Properties

Label 8016.2.a.bb.1.9
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: \(x^{9} - 29 x^{7} - 7 x^{6} + 266 x^{5} + 69 x^{4} - 901 x^{3} - 199 x^{2} + 875 x + 391\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2004)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-3.21153\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +4.21153 q^{5} +5.13720 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +4.21153 q^{5} +5.13720 q^{7} +1.00000 q^{9} +1.59181 q^{11} +5.05424 q^{13} -4.21153 q^{15} +0.754874 q^{17} +1.66875 q^{19} -5.13720 q^{21} -6.04218 q^{23} +12.7370 q^{25} -1.00000 q^{27} -4.24703 q^{29} +1.07105 q^{31} -1.59181 q^{33} +21.6355 q^{35} +4.94651 q^{37} -5.05424 q^{39} +3.04535 q^{41} -12.5625 q^{43} +4.21153 q^{45} -9.35819 q^{47} +19.3908 q^{49} -0.754874 q^{51} -8.52323 q^{53} +6.70395 q^{55} -1.66875 q^{57} +3.98059 q^{59} +8.18677 q^{61} +5.13720 q^{63} +21.2861 q^{65} +4.99504 q^{67} +6.04218 q^{69} -6.73050 q^{71} -6.06414 q^{73} -12.7370 q^{75} +8.17744 q^{77} +13.5695 q^{79} +1.00000 q^{81} +5.95533 q^{83} +3.17917 q^{85} +4.24703 q^{87} +7.83467 q^{89} +25.9646 q^{91} -1.07105 q^{93} +7.02797 q^{95} +3.42397 q^{97} +1.59181 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} + 9q^{5} - 2q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} + 9q^{5} - 2q^{7} + 9q^{9} - 7q^{11} + 6q^{13} - 9q^{15} + 7q^{17} - 2q^{19} + 2q^{21} - 19q^{23} + 22q^{25} - 9q^{27} + 13q^{29} - 12q^{31} + 7q^{33} - 4q^{35} + 15q^{37} - 6q^{39} + 18q^{41} + 6q^{43} + 9q^{45} - 25q^{47} + 19q^{49} - 7q^{51} + 17q^{53} + 3q^{55} + 2q^{57} - 3q^{59} + 14q^{61} - 2q^{63} + 14q^{65} + 4q^{67} + 19q^{69} - 17q^{71} - 20q^{73} - 22q^{75} + 14q^{77} + 8q^{79} + 9q^{81} + q^{83} + 5q^{85} - 13q^{87} + 36q^{89} + 41q^{91} + 12q^{93} - 5q^{95} + 31q^{97} - 7q^{99} + O(q^{100}) \)

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
\(5\) 4.21153 1.88345 0.941726 0.336380i \(-0.109203\pi\)
0.941726 + 0.336380i \(0.109203\pi\)
\(6\) 0 0
\(7\) 5.13720 1.94168 0.970840 0.239730i \(-0.0770590\pi\)
0.970840 + 0.239730i \(0.0770590\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.59181 0.479949 0.239974 0.970779i \(-0.422861\pi\)
0.239974 + 0.970779i \(0.422861\pi\)
\(12\) 0 0
\(13\) 5.05424 1.40179 0.700897 0.713263i \(-0.252783\pi\)
0.700897 + 0.713263i \(0.252783\pi\)
\(14\) 0 0
\(15\) −4.21153 −1.08741
\(16\) 0 0
\(17\) 0.754874 0.183084 0.0915419 0.995801i \(-0.470820\pi\)
0.0915419 + 0.995801i \(0.470820\pi\)
\(18\) 0 0
\(19\) 1.66875 0.382836 0.191418 0.981509i \(-0.438691\pi\)
0.191418 + 0.981509i \(0.438691\pi\)
\(20\) 0 0
\(21\) −5.13720 −1.12103
\(22\) 0 0
\(23\) −6.04218 −1.25988 −0.629941 0.776643i \(-0.716921\pi\)
−0.629941 + 0.776643i \(0.716921\pi\)
\(24\) 0 0
\(25\) 12.7370 2.54739
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.24703 −0.788654 −0.394327 0.918970i \(-0.629022\pi\)
−0.394327 + 0.918970i \(0.629022\pi\)
\(30\) 0 0
\(31\) 1.07105 0.192367 0.0961834 0.995364i \(-0.469336\pi\)
0.0961834 + 0.995364i \(0.469336\pi\)
\(32\) 0 0
\(33\) −1.59181 −0.277098
\(34\) 0 0
\(35\) 21.6355 3.65706
\(36\) 0 0
\(37\) 4.94651 0.813201 0.406600 0.913606i \(-0.366714\pi\)
0.406600 + 0.913606i \(0.366714\pi\)
\(38\) 0 0
\(39\) −5.05424 −0.809326
\(40\) 0 0
\(41\) 3.04535 0.475603 0.237802 0.971314i \(-0.423573\pi\)
0.237802 + 0.971314i \(0.423573\pi\)
\(42\) 0 0
\(43\) −12.5625 −1.91577 −0.957884 0.287154i \(-0.907291\pi\)
−0.957884 + 0.287154i \(0.907291\pi\)
\(44\) 0 0
\(45\) 4.21153 0.627818
\(46\) 0 0
\(47\) −9.35819 −1.36503 −0.682516 0.730871i \(-0.739114\pi\)
−0.682516 + 0.730871i \(0.739114\pi\)
\(48\) 0 0
\(49\) 19.3908 2.77012
\(50\) 0 0
\(51\) −0.754874 −0.105704
\(52\) 0 0
\(53\) −8.52323 −1.17076 −0.585378 0.810761i \(-0.699054\pi\)
−0.585378 + 0.810761i \(0.699054\pi\)
\(54\) 0 0
\(55\) 6.70395 0.903961
\(56\) 0 0
\(57\) −1.66875 −0.221031
\(58\) 0 0
\(59\) 3.98059 0.518229 0.259115 0.965847i \(-0.416569\pi\)
0.259115 + 0.965847i \(0.416569\pi\)
\(60\) 0 0
\(61\) 8.18677 1.04821 0.524104 0.851654i \(-0.324400\pi\)
0.524104 + 0.851654i \(0.324400\pi\)
\(62\) 0 0
\(63\) 5.13720 0.647226
\(64\) 0 0
\(65\) 21.2861 2.64021
\(66\) 0 0
\(67\) 4.99504 0.610242 0.305121 0.952314i \(-0.401303\pi\)
0.305121 + 0.952314i \(0.401303\pi\)
\(68\) 0 0
\(69\) 6.04218 0.727393
\(70\) 0 0
\(71\) −6.73050 −0.798763 −0.399382 0.916785i \(-0.630775\pi\)
−0.399382 + 0.916785i \(0.630775\pi\)
\(72\) 0 0
\(73\) −6.06414 −0.709754 −0.354877 0.934913i \(-0.615477\pi\)
−0.354877 + 0.934913i \(0.615477\pi\)
\(74\) 0 0
\(75\) −12.7370 −1.47074
\(76\) 0 0
\(77\) 8.17744 0.931906
\(78\) 0 0
\(79\) 13.5695 1.52669 0.763345 0.645991i \(-0.223556\pi\)
0.763345 + 0.645991i \(0.223556\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.95533 0.653683 0.326841 0.945079i \(-0.394016\pi\)
0.326841 + 0.945079i \(0.394016\pi\)
\(84\) 0 0
\(85\) 3.17917 0.344830
\(86\) 0 0
\(87\) 4.24703 0.455330
\(88\) 0 0
\(89\) 7.83467 0.830473 0.415237 0.909713i \(-0.363699\pi\)
0.415237 + 0.909713i \(0.363699\pi\)
\(90\) 0 0
\(91\) 25.9646 2.72183
\(92\) 0 0
\(93\) −1.07105 −0.111063
\(94\) 0 0
\(95\) 7.02797 0.721054
\(96\) 0 0
\(97\) 3.42397 0.347652 0.173826 0.984776i \(-0.444387\pi\)
0.173826 + 0.984776i \(0.444387\pi\)
\(98\) 0 0
\(99\) 1.59181 0.159983
\(100\) 0 0
\(101\) −17.3384 −1.72523 −0.862617 0.505858i \(-0.831176\pi\)
−0.862617 + 0.505858i \(0.831176\pi\)
\(102\) 0 0
\(103\) −10.2605 −1.01100 −0.505498 0.862828i \(-0.668691\pi\)
−0.505498 + 0.862828i \(0.668691\pi\)
\(104\) 0 0
\(105\) −21.6355 −2.11140
\(106\) 0 0
\(107\) −13.1324 −1.26955 −0.634777 0.772696i \(-0.718908\pi\)
−0.634777 + 0.772696i \(0.718908\pi\)
\(108\) 0 0
\(109\) −4.53965 −0.434820 −0.217410 0.976080i \(-0.569761\pi\)
−0.217410 + 0.976080i \(0.569761\pi\)
\(110\) 0 0
\(111\) −4.94651 −0.469502
\(112\) 0 0
\(113\) −20.4713 −1.92578 −0.962889 0.269897i \(-0.913010\pi\)
−0.962889 + 0.269897i \(0.913010\pi\)
\(114\) 0 0
\(115\) −25.4468 −2.37293
\(116\) 0 0
\(117\) 5.05424 0.467264
\(118\) 0 0
\(119\) 3.87794 0.355490
\(120\) 0 0
\(121\) −8.46614 −0.769649
\(122\) 0 0
\(123\) −3.04535 −0.274590
\(124\) 0 0
\(125\) 32.5845 2.91444
\(126\) 0 0
\(127\) 17.7560 1.57559 0.787795 0.615937i \(-0.211222\pi\)
0.787795 + 0.615937i \(0.211222\pi\)
\(128\) 0 0
\(129\) 12.5625 1.10607
\(130\) 0 0
\(131\) 19.4096 1.69582 0.847912 0.530137i \(-0.177859\pi\)
0.847912 + 0.530137i \(0.177859\pi\)
\(132\) 0 0
\(133\) 8.57268 0.743345
\(134\) 0 0
\(135\) −4.21153 −0.362471
\(136\) 0 0
\(137\) 9.29492 0.794118 0.397059 0.917793i \(-0.370031\pi\)
0.397059 + 0.917793i \(0.370031\pi\)
\(138\) 0 0
\(139\) −13.0428 −1.10628 −0.553140 0.833089i \(-0.686570\pi\)
−0.553140 + 0.833089i \(0.686570\pi\)
\(140\) 0 0
\(141\) 9.35819 0.788101
\(142\) 0 0
\(143\) 8.04538 0.672789
\(144\) 0 0
\(145\) −17.8865 −1.48539
\(146\) 0 0
\(147\) −19.3908 −1.59933
\(148\) 0 0
\(149\) 5.33359 0.436945 0.218473 0.975843i \(-0.429893\pi\)
0.218473 + 0.975843i \(0.429893\pi\)
\(150\) 0 0
\(151\) 18.2184 1.48259 0.741294 0.671180i \(-0.234212\pi\)
0.741294 + 0.671180i \(0.234212\pi\)
\(152\) 0 0
\(153\) 0.754874 0.0610280
\(154\) 0 0
\(155\) 4.51077 0.362314
\(156\) 0 0
\(157\) −9.86754 −0.787516 −0.393758 0.919214i \(-0.628825\pi\)
−0.393758 + 0.919214i \(0.628825\pi\)
\(158\) 0 0
\(159\) 8.52323 0.675936
\(160\) 0 0
\(161\) −31.0399 −2.44629
\(162\) 0 0
\(163\) 7.64582 0.598867 0.299433 0.954117i \(-0.403202\pi\)
0.299433 + 0.954117i \(0.403202\pi\)
\(164\) 0 0
\(165\) −6.70395 −0.521902
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 12.5453 0.965024
\(170\) 0 0
\(171\) 1.66875 0.127612
\(172\) 0 0
\(173\) 2.13111 0.162025 0.0810127 0.996713i \(-0.474185\pi\)
0.0810127 + 0.996713i \(0.474185\pi\)
\(174\) 0 0
\(175\) 65.4324 4.94622
\(176\) 0 0
\(177\) −3.98059 −0.299200
\(178\) 0 0
\(179\) −12.2841 −0.918155 −0.459078 0.888396i \(-0.651820\pi\)
−0.459078 + 0.888396i \(0.651820\pi\)
\(180\) 0 0
\(181\) −12.6588 −0.940924 −0.470462 0.882420i \(-0.655913\pi\)
−0.470462 + 0.882420i \(0.655913\pi\)
\(182\) 0 0
\(183\) −8.18677 −0.605183
\(184\) 0 0
\(185\) 20.8324 1.53163
\(186\) 0 0
\(187\) 1.20162 0.0878709
\(188\) 0 0
\(189\) −5.13720 −0.373676
\(190\) 0 0
\(191\) −14.9840 −1.08420 −0.542101 0.840313i \(-0.682371\pi\)
−0.542101 + 0.840313i \(0.682371\pi\)
\(192\) 0 0
\(193\) −19.2434 −1.38517 −0.692586 0.721335i \(-0.743529\pi\)
−0.692586 + 0.721335i \(0.743529\pi\)
\(194\) 0 0
\(195\) −21.2861 −1.52433
\(196\) 0 0
\(197\) −3.73605 −0.266182 −0.133091 0.991104i \(-0.542490\pi\)
−0.133091 + 0.991104i \(0.542490\pi\)
\(198\) 0 0
\(199\) −17.3082 −1.22694 −0.613472 0.789717i \(-0.710228\pi\)
−0.613472 + 0.789717i \(0.710228\pi\)
\(200\) 0 0
\(201\) −4.99504 −0.352323
\(202\) 0 0
\(203\) −21.8179 −1.53131
\(204\) 0 0
\(205\) 12.8256 0.895776
\(206\) 0 0
\(207\) −6.04218 −0.419961
\(208\) 0 0
\(209\) 2.65632 0.183742
\(210\) 0 0
\(211\) −10.3219 −0.710587 −0.355293 0.934755i \(-0.615619\pi\)
−0.355293 + 0.934755i \(0.615619\pi\)
\(212\) 0 0
\(213\) 6.73050 0.461166
\(214\) 0 0
\(215\) −52.9075 −3.60826
\(216\) 0 0
\(217\) 5.50221 0.373514
\(218\) 0 0
\(219\) 6.06414 0.409777
\(220\) 0 0
\(221\) 3.81531 0.256646
\(222\) 0 0
\(223\) −17.3854 −1.16421 −0.582107 0.813113i \(-0.697771\pi\)
−0.582107 + 0.813113i \(0.697771\pi\)
\(224\) 0 0
\(225\) 12.7370 0.849131
\(226\) 0 0
\(227\) −22.0068 −1.46065 −0.730323 0.683102i \(-0.760630\pi\)
−0.730323 + 0.683102i \(0.760630\pi\)
\(228\) 0 0
\(229\) −10.2875 −0.679815 −0.339907 0.940459i \(-0.610396\pi\)
−0.339907 + 0.940459i \(0.610396\pi\)
\(230\) 0 0
\(231\) −8.17744 −0.538036
\(232\) 0 0
\(233\) −5.06675 −0.331934 −0.165967 0.986131i \(-0.553074\pi\)
−0.165967 + 0.986131i \(0.553074\pi\)
\(234\) 0 0
\(235\) −39.4123 −2.57097
\(236\) 0 0
\(237\) −13.5695 −0.881434
\(238\) 0 0
\(239\) −2.30330 −0.148988 −0.0744939 0.997221i \(-0.523734\pi\)
−0.0744939 + 0.997221i \(0.523734\pi\)
\(240\) 0 0
\(241\) −0.625570 −0.0402965 −0.0201483 0.999797i \(-0.506414\pi\)
−0.0201483 + 0.999797i \(0.506414\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 81.6650 5.21739
\(246\) 0 0
\(247\) 8.43423 0.536657
\(248\) 0 0
\(249\) −5.95533 −0.377404
\(250\) 0 0
\(251\) 22.4644 1.41794 0.708970 0.705238i \(-0.249160\pi\)
0.708970 + 0.705238i \(0.249160\pi\)
\(252\) 0 0
\(253\) −9.61800 −0.604678
\(254\) 0 0
\(255\) −3.17917 −0.199088
\(256\) 0 0
\(257\) −13.6816 −0.853436 −0.426718 0.904385i \(-0.640330\pi\)
−0.426718 + 0.904385i \(0.640330\pi\)
\(258\) 0 0
\(259\) 25.4112 1.57897
\(260\) 0 0
\(261\) −4.24703 −0.262885
\(262\) 0 0
\(263\) −4.63785 −0.285982 −0.142991 0.989724i \(-0.545672\pi\)
−0.142991 + 0.989724i \(0.545672\pi\)
\(264\) 0 0
\(265\) −35.8958 −2.20506
\(266\) 0 0
\(267\) −7.83467 −0.479474
\(268\) 0 0
\(269\) 24.6489 1.50287 0.751434 0.659808i \(-0.229363\pi\)
0.751434 + 0.659808i \(0.229363\pi\)
\(270\) 0 0
\(271\) −11.6032 −0.704844 −0.352422 0.935841i \(-0.614642\pi\)
−0.352422 + 0.935841i \(0.614642\pi\)
\(272\) 0 0
\(273\) −25.9646 −1.57145
\(274\) 0 0
\(275\) 20.2748 1.22262
\(276\) 0 0
\(277\) −0.516066 −0.0310074 −0.0155037 0.999880i \(-0.504935\pi\)
−0.0155037 + 0.999880i \(0.504935\pi\)
\(278\) 0 0
\(279\) 1.07105 0.0641222
\(280\) 0 0
\(281\) −7.67914 −0.458099 −0.229050 0.973415i \(-0.573562\pi\)
−0.229050 + 0.973415i \(0.573562\pi\)
\(282\) 0 0
\(283\) 15.5410 0.923819 0.461909 0.886927i \(-0.347165\pi\)
0.461909 + 0.886927i \(0.347165\pi\)
\(284\) 0 0
\(285\) −7.02797 −0.416301
\(286\) 0 0
\(287\) 15.6445 0.923469
\(288\) 0 0
\(289\) −16.4302 −0.966480
\(290\) 0 0
\(291\) −3.42397 −0.200717
\(292\) 0 0
\(293\) 7.27759 0.425161 0.212581 0.977144i \(-0.431813\pi\)
0.212581 + 0.977144i \(0.431813\pi\)
\(294\) 0 0
\(295\) 16.7644 0.976060
\(296\) 0 0
\(297\) −1.59181 −0.0923662
\(298\) 0 0
\(299\) −30.5386 −1.76609
\(300\) 0 0
\(301\) −64.5363 −3.71981
\(302\) 0 0
\(303\) 17.3384 0.996064
\(304\) 0 0
\(305\) 34.4788 1.97425
\(306\) 0 0
\(307\) 0.121465 0.00693237 0.00346619 0.999994i \(-0.498897\pi\)
0.00346619 + 0.999994i \(0.498897\pi\)
\(308\) 0 0
\(309\) 10.2605 0.583699
\(310\) 0 0
\(311\) 16.6650 0.944984 0.472492 0.881335i \(-0.343355\pi\)
0.472492 + 0.881335i \(0.343355\pi\)
\(312\) 0 0
\(313\) −33.1709 −1.87493 −0.937464 0.348083i \(-0.886833\pi\)
−0.937464 + 0.348083i \(0.886833\pi\)
\(314\) 0 0
\(315\) 21.6355 1.21902
\(316\) 0 0
\(317\) 23.8173 1.33771 0.668856 0.743392i \(-0.266784\pi\)
0.668856 + 0.743392i \(0.266784\pi\)
\(318\) 0 0
\(319\) −6.76047 −0.378513
\(320\) 0 0
\(321\) 13.1324 0.732977
\(322\) 0 0
\(323\) 1.25969 0.0700912
\(324\) 0 0
\(325\) 64.3757 3.57092
\(326\) 0 0
\(327\) 4.53965 0.251043
\(328\) 0 0
\(329\) −48.0749 −2.65045
\(330\) 0 0
\(331\) −1.84876 −0.101617 −0.0508085 0.998708i \(-0.516180\pi\)
−0.0508085 + 0.998708i \(0.516180\pi\)
\(332\) 0 0
\(333\) 4.94651 0.271067
\(334\) 0 0
\(335\) 21.0368 1.14936
\(336\) 0 0
\(337\) 12.0656 0.657256 0.328628 0.944460i \(-0.393414\pi\)
0.328628 + 0.944460i \(0.393414\pi\)
\(338\) 0 0
\(339\) 20.4713 1.11185
\(340\) 0 0
\(341\) 1.70491 0.0923261
\(342\) 0 0
\(343\) 63.6541 3.43700
\(344\) 0 0
\(345\) 25.4468 1.37001
\(346\) 0 0
\(347\) 5.33156 0.286213 0.143106 0.989707i \(-0.454291\pi\)
0.143106 + 0.989707i \(0.454291\pi\)
\(348\) 0 0
\(349\) 17.1869 0.919994 0.459997 0.887920i \(-0.347850\pi\)
0.459997 + 0.887920i \(0.347850\pi\)
\(350\) 0 0
\(351\) −5.05424 −0.269775
\(352\) 0 0
\(353\) −36.7612 −1.95660 −0.978301 0.207189i \(-0.933568\pi\)
−0.978301 + 0.207189i \(0.933568\pi\)
\(354\) 0 0
\(355\) −28.3457 −1.50443
\(356\) 0 0
\(357\) −3.87794 −0.205242
\(358\) 0 0
\(359\) −16.2669 −0.858536 −0.429268 0.903177i \(-0.641228\pi\)
−0.429268 + 0.903177i \(0.641228\pi\)
\(360\) 0 0
\(361\) −16.2153 −0.853436
\(362\) 0 0
\(363\) 8.46614 0.444357
\(364\) 0 0
\(365\) −25.5393 −1.33679
\(366\) 0 0
\(367\) 24.3923 1.27327 0.636633 0.771167i \(-0.280327\pi\)
0.636633 + 0.771167i \(0.280327\pi\)
\(368\) 0 0
\(369\) 3.04535 0.158534
\(370\) 0 0
\(371\) −43.7855 −2.27323
\(372\) 0 0
\(373\) −19.3889 −1.00392 −0.501959 0.864892i \(-0.667387\pi\)
−0.501959 + 0.864892i \(0.667387\pi\)
\(374\) 0 0
\(375\) −32.5845 −1.68265
\(376\) 0 0
\(377\) −21.4655 −1.10553
\(378\) 0 0
\(379\) −12.6031 −0.647375 −0.323688 0.946164i \(-0.604923\pi\)
−0.323688 + 0.946164i \(0.604923\pi\)
\(380\) 0 0
\(381\) −17.7560 −0.909668
\(382\) 0 0
\(383\) 11.7144 0.598576 0.299288 0.954163i \(-0.403251\pi\)
0.299288 + 0.954163i \(0.403251\pi\)
\(384\) 0 0
\(385\) 34.4395 1.75520
\(386\) 0 0
\(387\) −12.5625 −0.638590
\(388\) 0 0
\(389\) −5.63603 −0.285758 −0.142879 0.989740i \(-0.545636\pi\)
−0.142879 + 0.989740i \(0.545636\pi\)
\(390\) 0 0
\(391\) −4.56109 −0.230664
\(392\) 0 0
\(393\) −19.4096 −0.979085
\(394\) 0 0
\(395\) 57.1484 2.87545
\(396\) 0 0
\(397\) −29.3054 −1.47080 −0.735399 0.677634i \(-0.763005\pi\)
−0.735399 + 0.677634i \(0.763005\pi\)
\(398\) 0 0
\(399\) −8.57268 −0.429171
\(400\) 0 0
\(401\) 21.4310 1.07021 0.535106 0.844785i \(-0.320271\pi\)
0.535106 + 0.844785i \(0.320271\pi\)
\(402\) 0 0
\(403\) 5.41335 0.269658
\(404\) 0 0
\(405\) 4.21153 0.209273
\(406\) 0 0
\(407\) 7.87390 0.390295
\(408\) 0 0
\(409\) 14.9959 0.741498 0.370749 0.928733i \(-0.379101\pi\)
0.370749 + 0.928733i \(0.379101\pi\)
\(410\) 0 0
\(411\) −9.29492 −0.458484
\(412\) 0 0
\(413\) 20.4491 1.00623
\(414\) 0 0
\(415\) 25.0811 1.23118
\(416\) 0 0
\(417\) 13.0428 0.638710
\(418\) 0 0
\(419\) 27.7911 1.35769 0.678843 0.734283i \(-0.262482\pi\)
0.678843 + 0.734283i \(0.262482\pi\)
\(420\) 0 0
\(421\) 18.1443 0.884300 0.442150 0.896941i \(-0.354216\pi\)
0.442150 + 0.896941i \(0.354216\pi\)
\(422\) 0 0
\(423\) −9.35819 −0.455011
\(424\) 0 0
\(425\) 9.61481 0.466387
\(426\) 0 0
\(427\) 42.0571 2.03528
\(428\) 0 0
\(429\) −8.04538 −0.388435
\(430\) 0 0
\(431\) 11.8731 0.571905 0.285952 0.958244i \(-0.407690\pi\)
0.285952 + 0.958244i \(0.407690\pi\)
\(432\) 0 0
\(433\) 7.57113 0.363845 0.181923 0.983313i \(-0.441768\pi\)
0.181923 + 0.983313i \(0.441768\pi\)
\(434\) 0 0
\(435\) 17.8865 0.857592
\(436\) 0 0
\(437\) −10.0829 −0.482328
\(438\) 0 0
\(439\) −26.3797 −1.25904 −0.629518 0.776986i \(-0.716748\pi\)
−0.629518 + 0.776986i \(0.716748\pi\)
\(440\) 0 0
\(441\) 19.3908 0.923372
\(442\) 0 0
\(443\) 22.2140 1.05542 0.527709 0.849425i \(-0.323051\pi\)
0.527709 + 0.849425i \(0.323051\pi\)
\(444\) 0 0
\(445\) 32.9959 1.56416
\(446\) 0 0
\(447\) −5.33359 −0.252270
\(448\) 0 0
\(449\) 11.2554 0.531176 0.265588 0.964087i \(-0.414434\pi\)
0.265588 + 0.964087i \(0.414434\pi\)
\(450\) 0 0
\(451\) 4.84761 0.228265
\(452\) 0 0
\(453\) −18.2184 −0.855973
\(454\) 0 0
\(455\) 109.351 5.12644
\(456\) 0 0
\(457\) 9.97107 0.466427 0.233213 0.972426i \(-0.425076\pi\)
0.233213 + 0.972426i \(0.425076\pi\)
\(458\) 0 0
\(459\) −0.754874 −0.0352345
\(460\) 0 0
\(461\) −19.6426 −0.914847 −0.457423 0.889249i \(-0.651228\pi\)
−0.457423 + 0.889249i \(0.651228\pi\)
\(462\) 0 0
\(463\) −24.5210 −1.13959 −0.569793 0.821788i \(-0.692977\pi\)
−0.569793 + 0.821788i \(0.692977\pi\)
\(464\) 0 0
\(465\) −4.51077 −0.209182
\(466\) 0 0
\(467\) −14.0943 −0.652205 −0.326102 0.945334i \(-0.605735\pi\)
−0.326102 + 0.945334i \(0.605735\pi\)
\(468\) 0 0
\(469\) 25.6605 1.18489
\(470\) 0 0
\(471\) 9.86754 0.454672
\(472\) 0 0
\(473\) −19.9972 −0.919471
\(474\) 0 0
\(475\) 21.2548 0.975235
\(476\) 0 0
\(477\) −8.52323 −0.390252
\(478\) 0 0
\(479\) −5.44493 −0.248785 −0.124393 0.992233i \(-0.539698\pi\)
−0.124393 + 0.992233i \(0.539698\pi\)
\(480\) 0 0
\(481\) 25.0008 1.13994
\(482\) 0 0
\(483\) 31.0399 1.41236
\(484\) 0 0
\(485\) 14.4201 0.654785
\(486\) 0 0
\(487\) 30.1077 1.36431 0.682155 0.731208i \(-0.261043\pi\)
0.682155 + 0.731208i \(0.261043\pi\)
\(488\) 0 0
\(489\) −7.64582 −0.345756
\(490\) 0 0
\(491\) −2.44218 −0.110214 −0.0551070 0.998480i \(-0.517550\pi\)
−0.0551070 + 0.998480i \(0.517550\pi\)
\(492\) 0 0
\(493\) −3.20597 −0.144390
\(494\) 0 0
\(495\) 6.70395 0.301320
\(496\) 0 0
\(497\) −34.5759 −1.55094
\(498\) 0 0
\(499\) 0.782658 0.0350366 0.0175183 0.999847i \(-0.494423\pi\)
0.0175183 + 0.999847i \(0.494423\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) −9.28226 −0.413876 −0.206938 0.978354i \(-0.566350\pi\)
−0.206938 + 0.978354i \(0.566350\pi\)
\(504\) 0 0
\(505\) −73.0211 −3.24940
\(506\) 0 0
\(507\) −12.5453 −0.557157
\(508\) 0 0
\(509\) −0.573049 −0.0254000 −0.0127000 0.999919i \(-0.504043\pi\)
−0.0127000 + 0.999919i \(0.504043\pi\)
\(510\) 0 0
\(511\) −31.1527 −1.37812
\(512\) 0 0
\(513\) −1.66875 −0.0736769
\(514\) 0 0
\(515\) −43.2123 −1.90416
\(516\) 0 0
\(517\) −14.8964 −0.655145
\(518\) 0 0
\(519\) −2.13111 −0.0935454
\(520\) 0 0
\(521\) −33.0780 −1.44918 −0.724588 0.689183i \(-0.757970\pi\)
−0.724588 + 0.689183i \(0.757970\pi\)
\(522\) 0 0
\(523\) 29.7761 1.30202 0.651010 0.759069i \(-0.274346\pi\)
0.651010 + 0.759069i \(0.274346\pi\)
\(524\) 0 0
\(525\) −65.4324 −2.85570
\(526\) 0 0
\(527\) 0.808510 0.0352192
\(528\) 0 0
\(529\) 13.5079 0.587301
\(530\) 0 0
\(531\) 3.98059 0.172743
\(532\) 0 0
\(533\) 15.3919 0.666697
\(534\) 0 0
\(535\) −55.3073 −2.39114
\(536\) 0 0
\(537\) 12.2841 0.530097
\(538\) 0 0
\(539\) 30.8665 1.32951
\(540\) 0 0
\(541\) 35.6923 1.53453 0.767266 0.641329i \(-0.221616\pi\)
0.767266 + 0.641329i \(0.221616\pi\)
\(542\) 0 0
\(543\) 12.6588 0.543243
\(544\) 0 0
\(545\) −19.1189 −0.818962
\(546\) 0 0
\(547\) −0.440380 −0.0188293 −0.00941464 0.999956i \(-0.502997\pi\)
−0.00941464 + 0.999956i \(0.502997\pi\)
\(548\) 0 0
\(549\) 8.18677 0.349403
\(550\) 0 0
\(551\) −7.08721 −0.301925
\(552\) 0 0
\(553\) 69.7093 2.96434
\(554\) 0 0
\(555\) −20.8324 −0.884284
\(556\) 0 0
\(557\) −5.81205 −0.246264 −0.123132 0.992390i \(-0.539294\pi\)
−0.123132 + 0.992390i \(0.539294\pi\)
\(558\) 0 0
\(559\) −63.4940 −2.68551
\(560\) 0 0
\(561\) −1.20162 −0.0507323
\(562\) 0 0
\(563\) −4.04845 −0.170622 −0.0853109 0.996354i \(-0.527188\pi\)
−0.0853109 + 0.996354i \(0.527188\pi\)
\(564\) 0 0
\(565\) −86.2155 −3.62711
\(566\) 0 0
\(567\) 5.13720 0.215742
\(568\) 0 0
\(569\) 23.2702 0.975538 0.487769 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(570\) 0 0
\(571\) 31.8512 1.33293 0.666466 0.745536i \(-0.267806\pi\)
0.666466 + 0.745536i \(0.267806\pi\)
\(572\) 0 0
\(573\) 14.9840 0.625964
\(574\) 0 0
\(575\) −76.9591 −3.20941
\(576\) 0 0
\(577\) −12.0290 −0.500773 −0.250387 0.968146i \(-0.580558\pi\)
−0.250387 + 0.968146i \(0.580558\pi\)
\(578\) 0 0
\(579\) 19.2434 0.799730
\(580\) 0 0
\(581\) 30.5937 1.26924
\(582\) 0 0
\(583\) −13.5674 −0.561902
\(584\) 0 0
\(585\) 21.2861 0.880070
\(586\) 0 0
\(587\) −4.88626 −0.201678 −0.100839 0.994903i \(-0.532153\pi\)
−0.100839 + 0.994903i \(0.532153\pi\)
\(588\) 0 0
\(589\) 1.78731 0.0736450
\(590\) 0 0
\(591\) 3.73605 0.153680
\(592\) 0 0
\(593\) −19.0611 −0.782746 −0.391373 0.920232i \(-0.628000\pi\)
−0.391373 + 0.920232i \(0.628000\pi\)
\(594\) 0 0
\(595\) 16.3321 0.669549
\(596\) 0 0
\(597\) 17.3082 0.708376
\(598\) 0 0
\(599\) 12.2482 0.500447 0.250224 0.968188i \(-0.419496\pi\)
0.250224 + 0.968188i \(0.419496\pi\)
\(600\) 0 0
\(601\) 3.87869 0.158215 0.0791076 0.996866i \(-0.474793\pi\)
0.0791076 + 0.996866i \(0.474793\pi\)
\(602\) 0 0
\(603\) 4.99504 0.203414
\(604\) 0 0
\(605\) −35.6554 −1.44960
\(606\) 0 0
\(607\) 12.6407 0.513071 0.256535 0.966535i \(-0.417419\pi\)
0.256535 + 0.966535i \(0.417419\pi\)
\(608\) 0 0
\(609\) 21.8179 0.884104
\(610\) 0 0
\(611\) −47.2985 −1.91349
\(612\) 0 0
\(613\) −4.03966 −0.163160 −0.0815802 0.996667i \(-0.525997\pi\)
−0.0815802 + 0.996667i \(0.525997\pi\)
\(614\) 0 0
\(615\) −12.8256 −0.517176
\(616\) 0 0
\(617\) 42.8979 1.72700 0.863502 0.504345i \(-0.168266\pi\)
0.863502 + 0.504345i \(0.168266\pi\)
\(618\) 0 0
\(619\) 9.22024 0.370593 0.185296 0.982683i \(-0.440676\pi\)
0.185296 + 0.982683i \(0.440676\pi\)
\(620\) 0 0
\(621\) 6.04218 0.242464
\(622\) 0 0
\(623\) 40.2483 1.61251
\(624\) 0 0
\(625\) 73.5456 2.94182
\(626\) 0 0
\(627\) −2.65632 −0.106083
\(628\) 0 0
\(629\) 3.73399 0.148884
\(630\) 0 0
\(631\) 3.58214 0.142603 0.0713014 0.997455i \(-0.477285\pi\)
0.0713014 + 0.997455i \(0.477285\pi\)
\(632\) 0 0
\(633\) 10.3219 0.410258
\(634\) 0 0
\(635\) 74.7799 2.96755
\(636\) 0 0
\(637\) 98.0058 3.88313
\(638\) 0 0
\(639\) −6.73050 −0.266254
\(640\) 0 0
\(641\) −3.55994 −0.140609 −0.0703046 0.997526i \(-0.522397\pi\)
−0.0703046 + 0.997526i \(0.522397\pi\)
\(642\) 0 0
\(643\) 15.3558 0.605573 0.302787 0.953058i \(-0.402083\pi\)
0.302787 + 0.953058i \(0.402083\pi\)
\(644\) 0 0
\(645\) 52.9075 2.08323
\(646\) 0 0
\(647\) 25.8947 1.01803 0.509013 0.860759i \(-0.330011\pi\)
0.509013 + 0.860759i \(0.330011\pi\)
\(648\) 0 0
\(649\) 6.33635 0.248723
\(650\) 0 0
\(651\) −5.50221 −0.215649
\(652\) 0 0
\(653\) 9.96529 0.389972 0.194986 0.980806i \(-0.437534\pi\)
0.194986 + 0.980806i \(0.437534\pi\)
\(654\) 0 0
\(655\) 81.7441 3.19400
\(656\) 0 0
\(657\) −6.06414 −0.236585
\(658\) 0 0
\(659\) −23.3366 −0.909065 −0.454533 0.890730i \(-0.650194\pi\)
−0.454533 + 0.890730i \(0.650194\pi\)
\(660\) 0 0
\(661\) 21.7943 0.847702 0.423851 0.905732i \(-0.360678\pi\)
0.423851 + 0.905732i \(0.360678\pi\)
\(662\) 0 0
\(663\) −3.81531 −0.148174
\(664\) 0 0
\(665\) 36.1041 1.40006
\(666\) 0 0
\(667\) 25.6613 0.993611
\(668\) 0 0
\(669\) 17.3854 0.672159
\(670\) 0 0
\(671\) 13.0318 0.503086
\(672\) 0 0
\(673\) −24.8535 −0.958034 −0.479017 0.877806i \(-0.659007\pi\)
−0.479017 + 0.877806i \(0.659007\pi\)
\(674\) 0 0
\(675\) −12.7370 −0.490246
\(676\) 0 0
\(677\) −11.6686 −0.448459 −0.224229 0.974536i \(-0.571987\pi\)
−0.224229 + 0.974536i \(0.571987\pi\)
\(678\) 0 0
\(679\) 17.5896 0.675028
\(680\) 0 0
\(681\) 22.0068 0.843304
\(682\) 0 0
\(683\) 40.3092 1.54239 0.771195 0.636599i \(-0.219659\pi\)
0.771195 + 0.636599i \(0.219659\pi\)
\(684\) 0 0
\(685\) 39.1458 1.49568
\(686\) 0 0
\(687\) 10.2875 0.392491
\(688\) 0 0
\(689\) −43.0784 −1.64116
\(690\) 0 0
\(691\) 0.648820 0.0246823 0.0123411 0.999924i \(-0.496072\pi\)
0.0123411 + 0.999924i \(0.496072\pi\)
\(692\) 0 0
\(693\) 8.17744 0.310635
\(694\) 0 0
\(695\) −54.9303 −2.08362
\(696\) 0 0
\(697\) 2.29885 0.0870753
\(698\) 0 0
\(699\) 5.06675 0.191642
\(700\) 0 0
\(701\) −14.2448 −0.538017 −0.269009 0.963138i \(-0.586696\pi\)
−0.269009 + 0.963138i \(0.586696\pi\)
\(702\) 0 0
\(703\) 8.25446 0.311323
\(704\) 0 0
\(705\) 39.4123 1.48435
\(706\) 0 0
\(707\) −89.0707 −3.34985
\(708\) 0 0
\(709\) 33.6480 1.26368 0.631839 0.775100i \(-0.282300\pi\)
0.631839 + 0.775100i \(0.282300\pi\)
\(710\) 0 0
\(711\) 13.5695 0.508896
\(712\) 0 0
\(713\) −6.47149 −0.242359
\(714\) 0 0
\(715\) 33.8834 1.26717
\(716\) 0 0
\(717\) 2.30330 0.0860182
\(718\) 0 0
\(719\) −17.0334 −0.635239 −0.317619 0.948218i \(-0.602883\pi\)
−0.317619 + 0.948218i \(0.602883\pi\)
\(720\) 0 0
\(721\) −52.7102 −1.96303
\(722\) 0 0
\(723\) 0.625570 0.0232652
\(724\) 0 0
\(725\) −54.0943 −2.00901
\(726\) 0 0
\(727\) −7.31178 −0.271179 −0.135589 0.990765i \(-0.543293\pi\)
−0.135589 + 0.990765i \(0.543293\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −9.48313 −0.350746
\(732\) 0 0
\(733\) 17.4158 0.643267 0.321634 0.946864i \(-0.395768\pi\)
0.321634 + 0.946864i \(0.395768\pi\)
\(734\) 0 0
\(735\) −81.6650 −3.01226
\(736\) 0 0
\(737\) 7.95116 0.292885
\(738\) 0 0
\(739\) 27.3162 1.00484 0.502421 0.864623i \(-0.332443\pi\)
0.502421 + 0.864623i \(0.332443\pi\)
\(740\) 0 0
\(741\) −8.43423 −0.309839
\(742\) 0 0
\(743\) −33.4574 −1.22743 −0.613716 0.789526i \(-0.710326\pi\)
−0.613716 + 0.789526i \(0.710326\pi\)
\(744\) 0 0
\(745\) 22.4626 0.822965
\(746\) 0 0
\(747\) 5.95533 0.217894
\(748\) 0 0
\(749\) −67.4636 −2.46506
\(750\) 0 0
\(751\) 9.04026 0.329884 0.164942 0.986303i \(-0.447256\pi\)
0.164942 + 0.986303i \(0.447256\pi\)
\(752\) 0 0
\(753\) −22.4644 −0.818649
\(754\) 0 0
\(755\) 76.7271 2.79239
\(756\) 0 0
\(757\) −29.0912 −1.05734 −0.528669 0.848828i \(-0.677309\pi\)
−0.528669 + 0.848828i \(0.677309\pi\)
\(758\) 0 0
\(759\) 9.61800 0.349111
\(760\) 0 0
\(761\) 25.4421 0.922274 0.461137 0.887329i \(-0.347442\pi\)
0.461137 + 0.887329i \(0.347442\pi\)
\(762\) 0 0
\(763\) −23.3211 −0.844280
\(764\) 0 0
\(765\) 3.17917 0.114943
\(766\) 0 0
\(767\) 20.1189 0.726450
\(768\) 0 0
\(769\) −16.4143 −0.591915 −0.295958 0.955201i \(-0.595639\pi\)
−0.295958 + 0.955201i \(0.595639\pi\)
\(770\) 0 0
\(771\) 13.6816 0.492731
\(772\) 0 0
\(773\) 46.6409 1.67756 0.838778 0.544474i \(-0.183271\pi\)
0.838778 + 0.544474i \(0.183271\pi\)
\(774\) 0 0
\(775\) 13.6420 0.490034
\(776\) 0 0
\(777\) −25.4112 −0.911621
\(778\) 0 0
\(779\) 5.08191 0.182078
\(780\) 0 0
\(781\) −10.7137 −0.383365
\(782\) 0 0
\(783\) 4.24703 0.151777
\(784\) 0 0
\(785\) −41.5574 −1.48325
\(786\) 0 0
\(787\) −4.52468 −0.161287 −0.0806437 0.996743i \(-0.525698\pi\)
−0.0806437 + 0.996743i \(0.525698\pi\)
\(788\) 0 0
\(789\) 4.63785 0.165112
\(790\) 0 0
\(791\) −105.165 −3.73924
\(792\) 0 0
\(793\) 41.3779 1.46937
\(794\) 0 0
\(795\) 35.8958 1.27309
\(796\) 0 0
\(797\) −5.76977 −0.204376 −0.102188 0.994765i \(-0.532584\pi\)
−0.102188 + 0.994765i \(0.532584\pi\)
\(798\) 0 0
\(799\) −7.06425 −0.249915
\(800\) 0 0
\(801\) 7.83467 0.276824
\(802\) 0 0
\(803\) −9.65296 −0.340646
\(804\) 0 0
\(805\) −130.725 −4.60746
\(806\) 0 0
\(807\) −24.6489 −0.867681
\(808\) 0 0
\(809\) −2.89016 −0.101613 −0.0508063 0.998709i \(-0.516179\pi\)
−0.0508063 + 0.998709i \(0.516179\pi\)
\(810\) 0 0
\(811\) 23.7963 0.835600 0.417800 0.908539i \(-0.362801\pi\)
0.417800 + 0.908539i \(0.362801\pi\)
\(812\) 0 0
\(813\) 11.6032 0.406942
\(814\) 0 0
\(815\) 32.2006 1.12794
\(816\) 0 0
\(817\) −20.9637 −0.733426
\(818\) 0 0
\(819\) 25.9646 0.907277
\(820\) 0 0
\(821\) 15.4083 0.537752 0.268876 0.963175i \(-0.413348\pi\)
0.268876 + 0.963175i \(0.413348\pi\)
\(822\) 0 0
\(823\) 53.0392 1.84883 0.924416 0.381386i \(-0.124553\pi\)
0.924416 + 0.381386i \(0.124553\pi\)
\(824\) 0 0
\(825\) −20.2748 −0.705879
\(826\) 0 0
\(827\) 37.1164 1.29066 0.645332 0.763902i \(-0.276719\pi\)
0.645332 + 0.763902i \(0.276719\pi\)
\(828\) 0 0
\(829\) −33.9601 −1.17948 −0.589742 0.807592i \(-0.700771\pi\)
−0.589742 + 0.807592i \(0.700771\pi\)
\(830\) 0 0
\(831\) 0.516066 0.0179021
\(832\) 0 0
\(833\) 14.6376 0.507164
\(834\) 0 0
\(835\) −4.21153 −0.145746
\(836\) 0 0
\(837\) −1.07105 −0.0370210
\(838\) 0 0
\(839\) −22.1451 −0.764533 −0.382267 0.924052i \(-0.624856\pi\)
−0.382267 + 0.924052i \(0.624856\pi\)
\(840\) 0 0
\(841\) −10.9627 −0.378025
\(842\) 0 0
\(843\) 7.67914 0.264484
\(844\) 0 0
\(845\) 52.8349 1.81758
\(846\) 0 0
\(847\) −43.4923 −1.49441
\(848\) 0 0
\(849\) −15.5410 −0.533367
\(850\) 0 0
\(851\) −29.8877 −1.02454
\(852\) 0 0
\(853\) 22.8091 0.780970 0.390485 0.920609i \(-0.372307\pi\)
0.390485 + 0.920609i \(0.372307\pi\)
\(854\) 0 0
\(855\) 7.02797 0.240351
\(856\) 0 0
\(857\) 53.9748 1.84374 0.921872 0.387493i \(-0.126659\pi\)
0.921872 + 0.387493i \(0.126659\pi\)
\(858\) 0 0
\(859\) −36.9731 −1.26151 −0.630754 0.775983i \(-0.717254\pi\)
−0.630754 + 0.775983i \(0.717254\pi\)
\(860\) 0 0
\(861\) −15.6445 −0.533165
\(862\) 0 0
\(863\) −20.2798 −0.690331 −0.345165 0.938542i \(-0.612177\pi\)
−0.345165 + 0.938542i \(0.612177\pi\)
\(864\) 0 0
\(865\) 8.97524 0.305167
\(866\) 0 0
\(867\) 16.4302 0.557998
\(868\) 0 0
\(869\) 21.6001 0.732732
\(870\) 0 0
\(871\) 25.2461 0.855432
\(872\) 0 0
\(873\) 3.42397 0.115884
\(874\) 0 0
\(875\) 167.393 5.65891
\(876\) 0 0
\(877\) 38.8159 1.31072 0.655360 0.755317i \(-0.272517\pi\)
0.655360 + 0.755317i \(0.272517\pi\)
\(878\) 0 0
\(879\) −7.27759 −0.245467
\(880\) 0 0
\(881\) −25.0400 −0.843618 −0.421809 0.906685i \(-0.638605\pi\)
−0.421809 + 0.906685i \(0.638605\pi\)
\(882\) 0 0
\(883\) 9.66634 0.325298 0.162649 0.986684i \(-0.447996\pi\)
0.162649 + 0.986684i \(0.447996\pi\)
\(884\) 0 0
\(885\) −16.7644 −0.563529
\(886\) 0 0
\(887\) 22.8139 0.766015 0.383008 0.923745i \(-0.374888\pi\)
0.383008 + 0.923745i \(0.374888\pi\)
\(888\) 0 0
\(889\) 91.2162 3.05929
\(890\) 0 0
\(891\) 1.59181 0.0533276
\(892\) 0 0
\(893\) −15.6164 −0.522584
\(894\) 0 0
\(895\) −51.7347 −1.72930
\(896\) 0 0
\(897\) 30.5386 1.01965
\(898\) 0 0
\(899\) −4.54879 −0.151711
\(900\) 0 0
\(901\) −6.43396 −0.214346
\(902\) 0 0
\(903\) 64.5363 2.14763
\(904\) 0 0
\(905\) −53.3130 −1.77219
\(906\) 0 0
\(907\) −43.6100 −1.44805 −0.724023 0.689775i \(-0.757709\pi\)
−0.724023 + 0.689775i \(0.757709\pi\)
\(908\) 0 0
\(909\) −17.3384 −0.575078
\(910\) 0 0
\(911\) −47.3364 −1.56832 −0.784162 0.620556i \(-0.786907\pi\)
−0.784162 + 0.620556i \(0.786907\pi\)
\(912\) 0 0
\(913\) 9.47976 0.313734
\(914\) 0 0
\(915\) −34.4788 −1.13983
\(916\) 0 0
\(917\) 99.7110 3.29275
\(918\) 0 0
\(919\) −3.69738 −0.121965 −0.0609826 0.998139i \(-0.519423\pi\)
−0.0609826 + 0.998139i \(0.519423\pi\)
\(920\) 0 0
\(921\) −0.121465 −0.00400241
\(922\) 0 0
\(923\) −34.0175 −1.11970
\(924\) 0 0
\(925\) 63.0035 2.07154
\(926\) 0 0
\(927\) −10.2605 −0.336999
\(928\) 0 0
\(929\) −19.5929 −0.642821 −0.321410 0.946940i \(-0.604157\pi\)
−0.321410 + 0.946940i \(0.604157\pi\)
\(930\) 0 0
\(931\) 32.3583 1.06050
\(932\) 0 0
\(933\) −16.6650 −0.545587
\(934\) 0 0
\(935\) 5.06064 0.165501
\(936\) 0 0
\(937\) −46.2138 −1.50974 −0.754869 0.655875i \(-0.772300\pi\)
−0.754869 + 0.655875i \(0.772300\pi\)
\(938\) 0 0
\(939\) 33.1709 1.08249
\(940\) 0 0
\(941\) −16.1732 −0.527231 −0.263616 0.964628i \(-0.584915\pi\)
−0.263616 + 0.964628i \(0.584915\pi\)
\(942\) 0 0
\(943\) −18.4005 −0.599204
\(944\) 0 0
\(945\) −21.6355 −0.703802
\(946\) 0 0
\(947\) 19.7305 0.641154 0.320577 0.947223i \(-0.396123\pi\)
0.320577 + 0.947223i \(0.396123\pi\)
\(948\) 0 0
\(949\) −30.6496 −0.994929
\(950\) 0 0
\(951\) −23.8173 −0.772329
\(952\) 0 0
\(953\) 44.2858 1.43456 0.717279 0.696786i \(-0.245387\pi\)
0.717279 + 0.696786i \(0.245387\pi\)
\(954\) 0 0
\(955\) −63.1054 −2.04204
\(956\) 0 0
\(957\) 6.76047 0.218535
\(958\) 0 0
\(959\) 47.7498 1.54192
\(960\) 0 0
\(961\) −29.8528 −0.962995
\(962\) 0 0
\(963\) −13.1324 −0.423184
\(964\) 0 0
\(965\) −81.0443 −2.60891
\(966\) 0 0
\(967\) 19.9140 0.640391 0.320195 0.947352i \(-0.396251\pi\)
0.320195 + 0.947352i \(0.396251\pi\)
\(968\) 0 0
\(969\) −1.25969 −0.0404672
\(970\) 0 0
\(971\) 30.1118 0.966334 0.483167 0.875528i \(-0.339486\pi\)
0.483167 + 0.875528i \(0.339486\pi\)
\(972\) 0 0
\(973\) −67.0037 −2.14804
\(974\) 0 0
\(975\) −64.3757 −2.06167
\(976\) 0 0
\(977\) 10.6796 0.341672 0.170836 0.985299i \(-0.445353\pi\)
0.170836 + 0.985299i \(0.445353\pi\)
\(978\) 0 0
\(979\) 12.4713 0.398585
\(980\) 0 0
\(981\) −4.53965 −0.144940
\(982\) 0 0
\(983\) 43.7358 1.39496 0.697478 0.716606i \(-0.254305\pi\)
0.697478 + 0.716606i \(0.254305\pi\)
\(984\) 0 0
\(985\) −15.7345 −0.501342
\(986\) 0 0
\(987\) 48.0749 1.53024
\(988\) 0 0
\(989\) 75.9051 2.41364
\(990\) 0 0
\(991\) 19.5304 0.620402 0.310201 0.950671i \(-0.399604\pi\)
0.310201 + 0.950671i \(0.399604\pi\)
\(992\) 0 0
\(993\) 1.84876 0.0586685
\(994\) 0 0
\(995\) −72.8938 −2.31089
\(996\) 0 0
\(997\) −5.72889 −0.181436 −0.0907179 0.995877i \(-0.528916\pi\)
−0.0907179 + 0.995877i \(0.528916\pi\)
\(998\) 0 0
\(999\) −4.94651 −0.156501
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.bb.1.9 9
4.3 odd 2 2004.2.a.d.1.9 9
12.11 even 2 6012.2.a.h.1.1 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2004.2.a.d.1.9 9 4.3 odd 2
6012.2.a.h.1.1 9 12.11 even 2
8016.2.a.bb.1.9 9 1.1 even 1 trivial