Properties

Label 8004.2.a.e.1.3
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $9$
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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 13 x^{7} + 32 x^{6} + 40 x^{5} - 79 x^{4} - 39 x^{3} + 58 x^{2} + 9 x - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.93556\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.36120 q^{5} +2.82157 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.36120 q^{5} +2.82157 q^{7} +1.00000 q^{9} +2.46337 q^{11} -2.24447 q^{13} +2.36120 q^{15} -1.17707 q^{17} +5.00771 q^{19} -2.82157 q^{21} -1.00000 q^{23} +0.575276 q^{25} -1.00000 q^{27} +1.00000 q^{29} -6.16676 q^{31} -2.46337 q^{33} -6.66229 q^{35} +4.43163 q^{37} +2.24447 q^{39} -3.21444 q^{41} -12.1113 q^{43} -2.36120 q^{45} -3.40978 q^{47} +0.961232 q^{49} +1.17707 q^{51} -3.20671 q^{53} -5.81651 q^{55} -5.00771 q^{57} -2.13703 q^{59} +5.45934 q^{61} +2.82157 q^{63} +5.29964 q^{65} +11.5503 q^{67} +1.00000 q^{69} +6.80571 q^{71} +0.535914 q^{73} -0.575276 q^{75} +6.95055 q^{77} -9.08449 q^{79} +1.00000 q^{81} -13.4663 q^{83} +2.77929 q^{85} -1.00000 q^{87} +10.9138 q^{89} -6.33291 q^{91} +6.16676 q^{93} -11.8242 q^{95} +18.3762 q^{97} +2.46337 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} - 2q^{11} - 7q^{13} + 3q^{15} - q^{19} - 7q^{21} - 9q^{23} + 2q^{25} - 9q^{27} + 9q^{29} + 8q^{31} + 2q^{33} - 5q^{35} - 8q^{37} + 7q^{39} - 19q^{41} - 3q^{43} - 3q^{45} - 3q^{47} - 18q^{49} - 17q^{53} + 9q^{55} + q^{57} - 10q^{59} + q^{61} + 7q^{63} - 16q^{65} + 12q^{67} + 9q^{69} - 7q^{71} + 13q^{73} - 2q^{75} - 15q^{77} - 10q^{79} + 9q^{81} + 9q^{83} - 6q^{85} - 9q^{87} - 5q^{89} - 18q^{91} - 8q^{93} + 31q^{95} - 7q^{97} - 2q^{99} + O(q^{100}) \)

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
\(5\) −2.36120 −1.05596 −0.527981 0.849256i \(-0.677051\pi\)
−0.527981 + 0.849256i \(0.677051\pi\)
\(6\) 0 0
\(7\) 2.82157 1.06645 0.533226 0.845973i \(-0.320980\pi\)
0.533226 + 0.845973i \(0.320980\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.46337 0.742733 0.371367 0.928486i \(-0.378889\pi\)
0.371367 + 0.928486i \(0.378889\pi\)
\(12\) 0 0
\(13\) −2.24447 −0.622503 −0.311252 0.950328i \(-0.600748\pi\)
−0.311252 + 0.950328i \(0.600748\pi\)
\(14\) 0 0
\(15\) 2.36120 0.609660
\(16\) 0 0
\(17\) −1.17707 −0.285480 −0.142740 0.989760i \(-0.545591\pi\)
−0.142740 + 0.989760i \(0.545591\pi\)
\(18\) 0 0
\(19\) 5.00771 1.14885 0.574424 0.818558i \(-0.305226\pi\)
0.574424 + 0.818558i \(0.305226\pi\)
\(20\) 0 0
\(21\) −2.82157 −0.615716
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0.575276 0.115055
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −6.16676 −1.10758 −0.553792 0.832655i \(-0.686820\pi\)
−0.553792 + 0.832655i \(0.686820\pi\)
\(32\) 0 0
\(33\) −2.46337 −0.428817
\(34\) 0 0
\(35\) −6.66229 −1.12613
\(36\) 0 0
\(37\) 4.43163 0.728555 0.364278 0.931290i \(-0.381316\pi\)
0.364278 + 0.931290i \(0.381316\pi\)
\(38\) 0 0
\(39\) 2.24447 0.359402
\(40\) 0 0
\(41\) −3.21444 −0.502011 −0.251006 0.967986i \(-0.580761\pi\)
−0.251006 + 0.967986i \(0.580761\pi\)
\(42\) 0 0
\(43\) −12.1113 −1.84696 −0.923478 0.383652i \(-0.874666\pi\)
−0.923478 + 0.383652i \(0.874666\pi\)
\(44\) 0 0
\(45\) −2.36120 −0.351987
\(46\) 0 0
\(47\) −3.40978 −0.497367 −0.248683 0.968585i \(-0.579998\pi\)
−0.248683 + 0.968585i \(0.579998\pi\)
\(48\) 0 0
\(49\) 0.961232 0.137319
\(50\) 0 0
\(51\) 1.17707 0.164822
\(52\) 0 0
\(53\) −3.20671 −0.440476 −0.220238 0.975446i \(-0.570683\pi\)
−0.220238 + 0.975446i \(0.570683\pi\)
\(54\) 0 0
\(55\) −5.81651 −0.784298
\(56\) 0 0
\(57\) −5.00771 −0.663288
\(58\) 0 0
\(59\) −2.13703 −0.278218 −0.139109 0.990277i \(-0.544424\pi\)
−0.139109 + 0.990277i \(0.544424\pi\)
\(60\) 0 0
\(61\) 5.45934 0.698997 0.349498 0.936937i \(-0.386352\pi\)
0.349498 + 0.936937i \(0.386352\pi\)
\(62\) 0 0
\(63\) 2.82157 0.355484
\(64\) 0 0
\(65\) 5.29964 0.657339
\(66\) 0 0
\(67\) 11.5503 1.41109 0.705546 0.708664i \(-0.250702\pi\)
0.705546 + 0.708664i \(0.250702\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 6.80571 0.807690 0.403845 0.914827i \(-0.367674\pi\)
0.403845 + 0.914827i \(0.367674\pi\)
\(72\) 0 0
\(73\) 0.535914 0.0627240 0.0313620 0.999508i \(-0.490016\pi\)
0.0313620 + 0.999508i \(0.490016\pi\)
\(74\) 0 0
\(75\) −0.575276 −0.0664271
\(76\) 0 0
\(77\) 6.95055 0.792089
\(78\) 0 0
\(79\) −9.08449 −1.02209 −0.511043 0.859555i \(-0.670741\pi\)
−0.511043 + 0.859555i \(0.670741\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.4663 −1.47811 −0.739057 0.673643i \(-0.764729\pi\)
−0.739057 + 0.673643i \(0.764729\pi\)
\(84\) 0 0
\(85\) 2.77929 0.301456
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 10.9138 1.15686 0.578431 0.815731i \(-0.303665\pi\)
0.578431 + 0.815731i \(0.303665\pi\)
\(90\) 0 0
\(91\) −6.33291 −0.663869
\(92\) 0 0
\(93\) 6.16676 0.639464
\(94\) 0 0
\(95\) −11.8242 −1.21314
\(96\) 0 0
\(97\) 18.3762 1.86582 0.932908 0.360115i \(-0.117262\pi\)
0.932908 + 0.360115i \(0.117262\pi\)
\(98\) 0 0
\(99\) 2.46337 0.247578
\(100\) 0 0
\(101\) −8.04365 −0.800374 −0.400187 0.916434i \(-0.631055\pi\)
−0.400187 + 0.916434i \(0.631055\pi\)
\(102\) 0 0
\(103\) 12.1332 1.19552 0.597758 0.801677i \(-0.296058\pi\)
0.597758 + 0.801677i \(0.296058\pi\)
\(104\) 0 0
\(105\) 6.66229 0.650173
\(106\) 0 0
\(107\) −5.53795 −0.535374 −0.267687 0.963506i \(-0.586259\pi\)
−0.267687 + 0.963506i \(0.586259\pi\)
\(108\) 0 0
\(109\) 3.56615 0.341575 0.170788 0.985308i \(-0.445369\pi\)
0.170788 + 0.985308i \(0.445369\pi\)
\(110\) 0 0
\(111\) −4.43163 −0.420632
\(112\) 0 0
\(113\) 0.165574 0.0155759 0.00778794 0.999970i \(-0.497521\pi\)
0.00778794 + 0.999970i \(0.497521\pi\)
\(114\) 0 0
\(115\) 2.36120 0.220183
\(116\) 0 0
\(117\) −2.24447 −0.207501
\(118\) 0 0
\(119\) −3.32117 −0.304451
\(120\) 0 0
\(121\) −4.93182 −0.448348
\(122\) 0 0
\(123\) 3.21444 0.289836
\(124\) 0 0
\(125\) 10.4477 0.934468
\(126\) 0 0
\(127\) 13.0244 1.15573 0.577863 0.816134i \(-0.303887\pi\)
0.577863 + 0.816134i \(0.303887\pi\)
\(128\) 0 0
\(129\) 12.1113 1.06634
\(130\) 0 0
\(131\) −20.4285 −1.78485 −0.892423 0.451199i \(-0.850996\pi\)
−0.892423 + 0.451199i \(0.850996\pi\)
\(132\) 0 0
\(133\) 14.1296 1.22519
\(134\) 0 0
\(135\) 2.36120 0.203220
\(136\) 0 0
\(137\) 12.2888 1.04990 0.524952 0.851132i \(-0.324083\pi\)
0.524952 + 0.851132i \(0.324083\pi\)
\(138\) 0 0
\(139\) 12.1965 1.03450 0.517248 0.855836i \(-0.326957\pi\)
0.517248 + 0.855836i \(0.326957\pi\)
\(140\) 0 0
\(141\) 3.40978 0.287155
\(142\) 0 0
\(143\) −5.52895 −0.462354
\(144\) 0 0
\(145\) −2.36120 −0.196087
\(146\) 0 0
\(147\) −0.961232 −0.0792811
\(148\) 0 0
\(149\) −21.7213 −1.77947 −0.889737 0.456473i \(-0.849113\pi\)
−0.889737 + 0.456473i \(0.849113\pi\)
\(150\) 0 0
\(151\) −1.74182 −0.141747 −0.0708737 0.997485i \(-0.522579\pi\)
−0.0708737 + 0.997485i \(0.522579\pi\)
\(152\) 0 0
\(153\) −1.17707 −0.0951601
\(154\) 0 0
\(155\) 14.5610 1.16957
\(156\) 0 0
\(157\) −18.6182 −1.48589 −0.742947 0.669350i \(-0.766573\pi\)
−0.742947 + 0.669350i \(0.766573\pi\)
\(158\) 0 0
\(159\) 3.20671 0.254309
\(160\) 0 0
\(161\) −2.82157 −0.222371
\(162\) 0 0
\(163\) −12.4124 −0.972214 −0.486107 0.873899i \(-0.661583\pi\)
−0.486107 + 0.873899i \(0.661583\pi\)
\(164\) 0 0
\(165\) 5.81651 0.452815
\(166\) 0 0
\(167\) 2.02150 0.156428 0.0782140 0.996937i \(-0.475078\pi\)
0.0782140 + 0.996937i \(0.475078\pi\)
\(168\) 0 0
\(169\) −7.96237 −0.612490
\(170\) 0 0
\(171\) 5.00771 0.382949
\(172\) 0 0
\(173\) −1.55676 −0.118358 −0.0591792 0.998247i \(-0.518848\pi\)
−0.0591792 + 0.998247i \(0.518848\pi\)
\(174\) 0 0
\(175\) 1.62318 0.122701
\(176\) 0 0
\(177\) 2.13703 0.160629
\(178\) 0 0
\(179\) −22.6826 −1.69538 −0.847689 0.530494i \(-0.822006\pi\)
−0.847689 + 0.530494i \(0.822006\pi\)
\(180\) 0 0
\(181\) −2.39045 −0.177680 −0.0888402 0.996046i \(-0.528316\pi\)
−0.0888402 + 0.996046i \(0.528316\pi\)
\(182\) 0 0
\(183\) −5.45934 −0.403566
\(184\) 0 0
\(185\) −10.4640 −0.769327
\(186\) 0 0
\(187\) −2.89954 −0.212036
\(188\) 0 0
\(189\) −2.82157 −0.205239
\(190\) 0 0
\(191\) 4.58330 0.331636 0.165818 0.986156i \(-0.446974\pi\)
0.165818 + 0.986156i \(0.446974\pi\)
\(192\) 0 0
\(193\) −17.3826 −1.25123 −0.625613 0.780133i \(-0.715151\pi\)
−0.625613 + 0.780133i \(0.715151\pi\)
\(194\) 0 0
\(195\) −5.29964 −0.379515
\(196\) 0 0
\(197\) 3.86402 0.275300 0.137650 0.990481i \(-0.456045\pi\)
0.137650 + 0.990481i \(0.456045\pi\)
\(198\) 0 0
\(199\) −1.66919 −0.118326 −0.0591628 0.998248i \(-0.518843\pi\)
−0.0591628 + 0.998248i \(0.518843\pi\)
\(200\) 0 0
\(201\) −11.5503 −0.814695
\(202\) 0 0
\(203\) 2.82157 0.198035
\(204\) 0 0
\(205\) 7.58995 0.530105
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 12.3358 0.853288
\(210\) 0 0
\(211\) 2.03379 0.140012 0.0700058 0.997547i \(-0.477698\pi\)
0.0700058 + 0.997547i \(0.477698\pi\)
\(212\) 0 0
\(213\) −6.80571 −0.466320
\(214\) 0 0
\(215\) 28.5972 1.95031
\(216\) 0 0
\(217\) −17.3999 −1.18118
\(218\) 0 0
\(219\) −0.535914 −0.0362137
\(220\) 0 0
\(221\) 2.64188 0.177712
\(222\) 0 0
\(223\) −7.15464 −0.479110 −0.239555 0.970883i \(-0.577002\pi\)
−0.239555 + 0.970883i \(0.577002\pi\)
\(224\) 0 0
\(225\) 0.575276 0.0383517
\(226\) 0 0
\(227\) 9.46474 0.628197 0.314099 0.949390i \(-0.398298\pi\)
0.314099 + 0.949390i \(0.398298\pi\)
\(228\) 0 0
\(229\) 19.9346 1.31731 0.658657 0.752443i \(-0.271125\pi\)
0.658657 + 0.752443i \(0.271125\pi\)
\(230\) 0 0
\(231\) −6.95055 −0.457313
\(232\) 0 0
\(233\) −20.0069 −1.31069 −0.655347 0.755328i \(-0.727478\pi\)
−0.655347 + 0.755328i \(0.727478\pi\)
\(234\) 0 0
\(235\) 8.05117 0.525200
\(236\) 0 0
\(237\) 9.08449 0.590101
\(238\) 0 0
\(239\) −24.1719 −1.56355 −0.781777 0.623559i \(-0.785686\pi\)
−0.781777 + 0.623559i \(0.785686\pi\)
\(240\) 0 0
\(241\) 24.6368 1.58700 0.793499 0.608571i \(-0.208257\pi\)
0.793499 + 0.608571i \(0.208257\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −2.26966 −0.145003
\(246\) 0 0
\(247\) −11.2396 −0.715162
\(248\) 0 0
\(249\) 13.4663 0.853390
\(250\) 0 0
\(251\) −21.1137 −1.33269 −0.666343 0.745645i \(-0.732141\pi\)
−0.666343 + 0.745645i \(0.732141\pi\)
\(252\) 0 0
\(253\) −2.46337 −0.154871
\(254\) 0 0
\(255\) −2.77929 −0.174046
\(256\) 0 0
\(257\) 15.9018 0.991929 0.495965 0.868343i \(-0.334815\pi\)
0.495965 + 0.868343i \(0.334815\pi\)
\(258\) 0 0
\(259\) 12.5041 0.776969
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −26.1463 −1.61225 −0.806124 0.591747i \(-0.798438\pi\)
−0.806124 + 0.591747i \(0.798438\pi\)
\(264\) 0 0
\(265\) 7.57169 0.465125
\(266\) 0 0
\(267\) −10.9138 −0.667915
\(268\) 0 0
\(269\) −2.39482 −0.146015 −0.0730073 0.997331i \(-0.523260\pi\)
−0.0730073 + 0.997331i \(0.523260\pi\)
\(270\) 0 0
\(271\) 4.23346 0.257164 0.128582 0.991699i \(-0.458957\pi\)
0.128582 + 0.991699i \(0.458957\pi\)
\(272\) 0 0
\(273\) 6.33291 0.383285
\(274\) 0 0
\(275\) 1.41712 0.0854553
\(276\) 0 0
\(277\) 14.8325 0.891199 0.445599 0.895233i \(-0.352991\pi\)
0.445599 + 0.895233i \(0.352991\pi\)
\(278\) 0 0
\(279\) −6.16676 −0.369195
\(280\) 0 0
\(281\) −27.3067 −1.62898 −0.814491 0.580176i \(-0.802984\pi\)
−0.814491 + 0.580176i \(0.802984\pi\)
\(282\) 0 0
\(283\) 13.7600 0.817950 0.408975 0.912546i \(-0.365886\pi\)
0.408975 + 0.912546i \(0.365886\pi\)
\(284\) 0 0
\(285\) 11.8242 0.700407
\(286\) 0 0
\(287\) −9.06976 −0.535371
\(288\) 0 0
\(289\) −15.6145 −0.918501
\(290\) 0 0
\(291\) −18.3762 −1.07723
\(292\) 0 0
\(293\) 14.0909 0.823201 0.411600 0.911364i \(-0.364970\pi\)
0.411600 + 0.911364i \(0.364970\pi\)
\(294\) 0 0
\(295\) 5.04596 0.293787
\(296\) 0 0
\(297\) −2.46337 −0.142939
\(298\) 0 0
\(299\) 2.24447 0.129801
\(300\) 0 0
\(301\) −34.1728 −1.96969
\(302\) 0 0
\(303\) 8.04365 0.462096
\(304\) 0 0
\(305\) −12.8906 −0.738114
\(306\) 0 0
\(307\) 16.7805 0.957716 0.478858 0.877893i \(-0.341051\pi\)
0.478858 + 0.877893i \(0.341051\pi\)
\(308\) 0 0
\(309\) −12.1332 −0.690231
\(310\) 0 0
\(311\) 32.9428 1.86802 0.934008 0.357251i \(-0.116286\pi\)
0.934008 + 0.357251i \(0.116286\pi\)
\(312\) 0 0
\(313\) 0.604859 0.0341886 0.0170943 0.999854i \(-0.494558\pi\)
0.0170943 + 0.999854i \(0.494558\pi\)
\(314\) 0 0
\(315\) −6.66229 −0.375377
\(316\) 0 0
\(317\) −31.1888 −1.75174 −0.875869 0.482549i \(-0.839711\pi\)
−0.875869 + 0.482549i \(0.839711\pi\)
\(318\) 0 0
\(319\) 2.46337 0.137922
\(320\) 0 0
\(321\) 5.53795 0.309098
\(322\) 0 0
\(323\) −5.89440 −0.327973
\(324\) 0 0
\(325\) −1.29119 −0.0716222
\(326\) 0 0
\(327\) −3.56615 −0.197209
\(328\) 0 0
\(329\) −9.62091 −0.530418
\(330\) 0 0
\(331\) −21.4478 −1.17888 −0.589440 0.807812i \(-0.700652\pi\)
−0.589440 + 0.807812i \(0.700652\pi\)
\(332\) 0 0
\(333\) 4.43163 0.242852
\(334\) 0 0
\(335\) −27.2726 −1.49006
\(336\) 0 0
\(337\) 25.7310 1.40166 0.700829 0.713330i \(-0.252814\pi\)
0.700829 + 0.713330i \(0.252814\pi\)
\(338\) 0 0
\(339\) −0.165574 −0.00899274
\(340\) 0 0
\(341\) −15.1910 −0.822639
\(342\) 0 0
\(343\) −17.0388 −0.920008
\(344\) 0 0
\(345\) −2.36120 −0.127123
\(346\) 0 0
\(347\) 7.72527 0.414714 0.207357 0.978265i \(-0.433514\pi\)
0.207357 + 0.978265i \(0.433514\pi\)
\(348\) 0 0
\(349\) 10.0404 0.537449 0.268725 0.963217i \(-0.413398\pi\)
0.268725 + 0.963217i \(0.413398\pi\)
\(350\) 0 0
\(351\) 2.24447 0.119801
\(352\) 0 0
\(353\) −17.0110 −0.905405 −0.452702 0.891662i \(-0.649540\pi\)
−0.452702 + 0.891662i \(0.649540\pi\)
\(354\) 0 0
\(355\) −16.0697 −0.852889
\(356\) 0 0
\(357\) 3.32117 0.175775
\(358\) 0 0
\(359\) 6.26101 0.330443 0.165222 0.986256i \(-0.447166\pi\)
0.165222 + 0.986256i \(0.447166\pi\)
\(360\) 0 0
\(361\) 6.07720 0.319852
\(362\) 0 0
\(363\) 4.93182 0.258854
\(364\) 0 0
\(365\) −1.26540 −0.0662341
\(366\) 0 0
\(367\) 15.2884 0.798049 0.399025 0.916940i \(-0.369349\pi\)
0.399025 + 0.916940i \(0.369349\pi\)
\(368\) 0 0
\(369\) −3.21444 −0.167337
\(370\) 0 0
\(371\) −9.04794 −0.469746
\(372\) 0 0
\(373\) 1.18321 0.0612643 0.0306321 0.999531i \(-0.490248\pi\)
0.0306321 + 0.999531i \(0.490248\pi\)
\(374\) 0 0
\(375\) −10.4477 −0.539515
\(376\) 0 0
\(377\) −2.24447 −0.115596
\(378\) 0 0
\(379\) −13.4193 −0.689301 −0.344650 0.938731i \(-0.612003\pi\)
−0.344650 + 0.938731i \(0.612003\pi\)
\(380\) 0 0
\(381\) −13.0244 −0.667258
\(382\) 0 0
\(383\) 3.52064 0.179896 0.0899481 0.995946i \(-0.471330\pi\)
0.0899481 + 0.995946i \(0.471330\pi\)
\(384\) 0 0
\(385\) −16.4117 −0.836416
\(386\) 0 0
\(387\) −12.1113 −0.615652
\(388\) 0 0
\(389\) −23.1423 −1.17336 −0.586680 0.809819i \(-0.699565\pi\)
−0.586680 + 0.809819i \(0.699565\pi\)
\(390\) 0 0
\(391\) 1.17707 0.0595267
\(392\) 0 0
\(393\) 20.4285 1.03048
\(394\) 0 0
\(395\) 21.4503 1.07928
\(396\) 0 0
\(397\) −35.7587 −1.79468 −0.897339 0.441342i \(-0.854502\pi\)
−0.897339 + 0.441342i \(0.854502\pi\)
\(398\) 0 0
\(399\) −14.1296 −0.707364
\(400\) 0 0
\(401\) −27.4606 −1.37132 −0.685659 0.727923i \(-0.740486\pi\)
−0.685659 + 0.727923i \(0.740486\pi\)
\(402\) 0 0
\(403\) 13.8411 0.689474
\(404\) 0 0
\(405\) −2.36120 −0.117329
\(406\) 0 0
\(407\) 10.9167 0.541122
\(408\) 0 0
\(409\) −34.4448 −1.70319 −0.851593 0.524204i \(-0.824363\pi\)
−0.851593 + 0.524204i \(0.824363\pi\)
\(410\) 0 0
\(411\) −12.2888 −0.606162
\(412\) 0 0
\(413\) −6.02977 −0.296706
\(414\) 0 0
\(415\) 31.7966 1.56083
\(416\) 0 0
\(417\) −12.1965 −0.597266
\(418\) 0 0
\(419\) −15.2570 −0.745354 −0.372677 0.927961i \(-0.621560\pi\)
−0.372677 + 0.927961i \(0.621560\pi\)
\(420\) 0 0
\(421\) −10.2774 −0.500888 −0.250444 0.968131i \(-0.580577\pi\)
−0.250444 + 0.968131i \(0.580577\pi\)
\(422\) 0 0
\(423\) −3.40978 −0.165789
\(424\) 0 0
\(425\) −0.677137 −0.0328460
\(426\) 0 0
\(427\) 15.4039 0.745446
\(428\) 0 0
\(429\) 5.52895 0.266940
\(430\) 0 0
\(431\) 13.9360 0.671271 0.335636 0.941992i \(-0.391049\pi\)
0.335636 + 0.941992i \(0.391049\pi\)
\(432\) 0 0
\(433\) −8.93234 −0.429261 −0.214630 0.976695i \(-0.568855\pi\)
−0.214630 + 0.976695i \(0.568855\pi\)
\(434\) 0 0
\(435\) 2.36120 0.113211
\(436\) 0 0
\(437\) −5.00771 −0.239551
\(438\) 0 0
\(439\) 15.8164 0.754877 0.377439 0.926035i \(-0.376805\pi\)
0.377439 + 0.926035i \(0.376805\pi\)
\(440\) 0 0
\(441\) 0.961232 0.0457729
\(442\) 0 0
\(443\) 0.957836 0.0455081 0.0227541 0.999741i \(-0.492757\pi\)
0.0227541 + 0.999741i \(0.492757\pi\)
\(444\) 0 0
\(445\) −25.7697 −1.22160
\(446\) 0 0
\(447\) 21.7213 1.02738
\(448\) 0 0
\(449\) −0.612674 −0.0289139 −0.0144569 0.999895i \(-0.504602\pi\)
−0.0144569 + 0.999895i \(0.504602\pi\)
\(450\) 0 0
\(451\) −7.91835 −0.372861
\(452\) 0 0
\(453\) 1.74182 0.0818380
\(454\) 0 0
\(455\) 14.9533 0.701021
\(456\) 0 0
\(457\) 13.7397 0.642715 0.321358 0.946958i \(-0.395861\pi\)
0.321358 + 0.946958i \(0.395861\pi\)
\(458\) 0 0
\(459\) 1.17707 0.0549407
\(460\) 0 0
\(461\) −27.6910 −1.28970 −0.644848 0.764311i \(-0.723079\pi\)
−0.644848 + 0.764311i \(0.723079\pi\)
\(462\) 0 0
\(463\) 16.0140 0.744232 0.372116 0.928186i \(-0.378632\pi\)
0.372116 + 0.928186i \(0.378632\pi\)
\(464\) 0 0
\(465\) −14.5610 −0.675249
\(466\) 0 0
\(467\) 21.1599 0.979162 0.489581 0.871958i \(-0.337150\pi\)
0.489581 + 0.871958i \(0.337150\pi\)
\(468\) 0 0
\(469\) 32.5899 1.50486
\(470\) 0 0
\(471\) 18.6182 0.857882
\(472\) 0 0
\(473\) −29.8346 −1.37180
\(474\) 0 0
\(475\) 2.88082 0.132181
\(476\) 0 0
\(477\) −3.20671 −0.146825
\(478\) 0 0
\(479\) −11.6140 −0.530657 −0.265329 0.964158i \(-0.585480\pi\)
−0.265329 + 0.964158i \(0.585480\pi\)
\(480\) 0 0
\(481\) −9.94665 −0.453528
\(482\) 0 0
\(483\) 2.82157 0.128386
\(484\) 0 0
\(485\) −43.3898 −1.97023
\(486\) 0 0
\(487\) −26.5870 −1.20477 −0.602385 0.798206i \(-0.705783\pi\)
−0.602385 + 0.798206i \(0.705783\pi\)
\(488\) 0 0
\(489\) 12.4124 0.561308
\(490\) 0 0
\(491\) −32.7224 −1.47674 −0.738371 0.674395i \(-0.764405\pi\)
−0.738371 + 0.674395i \(0.764405\pi\)
\(492\) 0 0
\(493\) −1.17707 −0.0530123
\(494\) 0 0
\(495\) −5.81651 −0.261433
\(496\) 0 0
\(497\) 19.2028 0.861362
\(498\) 0 0
\(499\) −36.0244 −1.61267 −0.806337 0.591456i \(-0.798553\pi\)
−0.806337 + 0.591456i \(0.798553\pi\)
\(500\) 0 0
\(501\) −2.02150 −0.0903138
\(502\) 0 0
\(503\) 32.6779 1.45704 0.728518 0.685026i \(-0.240209\pi\)
0.728518 + 0.685026i \(0.240209\pi\)
\(504\) 0 0
\(505\) 18.9927 0.845164
\(506\) 0 0
\(507\) 7.96237 0.353621
\(508\) 0 0
\(509\) −12.9207 −0.572700 −0.286350 0.958125i \(-0.592442\pi\)
−0.286350 + 0.958125i \(0.592442\pi\)
\(510\) 0 0
\(511\) 1.51212 0.0668921
\(512\) 0 0
\(513\) −5.00771 −0.221096
\(514\) 0 0
\(515\) −28.6488 −1.26242
\(516\) 0 0
\(517\) −8.39953 −0.369411
\(518\) 0 0
\(519\) 1.55676 0.0683343
\(520\) 0 0
\(521\) 0.00620199 0.000271714 0 0.000135857 1.00000i \(-0.499957\pi\)
0.000135857 1.00000i \(0.499957\pi\)
\(522\) 0 0
\(523\) 16.6212 0.726794 0.363397 0.931634i \(-0.381617\pi\)
0.363397 + 0.931634i \(0.381617\pi\)
\(524\) 0 0
\(525\) −1.62318 −0.0708413
\(526\) 0 0
\(527\) 7.25868 0.316193
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −2.13703 −0.0927392
\(532\) 0 0
\(533\) 7.21471 0.312504
\(534\) 0 0
\(535\) 13.0762 0.565334
\(536\) 0 0
\(537\) 22.6826 0.978827
\(538\) 0 0
\(539\) 2.36787 0.101991
\(540\) 0 0
\(541\) −34.2809 −1.47385 −0.736926 0.675973i \(-0.763724\pi\)
−0.736926 + 0.675973i \(0.763724\pi\)
\(542\) 0 0
\(543\) 2.39045 0.102584
\(544\) 0 0
\(545\) −8.42040 −0.360690
\(546\) 0 0
\(547\) −12.1453 −0.519294 −0.259647 0.965704i \(-0.583606\pi\)
−0.259647 + 0.965704i \(0.583606\pi\)
\(548\) 0 0
\(549\) 5.45934 0.232999
\(550\) 0 0
\(551\) 5.00771 0.213336
\(552\) 0 0
\(553\) −25.6325 −1.09000
\(554\) 0 0
\(555\) 10.4640 0.444171
\(556\) 0 0
\(557\) 34.6002 1.46606 0.733029 0.680198i \(-0.238106\pi\)
0.733029 + 0.680198i \(0.238106\pi\)
\(558\) 0 0
\(559\) 27.1834 1.14974
\(560\) 0 0
\(561\) 2.89954 0.122419
\(562\) 0 0
\(563\) −38.9129 −1.63998 −0.819992 0.572376i \(-0.806022\pi\)
−0.819992 + 0.572376i \(0.806022\pi\)
\(564\) 0 0
\(565\) −0.390953 −0.0164475
\(566\) 0 0
\(567\) 2.82157 0.118495
\(568\) 0 0
\(569\) −45.2467 −1.89684 −0.948421 0.317014i \(-0.897320\pi\)
−0.948421 + 0.317014i \(0.897320\pi\)
\(570\) 0 0
\(571\) −21.9622 −0.919088 −0.459544 0.888155i \(-0.651987\pi\)
−0.459544 + 0.888155i \(0.651987\pi\)
\(572\) 0 0
\(573\) −4.58330 −0.191470
\(574\) 0 0
\(575\) −0.575276 −0.0239907
\(576\) 0 0
\(577\) −27.5096 −1.14524 −0.572620 0.819821i \(-0.694073\pi\)
−0.572620 + 0.819821i \(0.694073\pi\)
\(578\) 0 0
\(579\) 17.3826 0.722396
\(580\) 0 0
\(581\) −37.9959 −1.57634
\(582\) 0 0
\(583\) −7.89930 −0.327156
\(584\) 0 0
\(585\) 5.29964 0.219113
\(586\) 0 0
\(587\) −4.60137 −0.189919 −0.0949595 0.995481i \(-0.530272\pi\)
−0.0949595 + 0.995481i \(0.530272\pi\)
\(588\) 0 0
\(589\) −30.8814 −1.27245
\(590\) 0 0
\(591\) −3.86402 −0.158944
\(592\) 0 0
\(593\) −24.9518 −1.02465 −0.512323 0.858793i \(-0.671215\pi\)
−0.512323 + 0.858793i \(0.671215\pi\)
\(594\) 0 0
\(595\) 7.84194 0.321488
\(596\) 0 0
\(597\) 1.66919 0.0683154
\(598\) 0 0
\(599\) −24.8140 −1.01387 −0.506937 0.861983i \(-0.669222\pi\)
−0.506937 + 0.861983i \(0.669222\pi\)
\(600\) 0 0
\(601\) 39.7618 1.62192 0.810959 0.585103i \(-0.198946\pi\)
0.810959 + 0.585103i \(0.198946\pi\)
\(602\) 0 0
\(603\) 11.5503 0.470364
\(604\) 0 0
\(605\) 11.6450 0.473438
\(606\) 0 0
\(607\) −18.6818 −0.758272 −0.379136 0.925341i \(-0.623779\pi\)
−0.379136 + 0.925341i \(0.623779\pi\)
\(608\) 0 0
\(609\) −2.82157 −0.114336
\(610\) 0 0
\(611\) 7.65313 0.309612
\(612\) 0 0
\(613\) −10.4059 −0.420292 −0.210146 0.977670i \(-0.567394\pi\)
−0.210146 + 0.977670i \(0.567394\pi\)
\(614\) 0 0
\(615\) −7.58995 −0.306056
\(616\) 0 0
\(617\) −26.8185 −1.07967 −0.539837 0.841770i \(-0.681514\pi\)
−0.539837 + 0.841770i \(0.681514\pi\)
\(618\) 0 0
\(619\) 1.86410 0.0749245 0.0374622 0.999298i \(-0.488073\pi\)
0.0374622 + 0.999298i \(0.488073\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 30.7940 1.23374
\(624\) 0 0
\(625\) −27.5454 −1.10182
\(626\) 0 0
\(627\) −12.3358 −0.492646
\(628\) 0 0
\(629\) −5.21632 −0.207988
\(630\) 0 0
\(631\) −8.76571 −0.348958 −0.174479 0.984661i \(-0.555824\pi\)
−0.174479 + 0.984661i \(0.555824\pi\)
\(632\) 0 0
\(633\) −2.03379 −0.0808357
\(634\) 0 0
\(635\) −30.7532 −1.22040
\(636\) 0 0
\(637\) −2.15745 −0.0854814
\(638\) 0 0
\(639\) 6.80571 0.269230
\(640\) 0 0
\(641\) 3.96717 0.156694 0.0783469 0.996926i \(-0.475036\pi\)
0.0783469 + 0.996926i \(0.475036\pi\)
\(642\) 0 0
\(643\) −18.1768 −0.716823 −0.358412 0.933564i \(-0.616682\pi\)
−0.358412 + 0.933564i \(0.616682\pi\)
\(644\) 0 0
\(645\) −28.5972 −1.12601
\(646\) 0 0
\(647\) 34.0048 1.33687 0.668433 0.743772i \(-0.266965\pi\)
0.668433 + 0.743772i \(0.266965\pi\)
\(648\) 0 0
\(649\) −5.26429 −0.206641
\(650\) 0 0
\(651\) 17.3999 0.681957
\(652\) 0 0
\(653\) −24.4414 −0.956466 −0.478233 0.878233i \(-0.658723\pi\)
−0.478233 + 0.878233i \(0.658723\pi\)
\(654\) 0 0
\(655\) 48.2358 1.88473
\(656\) 0 0
\(657\) 0.535914 0.0209080
\(658\) 0 0
\(659\) 13.8977 0.541379 0.270690 0.962667i \(-0.412748\pi\)
0.270690 + 0.962667i \(0.412748\pi\)
\(660\) 0 0
\(661\) −4.70133 −0.182860 −0.0914302 0.995811i \(-0.529144\pi\)
−0.0914302 + 0.995811i \(0.529144\pi\)
\(662\) 0 0
\(663\) −2.64188 −0.102602
\(664\) 0 0
\(665\) −33.3628 −1.29375
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) 7.15464 0.276614
\(670\) 0 0
\(671\) 13.4484 0.519168
\(672\) 0 0
\(673\) 5.03790 0.194197 0.0970983 0.995275i \(-0.469044\pi\)
0.0970983 + 0.995275i \(0.469044\pi\)
\(674\) 0 0
\(675\) −0.575276 −0.0221424
\(676\) 0 0
\(677\) −2.76911 −0.106426 −0.0532128 0.998583i \(-0.516946\pi\)
−0.0532128 + 0.998583i \(0.516946\pi\)
\(678\) 0 0
\(679\) 51.8495 1.98980
\(680\) 0 0
\(681\) −9.46474 −0.362690
\(682\) 0 0
\(683\) −8.25556 −0.315890 −0.157945 0.987448i \(-0.550487\pi\)
−0.157945 + 0.987448i \(0.550487\pi\)
\(684\) 0 0
\(685\) −29.0163 −1.10866
\(686\) 0 0
\(687\) −19.9346 −0.760552
\(688\) 0 0
\(689\) 7.19736 0.274197
\(690\) 0 0
\(691\) −31.7949 −1.20954 −0.604768 0.796401i \(-0.706734\pi\)
−0.604768 + 0.796401i \(0.706734\pi\)
\(692\) 0 0
\(693\) 6.95055 0.264030
\(694\) 0 0
\(695\) −28.7985 −1.09239
\(696\) 0 0
\(697\) 3.78361 0.143314
\(698\) 0 0
\(699\) 20.0069 0.756730
\(700\) 0 0
\(701\) 42.8623 1.61889 0.809444 0.587197i \(-0.199769\pi\)
0.809444 + 0.587197i \(0.199769\pi\)
\(702\) 0 0
\(703\) 22.1923 0.837000
\(704\) 0 0
\(705\) −8.05117 −0.303225
\(706\) 0 0
\(707\) −22.6957 −0.853560
\(708\) 0 0
\(709\) 27.2260 1.02249 0.511247 0.859434i \(-0.329184\pi\)
0.511247 + 0.859434i \(0.329184\pi\)
\(710\) 0 0
\(711\) −9.08449 −0.340695
\(712\) 0 0
\(713\) 6.16676 0.230947
\(714\) 0 0
\(715\) 13.0550 0.488228
\(716\) 0 0
\(717\) 24.1719 0.902718
\(718\) 0 0
\(719\) −8.21601 −0.306405 −0.153203 0.988195i \(-0.548959\pi\)
−0.153203 + 0.988195i \(0.548959\pi\)
\(720\) 0 0
\(721\) 34.2345 1.27496
\(722\) 0 0
\(723\) −24.6368 −0.916254
\(724\) 0 0
\(725\) 0.575276 0.0213652
\(726\) 0 0
\(727\) 36.2756 1.34539 0.672694 0.739921i \(-0.265137\pi\)
0.672694 + 0.739921i \(0.265137\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 14.2558 0.527269
\(732\) 0 0
\(733\) 16.1792 0.597592 0.298796 0.954317i \(-0.403415\pi\)
0.298796 + 0.954317i \(0.403415\pi\)
\(734\) 0 0
\(735\) 2.26966 0.0837178
\(736\) 0 0
\(737\) 28.4526 1.04807
\(738\) 0 0
\(739\) 23.8855 0.878642 0.439321 0.898330i \(-0.355219\pi\)
0.439321 + 0.898330i \(0.355219\pi\)
\(740\) 0 0
\(741\) 11.2396 0.412899
\(742\) 0 0
\(743\) −1.62100 −0.0594686 −0.0297343 0.999558i \(-0.509466\pi\)
−0.0297343 + 0.999558i \(0.509466\pi\)
\(744\) 0 0
\(745\) 51.2883 1.87906
\(746\) 0 0
\(747\) −13.4663 −0.492705
\(748\) 0 0
\(749\) −15.6257 −0.570950
\(750\) 0 0
\(751\) 15.7260 0.573849 0.286925 0.957953i \(-0.407367\pi\)
0.286925 + 0.957953i \(0.407367\pi\)
\(752\) 0 0
\(753\) 21.1137 0.769427
\(754\) 0 0
\(755\) 4.11279 0.149680
\(756\) 0 0
\(757\) 0.718152 0.0261017 0.0130508 0.999915i \(-0.495846\pi\)
0.0130508 + 0.999915i \(0.495846\pi\)
\(758\) 0 0
\(759\) 2.46337 0.0894146
\(760\) 0 0
\(761\) −18.6447 −0.675869 −0.337935 0.941170i \(-0.609728\pi\)
−0.337935 + 0.941170i \(0.609728\pi\)
\(762\) 0 0
\(763\) 10.0621 0.364273
\(764\) 0 0
\(765\) 2.77929 0.100485
\(766\) 0 0
\(767\) 4.79649 0.173191
\(768\) 0 0
\(769\) −51.6366 −1.86206 −0.931031 0.364940i \(-0.881089\pi\)
−0.931031 + 0.364940i \(0.881089\pi\)
\(770\) 0 0
\(771\) −15.9018 −0.572691
\(772\) 0 0
\(773\) 50.5927 1.81969 0.909846 0.414945i \(-0.136199\pi\)
0.909846 + 0.414945i \(0.136199\pi\)
\(774\) 0 0
\(775\) −3.54759 −0.127433
\(776\) 0 0
\(777\) −12.5041 −0.448583
\(778\) 0 0
\(779\) −16.0970 −0.576735
\(780\) 0 0
\(781\) 16.7650 0.599898
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 43.9614 1.56905
\(786\) 0 0
\(787\) −40.1277 −1.43040 −0.715199 0.698921i \(-0.753664\pi\)
−0.715199 + 0.698921i \(0.753664\pi\)
\(788\) 0 0
\(789\) 26.1463 0.930831
\(790\) 0 0
\(791\) 0.467177 0.0166109
\(792\) 0 0
\(793\) −12.2533 −0.435128
\(794\) 0 0
\(795\) −7.57169 −0.268540
\(796\) 0 0
\(797\) −22.2487 −0.788088 −0.394044 0.919092i \(-0.628924\pi\)
−0.394044 + 0.919092i \(0.628924\pi\)
\(798\) 0 0
\(799\) 4.01353 0.141988
\(800\) 0 0
\(801\) 10.9138 0.385621
\(802\) 0 0
\(803\) 1.32015 0.0465872
\(804\) 0 0
\(805\) 6.66229 0.234815
\(806\) 0 0
\(807\) 2.39482 0.0843015
\(808\) 0 0
\(809\) 52.6354 1.85056 0.925281 0.379282i \(-0.123829\pi\)
0.925281 + 0.379282i \(0.123829\pi\)
\(810\) 0 0
\(811\) 21.3831 0.750863 0.375431 0.926850i \(-0.377495\pi\)
0.375431 + 0.926850i \(0.377495\pi\)
\(812\) 0 0
\(813\) −4.23346 −0.148474
\(814\) 0 0
\(815\) 29.3082 1.02662
\(816\) 0 0
\(817\) −60.6499 −2.12187
\(818\) 0 0
\(819\) −6.33291 −0.221290
\(820\) 0 0
\(821\) 38.0037 1.32634 0.663169 0.748469i \(-0.269211\pi\)
0.663169 + 0.748469i \(0.269211\pi\)
\(822\) 0 0
\(823\) −22.2845 −0.776789 −0.388394 0.921493i \(-0.626970\pi\)
−0.388394 + 0.921493i \(0.626970\pi\)
\(824\) 0 0
\(825\) −1.41712 −0.0493376
\(826\) 0 0
\(827\) −17.9745 −0.625034 −0.312517 0.949912i \(-0.601172\pi\)
−0.312517 + 0.949912i \(0.601172\pi\)
\(828\) 0 0
\(829\) −39.6008 −1.37539 −0.687697 0.725998i \(-0.741378\pi\)
−0.687697 + 0.725998i \(0.741378\pi\)
\(830\) 0 0
\(831\) −14.8325 −0.514534
\(832\) 0 0
\(833\) −1.13143 −0.0392018
\(834\) 0 0
\(835\) −4.77316 −0.165182
\(836\) 0 0
\(837\) 6.16676 0.213155
\(838\) 0 0
\(839\) −36.8250 −1.27134 −0.635669 0.771961i \(-0.719276\pi\)
−0.635669 + 0.771961i \(0.719276\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 27.3067 0.940494
\(844\) 0 0
\(845\) 18.8008 0.646766
\(846\) 0 0
\(847\) −13.9155 −0.478141
\(848\) 0 0
\(849\) −13.7600 −0.472243
\(850\) 0 0
\(851\) −4.43163 −0.151914
\(852\) 0 0
\(853\) −53.9826 −1.84833 −0.924164 0.381995i \(-0.875237\pi\)
−0.924164 + 0.381995i \(0.875237\pi\)
\(854\) 0 0
\(855\) −11.8242 −0.404380
\(856\) 0 0
\(857\) 27.0718 0.924755 0.462378 0.886683i \(-0.346996\pi\)
0.462378 + 0.886683i \(0.346996\pi\)
\(858\) 0 0
\(859\) −29.5478 −1.00816 −0.504079 0.863658i \(-0.668168\pi\)
−0.504079 + 0.863658i \(0.668168\pi\)
\(860\) 0 0
\(861\) 9.06976 0.309097
\(862\) 0 0
\(863\) −33.1310 −1.12779 −0.563897 0.825845i \(-0.690698\pi\)
−0.563897 + 0.825845i \(0.690698\pi\)
\(864\) 0 0
\(865\) 3.67583 0.124982
\(866\) 0 0
\(867\) 15.6145 0.530297
\(868\) 0 0
\(869\) −22.3784 −0.759137
\(870\) 0 0
\(871\) −25.9242 −0.878409
\(872\) 0 0
\(873\) 18.3762 0.621939
\(874\) 0 0
\(875\) 29.4788 0.996565
\(876\) 0 0
\(877\) 22.4928 0.759527 0.379764 0.925084i \(-0.376005\pi\)
0.379764 + 0.925084i \(0.376005\pi\)
\(878\) 0 0
\(879\) −14.0909 −0.475275
\(880\) 0 0
\(881\) −20.2861 −0.683456 −0.341728 0.939799i \(-0.611012\pi\)
−0.341728 + 0.939799i \(0.611012\pi\)
\(882\) 0 0
\(883\) 28.1579 0.947588 0.473794 0.880636i \(-0.342884\pi\)
0.473794 + 0.880636i \(0.342884\pi\)
\(884\) 0 0
\(885\) −5.04596 −0.169618
\(886\) 0 0
\(887\) 23.7823 0.798532 0.399266 0.916835i \(-0.369265\pi\)
0.399266 + 0.916835i \(0.369265\pi\)
\(888\) 0 0
\(889\) 36.7491 1.23252
\(890\) 0 0
\(891\) 2.46337 0.0825259
\(892\) 0 0
\(893\) −17.0752 −0.571399
\(894\) 0 0
\(895\) 53.5582 1.79025
\(896\) 0 0
\(897\) −2.24447 −0.0749406
\(898\) 0 0
\(899\) −6.16676 −0.205673
\(900\) 0 0
\(901\) 3.77451 0.125747
\(902\) 0 0
\(903\) 34.1728 1.13720
\(904\) 0 0
\(905\) 5.64433 0.187624
\(906\) 0 0
\(907\) 31.7392 1.05388 0.526942 0.849901i \(-0.323339\pi\)
0.526942 + 0.849901i \(0.323339\pi\)
\(908\) 0 0
\(909\) −8.04365 −0.266791
\(910\) 0 0
\(911\) 36.1257 1.19690 0.598449 0.801161i \(-0.295784\pi\)
0.598449 + 0.801161i \(0.295784\pi\)
\(912\) 0 0
\(913\) −33.1724 −1.09784
\(914\) 0 0
\(915\) 12.8906 0.426150
\(916\) 0 0
\(917\) −57.6404 −1.90345
\(918\) 0 0
\(919\) 3.92619 0.129513 0.0647565 0.997901i \(-0.479373\pi\)
0.0647565 + 0.997901i \(0.479373\pi\)
\(920\) 0 0
\(921\) −16.7805 −0.552937
\(922\) 0 0
\(923\) −15.2752 −0.502789
\(924\) 0 0
\(925\) 2.54941 0.0838241
\(926\) 0 0
\(927\) 12.1332 0.398505
\(928\) 0 0
\(929\) 29.8059 0.977898 0.488949 0.872312i \(-0.337380\pi\)
0.488949 + 0.872312i \(0.337380\pi\)
\(930\) 0 0
\(931\) 4.81357 0.157758
\(932\) 0 0
\(933\) −32.9428 −1.07850
\(934\) 0 0
\(935\) 6.84641 0.223901
\(936\) 0 0
\(937\) 24.9343 0.814569 0.407285 0.913301i \(-0.366476\pi\)
0.407285 + 0.913301i \(0.366476\pi\)
\(938\) 0 0
\(939\) −0.604859 −0.0197388
\(940\) 0 0
\(941\) 4.77338 0.155608 0.0778038 0.996969i \(-0.475209\pi\)
0.0778038 + 0.996969i \(0.475209\pi\)
\(942\) 0 0
\(943\) 3.21444 0.104677
\(944\) 0 0
\(945\) 6.66229 0.216724
\(946\) 0 0
\(947\) 49.0605 1.59425 0.797126 0.603813i \(-0.206353\pi\)
0.797126 + 0.603813i \(0.206353\pi\)
\(948\) 0 0
\(949\) −1.20284 −0.0390459
\(950\) 0 0
\(951\) 31.1888 1.01137
\(952\) 0 0
\(953\) −42.5128 −1.37712 −0.688562 0.725177i \(-0.741758\pi\)
−0.688562 + 0.725177i \(0.741758\pi\)
\(954\) 0 0
\(955\) −10.8221 −0.350195
\(956\) 0 0
\(957\) −2.46337 −0.0796293
\(958\) 0 0
\(959\) 34.6737 1.11967
\(960\) 0 0
\(961\) 7.02899 0.226742
\(962\) 0 0
\(963\) −5.53795 −0.178458
\(964\) 0 0
\(965\) 41.0438 1.32125
\(966\) 0 0
\(967\) −16.6675 −0.535989 −0.267995 0.963420i \(-0.586361\pi\)
−0.267995 + 0.963420i \(0.586361\pi\)
\(968\) 0 0
\(969\) 5.89440 0.189356
\(970\) 0 0
\(971\) 37.6485 1.20820 0.604098 0.796910i \(-0.293533\pi\)
0.604098 + 0.796910i \(0.293533\pi\)
\(972\) 0 0
\(973\) 34.4133 1.10324
\(974\) 0 0
\(975\) 1.29119 0.0413511
\(976\) 0 0
\(977\) 48.0575 1.53750 0.768748 0.639552i \(-0.220880\pi\)
0.768748 + 0.639552i \(0.220880\pi\)
\(978\) 0 0
\(979\) 26.8847 0.859240
\(980\) 0 0
\(981\) 3.56615 0.113858
\(982\) 0 0
\(983\) −4.10065 −0.130790 −0.0653952 0.997859i \(-0.520831\pi\)
−0.0653952 + 0.997859i \(0.520831\pi\)
\(984\) 0 0
\(985\) −9.12372 −0.290706
\(986\) 0 0
\(987\) 9.62091 0.306237
\(988\) 0 0
\(989\) 12.1113 0.385117
\(990\) 0 0
\(991\) 20.8892 0.663567 0.331784 0.943356i \(-0.392350\pi\)
0.331784 + 0.943356i \(0.392350\pi\)
\(992\) 0 0
\(993\) 21.4478 0.680627
\(994\) 0 0
\(995\) 3.94129 0.124947
\(996\) 0 0
\(997\) −61.6324 −1.95192 −0.975958 0.217958i \(-0.930060\pi\)
−0.975958 + 0.217958i \(0.930060\pi\)
\(998\) 0 0
\(999\) −4.43163 −0.140211
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.e.1.3 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.e.1.3 9 1.1 even 1 trivial