Properties

Label 6048.2.c.c.3025.2
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.5
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.2
Root \(1.40294 - 0.178197i\)
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.c.3025.7

$q$-expansion

\(f(q)\) \(=\) \(q-2.80588i q^{5} +1.00000 q^{7} +O(q^{10})\) \(q-2.80588i q^{5} +1.00000 q^{7} -3.51867i q^{11} +4.87298i q^{13} +3.16228 q^{17} +7.87298i q^{19} +0.356394 q^{23} -2.87298 q^{25} -2.44949i q^{29} +7.00000 q^{31} -2.80588i q^{35} -5.87298i q^{37} +10.1544 q^{41} +8.87298i q^{43} -7.34847 q^{47} +1.00000 q^{49} -9.87298 q^{55} +12.9602i q^{59} -1.74597i q^{61} +13.6730 q^{65} +14.6190i q^{67} -3.51867 q^{71} -6.87298 q^{73} -3.51867i q^{77} +10.6190 q^{79} +12.9602i q^{83} -8.87298i q^{85} +10.1544 q^{89} +4.87298i q^{91} +22.0907 q^{95} +5.74597 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{7} + O(q^{10}) \) \( 8q + 8q^{7} + 8q^{25} + 56q^{31} + 8q^{49} - 48q^{55} - 24q^{73} - 8q^{79} - 16q^{97} + O(q^{100}) \)

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
\(5\) − 2.80588i − 1.25483i −0.778685 0.627415i \(-0.784113\pi\)
0.778685 0.627415i \(-0.215887\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.51867i − 1.06092i −0.847710 0.530460i \(-0.822019\pi\)
0.847710 0.530460i \(-0.177981\pi\)
\(12\) 0 0
\(13\) 4.87298i 1.35152i 0.737121 + 0.675761i \(0.236185\pi\)
−0.737121 + 0.675761i \(0.763815\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.16228 0.766965 0.383482 0.923548i \(-0.374725\pi\)
0.383482 + 0.923548i \(0.374725\pi\)
\(18\) 0 0
\(19\) 7.87298i 1.80619i 0.429445 + 0.903093i \(0.358709\pi\)
−0.429445 + 0.903093i \(0.641291\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.356394 0.0743133 0.0371566 0.999309i \(-0.488170\pi\)
0.0371566 + 0.999309i \(0.488170\pi\)
\(24\) 0 0
\(25\) −2.87298 −0.574597
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 2.44949i − 0.454859i −0.973795 0.227429i \(-0.926968\pi\)
0.973795 0.227429i \(-0.0730321\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 2.80588i − 0.474281i
\(36\) 0 0
\(37\) − 5.87298i − 0.965513i −0.875755 0.482756i \(-0.839636\pi\)
0.875755 0.482756i \(-0.160364\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.1544 1.58584 0.792922 0.609323i \(-0.208559\pi\)
0.792922 + 0.609323i \(0.208559\pi\)
\(42\) 0 0
\(43\) 8.87298i 1.35312i 0.736389 + 0.676559i \(0.236529\pi\)
−0.736389 + 0.676559i \(0.763471\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.34847 −1.07188 −0.535942 0.844255i \(-0.680044\pi\)
−0.535942 + 0.844255i \(0.680044\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −9.87298 −1.33127
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.9602i 1.68728i 0.536910 + 0.843640i \(0.319591\pi\)
−0.536910 + 0.843640i \(0.680409\pi\)
\(60\) 0 0
\(61\) − 1.74597i − 0.223548i −0.993734 0.111774i \(-0.964347\pi\)
0.993734 0.111774i \(-0.0356533\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13.6730 1.69593
\(66\) 0 0
\(67\) 14.6190i 1.78599i 0.450068 + 0.892995i \(0.351400\pi\)
−0.450068 + 0.892995i \(0.648600\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.51867 −0.417589 −0.208795 0.977959i \(-0.566954\pi\)
−0.208795 + 0.977959i \(0.566954\pi\)
\(72\) 0 0
\(73\) −6.87298 −0.804422 −0.402211 0.915547i \(-0.631758\pi\)
−0.402211 + 0.915547i \(0.631758\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.51867i − 0.400990i
\(78\) 0 0
\(79\) 10.6190 1.19473 0.597363 0.801971i \(-0.296215\pi\)
0.597363 + 0.801971i \(0.296215\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.9602i 1.42257i 0.702903 + 0.711285i \(0.251887\pi\)
−0.702903 + 0.711285i \(0.748113\pi\)
\(84\) 0 0
\(85\) − 8.87298i − 0.962410i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.1544 1.07636 0.538180 0.842830i \(-0.319112\pi\)
0.538180 + 0.842830i \(0.319112\pi\)
\(90\) 0 0
\(91\) 4.87298i 0.510827i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 22.0907 2.26646
\(96\) 0 0
\(97\) 5.74597 0.583415 0.291707 0.956508i \(-0.405777\pi\)
0.291707 + 0.956508i \(0.405777\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 17.8592i 1.77706i 0.458820 + 0.888529i \(0.348272\pi\)
−0.458820 + 0.888529i \(0.651728\pi\)
\(102\) 0 0
\(103\) −4.74597 −0.467634 −0.233817 0.972281i \(-0.575122\pi\)
−0.233817 + 0.972281i \(0.575122\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 4.58785i − 0.443525i −0.975101 0.221762i \(-0.928819\pi\)
0.975101 0.221762i \(-0.0711809\pi\)
\(108\) 0 0
\(109\) − 2.74597i − 0.263016i −0.991315 0.131508i \(-0.958018\pi\)
0.991315 0.131508i \(-0.0419819\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.13836 0.201160 0.100580 0.994929i \(-0.467930\pi\)
0.100580 + 0.994929i \(0.467930\pi\)
\(114\) 0 0
\(115\) − 1.00000i − 0.0932505i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.16228 0.289886
\(120\) 0 0
\(121\) −1.38105 −0.125550
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 5.96816i − 0.533809i
\(126\) 0 0
\(127\) −3.12702 −0.277478 −0.138739 0.990329i \(-0.544305\pi\)
−0.138739 + 0.990329i \(0.544305\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 3.87507i − 0.338566i −0.985567 0.169283i \(-0.945855\pi\)
0.985567 0.169283i \(-0.0541452\pi\)
\(132\) 0 0
\(133\) 7.87298i 0.682674i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.3858 −1.22906 −0.614531 0.788892i \(-0.710655\pi\)
−0.614531 + 0.788892i \(0.710655\pi\)
\(138\) 0 0
\(139\) − 14.0000i − 1.18746i −0.804663 0.593732i \(-0.797654\pi\)
0.804663 0.593732i \(-0.202346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 17.1464 1.43386
\(144\) 0 0
\(145\) −6.87298 −0.570770
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 9.79796i − 0.802680i −0.915929 0.401340i \(-0.868545\pi\)
0.915929 0.401340i \(-0.131455\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 19.6412i − 1.57762i
\(156\) 0 0
\(157\) − 14.8730i − 1.18699i −0.804836 0.593497i \(-0.797747\pi\)
0.804836 0.593497i \(-0.202253\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.356394 0.0280878
\(162\) 0 0
\(163\) − 15.1270i − 1.18484i −0.805630 0.592420i \(-0.798173\pi\)
0.805630 0.592420i \(-0.201827\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.2235 −0.868503 −0.434252 0.900792i \(-0.642987\pi\)
−0.434252 + 0.900792i \(0.642987\pi\)
\(168\) 0 0
\(169\) −10.7460 −0.826613
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.72878i 0.663637i 0.943343 + 0.331818i \(0.107662\pi\)
−0.943343 + 0.331818i \(0.892338\pi\)
\(174\) 0 0
\(175\) −2.87298 −0.217177
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 3.87507i − 0.289636i −0.989458 0.144818i \(-0.953740\pi\)
0.989458 0.144818i \(-0.0462597\pi\)
\(180\) 0 0
\(181\) 6.25403i 0.464859i 0.972613 + 0.232429i \(0.0746674\pi\)
−0.972613 + 0.232429i \(0.925333\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −16.4789 −1.21155
\(186\) 0 0
\(187\) − 11.1270i − 0.813688i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.3301 1.39867 0.699337 0.714792i \(-0.253479\pi\)
0.699337 + 0.714792i \(0.253479\pi\)
\(192\) 0 0
\(193\) 21.4919 1.54702 0.773512 0.633782i \(-0.218498\pi\)
0.773512 + 0.633782i \(0.218498\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 27.3460i − 1.94833i −0.225848 0.974163i \(-0.572515\pi\)
0.225848 0.974163i \(-0.427485\pi\)
\(198\) 0 0
\(199\) −7.87298 −0.558101 −0.279050 0.960276i \(-0.590020\pi\)
−0.279050 + 0.960276i \(0.590020\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 2.44949i − 0.171920i
\(204\) 0 0
\(205\) − 28.4919i − 1.98996i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 27.7024 1.91622
\(210\) 0 0
\(211\) 20.0000i 1.37686i 0.725304 + 0.688428i \(0.241699\pi\)
−0.725304 + 0.688428i \(0.758301\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 24.8966 1.69793
\(216\) 0 0
\(217\) 7.00000 0.475191
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 15.4097i 1.03657i
\(222\) 0 0
\(223\) 15.6190 1.04592 0.522961 0.852357i \(-0.324827\pi\)
0.522961 + 0.852357i \(0.324827\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 13.2714i − 0.880851i −0.897789 0.440426i \(-0.854828\pi\)
0.897789 0.440426i \(-0.145172\pi\)
\(228\) 0 0
\(229\) 0.254033i 0.0167870i 0.999965 + 0.00839350i \(0.00267176\pi\)
−0.999965 + 0.00839350i \(0.997328\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.712788 −0.0466963 −0.0233481 0.999727i \(-0.507433\pi\)
−0.0233481 + 0.999727i \(0.507433\pi\)
\(234\) 0 0
\(235\) 20.6190i 1.34503i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −12.2474 −0.792222 −0.396111 0.918203i \(-0.629640\pi\)
−0.396111 + 0.918203i \(0.629640\pi\)
\(240\) 0 0
\(241\) 5.38105 0.346624 0.173312 0.984867i \(-0.444553\pi\)
0.173312 + 0.984867i \(0.444553\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 2.80588i − 0.179261i
\(246\) 0 0
\(247\) −38.3649 −2.44110
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 2.13836i − 0.134972i −0.997720 0.0674862i \(-0.978502\pi\)
0.997720 0.0674862i \(-0.0214979\pi\)
\(252\) 0 0
\(253\) − 1.25403i − 0.0788404i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 27.3913 1.70862 0.854312 0.519761i \(-0.173979\pi\)
0.854312 + 0.519761i \(0.173979\pi\)
\(258\) 0 0
\(259\) − 5.87298i − 0.364929i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −0.356394 −0.0219762 −0.0109881 0.999940i \(-0.503498\pi\)
−0.0109881 + 0.999940i \(0.503498\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 17.9045i − 1.09166i −0.837897 0.545828i \(-0.816215\pi\)
0.837897 0.545828i \(-0.183785\pi\)
\(270\) 0 0
\(271\) 4.25403 0.258414 0.129207 0.991618i \(-0.458757\pi\)
0.129207 + 0.991618i \(0.458757\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.1091i 0.609601i
\(276\) 0 0
\(277\) − 1.87298i − 0.112537i −0.998416 0.0562683i \(-0.982080\pi\)
0.998416 0.0562683i \(-0.0179202\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.0454 1.31512 0.657559 0.753403i \(-0.271589\pi\)
0.657559 + 0.753403i \(0.271589\pi\)
\(282\) 0 0
\(283\) − 19.4919i − 1.15868i −0.815088 0.579338i \(-0.803311\pi\)
0.815088 0.579338i \(-0.196689\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.1544 0.599393
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 31.8434i 1.86031i 0.367168 + 0.930155i \(0.380327\pi\)
−0.367168 + 0.930155i \(0.619673\pi\)
\(294\) 0 0
\(295\) 36.3649 2.11725
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.73670i 0.100436i
\(300\) 0 0
\(301\) 8.87298i 0.511430i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −4.89898 −0.280515
\(306\) 0 0
\(307\) 6.38105i 0.364186i 0.983281 + 0.182093i \(0.0582872\pi\)
−0.983281 + 0.182093i \(0.941713\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.9124 0.618786 0.309393 0.950934i \(-0.399874\pi\)
0.309393 + 0.950934i \(0.399874\pi\)
\(312\) 0 0
\(313\) −4.87298 −0.275437 −0.137719 0.990471i \(-0.543977\pi\)
−0.137719 + 0.990471i \(0.543977\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.02391i 0.0575087i 0.999587 + 0.0287544i \(0.00915406\pi\)
−0.999587 + 0.0287544i \(0.990846\pi\)
\(318\) 0 0
\(319\) −8.61895 −0.482569
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 24.8966i 1.38528i
\(324\) 0 0
\(325\) − 14.0000i − 0.776580i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −7.34847 −0.405134
\(330\) 0 0
\(331\) − 10.0000i − 0.549650i −0.961494 0.274825i \(-0.911380\pi\)
0.961494 0.274825i \(-0.0886199\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 41.0191 2.24111
\(336\) 0 0
\(337\) −18.7460 −1.02116 −0.510579 0.859831i \(-0.670569\pi\)
−0.510579 + 0.859831i \(0.670569\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 24.6307i − 1.33383i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.63312i 0.248719i 0.992237 + 0.124359i \(0.0396876\pi\)
−0.992237 + 0.124359i \(0.960312\pi\)
\(348\) 0 0
\(349\) 7.49193i 0.401034i 0.979690 + 0.200517i \(0.0642622\pi\)
−0.979690 + 0.200517i \(0.935738\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3.82980 −0.203840 −0.101920 0.994793i \(-0.532499\pi\)
−0.101920 + 0.994793i \(0.532499\pi\)
\(354\) 0 0
\(355\) 9.87298i 0.524004i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −34.0723 −1.79827 −0.899133 0.437676i \(-0.855802\pi\)
−0.899133 + 0.437676i \(0.855802\pi\)
\(360\) 0 0
\(361\) −42.9839 −2.26231
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 19.2848i 1.00941i
\(366\) 0 0
\(367\) −14.1270 −0.737424 −0.368712 0.929544i \(-0.620201\pi\)
−0.368712 + 0.929544i \(0.620201\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 18.2379i − 0.944323i −0.881512 0.472161i \(-0.843474\pi\)
0.881512 0.472161i \(-0.156526\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11.9363 0.614752
\(378\) 0 0
\(379\) − 1.12702i − 0.0578910i −0.999581 0.0289455i \(-0.990785\pi\)
0.999581 0.0289455i \(-0.00921492\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 36.4312 1.86155 0.930774 0.365595i \(-0.119134\pi\)
0.930774 + 0.365595i \(0.119134\pi\)
\(384\) 0 0
\(385\) −9.87298 −0.503174
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9.08517i 0.460636i 0.973115 + 0.230318i \(0.0739767\pi\)
−0.973115 + 0.230318i \(0.926023\pi\)
\(390\) 0 0
\(391\) 1.12702 0.0569957
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 29.7955i − 1.49918i
\(396\) 0 0
\(397\) − 10.0000i − 0.501886i −0.968002 0.250943i \(-0.919259\pi\)
0.968002 0.250943i \(-0.0807406\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 21.4232 1.06982 0.534911 0.844909i \(-0.320345\pi\)
0.534911 + 0.844909i \(0.320345\pi\)
\(402\) 0 0
\(403\) 34.1109i 1.69918i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.6651 −1.02433
\(408\) 0 0
\(409\) 26.3649 1.30366 0.651831 0.758365i \(-0.274001\pi\)
0.651831 + 0.758365i \(0.274001\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 12.9602i 0.637732i
\(414\) 0 0
\(415\) 36.3649 1.78508
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 9.17571i − 0.448263i −0.974559 0.224131i \(-0.928046\pi\)
0.974559 0.224131i \(-0.0719545\pi\)
\(420\) 0 0
\(421\) 35.1109i 1.71120i 0.517638 + 0.855600i \(0.326811\pi\)
−0.517638 + 0.855600i \(0.673189\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −9.08517 −0.440696
\(426\) 0 0
\(427\) − 1.74597i − 0.0844933i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 13.6278 0.656426 0.328213 0.944604i \(-0.393554\pi\)
0.328213 + 0.944604i \(0.393554\pi\)
\(432\) 0 0
\(433\) 2.87298 0.138067 0.0690334 0.997614i \(-0.478009\pi\)
0.0690334 + 0.997614i \(0.478009\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.80588i 0.134224i
\(438\) 0 0
\(439\) −10.2540 −0.489398 −0.244699 0.969599i \(-0.578689\pi\)
−0.244699 + 0.969599i \(0.578689\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 0.758056i − 0.0360163i −0.999838 0.0180082i \(-0.994268\pi\)
0.999838 0.0180082i \(-0.00573249\pi\)
\(444\) 0 0
\(445\) − 28.4919i − 1.35065i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −26.0110 −1.22754 −0.613768 0.789487i \(-0.710347\pi\)
−0.613768 + 0.789487i \(0.710347\pi\)
\(450\) 0 0
\(451\) − 35.7298i − 1.68245i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 13.6730 0.641001
\(456\) 0 0
\(457\) 24.7460 1.15757 0.578784 0.815481i \(-0.303528\pi\)
0.578784 + 0.815481i \(0.303528\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 42.3994i − 1.97474i −0.158444 0.987368i \(-0.550648\pi\)
0.158444 0.987368i \(-0.449352\pi\)
\(462\) 0 0
\(463\) −31.4919 −1.46355 −0.731777 0.681544i \(-0.761309\pi\)
−0.731777 + 0.681544i \(0.761309\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 24.8966i − 1.15208i −0.817423 0.576038i \(-0.804598\pi\)
0.817423 0.576038i \(-0.195402\pi\)
\(468\) 0 0
\(469\) 14.6190i 0.675040i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 31.2211 1.43555
\(474\) 0 0
\(475\) − 22.6190i − 1.03783i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.61177 0.256408 0.128204 0.991748i \(-0.459079\pi\)
0.128204 + 0.991748i \(0.459079\pi\)
\(480\) 0 0
\(481\) 28.6190 1.30491
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 16.1225i − 0.732086i
\(486\) 0 0
\(487\) 28.1109 1.27383 0.636913 0.770936i \(-0.280211\pi\)
0.636913 + 0.770936i \(0.280211\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 40.3516i 1.82104i 0.413465 + 0.910520i \(0.364318\pi\)
−0.413465 + 0.910520i \(0.635682\pi\)
\(492\) 0 0
\(493\) − 7.74597i − 0.348861i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.51867 −0.157834
\(498\) 0 0
\(499\) − 4.36492i − 0.195401i −0.995216 0.0977003i \(-0.968851\pi\)
0.995216 0.0977003i \(-0.0311486\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 7.34847 0.327652 0.163826 0.986489i \(-0.447616\pi\)
0.163826 + 0.986489i \(0.447616\pi\)
\(504\) 0 0
\(505\) 50.1109 2.22991
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 19.2848i 0.854783i 0.904067 + 0.427392i \(0.140567\pi\)
−0.904067 + 0.427392i \(0.859433\pi\)
\(510\) 0 0
\(511\) −6.87298 −0.304043
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 13.3166i 0.586801i
\(516\) 0 0
\(517\) 25.8569i 1.13718i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −17.5934 −0.770779 −0.385390 0.922754i \(-0.625933\pi\)
−0.385390 + 0.922754i \(0.625933\pi\)
\(522\) 0 0
\(523\) − 19.2540i − 0.841920i −0.907079 0.420960i \(-0.861693\pi\)
0.907079 0.420960i \(-0.138307\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.1359 0.964257
\(528\) 0 0
\(529\) −22.8730 −0.994478
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 49.4820i 2.14330i
\(534\) 0 0
\(535\) −12.8730 −0.556548
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 3.51867i − 0.151560i
\(540\) 0 0
\(541\) 38.8569i 1.67059i 0.549805 + 0.835293i \(0.314702\pi\)
−0.549805 + 0.835293i \(0.685298\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −7.70486 −0.330040
\(546\) 0 0
\(547\) − 7.23790i − 0.309470i −0.987956 0.154735i \(-0.950548\pi\)
0.987956 0.154735i \(-0.0494524\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 19.2848 0.821560
\(552\) 0 0
\(553\) 10.6190 0.451564
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 28.7716i − 1.21909i −0.792750 0.609546i \(-0.791352\pi\)
0.792750 0.609546i \(-0.208648\pi\)
\(558\) 0 0
\(559\) −43.2379 −1.82877
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 7.34847i − 0.309701i −0.987938 0.154851i \(-0.950510\pi\)
0.987938 0.154851i \(-0.0494896\pi\)
\(564\) 0 0
\(565\) − 6.00000i − 0.252422i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 43.4686 1.82230 0.911148 0.412078i \(-0.135197\pi\)
0.911148 + 0.412078i \(0.135197\pi\)
\(570\) 0 0
\(571\) 28.9839i 1.21294i 0.795107 + 0.606469i \(0.207414\pi\)
−0.795107 + 0.606469i \(0.792586\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.02391 −0.0427002
\(576\) 0 0
\(577\) −37.7460 −1.57139 −0.785693 0.618617i \(-0.787693\pi\)
−0.785693 + 0.618617i \(0.787693\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 12.9602i 0.537681i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.23402i 0.257306i 0.991690 + 0.128653i \(0.0410653\pi\)
−0.991690 + 0.128653i \(0.958935\pi\)
\(588\) 0 0
\(589\) 55.1109i 2.27080i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −5.65704 −0.232307 −0.116153 0.993231i \(-0.537056\pi\)
−0.116153 + 0.993231i \(0.537056\pi\)
\(594\) 0 0
\(595\) − 8.87298i − 0.363757i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −0.265858 −0.0108627 −0.00543133 0.999985i \(-0.501729\pi\)
−0.00543133 + 0.999985i \(0.501729\pi\)
\(600\) 0 0
\(601\) −22.1109 −0.901922 −0.450961 0.892544i \(-0.648919\pi\)
−0.450961 + 0.892544i \(0.648919\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.87507i 0.157544i
\(606\) 0 0
\(607\) 31.2379 1.26791 0.633954 0.773371i \(-0.281431\pi\)
0.633954 + 0.773371i \(0.281431\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 35.8090i − 1.44868i
\(612\) 0 0
\(613\) − 6.23790i − 0.251946i −0.992034 0.125973i \(-0.959795\pi\)
0.992034 0.125973i \(-0.0402053\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −28.3700 −1.14213 −0.571066 0.820904i \(-0.693470\pi\)
−0.571066 + 0.820904i \(0.693470\pi\)
\(618\) 0 0
\(619\) − 30.4919i − 1.22557i −0.790248 0.612787i \(-0.790048\pi\)
0.790248 0.612787i \(-0.209952\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.1544 0.406826
\(624\) 0 0
\(625\) −31.1109 −1.24444
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 18.5720i − 0.740514i
\(630\) 0 0
\(631\) 1.12702 0.0448658 0.0224329 0.999748i \(-0.492859\pi\)
0.0224329 + 0.999748i \(0.492859\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8.77405i 0.348187i
\(636\) 0 0
\(637\) 4.87298i 0.193075i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30.8195 −1.21730 −0.608648 0.793441i \(-0.708288\pi\)
−0.608648 + 0.793441i \(0.708288\pi\)
\(642\) 0 0
\(643\) 31.6190i 1.24693i 0.781851 + 0.623465i \(0.214276\pi\)
−0.781851 + 0.623465i \(0.785724\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −16.5242 −0.649632 −0.324816 0.945777i \(-0.605302\pi\)
−0.324816 + 0.945777i \(0.605302\pi\)
\(648\) 0 0
\(649\) 45.6028 1.79007
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 19.2848i − 0.754672i −0.926076 0.377336i \(-0.876840\pi\)
0.926076 0.377336i \(-0.123160\pi\)
\(654\) 0 0
\(655\) −10.8730 −0.424843
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 25.8752i − 1.00795i −0.863717 0.503977i \(-0.831869\pi\)
0.863717 0.503977i \(-0.168131\pi\)
\(660\) 0 0
\(661\) 23.8569i 0.927924i 0.885855 + 0.463962i \(0.153573\pi\)
−0.885855 + 0.463962i \(0.846427\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 22.0907 0.856640
\(666\) 0 0
\(667\) − 0.872983i − 0.0338021i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.14348 −0.237167
\(672\) 0 0
\(673\) −1.49193 −0.0575098 −0.0287549 0.999586i \(-0.509154\pi\)
−0.0287549 + 0.999586i \(0.509154\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 10.5560i − 0.405701i −0.979210 0.202850i \(-0.934980\pi\)
0.979210 0.202850i \(-0.0650205\pi\)
\(678\) 0 0
\(679\) 5.74597 0.220510
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 12.6038i − 0.482273i −0.970491 0.241136i \(-0.922480\pi\)
0.970491 0.241136i \(-0.0775201\pi\)
\(684\) 0 0
\(685\) 40.3649i 1.54226i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 22.0000i 0.836919i 0.908235 + 0.418460i \(0.137430\pi\)
−0.908235 + 0.418460i \(0.862570\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −39.2824 −1.49007
\(696\) 0 0
\(697\) 32.1109 1.21629
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 14.4763i 0.546764i 0.961906 + 0.273382i \(0.0881423\pi\)
−0.961906 + 0.273382i \(0.911858\pi\)
\(702\) 0 0
\(703\) 46.2379 1.74390
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 17.8592i 0.671665i
\(708\) 0 0
\(709\) 36.7460i 1.38002i 0.723798 + 0.690012i \(0.242395\pi\)
−0.723798 + 0.690012i \(0.757605\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.49476 0.0934294
\(714\) 0 0
\(715\) − 48.1109i − 1.79925i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −7.43901 −0.277428 −0.138714 0.990332i \(-0.544297\pi\)
−0.138714 + 0.990332i \(0.544297\pi\)
\(720\) 0 0
\(721\) −4.74597 −0.176749
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.03734i 0.261360i
\(726\) 0 0
\(727\) 27.4919 1.01962 0.509810 0.860287i \(-0.329716\pi\)
0.509810 + 0.860287i \(0.329716\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 28.0588i 1.03779i
\(732\) 0 0
\(733\) − 30.3649i − 1.12155i −0.827967 0.560777i \(-0.810503\pi\)
0.827967 0.560777i \(-0.189497\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 51.4393 1.89479
\(738\) 0 0
\(739\) 15.7460i 0.579225i 0.957144 + 0.289612i \(0.0935264\pi\)
−0.957144 + 0.289612i \(0.906474\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −40.7532 −1.49509 −0.747545 0.664211i \(-0.768768\pi\)
−0.747545 + 0.664211i \(0.768768\pi\)
\(744\) 0 0
\(745\) −27.4919 −1.00723
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 4.58785i − 0.167637i
\(750\) 0 0
\(751\) −49.3488 −1.80076 −0.900381 0.435102i \(-0.856712\pi\)
−0.900381 + 0.435102i \(0.856712\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 33.6706i − 1.22540i
\(756\) 0 0
\(757\) − 10.2540i − 0.372689i −0.982484 0.186345i \(-0.940336\pi\)
0.982484 0.186345i \(-0.0596641\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −9.39630 −0.340616 −0.170308 0.985391i \(-0.554476\pi\)
−0.170308 + 0.985391i \(0.554476\pi\)
\(762\) 0 0
\(763\) − 2.74597i − 0.0994107i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −63.1550 −2.28040
\(768\) 0 0
\(769\) 50.3649 1.81621 0.908103 0.418748i \(-0.137531\pi\)
0.908103 + 0.418748i \(0.137531\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 44.1361i − 1.58746i −0.608267 0.793732i \(-0.708135\pi\)
0.608267 0.793732i \(-0.291865\pi\)
\(774\) 0 0
\(775\) −20.1109 −0.722404
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 79.9451i 2.86433i
\(780\) 0 0
\(781\) 12.3810i 0.443029i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −41.7319 −1.48947
\(786\) 0 0
\(787\) − 19.7460i − 0.703868i −0.936025 0.351934i \(-0.885524\pi\)
0.936025 0.351934i \(-0.114476\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.13836 0.0760315
\(792\) 0 0
\(793\) 8.50807 0.302130
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 28.8169i 1.02075i 0.859953 + 0.510373i \(0.170493\pi\)
−0.859953 + 0.510373i \(0.829507\pi\)
\(798\) 0 0
\(799\) −23.2379 −0.822098
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 24.1838i 0.853427i
\(804\) 0 0
\(805\) − 1.00000i − 0.0352454i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14.3858 0.505778 0.252889 0.967495i \(-0.418619\pi\)
0.252889 + 0.967495i \(0.418619\pi\)
\(810\) 0 0
\(811\) − 19.9839i − 0.701728i −0.936426 0.350864i \(-0.885888\pi\)
0.936426 0.350864i \(-0.114112\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −42.4446 −1.48677
\(816\) 0 0
\(817\) −69.8569 −2.44398
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 22.1359i 0.772550i 0.922384 + 0.386275i \(0.126238\pi\)
−0.922384 + 0.386275i \(0.873762\pi\)
\(822\) 0 0
\(823\) 6.50807 0.226857 0.113428 0.993546i \(-0.463817\pi\)
0.113428 + 0.993546i \(0.463817\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1.69143i − 0.0588169i −0.999567 0.0294085i \(-0.990638\pi\)
0.999567 0.0294085i \(-0.00936235\pi\)
\(828\) 0 0
\(829\) 36.3649i 1.26301i 0.775374 + 0.631503i \(0.217562\pi\)
−0.775374 + 0.631503i \(0.782438\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.16228 0.109566
\(834\) 0 0
\(835\) 31.4919i 1.08982i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −13.0508 −0.450563 −0.225281 0.974294i \(-0.572330\pi\)
−0.225281 + 0.974294i \(0.572330\pi\)
\(840\) 0 0
\(841\) 23.0000 0.793103
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 30.1519i 1.03726i
\(846\) 0 0
\(847\) −1.38105 −0.0474534
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 2.09310i − 0.0717504i
\(852\) 0 0
\(853\) 41.2379i 1.41196i 0.708232 + 0.705979i \(0.249493\pi\)
−0.708232 + 0.705979i \(0.750507\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −38.1227 −1.30225 −0.651123 0.758973i \(-0.725702\pi\)
−0.651123 + 0.758973i \(0.725702\pi\)
\(858\) 0 0
\(859\) − 24.1270i − 0.823203i −0.911364 0.411602i \(-0.864970\pi\)
0.911364 0.411602i \(-0.135030\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 11.6252 0.395726 0.197863 0.980230i \(-0.436600\pi\)
0.197863 + 0.980230i \(0.436600\pi\)
\(864\) 0 0
\(865\) 24.4919 0.832751
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 37.3646i − 1.26751i
\(870\) 0 0
\(871\) −71.2379 −2.41380
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 5.96816i − 0.201761i
\(876\) 0 0
\(877\) 17.4919i 0.590661i 0.955395 + 0.295330i \(0.0954297\pi\)
−0.955395 + 0.295330i \(0.904570\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 27.6119 0.930269 0.465134 0.885240i \(-0.346006\pi\)
0.465134 + 0.885240i \(0.346006\pi\)
\(882\) 0 0
\(883\) 31.2379i 1.05124i 0.850720 + 0.525620i \(0.176166\pi\)
−0.850720 + 0.525620i \(0.823834\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.86458 0.297643 0.148822 0.988864i \(-0.452452\pi\)
0.148822 + 0.988864i \(0.452452\pi\)
\(888\) 0 0
\(889\) −3.12702 −0.104877
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 57.8544i − 1.93602i
\(894\) 0 0
\(895\) −10.8730 −0.363444
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 17.1464i − 0.571865i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 17.5481 0.583318
\(906\) 0 0
\(907\) − 8.00000i − 0.265636i −0.991140 0.132818i \(-0.957597\pi\)
0.991140 0.132818i \(-0.0424025\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −23.5615 −0.780628 −0.390314 0.920682i \(-0.627634\pi\)
−0.390314 + 0.920682i \(0.627634\pi\)
\(912\) 0 0
\(913\) 45.6028 1.50923
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 3.87507i − 0.127966i
\(918\) 0 0
\(919\) 45.4919 1.50064 0.750320 0.661075i \(-0.229899\pi\)
0.750320 + 0.661075i \(0.229899\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 17.1464i − 0.564382i
\(924\) 0 0
\(925\) 16.8730i 0.554780i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 10.8219 0.355054 0.177527 0.984116i \(-0.443190\pi\)
0.177527 + 0.984116i \(0.443190\pi\)
\(930\) 0 0
\(931\) 7.87298i 0.258027i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −31.2211 −1.02104
\(936\) 0 0
\(937\) 10.5081 0.343284 0.171642 0.985159i \(-0.445093\pi\)
0.171642 + 0.985159i \(0.445093\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 39.9499i 1.30233i 0.758937 + 0.651165i \(0.225719\pi\)
−0.758937 + 0.651165i \(0.774281\pi\)
\(942\) 0 0
\(943\) 3.61895 0.117849
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 49.4367i 1.60648i 0.595657 + 0.803239i \(0.296892\pi\)
−0.595657 + 0.803239i \(0.703108\pi\)
\(948\) 0 0
\(949\) − 33.4919i − 1.08719i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −19.2848 −0.624696 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(954\) 0 0
\(955\) − 54.2379i − 1.75510i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −14.3858 −0.464542
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 60.3039i − 1.94125i
\(966\) 0 0
\(967\) 21.2379 0.682965 0.341482 0.939888i \(-0.389071\pi\)
0.341482 + 0.939888i \(0.389071\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 43.1574i − 1.38499i −0.721424 0.692494i \(-0.756512\pi\)
0.721424 0.692494i \(-0.243488\pi\)
\(972\) 0 0
\(973\) − 14.0000i − 0.448819i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −8.46292 −0.270753 −0.135376 0.990794i \(-0.543224\pi\)
−0.135376 + 0.990794i \(0.543224\pi\)
\(978\) 0 0
\(979\) − 35.7298i − 1.14193i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 9.57737 0.305471 0.152735 0.988267i \(-0.451192\pi\)
0.152735 + 0.988267i \(0.451192\pi\)
\(984\) 0 0
\(985\) −76.7298 −2.44482
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.16228i 0.100555i
\(990\) 0 0
\(991\) −40.6190 −1.29030 −0.645152 0.764054i \(-0.723206\pi\)
−0.645152 + 0.764054i \(0.723206\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 22.0907i 0.700321i
\(996\) 0 0
\(997\) − 18.0000i − 0.570066i −0.958518 0.285033i \(-0.907995\pi\)
0.958518 0.285033i \(-0.0920045\pi\)
\(998\) 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.c.c.3025.2 8
3.2 odd 2 inner 6048.2.c.c.3025.8 8
4.3 odd 2 1512.2.c.c.757.4 yes 8
8.3 odd 2 1512.2.c.c.757.3 8
8.5 even 2 inner 6048.2.c.c.3025.7 8
12.11 even 2 1512.2.c.c.757.5 yes 8
24.5 odd 2 inner 6048.2.c.c.3025.1 8
24.11 even 2 1512.2.c.c.757.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.c.757.3 8 8.3 odd 2
1512.2.c.c.757.4 yes 8 4.3 odd 2
1512.2.c.c.757.5 yes 8 12.11 even 2
1512.2.c.c.757.6 yes 8 24.11 even 2
6048.2.c.c.3025.1 8 24.5 odd 2 inner
6048.2.c.c.3025.2 8 1.1 even 1 trivial
6048.2.c.c.3025.7 8 8.5 even 2 inner
6048.2.c.c.3025.8 8 3.2 odd 2 inner