Properties

Label 1856.4.a.u.1.1
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $3$
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] = [1856,4,Mod(1,1856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, 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("1856.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.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: \(109.507544971\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 232)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.97021 q^{3} +12.4882 q^{5} +28.6773 q^{7} +8.64344 q^{9} +O(q^{10})\) \(q-5.97021 q^{3} +12.4882 q^{5} +28.6773 q^{7} +8.64344 q^{9} -18.2099 q^{11} -37.9978 q^{13} -74.5574 q^{15} -3.95162 q^{17} -36.8211 q^{19} -171.209 q^{21} -42.7480 q^{23} +30.9561 q^{25} +109.593 q^{27} +29.0000 q^{29} +160.731 q^{31} +108.717 q^{33} +358.129 q^{35} -313.040 q^{37} +226.855 q^{39} +496.787 q^{41} -139.195 q^{43} +107.941 q^{45} +417.656 q^{47} +479.386 q^{49} +23.5920 q^{51} +137.116 q^{53} -227.409 q^{55} +219.830 q^{57} +190.033 q^{59} -161.072 q^{61} +247.870 q^{63} -474.525 q^{65} +125.259 q^{67} +255.215 q^{69} +165.110 q^{71} -938.243 q^{73} -184.815 q^{75} -522.209 q^{77} +1315.60 q^{79} -887.664 q^{81} +505.294 q^{83} -49.3487 q^{85} -173.136 q^{87} -769.587 q^{89} -1089.67 q^{91} -959.599 q^{93} -459.830 q^{95} +1333.29 q^{97} -157.396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{3} - 4 q^{5} - 16 q^{7} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{3} - 4 q^{5} - 16 q^{7} + 45 q^{9} - 2 q^{11} - 28 q^{13} - 136 q^{15} - 66 q^{17} - 66 q^{19} - 472 q^{21} - 176 q^{23} - 9 q^{25} + 228 q^{27} + 87 q^{29} + 190 q^{31} + 154 q^{33} + 660 q^{35} - 442 q^{37} + 656 q^{39} + 1162 q^{41} + 30 q^{43} + 254 q^{45} + 738 q^{47} + 851 q^{49} - 576 q^{51} - 312 q^{53} - 464 q^{55} + 684 q^{57} + 44 q^{59} - 54 q^{61} - 964 q^{63} + 178 q^{65} - 116 q^{67} - 812 q^{69} + 1200 q^{71} - 1118 q^{73} - 1038 q^{75} - 792 q^{77} + 2262 q^{79} + 15 q^{81} - 1804 q^{83} + 8 q^{85} + 174 q^{87} + 1578 q^{89} - 1972 q^{91} + 706 q^{93} + 1052 q^{95} + 1450 q^{97} - 482 q^{99}+O(q^{100}) \) 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
\(3\) −5.97021 −1.14897 −0.574484 0.818516i \(-0.694797\pi\)
−0.574484 + 0.818516i \(0.694797\pi\)
\(4\) 0 0
\(5\) 12.4882 1.11698 0.558491 0.829511i \(-0.311381\pi\)
0.558491 + 0.829511i \(0.311381\pi\)
\(6\) 0 0
\(7\) 28.6773 1.54843 0.774213 0.632925i \(-0.218146\pi\)
0.774213 + 0.632925i \(0.218146\pi\)
\(8\) 0 0
\(9\) 8.64344 0.320127
\(10\) 0 0
\(11\) −18.2099 −0.499134 −0.249567 0.968358i \(-0.580288\pi\)
−0.249567 + 0.968358i \(0.580288\pi\)
\(12\) 0 0
\(13\) −37.9978 −0.810668 −0.405334 0.914169i \(-0.632845\pi\)
−0.405334 + 0.914169i \(0.632845\pi\)
\(14\) 0 0
\(15\) −74.5574 −1.28338
\(16\) 0 0
\(17\) −3.95162 −0.0563769 −0.0281885 0.999603i \(-0.508974\pi\)
−0.0281885 + 0.999603i \(0.508974\pi\)
\(18\) 0 0
\(19\) −36.8211 −0.444597 −0.222298 0.974979i \(-0.571356\pi\)
−0.222298 + 0.974979i \(0.571356\pi\)
\(20\) 0 0
\(21\) −171.209 −1.77909
\(22\) 0 0
\(23\) −42.7480 −0.387547 −0.193773 0.981046i \(-0.562073\pi\)
−0.193773 + 0.981046i \(0.562073\pi\)
\(24\) 0 0
\(25\) 30.9561 0.247649
\(26\) 0 0
\(27\) 109.593 0.781152
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) 160.731 0.931231 0.465616 0.884987i \(-0.345833\pi\)
0.465616 + 0.884987i \(0.345833\pi\)
\(32\) 0 0
\(33\) 108.717 0.573489
\(34\) 0 0
\(35\) 358.129 1.72957
\(36\) 0 0
\(37\) −313.040 −1.39090 −0.695451 0.718573i \(-0.744796\pi\)
−0.695451 + 0.718573i \(0.744796\pi\)
\(38\) 0 0
\(39\) 226.855 0.931432
\(40\) 0 0
\(41\) 496.787 1.89232 0.946159 0.323703i \(-0.104928\pi\)
0.946159 + 0.323703i \(0.104928\pi\)
\(42\) 0 0
\(43\) −139.195 −0.493653 −0.246827 0.969060i \(-0.579388\pi\)
−0.246827 + 0.969060i \(0.579388\pi\)
\(44\) 0 0
\(45\) 107.941 0.357577
\(46\) 0 0
\(47\) 417.656 1.29620 0.648100 0.761556i \(-0.275564\pi\)
0.648100 + 0.761556i \(0.275564\pi\)
\(48\) 0 0
\(49\) 479.386 1.39763
\(50\) 0 0
\(51\) 23.5920 0.0647753
\(52\) 0 0
\(53\) 137.116 0.355365 0.177683 0.984088i \(-0.443140\pi\)
0.177683 + 0.984088i \(0.443140\pi\)
\(54\) 0 0
\(55\) −227.409 −0.557524
\(56\) 0 0
\(57\) 219.830 0.510827
\(58\) 0 0
\(59\) 190.033 0.419326 0.209663 0.977774i \(-0.432763\pi\)
0.209663 + 0.977774i \(0.432763\pi\)
\(60\) 0 0
\(61\) −161.072 −0.338084 −0.169042 0.985609i \(-0.554067\pi\)
−0.169042 + 0.985609i \(0.554067\pi\)
\(62\) 0 0
\(63\) 247.870 0.495694
\(64\) 0 0
\(65\) −474.525 −0.905502
\(66\) 0 0
\(67\) 125.259 0.228400 0.114200 0.993458i \(-0.463570\pi\)
0.114200 + 0.993458i \(0.463570\pi\)
\(68\) 0 0
\(69\) 255.215 0.445279
\(70\) 0 0
\(71\) 165.110 0.275984 0.137992 0.990433i \(-0.455935\pi\)
0.137992 + 0.990433i \(0.455935\pi\)
\(72\) 0 0
\(73\) −938.243 −1.50429 −0.752144 0.658999i \(-0.770980\pi\)
−0.752144 + 0.658999i \(0.770980\pi\)
\(74\) 0 0
\(75\) −184.815 −0.284541
\(76\) 0 0
\(77\) −522.209 −0.772873
\(78\) 0 0
\(79\) 1315.60 1.87363 0.936816 0.349821i \(-0.113758\pi\)
0.936816 + 0.349821i \(0.113758\pi\)
\(80\) 0 0
\(81\) −887.664 −1.21765
\(82\) 0 0
\(83\) 505.294 0.668231 0.334116 0.942532i \(-0.391562\pi\)
0.334116 + 0.942532i \(0.391562\pi\)
\(84\) 0 0
\(85\) −49.3487 −0.0629720
\(86\) 0 0
\(87\) −173.136 −0.213358
\(88\) 0 0
\(89\) −769.587 −0.916584 −0.458292 0.888802i \(-0.651539\pi\)
−0.458292 + 0.888802i \(0.651539\pi\)
\(90\) 0 0
\(91\) −1089.67 −1.25526
\(92\) 0 0
\(93\) −959.599 −1.06996
\(94\) 0 0
\(95\) −459.830 −0.496606
\(96\) 0 0
\(97\) 1333.29 1.39561 0.697807 0.716286i \(-0.254159\pi\)
0.697807 + 0.716286i \(0.254159\pi\)
\(98\) 0 0
\(99\) −157.396 −0.159787
\(100\) 0 0
\(101\) −51.3094 −0.0505493 −0.0252746 0.999681i \(-0.508046\pi\)
−0.0252746 + 0.999681i \(0.508046\pi\)
\(102\) 0 0
\(103\) −1062.26 −1.01619 −0.508097 0.861300i \(-0.669651\pi\)
−0.508097 + 0.861300i \(0.669651\pi\)
\(104\) 0 0
\(105\) −2138.10 −1.98721
\(106\) 0 0
\(107\) 1414.85 1.27831 0.639155 0.769078i \(-0.279284\pi\)
0.639155 + 0.769078i \(0.279284\pi\)
\(108\) 0 0
\(109\) −495.330 −0.435266 −0.217633 0.976031i \(-0.569834\pi\)
−0.217633 + 0.976031i \(0.569834\pi\)
\(110\) 0 0
\(111\) 1868.91 1.59810
\(112\) 0 0
\(113\) −603.969 −0.502802 −0.251401 0.967883i \(-0.580891\pi\)
−0.251401 + 0.967883i \(0.580891\pi\)
\(114\) 0 0
\(115\) −533.848 −0.432883
\(116\) 0 0
\(117\) −328.432 −0.259517
\(118\) 0 0
\(119\) −113.322 −0.0872956
\(120\) 0 0
\(121\) −999.401 −0.750865
\(122\) 0 0
\(123\) −2965.92 −2.17421
\(124\) 0 0
\(125\) −1174.44 −0.840363
\(126\) 0 0
\(127\) 1282.23 0.895902 0.447951 0.894058i \(-0.352154\pi\)
0.447951 + 0.894058i \(0.352154\pi\)
\(128\) 0 0
\(129\) 831.026 0.567192
\(130\) 0 0
\(131\) 2446.25 1.63152 0.815762 0.578388i \(-0.196318\pi\)
0.815762 + 0.578388i \(0.196318\pi\)
\(132\) 0 0
\(133\) −1055.93 −0.688425
\(134\) 0 0
\(135\) 1368.62 0.872533
\(136\) 0 0
\(137\) 1259.91 0.785703 0.392852 0.919602i \(-0.371489\pi\)
0.392852 + 0.919602i \(0.371489\pi\)
\(138\) 0 0
\(139\) −822.786 −0.502070 −0.251035 0.967978i \(-0.580771\pi\)
−0.251035 + 0.967978i \(0.580771\pi\)
\(140\) 0 0
\(141\) −2493.49 −1.48929
\(142\) 0 0
\(143\) 691.934 0.404632
\(144\) 0 0
\(145\) 362.159 0.207418
\(146\) 0 0
\(147\) −2862.03 −1.60583
\(148\) 0 0
\(149\) 477.366 0.262466 0.131233 0.991352i \(-0.458106\pi\)
0.131233 + 0.991352i \(0.458106\pi\)
\(150\) 0 0
\(151\) 1740.92 0.938240 0.469120 0.883135i \(-0.344571\pi\)
0.469120 + 0.883135i \(0.344571\pi\)
\(152\) 0 0
\(153\) −34.1556 −0.0180478
\(154\) 0 0
\(155\) 2007.25 1.04017
\(156\) 0 0
\(157\) 1372.90 0.697892 0.348946 0.937143i \(-0.386540\pi\)
0.348946 + 0.937143i \(0.386540\pi\)
\(158\) 0 0
\(159\) −818.613 −0.408304
\(160\) 0 0
\(161\) −1225.90 −0.600088
\(162\) 0 0
\(163\) 3688.03 1.77220 0.886100 0.463495i \(-0.153405\pi\)
0.886100 + 0.463495i \(0.153405\pi\)
\(164\) 0 0
\(165\) 1357.68 0.640577
\(166\) 0 0
\(167\) −4059.56 −1.88107 −0.940534 0.339701i \(-0.889674\pi\)
−0.940534 + 0.339701i \(0.889674\pi\)
\(168\) 0 0
\(169\) −753.169 −0.342817
\(170\) 0 0
\(171\) −318.261 −0.142328
\(172\) 0 0
\(173\) −2811.92 −1.23576 −0.617879 0.786274i \(-0.712008\pi\)
−0.617879 + 0.786274i \(0.712008\pi\)
\(174\) 0 0
\(175\) 887.737 0.383466
\(176\) 0 0
\(177\) −1134.54 −0.481792
\(178\) 0 0
\(179\) 163.787 0.0683913 0.0341956 0.999415i \(-0.489113\pi\)
0.0341956 + 0.999415i \(0.489113\pi\)
\(180\) 0 0
\(181\) 3540.42 1.45391 0.726954 0.686686i \(-0.240935\pi\)
0.726954 + 0.686686i \(0.240935\pi\)
\(182\) 0 0
\(183\) 961.632 0.388447
\(184\) 0 0
\(185\) −3909.31 −1.55361
\(186\) 0 0
\(187\) 71.9584 0.0281397
\(188\) 0 0
\(189\) 3142.82 1.20956
\(190\) 0 0
\(191\) −2163.65 −0.819665 −0.409833 0.912161i \(-0.634413\pi\)
−0.409833 + 0.912161i \(0.634413\pi\)
\(192\) 0 0
\(193\) 4632.75 1.72784 0.863919 0.503630i \(-0.168003\pi\)
0.863919 + 0.503630i \(0.168003\pi\)
\(194\) 0 0
\(195\) 2833.02 1.04039
\(196\) 0 0
\(197\) 1239.91 0.448427 0.224214 0.974540i \(-0.428019\pi\)
0.224214 + 0.974540i \(0.428019\pi\)
\(198\) 0 0
\(199\) −4651.45 −1.65695 −0.828474 0.560028i \(-0.810790\pi\)
−0.828474 + 0.560028i \(0.810790\pi\)
\(200\) 0 0
\(201\) −747.820 −0.262424
\(202\) 0 0
\(203\) 831.641 0.287536
\(204\) 0 0
\(205\) 6203.99 2.11369
\(206\) 0 0
\(207\) −369.490 −0.124064
\(208\) 0 0
\(209\) 670.506 0.221913
\(210\) 0 0
\(211\) 3516.90 1.14746 0.573728 0.819046i \(-0.305497\pi\)
0.573728 + 0.819046i \(0.305497\pi\)
\(212\) 0 0
\(213\) −985.739 −0.317097
\(214\) 0 0
\(215\) −1738.30 −0.551402
\(216\) 0 0
\(217\) 4609.33 1.44194
\(218\) 0 0
\(219\) 5601.51 1.72838
\(220\) 0 0
\(221\) 150.153 0.0457030
\(222\) 0 0
\(223\) −3480.17 −1.04506 −0.522532 0.852620i \(-0.675012\pi\)
−0.522532 + 0.852620i \(0.675012\pi\)
\(224\) 0 0
\(225\) 267.567 0.0792792
\(226\) 0 0
\(227\) 1259.51 0.368266 0.184133 0.982901i \(-0.441052\pi\)
0.184133 + 0.982901i \(0.441052\pi\)
\(228\) 0 0
\(229\) 274.295 0.0791524 0.0395762 0.999217i \(-0.487399\pi\)
0.0395762 + 0.999217i \(0.487399\pi\)
\(230\) 0 0
\(231\) 3117.70 0.888006
\(232\) 0 0
\(233\) 6517.70 1.83257 0.916285 0.400526i \(-0.131173\pi\)
0.916285 + 0.400526i \(0.131173\pi\)
\(234\) 0 0
\(235\) 5215.79 1.44783
\(236\) 0 0
\(237\) −7854.43 −2.15274
\(238\) 0 0
\(239\) 6052.89 1.63820 0.819098 0.573654i \(-0.194474\pi\)
0.819098 + 0.573654i \(0.194474\pi\)
\(240\) 0 0
\(241\) 2265.19 0.605452 0.302726 0.953078i \(-0.402103\pi\)
0.302726 + 0.953078i \(0.402103\pi\)
\(242\) 0 0
\(243\) 2340.54 0.617884
\(244\) 0 0
\(245\) 5986.68 1.56112
\(246\) 0 0
\(247\) 1399.12 0.360420
\(248\) 0 0
\(249\) −3016.71 −0.767777
\(250\) 0 0
\(251\) 6653.27 1.67311 0.836555 0.547882i \(-0.184566\pi\)
0.836555 + 0.547882i \(0.184566\pi\)
\(252\) 0 0
\(253\) 778.435 0.193438
\(254\) 0 0
\(255\) 294.622 0.0723528
\(256\) 0 0
\(257\) −3144.58 −0.763244 −0.381622 0.924319i \(-0.624634\pi\)
−0.381622 + 0.924319i \(0.624634\pi\)
\(258\) 0 0
\(259\) −8977.12 −2.15371
\(260\) 0 0
\(261\) 250.660 0.0594462
\(262\) 0 0
\(263\) −2409.66 −0.564965 −0.282483 0.959272i \(-0.591158\pi\)
−0.282483 + 0.959272i \(0.591158\pi\)
\(264\) 0 0
\(265\) 1712.34 0.396937
\(266\) 0 0
\(267\) 4594.60 1.05313
\(268\) 0 0
\(269\) 4542.85 1.02968 0.514838 0.857288i \(-0.327852\pi\)
0.514838 + 0.857288i \(0.327852\pi\)
\(270\) 0 0
\(271\) 6519.35 1.46134 0.730668 0.682733i \(-0.239209\pi\)
0.730668 + 0.682733i \(0.239209\pi\)
\(272\) 0 0
\(273\) 6505.58 1.44225
\(274\) 0 0
\(275\) −563.706 −0.123610
\(276\) 0 0
\(277\) 7440.91 1.61401 0.807005 0.590544i \(-0.201087\pi\)
0.807005 + 0.590544i \(0.201087\pi\)
\(278\) 0 0
\(279\) 1389.27 0.298113
\(280\) 0 0
\(281\) 272.086 0.0577625 0.0288813 0.999583i \(-0.490806\pi\)
0.0288813 + 0.999583i \(0.490806\pi\)
\(282\) 0 0
\(283\) −1716.92 −0.360636 −0.180318 0.983608i \(-0.557713\pi\)
−0.180318 + 0.983608i \(0.557713\pi\)
\(284\) 0 0
\(285\) 2745.29 0.570585
\(286\) 0 0
\(287\) 14246.5 2.93012
\(288\) 0 0
\(289\) −4897.38 −0.996822
\(290\) 0 0
\(291\) −7960.00 −1.60352
\(292\) 0 0
\(293\) −5903.51 −1.17709 −0.588544 0.808465i \(-0.700299\pi\)
−0.588544 + 0.808465i \(0.700299\pi\)
\(294\) 0 0
\(295\) 2373.18 0.468379
\(296\) 0 0
\(297\) −1995.66 −0.389900
\(298\) 0 0
\(299\) 1624.33 0.314172
\(300\) 0 0
\(301\) −3991.74 −0.764386
\(302\) 0 0
\(303\) 306.328 0.0580795
\(304\) 0 0
\(305\) −2011.50 −0.377633
\(306\) 0 0
\(307\) −3769.05 −0.700688 −0.350344 0.936621i \(-0.613935\pi\)
−0.350344 + 0.936621i \(0.613935\pi\)
\(308\) 0 0
\(309\) 6341.94 1.16757
\(310\) 0 0
\(311\) −3141.76 −0.572839 −0.286419 0.958104i \(-0.592465\pi\)
−0.286419 + 0.958104i \(0.592465\pi\)
\(312\) 0 0
\(313\) −1743.33 −0.314820 −0.157410 0.987533i \(-0.550314\pi\)
−0.157410 + 0.987533i \(0.550314\pi\)
\(314\) 0 0
\(315\) 3095.46 0.553681
\(316\) 0 0
\(317\) 1836.62 0.325410 0.162705 0.986675i \(-0.447978\pi\)
0.162705 + 0.986675i \(0.447978\pi\)
\(318\) 0 0
\(319\) −528.086 −0.0926869
\(320\) 0 0
\(321\) −8446.99 −1.46874
\(322\) 0 0
\(323\) 145.503 0.0250650
\(324\) 0 0
\(325\) −1176.26 −0.200761
\(326\) 0 0
\(327\) 2957.23 0.500107
\(328\) 0 0
\(329\) 11977.2 2.00707
\(330\) 0 0
\(331\) −106.340 −0.0176585 −0.00882924 0.999961i \(-0.502810\pi\)
−0.00882924 + 0.999961i \(0.502810\pi\)
\(332\) 0 0
\(333\) −2705.74 −0.445266
\(334\) 0 0
\(335\) 1564.26 0.255118
\(336\) 0 0
\(337\) 347.401 0.0561547 0.0280774 0.999606i \(-0.491062\pi\)
0.0280774 + 0.999606i \(0.491062\pi\)
\(338\) 0 0
\(339\) 3605.82 0.577703
\(340\) 0 0
\(341\) −2926.89 −0.464809
\(342\) 0 0
\(343\) 3911.17 0.615694
\(344\) 0 0
\(345\) 3187.18 0.497369
\(346\) 0 0
\(347\) −4319.94 −0.668319 −0.334159 0.942517i \(-0.608452\pi\)
−0.334159 + 0.942517i \(0.608452\pi\)
\(348\) 0 0
\(349\) 9732.19 1.49270 0.746350 0.665554i \(-0.231805\pi\)
0.746350 + 0.665554i \(0.231805\pi\)
\(350\) 0 0
\(351\) −4164.27 −0.633255
\(352\) 0 0
\(353\) 8396.92 1.26607 0.633035 0.774123i \(-0.281809\pi\)
0.633035 + 0.774123i \(0.281809\pi\)
\(354\) 0 0
\(355\) 2061.93 0.308270
\(356\) 0 0
\(357\) 676.554 0.100300
\(358\) 0 0
\(359\) −1306.89 −0.192130 −0.0960652 0.995375i \(-0.530626\pi\)
−0.0960652 + 0.995375i \(0.530626\pi\)
\(360\) 0 0
\(361\) −5503.21 −0.802334
\(362\) 0 0
\(363\) 5966.64 0.862720
\(364\) 0 0
\(365\) −11717.0 −1.68026
\(366\) 0 0
\(367\) 9859.59 1.40236 0.701181 0.712984i \(-0.252657\pi\)
0.701181 + 0.712984i \(0.252657\pi\)
\(368\) 0 0
\(369\) 4293.95 0.605783
\(370\) 0 0
\(371\) 3932.12 0.550257
\(372\) 0 0
\(373\) 3393.01 0.471001 0.235500 0.971874i \(-0.424327\pi\)
0.235500 + 0.971874i \(0.424327\pi\)
\(374\) 0 0
\(375\) 7011.67 0.965550
\(376\) 0 0
\(377\) −1101.94 −0.150537
\(378\) 0 0
\(379\) −2030.82 −0.275240 −0.137620 0.990485i \(-0.543945\pi\)
−0.137620 + 0.990485i \(0.543945\pi\)
\(380\) 0 0
\(381\) −7655.19 −1.02936
\(382\) 0 0
\(383\) −4.81248 −0.000642052 0 −0.000321026 1.00000i \(-0.500102\pi\)
−0.000321026 1.00000i \(0.500102\pi\)
\(384\) 0 0
\(385\) −6521.47 −0.863285
\(386\) 0 0
\(387\) −1203.13 −0.158032
\(388\) 0 0
\(389\) −9031.93 −1.17722 −0.588608 0.808418i \(-0.700324\pi\)
−0.588608 + 0.808418i \(0.700324\pi\)
\(390\) 0 0
\(391\) 168.924 0.0218487
\(392\) 0 0
\(393\) −14604.6 −1.87457
\(394\) 0 0
\(395\) 16429.6 2.09281
\(396\) 0 0
\(397\) 1443.16 0.182443 0.0912217 0.995831i \(-0.470923\pi\)
0.0912217 + 0.995831i \(0.470923\pi\)
\(398\) 0 0
\(399\) 6304.11 0.790979
\(400\) 0 0
\(401\) −9877.59 −1.23008 −0.615042 0.788495i \(-0.710861\pi\)
−0.615042 + 0.788495i \(0.710861\pi\)
\(402\) 0 0
\(403\) −6107.43 −0.754920
\(404\) 0 0
\(405\) −11085.4 −1.36009
\(406\) 0 0
\(407\) 5700.41 0.694247
\(408\) 0 0
\(409\) 14530.7 1.75672 0.878358 0.478003i \(-0.158639\pi\)
0.878358 + 0.478003i \(0.158639\pi\)
\(410\) 0 0
\(411\) −7521.93 −0.902748
\(412\) 0 0
\(413\) 5449.63 0.649295
\(414\) 0 0
\(415\) 6310.23 0.746403
\(416\) 0 0
\(417\) 4912.21 0.576863
\(418\) 0 0
\(419\) 3312.84 0.386259 0.193130 0.981173i \(-0.438136\pi\)
0.193130 + 0.981173i \(0.438136\pi\)
\(420\) 0 0
\(421\) 6708.67 0.776629 0.388314 0.921527i \(-0.373057\pi\)
0.388314 + 0.921527i \(0.373057\pi\)
\(422\) 0 0
\(423\) 3609.98 0.414949
\(424\) 0 0
\(425\) −122.327 −0.0139617
\(426\) 0 0
\(427\) −4619.09 −0.523498
\(428\) 0 0
\(429\) −4130.99 −0.464909
\(430\) 0 0
\(431\) 14687.7 1.64148 0.820742 0.571299i \(-0.193560\pi\)
0.820742 + 0.571299i \(0.193560\pi\)
\(432\) 0 0
\(433\) 759.822 0.0843296 0.0421648 0.999111i \(-0.486575\pi\)
0.0421648 + 0.999111i \(0.486575\pi\)
\(434\) 0 0
\(435\) −2162.17 −0.238317
\(436\) 0 0
\(437\) 1574.03 0.172302
\(438\) 0 0
\(439\) −4803.78 −0.522260 −0.261130 0.965304i \(-0.584095\pi\)
−0.261130 + 0.965304i \(0.584095\pi\)
\(440\) 0 0
\(441\) 4143.54 0.447418
\(442\) 0 0
\(443\) 10123.3 1.08572 0.542858 0.839824i \(-0.317342\pi\)
0.542858 + 0.839824i \(0.317342\pi\)
\(444\) 0 0
\(445\) −9610.78 −1.02381
\(446\) 0 0
\(447\) −2849.98 −0.301565
\(448\) 0 0
\(449\) −636.749 −0.0669266 −0.0334633 0.999440i \(-0.510654\pi\)
−0.0334633 + 0.999440i \(0.510654\pi\)
\(450\) 0 0
\(451\) −9046.41 −0.944521
\(452\) 0 0
\(453\) −10393.7 −1.07801
\(454\) 0 0
\(455\) −13608.1 −1.40210
\(456\) 0 0
\(457\) 12598.2 1.28954 0.644770 0.764376i \(-0.276953\pi\)
0.644770 + 0.764376i \(0.276953\pi\)
\(458\) 0 0
\(459\) −433.068 −0.0440389
\(460\) 0 0
\(461\) −15728.3 −1.58902 −0.794512 0.607249i \(-0.792273\pi\)
−0.794512 + 0.607249i \(0.792273\pi\)
\(462\) 0 0
\(463\) −5504.55 −0.552523 −0.276262 0.961083i \(-0.589096\pi\)
−0.276262 + 0.961083i \(0.589096\pi\)
\(464\) 0 0
\(465\) −11983.7 −1.19512
\(466\) 0 0
\(467\) 14851.2 1.47159 0.735796 0.677203i \(-0.236808\pi\)
0.735796 + 0.677203i \(0.236808\pi\)
\(468\) 0 0
\(469\) 3592.07 0.353660
\(470\) 0 0
\(471\) −8196.48 −0.801855
\(472\) 0 0
\(473\) 2534.73 0.246399
\(474\) 0 0
\(475\) −1139.84 −0.110104
\(476\) 0 0
\(477\) 1185.16 0.113762
\(478\) 0 0
\(479\) −5778.20 −0.551175 −0.275587 0.961276i \(-0.588872\pi\)
−0.275587 + 0.961276i \(0.588872\pi\)
\(480\) 0 0
\(481\) 11894.8 1.12756
\(482\) 0 0
\(483\) 7318.86 0.689482
\(484\) 0 0
\(485\) 16650.4 1.55888
\(486\) 0 0
\(487\) 12639.2 1.17605 0.588027 0.808841i \(-0.299905\pi\)
0.588027 + 0.808841i \(0.299905\pi\)
\(488\) 0 0
\(489\) −22018.3 −2.03620
\(490\) 0 0
\(491\) −363.554 −0.0334154 −0.0167077 0.999860i \(-0.505318\pi\)
−0.0167077 + 0.999860i \(0.505318\pi\)
\(492\) 0 0
\(493\) −114.597 −0.0104689
\(494\) 0 0
\(495\) −1965.60 −0.178479
\(496\) 0 0
\(497\) 4734.89 0.427342
\(498\) 0 0
\(499\) −4327.18 −0.388198 −0.194099 0.980982i \(-0.562178\pi\)
−0.194099 + 0.980982i \(0.562178\pi\)
\(500\) 0 0
\(501\) 24236.4 2.16129
\(502\) 0 0
\(503\) −21621.8 −1.91664 −0.958318 0.285704i \(-0.907773\pi\)
−0.958318 + 0.285704i \(0.907773\pi\)
\(504\) 0 0
\(505\) −640.764 −0.0564626
\(506\) 0 0
\(507\) 4496.58 0.393886
\(508\) 0 0
\(509\) −12903.3 −1.12363 −0.561817 0.827261i \(-0.689898\pi\)
−0.561817 + 0.827261i \(0.689898\pi\)
\(510\) 0 0
\(511\) −26906.2 −2.32928
\(512\) 0 0
\(513\) −4035.32 −0.347297
\(514\) 0 0
\(515\) −13265.8 −1.13507
\(516\) 0 0
\(517\) −7605.45 −0.646977
\(518\) 0 0
\(519\) 16787.7 1.41985
\(520\) 0 0
\(521\) −5015.65 −0.421765 −0.210883 0.977511i \(-0.567634\pi\)
−0.210883 + 0.977511i \(0.567634\pi\)
\(522\) 0 0
\(523\) −10454.3 −0.874059 −0.437030 0.899447i \(-0.643970\pi\)
−0.437030 + 0.899447i \(0.643970\pi\)
\(524\) 0 0
\(525\) −5299.98 −0.440591
\(526\) 0 0
\(527\) −635.148 −0.0525000
\(528\) 0 0
\(529\) −10339.6 −0.849807
\(530\) 0 0
\(531\) 1642.54 0.134238
\(532\) 0 0
\(533\) −18876.8 −1.53404
\(534\) 0 0
\(535\) 17669.0 1.42785
\(536\) 0 0
\(537\) −977.845 −0.0785794
\(538\) 0 0
\(539\) −8729.54 −0.697603
\(540\) 0 0
\(541\) −16028.3 −1.27377 −0.636886 0.770958i \(-0.719778\pi\)
−0.636886 + 0.770958i \(0.719778\pi\)
\(542\) 0 0
\(543\) −21137.1 −1.67049
\(544\) 0 0
\(545\) −6185.80 −0.486184
\(546\) 0 0
\(547\) −7590.17 −0.593295 −0.296647 0.954987i \(-0.595869\pi\)
−0.296647 + 0.954987i \(0.595869\pi\)
\(548\) 0 0
\(549\) −1392.21 −0.108230
\(550\) 0 0
\(551\) −1067.81 −0.0825595
\(552\) 0 0
\(553\) 37727.9 2.90118
\(554\) 0 0
\(555\) 23339.4 1.78505
\(556\) 0 0
\(557\) 15598.4 1.18658 0.593291 0.804988i \(-0.297828\pi\)
0.593291 + 0.804988i \(0.297828\pi\)
\(558\) 0 0
\(559\) 5289.11 0.400189
\(560\) 0 0
\(561\) −429.607 −0.0323316
\(562\) 0 0
\(563\) −18275.4 −1.36806 −0.684030 0.729454i \(-0.739774\pi\)
−0.684030 + 0.729454i \(0.739774\pi\)
\(564\) 0 0
\(565\) −7542.51 −0.561621
\(566\) 0 0
\(567\) −25455.8 −1.88544
\(568\) 0 0
\(569\) −11103.6 −0.818081 −0.409040 0.912516i \(-0.634136\pi\)
−0.409040 + 0.912516i \(0.634136\pi\)
\(570\) 0 0
\(571\) 12505.8 0.916550 0.458275 0.888810i \(-0.348467\pi\)
0.458275 + 0.888810i \(0.348467\pi\)
\(572\) 0 0
\(573\) 12917.4 0.941769
\(574\) 0 0
\(575\) −1323.31 −0.0959756
\(576\) 0 0
\(577\) 16122.3 1.16322 0.581612 0.813466i \(-0.302422\pi\)
0.581612 + 0.813466i \(0.302422\pi\)
\(578\) 0 0
\(579\) −27658.5 −1.98523
\(580\) 0 0
\(581\) 14490.4 1.03471
\(582\) 0 0
\(583\) −2496.87 −0.177375
\(584\) 0 0
\(585\) −4101.53 −0.289876
\(586\) 0 0
\(587\) 82.7289 0.00581701 0.00290851 0.999996i \(-0.499074\pi\)
0.00290851 + 0.999996i \(0.499074\pi\)
\(588\) 0 0
\(589\) −5918.29 −0.414022
\(590\) 0 0
\(591\) −7402.55 −0.515229
\(592\) 0 0
\(593\) −5049.72 −0.349692 −0.174846 0.984596i \(-0.555943\pi\)
−0.174846 + 0.984596i \(0.555943\pi\)
\(594\) 0 0
\(595\) −1415.19 −0.0975076
\(596\) 0 0
\(597\) 27770.1 1.90378
\(598\) 0 0
\(599\) 16393.1 1.11820 0.559101 0.829100i \(-0.311147\pi\)
0.559101 + 0.829100i \(0.311147\pi\)
\(600\) 0 0
\(601\) 22564.2 1.53147 0.765733 0.643158i \(-0.222376\pi\)
0.765733 + 0.643158i \(0.222376\pi\)
\(602\) 0 0
\(603\) 1082.67 0.0731170
\(604\) 0 0
\(605\) −12480.8 −0.838703
\(606\) 0 0
\(607\) 10441.9 0.698230 0.349115 0.937080i \(-0.386482\pi\)
0.349115 + 0.937080i \(0.386482\pi\)
\(608\) 0 0
\(609\) −4965.07 −0.330369
\(610\) 0 0
\(611\) −15870.0 −1.05079
\(612\) 0 0
\(613\) 5153.31 0.339544 0.169772 0.985483i \(-0.445697\pi\)
0.169772 + 0.985483i \(0.445697\pi\)
\(614\) 0 0
\(615\) −37039.1 −2.42856
\(616\) 0 0
\(617\) 22893.5 1.49377 0.746886 0.664952i \(-0.231548\pi\)
0.746886 + 0.664952i \(0.231548\pi\)
\(618\) 0 0
\(619\) 1872.55 0.121590 0.0607949 0.998150i \(-0.480636\pi\)
0.0607949 + 0.998150i \(0.480636\pi\)
\(620\) 0 0
\(621\) −4684.87 −0.302733
\(622\) 0 0
\(623\) −22069.6 −1.41926
\(624\) 0 0
\(625\) −18536.2 −1.18632
\(626\) 0 0
\(627\) −4003.07 −0.254971
\(628\) 0 0
\(629\) 1237.01 0.0784148
\(630\) 0 0
\(631\) −12644.7 −0.797745 −0.398872 0.917006i \(-0.630598\pi\)
−0.398872 + 0.917006i \(0.630598\pi\)
\(632\) 0 0
\(633\) −20996.6 −1.31839
\(634\) 0 0
\(635\) 16012.8 1.00071
\(636\) 0 0
\(637\) −18215.6 −1.13301
\(638\) 0 0
\(639\) 1427.11 0.0883502
\(640\) 0 0
\(641\) −17232.7 −1.06186 −0.530929 0.847416i \(-0.678157\pi\)
−0.530929 + 0.847416i \(0.678157\pi\)
\(642\) 0 0
\(643\) 4194.66 0.257265 0.128632 0.991692i \(-0.458941\pi\)
0.128632 + 0.991692i \(0.458941\pi\)
\(644\) 0 0
\(645\) 10378.0 0.633543
\(646\) 0 0
\(647\) 29338.5 1.78271 0.891355 0.453305i \(-0.149755\pi\)
0.891355 + 0.453305i \(0.149755\pi\)
\(648\) 0 0
\(649\) −3460.48 −0.209300
\(650\) 0 0
\(651\) −27518.7 −1.65675
\(652\) 0 0
\(653\) 30988.6 1.85709 0.928543 0.371225i \(-0.121062\pi\)
0.928543 + 0.371225i \(0.121062\pi\)
\(654\) 0 0
\(655\) 30549.3 1.82238
\(656\) 0 0
\(657\) −8109.65 −0.481564
\(658\) 0 0
\(659\) 8159.70 0.482332 0.241166 0.970484i \(-0.422470\pi\)
0.241166 + 0.970484i \(0.422470\pi\)
\(660\) 0 0
\(661\) −24409.7 −1.43635 −0.718174 0.695864i \(-0.755022\pi\)
−0.718174 + 0.695864i \(0.755022\pi\)
\(662\) 0 0
\(663\) −896.443 −0.0525113
\(664\) 0 0
\(665\) −13186.7 −0.768959
\(666\) 0 0
\(667\) −1239.69 −0.0719657
\(668\) 0 0
\(669\) 20777.3 1.20074
\(670\) 0 0
\(671\) 2933.09 0.168749
\(672\) 0 0
\(673\) 371.652 0.0212870 0.0106435 0.999943i \(-0.496612\pi\)
0.0106435 + 0.999943i \(0.496612\pi\)
\(674\) 0 0
\(675\) 3392.56 0.193451
\(676\) 0 0
\(677\) −8788.32 −0.498911 −0.249455 0.968386i \(-0.580252\pi\)
−0.249455 + 0.968386i \(0.580252\pi\)
\(678\) 0 0
\(679\) 38235.0 2.16101
\(680\) 0 0
\(681\) −7519.51 −0.423125
\(682\) 0 0
\(683\) 8840.50 0.495274 0.247637 0.968853i \(-0.420346\pi\)
0.247637 + 0.968853i \(0.420346\pi\)
\(684\) 0 0
\(685\) 15734.0 0.877616
\(686\) 0 0
\(687\) −1637.60 −0.0909436
\(688\) 0 0
\(689\) −5210.11 −0.288083
\(690\) 0 0
\(691\) 34252.5 1.88571 0.942856 0.333199i \(-0.108128\pi\)
0.942856 + 0.333199i \(0.108128\pi\)
\(692\) 0 0
\(693\) −4513.68 −0.247418
\(694\) 0 0
\(695\) −10275.1 −0.560804
\(696\) 0 0
\(697\) −1963.11 −0.106683
\(698\) 0 0
\(699\) −38912.1 −2.10556
\(700\) 0 0
\(701\) 27978.5 1.50747 0.753733 0.657180i \(-0.228251\pi\)
0.753733 + 0.657180i \(0.228251\pi\)
\(702\) 0 0
\(703\) 11526.5 0.618391
\(704\) 0 0
\(705\) −31139.3 −1.66351
\(706\) 0 0
\(707\) −1471.41 −0.0782719
\(708\) 0 0
\(709\) 378.764 0.0200632 0.0100316 0.999950i \(-0.496807\pi\)
0.0100316 + 0.999950i \(0.496807\pi\)
\(710\) 0 0
\(711\) 11371.3 0.599801
\(712\) 0 0
\(713\) −6870.94 −0.360896
\(714\) 0 0
\(715\) 8641.03 0.451967
\(716\) 0 0
\(717\) −36137.0 −1.88223
\(718\) 0 0
\(719\) −9369.50 −0.485985 −0.242993 0.970028i \(-0.578129\pi\)
−0.242993 + 0.970028i \(0.578129\pi\)
\(720\) 0 0
\(721\) −30462.8 −1.57350
\(722\) 0 0
\(723\) −13523.7 −0.695645
\(724\) 0 0
\(725\) 897.728 0.0459873
\(726\) 0 0
\(727\) −14672.0 −0.748494 −0.374247 0.927329i \(-0.622099\pi\)
−0.374247 + 0.927329i \(0.622099\pi\)
\(728\) 0 0
\(729\) 9993.38 0.507717
\(730\) 0 0
\(731\) 550.047 0.0278307
\(732\) 0 0
\(733\) 9764.90 0.492053 0.246027 0.969263i \(-0.420875\pi\)
0.246027 + 0.969263i \(0.420875\pi\)
\(734\) 0 0
\(735\) −35741.8 −1.79368
\(736\) 0 0
\(737\) −2280.94 −0.114002
\(738\) 0 0
\(739\) 11150.5 0.555043 0.277521 0.960719i \(-0.410487\pi\)
0.277521 + 0.960719i \(0.410487\pi\)
\(740\) 0 0
\(741\) −8353.04 −0.414111
\(742\) 0 0
\(743\) −17461.6 −0.862186 −0.431093 0.902307i \(-0.641872\pi\)
−0.431093 + 0.902307i \(0.641872\pi\)
\(744\) 0 0
\(745\) 5961.47 0.293169
\(746\) 0 0
\(747\) 4367.48 0.213919
\(748\) 0 0
\(749\) 40574.2 1.97937
\(750\) 0 0
\(751\) −12692.5 −0.616717 −0.308358 0.951270i \(-0.599780\pi\)
−0.308358 + 0.951270i \(0.599780\pi\)
\(752\) 0 0
\(753\) −39721.5 −1.92235
\(754\) 0 0
\(755\) 21741.0 1.04800
\(756\) 0 0
\(757\) −35923.3 −1.72477 −0.862387 0.506250i \(-0.831031\pi\)
−0.862387 + 0.506250i \(0.831031\pi\)
\(758\) 0 0
\(759\) −4647.42 −0.222254
\(760\) 0 0
\(761\) −27302.4 −1.30054 −0.650270 0.759703i \(-0.725344\pi\)
−0.650270 + 0.759703i \(0.725344\pi\)
\(762\) 0 0
\(763\) −14204.7 −0.673978
\(764\) 0 0
\(765\) −426.543 −0.0201591
\(766\) 0 0
\(767\) −7220.84 −0.339934
\(768\) 0 0
\(769\) 17941.0 0.841314 0.420657 0.907220i \(-0.361800\pi\)
0.420657 + 0.907220i \(0.361800\pi\)
\(770\) 0 0
\(771\) 18773.8 0.876943
\(772\) 0 0
\(773\) −6566.84 −0.305553 −0.152777 0.988261i \(-0.548822\pi\)
−0.152777 + 0.988261i \(0.548822\pi\)
\(774\) 0 0
\(775\) 4975.61 0.230619
\(776\) 0 0
\(777\) 53595.3 2.47455
\(778\) 0 0
\(779\) −18292.2 −0.841318
\(780\) 0 0
\(781\) −3006.62 −0.137753
\(782\) 0 0
\(783\) 3178.18 0.145056
\(784\) 0 0
\(785\) 17145.1 0.779533
\(786\) 0 0
\(787\) −28734.0 −1.30147 −0.650736 0.759304i \(-0.725539\pi\)
−0.650736 + 0.759304i \(0.725539\pi\)
\(788\) 0 0
\(789\) 14386.2 0.649127
\(790\) 0 0
\(791\) −17320.2 −0.778552
\(792\) 0 0
\(793\) 6120.36 0.274074
\(794\) 0 0
\(795\) −10223.0 −0.456068
\(796\) 0 0
\(797\) 22553.9 1.00238 0.501192 0.865336i \(-0.332895\pi\)
0.501192 + 0.865336i \(0.332895\pi\)
\(798\) 0 0
\(799\) −1650.42 −0.0730757
\(800\) 0 0
\(801\) −6651.88 −0.293424
\(802\) 0 0
\(803\) 17085.3 0.750841
\(804\) 0 0
\(805\) −15309.3 −0.670288
\(806\) 0 0
\(807\) −27121.8 −1.18306
\(808\) 0 0
\(809\) 24921.7 1.08307 0.541533 0.840680i \(-0.317844\pi\)
0.541533 + 0.840680i \(0.317844\pi\)
\(810\) 0 0
\(811\) 41652.2 1.80346 0.901730 0.432300i \(-0.142298\pi\)
0.901730 + 0.432300i \(0.142298\pi\)
\(812\) 0 0
\(813\) −38921.9 −1.67903
\(814\) 0 0
\(815\) 46057.0 1.97952
\(816\) 0 0
\(817\) 5125.32 0.219477
\(818\) 0 0
\(819\) −9418.52 −0.401843
\(820\) 0 0
\(821\) 10969.1 0.466288 0.233144 0.972442i \(-0.425099\pi\)
0.233144 + 0.972442i \(0.425099\pi\)
\(822\) 0 0
\(823\) −16669.2 −0.706015 −0.353008 0.935620i \(-0.614841\pi\)
−0.353008 + 0.935620i \(0.614841\pi\)
\(824\) 0 0
\(825\) 3365.45 0.142024
\(826\) 0 0
\(827\) −9424.07 −0.396260 −0.198130 0.980176i \(-0.563487\pi\)
−0.198130 + 0.980176i \(0.563487\pi\)
\(828\) 0 0
\(829\) 23359.5 0.978658 0.489329 0.872099i \(-0.337242\pi\)
0.489329 + 0.872099i \(0.337242\pi\)
\(830\) 0 0
\(831\) −44423.8 −1.85445
\(832\) 0 0
\(833\) −1894.35 −0.0787938
\(834\) 0 0
\(835\) −50696.8 −2.10112
\(836\) 0 0
\(837\) 17614.9 0.727433
\(838\) 0 0
\(839\) 9260.03 0.381039 0.190520 0.981683i \(-0.438983\pi\)
0.190520 + 0.981683i \(0.438983\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −1624.41 −0.0663673
\(844\) 0 0
\(845\) −9405.76 −0.382921
\(846\) 0 0
\(847\) −28660.1 −1.16266
\(848\) 0 0
\(849\) 10250.4 0.414359
\(850\) 0 0
\(851\) 13381.8 0.539040
\(852\) 0 0
\(853\) 43938.1 1.76367 0.881836 0.471556i \(-0.156307\pi\)
0.881836 + 0.471556i \(0.156307\pi\)
\(854\) 0 0
\(855\) −3974.52 −0.158977
\(856\) 0 0
\(857\) 44540.6 1.77535 0.887676 0.460469i \(-0.152319\pi\)
0.887676 + 0.460469i \(0.152319\pi\)
\(858\) 0 0
\(859\) −5126.48 −0.203624 −0.101812 0.994804i \(-0.532464\pi\)
−0.101812 + 0.994804i \(0.532464\pi\)
\(860\) 0 0
\(861\) −85054.5 −3.36661
\(862\) 0 0
\(863\) −29711.0 −1.17193 −0.585964 0.810337i \(-0.699284\pi\)
−0.585964 + 0.810337i \(0.699284\pi\)
\(864\) 0 0
\(865\) −35115.9 −1.38032
\(866\) 0 0
\(867\) 29238.4 1.14532
\(868\) 0 0
\(869\) −23956.9 −0.935194
\(870\) 0 0
\(871\) −4759.55 −0.185156
\(872\) 0 0
\(873\) 11524.2 0.446775
\(874\) 0 0
\(875\) −33679.8 −1.30124
\(876\) 0 0
\(877\) 24925.9 0.959735 0.479867 0.877341i \(-0.340685\pi\)
0.479867 + 0.877341i \(0.340685\pi\)
\(878\) 0 0
\(879\) 35245.2 1.35244
\(880\) 0 0
\(881\) 13514.7 0.516825 0.258413 0.966035i \(-0.416801\pi\)
0.258413 + 0.966035i \(0.416801\pi\)
\(882\) 0 0
\(883\) 21759.9 0.829309 0.414654 0.909979i \(-0.363903\pi\)
0.414654 + 0.909979i \(0.363903\pi\)
\(884\) 0 0
\(885\) −14168.4 −0.538153
\(886\) 0 0
\(887\) 15556.4 0.588877 0.294439 0.955670i \(-0.404867\pi\)
0.294439 + 0.955670i \(0.404867\pi\)
\(888\) 0 0
\(889\) 36770.9 1.38724
\(890\) 0 0
\(891\) 16164.2 0.607769
\(892\) 0 0
\(893\) −15378.5 −0.576286
\(894\) 0 0
\(895\) 2045.42 0.0763919
\(896\) 0 0
\(897\) −9697.60 −0.360974
\(898\) 0 0
\(899\) 4661.20 0.172925
\(900\) 0 0
\(901\) −541.831 −0.0200344
\(902\) 0 0
\(903\) 23831.5 0.878255
\(904\) 0 0
\(905\) 44213.6 1.62399
\(906\) 0 0
\(907\) −36707.3 −1.34382 −0.671910 0.740633i \(-0.734526\pi\)
−0.671910 + 0.740633i \(0.734526\pi\)
\(908\) 0 0
\(909\) −443.490 −0.0161822
\(910\) 0 0
\(911\) 21608.1 0.785849 0.392924 0.919571i \(-0.371463\pi\)
0.392924 + 0.919571i \(0.371463\pi\)
\(912\) 0 0
\(913\) −9201.33 −0.333537
\(914\) 0 0
\(915\) 12009.1 0.433889
\(916\) 0 0
\(917\) 70151.7 2.52630
\(918\) 0 0
\(919\) 40069.5 1.43827 0.719135 0.694870i \(-0.244538\pi\)
0.719135 + 0.694870i \(0.244538\pi\)
\(920\) 0 0
\(921\) 22502.1 0.805068
\(922\) 0 0
\(923\) −6273.79 −0.223732
\(924\) 0 0
\(925\) −9690.49 −0.344456
\(926\) 0 0
\(927\) −9181.61 −0.325311
\(928\) 0 0
\(929\) −18663.0 −0.659112 −0.329556 0.944136i \(-0.606899\pi\)
−0.329556 + 0.944136i \(0.606899\pi\)
\(930\) 0 0
\(931\) −17651.5 −0.621380
\(932\) 0 0
\(933\) 18757.0 0.658173
\(934\) 0 0
\(935\) 898.633 0.0314315
\(936\) 0 0
\(937\) 13742.0 0.479115 0.239557 0.970882i \(-0.422998\pi\)
0.239557 + 0.970882i \(0.422998\pi\)
\(938\) 0 0
\(939\) 10408.0 0.361718
\(940\) 0 0
\(941\) −31953.9 −1.10698 −0.553489 0.832856i \(-0.686704\pi\)
−0.553489 + 0.832856i \(0.686704\pi\)
\(942\) 0 0
\(943\) −21236.6 −0.733362
\(944\) 0 0
\(945\) 39248.2 1.35105
\(946\) 0 0
\(947\) 11988.8 0.411386 0.205693 0.978617i \(-0.434055\pi\)
0.205693 + 0.978617i \(0.434055\pi\)
\(948\) 0 0
\(949\) 35651.1 1.21948
\(950\) 0 0
\(951\) −10965.0 −0.373886
\(952\) 0 0
\(953\) −6567.87 −0.223247 −0.111623 0.993751i \(-0.535605\pi\)
−0.111623 + 0.993751i \(0.535605\pi\)
\(954\) 0 0
\(955\) −27020.2 −0.915552
\(956\) 0 0
\(957\) 3152.78 0.106494
\(958\) 0 0
\(959\) 36130.8 1.21660
\(960\) 0 0
\(961\) −3956.49 −0.132808
\(962\) 0 0
\(963\) 12229.2 0.409222
\(964\) 0 0
\(965\) 57854.9 1.92997
\(966\) 0 0
\(967\) −40868.4 −1.35909 −0.679545 0.733634i \(-0.737823\pi\)
−0.679545 + 0.733634i \(0.737823\pi\)
\(968\) 0 0
\(969\) −868.683 −0.0287989
\(970\) 0 0
\(971\) −53372.2 −1.76395 −0.881976 0.471295i \(-0.843787\pi\)
−0.881976 + 0.471295i \(0.843787\pi\)
\(972\) 0 0
\(973\) −23595.3 −0.777419
\(974\) 0 0
\(975\) 7022.54 0.230668
\(976\) 0 0
\(977\) 32609.8 1.06784 0.533920 0.845535i \(-0.320718\pi\)
0.533920 + 0.845535i \(0.320718\pi\)
\(978\) 0 0
\(979\) 14014.1 0.457499
\(980\) 0 0
\(981\) −4281.36 −0.139341
\(982\) 0 0
\(983\) −23719.0 −0.769602 −0.384801 0.923000i \(-0.625730\pi\)
−0.384801 + 0.923000i \(0.625730\pi\)
\(984\) 0 0
\(985\) 15484.3 0.500885
\(986\) 0 0
\(987\) −71506.6 −2.30606
\(988\) 0 0
\(989\) 5950.33 0.191314
\(990\) 0 0
\(991\) 12328.2 0.395176 0.197588 0.980285i \(-0.436689\pi\)
0.197588 + 0.980285i \(0.436689\pi\)
\(992\) 0 0
\(993\) 634.871 0.0202890
\(994\) 0 0
\(995\) −58088.4 −1.85078
\(996\) 0 0
\(997\) −55140.7 −1.75158 −0.875790 0.482693i \(-0.839659\pi\)
−0.875790 + 0.482693i \(0.839659\pi\)
\(998\) 0 0
\(999\) −34306.8 −1.08651
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 1856.4.a.u.1.1 3
4.3 odd 2 1856.4.a.p.1.3 3
8.3 odd 2 232.4.a.b.1.1 3
8.5 even 2 464.4.a.g.1.3 3
24.11 even 2 2088.4.a.b.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.4.a.b.1.1 3 8.3 odd 2
464.4.a.g.1.3 3 8.5 even 2
1856.4.a.p.1.3 3 4.3 odd 2
1856.4.a.u.1.1 3 1.1 even 1 trivial
2088.4.a.b.1.3 3 24.11 even 2