Properties

Label 378.5.b.a.323.3
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 323.3
Root \(-3.56118i\) of defining polynomial
Character \(\chi\) \(=\) 378.323
Dual form 378.5.b.a.323.6

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843i q^{2} -8.00000 q^{4} -3.12468i q^{5} +18.5203 q^{7} +22.6274i q^{8} +O(q^{10})\) \(q-2.82843i q^{2} -8.00000 q^{4} -3.12468i q^{5} +18.5203 q^{7} +22.6274i q^{8} -8.83793 q^{10} +141.266i q^{11} -237.623 q^{13} -52.3832i q^{14} +64.0000 q^{16} -394.261i q^{17} +535.891 q^{19} +24.9974i q^{20} +399.561 q^{22} -487.858i q^{23} +615.236 q^{25} +672.101i q^{26} -148.162 q^{28} +1201.77i q^{29} +296.662 q^{31} -181.019i q^{32} -1115.14 q^{34} -57.8699i q^{35} -380.977 q^{37} -1515.73i q^{38} +70.7034 q^{40} +73.6638i q^{41} +2912.26 q^{43} -1130.13i q^{44} -1379.87 q^{46} -2845.13i q^{47} +343.000 q^{49} -1740.15i q^{50} +1900.99 q^{52} -2125.45i q^{53} +441.411 q^{55} +419.066i q^{56} +3399.11 q^{58} -5486.16i q^{59} -5032.38 q^{61} -839.086i q^{62} -512.000 q^{64} +742.497i q^{65} +7342.60 q^{67} +3154.09i q^{68} -163.681 q^{70} -8600.64i q^{71} +2165.28 q^{73} +1077.57i q^{74} -4287.13 q^{76} +2616.28i q^{77} -5796.36 q^{79} -199.979i q^{80} +208.353 q^{82} -2410.24i q^{83} -1231.94 q^{85} -8237.10i q^{86} -3196.49 q^{88} -13791.5i q^{89} -4400.85 q^{91} +3902.87i q^{92} -8047.25 q^{94} -1674.49i q^{95} +10613.0 q^{97} -970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 64 q^{4}+O(q^{10}) \) 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
\(5\) − 3.12468i − 0.124987i −0.998045 0.0624936i \(-0.980095\pi\)
0.998045 0.0624936i \(-0.0199053\pi\)
\(6\) 0 0
\(7\) 18.5203 0.377964
\(8\) 22.6274i 0.353553i
\(9\) 0 0
\(10\) −8.83793 −0.0883793
\(11\) 141.266i 1.16749i 0.811938 + 0.583744i \(0.198413\pi\)
−0.811938 + 0.583744i \(0.801587\pi\)
\(12\) 0 0
\(13\) −237.623 −1.40606 −0.703028 0.711162i \(-0.748169\pi\)
−0.703028 + 0.711162i \(0.748169\pi\)
\(14\) − 52.3832i − 0.267261i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) − 394.261i − 1.36423i −0.731247 0.682113i \(-0.761061\pi\)
0.731247 0.682113i \(-0.238939\pi\)
\(18\) 0 0
\(19\) 535.891 1.48446 0.742232 0.670143i \(-0.233767\pi\)
0.742232 + 0.670143i \(0.233767\pi\)
\(20\) 24.9974i 0.0624936i
\(21\) 0 0
\(22\) 399.561 0.825539
\(23\) − 487.858i − 0.922227i −0.887341 0.461114i \(-0.847450\pi\)
0.887341 0.461114i \(-0.152550\pi\)
\(24\) 0 0
\(25\) 615.236 0.984378
\(26\) 672.101i 0.994232i
\(27\) 0 0
\(28\) −148.162 −0.188982
\(29\) 1201.77i 1.42897i 0.699649 + 0.714487i \(0.253340\pi\)
−0.699649 + 0.714487i \(0.746660\pi\)
\(30\) 0 0
\(31\) 296.662 0.308701 0.154350 0.988016i \(-0.450672\pi\)
0.154350 + 0.988016i \(0.450672\pi\)
\(32\) − 181.019i − 0.176777i
\(33\) 0 0
\(34\) −1115.14 −0.964653
\(35\) − 57.8699i − 0.0472407i
\(36\) 0 0
\(37\) −380.977 −0.278289 −0.139144 0.990272i \(-0.544435\pi\)
−0.139144 + 0.990272i \(0.544435\pi\)
\(38\) − 1515.73i − 1.04967i
\(39\) 0 0
\(40\) 70.7034 0.0441896
\(41\) 73.6638i 0.0438214i 0.999760 + 0.0219107i \(0.00697495\pi\)
−0.999760 + 0.0219107i \(0.993025\pi\)
\(42\) 0 0
\(43\) 2912.26 1.57504 0.787522 0.616287i \(-0.211364\pi\)
0.787522 + 0.616287i \(0.211364\pi\)
\(44\) − 1130.13i − 0.583744i
\(45\) 0 0
\(46\) −1379.87 −0.652113
\(47\) − 2845.13i − 1.28797i −0.765037 0.643986i \(-0.777279\pi\)
0.765037 0.643986i \(-0.222721\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) − 1740.15i − 0.696061i
\(51\) 0 0
\(52\) 1900.99 0.703028
\(53\) − 2125.45i − 0.756657i −0.925671 0.378328i \(-0.876499\pi\)
0.925671 0.378328i \(-0.123501\pi\)
\(54\) 0 0
\(55\) 441.411 0.145921
\(56\) 419.066i 0.133631i
\(57\) 0 0
\(58\) 3399.11 1.01044
\(59\) − 5486.16i − 1.57603i −0.615656 0.788015i \(-0.711109\pi\)
0.615656 0.788015i \(-0.288891\pi\)
\(60\) 0 0
\(61\) −5032.38 −1.35243 −0.676214 0.736705i \(-0.736381\pi\)
−0.676214 + 0.736705i \(0.736381\pi\)
\(62\) − 839.086i − 0.218284i
\(63\) 0 0
\(64\) −512.000 −0.125000
\(65\) 742.497i 0.175739i
\(66\) 0 0
\(67\) 7342.60 1.63569 0.817844 0.575440i \(-0.195169\pi\)
0.817844 + 0.575440i \(0.195169\pi\)
\(68\) 3154.09i 0.682113i
\(69\) 0 0
\(70\) −163.681 −0.0334042
\(71\) − 8600.64i − 1.70614i −0.521799 0.853068i \(-0.674739\pi\)
0.521799 0.853068i \(-0.325261\pi\)
\(72\) 0 0
\(73\) 2165.28 0.406320 0.203160 0.979146i \(-0.434879\pi\)
0.203160 + 0.979146i \(0.434879\pi\)
\(74\) 1077.57i 0.196780i
\(75\) 0 0
\(76\) −4287.13 −0.742232
\(77\) 2616.28i 0.441269i
\(78\) 0 0
\(79\) −5796.36 −0.928756 −0.464378 0.885637i \(-0.653722\pi\)
−0.464378 + 0.885637i \(0.653722\pi\)
\(80\) − 199.979i − 0.0312468i
\(81\) 0 0
\(82\) 208.353 0.0309864
\(83\) − 2410.24i − 0.349868i −0.984580 0.174934i \(-0.944029\pi\)
0.984580 0.174934i \(-0.0559713\pi\)
\(84\) 0 0
\(85\) −1231.94 −0.170511
\(86\) − 8237.10i − 1.11372i
\(87\) 0 0
\(88\) −3196.49 −0.412769
\(89\) − 13791.5i − 1.74113i −0.492055 0.870564i \(-0.663754\pi\)
0.492055 0.870564i \(-0.336246\pi\)
\(90\) 0 0
\(91\) −4400.85 −0.531439
\(92\) 3902.87i 0.461114i
\(93\) 0 0
\(94\) −8047.25 −0.910734
\(95\) − 1674.49i − 0.185539i
\(96\) 0 0
\(97\) 10613.0 1.12797 0.563984 0.825786i \(-0.309268\pi\)
0.563984 + 0.825786i \(0.309268\pi\)
\(98\) − 970.151i − 0.101015i
\(99\) 0 0
\(100\) −4921.89 −0.492189
\(101\) 4424.36i 0.433718i 0.976203 + 0.216859i \(0.0695812\pi\)
−0.976203 + 0.216859i \(0.930419\pi\)
\(102\) 0 0
\(103\) 11498.7 1.08387 0.541933 0.840422i \(-0.317693\pi\)
0.541933 + 0.840422i \(0.317693\pi\)
\(104\) − 5376.81i − 0.497116i
\(105\) 0 0
\(106\) −6011.68 −0.535037
\(107\) 13254.7i 1.15771i 0.815429 + 0.578857i \(0.196501\pi\)
−0.815429 + 0.578857i \(0.803499\pi\)
\(108\) 0 0
\(109\) 7021.44 0.590981 0.295490 0.955346i \(-0.404517\pi\)
0.295490 + 0.955346i \(0.404517\pi\)
\(110\) − 1248.50i − 0.103182i
\(111\) 0 0
\(112\) 1185.30 0.0944911
\(113\) − 16163.8i − 1.26586i −0.774208 0.632932i \(-0.781851\pi\)
0.774208 0.632932i \(-0.218149\pi\)
\(114\) 0 0
\(115\) −1524.40 −0.115267
\(116\) − 9614.14i − 0.714487i
\(117\) 0 0
\(118\) −15517.2 −1.11442
\(119\) − 7301.82i − 0.515629i
\(120\) 0 0
\(121\) −5315.10 −0.363028
\(122\) 14233.7i 0.956311i
\(123\) 0 0
\(124\) −2373.29 −0.154350
\(125\) − 3875.34i − 0.248022i
\(126\) 0 0
\(127\) −29047.0 −1.80092 −0.900458 0.434942i \(-0.856769\pi\)
−0.900458 + 0.434942i \(0.856769\pi\)
\(128\) 1448.15i 0.0883883i
\(129\) 0 0
\(130\) 2100.10 0.124266
\(131\) 33352.2i 1.94349i 0.236032 + 0.971745i \(0.424153\pi\)
−0.236032 + 0.971745i \(0.575847\pi\)
\(132\) 0 0
\(133\) 9924.85 0.561075
\(134\) − 20768.0i − 1.15661i
\(135\) 0 0
\(136\) 8921.11 0.482326
\(137\) − 3627.12i − 0.193251i −0.995321 0.0966254i \(-0.969195\pi\)
0.995321 0.0966254i \(-0.0308049\pi\)
\(138\) 0 0
\(139\) 9107.22 0.471364 0.235682 0.971830i \(-0.424268\pi\)
0.235682 + 0.971830i \(0.424268\pi\)
\(140\) 462.959i 0.0236204i
\(141\) 0 0
\(142\) −24326.3 −1.20642
\(143\) − 33568.1i − 1.64155i
\(144\) 0 0
\(145\) 3755.14 0.178603
\(146\) − 6124.33i − 0.287312i
\(147\) 0 0
\(148\) 3047.82 0.139144
\(149\) − 9668.97i − 0.435520i −0.976002 0.217760i \(-0.930125\pi\)
0.976002 0.217760i \(-0.0698749\pi\)
\(150\) 0 0
\(151\) −21411.0 −0.939039 −0.469520 0.882922i \(-0.655573\pi\)
−0.469520 + 0.882922i \(0.655573\pi\)
\(152\) 12125.8i 0.524837i
\(153\) 0 0
\(154\) 7399.97 0.312024
\(155\) − 926.972i − 0.0385836i
\(156\) 0 0
\(157\) 26796.2 1.08711 0.543555 0.839374i \(-0.317078\pi\)
0.543555 + 0.839374i \(0.317078\pi\)
\(158\) 16394.6i 0.656729i
\(159\) 0 0
\(160\) −565.627 −0.0220948
\(161\) − 9035.26i − 0.348569i
\(162\) 0 0
\(163\) 32250.2 1.21383 0.606914 0.794768i \(-0.292407\pi\)
0.606914 + 0.794768i \(0.292407\pi\)
\(164\) − 589.310i − 0.0219107i
\(165\) 0 0
\(166\) −6817.20 −0.247394
\(167\) 6114.20i 0.219233i 0.993974 + 0.109617i \(0.0349623\pi\)
−0.993974 + 0.109617i \(0.965038\pi\)
\(168\) 0 0
\(169\) 27903.9 0.976994
\(170\) 3484.45i 0.120569i
\(171\) 0 0
\(172\) −23298.0 −0.787522
\(173\) 29551.1i 0.987372i 0.869640 + 0.493686i \(0.164351\pi\)
−0.869640 + 0.493686i \(0.835649\pi\)
\(174\) 0 0
\(175\) 11394.3 0.372060
\(176\) 9041.03i 0.291872i
\(177\) 0 0
\(178\) −39008.2 −1.23116
\(179\) − 3311.38i − 0.103348i −0.998664 0.0516741i \(-0.983544\pi\)
0.998664 0.0516741i \(-0.0164557\pi\)
\(180\) 0 0
\(181\) −38821.0 −1.18497 −0.592487 0.805580i \(-0.701854\pi\)
−0.592487 + 0.805580i \(0.701854\pi\)
\(182\) 12447.5i 0.375784i
\(183\) 0 0
\(184\) 11039.0 0.326057
\(185\) 1190.43i 0.0347825i
\(186\) 0 0
\(187\) 55695.7 1.59272
\(188\) 22761.1i 0.643986i
\(189\) 0 0
\(190\) −4736.17 −0.131196
\(191\) 40870.6i 1.12033i 0.828382 + 0.560163i \(0.189262\pi\)
−0.828382 + 0.560163i \(0.810738\pi\)
\(192\) 0 0
\(193\) 38211.7 1.02585 0.512923 0.858435i \(-0.328563\pi\)
0.512923 + 0.858435i \(0.328563\pi\)
\(194\) − 30018.2i − 0.797593i
\(195\) 0 0
\(196\) −2744.00 −0.0714286
\(197\) 21225.2i 0.546915i 0.961884 + 0.273457i \(0.0881673\pi\)
−0.961884 + 0.273457i \(0.911833\pi\)
\(198\) 0 0
\(199\) 44448.3 1.12240 0.561202 0.827679i \(-0.310339\pi\)
0.561202 + 0.827679i \(0.310339\pi\)
\(200\) 13921.2i 0.348030i
\(201\) 0 0
\(202\) 12514.0 0.306685
\(203\) 22257.0i 0.540102i
\(204\) 0 0
\(205\) 230.176 0.00547712
\(206\) − 32523.3i − 0.766409i
\(207\) 0 0
\(208\) −15207.9 −0.351514
\(209\) 75703.3i 1.73309i
\(210\) 0 0
\(211\) 902.367 0.0202683 0.0101342 0.999949i \(-0.496774\pi\)
0.0101342 + 0.999949i \(0.496774\pi\)
\(212\) 17003.6i 0.378328i
\(213\) 0 0
\(214\) 37489.9 0.818628
\(215\) − 9099.87i − 0.196860i
\(216\) 0 0
\(217\) 5494.25 0.116678
\(218\) − 19859.6i − 0.417886i
\(219\) 0 0
\(220\) −3531.29 −0.0729605
\(221\) 93685.7i 1.91818i
\(222\) 0 0
\(223\) 2439.54 0.0490566 0.0245283 0.999699i \(-0.492192\pi\)
0.0245283 + 0.999699i \(0.492192\pi\)
\(224\) − 3352.53i − 0.0668153i
\(225\) 0 0
\(226\) −45718.2 −0.895101
\(227\) 3408.89i 0.0661548i 0.999453 + 0.0330774i \(0.0105308\pi\)
−0.999453 + 0.0330774i \(0.989469\pi\)
\(228\) 0 0
\(229\) −10788.1 −0.205719 −0.102860 0.994696i \(-0.532799\pi\)
−0.102860 + 0.994696i \(0.532799\pi\)
\(230\) 4311.66i 0.0815058i
\(231\) 0 0
\(232\) −27192.9 −0.505219
\(233\) − 63204.8i − 1.16423i −0.813107 0.582114i \(-0.802226\pi\)
0.813107 0.582114i \(-0.197774\pi\)
\(234\) 0 0
\(235\) −8890.12 −0.160980
\(236\) 43889.3i 0.788015i
\(237\) 0 0
\(238\) −20652.7 −0.364604
\(239\) 42989.7i 0.752607i 0.926496 + 0.376304i \(0.122805\pi\)
−0.926496 + 0.376304i \(0.877195\pi\)
\(240\) 0 0
\(241\) 37709.8 0.649262 0.324631 0.945841i \(-0.394760\pi\)
0.324631 + 0.945841i \(0.394760\pi\)
\(242\) 15033.4i 0.256700i
\(243\) 0 0
\(244\) 40259.1 0.676214
\(245\) − 1071.76i − 0.0178553i
\(246\) 0 0
\(247\) −127340. −2.08724
\(248\) 6712.68i 0.109142i
\(249\) 0 0
\(250\) −10961.1 −0.175378
\(251\) − 113426.i − 1.80038i −0.435497 0.900190i \(-0.643427\pi\)
0.435497 0.900190i \(-0.356573\pi\)
\(252\) 0 0
\(253\) 68917.8 1.07669
\(254\) 82157.3i 1.27344i
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) 8447.17i 0.127892i 0.997953 + 0.0639462i \(0.0203686\pi\)
−0.997953 + 0.0639462i \(0.979631\pi\)
\(258\) 0 0
\(259\) −7055.80 −0.105183
\(260\) − 5939.98i − 0.0878695i
\(261\) 0 0
\(262\) 94334.4 1.37426
\(263\) 115681.i 1.67244i 0.548391 + 0.836222i \(0.315240\pi\)
−0.548391 + 0.836222i \(0.684760\pi\)
\(264\) 0 0
\(265\) −6641.35 −0.0945724
\(266\) − 28071.7i − 0.396740i
\(267\) 0 0
\(268\) −58740.8 −0.817844
\(269\) 34096.2i 0.471195i 0.971851 + 0.235598i \(0.0757048\pi\)
−0.971851 + 0.235598i \(0.924295\pi\)
\(270\) 0 0
\(271\) 68752.9 0.936165 0.468083 0.883685i \(-0.344945\pi\)
0.468083 + 0.883685i \(0.344945\pi\)
\(272\) − 25232.7i − 0.341056i
\(273\) 0 0
\(274\) −10259.1 −0.136649
\(275\) 86912.0i 1.14925i
\(276\) 0 0
\(277\) 115701. 1.50792 0.753958 0.656923i \(-0.228142\pi\)
0.753958 + 0.656923i \(0.228142\pi\)
\(278\) − 25759.1i − 0.333304i
\(279\) 0 0
\(280\) 1309.45 0.0167021
\(281\) − 9710.36i − 0.122977i −0.998108 0.0614883i \(-0.980415\pi\)
0.998108 0.0614883i \(-0.0195847\pi\)
\(282\) 0 0
\(283\) 16971.5 0.211907 0.105954 0.994371i \(-0.466210\pi\)
0.105954 + 0.994371i \(0.466210\pi\)
\(284\) 68805.1i 0.853068i
\(285\) 0 0
\(286\) −94945.0 −1.16075
\(287\) 1364.27i 0.0165629i
\(288\) 0 0
\(289\) −71920.8 −0.861110
\(290\) − 10621.1i − 0.126292i
\(291\) 0 0
\(292\) −17322.2 −0.203160
\(293\) − 46472.1i − 0.541324i −0.962674 0.270662i \(-0.912757\pi\)
0.962674 0.270662i \(-0.0872425\pi\)
\(294\) 0 0
\(295\) −17142.5 −0.196983
\(296\) − 8620.53i − 0.0983899i
\(297\) 0 0
\(298\) −27348.0 −0.307959
\(299\) 115927.i 1.29670i
\(300\) 0 0
\(301\) 53935.7 0.595311
\(302\) 60559.5i 0.664001i
\(303\) 0 0
\(304\) 34297.0 0.371116
\(305\) 15724.6i 0.169036i
\(306\) 0 0
\(307\) −104419. −1.10790 −0.553951 0.832549i \(-0.686880\pi\)
−0.553951 + 0.832549i \(0.686880\pi\)
\(308\) − 20930.3i − 0.220635i
\(309\) 0 0
\(310\) −2621.87 −0.0272828
\(311\) − 114779.i − 1.18671i −0.804943 0.593353i \(-0.797804\pi\)
0.804943 0.593353i \(-0.202196\pi\)
\(312\) 0 0
\(313\) −7552.66 −0.0770924 −0.0385462 0.999257i \(-0.512273\pi\)
−0.0385462 + 0.999257i \(0.512273\pi\)
\(314\) − 75791.0i − 0.768702i
\(315\) 0 0
\(316\) 46370.9 0.464378
\(317\) − 95323.4i − 0.948595i −0.880365 0.474298i \(-0.842702\pi\)
0.880365 0.474298i \(-0.157298\pi\)
\(318\) 0 0
\(319\) −169769. −1.66831
\(320\) 1599.84i 0.0156234i
\(321\) 0 0
\(322\) −25555.6 −0.246476
\(323\) − 211281.i − 2.02514i
\(324\) 0 0
\(325\) −146195. −1.38409
\(326\) − 91217.3i − 0.858306i
\(327\) 0 0
\(328\) −1666.82 −0.0154932
\(329\) − 52692.6i − 0.486808i
\(330\) 0 0
\(331\) −66158.3 −0.603849 −0.301924 0.953332i \(-0.597629\pi\)
−0.301924 + 0.953332i \(0.597629\pi\)
\(332\) 19281.9i 0.174934i
\(333\) 0 0
\(334\) 17293.6 0.155021
\(335\) − 22943.3i − 0.204440i
\(336\) 0 0
\(337\) −140142. −1.23398 −0.616990 0.786971i \(-0.711648\pi\)
−0.616990 + 0.786971i \(0.711648\pi\)
\(338\) − 78924.2i − 0.690839i
\(339\) 0 0
\(340\) 9855.51 0.0852553
\(341\) 41908.2i 0.360405i
\(342\) 0 0
\(343\) 6352.45 0.0539949
\(344\) 65896.8i 0.556862i
\(345\) 0 0
\(346\) 83583.0 0.698178
\(347\) − 28633.6i − 0.237803i −0.992906 0.118901i \(-0.962063\pi\)
0.992906 0.118901i \(-0.0379372\pi\)
\(348\) 0 0
\(349\) 82009.8 0.673310 0.336655 0.941628i \(-0.390704\pi\)
0.336655 + 0.941628i \(0.390704\pi\)
\(350\) − 32228.1i − 0.263086i
\(351\) 0 0
\(352\) 25571.9 0.206385
\(353\) 58668.4i 0.470820i 0.971896 + 0.235410i \(0.0756433\pi\)
−0.971896 + 0.235410i \(0.924357\pi\)
\(354\) 0 0
\(355\) −26874.2 −0.213245
\(356\) 110332.i 0.870564i
\(357\) 0 0
\(358\) −9365.99 −0.0730782
\(359\) 37840.3i 0.293606i 0.989166 + 0.146803i \(0.0468984\pi\)
−0.989166 + 0.146803i \(0.953102\pi\)
\(360\) 0 0
\(361\) 156859. 1.20363
\(362\) 109802.i 0.837904i
\(363\) 0 0
\(364\) 35206.8 0.265720
\(365\) − 6765.80i − 0.0507848i
\(366\) 0 0
\(367\) 76141.5 0.565313 0.282657 0.959221i \(-0.408784\pi\)
0.282657 + 0.959221i \(0.408784\pi\)
\(368\) − 31222.9i − 0.230557i
\(369\) 0 0
\(370\) 3367.05 0.0245950
\(371\) − 39363.9i − 0.285989i
\(372\) 0 0
\(373\) 85550.6 0.614901 0.307451 0.951564i \(-0.400524\pi\)
0.307451 + 0.951564i \(0.400524\pi\)
\(374\) − 157531.i − 1.12622i
\(375\) 0 0
\(376\) 64378.0 0.455367
\(377\) − 285568.i − 2.00922i
\(378\) 0 0
\(379\) −136931. −0.953287 −0.476644 0.879097i \(-0.658147\pi\)
−0.476644 + 0.879097i \(0.658147\pi\)
\(380\) 13395.9i 0.0927695i
\(381\) 0 0
\(382\) 115600. 0.792190
\(383\) − 42414.5i − 0.289146i −0.989494 0.144573i \(-0.953819\pi\)
0.989494 0.144573i \(-0.0461808\pi\)
\(384\) 0 0
\(385\) 8175.05 0.0551530
\(386\) − 108079.i − 0.725383i
\(387\) 0 0
\(388\) −84904.4 −0.563984
\(389\) 258455.i 1.70799i 0.520277 + 0.853997i \(0.325829\pi\)
−0.520277 + 0.853997i \(0.674171\pi\)
\(390\) 0 0
\(391\) −192343. −1.25813
\(392\) 7761.20i 0.0505076i
\(393\) 0 0
\(394\) 60034.0 0.386727
\(395\) 18111.8i 0.116083i
\(396\) 0 0
\(397\) 114157. 0.724303 0.362152 0.932119i \(-0.382042\pi\)
0.362152 + 0.932119i \(0.382042\pi\)
\(398\) − 125719.i − 0.793660i
\(399\) 0 0
\(400\) 39375.1 0.246095
\(401\) 140116.i 0.871362i 0.900101 + 0.435681i \(0.143492\pi\)
−0.900101 + 0.435681i \(0.856508\pi\)
\(402\) 0 0
\(403\) −70493.8 −0.434051
\(404\) − 35394.9i − 0.216859i
\(405\) 0 0
\(406\) 62952.4 0.381909
\(407\) − 53819.1i − 0.324899i
\(408\) 0 0
\(409\) −165912. −0.991817 −0.495908 0.868375i \(-0.665165\pi\)
−0.495908 + 0.868375i \(0.665165\pi\)
\(410\) − 651.035i − 0.00387291i
\(411\) 0 0
\(412\) −91989.9 −0.541933
\(413\) − 101605.i − 0.595683i
\(414\) 0 0
\(415\) −7531.24 −0.0437291
\(416\) 43014.4i 0.248558i
\(417\) 0 0
\(418\) 214121. 1.22548
\(419\) − 130600.i − 0.743903i −0.928252 0.371952i \(-0.878689\pi\)
0.928252 0.371952i \(-0.121311\pi\)
\(420\) 0 0
\(421\) −62110.0 −0.350427 −0.175213 0.984530i \(-0.556062\pi\)
−0.175213 + 0.984530i \(0.556062\pi\)
\(422\) − 2552.28i − 0.0143319i
\(423\) 0 0
\(424\) 48093.4 0.267519
\(425\) − 242564.i − 1.34291i
\(426\) 0 0
\(427\) −93201.1 −0.511170
\(428\) − 106037.i − 0.578857i
\(429\) 0 0
\(430\) −25738.3 −0.139201
\(431\) 112404.i 0.605100i 0.953133 + 0.302550i \(0.0978380\pi\)
−0.953133 + 0.302550i \(0.902162\pi\)
\(432\) 0 0
\(433\) 64860.0 0.345940 0.172970 0.984927i \(-0.444664\pi\)
0.172970 + 0.984927i \(0.444664\pi\)
\(434\) − 15540.1i − 0.0825038i
\(435\) 0 0
\(436\) −56171.5 −0.295490
\(437\) − 261439.i − 1.36901i
\(438\) 0 0
\(439\) 146510. 0.760218 0.380109 0.924942i \(-0.375887\pi\)
0.380109 + 0.924942i \(0.375887\pi\)
\(440\) 9987.99i 0.0515909i
\(441\) 0 0
\(442\) 264983. 1.35636
\(443\) − 130291.i − 0.663908i −0.943296 0.331954i \(-0.892292\pi\)
0.943296 0.331954i \(-0.107708\pi\)
\(444\) 0 0
\(445\) −43094.0 −0.217619
\(446\) − 6900.05i − 0.0346883i
\(447\) 0 0
\(448\) −9482.37 −0.0472456
\(449\) 93245.1i 0.462523i 0.972892 + 0.231262i \(0.0742853\pi\)
−0.972892 + 0.231262i \(0.925715\pi\)
\(450\) 0 0
\(451\) −10406.2 −0.0511610
\(452\) 129311.i 0.632932i
\(453\) 0 0
\(454\) 9641.80 0.0467785
\(455\) 13751.2i 0.0664231i
\(456\) 0 0
\(457\) −137893. −0.660252 −0.330126 0.943937i \(-0.607091\pi\)
−0.330126 + 0.943937i \(0.607091\pi\)
\(458\) 30513.4i 0.145465i
\(459\) 0 0
\(460\) 12195.2 0.0576333
\(461\) − 324498.i − 1.52690i −0.645869 0.763448i \(-0.723505\pi\)
0.645869 0.763448i \(-0.276495\pi\)
\(462\) 0 0
\(463\) −267434. −1.24754 −0.623770 0.781608i \(-0.714400\pi\)
−0.623770 + 0.781608i \(0.714400\pi\)
\(464\) 76913.1i 0.357244i
\(465\) 0 0
\(466\) −178770. −0.823234
\(467\) 77333.7i 0.354597i 0.984157 + 0.177299i \(0.0567358\pi\)
−0.984157 + 0.177299i \(0.943264\pi\)
\(468\) 0 0
\(469\) 135987. 0.618232
\(470\) 25145.1i 0.113830i
\(471\) 0 0
\(472\) 124138. 0.557211
\(473\) 411403.i 1.83884i
\(474\) 0 0
\(475\) 329700. 1.46127
\(476\) 58414.5i 0.257814i
\(477\) 0 0
\(478\) 121593. 0.532174
\(479\) − 291569.i − 1.27078i −0.772191 0.635390i \(-0.780839\pi\)
0.772191 0.635390i \(-0.219161\pi\)
\(480\) 0 0
\(481\) 90529.1 0.391290
\(482\) − 106659.i − 0.459098i
\(483\) 0 0
\(484\) 42520.8 0.181514
\(485\) − 33162.4i − 0.140981i
\(486\) 0 0
\(487\) −98706.0 −0.416184 −0.208092 0.978109i \(-0.566725\pi\)
−0.208092 + 0.978109i \(0.566725\pi\)
\(488\) − 113870.i − 0.478155i
\(489\) 0 0
\(490\) −3031.41 −0.0126256
\(491\) 68699.3i 0.284963i 0.989797 + 0.142482i \(0.0455082\pi\)
−0.989797 + 0.142482i \(0.954492\pi\)
\(492\) 0 0
\(493\) 473810. 1.94944
\(494\) 360173.i 1.47590i
\(495\) 0 0
\(496\) 18986.3 0.0771752
\(497\) − 159286.i − 0.644859i
\(498\) 0 0
\(499\) −180484. −0.724834 −0.362417 0.932016i \(-0.618048\pi\)
−0.362417 + 0.932016i \(0.618048\pi\)
\(500\) 31002.7i 0.124011i
\(501\) 0 0
\(502\) −320817. −1.27306
\(503\) 170691.i 0.674645i 0.941389 + 0.337323i \(0.109521\pi\)
−0.941389 + 0.337323i \(0.890479\pi\)
\(504\) 0 0
\(505\) 13824.7 0.0542092
\(506\) − 194929.i − 0.761334i
\(507\) 0 0
\(508\) 232376. 0.900458
\(509\) − 87777.9i − 0.338805i −0.985547 0.169403i \(-0.945816\pi\)
0.985547 0.169403i \(-0.0541838\pi\)
\(510\) 0 0
\(511\) 40101.5 0.153574
\(512\) − 11585.2i − 0.0441942i
\(513\) 0 0
\(514\) 23892.2 0.0904336
\(515\) − 35929.9i − 0.135469i
\(516\) 0 0
\(517\) 401921. 1.50369
\(518\) 19956.8i 0.0743758i
\(519\) 0 0
\(520\) −16800.8 −0.0621331
\(521\) − 433477.i − 1.59695i −0.602030 0.798473i \(-0.705641\pi\)
0.602030 0.798473i \(-0.294359\pi\)
\(522\) 0 0
\(523\) −380650. −1.39163 −0.695813 0.718223i \(-0.744956\pi\)
−0.695813 + 0.718223i \(0.744956\pi\)
\(524\) − 266818.i − 0.971745i
\(525\) 0 0
\(526\) 327196. 1.18260
\(527\) − 116962.i − 0.421137i
\(528\) 0 0
\(529\) 41835.4 0.149497
\(530\) 18784.6i 0.0668728i
\(531\) 0 0
\(532\) −79398.8 −0.280537
\(533\) − 17504.3i − 0.0616154i
\(534\) 0 0
\(535\) 41416.6 0.144699
\(536\) 166144.i 0.578303i
\(537\) 0 0
\(538\) 96438.5 0.333185
\(539\) 48454.3i 0.166784i
\(540\) 0 0
\(541\) −158944. −0.543061 −0.271531 0.962430i \(-0.587530\pi\)
−0.271531 + 0.962430i \(0.587530\pi\)
\(542\) − 194463.i − 0.661969i
\(543\) 0 0
\(544\) −71368.9 −0.241163
\(545\) − 21939.8i − 0.0738650i
\(546\) 0 0
\(547\) −500492. −1.67272 −0.836358 0.548183i \(-0.815320\pi\)
−0.836358 + 0.548183i \(0.815320\pi\)
\(548\) 29017.0i 0.0966254i
\(549\) 0 0
\(550\) 245824. 0.812642
\(551\) 644017.i 2.12126i
\(552\) 0 0
\(553\) −107350. −0.351037
\(554\) − 327251.i − 1.06626i
\(555\) 0 0
\(556\) −72857.7 −0.235682
\(557\) 411990.i 1.32793i 0.747762 + 0.663967i \(0.231128\pi\)
−0.747762 + 0.663967i \(0.768872\pi\)
\(558\) 0 0
\(559\) −692020. −2.21460
\(560\) − 3703.67i − 0.0118102i
\(561\) 0 0
\(562\) −27465.0 −0.0869576
\(563\) − 237439.i − 0.749091i −0.927209 0.374545i \(-0.877799\pi\)
0.927209 0.374545i \(-0.122201\pi\)
\(564\) 0 0
\(565\) −50506.7 −0.158217
\(566\) − 48002.5i − 0.149841i
\(567\) 0 0
\(568\) 194610. 0.603210
\(569\) 548677.i 1.69470i 0.531037 + 0.847349i \(0.321803\pi\)
−0.531037 + 0.847349i \(0.678197\pi\)
\(570\) 0 0
\(571\) 205016. 0.628806 0.314403 0.949290i \(-0.398196\pi\)
0.314403 + 0.949290i \(0.398196\pi\)
\(572\) 268545.i 0.820777i
\(573\) 0 0
\(574\) 3858.75 0.0117118
\(575\) − 300148.i − 0.907820i
\(576\) 0 0
\(577\) −292763. −0.879356 −0.439678 0.898155i \(-0.644907\pi\)
−0.439678 + 0.898155i \(0.644907\pi\)
\(578\) 203423.i 0.608897i
\(579\) 0 0
\(580\) −30041.1 −0.0893017
\(581\) − 44638.3i − 0.132238i
\(582\) 0 0
\(583\) 300254. 0.883388
\(584\) 48994.7i 0.143656i
\(585\) 0 0
\(586\) −131443. −0.382774
\(587\) 75489.1i 0.219083i 0.993982 + 0.109541i \(0.0349382\pi\)
−0.993982 + 0.109541i \(0.965062\pi\)
\(588\) 0 0
\(589\) 158978. 0.458255
\(590\) 48486.3i 0.139288i
\(591\) 0 0
\(592\) −24382.5 −0.0695722
\(593\) 664448.i 1.88952i 0.327764 + 0.944760i \(0.393705\pi\)
−0.327764 + 0.944760i \(0.606295\pi\)
\(594\) 0 0
\(595\) −22815.8 −0.0644470
\(596\) 77351.8i 0.217760i
\(597\) 0 0
\(598\) 327890. 0.916908
\(599\) 521123.i 1.45240i 0.687483 + 0.726201i \(0.258716\pi\)
−0.687483 + 0.726201i \(0.741284\pi\)
\(600\) 0 0
\(601\) 52148.0 0.144374 0.0721869 0.997391i \(-0.477002\pi\)
0.0721869 + 0.997391i \(0.477002\pi\)
\(602\) − 152553.i − 0.420948i
\(603\) 0 0
\(604\) 171288. 0.469520
\(605\) 16608.0i 0.0453739i
\(606\) 0 0
\(607\) −342719. −0.930168 −0.465084 0.885267i \(-0.653976\pi\)
−0.465084 + 0.885267i \(0.653976\pi\)
\(608\) − 97006.7i − 0.262419i
\(609\) 0 0
\(610\) 44475.8 0.119527
\(611\) 676070.i 1.81096i
\(612\) 0 0
\(613\) −702183. −1.86866 −0.934328 0.356415i \(-0.883999\pi\)
−0.934328 + 0.356415i \(0.883999\pi\)
\(614\) 295340.i 0.783405i
\(615\) 0 0
\(616\) −59199.7 −0.156012
\(617\) − 5722.23i − 0.0150312i −0.999972 0.00751562i \(-0.997608\pi\)
0.999972 0.00751562i \(-0.00239232\pi\)
\(618\) 0 0
\(619\) −493596. −1.28822 −0.644110 0.764933i \(-0.722772\pi\)
−0.644110 + 0.764933i \(0.722772\pi\)
\(620\) 7415.78i 0.0192918i
\(621\) 0 0
\(622\) −324645. −0.839127
\(623\) − 255422.i − 0.658085i
\(624\) 0 0
\(625\) 372414. 0.953379
\(626\) 21362.2i 0.0545126i
\(627\) 0 0
\(628\) −214369. −0.543555
\(629\) 150204.i 0.379648i
\(630\) 0 0
\(631\) 369529. 0.928089 0.464045 0.885812i \(-0.346398\pi\)
0.464045 + 0.885812i \(0.346398\pi\)
\(632\) − 131157.i − 0.328365i
\(633\) 0 0
\(634\) −269615. −0.670758
\(635\) 90762.5i 0.225091i
\(636\) 0 0
\(637\) −81504.9 −0.200865
\(638\) 480179.i 1.17967i
\(639\) 0 0
\(640\) 4525.02 0.0110474
\(641\) − 685619.i − 1.66866i −0.551269 0.834328i \(-0.685856\pi\)
0.551269 0.834328i \(-0.314144\pi\)
\(642\) 0 0
\(643\) 90659.6 0.219276 0.109638 0.993972i \(-0.465031\pi\)
0.109638 + 0.993972i \(0.465031\pi\)
\(644\) 72282.1i 0.174285i
\(645\) 0 0
\(646\) −597593. −1.43199
\(647\) 758270.i 1.81140i 0.423916 + 0.905702i \(0.360655\pi\)
−0.423916 + 0.905702i \(0.639345\pi\)
\(648\) 0 0
\(649\) 775008. 1.84000
\(650\) 413501.i 0.978700i
\(651\) 0 0
\(652\) −258001. −0.606914
\(653\) − 261552.i − 0.613383i −0.951809 0.306692i \(-0.900778\pi\)
0.951809 0.306692i \(-0.0992220\pi\)
\(654\) 0 0
\(655\) 104215. 0.242911
\(656\) 4714.48i 0.0109554i
\(657\) 0 0
\(658\) −149037. −0.344225
\(659\) − 517272.i − 1.19110i −0.803319 0.595549i \(-0.796934\pi\)
0.803319 0.595549i \(-0.203066\pi\)
\(660\) 0 0
\(661\) −626741. −1.43445 −0.717224 0.696842i \(-0.754588\pi\)
−0.717224 + 0.696842i \(0.754588\pi\)
\(662\) 187124.i 0.426985i
\(663\) 0 0
\(664\) 54537.6 0.123697
\(665\) − 31012.0i − 0.0701271i
\(666\) 0 0
\(667\) 586292. 1.31784
\(668\) − 48913.6i − 0.109617i
\(669\) 0 0
\(670\) −64893.4 −0.144561
\(671\) − 710905.i − 1.57894i
\(672\) 0 0
\(673\) 742305. 1.63890 0.819450 0.573151i \(-0.194279\pi\)
0.819450 + 0.573151i \(0.194279\pi\)
\(674\) 396381.i 0.872555i
\(675\) 0 0
\(676\) −223231. −0.488497
\(677\) 604523.i 1.31897i 0.751717 + 0.659486i \(0.229226\pi\)
−0.751717 + 0.659486i \(0.770774\pi\)
\(678\) 0 0
\(679\) 196556. 0.426332
\(680\) − 27875.6i − 0.0602846i
\(681\) 0 0
\(682\) 118534. 0.254845
\(683\) 720939.i 1.54546i 0.634737 + 0.772728i \(0.281109\pi\)
−0.634737 + 0.772728i \(0.718891\pi\)
\(684\) 0 0
\(685\) −11333.6 −0.0241539
\(686\) − 17967.4i − 0.0381802i
\(687\) 0 0
\(688\) 186384. 0.393761
\(689\) 505057.i 1.06390i
\(690\) 0 0
\(691\) −588601. −1.23272 −0.616360 0.787464i \(-0.711393\pi\)
−0.616360 + 0.787464i \(0.711393\pi\)
\(692\) − 236409.i − 0.493686i
\(693\) 0 0
\(694\) −80988.0 −0.168152
\(695\) − 28457.1i − 0.0589144i
\(696\) 0 0
\(697\) 29042.8 0.0597823
\(698\) − 231959.i − 0.476102i
\(699\) 0 0
\(700\) −91154.7 −0.186030
\(701\) 154697.i 0.314809i 0.987534 + 0.157404i \(0.0503126\pi\)
−0.987534 + 0.157404i \(0.949687\pi\)
\(702\) 0 0
\(703\) −204162. −0.413109
\(704\) − 72328.2i − 0.145936i
\(705\) 0 0
\(706\) 165939. 0.332920
\(707\) 81940.3i 0.163930i
\(708\) 0 0
\(709\) 715382. 1.42313 0.711567 0.702619i \(-0.247986\pi\)
0.711567 + 0.702619i \(0.247986\pi\)
\(710\) 76011.8i 0.150787i
\(711\) 0 0
\(712\) 312066. 0.615582
\(713\) − 144729.i − 0.284692i
\(714\) 0 0
\(715\) −104890. −0.205173
\(716\) 26491.0i 0.0516741i
\(717\) 0 0
\(718\) 107028. 0.207611
\(719\) − 371799.i − 0.719202i −0.933106 0.359601i \(-0.882913\pi\)
0.933106 0.359601i \(-0.117087\pi\)
\(720\) 0 0
\(721\) 212960. 0.409663
\(722\) − 443663.i − 0.851097i
\(723\) 0 0
\(724\) 310568. 0.592487
\(725\) 739371.i 1.40665i
\(726\) 0 0
\(727\) 52138.1 0.0986476 0.0493238 0.998783i \(-0.484293\pi\)
0.0493238 + 0.998783i \(0.484293\pi\)
\(728\) − 99579.8i − 0.187892i
\(729\) 0 0
\(730\) −19136.6 −0.0359103
\(731\) − 1.14819e6i − 2.14871i
\(732\) 0 0
\(733\) 324955. 0.604805 0.302403 0.953180i \(-0.402211\pi\)
0.302403 + 0.953180i \(0.402211\pi\)
\(734\) − 215361.i − 0.399737i
\(735\) 0 0
\(736\) −88311.8 −0.163028
\(737\) 1.03726e6i 1.90965i
\(738\) 0 0
\(739\) −971247. −1.77845 −0.889224 0.457473i \(-0.848755\pi\)
−0.889224 + 0.457473i \(0.848755\pi\)
\(740\) − 9523.45i − 0.0173913i
\(741\) 0 0
\(742\) −111338. −0.202225
\(743\) − 681732.i − 1.23491i −0.786605 0.617456i \(-0.788163\pi\)
0.786605 0.617456i \(-0.211837\pi\)
\(744\) 0 0
\(745\) −30212.4 −0.0544344
\(746\) − 241974.i − 0.434801i
\(747\) 0 0
\(748\) −445566. −0.796358
\(749\) 245480.i 0.437575i
\(750\) 0 0
\(751\) −317894. −0.563641 −0.281821 0.959467i \(-0.590938\pi\)
−0.281821 + 0.959467i \(0.590938\pi\)
\(752\) − 182088.i − 0.321993i
\(753\) 0 0
\(754\) −807709. −1.42073
\(755\) 66902.6i 0.117368i
\(756\) 0 0
\(757\) −700233. −1.22194 −0.610971 0.791653i \(-0.709221\pi\)
−0.610971 + 0.791653i \(0.709221\pi\)
\(758\) 387300.i 0.674076i
\(759\) 0 0
\(760\) 37889.4 0.0655979
\(761\) − 305818.i − 0.528072i −0.964513 0.264036i \(-0.914946\pi\)
0.964513 0.264036i \(-0.0850538\pi\)
\(762\) 0 0
\(763\) 130039. 0.223370
\(764\) − 326965.i − 0.560163i
\(765\) 0 0
\(766\) −119966. −0.204457
\(767\) 1.30364e6i 2.21599i
\(768\) 0 0
\(769\) 820424. 1.38735 0.693675 0.720289i \(-0.255991\pi\)
0.693675 + 0.720289i \(0.255991\pi\)
\(770\) − 23122.5i − 0.0389990i
\(771\) 0 0
\(772\) −305694. −0.512923
\(773\) − 483522.i − 0.809202i −0.914493 0.404601i \(-0.867410\pi\)
0.914493 0.404601i \(-0.132590\pi\)
\(774\) 0 0
\(775\) 182517. 0.303878
\(776\) 240146.i 0.398797i
\(777\) 0 0
\(778\) 731022. 1.20773
\(779\) 39475.8i 0.0650513i
\(780\) 0 0
\(781\) 1.21498e6 1.99189
\(782\) 544030.i 0.889629i
\(783\) 0 0
\(784\) 21952.0 0.0357143
\(785\) − 83729.4i − 0.135875i
\(786\) 0 0
\(787\) −185741. −0.299888 −0.149944 0.988695i \(-0.547909\pi\)
−0.149944 + 0.988695i \(0.547909\pi\)
\(788\) − 169802.i − 0.273457i
\(789\) 0 0
\(790\) 51227.8 0.0820827
\(791\) − 299358.i − 0.478452i
\(792\) 0 0
\(793\) 1.19581e6 1.90159
\(794\) − 322884.i − 0.512160i
\(795\) 0 0
\(796\) −355587. −0.561202
\(797\) − 720505.i − 1.13428i −0.823621 0.567140i \(-0.808050\pi\)
0.823621 0.567140i \(-0.191950\pi\)
\(798\) 0 0
\(799\) −1.12172e6 −1.75708
\(800\) − 111370.i − 0.174015i
\(801\) 0 0
\(802\) 396307. 0.616146
\(803\) 305880.i 0.474374i
\(804\) 0 0
\(805\) −28232.3 −0.0435667
\(806\) 199386.i 0.306920i
\(807\) 0 0
\(808\) −100112. −0.153343
\(809\) 686330.i 1.04866i 0.851514 + 0.524331i \(0.175685\pi\)
−0.851514 + 0.524331i \(0.824315\pi\)
\(810\) 0 0
\(811\) 1.24830e6 1.89791 0.948955 0.315412i \(-0.102143\pi\)
0.948955 + 0.315412i \(0.102143\pi\)
\(812\) − 178056.i − 0.270051i
\(813\) 0 0
\(814\) −152224. −0.229738
\(815\) − 100771.i − 0.151713i
\(816\) 0 0
\(817\) 1.56065e6 2.33810
\(818\) 469270.i 0.701320i
\(819\) 0 0
\(820\) −1841.41 −0.00273856
\(821\) − 842160.i − 1.24942i −0.780857 0.624710i \(-0.785217\pi\)
0.780857 0.624710i \(-0.214783\pi\)
\(822\) 0 0
\(823\) −105260. −0.155404 −0.0777019 0.996977i \(-0.524758\pi\)
−0.0777019 + 0.996977i \(0.524758\pi\)
\(824\) 260187.i 0.383204i
\(825\) 0 0
\(826\) −287383. −0.421212
\(827\) − 373318.i − 0.545844i −0.962036 0.272922i \(-0.912010\pi\)
0.962036 0.272922i \(-0.0879900\pi\)
\(828\) 0 0
\(829\) 867025. 1.26160 0.630801 0.775944i \(-0.282726\pi\)
0.630801 + 0.775944i \(0.282726\pi\)
\(830\) 21301.6i 0.0309211i
\(831\) 0 0
\(832\) 121663. 0.175757
\(833\) − 135232.i − 0.194889i
\(834\) 0 0
\(835\) 19104.9 0.0274014
\(836\) − 605626.i − 0.866547i
\(837\) 0 0
\(838\) −369394. −0.526019
\(839\) − 686493.i − 0.975242i −0.873055 0.487621i \(-0.837865\pi\)
0.873055 0.487621i \(-0.162135\pi\)
\(840\) 0 0
\(841\) −736964. −1.04197
\(842\) 175674.i 0.247789i
\(843\) 0 0
\(844\) −7218.94 −0.0101342
\(845\) − 87190.8i − 0.122112i
\(846\) 0 0
\(847\) −98437.0 −0.137212
\(848\) − 136029.i − 0.189164i
\(849\) 0 0
\(850\) −686074. −0.949583
\(851\) 185863.i 0.256645i
\(852\) 0 0
\(853\) 974427. 1.33922 0.669609 0.742714i \(-0.266462\pi\)
0.669609 + 0.742714i \(0.266462\pi\)
\(854\) 263612.i 0.361452i
\(855\) 0 0
\(856\) −299919. −0.409314
\(857\) − 275668.i − 0.375339i −0.982232 0.187670i \(-0.939907\pi\)
0.982232 0.187670i \(-0.0600934\pi\)
\(858\) 0 0
\(859\) 176529. 0.239237 0.119619 0.992820i \(-0.461833\pi\)
0.119619 + 0.992820i \(0.461833\pi\)
\(860\) 72798.9i 0.0984301i
\(861\) 0 0
\(862\) 317927. 0.427871
\(863\) − 856520.i − 1.15005i −0.818136 0.575024i \(-0.804993\pi\)
0.818136 0.575024i \(-0.195007\pi\)
\(864\) 0 0
\(865\) 92337.6 0.123409
\(866\) − 183452.i − 0.244617i
\(867\) 0 0
\(868\) −43954.0 −0.0583390
\(869\) − 818829.i − 1.08431i
\(870\) 0 0
\(871\) −1.74478e6 −2.29987
\(872\) 158877.i 0.208943i
\(873\) 0 0
\(874\) −739461. −0.968038
\(875\) − 71772.3i − 0.0937434i
\(876\) 0 0
\(877\) 1.00077e6 1.30118 0.650589 0.759430i \(-0.274522\pi\)
0.650589 + 0.759430i \(0.274522\pi\)
\(878\) − 414393.i − 0.537555i
\(879\) 0 0
\(880\) 28250.3 0.0364803
\(881\) − 775427.i − 0.999055i −0.866298 0.499527i \(-0.833507\pi\)
0.866298 0.499527i \(-0.166493\pi\)
\(882\) 0 0
\(883\) 160327. 0.205630 0.102815 0.994701i \(-0.467215\pi\)
0.102815 + 0.994701i \(0.467215\pi\)
\(884\) − 749485.i − 0.959089i
\(885\) 0 0
\(886\) −368519. −0.469454
\(887\) − 183914.i − 0.233759i −0.993146 0.116879i \(-0.962711\pi\)
0.993146 0.116879i \(-0.0372891\pi\)
\(888\) 0 0
\(889\) −537958. −0.680682
\(890\) 121888.i 0.153880i
\(891\) 0 0
\(892\) −19516.3 −0.0245283
\(893\) − 1.52468e6i − 1.91195i
\(894\) 0 0
\(895\) −10347.0 −0.0129172
\(896\) 26820.2i 0.0334077i
\(897\) 0 0
\(898\) 263737. 0.327053
\(899\) 356518.i 0.441126i
\(900\) 0 0
\(901\) −837982. −1.03225
\(902\) 29433.2i 0.0361763i
\(903\) 0 0
\(904\) 365745. 0.447550
\(905\) 121303.i 0.148107i
\(906\) 0 0
\(907\) 184492. 0.224266 0.112133 0.993693i \(-0.464232\pi\)
0.112133 + 0.993693i \(0.464232\pi\)
\(908\) − 27271.1i − 0.0330774i
\(909\) 0 0
\(910\) 38894.4 0.0469682
\(911\) 979147.i 1.17981i 0.807474 + 0.589904i \(0.200834\pi\)
−0.807474 + 0.589904i \(0.799166\pi\)
\(912\) 0 0
\(913\) 340486. 0.408467
\(914\) 390020.i 0.466868i
\(915\) 0 0
\(916\) 86304.9 0.102860
\(917\) 617692.i 0.734570i
\(918\) 0 0
\(919\) 1.32373e6 1.56736 0.783679 0.621166i \(-0.213341\pi\)
0.783679 + 0.621166i \(0.213341\pi\)
\(920\) − 34493.2i − 0.0407529i
\(921\) 0 0
\(922\) −917818. −1.07968
\(923\) 2.04371e6i 2.39892i
\(924\) 0 0
\(925\) −234391. −0.273941
\(926\) 756417.i 0.882143i
\(927\) 0 0
\(928\) 217543. 0.252609
\(929\) 1.15956e6i 1.34358i 0.740743 + 0.671789i \(0.234474\pi\)
−0.740743 + 0.671789i \(0.765526\pi\)
\(930\) 0 0
\(931\) 183811. 0.212066
\(932\) 505638.i 0.582114i
\(933\) 0 0
\(934\) 218733. 0.250738
\(935\) − 174031.i − 0.199069i
\(936\) 0 0
\(937\) −1.49483e6 −1.70260 −0.851300 0.524679i \(-0.824185\pi\)
−0.851300 + 0.524679i \(0.824185\pi\)
\(938\) − 384629.i − 0.437156i
\(939\) 0 0
\(940\) 71121.0 0.0804900
\(941\) − 1.18590e6i − 1.33927i −0.742689 0.669637i \(-0.766450\pi\)
0.742689 0.669637i \(-0.233550\pi\)
\(942\) 0 0
\(943\) 35937.5 0.0404133
\(944\) − 351114.i − 0.394007i
\(945\) 0 0
\(946\) 1.16362e6 1.30026
\(947\) − 865824.i − 0.965450i −0.875772 0.482725i \(-0.839647\pi\)
0.875772 0.482725i \(-0.160353\pi\)
\(948\) 0 0
\(949\) −514521. −0.571309
\(950\) − 932532.i − 1.03328i
\(951\) 0 0
\(952\) 165221. 0.182302
\(953\) − 388687.i − 0.427971i −0.976837 0.213986i \(-0.931355\pi\)
0.976837 0.213986i \(-0.0686446\pi\)
\(954\) 0 0
\(955\) 127708. 0.140026
\(956\) − 343917.i − 0.376304i
\(957\) 0 0
\(958\) −824682. −0.898577
\(959\) − 67175.3i − 0.0730419i
\(960\) 0 0
\(961\) −835513. −0.904704
\(962\) − 256055.i − 0.276683i
\(963\) 0 0
\(964\) −301678. −0.324631
\(965\) − 119399.i − 0.128218i
\(966\) 0 0
\(967\) 192495. 0.205858 0.102929 0.994689i \(-0.467179\pi\)
0.102929 + 0.994689i \(0.467179\pi\)
\(968\) − 120267.i − 0.128350i
\(969\) 0 0
\(970\) −93797.3 −0.0996889
\(971\) − 453950.i − 0.481470i −0.970591 0.240735i \(-0.922612\pi\)
0.970591 0.240735i \(-0.0773885\pi\)
\(972\) 0 0
\(973\) 168668. 0.178159
\(974\) 279183.i 0.294287i
\(975\) 0 0
\(976\) −322073. −0.338107
\(977\) − 727881.i − 0.762555i −0.924461 0.381278i \(-0.875484\pi\)
0.924461 0.381278i \(-0.124516\pi\)
\(978\) 0 0
\(979\) 1.94827e6 2.03275
\(980\) 8574.12i 0.00892765i
\(981\) 0 0
\(982\) 194311. 0.201500
\(983\) − 662759.i − 0.685881i −0.939357 0.342941i \(-0.888577\pi\)
0.939357 0.342941i \(-0.111423\pi\)
\(984\) 0 0
\(985\) 66322.0 0.0683573
\(986\) − 1.34014e6i − 1.37846i
\(987\) 0 0
\(988\) 1.01872e6 1.04362
\(989\) − 1.42077e6i − 1.45255i
\(990\) 0 0
\(991\) −1.16764e6 −1.18894 −0.594472 0.804116i \(-0.702639\pi\)
−0.594472 + 0.804116i \(0.702639\pi\)
\(992\) − 53701.5i − 0.0545711i
\(993\) 0 0
\(994\) −450529. −0.455984
\(995\) − 138887.i − 0.140286i
\(996\) 0 0
\(997\) 1.12634e6 1.13313 0.566563 0.824019i \(-0.308273\pi\)
0.566563 + 0.824019i \(0.308273\pi\)
\(998\) 510487.i 0.512535i
\(999\) 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.323.3 8
3.2 odd 2 inner 378.5.b.a.323.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.5.b.a.323.3 8 1.1 even 1 trivial
378.5.b.a.323.6 yes 8 3.2 odd 2 inner