Properties

Label 4600.2.a.bf.1.1
Level $4600$
Weight $2$
Character 4600.1
Self dual yes
Analytic conductor $36.731$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.521397.1
Defining polynomial: \(x^{5} - 9 x^{3} - 3 x^{2} + 18 x + 12\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.29544\) of defining polynomial
Character \(\chi\) \(=\) 4600.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.29544 q^{3} -1.61758 q^{7} +2.26905 q^{9} +O(q^{10})\) \(q-2.29544 q^{3} -1.61758 q^{7} +2.26905 q^{9} -2.79255 q^{11} +4.88663 q^{13} +3.54388 q^{17} -2.79255 q^{19} +3.71306 q^{21} -1.00000 q^{23} +1.67786 q^{27} -3.36282 q^{29} -4.84564 q^{31} +6.41013 q^{33} +3.87204 q^{37} -11.2170 q^{39} -0.327923 q^{41} -5.38343 q^{43} +0.0121087 q^{47} -4.38343 q^{49} -8.13476 q^{51} -0.866254 q^{53} +6.41013 q^{57} +2.96752 q^{59} -7.29816 q^{61} -3.67037 q^{63} +6.94082 q^{67} +2.29544 q^{69} -2.16864 q^{71} -6.02888 q^{73} +4.51718 q^{77} +6.50561 q^{79} -10.6586 q^{81} -7.88030 q^{83} +7.71915 q^{87} -9.71986 q^{89} -7.90452 q^{91} +11.1229 q^{93} +11.9952 q^{97} -6.33643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 4q^{7} + 3q^{9} + O(q^{10}) \) \( 5q + 4q^{7} + 3q^{9} + 4q^{13} + 6q^{17} - 5q^{23} + 9q^{27} + 12q^{29} - 18q^{31} + 6q^{33} + 10q^{37} + 9q^{39} - 6q^{41} + 10q^{43} + 22q^{47} + 15q^{49} - 6q^{51} + 10q^{53} + 6q^{57} - q^{59} + 10q^{61} + 8q^{67} + 8q^{71} + 6q^{73} - 27q^{81} - 2q^{83} + 39q^{87} + 14q^{89} - 46q^{91} - 3q^{93} + 6q^{97} - 6q^{99} + O(q^{100}) \)

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
\(3\) −2.29544 −1.32527 −0.662637 0.748941i \(-0.730563\pi\)
−0.662637 + 0.748941i \(0.730563\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.61758 −0.611388 −0.305694 0.952130i \(-0.598889\pi\)
−0.305694 + 0.952130i \(0.598889\pi\)
\(8\) 0 0
\(9\) 2.26905 0.756349
\(10\) 0 0
\(11\) −2.79255 −0.841985 −0.420993 0.907064i \(-0.638318\pi\)
−0.420993 + 0.907064i \(0.638318\pi\)
\(12\) 0 0
\(13\) 4.88663 1.35531 0.677653 0.735381i \(-0.262997\pi\)
0.677653 + 0.735381i \(0.262997\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.54388 0.859516 0.429758 0.902944i \(-0.358599\pi\)
0.429758 + 0.902944i \(0.358599\pi\)
\(18\) 0 0
\(19\) −2.79255 −0.640655 −0.320327 0.947307i \(-0.603793\pi\)
−0.320327 + 0.947307i \(0.603793\pi\)
\(20\) 0 0
\(21\) 3.71306 0.810257
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.67786 0.322904
\(28\) 0 0
\(29\) −3.36282 −0.624460 −0.312230 0.950007i \(-0.601076\pi\)
−0.312230 + 0.950007i \(0.601076\pi\)
\(30\) 0 0
\(31\) −4.84564 −0.870303 −0.435152 0.900357i \(-0.643305\pi\)
−0.435152 + 0.900357i \(0.643305\pi\)
\(32\) 0 0
\(33\) 6.41013 1.11586
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.87204 0.636559 0.318279 0.947997i \(-0.396895\pi\)
0.318279 + 0.947997i \(0.396895\pi\)
\(38\) 0 0
\(39\) −11.2170 −1.79615
\(40\) 0 0
\(41\) −0.327923 −0.0512130 −0.0256065 0.999672i \(-0.508152\pi\)
−0.0256065 + 0.999672i \(0.508152\pi\)
\(42\) 0 0
\(43\) −5.38343 −0.820965 −0.410483 0.911868i \(-0.634640\pi\)
−0.410483 + 0.911868i \(0.634640\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.0121087 0.00176623 0.000883117 1.00000i \(-0.499719\pi\)
0.000883117 1.00000i \(0.499719\pi\)
\(48\) 0 0
\(49\) −4.38343 −0.626204
\(50\) 0 0
\(51\) −8.13476 −1.13909
\(52\) 0 0
\(53\) −0.866254 −0.118989 −0.0594946 0.998229i \(-0.518949\pi\)
−0.0594946 + 0.998229i \(0.518949\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.41013 0.849043
\(58\) 0 0
\(59\) 2.96752 0.386338 0.193169 0.981166i \(-0.438123\pi\)
0.193169 + 0.981166i \(0.438123\pi\)
\(60\) 0 0
\(61\) −7.29816 −0.934434 −0.467217 0.884143i \(-0.654743\pi\)
−0.467217 + 0.884143i \(0.654743\pi\)
\(62\) 0 0
\(63\) −3.67037 −0.462423
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.94082 0.847956 0.423978 0.905673i \(-0.360633\pi\)
0.423978 + 0.905673i \(0.360633\pi\)
\(68\) 0 0
\(69\) 2.29544 0.276339
\(70\) 0 0
\(71\) −2.16864 −0.257370 −0.128685 0.991685i \(-0.541076\pi\)
−0.128685 + 0.991685i \(0.541076\pi\)
\(72\) 0 0
\(73\) −6.02888 −0.705627 −0.352813 0.935694i \(-0.614775\pi\)
−0.352813 + 0.935694i \(0.614775\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.51718 0.514780
\(78\) 0 0
\(79\) 6.50561 0.731938 0.365969 0.930627i \(-0.380738\pi\)
0.365969 + 0.930627i \(0.380738\pi\)
\(80\) 0 0
\(81\) −10.6586 −1.18429
\(82\) 0 0
\(83\) −7.88030 −0.864976 −0.432488 0.901640i \(-0.642364\pi\)
−0.432488 + 0.901640i \(0.642364\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.71915 0.827580
\(88\) 0 0
\(89\) −9.71986 −1.03030 −0.515151 0.857099i \(-0.672264\pi\)
−0.515151 + 0.857099i \(0.672264\pi\)
\(90\) 0 0
\(91\) −7.90452 −0.828619
\(92\) 0 0
\(93\) 11.1229 1.15339
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.9952 1.21793 0.608966 0.793197i \(-0.291585\pi\)
0.608966 + 0.793197i \(0.291585\pi\)
\(98\) 0 0
\(99\) −6.33643 −0.636835
\(100\) 0 0
\(101\) 11.4712 1.14143 0.570713 0.821150i \(-0.306667\pi\)
0.570713 + 0.821150i \(0.306667\pi\)
\(102\) 0 0
\(103\) −4.02030 −0.396132 −0.198066 0.980189i \(-0.563466\pi\)
−0.198066 + 0.980189i \(0.563466\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.8318 1.43385 0.716923 0.697152i \(-0.245550\pi\)
0.716923 + 0.697152i \(0.245550\pi\)
\(108\) 0 0
\(109\) 12.3573 1.18362 0.591809 0.806078i \(-0.298414\pi\)
0.591809 + 0.806078i \(0.298414\pi\)
\(110\) 0 0
\(111\) −8.88803 −0.843614
\(112\) 0 0
\(113\) −5.48632 −0.516109 −0.258055 0.966130i \(-0.583081\pi\)
−0.258055 + 0.966130i \(0.583081\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 11.0880 1.02509
\(118\) 0 0
\(119\) −5.73251 −0.525498
\(120\) 0 0
\(121\) −3.20167 −0.291061
\(122\) 0 0
\(123\) 0.752728 0.0678712
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −3.14959 −0.279481 −0.139740 0.990188i \(-0.544627\pi\)
−0.139740 + 0.990188i \(0.544627\pi\)
\(128\) 0 0
\(129\) 12.3573 1.08800
\(130\) 0 0
\(131\) −2.93231 −0.256197 −0.128099 0.991761i \(-0.540887\pi\)
−0.128099 + 0.991761i \(0.540887\pi\)
\(132\) 0 0
\(133\) 4.51718 0.391689
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.4054 1.23073 0.615366 0.788241i \(-0.289008\pi\)
0.615366 + 0.788241i \(0.289008\pi\)
\(138\) 0 0
\(139\) 18.2241 1.54575 0.772873 0.634561i \(-0.218819\pi\)
0.772873 + 0.634561i \(0.218819\pi\)
\(140\) 0 0
\(141\) −0.0277948 −0.00234074
\(142\) 0 0
\(143\) −13.6462 −1.14115
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 10.0619 0.829892
\(148\) 0 0
\(149\) −5.71986 −0.468589 −0.234294 0.972166i \(-0.575278\pi\)
−0.234294 + 0.972166i \(0.575278\pi\)
\(150\) 0 0
\(151\) −5.43082 −0.441954 −0.220977 0.975279i \(-0.570925\pi\)
−0.220977 + 0.975279i \(0.570925\pi\)
\(152\) 0 0
\(153\) 8.04122 0.650094
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −24.2004 −1.93140 −0.965701 0.259657i \(-0.916390\pi\)
−0.965701 + 0.259657i \(0.916390\pi\)
\(158\) 0 0
\(159\) 1.98844 0.157693
\(160\) 0 0
\(161\) 1.61758 0.127483
\(162\) 0 0
\(163\) −11.2970 −0.884849 −0.442425 0.896806i \(-0.645882\pi\)
−0.442425 + 0.896806i \(0.645882\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.6478 1.67516 0.837578 0.546318i \(-0.183971\pi\)
0.837578 + 0.546318i \(0.183971\pi\)
\(168\) 0 0
\(169\) 10.8791 0.836857
\(170\) 0 0
\(171\) −6.33643 −0.484559
\(172\) 0 0
\(173\) 1.39593 0.106130 0.0530652 0.998591i \(-0.483101\pi\)
0.0530652 + 0.998591i \(0.483101\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.81176 −0.512003
\(178\) 0 0
\(179\) 23.5665 1.76144 0.880722 0.473634i \(-0.157058\pi\)
0.880722 + 0.473634i \(0.157058\pi\)
\(180\) 0 0
\(181\) 8.33495 0.619532 0.309766 0.950813i \(-0.399749\pi\)
0.309766 + 0.950813i \(0.399749\pi\)
\(182\) 0 0
\(183\) 16.7525 1.23838
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −9.89645 −0.723700
\(188\) 0 0
\(189\) −2.71407 −0.197420
\(190\) 0 0
\(191\) 18.3074 1.32468 0.662340 0.749204i \(-0.269564\pi\)
0.662340 + 0.749204i \(0.269564\pi\)
\(192\) 0 0
\(193\) 15.4434 1.11164 0.555820 0.831303i \(-0.312404\pi\)
0.555820 + 0.831303i \(0.312404\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −0.876228 −0.0624287 −0.0312143 0.999513i \(-0.509937\pi\)
−0.0312143 + 0.999513i \(0.509937\pi\)
\(198\) 0 0
\(199\) −13.5818 −0.962788 −0.481394 0.876504i \(-0.659869\pi\)
−0.481394 + 0.876504i \(0.659869\pi\)
\(200\) 0 0
\(201\) −15.9322 −1.12377
\(202\) 0 0
\(203\) 5.43963 0.381787
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.26905 −0.157710
\(208\) 0 0
\(209\) 7.79833 0.539422
\(210\) 0 0
\(211\) 12.9100 0.888758 0.444379 0.895839i \(-0.353424\pi\)
0.444379 + 0.895839i \(0.353424\pi\)
\(212\) 0 0
\(213\) 4.97799 0.341086
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.83822 0.532093
\(218\) 0 0
\(219\) 13.8389 0.935148
\(220\) 0 0
\(221\) 17.3176 1.16491
\(222\) 0 0
\(223\) −8.05527 −0.539421 −0.269710 0.962941i \(-0.586928\pi\)
−0.269710 + 0.962941i \(0.586928\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.41490 0.293027 0.146514 0.989209i \(-0.453195\pi\)
0.146514 + 0.989209i \(0.453195\pi\)
\(228\) 0 0
\(229\) 17.5197 1.15773 0.578866 0.815423i \(-0.303495\pi\)
0.578866 + 0.815423i \(0.303495\pi\)
\(230\) 0 0
\(231\) −10.3689 −0.682224
\(232\) 0 0
\(233\) 15.8080 1.03562 0.517808 0.855497i \(-0.326748\pi\)
0.517808 + 0.855497i \(0.326748\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −14.9332 −0.970018
\(238\) 0 0
\(239\) −8.93972 −0.578263 −0.289131 0.957289i \(-0.593366\pi\)
−0.289131 + 0.957289i \(0.593366\pi\)
\(240\) 0 0
\(241\) 3.50992 0.226094 0.113047 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(242\) 0 0
\(243\) 19.4325 1.24660
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −13.6462 −0.868284
\(248\) 0 0
\(249\) 18.0888 1.14633
\(250\) 0 0
\(251\) 4.33837 0.273836 0.136918 0.990582i \(-0.456280\pi\)
0.136918 + 0.990582i \(0.456280\pi\)
\(252\) 0 0
\(253\) 2.79255 0.175566
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −16.6252 −1.03705 −0.518524 0.855063i \(-0.673518\pi\)
−0.518524 + 0.855063i \(0.673518\pi\)
\(258\) 0 0
\(259\) −6.26333 −0.389185
\(260\) 0 0
\(261\) −7.63039 −0.472310
\(262\) 0 0
\(263\) 13.0084 0.802134 0.401067 0.916049i \(-0.368640\pi\)
0.401067 + 0.916049i \(0.368640\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 22.3114 1.36543
\(268\) 0 0
\(269\) −8.30995 −0.506667 −0.253333 0.967379i \(-0.581527\pi\)
−0.253333 + 0.967379i \(0.581527\pi\)
\(270\) 0 0
\(271\) −8.35009 −0.507232 −0.253616 0.967305i \(-0.581620\pi\)
−0.253616 + 0.967305i \(0.581620\pi\)
\(272\) 0 0
\(273\) 18.1444 1.09815
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.23195 0.374441 0.187221 0.982318i \(-0.440052\pi\)
0.187221 + 0.982318i \(0.440052\pi\)
\(278\) 0 0
\(279\) −10.9950 −0.658253
\(280\) 0 0
\(281\) 8.01836 0.478335 0.239168 0.970978i \(-0.423125\pi\)
0.239168 + 0.970978i \(0.423125\pi\)
\(282\) 0 0
\(283\) −9.11741 −0.541974 −0.270987 0.962583i \(-0.587350\pi\)
−0.270987 + 0.962583i \(0.587350\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.530443 0.0313110
\(288\) 0 0
\(289\) −4.44094 −0.261232
\(290\) 0 0
\(291\) −27.5343 −1.61409
\(292\) 0 0
\(293\) 17.5323 1.02425 0.512124 0.858911i \(-0.328859\pi\)
0.512124 + 0.858911i \(0.328859\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −4.68550 −0.271881
\(298\) 0 0
\(299\) −4.88663 −0.282601
\(300\) 0 0
\(301\) 8.70814 0.501929
\(302\) 0 0
\(303\) −26.3314 −1.51270
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 30.3404 1.73162 0.865809 0.500375i \(-0.166804\pi\)
0.865809 + 0.500375i \(0.166804\pi\)
\(308\) 0 0
\(309\) 9.22837 0.524984
\(310\) 0 0
\(311\) −13.7703 −0.780840 −0.390420 0.920637i \(-0.627670\pi\)
−0.390420 + 0.920637i \(0.627670\pi\)
\(312\) 0 0
\(313\) 14.3715 0.812328 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 26.5726 1.49247 0.746233 0.665685i \(-0.231860\pi\)
0.746233 + 0.665685i \(0.231860\pi\)
\(318\) 0 0
\(319\) 9.39084 0.525786
\(320\) 0 0
\(321\) −34.0456 −1.90024
\(322\) 0 0
\(323\) −9.89645 −0.550653
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −28.3655 −1.56862
\(328\) 0 0
\(329\) −0.0195868 −0.00107986
\(330\) 0 0
\(331\) −12.2379 −0.672655 −0.336327 0.941745i \(-0.609185\pi\)
−0.336327 + 0.941745i \(0.609185\pi\)
\(332\) 0 0
\(333\) 8.78583 0.481461
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.7474 0.912290 0.456145 0.889906i \(-0.349230\pi\)
0.456145 + 0.889906i \(0.349230\pi\)
\(338\) 0 0
\(339\) 12.5935 0.683986
\(340\) 0 0
\(341\) 13.5317 0.732783
\(342\) 0 0
\(343\) 18.4136 0.994242
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.6122 1.10652 0.553260 0.833008i \(-0.313383\pi\)
0.553260 + 0.833008i \(0.313383\pi\)
\(348\) 0 0
\(349\) 35.5885 1.90501 0.952505 0.304523i \(-0.0984970\pi\)
0.952505 + 0.304523i \(0.0984970\pi\)
\(350\) 0 0
\(351\) 8.19907 0.437634
\(352\) 0 0
\(353\) −0.574309 −0.0305674 −0.0152837 0.999883i \(-0.504865\pi\)
−0.0152837 + 0.999883i \(0.504865\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 13.1586 0.696429
\(358\) 0 0
\(359\) 4.12882 0.217911 0.108955 0.994047i \(-0.465249\pi\)
0.108955 + 0.994047i \(0.465249\pi\)
\(360\) 0 0
\(361\) −11.2017 −0.589561
\(362\) 0 0
\(363\) 7.34924 0.385735
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.16724 0.217528 0.108764 0.994068i \(-0.465311\pi\)
0.108764 + 0.994068i \(0.465311\pi\)
\(368\) 0 0
\(369\) −0.744073 −0.0387349
\(370\) 0 0
\(371\) 1.40124 0.0727486
\(372\) 0 0
\(373\) −25.3579 −1.31298 −0.656491 0.754334i \(-0.727960\pi\)
−0.656491 + 0.754334i \(0.727960\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −16.4328 −0.846335
\(378\) 0 0
\(379\) 14.2580 0.732382 0.366191 0.930540i \(-0.380662\pi\)
0.366191 + 0.930540i \(0.380662\pi\)
\(380\) 0 0
\(381\) 7.22969 0.370388
\(382\) 0 0
\(383\) 16.6805 0.852334 0.426167 0.904645i \(-0.359864\pi\)
0.426167 + 0.904645i \(0.359864\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −12.2153 −0.620936
\(388\) 0 0
\(389\) 26.3853 1.33779 0.668894 0.743358i \(-0.266768\pi\)
0.668894 + 0.743358i \(0.266768\pi\)
\(390\) 0 0
\(391\) −3.54388 −0.179222
\(392\) 0 0
\(393\) 6.73095 0.339532
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −6.44231 −0.323330 −0.161665 0.986846i \(-0.551686\pi\)
−0.161665 + 0.986846i \(0.551686\pi\)
\(398\) 0 0
\(399\) −10.3689 −0.519095
\(400\) 0 0
\(401\) 21.6326 1.08028 0.540141 0.841574i \(-0.318371\pi\)
0.540141 + 0.841574i \(0.318371\pi\)
\(402\) 0 0
\(403\) −23.6789 −1.17953
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −10.8129 −0.535973
\(408\) 0 0
\(409\) 8.07200 0.399135 0.199567 0.979884i \(-0.436046\pi\)
0.199567 + 0.979884i \(0.436046\pi\)
\(410\) 0 0
\(411\) −33.0666 −1.63106
\(412\) 0 0
\(413\) −4.80020 −0.236202
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −41.8323 −2.04853
\(418\) 0 0
\(419\) −25.4538 −1.24350 −0.621751 0.783215i \(-0.713578\pi\)
−0.621751 + 0.783215i \(0.713578\pi\)
\(420\) 0 0
\(421\) 31.4111 1.53089 0.765443 0.643504i \(-0.222520\pi\)
0.765443 + 0.643504i \(0.222520\pi\)
\(422\) 0 0
\(423\) 0.0274752 0.00133589
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 11.8054 0.571302
\(428\) 0 0
\(429\) 31.3239 1.51233
\(430\) 0 0
\(431\) 3.35572 0.161639 0.0808196 0.996729i \(-0.474246\pi\)
0.0808196 + 0.996729i \(0.474246\pi\)
\(432\) 0 0
\(433\) 18.0365 0.866777 0.433388 0.901207i \(-0.357318\pi\)
0.433388 + 0.901207i \(0.357318\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.79255 0.133586
\(438\) 0 0
\(439\) 19.8080 0.945384 0.472692 0.881228i \(-0.343282\pi\)
0.472692 + 0.881228i \(0.343282\pi\)
\(440\) 0 0
\(441\) −9.94621 −0.473629
\(442\) 0 0
\(443\) 11.5628 0.549364 0.274682 0.961535i \(-0.411427\pi\)
0.274682 + 0.961535i \(0.411427\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 13.1296 0.621008
\(448\) 0 0
\(449\) −19.6506 −0.927368 −0.463684 0.886001i \(-0.653473\pi\)
−0.463684 + 0.886001i \(0.653473\pi\)
\(450\) 0 0
\(451\) 0.915742 0.0431206
\(452\) 0 0
\(453\) 12.4661 0.585710
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −23.0682 −1.07908 −0.539542 0.841959i \(-0.681402\pi\)
−0.539542 + 0.841959i \(0.681402\pi\)
\(458\) 0 0
\(459\) 5.94613 0.277541
\(460\) 0 0
\(461\) 1.86556 0.0868876 0.0434438 0.999056i \(-0.486167\pi\)
0.0434438 + 0.999056i \(0.486167\pi\)
\(462\) 0 0
\(463\) −12.9981 −0.604071 −0.302035 0.953297i \(-0.597666\pi\)
−0.302035 + 0.953297i \(0.597666\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.7712 1.28510 0.642548 0.766245i \(-0.277877\pi\)
0.642548 + 0.766245i \(0.277877\pi\)
\(468\) 0 0
\(469\) −11.2273 −0.518430
\(470\) 0 0
\(471\) 55.5506 2.55963
\(472\) 0 0
\(473\) 15.0335 0.691241
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.96557 −0.0899974
\(478\) 0 0
\(479\) −5.25003 −0.239880 −0.119940 0.992781i \(-0.538270\pi\)
−0.119940 + 0.992781i \(0.538270\pi\)
\(480\) 0 0
\(481\) 18.9212 0.862733
\(482\) 0 0
\(483\) −3.71306 −0.168950
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −9.90856 −0.449000 −0.224500 0.974474i \(-0.572075\pi\)
−0.224500 + 0.974474i \(0.572075\pi\)
\(488\) 0 0
\(489\) 25.9316 1.17267
\(490\) 0 0
\(491\) −24.2168 −1.09289 −0.546444 0.837495i \(-0.684019\pi\)
−0.546444 + 0.837495i \(0.684019\pi\)
\(492\) 0 0
\(493\) −11.9174 −0.536733
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.50795 0.157353
\(498\) 0 0
\(499\) −26.6722 −1.19401 −0.597007 0.802236i \(-0.703643\pi\)
−0.597007 + 0.802236i \(0.703643\pi\)
\(500\) 0 0
\(501\) −49.6912 −2.22004
\(502\) 0 0
\(503\) 2.38303 0.106254 0.0531271 0.998588i \(-0.483081\pi\)
0.0531271 + 0.998588i \(0.483081\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −24.9724 −1.10906
\(508\) 0 0
\(509\) 36.2807 1.60811 0.804056 0.594554i \(-0.202671\pi\)
0.804056 + 0.594554i \(0.202671\pi\)
\(510\) 0 0
\(511\) 9.75220 0.431412
\(512\) 0 0
\(513\) −4.68550 −0.206870
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −0.0338141 −0.00148714
\(518\) 0 0
\(519\) −3.20427 −0.140652
\(520\) 0 0
\(521\) −13.8498 −0.606773 −0.303386 0.952868i \(-0.598117\pi\)
−0.303386 + 0.952868i \(0.598117\pi\)
\(522\) 0 0
\(523\) −11.3982 −0.498411 −0.249205 0.968451i \(-0.580169\pi\)
−0.249205 + 0.968451i \(0.580169\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −17.1724 −0.748040
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 6.73344 0.292206
\(532\) 0 0
\(533\) −1.60244 −0.0694094
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −54.0955 −2.33439
\(538\) 0 0
\(539\) 12.2409 0.527255
\(540\) 0 0
\(541\) −15.6397 −0.672402 −0.336201 0.941790i \(-0.609142\pi\)
−0.336201 + 0.941790i \(0.609142\pi\)
\(542\) 0 0
\(543\) −19.1324 −0.821050
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 45.7911 1.95789 0.978944 0.204131i \(-0.0654368\pi\)
0.978944 + 0.204131i \(0.0654368\pi\)
\(548\) 0 0
\(549\) −16.5599 −0.706758
\(550\) 0 0
\(551\) 9.39084 0.400063
\(552\) 0 0
\(553\) −10.5234 −0.447499
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −30.8143 −1.30564 −0.652821 0.757512i \(-0.726415\pi\)
−0.652821 + 0.757512i \(0.726415\pi\)
\(558\) 0 0
\(559\) −26.3068 −1.11266
\(560\) 0 0
\(561\) 22.7167 0.959100
\(562\) 0 0
\(563\) 27.8036 1.17178 0.585891 0.810390i \(-0.300745\pi\)
0.585891 + 0.810390i \(0.300745\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 17.2411 0.724058
\(568\) 0 0
\(569\) 7.89706 0.331062 0.165531 0.986205i \(-0.447066\pi\)
0.165531 + 0.986205i \(0.447066\pi\)
\(570\) 0 0
\(571\) 6.12319 0.256248 0.128124 0.991758i \(-0.459105\pi\)
0.128124 + 0.991758i \(0.459105\pi\)
\(572\) 0 0
\(573\) −42.0236 −1.75556
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 27.2174 1.13307 0.566537 0.824036i \(-0.308283\pi\)
0.566537 + 0.824036i \(0.308283\pi\)
\(578\) 0 0
\(579\) −35.4494 −1.47323
\(580\) 0 0
\(581\) 12.7470 0.528836
\(582\) 0 0
\(583\) 2.41906 0.100187
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 26.7305 1.10329 0.551644 0.834080i \(-0.314001\pi\)
0.551644 + 0.834080i \(0.314001\pi\)
\(588\) 0 0
\(589\) 13.5317 0.557564
\(590\) 0 0
\(591\) 2.01133 0.0827350
\(592\) 0 0
\(593\) −0.0562817 −0.00231121 −0.00115561 0.999999i \(-0.500368\pi\)
−0.00115561 + 0.999999i \(0.500368\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 31.1762 1.27596
\(598\) 0 0
\(599\) −39.4760 −1.61295 −0.806473 0.591271i \(-0.798626\pi\)
−0.806473 + 0.591271i \(0.798626\pi\)
\(600\) 0 0
\(601\) 23.2118 0.946829 0.473415 0.880840i \(-0.343021\pi\)
0.473415 + 0.880840i \(0.343021\pi\)
\(602\) 0 0
\(603\) 15.7490 0.641350
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −19.7212 −0.800459 −0.400230 0.916415i \(-0.631070\pi\)
−0.400230 + 0.916415i \(0.631070\pi\)
\(608\) 0 0
\(609\) −12.4864 −0.505973
\(610\) 0 0
\(611\) 0.0591707 0.00239379
\(612\) 0 0
\(613\) −10.4609 −0.422512 −0.211256 0.977431i \(-0.567755\pi\)
−0.211256 + 0.977431i \(0.567755\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −14.4969 −0.583622 −0.291811 0.956476i \(-0.594258\pi\)
−0.291811 + 0.956476i \(0.594258\pi\)
\(618\) 0 0
\(619\) −1.30293 −0.0523692 −0.0261846 0.999657i \(-0.508336\pi\)
−0.0261846 + 0.999657i \(0.508336\pi\)
\(620\) 0 0
\(621\) −1.67786 −0.0673302
\(622\) 0 0
\(623\) 15.7227 0.629915
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −17.9006 −0.714881
\(628\) 0 0
\(629\) 13.7220 0.547133
\(630\) 0 0
\(631\) −0.653161 −0.0260019 −0.0130010 0.999915i \(-0.504138\pi\)
−0.0130010 + 0.999915i \(0.504138\pi\)
\(632\) 0 0
\(633\) −29.6340 −1.17785
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −21.4202 −0.848699
\(638\) 0 0
\(639\) −4.92075 −0.194662
\(640\) 0 0
\(641\) −13.2594 −0.523714 −0.261857 0.965107i \(-0.584335\pi\)
−0.261857 + 0.965107i \(0.584335\pi\)
\(642\) 0 0
\(643\) 45.9135 1.81065 0.905326 0.424716i \(-0.139626\pi\)
0.905326 + 0.424716i \(0.139626\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.36368 0.132240 0.0661199 0.997812i \(-0.478938\pi\)
0.0661199 + 0.997812i \(0.478938\pi\)
\(648\) 0 0
\(649\) −8.28694 −0.325291
\(650\) 0 0
\(651\) −17.9922 −0.705169
\(652\) 0 0
\(653\) −25.4704 −0.996734 −0.498367 0.866966i \(-0.666067\pi\)
−0.498367 + 0.866966i \(0.666067\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −13.6798 −0.533700
\(658\) 0 0
\(659\) 9.03335 0.351890 0.175945 0.984400i \(-0.443702\pi\)
0.175945 + 0.984400i \(0.443702\pi\)
\(660\) 0 0
\(661\) −3.20679 −0.124730 −0.0623648 0.998053i \(-0.519864\pi\)
−0.0623648 + 0.998053i \(0.519864\pi\)
\(662\) 0 0
\(663\) −39.7515 −1.54382
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.36282 0.130209
\(668\) 0 0
\(669\) 18.4904 0.714880
\(670\) 0 0
\(671\) 20.3805 0.786779
\(672\) 0 0
\(673\) −16.4923 −0.635733 −0.317866 0.948135i \(-0.602966\pi\)
−0.317866 + 0.948135i \(0.602966\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 34.9260 1.34232 0.671158 0.741314i \(-0.265797\pi\)
0.671158 + 0.741314i \(0.265797\pi\)
\(678\) 0 0
\(679\) −19.4033 −0.744629
\(680\) 0 0
\(681\) −10.1341 −0.388341
\(682\) 0 0
\(683\) 23.1448 0.885612 0.442806 0.896618i \(-0.353983\pi\)
0.442806 + 0.896618i \(0.353983\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −40.2153 −1.53431
\(688\) 0 0
\(689\) −4.23306 −0.161267
\(690\) 0 0
\(691\) 3.24273 0.123359 0.0616795 0.998096i \(-0.480354\pi\)
0.0616795 + 0.998096i \(0.480354\pi\)
\(692\) 0 0
\(693\) 10.2497 0.389353
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1.16212 −0.0440184
\(698\) 0 0
\(699\) −36.2863 −1.37247
\(700\) 0 0
\(701\) 19.9466 0.753373 0.376686 0.926341i \(-0.377063\pi\)
0.376686 + 0.926341i \(0.377063\pi\)
\(702\) 0 0
\(703\) −10.8129 −0.407814
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −18.5556 −0.697854
\(708\) 0 0
\(709\) 14.3774 0.539955 0.269977 0.962867i \(-0.412984\pi\)
0.269977 + 0.962867i \(0.412984\pi\)
\(710\) 0 0
\(711\) 14.7615 0.553601
\(712\) 0 0
\(713\) 4.84564 0.181471
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 20.5206 0.766356
\(718\) 0 0
\(719\) −51.4356 −1.91822 −0.959111 0.283029i \(-0.908661\pi\)
−0.959111 + 0.283029i \(0.908661\pi\)
\(720\) 0 0
\(721\) 6.50317 0.242191
\(722\) 0 0
\(723\) −8.05682 −0.299636
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17.4611 0.647597 0.323799 0.946126i \(-0.395040\pi\)
0.323799 + 0.946126i \(0.395040\pi\)
\(728\) 0 0
\(729\) −12.6305 −0.467797
\(730\) 0 0
\(731\) −19.0782 −0.705633
\(732\) 0 0
\(733\) −39.6443 −1.46429 −0.732147 0.681147i \(-0.761482\pi\)
−0.732147 + 0.681147i \(0.761482\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −19.3826 −0.713966
\(738\) 0 0
\(739\) −43.1149 −1.58601 −0.793003 0.609218i \(-0.791483\pi\)
−0.793003 + 0.609218i \(0.791483\pi\)
\(740\) 0 0
\(741\) 31.3239 1.15071
\(742\) 0 0
\(743\) −18.8929 −0.693112 −0.346556 0.938029i \(-0.612649\pi\)
−0.346556 + 0.938029i \(0.612649\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −17.8808 −0.654223
\(748\) 0 0
\(749\) −23.9917 −0.876637
\(750\) 0 0
\(751\) 27.5146 1.00402 0.502012 0.864861i \(-0.332593\pi\)
0.502012 + 0.864861i \(0.332593\pi\)
\(752\) 0 0
\(753\) −9.95847 −0.362907
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 14.6895 0.533899 0.266949 0.963711i \(-0.413984\pi\)
0.266949 + 0.963711i \(0.413984\pi\)
\(758\) 0 0
\(759\) −6.41013 −0.232673
\(760\) 0 0
\(761\) −37.5884 −1.36258 −0.681289 0.732015i \(-0.738580\pi\)
−0.681289 + 0.732015i \(0.738580\pi\)
\(762\) 0 0
\(763\) −19.9890 −0.723651
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14.5012 0.523606
\(768\) 0 0
\(769\) −43.6879 −1.57543 −0.787713 0.616042i \(-0.788735\pi\)
−0.787713 + 0.616042i \(0.788735\pi\)
\(770\) 0 0
\(771\) 38.1620 1.37437
\(772\) 0 0
\(773\) 7.01583 0.252342 0.126171 0.992009i \(-0.459731\pi\)
0.126171 + 0.992009i \(0.459731\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 14.3771 0.515776
\(778\) 0 0
\(779\) 0.915742 0.0328099
\(780\) 0 0
\(781\) 6.05604 0.216702
\(782\) 0 0
\(783\) −5.64234 −0.201641
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −23.6386 −0.842624 −0.421312 0.906916i \(-0.638430\pi\)
−0.421312 + 0.906916i \(0.638430\pi\)
\(788\) 0 0
\(789\) −29.8601 −1.06305
\(790\) 0 0
\(791\) 8.87457 0.315543
\(792\) 0 0
\(793\) −35.6634 −1.26644
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 31.9000 1.12996 0.564978 0.825106i \(-0.308885\pi\)
0.564978 + 0.825106i \(0.308885\pi\)
\(798\) 0 0
\(799\) 0.0429117 0.00151811
\(800\) 0 0
\(801\) −22.0548 −0.779268
\(802\) 0 0
\(803\) 16.8359 0.594127
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 19.0750 0.671472
\(808\) 0 0
\(809\) −5.68831 −0.199990 −0.0999951 0.994988i \(-0.531883\pi\)
−0.0999951 + 0.994988i \(0.531883\pi\)
\(810\) 0 0
\(811\) −16.1989 −0.568821 −0.284410 0.958703i \(-0.591798\pi\)
−0.284410 + 0.958703i \(0.591798\pi\)
\(812\) 0 0
\(813\) 19.1671 0.672221
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 15.0335 0.525955
\(818\) 0 0
\(819\) −17.9357 −0.626725
\(820\) 0 0
\(821\) −26.9329 −0.939965 −0.469982 0.882676i \(-0.655740\pi\)
−0.469982 + 0.882676i \(0.655740\pi\)
\(822\) 0 0
\(823\) 42.0280 1.46500 0.732502 0.680765i \(-0.238352\pi\)
0.732502 + 0.680765i \(0.238352\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11.1311 −0.387065 −0.193532 0.981094i \(-0.561994\pi\)
−0.193532 + 0.981094i \(0.561994\pi\)
\(828\) 0 0
\(829\) −20.1506 −0.699859 −0.349929 0.936776i \(-0.613794\pi\)
−0.349929 + 0.936776i \(0.613794\pi\)
\(830\) 0 0
\(831\) −14.3051 −0.496237
\(832\) 0 0
\(833\) −15.5343 −0.538233
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −8.13031 −0.281024
\(838\) 0 0
\(839\) −12.5573 −0.433526 −0.216763 0.976224i \(-0.569550\pi\)
−0.216763 + 0.976224i \(0.569550\pi\)
\(840\) 0 0
\(841\) −17.6914 −0.610050
\(842\) 0 0
\(843\) −18.4057 −0.633925
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 5.17896 0.177951
\(848\) 0 0
\(849\) 20.9285 0.718263
\(850\) 0 0
\(851\) −3.87204 −0.132732
\(852\) 0 0
\(853\) −50.2016 −1.71887 −0.859434 0.511246i \(-0.829184\pi\)
−0.859434 + 0.511246i \(0.829184\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −36.9126 −1.26091 −0.630456 0.776225i \(-0.717132\pi\)
−0.630456 + 0.776225i \(0.717132\pi\)
\(858\) 0 0
\(859\) 7.29638 0.248949 0.124475 0.992223i \(-0.460275\pi\)
0.124475 + 0.992223i \(0.460275\pi\)
\(860\) 0 0
\(861\) −1.21760 −0.0414957
\(862\) 0 0
\(863\) −52.3903 −1.78339 −0.891694 0.452640i \(-0.850482\pi\)
−0.891694 + 0.452640i \(0.850482\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 10.1939 0.346203
\(868\) 0 0
\(869\) −18.1672 −0.616281
\(870\) 0 0
\(871\) 33.9172 1.14924
\(872\) 0 0
\(873\) 27.2177 0.921181
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −28.1120 −0.949276 −0.474638 0.880181i \(-0.657421\pi\)
−0.474638 + 0.880181i \(0.657421\pi\)
\(878\) 0 0
\(879\) −40.2444 −1.35741
\(880\) 0 0
\(881\) 41.3936 1.39459 0.697293 0.716787i \(-0.254388\pi\)
0.697293 + 0.716787i \(0.254388\pi\)
\(882\) 0 0
\(883\) −36.7906 −1.23810 −0.619052 0.785350i \(-0.712483\pi\)
−0.619052 + 0.785350i \(0.712483\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 43.7836 1.47011 0.735055 0.678007i \(-0.237156\pi\)
0.735055 + 0.678007i \(0.237156\pi\)
\(888\) 0 0
\(889\) 5.09471 0.170871
\(890\) 0 0
\(891\) 29.7646 0.997151
\(892\) 0 0
\(893\) −0.0338141 −0.00113155
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 11.2170 0.374524
\(898\) 0 0
\(899\) 16.2950 0.543469
\(900\) 0 0
\(901\) −3.06990 −0.102273
\(902\) 0 0
\(903\) −19.9890 −0.665193
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −20.8370 −0.691882 −0.345941 0.938256i \(-0.612440\pi\)
−0.345941 + 0.938256i \(0.612440\pi\)
\(908\) 0 0
\(909\) 26.0287 0.863316
\(910\) 0 0
\(911\) 45.8193 1.51806 0.759031 0.651054i \(-0.225673\pi\)
0.759031 + 0.651054i \(0.225673\pi\)
\(912\) 0 0
\(913\) 22.0061 0.728297
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.74326 0.156636
\(918\) 0 0
\(919\) 30.3282 1.00044 0.500218 0.865900i \(-0.333253\pi\)
0.500218 + 0.865900i \(0.333253\pi\)
\(920\) 0 0
\(921\) −69.6446 −2.29487
\(922\) 0 0
\(923\) −10.5973 −0.348816
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −9.12226 −0.299614
\(928\) 0 0
\(929\) −35.7602 −1.17325 −0.586626 0.809858i \(-0.699544\pi\)
−0.586626 + 0.809858i \(0.699544\pi\)
\(930\) 0 0
\(931\) 12.2409 0.401181
\(932\) 0 0
\(933\) 31.6088 1.03483
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −26.6768 −0.871494 −0.435747 0.900069i \(-0.643516\pi\)
−0.435747 + 0.900069i \(0.643516\pi\)
\(938\) 0 0
\(939\) −32.9890 −1.07656
\(940\) 0 0
\(941\) −9.52987 −0.310665 −0.155332 0.987862i \(-0.549645\pi\)
−0.155332 + 0.987862i \(0.549645\pi\)
\(942\) 0 0
\(943\) 0.327923 0.0106787
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 32.5209 1.05679 0.528393 0.849000i \(-0.322795\pi\)
0.528393 + 0.849000i \(0.322795\pi\)
\(948\) 0 0
\(949\) −29.4609 −0.956341
\(950\) 0 0
\(951\) −60.9958 −1.97793
\(952\) 0 0
\(953\) 8.80252 0.285141 0.142571 0.989785i \(-0.454463\pi\)
0.142571 + 0.989785i \(0.454463\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −21.5561 −0.696810
\(958\) 0 0
\(959\) −23.3018 −0.752456
\(960\) 0 0
\(961\) −7.51974 −0.242572
\(962\) 0 0
\(963\) 33.6541 1.08449
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −47.0307 −1.51241 −0.756203 0.654337i \(-0.772948\pi\)
−0.756203 + 0.654337i \(0.772948\pi\)
\(968\) 0 0
\(969\) 22.7167 0.729766
\(970\) 0 0
\(971\) 33.3450 1.07009 0.535046 0.844823i \(-0.320294\pi\)
0.535046 + 0.844823i \(0.320294\pi\)
\(972\) 0 0
\(973\) −29.4789 −0.945050
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −55.8915 −1.78813 −0.894064 0.447939i \(-0.852158\pi\)
−0.894064 + 0.447939i \(0.852158\pi\)
\(978\) 0 0
\(979\) 27.1432 0.867500
\(980\) 0 0
\(981\) 28.0394 0.895229
\(982\) 0 0
\(983\) 44.2540 1.41148 0.705742 0.708469i \(-0.250614\pi\)
0.705742 + 0.708469i \(0.250614\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0.0449603 0.00143110
\(988\) 0 0
\(989\) 5.38343 0.171183
\(990\) 0 0
\(991\) −28.2426 −0.897157 −0.448579 0.893743i \(-0.648070\pi\)
−0.448579 + 0.893743i \(0.648070\pi\)
\(992\) 0 0
\(993\) 28.0913 0.891451
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 19.7730 0.626218 0.313109 0.949717i \(-0.398629\pi\)
0.313109 + 0.949717i \(0.398629\pi\)
\(998\) 0 0
\(999\) 6.49673 0.205547
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 4600.2.a.bf.1.1 yes 5
4.3 odd 2 9200.2.a.ct.1.5 5
5.2 odd 4 4600.2.e.w.4049.9 10
5.3 odd 4 4600.2.e.w.4049.2 10
5.4 even 2 4600.2.a.bd.1.5 5
20.19 odd 2 9200.2.a.cv.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.5 5 5.4 even 2
4600.2.a.bf.1.1 yes 5 1.1 even 1 trivial
4600.2.e.w.4049.2 10 5.3 odd 4
4600.2.e.w.4049.9 10 5.2 odd 4
9200.2.a.ct.1.5 5 4.3 odd 2
9200.2.a.cv.1.1 5 20.19 odd 2