Properties

Label 2304.4.a.br.1.1
Level $2304$
Weight $4$
Character 2304.1
Self dual yes
Analytic conductor $135.940$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,4,Mod(1,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2304.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2304.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: \(135.940400653\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1152)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 2304.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6.00000 q^{5} -21.1660 q^{7} +O(q^{10})\) \(q+6.00000 q^{5} -21.1660 q^{7} +42.3320 q^{11} -20.0000 q^{13} +8.00000 q^{17} -84.6640 q^{19} +169.328 q^{23} -89.0000 q^{25} -46.0000 q^{29} -21.1660 q^{31} -126.996 q^{35} +164.000 q^{37} +312.000 q^{41} -423.320 q^{43} +169.328 q^{47} +105.000 q^{49} +266.000 q^{53} +253.992 q^{55} -253.992 q^{59} +132.000 q^{61} -120.000 q^{65} -507.984 q^{67} -677.312 q^{71} +246.000 q^{73} -896.000 q^{77} +232.826 q^{79} +973.636 q^{83} +48.0000 q^{85} +1392.00 q^{89} +423.320 q^{91} -507.984 q^{95} -302.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 12 q^{5} - 40 q^{13} + 16 q^{17} - 178 q^{25} - 92 q^{29} + 328 q^{37} + 624 q^{41} + 210 q^{49} + 532 q^{53} + 264 q^{61} - 240 q^{65} + 492 q^{73} - 1792 q^{77} + 96 q^{85} + 2784 q^{89} - 604 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 6.00000 0.536656 0.268328 0.963328i \(-0.413529\pi\)
0.268328 + 0.963328i \(0.413529\pi\)
\(6\) 0 0
\(7\) −21.1660 −1.14286 −0.571429 0.820652i \(-0.693611\pi\)
−0.571429 + 0.820652i \(0.693611\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 42.3320 1.16033 0.580163 0.814500i \(-0.302989\pi\)
0.580163 + 0.814500i \(0.302989\pi\)
\(12\) 0 0
\(13\) −20.0000 −0.426692 −0.213346 0.976977i \(-0.568436\pi\)
−0.213346 + 0.976977i \(0.568436\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.00000 0.114134 0.0570672 0.998370i \(-0.481825\pi\)
0.0570672 + 0.998370i \(0.481825\pi\)
\(18\) 0 0
\(19\) −84.6640 −1.02228 −0.511139 0.859498i \(-0.670776\pi\)
−0.511139 + 0.859498i \(0.670776\pi\)
\(20\) 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
\(29\) −46.0000 −0.294551 −0.147276 0.989095i \(-0.547050\pi\)
−0.147276 + 0.989095i \(0.547050\pi\)
\(30\) 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
\(35\) −126.996 −0.613322
\(36\) 0 0
\(37\) 164.000 0.728687 0.364344 0.931265i \(-0.381293\pi\)
0.364344 + 0.931265i \(0.381293\pi\)
\(38\) 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
\(43\) −423.320 −1.50130 −0.750648 0.660702i \(-0.770259\pi\)
−0.750648 + 0.660702i \(0.770259\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 169.328 0.525511 0.262756 0.964862i \(-0.415369\pi\)
0.262756 + 0.964862i \(0.415369\pi\)
\(48\) 0 0
\(49\) 105.000 0.306122
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 266.000 0.689395 0.344697 0.938714i \(-0.387982\pi\)
0.344697 + 0.938714i \(0.387982\pi\)
\(54\) 0 0
\(55\) 253.992 0.622696
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −253.992 −0.560457 −0.280228 0.959933i \(-0.590410\pi\)
−0.280228 + 0.959933i \(0.590410\pi\)
\(60\) 0 0
\(61\) 132.000 0.277063 0.138532 0.990358i \(-0.455762\pi\)
0.138532 + 0.990358i \(0.455762\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −120.000 −0.228987
\(66\) 0 0
\(67\) −507.984 −0.926271 −0.463135 0.886287i \(-0.653276\pi\)
−0.463135 + 0.886287i \(0.653276\pi\)
\(68\) 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 \(-0.436813\pi\)
0.197206 + 0.980362i \(0.436813\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −896.000 −1.32609
\(78\) 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
\(83\) 973.636 1.28760 0.643798 0.765195i \(-0.277358\pi\)
0.643798 + 0.765195i \(0.277358\pi\)
\(84\) 0 0
\(85\) 48.0000 0.0612510
\(86\) 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
\(91\) 423.320 0.487649
\(92\) 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
\(101\) 1466.00 1.44428 0.722141 0.691746i \(-0.243158\pi\)
0.722141 + 0.691746i \(0.243158\pi\)
\(102\) 0 0
\(103\) −1248.79 −1.19463 −0.597317 0.802005i \(-0.703767\pi\)
−0.597317 + 0.802005i \(0.703767\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 423.320 0.382466 0.191233 0.981545i \(-0.438751\pi\)
0.191233 + 0.981545i \(0.438751\pi\)
\(108\) 0 0
\(109\) 1564.00 1.37435 0.687174 0.726492i \(-0.258851\pi\)
0.687174 + 0.726492i \(0.258851\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1552.00 −1.29203 −0.646017 0.763323i \(-0.723567\pi\)
−0.646017 + 0.763323i \(0.723567\pi\)
\(114\) 0 0
\(115\) 1015.97 0.823822
\(116\) 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
\(125\) −1284.00 −0.918756
\(126\) 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
\(131\) −2539.92 −1.69400 −0.847000 0.531593i \(-0.821594\pi\)
−0.847000 + 0.531593i \(0.821594\pi\)
\(132\) 0 0
\(133\) 1792.00 1.16832
\(134\) 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
\(139\) 1185.30 0.723277 0.361639 0.932318i \(-0.382217\pi\)
0.361639 + 0.932318i \(0.382217\pi\)
\(140\) 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
\(149\) 1974.00 1.08534 0.542672 0.839944i \(-0.317413\pi\)
0.542672 + 0.839944i \(0.317413\pi\)
\(150\) 0 0
\(151\) 3111.40 1.67684 0.838419 0.545027i \(-0.183481\pi\)
0.838419 + 0.545027i \(0.183481\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −126.996 −0.0658101
\(156\) 0 0
\(157\) 1796.00 0.912971 0.456485 0.889731i \(-0.349108\pi\)
0.456485 + 0.889731i \(0.349108\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3584.00 −1.75440
\(162\) 0 0
\(163\) 2624.59 1.26119 0.630593 0.776114i \(-0.282812\pi\)
0.630593 + 0.776114i \(0.282812\pi\)
\(164\) 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
\(173\) −4290.00 −1.88533 −0.942667 0.333736i \(-0.891691\pi\)
−0.942667 + 0.333736i \(0.891691\pi\)
\(174\) 0 0
\(175\) 1883.77 0.813714
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1608.62 0.671696 0.335848 0.941916i \(-0.390977\pi\)
0.335848 + 0.941916i \(0.390977\pi\)
\(180\) 0 0
\(181\) 3196.00 1.31247 0.656234 0.754557i \(-0.272148\pi\)
0.656234 + 0.754557i \(0.272148\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 984.000 0.391055
\(186\) 0 0
\(187\) 338.656 0.132433
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3217.23 1.21880 0.609400 0.792863i \(-0.291410\pi\)
0.609400 + 0.792863i \(0.291410\pi\)
\(192\) 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
\(197\) 3834.00 1.38661 0.693303 0.720647i \(-0.256155\pi\)
0.693303 + 0.720647i \(0.256155\pi\)
\(198\) 0 0
\(199\) −3788.72 −1.34962 −0.674811 0.737990i \(-0.735775\pi\)
−0.674811 + 0.737990i \(0.735775\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 973.636 0.336630
\(204\) 0 0
\(205\) 1872.00 0.637786
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3584.00 −1.18617
\(210\) 0 0
\(211\) −2878.58 −0.939192 −0.469596 0.882882i \(-0.655600\pi\)
−0.469596 + 0.882882i \(0.655600\pi\)
\(212\) 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
\(221\) −160.000 −0.0487003
\(222\) 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
\(227\) 3259.57 0.953062 0.476531 0.879158i \(-0.341894\pi\)
0.476531 + 0.879158i \(0.341894\pi\)
\(228\) 0 0
\(229\) 5260.00 1.51786 0.758931 0.651171i \(-0.225722\pi\)
0.758931 + 0.651171i \(0.225722\pi\)
\(230\) 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
\(235\) 1015.97 0.282019
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1523.95 0.412453 0.206227 0.978504i \(-0.433882\pi\)
0.206227 + 0.978504i \(0.433882\pi\)
\(240\) 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
\(245\) 630.000 0.164283
\(246\) 0 0
\(247\) 1693.28 0.436198
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2666.92 −0.670655 −0.335327 0.942102i \(-0.608847\pi\)
−0.335327 + 0.942102i \(0.608847\pi\)
\(252\) 0 0
\(253\) 7168.00 1.78122
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3104.00 −0.753394 −0.376697 0.926337i \(-0.622940\pi\)
−0.376697 + 0.926337i \(0.622940\pi\)
\(258\) 0 0
\(259\) −3471.23 −0.832786
\(260\) 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
\(269\) −4350.00 −0.985963 −0.492982 0.870040i \(-0.664093\pi\)
−0.492982 + 0.870040i \(0.664093\pi\)
\(270\) 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
\(275\) −3767.55 −0.826152
\(276\) 0 0
\(277\) 7532.00 1.63377 0.816885 0.576801i \(-0.195699\pi\)
0.816885 + 0.576801i \(0.195699\pi\)
\(278\) 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
\(283\) −8805.06 −1.84949 −0.924746 0.380584i \(-0.875723\pi\)
−0.924746 + 0.380584i \(0.875723\pi\)
\(284\) 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
\(293\) 858.000 0.171075 0.0855374 0.996335i \(-0.472739\pi\)
0.0855374 + 0.996335i \(0.472739\pi\)
\(294\) 0 0
\(295\) −1523.95 −0.300773
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3386.56 −0.655016
\(300\) 0 0
\(301\) 8960.00 1.71577
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 792.000 0.148688
\(306\) 0 0
\(307\) 1523.95 0.283311 0.141656 0.989916i \(-0.454757\pi\)
0.141656 + 0.989916i \(0.454757\pi\)
\(308\) 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 \(-0.496608\pi\)
0.0106546 + 0.999943i \(0.496608\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4594.00 0.813958 0.406979 0.913437i \(-0.366582\pi\)
0.406979 + 0.913437i \(0.366582\pi\)
\(318\) 0 0
\(319\) −1947.27 −0.341775
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −677.312 −0.116677
\(324\) 0 0
\(325\) 1780.00 0.303805
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3584.00 −0.600585
\(330\) 0 0
\(331\) −3725.22 −0.618600 −0.309300 0.950965i \(-0.600095\pi\)
−0.309300 + 0.950965i \(0.600095\pi\)
\(332\) 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
\(341\) −896.000 −0.142291
\(342\) 0 0
\(343\) 5037.51 0.793003
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11302.6 1.74858 0.874291 0.485402i \(-0.161327\pi\)
0.874291 + 0.485402i \(0.161327\pi\)
\(348\) 0 0
\(349\) 6388.00 0.979776 0.489888 0.871785i \(-0.337038\pi\)
0.489888 + 0.871785i \(0.337038\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8432.00 1.27136 0.635680 0.771953i \(-0.280720\pi\)
0.635680 + 0.771953i \(0.280720\pi\)
\(354\) 0 0
\(355\) −4063.87 −0.607572
\(356\) 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
\(365\) 1476.00 0.211664
\(366\) 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
\(371\) −5630.16 −0.787879
\(372\) 0 0
\(373\) 8276.00 1.14883 0.574417 0.818563i \(-0.305229\pi\)
0.574417 + 0.818563i \(0.305229\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 920.000 0.125683
\(378\) 0 0
\(379\) 13800.2 1.87037 0.935186 0.354158i \(-0.115233\pi\)
0.935186 + 0.354158i \(0.115233\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1862.61 0.248498 0.124249 0.992251i \(-0.460348\pi\)
0.124249 + 0.992251i \(0.460348\pi\)
\(384\) 0 0
\(385\) −5376.00 −0.711653
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6170.00 −0.804194 −0.402097 0.915597i \(-0.631719\pi\)
−0.402097 + 0.915597i \(0.631719\pi\)
\(390\) 0 0
\(391\) 1354.62 0.175208
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1396.96 0.177946
\(396\) 0 0
\(397\) 8644.00 1.09277 0.546385 0.837534i \(-0.316003\pi\)
0.546385 + 0.837534i \(0.316003\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8152.00 1.01519 0.507595 0.861596i \(-0.330535\pi\)
0.507595 + 0.861596i \(0.330535\pi\)
\(402\) 0 0
\(403\) 423.320 0.0523253
\(404\) 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 \(-0.483175\pi\)
0.0528319 + 0.998603i \(0.483175\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5376.00 0.640522
\(414\) 0 0
\(415\) 5841.82 0.690997
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14265.9 1.66333 0.831664 0.555279i \(-0.187389\pi\)
0.831664 + 0.555279i \(0.187389\pi\)
\(420\) 0 0
\(421\) 11564.0 1.33871 0.669353 0.742945i \(-0.266572\pi\)
0.669353 + 0.742945i \(0.266572\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −712.000 −0.0812637
\(426\) 0 0
\(427\) −2793.91 −0.316644
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7619.76 0.851580 0.425790 0.904822i \(-0.359996\pi\)
0.425790 + 0.904822i \(0.359996\pi\)
\(432\) 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
\(437\) −14336.0 −1.56930
\(438\) 0 0
\(439\) −17335.0 −1.88463 −0.942315 0.334727i \(-0.891356\pi\)
−0.942315 + 0.334727i \(0.891356\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −9016.72 −0.967037 −0.483518 0.875334i \(-0.660641\pi\)
−0.483518 + 0.875334i \(0.660641\pi\)
\(444\) 0 0
\(445\) 8352.00 0.889714
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6568.00 0.690341 0.345170 0.938540i \(-0.387821\pi\)
0.345170 + 0.938540i \(0.387821\pi\)
\(450\) 0 0
\(451\) 13207.6 1.37898
\(452\) 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 \(-0.661209\pi\)
−0.485079 + 0.874470i \(0.661209\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12606.0 1.27358 0.636790 0.771038i \(-0.280262\pi\)
0.636790 + 0.771038i \(0.280262\pi\)
\(462\) 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
\(467\) −4698.85 −0.465604 −0.232802 0.972524i \(-0.574789\pi\)
−0.232802 + 0.972524i \(0.574789\pi\)
\(468\) 0 0
\(469\) 10752.0 1.05860
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −17920.0 −1.74199
\(474\) 0 0
\(475\) 7535.10 0.727861
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10837.0 1.03373 0.516863 0.856068i \(-0.327100\pi\)
0.516863 + 0.856068i \(0.327100\pi\)
\(480\) 0 0
\(481\) −3280.00 −0.310925
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1812.00 −0.169647
\(486\) 0 0
\(487\) 13355.8 1.24272 0.621362 0.783523i \(-0.286580\pi\)
0.621362 + 0.783523i \(0.286580\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −13715.6 −1.26064 −0.630321 0.776335i \(-0.717077\pi\)
−0.630321 + 0.776335i \(0.717077\pi\)
\(492\) 0 0
\(493\) −368.000 −0.0336184
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14336.0 1.29388
\(498\) 0 0
\(499\) 6773.12 0.607629 0.303814 0.952731i \(-0.401740\pi\)
0.303814 + 0.952731i \(0.401740\pi\)
\(500\) 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
\(509\) 13458.0 1.17194 0.585968 0.810334i \(-0.300714\pi\)
0.585968 + 0.810334i \(0.300714\pi\)
\(510\) 0 0
\(511\) −5206.84 −0.450757
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7492.77 −0.641108
\(516\) 0 0
\(517\) 7168.00 0.609765
\(518\) 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
\(523\) 6011.15 0.502580 0.251290 0.967912i \(-0.419145\pi\)
0.251290 + 0.967912i \(0.419145\pi\)
\(524\) 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
\(533\) −6240.00 −0.507100
\(534\) 0 0
\(535\) 2539.92 0.205253
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4444.86 0.355202
\(540\) 0 0
\(541\) 2396.00 0.190411 0.0952053 0.995458i \(-0.469649\pi\)
0.0952053 + 0.995458i \(0.469649\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 9384.00 0.737553
\(546\) 0 0
\(547\) −1269.96 −0.0992680 −0.0496340 0.998767i \(-0.515805\pi\)
−0.0496340 + 0.998767i \(0.515805\pi\)
\(548\) 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
\(557\) 254.000 0.0193219 0.00966097 0.999953i \(-0.496925\pi\)
0.00966097 + 0.999953i \(0.496925\pi\)
\(558\) 0 0
\(559\) 8466.40 0.640592
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10032.7 0.751026 0.375513 0.926817i \(-0.377467\pi\)
0.375513 + 0.926817i \(0.377467\pi\)
\(564\) 0 0
\(565\) −9312.00 −0.693378
\(566\) 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
\(571\) −13376.9 −0.980397 −0.490198 0.871611i \(-0.663076\pi\)
−0.490198 + 0.871611i \(0.663076\pi\)
\(572\) 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
\(581\) −20608.0 −1.47154
\(582\) 0 0
\(583\) 11260.3 0.799922
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 26076.5 1.83355 0.916775 0.399405i \(-0.130783\pi\)
0.916775 + 0.399405i \(0.130783\pi\)
\(588\) 0 0
\(589\) 1792.00 0.125362
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4080.00 −0.282539 −0.141269 0.989971i \(-0.545118\pi\)
−0.141269 + 0.989971i \(0.545118\pi\)
\(594\) 0 0
\(595\) −1015.97 −0.0700011
\(596\) 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 \(-0.386361\pi\)
0.349471 + 0.936947i \(0.386361\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2766.00 0.185874
\(606\) 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
\(611\) −3386.56 −0.224232
\(612\) 0 0
\(613\) 5652.00 0.372402 0.186201 0.982512i \(-0.440383\pi\)
0.186201 + 0.982512i \(0.440383\pi\)
\(614\) 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
\(619\) 3725.22 0.241889 0.120944 0.992659i \(-0.461408\pi\)
0.120944 + 0.992659i \(0.461408\pi\)
\(620\) 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
\(629\) 1312.00 0.0831683
\(630\) 0 0
\(631\) −27452.3 −1.73195 −0.865974 0.500089i \(-0.833301\pi\)
−0.865974 + 0.500089i \(0.833301\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5968.81 −0.373016
\(636\) 0 0
\(637\) −2100.00 −0.130620
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6024.00 0.371191 0.185596 0.982626i \(-0.440579\pi\)
0.185596 + 0.982626i \(0.440579\pi\)
\(642\) 0 0
\(643\) −24637.2 −1.51104 −0.755519 0.655126i \(-0.772615\pi\)
−0.755519 + 0.655126i \(0.772615\pi\)
\(644\) 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
\(653\) 29054.0 1.74115 0.870575 0.492036i \(-0.163747\pi\)
0.870575 + 0.492036i \(0.163747\pi\)
\(654\) 0 0
\(655\) −15239.5 −0.909096
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3047.91 0.180166 0.0900831 0.995934i \(-0.471287\pi\)
0.0900831 + 0.995934i \(0.471287\pi\)
\(660\) 0 0
\(661\) −9628.00 −0.566544 −0.283272 0.959040i \(-0.591420\pi\)
−0.283272 + 0.959040i \(0.591420\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 10752.0 0.626984
\(666\) 0 0
\(667\) −7789.09 −0.452166
\(668\) 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
\(677\) −4922.00 −0.279421 −0.139710 0.990192i \(-0.544617\pi\)
−0.139710 + 0.990192i \(0.544617\pi\)
\(678\) 0 0
\(679\) 6392.14 0.361278
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 42.3320 0.00237158 0.00118579 0.999999i \(-0.499623\pi\)
0.00118579 + 0.999999i \(0.499623\pi\)
\(684\) 0 0
\(685\) 11952.0 0.666661
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5320.00 −0.294159
\(690\) 0 0
\(691\) −18372.1 −1.01144 −0.505722 0.862697i \(-0.668774\pi\)
−0.505722 + 0.862697i \(0.668774\pi\)
\(692\) 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
\(701\) −19310.0 −1.04041 −0.520206 0.854041i \(-0.674145\pi\)
−0.520206 + 0.854041i \(0.674145\pi\)
\(702\) 0 0
\(703\) −13884.9 −0.744920
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −31029.4 −1.65061
\(708\) 0 0
\(709\) −18820.0 −0.996897 −0.498448 0.866919i \(-0.666097\pi\)
−0.498448 + 0.866919i \(0.666097\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3584.00 −0.188249
\(714\) 0 0
\(715\) −5079.84 −0.265700
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 27431.1 1.42282 0.711411 0.702776i \(-0.248056\pi\)
0.711411 + 0.702776i \(0.248056\pi\)
\(720\) 0 0
\(721\) 26432.0 1.36530
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4094.00 0.209720
\(726\) 0 0
\(727\) 10561.8 0.538813 0.269406 0.963027i \(-0.413173\pi\)
0.269406 + 0.963027i \(0.413173\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3386.56 −0.171350
\(732\) 0 0
\(733\) −16468.0 −0.829822 −0.414911 0.909862i \(-0.636187\pi\)
−0.414911 + 0.909862i \(0.636187\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −21504.0 −1.07478
\(738\) 0 0
\(739\) 15070.2 0.750157 0.375079 0.926993i \(-0.377616\pi\)
0.375079 + 0.926993i \(0.377616\pi\)
\(740\) 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
\(749\) −8960.00 −0.437105
\(750\) 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
\(755\) 18668.4 0.899885
\(756\) 0 0
\(757\) −1572.00 −0.0754760 −0.0377380 0.999288i \(-0.512015\pi\)
−0.0377380 + 0.999288i \(0.512015\pi\)
\(758\) 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
\(763\) −33103.6 −1.57068
\(764\) 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
\(773\) 13978.0 0.650393 0.325196 0.945646i \(-0.394570\pi\)
0.325196 + 0.945646i \(0.394570\pi\)
\(774\) 0 0
\(775\) 1883.77 0.0873125
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −26415.2 −1.21492
\(780\) 0 0
\(781\) −28672.0 −1.31366
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 10776.0 0.489952
\(786\) 0 0
\(787\) 1777.94 0.0805297 0.0402649 0.999189i \(-0.487180\pi\)
0.0402649 + 0.999189i \(0.487180\pi\)
\(788\) 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
\(797\) −9650.00 −0.428884 −0.214442 0.976737i \(-0.568793\pi\)
−0.214442 + 0.976737i \(0.568793\pi\)
\(798\) 0 0
\(799\) 1354.62 0.0599789
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10413.7 0.457647
\(804\) 0 0
\(805\) −21504.0 −0.941511
\(806\) 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
\(811\) −931.304 −0.0403237 −0.0201619 0.999797i \(-0.506418\pi\)
−0.0201619 + 0.999797i \(0.506418\pi\)
\(812\) 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
\(821\) 6858.00 0.291530 0.145765 0.989319i \(-0.453436\pi\)
0.145765 + 0.989319i \(0.453436\pi\)
\(822\) 0 0
\(823\) −18816.6 −0.796968 −0.398484 0.917175i \(-0.630464\pi\)
−0.398484 + 0.917175i \(0.630464\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −8212.41 −0.345313 −0.172656 0.984982i \(-0.555235\pi\)
−0.172656 + 0.984982i \(0.555235\pi\)
\(828\) 0 0
\(829\) −25124.0 −1.05258 −0.526292 0.850304i \(-0.676418\pi\)
−0.526292 + 0.850304i \(0.676418\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 840.000 0.0349391
\(834\) 0 0
\(835\) 21335.3 0.884239
\(836\) 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
\(845\) −10782.0 −0.438949
\(846\) 0 0
\(847\) −9757.53 −0.395836
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 27769.8 1.11861
\(852\) 0 0
\(853\) −32812.0 −1.31707 −0.658535 0.752550i \(-0.728824\pi\)
−0.658535 + 0.752550i \(0.728824\pi\)
\(854\) 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
\(859\) 48173.8 1.91347 0.956735 0.290962i \(-0.0939752\pi\)
0.956735 + 0.290962i \(0.0939752\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 12361.0 0.487569 0.243784 0.969829i \(-0.421611\pi\)
0.243784 + 0.969829i \(0.421611\pi\)
\(864\) 0 0
\(865\) −25740.0 −1.01178
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 9856.00 0.384743
\(870\) 0 0
\(871\) 10159.7 0.395233
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 27177.2 1.05001
\(876\) 0 0
\(877\) −33212.0 −1.27878 −0.639390 0.768883i \(-0.720813\pi\)
−0.639390 + 0.768883i \(0.720813\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −34336.0 −1.31306 −0.656532 0.754298i \(-0.727977\pi\)
−0.656532 + 0.754298i \(0.727977\pi\)
\(882\) 0 0
\(883\) −16678.8 −0.635659 −0.317829 0.948148i \(-0.602954\pi\)
−0.317829 + 0.948148i \(0.602954\pi\)
\(884\) 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
\(893\) −14336.0 −0.537218
\(894\) 0 0
\(895\) 9651.70 0.360470
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 973.636 0.0361208
\(900\) 0 0
\(901\) 2128.00 0.0786836
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19176.0 0.704345
\(906\) 0 0
\(907\) 13292.3 0.486617 0.243309 0.969949i \(-0.421767\pi\)
0.243309 + 0.969949i \(0.421767\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −2539.92 −0.0923725 −0.0461862 0.998933i \(-0.514707\pi\)
−0.0461862 + 0.998933i \(0.514707\pi\)
\(912\) 0 0
\(913\) 41216.0 1.49403
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 53760.0 1.93600
\(918\) 0 0
\(919\) 35368.4 1.26953 0.634764 0.772706i \(-0.281097\pi\)
0.634764 + 0.772706i \(0.281097\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 13546.2 0.483077
\(924\) 0 0
\(925\) −14596.0 −0.518825
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −17576.0 −0.620721 −0.310361 0.950619i \(-0.600450\pi\)
−0.310361 + 0.950619i \(0.600450\pi\)
\(930\) 0 0
\(931\) −8889.72 −0.312942
\(932\) 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 \(-0.622444\pi\)
−0.375253 + 0.926923i \(0.622444\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −5218.00 −0.180767 −0.0903836 0.995907i \(-0.528809\pi\)
−0.0903836 + 0.995907i \(0.528809\pi\)
\(942\) 0 0
\(943\) 52830.4 1.82438
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −29717.1 −1.01972 −0.509860 0.860257i \(-0.670303\pi\)
−0.509860 + 0.860257i \(0.670303\pi\)
\(948\) 0 0
\(949\) −4920.00 −0.168293
\(950\) 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
\(955\) 19303.4 0.654077
\(956\) 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
\(965\) 2772.00 0.0924703
\(966\) 0 0
\(967\) −42988.2 −1.42958 −0.714791 0.699338i \(-0.753478\pi\)
−0.714791 + 0.699338i \(0.753478\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12403.3 0.409928 0.204964 0.978769i \(-0.434292\pi\)
0.204964 + 0.978769i \(0.434292\pi\)
\(972\) 0 0
\(973\) −25088.0 −0.826603
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2040.00 0.0668018 0.0334009 0.999442i \(-0.489366\pi\)
0.0334009 + 0.999442i \(0.489366\pi\)
\(978\) 0 0
\(979\) 58926.2 1.92369
\(980\) 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
\(989\) −71680.0 −2.30464
\(990\) 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
\(995\) −22732.3 −0.724284
\(996\) 0 0
\(997\) −62044.0 −1.97087 −0.985433 0.170065i \(-0.945602\pi\)
−0.985433 + 0.170065i \(0.945602\pi\)
\(998\) 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 2304.4.a.br.1.1 2
3.2 odd 2 2304.4.a.q.1.1 2
4.3 odd 2 inner 2304.4.a.br.1.2 2
8.3 odd 2 2304.4.a.r.1.2 2
8.5 even 2 2304.4.a.r.1.1 2
12.11 even 2 2304.4.a.q.1.2 2
16.3 odd 4 1152.4.d.m.577.1 yes 4
16.5 even 4 1152.4.d.m.577.4 yes 4
16.11 odd 4 1152.4.d.m.577.3 yes 4
16.13 even 4 1152.4.d.m.577.2 yes 4
24.5 odd 2 2304.4.a.bs.1.1 2
24.11 even 2 2304.4.a.bs.1.2 2
48.5 odd 4 1152.4.d.k.577.2 yes 4
48.11 even 4 1152.4.d.k.577.1 4
48.29 odd 4 1152.4.d.k.577.4 yes 4
48.35 even 4 1152.4.d.k.577.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.4.d.k.577.1 4 48.11 even 4
1152.4.d.k.577.2 yes 4 48.5 odd 4
1152.4.d.k.577.3 yes 4 48.35 even 4
1152.4.d.k.577.4 yes 4 48.29 odd 4
1152.4.d.m.577.1 yes 4 16.3 odd 4
1152.4.d.m.577.2 yes 4 16.13 even 4
1152.4.d.m.577.3 yes 4 16.11 odd 4
1152.4.d.m.577.4 yes 4 16.5 even 4
2304.4.a.q.1.1 2 3.2 odd 2
2304.4.a.q.1.2 2 12.11 even 2
2304.4.a.r.1.1 2 8.5 even 2
2304.4.a.r.1.2 2 8.3 odd 2
2304.4.a.br.1.1 2 1.1 even 1 trivial
2304.4.a.br.1.2 2 4.3 odd 2 inner
2304.4.a.bs.1.1 2 24.5 odd 2
2304.4.a.bs.1.2 2 24.11 even 2