Properties

Label 3420.2.f.a.1369.4
Level $3420$
Weight $2$
Character 3420.1369
Analytic conductor $27.309$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1369.4
Root \(2.28825i\) of defining polynomial
Character \(\chi\) \(=\) 3420.1369
Dual form 3420.2.f.a.1369.3

$q$-expansion

\(f(q)\) \(=\) \(q+2.23607 q^{5} +2.82843i q^{7} +O(q^{10})\) \(q+2.23607 q^{5} +2.82843i q^{7} +5.23607 q^{11} -4.03631i q^{13} +1.08036i q^{17} +1.00000 q^{19} -7.40492i q^{23} +5.00000 q^{25} +4.47214 q^{29} -4.00000 q^{31} +6.32456i q^{35} +6.86474i q^{37} +6.00000 q^{41} -8.48528i q^{43} -8.48528i q^{47} -1.00000 q^{49} -6.86474i q^{53} +11.7082 q^{55} -10.4721 q^{59} +1.70820 q^{61} -9.02546i q^{65} +1.62054i q^{67} +1.52786 q^{71} +13.7295i q^{73} +14.8098i q^{77} +11.4164 q^{79} -5.24419i q^{83} +2.41577i q^{85} -14.9443 q^{89} +11.4164 q^{91} +2.23607 q^{95} +4.44897i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{11} + 4 q^{19} + 20 q^{25} - 16 q^{31} + 24 q^{41} - 4 q^{49} + 20 q^{55} - 24 q^{59} - 20 q^{61} + 24 q^{71} - 8 q^{79} - 24 q^{89} - 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(5\) 2.23607 1.00000
\(6\) 0 0
\(7\) 2.82843i 1.06904i 0.845154 + 0.534522i \(0.179509\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.23607 1.57873 0.789367 0.613922i \(-0.210409\pi\)
0.789367 + 0.613922i \(0.210409\pi\)
\(12\) 0 0
\(13\) − 4.03631i − 1.11947i −0.828671 0.559735i \(-0.810903\pi\)
0.828671 0.559735i \(-0.189097\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.08036i 0.262027i 0.991381 + 0.131013i \(0.0418230\pi\)
−0.991381 + 0.131013i \(0.958177\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 7.40492i − 1.54403i −0.635603 0.772016i \(-0.719248\pi\)
0.635603 0.772016i \(-0.280752\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.32456i 1.06904i
\(36\) 0 0
\(37\) 6.86474i 1.12856i 0.825585 + 0.564278i \(0.190845\pi\)
−0.825585 + 0.564278i \(0.809155\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) − 8.48528i − 1.29399i −0.762493 0.646997i \(-0.776025\pi\)
0.762493 0.646997i \(-0.223975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 8.48528i − 1.23771i −0.785507 0.618853i \(-0.787598\pi\)
0.785507 0.618853i \(-0.212402\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.86474i − 0.942944i −0.881881 0.471472i \(-0.843723\pi\)
0.881881 0.471472i \(-0.156277\pi\)
\(54\) 0 0
\(55\) 11.7082 1.57873
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.4721 −1.36336 −0.681678 0.731652i \(-0.738749\pi\)
−0.681678 + 0.731652i \(0.738749\pi\)
\(60\) 0 0
\(61\) 1.70820 0.218713 0.109357 0.994003i \(-0.465121\pi\)
0.109357 + 0.994003i \(0.465121\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 9.02546i − 1.11947i
\(66\) 0 0
\(67\) 1.62054i 0.197981i 0.995088 + 0.0989905i \(0.0315613\pi\)
−0.995088 + 0.0989905i \(0.968439\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.52786 0.181324 0.0906621 0.995882i \(-0.471102\pi\)
0.0906621 + 0.995882i \(0.471102\pi\)
\(72\) 0 0
\(73\) 13.7295i 1.60691i 0.595363 + 0.803457i \(0.297008\pi\)
−0.595363 + 0.803457i \(0.702992\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 14.8098i 1.68774i
\(78\) 0 0
\(79\) 11.4164 1.28445 0.642223 0.766518i \(-0.278012\pi\)
0.642223 + 0.766518i \(0.278012\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 5.24419i − 0.575625i −0.957687 0.287812i \(-0.907072\pi\)
0.957687 0.287812i \(-0.0929280\pi\)
\(84\) 0 0
\(85\) 2.41577i 0.262027i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.9443 −1.58409 −0.792045 0.610463i \(-0.790983\pi\)
−0.792045 + 0.610463i \(0.790983\pi\)
\(90\) 0 0
\(91\) 11.4164 1.19676
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.23607 0.229416
\(96\) 0 0
\(97\) 4.44897i 0.451725i 0.974159 + 0.225862i \(0.0725199\pi\)
−0.974159 + 0.225862i \(0.927480\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.763932 −0.0760141 −0.0380070 0.999277i \(-0.512101\pi\)
−0.0380070 + 0.999277i \(0.512101\pi\)
\(102\) 0 0
\(103\) 15.3500i 1.51248i 0.654293 + 0.756241i \(0.272966\pi\)
−0.654293 + 0.756241i \(0.727034\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.4304i 1.58838i 0.607666 + 0.794192i \(0.292106\pi\)
−0.607666 + 0.794192i \(0.707894\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.1089i 1.13911i 0.821952 + 0.569556i \(0.192885\pi\)
−0.821952 + 0.569556i \(0.807115\pi\)
\(114\) 0 0
\(115\) − 16.5579i − 1.54403i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.05573 −0.280118
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) − 10.1058i − 0.896747i −0.893846 0.448374i \(-0.852003\pi\)
0.893846 0.448374i \(-0.147997\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 2.82843i 0.245256i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.08036i 0.0923016i 0.998934 + 0.0461508i \(0.0146955\pi\)
−0.998934 + 0.0461508i \(0.985305\pi\)
\(138\) 0 0
\(139\) −15.7082 −1.33235 −0.666176 0.745794i \(-0.732070\pi\)
−0.666176 + 0.745794i \(0.732070\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 21.1344i − 1.76735i
\(144\) 0 0
\(145\) 10.0000 0.830455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 18.6525 1.52807 0.764035 0.645175i \(-0.223215\pi\)
0.764035 + 0.645175i \(0.223215\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.94427 −0.718421
\(156\) 0 0
\(157\) − 19.3863i − 1.54720i −0.633676 0.773599i \(-0.718455\pi\)
0.633676 0.773599i \(-0.281545\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 20.9443 1.65064
\(162\) 0 0
\(163\) 8.48528i 0.664619i 0.943170 + 0.332309i \(0.107828\pi\)
−0.943170 + 0.332309i \(0.892172\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.70401i 0.364007i 0.983298 + 0.182004i \(0.0582583\pi\)
−0.983298 + 0.182004i \(0.941742\pi\)
\(168\) 0 0
\(169\) −3.29180 −0.253215
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 13.1893i − 1.00276i −0.865226 0.501382i \(-0.832825\pi\)
0.865226 0.501382i \(-0.167175\pi\)
\(174\) 0 0
\(175\) 14.1421i 1.06904i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −13.5279 −1.01112 −0.505560 0.862791i \(-0.668714\pi\)
−0.505560 + 0.862791i \(0.668714\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 15.3500i 1.12856i
\(186\) 0 0
\(187\) 5.65685i 0.413670i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.9443 1.51547 0.757737 0.652560i \(-0.226305\pi\)
0.757737 + 0.652560i \(0.226305\pi\)
\(192\) 0 0
\(193\) − 4.44897i − 0.320244i −0.987097 0.160122i \(-0.948811\pi\)
0.987097 0.160122i \(-0.0511888\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.6491i 0.901212i 0.892723 + 0.450606i \(0.148792\pi\)
−0.892723 + 0.450606i \(0.851208\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 12.6491i 0.887794i
\(204\) 0 0
\(205\) 13.4164 0.937043
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.23607 0.362186
\(210\) 0 0
\(211\) −11.4164 −0.785938 −0.392969 0.919552i \(-0.628552\pi\)
−0.392969 + 0.919552i \(0.628552\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 18.9737i − 1.29399i
\(216\) 0 0
\(217\) − 11.3137i − 0.768025i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.36068 0.293331
\(222\) 0 0
\(223\) 28.6668i 1.91967i 0.280560 + 0.959836i \(0.409480\pi\)
−0.280560 + 0.959836i \(0.590520\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 15.3500i − 1.01882i −0.860525 0.509408i \(-0.829864\pi\)
0.860525 0.509408i \(-0.170136\pi\)
\(228\) 0 0
\(229\) 5.70820 0.377209 0.188604 0.982053i \(-0.439604\pi\)
0.188604 + 0.982053i \(0.439604\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.6491i 0.828671i 0.910124 + 0.414335i \(0.135986\pi\)
−0.910124 + 0.414335i \(0.864014\pi\)
\(234\) 0 0
\(235\) − 18.9737i − 1.23771i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.88854 0.380898 0.190449 0.981697i \(-0.439006\pi\)
0.190449 + 0.981697i \(0.439006\pi\)
\(240\) 0 0
\(241\) 24.8328 1.59962 0.799811 0.600252i \(-0.204933\pi\)
0.799811 + 0.600252i \(0.204933\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.23607 −0.142857
\(246\) 0 0
\(247\) − 4.03631i − 0.256824i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −29.8885 −1.88655 −0.943274 0.332015i \(-0.892272\pi\)
−0.943274 + 0.332015i \(0.892272\pi\)
\(252\) 0 0
\(253\) − 38.7727i − 2.43762i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 4.70401i − 0.293428i −0.989179 0.146714i \(-0.953130\pi\)
0.989179 0.146714i \(-0.0468697\pi\)
\(258\) 0 0
\(259\) −19.4164 −1.20648
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.56564i 0.589843i 0.955522 + 0.294921i \(0.0952935\pi\)
−0.955522 + 0.294921i \(0.904707\pi\)
\(264\) 0 0
\(265\) − 15.3500i − 0.942944i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.94427 −0.179515 −0.0897577 0.995964i \(-0.528609\pi\)
−0.0897577 + 0.995964i \(0.528609\pi\)
\(270\) 0 0
\(271\) −19.1246 −1.16174 −0.580869 0.813997i \(-0.697287\pi\)
−0.580869 + 0.813997i \(0.697287\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 26.1803 1.57873
\(276\) 0 0
\(277\) − 3.24109i − 0.194738i −0.995248 0.0973691i \(-0.968957\pi\)
0.995248 0.0973691i \(-0.0310427\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.41641 −0.0844958 −0.0422479 0.999107i \(-0.513452\pi\)
−0.0422479 + 0.999107i \(0.513452\pi\)
\(282\) 0 0
\(283\) − 16.5579i − 0.984265i −0.870520 0.492133i \(-0.836218\pi\)
0.870520 0.492133i \(-0.163782\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 16.9706i 1.00174i
\(288\) 0 0
\(289\) 15.8328 0.931342
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 6.86474i − 0.401042i −0.979689 0.200521i \(-0.935736\pi\)
0.979689 0.200521i \(-0.0642635\pi\)
\(294\) 0 0
\(295\) −23.4164 −1.36336
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −29.8885 −1.72850
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.81966 0.218713
\(306\) 0 0
\(307\) 4.03631i 0.230364i 0.993344 + 0.115182i \(0.0367452\pi\)
−0.993344 + 0.115182i \(0.963255\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.70820 −0.210273 −0.105136 0.994458i \(-0.533528\pi\)
−0.105136 + 0.994458i \(0.533528\pi\)
\(312\) 0 0
\(313\) − 27.4589i − 1.55207i −0.630689 0.776036i \(-0.717228\pi\)
0.630689 0.776036i \(-0.282772\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 13.1893i − 0.740784i −0.928875 0.370392i \(-0.879223\pi\)
0.928875 0.370392i \(-0.120777\pi\)
\(318\) 0 0
\(319\) 23.4164 1.31107
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.08036i 0.0601130i
\(324\) 0 0
\(325\) − 20.1815i − 1.11947i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −19.4164 −1.06722 −0.533611 0.845730i \(-0.679165\pi\)
−0.533611 + 0.845730i \(0.679165\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.62365i 0.197981i
\(336\) 0 0
\(337\) 7.27740i 0.396425i 0.980159 + 0.198213i \(0.0635136\pi\)
−0.980159 + 0.198213i \(0.936486\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −20.9443 −1.13420
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 34.8639i − 1.87159i −0.352544 0.935795i \(-0.614683\pi\)
0.352544 0.935795i \(-0.385317\pi\)
\(348\) 0 0
\(349\) 1.41641 0.0758186 0.0379093 0.999281i \(-0.487930\pi\)
0.0379093 + 0.999281i \(0.487930\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 15.8902i 0.845750i 0.906188 + 0.422875i \(0.138979\pi\)
−0.906188 + 0.422875i \(0.861021\pi\)
\(354\) 0 0
\(355\) 3.41641 0.181324
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −11.1246 −0.587135 −0.293567 0.955938i \(-0.594842\pi\)
−0.293567 + 0.955938i \(0.594842\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 30.7000i 1.60691i
\(366\) 0 0
\(367\) 8.48528i 0.442928i 0.975169 + 0.221464i \(0.0710835\pi\)
−0.975169 + 0.221464i \(0.928916\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 19.4164 1.00805
\(372\) 0 0
\(373\) 34.3237i 1.77721i 0.458670 + 0.888606i \(0.348326\pi\)
−0.458670 + 0.888606i \(0.651674\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 18.0509i − 0.929670i
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 23.6777i 1.20987i 0.796274 + 0.604936i \(0.206801\pi\)
−0.796274 + 0.604936i \(0.793199\pi\)
\(384\) 0 0
\(385\) 33.1158i 1.68774i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.94427 0.149281 0.0746403 0.997211i \(-0.476219\pi\)
0.0746403 + 0.997211i \(0.476219\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 25.5279 1.28445
\(396\) 0 0
\(397\) 30.7000i 1.54079i 0.637566 + 0.770395i \(0.279941\pi\)
−0.637566 + 0.770395i \(0.720059\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −4.47214 −0.223328 −0.111664 0.993746i \(-0.535618\pi\)
−0.111664 + 0.993746i \(0.535618\pi\)
\(402\) 0 0
\(403\) 16.1452i 0.804252i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 35.9442i 1.78169i
\(408\) 0 0
\(409\) 9.41641 0.465611 0.232806 0.972523i \(-0.425209\pi\)
0.232806 + 0.972523i \(0.425209\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 29.6197i − 1.45749i
\(414\) 0 0
\(415\) − 11.7264i − 0.575625i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −3.05573 −0.149282 −0.0746410 0.997210i \(-0.523781\pi\)
−0.0746410 + 0.997210i \(0.523781\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5.40182i 0.262027i
\(426\) 0 0
\(427\) 4.83153i 0.233814i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −10.4721 −0.504425 −0.252213 0.967672i \(-0.581158\pi\)
−0.252213 + 0.967672i \(0.581158\pi\)
\(432\) 0 0
\(433\) 37.9774i 1.82508i 0.408989 + 0.912540i \(0.365882\pi\)
−0.408989 + 0.912540i \(0.634118\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 7.40492i − 0.354225i
\(438\) 0 0
\(439\) −34.8328 −1.66248 −0.831240 0.555914i \(-0.812368\pi\)
−0.831240 + 0.555914i \(0.812368\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5.24419i 0.249159i 0.992210 + 0.124580i \(0.0397582\pi\)
−0.992210 + 0.124580i \(0.960242\pi\)
\(444\) 0 0
\(445\) −33.4164 −1.58409
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −7.52786 −0.355262 −0.177631 0.984097i \(-0.556843\pi\)
−0.177631 + 0.984097i \(0.556843\pi\)
\(450\) 0 0
\(451\) 31.4164 1.47934
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 25.5279 1.19676
\(456\) 0 0
\(457\) − 37.9473i − 1.77510i −0.460710 0.887551i \(-0.652405\pi\)
0.460710 0.887551i \(-0.347595\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 11.8885 0.553705 0.276852 0.960912i \(-0.410709\pi\)
0.276852 + 0.960912i \(0.410709\pi\)
\(462\) 0 0
\(463\) − 33.5285i − 1.55820i −0.626900 0.779100i \(-0.715676\pi\)
0.626900 0.779100i \(-0.284324\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 11.7264i − 0.542632i −0.962490 0.271316i \(-0.912541\pi\)
0.962490 0.271316i \(-0.0874588\pi\)
\(468\) 0 0
\(469\) −4.58359 −0.211651
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 44.4295i − 2.04287i
\(474\) 0 0
\(475\) 5.00000 0.229416
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6.76393 0.309052 0.154526 0.987989i \(-0.450615\pi\)
0.154526 + 0.987989i \(0.450615\pi\)
\(480\) 0 0
\(481\) 27.7082 1.26339
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.94820i 0.451725i
\(486\) 0 0
\(487\) 17.7658i 0.805044i 0.915410 + 0.402522i \(0.131866\pi\)
−0.915410 + 0.402522i \(0.868134\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 29.8885 1.34885 0.674426 0.738343i \(-0.264391\pi\)
0.674426 + 0.738343i \(0.264391\pi\)
\(492\) 0 0
\(493\) 4.83153i 0.217601i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.32145i 0.193844i
\(498\) 0 0
\(499\) −23.1246 −1.03520 −0.517600 0.855623i \(-0.673174\pi\)
−0.517600 + 0.855623i \(0.673174\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 39.1853i 1.74719i 0.486656 + 0.873593i \(0.338216\pi\)
−0.486656 + 0.873593i \(0.661784\pi\)
\(504\) 0 0
\(505\) −1.70820 −0.0760141
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −37.4164 −1.65845 −0.829227 0.558913i \(-0.811219\pi\)
−0.829227 + 0.558913i \(0.811219\pi\)
\(510\) 0 0
\(511\) −38.8328 −1.71786
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 34.3237i 1.51248i
\(516\) 0 0
\(517\) − 44.4295i − 1.95401i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −35.8885 −1.57231 −0.786153 0.618032i \(-0.787930\pi\)
−0.786153 + 0.618032i \(0.787930\pi\)
\(522\) 0 0
\(523\) − 3.62365i − 0.158451i −0.996857 0.0792255i \(-0.974755\pi\)
0.996857 0.0792255i \(-0.0252447\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 4.32145i − 0.188245i
\(528\) 0 0
\(529\) −31.8328 −1.38404
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 24.2179i − 1.04899i
\(534\) 0 0
\(535\) 36.7394i 1.58838i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5.23607 −0.225533
\(540\) 0 0
\(541\) −1.70820 −0.0734414 −0.0367207 0.999326i \(-0.511691\pi\)
−0.0367207 + 0.999326i \(0.511691\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.47214 −0.191565
\(546\) 0 0
\(547\) 1.20788i 0.0516453i 0.999667 + 0.0258227i \(0.00822052\pi\)
−0.999667 + 0.0258227i \(0.991779\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.47214 0.190519
\(552\) 0 0
\(553\) 32.2905i 1.37313i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11.5687i 0.490184i 0.969500 + 0.245092i \(0.0788181\pi\)
−0.969500 + 0.245092i \(0.921182\pi\)
\(558\) 0 0
\(559\) −34.2492 −1.44859
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 17.3531i − 0.731347i −0.930743 0.365673i \(-0.880839\pi\)
0.930743 0.365673i \(-0.119161\pi\)
\(564\) 0 0
\(565\) 27.0764i 1.13911i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −21.0557 −0.882702 −0.441351 0.897335i \(-0.645501\pi\)
−0.441351 + 0.897335i \(0.645501\pi\)
\(570\) 0 0
\(571\) −15.1246 −0.632945 −0.316473 0.948602i \(-0.602499\pi\)
−0.316473 + 0.948602i \(0.602499\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 37.0246i − 1.54403i
\(576\) 0 0
\(577\) − 5.65685i − 0.235498i −0.993043 0.117749i \(-0.962432\pi\)
0.993043 0.117749i \(-0.0375678\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 14.8328 0.615369
\(582\) 0 0
\(583\) − 35.9442i − 1.48866i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 37.0246i − 1.52817i −0.645117 0.764084i \(-0.723191\pi\)
0.645117 0.764084i \(-0.276809\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 14.8098i − 0.608167i −0.952645 0.304084i \(-0.901650\pi\)
0.952645 0.304084i \(-0.0983502\pi\)
\(594\) 0 0
\(595\) −6.83282 −0.280118
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −36.8328 −1.50244 −0.751221 0.660051i \(-0.770535\pi\)
−0.751221 + 0.660051i \(0.770535\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 36.7082 1.49240
\(606\) 0 0
\(607\) 12.9343i 0.524985i 0.964934 + 0.262493i \(0.0845445\pi\)
−0.964934 + 0.262493i \(0.915455\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −34.2492 −1.38558
\(612\) 0 0
\(613\) 20.2117i 0.816341i 0.912906 + 0.408170i \(0.133833\pi\)
−0.912906 + 0.408170i \(0.866167\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 28.5393i 1.14895i 0.818522 + 0.574475i \(0.194794\pi\)
−0.818522 + 0.574475i \(0.805206\pi\)
\(618\) 0 0
\(619\) −0.875388 −0.0351848 −0.0175924 0.999845i \(-0.505600\pi\)
−0.0175924 + 0.999845i \(0.505600\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 42.2688i − 1.69346i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −7.41641 −0.295712
\(630\) 0 0
\(631\) 11.7082 0.466096 0.233048 0.972465i \(-0.425130\pi\)
0.233048 + 0.972465i \(0.425130\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 22.5973i − 0.896747i
\(636\) 0 0
\(637\) 4.03631i 0.159924i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 47.8885 1.89148 0.945742 0.324919i \(-0.105337\pi\)
0.945742 + 0.324919i \(0.105337\pi\)
\(642\) 0 0
\(643\) 11.7264i 0.462443i 0.972901 + 0.231221i \(0.0742722\pi\)
−0.972901 + 0.231221i \(0.925728\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 34.8639i − 1.37064i −0.728242 0.685320i \(-0.759662\pi\)
0.728242 0.685320i \(-0.240338\pi\)
\(648\) 0 0
\(649\) −54.8328 −2.15238
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 3.24109i − 0.126834i −0.997987 0.0634168i \(-0.979800\pi\)
0.997987 0.0634168i \(-0.0201998\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −27.0557 −1.05394 −0.526971 0.849883i \(-0.676672\pi\)
−0.526971 + 0.849883i \(0.676672\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.32456i 0.245256i
\(666\) 0 0
\(667\) − 33.1158i − 1.28225i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 8.94427 0.345290
\(672\) 0 0
\(673\) − 42.8090i − 1.65016i −0.565013 0.825082i \(-0.691129\pi\)
0.565013 0.825082i \(-0.308871\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 30.0022i 1.15308i 0.817069 + 0.576540i \(0.195597\pi\)
−0.817069 + 0.576540i \(0.804403\pi\)
\(678\) 0 0
\(679\) −12.5836 −0.482914
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10.1058i 0.386689i 0.981131 + 0.193344i \(0.0619334\pi\)
−0.981131 + 0.193344i \(0.938067\pi\)
\(684\) 0 0
\(685\) 2.41577i 0.0923016i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −27.7082 −1.05560
\(690\) 0 0
\(691\) −15.1246 −0.575367 −0.287684 0.957725i \(-0.592885\pi\)
−0.287684 + 0.957725i \(0.592885\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −35.1246 −1.33235
\(696\) 0 0
\(697\) 6.48218i 0.245530i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −41.1246 −1.55326 −0.776628 0.629960i \(-0.783071\pi\)
−0.776628 + 0.629960i \(0.783071\pi\)
\(702\) 0 0
\(703\) 6.86474i 0.258908i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 2.16073i − 0.0812625i
\(708\) 0 0
\(709\) −6.58359 −0.247252 −0.123626 0.992329i \(-0.539452\pi\)
−0.123626 + 0.992329i \(0.539452\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 29.6197i 1.10927i
\(714\) 0 0
\(715\) − 47.2579i − 1.76735i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −21.5967 −0.805423 −0.402711 0.915327i \(-0.631932\pi\)
−0.402711 + 0.915327i \(0.631932\pi\)
\(720\) 0 0
\(721\) −43.4164 −1.61691
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 22.3607 0.830455
\(726\) 0 0
\(727\) 8.48528i 0.314702i 0.987543 + 0.157351i \(0.0502953\pi\)
−0.987543 + 0.157351i \(0.949705\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 9.16718 0.339061
\(732\) 0 0
\(733\) 33.9411i 1.25364i 0.779162 + 0.626822i \(0.215645\pi\)
−0.779162 + 0.626822i \(0.784355\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.48528i 0.312559i
\(738\) 0 0
\(739\) −34.8328 −1.28135 −0.640673 0.767814i \(-0.721345\pi\)
−0.640673 + 0.767814i \(0.721345\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 6.86474i 0.251843i 0.992040 + 0.125921i \(0.0401887\pi\)
−0.992040 + 0.125921i \(0.959811\pi\)
\(744\) 0 0
\(745\) 41.7082 1.52807
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −46.4721 −1.69805
\(750\) 0 0
\(751\) 47.4164 1.73025 0.865125 0.501557i \(-0.167239\pi\)
0.865125 + 0.501557i \(0.167239\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −17.8885 −0.651031
\(756\) 0 0
\(757\) − 0.825324i − 0.0299969i −0.999888 0.0149985i \(-0.995226\pi\)
0.999888 0.0149985i \(-0.00477434\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −24.5410 −0.889611 −0.444806 0.895627i \(-0.646727\pi\)
−0.444806 + 0.895627i \(0.646727\pi\)
\(762\) 0 0
\(763\) − 5.65685i − 0.204792i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 42.2688i 1.52624i
\(768\) 0 0
\(769\) −28.5410 −1.02922 −0.514608 0.857426i \(-0.672062\pi\)
−0.514608 + 0.857426i \(0.672062\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 15.3500i 0.552102i 0.961143 + 0.276051i \(0.0890258\pi\)
−0.961143 + 0.276051i \(0.910974\pi\)
\(774\) 0 0
\(775\) −20.0000 −0.718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 43.3491i − 1.54720i
\(786\) 0 0
\(787\) − 3.62365i − 0.129169i −0.997912 0.0645845i \(-0.979428\pi\)
0.997912 0.0645845i \(-0.0205722\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −34.2492 −1.21776
\(792\) 0 0
\(793\) − 6.89484i − 0.244843i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 14.1120i − 0.499874i −0.968262 0.249937i \(-0.919590\pi\)
0.968262 0.249937i \(-0.0804099\pi\)
\(798\) 0 0
\(799\) 9.16718 0.324312
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 71.8885i 2.53689i
\(804\) 0 0
\(805\) 46.8328 1.65064
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16.4721 −0.579129 −0.289565 0.957158i \(-0.593511\pi\)
−0.289565 + 0.957158i \(0.593511\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 18.9737i 0.664619i
\(816\) 0 0
\(817\) − 8.48528i − 0.296862i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −21.0557 −0.734850 −0.367425 0.930053i \(-0.619761\pi\)
−0.367425 + 0.930053i \(0.619761\pi\)
\(822\) 0 0
\(823\) − 2.00310i − 0.0698238i −0.999390 0.0349119i \(-0.988885\pi\)
0.999390 0.0349119i \(-0.0111151\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 20.5942i − 0.716131i −0.933696 0.358065i \(-0.883436\pi\)
0.933696 0.358065i \(-0.116564\pi\)
\(828\) 0 0
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 1.08036i − 0.0374324i
\(834\) 0 0
\(835\) 10.5185i 0.364007i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −13.3050 −0.459338 −0.229669 0.973269i \(-0.573764\pi\)
−0.229669 + 0.973269i \(0.573764\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −7.36068 −0.253215
\(846\) 0 0
\(847\) 46.4326i 1.59544i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 50.8328 1.74253
\(852\) 0 0
\(853\) − 10.4884i − 0.359115i −0.983747 0.179558i \(-0.942533\pi\)
0.983747 0.179558i \(-0.0574667\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 40.8059i − 1.39390i −0.717119 0.696951i \(-0.754540\pi\)
0.717119 0.696951i \(-0.245460\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 20.5942i − 0.701035i −0.936556 0.350518i \(-0.886006\pi\)
0.936556 0.350518i \(-0.113994\pi\)
\(864\) 0 0
\(865\) − 29.4922i − 1.00276i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 59.7771 2.02780
\(870\) 0 0
\(871\) 6.54102 0.221634
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 31.6228i 1.06904i
\(876\) 0 0
\(877\) 4.44897i 0.150231i 0.997175 + 0.0751155i \(0.0239326\pi\)
−0.997175 + 0.0751155i \(0.976067\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 30.6525 1.03271 0.516354 0.856375i \(-0.327289\pi\)
0.516354 + 0.856375i \(0.327289\pi\)
\(882\) 0 0
\(883\) 14.9675i 0.503695i 0.967767 + 0.251848i \(0.0810382\pi\)
−0.967767 + 0.251848i \(0.918962\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 36.3268i − 1.21973i −0.792504 0.609867i \(-0.791223\pi\)
0.792504 0.609867i \(-0.208777\pi\)
\(888\) 0 0
\(889\) 28.5836 0.958663
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 8.48528i − 0.283949i
\(894\) 0 0
\(895\) −30.2492 −1.01112
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −17.8885 −0.596616
\(900\) 0 0
\(901\) 7.41641 0.247076
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 13.4164 0.445976
\(906\) 0 0
\(907\) 8.86784i 0.294452i 0.989103 + 0.147226i \(0.0470344\pi\)
−0.989103 + 0.147226i \(0.952966\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 37.5279 1.24335 0.621677 0.783274i \(-0.286452\pi\)
0.621677 + 0.783274i \(0.286452\pi\)
\(912\) 0 0
\(913\) − 27.4589i − 0.908759i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 26.8328 0.885133 0.442566 0.896736i \(-0.354068\pi\)
0.442566 + 0.896736i \(0.354068\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 6.16693i − 0.202987i
\(924\) 0 0
\(925\) 34.3237i 1.12856i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 16.4721 0.540433 0.270217 0.962800i \(-0.412905\pi\)
0.270217 + 0.962800i \(0.412905\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 12.6491i 0.413670i
\(936\) 0 0
\(937\) 51.6768i 1.68821i 0.536180 + 0.844104i \(0.319867\pi\)
−0.536180 + 0.844104i \(0.680133\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 21.0557 0.686397 0.343199 0.939263i \(-0.388490\pi\)
0.343199 + 0.939263i \(0.388490\pi\)
\(942\) 0 0
\(943\) − 44.4295i − 1.44682i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.6969i 0.932525i 0.884646 + 0.466263i \(0.154400\pi\)
−0.884646 + 0.466263i \(0.845600\pi\)
\(948\) 0 0
\(949\) 55.4164 1.79889
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 27.0764i − 0.877090i −0.898709 0.438545i \(-0.855494\pi\)
0.898709 0.438545i \(-0.144506\pi\)
\(954\) 0 0
\(955\) 46.8328 1.51547
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.05573 −0.0986746
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 9.94820i − 0.320244i
\(966\) 0 0
\(967\) 16.5579i 0.532466i 0.963909 + 0.266233i \(0.0857791\pi\)
−0.963909 + 0.266233i \(0.914221\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.36068 −0.139941 −0.0699704 0.997549i \(-0.522290\pi\)
−0.0699704 + 0.997549i \(0.522290\pi\)
\(972\) 0 0
\(973\) − 44.4295i − 1.42434i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 47.1304i − 1.50784i −0.656969 0.753918i \(-0.728162\pi\)
0.656969 0.753918i \(-0.271838\pi\)
\(978\) 0 0
\(979\) −78.2492 −2.50086
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 33.4009i − 1.06532i −0.846328 0.532662i \(-0.821192\pi\)
0.846328 0.532662i \(-0.178808\pi\)
\(984\) 0 0
\(985\) 28.2843i 0.901212i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −62.8328 −1.99797
\(990\) 0 0
\(991\) −14.8328 −0.471180 −0.235590 0.971853i \(-0.575702\pi\)
−0.235590 + 0.971853i \(0.575702\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 35.7771 1.13421
\(996\) 0 0
\(997\) − 30.7000i − 0.972280i −0.873881 0.486140i \(-0.838405\pi\)
0.873881 0.486140i \(-0.161595\pi\)
\(998\) 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 3420.2.f.a.1369.4 4
3.2 odd 2 380.2.c.a.229.4 yes 4
5.4 even 2 inner 3420.2.f.a.1369.3 4
12.11 even 2 1520.2.d.f.609.1 4
15.2 even 4 1900.2.a.j.1.4 4
15.8 even 4 1900.2.a.j.1.1 4
15.14 odd 2 380.2.c.a.229.1 4
60.23 odd 4 7600.2.a.ce.1.4 4
60.47 odd 4 7600.2.a.ce.1.1 4
60.59 even 2 1520.2.d.f.609.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.c.a.229.1 4 15.14 odd 2
380.2.c.a.229.4 yes 4 3.2 odd 2
1520.2.d.f.609.1 4 12.11 even 2
1520.2.d.f.609.4 4 60.59 even 2
1900.2.a.j.1.1 4 15.8 even 4
1900.2.a.j.1.4 4 15.2 even 4
3420.2.f.a.1369.3 4 5.4 even 2 inner
3420.2.f.a.1369.4 4 1.1 even 1 trivial
7600.2.a.ce.1.1 4 60.47 odd 4
7600.2.a.ce.1.4 4 60.23 odd 4