Properties

Label 378.5.b.b.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: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
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 323.3
Root \(-2.54824i\) of defining polynomial
Character \(\chi\) \(=\) 378.323
Dual form 378.5.b.b.323.6

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843i q^{2} -8.00000 q^{4} +17.8674i q^{5} -18.5203 q^{7} +22.6274i q^{8} +O(q^{10})\) \(q-2.82843i q^{2} -8.00000 q^{4} +17.8674i q^{5} -18.5203 q^{7} +22.6274i q^{8} +50.5366 q^{10} -45.4134i q^{11} -155.432 q^{13} +52.3832i q^{14} +64.0000 q^{16} +114.905i q^{17} -17.2296 q^{19} -142.939i q^{20} -128.449 q^{22} +55.0943i q^{23} +305.756 q^{25} +439.629i q^{26} +148.162 q^{28} -578.935i q^{29} +514.064 q^{31} -181.019i q^{32} +325.001 q^{34} -330.909i q^{35} +1287.01 q^{37} +48.7326i q^{38} -404.293 q^{40} -2151.69i q^{41} +2349.59 q^{43} +363.307i q^{44} +155.830 q^{46} -1965.66i q^{47} +343.000 q^{49} -864.809i q^{50} +1243.46 q^{52} -3259.56i q^{53} +811.419 q^{55} -419.066i q^{56} -1637.48 q^{58} +276.141i q^{59} -438.948 q^{61} -1453.99i q^{62} -512.000 q^{64} -2777.17i q^{65} -1761.95 q^{67} -919.241i q^{68} -935.951 q^{70} +1378.26i q^{71} +1873.02 q^{73} -3640.20i q^{74} +137.837 q^{76} +841.068i q^{77} +4585.41 q^{79} +1143.51i q^{80} -6085.89 q^{82} -7058.33i q^{83} -2053.05 q^{85} -6645.64i q^{86} +1027.59 q^{88} -7138.01i q^{89} +2878.64 q^{91} -440.755i q^{92} -5559.72 q^{94} -307.848i q^{95} -1109.01 q^{97} -970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 64 q^{4}+O(q^{10}) \) 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
\(5\) 17.8674i 0.714696i 0.933971 + 0.357348i \(0.116319\pi\)
−0.933971 + 0.357348i \(0.883681\pi\)
\(6\) 0 0
\(7\) −18.5203 −0.377964
\(8\) 22.6274i 0.353553i
\(9\) 0 0
\(10\) 50.5366 0.505366
\(11\) − 45.4134i − 0.375317i −0.982234 0.187659i \(-0.939910\pi\)
0.982234 0.187659i \(-0.0600899\pi\)
\(12\) 0 0
\(13\) −155.432 −0.919717 −0.459858 0.887992i \(-0.652100\pi\)
−0.459858 + 0.887992i \(0.652100\pi\)
\(14\) 52.3832i 0.267261i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) 114.905i 0.397596i 0.980041 + 0.198798i \(0.0637037\pi\)
−0.980041 + 0.198798i \(0.936296\pi\)
\(18\) 0 0
\(19\) −17.2296 −0.0477274 −0.0238637 0.999715i \(-0.507597\pi\)
−0.0238637 + 0.999715i \(0.507597\pi\)
\(20\) − 142.939i − 0.357348i
\(21\) 0 0
\(22\) −128.449 −0.265390
\(23\) 55.0943i 0.104148i 0.998643 + 0.0520740i \(0.0165832\pi\)
−0.998643 + 0.0520740i \(0.983417\pi\)
\(24\) 0 0
\(25\) 305.756 0.489210
\(26\) 439.629i 0.650338i
\(27\) 0 0
\(28\) 148.162 0.188982
\(29\) − 578.935i − 0.688389i −0.938898 0.344195i \(-0.888152\pi\)
0.938898 0.344195i \(-0.111848\pi\)
\(30\) 0 0
\(31\) 514.064 0.534926 0.267463 0.963568i \(-0.413815\pi\)
0.267463 + 0.963568i \(0.413815\pi\)
\(32\) − 181.019i − 0.176777i
\(33\) 0 0
\(34\) 325.001 0.281143
\(35\) − 330.909i − 0.270130i
\(36\) 0 0
\(37\) 1287.01 0.940107 0.470053 0.882638i \(-0.344235\pi\)
0.470053 + 0.882638i \(0.344235\pi\)
\(38\) 48.7326i 0.0337483i
\(39\) 0 0
\(40\) −404.293 −0.252683
\(41\) − 2151.69i − 1.28001i −0.768373 0.640003i \(-0.778933\pi\)
0.768373 0.640003i \(-0.221067\pi\)
\(42\) 0 0
\(43\) 2349.59 1.27073 0.635367 0.772210i \(-0.280849\pi\)
0.635367 + 0.772210i \(0.280849\pi\)
\(44\) 363.307i 0.187659i
\(45\) 0 0
\(46\) 155.830 0.0736438
\(47\) − 1965.66i − 0.889841i −0.895570 0.444920i \(-0.853232\pi\)
0.895570 0.444920i \(-0.146768\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) − 864.809i − 0.345924i
\(51\) 0 0
\(52\) 1243.46 0.459858
\(53\) − 3259.56i − 1.16040i −0.814474 0.580200i \(-0.802974\pi\)
0.814474 0.580200i \(-0.197026\pi\)
\(54\) 0 0
\(55\) 811.419 0.268238
\(56\) − 419.066i − 0.133631i
\(57\) 0 0
\(58\) −1637.48 −0.486765
\(59\) 276.141i 0.0793281i 0.999213 + 0.0396641i \(0.0126288\pi\)
−0.999213 + 0.0396641i \(0.987371\pi\)
\(60\) 0 0
\(61\) −438.948 −0.117965 −0.0589826 0.998259i \(-0.518786\pi\)
−0.0589826 + 0.998259i \(0.518786\pi\)
\(62\) − 1453.99i − 0.378250i
\(63\) 0 0
\(64\) −512.000 −0.125000
\(65\) − 2777.17i − 0.657318i
\(66\) 0 0
\(67\) −1761.95 −0.392503 −0.196251 0.980554i \(-0.562877\pi\)
−0.196251 + 0.980554i \(0.562877\pi\)
\(68\) − 919.241i − 0.198798i
\(69\) 0 0
\(70\) −935.951 −0.191010
\(71\) 1378.26i 0.273409i 0.990612 + 0.136705i \(0.0436511\pi\)
−0.990612 + 0.136705i \(0.956349\pi\)
\(72\) 0 0
\(73\) 1873.02 0.351477 0.175738 0.984437i \(-0.443769\pi\)
0.175738 + 0.984437i \(0.443769\pi\)
\(74\) − 3640.20i − 0.664756i
\(75\) 0 0
\(76\) 137.837 0.0238637
\(77\) 841.068i 0.141857i
\(78\) 0 0
\(79\) 4585.41 0.734724 0.367362 0.930078i \(-0.380261\pi\)
0.367362 + 0.930078i \(0.380261\pi\)
\(80\) 1143.51i 0.178674i
\(81\) 0 0
\(82\) −6085.89 −0.905100
\(83\) − 7058.33i − 1.02458i −0.858813 0.512290i \(-0.828797\pi\)
0.858813 0.512290i \(-0.171203\pi\)
\(84\) 0 0
\(85\) −2053.05 −0.284160
\(86\) − 6645.64i − 0.898545i
\(87\) 0 0
\(88\) 1027.59 0.132695
\(89\) − 7138.01i − 0.901150i −0.892739 0.450575i \(-0.851219\pi\)
0.892739 0.450575i \(-0.148781\pi\)
\(90\) 0 0
\(91\) 2878.64 0.347620
\(92\) − 440.755i − 0.0520740i
\(93\) 0 0
\(94\) −5559.72 −0.629213
\(95\) − 307.848i − 0.0341105i
\(96\) 0 0
\(97\) −1109.01 −0.117867 −0.0589336 0.998262i \(-0.518770\pi\)
−0.0589336 + 0.998262i \(0.518770\pi\)
\(98\) − 970.151i − 0.101015i
\(99\) 0 0
\(100\) −2446.05 −0.244605
\(101\) − 15526.5i − 1.52205i −0.648721 0.761027i \(-0.724696\pi\)
0.648721 0.761027i \(-0.275304\pi\)
\(102\) 0 0
\(103\) −17118.9 −1.61362 −0.806809 0.590813i \(-0.798807\pi\)
−0.806809 + 0.590813i \(0.798807\pi\)
\(104\) − 3517.03i − 0.325169i
\(105\) 0 0
\(106\) −9219.44 −0.820526
\(107\) 4086.42i 0.356924i 0.983947 + 0.178462i \(0.0571121\pi\)
−0.983947 + 0.178462i \(0.942888\pi\)
\(108\) 0 0
\(109\) 9630.93 0.810616 0.405308 0.914180i \(-0.367164\pi\)
0.405308 + 0.914180i \(0.367164\pi\)
\(110\) − 2295.04i − 0.189673i
\(111\) 0 0
\(112\) −1185.30 −0.0944911
\(113\) 5446.96i 0.426577i 0.976989 + 0.213288i \(0.0684174\pi\)
−0.976989 + 0.213288i \(0.931583\pi\)
\(114\) 0 0
\(115\) −984.392 −0.0744342
\(116\) 4631.48i 0.344195i
\(117\) 0 0
\(118\) 781.045 0.0560935
\(119\) − 2128.07i − 0.150277i
\(120\) 0 0
\(121\) 12578.6 0.859137
\(122\) 1241.53i 0.0834140i
\(123\) 0 0
\(124\) −4112.51 −0.267463
\(125\) 16630.2i 1.06433i
\(126\) 0 0
\(127\) 3615.47 0.224160 0.112080 0.993699i \(-0.464249\pi\)
0.112080 + 0.993699i \(0.464249\pi\)
\(128\) 1448.15i 0.0883883i
\(129\) 0 0
\(130\) −7855.02 −0.464794
\(131\) − 13601.5i − 0.792584i −0.918125 0.396292i \(-0.870297\pi\)
0.918125 0.396292i \(-0.129703\pi\)
\(132\) 0 0
\(133\) 319.096 0.0180392
\(134\) 4983.53i 0.277541i
\(135\) 0 0
\(136\) −2600.01 −0.140571
\(137\) − 1456.98i − 0.0776268i −0.999246 0.0388134i \(-0.987642\pi\)
0.999246 0.0388134i \(-0.0123578\pi\)
\(138\) 0 0
\(139\) −2883.53 −0.149243 −0.0746217 0.997212i \(-0.523775\pi\)
−0.0746217 + 0.997212i \(0.523775\pi\)
\(140\) 2647.27i 0.135065i
\(141\) 0 0
\(142\) 3898.29 0.193329
\(143\) 7058.71i 0.345186i
\(144\) 0 0
\(145\) 10344.1 0.491989
\(146\) − 5297.70i − 0.248531i
\(147\) 0 0
\(148\) −10296.0 −0.470053
\(149\) − 40158.5i − 1.80886i −0.426623 0.904430i \(-0.640297\pi\)
0.426623 0.904430i \(-0.359703\pi\)
\(150\) 0 0
\(151\) −3953.92 −0.173410 −0.0867051 0.996234i \(-0.527634\pi\)
−0.0867051 + 0.996234i \(0.527634\pi\)
\(152\) − 389.861i − 0.0168742i
\(153\) 0 0
\(154\) 2378.90 0.100308
\(155\) 9184.98i 0.382309i
\(156\) 0 0
\(157\) 15472.7 0.627722 0.313861 0.949469i \(-0.398377\pi\)
0.313861 + 0.949469i \(0.398377\pi\)
\(158\) − 12969.5i − 0.519528i
\(159\) 0 0
\(160\) 3234.34 0.126342
\(161\) − 1020.36i − 0.0393643i
\(162\) 0 0
\(163\) 37395.2 1.40748 0.703738 0.710460i \(-0.251513\pi\)
0.703738 + 0.710460i \(0.251513\pi\)
\(164\) 17213.5i 0.640003i
\(165\) 0 0
\(166\) −19964.0 −0.724487
\(167\) 5416.39i 0.194212i 0.995274 + 0.0971061i \(0.0309586\pi\)
−0.995274 + 0.0971061i \(0.969041\pi\)
\(168\) 0 0
\(169\) −4401.84 −0.154121
\(170\) 5806.92i 0.200931i
\(171\) 0 0
\(172\) −18796.7 −0.635367
\(173\) − 14162.2i − 0.473194i −0.971608 0.236597i \(-0.923968\pi\)
0.971608 0.236597i \(-0.0760321\pi\)
\(174\) 0 0
\(175\) −5662.69 −0.184904
\(176\) − 2906.46i − 0.0938294i
\(177\) 0 0
\(178\) −20189.3 −0.637209
\(179\) 16147.2i 0.503955i 0.967733 + 0.251977i \(0.0810809\pi\)
−0.967733 + 0.251977i \(0.918919\pi\)
\(180\) 0 0
\(181\) 26540.9 0.810138 0.405069 0.914286i \(-0.367248\pi\)
0.405069 + 0.914286i \(0.367248\pi\)
\(182\) − 8142.03i − 0.245805i
\(183\) 0 0
\(184\) −1246.64 −0.0368219
\(185\) 22995.4i 0.671890i
\(186\) 0 0
\(187\) 5218.23 0.149225
\(188\) 15725.3i 0.444920i
\(189\) 0 0
\(190\) −870.725 −0.0241198
\(191\) 2149.99i 0.0589344i 0.999566 + 0.0294672i \(0.00938106\pi\)
−0.999566 + 0.0294672i \(0.990619\pi\)
\(192\) 0 0
\(193\) 26344.9 0.707265 0.353633 0.935384i \(-0.384946\pi\)
0.353633 + 0.935384i \(0.384946\pi\)
\(194\) 3136.76i 0.0833448i
\(195\) 0 0
\(196\) −2744.00 −0.0714286
\(197\) 62745.0i 1.61676i 0.588658 + 0.808382i \(0.299656\pi\)
−0.588658 + 0.808382i \(0.700344\pi\)
\(198\) 0 0
\(199\) −41364.4 −1.04453 −0.522264 0.852784i \(-0.674912\pi\)
−0.522264 + 0.852784i \(0.674912\pi\)
\(200\) 6918.48i 0.172962i
\(201\) 0 0
\(202\) −43915.5 −1.07625
\(203\) 10722.0i 0.260187i
\(204\) 0 0
\(205\) 38445.1 0.914814
\(206\) 48419.5i 1.14100i
\(207\) 0 0
\(208\) −9947.66 −0.229929
\(209\) 782.454i 0.0179129i
\(210\) 0 0
\(211\) 60192.2 1.35200 0.675998 0.736904i \(-0.263713\pi\)
0.675998 + 0.736904i \(0.263713\pi\)
\(212\) 26076.5i 0.580200i
\(213\) 0 0
\(214\) 11558.1 0.252383
\(215\) 41981.0i 0.908188i
\(216\) 0 0
\(217\) −9520.60 −0.202183
\(218\) − 27240.4i − 0.573192i
\(219\) 0 0
\(220\) −6491.35 −0.134119
\(221\) − 17860.0i − 0.365675i
\(222\) 0 0
\(223\) 54766.3 1.10129 0.550647 0.834738i \(-0.314381\pi\)
0.550647 + 0.834738i \(0.314381\pi\)
\(224\) 3352.53i 0.0668153i
\(225\) 0 0
\(226\) 15406.3 0.301635
\(227\) 23659.5i 0.459149i 0.973291 + 0.229575i \(0.0737335\pi\)
−0.973291 + 0.229575i \(0.926267\pi\)
\(228\) 0 0
\(229\) −18364.7 −0.350198 −0.175099 0.984551i \(-0.556025\pi\)
−0.175099 + 0.984551i \(0.556025\pi\)
\(230\) 2784.28i 0.0526329i
\(231\) 0 0
\(232\) 13099.8 0.243382
\(233\) 71340.7i 1.31409i 0.753851 + 0.657046i \(0.228194\pi\)
−0.753851 + 0.657046i \(0.771806\pi\)
\(234\) 0 0
\(235\) 35121.2 0.635965
\(236\) − 2209.13i − 0.0396641i
\(237\) 0 0
\(238\) −6019.10 −0.106262
\(239\) 43626.2i 0.763751i 0.924214 + 0.381876i \(0.124722\pi\)
−0.924214 + 0.381876i \(0.875278\pi\)
\(240\) 0 0
\(241\) 12447.2 0.214308 0.107154 0.994242i \(-0.465826\pi\)
0.107154 + 0.994242i \(0.465826\pi\)
\(242\) − 35577.7i − 0.607501i
\(243\) 0 0
\(244\) 3511.59 0.0589826
\(245\) 6128.52i 0.102099i
\(246\) 0 0
\(247\) 2678.03 0.0438957
\(248\) 11631.9i 0.189125i
\(249\) 0 0
\(250\) 47037.3 0.752596
\(251\) 78794.7i 1.25069i 0.780348 + 0.625345i \(0.215042\pi\)
−0.780348 + 0.625345i \(0.784958\pi\)
\(252\) 0 0
\(253\) 2502.02 0.0390886
\(254\) − 10226.1i − 0.158505i
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) − 28911.2i − 0.437723i −0.975756 0.218861i \(-0.929766\pi\)
0.975756 0.218861i \(-0.0702343\pi\)
\(258\) 0 0
\(259\) −23835.7 −0.355327
\(260\) 22217.3i 0.328659i
\(261\) 0 0
\(262\) −38470.9 −0.560442
\(263\) − 65442.0i − 0.946118i −0.881031 0.473059i \(-0.843150\pi\)
0.881031 0.473059i \(-0.156850\pi\)
\(264\) 0 0
\(265\) 58239.9 0.829333
\(266\) − 902.540i − 0.0127557i
\(267\) 0 0
\(268\) 14095.6 0.196251
\(269\) 104321.i 1.44167i 0.693107 + 0.720835i \(0.256241\pi\)
−0.693107 + 0.720835i \(0.743759\pi\)
\(270\) 0 0
\(271\) −31635.6 −0.430762 −0.215381 0.976530i \(-0.569099\pi\)
−0.215381 + 0.976530i \(0.569099\pi\)
\(272\) 7353.93i 0.0993989i
\(273\) 0 0
\(274\) −4120.96 −0.0548905
\(275\) − 13885.4i − 0.183609i
\(276\) 0 0
\(277\) −116481. −1.51809 −0.759045 0.651039i \(-0.774334\pi\)
−0.759045 + 0.651039i \(0.774334\pi\)
\(278\) 8155.86i 0.105531i
\(279\) 0 0
\(280\) 7487.61 0.0955052
\(281\) − 17406.9i − 0.220449i −0.993907 0.110224i \(-0.964843\pi\)
0.993907 0.110224i \(-0.0351570\pi\)
\(282\) 0 0
\(283\) −10431.0 −0.130242 −0.0651211 0.997877i \(-0.520743\pi\)
−0.0651211 + 0.997877i \(0.520743\pi\)
\(284\) − 11026.0i − 0.136705i
\(285\) 0 0
\(286\) 19965.0 0.244083
\(287\) 39849.8i 0.483796i
\(288\) 0 0
\(289\) 70317.8 0.841918
\(290\) − 29257.4i − 0.347888i
\(291\) 0 0
\(292\) −14984.1 −0.175738
\(293\) 64199.7i 0.747822i 0.927465 + 0.373911i \(0.121983\pi\)
−0.927465 + 0.373911i \(0.878017\pi\)
\(294\) 0 0
\(295\) −4933.92 −0.0566955
\(296\) 29121.6i 0.332378i
\(297\) 0 0
\(298\) −113585. −1.27906
\(299\) − 8563.43i − 0.0957867i
\(300\) 0 0
\(301\) −43515.0 −0.480292
\(302\) 11183.4i 0.122619i
\(303\) 0 0
\(304\) −1102.69 −0.0119318
\(305\) − 7842.86i − 0.0843092i
\(306\) 0 0
\(307\) −65196.6 −0.691749 −0.345874 0.938281i \(-0.612418\pi\)
−0.345874 + 0.938281i \(0.612418\pi\)
\(308\) − 6728.55i − 0.0709283i
\(309\) 0 0
\(310\) 25979.1 0.270334
\(311\) − 151896.i − 1.57046i −0.619206 0.785229i \(-0.712545\pi\)
0.619206 0.785229i \(-0.287455\pi\)
\(312\) 0 0
\(313\) −54154.6 −0.552773 −0.276387 0.961047i \(-0.589137\pi\)
−0.276387 + 0.961047i \(0.589137\pi\)
\(314\) − 43763.5i − 0.443867i
\(315\) 0 0
\(316\) −36683.3 −0.367362
\(317\) − 137384.i − 1.36715i −0.729878 0.683577i \(-0.760423\pi\)
0.729878 0.683577i \(-0.239577\pi\)
\(318\) 0 0
\(319\) −26291.4 −0.258364
\(320\) − 9148.10i − 0.0893370i
\(321\) 0 0
\(322\) −2886.02 −0.0278347
\(323\) − 1979.77i − 0.0189762i
\(324\) 0 0
\(325\) −47524.4 −0.449935
\(326\) − 105770.i − 0.995236i
\(327\) 0 0
\(328\) 48687.2 0.452550
\(329\) 36404.5i 0.336328i
\(330\) 0 0
\(331\) −163805. −1.49510 −0.747550 0.664205i \(-0.768770\pi\)
−0.747550 + 0.664205i \(0.768770\pi\)
\(332\) 56466.6i 0.512290i
\(333\) 0 0
\(334\) 15319.9 0.137329
\(335\) − 31481.4i − 0.280520i
\(336\) 0 0
\(337\) 164077. 1.44473 0.722367 0.691510i \(-0.243054\pi\)
0.722367 + 0.691510i \(0.243054\pi\)
\(338\) 12450.3i 0.108980i
\(339\) 0 0
\(340\) 16424.4 0.142080
\(341\) − 23345.4i − 0.200767i
\(342\) 0 0
\(343\) −6352.45 −0.0539949
\(344\) 53165.1i 0.449272i
\(345\) 0 0
\(346\) −40056.8 −0.334599
\(347\) − 72809.2i − 0.604683i −0.953200 0.302341i \(-0.902232\pi\)
0.953200 0.302341i \(-0.0977682\pi\)
\(348\) 0 0
\(349\) 15848.6 0.130118 0.0650592 0.997881i \(-0.479276\pi\)
0.0650592 + 0.997881i \(0.479276\pi\)
\(350\) 16016.5i 0.130747i
\(351\) 0 0
\(352\) −8220.71 −0.0663474
\(353\) − 40491.9i − 0.324951i −0.986713 0.162476i \(-0.948052\pi\)
0.986713 0.162476i \(-0.0519479\pi\)
\(354\) 0 0
\(355\) −24625.8 −0.195404
\(356\) 57104.1i 0.450575i
\(357\) 0 0
\(358\) 45671.2 0.356350
\(359\) − 30479.4i − 0.236492i −0.992984 0.118246i \(-0.962273\pi\)
0.992984 0.118246i \(-0.0377272\pi\)
\(360\) 0 0
\(361\) −130024. −0.997722
\(362\) − 75069.1i − 0.572854i
\(363\) 0 0
\(364\) −23029.2 −0.173810
\(365\) 33466.0i 0.251199i
\(366\) 0 0
\(367\) −252948. −1.87801 −0.939007 0.343898i \(-0.888253\pi\)
−0.939007 + 0.343898i \(0.888253\pi\)
\(368\) 3526.04i 0.0260370i
\(369\) 0 0
\(370\) 65040.9 0.475098
\(371\) 60367.9i 0.438590i
\(372\) 0 0
\(373\) 131647. 0.946221 0.473110 0.881003i \(-0.343131\pi\)
0.473110 + 0.881003i \(0.343131\pi\)
\(374\) − 14759.4i − 0.105518i
\(375\) 0 0
\(376\) 44477.8 0.314606
\(377\) 89985.1i 0.633123i
\(378\) 0 0
\(379\) −64033.9 −0.445791 −0.222896 0.974842i \(-0.571551\pi\)
−0.222896 + 0.974842i \(0.571551\pi\)
\(380\) 2462.78i 0.0170553i
\(381\) 0 0
\(382\) 6081.08 0.0416729
\(383\) − 272341.i − 1.85659i −0.371849 0.928293i \(-0.621276\pi\)
0.371849 0.928293i \(-0.378724\pi\)
\(384\) 0 0
\(385\) −15027.7 −0.101384
\(386\) − 74514.7i − 0.500112i
\(387\) 0 0
\(388\) 8872.11 0.0589336
\(389\) − 285818.i − 1.88882i −0.328775 0.944408i \(-0.606636\pi\)
0.328775 0.944408i \(-0.393364\pi\)
\(390\) 0 0
\(391\) −6330.62 −0.0414088
\(392\) 7761.20i 0.0505076i
\(393\) 0 0
\(394\) 177470. 1.14323
\(395\) 81929.4i 0.525104i
\(396\) 0 0
\(397\) 201945. 1.28130 0.640652 0.767831i \(-0.278664\pi\)
0.640652 + 0.767831i \(0.278664\pi\)
\(398\) 116996.i 0.738593i
\(399\) 0 0
\(400\) 19568.4 0.122303
\(401\) − 58597.3i − 0.364409i −0.983261 0.182204i \(-0.941677\pi\)
0.983261 0.182204i \(-0.0583232\pi\)
\(402\) 0 0
\(403\) −79902.1 −0.491981
\(404\) 124212.i 0.761027i
\(405\) 0 0
\(406\) 30326.5 0.183980
\(407\) − 58447.3i − 0.352838i
\(408\) 0 0
\(409\) 179325. 1.07200 0.535999 0.844219i \(-0.319935\pi\)
0.535999 + 0.844219i \(0.319935\pi\)
\(410\) − 108739.i − 0.646871i
\(411\) 0 0
\(412\) 136951. 0.806809
\(413\) − 5114.21i − 0.0299832i
\(414\) 0 0
\(415\) 126114. 0.732263
\(416\) 28136.2i 0.162585i
\(417\) 0 0
\(418\) 2213.11 0.0126663
\(419\) 153806.i 0.876085i 0.898954 + 0.438042i \(0.144328\pi\)
−0.898954 + 0.438042i \(0.855672\pi\)
\(420\) 0 0
\(421\) −91254.8 −0.514863 −0.257432 0.966297i \(-0.582876\pi\)
−0.257432 + 0.966297i \(0.582876\pi\)
\(422\) − 170249.i − 0.956005i
\(423\) 0 0
\(424\) 73755.5 0.410263
\(425\) 35133.0i 0.194508i
\(426\) 0 0
\(427\) 8129.44 0.0445866
\(428\) − 32691.4i − 0.178462i
\(429\) 0 0
\(430\) 118740. 0.642186
\(431\) − 348521.i − 1.87618i −0.346394 0.938089i \(-0.612594\pi\)
0.346394 0.938089i \(-0.387406\pi\)
\(432\) 0 0
\(433\) −86162.8 −0.459562 −0.229781 0.973242i \(-0.573801\pi\)
−0.229781 + 0.973242i \(0.573801\pi\)
\(434\) 26928.3i 0.142965i
\(435\) 0 0
\(436\) −77047.4 −0.405308
\(437\) − 949.252i − 0.00497071i
\(438\) 0 0
\(439\) −284089. −1.47409 −0.737047 0.675842i \(-0.763780\pi\)
−0.737047 + 0.675842i \(0.763780\pi\)
\(440\) 18360.3i 0.0948364i
\(441\) 0 0
\(442\) −50515.6 −0.258572
\(443\) − 77877.1i − 0.396828i −0.980118 0.198414i \(-0.936421\pi\)
0.980118 0.198414i \(-0.0635790\pi\)
\(444\) 0 0
\(445\) 127538. 0.644048
\(446\) − 154902.i − 0.778733i
\(447\) 0 0
\(448\) 9482.37 0.0472456
\(449\) − 87226.0i − 0.432667i −0.976320 0.216333i \(-0.930590\pi\)
0.976320 0.216333i \(-0.0694098\pi\)
\(450\) 0 0
\(451\) −97715.5 −0.480408
\(452\) − 43575.7i − 0.213288i
\(453\) 0 0
\(454\) 66919.2 0.324667
\(455\) 51433.9i 0.248443i
\(456\) 0 0
\(457\) 60354.0 0.288984 0.144492 0.989506i \(-0.453845\pi\)
0.144492 + 0.989506i \(0.453845\pi\)
\(458\) 51943.3i 0.247627i
\(459\) 0 0
\(460\) 7875.14 0.0372171
\(461\) 136850.i 0.643938i 0.946750 + 0.321969i \(0.104345\pi\)
−0.946750 + 0.321969i \(0.895655\pi\)
\(462\) 0 0
\(463\) 167651. 0.782068 0.391034 0.920376i \(-0.372118\pi\)
0.391034 + 0.920376i \(0.372118\pi\)
\(464\) − 37051.8i − 0.172097i
\(465\) 0 0
\(466\) 201782. 0.929203
\(467\) − 188881.i − 0.866074i −0.901376 0.433037i \(-0.857442\pi\)
0.901376 0.433037i \(-0.142558\pi\)
\(468\) 0 0
\(469\) 32631.7 0.148352
\(470\) − 99337.7i − 0.449696i
\(471\) 0 0
\(472\) −6248.36 −0.0280467
\(473\) − 106703.i − 0.476929i
\(474\) 0 0
\(475\) −5268.05 −0.0233487
\(476\) 17024.6i 0.0751385i
\(477\) 0 0
\(478\) 123394. 0.540054
\(479\) 396256.i 1.72705i 0.504305 + 0.863526i \(0.331749\pi\)
−0.504305 + 0.863526i \(0.668251\pi\)
\(480\) 0 0
\(481\) −200042. −0.864632
\(482\) − 35206.1i − 0.151539i
\(483\) 0 0
\(484\) −100629. −0.429568
\(485\) − 19815.2i − 0.0842392i
\(486\) 0 0
\(487\) −205617. −0.866962 −0.433481 0.901163i \(-0.642715\pi\)
−0.433481 + 0.901163i \(0.642715\pi\)
\(488\) − 9932.27i − 0.0417070i
\(489\) 0 0
\(490\) 17334.1 0.0721952
\(491\) − 146443.i − 0.607445i −0.952761 0.303723i \(-0.901770\pi\)
0.952761 0.303723i \(-0.0982296\pi\)
\(492\) 0 0
\(493\) 66522.6 0.273700
\(494\) − 7574.61i − 0.0310389i
\(495\) 0 0
\(496\) 32900.1 0.133732
\(497\) − 25525.6i − 0.103339i
\(498\) 0 0
\(499\) −266336. −1.06962 −0.534810 0.844973i \(-0.679617\pi\)
−0.534810 + 0.844973i \(0.679617\pi\)
\(500\) − 133042.i − 0.532166i
\(501\) 0 0
\(502\) 222865. 0.884371
\(503\) 8405.60i 0.0332225i 0.999862 + 0.0166113i \(0.00528777\pi\)
−0.999862 + 0.0166113i \(0.994712\pi\)
\(504\) 0 0
\(505\) 277417. 1.08780
\(506\) − 7076.79i − 0.0276398i
\(507\) 0 0
\(508\) −28923.8 −0.112080
\(509\) 106126.i 0.409624i 0.978801 + 0.204812i \(0.0656584\pi\)
−0.978801 + 0.204812i \(0.934342\pi\)
\(510\) 0 0
\(511\) −34688.8 −0.132846
\(512\) − 11585.2i − 0.0441942i
\(513\) 0 0
\(514\) −81773.1 −0.309517
\(515\) − 305870.i − 1.15325i
\(516\) 0 0
\(517\) −89267.3 −0.333973
\(518\) 67417.5i 0.251254i
\(519\) 0 0
\(520\) 62840.1 0.232397
\(521\) 68150.9i 0.251071i 0.992089 + 0.125535i \(0.0400648\pi\)
−0.992089 + 0.125535i \(0.959935\pi\)
\(522\) 0 0
\(523\) −117480. −0.429496 −0.214748 0.976669i \(-0.568893\pi\)
−0.214748 + 0.976669i \(0.568893\pi\)
\(524\) 108812.i 0.396292i
\(525\) 0 0
\(526\) −185098. −0.669006
\(527\) 59068.6i 0.212684i
\(528\) 0 0
\(529\) 276806. 0.989153
\(530\) − 164727.i − 0.586427i
\(531\) 0 0
\(532\) −2552.77 −0.00901962
\(533\) 334442.i 1.17724i
\(534\) 0 0
\(535\) −73013.7 −0.255092
\(536\) − 39868.3i − 0.138771i
\(537\) 0 0
\(538\) 295063. 1.01941
\(539\) − 15576.8i − 0.0536168i
\(540\) 0 0
\(541\) 78291.9 0.267499 0.133750 0.991015i \(-0.457298\pi\)
0.133750 + 0.991015i \(0.457298\pi\)
\(542\) 89478.9i 0.304594i
\(543\) 0 0
\(544\) 20800.0 0.0702856
\(545\) 172080.i 0.579344i
\(546\) 0 0
\(547\) −294769. −0.985160 −0.492580 0.870267i \(-0.663946\pi\)
−0.492580 + 0.870267i \(0.663946\pi\)
\(548\) 11655.8i 0.0388134i
\(549\) 0 0
\(550\) −39273.9 −0.129831
\(551\) 9974.81i 0.0328550i
\(552\) 0 0
\(553\) −84923.0 −0.277700
\(554\) 329459.i 1.07345i
\(555\) 0 0
\(556\) 23068.3 0.0746217
\(557\) − 477782.i − 1.54000i −0.638047 0.769998i \(-0.720257\pi\)
0.638047 0.769998i \(-0.279743\pi\)
\(558\) 0 0
\(559\) −365202. −1.16872
\(560\) − 21178.2i − 0.0675324i
\(561\) 0 0
\(562\) −49234.1 −0.155881
\(563\) 450077.i 1.41994i 0.704232 + 0.709970i \(0.251292\pi\)
−0.704232 + 0.709970i \(0.748708\pi\)
\(564\) 0 0
\(565\) −97323.0 −0.304873
\(566\) 29503.2i 0.0920951i
\(567\) 0 0
\(568\) −31186.4 −0.0966647
\(569\) 2913.20i 0.00899800i 0.999990 + 0.00449900i \(0.00143208\pi\)
−0.999990 + 0.00449900i \(0.998568\pi\)
\(570\) 0 0
\(571\) 108683. 0.333342 0.166671 0.986013i \(-0.446698\pi\)
0.166671 + 0.986013i \(0.446698\pi\)
\(572\) − 56469.6i − 0.172593i
\(573\) 0 0
\(574\) 112712. 0.342096
\(575\) 16845.4i 0.0509503i
\(576\) 0 0
\(577\) 388049. 1.16556 0.582780 0.812630i \(-0.301965\pi\)
0.582780 + 0.812630i \(0.301965\pi\)
\(578\) − 198889.i − 0.595326i
\(579\) 0 0
\(580\) −82752.5 −0.245994
\(581\) 130722.i 0.387255i
\(582\) 0 0
\(583\) −148028. −0.435518
\(584\) 42381.6i 0.124266i
\(585\) 0 0
\(586\) 181584. 0.528790
\(587\) 541195.i 1.57064i 0.619087 + 0.785322i \(0.287503\pi\)
−0.619087 + 0.785322i \(0.712497\pi\)
\(588\) 0 0
\(589\) −8857.11 −0.0255306
\(590\) 13955.2i 0.0400897i
\(591\) 0 0
\(592\) 82368.4 0.235027
\(593\) − 545815.i − 1.55216i −0.630636 0.776079i \(-0.717206\pi\)
0.630636 0.776079i \(-0.282794\pi\)
\(594\) 0 0
\(595\) 38023.1 0.107402
\(596\) 321268.i 0.904430i
\(597\) 0 0
\(598\) −24221.0 −0.0677315
\(599\) − 30038.9i − 0.0837202i −0.999123 0.0418601i \(-0.986672\pi\)
0.999123 0.0418601i \(-0.0133284\pi\)
\(600\) 0 0
\(601\) −250122. −0.692473 −0.346237 0.938147i \(-0.612541\pi\)
−0.346237 + 0.938147i \(0.612541\pi\)
\(602\) 123079.i 0.339618i
\(603\) 0 0
\(604\) 31631.4 0.0867051
\(605\) 224747.i 0.614021i
\(606\) 0 0
\(607\) −270171. −0.733265 −0.366632 0.930366i \(-0.619489\pi\)
−0.366632 + 0.930366i \(0.619489\pi\)
\(608\) 3118.89i 0.00843709i
\(609\) 0 0
\(610\) −22183.0 −0.0596156
\(611\) 305527.i 0.818402i
\(612\) 0 0
\(613\) 378425. 1.00707 0.503535 0.863975i \(-0.332033\pi\)
0.503535 + 0.863975i \(0.332033\pi\)
\(614\) 184404.i 0.489140i
\(615\) 0 0
\(616\) −19031.2 −0.0501539
\(617\) 580490.i 1.52484i 0.647083 + 0.762420i \(0.275989\pi\)
−0.647083 + 0.762420i \(0.724011\pi\)
\(618\) 0 0
\(619\) 196149. 0.511922 0.255961 0.966687i \(-0.417608\pi\)
0.255961 + 0.966687i \(0.417608\pi\)
\(620\) − 73479.9i − 0.191155i
\(621\) 0 0
\(622\) −429627. −1.11048
\(623\) 132198.i 0.340603i
\(624\) 0 0
\(625\) −106040. −0.271463
\(626\) 153172.i 0.390870i
\(627\) 0 0
\(628\) −123782. −0.313861
\(629\) 147884.i 0.373782i
\(630\) 0 0
\(631\) 69136.2 0.173639 0.0868194 0.996224i \(-0.472330\pi\)
0.0868194 + 0.996224i \(0.472330\pi\)
\(632\) 103756.i 0.259764i
\(633\) 0 0
\(634\) −388581. −0.966724
\(635\) 64599.1i 0.160206i
\(636\) 0 0
\(637\) −53313.2 −0.131388
\(638\) 74363.4i 0.182691i
\(639\) 0 0
\(640\) −25874.7 −0.0631708
\(641\) − 426696.i − 1.03849i −0.854625 0.519245i \(-0.826213\pi\)
0.854625 0.519245i \(-0.173787\pi\)
\(642\) 0 0
\(643\) −129308. −0.312755 −0.156377 0.987697i \(-0.549982\pi\)
−0.156377 + 0.987697i \(0.549982\pi\)
\(644\) 8162.89i 0.0196821i
\(645\) 0 0
\(646\) −5599.63 −0.0134182
\(647\) 71367.5i 0.170487i 0.996360 + 0.0852437i \(0.0271669\pi\)
−0.996360 + 0.0852437i \(0.972833\pi\)
\(648\) 0 0
\(649\) 12540.5 0.0297732
\(650\) 134419.i 0.318152i
\(651\) 0 0
\(652\) −299162. −0.703738
\(653\) − 308085.i − 0.722511i −0.932467 0.361256i \(-0.882348\pi\)
0.932467 0.361256i \(-0.117652\pi\)
\(654\) 0 0
\(655\) 243024. 0.566456
\(656\) − 137708.i − 0.320001i
\(657\) 0 0
\(658\) 102967. 0.237820
\(659\) 576984.i 1.32860i 0.747468 + 0.664298i \(0.231269\pi\)
−0.747468 + 0.664298i \(0.768731\pi\)
\(660\) 0 0
\(661\) 13885.9 0.0317813 0.0158907 0.999874i \(-0.494942\pi\)
0.0158907 + 0.999874i \(0.494942\pi\)
\(662\) 463310.i 1.05720i
\(663\) 0 0
\(664\) 159712. 0.362244
\(665\) 5701.42i 0.0128926i
\(666\) 0 0
\(667\) 31896.0 0.0716944
\(668\) − 43331.1i − 0.0971061i
\(669\) 0 0
\(670\) −89042.8 −0.198358
\(671\) 19934.1i 0.0442744i
\(672\) 0 0
\(673\) 61470.0 0.135717 0.0678583 0.997695i \(-0.478383\pi\)
0.0678583 + 0.997695i \(0.478383\pi\)
\(674\) − 464080.i − 1.02158i
\(675\) 0 0
\(676\) 35214.7 0.0770603
\(677\) 553915.i 1.20855i 0.796774 + 0.604277i \(0.206538\pi\)
−0.796774 + 0.604277i \(0.793462\pi\)
\(678\) 0 0
\(679\) 20539.2 0.0445497
\(680\) − 46455.3i − 0.100466i
\(681\) 0 0
\(682\) −66030.8 −0.141964
\(683\) − 111142.i − 0.238253i −0.992879 0.119127i \(-0.961991\pi\)
0.992879 0.119127i \(-0.0380094\pi\)
\(684\) 0 0
\(685\) 26032.4 0.0554796
\(686\) 17967.4i 0.0381802i
\(687\) 0 0
\(688\) 150374. 0.317684
\(689\) 506641.i 1.06724i
\(690\) 0 0
\(691\) −284092. −0.594981 −0.297491 0.954725i \(-0.596150\pi\)
−0.297491 + 0.954725i \(0.596150\pi\)
\(692\) 113298.i 0.236597i
\(693\) 0 0
\(694\) −205936. −0.427575
\(695\) − 51521.2i − 0.106664i
\(696\) 0 0
\(697\) 247240. 0.508924
\(698\) − 44826.5i − 0.0920076i
\(699\) 0 0
\(700\) 45301.5 0.0924520
\(701\) 125819.i 0.256041i 0.991772 + 0.128020i \(0.0408623\pi\)
−0.991772 + 0.128020i \(0.959138\pi\)
\(702\) 0 0
\(703\) −22174.6 −0.0448688
\(704\) 23251.7i 0.0469147i
\(705\) 0 0
\(706\) −114528. −0.229775
\(707\) 287554.i 0.575282i
\(708\) 0 0
\(709\) −864378. −1.71954 −0.859768 0.510686i \(-0.829392\pi\)
−0.859768 + 0.510686i \(0.829392\pi\)
\(710\) 69652.3i 0.138172i
\(711\) 0 0
\(712\) 161515. 0.318605
\(713\) 28322.0i 0.0557115i
\(714\) 0 0
\(715\) −126121. −0.246703
\(716\) − 129178.i − 0.251977i
\(717\) 0 0
\(718\) −86208.6 −0.167225
\(719\) − 537857.i − 1.04042i −0.854038 0.520210i \(-0.825854\pi\)
0.854038 0.520210i \(-0.174146\pi\)
\(720\) 0 0
\(721\) 317046. 0.609890
\(722\) 367764.i 0.705496i
\(723\) 0 0
\(724\) −212327. −0.405069
\(725\) − 177013.i − 0.336767i
\(726\) 0 0
\(727\) −182656. −0.345592 −0.172796 0.984958i \(-0.555280\pi\)
−0.172796 + 0.984958i \(0.555280\pi\)
\(728\) 65136.3i 0.122902i
\(729\) 0 0
\(730\) 94656.0 0.177624
\(731\) 269980.i 0.505238i
\(732\) 0 0
\(733\) 690415. 1.28500 0.642499 0.766287i \(-0.277898\pi\)
0.642499 + 0.766287i \(0.277898\pi\)
\(734\) 715445.i 1.32796i
\(735\) 0 0
\(736\) 9973.14 0.0184110
\(737\) 80015.9i 0.147313i
\(738\) 0 0
\(739\) −447899. −0.820146 −0.410073 0.912053i \(-0.634497\pi\)
−0.410073 + 0.912053i \(0.634497\pi\)
\(740\) − 183964.i − 0.335945i
\(741\) 0 0
\(742\) 170746. 0.310130
\(743\) − 156899.i − 0.284212i −0.989851 0.142106i \(-0.954613\pi\)
0.989851 0.142106i \(-0.0453873\pi\)
\(744\) 0 0
\(745\) 717527. 1.29278
\(746\) − 372353.i − 0.669079i
\(747\) 0 0
\(748\) −41745.9 −0.0746123
\(749\) − 75681.6i − 0.134905i
\(750\) 0 0
\(751\) −733687. −1.30086 −0.650431 0.759565i \(-0.725412\pi\)
−0.650431 + 0.759565i \(0.725412\pi\)
\(752\) − 125802.i − 0.222460i
\(753\) 0 0
\(754\) 254516. 0.447686
\(755\) − 70646.3i − 0.123935i
\(756\) 0 0
\(757\) 1.11424e6 1.94440 0.972202 0.234142i \(-0.0752281\pi\)
0.972202 + 0.234142i \(0.0752281\pi\)
\(758\) 181115.i 0.315222i
\(759\) 0 0
\(760\) 6965.80 0.0120599
\(761\) − 747308.i − 1.29042i −0.764006 0.645209i \(-0.776770\pi\)
0.764006 0.645209i \(-0.223230\pi\)
\(762\) 0 0
\(763\) −178367. −0.306384
\(764\) − 17199.9i − 0.0294672i
\(765\) 0 0
\(766\) −770296. −1.31280
\(767\) − 42921.2i − 0.0729594i
\(768\) 0 0
\(769\) 965130. 1.63205 0.816024 0.578017i \(-0.196173\pi\)
0.816024 + 0.578017i \(0.196173\pi\)
\(770\) 42504.7i 0.0716896i
\(771\) 0 0
\(772\) −210759. −0.353633
\(773\) 195019.i 0.326376i 0.986595 + 0.163188i \(0.0521776\pi\)
−0.986595 + 0.163188i \(0.947822\pi\)
\(774\) 0 0
\(775\) 157178. 0.261691
\(776\) − 25094.1i − 0.0416724i
\(777\) 0 0
\(778\) −808414. −1.33560
\(779\) 37072.7i 0.0610913i
\(780\) 0 0
\(781\) 62591.3 0.102615
\(782\) 17905.7i 0.0292804i
\(783\) 0 0
\(784\) 21952.0 0.0357143
\(785\) 276457.i 0.448630i
\(786\) 0 0
\(787\) 700518. 1.13102 0.565510 0.824742i \(-0.308679\pi\)
0.565510 + 0.824742i \(0.308679\pi\)
\(788\) − 501960.i − 0.808382i
\(789\) 0 0
\(790\) 231731. 0.371305
\(791\) − 100879.i − 0.161231i
\(792\) 0 0
\(793\) 68226.7 0.108495
\(794\) − 571187.i − 0.906019i
\(795\) 0 0
\(796\) 330915. 0.522264
\(797\) − 6535.45i − 0.0102887i −0.999987 0.00514433i \(-0.998363\pi\)
0.999987 0.00514433i \(-0.00163750\pi\)
\(798\) 0 0
\(799\) 225864. 0.353797
\(800\) − 55347.8i − 0.0864809i
\(801\) 0 0
\(802\) −165738. −0.257676
\(803\) − 85060.2i − 0.131915i
\(804\) 0 0
\(805\) 18231.2 0.0281335
\(806\) 225997.i 0.347883i
\(807\) 0 0
\(808\) 351324. 0.538127
\(809\) − 1.10789e6i − 1.69278i −0.532566 0.846388i \(-0.678772\pi\)
0.532566 0.846388i \(-0.321228\pi\)
\(810\) 0 0
\(811\) −961969. −1.46258 −0.731289 0.682067i \(-0.761081\pi\)
−0.731289 + 0.682067i \(0.761081\pi\)
\(812\) − 85776.2i − 0.130093i
\(813\) 0 0
\(814\) −165314. −0.249494
\(815\) 668155.i 1.00592i
\(816\) 0 0
\(817\) −40482.4 −0.0606488
\(818\) − 507208.i − 0.758017i
\(819\) 0 0
\(820\) −307560. −0.457407
\(821\) − 926435.i − 1.37445i −0.726445 0.687224i \(-0.758829\pi\)
0.726445 0.687224i \(-0.241171\pi\)
\(822\) 0 0
\(823\) 608329. 0.898129 0.449064 0.893499i \(-0.351757\pi\)
0.449064 + 0.893499i \(0.351757\pi\)
\(824\) − 387356.i − 0.570500i
\(825\) 0 0
\(826\) −14465.2 −0.0212013
\(827\) 26569.8i 0.0388487i 0.999811 + 0.0194244i \(0.00618335\pi\)
−0.999811 + 0.0194244i \(0.993817\pi\)
\(828\) 0 0
\(829\) 785642. 1.14318 0.571591 0.820538i \(-0.306326\pi\)
0.571591 + 0.820538i \(0.306326\pi\)
\(830\) − 356704.i − 0.517788i
\(831\) 0 0
\(832\) 79581.3 0.114965
\(833\) 39412.5i 0.0567994i
\(834\) 0 0
\(835\) −96776.7 −0.138803
\(836\) − 6259.63i − 0.00895646i
\(837\) 0 0
\(838\) 435030. 0.619485
\(839\) 338072.i 0.480270i 0.970739 + 0.240135i \(0.0771918\pi\)
−0.970739 + 0.240135i \(0.922808\pi\)
\(840\) 0 0
\(841\) 372115. 0.526121
\(842\) 258108.i 0.364063i
\(843\) 0 0
\(844\) −481538. −0.675998
\(845\) − 78649.4i − 0.110149i
\(846\) 0 0
\(847\) −232959. −0.324723
\(848\) − 208612.i − 0.290100i
\(849\) 0 0
\(850\) 99371.0 0.137538
\(851\) 70906.7i 0.0979103i
\(852\) 0 0
\(853\) 742542. 1.02052 0.510262 0.860019i \(-0.329548\pi\)
0.510262 + 0.860019i \(0.329548\pi\)
\(854\) − 22993.5i − 0.0315275i
\(855\) 0 0
\(856\) −92465.2 −0.126192
\(857\) 419964.i 0.571808i 0.958258 + 0.285904i \(0.0922938\pi\)
−0.958258 + 0.285904i \(0.907706\pi\)
\(858\) 0 0
\(859\) 843636. 1.14332 0.571662 0.820490i \(-0.306299\pi\)
0.571662 + 0.820490i \(0.306299\pi\)
\(860\) − 335848.i − 0.454094i
\(861\) 0 0
\(862\) −985766. −1.32666
\(863\) − 346514.i − 0.465264i −0.972565 0.232632i \(-0.925266\pi\)
0.972565 0.232632i \(-0.0747338\pi\)
\(864\) 0 0
\(865\) 253042. 0.338190
\(866\) 243705.i 0.324959i
\(867\) 0 0
\(868\) 76164.8 0.101092
\(869\) − 208239.i − 0.275755i
\(870\) 0 0
\(871\) 273863. 0.360992
\(872\) 217923.i 0.286596i
\(873\) 0 0
\(874\) −2684.89 −0.00351482
\(875\) − 307995.i − 0.402280i
\(876\) 0 0
\(877\) 316397. 0.411370 0.205685 0.978618i \(-0.434058\pi\)
0.205685 + 0.978618i \(0.434058\pi\)
\(878\) 803524.i 1.04234i
\(879\) 0 0
\(880\) 51930.8 0.0670594
\(881\) 67739.5i 0.0872751i 0.999047 + 0.0436375i \(0.0138947\pi\)
−0.999047 + 0.0436375i \(0.986105\pi\)
\(882\) 0 0
\(883\) 1.13315e6 1.45333 0.726666 0.686991i \(-0.241069\pi\)
0.726666 + 0.686991i \(0.241069\pi\)
\(884\) 142880.i 0.182838i
\(885\) 0 0
\(886\) −220270. −0.280600
\(887\) − 925863.i − 1.17679i −0.808573 0.588396i \(-0.799760\pi\)
0.808573 0.588396i \(-0.200240\pi\)
\(888\) 0 0
\(889\) −66959.5 −0.0847244
\(890\) − 360731.i − 0.455411i
\(891\) 0 0
\(892\) −438130. −0.550647
\(893\) 33867.5i 0.0424698i
\(894\) 0 0
\(895\) −288509. −0.360174
\(896\) − 26820.2i − 0.0334077i
\(897\) 0 0
\(898\) −246712. −0.305942
\(899\) − 297610.i − 0.368237i
\(900\) 0 0
\(901\) 374540. 0.461370
\(902\) 276381.i 0.339700i
\(903\) 0 0
\(904\) −123251. −0.150818
\(905\) 474217.i 0.579002i
\(906\) 0 0
\(907\) 478626. 0.581811 0.290905 0.956752i \(-0.406043\pi\)
0.290905 + 0.956752i \(0.406043\pi\)
\(908\) − 189276.i − 0.229575i
\(909\) 0 0
\(910\) 145477. 0.175676
\(911\) − 388418.i − 0.468018i −0.972234 0.234009i \(-0.924815\pi\)
0.972234 0.234009i \(-0.0751845\pi\)
\(912\) 0 0
\(913\) −320543. −0.384543
\(914\) − 170707.i − 0.204342i
\(915\) 0 0
\(916\) 146918. 0.175099
\(917\) 251904.i 0.299569i
\(918\) 0 0
\(919\) −807823. −0.956501 −0.478250 0.878223i \(-0.658729\pi\)
−0.478250 + 0.878223i \(0.658729\pi\)
\(920\) − 22274.2i − 0.0263165i
\(921\) 0 0
\(922\) 387071. 0.455333
\(923\) − 214225.i − 0.251459i
\(924\) 0 0
\(925\) 393510. 0.459910
\(926\) − 474189.i − 0.553005i
\(927\) 0 0
\(928\) −104798. −0.121691
\(929\) 1.62506e6i 1.88294i 0.337094 + 0.941471i \(0.390556\pi\)
−0.337094 + 0.941471i \(0.609444\pi\)
\(930\) 0 0
\(931\) −5909.75 −0.00681819
\(932\) − 570726.i − 0.657046i
\(933\) 0 0
\(934\) −534237. −0.612407
\(935\) 93236.2i 0.106650i
\(936\) 0 0
\(937\) −1.17872e6 −1.34255 −0.671275 0.741208i \(-0.734253\pi\)
−0.671275 + 0.741208i \(0.734253\pi\)
\(938\) − 92296.3i − 0.104901i
\(939\) 0 0
\(940\) −280970. −0.317983
\(941\) − 418331.i − 0.472433i −0.971700 0.236217i \(-0.924093\pi\)
0.971700 0.236217i \(-0.0759075\pi\)
\(942\) 0 0
\(943\) 118546. 0.133310
\(944\) 17673.0i 0.0198320i
\(945\) 0 0
\(946\) −301801. −0.337240
\(947\) − 860451.i − 0.959458i −0.877417 0.479729i \(-0.840735\pi\)
0.877417 0.479729i \(-0.159265\pi\)
\(948\) 0 0
\(949\) −291127. −0.323259
\(950\) 14900.3i 0.0165100i
\(951\) 0 0
\(952\) 48152.8 0.0531309
\(953\) − 555766.i − 0.611937i −0.952042 0.305968i \(-0.901020\pi\)
0.952042 0.305968i \(-0.0989801\pi\)
\(954\) 0 0
\(955\) −38414.7 −0.0421202
\(956\) − 349010.i − 0.381876i
\(957\) 0 0
\(958\) 1.12078e6 1.22121
\(959\) 26983.6i 0.0293402i
\(960\) 0 0
\(961\) −659259. −0.713854
\(962\) 565805.i 0.611387i
\(963\) 0 0
\(964\) −99577.9 −0.107154
\(965\) 470715.i 0.505479i
\(966\) 0 0
\(967\) −862383. −0.922247 −0.461124 0.887336i \(-0.652554\pi\)
−0.461124 + 0.887336i \(0.652554\pi\)
\(968\) 284622.i 0.303751i
\(969\) 0 0
\(970\) −56045.8 −0.0595661
\(971\) − 1.69499e6i − 1.79775i −0.438209 0.898873i \(-0.644387\pi\)
0.438209 0.898873i \(-0.355613\pi\)
\(972\) 0 0
\(973\) 53403.8 0.0564087
\(974\) 581572.i 0.613035i
\(975\) 0 0
\(976\) −28092.7 −0.0294913
\(977\) − 1.20992e6i − 1.26755i −0.773516 0.633777i \(-0.781504\pi\)
0.773516 0.633777i \(-0.218496\pi\)
\(978\) 0 0
\(979\) −324161. −0.338217
\(980\) − 49028.1i − 0.0510497i
\(981\) 0 0
\(982\) −414205. −0.429529
\(983\) − 1.28838e6i − 1.33332i −0.745360 0.666662i \(-0.767723\pi\)
0.745360 0.666662i \(-0.232277\pi\)
\(984\) 0 0
\(985\) −1.12109e6 −1.15549
\(986\) − 188154.i − 0.193535i
\(987\) 0 0
\(988\) −21424.2 −0.0219478
\(989\) 129449.i 0.132345i
\(990\) 0 0
\(991\) −1.23803e6 −1.26062 −0.630312 0.776342i \(-0.717073\pi\)
−0.630312 + 0.776342i \(0.717073\pi\)
\(992\) − 93055.5i − 0.0945625i
\(993\) 0 0
\(994\) −72197.4 −0.0730716
\(995\) − 739073.i − 0.746520i
\(996\) 0 0
\(997\) −288690. −0.290430 −0.145215 0.989400i \(-0.546387\pi\)
−0.145215 + 0.989400i \(0.546387\pi\)
\(998\) 753313.i 0.756335i
\(999\) 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.323.3 8
3.2 odd 2 inner 378.5.b.b.323.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.5.b.b.323.3 8 1.1 even 1 trivial
378.5.b.b.323.6 yes 8 3.2 odd 2 inner