Properties

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

$q$-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 \(0.244965\pi\)
−0.718202 + 0.695834i \(0.755035\pi\)
\(6\) 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 \(0.0397308\pi\)
−0.992220 + 0.124494i \(0.960269\pi\)
\(12\) 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 \(-0.935582\pi\)
0.979592 0.200995i \(-0.0644176\pi\)
\(18\) 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 \(-0.992879\pi\)
0.999750 0.0223697i \(-0.00712108\pi\)
\(24\) 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 \(0.247629\pi\)
−0.712353 + 0.701821i \(0.752371\pi\)
\(30\) 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 \(0.0396175\pi\)
−0.992265 + 0.124141i \(0.960382\pi\)
\(42\) 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 \(0.233797\pi\)
−0.742169 + 0.670213i \(0.766203\pi\)
\(48\) 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 \(0.131064\pi\)
−0.916422 + 0.400214i \(0.868936\pi\)
\(54\) 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 \(-0.962088\pi\)
0.992915 0.118824i \(-0.0379123\pi\)
\(60\) 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 \(0.183221\pi\)
−0.838862 + 0.544344i \(0.816779\pi\)
\(72\) 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 \(-0.980892\pi\)
0.998199 0.0599927i \(-0.0191077\pi\)
\(84\) 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 \(-0.978777\pi\)
0.997778 0.0666231i \(-0.0212225\pi\)
\(90\) 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 \(0.175183\pi\)
−0.852339 + 0.522990i \(0.824817\pi\)
\(102\) 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 \(-0.811180\pi\)
0.829159 0.559013i \(-0.188820\pi\)
\(108\) 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 \(0.276456\pi\)
−0.645964 + 0.763368i \(0.723544\pi\)
\(114\) 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 \(-0.824653\pi\)
0.852069 0.523429i \(-0.175347\pi\)
\(132\) 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 \(0.290891\pi\)
−0.610693 + 0.791867i \(0.709109\pi\)
\(138\) 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 \(-0.981592\pi\)
0.998328 0.0577978i \(-0.0184079\pi\)
\(150\) 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 \(0.262492\pi\)
−0.678819 + 0.734306i \(0.737508\pi\)
\(168\) 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 \(0.0698188\pi\)
−0.976041 + 0.217588i \(0.930181\pi\)
\(174\) 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 \(-0.815238\pi\)
0.836217 0.548399i \(-0.184762\pi\)
\(180\) 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 \(-0.821574\pi\)
0.846966 0.531647i \(-0.178426\pi\)
\(192\) 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 \(-0.928989\pi\)
0.975219 0.221241i \(-0.0710106\pi\)
\(198\) 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 \(-0.877498\pi\)
0.926854 0.375422i \(-0.122502\pi\)
\(228\) 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 \(-0.802087\pi\)
0.812853 0.582468i \(-0.197913\pi\)
\(234\) 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 \(0.138341\pi\)
−0.907033 + 0.421059i \(0.861659\pi\)
\(240\) 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 \(-0.855768\pi\)
0.899087 0.437771i \(-0.144232\pi\)
\(252\) 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 \(-0.945736\pi\)
0.985504 0.169652i \(-0.0542643\pi\)
\(258\) 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 \(0.0131963\pi\)
−0.999141 + 0.0414456i \(0.986804\pi\)
\(264\) 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 \(-0.680602\pi\)
0.537422 0.843314i \(-0.319398\pi\)
\(270\) 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 \(0.215847\pi\)
−0.778764 + 0.627317i \(0.784153\pi\)
\(282\) 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 \(0.0624342\pi\)
−0.980826 + 0.194888i \(0.937566\pi\)
\(294\) 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 \(0.151073\pi\)
−0.889471 + 0.456991i \(0.848927\pi\)
\(312\) 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 \(0.0531539\pi\)
−0.986090 + 0.166213i \(0.946846\pi\)
\(318\) 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 \(0.0573703\pi\)
−0.983802 + 0.179260i \(0.942630\pi\)
\(348\) 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 \(0.245608\pi\)
−0.716796 + 0.697283i \(0.754392\pi\)
\(354\) 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 \(0.264282\pi\)
−0.674680 + 0.738110i \(0.735718\pi\)
\(360\) 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 \(0.124051\pi\)
−0.925016 + 0.379927i \(0.875949\pi\)
\(384\) 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 \(-0.758739\pi\)
0.726250 0.687430i \(-0.241261\pi\)
\(390\) 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 \(-0.791123\pi\)
0.792312 0.610117i \(-0.208877\pi\)
\(402\) 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 \(0.106327\pi\)
−0.944727 + 0.327858i \(0.893673\pi\)
\(420\) 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 \(0.0599097\pi\)
−0.982340 + 0.187103i \(0.940090\pi\)
\(432\) 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 \(-0.860440\pi\)
0.905414 0.424529i \(-0.139560\pi\)
\(444\) 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 \(0.00757192\pi\)
−0.999717 + 0.0237856i \(0.992428\pi\)
\(450\) 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 \(0.332660\pi\)
−0.501831 + 0.864966i \(0.667340\pi\)
\(462\) 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 \(-0.760987\pi\)
0.731087 0.682284i \(-0.239013\pi\)
\(468\) 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 \(0.146022\pi\)
−0.896611 + 0.442819i \(0.853978\pi\)
\(480\) 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 \(0.0173354\pi\)
−0.998517 + 0.0544337i \(0.982665\pi\)
\(492\) 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 \(0.231985\pi\)
−0.745973 + 0.665977i \(0.768015\pi\)
\(504\) 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 \(-0.804067\pi\)
0.816460 0.577402i \(-0.195933\pi\)
\(510\) 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 \(0.151428\pi\)
−0.888961 + 0.457983i \(0.848572\pi\)
\(522\) 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 \(-0.701700\pi\)
0.592099 0.805866i \(-0.298300\pi\)
\(558\) 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 \(-0.979527\pi\)
0.997932 0.0642725i \(-0.0204727\pi\)
\(564\) 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 \(-0.697325\pi\)
0.580966 0.813928i \(-0.302675\pi\)
\(570\) 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 \(-0.616579\pi\)
0.358112 0.933679i \(-0.383421\pi\)
\(588\) 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 \(0.166490\pi\)
−0.866303 + 0.499519i \(0.833510\pi\)
\(594\) 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 \(-0.885900\pi\)
0.936440 0.350828i \(-0.114100\pi\)
\(600\) 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 \(0.0332010\pi\)
−0.994565 + 0.104115i \(0.966799\pi\)
\(618\) 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 \(-0.596573\pi\)
0.298759 0.954329i \(-0.403427\pi\)
\(642\) 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 \(0.290087\pi\)
−0.612691 + 0.790322i \(0.709913\pi\)
\(648\) 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 \(-0.998512\pi\)
0.999989 0.00467410i \(-0.00148782\pi\)
\(654\) 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 \(-0.954283\pi\)
0.989704 0.143130i \(-0.0457168\pi\)
\(660\) 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 \(0.170928\pi\)
−0.859254 + 0.511549i \(0.829072\pi\)
\(678\) 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 \(-0.825595\pi\)
0.853616 0.520903i \(-0.174405\pi\)
\(684\) 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 \(0.245700\pi\)
−0.716594 + 0.697491i \(0.754300\pi\)
\(702\) 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 \(-0.602063\pi\)
0.315174 0.949034i \(-0.397937\pi\)
\(720\) 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 \(-0.972202\pi\)
0.996189 0.0872181i \(-0.0277977\pi\)
\(744\) 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 \(0.100372\pi\)
−0.950694 + 0.310129i \(0.899628\pi\)
\(762\) 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 \(-0.915140\pi\)
0.964673 0.263450i \(-0.0848604\pi\)
\(774\) 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 \(0.176210\pi\)
−0.850647 + 0.525737i \(0.823790\pi\)
\(798\) 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 \(0.0452118\pi\)
−0.989930 + 0.141560i \(0.954788\pi\)
\(810\) 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 \(-0.826813\pi\)
0.855603 0.517632i \(-0.173187\pi\)
\(822\) 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 \(-0.947536\pi\)
0.986448 0.164076i \(-0.0524643\pi\)
\(828\) 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 \(0.0556679\pi\)
−0.984746 + 0.173996i \(0.944332\pi\)
\(840\) 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 \(-0.676164\pi\)
0.525613 0.850724i \(-0.323836\pi\)
\(858\) 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 \(0.187828\pi\)
−0.830896 + 0.556428i \(0.812172\pi\)
\(864\) 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 \(0.271221\pi\)
−0.658429 + 0.752643i \(0.728779\pi\)
\(882\) 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 \(-0.849740\pi\)
0.890635 0.454719i \(-0.150260\pi\)
\(888\) 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 \(0.243713\pi\)
−0.720934 + 0.693003i \(0.756287\pi\)
\(912\) 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 \(0.258478\pi\)
−0.688025 + 0.725687i \(0.741522\pi\)
\(930\) 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 \(0.228050\pi\)
−0.754148 + 0.656704i \(0.771950\pi\)
\(942\) 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 \(-0.988780\pi\)
0.999379 0.0352399i \(-0.0112195\pi\)
\(948\) 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 \(0.171725\pi\)
−0.857971 + 0.513699i \(0.828275\pi\)
\(954\) 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 \(-0.911997\pi\)
0.962025 0.272961i \(-0.0880029\pi\)
\(972\) 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 \(0.175542\pi\)
−0.851749 + 0.523950i \(0.824458\pi\)
\(978\) 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 \(-0.944205\pi\)
0.984677 0.174389i \(-0.0557951\pi\)
\(984\) 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.323.4 8
3.2 odd 2 inner 378.5.b.b.323.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.5.b.b.323.4 8 1.1 even 1 trivial
378.5.b.b.323.5 yes 8 3.2 odd 2 inner