Properties

Label 256.12.b.e.129.1
Level $256$
Weight $12$
Character 256.129
Analytic conductor $196.696$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 256.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(196.695854223\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 256.129
Dual form 256.12.b.e.129.2

$q$-expansion

\(f(q)\) \(=\) \(q-252.000i q^{3} +4830.00i q^{5} +16744.0 q^{7} +113643. q^{9} +O(q^{10})\) \(q-252.000i q^{3} +4830.00i q^{5} +16744.0 q^{7} +113643. q^{9} +534612. i q^{11} +577738. i q^{13} +1.21716e6 q^{15} -6.90593e6 q^{17} -1.06614e7i q^{19} -4.21949e6i q^{21} -1.86433e7 q^{23} +2.54992e7 q^{25} -7.32791e7i q^{27} -1.28407e8i q^{29} -5.28432e7 q^{31} +1.34722e8 q^{33} +8.08735e7i q^{35} -1.82213e8i q^{37} +1.45590e8 q^{39} -3.08120e8 q^{41} -1.71257e7i q^{43} +5.48896e8i q^{45} +2.68735e9 q^{47} -1.69697e9 q^{49} +1.74030e9i q^{51} -1.59606e9i q^{53} -2.58218e9 q^{55} -2.68668e9 q^{57} -5.18920e9i q^{59} -6.95648e9i q^{61} +1.90284e9 q^{63} -2.79047e9 q^{65} +1.54818e10i q^{67} +4.69810e9i q^{69} -9.79149e9 q^{71} -1.46379e9 q^{73} -6.42580e9i q^{75} +8.95154e9i q^{77} +3.81168e10 q^{79} +1.66519e9 q^{81} +2.93351e10i q^{83} -3.33557e10i q^{85} -3.23585e10 q^{87} +2.49929e10 q^{89} +9.67365e9i q^{91} +1.33165e10i q^{93} +5.14947e10 q^{95} +7.50136e10 q^{97} +6.07549e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 33488q^{7} + 227286q^{9} + O(q^{10}) \) \( 2q + 33488q^{7} + 227286q^{9} + 2434320q^{15} - 13811868q^{17} - 37286544q^{23} + 50998450q^{25} - 105686336q^{31} + 269444448q^{33} + 291179952q^{39} - 616240884q^{41} + 5374696992q^{47} - 3393930414q^{49} - 5164351920q^{55} - 5373355680q^{57} + 3805676784q^{63} - 5580949080q^{65} - 19582970544q^{71} - 2927582644q^{73} + 76233691360q^{79} + 3330376722q^{81} - 64716941520q^{87} + 49985834220q^{89} + 102989317200q^{95} + 150027137092q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) − 252.000i − 0.598734i −0.954138 0.299367i \(-0.903225\pi\)
0.954138 0.299367i \(-0.0967754\pi\)
\(4\) 0 0
\(5\) 4830.00i 0.691213i 0.938379 + 0.345607i \(0.112327\pi\)
−0.938379 + 0.345607i \(0.887673\pi\)
\(6\) 0 0
\(7\) 16744.0 0.376548 0.188274 0.982117i \(-0.439711\pi\)
0.188274 + 0.982117i \(0.439711\pi\)
\(8\) 0 0
\(9\) 113643. 0.641518
\(10\) 0 0
\(11\) 534612.i 1.00087i 0.865773 + 0.500436i \(0.166827\pi\)
−0.865773 + 0.500436i \(0.833173\pi\)
\(12\) 0 0
\(13\) 577738.i 0.431561i 0.976442 + 0.215781i \(0.0692296\pi\)
−0.976442 + 0.215781i \(0.930770\pi\)
\(14\) 0 0
\(15\) 1.21716e6 0.413853
\(16\) 0 0
\(17\) −6.90593e6 −1.17965 −0.589825 0.807531i \(-0.700803\pi\)
−0.589825 + 0.807531i \(0.700803\pi\)
\(18\) 0 0
\(19\) − 1.06614e7i − 0.987803i −0.869518 0.493901i \(-0.835570\pi\)
0.869518 0.493901i \(-0.164430\pi\)
\(20\) 0 0
\(21\) − 4.21949e6i − 0.225452i
\(22\) 0 0
\(23\) −1.86433e7 −0.603975 −0.301988 0.953312i \(-0.597650\pi\)
−0.301988 + 0.953312i \(0.597650\pi\)
\(24\) 0 0
\(25\) 2.54992e7 0.522224
\(26\) 0 0
\(27\) − 7.32791e7i − 0.982832i
\(28\) 0 0
\(29\) − 1.28407e8i − 1.16251i −0.813720 0.581257i \(-0.802561\pi\)
0.813720 0.581257i \(-0.197439\pi\)
\(30\) 0 0
\(31\) −5.28432e7 −0.331512 −0.165756 0.986167i \(-0.553006\pi\)
−0.165756 + 0.986167i \(0.553006\pi\)
\(32\) 0 0
\(33\) 1.34722e8 0.599256
\(34\) 0 0
\(35\) 8.08735e7i 0.260275i
\(36\) 0 0
\(37\) − 1.82213e8i − 0.431987i −0.976395 0.215993i \(-0.930701\pi\)
0.976395 0.215993i \(-0.0692990\pi\)
\(38\) 0 0
\(39\) 1.45590e8 0.258390
\(40\) 0 0
\(41\) −3.08120e8 −0.415345 −0.207673 0.978198i \(-0.566589\pi\)
−0.207673 + 0.978198i \(0.566589\pi\)
\(42\) 0 0
\(43\) − 1.71257e7i − 0.0177653i −0.999961 0.00888264i \(-0.997173\pi\)
0.999961 0.00888264i \(-0.00282747\pi\)
\(44\) 0 0
\(45\) 5.48896e8i 0.443426i
\(46\) 0 0
\(47\) 2.68735e9 1.70917 0.854586 0.519310i \(-0.173811\pi\)
0.854586 + 0.519310i \(0.173811\pi\)
\(48\) 0 0
\(49\) −1.69697e9 −0.858212
\(50\) 0 0
\(51\) 1.74030e9i 0.706296i
\(52\) 0 0
\(53\) − 1.59606e9i − 0.524241i −0.965035 0.262120i \(-0.915578\pi\)
0.965035 0.262120i \(-0.0844217\pi\)
\(54\) 0 0
\(55\) −2.58218e9 −0.691817
\(56\) 0 0
\(57\) −2.68668e9 −0.591431
\(58\) 0 0
\(59\) − 5.18920e9i − 0.944963i −0.881341 0.472481i \(-0.843358\pi\)
0.881341 0.472481i \(-0.156642\pi\)
\(60\) 0 0
\(61\) − 6.95648e9i − 1.05457i −0.849689 0.527285i \(-0.823210\pi\)
0.849689 0.527285i \(-0.176790\pi\)
\(62\) 0 0
\(63\) 1.90284e9 0.241562
\(64\) 0 0
\(65\) −2.79047e9 −0.298301
\(66\) 0 0
\(67\) 1.54818e10i 1.40091i 0.713696 + 0.700456i \(0.247020\pi\)
−0.713696 + 0.700456i \(0.752980\pi\)
\(68\) 0 0
\(69\) 4.69810e9i 0.361620i
\(70\) 0 0
\(71\) −9.79149e9 −0.644062 −0.322031 0.946729i \(-0.604366\pi\)
−0.322031 + 0.946729i \(0.604366\pi\)
\(72\) 0 0
\(73\) −1.46379e9 −0.0826425 −0.0413212 0.999146i \(-0.513157\pi\)
−0.0413212 + 0.999146i \(0.513157\pi\)
\(74\) 0 0
\(75\) − 6.42580e9i − 0.312673i
\(76\) 0 0
\(77\) 8.95154e9i 0.376876i
\(78\) 0 0
\(79\) 3.81168e10 1.39370 0.696848 0.717219i \(-0.254585\pi\)
0.696848 + 0.717219i \(0.254585\pi\)
\(80\) 0 0
\(81\) 1.66519e9 0.0530635
\(82\) 0 0
\(83\) 2.93351e10i 0.817444i 0.912659 + 0.408722i \(0.134025\pi\)
−0.912659 + 0.408722i \(0.865975\pi\)
\(84\) 0 0
\(85\) − 3.33557e10i − 0.815390i
\(86\) 0 0
\(87\) −3.23585e10 −0.696037
\(88\) 0 0
\(89\) 2.49929e10 0.474430 0.237215 0.971457i \(-0.423765\pi\)
0.237215 + 0.971457i \(0.423765\pi\)
\(90\) 0 0
\(91\) 9.67365e9i 0.162503i
\(92\) 0 0
\(93\) 1.33165e10i 0.198488i
\(94\) 0 0
\(95\) 5.14947e10 0.682782
\(96\) 0 0
\(97\) 7.50136e10 0.886942 0.443471 0.896289i \(-0.353747\pi\)
0.443471 + 0.896289i \(0.353747\pi\)
\(98\) 0 0
\(99\) 6.07549e10i 0.642078i
\(100\) 0 0
\(101\) 8.17430e10i 0.773896i 0.922101 + 0.386948i \(0.126471\pi\)
−0.922101 + 0.386948i \(0.873529\pi\)
\(102\) 0 0
\(103\) 2.25755e11 1.91881 0.959407 0.282025i \(-0.0910061\pi\)
0.959407 + 0.282025i \(0.0910061\pi\)
\(104\) 0 0
\(105\) 2.03801e10 0.155835
\(106\) 0 0
\(107\) 9.02413e10i 0.622006i 0.950409 + 0.311003i \(0.100665\pi\)
−0.950409 + 0.311003i \(0.899335\pi\)
\(108\) 0 0
\(109\) − 7.34827e10i − 0.457445i −0.973492 0.228723i \(-0.926545\pi\)
0.973492 0.228723i \(-0.0734549\pi\)
\(110\) 0 0
\(111\) −4.59178e10 −0.258645
\(112\) 0 0
\(113\) −8.51469e10 −0.434748 −0.217374 0.976088i \(-0.569749\pi\)
−0.217374 + 0.976088i \(0.569749\pi\)
\(114\) 0 0
\(115\) − 9.00470e10i − 0.417476i
\(116\) 0 0
\(117\) 6.56559e10i 0.276854i
\(118\) 0 0
\(119\) −1.15633e11 −0.444195
\(120\) 0 0
\(121\) −4.98320e8 −0.00174658
\(122\) 0 0
\(123\) 7.76464e10i 0.248681i
\(124\) 0 0
\(125\) 3.59001e11i 1.05218i
\(126\) 0 0
\(127\) −2.62717e11 −0.705615 −0.352808 0.935696i \(-0.614773\pi\)
−0.352808 + 0.935696i \(0.614773\pi\)
\(128\) 0 0
\(129\) −4.31568e9 −0.0106367
\(130\) 0 0
\(131\) − 6.31529e11i − 1.43021i −0.699015 0.715107i \(-0.746378\pi\)
0.699015 0.715107i \(-0.253622\pi\)
\(132\) 0 0
\(133\) − 1.78515e11i − 0.371955i
\(134\) 0 0
\(135\) 3.53938e11 0.679347
\(136\) 0 0
\(137\) 2.97199e11 0.526119 0.263059 0.964780i \(-0.415268\pi\)
0.263059 + 0.964780i \(0.415268\pi\)
\(138\) 0 0
\(139\) 5.96794e11i 0.975535i 0.872974 + 0.487767i \(0.162189\pi\)
−0.872974 + 0.487767i \(0.837811\pi\)
\(140\) 0 0
\(141\) − 6.77212e11i − 1.02334i
\(142\) 0 0
\(143\) −3.08866e11 −0.431938
\(144\) 0 0
\(145\) 6.20204e11 0.803546
\(146\) 0 0
\(147\) 4.27635e11i 0.513840i
\(148\) 0 0
\(149\) − 1.11543e12i − 1.24428i −0.782905 0.622142i \(-0.786263\pi\)
0.782905 0.622142i \(-0.213737\pi\)
\(150\) 0 0
\(151\) 8.24447e11 0.854653 0.427326 0.904097i \(-0.359456\pi\)
0.427326 + 0.904097i \(0.359456\pi\)
\(152\) 0 0
\(153\) −7.84811e11 −0.756767
\(154\) 0 0
\(155\) − 2.55233e11i − 0.229146i
\(156\) 0 0
\(157\) − 1.31512e12i − 1.10031i −0.835062 0.550156i \(-0.814568\pi\)
0.835062 0.550156i \(-0.185432\pi\)
\(158\) 0 0
\(159\) −4.02206e11 −0.313881
\(160\) 0 0
\(161\) −3.12163e11 −0.227425
\(162\) 0 0
\(163\) 3.57833e11i 0.243584i 0.992556 + 0.121792i \(0.0388640\pi\)
−0.992556 + 0.121792i \(0.961136\pi\)
\(164\) 0 0
\(165\) 6.50708e11i 0.414214i
\(166\) 0 0
\(167\) −2.75483e12 −1.64117 −0.820587 0.571521i \(-0.806354\pi\)
−0.820587 + 0.571521i \(0.806354\pi\)
\(168\) 0 0
\(169\) 1.45838e12 0.813755
\(170\) 0 0
\(171\) − 1.21160e12i − 0.633693i
\(172\) 0 0
\(173\) 9.50387e11i 0.466280i 0.972443 + 0.233140i \(0.0749001\pi\)
−0.972443 + 0.233140i \(0.925100\pi\)
\(174\) 0 0
\(175\) 4.26959e11 0.196642
\(176\) 0 0
\(177\) −1.30768e12 −0.565781
\(178\) 0 0
\(179\) − 1.68138e12i − 0.683873i −0.939723 0.341936i \(-0.888917\pi\)
0.939723 0.341936i \(-0.111083\pi\)
\(180\) 0 0
\(181\) − 9.96774e11i − 0.381386i −0.981650 0.190693i \(-0.938927\pi\)
0.981650 0.190693i \(-0.0610735\pi\)
\(182\) 0 0
\(183\) −1.75303e12 −0.631406
\(184\) 0 0
\(185\) 8.80090e11 0.298595
\(186\) 0 0
\(187\) − 3.69200e12i − 1.18068i
\(188\) 0 0
\(189\) − 1.22698e12i − 0.370083i
\(190\) 0 0
\(191\) 2.76240e12 0.786328 0.393164 0.919468i \(-0.371381\pi\)
0.393164 + 0.919468i \(0.371381\pi\)
\(192\) 0 0
\(193\) 5.44239e12 1.46293 0.731466 0.681878i \(-0.238836\pi\)
0.731466 + 0.681878i \(0.238836\pi\)
\(194\) 0 0
\(195\) 7.03200e11i 0.178603i
\(196\) 0 0
\(197\) − 2.87609e12i − 0.690619i −0.938489 0.345309i \(-0.887774\pi\)
0.938489 0.345309i \(-0.112226\pi\)
\(198\) 0 0
\(199\) −7.28391e11 −0.165452 −0.0827262 0.996572i \(-0.526363\pi\)
−0.0827262 + 0.996572i \(0.526363\pi\)
\(200\) 0 0
\(201\) 3.90142e12 0.838773
\(202\) 0 0
\(203\) − 2.15004e12i − 0.437742i
\(204\) 0 0
\(205\) − 1.48822e12i − 0.287092i
\(206\) 0 0
\(207\) −2.11868e12 −0.387461
\(208\) 0 0
\(209\) 5.69972e12 0.988665
\(210\) 0 0
\(211\) 6.79317e12i 1.11820i 0.829101 + 0.559099i \(0.188853\pi\)
−0.829101 + 0.559099i \(0.811147\pi\)
\(212\) 0 0
\(213\) 2.46745e12i 0.385622i
\(214\) 0 0
\(215\) 8.27172e10 0.0122796
\(216\) 0 0
\(217\) −8.84806e11 −0.124830
\(218\) 0 0
\(219\) 3.68875e11i 0.0494808i
\(220\) 0 0
\(221\) − 3.98982e12i − 0.509092i
\(222\) 0 0
\(223\) 7.33486e12 0.890667 0.445333 0.895365i \(-0.353085\pi\)
0.445333 + 0.895365i \(0.353085\pi\)
\(224\) 0 0
\(225\) 2.89781e12 0.335016
\(226\) 0 0
\(227\) 1.35984e12i 0.149743i 0.997193 + 0.0748713i \(0.0238546\pi\)
−0.997193 + 0.0748713i \(0.976145\pi\)
\(228\) 0 0
\(229\) − 1.18244e13i − 1.24075i −0.784305 0.620375i \(-0.786980\pi\)
0.784305 0.620375i \(-0.213020\pi\)
\(230\) 0 0
\(231\) 2.25579e12 0.225649
\(232\) 0 0
\(233\) 1.75634e13 1.67552 0.837761 0.546038i \(-0.183865\pi\)
0.837761 + 0.546038i \(0.183865\pi\)
\(234\) 0 0
\(235\) 1.29799e13i 1.18140i
\(236\) 0 0
\(237\) − 9.60545e12i − 0.834452i
\(238\) 0 0
\(239\) −7.13958e12 −0.592221 −0.296111 0.955154i \(-0.595690\pi\)
−0.296111 + 0.955154i \(0.595690\pi\)
\(240\) 0 0
\(241\) −2.31307e11 −0.0183271 −0.00916357 0.999958i \(-0.502917\pi\)
−0.00916357 + 0.999958i \(0.502917\pi\)
\(242\) 0 0
\(243\) − 1.34008e13i − 1.01460i
\(244\) 0 0
\(245\) − 8.19634e12i − 0.593207i
\(246\) 0 0
\(247\) 6.15951e12 0.426297
\(248\) 0 0
\(249\) 7.39245e12 0.489431
\(250\) 0 0
\(251\) 1.29831e13i 0.822567i 0.911507 + 0.411284i \(0.134919\pi\)
−0.911507 + 0.411284i \(0.865081\pi\)
\(252\) 0 0
\(253\) − 9.96692e12i − 0.604502i
\(254\) 0 0
\(255\) −8.40563e12 −0.488201
\(256\) 0 0
\(257\) 2.39612e13 1.33314 0.666571 0.745442i \(-0.267761\pi\)
0.666571 + 0.745442i \(0.267761\pi\)
\(258\) 0 0
\(259\) − 3.05098e12i − 0.162664i
\(260\) 0 0
\(261\) − 1.45925e13i − 0.745774i
\(262\) 0 0
\(263\) 2.42737e13 1.18954 0.594771 0.803895i \(-0.297243\pi\)
0.594771 + 0.803895i \(0.297243\pi\)
\(264\) 0 0
\(265\) 7.70895e12 0.362362
\(266\) 0 0
\(267\) − 6.29822e12i − 0.284057i
\(268\) 0 0
\(269\) − 2.58377e13i − 1.11845i −0.829016 0.559225i \(-0.811099\pi\)
0.829016 0.559225i \(-0.188901\pi\)
\(270\) 0 0
\(271\) −3.76793e12 −0.156593 −0.0782964 0.996930i \(-0.524948\pi\)
−0.0782964 + 0.996930i \(0.524948\pi\)
\(272\) 0 0
\(273\) 2.43776e12 0.0972963
\(274\) 0 0
\(275\) 1.36322e13i 0.522680i
\(276\) 0 0
\(277\) − 1.64189e13i − 0.604931i −0.953160 0.302466i \(-0.902190\pi\)
0.953160 0.302466i \(-0.0978098\pi\)
\(278\) 0 0
\(279\) −6.00526e12 −0.212671
\(280\) 0 0
\(281\) −2.10357e13 −0.716263 −0.358132 0.933671i \(-0.616586\pi\)
−0.358132 + 0.933671i \(0.616586\pi\)
\(282\) 0 0
\(283\) 1.67132e13i 0.547310i 0.961828 + 0.273655i \(0.0882327\pi\)
−0.961828 + 0.273655i \(0.911767\pi\)
\(284\) 0 0
\(285\) − 1.29767e13i − 0.408805i
\(286\) 0 0
\(287\) −5.15917e12 −0.156397
\(288\) 0 0
\(289\) 1.34200e13 0.391575
\(290\) 0 0
\(291\) − 1.89034e13i − 0.531042i
\(292\) 0 0
\(293\) − 2.39269e13i − 0.647312i −0.946175 0.323656i \(-0.895088\pi\)
0.946175 0.323656i \(-0.104912\pi\)
\(294\) 0 0
\(295\) 2.50639e13 0.653171
\(296\) 0 0
\(297\) 3.91759e13 0.983690
\(298\) 0 0
\(299\) − 1.07709e13i − 0.260652i
\(300\) 0 0
\(301\) − 2.86753e11i − 0.00668947i
\(302\) 0 0
\(303\) 2.05992e13 0.463358
\(304\) 0 0
\(305\) 3.35998e13 0.728933
\(306\) 0 0
\(307\) − 1.53111e13i − 0.320439i −0.987081 0.160219i \(-0.948780\pi\)
0.987081 0.160219i \(-0.0512202\pi\)
\(308\) 0 0
\(309\) − 5.68903e13i − 1.14886i
\(310\) 0 0
\(311\) −4.98752e13 −0.972080 −0.486040 0.873936i \(-0.661559\pi\)
−0.486040 + 0.873936i \(0.661559\pi\)
\(312\) 0 0
\(313\) 9.94808e13 1.87174 0.935870 0.352345i \(-0.114616\pi\)
0.935870 + 0.352345i \(0.114616\pi\)
\(314\) 0 0
\(315\) 9.19071e12i 0.166971i
\(316\) 0 0
\(317\) − 8.33692e13i − 1.46278i −0.681958 0.731392i \(-0.738871\pi\)
0.681958 0.731392i \(-0.261129\pi\)
\(318\) 0 0
\(319\) 6.86477e13 1.16353
\(320\) 0 0
\(321\) 2.27408e13 0.372416
\(322\) 0 0
\(323\) 7.36271e13i 1.16526i
\(324\) 0 0
\(325\) 1.47319e13i 0.225372i
\(326\) 0 0
\(327\) −1.85176e13 −0.273888
\(328\) 0 0
\(329\) 4.49970e13 0.643585
\(330\) 0 0
\(331\) − 6.35840e13i − 0.879618i −0.898091 0.439809i \(-0.855046\pi\)
0.898091 0.439809i \(-0.144954\pi\)
\(332\) 0 0
\(333\) − 2.07073e13i − 0.277127i
\(334\) 0 0
\(335\) −7.47772e13 −0.968329
\(336\) 0 0
\(337\) 1.21001e14 1.51644 0.758221 0.651997i \(-0.226069\pi\)
0.758221 + 0.651997i \(0.226069\pi\)
\(338\) 0 0
\(339\) 2.14570e13i 0.260298i
\(340\) 0 0
\(341\) − 2.82506e13i − 0.331802i
\(342\) 0 0
\(343\) −6.15223e13 −0.699705
\(344\) 0 0
\(345\) −2.26918e13 −0.249957
\(346\) 0 0
\(347\) − 1.55662e14i − 1.66100i −0.557020 0.830499i \(-0.688055\pi\)
0.557020 0.830499i \(-0.311945\pi\)
\(348\) 0 0
\(349\) 2.56430e13i 0.265112i 0.991176 + 0.132556i \(0.0423184\pi\)
−0.991176 + 0.132556i \(0.957682\pi\)
\(350\) 0 0
\(351\) 4.23361e13 0.424152
\(352\) 0 0
\(353\) 2.49098e13 0.241885 0.120943 0.992659i \(-0.461408\pi\)
0.120943 + 0.992659i \(0.461408\pi\)
\(354\) 0 0
\(355\) − 4.72929e13i − 0.445184i
\(356\) 0 0
\(357\) 2.91395e13i 0.265954i
\(358\) 0 0
\(359\) −1.57584e14 −1.39474 −0.697370 0.716712i \(-0.745646\pi\)
−0.697370 + 0.716712i \(0.745646\pi\)
\(360\) 0 0
\(361\) 2.82438e12 0.0242457
\(362\) 0 0
\(363\) 1.25577e11i 0.00104574i
\(364\) 0 0
\(365\) − 7.07011e12i − 0.0571236i
\(366\) 0 0
\(367\) −1.77901e14 −1.39481 −0.697406 0.716676i \(-0.745662\pi\)
−0.697406 + 0.716676i \(0.745662\pi\)
\(368\) 0 0
\(369\) −3.50157e13 −0.266452
\(370\) 0 0
\(371\) − 2.67244e13i − 0.197402i
\(372\) 0 0
\(373\) − 5.51617e13i − 0.395585i −0.980244 0.197792i \(-0.936623\pi\)
0.980244 0.197792i \(-0.0633772\pi\)
\(374\) 0 0
\(375\) 9.04683e13 0.629976
\(376\) 0 0
\(377\) 7.41854e13 0.501696
\(378\) 0 0
\(379\) 1.46463e14i 0.962083i 0.876698 + 0.481042i \(0.159741\pi\)
−0.876698 + 0.481042i \(0.840259\pi\)
\(380\) 0 0
\(381\) 6.62047e13i 0.422476i
\(382\) 0 0
\(383\) 2.31450e14 1.43504 0.717519 0.696539i \(-0.245278\pi\)
0.717519 + 0.696539i \(0.245278\pi\)
\(384\) 0 0
\(385\) −4.32360e13 −0.260502
\(386\) 0 0
\(387\) − 1.94622e12i − 0.0113967i
\(388\) 0 0
\(389\) − 1.49872e14i − 0.853093i −0.904466 0.426547i \(-0.859730\pi\)
0.904466 0.426547i \(-0.140270\pi\)
\(390\) 0 0
\(391\) 1.28749e14 0.712480
\(392\) 0 0
\(393\) −1.59145e14 −0.856317
\(394\) 0 0
\(395\) 1.84104e14i 0.963341i
\(396\) 0 0
\(397\) − 2.08111e14i − 1.05912i −0.848271 0.529562i \(-0.822356\pi\)
0.848271 0.529562i \(-0.177644\pi\)
\(398\) 0 0
\(399\) −4.49857e13 −0.222702
\(400\) 0 0
\(401\) −1.33408e14 −0.642521 −0.321261 0.946991i \(-0.604107\pi\)
−0.321261 + 0.946991i \(0.604107\pi\)
\(402\) 0 0
\(403\) − 3.05295e13i − 0.143068i
\(404\) 0 0
\(405\) 8.04286e12i 0.0366782i
\(406\) 0 0
\(407\) 9.74134e13 0.432364
\(408\) 0 0
\(409\) 2.06168e14 0.890722 0.445361 0.895351i \(-0.353075\pi\)
0.445361 + 0.895351i \(0.353075\pi\)
\(410\) 0 0
\(411\) − 7.48941e13i − 0.315005i
\(412\) 0 0
\(413\) − 8.68880e13i − 0.355824i
\(414\) 0 0
\(415\) −1.41689e14 −0.565028
\(416\) 0 0
\(417\) 1.50392e14 0.584085
\(418\) 0 0
\(419\) − 7.34035e13i − 0.277677i −0.990315 0.138838i \(-0.955663\pi\)
0.990315 0.138838i \(-0.0443369\pi\)
\(420\) 0 0
\(421\) 1.71112e14i 0.630563i 0.948998 + 0.315282i \(0.102099\pi\)
−0.948998 + 0.315282i \(0.897901\pi\)
\(422\) 0 0
\(423\) 3.05398e14 1.09646
\(424\) 0 0
\(425\) −1.76096e14 −0.616042
\(426\) 0 0
\(427\) − 1.16479e14i − 0.397096i
\(428\) 0 0
\(429\) 7.78341e13i 0.258616i
\(430\) 0 0
\(431\) −7.17758e13 −0.232463 −0.116231 0.993222i \(-0.537081\pi\)
−0.116231 + 0.993222i \(0.537081\pi\)
\(432\) 0 0
\(433\) 9.98812e13 0.315356 0.157678 0.987491i \(-0.449599\pi\)
0.157678 + 0.987491i \(0.449599\pi\)
\(434\) 0 0
\(435\) − 1.56291e14i − 0.481110i
\(436\) 0 0
\(437\) 1.98764e14i 0.596608i
\(438\) 0 0
\(439\) 2.90312e13 0.0849788 0.0424894 0.999097i \(-0.486471\pi\)
0.0424894 + 0.999097i \(0.486471\pi\)
\(440\) 0 0
\(441\) −1.92848e14 −0.550558
\(442\) 0 0
\(443\) 3.28370e14i 0.914414i 0.889360 + 0.457207i \(0.151150\pi\)
−0.889360 + 0.457207i \(0.848850\pi\)
\(444\) 0 0
\(445\) 1.20716e14i 0.327932i
\(446\) 0 0
\(447\) −2.81089e14 −0.744994
\(448\) 0 0
\(449\) −6.12368e14 −1.58364 −0.791822 0.610752i \(-0.790867\pi\)
−0.791822 + 0.610752i \(0.790867\pi\)
\(450\) 0 0
\(451\) − 1.64725e14i − 0.415708i
\(452\) 0 0
\(453\) − 2.07761e14i − 0.511709i
\(454\) 0 0
\(455\) −4.67237e13 −0.112325
\(456\) 0 0
\(457\) −3.03483e14 −0.712189 −0.356095 0.934450i \(-0.615892\pi\)
−0.356095 + 0.934450i \(0.615892\pi\)
\(458\) 0 0
\(459\) 5.06060e14i 1.15940i
\(460\) 0 0
\(461\) 7.29308e14i 1.63138i 0.578487 + 0.815691i \(0.303643\pi\)
−0.578487 + 0.815691i \(0.696357\pi\)
\(462\) 0 0
\(463\) 1.22188e14 0.266891 0.133445 0.991056i \(-0.457396\pi\)
0.133445 + 0.991056i \(0.457396\pi\)
\(464\) 0 0
\(465\) −6.43186e13 −0.137197
\(466\) 0 0
\(467\) 6.17381e14i 1.28621i 0.765780 + 0.643103i \(0.222353\pi\)
−0.765780 + 0.643103i \(0.777647\pi\)
\(468\) 0 0
\(469\) 2.59228e14i 0.527510i
\(470\) 0 0
\(471\) −3.31409e14 −0.658794
\(472\) 0 0
\(473\) 9.15561e12 0.0177808
\(474\) 0 0
\(475\) − 2.71858e14i − 0.515854i
\(476\) 0 0
\(477\) − 1.81381e14i − 0.336310i
\(478\) 0 0
\(479\) 1.05084e15 1.90410 0.952052 0.305938i \(-0.0989700\pi\)
0.952052 + 0.305938i \(0.0989700\pi\)
\(480\) 0 0
\(481\) 1.05272e14 0.186429
\(482\) 0 0
\(483\) 7.86651e13i 0.136167i
\(484\) 0 0
\(485\) 3.62316e14i 0.613066i
\(486\) 0 0
\(487\) 2.19910e14 0.363777 0.181889 0.983319i \(-0.441779\pi\)
0.181889 + 0.983319i \(0.441779\pi\)
\(488\) 0 0
\(489\) 9.01739e13 0.145842
\(490\) 0 0
\(491\) − 4.83863e14i − 0.765199i −0.923914 0.382599i \(-0.875029\pi\)
0.923914 0.382599i \(-0.124971\pi\)
\(492\) 0 0
\(493\) 8.86768e14i 1.37136i
\(494\) 0 0
\(495\) −2.93446e14 −0.443813
\(496\) 0 0
\(497\) −1.63949e14 −0.242520
\(498\) 0 0
\(499\) 1.08878e14i 0.157538i 0.996893 + 0.0787691i \(0.0250990\pi\)
−0.996893 + 0.0787691i \(0.974901\pi\)
\(500\) 0 0
\(501\) 6.94218e14i 0.982626i
\(502\) 0 0
\(503\) −5.06588e14 −0.701506 −0.350753 0.936468i \(-0.614074\pi\)
−0.350753 + 0.936468i \(0.614074\pi\)
\(504\) 0 0
\(505\) −3.94818e14 −0.534927
\(506\) 0 0
\(507\) − 3.67512e14i − 0.487222i
\(508\) 0 0
\(509\) − 8.57534e13i − 0.111251i −0.998452 0.0556254i \(-0.982285\pi\)
0.998452 0.0556254i \(-0.0177153\pi\)
\(510\) 0 0
\(511\) −2.45097e13 −0.0311188
\(512\) 0 0
\(513\) −7.81259e14 −0.970844
\(514\) 0 0
\(515\) 1.09040e15i 1.32631i
\(516\) 0 0
\(517\) 1.43669e15i 1.71066i
\(518\) 0 0
\(519\) 2.39498e14 0.279178
\(520\) 0 0
\(521\) −9.27575e14 −1.05862 −0.529312 0.848428i \(-0.677550\pi\)
−0.529312 + 0.848428i \(0.677550\pi\)
\(522\) 0 0
\(523\) − 2.18187e13i − 0.0243820i −0.999926 0.0121910i \(-0.996119\pi\)
0.999926 0.0121910i \(-0.00388061\pi\)
\(524\) 0 0
\(525\) − 1.07594e14i − 0.117736i
\(526\) 0 0
\(527\) 3.64931e14 0.391069
\(528\) 0 0
\(529\) −6.05238e14 −0.635214
\(530\) 0 0
\(531\) − 5.89717e14i − 0.606211i
\(532\) 0 0
\(533\) − 1.78013e14i − 0.179247i
\(534\) 0 0
\(535\) −4.35865e14 −0.429939
\(536\) 0 0
\(537\) −4.23709e14 −0.409458
\(538\) 0 0
\(539\) − 9.07218e14i − 0.858961i
\(540\) 0 0
\(541\) 1.69527e15i 1.57273i 0.617765 + 0.786363i \(0.288038\pi\)
−0.617765 + 0.786363i \(0.711962\pi\)
\(542\) 0 0
\(543\) −2.51187e14 −0.228349
\(544\) 0 0
\(545\) 3.54921e14 0.316192
\(546\) 0 0
\(547\) − 7.52145e14i − 0.656706i −0.944555 0.328353i \(-0.893506\pi\)
0.944555 0.328353i \(-0.106494\pi\)
\(548\) 0 0
\(549\) − 7.90555e14i − 0.676526i
\(550\) 0 0
\(551\) −1.36900e15 −1.14834
\(552\) 0 0
\(553\) 6.38228e14 0.524793
\(554\) 0 0
\(555\) − 2.21783e14i − 0.178779i
\(556\) 0 0
\(557\) − 1.87489e14i − 0.148174i −0.997252 0.0740870i \(-0.976396\pi\)
0.997252 0.0740870i \(-0.0236043\pi\)
\(558\) 0 0
\(559\) 9.89417e12 0.00766681
\(560\) 0 0
\(561\) −9.30383e14 −0.706913
\(562\) 0 0
\(563\) − 2.44971e14i − 0.182524i −0.995827 0.0912618i \(-0.970910\pi\)
0.995827 0.0912618i \(-0.0290900\pi\)
\(564\) 0 0
\(565\) − 4.11259e14i − 0.300503i
\(566\) 0 0
\(567\) 2.78819e13 0.0199809
\(568\) 0 0
\(569\) −1.35243e15 −0.950596 −0.475298 0.879825i \(-0.657660\pi\)
−0.475298 + 0.879825i \(0.657660\pi\)
\(570\) 0 0
\(571\) 1.43223e15i 0.987447i 0.869619 + 0.493723i \(0.164364\pi\)
−0.869619 + 0.493723i \(0.835636\pi\)
\(572\) 0 0
\(573\) − 6.96126e14i − 0.470801i
\(574\) 0 0
\(575\) −4.75389e14 −0.315410
\(576\) 0 0
\(577\) −8.77659e14 −0.571293 −0.285647 0.958335i \(-0.592208\pi\)
−0.285647 + 0.958335i \(0.592208\pi\)
\(578\) 0 0
\(579\) − 1.37148e15i − 0.875907i
\(580\) 0 0
\(581\) 4.91187e14i 0.307807i
\(582\) 0 0
\(583\) 8.53271e14 0.524698
\(584\) 0 0
\(585\) −3.17118e14 −0.191365
\(586\) 0 0
\(587\) − 2.43425e15i − 1.44164i −0.693124 0.720818i \(-0.743766\pi\)
0.693124 0.720818i \(-0.256234\pi\)
\(588\) 0 0
\(589\) 5.63383e14i 0.327469i
\(590\) 0 0
\(591\) −7.24775e14 −0.413497
\(592\) 0 0
\(593\) −3.03318e14 −0.169863 −0.0849313 0.996387i \(-0.527067\pi\)
−0.0849313 + 0.996387i \(0.527067\pi\)
\(594\) 0 0
\(595\) − 5.58507e14i − 0.307033i
\(596\) 0 0
\(597\) 1.83555e14i 0.0990619i
\(598\) 0 0
\(599\) 1.70198e15 0.901795 0.450898 0.892576i \(-0.351104\pi\)
0.450898 + 0.892576i \(0.351104\pi\)
\(600\) 0 0
\(601\) −2.33922e15 −1.21692 −0.608458 0.793586i \(-0.708212\pi\)
−0.608458 + 0.793586i \(0.708212\pi\)
\(602\) 0 0
\(603\) 1.75940e15i 0.898710i
\(604\) 0 0
\(605\) − 2.40689e12i − 0.00120726i
\(606\) 0 0
\(607\) −2.49607e15 −1.22947 −0.614737 0.788732i \(-0.710738\pi\)
−0.614737 + 0.788732i \(0.710738\pi\)
\(608\) 0 0
\(609\) −5.41810e14 −0.262091
\(610\) 0 0
\(611\) 1.55258e15i 0.737612i
\(612\) 0 0
\(613\) 2.47301e15i 1.15397i 0.816756 + 0.576983i \(0.195770\pi\)
−0.816756 + 0.576983i \(0.804230\pi\)
\(614\) 0 0
\(615\) −3.75032e14 −0.171892
\(616\) 0 0
\(617\) −2.43368e13 −0.0109571 −0.00547854 0.999985i \(-0.501744\pi\)
−0.00547854 + 0.999985i \(0.501744\pi\)
\(618\) 0 0
\(619\) 4.22545e15i 1.86885i 0.356160 + 0.934425i \(0.384086\pi\)
−0.356160 + 0.934425i \(0.615914\pi\)
\(620\) 0 0
\(621\) 1.36616e15i 0.593606i
\(622\) 0 0
\(623\) 4.18481e14 0.178645
\(624\) 0 0
\(625\) −4.88896e14 −0.205058
\(626\) 0 0
\(627\) − 1.43633e15i − 0.591947i
\(628\) 0 0
\(629\) 1.25835e15i 0.509594i
\(630\) 0 0
\(631\) 4.26326e15 1.69660 0.848302 0.529513i \(-0.177625\pi\)
0.848302 + 0.529513i \(0.177625\pi\)
\(632\) 0 0
\(633\) 1.71188e15 0.669503
\(634\) 0 0
\(635\) − 1.26892e15i − 0.487731i
\(636\) 0 0
\(637\) − 9.80401e14i − 0.370371i
\(638\) 0 0
\(639\) −1.11273e15 −0.413177
\(640\) 0 0
\(641\) 1.00830e15 0.368018 0.184009 0.982925i \(-0.441092\pi\)
0.184009 + 0.982925i \(0.441092\pi\)
\(642\) 0 0
\(643\) − 3.03982e14i − 0.109066i −0.998512 0.0545328i \(-0.982633\pi\)
0.998512 0.0545328i \(-0.0173670\pi\)
\(644\) 0 0
\(645\) − 2.08447e13i − 0.00735221i
\(646\) 0 0
\(647\) −3.43583e15 −1.19140 −0.595700 0.803207i \(-0.703125\pi\)
−0.595700 + 0.803207i \(0.703125\pi\)
\(648\) 0 0
\(649\) 2.77421e15 0.945788
\(650\) 0 0
\(651\) 2.22971e14i 0.0747400i
\(652\) 0 0
\(653\) 1.18539e15i 0.390695i 0.980734 + 0.195347i \(0.0625834\pi\)
−0.980734 + 0.195347i \(0.937417\pi\)
\(654\) 0 0
\(655\) 3.05028e15 0.988583
\(656\) 0 0
\(657\) −1.66350e14 −0.0530167
\(658\) 0 0
\(659\) 2.26510e15i 0.709934i 0.934879 + 0.354967i \(0.115508\pi\)
−0.934879 + 0.354967i \(0.884492\pi\)
\(660\) 0 0
\(661\) − 5.33012e15i − 1.64297i −0.570232 0.821484i \(-0.693147\pi\)
0.570232 0.821484i \(-0.306853\pi\)
\(662\) 0 0
\(663\) −1.00543e15 −0.304810
\(664\) 0 0
\(665\) 8.62227e14 0.257100
\(666\) 0 0
\(667\) 2.39392e15i 0.702130i
\(668\) 0 0
\(669\) − 1.84839e15i − 0.533272i
\(670\) 0 0
\(671\) 3.71902e15 1.05549
\(672\) 0 0
\(673\) 4.74120e15 1.32375 0.661874 0.749615i \(-0.269761\pi\)
0.661874 + 0.749615i \(0.269761\pi\)
\(674\) 0 0
\(675\) − 1.86856e15i − 0.513259i
\(676\) 0 0
\(677\) − 1.41307e15i − 0.381880i −0.981602 0.190940i \(-0.938846\pi\)
0.981602 0.190940i \(-0.0611535\pi\)
\(678\) 0 0
\(679\) 1.25603e15 0.333976
\(680\) 0 0
\(681\) 3.42680e14 0.0896559
\(682\) 0 0
\(683\) − 3.03116e15i − 0.780359i −0.920739 0.390180i \(-0.872413\pi\)
0.920739 0.390180i \(-0.127587\pi\)
\(684\) 0 0
\(685\) 1.43547e15i 0.363660i
\(686\) 0 0
\(687\) −2.97975e15 −0.742879
\(688\) 0 0
\(689\) 9.22102e14 0.226242
\(690\) 0 0
\(691\) 2.74731e15i 0.663405i 0.943384 + 0.331703i \(0.107623\pi\)
−0.943384 + 0.331703i \(0.892377\pi\)
\(692\) 0 0
\(693\) 1.01728e15i 0.241773i
\(694\) 0 0
\(695\) −2.88251e15 −0.674303
\(696\) 0 0
\(697\) 2.12786e15 0.489962
\(698\) 0 0
\(699\) − 4.42597e15i − 1.00319i
\(700\) 0 0
\(701\) − 5.72747e15i − 1.27795i −0.769228 0.638974i \(-0.779359\pi\)
0.769228 0.638974i \(-0.220641\pi\)
\(702\) 0 0
\(703\) −1.94265e15 −0.426718
\(704\) 0 0
\(705\) 3.27093e15 0.707345
\(706\) 0 0
\(707\) 1.36870e15i 0.291409i
\(708\) 0 0
\(709\) 6.98326e14i 0.146388i 0.997318 + 0.0731938i \(0.0233192\pi\)
−0.997318 + 0.0731938i \(0.976681\pi\)
\(710\) 0 0
\(711\) 4.33171e15 0.894081
\(712\) 0 0
\(713\) 9.85170e14 0.200225
\(714\) 0 0
\(715\) − 1.49182e15i − 0.298561i
\(716\) 0 0
\(717\) 1.79917e15i 0.354583i
\(718\) 0 0
\(719\) 9.70979e15 1.88452 0.942260 0.334882i \(-0.108696\pi\)
0.942260 + 0.334882i \(0.108696\pi\)
\(720\) 0 0
\(721\) 3.78004e15 0.722525
\(722\) 0 0
\(723\) 5.82893e13i 0.0109731i
\(724\) 0 0
\(725\) − 3.27427e15i − 0.607093i
\(726\) 0 0
\(727\) −2.46469e15 −0.450114 −0.225057 0.974346i \(-0.572257\pi\)
−0.225057 + 0.974346i \(0.572257\pi\)
\(728\) 0 0
\(729\) −3.08202e15 −0.554413
\(730\) 0 0
\(731\) 1.18269e14i 0.0209568i
\(732\) 0 0
\(733\) − 7.91285e15i − 1.38121i −0.723230 0.690607i \(-0.757343\pi\)
0.723230 0.690607i \(-0.242657\pi\)
\(734\) 0 0
\(735\) −2.06548e15 −0.355173
\(736\) 0 0
\(737\) −8.27677e15 −1.40213
\(738\) 0 0
\(739\) 8.40694e15i 1.40312i 0.712613 + 0.701558i \(0.247512\pi\)
−0.712613 + 0.701558i \(0.752488\pi\)
\(740\) 0 0
\(741\) − 1.55220e15i − 0.255239i
\(742\) 0 0
\(743\) −1.36287e15 −0.220809 −0.110404 0.993887i \(-0.535215\pi\)
−0.110404 + 0.993887i \(0.535215\pi\)
\(744\) 0 0
\(745\) 5.38754e15 0.860065
\(746\) 0 0
\(747\) 3.33373e15i 0.524405i
\(748\) 0 0
\(749\) 1.51100e15i 0.234215i
\(750\) 0 0
\(751\) 6.81722e15 1.04133 0.520664 0.853762i \(-0.325684\pi\)
0.520664 + 0.853762i \(0.325684\pi\)
\(752\) 0 0
\(753\) 3.27173e15 0.492499
\(754\) 0 0
\(755\) 3.98208e15i 0.590747i
\(756\) 0 0
\(757\) − 6.67049e14i − 0.0975282i −0.998810 0.0487641i \(-0.984472\pi\)
0.998810 0.0487641i \(-0.0155283\pi\)
\(758\) 0 0
\(759\) −2.51166e15 −0.361936
\(760\) 0 0
\(761\) 7.74408e15 1.09990 0.549951 0.835197i \(-0.314646\pi\)
0.549951 + 0.835197i \(0.314646\pi\)
\(762\) 0 0
\(763\) − 1.23039e15i − 0.172250i
\(764\) 0 0
\(765\) − 3.79064e15i − 0.523088i
\(766\) 0 0
\(767\) 2.99800e15 0.407809
\(768\) 0 0
\(769\) 2.52411e15 0.338465 0.169232 0.985576i \(-0.445871\pi\)
0.169232 + 0.985576i \(0.445871\pi\)
\(770\) 0 0
\(771\) − 6.03822e15i − 0.798197i
\(772\) 0 0
\(773\) − 1.11453e16i − 1.45246i −0.687453 0.726229i \(-0.741271\pi\)
0.687453 0.726229i \(-0.258729\pi\)
\(774\) 0 0
\(775\) −1.34746e15 −0.173124
\(776\) 0 0
\(777\) −7.68847e14 −0.0973922
\(778\) 0 0
\(779\) 3.28500e15i 0.410279i
\(780\) 0 0
\(781\) − 5.23465e15i − 0.644624i
\(782\) 0 0
\(783\) −9.40952e15 −1.14256
\(784\) 0 0
\(785\) 6.35201e15 0.760551
\(786\) 0 0
\(787\) − 1.32271e16i − 1.56172i −0.624705 0.780861i \(-0.714781\pi\)
0.624705 0.780861i \(-0.285219\pi\)
\(788\) 0 0
\(789\) − 6.11698e15i − 0.712219i
\(790\) 0 0
\(791\) −1.42570e15 −0.163703
\(792\) 0 0
\(793\) 4.01902e15 0.455112
\(794\) 0 0
\(795\) − 1.94266e15i − 0.216958i
\(796\) 0 0
\(797\) − 2.30248e15i − 0.253615i −0.991927 0.126807i \(-0.959527\pi\)
0.991927 0.126807i \(-0.0404730\pi\)
\(798\) 0 0
\(799\) −1.85587e16 −2.01623
\(800\) 0 0
\(801\) 2.84027e15 0.304355
\(802\) 0 0
\(803\) − 7.82560e14i − 0.0827146i
\(804\) 0 0
\(805\) − 1.50775e15i − 0.157199i
\(806\) 0 0
\(807\) −6.51110e15 −0.669653
\(808\) 0 0
\(809\) −5.60472e15 −0.568639 −0.284320 0.958730i \(-0.591768\pi\)
−0.284320 + 0.958730i \(0.591768\pi\)
\(810\) 0 0
\(811\) − 5.08516e15i − 0.508968i −0.967077 0.254484i \(-0.918094\pi\)
0.967077 0.254484i \(-0.0819056\pi\)
\(812\) 0 0
\(813\) 9.49519e14i 0.0937574i
\(814\) 0 0
\(815\) −1.72833e15 −0.168368
\(816\) 0 0
\(817\) −1.82584e14 −0.0175486
\(818\) 0 0
\(819\) 1.09934e15i 0.104249i
\(820\) 0 0
\(821\) 2.79111e14i 0.0261150i 0.999915 + 0.0130575i \(0.00415644\pi\)
−0.999915 + 0.0130575i \(0.995844\pi\)
\(822\) 0 0
\(823\) 1.35265e16 1.24878 0.624391 0.781112i \(-0.285347\pi\)
0.624391 + 0.781112i \(0.285347\pi\)
\(824\) 0 0
\(825\) 3.43531e15 0.312946
\(826\) 0 0
\(827\) 2.72544e14i 0.0244994i 0.999925 + 0.0122497i \(0.00389930\pi\)
−0.999925 + 0.0122497i \(0.996101\pi\)
\(828\) 0 0
\(829\) − 1.80459e16i − 1.60077i −0.599486 0.800385i \(-0.704628\pi\)
0.599486 0.800385i \(-0.295372\pi\)
\(830\) 0 0
\(831\) −4.13757e15 −0.362193
\(832\) 0 0
\(833\) 1.17191e16 1.01239
\(834\) 0 0
\(835\) − 1.33058e16i − 1.13440i
\(836\) 0 0
\(837\) 3.87230e15i 0.325821i
\(838\) 0 0
\(839\) 7.96183e15 0.661184 0.330592 0.943774i \(-0.392752\pi\)
0.330592 + 0.943774i \(0.392752\pi\)
\(840\) 0 0
\(841\) −4.28775e15 −0.351440
\(842\) 0 0
\(843\) 5.30100e15i 0.428851i
\(844\) 0 0
\(845\) 7.04397e15i 0.562478i
\(846\) 0 0
\(847\) −8.34387e12 −0.000657671 0
\(848\) 0 0
\(849\) 4.21172e15 0.327693
\(850\) 0 0
\(851\) 3.39705e15i 0.260909i
\(852\) 0 0
\(853\) − 1.49826e16i − 1.13598i −0.823037 0.567988i \(-0.807722\pi\)
0.823037 0.567988i \(-0.192278\pi\)
\(854\) 0 0
\(855\) 5.85201e15 0.438017
\(856\) 0 0
\(857\) 2.22561e16 1.64458 0.822290 0.569068i \(-0.192696\pi\)
0.822290 + 0.569068i \(0.192696\pi\)
\(858\) 0 0
\(859\) 5.44237e15i 0.397032i 0.980098 + 0.198516i \(0.0636122\pi\)
−0.980098 + 0.198516i \(0.936388\pi\)
\(860\) 0 0
\(861\) 1.30011e15i 0.0936403i
\(862\) 0 0
\(863\) 1.08110e16 0.768787 0.384393 0.923169i \(-0.374411\pi\)
0.384393 + 0.923169i \(0.374411\pi\)
\(864\) 0 0
\(865\) −4.59037e15 −0.322299
\(866\) 0 0
\(867\) − 3.38185e15i − 0.234449i
\(868\) 0 0
\(869\) 2.03777e16i 1.39491i
\(870\) 0 0
\(871\) −8.94444e15 −0.604579
\(872\) 0 0
\(873\) 8.52477e15 0.568989
\(874\) 0 0
\(875\) 6.01111e15i 0.396197i
\(876\) 0 0
\(877\) 2.81024e16i 1.82914i 0.404431 + 0.914568i \(0.367470\pi\)
−0.404431 + 0.914568i \(0.632530\pi\)
\(878\) 0 0
\(879\) −6.02957e15 −0.387568
\(880\) 0 0
\(881\) 4.22209e15 0.268016 0.134008 0.990980i \(-0.457215\pi\)
0.134008 + 0.990980i \(0.457215\pi\)
\(882\) 0 0
\(883\) − 5.16092e14i − 0.0323551i −0.999869 0.0161776i \(-0.994850\pi\)
0.999869 0.0161776i \(-0.00514970\pi\)
\(884\) 0 0
\(885\) − 6.31609e15i − 0.391075i
\(886\) 0 0
\(887\) −5.71906e15 −0.349740 −0.174870 0.984592i \(-0.555950\pi\)
−0.174870 + 0.984592i \(0.555950\pi\)
\(888\) 0 0
\(889\) −4.39894e15 −0.265698
\(890\) 0 0
\(891\) 8.90230e14i 0.0531098i
\(892\) 0 0
\(893\) − 2.86510e16i − 1.68832i
\(894\) 0 0
\(895\) 8.12109e15 0.472702
\(896\) 0 0
\(897\) −2.71427e15 −0.156061
\(898\) 0 0
\(899\) 6.78541e15i 0.385388i
\(900\) 0 0
\(901\) 1.10223e16i 0.618421i
\(902\) 0 0
\(903\) −7.22617e13 −0.00400521
\(904\) 0 0
\(905\) 4.81442e15 0.263619
\(906\) 0 0
\(907\) − 8.43778e13i − 0.00456445i −0.999997 0.00228222i \(-0.999274\pi\)
0.999997 0.00228222i \(-0.000726455\pi\)
\(908\) 0 0
\(909\) 9.28952e15i 0.496468i
\(910\) 0 0
\(911\) −1.10091e16 −0.581298 −0.290649 0.956830i \(-0.593871\pi\)
−0.290649 + 0.956830i \(0.593871\pi\)
\(912\) 0 0
\(913\) −1.56829e16 −0.818158
\(914\) 0 0
\(915\) − 8.46715e15i − 0.436437i
\(916\) 0 0
\(917\) − 1.05743e16i − 0.538544i
\(918\) 0 0
\(919\) 4.86351e15 0.244746 0.122373 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(920\) 0 0
\(921\) −3.85840e15 −0.191857
\(922\) 0 0
\(923\) − 5.65691e15i − 0.277952i
\(924\) 0 0
\(925\) − 4.64630e15i − 0.225594i
\(926\) 0 0
\(927\) 2.56555e16 1.23095
\(928\) 0 0
\(929\) 3.57534e15 0.169524 0.0847620 0.996401i \(-0.472987\pi\)
0.0847620 + 0.996401i \(0.472987\pi\)
\(930\) 0 0
\(931\) 1.80921e16i 0.847744i
\(932\) 0 0
\(933\) 1.25685e16i 0.582017i
\(934\) 0 0
\(935\) 1.78323e16 0.816102
\(936\) 0 0
\(937\) −3.86373e16 −1.74759 −0.873795 0.486295i \(-0.838348\pi\)
−0.873795 + 0.486295i \(0.838348\pi\)
\(938\) 0 0
\(939\) − 2.50692e16i − 1.12067i
\(940\) 0 0
\(941\) 3.48997e16i 1.54198i 0.636846 + 0.770991i \(0.280239\pi\)
−0.636846 + 0.770991i \(0.719761\pi\)
\(942\) 0 0
\(943\) 5.74437e15 0.250858
\(944\) 0 0
\(945\) 5.92634e15 0.255806
\(946\) 0 0
\(947\) 2.85123e16i 1.21649i 0.793751 + 0.608243i \(0.208125\pi\)
−0.793751 + 0.608243i \(0.791875\pi\)
\(948\) 0 0
\(949\) − 8.45688e14i − 0.0356653i
\(950\) 0 0
\(951\) −2.10091e16 −0.875817
\(952\) 0 0
\(953\) −4.00334e16 −1.64973 −0.824863 0.565332i \(-0.808748\pi\)
−0.824863 + 0.565332i \(0.808748\pi\)
\(954\) 0 0
\(955\) 1.33424e16i 0.543520i
\(956\) 0 0
\(957\) − 1.72992e16i − 0.696644i
\(958\) 0 0
\(959\) 4.97630e15 0.198109
\(960\) 0 0
\(961\) −2.26161e16 −0.890100
\(962\) 0 0
\(963\) 1.02553e16i 0.399028i
\(964\) 0 0
\(965\) 2.62867e16i 1.01120i
\(966\) 0 0
\(967\) −1.84953e16 −0.703422 −0.351711 0.936109i \(-0.614400\pi\)
−0.351711 + 0.936109i \(0.614400\pi\)
\(968\) 0 0
\(969\) 1.85540e16 0.697682
\(970\) 0 0
\(971\) − 2.14877e16i − 0.798884i −0.916759 0.399442i \(-0.869204\pi\)
0.916759 0.399442i \(-0.130796\pi\)
\(972\) 0 0
\(973\) 9.99271e15i 0.367335i
\(974\) 0 0
\(975\) 3.71243e15 0.134938
\(976\) 0 0
\(977\) −8.73880e15 −0.314074 −0.157037 0.987593i \(-0.550194\pi\)
−0.157037 + 0.987593i \(0.550194\pi\)
\(978\) 0 0
\(979\) 1.33615e16i 0.474844i
\(980\) 0 0
\(981\) − 8.35079e15i − 0.293459i
\(982\) 0 0
\(983\) −1.18924e16 −0.413263 −0.206631 0.978419i \(-0.566250\pi\)
−0.206631 + 0.978419i \(0.566250\pi\)
\(984\) 0 0
\(985\) 1.38915e16 0.477365
\(986\) 0 0
\(987\) − 1.13392e16i − 0.385336i
\(988\) 0 0
\(989\) 3.19279e14i 0.0107298i
\(990\) 0 0
\(991\) 2.34409e16 0.779056 0.389528 0.921015i \(-0.372638\pi\)
0.389528 + 0.921015i \(0.372638\pi\)
\(992\) 0 0
\(993\) −1.60232e16 −0.526657
\(994\) 0 0
\(995\) − 3.51813e15i − 0.114363i
\(996\) 0 0
\(997\) − 2.14004e16i − 0.688016i −0.938967 0.344008i \(-0.888215\pi\)
0.938967 0.344008i \(-0.111785\pi\)
\(998\) 0 0
\(999\) −1.33524e16 −0.424571
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 256.12.b.e.129.1 2
4.3 odd 2 256.12.b.c.129.2 2
8.3 odd 2 256.12.b.c.129.1 2
8.5 even 2 inner 256.12.b.e.129.2 2
16.3 odd 4 16.12.a.a.1.1 1
16.5 even 4 64.12.a.b.1.1 1
16.11 odd 4 64.12.a.f.1.1 1
16.13 even 4 1.12.a.a.1.1 1
48.29 odd 4 9.12.a.b.1.1 1
48.35 even 4 144.12.a.d.1.1 1
80.13 odd 4 25.12.b.b.24.2 2
80.29 even 4 25.12.a.b.1.1 1
80.77 odd 4 25.12.b.b.24.1 2
112.13 odd 4 49.12.a.a.1.1 1
112.45 odd 12 49.12.c.c.30.1 2
112.61 odd 12 49.12.c.c.18.1 2
112.93 even 12 49.12.c.b.18.1 2
112.109 even 12 49.12.c.b.30.1 2
144.13 even 12 81.12.c.d.55.1 2
144.29 odd 12 81.12.c.b.28.1 2
144.61 even 12 81.12.c.d.28.1 2
144.77 odd 12 81.12.c.b.55.1 2
176.109 odd 4 121.12.a.b.1.1 1
208.77 even 4 169.12.a.a.1.1 1
240.29 odd 4 225.12.a.b.1.1 1
240.77 even 4 225.12.b.d.199.2 2
240.173 even 4 225.12.b.d.199.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.12.a.a.1.1 1 16.13 even 4
9.12.a.b.1.1 1 48.29 odd 4
16.12.a.a.1.1 1 16.3 odd 4
25.12.a.b.1.1 1 80.29 even 4
25.12.b.b.24.1 2 80.77 odd 4
25.12.b.b.24.2 2 80.13 odd 4
49.12.a.a.1.1 1 112.13 odd 4
49.12.c.b.18.1 2 112.93 even 12
49.12.c.b.30.1 2 112.109 even 12
49.12.c.c.18.1 2 112.61 odd 12
49.12.c.c.30.1 2 112.45 odd 12
64.12.a.b.1.1 1 16.5 even 4
64.12.a.f.1.1 1 16.11 odd 4
81.12.c.b.28.1 2 144.29 odd 12
81.12.c.b.55.1 2 144.77 odd 12
81.12.c.d.28.1 2 144.61 even 12
81.12.c.d.55.1 2 144.13 even 12
121.12.a.b.1.1 1 176.109 odd 4
144.12.a.d.1.1 1 48.35 even 4
169.12.a.a.1.1 1 208.77 even 4
225.12.a.b.1.1 1 240.29 odd 4
225.12.b.d.199.1 2 240.173 even 4
225.12.b.d.199.2 2 240.77 even 4
256.12.b.c.129.1 2 8.3 odd 2
256.12.b.c.129.2 2 4.3 odd 2
256.12.b.e.129.1 2 1.1 even 1 trivial
256.12.b.e.129.2 2 8.5 even 2 inner