Properties

Label 2304.4.a.bn.1.2
Level $2304$
Weight $4$
Character 2304.1
Self dual yes
Analytic conductor $135.940$
Analytic rank $1$
Dimension $2$
CM no
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: \(1\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
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)
 
\(f(q)\) \(=\) \(q+10.5830 q^{5} +8.00000 q^{7} +O(q^{10})\) \(q+10.5830 q^{5} +8.00000 q^{7} +15.8745 q^{11} -52.9150 q^{13} +14.0000 q^{17} +37.0405 q^{19} -152.000 q^{23} -13.0000 q^{25} -158.745 q^{29} +224.000 q^{31} +84.6640 q^{35} +243.409 q^{37} -70.0000 q^{41} -439.195 q^{43} -336.000 q^{47} -279.000 q^{49} -31.7490 q^{53} +168.000 q^{55} -534.442 q^{59} -95.2470 q^{61} -560.000 q^{65} -174.620 q^{67} -72.0000 q^{71} +294.000 q^{73} +126.996 q^{77} -464.000 q^{79} -545.025 q^{83} +148.162 q^{85} +266.000 q^{89} -423.320 q^{91} +392.000 q^{95} +994.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{7} + 28 q^{17} - 304 q^{23} - 26 q^{25} + 448 q^{31} - 140 q^{41} - 672 q^{47} - 558 q^{49} + 336 q^{55} - 1120 q^{65} - 144 q^{71} + 588 q^{73} - 928 q^{79} + 532 q^{89} + 784 q^{95} + 1988 q^{97}+O(q^{100}) \) 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
\(5\) 10.5830 0.946573 0.473286 0.880909i \(-0.343068\pi\)
0.473286 + 0.880909i \(0.343068\pi\)
\(6\) 0 0
\(7\) 8.00000 0.431959 0.215980 0.976398i \(-0.430705\pi\)
0.215980 + 0.976398i \(0.430705\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 15.8745 0.435122 0.217561 0.976047i \(-0.430190\pi\)
0.217561 + 0.976047i \(0.430190\pi\)
\(12\) 0 0
\(13\) −52.9150 −1.12892 −0.564461 0.825460i \(-0.690916\pi\)
−0.564461 + 0.825460i \(0.690916\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 14.0000 0.199735 0.0998676 0.995001i \(-0.468158\pi\)
0.0998676 + 0.995001i \(0.468158\pi\)
\(18\) 0 0
\(19\) 37.0405 0.447246 0.223623 0.974676i \(-0.428212\pi\)
0.223623 + 0.974676i \(0.428212\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −152.000 −1.37801 −0.689004 0.724757i \(-0.741952\pi\)
−0.689004 + 0.724757i \(0.741952\pi\)
\(24\) 0 0
\(25\) −13.0000 −0.104000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −158.745 −1.01649 −0.508245 0.861212i \(-0.669706\pi\)
−0.508245 + 0.861212i \(0.669706\pi\)
\(30\) 0 0
\(31\) 224.000 1.29779 0.648897 0.760877i \(-0.275231\pi\)
0.648897 + 0.760877i \(0.275231\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 84.6640 0.408881
\(36\) 0 0
\(37\) 243.409 1.08152 0.540760 0.841177i \(-0.318137\pi\)
0.540760 + 0.841177i \(0.318137\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −70.0000 −0.266638 −0.133319 0.991073i \(-0.542564\pi\)
−0.133319 + 0.991073i \(0.542564\pi\)
\(42\) 0 0
\(43\) −439.195 −1.55759 −0.778797 0.627276i \(-0.784170\pi\)
−0.778797 + 0.627276i \(0.784170\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −336.000 −1.04278 −0.521390 0.853319i \(-0.674586\pi\)
−0.521390 + 0.853319i \(0.674586\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −31.7490 −0.0822842 −0.0411421 0.999153i \(-0.513100\pi\)
−0.0411421 + 0.999153i \(0.513100\pi\)
\(54\) 0 0
\(55\) 168.000 0.411875
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −534.442 −1.17929 −0.589647 0.807661i \(-0.700733\pi\)
−0.589647 + 0.807661i \(0.700733\pi\)
\(60\) 0 0
\(61\) −95.2470 −0.199920 −0.0999601 0.994991i \(-0.531872\pi\)
−0.0999601 + 0.994991i \(0.531872\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −560.000 −1.06861
\(66\) 0 0
\(67\) −174.620 −0.318406 −0.159203 0.987246i \(-0.550892\pi\)
−0.159203 + 0.987246i \(0.550892\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −72.0000 −0.120350 −0.0601748 0.998188i \(-0.519166\pi\)
−0.0601748 + 0.998188i \(0.519166\pi\)
\(72\) 0 0
\(73\) 294.000 0.471371 0.235686 0.971829i \(-0.424266\pi\)
0.235686 + 0.971829i \(0.424266\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 126.996 0.187955
\(78\) 0 0
\(79\) −464.000 −0.660811 −0.330406 0.943839i \(-0.607186\pi\)
−0.330406 + 0.943839i \(0.607186\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −545.025 −0.720774 −0.360387 0.932803i \(-0.617355\pi\)
−0.360387 + 0.932803i \(0.617355\pi\)
\(84\) 0 0
\(85\) 148.162 0.189064
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 266.000 0.316808 0.158404 0.987374i \(-0.449365\pi\)
0.158404 + 0.987374i \(0.449365\pi\)
\(90\) 0 0
\(91\) −423.320 −0.487649
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 392.000 0.423351
\(96\) 0 0
\(97\) 994.000 1.04047 0.520234 0.854024i \(-0.325845\pi\)
0.520234 + 0.854024i \(0.325845\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −751.393 −0.740262 −0.370131 0.928980i \(-0.620687\pi\)
−0.370131 + 0.928980i \(0.620687\pi\)
\(102\) 0 0
\(103\) −1176.00 −1.12500 −0.562499 0.826798i \(-0.690160\pi\)
−0.562499 + 0.826798i \(0.690160\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 269.867 0.243822 0.121911 0.992541i \(-0.461098\pi\)
0.121911 + 0.992541i \(0.461098\pi\)
\(108\) 0 0
\(109\) 1894.36 1.66465 0.832324 0.554290i \(-0.187010\pi\)
0.832324 + 0.554290i \(0.187010\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1710.00 1.42357 0.711784 0.702398i \(-0.247887\pi\)
0.711784 + 0.702398i \(0.247887\pi\)
\(114\) 0 0
\(115\) −1608.62 −1.30439
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 112.000 0.0862775
\(120\) 0 0
\(121\) −1079.00 −0.810669
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1460.45 −1.04502
\(126\) 0 0
\(127\) −1664.00 −1.16265 −0.581323 0.813673i \(-0.697465\pi\)
−0.581323 + 0.813673i \(0.697465\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −672.021 −0.448204 −0.224102 0.974566i \(-0.571945\pi\)
−0.224102 + 0.974566i \(0.571945\pi\)
\(132\) 0 0
\(133\) 296.324 0.193192
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1062.00 −0.662283 −0.331142 0.943581i \(-0.607434\pi\)
−0.331142 + 0.943581i \(0.607434\pi\)
\(138\) 0 0
\(139\) 2693.37 1.64352 0.821759 0.569835i \(-0.192993\pi\)
0.821759 + 0.569835i \(0.192993\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −840.000 −0.491219
\(144\) 0 0
\(145\) −1680.00 −0.962182
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −793.725 −0.436406 −0.218203 0.975903i \(-0.570020\pi\)
−0.218203 + 0.975903i \(0.570020\pi\)
\(150\) 0 0
\(151\) −744.000 −0.400966 −0.200483 0.979697i \(-0.564251\pi\)
−0.200483 + 0.979697i \(0.564251\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2370.59 1.22846
\(156\) 0 0
\(157\) −179.911 −0.0914552 −0.0457276 0.998954i \(-0.514561\pi\)
−0.0457276 + 0.998954i \(0.514561\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1216.00 −0.595244
\(162\) 0 0
\(163\) 1772.65 0.851809 0.425905 0.904768i \(-0.359956\pi\)
0.425905 + 0.904768i \(0.359956\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1960.00 −0.908200 −0.454100 0.890951i \(-0.650039\pi\)
−0.454100 + 0.890951i \(0.650039\pi\)
\(168\) 0 0
\(169\) 603.000 0.274465
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2000.19 0.879026 0.439513 0.898236i \(-0.355151\pi\)
0.439513 + 0.898236i \(0.355151\pi\)
\(174\) 0 0
\(175\) −104.000 −0.0449238
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3264.86 1.36328 0.681639 0.731688i \(-0.261267\pi\)
0.681639 + 0.731688i \(0.261267\pi\)
\(180\) 0 0
\(181\) −2338.84 −0.960469 −0.480235 0.877140i \(-0.659448\pi\)
−0.480235 + 0.877140i \(0.659448\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2576.00 1.02374
\(186\) 0 0
\(187\) 222.243 0.0869092
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3904.00 −1.47897 −0.739486 0.673172i \(-0.764931\pi\)
−0.739486 + 0.673172i \(0.764931\pi\)
\(192\) 0 0
\(193\) 3330.00 1.24196 0.620981 0.783826i \(-0.286734\pi\)
0.620981 + 0.783826i \(0.286734\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1195.88 0.432502 0.216251 0.976338i \(-0.430617\pi\)
0.216251 + 0.976338i \(0.430617\pi\)
\(198\) 0 0
\(199\) 1736.00 0.618401 0.309200 0.950997i \(-0.399939\pi\)
0.309200 + 0.950997i \(0.399939\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1269.96 −0.439083
\(204\) 0 0
\(205\) −740.810 −0.252392
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 588.000 0.194607
\(210\) 0 0
\(211\) 2915.62 0.951277 0.475638 0.879641i \(-0.342217\pi\)
0.475638 + 0.879641i \(0.342217\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4648.00 −1.47438
\(216\) 0 0
\(217\) 1792.00 0.560594
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −740.810 −0.225486
\(222\) 0 0
\(223\) 1568.00 0.470857 0.235428 0.971892i \(-0.424351\pi\)
0.235428 + 0.971892i \(0.424351\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1264.67 −0.369775 −0.184888 0.982760i \(-0.559192\pi\)
−0.184888 + 0.982760i \(0.559192\pi\)
\(228\) 0 0
\(229\) 5153.92 1.48725 0.743626 0.668595i \(-0.233104\pi\)
0.743626 + 0.668595i \(0.233104\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −838.000 −0.235619 −0.117809 0.993036i \(-0.537587\pi\)
−0.117809 + 0.993036i \(0.537587\pi\)
\(234\) 0 0
\(235\) −3555.89 −0.987067
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6288.00 −1.70183 −0.850914 0.525305i \(-0.823951\pi\)
−0.850914 + 0.525305i \(0.823951\pi\)
\(240\) 0 0
\(241\) −2926.00 −0.782076 −0.391038 0.920375i \(-0.627884\pi\)
−0.391038 + 0.920375i \(0.627884\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2952.66 −0.769953
\(246\) 0 0
\(247\) −1960.00 −0.504906
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5444.96 −1.36925 −0.684627 0.728894i \(-0.740035\pi\)
−0.684627 + 0.728894i \(0.740035\pi\)
\(252\) 0 0
\(253\) −2412.93 −0.599602
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2562.00 −0.621841 −0.310921 0.950436i \(-0.600637\pi\)
−0.310921 + 0.950436i \(0.600637\pi\)
\(258\) 0 0
\(259\) 1947.27 0.467172
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −5896.00 −1.38237 −0.691184 0.722679i \(-0.742911\pi\)
−0.691184 + 0.722679i \(0.742911\pi\)
\(264\) 0 0
\(265\) −336.000 −0.0778880
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5365.58 −1.21615 −0.608077 0.793878i \(-0.708059\pi\)
−0.608077 + 0.793878i \(0.708059\pi\)
\(270\) 0 0
\(271\) −1680.00 −0.376578 −0.188289 0.982114i \(-0.560294\pi\)
−0.188289 + 0.982114i \(0.560294\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −206.369 −0.0452527
\(276\) 0 0
\(277\) −1576.87 −0.342039 −0.171019 0.985268i \(-0.554706\pi\)
−0.171019 + 0.985268i \(0.554706\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2742.00 −0.582114 −0.291057 0.956706i \(-0.594007\pi\)
−0.291057 + 0.956706i \(0.594007\pi\)
\(282\) 0 0
\(283\) 2989.70 0.627983 0.313991 0.949426i \(-0.398334\pi\)
0.313991 + 0.949426i \(0.398334\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −560.000 −0.115177
\(288\) 0 0
\(289\) −4717.00 −0.960106
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9238.96 1.84214 0.921068 0.389401i \(-0.127318\pi\)
0.921068 + 0.389401i \(0.127318\pi\)
\(294\) 0 0
\(295\) −5656.00 −1.11629
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8043.08 1.55566
\(300\) 0 0
\(301\) −3513.56 −0.672818
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1008.00 −0.189239
\(306\) 0 0
\(307\) −2587.54 −0.481039 −0.240520 0.970644i \(-0.577318\pi\)
−0.240520 + 0.970644i \(0.577318\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2744.00 −0.500315 −0.250157 0.968205i \(-0.580482\pi\)
−0.250157 + 0.968205i \(0.580482\pi\)
\(312\) 0 0
\(313\) −2282.00 −0.412097 −0.206048 0.978542i \(-0.566060\pi\)
−0.206048 + 0.978542i \(0.566060\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9577.62 1.69695 0.848474 0.529237i \(-0.177522\pi\)
0.848474 + 0.529237i \(0.177522\pi\)
\(318\) 0 0
\(319\) −2520.00 −0.442298
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 518.567 0.0893308
\(324\) 0 0
\(325\) 687.895 0.117408
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2688.00 −0.450438
\(330\) 0 0
\(331\) −4249.08 −0.705590 −0.352795 0.935701i \(-0.614769\pi\)
−0.352795 + 0.935701i \(0.614769\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1848.00 −0.301394
\(336\) 0 0
\(337\) 6130.00 0.990868 0.495434 0.868646i \(-0.335009\pi\)
0.495434 + 0.868646i \(0.335009\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3555.89 0.564699
\(342\) 0 0
\(343\) −4976.00 −0.783320
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2481.71 −0.383935 −0.191967 0.981401i \(-0.561487\pi\)
−0.191967 + 0.981401i \(0.561487\pi\)
\(348\) 0 0
\(349\) 328.073 0.0503191 0.0251595 0.999683i \(-0.491991\pi\)
0.0251595 + 0.999683i \(0.491991\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10206.0 1.53884 0.769420 0.638743i \(-0.220545\pi\)
0.769420 + 0.638743i \(0.220545\pi\)
\(354\) 0 0
\(355\) −761.976 −0.113920
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3176.00 −0.466916 −0.233458 0.972367i \(-0.575004\pi\)
−0.233458 + 0.972367i \(0.575004\pi\)
\(360\) 0 0
\(361\) −5487.00 −0.799971
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3111.40 0.446187
\(366\) 0 0
\(367\) −11760.0 −1.67266 −0.836331 0.548225i \(-0.815304\pi\)
−0.836331 + 0.548225i \(0.815304\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −253.992 −0.0355434
\(372\) 0 0
\(373\) −10974.6 −1.52344 −0.761719 0.647908i \(-0.775644\pi\)
−0.761719 + 0.647908i \(0.775644\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8400.00 1.14754
\(378\) 0 0
\(379\) 3074.36 0.416674 0.208337 0.978057i \(-0.433195\pi\)
0.208337 + 0.978057i \(0.433195\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2688.00 −0.358617 −0.179309 0.983793i \(-0.557386\pi\)
−0.179309 + 0.983793i \(0.557386\pi\)
\(384\) 0 0
\(385\) 1344.00 0.177913
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10487.8 −1.36697 −0.683484 0.729966i \(-0.739536\pi\)
−0.683484 + 0.729966i \(0.739536\pi\)
\(390\) 0 0
\(391\) −2128.00 −0.275237
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4910.51 −0.625506
\(396\) 0 0
\(397\) 5704.24 0.721127 0.360564 0.932735i \(-0.382584\pi\)
0.360564 + 0.932735i \(0.382584\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12402.0 −1.54445 −0.772227 0.635346i \(-0.780857\pi\)
−0.772227 + 0.635346i \(0.780857\pi\)
\(402\) 0 0
\(403\) −11853.0 −1.46511
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3864.00 0.470593
\(408\) 0 0
\(409\) 12278.0 1.48437 0.742186 0.670194i \(-0.233789\pi\)
0.742186 + 0.670194i \(0.233789\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4275.53 −0.509407
\(414\) 0 0
\(415\) −5768.00 −0.682265
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −8207.12 −0.956907 −0.478454 0.878113i \(-0.658802\pi\)
−0.478454 + 0.878113i \(0.658802\pi\)
\(420\) 0 0
\(421\) −1449.87 −0.167844 −0.0839221 0.996472i \(-0.526745\pi\)
−0.0839221 + 0.996472i \(0.526745\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −182.000 −0.0207725
\(426\) 0 0
\(427\) −761.976 −0.0863574
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7632.00 −0.852948 −0.426474 0.904500i \(-0.640244\pi\)
−0.426474 + 0.904500i \(0.640244\pi\)
\(432\) 0 0
\(433\) 3794.00 0.421081 0.210540 0.977585i \(-0.432478\pi\)
0.210540 + 0.977585i \(0.432478\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5630.16 −0.616309
\(438\) 0 0
\(439\) 1848.00 0.200912 0.100456 0.994942i \(-0.467970\pi\)
0.100456 + 0.994942i \(0.467970\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12334.5 1.32287 0.661433 0.750004i \(-0.269949\pi\)
0.661433 + 0.750004i \(0.269949\pi\)
\(444\) 0 0
\(445\) 2815.08 0.299882
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3582.00 0.376492 0.188246 0.982122i \(-0.439720\pi\)
0.188246 + 0.982122i \(0.439720\pi\)
\(450\) 0 0
\(451\) −1111.22 −0.116020
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4480.00 −0.461595
\(456\) 0 0
\(457\) −2714.00 −0.277802 −0.138901 0.990306i \(-0.544357\pi\)
−0.138901 + 0.990306i \(0.544357\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8349.99 0.843596 0.421798 0.906690i \(-0.361399\pi\)
0.421798 + 0.906690i \(0.361399\pi\)
\(462\) 0 0
\(463\) 2224.00 0.223236 0.111618 0.993751i \(-0.464397\pi\)
0.111618 + 0.993751i \(0.464397\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10292.0 1.01982 0.509910 0.860228i \(-0.329679\pi\)
0.509910 + 0.860228i \(0.329679\pi\)
\(468\) 0 0
\(469\) −1396.96 −0.137538
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6972.00 −0.677744
\(474\) 0 0
\(475\) −481.527 −0.0465136
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −17696.0 −1.68800 −0.843999 0.536345i \(-0.819805\pi\)
−0.843999 + 0.536345i \(0.819805\pi\)
\(480\) 0 0
\(481\) −12880.0 −1.22095
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 10519.5 0.984879
\(486\) 0 0
\(487\) −1304.00 −0.121334 −0.0606672 0.998158i \(-0.519323\pi\)
−0.0606672 + 0.998158i \(0.519323\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −16662.9 −1.53154 −0.765772 0.643112i \(-0.777643\pi\)
−0.765772 + 0.643112i \(0.777643\pi\)
\(492\) 0 0
\(493\) −2222.43 −0.203029
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −576.000 −0.0519862
\(498\) 0 0
\(499\) −3095.53 −0.277705 −0.138853 0.990313i \(-0.544341\pi\)
−0.138853 + 0.990313i \(0.544341\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −19320.0 −1.71260 −0.856298 0.516481i \(-0.827242\pi\)
−0.856298 + 0.516481i \(0.827242\pi\)
\(504\) 0 0
\(505\) −7952.00 −0.700712
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4476.61 −0.389828 −0.194914 0.980820i \(-0.562443\pi\)
−0.194914 + 0.980820i \(0.562443\pi\)
\(510\) 0 0
\(511\) 2352.00 0.203613
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −12445.6 −1.06489
\(516\) 0 0
\(517\) −5333.83 −0.453737
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2982.00 −0.250756 −0.125378 0.992109i \(-0.540014\pi\)
−0.125378 + 0.992109i \(0.540014\pi\)
\(522\) 0 0
\(523\) 2016.06 0.168559 0.0842794 0.996442i \(-0.473141\pi\)
0.0842794 + 0.996442i \(0.473141\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3136.00 0.259215
\(528\) 0 0
\(529\) 10937.0 0.898907
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3704.05 0.301014
\(534\) 0 0
\(535\) 2856.00 0.230796
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4428.99 −0.353933
\(540\) 0 0
\(541\) −15419.4 −1.22539 −0.612693 0.790321i \(-0.709914\pi\)
−0.612693 + 0.790321i \(0.709914\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 20048.0 1.57571
\(546\) 0 0
\(547\) 12609.7 0.985649 0.492824 0.870129i \(-0.335965\pi\)
0.492824 + 0.870129i \(0.335965\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5880.00 −0.454621
\(552\) 0 0
\(553\) −3712.00 −0.285444
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7143.53 −0.543413 −0.271706 0.962380i \(-0.587588\pi\)
−0.271706 + 0.962380i \(0.587588\pi\)
\(558\) 0 0
\(559\) 23240.0 1.75840
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −7572.14 −0.566834 −0.283417 0.958997i \(-0.591468\pi\)
−0.283417 + 0.958997i \(0.591468\pi\)
\(564\) 0 0
\(565\) 18096.9 1.34751
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15594.0 1.14892 0.574459 0.818533i \(-0.305212\pi\)
0.574459 + 0.818533i \(0.305212\pi\)
\(570\) 0 0
\(571\) −16737.0 −1.22666 −0.613330 0.789827i \(-0.710170\pi\)
−0.613330 + 0.789827i \(0.710170\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1976.00 0.143313
\(576\) 0 0
\(577\) 6594.00 0.475757 0.237879 0.971295i \(-0.423548\pi\)
0.237879 + 0.971295i \(0.423548\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −4360.20 −0.311345
\(582\) 0 0
\(583\) −504.000 −0.0358037
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23213.8 1.63226 0.816130 0.577868i \(-0.196115\pi\)
0.816130 + 0.577868i \(0.196115\pi\)
\(588\) 0 0
\(589\) 8297.08 0.580433
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −14322.0 −0.991794 −0.495897 0.868381i \(-0.665161\pi\)
−0.495897 + 0.868381i \(0.665161\pi\)
\(594\) 0 0
\(595\) 1185.30 0.0816679
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −16088.0 −1.09739 −0.548696 0.836022i \(-0.684876\pi\)
−0.548696 + 0.836022i \(0.684876\pi\)
\(600\) 0 0
\(601\) 21238.0 1.44146 0.720729 0.693217i \(-0.243807\pi\)
0.720729 + 0.693217i \(0.243807\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −11419.1 −0.767357
\(606\) 0 0
\(607\) −13664.0 −0.913681 −0.456841 0.889549i \(-0.651019\pi\)
−0.456841 + 0.889549i \(0.651019\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 17779.4 1.17722
\(612\) 0 0
\(613\) 20393.5 1.34369 0.671846 0.740690i \(-0.265501\pi\)
0.671846 + 0.740690i \(0.265501\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3782.00 −0.246771 −0.123385 0.992359i \(-0.539375\pi\)
−0.123385 + 0.992359i \(0.539375\pi\)
\(618\) 0 0
\(619\) 5825.94 0.378295 0.189147 0.981949i \(-0.439428\pi\)
0.189147 + 0.981949i \(0.439428\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2128.00 0.136848
\(624\) 0 0
\(625\) −13831.0 −0.885184
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3407.73 0.216017
\(630\) 0 0
\(631\) −2056.00 −0.129712 −0.0648558 0.997895i \(-0.520659\pi\)
−0.0648558 + 0.997895i \(0.520659\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −17610.1 −1.10053
\(636\) 0 0
\(637\) 14763.3 0.918278
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −11842.0 −0.729689 −0.364845 0.931068i \(-0.618878\pi\)
−0.364845 + 0.931068i \(0.618878\pi\)
\(642\) 0 0
\(643\) 16250.2 0.996649 0.498325 0.866991i \(-0.333949\pi\)
0.498325 + 0.866991i \(0.333949\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19320.0 1.17395 0.586976 0.809604i \(-0.300318\pi\)
0.586976 + 0.809604i \(0.300318\pi\)
\(648\) 0 0
\(649\) −8484.00 −0.513137
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2317.68 −0.138894 −0.0694470 0.997586i \(-0.522123\pi\)
−0.0694470 + 0.997586i \(0.522123\pi\)
\(654\) 0 0
\(655\) −7112.00 −0.424258
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 27732.8 1.63932 0.819662 0.572847i \(-0.194161\pi\)
0.819662 + 0.572847i \(0.194161\pi\)
\(660\) 0 0
\(661\) 22467.7 1.32208 0.661039 0.750352i \(-0.270116\pi\)
0.661039 + 0.750352i \(0.270116\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3136.00 0.182870
\(666\) 0 0
\(667\) 24129.3 1.40073
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1512.00 −0.0869897
\(672\) 0 0
\(673\) −10078.0 −0.577234 −0.288617 0.957445i \(-0.593195\pi\)
−0.288617 + 0.957445i \(0.593195\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −16160.2 −0.917413 −0.458707 0.888588i \(-0.651687\pi\)
−0.458707 + 0.888588i \(0.651687\pi\)
\(678\) 0 0
\(679\) 7952.00 0.449440
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 16356.0 0.916320 0.458160 0.888870i \(-0.348509\pi\)
0.458160 + 0.888870i \(0.348509\pi\)
\(684\) 0 0
\(685\) −11239.2 −0.626899
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1680.00 0.0928925
\(690\) 0 0
\(691\) 29246.1 1.61009 0.805047 0.593211i \(-0.202140\pi\)
0.805047 + 0.593211i \(0.202140\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 28504.0 1.55571
\(696\) 0 0
\(697\) −980.000 −0.0532570
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2465.84 0.132858 0.0664290 0.997791i \(-0.478839\pi\)
0.0664290 + 0.997791i \(0.478839\pi\)
\(702\) 0 0
\(703\) 9016.00 0.483705
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6011.15 −0.319763
\(708\) 0 0
\(709\) −31674.9 −1.67782 −0.838912 0.544267i \(-0.816808\pi\)
−0.838912 + 0.544267i \(0.816808\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −34048.0 −1.78837
\(714\) 0 0
\(715\) −8889.72 −0.464975
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 9296.00 0.482173 0.241086 0.970504i \(-0.422496\pi\)
0.241086 + 0.970504i \(0.422496\pi\)
\(720\) 0 0
\(721\) −9408.00 −0.485953
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2063.69 0.105715
\(726\) 0 0
\(727\) −21672.0 −1.10560 −0.552799 0.833315i \(-0.686440\pi\)
−0.552799 + 0.833315i \(0.686440\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −6148.73 −0.311106
\(732\) 0 0
\(733\) 9471.79 0.477283 0.238642 0.971108i \(-0.423298\pi\)
0.238642 + 0.971108i \(0.423298\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2772.00 −0.138545
\(738\) 0 0
\(739\) −6863.08 −0.341627 −0.170814 0.985303i \(-0.554640\pi\)
−0.170814 + 0.985303i \(0.554640\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 17432.0 0.860724 0.430362 0.902656i \(-0.358386\pi\)
0.430362 + 0.902656i \(0.358386\pi\)
\(744\) 0 0
\(745\) −8400.00 −0.413090
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2158.93 0.105321
\(750\) 0 0
\(751\) −11632.0 −0.565190 −0.282595 0.959239i \(-0.591195\pi\)
−0.282595 + 0.959239i \(0.591195\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −7873.76 −0.379543
\(756\) 0 0
\(757\) −16731.7 −0.803336 −0.401668 0.915785i \(-0.631569\pi\)
−0.401668 + 0.915785i \(0.631569\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 39466.0 1.87995 0.939975 0.341244i \(-0.110848\pi\)
0.939975 + 0.341244i \(0.110848\pi\)
\(762\) 0 0
\(763\) 15154.9 0.719060
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 28280.0 1.33133
\(768\) 0 0
\(769\) 35266.0 1.65374 0.826869 0.562395i \(-0.190120\pi\)
0.826869 + 0.562395i \(0.190120\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −16244.9 −0.755872 −0.377936 0.925832i \(-0.623366\pi\)
−0.377936 + 0.925832i \(0.623366\pi\)
\(774\) 0 0
\(775\) −2912.00 −0.134970
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2592.84 −0.119253
\(780\) 0 0
\(781\) −1142.96 −0.0523668
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1904.00 −0.0865690
\(786\) 0 0
\(787\) −34844.5 −1.57824 −0.789119 0.614240i \(-0.789463\pi\)
−0.789119 + 0.614240i \(0.789463\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 13680.0 0.614924
\(792\) 0 0
\(793\) 5040.00 0.225694
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2550.50 0.113354 0.0566772 0.998393i \(-0.481949\pi\)
0.0566772 + 0.998393i \(0.481949\pi\)
\(798\) 0 0
\(799\) −4704.00 −0.208280
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4667.11 0.205104
\(804\) 0 0
\(805\) −12868.9 −0.563441
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −24390.0 −1.05996 −0.529979 0.848010i \(-0.677800\pi\)
−0.529979 + 0.848010i \(0.677800\pi\)
\(810\) 0 0
\(811\) −9582.91 −0.414922 −0.207461 0.978243i \(-0.566520\pi\)
−0.207461 + 0.978243i \(0.566520\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 18760.0 0.806300
\(816\) 0 0
\(817\) −16268.0 −0.696628
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8773.31 0.372948 0.186474 0.982460i \(-0.440294\pi\)
0.186474 + 0.982460i \(0.440294\pi\)
\(822\) 0 0
\(823\) 21688.0 0.918586 0.459293 0.888285i \(-0.348103\pi\)
0.459293 + 0.888285i \(0.348103\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 19446.3 0.817670 0.408835 0.912608i \(-0.365935\pi\)
0.408835 + 0.912608i \(0.365935\pi\)
\(828\) 0 0
\(829\) 19546.8 0.818925 0.409462 0.912327i \(-0.365716\pi\)
0.409462 + 0.912327i \(0.365716\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3906.00 −0.162467
\(834\) 0 0
\(835\) −20742.7 −0.859677
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −18760.0 −0.771951 −0.385976 0.922509i \(-0.626135\pi\)
−0.385976 + 0.922509i \(0.626135\pi\)
\(840\) 0 0
\(841\) 811.000 0.0332527
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6381.55 0.259801
\(846\) 0 0
\(847\) −8632.00 −0.350176
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −36998.2 −1.49034
\(852\) 0 0
\(853\) 28732.9 1.15333 0.576667 0.816979i \(-0.304353\pi\)
0.576667 + 0.816979i \(0.304353\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 8778.00 0.349884 0.174942 0.984579i \(-0.444026\pi\)
0.174942 + 0.984579i \(0.444026\pi\)
\(858\) 0 0
\(859\) −5646.03 −0.224261 −0.112130 0.993693i \(-0.535767\pi\)
−0.112130 + 0.993693i \(0.535767\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9312.00 0.367305 0.183652 0.982991i \(-0.441208\pi\)
0.183652 + 0.982991i \(0.441208\pi\)
\(864\) 0 0
\(865\) 21168.0 0.832062
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −7365.77 −0.287534
\(870\) 0 0
\(871\) 9240.00 0.359455
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −11683.6 −0.451405
\(876\) 0 0
\(877\) −137.579 −0.00529728 −0.00264864 0.999996i \(-0.500843\pi\)
−0.00264864 + 0.999996i \(0.500843\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 31150.0 1.19123 0.595613 0.803272i \(-0.296909\pi\)
0.595613 + 0.803272i \(0.296909\pi\)
\(882\) 0 0
\(883\) −12577.9 −0.479366 −0.239683 0.970851i \(-0.577043\pi\)
−0.239683 + 0.970851i \(0.577043\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 37128.0 1.40545 0.702726 0.711460i \(-0.251966\pi\)
0.702726 + 0.711460i \(0.251966\pi\)
\(888\) 0 0
\(889\) −13312.0 −0.502216
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −12445.6 −0.466379
\(894\) 0 0
\(895\) 34552.0 1.29044
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −35558.9 −1.31919
\(900\) 0 0
\(901\) −444.486 −0.0164351
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −24752.0 −0.909154
\(906\) 0 0
\(907\) 35204.4 1.28880 0.644400 0.764688i \(-0.277107\pi\)
0.644400 + 0.764688i \(0.277107\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 10512.0 0.382303 0.191152 0.981561i \(-0.438778\pi\)
0.191152 + 0.981561i \(0.438778\pi\)
\(912\) 0 0
\(913\) −8652.00 −0.313625
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −5376.17 −0.193606
\(918\) 0 0
\(919\) 46104.0 1.65488 0.827438 0.561557i \(-0.189798\pi\)
0.827438 + 0.561557i \(0.189798\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3809.88 0.135865
\(924\) 0 0
\(925\) −3164.32 −0.112478
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 5726.00 0.202222 0.101111 0.994875i \(-0.467760\pi\)
0.101111 + 0.994875i \(0.467760\pi\)
\(930\) 0 0
\(931\) −10334.3 −0.363795
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2352.00 0.0822659
\(936\) 0 0
\(937\) −1274.00 −0.0444181 −0.0222091 0.999753i \(-0.507070\pi\)
−0.0222091 + 0.999753i \(0.507070\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −26446.9 −0.916201 −0.458101 0.888900i \(-0.651470\pi\)
−0.458101 + 0.888900i \(0.651470\pi\)
\(942\) 0 0
\(943\) 10640.0 0.367430
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 23922.9 0.820897 0.410448 0.911884i \(-0.365372\pi\)
0.410448 + 0.911884i \(0.365372\pi\)
\(948\) 0 0
\(949\) −15557.0 −0.532141
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 38250.0 1.30015 0.650073 0.759872i \(-0.274738\pi\)
0.650073 + 0.759872i \(0.274738\pi\)
\(954\) 0 0
\(955\) −41316.1 −1.39995
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8496.00 −0.286079
\(960\) 0 0
\(961\) 20385.0 0.684267
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 35241.4 1.17561
\(966\) 0 0
\(967\) −4664.00 −0.155103 −0.0775513 0.996988i \(-0.524710\pi\)
−0.0775513 + 0.996988i \(0.524710\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −30971.2 −1.02360 −0.511798 0.859106i \(-0.671020\pi\)
−0.511798 + 0.859106i \(0.671020\pi\)
\(972\) 0 0
\(973\) 21547.0 0.709933
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4814.00 0.157639 0.0788196 0.996889i \(-0.474885\pi\)
0.0788196 + 0.996889i \(0.474885\pi\)
\(978\) 0 0
\(979\) 4222.62 0.137850
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −12376.0 −0.401560 −0.200780 0.979636i \(-0.564348\pi\)
−0.200780 + 0.979636i \(0.564348\pi\)
\(984\) 0 0
\(985\) 12656.0 0.409395
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 66757.6 2.14638
\(990\) 0 0
\(991\) 45344.0 1.45348 0.726740 0.686912i \(-0.241034\pi\)
0.726740 + 0.686912i \(0.241034\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 18372.1 0.585361
\(996\) 0 0
\(997\) −26002.4 −0.825984 −0.412992 0.910735i \(-0.635516\pi\)
−0.412992 + 0.910735i \(0.635516\pi\)
\(998\) 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.bn.1.2 2
3.2 odd 2 256.4.a.l.1.1 2
4.3 odd 2 2304.4.a.v.1.2 2
8.3 odd 2 2304.4.a.v.1.1 2
8.5 even 2 inner 2304.4.a.bn.1.1 2
12.11 even 2 256.4.a.j.1.2 2
16.3 odd 4 288.4.d.a.145.1 2
16.5 even 4 72.4.d.b.37.2 2
16.11 odd 4 288.4.d.a.145.2 2
16.13 even 4 72.4.d.b.37.1 2
24.5 odd 2 256.4.a.l.1.2 2
24.11 even 2 256.4.a.j.1.1 2
48.5 odd 4 8.4.b.a.5.1 2
48.11 even 4 32.4.b.a.17.1 2
48.29 odd 4 8.4.b.a.5.2 yes 2
48.35 even 4 32.4.b.a.17.2 2
240.29 odd 4 200.4.d.a.101.1 2
240.53 even 4 200.4.f.a.149.2 4
240.59 even 4 800.4.d.a.401.2 2
240.77 even 4 200.4.f.a.149.1 4
240.83 odd 4 800.4.f.a.49.1 4
240.107 odd 4 800.4.f.a.49.2 4
240.149 odd 4 200.4.d.a.101.2 2
240.173 even 4 200.4.f.a.149.4 4
240.179 even 4 800.4.d.a.401.1 2
240.197 even 4 200.4.f.a.149.3 4
240.203 odd 4 800.4.f.a.49.3 4
240.227 odd 4 800.4.f.a.49.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.4.b.a.5.1 2 48.5 odd 4
8.4.b.a.5.2 yes 2 48.29 odd 4
32.4.b.a.17.1 2 48.11 even 4
32.4.b.a.17.2 2 48.35 even 4
72.4.d.b.37.1 2 16.13 even 4
72.4.d.b.37.2 2 16.5 even 4
200.4.d.a.101.1 2 240.29 odd 4
200.4.d.a.101.2 2 240.149 odd 4
200.4.f.a.149.1 4 240.77 even 4
200.4.f.a.149.2 4 240.53 even 4
200.4.f.a.149.3 4 240.197 even 4
200.4.f.a.149.4 4 240.173 even 4
256.4.a.j.1.1 2 24.11 even 2
256.4.a.j.1.2 2 12.11 even 2
256.4.a.l.1.1 2 3.2 odd 2
256.4.a.l.1.2 2 24.5 odd 2
288.4.d.a.145.1 2 16.3 odd 4
288.4.d.a.145.2 2 16.11 odd 4
800.4.d.a.401.1 2 240.179 even 4
800.4.d.a.401.2 2 240.59 even 4
800.4.f.a.49.1 4 240.83 odd 4
800.4.f.a.49.2 4 240.107 odd 4
800.4.f.a.49.3 4 240.203 odd 4
800.4.f.a.49.4 4 240.227 odd 4
2304.4.a.v.1.1 2 8.3 odd 2
2304.4.a.v.1.2 2 4.3 odd 2
2304.4.a.bn.1.1 2 8.5 even 2 inner
2304.4.a.bn.1.2 2 1.1 even 1 trivial