Properties

Label 2070.4.a.bg.1.3
Level $2070$
Weight $4$
Character 2070.1
Self dual yes
Analytic conductor $122.134$
Analytic rank $0$
Dimension $4$
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] = [2070,4,Mod(1,2070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2070, 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("2070.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2070.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: \(122.133953712\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 60x^{2} - 45x + 108 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(9.04090\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} +5.03071 q^{7} -8.00000 q^{8} +10.0000 q^{10} +5.58219 q^{11} +62.7277 q^{13} -10.0614 q^{14} +16.0000 q^{16} +19.7435 q^{17} +158.545 q^{19} -20.0000 q^{20} -11.1644 q^{22} +23.0000 q^{23} +25.0000 q^{25} -125.455 q^{26} +20.1228 q^{28} +35.5033 q^{29} +282.041 q^{31} -32.0000 q^{32} -39.4870 q^{34} -25.1536 q^{35} -139.981 q^{37} -317.090 q^{38} +40.0000 q^{40} -227.680 q^{41} +436.962 q^{43} +22.3288 q^{44} -46.0000 q^{46} -90.2701 q^{47} -317.692 q^{49} -50.0000 q^{50} +250.911 q^{52} -330.183 q^{53} -27.9109 q^{55} -40.2457 q^{56} -71.0066 q^{58} +796.203 q^{59} -568.580 q^{61} -564.081 q^{62} +64.0000 q^{64} -313.638 q^{65} +85.1419 q^{67} +78.9740 q^{68} +50.3071 q^{70} +369.578 q^{71} -310.188 q^{73} +279.963 q^{74} +634.180 q^{76} +28.0824 q^{77} -1325.46 q^{79} -80.0000 q^{80} +455.360 q^{82} +158.806 q^{83} -98.7175 q^{85} -873.924 q^{86} -44.6575 q^{88} +1233.89 q^{89} +315.565 q^{91} +92.0000 q^{92} +180.540 q^{94} -792.725 q^{95} -106.389 q^{97} +635.384 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} + 16 q^{4} - 20 q^{5} + 8 q^{7} - 32 q^{8} + 40 q^{10} - 21 q^{11} + 70 q^{13} - 16 q^{14} + 64 q^{16} - 56 q^{17} + 173 q^{19} - 80 q^{20} + 42 q^{22} + 92 q^{23} + 100 q^{25} - 140 q^{26}+ \cdots + 248 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 5.03071 0.271633 0.135816 0.990734i \(-0.456634\pi\)
0.135816 + 0.990734i \(0.456634\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 10.0000 0.316228
\(11\) 5.58219 0.153008 0.0765042 0.997069i \(-0.475624\pi\)
0.0765042 + 0.997069i \(0.475624\pi\)
\(12\) 0 0
\(13\) 62.7277 1.33827 0.669136 0.743140i \(-0.266664\pi\)
0.669136 + 0.743140i \(0.266664\pi\)
\(14\) −10.0614 −0.192073
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 19.7435 0.281677 0.140838 0.990033i \(-0.455020\pi\)
0.140838 + 0.990033i \(0.455020\pi\)
\(18\) 0 0
\(19\) 158.545 1.91435 0.957177 0.289505i \(-0.0934907\pi\)
0.957177 + 0.289505i \(0.0934907\pi\)
\(20\) −20.0000 −0.223607
\(21\) 0 0
\(22\) −11.1644 −0.108193
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −125.455 −0.946301
\(27\) 0 0
\(28\) 20.1228 0.135816
\(29\) 35.5033 0.227338 0.113669 0.993519i \(-0.463740\pi\)
0.113669 + 0.993519i \(0.463740\pi\)
\(30\) 0 0
\(31\) 282.041 1.63406 0.817032 0.576592i \(-0.195618\pi\)
0.817032 + 0.576592i \(0.195618\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −39.4870 −0.199175
\(35\) −25.1536 −0.121478
\(36\) 0 0
\(37\) −139.981 −0.621967 −0.310984 0.950415i \(-0.600658\pi\)
−0.310984 + 0.950415i \(0.600658\pi\)
\(38\) −317.090 −1.35365
\(39\) 0 0
\(40\) 40.0000 0.158114
\(41\) −227.680 −0.867260 −0.433630 0.901091i \(-0.642768\pi\)
−0.433630 + 0.901091i \(0.642768\pi\)
\(42\) 0 0
\(43\) 436.962 1.54968 0.774838 0.632159i \(-0.217831\pi\)
0.774838 + 0.632159i \(0.217831\pi\)
\(44\) 22.3288 0.0765042
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) −90.2701 −0.280154 −0.140077 0.990141i \(-0.544735\pi\)
−0.140077 + 0.990141i \(0.544735\pi\)
\(48\) 0 0
\(49\) −317.692 −0.926216
\(50\) −50.0000 −0.141421
\(51\) 0 0
\(52\) 250.911 0.669136
\(53\) −330.183 −0.855737 −0.427869 0.903841i \(-0.640735\pi\)
−0.427869 + 0.903841i \(0.640735\pi\)
\(54\) 0 0
\(55\) −27.9109 −0.0684275
\(56\) −40.2457 −0.0960367
\(57\) 0 0
\(58\) −71.0066 −0.160752
\(59\) 796.203 1.75689 0.878447 0.477839i \(-0.158580\pi\)
0.878447 + 0.477839i \(0.158580\pi\)
\(60\) 0 0
\(61\) −568.580 −1.19343 −0.596715 0.802453i \(-0.703528\pi\)
−0.596715 + 0.802453i \(0.703528\pi\)
\(62\) −564.081 −1.15546
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −313.638 −0.598493
\(66\) 0 0
\(67\) 85.1419 0.155250 0.0776250 0.996983i \(-0.475266\pi\)
0.0776250 + 0.996983i \(0.475266\pi\)
\(68\) 78.9740 0.140838
\(69\) 0 0
\(70\) 50.3071 0.0858978
\(71\) 369.578 0.617758 0.308879 0.951101i \(-0.400046\pi\)
0.308879 + 0.951101i \(0.400046\pi\)
\(72\) 0 0
\(73\) −310.188 −0.497325 −0.248662 0.968590i \(-0.579991\pi\)
−0.248662 + 0.968590i \(0.579991\pi\)
\(74\) 279.963 0.439797
\(75\) 0 0
\(76\) 634.180 0.957177
\(77\) 28.0824 0.0415621
\(78\) 0 0
\(79\) −1325.46 −1.88766 −0.943832 0.330427i \(-0.892807\pi\)
−0.943832 + 0.330427i \(0.892807\pi\)
\(80\) −80.0000 −0.111803
\(81\) 0 0
\(82\) 455.360 0.613246
\(83\) 158.806 0.210014 0.105007 0.994471i \(-0.466513\pi\)
0.105007 + 0.994471i \(0.466513\pi\)
\(84\) 0 0
\(85\) −98.7175 −0.125970
\(86\) −873.924 −1.09579
\(87\) 0 0
\(88\) −44.6575 −0.0540967
\(89\) 1233.89 1.46957 0.734787 0.678298i \(-0.237282\pi\)
0.734787 + 0.678298i \(0.237282\pi\)
\(90\) 0 0
\(91\) 315.565 0.363519
\(92\) 92.0000 0.104257
\(93\) 0 0
\(94\) 180.540 0.198099
\(95\) −792.725 −0.856125
\(96\) 0 0
\(97\) −106.389 −0.111363 −0.0556814 0.998449i \(-0.517733\pi\)
−0.0556814 + 0.998449i \(0.517733\pi\)
\(98\) 635.384 0.654933
\(99\) 0 0
\(100\) 100.000 0.100000
\(101\) −642.676 −0.633155 −0.316578 0.948567i \(-0.602534\pi\)
−0.316578 + 0.948567i \(0.602534\pi\)
\(102\) 0 0
\(103\) 1621.99 1.55165 0.775823 0.630951i \(-0.217335\pi\)
0.775823 + 0.630951i \(0.217335\pi\)
\(104\) −501.822 −0.473150
\(105\) 0 0
\(106\) 660.365 0.605098
\(107\) −1490.86 −1.34698 −0.673490 0.739196i \(-0.735205\pi\)
−0.673490 + 0.739196i \(0.735205\pi\)
\(108\) 0 0
\(109\) 1204.00 1.05800 0.529001 0.848621i \(-0.322567\pi\)
0.529001 + 0.848621i \(0.322567\pi\)
\(110\) 55.8219 0.0483855
\(111\) 0 0
\(112\) 80.4914 0.0679082
\(113\) −276.771 −0.230411 −0.115206 0.993342i \(-0.536753\pi\)
−0.115206 + 0.993342i \(0.536753\pi\)
\(114\) 0 0
\(115\) −115.000 −0.0932505
\(116\) 142.013 0.113669
\(117\) 0 0
\(118\) −1592.41 −1.24231
\(119\) 99.3239 0.0765126
\(120\) 0 0
\(121\) −1299.84 −0.976588
\(122\) 1137.16 0.843883
\(123\) 0 0
\(124\) 1128.16 0.817032
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1552.79 1.08494 0.542471 0.840075i \(-0.317489\pi\)
0.542471 + 0.840075i \(0.317489\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) 627.277 0.423199
\(131\) 313.873 0.209337 0.104669 0.994507i \(-0.466622\pi\)
0.104669 + 0.994507i \(0.466622\pi\)
\(132\) 0 0
\(133\) 797.594 0.520001
\(134\) −170.284 −0.109778
\(135\) 0 0
\(136\) −157.948 −0.0995877
\(137\) 1066.09 0.664835 0.332417 0.943132i \(-0.392136\pi\)
0.332417 + 0.943132i \(0.392136\pi\)
\(138\) 0 0
\(139\) −1594.79 −0.973152 −0.486576 0.873638i \(-0.661754\pi\)
−0.486576 + 0.873638i \(0.661754\pi\)
\(140\) −100.614 −0.0607389
\(141\) 0 0
\(142\) −739.155 −0.436821
\(143\) 350.158 0.204767
\(144\) 0 0
\(145\) −177.517 −0.101669
\(146\) 620.375 0.351662
\(147\) 0 0
\(148\) −559.925 −0.310984
\(149\) 1096.91 0.603102 0.301551 0.953450i \(-0.402496\pi\)
0.301551 + 0.953450i \(0.402496\pi\)
\(150\) 0 0
\(151\) 2734.41 1.47366 0.736832 0.676076i \(-0.236321\pi\)
0.736832 + 0.676076i \(0.236321\pi\)
\(152\) −1268.36 −0.676826
\(153\) 0 0
\(154\) −56.1647 −0.0293889
\(155\) −1410.20 −0.730776
\(156\) 0 0
\(157\) −1439.63 −0.731813 −0.365907 0.930652i \(-0.619241\pi\)
−0.365907 + 0.930652i \(0.619241\pi\)
\(158\) 2650.91 1.33478
\(159\) 0 0
\(160\) 160.000 0.0790569
\(161\) 115.706 0.0566394
\(162\) 0 0
\(163\) −2995.82 −1.43957 −0.719787 0.694195i \(-0.755760\pi\)
−0.719787 + 0.694195i \(0.755760\pi\)
\(164\) −910.721 −0.433630
\(165\) 0 0
\(166\) −317.611 −0.148503
\(167\) 351.532 0.162888 0.0814442 0.996678i \(-0.474047\pi\)
0.0814442 + 0.996678i \(0.474047\pi\)
\(168\) 0 0
\(169\) 1737.76 0.790971
\(170\) 197.435 0.0890740
\(171\) 0 0
\(172\) 1747.85 0.774838
\(173\) 3701.77 1.62682 0.813411 0.581690i \(-0.197608\pi\)
0.813411 + 0.581690i \(0.197608\pi\)
\(174\) 0 0
\(175\) 125.768 0.0543266
\(176\) 89.3150 0.0382521
\(177\) 0 0
\(178\) −2467.78 −1.03915
\(179\) −3584.09 −1.49658 −0.748290 0.663372i \(-0.769125\pi\)
−0.748290 + 0.663372i \(0.769125\pi\)
\(180\) 0 0
\(181\) 783.672 0.321822 0.160911 0.986969i \(-0.448557\pi\)
0.160911 + 0.986969i \(0.448557\pi\)
\(182\) −631.130 −0.257046
\(183\) 0 0
\(184\) −184.000 −0.0737210
\(185\) 699.907 0.278152
\(186\) 0 0
\(187\) 110.212 0.0430989
\(188\) −361.080 −0.140077
\(189\) 0 0
\(190\) 1585.45 0.605372
\(191\) −1491.95 −0.565201 −0.282601 0.959238i \(-0.591197\pi\)
−0.282601 + 0.959238i \(0.591197\pi\)
\(192\) 0 0
\(193\) 3091.09 1.15286 0.576430 0.817147i \(-0.304446\pi\)
0.576430 + 0.817147i \(0.304446\pi\)
\(194\) 212.778 0.0787454
\(195\) 0 0
\(196\) −1270.77 −0.463108
\(197\) −2820.30 −1.01999 −0.509995 0.860178i \(-0.670353\pi\)
−0.509995 + 0.860178i \(0.670353\pi\)
\(198\) 0 0
\(199\) 1316.11 0.468825 0.234413 0.972137i \(-0.424683\pi\)
0.234413 + 0.972137i \(0.424683\pi\)
\(200\) −200.000 −0.0707107
\(201\) 0 0
\(202\) 1285.35 0.447708
\(203\) 178.607 0.0617524
\(204\) 0 0
\(205\) 1138.40 0.387851
\(206\) −3243.98 −1.09718
\(207\) 0 0
\(208\) 1003.64 0.334568
\(209\) 885.028 0.292912
\(210\) 0 0
\(211\) −3399.19 −1.10905 −0.554525 0.832167i \(-0.687100\pi\)
−0.554525 + 0.832167i \(0.687100\pi\)
\(212\) −1320.73 −0.427869
\(213\) 0 0
\(214\) 2981.72 0.952458
\(215\) −2184.81 −0.693037
\(216\) 0 0
\(217\) 1418.86 0.443865
\(218\) −2408.00 −0.748121
\(219\) 0 0
\(220\) −111.644 −0.0342137
\(221\) 1238.46 0.376960
\(222\) 0 0
\(223\) 864.660 0.259650 0.129825 0.991537i \(-0.458558\pi\)
0.129825 + 0.991537i \(0.458558\pi\)
\(224\) −160.983 −0.0480184
\(225\) 0 0
\(226\) 553.543 0.162925
\(227\) 1979.40 0.578754 0.289377 0.957215i \(-0.406552\pi\)
0.289377 + 0.957215i \(0.406552\pi\)
\(228\) 0 0
\(229\) −3113.53 −0.898462 −0.449231 0.893416i \(-0.648302\pi\)
−0.449231 + 0.893416i \(0.648302\pi\)
\(230\) 230.000 0.0659380
\(231\) 0 0
\(232\) −284.026 −0.0803761
\(233\) −6018.65 −1.69225 −0.846126 0.532983i \(-0.821071\pi\)
−0.846126 + 0.532983i \(0.821071\pi\)
\(234\) 0 0
\(235\) 451.350 0.125289
\(236\) 3184.81 0.878447
\(237\) 0 0
\(238\) −198.648 −0.0541026
\(239\) 4224.88 1.14345 0.571725 0.820445i \(-0.306274\pi\)
0.571725 + 0.820445i \(0.306274\pi\)
\(240\) 0 0
\(241\) 1394.24 0.372658 0.186329 0.982487i \(-0.440341\pi\)
0.186329 + 0.982487i \(0.440341\pi\)
\(242\) 2599.68 0.690552
\(243\) 0 0
\(244\) −2274.32 −0.596715
\(245\) 1588.46 0.414216
\(246\) 0 0
\(247\) 9945.16 2.56192
\(248\) −2256.32 −0.577729
\(249\) 0 0
\(250\) 250.000 0.0632456
\(251\) 5968.83 1.50099 0.750497 0.660874i \(-0.229814\pi\)
0.750497 + 0.660874i \(0.229814\pi\)
\(252\) 0 0
\(253\) 128.390 0.0319045
\(254\) −3105.57 −0.767169
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 1636.06 0.397100 0.198550 0.980091i \(-0.436377\pi\)
0.198550 + 0.980091i \(0.436377\pi\)
\(258\) 0 0
\(259\) −704.206 −0.168947
\(260\) −1254.55 −0.299247
\(261\) 0 0
\(262\) −627.745 −0.148024
\(263\) −2995.55 −0.702333 −0.351166 0.936313i \(-0.614215\pi\)
−0.351166 + 0.936313i \(0.614215\pi\)
\(264\) 0 0
\(265\) 1650.91 0.382697
\(266\) −1595.19 −0.367696
\(267\) 0 0
\(268\) 340.568 0.0776250
\(269\) 5190.17 1.17640 0.588198 0.808717i \(-0.299838\pi\)
0.588198 + 0.808717i \(0.299838\pi\)
\(270\) 0 0
\(271\) 6720.43 1.50641 0.753205 0.657786i \(-0.228507\pi\)
0.753205 + 0.657786i \(0.228507\pi\)
\(272\) 315.896 0.0704192
\(273\) 0 0
\(274\) −2132.18 −0.470109
\(275\) 139.555 0.0306017
\(276\) 0 0
\(277\) −6114.74 −1.32635 −0.663176 0.748464i \(-0.730792\pi\)
−0.663176 + 0.748464i \(0.730792\pi\)
\(278\) 3189.58 0.688123
\(279\) 0 0
\(280\) 201.228 0.0429489
\(281\) 3549.07 0.753450 0.376725 0.926325i \(-0.377050\pi\)
0.376725 + 0.926325i \(0.377050\pi\)
\(282\) 0 0
\(283\) 3103.11 0.651806 0.325903 0.945403i \(-0.394332\pi\)
0.325903 + 0.945403i \(0.394332\pi\)
\(284\) 1478.31 0.308879
\(285\) 0 0
\(286\) −700.316 −0.144792
\(287\) −1145.39 −0.235576
\(288\) 0 0
\(289\) −4523.19 −0.920658
\(290\) 355.033 0.0718905
\(291\) 0 0
\(292\) −1240.75 −0.248662
\(293\) −1218.51 −0.242957 −0.121478 0.992594i \(-0.538763\pi\)
−0.121478 + 0.992594i \(0.538763\pi\)
\(294\) 0 0
\(295\) −3981.01 −0.785707
\(296\) 1119.85 0.219899
\(297\) 0 0
\(298\) −2193.82 −0.426458
\(299\) 1442.74 0.279049
\(300\) 0 0
\(301\) 2198.23 0.420943
\(302\) −5468.82 −1.04204
\(303\) 0 0
\(304\) 2536.72 0.478588
\(305\) 2842.90 0.533718
\(306\) 0 0
\(307\) 2564.45 0.476745 0.238373 0.971174i \(-0.423386\pi\)
0.238373 + 0.971174i \(0.423386\pi\)
\(308\) 112.329 0.0207811
\(309\) 0 0
\(310\) 2820.41 0.516736
\(311\) 92.6253 0.0168884 0.00844421 0.999964i \(-0.497312\pi\)
0.00844421 + 0.999964i \(0.497312\pi\)
\(312\) 0 0
\(313\) −4276.44 −0.772263 −0.386132 0.922444i \(-0.626189\pi\)
−0.386132 + 0.922444i \(0.626189\pi\)
\(314\) 2879.25 0.517470
\(315\) 0 0
\(316\) −5301.82 −0.943832
\(317\) 2964.24 0.525199 0.262600 0.964905i \(-0.415420\pi\)
0.262600 + 0.964905i \(0.415420\pi\)
\(318\) 0 0
\(319\) 198.186 0.0347846
\(320\) −320.000 −0.0559017
\(321\) 0 0
\(322\) −231.413 −0.0400501
\(323\) 3130.23 0.539229
\(324\) 0 0
\(325\) 1568.19 0.267654
\(326\) 5991.63 1.01793
\(327\) 0 0
\(328\) 1821.44 0.306623
\(329\) −454.123 −0.0760991
\(330\) 0 0
\(331\) −1348.81 −0.223979 −0.111989 0.993709i \(-0.535722\pi\)
−0.111989 + 0.993709i \(0.535722\pi\)
\(332\) 635.223 0.105007
\(333\) 0 0
\(334\) −703.064 −0.115180
\(335\) −425.710 −0.0694299
\(336\) 0 0
\(337\) −5764.30 −0.931755 −0.465878 0.884849i \(-0.654261\pi\)
−0.465878 + 0.884849i \(0.654261\pi\)
\(338\) −3475.53 −0.559301
\(339\) 0 0
\(340\) −394.870 −0.0629848
\(341\) 1574.40 0.250026
\(342\) 0 0
\(343\) −3323.75 −0.523223
\(344\) −3495.70 −0.547894
\(345\) 0 0
\(346\) −7403.53 −1.15034
\(347\) −4066.16 −0.629057 −0.314528 0.949248i \(-0.601846\pi\)
−0.314528 + 0.949248i \(0.601846\pi\)
\(348\) 0 0
\(349\) 8407.82 1.28957 0.644786 0.764363i \(-0.276947\pi\)
0.644786 + 0.764363i \(0.276947\pi\)
\(350\) −251.536 −0.0384147
\(351\) 0 0
\(352\) −178.630 −0.0270483
\(353\) −256.068 −0.0386095 −0.0193047 0.999814i \(-0.506145\pi\)
−0.0193047 + 0.999814i \(0.506145\pi\)
\(354\) 0 0
\(355\) −1847.89 −0.276270
\(356\) 4935.56 0.734787
\(357\) 0 0
\(358\) 7168.19 1.05824
\(359\) 12052.5 1.77189 0.885943 0.463794i \(-0.153512\pi\)
0.885943 + 0.463794i \(0.153512\pi\)
\(360\) 0 0
\(361\) 18277.5 2.66475
\(362\) −1567.34 −0.227563
\(363\) 0 0
\(364\) 1262.26 0.181759
\(365\) 1550.94 0.222410
\(366\) 0 0
\(367\) −9245.40 −1.31500 −0.657502 0.753453i \(-0.728387\pi\)
−0.657502 + 0.753453i \(0.728387\pi\)
\(368\) 368.000 0.0521286
\(369\) 0 0
\(370\) −1399.81 −0.196683
\(371\) −1661.05 −0.232446
\(372\) 0 0
\(373\) −869.265 −0.120667 −0.0603335 0.998178i \(-0.519216\pi\)
−0.0603335 + 0.998178i \(0.519216\pi\)
\(374\) −220.424 −0.0304755
\(375\) 0 0
\(376\) 722.161 0.0990495
\(377\) 2227.04 0.304240
\(378\) 0 0
\(379\) 1278.76 0.173313 0.0866566 0.996238i \(-0.472382\pi\)
0.0866566 + 0.996238i \(0.472382\pi\)
\(380\) −3170.90 −0.428062
\(381\) 0 0
\(382\) 2983.89 0.399658
\(383\) 5343.75 0.712932 0.356466 0.934308i \(-0.383982\pi\)
0.356466 + 0.934308i \(0.383982\pi\)
\(384\) 0 0
\(385\) −140.412 −0.0185871
\(386\) −6182.19 −0.815194
\(387\) 0 0
\(388\) −425.557 −0.0556814
\(389\) 708.346 0.0923254 0.0461627 0.998934i \(-0.485301\pi\)
0.0461627 + 0.998934i \(0.485301\pi\)
\(390\) 0 0
\(391\) 454.101 0.0587336
\(392\) 2541.54 0.327467
\(393\) 0 0
\(394\) 5640.60 0.721241
\(395\) 6627.28 0.844189
\(396\) 0 0
\(397\) −5924.01 −0.748911 −0.374455 0.927245i \(-0.622170\pi\)
−0.374455 + 0.927245i \(0.622170\pi\)
\(398\) −2632.21 −0.331510
\(399\) 0 0
\(400\) 400.000 0.0500000
\(401\) 11393.6 1.41888 0.709441 0.704765i \(-0.248948\pi\)
0.709441 + 0.704765i \(0.248948\pi\)
\(402\) 0 0
\(403\) 17691.8 2.18682
\(404\) −2570.71 −0.316578
\(405\) 0 0
\(406\) −357.214 −0.0436656
\(407\) −781.402 −0.0951663
\(408\) 0 0
\(409\) 13804.4 1.66891 0.834457 0.551073i \(-0.185782\pi\)
0.834457 + 0.551073i \(0.185782\pi\)
\(410\) −2276.80 −0.274252
\(411\) 0 0
\(412\) 6487.96 0.775823
\(413\) 4005.47 0.477230
\(414\) 0 0
\(415\) −794.028 −0.0939212
\(416\) −2007.29 −0.236575
\(417\) 0 0
\(418\) −1770.06 −0.207120
\(419\) 14795.8 1.72512 0.862558 0.505958i \(-0.168861\pi\)
0.862558 + 0.505958i \(0.168861\pi\)
\(420\) 0 0
\(421\) 5804.66 0.671976 0.335988 0.941866i \(-0.390930\pi\)
0.335988 + 0.941866i \(0.390930\pi\)
\(422\) 6798.37 0.784217
\(423\) 0 0
\(424\) 2641.46 0.302549
\(425\) 493.588 0.0563353
\(426\) 0 0
\(427\) −2860.36 −0.324175
\(428\) −5963.44 −0.673490
\(429\) 0 0
\(430\) 4369.62 0.490051
\(431\) −15268.2 −1.70636 −0.853181 0.521615i \(-0.825330\pi\)
−0.853181 + 0.521615i \(0.825330\pi\)
\(432\) 0 0
\(433\) 2250.62 0.249788 0.124894 0.992170i \(-0.460141\pi\)
0.124894 + 0.992170i \(0.460141\pi\)
\(434\) −2837.73 −0.313860
\(435\) 0 0
\(436\) 4816.00 0.529001
\(437\) 3646.53 0.399170
\(438\) 0 0
\(439\) 10366.2 1.12700 0.563499 0.826117i \(-0.309455\pi\)
0.563499 + 0.826117i \(0.309455\pi\)
\(440\) 223.288 0.0241928
\(441\) 0 0
\(442\) −2476.93 −0.266551
\(443\) 3881.11 0.416246 0.208123 0.978103i \(-0.433264\pi\)
0.208123 + 0.978103i \(0.433264\pi\)
\(444\) 0 0
\(445\) −6169.45 −0.657214
\(446\) −1729.32 −0.183600
\(447\) 0 0
\(448\) 321.965 0.0339541
\(449\) 10146.5 1.06647 0.533233 0.845968i \(-0.320977\pi\)
0.533233 + 0.845968i \(0.320977\pi\)
\(450\) 0 0
\(451\) −1270.95 −0.132698
\(452\) −1107.09 −0.115206
\(453\) 0 0
\(454\) −3958.79 −0.409241
\(455\) −1577.82 −0.162570
\(456\) 0 0
\(457\) −4728.87 −0.484042 −0.242021 0.970271i \(-0.577810\pi\)
−0.242021 + 0.970271i \(0.577810\pi\)
\(458\) 6227.06 0.635308
\(459\) 0 0
\(460\) −460.000 −0.0466252
\(461\) −8380.69 −0.846698 −0.423349 0.905967i \(-0.639146\pi\)
−0.423349 + 0.905967i \(0.639146\pi\)
\(462\) 0 0
\(463\) 6845.17 0.687089 0.343545 0.939136i \(-0.388372\pi\)
0.343545 + 0.939136i \(0.388372\pi\)
\(464\) 568.053 0.0568345
\(465\) 0 0
\(466\) 12037.3 1.19660
\(467\) 14884.0 1.47484 0.737418 0.675436i \(-0.236045\pi\)
0.737418 + 0.675436i \(0.236045\pi\)
\(468\) 0 0
\(469\) 428.324 0.0421710
\(470\) −902.701 −0.0885925
\(471\) 0 0
\(472\) −6369.62 −0.621156
\(473\) 2439.20 0.237114
\(474\) 0 0
\(475\) 3963.62 0.382871
\(476\) 397.295 0.0382563
\(477\) 0 0
\(478\) −8449.76 −0.808542
\(479\) −6048.69 −0.576976 −0.288488 0.957483i \(-0.593153\pi\)
−0.288488 + 0.957483i \(0.593153\pi\)
\(480\) 0 0
\(481\) −8780.71 −0.832361
\(482\) −2788.47 −0.263509
\(483\) 0 0
\(484\) −5199.36 −0.488294
\(485\) 531.946 0.0498029
\(486\) 0 0
\(487\) 3199.35 0.297693 0.148846 0.988860i \(-0.452444\pi\)
0.148846 + 0.988860i \(0.452444\pi\)
\(488\) 4548.64 0.421941
\(489\) 0 0
\(490\) −3176.92 −0.292895
\(491\) 1614.41 0.148385 0.0741926 0.997244i \(-0.476362\pi\)
0.0741926 + 0.997244i \(0.476362\pi\)
\(492\) 0 0
\(493\) 700.960 0.0640358
\(494\) −19890.3 −1.81155
\(495\) 0 0
\(496\) 4512.65 0.408516
\(497\) 1859.24 0.167803
\(498\) 0 0
\(499\) −5392.04 −0.483729 −0.241864 0.970310i \(-0.577759\pi\)
−0.241864 + 0.970310i \(0.577759\pi\)
\(500\) −500.000 −0.0447214
\(501\) 0 0
\(502\) −11937.7 −1.06136
\(503\) −12625.4 −1.11917 −0.559583 0.828775i \(-0.689039\pi\)
−0.559583 + 0.828775i \(0.689039\pi\)
\(504\) 0 0
\(505\) 3213.38 0.283156
\(506\) −256.781 −0.0225599
\(507\) 0 0
\(508\) 6211.15 0.542471
\(509\) −7065.76 −0.615294 −0.307647 0.951501i \(-0.599541\pi\)
−0.307647 + 0.951501i \(0.599541\pi\)
\(510\) 0 0
\(511\) −1560.46 −0.135090
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) −3272.12 −0.280792
\(515\) −8109.96 −0.693917
\(516\) 0 0
\(517\) −503.905 −0.0428660
\(518\) 1408.41 0.119463
\(519\) 0 0
\(520\) 2509.11 0.211599
\(521\) 8399.20 0.706287 0.353144 0.935569i \(-0.385113\pi\)
0.353144 + 0.935569i \(0.385113\pi\)
\(522\) 0 0
\(523\) −7002.06 −0.585428 −0.292714 0.956200i \(-0.594558\pi\)
−0.292714 + 0.956200i \(0.594558\pi\)
\(524\) 1255.49 0.104669
\(525\) 0 0
\(526\) 5991.10 0.496624
\(527\) 5568.47 0.460278
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) −3301.83 −0.270608
\(531\) 0 0
\(532\) 3190.37 0.260001
\(533\) −14281.9 −1.16063
\(534\) 0 0
\(535\) 7454.30 0.602388
\(536\) −681.136 −0.0548891
\(537\) 0 0
\(538\) −10380.3 −0.831838
\(539\) −1773.42 −0.141719
\(540\) 0 0
\(541\) 3338.13 0.265282 0.132641 0.991164i \(-0.457654\pi\)
0.132641 + 0.991164i \(0.457654\pi\)
\(542\) −13440.9 −1.06519
\(543\) 0 0
\(544\) −631.792 −0.0497939
\(545\) −6020.00 −0.473153
\(546\) 0 0
\(547\) 9609.09 0.751106 0.375553 0.926801i \(-0.377453\pi\)
0.375553 + 0.926801i \(0.377453\pi\)
\(548\) 4264.37 0.332417
\(549\) 0 0
\(550\) −279.109 −0.0216387
\(551\) 5628.87 0.435205
\(552\) 0 0
\(553\) −6667.98 −0.512751
\(554\) 12229.5 0.937872
\(555\) 0 0
\(556\) −6379.15 −0.486576
\(557\) 11931.1 0.907608 0.453804 0.891102i \(-0.350067\pi\)
0.453804 + 0.891102i \(0.350067\pi\)
\(558\) 0 0
\(559\) 27409.6 2.07389
\(560\) −402.457 −0.0303695
\(561\) 0 0
\(562\) −7098.13 −0.532770
\(563\) 10082.4 0.754746 0.377373 0.926061i \(-0.376827\pi\)
0.377373 + 0.926061i \(0.376827\pi\)
\(564\) 0 0
\(565\) 1383.86 0.103043
\(566\) −6206.23 −0.460896
\(567\) 0 0
\(568\) −2956.62 −0.218410
\(569\) 18710.5 1.37853 0.689267 0.724507i \(-0.257933\pi\)
0.689267 + 0.724507i \(0.257933\pi\)
\(570\) 0 0
\(571\) 13510.1 0.990159 0.495079 0.868848i \(-0.335139\pi\)
0.495079 + 0.868848i \(0.335139\pi\)
\(572\) 1400.63 0.102383
\(573\) 0 0
\(574\) 2290.79 0.166578
\(575\) 575.000 0.0417029
\(576\) 0 0
\(577\) 11582.3 0.835662 0.417831 0.908525i \(-0.362790\pi\)
0.417831 + 0.908525i \(0.362790\pi\)
\(578\) 9046.39 0.651004
\(579\) 0 0
\(580\) −710.066 −0.0508343
\(581\) 798.905 0.0570468
\(582\) 0 0
\(583\) −1843.14 −0.130935
\(584\) 2481.50 0.175831
\(585\) 0 0
\(586\) 2437.03 0.171796
\(587\) 6019.34 0.423245 0.211623 0.977351i \(-0.432125\pi\)
0.211623 + 0.977351i \(0.432125\pi\)
\(588\) 0 0
\(589\) 44716.1 3.12818
\(590\) 7962.03 0.555579
\(591\) 0 0
\(592\) −2239.70 −0.155492
\(593\) −26299.3 −1.82122 −0.910608 0.413270i \(-0.864387\pi\)
−0.910608 + 0.413270i \(0.864387\pi\)
\(594\) 0 0
\(595\) −496.619 −0.0342175
\(596\) 4387.63 0.301551
\(597\) 0 0
\(598\) −2885.47 −0.197317
\(599\) 27368.0 1.86682 0.933409 0.358813i \(-0.116819\pi\)
0.933409 + 0.358813i \(0.116819\pi\)
\(600\) 0 0
\(601\) −3472.33 −0.235673 −0.117836 0.993033i \(-0.537596\pi\)
−0.117836 + 0.993033i \(0.537596\pi\)
\(602\) −4396.46 −0.297652
\(603\) 0 0
\(604\) 10937.6 0.736832
\(605\) 6499.20 0.436744
\(606\) 0 0
\(607\) 5061.10 0.338425 0.169212 0.985580i \(-0.445878\pi\)
0.169212 + 0.985580i \(0.445878\pi\)
\(608\) −5073.44 −0.338413
\(609\) 0 0
\(610\) −5685.80 −0.377396
\(611\) −5662.44 −0.374922
\(612\) 0 0
\(613\) −28272.6 −1.86283 −0.931417 0.363953i \(-0.881427\pi\)
−0.931417 + 0.363953i \(0.881427\pi\)
\(614\) −5128.90 −0.337110
\(615\) 0 0
\(616\) −224.659 −0.0146944
\(617\) 2222.47 0.145013 0.0725066 0.997368i \(-0.476900\pi\)
0.0725066 + 0.997368i \(0.476900\pi\)
\(618\) 0 0
\(619\) −10498.2 −0.681679 −0.340840 0.940121i \(-0.610711\pi\)
−0.340840 + 0.940121i \(0.610711\pi\)
\(620\) −5640.81 −0.365388
\(621\) 0 0
\(622\) −185.251 −0.0119419
\(623\) 6207.34 0.399185
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 8552.87 0.546073
\(627\) 0 0
\(628\) −5758.51 −0.365907
\(629\) −2763.72 −0.175194
\(630\) 0 0
\(631\) −11402.2 −0.719355 −0.359678 0.933077i \(-0.617113\pi\)
−0.359678 + 0.933077i \(0.617113\pi\)
\(632\) 10603.6 0.667390
\(633\) 0 0
\(634\) −5928.47 −0.371372
\(635\) −7763.93 −0.485200
\(636\) 0 0
\(637\) −19928.1 −1.23953
\(638\) −396.372 −0.0245964
\(639\) 0 0
\(640\) 640.000 0.0395285
\(641\) −26128.3 −1.60999 −0.804996 0.593280i \(-0.797833\pi\)
−0.804996 + 0.593280i \(0.797833\pi\)
\(642\) 0 0
\(643\) 863.502 0.0529599 0.0264799 0.999649i \(-0.491570\pi\)
0.0264799 + 0.999649i \(0.491570\pi\)
\(644\) 462.825 0.0283197
\(645\) 0 0
\(646\) −6260.47 −0.381292
\(647\) 5477.43 0.332828 0.166414 0.986056i \(-0.446781\pi\)
0.166414 + 0.986056i \(0.446781\pi\)
\(648\) 0 0
\(649\) 4444.55 0.268820
\(650\) −3136.38 −0.189260
\(651\) 0 0
\(652\) −11983.3 −0.719787
\(653\) −19233.2 −1.15261 −0.576305 0.817235i \(-0.695506\pi\)
−0.576305 + 0.817235i \(0.695506\pi\)
\(654\) 0 0
\(655\) −1569.36 −0.0936185
\(656\) −3642.88 −0.216815
\(657\) 0 0
\(658\) 908.245 0.0538102
\(659\) −9952.51 −0.588307 −0.294154 0.955758i \(-0.595038\pi\)
−0.294154 + 0.955758i \(0.595038\pi\)
\(660\) 0 0
\(661\) 3047.96 0.179352 0.0896762 0.995971i \(-0.471417\pi\)
0.0896762 + 0.995971i \(0.471417\pi\)
\(662\) 2697.61 0.158377
\(663\) 0 0
\(664\) −1270.45 −0.0742513
\(665\) −3987.97 −0.232552
\(666\) 0 0
\(667\) 816.576 0.0474032
\(668\) 1406.13 0.0814442
\(669\) 0 0
\(670\) 851.419 0.0490943
\(671\) −3173.92 −0.182605
\(672\) 0 0
\(673\) 16245.9 0.930508 0.465254 0.885177i \(-0.345963\pi\)
0.465254 + 0.885177i \(0.345963\pi\)
\(674\) 11528.6 0.658850
\(675\) 0 0
\(676\) 6951.05 0.395486
\(677\) 30056.6 1.70631 0.853154 0.521659i \(-0.174687\pi\)
0.853154 + 0.521659i \(0.174687\pi\)
\(678\) 0 0
\(679\) −535.213 −0.0302498
\(680\) 789.740 0.0445370
\(681\) 0 0
\(682\) −3148.81 −0.176795
\(683\) −10064.8 −0.563866 −0.281933 0.959434i \(-0.590976\pi\)
−0.281933 + 0.959434i \(0.590976\pi\)
\(684\) 0 0
\(685\) −5330.46 −0.297323
\(686\) 6647.50 0.369975
\(687\) 0 0
\(688\) 6991.39 0.387419
\(689\) −20711.6 −1.14521
\(690\) 0 0
\(691\) 10260.7 0.564886 0.282443 0.959284i \(-0.408855\pi\)
0.282443 + 0.959284i \(0.408855\pi\)
\(692\) 14807.1 0.813411
\(693\) 0 0
\(694\) 8132.31 0.444810
\(695\) 7973.94 0.435207
\(696\) 0 0
\(697\) −4495.21 −0.244287
\(698\) −16815.6 −0.911865
\(699\) 0 0
\(700\) 503.071 0.0271633
\(701\) 19815.6 1.06766 0.533828 0.845593i \(-0.320753\pi\)
0.533828 + 0.845593i \(0.320753\pi\)
\(702\) 0 0
\(703\) −22193.3 −1.19067
\(704\) 357.260 0.0191261
\(705\) 0 0
\(706\) 512.136 0.0273010
\(707\) −3233.12 −0.171986
\(708\) 0 0
\(709\) −30533.5 −1.61736 −0.808681 0.588247i \(-0.799818\pi\)
−0.808681 + 0.588247i \(0.799818\pi\)
\(710\) 3695.78 0.195352
\(711\) 0 0
\(712\) −9871.12 −0.519573
\(713\) 6486.93 0.340726
\(714\) 0 0
\(715\) −1750.79 −0.0915745
\(716\) −14336.4 −0.748290
\(717\) 0 0
\(718\) −24105.0 −1.25291
\(719\) −9125.45 −0.473327 −0.236663 0.971592i \(-0.576054\pi\)
−0.236663 + 0.971592i \(0.576054\pi\)
\(720\) 0 0
\(721\) 8159.77 0.421478
\(722\) −36555.0 −1.88426
\(723\) 0 0
\(724\) 3134.69 0.160911
\(725\) 887.583 0.0454676
\(726\) 0 0
\(727\) 2635.53 0.134452 0.0672259 0.997738i \(-0.478585\pi\)
0.0672259 + 0.997738i \(0.478585\pi\)
\(728\) −2524.52 −0.128523
\(729\) 0 0
\(730\) −3101.88 −0.157268
\(731\) 8627.17 0.436508
\(732\) 0 0
\(733\) 10778.0 0.543102 0.271551 0.962424i \(-0.412463\pi\)
0.271551 + 0.962424i \(0.412463\pi\)
\(734\) 18490.8 0.929848
\(735\) 0 0
\(736\) −736.000 −0.0368605
\(737\) 475.278 0.0237545
\(738\) 0 0
\(739\) −1758.72 −0.0875449 −0.0437725 0.999042i \(-0.513938\pi\)
−0.0437725 + 0.999042i \(0.513938\pi\)
\(740\) 2799.63 0.139076
\(741\) 0 0
\(742\) 3322.11 0.164364
\(743\) −9737.97 −0.480823 −0.240412 0.970671i \(-0.577282\pi\)
−0.240412 + 0.970671i \(0.577282\pi\)
\(744\) 0 0
\(745\) −5484.54 −0.269716
\(746\) 1738.53 0.0853245
\(747\) 0 0
\(748\) 440.848 0.0215495
\(749\) −7500.08 −0.365884
\(750\) 0 0
\(751\) −23795.7 −1.15621 −0.578107 0.815961i \(-0.696208\pi\)
−0.578107 + 0.815961i \(0.696208\pi\)
\(752\) −1444.32 −0.0700385
\(753\) 0 0
\(754\) −4454.08 −0.215130
\(755\) −13672.1 −0.659043
\(756\) 0 0
\(757\) 23647.6 1.13538 0.567692 0.823241i \(-0.307837\pi\)
0.567692 + 0.823241i \(0.307837\pi\)
\(758\) −2557.53 −0.122551
\(759\) 0 0
\(760\) 6341.80 0.302686
\(761\) 4606.09 0.219410 0.109705 0.993964i \(-0.465009\pi\)
0.109705 + 0.993964i \(0.465009\pi\)
\(762\) 0 0
\(763\) 6056.98 0.287388
\(764\) −5967.79 −0.282601
\(765\) 0 0
\(766\) −10687.5 −0.504119
\(767\) 49944.0 2.35120
\(768\) 0 0
\(769\) 17760.9 0.832867 0.416433 0.909166i \(-0.363280\pi\)
0.416433 + 0.909166i \(0.363280\pi\)
\(770\) 280.824 0.0131431
\(771\) 0 0
\(772\) 12364.4 0.576430
\(773\) −7147.47 −0.332570 −0.166285 0.986078i \(-0.553177\pi\)
−0.166285 + 0.986078i \(0.553177\pi\)
\(774\) 0 0
\(775\) 7051.02 0.326813
\(776\) 851.114 0.0393727
\(777\) 0 0
\(778\) −1416.69 −0.0652839
\(779\) −36097.6 −1.66024
\(780\) 0 0
\(781\) 2063.05 0.0945222
\(782\) −908.201 −0.0415310
\(783\) 0 0
\(784\) −5083.07 −0.231554
\(785\) 7198.13 0.327277
\(786\) 0 0
\(787\) −31719.1 −1.43668 −0.718338 0.695694i \(-0.755097\pi\)
−0.718338 + 0.695694i \(0.755097\pi\)
\(788\) −11281.2 −0.509995
\(789\) 0 0
\(790\) −13254.6 −0.596932
\(791\) −1392.36 −0.0625872
\(792\) 0 0
\(793\) −35665.7 −1.59713
\(794\) 11848.0 0.529560
\(795\) 0 0
\(796\) 5264.42 0.234413
\(797\) −30828.3 −1.37013 −0.685066 0.728481i \(-0.740227\pi\)
−0.685066 + 0.728481i \(0.740227\pi\)
\(798\) 0 0
\(799\) −1782.25 −0.0789129
\(800\) −800.000 −0.0353553
\(801\) 0 0
\(802\) −22787.3 −1.00330
\(803\) −1731.53 −0.0760949
\(804\) 0 0
\(805\) −578.532 −0.0253299
\(806\) −35383.5 −1.54632
\(807\) 0 0
\(808\) 5141.41 0.223854
\(809\) −36679.2 −1.59403 −0.797015 0.603959i \(-0.793589\pi\)
−0.797015 + 0.603959i \(0.793589\pi\)
\(810\) 0 0
\(811\) −12210.0 −0.528672 −0.264336 0.964431i \(-0.585153\pi\)
−0.264336 + 0.964431i \(0.585153\pi\)
\(812\) 714.427 0.0308762
\(813\) 0 0
\(814\) 1562.80 0.0672927
\(815\) 14979.1 0.643797
\(816\) 0 0
\(817\) 69278.2 2.96663
\(818\) −27608.9 −1.18010
\(819\) 0 0
\(820\) 4553.60 0.193925
\(821\) 9996.47 0.424944 0.212472 0.977167i \(-0.431849\pi\)
0.212472 + 0.977167i \(0.431849\pi\)
\(822\) 0 0
\(823\) −42868.8 −1.81569 −0.907845 0.419306i \(-0.862273\pi\)
−0.907845 + 0.419306i \(0.862273\pi\)
\(824\) −12975.9 −0.548590
\(825\) 0 0
\(826\) −8010.93 −0.337453
\(827\) 15833.4 0.665759 0.332880 0.942969i \(-0.391980\pi\)
0.332880 + 0.942969i \(0.391980\pi\)
\(828\) 0 0
\(829\) −17722.3 −0.742484 −0.371242 0.928536i \(-0.621068\pi\)
−0.371242 + 0.928536i \(0.621068\pi\)
\(830\) 1588.06 0.0664123
\(831\) 0 0
\(832\) 4014.57 0.167284
\(833\) −6272.35 −0.260893
\(834\) 0 0
\(835\) −1757.66 −0.0728459
\(836\) 3540.11 0.146456
\(837\) 0 0
\(838\) −29591.7 −1.21984
\(839\) 40528.4 1.66769 0.833847 0.551996i \(-0.186134\pi\)
0.833847 + 0.551996i \(0.186134\pi\)
\(840\) 0 0
\(841\) −23128.5 −0.948317
\(842\) −11609.3 −0.475159
\(843\) 0 0
\(844\) −13596.7 −0.554525
\(845\) −8688.82 −0.353733
\(846\) 0 0
\(847\) −6539.11 −0.265273
\(848\) −5282.92 −0.213934
\(849\) 0 0
\(850\) −987.175 −0.0398351
\(851\) −3219.57 −0.129689
\(852\) 0 0
\(853\) 8116.97 0.325814 0.162907 0.986641i \(-0.447913\pi\)
0.162907 + 0.986641i \(0.447913\pi\)
\(854\) 5720.73 0.229226
\(855\) 0 0
\(856\) 11926.9 0.476229
\(857\) −16285.5 −0.649126 −0.324563 0.945864i \(-0.605217\pi\)
−0.324563 + 0.945864i \(0.605217\pi\)
\(858\) 0 0
\(859\) −26487.5 −1.05209 −0.526043 0.850458i \(-0.676325\pi\)
−0.526043 + 0.850458i \(0.676325\pi\)
\(860\) −8739.24 −0.346518
\(861\) 0 0
\(862\) 30536.3 1.20658
\(863\) −40396.2 −1.59340 −0.796699 0.604376i \(-0.793422\pi\)
−0.796699 + 0.604376i \(0.793422\pi\)
\(864\) 0 0
\(865\) −18508.8 −0.727537
\(866\) −4501.25 −0.176627
\(867\) 0 0
\(868\) 5675.46 0.221933
\(869\) −7398.94 −0.288828
\(870\) 0 0
\(871\) 5340.76 0.207767
\(872\) −9632.00 −0.374060
\(873\) 0 0
\(874\) −7293.07 −0.282256
\(875\) −628.839 −0.0242956
\(876\) 0 0
\(877\) 10962.9 0.422109 0.211054 0.977474i \(-0.432310\pi\)
0.211054 + 0.977474i \(0.432310\pi\)
\(878\) −20732.4 −0.796908
\(879\) 0 0
\(880\) −446.575 −0.0171069
\(881\) −23906.5 −0.914222 −0.457111 0.889410i \(-0.651116\pi\)
−0.457111 + 0.889410i \(0.651116\pi\)
\(882\) 0 0
\(883\) −30509.1 −1.16275 −0.581377 0.813634i \(-0.697486\pi\)
−0.581377 + 0.813634i \(0.697486\pi\)
\(884\) 4953.86 0.188480
\(885\) 0 0
\(886\) −7762.22 −0.294331
\(887\) 40034.3 1.51547 0.757733 0.652565i \(-0.226307\pi\)
0.757733 + 0.652565i \(0.226307\pi\)
\(888\) 0 0
\(889\) 7811.62 0.294706
\(890\) 12338.9 0.464720
\(891\) 0 0
\(892\) 3458.64 0.129825
\(893\) −14311.9 −0.536314
\(894\) 0 0
\(895\) 17920.5 0.669291
\(896\) −643.931 −0.0240092
\(897\) 0 0
\(898\) −20293.0 −0.754106
\(899\) 10013.4 0.371485
\(900\) 0 0
\(901\) −6518.96 −0.241041
\(902\) 2541.91 0.0938318
\(903\) 0 0
\(904\) 2214.17 0.0814626
\(905\) −3918.36 −0.143923
\(906\) 0 0
\(907\) −2843.48 −0.104097 −0.0520486 0.998645i \(-0.516575\pi\)
−0.0520486 + 0.998645i \(0.516575\pi\)
\(908\) 7917.58 0.289377
\(909\) 0 0
\(910\) 3155.65 0.114955
\(911\) −42268.6 −1.53723 −0.768617 0.639710i \(-0.779055\pi\)
−0.768617 + 0.639710i \(0.779055\pi\)
\(912\) 0 0
\(913\) 886.483 0.0321340
\(914\) 9457.74 0.342269
\(915\) 0 0
\(916\) −12454.1 −0.449231
\(917\) 1579.00 0.0568629
\(918\) 0 0
\(919\) −12699.6 −0.455844 −0.227922 0.973679i \(-0.573193\pi\)
−0.227922 + 0.973679i \(0.573193\pi\)
\(920\) 920.000 0.0329690
\(921\) 0 0
\(922\) 16761.4 0.598706
\(923\) 23182.8 0.826728
\(924\) 0 0
\(925\) −3499.53 −0.124393
\(926\) −13690.3 −0.485845
\(927\) 0 0
\(928\) −1136.11 −0.0401880
\(929\) 24889.6 0.879011 0.439505 0.898240i \(-0.355154\pi\)
0.439505 + 0.898240i \(0.355154\pi\)
\(930\) 0 0
\(931\) −50368.5 −1.77310
\(932\) −24074.6 −0.846126
\(933\) 0 0
\(934\) −29768.0 −1.04287
\(935\) −551.060 −0.0192744
\(936\) 0 0
\(937\) 9585.48 0.334199 0.167099 0.985940i \(-0.446560\pi\)
0.167099 + 0.985940i \(0.446560\pi\)
\(938\) −856.649 −0.0298194
\(939\) 0 0
\(940\) 1805.40 0.0626444
\(941\) 21146.9 0.732592 0.366296 0.930498i \(-0.380626\pi\)
0.366296 + 0.930498i \(0.380626\pi\)
\(942\) 0 0
\(943\) −5236.65 −0.180836
\(944\) 12739.2 0.439224
\(945\) 0 0
\(946\) −4878.41 −0.167665
\(947\) −48747.2 −1.67272 −0.836362 0.548177i \(-0.815322\pi\)
−0.836362 + 0.548177i \(0.815322\pi\)
\(948\) 0 0
\(949\) −19457.3 −0.665556
\(950\) −7927.25 −0.270730
\(951\) 0 0
\(952\) −794.591 −0.0270513
\(953\) −24925.9 −0.847249 −0.423625 0.905838i \(-0.639242\pi\)
−0.423625 + 0.905838i \(0.639242\pi\)
\(954\) 0 0
\(955\) 7459.73 0.252766
\(956\) 16899.5 0.571725
\(957\) 0 0
\(958\) 12097.4 0.407984
\(959\) 5363.20 0.180591
\(960\) 0 0
\(961\) 49755.9 1.67017
\(962\) 17561.4 0.588568
\(963\) 0 0
\(964\) 5576.95 0.186329
\(965\) −15455.5 −0.515574
\(966\) 0 0
\(967\) −1881.92 −0.0625838 −0.0312919 0.999510i \(-0.509962\pi\)
−0.0312919 + 0.999510i \(0.509962\pi\)
\(968\) 10398.7 0.345276
\(969\) 0 0
\(970\) −1063.89 −0.0352160
\(971\) 14637.3 0.483763 0.241881 0.970306i \(-0.422236\pi\)
0.241881 + 0.970306i \(0.422236\pi\)
\(972\) 0 0
\(973\) −8022.92 −0.264340
\(974\) −6398.70 −0.210501
\(975\) 0 0
\(976\) −9097.28 −0.298358
\(977\) −42591.4 −1.39470 −0.697348 0.716733i \(-0.745637\pi\)
−0.697348 + 0.716733i \(0.745637\pi\)
\(978\) 0 0
\(979\) 6887.81 0.224857
\(980\) 6353.84 0.207108
\(981\) 0 0
\(982\) −3228.81 −0.104924
\(983\) 48858.3 1.58529 0.792644 0.609684i \(-0.208704\pi\)
0.792644 + 0.609684i \(0.208704\pi\)
\(984\) 0 0
\(985\) 14101.5 0.456153
\(986\) −1401.92 −0.0452801
\(987\) 0 0
\(988\) 39780.6 1.28096
\(989\) 10050.1 0.323130
\(990\) 0 0
\(991\) −56522.1 −1.81179 −0.905895 0.423502i \(-0.860801\pi\)
−0.905895 + 0.423502i \(0.860801\pi\)
\(992\) −9025.30 −0.288864
\(993\) 0 0
\(994\) −3718.48 −0.118655
\(995\) −6580.53 −0.209665
\(996\) 0 0
\(997\) −20767.9 −0.659706 −0.329853 0.944032i \(-0.606999\pi\)
−0.329853 + 0.944032i \(0.606999\pi\)
\(998\) 10784.1 0.342048
\(999\) 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 2070.4.a.bg.1.3 4
3.2 odd 2 230.4.a.j.1.1 4
12.11 even 2 1840.4.a.k.1.4 4
15.2 even 4 1150.4.b.o.599.8 8
15.8 even 4 1150.4.b.o.599.1 8
15.14 odd 2 1150.4.a.n.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.j.1.1 4 3.2 odd 2
1150.4.a.n.1.4 4 15.14 odd 2
1150.4.b.o.599.1 8 15.8 even 4
1150.4.b.o.599.8 8 15.2 even 4
1840.4.a.k.1.4 4 12.11 even 2
2070.4.a.bg.1.3 4 1.1 even 1 trivial