Properties

Label 8016.2.a.ba.1.5
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 16 x^{7} + 45 x^{6} + 67 x^{5} - 166 x^{4} - 83 x^{3} + 152 x^{2} + 51 x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.141931\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.858069 q^{5} -2.33225 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.858069 q^{5} -2.33225 q^{7} +1.00000 q^{9} -0.664078 q^{11} -2.80424 q^{13} +0.858069 q^{15} -4.73771 q^{17} +7.58466 q^{19} +2.33225 q^{21} -6.58999 q^{23} -4.26372 q^{25} -1.00000 q^{27} -4.75833 q^{29} +2.96038 q^{31} +0.664078 q^{33} +2.00123 q^{35} +4.12480 q^{37} +2.80424 q^{39} -7.76756 q^{41} -4.13938 q^{43} -0.858069 q^{45} +1.39111 q^{47} -1.56060 q^{49} +4.73771 q^{51} -1.74469 q^{53} +0.569824 q^{55} -7.58466 q^{57} +10.2363 q^{59} -0.762183 q^{61} -2.33225 q^{63} +2.40623 q^{65} -0.00130165 q^{67} +6.58999 q^{69} +2.99630 q^{71} -16.6717 q^{73} +4.26372 q^{75} +1.54880 q^{77} +10.5928 q^{79} +1.00000 q^{81} -17.4041 q^{83} +4.06528 q^{85} +4.75833 q^{87} -4.94100 q^{89} +6.54021 q^{91} -2.96038 q^{93} -6.50816 q^{95} +17.4544 q^{97} -0.664078 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} - 6q^{5} + 11q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} - 6q^{5} + 11q^{7} + 9q^{9} - q^{11} - 4q^{13} + 6q^{15} - 9q^{17} + 8q^{19} - 11q^{21} + 7q^{23} - q^{25} - 9q^{27} - 9q^{29} + 25q^{31} + q^{33} - 5q^{35} - 6q^{37} + 4q^{39} - 4q^{41} + 24q^{43} - 6q^{45} + 16q^{47} + 4q^{49} + 9q^{51} - 26q^{53} + 29q^{55} - 8q^{57} + 4q^{59} - 20q^{61} + 11q^{63} - 8q^{65} + 25q^{67} - 7q^{69} + 15q^{71} - 10q^{73} + q^{75} - 20q^{77} + 34q^{79} + 9q^{81} + 4q^{83} - 13q^{85} + 9q^{87} + 13q^{89} + 21q^{91} - 25q^{93} + 7q^{95} - 4q^{97} - q^{99} + O(q^{100}) \)

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
\(5\) −0.858069 −0.383740 −0.191870 0.981420i \(-0.561455\pi\)
−0.191870 + 0.981420i \(0.561455\pi\)
\(6\) 0 0
\(7\) −2.33225 −0.881509 −0.440754 0.897628i \(-0.645289\pi\)
−0.440754 + 0.897628i \(0.645289\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.664078 −0.200227 −0.100113 0.994976i \(-0.531921\pi\)
−0.100113 + 0.994976i \(0.531921\pi\)
\(12\) 0 0
\(13\) −2.80424 −0.777757 −0.388879 0.921289i \(-0.627137\pi\)
−0.388879 + 0.921289i \(0.627137\pi\)
\(14\) 0 0
\(15\) 0.858069 0.221552
\(16\) 0 0
\(17\) −4.73771 −1.14906 −0.574532 0.818482i \(-0.694816\pi\)
−0.574532 + 0.818482i \(0.694816\pi\)
\(18\) 0 0
\(19\) 7.58466 1.74004 0.870021 0.493015i \(-0.164105\pi\)
0.870021 + 0.493015i \(0.164105\pi\)
\(20\) 0 0
\(21\) 2.33225 0.508939
\(22\) 0 0
\(23\) −6.58999 −1.37411 −0.687054 0.726606i \(-0.741097\pi\)
−0.687054 + 0.726606i \(0.741097\pi\)
\(24\) 0 0
\(25\) −4.26372 −0.852744
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.75833 −0.883600 −0.441800 0.897114i \(-0.645660\pi\)
−0.441800 + 0.897114i \(0.645660\pi\)
\(30\) 0 0
\(31\) 2.96038 0.531700 0.265850 0.964014i \(-0.414347\pi\)
0.265850 + 0.964014i \(0.414347\pi\)
\(32\) 0 0
\(33\) 0.664078 0.115601
\(34\) 0 0
\(35\) 2.00123 0.338270
\(36\) 0 0
\(37\) 4.12480 0.678113 0.339056 0.940766i \(-0.389892\pi\)
0.339056 + 0.940766i \(0.389892\pi\)
\(38\) 0 0
\(39\) 2.80424 0.449038
\(40\) 0 0
\(41\) −7.76756 −1.21309 −0.606544 0.795050i \(-0.707445\pi\)
−0.606544 + 0.795050i \(0.707445\pi\)
\(42\) 0 0
\(43\) −4.13938 −0.631249 −0.315624 0.948884i \(-0.602214\pi\)
−0.315624 + 0.948884i \(0.602214\pi\)
\(44\) 0 0
\(45\) −0.858069 −0.127913
\(46\) 0 0
\(47\) 1.39111 0.202914 0.101457 0.994840i \(-0.467650\pi\)
0.101457 + 0.994840i \(0.467650\pi\)
\(48\) 0 0
\(49\) −1.56060 −0.222942
\(50\) 0 0
\(51\) 4.73771 0.663412
\(52\) 0 0
\(53\) −1.74469 −0.239651 −0.119826 0.992795i \(-0.538234\pi\)
−0.119826 + 0.992795i \(0.538234\pi\)
\(54\) 0 0
\(55\) 0.569824 0.0768351
\(56\) 0 0
\(57\) −7.58466 −1.00461
\(58\) 0 0
\(59\) 10.2363 1.33265 0.666325 0.745661i \(-0.267866\pi\)
0.666325 + 0.745661i \(0.267866\pi\)
\(60\) 0 0
\(61\) −0.762183 −0.0975875 −0.0487937 0.998809i \(-0.515538\pi\)
−0.0487937 + 0.998809i \(0.515538\pi\)
\(62\) 0 0
\(63\) −2.33225 −0.293836
\(64\) 0 0
\(65\) 2.40623 0.298457
\(66\) 0 0
\(67\) −0.00130165 −0.000159022 0 −7.95109e−5 1.00000i \(-0.500025\pi\)
−7.95109e−5 1.00000i \(0.500025\pi\)
\(68\) 0 0
\(69\) 6.58999 0.793342
\(70\) 0 0
\(71\) 2.99630 0.355596 0.177798 0.984067i \(-0.443103\pi\)
0.177798 + 0.984067i \(0.443103\pi\)
\(72\) 0 0
\(73\) −16.6717 −1.95128 −0.975640 0.219379i \(-0.929597\pi\)
−0.975640 + 0.219379i \(0.929597\pi\)
\(74\) 0 0
\(75\) 4.26372 0.492332
\(76\) 0 0
\(77\) 1.54880 0.176502
\(78\) 0 0
\(79\) 10.5928 1.19178 0.595891 0.803065i \(-0.296799\pi\)
0.595891 + 0.803065i \(0.296799\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −17.4041 −1.91035 −0.955177 0.296037i \(-0.904335\pi\)
−0.955177 + 0.296037i \(0.904335\pi\)
\(84\) 0 0
\(85\) 4.06528 0.440941
\(86\) 0 0
\(87\) 4.75833 0.510147
\(88\) 0 0
\(89\) −4.94100 −0.523745 −0.261872 0.965103i \(-0.584340\pi\)
−0.261872 + 0.965103i \(0.584340\pi\)
\(90\) 0 0
\(91\) 6.54021 0.685600
\(92\) 0 0
\(93\) −2.96038 −0.306977
\(94\) 0 0
\(95\) −6.50816 −0.667723
\(96\) 0 0
\(97\) 17.4544 1.77222 0.886112 0.463471i \(-0.153396\pi\)
0.886112 + 0.463471i \(0.153396\pi\)
\(98\) 0 0
\(99\) −0.664078 −0.0667423
\(100\) 0 0
\(101\) 2.83624 0.282216 0.141108 0.989994i \(-0.454933\pi\)
0.141108 + 0.989994i \(0.454933\pi\)
\(102\) 0 0
\(103\) −1.68361 −0.165891 −0.0829454 0.996554i \(-0.526433\pi\)
−0.0829454 + 0.996554i \(0.526433\pi\)
\(104\) 0 0
\(105\) −2.00123 −0.195300
\(106\) 0 0
\(107\) −10.4960 −1.01468 −0.507341 0.861745i \(-0.669372\pi\)
−0.507341 + 0.861745i \(0.669372\pi\)
\(108\) 0 0
\(109\) −3.61751 −0.346494 −0.173247 0.984878i \(-0.555426\pi\)
−0.173247 + 0.984878i \(0.555426\pi\)
\(110\) 0 0
\(111\) −4.12480 −0.391509
\(112\) 0 0
\(113\) 2.15002 0.202257 0.101129 0.994873i \(-0.467755\pi\)
0.101129 + 0.994873i \(0.467755\pi\)
\(114\) 0 0
\(115\) 5.65467 0.527300
\(116\) 0 0
\(117\) −2.80424 −0.259252
\(118\) 0 0
\(119\) 11.0495 1.01291
\(120\) 0 0
\(121\) −10.5590 −0.959909
\(122\) 0 0
\(123\) 7.76756 0.700377
\(124\) 0 0
\(125\) 7.94891 0.710972
\(126\) 0 0
\(127\) 12.3626 1.09701 0.548504 0.836148i \(-0.315198\pi\)
0.548504 + 0.836148i \(0.315198\pi\)
\(128\) 0 0
\(129\) 4.13938 0.364452
\(130\) 0 0
\(131\) −20.5805 −1.79812 −0.899061 0.437822i \(-0.855750\pi\)
−0.899061 + 0.437822i \(0.855750\pi\)
\(132\) 0 0
\(133\) −17.6894 −1.53386
\(134\) 0 0
\(135\) 0.858069 0.0738508
\(136\) 0 0
\(137\) 10.1249 0.865031 0.432515 0.901627i \(-0.357626\pi\)
0.432515 + 0.901627i \(0.357626\pi\)
\(138\) 0 0
\(139\) −1.84695 −0.156656 −0.0783280 0.996928i \(-0.524958\pi\)
−0.0783280 + 0.996928i \(0.524958\pi\)
\(140\) 0 0
\(141\) −1.39111 −0.117153
\(142\) 0 0
\(143\) 1.86224 0.155728
\(144\) 0 0
\(145\) 4.08297 0.339073
\(146\) 0 0
\(147\) 1.56060 0.128716
\(148\) 0 0
\(149\) 5.36994 0.439922 0.219961 0.975509i \(-0.429407\pi\)
0.219961 + 0.975509i \(0.429407\pi\)
\(150\) 0 0
\(151\) 13.0563 1.06251 0.531254 0.847213i \(-0.321721\pi\)
0.531254 + 0.847213i \(0.321721\pi\)
\(152\) 0 0
\(153\) −4.73771 −0.383021
\(154\) 0 0
\(155\) −2.54021 −0.204034
\(156\) 0 0
\(157\) 5.47662 0.437082 0.218541 0.975828i \(-0.429870\pi\)
0.218541 + 0.975828i \(0.429870\pi\)
\(158\) 0 0
\(159\) 1.74469 0.138363
\(160\) 0 0
\(161\) 15.3695 1.21129
\(162\) 0 0
\(163\) −18.0321 −1.41239 −0.706193 0.708019i \(-0.749589\pi\)
−0.706193 + 0.708019i \(0.749589\pi\)
\(164\) 0 0
\(165\) −0.569824 −0.0443607
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −5.13621 −0.395093
\(170\) 0 0
\(171\) 7.58466 0.580014
\(172\) 0 0
\(173\) 11.9069 0.905266 0.452633 0.891697i \(-0.350485\pi\)
0.452633 + 0.891697i \(0.350485\pi\)
\(174\) 0 0
\(175\) 9.94407 0.751701
\(176\) 0 0
\(177\) −10.2363 −0.769406
\(178\) 0 0
\(179\) 9.99610 0.747144 0.373572 0.927601i \(-0.378133\pi\)
0.373572 + 0.927601i \(0.378133\pi\)
\(180\) 0 0
\(181\) −12.9764 −0.964526 −0.482263 0.876027i \(-0.660185\pi\)
−0.482263 + 0.876027i \(0.660185\pi\)
\(182\) 0 0
\(183\) 0.762183 0.0563422
\(184\) 0 0
\(185\) −3.53936 −0.260219
\(186\) 0 0
\(187\) 3.14621 0.230073
\(188\) 0 0
\(189\) 2.33225 0.169646
\(190\) 0 0
\(191\) 0.780931 0.0565062 0.0282531 0.999601i \(-0.491006\pi\)
0.0282531 + 0.999601i \(0.491006\pi\)
\(192\) 0 0
\(193\) −3.18460 −0.229232 −0.114616 0.993410i \(-0.536564\pi\)
−0.114616 + 0.993410i \(0.536564\pi\)
\(194\) 0 0
\(195\) −2.40623 −0.172314
\(196\) 0 0
\(197\) −7.29302 −0.519606 −0.259803 0.965662i \(-0.583658\pi\)
−0.259803 + 0.965662i \(0.583658\pi\)
\(198\) 0 0
\(199\) 20.4949 1.45284 0.726422 0.687249i \(-0.241182\pi\)
0.726422 + 0.687249i \(0.241182\pi\)
\(200\) 0 0
\(201\) 0.00130165 9.18113e−5 0
\(202\) 0 0
\(203\) 11.0976 0.778901
\(204\) 0 0
\(205\) 6.66510 0.465511
\(206\) 0 0
\(207\) −6.58999 −0.458036
\(208\) 0 0
\(209\) −5.03681 −0.348403
\(210\) 0 0
\(211\) −2.79418 −0.192359 −0.0961795 0.995364i \(-0.530662\pi\)
−0.0961795 + 0.995364i \(0.530662\pi\)
\(212\) 0 0
\(213\) −2.99630 −0.205303
\(214\) 0 0
\(215\) 3.55187 0.242235
\(216\) 0 0
\(217\) −6.90435 −0.468698
\(218\) 0 0
\(219\) 16.6717 1.12657
\(220\) 0 0
\(221\) 13.2857 0.893692
\(222\) 0 0
\(223\) 1.07046 0.0716830 0.0358415 0.999357i \(-0.488589\pi\)
0.0358415 + 0.999357i \(0.488589\pi\)
\(224\) 0 0
\(225\) −4.26372 −0.284248
\(226\) 0 0
\(227\) 3.06628 0.203516 0.101758 0.994809i \(-0.467553\pi\)
0.101758 + 0.994809i \(0.467553\pi\)
\(228\) 0 0
\(229\) −0.166978 −0.0110342 −0.00551710 0.999985i \(-0.501756\pi\)
−0.00551710 + 0.999985i \(0.501756\pi\)
\(230\) 0 0
\(231\) −1.54880 −0.101903
\(232\) 0 0
\(233\) 9.75862 0.639309 0.319654 0.947534i \(-0.396433\pi\)
0.319654 + 0.947534i \(0.396433\pi\)
\(234\) 0 0
\(235\) −1.19367 −0.0778663
\(236\) 0 0
\(237\) −10.5928 −0.688076
\(238\) 0 0
\(239\) 27.8243 1.79980 0.899902 0.436093i \(-0.143638\pi\)
0.899902 + 0.436093i \(0.143638\pi\)
\(240\) 0 0
\(241\) −6.03419 −0.388696 −0.194348 0.980933i \(-0.562259\pi\)
−0.194348 + 0.980933i \(0.562259\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 1.33910 0.0855519
\(246\) 0 0
\(247\) −21.2693 −1.35333
\(248\) 0 0
\(249\) 17.4041 1.10294
\(250\) 0 0
\(251\) 19.2169 1.21296 0.606481 0.795098i \(-0.292581\pi\)
0.606481 + 0.795098i \(0.292581\pi\)
\(252\) 0 0
\(253\) 4.37627 0.275133
\(254\) 0 0
\(255\) −4.06528 −0.254578
\(256\) 0 0
\(257\) −8.38942 −0.523318 −0.261659 0.965160i \(-0.584270\pi\)
−0.261659 + 0.965160i \(0.584270\pi\)
\(258\) 0 0
\(259\) −9.62007 −0.597762
\(260\) 0 0
\(261\) −4.75833 −0.294533
\(262\) 0 0
\(263\) 16.9678 1.04628 0.523139 0.852247i \(-0.324761\pi\)
0.523139 + 0.852247i \(0.324761\pi\)
\(264\) 0 0
\(265\) 1.49706 0.0919638
\(266\) 0 0
\(267\) 4.94100 0.302384
\(268\) 0 0
\(269\) −13.6797 −0.834068 −0.417034 0.908891i \(-0.636930\pi\)
−0.417034 + 0.908891i \(0.636930\pi\)
\(270\) 0 0
\(271\) 16.2857 0.989284 0.494642 0.869097i \(-0.335299\pi\)
0.494642 + 0.869097i \(0.335299\pi\)
\(272\) 0 0
\(273\) −6.54021 −0.395831
\(274\) 0 0
\(275\) 2.83144 0.170742
\(276\) 0 0
\(277\) −4.11423 −0.247200 −0.123600 0.992332i \(-0.539444\pi\)
−0.123600 + 0.992332i \(0.539444\pi\)
\(278\) 0 0
\(279\) 2.96038 0.177233
\(280\) 0 0
\(281\) 29.3388 1.75020 0.875102 0.483939i \(-0.160794\pi\)
0.875102 + 0.483939i \(0.160794\pi\)
\(282\) 0 0
\(283\) 16.5471 0.983621 0.491810 0.870702i \(-0.336335\pi\)
0.491810 + 0.870702i \(0.336335\pi\)
\(284\) 0 0
\(285\) 6.50816 0.385510
\(286\) 0 0
\(287\) 18.1159 1.06935
\(288\) 0 0
\(289\) 5.44589 0.320346
\(290\) 0 0
\(291\) −17.4544 −1.02319
\(292\) 0 0
\(293\) −31.9181 −1.86468 −0.932338 0.361588i \(-0.882235\pi\)
−0.932338 + 0.361588i \(0.882235\pi\)
\(294\) 0 0
\(295\) −8.78344 −0.511391
\(296\) 0 0
\(297\) 0.664078 0.0385337
\(298\) 0 0
\(299\) 18.4799 1.06872
\(300\) 0 0
\(301\) 9.65407 0.556451
\(302\) 0 0
\(303\) −2.83624 −0.162938
\(304\) 0 0
\(305\) 0.654005 0.0374482
\(306\) 0 0
\(307\) 12.7567 0.728063 0.364031 0.931387i \(-0.381400\pi\)
0.364031 + 0.931387i \(0.381400\pi\)
\(308\) 0 0
\(309\) 1.68361 0.0957771
\(310\) 0 0
\(311\) 4.06532 0.230523 0.115262 0.993335i \(-0.463229\pi\)
0.115262 + 0.993335i \(0.463229\pi\)
\(312\) 0 0
\(313\) −9.87994 −0.558447 −0.279224 0.960226i \(-0.590077\pi\)
−0.279224 + 0.960226i \(0.590077\pi\)
\(314\) 0 0
\(315\) 2.00123 0.112757
\(316\) 0 0
\(317\) 11.2409 0.631353 0.315676 0.948867i \(-0.397769\pi\)
0.315676 + 0.948867i \(0.397769\pi\)
\(318\) 0 0
\(319\) 3.15990 0.176920
\(320\) 0 0
\(321\) 10.4960 0.585827
\(322\) 0 0
\(323\) −35.9339 −1.99942
\(324\) 0 0
\(325\) 11.9565 0.663228
\(326\) 0 0
\(327\) 3.61751 0.200049
\(328\) 0 0
\(329\) −3.24442 −0.178871
\(330\) 0 0
\(331\) 14.6160 0.803370 0.401685 0.915778i \(-0.368425\pi\)
0.401685 + 0.915778i \(0.368425\pi\)
\(332\) 0 0
\(333\) 4.12480 0.226038
\(334\) 0 0
\(335\) 0.00111690 6.10230e−5 0
\(336\) 0 0
\(337\) 12.7052 0.692094 0.346047 0.938217i \(-0.387524\pi\)
0.346047 + 0.938217i \(0.387524\pi\)
\(338\) 0 0
\(339\) −2.15002 −0.116773
\(340\) 0 0
\(341\) −1.96592 −0.106461
\(342\) 0 0
\(343\) 19.9655 1.07803
\(344\) 0 0
\(345\) −5.65467 −0.304437
\(346\) 0 0
\(347\) −34.7085 −1.86325 −0.931625 0.363422i \(-0.881608\pi\)
−0.931625 + 0.363422i \(0.881608\pi\)
\(348\) 0 0
\(349\) −0.432938 −0.0231746 −0.0115873 0.999933i \(-0.503688\pi\)
−0.0115873 + 0.999933i \(0.503688\pi\)
\(350\) 0 0
\(351\) 2.80424 0.149679
\(352\) 0 0
\(353\) 15.5524 0.827769 0.413884 0.910329i \(-0.364172\pi\)
0.413884 + 0.910329i \(0.364172\pi\)
\(354\) 0 0
\(355\) −2.57103 −0.136456
\(356\) 0 0
\(357\) −11.0495 −0.584803
\(358\) 0 0
\(359\) −12.9062 −0.681163 −0.340581 0.940215i \(-0.610624\pi\)
−0.340581 + 0.940215i \(0.610624\pi\)
\(360\) 0 0
\(361\) 38.5271 2.02774
\(362\) 0 0
\(363\) 10.5590 0.554204
\(364\) 0 0
\(365\) 14.3055 0.748784
\(366\) 0 0
\(367\) 24.5614 1.28210 0.641048 0.767501i \(-0.278500\pi\)
0.641048 + 0.767501i \(0.278500\pi\)
\(368\) 0 0
\(369\) −7.76756 −0.404363
\(370\) 0 0
\(371\) 4.06905 0.211255
\(372\) 0 0
\(373\) 25.7222 1.33184 0.665922 0.746021i \(-0.268038\pi\)
0.665922 + 0.746021i \(0.268038\pi\)
\(374\) 0 0
\(375\) −7.94891 −0.410480
\(376\) 0 0
\(377\) 13.3435 0.687226
\(378\) 0 0
\(379\) −34.4480 −1.76947 −0.884737 0.466090i \(-0.845662\pi\)
−0.884737 + 0.466090i \(0.845662\pi\)
\(380\) 0 0
\(381\) −12.3626 −0.633358
\(382\) 0 0
\(383\) −19.7939 −1.01142 −0.505711 0.862703i \(-0.668770\pi\)
−0.505711 + 0.862703i \(0.668770\pi\)
\(384\) 0 0
\(385\) −1.32897 −0.0677308
\(386\) 0 0
\(387\) −4.13938 −0.210416
\(388\) 0 0
\(389\) 18.5357 0.939798 0.469899 0.882720i \(-0.344290\pi\)
0.469899 + 0.882720i \(0.344290\pi\)
\(390\) 0 0
\(391\) 31.2215 1.57894
\(392\) 0 0
\(393\) 20.5805 1.03815
\(394\) 0 0
\(395\) −9.08935 −0.457335
\(396\) 0 0
\(397\) −5.39197 −0.270615 −0.135308 0.990804i \(-0.543202\pi\)
−0.135308 + 0.990804i \(0.543202\pi\)
\(398\) 0 0
\(399\) 17.6894 0.885575
\(400\) 0 0
\(401\) −13.8183 −0.690055 −0.345027 0.938593i \(-0.612130\pi\)
−0.345027 + 0.938593i \(0.612130\pi\)
\(402\) 0 0
\(403\) −8.30163 −0.413534
\(404\) 0 0
\(405\) −0.858069 −0.0426378
\(406\) 0 0
\(407\) −2.73919 −0.135776
\(408\) 0 0
\(409\) 6.26519 0.309794 0.154897 0.987931i \(-0.450495\pi\)
0.154897 + 0.987931i \(0.450495\pi\)
\(410\) 0 0
\(411\) −10.1249 −0.499426
\(412\) 0 0
\(413\) −23.8736 −1.17474
\(414\) 0 0
\(415\) 14.9340 0.733079
\(416\) 0 0
\(417\) 1.84695 0.0904454
\(418\) 0 0
\(419\) −0.914237 −0.0446634 −0.0223317 0.999751i \(-0.507109\pi\)
−0.0223317 + 0.999751i \(0.507109\pi\)
\(420\) 0 0
\(421\) 31.5414 1.53723 0.768617 0.639709i \(-0.220945\pi\)
0.768617 + 0.639709i \(0.220945\pi\)
\(422\) 0 0
\(423\) 1.39111 0.0676381
\(424\) 0 0
\(425\) 20.2003 0.979856
\(426\) 0 0
\(427\) 1.77760 0.0860242
\(428\) 0 0
\(429\) −1.86224 −0.0899096
\(430\) 0 0
\(431\) −38.5807 −1.85837 −0.929184 0.369619i \(-0.879488\pi\)
−0.929184 + 0.369619i \(0.879488\pi\)
\(432\) 0 0
\(433\) 8.12866 0.390639 0.195319 0.980740i \(-0.437426\pi\)
0.195319 + 0.980740i \(0.437426\pi\)
\(434\) 0 0
\(435\) −4.08297 −0.195764
\(436\) 0 0
\(437\) −49.9829 −2.39101
\(438\) 0 0
\(439\) 28.3001 1.35069 0.675344 0.737503i \(-0.263995\pi\)
0.675344 + 0.737503i \(0.263995\pi\)
\(440\) 0 0
\(441\) −1.56060 −0.0743141
\(442\) 0 0
\(443\) 38.5945 1.83368 0.916841 0.399253i \(-0.130731\pi\)
0.916841 + 0.399253i \(0.130731\pi\)
\(444\) 0 0
\(445\) 4.23971 0.200982
\(446\) 0 0
\(447\) −5.36994 −0.253989
\(448\) 0 0
\(449\) 9.70687 0.458096 0.229048 0.973415i \(-0.426439\pi\)
0.229048 + 0.973415i \(0.426439\pi\)
\(450\) 0 0
\(451\) 5.15826 0.242893
\(452\) 0 0
\(453\) −13.0563 −0.613439
\(454\) 0 0
\(455\) −5.61195 −0.263092
\(456\) 0 0
\(457\) −25.1697 −1.17739 −0.588694 0.808356i \(-0.700358\pi\)
−0.588694 + 0.808356i \(0.700358\pi\)
\(458\) 0 0
\(459\) 4.73771 0.221137
\(460\) 0 0
\(461\) 33.4704 1.55887 0.779435 0.626483i \(-0.215506\pi\)
0.779435 + 0.626483i \(0.215506\pi\)
\(462\) 0 0
\(463\) −23.8185 −1.10694 −0.553469 0.832870i \(-0.686696\pi\)
−0.553469 + 0.832870i \(0.686696\pi\)
\(464\) 0 0
\(465\) 2.54021 0.117799
\(466\) 0 0
\(467\) −5.42414 −0.250999 −0.125500 0.992094i \(-0.540053\pi\)
−0.125500 + 0.992094i \(0.540053\pi\)
\(468\) 0 0
\(469\) 0.00303578 0.000140179 0
\(470\) 0 0
\(471\) −5.47662 −0.252349
\(472\) 0 0
\(473\) 2.74887 0.126393
\(474\) 0 0
\(475\) −32.3389 −1.48381
\(476\) 0 0
\(477\) −1.74469 −0.0798838
\(478\) 0 0
\(479\) −12.8509 −0.587172 −0.293586 0.955933i \(-0.594849\pi\)
−0.293586 + 0.955933i \(0.594849\pi\)
\(480\) 0 0
\(481\) −11.5669 −0.527407
\(482\) 0 0
\(483\) −15.3695 −0.699338
\(484\) 0 0
\(485\) −14.9771 −0.680073
\(486\) 0 0
\(487\) 32.9193 1.49171 0.745857 0.666106i \(-0.232040\pi\)
0.745857 + 0.666106i \(0.232040\pi\)
\(488\) 0 0
\(489\) 18.0321 0.815442
\(490\) 0 0
\(491\) 23.3929 1.05571 0.527854 0.849335i \(-0.322997\pi\)
0.527854 + 0.849335i \(0.322997\pi\)
\(492\) 0 0
\(493\) 22.5436 1.01531
\(494\) 0 0
\(495\) 0.569824 0.0256117
\(496\) 0 0
\(497\) −6.98813 −0.313461
\(498\) 0 0
\(499\) −7.77468 −0.348042 −0.174021 0.984742i \(-0.555676\pi\)
−0.174021 + 0.984742i \(0.555676\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) 11.5995 0.517195 0.258597 0.965985i \(-0.416740\pi\)
0.258597 + 0.965985i \(0.416740\pi\)
\(504\) 0 0
\(505\) −2.43369 −0.108298
\(506\) 0 0
\(507\) 5.13621 0.228107
\(508\) 0 0
\(509\) −14.3183 −0.634646 −0.317323 0.948317i \(-0.602784\pi\)
−0.317323 + 0.948317i \(0.602784\pi\)
\(510\) 0 0
\(511\) 38.8827 1.72007
\(512\) 0 0
\(513\) −7.58466 −0.334871
\(514\) 0 0
\(515\) 1.44465 0.0636590
\(516\) 0 0
\(517\) −0.923805 −0.0406289
\(518\) 0 0
\(519\) −11.9069 −0.522656
\(520\) 0 0
\(521\) 24.7954 1.08631 0.543153 0.839633i \(-0.317230\pi\)
0.543153 + 0.839633i \(0.317230\pi\)
\(522\) 0 0
\(523\) −15.8863 −0.694660 −0.347330 0.937743i \(-0.612912\pi\)
−0.347330 + 0.937743i \(0.612912\pi\)
\(524\) 0 0
\(525\) −9.94407 −0.433995
\(526\) 0 0
\(527\) −14.0254 −0.610957
\(528\) 0 0
\(529\) 20.4280 0.888174
\(530\) 0 0
\(531\) 10.2363 0.444217
\(532\) 0 0
\(533\) 21.7821 0.943489
\(534\) 0 0
\(535\) 9.00625 0.389374
\(536\) 0 0
\(537\) −9.99610 −0.431364
\(538\) 0 0
\(539\) 1.03636 0.0446391
\(540\) 0 0
\(541\) −34.8029 −1.49629 −0.748147 0.663532i \(-0.769056\pi\)
−0.748147 + 0.663532i \(0.769056\pi\)
\(542\) 0 0
\(543\) 12.9764 0.556869
\(544\) 0 0
\(545\) 3.10407 0.132964
\(546\) 0 0
\(547\) 16.7384 0.715682 0.357841 0.933782i \(-0.383513\pi\)
0.357841 + 0.933782i \(0.383513\pi\)
\(548\) 0 0
\(549\) −0.762183 −0.0325292
\(550\) 0 0
\(551\) −36.0903 −1.53750
\(552\) 0 0
\(553\) −24.7051 −1.05057
\(554\) 0 0
\(555\) 3.53936 0.150237
\(556\) 0 0
\(557\) −4.14278 −0.175535 −0.0877677 0.996141i \(-0.527973\pi\)
−0.0877677 + 0.996141i \(0.527973\pi\)
\(558\) 0 0
\(559\) 11.6078 0.490959
\(560\) 0 0
\(561\) −3.14621 −0.132833
\(562\) 0 0
\(563\) −31.1358 −1.31222 −0.656109 0.754666i \(-0.727799\pi\)
−0.656109 + 0.754666i \(0.727799\pi\)
\(564\) 0 0
\(565\) −1.84487 −0.0776142
\(566\) 0 0
\(567\) −2.33225 −0.0979454
\(568\) 0 0
\(569\) −16.2278 −0.680305 −0.340153 0.940370i \(-0.610479\pi\)
−0.340153 + 0.940370i \(0.610479\pi\)
\(570\) 0 0
\(571\) 15.7948 0.660991 0.330495 0.943808i \(-0.392784\pi\)
0.330495 + 0.943808i \(0.392784\pi\)
\(572\) 0 0
\(573\) −0.780931 −0.0326239
\(574\) 0 0
\(575\) 28.0979 1.17176
\(576\) 0 0
\(577\) 29.3450 1.22165 0.610825 0.791766i \(-0.290838\pi\)
0.610825 + 0.791766i \(0.290838\pi\)
\(578\) 0 0
\(579\) 3.18460 0.132347
\(580\) 0 0
\(581\) 40.5909 1.68399
\(582\) 0 0
\(583\) 1.15861 0.0479847
\(584\) 0 0
\(585\) 2.40623 0.0994855
\(586\) 0 0
\(587\) −3.97172 −0.163931 −0.0819653 0.996635i \(-0.526120\pi\)
−0.0819653 + 0.996635i \(0.526120\pi\)
\(588\) 0 0
\(589\) 22.4535 0.925180
\(590\) 0 0
\(591\) 7.29302 0.299995
\(592\) 0 0
\(593\) 13.0664 0.536573 0.268286 0.963339i \(-0.413543\pi\)
0.268286 + 0.963339i \(0.413543\pi\)
\(594\) 0 0
\(595\) −9.48126 −0.388694
\(596\) 0 0
\(597\) −20.4949 −0.838800
\(598\) 0 0
\(599\) −26.1897 −1.07008 −0.535042 0.844826i \(-0.679704\pi\)
−0.535042 + 0.844826i \(0.679704\pi\)
\(600\) 0 0
\(601\) 31.0894 1.26816 0.634082 0.773266i \(-0.281378\pi\)
0.634082 + 0.773266i \(0.281378\pi\)
\(602\) 0 0
\(603\) −0.00130165 −5.30073e−5 0
\(604\) 0 0
\(605\) 9.06035 0.368356
\(606\) 0 0
\(607\) −7.46967 −0.303185 −0.151592 0.988443i \(-0.548440\pi\)
−0.151592 + 0.988443i \(0.548440\pi\)
\(608\) 0 0
\(609\) −11.0976 −0.449699
\(610\) 0 0
\(611\) −3.90101 −0.157818
\(612\) 0 0
\(613\) −7.66230 −0.309477 −0.154739 0.987955i \(-0.549454\pi\)
−0.154739 + 0.987955i \(0.549454\pi\)
\(614\) 0 0
\(615\) −6.66510 −0.268763
\(616\) 0 0
\(617\) 11.5871 0.466479 0.233239 0.972419i \(-0.425067\pi\)
0.233239 + 0.972419i \(0.425067\pi\)
\(618\) 0 0
\(619\) −4.39728 −0.176741 −0.0883707 0.996088i \(-0.528166\pi\)
−0.0883707 + 0.996088i \(0.528166\pi\)
\(620\) 0 0
\(621\) 6.58999 0.264447
\(622\) 0 0
\(623\) 11.5237 0.461685
\(624\) 0 0
\(625\) 14.4979 0.579915
\(626\) 0 0
\(627\) 5.03681 0.201151
\(628\) 0 0
\(629\) −19.5421 −0.779194
\(630\) 0 0
\(631\) 25.3849 1.01056 0.505279 0.862956i \(-0.331390\pi\)
0.505279 + 0.862956i \(0.331390\pi\)
\(632\) 0 0
\(633\) 2.79418 0.111059
\(634\) 0 0
\(635\) −10.6080 −0.420966
\(636\) 0 0
\(637\) 4.37629 0.173395
\(638\) 0 0
\(639\) 2.99630 0.118532
\(640\) 0 0
\(641\) 23.9833 0.947283 0.473642 0.880718i \(-0.342939\pi\)
0.473642 + 0.880718i \(0.342939\pi\)
\(642\) 0 0
\(643\) 27.3549 1.07877 0.539386 0.842059i \(-0.318656\pi\)
0.539386 + 0.842059i \(0.318656\pi\)
\(644\) 0 0
\(645\) −3.55187 −0.139855
\(646\) 0 0
\(647\) 8.92955 0.351057 0.175528 0.984474i \(-0.443837\pi\)
0.175528 + 0.984474i \(0.443837\pi\)
\(648\) 0 0
\(649\) −6.79769 −0.266833
\(650\) 0 0
\(651\) 6.90435 0.270603
\(652\) 0 0
\(653\) 1.42815 0.0558877 0.0279438 0.999609i \(-0.491104\pi\)
0.0279438 + 0.999609i \(0.491104\pi\)
\(654\) 0 0
\(655\) 17.6594 0.690012
\(656\) 0 0
\(657\) −16.6717 −0.650426
\(658\) 0 0
\(659\) 20.7009 0.806394 0.403197 0.915113i \(-0.367899\pi\)
0.403197 + 0.915113i \(0.367899\pi\)
\(660\) 0 0
\(661\) 13.2990 0.517272 0.258636 0.965975i \(-0.416727\pi\)
0.258636 + 0.965975i \(0.416727\pi\)
\(662\) 0 0
\(663\) −13.2857 −0.515974
\(664\) 0 0
\(665\) 15.1787 0.588604
\(666\) 0 0
\(667\) 31.3574 1.21416
\(668\) 0 0
\(669\) −1.07046 −0.0413862
\(670\) 0 0
\(671\) 0.506148 0.0195396
\(672\) 0 0
\(673\) 12.6827 0.488882 0.244441 0.969664i \(-0.421396\pi\)
0.244441 + 0.969664i \(0.421396\pi\)
\(674\) 0 0
\(675\) 4.26372 0.164111
\(676\) 0 0
\(677\) 2.60520 0.100126 0.0500629 0.998746i \(-0.484058\pi\)
0.0500629 + 0.998746i \(0.484058\pi\)
\(678\) 0 0
\(679\) −40.7080 −1.56223
\(680\) 0 0
\(681\) −3.06628 −0.117500
\(682\) 0 0
\(683\) −24.5138 −0.937995 −0.468998 0.883199i \(-0.655385\pi\)
−0.468998 + 0.883199i \(0.655385\pi\)
\(684\) 0 0
\(685\) −8.68788 −0.331947
\(686\) 0 0
\(687\) 0.166978 0.00637060
\(688\) 0 0
\(689\) 4.89253 0.186391
\(690\) 0 0
\(691\) −32.0143 −1.21788 −0.608941 0.793215i \(-0.708405\pi\)
−0.608941 + 0.793215i \(0.708405\pi\)
\(692\) 0 0
\(693\) 1.54880 0.0588339
\(694\) 0 0
\(695\) 1.58481 0.0601152
\(696\) 0 0
\(697\) 36.8004 1.39392
\(698\) 0 0
\(699\) −9.75862 −0.369105
\(700\) 0 0
\(701\) −4.88152 −0.184372 −0.0921862 0.995742i \(-0.529386\pi\)
−0.0921862 + 0.995742i \(0.529386\pi\)
\(702\) 0 0
\(703\) 31.2852 1.17994
\(704\) 0 0
\(705\) 1.19367 0.0449561
\(706\) 0 0
\(707\) −6.61482 −0.248776
\(708\) 0 0
\(709\) 42.2523 1.58682 0.793408 0.608690i \(-0.208305\pi\)
0.793408 + 0.608690i \(0.208305\pi\)
\(710\) 0 0
\(711\) 10.5928 0.397261
\(712\) 0 0
\(713\) −19.5089 −0.730613
\(714\) 0 0
\(715\) −1.59793 −0.0597590
\(716\) 0 0
\(717\) −27.8243 −1.03912
\(718\) 0 0
\(719\) 42.4125 1.58172 0.790860 0.611997i \(-0.209634\pi\)
0.790860 + 0.611997i \(0.209634\pi\)
\(720\) 0 0
\(721\) 3.92660 0.146234
\(722\) 0 0
\(723\) 6.03419 0.224414
\(724\) 0 0
\(725\) 20.2882 0.753484
\(726\) 0 0
\(727\) −22.1250 −0.820570 −0.410285 0.911957i \(-0.634571\pi\)
−0.410285 + 0.911957i \(0.634571\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 19.6112 0.725345
\(732\) 0 0
\(733\) 35.9161 1.32659 0.663297 0.748357i \(-0.269157\pi\)
0.663297 + 0.748357i \(0.269157\pi\)
\(734\) 0 0
\(735\) −1.33910 −0.0493934
\(736\) 0 0
\(737\) 0.000864396 0 3.18404e−5 0
\(738\) 0 0
\(739\) 20.3434 0.748344 0.374172 0.927359i \(-0.377927\pi\)
0.374172 + 0.927359i \(0.377927\pi\)
\(740\) 0 0
\(741\) 21.2693 0.781345
\(742\) 0 0
\(743\) −47.9841 −1.76036 −0.880182 0.474636i \(-0.842580\pi\)
−0.880182 + 0.474636i \(0.842580\pi\)
\(744\) 0 0
\(745\) −4.60778 −0.168816
\(746\) 0 0
\(747\) −17.4041 −0.636784
\(748\) 0 0
\(749\) 24.4792 0.894451
\(750\) 0 0
\(751\) −40.8830 −1.49184 −0.745922 0.666034i \(-0.767991\pi\)
−0.745922 + 0.666034i \(0.767991\pi\)
\(752\) 0 0
\(753\) −19.2169 −0.700304
\(754\) 0 0
\(755\) −11.2032 −0.407727
\(756\) 0 0
\(757\) 52.4727 1.90715 0.953576 0.301151i \(-0.0973708\pi\)
0.953576 + 0.301151i \(0.0973708\pi\)
\(758\) 0 0
\(759\) −4.37627 −0.158848
\(760\) 0 0
\(761\) −30.9663 −1.12253 −0.561264 0.827636i \(-0.689685\pi\)
−0.561264 + 0.827636i \(0.689685\pi\)
\(762\) 0 0
\(763\) 8.43694 0.305438
\(764\) 0 0
\(765\) 4.06528 0.146980
\(766\) 0 0
\(767\) −28.7050 −1.03648
\(768\) 0 0
\(769\) −24.2091 −0.873003 −0.436501 0.899704i \(-0.643783\pi\)
−0.436501 + 0.899704i \(0.643783\pi\)
\(770\) 0 0
\(771\) 8.38942 0.302138
\(772\) 0 0
\(773\) −13.4368 −0.483289 −0.241644 0.970365i \(-0.577687\pi\)
−0.241644 + 0.970365i \(0.577687\pi\)
\(774\) 0 0
\(775\) −12.6222 −0.453404
\(776\) 0 0
\(777\) 9.62007 0.345118
\(778\) 0 0
\(779\) −58.9143 −2.11082
\(780\) 0 0
\(781\) −1.98978 −0.0711998
\(782\) 0 0
\(783\) 4.75833 0.170049
\(784\) 0 0
\(785\) −4.69932 −0.167726
\(786\) 0 0
\(787\) 38.7500 1.38129 0.690644 0.723195i \(-0.257327\pi\)
0.690644 + 0.723195i \(0.257327\pi\)
\(788\) 0 0
\(789\) −16.9678 −0.604069
\(790\) 0 0
\(791\) −5.01440 −0.178292
\(792\) 0 0
\(793\) 2.13735 0.0758994
\(794\) 0 0
\(795\) −1.49706 −0.0530953
\(796\) 0 0
\(797\) −4.12743 −0.146201 −0.0731006 0.997325i \(-0.523289\pi\)
−0.0731006 + 0.997325i \(0.523289\pi\)
\(798\) 0 0
\(799\) −6.59067 −0.233161
\(800\) 0 0
\(801\) −4.94100 −0.174582
\(802\) 0 0
\(803\) 11.0713 0.390699
\(804\) 0 0
\(805\) −13.1881 −0.464820
\(806\) 0 0
\(807\) 13.6797 0.481549
\(808\) 0 0
\(809\) −43.5841 −1.53233 −0.766167 0.642641i \(-0.777839\pi\)
−0.766167 + 0.642641i \(0.777839\pi\)
\(810\) 0 0
\(811\) 6.11071 0.214576 0.107288 0.994228i \(-0.465783\pi\)
0.107288 + 0.994228i \(0.465783\pi\)
\(812\) 0 0
\(813\) −16.2857 −0.571163
\(814\) 0 0
\(815\) 15.4728 0.541989
\(816\) 0 0
\(817\) −31.3958 −1.09840
\(818\) 0 0
\(819\) 6.54021 0.228533
\(820\) 0 0
\(821\) −44.6483 −1.55824 −0.779118 0.626878i \(-0.784333\pi\)
−0.779118 + 0.626878i \(0.784333\pi\)
\(822\) 0 0
\(823\) 52.0320 1.81372 0.906861 0.421429i \(-0.138471\pi\)
0.906861 + 0.421429i \(0.138471\pi\)
\(824\) 0 0
\(825\) −2.83144 −0.0985781
\(826\) 0 0
\(827\) −0.434728 −0.0151170 −0.00755849 0.999971i \(-0.502406\pi\)
−0.00755849 + 0.999971i \(0.502406\pi\)
\(828\) 0 0
\(829\) 45.0928 1.56614 0.783069 0.621935i \(-0.213653\pi\)
0.783069 + 0.621935i \(0.213653\pi\)
\(830\) 0 0
\(831\) 4.11423 0.142721
\(832\) 0 0
\(833\) 7.39365 0.256175
\(834\) 0 0
\(835\) −0.858069 −0.0296947
\(836\) 0 0
\(837\) −2.96038 −0.102326
\(838\) 0 0
\(839\) 28.8473 0.995919 0.497960 0.867200i \(-0.334083\pi\)
0.497960 + 0.867200i \(0.334083\pi\)
\(840\) 0 0
\(841\) −6.35829 −0.219251
\(842\) 0 0
\(843\) −29.3388 −1.01048
\(844\) 0 0
\(845\) 4.40722 0.151613
\(846\) 0 0
\(847\) 24.6263 0.846168
\(848\) 0 0
\(849\) −16.5471 −0.567894
\(850\) 0 0
\(851\) −27.1824 −0.931800
\(852\) 0 0
\(853\) −34.0205 −1.16484 −0.582420 0.812888i \(-0.697894\pi\)
−0.582420 + 0.812888i \(0.697894\pi\)
\(854\) 0 0
\(855\) −6.50816 −0.222574
\(856\) 0 0
\(857\) −4.41103 −0.150678 −0.0753389 0.997158i \(-0.524004\pi\)
−0.0753389 + 0.997158i \(0.524004\pi\)
\(858\) 0 0
\(859\) −0.717858 −0.0244930 −0.0122465 0.999925i \(-0.503898\pi\)
−0.0122465 + 0.999925i \(0.503898\pi\)
\(860\) 0 0
\(861\) −18.1159 −0.617388
\(862\) 0 0
\(863\) −37.7870 −1.28628 −0.643142 0.765747i \(-0.722370\pi\)
−0.643142 + 0.765747i \(0.722370\pi\)
\(864\) 0 0
\(865\) −10.2170 −0.347387
\(866\) 0 0
\(867\) −5.44589 −0.184952
\(868\) 0 0
\(869\) −7.03444 −0.238627
\(870\) 0 0
\(871\) 0.00365014 0.000123680 0
\(872\) 0 0
\(873\) 17.4544 0.590741
\(874\) 0 0
\(875\) −18.5389 −0.626728
\(876\) 0 0
\(877\) −33.0599 −1.11635 −0.558177 0.829722i \(-0.688499\pi\)
−0.558177 + 0.829722i \(0.688499\pi\)
\(878\) 0 0
\(879\) 31.9181 1.07657
\(880\) 0 0
\(881\) −33.8792 −1.14142 −0.570710 0.821152i \(-0.693332\pi\)
−0.570710 + 0.821152i \(0.693332\pi\)
\(882\) 0 0
\(883\) 36.9683 1.24408 0.622041 0.782985i \(-0.286304\pi\)
0.622041 + 0.782985i \(0.286304\pi\)
\(884\) 0 0
\(885\) 8.78344 0.295252
\(886\) 0 0
\(887\) −27.6020 −0.926786 −0.463393 0.886153i \(-0.653368\pi\)
−0.463393 + 0.886153i \(0.653368\pi\)
\(888\) 0 0
\(889\) −28.8328 −0.967022
\(890\) 0 0
\(891\) −0.664078 −0.0222474
\(892\) 0 0
\(893\) 10.5511 0.353079
\(894\) 0 0
\(895\) −8.57734 −0.286709
\(896\) 0 0
\(897\) −18.4799 −0.617027
\(898\) 0 0
\(899\) −14.0865 −0.469810
\(900\) 0 0
\(901\) 8.26583 0.275375
\(902\) 0 0
\(903\) −9.65407 −0.321267
\(904\) 0 0
\(905\) 11.1346 0.370127
\(906\) 0 0
\(907\) −35.2051 −1.16897 −0.584484 0.811406i \(-0.698703\pi\)
−0.584484 + 0.811406i \(0.698703\pi\)
\(908\) 0 0
\(909\) 2.83624 0.0940720
\(910\) 0 0
\(911\) −3.23223 −0.107088 −0.0535442 0.998565i \(-0.517052\pi\)
−0.0535442 + 0.998565i \(0.517052\pi\)
\(912\) 0 0
\(913\) 11.5577 0.382504
\(914\) 0 0
\(915\) −0.654005 −0.0216207
\(916\) 0 0
\(917\) 47.9988 1.58506
\(918\) 0 0
\(919\) −11.7314 −0.386984 −0.193492 0.981102i \(-0.561981\pi\)
−0.193492 + 0.981102i \(0.561981\pi\)
\(920\) 0 0
\(921\) −12.7567 −0.420347
\(922\) 0 0
\(923\) −8.40236 −0.276567
\(924\) 0 0
\(925\) −17.5870 −0.578256
\(926\) 0 0
\(927\) −1.68361 −0.0552970
\(928\) 0 0
\(929\) 36.8679 1.20960 0.604798 0.796379i \(-0.293254\pi\)
0.604798 + 0.796379i \(0.293254\pi\)
\(930\) 0 0
\(931\) −11.8366 −0.387929
\(932\) 0 0
\(933\) −4.06532 −0.133093
\(934\) 0 0
\(935\) −2.69966 −0.0882884
\(936\) 0 0
\(937\) −10.7009 −0.349583 −0.174792 0.984605i \(-0.555925\pi\)
−0.174792 + 0.984605i \(0.555925\pi\)
\(938\) 0 0
\(939\) 9.87994 0.322420
\(940\) 0 0
\(941\) −4.86481 −0.158588 −0.0792942 0.996851i \(-0.525267\pi\)
−0.0792942 + 0.996851i \(0.525267\pi\)
\(942\) 0 0
\(943\) 51.1881 1.66692
\(944\) 0 0
\(945\) −2.00123 −0.0651001
\(946\) 0 0
\(947\) 4.67872 0.152038 0.0760190 0.997106i \(-0.475779\pi\)
0.0760190 + 0.997106i \(0.475779\pi\)
\(948\) 0 0
\(949\) 46.7516 1.51762
\(950\) 0 0
\(951\) −11.2409 −0.364512
\(952\) 0 0
\(953\) −38.4497 −1.24551 −0.622754 0.782418i \(-0.713986\pi\)
−0.622754 + 0.782418i \(0.713986\pi\)
\(954\) 0 0
\(955\) −0.670092 −0.0216837
\(956\) 0 0
\(957\) −3.15990 −0.102145
\(958\) 0 0
\(959\) −23.6139 −0.762532
\(960\) 0 0
\(961\) −22.2362 −0.717295
\(962\) 0 0
\(963\) −10.4960 −0.338227
\(964\) 0 0
\(965\) 2.73260 0.0879656
\(966\) 0 0
\(967\) −30.5693 −0.983044 −0.491522 0.870865i \(-0.663559\pi\)
−0.491522 + 0.870865i \(0.663559\pi\)
\(968\) 0 0
\(969\) 35.9339 1.15436
\(970\) 0 0
\(971\) −0.810886 −0.0260226 −0.0130113 0.999915i \(-0.504142\pi\)
−0.0130113 + 0.999915i \(0.504142\pi\)
\(972\) 0 0
\(973\) 4.30755 0.138094
\(974\) 0 0
\(975\) −11.9565 −0.382915
\(976\) 0 0
\(977\) 9.23855 0.295567 0.147784 0.989020i \(-0.452786\pi\)
0.147784 + 0.989020i \(0.452786\pi\)
\(978\) 0 0
\(979\) 3.28120 0.104868
\(980\) 0 0
\(981\) −3.61751 −0.115498
\(982\) 0 0
\(983\) 1.30384 0.0415861 0.0207931 0.999784i \(-0.493381\pi\)
0.0207931 + 0.999784i \(0.493381\pi\)
\(984\) 0 0
\(985\) 6.25791 0.199394
\(986\) 0 0
\(987\) 3.24442 0.103271
\(988\) 0 0
\(989\) 27.2785 0.867405
\(990\) 0 0
\(991\) −53.9125 −1.71259 −0.856293 0.516490i \(-0.827238\pi\)
−0.856293 + 0.516490i \(0.827238\pi\)
\(992\) 0 0
\(993\) −14.6160 −0.463826
\(994\) 0 0
\(995\) −17.5860 −0.557515
\(996\) 0 0
\(997\) −47.6636 −1.50952 −0.754760 0.656001i \(-0.772247\pi\)
−0.754760 + 0.656001i \(0.772247\pi\)
\(998\) 0 0
\(999\) −4.12480 −0.130503
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 8016.2.a.ba.1.5 9
4.3 odd 2 4008.2.a.i.1.5 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.i.1.5 9 4.3 odd 2
8016.2.a.ba.1.5 9 1.1 even 1 trivial