Properties

Label 256.4.a.n.1.4
Level $256$
Weight $4$
Character 256.1
Self dual yes
Analytic conductor $15.104$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,4,Mod(1,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 256.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(15.1044889615\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.874032\) of defining polynomial
Character \(\chi\) \(=\) 256.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6.32456 q^{3} +17.8885 q^{5} -22.6274 q^{7} +13.0000 q^{9} +O(q^{10})\) \(q+6.32456 q^{3} +17.8885 q^{5} -22.6274 q^{7} +13.0000 q^{9} +44.2719 q^{11} +17.8885 q^{13} +113.137 q^{15} +70.0000 q^{17} +82.2192 q^{19} -143.108 q^{21} -158.392 q^{23} +195.000 q^{25} -88.5438 q^{27} -125.220 q^{29} +280.000 q^{33} -404.772 q^{35} +375.659 q^{37} +113.137 q^{39} +182.000 q^{41} -132.816 q^{43} +232.551 q^{45} -316.784 q^{47} +169.000 q^{49} +442.719 q^{51} -125.220 q^{53} +791.960 q^{55} +520.000 q^{57} -82.2192 q^{59} +232.551 q^{61} -294.156 q^{63} +320.000 q^{65} -221.359 q^{67} -1001.76 q^{69} -113.137 q^{71} -910.000 q^{73} +1233.29 q^{75} -1001.76 q^{77} -678.823 q^{79} -911.000 q^{81} +714.675 q^{83} +1252.20 q^{85} -791.960 q^{87} +546.000 q^{89} -404.772 q^{91} +1470.78 q^{95} -490.000 q^{97} +575.535 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 52 q^{9} + 280 q^{17} + 780 q^{25} + 1120 q^{33} + 728 q^{41} + 676 q^{49} + 2080 q^{57} + 1280 q^{65} - 3640 q^{73} - 3644 q^{81} + 2184 q^{89} - 1960 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 6.32456 1.21716 0.608581 0.793492i \(-0.291739\pi\)
0.608581 + 0.793492i \(0.291739\pi\)
\(4\) 0 0
\(5\) 17.8885 1.60000 0.800000 0.600000i \(-0.204833\pi\)
0.800000 + 0.600000i \(0.204833\pi\)
\(6\) 0 0
\(7\) −22.6274 −1.22177 −0.610883 0.791721i \(-0.709185\pi\)
−0.610883 + 0.791721i \(0.709185\pi\)
\(8\) 0 0
\(9\) 13.0000 0.481481
\(10\) 0 0
\(11\) 44.2719 1.21350 0.606749 0.794894i \(-0.292473\pi\)
0.606749 + 0.794894i \(0.292473\pi\)
\(12\) 0 0
\(13\) 17.8885 0.381645 0.190823 0.981625i \(-0.438884\pi\)
0.190823 + 0.981625i \(0.438884\pi\)
\(14\) 0 0
\(15\) 113.137 1.94746
\(16\) 0 0
\(17\) 70.0000 0.998676 0.499338 0.866407i \(-0.333577\pi\)
0.499338 + 0.866407i \(0.333577\pi\)
\(18\) 0 0
\(19\) 82.2192 0.992757 0.496378 0.868106i \(-0.334663\pi\)
0.496378 + 0.868106i \(0.334663\pi\)
\(20\) 0 0
\(21\) −143.108 −1.48709
\(22\) 0 0
\(23\) −158.392 −1.43596 −0.717978 0.696066i \(-0.754932\pi\)
−0.717978 + 0.696066i \(0.754932\pi\)
\(24\) 0 0
\(25\) 195.000 1.56000
\(26\) 0 0
\(27\) −88.5438 −0.631121
\(28\) 0 0
\(29\) −125.220 −0.801818 −0.400909 0.916118i \(-0.631306\pi\)
−0.400909 + 0.916118i \(0.631306\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 280.000 1.47702
\(34\) 0 0
\(35\) −404.772 −1.95483
\(36\) 0 0
\(37\) 375.659 1.66914 0.834568 0.550905i \(-0.185717\pi\)
0.834568 + 0.550905i \(0.185717\pi\)
\(38\) 0 0
\(39\) 113.137 0.464524
\(40\) 0 0
\(41\) 182.000 0.693259 0.346630 0.938002i \(-0.387326\pi\)
0.346630 + 0.938002i \(0.387326\pi\)
\(42\) 0 0
\(43\) −132.816 −0.471028 −0.235514 0.971871i \(-0.575677\pi\)
−0.235514 + 0.971871i \(0.575677\pi\)
\(44\) 0 0
\(45\) 232.551 0.770370
\(46\) 0 0
\(47\) −316.784 −0.983142 −0.491571 0.870838i \(-0.663577\pi\)
−0.491571 + 0.870838i \(0.663577\pi\)
\(48\) 0 0
\(49\) 169.000 0.492711
\(50\) 0 0
\(51\) 442.719 1.21555
\(52\) 0 0
\(53\) −125.220 −0.324533 −0.162267 0.986747i \(-0.551880\pi\)
−0.162267 + 0.986747i \(0.551880\pi\)
\(54\) 0 0
\(55\) 791.960 1.94160
\(56\) 0 0
\(57\) 520.000 1.20835
\(58\) 0 0
\(59\) −82.2192 −0.181424 −0.0907121 0.995877i \(-0.528914\pi\)
−0.0907121 + 0.995877i \(0.528914\pi\)
\(60\) 0 0
\(61\) 232.551 0.488117 0.244058 0.969761i \(-0.421521\pi\)
0.244058 + 0.969761i \(0.421521\pi\)
\(62\) 0 0
\(63\) −294.156 −0.588258
\(64\) 0 0
\(65\) 320.000 0.610633
\(66\) 0 0
\(67\) −221.359 −0.403632 −0.201816 0.979423i \(-0.564684\pi\)
−0.201816 + 0.979423i \(0.564684\pi\)
\(68\) 0 0
\(69\) −1001.76 −1.74779
\(70\) 0 0
\(71\) −113.137 −0.189111 −0.0945556 0.995520i \(-0.530143\pi\)
−0.0945556 + 0.995520i \(0.530143\pi\)
\(72\) 0 0
\(73\) −910.000 −1.45901 −0.729503 0.683978i \(-0.760249\pi\)
−0.729503 + 0.683978i \(0.760249\pi\)
\(74\) 0 0
\(75\) 1233.29 1.89877
\(76\) 0 0
\(77\) −1001.76 −1.48261
\(78\) 0 0
\(79\) −678.823 −0.966753 −0.483377 0.875413i \(-0.660590\pi\)
−0.483377 + 0.875413i \(0.660590\pi\)
\(80\) 0 0
\(81\) −911.000 −1.24966
\(82\) 0 0
\(83\) 714.675 0.945129 0.472565 0.881296i \(-0.343328\pi\)
0.472565 + 0.881296i \(0.343328\pi\)
\(84\) 0 0
\(85\) 1252.20 1.59788
\(86\) 0 0
\(87\) −791.960 −0.975942
\(88\) 0 0
\(89\) 546.000 0.650291 0.325145 0.945664i \(-0.394587\pi\)
0.325145 + 0.945664i \(0.394587\pi\)
\(90\) 0 0
\(91\) −404.772 −0.466281
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1470.78 1.58841
\(96\) 0 0
\(97\) −490.000 −0.512907 −0.256453 0.966557i \(-0.582554\pi\)
−0.256453 + 0.966557i \(0.582554\pi\)
\(98\) 0 0
\(99\) 575.535 0.584277
\(100\) 0 0
\(101\) 232.551 0.229106 0.114553 0.993417i \(-0.463456\pi\)
0.114553 + 0.993417i \(0.463456\pi\)
\(102\) 0 0
\(103\) 158.392 0.151523 0.0757613 0.997126i \(-0.475861\pi\)
0.0757613 + 0.997126i \(0.475861\pi\)
\(104\) 0 0
\(105\) −2560.00 −2.37934
\(106\) 0 0
\(107\) −1726.60 −1.55997 −0.779986 0.625797i \(-0.784774\pi\)
−0.779986 + 0.625797i \(0.784774\pi\)
\(108\) 0 0
\(109\) 1377.42 1.21039 0.605196 0.796077i \(-0.293095\pi\)
0.605196 + 0.796077i \(0.293095\pi\)
\(110\) 0 0
\(111\) 2375.88 2.03161
\(112\) 0 0
\(113\) −910.000 −0.757572 −0.378786 0.925484i \(-0.623658\pi\)
−0.378786 + 0.925484i \(0.623658\pi\)
\(114\) 0 0
\(115\) −2833.40 −2.29753
\(116\) 0 0
\(117\) 232.551 0.183755
\(118\) 0 0
\(119\) −1583.92 −1.22015
\(120\) 0 0
\(121\) 629.000 0.472577
\(122\) 0 0
\(123\) 1151.07 0.843808
\(124\) 0 0
\(125\) 1252.20 0.896000
\(126\) 0 0
\(127\) −1900.70 −1.32803 −0.664016 0.747718i \(-0.731149\pi\)
−0.664016 + 0.747718i \(0.731149\pi\)
\(128\) 0 0
\(129\) −840.000 −0.573317
\(130\) 0 0
\(131\) −170.763 −0.113890 −0.0569452 0.998377i \(-0.518136\pi\)
−0.0569452 + 0.998377i \(0.518136\pi\)
\(132\) 0 0
\(133\) −1860.41 −1.21292
\(134\) 0 0
\(135\) −1583.92 −1.00979
\(136\) 0 0
\(137\) −1930.00 −1.20358 −0.601792 0.798653i \(-0.705546\pi\)
−0.601792 + 0.798653i \(0.705546\pi\)
\(138\) 0 0
\(139\) −1144.74 −0.698532 −0.349266 0.937024i \(-0.613569\pi\)
−0.349266 + 0.937024i \(0.613569\pi\)
\(140\) 0 0
\(141\) −2003.52 −1.19664
\(142\) 0 0
\(143\) 791.960 0.463126
\(144\) 0 0
\(145\) −2240.00 −1.28291
\(146\) 0 0
\(147\) 1068.85 0.599709
\(148\) 0 0
\(149\) −1627.86 −0.895029 −0.447514 0.894277i \(-0.647691\pi\)
−0.447514 + 0.894277i \(0.647691\pi\)
\(150\) 0 0
\(151\) 2375.88 1.28044 0.640219 0.768192i \(-0.278843\pi\)
0.640219 + 0.768192i \(0.278843\pi\)
\(152\) 0 0
\(153\) 910.000 0.480844
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −3488.27 −1.77321 −0.886605 0.462528i \(-0.846943\pi\)
−0.886605 + 0.462528i \(0.846943\pi\)
\(158\) 0 0
\(159\) −791.960 −0.395009
\(160\) 0 0
\(161\) 3584.00 1.75440
\(162\) 0 0
\(163\) 3497.48 1.68064 0.840318 0.542094i \(-0.182368\pi\)
0.840318 + 0.542094i \(0.182368\pi\)
\(164\) 0 0
\(165\) 5008.79 2.36324
\(166\) 0 0
\(167\) 2059.09 0.954117 0.477059 0.878872i \(-0.341703\pi\)
0.477059 + 0.878872i \(0.341703\pi\)
\(168\) 0 0
\(169\) −1877.00 −0.854347
\(170\) 0 0
\(171\) 1068.85 0.477994
\(172\) 0 0
\(173\) 2021.41 0.888350 0.444175 0.895940i \(-0.353497\pi\)
0.444175 + 0.895940i \(0.353497\pi\)
\(174\) 0 0
\(175\) −4412.35 −1.90595
\(176\) 0 0
\(177\) −520.000 −0.220823
\(178\) 0 0
\(179\) −3586.02 −1.49739 −0.748693 0.662917i \(-0.769318\pi\)
−0.748693 + 0.662917i \(0.769318\pi\)
\(180\) 0 0
\(181\) −2486.51 −1.02111 −0.510554 0.859846i \(-0.670560\pi\)
−0.510554 + 0.859846i \(0.670560\pi\)
\(182\) 0 0
\(183\) 1470.78 0.594117
\(184\) 0 0
\(185\) 6720.00 2.67062
\(186\) 0 0
\(187\) 3099.03 1.21189
\(188\) 0 0
\(189\) 2003.52 0.771082
\(190\) 0 0
\(191\) 2262.74 0.857205 0.428603 0.903493i \(-0.359006\pi\)
0.428603 + 0.903493i \(0.359006\pi\)
\(192\) 0 0
\(193\) 630.000 0.234966 0.117483 0.993075i \(-0.462517\pi\)
0.117483 + 0.993075i \(0.462517\pi\)
\(194\) 0 0
\(195\) 2023.86 0.743238
\(196\) 0 0
\(197\) 1878.30 0.679305 0.339653 0.940551i \(-0.389690\pi\)
0.339653 + 0.940551i \(0.389690\pi\)
\(198\) 0 0
\(199\) 3959.80 1.41057 0.705283 0.708926i \(-0.250820\pi\)
0.705283 + 0.708926i \(0.250820\pi\)
\(200\) 0 0
\(201\) −1400.00 −0.491286
\(202\) 0 0
\(203\) 2833.40 0.979634
\(204\) 0 0
\(205\) 3255.71 1.10921
\(206\) 0 0
\(207\) −2059.09 −0.691386
\(208\) 0 0
\(209\) 3640.00 1.20471
\(210\) 0 0
\(211\) 2789.13 0.910007 0.455004 0.890490i \(-0.349638\pi\)
0.455004 + 0.890490i \(0.349638\pi\)
\(212\) 0 0
\(213\) −715.542 −0.230179
\(214\) 0 0
\(215\) −2375.88 −0.753645
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −5755.35 −1.77585
\(220\) 0 0
\(221\) 1252.20 0.381140
\(222\) 0 0
\(223\) −2534.27 −0.761019 −0.380510 0.924777i \(-0.624251\pi\)
−0.380510 + 0.924777i \(0.624251\pi\)
\(224\) 0 0
\(225\) 2535.00 0.751111
\(226\) 0 0
\(227\) −2219.92 −0.649080 −0.324540 0.945872i \(-0.605210\pi\)
−0.324540 + 0.945872i \(0.605210\pi\)
\(228\) 0 0
\(229\) −4275.36 −1.23373 −0.616864 0.787069i \(-0.711597\pi\)
−0.616864 + 0.787069i \(0.711597\pi\)
\(230\) 0 0
\(231\) −6335.68 −1.80458
\(232\) 0 0
\(233\) 5010.00 1.40865 0.704326 0.709876i \(-0.251249\pi\)
0.704326 + 0.709876i \(0.251249\pi\)
\(234\) 0 0
\(235\) −5666.80 −1.57303
\(236\) 0 0
\(237\) −4293.25 −1.17669
\(238\) 0 0
\(239\) 1583.92 0.428683 0.214341 0.976759i \(-0.431239\pi\)
0.214341 + 0.976759i \(0.431239\pi\)
\(240\) 0 0
\(241\) 1638.00 0.437813 0.218906 0.975746i \(-0.429751\pi\)
0.218906 + 0.975746i \(0.429751\pi\)
\(242\) 0 0
\(243\) −3370.99 −0.889913
\(244\) 0 0
\(245\) 3023.16 0.788338
\(246\) 0 0
\(247\) 1470.78 0.378881
\(248\) 0 0
\(249\) 4520.00 1.15037
\(250\) 0 0
\(251\) 1233.29 0.310137 0.155069 0.987904i \(-0.450440\pi\)
0.155069 + 0.987904i \(0.450440\pi\)
\(252\) 0 0
\(253\) −7012.31 −1.74253
\(254\) 0 0
\(255\) 7919.60 1.94488
\(256\) 0 0
\(257\) 1890.00 0.458735 0.229368 0.973340i \(-0.426334\pi\)
0.229368 + 0.973340i \(0.426334\pi\)
\(258\) 0 0
\(259\) −8500.20 −2.03929
\(260\) 0 0
\(261\) −1627.86 −0.386061
\(262\) 0 0
\(263\) −2647.41 −0.620708 −0.310354 0.950621i \(-0.600448\pi\)
−0.310354 + 0.950621i \(0.600448\pi\)
\(264\) 0 0
\(265\) −2240.00 −0.519253
\(266\) 0 0
\(267\) 3453.21 0.791509
\(268\) 0 0
\(269\) −3488.27 −0.790644 −0.395322 0.918543i \(-0.629367\pi\)
−0.395322 + 0.918543i \(0.629367\pi\)
\(270\) 0 0
\(271\) 7919.60 1.77521 0.887604 0.460608i \(-0.152369\pi\)
0.887604 + 0.460608i \(0.152369\pi\)
\(272\) 0 0
\(273\) −2560.00 −0.567539
\(274\) 0 0
\(275\) 8633.02 1.89306
\(276\) 0 0
\(277\) 4883.57 1.05930 0.529649 0.848217i \(-0.322324\pi\)
0.529649 + 0.848217i \(0.322324\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1118.00 −0.237346 −0.118673 0.992933i \(-0.537864\pi\)
−0.118673 + 0.992933i \(0.537864\pi\)
\(282\) 0 0
\(283\) 2118.73 0.445036 0.222518 0.974929i \(-0.428572\pi\)
0.222518 + 0.974929i \(0.428572\pi\)
\(284\) 0 0
\(285\) 9302.04 1.93335
\(286\) 0 0
\(287\) −4118.19 −0.847000
\(288\) 0 0
\(289\) −13.0000 −0.00264604
\(290\) 0 0
\(291\) −3099.03 −0.624290
\(292\) 0 0
\(293\) −4275.36 −0.852455 −0.426227 0.904616i \(-0.640158\pi\)
−0.426227 + 0.904616i \(0.640158\pi\)
\(294\) 0 0
\(295\) −1470.78 −0.290279
\(296\) 0 0
\(297\) −3920.00 −0.765864
\(298\) 0 0
\(299\) −2833.40 −0.548026
\(300\) 0 0
\(301\) 3005.28 0.575486
\(302\) 0 0
\(303\) 1470.78 0.278859
\(304\) 0 0
\(305\) 4160.00 0.780987
\(306\) 0 0
\(307\) 7975.26 1.48265 0.741323 0.671148i \(-0.234199\pi\)
0.741323 + 0.671148i \(0.234199\pi\)
\(308\) 0 0
\(309\) 1001.76 0.184427
\(310\) 0 0
\(311\) −10295.5 −1.87718 −0.938590 0.345035i \(-0.887867\pi\)
−0.938590 + 0.345035i \(0.887867\pi\)
\(312\) 0 0
\(313\) −2170.00 −0.391871 −0.195936 0.980617i \(-0.562774\pi\)
−0.195936 + 0.980617i \(0.562774\pi\)
\(314\) 0 0
\(315\) −5262.03 −0.941212
\(316\) 0 0
\(317\) 4883.57 0.865264 0.432632 0.901571i \(-0.357585\pi\)
0.432632 + 0.901571i \(0.357585\pi\)
\(318\) 0 0
\(319\) −5543.72 −0.973005
\(320\) 0 0
\(321\) −10920.0 −1.89874
\(322\) 0 0
\(323\) 5755.35 0.991443
\(324\) 0 0
\(325\) 3488.27 0.595367
\(326\) 0 0
\(327\) 8711.56 1.47324
\(328\) 0 0
\(329\) 7168.00 1.20117
\(330\) 0 0
\(331\) 11732.1 1.94819 0.974096 0.226133i \(-0.0726085\pi\)
0.974096 + 0.226133i \(0.0726085\pi\)
\(332\) 0 0
\(333\) 4883.57 0.803658
\(334\) 0 0
\(335\) −3959.80 −0.645812
\(336\) 0 0
\(337\) −10990.0 −1.77645 −0.888225 0.459409i \(-0.848061\pi\)
−0.888225 + 0.459409i \(0.848061\pi\)
\(338\) 0 0
\(339\) −5755.35 −0.922087
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 3937.17 0.619788
\(344\) 0 0
\(345\) −17920.0 −2.79646
\(346\) 0 0
\(347\) 4117.29 0.636967 0.318483 0.947928i \(-0.396826\pi\)
0.318483 + 0.947928i \(0.396826\pi\)
\(348\) 0 0
\(349\) −4275.36 −0.655745 −0.327872 0.944722i \(-0.606332\pi\)
−0.327872 + 0.944722i \(0.606332\pi\)
\(350\) 0 0
\(351\) −1583.92 −0.240864
\(352\) 0 0
\(353\) 8610.00 1.29820 0.649099 0.760704i \(-0.275146\pi\)
0.649099 + 0.760704i \(0.275146\pi\)
\(354\) 0 0
\(355\) −2023.86 −0.302578
\(356\) 0 0
\(357\) −10017.6 −1.48512
\(358\) 0 0
\(359\) 10295.5 1.51358 0.756789 0.653659i \(-0.226767\pi\)
0.756789 + 0.653659i \(0.226767\pi\)
\(360\) 0 0
\(361\) −99.0000 −0.0144336
\(362\) 0 0
\(363\) 3978.15 0.575202
\(364\) 0 0
\(365\) −16278.6 −2.33441
\(366\) 0 0
\(367\) −2851.05 −0.405515 −0.202757 0.979229i \(-0.564990\pi\)
−0.202757 + 0.979229i \(0.564990\pi\)
\(368\) 0 0
\(369\) 2366.00 0.333791
\(370\) 0 0
\(371\) 2833.40 0.396504
\(372\) 0 0
\(373\) 4883.57 0.677914 0.338957 0.940802i \(-0.389926\pi\)
0.338957 + 0.940802i \(0.389926\pi\)
\(374\) 0 0
\(375\) 7919.60 1.09058
\(376\) 0 0
\(377\) −2240.00 −0.306010
\(378\) 0 0
\(379\) −3674.57 −0.498021 −0.249010 0.968501i \(-0.580105\pi\)
−0.249010 + 0.968501i \(0.580105\pi\)
\(380\) 0 0
\(381\) −12021.1 −1.61643
\(382\) 0 0
\(383\) −2534.27 −0.338108 −0.169054 0.985607i \(-0.554071\pi\)
−0.169054 + 0.985607i \(0.554071\pi\)
\(384\) 0 0
\(385\) −17920.0 −2.37218
\(386\) 0 0
\(387\) −1726.60 −0.226791
\(388\) 0 0
\(389\) 11395.0 1.48522 0.742609 0.669726i \(-0.233588\pi\)
0.742609 + 0.669726i \(0.233588\pi\)
\(390\) 0 0
\(391\) −11087.4 −1.43406
\(392\) 0 0
\(393\) −1080.00 −0.138623
\(394\) 0 0
\(395\) −12143.1 −1.54681
\(396\) 0 0
\(397\) 13255.4 1.67574 0.837872 0.545867i \(-0.183800\pi\)
0.837872 + 0.545867i \(0.183800\pi\)
\(398\) 0 0
\(399\) −11766.3 −1.47631
\(400\) 0 0
\(401\) −1722.00 −0.214445 −0.107223 0.994235i \(-0.534196\pi\)
−0.107223 + 0.994235i \(0.534196\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −16296.5 −1.99945
\(406\) 0 0
\(407\) 16631.2 2.02549
\(408\) 0 0
\(409\) −13594.0 −1.64347 −0.821736 0.569868i \(-0.806994\pi\)
−0.821736 + 0.569868i \(0.806994\pi\)
\(410\) 0 0
\(411\) −12206.4 −1.46496
\(412\) 0 0
\(413\) 1860.41 0.221658
\(414\) 0 0
\(415\) 12784.5 1.51221
\(416\) 0 0
\(417\) −7240.00 −0.850226
\(418\) 0 0
\(419\) −4775.04 −0.556744 −0.278372 0.960473i \(-0.589795\pi\)
−0.278372 + 0.960473i \(0.589795\pi\)
\(420\) 0 0
\(421\) 7888.85 0.913252 0.456626 0.889659i \(-0.349058\pi\)
0.456626 + 0.889659i \(0.349058\pi\)
\(422\) 0 0
\(423\) −4118.19 −0.473365
\(424\) 0 0
\(425\) 13650.0 1.55793
\(426\) 0 0
\(427\) −5262.03 −0.596364
\(428\) 0 0
\(429\) 5008.79 0.563699
\(430\) 0 0
\(431\) 1583.92 0.177018 0.0885089 0.996075i \(-0.471790\pi\)
0.0885089 + 0.996075i \(0.471790\pi\)
\(432\) 0 0
\(433\) 14630.0 1.62373 0.811863 0.583849i \(-0.198454\pi\)
0.811863 + 0.583849i \(0.198454\pi\)
\(434\) 0 0
\(435\) −14167.0 −1.56151
\(436\) 0 0
\(437\) −13022.9 −1.42556
\(438\) 0 0
\(439\) −10295.5 −1.11931 −0.559654 0.828726i \(-0.689066\pi\)
−0.559654 + 0.828726i \(0.689066\pi\)
\(440\) 0 0
\(441\) 2197.00 0.237231
\(442\) 0 0
\(443\) 17753.0 1.90400 0.952000 0.306099i \(-0.0990238\pi\)
0.952000 + 0.306099i \(0.0990238\pi\)
\(444\) 0 0
\(445\) 9767.14 1.04047
\(446\) 0 0
\(447\) −10295.5 −1.08939
\(448\) 0 0
\(449\) 3894.00 0.409286 0.204643 0.978837i \(-0.434397\pi\)
0.204643 + 0.978837i \(0.434397\pi\)
\(450\) 0 0
\(451\) 8057.48 0.841268
\(452\) 0 0
\(453\) 15026.4 1.55850
\(454\) 0 0
\(455\) −7240.77 −0.746050
\(456\) 0 0
\(457\) −2730.00 −0.279440 −0.139720 0.990191i \(-0.544620\pi\)
−0.139720 + 0.990191i \(0.544620\pi\)
\(458\) 0 0
\(459\) −6198.06 −0.630285
\(460\) 0 0
\(461\) 10250.1 1.03557 0.517784 0.855512i \(-0.326757\pi\)
0.517784 + 0.855512i \(0.326757\pi\)
\(462\) 0 0
\(463\) 7648.07 0.767680 0.383840 0.923400i \(-0.374601\pi\)
0.383840 + 0.923400i \(0.374601\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −13730.6 −1.36055 −0.680275 0.732957i \(-0.738140\pi\)
−0.680275 + 0.732957i \(0.738140\pi\)
\(468\) 0 0
\(469\) 5008.79 0.493144
\(470\) 0 0
\(471\) −22061.7 −2.15828
\(472\) 0 0
\(473\) −5880.00 −0.571591
\(474\) 0 0
\(475\) 16032.7 1.54870
\(476\) 0 0
\(477\) −1627.86 −0.156257
\(478\) 0 0
\(479\) 12671.4 1.20870 0.604352 0.796718i \(-0.293432\pi\)
0.604352 + 0.796718i \(0.293432\pi\)
\(480\) 0 0
\(481\) 6720.00 0.637018
\(482\) 0 0
\(483\) 22667.2 2.13539
\(484\) 0 0
\(485\) −8765.39 −0.820651
\(486\) 0 0
\(487\) 2059.09 0.191594 0.0957972 0.995401i \(-0.469460\pi\)
0.0957972 + 0.995401i \(0.469460\pi\)
\(488\) 0 0
\(489\) 22120.0 2.04561
\(490\) 0 0
\(491\) 5888.16 0.541200 0.270600 0.962692i \(-0.412778\pi\)
0.270600 + 0.962692i \(0.412778\pi\)
\(492\) 0 0
\(493\) −8765.39 −0.800757
\(494\) 0 0
\(495\) 10295.5 0.934843
\(496\) 0 0
\(497\) 2560.00 0.231050
\(498\) 0 0
\(499\) 10935.2 0.981012 0.490506 0.871438i \(-0.336812\pi\)
0.490506 + 0.871438i \(0.336812\pi\)
\(500\) 0 0
\(501\) 13022.9 1.16131
\(502\) 0 0
\(503\) 14413.7 1.27768 0.638840 0.769339i \(-0.279414\pi\)
0.638840 + 0.769339i \(0.279414\pi\)
\(504\) 0 0
\(505\) 4160.00 0.366569
\(506\) 0 0
\(507\) −11871.2 −1.03988
\(508\) 0 0
\(509\) −12790.3 −1.11379 −0.556896 0.830582i \(-0.688008\pi\)
−0.556896 + 0.830582i \(0.688008\pi\)
\(510\) 0 0
\(511\) 20590.9 1.78256
\(512\) 0 0
\(513\) −7280.00 −0.626549
\(514\) 0 0
\(515\) 2833.40 0.242436
\(516\) 0 0
\(517\) −14024.6 −1.19304
\(518\) 0 0
\(519\) 12784.5 1.08127
\(520\) 0 0
\(521\) 16758.0 1.40918 0.704589 0.709616i \(-0.251132\pi\)
0.704589 + 0.709616i \(0.251132\pi\)
\(522\) 0 0
\(523\) −16197.2 −1.35421 −0.677107 0.735885i \(-0.736766\pi\)
−0.677107 + 0.735885i \(0.736766\pi\)
\(524\) 0 0
\(525\) −27906.1 −2.31985
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 12921.0 1.06197
\(530\) 0 0
\(531\) −1068.85 −0.0873524
\(532\) 0 0
\(533\) 3255.71 0.264579
\(534\) 0 0
\(535\) −30886.4 −2.49596
\(536\) 0 0
\(537\) −22680.0 −1.82256
\(538\) 0 0
\(539\) 7481.95 0.597904
\(540\) 0 0
\(541\) 375.659 0.0298537 0.0149269 0.999889i \(-0.495248\pi\)
0.0149269 + 0.999889i \(0.495248\pi\)
\(542\) 0 0
\(543\) −15726.1 −1.24285
\(544\) 0 0
\(545\) 24640.0 1.93663
\(546\) 0 0
\(547\) −1638.06 −0.128041 −0.0640205 0.997949i \(-0.520392\pi\)
−0.0640205 + 0.997949i \(0.520392\pi\)
\(548\) 0 0
\(549\) 3023.16 0.235019
\(550\) 0 0
\(551\) −10295.5 −0.796011
\(552\) 0 0
\(553\) 15360.0 1.18115
\(554\) 0 0
\(555\) 42501.0 3.25057
\(556\) 0 0
\(557\) 19909.9 1.51456 0.757282 0.653089i \(-0.226527\pi\)
0.757282 + 0.653089i \(0.226527\pi\)
\(558\) 0 0
\(559\) −2375.88 −0.179766
\(560\) 0 0
\(561\) 19600.0 1.47507
\(562\) 0 0
\(563\) 790.569 0.0591803 0.0295902 0.999562i \(-0.490580\pi\)
0.0295902 + 0.999562i \(0.490580\pi\)
\(564\) 0 0
\(565\) −16278.6 −1.21211
\(566\) 0 0
\(567\) 20613.6 1.52679
\(568\) 0 0
\(569\) 20454.0 1.50699 0.753494 0.657455i \(-0.228367\pi\)
0.753494 + 0.657455i \(0.228367\pi\)
\(570\) 0 0
\(571\) −5622.53 −0.412076 −0.206038 0.978544i \(-0.566057\pi\)
−0.206038 + 0.978544i \(0.566057\pi\)
\(572\) 0 0
\(573\) 14310.8 1.04336
\(574\) 0 0
\(575\) −30886.4 −2.24009
\(576\) 0 0
\(577\) −22750.0 −1.64141 −0.820706 0.571351i \(-0.806420\pi\)
−0.820706 + 0.571351i \(0.806420\pi\)
\(578\) 0 0
\(579\) 3984.47 0.285991
\(580\) 0 0
\(581\) −16171.2 −1.15473
\(582\) 0 0
\(583\) −5543.72 −0.393820
\(584\) 0 0
\(585\) 4160.00 0.294008
\(586\) 0 0
\(587\) −15843.0 −1.11399 −0.556994 0.830516i \(-0.688045\pi\)
−0.556994 + 0.830516i \(0.688045\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 11879.4 0.826824
\(592\) 0 0
\(593\) −12110.0 −0.838614 −0.419307 0.907845i \(-0.637727\pi\)
−0.419307 + 0.907845i \(0.637727\pi\)
\(594\) 0 0
\(595\) −28334.0 −1.95224
\(596\) 0 0
\(597\) 25044.0 1.71689
\(598\) 0 0
\(599\) 19120.2 1.30422 0.652111 0.758124i \(-0.273884\pi\)
0.652111 + 0.758124i \(0.273884\pi\)
\(600\) 0 0
\(601\) −4382.00 −0.297413 −0.148707 0.988881i \(-0.547511\pi\)
−0.148707 + 0.988881i \(0.547511\pi\)
\(602\) 0 0
\(603\) −2877.67 −0.194341
\(604\) 0 0
\(605\) 11251.9 0.756123
\(606\) 0 0
\(607\) −8236.38 −0.550749 −0.275374 0.961337i \(-0.588802\pi\)
−0.275374 + 0.961337i \(0.588802\pi\)
\(608\) 0 0
\(609\) 17920.0 1.19237
\(610\) 0 0
\(611\) −5666.80 −0.375212
\(612\) 0 0
\(613\) −2128.74 −0.140259 −0.0701296 0.997538i \(-0.522341\pi\)
−0.0701296 + 0.997538i \(0.522341\pi\)
\(614\) 0 0
\(615\) 20590.9 1.35009
\(616\) 0 0
\(617\) −15470.0 −1.00940 −0.504699 0.863295i \(-0.668397\pi\)
−0.504699 + 0.863295i \(0.668397\pi\)
\(618\) 0 0
\(619\) −19486.0 −1.26528 −0.632639 0.774447i \(-0.718028\pi\)
−0.632639 + 0.774447i \(0.718028\pi\)
\(620\) 0 0
\(621\) 14024.6 0.906262
\(622\) 0 0
\(623\) −12354.6 −0.794503
\(624\) 0 0
\(625\) −1975.00 −0.126400
\(626\) 0 0
\(627\) 23021.4 1.46632
\(628\) 0 0
\(629\) 26296.2 1.66693
\(630\) 0 0
\(631\) 6448.81 0.406851 0.203426 0.979090i \(-0.434792\pi\)
0.203426 + 0.979090i \(0.434792\pi\)
\(632\) 0 0
\(633\) 17640.0 1.10763
\(634\) 0 0
\(635\) −34000.8 −2.12485
\(636\) 0 0
\(637\) 3023.16 0.188041
\(638\) 0 0
\(639\) −1470.78 −0.0910536
\(640\) 0 0
\(641\) 182.000 0.0112146 0.00560731 0.999984i \(-0.498215\pi\)
0.00560731 + 0.999984i \(0.498215\pi\)
\(642\) 0 0
\(643\) −23293.3 −1.42862 −0.714308 0.699832i \(-0.753258\pi\)
−0.714308 + 0.699832i \(0.753258\pi\)
\(644\) 0 0
\(645\) −15026.4 −0.917307
\(646\) 0 0
\(647\) 2059.09 0.125118 0.0625590 0.998041i \(-0.480074\pi\)
0.0625590 + 0.998041i \(0.480074\pi\)
\(648\) 0 0
\(649\) −3640.00 −0.220158
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −14650.7 −0.877989 −0.438995 0.898490i \(-0.644665\pi\)
−0.438995 + 0.898490i \(0.644665\pi\)
\(654\) 0 0
\(655\) −3054.70 −0.182225
\(656\) 0 0
\(657\) −11830.0 −0.702484
\(658\) 0 0
\(659\) 10935.2 0.646393 0.323197 0.946332i \(-0.395242\pi\)
0.323197 + 0.946332i \(0.395242\pi\)
\(660\) 0 0
\(661\) −3774.48 −0.222103 −0.111052 0.993815i \(-0.535422\pi\)
−0.111052 + 0.993815i \(0.535422\pi\)
\(662\) 0 0
\(663\) 7919.60 0.463909
\(664\) 0 0
\(665\) −33280.0 −1.94067
\(666\) 0 0
\(667\) 19833.8 1.15138
\(668\) 0 0
\(669\) −16028.1 −0.926283
\(670\) 0 0
\(671\) 10295.5 0.592328
\(672\) 0 0
\(673\) −2410.00 −0.138037 −0.0690183 0.997615i \(-0.521987\pi\)
−0.0690183 + 0.997615i \(0.521987\pi\)
\(674\) 0 0
\(675\) −17266.0 −0.984548
\(676\) 0 0
\(677\) 27566.2 1.56493 0.782464 0.622695i \(-0.213962\pi\)
0.782464 + 0.622695i \(0.213962\pi\)
\(678\) 0 0
\(679\) 11087.4 0.626652
\(680\) 0 0
\(681\) −14040.0 −0.790035
\(682\) 0 0
\(683\) 2346.41 0.131454 0.0657269 0.997838i \(-0.479063\pi\)
0.0657269 + 0.997838i \(0.479063\pi\)
\(684\) 0 0
\(685\) −34524.9 −1.92573
\(686\) 0 0
\(687\) −27039.8 −1.50165
\(688\) 0 0
\(689\) −2240.00 −0.123857
\(690\) 0 0
\(691\) 10277.4 0.565804 0.282902 0.959149i \(-0.408703\pi\)
0.282902 + 0.959149i \(0.408703\pi\)
\(692\) 0 0
\(693\) −13022.9 −0.713849
\(694\) 0 0
\(695\) −20477.8 −1.11765
\(696\) 0 0
\(697\) 12740.0 0.692341
\(698\) 0 0
\(699\) 31686.0 1.71456
\(700\) 0 0
\(701\) −626.099 −0.0337339 −0.0168669 0.999858i \(-0.505369\pi\)
−0.0168669 + 0.999858i \(0.505369\pi\)
\(702\) 0 0
\(703\) 30886.4 1.65705
\(704\) 0 0
\(705\) −35840.0 −1.91463
\(706\) 0 0
\(707\) −5262.03 −0.279914
\(708\) 0 0
\(709\) −14650.7 −0.776050 −0.388025 0.921649i \(-0.626843\pi\)
−0.388025 + 0.921649i \(0.626843\pi\)
\(710\) 0 0
\(711\) −8824.69 −0.465474
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 14167.0 0.741001
\(716\) 0 0
\(717\) 10017.6 0.521776
\(718\) 0 0
\(719\) −14255.3 −0.739405 −0.369702 0.929150i \(-0.620540\pi\)
−0.369702 + 0.929150i \(0.620540\pi\)
\(720\) 0 0
\(721\) −3584.00 −0.185125
\(722\) 0 0
\(723\) 10359.6 0.532889
\(724\) 0 0
\(725\) −24417.9 −1.25084
\(726\) 0 0
\(727\) 4276.58 0.218170 0.109085 0.994032i \(-0.465208\pi\)
0.109085 + 0.994032i \(0.465208\pi\)
\(728\) 0 0
\(729\) 3277.00 0.166489
\(730\) 0 0
\(731\) −9297.10 −0.470404
\(732\) 0 0
\(733\) −9284.15 −0.467828 −0.233914 0.972257i \(-0.575153\pi\)
−0.233914 + 0.972257i \(0.575153\pi\)
\(734\) 0 0
\(735\) 19120.2 0.959535
\(736\) 0 0
\(737\) −9800.00 −0.489807
\(738\) 0 0
\(739\) 1726.60 0.0859461 0.0429730 0.999076i \(-0.486317\pi\)
0.0429730 + 0.999076i \(0.486317\pi\)
\(740\) 0 0
\(741\) 9302.04 0.461159
\(742\) 0 0
\(743\) −28352.2 −1.39992 −0.699959 0.714183i \(-0.746799\pi\)
−0.699959 + 0.714183i \(0.746799\pi\)
\(744\) 0 0
\(745\) −29120.0 −1.43205
\(746\) 0 0
\(747\) 9290.77 0.455062
\(748\) 0 0
\(749\) 39068.6 1.90592
\(750\) 0 0
\(751\) 20590.9 1.00050 0.500249 0.865881i \(-0.333242\pi\)
0.500249 + 0.865881i \(0.333242\pi\)
\(752\) 0 0
\(753\) 7800.00 0.377487
\(754\) 0 0
\(755\) 42501.0 2.04870
\(756\) 0 0
\(757\) −22664.8 −1.08820 −0.544099 0.839021i \(-0.683128\pi\)
−0.544099 + 0.839021i \(0.683128\pi\)
\(758\) 0 0
\(759\) −44349.7 −2.12094
\(760\) 0 0
\(761\) −7098.00 −0.338111 −0.169055 0.985607i \(-0.554072\pi\)
−0.169055 + 0.985607i \(0.554072\pi\)
\(762\) 0 0
\(763\) −31167.4 −1.47882
\(764\) 0 0
\(765\) 16278.6 0.769350
\(766\) 0 0
\(767\) −1470.78 −0.0692397
\(768\) 0 0
\(769\) 17654.0 0.827854 0.413927 0.910310i \(-0.364157\pi\)
0.413927 + 0.910310i \(0.364157\pi\)
\(770\) 0 0
\(771\) 11953.4 0.558355
\(772\) 0 0
\(773\) 6529.32 0.303808 0.151904 0.988395i \(-0.451460\pi\)
0.151904 + 0.988395i \(0.451460\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −53760.0 −2.48215
\(778\) 0 0
\(779\) 14963.9 0.688238
\(780\) 0 0
\(781\) −5008.79 −0.229486
\(782\) 0 0
\(783\) 11087.4 0.506044
\(784\) 0 0
\(785\) −62400.0 −2.83714
\(786\) 0 0
\(787\) 2384.36 0.107996 0.0539982 0.998541i \(-0.482803\pi\)
0.0539982 + 0.998541i \(0.482803\pi\)
\(788\) 0 0
\(789\) −16743.7 −0.755502
\(790\) 0 0
\(791\) 20590.9 0.925575
\(792\) 0 0
\(793\) 4160.00 0.186287
\(794\) 0 0
\(795\) −14167.0 −0.632015
\(796\) 0 0
\(797\) −29534.0 −1.31261 −0.656303 0.754497i \(-0.727881\pi\)
−0.656303 + 0.754497i \(0.727881\pi\)
\(798\) 0 0
\(799\) −22174.9 −0.981840
\(800\) 0 0
\(801\) 7098.00 0.313103
\(802\) 0 0
\(803\) −40287.4 −1.77050
\(804\) 0 0
\(805\) 64112.5 2.80704
\(806\) 0 0
\(807\) −22061.7 −0.962342
\(808\) 0 0
\(809\) 10934.0 0.475178 0.237589 0.971366i \(-0.423643\pi\)
0.237589 + 0.971366i \(0.423643\pi\)
\(810\) 0 0
\(811\) 17348.3 0.751146 0.375573 0.926793i \(-0.377446\pi\)
0.375573 + 0.926793i \(0.377446\pi\)
\(812\) 0 0
\(813\) 50087.9 2.16071
\(814\) 0 0
\(815\) 62564.8 2.68902
\(816\) 0 0
\(817\) −10920.0 −0.467616
\(818\) 0 0
\(819\) −5262.03 −0.224506
\(820\) 0 0
\(821\) 14901.2 0.633440 0.316720 0.948519i \(-0.397419\pi\)
0.316720 + 0.948519i \(0.397419\pi\)
\(822\) 0 0
\(823\) 17943.5 0.759991 0.379995 0.924988i \(-0.375926\pi\)
0.379995 + 0.924988i \(0.375926\pi\)
\(824\) 0 0
\(825\) 54600.0 2.30416
\(826\) 0 0
\(827\) 25899.1 1.08899 0.544497 0.838763i \(-0.316721\pi\)
0.544497 + 0.838763i \(0.316721\pi\)
\(828\) 0 0
\(829\) 25276.5 1.05897 0.529487 0.848318i \(-0.322384\pi\)
0.529487 + 0.848318i \(0.322384\pi\)
\(830\) 0 0
\(831\) 30886.4 1.28934
\(832\) 0 0
\(833\) 11830.0 0.492059
\(834\) 0 0
\(835\) 36834.2 1.52659
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 10295.5 0.423646 0.211823 0.977308i \(-0.432060\pi\)
0.211823 + 0.977308i \(0.432060\pi\)
\(840\) 0 0
\(841\) −8709.00 −0.357087
\(842\) 0 0
\(843\) −7070.85 −0.288889
\(844\) 0 0
\(845\) −33576.8 −1.36695
\(846\) 0 0
\(847\) −14232.6 −0.577378
\(848\) 0 0
\(849\) 13400.0 0.541681
\(850\) 0 0
\(851\) −59501.4 −2.39681
\(852\) 0 0
\(853\) 21770.4 0.873860 0.436930 0.899495i \(-0.356066\pi\)
0.436930 + 0.899495i \(0.356066\pi\)
\(854\) 0 0
\(855\) 19120.2 0.764791
\(856\) 0 0
\(857\) −24570.0 −0.979341 −0.489670 0.871908i \(-0.662883\pi\)
−0.489670 + 0.871908i \(0.662883\pi\)
\(858\) 0 0
\(859\) 5837.56 0.231869 0.115934 0.993257i \(-0.463014\pi\)
0.115934 + 0.993257i \(0.463014\pi\)
\(860\) 0 0
\(861\) −26045.7 −1.03094
\(862\) 0 0
\(863\) −23713.5 −0.935363 −0.467681 0.883897i \(-0.654911\pi\)
−0.467681 + 0.883897i \(0.654911\pi\)
\(864\) 0 0
\(865\) 36160.0 1.42136
\(866\) 0 0
\(867\) −82.2192 −0.00322066
\(868\) 0 0
\(869\) −30052.8 −1.17315
\(870\) 0 0
\(871\) −3959.80 −0.154044
\(872\) 0 0
\(873\) −6370.00 −0.246955
\(874\) 0 0
\(875\) −28334.0 −1.09470
\(876\) 0 0
\(877\) −43701.7 −1.68267 −0.841335 0.540514i \(-0.818230\pi\)
−0.841335 + 0.540514i \(0.818230\pi\)
\(878\) 0 0
\(879\) −27039.8 −1.03758
\(880\) 0 0
\(881\) −17038.0 −0.651561 −0.325780 0.945446i \(-0.605627\pi\)
−0.325780 + 0.945446i \(0.605627\pi\)
\(882\) 0 0
\(883\) −27492.8 −1.04780 −0.523900 0.851780i \(-0.675524\pi\)
−0.523900 + 0.851780i \(0.675524\pi\)
\(884\) 0 0
\(885\) −9302.04 −0.353316
\(886\) 0 0
\(887\) −43241.0 −1.63686 −0.818428 0.574610i \(-0.805154\pi\)
−0.818428 + 0.574610i \(0.805154\pi\)
\(888\) 0 0
\(889\) 43008.0 1.62254
\(890\) 0 0
\(891\) −40331.7 −1.51646
\(892\) 0 0
\(893\) −26045.7 −0.976021
\(894\) 0 0
\(895\) −64148.7 −2.39582
\(896\) 0 0
\(897\) −17920.0 −0.667036
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −8765.39 −0.324104
\(902\) 0 0
\(903\) 19007.0 0.700459
\(904\) 0 0
\(905\) −44480.0 −1.63377
\(906\) 0 0
\(907\) −40154.6 −1.47002 −0.735012 0.678054i \(-0.762823\pi\)
−0.735012 + 0.678054i \(0.762823\pi\)
\(908\) 0 0
\(909\) 3023.16 0.110310
\(910\) 0 0
\(911\) −11087.4 −0.403231 −0.201615 0.979465i \(-0.564619\pi\)
−0.201615 + 0.979465i \(0.564619\pi\)
\(912\) 0 0
\(913\) 31640.0 1.14691
\(914\) 0 0
\(915\) 26310.2 0.950587
\(916\) 0 0
\(917\) 3863.93 0.139147
\(918\) 0 0
\(919\) 1470.78 0.0527928 0.0263964 0.999652i \(-0.491597\pi\)
0.0263964 + 0.999652i \(0.491597\pi\)
\(920\) 0 0
\(921\) 50440.0 1.80462
\(922\) 0 0
\(923\) −2023.86 −0.0721734
\(924\) 0 0
\(925\) 73253.6 2.60385
\(926\) 0 0
\(927\) 2059.09 0.0729553
\(928\) 0 0
\(929\) 32214.0 1.13768 0.568841 0.822447i \(-0.307392\pi\)
0.568841 + 0.822447i \(0.307392\pi\)
\(930\) 0 0
\(931\) 13895.0 0.489143
\(932\) 0 0
\(933\) −65114.3 −2.28483
\(934\) 0 0
\(935\) 55437.2 1.93903
\(936\) 0 0
\(937\) 13650.0 0.475908 0.237954 0.971276i \(-0.423523\pi\)
0.237954 + 0.971276i \(0.423523\pi\)
\(938\) 0 0
\(939\) −13724.3 −0.476970
\(940\) 0 0
\(941\) 6529.32 0.226195 0.113098 0.993584i \(-0.463923\pi\)
0.113098 + 0.993584i \(0.463923\pi\)
\(942\) 0 0
\(943\) −28827.3 −0.995490
\(944\) 0 0
\(945\) 35840.0 1.23373
\(946\) 0 0
\(947\) −11555.0 −0.396500 −0.198250 0.980151i \(-0.563526\pi\)
−0.198250 + 0.980151i \(0.563526\pi\)
\(948\) 0 0
\(949\) −16278.6 −0.556823
\(950\) 0 0
\(951\) 30886.4 1.05317
\(952\) 0 0
\(953\) 10470.0 0.355883 0.177942 0.984041i \(-0.443056\pi\)
0.177942 + 0.984041i \(0.443056\pi\)
\(954\) 0 0
\(955\) 40477.2 1.37153
\(956\) 0 0
\(957\) −35061.5 −1.18430
\(958\) 0 0
\(959\) 43670.9 1.47050
\(960\) 0 0
\(961\) −29791.0 −1.00000
\(962\) 0 0
\(963\) −22445.8 −0.751098
\(964\) 0 0
\(965\) 11269.8 0.375945
\(966\) 0 0
\(967\) 40706.7 1.35371 0.676856 0.736115i \(-0.263342\pi\)
0.676856 + 0.736115i \(0.263342\pi\)
\(968\) 0 0
\(969\) 36400.0 1.20675
\(970\) 0 0
\(971\) 49502.3 1.63605 0.818025 0.575183i \(-0.195069\pi\)
0.818025 + 0.575183i \(0.195069\pi\)
\(972\) 0 0
\(973\) 25902.6 0.853443
\(974\) 0 0
\(975\) 22061.7 0.724657
\(976\) 0 0
\(977\) −51770.0 −1.69526 −0.847630 0.530588i \(-0.821971\pi\)
−0.847630 + 0.530588i \(0.821971\pi\)
\(978\) 0 0
\(979\) 24172.5 0.789127
\(980\) 0 0
\(981\) 17906.4 0.582781
\(982\) 0 0
\(983\) −22333.3 −0.724639 −0.362320 0.932054i \(-0.618015\pi\)
−0.362320 + 0.932054i \(0.618015\pi\)
\(984\) 0 0
\(985\) 33600.0 1.08689
\(986\) 0 0
\(987\) 45334.4 1.46202
\(988\) 0 0
\(989\) 21036.9 0.676376
\(990\) 0 0
\(991\) −52948.2 −1.69723 −0.848614 0.529012i \(-0.822563\pi\)
−0.848614 + 0.529012i \(0.822563\pi\)
\(992\) 0 0
\(993\) 74200.0 2.37126
\(994\) 0 0
\(995\) 70835.0 2.25691
\(996\) 0 0
\(997\) 17548.7 0.557444 0.278722 0.960372i \(-0.410089\pi\)
0.278722 + 0.960372i \(0.410089\pi\)
\(998\) 0 0
\(999\) −33262.3 −1.05343
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 256.4.a.n.1.4 4
3.2 odd 2 2304.4.a.bz.1.1 4
4.3 odd 2 inner 256.4.a.n.1.2 4
8.3 odd 2 inner 256.4.a.n.1.3 4
8.5 even 2 inner 256.4.a.n.1.1 4
12.11 even 2 2304.4.a.bz.1.2 4
16.3 odd 4 128.4.b.e.65.1 4
16.5 even 4 128.4.b.e.65.2 yes 4
16.11 odd 4 128.4.b.e.65.4 yes 4
16.13 even 4 128.4.b.e.65.3 yes 4
24.5 odd 2 2304.4.a.bz.1.3 4
24.11 even 2 2304.4.a.bz.1.4 4
48.5 odd 4 1152.4.d.j.577.2 4
48.11 even 4 1152.4.d.j.577.1 4
48.29 odd 4 1152.4.d.j.577.4 4
48.35 even 4 1152.4.d.j.577.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.b.e.65.1 4 16.3 odd 4
128.4.b.e.65.2 yes 4 16.5 even 4
128.4.b.e.65.3 yes 4 16.13 even 4
128.4.b.e.65.4 yes 4 16.11 odd 4
256.4.a.n.1.1 4 8.5 even 2 inner
256.4.a.n.1.2 4 4.3 odd 2 inner
256.4.a.n.1.3 4 8.3 odd 2 inner
256.4.a.n.1.4 4 1.1 even 1 trivial
1152.4.d.j.577.1 4 48.11 even 4
1152.4.d.j.577.2 4 48.5 odd 4
1152.4.d.j.577.3 4 48.35 even 4
1152.4.d.j.577.4 4 48.29 odd 4
2304.4.a.bz.1.1 4 3.2 odd 2
2304.4.a.bz.1.2 4 12.11 even 2
2304.4.a.bz.1.3 4 24.5 odd 2
2304.4.a.bz.1.4 4 24.11 even 2