Properties

Label 380.2.c.a.229.1
Level $380$
Weight $2$
Character 380.229
Analytic conductor $3.034$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.03431527681\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 229.1
Root \(-2.28825i\) of defining polynomial
Character \(\chi\) \(=\) 380.229
Dual form 380.2.c.a.229.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.28825i q^{3} -2.23607 q^{5} -2.82843i q^{7} -2.23607 q^{9} +O(q^{10})\) \(q-2.28825i q^{3} -2.23607 q^{5} -2.82843i q^{7} -2.23607 q^{9} -5.23607 q^{11} +4.03631i q^{13} +5.11667i q^{15} +1.08036i q^{17} +1.00000 q^{19} -6.47214 q^{21} -7.40492i q^{23} +5.00000 q^{25} -1.74806i q^{27} -4.47214 q^{29} -4.00000 q^{31} +11.9814i q^{33} +6.32456i q^{35} -6.86474i q^{37} +9.23607 q^{39} -6.00000 q^{41} +8.48528i q^{43} +5.00000 q^{45} -8.48528i q^{47} -1.00000 q^{49} +2.47214 q^{51} -6.86474i q^{53} +11.7082 q^{55} -2.28825i q^{57} +10.4721 q^{59} +1.70820 q^{61} +6.32456i q^{63} -9.02546i q^{65} -1.62054i q^{67} -16.9443 q^{69} -1.52786 q^{71} -13.7295i q^{73} -11.4412i q^{75} +14.8098i q^{77} +11.4164 q^{79} -10.7082 q^{81} -5.24419i q^{83} -2.41577i q^{85} +10.2333i q^{87} +14.9443 q^{89} +11.4164 q^{91} +9.15298i q^{93} -2.23607 q^{95} -4.44897i q^{97} +11.7082 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{11} + 4 q^{19} - 8 q^{21} + 20 q^{25} - 16 q^{31} + 28 q^{39} - 24 q^{41} + 20 q^{45} - 4 q^{49} - 8 q^{51} + 20 q^{55} + 24 q^{59} - 20 q^{61} - 32 q^{69} - 24 q^{71} - 8 q^{79} - 16 q^{81} + 24 q^{89} - 8 q^{91} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 2.28825i − 1.32112i −0.750774 0.660560i \(-0.770319\pi\)
0.750774 0.660560i \(-0.229681\pi\)
\(4\) 0 0
\(5\) −2.23607 −1.00000
\(6\) 0 0
\(7\) − 2.82843i − 1.06904i −0.845154 0.534522i \(-0.820491\pi\)
0.845154 0.534522i \(-0.179509\pi\)
\(8\) 0 0
\(9\) −2.23607 −0.745356
\(10\) 0 0
\(11\) −5.23607 −1.57873 −0.789367 0.613922i \(-0.789591\pi\)
−0.789367 + 0.613922i \(0.789591\pi\)
\(12\) 0 0
\(13\) 4.03631i 1.11947i 0.828671 + 0.559735i \(0.189097\pi\)
−0.828671 + 0.559735i \(0.810903\pi\)
\(14\) 0 0
\(15\) 5.11667i 1.32112i
\(16\) 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
\(21\) −6.47214 −1.41234
\(22\) 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
\(27\) − 1.74806i − 0.336415i
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 11.9814i 2.08570i
\(34\) 0 0
\(35\) 6.32456i 1.06904i
\(36\) 0 0
\(37\) − 6.86474i − 1.12856i −0.825585 0.564278i \(-0.809155\pi\)
0.825585 0.564278i \(-0.190845\pi\)
\(38\) 0 0
\(39\) 9.23607 1.47895
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.48528i 1.29399i 0.762493 + 0.646997i \(0.223975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 5.00000 0.745356
\(46\) 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
\(51\) 2.47214 0.346168
\(52\) 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
\(57\) − 2.28825i − 0.303086i
\(58\) 0 0
\(59\) 10.4721 1.36336 0.681678 0.731652i \(-0.261251\pi\)
0.681678 + 0.731652i \(0.261251\pi\)
\(60\) 0 0
\(61\) 1.70820 0.218713 0.109357 0.994003i \(-0.465121\pi\)
0.109357 + 0.994003i \(0.465121\pi\)
\(62\) 0 0
\(63\) 6.32456i 0.796819i
\(64\) 0 0
\(65\) − 9.02546i − 1.11947i
\(66\) 0 0
\(67\) − 1.62054i − 0.197981i −0.995088 0.0989905i \(-0.968439\pi\)
0.995088 0.0989905i \(-0.0315613\pi\)
\(68\) 0 0
\(69\) −16.9443 −2.03985
\(70\) 0 0
\(71\) −1.52786 −0.181324 −0.0906621 0.995882i \(-0.528898\pi\)
−0.0906621 + 0.995882i \(0.528898\pi\)
\(72\) 0 0
\(73\) − 13.7295i − 1.60691i −0.595363 0.803457i \(-0.702992\pi\)
0.595363 0.803457i \(-0.297008\pi\)
\(74\) 0 0
\(75\) − 11.4412i − 1.32112i
\(76\) 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
\(81\) −10.7082 −1.18980
\(82\) 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
\(87\) 10.2333i 1.09713i
\(88\) 0 0
\(89\) 14.9443 1.58409 0.792045 0.610463i \(-0.209017\pi\)
0.792045 + 0.610463i \(0.209017\pi\)
\(90\) 0 0
\(91\) 11.4164 1.19676
\(92\) 0 0
\(93\) 9.15298i 0.949120i
\(94\) 0 0
\(95\) −2.23607 −0.229416
\(96\) 0 0
\(97\) − 4.44897i − 0.451725i −0.974159 0.225862i \(-0.927480\pi\)
0.974159 0.225862i \(-0.0725199\pi\)
\(98\) 0 0
\(99\) 11.7082 1.17672
\(100\) 0 0
\(101\) 0.763932 0.0760141 0.0380070 0.999277i \(-0.487899\pi\)
0.0380070 + 0.999277i \(0.487899\pi\)
\(102\) 0 0
\(103\) − 15.3500i − 1.51248i −0.654293 0.756241i \(-0.727034\pi\)
0.654293 0.756241i \(-0.272966\pi\)
\(104\) 0 0
\(105\) 14.4721 1.41234
\(106\) 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
\(111\) −15.7082 −1.49096
\(112\) 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
\(117\) − 9.02546i − 0.834404i
\(118\) 0 0
\(119\) 3.05573 0.280118
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) 0 0
\(123\) 13.7295i 1.23794i
\(124\) 0 0
\(125\) −11.1803 −1.00000
\(126\) 0 0
\(127\) 10.1058i 0.896747i 0.893846 + 0.448374i \(0.147997\pi\)
−0.893846 + 0.448374i \(0.852003\pi\)
\(128\) 0 0
\(129\) 19.4164 1.70952
\(130\) 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
\(135\) 3.90879i 0.336415i
\(136\) 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
\(141\) −19.4164 −1.63516
\(142\) 0 0
\(143\) − 21.1344i − 1.76735i
\(144\) 0 0
\(145\) 10.0000 0.830455
\(146\) 0 0
\(147\) 2.28825i 0.188731i
\(148\) 0 0
\(149\) −18.6525 −1.52807 −0.764035 0.645175i \(-0.776785\pi\)
−0.764035 + 0.645175i \(0.776785\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) − 2.41577i − 0.195303i
\(154\) 0 0
\(155\) 8.94427 0.718421
\(156\) 0 0
\(157\) 19.3863i 1.54720i 0.633676 + 0.773599i \(0.281545\pi\)
−0.633676 + 0.773599i \(0.718455\pi\)
\(158\) 0 0
\(159\) −15.7082 −1.24574
\(160\) 0 0
\(161\) −20.9443 −1.65064
\(162\) 0 0
\(163\) − 8.48528i − 0.664619i −0.943170 0.332309i \(-0.892172\pi\)
0.943170 0.332309i \(-0.107828\pi\)
\(164\) 0 0
\(165\) − 26.7912i − 2.08570i
\(166\) 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
\(171\) −2.23607 −0.170996
\(172\) 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
\(177\) − 23.9628i − 1.80116i
\(178\) 0 0
\(179\) 13.5279 1.01112 0.505560 0.862791i \(-0.331286\pi\)
0.505560 + 0.862791i \(0.331286\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) − 3.90879i − 0.288946i
\(184\) 0 0
\(185\) 15.3500i 1.12856i
\(186\) 0 0
\(187\) − 5.65685i − 0.413670i
\(188\) 0 0
\(189\) −4.94427 −0.359643
\(190\) 0 0
\(191\) −20.9443 −1.51547 −0.757737 0.652560i \(-0.773695\pi\)
−0.757737 + 0.652560i \(0.773695\pi\)
\(192\) 0 0
\(193\) 4.44897i 0.320244i 0.987097 + 0.160122i \(0.0511888\pi\)
−0.987097 + 0.160122i \(0.948811\pi\)
\(194\) 0 0
\(195\) −20.6525 −1.47895
\(196\) 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
\(201\) −3.70820 −0.261557
\(202\) 0 0
\(203\) 12.6491i 0.887794i
\(204\) 0 0
\(205\) 13.4164 0.937043
\(206\) 0 0
\(207\) 16.5579i 1.15085i
\(208\) 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
\(213\) 3.49613i 0.239551i
\(214\) 0 0
\(215\) − 18.9737i − 1.29399i
\(216\) 0 0
\(217\) 11.3137i 0.768025i
\(218\) 0 0
\(219\) −31.4164 −2.12292
\(220\) 0 0
\(221\) −4.36068 −0.293331
\(222\) 0 0
\(223\) − 28.6668i − 1.91967i −0.280560 0.959836i \(-0.590520\pi\)
0.280560 0.959836i \(-0.409480\pi\)
\(224\) 0 0
\(225\) −11.1803 −0.745356
\(226\) 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
\(231\) 33.8885 2.22970
\(232\) 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
\(237\) − 26.1235i − 1.69691i
\(238\) 0 0
\(239\) −5.88854 −0.380898 −0.190449 0.981697i \(-0.560994\pi\)
−0.190449 + 0.981697i \(0.560994\pi\)
\(240\) 0 0
\(241\) 24.8328 1.59962 0.799811 0.600252i \(-0.204933\pi\)
0.799811 + 0.600252i \(0.204933\pi\)
\(242\) 0 0
\(243\) 19.2588i 1.23545i
\(244\) 0 0
\(245\) 2.23607 0.142857
\(246\) 0 0
\(247\) 4.03631i 0.256824i
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 29.8885 1.88655 0.943274 0.332015i \(-0.107728\pi\)
0.943274 + 0.332015i \(0.107728\pi\)
\(252\) 0 0
\(253\) 38.7727i 2.43762i
\(254\) 0 0
\(255\) −5.52786 −0.346168
\(256\) 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
\(261\) 10.0000 0.618984
\(262\) 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
\(267\) − 34.1962i − 2.09277i
\(268\) 0 0
\(269\) 2.94427 0.179515 0.0897577 0.995964i \(-0.471391\pi\)
0.0897577 + 0.995964i \(0.471391\pi\)
\(270\) 0 0
\(271\) −19.1246 −1.16174 −0.580869 0.813997i \(-0.697287\pi\)
−0.580869 + 0.813997i \(0.697287\pi\)
\(272\) 0 0
\(273\) − 26.1235i − 1.58107i
\(274\) 0 0
\(275\) −26.1803 −1.57873
\(276\) 0 0
\(277\) 3.24109i 0.194738i 0.995248 + 0.0973691i \(0.0310427\pi\)
−0.995248 + 0.0973691i \(0.968957\pi\)
\(278\) 0 0
\(279\) 8.94427 0.535480
\(280\) 0 0
\(281\) 1.41641 0.0844958 0.0422479 0.999107i \(-0.486548\pi\)
0.0422479 + 0.999107i \(0.486548\pi\)
\(282\) 0 0
\(283\) 16.5579i 0.984265i 0.870520 + 0.492133i \(0.163782\pi\)
−0.870520 + 0.492133i \(0.836218\pi\)
\(284\) 0 0
\(285\) 5.11667i 0.303086i
\(286\) 0 0
\(287\) 16.9706i 1.00174i
\(288\) 0 0
\(289\) 15.8328 0.931342
\(290\) 0 0
\(291\) −10.1803 −0.596782
\(292\) 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
\(297\) 9.15298i 0.531110i
\(298\) 0 0
\(299\) 29.8885 1.72850
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 0 0
\(303\) − 1.74806i − 0.100424i
\(304\) 0 0
\(305\) −3.81966 −0.218713
\(306\) 0 0
\(307\) − 4.03631i − 0.230364i −0.993344 0.115182i \(-0.963255\pi\)
0.993344 0.115182i \(-0.0367452\pi\)
\(308\) 0 0
\(309\) −35.1246 −1.99817
\(310\) 0 0
\(311\) 3.70820 0.210273 0.105136 0.994458i \(-0.466472\pi\)
0.105136 + 0.994458i \(0.466472\pi\)
\(312\) 0 0
\(313\) 27.4589i 1.55207i 0.630689 + 0.776036i \(0.282772\pi\)
−0.630689 + 0.776036i \(0.717228\pi\)
\(314\) 0 0
\(315\) − 14.1421i − 0.796819i
\(316\) 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
\(321\) 37.5967 2.09845
\(322\) 0 0
\(323\) 1.08036i 0.0601130i
\(324\) 0 0
\(325\) 20.1815i 1.11947i
\(326\) 0 0
\(327\) 4.57649i 0.253081i
\(328\) 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
\(333\) 15.3500i 0.841176i
\(334\) 0 0
\(335\) 3.62365i 0.197981i
\(336\) 0 0
\(337\) − 7.27740i − 0.396425i −0.980159 0.198213i \(-0.936486\pi\)
0.980159 0.198213i \(-0.0635136\pi\)
\(338\) 0 0
\(339\) 27.7082 1.50490
\(340\) 0 0
\(341\) 20.9443 1.13420
\(342\) 0 0
\(343\) − 16.9706i − 0.916324i
\(344\) 0 0
\(345\) 37.8885 2.03985
\(346\) 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
\(351\) 7.05573 0.376607
\(352\) 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
\(357\) − 6.99226i − 0.370069i
\(358\) 0 0
\(359\) 11.1246 0.587135 0.293567 0.955938i \(-0.405158\pi\)
0.293567 + 0.955938i \(0.405158\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) − 37.5648i − 1.97164i
\(364\) 0 0
\(365\) 30.7000i 1.60691i
\(366\) 0 0
\(367\) − 8.48528i − 0.442928i −0.975169 0.221464i \(-0.928916\pi\)
0.975169 0.221464i \(-0.0710835\pi\)
\(368\) 0 0
\(369\) 13.4164 0.698430
\(370\) 0 0
\(371\) −19.4164 −1.00805
\(372\) 0 0
\(373\) − 34.3237i − 1.77721i −0.458670 0.888606i \(-0.651674\pi\)
0.458670 0.888606i \(-0.348326\pi\)
\(374\) 0 0
\(375\) 25.5834i 1.32112i
\(376\) 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
\(381\) 23.1246 1.18471
\(382\) 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
\(387\) − 18.9737i − 0.964486i
\(388\) 0 0
\(389\) −2.94427 −0.149281 −0.0746403 0.997211i \(-0.523781\pi\)
−0.0746403 + 0.997211i \(0.523781\pi\)
\(390\) 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 \(-0.720059\pi\)
0.637566 0.770395i \(-0.279941\pi\)
\(398\) 0 0
\(399\) −6.47214 −0.324012
\(400\) 0 0
\(401\) 4.47214 0.223328 0.111664 0.993746i \(-0.464382\pi\)
0.111664 + 0.993746i \(0.464382\pi\)
\(402\) 0 0
\(403\) − 16.1452i − 0.804252i
\(404\) 0 0
\(405\) 23.9443 1.18980
\(406\) 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
\(411\) 2.47214 0.121941
\(412\) 0 0
\(413\) − 29.6197i − 1.45749i
\(414\) 0 0
\(415\) 11.7264i 0.575625i
\(416\) 0 0
\(417\) 35.9442i 1.76020i
\(418\) 0 0
\(419\) 3.05573 0.149282 0.0746410 0.997210i \(-0.476219\pi\)
0.0746410 + 0.997210i \(0.476219\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 0 0
\(423\) 18.9737i 0.922531i
\(424\) 0 0
\(425\) 5.40182i 0.262027i
\(426\) 0 0
\(427\) − 4.83153i − 0.233814i
\(428\) 0 0
\(429\) −48.3607 −2.33488
\(430\) 0 0
\(431\) 10.4721 0.504425 0.252213 0.967672i \(-0.418842\pi\)
0.252213 + 0.967672i \(0.418842\pi\)
\(432\) 0 0
\(433\) − 37.9774i − 1.82508i −0.408989 0.912540i \(-0.634118\pi\)
0.408989 0.912540i \(-0.365882\pi\)
\(434\) 0 0
\(435\) − 22.8825i − 1.09713i
\(436\) 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
\(441\) 2.23607 0.106479
\(442\) 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
\(447\) 42.6814i 2.01876i
\(448\) 0 0
\(449\) 7.52786 0.355262 0.177631 0.984097i \(-0.443157\pi\)
0.177631 + 0.984097i \(0.443157\pi\)
\(450\) 0 0
\(451\) 31.4164 1.47934
\(452\) 0 0
\(453\) 18.3060i 0.860089i
\(454\) 0 0
\(455\) −25.5279 −1.19676
\(456\) 0 0
\(457\) 37.9473i 1.77510i 0.460710 + 0.887551i \(0.347595\pi\)
−0.460710 + 0.887551i \(0.652405\pi\)
\(458\) 0 0
\(459\) 1.88854 0.0881497
\(460\) 0 0
\(461\) −11.8885 −0.553705 −0.276852 0.960912i \(-0.589291\pi\)
−0.276852 + 0.960912i \(0.589291\pi\)
\(462\) 0 0
\(463\) 33.5285i 1.55820i 0.626900 + 0.779100i \(0.284324\pi\)
−0.626900 + 0.779100i \(0.715676\pi\)
\(464\) 0 0
\(465\) − 20.4667i − 0.949120i
\(466\) 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
\(471\) 44.3607 2.04403
\(472\) 0 0
\(473\) − 44.4295i − 2.04287i
\(474\) 0 0
\(475\) 5.00000 0.229416
\(476\) 0 0
\(477\) 15.3500i 0.702829i
\(478\) 0 0
\(479\) −6.76393 −0.309052 −0.154526 0.987989i \(-0.549385\pi\)
−0.154526 + 0.987989i \(0.549385\pi\)
\(480\) 0 0
\(481\) 27.7082 1.26339
\(482\) 0 0
\(483\) 47.9256i 2.18069i
\(484\) 0 0
\(485\) 9.94820i 0.451725i
\(486\) 0 0
\(487\) − 17.7658i − 0.805044i −0.915410 0.402522i \(-0.868134\pi\)
0.915410 0.402522i \(-0.131866\pi\)
\(488\) 0 0
\(489\) −19.4164 −0.878040
\(490\) 0 0
\(491\) −29.8885 −1.34885 −0.674426 0.738343i \(-0.735609\pi\)
−0.674426 + 0.738343i \(0.735609\pi\)
\(492\) 0 0
\(493\) − 4.83153i − 0.217601i
\(494\) 0 0
\(495\) −26.1803 −1.17672
\(496\) 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
\(501\) 10.7639 0.480897
\(502\) 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
\(507\) 7.53244i 0.334527i
\(508\) 0 0
\(509\) 37.4164 1.65845 0.829227 0.558913i \(-0.188781\pi\)
0.829227 + 0.558913i \(0.188781\pi\)
\(510\) 0 0
\(511\) −38.8328 −1.71786
\(512\) 0 0
\(513\) − 1.74806i − 0.0771789i
\(514\) 0 0
\(515\) 34.3237i 1.51248i
\(516\) 0 0
\(517\) 44.4295i 1.95401i
\(518\) 0 0
\(519\) −30.1803 −1.32477
\(520\) 0 0
\(521\) 35.8885 1.57231 0.786153 0.618032i \(-0.212070\pi\)
0.786153 + 0.618032i \(0.212070\pi\)
\(522\) 0 0
\(523\) 3.62365i 0.158451i 0.996857 + 0.0792255i \(0.0252447\pi\)
−0.996857 + 0.0792255i \(0.974755\pi\)
\(524\) 0 0
\(525\) −32.3607 −1.41234
\(526\) 0 0
\(527\) − 4.32145i − 0.188245i
\(528\) 0 0
\(529\) −31.8328 −1.38404
\(530\) 0 0
\(531\) −23.4164 −1.01619
\(532\) 0 0
\(533\) − 24.2179i − 1.04899i
\(534\) 0 0
\(535\) − 36.7394i − 1.58838i
\(536\) 0 0
\(537\) − 30.9551i − 1.33581i
\(538\) 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
\(543\) − 13.7295i − 0.589188i
\(544\) 0 0
\(545\) 4.47214 0.191565
\(546\) 0 0
\(547\) − 1.20788i − 0.0516453i −0.999667 0.0258227i \(-0.991779\pi\)
0.999667 0.0258227i \(-0.00822052\pi\)
\(548\) 0 0
\(549\) −3.81966 −0.163019
\(550\) 0 0
\(551\) −4.47214 −0.190519
\(552\) 0 0
\(553\) − 32.2905i − 1.37313i
\(554\) 0 0
\(555\) 35.1246 1.49096
\(556\) 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
\(561\) −12.9443 −0.546508
\(562\) 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
\(567\) 30.2874i 1.27195i
\(568\) 0 0
\(569\) 21.0557 0.882702 0.441351 0.897335i \(-0.354499\pi\)
0.441351 + 0.897335i \(0.354499\pi\)
\(570\) 0 0
\(571\) −15.1246 −0.632945 −0.316473 0.948602i \(-0.602499\pi\)
−0.316473 + 0.948602i \(0.602499\pi\)
\(572\) 0 0
\(573\) 47.9256i 2.00212i
\(574\) 0 0
\(575\) − 37.0246i − 1.54403i
\(576\) 0 0
\(577\) 5.65685i 0.235498i 0.993043 + 0.117749i \(0.0375678\pi\)
−0.993043 + 0.117749i \(0.962432\pi\)
\(578\) 0 0
\(579\) 10.1803 0.423080
\(580\) 0 0
\(581\) −14.8328 −0.615369
\(582\) 0 0
\(583\) 35.9442i 1.48866i
\(584\) 0 0
\(585\) 20.1815i 0.834404i
\(586\) 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
\(591\) 28.9443 1.19061
\(592\) 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
\(597\) − 36.6119i − 1.49843i
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −36.8328 −1.50244 −0.751221 0.660051i \(-0.770535\pi\)
−0.751221 + 0.660051i \(0.770535\pi\)
\(602\) 0 0
\(603\) 3.62365i 0.147566i
\(604\) 0 0
\(605\) −36.7082 −1.49240
\(606\) 0 0
\(607\) − 12.9343i − 0.524985i −0.964934 0.262493i \(-0.915455\pi\)
0.964934 0.262493i \(-0.0845445\pi\)
\(608\) 0 0
\(609\) 28.9443 1.17288
\(610\) 0 0
\(611\) 34.2492 1.38558
\(612\) 0 0
\(613\) − 20.2117i − 0.816341i −0.912906 0.408170i \(-0.866167\pi\)
0.912906 0.408170i \(-0.133833\pi\)
\(614\) 0 0
\(615\) − 30.7000i − 1.23794i
\(616\) 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
\(621\) −12.9443 −0.519436
\(622\) 0 0
\(623\) − 42.2688i − 1.69346i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 11.9814i 0.478491i
\(628\) 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
\(633\) 26.1235i 1.03832i
\(634\) 0 0
\(635\) − 22.5973i − 0.896747i
\(636\) 0 0
\(637\) − 4.03631i − 0.159924i
\(638\) 0 0
\(639\) 3.41641 0.135151
\(640\) 0 0
\(641\) −47.8885 −1.89148 −0.945742 0.324919i \(-0.894663\pi\)
−0.945742 + 0.324919i \(0.894663\pi\)
\(642\) 0 0
\(643\) − 11.7264i − 0.462443i −0.972901 0.231221i \(-0.925728\pi\)
0.972901 0.231221i \(-0.0742722\pi\)
\(644\) 0 0
\(645\) −43.4164 −1.70952
\(646\) 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
\(651\) 25.8885 1.01465
\(652\) 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
\(657\) 30.7000i 1.19772i
\(658\) 0 0
\(659\) 27.0557 1.05394 0.526971 0.849883i \(-0.323328\pi\)
0.526971 + 0.849883i \(0.323328\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 0 0
\(663\) 9.97831i 0.387525i
\(664\) 0 0
\(665\) 6.32456i 0.245256i
\(666\) 0 0
\(667\) 33.1158i 1.28225i
\(668\) 0 0
\(669\) −65.5967 −2.53612
\(670\) 0 0
\(671\) −8.94427 −0.345290
\(672\) 0 0
\(673\) 42.8090i 1.65016i 0.565013 + 0.825082i \(0.308871\pi\)
−0.565013 + 0.825082i \(0.691129\pi\)
\(674\) 0 0
\(675\) − 8.74032i − 0.336415i
\(676\) 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
\(681\) −35.1246 −1.34598
\(682\) 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
\(687\) − 13.0618i − 0.498338i
\(688\) 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
\(693\) − 33.1158i − 1.25797i
\(694\) 0 0
\(695\) 35.1246 1.33235
\(696\) 0 0
\(697\) − 6.48218i − 0.245530i
\(698\) 0 0
\(699\) 28.9443 1.09477
\(700\) 0 0
\(701\) 41.1246 1.55326 0.776628 0.629960i \(-0.216929\pi\)
0.776628 + 0.629960i \(0.216929\pi\)
\(702\) 0 0
\(703\) − 6.86474i − 0.258908i
\(704\) 0 0
\(705\) 43.4164 1.63516
\(706\) 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
\(711\) −25.5279 −0.957370
\(712\) 0 0
\(713\) 29.6197i 1.10927i
\(714\) 0 0
\(715\) 47.2579i 1.76735i
\(716\) 0 0
\(717\) 13.4744i 0.503212i
\(718\) 0 0
\(719\) 21.5967 0.805423 0.402711 0.915327i \(-0.368068\pi\)
0.402711 + 0.915327i \(0.368068\pi\)
\(720\) 0 0
\(721\) −43.4164 −1.61691
\(722\) 0 0
\(723\) − 56.8236i − 2.11329i
\(724\) 0 0
\(725\) −22.3607 −0.830455
\(726\) 0 0
\(727\) − 8.48528i − 0.314702i −0.987543 0.157351i \(-0.949705\pi\)
0.987543 0.157351i \(-0.0502953\pi\)
\(728\) 0 0
\(729\) 11.9443 0.442380
\(730\) 0 0
\(731\) −9.16718 −0.339061
\(732\) 0 0
\(733\) − 33.9411i − 1.25364i −0.779162 0.626822i \(-0.784355\pi\)
0.779162 0.626822i \(-0.215645\pi\)
\(734\) 0 0
\(735\) − 5.11667i − 0.188731i
\(736\) 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
\(741\) 9.23607 0.339295
\(742\) 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
\(747\) 11.7264i 0.429045i
\(748\) 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
\(753\) − 68.3923i − 2.49236i
\(754\) 0 0
\(755\) 17.8885 0.651031
\(756\) 0 0
\(757\) 0.825324i 0.0299969i 0.999888 + 0.0149985i \(0.00477434\pi\)
−0.999888 + 0.0149985i \(0.995226\pi\)
\(758\) 0 0
\(759\) 88.7214 3.22038
\(760\) 0 0
\(761\) 24.5410 0.889611 0.444806 0.895627i \(-0.353273\pi\)
0.444806 + 0.895627i \(0.353273\pi\)
\(762\) 0 0
\(763\) 5.65685i 0.204792i
\(764\) 0 0
\(765\) 5.40182i 0.195303i
\(766\) 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
\(771\) −10.7639 −0.387654
\(772\) 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
\(777\) 44.4295i 1.59390i
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) 7.81758i 0.279378i
\(784\) 0 0
\(785\) − 43.3491i − 1.54720i
\(786\) 0 0
\(787\) 3.62365i 0.129169i 0.997912 + 0.0645845i \(0.0205722\pi\)
−0.997912 + 0.0645845i \(0.979428\pi\)
\(788\) 0 0
\(789\) 21.8885 0.779253
\(790\) 0 0
\(791\) 34.2492 1.21776
\(792\) 0 0
\(793\) 6.89484i 0.244843i
\(794\) 0 0
\(795\) 35.1246 1.24574
\(796\) 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
\(801\) −33.4164 −1.18071
\(802\) 0 0
\(803\) 71.8885i 2.53689i
\(804\) 0 0
\(805\) 46.8328 1.65064
\(806\) 0 0
\(807\) − 6.73722i − 0.237161i
\(808\) 0 0
\(809\) 16.4721 0.579129 0.289565 0.957158i \(-0.406489\pi\)
0.289565 + 0.957158i \(0.406489\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 0 0
\(813\) 43.7618i 1.53479i
\(814\) 0 0
\(815\) 18.9737i 0.664619i
\(816\) 0 0
\(817\) 8.48528i 0.296862i
\(818\) 0 0
\(819\) −25.5279 −0.892016
\(820\) 0 0
\(821\) 21.0557 0.734850 0.367425 0.930053i \(-0.380239\pi\)
0.367425 + 0.930053i \(0.380239\pi\)
\(822\) 0 0
\(823\) 2.00310i 0.0698238i 0.999390 + 0.0349119i \(0.0111151\pi\)
−0.999390 + 0.0349119i \(0.988885\pi\)
\(824\) 0 0
\(825\) 59.9070i 2.08570i
\(826\) 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
\(831\) 7.41641 0.257272
\(832\) 0 0
\(833\) − 1.08036i − 0.0374324i
\(834\) 0 0
\(835\) − 10.5185i − 0.364007i
\(836\) 0 0
\(837\) 6.99226i 0.241688i
\(838\) 0 0
\(839\) 13.3050 0.459338 0.229669 0.973269i \(-0.426236\pi\)
0.229669 + 0.973269i \(0.426236\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) − 3.24109i − 0.111629i
\(844\) 0 0
\(845\) 7.36068 0.253215
\(846\) 0 0
\(847\) − 46.4326i − 1.59544i
\(848\) 0 0
\(849\) 37.8885 1.30033
\(850\) 0 0
\(851\) −50.8328 −1.74253
\(852\) 0 0
\(853\) 10.4884i 0.359115i 0.983747 + 0.179558i \(0.0574667\pi\)
−0.983747 + 0.179558i \(0.942533\pi\)
\(854\) 0 0
\(855\) 5.00000 0.170996
\(856\) 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
\(861\) 38.8328 1.32342
\(862\) 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
\(867\) − 36.2294i − 1.23041i
\(868\) 0 0
\(869\) −59.7771 −2.02780
\(870\) 0 0
\(871\) 6.54102 0.221634
\(872\) 0 0
\(873\) 9.94820i 0.336696i
\(874\) 0 0
\(875\) 31.6228i 1.06904i
\(876\) 0 0
\(877\) − 4.44897i − 0.150231i −0.997175 0.0751155i \(-0.976067\pi\)
0.997175 0.0751155i \(-0.0239326\pi\)
\(878\) 0 0
\(879\) −15.7082 −0.529825
\(880\) 0 0
\(881\) −30.6525 −1.03271 −0.516354 0.856375i \(-0.672711\pi\)
−0.516354 + 0.856375i \(0.672711\pi\)
\(882\) 0 0
\(883\) − 14.9675i − 0.503695i −0.967767 0.251848i \(-0.918962\pi\)
0.967767 0.251848i \(-0.0810382\pi\)
\(884\) 0 0
\(885\) 53.5825i 1.80116i
\(886\) 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
\(891\) 56.0689 1.87838
\(892\) 0 0
\(893\) − 8.48528i − 0.283949i
\(894\) 0 0
\(895\) −30.2492 −1.01112
\(896\) 0 0
\(897\) − 68.3923i − 2.28355i
\(898\) 0 0
\(899\) 17.8885 0.596616
\(900\) 0 0
\(901\) 7.41641 0.247076
\(902\) 0 0
\(903\) − 54.9179i − 1.82755i
\(904\) 0 0
\(905\) −13.4164 −0.445976
\(906\) 0 0
\(907\) − 8.86784i − 0.294452i −0.989103 0.147226i \(-0.952966\pi\)
0.989103 0.147226i \(-0.0470344\pi\)
\(908\) 0 0
\(909\) −1.70820 −0.0566575
\(910\) 0 0
\(911\) −37.5279 −1.24335 −0.621677 0.783274i \(-0.713548\pi\)
−0.621677 + 0.783274i \(0.713548\pi\)
\(912\) 0 0
\(913\) 27.4589i 0.908759i
\(914\) 0 0
\(915\) 8.74032i 0.288946i
\(916\) 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
\(921\) −9.23607 −0.304339
\(922\) 0 0
\(923\) − 6.16693i − 0.202987i
\(924\) 0 0
\(925\) − 34.3237i − 1.12856i
\(926\) 0 0
\(927\) 34.3237i 1.12734i
\(928\) 0 0
\(929\) −16.4721 −0.540433 −0.270217 0.962800i \(-0.587095\pi\)
−0.270217 + 0.962800i \(0.587095\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) − 8.48528i − 0.277796i
\(934\) 0 0
\(935\) 12.6491i 0.413670i
\(936\) 0 0
\(937\) − 51.6768i − 1.68821i −0.536180 0.844104i \(-0.680133\pi\)
0.536180 0.844104i \(-0.319867\pi\)
\(938\) 0 0
\(939\) 62.8328 2.05047
\(940\) 0 0
\(941\) −21.0557 −0.686397 −0.343199 0.939263i \(-0.611510\pi\)
−0.343199 + 0.939263i \(0.611510\pi\)
\(942\) 0 0
\(943\) 44.4295i 1.44682i
\(944\) 0 0
\(945\) 11.0557 0.359643
\(946\) 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
\(951\) −30.1803 −0.978665
\(952\) 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
\(957\) − 53.5825i − 1.73208i
\(958\) 0 0
\(959\) 3.05573 0.0986746
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) − 36.7394i − 1.18391i
\(964\) 0 0
\(965\) − 9.94820i − 0.320244i
\(966\) 0 0
\(967\) − 16.5579i − 0.532466i −0.963909 0.266233i \(-0.914221\pi\)
0.963909 0.266233i \(-0.0857791\pi\)
\(968\) 0 0
\(969\) 2.47214 0.0794164
\(970\) 0 0
\(971\) 4.36068 0.139941 0.0699704 0.997549i \(-0.477710\pi\)
0.0699704 + 0.997549i \(0.477710\pi\)
\(972\) 0 0
\(973\) 44.4295i 1.42434i
\(974\) 0 0
\(975\) 46.1803 1.47895
\(976\) 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
\(981\) 4.47214 0.142784
\(982\) 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
\(987\) 54.9179i 1.74806i
\(988\) 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
\(993\) 44.4295i 1.40993i
\(994\) 0 0
\(995\) −35.7771 −1.13421
\(996\) 0 0
\(997\) 30.7000i 0.972280i 0.873881 + 0.486140i \(0.161595\pi\)
−0.873881 + 0.486140i \(0.838405\pi\)
\(998\) 0 0
\(999\) −12.0000 −0.379663
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 380.2.c.a.229.1 4
3.2 odd 2 3420.2.f.a.1369.3 4
4.3 odd 2 1520.2.d.f.609.4 4
5.2 odd 4 1900.2.a.j.1.1 4
5.3 odd 4 1900.2.a.j.1.4 4
5.4 even 2 inner 380.2.c.a.229.4 yes 4
15.14 odd 2 3420.2.f.a.1369.4 4
20.3 even 4 7600.2.a.ce.1.1 4
20.7 even 4 7600.2.a.ce.1.4 4
20.19 odd 2 1520.2.d.f.609.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.c.a.229.1 4 1.1 even 1 trivial
380.2.c.a.229.4 yes 4 5.4 even 2 inner
1520.2.d.f.609.1 4 20.19 odd 2
1520.2.d.f.609.4 4 4.3 odd 2
1900.2.a.j.1.1 4 5.2 odd 4
1900.2.a.j.1.4 4 5.3 odd 4
3420.2.f.a.1369.3 4 3.2 odd 2
3420.2.f.a.1369.4 4 15.14 odd 2
7600.2.a.ce.1.1 4 20.3 even 4
7600.2.a.ce.1.4 4 20.7 even 4