Properties

Label 6048.2.j.a.5615.2
Level 6048
Weight 2
Character 6048.5615
Analytic conductor 48.294
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.56070144.2
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5615.2
Root \(0.500000 + 1.19293i\)
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.a.5615.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.38587 q^{5} +1.00000i q^{7} +O(q^{10})\) \(q-4.38587 q^{5} +1.00000i q^{7} -5.11792i q^{11} -5.21068i q^{13} -2.00000i q^{17} +6.50379 q^{19} -4.63929 q^{23} +14.2358 q^{25} -1.26795 q^{29} -4.21068i q^{31} -4.38587i q^{35} -1.98547i q^{37} +7.37134i q^{41} -3.71753 q^{43} +2.57558 q^{47} -1.00000 q^{49} +10.2358 q^{53} +22.4465i q^{55} +3.50379i q^{59} -4.53590i q^{61} +22.8533i q^{65} -5.21068 q^{67} -1.36071 q^{71} +15.4465 q^{73} +5.11792 q^{77} -3.98241i q^{79} -11.6893i q^{83} +8.77174i q^{85} -14.8645i q^{89} +5.21068 q^{91} -28.5247 q^{95} +4.42136 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + O(q^{10}) \) \( 8q - 8q^{5} - 16q^{19} - 16q^{23} + 32q^{25} - 24q^{29} - 8q^{43} + 8q^{47} - 8q^{49} - 8q^{67} - 32q^{71} + 8q^{73} + 8q^{91} - 112q^{95} - 32q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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
\(5\) −4.38587 −1.96142 −0.980710 0.195469i \(-0.937377\pi\)
−0.980710 + 0.195469i \(0.937377\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 5.11792i − 1.54311i −0.636162 0.771555i \(-0.719479\pi\)
0.636162 0.771555i \(-0.280521\pi\)
\(12\) 0 0
\(13\) − 5.21068i − 1.44518i −0.691276 0.722591i \(-0.742951\pi\)
0.691276 0.722591i \(-0.257049\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) 6.50379 1.49207 0.746035 0.665906i \(-0.231955\pi\)
0.746035 + 0.665906i \(0.231955\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.63929 −0.967359 −0.483680 0.875245i \(-0.660700\pi\)
−0.483680 + 0.875245i \(0.660700\pi\)
\(24\) 0 0
\(25\) 14.2358 2.84717
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.26795 −0.235452 −0.117726 0.993046i \(-0.537560\pi\)
−0.117726 + 0.993046i \(0.537560\pi\)
\(30\) 0 0
\(31\) − 4.21068i − 0.756260i −0.925752 0.378130i \(-0.876567\pi\)
0.925752 0.378130i \(-0.123433\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 4.38587i − 0.741347i
\(36\) 0 0
\(37\) − 1.98547i − 0.326410i −0.986592 0.163205i \(-0.947817\pi\)
0.986592 0.163205i \(-0.0521832\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.37134i 1.15121i 0.817728 + 0.575605i \(0.195233\pi\)
−0.817728 + 0.575605i \(0.804767\pi\)
\(42\) 0 0
\(43\) −3.71753 −0.566917 −0.283459 0.958984i \(-0.591482\pi\)
−0.283459 + 0.958984i \(0.591482\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.57558 0.375687 0.187844 0.982199i \(-0.439850\pi\)
0.187844 + 0.982199i \(0.439850\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.2358 1.40600 0.703000 0.711190i \(-0.251843\pi\)
0.703000 + 0.711190i \(0.251843\pi\)
\(54\) 0 0
\(55\) 22.4465i 3.02669i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.50379i 0.456154i 0.973643 + 0.228077i \(0.0732438\pi\)
−0.973643 + 0.228077i \(0.926756\pi\)
\(60\) 0 0
\(61\) − 4.53590i − 0.580762i −0.956911 0.290381i \(-0.906218\pi\)
0.956911 0.290381i \(-0.0937821\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 22.8533i 2.83461i
\(66\) 0 0
\(67\) −5.21068 −0.636586 −0.318293 0.947992i \(-0.603110\pi\)
−0.318293 + 0.947992i \(0.603110\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.36071 −0.161486 −0.0807432 0.996735i \(-0.525729\pi\)
−0.0807432 + 0.996735i \(0.525729\pi\)
\(72\) 0 0
\(73\) 15.4465 1.80788 0.903939 0.427662i \(-0.140662\pi\)
0.903939 + 0.427662i \(0.140662\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.11792 0.583241
\(78\) 0 0
\(79\) − 3.98241i − 0.448057i −0.974583 0.224028i \(-0.928079\pi\)
0.974583 0.224028i \(-0.0719208\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 11.6893i − 1.28307i −0.767095 0.641534i \(-0.778298\pi\)
0.767095 0.641534i \(-0.221702\pi\)
\(84\) 0 0
\(85\) 8.77174i 0.951428i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 14.8645i − 1.57563i −0.615910 0.787817i \(-0.711211\pi\)
0.615910 0.787817i \(-0.288789\pi\)
\(90\) 0 0
\(91\) 5.21068 0.546227
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −28.5247 −2.92658
\(96\) 0 0
\(97\) 4.42136 0.448921 0.224460 0.974483i \(-0.427938\pi\)
0.224460 + 0.974483i \(0.427938\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) − 1.67478i − 0.165021i −0.996590 0.0825105i \(-0.973706\pi\)
0.996590 0.0825105i \(-0.0262938\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.2649i 0.992344i 0.868224 + 0.496172i \(0.165262\pi\)
−0.868224 + 0.496172i \(0.834738\pi\)
\(108\) 0 0
\(109\) 1.29311i 0.123857i 0.998081 + 0.0619287i \(0.0197251\pi\)
−0.998081 + 0.0619287i \(0.980275\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 8.07937i − 0.760043i −0.924978 0.380022i \(-0.875917\pi\)
0.924978 0.380022i \(-0.124083\pi\)
\(114\) 0 0
\(115\) 20.3473 1.89740
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) −15.1931 −1.38119
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −40.5072 −3.62307
\(126\) 0 0
\(127\) 13.4465i 1.19319i 0.802544 + 0.596593i \(0.203479\pi\)
−0.802544 + 0.596593i \(0.796521\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 1.77480i − 0.155065i −0.996990 0.0775323i \(-0.975296\pi\)
0.996990 0.0775323i \(-0.0247041\pi\)
\(132\) 0 0
\(133\) 6.50379i 0.563950i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 14.5756i − 1.24528i −0.782510 0.622638i \(-0.786061\pi\)
0.782510 0.622638i \(-0.213939\pi\)
\(138\) 0 0
\(139\) −15.0076 −1.27293 −0.636463 0.771307i \(-0.719603\pi\)
−0.636463 + 0.771307i \(0.719603\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −26.6678 −2.23008
\(144\) 0 0
\(145\) 5.56106 0.461821
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.33669 0.437198 0.218599 0.975815i \(-0.429851\pi\)
0.218599 + 0.975815i \(0.429851\pi\)
\(150\) 0 0
\(151\) 17.5725i 1.43003i 0.699108 + 0.715016i \(0.253581\pi\)
−0.699108 + 0.715016i \(0.746419\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 18.4675i 1.48334i
\(156\) 0 0
\(157\) − 13.8259i − 1.10343i −0.834032 0.551715i \(-0.813973\pi\)
0.834032 0.551715i \(-0.186027\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 4.63929i − 0.365627i
\(162\) 0 0
\(163\) −8.67478 −0.679461 −0.339731 0.940523i \(-0.610336\pi\)
−0.339731 + 0.940523i \(0.610336\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.8855 −1.22925 −0.614627 0.788818i \(-0.710693\pi\)
−0.614627 + 0.788818i \(0.710693\pi\)
\(168\) 0 0
\(169\) −14.1512 −1.08855
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.22940 −0.321555 −0.160778 0.986991i \(-0.551400\pi\)
−0.160778 + 0.986991i \(0.551400\pi\)
\(174\) 0 0
\(175\) 14.2358i 1.07613i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 6.45041i − 0.482126i −0.970509 0.241063i \(-0.922504\pi\)
0.970509 0.241063i \(-0.0774961\pi\)
\(180\) 0 0
\(181\) − 10.5068i − 0.780968i −0.920610 0.390484i \(-0.872308\pi\)
0.920610 0.390484i \(-0.127692\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.70803i 0.640227i
\(186\) 0 0
\(187\) −10.2358 −0.748519
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.25986 0.597663 0.298831 0.954306i \(-0.403403\pi\)
0.298831 + 0.954306i \(0.403403\pi\)
\(192\) 0 0
\(193\) −8.07937 −0.581566 −0.290783 0.956789i \(-0.593916\pi\)
−0.290783 + 0.956789i \(0.593916\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −7.04275 −0.501775 −0.250887 0.968016i \(-0.580722\pi\)
−0.250887 + 0.968016i \(0.580722\pi\)
\(198\) 0 0
\(199\) − 20.1336i − 1.42723i −0.700537 0.713616i \(-0.747056\pi\)
0.700537 0.713616i \(-0.252944\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 1.26795i − 0.0889926i
\(204\) 0 0
\(205\) − 32.3297i − 2.25801i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 33.2859i − 2.30243i
\(210\) 0 0
\(211\) −7.54347 −0.519314 −0.259657 0.965701i \(-0.583610\pi\)
−0.259657 + 0.965701i \(0.583610\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 16.3046 1.11196
\(216\) 0 0
\(217\) 4.21068 0.285839
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −10.4214 −0.701016
\(222\) 0 0
\(223\) − 5.48926i − 0.367588i −0.982965 0.183794i \(-0.941162\pi\)
0.982965 0.183794i \(-0.0588380\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.127416i 0.00845689i 0.999991 + 0.00422845i \(0.00134596\pi\)
−0.999991 + 0.00422845i \(0.998654\pi\)
\(228\) 0 0
\(229\) − 6.53590i − 0.431904i −0.976404 0.215952i \(-0.930714\pi\)
0.976404 0.215952i \(-0.0692855\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 10.5359i − 0.690230i −0.938560 0.345115i \(-0.887840\pi\)
0.938560 0.345115i \(-0.112160\pi\)
\(234\) 0 0
\(235\) −11.2962 −0.736881
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 24.3862 1.57741 0.788706 0.614771i \(-0.210752\pi\)
0.788706 + 0.614771i \(0.210752\pi\)
\(240\) 0 0
\(241\) −10.8603 −0.699573 −0.349787 0.936829i \(-0.613746\pi\)
−0.349787 + 0.936829i \(0.613746\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.38587 0.280203
\(246\) 0 0
\(247\) − 33.8891i − 2.15631i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 3.73511i − 0.235758i −0.993028 0.117879i \(-0.962390\pi\)
0.993028 0.117879i \(-0.0376095\pi\)
\(252\) 0 0
\(253\) 23.7435i 1.49274i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 7.52781i − 0.469572i −0.972047 0.234786i \(-0.924561\pi\)
0.972047 0.234786i \(-0.0754389\pi\)
\(258\) 0 0
\(259\) 1.98547 0.123371
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −16.7187 −1.03092 −0.515458 0.856915i \(-0.672378\pi\)
−0.515458 + 0.856915i \(0.672378\pi\)
\(264\) 0 0
\(265\) −44.8930 −2.75776
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.11486 0.250887 0.125444 0.992101i \(-0.459965\pi\)
0.125444 + 0.992101i \(0.459965\pi\)
\(270\) 0 0
\(271\) 4.31293i 0.261992i 0.991383 + 0.130996i \(0.0418175\pi\)
−0.991383 + 0.130996i \(0.958182\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 72.8579i − 4.39349i
\(276\) 0 0
\(277\) 27.7564i 1.66772i 0.551976 + 0.833860i \(0.313874\pi\)
−0.551976 + 0.833860i \(0.686126\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 21.8961i 1.30621i 0.757267 + 0.653106i \(0.226534\pi\)
−0.757267 + 0.653106i \(0.773466\pi\)
\(282\) 0 0
\(283\) −15.3786 −0.914164 −0.457082 0.889425i \(-0.651105\pi\)
−0.457082 + 0.889425i \(0.651105\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −7.37134 −0.435117
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.408482 0.0238638 0.0119319 0.999929i \(-0.496202\pi\)
0.0119319 + 0.999929i \(0.496202\pi\)
\(294\) 0 0
\(295\) − 15.3671i − 0.894710i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 24.1739i 1.39801i
\(300\) 0 0
\(301\) − 3.71753i − 0.214275i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 19.8939i 1.13912i
\(306\) 0 0
\(307\) −9.99694 −0.570555 −0.285278 0.958445i \(-0.592086\pi\)
−0.285278 + 0.958445i \(0.592086\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −19.9603 −1.13185 −0.565923 0.824458i \(-0.691480\pi\)
−0.565923 + 0.824458i \(0.691480\pi\)
\(312\) 0 0
\(313\) 23.4465 1.32528 0.662638 0.748940i \(-0.269437\pi\)
0.662638 + 0.748940i \(0.269437\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −20.7740 −1.16678 −0.583391 0.812191i \(-0.698275\pi\)
−0.583391 + 0.812191i \(0.698275\pi\)
\(318\) 0 0
\(319\) 6.48926i 0.363329i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 13.0076i − 0.723761i
\(324\) 0 0
\(325\) − 74.1784i − 4.11468i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.57558i 0.141997i
\(330\) 0 0
\(331\) −19.0366 −1.04635 −0.523174 0.852226i \(-0.675252\pi\)
−0.523174 + 0.852226i \(0.675252\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 22.8533 1.24861
\(336\) 0 0
\(337\) 2.88546 0.157181 0.0785905 0.996907i \(-0.474958\pi\)
0.0785905 + 0.996907i \(0.474958\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −21.5499 −1.16699
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 30.7033i 1.64824i 0.566415 + 0.824120i \(0.308330\pi\)
−0.566415 + 0.824120i \(0.691670\pi\)
\(348\) 0 0
\(349\) 26.5007i 1.41855i 0.704931 + 0.709276i \(0.250978\pi\)
−0.704931 + 0.709276i \(0.749022\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.29955i 0.335291i 0.985847 + 0.167645i \(0.0536164\pi\)
−0.985847 + 0.167645i \(0.946384\pi\)
\(354\) 0 0
\(355\) 5.96789 0.316743
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.60770 −0.0848509 −0.0424255 0.999100i \(-0.513508\pi\)
−0.0424255 + 0.999100i \(0.513508\pi\)
\(360\) 0 0
\(361\) 23.2992 1.22628
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −67.7464 −3.54601
\(366\) 0 0
\(367\) − 17.1763i − 0.896597i −0.893884 0.448298i \(-0.852030\pi\)
0.893884 0.448298i \(-0.147970\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10.2358i 0.531418i
\(372\) 0 0
\(373\) − 2.27941i − 0.118024i −0.998257 0.0590118i \(-0.981205\pi\)
0.998257 0.0590118i \(-0.0187950\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.60688i 0.340271i
\(378\) 0 0
\(379\) −20.1389 −1.03446 −0.517232 0.855845i \(-0.673038\pi\)
−0.517232 + 0.855845i \(0.673038\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −22.3068 −1.13982 −0.569912 0.821705i \(-0.693023\pi\)
−0.569912 + 0.821705i \(0.693023\pi\)
\(384\) 0 0
\(385\) −22.4465 −1.14398
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.57864 −0.0800404 −0.0400202 0.999199i \(-0.512742\pi\)
−0.0400202 + 0.999199i \(0.512742\pi\)
\(390\) 0 0
\(391\) 9.27858i 0.469238i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 17.4663i 0.878827i
\(396\) 0 0
\(397\) − 31.4289i − 1.57737i −0.614796 0.788686i \(-0.710762\pi\)
0.614796 0.788686i \(-0.289238\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 26.5404i − 1.32536i −0.748901 0.662682i \(-0.769418\pi\)
0.748901 0.662682i \(-0.230582\pi\)
\(402\) 0 0
\(403\) −21.9405 −1.09293
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −10.1615 −0.503687
\(408\) 0 0
\(409\) 32.0037 1.58248 0.791240 0.611506i \(-0.209436\pi\)
0.791240 + 0.611506i \(0.209436\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.50379 −0.172410
\(414\) 0 0
\(415\) 51.2678i 2.51663i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 20.6572i − 1.00917i −0.863362 0.504585i \(-0.831646\pi\)
0.863362 0.504585i \(-0.168354\pi\)
\(420\) 0 0
\(421\) 32.9930i 1.60798i 0.594641 + 0.803991i \(0.297294\pi\)
−0.594641 + 0.803991i \(0.702706\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 28.4717i − 1.38108i
\(426\) 0 0
\(427\) 4.53590 0.219508
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.48282 0.312267 0.156133 0.987736i \(-0.450097\pi\)
0.156133 + 0.987736i \(0.450097\pi\)
\(432\) 0 0
\(433\) −18.4038 −0.884429 −0.442214 0.896909i \(-0.645807\pi\)
−0.442214 + 0.896909i \(0.645807\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −30.1730 −1.44337
\(438\) 0 0
\(439\) − 4.31293i − 0.205845i −0.994689 0.102923i \(-0.967181\pi\)
0.994689 0.102923i \(-0.0328194\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 5.54540i − 0.263470i −0.991285 0.131735i \(-0.957945\pi\)
0.991285 0.131735i \(-0.0420547\pi\)
\(444\) 0 0
\(445\) 65.1937i 3.09048i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 30.2007i 1.42526i 0.701541 + 0.712629i \(0.252496\pi\)
−0.701541 + 0.712629i \(0.747504\pi\)
\(450\) 0 0
\(451\) 37.7259 1.77644
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −22.8533 −1.07138
\(456\) 0 0
\(457\) −24.3205 −1.13767 −0.568833 0.822453i \(-0.692605\pi\)
−0.568833 + 0.822453i \(0.692605\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 37.3141 1.73789 0.868945 0.494909i \(-0.164799\pi\)
0.868945 + 0.494909i \(0.164799\pi\)
\(462\) 0 0
\(463\) 35.2853i 1.63985i 0.572472 + 0.819924i \(0.305985\pi\)
−0.572472 + 0.819924i \(0.694015\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.7824i 0.961693i 0.876805 + 0.480847i \(0.159671\pi\)
−0.876805 + 0.480847i \(0.840329\pi\)
\(468\) 0 0
\(469\) − 5.21068i − 0.240607i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 19.0260i 0.874816i
\(474\) 0 0
\(475\) 92.5868 4.24818
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 17.1640 0.784245 0.392123 0.919913i \(-0.371741\pi\)
0.392123 + 0.919913i \(0.371741\pi\)
\(480\) 0 0
\(481\) −10.3457 −0.471722
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −19.3915 −0.880522
\(486\) 0 0
\(487\) − 38.7961i − 1.75802i −0.476805 0.879009i \(-0.658205\pi\)
0.476805 0.879009i \(-0.341795\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 31.1973i 1.40791i 0.710243 + 0.703957i \(0.248585\pi\)
−0.710243 + 0.703957i \(0.751415\pi\)
\(492\) 0 0
\(493\) 2.53590i 0.114211i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 1.36071i − 0.0610361i
\(498\) 0 0
\(499\) 8.78320 0.393190 0.196595 0.980485i \(-0.437012\pi\)
0.196595 + 0.980485i \(0.437012\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 11.2679 0.502413 0.251207 0.967934i \(-0.419173\pi\)
0.251207 + 0.967934i \(0.419173\pi\)
\(504\) 0 0
\(505\) 8.77174 0.390337
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.99388 0.354322 0.177161 0.984182i \(-0.443309\pi\)
0.177161 + 0.984182i \(0.443309\pi\)
\(510\) 0 0
\(511\) 15.4465i 0.683314i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7.34536i 0.323675i
\(516\) 0 0
\(517\) − 13.1816i − 0.579727i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 15.1797i − 0.665035i −0.943097 0.332517i \(-0.892102\pi\)
0.943097 0.332517i \(-0.107898\pi\)
\(522\) 0 0
\(523\) 5.58846 0.244366 0.122183 0.992508i \(-0.461010\pi\)
0.122183 + 0.992508i \(0.461010\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.42136 −0.366840
\(528\) 0 0
\(529\) −1.47698 −0.0642163
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 38.4097 1.66371
\(534\) 0 0
\(535\) − 45.0204i − 1.94640i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.11792i 0.220444i
\(540\) 0 0
\(541\) 10.1229i 0.435219i 0.976036 + 0.217610i \(0.0698260\pi\)
−0.976036 + 0.217610i \(0.930174\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 5.67140i − 0.242936i
\(546\) 0 0
\(547\) −7.03662 −0.300864 −0.150432 0.988620i \(-0.548067\pi\)
−0.150432 + 0.988620i \(0.548067\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −8.24647 −0.351311
\(552\) 0 0
\(553\) 3.98241 0.169349
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −11.0427 −0.467896 −0.233948 0.972249i \(-0.575165\pi\)
−0.233948 + 0.972249i \(0.575165\pi\)
\(558\) 0 0
\(559\) 19.3708i 0.819299i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19.1115i 0.805453i 0.915320 + 0.402726i \(0.131937\pi\)
−0.915320 + 0.402726i \(0.868063\pi\)
\(564\) 0 0
\(565\) 35.4351i 1.49076i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 12.6443i − 0.530077i −0.964238 0.265039i \(-0.914615\pi\)
0.964238 0.265039i \(-0.0853847\pi\)
\(570\) 0 0
\(571\) −19.4351 −0.813332 −0.406666 0.913577i \(-0.633309\pi\)
−0.406666 + 0.913577i \(0.633309\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −66.0442 −2.75423
\(576\) 0 0
\(577\) −42.8137 −1.78236 −0.891178 0.453654i \(-0.850120\pi\)
−0.891178 + 0.453654i \(0.850120\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 11.6893 0.484954
\(582\) 0 0
\(583\) − 52.3862i − 2.16961i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.9221i 1.19374i 0.802337 + 0.596871i \(0.203590\pi\)
−0.802337 + 0.596871i \(0.796410\pi\)
\(588\) 0 0
\(589\) − 27.3854i − 1.12839i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 41.9140i − 1.72120i −0.509280 0.860601i \(-0.670088\pi\)
0.509280 0.860601i \(-0.329912\pi\)
\(594\) 0 0
\(595\) −8.77174 −0.359606
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 20.5829 0.840993 0.420496 0.907294i \(-0.361856\pi\)
0.420496 + 0.907294i \(0.361856\pi\)
\(600\) 0 0
\(601\) 5.44652 0.222168 0.111084 0.993811i \(-0.464568\pi\)
0.111084 + 0.993811i \(0.464568\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 66.6349 2.70909
\(606\) 0 0
\(607\) − 3.40600i − 0.138245i −0.997608 0.0691226i \(-0.977980\pi\)
0.997608 0.0691226i \(-0.0220200\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 13.4205i − 0.542937i
\(612\) 0 0
\(613\) 23.0992i 0.932968i 0.884530 + 0.466484i \(0.154479\pi\)
−0.884530 + 0.466484i \(0.845521\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 25.4457i 1.02440i 0.858865 + 0.512202i \(0.171170\pi\)
−0.858865 + 0.512202i \(0.828830\pi\)
\(618\) 0 0
\(619\) 18.7167 0.752287 0.376144 0.926561i \(-0.377250\pi\)
0.376144 + 0.926561i \(0.377250\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 14.8645 0.595533
\(624\) 0 0
\(625\) 106.480 4.25920
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.97095 −0.158332
\(630\) 0 0
\(631\) 31.5549i 1.25618i 0.778140 + 0.628091i \(0.216163\pi\)
−0.778140 + 0.628091i \(0.783837\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 58.9746i − 2.34034i
\(636\) 0 0
\(637\) 5.21068i 0.206455i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.53508i 0.376613i 0.982110 + 0.188306i \(0.0602998\pi\)
−0.982110 + 0.188306i \(0.939700\pi\)
\(642\) 0 0
\(643\) 37.8243 1.49164 0.745822 0.666145i \(-0.232057\pi\)
0.745822 + 0.666145i \(0.232057\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −19.4663 −0.765301 −0.382650 0.923893i \(-0.624989\pi\)
−0.382650 + 0.923893i \(0.624989\pi\)
\(648\) 0 0
\(649\) 17.9321 0.703896
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9.58928 −0.375257 −0.187629 0.982240i \(-0.560080\pi\)
−0.187629 + 0.982240i \(0.560080\pi\)
\(654\) 0 0
\(655\) 7.78402i 0.304147i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 13.0615i − 0.508803i −0.967099 0.254402i \(-0.918122\pi\)
0.967099 0.254402i \(-0.0818785\pi\)
\(660\) 0 0
\(661\) 39.8807i 1.55118i 0.631236 + 0.775591i \(0.282548\pi\)
−0.631236 + 0.775591i \(0.717452\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 28.5247i − 1.10614i
\(666\) 0 0
\(667\) 5.88239 0.227767
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −23.2144 −0.896180
\(672\) 0 0
\(673\) −12.8992 −0.497226 −0.248613 0.968603i \(-0.579975\pi\)
−0.248613 + 0.968603i \(0.579975\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 30.1713 1.15958 0.579789 0.814767i \(-0.303135\pi\)
0.579789 + 0.814767i \(0.303135\pi\)
\(678\) 0 0
\(679\) 4.42136i 0.169676i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 6.61719i − 0.253200i −0.991954 0.126600i \(-0.959594\pi\)
0.991954 0.126600i \(-0.0404064\pi\)
\(684\) 0 0
\(685\) 63.9266i 2.44251i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 53.3357i − 2.03193i
\(690\) 0 0
\(691\) −40.6701 −1.54716 −0.773581 0.633697i \(-0.781537\pi\)
−0.773581 + 0.633697i \(0.781537\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 65.8212 2.49674
\(696\) 0 0
\(697\) 14.7427 0.558419
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 18.4320 0.696167 0.348083 0.937464i \(-0.386833\pi\)
0.348083 + 0.937464i \(0.386833\pi\)
\(702\) 0 0
\(703\) − 12.9131i − 0.487027i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 2.00000i − 0.0752177i
\(708\) 0 0
\(709\) − 21.1221i − 0.793258i −0.917979 0.396629i \(-0.870180\pi\)
0.917979 0.396629i \(-0.129820\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 19.5346i 0.731575i
\(714\) 0 0
\(715\) 116.962 4.37411
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −24.1319 −0.899969 −0.449985 0.893036i \(-0.648571\pi\)
−0.449985 + 0.893036i \(0.648571\pi\)
\(720\) 0 0
\(721\) 1.67478 0.0623721
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −18.0503 −0.670372
\(726\) 0 0
\(727\) − 18.3923i − 0.682133i −0.940039 0.341066i \(-0.889212\pi\)
0.940039 0.341066i \(-0.110788\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7.43505i 0.274995i
\(732\) 0 0
\(733\) 34.5602i 1.27651i 0.769824 + 0.638256i \(0.220344\pi\)
−0.769824 + 0.638256i \(0.779656\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 26.6678i 0.982322i
\(738\) 0 0
\(739\) −45.0366 −1.65670 −0.828350 0.560212i \(-0.810720\pi\)
−0.828350 + 0.560212i \(0.810720\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −7.83850 −0.287567 −0.143783 0.989609i \(-0.545927\pi\)
−0.143783 + 0.989609i \(0.545927\pi\)
\(744\) 0 0
\(745\) −23.4060 −0.857529
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −10.2649 −0.375071
\(750\) 0 0
\(751\) − 38.4328i − 1.40243i −0.712948 0.701217i \(-0.752641\pi\)
0.712948 0.701217i \(-0.247359\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 77.0708i − 2.80489i
\(756\) 0 0
\(757\) 2.24423i 0.0815680i 0.999168 + 0.0407840i \(0.0129855\pi\)
−0.999168 + 0.0407840i \(0.987014\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 28.3946i 1.02930i 0.857400 + 0.514651i \(0.172079\pi\)
−0.857400 + 0.514651i \(0.827921\pi\)
\(762\) 0 0
\(763\) −1.29311 −0.0468137
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.2571 0.659226
\(768\) 0 0
\(769\) 27.2968 0.984348 0.492174 0.870497i \(-0.336202\pi\)
0.492174 + 0.870497i \(0.336202\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −31.6583 −1.13867 −0.569335 0.822105i \(-0.692799\pi\)
−0.569335 + 0.822105i \(0.692799\pi\)
\(774\) 0 0
\(775\) − 59.9425i − 2.15320i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 47.9416i 1.71769i
\(780\) 0 0
\(781\) 6.96400i 0.249191i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 60.6388i 2.16429i
\(786\) 0 0
\(787\) 24.1375 0.860408 0.430204 0.902732i \(-0.358442\pi\)
0.430204 + 0.902732i \(0.358442\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 8.07937 0.287269
\(792\) 0 0
\(793\) −23.6351 −0.839307
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 23.1867 0.821313 0.410657 0.911790i \(-0.365299\pi\)
0.410657 + 0.911790i \(0.365299\pi\)
\(798\) 0 0
\(799\) − 5.15117i − 0.182235i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 79.0540i − 2.78976i
\(804\) 0 0
\(805\) 20.3473i 0.717149i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 24.2823i − 0.853719i −0.904318 0.426860i \(-0.859620\pi\)
0.904318 0.426860i \(-0.140380\pi\)
\(810\) 0 0
\(811\) −29.1610 −1.02398 −0.511990 0.858991i \(-0.671092\pi\)
−0.511990 + 0.858991i \(0.671092\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 38.0464 1.33271
\(816\) 0 0
\(817\) −24.1780 −0.845881
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −42.6701 −1.48920 −0.744598 0.667513i \(-0.767359\pi\)
−0.744598 + 0.667513i \(0.767359\pi\)
\(822\) 0 0
\(823\) − 18.4487i − 0.643083i −0.946896 0.321541i \(-0.895799\pi\)
0.946896 0.321541i \(-0.104201\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37.8393i 1.31580i 0.753104 + 0.657901i \(0.228556\pi\)
−0.753104 + 0.657901i \(0.771444\pi\)
\(828\) 0 0
\(829\) − 46.5463i − 1.61662i −0.588756 0.808310i \(-0.700382\pi\)
0.588756 0.808310i \(-0.299618\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.00000i 0.0692959i
\(834\) 0 0
\(835\) 69.6715 2.41108
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 12.9388 0.446698 0.223349 0.974739i \(-0.428301\pi\)
0.223349 + 0.974739i \(0.428301\pi\)
\(840\) 0 0
\(841\) −27.3923 −0.944562
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 62.0651 2.13511
\(846\) 0 0
\(847\) − 15.1931i − 0.522041i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 9.21119i 0.315756i
\(852\) 0 0
\(853\) − 49.9509i − 1.71029i −0.518391 0.855144i \(-0.673469\pi\)
0.518391 0.855144i \(-0.326531\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.4208i 1.10747i 0.832691 + 0.553737i \(0.186799\pi\)
−0.832691 + 0.553737i \(0.813201\pi\)
\(858\) 0 0
\(859\) 18.3311 0.625450 0.312725 0.949844i \(-0.398758\pi\)
0.312725 + 0.949844i \(0.398758\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −3.77704 −0.128572 −0.0642859 0.997932i \(-0.520477\pi\)
−0.0642859 + 0.997932i \(0.520477\pi\)
\(864\) 0 0
\(865\) 18.5496 0.630705
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −20.3817 −0.691401
\(870\) 0 0
\(871\) 27.1512i 0.919982i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 40.5072i − 1.36939i
\(876\) 0 0
\(877\) 9.61218i 0.324580i 0.986743 + 0.162290i \(0.0518880\pi\)
−0.986743 + 0.162290i \(0.948112\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 22.4722i 0.757107i 0.925579 + 0.378554i \(0.123578\pi\)
−0.925579 + 0.378554i \(0.876422\pi\)
\(882\) 0 0
\(883\) 9.77704 0.329023 0.164512 0.986375i \(-0.447395\pi\)
0.164512 + 0.986375i \(0.447395\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −59.1043 −1.98453 −0.992265 0.124141i \(-0.960383\pi\)
−0.992265 + 0.124141i \(0.960383\pi\)
\(888\) 0 0
\(889\) −13.4465 −0.450982
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 16.7510 0.560552
\(894\) 0 0
\(895\) 28.2906i 0.945652i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.33893i 0.178063i
\(900\) 0 0
\(901\) − 20.4717i − 0.682010i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 46.0816i 1.53181i
\(906\) 0 0
\(907\) 31.9861 1.06208 0.531040 0.847346i \(-0.321801\pi\)
0.531040 + 0.847346i \(0.321801\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 26.4052 0.874843 0.437421 0.899257i \(-0.355892\pi\)
0.437421 + 0.899257i \(0.355892\pi\)
\(912\) 0 0
\(913\) −59.8249 −1.97992
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.77480 0.0586089
\(918\) 0 0
\(919\) 46.4656i 1.53276i 0.642389 + 0.766379i \(0.277943\pi\)
−0.642389 + 0.766379i \(0.722057\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 7.09021i 0.233377i
\(924\) 0 0
\(925\) − 28.2649i − 0.929344i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 16.7656i 0.550062i 0.961435 + 0.275031i \(0.0886881\pi\)
−0.961435 + 0.275031i \(0.911312\pi\)
\(930\) 0 0
\(931\) −6.50379 −0.213153
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 44.8930 1.46816
\(936\) 0 0
\(937\) −32.3923 −1.05821 −0.529105 0.848556i \(-0.677472\pi\)
−0.529105 + 0.848556i \(0.677472\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 40.1216 1.30793 0.653964 0.756526i \(-0.273105\pi\)
0.653964 + 0.756526i \(0.273105\pi\)
\(942\) 0 0
\(943\) − 34.1978i − 1.11363i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 38.9905i − 1.26702i −0.773734 0.633511i \(-0.781613\pi\)
0.773734 0.633511i \(-0.218387\pi\)
\(948\) 0 0
\(949\) − 80.4868i − 2.61271i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 1.39764i − 0.0452739i −0.999744 0.0226370i \(-0.992794\pi\)
0.999744 0.0226370i \(-0.00720619\pi\)
\(954\) 0 0
\(955\) −36.2267 −1.17227
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 14.5756 0.470670
\(960\) 0 0
\(961\) 13.2702 0.428071
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 35.4351 1.14069
\(966\) 0 0
\(967\) 21.9709i 0.706538i 0.935522 + 0.353269i \(0.114930\pi\)
−0.935522 + 0.353269i \(0.885070\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 42.7098i − 1.37062i −0.728251 0.685311i \(-0.759666\pi\)
0.728251 0.685311i \(-0.240334\pi\)
\(972\) 0 0
\(973\) − 15.0076i − 0.481121i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 60.3885i − 1.93200i −0.258546 0.965999i \(-0.583243\pi\)
0.258546 0.965999i \(-0.416757\pi\)
\(978\) 0 0
\(979\) −76.0753 −2.43138
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −19.6251 −0.625943 −0.312971 0.949763i \(-0.601324\pi\)
−0.312971 + 0.949763i \(0.601324\pi\)
\(984\) 0 0
\(985\) 30.8886 0.984191
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 17.2467 0.548413
\(990\) 0 0
\(991\) 7.84660i 0.249256i 0.992204 + 0.124628i \(0.0397737\pi\)
−0.992204 + 0.124628i \(0.960226\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 88.3032i 2.79940i
\(996\) 0 0
\(997\) − 19.9128i − 0.630646i −0.948984 0.315323i \(-0.897887\pi\)
0.948984 0.315323i \(-0.102113\pi\)
\(998\) 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 6048.2.j.a.5615.2 8
3.2 odd 2 6048.2.j.b.5615.8 8
4.3 odd 2 1512.2.j.a.323.7 yes 8
8.3 odd 2 6048.2.j.b.5615.7 8
8.5 even 2 1512.2.j.b.323.4 yes 8
12.11 even 2 1512.2.j.b.323.2 yes 8
24.5 odd 2 1512.2.j.a.323.5 8
24.11 even 2 inner 6048.2.j.a.5615.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.a.323.5 8 24.5 odd 2
1512.2.j.a.323.7 yes 8 4.3 odd 2
1512.2.j.b.323.2 yes 8 12.11 even 2
1512.2.j.b.323.4 yes 8 8.5 even 2
6048.2.j.a.5615.1 8 24.11 even 2 inner
6048.2.j.a.5615.2 8 1.1 even 1 trivial
6048.2.j.b.5615.7 8 8.3 odd 2
6048.2.j.b.5615.8 8 3.2 odd 2