Properties

Label 378.5.b.b.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: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 92x^{6} + 2949x^{4} + 37548x^{2} + 142884 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.5
Root \(5.99257i\) of defining polynomial
Character \(\chi\) \(=\) 378.323
Dual form 378.5.b.b.323.4

$q$-expansion

\(f(q)\) \(=\) \(q+2.82843i q^{2} -8.00000 q^{4} -34.7917i q^{5} +18.5203 q^{7} -22.6274i q^{8} +O(q^{10})\) \(q+2.82843i q^{2} -8.00000 q^{4} -34.7917i q^{5} +18.5203 q^{7} -22.6274i q^{8} +98.4059 q^{10} -30.1276i q^{11} +143.140 q^{13} +52.3832i q^{14} +64.0000 q^{16} +116.175i q^{17} -89.0628 q^{19} +278.334i q^{20} +85.2137 q^{22} +23.6671i q^{23} -585.464 q^{25} +404.860i q^{26} -148.162 q^{28} -1180.46i q^{29} +171.591 q^{31} +181.019i q^{32} -328.593 q^{34} -644.352i q^{35} -974.714 q^{37} -251.908i q^{38} -787.247 q^{40} -417.362i q^{41} -3148.03 q^{43} +241.021i q^{44} -66.9407 q^{46} -2961.00i q^{47} +343.000 q^{49} -1655.94i q^{50} -1145.12 q^{52} -2248.40i q^{53} -1048.19 q^{55} -419.066i q^{56} +3338.85 q^{58} +827.250i q^{59} -6889.71 q^{61} +485.332i q^{62} -512.000 q^{64} -4980.08i q^{65} +4866.31 q^{67} -929.402i q^{68} +1822.50 q^{70} -5488.08i q^{71} +777.420 q^{73} -2756.91i q^{74} +712.502 q^{76} -557.971i q^{77} +3899.43 q^{79} -2226.67i q^{80} +1180.48 q^{82} +826.579i q^{83} +4041.94 q^{85} -8903.96i q^{86} -681.709 q^{88} +1055.44i q^{89} +2650.99 q^{91} -189.337i q^{92} +8374.98 q^{94} +3098.65i q^{95} +5713.52 q^{97} +970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 64 q^{4} - 32 q^{10} + 296 q^{13} + 512 q^{16} - 80 q^{19} + 800 q^{22} - 1464 q^{25} - 1024 q^{31} + 896 q^{34} + 904 q^{37} + 256 q^{40} - 3224 q^{43} - 4384 q^{46} + 2744 q^{49} - 2368 q^{52} + 8312 q^{55} + 2944 q^{58} + 128 q^{61} - 4096 q^{64} - 10152 q^{67} + 1568 q^{70} + 10632 q^{73} + 640 q^{76} - 4072 q^{79} - 3552 q^{82} + 13448 q^{85} - 6400 q^{88} - 6664 q^{91} - 384 q^{94} + 35616 q^{97}+O(q^{100}) \) 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
\(5\) − 34.7917i − 1.39167i −0.718202 0.695834i \(-0.755035\pi\)
0.718202 0.695834i \(-0.244965\pi\)
\(6\) 0 0
\(7\) 18.5203 0.377964
\(8\) − 22.6274i − 0.353553i
\(9\) 0 0
\(10\) 98.4059 0.984059
\(11\) − 30.1276i − 0.248988i −0.992220 0.124494i \(-0.960269\pi\)
0.992220 0.124494i \(-0.0397308\pi\)
\(12\) 0 0
\(13\) 143.140 0.846981 0.423491 0.905901i \(-0.360805\pi\)
0.423491 + 0.905901i \(0.360805\pi\)
\(14\) 52.3832i 0.267261i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) 116.175i 0.401991i 0.979592 + 0.200995i \(0.0644176\pi\)
−0.979592 + 0.200995i \(0.935582\pi\)
\(18\) 0 0
\(19\) −89.0628 −0.246711 −0.123356 0.992363i \(-0.539366\pi\)
−0.123356 + 0.992363i \(0.539366\pi\)
\(20\) 278.334i 0.695834i
\(21\) 0 0
\(22\) 85.2137 0.176061
\(23\) 23.6671i 0.0447393i 0.999750 + 0.0223697i \(0.00712108\pi\)
−0.999750 + 0.0223697i \(0.992879\pi\)
\(24\) 0 0
\(25\) −585.464 −0.936742
\(26\) 404.860i 0.598906i
\(27\) 0 0
\(28\) −148.162 −0.188982
\(29\) − 1180.46i − 1.40364i −0.712353 0.701821i \(-0.752371\pi\)
0.712353 0.701821i \(-0.247629\pi\)
\(30\) 0 0
\(31\) 171.591 0.178554 0.0892772 0.996007i \(-0.471544\pi\)
0.0892772 + 0.996007i \(0.471544\pi\)
\(32\) 181.019i 0.176777i
\(33\) 0 0
\(34\) −328.593 −0.284250
\(35\) − 644.352i − 0.526001i
\(36\) 0 0
\(37\) −974.714 −0.711989 −0.355995 0.934488i \(-0.615858\pi\)
−0.355995 + 0.934488i \(0.615858\pi\)
\(38\) − 251.908i − 0.174451i
\(39\) 0 0
\(40\) −787.247 −0.492029
\(41\) − 417.362i − 0.248282i −0.992265 0.124141i \(-0.960382\pi\)
0.992265 0.124141i \(-0.0396175\pi\)
\(42\) 0 0
\(43\) −3148.03 −1.70256 −0.851278 0.524715i \(-0.824172\pi\)
−0.851278 + 0.524715i \(0.824172\pi\)
\(44\) 241.021i 0.124494i
\(45\) 0 0
\(46\) −66.9407 −0.0316355
\(47\) − 2961.00i − 1.34043i −0.742169 0.670213i \(-0.766203\pi\)
0.742169 0.670213i \(-0.233797\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) − 1655.94i − 0.662377i
\(51\) 0 0
\(52\) −1145.12 −0.423491
\(53\) − 2248.40i − 0.800427i −0.916422 0.400214i \(-0.868936\pi\)
0.916422 0.400214i \(-0.131064\pi\)
\(54\) 0 0
\(55\) −1048.19 −0.346509
\(56\) − 419.066i − 0.133631i
\(57\) 0 0
\(58\) 3338.85 0.992525
\(59\) 827.250i 0.237647i 0.992915 + 0.118824i \(0.0379123\pi\)
−0.992915 + 0.118824i \(0.962088\pi\)
\(60\) 0 0
\(61\) −6889.71 −1.85157 −0.925787 0.378046i \(-0.876596\pi\)
−0.925787 + 0.378046i \(0.876596\pi\)
\(62\) 485.332i 0.126257i
\(63\) 0 0
\(64\) −512.000 −0.125000
\(65\) − 4980.08i − 1.17872i
\(66\) 0 0
\(67\) 4866.31 1.08405 0.542026 0.840361i \(-0.317657\pi\)
0.542026 + 0.840361i \(0.317657\pi\)
\(68\) − 929.402i − 0.200995i
\(69\) 0 0
\(70\) 1822.50 0.371939
\(71\) − 5488.08i − 1.08869i −0.838862 0.544344i \(-0.816779\pi\)
0.838862 0.544344i \(-0.183221\pi\)
\(72\) 0 0
\(73\) 777.420 0.145885 0.0729424 0.997336i \(-0.476761\pi\)
0.0729424 + 0.997336i \(0.476761\pi\)
\(74\) − 2756.91i − 0.503453i
\(75\) 0 0
\(76\) 712.502 0.123356
\(77\) − 557.971i − 0.0941087i
\(78\) 0 0
\(79\) 3899.43 0.624808 0.312404 0.949949i \(-0.398866\pi\)
0.312404 + 0.949949i \(0.398866\pi\)
\(80\) − 2226.67i − 0.347917i
\(81\) 0 0
\(82\) 1180.48 0.175562
\(83\) 826.579i 0.119985i 0.998199 + 0.0599927i \(0.0191077\pi\)
−0.998199 + 0.0599927i \(0.980892\pi\)
\(84\) 0 0
\(85\) 4041.94 0.559438
\(86\) − 8903.96i − 1.20389i
\(87\) 0 0
\(88\) −681.709 −0.0880307
\(89\) 1055.44i 0.133246i 0.997778 + 0.0666231i \(0.0212225\pi\)
−0.997778 + 0.0666231i \(0.978777\pi\)
\(90\) 0 0
\(91\) 2650.99 0.320129
\(92\) − 189.337i − 0.0223697i
\(93\) 0 0
\(94\) 8374.98 0.947825
\(95\) 3098.65i 0.343341i
\(96\) 0 0
\(97\) 5713.52 0.607240 0.303620 0.952793i \(-0.401805\pi\)
0.303620 + 0.952793i \(0.401805\pi\)
\(98\) 970.151i 0.101015i
\(99\) 0 0
\(100\) 4683.71 0.468371
\(101\) − 10670.0i − 1.04598i −0.852339 0.522990i \(-0.824817\pi\)
0.852339 0.522990i \(-0.175183\pi\)
\(102\) 0 0
\(103\) −16761.8 −1.57996 −0.789981 0.613132i \(-0.789910\pi\)
−0.789981 + 0.613132i \(0.789910\pi\)
\(104\) − 3238.88i − 0.299453i
\(105\) 0 0
\(106\) 6359.44 0.565987
\(107\) 12800.3i 1.11803i 0.829159 + 0.559013i \(0.188820\pi\)
−0.829159 + 0.559013i \(0.811180\pi\)
\(108\) 0 0
\(109\) −1931.09 −0.162536 −0.0812680 0.996692i \(-0.525897\pi\)
−0.0812680 + 0.996692i \(0.525897\pi\)
\(110\) − 2964.73i − 0.245019i
\(111\) 0 0
\(112\) 1185.30 0.0944911
\(113\) − 19494.9i − 1.52674i −0.645964 0.763368i \(-0.723544\pi\)
0.645964 0.763368i \(-0.276456\pi\)
\(114\) 0 0
\(115\) 823.419 0.0622623
\(116\) 9443.71i 0.701821i
\(117\) 0 0
\(118\) −2339.82 −0.168042
\(119\) 2151.60i 0.151938i
\(120\) 0 0
\(121\) 13733.3 0.938005
\(122\) − 19487.0i − 1.30926i
\(123\) 0 0
\(124\) −1372.73 −0.0892772
\(125\) − 1375.53i − 0.0880338i
\(126\) 0 0
\(127\) −27254.3 −1.68977 −0.844885 0.534949i \(-0.820331\pi\)
−0.844885 + 0.534949i \(0.820331\pi\)
\(128\) − 1448.15i − 0.0883883i
\(129\) 0 0
\(130\) 14085.8 0.833479
\(131\) 17965.1i 1.04686i 0.852069 + 0.523429i \(0.175347\pi\)
−0.852069 + 0.523429i \(0.824653\pi\)
\(132\) 0 0
\(133\) −1649.47 −0.0932481
\(134\) 13764.0i 0.766541i
\(135\) 0 0
\(136\) 2628.75 0.142125
\(137\) − 29725.1i − 1.58373i −0.610693 0.791867i \(-0.709109\pi\)
0.610693 0.791867i \(-0.290891\pi\)
\(138\) 0 0
\(139\) −12266.3 −0.634871 −0.317435 0.948280i \(-0.602822\pi\)
−0.317435 + 0.948280i \(0.602822\pi\)
\(140\) 5154.81i 0.263001i
\(141\) 0 0
\(142\) 15522.6 0.769819
\(143\) − 4312.46i − 0.210888i
\(144\) 0 0
\(145\) −41070.3 −1.95341
\(146\) 2198.88i 0.103156i
\(147\) 0 0
\(148\) 7797.71 0.355995
\(149\) 2566.34i 0.115596i 0.998328 + 0.0577978i \(0.0184079\pi\)
−0.998328 + 0.0577978i \(0.981592\pi\)
\(150\) 0 0
\(151\) 6919.18 0.303459 0.151730 0.988422i \(-0.451516\pi\)
0.151730 + 0.988422i \(0.451516\pi\)
\(152\) 2015.26i 0.0872256i
\(153\) 0 0
\(154\) 1578.18 0.0665449
\(155\) − 5969.94i − 0.248488i
\(156\) 0 0
\(157\) −1569.92 −0.0636912 −0.0318456 0.999493i \(-0.510138\pi\)
−0.0318456 + 0.999493i \(0.510138\pi\)
\(158\) 11029.2i 0.441806i
\(159\) 0 0
\(160\) 6297.97 0.246015
\(161\) 438.321i 0.0169099i
\(162\) 0 0
\(163\) 27851.1 1.04826 0.524128 0.851639i \(-0.324391\pi\)
0.524128 + 0.851639i \(0.324391\pi\)
\(164\) 3338.90i 0.124141i
\(165\) 0 0
\(166\) −2337.92 −0.0848425
\(167\) − 40958.1i − 1.46861i −0.678819 0.734306i \(-0.737508\pi\)
0.678819 0.734306i \(-0.262492\pi\)
\(168\) 0 0
\(169\) −8072.00 −0.282623
\(170\) 11432.3i 0.395582i
\(171\) 0 0
\(172\) 25184.2 0.851278
\(173\) − 13024.4i − 0.435175i −0.976041 0.217588i \(-0.930181\pi\)
0.976041 0.217588i \(-0.0698188\pi\)
\(174\) 0 0
\(175\) −10842.9 −0.354055
\(176\) − 1928.17i − 0.0622471i
\(177\) 0 0
\(178\) −2985.24 −0.0942193
\(179\) 35142.5i 1.09680i 0.836217 + 0.548399i \(0.184762\pi\)
−0.836217 + 0.548399i \(0.815238\pi\)
\(180\) 0 0
\(181\) −14046.6 −0.428760 −0.214380 0.976750i \(-0.568773\pi\)
−0.214380 + 0.976750i \(0.568773\pi\)
\(182\) 7498.12i 0.226365i
\(183\) 0 0
\(184\) 535.525 0.0158177
\(185\) 33912.0i 0.990854i
\(186\) 0 0
\(187\) 3500.08 0.100091
\(188\) 23688.0i 0.670213i
\(189\) 0 0
\(190\) −8764.30 −0.242778
\(191\) 38790.0i 1.06329i 0.846966 + 0.531647i \(0.178426\pi\)
−0.846966 + 0.531647i \(0.821574\pi\)
\(192\) 0 0
\(193\) 3790.94 0.101773 0.0508864 0.998704i \(-0.483795\pi\)
0.0508864 + 0.998704i \(0.483795\pi\)
\(194\) 16160.3i 0.429383i
\(195\) 0 0
\(196\) −2744.00 −0.0714286
\(197\) 17172.3i 0.442481i 0.975219 + 0.221241i \(0.0710106\pi\)
−0.975219 + 0.221241i \(0.928989\pi\)
\(198\) 0 0
\(199\) −33411.7 −0.843708 −0.421854 0.906664i \(-0.638621\pi\)
−0.421854 + 0.906664i \(0.638621\pi\)
\(200\) 13247.5i 0.331188i
\(201\) 0 0
\(202\) 30179.4 0.739619
\(203\) − 21862.5i − 0.530527i
\(204\) 0 0
\(205\) −14520.7 −0.345526
\(206\) − 47409.6i − 1.11720i
\(207\) 0 0
\(208\) 9160.95 0.211745
\(209\) 2683.25i 0.0614282i
\(210\) 0 0
\(211\) 44323.4 0.995563 0.497781 0.867303i \(-0.334148\pi\)
0.497781 + 0.867303i \(0.334148\pi\)
\(212\) 17987.2i 0.400214i
\(213\) 0 0
\(214\) −36204.7 −0.790564
\(215\) 109525.i 2.36939i
\(216\) 0 0
\(217\) 3177.90 0.0674872
\(218\) − 5461.95i − 0.114930i
\(219\) 0 0
\(220\) 8385.53 0.173255
\(221\) 16629.3i 0.340478i
\(222\) 0 0
\(223\) 12601.5 0.253404 0.126702 0.991941i \(-0.459561\pi\)
0.126702 + 0.991941i \(0.459561\pi\)
\(224\) 3352.53i 0.0668153i
\(225\) 0 0
\(226\) 55139.9 1.07957
\(227\) 38690.2i 0.750844i 0.926854 + 0.375422i \(0.122502\pi\)
−0.926854 + 0.375422i \(0.877498\pi\)
\(228\) 0 0
\(229\) −72276.5 −1.37824 −0.689122 0.724645i \(-0.742004\pi\)
−0.689122 + 0.724645i \(0.742004\pi\)
\(230\) 2328.98i 0.0440261i
\(231\) 0 0
\(232\) −26710.8 −0.496262
\(233\) 63243.2i 1.16494i 0.812853 + 0.582468i \(0.197913\pi\)
−0.812853 + 0.582468i \(0.802087\pi\)
\(234\) 0 0
\(235\) −103018. −1.86543
\(236\) − 6618.00i − 0.118824i
\(237\) 0 0
\(238\) −6085.63 −0.107437
\(239\) − 48102.6i − 0.842118i −0.907033 0.421059i \(-0.861659\pi\)
0.907033 0.421059i \(-0.138341\pi\)
\(240\) 0 0
\(241\) −103824. −1.78757 −0.893786 0.448494i \(-0.851961\pi\)
−0.893786 + 0.448494i \(0.851961\pi\)
\(242\) 38843.7i 0.663270i
\(243\) 0 0
\(244\) 55117.6 0.925787
\(245\) − 11933.6i − 0.198810i
\(246\) 0 0
\(247\) −12748.4 −0.208960
\(248\) − 3882.65i − 0.0631285i
\(249\) 0 0
\(250\) 3890.58 0.0622493
\(251\) 55160.0i 0.875542i 0.899087 + 0.437771i \(0.144232\pi\)
−0.899087 + 0.437771i \(0.855768\pi\)
\(252\) 0 0
\(253\) 713.032 0.0111396
\(254\) − 77086.8i − 1.19485i
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) 22410.6i 0.339303i 0.985504 + 0.169652i \(0.0542643\pi\)
−0.985504 + 0.169652i \(0.945736\pi\)
\(258\) 0 0
\(259\) −18051.9 −0.269107
\(260\) 39840.6i 0.589359i
\(261\) 0 0
\(262\) −50813.0 −0.740240
\(263\) − 5733.50i − 0.0828911i −0.999141 0.0414456i \(-0.986804\pi\)
0.999141 0.0414456i \(-0.0131963\pi\)
\(264\) 0 0
\(265\) −78225.7 −1.11393
\(266\) − 4665.39i − 0.0659364i
\(267\) 0 0
\(268\) −38930.5 −0.542026
\(269\) 122046.i 1.68663i 0.537422 + 0.843314i \(0.319398\pi\)
−0.537422 + 0.843314i \(0.680602\pi\)
\(270\) 0 0
\(271\) 138859. 1.89076 0.945378 0.325975i \(-0.105693\pi\)
0.945378 + 0.325975i \(0.105693\pi\)
\(272\) 7435.22i 0.100498i
\(273\) 0 0
\(274\) 84075.3 1.11987
\(275\) 17638.6i 0.233238i
\(276\) 0 0
\(277\) −6187.42 −0.0806400 −0.0403200 0.999187i \(-0.512838\pi\)
−0.0403200 + 0.999187i \(0.512838\pi\)
\(278\) − 34694.4i − 0.448921i
\(279\) 0 0
\(280\) −14580.0 −0.185970
\(281\) − 99067.2i − 1.25463i −0.778764 0.627317i \(-0.784153\pi\)
0.778764 0.627317i \(-0.215847\pi\)
\(282\) 0 0
\(283\) 20879.0 0.260697 0.130349 0.991468i \(-0.458390\pi\)
0.130349 + 0.991468i \(0.458390\pi\)
\(284\) 43904.6i 0.544344i
\(285\) 0 0
\(286\) 12197.5 0.149121
\(287\) − 7729.66i − 0.0938418i
\(288\) 0 0
\(289\) 70024.3 0.838404
\(290\) − 116164.i − 1.38127i
\(291\) 0 0
\(292\) −6219.36 −0.0729424
\(293\) − 33461.8i − 0.389775i −0.980826 0.194888i \(-0.937566\pi\)
0.980826 0.194888i \(-0.0624342\pi\)
\(294\) 0 0
\(295\) 28781.5 0.330726
\(296\) 22055.2i 0.251726i
\(297\) 0 0
\(298\) −7258.70 −0.0817384
\(299\) 3387.70i 0.0378933i
\(300\) 0 0
\(301\) −58302.3 −0.643506
\(302\) 19570.4i 0.214578i
\(303\) 0 0
\(304\) −5700.02 −0.0616778
\(305\) 239705.i 2.57678i
\(306\) 0 0
\(307\) 151671. 1.60926 0.804630 0.593776i \(-0.202364\pi\)
0.804630 + 0.593776i \(0.202364\pi\)
\(308\) 4463.77i 0.0470544i
\(309\) 0 0
\(310\) 16885.5 0.175708
\(311\) − 88401.3i − 0.913982i −0.889471 0.456991i \(-0.848927\pi\)
0.889471 0.456991i \(-0.151073\pi\)
\(312\) 0 0
\(313\) 152618. 1.55782 0.778911 0.627135i \(-0.215772\pi\)
0.778911 + 0.627135i \(0.215772\pi\)
\(314\) − 4440.41i − 0.0450364i
\(315\) 0 0
\(316\) −31195.4 −0.312404
\(317\) − 33405.1i − 0.332426i −0.986090 0.166213i \(-0.946846\pi\)
0.986090 0.166213i \(-0.0531539\pi\)
\(318\) 0 0
\(319\) −35564.5 −0.349491
\(320\) 17813.4i 0.173959i
\(321\) 0 0
\(322\) −1239.76 −0.0119571
\(323\) − 10346.9i − 0.0991756i
\(324\) 0 0
\(325\) −83803.2 −0.793403
\(326\) 78774.9i 0.741229i
\(327\) 0 0
\(328\) −9443.83 −0.0877810
\(329\) − 54838.5i − 0.506634i
\(330\) 0 0
\(331\) 208993. 1.90755 0.953773 0.300529i \(-0.0971631\pi\)
0.953773 + 0.300529i \(0.0971631\pi\)
\(332\) − 6612.63i − 0.0599927i
\(333\) 0 0
\(334\) 115847. 1.03847
\(335\) − 169307.i − 1.50864i
\(336\) 0 0
\(337\) 203893. 1.79532 0.897659 0.440690i \(-0.145266\pi\)
0.897659 + 0.440690i \(0.145266\pi\)
\(338\) − 22831.1i − 0.199845i
\(339\) 0 0
\(340\) −32335.5 −0.279719
\(341\) − 5169.61i − 0.0444579i
\(342\) 0 0
\(343\) 6352.45 0.0539949
\(344\) 71231.7i 0.601945i
\(345\) 0 0
\(346\) 36838.4 0.307715
\(347\) − 43169.0i − 0.358520i −0.983802 0.179260i \(-0.942630\pi\)
0.983802 0.179260i \(-0.0573703\pi\)
\(348\) 0 0
\(349\) 203110. 1.66755 0.833777 0.552101i \(-0.186174\pi\)
0.833777 + 0.552101i \(0.186174\pi\)
\(350\) − 30668.5i − 0.250355i
\(351\) 0 0
\(352\) 5453.68 0.0440153
\(353\) − 173776.i − 1.39457i −0.716796 0.697283i \(-0.754392\pi\)
0.716796 0.697283i \(-0.245608\pi\)
\(354\) 0 0
\(355\) −190940. −1.51509
\(356\) − 8443.54i − 0.0666231i
\(357\) 0 0
\(358\) −99398.0 −0.775553
\(359\) − 190257.i − 1.47622i −0.674680 0.738110i \(-0.735718\pi\)
0.674680 0.738110i \(-0.264282\pi\)
\(360\) 0 0
\(361\) −122389. −0.939134
\(362\) − 39729.8i − 0.303179i
\(363\) 0 0
\(364\) −21207.9 −0.160064
\(365\) − 27047.8i − 0.203023i
\(366\) 0 0
\(367\) 147637. 1.09613 0.548067 0.836434i \(-0.315364\pi\)
0.548067 + 0.836434i \(0.315364\pi\)
\(368\) 1514.69i 0.0111848i
\(369\) 0 0
\(370\) −95917.5 −0.700639
\(371\) − 41640.9i − 0.302533i
\(372\) 0 0
\(373\) −221469. −1.59182 −0.795912 0.605413i \(-0.793008\pi\)
−0.795912 + 0.605413i \(0.793008\pi\)
\(374\) 9899.72i 0.0707750i
\(375\) 0 0
\(376\) −66999.8 −0.473912
\(377\) − 168971.i − 1.18886i
\(378\) 0 0
\(379\) 100936. 0.702693 0.351347 0.936245i \(-0.385724\pi\)
0.351347 + 0.936245i \(0.385724\pi\)
\(380\) − 24789.2i − 0.171670i
\(381\) 0 0
\(382\) −109715. −0.751862
\(383\) − 111462.i − 0.759854i −0.925016 0.379927i \(-0.875949\pi\)
0.925016 0.379927i \(-0.124051\pi\)
\(384\) 0 0
\(385\) −19412.8 −0.130968
\(386\) 10722.4i 0.0719643i
\(387\) 0 0
\(388\) −45708.1 −0.303620
\(389\) 208045.i 1.37486i 0.726250 + 0.687430i \(0.241261\pi\)
−0.726250 + 0.687430i \(0.758739\pi\)
\(390\) 0 0
\(391\) −2749.53 −0.0179848
\(392\) − 7761.20i − 0.0505076i
\(393\) 0 0
\(394\) −48570.5 −0.312882
\(395\) − 135668.i − 0.869526i
\(396\) 0 0
\(397\) −83732.9 −0.531270 −0.265635 0.964074i \(-0.585582\pi\)
−0.265635 + 0.964074i \(0.585582\pi\)
\(398\) − 94502.5i − 0.596592i
\(399\) 0 0
\(400\) −37469.7 −0.234186
\(401\) 196215.i 1.22023i 0.792312 + 0.610117i \(0.208877\pi\)
−0.792312 + 0.610117i \(0.791123\pi\)
\(402\) 0 0
\(403\) 24561.5 0.151232
\(404\) 85360.3i 0.522990i
\(405\) 0 0
\(406\) 61836.4 0.375139
\(407\) 29365.8i 0.177277i
\(408\) 0 0
\(409\) −109093. −0.652153 −0.326077 0.945343i \(-0.605727\pi\)
−0.326077 + 0.945343i \(0.605727\pi\)
\(410\) − 41070.9i − 0.244324i
\(411\) 0 0
\(412\) 134094. 0.789981
\(413\) 15320.9i 0.0898222i
\(414\) 0 0
\(415\) 28758.1 0.166980
\(416\) 25911.1i 0.149727i
\(417\) 0 0
\(418\) −7589.37 −0.0434363
\(419\) − 115118.i − 0.655715i −0.944727 0.327858i \(-0.893673\pi\)
0.944727 0.327858i \(-0.106327\pi\)
\(420\) 0 0
\(421\) 41327.3 0.233170 0.116585 0.993181i \(-0.462805\pi\)
0.116585 + 0.993181i \(0.462805\pi\)
\(422\) 125366.i 0.703969i
\(423\) 0 0
\(424\) −50875.5 −0.282994
\(425\) − 68016.4i − 0.376562i
\(426\) 0 0
\(427\) −127599. −0.699829
\(428\) − 102402.i − 0.559013i
\(429\) 0 0
\(430\) −309784. −1.67541
\(431\) − 69512.7i − 0.374205i −0.982340 0.187103i \(-0.940090\pi\)
0.982340 0.187103i \(-0.0599097\pi\)
\(432\) 0 0
\(433\) −93109.8 −0.496615 −0.248307 0.968681i \(-0.579874\pi\)
−0.248307 + 0.968681i \(0.579874\pi\)
\(434\) 8988.47i 0.0477206i
\(435\) 0 0
\(436\) 15448.7 0.0812680
\(437\) − 2107.86i − 0.0110377i
\(438\) 0 0
\(439\) 77844.3 0.403922 0.201961 0.979394i \(-0.435269\pi\)
0.201961 + 0.979394i \(0.435269\pi\)
\(440\) 23717.8i 0.122510i
\(441\) 0 0
\(442\) −47034.8 −0.240755
\(443\) 166627.i 0.849058i 0.905414 + 0.424529i \(0.139560\pi\)
−0.905414 + 0.424529i \(0.860440\pi\)
\(444\) 0 0
\(445\) 36720.7 0.185435
\(446\) 35642.5i 0.179184i
\(447\) 0 0
\(448\) −9482.37 −0.0472456
\(449\) − 9590.42i − 0.0475713i −0.999717 0.0237856i \(-0.992428\pi\)
0.999717 0.0237856i \(-0.00757192\pi\)
\(450\) 0 0
\(451\) −12574.1 −0.0618193
\(452\) 155959.i 0.763368i
\(453\) 0 0
\(454\) −109432. −0.530927
\(455\) − 92232.4i − 0.445513i
\(456\) 0 0
\(457\) 20636.1 0.0988086 0.0494043 0.998779i \(-0.484268\pi\)
0.0494043 + 0.998779i \(0.484268\pi\)
\(458\) − 204429.i − 0.974566i
\(459\) 0 0
\(460\) −6587.35 −0.0311312
\(461\) − 367647.i − 1.72993i −0.501831 0.864966i \(-0.667340\pi\)
0.501831 0.864966i \(-0.332660\pi\)
\(462\) 0 0
\(463\) 155758. 0.726588 0.363294 0.931675i \(-0.381652\pi\)
0.363294 + 0.931675i \(0.381652\pi\)
\(464\) − 75549.6i − 0.350911i
\(465\) 0 0
\(466\) −178879. −0.823734
\(467\) 297597.i 1.36457i 0.731087 + 0.682284i \(0.239013\pi\)
−0.731087 + 0.682284i \(0.760987\pi\)
\(468\) 0 0
\(469\) 90125.4 0.409733
\(470\) − 291380.i − 1.31906i
\(471\) 0 0
\(472\) 18718.5 0.0840210
\(473\) 94842.4i 0.423917i
\(474\) 0 0
\(475\) 52143.1 0.231105
\(476\) − 17212.8i − 0.0759691i
\(477\) 0 0
\(478\) 136055. 0.595467
\(479\) − 203202.i − 0.885639i −0.896611 0.442819i \(-0.853978\pi\)
0.896611 0.442819i \(-0.146022\pi\)
\(480\) 0 0
\(481\) −139520. −0.603042
\(482\) − 293658.i − 1.26400i
\(483\) 0 0
\(484\) −109867. −0.469002
\(485\) − 198783.i − 0.845076i
\(486\) 0 0
\(487\) 312755. 1.31870 0.659351 0.751835i \(-0.270831\pi\)
0.659351 + 0.751835i \(0.270831\pi\)
\(488\) 155896.i 0.654630i
\(489\) 0 0
\(490\) 33753.2 0.140580
\(491\) − 26245.9i − 0.108867i −0.998517 0.0544337i \(-0.982665\pi\)
0.998517 0.0544337i \(-0.0173354\pi\)
\(492\) 0 0
\(493\) 137141. 0.564251
\(494\) − 36058.0i − 0.147757i
\(495\) 0 0
\(496\) 10981.8 0.0446386
\(497\) − 101641.i − 0.411485i
\(498\) 0 0
\(499\) 137671. 0.552891 0.276446 0.961030i \(-0.410843\pi\)
0.276446 + 0.961030i \(0.410843\pi\)
\(500\) 11004.2i 0.0440169i
\(501\) 0 0
\(502\) −156016. −0.619102
\(503\) − 336996.i − 1.33195i −0.745973 0.665977i \(-0.768015\pi\)
0.745973 0.665977i \(-0.231985\pi\)
\(504\) 0 0
\(505\) −371229. −1.45566
\(506\) 2016.76i 0.00787686i
\(507\) 0 0
\(508\) 218034. 0.844885
\(509\) 299188.i 1.15480i 0.816460 + 0.577402i \(0.195933\pi\)
−0.816460 + 0.577402i \(0.804067\pi\)
\(510\) 0 0
\(511\) 14398.0 0.0551393
\(512\) 11585.2i 0.0441942i
\(513\) 0 0
\(514\) −63386.9 −0.239924
\(515\) 583172.i 2.19878i
\(516\) 0 0
\(517\) −89207.9 −0.333751
\(518\) − 51058.6i − 0.190287i
\(519\) 0 0
\(520\) −112686. −0.416739
\(521\) − 248631.i − 0.915967i −0.888961 0.457983i \(-0.848572\pi\)
0.888961 0.457983i \(-0.151428\pi\)
\(522\) 0 0
\(523\) 45984.3 0.168115 0.0840574 0.996461i \(-0.473212\pi\)
0.0840574 + 0.996461i \(0.473212\pi\)
\(524\) − 143721.i − 0.523429i
\(525\) 0 0
\(526\) 16216.8 0.0586129
\(527\) 19934.6i 0.0717772i
\(528\) 0 0
\(529\) 279281. 0.997998
\(530\) − 221256.i − 0.787667i
\(531\) 0 0
\(532\) 13195.7 0.0466241
\(533\) − 59741.1i − 0.210290i
\(534\) 0 0
\(535\) 445344. 1.55592
\(536\) − 110112.i − 0.383271i
\(537\) 0 0
\(538\) −345198. −1.19263
\(539\) − 10333.8i − 0.0355698i
\(540\) 0 0
\(541\) 306453. 1.04705 0.523527 0.852009i \(-0.324616\pi\)
0.523527 + 0.852009i \(0.324616\pi\)
\(542\) 392753.i 1.33697i
\(543\) 0 0
\(544\) −21030.0 −0.0710626
\(545\) 67185.9i 0.226196i
\(546\) 0 0
\(547\) −17480.8 −0.0584234 −0.0292117 0.999573i \(-0.509300\pi\)
−0.0292117 + 0.999573i \(0.509300\pi\)
\(548\) 237801.i 0.791867i
\(549\) 0 0
\(550\) −49889.5 −0.164924
\(551\) 105135.i 0.346294i
\(552\) 0 0
\(553\) 72218.4 0.236155
\(554\) − 17500.7i − 0.0570211i
\(555\) 0 0
\(556\) 98130.7 0.317435
\(557\) 500038.i 1.61173i 0.592099 + 0.805866i \(0.298300\pi\)
−0.592099 + 0.805866i \(0.701700\pi\)
\(558\) 0 0
\(559\) −450608. −1.44203
\(560\) − 41238.5i − 0.131500i
\(561\) 0 0
\(562\) 280204. 0.887160
\(563\) 40744.8i 0.128545i 0.997932 + 0.0642725i \(0.0204727\pi\)
−0.997932 + 0.0642725i \(0.979527\pi\)
\(564\) 0 0
\(565\) −678261. −2.12471
\(566\) 59054.7i 0.184341i
\(567\) 0 0
\(568\) −124181. −0.384909
\(569\) 527036.i 1.62786i 0.580966 + 0.813928i \(0.302675\pi\)
−0.580966 + 0.813928i \(0.697325\pi\)
\(570\) 0 0
\(571\) 393264. 1.20618 0.603090 0.797673i \(-0.293936\pi\)
0.603090 + 0.797673i \(0.293936\pi\)
\(572\) 34499.7i 0.105444i
\(573\) 0 0
\(574\) 21862.8 0.0663562
\(575\) − 13856.2i − 0.0419092i
\(576\) 0 0
\(577\) −92157.9 −0.276810 −0.138405 0.990376i \(-0.544197\pi\)
−0.138405 + 0.990376i \(0.544197\pi\)
\(578\) 198059.i 0.592841i
\(579\) 0 0
\(580\) 328563. 0.976703
\(581\) 15308.5i 0.0453502i
\(582\) 0 0
\(583\) −67738.9 −0.199297
\(584\) − 17591.0i − 0.0515781i
\(585\) 0 0
\(586\) 94644.3 0.275613
\(587\) 643433.i 1.86736i 0.358112 + 0.933679i \(0.383421\pi\)
−0.358112 + 0.933679i \(0.616579\pi\)
\(588\) 0 0
\(589\) −15282.3 −0.0440514
\(590\) 81406.3i 0.233859i
\(591\) 0 0
\(592\) −62381.7 −0.177997
\(593\) − 351311.i − 0.999038i −0.866303 0.499519i \(-0.833510\pi\)
0.866303 0.499519i \(-0.166490\pi\)
\(594\) 0 0
\(595\) 74857.7 0.211448
\(596\) − 20530.7i − 0.0577978i
\(597\) 0 0
\(598\) −9581.87 −0.0267946
\(599\) 251755.i 0.701656i 0.936440 + 0.350828i \(0.114100\pi\)
−0.936440 + 0.350828i \(0.885900\pi\)
\(600\) 0 0
\(601\) 195702. 0.541810 0.270905 0.962606i \(-0.412677\pi\)
0.270905 + 0.962606i \(0.412677\pi\)
\(602\) − 164904.i − 0.455027i
\(603\) 0 0
\(604\) −55353.4 −0.151730
\(605\) − 477806.i − 1.30539i
\(606\) 0 0
\(607\) 15312.0 0.0415581 0.0207790 0.999784i \(-0.493385\pi\)
0.0207790 + 0.999784i \(0.493385\pi\)
\(608\) − 16122.1i − 0.0436128i
\(609\) 0 0
\(610\) −677987. −1.82206
\(611\) − 423837.i − 1.13532i
\(612\) 0 0
\(613\) −517619. −1.37749 −0.688747 0.725002i \(-0.741839\pi\)
−0.688747 + 0.725002i \(0.741839\pi\)
\(614\) 428991.i 1.13792i
\(615\) 0 0
\(616\) −12625.4 −0.0332725
\(617\) − 79270.8i − 0.208230i −0.994565 0.104115i \(-0.966799\pi\)
0.994565 0.104115i \(-0.0332010\pi\)
\(618\) 0 0
\(619\) 454097. 1.18513 0.592567 0.805521i \(-0.298115\pi\)
0.592567 + 0.805521i \(0.298115\pi\)
\(620\) 47759.5i 0.124244i
\(621\) 0 0
\(622\) 250037. 0.646283
\(623\) 19547.1i 0.0503623i
\(624\) 0 0
\(625\) −413772. −1.05926
\(626\) 431670.i 1.10155i
\(627\) 0 0
\(628\) 12559.4 0.0318456
\(629\) − 113238.i − 0.286213i
\(630\) 0 0
\(631\) 589600. 1.48081 0.740404 0.672162i \(-0.234634\pi\)
0.740404 + 0.672162i \(0.234634\pi\)
\(632\) − 88234.0i − 0.220903i
\(633\) 0 0
\(634\) 94483.9 0.235060
\(635\) 948224.i 2.35160i
\(636\) 0 0
\(637\) 49097.0 0.120997
\(638\) − 100592.i − 0.247127i
\(639\) 0 0
\(640\) −50383.8 −0.123007
\(641\) 784231.i 1.90866i 0.298759 + 0.954329i \(0.403427\pi\)
−0.298759 + 0.954329i \(0.596573\pi\)
\(642\) 0 0
\(643\) −2503.23 −0.00605452 −0.00302726 0.999995i \(-0.500964\pi\)
−0.00302726 + 0.999995i \(0.500964\pi\)
\(644\) − 3506.57i − 0.00845493i
\(645\) 0 0
\(646\) 29265.4 0.0701278
\(647\) − 661672.i − 1.58064i −0.612691 0.790322i \(-0.709913\pi\)
0.612691 0.790322i \(-0.290087\pi\)
\(648\) 0 0
\(649\) 24923.1 0.0591714
\(650\) − 237031.i − 0.561021i
\(651\) 0 0
\(652\) −222809. −0.524128
\(653\) 3986.16i 0.00934821i 0.999989 + 0.00467410i \(0.00148782\pi\)
−0.999989 + 0.00467410i \(0.998512\pi\)
\(654\) 0 0
\(655\) 625038. 1.45688
\(656\) − 26711.2i − 0.0620705i
\(657\) 0 0
\(658\) 155107. 0.358244
\(659\) 124318.i 0.286261i 0.989704 + 0.143130i \(0.0457168\pi\)
−0.989704 + 0.143130i \(0.954283\pi\)
\(660\) 0 0
\(661\) 112154. 0.256691 0.128346 0.991729i \(-0.459033\pi\)
0.128346 + 0.991729i \(0.459033\pi\)
\(662\) 591120.i 1.34884i
\(663\) 0 0
\(664\) 18703.4 0.0424212
\(665\) 57387.8i 0.129771i
\(666\) 0 0
\(667\) 27938.1 0.0627980
\(668\) 327665.i 0.734306i
\(669\) 0 0
\(670\) 478874. 1.06677
\(671\) 207570.i 0.461020i
\(672\) 0 0
\(673\) 1349.83 0.00298023 0.00149012 0.999999i \(-0.499526\pi\)
0.00149012 + 0.999999i \(0.499526\pi\)
\(674\) 576695.i 1.26948i
\(675\) 0 0
\(676\) 64576.0 0.141312
\(677\) − 468915.i − 1.02310i −0.859254 0.511549i \(-0.829072\pi\)
0.859254 0.511549i \(-0.170928\pi\)
\(678\) 0 0
\(679\) 105816. 0.229515
\(680\) − 91458.6i − 0.197791i
\(681\) 0 0
\(682\) 14621.9 0.0314365
\(683\) 485991.i 1.04181i 0.853616 + 0.520903i \(0.174405\pi\)
−0.853616 + 0.520903i \(0.825595\pi\)
\(684\) 0 0
\(685\) −1.03419e6 −2.20403
\(686\) 17967.4i 0.0381802i
\(687\) 0 0
\(688\) −201474. −0.425639
\(689\) − 321836.i − 0.677947i
\(690\) 0 0
\(691\) −7451.91 −0.0156067 −0.00780335 0.999970i \(-0.502484\pi\)
−0.00780335 + 0.999970i \(0.502484\pi\)
\(692\) 104195.i 0.217588i
\(693\) 0 0
\(694\) 122100. 0.253512
\(695\) 426767.i 0.883530i
\(696\) 0 0
\(697\) 48487.2 0.0998071
\(698\) 574481.i 1.17914i
\(699\) 0 0
\(700\) 86743.6 0.177028
\(701\) − 685495.i − 1.39498i −0.716594 0.697491i \(-0.754300\pi\)
0.716594 0.697491i \(-0.245700\pi\)
\(702\) 0 0
\(703\) 86810.7 0.175656
\(704\) 15425.3i 0.0311235i
\(705\) 0 0
\(706\) 491511. 0.986107
\(707\) − 197612.i − 0.395343i
\(708\) 0 0
\(709\) −372506. −0.741039 −0.370520 0.928825i \(-0.620820\pi\)
−0.370520 + 0.928825i \(0.620820\pi\)
\(710\) − 540059.i − 1.07133i
\(711\) 0 0
\(712\) 23881.9 0.0471096
\(713\) 4061.05i 0.00798840i
\(714\) 0 0
\(715\) −150038. −0.293487
\(716\) − 281140.i − 0.548399i
\(717\) 0 0
\(718\) 538127. 1.04385
\(719\) 981227.i 1.89807i 0.315174 + 0.949034i \(0.397937\pi\)
−0.315174 + 0.949034i \(0.602063\pi\)
\(720\) 0 0
\(721\) −310433. −0.597169
\(722\) − 346168.i − 0.664068i
\(723\) 0 0
\(724\) 112373. 0.214380
\(725\) 691119.i 1.31485i
\(726\) 0 0
\(727\) 290468. 0.549578 0.274789 0.961505i \(-0.411392\pi\)
0.274789 + 0.961505i \(0.411392\pi\)
\(728\) − 59985.0i − 0.113183i
\(729\) 0 0
\(730\) 76502.7 0.143559
\(731\) − 365723.i − 0.684412i
\(732\) 0 0
\(733\) −243787. −0.453736 −0.226868 0.973926i \(-0.572849\pi\)
−0.226868 + 0.973926i \(0.572849\pi\)
\(734\) 417581.i 0.775084i
\(735\) 0 0
\(736\) −4284.20 −0.00790887
\(737\) − 146610.i − 0.269916i
\(738\) 0 0
\(739\) 133565. 0.244570 0.122285 0.992495i \(-0.460978\pi\)
0.122285 + 0.992495i \(0.460978\pi\)
\(740\) − 271296.i − 0.495427i
\(741\) 0 0
\(742\) 117778. 0.213923
\(743\) 96297.3i 0.174436i 0.996189 + 0.0872181i \(0.0277977\pi\)
−0.996189 + 0.0872181i \(0.972202\pi\)
\(744\) 0 0
\(745\) 89287.3 0.160871
\(746\) − 626408.i − 1.12559i
\(747\) 0 0
\(748\) −28000.6 −0.0500455
\(749\) 237065.i 0.422574i
\(750\) 0 0
\(751\) −123750. −0.219415 −0.109707 0.993964i \(-0.534991\pi\)
−0.109707 + 0.993964i \(0.534991\pi\)
\(752\) − 189504.i − 0.335107i
\(753\) 0 0
\(754\) 477923. 0.840650
\(755\) − 240730.i − 0.422315i
\(756\) 0 0
\(757\) 644846. 1.12529 0.562645 0.826699i \(-0.309784\pi\)
0.562645 + 0.826699i \(0.309784\pi\)
\(758\) 285489.i 0.496879i
\(759\) 0 0
\(760\) 70114.4 0.121389
\(761\) − 359205.i − 0.620259i −0.950694 0.310129i \(-0.899628\pi\)
0.950694 0.310129i \(-0.100372\pi\)
\(762\) 0 0
\(763\) −35764.3 −0.0614328
\(764\) − 310320.i − 0.531647i
\(765\) 0 0
\(766\) 315263. 0.537298
\(767\) 118412.i 0.201283i
\(768\) 0 0
\(769\) −390626. −0.660554 −0.330277 0.943884i \(-0.607142\pi\)
−0.330277 + 0.943884i \(0.607142\pi\)
\(770\) − 54907.6i − 0.0926085i
\(771\) 0 0
\(772\) −30327.5 −0.0508864
\(773\) 314838.i 0.526900i 0.964673 + 0.263450i \(0.0848604\pi\)
−0.964673 + 0.263450i \(0.915140\pi\)
\(774\) 0 0
\(775\) −100460. −0.167259
\(776\) − 129282.i − 0.214692i
\(777\) 0 0
\(778\) −588441. −0.972174
\(779\) 37171.4i 0.0612540i
\(780\) 0 0
\(781\) −165343. −0.271071
\(782\) − 7776.85i − 0.0127172i
\(783\) 0 0
\(784\) 21952.0 0.0357143
\(785\) 54620.3i 0.0886370i
\(786\) 0 0
\(787\) −691316. −1.11616 −0.558081 0.829787i \(-0.688462\pi\)
−0.558081 + 0.829787i \(0.688462\pi\)
\(788\) − 137378.i − 0.221241i
\(789\) 0 0
\(790\) 383727. 0.614848
\(791\) − 361050.i − 0.577052i
\(792\) 0 0
\(793\) −986191. −1.56825
\(794\) − 236832.i − 0.375664i
\(795\) 0 0
\(796\) 267294. 0.421854
\(797\) − 667905.i − 1.05147i −0.850647 0.525737i \(-0.823790\pi\)
0.850647 0.525737i \(-0.176210\pi\)
\(798\) 0 0
\(799\) 343995. 0.538839
\(800\) − 105980.i − 0.165594i
\(801\) 0 0
\(802\) −554979. −0.862835
\(803\) − 23421.8i − 0.0363236i
\(804\) 0 0
\(805\) 15249.9 0.0235329
\(806\) 69470.3i 0.106937i
\(807\) 0 0
\(808\) −241435. −0.369810
\(809\) − 185296.i − 0.283120i −0.989930 0.141560i \(-0.954788\pi\)
0.989930 0.141560i \(-0.0452118\pi\)
\(810\) 0 0
\(811\) −902916. −1.37279 −0.686397 0.727227i \(-0.740809\pi\)
−0.686397 + 0.727227i \(0.740809\pi\)
\(812\) 174900.i 0.265263i
\(813\) 0 0
\(814\) −83058.9 −0.125354
\(815\) − 968989.i − 1.45883i
\(816\) 0 0
\(817\) 280372. 0.420040
\(818\) − 308561.i − 0.461142i
\(819\) 0 0
\(820\) 116166. 0.172763
\(821\) 697811.i 1.03526i 0.855603 + 0.517632i \(0.173187\pi\)
−0.855603 + 0.517632i \(0.826813\pi\)
\(822\) 0 0
\(823\) −1.07628e6 −1.58900 −0.794501 0.607263i \(-0.792268\pi\)
−0.794501 + 0.607263i \(0.792268\pi\)
\(824\) 379276.i 0.558601i
\(825\) 0 0
\(826\) −43334.0 −0.0635139
\(827\) 224433.i 0.328153i 0.986448 + 0.164076i \(0.0524643\pi\)
−0.986448 + 0.164076i \(0.947536\pi\)
\(828\) 0 0
\(829\) 211296. 0.307455 0.153727 0.988113i \(-0.450872\pi\)
0.153727 + 0.988113i \(0.450872\pi\)
\(830\) 81340.2i 0.118073i
\(831\) 0 0
\(832\) −73287.6 −0.105873
\(833\) 39848.1i 0.0574272i
\(834\) 0 0
\(835\) −1.42500e6 −2.04382
\(836\) − 21466.0i − 0.0307141i
\(837\) 0 0
\(838\) 325603. 0.463661
\(839\) − 244959.i − 0.347992i −0.984746 0.173996i \(-0.944332\pi\)
0.984746 0.173996i \(-0.0556679\pi\)
\(840\) 0 0
\(841\) −686212. −0.970212
\(842\) 116891.i 0.164876i
\(843\) 0 0
\(844\) −354588. −0.497781
\(845\) 280839.i 0.393318i
\(846\) 0 0
\(847\) 254345. 0.354532
\(848\) − 143898.i − 0.200107i
\(849\) 0 0
\(850\) 192380. 0.266269
\(851\) − 23068.6i − 0.0318539i
\(852\) 0 0
\(853\) −608980. −0.836961 −0.418480 0.908226i \(-0.637437\pi\)
−0.418480 + 0.908226i \(0.637437\pi\)
\(854\) − 360905.i − 0.494854i
\(855\) 0 0
\(856\) 289638. 0.395282
\(857\) 1.24963e6i 1.70145i 0.525613 + 0.850724i \(0.323836\pi\)
−0.525613 + 0.850724i \(0.676164\pi\)
\(858\) 0 0
\(859\) 510641. 0.692037 0.346019 0.938228i \(-0.387533\pi\)
0.346019 + 0.938228i \(0.387533\pi\)
\(860\) − 876202.i − 1.18470i
\(861\) 0 0
\(862\) 196612. 0.264603
\(863\) − 828820.i − 1.11286i −0.830896 0.556428i \(-0.812172\pi\)
0.830896 0.556428i \(-0.187828\pi\)
\(864\) 0 0
\(865\) −453140. −0.605620
\(866\) − 263354.i − 0.351160i
\(867\) 0 0
\(868\) −25423.2 −0.0337436
\(869\) − 117480.i − 0.155570i
\(870\) 0 0
\(871\) 696563. 0.918172
\(872\) 43695.6i 0.0574651i
\(873\) 0 0
\(874\) 5961.92 0.00780483
\(875\) − 25475.1i − 0.0332736i
\(876\) 0 0
\(877\) −1.16328e6 −1.51247 −0.756233 0.654302i \(-0.772963\pi\)
−0.756233 + 0.654302i \(0.772963\pi\)
\(878\) 220177.i 0.285616i
\(879\) 0 0
\(880\) −67084.2 −0.0866273
\(881\) − 1.16834e6i − 1.50529i −0.658429 0.752643i \(-0.728779\pi\)
0.658429 0.752643i \(-0.271221\pi\)
\(882\) 0 0
\(883\) 879010. 1.12739 0.563693 0.825985i \(-0.309380\pi\)
0.563693 + 0.825985i \(0.309380\pi\)
\(884\) − 133034.i − 0.170239i
\(885\) 0 0
\(886\) −471292. −0.600375
\(887\) 715518.i 0.909438i 0.890635 + 0.454719i \(0.150260\pi\)
−0.890635 + 0.454719i \(0.849740\pi\)
\(888\) 0 0
\(889\) −504756. −0.638673
\(890\) 103862.i 0.131122i
\(891\) 0 0
\(892\) −100812. −0.126702
\(893\) 263715.i 0.330698i
\(894\) 0 0
\(895\) 1.22267e6 1.52638
\(896\) − 26820.2i − 0.0334077i
\(897\) 0 0
\(898\) 27125.8 0.0336380
\(899\) − 202556.i − 0.250626i
\(900\) 0 0
\(901\) 261208. 0.321764
\(902\) − 35565.0i − 0.0437129i
\(903\) 0 0
\(904\) −441119. −0.539783
\(905\) 488706.i 0.596692i
\(906\) 0 0
\(907\) −319774. −0.388713 −0.194357 0.980931i \(-0.562262\pi\)
−0.194357 + 0.980931i \(0.562262\pi\)
\(908\) − 309522.i − 0.375422i
\(909\) 0 0
\(910\) 260873. 0.315025
\(911\) − 1.15028e6i − 1.38601i −0.720934 0.693003i \(-0.756287\pi\)
0.720934 0.693003i \(-0.243713\pi\)
\(912\) 0 0
\(913\) 24902.8 0.0298750
\(914\) 58367.7i 0.0698683i
\(915\) 0 0
\(916\) 578212. 0.689122
\(917\) 332719.i 0.395675i
\(918\) 0 0
\(919\) 276537. 0.327433 0.163716 0.986507i \(-0.447652\pi\)
0.163716 + 0.986507i \(0.447652\pi\)
\(920\) − 18631.8i − 0.0220130i
\(921\) 0 0
\(922\) 1.03986e6 1.22325
\(923\) − 785562.i − 0.922098i
\(924\) 0 0
\(925\) 570660. 0.666951
\(926\) 440550.i 0.513775i
\(927\) 0 0
\(928\) 213687. 0.248131
\(929\) − 1.25259e6i − 1.45137i −0.688025 0.725687i \(-0.741522\pi\)
0.688025 0.725687i \(-0.258478\pi\)
\(930\) 0 0
\(931\) −30548.5 −0.0352445
\(932\) − 505946.i − 0.582468i
\(933\) 0 0
\(934\) −841732. −0.964895
\(935\) − 121774.i − 0.139293i
\(936\) 0 0
\(937\) −1.25465e6 −1.42904 −0.714520 0.699615i \(-0.753355\pi\)
−0.714520 + 0.699615i \(0.753355\pi\)
\(938\) 254913.i 0.289725i
\(939\) 0 0
\(940\) 824147. 0.932715
\(941\) − 1.16300e6i − 1.31341i −0.754148 0.656704i \(-0.771950\pi\)
0.754148 0.656704i \(-0.228050\pi\)
\(942\) 0 0
\(943\) 9877.75 0.0111080
\(944\) 52944.0i 0.0594118i
\(945\) 0 0
\(946\) −268255. −0.299754
\(947\) 63206.8i 0.0704797i 0.999379 + 0.0352399i \(0.0112195\pi\)
−0.999379 + 0.0352399i \(0.988780\pi\)
\(948\) 0 0
\(949\) 111280. 0.123562
\(950\) 147483.i 0.163416i
\(951\) 0 0
\(952\) 48685.1 0.0537183
\(953\) − 933092.i − 1.02740i −0.857971 0.513699i \(-0.828275\pi\)
0.857971 0.513699i \(-0.171725\pi\)
\(954\) 0 0
\(955\) 1.34957e6 1.47975
\(956\) 384821.i 0.421059i
\(957\) 0 0
\(958\) 574741. 0.626241
\(959\) − 550517.i − 0.598595i
\(960\) 0 0
\(961\) −894078. −0.968118
\(962\) − 394623.i − 0.426415i
\(963\) 0 0
\(964\) 830592. 0.893786
\(965\) − 131893.i − 0.141634i
\(966\) 0 0
\(967\) 124623. 0.133274 0.0666370 0.997777i \(-0.478773\pi\)
0.0666370 + 0.997777i \(0.478773\pi\)
\(968\) − 310750.i − 0.331635i
\(969\) 0 0
\(970\) 562244. 0.597559
\(971\) 514717.i 0.545922i 0.962025 + 0.272961i \(0.0880029\pi\)
−0.962025 + 0.272961i \(0.911997\pi\)
\(972\) 0 0
\(973\) −227176. −0.239959
\(974\) 884605.i 0.932463i
\(975\) 0 0
\(976\) −440941. −0.462893
\(977\) − 1.00025e6i − 1.04790i −0.851749 0.523950i \(-0.824458\pi\)
0.851749 0.523950i \(-0.175542\pi\)
\(978\) 0 0
\(979\) 31797.9 0.0331767
\(980\) 95468.5i 0.0994049i
\(981\) 0 0
\(982\) 74234.6 0.0769809
\(983\) 337021.i 0.348779i 0.984677 + 0.174389i \(0.0557951\pi\)
−0.984677 + 0.174389i \(0.944205\pi\)
\(984\) 0 0
\(985\) 597452. 0.615788
\(986\) 387892.i 0.398986i
\(987\) 0 0
\(988\) 101987. 0.104480
\(989\) − 74504.6i − 0.0761712i
\(990\) 0 0
\(991\) 1.26960e6 1.29277 0.646385 0.763012i \(-0.276280\pi\)
0.646385 + 0.763012i \(0.276280\pi\)
\(992\) 31061.2i 0.0315642i
\(993\) 0 0
\(994\) 287483. 0.290964
\(995\) 1.16245e6i 1.17416i
\(996\) 0 0
\(997\) 750408. 0.754931 0.377465 0.926024i \(-0.376796\pi\)
0.377465 + 0.926024i \(0.376796\pi\)
\(998\) 389391.i 0.390953i
\(999\) 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.b.323.5 yes 8
3.2 odd 2 inner 378.5.b.b.323.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.5.b.b.323.4 8 3.2 odd 2 inner
378.5.b.b.323.5 yes 8 1.1 even 1 trivial