Properties

Label 1840.4.a.p.1.4
Level $1840$
Weight $4$
Character 1840.1
Self dual yes
Analytic conductor $108.564$
Analytic rank $1$
Dimension $5$
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] = [1840,4,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1840.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(108.563514411\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 34x^{3} - 9x^{2} + 260x + 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(4.98640\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.39409 q^{3} -5.00000 q^{5} +3.57544 q^{7} +27.6726 q^{9} +O(q^{10})\) \(q+7.39409 q^{3} -5.00000 q^{5} +3.57544 q^{7} +27.6726 q^{9} +43.8100 q^{11} -65.2647 q^{13} -36.9705 q^{15} +25.0257 q^{17} -93.1311 q^{19} +26.4371 q^{21} -23.0000 q^{23} +25.0000 q^{25} +4.97335 q^{27} -84.6447 q^{29} -275.951 q^{31} +323.935 q^{33} -17.8772 q^{35} -270.136 q^{37} -482.574 q^{39} +178.492 q^{41} +46.0544 q^{43} -138.363 q^{45} +390.733 q^{47} -330.216 q^{49} +185.043 q^{51} -460.760 q^{53} -219.050 q^{55} -688.620 q^{57} +171.714 q^{59} -432.337 q^{61} +98.9417 q^{63} +326.324 q^{65} -234.236 q^{67} -170.064 q^{69} +465.620 q^{71} +363.534 q^{73} +184.852 q^{75} +156.640 q^{77} -430.514 q^{79} -710.387 q^{81} -318.020 q^{83} -125.129 q^{85} -625.871 q^{87} +1453.20 q^{89} -233.350 q^{91} -2040.41 q^{93} +465.655 q^{95} -919.515 q^{97} +1212.34 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{3} - 25 q^{5} + 15 q^{7} + 79 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 6 q^{3} - 25 q^{5} + 15 q^{7} + 79 q^{9} + 153 q^{11} + 28 q^{13} - 30 q^{15} - 341 q^{17} - 3 q^{19} - 212 q^{21} - 115 q^{23} + 125 q^{25} + 243 q^{27} - 583 q^{29} - 662 q^{31} - 457 q^{33} - 75 q^{35} - 172 q^{37} - 83 q^{39} + 344 q^{41} + 230 q^{43} - 395 q^{45} + 337 q^{47} - 4 q^{49} + 205 q^{51} - 942 q^{53} - 765 q^{55} - 890 q^{57} + 1166 q^{59} + 499 q^{61} + 1228 q^{63} - 140 q^{65} + 972 q^{67} - 138 q^{69} + 14 q^{71} - 229 q^{73} + 150 q^{75} + 312 q^{77} + 88 q^{79} + 897 q^{81} + 72 q^{83} + 1705 q^{85} - 1157 q^{87} - 90 q^{89} + 1309 q^{91} - 3071 q^{93} + 15 q^{95} - 1765 q^{97} + 91 q^{99}+O(q^{100}) \) 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
\(3\) 7.39409 1.42299 0.711497 0.702689i \(-0.248018\pi\)
0.711497 + 0.702689i \(0.248018\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 3.57544 0.193056 0.0965278 0.995330i \(-0.469226\pi\)
0.0965278 + 0.995330i \(0.469226\pi\)
\(8\) 0 0
\(9\) 27.6726 1.02491
\(10\) 0 0
\(11\) 43.8100 1.20084 0.600419 0.799686i \(-0.295000\pi\)
0.600419 + 0.799686i \(0.295000\pi\)
\(12\) 0 0
\(13\) −65.2647 −1.39240 −0.696199 0.717848i \(-0.745127\pi\)
−0.696199 + 0.717848i \(0.745127\pi\)
\(14\) 0 0
\(15\) −36.9705 −0.636382
\(16\) 0 0
\(17\) 25.0257 0.357037 0.178519 0.983937i \(-0.442870\pi\)
0.178519 + 0.983937i \(0.442870\pi\)
\(18\) 0 0
\(19\) −93.1311 −1.12451 −0.562256 0.826963i \(-0.690066\pi\)
−0.562256 + 0.826963i \(0.690066\pi\)
\(20\) 0 0
\(21\) 26.4371 0.274717
\(22\) 0 0
\(23\) −23.0000 −0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 4.97335 0.0354490
\(28\) 0 0
\(29\) −84.6447 −0.542004 −0.271002 0.962579i \(-0.587355\pi\)
−0.271002 + 0.962579i \(0.587355\pi\)
\(30\) 0 0
\(31\) −275.951 −1.59879 −0.799393 0.600809i \(-0.794845\pi\)
−0.799393 + 0.600809i \(0.794845\pi\)
\(32\) 0 0
\(33\) 323.935 1.70879
\(34\) 0 0
\(35\) −17.8772 −0.0863371
\(36\) 0 0
\(37\) −270.136 −1.20027 −0.600137 0.799897i \(-0.704887\pi\)
−0.600137 + 0.799897i \(0.704887\pi\)
\(38\) 0 0
\(39\) −482.574 −1.98137
\(40\) 0 0
\(41\) 178.492 0.679898 0.339949 0.940444i \(-0.389590\pi\)
0.339949 + 0.940444i \(0.389590\pi\)
\(42\) 0 0
\(43\) 46.0544 0.163331 0.0816654 0.996660i \(-0.473976\pi\)
0.0816654 + 0.996660i \(0.473976\pi\)
\(44\) 0 0
\(45\) −138.363 −0.458354
\(46\) 0 0
\(47\) 390.733 1.21265 0.606323 0.795219i \(-0.292644\pi\)
0.606323 + 0.795219i \(0.292644\pi\)
\(48\) 0 0
\(49\) −330.216 −0.962730
\(50\) 0 0
\(51\) 185.043 0.508062
\(52\) 0 0
\(53\) −460.760 −1.19415 −0.597077 0.802184i \(-0.703671\pi\)
−0.597077 + 0.802184i \(0.703671\pi\)
\(54\) 0 0
\(55\) −219.050 −0.537031
\(56\) 0 0
\(57\) −688.620 −1.60017
\(58\) 0 0
\(59\) 171.714 0.378902 0.189451 0.981890i \(-0.439329\pi\)
0.189451 + 0.981890i \(0.439329\pi\)
\(60\) 0 0
\(61\) −432.337 −0.907460 −0.453730 0.891139i \(-0.649907\pi\)
−0.453730 + 0.891139i \(0.649907\pi\)
\(62\) 0 0
\(63\) 98.9417 0.197865
\(64\) 0 0
\(65\) 326.324 0.622700
\(66\) 0 0
\(67\) −234.236 −0.427112 −0.213556 0.976931i \(-0.568505\pi\)
−0.213556 + 0.976931i \(0.568505\pi\)
\(68\) 0 0
\(69\) −170.064 −0.296715
\(70\) 0 0
\(71\) 465.620 0.778295 0.389147 0.921176i \(-0.372770\pi\)
0.389147 + 0.921176i \(0.372770\pi\)
\(72\) 0 0
\(73\) 363.534 0.582856 0.291428 0.956593i \(-0.405870\pi\)
0.291428 + 0.956593i \(0.405870\pi\)
\(74\) 0 0
\(75\) 184.852 0.284599
\(76\) 0 0
\(77\) 156.640 0.231828
\(78\) 0 0
\(79\) −430.514 −0.613122 −0.306561 0.951851i \(-0.599178\pi\)
−0.306561 + 0.951851i \(0.599178\pi\)
\(80\) 0 0
\(81\) −710.387 −0.974468
\(82\) 0 0
\(83\) −318.020 −0.420569 −0.210285 0.977640i \(-0.567439\pi\)
−0.210285 + 0.977640i \(0.567439\pi\)
\(84\) 0 0
\(85\) −125.129 −0.159672
\(86\) 0 0
\(87\) −625.871 −0.771269
\(88\) 0 0
\(89\) 1453.20 1.73077 0.865385 0.501108i \(-0.167074\pi\)
0.865385 + 0.501108i \(0.167074\pi\)
\(90\) 0 0
\(91\) −233.350 −0.268810
\(92\) 0 0
\(93\) −2040.41 −2.27506
\(94\) 0 0
\(95\) 465.655 0.502897
\(96\) 0 0
\(97\) −919.515 −0.962501 −0.481250 0.876583i \(-0.659817\pi\)
−0.481250 + 0.876583i \(0.659817\pi\)
\(98\) 0 0
\(99\) 1212.34 1.23075
\(100\) 0 0
\(101\) 773.116 0.761662 0.380831 0.924645i \(-0.375638\pi\)
0.380831 + 0.924645i \(0.375638\pi\)
\(102\) 0 0
\(103\) 2025.62 1.93776 0.968882 0.247521i \(-0.0796160\pi\)
0.968882 + 0.247521i \(0.0796160\pi\)
\(104\) 0 0
\(105\) −132.186 −0.122857
\(106\) 0 0
\(107\) −456.342 −0.412302 −0.206151 0.978520i \(-0.566094\pi\)
−0.206151 + 0.978520i \(0.566094\pi\)
\(108\) 0 0
\(109\) 287.101 0.252287 0.126144 0.992012i \(-0.459740\pi\)
0.126144 + 0.992012i \(0.459740\pi\)
\(110\) 0 0
\(111\) −1997.41 −1.70798
\(112\) 0 0
\(113\) −1519.75 −1.26519 −0.632595 0.774483i \(-0.718010\pi\)
−0.632595 + 0.774483i \(0.718010\pi\)
\(114\) 0 0
\(115\) 115.000 0.0932505
\(116\) 0 0
\(117\) −1806.05 −1.42709
\(118\) 0 0
\(119\) 89.4780 0.0689280
\(120\) 0 0
\(121\) 588.318 0.442012
\(122\) 0 0
\(123\) 1319.79 0.967490
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −130.296 −0.0910385 −0.0455193 0.998963i \(-0.514494\pi\)
−0.0455193 + 0.998963i \(0.514494\pi\)
\(128\) 0 0
\(129\) 340.530 0.232419
\(130\) 0 0
\(131\) 1380.44 0.920682 0.460341 0.887742i \(-0.347727\pi\)
0.460341 + 0.887742i \(0.347727\pi\)
\(132\) 0 0
\(133\) −332.984 −0.217093
\(134\) 0 0
\(135\) −24.8668 −0.0158533
\(136\) 0 0
\(137\) −975.666 −0.608444 −0.304222 0.952601i \(-0.598396\pi\)
−0.304222 + 0.952601i \(0.598396\pi\)
\(138\) 0 0
\(139\) −2733.61 −1.66807 −0.834034 0.551713i \(-0.813974\pi\)
−0.834034 + 0.551713i \(0.813974\pi\)
\(140\) 0 0
\(141\) 2889.12 1.72559
\(142\) 0 0
\(143\) −2859.25 −1.67205
\(144\) 0 0
\(145\) 423.224 0.242392
\(146\) 0 0
\(147\) −2441.65 −1.36996
\(148\) 0 0
\(149\) −1040.01 −0.571820 −0.285910 0.958256i \(-0.592296\pi\)
−0.285910 + 0.958256i \(0.592296\pi\)
\(150\) 0 0
\(151\) −2822.72 −1.52125 −0.760627 0.649189i \(-0.775108\pi\)
−0.760627 + 0.649189i \(0.775108\pi\)
\(152\) 0 0
\(153\) 692.527 0.365931
\(154\) 0 0
\(155\) 1379.76 0.714999
\(156\) 0 0
\(157\) −3318.49 −1.68691 −0.843454 0.537202i \(-0.819481\pi\)
−0.843454 + 0.537202i \(0.819481\pi\)
\(158\) 0 0
\(159\) −3406.90 −1.69927
\(160\) 0 0
\(161\) −82.2351 −0.0402549
\(162\) 0 0
\(163\) −954.290 −0.458563 −0.229282 0.973360i \(-0.573638\pi\)
−0.229282 + 0.973360i \(0.573638\pi\)
\(164\) 0 0
\(165\) −1619.68 −0.764192
\(166\) 0 0
\(167\) −1137.20 −0.526942 −0.263471 0.964667i \(-0.584867\pi\)
−0.263471 + 0.964667i \(0.584867\pi\)
\(168\) 0 0
\(169\) 2062.49 0.938774
\(170\) 0 0
\(171\) −2577.18 −1.15253
\(172\) 0 0
\(173\) −4348.83 −1.91119 −0.955593 0.294689i \(-0.904784\pi\)
−0.955593 + 0.294689i \(0.904784\pi\)
\(174\) 0 0
\(175\) 89.3860 0.0386111
\(176\) 0 0
\(177\) 1269.67 0.539175
\(178\) 0 0
\(179\) 2627.76 1.09725 0.548625 0.836069i \(-0.315152\pi\)
0.548625 + 0.836069i \(0.315152\pi\)
\(180\) 0 0
\(181\) −1985.85 −0.815507 −0.407754 0.913092i \(-0.633688\pi\)
−0.407754 + 0.913092i \(0.633688\pi\)
\(182\) 0 0
\(183\) −3196.74 −1.29131
\(184\) 0 0
\(185\) 1350.68 0.536779
\(186\) 0 0
\(187\) 1096.38 0.428744
\(188\) 0 0
\(189\) 17.7819 0.00684362
\(190\) 0 0
\(191\) 4129.34 1.56434 0.782169 0.623066i \(-0.214113\pi\)
0.782169 + 0.623066i \(0.214113\pi\)
\(192\) 0 0
\(193\) 2623.33 0.978401 0.489201 0.872171i \(-0.337289\pi\)
0.489201 + 0.872171i \(0.337289\pi\)
\(194\) 0 0
\(195\) 2412.87 0.886098
\(196\) 0 0
\(197\) 3297.25 1.19248 0.596242 0.802805i \(-0.296660\pi\)
0.596242 + 0.802805i \(0.296660\pi\)
\(198\) 0 0
\(199\) −3608.58 −1.28545 −0.642726 0.766096i \(-0.722197\pi\)
−0.642726 + 0.766096i \(0.722197\pi\)
\(200\) 0 0
\(201\) −1731.97 −0.607778
\(202\) 0 0
\(203\) −302.642 −0.104637
\(204\) 0 0
\(205\) −892.461 −0.304059
\(206\) 0 0
\(207\) −636.470 −0.213709
\(208\) 0 0
\(209\) −4080.07 −1.35036
\(210\) 0 0
\(211\) −4486.91 −1.46394 −0.731971 0.681336i \(-0.761399\pi\)
−0.731971 + 0.681336i \(0.761399\pi\)
\(212\) 0 0
\(213\) 3442.84 1.10751
\(214\) 0 0
\(215\) −230.272 −0.0730438
\(216\) 0 0
\(217\) −986.648 −0.308654
\(218\) 0 0
\(219\) 2688.01 0.829400
\(220\) 0 0
\(221\) −1633.30 −0.497138
\(222\) 0 0
\(223\) −5279.87 −1.58550 −0.792749 0.609548i \(-0.791351\pi\)
−0.792749 + 0.609548i \(0.791351\pi\)
\(224\) 0 0
\(225\) 691.815 0.204982
\(226\) 0 0
\(227\) 1784.77 0.521846 0.260923 0.965360i \(-0.415973\pi\)
0.260923 + 0.965360i \(0.415973\pi\)
\(228\) 0 0
\(229\) 3938.09 1.13640 0.568201 0.822890i \(-0.307640\pi\)
0.568201 + 0.822890i \(0.307640\pi\)
\(230\) 0 0
\(231\) 1158.21 0.329890
\(232\) 0 0
\(233\) −991.889 −0.278888 −0.139444 0.990230i \(-0.544531\pi\)
−0.139444 + 0.990230i \(0.544531\pi\)
\(234\) 0 0
\(235\) −1953.67 −0.542311
\(236\) 0 0
\(237\) −3183.26 −0.872469
\(238\) 0 0
\(239\) −4989.81 −1.35048 −0.675239 0.737599i \(-0.735959\pi\)
−0.675239 + 0.737599i \(0.735959\pi\)
\(240\) 0 0
\(241\) 3317.95 0.886838 0.443419 0.896314i \(-0.353765\pi\)
0.443419 + 0.896314i \(0.353765\pi\)
\(242\) 0 0
\(243\) −5386.95 −1.42211
\(244\) 0 0
\(245\) 1651.08 0.430546
\(246\) 0 0
\(247\) 6078.18 1.56577
\(248\) 0 0
\(249\) −2351.47 −0.598467
\(250\) 0 0
\(251\) 3640.41 0.915460 0.457730 0.889091i \(-0.348663\pi\)
0.457730 + 0.889091i \(0.348663\pi\)
\(252\) 0 0
\(253\) −1007.63 −0.250392
\(254\) 0 0
\(255\) −925.213 −0.227212
\(256\) 0 0
\(257\) 7166.41 1.73941 0.869705 0.493571i \(-0.164309\pi\)
0.869705 + 0.493571i \(0.164309\pi\)
\(258\) 0 0
\(259\) −965.856 −0.231719
\(260\) 0 0
\(261\) −2342.34 −0.555507
\(262\) 0 0
\(263\) 8092.80 1.89743 0.948714 0.316137i \(-0.102386\pi\)
0.948714 + 0.316137i \(0.102386\pi\)
\(264\) 0 0
\(265\) 2303.80 0.534042
\(266\) 0 0
\(267\) 10745.1 2.46288
\(268\) 0 0
\(269\) −1356.94 −0.307562 −0.153781 0.988105i \(-0.549145\pi\)
−0.153781 + 0.988105i \(0.549145\pi\)
\(270\) 0 0
\(271\) 6907.18 1.54827 0.774135 0.633021i \(-0.218185\pi\)
0.774135 + 0.633021i \(0.218185\pi\)
\(272\) 0 0
\(273\) −1725.41 −0.382515
\(274\) 0 0
\(275\) 1095.25 0.240168
\(276\) 0 0
\(277\) 7889.76 1.71137 0.855685 0.517497i \(-0.173136\pi\)
0.855685 + 0.517497i \(0.173136\pi\)
\(278\) 0 0
\(279\) −7636.30 −1.63861
\(280\) 0 0
\(281\) −7569.89 −1.60705 −0.803527 0.595269i \(-0.797046\pi\)
−0.803527 + 0.595269i \(0.797046\pi\)
\(282\) 0 0
\(283\) −2541.02 −0.533739 −0.266869 0.963733i \(-0.585989\pi\)
−0.266869 + 0.963733i \(0.585989\pi\)
\(284\) 0 0
\(285\) 3443.10 0.715619
\(286\) 0 0
\(287\) 638.188 0.131258
\(288\) 0 0
\(289\) −4286.71 −0.872525
\(290\) 0 0
\(291\) −6798.98 −1.36963
\(292\) 0 0
\(293\) −1187.01 −0.236676 −0.118338 0.992973i \(-0.537757\pi\)
−0.118338 + 0.992973i \(0.537757\pi\)
\(294\) 0 0
\(295\) −858.569 −0.169450
\(296\) 0 0
\(297\) 217.883 0.0425684
\(298\) 0 0
\(299\) 1501.09 0.290335
\(300\) 0 0
\(301\) 164.665 0.0315319
\(302\) 0 0
\(303\) 5716.49 1.08384
\(304\) 0 0
\(305\) 2161.69 0.405829
\(306\) 0 0
\(307\) 2865.01 0.532621 0.266310 0.963887i \(-0.414195\pi\)
0.266310 + 0.963887i \(0.414195\pi\)
\(308\) 0 0
\(309\) 14977.6 2.75743
\(310\) 0 0
\(311\) 952.727 0.173711 0.0868556 0.996221i \(-0.472318\pi\)
0.0868556 + 0.996221i \(0.472318\pi\)
\(312\) 0 0
\(313\) −708.750 −0.127990 −0.0639951 0.997950i \(-0.520384\pi\)
−0.0639951 + 0.997950i \(0.520384\pi\)
\(314\) 0 0
\(315\) −494.709 −0.0884879
\(316\) 0 0
\(317\) 4406.36 0.780712 0.390356 0.920664i \(-0.372352\pi\)
0.390356 + 0.920664i \(0.372352\pi\)
\(318\) 0 0
\(319\) −3708.29 −0.650860
\(320\) 0 0
\(321\) −3374.24 −0.586703
\(322\) 0 0
\(323\) −2330.67 −0.401493
\(324\) 0 0
\(325\) −1631.62 −0.278480
\(326\) 0 0
\(327\) 2122.85 0.359003
\(328\) 0 0
\(329\) 1397.04 0.234108
\(330\) 0 0
\(331\) 10068.7 1.67198 0.835990 0.548744i \(-0.184894\pi\)
0.835990 + 0.548744i \(0.184894\pi\)
\(332\) 0 0
\(333\) −7475.37 −1.23017
\(334\) 0 0
\(335\) 1171.18 0.191010
\(336\) 0 0
\(337\) −5782.71 −0.934732 −0.467366 0.884064i \(-0.654797\pi\)
−0.467366 + 0.884064i \(0.654797\pi\)
\(338\) 0 0
\(339\) −11237.2 −1.80036
\(340\) 0 0
\(341\) −12089.4 −1.91988
\(342\) 0 0
\(343\) −2407.04 −0.378916
\(344\) 0 0
\(345\) 850.321 0.132695
\(346\) 0 0
\(347\) 2808.27 0.434454 0.217227 0.976121i \(-0.430299\pi\)
0.217227 + 0.976121i \(0.430299\pi\)
\(348\) 0 0
\(349\) −1990.62 −0.305316 −0.152658 0.988279i \(-0.548783\pi\)
−0.152658 + 0.988279i \(0.548783\pi\)
\(350\) 0 0
\(351\) −324.584 −0.0493591
\(352\) 0 0
\(353\) −9719.18 −1.46544 −0.732719 0.680531i \(-0.761749\pi\)
−0.732719 + 0.680531i \(0.761749\pi\)
\(354\) 0 0
\(355\) −2328.10 −0.348064
\(356\) 0 0
\(357\) 661.608 0.0980841
\(358\) 0 0
\(359\) 3571.57 0.525070 0.262535 0.964922i \(-0.415441\pi\)
0.262535 + 0.964922i \(0.415441\pi\)
\(360\) 0 0
\(361\) 1814.39 0.264528
\(362\) 0 0
\(363\) 4350.08 0.628980
\(364\) 0 0
\(365\) −1817.67 −0.260661
\(366\) 0 0
\(367\) −6150.13 −0.874752 −0.437376 0.899279i \(-0.644092\pi\)
−0.437376 + 0.899279i \(0.644092\pi\)
\(368\) 0 0
\(369\) 4939.35 0.696835
\(370\) 0 0
\(371\) −1647.42 −0.230538
\(372\) 0 0
\(373\) −2039.11 −0.283059 −0.141529 0.989934i \(-0.545202\pi\)
−0.141529 + 0.989934i \(0.545202\pi\)
\(374\) 0 0
\(375\) −924.262 −0.127276
\(376\) 0 0
\(377\) 5524.32 0.754686
\(378\) 0 0
\(379\) −4989.43 −0.676227 −0.338113 0.941105i \(-0.609789\pi\)
−0.338113 + 0.941105i \(0.609789\pi\)
\(380\) 0 0
\(381\) −963.420 −0.129547
\(382\) 0 0
\(383\) −10033.8 −1.33865 −0.669323 0.742972i \(-0.733416\pi\)
−0.669323 + 0.742972i \(0.733416\pi\)
\(384\) 0 0
\(385\) −783.200 −0.103677
\(386\) 0 0
\(387\) 1274.44 0.167400
\(388\) 0 0
\(389\) −9316.20 −1.21427 −0.607134 0.794599i \(-0.707681\pi\)
−0.607134 + 0.794599i \(0.707681\pi\)
\(390\) 0 0
\(391\) −575.592 −0.0744474
\(392\) 0 0
\(393\) 10207.1 1.31013
\(394\) 0 0
\(395\) 2152.57 0.274196
\(396\) 0 0
\(397\) 5744.41 0.726205 0.363103 0.931749i \(-0.381717\pi\)
0.363103 + 0.931749i \(0.381717\pi\)
\(398\) 0 0
\(399\) −2462.12 −0.308922
\(400\) 0 0
\(401\) 9605.68 1.19622 0.598111 0.801414i \(-0.295918\pi\)
0.598111 + 0.801414i \(0.295918\pi\)
\(402\) 0 0
\(403\) 18009.9 2.22615
\(404\) 0 0
\(405\) 3551.94 0.435795
\(406\) 0 0
\(407\) −11834.7 −1.44133
\(408\) 0 0
\(409\) −10013.5 −1.21060 −0.605298 0.795999i \(-0.706946\pi\)
−0.605298 + 0.795999i \(0.706946\pi\)
\(410\) 0 0
\(411\) −7214.16 −0.865811
\(412\) 0 0
\(413\) 613.952 0.0731491
\(414\) 0 0
\(415\) 1590.10 0.188084
\(416\) 0 0
\(417\) −20212.5 −2.37365
\(418\) 0 0
\(419\) 674.948 0.0786954 0.0393477 0.999226i \(-0.487472\pi\)
0.0393477 + 0.999226i \(0.487472\pi\)
\(420\) 0 0
\(421\) 900.392 0.104234 0.0521169 0.998641i \(-0.483403\pi\)
0.0521169 + 0.998641i \(0.483403\pi\)
\(422\) 0 0
\(423\) 10812.6 1.24285
\(424\) 0 0
\(425\) 625.643 0.0714074
\(426\) 0 0
\(427\) −1545.79 −0.175190
\(428\) 0 0
\(429\) −21141.6 −2.37931
\(430\) 0 0
\(431\) 10830.3 1.21039 0.605193 0.796079i \(-0.293096\pi\)
0.605193 + 0.796079i \(0.293096\pi\)
\(432\) 0 0
\(433\) 11231.0 1.24649 0.623243 0.782029i \(-0.285815\pi\)
0.623243 + 0.782029i \(0.285815\pi\)
\(434\) 0 0
\(435\) 3129.35 0.344922
\(436\) 0 0
\(437\) 2142.01 0.234477
\(438\) 0 0
\(439\) −13988.9 −1.52085 −0.760425 0.649426i \(-0.775009\pi\)
−0.760425 + 0.649426i \(0.775009\pi\)
\(440\) 0 0
\(441\) −9137.95 −0.986713
\(442\) 0 0
\(443\) −298.267 −0.0319890 −0.0159945 0.999872i \(-0.505091\pi\)
−0.0159945 + 0.999872i \(0.505091\pi\)
\(444\) 0 0
\(445\) −7265.98 −0.774024
\(446\) 0 0
\(447\) −7689.96 −0.813697
\(448\) 0 0
\(449\) −13606.7 −1.43016 −0.715079 0.699044i \(-0.753609\pi\)
−0.715079 + 0.699044i \(0.753609\pi\)
\(450\) 0 0
\(451\) 7819.75 0.816447
\(452\) 0 0
\(453\) −20871.4 −2.16474
\(454\) 0 0
\(455\) 1166.75 0.120216
\(456\) 0 0
\(457\) 4374.49 0.447768 0.223884 0.974616i \(-0.428126\pi\)
0.223884 + 0.974616i \(0.428126\pi\)
\(458\) 0 0
\(459\) 124.462 0.0126566
\(460\) 0 0
\(461\) 1026.64 0.103721 0.0518604 0.998654i \(-0.483485\pi\)
0.0518604 + 0.998654i \(0.483485\pi\)
\(462\) 0 0
\(463\) −10398.6 −1.04377 −0.521884 0.853016i \(-0.674771\pi\)
−0.521884 + 0.853016i \(0.674771\pi\)
\(464\) 0 0
\(465\) 10202.1 1.01744
\(466\) 0 0
\(467\) 5755.27 0.570282 0.285141 0.958486i \(-0.407959\pi\)
0.285141 + 0.958486i \(0.407959\pi\)
\(468\) 0 0
\(469\) −837.498 −0.0824564
\(470\) 0 0
\(471\) −24537.2 −2.40046
\(472\) 0 0
\(473\) 2017.64 0.196134
\(474\) 0 0
\(475\) −2328.28 −0.224902
\(476\) 0 0
\(477\) −12750.4 −1.22390
\(478\) 0 0
\(479\) 7944.26 0.757793 0.378896 0.925439i \(-0.376304\pi\)
0.378896 + 0.925439i \(0.376304\pi\)
\(480\) 0 0
\(481\) 17630.4 1.67126
\(482\) 0 0
\(483\) −608.054 −0.0572824
\(484\) 0 0
\(485\) 4597.57 0.430443
\(486\) 0 0
\(487\) 13904.3 1.29376 0.646881 0.762591i \(-0.276073\pi\)
0.646881 + 0.762591i \(0.276073\pi\)
\(488\) 0 0
\(489\) −7056.11 −0.652532
\(490\) 0 0
\(491\) −3935.58 −0.361731 −0.180866 0.983508i \(-0.557890\pi\)
−0.180866 + 0.983508i \(0.557890\pi\)
\(492\) 0 0
\(493\) −2118.30 −0.193516
\(494\) 0 0
\(495\) −6061.69 −0.550409
\(496\) 0 0
\(497\) 1664.80 0.150254
\(498\) 0 0
\(499\) 7865.34 0.705613 0.352807 0.935696i \(-0.385227\pi\)
0.352807 + 0.935696i \(0.385227\pi\)
\(500\) 0 0
\(501\) −8408.58 −0.749835
\(502\) 0 0
\(503\) −2435.58 −0.215899 −0.107950 0.994156i \(-0.534429\pi\)
−0.107950 + 0.994156i \(0.534429\pi\)
\(504\) 0 0
\(505\) −3865.58 −0.340626
\(506\) 0 0
\(507\) 15250.2 1.33587
\(508\) 0 0
\(509\) −19061.4 −1.65988 −0.829942 0.557850i \(-0.811627\pi\)
−0.829942 + 0.557850i \(0.811627\pi\)
\(510\) 0 0
\(511\) 1299.79 0.112524
\(512\) 0 0
\(513\) −463.173 −0.0398628
\(514\) 0 0
\(515\) −10128.1 −0.866595
\(516\) 0 0
\(517\) 17118.0 1.45619
\(518\) 0 0
\(519\) −32155.6 −2.71961
\(520\) 0 0
\(521\) 8446.00 0.710222 0.355111 0.934824i \(-0.384443\pi\)
0.355111 + 0.934824i \(0.384443\pi\)
\(522\) 0 0
\(523\) 20207.4 1.68950 0.844749 0.535162i \(-0.179749\pi\)
0.844749 + 0.535162i \(0.179749\pi\)
\(524\) 0 0
\(525\) 660.928 0.0549434
\(526\) 0 0
\(527\) −6905.89 −0.570826
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) 4751.77 0.388341
\(532\) 0 0
\(533\) −11649.3 −0.946689
\(534\) 0 0
\(535\) 2281.71 0.184387
\(536\) 0 0
\(537\) 19429.9 1.56138
\(538\) 0 0
\(539\) −14466.8 −1.15608
\(540\) 0 0
\(541\) 14298.5 1.13631 0.568153 0.822923i \(-0.307658\pi\)
0.568153 + 0.822923i \(0.307658\pi\)
\(542\) 0 0
\(543\) −14683.5 −1.16046
\(544\) 0 0
\(545\) −1435.51 −0.112826
\(546\) 0 0
\(547\) 1820.52 0.142303 0.0711515 0.997466i \(-0.477333\pi\)
0.0711515 + 0.997466i \(0.477333\pi\)
\(548\) 0 0
\(549\) −11963.9 −0.930067
\(550\) 0 0
\(551\) 7883.05 0.609491
\(552\) 0 0
\(553\) −1539.28 −0.118367
\(554\) 0 0
\(555\) 9987.06 0.763833
\(556\) 0 0
\(557\) 19947.4 1.51741 0.758707 0.651432i \(-0.225832\pi\)
0.758707 + 0.651432i \(0.225832\pi\)
\(558\) 0 0
\(559\) −3005.73 −0.227422
\(560\) 0 0
\(561\) 8106.72 0.610100
\(562\) 0 0
\(563\) −6202.95 −0.464339 −0.232170 0.972675i \(-0.574582\pi\)
−0.232170 + 0.972675i \(0.574582\pi\)
\(564\) 0 0
\(565\) 7598.77 0.565810
\(566\) 0 0
\(567\) −2539.95 −0.188126
\(568\) 0 0
\(569\) 15907.6 1.17203 0.586013 0.810302i \(-0.300697\pi\)
0.586013 + 0.810302i \(0.300697\pi\)
\(570\) 0 0
\(571\) 4501.17 0.329892 0.164946 0.986303i \(-0.447255\pi\)
0.164946 + 0.986303i \(0.447255\pi\)
\(572\) 0 0
\(573\) 30532.7 2.22604
\(574\) 0 0
\(575\) −575.000 −0.0417029
\(576\) 0 0
\(577\) 8735.09 0.630237 0.315119 0.949052i \(-0.397956\pi\)
0.315119 + 0.949052i \(0.397956\pi\)
\(578\) 0 0
\(579\) 19397.1 1.39226
\(580\) 0 0
\(581\) −1137.06 −0.0811932
\(582\) 0 0
\(583\) −20185.9 −1.43399
\(584\) 0 0
\(585\) 9030.23 0.638212
\(586\) 0 0
\(587\) −17132.8 −1.20468 −0.602340 0.798240i \(-0.705765\pi\)
−0.602340 + 0.798240i \(0.705765\pi\)
\(588\) 0 0
\(589\) 25699.7 1.79785
\(590\) 0 0
\(591\) 24380.2 1.69690
\(592\) 0 0
\(593\) −7756.71 −0.537150 −0.268575 0.963259i \(-0.586553\pi\)
−0.268575 + 0.963259i \(0.586553\pi\)
\(594\) 0 0
\(595\) −447.390 −0.0308255
\(596\) 0 0
\(597\) −26682.1 −1.82919
\(598\) 0 0
\(599\) −11689.1 −0.797338 −0.398669 0.917095i \(-0.630528\pi\)
−0.398669 + 0.917095i \(0.630528\pi\)
\(600\) 0 0
\(601\) −5361.70 −0.363907 −0.181954 0.983307i \(-0.558242\pi\)
−0.181954 + 0.983307i \(0.558242\pi\)
\(602\) 0 0
\(603\) −6481.93 −0.437752
\(604\) 0 0
\(605\) −2941.59 −0.197674
\(606\) 0 0
\(607\) −6585.62 −0.440366 −0.220183 0.975459i \(-0.570665\pi\)
−0.220183 + 0.975459i \(0.570665\pi\)
\(608\) 0 0
\(609\) −2237.76 −0.148898
\(610\) 0 0
\(611\) −25501.1 −1.68849
\(612\) 0 0
\(613\) −10187.9 −0.671265 −0.335633 0.941993i \(-0.608950\pi\)
−0.335633 + 0.941993i \(0.608950\pi\)
\(614\) 0 0
\(615\) −6598.94 −0.432675
\(616\) 0 0
\(617\) −14025.9 −0.915174 −0.457587 0.889165i \(-0.651286\pi\)
−0.457587 + 0.889165i \(0.651286\pi\)
\(618\) 0 0
\(619\) 23123.6 1.50148 0.750741 0.660597i \(-0.229697\pi\)
0.750741 + 0.660597i \(0.229697\pi\)
\(620\) 0 0
\(621\) −114.387 −0.00739162
\(622\) 0 0
\(623\) 5195.82 0.334135
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −30168.4 −1.92155
\(628\) 0 0
\(629\) −6760.35 −0.428542
\(630\) 0 0
\(631\) 21569.0 1.36078 0.680389 0.732852i \(-0.261811\pi\)
0.680389 + 0.732852i \(0.261811\pi\)
\(632\) 0 0
\(633\) −33176.6 −2.08318
\(634\) 0 0
\(635\) 651.480 0.0407137
\(636\) 0 0
\(637\) 21551.5 1.34050
\(638\) 0 0
\(639\) 12884.9 0.797683
\(640\) 0 0
\(641\) −14908.3 −0.918633 −0.459317 0.888273i \(-0.651906\pi\)
−0.459317 + 0.888273i \(0.651906\pi\)
\(642\) 0 0
\(643\) −20215.7 −1.23986 −0.619930 0.784657i \(-0.712839\pi\)
−0.619930 + 0.784657i \(0.712839\pi\)
\(644\) 0 0
\(645\) −1702.65 −0.103941
\(646\) 0 0
\(647\) 2941.45 0.178733 0.0893666 0.995999i \(-0.471516\pi\)
0.0893666 + 0.995999i \(0.471516\pi\)
\(648\) 0 0
\(649\) 7522.78 0.455000
\(650\) 0 0
\(651\) −7295.36 −0.439213
\(652\) 0 0
\(653\) 5575.27 0.334115 0.167058 0.985947i \(-0.446573\pi\)
0.167058 + 0.985947i \(0.446573\pi\)
\(654\) 0 0
\(655\) −6902.19 −0.411742
\(656\) 0 0
\(657\) 10059.9 0.597376
\(658\) 0 0
\(659\) 20945.0 1.23809 0.619046 0.785354i \(-0.287519\pi\)
0.619046 + 0.785354i \(0.287519\pi\)
\(660\) 0 0
\(661\) −18339.9 −1.07919 −0.539593 0.841926i \(-0.681422\pi\)
−0.539593 + 0.841926i \(0.681422\pi\)
\(662\) 0 0
\(663\) −12076.8 −0.707424
\(664\) 0 0
\(665\) 1664.92 0.0970871
\(666\) 0 0
\(667\) 1946.83 0.113016
\(668\) 0 0
\(669\) −39039.8 −2.25615
\(670\) 0 0
\(671\) −18940.7 −1.08971
\(672\) 0 0
\(673\) 8682.73 0.497318 0.248659 0.968591i \(-0.420010\pi\)
0.248659 + 0.968591i \(0.420010\pi\)
\(674\) 0 0
\(675\) 124.334 0.00708979
\(676\) 0 0
\(677\) −24621.6 −1.39776 −0.698880 0.715238i \(-0.746318\pi\)
−0.698880 + 0.715238i \(0.746318\pi\)
\(678\) 0 0
\(679\) −3287.67 −0.185816
\(680\) 0 0
\(681\) 13196.7 0.742584
\(682\) 0 0
\(683\) 12701.5 0.711582 0.355791 0.934565i \(-0.384211\pi\)
0.355791 + 0.934565i \(0.384211\pi\)
\(684\) 0 0
\(685\) 4878.33 0.272104
\(686\) 0 0
\(687\) 29118.6 1.61709
\(688\) 0 0
\(689\) 30071.4 1.66274
\(690\) 0 0
\(691\) 12914.2 0.710968 0.355484 0.934682i \(-0.384316\pi\)
0.355484 + 0.934682i \(0.384316\pi\)
\(692\) 0 0
\(693\) 4334.64 0.237604
\(694\) 0 0
\(695\) 13668.0 0.745983
\(696\) 0 0
\(697\) 4466.90 0.242749
\(698\) 0 0
\(699\) −7334.12 −0.396855
\(700\) 0 0
\(701\) 13188.4 0.710581 0.355290 0.934756i \(-0.384382\pi\)
0.355290 + 0.934756i \(0.384382\pi\)
\(702\) 0 0
\(703\) 25158.1 1.34972
\(704\) 0 0
\(705\) −14445.6 −0.771706
\(706\) 0 0
\(707\) 2764.23 0.147043
\(708\) 0 0
\(709\) −13435.5 −0.711680 −0.355840 0.934547i \(-0.615805\pi\)
−0.355840 + 0.934547i \(0.615805\pi\)
\(710\) 0 0
\(711\) −11913.5 −0.628396
\(712\) 0 0
\(713\) 6346.88 0.333370
\(714\) 0 0
\(715\) 14296.3 0.747761
\(716\) 0 0
\(717\) −36895.1 −1.92172
\(718\) 0 0
\(719\) 17486.8 0.907020 0.453510 0.891251i \(-0.350172\pi\)
0.453510 + 0.891251i \(0.350172\pi\)
\(720\) 0 0
\(721\) 7242.46 0.374096
\(722\) 0 0
\(723\) 24533.2 1.26197
\(724\) 0 0
\(725\) −2116.12 −0.108401
\(726\) 0 0
\(727\) 13036.5 0.665060 0.332530 0.943093i \(-0.392098\pi\)
0.332530 + 0.943093i \(0.392098\pi\)
\(728\) 0 0
\(729\) −20651.1 −1.04919
\(730\) 0 0
\(731\) 1152.54 0.0583152
\(732\) 0 0
\(733\) −35051.0 −1.76622 −0.883108 0.469169i \(-0.844554\pi\)
−0.883108 + 0.469169i \(0.844554\pi\)
\(734\) 0 0
\(735\) 12208.2 0.612664
\(736\) 0 0
\(737\) −10261.9 −0.512893
\(738\) 0 0
\(739\) −25898.2 −1.28915 −0.644575 0.764541i \(-0.722965\pi\)
−0.644575 + 0.764541i \(0.722965\pi\)
\(740\) 0 0
\(741\) 44942.6 2.22808
\(742\) 0 0
\(743\) 25133.5 1.24100 0.620498 0.784208i \(-0.286931\pi\)
0.620498 + 0.784208i \(0.286931\pi\)
\(744\) 0 0
\(745\) 5200.07 0.255726
\(746\) 0 0
\(747\) −8800.45 −0.431046
\(748\) 0 0
\(749\) −1631.62 −0.0795971
\(750\) 0 0
\(751\) −20368.0 −0.989667 −0.494833 0.868988i \(-0.664771\pi\)
−0.494833 + 0.868988i \(0.664771\pi\)
\(752\) 0 0
\(753\) 26917.5 1.30269
\(754\) 0 0
\(755\) 14113.6 0.680326
\(756\) 0 0
\(757\) 25262.7 1.21293 0.606465 0.795110i \(-0.292587\pi\)
0.606465 + 0.795110i \(0.292587\pi\)
\(758\) 0 0
\(759\) −7450.51 −0.356306
\(760\) 0 0
\(761\) 14926.1 0.711001 0.355500 0.934676i \(-0.384310\pi\)
0.355500 + 0.934676i \(0.384310\pi\)
\(762\) 0 0
\(763\) 1026.51 0.0487055
\(764\) 0 0
\(765\) −3462.64 −0.163650
\(766\) 0 0
\(767\) −11206.9 −0.527583
\(768\) 0 0
\(769\) 32390.7 1.51891 0.759453 0.650562i \(-0.225467\pi\)
0.759453 + 0.650562i \(0.225467\pi\)
\(770\) 0 0
\(771\) 52989.1 2.47517
\(772\) 0 0
\(773\) −28163.3 −1.31043 −0.655215 0.755443i \(-0.727422\pi\)
−0.655215 + 0.755443i \(0.727422\pi\)
\(774\) 0 0
\(775\) −6898.79 −0.319757
\(776\) 0 0
\(777\) −7141.63 −0.329735
\(778\) 0 0
\(779\) −16623.2 −0.764553
\(780\) 0 0
\(781\) 20398.8 0.934606
\(782\) 0 0
\(783\) −420.968 −0.0192135
\(784\) 0 0
\(785\) 16592.5 0.754408
\(786\) 0 0
\(787\) −34922.9 −1.58179 −0.790893 0.611954i \(-0.790384\pi\)
−0.790893 + 0.611954i \(0.790384\pi\)
\(788\) 0 0
\(789\) 59838.9 2.70003
\(790\) 0 0
\(791\) −5433.79 −0.244252
\(792\) 0 0
\(793\) 28216.4 1.26355
\(794\) 0 0
\(795\) 17034.5 0.759939
\(796\) 0 0
\(797\) 26096.8 1.15985 0.579923 0.814671i \(-0.303083\pi\)
0.579923 + 0.814671i \(0.303083\pi\)
\(798\) 0 0
\(799\) 9778.39 0.432959
\(800\) 0 0
\(801\) 40213.7 1.77389
\(802\) 0 0
\(803\) 15926.4 0.699915
\(804\) 0 0
\(805\) 411.176 0.0180025
\(806\) 0 0
\(807\) −10033.3 −0.437658
\(808\) 0 0
\(809\) −236.848 −0.0102931 −0.00514656 0.999987i \(-0.501638\pi\)
−0.00514656 + 0.999987i \(0.501638\pi\)
\(810\) 0 0
\(811\) 36386.9 1.57548 0.787742 0.616005i \(-0.211250\pi\)
0.787742 + 0.616005i \(0.211250\pi\)
\(812\) 0 0
\(813\) 51072.3 2.20318
\(814\) 0 0
\(815\) 4771.45 0.205076
\(816\) 0 0
\(817\) −4289.09 −0.183668
\(818\) 0 0
\(819\) −6457.41 −0.275507
\(820\) 0 0
\(821\) −27347.7 −1.16254 −0.581268 0.813712i \(-0.697443\pi\)
−0.581268 + 0.813712i \(0.697443\pi\)
\(822\) 0 0
\(823\) −2368.01 −0.100296 −0.0501480 0.998742i \(-0.515969\pi\)
−0.0501480 + 0.998742i \(0.515969\pi\)
\(824\) 0 0
\(825\) 8098.38 0.341757
\(826\) 0 0
\(827\) −19465.1 −0.818463 −0.409231 0.912431i \(-0.634203\pi\)
−0.409231 + 0.912431i \(0.634203\pi\)
\(828\) 0 0
\(829\) 33035.4 1.38404 0.692020 0.721879i \(-0.256721\pi\)
0.692020 + 0.721879i \(0.256721\pi\)
\(830\) 0 0
\(831\) 58337.6 2.43527
\(832\) 0 0
\(833\) −8263.90 −0.343730
\(834\) 0 0
\(835\) 5686.01 0.235656
\(836\) 0 0
\(837\) −1372.40 −0.0566753
\(838\) 0 0
\(839\) 36015.4 1.48199 0.740995 0.671511i \(-0.234354\pi\)
0.740995 + 0.671511i \(0.234354\pi\)
\(840\) 0 0
\(841\) −17224.3 −0.706231
\(842\) 0 0
\(843\) −55972.5 −2.28683
\(844\) 0 0
\(845\) −10312.4 −0.419833
\(846\) 0 0
\(847\) 2103.50 0.0853329
\(848\) 0 0
\(849\) −18788.6 −0.759507
\(850\) 0 0
\(851\) 6213.13 0.250274
\(852\) 0 0
\(853\) 9941.26 0.399041 0.199521 0.979894i \(-0.436062\pi\)
0.199521 + 0.979894i \(0.436062\pi\)
\(854\) 0 0
\(855\) 12885.9 0.515425
\(856\) 0 0
\(857\) −11884.7 −0.473716 −0.236858 0.971544i \(-0.576118\pi\)
−0.236858 + 0.971544i \(0.576118\pi\)
\(858\) 0 0
\(859\) −24824.3 −0.986022 −0.493011 0.870023i \(-0.664104\pi\)
−0.493011 + 0.870023i \(0.664104\pi\)
\(860\) 0 0
\(861\) 4718.82 0.186779
\(862\) 0 0
\(863\) 18701.1 0.737653 0.368826 0.929498i \(-0.379760\pi\)
0.368826 + 0.929498i \(0.379760\pi\)
\(864\) 0 0
\(865\) 21744.1 0.854709
\(866\) 0 0
\(867\) −31696.4 −1.24160
\(868\) 0 0
\(869\) −18860.8 −0.736260
\(870\) 0 0
\(871\) 15287.4 0.594711
\(872\) 0 0
\(873\) −25445.4 −0.986478
\(874\) 0 0
\(875\) −446.930 −0.0172674
\(876\) 0 0
\(877\) −2743.01 −0.105615 −0.0528077 0.998605i \(-0.516817\pi\)
−0.0528077 + 0.998605i \(0.516817\pi\)
\(878\) 0 0
\(879\) −8776.90 −0.336789
\(880\) 0 0
\(881\) 39165.3 1.49774 0.748872 0.662715i \(-0.230596\pi\)
0.748872 + 0.662715i \(0.230596\pi\)
\(882\) 0 0
\(883\) 27421.4 1.04508 0.522539 0.852615i \(-0.324985\pi\)
0.522539 + 0.852615i \(0.324985\pi\)
\(884\) 0 0
\(885\) −6348.34 −0.241127
\(886\) 0 0
\(887\) 22388.7 0.847509 0.423754 0.905777i \(-0.360712\pi\)
0.423754 + 0.905777i \(0.360712\pi\)
\(888\) 0 0
\(889\) −465.865 −0.0175755
\(890\) 0 0
\(891\) −31122.1 −1.17018
\(892\) 0 0
\(893\) −36389.4 −1.36363
\(894\) 0 0
\(895\) −13138.8 −0.490705
\(896\) 0 0
\(897\) 11099.2 0.413145
\(898\) 0 0
\(899\) 23357.8 0.866549
\(900\) 0 0
\(901\) −11530.8 −0.426358
\(902\) 0 0
\(903\) 1217.55 0.0448697
\(904\) 0 0
\(905\) 9929.23 0.364706
\(906\) 0 0
\(907\) 8406.44 0.307752 0.153876 0.988090i \(-0.450824\pi\)
0.153876 + 0.988090i \(0.450824\pi\)
\(908\) 0 0
\(909\) 21394.1 0.780636
\(910\) 0 0
\(911\) 23147.0 0.841816 0.420908 0.907103i \(-0.361712\pi\)
0.420908 + 0.907103i \(0.361712\pi\)
\(912\) 0 0
\(913\) −13932.5 −0.505036
\(914\) 0 0
\(915\) 15983.7 0.577492
\(916\) 0 0
\(917\) 4935.67 0.177743
\(918\) 0 0
\(919\) 8915.09 0.320002 0.160001 0.987117i \(-0.448850\pi\)
0.160001 + 0.987117i \(0.448850\pi\)
\(920\) 0 0
\(921\) 21184.1 0.757916
\(922\) 0 0
\(923\) −30388.6 −1.08370
\(924\) 0 0
\(925\) −6753.40 −0.240055
\(926\) 0 0
\(927\) 56054.1 1.98604
\(928\) 0 0
\(929\) 29110.6 1.02808 0.514041 0.857766i \(-0.328148\pi\)
0.514041 + 0.857766i \(0.328148\pi\)
\(930\) 0 0
\(931\) 30753.4 1.08260
\(932\) 0 0
\(933\) 7044.55 0.247190
\(934\) 0 0
\(935\) −5481.89 −0.191740
\(936\) 0 0
\(937\) 47055.8 1.64061 0.820303 0.571930i \(-0.193805\pi\)
0.820303 + 0.571930i \(0.193805\pi\)
\(938\) 0 0
\(939\) −5240.56 −0.182129
\(940\) 0 0
\(941\) 36217.3 1.25468 0.627339 0.778747i \(-0.284144\pi\)
0.627339 + 0.778747i \(0.284144\pi\)
\(942\) 0 0
\(943\) −4105.32 −0.141768
\(944\) 0 0
\(945\) −88.9096 −0.00306056
\(946\) 0 0
\(947\) −13445.6 −0.461376 −0.230688 0.973028i \(-0.574098\pi\)
−0.230688 + 0.973028i \(0.574098\pi\)
\(948\) 0 0
\(949\) −23726.0 −0.811568
\(950\) 0 0
\(951\) 32581.0 1.11095
\(952\) 0 0
\(953\) 25975.6 0.882929 0.441465 0.897279i \(-0.354459\pi\)
0.441465 + 0.897279i \(0.354459\pi\)
\(954\) 0 0
\(955\) −20646.7 −0.699594
\(956\) 0 0
\(957\) −27419.4 −0.926169
\(958\) 0 0
\(959\) −3488.43 −0.117463
\(960\) 0 0
\(961\) 46358.2 1.55611
\(962\) 0 0
\(963\) −12628.2 −0.422573
\(964\) 0 0
\(965\) −13116.6 −0.437554
\(966\) 0 0
\(967\) 7598.70 0.252697 0.126348 0.991986i \(-0.459674\pi\)
0.126348 + 0.991986i \(0.459674\pi\)
\(968\) 0 0
\(969\) −17233.2 −0.571321
\(970\) 0 0
\(971\) −11186.0 −0.369696 −0.184848 0.982767i \(-0.559179\pi\)
−0.184848 + 0.982767i \(0.559179\pi\)
\(972\) 0 0
\(973\) −9773.84 −0.322030
\(974\) 0 0
\(975\) −12064.3 −0.396275
\(976\) 0 0
\(977\) −2597.14 −0.0850460 −0.0425230 0.999095i \(-0.513540\pi\)
−0.0425230 + 0.999095i \(0.513540\pi\)
\(978\) 0 0
\(979\) 63664.6 2.07837
\(980\) 0 0
\(981\) 7944.84 0.258572
\(982\) 0 0
\(983\) 33005.1 1.07090 0.535452 0.844566i \(-0.320141\pi\)
0.535452 + 0.844566i \(0.320141\pi\)
\(984\) 0 0
\(985\) −16486.3 −0.533295
\(986\) 0 0
\(987\) 10329.9 0.333134
\(988\) 0 0
\(989\) −1059.25 −0.0340568
\(990\) 0 0
\(991\) −4986.74 −0.159848 −0.0799239 0.996801i \(-0.525468\pi\)
−0.0799239 + 0.996801i \(0.525468\pi\)
\(992\) 0 0
\(993\) 74448.9 2.37922
\(994\) 0 0
\(995\) 18042.9 0.574872
\(996\) 0 0
\(997\) −58754.2 −1.86636 −0.933182 0.359405i \(-0.882980\pi\)
−0.933182 + 0.359405i \(0.882980\pi\)
\(998\) 0 0
\(999\) −1343.48 −0.0425484
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 1840.4.a.p.1.4 5
4.3 odd 2 115.4.a.d.1.5 5
12.11 even 2 1035.4.a.m.1.1 5
20.3 even 4 575.4.b.h.24.3 10
20.7 even 4 575.4.b.h.24.8 10
20.19 odd 2 575.4.a.k.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.d.1.5 5 4.3 odd 2
575.4.a.k.1.1 5 20.19 odd 2
575.4.b.h.24.3 10 20.3 even 4
575.4.b.h.24.8 10 20.7 even 4
1035.4.a.m.1.1 5 12.11 even 2
1840.4.a.p.1.4 5 1.1 even 1 trivial