Properties

Label 384.4.d.d.193.1
Level $384$
Weight $4$
Character 384.193
Analytic conductor $22.657$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.4.d.d.193.4

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} -2.82843i q^{5} -14.1421 q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} -2.82843i q^{5} -14.1421 q^{7} -9.00000 q^{9} -20.0000i q^{11} +39.5980i q^{13} -8.48528 q^{15} -34.0000 q^{17} +52.0000i q^{19} +42.4264i q^{21} -62.2254 q^{23} +117.000 q^{25} +27.0000i q^{27} +200.818i q^{29} -110.309 q^{31} -60.0000 q^{33} +40.0000i q^{35} +271.529i q^{37} +118.794 q^{39} +26.0000 q^{41} -252.000i q^{43} +25.4558i q^{45} +345.068 q^{47} -143.000 q^{49} +102.000i q^{51} +681.651i q^{53} -56.5685 q^{55} +156.000 q^{57} -364.000i q^{59} +735.391i q^{61} +127.279 q^{63} +112.000 q^{65} +628.000i q^{67} +186.676i q^{69} +333.754 q^{71} -338.000 q^{73} -351.000i q^{75} +282.843i q^{77} -789.131 q^{79} +81.0000 q^{81} +1036.00i q^{83} +96.1665i q^{85} +602.455 q^{87} -234.000 q^{89} -560.000i q^{91} +330.926i q^{93} +147.078 q^{95} -178.000 q^{97} +180.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{9} - 136 q^{17} + 468 q^{25} - 240 q^{33} + 104 q^{41} - 572 q^{49} + 624 q^{57} + 448 q^{65} - 1352 q^{73} + 324 q^{81} - 936 q^{89} - 712 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 3.00000i − 0.577350i
\(4\) 0 0
\(5\) − 2.82843i − 0.252982i −0.991968 0.126491i \(-0.959628\pi\)
0.991968 0.126491i \(-0.0403715\pi\)
\(6\) 0 0
\(7\) −14.1421 −0.763604 −0.381802 0.924244i \(-0.624696\pi\)
−0.381802 + 0.924244i \(0.624696\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) − 20.0000i − 0.548202i −0.961701 0.274101i \(-0.911620\pi\)
0.961701 0.274101i \(-0.0883803\pi\)
\(12\) 0 0
\(13\) 39.5980i 0.844808i 0.906408 + 0.422404i \(0.138814\pi\)
−0.906408 + 0.422404i \(0.861186\pi\)
\(14\) 0 0
\(15\) −8.48528 −0.146059
\(16\) 0 0
\(17\) −34.0000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 52.0000i 0.627875i 0.949444 + 0.313937i \(0.101648\pi\)
−0.949444 + 0.313937i \(0.898352\pi\)
\(20\) 0 0
\(21\) 42.4264i 0.440867i
\(22\) 0 0
\(23\) −62.2254 −0.564126 −0.282063 0.959396i \(-0.591019\pi\)
−0.282063 + 0.959396i \(0.591019\pi\)
\(24\) 0 0
\(25\) 117.000 0.936000
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 200.818i 1.28590i 0.765909 + 0.642949i \(0.222289\pi\)
−0.765909 + 0.642949i \(0.777711\pi\)
\(30\) 0 0
\(31\) −110.309 −0.639097 −0.319549 0.947570i \(-0.603531\pi\)
−0.319549 + 0.947570i \(0.603531\pi\)
\(32\) 0 0
\(33\) −60.0000 −0.316505
\(34\) 0 0
\(35\) 40.0000i 0.193178i
\(36\) 0 0
\(37\) 271.529i 1.20646i 0.797567 + 0.603231i \(0.206120\pi\)
−0.797567 + 0.603231i \(0.793880\pi\)
\(38\) 0 0
\(39\) 118.794 0.487750
\(40\) 0 0
\(41\) 26.0000 0.0990370 0.0495185 0.998773i \(-0.484231\pi\)
0.0495185 + 0.998773i \(0.484231\pi\)
\(42\) 0 0
\(43\) − 252.000i − 0.893713i −0.894606 0.446856i \(-0.852544\pi\)
0.894606 0.446856i \(-0.147456\pi\)
\(44\) 0 0
\(45\) 25.4558i 0.0843274i
\(46\) 0 0
\(47\) 345.068 1.07092 0.535461 0.844560i \(-0.320138\pi\)
0.535461 + 0.844560i \(0.320138\pi\)
\(48\) 0 0
\(49\) −143.000 −0.416910
\(50\) 0 0
\(51\) 102.000i 0.280056i
\(52\) 0 0
\(53\) 681.651i 1.76664i 0.468770 + 0.883320i \(0.344697\pi\)
−0.468770 + 0.883320i \(0.655303\pi\)
\(54\) 0 0
\(55\) −56.5685 −0.138685
\(56\) 0 0
\(57\) 156.000 0.362504
\(58\) 0 0
\(59\) − 364.000i − 0.803199i −0.915815 0.401600i \(-0.868454\pi\)
0.915815 0.401600i \(-0.131546\pi\)
\(60\) 0 0
\(61\) 735.391i 1.54356i 0.635889 + 0.771780i \(0.280633\pi\)
−0.635889 + 0.771780i \(0.719367\pi\)
\(62\) 0 0
\(63\) 127.279 0.254535
\(64\) 0 0
\(65\) 112.000 0.213721
\(66\) 0 0
\(67\) 628.000i 1.14511i 0.819866 + 0.572555i \(0.194048\pi\)
−0.819866 + 0.572555i \(0.805952\pi\)
\(68\) 0 0
\(69\) 186.676i 0.325698i
\(70\) 0 0
\(71\) 333.754 0.557878 0.278939 0.960309i \(-0.410017\pi\)
0.278939 + 0.960309i \(0.410017\pi\)
\(72\) 0 0
\(73\) −338.000 −0.541917 −0.270958 0.962591i \(-0.587341\pi\)
−0.270958 + 0.962591i \(0.587341\pi\)
\(74\) 0 0
\(75\) − 351.000i − 0.540400i
\(76\) 0 0
\(77\) 282.843i 0.418609i
\(78\) 0 0
\(79\) −789.131 −1.12385 −0.561925 0.827188i \(-0.689939\pi\)
−0.561925 + 0.827188i \(0.689939\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1036.00i 1.37007i 0.728510 + 0.685035i \(0.240213\pi\)
−0.728510 + 0.685035i \(0.759787\pi\)
\(84\) 0 0
\(85\) 96.1665i 0.122714i
\(86\) 0 0
\(87\) 602.455 0.742413
\(88\) 0 0
\(89\) −234.000 −0.278696 −0.139348 0.990243i \(-0.544501\pi\)
−0.139348 + 0.990243i \(0.544501\pi\)
\(90\) 0 0
\(91\) − 560.000i − 0.645098i
\(92\) 0 0
\(93\) 330.926i 0.368983i
\(94\) 0 0
\(95\) 147.078 0.158841
\(96\) 0 0
\(97\) −178.000 −0.186321 −0.0931606 0.995651i \(-0.529697\pi\)
−0.0931606 + 0.995651i \(0.529697\pi\)
\(98\) 0 0
\(99\) 180.000i 0.182734i
\(100\) 0 0
\(101\) − 257.387i − 0.253574i −0.991930 0.126787i \(-0.959534\pi\)
0.991930 0.126787i \(-0.0404664\pi\)
\(102\) 0 0
\(103\) −1886.56 −1.80474 −0.902371 0.430961i \(-0.858175\pi\)
−0.902371 + 0.430961i \(0.858175\pi\)
\(104\) 0 0
\(105\) 120.000 0.111531
\(106\) 0 0
\(107\) − 1404.00i − 1.26850i −0.773127 0.634251i \(-0.781308\pi\)
0.773127 0.634251i \(-0.218692\pi\)
\(108\) 0 0
\(109\) − 39.5980i − 0.0347963i −0.999849 0.0173982i \(-0.994462\pi\)
0.999849 0.0173982i \(-0.00553829\pi\)
\(110\) 0 0
\(111\) 814.587 0.696551
\(112\) 0 0
\(113\) 1378.00 1.14718 0.573590 0.819143i \(-0.305550\pi\)
0.573590 + 0.819143i \(0.305550\pi\)
\(114\) 0 0
\(115\) 176.000i 0.142714i
\(116\) 0 0
\(117\) − 356.382i − 0.281603i
\(118\) 0 0
\(119\) 480.833 0.370402
\(120\) 0 0
\(121\) 931.000 0.699474
\(122\) 0 0
\(123\) − 78.0000i − 0.0571791i
\(124\) 0 0
\(125\) − 684.479i − 0.489774i
\(126\) 0 0
\(127\) −1790.39 −1.25096 −0.625480 0.780241i \(-0.715097\pi\)
−0.625480 + 0.780241i \(0.715097\pi\)
\(128\) 0 0
\(129\) −756.000 −0.515985
\(130\) 0 0
\(131\) 1572.00i 1.04844i 0.851581 + 0.524222i \(0.175644\pi\)
−0.851581 + 0.524222i \(0.824356\pi\)
\(132\) 0 0
\(133\) − 735.391i − 0.479447i
\(134\) 0 0
\(135\) 76.3675 0.0486864
\(136\) 0 0
\(137\) −2854.00 −1.77981 −0.889904 0.456148i \(-0.849229\pi\)
−0.889904 + 0.456148i \(0.849229\pi\)
\(138\) 0 0
\(139\) − 1964.00i − 1.19845i −0.800581 0.599224i \(-0.795476\pi\)
0.800581 0.599224i \(-0.204524\pi\)
\(140\) 0 0
\(141\) − 1035.20i − 0.618297i
\(142\) 0 0
\(143\) 791.960 0.463126
\(144\) 0 0
\(145\) 568.000 0.325309
\(146\) 0 0
\(147\) 429.000i 0.240703i
\(148\) 0 0
\(149\) − 1507.55i − 0.828882i −0.910076 0.414441i \(-0.863977\pi\)
0.910076 0.414441i \(-0.136023\pi\)
\(150\) 0 0
\(151\) −2265.57 −1.22099 −0.610495 0.792020i \(-0.709029\pi\)
−0.610495 + 0.792020i \(0.709029\pi\)
\(152\) 0 0
\(153\) 306.000 0.161690
\(154\) 0 0
\(155\) 312.000i 0.161680i
\(156\) 0 0
\(157\) − 3529.88i − 1.79436i −0.441663 0.897181i \(-0.645611\pi\)
0.441663 0.897181i \(-0.354389\pi\)
\(158\) 0 0
\(159\) 2044.95 1.01997
\(160\) 0 0
\(161\) 880.000 0.430768
\(162\) 0 0
\(163\) 2932.00i 1.40891i 0.709750 + 0.704454i \(0.248808\pi\)
−0.709750 + 0.704454i \(0.751192\pi\)
\(164\) 0 0
\(165\) 169.706i 0.0800701i
\(166\) 0 0
\(167\) 3676.96 1.70378 0.851890 0.523720i \(-0.175456\pi\)
0.851890 + 0.523720i \(0.175456\pi\)
\(168\) 0 0
\(169\) 629.000 0.286299
\(170\) 0 0
\(171\) − 468.000i − 0.209292i
\(172\) 0 0
\(173\) − 1445.33i − 0.635180i −0.948228 0.317590i \(-0.897126\pi\)
0.948228 0.317590i \(-0.102874\pi\)
\(174\) 0 0
\(175\) −1654.63 −0.714733
\(176\) 0 0
\(177\) −1092.00 −0.463727
\(178\) 0 0
\(179\) − 1308.00i − 0.546170i −0.961990 0.273085i \(-0.911956\pi\)
0.961990 0.273085i \(-0.0880441\pi\)
\(180\) 0 0
\(181\) − 1996.87i − 0.820034i −0.912078 0.410017i \(-0.865523\pi\)
0.912078 0.410017i \(-0.134477\pi\)
\(182\) 0 0
\(183\) 2206.17 0.891175
\(184\) 0 0
\(185\) 768.000 0.305213
\(186\) 0 0
\(187\) 680.000i 0.265917i
\(188\) 0 0
\(189\) − 381.838i − 0.146956i
\(190\) 0 0
\(191\) 939.038 0.355740 0.177870 0.984054i \(-0.443079\pi\)
0.177870 + 0.984054i \(0.443079\pi\)
\(192\) 0 0
\(193\) −2490.00 −0.928674 −0.464337 0.885659i \(-0.653707\pi\)
−0.464337 + 0.885659i \(0.653707\pi\)
\(194\) 0 0
\(195\) − 336.000i − 0.123392i
\(196\) 0 0
\(197\) − 2723.78i − 0.985081i −0.870290 0.492540i \(-0.836068\pi\)
0.870290 0.492540i \(-0.163932\pi\)
\(198\) 0 0
\(199\) −2158.09 −0.768758 −0.384379 0.923175i \(-0.625584\pi\)
−0.384379 + 0.923175i \(0.625584\pi\)
\(200\) 0 0
\(201\) 1884.00 0.661130
\(202\) 0 0
\(203\) − 2840.00i − 0.981916i
\(204\) 0 0
\(205\) − 73.5391i − 0.0250546i
\(206\) 0 0
\(207\) 560.029 0.188042
\(208\) 0 0
\(209\) 1040.00 0.344202
\(210\) 0 0
\(211\) − 924.000i − 0.301473i −0.988574 0.150736i \(-0.951836\pi\)
0.988574 0.150736i \(-0.0481645\pi\)
\(212\) 0 0
\(213\) − 1001.26i − 0.322091i
\(214\) 0 0
\(215\) −712.764 −0.226093
\(216\) 0 0
\(217\) 1560.00 0.488017
\(218\) 0 0
\(219\) 1014.00i 0.312876i
\(220\) 0 0
\(221\) − 1346.33i − 0.409792i
\(222\) 0 0
\(223\) −2276.88 −0.683728 −0.341864 0.939749i \(-0.611058\pi\)
−0.341864 + 0.939749i \(0.611058\pi\)
\(224\) 0 0
\(225\) −1053.00 −0.312000
\(226\) 0 0
\(227\) 156.000i 0.0456127i 0.999740 + 0.0228064i \(0.00726012\pi\)
−0.999740 + 0.0228064i \(0.992740\pi\)
\(228\) 0 0
\(229\) 639.225i 0.184459i 0.995738 + 0.0922296i \(0.0293994\pi\)
−0.995738 + 0.0922296i \(0.970601\pi\)
\(230\) 0 0
\(231\) 848.528 0.241684
\(232\) 0 0
\(233\) −2826.00 −0.794581 −0.397291 0.917693i \(-0.630049\pi\)
−0.397291 + 0.917693i \(0.630049\pi\)
\(234\) 0 0
\(235\) − 976.000i − 0.270924i
\(236\) 0 0
\(237\) 2367.39i 0.648855i
\(238\) 0 0
\(239\) 2466.39 0.667521 0.333760 0.942658i \(-0.391682\pi\)
0.333760 + 0.942658i \(0.391682\pi\)
\(240\) 0 0
\(241\) −3354.00 −0.896474 −0.448237 0.893915i \(-0.647948\pi\)
−0.448237 + 0.893915i \(0.647948\pi\)
\(242\) 0 0
\(243\) − 243.000i − 0.0641500i
\(244\) 0 0
\(245\) 404.465i 0.105471i
\(246\) 0 0
\(247\) −2059.09 −0.530433
\(248\) 0 0
\(249\) 3108.00 0.791010
\(250\) 0 0
\(251\) 6396.00i 1.60841i 0.594349 + 0.804207i \(0.297410\pi\)
−0.594349 + 0.804207i \(0.702590\pi\)
\(252\) 0 0
\(253\) 1244.51i 0.309255i
\(254\) 0 0
\(255\) 288.500 0.0708492
\(256\) 0 0
\(257\) 6882.00 1.67038 0.835189 0.549962i \(-0.185358\pi\)
0.835189 + 0.549962i \(0.185358\pi\)
\(258\) 0 0
\(259\) − 3840.00i − 0.921259i
\(260\) 0 0
\(261\) − 1807.36i − 0.428632i
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 1928.00 0.446929
\(266\) 0 0
\(267\) 702.000i 0.160905i
\(268\) 0 0
\(269\) 1434.01i 0.325031i 0.986706 + 0.162515i \(0.0519607\pi\)
−0.986706 + 0.162515i \(0.948039\pi\)
\(270\) 0 0
\(271\) −5942.53 −1.33204 −0.666020 0.745934i \(-0.732003\pi\)
−0.666020 + 0.745934i \(0.732003\pi\)
\(272\) 0 0
\(273\) −1680.00 −0.372448
\(274\) 0 0
\(275\) − 2340.00i − 0.513117i
\(276\) 0 0
\(277\) 1103.09i 0.239271i 0.992818 + 0.119635i \(0.0381726\pi\)
−0.992818 + 0.119635i \(0.961827\pi\)
\(278\) 0 0
\(279\) 992.778 0.213032
\(280\) 0 0
\(281\) −6266.00 −1.33024 −0.665121 0.746735i \(-0.731620\pi\)
−0.665121 + 0.746735i \(0.731620\pi\)
\(282\) 0 0
\(283\) 8596.00i 1.80558i 0.430082 + 0.902790i \(0.358485\pi\)
−0.430082 + 0.902790i \(0.641515\pi\)
\(284\) 0 0
\(285\) − 441.235i − 0.0917070i
\(286\) 0 0
\(287\) −367.696 −0.0756250
\(288\) 0 0
\(289\) −3757.00 −0.764706
\(290\) 0 0
\(291\) 534.000i 0.107573i
\(292\) 0 0
\(293\) 8397.60i 1.67438i 0.546913 + 0.837189i \(0.315803\pi\)
−0.546913 + 0.837189i \(0.684197\pi\)
\(294\) 0 0
\(295\) −1029.55 −0.203195
\(296\) 0 0
\(297\) 540.000 0.105502
\(298\) 0 0
\(299\) − 2464.00i − 0.476578i
\(300\) 0 0
\(301\) 3563.82i 0.682442i
\(302\) 0 0
\(303\) −772.161 −0.146401
\(304\) 0 0
\(305\) 2080.00 0.390493
\(306\) 0 0
\(307\) − 4940.00i − 0.918374i −0.888340 0.459187i \(-0.848141\pi\)
0.888340 0.459187i \(-0.151859\pi\)
\(308\) 0 0
\(309\) 5659.68i 1.04197i
\(310\) 0 0
\(311\) −3382.80 −0.616788 −0.308394 0.951259i \(-0.599791\pi\)
−0.308394 + 0.951259i \(0.599791\pi\)
\(312\) 0 0
\(313\) 3106.00 0.560899 0.280450 0.959869i \(-0.409516\pi\)
0.280450 + 0.959869i \(0.409516\pi\)
\(314\) 0 0
\(315\) − 360.000i − 0.0643927i
\(316\) 0 0
\(317\) 6728.83i 1.19220i 0.802909 + 0.596102i \(0.203285\pi\)
−0.802909 + 0.596102i \(0.796715\pi\)
\(318\) 0 0
\(319\) 4016.37 0.704932
\(320\) 0 0
\(321\) −4212.00 −0.732370
\(322\) 0 0
\(323\) − 1768.00i − 0.304564i
\(324\) 0 0
\(325\) 4632.96i 0.790740i
\(326\) 0 0
\(327\) −118.794 −0.0200897
\(328\) 0 0
\(329\) −4880.00 −0.817760
\(330\) 0 0
\(331\) − 2908.00i − 0.482895i −0.970414 0.241447i \(-0.922378\pi\)
0.970414 0.241447i \(-0.0776221\pi\)
\(332\) 0 0
\(333\) − 2443.76i − 0.402154i
\(334\) 0 0
\(335\) 1776.25 0.289693
\(336\) 0 0
\(337\) 4298.00 0.694739 0.347369 0.937728i \(-0.387075\pi\)
0.347369 + 0.937728i \(0.387075\pi\)
\(338\) 0 0
\(339\) − 4134.00i − 0.662325i
\(340\) 0 0
\(341\) 2206.17i 0.350355i
\(342\) 0 0
\(343\) 6873.08 1.08196
\(344\) 0 0
\(345\) 528.000 0.0823958
\(346\) 0 0
\(347\) 9996.00i 1.54644i 0.634140 + 0.773218i \(0.281354\pi\)
−0.634140 + 0.773218i \(0.718646\pi\)
\(348\) 0 0
\(349\) 3993.74i 0.612550i 0.951943 + 0.306275i \(0.0990827\pi\)
−0.951943 + 0.306275i \(0.900917\pi\)
\(350\) 0 0
\(351\) −1069.15 −0.162583
\(352\) 0 0
\(353\) 6738.00 1.01594 0.507971 0.861374i \(-0.330396\pi\)
0.507971 + 0.861374i \(0.330396\pi\)
\(354\) 0 0
\(355\) − 944.000i − 0.141133i
\(356\) 0 0
\(357\) − 1442.50i − 0.213852i
\(358\) 0 0
\(359\) −2132.63 −0.313527 −0.156763 0.987636i \(-0.550106\pi\)
−0.156763 + 0.987636i \(0.550106\pi\)
\(360\) 0 0
\(361\) 4155.00 0.605773
\(362\) 0 0
\(363\) − 2793.00i − 0.403842i
\(364\) 0 0
\(365\) 956.008i 0.137095i
\(366\) 0 0
\(367\) 7628.27 1.08499 0.542496 0.840058i \(-0.317479\pi\)
0.542496 + 0.840058i \(0.317479\pi\)
\(368\) 0 0
\(369\) −234.000 −0.0330123
\(370\) 0 0
\(371\) − 9640.00i − 1.34901i
\(372\) 0 0
\(373\) 8383.46i 1.16375i 0.813278 + 0.581875i \(0.197681\pi\)
−0.813278 + 0.581875i \(0.802319\pi\)
\(374\) 0 0
\(375\) −2053.44 −0.282771
\(376\) 0 0
\(377\) −7952.00 −1.08634
\(378\) 0 0
\(379\) 12788.0i 1.73318i 0.499020 + 0.866590i \(0.333693\pi\)
−0.499020 + 0.866590i \(0.666307\pi\)
\(380\) 0 0
\(381\) 5371.18i 0.722242i
\(382\) 0 0
\(383\) 2319.31 0.309429 0.154714 0.987959i \(-0.450554\pi\)
0.154714 + 0.987959i \(0.450554\pi\)
\(384\) 0 0
\(385\) 800.000 0.105901
\(386\) 0 0
\(387\) 2268.00i 0.297904i
\(388\) 0 0
\(389\) − 2684.18i − 0.349854i −0.984581 0.174927i \(-0.944031\pi\)
0.984581 0.174927i \(-0.0559689\pi\)
\(390\) 0 0
\(391\) 2115.66 0.273641
\(392\) 0 0
\(393\) 4716.00 0.605320
\(394\) 0 0
\(395\) 2232.00i 0.284314i
\(396\) 0 0
\(397\) 2206.17i 0.278903i 0.990229 + 0.139452i \(0.0445340\pi\)
−0.990229 + 0.139452i \(0.955466\pi\)
\(398\) 0 0
\(399\) −2206.17 −0.276809
\(400\) 0 0
\(401\) 3582.00 0.446076 0.223038 0.974810i \(-0.428403\pi\)
0.223038 + 0.974810i \(0.428403\pi\)
\(402\) 0 0
\(403\) − 4368.00i − 0.539915i
\(404\) 0 0
\(405\) − 229.103i − 0.0281091i
\(406\) 0 0
\(407\) 5430.58 0.661385
\(408\) 0 0
\(409\) −5126.00 −0.619717 −0.309859 0.950783i \(-0.600282\pi\)
−0.309859 + 0.950783i \(0.600282\pi\)
\(410\) 0 0
\(411\) 8562.00i 1.02757i
\(412\) 0 0
\(413\) 5147.74i 0.613326i
\(414\) 0 0
\(415\) 2930.25 0.346603
\(416\) 0 0
\(417\) −5892.00 −0.691924
\(418\) 0 0
\(419\) 2924.00i 0.340923i 0.985364 + 0.170462i \(0.0545259\pi\)
−0.985364 + 0.170462i \(0.945474\pi\)
\(420\) 0 0
\(421\) − 7314.31i − 0.846741i −0.905957 0.423370i \(-0.860847\pi\)
0.905957 0.423370i \(-0.139153\pi\)
\(422\) 0 0
\(423\) −3105.61 −0.356974
\(424\) 0 0
\(425\) −3978.00 −0.454027
\(426\) 0 0
\(427\) − 10400.0i − 1.17867i
\(428\) 0 0
\(429\) − 2375.88i − 0.267386i
\(430\) 0 0
\(431\) −15844.8 −1.77081 −0.885405 0.464819i \(-0.846119\pi\)
−0.885405 + 0.464819i \(0.846119\pi\)
\(432\) 0 0
\(433\) −6274.00 −0.696326 −0.348163 0.937434i \(-0.613194\pi\)
−0.348163 + 0.937434i \(0.613194\pi\)
\(434\) 0 0
\(435\) − 1704.00i − 0.187817i
\(436\) 0 0
\(437\) − 3235.72i − 0.354200i
\(438\) 0 0
\(439\) 4596.19 0.499691 0.249846 0.968286i \(-0.419620\pi\)
0.249846 + 0.968286i \(0.419620\pi\)
\(440\) 0 0
\(441\) 1287.00 0.138970
\(442\) 0 0
\(443\) 5084.00i 0.545255i 0.962120 + 0.272628i \(0.0878927\pi\)
−0.962120 + 0.272628i \(0.912107\pi\)
\(444\) 0 0
\(445\) 661.852i 0.0705051i
\(446\) 0 0
\(447\) −4522.65 −0.478555
\(448\) 0 0
\(449\) 14190.0 1.49146 0.745732 0.666246i \(-0.232100\pi\)
0.745732 + 0.666246i \(0.232100\pi\)
\(450\) 0 0
\(451\) − 520.000i − 0.0542923i
\(452\) 0 0
\(453\) 6796.71i 0.704939i
\(454\) 0 0
\(455\) −1583.92 −0.163198
\(456\) 0 0
\(457\) 6474.00 0.662672 0.331336 0.943513i \(-0.392501\pi\)
0.331336 + 0.943513i \(0.392501\pi\)
\(458\) 0 0
\(459\) − 918.000i − 0.0933520i
\(460\) 0 0
\(461\) − 6321.53i − 0.638662i −0.947643 0.319331i \(-0.896542\pi\)
0.947643 0.319331i \(-0.103458\pi\)
\(462\) 0 0
\(463\) 11435.3 1.14783 0.573915 0.818915i \(-0.305424\pi\)
0.573915 + 0.818915i \(0.305424\pi\)
\(464\) 0 0
\(465\) 936.000 0.0933462
\(466\) 0 0
\(467\) − 3796.00i − 0.376141i −0.982156 0.188071i \(-0.939777\pi\)
0.982156 0.188071i \(-0.0602234\pi\)
\(468\) 0 0
\(469\) − 8881.26i − 0.874411i
\(470\) 0 0
\(471\) −10589.6 −1.03598
\(472\) 0 0
\(473\) −5040.00 −0.489935
\(474\) 0 0
\(475\) 6084.00i 0.587691i
\(476\) 0 0
\(477\) − 6134.86i − 0.588880i
\(478\) 0 0
\(479\) −10493.5 −1.00096 −0.500479 0.865749i \(-0.666843\pi\)
−0.500479 + 0.865749i \(0.666843\pi\)
\(480\) 0 0
\(481\) −10752.0 −1.01923
\(482\) 0 0
\(483\) − 2640.00i − 0.248704i
\(484\) 0 0
\(485\) 503.460i 0.0471360i
\(486\) 0 0
\(487\) 15406.4 1.43354 0.716769 0.697311i \(-0.245620\pi\)
0.716769 + 0.697311i \(0.245620\pi\)
\(488\) 0 0
\(489\) 8796.00 0.813433
\(490\) 0 0
\(491\) − 15452.0i − 1.42024i −0.704079 0.710121i \(-0.748640\pi\)
0.704079 0.710121i \(-0.251360\pi\)
\(492\) 0 0
\(493\) − 6827.82i − 0.623752i
\(494\) 0 0
\(495\) 509.117 0.0462285
\(496\) 0 0
\(497\) −4720.00 −0.425998
\(498\) 0 0
\(499\) 52.0000i 0.00466501i 0.999997 + 0.00233250i \(0.000742460\pi\)
−0.999997 + 0.00233250i \(0.999258\pi\)
\(500\) 0 0
\(501\) − 11030.9i − 0.983678i
\(502\) 0 0
\(503\) −12428.1 −1.10167 −0.550837 0.834613i \(-0.685691\pi\)
−0.550837 + 0.834613i \(0.685691\pi\)
\(504\) 0 0
\(505\) −728.000 −0.0641497
\(506\) 0 0
\(507\) − 1887.00i − 0.165295i
\(508\) 0 0
\(509\) − 16362.5i − 1.42486i −0.701744 0.712429i \(-0.747595\pi\)
0.701744 0.712429i \(-0.252405\pi\)
\(510\) 0 0
\(511\) 4780.04 0.413809
\(512\) 0 0
\(513\) −1404.00 −0.120835
\(514\) 0 0
\(515\) 5336.00i 0.456567i
\(516\) 0 0
\(517\) − 6901.36i − 0.587082i
\(518\) 0 0
\(519\) −4335.98 −0.366721
\(520\) 0 0
\(521\) 714.000 0.0600401 0.0300201 0.999549i \(-0.490443\pi\)
0.0300201 + 0.999549i \(0.490443\pi\)
\(522\) 0 0
\(523\) − 5980.00i − 0.499975i −0.968249 0.249988i \(-0.919573\pi\)
0.968249 0.249988i \(-0.0804266\pi\)
\(524\) 0 0
\(525\) 4963.89i 0.412651i
\(526\) 0 0
\(527\) 3750.49 0.310008
\(528\) 0 0
\(529\) −8295.00 −0.681762
\(530\) 0 0
\(531\) 3276.00i 0.267733i
\(532\) 0 0
\(533\) 1029.55i 0.0836673i
\(534\) 0 0
\(535\) −3971.11 −0.320909
\(536\) 0 0
\(537\) −3924.00 −0.315332
\(538\) 0 0
\(539\) 2860.00i 0.228551i
\(540\) 0 0
\(541\) − 13729.2i − 1.09106i −0.838091 0.545530i \(-0.816328\pi\)
0.838091 0.545530i \(-0.183672\pi\)
\(542\) 0 0
\(543\) −5990.61 −0.473447
\(544\) 0 0
\(545\) −112.000 −0.00880285
\(546\) 0 0
\(547\) 18500.0i 1.44607i 0.690809 + 0.723037i \(0.257255\pi\)
−0.690809 + 0.723037i \(0.742745\pi\)
\(548\) 0 0
\(549\) − 6618.52i − 0.514520i
\(550\) 0 0
\(551\) −10442.6 −0.807382
\(552\) 0 0
\(553\) 11160.0 0.858176
\(554\) 0 0
\(555\) − 2304.00i − 0.176215i
\(556\) 0 0
\(557\) − 8765.30i − 0.666782i −0.942789 0.333391i \(-0.891807\pi\)
0.942789 0.333391i \(-0.108193\pi\)
\(558\) 0 0
\(559\) 9978.69 0.755015
\(560\) 0 0
\(561\) 2040.00 0.153527
\(562\) 0 0
\(563\) 268.000i 0.0200619i 0.999950 + 0.0100310i \(0.00319301\pi\)
−0.999950 + 0.0100310i \(0.996807\pi\)
\(564\) 0 0
\(565\) − 3897.57i − 0.290216i
\(566\) 0 0
\(567\) −1145.51 −0.0848448
\(568\) 0 0
\(569\) 13866.0 1.02160 0.510802 0.859698i \(-0.329348\pi\)
0.510802 + 0.859698i \(0.329348\pi\)
\(570\) 0 0
\(571\) 5140.00i 0.376712i 0.982101 + 0.188356i \(0.0603158\pi\)
−0.982101 + 0.188356i \(0.939684\pi\)
\(572\) 0 0
\(573\) − 2817.11i − 0.205387i
\(574\) 0 0
\(575\) −7280.37 −0.528022
\(576\) 0 0
\(577\) 9386.00 0.677200 0.338600 0.940930i \(-0.390047\pi\)
0.338600 + 0.940930i \(0.390047\pi\)
\(578\) 0 0
\(579\) 7470.00i 0.536170i
\(580\) 0 0
\(581\) − 14651.3i − 1.04619i
\(582\) 0 0
\(583\) 13633.0 0.968477
\(584\) 0 0
\(585\) −1008.00 −0.0712405
\(586\) 0 0
\(587\) − 8844.00i − 0.621859i −0.950433 0.310929i \(-0.899360\pi\)
0.950433 0.310929i \(-0.100640\pi\)
\(588\) 0 0
\(589\) − 5736.05i − 0.401273i
\(590\) 0 0
\(591\) −8171.33 −0.568737
\(592\) 0 0
\(593\) −9406.00 −0.651363 −0.325681 0.945480i \(-0.605594\pi\)
−0.325681 + 0.945480i \(0.605594\pi\)
\(594\) 0 0
\(595\) − 1360.00i − 0.0937051i
\(596\) 0 0
\(597\) 6474.27i 0.443843i
\(598\) 0 0
\(599\) 23459.0 1.60018 0.800090 0.599880i \(-0.204785\pi\)
0.800090 + 0.599880i \(0.204785\pi\)
\(600\) 0 0
\(601\) 1262.00 0.0856540 0.0428270 0.999083i \(-0.486364\pi\)
0.0428270 + 0.999083i \(0.486364\pi\)
\(602\) 0 0
\(603\) − 5652.00i − 0.381704i
\(604\) 0 0
\(605\) − 2633.27i − 0.176955i
\(606\) 0 0
\(607\) 16288.9 1.08920 0.544602 0.838695i \(-0.316681\pi\)
0.544602 + 0.838695i \(0.316681\pi\)
\(608\) 0 0
\(609\) −8520.00 −0.566909
\(610\) 0 0
\(611\) 13664.0i 0.904724i
\(612\) 0 0
\(613\) 7138.95i 0.470374i 0.971950 + 0.235187i \(0.0755703\pi\)
−0.971950 + 0.235187i \(0.924430\pi\)
\(614\) 0 0
\(615\) −220.617 −0.0144653
\(616\) 0 0
\(617\) −16874.0 −1.10101 −0.550504 0.834833i \(-0.685564\pi\)
−0.550504 + 0.834833i \(0.685564\pi\)
\(618\) 0 0
\(619\) − 20748.0i − 1.34723i −0.739085 0.673613i \(-0.764742\pi\)
0.739085 0.673613i \(-0.235258\pi\)
\(620\) 0 0
\(621\) − 1680.09i − 0.108566i
\(622\) 0 0
\(623\) 3309.26 0.212813
\(624\) 0 0
\(625\) 12689.0 0.812096
\(626\) 0 0
\(627\) − 3120.00i − 0.198725i
\(628\) 0 0
\(629\) − 9231.99i − 0.585220i
\(630\) 0 0
\(631\) 14840.8 0.936294 0.468147 0.883651i \(-0.344922\pi\)
0.468147 + 0.883651i \(0.344922\pi\)
\(632\) 0 0
\(633\) −2772.00 −0.174055
\(634\) 0 0
\(635\) 5064.00i 0.316470i
\(636\) 0 0
\(637\) − 5662.51i − 0.352209i
\(638\) 0 0
\(639\) −3003.79 −0.185959
\(640\) 0 0
\(641\) 17758.0 1.09423 0.547113 0.837059i \(-0.315727\pi\)
0.547113 + 0.837059i \(0.315727\pi\)
\(642\) 0 0
\(643\) − 1148.00i − 0.0704086i −0.999380 0.0352043i \(-0.988792\pi\)
0.999380 0.0352043i \(-0.0112082\pi\)
\(644\) 0 0
\(645\) 2138.29i 0.130535i
\(646\) 0 0
\(647\) −26988.9 −1.63994 −0.819970 0.572406i \(-0.806010\pi\)
−0.819970 + 0.572406i \(0.806010\pi\)
\(648\) 0 0
\(649\) −7280.00 −0.440316
\(650\) 0 0
\(651\) − 4680.00i − 0.281757i
\(652\) 0 0
\(653\) 21069.0i 1.26262i 0.775530 + 0.631311i \(0.217483\pi\)
−0.775530 + 0.631311i \(0.782517\pi\)
\(654\) 0 0
\(655\) 4446.29 0.265238
\(656\) 0 0
\(657\) 3042.00 0.180639
\(658\) 0 0
\(659\) 18356.0i 1.08505i 0.840040 + 0.542525i \(0.182532\pi\)
−0.840040 + 0.542525i \(0.817468\pi\)
\(660\) 0 0
\(661\) − 15250.9i − 0.897414i −0.893679 0.448707i \(-0.851885\pi\)
0.893679 0.448707i \(-0.148115\pi\)
\(662\) 0 0
\(663\) −4038.99 −0.236594
\(664\) 0 0
\(665\) −2080.00 −0.121292
\(666\) 0 0
\(667\) − 12496.0i − 0.725408i
\(668\) 0 0
\(669\) 6830.65i 0.394751i
\(670\) 0 0
\(671\) 14707.8 0.846184
\(672\) 0 0
\(673\) −12082.0 −0.692016 −0.346008 0.938232i \(-0.612463\pi\)
−0.346008 + 0.938232i \(0.612463\pi\)
\(674\) 0 0
\(675\) 3159.00i 0.180133i
\(676\) 0 0
\(677\) − 12742.1i − 0.723364i −0.932302 0.361682i \(-0.882203\pi\)
0.932302 0.361682i \(-0.117797\pi\)
\(678\) 0 0
\(679\) 2517.30 0.142276
\(680\) 0 0
\(681\) 468.000 0.0263345
\(682\) 0 0
\(683\) − 33508.0i − 1.87723i −0.344967 0.938615i \(-0.612110\pi\)
0.344967 0.938615i \(-0.387890\pi\)
\(684\) 0 0
\(685\) 8072.33i 0.450260i
\(686\) 0 0
\(687\) 1917.67 0.106498
\(688\) 0 0
\(689\) −26992.0 −1.49247
\(690\) 0 0
\(691\) − 364.000i − 0.0200394i −0.999950 0.0100197i \(-0.996811\pi\)
0.999950 0.0100197i \(-0.00318942\pi\)
\(692\) 0 0
\(693\) − 2545.58i − 0.139536i
\(694\) 0 0
\(695\) −5555.03 −0.303186
\(696\) 0 0
\(697\) −884.000 −0.0480400
\(698\) 0 0
\(699\) 8478.00i 0.458752i
\(700\) 0 0
\(701\) − 3849.49i − 0.207408i −0.994608 0.103704i \(-0.966930\pi\)
0.994608 0.103704i \(-0.0330695\pi\)
\(702\) 0 0
\(703\) −14119.5 −0.757507
\(704\) 0 0
\(705\) −2928.00 −0.156418
\(706\) 0 0
\(707\) 3640.00i 0.193630i
\(708\) 0 0
\(709\) 23606.1i 1.25041i 0.780459 + 0.625207i \(0.214986\pi\)
−0.780459 + 0.625207i \(0.785014\pi\)
\(710\) 0 0
\(711\) 7102.18 0.374617
\(712\) 0 0
\(713\) 6864.00 0.360531
\(714\) 0 0
\(715\) − 2240.00i − 0.117163i
\(716\) 0 0
\(717\) − 7399.17i − 0.385393i
\(718\) 0 0
\(719\) −15799.6 −0.819507 −0.409753 0.912196i \(-0.634385\pi\)
−0.409753 + 0.912196i \(0.634385\pi\)
\(720\) 0 0
\(721\) 26680.0 1.37811
\(722\) 0 0
\(723\) 10062.0i 0.517579i
\(724\) 0 0
\(725\) 23495.7i 1.20360i
\(726\) 0 0
\(727\) −4607.51 −0.235052 −0.117526 0.993070i \(-0.537496\pi\)
−0.117526 + 0.993070i \(0.537496\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 8568.00i 0.433514i
\(732\) 0 0
\(733\) 26219.5i 1.32120i 0.750738 + 0.660600i \(0.229698\pi\)
−0.750738 + 0.660600i \(0.770302\pi\)
\(734\) 0 0
\(735\) 1213.40 0.0608935
\(736\) 0 0
\(737\) 12560.0 0.627752
\(738\) 0 0
\(739\) 27924.0i 1.38999i 0.719016 + 0.694994i \(0.244593\pi\)
−0.719016 + 0.694994i \(0.755407\pi\)
\(740\) 0 0
\(741\) 6177.28i 0.306246i
\(742\) 0 0
\(743\) 8937.83 0.441315 0.220658 0.975351i \(-0.429180\pi\)
0.220658 + 0.975351i \(0.429180\pi\)
\(744\) 0 0
\(745\) −4264.00 −0.209692
\(746\) 0 0
\(747\) − 9324.00i − 0.456690i
\(748\) 0 0
\(749\) 19855.6i 0.968633i
\(750\) 0 0
\(751\) −14082.7 −0.684270 −0.342135 0.939651i \(-0.611150\pi\)
−0.342135 + 0.939651i \(0.611150\pi\)
\(752\) 0 0
\(753\) 19188.0 0.928618
\(754\) 0 0
\(755\) 6408.00i 0.308889i
\(756\) 0 0
\(757\) 14871.9i 0.714039i 0.934097 + 0.357019i \(0.116207\pi\)
−0.934097 + 0.357019i \(0.883793\pi\)
\(758\) 0 0
\(759\) 3733.52 0.178549
\(760\) 0 0
\(761\) 15834.0 0.754247 0.377124 0.926163i \(-0.376913\pi\)
0.377124 + 0.926163i \(0.376913\pi\)
\(762\) 0 0
\(763\) 560.000i 0.0265706i
\(764\) 0 0
\(765\) − 865.499i − 0.0409048i
\(766\) 0 0
\(767\) 14413.7 0.678549
\(768\) 0 0
\(769\) −16666.0 −0.781523 −0.390762 0.920492i \(-0.627788\pi\)
−0.390762 + 0.920492i \(0.627788\pi\)
\(770\) 0 0
\(771\) − 20646.0i − 0.964394i
\(772\) 0 0
\(773\) 30957.1i 1.44043i 0.693752 + 0.720214i \(0.255956\pi\)
−0.693752 + 0.720214i \(0.744044\pi\)
\(774\) 0 0
\(775\) −12906.1 −0.598195
\(776\) 0 0
\(777\) −11520.0 −0.531889
\(778\) 0 0
\(779\) 1352.00i 0.0621828i
\(780\) 0 0
\(781\) − 6675.09i − 0.305830i
\(782\) 0 0
\(783\) −5422.09 −0.247471
\(784\) 0 0
\(785\) −9984.00 −0.453942
\(786\) 0 0
\(787\) 20228.0i 0.916201i 0.888900 + 0.458101i \(0.151470\pi\)
−0.888900 + 0.458101i \(0.848530\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −19487.9 −0.875991
\(792\) 0 0
\(793\) −29120.0 −1.30401
\(794\) 0 0
\(795\) − 5784.00i − 0.258034i
\(796\) 0 0
\(797\) 9008.54i 0.400375i 0.979758 + 0.200187i \(0.0641551\pi\)
−0.979758 + 0.200187i \(0.935845\pi\)
\(798\) 0 0
\(799\) −11732.3 −0.519474
\(800\) 0 0
\(801\) 2106.00 0.0928987
\(802\) 0 0
\(803\) 6760.00i 0.297080i
\(804\) 0 0
\(805\) − 2489.02i − 0.108977i
\(806\) 0 0
\(807\) 4302.04 0.187657
\(808\) 0 0
\(809\) 9242.00 0.401646 0.200823 0.979628i \(-0.435638\pi\)
0.200823 + 0.979628i \(0.435638\pi\)
\(810\) 0 0
\(811\) − 10972.0i − 0.475067i −0.971379 0.237533i \(-0.923661\pi\)
0.971379 0.237533i \(-0.0763389\pi\)
\(812\) 0 0
\(813\) 17827.6i 0.769053i
\(814\) 0 0
\(815\) 8292.95 0.356429
\(816\) 0 0
\(817\) 13104.0 0.561139
\(818\) 0 0
\(819\) 5040.00i 0.215033i
\(820\) 0 0
\(821\) 9336.64i 0.396895i 0.980112 + 0.198448i \(0.0635900\pi\)
−0.980112 + 0.198448i \(0.936410\pi\)
\(822\) 0 0
\(823\) −3566.65 −0.151064 −0.0755319 0.997143i \(-0.524065\pi\)
−0.0755319 + 0.997143i \(0.524065\pi\)
\(824\) 0 0
\(825\) −7020.00 −0.296249
\(826\) 0 0
\(827\) − 18876.0i − 0.793691i −0.917885 0.396846i \(-0.870105\pi\)
0.917885 0.396846i \(-0.129895\pi\)
\(828\) 0 0
\(829\) 6974.90i 0.292218i 0.989269 + 0.146109i \(0.0466749\pi\)
−0.989269 + 0.146109i \(0.953325\pi\)
\(830\) 0 0
\(831\) 3309.26 0.138143
\(832\) 0 0
\(833\) 4862.00 0.202231
\(834\) 0 0
\(835\) − 10400.0i − 0.431026i
\(836\) 0 0
\(837\) − 2978.33i − 0.122994i
\(838\) 0 0
\(839\) 30077.5 1.23765 0.618826 0.785528i \(-0.287608\pi\)
0.618826 + 0.785528i \(0.287608\pi\)
\(840\) 0 0
\(841\) −15939.0 −0.653532
\(842\) 0 0
\(843\) 18798.0i 0.768016i
\(844\) 0 0
\(845\) − 1779.08i − 0.0724287i
\(846\) 0 0
\(847\) −13166.3 −0.534121
\(848\) 0 0
\(849\) 25788.0 1.04245
\(850\) 0 0
\(851\) − 16896.0i − 0.680596i
\(852\) 0 0
\(853\) 41159.3i 1.65213i 0.563575 + 0.826065i \(0.309426\pi\)
−0.563575 + 0.826065i \(0.690574\pi\)
\(854\) 0 0
\(855\) −1323.70 −0.0529470
\(856\) 0 0
\(857\) 25194.0 1.00421 0.502107 0.864806i \(-0.332559\pi\)
0.502107 + 0.864806i \(0.332559\pi\)
\(858\) 0 0
\(859\) − 9308.00i − 0.369715i −0.982765 0.184857i \(-0.940818\pi\)
0.982765 0.184857i \(-0.0591823\pi\)
\(860\) 0 0
\(861\) 1103.09i 0.0436621i
\(862\) 0 0
\(863\) −26802.2 −1.05719 −0.528596 0.848874i \(-0.677281\pi\)
−0.528596 + 0.848874i \(0.677281\pi\)
\(864\) 0 0
\(865\) −4088.00 −0.160689
\(866\) 0 0
\(867\) 11271.0i 0.441503i
\(868\) 0 0
\(869\) 15782.6i 0.616098i
\(870\) 0 0
\(871\) −24867.5 −0.967399
\(872\) 0 0
\(873\) 1602.00 0.0621071
\(874\) 0 0
\(875\) 9680.00i 0.373993i
\(876\) 0 0
\(877\) 1436.84i 0.0553235i 0.999617 + 0.0276617i \(0.00880613\pi\)
−0.999617 + 0.0276617i \(0.991194\pi\)
\(878\) 0 0
\(879\) 25192.8 0.966703
\(880\) 0 0
\(881\) −42830.0 −1.63789 −0.818944 0.573873i \(-0.805440\pi\)
−0.818944 + 0.573873i \(0.805440\pi\)
\(882\) 0 0
\(883\) − 23964.0i − 0.913310i −0.889644 0.456655i \(-0.849047\pi\)
0.889644 0.456655i \(-0.150953\pi\)
\(884\) 0 0
\(885\) 3088.64i 0.117315i
\(886\) 0 0
\(887\) 28239.0 1.06897 0.534483 0.845179i \(-0.320506\pi\)
0.534483 + 0.845179i \(0.320506\pi\)
\(888\) 0 0
\(889\) 25320.0 0.955237
\(890\) 0 0
\(891\) − 1620.00i − 0.0609114i
\(892\) 0 0
\(893\) 17943.5i 0.672405i
\(894\) 0 0
\(895\) −3699.58 −0.138171
\(896\) 0 0
\(897\) −7392.00 −0.275152
\(898\) 0 0
\(899\) − 22152.0i − 0.821814i
\(900\) 0 0
\(901\) − 23176.1i − 0.856947i
\(902\) 0 0
\(903\) 10691.5 0.394008
\(904\) 0 0
\(905\) −5648.00 −0.207454
\(906\) 0 0
\(907\) 31972.0i 1.17047i 0.810865 + 0.585233i \(0.198997\pi\)
−0.810865 + 0.585233i \(0.801003\pi\)
\(908\) 0 0
\(909\) 2316.48i 0.0845246i
\(910\) 0 0
\(911\) 26858.7 0.976806 0.488403 0.872618i \(-0.337580\pi\)
0.488403 + 0.872618i \(0.337580\pi\)
\(912\) 0 0
\(913\) 20720.0 0.751075
\(914\) 0 0
\(915\) − 6240.00i − 0.225451i
\(916\) 0 0
\(917\) − 22231.4i − 0.800596i
\(918\) 0 0
\(919\) −40336.2 −1.44784 −0.723922 0.689882i \(-0.757662\pi\)
−0.723922 + 0.689882i \(0.757662\pi\)
\(920\) 0 0
\(921\) −14820.0 −0.530223
\(922\) 0 0
\(923\) 13216.0i 0.471300i
\(924\) 0 0
\(925\) 31768.9i 1.12925i
\(926\) 0 0
\(927\) 16979.0 0.601580
\(928\) 0 0
\(929\) −13650.0 −0.482069 −0.241034 0.970517i \(-0.577487\pi\)
−0.241034 + 0.970517i \(0.577487\pi\)
\(930\) 0 0
\(931\) − 7436.00i − 0.261767i
\(932\) 0 0
\(933\) 10148.4i 0.356102i
\(934\) 0 0
\(935\) 1923.33 0.0672723
\(936\) 0 0
\(937\) −7098.00 −0.247472 −0.123736 0.992315i \(-0.539488\pi\)
−0.123736 + 0.992315i \(0.539488\pi\)
\(938\) 0 0
\(939\) − 9318.00i − 0.323835i
\(940\) 0 0
\(941\) − 41326.1i − 1.43166i −0.698274 0.715831i \(-0.746048\pi\)
0.698274 0.715831i \(-0.253952\pi\)
\(942\) 0 0
\(943\) −1617.86 −0.0558693
\(944\) 0 0
\(945\) −1080.00 −0.0371771
\(946\) 0 0
\(947\) − 9900.00i − 0.339711i −0.985469 0.169856i \(-0.945670\pi\)
0.985469 0.169856i \(-0.0543302\pi\)
\(948\) 0 0
\(949\) − 13384.1i − 0.457815i
\(950\) 0 0
\(951\) 20186.5 0.688319
\(952\) 0 0
\(953\) 46938.0 1.59546 0.797729 0.603016i \(-0.206035\pi\)
0.797729 + 0.603016i \(0.206035\pi\)
\(954\) 0 0
\(955\) − 2656.00i − 0.0899960i
\(956\) 0 0
\(957\) − 12049.1i − 0.406993i
\(958\) 0 0
\(959\) 40361.7 1.35907
\(960\) 0 0
\(961\) −17623.0 −0.591554
\(962\) 0 0
\(963\) 12636.0i 0.422834i
\(964\) 0 0
\(965\) 7042.78i 0.234938i
\(966\) 0 0
\(967\) −6989.04 −0.232422 −0.116211 0.993225i \(-0.537075\pi\)
−0.116211 + 0.993225i \(0.537075\pi\)
\(968\) 0 0
\(969\) −5304.00 −0.175840
\(970\) 0 0
\(971\) 53052.0i 1.75337i 0.481067 + 0.876684i \(0.340249\pi\)
−0.481067 + 0.876684i \(0.659751\pi\)
\(972\) 0 0
\(973\) 27775.2i 0.915139i
\(974\) 0 0
\(975\) 13898.9 0.456534
\(976\) 0 0
\(977\) −41890.0 −1.37173 −0.685865 0.727729i \(-0.740576\pi\)
−0.685865 + 0.727729i \(0.740576\pi\)
\(978\) 0 0
\(979\) 4680.00i 0.152782i
\(980\) 0 0
\(981\) 356.382i 0.0115988i
\(982\) 0 0
\(983\) 10861.2 0.352408 0.176204 0.984354i \(-0.443618\pi\)
0.176204 + 0.984354i \(0.443618\pi\)
\(984\) 0 0
\(985\) −7704.00 −0.249208
\(986\) 0 0
\(987\) 14640.0i 0.472134i
\(988\) 0 0
\(989\) 15680.8i 0.504166i
\(990\) 0 0
\(991\) 330.926 0.0106077 0.00530384 0.999986i \(-0.498312\pi\)
0.00530384 + 0.999986i \(0.498312\pi\)
\(992\) 0 0
\(993\) −8724.00 −0.278799
\(994\) 0 0
\(995\) 6104.00i 0.194482i
\(996\) 0 0
\(997\) − 39948.7i − 1.26900i −0.772925 0.634498i \(-0.781207\pi\)
0.772925 0.634498i \(-0.218793\pi\)
\(998\) 0 0
\(999\) −7331.28 −0.232184
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 384.4.d.d.193.1 4
3.2 odd 2 1152.4.d.n.577.3 4
4.3 odd 2 inner 384.4.d.d.193.3 yes 4
8.3 odd 2 inner 384.4.d.d.193.2 yes 4
8.5 even 2 inner 384.4.d.d.193.4 yes 4
12.11 even 2 1152.4.d.n.577.4 4
16.3 odd 4 768.4.a.h.1.1 2
16.5 even 4 768.4.a.h.1.2 2
16.11 odd 4 768.4.a.m.1.2 2
16.13 even 4 768.4.a.m.1.1 2
24.5 odd 2 1152.4.d.n.577.1 4
24.11 even 2 1152.4.d.n.577.2 4
48.5 odd 4 2304.4.a.bb.1.1 2
48.11 even 4 2304.4.a.bh.1.1 2
48.29 odd 4 2304.4.a.bh.1.2 2
48.35 even 4 2304.4.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.d.193.1 4 1.1 even 1 trivial
384.4.d.d.193.2 yes 4 8.3 odd 2 inner
384.4.d.d.193.3 yes 4 4.3 odd 2 inner
384.4.d.d.193.4 yes 4 8.5 even 2 inner
768.4.a.h.1.1 2 16.3 odd 4
768.4.a.h.1.2 2 16.5 even 4
768.4.a.m.1.1 2 16.13 even 4
768.4.a.m.1.2 2 16.11 odd 4
1152.4.d.n.577.1 4 24.5 odd 2
1152.4.d.n.577.2 4 24.11 even 2
1152.4.d.n.577.3 4 3.2 odd 2
1152.4.d.n.577.4 4 12.11 even 2
2304.4.a.bb.1.1 2 48.5 odd 4
2304.4.a.bb.1.2 2 48.35 even 4
2304.4.a.bh.1.1 2 48.11 even 4
2304.4.a.bh.1.2 2 48.29 odd 4