Properties

Label 378.5.b.a.323.5
Level $378$
Weight $5$
Character 378.323
Analytic conductor $39.074$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 378.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.0738460457\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.5443747577856.29
Defining polynomial: \( x^{8} + 24x^{6} + 180x^{4} + 488x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.5
Root \(2.39656i\) of defining polynomial
Character \(\chi\) \(=\) 378.323
Dual form 378.5.b.a.323.4

$q$-expansion

\(f(q)\) \(=\) \(q+2.82843i q^{2} -8.00000 q^{4} -14.9735i q^{5} -18.5203 q^{7} -22.6274i q^{8} +O(q^{10})\) \(q+2.82843i q^{2} -8.00000 q^{4} -14.9735i q^{5} -18.5203 q^{7} -22.6274i q^{8} +42.3515 q^{10} +207.672i q^{11} -62.8805 q^{13} -52.3832i q^{14} +64.0000 q^{16} -187.297i q^{17} +462.879 q^{19} +119.788i q^{20} -587.385 q^{22} +455.668i q^{23} +400.793 q^{25} -177.853i q^{26} +148.162 q^{28} -1181.31i q^{29} -1516.38 q^{31} +181.019i q^{32} +529.755 q^{34} +277.314i q^{35} -2092.12 q^{37} +1309.22i q^{38} -338.812 q^{40} -2995.00i q^{41} -3082.43 q^{43} -1661.38i q^{44} -1288.82 q^{46} -1652.15i q^{47} +343.000 q^{49} +1133.62i q^{50} +503.044 q^{52} -2863.40i q^{53} +3109.58 q^{55} +419.066i q^{56} +3341.26 q^{58} -3120.03i q^{59} +3220.68 q^{61} -4288.97i q^{62} -512.000 q^{64} +941.542i q^{65} -5231.48 q^{67} +1498.37i q^{68} -784.361 q^{70} +2620.11i q^{71} -6236.52 q^{73} -5917.41i q^{74} -3703.04 q^{76} -3846.14i q^{77} +3408.06 q^{79} -958.306i q^{80} +8471.13 q^{82} +6087.85i q^{83} -2804.49 q^{85} -8718.43i q^{86} +4699.08 q^{88} -13322.1i q^{89} +1164.56 q^{91} -3645.34i q^{92} +4672.99 q^{94} -6930.94i q^{95} -6073.09 q^{97} +970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 64 q^{4} - 224 q^{10} - 376 q^{13} + 512 q^{16} + 1120 q^{19} - 1120 q^{22} + 792 q^{25} - 880 q^{31} - 1792 q^{34} + 1576 q^{37} + 1792 q^{40} - 5768 q^{43} - 160 q^{46} + 2744 q^{49} + 3008 q^{52} + 488 q^{55} + 7552 q^{58} - 2560 q^{61} - 4096 q^{64} + 23784 q^{67} + 1568 q^{70} - 13176 q^{73} - 8960 q^{76} - 9592 q^{79} - 3360 q^{82} - 39880 q^{85} + 8960 q^{88} + 5096 q^{91} + 4608 q^{94} - 14016 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.82843i 0.707107i
\(3\) 0 0
\(4\) −8.00000 −0.500000
\(5\) − 14.9735i − 0.598941i −0.954106 0.299471i \(-0.903190\pi\)
0.954106 0.299471i \(-0.0968100\pi\)
\(6\) 0 0
\(7\) −18.5203 −0.377964
\(8\) − 22.6274i − 0.353553i
\(9\) 0 0
\(10\) 42.3515 0.423515
\(11\) 207.672i 1.71630i 0.513401 + 0.858149i \(0.328385\pi\)
−0.513401 + 0.858149i \(0.671615\pi\)
\(12\) 0 0
\(13\) −62.8805 −0.372074 −0.186037 0.982543i \(-0.559564\pi\)
−0.186037 + 0.982543i \(0.559564\pi\)
\(14\) − 52.3832i − 0.267261i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) − 187.297i − 0.648085i −0.946042 0.324043i \(-0.894958\pi\)
0.946042 0.324043i \(-0.105042\pi\)
\(18\) 0 0
\(19\) 462.879 1.28221 0.641107 0.767451i \(-0.278475\pi\)
0.641107 + 0.767451i \(0.278475\pi\)
\(20\) 119.788i 0.299471i
\(21\) 0 0
\(22\) −587.385 −1.21361
\(23\) 455.668i 0.861376i 0.902501 + 0.430688i \(0.141729\pi\)
−0.902501 + 0.430688i \(0.858271\pi\)
\(24\) 0 0
\(25\) 400.793 0.641270
\(26\) − 177.853i − 0.263096i
\(27\) 0 0
\(28\) 148.162 0.188982
\(29\) − 1181.31i − 1.40465i −0.711855 0.702326i \(-0.752145\pi\)
0.711855 0.702326i \(-0.247855\pi\)
\(30\) 0 0
\(31\) −1516.38 −1.57792 −0.788960 0.614445i \(-0.789380\pi\)
−0.788960 + 0.614445i \(0.789380\pi\)
\(32\) 181.019i 0.176777i
\(33\) 0 0
\(34\) 529.755 0.458265
\(35\) 277.314i 0.226378i
\(36\) 0 0
\(37\) −2092.12 −1.52821 −0.764105 0.645092i \(-0.776819\pi\)
−0.764105 + 0.645092i \(0.776819\pi\)
\(38\) 1309.22i 0.906663i
\(39\) 0 0
\(40\) −338.812 −0.211758
\(41\) − 2995.00i − 1.78168i −0.454321 0.890838i \(-0.650118\pi\)
0.454321 0.890838i \(-0.349882\pi\)
\(42\) 0 0
\(43\) −3082.43 −1.66708 −0.833540 0.552459i \(-0.813690\pi\)
−0.833540 + 0.552459i \(0.813690\pi\)
\(44\) − 1661.38i − 0.858149i
\(45\) 0 0
\(46\) −1288.82 −0.609085
\(47\) − 1652.15i − 0.747918i −0.927445 0.373959i \(-0.878000\pi\)
0.927445 0.373959i \(-0.122000\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) 1133.62i 0.453446i
\(51\) 0 0
\(52\) 503.044 0.186037
\(53\) − 2863.40i − 1.01936i −0.860363 0.509682i \(-0.829763\pi\)
0.860363 0.509682i \(-0.170237\pi\)
\(54\) 0 0
\(55\) 3109.58 1.02796
\(56\) 419.066i 0.133631i
\(57\) 0 0
\(58\) 3341.26 0.993239
\(59\) − 3120.03i − 0.896302i −0.893958 0.448151i \(-0.852083\pi\)
0.893958 0.448151i \(-0.147917\pi\)
\(60\) 0 0
\(61\) 3220.68 0.865540 0.432770 0.901504i \(-0.357536\pi\)
0.432770 + 0.901504i \(0.357536\pi\)
\(62\) − 4288.97i − 1.11576i
\(63\) 0 0
\(64\) −512.000 −0.125000
\(65\) 941.542i 0.222850i
\(66\) 0 0
\(67\) −5231.48 −1.16540 −0.582700 0.812688i \(-0.698004\pi\)
−0.582700 + 0.812688i \(0.698004\pi\)
\(68\) 1498.37i 0.324043i
\(69\) 0 0
\(70\) −784.361 −0.160074
\(71\) 2620.11i 0.519759i 0.965641 + 0.259880i \(0.0836828\pi\)
−0.965641 + 0.259880i \(0.916317\pi\)
\(72\) 0 0
\(73\) −6236.52 −1.17030 −0.585149 0.810926i \(-0.698964\pi\)
−0.585149 + 0.810926i \(0.698964\pi\)
\(74\) − 5917.41i − 1.08061i
\(75\) 0 0
\(76\) −3703.04 −0.641107
\(77\) − 3846.14i − 0.648700i
\(78\) 0 0
\(79\) 3408.06 0.546076 0.273038 0.962003i \(-0.411971\pi\)
0.273038 + 0.962003i \(0.411971\pi\)
\(80\) − 958.306i − 0.149735i
\(81\) 0 0
\(82\) 8471.13 1.25984
\(83\) 6087.85i 0.883706i 0.897087 + 0.441853i \(0.145679\pi\)
−0.897087 + 0.441853i \(0.854321\pi\)
\(84\) 0 0
\(85\) −2804.49 −0.388165
\(86\) − 8718.43i − 1.17880i
\(87\) 0 0
\(88\) 4699.08 0.606803
\(89\) − 13322.1i − 1.68187i −0.541134 0.840936i \(-0.682005\pi\)
0.541134 0.840936i \(-0.317995\pi\)
\(90\) 0 0
\(91\) 1164.56 0.140631
\(92\) − 3645.34i − 0.430688i
\(93\) 0 0
\(94\) 4672.99 0.528858
\(95\) − 6930.94i − 0.767971i
\(96\) 0 0
\(97\) −6073.09 −0.645455 −0.322728 0.946492i \(-0.604600\pi\)
−0.322728 + 0.946492i \(0.604600\pi\)
\(98\) 970.151i 0.101015i
\(99\) 0 0
\(100\) −3206.35 −0.320635
\(101\) − 5450.61i − 0.534321i −0.963652 0.267160i \(-0.913915\pi\)
0.963652 0.267160i \(-0.0860854\pi\)
\(102\) 0 0
\(103\) 4836.88 0.455922 0.227961 0.973670i \(-0.426794\pi\)
0.227961 + 0.973670i \(0.426794\pi\)
\(104\) 1422.82i 0.131548i
\(105\) 0 0
\(106\) 8098.91 0.720800
\(107\) − 7429.22i − 0.648897i −0.945903 0.324448i \(-0.894821\pi\)
0.945903 0.324448i \(-0.105179\pi\)
\(108\) 0 0
\(109\) 15978.9 1.34492 0.672458 0.740136i \(-0.265239\pi\)
0.672458 + 0.740136i \(0.265239\pi\)
\(110\) 8795.23i 0.726878i
\(111\) 0 0
\(112\) −1185.30 −0.0944911
\(113\) 8283.62i 0.648729i 0.945932 + 0.324364i \(0.105150\pi\)
−0.945932 + 0.324364i \(0.894850\pi\)
\(114\) 0 0
\(115\) 6822.96 0.515914
\(116\) 9450.50i 0.702326i
\(117\) 0 0
\(118\) 8824.77 0.633781
\(119\) 3468.78i 0.244953i
\(120\) 0 0
\(121\) −28486.7 −1.94568
\(122\) 9109.45i 0.612029i
\(123\) 0 0
\(124\) 12131.0 0.788960
\(125\) − 15359.7i − 0.983024i
\(126\) 0 0
\(127\) 25189.7 1.56176 0.780882 0.624679i \(-0.214770\pi\)
0.780882 + 0.624679i \(0.214770\pi\)
\(128\) − 1448.15i − 0.0883883i
\(129\) 0 0
\(130\) −2663.08 −0.157579
\(131\) − 10668.0i − 0.621644i −0.950468 0.310822i \(-0.899396\pi\)
0.950468 0.310822i \(-0.100604\pi\)
\(132\) 0 0
\(133\) −8572.65 −0.484632
\(134\) − 14796.9i − 0.824062i
\(135\) 0 0
\(136\) −4238.04 −0.229133
\(137\) 9682.43i 0.515873i 0.966162 + 0.257937i \(0.0830426\pi\)
−0.966162 + 0.257937i \(0.916957\pi\)
\(138\) 0 0
\(139\) −17916.1 −0.927285 −0.463643 0.886022i \(-0.653458\pi\)
−0.463643 + 0.886022i \(0.653458\pi\)
\(140\) − 2218.51i − 0.113189i
\(141\) 0 0
\(142\) −7410.78 −0.367525
\(143\) − 13058.5i − 0.638589i
\(144\) 0 0
\(145\) −17688.4 −0.841304
\(146\) − 17639.5i − 0.827526i
\(147\) 0 0
\(148\) 16737.0 0.764105
\(149\) 17588.5i 0.792240i 0.918199 + 0.396120i \(0.129644\pi\)
−0.918199 + 0.396120i \(0.870356\pi\)
\(150\) 0 0
\(151\) −35597.0 −1.56120 −0.780602 0.625028i \(-0.785088\pi\)
−0.780602 + 0.625028i \(0.785088\pi\)
\(152\) − 10473.8i − 0.453331i
\(153\) 0 0
\(154\) 10878.5 0.458700
\(155\) 22705.6i 0.945081i
\(156\) 0 0
\(157\) 32966.9 1.33745 0.668726 0.743509i \(-0.266840\pi\)
0.668726 + 0.743509i \(0.266840\pi\)
\(158\) 9639.46i 0.386134i
\(159\) 0 0
\(160\) 2710.50 0.105879
\(161\) − 8439.09i − 0.325570i
\(162\) 0 0
\(163\) −20460.9 −0.770102 −0.385051 0.922895i \(-0.625816\pi\)
−0.385051 + 0.922895i \(0.625816\pi\)
\(164\) 23960.0i 0.890838i
\(165\) 0 0
\(166\) −17219.1 −0.624875
\(167\) 19104.1i 0.685006i 0.939517 + 0.342503i \(0.111275\pi\)
−0.939517 + 0.342503i \(0.888725\pi\)
\(168\) 0 0
\(169\) −24607.0 −0.861561
\(170\) − 7932.30i − 0.274474i
\(171\) 0 0
\(172\) 24659.5 0.833540
\(173\) − 27932.8i − 0.933302i −0.884442 0.466651i \(-0.845460\pi\)
0.884442 0.466651i \(-0.154540\pi\)
\(174\) 0 0
\(175\) −7422.80 −0.242377
\(176\) 13291.0i 0.429074i
\(177\) 0 0
\(178\) 37680.6 1.18926
\(179\) − 26824.0i − 0.837179i −0.908176 0.418589i \(-0.862525\pi\)
0.908176 0.418589i \(-0.137475\pi\)
\(180\) 0 0
\(181\) −8214.15 −0.250729 −0.125365 0.992111i \(-0.540010\pi\)
−0.125365 + 0.992111i \(0.540010\pi\)
\(182\) 3293.88i 0.0994409i
\(183\) 0 0
\(184\) 10310.6 0.304543
\(185\) 31326.4i 0.915308i
\(186\) 0 0
\(187\) 38896.3 1.11231
\(188\) 13217.2i 0.373959i
\(189\) 0 0
\(190\) 19603.7 0.543038
\(191\) − 19528.1i − 0.535296i −0.963517 0.267648i \(-0.913754\pi\)
0.963517 0.267648i \(-0.0862464\pi\)
\(192\) 0 0
\(193\) 1684.59 0.0452252 0.0226126 0.999744i \(-0.492802\pi\)
0.0226126 + 0.999744i \(0.492802\pi\)
\(194\) − 17177.3i − 0.456406i
\(195\) 0 0
\(196\) −2744.00 −0.0714286
\(197\) 25114.6i 0.647134i 0.946205 + 0.323567i \(0.104882\pi\)
−0.946205 + 0.323567i \(0.895118\pi\)
\(198\) 0 0
\(199\) −48183.6 −1.21673 −0.608363 0.793659i \(-0.708173\pi\)
−0.608363 + 0.793659i \(0.708173\pi\)
\(200\) − 9068.92i − 0.226723i
\(201\) 0 0
\(202\) 15416.6 0.377822
\(203\) 21878.2i 0.530909i
\(204\) 0 0
\(205\) −44845.7 −1.06712
\(206\) 13680.8i 0.322386i
\(207\) 0 0
\(208\) −4024.35 −0.0930184
\(209\) 96127.1i 2.20066i
\(210\) 0 0
\(211\) −37374.5 −0.839480 −0.419740 0.907644i \(-0.637879\pi\)
−0.419740 + 0.907644i \(0.637879\pi\)
\(212\) 22907.2i 0.509682i
\(213\) 0 0
\(214\) 21013.0 0.458839
\(215\) 46154.9i 0.998483i
\(216\) 0 0
\(217\) 28083.8 0.596397
\(218\) 45195.3i 0.950999i
\(219\) 0 0
\(220\) −24876.7 −0.513981
\(221\) 11777.3i 0.241135i
\(222\) 0 0
\(223\) −14929.9 −0.300225 −0.150113 0.988669i \(-0.547964\pi\)
−0.150113 + 0.988669i \(0.547964\pi\)
\(224\) − 3352.53i − 0.0668153i
\(225\) 0 0
\(226\) −23429.6 −0.458720
\(227\) 81222.3i 1.57625i 0.615518 + 0.788123i \(0.288947\pi\)
−0.615518 + 0.788123i \(0.711053\pi\)
\(228\) 0 0
\(229\) −26288.7 −0.501300 −0.250650 0.968078i \(-0.580644\pi\)
−0.250650 + 0.968078i \(0.580644\pi\)
\(230\) 19298.2i 0.364806i
\(231\) 0 0
\(232\) −26730.1 −0.496620
\(233\) − 77377.7i − 1.42529i −0.701523 0.712647i \(-0.747496\pi\)
0.701523 0.712647i \(-0.252504\pi\)
\(234\) 0 0
\(235\) −24738.5 −0.447959
\(236\) 24960.2i 0.448151i
\(237\) 0 0
\(238\) −9811.20 −0.173208
\(239\) 214.251i 0.00375083i 0.999998 + 0.00187541i \(0.000596963\pi\)
−0.999998 + 0.00187541i \(0.999403\pi\)
\(240\) 0 0
\(241\) 3165.30 0.0544980 0.0272490 0.999629i \(-0.491325\pi\)
0.0272490 + 0.999629i \(0.491325\pi\)
\(242\) − 80572.4i − 1.37580i
\(243\) 0 0
\(244\) −25765.4 −0.432770
\(245\) − 5135.92i − 0.0855630i
\(246\) 0 0
\(247\) −29106.1 −0.477078
\(248\) 34311.8i 0.557879i
\(249\) 0 0
\(250\) 43443.9 0.695103
\(251\) − 31889.6i − 0.506177i −0.967443 0.253088i \(-0.918554\pi\)
0.967443 0.253088i \(-0.0814464\pi\)
\(252\) 0 0
\(253\) −94629.5 −1.47838
\(254\) 71247.2i 1.10433i
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) 89826.0i 1.35999i 0.733217 + 0.679995i \(0.238018\pi\)
−0.733217 + 0.679995i \(0.761982\pi\)
\(258\) 0 0
\(259\) 38746.6 0.577609
\(260\) − 7532.34i − 0.111425i
\(261\) 0 0
\(262\) 30173.8 0.439569
\(263\) 23175.4i 0.335055i 0.985867 + 0.167528i \(0.0535783\pi\)
−0.985867 + 0.167528i \(0.946422\pi\)
\(264\) 0 0
\(265\) −42875.1 −0.610539
\(266\) − 24247.1i − 0.342686i
\(267\) 0 0
\(268\) 41851.8 0.582700
\(269\) − 56937.5i − 0.786853i −0.919356 0.393427i \(-0.871290\pi\)
0.919356 0.393427i \(-0.128710\pi\)
\(270\) 0 0
\(271\) 123059. 1.67561 0.837807 0.545967i \(-0.183838\pi\)
0.837807 + 0.545967i \(0.183838\pi\)
\(272\) − 11987.0i − 0.162021i
\(273\) 0 0
\(274\) −27386.0 −0.364778
\(275\) 83233.6i 1.10061i
\(276\) 0 0
\(277\) −10073.2 −0.131283 −0.0656415 0.997843i \(-0.520909\pi\)
−0.0656415 + 0.997843i \(0.520909\pi\)
\(278\) − 50674.3i − 0.655690i
\(279\) 0 0
\(280\) 6274.89 0.0800369
\(281\) 123794.i 1.56778i 0.620898 + 0.783892i \(0.286768\pi\)
−0.620898 + 0.783892i \(0.713232\pi\)
\(282\) 0 0
\(283\) 112317. 1.40240 0.701201 0.712964i \(-0.252648\pi\)
0.701201 + 0.712964i \(0.252648\pi\)
\(284\) − 20960.8i − 0.259880i
\(285\) 0 0
\(286\) 36935.0 0.451551
\(287\) 55468.1i 0.673410i
\(288\) 0 0
\(289\) 48441.0 0.579986
\(290\) − 50030.4i − 0.594892i
\(291\) 0 0
\(292\) 49892.2 0.585149
\(293\) − 64511.2i − 0.751449i −0.926731 0.375725i \(-0.877394\pi\)
0.926731 0.375725i \(-0.122606\pi\)
\(294\) 0 0
\(295\) −46717.8 −0.536832
\(296\) 47339.3i 0.540304i
\(297\) 0 0
\(298\) −49747.9 −0.560198
\(299\) − 28652.6i − 0.320496i
\(300\) 0 0
\(301\) 57087.4 0.630097
\(302\) − 100684.i − 1.10394i
\(303\) 0 0
\(304\) 29624.3 0.320554
\(305\) − 48224.9i − 0.518408i
\(306\) 0 0
\(307\) 11483.3 0.121840 0.0609199 0.998143i \(-0.480597\pi\)
0.0609199 + 0.998143i \(0.480597\pi\)
\(308\) 30769.1i 0.324350i
\(309\) 0 0
\(310\) −64221.0 −0.668273
\(311\) 11256.9i 0.116385i 0.998305 + 0.0581926i \(0.0185337\pi\)
−0.998305 + 0.0581926i \(0.981466\pi\)
\(312\) 0 0
\(313\) −112068. −1.14392 −0.571959 0.820282i \(-0.693816\pi\)
−0.571959 + 0.820282i \(0.693816\pi\)
\(314\) 93244.4i 0.945722i
\(315\) 0 0
\(316\) −27264.5 −0.273038
\(317\) − 47792.1i − 0.475595i −0.971315 0.237798i \(-0.923574\pi\)
0.971315 0.237798i \(-0.0764255\pi\)
\(318\) 0 0
\(319\) 245326. 2.41080
\(320\) 7666.45i 0.0748676i
\(321\) 0 0
\(322\) 23869.4 0.230213
\(323\) − 86695.8i − 0.830984i
\(324\) 0 0
\(325\) −25202.1 −0.238600
\(326\) − 57872.0i − 0.544545i
\(327\) 0 0
\(328\) −67769.0 −0.629918
\(329\) 30598.3i 0.282686i
\(330\) 0 0
\(331\) −108343. −0.988885 −0.494443 0.869210i \(-0.664628\pi\)
−0.494443 + 0.869210i \(0.664628\pi\)
\(332\) − 48702.8i − 0.441853i
\(333\) 0 0
\(334\) −54034.7 −0.484373
\(335\) 78333.7i 0.698006i
\(336\) 0 0
\(337\) 163814. 1.44241 0.721207 0.692719i \(-0.243588\pi\)
0.721207 + 0.692719i \(0.243588\pi\)
\(338\) − 69599.2i − 0.609216i
\(339\) 0 0
\(340\) 22435.9 0.194082
\(341\) − 314910.i − 2.70818i
\(342\) 0 0
\(343\) −6352.45 −0.0539949
\(344\) 69747.5i 0.589402i
\(345\) 0 0
\(346\) 79005.8 0.659944
\(347\) − 199534.i − 1.65713i −0.559892 0.828566i \(-0.689157\pi\)
0.559892 0.828566i \(-0.310843\pi\)
\(348\) 0 0
\(349\) 28305.5 0.232391 0.116196 0.993226i \(-0.462930\pi\)
0.116196 + 0.993226i \(0.462930\pi\)
\(350\) − 20994.8i − 0.171386i
\(351\) 0 0
\(352\) −37592.6 −0.303401
\(353\) 134579.i 1.08001i 0.841661 + 0.540007i \(0.181578\pi\)
−0.841661 + 0.540007i \(0.818422\pi\)
\(354\) 0 0
\(355\) 39232.2 0.311305
\(356\) 106577.i 0.840936i
\(357\) 0 0
\(358\) 75869.8 0.591975
\(359\) − 125745.i − 0.975666i −0.872937 0.487833i \(-0.837787\pi\)
0.872937 0.487833i \(-0.162213\pi\)
\(360\) 0 0
\(361\) 83936.4 0.644074
\(362\) − 23233.1i − 0.177292i
\(363\) 0 0
\(364\) −9316.50 −0.0703153
\(365\) 93382.7i 0.700940i
\(366\) 0 0
\(367\) −22221.8 −0.164986 −0.0824931 0.996592i \(-0.526288\pi\)
−0.0824931 + 0.996592i \(0.526288\pi\)
\(368\) 29162.8i 0.215344i
\(369\) 0 0
\(370\) −88604.5 −0.647221
\(371\) 53030.8i 0.385284i
\(372\) 0 0
\(373\) 34876.0 0.250674 0.125337 0.992114i \(-0.459999\pi\)
0.125337 + 0.992114i \(0.459999\pi\)
\(374\) 110015.i 0.786520i
\(375\) 0 0
\(376\) −37383.9 −0.264429
\(377\) 74281.5i 0.522634i
\(378\) 0 0
\(379\) −102131. −0.711016 −0.355508 0.934673i \(-0.615692\pi\)
−0.355508 + 0.934673i \(0.615692\pi\)
\(380\) 55447.5i 0.383986i
\(381\) 0 0
\(382\) 55233.9 0.378511
\(383\) − 12013.5i − 0.0818981i −0.999161 0.0409490i \(-0.986962\pi\)
0.999161 0.0409490i \(-0.0130381\pi\)
\(384\) 0 0
\(385\) −57590.3 −0.388533
\(386\) 4764.75i 0.0319791i
\(387\) 0 0
\(388\) 48584.7 0.322728
\(389\) − 102479.i − 0.677227i −0.940926 0.338614i \(-0.890042\pi\)
0.940926 0.338614i \(-0.109958\pi\)
\(390\) 0 0
\(391\) 85345.1 0.558245
\(392\) − 7761.20i − 0.0505076i
\(393\) 0 0
\(394\) −71034.9 −0.457593
\(395\) − 51030.7i − 0.327068i
\(396\) 0 0
\(397\) 102844. 0.652528 0.326264 0.945279i \(-0.394210\pi\)
0.326264 + 0.945279i \(0.394210\pi\)
\(398\) − 136284.i − 0.860355i
\(399\) 0 0
\(400\) 25650.8 0.160317
\(401\) − 252453.i − 1.56997i −0.619514 0.784986i \(-0.712670\pi\)
0.619514 0.784986i \(-0.287330\pi\)
\(402\) 0 0
\(403\) 95350.7 0.587102
\(404\) 43604.9i 0.267160i
\(405\) 0 0
\(406\) −61881.0 −0.375409
\(407\) − 434475.i − 2.62286i
\(408\) 0 0
\(409\) −42140.0 −0.251911 −0.125956 0.992036i \(-0.540200\pi\)
−0.125956 + 0.992036i \(0.540200\pi\)
\(410\) − 126843.i − 0.754567i
\(411\) 0 0
\(412\) −38695.0 −0.227961
\(413\) 57783.7i 0.338770i
\(414\) 0 0
\(415\) 91156.7 0.529288
\(416\) − 11382.6i − 0.0657740i
\(417\) 0 0
\(418\) −271889. −1.55610
\(419\) 219896.i 1.25254i 0.779608 + 0.626268i \(0.215418\pi\)
−0.779608 + 0.626268i \(0.784582\pi\)
\(420\) 0 0
\(421\) −148584. −0.838316 −0.419158 0.907913i \(-0.637675\pi\)
−0.419158 + 0.907913i \(0.637675\pi\)
\(422\) − 105711.i − 0.593602i
\(423\) 0 0
\(424\) −64791.2 −0.360400
\(425\) − 75067.3i − 0.415597i
\(426\) 0 0
\(427\) −59647.7 −0.327143
\(428\) 59433.7i 0.324448i
\(429\) 0 0
\(430\) −130546. −0.706034
\(431\) − 340490.i − 1.83295i −0.400097 0.916473i \(-0.631024\pi\)
0.400097 0.916473i \(-0.368976\pi\)
\(432\) 0 0
\(433\) −341800. −1.82304 −0.911519 0.411257i \(-0.865090\pi\)
−0.911519 + 0.411257i \(0.865090\pi\)
\(434\) 79432.9i 0.421717i
\(435\) 0 0
\(436\) −127831. −0.672458
\(437\) 210919.i 1.10447i
\(438\) 0 0
\(439\) 34198.2 0.177449 0.0887246 0.996056i \(-0.471721\pi\)
0.0887246 + 0.996056i \(0.471721\pi\)
\(440\) − 70361.8i − 0.363439i
\(441\) 0 0
\(442\) −33311.2 −0.170509
\(443\) 127446.i 0.649409i 0.945815 + 0.324705i \(0.105265\pi\)
−0.945815 + 0.324705i \(0.894735\pi\)
\(444\) 0 0
\(445\) −199479. −1.00734
\(446\) − 42228.1i − 0.212291i
\(447\) 0 0
\(448\) 9482.37 0.0472456
\(449\) 250976.i 1.24491i 0.782655 + 0.622456i \(0.213865\pi\)
−0.782655 + 0.622456i \(0.786135\pi\)
\(450\) 0 0
\(451\) 621977. 3.05789
\(452\) − 66268.9i − 0.324364i
\(453\) 0 0
\(454\) −229731. −1.11457
\(455\) − 17437.6i − 0.0842295i
\(456\) 0 0
\(457\) −311362. −1.49085 −0.745425 0.666590i \(-0.767753\pi\)
−0.745425 + 0.666590i \(0.767753\pi\)
\(458\) − 74355.6i − 0.354473i
\(459\) 0 0
\(460\) −54583.7 −0.257957
\(461\) 299717.i 1.41029i 0.709061 + 0.705147i \(0.249119\pi\)
−0.709061 + 0.705147i \(0.750881\pi\)
\(462\) 0 0
\(463\) 350647. 1.63572 0.817859 0.575418i \(-0.195161\pi\)
0.817859 + 0.575418i \(0.195161\pi\)
\(464\) − 75604.0i − 0.351163i
\(465\) 0 0
\(466\) 218857. 1.00783
\(467\) 187935.i 0.861734i 0.902416 + 0.430867i \(0.141792\pi\)
−0.902416 + 0.430867i \(0.858208\pi\)
\(468\) 0 0
\(469\) 96888.3 0.440479
\(470\) − 69971.1i − 0.316755i
\(471\) 0 0
\(472\) −70598.1 −0.316891
\(473\) − 640135.i − 2.86121i
\(474\) 0 0
\(475\) 185519. 0.822245
\(476\) − 27750.3i − 0.122477i
\(477\) 0 0
\(478\) −605.993 −0.00265223
\(479\) − 440865.i − 1.92148i −0.277459 0.960738i \(-0.589492\pi\)
0.277459 0.960738i \(-0.410508\pi\)
\(480\) 0 0
\(481\) 131553. 0.568607
\(482\) 8952.81i 0.0385359i
\(483\) 0 0
\(484\) 227893. 0.972839
\(485\) 90935.6i 0.386590i
\(486\) 0 0
\(487\) −309791. −1.30620 −0.653101 0.757270i \(-0.726532\pi\)
−0.653101 + 0.757270i \(0.726532\pi\)
\(488\) − 72875.6i − 0.306015i
\(489\) 0 0
\(490\) 14526.6 0.0605022
\(491\) 43693.9i 0.181241i 0.995885 + 0.0906207i \(0.0288851\pi\)
−0.995885 + 0.0906207i \(0.971115\pi\)
\(492\) 0 0
\(493\) −221256. −0.910335
\(494\) − 82324.4i − 0.337345i
\(495\) 0 0
\(496\) −97048.4 −0.394480
\(497\) − 48525.0i − 0.196450i
\(498\) 0 0
\(499\) −18463.9 −0.0741521 −0.0370760 0.999312i \(-0.511804\pi\)
−0.0370760 + 0.999312i \(0.511804\pi\)
\(500\) 122878.i 0.491512i
\(501\) 0 0
\(502\) 90197.5 0.357921
\(503\) 186650.i 0.737722i 0.929484 + 0.368861i \(0.120252\pi\)
−0.929484 + 0.368861i \(0.879748\pi\)
\(504\) 0 0
\(505\) −81614.8 −0.320027
\(506\) − 267653.i − 1.04537i
\(507\) 0 0
\(508\) −201518. −0.780882
\(509\) − 158194.i − 0.610595i −0.952257 0.305298i \(-0.901244\pi\)
0.952257 0.305298i \(-0.0987559\pi\)
\(510\) 0 0
\(511\) 115502. 0.442331
\(512\) 11585.2i 0.0441942i
\(513\) 0 0
\(514\) −254066. −0.961658
\(515\) − 72425.1i − 0.273070i
\(516\) 0 0
\(517\) 343106. 1.28365
\(518\) 109592.i 0.408431i
\(519\) 0 0
\(520\) 21304.7 0.0787895
\(521\) 352048.i 1.29696i 0.761232 + 0.648479i \(0.224595\pi\)
−0.761232 + 0.648479i \(0.775405\pi\)
\(522\) 0 0
\(523\) 182994. 0.669013 0.334506 0.942393i \(-0.391430\pi\)
0.334506 + 0.942393i \(0.391430\pi\)
\(524\) 85344.3i 0.310822i
\(525\) 0 0
\(526\) −65550.0 −0.236920
\(527\) 284013.i 1.02263i
\(528\) 0 0
\(529\) 72207.6 0.258031
\(530\) − 121269.i − 0.431717i
\(531\) 0 0
\(532\) 68581.2 0.242316
\(533\) 188327.i 0.662915i
\(534\) 0 0
\(535\) −111242. −0.388651
\(536\) 118375.i 0.412031i
\(537\) 0 0
\(538\) 161044. 0.556389
\(539\) 71231.5i 0.245185i
\(540\) 0 0
\(541\) 216902. 0.741087 0.370543 0.928815i \(-0.379172\pi\)
0.370543 + 0.928815i \(0.379172\pi\)
\(542\) 348063.i 1.18484i
\(543\) 0 0
\(544\) 33904.3 0.114566
\(545\) − 239261.i − 0.805525i
\(546\) 0 0
\(547\) −307754. −1.02856 −0.514279 0.857623i \(-0.671940\pi\)
−0.514279 + 0.857623i \(0.671940\pi\)
\(548\) − 77459.4i − 0.257937i
\(549\) 0 0
\(550\) −235420. −0.778248
\(551\) − 546805.i − 1.80107i
\(552\) 0 0
\(553\) −63118.2 −0.206397
\(554\) − 28491.4i − 0.0928311i
\(555\) 0 0
\(556\) 143329. 0.463643
\(557\) 74238.2i 0.239286i 0.992817 + 0.119643i \(0.0381750\pi\)
−0.992817 + 0.119643i \(0.961825\pi\)
\(558\) 0 0
\(559\) 193825. 0.620277
\(560\) 17748.1i 0.0565946i
\(561\) 0 0
\(562\) −350142. −1.10859
\(563\) 279024.i 0.880286i 0.897928 + 0.440143i \(0.145072\pi\)
−0.897928 + 0.440143i \(0.854928\pi\)
\(564\) 0 0
\(565\) 124035. 0.388550
\(566\) 317680.i 0.991648i
\(567\) 0 0
\(568\) 59286.2 0.183763
\(569\) − 154079.i − 0.475902i −0.971277 0.237951i \(-0.923524\pi\)
0.971277 0.237951i \(-0.0764758\pi\)
\(570\) 0 0
\(571\) 202500. 0.621087 0.310543 0.950559i \(-0.399489\pi\)
0.310543 + 0.950559i \(0.399489\pi\)
\(572\) 104468.i 0.319295i
\(573\) 0 0
\(574\) −156888. −0.476173
\(575\) 182629.i 0.552374i
\(576\) 0 0
\(577\) −404034. −1.21358 −0.606788 0.794864i \(-0.707542\pi\)
−0.606788 + 0.794864i \(0.707542\pi\)
\(578\) 137012.i 0.410112i
\(579\) 0 0
\(580\) 141507. 0.420652
\(581\) − 112749.i − 0.334010i
\(582\) 0 0
\(583\) 594647. 1.74953
\(584\) 141116.i 0.413763i
\(585\) 0 0
\(586\) 182465. 0.531355
\(587\) 204467.i 0.593399i 0.954971 + 0.296700i \(0.0958860\pi\)
−0.954971 + 0.296700i \(0.904114\pi\)
\(588\) 0 0
\(589\) −701901. −2.02323
\(590\) − 132138.i − 0.379598i
\(591\) 0 0
\(592\) −133896. −0.382053
\(593\) 226013.i 0.642723i 0.946957 + 0.321361i \(0.104140\pi\)
−0.946957 + 0.321361i \(0.895860\pi\)
\(594\) 0 0
\(595\) 51939.9 0.146713
\(596\) − 140708.i − 0.396120i
\(597\) 0 0
\(598\) 81041.8 0.226625
\(599\) − 45246.9i − 0.126106i −0.998010 0.0630530i \(-0.979916\pi\)
0.998010 0.0630530i \(-0.0200837\pi\)
\(600\) 0 0
\(601\) 24718.9 0.0684352 0.0342176 0.999414i \(-0.489106\pi\)
0.0342176 + 0.999414i \(0.489106\pi\)
\(602\) 161468.i 0.445546i
\(603\) 0 0
\(604\) 284776. 0.780602
\(605\) 426546.i 1.16535i
\(606\) 0 0
\(607\) 34864.1 0.0946241 0.0473120 0.998880i \(-0.484934\pi\)
0.0473120 + 0.998880i \(0.484934\pi\)
\(608\) 83790.1i 0.226666i
\(609\) 0 0
\(610\) 136401. 0.366570
\(611\) 103888.i 0.278281i
\(612\) 0 0
\(613\) −175378. −0.466717 −0.233359 0.972391i \(-0.574972\pi\)
−0.233359 + 0.972391i \(0.574972\pi\)
\(614\) 32479.6i 0.0861537i
\(615\) 0 0
\(616\) −87028.2 −0.229350
\(617\) 192394.i 0.505383i 0.967547 + 0.252691i \(0.0813158\pi\)
−0.967547 + 0.252691i \(0.918684\pi\)
\(618\) 0 0
\(619\) −235328. −0.614174 −0.307087 0.951681i \(-0.599354\pi\)
−0.307087 + 0.951681i \(0.599354\pi\)
\(620\) − 181645.i − 0.472540i
\(621\) 0 0
\(622\) −31839.3 −0.0822967
\(623\) 246729.i 0.635688i
\(624\) 0 0
\(625\) 20506.3 0.0524961
\(626\) − 316977.i − 0.808872i
\(627\) 0 0
\(628\) −263735. −0.668726
\(629\) 391847.i 0.990411i
\(630\) 0 0
\(631\) 534962. 1.34358 0.671791 0.740741i \(-0.265525\pi\)
0.671791 + 0.740741i \(0.265525\pi\)
\(632\) − 77115.7i − 0.193067i
\(633\) 0 0
\(634\) 135176. 0.336297
\(635\) − 377179.i − 0.935405i
\(636\) 0 0
\(637\) −21568.0 −0.0531534
\(638\) 693886.i 1.70469i
\(639\) 0 0
\(640\) −21684.0 −0.0529394
\(641\) − 715669.i − 1.74179i −0.491469 0.870895i \(-0.663540\pi\)
0.491469 0.870895i \(-0.336460\pi\)
\(642\) 0 0
\(643\) 401264. 0.970528 0.485264 0.874368i \(-0.338723\pi\)
0.485264 + 0.874368i \(0.338723\pi\)
\(644\) 67512.7i 0.162785i
\(645\) 0 0
\(646\) 245213. 0.587595
\(647\) 761674.i 1.81953i 0.415119 + 0.909767i \(0.363740\pi\)
−0.415119 + 0.909767i \(0.636260\pi\)
\(648\) 0 0
\(649\) 647942. 1.53832
\(650\) − 71282.2i − 0.168715i
\(651\) 0 0
\(652\) 163687. 0.385051
\(653\) 225493.i 0.528819i 0.964411 + 0.264409i \(0.0851770\pi\)
−0.964411 + 0.264409i \(0.914823\pi\)
\(654\) 0 0
\(655\) −159738. −0.372328
\(656\) − 191680.i − 0.445419i
\(657\) 0 0
\(658\) −86545.0 −0.199890
\(659\) 844001.i 1.94344i 0.236127 + 0.971722i \(0.424122\pi\)
−0.236127 + 0.971722i \(0.575878\pi\)
\(660\) 0 0
\(661\) −233508. −0.534441 −0.267220 0.963635i \(-0.586105\pi\)
−0.267220 + 0.963635i \(0.586105\pi\)
\(662\) − 306441.i − 0.699247i
\(663\) 0 0
\(664\) 137752. 0.312437
\(665\) 128363.i 0.290266i
\(666\) 0 0
\(667\) 538287. 1.20993
\(668\) − 152833.i − 0.342503i
\(669\) 0 0
\(670\) −221561. −0.493564
\(671\) 668844.i 1.48552i
\(672\) 0 0
\(673\) −380168. −0.839355 −0.419677 0.907673i \(-0.637857\pi\)
−0.419677 + 0.907673i \(0.637857\pi\)
\(674\) 463335.i 1.01994i
\(675\) 0 0
\(676\) 196856. 0.430781
\(677\) − 390571.i − 0.852163i −0.904685 0.426082i \(-0.859894\pi\)
0.904685 0.426082i \(-0.140106\pi\)
\(678\) 0 0
\(679\) 112475. 0.243959
\(680\) 63458.4i 0.137237i
\(681\) 0 0
\(682\) 890699. 1.91497
\(683\) − 483186.i − 1.03579i −0.855443 0.517896i \(-0.826715\pi\)
0.855443 0.517896i \(-0.173285\pi\)
\(684\) 0 0
\(685\) 144980. 0.308978
\(686\) − 17967.4i − 0.0381802i
\(687\) 0 0
\(688\) −197276. −0.416770
\(689\) 180052.i 0.379279i
\(690\) 0 0
\(691\) −145791. −0.305334 −0.152667 0.988278i \(-0.548786\pi\)
−0.152667 + 0.988278i \(0.548786\pi\)
\(692\) 223462.i 0.466651i
\(693\) 0 0
\(694\) 564366. 1.17177
\(695\) 268267.i 0.555389i
\(696\) 0 0
\(697\) −560953. −1.15468
\(698\) 80060.0i 0.164325i
\(699\) 0 0
\(700\) 59382.4 0.121189
\(701\) 427557.i 0.870078i 0.900412 + 0.435039i \(0.143265\pi\)
−0.900412 + 0.435039i \(0.856735\pi\)
\(702\) 0 0
\(703\) −968400. −1.95949
\(704\) − 106328.i − 0.214537i
\(705\) 0 0
\(706\) −380648. −0.763685
\(707\) 100947.i 0.201954i
\(708\) 0 0
\(709\) −568967. −1.13186 −0.565932 0.824452i \(-0.691484\pi\)
−0.565932 + 0.824452i \(0.691484\pi\)
\(710\) 110965.i 0.220126i
\(711\) 0 0
\(712\) −301445. −0.594632
\(713\) − 690966.i − 1.35918i
\(714\) 0 0
\(715\) −195532. −0.382477
\(716\) 214592.i 0.418589i
\(717\) 0 0
\(718\) 355660. 0.689900
\(719\) 460546.i 0.890873i 0.895314 + 0.445436i \(0.146951\pi\)
−0.895314 + 0.445436i \(0.853049\pi\)
\(720\) 0 0
\(721\) −89580.2 −0.172322
\(722\) 237408.i 0.455429i
\(723\) 0 0
\(724\) 65713.2 0.125365
\(725\) − 473462.i − 0.900761i
\(726\) 0 0
\(727\) 551066. 1.04264 0.521321 0.853361i \(-0.325439\pi\)
0.521321 + 0.853361i \(0.325439\pi\)
\(728\) − 26351.0i − 0.0497204i
\(729\) 0 0
\(730\) −264126. −0.495639
\(731\) 577329.i 1.08041i
\(732\) 0 0
\(733\) −280002. −0.521138 −0.260569 0.965455i \(-0.583910\pi\)
−0.260569 + 0.965455i \(0.583910\pi\)
\(734\) − 62852.8i − 0.116663i
\(735\) 0 0
\(736\) −82484.7 −0.152271
\(737\) − 1.08643e6i − 2.00017i
\(738\) 0 0
\(739\) 8667.21 0.0158705 0.00793525 0.999969i \(-0.497474\pi\)
0.00793525 + 0.999969i \(0.497474\pi\)
\(740\) − 250611.i − 0.457654i
\(741\) 0 0
\(742\) −149994. −0.272437
\(743\) − 370104.i − 0.670419i −0.942144 0.335210i \(-0.891193\pi\)
0.942144 0.335210i \(-0.108807\pi\)
\(744\) 0 0
\(745\) 263362. 0.474505
\(746\) 98644.2i 0.177253i
\(747\) 0 0
\(748\) −311170. −0.556154
\(749\) 137591.i 0.245260i
\(750\) 0 0
\(751\) 508370. 0.901363 0.450681 0.892685i \(-0.351181\pi\)
0.450681 + 0.892685i \(0.351181\pi\)
\(752\) − 105738.i − 0.186980i
\(753\) 0 0
\(754\) −210100. −0.369558
\(755\) 533013.i 0.935070i
\(756\) 0 0
\(757\) 832192. 1.45222 0.726109 0.687580i \(-0.241327\pi\)
0.726109 + 0.687580i \(0.241327\pi\)
\(758\) − 288870.i − 0.502765i
\(759\) 0 0
\(760\) −156829. −0.271519
\(761\) 599649.i 1.03545i 0.855548 + 0.517724i \(0.173220\pi\)
−0.855548 + 0.517724i \(0.826780\pi\)
\(762\) 0 0
\(763\) −295934. −0.508330
\(764\) 156225.i 0.267648i
\(765\) 0 0
\(766\) 33979.4 0.0579107
\(767\) 196189.i 0.333490i
\(768\) 0 0
\(769\) 122085. 0.206447 0.103224 0.994658i \(-0.467084\pi\)
0.103224 + 0.994658i \(0.467084\pi\)
\(770\) − 162890.i − 0.274734i
\(771\) 0 0
\(772\) −13476.8 −0.0226126
\(773\) − 655751.i − 1.09744i −0.836007 0.548719i \(-0.815116\pi\)
0.836007 0.548719i \(-0.184884\pi\)
\(774\) 0 0
\(775\) −607755. −1.01187
\(776\) 137418.i 0.228203i
\(777\) 0 0
\(778\) 289854. 0.478872
\(779\) − 1.38632e6i − 2.28449i
\(780\) 0 0
\(781\) −544123. −0.892061
\(782\) 241392.i 0.394739i
\(783\) 0 0
\(784\) 21952.0 0.0357143
\(785\) − 493630.i − 0.801055i
\(786\) 0 0
\(787\) 208274. 0.336268 0.168134 0.985764i \(-0.446226\pi\)
0.168134 + 0.985764i \(0.446226\pi\)
\(788\) − 200917.i − 0.323567i
\(789\) 0 0
\(790\) 144337. 0.231272
\(791\) − 153415.i − 0.245196i
\(792\) 0 0
\(793\) −202518. −0.322045
\(794\) 290888.i 0.461407i
\(795\) 0 0
\(796\) 385468. 0.608363
\(797\) 464263.i 0.730883i 0.930834 + 0.365442i \(0.119082\pi\)
−0.930834 + 0.365442i \(0.880918\pi\)
\(798\) 0 0
\(799\) −309442. −0.484715
\(800\) 72551.4i 0.113362i
\(801\) 0 0
\(802\) 714045. 1.11014
\(803\) − 1.29515e6i − 2.00858i
\(804\) 0 0
\(805\) −126363. −0.194997
\(806\) 269693.i 0.415144i
\(807\) 0 0
\(808\) −123333. −0.188911
\(809\) 653300.i 0.998195i 0.866546 + 0.499097i \(0.166335\pi\)
−0.866546 + 0.499097i \(0.833665\pi\)
\(810\) 0 0
\(811\) −1.15300e6 −1.75302 −0.876510 0.481384i \(-0.840134\pi\)
−0.876510 + 0.481384i \(0.840134\pi\)
\(812\) − 175026.i − 0.265454i
\(813\) 0 0
\(814\) 1.22888e6 1.85464
\(815\) 306371.i 0.461246i
\(816\) 0 0
\(817\) −1.42679e6 −2.13755
\(818\) − 119190.i − 0.178128i
\(819\) 0 0
\(820\) 358765. 0.533559
\(821\) 440108.i 0.652939i 0.945208 + 0.326470i \(0.105859\pi\)
−0.945208 + 0.326470i \(0.894141\pi\)
\(822\) 0 0
\(823\) 628536. 0.927963 0.463981 0.885845i \(-0.346420\pi\)
0.463981 + 0.885845i \(0.346420\pi\)
\(824\) − 109446.i − 0.161193i
\(825\) 0 0
\(826\) −163437. −0.239547
\(827\) − 92273.2i − 0.134916i −0.997722 0.0674582i \(-0.978511\pi\)
0.997722 0.0674582i \(-0.0214889\pi\)
\(828\) 0 0
\(829\) −745793. −1.08520 −0.542599 0.839992i \(-0.682560\pi\)
−0.542599 + 0.839992i \(0.682560\pi\)
\(830\) 257830.i 0.374263i
\(831\) 0 0
\(832\) 32194.8 0.0465092
\(833\) − 64242.7i − 0.0925836i
\(834\) 0 0
\(835\) 286056. 0.410278
\(836\) − 769017.i − 1.10033i
\(837\) 0 0
\(838\) −621961. −0.885677
\(839\) − 957819.i − 1.36069i −0.732891 0.680346i \(-0.761830\pi\)
0.732891 0.680346i \(-0.238170\pi\)
\(840\) 0 0
\(841\) −688219. −0.973049
\(842\) − 420259.i − 0.592779i
\(843\) 0 0
\(844\) 298996. 0.419740
\(845\) 368454.i 0.516024i
\(846\) 0 0
\(847\) 527580. 0.735397
\(848\) − 183257.i − 0.254841i
\(849\) 0 0
\(850\) 212322. 0.293872
\(851\) − 953312.i − 1.31636i
\(852\) 0 0
\(853\) −943000. −1.29603 −0.648013 0.761629i \(-0.724400\pi\)
−0.648013 + 0.761629i \(0.724400\pi\)
\(854\) − 168709.i − 0.231325i
\(855\) 0 0
\(856\) −168104. −0.229420
\(857\) 601580.i 0.819090i 0.912290 + 0.409545i \(0.134313\pi\)
−0.912290 + 0.409545i \(0.865687\pi\)
\(858\) 0 0
\(859\) 456174. 0.618221 0.309111 0.951026i \(-0.399969\pi\)
0.309111 + 0.951026i \(0.399969\pi\)
\(860\) − 369239.i − 0.499241i
\(861\) 0 0
\(862\) 963051. 1.29609
\(863\) 387585.i 0.520410i 0.965553 + 0.260205i \(0.0837901\pi\)
−0.965553 + 0.260205i \(0.916210\pi\)
\(864\) 0 0
\(865\) −418252. −0.558993
\(866\) − 966756.i − 1.28908i
\(867\) 0 0
\(868\) −224670. −0.298199
\(869\) 707759.i 0.937230i
\(870\) 0 0
\(871\) 328958. 0.433614
\(872\) − 361562.i − 0.475499i
\(873\) 0 0
\(874\) −596570. −0.780978
\(875\) 284467.i 0.371548i
\(876\) 0 0
\(877\) 382540. 0.497368 0.248684 0.968585i \(-0.420002\pi\)
0.248684 + 0.968585i \(0.420002\pi\)
\(878\) 96727.1i 0.125476i
\(879\) 0 0
\(880\) 199013. 0.256990
\(881\) 166044.i 0.213930i 0.994263 + 0.106965i \(0.0341132\pi\)
−0.994263 + 0.106965i \(0.965887\pi\)
\(882\) 0 0
\(883\) −882718. −1.13214 −0.566071 0.824357i \(-0.691537\pi\)
−0.566071 + 0.824357i \(0.691537\pi\)
\(884\) − 94218.4i − 0.120568i
\(885\) 0 0
\(886\) −360472. −0.459202
\(887\) − 233588.i − 0.296896i −0.988920 0.148448i \(-0.952572\pi\)
0.988920 0.148448i \(-0.0474277\pi\)
\(888\) 0 0
\(889\) −466520. −0.590291
\(890\) − 564212.i − 0.712299i
\(891\) 0 0
\(892\) 119439. 0.150113
\(893\) − 764747.i − 0.958992i
\(894\) 0 0
\(895\) −401651. −0.501421
\(896\) 26820.2i 0.0334077i
\(897\) 0 0
\(898\) −709866. −0.880286
\(899\) 1.79132e6i 2.21643i
\(900\) 0 0
\(901\) −536304. −0.660635
\(902\) 1.75922e6i 2.16225i
\(903\) 0 0
\(904\) 187437. 0.229360
\(905\) 122995.i 0.150172i
\(906\) 0 0
\(907\) −192790. −0.234353 −0.117176 0.993111i \(-0.537384\pi\)
−0.117176 + 0.993111i \(0.537384\pi\)
\(908\) − 649779.i − 0.788123i
\(909\) 0 0
\(910\) 49321.0 0.0595592
\(911\) − 1.15573e6i − 1.39258i −0.717759 0.696292i \(-0.754832\pi\)
0.717759 0.696292i \(-0.245168\pi\)
\(912\) 0 0
\(913\) −1.26428e6 −1.51670
\(914\) − 880666.i − 1.05419i
\(915\) 0 0
\(916\) 210309. 0.250650
\(917\) 197575.i 0.234959i
\(918\) 0 0
\(919\) 1.17909e6 1.39610 0.698049 0.716050i \(-0.254052\pi\)
0.698049 + 0.716050i \(0.254052\pi\)
\(920\) − 154386.i − 0.182403i
\(921\) 0 0
\(922\) −847728. −0.997229
\(923\) − 164753.i − 0.193389i
\(924\) 0 0
\(925\) −838508. −0.979995
\(926\) 991780.i 1.15663i
\(927\) 0 0
\(928\) 213840. 0.248310
\(929\) − 630809.i − 0.730915i −0.930828 0.365457i \(-0.880913\pi\)
0.930828 0.365457i \(-0.119087\pi\)
\(930\) 0 0
\(931\) 158768. 0.183174
\(932\) 619022.i 0.712647i
\(933\) 0 0
\(934\) −531559. −0.609338
\(935\) − 582414.i − 0.666206i
\(936\) 0 0
\(937\) 936497. 1.06666 0.533331 0.845906i \(-0.320940\pi\)
0.533331 + 0.845906i \(0.320940\pi\)
\(938\) 274042.i 0.311466i
\(939\) 0 0
\(940\) 197908. 0.223979
\(941\) − 1.58769e6i − 1.79303i −0.443016 0.896514i \(-0.646092\pi\)
0.443016 0.896514i \(-0.353908\pi\)
\(942\) 0 0
\(943\) 1.36472e6 1.53469
\(944\) − 199682.i − 0.224075i
\(945\) 0 0
\(946\) 1.81057e6 2.02318
\(947\) − 75297.7i − 0.0839618i −0.999118 0.0419809i \(-0.986633\pi\)
0.999118 0.0419809i \(-0.0133669\pi\)
\(948\) 0 0
\(949\) 392155. 0.435437
\(950\) 524727.i 0.581415i
\(951\) 0 0
\(952\) 78489.6 0.0866040
\(953\) 20401.2i 0.0224632i 0.999937 + 0.0112316i \(0.00357520\pi\)
−0.999937 + 0.0112316i \(0.996425\pi\)
\(954\) 0 0
\(955\) −292405. −0.320611
\(956\) − 1714.01i − 0.00187541i
\(957\) 0 0
\(958\) 1.24696e6 1.35869
\(959\) − 179321.i − 0.194982i
\(960\) 0 0
\(961\) 1.37589e6 1.48983
\(962\) 372089.i 0.402066i
\(963\) 0 0
\(964\) −25322.4 −0.0272490
\(965\) − 25224.3i − 0.0270873i
\(966\) 0 0
\(967\) −619443. −0.662443 −0.331222 0.943553i \(-0.607461\pi\)
−0.331222 + 0.943553i \(0.607461\pi\)
\(968\) 644580.i 0.687901i
\(969\) 0 0
\(970\) −257205. −0.273360
\(971\) − 1.61783e6i − 1.71591i −0.513725 0.857955i \(-0.671735\pi\)
0.513725 0.857955i \(-0.328265\pi\)
\(972\) 0 0
\(973\) 331810. 0.350481
\(974\) − 876221.i − 0.923625i
\(975\) 0 0
\(976\) 206123. 0.216385
\(977\) 320209.i 0.335463i 0.985833 + 0.167732i \(0.0536442\pi\)
−0.985833 + 0.167732i \(0.946356\pi\)
\(978\) 0 0
\(979\) 2.76663e6 2.88659
\(980\) 41087.4i 0.0427815i
\(981\) 0 0
\(982\) −123585. −0.128157
\(983\) − 947603.i − 0.980662i −0.871536 0.490331i \(-0.836876\pi\)
0.871536 0.490331i \(-0.163124\pi\)
\(984\) 0 0
\(985\) 376055. 0.387595
\(986\) − 625806.i − 0.643704i
\(987\) 0 0
\(988\) 232849. 0.238539
\(989\) − 1.40457e6i − 1.43598i
\(990\) 0 0
\(991\) −309271. −0.314914 −0.157457 0.987526i \(-0.550330\pi\)
−0.157457 + 0.987526i \(0.550330\pi\)
\(992\) − 274494.i − 0.278939i
\(993\) 0 0
\(994\) 137250. 0.138911
\(995\) 721478.i 0.728747i
\(996\) 0 0
\(997\) 149332. 0.150232 0.0751159 0.997175i \(-0.476067\pi\)
0.0751159 + 0.997175i \(0.476067\pi\)
\(998\) − 52223.9i − 0.0524334i
\(999\) 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 378.5.b.a.323.5 yes 8
3.2 odd 2 inner 378.5.b.a.323.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.5.b.a.323.4 8 3.2 odd 2 inner
378.5.b.a.323.5 yes 8 1.1 even 1 trivial