Properties

Label 507.4.b.e.337.3
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-17})\)
Defining polynomial: \( x^{4} - 17x^{2} + 289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.3
Root \(-3.57071 + 2.06155i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.e.337.2

$q$-expansion

\(f(q)\) \(=\) \(q+4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} -13.4424i q^{5} -12.3693i q^{6} -31.4219i q^{7} -4.12311i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} -13.4424i q^{5} -12.3693i q^{6} -31.4219i q^{7} -4.12311i q^{8} +9.00000 q^{9} +55.4243 q^{10} +40.4962i q^{11} +27.0000 q^{12} +129.556 q^{14} +40.3271i q^{15} -55.0000 q^{16} +43.1414 q^{17} +37.1080i q^{18} +26.9779i q^{19} +120.981i q^{20} +94.2656i q^{21} -166.970 q^{22} +19.0100 q^{23} +12.3693i q^{24} -55.6971 q^{25} -27.0000 q^{27} +282.797i q^{28} -154.111 q^{29} -166.273 q^{30} -308.270i q^{31} -259.756i q^{32} -121.489i q^{33} +177.877i q^{34} -422.384 q^{35} -81.0000 q^{36} +43.5116i q^{37} -111.233 q^{38} -55.4243 q^{40} -47.8384i q^{41} -388.667 q^{42} -342.121 q^{43} -364.466i q^{44} -120.981i q^{45} +78.3802i q^{46} +133.468i q^{47} +165.000 q^{48} -644.334 q^{49} -229.645i q^{50} -129.424 q^{51} -438.454 q^{53} -111.324i q^{54} +544.364 q^{55} -129.556 q^{56} -80.9338i q^{57} -635.418i q^{58} +590.553i q^{59} -362.944i q^{60} -541.304 q^{61} +1271.03 q^{62} -282.797i q^{63} +631.000 q^{64} +500.910 q^{66} +230.345i q^{67} -388.273 q^{68} -57.0300 q^{69} -1741.54i q^{70} -449.412i q^{71} -37.1080i q^{72} -389.711i q^{73} -179.403 q^{74} +167.091 q^{75} -242.801i q^{76} +1272.47 q^{77} -897.820 q^{79} +739.330i q^{80} +81.0000 q^{81} +197.243 q^{82} +1300.24i q^{83} -848.391i q^{84} -579.923i q^{85} -1410.60i q^{86} +462.334 q^{87} +166.970 q^{88} -925.045i q^{89} +498.819 q^{90} -171.090 q^{92} +924.811i q^{93} -550.301 q^{94} +362.647 q^{95} +779.267i q^{96} +1560.49i q^{97} -2656.66i q^{98} +364.466i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} - 36 q^{4} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} - 36 q^{4} + 36 q^{9} + 136 q^{10} + 108 q^{12} + 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} + 120 q^{25} - 108 q^{27} + 12 q^{29} - 408 q^{30} - 804 q^{35} - 324 q^{36} + 612 q^{38} - 136 q^{40} - 612 q^{42} - 940 q^{43} + 660 q^{48} - 692 q^{49} - 432 q^{51} - 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} + 2524 q^{64} + 204 q^{66} - 1296 q^{68} - 828 q^{69} - 3060 q^{74} - 360 q^{75} + 2976 q^{77} + 8 q^{79} + 324 q^{81} - 68 q^{82} - 36 q^{87} + 68 q^{88} + 1224 q^{90} - 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.12311i 1.45774i 0.684653 + 0.728869i \(0.259954\pi\)
−0.684653 + 0.728869i \(0.740046\pi\)
\(3\) −3.00000 −0.577350
\(4\) −9.00000 −1.12500
\(5\) − 13.4424i − 1.20232i −0.799128 0.601161i \(-0.794705\pi\)
0.799128 0.601161i \(-0.205295\pi\)
\(6\) − 12.3693i − 0.841625i
\(7\) − 31.4219i − 1.69662i −0.529498 0.848311i \(-0.677620\pi\)
0.529498 0.848311i \(-0.322380\pi\)
\(8\) − 4.12311i − 0.182217i
\(9\) 9.00000 0.333333
\(10\) 55.4243 1.75267
\(11\) 40.4962i 1.11001i 0.831849 + 0.555003i \(0.187283\pi\)
−0.831849 + 0.555003i \(0.812717\pi\)
\(12\) 27.0000 0.649519
\(13\) 0 0
\(14\) 129.556 2.47323
\(15\) 40.3271i 0.694161i
\(16\) −55.0000 −0.859375
\(17\) 43.1414 0.615490 0.307745 0.951469i \(-0.400426\pi\)
0.307745 + 0.951469i \(0.400426\pi\)
\(18\) 37.1080i 0.485913i
\(19\) 26.9779i 0.325745i 0.986647 + 0.162873i \(0.0520760\pi\)
−0.986647 + 0.162873i \(0.947924\pi\)
\(20\) 120.981i 1.35261i
\(21\) 94.2656i 0.979545i
\(22\) −166.970 −1.61810
\(23\) 19.0100 0.172342 0.0861709 0.996280i \(-0.472537\pi\)
0.0861709 + 0.996280i \(0.472537\pi\)
\(24\) 12.3693i 0.105203i
\(25\) −55.6971 −0.445577
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 282.797i 1.90870i
\(29\) −154.111 −0.986820 −0.493410 0.869797i \(-0.664250\pi\)
−0.493410 + 0.869797i \(0.664250\pi\)
\(30\) −166.273 −1.01190
\(31\) − 308.270i − 1.78603i −0.450025 0.893016i \(-0.648585\pi\)
0.450025 0.893016i \(-0.351415\pi\)
\(32\) − 259.756i − 1.43496i
\(33\) − 121.489i − 0.640862i
\(34\) 177.877i 0.897223i
\(35\) −422.384 −2.03988
\(36\) −81.0000 −0.375000
\(37\) 43.5116i 0.193331i 0.995317 + 0.0966657i \(0.0308178\pi\)
−0.995317 + 0.0966657i \(0.969182\pi\)
\(38\) −111.233 −0.474851
\(39\) 0 0
\(40\) −55.4243 −0.219084
\(41\) − 47.8384i − 0.182222i −0.995841 0.0911110i \(-0.970958\pi\)
0.995841 0.0911110i \(-0.0290418\pi\)
\(42\) −388.667 −1.42792
\(43\) −342.121 −1.21333 −0.606663 0.794959i \(-0.707492\pi\)
−0.606663 + 0.794959i \(0.707492\pi\)
\(44\) − 364.466i − 1.24876i
\(45\) − 120.981i − 0.400774i
\(46\) 78.3802i 0.251229i
\(47\) 133.468i 0.414218i 0.978318 + 0.207109i \(0.0664055\pi\)
−0.978318 + 0.207109i \(0.933594\pi\)
\(48\) 165.000 0.496160
\(49\) −644.334 −1.87853
\(50\) − 229.645i − 0.649535i
\(51\) −129.424 −0.355353
\(52\) 0 0
\(53\) −438.454 −1.13635 −0.568173 0.822909i \(-0.692350\pi\)
−0.568173 + 0.822909i \(0.692350\pi\)
\(54\) − 111.324i − 0.280542i
\(55\) 544.364 1.33458
\(56\) −129.556 −0.309154
\(57\) − 80.9338i − 0.188069i
\(58\) − 635.418i − 1.43852i
\(59\) 590.553i 1.30311i 0.758601 + 0.651555i \(0.225883\pi\)
−0.758601 + 0.651555i \(0.774117\pi\)
\(60\) − 362.944i − 0.780931i
\(61\) −541.304 −1.13618 −0.568089 0.822967i \(-0.692317\pi\)
−0.568089 + 0.822967i \(0.692317\pi\)
\(62\) 1271.03 2.60357
\(63\) − 282.797i − 0.565541i
\(64\) 631.000 1.23242
\(65\) 0 0
\(66\) 500.910 0.934208
\(67\) 230.345i 0.420018i 0.977700 + 0.210009i \(0.0673493\pi\)
−0.977700 + 0.210009i \(0.932651\pi\)
\(68\) −388.273 −0.692426
\(69\) −57.0300 −0.0995015
\(70\) − 1741.54i − 2.97362i
\(71\) − 449.412i − 0.751203i −0.926781 0.375601i \(-0.877436\pi\)
0.926781 0.375601i \(-0.122564\pi\)
\(72\) − 37.1080i − 0.0607391i
\(73\) − 389.711i − 0.624826i −0.949946 0.312413i \(-0.898863\pi\)
0.949946 0.312413i \(-0.101137\pi\)
\(74\) −179.403 −0.281826
\(75\) 167.091 0.257254
\(76\) − 242.801i − 0.366464i
\(77\) 1272.47 1.88326
\(78\) 0 0
\(79\) −897.820 −1.27864 −0.639321 0.768940i \(-0.720784\pi\)
−0.639321 + 0.768940i \(0.720784\pi\)
\(80\) 739.330i 1.03325i
\(81\) 81.0000 0.111111
\(82\) 197.243 0.265632
\(83\) 1300.24i 1.71952i 0.510700 + 0.859759i \(0.329386\pi\)
−0.510700 + 0.859759i \(0.670614\pi\)
\(84\) − 848.391i − 1.10199i
\(85\) − 579.923i − 0.740017i
\(86\) − 1410.60i − 1.76871i
\(87\) 462.334 0.569741
\(88\) 166.970 0.202262
\(89\) − 925.045i − 1.10174i −0.834592 0.550869i \(-0.814297\pi\)
0.834592 0.550869i \(-0.185703\pi\)
\(90\) 498.819 0.584223
\(91\) 0 0
\(92\) −171.090 −0.193884
\(93\) 924.811i 1.03117i
\(94\) −550.301 −0.603822
\(95\) 362.647 0.391651
\(96\) 779.267i 0.828475i
\(97\) 1560.49i 1.63344i 0.577031 + 0.816722i \(0.304211\pi\)
−0.577031 + 0.816722i \(0.695789\pi\)
\(98\) − 2656.66i − 2.73840i
\(99\) 364.466i 0.370002i
\(100\) 501.274 0.501274
\(101\) −958.840 −0.944635 −0.472318 0.881428i \(-0.656582\pi\)
−0.472318 + 0.881428i \(0.656582\pi\)
\(102\) − 533.630i − 0.518012i
\(103\) −635.153 −0.607606 −0.303803 0.952735i \(-0.598257\pi\)
−0.303803 + 0.952735i \(0.598257\pi\)
\(104\) 0 0
\(105\) 1267.15 1.17773
\(106\) − 1807.79i − 1.65649i
\(107\) −1448.32 −1.30855 −0.654275 0.756257i \(-0.727026\pi\)
−0.654275 + 0.756257i \(0.727026\pi\)
\(108\) 243.000 0.216506
\(109\) 331.084i 0.290937i 0.989363 + 0.145468i \(0.0464689\pi\)
−0.989363 + 0.145468i \(0.953531\pi\)
\(110\) 2244.47i 1.94547i
\(111\) − 130.535i − 0.111620i
\(112\) 1728.20i 1.45803i
\(113\) 695.204 0.578755 0.289378 0.957215i \(-0.406552\pi\)
0.289378 + 0.957215i \(0.406552\pi\)
\(114\) 333.699 0.274156
\(115\) − 255.539i − 0.207210i
\(116\) 1387.00 1.11017
\(117\) 0 0
\(118\) −2434.91 −1.89959
\(119\) − 1355.58i − 1.04425i
\(120\) 166.273 0.126488
\(121\) −308.940 −0.232111
\(122\) − 2231.85i − 1.65625i
\(123\) 143.515i 0.105206i
\(124\) 2774.43i 2.00929i
\(125\) − 931.594i − 0.666595i
\(126\) 1166.00 0.824410
\(127\) −247.154 −0.172688 −0.0863441 0.996265i \(-0.527518\pi\)
−0.0863441 + 0.996265i \(0.527518\pi\)
\(128\) 523.634i 0.361587i
\(129\) 1026.36 0.700514
\(130\) 0 0
\(131\) 472.243 0.314962 0.157481 0.987522i \(-0.449663\pi\)
0.157481 + 0.987522i \(0.449663\pi\)
\(132\) 1093.40i 0.720969i
\(133\) 847.697 0.552667
\(134\) −949.739 −0.612275
\(135\) 362.944i 0.231387i
\(136\) − 177.877i − 0.112153i
\(137\) − 1830.70i − 1.14166i −0.821069 0.570829i \(-0.806622\pi\)
0.821069 0.570829i \(-0.193378\pi\)
\(138\) − 235.141i − 0.145047i
\(139\) −100.000 −0.0610208 −0.0305104 0.999534i \(-0.509713\pi\)
−0.0305104 + 0.999534i \(0.509713\pi\)
\(140\) 3801.46 2.29487
\(141\) − 400.403i − 0.239149i
\(142\) 1852.97 1.09506
\(143\) 0 0
\(144\) −495.000 −0.286458
\(145\) 2071.62i 1.18647i
\(146\) 1606.82 0.910832
\(147\) 1933.00 1.08457
\(148\) − 391.604i − 0.217498i
\(149\) − 149.557i − 0.0822293i −0.999154 0.0411147i \(-0.986909\pi\)
0.999154 0.0411147i \(-0.0130909\pi\)
\(150\) 688.936i 0.375009i
\(151\) − 800.032i − 0.431163i −0.976486 0.215582i \(-0.930835\pi\)
0.976486 0.215582i \(-0.0691647\pi\)
\(152\) 111.233 0.0593564
\(153\) 388.273 0.205163
\(154\) 5246.51i 2.74530i
\(155\) −4143.88 −2.14739
\(156\) 0 0
\(157\) −2706.16 −1.37564 −0.687818 0.725884i \(-0.741431\pi\)
−0.687818 + 0.725884i \(0.741431\pi\)
\(158\) − 3701.81i − 1.86392i
\(159\) 1315.36 0.656070
\(160\) −3491.73 −1.72528
\(161\) − 597.330i − 0.292399i
\(162\) 333.972i 0.161971i
\(163\) 3678.25i 1.76750i 0.467959 + 0.883750i \(0.344990\pi\)
−0.467959 + 0.883750i \(0.655010\pi\)
\(164\) 430.546i 0.205000i
\(165\) −1633.09 −0.770522
\(166\) −5361.03 −2.50661
\(167\) − 3223.11i − 1.49348i −0.665114 0.746742i \(-0.731617\pi\)
0.665114 0.746742i \(-0.268383\pi\)
\(168\) 388.667 0.178490
\(169\) 0 0
\(170\) 2391.08 1.07875
\(171\) 242.801i 0.108582i
\(172\) 3079.09 1.36499
\(173\) 2689.54 1.18198 0.590988 0.806680i \(-0.298738\pi\)
0.590988 + 0.806680i \(0.298738\pi\)
\(174\) 1906.25i 0.830533i
\(175\) 1750.11i 0.755976i
\(176\) − 2227.29i − 0.953911i
\(177\) − 1771.66i − 0.752351i
\(178\) 3814.06 1.60604
\(179\) 1524.04 0.636381 0.318191 0.948027i \(-0.396925\pi\)
0.318191 + 0.948027i \(0.396925\pi\)
\(180\) 1088.83i 0.450871i
\(181\) −476.881 −0.195836 −0.0979180 0.995194i \(-0.531218\pi\)
−0.0979180 + 0.995194i \(0.531218\pi\)
\(182\) 0 0
\(183\) 1623.91 0.655973
\(184\) − 78.3802i − 0.0314036i
\(185\) 584.899 0.232446
\(186\) −3813.09 −1.50317
\(187\) 1747.06i 0.683197i
\(188\) − 1201.21i − 0.465996i
\(189\) 848.391i 0.326515i
\(190\) 1495.23i 0.570924i
\(191\) −1369.74 −0.518906 −0.259453 0.965756i \(-0.583542\pi\)
−0.259453 + 0.965756i \(0.583542\pi\)
\(192\) −1893.00 −0.711539
\(193\) − 2144.72i − 0.799898i −0.916537 0.399949i \(-0.869028\pi\)
0.916537 0.399949i \(-0.130972\pi\)
\(194\) −6434.08 −2.38113
\(195\) 0 0
\(196\) 5799.01 2.11334
\(197\) 239.739i 0.0867040i 0.999060 + 0.0433520i \(0.0138037\pi\)
−0.999060 + 0.0433520i \(0.986196\pi\)
\(198\) −1502.73 −0.539365
\(199\) 1589.94 0.566371 0.283185 0.959065i \(-0.408609\pi\)
0.283185 + 0.959065i \(0.408609\pi\)
\(200\) 229.645i 0.0811918i
\(201\) − 691.036i − 0.242497i
\(202\) − 3953.40i − 1.37703i
\(203\) 4842.47i 1.67426i
\(204\) 1164.82 0.399773
\(205\) −643.061 −0.219090
\(206\) − 2618.80i − 0.885731i
\(207\) 171.090 0.0574472
\(208\) 0 0
\(209\) −1092.50 −0.361579
\(210\) 5224.61i 1.71682i
\(211\) −1872.85 −0.611055 −0.305527 0.952183i \(-0.598833\pi\)
−0.305527 + 0.952183i \(0.598833\pi\)
\(212\) 3946.09 1.27839
\(213\) 1348.24i 0.433707i
\(214\) − 5971.59i − 1.90752i
\(215\) 4598.92i 1.45881i
\(216\) 111.324i 0.0350677i
\(217\) −9686.43 −3.03022
\(218\) −1365.09 −0.424109
\(219\) 1169.13i 0.360743i
\(220\) −4899.28 −1.50141
\(221\) 0 0
\(222\) 538.209 0.162713
\(223\) 56.1283i 0.0168548i 0.999964 + 0.00842742i \(0.00268256\pi\)
−0.999964 + 0.00842742i \(0.997317\pi\)
\(224\) −8162.01 −2.43459
\(225\) −501.274 −0.148526
\(226\) 2866.40i 0.843673i
\(227\) 667.390i 0.195137i 0.995229 + 0.0975687i \(0.0311066\pi\)
−0.995229 + 0.0975687i \(0.968893\pi\)
\(228\) 728.404i 0.211578i
\(229\) 723.299i 0.208720i 0.994540 + 0.104360i \(0.0332795\pi\)
−0.994540 + 0.104360i \(0.966721\pi\)
\(230\) 1053.62 0.302058
\(231\) −3817.40 −1.08730
\(232\) 635.418i 0.179816i
\(233\) 275.451 0.0774482 0.0387241 0.999250i \(-0.487671\pi\)
0.0387241 + 0.999250i \(0.487671\pi\)
\(234\) 0 0
\(235\) 1794.12 0.498024
\(236\) − 5314.98i − 1.46600i
\(237\) 2693.46 0.738224
\(238\) 5589.22 1.52225
\(239\) 1529.39i 0.413925i 0.978349 + 0.206963i \(0.0663579\pi\)
−0.978349 + 0.206963i \(0.933642\pi\)
\(240\) − 2217.99i − 0.596544i
\(241\) 975.526i 0.260743i 0.991465 + 0.130372i \(0.0416170\pi\)
−0.991465 + 0.130372i \(0.958383\pi\)
\(242\) − 1273.79i − 0.338357i
\(243\) −243.000 −0.0641500
\(244\) 4871.74 1.27820
\(245\) 8661.38i 2.25859i
\(246\) −591.729 −0.153363
\(247\) 0 0
\(248\) −1271.03 −0.325446
\(249\) − 3900.72i − 0.992765i
\(250\) 3841.06 0.971720
\(251\) 1874.14 0.471294 0.235647 0.971839i \(-0.424279\pi\)
0.235647 + 0.971839i \(0.424279\pi\)
\(252\) 2545.17i 0.636233i
\(253\) 769.832i 0.191300i
\(254\) − 1019.04i − 0.251734i
\(255\) 1739.77i 0.427249i
\(256\) 2889.00 0.705322
\(257\) 1818.19 0.441305 0.220653 0.975352i \(-0.429181\pi\)
0.220653 + 0.975352i \(0.429181\pi\)
\(258\) 4231.81i 1.02117i
\(259\) 1367.22 0.328010
\(260\) 0 0
\(261\) −1387.00 −0.328940
\(262\) 1947.11i 0.459132i
\(263\) 673.799 0.157978 0.0789890 0.996875i \(-0.474831\pi\)
0.0789890 + 0.996875i \(0.474831\pi\)
\(264\) −500.910 −0.116776
\(265\) 5893.86i 1.36625i
\(266\) 3495.14i 0.805643i
\(267\) 2775.14i 0.636088i
\(268\) − 2073.11i − 0.472520i
\(269\) −3356.40 −0.760756 −0.380378 0.924831i \(-0.624206\pi\)
−0.380378 + 0.924831i \(0.624206\pi\)
\(270\) −1496.46 −0.337301
\(271\) 8915.55i 1.99845i 0.0393133 + 0.999227i \(0.487483\pi\)
−0.0393133 + 0.999227i \(0.512517\pi\)
\(272\) −2372.78 −0.528937
\(273\) 0 0
\(274\) 7548.16 1.66424
\(275\) − 2255.52i − 0.494593i
\(276\) 513.270 0.111939
\(277\) −4017.31 −0.871396 −0.435698 0.900093i \(-0.643498\pi\)
−0.435698 + 0.900093i \(0.643498\pi\)
\(278\) − 412.311i − 0.0889523i
\(279\) − 2774.43i − 0.595344i
\(280\) 1741.54i 0.371702i
\(281\) 1841.12i 0.390860i 0.980718 + 0.195430i \(0.0626103\pi\)
−0.980718 + 0.195430i \(0.937390\pi\)
\(282\) 1650.90 0.348617
\(283\) −4849.40 −1.01861 −0.509305 0.860586i \(-0.670098\pi\)
−0.509305 + 0.860586i \(0.670098\pi\)
\(284\) 4044.71i 0.845103i
\(285\) −1087.94 −0.226120
\(286\) 0 0
\(287\) −1503.17 −0.309162
\(288\) − 2337.80i − 0.478320i
\(289\) −3051.82 −0.621172
\(290\) −8541.52 −1.72957
\(291\) − 4681.48i − 0.943070i
\(292\) 3507.40i 0.702929i
\(293\) − 1413.85i − 0.281905i −0.990016 0.140953i \(-0.954983\pi\)
0.990016 0.140953i \(-0.0450165\pi\)
\(294\) 7969.97i 1.58101i
\(295\) 7938.43 1.56676
\(296\) 179.403 0.0352283
\(297\) − 1093.40i − 0.213621i
\(298\) 616.639 0.119869
\(299\) 0 0
\(300\) −1503.82 −0.289411
\(301\) 10750.1i 2.05856i
\(302\) 3298.62 0.628523
\(303\) 2876.52 0.545385
\(304\) − 1483.79i − 0.279937i
\(305\) 7276.41i 1.36605i
\(306\) 1600.89i 0.299074i
\(307\) − 4625.64i − 0.859932i −0.902845 0.429966i \(-0.858526\pi\)
0.902845 0.429966i \(-0.141474\pi\)
\(308\) −11452.2 −2.11867
\(309\) 1905.46 0.350802
\(310\) − 17085.7i − 3.13032i
\(311\) 6060.79 1.10507 0.552534 0.833490i \(-0.313661\pi\)
0.552534 + 0.833490i \(0.313661\pi\)
\(312\) 0 0
\(313\) 969.946 0.175158 0.0875792 0.996158i \(-0.472087\pi\)
0.0875792 + 0.996158i \(0.472087\pi\)
\(314\) − 11157.8i − 2.00532i
\(315\) −3801.46 −0.679962
\(316\) 8080.38 1.43847
\(317\) − 8741.63i − 1.54883i −0.632679 0.774414i \(-0.718045\pi\)
0.632679 0.774414i \(-0.281955\pi\)
\(318\) 5423.38i 0.956378i
\(319\) − 6240.92i − 1.09537i
\(320\) − 8482.13i − 1.48177i
\(321\) 4344.97 0.755491
\(322\) 2462.85 0.426241
\(323\) 1163.87i 0.200493i
\(324\) −729.000 −0.125000
\(325\) 0 0
\(326\) −15165.8 −2.57655
\(327\) − 993.252i − 0.167972i
\(328\) −197.243 −0.0332040
\(329\) 4193.81 0.702772
\(330\) − 6733.41i − 1.12322i
\(331\) − 6987.76i − 1.16037i −0.814485 0.580184i \(-0.802981\pi\)
0.814485 0.580184i \(-0.197019\pi\)
\(332\) − 11702.2i − 1.93446i
\(333\) 391.604i 0.0644438i
\(334\) 13289.2 2.17711
\(335\) 3096.39 0.504996
\(336\) − 5184.61i − 0.841797i
\(337\) 4156.59 0.671881 0.335940 0.941883i \(-0.390946\pi\)
0.335940 + 0.941883i \(0.390946\pi\)
\(338\) 0 0
\(339\) −2085.61 −0.334144
\(340\) 5219.30i 0.832519i
\(341\) 12483.8 1.98250
\(342\) −1001.10 −0.158284
\(343\) 9468.49i 1.49053i
\(344\) 1410.60i 0.221089i
\(345\) 766.618i 0.119633i
\(346\) 11089.3i 1.72301i
\(347\) −312.513 −0.0483475 −0.0241737 0.999708i \(-0.507695\pi\)
−0.0241737 + 0.999708i \(0.507695\pi\)
\(348\) −4161.01 −0.640958
\(349\) − 4458.75i − 0.683872i −0.939723 0.341936i \(-0.888917\pi\)
0.939723 0.341936i \(-0.111083\pi\)
\(350\) −7215.88 −1.10201
\(351\) 0 0
\(352\) 10519.1 1.59281
\(353\) 2249.17i 0.339126i 0.985519 + 0.169563i \(0.0542356\pi\)
−0.985519 + 0.169563i \(0.945764\pi\)
\(354\) 7304.74 1.09673
\(355\) −6041.16 −0.903187
\(356\) 8325.41i 1.23945i
\(357\) 4066.75i 0.602900i
\(358\) 6283.78i 0.927677i
\(359\) − 7842.79i − 1.15300i −0.817098 0.576499i \(-0.804418\pi\)
0.817098 0.576499i \(-0.195582\pi\)
\(360\) −498.819 −0.0730279
\(361\) 6131.19 0.893890
\(362\) − 1966.23i − 0.285478i
\(363\) 926.820 0.134009
\(364\) 0 0
\(365\) −5238.64 −0.751241
\(366\) 6695.56i 0.956237i
\(367\) 6660.24 0.947307 0.473653 0.880711i \(-0.342935\pi\)
0.473653 + 0.880711i \(0.342935\pi\)
\(368\) −1045.55 −0.148106
\(369\) − 430.546i − 0.0607407i
\(370\) 2411.60i 0.338846i
\(371\) 13777.1i 1.92795i
\(372\) − 8323.30i − 1.16006i
\(373\) −36.9873 −0.00513439 −0.00256720 0.999997i \(-0.500817\pi\)
−0.00256720 + 0.999997i \(0.500817\pi\)
\(374\) −7203.32 −0.995923
\(375\) 2794.78i 0.384859i
\(376\) 550.301 0.0754777
\(377\) 0 0
\(378\) −3498.00 −0.475973
\(379\) − 12079.9i − 1.63721i −0.574360 0.818603i \(-0.694749\pi\)
0.574360 0.818603i \(-0.305251\pi\)
\(380\) −3263.82 −0.440607
\(381\) 741.463 0.0997015
\(382\) − 5647.59i − 0.756429i
\(383\) 10567.4i 1.40984i 0.709287 + 0.704919i \(0.249017\pi\)
−0.709287 + 0.704919i \(0.750983\pi\)
\(384\) − 1570.90i − 0.208763i
\(385\) − 17104.9i − 2.26428i
\(386\) 8842.90 1.16604
\(387\) −3079.09 −0.404442
\(388\) − 14044.4i − 1.83763i
\(389\) 9757.49 1.27179 0.635893 0.771778i \(-0.280632\pi\)
0.635893 + 0.771778i \(0.280632\pi\)
\(390\) 0 0
\(391\) 820.119 0.106075
\(392\) 2656.66i 0.342300i
\(393\) −1416.73 −0.181844
\(394\) −988.469 −0.126392
\(395\) 12068.8i 1.53734i
\(396\) − 3280.19i − 0.416252i
\(397\) − 14200.5i − 1.79522i −0.440786 0.897612i \(-0.645300\pi\)
0.440786 0.897612i \(-0.354700\pi\)
\(398\) 6555.48i 0.825620i
\(399\) −2543.09 −0.319082
\(400\) 3063.34 0.382918
\(401\) − 12676.4i − 1.57863i −0.613992 0.789313i \(-0.710437\pi\)
0.613992 0.789313i \(-0.289563\pi\)
\(402\) 2849.22 0.353497
\(403\) 0 0
\(404\) 8629.56 1.06271
\(405\) − 1088.83i − 0.133591i
\(406\) −19966.0 −2.44063
\(407\) −1762.05 −0.214599
\(408\) 533.630i 0.0647515i
\(409\) − 1533.68i − 0.185417i −0.995693 0.0927083i \(-0.970448\pi\)
0.995693 0.0927083i \(-0.0295524\pi\)
\(410\) − 2651.41i − 0.319375i
\(411\) 5492.09i 0.659136i
\(412\) 5716.38 0.683557
\(413\) 18556.3 2.21089
\(414\) 705.422i 0.0837430i
\(415\) 17478.3 2.06741
\(416\) 0 0
\(417\) 300.000 0.0352304
\(418\) − 4504.50i − 0.527087i
\(419\) −2165.89 −0.252532 −0.126266 0.991996i \(-0.540299\pi\)
−0.126266 + 0.991996i \(0.540299\pi\)
\(420\) −11404.4 −1.32494
\(421\) − 734.575i − 0.0850380i −0.999096 0.0425190i \(-0.986462\pi\)
0.999096 0.0425190i \(-0.0135383\pi\)
\(422\) − 7721.98i − 0.890758i
\(423\) 1201.21i 0.138073i
\(424\) 1807.79i 0.207062i
\(425\) −2402.85 −0.274248
\(426\) −5558.92 −0.632231
\(427\) 17008.8i 1.92767i
\(428\) 13034.9 1.47212
\(429\) 0 0
\(430\) −18961.8 −2.12656
\(431\) 13709.3i 1.53215i 0.642754 + 0.766073i \(0.277792\pi\)
−0.642754 + 0.766073i \(0.722208\pi\)
\(432\) 1485.00 0.165387
\(433\) −10049.9 −1.11540 −0.557701 0.830042i \(-0.688316\pi\)
−0.557701 + 0.830042i \(0.688316\pi\)
\(434\) − 39938.2i − 4.41727i
\(435\) − 6214.87i − 0.685011i
\(436\) − 2979.76i − 0.327304i
\(437\) 512.850i 0.0561395i
\(438\) −4820.46 −0.525869
\(439\) 8133.47 0.884258 0.442129 0.896951i \(-0.354223\pi\)
0.442129 + 0.896951i \(0.354223\pi\)
\(440\) − 2244.47i − 0.243184i
\(441\) −5799.01 −0.626175
\(442\) 0 0
\(443\) −2370.78 −0.254264 −0.127132 0.991886i \(-0.540577\pi\)
−0.127132 + 0.991886i \(0.540577\pi\)
\(444\) 1174.81i 0.125572i
\(445\) −12434.8 −1.32464
\(446\) −231.423 −0.0245699
\(447\) 448.670i 0.0474751i
\(448\) − 19827.2i − 2.09095i
\(449\) 12923.2i 1.35832i 0.733992 + 0.679158i \(0.237655\pi\)
−0.733992 + 0.679158i \(0.762345\pi\)
\(450\) − 2066.81i − 0.216512i
\(451\) 1937.27 0.202267
\(452\) −6256.84 −0.651099
\(453\) 2400.10i 0.248932i
\(454\) −2751.72 −0.284459
\(455\) 0 0
\(456\) −333.699 −0.0342694
\(457\) − 8401.26i − 0.859944i −0.902842 0.429972i \(-0.858523\pi\)
0.902842 0.429972i \(-0.141477\pi\)
\(458\) −2982.24 −0.304260
\(459\) −1164.82 −0.118451
\(460\) 2299.85i 0.233111i
\(461\) − 17627.3i − 1.78088i −0.455098 0.890441i \(-0.650396\pi\)
0.455098 0.890441i \(-0.349604\pi\)
\(462\) − 15739.5i − 1.58500i
\(463\) − 5461.81i − 0.548233i −0.961697 0.274116i \(-0.911615\pi\)
0.961697 0.274116i \(-0.0883853\pi\)
\(464\) 8476.13 0.848048
\(465\) 12431.6 1.23979
\(466\) 1135.72i 0.112899i
\(467\) −8262.19 −0.818691 −0.409345 0.912379i \(-0.634243\pi\)
−0.409345 + 0.912379i \(0.634243\pi\)
\(468\) 0 0
\(469\) 7237.89 0.712611
\(470\) 7397.35i 0.725988i
\(471\) 8118.47 0.794223
\(472\) 2434.91 0.237449
\(473\) − 13854.6i − 1.34680i
\(474\) 11105.4i 1.07614i
\(475\) − 1502.59i − 0.145145i
\(476\) 12200.3i 1.17479i
\(477\) −3946.09 −0.378782
\(478\) −6305.84 −0.603395
\(479\) − 1575.87i − 0.150320i −0.997171 0.0751601i \(-0.976053\pi\)
0.997171 0.0751601i \(-0.0239468\pi\)
\(480\) 10475.2 0.996093
\(481\) 0 0
\(482\) −4022.20 −0.380095
\(483\) 1791.99i 0.168816i
\(484\) 2780.46 0.261125
\(485\) 20976.7 1.96393
\(486\) − 1001.91i − 0.0935139i
\(487\) 12595.7i 1.17200i 0.810310 + 0.586001i \(0.199299\pi\)
−0.810310 + 0.586001i \(0.800701\pi\)
\(488\) 2231.85i 0.207031i
\(489\) − 11034.7i − 1.02047i
\(490\) −35711.8 −3.29244
\(491\) 1071.21 0.0984586 0.0492293 0.998788i \(-0.484323\pi\)
0.0492293 + 0.998788i \(0.484323\pi\)
\(492\) − 1291.64i − 0.118357i
\(493\) −6648.59 −0.607378
\(494\) 0 0
\(495\) 4899.28 0.444861
\(496\) 16954.9i 1.53487i
\(497\) −14121.4 −1.27451
\(498\) 16083.1 1.44719
\(499\) 1422.30i 0.127597i 0.997963 + 0.0637985i \(0.0203215\pi\)
−0.997963 + 0.0637985i \(0.979678\pi\)
\(500\) 8384.35i 0.749919i
\(501\) 9669.33i 0.862263i
\(502\) 7727.28i 0.687022i
\(503\) −9349.34 −0.828760 −0.414380 0.910104i \(-0.636002\pi\)
−0.414380 + 0.910104i \(0.636002\pi\)
\(504\) −1166.00 −0.103051
\(505\) 12889.1i 1.13576i
\(506\) −3174.10 −0.278866
\(507\) 0 0
\(508\) 2224.39 0.194274
\(509\) − 13736.3i − 1.19617i −0.801432 0.598086i \(-0.795928\pi\)
0.801432 0.598086i \(-0.204072\pi\)
\(510\) −7173.25 −0.622817
\(511\) −12245.5 −1.06009
\(512\) 16100.7i 1.38976i
\(513\) − 728.404i − 0.0626897i
\(514\) 7496.58i 0.643307i
\(515\) 8537.96i 0.730538i
\(516\) −9237.28 −0.788079
\(517\) −5404.93 −0.459785
\(518\) 5637.17i 0.478153i
\(519\) −8068.62 −0.682414
\(520\) 0 0
\(521\) 11052.3 0.929386 0.464693 0.885472i \(-0.346165\pi\)
0.464693 + 0.885472i \(0.346165\pi\)
\(522\) − 5718.76i − 0.479508i
\(523\) 6477.04 0.541532 0.270766 0.962645i \(-0.412723\pi\)
0.270766 + 0.962645i \(0.412723\pi\)
\(524\) −4250.19 −0.354332
\(525\) − 5250.33i − 0.436463i
\(526\) 2778.14i 0.230290i
\(527\) − 13299.2i − 1.09929i
\(528\) 6681.87i 0.550741i
\(529\) −11805.6 −0.970298
\(530\) −24301.0 −1.99164
\(531\) 5314.98i 0.434370i
\(532\) −7629.27 −0.621750
\(533\) 0 0
\(534\) −11442.2 −0.927250
\(535\) 19468.9i 1.57330i
\(536\) 949.739 0.0765344
\(537\) −4572.12 −0.367415
\(538\) − 13838.8i − 1.10898i
\(539\) − 26093.1i − 2.08517i
\(540\) − 3266.49i − 0.260310i
\(541\) 18341.5i 1.45761i 0.684723 + 0.728803i \(0.259923\pi\)
−0.684723 + 0.728803i \(0.740077\pi\)
\(542\) −36759.7 −2.91322
\(543\) 1430.64 0.113066
\(544\) − 11206.2i − 0.883204i
\(545\) 4450.55 0.349799
\(546\) 0 0
\(547\) −18943.1 −1.48071 −0.740356 0.672215i \(-0.765343\pi\)
−0.740356 + 0.672215i \(0.765343\pi\)
\(548\) 16476.3i 1.28436i
\(549\) −4871.74 −0.378726
\(550\) 9299.75 0.720987
\(551\) − 4157.61i − 0.321452i
\(552\) 235.141i 0.0181309i
\(553\) 28211.2i 2.16937i
\(554\) − 16563.8i − 1.27027i
\(555\) −1754.70 −0.134203
\(556\) 900.000 0.0686484
\(557\) 415.532i 0.0316098i 0.999875 + 0.0158049i \(0.00503106\pi\)
−0.999875 + 0.0158049i \(0.994969\pi\)
\(558\) 11439.3 0.867856
\(559\) 0 0
\(560\) 23231.1 1.75303
\(561\) − 5241.19i − 0.394444i
\(562\) −7591.12 −0.569772
\(563\) −18291.8 −1.36929 −0.684643 0.728879i \(-0.740042\pi\)
−0.684643 + 0.728879i \(0.740042\pi\)
\(564\) 3603.63i 0.269043i
\(565\) − 9345.19i − 0.695850i
\(566\) − 19994.6i − 1.48487i
\(567\) − 2545.17i − 0.188514i
\(568\) −1852.97 −0.136882
\(569\) −4347.47 −0.320308 −0.160154 0.987092i \(-0.551199\pi\)
−0.160154 + 0.987092i \(0.551199\pi\)
\(570\) − 4485.70i − 0.329623i
\(571\) 16756.0 1.22805 0.614024 0.789288i \(-0.289550\pi\)
0.614024 + 0.789288i \(0.289550\pi\)
\(572\) 0 0
\(573\) 4109.23 0.299591
\(574\) − 6197.74i − 0.450677i
\(575\) −1058.80 −0.0767915
\(576\) 5679.00 0.410807
\(577\) − 19974.7i − 1.44117i −0.693364 0.720587i \(-0.743872\pi\)
0.693364 0.720587i \(-0.256128\pi\)
\(578\) − 12583.0i − 0.905506i
\(579\) 6434.16i 0.461821i
\(580\) − 18644.6i − 1.33478i
\(581\) 40856.0 2.91737
\(582\) 19302.2 1.37475
\(583\) − 17755.7i − 1.26135i
\(584\) −1606.82 −0.113854
\(585\) 0 0
\(586\) 5829.47 0.410944
\(587\) 15748.7i 1.10735i 0.832732 + 0.553677i \(0.186776\pi\)
−0.832732 + 0.553677i \(0.813224\pi\)
\(588\) −17397.0 −1.22014
\(589\) 8316.50 0.581792
\(590\) 32731.0i 2.28392i
\(591\) − 719.217i − 0.0500586i
\(592\) − 2393.14i − 0.166144i
\(593\) 13318.4i 0.922297i 0.887323 + 0.461148i \(0.152562\pi\)
−0.887323 + 0.461148i \(0.847438\pi\)
\(594\) 4508.19 0.311403
\(595\) −18222.3 −1.25553
\(596\) 1346.01i 0.0925080i
\(597\) −4769.82 −0.326994
\(598\) 0 0
\(599\) 2970.80 0.202644 0.101322 0.994854i \(-0.467693\pi\)
0.101322 + 0.994854i \(0.467693\pi\)
\(600\) − 688.936i − 0.0468761i
\(601\) 10632.6 0.721654 0.360827 0.932633i \(-0.382495\pi\)
0.360827 + 0.932633i \(0.382495\pi\)
\(602\) −44323.8 −3.00083
\(603\) 2073.11i 0.140006i
\(604\) 7200.29i 0.485059i
\(605\) 4152.88i 0.279072i
\(606\) 11860.2i 0.795029i
\(607\) 11587.9 0.774856 0.387428 0.921900i \(-0.373364\pi\)
0.387428 + 0.921900i \(0.373364\pi\)
\(608\) 7007.67 0.467432
\(609\) − 14527.4i − 0.966634i
\(610\) −30001.4 −1.99135
\(611\) 0 0
\(612\) −3494.46 −0.230809
\(613\) − 20792.3i − 1.36998i −0.728555 0.684988i \(-0.759808\pi\)
0.728555 0.684988i \(-0.240192\pi\)
\(614\) 19072.0 1.25356
\(615\) 1929.18 0.126491
\(616\) − 5246.51i − 0.343162i
\(617\) 1562.78i 0.101969i 0.998699 + 0.0509846i \(0.0162359\pi\)
−0.998699 + 0.0509846i \(0.983764\pi\)
\(618\) 7856.41i 0.511377i
\(619\) 758.406i 0.0492454i 0.999697 + 0.0246227i \(0.00783844\pi\)
−0.999697 + 0.0246227i \(0.992162\pi\)
\(620\) 37294.9 2.41581
\(621\) −513.270 −0.0331672
\(622\) 24989.3i 1.61090i
\(623\) −29066.7 −1.86923
\(624\) 0 0
\(625\) −19485.0 −1.24704
\(626\) 3999.19i 0.255335i
\(627\) 3277.51 0.208758
\(628\) 24355.4 1.54759
\(629\) 1877.15i 0.118994i
\(630\) − 15673.8i − 0.991206i
\(631\) − 14265.2i − 0.899981i −0.893033 0.449990i \(-0.851427\pi\)
0.893033 0.449990i \(-0.148573\pi\)
\(632\) 3701.81i 0.232990i
\(633\) 5618.56 0.352793
\(634\) 36042.7 2.25779
\(635\) 3322.34i 0.207627i
\(636\) −11838.3 −0.738078
\(637\) 0 0
\(638\) 25732.0 1.59677
\(639\) − 4044.71i − 0.250401i
\(640\) 7038.88 0.434744
\(641\) −3985.64 −0.245590 −0.122795 0.992432i \(-0.539186\pi\)
−0.122795 + 0.992432i \(0.539186\pi\)
\(642\) 17914.8i 1.10131i
\(643\) 8156.55i 0.500254i 0.968213 + 0.250127i \(0.0804723\pi\)
−0.968213 + 0.250127i \(0.919528\pi\)
\(644\) 5375.97i 0.328949i
\(645\) − 13796.8i − 0.842243i
\(646\) −4798.74 −0.292266
\(647\) −11279.2 −0.685368 −0.342684 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(648\) − 333.972i − 0.0202464i
\(649\) −23915.2 −1.44646
\(650\) 0 0
\(651\) 29059.3 1.74950
\(652\) − 33104.2i − 1.98844i
\(653\) 6565.75 0.393473 0.196736 0.980456i \(-0.436966\pi\)
0.196736 + 0.980456i \(0.436966\pi\)
\(654\) 4095.28 0.244860
\(655\) − 6348.06i − 0.378686i
\(656\) 2631.11i 0.156597i
\(657\) − 3507.40i − 0.208275i
\(658\) 17291.5i 1.02446i
\(659\) 4799.35 0.283696 0.141848 0.989888i \(-0.454696\pi\)
0.141848 + 0.989888i \(0.454696\pi\)
\(660\) 14697.8 0.866837
\(661\) 15593.6i 0.917581i 0.888545 + 0.458790i \(0.151717\pi\)
−0.888545 + 0.458790i \(0.848283\pi\)
\(662\) 28811.3 1.69151
\(663\) 0 0
\(664\) 5361.03 0.313326
\(665\) − 11395.1i − 0.664483i
\(666\) −1614.63 −0.0939422
\(667\) −2929.66 −0.170070
\(668\) 29008.0i 1.68017i
\(669\) − 168.385i − 0.00973115i
\(670\) 12766.7i 0.736152i
\(671\) − 21920.8i − 1.26116i
\(672\) 24486.0 1.40561
\(673\) 2205.54 0.126326 0.0631630 0.998003i \(-0.479881\pi\)
0.0631630 + 0.998003i \(0.479881\pi\)
\(674\) 17138.1i 0.979426i
\(675\) 1503.82 0.0857514
\(676\) 0 0
\(677\) −15046.4 −0.854182 −0.427091 0.904209i \(-0.640462\pi\)
−0.427091 + 0.904209i \(0.640462\pi\)
\(678\) − 8599.20i − 0.487095i
\(679\) 49033.6 2.77134
\(680\) −2391.08 −0.134844
\(681\) − 2002.17i − 0.112663i
\(682\) 51471.9i 2.88997i
\(683\) 30632.5i 1.71614i 0.513537 + 0.858068i \(0.328335\pi\)
−0.513537 + 0.858068i \(0.671665\pi\)
\(684\) − 2185.21i − 0.122155i
\(685\) −24608.9 −1.37264
\(686\) −39039.6 −2.17280
\(687\) − 2169.90i − 0.120505i
\(688\) 18816.7 1.04270
\(689\) 0 0
\(690\) −3160.85 −0.174393
\(691\) − 2175.72i − 0.119780i −0.998205 0.0598901i \(-0.980925\pi\)
0.998205 0.0598901i \(-0.0190750\pi\)
\(692\) −24205.9 −1.32972
\(693\) 11452.2 0.627753
\(694\) − 1288.52i − 0.0704780i
\(695\) 1344.24i 0.0733666i
\(696\) − 1906.25i − 0.103817i
\(697\) − 2063.82i − 0.112156i
\(698\) 18383.9 0.996906
\(699\) −826.354 −0.0447147
\(700\) − 15751.0i − 0.850473i
\(701\) 32718.2 1.76284 0.881419 0.472335i \(-0.156589\pi\)
0.881419 + 0.472335i \(0.156589\pi\)
\(702\) 0 0
\(703\) −1173.85 −0.0629768
\(704\) 25553.1i 1.36799i
\(705\) −5382.36 −0.287534
\(706\) −9273.58 −0.494356
\(707\) 30128.6i 1.60269i
\(708\) 15944.9i 0.846395i
\(709\) − 25219.7i − 1.33589i −0.744211 0.667945i \(-0.767174\pi\)
0.744211 0.667945i \(-0.232826\pi\)
\(710\) − 24908.3i − 1.31661i
\(711\) −8080.38 −0.426214
\(712\) −3814.06 −0.200756
\(713\) − 5860.22i − 0.307808i
\(714\) −16767.7 −0.878871
\(715\) 0 0
\(716\) −13716.4 −0.715929
\(717\) − 4588.18i − 0.238980i
\(718\) 32336.6 1.68077
\(719\) 35466.2 1.83959 0.919796 0.392398i \(-0.128354\pi\)
0.919796 + 0.392398i \(0.128354\pi\)
\(720\) 6653.97i 0.344415i
\(721\) 19957.7i 1.03088i
\(722\) 25279.5i 1.30306i
\(723\) − 2926.58i − 0.150540i
\(724\) 4291.93 0.220315
\(725\) 8583.57 0.439704
\(726\) 3821.38i 0.195351i
\(727\) 14262.2 0.727588 0.363794 0.931479i \(-0.381481\pi\)
0.363794 + 0.931479i \(0.381481\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) − 21599.5i − 1.09511i
\(731\) −14759.6 −0.746790
\(732\) −14615.2 −0.737970
\(733\) − 16022.5i − 0.807371i −0.914898 0.403685i \(-0.867729\pi\)
0.914898 0.403685i \(-0.132271\pi\)
\(734\) 27460.9i 1.38093i
\(735\) − 25984.1i − 1.30400i
\(736\) − 4937.96i − 0.247304i
\(737\) −9328.11 −0.466222
\(738\) 1775.19 0.0885440
\(739\) 3796.21i 0.188966i 0.995526 + 0.0944830i \(0.0301198\pi\)
−0.995526 + 0.0944830i \(0.969880\pi\)
\(740\) −5264.09 −0.261502
\(741\) 0 0
\(742\) −56804.3 −2.81044
\(743\) 30329.3i 1.49754i 0.662827 + 0.748772i \(0.269356\pi\)
−0.662827 + 0.748772i \(0.730644\pi\)
\(744\) 3813.09 0.187896
\(745\) −2010.40 −0.0988661
\(746\) − 152.502i − 0.00748460i
\(747\) 11702.2i 0.573173i
\(748\) − 15723.6i − 0.768597i
\(749\) 45509.1i 2.22011i
\(750\) −11523.2 −0.561023
\(751\) −21551.9 −1.04719 −0.523595 0.851967i \(-0.675409\pi\)
−0.523595 + 0.851967i \(0.675409\pi\)
\(752\) − 7340.72i − 0.355969i
\(753\) −5622.42 −0.272101
\(754\) 0 0
\(755\) −10754.3 −0.518397
\(756\) − 7635.52i − 0.367329i
\(757\) −20417.6 −0.980306 −0.490153 0.871637i \(-0.663059\pi\)
−0.490153 + 0.871637i \(0.663059\pi\)
\(758\) 49806.5 2.38662
\(759\) − 2309.50i − 0.110447i
\(760\) − 1495.23i − 0.0713655i
\(761\) − 31375.4i − 1.49456i −0.664512 0.747278i \(-0.731360\pi\)
0.664512 0.747278i \(-0.268640\pi\)
\(762\) 3057.13i 0.145339i
\(763\) 10403.3 0.493609
\(764\) 12327.7 0.583770
\(765\) − 5219.30i − 0.246672i
\(766\) −43570.5 −2.05518
\(767\) 0 0
\(768\) −8667.00 −0.407218
\(769\) − 12452.7i − 0.583946i −0.956427 0.291973i \(-0.905688\pi\)
0.956427 0.291973i \(-0.0943118\pi\)
\(770\) 70525.5 3.30073
\(771\) −5454.56 −0.254788
\(772\) 19302.5i 0.899885i
\(773\) 37449.8i 1.74253i 0.490815 + 0.871264i \(0.336699\pi\)
−0.490815 + 0.871264i \(0.663301\pi\)
\(774\) − 12695.4i − 0.589571i
\(775\) 17169.8i 0.795815i
\(776\) 6434.08 0.297642
\(777\) −4101.65 −0.189377
\(778\) 40231.2i 1.85393i
\(779\) 1290.58 0.0593580
\(780\) 0 0
\(781\) 18199.5 0.833839
\(782\) 3381.44i 0.154629i
\(783\) 4161.01 0.189914
\(784\) 35438.4 1.61436
\(785\) 36377.1i 1.65396i
\(786\) − 5841.32i − 0.265080i
\(787\) − 26460.2i − 1.19848i −0.800570 0.599240i \(-0.795470\pi\)
0.800570 0.599240i \(-0.204530\pi\)
\(788\) − 2157.65i − 0.0975420i
\(789\) −2021.40 −0.0912086
\(790\) −49761.0 −2.24104
\(791\) − 21844.6i − 0.981928i
\(792\) 1502.73 0.0674207
\(793\) 0 0
\(794\) 58550.3 2.61697
\(795\) − 17681.6i − 0.788807i
\(796\) −14309.4 −0.637167
\(797\) −4749.47 −0.211085 −0.105543 0.994415i \(-0.533658\pi\)
−0.105543 + 0.994415i \(0.533658\pi\)
\(798\) − 10485.4i − 0.465138i
\(799\) 5757.99i 0.254947i
\(800\) 14467.6i 0.639386i
\(801\) − 8325.41i − 0.367246i
\(802\) 52266.1 2.30122
\(803\) 15781.8 0.693560
\(804\) 6219.33i 0.272809i
\(805\) −8029.53 −0.351557
\(806\) 0 0
\(807\) 10069.2 0.439223
\(808\) 3953.40i 0.172129i
\(809\) −4464.04 −0.194002 −0.0970009 0.995284i \(-0.530925\pi\)
−0.0970009 + 0.995284i \(0.530925\pi\)
\(810\) 4489.37 0.194741
\(811\) 20774.6i 0.899499i 0.893155 + 0.449749i \(0.148487\pi\)
−0.893155 + 0.449749i \(0.851513\pi\)
\(812\) − 43582.2i − 1.88354i
\(813\) − 26746.6i − 1.15381i
\(814\) − 7265.13i − 0.312829i
\(815\) 49444.3 2.12510
\(816\) 7118.34 0.305382
\(817\) − 9229.73i − 0.395235i
\(818\) 6323.51 0.270289
\(819\) 0 0
\(820\) 5787.55 0.246476
\(821\) − 35769.2i − 1.52053i −0.649614 0.760264i \(-0.725070\pi\)
0.649614 0.760264i \(-0.274930\pi\)
\(822\) −22644.5 −0.960848
\(823\) −2945.44 −0.124753 −0.0623764 0.998053i \(-0.519868\pi\)
−0.0623764 + 0.998053i \(0.519868\pi\)
\(824\) 2618.80i 0.110716i
\(825\) 6766.56i 0.285553i
\(826\) 76509.6i 3.22289i
\(827\) 17878.6i 0.751753i 0.926670 + 0.375877i \(0.122658\pi\)
−0.926670 + 0.375877i \(0.877342\pi\)
\(828\) −1539.81 −0.0646281
\(829\) −13423.8 −0.562398 −0.281199 0.959649i \(-0.590732\pi\)
−0.281199 + 0.959649i \(0.590732\pi\)
\(830\) 72065.0i 3.01375i
\(831\) 12051.9 0.503101
\(832\) 0 0
\(833\) −27797.5 −1.15621
\(834\) 1236.93i 0.0513566i
\(835\) −43326.2 −1.79565
\(836\) 9832.53 0.406776
\(837\) 8323.30i 0.343722i
\(838\) − 8930.21i − 0.368125i
\(839\) 31542.7i 1.29794i 0.760813 + 0.648971i \(0.224800\pi\)
−0.760813 + 0.648971i \(0.775200\pi\)
\(840\) − 5224.61i − 0.214602i
\(841\) −638.669 −0.0261867
\(842\) 3028.73 0.123963
\(843\) − 5523.35i − 0.225663i
\(844\) 16855.7 0.687437
\(845\) 0 0
\(846\) −4952.71 −0.201274
\(847\) 9707.47i 0.393805i
\(848\) 24115.0 0.976547
\(849\) 14548.2 0.588095
\(850\) − 9907.22i − 0.399782i
\(851\) 827.155i 0.0333191i
\(852\) − 12134.1i − 0.487920i
\(853\) − 21810.6i − 0.875476i −0.899103 0.437738i \(-0.855780\pi\)
0.899103 0.437738i \(-0.144220\pi\)
\(854\) −70129.1 −2.81003
\(855\) 3263.82 0.130550
\(856\) 5971.59i 0.238440i
\(857\) −33234.6 −1.32470 −0.662352 0.749193i \(-0.730442\pi\)
−0.662352 + 0.749193i \(0.730442\pi\)
\(858\) 0 0
\(859\) 42697.7 1.69596 0.847978 0.530032i \(-0.177820\pi\)
0.847978 + 0.530032i \(0.177820\pi\)
\(860\) − 41390.3i − 1.64116i
\(861\) 4509.52 0.178495
\(862\) −56525.0 −2.23347
\(863\) − 27419.2i − 1.08153i −0.841174 0.540765i \(-0.818135\pi\)
0.841174 0.540765i \(-0.181865\pi\)
\(864\) 7013.40i 0.276158i
\(865\) − 36153.8i − 1.42112i
\(866\) − 41437.0i − 1.62596i
\(867\) 9155.45 0.358634
\(868\) 87177.9 3.40900
\(869\) − 36358.3i − 1.41930i
\(870\) 25624.5 0.998567
\(871\) 0 0
\(872\) 1365.09 0.0530137
\(873\) 14044.4i 0.544482i
\(874\) −2114.54 −0.0818367
\(875\) −29272.4 −1.13096
\(876\) − 10522.2i − 0.405836i
\(877\) − 42604.2i − 1.64041i −0.572068 0.820206i \(-0.693859\pi\)
0.572068 0.820206i \(-0.306141\pi\)
\(878\) 33535.2i 1.28902i
\(879\) 4241.56i 0.162758i
\(880\) −29940.0 −1.14691
\(881\) 4699.40 0.179712 0.0898562 0.995955i \(-0.471359\pi\)
0.0898562 + 0.995955i \(0.471359\pi\)
\(882\) − 23909.9i − 0.912799i
\(883\) −22233.5 −0.847358 −0.423679 0.905812i \(-0.639262\pi\)
−0.423679 + 0.905812i \(0.639262\pi\)
\(884\) 0 0
\(885\) −23815.3 −0.904568
\(886\) − 9774.98i − 0.370651i
\(887\) −8852.46 −0.335103 −0.167552 0.985863i \(-0.553586\pi\)
−0.167552 + 0.985863i \(0.553586\pi\)
\(888\) −538.209 −0.0203391
\(889\) 7766.05i 0.292986i
\(890\) − 51270.0i − 1.93098i
\(891\) 3280.19i 0.123334i
\(892\) − 505.155i − 0.0189617i
\(893\) −3600.68 −0.134930
\(894\) −1849.92 −0.0692063
\(895\) − 20486.7i − 0.765135i
\(896\) 16453.6 0.613477
\(897\) 0 0
\(898\) −53283.7 −1.98007
\(899\) 47508.0i 1.76249i
\(900\) 4511.47 0.167091
\(901\) −18915.5 −0.699410
\(902\) 7987.58i 0.294853i
\(903\) − 32250.3i − 1.18851i
\(904\) − 2866.40i − 0.105459i
\(905\) 6410.41i 0.235458i
\(906\) −9895.85 −0.362878
\(907\) −37172.4 −1.36085 −0.680424 0.732818i \(-0.738205\pi\)
−0.680424 + 0.732818i \(0.738205\pi\)
\(908\) − 6006.51i − 0.219530i
\(909\) −8629.56 −0.314878
\(910\) 0 0
\(911\) 38035.1 1.38327 0.691635 0.722247i \(-0.256891\pi\)
0.691635 + 0.722247i \(0.256891\pi\)
\(912\) 4451.36i 0.161622i
\(913\) −52654.8 −1.90867
\(914\) 34639.3 1.25357
\(915\) − 21829.2i − 0.788691i
\(916\) − 6509.70i − 0.234810i
\(917\) − 14838.8i − 0.534372i
\(918\) − 4802.67i − 0.172671i
\(919\) −8352.27 −0.299800 −0.149900 0.988701i \(-0.547895\pi\)
−0.149900 + 0.988701i \(0.547895\pi\)
\(920\) −1053.62 −0.0377573
\(921\) 13876.9i 0.496482i
\(922\) 72679.4 2.59606
\(923\) 0 0
\(924\) 34356.6 1.22321
\(925\) − 2423.47i − 0.0861440i
\(926\) 22519.6 0.799179
\(927\) −5716.38 −0.202535
\(928\) 40031.3i 1.41605i
\(929\) − 20232.1i − 0.714524i −0.934004 0.357262i \(-0.883710\pi\)
0.934004 0.357262i \(-0.116290\pi\)
\(930\) 51257.0i 1.80729i
\(931\) − 17382.8i − 0.611921i
\(932\) −2479.06 −0.0871292
\(933\) −18182.4 −0.638011
\(934\) − 34065.9i − 1.19344i
\(935\) 23484.7 0.821423
\(936\) 0 0
\(937\) −27766.9 −0.968095 −0.484048 0.875042i \(-0.660834\pi\)
−0.484048 + 0.875042i \(0.660834\pi\)
\(938\) 29842.6i 1.03880i
\(939\) −2909.84 −0.101128
\(940\) −16147.1 −0.560277
\(941\) 400.765i 0.0138837i 0.999976 + 0.00694185i \(0.00220968\pi\)
−0.999976 + 0.00694185i \(0.997790\pi\)
\(942\) 33473.3i 1.15777i
\(943\) − 909.408i − 0.0314045i
\(944\) − 32480.4i − 1.11986i
\(945\) 11404.4 0.392576
\(946\) 57124.0 1.96328
\(947\) 21804.4i 0.748201i 0.927388 + 0.374101i \(0.122049\pi\)
−0.927388 + 0.374101i \(0.877951\pi\)
\(948\) −24241.1 −0.830502
\(949\) 0 0
\(950\) 6195.35 0.211583
\(951\) 26224.9i 0.894217i
\(952\) −5589.22 −0.190281
\(953\) −30480.4 −1.03605 −0.518026 0.855365i \(-0.673333\pi\)
−0.518026 + 0.855365i \(0.673333\pi\)
\(954\) − 16270.1i − 0.552165i
\(955\) 18412.6i 0.623892i
\(956\) − 13764.5i − 0.465666i
\(957\) 18722.8i 0.632415i
\(958\) 6497.48 0.219127
\(959\) −57524.0 −1.93696
\(960\) 25446.4i 0.855499i
\(961\) −65239.6 −2.18991
\(962\) 0 0
\(963\) −13034.9 −0.436183
\(964\) − 8779.73i − 0.293336i
\(965\) −28830.1 −0.961734
\(966\) −7388.56 −0.246090
\(967\) − 23864.1i − 0.793608i −0.917903 0.396804i \(-0.870119\pi\)
0.917903 0.396804i \(-0.129881\pi\)
\(968\) 1273.79i 0.0422947i
\(969\) − 3491.60i − 0.115755i
\(970\) 86489.2i 2.86289i
\(971\) −14010.3 −0.463040 −0.231520 0.972830i \(-0.574370\pi\)
−0.231520 + 0.972830i \(0.574370\pi\)
\(972\) 2187.00 0.0721688
\(973\) 3142.19i 0.103529i
\(974\) −51933.3 −1.70847
\(975\) 0 0
\(976\) 29771.7 0.976404
\(977\) 25448.2i 0.833326i 0.909061 + 0.416663i \(0.136800\pi\)
−0.909061 + 0.416663i \(0.863200\pi\)
\(978\) 45497.4 1.48757
\(979\) 37460.8 1.22293
\(980\) − 77952.4i − 2.54092i
\(981\) 2979.76i 0.0969789i
\(982\) 4416.72i 0.143527i
\(983\) 52479.9i 1.70280i 0.524519 + 0.851399i \(0.324245\pi\)
−0.524519 + 0.851399i \(0.675755\pi\)
\(984\) 591.729 0.0191703
\(985\) 3222.66 0.104246
\(986\) − 27412.8i − 0.885398i
\(987\) −12581.4 −0.405746
\(988\) 0 0
\(989\) −6503.73 −0.209107
\(990\) 20200.2i 0.648491i
\(991\) 38048.5 1.21963 0.609814 0.792545i \(-0.291244\pi\)
0.609814 + 0.792545i \(0.291244\pi\)
\(992\) −80075.0 −2.56289
\(993\) 20963.3i 0.669939i
\(994\) − 58223.9i − 1.85790i
\(995\) − 21372.5i − 0.680960i
\(996\) 35106.5i 1.11686i
\(997\) −31236.6 −0.992251 −0.496125 0.868251i \(-0.665244\pi\)
−0.496125 + 0.868251i \(0.665244\pi\)
\(998\) −5864.30 −0.186003
\(999\) − 1174.81i − 0.0372066i
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 507.4.b.e.337.3 4
13.3 even 3 39.4.j.b.4.1 4
13.4 even 6 39.4.j.b.10.1 yes 4
13.5 odd 4 507.4.a.k.1.3 4
13.8 odd 4 507.4.a.k.1.2 4
13.12 even 2 inner 507.4.b.e.337.2 4
39.5 even 4 1521.4.a.z.1.2 4
39.8 even 4 1521.4.a.z.1.3 4
39.17 odd 6 117.4.q.d.10.2 4
39.29 odd 6 117.4.q.d.82.2 4
52.3 odd 6 624.4.bv.c.433.1 4
52.43 odd 6 624.4.bv.c.49.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.b.4.1 4 13.3 even 3
39.4.j.b.10.1 yes 4 13.4 even 6
117.4.q.d.10.2 4 39.17 odd 6
117.4.q.d.82.2 4 39.29 odd 6
507.4.a.k.1.2 4 13.8 odd 4
507.4.a.k.1.3 4 13.5 odd 4
507.4.b.e.337.2 4 13.12 even 2 inner
507.4.b.e.337.3 4 1.1 even 1 trivial
624.4.bv.c.49.2 4 52.43 odd 6
624.4.bv.c.433.1 4 52.3 odd 6
1521.4.a.z.1.2 4 39.5 even 4
1521.4.a.z.1.3 4 39.8 even 4