Properties

Label 128.4.b.e.65.1
Level $128$
Weight $4$
Character 128.65
Analytic conductor $7.552$
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] = [128,4,Mod(65,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.65");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(7.55224448073\)
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: \( x^{4} + 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 65.1
Root \(-0.707107 + 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 128.65
Dual form 128.4.b.e.65.4

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) − 6.32456i − 1.21716i −0.793492 0.608581i \(-0.791739\pi\)
0.793492 0.608581i \(-0.208261\pi\)
\(4\) 0 0
\(5\) − 17.8885i − 1.60000i −0.600000 0.800000i \(-0.704833\pi\)
0.600000 0.800000i \(-0.295167\pi\)
\(6\) 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
\(11\) 44.2719i 1.21350i 0.794894 + 0.606749i \(0.207527\pi\)
−0.794894 + 0.606749i \(0.792473\pi\)
\(12\) 0 0
\(13\) 17.8885i 0.381645i 0.981625 + 0.190823i \(0.0611155\pi\)
−0.981625 + 0.190823i \(0.938884\pi\)
\(14\) 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
\(19\) − 82.2192i − 0.992757i −0.868106 0.496378i \(-0.834663\pi\)
0.868106 0.496378i \(-0.165337\pi\)
\(20\) 0 0
\(21\) 143.108i 1.48709i
\(22\) 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
\(27\) − 88.5438i − 0.631121i
\(28\) 0 0
\(29\) − 125.220i − 0.801818i −0.916118 0.400909i \(-0.868694\pi\)
0.916118 0.400909i \(-0.131306\pi\)
\(30\) 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
\(35\) 404.772i 1.95483i
\(36\) 0 0
\(37\) − 375.659i − 1.66914i −0.550905 0.834568i \(-0.685717\pi\)
0.550905 0.834568i \(-0.314283\pi\)
\(38\) 0 0
\(39\) 113.137 0.464524
\(40\) 0 0
\(41\) −182.000 −0.693259 −0.346630 0.938002i \(-0.612674\pi\)
−0.346630 + 0.938002i \(0.612674\pi\)
\(42\) 0 0
\(43\) − 132.816i − 0.471028i −0.971871 0.235514i \(-0.924323\pi\)
0.971871 0.235514i \(-0.0756773\pi\)
\(44\) 0 0
\(45\) 232.551i 0.770370i
\(46\) 0 0
\(47\) 316.784 0.983142 0.491571 0.870838i \(-0.336423\pi\)
0.491571 + 0.870838i \(0.336423\pi\)
\(48\) 0 0
\(49\) 169.000 0.492711
\(50\) 0 0
\(51\) − 442.719i − 1.21555i
\(52\) 0 0
\(53\) 125.220i 0.324533i 0.986747 + 0.162267i \(0.0518805\pi\)
−0.986747 + 0.162267i \(0.948120\pi\)
\(54\) 0 0
\(55\) 791.960 1.94160
\(56\) 0 0
\(57\) −520.000 −1.20835
\(58\) 0 0
\(59\) − 82.2192i − 0.181424i −0.995877 0.0907121i \(-0.971086\pi\)
0.995877 0.0907121i \(-0.0289143\pi\)
\(60\) 0 0
\(61\) 232.551i 0.488117i 0.969761 + 0.244058i \(0.0784788\pi\)
−0.969761 + 0.244058i \(0.921521\pi\)
\(62\) 0 0
\(63\) 294.156 0.588258
\(64\) 0 0
\(65\) 320.000 0.610633
\(66\) 0 0
\(67\) 221.359i 0.403632i 0.979423 + 0.201816i \(0.0646843\pi\)
−0.979423 + 0.201816i \(0.935316\pi\)
\(68\) 0 0
\(69\) 1001.76i 1.74779i
\(70\) 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 \(-0.239751\pi\)
0.729503 + 0.683978i \(0.239751\pi\)
\(74\) 0 0
\(75\) 1233.29i 1.89877i
\(76\) 0 0
\(77\) − 1001.76i − 1.48261i
\(78\) 0 0
\(79\) 678.823 0.966753 0.483377 0.875413i \(-0.339410\pi\)
0.483377 + 0.875413i \(0.339410\pi\)
\(80\) 0 0
\(81\) −911.000 −1.24966
\(82\) 0 0
\(83\) − 714.675i − 0.945129i −0.881296 0.472565i \(-0.843328\pi\)
0.881296 0.472565i \(-0.156672\pi\)
\(84\) 0 0
\(85\) − 1252.20i − 1.59788i
\(86\) 0 0
\(87\) −791.960 −0.975942
\(88\) 0 0
\(89\) −546.000 −0.650291 −0.325145 0.945664i \(-0.605413\pi\)
−0.325145 + 0.945664i \(0.605413\pi\)
\(90\) 0 0
\(91\) − 404.772i − 0.466281i
\(92\) 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
\(99\) − 575.535i − 0.584277i
\(100\) 0 0
\(101\) − 232.551i − 0.229106i −0.993417 0.114553i \(-0.963456\pi\)
0.993417 0.114553i \(-0.0365436\pi\)
\(102\) 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
\(107\) − 1726.60i − 1.55997i −0.625797 0.779986i \(-0.715226\pi\)
0.625797 0.779986i \(-0.284774\pi\)
\(108\) 0 0
\(109\) 1377.42i 1.21039i 0.796077 + 0.605196i \(0.206905\pi\)
−0.796077 + 0.605196i \(0.793095\pi\)
\(110\) 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
\(115\) 2833.40i 2.29753i
\(116\) 0 0
\(117\) − 232.551i − 0.183755i
\(118\) 0 0
\(119\) −1583.92 −1.22015
\(120\) 0 0
\(121\) −629.000 −0.472577
\(122\) 0 0
\(123\) 1151.07i 0.843808i
\(124\) 0 0
\(125\) 1252.20i 0.896000i
\(126\) 0 0
\(127\) 1900.70 1.32803 0.664016 0.747718i \(-0.268851\pi\)
0.664016 + 0.747718i \(0.268851\pi\)
\(128\) 0 0
\(129\) −840.000 −0.573317
\(130\) 0 0
\(131\) 170.763i 0.113890i 0.998377 + 0.0569452i \(0.0181360\pi\)
−0.998377 + 0.0569452i \(0.981864\pi\)
\(132\) 0 0
\(133\) 1860.41i 1.21292i
\(134\) 0 0
\(135\) −1583.92 −1.00979
\(136\) 0 0
\(137\) 1930.00 1.20358 0.601792 0.798653i \(-0.294454\pi\)
0.601792 + 0.798653i \(0.294454\pi\)
\(138\) 0 0
\(139\) − 1144.74i − 0.698532i −0.937024 0.349266i \(-0.886431\pi\)
0.937024 0.349266i \(-0.113569\pi\)
\(140\) 0 0
\(141\) − 2003.52i − 1.19664i
\(142\) 0 0
\(143\) −791.960 −0.463126
\(144\) 0 0
\(145\) −2240.00 −1.28291
\(146\) 0 0
\(147\) − 1068.85i − 0.599709i
\(148\) 0 0
\(149\) 1627.86i 0.895029i 0.894277 + 0.447514i \(0.147691\pi\)
−0.894277 + 0.447514i \(0.852309\pi\)
\(150\) 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
\(157\) − 3488.27i − 1.77321i −0.462528 0.886605i \(-0.653057\pi\)
0.462528 0.886605i \(-0.346943\pi\)
\(158\) 0 0
\(159\) 791.960 0.395009
\(160\) 0 0
\(161\) 3584.00 1.75440
\(162\) 0 0
\(163\) − 3497.48i − 1.68064i −0.542094 0.840318i \(-0.682368\pi\)
0.542094 0.840318i \(-0.317632\pi\)
\(164\) 0 0
\(165\) − 5008.79i − 2.36324i
\(166\) 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
\(171\) 1068.85i 0.477994i
\(172\) 0 0
\(173\) 2021.41i 0.888350i 0.895940 + 0.444175i \(0.146503\pi\)
−0.895940 + 0.444175i \(0.853497\pi\)
\(174\) 0 0
\(175\) 4412.35 1.90595
\(176\) 0 0
\(177\) −520.000 −0.220823
\(178\) 0 0
\(179\) 3586.02i 1.49739i 0.662917 + 0.748693i \(0.269318\pi\)
−0.662917 + 0.748693i \(0.730682\pi\)
\(180\) 0 0
\(181\) 2486.51i 1.02111i 0.859846 + 0.510554i \(0.170560\pi\)
−0.859846 + 0.510554i \(0.829440\pi\)
\(182\) 0 0
\(183\) 1470.78 0.594117
\(184\) 0 0
\(185\) −6720.00 −2.67062
\(186\) 0 0
\(187\) 3099.03i 1.21189i
\(188\) 0 0
\(189\) 2003.52i 0.771082i
\(190\) 0 0
\(191\) −2262.74 −0.857205 −0.428603 0.903493i \(-0.640994\pi\)
−0.428603 + 0.903493i \(0.640994\pi\)
\(192\) 0 0
\(193\) 630.000 0.234966 0.117483 0.993075i \(-0.462517\pi\)
0.117483 + 0.993075i \(0.462517\pi\)
\(194\) 0 0
\(195\) − 2023.86i − 0.743238i
\(196\) 0 0
\(197\) − 1878.30i − 0.679305i −0.940551 0.339653i \(-0.889690\pi\)
0.940551 0.339653i \(-0.110310\pi\)
\(198\) 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
\(203\) 2833.40i 0.979634i
\(204\) 0 0
\(205\) 3255.71i 1.10921i
\(206\) 0 0
\(207\) 2059.09 0.691386
\(208\) 0 0
\(209\) 3640.00 1.20471
\(210\) 0 0
\(211\) − 2789.13i − 0.910007i −0.890490 0.455004i \(-0.849638\pi\)
0.890490 0.455004i \(-0.150362\pi\)
\(212\) 0 0
\(213\) 715.542i 0.230179i
\(214\) 0 0
\(215\) −2375.88 −0.753645
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 5755.35i − 1.77585i
\(220\) 0 0
\(221\) 1252.20i 0.381140i
\(222\) 0 0
\(223\) 2534.27 0.761019 0.380510 0.924777i \(-0.375749\pi\)
0.380510 + 0.924777i \(0.375749\pi\)
\(224\) 0 0
\(225\) 2535.00 0.751111
\(226\) 0 0
\(227\) 2219.92i 0.649080i 0.945872 + 0.324540i \(0.105210\pi\)
−0.945872 + 0.324540i \(0.894790\pi\)
\(228\) 0 0
\(229\) 4275.36i 1.23373i 0.787069 + 0.616864i \(0.211597\pi\)
−0.787069 + 0.616864i \(0.788403\pi\)
\(230\) 0 0
\(231\) −6335.68 −1.80458
\(232\) 0 0
\(233\) −5010.00 −1.40865 −0.704326 0.709876i \(-0.748751\pi\)
−0.704326 + 0.709876i \(0.748751\pi\)
\(234\) 0 0
\(235\) − 5666.80i − 1.57303i
\(236\) 0 0
\(237\) − 4293.25i − 1.17669i
\(238\) 0 0
\(239\) −1583.92 −0.428683 −0.214341 0.976759i \(-0.568761\pi\)
−0.214341 + 0.976759i \(0.568761\pi\)
\(240\) 0 0
\(241\) 1638.00 0.437813 0.218906 0.975746i \(-0.429751\pi\)
0.218906 + 0.975746i \(0.429751\pi\)
\(242\) 0 0
\(243\) 3370.99i 0.889913i
\(244\) 0 0
\(245\) − 3023.16i − 0.788338i
\(246\) 0 0
\(247\) 1470.78 0.378881
\(248\) 0 0
\(249\) −4520.00 −1.15037
\(250\) 0 0
\(251\) 1233.29i 0.310137i 0.987904 + 0.155069i \(0.0495599\pi\)
−0.987904 + 0.155069i \(0.950440\pi\)
\(252\) 0 0
\(253\) − 7012.31i − 1.74253i
\(254\) 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
\(259\) 8500.20i 2.03929i
\(260\) 0 0
\(261\) 1627.86i 0.386061i
\(262\) 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
\(267\) 3453.21i 0.791509i
\(268\) 0 0
\(269\) − 3488.27i − 0.790644i −0.918543 0.395322i \(-0.870633\pi\)
0.918543 0.395322i \(-0.129367\pi\)
\(270\) 0 0
\(271\) −7919.60 −1.77521 −0.887604 0.460608i \(-0.847631\pi\)
−0.887604 + 0.460608i \(0.847631\pi\)
\(272\) 0 0
\(273\) −2560.00 −0.567539
\(274\) 0 0
\(275\) − 8633.02i − 1.89306i
\(276\) 0 0
\(277\) − 4883.57i − 1.05930i −0.848217 0.529649i \(-0.822324\pi\)
0.848217 0.529649i \(-0.177676\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1118.00 0.237346 0.118673 0.992933i \(-0.462136\pi\)
0.118673 + 0.992933i \(0.462136\pi\)
\(282\) 0 0
\(283\) 2118.73i 0.445036i 0.974929 + 0.222518i \(0.0714276\pi\)
−0.974929 + 0.222518i \(0.928572\pi\)
\(284\) 0 0
\(285\) 9302.04i 1.93335i
\(286\) 0 0
\(287\) 4118.19 0.847000
\(288\) 0 0
\(289\) −13.0000 −0.00264604
\(290\) 0 0
\(291\) 3099.03i 0.624290i
\(292\) 0 0
\(293\) 4275.36i 0.852455i 0.904616 + 0.426227i \(0.140158\pi\)
−0.904616 + 0.426227i \(0.859842\pi\)
\(294\) 0 0
\(295\) −1470.78 −0.290279
\(296\) 0 0
\(297\) 3920.00 0.765864
\(298\) 0 0
\(299\) − 2833.40i − 0.548026i
\(300\) 0 0
\(301\) 3005.28i 0.575486i
\(302\) 0 0
\(303\) −1470.78 −0.278859
\(304\) 0 0
\(305\) 4160.00 0.780987
\(306\) 0 0
\(307\) − 7975.26i − 1.48265i −0.671148 0.741323i \(-0.734199\pi\)
0.671148 0.741323i \(-0.265801\pi\)
\(308\) 0 0
\(309\) − 1001.76i − 0.184427i
\(310\) 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 \(-0.437226\pi\)
0.195936 + 0.980617i \(0.437226\pi\)
\(314\) 0 0
\(315\) − 5262.03i − 0.941212i
\(316\) 0 0
\(317\) 4883.57i 0.865264i 0.901571 + 0.432632i \(0.142415\pi\)
−0.901571 + 0.432632i \(0.857585\pi\)
\(318\) 0 0
\(319\) 5543.72 0.973005
\(320\) 0 0
\(321\) −10920.0 −1.89874
\(322\) 0 0
\(323\) − 5755.35i − 0.991443i
\(324\) 0 0
\(325\) − 3488.27i − 0.595367i
\(326\) 0 0
\(327\) 8711.56 1.47324
\(328\) 0 0
\(329\) −7168.00 −1.20117
\(330\) 0 0
\(331\) 11732.1i 1.94819i 0.226133 + 0.974096i \(0.427392\pi\)
−0.226133 + 0.974096i \(0.572608\pi\)
\(332\) 0 0
\(333\) 4883.57i 0.803658i
\(334\) 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
\(339\) 5755.35i 0.922087i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 3937.17 0.619788
\(344\) 0 0
\(345\) 17920.0 2.79646
\(346\) 0 0
\(347\) 4117.29i 0.636967i 0.947928 + 0.318483i \(0.103174\pi\)
−0.947928 + 0.318483i \(0.896826\pi\)
\(348\) 0 0
\(349\) − 4275.36i − 0.655745i −0.944722 0.327872i \(-0.893668\pi\)
0.944722 0.327872i \(-0.106332\pi\)
\(350\) 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
\(355\) 2023.86i 0.302578i
\(356\) 0 0
\(357\) 10017.6i 1.48512i
\(358\) 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
\(363\) 3978.15i 0.575202i
\(364\) 0 0
\(365\) − 16278.6i − 2.33441i
\(366\) 0 0
\(367\) 2851.05 0.405515 0.202757 0.979229i \(-0.435010\pi\)
0.202757 + 0.979229i \(0.435010\pi\)
\(368\) 0 0
\(369\) 2366.00 0.333791
\(370\) 0 0
\(371\) − 2833.40i − 0.396504i
\(372\) 0 0
\(373\) − 4883.57i − 0.677914i −0.940802 0.338957i \(-0.889926\pi\)
0.940802 0.338957i \(-0.110074\pi\)
\(374\) 0 0
\(375\) 7919.60 1.09058
\(376\) 0 0
\(377\) 2240.00 0.306010
\(378\) 0 0
\(379\) − 3674.57i − 0.498021i −0.968501 0.249010i \(-0.919895\pi\)
0.968501 0.249010i \(-0.0801053\pi\)
\(380\) 0 0
\(381\) − 12021.1i − 1.61643i
\(382\) 0 0
\(383\) 2534.27 0.338108 0.169054 0.985607i \(-0.445929\pi\)
0.169054 + 0.985607i \(0.445929\pi\)
\(384\) 0 0
\(385\) −17920.0 −2.37218
\(386\) 0 0
\(387\) 1726.60i 0.226791i
\(388\) 0 0
\(389\) − 11395.0i − 1.48522i −0.669726 0.742609i \(-0.733588\pi\)
0.669726 0.742609i \(-0.266412\pi\)
\(390\) 0 0
\(391\) −11087.4 −1.43406
\(392\) 0 0
\(393\) 1080.00 0.138623
\(394\) 0 0
\(395\) − 12143.1i − 1.54681i
\(396\) 0 0
\(397\) 13255.4i 1.67574i 0.545867 + 0.837872i \(0.316200\pi\)
−0.545867 + 0.837872i \(0.683800\pi\)
\(398\) 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
\(405\) 16296.5i 1.99945i
\(406\) 0 0
\(407\) 16631.2 2.02549
\(408\) 0 0
\(409\) 13594.0 1.64347 0.821736 0.569868i \(-0.193006\pi\)
0.821736 + 0.569868i \(0.193006\pi\)
\(410\) 0 0
\(411\) − 12206.4i − 1.46496i
\(412\) 0 0
\(413\) 1860.41i 0.221658i
\(414\) 0 0
\(415\) −12784.5 −1.51221
\(416\) 0 0
\(417\) −7240.00 −0.850226
\(418\) 0 0
\(419\) 4775.04i 0.556744i 0.960473 + 0.278372i \(0.0897949\pi\)
−0.960473 + 0.278372i \(0.910205\pi\)
\(420\) 0 0
\(421\) − 7888.85i − 0.913252i −0.889659 0.456626i \(-0.849058\pi\)
0.889659 0.456626i \(-0.150942\pi\)
\(422\) 0 0
\(423\) −4118.19 −0.473365
\(424\) 0 0
\(425\) −13650.0 −1.55793
\(426\) 0 0
\(427\) − 5262.03i − 0.596364i
\(428\) 0 0
\(429\) 5008.79i 0.563699i
\(430\) 0 0
\(431\) −1583.92 −0.177018 −0.0885089 0.996075i \(-0.528210\pi\)
−0.0885089 + 0.996075i \(0.528210\pi\)
\(432\) 0 0
\(433\) 14630.0 1.62373 0.811863 0.583849i \(-0.198454\pi\)
0.811863 + 0.583849i \(0.198454\pi\)
\(434\) 0 0
\(435\) 14167.0i 1.56151i
\(436\) 0 0
\(437\) 13022.9i 1.42556i
\(438\) 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
\(443\) 17753.0i 1.90400i 0.306099 + 0.952000i \(0.400976\pi\)
−0.306099 + 0.952000i \(0.599024\pi\)
\(444\) 0 0
\(445\) 9767.14i 1.04047i
\(446\) 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
\(451\) − 8057.48i − 0.841268i
\(452\) 0 0
\(453\) − 15026.4i − 1.55850i
\(454\) 0 0
\(455\) −7240.77 −0.746050
\(456\) 0 0
\(457\) 2730.00 0.279440 0.139720 0.990191i \(-0.455380\pi\)
0.139720 + 0.990191i \(0.455380\pi\)
\(458\) 0 0
\(459\) − 6198.06i − 0.630285i
\(460\) 0 0
\(461\) 10250.1i 1.03557i 0.855512 + 0.517784i \(0.173243\pi\)
−0.855512 + 0.517784i \(0.826757\pi\)
\(462\) 0 0
\(463\) −7648.07 −0.767680 −0.383840 0.923400i \(-0.625399\pi\)
−0.383840 + 0.923400i \(0.625399\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13730.6i 1.36055i 0.732957 + 0.680275i \(0.238140\pi\)
−0.732957 + 0.680275i \(0.761860\pi\)
\(468\) 0 0
\(469\) − 5008.79i − 0.493144i
\(470\) 0 0
\(471\) −22061.7 −2.15828
\(472\) 0 0
\(473\) 5880.00 0.571591
\(474\) 0 0
\(475\) 16032.7i 1.54870i
\(476\) 0 0
\(477\) − 1627.86i − 0.156257i
\(478\) 0 0
\(479\) −12671.4 −1.20870 −0.604352 0.796718i \(-0.706568\pi\)
−0.604352 + 0.796718i \(0.706568\pi\)
\(480\) 0 0
\(481\) 6720.00 0.637018
\(482\) 0 0
\(483\) − 22667.2i − 2.13539i
\(484\) 0 0
\(485\) 8765.39i 0.820651i
\(486\) 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
\(491\) 5888.16i 0.541200i 0.962692 + 0.270600i \(0.0872220\pi\)
−0.962692 + 0.270600i \(0.912778\pi\)
\(492\) 0 0
\(493\) − 8765.39i − 0.800757i
\(494\) 0 0
\(495\) −10295.5 −0.934843
\(496\) 0 0
\(497\) 2560.00 0.231050
\(498\) 0 0
\(499\) − 10935.2i − 0.981012i −0.871438 0.490506i \(-0.836812\pi\)
0.871438 0.490506i \(-0.163188\pi\)
\(500\) 0 0
\(501\) − 13022.9i − 1.16131i
\(502\) 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
\(507\) − 11871.2i − 1.03988i
\(508\) 0 0
\(509\) − 12790.3i − 1.11379i −0.830582 0.556896i \(-0.811992\pi\)
0.830582 0.556896i \(-0.188008\pi\)
\(510\) 0 0
\(511\) −20590.9 −1.78256
\(512\) 0 0
\(513\) −7280.00 −0.626549
\(514\) 0 0
\(515\) − 2833.40i − 0.242436i
\(516\) 0 0
\(517\) 14024.6i 1.19304i
\(518\) 0 0
\(519\) 12784.5 1.08127
\(520\) 0 0
\(521\) −16758.0 −1.40918 −0.704589 0.709616i \(-0.748868\pi\)
−0.704589 + 0.709616i \(0.748868\pi\)
\(522\) 0 0
\(523\) − 16197.2i − 1.35421i −0.735885 0.677107i \(-0.763234\pi\)
0.735885 0.677107i \(-0.236766\pi\)
\(524\) 0 0
\(525\) − 27906.1i − 2.31985i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 12921.0 1.06197
\(530\) 0 0
\(531\) 1068.85i 0.0873524i
\(532\) 0 0
\(533\) − 3255.71i − 0.264579i
\(534\) 0 0
\(535\) −30886.4 −2.49596
\(536\) 0 0
\(537\) 22680.0 1.82256
\(538\) 0 0
\(539\) 7481.95i 0.597904i
\(540\) 0 0
\(541\) 375.659i 0.0298537i 0.999889 + 0.0149269i \(0.00475154\pi\)
−0.999889 + 0.0149269i \(0.995248\pi\)
\(542\) 0 0
\(543\) 15726.1 1.24285
\(544\) 0 0
\(545\) 24640.0 1.93663
\(546\) 0 0
\(547\) 1638.06i 0.128041i 0.997949 + 0.0640205i \(0.0203923\pi\)
−0.997949 + 0.0640205i \(0.979608\pi\)
\(548\) 0 0
\(549\) − 3023.16i − 0.235019i
\(550\) 0 0
\(551\) −10295.5 −0.796011
\(552\) 0 0
\(553\) −15360.0 −1.18115
\(554\) 0 0
\(555\) 42501.0i 3.25057i
\(556\) 0 0
\(557\) 19909.9i 1.51456i 0.653089 + 0.757282i \(0.273473\pi\)
−0.653089 + 0.757282i \(0.726527\pi\)
\(558\) 0 0
\(559\) 2375.88 0.179766
\(560\) 0 0
\(561\) 19600.0 1.47507
\(562\) 0 0
\(563\) − 790.569i − 0.0591803i −0.999562 0.0295902i \(-0.990580\pi\)
0.999562 0.0295902i \(-0.00942022\pi\)
\(564\) 0 0
\(565\) 16278.6i 1.21211i
\(566\) 0 0
\(567\) 20613.6 1.52679
\(568\) 0 0
\(569\) −20454.0 −1.50699 −0.753494 0.657455i \(-0.771633\pi\)
−0.753494 + 0.657455i \(0.771633\pi\)
\(570\) 0 0
\(571\) − 5622.53i − 0.412076i −0.978544 0.206038i \(-0.933943\pi\)
0.978544 0.206038i \(-0.0660571\pi\)
\(572\) 0 0
\(573\) 14310.8i 1.04336i
\(574\) 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
\(579\) − 3984.47i − 0.285991i
\(580\) 0 0
\(581\) 16171.2i 1.15473i
\(582\) 0 0
\(583\) −5543.72 −0.393820
\(584\) 0 0
\(585\) −4160.00 −0.294008
\(586\) 0 0
\(587\) − 15843.0i − 1.11399i −0.830516 0.556994i \(-0.811955\pi\)
0.830516 0.556994i \(-0.188045\pi\)
\(588\) 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
\(595\) 28334.0i 1.95224i
\(596\) 0 0
\(597\) − 25044.0i − 1.71689i
\(598\) 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 \(-0.452489\pi\)
0.148707 + 0.988881i \(0.452489\pi\)
\(602\) 0 0
\(603\) − 2877.67i − 0.194341i
\(604\) 0 0
\(605\) 11251.9i 0.756123i
\(606\) 0 0
\(607\) 8236.38 0.550749 0.275374 0.961337i \(-0.411198\pi\)
0.275374 + 0.961337i \(0.411198\pi\)
\(608\) 0 0
\(609\) 17920.0 1.19237
\(610\) 0 0
\(611\) 5666.80i 0.375212i
\(612\) 0 0
\(613\) 2128.74i 0.140259i 0.997538 + 0.0701296i \(0.0223413\pi\)
−0.997538 + 0.0701296i \(0.977659\pi\)
\(614\) 0 0
\(615\) 20590.9 1.35009
\(616\) 0 0
\(617\) 15470.0 1.00940 0.504699 0.863295i \(-0.331603\pi\)
0.504699 + 0.863295i \(0.331603\pi\)
\(618\) 0 0
\(619\) − 19486.0i − 1.26528i −0.774447 0.632639i \(-0.781972\pi\)
0.774447 0.632639i \(-0.218028\pi\)
\(620\) 0 0
\(621\) 14024.6i 0.906262i
\(622\) 0 0
\(623\) 12354.6 0.794503
\(624\) 0 0
\(625\) −1975.00 −0.126400
\(626\) 0 0
\(627\) − 23021.4i − 1.46632i
\(628\) 0 0
\(629\) − 26296.2i − 1.66693i
\(630\) 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
\(635\) − 34000.8i − 2.12485i
\(636\) 0 0
\(637\) 3023.16i 0.188041i
\(638\) 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
\(643\) 23293.3i 1.42862i 0.699832 + 0.714308i \(0.253258\pi\)
−0.699832 + 0.714308i \(0.746742\pi\)
\(644\) 0 0
\(645\) 15026.4i 0.917307i
\(646\) 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
\(653\) − 14650.7i − 0.877989i −0.898490 0.438995i \(-0.855335\pi\)
0.898490 0.438995i \(-0.144665\pi\)
\(654\) 0 0
\(655\) 3054.70 0.182225
\(656\) 0 0
\(657\) −11830.0 −0.702484
\(658\) 0 0
\(659\) − 10935.2i − 0.646393i −0.946332 0.323197i \(-0.895242\pi\)
0.946332 0.323197i \(-0.104758\pi\)
\(660\) 0 0
\(661\) 3774.48i 0.222103i 0.993815 + 0.111052i \(0.0354219\pi\)
−0.993815 + 0.111052i \(0.964578\pi\)
\(662\) 0 0
\(663\) 7919.60 0.463909
\(664\) 0 0
\(665\) 33280.0 1.94067
\(666\) 0 0
\(667\) 19833.8i 1.15138i
\(668\) 0 0
\(669\) − 16028.1i − 0.926283i
\(670\) 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
\(675\) 17266.0i 0.984548i
\(676\) 0 0
\(677\) − 27566.2i − 1.56493i −0.622695 0.782464i \(-0.713962\pi\)
0.622695 0.782464i \(-0.286038\pi\)
\(678\) 0 0
\(679\) 11087.4 0.626652
\(680\) 0 0
\(681\) 14040.0 0.790035
\(682\) 0 0
\(683\) 2346.41i 0.131454i 0.997838 + 0.0657269i \(0.0209366\pi\)
−0.997838 + 0.0657269i \(0.979063\pi\)
\(684\) 0 0
\(685\) − 34524.9i − 1.92573i
\(686\) 0 0
\(687\) 27039.8 1.50165
\(688\) 0 0
\(689\) −2240.00 −0.123857
\(690\) 0 0
\(691\) − 10277.4i − 0.565804i −0.959149 0.282902i \(-0.908703\pi\)
0.959149 0.282902i \(-0.0912972\pi\)
\(692\) 0 0
\(693\) 13022.9i 0.713849i
\(694\) 0 0
\(695\) −20477.8 −1.11765
\(696\) 0 0
\(697\) −12740.0 −0.692341
\(698\) 0 0
\(699\) 31686.0i 1.71456i
\(700\) 0 0
\(701\) − 626.099i − 0.0337339i −0.999858 0.0168669i \(-0.994631\pi\)
0.999858 0.0168669i \(-0.00536916\pi\)
\(702\) 0 0
\(703\) −30886.4 −1.65705
\(704\) 0 0
\(705\) −35840.0 −1.91463
\(706\) 0 0
\(707\) 5262.03i 0.279914i
\(708\) 0 0
\(709\) 14650.7i 0.776050i 0.921649 + 0.388025i \(0.126843\pi\)
−0.921649 + 0.388025i \(0.873157\pi\)
\(710\) 0 0
\(711\) −8824.69 −0.465474
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 14167.0i 0.741001i
\(716\) 0 0
\(717\) 10017.6i 0.521776i
\(718\) 0 0
\(719\) 14255.3 0.739405 0.369702 0.929150i \(-0.379460\pi\)
0.369702 + 0.929150i \(0.379460\pi\)
\(720\) 0 0
\(721\) −3584.00 −0.185125
\(722\) 0 0
\(723\) − 10359.6i − 0.532889i
\(724\) 0 0
\(725\) 24417.9i 1.25084i
\(726\) 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
\(731\) − 9297.10i − 0.470404i
\(732\) 0 0
\(733\) − 9284.15i − 0.467828i −0.972257 0.233914i \(-0.924847\pi\)
0.972257 0.233914i \(-0.0751534\pi\)
\(734\) 0 0
\(735\) −19120.2 −0.959535
\(736\) 0 0
\(737\) −9800.00 −0.489807
\(738\) 0 0
\(739\) − 1726.60i − 0.0859461i −0.999076 0.0429730i \(-0.986317\pi\)
0.999076 0.0429730i \(-0.0136830\pi\)
\(740\) 0 0
\(741\) − 9302.04i − 0.461159i
\(742\) 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
\(747\) 9290.77i 0.455062i
\(748\) 0 0
\(749\) 39068.6i 1.90592i
\(750\) 0 0
\(751\) −20590.9 −1.00050 −0.500249 0.865881i \(-0.666758\pi\)
−0.500249 + 0.865881i \(0.666758\pi\)
\(752\) 0 0
\(753\) 7800.00 0.377487
\(754\) 0 0
\(755\) − 42501.0i − 2.04870i
\(756\) 0 0
\(757\) 22664.8i 1.08820i 0.839021 + 0.544099i \(0.183128\pi\)
−0.839021 + 0.544099i \(0.816872\pi\)
\(758\) 0 0
\(759\) −44349.7 −2.12094
\(760\) 0 0
\(761\) 7098.00 0.338111 0.169055 0.985607i \(-0.445928\pi\)
0.169055 + 0.985607i \(0.445928\pi\)
\(762\) 0 0
\(763\) − 31167.4i − 1.47882i
\(764\) 0 0
\(765\) 16278.6i 0.769350i
\(766\) 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
\(771\) − 11953.4i − 0.558355i
\(772\) 0 0
\(773\) − 6529.32i − 0.303808i −0.988395 0.151904i \(-0.951460\pi\)
0.988395 0.151904i \(-0.0485404\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 53760.0 2.48215
\(778\) 0 0
\(779\) 14963.9i 0.688238i
\(780\) 0 0
\(781\) − 5008.79i − 0.229486i
\(782\) 0 0
\(783\) −11087.4 −0.506044
\(784\) 0 0
\(785\) −62400.0 −2.83714
\(786\) 0 0
\(787\) − 2384.36i − 0.107996i −0.998541 0.0539982i \(-0.982803\pi\)
0.998541 0.0539982i \(-0.0171965\pi\)
\(788\) 0 0
\(789\) 16743.7i 0.755502i
\(790\) 0 0
\(791\) 20590.9 0.925575
\(792\) 0 0
\(793\) −4160.00 −0.186287
\(794\) 0 0
\(795\) − 14167.0i − 0.632015i
\(796\) 0 0
\(797\) − 29534.0i − 1.31261i −0.754497 0.656303i \(-0.772119\pi\)
0.754497 0.656303i \(-0.227881\pi\)
\(798\) 0 0
\(799\) 22174.9 0.981840
\(800\) 0 0
\(801\) 7098.00 0.313103
\(802\) 0 0
\(803\) 40287.4i 1.77050i
\(804\) 0 0
\(805\) − 64112.5i − 2.80704i
\(806\) 0 0
\(807\) −22061.7 −0.962342
\(808\) 0 0
\(809\) −10934.0 −0.475178 −0.237589 0.971366i \(-0.576357\pi\)
−0.237589 + 0.971366i \(0.576357\pi\)
\(810\) 0 0
\(811\) 17348.3i 0.751146i 0.926793 + 0.375573i \(0.122554\pi\)
−0.926793 + 0.375573i \(0.877446\pi\)
\(812\) 0 0
\(813\) 50087.9i 2.16071i
\(814\) 0 0
\(815\) −62564.8 −2.68902
\(816\) 0 0
\(817\) −10920.0 −0.467616
\(818\) 0 0
\(819\) 5262.03i 0.224506i
\(820\) 0 0
\(821\) − 14901.2i − 0.633440i −0.948519 0.316720i \(-0.897419\pi\)
0.948519 0.316720i \(-0.102581\pi\)
\(822\) 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
\(827\) 25899.1i 1.08899i 0.838763 + 0.544497i \(0.183279\pi\)
−0.838763 + 0.544497i \(0.816721\pi\)
\(828\) 0 0
\(829\) 25276.5i 1.05897i 0.848318 + 0.529487i \(0.177616\pi\)
−0.848318 + 0.529487i \(0.822384\pi\)
\(830\) 0 0
\(831\) −30886.4 −1.28934
\(832\) 0 0
\(833\) 11830.0 0.492059
\(834\) 0 0
\(835\) − 36834.2i − 1.52659i
\(836\) 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
\(843\) − 7070.85i − 0.288889i
\(844\) 0 0
\(845\) − 33576.8i − 1.36695i
\(846\) 0 0
\(847\) 14232.6 0.577378
\(848\) 0 0
\(849\) 13400.0 0.541681
\(850\) 0 0
\(851\) 59501.4i 2.39681i
\(852\) 0 0
\(853\) − 21770.4i − 0.873860i −0.899495 0.436930i \(-0.856066\pi\)
0.899495 0.436930i \(-0.143934\pi\)
\(854\) 0 0
\(855\) 19120.2 0.764791
\(856\) 0 0
\(857\) 24570.0 0.979341 0.489670 0.871908i \(-0.337117\pi\)
0.489670 + 0.871908i \(0.337117\pi\)
\(858\) 0 0
\(859\) 5837.56i 0.231869i 0.993257 + 0.115934i \(0.0369862\pi\)
−0.993257 + 0.115934i \(0.963014\pi\)
\(860\) 0 0
\(861\) − 26045.7i − 1.03094i
\(862\) 0 0
\(863\) 23713.5 0.935363 0.467681 0.883897i \(-0.345089\pi\)
0.467681 + 0.883897i \(0.345089\pi\)
\(864\) 0 0
\(865\) 36160.0 1.42136
\(866\) 0 0
\(867\) 82.2192i 0.00322066i
\(868\) 0 0
\(869\) 30052.8i 1.17315i
\(870\) 0 0
\(871\) −3959.80 −0.154044
\(872\) 0 0
\(873\) 6370.00 0.246955
\(874\) 0 0
\(875\) − 28334.0i − 1.09470i
\(876\) 0 0
\(877\) − 43701.7i − 1.68267i −0.540514 0.841335i \(-0.681770\pi\)
0.540514 0.841335i \(-0.318230\pi\)
\(878\) 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
\(883\) 27492.8i 1.04780i 0.851780 + 0.523900i \(0.175524\pi\)
−0.851780 + 0.523900i \(0.824476\pi\)
\(884\) 0 0
\(885\) 9302.04i 0.353316i
\(886\) 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
\(891\) − 40331.7i − 1.51646i
\(892\) 0 0
\(893\) − 26045.7i − 0.976021i
\(894\) 0 0
\(895\) 64148.7 2.39582
\(896\) 0 0
\(897\) −17920.0 −0.667036
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 8765.39i 0.324104i
\(902\) 0 0
\(903\) 19007.0 0.700459
\(904\) 0 0
\(905\) 44480.0 1.63377
\(906\) 0 0
\(907\) − 40154.6i − 1.47002i −0.678054 0.735012i \(-0.737177\pi\)
0.678054 0.735012i \(-0.262823\pi\)
\(908\) 0 0
\(909\) 3023.16i 0.110310i
\(910\) 0 0
\(911\) 11087.4 0.403231 0.201615 0.979465i \(-0.435381\pi\)
0.201615 + 0.979465i \(0.435381\pi\)
\(912\) 0 0
\(913\) 31640.0 1.14691
\(914\) 0 0
\(915\) − 26310.2i − 0.950587i
\(916\) 0 0
\(917\) − 3863.93i − 0.139147i
\(918\) 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
\(923\) − 2023.86i − 0.0721734i
\(924\) 0 0
\(925\) 73253.6i 2.60385i
\(926\) 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
\(931\) − 13895.0i − 0.489143i
\(932\) 0 0
\(933\) 65114.3i 2.28483i
\(934\) 0 0
\(935\) 55437.2 1.93903
\(936\) 0 0
\(937\) −13650.0 −0.475908 −0.237954 0.971276i \(-0.576477\pi\)
−0.237954 + 0.971276i \(0.576477\pi\)
\(938\) 0 0
\(939\) − 13724.3i − 0.476970i
\(940\) 0 0
\(941\) 6529.32i 0.226195i 0.993584 + 0.113098i \(0.0360773\pi\)
−0.993584 + 0.113098i \(0.963923\pi\)
\(942\) 0 0
\(943\) 28827.3 0.995490
\(944\) 0 0
\(945\) 35840.0 1.23373
\(946\) 0 0
\(947\) 11555.0i 0.396500i 0.980151 + 0.198250i \(0.0635258\pi\)
−0.980151 + 0.198250i \(0.936474\pi\)
\(948\) 0 0
\(949\) 16278.6i 0.556823i
\(950\) 0 0
\(951\) 30886.4 1.05317
\(952\) 0 0
\(953\) −10470.0 −0.355883 −0.177942 0.984041i \(-0.556944\pi\)
−0.177942 + 0.984041i \(0.556944\pi\)
\(954\) 0 0
\(955\) 40477.2i 1.37153i
\(956\) 0 0
\(957\) − 35061.5i − 1.18430i
\(958\) 0 0
\(959\) −43670.9 −1.47050
\(960\) 0 0
\(961\) −29791.0 −1.00000
\(962\) 0 0
\(963\) 22445.8i 0.751098i
\(964\) 0 0
\(965\) − 11269.8i − 0.375945i
\(966\) 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
\(971\) 49502.3i 1.63605i 0.575183 + 0.818025i \(0.304931\pi\)
−0.575183 + 0.818025i \(0.695069\pi\)
\(972\) 0 0
\(973\) 25902.6i 0.853443i
\(974\) 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
\(979\) − 24172.5i − 0.789127i
\(980\) 0 0
\(981\) − 17906.4i − 0.582781i
\(982\) 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
\(987\) 45334.4i 1.46202i
\(988\) 0 0
\(989\) 21036.9i 0.676376i
\(990\) 0 0
\(991\) 52948.2 1.69723 0.848614 0.529012i \(-0.177437\pi\)
0.848614 + 0.529012i \(0.177437\pi\)
\(992\) 0 0
\(993\) 74200.0 2.37126
\(994\) 0 0
\(995\) − 70835.0i − 2.25691i
\(996\) 0 0
\(997\) − 17548.7i − 0.557444i −0.960372 0.278722i \(-0.910089\pi\)
0.960372 0.278722i \(-0.0899108\pi\)
\(998\) 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 128.4.b.e.65.1 4
3.2 odd 2 1152.4.d.j.577.3 4
4.3 odd 2 inner 128.4.b.e.65.3 yes 4
8.3 odd 2 inner 128.4.b.e.65.2 yes 4
8.5 even 2 inner 128.4.b.e.65.4 yes 4
12.11 even 2 1152.4.d.j.577.4 4
16.3 odd 4 256.4.a.n.1.1 4
16.5 even 4 256.4.a.n.1.2 4
16.11 odd 4 256.4.a.n.1.4 4
16.13 even 4 256.4.a.n.1.3 4
24.5 odd 2 1152.4.d.j.577.1 4
24.11 even 2 1152.4.d.j.577.2 4
48.5 odd 4 2304.4.a.bz.1.2 4
48.11 even 4 2304.4.a.bz.1.1 4
48.29 odd 4 2304.4.a.bz.1.4 4
48.35 even 4 2304.4.a.bz.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.b.e.65.1 4 1.1 even 1 trivial
128.4.b.e.65.2 yes 4 8.3 odd 2 inner
128.4.b.e.65.3 yes 4 4.3 odd 2 inner
128.4.b.e.65.4 yes 4 8.5 even 2 inner
256.4.a.n.1.1 4 16.3 odd 4
256.4.a.n.1.2 4 16.5 even 4
256.4.a.n.1.3 4 16.13 even 4
256.4.a.n.1.4 4 16.11 odd 4
1152.4.d.j.577.1 4 24.5 odd 2
1152.4.d.j.577.2 4 24.11 even 2
1152.4.d.j.577.3 4 3.2 odd 2
1152.4.d.j.577.4 4 12.11 even 2
2304.4.a.bz.1.1 4 48.11 even 4
2304.4.a.bz.1.2 4 48.5 odd 4
2304.4.a.bz.1.3 4 48.35 even 4
2304.4.a.bz.1.4 4 48.29 odd 4