Properties

Label 1440.2.h.b.1151.4
Level $1440$
Weight $2$
Character 1440.1151
Analytic conductor $11.498$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1151.4
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1440.1151
Dual form 1440.2.h.b.1151.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{5} +3.41421i q^{7} +O(q^{10})\) \(q+1.00000i q^{5} +3.41421i q^{7} +1.41421 q^{11} -6.24264 q^{13} +6.82843i q^{17} -5.65685i q^{19} -8.82843 q^{23} -1.00000 q^{25} +2.00000i q^{29} -8.82843i q^{31} -3.41421 q^{35} +7.41421 q^{37} -0.242641i q^{41} +1.17157i q^{43} -8.00000 q^{47} -4.65685 q^{49} +1.17157i q^{53} +1.41421i q^{55} -2.58579 q^{59} -8.82843 q^{61} -6.24264i q^{65} -11.3137i q^{67} -2.34315 q^{71} -4.82843 q^{73} +4.82843i q^{77} +10.4853i q^{79} +6.82843 q^{83} -6.82843 q^{85} +15.0711i q^{89} -21.3137i q^{91} +5.65685 q^{95} -6.48528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q - 8 q^{13} - 24 q^{23} - 4 q^{25} - 8 q^{35} + 24 q^{37} - 32 q^{47} + 4 q^{49} - 16 q^{59} - 24 q^{61} - 32 q^{71} - 8 q^{73} + 16 q^{83} - 16 q^{85} + 8 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 3.41421i 1.29045i 0.763992 + 0.645226i \(0.223237\pi\)
−0.763992 + 0.645226i \(0.776763\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421 0.426401 0.213201 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) 0 0
\(13\) −6.24264 −1.73140 −0.865699 0.500566i \(-0.833125\pi\)
−0.865699 + 0.500566i \(0.833125\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.82843i 1.65614i 0.560627 + 0.828068i \(0.310560\pi\)
−0.560627 + 0.828068i \(0.689440\pi\)
\(18\) 0 0
\(19\) − 5.65685i − 1.29777i −0.760886 0.648886i \(-0.775235\pi\)
0.760886 0.648886i \(-0.224765\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.82843 −1.84085 −0.920427 0.390914i \(-0.872159\pi\)
−0.920427 + 0.390914i \(0.872159\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) − 8.82843i − 1.58563i −0.609461 0.792816i \(-0.708614\pi\)
0.609461 0.792816i \(-0.291386\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.41421 −0.577107
\(36\) 0 0
\(37\) 7.41421 1.21889 0.609445 0.792829i \(-0.291392\pi\)
0.609445 + 0.792829i \(0.291392\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 0.242641i − 0.0378941i −0.999820 0.0189471i \(-0.993969\pi\)
0.999820 0.0189471i \(-0.00603140\pi\)
\(42\) 0 0
\(43\) 1.17157i 0.178663i 0.996002 + 0.0893316i \(0.0284731\pi\)
−0.996002 + 0.0893316i \(0.971527\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −4.65685 −0.665265
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.17157i 0.160928i 0.996758 + 0.0804640i \(0.0256402\pi\)
−0.996758 + 0.0804640i \(0.974360\pi\)
\(54\) 0 0
\(55\) 1.41421i 0.190693i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.58579 −0.336641 −0.168320 0.985732i \(-0.553834\pi\)
−0.168320 + 0.985732i \(0.553834\pi\)
\(60\) 0 0
\(61\) −8.82843 −1.13036 −0.565182 0.824966i \(-0.691194\pi\)
−0.565182 + 0.824966i \(0.691194\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 6.24264i − 0.774304i
\(66\) 0 0
\(67\) − 11.3137i − 1.38219i −0.722764 0.691095i \(-0.757129\pi\)
0.722764 0.691095i \(-0.242871\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.34315 −0.278080 −0.139040 0.990287i \(-0.544402\pi\)
−0.139040 + 0.990287i \(0.544402\pi\)
\(72\) 0 0
\(73\) −4.82843 −0.565125 −0.282562 0.959249i \(-0.591184\pi\)
−0.282562 + 0.959249i \(0.591184\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.82843i 0.550250i
\(78\) 0 0
\(79\) 10.4853i 1.17969i 0.807518 + 0.589843i \(0.200810\pi\)
−0.807518 + 0.589843i \(0.799190\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.82843 0.749517 0.374759 0.927122i \(-0.377726\pi\)
0.374759 + 0.927122i \(0.377726\pi\)
\(84\) 0 0
\(85\) −6.82843 −0.740647
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.0711i 1.59753i 0.601643 + 0.798765i \(0.294513\pi\)
−0.601643 + 0.798765i \(0.705487\pi\)
\(90\) 0 0
\(91\) − 21.3137i − 2.23428i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.65685 0.580381
\(96\) 0 0
\(97\) −6.48528 −0.658481 −0.329240 0.944246i \(-0.606793\pi\)
−0.329240 + 0.944246i \(0.606793\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.8284i 1.67449i 0.546827 + 0.837246i \(0.315835\pi\)
−0.546827 + 0.837246i \(0.684165\pi\)
\(102\) 0 0
\(103\) 6.24264i 0.615106i 0.951531 + 0.307553i \(0.0995101\pi\)
−0.951531 + 0.307553i \(0.900490\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.1421 −0.980477 −0.490239 0.871588i \(-0.663090\pi\)
−0.490239 + 0.871588i \(0.663090\pi\)
\(108\) 0 0
\(109\) 3.17157 0.303782 0.151891 0.988397i \(-0.451464\pi\)
0.151891 + 0.988397i \(0.451464\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.00000i 0.376288i 0.982141 + 0.188144i \(0.0602472\pi\)
−0.982141 + 0.188144i \(0.939753\pi\)
\(114\) 0 0
\(115\) − 8.82843i − 0.823255i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −23.3137 −2.13716
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 1.00000i − 0.0894427i
\(126\) 0 0
\(127\) 15.4142i 1.36779i 0.729580 + 0.683895i \(0.239715\pi\)
−0.729580 + 0.683895i \(0.760285\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 13.8995 1.21440 0.607202 0.794547i \(-0.292292\pi\)
0.607202 + 0.794547i \(0.292292\pi\)
\(132\) 0 0
\(133\) 19.3137 1.67471
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.65685i 0.483298i 0.970364 + 0.241649i \(0.0776882\pi\)
−0.970364 + 0.241649i \(0.922312\pi\)
\(138\) 0 0
\(139\) 8.34315i 0.707656i 0.935310 + 0.353828i \(0.115120\pi\)
−0.935310 + 0.353828i \(0.884880\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.82843 −0.738270
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 4.82843i − 0.395560i −0.980246 0.197780i \(-0.936627\pi\)
0.980246 0.197780i \(-0.0633732\pi\)
\(150\) 0 0
\(151\) 10.9706i 0.892772i 0.894841 + 0.446386i \(0.147289\pi\)
−0.894841 + 0.446386i \(0.852711\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.82843 0.709116
\(156\) 0 0
\(157\) 0.100505 0.00802118 0.00401059 0.999992i \(-0.498723\pi\)
0.00401059 + 0.999992i \(0.498723\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 30.1421i − 2.37553i
\(162\) 0 0
\(163\) − 8.48528i − 0.664619i −0.943170 0.332309i \(-0.892172\pi\)
0.943170 0.332309i \(-0.107828\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.4853 1.12090 0.560452 0.828187i \(-0.310627\pi\)
0.560452 + 0.828187i \(0.310627\pi\)
\(168\) 0 0
\(169\) 25.9706 1.99774
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.485281i 0.0368953i 0.999830 + 0.0184476i \(0.00587240\pi\)
−0.999830 + 0.0184476i \(0.994128\pi\)
\(174\) 0 0
\(175\) − 3.41421i − 0.258090i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.4142 1.00263 0.501313 0.865266i \(-0.332851\pi\)
0.501313 + 0.865266i \(0.332851\pi\)
\(180\) 0 0
\(181\) −24.8284 −1.84548 −0.922741 0.385420i \(-0.874057\pi\)
−0.922741 + 0.385420i \(0.874057\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.41421i 0.545104i
\(186\) 0 0
\(187\) 9.65685i 0.706179i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.17157 −0.0847720 −0.0423860 0.999101i \(-0.513496\pi\)
−0.0423860 + 0.999101i \(0.513496\pi\)
\(192\) 0 0
\(193\) 4.34315 0.312626 0.156313 0.987708i \(-0.450039\pi\)
0.156313 + 0.987708i \(0.450039\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) − 15.6569i − 1.10988i −0.831889 0.554942i \(-0.812740\pi\)
0.831889 0.554942i \(-0.187260\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.82843 −0.479262
\(204\) 0 0
\(205\) 0.242641 0.0169468
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 8.00000i − 0.553372i
\(210\) 0 0
\(211\) − 6.00000i − 0.413057i −0.978441 0.206529i \(-0.933783\pi\)
0.978441 0.206529i \(-0.0662166\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.17157 −0.0799006
\(216\) 0 0
\(217\) 30.1421 2.04618
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 42.6274i − 2.86743i
\(222\) 0 0
\(223\) − 4.10051i − 0.274590i −0.990530 0.137295i \(-0.956159\pi\)
0.990530 0.137295i \(-0.0438409\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −28.2843 −1.87729 −0.938647 0.344881i \(-0.887919\pi\)
−0.938647 + 0.344881i \(0.887919\pi\)
\(228\) 0 0
\(229\) 7.65685 0.505979 0.252990 0.967469i \(-0.418586\pi\)
0.252990 + 0.967469i \(0.418586\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.4853i 0.817938i 0.912548 + 0.408969i \(0.134112\pi\)
−0.912548 + 0.408969i \(0.865888\pi\)
\(234\) 0 0
\(235\) − 8.00000i − 0.521862i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.14214 −0.397302 −0.198651 0.980070i \(-0.563656\pi\)
−0.198651 + 0.980070i \(0.563656\pi\)
\(240\) 0 0
\(241\) −14.3431 −0.923923 −0.461962 0.886900i \(-0.652854\pi\)
−0.461962 + 0.886900i \(0.652854\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 4.65685i − 0.297516i
\(246\) 0 0
\(247\) 35.3137i 2.24696i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.7279 1.30833 0.654167 0.756350i \(-0.273019\pi\)
0.654167 + 0.756350i \(0.273019\pi\)
\(252\) 0 0
\(253\) −12.4853 −0.784943
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.97056i 0.559568i 0.960063 + 0.279784i \(0.0902629\pi\)
−0.960063 + 0.279784i \(0.909737\pi\)
\(258\) 0 0
\(259\) 25.3137i 1.57292i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 24.8284 1.53099 0.765493 0.643444i \(-0.222495\pi\)
0.765493 + 0.643444i \(0.222495\pi\)
\(264\) 0 0
\(265\) −1.17157 −0.0719691
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.9706i 0.668887i 0.942416 + 0.334444i \(0.108548\pi\)
−0.942416 + 0.334444i \(0.891452\pi\)
\(270\) 0 0
\(271\) 1.31371i 0.0798021i 0.999204 + 0.0399011i \(0.0127043\pi\)
−0.999204 + 0.0399011i \(0.987296\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.41421 −0.0852803
\(276\) 0 0
\(277\) 32.8701 1.97497 0.987485 0.157712i \(-0.0504117\pi\)
0.987485 + 0.157712i \(0.0504117\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 1.89949i − 0.113314i −0.998394 0.0566572i \(-0.981956\pi\)
0.998394 0.0566572i \(-0.0180442\pi\)
\(282\) 0 0
\(283\) − 15.3137i − 0.910305i −0.890413 0.455153i \(-0.849585\pi\)
0.890413 0.455153i \(-0.150415\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.828427 0.0489005
\(288\) 0 0
\(289\) −29.6274 −1.74279
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.00000i 0.116841i 0.998292 + 0.0584206i \(0.0186065\pi\)
−0.998292 + 0.0584206i \(0.981394\pi\)
\(294\) 0 0
\(295\) − 2.58579i − 0.150550i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 55.1127 3.18725
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 8.82843i − 0.505514i
\(306\) 0 0
\(307\) 19.3137i 1.10229i 0.834409 + 0.551146i \(0.185809\pi\)
−0.834409 + 0.551146i \(0.814191\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.4853 1.61525 0.807626 0.589695i \(-0.200752\pi\)
0.807626 + 0.589695i \(0.200752\pi\)
\(312\) 0 0
\(313\) 22.2843 1.25958 0.629791 0.776765i \(-0.283141\pi\)
0.629791 + 0.776765i \(0.283141\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.31371i 0.298448i 0.988803 + 0.149224i \(0.0476775\pi\)
−0.988803 + 0.149224i \(0.952323\pi\)
\(318\) 0 0
\(319\) 2.82843i 0.158362i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 38.6274 2.14929
\(324\) 0 0
\(325\) 6.24264 0.346279
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 27.3137i − 1.50585i
\(330\) 0 0
\(331\) − 8.68629i − 0.477442i −0.971088 0.238721i \(-0.923272\pi\)
0.971088 0.238721i \(-0.0767281\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 11.3137 0.618134
\(336\) 0 0
\(337\) 11.1716 0.608554 0.304277 0.952584i \(-0.401585\pi\)
0.304277 + 0.952584i \(0.401585\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 12.4853i − 0.676116i
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.9706 −0.696296 −0.348148 0.937440i \(-0.613189\pi\)
−0.348148 + 0.937440i \(0.613189\pi\)
\(348\) 0 0
\(349\) 3.17157 0.169770 0.0848852 0.996391i \(-0.472948\pi\)
0.0848852 + 0.996391i \(0.472948\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 32.4853i − 1.72902i −0.502618 0.864509i \(-0.667630\pi\)
0.502618 0.864509i \(-0.332370\pi\)
\(354\) 0 0
\(355\) − 2.34315i − 0.124361i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −31.1127 −1.64207 −0.821033 0.570881i \(-0.806602\pi\)
−0.821033 + 0.570881i \(0.806602\pi\)
\(360\) 0 0
\(361\) −13.0000 −0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 4.82843i − 0.252731i
\(366\) 0 0
\(367\) 33.0711i 1.72630i 0.504951 + 0.863148i \(0.331510\pi\)
−0.504951 + 0.863148i \(0.668490\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) 17.0711 0.883906 0.441953 0.897038i \(-0.354286\pi\)
0.441953 + 0.897038i \(0.354286\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 12.4853i − 0.643025i
\(378\) 0 0
\(379\) 29.3137i 1.50574i 0.658167 + 0.752872i \(0.271332\pi\)
−0.658167 + 0.752872i \(0.728668\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −14.6274 −0.747426 −0.373713 0.927544i \(-0.621916\pi\)
−0.373713 + 0.927544i \(0.621916\pi\)
\(384\) 0 0
\(385\) −4.82843 −0.246079
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.65685i 0.388218i 0.980980 + 0.194109i \(0.0621815\pi\)
−0.980980 + 0.194109i \(0.937818\pi\)
\(390\) 0 0
\(391\) − 60.2843i − 3.04871i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −10.4853 −0.527572
\(396\) 0 0
\(397\) −16.3848 −0.822328 −0.411164 0.911561i \(-0.634878\pi\)
−0.411164 + 0.911561i \(0.634878\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 16.7279i 0.835353i 0.908596 + 0.417676i \(0.137155\pi\)
−0.908596 + 0.417676i \(0.862845\pi\)
\(402\) 0 0
\(403\) 55.1127i 2.74536i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.4853 0.519736
\(408\) 0 0
\(409\) 11.6569 0.576394 0.288197 0.957571i \(-0.406944\pi\)
0.288197 + 0.957571i \(0.406944\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 8.82843i − 0.434418i
\(414\) 0 0
\(415\) 6.82843i 0.335194i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −36.2426 −1.77057 −0.885284 0.465050i \(-0.846036\pi\)
−0.885284 + 0.465050i \(0.846036\pi\)
\(420\) 0 0
\(421\) 27.9411 1.36177 0.680884 0.732392i \(-0.261596\pi\)
0.680884 + 0.732392i \(0.261596\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 6.82843i − 0.331227i
\(426\) 0 0
\(427\) − 30.1421i − 1.45868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.4853 0.601395 0.300697 0.953720i \(-0.402781\pi\)
0.300697 + 0.953720i \(0.402781\pi\)
\(432\) 0 0
\(433\) 7.17157 0.344644 0.172322 0.985041i \(-0.444873\pi\)
0.172322 + 0.985041i \(0.444873\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 49.9411i 2.38901i
\(438\) 0 0
\(439\) − 4.34315i − 0.207287i −0.994615 0.103644i \(-0.966950\pi\)
0.994615 0.103644i \(-0.0330501\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −15.3137 −0.727576 −0.363788 0.931482i \(-0.618517\pi\)
−0.363788 + 0.931482i \(0.618517\pi\)
\(444\) 0 0
\(445\) −15.0711 −0.714437
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 17.2132i 0.812341i 0.913797 + 0.406171i \(0.133136\pi\)
−0.913797 + 0.406171i \(0.866864\pi\)
\(450\) 0 0
\(451\) − 0.343146i − 0.0161581i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 21.3137 0.999202
\(456\) 0 0
\(457\) 32.1421 1.50355 0.751773 0.659422i \(-0.229199\pi\)
0.751773 + 0.659422i \(0.229199\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 33.7990i − 1.57418i −0.616841 0.787088i \(-0.711588\pi\)
0.616841 0.787088i \(-0.288412\pi\)
\(462\) 0 0
\(463\) − 20.3848i − 0.947361i −0.880697 0.473680i \(-0.842925\pi\)
0.880697 0.473680i \(-0.157075\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −11.7990 −0.545992 −0.272996 0.962015i \(-0.588015\pi\)
−0.272996 + 0.962015i \(0.588015\pi\)
\(468\) 0 0
\(469\) 38.6274 1.78365
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.65685i 0.0761822i
\(474\) 0 0
\(475\) 5.65685i 0.259554i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) −46.2843 −2.11038
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 6.48528i − 0.294481i
\(486\) 0 0
\(487\) − 6.04163i − 0.273772i −0.990587 0.136886i \(-0.956291\pi\)
0.990587 0.136886i \(-0.0437095\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −12.9289 −0.583475 −0.291737 0.956498i \(-0.594233\pi\)
−0.291737 + 0.956498i \(0.594233\pi\)
\(492\) 0 0
\(493\) −13.6569 −0.615074
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 8.00000i − 0.358849i
\(498\) 0 0
\(499\) − 12.9706i − 0.580642i −0.956929 0.290321i \(-0.906238\pi\)
0.956929 0.290321i \(-0.0937621\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −16.8284 −0.748855
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 20.1421i − 0.892784i −0.894837 0.446392i \(-0.852709\pi\)
0.894837 0.446392i \(-0.147291\pi\)
\(510\) 0 0
\(511\) − 16.4853i − 0.729266i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.24264 −0.275084
\(516\) 0 0
\(517\) −11.3137 −0.497576
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.27208i 0.318596i 0.987231 + 0.159298i \(0.0509230\pi\)
−0.987231 + 0.159298i \(0.949077\pi\)
\(522\) 0 0
\(523\) 27.7990i 1.21556i 0.794104 + 0.607782i \(0.207941\pi\)
−0.794104 + 0.607782i \(0.792059\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 60.2843 2.62602
\(528\) 0 0
\(529\) 54.9411 2.38874
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.51472i 0.0656097i
\(534\) 0 0
\(535\) − 10.1421i − 0.438483i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.58579 −0.283670
\(540\) 0 0
\(541\) −4.14214 −0.178084 −0.0890422 0.996028i \(-0.528381\pi\)
−0.0890422 + 0.996028i \(0.528381\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.17157i 0.135855i
\(546\) 0 0
\(547\) − 4.20101i − 0.179622i −0.995959 0.0898111i \(-0.971374\pi\)
0.995959 0.0898111i \(-0.0286263\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 11.3137 0.481980
\(552\) 0 0
\(553\) −35.7990 −1.52233
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 20.4853i − 0.867989i −0.900915 0.433995i \(-0.857104\pi\)
0.900915 0.433995i \(-0.142896\pi\)
\(558\) 0 0
\(559\) − 7.31371i − 0.309337i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −22.3431 −0.941651 −0.470826 0.882226i \(-0.656044\pi\)
−0.470826 + 0.882226i \(0.656044\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 34.5858i 1.44991i 0.688795 + 0.724956i \(0.258140\pi\)
−0.688795 + 0.724956i \(0.741860\pi\)
\(570\) 0 0
\(571\) 34.6274i 1.44911i 0.689216 + 0.724556i \(0.257955\pi\)
−0.689216 + 0.724556i \(0.742045\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8.82843 0.368171
\(576\) 0 0
\(577\) −27.4558 −1.14300 −0.571501 0.820601i \(-0.693639\pi\)
−0.571501 + 0.820601i \(0.693639\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 23.3137i 0.967216i
\(582\) 0 0
\(583\) 1.65685i 0.0686199i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −37.4558 −1.54597 −0.772984 0.634425i \(-0.781237\pi\)
−0.772984 + 0.634425i \(0.781237\pi\)
\(588\) 0 0
\(589\) −49.9411 −2.05779
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 14.6274i − 0.600676i −0.953833 0.300338i \(-0.902901\pi\)
0.953833 0.300338i \(-0.0970995\pi\)
\(594\) 0 0
\(595\) − 23.3137i − 0.955769i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 13.8579 0.566217 0.283108 0.959088i \(-0.408634\pi\)
0.283108 + 0.959088i \(0.408634\pi\)
\(600\) 0 0
\(601\) 45.9411 1.87398 0.936989 0.349359i \(-0.113601\pi\)
0.936989 + 0.349359i \(0.113601\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 9.00000i − 0.365902i
\(606\) 0 0
\(607\) − 10.0416i − 0.407577i −0.979015 0.203789i \(-0.934674\pi\)
0.979015 0.203789i \(-0.0653255\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 49.9411 2.02040
\(612\) 0 0
\(613\) −21.0711 −0.851052 −0.425526 0.904946i \(-0.639911\pi\)
−0.425526 + 0.904946i \(0.639911\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13.1716i 0.530268i 0.964212 + 0.265134i \(0.0854161\pi\)
−0.964212 + 0.265134i \(0.914584\pi\)
\(618\) 0 0
\(619\) 26.9706i 1.08404i 0.840366 + 0.542019i \(0.182340\pi\)
−0.840366 + 0.542019i \(0.817660\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −51.4558 −2.06153
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 50.6274i 2.01865i
\(630\) 0 0
\(631\) − 16.1421i − 0.642608i −0.946976 0.321304i \(-0.895879\pi\)
0.946976 0.321304i \(-0.104121\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −15.4142 −0.611694
\(636\) 0 0
\(637\) 29.0711 1.15184
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 14.8701i 0.587332i 0.955908 + 0.293666i \(0.0948753\pi\)
−0.955908 + 0.293666i \(0.905125\pi\)
\(642\) 0 0
\(643\) 1.17157i 0.0462023i 0.999733 + 0.0231012i \(0.00735398\pi\)
−0.999733 + 0.0231012i \(0.992646\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.0294 −0.590868 −0.295434 0.955363i \(-0.595464\pi\)
−0.295434 + 0.955363i \(0.595464\pi\)
\(648\) 0 0
\(649\) −3.65685 −0.143544
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 30.2843i 1.18512i 0.805528 + 0.592558i \(0.201882\pi\)
−0.805528 + 0.592558i \(0.798118\pi\)
\(654\) 0 0
\(655\) 13.8995i 0.543098i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.04163 0.157440 0.0787198 0.996897i \(-0.474917\pi\)
0.0787198 + 0.996897i \(0.474917\pi\)
\(660\) 0 0
\(661\) −28.1421 −1.09460 −0.547301 0.836936i \(-0.684345\pi\)
−0.547301 + 0.836936i \(0.684345\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 19.3137i 0.748953i
\(666\) 0 0
\(667\) − 17.6569i − 0.683676i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −12.4853 −0.481989
\(672\) 0 0
\(673\) −42.7696 −1.64865 −0.824323 0.566120i \(-0.808444\pi\)
−0.824323 + 0.566120i \(0.808444\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.68629i 0.256975i 0.991711 + 0.128488i \(0.0410122\pi\)
−0.991711 + 0.128488i \(0.958988\pi\)
\(678\) 0 0
\(679\) − 22.1421i − 0.849737i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 19.7990 0.757587 0.378794 0.925481i \(-0.376339\pi\)
0.378794 + 0.925481i \(0.376339\pi\)
\(684\) 0 0
\(685\) −5.65685 −0.216137
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 7.31371i − 0.278630i
\(690\) 0 0
\(691\) − 29.6569i − 1.12820i −0.825707 0.564100i \(-0.809223\pi\)
0.825707 0.564100i \(-0.190777\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.34315 −0.316474
\(696\) 0 0
\(697\) 1.65685 0.0627578
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 21.5147i 0.812600i 0.913740 + 0.406300i \(0.133181\pi\)
−0.913740 + 0.406300i \(0.866819\pi\)
\(702\) 0 0
\(703\) − 41.9411i − 1.58184i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −57.4558 −2.16085
\(708\) 0 0
\(709\) −31.6569 −1.18890 −0.594449 0.804133i \(-0.702630\pi\)
−0.594449 + 0.804133i \(0.702630\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 77.9411i 2.91892i
\(714\) 0 0
\(715\) − 8.82843i − 0.330164i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −27.7990 −1.03673 −0.518364 0.855160i \(-0.673459\pi\)
−0.518364 + 0.855160i \(0.673459\pi\)
\(720\) 0 0
\(721\) −21.3137 −0.793764
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 2.00000i − 0.0742781i
\(726\) 0 0
\(727\) 9.75736i 0.361880i 0.983494 + 0.180940i \(0.0579141\pi\)
−0.983494 + 0.180940i \(0.942086\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −2.24264 −0.0828338 −0.0414169 0.999142i \(-0.513187\pi\)
−0.0414169 + 0.999142i \(0.513187\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 16.0000i − 0.589368i
\(738\) 0 0
\(739\) 12.0000i 0.441427i 0.975339 + 0.220714i \(0.0708386\pi\)
−0.975339 + 0.220714i \(0.929161\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.02944 0.257885 0.128943 0.991652i \(-0.458842\pi\)
0.128943 + 0.991652i \(0.458842\pi\)
\(744\) 0 0
\(745\) 4.82843 0.176900
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 34.6274i − 1.26526i
\(750\) 0 0
\(751\) − 21.7990i − 0.795456i −0.917503 0.397728i \(-0.869799\pi\)
0.917503 0.397728i \(-0.130201\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −10.9706 −0.399260
\(756\) 0 0
\(757\) −3.41421 −0.124092 −0.0620459 0.998073i \(-0.519763\pi\)
−0.0620459 + 0.998073i \(0.519763\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 38.8701i − 1.40904i −0.709685 0.704519i \(-0.751163\pi\)
0.709685 0.704519i \(-0.248837\pi\)
\(762\) 0 0
\(763\) 10.8284i 0.392015i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 16.1421 0.582859
\(768\) 0 0
\(769\) −31.5980 −1.13945 −0.569726 0.821835i \(-0.692951\pi\)
−0.569726 + 0.821835i \(0.692951\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 43.2548i − 1.55577i −0.628408 0.777884i \(-0.716293\pi\)
0.628408 0.777884i \(-0.283707\pi\)
\(774\) 0 0
\(775\) 8.82843i 0.317126i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.37258 −0.0491779
\(780\) 0 0
\(781\) −3.31371 −0.118574
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.100505i 0.00358718i
\(786\) 0 0
\(787\) 26.1421i 0.931866i 0.884820 + 0.465933i \(0.154281\pi\)
−0.884820 + 0.465933i \(0.845719\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −13.6569 −0.485582
\(792\) 0 0
\(793\) 55.1127 1.95711
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 0.485281i − 0.0171895i −0.999963 0.00859477i \(-0.997264\pi\)
0.999963 0.00859477i \(-0.00273584\pi\)
\(798\) 0 0
\(799\) − 54.6274i − 1.93258i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.82843 −0.240970
\(804\) 0 0
\(805\) 30.1421 1.06237
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 10.1005i − 0.355115i −0.984110 0.177557i \(-0.943180\pi\)
0.984110 0.177557i \(-0.0568195\pi\)
\(810\) 0 0
\(811\) − 12.6274i − 0.443409i −0.975114 0.221704i \(-0.928838\pi\)
0.975114 0.221704i \(-0.0711620\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8.48528 0.297226
\(816\) 0 0
\(817\) 6.62742 0.231864
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 5.31371i 0.185450i 0.995692 + 0.0927249i \(0.0295577\pi\)
−0.995692 + 0.0927249i \(0.970442\pi\)
\(822\) 0 0
\(823\) − 36.8701i − 1.28521i −0.766198 0.642605i \(-0.777854\pi\)
0.766198 0.642605i \(-0.222146\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −50.6274 −1.76049 −0.880244 0.474522i \(-0.842621\pi\)
−0.880244 + 0.474522i \(0.842621\pi\)
\(828\) 0 0
\(829\) 13.5147 0.469386 0.234693 0.972070i \(-0.424592\pi\)
0.234693 + 0.972070i \(0.424592\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 31.7990i − 1.10177i
\(834\) 0 0
\(835\) 14.4853i 0.501284i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −24.4853 −0.845326 −0.422663 0.906287i \(-0.638905\pi\)
−0.422663 + 0.906287i \(0.638905\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 25.9706i 0.893415i
\(846\) 0 0
\(847\) − 30.7279i − 1.05582i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −65.4558 −2.24380
\(852\) 0 0
\(853\) 47.2132 1.61655 0.808275 0.588806i \(-0.200402\pi\)
0.808275 + 0.588806i \(0.200402\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 48.4853i 1.65623i 0.560561 + 0.828113i \(0.310585\pi\)
−0.560561 + 0.828113i \(0.689415\pi\)
\(858\) 0 0
\(859\) 10.0000i 0.341196i 0.985341 + 0.170598i \(0.0545699\pi\)
−0.985341 + 0.170598i \(0.945430\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 50.7696 1.72822 0.864108 0.503307i \(-0.167883\pi\)
0.864108 + 0.503307i \(0.167883\pi\)
\(864\) 0 0
\(865\) −0.485281 −0.0165001
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 14.8284i 0.503020i
\(870\) 0 0
\(871\) 70.6274i 2.39312i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.41421 0.115421
\(876\) 0 0
\(877\) 4.58579 0.154851 0.0774255 0.996998i \(-0.475330\pi\)
0.0774255 + 0.996998i \(0.475330\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 25.4142i 0.856227i 0.903725 + 0.428113i \(0.140822\pi\)
−0.903725 + 0.428113i \(0.859178\pi\)
\(882\) 0 0
\(883\) − 33.4558i − 1.12588i −0.826498 0.562939i \(-0.809670\pi\)
0.826498 0.562939i \(-0.190330\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 28.1421 0.944920 0.472460 0.881352i \(-0.343366\pi\)
0.472460 + 0.881352i \(0.343366\pi\)
\(888\) 0 0
\(889\) −52.6274 −1.76507
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 45.2548i 1.51440i
\(894\) 0 0
\(895\) 13.4142i 0.448388i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 17.6569 0.588889
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 24.8284i − 0.825325i
\(906\) 0 0
\(907\) 31.1127i 1.03308i 0.856263 + 0.516540i \(0.172780\pi\)
−0.856263 + 0.516540i \(0.827220\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −36.2843 −1.20215 −0.601076 0.799192i \(-0.705261\pi\)
−0.601076 + 0.799192i \(0.705261\pi\)
\(912\) 0 0
\(913\) 9.65685 0.319595
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 47.4558i 1.56713i
\(918\) 0 0
\(919\) − 26.4853i − 0.873669i −0.899542 0.436834i \(-0.856100\pi\)
0.899542 0.436834i \(-0.143900\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 14.6274 0.481467
\(924\) 0 0
\(925\) −7.41421 −0.243778
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 32.7279i − 1.07377i −0.843656 0.536884i \(-0.819601\pi\)
0.843656 0.536884i \(-0.180399\pi\)
\(930\) 0 0
\(931\) 26.3431i 0.863362i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −9.65685 −0.315813
\(936\) 0 0
\(937\) −42.2843 −1.38137 −0.690683 0.723157i \(-0.742690\pi\)
−0.690683 + 0.723157i \(0.742690\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 46.9706i 1.53120i 0.643319 + 0.765598i \(0.277557\pi\)
−0.643319 + 0.765598i \(0.722443\pi\)
\(942\) 0 0
\(943\) 2.14214i 0.0697575i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 44.9706 1.46135 0.730673 0.682727i \(-0.239206\pi\)
0.730673 + 0.682727i \(0.239206\pi\)
\(948\) 0 0
\(949\) 30.1421 0.978455
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 2.34315i − 0.0759019i −0.999280 0.0379510i \(-0.987917\pi\)
0.999280 0.0379510i \(-0.0120831\pi\)
\(954\) 0 0
\(955\) − 1.17157i − 0.0379112i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −19.3137 −0.623672
\(960\) 0 0
\(961\) −46.9411 −1.51423
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.34315i 0.139811i
\(966\) 0 0
\(967\) 19.8995i 0.639925i 0.947430 + 0.319962i \(0.103670\pi\)
−0.947430 + 0.319962i \(0.896330\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −29.6985 −0.953070 −0.476535 0.879156i \(-0.658107\pi\)
−0.476535 + 0.879156i \(0.658107\pi\)
\(972\) 0 0
\(973\) −28.4853 −0.913196
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24.4853i 0.783354i 0.920103 + 0.391677i \(0.128105\pi\)
−0.920103 + 0.391677i \(0.871895\pi\)
\(978\) 0 0
\(979\) 21.3137i 0.681189i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −19.3137 −0.616012 −0.308006 0.951384i \(-0.599662\pi\)
−0.308006 + 0.951384i \(0.599662\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 10.3431i − 0.328893i
\(990\) 0 0
\(991\) − 0.343146i − 0.0109004i −0.999985 0.00545019i \(-0.998265\pi\)
0.999985 0.00545019i \(-0.00173486\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 15.6569 0.496356
\(996\) 0 0
\(997\) −34.2426 −1.08448 −0.542238 0.840225i \(-0.682423\pi\)
−0.542238 + 0.840225i \(0.682423\pi\)
\(998\) 0 0
\(999\) 0 0
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 1440.2.h.b.1151.4 yes 4
3.2 odd 2 1440.2.h.c.1151.2 yes 4
4.3 odd 2 1440.2.h.c.1151.3 yes 4
5.2 odd 4 7200.2.o.c.7199.2 4
5.3 odd 4 7200.2.o.l.7199.3 4
5.4 even 2 7200.2.h.f.1151.1 4
8.3 odd 2 2880.2.h.c.1151.1 4
8.5 even 2 2880.2.h.b.1151.2 4
12.11 even 2 inner 1440.2.h.b.1151.1 4
15.2 even 4 7200.2.o.d.7199.2 4
15.8 even 4 7200.2.o.k.7199.3 4
15.14 odd 2 7200.2.h.e.1151.1 4
20.3 even 4 7200.2.o.d.7199.1 4
20.7 even 4 7200.2.o.k.7199.4 4
20.19 odd 2 7200.2.h.e.1151.4 4
24.5 odd 2 2880.2.h.c.1151.4 4
24.11 even 2 2880.2.h.b.1151.3 4
60.23 odd 4 7200.2.o.c.7199.1 4
60.47 odd 4 7200.2.o.l.7199.4 4
60.59 even 2 7200.2.h.f.1151.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1440.2.h.b.1151.1 4 12.11 even 2 inner
1440.2.h.b.1151.4 yes 4 1.1 even 1 trivial
1440.2.h.c.1151.2 yes 4 3.2 odd 2
1440.2.h.c.1151.3 yes 4 4.3 odd 2
2880.2.h.b.1151.2 4 8.5 even 2
2880.2.h.b.1151.3 4 24.11 even 2
2880.2.h.c.1151.1 4 8.3 odd 2
2880.2.h.c.1151.4 4 24.5 odd 2
7200.2.h.e.1151.1 4 15.14 odd 2
7200.2.h.e.1151.4 4 20.19 odd 2
7200.2.h.f.1151.1 4 5.4 even 2
7200.2.h.f.1151.4 4 60.59 even 2
7200.2.o.c.7199.1 4 60.23 odd 4
7200.2.o.c.7199.2 4 5.2 odd 4
7200.2.o.d.7199.1 4 20.3 even 4
7200.2.o.d.7199.2 4 15.2 even 4
7200.2.o.k.7199.3 4 15.8 even 4
7200.2.o.k.7199.4 4 20.7 even 4
7200.2.o.l.7199.3 4 5.3 odd 4
7200.2.o.l.7199.4 4 60.47 odd 4