Properties

Label 169.4.b.a.168.1
Level $169$
Weight $4$
Character 169.168
Analytic conductor $9.971$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 168.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 169.168
Dual form 169.4.b.a.168.2

$q$-expansion

\(f(q)\) \(=\) \(q-5.00000i q^{2} -7.00000 q^{3} -17.0000 q^{4} -7.00000i q^{5} +35.0000i q^{6} +13.0000i q^{7} +45.0000i q^{8} +22.0000 q^{9} +O(q^{10})\) \(q-5.00000i q^{2} -7.00000 q^{3} -17.0000 q^{4} -7.00000i q^{5} +35.0000i q^{6} +13.0000i q^{7} +45.0000i q^{8} +22.0000 q^{9} -35.0000 q^{10} +26.0000i q^{11} +119.000 q^{12} +65.0000 q^{14} +49.0000i q^{15} +89.0000 q^{16} -77.0000 q^{17} -110.000i q^{18} -126.000i q^{19} +119.000i q^{20} -91.0000i q^{21} +130.000 q^{22} +96.0000 q^{23} -315.000i q^{24} +76.0000 q^{25} +35.0000 q^{27} -221.000i q^{28} -82.0000 q^{29} +245.000 q^{30} +196.000i q^{31} -85.0000i q^{32} -182.000i q^{33} +385.000i q^{34} +91.0000 q^{35} -374.000 q^{36} +131.000i q^{37} -630.000 q^{38} +315.000 q^{40} +336.000i q^{41} -455.000 q^{42} +201.000 q^{43} -442.000i q^{44} -154.000i q^{45} -480.000i q^{46} +105.000i q^{47} -623.000 q^{48} +174.000 q^{49} -380.000i q^{50} +539.000 q^{51} -432.000 q^{53} -175.000i q^{54} +182.000 q^{55} -585.000 q^{56} +882.000i q^{57} +410.000i q^{58} +294.000i q^{59} -833.000i q^{60} -56.0000 q^{61} +980.000 q^{62} +286.000i q^{63} +287.000 q^{64} -910.000 q^{66} +478.000i q^{67} +1309.00 q^{68} -672.000 q^{69} -455.000i q^{70} +9.00000i q^{71} +990.000i q^{72} -98.0000i q^{73} +655.000 q^{74} -532.000 q^{75} +2142.00i q^{76} -338.000 q^{77} +1304.00 q^{79} -623.000i q^{80} -839.000 q^{81} +1680.00 q^{82} -308.000i q^{83} +1547.00i q^{84} +539.000i q^{85} -1005.00i q^{86} +574.000 q^{87} -1170.00 q^{88} +1190.00i q^{89} -770.000 q^{90} -1632.00 q^{92} -1372.00i q^{93} +525.000 q^{94} -882.000 q^{95} +595.000i q^{96} +70.0000i q^{97} -870.000i q^{98} +572.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 14q^{3} - 34q^{4} + 44q^{9} + O(q^{10}) \) \( 2q - 14q^{3} - 34q^{4} + 44q^{9} - 70q^{10} + 238q^{12} + 130q^{14} + 178q^{16} - 154q^{17} + 260q^{22} + 192q^{23} + 152q^{25} + 70q^{27} - 164q^{29} + 490q^{30} + 182q^{35} - 748q^{36} - 1260q^{38} + 630q^{40} - 910q^{42} + 402q^{43} - 1246q^{48} + 348q^{49} + 1078q^{51} - 864q^{53} + 364q^{55} - 1170q^{56} - 112q^{61} + 1960q^{62} + 574q^{64} - 1820q^{66} + 2618q^{68} - 1344q^{69} + 1310q^{74} - 1064q^{75} - 676q^{77} + 2608q^{79} - 1678q^{81} + 3360q^{82} + 1148q^{87} - 2340q^{88} - 1540q^{90} - 3264q^{92} + 1050q^{94} - 1764q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 5.00000i − 1.76777i −0.467707 0.883883i \(-0.654920\pi\)
0.467707 0.883883i \(-0.345080\pi\)
\(3\) −7.00000 −1.34715 −0.673575 0.739119i \(-0.735242\pi\)
−0.673575 + 0.739119i \(0.735242\pi\)
\(4\) −17.0000 −2.12500
\(5\) − 7.00000i − 0.626099i −0.949737 0.313050i \(-0.898649\pi\)
0.949737 0.313050i \(-0.101351\pi\)
\(6\) 35.0000i 2.38145i
\(7\) 13.0000i 0.701934i 0.936388 + 0.350967i \(0.114147\pi\)
−0.936388 + 0.350967i \(0.885853\pi\)
\(8\) 45.0000i 1.98874i
\(9\) 22.0000 0.814815
\(10\) −35.0000 −1.10680
\(11\) 26.0000i 0.712663i 0.934360 + 0.356332i \(0.115973\pi\)
−0.934360 + 0.356332i \(0.884027\pi\)
\(12\) 119.000 2.86270
\(13\) 0 0
\(14\) 65.0000 1.24086
\(15\) 49.0000i 0.843450i
\(16\) 89.0000 1.39062
\(17\) −77.0000 −1.09854 −0.549272 0.835644i \(-0.685095\pi\)
−0.549272 + 0.835644i \(0.685095\pi\)
\(18\) − 110.000i − 1.44040i
\(19\) − 126.000i − 1.52139i −0.649110 0.760694i \(-0.724859\pi\)
0.649110 0.760694i \(-0.275141\pi\)
\(20\) 119.000i 1.33046i
\(21\) − 91.0000i − 0.945611i
\(22\) 130.000 1.25982
\(23\) 96.0000 0.870321 0.435161 0.900353i \(-0.356692\pi\)
0.435161 + 0.900353i \(0.356692\pi\)
\(24\) − 315.000i − 2.67913i
\(25\) 76.0000 0.608000
\(26\) 0 0
\(27\) 35.0000 0.249472
\(28\) − 221.000i − 1.49161i
\(29\) −82.0000 −0.525070 −0.262535 0.964923i \(-0.584558\pi\)
−0.262535 + 0.964923i \(0.584558\pi\)
\(30\) 245.000 1.49102
\(31\) 196.000i 1.13557i 0.823177 + 0.567785i \(0.192199\pi\)
−0.823177 + 0.567785i \(0.807801\pi\)
\(32\) − 85.0000i − 0.469563i
\(33\) − 182.000i − 0.960065i
\(34\) 385.000i 1.94197i
\(35\) 91.0000 0.439480
\(36\) −374.000 −1.73148
\(37\) 131.000i 0.582061i 0.956714 + 0.291031i \(0.0939982\pi\)
−0.956714 + 0.291031i \(0.906002\pi\)
\(38\) −630.000 −2.68946
\(39\) 0 0
\(40\) 315.000 1.24515
\(41\) 336.000i 1.27986i 0.768432 + 0.639932i \(0.221037\pi\)
−0.768432 + 0.639932i \(0.778963\pi\)
\(42\) −455.000 −1.67162
\(43\) 201.000 0.712842 0.356421 0.934325i \(-0.383997\pi\)
0.356421 + 0.934325i \(0.383997\pi\)
\(44\) − 442.000i − 1.51441i
\(45\) − 154.000i − 0.510155i
\(46\) − 480.000i − 1.53852i
\(47\) 105.000i 0.325869i 0.986637 + 0.162934i \(0.0520959\pi\)
−0.986637 + 0.162934i \(0.947904\pi\)
\(48\) −623.000 −1.87338
\(49\) 174.000 0.507289
\(50\) − 380.000i − 1.07480i
\(51\) 539.000 1.47990
\(52\) 0 0
\(53\) −432.000 −1.11962 −0.559809 0.828622i \(-0.689126\pi\)
−0.559809 + 0.828622i \(0.689126\pi\)
\(54\) − 175.000i − 0.441009i
\(55\) 182.000 0.446198
\(56\) −585.000 −1.39596
\(57\) 882.000i 2.04954i
\(58\) 410.000i 0.928201i
\(59\) 294.000i 0.648738i 0.945931 + 0.324369i \(0.105152\pi\)
−0.945931 + 0.324369i \(0.894848\pi\)
\(60\) − 833.000i − 1.79233i
\(61\) −56.0000 −0.117542 −0.0587710 0.998271i \(-0.518718\pi\)
−0.0587710 + 0.998271i \(0.518718\pi\)
\(62\) 980.000 2.00742
\(63\) 286.000i 0.571946i
\(64\) 287.000 0.560547
\(65\) 0 0
\(66\) −910.000 −1.69717
\(67\) 478.000i 0.871597i 0.900044 + 0.435798i \(0.143534\pi\)
−0.900044 + 0.435798i \(0.856466\pi\)
\(68\) 1309.00 2.33441
\(69\) −672.000 −1.17245
\(70\) − 455.000i − 0.776899i
\(71\) 9.00000i 0.0150437i 0.999972 + 0.00752186i \(0.00239430\pi\)
−0.999972 + 0.00752186i \(0.997606\pi\)
\(72\) 990.000i 1.62045i
\(73\) − 98.0000i − 0.157124i −0.996909 0.0785619i \(-0.974967\pi\)
0.996909 0.0785619i \(-0.0250328\pi\)
\(74\) 655.000 1.02895
\(75\) −532.000 −0.819068
\(76\) 2142.00i 3.23295i
\(77\) −338.000 −0.500243
\(78\) 0 0
\(79\) 1304.00 1.85711 0.928554 0.371198i \(-0.121053\pi\)
0.928554 + 0.371198i \(0.121053\pi\)
\(80\) − 623.000i − 0.870669i
\(81\) −839.000 −1.15089
\(82\) 1680.00 2.26250
\(83\) − 308.000i − 0.407318i −0.979042 0.203659i \(-0.934717\pi\)
0.979042 0.203659i \(-0.0652834\pi\)
\(84\) 1547.00i 2.00942i
\(85\) 539.000i 0.687797i
\(86\) − 1005.00i − 1.26014i
\(87\) 574.000 0.707348
\(88\) −1170.00 −1.41730
\(89\) 1190.00i 1.41730i 0.705560 + 0.708650i \(0.250696\pi\)
−0.705560 + 0.708650i \(0.749304\pi\)
\(90\) −770.000 −0.901835
\(91\) 0 0
\(92\) −1632.00 −1.84943
\(93\) − 1372.00i − 1.52978i
\(94\) 525.000 0.576060
\(95\) −882.000 −0.952540
\(96\) 595.000i 0.632572i
\(97\) 70.0000i 0.0732724i 0.999329 + 0.0366362i \(0.0116643\pi\)
−0.999329 + 0.0366362i \(0.988336\pi\)
\(98\) − 870.000i − 0.896768i
\(99\) 572.000i 0.580689i
\(100\) −1292.00 −1.29200
\(101\) −420.000 −0.413778 −0.206889 0.978364i \(-0.566334\pi\)
−0.206889 + 0.978364i \(0.566334\pi\)
\(102\) − 2695.00i − 2.61613i
\(103\) −588.000 −0.562499 −0.281249 0.959635i \(-0.590749\pi\)
−0.281249 + 0.959635i \(0.590749\pi\)
\(104\) 0 0
\(105\) −637.000 −0.592046
\(106\) 2160.00i 1.97922i
\(107\) −684.000 −0.617989 −0.308994 0.951064i \(-0.599992\pi\)
−0.308994 + 0.951064i \(0.599992\pi\)
\(108\) −595.000 −0.530129
\(109\) 373.000i 0.327770i 0.986479 + 0.163885i \(0.0524026\pi\)
−0.986479 + 0.163885i \(0.947597\pi\)
\(110\) − 910.000i − 0.788774i
\(111\) − 917.000i − 0.784124i
\(112\) 1157.00i 0.976127i
\(113\) −1734.00 −1.44355 −0.721774 0.692128i \(-0.756673\pi\)
−0.721774 + 0.692128i \(0.756673\pi\)
\(114\) 4410.00 3.62311
\(115\) − 672.000i − 0.544907i
\(116\) 1394.00 1.11577
\(117\) 0 0
\(118\) 1470.00 1.14682
\(119\) − 1001.00i − 0.771105i
\(120\) −2205.00 −1.67740
\(121\) 655.000 0.492111
\(122\) 280.000i 0.207787i
\(123\) − 2352.00i − 1.72417i
\(124\) − 3332.00i − 2.41308i
\(125\) − 1407.00i − 1.00677i
\(126\) 1430.00 1.01107
\(127\) −1892.00 −1.32195 −0.660976 0.750407i \(-0.729857\pi\)
−0.660976 + 0.750407i \(0.729857\pi\)
\(128\) − 2115.00i − 1.46048i
\(129\) −1407.00 −0.960306
\(130\) 0 0
\(131\) 1435.00 0.957073 0.478536 0.878068i \(-0.341167\pi\)
0.478536 + 0.878068i \(0.341167\pi\)
\(132\) 3094.00i 2.04014i
\(133\) 1638.00 1.06791
\(134\) 2390.00 1.54078
\(135\) − 245.000i − 0.156194i
\(136\) − 3465.00i − 2.18472i
\(137\) 1776.00i 1.10755i 0.832667 + 0.553773i \(0.186813\pi\)
−0.832667 + 0.553773i \(0.813187\pi\)
\(138\) 3360.00i 2.07262i
\(139\) −1869.00 −1.14048 −0.570239 0.821479i \(-0.693150\pi\)
−0.570239 + 0.821479i \(0.693150\pi\)
\(140\) −1547.00 −0.933895
\(141\) − 735.000i − 0.438994i
\(142\) 45.0000 0.0265938
\(143\) 0 0
\(144\) 1958.00 1.13310
\(145\) 574.000i 0.328746i
\(146\) −490.000 −0.277758
\(147\) −1218.00 −0.683394
\(148\) − 2227.00i − 1.23688i
\(149\) 2466.00i 1.35586i 0.735128 + 0.677928i \(0.237122\pi\)
−0.735128 + 0.677928i \(0.762878\pi\)
\(150\) 2660.00i 1.44792i
\(151\) 3323.00i 1.79087i 0.445189 + 0.895437i \(0.353137\pi\)
−0.445189 + 0.895437i \(0.646863\pi\)
\(152\) 5670.00 3.02564
\(153\) −1694.00 −0.895110
\(154\) 1690.00i 0.884312i
\(155\) 1372.00 0.710979
\(156\) 0 0
\(157\) −2730.00 −1.38776 −0.693878 0.720092i \(-0.744099\pi\)
−0.693878 + 0.720092i \(0.744099\pi\)
\(158\) − 6520.00i − 3.28293i
\(159\) 3024.00 1.50829
\(160\) −595.000 −0.293993
\(161\) 1248.00i 0.610908i
\(162\) 4195.00i 2.03451i
\(163\) 544.000i 0.261407i 0.991421 + 0.130704i \(0.0417236\pi\)
−0.991421 + 0.130704i \(0.958276\pi\)
\(164\) − 5712.00i − 2.71971i
\(165\) −1274.00 −0.601096
\(166\) −1540.00 −0.720043
\(167\) − 1624.00i − 0.752508i −0.926516 0.376254i \(-0.877212\pi\)
0.926516 0.376254i \(-0.122788\pi\)
\(168\) 4095.00 1.88057
\(169\) 0 0
\(170\) 2695.00 1.21587
\(171\) − 2772.00i − 1.23965i
\(172\) −3417.00 −1.51479
\(173\) 336.000 0.147662 0.0738312 0.997271i \(-0.476477\pi\)
0.0738312 + 0.997271i \(0.476477\pi\)
\(174\) − 2870.00i − 1.25043i
\(175\) 988.000i 0.426776i
\(176\) 2314.00i 0.991047i
\(177\) − 2058.00i − 0.873948i
\(178\) 5950.00 2.50546
\(179\) 3029.00 1.26479 0.632397 0.774645i \(-0.282071\pi\)
0.632397 + 0.774645i \(0.282071\pi\)
\(180\) 2618.00i 1.08408i
\(181\) 28.0000 0.0114985 0.00574924 0.999983i \(-0.498170\pi\)
0.00574924 + 0.999983i \(0.498170\pi\)
\(182\) 0 0
\(183\) 392.000 0.158347
\(184\) 4320.00i 1.73084i
\(185\) 917.000 0.364428
\(186\) −6860.00 −2.70430
\(187\) − 2002.00i − 0.782892i
\(188\) − 1785.00i − 0.692471i
\(189\) 455.000i 0.175113i
\(190\) 4410.00i 1.68387i
\(191\) 422.000 0.159868 0.0799342 0.996800i \(-0.474529\pi\)
0.0799342 + 0.996800i \(0.474529\pi\)
\(192\) −2009.00 −0.755141
\(193\) − 492.000i − 0.183497i −0.995782 0.0917485i \(-0.970754\pi\)
0.995782 0.0917485i \(-0.0292456\pi\)
\(194\) 350.000 0.129529
\(195\) 0 0
\(196\) −2958.00 −1.07799
\(197\) 2991.00i 1.08173i 0.841111 + 0.540863i \(0.181902\pi\)
−0.841111 + 0.540863i \(0.818098\pi\)
\(198\) 2860.00 1.02652
\(199\) 70.0000 0.0249355 0.0124678 0.999922i \(-0.496031\pi\)
0.0124678 + 0.999922i \(0.496031\pi\)
\(200\) 3420.00i 1.20915i
\(201\) − 3346.00i − 1.17417i
\(202\) 2100.00i 0.731463i
\(203\) − 1066.00i − 0.368564i
\(204\) −9163.00 −3.14480
\(205\) 2352.00 0.801321
\(206\) 2940.00i 0.994367i
\(207\) 2112.00 0.709150
\(208\) 0 0
\(209\) 3276.00 1.08424
\(210\) 3185.00i 1.04660i
\(211\) 2851.00 0.930194 0.465097 0.885260i \(-0.346019\pi\)
0.465097 + 0.885260i \(0.346019\pi\)
\(212\) 7344.00 2.37919
\(213\) − 63.0000i − 0.0202661i
\(214\) 3420.00i 1.09246i
\(215\) − 1407.00i − 0.446310i
\(216\) 1575.00i 0.496135i
\(217\) −2548.00 −0.797095
\(218\) 1865.00 0.579421
\(219\) 686.000i 0.211669i
\(220\) −3094.00 −0.948170
\(221\) 0 0
\(222\) −4585.00 −1.38615
\(223\) 217.000i 0.0651632i 0.999469 + 0.0325816i \(0.0103729\pi\)
−0.999469 + 0.0325816i \(0.989627\pi\)
\(224\) 1105.00 0.329602
\(225\) 1672.00 0.495407
\(226\) 8670.00i 2.55186i
\(227\) − 2576.00i − 0.753194i −0.926377 0.376597i \(-0.877094\pi\)
0.926377 0.376597i \(-0.122906\pi\)
\(228\) − 14994.0i − 4.35527i
\(229\) − 455.000i − 0.131298i −0.997843 0.0656490i \(-0.979088\pi\)
0.997843 0.0656490i \(-0.0209118\pi\)
\(230\) −3360.00 −0.963269
\(231\) 2366.00 0.673902
\(232\) − 3690.00i − 1.04423i
\(233\) −3061.00 −0.860656 −0.430328 0.902673i \(-0.641602\pi\)
−0.430328 + 0.902673i \(0.641602\pi\)
\(234\) 0 0
\(235\) 735.000 0.204026
\(236\) − 4998.00i − 1.37857i
\(237\) −9128.00 −2.50180
\(238\) −5005.00 −1.36313
\(239\) − 3477.00i − 0.941039i −0.882389 0.470520i \(-0.844066\pi\)
0.882389 0.470520i \(-0.155934\pi\)
\(240\) 4361.00i 1.17292i
\(241\) 1610.00i 0.430329i 0.976578 + 0.215164i \(0.0690287\pi\)
−0.976578 + 0.215164i \(0.930971\pi\)
\(242\) − 3275.00i − 0.869938i
\(243\) 4928.00 1.30095
\(244\) 952.000 0.249777
\(245\) − 1218.00i − 0.317613i
\(246\) −11760.0 −3.04793
\(247\) 0 0
\(248\) −8820.00 −2.25835
\(249\) 2156.00i 0.548719i
\(250\) −7035.00 −1.77973
\(251\) −1008.00 −0.253484 −0.126742 0.991936i \(-0.540452\pi\)
−0.126742 + 0.991936i \(0.540452\pi\)
\(252\) − 4862.00i − 1.21539i
\(253\) 2496.00i 0.620246i
\(254\) 9460.00i 2.33690i
\(255\) − 3773.00i − 0.926566i
\(256\) −8279.00 −2.02124
\(257\) −6041.00 −1.46625 −0.733127 0.680092i \(-0.761940\pi\)
−0.733127 + 0.680092i \(0.761940\pi\)
\(258\) 7035.00i 1.69760i
\(259\) −1703.00 −0.408569
\(260\) 0 0
\(261\) −1804.00 −0.427834
\(262\) − 7175.00i − 1.69188i
\(263\) −3708.00 −0.869373 −0.434686 0.900582i \(-0.643141\pi\)
−0.434686 + 0.900582i \(0.643141\pi\)
\(264\) 8190.00 1.90932
\(265\) 3024.00i 0.700992i
\(266\) − 8190.00i − 1.88782i
\(267\) − 8330.00i − 1.90932i
\(268\) − 8126.00i − 1.85214i
\(269\) 8344.00 1.89124 0.945618 0.325278i \(-0.105458\pi\)
0.945618 + 0.325278i \(0.105458\pi\)
\(270\) −1225.00 −0.276115
\(271\) 1617.00i 0.362457i 0.983441 + 0.181228i \(0.0580073\pi\)
−0.983441 + 0.181228i \(0.941993\pi\)
\(272\) −6853.00 −1.52766
\(273\) 0 0
\(274\) 8880.00 1.95788
\(275\) 1976.00i 0.433299i
\(276\) 11424.0 2.49146
\(277\) 3820.00 0.828598 0.414299 0.910141i \(-0.364027\pi\)
0.414299 + 0.910141i \(0.364027\pi\)
\(278\) 9345.00i 2.01610i
\(279\) 4312.00i 0.925278i
\(280\) 4095.00i 0.874011i
\(281\) 6214.00i 1.31920i 0.751615 + 0.659602i \(0.229275\pi\)
−0.751615 + 0.659602i \(0.770725\pi\)
\(282\) −3675.00 −0.776039
\(283\) 5292.00 1.11158 0.555789 0.831323i \(-0.312416\pi\)
0.555789 + 0.831323i \(0.312416\pi\)
\(284\) − 153.000i − 0.0319679i
\(285\) 6174.00 1.28321
\(286\) 0 0
\(287\) −4368.00 −0.898379
\(288\) − 1870.00i − 0.382607i
\(289\) 1016.00 0.206798
\(290\) 2870.00 0.581146
\(291\) − 490.000i − 0.0987090i
\(292\) 1666.00i 0.333888i
\(293\) 903.000i 0.180047i 0.995940 + 0.0900236i \(0.0286942\pi\)
−0.995940 + 0.0900236i \(0.971306\pi\)
\(294\) 6090.00i 1.20808i
\(295\) 2058.00 0.406174
\(296\) −5895.00 −1.15757
\(297\) 910.000i 0.177790i
\(298\) 12330.0 2.39684
\(299\) 0 0
\(300\) 9044.00 1.74052
\(301\) 2613.00i 0.500368i
\(302\) 16615.0 3.16585
\(303\) 2940.00 0.557421
\(304\) − 11214.0i − 2.11568i
\(305\) 392.000i 0.0735930i
\(306\) 8470.00i 1.58235i
\(307\) − 2114.00i − 0.393004i −0.980503 0.196502i \(-0.937042\pi\)
0.980503 0.196502i \(-0.0629583\pi\)
\(308\) 5746.00 1.06302
\(309\) 4116.00 0.757770
\(310\) − 6860.00i − 1.25684i
\(311\) −3402.00 −0.620288 −0.310144 0.950690i \(-0.600377\pi\)
−0.310144 + 0.950690i \(0.600377\pi\)
\(312\) 0 0
\(313\) −10689.0 −1.93028 −0.965141 0.261732i \(-0.915706\pi\)
−0.965141 + 0.261732i \(0.915706\pi\)
\(314\) 13650.0i 2.45323i
\(315\) 2002.00 0.358095
\(316\) −22168.0 −3.94635
\(317\) − 7054.00i − 1.24982i −0.780698 0.624909i \(-0.785136\pi\)
0.780698 0.624909i \(-0.214864\pi\)
\(318\) − 15120.0i − 2.66631i
\(319\) − 2132.00i − 0.374198i
\(320\) − 2009.00i − 0.350958i
\(321\) 4788.00 0.832524
\(322\) 6240.00 1.07994
\(323\) 9702.00i 1.67131i
\(324\) 14263.0 2.44564
\(325\) 0 0
\(326\) 2720.00 0.462107
\(327\) − 2611.00i − 0.441555i
\(328\) −15120.0 −2.54531
\(329\) −1365.00 −0.228738
\(330\) 6370.00i 1.06260i
\(331\) 9704.00i 1.61142i 0.592310 + 0.805710i \(0.298216\pi\)
−0.592310 + 0.805710i \(0.701784\pi\)
\(332\) 5236.00i 0.865551i
\(333\) 2882.00i 0.474272i
\(334\) −8120.00 −1.33026
\(335\) 3346.00 0.545706
\(336\) − 8099.00i − 1.31499i
\(337\) 10449.0 1.68900 0.844500 0.535555i \(-0.179897\pi\)
0.844500 + 0.535555i \(0.179897\pi\)
\(338\) 0 0
\(339\) 12138.0 1.94468
\(340\) − 9163.00i − 1.46157i
\(341\) −5096.00 −0.809278
\(342\) −13860.0 −2.19141
\(343\) 6721.00i 1.05802i
\(344\) 9045.00i 1.41766i
\(345\) 4704.00i 0.734072i
\(346\) − 1680.00i − 0.261033i
\(347\) −621.000 −0.0960721 −0.0480361 0.998846i \(-0.515296\pi\)
−0.0480361 + 0.998846i \(0.515296\pi\)
\(348\) −9758.00 −1.50311
\(349\) − 12481.0i − 1.91431i −0.289584 0.957153i \(-0.593517\pi\)
0.289584 0.957153i \(-0.406483\pi\)
\(350\) 4940.00 0.754440
\(351\) 0 0
\(352\) 2210.00 0.334640
\(353\) − 1400.00i − 0.211089i −0.994415 0.105545i \(-0.966341\pi\)
0.994415 0.105545i \(-0.0336586\pi\)
\(354\) −10290.0 −1.54494
\(355\) 63.0000 0.00941885
\(356\) − 20230.0i − 3.01176i
\(357\) 7007.00i 1.03879i
\(358\) − 15145.0i − 2.23586i
\(359\) 4968.00i 0.730365i 0.930936 + 0.365182i \(0.118993\pi\)
−0.930936 + 0.365182i \(0.881007\pi\)
\(360\) 6930.00 1.01456
\(361\) −9017.00 −1.31462
\(362\) − 140.000i − 0.0203266i
\(363\) −4585.00 −0.662948
\(364\) 0 0
\(365\) −686.000 −0.0983750
\(366\) − 1960.00i − 0.279920i
\(367\) 8722.00 1.24056 0.620279 0.784381i \(-0.287019\pi\)
0.620279 + 0.784381i \(0.287019\pi\)
\(368\) 8544.00 1.21029
\(369\) 7392.00i 1.04285i
\(370\) − 4585.00i − 0.644224i
\(371\) − 5616.00i − 0.785898i
\(372\) 23324.0i 3.25079i
\(373\) 10012.0 1.38982 0.694908 0.719098i \(-0.255445\pi\)
0.694908 + 0.719098i \(0.255445\pi\)
\(374\) −10010.0 −1.38397
\(375\) 9849.00i 1.35627i
\(376\) −4725.00 −0.648067
\(377\) 0 0
\(378\) 2275.00 0.309559
\(379\) − 3372.00i − 0.457013i −0.973542 0.228507i \(-0.926616\pi\)
0.973542 0.228507i \(-0.0733843\pi\)
\(380\) 14994.0 2.02415
\(381\) 13244.0 1.78087
\(382\) − 2110.00i − 0.282610i
\(383\) − 847.000i − 0.113002i −0.998403 0.0565009i \(-0.982006\pi\)
0.998403 0.0565009i \(-0.0179944\pi\)
\(384\) 14805.0i 1.96749i
\(385\) 2366.00i 0.313201i
\(386\) −2460.00 −0.324380
\(387\) 4422.00 0.580834
\(388\) − 1190.00i − 0.155704i
\(389\) −11314.0 −1.47466 −0.737330 0.675533i \(-0.763914\pi\)
−0.737330 + 0.675533i \(0.763914\pi\)
\(390\) 0 0
\(391\) −7392.00 −0.956086
\(392\) 7830.00i 1.00886i
\(393\) −10045.0 −1.28932
\(394\) 14955.0 1.91224
\(395\) − 9128.00i − 1.16273i
\(396\) − 9724.00i − 1.23396i
\(397\) − 1862.00i − 0.235393i −0.993050 0.117697i \(-0.962449\pi\)
0.993050 0.117697i \(-0.0375510\pi\)
\(398\) − 350.000i − 0.0440802i
\(399\) −11466.0 −1.43864
\(400\) 6764.00 0.845500
\(401\) − 6820.00i − 0.849313i −0.905355 0.424657i \(-0.860395\pi\)
0.905355 0.424657i \(-0.139605\pi\)
\(402\) −16730.0 −2.07566
\(403\) 0 0
\(404\) 7140.00 0.879278
\(405\) 5873.00i 0.720572i
\(406\) −5330.00 −0.651536
\(407\) −3406.00 −0.414814
\(408\) 24255.0i 2.94314i
\(409\) − 12992.0i − 1.57069i −0.619057 0.785346i \(-0.712485\pi\)
0.619057 0.785346i \(-0.287515\pi\)
\(410\) − 11760.0i − 1.41655i
\(411\) − 12432.0i − 1.49203i
\(412\) 9996.00 1.19531
\(413\) −3822.00 −0.455371
\(414\) − 10560.0i − 1.25361i
\(415\) −2156.00 −0.255021
\(416\) 0 0
\(417\) 13083.0 1.53640
\(418\) − 16380.0i − 1.91668i
\(419\) −7343.00 −0.856155 −0.428078 0.903742i \(-0.640809\pi\)
−0.428078 + 0.903742i \(0.640809\pi\)
\(420\) 10829.0 1.25810
\(421\) − 5059.00i − 0.585655i −0.956165 0.292827i \(-0.905404\pi\)
0.956165 0.292827i \(-0.0945961\pi\)
\(422\) − 14255.0i − 1.64437i
\(423\) 2310.00i 0.265523i
\(424\) − 19440.0i − 2.22663i
\(425\) −5852.00 −0.667915
\(426\) −315.000 −0.0358258
\(427\) − 728.000i − 0.0825068i
\(428\) 11628.0 1.31323
\(429\) 0 0
\(430\) −7035.00 −0.788972
\(431\) 3243.00i 0.362436i 0.983443 + 0.181218i \(0.0580039\pi\)
−0.983443 + 0.181218i \(0.941996\pi\)
\(432\) 3115.00 0.346922
\(433\) −11599.0 −1.28733 −0.643663 0.765309i \(-0.722586\pi\)
−0.643663 + 0.765309i \(0.722586\pi\)
\(434\) 12740.0i 1.40908i
\(435\) − 4018.00i − 0.442870i
\(436\) − 6341.00i − 0.696511i
\(437\) − 12096.0i − 1.32410i
\(438\) 3430.00 0.374182
\(439\) 17374.0 1.88887 0.944437 0.328692i \(-0.106608\pi\)
0.944437 + 0.328692i \(0.106608\pi\)
\(440\) 8190.00i 0.887370i
\(441\) 3828.00 0.413346
\(442\) 0 0
\(443\) 989.000 0.106070 0.0530348 0.998593i \(-0.483111\pi\)
0.0530348 + 0.998593i \(0.483111\pi\)
\(444\) 15589.0i 1.66626i
\(445\) 8330.00 0.887370
\(446\) 1085.00 0.115193
\(447\) − 17262.0i − 1.82654i
\(448\) 3731.00i 0.393467i
\(449\) 14474.0i 1.52131i 0.649154 + 0.760657i \(0.275123\pi\)
−0.649154 + 0.760657i \(0.724877\pi\)
\(450\) − 8360.00i − 0.875765i
\(451\) −8736.00 −0.912111
\(452\) 29478.0 3.06754
\(453\) − 23261.0i − 2.41258i
\(454\) −12880.0 −1.33147
\(455\) 0 0
\(456\) −39690.0 −4.07600
\(457\) − 1594.00i − 0.163160i −0.996667 0.0815801i \(-0.974003\pi\)
0.996667 0.0815801i \(-0.0259966\pi\)
\(458\) −2275.00 −0.232104
\(459\) −2695.00 −0.274056
\(460\) 11424.0i 1.15793i
\(461\) − 5915.00i − 0.597590i −0.954317 0.298795i \(-0.903415\pi\)
0.954317 0.298795i \(-0.0965847\pi\)
\(462\) − 11830.0i − 1.19130i
\(463\) 11072.0i 1.11136i 0.831396 + 0.555680i \(0.187542\pi\)
−0.831396 + 0.555680i \(0.812458\pi\)
\(464\) −7298.00 −0.730175
\(465\) −9604.00 −0.957795
\(466\) 15305.0i 1.52144i
\(467\) −1260.00 −0.124852 −0.0624260 0.998050i \(-0.519884\pi\)
−0.0624260 + 0.998050i \(0.519884\pi\)
\(468\) 0 0
\(469\) −6214.00 −0.611804
\(470\) − 3675.00i − 0.360670i
\(471\) 19110.0 1.86952
\(472\) −13230.0 −1.29017
\(473\) 5226.00i 0.508016i
\(474\) 45640.0i 4.42260i
\(475\) − 9576.00i − 0.925004i
\(476\) 17017.0i 1.63860i
\(477\) −9504.00 −0.912281
\(478\) −17385.0 −1.66354
\(479\) 12033.0i 1.14781i 0.818921 + 0.573906i \(0.194572\pi\)
−0.818921 + 0.573906i \(0.805428\pi\)
\(480\) 4165.00 0.396053
\(481\) 0 0
\(482\) 8050.00 0.760721
\(483\) − 8736.00i − 0.822985i
\(484\) −11135.0 −1.04574
\(485\) 490.000 0.0458758
\(486\) − 24640.0i − 2.29978i
\(487\) − 2280.00i − 0.212149i −0.994358 0.106075i \(-0.966172\pi\)
0.994358 0.106075i \(-0.0338282\pi\)
\(488\) − 2520.00i − 0.233760i
\(489\) − 3808.00i − 0.352155i
\(490\) −6090.00 −0.561466
\(491\) −16767.0 −1.54111 −0.770554 0.637375i \(-0.780020\pi\)
−0.770554 + 0.637375i \(0.780020\pi\)
\(492\) 39984.0i 3.66386i
\(493\) 6314.00 0.576812
\(494\) 0 0
\(495\) 4004.00 0.363569
\(496\) 17444.0i 1.57915i
\(497\) −117.000 −0.0105597
\(498\) 10780.0 0.970007
\(499\) 12840.0i 1.15190i 0.817485 + 0.575949i \(0.195367\pi\)
−0.817485 + 0.575949i \(0.804633\pi\)
\(500\) 23919.0i 2.13938i
\(501\) 11368.0i 1.01374i
\(502\) 5040.00i 0.448100i
\(503\) −2198.00 −0.194839 −0.0974195 0.995243i \(-0.531059\pi\)
−0.0974195 + 0.995243i \(0.531059\pi\)
\(504\) −12870.0 −1.13745
\(505\) 2940.00i 0.259066i
\(506\) 12480.0 1.09645
\(507\) 0 0
\(508\) 32164.0 2.80915
\(509\) − 17066.0i − 1.48612i −0.669223 0.743062i \(-0.733373\pi\)
0.669223 0.743062i \(-0.266627\pi\)
\(510\) −18865.0 −1.63795
\(511\) 1274.00 0.110290
\(512\) 24475.0i 2.11260i
\(513\) − 4410.00i − 0.379544i
\(514\) 30205.0i 2.59200i
\(515\) 4116.00i 0.352180i
\(516\) 23919.0 2.04065
\(517\) −2730.00 −0.232235
\(518\) 8515.00i 0.722254i
\(519\) −2352.00 −0.198924
\(520\) 0 0
\(521\) 2583.00 0.217204 0.108602 0.994085i \(-0.465363\pi\)
0.108602 + 0.994085i \(0.465363\pi\)
\(522\) 9020.00i 0.756312i
\(523\) 18620.0 1.55678 0.778390 0.627781i \(-0.216037\pi\)
0.778390 + 0.627781i \(0.216037\pi\)
\(524\) −24395.0 −2.03378
\(525\) − 6916.00i − 0.574931i
\(526\) 18540.0i 1.53685i
\(527\) − 15092.0i − 1.24747i
\(528\) − 16198.0i − 1.33509i
\(529\) −2951.00 −0.242541
\(530\) 15120.0 1.23919
\(531\) 6468.00i 0.528601i
\(532\) −27846.0 −2.26932
\(533\) 0 0
\(534\) −41650.0 −3.37523
\(535\) 4788.00i 0.386922i
\(536\) −21510.0 −1.73338
\(537\) −21203.0 −1.70387
\(538\) − 41720.0i − 3.34327i
\(539\) 4524.00i 0.361526i
\(540\) 4165.00i 0.331913i
\(541\) 16833.0i 1.33772i 0.743388 + 0.668861i \(0.233218\pi\)
−0.743388 + 0.668861i \(0.766782\pi\)
\(542\) 8085.00 0.640739
\(543\) −196.000 −0.0154902
\(544\) 6545.00i 0.515836i
\(545\) 2611.00 0.205216
\(546\) 0 0
\(547\) −8615.00 −0.673402 −0.336701 0.941612i \(-0.609311\pi\)
−0.336701 + 0.941612i \(0.609311\pi\)
\(548\) − 30192.0i − 2.35354i
\(549\) −1232.00 −0.0957750
\(550\) 9880.00 0.765972
\(551\) 10332.0i 0.798835i
\(552\) − 30240.0i − 2.33170i
\(553\) 16952.0i 1.30357i
\(554\) − 19100.0i − 1.46477i
\(555\) −6419.00 −0.490939
\(556\) 31773.0 2.42352
\(557\) − 8535.00i − 0.649263i −0.945841 0.324632i \(-0.894760\pi\)
0.945841 0.324632i \(-0.105240\pi\)
\(558\) 21560.0 1.63568
\(559\) 0 0
\(560\) 8099.00 0.611152
\(561\) 14014.0i 1.05467i
\(562\) 31070.0 2.33204
\(563\) 4641.00 0.347415 0.173708 0.984797i \(-0.444425\pi\)
0.173708 + 0.984797i \(0.444425\pi\)
\(564\) 12495.0i 0.932862i
\(565\) 12138.0i 0.903804i
\(566\) − 26460.0i − 1.96501i
\(567\) − 10907.0i − 0.807850i
\(568\) −405.000 −0.0299180
\(569\) 4793.00 0.353134 0.176567 0.984289i \(-0.443501\pi\)
0.176567 + 0.984289i \(0.443501\pi\)
\(570\) − 30870.0i − 2.26842i
\(571\) 5563.00 0.407713 0.203857 0.979001i \(-0.434652\pi\)
0.203857 + 0.979001i \(0.434652\pi\)
\(572\) 0 0
\(573\) −2954.00 −0.215367
\(574\) 21840.0i 1.58813i
\(575\) 7296.00 0.529155
\(576\) 6314.00 0.456742
\(577\) 24038.0i 1.73434i 0.498011 + 0.867171i \(0.334064\pi\)
−0.498011 + 0.867171i \(0.665936\pi\)
\(578\) − 5080.00i − 0.365571i
\(579\) 3444.00i 0.247198i
\(580\) − 9758.00i − 0.698584i
\(581\) 4004.00 0.285910
\(582\) −2450.00 −0.174494
\(583\) − 11232.0i − 0.797911i
\(584\) 4410.00 0.312478
\(585\) 0 0
\(586\) 4515.00 0.318281
\(587\) − 21224.0i − 1.49235i −0.665751 0.746174i \(-0.731889\pi\)
0.665751 0.746174i \(-0.268111\pi\)
\(588\) 20706.0 1.45221
\(589\) 24696.0 1.72764
\(590\) − 10290.0i − 0.718021i
\(591\) − 20937.0i − 1.45725i
\(592\) 11659.0i 0.809429i
\(593\) − 4354.00i − 0.301513i −0.988571 0.150757i \(-0.951829\pi\)
0.988571 0.150757i \(-0.0481710\pi\)
\(594\) 4550.00 0.314291
\(595\) −7007.00 −0.482788
\(596\) − 41922.0i − 2.88119i
\(597\) −490.000 −0.0335919
\(598\) 0 0
\(599\) 7310.00 0.498629 0.249314 0.968423i \(-0.419795\pi\)
0.249314 + 0.968423i \(0.419795\pi\)
\(600\) − 23940.0i − 1.62891i
\(601\) −7595.00 −0.515485 −0.257743 0.966214i \(-0.582979\pi\)
−0.257743 + 0.966214i \(0.582979\pi\)
\(602\) 13065.0 0.884534
\(603\) 10516.0i 0.710190i
\(604\) − 56491.0i − 3.80561i
\(605\) − 4585.00i − 0.308110i
\(606\) − 14700.0i − 0.985391i
\(607\) −826.000 −0.0552328 −0.0276164 0.999619i \(-0.508792\pi\)
−0.0276164 + 0.999619i \(0.508792\pi\)
\(608\) −10710.0 −0.714388
\(609\) 7462.00i 0.496511i
\(610\) 1960.00 0.130095
\(611\) 0 0
\(612\) 28798.0 1.90211
\(613\) 14590.0i 0.961312i 0.876909 + 0.480656i \(0.159602\pi\)
−0.876909 + 0.480656i \(0.840398\pi\)
\(614\) −10570.0 −0.694740
\(615\) −16464.0 −1.07950
\(616\) − 15210.0i − 0.994851i
\(617\) 4888.00i 0.318936i 0.987203 + 0.159468i \(0.0509779\pi\)
−0.987203 + 0.159468i \(0.949022\pi\)
\(618\) − 20580.0i − 1.33956i
\(619\) 11004.0i 0.714520i 0.934005 + 0.357260i \(0.116289\pi\)
−0.934005 + 0.357260i \(0.883711\pi\)
\(620\) −23324.0 −1.51083
\(621\) 3360.00 0.217121
\(622\) 17010.0i 1.09653i
\(623\) −15470.0 −0.994851
\(624\) 0 0
\(625\) −349.000 −0.0223360
\(626\) 53445.0i 3.41229i
\(627\) −22932.0 −1.46063
\(628\) 46410.0 2.94898
\(629\) − 10087.0i − 0.639420i
\(630\) − 10010.0i − 0.633028i
\(631\) 4975.00i 0.313869i 0.987609 + 0.156935i \(0.0501612\pi\)
−0.987609 + 0.156935i \(0.949839\pi\)
\(632\) 58680.0i 3.69330i
\(633\) −19957.0 −1.25311
\(634\) −35270.0 −2.20939
\(635\) 13244.0i 0.827673i
\(636\) −51408.0 −3.20513
\(637\) 0 0
\(638\) −10660.0 −0.661494
\(639\) 198.000i 0.0122578i
\(640\) −14805.0 −0.914405
\(641\) −3950.00 −0.243394 −0.121697 0.992567i \(-0.538834\pi\)
−0.121697 + 0.992567i \(0.538834\pi\)
\(642\) − 23940.0i − 1.47171i
\(643\) − 3682.00i − 0.225823i −0.993605 0.112911i \(-0.963982\pi\)
0.993605 0.112911i \(-0.0360176\pi\)
\(644\) − 21216.0i − 1.29818i
\(645\) 9849.00i 0.601247i
\(646\) 48510.0 2.95449
\(647\) −10402.0 −0.632063 −0.316032 0.948749i \(-0.602351\pi\)
−0.316032 + 0.948749i \(0.602351\pi\)
\(648\) − 37755.0i − 2.28882i
\(649\) −7644.00 −0.462332
\(650\) 0 0
\(651\) 17836.0 1.07381
\(652\) − 9248.00i − 0.555490i
\(653\) −31680.0 −1.89852 −0.949260 0.314491i \(-0.898166\pi\)
−0.949260 + 0.314491i \(0.898166\pi\)
\(654\) −13055.0 −0.780567
\(655\) − 10045.0i − 0.599222i
\(656\) 29904.0i 1.77981i
\(657\) − 2156.00i − 0.128027i
\(658\) 6825.00i 0.404356i
\(659\) 21940.0 1.29691 0.648453 0.761255i \(-0.275416\pi\)
0.648453 + 0.761255i \(0.275416\pi\)
\(660\) 21658.0 1.27733
\(661\) 31374.0i 1.84615i 0.384616 + 0.923077i \(0.374334\pi\)
−0.384616 + 0.923077i \(0.625666\pi\)
\(662\) 48520.0 2.84862
\(663\) 0 0
\(664\) 13860.0 0.810049
\(665\) − 11466.0i − 0.668620i
\(666\) 14410.0 0.838403
\(667\) −7872.00 −0.456979
\(668\) 27608.0i 1.59908i
\(669\) − 1519.00i − 0.0877847i
\(670\) − 16730.0i − 0.964681i
\(671\) − 1456.00i − 0.0837679i
\(672\) −7735.00 −0.444024
\(673\) −18013.0 −1.03172 −0.515862 0.856672i \(-0.672528\pi\)
−0.515862 + 0.856672i \(0.672528\pi\)
\(674\) − 52245.0i − 2.98576i
\(675\) 2660.00 0.151679
\(676\) 0 0
\(677\) −10640.0 −0.604030 −0.302015 0.953303i \(-0.597659\pi\)
−0.302015 + 0.953303i \(0.597659\pi\)
\(678\) − 60690.0i − 3.43774i
\(679\) −910.000 −0.0514324
\(680\) −24255.0 −1.36785
\(681\) 18032.0i 1.01467i
\(682\) 25480.0i 1.43062i
\(683\) 9336.00i 0.523034i 0.965199 + 0.261517i \(0.0842227\pi\)
−0.965199 + 0.261517i \(0.915777\pi\)
\(684\) 47124.0i 2.63426i
\(685\) 12432.0 0.693434
\(686\) 33605.0 1.87033
\(687\) 3185.00i 0.176878i
\(688\) 17889.0 0.991296
\(689\) 0 0
\(690\) 23520.0 1.29767
\(691\) 4200.00i 0.231224i 0.993294 + 0.115612i \(0.0368829\pi\)
−0.993294 + 0.115612i \(0.963117\pi\)
\(692\) −5712.00 −0.313783
\(693\) −7436.00 −0.407605
\(694\) 3105.00i 0.169833i
\(695\) 13083.0i 0.714052i
\(696\) 25830.0i 1.40673i
\(697\) − 25872.0i − 1.40599i
\(698\) −62405.0 −3.38405
\(699\) 21427.0 1.15943
\(700\) − 16796.0i − 0.906899i
\(701\) −9872.00 −0.531898 −0.265949 0.963987i \(-0.585685\pi\)
−0.265949 + 0.963987i \(0.585685\pi\)
\(702\) 0 0
\(703\) 16506.0 0.885541
\(704\) 7462.00i 0.399481i
\(705\) −5145.00 −0.274854
\(706\) −7000.00 −0.373156
\(707\) − 5460.00i − 0.290445i
\(708\) 34986.0i 1.85714i
\(709\) − 28450.0i − 1.50700i −0.657449 0.753499i \(-0.728364\pi\)
0.657449 0.753499i \(-0.271636\pi\)
\(710\) − 315.000i − 0.0166503i
\(711\) 28688.0 1.51320
\(712\) −53550.0 −2.81864
\(713\) 18816.0i 0.988310i
\(714\) 35035.0 1.83635
\(715\) 0 0
\(716\) −51493.0 −2.68769
\(717\) 24339.0i 1.26772i
\(718\) 24840.0 1.29111
\(719\) −32718.0 −1.69705 −0.848523 0.529159i \(-0.822507\pi\)
−0.848523 + 0.529159i \(0.822507\pi\)
\(720\) − 13706.0i − 0.709434i
\(721\) − 7644.00i − 0.394837i
\(722\) 45085.0i 2.32395i
\(723\) − 11270.0i − 0.579718i
\(724\) −476.000 −0.0244343
\(725\) −6232.00 −0.319242
\(726\) 22925.0i 1.17194i
\(727\) 22834.0 1.16488 0.582439 0.812874i \(-0.302099\pi\)
0.582439 + 0.812874i \(0.302099\pi\)
\(728\) 0 0
\(729\) −11843.0 −0.601687
\(730\) 3430.00i 0.173904i
\(731\) −15477.0 −0.783088
\(732\) −6664.00 −0.336487
\(733\) 7875.00i 0.396821i 0.980119 + 0.198410i \(0.0635779\pi\)
−0.980119 + 0.198410i \(0.936422\pi\)
\(734\) − 43610.0i − 2.19302i
\(735\) 8526.00i 0.427872i
\(736\) − 8160.00i − 0.408671i
\(737\) −12428.0 −0.621155
\(738\) 36960.0 1.84352
\(739\) 2140.00i 0.106524i 0.998581 + 0.0532620i \(0.0169618\pi\)
−0.998581 + 0.0532620i \(0.983038\pi\)
\(740\) −15589.0 −0.774410
\(741\) 0 0
\(742\) −28080.0 −1.38928
\(743\) 31971.0i 1.57860i 0.614006 + 0.789302i \(0.289557\pi\)
−0.614006 + 0.789302i \(0.710443\pi\)
\(744\) 61740.0 3.04234
\(745\) 17262.0 0.848900
\(746\) − 50060.0i − 2.45687i
\(747\) − 6776.00i − 0.331889i
\(748\) 34034.0i 1.66364i
\(749\) − 8892.00i − 0.433787i
\(750\) 49245.0 2.39756
\(751\) 7432.00 0.361115 0.180558 0.983564i \(-0.442210\pi\)
0.180558 + 0.983564i \(0.442210\pi\)
\(752\) 9345.00i 0.453161i
\(753\) 7056.00 0.341481
\(754\) 0 0
\(755\) 23261.0 1.12126
\(756\) − 7735.00i − 0.372115i
\(757\) 20176.0 0.968704 0.484352 0.874873i \(-0.339055\pi\)
0.484352 + 0.874873i \(0.339055\pi\)
\(758\) −16860.0 −0.807893
\(759\) − 17472.0i − 0.835564i
\(760\) − 39690.0i − 1.89435i
\(761\) 9478.00i 0.451481i 0.974187 + 0.225741i \(0.0724802\pi\)
−0.974187 + 0.225741i \(0.927520\pi\)
\(762\) − 66220.0i − 3.14816i
\(763\) −4849.00 −0.230073
\(764\) −7174.00 −0.339720
\(765\) 11858.0i 0.560427i
\(766\) −4235.00 −0.199761
\(767\) 0 0
\(768\) 57953.0 2.72292
\(769\) − 12096.0i − 0.567221i −0.958940 0.283610i \(-0.908468\pi\)
0.958940 0.283610i \(-0.0915323\pi\)
\(770\) 11830.0 0.553667
\(771\) 42287.0 1.97526
\(772\) 8364.00i 0.389931i
\(773\) 17941.0i 0.834790i 0.908725 + 0.417395i \(0.137057\pi\)
−0.908725 + 0.417395i \(0.862943\pi\)
\(774\) − 22110.0i − 1.02678i
\(775\) 14896.0i 0.690426i
\(776\) −3150.00 −0.145720
\(777\) 11921.0 0.550403
\(778\) 56570.0i 2.60685i
\(779\) 42336.0 1.94717
\(780\) 0 0
\(781\) −234.000 −0.0107211
\(782\) 36960.0i 1.69014i
\(783\) −2870.00 −0.130990
\(784\) 15486.0 0.705448
\(785\) 19110.0i 0.868873i
\(786\) 50225.0i 2.27922i
\(787\) − 6664.00i − 0.301837i −0.988546 0.150919i \(-0.951777\pi\)
0.988546 0.150919i \(-0.0482232\pi\)
\(788\) − 50847.0i − 2.29867i
\(789\) 25956.0 1.17118
\(790\) −45640.0 −2.05544
\(791\) − 22542.0i − 1.01328i
\(792\) −25740.0 −1.15484
\(793\) 0 0
\(794\) −9310.00 −0.416120
\(795\) − 21168.0i − 0.944342i
\(796\) −1190.00 −0.0529880
\(797\) 1442.00 0.0640882 0.0320441 0.999486i \(-0.489798\pi\)
0.0320441 + 0.999486i \(0.489798\pi\)
\(798\) 57330.0i 2.54318i
\(799\) − 8085.00i − 0.357981i
\(800\) − 6460.00i − 0.285494i
\(801\) 26180.0i 1.15484i
\(802\) −34100.0 −1.50139
\(803\) 2548.00 0.111976
\(804\) 56882.0i 2.49512i
\(805\) 8736.00 0.382489
\(806\) 0 0
\(807\) −58408.0 −2.54778
\(808\) − 18900.0i − 0.822896i
\(809\) 30207.0 1.31276 0.656379 0.754431i \(-0.272087\pi\)
0.656379 + 0.754431i \(0.272087\pi\)
\(810\) 29365.0 1.27380
\(811\) 21140.0i 0.915322i 0.889127 + 0.457661i \(0.151313\pi\)
−0.889127 + 0.457661i \(0.848687\pi\)
\(812\) 18122.0i 0.783199i
\(813\) − 11319.0i − 0.488284i
\(814\) 17030.0i 0.733294i
\(815\) 3808.00 0.163667
\(816\) 47971.0 2.05799
\(817\) − 25326.0i − 1.08451i
\(818\) −64960.0 −2.77662
\(819\) 0 0
\(820\) −39984.0 −1.70281
\(821\) 569.000i 0.0241879i 0.999927 + 0.0120939i \(0.00384971\pi\)
−0.999927 + 0.0120939i \(0.996150\pi\)
\(822\) −62160.0 −2.63757
\(823\) 8538.00 0.361623 0.180812 0.983518i \(-0.442128\pi\)
0.180812 + 0.983518i \(0.442128\pi\)
\(824\) − 26460.0i − 1.11866i
\(825\) − 13832.0i − 0.583719i
\(826\) 19110.0i 0.804990i
\(827\) 32702.0i 1.37504i 0.726164 + 0.687521i \(0.241301\pi\)
−0.726164 + 0.687521i \(0.758699\pi\)
\(828\) −35904.0 −1.50694
\(829\) 21154.0 0.886259 0.443130 0.896458i \(-0.353868\pi\)
0.443130 + 0.896458i \(0.353868\pi\)
\(830\) 10780.0i 0.450818i
\(831\) −26740.0 −1.11625
\(832\) 0 0
\(833\) −13398.0 −0.557279
\(834\) − 65415.0i − 2.71599i
\(835\) −11368.0 −0.471145
\(836\) −55692.0 −2.30400
\(837\) 6860.00i 0.283293i
\(838\) 36715.0i 1.51348i
\(839\) 2184.00i 0.0898690i 0.998990 + 0.0449345i \(0.0143079\pi\)
−0.998990 + 0.0449345i \(0.985692\pi\)
\(840\) − 28665.0i − 1.17742i
\(841\) −17665.0 −0.724302
\(842\) −25295.0 −1.03530
\(843\) − 43498.0i − 1.77717i
\(844\) −48467.0 −1.97666
\(845\) 0 0
\(846\) 11550.0 0.469382
\(847\) 8515.00i 0.345430i
\(848\) −38448.0 −1.55697
\(849\) −37044.0 −1.49746
\(850\) 29260.0i 1.18072i
\(851\) 12576.0i 0.506580i
\(852\) 1071.00i 0.0430656i
\(853\) − 36687.0i − 1.47261i −0.676648 0.736307i \(-0.736568\pi\)
0.676648 0.736307i \(-0.263432\pi\)
\(854\) −3640.00 −0.145853
\(855\) −19404.0 −0.776144
\(856\) − 30780.0i − 1.22902i
\(857\) −36806.0 −1.46706 −0.733529 0.679658i \(-0.762128\pi\)
−0.733529 + 0.679658i \(0.762128\pi\)
\(858\) 0 0
\(859\) 4900.00 0.194628 0.0973142 0.995254i \(-0.468975\pi\)
0.0973142 + 0.995254i \(0.468975\pi\)
\(860\) 23919.0i 0.948408i
\(861\) 30576.0 1.21025
\(862\) 16215.0 0.640702
\(863\) − 13697.0i − 0.540268i −0.962823 0.270134i \(-0.912932\pi\)
0.962823 0.270134i \(-0.0870680\pi\)
\(864\) − 2975.00i − 0.117143i
\(865\) − 2352.00i − 0.0924513i
\(866\) 57995.0i 2.27569i
\(867\) −7112.00 −0.278588
\(868\) 43316.0 1.69383
\(869\) 33904.0i 1.32349i
\(870\) −20090.0 −0.782891
\(871\) 0 0
\(872\) −16785.0 −0.651848
\(873\) 1540.00i 0.0597034i
\(874\) −60480.0 −2.34069
\(875\) 18291.0 0.706684
\(876\) − 11662.0i − 0.449797i
\(877\) 6239.00i 0.240224i 0.992760 + 0.120112i \(0.0383253\pi\)
−0.992760 + 0.120112i \(0.961675\pi\)
\(878\) − 86870.0i − 3.33909i
\(879\) − 6321.00i − 0.242551i
\(880\) 16198.0 0.620494
\(881\) −133.000 −0.00508613 −0.00254307 0.999997i \(-0.500809\pi\)
−0.00254307 + 0.999997i \(0.500809\pi\)
\(882\) − 19140.0i − 0.730700i
\(883\) 26003.0 0.991020 0.495510 0.868602i \(-0.334981\pi\)
0.495510 + 0.868602i \(0.334981\pi\)
\(884\) 0 0
\(885\) −14406.0 −0.547178
\(886\) − 4945.00i − 0.187506i
\(887\) −31248.0 −1.18287 −0.591435 0.806353i \(-0.701438\pi\)
−0.591435 + 0.806353i \(0.701438\pi\)
\(888\) 41265.0 1.55942
\(889\) − 24596.0i − 0.927923i
\(890\) − 41650.0i − 1.56866i
\(891\) − 21814.0i − 0.820198i
\(892\) − 3689.00i − 0.138472i
\(893\) 13230.0 0.495773
\(894\) −86310.0 −3.22890
\(895\) − 21203.0i − 0.791886i
\(896\) 27495.0 1.02516
\(897\) 0 0
\(898\) 72370.0 2.68933
\(899\) − 16072.0i − 0.596253i
\(900\) −28424.0 −1.05274
\(901\) 33264.0 1.22995
\(902\) 43680.0i 1.61240i
\(903\) − 18291.0i − 0.674071i
\(904\) − 78030.0i − 2.87084i
\(905\) − 196.000i − 0.00719918i
\(906\) −116305. −4.26487
\(907\) 38253.0 1.40041 0.700204 0.713943i \(-0.253092\pi\)
0.700204 + 0.713943i \(0.253092\pi\)
\(908\) 43792.0i 1.60054i
\(909\) −9240.00 −0.337152
\(910\) 0 0
\(911\) 36374.0 1.32286 0.661429 0.750007i \(-0.269950\pi\)
0.661429 + 0.750007i \(0.269950\pi\)
\(912\) 78498.0i 2.85014i
\(913\) 8008.00 0.290281
\(914\) −7970.00 −0.288429
\(915\) − 2744.00i − 0.0991408i
\(916\) 7735.00i 0.279008i
\(917\) 18655.0i 0.671802i
\(918\) 13475.0i 0.484468i
\(919\) −27648.0 −0.992408 −0.496204 0.868206i \(-0.665273\pi\)
−0.496204 + 0.868206i \(0.665273\pi\)
\(920\) 30240.0 1.08368
\(921\) 14798.0i 0.529436i
\(922\) −29575.0 −1.05640
\(923\) 0 0
\(924\) −40222.0 −1.43204
\(925\) 9956.00i 0.353893i
\(926\) 55360.0 1.96462
\(927\) −12936.0 −0.458332
\(928\) 6970.00i 0.246553i
\(929\) 756.000i 0.0266992i 0.999911 + 0.0133496i \(0.00424944\pi\)
−0.999911 + 0.0133496i \(0.995751\pi\)
\(930\) 48020.0i 1.69316i
\(931\) − 21924.0i − 0.771783i
\(932\) 52037.0 1.82889
\(933\) 23814.0 0.835622
\(934\) 6300.00i 0.220709i
\(935\) −14014.0 −0.490168
\(936\) 0 0
\(937\) 20846.0 0.726797 0.363399 0.931634i \(-0.381616\pi\)
0.363399 + 0.931634i \(0.381616\pi\)
\(938\) 31070.0i 1.08153i
\(939\) 74823.0 2.60038
\(940\) −12495.0 −0.433555
\(941\) − 41321.0i − 1.43148i −0.698365 0.715742i \(-0.746089\pi\)
0.698365 0.715742i \(-0.253911\pi\)
\(942\) − 95550.0i − 3.30487i
\(943\) 32256.0i 1.11389i
\(944\) 26166.0i 0.902151i
\(945\) 3185.00 0.109638
\(946\) 26130.0 0.898055
\(947\) − 54966.0i − 1.88612i −0.332624 0.943060i \(-0.607934\pi\)
0.332624 0.943060i \(-0.392066\pi\)
\(948\) 155176. 5.31633
\(949\) 0 0
\(950\) −47880.0 −1.63519
\(951\) 49378.0i 1.68369i
\(952\) 45045.0 1.53353
\(953\) 44553.0 1.51439 0.757195 0.653189i \(-0.226569\pi\)
0.757195 + 0.653189i \(0.226569\pi\)
\(954\) 47520.0i 1.61270i
\(955\) − 2954.00i − 0.100093i
\(956\) 59109.0i 1.99971i
\(957\) 14924.0i 0.504101i
\(958\) 60165.0 2.02906
\(959\) −23088.0 −0.777425
\(960\) 14063.0i 0.472793i
\(961\) −8625.00 −0.289517
\(962\) 0 0
\(963\) −15048.0 −0.503546
\(964\) − 27370.0i − 0.914448i
\(965\) −3444.00 −0.114887
\(966\) −43680.0 −1.45485
\(967\) − 27907.0i − 0.928054i −0.885821 0.464027i \(-0.846404\pi\)
0.885821 0.464027i \(-0.153596\pi\)
\(968\) 29475.0i 0.978680i
\(969\) − 67914.0i − 2.25151i
\(970\) − 2450.00i − 0.0810977i
\(971\) −16443.0 −0.543441 −0.271720 0.962376i \(-0.587593\pi\)
−0.271720 + 0.962376i \(0.587593\pi\)
\(972\) −83776.0 −2.76452
\(973\) − 24297.0i − 0.800541i
\(974\) −11400.0 −0.375030
\(975\) 0 0
\(976\) −4984.00 −0.163457
\(977\) − 45414.0i − 1.48713i −0.668666 0.743563i \(-0.733134\pi\)
0.668666 0.743563i \(-0.266866\pi\)
\(978\) −19040.0 −0.622528
\(979\) −30940.0 −1.01006
\(980\) 20706.0i 0.674927i
\(981\) 8206.00i 0.267072i
\(982\) 83835.0i 2.72432i
\(983\) 8981.00i 0.291403i 0.989329 + 0.145702i \(0.0465440\pi\)
−0.989329 + 0.145702i \(0.953456\pi\)
\(984\) 105840. 3.42892
\(985\) 20937.0 0.677267
\(986\) − 31570.0i − 1.01967i
\(987\) 9555.00 0.308145
\(988\) 0 0
\(989\) 19296.0 0.620402
\(990\) − 20020.0i − 0.642704i
\(991\) −17414.0 −0.558198 −0.279099 0.960262i \(-0.590036\pi\)
−0.279099 + 0.960262i \(0.590036\pi\)
\(992\) 16660.0 0.533221
\(993\) − 67928.0i − 2.17083i
\(994\) 585.000i 0.0186671i
\(995\) − 490.000i − 0.0156121i
\(996\) − 36652.0i − 1.16603i
\(997\) −23702.0 −0.752909 −0.376454 0.926435i \(-0.622857\pi\)
−0.376454 + 0.926435i \(0.622857\pi\)
\(998\) 64200.0 2.03629
\(999\) 4585.00i 0.145208i
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 169.4.b.a.168.1 2
13.2 odd 12 169.4.c.e.22.1 2
13.3 even 3 169.4.e.e.147.2 4
13.4 even 6 169.4.e.e.23.2 4
13.5 odd 4 13.4.a.a.1.1 1
13.6 odd 12 169.4.c.e.146.1 2
13.7 odd 12 169.4.c.a.146.1 2
13.8 odd 4 169.4.a.e.1.1 1
13.9 even 3 169.4.e.e.23.1 4
13.10 even 6 169.4.e.e.147.1 4
13.11 odd 12 169.4.c.a.22.1 2
13.12 even 2 inner 169.4.b.a.168.2 2
39.5 even 4 117.4.a.b.1.1 1
39.8 even 4 1521.4.a.a.1.1 1
52.31 even 4 208.4.a.g.1.1 1
65.18 even 4 325.4.b.b.274.2 2
65.44 odd 4 325.4.a.d.1.1 1
65.57 even 4 325.4.b.b.274.1 2
91.83 even 4 637.4.a.a.1.1 1
104.5 odd 4 832.4.a.r.1.1 1
104.83 even 4 832.4.a.a.1.1 1
143.109 even 4 1573.4.a.a.1.1 1
156.83 odd 4 1872.4.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.a.a.1.1 1 13.5 odd 4
117.4.a.b.1.1 1 39.5 even 4
169.4.a.e.1.1 1 13.8 odd 4
169.4.b.a.168.1 2 1.1 even 1 trivial
169.4.b.a.168.2 2 13.12 even 2 inner
169.4.c.a.22.1 2 13.11 odd 12
169.4.c.a.146.1 2 13.7 odd 12
169.4.c.e.22.1 2 13.2 odd 12
169.4.c.e.146.1 2 13.6 odd 12
169.4.e.e.23.1 4 13.9 even 3
169.4.e.e.23.2 4 13.4 even 6
169.4.e.e.147.1 4 13.10 even 6
169.4.e.e.147.2 4 13.3 even 3
208.4.a.g.1.1 1 52.31 even 4
325.4.a.d.1.1 1 65.44 odd 4
325.4.b.b.274.1 2 65.57 even 4
325.4.b.b.274.2 2 65.18 even 4
637.4.a.a.1.1 1 91.83 even 4
832.4.a.a.1.1 1 104.83 even 4
832.4.a.r.1.1 1 104.5 odd 4
1521.4.a.a.1.1 1 39.8 even 4
1573.4.a.a.1.1 1 143.109 even 4
1872.4.a.k.1.1 1 156.83 odd 4