Properties

Label 7448.2.a.bq.1.3
Level $7448$
Weight $2$
Character 7448.1
Self dual yes
Analytic conductor $59.473$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.4725794254\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 15 x^{6} - x^{5} + 66 x^{4} + 4 x^{3} - 76 x^{2} + 8 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1064)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.34151\) of defining polynomial
Character \(\chi\) \(=\) 7448.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.34151 q^{3} +2.90869 q^{5} -1.20034 q^{9} +O(q^{10})\) \(q-1.34151 q^{3} +2.90869 q^{5} -1.20034 q^{9} +4.99462 q^{11} +1.55217 q^{13} -3.90205 q^{15} +6.19129 q^{17} +1.00000 q^{19} +6.78397 q^{23} +3.46047 q^{25} +5.63482 q^{27} +0.707720 q^{29} +9.56989 q^{31} -6.70036 q^{33} +4.24407 q^{37} -2.08226 q^{39} -2.00000 q^{41} -1.25818 q^{43} -3.49141 q^{45} -12.3543 q^{47} -8.30570 q^{51} +0.891734 q^{53} +14.5278 q^{55} -1.34151 q^{57} +10.5769 q^{59} -11.8291 q^{61} +4.51478 q^{65} +1.98968 q^{67} -9.10079 q^{69} -12.4174 q^{71} -7.00688 q^{73} -4.64228 q^{75} +16.9359 q^{79} -3.95817 q^{81} -0.494217 q^{83} +18.0085 q^{85} -0.949417 q^{87} -3.38043 q^{89} -12.8381 q^{93} +2.90869 q^{95} -10.0685 q^{97} -5.99524 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{5} + 6 q^{9} + O(q^{10}) \) \( 8 q - q^{5} + 6 q^{9} + 9 q^{11} + 8 q^{15} + 4 q^{17} + 8 q^{19} + 25 q^{23} + 15 q^{25} + 3 q^{27} + 6 q^{29} + 8 q^{33} + 13 q^{37} - 11 q^{39} - 16 q^{41} + 17 q^{43} + 17 q^{45} - 24 q^{47} + 5 q^{51} + 2 q^{53} + 5 q^{55} + 2 q^{59} - 13 q^{61} - 26 q^{65} + 2 q^{67} - 11 q^{69} + 10 q^{71} + 5 q^{73} - 20 q^{75} + 16 q^{79} - 12 q^{81} - 43 q^{83} + 24 q^{85} + 20 q^{87} + 8 q^{89} + 2 q^{93} - q^{95} - 12 q^{97} + 37 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.34151 −0.774524 −0.387262 0.921970i \(-0.626579\pi\)
−0.387262 + 0.921970i \(0.626579\pi\)
\(4\) 0 0
\(5\) 2.90869 1.30081 0.650403 0.759590i \(-0.274600\pi\)
0.650403 + 0.759590i \(0.274600\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.20034 −0.400113
\(10\) 0 0
\(11\) 4.99462 1.50594 0.752968 0.658057i \(-0.228621\pi\)
0.752968 + 0.658057i \(0.228621\pi\)
\(12\) 0 0
\(13\) 1.55217 0.430495 0.215247 0.976560i \(-0.430944\pi\)
0.215247 + 0.976560i \(0.430944\pi\)
\(14\) 0 0
\(15\) −3.90205 −1.00750
\(16\) 0 0
\(17\) 6.19129 1.50161 0.750804 0.660525i \(-0.229666\pi\)
0.750804 + 0.660525i \(0.229666\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.78397 1.41456 0.707278 0.706936i \(-0.249923\pi\)
0.707278 + 0.706936i \(0.249923\pi\)
\(24\) 0 0
\(25\) 3.46047 0.692095
\(26\) 0 0
\(27\) 5.63482 1.08442
\(28\) 0 0
\(29\) 0.707720 0.131420 0.0657102 0.997839i \(-0.479069\pi\)
0.0657102 + 0.997839i \(0.479069\pi\)
\(30\) 0 0
\(31\) 9.56989 1.71880 0.859401 0.511302i \(-0.170836\pi\)
0.859401 + 0.511302i \(0.170836\pi\)
\(32\) 0 0
\(33\) −6.70036 −1.16638
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.24407 0.697720 0.348860 0.937175i \(-0.386569\pi\)
0.348860 + 0.937175i \(0.386569\pi\)
\(38\) 0 0
\(39\) −2.08226 −0.333428
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −1.25818 −0.191871 −0.0959355 0.995388i \(-0.530584\pi\)
−0.0959355 + 0.995388i \(0.530584\pi\)
\(44\) 0 0
\(45\) −3.49141 −0.520469
\(46\) 0 0
\(47\) −12.3543 −1.80206 −0.901030 0.433757i \(-0.857188\pi\)
−0.901030 + 0.433757i \(0.857188\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −8.30570 −1.16303
\(52\) 0 0
\(53\) 0.891734 0.122489 0.0612445 0.998123i \(-0.480493\pi\)
0.0612445 + 0.998123i \(0.480493\pi\)
\(54\) 0 0
\(55\) 14.5278 1.95893
\(56\) 0 0
\(57\) −1.34151 −0.177688
\(58\) 0 0
\(59\) 10.5769 1.37700 0.688498 0.725238i \(-0.258270\pi\)
0.688498 + 0.725238i \(0.258270\pi\)
\(60\) 0 0
\(61\) −11.8291 −1.51457 −0.757283 0.653086i \(-0.773474\pi\)
−0.757283 + 0.653086i \(0.773474\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.51478 0.559990
\(66\) 0 0
\(67\) 1.98968 0.243078 0.121539 0.992587i \(-0.461217\pi\)
0.121539 + 0.992587i \(0.461217\pi\)
\(68\) 0 0
\(69\) −9.10079 −1.09561
\(70\) 0 0
\(71\) −12.4174 −1.47368 −0.736839 0.676068i \(-0.763683\pi\)
−0.736839 + 0.676068i \(0.763683\pi\)
\(72\) 0 0
\(73\) −7.00688 −0.820094 −0.410047 0.912064i \(-0.634488\pi\)
−0.410047 + 0.912064i \(0.634488\pi\)
\(74\) 0 0
\(75\) −4.64228 −0.536044
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 16.9359 1.90544 0.952721 0.303845i \(-0.0982706\pi\)
0.952721 + 0.303845i \(0.0982706\pi\)
\(80\) 0 0
\(81\) −3.95817 −0.439796
\(82\) 0 0
\(83\) −0.494217 −0.0542474 −0.0271237 0.999632i \(-0.508635\pi\)
−0.0271237 + 0.999632i \(0.508635\pi\)
\(84\) 0 0
\(85\) 18.0085 1.95330
\(86\) 0 0
\(87\) −0.949417 −0.101788
\(88\) 0 0
\(89\) −3.38043 −0.358325 −0.179163 0.983819i \(-0.557339\pi\)
−0.179163 + 0.983819i \(0.557339\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −12.8381 −1.33125
\(94\) 0 0
\(95\) 2.90869 0.298425
\(96\) 0 0
\(97\) −10.0685 −1.02230 −0.511152 0.859491i \(-0.670781\pi\)
−0.511152 + 0.859491i \(0.670781\pi\)
\(98\) 0 0
\(99\) −5.99524 −0.602545
\(100\) 0 0
\(101\) 2.57231 0.255954 0.127977 0.991777i \(-0.459152\pi\)
0.127977 + 0.991777i \(0.459152\pi\)
\(102\) 0 0
\(103\) 6.43706 0.634263 0.317131 0.948382i \(-0.397280\pi\)
0.317131 + 0.948382i \(0.397280\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.39859 0.231881 0.115940 0.993256i \(-0.463012\pi\)
0.115940 + 0.993256i \(0.463012\pi\)
\(108\) 0 0
\(109\) −14.2546 −1.36534 −0.682670 0.730727i \(-0.739181\pi\)
−0.682670 + 0.730727i \(0.739181\pi\)
\(110\) 0 0
\(111\) −5.69348 −0.540401
\(112\) 0 0
\(113\) −0.713643 −0.0671338 −0.0335669 0.999436i \(-0.510687\pi\)
−0.0335669 + 0.999436i \(0.510687\pi\)
\(114\) 0 0
\(115\) 19.7325 1.84006
\(116\) 0 0
\(117\) −1.86313 −0.172247
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 13.9463 1.26784
\(122\) 0 0
\(123\) 2.68303 0.241921
\(124\) 0 0
\(125\) −4.47800 −0.400525
\(126\) 0 0
\(127\) −8.73364 −0.774985 −0.387493 0.921873i \(-0.626659\pi\)
−0.387493 + 0.921873i \(0.626659\pi\)
\(128\) 0 0
\(129\) 1.68787 0.148609
\(130\) 0 0
\(131\) −6.20006 −0.541702 −0.270851 0.962621i \(-0.587305\pi\)
−0.270851 + 0.962621i \(0.587305\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 16.3899 1.41062
\(136\) 0 0
\(137\) 17.8363 1.52386 0.761928 0.647662i \(-0.224253\pi\)
0.761928 + 0.647662i \(0.224253\pi\)
\(138\) 0 0
\(139\) −15.1484 −1.28487 −0.642435 0.766340i \(-0.722076\pi\)
−0.642435 + 0.766340i \(0.722076\pi\)
\(140\) 0 0
\(141\) 16.5735 1.39574
\(142\) 0 0
\(143\) 7.75251 0.648297
\(144\) 0 0
\(145\) 2.05854 0.170952
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.61171 0.705499 0.352749 0.935718i \(-0.385247\pi\)
0.352749 + 0.935718i \(0.385247\pi\)
\(150\) 0 0
\(151\) 17.2640 1.40492 0.702461 0.711722i \(-0.252084\pi\)
0.702461 + 0.711722i \(0.252084\pi\)
\(152\) 0 0
\(153\) −7.43165 −0.600813
\(154\) 0 0
\(155\) 27.8358 2.23583
\(156\) 0 0
\(157\) −16.9496 −1.35273 −0.676364 0.736567i \(-0.736445\pi\)
−0.676364 + 0.736567i \(0.736445\pi\)
\(158\) 0 0
\(159\) −1.19627 −0.0948707
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −17.2680 −1.35253 −0.676266 0.736657i \(-0.736403\pi\)
−0.676266 + 0.736657i \(0.736403\pi\)
\(164\) 0 0
\(165\) −19.4893 −1.51724
\(166\) 0 0
\(167\) 8.74790 0.676933 0.338466 0.940979i \(-0.390092\pi\)
0.338466 + 0.940979i \(0.390092\pi\)
\(168\) 0 0
\(169\) −10.5908 −0.814674
\(170\) 0 0
\(171\) −1.20034 −0.0917922
\(172\) 0 0
\(173\) 6.99381 0.531730 0.265865 0.964010i \(-0.414343\pi\)
0.265865 + 0.964010i \(0.414343\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −14.1891 −1.06652
\(178\) 0 0
\(179\) 20.4426 1.52795 0.763977 0.645243i \(-0.223244\pi\)
0.763977 + 0.645243i \(0.223244\pi\)
\(180\) 0 0
\(181\) 9.96247 0.740505 0.370252 0.928931i \(-0.379271\pi\)
0.370252 + 0.928931i \(0.379271\pi\)
\(182\) 0 0
\(183\) 15.8690 1.17307
\(184\) 0 0
\(185\) 12.3447 0.907598
\(186\) 0 0
\(187\) 30.9232 2.26133
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 14.5416 1.05219 0.526097 0.850425i \(-0.323655\pi\)
0.526097 + 0.850425i \(0.323655\pi\)
\(192\) 0 0
\(193\) −15.7083 −1.13071 −0.565353 0.824849i \(-0.691260\pi\)
−0.565353 + 0.824849i \(0.691260\pi\)
\(194\) 0 0
\(195\) −6.05665 −0.433725
\(196\) 0 0
\(197\) 0.726728 0.0517772 0.0258886 0.999665i \(-0.491758\pi\)
0.0258886 + 0.999665i \(0.491758\pi\)
\(198\) 0 0
\(199\) −14.3804 −1.01940 −0.509699 0.860353i \(-0.670243\pi\)
−0.509699 + 0.860353i \(0.670243\pi\)
\(200\) 0 0
\(201\) −2.66919 −0.188270
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −5.81738 −0.406303
\(206\) 0 0
\(207\) −8.14306 −0.565982
\(208\) 0 0
\(209\) 4.99462 0.345485
\(210\) 0 0
\(211\) −20.8134 −1.43285 −0.716427 0.697662i \(-0.754224\pi\)
−0.716427 + 0.697662i \(0.754224\pi\)
\(212\) 0 0
\(213\) 16.6582 1.14140
\(214\) 0 0
\(215\) −3.65966 −0.249587
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 9.39984 0.635182
\(220\) 0 0
\(221\) 9.60993 0.646434
\(222\) 0 0
\(223\) 6.02432 0.403418 0.201709 0.979445i \(-0.435350\pi\)
0.201709 + 0.979445i \(0.435350\pi\)
\(224\) 0 0
\(225\) −4.15374 −0.276916
\(226\) 0 0
\(227\) 7.86625 0.522102 0.261051 0.965325i \(-0.415931\pi\)
0.261051 + 0.965325i \(0.415931\pi\)
\(228\) 0 0
\(229\) −28.6691 −1.89450 −0.947252 0.320488i \(-0.896153\pi\)
−0.947252 + 0.320488i \(0.896153\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.04304 0.199356 0.0996781 0.995020i \(-0.468219\pi\)
0.0996781 + 0.995020i \(0.468219\pi\)
\(234\) 0 0
\(235\) −35.9348 −2.34413
\(236\) 0 0
\(237\) −22.7198 −1.47581
\(238\) 0 0
\(239\) 9.23545 0.597392 0.298696 0.954348i \(-0.403448\pi\)
0.298696 + 0.954348i \(0.403448\pi\)
\(240\) 0 0
\(241\) −9.03850 −0.582221 −0.291110 0.956689i \(-0.594025\pi\)
−0.291110 + 0.956689i \(0.594025\pi\)
\(242\) 0 0
\(243\) −11.5945 −0.743788
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.55217 0.0987622
\(248\) 0 0
\(249\) 0.663000 0.0420159
\(250\) 0 0
\(251\) −2.91027 −0.183695 −0.0918473 0.995773i \(-0.529277\pi\)
−0.0918473 + 0.995773i \(0.529277\pi\)
\(252\) 0 0
\(253\) 33.8834 2.13023
\(254\) 0 0
\(255\) −24.1587 −1.51288
\(256\) 0 0
\(257\) 22.9088 1.42901 0.714505 0.699631i \(-0.246652\pi\)
0.714505 + 0.699631i \(0.246652\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −0.849504 −0.0525830
\(262\) 0 0
\(263\) 17.6699 1.08957 0.544787 0.838574i \(-0.316610\pi\)
0.544787 + 0.838574i \(0.316610\pi\)
\(264\) 0 0
\(265\) 2.59378 0.159334
\(266\) 0 0
\(267\) 4.53490 0.277531
\(268\) 0 0
\(269\) −11.4234 −0.696497 −0.348248 0.937402i \(-0.613223\pi\)
−0.348248 + 0.937402i \(0.613223\pi\)
\(270\) 0 0
\(271\) −5.04922 −0.306719 −0.153359 0.988170i \(-0.549009\pi\)
−0.153359 + 0.988170i \(0.549009\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 17.2838 1.04225
\(276\) 0 0
\(277\) 6.45687 0.387956 0.193978 0.981006i \(-0.437861\pi\)
0.193978 + 0.981006i \(0.437861\pi\)
\(278\) 0 0
\(279\) −11.4871 −0.687715
\(280\) 0 0
\(281\) −10.2877 −0.613711 −0.306856 0.951756i \(-0.599277\pi\)
−0.306856 + 0.951756i \(0.599277\pi\)
\(282\) 0 0
\(283\) −22.0500 −1.31074 −0.655368 0.755310i \(-0.727486\pi\)
−0.655368 + 0.755310i \(0.727486\pi\)
\(284\) 0 0
\(285\) −3.90205 −0.231137
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 21.3320 1.25483
\(290\) 0 0
\(291\) 13.5071 0.791798
\(292\) 0 0
\(293\) −20.5297 −1.19936 −0.599680 0.800240i \(-0.704705\pi\)
−0.599680 + 0.800240i \(0.704705\pi\)
\(294\) 0 0
\(295\) 30.7649 1.79120
\(296\) 0 0
\(297\) 28.1438 1.63307
\(298\) 0 0
\(299\) 10.5299 0.608958
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −3.45079 −0.198242
\(304\) 0 0
\(305\) −34.4073 −1.97016
\(306\) 0 0
\(307\) 21.5830 1.23181 0.615904 0.787821i \(-0.288791\pi\)
0.615904 + 0.787821i \(0.288791\pi\)
\(308\) 0 0
\(309\) −8.63541 −0.491251
\(310\) 0 0
\(311\) −23.6510 −1.34112 −0.670562 0.741853i \(-0.733947\pi\)
−0.670562 + 0.741853i \(0.733947\pi\)
\(312\) 0 0
\(313\) −26.7120 −1.50985 −0.754927 0.655809i \(-0.772328\pi\)
−0.754927 + 0.655809i \(0.772328\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −27.0740 −1.52063 −0.760314 0.649555i \(-0.774955\pi\)
−0.760314 + 0.649555i \(0.774955\pi\)
\(318\) 0 0
\(319\) 3.53480 0.197911
\(320\) 0 0
\(321\) −3.21775 −0.179597
\(322\) 0 0
\(323\) 6.19129 0.344492
\(324\) 0 0
\(325\) 5.37125 0.297943
\(326\) 0 0
\(327\) 19.1227 1.05749
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3.14003 0.172591 0.0862957 0.996270i \(-0.472497\pi\)
0.0862957 + 0.996270i \(0.472497\pi\)
\(332\) 0 0
\(333\) −5.09432 −0.279167
\(334\) 0 0
\(335\) 5.78737 0.316198
\(336\) 0 0
\(337\) 21.6417 1.17890 0.589449 0.807805i \(-0.299345\pi\)
0.589449 + 0.807805i \(0.299345\pi\)
\(338\) 0 0
\(339\) 0.957362 0.0519968
\(340\) 0 0
\(341\) 47.7980 2.58841
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −26.4714 −1.42517
\(346\) 0 0
\(347\) −2.56676 −0.137791 −0.0688955 0.997624i \(-0.521948\pi\)
−0.0688955 + 0.997624i \(0.521948\pi\)
\(348\) 0 0
\(349\) 1.91567 0.102543 0.0512716 0.998685i \(-0.483673\pi\)
0.0512716 + 0.998685i \(0.483673\pi\)
\(350\) 0 0
\(351\) 8.74619 0.466837
\(352\) 0 0
\(353\) −15.8777 −0.845087 −0.422543 0.906343i \(-0.638863\pi\)
−0.422543 + 0.906343i \(0.638863\pi\)
\(354\) 0 0
\(355\) −36.1185 −1.91697
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.0794 1.05975 0.529875 0.848076i \(-0.322239\pi\)
0.529875 + 0.848076i \(0.322239\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −18.7091 −0.981975
\(364\) 0 0
\(365\) −20.3808 −1.06678
\(366\) 0 0
\(367\) 3.58480 0.187125 0.0935626 0.995613i \(-0.470174\pi\)
0.0935626 + 0.995613i \(0.470174\pi\)
\(368\) 0 0
\(369\) 2.40068 0.124974
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 26.6745 1.38115 0.690576 0.723260i \(-0.257357\pi\)
0.690576 + 0.723260i \(0.257357\pi\)
\(374\) 0 0
\(375\) 6.00730 0.310216
\(376\) 0 0
\(377\) 1.09850 0.0565758
\(378\) 0 0
\(379\) 18.9451 0.973146 0.486573 0.873640i \(-0.338247\pi\)
0.486573 + 0.873640i \(0.338247\pi\)
\(380\) 0 0
\(381\) 11.7163 0.600244
\(382\) 0 0
\(383\) −14.9515 −0.763988 −0.381994 0.924165i \(-0.624762\pi\)
−0.381994 + 0.924165i \(0.624762\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.51025 0.0767701
\(388\) 0 0
\(389\) 36.6168 1.85655 0.928273 0.371900i \(-0.121294\pi\)
0.928273 + 0.371900i \(0.121294\pi\)
\(390\) 0 0
\(391\) 42.0015 2.12411
\(392\) 0 0
\(393\) 8.31748 0.419561
\(394\) 0 0
\(395\) 49.2614 2.47861
\(396\) 0 0
\(397\) −19.8520 −0.996346 −0.498173 0.867078i \(-0.665996\pi\)
−0.498173 + 0.867078i \(0.665996\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.63008 0.231215 0.115608 0.993295i \(-0.463118\pi\)
0.115608 + 0.993295i \(0.463118\pi\)
\(402\) 0 0
\(403\) 14.8541 0.739935
\(404\) 0 0
\(405\) −11.5131 −0.572090
\(406\) 0 0
\(407\) 21.1975 1.05072
\(408\) 0 0
\(409\) 4.41274 0.218196 0.109098 0.994031i \(-0.465204\pi\)
0.109098 + 0.994031i \(0.465204\pi\)
\(410\) 0 0
\(411\) −23.9276 −1.18026
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1.43752 −0.0705653
\(416\) 0 0
\(417\) 20.3218 0.995162
\(418\) 0 0
\(419\) 32.8148 1.60311 0.801554 0.597922i \(-0.204007\pi\)
0.801554 + 0.597922i \(0.204007\pi\)
\(420\) 0 0
\(421\) 10.3765 0.505721 0.252861 0.967503i \(-0.418629\pi\)
0.252861 + 0.967503i \(0.418629\pi\)
\(422\) 0 0
\(423\) 14.8294 0.721028
\(424\) 0 0
\(425\) 21.4248 1.03926
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −10.4001 −0.502122
\(430\) 0 0
\(431\) −2.11967 −0.102101 −0.0510505 0.998696i \(-0.516257\pi\)
−0.0510505 + 0.998696i \(0.516257\pi\)
\(432\) 0 0
\(433\) −4.23274 −0.203413 −0.101706 0.994814i \(-0.532430\pi\)
−0.101706 + 0.994814i \(0.532430\pi\)
\(434\) 0 0
\(435\) −2.76156 −0.132407
\(436\) 0 0
\(437\) 6.78397 0.324521
\(438\) 0 0
\(439\) −23.7380 −1.13295 −0.566476 0.824078i \(-0.691694\pi\)
−0.566476 + 0.824078i \(0.691694\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.6401 −0.505529 −0.252764 0.967528i \(-0.581340\pi\)
−0.252764 + 0.967528i \(0.581340\pi\)
\(444\) 0 0
\(445\) −9.83263 −0.466111
\(446\) 0 0
\(447\) −11.5527 −0.546426
\(448\) 0 0
\(449\) 18.4650 0.871418 0.435709 0.900088i \(-0.356498\pi\)
0.435709 + 0.900088i \(0.356498\pi\)
\(450\) 0 0
\(451\) −9.98925 −0.470375
\(452\) 0 0
\(453\) −23.1599 −1.08815
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.43795 −0.394711 −0.197355 0.980332i \(-0.563235\pi\)
−0.197355 + 0.980332i \(0.563235\pi\)
\(458\) 0 0
\(459\) 34.8868 1.62837
\(460\) 0 0
\(461\) −18.8951 −0.880034 −0.440017 0.897990i \(-0.645028\pi\)
−0.440017 + 0.897990i \(0.645028\pi\)
\(462\) 0 0
\(463\) 27.5602 1.28083 0.640415 0.768029i \(-0.278762\pi\)
0.640415 + 0.768029i \(0.278762\pi\)
\(464\) 0 0
\(465\) −37.3422 −1.73170
\(466\) 0 0
\(467\) 23.2741 1.07700 0.538498 0.842627i \(-0.318992\pi\)
0.538498 + 0.842627i \(0.318992\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 22.7382 1.04772
\(472\) 0 0
\(473\) −6.28415 −0.288945
\(474\) 0 0
\(475\) 3.46047 0.158777
\(476\) 0 0
\(477\) −1.07038 −0.0490095
\(478\) 0 0
\(479\) 4.40293 0.201175 0.100588 0.994928i \(-0.467928\pi\)
0.100588 + 0.994928i \(0.467928\pi\)
\(480\) 0 0
\(481\) 6.58751 0.300365
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −29.2862 −1.32982
\(486\) 0 0
\(487\) −24.2497 −1.09886 −0.549429 0.835541i \(-0.685155\pi\)
−0.549429 + 0.835541i \(0.685155\pi\)
\(488\) 0 0
\(489\) 23.1652 1.04757
\(490\) 0 0
\(491\) −27.0129 −1.21908 −0.609538 0.792757i \(-0.708645\pi\)
−0.609538 + 0.792757i \(0.708645\pi\)
\(492\) 0 0
\(493\) 4.38170 0.197342
\(494\) 0 0
\(495\) −17.4383 −0.783793
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 19.6939 0.881621 0.440810 0.897600i \(-0.354691\pi\)
0.440810 + 0.897600i \(0.354691\pi\)
\(500\) 0 0
\(501\) −11.7354 −0.524300
\(502\) 0 0
\(503\) 0.561980 0.0250574 0.0125287 0.999922i \(-0.496012\pi\)
0.0125287 + 0.999922i \(0.496012\pi\)
\(504\) 0 0
\(505\) 7.48204 0.332946
\(506\) 0 0
\(507\) 14.2077 0.630985
\(508\) 0 0
\(509\) −17.6989 −0.784490 −0.392245 0.919861i \(-0.628301\pi\)
−0.392245 + 0.919861i \(0.628301\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 5.63482 0.248783
\(514\) 0 0
\(515\) 18.7234 0.825052
\(516\) 0 0
\(517\) −61.7051 −2.71379
\(518\) 0 0
\(519\) −9.38230 −0.411837
\(520\) 0 0
\(521\) 40.3035 1.76573 0.882863 0.469630i \(-0.155613\pi\)
0.882863 + 0.469630i \(0.155613\pi\)
\(522\) 0 0
\(523\) 1.95670 0.0855606 0.0427803 0.999085i \(-0.486378\pi\)
0.0427803 + 0.999085i \(0.486378\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 59.2499 2.58097
\(528\) 0 0
\(529\) 23.0222 1.00097
\(530\) 0 0
\(531\) −12.6959 −0.550954
\(532\) 0 0
\(533\) −3.10434 −0.134464
\(534\) 0 0
\(535\) 6.97677 0.301632
\(536\) 0 0
\(537\) −27.4241 −1.18344
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −31.0688 −1.33575 −0.667877 0.744272i \(-0.732797\pi\)
−0.667877 + 0.744272i \(0.732797\pi\)
\(542\) 0 0
\(543\) −13.3648 −0.573538
\(544\) 0 0
\(545\) −41.4621 −1.77604
\(546\) 0 0
\(547\) 36.8556 1.57583 0.787917 0.615782i \(-0.211160\pi\)
0.787917 + 0.615782i \(0.211160\pi\)
\(548\) 0 0
\(549\) 14.1990 0.605998
\(550\) 0 0
\(551\) 0.707720 0.0301499
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −16.5606 −0.702956
\(556\) 0 0
\(557\) 23.2272 0.984167 0.492083 0.870548i \(-0.336236\pi\)
0.492083 + 0.870548i \(0.336236\pi\)
\(558\) 0 0
\(559\) −1.95291 −0.0825994
\(560\) 0 0
\(561\) −41.4839 −1.75145
\(562\) 0 0
\(563\) 24.0547 1.01378 0.506891 0.862010i \(-0.330794\pi\)
0.506891 + 0.862010i \(0.330794\pi\)
\(564\) 0 0
\(565\) −2.07576 −0.0873281
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −29.5761 −1.23989 −0.619946 0.784644i \(-0.712846\pi\)
−0.619946 + 0.784644i \(0.712846\pi\)
\(570\) 0 0
\(571\) −26.4925 −1.10867 −0.554337 0.832292i \(-0.687028\pi\)
−0.554337 + 0.832292i \(0.687028\pi\)
\(572\) 0 0
\(573\) −19.5078 −0.814949
\(574\) 0 0
\(575\) 23.4757 0.979006
\(576\) 0 0
\(577\) 25.5816 1.06497 0.532487 0.846438i \(-0.321257\pi\)
0.532487 + 0.846438i \(0.321257\pi\)
\(578\) 0 0
\(579\) 21.0729 0.875759
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 4.45388 0.184461
\(584\) 0 0
\(585\) −5.41927 −0.224059
\(586\) 0 0
\(587\) 27.2213 1.12354 0.561771 0.827293i \(-0.310120\pi\)
0.561771 + 0.827293i \(0.310120\pi\)
\(588\) 0 0
\(589\) 9.56989 0.394320
\(590\) 0 0
\(591\) −0.974916 −0.0401027
\(592\) 0 0
\(593\) 1.90598 0.0782690 0.0391345 0.999234i \(-0.487540\pi\)
0.0391345 + 0.999234i \(0.487540\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 19.2915 0.789548
\(598\) 0 0
\(599\) −20.0500 −0.819221 −0.409610 0.912261i \(-0.634335\pi\)
−0.409610 + 0.912261i \(0.634335\pi\)
\(600\) 0 0
\(601\) −14.6961 −0.599466 −0.299733 0.954023i \(-0.596898\pi\)
−0.299733 + 0.954023i \(0.596898\pi\)
\(602\) 0 0
\(603\) −2.38829 −0.0972589
\(604\) 0 0
\(605\) 40.5654 1.64922
\(606\) 0 0
\(607\) 23.4800 0.953025 0.476512 0.879168i \(-0.341901\pi\)
0.476512 + 0.879168i \(0.341901\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −19.1760 −0.775777
\(612\) 0 0
\(613\) 27.3278 1.10376 0.551880 0.833924i \(-0.313911\pi\)
0.551880 + 0.833924i \(0.313911\pi\)
\(614\) 0 0
\(615\) 7.80410 0.314692
\(616\) 0 0
\(617\) −25.5796 −1.02980 −0.514898 0.857252i \(-0.672170\pi\)
−0.514898 + 0.857252i \(0.672170\pi\)
\(618\) 0 0
\(619\) 1.49116 0.0599350 0.0299675 0.999551i \(-0.490460\pi\)
0.0299675 + 0.999551i \(0.490460\pi\)
\(620\) 0 0
\(621\) 38.2264 1.53397
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −30.3275 −1.21310
\(626\) 0 0
\(627\) −6.70036 −0.267587
\(628\) 0 0
\(629\) 26.2762 1.04770
\(630\) 0 0
\(631\) −22.0819 −0.879066 −0.439533 0.898226i \(-0.644856\pi\)
−0.439533 + 0.898226i \(0.644856\pi\)
\(632\) 0 0
\(633\) 27.9215 1.10978
\(634\) 0 0
\(635\) −25.4034 −1.00810
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 14.9051 0.589638
\(640\) 0 0
\(641\) −2.80334 −0.110725 −0.0553627 0.998466i \(-0.517632\pi\)
−0.0553627 + 0.998466i \(0.517632\pi\)
\(642\) 0 0
\(643\) −27.0640 −1.06730 −0.533649 0.845706i \(-0.679180\pi\)
−0.533649 + 0.845706i \(0.679180\pi\)
\(644\) 0 0
\(645\) 4.90949 0.193311
\(646\) 0 0
\(647\) 5.37572 0.211341 0.105671 0.994401i \(-0.466301\pi\)
0.105671 + 0.994401i \(0.466301\pi\)
\(648\) 0 0
\(649\) 52.8277 2.07367
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −28.1250 −1.10062 −0.550308 0.834962i \(-0.685490\pi\)
−0.550308 + 0.834962i \(0.685490\pi\)
\(654\) 0 0
\(655\) −18.0341 −0.704649
\(656\) 0 0
\(657\) 8.41064 0.328130
\(658\) 0 0
\(659\) 35.3753 1.37802 0.689012 0.724750i \(-0.258045\pi\)
0.689012 + 0.724750i \(0.258045\pi\)
\(660\) 0 0
\(661\) 43.4717 1.69085 0.845426 0.534093i \(-0.179347\pi\)
0.845426 + 0.534093i \(0.179347\pi\)
\(662\) 0 0
\(663\) −12.8919 −0.500679
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.80115 0.185901
\(668\) 0 0
\(669\) −8.08171 −0.312457
\(670\) 0 0
\(671\) −59.0821 −2.28084
\(672\) 0 0
\(673\) 17.7175 0.682961 0.341481 0.939889i \(-0.389072\pi\)
0.341481 + 0.939889i \(0.389072\pi\)
\(674\) 0 0
\(675\) 19.4991 0.750522
\(676\) 0 0
\(677\) 44.8447 1.72352 0.861761 0.507314i \(-0.169362\pi\)
0.861761 + 0.507314i \(0.169362\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −10.5527 −0.404380
\(682\) 0 0
\(683\) −30.0471 −1.14972 −0.574861 0.818251i \(-0.694944\pi\)
−0.574861 + 0.818251i \(0.694944\pi\)
\(684\) 0 0
\(685\) 51.8802 1.98224
\(686\) 0 0
\(687\) 38.4600 1.46734
\(688\) 0 0
\(689\) 1.38412 0.0527309
\(690\) 0 0
\(691\) −18.4930 −0.703508 −0.351754 0.936092i \(-0.614415\pi\)
−0.351754 + 0.936092i \(0.614415\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −44.0620 −1.67137
\(696\) 0 0
\(697\) −12.3826 −0.469023
\(698\) 0 0
\(699\) −4.08228 −0.154406
\(700\) 0 0
\(701\) −26.5282 −1.00196 −0.500978 0.865460i \(-0.667026\pi\)
−0.500978 + 0.865460i \(0.667026\pi\)
\(702\) 0 0
\(703\) 4.24407 0.160068
\(704\) 0 0
\(705\) 48.2071 1.81558
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 37.9319 1.42456 0.712281 0.701894i \(-0.247662\pi\)
0.712281 + 0.701894i \(0.247662\pi\)
\(710\) 0 0
\(711\) −20.3289 −0.762393
\(712\) 0 0
\(713\) 64.9218 2.43134
\(714\) 0 0
\(715\) 22.5496 0.843309
\(716\) 0 0
\(717\) −12.3895 −0.462694
\(718\) 0 0
\(719\) −0.431495 −0.0160921 −0.00804603 0.999968i \(-0.502561\pi\)
−0.00804603 + 0.999968i \(0.502561\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 12.1253 0.450944
\(724\) 0 0
\(725\) 2.44905 0.0909554
\(726\) 0 0
\(727\) −3.14690 −0.116712 −0.0583560 0.998296i \(-0.518586\pi\)
−0.0583560 + 0.998296i \(0.518586\pi\)
\(728\) 0 0
\(729\) 27.4287 1.01588
\(730\) 0 0
\(731\) −7.78977 −0.288115
\(732\) 0 0
\(733\) −14.7044 −0.543119 −0.271559 0.962422i \(-0.587539\pi\)
−0.271559 + 0.962422i \(0.587539\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.93772 0.366061
\(738\) 0 0
\(739\) −14.9990 −0.551747 −0.275873 0.961194i \(-0.588967\pi\)
−0.275873 + 0.961194i \(0.588967\pi\)
\(740\) 0 0
\(741\) −2.08226 −0.0764937
\(742\) 0 0
\(743\) −31.3454 −1.14995 −0.574975 0.818171i \(-0.694988\pi\)
−0.574975 + 0.818171i \(0.694988\pi\)
\(744\) 0 0
\(745\) 25.0488 0.917717
\(746\) 0 0
\(747\) 0.593228 0.0217051
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4.43909 0.161985 0.0809923 0.996715i \(-0.474191\pi\)
0.0809923 + 0.996715i \(0.474191\pi\)
\(752\) 0 0
\(753\) 3.90417 0.142276
\(754\) 0 0
\(755\) 50.2155 1.82753
\(756\) 0 0
\(757\) 7.60936 0.276567 0.138283 0.990393i \(-0.455841\pi\)
0.138283 + 0.990393i \(0.455841\pi\)
\(758\) 0 0
\(759\) −45.4550 −1.64991
\(760\) 0 0
\(761\) 13.4829 0.488755 0.244378 0.969680i \(-0.421416\pi\)
0.244378 + 0.969680i \(0.421416\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −21.6163 −0.781541
\(766\) 0 0
\(767\) 16.4172 0.592789
\(768\) 0 0
\(769\) 21.3561 0.770121 0.385061 0.922891i \(-0.374181\pi\)
0.385061 + 0.922891i \(0.374181\pi\)
\(770\) 0 0
\(771\) −30.7324 −1.10680
\(772\) 0 0
\(773\) −41.4207 −1.48980 −0.744900 0.667176i \(-0.767503\pi\)
−0.744900 + 0.667176i \(0.767503\pi\)
\(774\) 0 0
\(775\) 33.1164 1.18957
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) −62.0204 −2.21927
\(782\) 0 0
\(783\) 3.98787 0.142515
\(784\) 0 0
\(785\) −49.3012 −1.75964
\(786\) 0 0
\(787\) −36.3559 −1.29595 −0.647974 0.761662i \(-0.724384\pi\)
−0.647974 + 0.761662i \(0.724384\pi\)
\(788\) 0 0
\(789\) −23.7045 −0.843901
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −18.3608 −0.652013
\(794\) 0 0
\(795\) −3.47959 −0.123408
\(796\) 0 0
\(797\) 8.65901 0.306718 0.153359 0.988171i \(-0.450991\pi\)
0.153359 + 0.988171i \(0.450991\pi\)
\(798\) 0 0
\(799\) −76.4890 −2.70599
\(800\) 0 0
\(801\) 4.05766 0.143371
\(802\) 0 0
\(803\) −34.9968 −1.23501
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 15.3247 0.539453
\(808\) 0 0
\(809\) −36.1306 −1.27028 −0.635142 0.772395i \(-0.719059\pi\)
−0.635142 + 0.772395i \(0.719059\pi\)
\(810\) 0 0
\(811\) 30.7940 1.08133 0.540663 0.841240i \(-0.318174\pi\)
0.540663 + 0.841240i \(0.318174\pi\)
\(812\) 0 0
\(813\) 6.77361 0.237561
\(814\) 0 0
\(815\) −50.2272 −1.75938
\(816\) 0 0
\(817\) −1.25818 −0.0440182
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.3877 0.432333 0.216166 0.976357i \(-0.430645\pi\)
0.216166 + 0.976357i \(0.430645\pi\)
\(822\) 0 0
\(823\) −0.475455 −0.0165733 −0.00828665 0.999966i \(-0.502638\pi\)
−0.00828665 + 0.999966i \(0.502638\pi\)
\(824\) 0 0
\(825\) −23.1864 −0.807248
\(826\) 0 0
\(827\) 29.8510 1.03802 0.519010 0.854768i \(-0.326301\pi\)
0.519010 + 0.854768i \(0.326301\pi\)
\(828\) 0 0
\(829\) 30.0129 1.04239 0.521196 0.853437i \(-0.325486\pi\)
0.521196 + 0.853437i \(0.325486\pi\)
\(830\) 0 0
\(831\) −8.66199 −0.300481
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 25.4449 0.880558
\(836\) 0 0
\(837\) 53.9246 1.86391
\(838\) 0 0
\(839\) −34.3155 −1.18470 −0.592351 0.805680i \(-0.701800\pi\)
−0.592351 + 0.805680i \(0.701800\pi\)
\(840\) 0 0
\(841\) −28.4991 −0.982729
\(842\) 0 0
\(843\) 13.8011 0.475334
\(844\) 0 0
\(845\) −30.8053 −1.05973
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 29.5804 1.01520
\(850\) 0 0
\(851\) 28.7916 0.986964
\(852\) 0 0
\(853\) −6.59556 −0.225828 −0.112914 0.993605i \(-0.536018\pi\)
−0.112914 + 0.993605i \(0.536018\pi\)
\(854\) 0 0
\(855\) −3.49141 −0.119404
\(856\) 0 0
\(857\) 44.7037 1.52705 0.763525 0.645779i \(-0.223467\pi\)
0.763525 + 0.645779i \(0.223467\pi\)
\(858\) 0 0
\(859\) −53.0696 −1.81071 −0.905356 0.424653i \(-0.860396\pi\)
−0.905356 + 0.424653i \(0.860396\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −38.4050 −1.30732 −0.653661 0.756788i \(-0.726768\pi\)
−0.653661 + 0.756788i \(0.726768\pi\)
\(864\) 0 0
\(865\) 20.3428 0.691677
\(866\) 0 0
\(867\) −28.6172 −0.971893
\(868\) 0 0
\(869\) 84.5887 2.86948
\(870\) 0 0
\(871\) 3.08833 0.104644
\(872\) 0 0
\(873\) 12.0856 0.409037
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 20.7329 0.700099 0.350050 0.936731i \(-0.386165\pi\)
0.350050 + 0.936731i \(0.386165\pi\)
\(878\) 0 0
\(879\) 27.5409 0.928933
\(880\) 0 0
\(881\) 39.3452 1.32557 0.662786 0.748809i \(-0.269374\pi\)
0.662786 + 0.748809i \(0.269374\pi\)
\(882\) 0 0
\(883\) 26.7816 0.901272 0.450636 0.892708i \(-0.351197\pi\)
0.450636 + 0.892708i \(0.351197\pi\)
\(884\) 0 0
\(885\) −41.2716 −1.38733
\(886\) 0 0
\(887\) 9.93418 0.333557 0.166779 0.985994i \(-0.446663\pi\)
0.166779 + 0.985994i \(0.446663\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −19.7696 −0.662305
\(892\) 0 0
\(893\) −12.3543 −0.413421
\(894\) 0 0
\(895\) 59.4613 1.98757
\(896\) 0 0
\(897\) −14.1260 −0.471653
\(898\) 0 0
\(899\) 6.77280 0.225886
\(900\) 0 0
\(901\) 5.52098 0.183931
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 28.9777 0.963252
\(906\) 0 0
\(907\) 19.1589 0.636162 0.318081 0.948064i \(-0.396962\pi\)
0.318081 + 0.948064i \(0.396962\pi\)
\(908\) 0 0
\(909\) −3.08764 −0.102411
\(910\) 0 0
\(911\) −53.0363 −1.75717 −0.878586 0.477584i \(-0.841513\pi\)
−0.878586 + 0.477584i \(0.841513\pi\)
\(912\) 0 0
\(913\) −2.46843 −0.0816931
\(914\) 0 0
\(915\) 46.1579 1.52593
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 42.7974 1.41176 0.705878 0.708334i \(-0.250553\pi\)
0.705878 + 0.708334i \(0.250553\pi\)
\(920\) 0 0
\(921\) −28.9539 −0.954065
\(922\) 0 0
\(923\) −19.2740 −0.634411
\(924\) 0 0
\(925\) 14.6865 0.482889
\(926\) 0 0
\(927\) −7.72666 −0.253777
\(928\) 0 0
\(929\) 20.3584 0.667938 0.333969 0.942584i \(-0.391612\pi\)
0.333969 + 0.942584i \(0.391612\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 31.7282 1.03873
\(934\) 0 0
\(935\) 89.9459 2.94154
\(936\) 0 0
\(937\) 46.5032 1.51919 0.759596 0.650395i \(-0.225396\pi\)
0.759596 + 0.650395i \(0.225396\pi\)
\(938\) 0 0
\(939\) 35.8346 1.16942
\(940\) 0 0
\(941\) −39.8300 −1.29842 −0.649211 0.760609i \(-0.724901\pi\)
−0.649211 + 0.760609i \(0.724901\pi\)
\(942\) 0 0
\(943\) −13.5679 −0.441833
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −7.34847 −0.238793 −0.119397 0.992847i \(-0.538096\pi\)
−0.119397 + 0.992847i \(0.538096\pi\)
\(948\) 0 0
\(949\) −10.8759 −0.353046
\(950\) 0 0
\(951\) 36.3202 1.17776
\(952\) 0 0
\(953\) 27.7557 0.899094 0.449547 0.893257i \(-0.351585\pi\)
0.449547 + 0.893257i \(0.351585\pi\)
\(954\) 0 0
\(955\) 42.2970 1.36870
\(956\) 0 0
\(957\) −4.74198 −0.153286
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 60.5828 1.95428
\(962\) 0 0
\(963\) −2.87913 −0.0927786
\(964\) 0 0
\(965\) −45.6905 −1.47083
\(966\) 0 0
\(967\) −42.6002 −1.36993 −0.684965 0.728576i \(-0.740183\pi\)
−0.684965 + 0.728576i \(0.740183\pi\)
\(968\) 0 0
\(969\) −8.30570 −0.266818
\(970\) 0 0
\(971\) 43.4695 1.39500 0.697501 0.716584i \(-0.254295\pi\)
0.697501 + 0.716584i \(0.254295\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −7.20560 −0.230764
\(976\) 0 0
\(977\) −25.1705 −0.805275 −0.402637 0.915360i \(-0.631906\pi\)
−0.402637 + 0.915360i \(0.631906\pi\)
\(978\) 0 0
\(979\) −16.8840 −0.539615
\(980\) 0 0
\(981\) 17.1103 0.546291
\(982\) 0 0
\(983\) −32.9857 −1.05208 −0.526041 0.850459i \(-0.676324\pi\)
−0.526041 + 0.850459i \(0.676324\pi\)
\(984\) 0 0
\(985\) 2.11383 0.0673521
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.53547 −0.271412
\(990\) 0 0
\(991\) 41.2891 1.31159 0.655796 0.754938i \(-0.272333\pi\)
0.655796 + 0.754938i \(0.272333\pi\)
\(992\) 0 0
\(993\) −4.21239 −0.133676
\(994\) 0 0
\(995\) −41.8281 −1.32604
\(996\) 0 0
\(997\) −10.5184 −0.333122 −0.166561 0.986031i \(-0.553266\pi\)
−0.166561 + 0.986031i \(0.553266\pi\)
\(998\) 0 0
\(999\) 23.9145 0.756622
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 7448.2.a.bq.1.3 8
7.2 even 3 1064.2.q.n.305.6 16
7.4 even 3 1064.2.q.n.457.6 yes 16
7.6 odd 2 7448.2.a.br.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1064.2.q.n.305.6 16 7.2 even 3
1064.2.q.n.457.6 yes 16 7.4 even 3
7448.2.a.bq.1.3 8 1.1 even 1 trivial
7448.2.a.br.1.6 8 7.6 odd 2