Properties

Label 1152.4.d.k.577.2
Level $1152$
Weight $4$
Character 1152.577
Analytic conductor $67.970$
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] = [1152,4,Mod(577,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.577");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1152.d (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: \(67.9702003266\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 3x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 577.2
Root \(1.32288 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1152.577
Dual form 1152.4.d.k.577.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.00000i q^{5} +21.1660 q^{7} +O(q^{10})\) \(q-6.00000i q^{5} +21.1660 q^{7} -42.3320i q^{11} +20.0000i q^{13} -8.00000 q^{17} +84.6640i q^{19} +169.328 q^{23} +89.0000 q^{25} -46.0000i q^{29} -21.1660 q^{31} -126.996i q^{35} +164.000i q^{37} +312.000 q^{41} -423.320i q^{43} -169.328 q^{47} +105.000 q^{49} -266.000i q^{53} -253.992 q^{55} +253.992i q^{59} -132.000i q^{61} +120.000 q^{65} +507.984i q^{67} -677.312 q^{71} -246.000 q^{73} -896.000i q^{77} +232.826 q^{79} +973.636i q^{83} +48.0000i q^{85} +1392.00 q^{89} +423.320i q^{91} +507.984 q^{95} -302.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{17} + 356 q^{25} + 1248 q^{41} + 420 q^{49} + 480 q^{65} - 984 q^{73} + 5568 q^{89} - 1208 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 6.00000i − 0.536656i −0.963328 0.268328i \(-0.913529\pi\)
0.963328 0.268328i \(-0.0864711\pi\)
\(6\) 0 0
\(7\) 21.1660 1.14286 0.571429 0.820652i \(-0.306389\pi\)
0.571429 + 0.820652i \(0.306389\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 42.3320i − 1.16033i −0.814500 0.580163i \(-0.802989\pi\)
0.814500 0.580163i \(-0.197011\pi\)
\(12\) 0 0
\(13\) 20.0000i 0.426692i 0.976977 + 0.213346i \(0.0684362\pi\)
−0.976977 + 0.213346i \(0.931564\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −8.00000 −0.114134 −0.0570672 0.998370i \(-0.518175\pi\)
−0.0570672 + 0.998370i \(0.518175\pi\)
\(18\) 0 0
\(19\) 84.6640i 1.02228i 0.859498 + 0.511139i \(0.170776\pi\)
−0.859498 + 0.511139i \(0.829224\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 169.328 1.53510 0.767551 0.640988i \(-0.221475\pi\)
0.767551 + 0.640988i \(0.221475\pi\)
\(24\) 0 0
\(25\) 89.0000 0.712000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 46.0000i − 0.294551i −0.989095 0.147276i \(-0.952950\pi\)
0.989095 0.147276i \(-0.0470504\pi\)
\(30\) 0 0
\(31\) −21.1660 −0.122630 −0.0613150 0.998118i \(-0.519529\pi\)
−0.0613150 + 0.998118i \(0.519529\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 126.996i − 0.613322i
\(36\) 0 0
\(37\) 164.000i 0.728687i 0.931265 + 0.364344i \(0.118707\pi\)
−0.931265 + 0.364344i \(0.881293\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 312.000 1.18844 0.594222 0.804301i \(-0.297460\pi\)
0.594222 + 0.804301i \(0.297460\pi\)
\(42\) 0 0
\(43\) − 423.320i − 1.50130i −0.660702 0.750648i \(-0.729741\pi\)
0.660702 0.750648i \(-0.270259\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −169.328 −0.525511 −0.262756 0.964862i \(-0.584631\pi\)
−0.262756 + 0.964862i \(0.584631\pi\)
\(48\) 0 0
\(49\) 105.000 0.306122
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 266.000i − 0.689395i −0.938714 0.344697i \(-0.887982\pi\)
0.938714 0.344697i \(-0.112018\pi\)
\(54\) 0 0
\(55\) −253.992 −0.622696
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 253.992i 0.560457i 0.959933 + 0.280228i \(0.0904102\pi\)
−0.959933 + 0.280228i \(0.909590\pi\)
\(60\) 0 0
\(61\) − 132.000i − 0.277063i −0.990358 0.138532i \(-0.955762\pi\)
0.990358 0.138532i \(-0.0442383\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 120.000 0.228987
\(66\) 0 0
\(67\) 507.984i 0.926271i 0.886287 + 0.463135i \(0.153276\pi\)
−0.886287 + 0.463135i \(0.846724\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −677.312 −1.13214 −0.566072 0.824356i \(-0.691537\pi\)
−0.566072 + 0.824356i \(0.691537\pi\)
\(72\) 0 0
\(73\) −246.000 −0.394413 −0.197206 0.980362i \(-0.563187\pi\)
−0.197206 + 0.980362i \(0.563187\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 896.000i − 1.32609i
\(78\) 0 0
\(79\) 232.826 0.331582 0.165791 0.986161i \(-0.446982\pi\)
0.165791 + 0.986161i \(0.446982\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 973.636i 1.28760i 0.765195 + 0.643798i \(0.222642\pi\)
−0.765195 + 0.643798i \(0.777358\pi\)
\(84\) 0 0
\(85\) 48.0000i 0.0612510i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1392.00 1.65788 0.828942 0.559334i \(-0.188943\pi\)
0.828942 + 0.559334i \(0.188943\pi\)
\(90\) 0 0
\(91\) 423.320i 0.487649i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 507.984 0.548611
\(96\) 0 0
\(97\) −302.000 −0.316118 −0.158059 0.987430i \(-0.550524\pi\)
−0.158059 + 0.987430i \(0.550524\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 1466.00i − 1.44428i −0.691746 0.722141i \(-0.743158\pi\)
0.691746 0.722141i \(-0.256842\pi\)
\(102\) 0 0
\(103\) 1248.79 1.19463 0.597317 0.802005i \(-0.296233\pi\)
0.597317 + 0.802005i \(0.296233\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 423.320i − 0.382466i −0.981545 0.191233i \(-0.938751\pi\)
0.981545 0.191233i \(-0.0612487\pi\)
\(108\) 0 0
\(109\) − 1564.00i − 1.37435i −0.726492 0.687174i \(-0.758851\pi\)
0.726492 0.687174i \(-0.241149\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1552.00 1.29203 0.646017 0.763323i \(-0.276433\pi\)
0.646017 + 0.763323i \(0.276433\pi\)
\(114\) 0 0
\(115\) − 1015.97i − 0.823822i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −169.328 −0.130439
\(120\) 0 0
\(121\) −461.000 −0.346356
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 1284.00i − 0.918756i
\(126\) 0 0
\(127\) −994.802 −0.695074 −0.347537 0.937666i \(-0.612982\pi\)
−0.347537 + 0.937666i \(0.612982\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 2539.92i − 1.69400i −0.531593 0.847000i \(-0.678406\pi\)
0.531593 0.847000i \(-0.321594\pi\)
\(132\) 0 0
\(133\) 1792.00i 1.16832i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1992.00 1.24225 0.621124 0.783712i \(-0.286676\pi\)
0.621124 + 0.783712i \(0.286676\pi\)
\(138\) 0 0
\(139\) 1185.30i 0.723277i 0.932318 + 0.361639i \(0.117783\pi\)
−0.932318 + 0.361639i \(0.882217\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 846.640 0.495102
\(144\) 0 0
\(145\) −276.000 −0.158073
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 1974.00i − 1.08534i −0.839944 0.542672i \(-0.817413\pi\)
0.839944 0.542672i \(-0.182587\pi\)
\(150\) 0 0
\(151\) −3111.40 −1.67684 −0.838419 0.545027i \(-0.816519\pi\)
−0.838419 + 0.545027i \(0.816519\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 126.996i 0.0658101i
\(156\) 0 0
\(157\) − 1796.00i − 0.912971i −0.889731 0.456485i \(-0.849108\pi\)
0.889731 0.456485i \(-0.150892\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3584.00 1.75440
\(162\) 0 0
\(163\) − 2624.59i − 1.26119i −0.776114 0.630593i \(-0.782812\pi\)
0.776114 0.630593i \(-0.217188\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3555.89 1.64768 0.823841 0.566820i \(-0.191827\pi\)
0.823841 + 0.566820i \(0.191827\pi\)
\(168\) 0 0
\(169\) 1797.00 0.817934
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 4290.00i − 1.88533i −0.333736 0.942667i \(-0.608309\pi\)
0.333736 0.942667i \(-0.391691\pi\)
\(174\) 0 0
\(175\) 1883.77 0.813714
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1608.62i 0.671696i 0.941916 + 0.335848i \(0.109023\pi\)
−0.941916 + 0.335848i \(0.890977\pi\)
\(180\) 0 0
\(181\) 3196.00i 1.31247i 0.754557 + 0.656234i \(0.227852\pi\)
−0.754557 + 0.656234i \(0.772148\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 984.000 0.391055
\(186\) 0 0
\(187\) 338.656i 0.132433i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3217.23 −1.21880 −0.609400 0.792863i \(-0.708590\pi\)
−0.609400 + 0.792863i \(0.708590\pi\)
\(192\) 0 0
\(193\) 462.000 0.172308 0.0861541 0.996282i \(-0.472542\pi\)
0.0861541 + 0.996282i \(0.472542\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 3834.00i − 1.38661i −0.720647 0.693303i \(-0.756155\pi\)
0.720647 0.693303i \(-0.243845\pi\)
\(198\) 0 0
\(199\) 3788.72 1.34962 0.674811 0.737990i \(-0.264225\pi\)
0.674811 + 0.737990i \(0.264225\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 973.636i − 0.336630i
\(204\) 0 0
\(205\) − 1872.00i − 0.637786i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3584.00 1.18617
\(210\) 0 0
\(211\) 2878.58i 0.939192i 0.882882 + 0.469596i \(0.155600\pi\)
−0.882882 + 0.469596i \(0.844400\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2539.92 −0.805680
\(216\) 0 0
\(217\) −448.000 −0.140148
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 160.000i − 0.0487003i
\(222\) 0 0
\(223\) −910.138 −0.273307 −0.136653 0.990619i \(-0.543635\pi\)
−0.136653 + 0.990619i \(0.543635\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3259.57i 0.953062i 0.879158 + 0.476531i \(0.158106\pi\)
−0.879158 + 0.476531i \(0.841894\pi\)
\(228\) 0 0
\(229\) 5260.00i 1.51786i 0.651171 + 0.758931i \(0.274278\pi\)
−0.651171 + 0.758931i \(0.725722\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1936.00 0.544342 0.272171 0.962249i \(-0.412258\pi\)
0.272171 + 0.962249i \(0.412258\pi\)
\(234\) 0 0
\(235\) 1015.97i 0.282019i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1523.95 −0.412453 −0.206227 0.978504i \(-0.566118\pi\)
−0.206227 + 0.978504i \(0.566118\pi\)
\(240\) 0 0
\(241\) 3202.00 0.855846 0.427923 0.903815i \(-0.359245\pi\)
0.427923 + 0.903815i \(0.359245\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 630.000i − 0.164283i
\(246\) 0 0
\(247\) −1693.28 −0.436198
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2666.92i 0.670655i 0.942102 + 0.335327i \(0.108847\pi\)
−0.942102 + 0.335327i \(0.891153\pi\)
\(252\) 0 0
\(253\) − 7168.00i − 1.78122i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3104.00 0.753394 0.376697 0.926337i \(-0.377060\pi\)
0.376697 + 0.926337i \(0.377060\pi\)
\(258\) 0 0
\(259\) 3471.23i 0.832786i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4741.19 −1.11161 −0.555806 0.831312i \(-0.687590\pi\)
−0.555806 + 0.831312i \(0.687590\pi\)
\(264\) 0 0
\(265\) −1596.00 −0.369968
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 4350.00i − 0.985963i −0.870040 0.492982i \(-0.835907\pi\)
0.870040 0.492982i \(-0.164093\pi\)
\(270\) 0 0
\(271\) −1079.47 −0.241967 −0.120983 0.992655i \(-0.538605\pi\)
−0.120983 + 0.992655i \(0.538605\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 3767.55i − 0.826152i
\(276\) 0 0
\(277\) 7532.00i 1.63377i 0.576801 + 0.816885i \(0.304301\pi\)
−0.576801 + 0.816885i \(0.695699\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1824.00 0.387227 0.193613 0.981078i \(-0.437979\pi\)
0.193613 + 0.981078i \(0.437979\pi\)
\(282\) 0 0
\(283\) − 8805.06i − 1.84949i −0.380584 0.924746i \(-0.624277\pi\)
0.380584 0.924746i \(-0.375723\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6603.80 1.35822
\(288\) 0 0
\(289\) −4849.00 −0.986973
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 858.000i − 0.171075i −0.996335 0.0855374i \(-0.972739\pi\)
0.996335 0.0855374i \(-0.0272607\pi\)
\(294\) 0 0
\(295\) 1523.95 0.300773
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3386.56i 0.655016i
\(300\) 0 0
\(301\) − 8960.00i − 1.71577i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −792.000 −0.148688
\(306\) 0 0
\(307\) − 1523.95i − 0.283311i −0.989916 0.141656i \(-0.954757\pi\)
0.989916 0.141656i \(-0.0452426\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4402.53 0.802716 0.401358 0.915921i \(-0.368538\pi\)
0.401358 + 0.915921i \(0.368538\pi\)
\(312\) 0 0
\(313\) −118.000 −0.0213091 −0.0106546 0.999943i \(-0.503392\pi\)
−0.0106546 + 0.999943i \(0.503392\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4594.00i 0.813958i 0.913437 + 0.406979i \(0.133418\pi\)
−0.913437 + 0.406979i \(0.866582\pi\)
\(318\) 0 0
\(319\) −1947.27 −0.341775
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 677.312i − 0.116677i
\(324\) 0 0
\(325\) 1780.00i 0.303805i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3584.00 −0.600585
\(330\) 0 0
\(331\) − 3725.22i − 0.618600i −0.950965 0.309300i \(-0.899905\pi\)
0.950965 0.309300i \(-0.100095\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3047.91 0.497089
\(336\) 0 0
\(337\) 1166.00 0.188475 0.0942375 0.995550i \(-0.469959\pi\)
0.0942375 + 0.995550i \(0.469959\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 896.000i 0.142291i
\(342\) 0 0
\(343\) −5037.51 −0.793003
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 11302.6i − 1.74858i −0.485402 0.874291i \(-0.661327\pi\)
0.485402 0.874291i \(-0.338673\pi\)
\(348\) 0 0
\(349\) − 6388.00i − 0.979776i −0.871785 0.489888i \(-0.837038\pi\)
0.871785 0.489888i \(-0.162962\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8432.00 −1.27136 −0.635680 0.771953i \(-0.719280\pi\)
−0.635680 + 0.771953i \(0.719280\pi\)
\(354\) 0 0
\(355\) 4063.87i 0.607572i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2031.94 −0.298723 −0.149361 0.988783i \(-0.547722\pi\)
−0.149361 + 0.988783i \(0.547722\pi\)
\(360\) 0 0
\(361\) −309.000 −0.0450503
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1476.00i 0.211664i
\(366\) 0 0
\(367\) 5397.33 0.767680 0.383840 0.923400i \(-0.374601\pi\)
0.383840 + 0.923400i \(0.374601\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 5630.16i − 0.787879i
\(372\) 0 0
\(373\) 8276.00i 1.14883i 0.818563 + 0.574417i \(0.194771\pi\)
−0.818563 + 0.574417i \(0.805229\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 920.000 0.125683
\(378\) 0 0
\(379\) 13800.2i 1.87037i 0.354158 + 0.935186i \(0.384767\pi\)
−0.354158 + 0.935186i \(0.615233\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1862.61 −0.248498 −0.124249 0.992251i \(-0.539652\pi\)
−0.124249 + 0.992251i \(0.539652\pi\)
\(384\) 0 0
\(385\) −5376.00 −0.711653
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6170.00i 0.804194i 0.915597 + 0.402097i \(0.131719\pi\)
−0.915597 + 0.402097i \(0.868281\pi\)
\(390\) 0 0
\(391\) −1354.62 −0.175208
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 1396.96i − 0.177946i
\(396\) 0 0
\(397\) − 8644.00i − 1.09277i −0.837534 0.546385i \(-0.816003\pi\)
0.837534 0.546385i \(-0.183997\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8152.00 −1.01519 −0.507595 0.861596i \(-0.669465\pi\)
−0.507595 + 0.861596i \(0.669465\pi\)
\(402\) 0 0
\(403\) − 423.320i − 0.0523253i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6942.45 0.845515
\(408\) 0 0
\(409\) −874.000 −0.105664 −0.0528319 0.998603i \(-0.516825\pi\)
−0.0528319 + 0.998603i \(0.516825\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5376.00i 0.640522i
\(414\) 0 0
\(415\) 5841.82 0.690997
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14265.9i 1.66333i 0.555279 + 0.831664i \(0.312611\pi\)
−0.555279 + 0.831664i \(0.687389\pi\)
\(420\) 0 0
\(421\) 11564.0i 1.33871i 0.742945 + 0.669353i \(0.233428\pi\)
−0.742945 + 0.669353i \(0.766572\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −712.000 −0.0812637
\(426\) 0 0
\(427\) − 2793.91i − 0.316644i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7619.76 −0.851580 −0.425790 0.904822i \(-0.640004\pi\)
−0.425790 + 0.904822i \(0.640004\pi\)
\(432\) 0 0
\(433\) 7550.00 0.837944 0.418972 0.907999i \(-0.362391\pi\)
0.418972 + 0.907999i \(0.362391\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14336.0i 1.56930i
\(438\) 0 0
\(439\) 17335.0 1.88463 0.942315 0.334727i \(-0.108644\pi\)
0.942315 + 0.334727i \(0.108644\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9016.72i 0.967037i 0.875334 + 0.483518i \(0.160641\pi\)
−0.875334 + 0.483518i \(0.839359\pi\)
\(444\) 0 0
\(445\) − 8352.00i − 0.889714i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6568.00 −0.690341 −0.345170 0.938540i \(-0.612179\pi\)
−0.345170 + 0.938540i \(0.612179\pi\)
\(450\) 0 0
\(451\) − 13207.6i − 1.37898i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2539.92 0.261700
\(456\) 0 0
\(457\) 9478.00 0.970158 0.485079 0.874470i \(-0.338791\pi\)
0.485079 + 0.874470i \(0.338791\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12606.0i 1.27358i 0.771038 + 0.636790i \(0.219738\pi\)
−0.771038 + 0.636790i \(0.780262\pi\)
\(462\) 0 0
\(463\) −14371.7 −1.44257 −0.721286 0.692638i \(-0.756448\pi\)
−0.721286 + 0.692638i \(0.756448\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 4698.85i − 0.465604i −0.972524 0.232802i \(-0.925211\pi\)
0.972524 0.232802i \(-0.0747894\pi\)
\(468\) 0 0
\(469\) 10752.0i 1.05860i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −17920.0 −1.74199
\(474\) 0 0
\(475\) 7535.10i 0.727861i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −10837.0 −1.03373 −0.516863 0.856068i \(-0.672900\pi\)
−0.516863 + 0.856068i \(0.672900\pi\)
\(480\) 0 0
\(481\) −3280.00 −0.310925
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1812.00i 0.169647i
\(486\) 0 0
\(487\) −13355.8 −1.24272 −0.621362 0.783523i \(-0.713420\pi\)
−0.621362 + 0.783523i \(0.713420\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 13715.6i 1.26064i 0.776335 + 0.630321i \(0.217077\pi\)
−0.776335 + 0.630321i \(0.782923\pi\)
\(492\) 0 0
\(493\) 368.000i 0.0336184i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −14336.0 −1.29388
\(498\) 0 0
\(499\) − 6773.12i − 0.607629i −0.952731 0.303814i \(-0.901740\pi\)
0.952731 0.303814i \(-0.0982602\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −11175.7 −0.990652 −0.495326 0.868707i \(-0.664951\pi\)
−0.495326 + 0.868707i \(0.664951\pi\)
\(504\) 0 0
\(505\) −8796.00 −0.775083
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 13458.0i 1.17194i 0.810334 + 0.585968i \(0.199286\pi\)
−0.810334 + 0.585968i \(0.800714\pi\)
\(510\) 0 0
\(511\) −5206.84 −0.450757
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 7492.77i − 0.641108i
\(516\) 0 0
\(517\) 7168.00i 0.609765i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5352.00 −0.450049 −0.225024 0.974353i \(-0.572246\pi\)
−0.225024 + 0.974353i \(0.572246\pi\)
\(522\) 0 0
\(523\) 6011.15i 0.502580i 0.967912 + 0.251290i \(0.0808547\pi\)
−0.967912 + 0.251290i \(0.919145\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 169.328 0.0139963
\(528\) 0 0
\(529\) 16505.0 1.35654
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6240.00i 0.507100i
\(534\) 0 0
\(535\) −2539.92 −0.205253
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 4444.86i − 0.355202i
\(540\) 0 0
\(541\) − 2396.00i − 0.190411i −0.995458 0.0952053i \(-0.969649\pi\)
0.995458 0.0952053i \(-0.0303507\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9384.00 −0.737553
\(546\) 0 0
\(547\) 1269.96i 0.0992680i 0.998767 + 0.0496340i \(0.0158055\pi\)
−0.998767 + 0.0496340i \(0.984195\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3894.55 0.301113
\(552\) 0 0
\(553\) 4928.00 0.378951
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 254.000i 0.0193219i 0.999953 + 0.00966097i \(0.00307523\pi\)
−0.999953 + 0.00966097i \(0.996925\pi\)
\(558\) 0 0
\(559\) 8466.40 0.640592
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10032.7i 0.751026i 0.926817 + 0.375513i \(0.122533\pi\)
−0.926817 + 0.375513i \(0.877467\pi\)
\(564\) 0 0
\(565\) − 9312.00i − 0.693378i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9320.00 −0.686669 −0.343335 0.939213i \(-0.611556\pi\)
−0.343335 + 0.939213i \(0.611556\pi\)
\(570\) 0 0
\(571\) − 13376.9i − 0.980397i −0.871611 0.490198i \(-0.836924\pi\)
0.871611 0.490198i \(-0.163076\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 15070.2 1.09299
\(576\) 0 0
\(577\) −13758.0 −0.992640 −0.496320 0.868140i \(-0.665316\pi\)
−0.496320 + 0.868140i \(0.665316\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 20608.0i 1.47154i
\(582\) 0 0
\(583\) −11260.3 −0.799922
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 26076.5i − 1.83355i −0.399405 0.916775i \(-0.630783\pi\)
0.399405 0.916775i \(-0.369217\pi\)
\(588\) 0 0
\(589\) − 1792.00i − 0.125362i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4080.00 0.282539 0.141269 0.989971i \(-0.454882\pi\)
0.141269 + 0.989971i \(0.454882\pi\)
\(594\) 0 0
\(595\) 1015.97i 0.0700011i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −15747.5 −1.07417 −0.537083 0.843529i \(-0.680474\pi\)
−0.537083 + 0.843529i \(0.680474\pi\)
\(600\) 0 0
\(601\) −10298.0 −0.698942 −0.349471 0.936947i \(-0.613639\pi\)
−0.349471 + 0.936947i \(0.613639\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2766.00i 0.185874i
\(606\) 0 0
\(607\) 23219.1 1.55261 0.776305 0.630357i \(-0.217091\pi\)
0.776305 + 0.630357i \(0.217091\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 3386.56i − 0.224232i
\(612\) 0 0
\(613\) 5652.00i 0.372402i 0.982512 + 0.186201i \(0.0596175\pi\)
−0.982512 + 0.186201i \(0.940383\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −26848.0 −1.75180 −0.875899 0.482494i \(-0.839731\pi\)
−0.875899 + 0.482494i \(0.839731\pi\)
\(618\) 0 0
\(619\) 3725.22i 0.241889i 0.992659 + 0.120944i \(0.0385923\pi\)
−0.992659 + 0.120944i \(0.961408\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 29463.1 1.89472
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 1312.00i − 0.0831683i
\(630\) 0 0
\(631\) 27452.3 1.73195 0.865974 0.500089i \(-0.166699\pi\)
0.865974 + 0.500089i \(0.166699\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 5968.81i 0.373016i
\(636\) 0 0
\(637\) 2100.00i 0.130620i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −6024.00 −0.371191 −0.185596 0.982626i \(-0.559421\pi\)
−0.185596 + 0.982626i \(0.559421\pi\)
\(642\) 0 0
\(643\) 24637.2i 1.51104i 0.655126 + 0.755519i \(0.272615\pi\)
−0.655126 + 0.755519i \(0.727385\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20827.4 1.26555 0.632773 0.774338i \(-0.281917\pi\)
0.632773 + 0.774338i \(0.281917\pi\)
\(648\) 0 0
\(649\) 10752.0 0.650313
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 29054.0i 1.74115i 0.492036 + 0.870575i \(0.336253\pi\)
−0.492036 + 0.870575i \(0.663747\pi\)
\(654\) 0 0
\(655\) −15239.5 −0.909096
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3047.91i 0.180166i 0.995934 + 0.0900831i \(0.0287133\pi\)
−0.995934 + 0.0900831i \(0.971287\pi\)
\(660\) 0 0
\(661\) − 9628.00i − 0.566544i −0.959040 0.283272i \(-0.908580\pi\)
0.959040 0.283272i \(-0.0914200\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 10752.0 0.626984
\(666\) 0 0
\(667\) − 7789.09i − 0.452166i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5587.83 −0.321484
\(672\) 0 0
\(673\) −28690.0 −1.64327 −0.821633 0.570017i \(-0.806937\pi\)
−0.821633 + 0.570017i \(0.806937\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4922.00i 0.279421i 0.990192 + 0.139710i \(0.0446171\pi\)
−0.990192 + 0.139710i \(0.955383\pi\)
\(678\) 0 0
\(679\) −6392.14 −0.361278
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 42.3320i − 0.00237158i −0.999999 0.00118579i \(-0.999623\pi\)
0.999999 0.00118579i \(-0.000377449\pi\)
\(684\) 0 0
\(685\) − 11952.0i − 0.666661i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 5320.00 0.294159
\(690\) 0 0
\(691\) 18372.1i 1.01144i 0.862697 + 0.505722i \(0.168774\pi\)
−0.862697 + 0.505722i \(0.831226\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7111.78 0.388151
\(696\) 0 0
\(697\) −2496.00 −0.135642
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 19310.0i − 1.04041i −0.854041 0.520206i \(-0.825855\pi\)
0.854041 0.520206i \(-0.174145\pi\)
\(702\) 0 0
\(703\) −13884.9 −0.744920
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 31029.4i − 1.65061i
\(708\) 0 0
\(709\) − 18820.0i − 0.996897i −0.866919 0.498448i \(-0.833903\pi\)
0.866919 0.498448i \(-0.166097\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3584.00 −0.188249
\(714\) 0 0
\(715\) − 5079.84i − 0.265700i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −27431.1 −1.42282 −0.711411 0.702776i \(-0.751944\pi\)
−0.711411 + 0.702776i \(0.751944\pi\)
\(720\) 0 0
\(721\) 26432.0 1.36530
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 4094.00i − 0.209720i
\(726\) 0 0
\(727\) −10561.8 −0.538813 −0.269406 0.963027i \(-0.586827\pi\)
−0.269406 + 0.963027i \(0.586827\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3386.56i 0.171350i
\(732\) 0 0
\(733\) 16468.0i 0.829822i 0.909862 + 0.414911i \(0.136187\pi\)
−0.909862 + 0.414911i \(0.863813\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 21504.0 1.07478
\(738\) 0 0
\(739\) − 15070.2i − 0.750157i −0.926993 0.375079i \(-0.877616\pi\)
0.926993 0.375079i \(-0.122384\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −34373.6 −1.69723 −0.848617 0.529007i \(-0.822564\pi\)
−0.848617 + 0.529007i \(0.822564\pi\)
\(744\) 0 0
\(745\) −11844.0 −0.582457
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 8960.00i − 0.437105i
\(750\) 0 0
\(751\) 1756.78 0.0853605 0.0426803 0.999089i \(-0.486410\pi\)
0.0426803 + 0.999089i \(0.486410\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 18668.4i 0.899885i
\(756\) 0 0
\(757\) − 1572.00i − 0.0754760i −0.999288 0.0377380i \(-0.987985\pi\)
0.999288 0.0377380i \(-0.0120152\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −17816.0 −0.848659 −0.424329 0.905508i \(-0.639490\pi\)
−0.424329 + 0.905508i \(0.639490\pi\)
\(762\) 0 0
\(763\) − 33103.6i − 1.57068i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5079.84 −0.239143
\(768\) 0 0
\(769\) −21746.0 −1.01974 −0.509870 0.860251i \(-0.670307\pi\)
−0.509870 + 0.860251i \(0.670307\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 13978.0i − 0.650393i −0.945646 0.325196i \(-0.894570\pi\)
0.945646 0.325196i \(-0.105430\pi\)
\(774\) 0 0
\(775\) −1883.77 −0.0873125
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 26415.2i 1.21492i
\(780\) 0 0
\(781\) 28672.0i 1.31366i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −10776.0 −0.489952
\(786\) 0 0
\(787\) − 1777.94i − 0.0805297i −0.999189 0.0402649i \(-0.987180\pi\)
0.999189 0.0402649i \(-0.0128202\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 32849.6 1.47661
\(792\) 0 0
\(793\) 2640.00 0.118221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 9650.00i − 0.428884i −0.976737 0.214442i \(-0.931207\pi\)
0.976737 0.214442i \(-0.0687933\pi\)
\(798\) 0 0
\(799\) 1354.62 0.0599789
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10413.7i 0.457647i
\(804\) 0 0
\(805\) − 21504.0i − 0.941511i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 38024.0 1.65248 0.826238 0.563322i \(-0.190477\pi\)
0.826238 + 0.563322i \(0.190477\pi\)
\(810\) 0 0
\(811\) − 931.304i − 0.0403237i −0.999797 0.0201619i \(-0.993582\pi\)
0.999797 0.0201619i \(-0.00641815\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −15747.5 −0.676824
\(816\) 0 0
\(817\) 35840.0 1.53474
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 6858.00i − 0.291530i −0.989319 0.145765i \(-0.953436\pi\)
0.989319 0.145765i \(-0.0465643\pi\)
\(822\) 0 0
\(823\) 18816.6 0.796968 0.398484 0.917175i \(-0.369536\pi\)
0.398484 + 0.917175i \(0.369536\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8212.41i 0.345313i 0.984982 + 0.172656i \(0.0552350\pi\)
−0.984982 + 0.172656i \(0.944765\pi\)
\(828\) 0 0
\(829\) 25124.0i 1.05258i 0.850304 + 0.526292i \(0.176418\pi\)
−0.850304 + 0.526292i \(0.823582\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −840.000 −0.0349391
\(834\) 0 0
\(835\) − 21335.3i − 0.884239i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −6095.81 −0.250835 −0.125418 0.992104i \(-0.540027\pi\)
−0.125418 + 0.992104i \(0.540027\pi\)
\(840\) 0 0
\(841\) 22273.0 0.913240
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 10782.0i − 0.438949i
\(846\) 0 0
\(847\) −9757.53 −0.395836
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 27769.8i 1.11861i
\(852\) 0 0
\(853\) − 32812.0i − 1.31707i −0.752550 0.658535i \(-0.771176\pi\)
0.752550 0.658535i \(-0.228824\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21944.0 0.874671 0.437335 0.899299i \(-0.355922\pi\)
0.437335 + 0.899299i \(0.355922\pi\)
\(858\) 0 0
\(859\) 48173.8i 1.91347i 0.290962 + 0.956735i \(0.406025\pi\)
−0.290962 + 0.956735i \(0.593975\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −12361.0 −0.487569 −0.243784 0.969829i \(-0.578389\pi\)
−0.243784 + 0.969829i \(0.578389\pi\)
\(864\) 0 0
\(865\) −25740.0 −1.01178
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 9856.00i − 0.384743i
\(870\) 0 0
\(871\) −10159.7 −0.395233
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 27177.2i − 1.05001i
\(876\) 0 0
\(877\) 33212.0i 1.27878i 0.768883 + 0.639390i \(0.220813\pi\)
−0.768883 + 0.639390i \(0.779187\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 34336.0 1.31306 0.656532 0.754298i \(-0.272023\pi\)
0.656532 + 0.754298i \(0.272023\pi\)
\(882\) 0 0
\(883\) 16678.8i 0.635659i 0.948148 + 0.317829i \(0.102954\pi\)
−0.948148 + 0.317829i \(0.897046\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 5079.84 0.192294 0.0961468 0.995367i \(-0.469348\pi\)
0.0961468 + 0.995367i \(0.469348\pi\)
\(888\) 0 0
\(889\) −21056.0 −0.794371
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 14336.0i − 0.537218i
\(894\) 0 0
\(895\) 9651.70 0.360470
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 973.636i 0.0361208i
\(900\) 0 0
\(901\) 2128.00i 0.0786836i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19176.0 0.704345
\(906\) 0 0
\(907\) 13292.3i 0.486617i 0.969949 + 0.243309i \(0.0782328\pi\)
−0.969949 + 0.243309i \(0.921767\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 2539.92 0.0923725 0.0461862 0.998933i \(-0.485293\pi\)
0.0461862 + 0.998933i \(0.485293\pi\)
\(912\) 0 0
\(913\) 41216.0 1.49403
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 53760.0i − 1.93600i
\(918\) 0 0
\(919\) −35368.4 −1.26953 −0.634764 0.772706i \(-0.718903\pi\)
−0.634764 + 0.772706i \(0.718903\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 13546.2i − 0.483077i
\(924\) 0 0
\(925\) 14596.0i 0.518825i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 17576.0 0.620721 0.310361 0.950619i \(-0.399550\pi\)
0.310361 + 0.950619i \(0.399550\pi\)
\(930\) 0 0
\(931\) 8889.72i 0.312942i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2031.94 0.0710711
\(936\) 0 0
\(937\) 21526.0 0.750506 0.375253 0.926923i \(-0.377556\pi\)
0.375253 + 0.926923i \(0.377556\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 5218.00i − 0.180767i −0.995907 0.0903836i \(-0.971191\pi\)
0.995907 0.0903836i \(-0.0288093\pi\)
\(942\) 0 0
\(943\) 52830.4 1.82438
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 29717.1i − 1.01972i −0.860257 0.509860i \(-0.829697\pi\)
0.860257 0.509860i \(-0.170303\pi\)
\(948\) 0 0
\(949\) − 4920.00i − 0.168293i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 48888.0 1.66174 0.830870 0.556467i \(-0.187843\pi\)
0.830870 + 0.556467i \(0.187843\pi\)
\(954\) 0 0
\(955\) 19303.4i 0.654077i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 42162.7 1.41971
\(960\) 0 0
\(961\) −29343.0 −0.984962
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 2772.00i − 0.0924703i
\(966\) 0 0
\(967\) 42988.2 1.42958 0.714791 0.699338i \(-0.246522\pi\)
0.714791 + 0.699338i \(0.246522\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 12403.3i − 0.409928i −0.978769 0.204964i \(-0.934292\pi\)
0.978769 0.204964i \(-0.0657078\pi\)
\(972\) 0 0
\(973\) 25088.0i 0.826603i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2040.00 −0.0668018 −0.0334009 0.999442i \(-0.510634\pi\)
−0.0334009 + 0.999442i \(0.510634\pi\)
\(978\) 0 0
\(979\) − 58926.2i − 1.92369i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −20658.0 −0.670284 −0.335142 0.942168i \(-0.608784\pi\)
−0.335142 + 0.942168i \(0.608784\pi\)
\(984\) 0 0
\(985\) −23004.0 −0.744130
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 71680.0i − 2.30464i
\(990\) 0 0
\(991\) −10773.5 −0.345340 −0.172670 0.984980i \(-0.555239\pi\)
−0.172670 + 0.984980i \(0.555239\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 22732.3i − 0.724284i
\(996\) 0 0
\(997\) − 62044.0i − 1.97087i −0.170065 0.985433i \(-0.554398\pi\)
0.170065 0.985433i \(-0.445602\pi\)
\(998\) 0 0
\(999\) 0 0
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 1152.4.d.k.577.2 yes 4
3.2 odd 2 1152.4.d.m.577.4 yes 4
4.3 odd 2 inner 1152.4.d.k.577.1 4
8.3 odd 2 inner 1152.4.d.k.577.3 yes 4
8.5 even 2 inner 1152.4.d.k.577.4 yes 4
12.11 even 2 1152.4.d.m.577.3 yes 4
16.3 odd 4 2304.4.a.q.1.2 2
16.5 even 4 2304.4.a.bs.1.1 2
16.11 odd 4 2304.4.a.bs.1.2 2
16.13 even 4 2304.4.a.q.1.1 2
24.5 odd 2 1152.4.d.m.577.2 yes 4
24.11 even 2 1152.4.d.m.577.1 yes 4
48.5 odd 4 2304.4.a.r.1.1 2
48.11 even 4 2304.4.a.r.1.2 2
48.29 odd 4 2304.4.a.br.1.1 2
48.35 even 4 2304.4.a.br.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.4.d.k.577.1 4 4.3 odd 2 inner
1152.4.d.k.577.2 yes 4 1.1 even 1 trivial
1152.4.d.k.577.3 yes 4 8.3 odd 2 inner
1152.4.d.k.577.4 yes 4 8.5 even 2 inner
1152.4.d.m.577.1 yes 4 24.11 even 2
1152.4.d.m.577.2 yes 4 24.5 odd 2
1152.4.d.m.577.3 yes 4 12.11 even 2
1152.4.d.m.577.4 yes 4 3.2 odd 2
2304.4.a.q.1.1 2 16.13 even 4
2304.4.a.q.1.2 2 16.3 odd 4
2304.4.a.r.1.1 2 48.5 odd 4
2304.4.a.r.1.2 2 48.11 even 4
2304.4.a.br.1.1 2 48.29 odd 4
2304.4.a.br.1.2 2 48.35 even 4
2304.4.a.bs.1.1 2 16.5 even 4
2304.4.a.bs.1.2 2 16.11 odd 4