 0.616579i
\(388\) − 7201.88i − 0.942320i
\(389\) 3844.51 0.501091 0.250545 0.968105i \(-0.419390\pi\)
0.250545 + 0.968105i \(0.419390\pi\)
\(390\) 0 0
\(391\) 2852.21 0.368907
\(392\) − 578.580i − 0.0745478i
\(393\) − 1794.43i − 0.230323i
\(394\) 18170.7 2.32341
\(395\) 0 0
\(396\) −218.915 −0.0277800
\(397\) − 8046.40i − 1.01722i −0.860996 0.508611i \(-0.830159\pi\)
0.860996 0.508611i \(-0.169841\pi\)
\(398\) − 17899.9i − 2.25437i
\(399\) −1836.39 −0.230412
\(400\) 0 0
\(401\) 7741.38 0.964055 0.482027 0.876156i \(-0.339901\pi\)
0.482027 + 0.876156i \(0.339901\pi\)
\(402\) − 3471.76i − 0.430735i
\(403\) 5143.74i 0.635801i
\(404\) 7448.58 0.917279
\(405\) 0 0
\(406\) −3544.43 −0.433269
\(407\) − 89.9840i − 0.0109591i
\(408\) − 572.509i − 0.0694691i
\(409\) 8966.94 1.08407 0.542037 0.840354i \(-0.317653\pi\)
0.542037 + 0.840354i \(0.317653\pi\)
\(410\) 0 0
\(411\) 3015.76 0.361937
\(412\) − 771.997i − 0.0923145i
\(413\) 1472.55i 0.175447i
\(414\) −5656.80 −0.671537
\(415\) 0 0
\(416\) −10298.0 −1.21370
\(417\) 5625.02i 0.660571i
\(418\) 1617.11i 0.189223i
\(419\) 12413.6 1.44736 0.723681 0.690135i \(-0.242449\pi\)
0.723681 + 0.690135i \(0.242449\pi\)
\(420\) 0 0
\(421\) −1672.14 −0.193575 −0.0967875 0.995305i \(-0.530857\pi\)
−0.0967875 + 0.995305i \(0.530857\pi\)
\(422\) − 11639.1i − 1.34261i
\(423\) 948.750i 0.109054i
\(424\) −1286.79 −0.147387
\(425\) 0 0
\(426\) −8483.24 −0.964823
\(427\) 4720.13i 0.534948i
\(428\) − 5547.02i − 0.626461i
\(429\) −849.653 −0.0956216
\(430\) 0 0
\(431\) 16021.8 1.79059 0.895296 0.445472i \(-0.146964\pi\)
0.895296 + 0.445472i \(0.146964\pi\)
\(432\) 2147.34i 0.239153i
\(433\) − 10882.7i − 1.20782i −0.797051 0.603912i \(-0.793608\pi\)
0.797051 0.603912i \(-0.206392\pi\)
\(434\) −2350.99 −0.260025
\(435\) 0 0
\(436\) 1561.15 0.171480
\(437\) 15432.4i 1.68932i
\(438\) 3370.12i 0.367650i
\(439\) −7738.40 −0.841307 −0.420653 0.907221i \(-0.638199\pi\)
−0.420653 + 0.907221i \(0.638199\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 3139.78i 0.337882i
\(443\) − 8766.56i − 0.940207i −0.882611 0.470103i \(-0.844217\pi\)
0.882611 0.470103i \(-0.155783\pi\)
\(444\) 243.562 0.0260337
\(445\) 0 0
\(446\) 19437.7 2.06368
\(447\) 3155.39i 0.333881i
\(448\) − 253.014i − 0.0266826i
\(449\) −3099.58 −0.325786 −0.162893 0.986644i \(-0.552083\pi\)
−0.162893 + 0.986644i \(0.552083\pi\)
\(450\) 0 0
\(451\) −1091.77 −0.113990
\(452\) 8814.15i 0.917219i
\(453\) − 2251.15i − 0.233484i
\(454\) −1000.88 −0.103466
\(455\) 0 0
\(456\) 3097.66 0.318117
\(457\) − 6122.94i − 0.626737i −0.949632 0.313369i \(-0.898542\pi\)
0.949632 0.313369i \(-0.101458\pi\)
\(458\) − 9889.16i − 1.00893i
\(459\) 436.372 0.0443749
\(460\) 0 0
\(461\) −10412.2 −1.05194 −0.525970 0.850503i \(-0.676297\pi\)
−0.525970 + 0.850503i \(0.676297\pi\)
\(462\) − 388.341i − 0.0391066i
\(463\) 11278.5i 1.13209i 0.824376 + 0.566043i \(0.191526\pi\)
−0.824376 + 0.566043i \(0.808474\pi\)
\(464\) 11307.0 1.13128
\(465\) 0 0
\(466\) 20589.6 2.04677
\(467\) 14923.2i 1.47872i 0.673311 + 0.739359i \(0.264872\pi\)
−0.673311 + 0.739359i \(0.735128\pi\)
\(468\) − 2299.78i − 0.227153i
\(469\) 2274.50 0.223938
\(470\) 0 0
\(471\) 4361.69 0.426701
\(472\) − 2483.93i − 0.242230i
\(473\) 2708.11i 0.263254i
\(474\) −4549.95 −0.440899
\(475\) 0 0
\(476\) −529.992 −0.0510339
\(477\) − 980.804i − 0.0941466i
\(478\) − 5656.34i − 0.541245i
\(479\) 4674.21 0.445867 0.222933 0.974834i \(-0.428437\pi\)
0.222933 + 0.974834i \(0.428437\pi\)
\(480\) 0 0
\(481\) 945.316 0.0896106
\(482\) − 15421.6i − 1.45733i
\(483\) − 3706.02i − 0.349130i
\(484\) 6108.99 0.573721
\(485\) 0 0
\(486\) −865.457 −0.0807777
\(487\) 17081.7i 1.58941i 0.606994 + 0.794706i \(0.292375\pi\)
−0.606994 + 0.794706i \(0.707625\pi\)
\(488\) − 7962.02i − 0.738573i
\(489\) 3901.47 0.360799
\(490\) 0 0
\(491\) 18203.9 1.67318 0.836588 0.547832i \(-0.184547\pi\)
0.836588 + 0.547832i \(0.184547\pi\)
\(492\) − 2955.13i − 0.270787i
\(493\) − 2297.75i − 0.209909i
\(494\) −16988.3 −1.54725
\(495\) 0 0
\(496\) 7499.81 0.678934
\(497\) − 5557.75i − 0.501608i
\(498\) − 3024.07i − 0.272112i
\(499\) −7109.47 −0.637803 −0.318901 0.947788i \(-0.603314\pi\)
−0.318901 + 0.947788i \(0.603314\pi\)
\(500\) 0 0
\(501\) 6334.38 0.564869
\(502\) 4988.07i 0.443483i
\(503\) 15402.0i 1.36529i 0.730748 + 0.682647i \(0.239171\pi\)
−0.730748 + 0.682647i \(0.760829\pi\)
\(504\) −743.889 −0.0657450
\(505\) 0 0
\(506\) −3263.49 −0.286719
\(507\) − 2334.93i − 0.204532i
\(508\) − 7677.17i − 0.670511i
\(509\) −6404.72 −0.557730 −0.278865 0.960330i \(-0.589958\pi\)
−0.278865 + 0.960330i \(0.589958\pi\)
\(510\) 0 0
\(511\) −2207.92 −0.191140
\(512\) 7502.22i 0.647567i
\(513\) 2361.07i 0.203204i
\(514\) −15332.0 −1.31569
\(515\) 0 0
\(516\) −7330.13 −0.625370
\(517\) 547.348i 0.0465616i
\(518\) 432.064i 0.0366483i
\(519\) 1005.88 0.0850732
\(520\) 0 0
\(521\) −8916.72 −0.749806 −0.374903 0.927064i \(-0.622324\pi\)
−0.374903 + 0.927064i \(0.622324\pi\)
\(522\) 4557.13i 0.382107i
\(523\) 6929.40i 0.579353i 0.957125 + 0.289677i \(0.0935478\pi\)
−0.957125 + 0.289677i \(0.906452\pi\)
\(524\) 2802.09 0.233607
\(525\) 0 0
\(526\) −6142.58 −0.509181
\(527\) − 1524.07i − 0.125977i
\(528\) 1238.83i 0.102109i
\(529\) −18977.2 −1.55973
\(530\) 0 0
\(531\) 1893.28 0.154730
\(532\) − 2867.61i − 0.233697i
\(533\) − 11469.5i − 0.932078i
\(534\) −9008.56 −0.730035
\(535\) 0 0
\(536\) −3836.69 −0.309178
\(537\) − 6966.68i − 0.559840i
\(538\) − 28509.6i − 2.28464i
\(539\) 254.420 0.0203314
\(540\) 0 0
\(541\) −6929.23 −0.550667 −0.275334 0.961349i \(-0.588788\pi\)
−0.275334 + 0.961349i \(0.588788\pi\)
\(542\) − 6993.65i − 0.554249i
\(543\) − 4576.78i − 0.361710i
\(544\) 3051.25 0.240480
\(545\) 0 0
\(546\) 4079.67 0.319769
\(547\) − 8509.95i − 0.665190i −0.943070 0.332595i \(-0.892076\pi\)
0.943070 0.332595i \(-0.107924\pi\)
\(548\) 4709.26i 0.367098i
\(549\) 6068.74 0.471780
\(550\) 0 0
\(551\) 12432.4 0.961228
\(552\) 6251.41i 0.482024i
\(553\) − 2980.88i − 0.229222i
\(554\) 11676.0 0.895422
\(555\) 0 0
\(556\) −8783.76 −0.669990
\(557\) − 4043.94i − 0.307625i −0.988100 0.153813i \(-0.950845\pi\)
0.988100 0.153813i \(-0.0491552\pi\)
\(558\) 3022.70i 0.229321i
\(559\) −28449.8 −2.15259
\(560\) 0 0
\(561\) 251.750 0.0189463
\(562\) 10182.6i 0.764285i
\(563\) − 15878.9i − 1.18866i −0.804221 0.594331i \(-0.797417\pi\)
0.804221 0.594331i \(-0.202583\pi\)
\(564\) −1481.52 −0.110609
\(565\) 0 0
\(566\) −19352.3 −1.43717
\(567\) − 567.000i − 0.0419961i
\(568\) 9374.95i 0.692543i
\(569\) −11611.6 −0.855510 −0.427755 0.903895i \(-0.640695\pi\)
−0.427755 + 0.903895i \(0.640695\pi\)
\(570\) 0 0
\(571\) 17395.7 1.27493 0.637466 0.770478i \(-0.279982\pi\)
0.637466 + 0.770478i \(0.279982\pi\)
\(572\) − 1326.78i − 0.0969850i
\(573\) 881.736i 0.0642846i
\(574\) 5242.21 0.381195
\(575\) 0 0
\(576\) −325.304 −0.0235318
\(577\) − 11474.0i − 0.827848i −0.910311 0.413924i \(-0.864158\pi\)
0.910311 0.413924i \(-0.135842\pi\)
\(578\) 16567.6i 1.19225i
\(579\) −10994.7 −0.789162
\(580\) 0 0
\(581\) 1981.20 0.141470
\(582\) − 16425.9i − 1.16989i
\(583\) − 565.841i − 0.0401968i
\(584\) 3724.37 0.263896
\(585\) 0 0
\(586\) 30570.3 2.15503
\(587\) 11870.4i 0.834659i 0.908755 + 0.417330i \(0.137034\pi\)
−0.908755 + 0.417330i \(0.862966\pi\)
\(588\) 688.645i 0.0482980i
\(589\) 8246.26 0.576878
\(590\) 0 0
\(591\) 15305.7 1.06530
\(592\) − 1378.32i − 0.0956899i
\(593\) 5760.65i 0.398923i 0.979906 + 0.199462i \(0.0639193\pi\)
−0.979906 + 0.199462i \(0.936081\pi\)
\(594\) −499.295 −0.0344888
\(595\) 0 0
\(596\) −4927.31 −0.338642
\(597\) − 15077.6i − 1.03364i
\(598\) − 34284.2i − 2.34446i
\(599\) 21696.5 1.47996 0.739978 0.672631i \(-0.234836\pi\)
0.739978 + 0.672631i \(0.234836\pi\)
\(600\) 0 0
\(601\) −12403.0 −0.841812 −0.420906 0.907104i \(-0.638288\pi\)
−0.420906 + 0.907104i \(0.638288\pi\)
\(602\) − 13003.2i − 0.880350i
\(603\) − 2924.36i − 0.197495i
\(604\) 3515.29 0.236813
\(605\) 0 0
\(606\) 16988.5 1.13880
\(607\) 17066.5i 1.14120i 0.821228 + 0.570600i \(0.193289\pi\)
−0.821228 + 0.570600i \(0.806711\pi\)
\(608\) 16509.3i 1.10122i
\(609\) −2985.58 −0.198656
\(610\) 0 0
\(611\) −5750.10 −0.380727
\(612\) 681.418i 0.0450077i
\(613\) − 2707.50i − 0.178393i −0.996014 0.0891965i \(-0.971570\pi\)
0.996014 0.0891965i \(-0.0284299\pi\)
\(614\) 18768.8 1.23363
\(615\) 0 0
\(616\) −429.161 −0.0280704
\(617\) − 23226.3i − 1.51549i −0.652553 0.757743i \(-0.726302\pi\)
0.652553 0.757743i \(-0.273698\pi\)
\(618\) − 1760.75i − 0.114608i
\(619\) 2298.43 0.149243 0.0746216 0.997212i \(-0.476225\pi\)
0.0746216 + 0.997212i \(0.476225\pi\)
\(620\) 0 0
\(621\) −4764.89 −0.307904
\(622\) − 16958.1i − 1.09318i
\(623\) − 5901.91i − 0.379543i
\(624\) −13014.4 −0.834926
\(625\) 0 0
\(626\) −27078.3 −1.72886
\(627\) 1362.14i 0.0867599i
\(628\) 6811.01i 0.432785i
\(629\) −280.094 −0.0177553
\(630\) 0 0
\(631\) −663.913 −0.0418858 −0.0209429 0.999781i \(-0.506667\pi\)
−0.0209429 + 0.999781i \(0.506667\pi\)
\(632\) 5028.22i 0.316474i
\(633\) − 9803.95i − 0.615596i
\(634\) −28722.3 −1.79923
\(635\) 0 0
\(636\) 1531.58 0.0954890
\(637\) 2672.77i 0.166247i
\(638\) 2629.08i 0.163144i
\(639\) −7145.68 −0.442377
\(640\) 0 0
\(641\) 15215.6 0.937566 0.468783 0.883313i \(-0.344693\pi\)
0.468783 + 0.883313i \(0.344693\pi\)
\(642\) − 12651.5i − 0.777749i
\(643\) − 12904.0i − 0.791420i −0.918375 0.395710i \(-0.870498\pi\)
0.918375 0.395710i \(-0.129502\pi\)
\(644\) 5787.15 0.354108
\(645\) 0 0
\(646\) 5033.58 0.306569
\(647\) − 9425.54i − 0.572730i −0.958121 0.286365i \(-0.907553\pi\)
0.958121 0.286365i \(-0.0924470\pi\)
\(648\) 956.429i 0.0579816i
\(649\) 1092.26 0.0660632
\(650\) 0 0
\(651\) −1980.30 −0.119223
\(652\) 6092.35i 0.365943i
\(653\) 29894.7i 1.79153i 0.444528 + 0.895765i \(0.353372\pi\)
−0.444528 + 0.895765i \(0.646628\pi\)
\(654\) 3560.63 0.212892
\(655\) 0 0
\(656\) −16723.0 −0.995312
\(657\) 2838.75i 0.168570i
\(658\) − 2628.13i − 0.155707i
\(659\) 11593.6 0.685313 0.342656 0.939461i \(-0.388673\pi\)
0.342656 + 0.939461i \(0.388673\pi\)
\(660\) 0 0
\(661\) −17149.2 −1.00911 −0.504557 0.863378i \(-0.668344\pi\)
−0.504557 + 0.863378i \(0.668344\pi\)
\(662\) 24791.2i 1.45550i
\(663\) 2644.72i 0.154921i
\(664\) −3341.94 −0.195320
\(665\) 0 0
\(666\) 555.511 0.0323208
\(667\) 25089.9i 1.45650i
\(668\) 9891.47i 0.572923i
\(669\) 16372.9 0.946210
\(670\) 0 0
\(671\) 3501.15 0.201431
\(672\) − 3964.64i − 0.227589i
\(673\) 16475.0i 0.943633i 0.881697 + 0.471817i \(0.156402\pi\)
−0.881697 + 0.471817i \(0.843598\pi\)
\(674\) −16851.9 −0.963071
\(675\) 0 0
\(676\) 3646.11 0.207448
\(677\) − 4559.89i − 0.258864i −0.991588 0.129432i \(-0.958685\pi\)
0.991588 0.129432i \(-0.0413154\pi\)
\(678\) 20103.1i 1.13872i
\(679\) 10761.3 0.608221
\(680\) 0 0
\(681\) −843.070 −0.0474398
\(682\) 1743.84i 0.0979106i
\(683\) 27895.9i 1.56282i 0.624017 + 0.781411i \(0.285500\pi\)
−0.624017 + 0.781411i \(0.714500\pi\)
\(684\) −3686.93 −0.206101
\(685\) 0 0
\(686\) −1221.61 −0.0679904
\(687\) − 8329.93i − 0.462601i
\(688\) 41481.1i 2.29862i
\(689\) 5944.37 0.328683
\(690\) 0 0
\(691\) −28178.5 −1.55132 −0.775659 0.631152i \(-0.782582\pi\)
−0.775659 + 0.631152i \(0.782582\pi\)
\(692\) 1570.73i 0.0862862i
\(693\) − 327.111i − 0.0179306i
\(694\) −34891.4 −1.90844
\(695\) 0 0
\(696\) 5036.14 0.274274
\(697\) 3398.37i 0.184680i
\(698\) − 45240.4i − 2.45326i
\(699\) 17343.3 0.938458
\(700\) 0 0
\(701\) 3912.96 0.210828 0.105414 0.994428i \(-0.466383\pi\)
0.105414 + 0.994428i \(0.466383\pi\)
\(702\) − 5245.29i − 0.282009i
\(703\) − 1515.50i − 0.0813060i
\(704\) −187.673 −0.0100471
\(705\) 0 0
\(706\) −35509.4 −1.89294
\(707\) 11130.0i 0.592058i
\(708\) 2956.46i 0.156936i
\(709\) 7782.72 0.412251 0.206126 0.978526i \(-0.433914\pi\)
0.206126 + 0.978526i \(0.433914\pi\)
\(710\) 0 0
\(711\) −3832.56 −0.202155
\(712\) 9955.49i 0.524014i
\(713\) 16641.8i 0.874112i
\(714\) −1208.79 −0.0633584
\(715\) 0 0
\(716\) 10878.8 0.567823
\(717\) − 4764.50i − 0.248164i
\(718\) − 15628.9i − 0.812345i
\(719\) 27868.8 1.44552 0.722762 0.691097i \(-0.242872\pi\)
0.722762 + 0.691097i \(0.242872\pi\)
\(720\) 0 0
\(721\) 1153.55 0.0595844
\(722\) 2806.40i 0.144658i
\(723\) − 12990.0i − 0.668194i
\(724\) 7146.89 0.366868
\(725\) 0 0
\(726\) 13933.2 0.712274
\(727\) − 34202.4i − 1.74484i −0.488759 0.872419i \(-0.662550\pi\)
0.488759 0.872419i \(-0.337450\pi\)
\(728\) − 4508.50i − 0.229527i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 8429.58 0.426510
\(732\) 9476.65i 0.478507i
\(733\) 12544.2i 0.632101i 0.948742 + 0.316051i \(0.102357\pi\)
−0.948742 + 0.316051i \(0.897643\pi\)
\(734\) 33625.7 1.69094
\(735\) 0 0
\(736\) −33317.6 −1.66862
\(737\) − 1687.11i − 0.0843221i
\(738\) − 6739.99i − 0.336182i
\(739\) −4563.19 −0.227144 −0.113572 0.993530i \(-0.536229\pi\)
−0.113572 + 0.993530i \(0.536229\pi\)
\(740\) 0 0
\(741\) −14309.7 −0.709422
\(742\) 2716.92i 0.134422i
\(743\) − 10369.7i − 0.512017i −0.966674 0.256009i \(-0.917592\pi\)
0.966674 0.256009i \(-0.0824076\pi\)
\(744\) 3340.42 0.164605
\(745\) 0 0
\(746\) −11466.1 −0.562738
\(747\) − 2547.26i − 0.124765i
\(748\) 393.120i 0.0192164i
\(749\) 8288.57 0.404349
\(750\) 0 0
\(751\) −36808.0 −1.78847 −0.894237 0.447595i \(-0.852281\pi\)
−0.894237 + 0.447595i \(0.852281\pi\)
\(752\) 8383.92i 0.406556i
\(753\) 4201.60i 0.203340i
\(754\) −27619.4 −1.33401
\(755\) 0 0
\(756\) 885.400 0.0425948
\(757\) − 12516.6i − 0.600955i −0.953789 0.300477i \(-0.902854\pi\)
0.953789 0.300477i \(-0.0971460\pi\)
\(758\) 49902.3i 2.39121i
\(759\) −2748.93 −0.131462
\(760\) 0 0
\(761\) 11745.1 0.559473 0.279736 0.960077i \(-0.409753\pi\)
0.279736 + 0.960077i \(0.409753\pi\)
\(762\) − 17509.9i − 0.832438i
\(763\) 2332.73i 0.110682i
\(764\) −1376.88 −0.0652011
\(765\) 0 0
\(766\) 18957.7 0.894216
\(767\) 11474.6i 0.540189i
\(768\) 15629.5i 0.734350i
\(769\) −36497.1 −1.71147 −0.855735 0.517414i \(-0.826895\pi\)
−0.855735 + 0.517414i \(0.826895\pi\)
\(770\) 0 0
\(771\) −12914.6 −0.603252
\(772\) − 17168.8i − 0.800414i
\(773\) 29858.1i 1.38929i 0.719353 + 0.694644i \(0.244438\pi\)
−0.719353 + 0.694644i \(0.755562\pi\)
\(774\) −16718.4 −0.776396
\(775\) 0 0
\(776\) −18152.5 −0.839737
\(777\) 363.940i 0.0168035i
\(778\) 13692.4i 0.630973i
\(779\) −18387.5 −0.845699
\(780\) 0 0
\(781\) −4122.45 −0.188877
\(782\) 10158.3i 0.464527i
\(783\) 3838.60i 0.175199i
\(784\) 3897.03 0.177525
\(785\) 0 0
\(786\) 6390.94 0.290022
\(787\) 3168.00i 0.143491i 0.997423 + 0.0717453i \(0.0228569\pi\)
−0.997423 + 0.0717453i \(0.977143\pi\)
\(788\) 23900.6i 1.08049i
\(789\) −5174.07 −0.233463
\(790\) 0 0
\(791\) −13170.5 −0.592019
\(792\) 551.778i 0.0247558i
\(793\) 36780.8i 1.64707i
\(794\) 28657.7 1.28089
\(795\) 0 0
\(796\) 23544.5 1.04838
\(797\) 29317.0i 1.30296i 0.758665 + 0.651481i \(0.225852\pi\)
−0.758665 + 0.651481i \(0.774148\pi\)
\(798\) − 6540.39i − 0.290134i
\(799\) 1703.74 0.0754366
\(800\) 0 0
\(801\) −7588.18 −0.334725
\(802\) 27571.3i 1.21394i
\(803\) 1637.72i 0.0719724i
\(804\) 4566.54 0.200310
\(805\) 0 0
\(806\) −18319.7 −0.800600
\(807\) − 24014.5i − 1.04752i
\(808\) − 18774.3i − 0.817422i
\(809\) −16657.3 −0.723904 −0.361952 0.932197i \(-0.617890\pi\)
−0.361952 + 0.932197i \(0.617890\pi\)
\(810\) 0 0
\(811\) 5144.55 0.222749 0.111375 0.993779i \(-0.464475\pi\)
0.111375 + 0.993779i \(0.464475\pi\)
\(812\) − 4662.14i − 0.201489i
\(813\) − 5890.95i − 0.254126i
\(814\) 320.483 0.0137997
\(815\) 0 0
\(816\) 3856.13 0.165431
\(817\) 45609.7i 1.95310i
\(818\) 31936.2i 1.36507i
\(819\) 3436.42 0.146616
\(820\) 0 0
\(821\) −5217.18 −0.221779 −0.110890 0.993833i \(-0.535370\pi\)
−0.110890 + 0.993833i \(0.535370\pi\)
\(822\) 10740.8i 0.455751i
\(823\) − 42326.5i − 1.79272i −0.443327 0.896360i \(-0.646202\pi\)
0.443327 0.896360i \(-0.353798\pi\)
\(824\) −1945.83 −0.0822649
\(825\) 0 0
\(826\) −5244.57 −0.220922
\(827\) − 31675.8i − 1.33189i −0.746000 0.665946i \(-0.768028\pi\)
0.746000 0.665946i \(-0.231972\pi\)
\(828\) − 7440.62i − 0.312294i
\(829\) 3471.22 0.145429 0.0727144 0.997353i \(-0.476834\pi\)
0.0727144 + 0.997353i \(0.476834\pi\)
\(830\) 0 0
\(831\) 9835.00 0.410556
\(832\) − 1971.57i − 0.0821539i
\(833\) − 791.934i − 0.0329399i
\(834\) −20033.8 −0.831791
\(835\) 0 0
\(836\) −2127.05 −0.0879970
\(837\) 2546.11i 0.105145i
\(838\) 44211.7i 1.82252i
\(839\) 20964.9 0.862682 0.431341 0.902189i \(-0.358041\pi\)
0.431341 + 0.902189i \(0.358041\pi\)
\(840\) 0 0
\(841\) −4176.57 −0.171248
\(842\) − 5955.41i − 0.243749i
\(843\) 8577.12i 0.350429i
\(844\) 15309.4 0.624373
\(845\) 0 0
\(846\) −3379.02 −0.137321
\(847\) 9128.28i 0.370309i
\(848\) − 8667.17i − 0.350981i
\(849\) −16301.0 −0.658950
\(850\) 0 0
\(851\) 3058.44 0.123199
\(852\) − 11158.4i − 0.448685i
\(853\) − 1084.77i − 0.0435426i −0.999763 0.0217713i \(-0.993069\pi\)
0.999763 0.0217713i \(-0.00693057\pi\)
\(854\) −16811.0 −0.673607
\(855\) 0 0
\(856\) −13981.4 −0.558263
\(857\) − 12661.6i − 0.504679i −0.967639 0.252340i \(-0.918800\pi\)
0.967639 0.252340i \(-0.0812001\pi\)
\(858\) − 3026.09i − 0.120407i
\(859\) 39678.7 1.57604 0.788021 0.615648i \(-0.211106\pi\)
0.788021 + 0.615648i \(0.211106\pi\)
\(860\) 0 0
\(861\) 4415.67 0.174780
\(862\) 57062.6i 2.25471i
\(863\) 41614.0i 1.64143i 0.571334 + 0.820717i \(0.306426\pi\)
−0.571334 + 0.820717i \(0.693574\pi\)
\(864\) −5097.40 −0.200714
\(865\) 0 0
\(866\) 38759.2 1.52089
\(867\) 13955.4i 0.546654i
\(868\) − 3092.35i − 0.120923i
\(869\) −2211.06 −0.0863120
\(870\) 0 0
\(871\) 17723.7 0.689489
\(872\) − 3934.90i − 0.152813i
\(873\) − 13836.0i − 0.536400i
\(874\) −54963.3 −2.12719
\(875\) 0 0
\(876\) −4432.86 −0.170973
\(877\) − 37061.0i − 1.42698i −0.700665 0.713490i \(-0.747113\pi\)
0.700665 0.713490i \(-0.252887\pi\)
\(878\) − 27560.7i − 1.05937i
\(879\) 25750.3 0.988095
\(880\) 0 0
\(881\) −25468.7 −0.973962 −0.486981 0.873412i \(-0.661902\pi\)
−0.486981 + 0.873412i \(0.661902\pi\)
\(882\) 1570.64i 0.0599619i
\(883\) − 34428.3i − 1.31212i −0.754707 0.656062i \(-0.772221\pi\)
0.754707 0.656062i \(-0.227779\pi\)
\(884\) −4129.88 −0.157130
\(885\) 0 0
\(886\) 31222.6 1.18391
\(887\) 41295.4i 1.56321i 0.623777 + 0.781603i \(0.285597\pi\)
−0.623777 + 0.781603i \(0.714403\pi\)
\(888\) − 613.903i − 0.0231996i
\(889\) 11471.5 0.432782
\(890\) 0 0
\(891\) −420.571 −0.0158133
\(892\) 25567.2i 0.959702i
\(893\) 9218.37i 0.345443i
\(894\) −11238.1 −0.420423
\(895\) 0 0
\(896\) −9671.26 −0.360596
\(897\) − 28878.6i − 1.07495i
\(898\) − 11039.3i − 0.410230i
\(899\) 13406.7 0.497373
\(900\) 0 0
\(901\) −1761.30 −0.0651247
\(902\) − 3888.40i − 0.143536i
\(903\) − 10953.0i − 0.403646i
\(904\) 22216.2 0.817368
\(905\) 0 0
\(906\) 8017.59 0.294003
\(907\) − 53733.8i − 1.96715i −0.180509 0.983573i \(-0.557775\pi\)
0.180509 0.983573i \(-0.442225\pi\)
\(908\) − 1316.50i − 0.0481162i
\(909\) 14309.9 0.522146
\(910\) 0 0
\(911\) 24296.8 0.883634 0.441817 0.897105i \(-0.354334\pi\)
0.441817 + 0.897105i \(0.354334\pi\)
\(912\) 20864.3i 0.757550i
\(913\) − 1469.55i − 0.0532696i
\(914\) 21807.2 0.789187
\(915\) 0 0
\(916\) 13007.6 0.469197
\(917\) 4186.99i 0.150782i
\(918\) 1554.16i 0.0558769i
\(919\) −4280.46 −0.153645 −0.0768223 0.997045i \(-0.524477\pi\)
−0.0768223 + 0.997045i \(0.524477\pi\)
\(920\) 0 0
\(921\) 15809.5 0.565625
\(922\) − 37083.6i − 1.32460i
\(923\) − 43307.9i − 1.54442i
\(924\) 510.801 0.0181863
\(925\) 0 0
\(926\) −40168.9 −1.42552
\(927\) − 1483.13i − 0.0525485i
\(928\) 26840.7i 0.949450i
\(929\) 31884.5 1.12604 0.563022 0.826442i \(-0.309639\pi\)
0.563022 + 0.826442i \(0.309639\pi\)
\(930\) 0 0
\(931\) 4284.90 0.150840
\(932\) 27082.4i 0.951839i
\(933\) − 14284.3i − 0.501229i
\(934\) −53149.6 −1.86200
\(935\) 0 0
\(936\) −5796.64 −0.202424
\(937\) − 44523.1i − 1.55230i −0.630548 0.776151i \(-0.717170\pi\)
0.630548 0.776151i \(-0.282830\pi\)
\(938\) 8100.76i 0.281982i
\(939\) −22808.8 −0.792693
\(940\) 0 0
\(941\) 46374.9 1.60657 0.803283 0.595598i \(-0.203085\pi\)
0.803283 + 0.595598i \(0.203085\pi\)
\(942\) 15534.4i 0.537301i
\(943\) − 37107.9i − 1.28144i
\(944\) 16730.6 0.576836
\(945\) 0 0
\(946\) −9645.09 −0.331489
\(947\) − 20348.2i − 0.698234i −0.937079 0.349117i \(-0.886482\pi\)
0.937079 0.349117i \(-0.113518\pi\)
\(948\) − 5984.74i − 0.205037i
\(949\) −17204.8 −0.588507
\(950\) 0 0
\(951\) −24193.6 −0.824956
\(952\) 1335.85i 0.0454782i
\(953\) 45012.9i 1.53002i 0.644018 + 0.765010i \(0.277266\pi\)
−0.644018 + 0.765010i \(0.722734\pi\)
\(954\) 3493.18 0.118549
\(955\) 0 0
\(956\) 7440.02 0.251702
\(957\) 2214.55i 0.0748027i
\(958\) 16647.4i 0.561435i
\(959\) −7036.76 −0.236944
\(960\) 0 0
\(961\) −20898.5 −0.701503
\(962\) 3366.79i 0.112838i
\(963\) − 10656.7i − 0.356603i
\(964\) 20284.6 0.677721
\(965\) 0 0
\(966\) 13199.2 0.439624
\(967\) − 40305.8i − 1.34038i −0.742190 0.670190i \(-0.766213\pi\)
0.742190 0.670190i \(-0.233787\pi\)
\(968\) − 15397.8i − 0.511265i
\(969\) 4239.93 0.140564
\(970\) 0 0
\(971\) 33991.8 1.12343 0.561713 0.827332i \(-0.310142\pi\)
0.561713 + 0.827332i \(0.310142\pi\)
\(972\) − 1138.37i − 0.0375651i
\(973\) − 13125.0i − 0.432445i
\(974\) −60837.2 −2.00139
\(975\) 0 0
\(976\) 53628.2 1.75881
\(977\) − 18219.0i − 0.596600i −0.954472 0.298300i \(-0.903580\pi\)
0.954472 0.298300i \(-0.0964196\pi\)
\(978\) 13895.3i 0.454317i
\(979\) −4377.73 −0.142914
\(980\) 0 0
\(981\) 2999.22 0.0976125
\(982\) 64834.1i 2.10686i
\(983\) − 7676.89i − 0.249089i −0.992214 0.124545i \(-0.960253\pi\)
0.992214 0.124545i \(-0.0397470\pi\)
\(984\) −7448.45 −0.241309
\(985\) 0 0
\(986\) 8183.55 0.264318
\(987\) − 2213.75i − 0.0713925i
\(988\) − 22345.4i − 0.719537i
\(989\) −92045.3 −2.95942
\(990\) 0 0
\(991\) 50585.6 1.62150 0.810748 0.585395i \(-0.199061\pi\)
0.810748 + 0.585395i \(0.199061\pi\)
\(992\) 17803.2i 0.569810i
\(993\) 20882.4i 0.667354i
\(994\) 19794.2 0.631625
\(995\) 0 0
\(996\) 3977.69 0.126544
\(997\) − 53060.5i − 1.68550i −0.538305 0.842750i \(-0.680935\pi\)
0.538305 0.842750i \(-0.319065\pi\)
\(998\) − 25320.8i − 0.803121i
\(999\) 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.d.m.274.4 4
5.2 odd 4 525.4.a.j.1.1 2
5.3 odd 4 525.4.a.m.1.2 yes 2
5.4 even 2 inner 525.4.d.m.274.1 4
15.2 even 4 1575.4.a.x.1.2 2
15.8 even 4 1575.4.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.j.1.1 2 5.2 odd 4
525.4.a.m.1.2 yes 2 5.3 odd 4
525.4.d.m.274.1 4 5.4 even 2 inner
525.4.d.m.274.4 4 1.1 even 1 trivial
1575.4.a.o.1.1 2 15.8 even 4
1575.4.a.x.1.2 2 15.2 even 4