Properties

Label 1152.4.a.s.1.1
Level $1152$
Weight $4$
Character 1152.1
Self dual yes
Analytic conductor $67.970$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,4,Mod(1,1152)]
 
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, 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("1152.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1152.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: \(67.9702003266\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1152.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-11.8564 q^{5} +9.85641 q^{7} +O(q^{10})\) \(q-11.8564 q^{5} +9.85641 q^{7} +39.0718 q^{11} -91.5692 q^{13} +37.1384 q^{17} +46.4974 q^{19} +120.708 q^{23} +15.5744 q^{25} -27.2820 q^{29} -81.1487 q^{31} -116.862 q^{35} -10.9948 q^{37} +205.426 q^{41} -115.359 q^{43} +312.841 q^{47} -245.851 q^{49} +90.9948 q^{53} -463.251 q^{55} -550.631 q^{59} -630.974 q^{61} +1085.68 q^{65} +661.041 q^{67} -494.985 q^{71} +566.267 q^{73} +385.108 q^{77} +49.4153 q^{79} +564.067 q^{83} -440.328 q^{85} -1089.67 q^{89} -902.543 q^{91} -551.292 q^{95} +464.605 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} - 8 q^{7} + 92 q^{11} - 100 q^{13} - 92 q^{17} - 4 q^{19} - 8 q^{23} + 142 q^{25} + 84 q^{29} - 384 q^{31} - 400 q^{35} + 172 q^{37} + 300 q^{41} - 300 q^{43} + 16 q^{47} - 270 q^{49} - 12 q^{53} + 376 q^{55} - 644 q^{59} - 292 q^{61} + 952 q^{65} + 172 q^{67} - 408 q^{71} + 412 q^{73} - 560 q^{77} - 400 q^{79} + 948 q^{83} - 2488 q^{85} - 572 q^{89} - 752 q^{91} - 1352 q^{95} + 2204 q^{97}+O(q^{100}) \) 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
\(5\) −11.8564 −1.06047 −0.530235 0.847851i \(-0.677896\pi\)
−0.530235 + 0.847851i \(0.677896\pi\)
\(6\) 0 0
\(7\) 9.85641 0.532196 0.266098 0.963946i \(-0.414266\pi\)
0.266098 + 0.963946i \(0.414266\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 39.0718 1.07096 0.535481 0.844547i \(-0.320130\pi\)
0.535481 + 0.844547i \(0.320130\pi\)
\(12\) 0 0
\(13\) −91.5692 −1.95359 −0.976797 0.214166i \(-0.931297\pi\)
−0.976797 + 0.214166i \(0.931297\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 37.1384 0.529847 0.264923 0.964269i \(-0.414653\pi\)
0.264923 + 0.964269i \(0.414653\pi\)
\(18\) 0 0
\(19\) 46.4974 0.561434 0.280717 0.959791i \(-0.409428\pi\)
0.280717 + 0.959791i \(0.409428\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 120.708 1.09432 0.547158 0.837029i \(-0.315710\pi\)
0.547158 + 0.837029i \(0.315710\pi\)
\(24\) 0 0
\(25\) 15.5744 0.124595
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −27.2820 −0.174695 −0.0873473 0.996178i \(-0.527839\pi\)
−0.0873473 + 0.996178i \(0.527839\pi\)
\(30\) 0 0
\(31\) −81.1487 −0.470153 −0.235077 0.971977i \(-0.575534\pi\)
−0.235077 + 0.971977i \(0.575534\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −116.862 −0.564377
\(36\) 0 0
\(37\) −10.9948 −0.0488525 −0.0244262 0.999702i \(-0.507776\pi\)
−0.0244262 + 0.999702i \(0.507776\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 205.426 0.782490 0.391245 0.920287i \(-0.372044\pi\)
0.391245 + 0.920287i \(0.372044\pi\)
\(42\) 0 0
\(43\) −115.359 −0.409118 −0.204559 0.978854i \(-0.565576\pi\)
−0.204559 + 0.978854i \(0.565576\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 312.841 0.970905 0.485453 0.874263i \(-0.338655\pi\)
0.485453 + 0.874263i \(0.338655\pi\)
\(48\) 0 0
\(49\) −245.851 −0.716767
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 90.9948 0.235832 0.117916 0.993024i \(-0.462379\pi\)
0.117916 + 0.993024i \(0.462379\pi\)
\(54\) 0 0
\(55\) −463.251 −1.13572
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −550.631 −1.21502 −0.607509 0.794313i \(-0.707831\pi\)
−0.607509 + 0.794313i \(0.707831\pi\)
\(60\) 0 0
\(61\) −630.974 −1.32439 −0.662196 0.749330i \(-0.730376\pi\)
−0.662196 + 0.749330i \(0.730376\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1085.68 2.07173
\(66\) 0 0
\(67\) 661.041 1.20536 0.602679 0.797984i \(-0.294100\pi\)
0.602679 + 0.797984i \(0.294100\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −494.985 −0.827378 −0.413689 0.910418i \(-0.635760\pi\)
−0.413689 + 0.910418i \(0.635760\pi\)
\(72\) 0 0
\(73\) 566.267 0.907897 0.453949 0.891028i \(-0.350015\pi\)
0.453949 + 0.891028i \(0.350015\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 385.108 0.569962
\(78\) 0 0
\(79\) 49.4153 0.0703754 0.0351877 0.999381i \(-0.488797\pi\)
0.0351877 + 0.999381i \(0.488797\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 564.067 0.745956 0.372978 0.927840i \(-0.378337\pi\)
0.372978 + 0.927840i \(0.378337\pi\)
\(84\) 0 0
\(85\) −440.328 −0.561886
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1089.67 −1.29781 −0.648904 0.760870i \(-0.724772\pi\)
−0.648904 + 0.760870i \(0.724772\pi\)
\(90\) 0 0
\(91\) −902.543 −1.03970
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −551.292 −0.595383
\(96\) 0 0
\(97\) 464.605 0.486325 0.243162 0.969986i \(-0.421815\pi\)
0.243162 + 0.969986i \(0.421815\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −965.005 −0.950709 −0.475354 0.879794i \(-0.657680\pi\)
−0.475354 + 0.879794i \(0.657680\pi\)
\(102\) 0 0
\(103\) −1828.99 −1.74967 −0.874836 0.484419i \(-0.839031\pi\)
−0.874836 + 0.484419i \(0.839031\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1726.59 −1.55996 −0.779980 0.625805i \(-0.784771\pi\)
−0.779980 + 0.625805i \(0.784771\pi\)
\(108\) 0 0
\(109\) −423.015 −0.371720 −0.185860 0.982576i \(-0.559507\pi\)
−0.185860 + 0.982576i \(0.559507\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −763.108 −0.635284 −0.317642 0.948211i \(-0.602891\pi\)
−0.317642 + 0.948211i \(0.602891\pi\)
\(114\) 0 0
\(115\) −1431.16 −1.16049
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 366.052 0.281982
\(120\) 0 0
\(121\) 195.605 0.146961
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1297.39 0.928340
\(126\) 0 0
\(127\) −2159.38 −1.50877 −0.754387 0.656430i \(-0.772066\pi\)
−0.754387 + 0.656430i \(0.772066\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −710.723 −0.474016 −0.237008 0.971508i \(-0.576167\pi\)
−0.237008 + 0.971508i \(0.576167\pi\)
\(132\) 0 0
\(133\) 458.297 0.298793
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 987.128 0.615592 0.307796 0.951452i \(-0.400409\pi\)
0.307796 + 0.951452i \(0.400409\pi\)
\(138\) 0 0
\(139\) −2334.15 −1.42432 −0.712158 0.702019i \(-0.752282\pi\)
−0.712158 + 0.702019i \(0.752282\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3577.77 −2.09223
\(144\) 0 0
\(145\) 323.467 0.185258
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1792.29 −0.985435 −0.492718 0.870189i \(-0.663996\pi\)
−0.492718 + 0.870189i \(0.663996\pi\)
\(150\) 0 0
\(151\) 2864.32 1.54367 0.771837 0.635820i \(-0.219338\pi\)
0.771837 + 0.635820i \(0.219338\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 962.133 0.498583
\(156\) 0 0
\(157\) −2123.57 −1.07949 −0.539743 0.841830i \(-0.681479\pi\)
−0.539743 + 0.841830i \(0.681479\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1189.74 0.582391
\(162\) 0 0
\(163\) −3164.21 −1.52049 −0.760245 0.649636i \(-0.774921\pi\)
−0.760245 + 0.649636i \(0.774921\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1340.34 0.621069 0.310534 0.950562i \(-0.399492\pi\)
0.310534 + 0.950562i \(0.399492\pi\)
\(168\) 0 0
\(169\) 6187.92 2.81653
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2802.13 −1.23146 −0.615729 0.787958i \(-0.711138\pi\)
−0.615729 + 0.787958i \(0.711138\pi\)
\(174\) 0 0
\(175\) 153.507 0.0663089
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1074.84 0.448810 0.224405 0.974496i \(-0.427956\pi\)
0.224405 + 0.974496i \(0.427956\pi\)
\(180\) 0 0
\(181\) −571.036 −0.234502 −0.117251 0.993102i \(-0.537408\pi\)
−0.117251 + 0.993102i \(0.537408\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 130.359 0.0518065
\(186\) 0 0
\(187\) 1451.07 0.567446
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −872.657 −0.330593 −0.165296 0.986244i \(-0.552858\pi\)
−0.165296 + 0.986244i \(0.552858\pi\)
\(192\) 0 0
\(193\) 672.523 0.250825 0.125413 0.992105i \(-0.459975\pi\)
0.125413 + 0.992105i \(0.459975\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3129.56 −1.13184 −0.565918 0.824461i \(-0.691478\pi\)
−0.565918 + 0.824461i \(0.691478\pi\)
\(198\) 0 0
\(199\) 1502.37 0.535176 0.267588 0.963533i \(-0.413773\pi\)
0.267588 + 0.963533i \(0.413773\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −268.903 −0.0929718
\(204\) 0 0
\(205\) −2435.61 −0.829807
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1816.74 0.601275
\(210\) 0 0
\(211\) 1773.70 0.578703 0.289352 0.957223i \(-0.406560\pi\)
0.289352 + 0.957223i \(0.406560\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1367.74 0.433857
\(216\) 0 0
\(217\) −799.835 −0.250214
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3400.74 −1.03511
\(222\) 0 0
\(223\) −5413.66 −1.62568 −0.812838 0.582490i \(-0.802078\pi\)
−0.812838 + 0.582490i \(0.802078\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3744.27 1.09478 0.547392 0.836876i \(-0.315621\pi\)
0.547392 + 0.836876i \(0.315621\pi\)
\(228\) 0 0
\(229\) −680.739 −0.196439 −0.0982194 0.995165i \(-0.531315\pi\)
−0.0982194 + 0.995165i \(0.531315\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 679.271 0.190989 0.0954947 0.995430i \(-0.469557\pi\)
0.0954947 + 0.995430i \(0.469557\pi\)
\(234\) 0 0
\(235\) −3709.17 −1.02962
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3901.27 −1.05587 −0.527934 0.849286i \(-0.677033\pi\)
−0.527934 + 0.849286i \(0.677033\pi\)
\(240\) 0 0
\(241\) 5859.72 1.56622 0.783108 0.621886i \(-0.213633\pi\)
0.783108 + 0.621886i \(0.213633\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2914.91 0.760110
\(246\) 0 0
\(247\) −4257.73 −1.09681
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3381.48 −0.850348 −0.425174 0.905112i \(-0.639787\pi\)
−0.425174 + 0.905112i \(0.639787\pi\)
\(252\) 0 0
\(253\) 4716.27 1.17197
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3652.22 −0.886455 −0.443227 0.896409i \(-0.646167\pi\)
−0.443227 + 0.896409i \(0.646167\pi\)
\(258\) 0 0
\(259\) −108.370 −0.0259991
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2462.74 −0.577410 −0.288705 0.957418i \(-0.593225\pi\)
−0.288705 + 0.957418i \(0.593225\pi\)
\(264\) 0 0
\(265\) −1078.87 −0.250093
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 482.503 0.109363 0.0546816 0.998504i \(-0.482586\pi\)
0.0546816 + 0.998504i \(0.482586\pi\)
\(270\) 0 0
\(271\) −3602.48 −0.807510 −0.403755 0.914867i \(-0.632295\pi\)
−0.403755 + 0.914867i \(0.632295\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 608.519 0.133437
\(276\) 0 0
\(277\) −6177.72 −1.34001 −0.670006 0.742356i \(-0.733709\pi\)
−0.670006 + 0.742356i \(0.733709\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 543.477 0.115378 0.0576888 0.998335i \(-0.481627\pi\)
0.0576888 + 0.998335i \(0.481627\pi\)
\(282\) 0 0
\(283\) −6582.11 −1.38256 −0.691282 0.722585i \(-0.742954\pi\)
−0.691282 + 0.722585i \(0.742954\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2024.76 0.416438
\(288\) 0 0
\(289\) −3533.74 −0.719262
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6753.68 1.34660 0.673301 0.739369i \(-0.264876\pi\)
0.673301 + 0.739369i \(0.264876\pi\)
\(294\) 0 0
\(295\) 6528.50 1.28849
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −11053.1 −2.13785
\(300\) 0 0
\(301\) −1137.03 −0.217731
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7481.09 1.40448
\(306\) 0 0
\(307\) 1821.88 0.338698 0.169349 0.985556i \(-0.445833\pi\)
0.169349 + 0.985556i \(0.445833\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10598.2 1.93237 0.966184 0.257852i \(-0.0830148\pi\)
0.966184 + 0.257852i \(0.0830148\pi\)
\(312\) 0 0
\(313\) −7969.69 −1.43921 −0.719606 0.694382i \(-0.755678\pi\)
−0.719606 + 0.694382i \(0.755678\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7773.66 1.37733 0.688663 0.725082i \(-0.258198\pi\)
0.688663 + 0.725082i \(0.258198\pi\)
\(318\) 0 0
\(319\) −1065.96 −0.187092
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1726.84 0.297474
\(324\) 0 0
\(325\) −1426.13 −0.243408
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3083.49 0.516712
\(330\) 0 0
\(331\) 8212.87 1.36381 0.681903 0.731442i \(-0.261152\pi\)
0.681903 + 0.731442i \(0.261152\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7837.57 −1.27825
\(336\) 0 0
\(337\) 5260.17 0.850267 0.425133 0.905131i \(-0.360227\pi\)
0.425133 + 0.905131i \(0.360227\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3170.63 −0.503516
\(342\) 0 0
\(343\) −5803.96 −0.913657
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2162.12 −0.334492 −0.167246 0.985915i \(-0.553487\pi\)
−0.167246 + 0.985915i \(0.553487\pi\)
\(348\) 0 0
\(349\) −2888.90 −0.443093 −0.221546 0.975150i \(-0.571110\pi\)
−0.221546 + 0.975150i \(0.571110\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −11550.2 −1.74152 −0.870760 0.491709i \(-0.836372\pi\)
−0.870760 + 0.491709i \(0.836372\pi\)
\(354\) 0 0
\(355\) 5868.74 0.877409
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6523.97 0.959114 0.479557 0.877511i \(-0.340797\pi\)
0.479557 + 0.877511i \(0.340797\pi\)
\(360\) 0 0
\(361\) −4696.99 −0.684792
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6713.89 −0.962797
\(366\) 0 0
\(367\) 6856.96 0.975287 0.487644 0.873043i \(-0.337857\pi\)
0.487644 + 0.873043i \(0.337857\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 896.882 0.125509
\(372\) 0 0
\(373\) 5001.68 0.694309 0.347155 0.937808i \(-0.387148\pi\)
0.347155 + 0.937808i \(0.387148\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2498.19 0.341283
\(378\) 0 0
\(379\) −84.9775 −0.0115172 −0.00575858 0.999983i \(-0.501833\pi\)
−0.00575858 + 0.999983i \(0.501833\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2596.76 −0.346444 −0.173222 0.984883i \(-0.555418\pi\)
−0.173222 + 0.984883i \(0.555418\pi\)
\(384\) 0 0
\(385\) −4565.99 −0.604427
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9213.21 1.20084 0.600422 0.799683i \(-0.294999\pi\)
0.600422 + 0.799683i \(0.294999\pi\)
\(390\) 0 0
\(391\) 4482.89 0.579820
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −585.888 −0.0746310
\(396\) 0 0
\(397\) 6614.19 0.836163 0.418082 0.908409i \(-0.362703\pi\)
0.418082 + 0.908409i \(0.362703\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3434.94 −0.427763 −0.213881 0.976860i \(-0.568611\pi\)
−0.213881 + 0.976860i \(0.568611\pi\)
\(402\) 0 0
\(403\) 7430.73 0.918489
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −429.588 −0.0523192
\(408\) 0 0
\(409\) −6435.82 −0.778071 −0.389035 0.921223i \(-0.627192\pi\)
−0.389035 + 0.921223i \(0.627192\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5427.24 −0.646627
\(414\) 0 0
\(415\) −6687.80 −0.791064
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4222.04 0.492267 0.246134 0.969236i \(-0.420840\pi\)
0.246134 + 0.969236i \(0.420840\pi\)
\(420\) 0 0
\(421\) −5164.92 −0.597917 −0.298958 0.954266i \(-0.596639\pi\)
−0.298958 + 0.954266i \(0.596639\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 578.408 0.0660163
\(426\) 0 0
\(427\) −6219.14 −0.704837
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7476.41 −0.835559 −0.417780 0.908548i \(-0.637192\pi\)
−0.417780 + 0.908548i \(0.637192\pi\)
\(432\) 0 0
\(433\) −11289.7 −1.25299 −0.626497 0.779424i \(-0.715512\pi\)
−0.626497 + 0.779424i \(0.715512\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5612.59 0.614386
\(438\) 0 0
\(439\) −1800.79 −0.195779 −0.0978895 0.995197i \(-0.531209\pi\)
−0.0978895 + 0.995197i \(0.531209\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7020.76 −0.752972 −0.376486 0.926422i \(-0.622868\pi\)
−0.376486 + 0.926422i \(0.622868\pi\)
\(444\) 0 0
\(445\) 12919.6 1.37629
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3318.29 0.348774 0.174387 0.984677i \(-0.444206\pi\)
0.174387 + 0.984677i \(0.444206\pi\)
\(450\) 0 0
\(451\) 8026.35 0.838018
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 10700.9 1.10256
\(456\) 0 0
\(457\) 4468.38 0.457378 0.228689 0.973500i \(-0.426556\pi\)
0.228689 + 0.973500i \(0.426556\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1179.45 0.119159 0.0595795 0.998224i \(-0.481024\pi\)
0.0595795 + 0.998224i \(0.481024\pi\)
\(462\) 0 0
\(463\) 13763.1 1.38148 0.690742 0.723101i \(-0.257284\pi\)
0.690742 + 0.723101i \(0.257284\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17014.1 1.68591 0.842955 0.537984i \(-0.180814\pi\)
0.842955 + 0.537984i \(0.180814\pi\)
\(468\) 0 0
\(469\) 6515.49 0.641487
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4507.28 −0.438150
\(474\) 0 0
\(475\) 724.168 0.0699518
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 18976.7 1.81017 0.905083 0.425236i \(-0.139809\pi\)
0.905083 + 0.425236i \(0.139809\pi\)
\(480\) 0 0
\(481\) 1006.79 0.0954379
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −5508.55 −0.515733
\(486\) 0 0
\(487\) 10579.6 0.984408 0.492204 0.870480i \(-0.336191\pi\)
0.492204 + 0.870480i \(0.336191\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15200.9 1.39716 0.698579 0.715533i \(-0.253816\pi\)
0.698579 + 0.715533i \(0.253816\pi\)
\(492\) 0 0
\(493\) −1013.21 −0.0925614
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4878.77 −0.440327
\(498\) 0 0
\(499\) 2553.92 0.229117 0.114558 0.993417i \(-0.463455\pi\)
0.114558 + 0.993417i \(0.463455\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4863.70 −0.431137 −0.215568 0.976489i \(-0.569160\pi\)
−0.215568 + 0.976489i \(0.569160\pi\)
\(504\) 0 0
\(505\) 11441.5 1.00820
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17659.5 1.53781 0.768903 0.639366i \(-0.220803\pi\)
0.768903 + 0.639366i \(0.220803\pi\)
\(510\) 0 0
\(511\) 5581.35 0.483179
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 21685.3 1.85547
\(516\) 0 0
\(517\) 12223.3 1.03980
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3778.19 0.317707 0.158853 0.987302i \(-0.449220\pi\)
0.158853 + 0.987302i \(0.449220\pi\)
\(522\) 0 0
\(523\) −2792.67 −0.233489 −0.116745 0.993162i \(-0.537246\pi\)
−0.116745 + 0.993162i \(0.537246\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3013.74 −0.249109
\(528\) 0 0
\(529\) 2403.34 0.197529
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −18810.7 −1.52867
\(534\) 0 0
\(535\) 20471.1 1.65429
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −9605.85 −0.767631
\(540\) 0 0
\(541\) 2062.12 0.163877 0.0819383 0.996637i \(-0.473889\pi\)
0.0819383 + 0.996637i \(0.473889\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5015.44 0.394198
\(546\) 0 0
\(547\) 16562.0 1.29459 0.647296 0.762239i \(-0.275900\pi\)
0.647296 + 0.762239i \(0.275900\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1268.54 −0.0980795
\(552\) 0 0
\(553\) 487.057 0.0374535
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9272.94 −0.705399 −0.352699 0.935737i \(-0.614736\pi\)
−0.352699 + 0.935737i \(0.614736\pi\)
\(558\) 0 0
\(559\) 10563.3 0.799251
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 62.0181 0.00464254 0.00232127 0.999997i \(-0.499261\pi\)
0.00232127 + 0.999997i \(0.499261\pi\)
\(564\) 0 0
\(565\) 9047.71 0.673699
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10015.9 −0.737938 −0.368969 0.929442i \(-0.620289\pi\)
−0.368969 + 0.929442i \(0.620289\pi\)
\(570\) 0 0
\(571\) 15543.3 1.13917 0.569587 0.821931i \(-0.307103\pi\)
0.569587 + 0.821931i \(0.307103\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1879.95 0.136346
\(576\) 0 0
\(577\) 3776.77 0.272494 0.136247 0.990675i \(-0.456496\pi\)
0.136247 + 0.990675i \(0.456496\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5559.67 0.396995
\(582\) 0 0
\(583\) 3555.33 0.252567
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 596.285 0.0419273 0.0209637 0.999780i \(-0.493327\pi\)
0.0209637 + 0.999780i \(0.493327\pi\)
\(588\) 0 0
\(589\) −3773.21 −0.263960
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −20179.6 −1.39743 −0.698717 0.715398i \(-0.746245\pi\)
−0.698717 + 0.715398i \(0.746245\pi\)
\(594\) 0 0
\(595\) −4340.06 −0.299034
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 13076.3 0.891960 0.445980 0.895043i \(-0.352855\pi\)
0.445980 + 0.895043i \(0.352855\pi\)
\(600\) 0 0
\(601\) −16866.9 −1.14479 −0.572393 0.819980i \(-0.693985\pi\)
−0.572393 + 0.819980i \(0.693985\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2319.18 −0.155848
\(606\) 0 0
\(607\) −19953.0 −1.33421 −0.667107 0.744962i \(-0.732468\pi\)
−0.667107 + 0.744962i \(0.732468\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −28646.6 −1.89676
\(612\) 0 0
\(613\) 7214.32 0.475340 0.237670 0.971346i \(-0.423616\pi\)
0.237670 + 0.971346i \(0.423616\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −22212.9 −1.44936 −0.724681 0.689084i \(-0.758013\pi\)
−0.724681 + 0.689084i \(0.758013\pi\)
\(618\) 0 0
\(619\) −4349.02 −0.282394 −0.141197 0.989982i \(-0.545095\pi\)
−0.141197 + 0.989982i \(0.545095\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10740.2 −0.690688
\(624\) 0 0
\(625\) −17329.2 −1.10907
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −408.331 −0.0258843
\(630\) 0 0
\(631\) 10127.9 0.638966 0.319483 0.947592i \(-0.396491\pi\)
0.319483 + 0.947592i \(0.396491\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 25602.5 1.60001
\(636\) 0 0
\(637\) 22512.4 1.40027
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17624.6 1.08601 0.543003 0.839731i \(-0.317287\pi\)
0.543003 + 0.839731i \(0.317287\pi\)
\(642\) 0 0
\(643\) −27238.1 −1.67055 −0.835277 0.549829i \(-0.814693\pi\)
−0.835277 + 0.549829i \(0.814693\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −28702.1 −1.74404 −0.872021 0.489469i \(-0.837191\pi\)
−0.872021 + 0.489469i \(0.837191\pi\)
\(648\) 0 0
\(649\) −21514.1 −1.30124
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7937.84 0.475700 0.237850 0.971302i \(-0.423557\pi\)
0.237850 + 0.971302i \(0.423557\pi\)
\(654\) 0 0
\(655\) 8426.62 0.502680
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3675.29 0.217252 0.108626 0.994083i \(-0.465355\pi\)
0.108626 + 0.994083i \(0.465355\pi\)
\(660\) 0 0
\(661\) 4205.49 0.247466 0.123733 0.992316i \(-0.460513\pi\)
0.123733 + 0.992316i \(0.460513\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5433.76 −0.316860
\(666\) 0 0
\(667\) −3293.15 −0.191171
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −24653.3 −1.41838
\(672\) 0 0
\(673\) −4634.55 −0.265452 −0.132726 0.991153i \(-0.542373\pi\)
−0.132726 + 0.991153i \(0.542373\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5327.42 0.302437 0.151218 0.988500i \(-0.451680\pi\)
0.151218 + 0.988500i \(0.451680\pi\)
\(678\) 0 0
\(679\) 4579.34 0.258820
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −2816.05 −0.157764 −0.0788822 0.996884i \(-0.525135\pi\)
−0.0788822 + 0.996884i \(0.525135\pi\)
\(684\) 0 0
\(685\) −11703.8 −0.652816
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8332.33 −0.460720
\(690\) 0 0
\(691\) −16910.1 −0.930955 −0.465477 0.885060i \(-0.654117\pi\)
−0.465477 + 0.885060i \(0.654117\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 27674.6 1.51044
\(696\) 0 0
\(697\) 7629.19 0.414600
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 30588.4 1.64808 0.824042 0.566528i \(-0.191714\pi\)
0.824042 + 0.566528i \(0.191714\pi\)
\(702\) 0 0
\(703\) −511.232 −0.0274274
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −9511.48 −0.505963
\(708\) 0 0
\(709\) 21242.9 1.12524 0.562619 0.826716i \(-0.309794\pi\)
0.562619 + 0.826716i \(0.309794\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −9795.28 −0.514496
\(714\) 0 0
\(715\) 42419.5 2.21874
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −12452.7 −0.645907 −0.322954 0.946415i \(-0.604676\pi\)
−0.322954 + 0.946415i \(0.604676\pi\)
\(720\) 0 0
\(721\) −18027.3 −0.931168
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −424.901 −0.0217661
\(726\) 0 0
\(727\) −27178.3 −1.38650 −0.693251 0.720696i \(-0.743822\pi\)
−0.693251 + 0.720696i \(0.743822\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −4284.25 −0.216770
\(732\) 0 0
\(733\) −21999.7 −1.10856 −0.554282 0.832329i \(-0.687007\pi\)
−0.554282 + 0.832329i \(0.687007\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 25828.1 1.29089
\(738\) 0 0
\(739\) 13644.0 0.679166 0.339583 0.940576i \(-0.389714\pi\)
0.339583 + 0.940576i \(0.389714\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13980.2 0.690286 0.345143 0.938550i \(-0.387830\pi\)
0.345143 + 0.938550i \(0.387830\pi\)
\(744\) 0 0
\(745\) 21250.1 1.04502
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −17018.0 −0.830204
\(750\) 0 0
\(751\) −21138.7 −1.02711 −0.513557 0.858055i \(-0.671673\pi\)
−0.513557 + 0.858055i \(0.671673\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −33960.5 −1.63702
\(756\) 0 0
\(757\) −28328.1 −1.36011 −0.680054 0.733162i \(-0.738044\pi\)
−0.680054 + 0.733162i \(0.738044\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1093.79 0.0521023 0.0260512 0.999661i \(-0.491707\pi\)
0.0260512 + 0.999661i \(0.491707\pi\)
\(762\) 0 0
\(763\) −4169.41 −0.197828
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 50420.8 2.37365
\(768\) 0 0
\(769\) −1659.21 −0.0778059 −0.0389029 0.999243i \(-0.512386\pi\)
−0.0389029 + 0.999243i \(0.512386\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 7537.89 0.350736 0.175368 0.984503i \(-0.443888\pi\)
0.175368 + 0.984503i \(0.443888\pi\)
\(774\) 0 0
\(775\) −1263.84 −0.0585787
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9551.76 0.439316
\(780\) 0 0
\(781\) −19339.9 −0.886091
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 25177.9 1.14476
\(786\) 0 0
\(787\) 9304.20 0.421422 0.210711 0.977548i \(-0.432422\pi\)
0.210711 + 0.977548i \(0.432422\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −7521.50 −0.338096
\(792\) 0 0
\(793\) 57777.8 2.58733
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 38619.5 1.71640 0.858201 0.513314i \(-0.171582\pi\)
0.858201 + 0.513314i \(0.171582\pi\)
\(798\) 0 0
\(799\) 11618.4 0.514431
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 22125.1 0.972324
\(804\) 0 0
\(805\) −14106.1 −0.617608
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18550.5 0.806179 0.403090 0.915160i \(-0.367936\pi\)
0.403090 + 0.915160i \(0.367936\pi\)
\(810\) 0 0
\(811\) 3704.80 0.160411 0.0802054 0.996778i \(-0.474442\pi\)
0.0802054 + 0.996778i \(0.474442\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 37516.2 1.61243
\(816\) 0 0
\(817\) −5363.90 −0.229693
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5544.86 −0.235709 −0.117854 0.993031i \(-0.537602\pi\)
−0.117854 + 0.993031i \(0.537602\pi\)
\(822\) 0 0
\(823\) 3915.39 0.165835 0.0829174 0.996556i \(-0.473576\pi\)
0.0829174 + 0.996556i \(0.473576\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −35091.2 −1.47550 −0.737751 0.675073i \(-0.764112\pi\)
−0.737751 + 0.675073i \(0.764112\pi\)
\(828\) 0 0
\(829\) 3532.71 0.148005 0.0740025 0.997258i \(-0.476423\pi\)
0.0740025 + 0.997258i \(0.476423\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −9130.53 −0.379777
\(834\) 0 0
\(835\) −15891.6 −0.658624
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −30081.1 −1.23780 −0.618900 0.785470i \(-0.712421\pi\)
−0.618900 + 0.785470i \(0.712421\pi\)
\(840\) 0 0
\(841\) −23644.7 −0.969482
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −73366.5 −2.98685
\(846\) 0 0
\(847\) 1927.97 0.0782121
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1327.16 −0.0534601
\(852\) 0 0
\(853\) 11222.1 0.450453 0.225226 0.974306i \(-0.427688\pi\)
0.225226 + 0.974306i \(0.427688\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 27486.0 1.09557 0.547785 0.836619i \(-0.315471\pi\)
0.547785 + 0.836619i \(0.315471\pi\)
\(858\) 0 0
\(859\) −14976.2 −0.594855 −0.297427 0.954744i \(-0.596129\pi\)
−0.297427 + 0.954744i \(0.596129\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −9419.49 −0.371545 −0.185772 0.982593i \(-0.559479\pi\)
−0.185772 + 0.982593i \(0.559479\pi\)
\(864\) 0 0
\(865\) 33223.2 1.30592
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1930.75 0.0753694
\(870\) 0 0
\(871\) −60531.0 −2.35478
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12787.6 0.494059
\(876\) 0 0
\(877\) −21069.8 −0.811263 −0.405631 0.914037i \(-0.632948\pi\)
−0.405631 + 0.914037i \(0.632948\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 14511.7 0.554952 0.277476 0.960733i \(-0.410502\pi\)
0.277476 + 0.960733i \(0.410502\pi\)
\(882\) 0 0
\(883\) 26515.9 1.01057 0.505284 0.862953i \(-0.331388\pi\)
0.505284 + 0.862953i \(0.331388\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −29223.6 −1.10624 −0.553119 0.833102i \(-0.686563\pi\)
−0.553119 + 0.833102i \(0.686563\pi\)
\(888\) 0 0
\(889\) −21283.8 −0.802964
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 14546.3 0.545099
\(894\) 0 0
\(895\) −12743.7 −0.475949
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2213.90 0.0821332
\(900\) 0 0
\(901\) 3379.41 0.124955
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6770.44 0.248682
\(906\) 0 0
\(907\) −32139.3 −1.17659 −0.588296 0.808646i \(-0.700201\pi\)
−0.588296 + 0.808646i \(0.700201\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 21430.7 0.779396 0.389698 0.920943i \(-0.372579\pi\)
0.389698 + 0.920943i \(0.372579\pi\)
\(912\) 0 0
\(913\) 22039.1 0.798891
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −7005.17 −0.252270
\(918\) 0 0
\(919\) −11676.1 −0.419106 −0.209553 0.977797i \(-0.567201\pi\)
−0.209553 + 0.977797i \(0.567201\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 45325.3 1.61636
\(924\) 0 0
\(925\) −171.238 −0.00608677
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −41969.7 −1.48222 −0.741110 0.671383i \(-0.765700\pi\)
−0.741110 + 0.671383i \(0.765700\pi\)
\(930\) 0 0
\(931\) −11431.4 −0.402417
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −17204.4 −0.601759
\(936\) 0 0
\(937\) −45948.6 −1.60200 −0.801001 0.598663i \(-0.795699\pi\)
−0.801001 + 0.598663i \(0.795699\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 15000.1 0.519648 0.259824 0.965656i \(-0.416335\pi\)
0.259824 + 0.965656i \(0.416335\pi\)
\(942\) 0 0
\(943\) 24796.4 0.856292
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −37423.2 −1.28415 −0.642075 0.766642i \(-0.721926\pi\)
−0.642075 + 0.766642i \(0.721926\pi\)
\(948\) 0 0
\(949\) −51852.6 −1.77366
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 55403.0 1.88319 0.941594 0.336750i \(-0.109328\pi\)
0.941594 + 0.336750i \(0.109328\pi\)
\(954\) 0 0
\(955\) 10346.6 0.350584
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 9729.54 0.327615
\(960\) 0 0
\(961\) −23205.9 −0.778956
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −7973.70 −0.265992
\(966\) 0 0
\(967\) −31440.5 −1.04556 −0.522780 0.852467i \(-0.675105\pi\)
−0.522780 + 0.852467i \(0.675105\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −19010.3 −0.628290 −0.314145 0.949375i \(-0.601718\pi\)
−0.314145 + 0.949375i \(0.601718\pi\)
\(972\) 0 0
\(973\) −23006.3 −0.758015
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8504.01 0.278472 0.139236 0.990259i \(-0.455535\pi\)
0.139236 + 0.990259i \(0.455535\pi\)
\(978\) 0 0
\(979\) −42575.4 −1.38990
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −52828.3 −1.71410 −0.857051 0.515232i \(-0.827706\pi\)
−0.857051 + 0.515232i \(0.827706\pi\)
\(984\) 0 0
\(985\) 37105.3 1.20028
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −13924.7 −0.447705
\(990\) 0 0
\(991\) 13275.5 0.425541 0.212770 0.977102i \(-0.431751\pi\)
0.212770 + 0.977102i \(0.431751\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −17812.7 −0.567538
\(996\) 0 0
\(997\) −20858.1 −0.662570 −0.331285 0.943531i \(-0.607482\pi\)
−0.331285 + 0.943531i \(0.607482\pi\)
\(998\) 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.a.s.1.1 2
3.2 odd 2 128.4.a.e.1.1 2
4.3 odd 2 1152.4.a.t.1.1 2
8.3 odd 2 1152.4.a.r.1.2 2
8.5 even 2 1152.4.a.q.1.2 2
12.11 even 2 128.4.a.g.1.2 yes 2
24.5 odd 2 128.4.a.h.1.2 yes 2
24.11 even 2 128.4.a.f.1.1 yes 2
48.5 odd 4 256.4.b.i.129.4 4
48.11 even 4 256.4.b.h.129.1 4
48.29 odd 4 256.4.b.i.129.1 4
48.35 even 4 256.4.b.h.129.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.a.e.1.1 2 3.2 odd 2
128.4.a.f.1.1 yes 2 24.11 even 2
128.4.a.g.1.2 yes 2 12.11 even 2
128.4.a.h.1.2 yes 2 24.5 odd 2
256.4.b.h.129.1 4 48.11 even 4
256.4.b.h.129.4 4 48.35 even 4
256.4.b.i.129.1 4 48.29 odd 4
256.4.b.i.129.4 4 48.5 odd 4
1152.4.a.q.1.2 2 8.5 even 2
1152.4.a.r.1.2 2 8.3 odd 2
1152.4.a.s.1.1 2 1.1 even 1 trivial
1152.4.a.t.1.1 2 4.3 odd 2