Properties

Label 8004.2.a.i.1.2
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.33956\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.33956 q^{5} -2.20985 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.33956 q^{5} -2.20985 q^{7} +1.00000 q^{9} +4.06495 q^{11} +6.44461 q^{13} +3.33956 q^{15} +6.35391 q^{17} +6.75414 q^{19} +2.20985 q^{21} +1.00000 q^{23} +6.15268 q^{25} -1.00000 q^{27} +1.00000 q^{29} +3.50912 q^{31} -4.06495 q^{33} +7.37993 q^{35} +1.43441 q^{37} -6.44461 q^{39} +10.9947 q^{41} -11.9008 q^{43} -3.33956 q^{45} -5.99574 q^{47} -2.11656 q^{49} -6.35391 q^{51} -4.14466 q^{53} -13.5752 q^{55} -6.75414 q^{57} +1.75491 q^{59} +15.3917 q^{61} -2.20985 q^{63} -21.5222 q^{65} -2.00839 q^{67} -1.00000 q^{69} +16.1058 q^{71} -4.96314 q^{73} -6.15268 q^{75} -8.98293 q^{77} -6.38974 q^{79} +1.00000 q^{81} +13.0549 q^{83} -21.2193 q^{85} -1.00000 q^{87} +6.82075 q^{89} -14.2416 q^{91} -3.50912 q^{93} -22.5559 q^{95} +3.18129 q^{97} +4.06495 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + 5q^{11} + 8q^{13} - 5q^{15} + 7q^{17} + q^{19} + 4q^{21} + 16q^{23} + 31q^{25} - 16q^{27} + 16q^{29} - 2q^{31} - 5q^{33} + 5q^{35} + 14q^{37} - 8q^{39} - q^{41} - 13q^{43} + 5q^{45} - 4q^{47} + 30q^{49} - 7q^{51} + 19q^{53} - 37q^{55} - q^{57} + 12q^{59} + 21q^{61} - 4q^{63} + 26q^{65} - 11q^{67} - 16q^{69} + 7q^{71} - 13q^{73} - 31q^{75} + 4q^{77} - 18q^{79} + 16q^{81} + 25q^{83} + 48q^{85} - 16q^{87} + 12q^{89} - 11q^{91} + 2q^{93} - q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −3.33956 −1.49350 −0.746749 0.665106i \(-0.768386\pi\)
−0.746749 + 0.665106i \(0.768386\pi\)
\(6\) 0 0
\(7\) −2.20985 −0.835245 −0.417622 0.908621i \(-0.637136\pi\)
−0.417622 + 0.908621i \(0.637136\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.06495 1.22563 0.612814 0.790227i \(-0.290037\pi\)
0.612814 + 0.790227i \(0.290037\pi\)
\(12\) 0 0
\(13\) 6.44461 1.78741 0.893707 0.448651i \(-0.148095\pi\)
0.893707 + 0.448651i \(0.148095\pi\)
\(14\) 0 0
\(15\) 3.33956 0.862271
\(16\) 0 0
\(17\) 6.35391 1.54105 0.770525 0.637410i \(-0.219994\pi\)
0.770525 + 0.637410i \(0.219994\pi\)
\(18\) 0 0
\(19\) 6.75414 1.54951 0.774753 0.632264i \(-0.217874\pi\)
0.774753 + 0.632264i \(0.217874\pi\)
\(20\) 0 0
\(21\) 2.20985 0.482229
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 6.15268 1.23054
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 3.50912 0.630256 0.315128 0.949049i \(-0.397953\pi\)
0.315128 + 0.949049i \(0.397953\pi\)
\(32\) 0 0
\(33\) −4.06495 −0.707617
\(34\) 0 0
\(35\) 7.37993 1.24744
\(36\) 0 0
\(37\) 1.43441 0.235816 0.117908 0.993025i \(-0.462381\pi\)
0.117908 + 0.993025i \(0.462381\pi\)
\(38\) 0 0
\(39\) −6.44461 −1.03196
\(40\) 0 0
\(41\) 10.9947 1.71708 0.858541 0.512746i \(-0.171372\pi\)
0.858541 + 0.512746i \(0.171372\pi\)
\(42\) 0 0
\(43\) −11.9008 −1.81485 −0.907425 0.420214i \(-0.861955\pi\)
−0.907425 + 0.420214i \(0.861955\pi\)
\(44\) 0 0
\(45\) −3.33956 −0.497833
\(46\) 0 0
\(47\) −5.99574 −0.874568 −0.437284 0.899323i \(-0.644060\pi\)
−0.437284 + 0.899323i \(0.644060\pi\)
\(48\) 0 0
\(49\) −2.11656 −0.302366
\(50\) 0 0
\(51\) −6.35391 −0.889726
\(52\) 0 0
\(53\) −4.14466 −0.569313 −0.284656 0.958630i \(-0.591879\pi\)
−0.284656 + 0.958630i \(0.591879\pi\)
\(54\) 0 0
\(55\) −13.5752 −1.83047
\(56\) 0 0
\(57\) −6.75414 −0.894608
\(58\) 0 0
\(59\) 1.75491 0.228470 0.114235 0.993454i \(-0.463558\pi\)
0.114235 + 0.993454i \(0.463558\pi\)
\(60\) 0 0
\(61\) 15.3917 1.97071 0.985355 0.170514i \(-0.0545427\pi\)
0.985355 + 0.170514i \(0.0545427\pi\)
\(62\) 0 0
\(63\) −2.20985 −0.278415
\(64\) 0 0
\(65\) −21.5222 −2.66950
\(66\) 0 0
\(67\) −2.00839 −0.245364 −0.122682 0.992446i \(-0.539150\pi\)
−0.122682 + 0.992446i \(0.539150\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 16.1058 1.91141 0.955705 0.294326i \(-0.0950951\pi\)
0.955705 + 0.294326i \(0.0950951\pi\)
\(72\) 0 0
\(73\) −4.96314 −0.580891 −0.290446 0.956891i \(-0.593804\pi\)
−0.290446 + 0.956891i \(0.593804\pi\)
\(74\) 0 0
\(75\) −6.15268 −0.710450
\(76\) 0 0
\(77\) −8.98293 −1.02370
\(78\) 0 0
\(79\) −6.38974 −0.718902 −0.359451 0.933164i \(-0.617036\pi\)
−0.359451 + 0.933164i \(0.617036\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.0549 1.43296 0.716482 0.697605i \(-0.245751\pi\)
0.716482 + 0.697605i \(0.245751\pi\)
\(84\) 0 0
\(85\) −21.2193 −2.30155
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 6.82075 0.722998 0.361499 0.932373i \(-0.382265\pi\)
0.361499 + 0.932373i \(0.382265\pi\)
\(90\) 0 0
\(91\) −14.2416 −1.49293
\(92\) 0 0
\(93\) −3.50912 −0.363879
\(94\) 0 0
\(95\) −22.5559 −2.31418
\(96\) 0 0
\(97\) 3.18129 0.323011 0.161505 0.986872i \(-0.448365\pi\)
0.161505 + 0.986872i \(0.448365\pi\)
\(98\) 0 0
\(99\) 4.06495 0.408543
\(100\) 0 0
\(101\) −7.11345 −0.707815 −0.353908 0.935280i \(-0.615147\pi\)
−0.353908 + 0.935280i \(0.615147\pi\)
\(102\) 0 0
\(103\) −4.76739 −0.469745 −0.234872 0.972026i \(-0.575467\pi\)
−0.234872 + 0.972026i \(0.575467\pi\)
\(104\) 0 0
\(105\) −7.37993 −0.720208
\(106\) 0 0
\(107\) 2.27498 0.219930 0.109965 0.993935i \(-0.464926\pi\)
0.109965 + 0.993935i \(0.464926\pi\)
\(108\) 0 0
\(109\) −7.62414 −0.730260 −0.365130 0.930957i \(-0.618975\pi\)
−0.365130 + 0.930957i \(0.618975\pi\)
\(110\) 0 0
\(111\) −1.43441 −0.136148
\(112\) 0 0
\(113\) −4.87542 −0.458641 −0.229321 0.973351i \(-0.573650\pi\)
−0.229321 + 0.973351i \(0.573650\pi\)
\(114\) 0 0
\(115\) −3.33956 −0.311416
\(116\) 0 0
\(117\) 6.44461 0.595805
\(118\) 0 0
\(119\) −14.0412 −1.28715
\(120\) 0 0
\(121\) 5.52383 0.502166
\(122\) 0 0
\(123\) −10.9947 −0.991357
\(124\) 0 0
\(125\) −3.84944 −0.344304
\(126\) 0 0
\(127\) −7.55819 −0.670681 −0.335340 0.942097i \(-0.608851\pi\)
−0.335340 + 0.942097i \(0.608851\pi\)
\(128\) 0 0
\(129\) 11.9008 1.04780
\(130\) 0 0
\(131\) −3.40700 −0.297671 −0.148836 0.988862i \(-0.547553\pi\)
−0.148836 + 0.988862i \(0.547553\pi\)
\(132\) 0 0
\(133\) −14.9256 −1.29422
\(134\) 0 0
\(135\) 3.33956 0.287424
\(136\) 0 0
\(137\) 7.48240 0.639264 0.319632 0.947542i \(-0.396441\pi\)
0.319632 + 0.947542i \(0.396441\pi\)
\(138\) 0 0
\(139\) −18.1783 −1.54187 −0.770933 0.636917i \(-0.780210\pi\)
−0.770933 + 0.636917i \(0.780210\pi\)
\(140\) 0 0
\(141\) 5.99574 0.504932
\(142\) 0 0
\(143\) 26.1970 2.19071
\(144\) 0 0
\(145\) −3.33956 −0.277336
\(146\) 0 0
\(147\) 2.11656 0.174571
\(148\) 0 0
\(149\) 12.2182 1.00096 0.500479 0.865749i \(-0.333157\pi\)
0.500479 + 0.865749i \(0.333157\pi\)
\(150\) 0 0
\(151\) 9.96548 0.810979 0.405489 0.914100i \(-0.367101\pi\)
0.405489 + 0.914100i \(0.367101\pi\)
\(152\) 0 0
\(153\) 6.35391 0.513683
\(154\) 0 0
\(155\) −11.7189 −0.941286
\(156\) 0 0
\(157\) 20.5997 1.64403 0.822017 0.569463i \(-0.192849\pi\)
0.822017 + 0.569463i \(0.192849\pi\)
\(158\) 0 0
\(159\) 4.14466 0.328693
\(160\) 0 0
\(161\) −2.20985 −0.174161
\(162\) 0 0
\(163\) 17.1951 1.34682 0.673411 0.739268i \(-0.264829\pi\)
0.673411 + 0.739268i \(0.264829\pi\)
\(164\) 0 0
\(165\) 13.5752 1.05682
\(166\) 0 0
\(167\) 23.5313 1.82091 0.910453 0.413613i \(-0.135733\pi\)
0.910453 + 0.413613i \(0.135733\pi\)
\(168\) 0 0
\(169\) 28.5331 2.19485
\(170\) 0 0
\(171\) 6.75414 0.516502
\(172\) 0 0
\(173\) −24.7699 −1.88322 −0.941612 0.336700i \(-0.890689\pi\)
−0.941612 + 0.336700i \(0.890689\pi\)
\(174\) 0 0
\(175\) −13.5965 −1.02780
\(176\) 0 0
\(177\) −1.75491 −0.131907
\(178\) 0 0
\(179\) 0.379109 0.0283359 0.0141680 0.999900i \(-0.495490\pi\)
0.0141680 + 0.999900i \(0.495490\pi\)
\(180\) 0 0
\(181\) −10.6216 −0.789499 −0.394749 0.918789i \(-0.629169\pi\)
−0.394749 + 0.918789i \(0.629169\pi\)
\(182\) 0 0
\(183\) −15.3917 −1.13779
\(184\) 0 0
\(185\) −4.79030 −0.352190
\(186\) 0 0
\(187\) 25.8283 1.88876
\(188\) 0 0
\(189\) 2.20985 0.160743
\(190\) 0 0
\(191\) 3.59888 0.260406 0.130203 0.991487i \(-0.458437\pi\)
0.130203 + 0.991487i \(0.458437\pi\)
\(192\) 0 0
\(193\) −0.135250 −0.00973548 −0.00486774 0.999988i \(-0.501549\pi\)
−0.00486774 + 0.999988i \(0.501549\pi\)
\(194\) 0 0
\(195\) 21.5222 1.54124
\(196\) 0 0
\(197\) −21.5135 −1.53277 −0.766387 0.642379i \(-0.777948\pi\)
−0.766387 + 0.642379i \(0.777948\pi\)
\(198\) 0 0
\(199\) −13.9483 −0.988771 −0.494385 0.869243i \(-0.664607\pi\)
−0.494385 + 0.869243i \(0.664607\pi\)
\(200\) 0 0
\(201\) 2.00839 0.141661
\(202\) 0 0
\(203\) −2.20985 −0.155101
\(204\) 0 0
\(205\) −36.7174 −2.56446
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 27.4553 1.89912
\(210\) 0 0
\(211\) 2.41989 0.166592 0.0832960 0.996525i \(-0.473455\pi\)
0.0832960 + 0.996525i \(0.473455\pi\)
\(212\) 0 0
\(213\) −16.1058 −1.10355
\(214\) 0 0
\(215\) 39.7434 2.71048
\(216\) 0 0
\(217\) −7.75462 −0.526418
\(218\) 0 0
\(219\) 4.96314 0.335378
\(220\) 0 0
\(221\) 40.9485 2.75450
\(222\) 0 0
\(223\) −7.01623 −0.469842 −0.234921 0.972015i \(-0.575483\pi\)
−0.234921 + 0.972015i \(0.575483\pi\)
\(224\) 0 0
\(225\) 6.15268 0.410178
\(226\) 0 0
\(227\) −29.3325 −1.94687 −0.973434 0.228967i \(-0.926465\pi\)
−0.973434 + 0.228967i \(0.926465\pi\)
\(228\) 0 0
\(229\) −4.07873 −0.269530 −0.134765 0.990878i \(-0.543028\pi\)
−0.134765 + 0.990878i \(0.543028\pi\)
\(230\) 0 0
\(231\) 8.98293 0.591034
\(232\) 0 0
\(233\) −17.7597 −1.16348 −0.581739 0.813375i \(-0.697628\pi\)
−0.581739 + 0.813375i \(0.697628\pi\)
\(234\) 0 0
\(235\) 20.0231 1.30617
\(236\) 0 0
\(237\) 6.38974 0.415058
\(238\) 0 0
\(239\) 6.71824 0.434567 0.217283 0.976109i \(-0.430280\pi\)
0.217283 + 0.976109i \(0.430280\pi\)
\(240\) 0 0
\(241\) 1.59497 0.102741 0.0513706 0.998680i \(-0.483641\pi\)
0.0513706 + 0.998680i \(0.483641\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 7.06840 0.451583
\(246\) 0 0
\(247\) 43.5278 2.76961
\(248\) 0 0
\(249\) −13.0549 −0.827323
\(250\) 0 0
\(251\) 10.5729 0.667354 0.333677 0.942687i \(-0.391710\pi\)
0.333677 + 0.942687i \(0.391710\pi\)
\(252\) 0 0
\(253\) 4.06495 0.255561
\(254\) 0 0
\(255\) 21.2193 1.32880
\(256\) 0 0
\(257\) 20.3609 1.27008 0.635038 0.772481i \(-0.280984\pi\)
0.635038 + 0.772481i \(0.280984\pi\)
\(258\) 0 0
\(259\) −3.16983 −0.196964
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −5.97404 −0.368375 −0.184187 0.982891i \(-0.558965\pi\)
−0.184187 + 0.982891i \(0.558965\pi\)
\(264\) 0 0
\(265\) 13.8413 0.850267
\(266\) 0 0
\(267\) −6.82075 −0.417423
\(268\) 0 0
\(269\) 29.5777 1.80338 0.901692 0.432380i \(-0.142326\pi\)
0.901692 + 0.432380i \(0.142326\pi\)
\(270\) 0 0
\(271\) −3.19390 −0.194016 −0.0970079 0.995284i \(-0.530927\pi\)
−0.0970079 + 0.995284i \(0.530927\pi\)
\(272\) 0 0
\(273\) 14.2416 0.861943
\(274\) 0 0
\(275\) 25.0103 1.50818
\(276\) 0 0
\(277\) −12.4087 −0.745564 −0.372782 0.927919i \(-0.621596\pi\)
−0.372782 + 0.927919i \(0.621596\pi\)
\(278\) 0 0
\(279\) 3.50912 0.210085
\(280\) 0 0
\(281\) −7.15180 −0.426641 −0.213320 0.976982i \(-0.568428\pi\)
−0.213320 + 0.976982i \(0.568428\pi\)
\(282\) 0 0
\(283\) −20.2325 −1.20270 −0.601349 0.798986i \(-0.705370\pi\)
−0.601349 + 0.798986i \(0.705370\pi\)
\(284\) 0 0
\(285\) 22.5559 1.33609
\(286\) 0 0
\(287\) −24.2966 −1.43418
\(288\) 0 0
\(289\) 23.3722 1.37484
\(290\) 0 0
\(291\) −3.18129 −0.186490
\(292\) 0 0
\(293\) −5.13032 −0.299716 −0.149858 0.988708i \(-0.547882\pi\)
−0.149858 + 0.988708i \(0.547882\pi\)
\(294\) 0 0
\(295\) −5.86064 −0.341220
\(296\) 0 0
\(297\) −4.06495 −0.235872
\(298\) 0 0
\(299\) 6.44461 0.372702
\(300\) 0 0
\(301\) 26.2989 1.51584
\(302\) 0 0
\(303\) 7.11345 0.408657
\(304\) 0 0
\(305\) −51.4017 −2.94325
\(306\) 0 0
\(307\) 16.6305 0.949156 0.474578 0.880214i \(-0.342601\pi\)
0.474578 + 0.880214i \(0.342601\pi\)
\(308\) 0 0
\(309\) 4.76739 0.271207
\(310\) 0 0
\(311\) −26.2790 −1.49015 −0.745074 0.666982i \(-0.767586\pi\)
−0.745074 + 0.666982i \(0.767586\pi\)
\(312\) 0 0
\(313\) 5.80775 0.328274 0.164137 0.986438i \(-0.447516\pi\)
0.164137 + 0.986438i \(0.447516\pi\)
\(314\) 0 0
\(315\) 7.37993 0.415812
\(316\) 0 0
\(317\) 25.3663 1.42472 0.712358 0.701816i \(-0.247627\pi\)
0.712358 + 0.701816i \(0.247627\pi\)
\(318\) 0 0
\(319\) 4.06495 0.227594
\(320\) 0 0
\(321\) −2.27498 −0.126977
\(322\) 0 0
\(323\) 42.9152 2.38787
\(324\) 0 0
\(325\) 39.6516 2.19948
\(326\) 0 0
\(327\) 7.62414 0.421616
\(328\) 0 0
\(329\) 13.2497 0.730479
\(330\) 0 0
\(331\) 5.39064 0.296296 0.148148 0.988965i \(-0.452669\pi\)
0.148148 + 0.988965i \(0.452669\pi\)
\(332\) 0 0
\(333\) 1.43441 0.0786052
\(334\) 0 0
\(335\) 6.70714 0.366450
\(336\) 0 0
\(337\) 2.36702 0.128940 0.0644700 0.997920i \(-0.479464\pi\)
0.0644700 + 0.997920i \(0.479464\pi\)
\(338\) 0 0
\(339\) 4.87542 0.264797
\(340\) 0 0
\(341\) 14.2644 0.772460
\(342\) 0 0
\(343\) 20.1462 1.08779
\(344\) 0 0
\(345\) 3.33956 0.179796
\(346\) 0 0
\(347\) 15.8636 0.851601 0.425800 0.904817i \(-0.359993\pi\)
0.425800 + 0.904817i \(0.359993\pi\)
\(348\) 0 0
\(349\) 4.17563 0.223516 0.111758 0.993735i \(-0.464352\pi\)
0.111758 + 0.993735i \(0.464352\pi\)
\(350\) 0 0
\(351\) −6.44461 −0.343988
\(352\) 0 0
\(353\) 32.0996 1.70849 0.854245 0.519871i \(-0.174020\pi\)
0.854245 + 0.519871i \(0.174020\pi\)
\(354\) 0 0
\(355\) −53.7864 −2.85469
\(356\) 0 0
\(357\) 14.0412 0.743139
\(358\) 0 0
\(359\) −7.60469 −0.401360 −0.200680 0.979657i \(-0.564315\pi\)
−0.200680 + 0.979657i \(0.564315\pi\)
\(360\) 0 0
\(361\) 26.6184 1.40097
\(362\) 0 0
\(363\) −5.52383 −0.289926
\(364\) 0 0
\(365\) 16.5747 0.867560
\(366\) 0 0
\(367\) −21.8342 −1.13974 −0.569868 0.821736i \(-0.693006\pi\)
−0.569868 + 0.821736i \(0.693006\pi\)
\(368\) 0 0
\(369\) 10.9947 0.572360
\(370\) 0 0
\(371\) 9.15907 0.475515
\(372\) 0 0
\(373\) −24.2823 −1.25729 −0.628646 0.777692i \(-0.716391\pi\)
−0.628646 + 0.777692i \(0.716391\pi\)
\(374\) 0 0
\(375\) 3.84944 0.198784
\(376\) 0 0
\(377\) 6.44461 0.331915
\(378\) 0 0
\(379\) 37.0281 1.90200 0.951002 0.309183i \(-0.100056\pi\)
0.951002 + 0.309183i \(0.100056\pi\)
\(380\) 0 0
\(381\) 7.55819 0.387218
\(382\) 0 0
\(383\) −32.5243 −1.66192 −0.830958 0.556335i \(-0.812207\pi\)
−0.830958 + 0.556335i \(0.812207\pi\)
\(384\) 0 0
\(385\) 29.9991 1.52889
\(386\) 0 0
\(387\) −11.9008 −0.604950
\(388\) 0 0
\(389\) −11.3820 −0.577091 −0.288545 0.957466i \(-0.593172\pi\)
−0.288545 + 0.957466i \(0.593172\pi\)
\(390\) 0 0
\(391\) 6.35391 0.321331
\(392\) 0 0
\(393\) 3.40700 0.171861
\(394\) 0 0
\(395\) 21.3389 1.07368
\(396\) 0 0
\(397\) 5.65412 0.283772 0.141886 0.989883i \(-0.454683\pi\)
0.141886 + 0.989883i \(0.454683\pi\)
\(398\) 0 0
\(399\) 14.9256 0.747217
\(400\) 0 0
\(401\) −1.13254 −0.0565563 −0.0282782 0.999600i \(-0.509002\pi\)
−0.0282782 + 0.999600i \(0.509002\pi\)
\(402\) 0 0
\(403\) 22.6149 1.12653
\(404\) 0 0
\(405\) −3.33956 −0.165944
\(406\) 0 0
\(407\) 5.83081 0.289023
\(408\) 0 0
\(409\) 6.07242 0.300262 0.150131 0.988666i \(-0.452030\pi\)
0.150131 + 0.988666i \(0.452030\pi\)
\(410\) 0 0
\(411\) −7.48240 −0.369080
\(412\) 0 0
\(413\) −3.87809 −0.190829
\(414\) 0 0
\(415\) −43.5978 −2.14013
\(416\) 0 0
\(417\) 18.1783 0.890196
\(418\) 0 0
\(419\) 9.05737 0.442481 0.221241 0.975219i \(-0.428989\pi\)
0.221241 + 0.975219i \(0.428989\pi\)
\(420\) 0 0
\(421\) −14.9464 −0.728444 −0.364222 0.931312i \(-0.618665\pi\)
−0.364222 + 0.931312i \(0.618665\pi\)
\(422\) 0 0
\(423\) −5.99574 −0.291523
\(424\) 0 0
\(425\) 39.0936 1.89632
\(426\) 0 0
\(427\) −34.0134 −1.64603
\(428\) 0 0
\(429\) −26.1970 −1.26481
\(430\) 0 0
\(431\) −9.32236 −0.449042 −0.224521 0.974469i \(-0.572082\pi\)
−0.224521 + 0.974469i \(0.572082\pi\)
\(432\) 0 0
\(433\) 10.0577 0.483343 0.241672 0.970358i \(-0.422304\pi\)
0.241672 + 0.970358i \(0.422304\pi\)
\(434\) 0 0
\(435\) 3.33956 0.160120
\(436\) 0 0
\(437\) 6.75414 0.323094
\(438\) 0 0
\(439\) −33.0296 −1.57642 −0.788208 0.615409i \(-0.788991\pi\)
−0.788208 + 0.615409i \(0.788991\pi\)
\(440\) 0 0
\(441\) −2.11656 −0.100789
\(442\) 0 0
\(443\) −17.4678 −0.829920 −0.414960 0.909840i \(-0.636204\pi\)
−0.414960 + 0.909840i \(0.636204\pi\)
\(444\) 0 0
\(445\) −22.7783 −1.07980
\(446\) 0 0
\(447\) −12.2182 −0.577903
\(448\) 0 0
\(449\) 17.7256 0.836525 0.418262 0.908326i \(-0.362639\pi\)
0.418262 + 0.908326i \(0.362639\pi\)
\(450\) 0 0
\(451\) 44.6929 2.10450
\(452\) 0 0
\(453\) −9.96548 −0.468219
\(454\) 0 0
\(455\) 47.5608 2.22969
\(456\) 0 0
\(457\) −30.5628 −1.42967 −0.714835 0.699294i \(-0.753498\pi\)
−0.714835 + 0.699294i \(0.753498\pi\)
\(458\) 0 0
\(459\) −6.35391 −0.296575
\(460\) 0 0
\(461\) 25.2845 1.17762 0.588808 0.808273i \(-0.299597\pi\)
0.588808 + 0.808273i \(0.299597\pi\)
\(462\) 0 0
\(463\) −37.9300 −1.76276 −0.881379 0.472409i \(-0.843384\pi\)
−0.881379 + 0.472409i \(0.843384\pi\)
\(464\) 0 0
\(465\) 11.7189 0.543452
\(466\) 0 0
\(467\) −5.04272 −0.233349 −0.116675 0.993170i \(-0.537223\pi\)
−0.116675 + 0.993170i \(0.537223\pi\)
\(468\) 0 0
\(469\) 4.43824 0.204939
\(470\) 0 0
\(471\) −20.5997 −0.949183
\(472\) 0 0
\(473\) −48.3761 −2.22433
\(474\) 0 0
\(475\) 41.5561 1.90672
\(476\) 0 0
\(477\) −4.14466 −0.189771
\(478\) 0 0
\(479\) −3.70719 −0.169386 −0.0846929 0.996407i \(-0.526991\pi\)
−0.0846929 + 0.996407i \(0.526991\pi\)
\(480\) 0 0
\(481\) 9.24422 0.421500
\(482\) 0 0
\(483\) 2.20985 0.100552
\(484\) 0 0
\(485\) −10.6241 −0.482416
\(486\) 0 0
\(487\) −22.2968 −1.01037 −0.505183 0.863012i \(-0.668575\pi\)
−0.505183 + 0.863012i \(0.668575\pi\)
\(488\) 0 0
\(489\) −17.1951 −0.777588
\(490\) 0 0
\(491\) 22.7769 1.02791 0.513953 0.857818i \(-0.328180\pi\)
0.513953 + 0.857818i \(0.328180\pi\)
\(492\) 0 0
\(493\) 6.35391 0.286166
\(494\) 0 0
\(495\) −13.5752 −0.610158
\(496\) 0 0
\(497\) −35.5915 −1.59650
\(498\) 0 0
\(499\) −1.10096 −0.0492856 −0.0246428 0.999696i \(-0.507845\pi\)
−0.0246428 + 0.999696i \(0.507845\pi\)
\(500\) 0 0
\(501\) −23.5313 −1.05130
\(502\) 0 0
\(503\) 17.8060 0.793931 0.396965 0.917834i \(-0.370063\pi\)
0.396965 + 0.917834i \(0.370063\pi\)
\(504\) 0 0
\(505\) 23.7558 1.05712
\(506\) 0 0
\(507\) −28.5331 −1.26720
\(508\) 0 0
\(509\) −43.6932 −1.93667 −0.968334 0.249660i \(-0.919681\pi\)
−0.968334 + 0.249660i \(0.919681\pi\)
\(510\) 0 0
\(511\) 10.9678 0.485186
\(512\) 0 0
\(513\) −6.75414 −0.298203
\(514\) 0 0
\(515\) 15.9210 0.701563
\(516\) 0 0
\(517\) −24.3724 −1.07190
\(518\) 0 0
\(519\) 24.7699 1.08728
\(520\) 0 0
\(521\) 9.49544 0.416003 0.208001 0.978129i \(-0.433304\pi\)
0.208001 + 0.978129i \(0.433304\pi\)
\(522\) 0 0
\(523\) 16.2479 0.710471 0.355236 0.934777i \(-0.384401\pi\)
0.355236 + 0.934777i \(0.384401\pi\)
\(524\) 0 0
\(525\) 13.5965 0.593400
\(526\) 0 0
\(527\) 22.2966 0.971256
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 1.75491 0.0761567
\(532\) 0 0
\(533\) 70.8565 3.06914
\(534\) 0 0
\(535\) −7.59742 −0.328465
\(536\) 0 0
\(537\) −0.379109 −0.0163597
\(538\) 0 0
\(539\) −8.60373 −0.370589
\(540\) 0 0
\(541\) 1.62428 0.0698332 0.0349166 0.999390i \(-0.488883\pi\)
0.0349166 + 0.999390i \(0.488883\pi\)
\(542\) 0 0
\(543\) 10.6216 0.455817
\(544\) 0 0
\(545\) 25.4613 1.09064
\(546\) 0 0
\(547\) 20.0252 0.856214 0.428107 0.903728i \(-0.359181\pi\)
0.428107 + 0.903728i \(0.359181\pi\)
\(548\) 0 0
\(549\) 15.3917 0.656904
\(550\) 0 0
\(551\) 6.75414 0.287736
\(552\) 0 0
\(553\) 14.1204 0.600459
\(554\) 0 0
\(555\) 4.79030 0.203337
\(556\) 0 0
\(557\) 29.5276 1.25112 0.625561 0.780175i \(-0.284870\pi\)
0.625561 + 0.780175i \(0.284870\pi\)
\(558\) 0 0
\(559\) −76.6959 −3.24389
\(560\) 0 0
\(561\) −25.8283 −1.09047
\(562\) 0 0
\(563\) −6.91811 −0.291564 −0.145782 0.989317i \(-0.546570\pi\)
−0.145782 + 0.989317i \(0.546570\pi\)
\(564\) 0 0
\(565\) 16.2818 0.684979
\(566\) 0 0
\(567\) −2.20985 −0.0928050
\(568\) 0 0
\(569\) −36.8975 −1.54682 −0.773412 0.633904i \(-0.781452\pi\)
−0.773412 + 0.633904i \(0.781452\pi\)
\(570\) 0 0
\(571\) −27.0361 −1.13143 −0.565713 0.824602i \(-0.691399\pi\)
−0.565713 + 0.824602i \(0.691399\pi\)
\(572\) 0 0
\(573\) −3.59888 −0.150346
\(574\) 0 0
\(575\) 6.15268 0.256584
\(576\) 0 0
\(577\) −13.3904 −0.557448 −0.278724 0.960371i \(-0.589911\pi\)
−0.278724 + 0.960371i \(0.589911\pi\)
\(578\) 0 0
\(579\) 0.135250 0.00562078
\(580\) 0 0
\(581\) −28.8494 −1.19688
\(582\) 0 0
\(583\) −16.8478 −0.697766
\(584\) 0 0
\(585\) −21.5222 −0.889833
\(586\) 0 0
\(587\) 39.2870 1.62155 0.810774 0.585359i \(-0.199046\pi\)
0.810774 + 0.585359i \(0.199046\pi\)
\(588\) 0 0
\(589\) 23.7011 0.976586
\(590\) 0 0
\(591\) 21.5135 0.884948
\(592\) 0 0
\(593\) 2.03447 0.0835455 0.0417727 0.999127i \(-0.486699\pi\)
0.0417727 + 0.999127i \(0.486699\pi\)
\(594\) 0 0
\(595\) 46.8914 1.92236
\(596\) 0 0
\(597\) 13.9483 0.570867
\(598\) 0 0
\(599\) 18.1245 0.740549 0.370274 0.928922i \(-0.379264\pi\)
0.370274 + 0.928922i \(0.379264\pi\)
\(600\) 0 0
\(601\) −24.9335 −1.01706 −0.508530 0.861045i \(-0.669811\pi\)
−0.508530 + 0.861045i \(0.669811\pi\)
\(602\) 0 0
\(603\) −2.00839 −0.0817880
\(604\) 0 0
\(605\) −18.4472 −0.749984
\(606\) 0 0
\(607\) −2.49290 −0.101184 −0.0505918 0.998719i \(-0.516111\pi\)
−0.0505918 + 0.998719i \(0.516111\pi\)
\(608\) 0 0
\(609\) 2.20985 0.0895476
\(610\) 0 0
\(611\) −38.6402 −1.56322
\(612\) 0 0
\(613\) −36.6838 −1.48165 −0.740823 0.671701i \(-0.765564\pi\)
−0.740823 + 0.671701i \(0.765564\pi\)
\(614\) 0 0
\(615\) 36.7174 1.48059
\(616\) 0 0
\(617\) 19.8860 0.800579 0.400289 0.916389i \(-0.368910\pi\)
0.400289 + 0.916389i \(0.368910\pi\)
\(618\) 0 0
\(619\) −26.1058 −1.04928 −0.524640 0.851324i \(-0.675800\pi\)
−0.524640 + 0.851324i \(0.675800\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −15.0728 −0.603880
\(624\) 0 0
\(625\) −17.9079 −0.716318
\(626\) 0 0
\(627\) −27.4553 −1.09646
\(628\) 0 0
\(629\) 9.11412 0.363404
\(630\) 0 0
\(631\) −36.5996 −1.45701 −0.728503 0.685043i \(-0.759784\pi\)
−0.728503 + 0.685043i \(0.759784\pi\)
\(632\) 0 0
\(633\) −2.41989 −0.0961819
\(634\) 0 0
\(635\) 25.2410 1.00166
\(636\) 0 0
\(637\) −13.6404 −0.540454
\(638\) 0 0
\(639\) 16.1058 0.637137
\(640\) 0 0
\(641\) 25.7095 1.01547 0.507733 0.861515i \(-0.330484\pi\)
0.507733 + 0.861515i \(0.330484\pi\)
\(642\) 0 0
\(643\) −36.1833 −1.42693 −0.713466 0.700690i \(-0.752876\pi\)
−0.713466 + 0.700690i \(0.752876\pi\)
\(644\) 0 0
\(645\) −39.7434 −1.56489
\(646\) 0 0
\(647\) −0.782056 −0.0307458 −0.0153729 0.999882i \(-0.504894\pi\)
−0.0153729 + 0.999882i \(0.504894\pi\)
\(648\) 0 0
\(649\) 7.13364 0.280020
\(650\) 0 0
\(651\) 7.75462 0.303928
\(652\) 0 0
\(653\) 38.3634 1.50128 0.750638 0.660714i \(-0.229746\pi\)
0.750638 + 0.660714i \(0.229746\pi\)
\(654\) 0 0
\(655\) 11.3779 0.444571
\(656\) 0 0
\(657\) −4.96314 −0.193630
\(658\) 0 0
\(659\) 14.6840 0.572006 0.286003 0.958229i \(-0.407673\pi\)
0.286003 + 0.958229i \(0.407673\pi\)
\(660\) 0 0
\(661\) −22.0207 −0.856506 −0.428253 0.903659i \(-0.640871\pi\)
−0.428253 + 0.903659i \(0.640871\pi\)
\(662\) 0 0
\(663\) −40.9485 −1.59031
\(664\) 0 0
\(665\) 49.8451 1.93291
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) 7.01623 0.271263
\(670\) 0 0
\(671\) 62.5667 2.41536
\(672\) 0 0
\(673\) 3.33236 0.128453 0.0642265 0.997935i \(-0.479542\pi\)
0.0642265 + 0.997935i \(0.479542\pi\)
\(674\) 0 0
\(675\) −6.15268 −0.236817
\(676\) 0 0
\(677\) −13.2110 −0.507738 −0.253869 0.967239i \(-0.581703\pi\)
−0.253869 + 0.967239i \(0.581703\pi\)
\(678\) 0 0
\(679\) −7.03017 −0.269793
\(680\) 0 0
\(681\) 29.3325 1.12403
\(682\) 0 0
\(683\) 24.9764 0.955695 0.477847 0.878443i \(-0.341417\pi\)
0.477847 + 0.878443i \(0.341417\pi\)
\(684\) 0 0
\(685\) −24.9879 −0.954740
\(686\) 0 0
\(687\) 4.07873 0.155613
\(688\) 0 0
\(689\) −26.7107 −1.01760
\(690\) 0 0
\(691\) 37.2704 1.41783 0.708916 0.705293i \(-0.249185\pi\)
0.708916 + 0.705293i \(0.249185\pi\)
\(692\) 0 0
\(693\) −8.98293 −0.341233
\(694\) 0 0
\(695\) 60.7076 2.30277
\(696\) 0 0
\(697\) 69.8593 2.64611
\(698\) 0 0
\(699\) 17.7597 0.671735
\(700\) 0 0
\(701\) −31.0146 −1.17140 −0.585702 0.810526i \(-0.699181\pi\)
−0.585702 + 0.810526i \(0.699181\pi\)
\(702\) 0 0
\(703\) 9.68822 0.365398
\(704\) 0 0
\(705\) −20.0231 −0.754115
\(706\) 0 0
\(707\) 15.7197 0.591199
\(708\) 0 0
\(709\) −0.379793 −0.0142634 −0.00713171 0.999975i \(-0.502270\pi\)
−0.00713171 + 0.999975i \(0.502270\pi\)
\(710\) 0 0
\(711\) −6.38974 −0.239634
\(712\) 0 0
\(713\) 3.50912 0.131417
\(714\) 0 0
\(715\) −87.4867 −3.27182
\(716\) 0 0
\(717\) −6.71824 −0.250897
\(718\) 0 0
\(719\) −7.46602 −0.278436 −0.139218 0.990262i \(-0.544459\pi\)
−0.139218 + 0.990262i \(0.544459\pi\)
\(720\) 0 0
\(721\) 10.5352 0.392352
\(722\) 0 0
\(723\) −1.59497 −0.0593177
\(724\) 0 0
\(725\) 6.15268 0.228505
\(726\) 0 0
\(727\) −27.7893 −1.03065 −0.515323 0.856996i \(-0.672328\pi\)
−0.515323 + 0.856996i \(0.672328\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −75.6164 −2.79678
\(732\) 0 0
\(733\) −2.16486 −0.0799611 −0.0399806 0.999200i \(-0.512730\pi\)
−0.0399806 + 0.999200i \(0.512730\pi\)
\(734\) 0 0
\(735\) −7.06840 −0.260722
\(736\) 0 0
\(737\) −8.16401 −0.300725
\(738\) 0 0
\(739\) 6.59644 0.242654 0.121327 0.992613i \(-0.461285\pi\)
0.121327 + 0.992613i \(0.461285\pi\)
\(740\) 0 0
\(741\) −43.5278 −1.59904
\(742\) 0 0
\(743\) −3.49482 −0.128213 −0.0641063 0.997943i \(-0.520420\pi\)
−0.0641063 + 0.997943i \(0.520420\pi\)
\(744\) 0 0
\(745\) −40.8036 −1.49493
\(746\) 0 0
\(747\) 13.0549 0.477655
\(748\) 0 0
\(749\) −5.02735 −0.183695
\(750\) 0 0
\(751\) 14.9339 0.544948 0.272474 0.962163i \(-0.412158\pi\)
0.272474 + 0.962163i \(0.412158\pi\)
\(752\) 0 0
\(753\) −10.5729 −0.385297
\(754\) 0 0
\(755\) −33.2803 −1.21120
\(756\) 0 0
\(757\) −10.0381 −0.364842 −0.182421 0.983221i \(-0.558393\pi\)
−0.182421 + 0.983221i \(0.558393\pi\)
\(758\) 0 0
\(759\) −4.06495 −0.147548
\(760\) 0 0
\(761\) −34.2606 −1.24195 −0.620973 0.783832i \(-0.713262\pi\)
−0.620973 + 0.783832i \(0.713262\pi\)
\(762\) 0 0
\(763\) 16.8482 0.609946
\(764\) 0 0
\(765\) −21.2193 −0.767185
\(766\) 0 0
\(767\) 11.3097 0.408371
\(768\) 0 0
\(769\) 32.1636 1.15985 0.579924 0.814670i \(-0.303082\pi\)
0.579924 + 0.814670i \(0.303082\pi\)
\(770\) 0 0
\(771\) −20.3609 −0.733279
\(772\) 0 0
\(773\) 33.8637 1.21799 0.608996 0.793173i \(-0.291572\pi\)
0.608996 + 0.793173i \(0.291572\pi\)
\(774\) 0 0
\(775\) 21.5905 0.775553
\(776\) 0 0
\(777\) 3.16983 0.113717
\(778\) 0 0
\(779\) 74.2597 2.66063
\(780\) 0 0
\(781\) 65.4694 2.34268
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) −68.7939 −2.45536
\(786\) 0 0
\(787\) 12.8749 0.458941 0.229470 0.973316i \(-0.426301\pi\)
0.229470 + 0.973316i \(0.426301\pi\)
\(788\) 0 0
\(789\) 5.97404 0.212681
\(790\) 0 0
\(791\) 10.7739 0.383078
\(792\) 0 0
\(793\) 99.1938 3.52248
\(794\) 0 0
\(795\) −13.8413 −0.490902
\(796\) 0 0
\(797\) 35.2384 1.24821 0.624103 0.781342i \(-0.285464\pi\)
0.624103 + 0.781342i \(0.285464\pi\)
\(798\) 0 0
\(799\) −38.0964 −1.34775
\(800\) 0 0
\(801\) 6.82075 0.240999
\(802\) 0 0
\(803\) −20.1749 −0.711957
\(804\) 0 0
\(805\) 7.37993 0.260108
\(806\) 0 0
\(807\) −29.5777 −1.04118
\(808\) 0 0
\(809\) −23.4325 −0.823843 −0.411922 0.911219i \(-0.635142\pi\)
−0.411922 + 0.911219i \(0.635142\pi\)
\(810\) 0 0
\(811\) 6.62169 0.232519 0.116259 0.993219i \(-0.462910\pi\)
0.116259 + 0.993219i \(0.462910\pi\)
\(812\) 0 0
\(813\) 3.19390 0.112015
\(814\) 0 0
\(815\) −57.4240 −2.01148
\(816\) 0 0
\(817\) −80.3795 −2.81212
\(818\) 0 0
\(819\) −14.2416 −0.497643
\(820\) 0 0
\(821\) 10.2101 0.356335 0.178167 0.984000i \(-0.442983\pi\)
0.178167 + 0.984000i \(0.442983\pi\)
\(822\) 0 0
\(823\) 16.0609 0.559847 0.279923 0.960022i \(-0.409691\pi\)
0.279923 + 0.960022i \(0.409691\pi\)
\(824\) 0 0
\(825\) −25.0103 −0.870748
\(826\) 0 0
\(827\) 15.8698 0.551848 0.275924 0.961179i \(-0.411016\pi\)
0.275924 + 0.961179i \(0.411016\pi\)
\(828\) 0 0
\(829\) 52.5195 1.82408 0.912039 0.410103i \(-0.134507\pi\)
0.912039 + 0.410103i \(0.134507\pi\)
\(830\) 0 0
\(831\) 12.4087 0.430452
\(832\) 0 0
\(833\) −13.4485 −0.465962
\(834\) 0 0
\(835\) −78.5842 −2.71952
\(836\) 0 0
\(837\) −3.50912 −0.121293
\(838\) 0 0
\(839\) 2.33389 0.0805747 0.0402874 0.999188i \(-0.487173\pi\)
0.0402874 + 0.999188i \(0.487173\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 7.15180 0.246321
\(844\) 0 0
\(845\) −95.2879 −3.27800
\(846\) 0 0
\(847\) −12.2068 −0.419432
\(848\) 0 0
\(849\) 20.2325 0.694378
\(850\) 0 0
\(851\) 1.43441 0.0491710
\(852\) 0 0
\(853\) 54.3172 1.85979 0.929893 0.367830i \(-0.119899\pi\)
0.929893 + 0.367830i \(0.119899\pi\)
\(854\) 0 0
\(855\) −22.5559 −0.771395
\(856\) 0 0
\(857\) −32.7015 −1.11706 −0.558531 0.829484i \(-0.688635\pi\)
−0.558531 + 0.829484i \(0.688635\pi\)
\(858\) 0 0
\(859\) 1.04481 0.0356485 0.0178242 0.999841i \(-0.494326\pi\)
0.0178242 + 0.999841i \(0.494326\pi\)
\(860\) 0 0
\(861\) 24.2966 0.828026
\(862\) 0 0
\(863\) −34.7540 −1.18304 −0.591519 0.806291i \(-0.701472\pi\)
−0.591519 + 0.806291i \(0.701472\pi\)
\(864\) 0 0
\(865\) 82.7208 2.81259
\(866\) 0 0
\(867\) −23.3722 −0.793761
\(868\) 0 0
\(869\) −25.9740 −0.881107
\(870\) 0 0
\(871\) −12.9433 −0.438567
\(872\) 0 0
\(873\) 3.18129 0.107670
\(874\) 0 0
\(875\) 8.50668 0.287578
\(876\) 0 0
\(877\) 57.9636 1.95729 0.978646 0.205552i \(-0.0658989\pi\)
0.978646 + 0.205552i \(0.0658989\pi\)
\(878\) 0 0
\(879\) 5.13032 0.173041
\(880\) 0 0
\(881\) −14.0577 −0.473614 −0.236807 0.971557i \(-0.576101\pi\)
−0.236807 + 0.971557i \(0.576101\pi\)
\(882\) 0 0
\(883\) −15.9333 −0.536197 −0.268098 0.963392i \(-0.586395\pi\)
−0.268098 + 0.963392i \(0.586395\pi\)
\(884\) 0 0
\(885\) 5.86064 0.197003
\(886\) 0 0
\(887\) 33.2109 1.11511 0.557557 0.830139i \(-0.311739\pi\)
0.557557 + 0.830139i \(0.311739\pi\)
\(888\) 0 0
\(889\) 16.7025 0.560183
\(890\) 0 0
\(891\) 4.06495 0.136181
\(892\) 0 0
\(893\) −40.4961 −1.35515
\(894\) 0 0
\(895\) −1.26606 −0.0423196
\(896\) 0 0
\(897\) −6.44461 −0.215179
\(898\) 0 0
\(899\) 3.50912 0.117036
\(900\) 0 0
\(901\) −26.3348 −0.877339
\(902\) 0 0
\(903\) −26.2989 −0.875173
\(904\) 0 0
\(905\) 35.4716 1.17911
\(906\) 0 0
\(907\) −14.7888 −0.491055 −0.245528 0.969390i \(-0.578961\pi\)
−0.245528 + 0.969390i \(0.578961\pi\)
\(908\) 0 0
\(909\) −7.11345 −0.235938
\(910\) 0 0
\(911\) −43.5333 −1.44232 −0.721160 0.692768i \(-0.756391\pi\)
−0.721160 + 0.692768i \(0.756391\pi\)
\(912\) 0 0
\(913\) 53.0677 1.75628
\(914\) 0 0
\(915\) 51.4017 1.69929
\(916\) 0 0
\(917\) 7.52897 0.248628
\(918\) 0 0
\(919\) −18.1217 −0.597778 −0.298889 0.954288i \(-0.596616\pi\)
−0.298889 + 0.954288i \(0.596616\pi\)
\(920\) 0 0
\(921\) −16.6305 −0.547995
\(922\) 0 0
\(923\) 103.796 3.41648
\(924\) 0 0
\(925\) 8.82547 0.290180
\(926\) 0 0
\(927\) −4.76739 −0.156582
\(928\) 0 0
\(929\) −15.3507 −0.503640 −0.251820 0.967774i \(-0.581029\pi\)
−0.251820 + 0.967774i \(0.581029\pi\)
\(930\) 0 0
\(931\) −14.2956 −0.468518
\(932\) 0 0
\(933\) 26.2790 0.860337
\(934\) 0 0
\(935\) −86.2554 −2.82085
\(936\) 0 0
\(937\) 28.5415 0.932411 0.466205 0.884677i \(-0.345621\pi\)
0.466205 + 0.884677i \(0.345621\pi\)
\(938\) 0 0
\(939\) −5.80775 −0.189529
\(940\) 0 0
\(941\) −32.1719 −1.04877 −0.524387 0.851480i \(-0.675706\pi\)
−0.524387 + 0.851480i \(0.675706\pi\)
\(942\) 0 0
\(943\) 10.9947 0.358036
\(944\) 0 0
\(945\) −7.37993 −0.240069
\(946\) 0 0
\(947\) 27.5333 0.894713 0.447357 0.894356i \(-0.352365\pi\)
0.447357 + 0.894356i \(0.352365\pi\)
\(948\) 0 0
\(949\) −31.9855 −1.03829
\(950\) 0 0
\(951\) −25.3663 −0.822560
\(952\) 0 0
\(953\) 25.3022 0.819619 0.409809 0.912171i \(-0.365595\pi\)
0.409809 + 0.912171i \(0.365595\pi\)
\(954\) 0 0
\(955\) −12.0187 −0.388916
\(956\) 0 0
\(957\) −4.06495 −0.131401
\(958\) 0 0
\(959\) −16.5350 −0.533942
\(960\) 0 0
\(961\) −18.6861 −0.602777
\(962\) 0 0
\(963\) 2.27498 0.0733101
\(964\) 0 0
\(965\) 0.451675 0.0145399
\(966\) 0 0
\(967\) −16.0495 −0.516116 −0.258058 0.966129i \(-0.583083\pi\)
−0.258058 + 0.966129i \(0.583083\pi\)
\(968\) 0 0
\(969\) −42.9152 −1.37864
\(970\) 0 0
\(971\) 5.75466 0.184676 0.0923379 0.995728i \(-0.470566\pi\)
0.0923379 + 0.995728i \(0.470566\pi\)
\(972\) 0 0
\(973\) 40.1714 1.28783
\(974\) 0 0
\(975\) −39.6516 −1.26987
\(976\) 0 0
\(977\) 54.2157 1.73451 0.867257 0.497861i \(-0.165881\pi\)
0.867257 + 0.497861i \(0.165881\pi\)
\(978\) 0 0
\(979\) 27.7260 0.886127
\(980\) 0 0
\(981\) −7.62414 −0.243420
\(982\) 0 0
\(983\) −59.3329 −1.89242 −0.946212 0.323546i \(-0.895125\pi\)
−0.946212 + 0.323546i \(0.895125\pi\)
\(984\) 0 0
\(985\) 71.8457 2.28920
\(986\) 0 0
\(987\) −13.2497 −0.421742
\(988\) 0 0
\(989\) −11.9008 −0.378422
\(990\) 0 0
\(991\) 21.1924 0.673200 0.336600 0.941648i \(-0.390723\pi\)
0.336600 + 0.941648i \(0.390723\pi\)
\(992\) 0 0
\(993\) −5.39064 −0.171067
\(994\) 0 0
\(995\) 46.5813 1.47673
\(996\) 0 0
\(997\) 46.1433 1.46137 0.730687 0.682713i \(-0.239200\pi\)
0.730687 + 0.682713i \(0.239200\pi\)
\(998\) 0 0
\(999\) −1.43441 −0.0453828
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 8004.2.a.i.1.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.i.1.2 16 1.1 even 1 trivial