Properties

Label 7098.2.a.ch.1.1
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.80194 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.80194 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.80194 q^{10} +1.33513 q^{11} -1.00000 q^{12} +1.00000 q^{14} +2.80194 q^{15} +1.00000 q^{16} +0.356896 q^{17} +1.00000 q^{18} -2.44504 q^{19} -2.80194 q^{20} -1.00000 q^{21} +1.33513 q^{22} +2.51573 q^{23} -1.00000 q^{24} +2.85086 q^{25} -1.00000 q^{27} +1.00000 q^{28} -5.63102 q^{29} +2.80194 q^{30} -9.54288 q^{31} +1.00000 q^{32} -1.33513 q^{33} +0.356896 q^{34} -2.80194 q^{35} +1.00000 q^{36} +1.69202 q^{37} -2.44504 q^{38} -2.80194 q^{40} +2.80194 q^{41} -1.00000 q^{42} +6.51573 q^{43} +1.33513 q^{44} -2.80194 q^{45} +2.51573 q^{46} +5.09783 q^{47} -1.00000 q^{48} +1.00000 q^{49} +2.85086 q^{50} -0.356896 q^{51} +8.25667 q^{53} -1.00000 q^{54} -3.74094 q^{55} +1.00000 q^{56} +2.44504 q^{57} -5.63102 q^{58} -4.18598 q^{59} +2.80194 q^{60} -7.00000 q^{61} -9.54288 q^{62} +1.00000 q^{63} +1.00000 q^{64} -1.33513 q^{66} +8.83877 q^{67} +0.356896 q^{68} -2.51573 q^{69} -2.80194 q^{70} -5.20775 q^{71} +1.00000 q^{72} +0.850855 q^{73} +1.69202 q^{74} -2.85086 q^{75} -2.44504 q^{76} +1.33513 q^{77} -8.80194 q^{79} -2.80194 q^{80} +1.00000 q^{81} +2.80194 q^{82} -10.4450 q^{83} -1.00000 q^{84} -1.00000 q^{85} +6.51573 q^{86} +5.63102 q^{87} +1.33513 q^{88} -3.40044 q^{89} -2.80194 q^{90} +2.51573 q^{92} +9.54288 q^{93} +5.09783 q^{94} +6.85086 q^{95} -1.00000 q^{96} -8.08815 q^{97} +1.00000 q^{98} +1.33513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 4q^{5} - 3q^{6} + 3q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 4q^{5} - 3q^{6} + 3q^{7} + 3q^{8} + 3q^{9} - 4q^{10} + 3q^{11} - 3q^{12} + 3q^{14} + 4q^{15} + 3q^{16} - 3q^{17} + 3q^{18} - 7q^{19} - 4q^{20} - 3q^{21} + 3q^{22} - 5q^{23} - 3q^{24} - 5q^{25} - 3q^{27} + 3q^{28} - 2q^{29} + 4q^{30} - 10q^{31} + 3q^{32} - 3q^{33} - 3q^{34} - 4q^{35} + 3q^{36} - 7q^{38} - 4q^{40} + 4q^{41} - 3q^{42} + 7q^{43} + 3q^{44} - 4q^{45} - 5q^{46} - 3q^{47} - 3q^{48} + 3q^{49} - 5q^{50} + 3q^{51} - 2q^{53} - 3q^{54} + 3q^{55} + 3q^{56} + 7q^{57} - 2q^{58} + 2q^{59} + 4q^{60} - 21q^{61} - 10q^{62} + 3q^{63} + 3q^{64} - 3q^{66} - 6q^{67} - 3q^{68} + 5q^{69} - 4q^{70} + 2q^{71} + 3q^{72} - 11q^{73} + 5q^{75} - 7q^{76} + 3q^{77} - 22q^{79} - 4q^{80} + 3q^{81} + 4q^{82} - 31q^{83} - 3q^{84} - 3q^{85} + 7q^{86} + 2q^{87} + 3q^{88} - 4q^{90} - 5q^{92} + 10q^{93} - 3q^{94} + 7q^{95} - 3q^{96} - 28q^{97} + 3q^{98} + 3q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −2.80194 −1.25306 −0.626532 0.779395i \(-0.715526\pi\)
−0.626532 + 0.779395i \(0.715526\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.80194 −0.886051
\(11\) 1.33513 0.402556 0.201278 0.979534i \(-0.435491\pi\)
0.201278 + 0.979534i \(0.435491\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) 2.80194 0.723457
\(16\) 1.00000 0.250000
\(17\) 0.356896 0.0865600 0.0432800 0.999063i \(-0.486219\pi\)
0.0432800 + 0.999063i \(0.486219\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.44504 −0.560931 −0.280466 0.959864i \(-0.590489\pi\)
−0.280466 + 0.959864i \(0.590489\pi\)
\(20\) −2.80194 −0.626532
\(21\) −1.00000 −0.218218
\(22\) 1.33513 0.284650
\(23\) 2.51573 0.524566 0.262283 0.964991i \(-0.415525\pi\)
0.262283 + 0.964991i \(0.415525\pi\)
\(24\) −1.00000 −0.204124
\(25\) 2.85086 0.570171
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −5.63102 −1.04565 −0.522827 0.852439i \(-0.675123\pi\)
−0.522827 + 0.852439i \(0.675123\pi\)
\(30\) 2.80194 0.511562
\(31\) −9.54288 −1.71395 −0.856976 0.515357i \(-0.827659\pi\)
−0.856976 + 0.515357i \(0.827659\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.33513 −0.232416
\(34\) 0.356896 0.0612071
\(35\) −2.80194 −0.473614
\(36\) 1.00000 0.166667
\(37\) 1.69202 0.278167 0.139083 0.990281i \(-0.455584\pi\)
0.139083 + 0.990281i \(0.455584\pi\)
\(38\) −2.44504 −0.396638
\(39\) 0 0
\(40\) −2.80194 −0.443025
\(41\) 2.80194 0.437589 0.218795 0.975771i \(-0.429788\pi\)
0.218795 + 0.975771i \(0.429788\pi\)
\(42\) −1.00000 −0.154303
\(43\) 6.51573 0.993639 0.496820 0.867854i \(-0.334501\pi\)
0.496820 + 0.867854i \(0.334501\pi\)
\(44\) 1.33513 0.201278
\(45\) −2.80194 −0.417688
\(46\) 2.51573 0.370924
\(47\) 5.09783 0.743596 0.371798 0.928314i \(-0.378741\pi\)
0.371798 + 0.928314i \(0.378741\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 2.85086 0.403172
\(51\) −0.356896 −0.0499754
\(52\) 0 0
\(53\) 8.25667 1.13414 0.567070 0.823669i \(-0.308077\pi\)
0.567070 + 0.823669i \(0.308077\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.74094 −0.504428
\(56\) 1.00000 0.133631
\(57\) 2.44504 0.323854
\(58\) −5.63102 −0.739389
\(59\) −4.18598 −0.544968 −0.272484 0.962160i \(-0.587845\pi\)
−0.272484 + 0.962160i \(0.587845\pi\)
\(60\) 2.80194 0.361729
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) −9.54288 −1.21195
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.33513 −0.164343
\(67\) 8.83877 1.07983 0.539914 0.841720i \(-0.318457\pi\)
0.539914 + 0.841720i \(0.318457\pi\)
\(68\) 0.356896 0.0432800
\(69\) −2.51573 −0.302858
\(70\) −2.80194 −0.334896
\(71\) −5.20775 −0.618046 −0.309023 0.951055i \(-0.600002\pi\)
−0.309023 + 0.951055i \(0.600002\pi\)
\(72\) 1.00000 0.117851
\(73\) 0.850855 0.0995851 0.0497925 0.998760i \(-0.484144\pi\)
0.0497925 + 0.998760i \(0.484144\pi\)
\(74\) 1.69202 0.196694
\(75\) −2.85086 −0.329188
\(76\) −2.44504 −0.280466
\(77\) 1.33513 0.152152
\(78\) 0 0
\(79\) −8.80194 −0.990295 −0.495148 0.868809i \(-0.664886\pi\)
−0.495148 + 0.868809i \(0.664886\pi\)
\(80\) −2.80194 −0.313266
\(81\) 1.00000 0.111111
\(82\) 2.80194 0.309422
\(83\) −10.4450 −1.14649 −0.573246 0.819383i \(-0.694316\pi\)
−0.573246 + 0.819383i \(0.694316\pi\)
\(84\) −1.00000 −0.109109
\(85\) −1.00000 −0.108465
\(86\) 6.51573 0.702609
\(87\) 5.63102 0.603709
\(88\) 1.33513 0.142325
\(89\) −3.40044 −0.360446 −0.180223 0.983626i \(-0.557682\pi\)
−0.180223 + 0.983626i \(0.557682\pi\)
\(90\) −2.80194 −0.295350
\(91\) 0 0
\(92\) 2.51573 0.262283
\(93\) 9.54288 0.989550
\(94\) 5.09783 0.525801
\(95\) 6.85086 0.702883
\(96\) −1.00000 −0.102062
\(97\) −8.08815 −0.821227 −0.410613 0.911810i \(-0.634685\pi\)
−0.410613 + 0.911810i \(0.634685\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.33513 0.134185
\(100\) 2.85086 0.285086
\(101\) −6.53319 −0.650076 −0.325038 0.945701i \(-0.605377\pi\)
−0.325038 + 0.945701i \(0.605377\pi\)
\(102\) −0.356896 −0.0353380
\(103\) 0.225209 0.0221905 0.0110953 0.999938i \(-0.496468\pi\)
0.0110953 + 0.999938i \(0.496468\pi\)
\(104\) 0 0
\(105\) 2.80194 0.273441
\(106\) 8.25667 0.801959
\(107\) 4.50604 0.435615 0.217808 0.975992i \(-0.430109\pi\)
0.217808 + 0.975992i \(0.430109\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −7.10752 −0.680777 −0.340389 0.940285i \(-0.610559\pi\)
−0.340389 + 0.940285i \(0.610559\pi\)
\(110\) −3.74094 −0.356685
\(111\) −1.69202 −0.160600
\(112\) 1.00000 0.0944911
\(113\) 2.86831 0.269828 0.134914 0.990857i \(-0.456924\pi\)
0.134914 + 0.990857i \(0.456924\pi\)
\(114\) 2.44504 0.228999
\(115\) −7.04892 −0.657315
\(116\) −5.63102 −0.522827
\(117\) 0 0
\(118\) −4.18598 −0.385351
\(119\) 0.356896 0.0327166
\(120\) 2.80194 0.255781
\(121\) −9.21744 −0.837949
\(122\) −7.00000 −0.633750
\(123\) −2.80194 −0.252642
\(124\) −9.54288 −0.856976
\(125\) 6.02177 0.538604
\(126\) 1.00000 0.0890871
\(127\) 2.57673 0.228648 0.114324 0.993444i \(-0.463530\pi\)
0.114324 + 0.993444i \(0.463530\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.51573 −0.573678
\(130\) 0 0
\(131\) 12.7724 1.11593 0.557965 0.829865i \(-0.311582\pi\)
0.557965 + 0.829865i \(0.311582\pi\)
\(132\) −1.33513 −0.116208
\(133\) −2.44504 −0.212012
\(134\) 8.83877 0.763554
\(135\) 2.80194 0.241152
\(136\) 0.356896 0.0306036
\(137\) −7.30798 −0.624363 −0.312181 0.950023i \(-0.601060\pi\)
−0.312181 + 0.950023i \(0.601060\pi\)
\(138\) −2.51573 −0.214153
\(139\) 22.9812 1.94924 0.974621 0.223863i \(-0.0718669\pi\)
0.974621 + 0.223863i \(0.0718669\pi\)
\(140\) −2.80194 −0.236807
\(141\) −5.09783 −0.429315
\(142\) −5.20775 −0.437025
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 15.7778 1.31027
\(146\) 0.850855 0.0704173
\(147\) −1.00000 −0.0824786
\(148\) 1.69202 0.139083
\(149\) 3.95108 0.323685 0.161843 0.986817i \(-0.448256\pi\)
0.161843 + 0.986817i \(0.448256\pi\)
\(150\) −2.85086 −0.232771
\(151\) −5.26875 −0.428765 −0.214382 0.976750i \(-0.568774\pi\)
−0.214382 + 0.976750i \(0.568774\pi\)
\(152\) −2.44504 −0.198319
\(153\) 0.356896 0.0288533
\(154\) 1.33513 0.107587
\(155\) 26.7385 2.14769
\(156\) 0 0
\(157\) −16.1521 −1.28908 −0.644540 0.764570i \(-0.722951\pi\)
−0.644540 + 0.764570i \(0.722951\pi\)
\(158\) −8.80194 −0.700245
\(159\) −8.25667 −0.654796
\(160\) −2.80194 −0.221513
\(161\) 2.51573 0.198267
\(162\) 1.00000 0.0785674
\(163\) −11.9825 −0.938545 −0.469273 0.883053i \(-0.655484\pi\)
−0.469273 + 0.883053i \(0.655484\pi\)
\(164\) 2.80194 0.218795
\(165\) 3.74094 0.291232
\(166\) −10.4450 −0.810692
\(167\) −18.6069 −1.43984 −0.719921 0.694056i \(-0.755822\pi\)
−0.719921 + 0.694056i \(0.755822\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) −1.00000 −0.0766965
\(171\) −2.44504 −0.186977
\(172\) 6.51573 0.496820
\(173\) 15.0422 1.14364 0.571819 0.820380i \(-0.306238\pi\)
0.571819 + 0.820380i \(0.306238\pi\)
\(174\) 5.63102 0.426887
\(175\) 2.85086 0.215504
\(176\) 1.33513 0.100639
\(177\) 4.18598 0.314638
\(178\) −3.40044 −0.254873
\(179\) −23.6112 −1.76478 −0.882391 0.470517i \(-0.844067\pi\)
−0.882391 + 0.470517i \(0.844067\pi\)
\(180\) −2.80194 −0.208844
\(181\) 15.6679 1.16458 0.582291 0.812980i \(-0.302156\pi\)
0.582291 + 0.812980i \(0.302156\pi\)
\(182\) 0 0
\(183\) 7.00000 0.517455
\(184\) 2.51573 0.185462
\(185\) −4.74094 −0.348561
\(186\) 9.54288 0.699718
\(187\) 0.476501 0.0348452
\(188\) 5.09783 0.371798
\(189\) −1.00000 −0.0727393
\(190\) 6.85086 0.497013
\(191\) −22.5526 −1.63185 −0.815923 0.578160i \(-0.803771\pi\)
−0.815923 + 0.578160i \(0.803771\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −24.2610 −1.74634 −0.873172 0.487413i \(-0.837941\pi\)
−0.873172 + 0.487413i \(0.837941\pi\)
\(194\) −8.08815 −0.580695
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −0.411190 −0.0292961 −0.0146480 0.999893i \(-0.504663\pi\)
−0.0146480 + 0.999893i \(0.504663\pi\)
\(198\) 1.33513 0.0948832
\(199\) −5.73125 −0.406278 −0.203139 0.979150i \(-0.565114\pi\)
−0.203139 + 0.979150i \(0.565114\pi\)
\(200\) 2.85086 0.201586
\(201\) −8.83877 −0.623439
\(202\) −6.53319 −0.459673
\(203\) −5.63102 −0.395220
\(204\) −0.356896 −0.0249877
\(205\) −7.85086 −0.548328
\(206\) 0.225209 0.0156911
\(207\) 2.51573 0.174855
\(208\) 0 0
\(209\) −3.26444 −0.225806
\(210\) 2.80194 0.193352
\(211\) −24.5284 −1.68860 −0.844302 0.535867i \(-0.819985\pi\)
−0.844302 + 0.535867i \(0.819985\pi\)
\(212\) 8.25667 0.567070
\(213\) 5.20775 0.356829
\(214\) 4.50604 0.308027
\(215\) −18.2567 −1.24509
\(216\) −1.00000 −0.0680414
\(217\) −9.54288 −0.647813
\(218\) −7.10752 −0.481382
\(219\) −0.850855 −0.0574955
\(220\) −3.74094 −0.252214
\(221\) 0 0
\(222\) −1.69202 −0.113561
\(223\) −21.0532 −1.40983 −0.704914 0.709293i \(-0.749015\pi\)
−0.704914 + 0.709293i \(0.749015\pi\)
\(224\) 1.00000 0.0668153
\(225\) 2.85086 0.190057
\(226\) 2.86831 0.190797
\(227\) 3.49827 0.232188 0.116094 0.993238i \(-0.462963\pi\)
0.116094 + 0.993238i \(0.462963\pi\)
\(228\) 2.44504 0.161927
\(229\) −22.0097 −1.45444 −0.727221 0.686404i \(-0.759188\pi\)
−0.727221 + 0.686404i \(0.759188\pi\)
\(230\) −7.04892 −0.464792
\(231\) −1.33513 −0.0878448
\(232\) −5.63102 −0.369695
\(233\) 21.2204 1.39020 0.695098 0.718915i \(-0.255361\pi\)
0.695098 + 0.718915i \(0.255361\pi\)
\(234\) 0 0
\(235\) −14.2838 −0.931773
\(236\) −4.18598 −0.272484
\(237\) 8.80194 0.571747
\(238\) 0.356896 0.0231341
\(239\) −2.09783 −0.135698 −0.0678488 0.997696i \(-0.521614\pi\)
−0.0678488 + 0.997696i \(0.521614\pi\)
\(240\) 2.80194 0.180864
\(241\) −5.22952 −0.336863 −0.168432 0.985713i \(-0.553870\pi\)
−0.168432 + 0.985713i \(0.553870\pi\)
\(242\) −9.21744 −0.592519
\(243\) −1.00000 −0.0641500
\(244\) −7.00000 −0.448129
\(245\) −2.80194 −0.179009
\(246\) −2.80194 −0.178645
\(247\) 0 0
\(248\) −9.54288 −0.605973
\(249\) 10.4450 0.661928
\(250\) 6.02177 0.380850
\(251\) −11.4426 −0.722254 −0.361127 0.932517i \(-0.617608\pi\)
−0.361127 + 0.932517i \(0.617608\pi\)
\(252\) 1.00000 0.0629941
\(253\) 3.35881 0.211167
\(254\) 2.57673 0.161678
\(255\) 1.00000 0.0626224
\(256\) 1.00000 0.0625000
\(257\) −21.5623 −1.34502 −0.672508 0.740090i \(-0.734783\pi\)
−0.672508 + 0.740090i \(0.734783\pi\)
\(258\) −6.51573 −0.405652
\(259\) 1.69202 0.105137
\(260\) 0 0
\(261\) −5.63102 −0.348552
\(262\) 12.7724 0.789081
\(263\) −11.5284 −0.710872 −0.355436 0.934701i \(-0.615668\pi\)
−0.355436 + 0.934701i \(0.615668\pi\)
\(264\) −1.33513 −0.0821713
\(265\) −23.1347 −1.42115
\(266\) −2.44504 −0.149915
\(267\) 3.40044 0.208103
\(268\) 8.83877 0.539914
\(269\) −30.5211 −1.86090 −0.930452 0.366413i \(-0.880586\pi\)
−0.930452 + 0.366413i \(0.880586\pi\)
\(270\) 2.80194 0.170521
\(271\) −12.6160 −0.766365 −0.383182 0.923673i \(-0.625172\pi\)
−0.383182 + 0.923673i \(0.625172\pi\)
\(272\) 0.356896 0.0216400
\(273\) 0 0
\(274\) −7.30798 −0.441491
\(275\) 3.80625 0.229525
\(276\) −2.51573 −0.151429
\(277\) 14.7899 0.888636 0.444318 0.895869i \(-0.353446\pi\)
0.444318 + 0.895869i \(0.353446\pi\)
\(278\) 22.9812 1.37832
\(279\) −9.54288 −0.571317
\(280\) −2.80194 −0.167448
\(281\) 15.5623 0.928366 0.464183 0.885739i \(-0.346348\pi\)
0.464183 + 0.885739i \(0.346348\pi\)
\(282\) −5.09783 −0.303572
\(283\) 17.5894 1.04558 0.522791 0.852461i \(-0.324891\pi\)
0.522791 + 0.852461i \(0.324891\pi\)
\(284\) −5.20775 −0.309023
\(285\) −6.85086 −0.405810
\(286\) 0 0
\(287\) 2.80194 0.165393
\(288\) 1.00000 0.0589256
\(289\) −16.8726 −0.992507
\(290\) 15.7778 0.926503
\(291\) 8.08815 0.474136
\(292\) 0.850855 0.0497925
\(293\) 11.4263 0.667529 0.333764 0.942656i \(-0.391681\pi\)
0.333764 + 0.942656i \(0.391681\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 11.7289 0.682880
\(296\) 1.69202 0.0983468
\(297\) −1.33513 −0.0774718
\(298\) 3.95108 0.228880
\(299\) 0 0
\(300\) −2.85086 −0.164594
\(301\) 6.51573 0.375560
\(302\) −5.26875 −0.303182
\(303\) 6.53319 0.375322
\(304\) −2.44504 −0.140233
\(305\) 19.6136 1.12307
\(306\) 0.356896 0.0204024
\(307\) −10.5670 −0.603093 −0.301546 0.953452i \(-0.597503\pi\)
−0.301546 + 0.953452i \(0.597503\pi\)
\(308\) 1.33513 0.0760758
\(309\) −0.225209 −0.0128117
\(310\) 26.7385 1.51865
\(311\) −7.40821 −0.420081 −0.210040 0.977693i \(-0.567360\pi\)
−0.210040 + 0.977693i \(0.567360\pi\)
\(312\) 0 0
\(313\) −2.12737 −0.120246 −0.0601232 0.998191i \(-0.519149\pi\)
−0.0601232 + 0.998191i \(0.519149\pi\)
\(314\) −16.1521 −0.911517
\(315\) −2.80194 −0.157871
\(316\) −8.80194 −0.495148
\(317\) 32.6383 1.83315 0.916575 0.399862i \(-0.130942\pi\)
0.916575 + 0.399862i \(0.130942\pi\)
\(318\) −8.25667 −0.463011
\(319\) −7.51812 −0.420934
\(320\) −2.80194 −0.156633
\(321\) −4.50604 −0.251503
\(322\) 2.51573 0.140196
\(323\) −0.872625 −0.0485542
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −11.9825 −0.663652
\(327\) 7.10752 0.393047
\(328\) 2.80194 0.154711
\(329\) 5.09783 0.281053
\(330\) 3.74094 0.205932
\(331\) −30.0006 −1.64898 −0.824491 0.565875i \(-0.808538\pi\)
−0.824491 + 0.565875i \(0.808538\pi\)
\(332\) −10.4450 −0.573246
\(333\) 1.69202 0.0927222
\(334\) −18.6069 −1.01812
\(335\) −24.7657 −1.35309
\(336\) −1.00000 −0.0545545
\(337\) 12.5472 0.683489 0.341744 0.939793i \(-0.388982\pi\)
0.341744 + 0.939793i \(0.388982\pi\)
\(338\) 0 0
\(339\) −2.86831 −0.155785
\(340\) −1.00000 −0.0542326
\(341\) −12.7409 −0.689961
\(342\) −2.44504 −0.132213
\(343\) 1.00000 0.0539949
\(344\) 6.51573 0.351305
\(345\) 7.04892 0.379501
\(346\) 15.0422 0.808674
\(347\) 25.5719 1.37277 0.686387 0.727237i \(-0.259196\pi\)
0.686387 + 0.727237i \(0.259196\pi\)
\(348\) 5.63102 0.301854
\(349\) −22.9855 −1.23039 −0.615193 0.788376i \(-0.710922\pi\)
−0.615193 + 0.788376i \(0.710922\pi\)
\(350\) 2.85086 0.152385
\(351\) 0 0
\(352\) 1.33513 0.0711624
\(353\) 22.4397 1.19434 0.597171 0.802114i \(-0.296291\pi\)
0.597171 + 0.802114i \(0.296291\pi\)
\(354\) 4.18598 0.222482
\(355\) 14.5918 0.774452
\(356\) −3.40044 −0.180223
\(357\) −0.356896 −0.0188889
\(358\) −23.6112 −1.24789
\(359\) 6.80492 0.359150 0.179575 0.983744i \(-0.442528\pi\)
0.179575 + 0.983744i \(0.442528\pi\)
\(360\) −2.80194 −0.147675
\(361\) −13.0218 −0.685356
\(362\) 15.6679 0.823484
\(363\) 9.21744 0.483790
\(364\) 0 0
\(365\) −2.38404 −0.124787
\(366\) 7.00000 0.365896
\(367\) −21.9409 −1.14531 −0.572653 0.819798i \(-0.694086\pi\)
−0.572653 + 0.819798i \(0.694086\pi\)
\(368\) 2.51573 0.131141
\(369\) 2.80194 0.145863
\(370\) −4.74094 −0.246470
\(371\) 8.25667 0.428665
\(372\) 9.54288 0.494775
\(373\) −18.3207 −0.948607 −0.474304 0.880361i \(-0.657300\pi\)
−0.474304 + 0.880361i \(0.657300\pi\)
\(374\) 0.476501 0.0246393
\(375\) −6.02177 −0.310963
\(376\) 5.09783 0.262901
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) −25.8950 −1.33014 −0.665068 0.746783i \(-0.731597\pi\)
−0.665068 + 0.746783i \(0.731597\pi\)
\(380\) 6.85086 0.351441
\(381\) −2.57673 −0.132010
\(382\) −22.5526 −1.15389
\(383\) −20.0218 −1.02306 −0.511532 0.859264i \(-0.670922\pi\)
−0.511532 + 0.859264i \(0.670922\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −3.74094 −0.190656
\(386\) −24.2610 −1.23485
\(387\) 6.51573 0.331213
\(388\) −8.08815 −0.410613
\(389\) 13.7952 0.699446 0.349723 0.936853i \(-0.386276\pi\)
0.349723 + 0.936853i \(0.386276\pi\)
\(390\) 0 0
\(391\) 0.897853 0.0454064
\(392\) 1.00000 0.0505076
\(393\) −12.7724 −0.644282
\(394\) −0.411190 −0.0207155
\(395\) 24.6625 1.24090
\(396\) 1.33513 0.0670926
\(397\) −7.07846 −0.355258 −0.177629 0.984098i \(-0.556843\pi\)
−0.177629 + 0.984098i \(0.556843\pi\)
\(398\) −5.73125 −0.287282
\(399\) 2.44504 0.122405
\(400\) 2.85086 0.142543
\(401\) −2.94869 −0.147251 −0.0736253 0.997286i \(-0.523457\pi\)
−0.0736253 + 0.997286i \(0.523457\pi\)
\(402\) −8.83877 −0.440838
\(403\) 0 0
\(404\) −6.53319 −0.325038
\(405\) −2.80194 −0.139229
\(406\) −5.63102 −0.279463
\(407\) 2.25906 0.111978
\(408\) −0.356896 −0.0176690
\(409\) −8.41358 −0.416025 −0.208012 0.978126i \(-0.566699\pi\)
−0.208012 + 0.978126i \(0.566699\pi\)
\(410\) −7.85086 −0.387726
\(411\) 7.30798 0.360476
\(412\) 0.225209 0.0110953
\(413\) −4.18598 −0.205979
\(414\) 2.51573 0.123641
\(415\) 29.2664 1.43663
\(416\) 0 0
\(417\) −22.9812 −1.12539
\(418\) −3.26444 −0.159669
\(419\) 36.9168 1.80350 0.901751 0.432256i \(-0.142282\pi\)
0.901751 + 0.432256i \(0.142282\pi\)
\(420\) 2.80194 0.136721
\(421\) 5.42221 0.264262 0.132131 0.991232i \(-0.457818\pi\)
0.132131 + 0.991232i \(0.457818\pi\)
\(422\) −24.5284 −1.19402
\(423\) 5.09783 0.247865
\(424\) 8.25667 0.400979
\(425\) 1.01746 0.0493540
\(426\) 5.20775 0.252316
\(427\) −7.00000 −0.338754
\(428\) 4.50604 0.217808
\(429\) 0 0
\(430\) −18.2567 −0.880415
\(431\) −29.8442 −1.43754 −0.718771 0.695247i \(-0.755295\pi\)
−0.718771 + 0.695247i \(0.755295\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −19.4873 −0.936498 −0.468249 0.883597i \(-0.655115\pi\)
−0.468249 + 0.883597i \(0.655115\pi\)
\(434\) −9.54288 −0.458073
\(435\) −15.7778 −0.756486
\(436\) −7.10752 −0.340389
\(437\) −6.15106 −0.294245
\(438\) −0.850855 −0.0406554
\(439\) 36.6450 1.74897 0.874486 0.485051i \(-0.161199\pi\)
0.874486 + 0.485051i \(0.161199\pi\)
\(440\) −3.74094 −0.178342
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 34.6679 1.64712 0.823560 0.567229i \(-0.191985\pi\)
0.823560 + 0.567229i \(0.191985\pi\)
\(444\) −1.69202 −0.0802998
\(445\) 9.52781 0.451662
\(446\) −21.0532 −0.996899
\(447\) −3.95108 −0.186880
\(448\) 1.00000 0.0472456
\(449\) 8.72455 0.411737 0.205868 0.978580i \(-0.433998\pi\)
0.205868 + 0.978580i \(0.433998\pi\)
\(450\) 2.85086 0.134391
\(451\) 3.74094 0.176154
\(452\) 2.86831 0.134914
\(453\) 5.26875 0.247547
\(454\) 3.49827 0.164182
\(455\) 0 0
\(456\) 2.44504 0.114500
\(457\) −19.4300 −0.908896 −0.454448 0.890773i \(-0.650163\pi\)
−0.454448 + 0.890773i \(0.650163\pi\)
\(458\) −22.0097 −1.02845
\(459\) −0.356896 −0.0166585
\(460\) −7.04892 −0.328657
\(461\) 35.0573 1.63278 0.816390 0.577501i \(-0.195972\pi\)
0.816390 + 0.577501i \(0.195972\pi\)
\(462\) −1.33513 −0.0621157
\(463\) 2.20105 0.102291 0.0511456 0.998691i \(-0.483713\pi\)
0.0511456 + 0.998691i \(0.483713\pi\)
\(464\) −5.63102 −0.261414
\(465\) −26.7385 −1.23997
\(466\) 21.2204 0.983017
\(467\) −8.25667 −0.382073 −0.191037 0.981583i \(-0.561185\pi\)
−0.191037 + 0.981583i \(0.561185\pi\)
\(468\) 0 0
\(469\) 8.83877 0.408137
\(470\) −14.2838 −0.658863
\(471\) 16.1521 0.744251
\(472\) −4.18598 −0.192675
\(473\) 8.69932 0.399995
\(474\) 8.80194 0.404286
\(475\) −6.97046 −0.319827
\(476\) 0.356896 0.0163583
\(477\) 8.25667 0.378047
\(478\) −2.09783 −0.0959527
\(479\) −28.9439 −1.32248 −0.661240 0.750174i \(-0.729970\pi\)
−0.661240 + 0.750174i \(0.729970\pi\)
\(480\) 2.80194 0.127890
\(481\) 0 0
\(482\) −5.22952 −0.238198
\(483\) −2.51573 −0.114470
\(484\) −9.21744 −0.418975
\(485\) 22.6625 1.02905
\(486\) −1.00000 −0.0453609
\(487\) 3.39804 0.153980 0.0769900 0.997032i \(-0.475469\pi\)
0.0769900 + 0.997032i \(0.475469\pi\)
\(488\) −7.00000 −0.316875
\(489\) 11.9825 0.541869
\(490\) −2.80194 −0.126579
\(491\) 28.3327 1.27864 0.639319 0.768941i \(-0.279216\pi\)
0.639319 + 0.768941i \(0.279216\pi\)
\(492\) −2.80194 −0.126321
\(493\) −2.00969 −0.0905118
\(494\) 0 0
\(495\) −3.74094 −0.168143
\(496\) −9.54288 −0.428488
\(497\) −5.20775 −0.233600
\(498\) 10.4450 0.468054
\(499\) 11.2664 0.504351 0.252176 0.967681i \(-0.418854\pi\)
0.252176 + 0.967681i \(0.418854\pi\)
\(500\) 6.02177 0.269302
\(501\) 18.6069 0.831293
\(502\) −11.4426 −0.510710
\(503\) −3.75063 −0.167232 −0.0836161 0.996498i \(-0.526647\pi\)
−0.0836161 + 0.996498i \(0.526647\pi\)
\(504\) 1.00000 0.0445435
\(505\) 18.3056 0.814588
\(506\) 3.35881 0.149318
\(507\) 0 0
\(508\) 2.57673 0.114324
\(509\) −0.450419 −0.0199645 −0.00998223 0.999950i \(-0.503177\pi\)
−0.00998223 + 0.999950i \(0.503177\pi\)
\(510\) 1.00000 0.0442807
\(511\) 0.850855 0.0376396
\(512\) 1.00000 0.0441942
\(513\) 2.44504 0.107951
\(514\) −21.5623 −0.951070
\(515\) −0.631023 −0.0278062
\(516\) −6.51573 −0.286839
\(517\) 6.80625 0.299339
\(518\) 1.69202 0.0743432
\(519\) −15.0422 −0.660280
\(520\) 0 0
\(521\) −22.8485 −1.00101 −0.500505 0.865734i \(-0.666852\pi\)
−0.500505 + 0.865734i \(0.666852\pi\)
\(522\) −5.63102 −0.246463
\(523\) −14.3478 −0.627385 −0.313693 0.949525i \(-0.601566\pi\)
−0.313693 + 0.949525i \(0.601566\pi\)
\(524\) 12.7724 0.557965
\(525\) −2.85086 −0.124422
\(526\) −11.5284 −0.502662
\(527\) −3.40581 −0.148360
\(528\) −1.33513 −0.0581039
\(529\) −16.6711 −0.724831
\(530\) −23.1347 −1.00491
\(531\) −4.18598 −0.181656
\(532\) −2.44504 −0.106006
\(533\) 0 0
\(534\) 3.40044 0.147151
\(535\) −12.6256 −0.545854
\(536\) 8.83877 0.381777
\(537\) 23.6112 1.01890
\(538\) −30.5211 −1.31586
\(539\) 1.33513 0.0575079
\(540\) 2.80194 0.120576
\(541\) −43.7342 −1.88028 −0.940141 0.340786i \(-0.889307\pi\)
−0.940141 + 0.340786i \(0.889307\pi\)
\(542\) −12.6160 −0.541902
\(543\) −15.6679 −0.672372
\(544\) 0.356896 0.0153018
\(545\) 19.9148 0.853058
\(546\) 0 0
\(547\) 13.8224 0.591002 0.295501 0.955342i \(-0.404513\pi\)
0.295501 + 0.955342i \(0.404513\pi\)
\(548\) −7.30798 −0.312181
\(549\) −7.00000 −0.298753
\(550\) 3.80625 0.162299
\(551\) 13.7681 0.586540
\(552\) −2.51573 −0.107077
\(553\) −8.80194 −0.374296
\(554\) 14.7899 0.628361
\(555\) 4.74094 0.201242
\(556\) 22.9812 0.974621
\(557\) −42.5846 −1.80437 −0.902184 0.431351i \(-0.858037\pi\)
−0.902184 + 0.431351i \(0.858037\pi\)
\(558\) −9.54288 −0.403982
\(559\) 0 0
\(560\) −2.80194 −0.118403
\(561\) −0.476501 −0.0201179
\(562\) 15.5623 0.656454
\(563\) −11.3491 −0.478309 −0.239154 0.970982i \(-0.576870\pi\)
−0.239154 + 0.970982i \(0.576870\pi\)
\(564\) −5.09783 −0.214658
\(565\) −8.03684 −0.338112
\(566\) 17.5894 0.739338
\(567\) 1.00000 0.0419961
\(568\) −5.20775 −0.218512
\(569\) 3.52052 0.147588 0.0737938 0.997274i \(-0.476489\pi\)
0.0737938 + 0.997274i \(0.476489\pi\)
\(570\) −6.85086 −0.286951
\(571\) 19.8847 0.832149 0.416075 0.909331i \(-0.363406\pi\)
0.416075 + 0.909331i \(0.363406\pi\)
\(572\) 0 0
\(573\) 22.5526 0.942147
\(574\) 2.80194 0.116951
\(575\) 7.17198 0.299092
\(576\) 1.00000 0.0416667
\(577\) −5.61356 −0.233696 −0.116848 0.993150i \(-0.537279\pi\)
−0.116848 + 0.993150i \(0.537279\pi\)
\(578\) −16.8726 −0.701809
\(579\) 24.2610 1.00825
\(580\) 15.7778 0.655136
\(581\) −10.4450 −0.433333
\(582\) 8.08815 0.335264
\(583\) 11.0237 0.456555
\(584\) 0.850855 0.0352086
\(585\) 0 0
\(586\) 11.4263 0.472014
\(587\) 4.46011 0.184088 0.0920442 0.995755i \(-0.470660\pi\)
0.0920442 + 0.995755i \(0.470660\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 23.3327 0.961409
\(590\) 11.7289 0.482869
\(591\) 0.411190 0.0169141
\(592\) 1.69202 0.0695417
\(593\) −1.06100 −0.0435700 −0.0217850 0.999763i \(-0.506935\pi\)
−0.0217850 + 0.999763i \(0.506935\pi\)
\(594\) −1.33513 −0.0547809
\(595\) −1.00000 −0.0409960
\(596\) 3.95108 0.161843
\(597\) 5.73125 0.234564
\(598\) 0 0
\(599\) −3.22282 −0.131681 −0.0658404 0.997830i \(-0.520973\pi\)
−0.0658404 + 0.997830i \(0.520973\pi\)
\(600\) −2.85086 −0.116386
\(601\) 22.8616 0.932544 0.466272 0.884641i \(-0.345597\pi\)
0.466272 + 0.884641i \(0.345597\pi\)
\(602\) 6.51573 0.265561
\(603\) 8.83877 0.359943
\(604\) −5.26875 −0.214382
\(605\) 25.8267 1.05000
\(606\) 6.53319 0.265393
\(607\) 43.4892 1.76517 0.882586 0.470152i \(-0.155801\pi\)
0.882586 + 0.470152i \(0.155801\pi\)
\(608\) −2.44504 −0.0991595
\(609\) 5.63102 0.228181
\(610\) 19.6136 0.794130
\(611\) 0 0
\(612\) 0.356896 0.0144267
\(613\) 1.81833 0.0734417 0.0367209 0.999326i \(-0.488309\pi\)
0.0367209 + 0.999326i \(0.488309\pi\)
\(614\) −10.5670 −0.426451
\(615\) 7.85086 0.316577
\(616\) 1.33513 0.0537937
\(617\) 9.77777 0.393638 0.196819 0.980440i \(-0.436939\pi\)
0.196819 + 0.980440i \(0.436939\pi\)
\(618\) −0.225209 −0.00905925
\(619\) 31.8079 1.27847 0.639234 0.769012i \(-0.279252\pi\)
0.639234 + 0.769012i \(0.279252\pi\)
\(620\) 26.7385 1.07385
\(621\) −2.51573 −0.100953
\(622\) −7.40821 −0.297042
\(623\) −3.40044 −0.136236
\(624\) 0 0
\(625\) −31.1269 −1.24508
\(626\) −2.12737 −0.0850270
\(627\) 3.26444 0.130369
\(628\) −16.1521 −0.644540
\(629\) 0.603875 0.0240781
\(630\) −2.80194 −0.111632
\(631\) 18.0030 0.716687 0.358344 0.933590i \(-0.383342\pi\)
0.358344 + 0.933590i \(0.383342\pi\)
\(632\) −8.80194 −0.350122
\(633\) 24.5284 0.974916
\(634\) 32.6383 1.29623
\(635\) −7.21983 −0.286510
\(636\) −8.25667 −0.327398
\(637\) 0 0
\(638\) −7.51812 −0.297645
\(639\) −5.20775 −0.206015
\(640\) −2.80194 −0.110756
\(641\) 20.0006 0.789976 0.394988 0.918686i \(-0.370749\pi\)
0.394988 + 0.918686i \(0.370749\pi\)
\(642\) −4.50604 −0.177839
\(643\) −40.5719 −1.60000 −0.800001 0.599999i \(-0.795168\pi\)
−0.800001 + 0.599999i \(0.795168\pi\)
\(644\) 2.51573 0.0991336
\(645\) 18.2567 0.718856
\(646\) −0.872625 −0.0343330
\(647\) 12.9226 0.508040 0.254020 0.967199i \(-0.418247\pi\)
0.254020 + 0.967199i \(0.418247\pi\)
\(648\) 1.00000 0.0392837
\(649\) −5.58881 −0.219380
\(650\) 0 0
\(651\) 9.54288 0.374015
\(652\) −11.9825 −0.469273
\(653\) 30.5187 1.19429 0.597145 0.802133i \(-0.296302\pi\)
0.597145 + 0.802133i \(0.296302\pi\)
\(654\) 7.10752 0.277926
\(655\) −35.7875 −1.39833
\(656\) 2.80194 0.109397
\(657\) 0.850855 0.0331950
\(658\) 5.09783 0.198734
\(659\) 4.65950 0.181508 0.0907541 0.995873i \(-0.471072\pi\)
0.0907541 + 0.995873i \(0.471072\pi\)
\(660\) 3.74094 0.145616
\(661\) −27.0097 −1.05056 −0.525278 0.850931i \(-0.676039\pi\)
−0.525278 + 0.850931i \(0.676039\pi\)
\(662\) −30.0006 −1.16601
\(663\) 0 0
\(664\) −10.4450 −0.405346
\(665\) 6.85086 0.265665
\(666\) 1.69202 0.0655645
\(667\) −14.1661 −0.548515
\(668\) −18.6069 −0.719921
\(669\) 21.0532 0.813965
\(670\) −24.7657 −0.956782
\(671\) −9.34588 −0.360794
\(672\) −1.00000 −0.0385758
\(673\) 21.1491 0.815240 0.407620 0.913152i \(-0.366359\pi\)
0.407620 + 0.913152i \(0.366359\pi\)
\(674\) 12.5472 0.483300
\(675\) −2.85086 −0.109729
\(676\) 0 0
\(677\) 22.0968 0.849248 0.424624 0.905370i \(-0.360406\pi\)
0.424624 + 0.905370i \(0.360406\pi\)
\(678\) −2.86831 −0.110157
\(679\) −8.08815 −0.310395
\(680\) −1.00000 −0.0383482
\(681\) −3.49827 −0.134054
\(682\) −12.7409 −0.487876
\(683\) −28.0170 −1.07204 −0.536020 0.844205i \(-0.680073\pi\)
−0.536020 + 0.844205i \(0.680073\pi\)
\(684\) −2.44504 −0.0934885
\(685\) 20.4765 0.782367
\(686\) 1.00000 0.0381802
\(687\) 22.0097 0.839722
\(688\) 6.51573 0.248410
\(689\) 0 0
\(690\) 7.04892 0.268348
\(691\) 16.2422 0.617882 0.308941 0.951081i \(-0.400025\pi\)
0.308941 + 0.951081i \(0.400025\pi\)
\(692\) 15.0422 0.571819
\(693\) 1.33513 0.0507172
\(694\) 25.5719 0.970698
\(695\) −64.3919 −2.44253
\(696\) 5.63102 0.213443
\(697\) 1.00000 0.0378777
\(698\) −22.9855 −0.870015
\(699\) −21.2204 −0.802630
\(700\) 2.85086 0.107752
\(701\) 0.0502453 0.00189774 0.000948869 1.00000i \(-0.499698\pi\)
0.000948869 1.00000i \(0.499698\pi\)
\(702\) 0 0
\(703\) −4.13706 −0.156032
\(704\) 1.33513 0.0503194
\(705\) 14.2838 0.537960
\(706\) 22.4397 0.844528
\(707\) −6.53319 −0.245706
\(708\) 4.18598 0.157319
\(709\) 39.6544 1.48925 0.744627 0.667481i \(-0.232627\pi\)
0.744627 + 0.667481i \(0.232627\pi\)
\(710\) 14.5918 0.547620
\(711\) −8.80194 −0.330098
\(712\) −3.40044 −0.127437
\(713\) −24.0073 −0.899080
\(714\) −0.356896 −0.0133565
\(715\) 0 0
\(716\) −23.6112 −0.882391
\(717\) 2.09783 0.0783451
\(718\) 6.80492 0.253957
\(719\) 17.0694 0.636580 0.318290 0.947993i \(-0.396891\pi\)
0.318290 + 0.947993i \(0.396891\pi\)
\(720\) −2.80194 −0.104422
\(721\) 0.225209 0.00838723
\(722\) −13.0218 −0.484620
\(723\) 5.22952 0.194488
\(724\) 15.6679 0.582291
\(725\) −16.0532 −0.596202
\(726\) 9.21744 0.342091
\(727\) −0.154048 −0.00571332 −0.00285666 0.999996i \(-0.500909\pi\)
−0.00285666 + 0.999996i \(0.500909\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −2.38404 −0.0882374
\(731\) 2.32544 0.0860094
\(732\) 7.00000 0.258727
\(733\) −31.5415 −1.16501 −0.582507 0.812826i \(-0.697928\pi\)
−0.582507 + 0.812826i \(0.697928\pi\)
\(734\) −21.9409 −0.809854
\(735\) 2.80194 0.103351
\(736\) 2.51573 0.0927310
\(737\) 11.8009 0.434691
\(738\) 2.80194 0.103141
\(739\) 15.6679 0.576351 0.288176 0.957578i \(-0.406951\pi\)
0.288176 + 0.957578i \(0.406951\pi\)
\(740\) −4.74094 −0.174280
\(741\) 0 0
\(742\) 8.25667 0.303112
\(743\) 7.94677 0.291539 0.145769 0.989319i \(-0.453434\pi\)
0.145769 + 0.989319i \(0.453434\pi\)
\(744\) 9.54288 0.349859
\(745\) −11.0707 −0.405599
\(746\) −18.3207 −0.670767
\(747\) −10.4450 −0.382164
\(748\) 0.476501 0.0174226
\(749\) 4.50604 0.164647
\(750\) −6.02177 −0.219884
\(751\) 34.5026 1.25902 0.629509 0.776994i \(-0.283256\pi\)
0.629509 + 0.776994i \(0.283256\pi\)
\(752\) 5.09783 0.185899
\(753\) 11.4426 0.416993
\(754\) 0 0
\(755\) 14.7627 0.537270
\(756\) −1.00000 −0.0363696
\(757\) −20.6732 −0.751382 −0.375691 0.926745i \(-0.622595\pi\)
−0.375691 + 0.926745i \(0.622595\pi\)
\(758\) −25.8950 −0.940548
\(759\) −3.35881 −0.121917
\(760\) 6.85086 0.248507
\(761\) −7.67994 −0.278398 −0.139199 0.990264i \(-0.544453\pi\)
−0.139199 + 0.990264i \(0.544453\pi\)
\(762\) −2.57673 −0.0933450
\(763\) −7.10752 −0.257310
\(764\) −22.5526 −0.815923
\(765\) −1.00000 −0.0361551
\(766\) −20.0218 −0.723416
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 12.3086 0.443858 0.221929 0.975063i \(-0.428765\pi\)
0.221929 + 0.975063i \(0.428765\pi\)
\(770\) −3.74094 −0.134814
\(771\) 21.5623 0.776546
\(772\) −24.2610 −0.873172
\(773\) 25.8291 0.929008 0.464504 0.885571i \(-0.346233\pi\)
0.464504 + 0.885571i \(0.346233\pi\)
\(774\) 6.51573 0.234203
\(775\) −27.2054 −0.977245
\(776\) −8.08815 −0.290348
\(777\) −1.69202 −0.0607009
\(778\) 13.7952 0.494583
\(779\) −6.85086 −0.245457
\(780\) 0 0
\(781\) −6.95300 −0.248798
\(782\) 0.897853 0.0321072
\(783\) 5.63102 0.201236
\(784\) 1.00000 0.0357143
\(785\) 45.2573 1.61530
\(786\) −12.7724 −0.455576
\(787\) −17.0248 −0.606867 −0.303433 0.952853i \(-0.598133\pi\)
−0.303433 + 0.952853i \(0.598133\pi\)
\(788\) −0.411190 −0.0146480
\(789\) 11.5284 0.410422
\(790\) 24.6625 0.877452
\(791\) 2.86831 0.101985
\(792\) 1.33513 0.0474416
\(793\) 0 0
\(794\) −7.07846 −0.251205
\(795\) 23.1347 0.820502
\(796\) −5.73125 −0.203139
\(797\) −48.3919 −1.71413 −0.857065 0.515208i \(-0.827715\pi\)
−0.857065 + 0.515208i \(0.827715\pi\)
\(798\) 2.44504 0.0865535
\(799\) 1.81940 0.0643656
\(800\) 2.85086 0.100793
\(801\) −3.40044 −0.120149
\(802\) −2.94869 −0.104122
\(803\) 1.13600 0.0400885
\(804\) −8.83877 −0.311720
\(805\) −7.04892 −0.248442
\(806\) 0 0
\(807\) 30.5211 1.07439
\(808\) −6.53319 −0.229837
\(809\) −1.51440 −0.0532435 −0.0266218 0.999646i \(-0.508475\pi\)
−0.0266218 + 0.999646i \(0.508475\pi\)
\(810\) −2.80194 −0.0984501
\(811\) −23.0170 −0.808236 −0.404118 0.914707i \(-0.632421\pi\)
−0.404118 + 0.914707i \(0.632421\pi\)
\(812\) −5.63102 −0.197610
\(813\) 12.6160 0.442461
\(814\) 2.25906 0.0791801
\(815\) 33.5743 1.17606
\(816\) −0.356896 −0.0124939
\(817\) −15.9312 −0.557363
\(818\) −8.41358 −0.294174
\(819\) 0 0
\(820\) −7.85086 −0.274164
\(821\) −5.90754 −0.206175 −0.103087 0.994672i \(-0.532872\pi\)
−0.103087 + 0.994672i \(0.532872\pi\)
\(822\) 7.30798 0.254895
\(823\) 45.2567 1.57755 0.788774 0.614683i \(-0.210716\pi\)
0.788774 + 0.614683i \(0.210716\pi\)
\(824\) 0.225209 0.00784554
\(825\) −3.80625 −0.132517
\(826\) −4.18598 −0.145649
\(827\) 28.6534 0.996376 0.498188 0.867069i \(-0.333999\pi\)
0.498188 + 0.867069i \(0.333999\pi\)
\(828\) 2.51573 0.0874276
\(829\) −20.4257 −0.709413 −0.354706 0.934978i \(-0.615419\pi\)
−0.354706 + 0.934978i \(0.615419\pi\)
\(830\) 29.2664 1.01585
\(831\) −14.7899 −0.513054
\(832\) 0 0
\(833\) 0.356896 0.0123657
\(834\) −22.9812 −0.795774
\(835\) 52.1353 1.80422
\(836\) −3.26444 −0.112903
\(837\) 9.54288 0.329850
\(838\) 36.9168 1.27527
\(839\) 43.5623 1.50394 0.751968 0.659200i \(-0.229105\pi\)
0.751968 + 0.659200i \(0.229105\pi\)
\(840\) 2.80194 0.0966760
\(841\) 2.70841 0.0933936
\(842\) 5.42221 0.186862
\(843\) −15.5623 −0.535992
\(844\) −24.5284 −0.844302
\(845\) 0 0
\(846\) 5.09783 0.175267
\(847\) −9.21744 −0.316715
\(848\) 8.25667 0.283535
\(849\) −17.5894 −0.603667
\(850\) 1.01746 0.0348985
\(851\) 4.25667 0.145917
\(852\) 5.20775 0.178415
\(853\) 2.45473 0.0840484 0.0420242 0.999117i \(-0.486619\pi\)
0.0420242 + 0.999117i \(0.486619\pi\)
\(854\) −7.00000 −0.239535
\(855\) 6.85086 0.234294
\(856\) 4.50604 0.154013
\(857\) −1.14616 −0.0391521 −0.0195761 0.999808i \(-0.506232\pi\)
−0.0195761 + 0.999808i \(0.506232\pi\)
\(858\) 0 0
\(859\) 33.7162 1.15038 0.575191 0.818019i \(-0.304928\pi\)
0.575191 + 0.818019i \(0.304928\pi\)
\(860\) −18.2567 −0.622547
\(861\) −2.80194 −0.0954898
\(862\) −29.8442 −1.01650
\(863\) 40.5924 1.38178 0.690890 0.722959i \(-0.257219\pi\)
0.690890 + 0.722959i \(0.257219\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −42.1473 −1.43305
\(866\) −19.4873 −0.662204
\(867\) 16.8726 0.573024
\(868\) −9.54288 −0.323906
\(869\) −11.7517 −0.398649
\(870\) −15.7778 −0.534917
\(871\) 0 0
\(872\) −7.10752 −0.240691
\(873\) −8.08815 −0.273742
\(874\) −6.15106 −0.208063
\(875\) 6.02177 0.203573
\(876\) −0.850855 −0.0287477
\(877\) −28.4969 −0.962273 −0.481137 0.876646i \(-0.659776\pi\)
−0.481137 + 0.876646i \(0.659776\pi\)
\(878\) 36.6450 1.23671
\(879\) −11.4263 −0.385398
\(880\) −3.74094 −0.126107
\(881\) 30.6883 1.03392 0.516958 0.856011i \(-0.327064\pi\)
0.516958 + 0.856011i \(0.327064\pi\)
\(882\) 1.00000 0.0336718
\(883\) 44.5230 1.49832 0.749160 0.662390i \(-0.230458\pi\)
0.749160 + 0.662390i \(0.230458\pi\)
\(884\) 0 0
\(885\) −11.7289 −0.394261
\(886\) 34.6679 1.16469
\(887\) −2.76749 −0.0929234 −0.0464617 0.998920i \(-0.514795\pi\)
−0.0464617 + 0.998920i \(0.514795\pi\)
\(888\) −1.69202 −0.0567805
\(889\) 2.57673 0.0864207
\(890\) 9.52781 0.319373
\(891\) 1.33513 0.0447284
\(892\) −21.0532 −0.704914
\(893\) −12.4644 −0.417106
\(894\) −3.95108 −0.132144
\(895\) 66.1570 2.21139
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 8.72455 0.291142
\(899\) 53.7362 1.79220
\(900\) 2.85086 0.0950285
\(901\) 2.94677 0.0981712
\(902\) 3.74094 0.124560
\(903\) −6.51573 −0.216830
\(904\) 2.86831 0.0953987
\(905\) −43.9004 −1.45930
\(906\) 5.26875 0.175042
\(907\) 16.7496 0.556160 0.278080 0.960558i \(-0.410302\pi\)
0.278080 + 0.960558i \(0.410302\pi\)
\(908\) 3.49827 0.116094
\(909\) −6.53319 −0.216692
\(910\) 0 0
\(911\) −34.1866 −1.13265 −0.566326 0.824181i \(-0.691636\pi\)
−0.566326 + 0.824181i \(0.691636\pi\)
\(912\) 2.44504 0.0809634
\(913\) −13.9454 −0.461527
\(914\) −19.4300 −0.642686
\(915\) −19.6136 −0.648404
\(916\) −22.0097 −0.727221
\(917\) 12.7724 0.421782
\(918\) −0.356896 −0.0117793
\(919\) −28.4118 −0.937218 −0.468609 0.883406i \(-0.655245\pi\)
−0.468609 + 0.883406i \(0.655245\pi\)
\(920\) −7.04892 −0.232396
\(921\) 10.5670 0.348196
\(922\) 35.0573 1.15455
\(923\) 0 0
\(924\) −1.33513 −0.0439224
\(925\) 4.82371 0.158603
\(926\) 2.20105 0.0723309
\(927\) 0.225209 0.00739685
\(928\) −5.63102 −0.184847
\(929\) −23.9366 −0.785335 −0.392667 0.919681i \(-0.628448\pi\)
−0.392667 + 0.919681i \(0.628448\pi\)
\(930\) −26.7385 −0.876791
\(931\) −2.44504 −0.0801330
\(932\) 21.2204 0.695098
\(933\) 7.40821 0.242534
\(934\) −8.25667 −0.270166
\(935\) −1.33513 −0.0436633
\(936\) 0 0
\(937\) 18.6219 0.608352 0.304176 0.952616i \(-0.401619\pi\)
0.304176 + 0.952616i \(0.401619\pi\)
\(938\) 8.83877 0.288596
\(939\) 2.12737 0.0694242
\(940\) −14.2838 −0.465887
\(941\) 37.4677 1.22141 0.610706 0.791858i \(-0.290886\pi\)
0.610706 + 0.791858i \(0.290886\pi\)
\(942\) 16.1521 0.526265
\(943\) 7.04892 0.229544
\(944\) −4.18598 −0.136242
\(945\) 2.80194 0.0911470
\(946\) 8.69932 0.282839
\(947\) −15.4588 −0.502343 −0.251171 0.967943i \(-0.580816\pi\)
−0.251171 + 0.967943i \(0.580816\pi\)
\(948\) 8.80194 0.285874
\(949\) 0 0
\(950\) −6.97046 −0.226152
\(951\) −32.6383 −1.05837
\(952\) 0.356896 0.0115671
\(953\) 0.151064 0.00489344 0.00244672 0.999997i \(-0.499221\pi\)
0.00244672 + 0.999997i \(0.499221\pi\)
\(954\) 8.25667 0.267320
\(955\) 63.1909 2.04481
\(956\) −2.09783 −0.0678488
\(957\) 7.51812 0.243026
\(958\) −28.9439 −0.935135
\(959\) −7.30798 −0.235987
\(960\) 2.80194 0.0904322
\(961\) 60.0665 1.93763
\(962\) 0 0
\(963\) 4.50604 0.145205
\(964\) −5.22952 −0.168432
\(965\) 67.9778 2.18828
\(966\) −2.51573 −0.0809423
\(967\) −17.5532 −0.564471 −0.282236 0.959345i \(-0.591076\pi\)
−0.282236 + 0.959345i \(0.591076\pi\)
\(968\) −9.21744 −0.296260
\(969\) 0.872625 0.0280328
\(970\) 22.6625 0.727648
\(971\) 30.2228 0.969896 0.484948 0.874543i \(-0.338838\pi\)
0.484948 + 0.874543i \(0.338838\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 22.9812 0.736744
\(974\) 3.39804 0.108880
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) −47.7230 −1.52679 −0.763397 0.645929i \(-0.776470\pi\)
−0.763397 + 0.645929i \(0.776470\pi\)
\(978\) 11.9825 0.383159
\(979\) −4.54001 −0.145099
\(980\) −2.80194 −0.0895046
\(981\) −7.10752 −0.226926
\(982\) 28.3327 0.904134
\(983\) −23.3317 −0.744165 −0.372082 0.928200i \(-0.621356\pi\)
−0.372082 + 0.928200i \(0.621356\pi\)
\(984\) −2.80194 −0.0893225
\(985\) 1.15213 0.0367099
\(986\) −2.00969 −0.0640015
\(987\) −5.09783 −0.162266
\(988\) 0 0
\(989\) 16.3918 0.521229
\(990\) −3.74094 −0.118895
\(991\) −7.16229 −0.227518 −0.113759 0.993508i \(-0.536289\pi\)
−0.113759 + 0.993508i \(0.536289\pi\)
\(992\) −9.54288 −0.302987
\(993\) 30.0006 0.952040
\(994\) −5.20775 −0.165180
\(995\) 16.0586 0.509092
\(996\) 10.4450 0.330964
\(997\) 24.4537 0.774455 0.387228 0.921984i \(-0.373433\pi\)
0.387228 + 0.921984i \(0.373433\pi\)
\(998\) 11.2664 0.356630
\(999\) −1.69202 −0.0535332
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 7098.2.a.ch.1.1 yes 3
13.12 even 2 7098.2.a.ce.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7098.2.a.ce.1.3 3 13.12 even 2
7098.2.a.ch.1.1 yes 3 1.1 even 1 trivial