Properties

Label 6384.2.a.cb.1.2
Level $6384$
Weight $2$
Character 6384.1
Self dual yes
Analytic conductor $50.976$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6384.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(50.9764966504\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.9248.1
Defining polynomial: \( x^{4} - 5x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3192)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.662153\) of defining polynomial
Character \(\chi\) \(=\) 6384.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.69614 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.69614 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.32431 q^{11} -1.69614 q^{15} +6.14355 q^{17} +1.00000 q^{19} +1.00000 q^{21} -2.91779 q^{23} -2.12311 q^{25} +1.00000 q^{27} -1.42696 q^{29} +3.32431 q^{31} +1.32431 q^{33} -1.69614 q^{35} +9.83969 q^{37} -8.04090 q^{41} -7.16400 q^{43} -1.69614 q^{45} +7.66906 q^{47} +1.00000 q^{49} +6.14355 q^{51} +8.81925 q^{53} -2.24621 q^{55} +1.00000 q^{57} -9.43318 q^{59} +4.64861 q^{61} +1.00000 q^{63} +11.5705 q^{67} -2.91779 q^{69} +3.22165 q^{71} -8.04090 q^{73} -2.12311 q^{75} +1.32431 q^{77} +10.3101 q^{79} +1.00000 q^{81} +4.77835 q^{83} -10.4203 q^{85} -1.42696 q^{87} -11.6385 q^{89} +3.32431 q^{93} -1.69614 q^{95} -6.71659 q^{97} +1.32431 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{7} + 4 q^{9} - 4 q^{17} + 4 q^{19} + 4 q^{21} - 4 q^{23} + 8 q^{25} + 4 q^{27} + 4 q^{29} + 8 q^{31} + 4 q^{37} - 8 q^{41} + 12 q^{43} + 8 q^{47} + 4 q^{49} - 4 q^{51} + 12 q^{53} + 24 q^{55} + 4 q^{57} + 8 q^{61} + 4 q^{63} + 8 q^{67} - 4 q^{69} + 12 q^{71} - 8 q^{73} + 8 q^{75} + 20 q^{79} + 4 q^{81} + 20 q^{83} - 4 q^{85} + 4 q^{87} + 8 q^{93} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −1.69614 −0.758537 −0.379269 0.925287i \(-0.623824\pi\)
−0.379269 + 0.925287i \(0.623824\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.32431 0.399294 0.199647 0.979868i \(-0.436021\pi\)
0.199647 + 0.979868i \(0.436021\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −1.69614 −0.437942
\(16\) 0 0
\(17\) 6.14355 1.49003 0.745015 0.667047i \(-0.232442\pi\)
0.745015 + 0.667047i \(0.232442\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −2.91779 −0.608401 −0.304201 0.952608i \(-0.598389\pi\)
−0.304201 + 0.952608i \(0.598389\pi\)
\(24\) 0 0
\(25\) −2.12311 −0.424621
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.42696 −0.264981 −0.132490 0.991184i \(-0.542297\pi\)
−0.132490 + 0.991184i \(0.542297\pi\)
\(30\) 0 0
\(31\) 3.32431 0.597063 0.298532 0.954400i \(-0.403503\pi\)
0.298532 + 0.954400i \(0.403503\pi\)
\(32\) 0 0
\(33\) 1.32431 0.230532
\(34\) 0 0
\(35\) −1.69614 −0.286700
\(36\) 0 0
\(37\) 9.83969 1.61764 0.808818 0.588059i \(-0.200108\pi\)
0.808818 + 0.588059i \(0.200108\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.04090 −1.25578 −0.627888 0.778303i \(-0.716081\pi\)
−0.627888 + 0.778303i \(0.716081\pi\)
\(42\) 0 0
\(43\) −7.16400 −1.09250 −0.546250 0.837622i \(-0.683945\pi\)
−0.546250 + 0.837622i \(0.683945\pi\)
\(44\) 0 0
\(45\) −1.69614 −0.252846
\(46\) 0 0
\(47\) 7.66906 1.11865 0.559324 0.828949i \(-0.311061\pi\)
0.559324 + 0.828949i \(0.311061\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.14355 0.860270
\(52\) 0 0
\(53\) 8.81925 1.21142 0.605708 0.795687i \(-0.292890\pi\)
0.605708 + 0.795687i \(0.292890\pi\)
\(54\) 0 0
\(55\) −2.24621 −0.302879
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −9.43318 −1.22810 −0.614048 0.789269i \(-0.710460\pi\)
−0.614048 + 0.789269i \(0.710460\pi\)
\(60\) 0 0
\(61\) 4.64861 0.595194 0.297597 0.954692i \(-0.403815\pi\)
0.297597 + 0.954692i \(0.403815\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.5705 1.41356 0.706782 0.707432i \(-0.250146\pi\)
0.706782 + 0.707432i \(0.250146\pi\)
\(68\) 0 0
\(69\) −2.91779 −0.351261
\(70\) 0 0
\(71\) 3.22165 0.382339 0.191170 0.981557i \(-0.438772\pi\)
0.191170 + 0.981557i \(0.438772\pi\)
\(72\) 0 0
\(73\) −8.04090 −0.941116 −0.470558 0.882369i \(-0.655947\pi\)
−0.470558 + 0.882369i \(0.655947\pi\)
\(74\) 0 0
\(75\) −2.12311 −0.245155
\(76\) 0 0
\(77\) 1.32431 0.150919
\(78\) 0 0
\(79\) 10.3101 1.15997 0.579987 0.814626i \(-0.303058\pi\)
0.579987 + 0.814626i \(0.303058\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.77835 0.524492 0.262246 0.965001i \(-0.415537\pi\)
0.262246 + 0.965001i \(0.415537\pi\)
\(84\) 0 0
\(85\) −10.4203 −1.13024
\(86\) 0 0
\(87\) −1.42696 −0.152987
\(88\) 0 0
\(89\) −11.6385 −1.23368 −0.616839 0.787089i \(-0.711587\pi\)
−0.616839 + 0.787089i \(0.711587\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.32431 0.344715
\(94\) 0 0
\(95\) −1.69614 −0.174020
\(96\) 0 0
\(97\) −6.71659 −0.681966 −0.340983 0.940069i \(-0.610760\pi\)
−0.340983 + 0.940069i \(0.610760\pi\)
\(98\) 0 0
\(99\) 1.32431 0.133098
\(100\) 0 0
\(101\) 13.7779 1.37096 0.685478 0.728094i \(-0.259593\pi\)
0.685478 + 0.728094i \(0.259593\pi\)
\(102\) 0 0
\(103\) 13.1911 1.29976 0.649878 0.760039i \(-0.274820\pi\)
0.649878 + 0.760039i \(0.274820\pi\)
\(104\) 0 0
\(105\) −1.69614 −0.165526
\(106\) 0 0
\(107\) −5.46786 −0.528598 −0.264299 0.964441i \(-0.585141\pi\)
−0.264299 + 0.964441i \(0.585141\pi\)
\(108\) 0 0
\(109\) −14.5292 −1.39165 −0.695823 0.718214i \(-0.744960\pi\)
−0.695823 + 0.718214i \(0.744960\pi\)
\(110\) 0 0
\(111\) 9.83969 0.933942
\(112\) 0 0
\(113\) −2.77835 −0.261365 −0.130683 0.991424i \(-0.541717\pi\)
−0.130683 + 0.991424i \(0.541717\pi\)
\(114\) 0 0
\(115\) 4.94898 0.461495
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.14355 0.563179
\(120\) 0 0
\(121\) −9.24621 −0.840565
\(122\) 0 0
\(123\) −8.04090 −0.725023
\(124\) 0 0
\(125\) 12.0818 1.08063
\(126\) 0 0
\(127\) −1.16400 −0.103288 −0.0516442 0.998666i \(-0.516446\pi\)
−0.0516442 + 0.998666i \(0.516446\pi\)
\(128\) 0 0
\(129\) −7.16400 −0.630755
\(130\) 0 0
\(131\) 0.573035 0.0500663 0.0250332 0.999687i \(-0.492031\pi\)
0.0250332 + 0.999687i \(0.492031\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) −1.69614 −0.145981
\(136\) 0 0
\(137\) −3.83558 −0.327696 −0.163848 0.986486i \(-0.552391\pi\)
−0.163848 + 0.986486i \(0.552391\pi\)
\(138\) 0 0
\(139\) 20.8255 1.76639 0.883196 0.469004i \(-0.155387\pi\)
0.883196 + 0.469004i \(0.155387\pi\)
\(140\) 0 0
\(141\) 7.66906 0.645852
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2.42033 0.200998
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −1.63438 −0.133894 −0.0669468 0.997757i \(-0.521326\pi\)
−0.0669468 + 0.997757i \(0.521326\pi\)
\(150\) 0 0
\(151\) 5.08221 0.413584 0.206792 0.978385i \(-0.433698\pi\)
0.206792 + 0.978385i \(0.433698\pi\)
\(152\) 0 0
\(153\) 6.14355 0.496677
\(154\) 0 0
\(155\) −5.63849 −0.452895
\(156\) 0 0
\(157\) 4.24621 0.338885 0.169442 0.985540i \(-0.445803\pi\)
0.169442 + 0.985540i \(0.445803\pi\)
\(158\) 0 0
\(159\) 8.81925 0.699412
\(160\) 0 0
\(161\) −2.91779 −0.229954
\(162\) 0 0
\(163\) 21.9486 1.71914 0.859572 0.511014i \(-0.170730\pi\)
0.859572 + 0.511014i \(0.170730\pi\)
\(164\) 0 0
\(165\) −2.24621 −0.174867
\(166\) 0 0
\(167\) 8.20532 0.634946 0.317473 0.948267i \(-0.397166\pi\)
0.317473 + 0.948267i \(0.397166\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 1.39228 0.105853 0.0529266 0.998598i \(-0.483145\pi\)
0.0529266 + 0.998598i \(0.483145\pi\)
\(174\) 0 0
\(175\) −2.12311 −0.160492
\(176\) 0 0
\(177\) −9.43318 −0.709041
\(178\) 0 0
\(179\) −19.1064 −1.42808 −0.714038 0.700107i \(-0.753136\pi\)
−0.714038 + 0.700107i \(0.753136\pi\)
\(180\) 0 0
\(181\) 19.9895 1.48580 0.742902 0.669400i \(-0.233449\pi\)
0.742902 + 0.669400i \(0.233449\pi\)
\(182\) 0 0
\(183\) 4.64861 0.343635
\(184\) 0 0
\(185\) −16.6895 −1.22704
\(186\) 0 0
\(187\) 8.13595 0.594960
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 5.70235 0.412608 0.206304 0.978488i \(-0.433856\pi\)
0.206304 + 0.978488i \(0.433856\pi\)
\(192\) 0 0
\(193\) 22.9766 1.65389 0.826947 0.562281i \(-0.190076\pi\)
0.826947 + 0.562281i \(0.190076\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.24210 −0.444731 −0.222366 0.974963i \(-0.571378\pi\)
−0.222366 + 0.974963i \(0.571378\pi\)
\(198\) 0 0
\(199\) −17.7203 −1.25616 −0.628079 0.778150i \(-0.716158\pi\)
−0.628079 + 0.778150i \(0.716158\pi\)
\(200\) 0 0
\(201\) 11.5705 0.816121
\(202\) 0 0
\(203\) −1.42696 −0.100153
\(204\) 0 0
\(205\) 13.6385 0.952554
\(206\) 0 0
\(207\) −2.91779 −0.202800
\(208\) 0 0
\(209\) 1.32431 0.0916042
\(210\) 0 0
\(211\) 14.8268 1.02072 0.510361 0.859960i \(-0.329512\pi\)
0.510361 + 0.859960i \(0.329512\pi\)
\(212\) 0 0
\(213\) 3.22165 0.220744
\(214\) 0 0
\(215\) 12.1512 0.828702
\(216\) 0 0
\(217\) 3.32431 0.225669
\(218\) 0 0
\(219\) −8.04090 −0.543353
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 18.9219 1.26710 0.633552 0.773700i \(-0.281596\pi\)
0.633552 + 0.773700i \(0.281596\pi\)
\(224\) 0 0
\(225\) −2.12311 −0.141540
\(226\) 0 0
\(227\) 11.3381 0.752538 0.376269 0.926511i \(-0.377207\pi\)
0.376269 + 0.926511i \(0.377207\pi\)
\(228\) 0 0
\(229\) −15.9867 −1.05643 −0.528217 0.849110i \(-0.677139\pi\)
−0.528217 + 0.849110i \(0.677139\pi\)
\(230\) 0 0
\(231\) 1.32431 0.0871330
\(232\) 0 0
\(233\) 9.87648 0.647029 0.323515 0.946223i \(-0.395135\pi\)
0.323515 + 0.946223i \(0.395135\pi\)
\(234\) 0 0
\(235\) −13.0078 −0.848536
\(236\) 0 0
\(237\) 10.3101 0.669711
\(238\) 0 0
\(239\) 11.6155 0.751346 0.375673 0.926752i \(-0.377412\pi\)
0.375673 + 0.926752i \(0.377412\pi\)
\(240\) 0 0
\(241\) −14.3142 −0.922058 −0.461029 0.887385i \(-0.652520\pi\)
−0.461029 + 0.887385i \(0.652520\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −1.69614 −0.108362
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 4.77835 0.302816
\(250\) 0 0
\(251\) 25.0779 1.58290 0.791451 0.611233i \(-0.209326\pi\)
0.791451 + 0.611233i \(0.209326\pi\)
\(252\) 0 0
\(253\) −3.86405 −0.242931
\(254\) 0 0
\(255\) −10.4203 −0.652547
\(256\) 0 0
\(257\) −28.7304 −1.79215 −0.896077 0.443899i \(-0.853595\pi\)
−0.896077 + 0.443899i \(0.853595\pi\)
\(258\) 0 0
\(259\) 9.83969 0.611409
\(260\) 0 0
\(261\) −1.42696 −0.0883269
\(262\) 0 0
\(263\) 5.09464 0.314149 0.157074 0.987587i \(-0.449794\pi\)
0.157074 + 0.987587i \(0.449794\pi\)
\(264\) 0 0
\(265\) −14.9587 −0.918905
\(266\) 0 0
\(267\) −11.6385 −0.712264
\(268\) 0 0
\(269\) −13.1410 −0.801223 −0.400612 0.916248i \(-0.631202\pi\)
−0.400612 + 0.916248i \(0.631202\pi\)
\(270\) 0 0
\(271\) 13.2972 0.807749 0.403875 0.914814i \(-0.367663\pi\)
0.403875 + 0.914814i \(0.367663\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.81164 −0.169548
\(276\) 0 0
\(277\) 23.2049 1.39425 0.697124 0.716951i \(-0.254463\pi\)
0.697124 + 0.716951i \(0.254463\pi\)
\(278\) 0 0
\(279\) 3.32431 0.199021
\(280\) 0 0
\(281\) 26.4036 1.57511 0.787553 0.616247i \(-0.211348\pi\)
0.787553 + 0.616247i \(0.211348\pi\)
\(282\) 0 0
\(283\) 28.7795 1.71077 0.855383 0.517997i \(-0.173322\pi\)
0.855383 + 0.517997i \(0.173322\pi\)
\(284\) 0 0
\(285\) −1.69614 −0.100471
\(286\) 0 0
\(287\) −8.04090 −0.474639
\(288\) 0 0
\(289\) 20.7433 1.22019
\(290\) 0 0
\(291\) −6.71659 −0.393733
\(292\) 0 0
\(293\) −26.3689 −1.54049 −0.770244 0.637750i \(-0.779865\pi\)
−0.770244 + 0.637750i \(0.779865\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) 0 0
\(297\) 1.32431 0.0768441
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −7.16400 −0.412926
\(302\) 0 0
\(303\) 13.7779 0.791522
\(304\) 0 0
\(305\) −7.88470 −0.451477
\(306\) 0 0
\(307\) 9.73082 0.555367 0.277684 0.960673i \(-0.410433\pi\)
0.277684 + 0.960673i \(0.410433\pi\)
\(308\) 0 0
\(309\) 13.1911 0.750414
\(310\) 0 0
\(311\) −27.4844 −1.55850 −0.779249 0.626715i \(-0.784399\pi\)
−0.779249 + 0.626715i \(0.784399\pi\)
\(312\) 0 0
\(313\) 22.9766 1.29872 0.649358 0.760483i \(-0.275038\pi\)
0.649358 + 0.760483i \(0.275038\pi\)
\(314\) 0 0
\(315\) −1.69614 −0.0955667
\(316\) 0 0
\(317\) 30.1983 1.69610 0.848052 0.529913i \(-0.177776\pi\)
0.848052 + 0.529913i \(0.177776\pi\)
\(318\) 0 0
\(319\) −1.88974 −0.105805
\(320\) 0 0
\(321\) −5.46786 −0.305186
\(322\) 0 0
\(323\) 6.14355 0.341836
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −14.5292 −0.803467
\(328\) 0 0
\(329\) 7.66906 0.422809
\(330\) 0 0
\(331\) 24.3266 1.33711 0.668556 0.743662i \(-0.266913\pi\)
0.668556 + 0.743662i \(0.266913\pi\)
\(332\) 0 0
\(333\) 9.83969 0.539212
\(334\) 0 0
\(335\) −19.6252 −1.07224
\(336\) 0 0
\(337\) 14.3689 0.782724 0.391362 0.920237i \(-0.372004\pi\)
0.391362 + 0.920237i \(0.372004\pi\)
\(338\) 0 0
\(339\) −2.77835 −0.150899
\(340\) 0 0
\(341\) 4.40240 0.238403
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 4.94898 0.266444
\(346\) 0 0
\(347\) −18.0138 −0.967032 −0.483516 0.875335i \(-0.660641\pi\)
−0.483516 + 0.875335i \(0.660641\pi\)
\(348\) 0 0
\(349\) −10.4924 −0.561646 −0.280823 0.959760i \(-0.590607\pi\)
−0.280823 + 0.959760i \(0.590607\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.6562 −0.673622 −0.336811 0.941572i \(-0.609348\pi\)
−0.336811 + 0.941572i \(0.609348\pi\)
\(354\) 0 0
\(355\) −5.46437 −0.290019
\(356\) 0 0
\(357\) 6.14355 0.325151
\(358\) 0 0
\(359\) −36.3050 −1.91611 −0.958053 0.286590i \(-0.907478\pi\)
−0.958053 + 0.286590i \(0.907478\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −9.24621 −0.485300
\(364\) 0 0
\(365\) 13.6385 0.713871
\(366\) 0 0
\(367\) 23.0634 1.20390 0.601951 0.798533i \(-0.294390\pi\)
0.601951 + 0.798533i \(0.294390\pi\)
\(368\) 0 0
\(369\) −8.04090 −0.418592
\(370\) 0 0
\(371\) 8.81925 0.457872
\(372\) 0 0
\(373\) −15.1369 −0.783760 −0.391880 0.920016i \(-0.628175\pi\)
−0.391880 + 0.920016i \(0.628175\pi\)
\(374\) 0 0
\(375\) 12.0818 0.623901
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 4.42948 0.227527 0.113764 0.993508i \(-0.463709\pi\)
0.113764 + 0.993508i \(0.463709\pi\)
\(380\) 0 0
\(381\) −1.16400 −0.0596336
\(382\) 0 0
\(383\) −5.98674 −0.305908 −0.152954 0.988233i \(-0.548879\pi\)
−0.152954 + 0.988233i \(0.548879\pi\)
\(384\) 0 0
\(385\) −2.24621 −0.114478
\(386\) 0 0
\(387\) −7.16400 −0.364167
\(388\) 0 0
\(389\) 26.9858 1.36823 0.684116 0.729373i \(-0.260188\pi\)
0.684116 + 0.729373i \(0.260188\pi\)
\(390\) 0 0
\(391\) −17.9256 −0.906537
\(392\) 0 0
\(393\) 0.573035 0.0289058
\(394\) 0 0
\(395\) −17.4873 −0.879883
\(396\) 0 0
\(397\) −14.1360 −0.709463 −0.354732 0.934968i \(-0.615428\pi\)
−0.354732 + 0.934968i \(0.615428\pi\)
\(398\) 0 0
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) 4.33505 0.216482 0.108241 0.994125i \(-0.465478\pi\)
0.108241 + 0.994125i \(0.465478\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.69614 −0.0842819
\(406\) 0 0
\(407\) 13.0308 0.645912
\(408\) 0 0
\(409\) 19.5191 0.965157 0.482578 0.875853i \(-0.339700\pi\)
0.482578 + 0.875853i \(0.339700\pi\)
\(410\) 0 0
\(411\) −3.83558 −0.189195
\(412\) 0 0
\(413\) −9.43318 −0.464176
\(414\) 0 0
\(415\) −8.10476 −0.397847
\(416\) 0 0
\(417\) 20.8255 1.01983
\(418\) 0 0
\(419\) 10.9552 0.535196 0.267598 0.963531i \(-0.413770\pi\)
0.267598 + 0.963531i \(0.413770\pi\)
\(420\) 0 0
\(421\) −1.49843 −0.0730290 −0.0365145 0.999333i \(-0.511626\pi\)
−0.0365145 + 0.999333i \(0.511626\pi\)
\(422\) 0 0
\(423\) 7.66906 0.372883
\(424\) 0 0
\(425\) −13.0434 −0.632698
\(426\) 0 0
\(427\) 4.64861 0.224962
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.64471 0.368233 0.184116 0.982904i \(-0.441058\pi\)
0.184116 + 0.982904i \(0.441058\pi\)
\(432\) 0 0
\(433\) −26.3418 −1.26591 −0.632954 0.774190i \(-0.718158\pi\)
−0.632954 + 0.774190i \(0.718158\pi\)
\(434\) 0 0
\(435\) 2.42033 0.116046
\(436\) 0 0
\(437\) −2.91779 −0.139577
\(438\) 0 0
\(439\) −5.29584 −0.252757 −0.126378 0.991982i \(-0.540335\pi\)
−0.126378 + 0.991982i \(0.540335\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 11.9118 0.565946 0.282973 0.959128i \(-0.408679\pi\)
0.282973 + 0.959128i \(0.408679\pi\)
\(444\) 0 0
\(445\) 19.7405 0.935791
\(446\) 0 0
\(447\) −1.63438 −0.0773035
\(448\) 0 0
\(449\) −19.8091 −0.934850 −0.467425 0.884033i \(-0.654818\pi\)
−0.467425 + 0.884033i \(0.654818\pi\)
\(450\) 0 0
\(451\) −10.6486 −0.501424
\(452\) 0 0
\(453\) 5.08221 0.238783
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 17.9077 0.837685 0.418843 0.908059i \(-0.362436\pi\)
0.418843 + 0.908059i \(0.362436\pi\)
\(458\) 0 0
\(459\) 6.14355 0.286757
\(460\) 0 0
\(461\) −1.49083 −0.0694347 −0.0347173 0.999397i \(-0.511053\pi\)
−0.0347173 + 0.999397i \(0.511053\pi\)
\(462\) 0 0
\(463\) 18.7028 0.869192 0.434596 0.900626i \(-0.356891\pi\)
0.434596 + 0.900626i \(0.356891\pi\)
\(464\) 0 0
\(465\) −5.63849 −0.261479
\(466\) 0 0
\(467\) −22.1574 −1.02532 −0.512660 0.858591i \(-0.671340\pi\)
−0.512660 + 0.858591i \(0.671340\pi\)
\(468\) 0 0
\(469\) 11.5705 0.534277
\(470\) 0 0
\(471\) 4.24621 0.195655
\(472\) 0 0
\(473\) −9.48734 −0.436228
\(474\) 0 0
\(475\) −2.12311 −0.0974148
\(476\) 0 0
\(477\) 8.81925 0.403806
\(478\) 0 0
\(479\) −16.9387 −0.773947 −0.386973 0.922091i \(-0.626480\pi\)
−0.386973 + 0.922091i \(0.626480\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −2.91779 −0.132764
\(484\) 0 0
\(485\) 11.3923 0.517297
\(486\) 0 0
\(487\) −10.4460 −0.473354 −0.236677 0.971588i \(-0.576058\pi\)
−0.236677 + 0.971588i \(0.576058\pi\)
\(488\) 0 0
\(489\) 21.9486 0.992548
\(490\) 0 0
\(491\) 5.66557 0.255684 0.127842 0.991795i \(-0.459195\pi\)
0.127842 + 0.991795i \(0.459195\pi\)
\(492\) 0 0
\(493\) −8.76663 −0.394829
\(494\) 0 0
\(495\) −2.24621 −0.100960
\(496\) 0 0
\(497\) 3.22165 0.144511
\(498\) 0 0
\(499\) −29.8438 −1.33599 −0.667996 0.744165i \(-0.732848\pi\)
−0.667996 + 0.744165i \(0.732848\pi\)
\(500\) 0 0
\(501\) 8.20532 0.366586
\(502\) 0 0
\(503\) −9.02868 −0.402569 −0.201284 0.979533i \(-0.564512\pi\)
−0.201284 + 0.979533i \(0.564512\pi\)
\(504\) 0 0
\(505\) −23.3693 −1.03992
\(506\) 0 0
\(507\) −13.0000 −0.577350
\(508\) 0 0
\(509\) 23.9256 1.06048 0.530242 0.847846i \(-0.322101\pi\)
0.530242 + 0.847846i \(0.322101\pi\)
\(510\) 0 0
\(511\) −8.04090 −0.355708
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) −22.3739 −0.985913
\(516\) 0 0
\(517\) 10.1562 0.446669
\(518\) 0 0
\(519\) 1.39228 0.0611144
\(520\) 0 0
\(521\) 26.5742 1.16424 0.582119 0.813104i \(-0.302224\pi\)
0.582119 + 0.813104i \(0.302224\pi\)
\(522\) 0 0
\(523\) 9.73082 0.425499 0.212750 0.977107i \(-0.431758\pi\)
0.212750 + 0.977107i \(0.431758\pi\)
\(524\) 0 0
\(525\) −2.12311 −0.0926599
\(526\) 0 0
\(527\) 20.4231 0.889642
\(528\) 0 0
\(529\) −14.4865 −0.629848
\(530\) 0 0
\(531\) −9.43318 −0.409365
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 9.27426 0.400961
\(536\) 0 0
\(537\) −19.1064 −0.824500
\(538\) 0 0
\(539\) 1.32431 0.0570419
\(540\) 0 0
\(541\) −2.49242 −0.107158 −0.0535788 0.998564i \(-0.517063\pi\)
−0.0535788 + 0.998564i \(0.517063\pi\)
\(542\) 0 0
\(543\) 19.9895 0.857830
\(544\) 0 0
\(545\) 24.6436 1.05561
\(546\) 0 0
\(547\) 5.12722 0.219224 0.109612 0.993974i \(-0.465039\pi\)
0.109612 + 0.993974i \(0.465039\pi\)
\(548\) 0 0
\(549\) 4.64861 0.198398
\(550\) 0 0
\(551\) −1.42696 −0.0607907
\(552\) 0 0
\(553\) 10.3101 0.438429
\(554\) 0 0
\(555\) −16.6895 −0.708430
\(556\) 0 0
\(557\) −7.96238 −0.337377 −0.168688 0.985669i \(-0.553953\pi\)
−0.168688 + 0.985669i \(0.553953\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 8.13595 0.343500
\(562\) 0 0
\(563\) −18.9899 −0.800328 −0.400164 0.916444i \(-0.631047\pi\)
−0.400164 + 0.916444i \(0.631047\pi\)
\(564\) 0 0
\(565\) 4.71247 0.198255
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −9.71407 −0.407235 −0.203618 0.979051i \(-0.565270\pi\)
−0.203618 + 0.979051i \(0.565270\pi\)
\(570\) 0 0
\(571\) −17.7079 −0.741051 −0.370525 0.928822i \(-0.620822\pi\)
−0.370525 + 0.928822i \(0.620822\pi\)
\(572\) 0 0
\(573\) 5.70235 0.238219
\(574\) 0 0
\(575\) 6.19478 0.258340
\(576\) 0 0
\(577\) −38.1636 −1.58877 −0.794385 0.607414i \(-0.792207\pi\)
−0.794385 + 0.607414i \(0.792207\pi\)
\(578\) 0 0
\(579\) 22.9766 0.954876
\(580\) 0 0
\(581\) 4.77835 0.198239
\(582\) 0 0
\(583\) 11.6794 0.483711
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 41.9186 1.73016 0.865082 0.501631i \(-0.167266\pi\)
0.865082 + 0.501631i \(0.167266\pi\)
\(588\) 0 0
\(589\) 3.32431 0.136976
\(590\) 0 0
\(591\) −6.24210 −0.256766
\(592\) 0 0
\(593\) −22.0200 −0.904254 −0.452127 0.891954i \(-0.649335\pi\)
−0.452127 + 0.891954i \(0.649335\pi\)
\(594\) 0 0
\(595\) −10.4203 −0.427192
\(596\) 0 0
\(597\) −17.7203 −0.725243
\(598\) 0 0
\(599\) 14.0880 0.575620 0.287810 0.957687i \(-0.407073\pi\)
0.287810 + 0.957687i \(0.407073\pi\)
\(600\) 0 0
\(601\) −5.01696 −0.204646 −0.102323 0.994751i \(-0.532628\pi\)
−0.102323 + 0.994751i \(0.532628\pi\)
\(602\) 0 0
\(603\) 11.5705 0.471188
\(604\) 0 0
\(605\) 15.6829 0.637600
\(606\) 0 0
\(607\) 35.9215 1.45801 0.729004 0.684509i \(-0.239984\pi\)
0.729004 + 0.684509i \(0.239984\pi\)
\(608\) 0 0
\(609\) −1.42696 −0.0578235
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 29.5053 1.19171 0.595853 0.803093i \(-0.296814\pi\)
0.595853 + 0.803093i \(0.296814\pi\)
\(614\) 0 0
\(615\) 13.6385 0.549957
\(616\) 0 0
\(617\) −12.3822 −0.498487 −0.249244 0.968441i \(-0.580182\pi\)
−0.249244 + 0.968441i \(0.580182\pi\)
\(618\) 0 0
\(619\) −26.5333 −1.06646 −0.533232 0.845969i \(-0.679023\pi\)
−0.533232 + 0.845969i \(0.679023\pi\)
\(620\) 0 0
\(621\) −2.91779 −0.117087
\(622\) 0 0
\(623\) −11.6385 −0.466286
\(624\) 0 0
\(625\) −9.87689 −0.395076
\(626\) 0 0
\(627\) 1.32431 0.0528877
\(628\) 0 0
\(629\) 60.4507 2.41033
\(630\) 0 0
\(631\) −20.0588 −0.798529 −0.399265 0.916836i \(-0.630735\pi\)
−0.399265 + 0.916836i \(0.630735\pi\)
\(632\) 0 0
\(633\) 14.8268 0.589314
\(634\) 0 0
\(635\) 1.97431 0.0783481
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3.22165 0.127446
\(640\) 0 0
\(641\) 43.0237 1.69934 0.849668 0.527319i \(-0.176803\pi\)
0.849668 + 0.527319i \(0.176803\pi\)
\(642\) 0 0
\(643\) −31.9665 −1.26064 −0.630318 0.776337i \(-0.717075\pi\)
−0.630318 + 0.776337i \(0.717075\pi\)
\(644\) 0 0
\(645\) 12.1512 0.478451
\(646\) 0 0
\(647\) −15.3820 −0.604727 −0.302364 0.953193i \(-0.597776\pi\)
−0.302364 + 0.953193i \(0.597776\pi\)
\(648\) 0 0
\(649\) −12.4924 −0.490370
\(650\) 0 0
\(651\) 3.32431 0.130290
\(652\) 0 0
\(653\) −17.3013 −0.677054 −0.338527 0.940957i \(-0.609929\pi\)
−0.338527 + 0.940957i \(0.609929\pi\)
\(654\) 0 0
\(655\) −0.971949 −0.0379772
\(656\) 0 0
\(657\) −8.04090 −0.313705
\(658\) 0 0
\(659\) −46.0421 −1.79354 −0.896772 0.442492i \(-0.854094\pi\)
−0.896772 + 0.442492i \(0.854094\pi\)
\(660\) 0 0
\(661\) 5.69963 0.221690 0.110845 0.993838i \(-0.464644\pi\)
0.110845 + 0.993838i \(0.464644\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.69614 −0.0657735
\(666\) 0 0
\(667\) 4.16358 0.161215
\(668\) 0 0
\(669\) 18.9219 0.731563
\(670\) 0 0
\(671\) 6.15619 0.237657
\(672\) 0 0
\(673\) −18.2095 −0.701925 −0.350963 0.936389i \(-0.614146\pi\)
−0.350963 + 0.936389i \(0.614146\pi\)
\(674\) 0 0
\(675\) −2.12311 −0.0817184
\(676\) 0 0
\(677\) −43.1126 −1.65695 −0.828475 0.560026i \(-0.810791\pi\)
−0.828475 + 0.560026i \(0.810791\pi\)
\(678\) 0 0
\(679\) −6.71659 −0.257759
\(680\) 0 0
\(681\) 11.3381 0.434478
\(682\) 0 0
\(683\) 1.60381 0.0613681 0.0306841 0.999529i \(-0.490231\pi\)
0.0306841 + 0.999529i \(0.490231\pi\)
\(684\) 0 0
\(685\) 6.50569 0.248569
\(686\) 0 0
\(687\) −15.9867 −0.609932
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 18.3739 0.698977 0.349489 0.936941i \(-0.386355\pi\)
0.349489 + 0.936941i \(0.386355\pi\)
\(692\) 0 0
\(693\) 1.32431 0.0503063
\(694\) 0 0
\(695\) −35.3229 −1.33987
\(696\) 0 0
\(697\) −49.3997 −1.87115
\(698\) 0 0
\(699\) 9.87648 0.373563
\(700\) 0 0
\(701\) −30.5958 −1.15559 −0.577794 0.816183i \(-0.696086\pi\)
−0.577794 + 0.816183i \(0.696086\pi\)
\(702\) 0 0
\(703\) 9.83969 0.371111
\(704\) 0 0
\(705\) −13.0078 −0.489902
\(706\) 0 0
\(707\) 13.7779 0.518172
\(708\) 0 0
\(709\) 25.2334 0.947659 0.473829 0.880617i \(-0.342871\pi\)
0.473829 + 0.880617i \(0.342871\pi\)
\(710\) 0 0
\(711\) 10.3101 0.386658
\(712\) 0 0
\(713\) −9.69963 −0.363254
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 11.6155 0.433790
\(718\) 0 0
\(719\) −21.8459 −0.814715 −0.407357 0.913269i \(-0.633550\pi\)
−0.407357 + 0.913269i \(0.633550\pi\)
\(720\) 0 0
\(721\) 13.1911 0.491262
\(722\) 0 0
\(723\) −14.3142 −0.532350
\(724\) 0 0
\(725\) 3.02960 0.112516
\(726\) 0 0
\(727\) −39.8889 −1.47940 −0.739699 0.672938i \(-0.765032\pi\)
−0.739699 + 0.672938i \(0.765032\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −44.0124 −1.62786
\(732\) 0 0
\(733\) −34.7713 −1.28431 −0.642154 0.766576i \(-0.721959\pi\)
−0.642154 + 0.766576i \(0.721959\pi\)
\(734\) 0 0
\(735\) −1.69614 −0.0625631
\(736\) 0 0
\(737\) 15.3229 0.564427
\(738\) 0 0
\(739\) 3.02805 0.111389 0.0556943 0.998448i \(-0.482263\pi\)
0.0556943 + 0.998448i \(0.482263\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −25.0737 −0.919864 −0.459932 0.887954i \(-0.652126\pi\)
−0.459932 + 0.887954i \(0.652126\pi\)
\(744\) 0 0
\(745\) 2.77214 0.101563
\(746\) 0 0
\(747\) 4.77835 0.174831
\(748\) 0 0
\(749\) −5.46786 −0.199791
\(750\) 0 0
\(751\) −12.7616 −0.465677 −0.232839 0.972515i \(-0.574801\pi\)
−0.232839 + 0.972515i \(0.574801\pi\)
\(752\) 0 0
\(753\) 25.0779 0.913889
\(754\) 0 0
\(755\) −8.62014 −0.313719
\(756\) 0 0
\(757\) 16.3198 0.593152 0.296576 0.955009i \(-0.404155\pi\)
0.296576 + 0.955009i \(0.404155\pi\)
\(758\) 0 0
\(759\) −3.86405 −0.140256
\(760\) 0 0
\(761\) −24.6492 −0.893534 −0.446767 0.894650i \(-0.647425\pi\)
−0.446767 + 0.894650i \(0.647425\pi\)
\(762\) 0 0
\(763\) −14.5292 −0.525993
\(764\) 0 0
\(765\) −10.4203 −0.376748
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 12.1103 0.436707 0.218354 0.975870i \(-0.429931\pi\)
0.218354 + 0.975870i \(0.429931\pi\)
\(770\) 0 0
\(771\) −28.7304 −1.03470
\(772\) 0 0
\(773\) −35.4456 −1.27489 −0.637445 0.770496i \(-0.720009\pi\)
−0.637445 + 0.770496i \(0.720009\pi\)
\(774\) 0 0
\(775\) −7.05785 −0.253526
\(776\) 0 0
\(777\) 9.83969 0.352997
\(778\) 0 0
\(779\) −8.04090 −0.288095
\(780\) 0 0
\(781\) 4.26645 0.152666
\(782\) 0 0
\(783\) −1.42696 −0.0509956
\(784\) 0 0
\(785\) −7.20217 −0.257057
\(786\) 0 0
\(787\) −24.5890 −0.876501 −0.438251 0.898853i \(-0.644402\pi\)
−0.438251 + 0.898853i \(0.644402\pi\)
\(788\) 0 0
\(789\) 5.09464 0.181374
\(790\) 0 0
\(791\) −2.77835 −0.0987868
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −14.9587 −0.530530
\(796\) 0 0
\(797\) −8.85393 −0.313622 −0.156811 0.987629i \(-0.550121\pi\)
−0.156811 + 0.987629i \(0.550121\pi\)
\(798\) 0 0
\(799\) 47.1153 1.66682
\(800\) 0 0
\(801\) −11.6385 −0.411226
\(802\) 0 0
\(803\) −10.6486 −0.375781
\(804\) 0 0
\(805\) 4.94898 0.174429
\(806\) 0 0
\(807\) −13.1410 −0.462586
\(808\) 0 0
\(809\) −5.05924 −0.177874 −0.0889368 0.996037i \(-0.528347\pi\)
−0.0889368 + 0.996037i \(0.528347\pi\)
\(810\) 0 0
\(811\) −24.2095 −0.850111 −0.425056 0.905167i \(-0.639745\pi\)
−0.425056 + 0.905167i \(0.639745\pi\)
\(812\) 0 0
\(813\) 13.2972 0.466354
\(814\) 0 0
\(815\) −37.2279 −1.30404
\(816\) 0 0
\(817\) −7.16400 −0.250637
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 32.3372 1.12857 0.564287 0.825579i \(-0.309151\pi\)
0.564287 + 0.825579i \(0.309151\pi\)
\(822\) 0 0
\(823\) −43.2734 −1.50842 −0.754208 0.656635i \(-0.771979\pi\)
−0.754208 + 0.656635i \(0.771979\pi\)
\(824\) 0 0
\(825\) −2.81164 −0.0978889
\(826\) 0 0
\(827\) 0.975438 0.0339193 0.0169597 0.999856i \(-0.494601\pi\)
0.0169597 + 0.999856i \(0.494601\pi\)
\(828\) 0 0
\(829\) −20.0436 −0.696144 −0.348072 0.937468i \(-0.613163\pi\)
−0.348072 + 0.937468i \(0.613163\pi\)
\(830\) 0 0
\(831\) 23.2049 0.804969
\(832\) 0 0
\(833\) 6.14355 0.212862
\(834\) 0 0
\(835\) −13.9174 −0.481631
\(836\) 0 0
\(837\) 3.32431 0.114905
\(838\) 0 0
\(839\) 47.4947 1.63970 0.819850 0.572578i \(-0.194057\pi\)
0.819850 + 0.572578i \(0.194057\pi\)
\(840\) 0 0
\(841\) −26.9638 −0.929785
\(842\) 0 0
\(843\) 26.4036 0.909388
\(844\) 0 0
\(845\) 22.0498 0.758537
\(846\) 0 0
\(847\) −9.24621 −0.317704
\(848\) 0 0
\(849\) 28.7795 0.987711
\(850\) 0 0
\(851\) −28.7102 −0.984172
\(852\) 0 0
\(853\) 4.64861 0.159166 0.0795828 0.996828i \(-0.474641\pi\)
0.0795828 + 0.996828i \(0.474641\pi\)
\(854\) 0 0
\(855\) −1.69614 −0.0580068
\(856\) 0 0
\(857\) −20.3974 −0.696761 −0.348380 0.937353i \(-0.613268\pi\)
−0.348380 + 0.937353i \(0.613268\pi\)
\(858\) 0 0
\(859\) −5.07949 −0.173310 −0.0866549 0.996238i \(-0.527618\pi\)
−0.0866549 + 0.996238i \(0.527618\pi\)
\(860\) 0 0
\(861\) −8.04090 −0.274033
\(862\) 0 0
\(863\) −12.5119 −0.425910 −0.212955 0.977062i \(-0.568309\pi\)
−0.212955 + 0.977062i \(0.568309\pi\)
\(864\) 0 0
\(865\) −2.36151 −0.0802936
\(866\) 0 0
\(867\) 20.7433 0.704478
\(868\) 0 0
\(869\) 13.6537 0.463170
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −6.71659 −0.227322
\(874\) 0 0
\(875\) 12.0818 0.408439
\(876\) 0 0
\(877\) 52.6601 1.77821 0.889103 0.457707i \(-0.151329\pi\)
0.889103 + 0.457707i \(0.151329\pi\)
\(878\) 0 0
\(879\) −26.3689 −0.889401
\(880\) 0 0
\(881\) 23.5892 0.794739 0.397369 0.917659i \(-0.369923\pi\)
0.397369 + 0.917659i \(0.369923\pi\)
\(882\) 0 0
\(883\) 41.9256 1.41091 0.705454 0.708755i \(-0.250743\pi\)
0.705454 + 0.708755i \(0.250743\pi\)
\(884\) 0 0
\(885\) 16.0000 0.537834
\(886\) 0 0
\(887\) −2.31280 −0.0776561 −0.0388281 0.999246i \(-0.512362\pi\)
−0.0388281 + 0.999246i \(0.512362\pi\)
\(888\) 0 0
\(889\) −1.16400 −0.0390394
\(890\) 0 0
\(891\) 1.32431 0.0443660
\(892\) 0 0
\(893\) 7.66906 0.256635
\(894\) 0 0
\(895\) 32.4071 1.08325
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4.74367 −0.158210
\(900\) 0 0
\(901\) 54.1815 1.80505
\(902\) 0 0
\(903\) −7.16400 −0.238403
\(904\) 0 0
\(905\) −33.9049 −1.12704
\(906\) 0 0
\(907\) 14.4980 0.481399 0.240699 0.970600i \(-0.422623\pi\)
0.240699 + 0.970600i \(0.422623\pi\)
\(908\) 0 0
\(909\) 13.7779 0.456985
\(910\) 0 0
\(911\) 25.1320 0.832662 0.416331 0.909213i \(-0.363316\pi\)
0.416331 + 0.909213i \(0.363316\pi\)
\(912\) 0 0
\(913\) 6.32800 0.209426
\(914\) 0 0
\(915\) −7.88470 −0.260660
\(916\) 0 0
\(917\) 0.573035 0.0189233
\(918\) 0 0
\(919\) −7.17874 −0.236805 −0.118402 0.992966i \(-0.537777\pi\)
−0.118402 + 0.992966i \(0.537777\pi\)
\(920\) 0 0
\(921\) 9.73082 0.320642
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −20.8907 −0.686882
\(926\) 0 0
\(927\) 13.1911 0.433252
\(928\) 0 0
\(929\) −45.1334 −1.48078 −0.740390 0.672178i \(-0.765359\pi\)
−0.740390 + 0.672178i \(0.765359\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) −27.4844 −0.899799
\(934\) 0 0
\(935\) −13.7997 −0.451299
\(936\) 0 0
\(937\) 52.4916 1.71483 0.857413 0.514629i \(-0.172071\pi\)
0.857413 + 0.514629i \(0.172071\pi\)
\(938\) 0 0
\(939\) 22.9766 0.749814
\(940\) 0 0
\(941\) −35.3055 −1.15092 −0.575462 0.817828i \(-0.695178\pi\)
−0.575462 + 0.817828i \(0.695178\pi\)
\(942\) 0 0
\(943\) 23.4616 0.764016
\(944\) 0 0
\(945\) −1.69614 −0.0551755
\(946\) 0 0
\(947\) 23.7800 0.772747 0.386374 0.922342i \(-0.373728\pi\)
0.386374 + 0.922342i \(0.373728\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 30.1983 0.979246
\(952\) 0 0
\(953\) −16.6957 −0.540828 −0.270414 0.962744i \(-0.587160\pi\)
−0.270414 + 0.962744i \(0.587160\pi\)
\(954\) 0 0
\(955\) −9.67200 −0.312978
\(956\) 0 0
\(957\) −1.88974 −0.0610866
\(958\) 0 0
\(959\) −3.83558 −0.123857
\(960\) 0 0
\(961\) −19.9490 −0.643516
\(962\) 0 0
\(963\) −5.46786 −0.176199
\(964\) 0 0
\(965\) −38.9716 −1.25454
\(966\) 0 0
\(967\) 47.7300 1.53489 0.767446 0.641113i \(-0.221527\pi\)
0.767446 + 0.641113i \(0.221527\pi\)
\(968\) 0 0
\(969\) 6.14355 0.197359
\(970\) 0 0
\(971\) −18.6486 −0.598462 −0.299231 0.954181i \(-0.596730\pi\)
−0.299231 + 0.954181i \(0.596730\pi\)
\(972\) 0 0
\(973\) 20.8255 0.667634
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −25.3728 −0.811748 −0.405874 0.913929i \(-0.633033\pi\)
−0.405874 + 0.913929i \(0.633033\pi\)
\(978\) 0 0
\(979\) −15.4129 −0.492600
\(980\) 0 0
\(981\) −14.5292 −0.463882
\(982\) 0 0
\(983\) −44.3841 −1.41563 −0.707817 0.706396i \(-0.750320\pi\)
−0.707817 + 0.706396i \(0.750320\pi\)
\(984\) 0 0
\(985\) 10.5875 0.337345
\(986\) 0 0
\(987\) 7.66906 0.244109
\(988\) 0 0
\(989\) 20.9031 0.664678
\(990\) 0 0
\(991\) 34.7483 1.10382 0.551909 0.833905i \(-0.313900\pi\)
0.551909 + 0.833905i \(0.313900\pi\)
\(992\) 0 0
\(993\) 24.3266 0.771982
\(994\) 0 0
\(995\) 30.0561 0.952843
\(996\) 0 0
\(997\) −1.32292 −0.0418972 −0.0209486 0.999781i \(-0.506669\pi\)
−0.0209486 + 0.999781i \(0.506669\pi\)
\(998\) 0 0
\(999\) 9.83969 0.311314
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 6384.2.a.cb.1.2 4
4.3 odd 2 3192.2.a.x.1.2 4
12.11 even 2 9576.2.a.cj.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.x.1.2 4 4.3 odd 2
6384.2.a.cb.1.2 4 1.1 even 1 trivial
9576.2.a.cj.1.3 4 12.11 even 2