Properties

Label 320.6.a.x.1.3
Level 320
Weight 6
Character 320.1
Self dual yes
Analytic conductor 51.323
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.39180.1
Defining polynomial: \(x^{3} - x^{2} - 36 x - 24\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-5.11788\) of defining polynomial
Character \(\chi\) \(=\) 320.1

$q$-expansion

\(f(q)\) \(=\) \(q+18.4715 q^{3} +25.0000 q^{5} +121.899 q^{7} +98.1968 q^{9} +O(q^{10})\) \(q+18.4715 q^{3} +25.0000 q^{5} +121.899 q^{7} +98.1968 q^{9} +438.282 q^{11} +758.285 q^{13} +461.788 q^{15} -1534.86 q^{17} -75.8518 q^{19} +2251.66 q^{21} +3694.27 q^{23} +625.000 q^{25} -2674.73 q^{27} -6323.09 q^{29} +2691.35 q^{31} +8095.74 q^{33} +3047.47 q^{35} +7252.19 q^{37} +14006.7 q^{39} +4913.34 q^{41} -2533.81 q^{43} +2454.92 q^{45} +11380.6 q^{47} -1947.64 q^{49} -28351.1 q^{51} -29442.3 q^{53} +10957.1 q^{55} -1401.10 q^{57} +5681.08 q^{59} +48039.8 q^{61} +11970.1 q^{63} +18957.1 q^{65} -39570.6 q^{67} +68238.7 q^{69} +12622.3 q^{71} +57959.2 q^{73} +11544.7 q^{75} +53426.2 q^{77} -29504.0 q^{79} -73268.2 q^{81} -112209. q^{83} -38371.4 q^{85} -116797. q^{87} +66969.5 q^{89} +92434.2 q^{91} +49713.4 q^{93} -1896.30 q^{95} +131833. q^{97} +43037.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 10q^{3} + 75q^{5} - 6q^{7} + 467q^{9} + O(q^{10}) \) \( 3q - 10q^{3} + 75q^{5} - 6q^{7} + 467q^{9} + 396q^{11} + 354q^{13} - 250q^{15} + 1158q^{17} - 3192q^{19} + 820q^{21} + 6126q^{23} + 1875q^{25} - 13804q^{27} - 426q^{29} + 3276q^{31} - 2408q^{33} - 150q^{35} + 11562q^{37} + 21348q^{39} + 12450q^{41} - 26346q^{43} + 11675q^{45} + 36762q^{47} - 3849q^{49} - 71444q^{51} + 21162q^{53} + 9900q^{55} + 69136q^{57} - 35040q^{59} + 24138q^{61} + 80986q^{63} + 8850q^{65} + 9570q^{67} + 27036q^{69} + 88092q^{71} + 66750q^{73} - 6250q^{75} + 136488q^{77} - 92952q^{79} + 151391q^{81} - 30258q^{83} + 28950q^{85} - 26228q^{87} + 172686q^{89} + 106812q^{91} + 318232q^{93} - 79800q^{95} + 170910q^{97} + 351436q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 18.4715 1.18495 0.592474 0.805590i \(-0.298151\pi\)
0.592474 + 0.805590i \(0.298151\pi\)
\(4\) 0 0
\(5\) 25.0000 0.447214
\(6\) 0 0
\(7\) 121.899 0.940275 0.470138 0.882593i \(-0.344204\pi\)
0.470138 + 0.882593i \(0.344204\pi\)
\(8\) 0 0
\(9\) 98.1968 0.404102
\(10\) 0 0
\(11\) 438.282 1.09213 0.546063 0.837744i \(-0.316126\pi\)
0.546063 + 0.837744i \(0.316126\pi\)
\(12\) 0 0
\(13\) 758.285 1.24444 0.622220 0.782842i \(-0.286231\pi\)
0.622220 + 0.782842i \(0.286231\pi\)
\(14\) 0 0
\(15\) 461.788 0.529925
\(16\) 0 0
\(17\) −1534.86 −1.28809 −0.644044 0.764989i \(-0.722744\pi\)
−0.644044 + 0.764989i \(0.722744\pi\)
\(18\) 0 0
\(19\) −75.8518 −0.0482039 −0.0241019 0.999710i \(-0.507673\pi\)
−0.0241019 + 0.999710i \(0.507673\pi\)
\(20\) 0 0
\(21\) 2251.66 1.11418
\(22\) 0 0
\(23\) 3694.27 1.45616 0.728079 0.685493i \(-0.240413\pi\)
0.728079 + 0.685493i \(0.240413\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) −2674.73 −0.706108
\(28\) 0 0
\(29\) −6323.09 −1.39616 −0.698079 0.716021i \(-0.745961\pi\)
−0.698079 + 0.716021i \(0.745961\pi\)
\(30\) 0 0
\(31\) 2691.35 0.502998 0.251499 0.967857i \(-0.419076\pi\)
0.251499 + 0.967857i \(0.419076\pi\)
\(32\) 0 0
\(33\) 8095.74 1.29411
\(34\) 0 0
\(35\) 3047.47 0.420504
\(36\) 0 0
\(37\) 7252.19 0.870893 0.435446 0.900215i \(-0.356591\pi\)
0.435446 + 0.900215i \(0.356591\pi\)
\(38\) 0 0
\(39\) 14006.7 1.47460
\(40\) 0 0
\(41\) 4913.34 0.456475 0.228238 0.973605i \(-0.426704\pi\)
0.228238 + 0.973605i \(0.426704\pi\)
\(42\) 0 0
\(43\) −2533.81 −0.208979 −0.104490 0.994526i \(-0.533321\pi\)
−0.104490 + 0.994526i \(0.533321\pi\)
\(44\) 0 0
\(45\) 2454.92 0.180720
\(46\) 0 0
\(47\) 11380.6 0.751487 0.375743 0.926724i \(-0.377387\pi\)
0.375743 + 0.926724i \(0.377387\pi\)
\(48\) 0 0
\(49\) −1947.64 −0.115882
\(50\) 0 0
\(51\) −28351.1 −1.52632
\(52\) 0 0
\(53\) −29442.3 −1.43973 −0.719866 0.694113i \(-0.755797\pi\)
−0.719866 + 0.694113i \(0.755797\pi\)
\(54\) 0 0
\(55\) 10957.1 0.488413
\(56\) 0 0
\(57\) −1401.10 −0.0571191
\(58\) 0 0
\(59\) 5681.08 0.212472 0.106236 0.994341i \(-0.466120\pi\)
0.106236 + 0.994341i \(0.466120\pi\)
\(60\) 0 0
\(61\) 48039.8 1.65301 0.826507 0.562926i \(-0.190324\pi\)
0.826507 + 0.562926i \(0.190324\pi\)
\(62\) 0 0
\(63\) 11970.1 0.379967
\(64\) 0 0
\(65\) 18957.1 0.556531
\(66\) 0 0
\(67\) −39570.6 −1.07693 −0.538463 0.842649i \(-0.680995\pi\)
−0.538463 + 0.842649i \(0.680995\pi\)
\(68\) 0 0
\(69\) 68238.7 1.72547
\(70\) 0 0
\(71\) 12622.3 0.297161 0.148581 0.988900i \(-0.452530\pi\)
0.148581 + 0.988900i \(0.452530\pi\)
\(72\) 0 0
\(73\) 57959.2 1.27296 0.636481 0.771292i \(-0.280389\pi\)
0.636481 + 0.771292i \(0.280389\pi\)
\(74\) 0 0
\(75\) 11544.7 0.236990
\(76\) 0 0
\(77\) 53426.2 1.02690
\(78\) 0 0
\(79\) −29504.0 −0.531879 −0.265939 0.963990i \(-0.585682\pi\)
−0.265939 + 0.963990i \(0.585682\pi\)
\(80\) 0 0
\(81\) −73268.2 −1.24080
\(82\) 0 0
\(83\) −112209. −1.78786 −0.893928 0.448211i \(-0.852061\pi\)
−0.893928 + 0.448211i \(0.852061\pi\)
\(84\) 0 0
\(85\) −38371.4 −0.576050
\(86\) 0 0
\(87\) −116797. −1.65437
\(88\) 0 0
\(89\) 66969.5 0.896194 0.448097 0.893985i \(-0.352102\pi\)
0.448097 + 0.893985i \(0.352102\pi\)
\(90\) 0 0
\(91\) 92434.2 1.17012
\(92\) 0 0
\(93\) 49713.4 0.596027
\(94\) 0 0
\(95\) −1896.30 −0.0215574
\(96\) 0 0
\(97\) 131833. 1.42264 0.711322 0.702867i \(-0.248097\pi\)
0.711322 + 0.702867i \(0.248097\pi\)
\(98\) 0 0
\(99\) 43037.9 0.441330
\(100\) 0 0
\(101\) 30688.2 0.299343 0.149671 0.988736i \(-0.452178\pi\)
0.149671 + 0.988736i \(0.452178\pi\)
\(102\) 0 0
\(103\) −140744. −1.30718 −0.653591 0.756848i \(-0.726738\pi\)
−0.653591 + 0.756848i \(0.726738\pi\)
\(104\) 0 0
\(105\) 56291.5 0.498275
\(106\) 0 0
\(107\) −14196.9 −0.119876 −0.0599381 0.998202i \(-0.519090\pi\)
−0.0599381 + 0.998202i \(0.519090\pi\)
\(108\) 0 0
\(109\) −9873.09 −0.0795952 −0.0397976 0.999208i \(-0.512671\pi\)
−0.0397976 + 0.999208i \(0.512671\pi\)
\(110\) 0 0
\(111\) 133959. 1.03196
\(112\) 0 0
\(113\) 245866. 1.81135 0.905675 0.423973i \(-0.139365\pi\)
0.905675 + 0.423973i \(0.139365\pi\)
\(114\) 0 0
\(115\) 92356.7 0.651214
\(116\) 0 0
\(117\) 74461.2 0.502881
\(118\) 0 0
\(119\) −187097. −1.21116
\(120\) 0 0
\(121\) 31040.5 0.192737
\(122\) 0 0
\(123\) 90756.9 0.540900
\(124\) 0 0
\(125\) 15625.0 0.0894427
\(126\) 0 0
\(127\) 88403.3 0.486361 0.243181 0.969981i \(-0.421809\pi\)
0.243181 + 0.969981i \(0.421809\pi\)
\(128\) 0 0
\(129\) −46803.3 −0.247630
\(130\) 0 0
\(131\) 274297. 1.39651 0.698253 0.715851i \(-0.253961\pi\)
0.698253 + 0.715851i \(0.253961\pi\)
\(132\) 0 0
\(133\) −9246.26 −0.0453249
\(134\) 0 0
\(135\) −66868.4 −0.315781
\(136\) 0 0
\(137\) −309183. −1.40739 −0.703694 0.710503i \(-0.748467\pi\)
−0.703694 + 0.710503i \(0.748467\pi\)
\(138\) 0 0
\(139\) 381380. 1.67425 0.837125 0.547011i \(-0.184235\pi\)
0.837125 + 0.547011i \(0.184235\pi\)
\(140\) 0 0
\(141\) 210217. 0.890473
\(142\) 0 0
\(143\) 332343. 1.35908
\(144\) 0 0
\(145\) −158077. −0.624381
\(146\) 0 0
\(147\) −35975.8 −0.137315
\(148\) 0 0
\(149\) 101742. 0.375436 0.187718 0.982223i \(-0.439891\pi\)
0.187718 + 0.982223i \(0.439891\pi\)
\(150\) 0 0
\(151\) 338843. 1.20936 0.604680 0.796469i \(-0.293301\pi\)
0.604680 + 0.796469i \(0.293301\pi\)
\(152\) 0 0
\(153\) −150718. −0.520519
\(154\) 0 0
\(155\) 67283.9 0.224948
\(156\) 0 0
\(157\) −101650. −0.329123 −0.164561 0.986367i \(-0.552621\pi\)
−0.164561 + 0.986367i \(0.552621\pi\)
\(158\) 0 0
\(159\) −543843. −1.70601
\(160\) 0 0
\(161\) 450327. 1.36919
\(162\) 0 0
\(163\) −214136. −0.631279 −0.315640 0.948879i \(-0.602219\pi\)
−0.315640 + 0.948879i \(0.602219\pi\)
\(164\) 0 0
\(165\) 202393. 0.578744
\(166\) 0 0
\(167\) −606370. −1.68247 −0.841234 0.540672i \(-0.818170\pi\)
−0.841234 + 0.540672i \(0.818170\pi\)
\(168\) 0 0
\(169\) 203703. 0.548633
\(170\) 0 0
\(171\) −7448.41 −0.0194793
\(172\) 0 0
\(173\) 426958. 1.08460 0.542301 0.840184i \(-0.317553\pi\)
0.542301 + 0.840184i \(0.317553\pi\)
\(174\) 0 0
\(175\) 76186.9 0.188055
\(176\) 0 0
\(177\) 104938. 0.251768
\(178\) 0 0
\(179\) −682764. −1.59271 −0.796357 0.604826i \(-0.793243\pi\)
−0.796357 + 0.604826i \(0.793243\pi\)
\(180\) 0 0
\(181\) −725666. −1.64642 −0.823209 0.567738i \(-0.807819\pi\)
−0.823209 + 0.567738i \(0.807819\pi\)
\(182\) 0 0
\(183\) 887368. 1.95874
\(184\) 0 0
\(185\) 181305. 0.389475
\(186\) 0 0
\(187\) −672700. −1.40675
\(188\) 0 0
\(189\) −326047. −0.663936
\(190\) 0 0
\(191\) 662000. 1.31303 0.656515 0.754313i \(-0.272030\pi\)
0.656515 + 0.754313i \(0.272030\pi\)
\(192\) 0 0
\(193\) −481557. −0.930582 −0.465291 0.885158i \(-0.654050\pi\)
−0.465291 + 0.885158i \(0.654050\pi\)
\(194\) 0 0
\(195\) 350167. 0.659460
\(196\) 0 0
\(197\) −569264. −1.04508 −0.522538 0.852616i \(-0.675015\pi\)
−0.522538 + 0.852616i \(0.675015\pi\)
\(198\) 0 0
\(199\) −850151. −1.52182 −0.760911 0.648857i \(-0.775247\pi\)
−0.760911 + 0.648857i \(0.775247\pi\)
\(200\) 0 0
\(201\) −730930. −1.27610
\(202\) 0 0
\(203\) −770778. −1.31277
\(204\) 0 0
\(205\) 122834. 0.204142
\(206\) 0 0
\(207\) 362765. 0.588437
\(208\) 0 0
\(209\) −33244.5 −0.0526447
\(210\) 0 0
\(211\) −285552. −0.441548 −0.220774 0.975325i \(-0.570858\pi\)
−0.220774 + 0.975325i \(0.570858\pi\)
\(212\) 0 0
\(213\) 233153. 0.352120
\(214\) 0 0
\(215\) −63345.3 −0.0934583
\(216\) 0 0
\(217\) 328073. 0.472957
\(218\) 0 0
\(219\) 1.07059e6 1.50839
\(220\) 0 0
\(221\) −1.16386e6 −1.60295
\(222\) 0 0
\(223\) 279669. 0.376602 0.188301 0.982111i \(-0.439702\pi\)
0.188301 + 0.982111i \(0.439702\pi\)
\(224\) 0 0
\(225\) 61373.0 0.0808204
\(226\) 0 0
\(227\) −1.27295e6 −1.63964 −0.819819 0.572623i \(-0.805926\pi\)
−0.819819 + 0.572623i \(0.805926\pi\)
\(228\) 0 0
\(229\) −584351. −0.736352 −0.368176 0.929756i \(-0.620018\pi\)
−0.368176 + 0.929756i \(0.620018\pi\)
\(230\) 0 0
\(231\) 986863. 1.21682
\(232\) 0 0
\(233\) −658722. −0.794900 −0.397450 0.917624i \(-0.630105\pi\)
−0.397450 + 0.917624i \(0.630105\pi\)
\(234\) 0 0
\(235\) 284515. 0.336075
\(236\) 0 0
\(237\) −544983. −0.630248
\(238\) 0 0
\(239\) −664166. −0.752111 −0.376055 0.926597i \(-0.622720\pi\)
−0.376055 + 0.926597i \(0.622720\pi\)
\(240\) 0 0
\(241\) −192140. −0.213095 −0.106548 0.994308i \(-0.533980\pi\)
−0.106548 + 0.994308i \(0.533980\pi\)
\(242\) 0 0
\(243\) −703414. −0.764180
\(244\) 0 0
\(245\) −48690.9 −0.0518242
\(246\) 0 0
\(247\) −57517.3 −0.0599869
\(248\) 0 0
\(249\) −2.07267e6 −2.11852
\(250\) 0 0
\(251\) −1.20021e6 −1.20246 −0.601231 0.799075i \(-0.705323\pi\)
−0.601231 + 0.799075i \(0.705323\pi\)
\(252\) 0 0
\(253\) 1.61913e6 1.59031
\(254\) 0 0
\(255\) −708778. −0.682589
\(256\) 0 0
\(257\) −494440. −0.466961 −0.233481 0.972361i \(-0.575012\pi\)
−0.233481 + 0.972361i \(0.575012\pi\)
\(258\) 0 0
\(259\) 884034. 0.818879
\(260\) 0 0
\(261\) −620907. −0.564190
\(262\) 0 0
\(263\) 535899. 0.477742 0.238871 0.971051i \(-0.423223\pi\)
0.238871 + 0.971051i \(0.423223\pi\)
\(264\) 0 0
\(265\) −736056. −0.643867
\(266\) 0 0
\(267\) 1.23703e6 1.06194
\(268\) 0 0
\(269\) 886621. 0.747063 0.373532 0.927617i \(-0.378147\pi\)
0.373532 + 0.927617i \(0.378147\pi\)
\(270\) 0 0
\(271\) −1.31305e6 −1.08607 −0.543036 0.839710i \(-0.682725\pi\)
−0.543036 + 0.839710i \(0.682725\pi\)
\(272\) 0 0
\(273\) 1.70740e6 1.38653
\(274\) 0 0
\(275\) 273927. 0.218425
\(276\) 0 0
\(277\) −175211. −0.137203 −0.0686013 0.997644i \(-0.521854\pi\)
−0.0686013 + 0.997644i \(0.521854\pi\)
\(278\) 0 0
\(279\) 264282. 0.203263
\(280\) 0 0
\(281\) −1.78398e6 −1.34779 −0.673897 0.738826i \(-0.735381\pi\)
−0.673897 + 0.738826i \(0.735381\pi\)
\(282\) 0 0
\(283\) −1.98478e6 −1.47315 −0.736573 0.676358i \(-0.763557\pi\)
−0.736573 + 0.676358i \(0.763557\pi\)
\(284\) 0 0
\(285\) −35027.4 −0.0255444
\(286\) 0 0
\(287\) 598931. 0.429212
\(288\) 0 0
\(289\) 935925. 0.659168
\(290\) 0 0
\(291\) 2.43516e6 1.68576
\(292\) 0 0
\(293\) −2.44146e6 −1.66143 −0.830714 0.556700i \(-0.812067\pi\)
−0.830714 + 0.556700i \(0.812067\pi\)
\(294\) 0 0
\(295\) 142027. 0.0950202
\(296\) 0 0
\(297\) −1.17229e6 −0.771158
\(298\) 0 0
\(299\) 2.80131e6 1.81210
\(300\) 0 0
\(301\) −308869. −0.196498
\(302\) 0 0
\(303\) 566858. 0.354706
\(304\) 0 0
\(305\) 1.20100e6 0.739251
\(306\) 0 0
\(307\) 2.53943e6 1.53777 0.768883 0.639390i \(-0.220813\pi\)
0.768883 + 0.639390i \(0.220813\pi\)
\(308\) 0 0
\(309\) −2.59975e6 −1.54894
\(310\) 0 0
\(311\) 1.86583e6 1.09388 0.546941 0.837171i \(-0.315792\pi\)
0.546941 + 0.837171i \(0.315792\pi\)
\(312\) 0 0
\(313\) 2.17459e6 1.25463 0.627316 0.778765i \(-0.284153\pi\)
0.627316 + 0.778765i \(0.284153\pi\)
\(314\) 0 0
\(315\) 299252. 0.169926
\(316\) 0 0
\(317\) 1.48421e6 0.829557 0.414779 0.909922i \(-0.363859\pi\)
0.414779 + 0.909922i \(0.363859\pi\)
\(318\) 0 0
\(319\) −2.77130e6 −1.52478
\(320\) 0 0
\(321\) −262238. −0.142047
\(322\) 0 0
\(323\) 116422. 0.0620908
\(324\) 0 0
\(325\) 473928. 0.248888
\(326\) 0 0
\(327\) −182371. −0.0943162
\(328\) 0 0
\(329\) 1.38729e6 0.706604
\(330\) 0 0
\(331\) −2.02880e6 −1.01782 −0.508908 0.860821i \(-0.669950\pi\)
−0.508908 + 0.860821i \(0.669950\pi\)
\(332\) 0 0
\(333\) 712141. 0.351930
\(334\) 0 0
\(335\) −989266. −0.481616
\(336\) 0 0
\(337\) −1.34355e6 −0.644435 −0.322218 0.946666i \(-0.604428\pi\)
−0.322218 + 0.946666i \(0.604428\pi\)
\(338\) 0 0
\(339\) 4.54152e6 2.14636
\(340\) 0 0
\(341\) 1.17957e6 0.549337
\(342\) 0 0
\(343\) −2.28617e6 −1.04924
\(344\) 0 0
\(345\) 1.70597e6 0.771655
\(346\) 0 0
\(347\) −1.52038e6 −0.677843 −0.338921 0.940815i \(-0.610062\pi\)
−0.338921 + 0.940815i \(0.610062\pi\)
\(348\) 0 0
\(349\) 1.80948e6 0.795224 0.397612 0.917554i \(-0.369839\pi\)
0.397612 + 0.917554i \(0.369839\pi\)
\(350\) 0 0
\(351\) −2.02821e6 −0.878710
\(352\) 0 0
\(353\) 2.48843e6 1.06289 0.531446 0.847092i \(-0.321649\pi\)
0.531446 + 0.847092i \(0.321649\pi\)
\(354\) 0 0
\(355\) 315557. 0.132894
\(356\) 0 0
\(357\) −3.45597e6 −1.43516
\(358\) 0 0
\(359\) 1.44154e6 0.590324 0.295162 0.955447i \(-0.404626\pi\)
0.295162 + 0.955447i \(0.404626\pi\)
\(360\) 0 0
\(361\) −2.47035e6 −0.997676
\(362\) 0 0
\(363\) 573365. 0.228383
\(364\) 0 0
\(365\) 1.44898e6 0.569286
\(366\) 0 0
\(367\) 257353. 0.0997388 0.0498694 0.998756i \(-0.484119\pi\)
0.0498694 + 0.998756i \(0.484119\pi\)
\(368\) 0 0
\(369\) 482474. 0.184463
\(370\) 0 0
\(371\) −3.58898e6 −1.35374
\(372\) 0 0
\(373\) 4.55163e6 1.69393 0.846964 0.531651i \(-0.178428\pi\)
0.846964 + 0.531651i \(0.178428\pi\)
\(374\) 0 0
\(375\) 288617. 0.105985
\(376\) 0 0
\(377\) −4.79471e6 −1.73743
\(378\) 0 0
\(379\) −2.96634e6 −1.06077 −0.530387 0.847756i \(-0.677953\pi\)
−0.530387 + 0.847756i \(0.677953\pi\)
\(380\) 0 0
\(381\) 1.63294e6 0.576313
\(382\) 0 0
\(383\) 3.57107e6 1.24395 0.621973 0.783039i \(-0.286331\pi\)
0.621973 + 0.783039i \(0.286331\pi\)
\(384\) 0 0
\(385\) 1.33565e6 0.459243
\(386\) 0 0
\(387\) −248812. −0.0844489
\(388\) 0 0
\(389\) 881685. 0.295420 0.147710 0.989031i \(-0.452810\pi\)
0.147710 + 0.989031i \(0.452810\pi\)
\(390\) 0 0
\(391\) −5.67017e6 −1.87566
\(392\) 0 0
\(393\) 5.06668e6 1.65479
\(394\) 0 0
\(395\) −737599. −0.237863
\(396\) 0 0
\(397\) 2.56117e6 0.815570 0.407785 0.913078i \(-0.366301\pi\)
0.407785 + 0.913078i \(0.366301\pi\)
\(398\) 0 0
\(399\) −170792. −0.0537077
\(400\) 0 0
\(401\) −3.37596e6 −1.04842 −0.524211 0.851589i \(-0.675640\pi\)
−0.524211 + 0.851589i \(0.675640\pi\)
\(402\) 0 0
\(403\) 2.04081e6 0.625952
\(404\) 0 0
\(405\) −1.83171e6 −0.554904
\(406\) 0 0
\(407\) 3.17851e6 0.951124
\(408\) 0 0
\(409\) −2.56437e6 −0.758006 −0.379003 0.925395i \(-0.623733\pi\)
−0.379003 + 0.925395i \(0.623733\pi\)
\(410\) 0 0
\(411\) −5.71107e6 −1.66768
\(412\) 0 0
\(413\) 692518. 0.199782
\(414\) 0 0
\(415\) −2.80522e6 −0.799553
\(416\) 0 0
\(417\) 7.04466e6 1.98390
\(418\) 0 0
\(419\) −6.64881e6 −1.85016 −0.925079 0.379775i \(-0.876001\pi\)
−0.925079 + 0.379775i \(0.876001\pi\)
\(420\) 0 0
\(421\) −4.98840e6 −1.37169 −0.685846 0.727747i \(-0.740568\pi\)
−0.685846 + 0.727747i \(0.740568\pi\)
\(422\) 0 0
\(423\) 1.11754e6 0.303677
\(424\) 0 0
\(425\) −959285. −0.257617
\(426\) 0 0
\(427\) 5.85600e6 1.55429
\(428\) 0 0
\(429\) 6.13888e6 1.61045
\(430\) 0 0
\(431\) 802200. 0.208012 0.104006 0.994577i \(-0.466834\pi\)
0.104006 + 0.994577i \(0.466834\pi\)
\(432\) 0 0
\(433\) 5.80978e6 1.48916 0.744578 0.667535i \(-0.232651\pi\)
0.744578 + 0.667535i \(0.232651\pi\)
\(434\) 0 0
\(435\) −2.91993e6 −0.739859
\(436\) 0 0
\(437\) −280217. −0.0701925
\(438\) 0 0
\(439\) 585972. 0.145116 0.0725581 0.997364i \(-0.476884\pi\)
0.0725581 + 0.997364i \(0.476884\pi\)
\(440\) 0 0
\(441\) −191252. −0.0468283
\(442\) 0 0
\(443\) −4.41675e6 −1.06929 −0.534643 0.845078i \(-0.679554\pi\)
−0.534643 + 0.845078i \(0.679554\pi\)
\(444\) 0 0
\(445\) 1.67424e6 0.400790
\(446\) 0 0
\(447\) 1.87934e6 0.444872
\(448\) 0 0
\(449\) 2.28408e6 0.534681 0.267340 0.963602i \(-0.413855\pi\)
0.267340 + 0.963602i \(0.413855\pi\)
\(450\) 0 0
\(451\) 2.15343e6 0.498528
\(452\) 0 0
\(453\) 6.25893e6 1.43303
\(454\) 0 0
\(455\) 2.31086e6 0.523292
\(456\) 0 0
\(457\) 3.43086e6 0.768444 0.384222 0.923241i \(-0.374470\pi\)
0.384222 + 0.923241i \(0.374470\pi\)
\(458\) 0 0
\(459\) 4.10533e6 0.909529
\(460\) 0 0
\(461\) −674699. −0.147862 −0.0739311 0.997263i \(-0.523555\pi\)
−0.0739311 + 0.997263i \(0.523555\pi\)
\(462\) 0 0
\(463\) 5.78627e6 1.25443 0.627215 0.778846i \(-0.284195\pi\)
0.627215 + 0.778846i \(0.284195\pi\)
\(464\) 0 0
\(465\) 1.24283e6 0.266551
\(466\) 0 0
\(467\) −6.88097e6 −1.46002 −0.730008 0.683439i \(-0.760484\pi\)
−0.730008 + 0.683439i \(0.760484\pi\)
\(468\) 0 0
\(469\) −4.82362e6 −1.01261
\(470\) 0 0
\(471\) −1.87763e6 −0.389993
\(472\) 0 0
\(473\) −1.11052e6 −0.228231
\(474\) 0 0
\(475\) −47407.4 −0.00964078
\(476\) 0 0
\(477\) −2.89114e6 −0.581798
\(478\) 0 0
\(479\) 4.94000e6 0.983757 0.491879 0.870664i \(-0.336310\pi\)
0.491879 + 0.870664i \(0.336310\pi\)
\(480\) 0 0
\(481\) 5.49923e6 1.08377
\(482\) 0 0
\(483\) 8.31823e6 1.62242
\(484\) 0 0
\(485\) 3.29583e6 0.636225
\(486\) 0 0
\(487\) 5.21802e6 0.996973 0.498487 0.866897i \(-0.333889\pi\)
0.498487 + 0.866897i \(0.333889\pi\)
\(488\) 0 0
\(489\) −3.95542e6 −0.748033
\(490\) 0 0
\(491\) 4.20805e6 0.787730 0.393865 0.919168i \(-0.371138\pi\)
0.393865 + 0.919168i \(0.371138\pi\)
\(492\) 0 0
\(493\) 9.70503e6 1.79837
\(494\) 0 0
\(495\) 1.07595e6 0.197369
\(496\) 0 0
\(497\) 1.53864e6 0.279413
\(498\) 0 0
\(499\) −7.74013e6 −1.39154 −0.695771 0.718263i \(-0.744937\pi\)
−0.695771 + 0.718263i \(0.744937\pi\)
\(500\) 0 0
\(501\) −1.12006e7 −1.99364
\(502\) 0 0
\(503\) −6.20893e6 −1.09420 −0.547100 0.837067i \(-0.684268\pi\)
−0.547100 + 0.837067i \(0.684268\pi\)
\(504\) 0 0
\(505\) 767206. 0.133870
\(506\) 0 0
\(507\) 3.76271e6 0.650101
\(508\) 0 0
\(509\) −8.43892e6 −1.44375 −0.721876 0.692022i \(-0.756720\pi\)
−0.721876 + 0.692022i \(0.756720\pi\)
\(510\) 0 0
\(511\) 7.06517e6 1.19693
\(512\) 0 0
\(513\) 202883. 0.0340372
\(514\) 0 0
\(515\) −3.51859e6 −0.584590
\(516\) 0 0
\(517\) 4.98793e6 0.820717
\(518\) 0 0
\(519\) 7.88657e6 1.28520
\(520\) 0 0
\(521\) −6.61888e6 −1.06829 −0.534146 0.845392i \(-0.679367\pi\)
−0.534146 + 0.845392i \(0.679367\pi\)
\(522\) 0 0
\(523\) 5.48657e6 0.877095 0.438548 0.898708i \(-0.355493\pi\)
0.438548 + 0.898708i \(0.355493\pi\)
\(524\) 0 0
\(525\) 1.40729e6 0.222835
\(526\) 0 0
\(527\) −4.13084e6 −0.647906
\(528\) 0 0
\(529\) 7.21127e6 1.12040
\(530\) 0 0
\(531\) 557864. 0.0858602
\(532\) 0 0
\(533\) 3.72571e6 0.568057
\(534\) 0 0
\(535\) −354922. −0.0536103
\(536\) 0 0
\(537\) −1.26117e7 −1.88728
\(538\) 0 0
\(539\) −853614. −0.126558
\(540\) 0 0
\(541\) 427673. 0.0628230 0.0314115 0.999507i \(-0.490000\pi\)
0.0314115 + 0.999507i \(0.490000\pi\)
\(542\) 0 0
\(543\) −1.34041e7 −1.95092
\(544\) 0 0
\(545\) −246827. −0.0355960
\(546\) 0 0
\(547\) 8.94393e6 1.27809 0.639043 0.769171i \(-0.279331\pi\)
0.639043 + 0.769171i \(0.279331\pi\)
\(548\) 0 0
\(549\) 4.71736e6 0.667987
\(550\) 0 0
\(551\) 479618. 0.0673002
\(552\) 0 0
\(553\) −3.59650e6 −0.500112
\(554\) 0 0
\(555\) 3.34897e6 0.461508
\(556\) 0 0
\(557\) −7.93887e6 −1.08423 −0.542114 0.840305i \(-0.682376\pi\)
−0.542114 + 0.840305i \(0.682376\pi\)
\(558\) 0 0
\(559\) −1.92135e6 −0.260062
\(560\) 0 0
\(561\) −1.24258e7 −1.66693
\(562\) 0 0
\(563\) 1.61543e6 0.214792 0.107396 0.994216i \(-0.465749\pi\)
0.107396 + 0.994216i \(0.465749\pi\)
\(564\) 0 0
\(565\) 6.14665e6 0.810060
\(566\) 0 0
\(567\) −8.93132e6 −1.16670
\(568\) 0 0
\(569\) 1.97353e6 0.255543 0.127771 0.991804i \(-0.459218\pi\)
0.127771 + 0.991804i \(0.459218\pi\)
\(570\) 0 0
\(571\) 7.18140e6 0.921762 0.460881 0.887462i \(-0.347533\pi\)
0.460881 + 0.887462i \(0.347533\pi\)
\(572\) 0 0
\(573\) 1.22281e7 1.55587
\(574\) 0 0
\(575\) 2.30892e6 0.291232
\(576\) 0 0
\(577\) −9.61498e6 −1.20229 −0.601145 0.799140i \(-0.705288\pi\)
−0.601145 + 0.799140i \(0.705288\pi\)
\(578\) 0 0
\(579\) −8.89509e6 −1.10269
\(580\) 0 0
\(581\) −1.36782e7 −1.68108
\(582\) 0 0
\(583\) −1.29040e7 −1.57237
\(584\) 0 0
\(585\) 1.86153e6 0.224895
\(586\) 0 0
\(587\) −2.66522e6 −0.319255 −0.159627 0.987177i \(-0.551029\pi\)
−0.159627 + 0.987177i \(0.551029\pi\)
\(588\) 0 0
\(589\) −204144. −0.0242465
\(590\) 0 0
\(591\) −1.05152e7 −1.23836
\(592\) 0 0
\(593\) 5.83235e6 0.681093 0.340547 0.940228i \(-0.389388\pi\)
0.340547 + 0.940228i \(0.389388\pi\)
\(594\) 0 0
\(595\) −4.67743e6 −0.541646
\(596\) 0 0
\(597\) −1.57036e7 −1.80328
\(598\) 0 0
\(599\) 6.39890e6 0.728683 0.364341 0.931265i \(-0.381294\pi\)
0.364341 + 0.931265i \(0.381294\pi\)
\(600\) 0 0
\(601\) 1.00505e7 1.13502 0.567509 0.823367i \(-0.307907\pi\)
0.567509 + 0.823367i \(0.307907\pi\)
\(602\) 0 0
\(603\) −3.88571e6 −0.435188
\(604\) 0 0
\(605\) 776012. 0.0861946
\(606\) 0 0
\(607\) 1.83636e6 0.202295 0.101148 0.994871i \(-0.467749\pi\)
0.101148 + 0.994871i \(0.467749\pi\)
\(608\) 0 0
\(609\) −1.42374e7 −1.55557
\(610\) 0 0
\(611\) 8.62976e6 0.935181
\(612\) 0 0
\(613\) −1.33155e7 −1.43122 −0.715611 0.698499i \(-0.753852\pi\)
−0.715611 + 0.698499i \(0.753852\pi\)
\(614\) 0 0
\(615\) 2.26892e6 0.241898
\(616\) 0 0
\(617\) 3.09296e6 0.327085 0.163543 0.986536i \(-0.447708\pi\)
0.163543 + 0.986536i \(0.447708\pi\)
\(618\) 0 0
\(619\) 9.64802e6 1.01207 0.506036 0.862512i \(-0.331110\pi\)
0.506036 + 0.862512i \(0.331110\pi\)
\(620\) 0 0
\(621\) −9.88118e6 −1.02821
\(622\) 0 0
\(623\) 8.16352e6 0.842669
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) −614077. −0.0623812
\(628\) 0 0
\(629\) −1.11311e7 −1.12179
\(630\) 0 0
\(631\) −1.72048e7 −1.72019 −0.860096 0.510132i \(-0.829597\pi\)
−0.860096 + 0.510132i \(0.829597\pi\)
\(632\) 0 0
\(633\) −5.27457e6 −0.523212
\(634\) 0 0
\(635\) 2.21008e6 0.217507
\(636\) 0 0
\(637\) −1.47686e6 −0.144209
\(638\) 0 0
\(639\) 1.23947e6 0.120083
\(640\) 0 0
\(641\) 1.86292e6 0.179081 0.0895403 0.995983i \(-0.471460\pi\)
0.0895403 + 0.995983i \(0.471460\pi\)
\(642\) 0 0
\(643\) 1.39819e7 1.33364 0.666821 0.745218i \(-0.267655\pi\)
0.666821 + 0.745218i \(0.267655\pi\)
\(644\) 0 0
\(645\) −1.17008e6 −0.110743
\(646\) 0 0
\(647\) −1.28310e7 −1.20504 −0.602520 0.798104i \(-0.705837\pi\)
−0.602520 + 0.798104i \(0.705837\pi\)
\(648\) 0 0
\(649\) 2.48992e6 0.232045
\(650\) 0 0
\(651\) 6.06001e6 0.560430
\(652\) 0 0
\(653\) −423926. −0.0389051 −0.0194526 0.999811i \(-0.506192\pi\)
−0.0194526 + 0.999811i \(0.506192\pi\)
\(654\) 0 0
\(655\) 6.85743e6 0.624537
\(656\) 0 0
\(657\) 5.69141e6 0.514406
\(658\) 0 0
\(659\) −1.64781e7 −1.47807 −0.739035 0.673668i \(-0.764718\pi\)
−0.739035 + 0.673668i \(0.764718\pi\)
\(660\) 0 0
\(661\) 1.71556e7 1.52723 0.763613 0.645674i \(-0.223423\pi\)
0.763613 + 0.645674i \(0.223423\pi\)
\(662\) 0 0
\(663\) −2.14982e7 −1.89941
\(664\) 0 0
\(665\) −231157. −0.0202699
\(666\) 0 0
\(667\) −2.33592e7 −2.03303
\(668\) 0 0
\(669\) 5.16592e6 0.446254
\(670\) 0 0
\(671\) 2.10550e7 1.80530
\(672\) 0 0
\(673\) 2.24172e7 1.90785 0.953923 0.300053i \(-0.0970043\pi\)
0.953923 + 0.300053i \(0.0970043\pi\)
\(674\) 0 0
\(675\) −1.67171e6 −0.141222
\(676\) 0 0
\(677\) 5.63329e6 0.472379 0.236190 0.971707i \(-0.424101\pi\)
0.236190 + 0.971707i \(0.424101\pi\)
\(678\) 0 0
\(679\) 1.60704e7 1.33768
\(680\) 0 0
\(681\) −2.35134e7 −1.94289
\(682\) 0 0
\(683\) 1.27876e6 0.104891 0.0524453 0.998624i \(-0.483298\pi\)
0.0524453 + 0.998624i \(0.483298\pi\)
\(684\) 0 0
\(685\) −7.72957e6 −0.629403
\(686\) 0 0
\(687\) −1.07939e7 −0.872539
\(688\) 0 0
\(689\) −2.23256e7 −1.79166
\(690\) 0 0
\(691\) 712345. 0.0567539 0.0283769 0.999597i \(-0.490966\pi\)
0.0283769 + 0.999597i \(0.490966\pi\)
\(692\) 0 0
\(693\) 5.24628e6 0.414972
\(694\) 0 0
\(695\) 9.53449e6 0.748748
\(696\) 0 0
\(697\) −7.54127e6 −0.587980
\(698\) 0 0
\(699\) −1.21676e7 −0.941915
\(700\) 0 0
\(701\) −7.89586e6 −0.606882 −0.303441 0.952850i \(-0.598136\pi\)
−0.303441 + 0.952850i \(0.598136\pi\)
\(702\) 0 0
\(703\) −550091. −0.0419804
\(704\) 0 0
\(705\) 5.25543e6 0.398231
\(706\) 0 0
\(707\) 3.74087e6 0.281464
\(708\) 0 0
\(709\) 812363. 0.0606924 0.0303462 0.999539i \(-0.490339\pi\)
0.0303462 + 0.999539i \(0.490339\pi\)
\(710\) 0 0
\(711\) −2.89719e6 −0.214933
\(712\) 0 0
\(713\) 9.94258e6 0.732446
\(714\) 0 0
\(715\) 8.30858e6 0.607801
\(716\) 0 0
\(717\) −1.22681e7 −0.891212
\(718\) 0 0
\(719\) −1.46975e7 −1.06028 −0.530142 0.847909i \(-0.677861\pi\)
−0.530142 + 0.847909i \(0.677861\pi\)
\(720\) 0 0
\(721\) −1.71565e7 −1.22911
\(722\) 0 0
\(723\) −3.54911e6 −0.252507
\(724\) 0 0
\(725\) −3.95193e6 −0.279231
\(726\) 0 0
\(727\) 946478. 0.0664163 0.0332081 0.999448i \(-0.489428\pi\)
0.0332081 + 0.999448i \(0.489428\pi\)
\(728\) 0 0
\(729\) 4.81105e6 0.335290
\(730\) 0 0
\(731\) 3.88903e6 0.269183
\(732\) 0 0
\(733\) −2.33370e6 −0.160430 −0.0802150 0.996778i \(-0.525561\pi\)
−0.0802150 + 0.996778i \(0.525561\pi\)
\(734\) 0 0
\(735\) −899394. −0.0614090
\(736\) 0 0
\(737\) −1.73431e7 −1.17614
\(738\) 0 0
\(739\) −1.17842e7 −0.793757 −0.396878 0.917871i \(-0.629906\pi\)
−0.396878 + 0.917871i \(0.629906\pi\)
\(740\) 0 0
\(741\) −1.06243e6 −0.0710813
\(742\) 0 0
\(743\) 1.02537e7 0.681413 0.340706 0.940170i \(-0.389334\pi\)
0.340706 + 0.940170i \(0.389334\pi\)
\(744\) 0 0
\(745\) 2.54356e6 0.167900
\(746\) 0 0
\(747\) −1.10186e7 −0.722476
\(748\) 0 0
\(749\) −1.73058e6 −0.112717
\(750\) 0 0
\(751\) 2.02684e7 1.31135 0.655676 0.755043i \(-0.272384\pi\)
0.655676 + 0.755043i \(0.272384\pi\)
\(752\) 0 0
\(753\) −2.21696e7 −1.42486
\(754\) 0 0
\(755\) 8.47106e6 0.540842
\(756\) 0 0
\(757\) 1.54478e7 0.979776 0.489888 0.871785i \(-0.337038\pi\)
0.489888 + 0.871785i \(0.337038\pi\)
\(758\) 0 0
\(759\) 2.99078e7 1.88443
\(760\) 0 0
\(761\) −1.68238e7 −1.05308 −0.526540 0.850150i \(-0.676511\pi\)
−0.526540 + 0.850150i \(0.676511\pi\)
\(762\) 0 0
\(763\) −1.20352e6 −0.0748414
\(764\) 0 0
\(765\) −3.76795e6 −0.232783
\(766\) 0 0
\(767\) 4.30788e6 0.264408
\(768\) 0 0
\(769\) 2.25563e7 1.37547 0.687737 0.725960i \(-0.258604\pi\)
0.687737 + 0.725960i \(0.258604\pi\)
\(770\) 0 0
\(771\) −9.13305e6 −0.553325
\(772\) 0 0
\(773\) 5.87038e6 0.353360 0.176680 0.984268i \(-0.443464\pi\)
0.176680 + 0.984268i \(0.443464\pi\)
\(774\) 0 0
\(775\) 1.68210e6 0.100600
\(776\) 0 0
\(777\) 1.63294e7 0.970329
\(778\) 0 0
\(779\) −372686. −0.0220039
\(780\) 0 0
\(781\) 5.53212e6 0.324537
\(782\) 0 0
\(783\) 1.69126e7 0.985838
\(784\) 0 0
\(785\) −2.54125e6 −0.147188
\(786\) 0 0
\(787\) 2.45154e6 0.141092 0.0705459 0.997509i \(-0.477526\pi\)
0.0705459 + 0.997509i \(0.477526\pi\)
\(788\) 0 0
\(789\) 9.89886e6 0.566099
\(790\) 0 0
\(791\) 2.99708e7 1.70317
\(792\) 0 0
\(793\) 3.64279e7 2.05708
\(794\) 0 0
\(795\) −1.35961e7 −0.762949
\(796\) 0 0
\(797\) 2.18622e7 1.21913 0.609564 0.792737i \(-0.291345\pi\)
0.609564 + 0.792737i \(0.291345\pi\)
\(798\) 0 0
\(799\) −1.74676e7 −0.967980
\(800\) 0 0
\(801\) 6.57619e6 0.362154
\(802\) 0 0
\(803\) 2.54025e7 1.39023
\(804\) 0 0
\(805\) 1.12582e7 0.612320
\(806\) 0 0
\(807\) 1.63772e7 0.885231
\(808\) 0 0
\(809\) 2.66048e7 1.42919 0.714593 0.699541i \(-0.246612\pi\)
0.714593 + 0.699541i \(0.246612\pi\)
\(810\) 0 0
\(811\) 1.44067e7 0.769152 0.384576 0.923093i \(-0.374348\pi\)
0.384576 + 0.923093i \(0.374348\pi\)
\(812\) 0 0
\(813\) −2.42540e7 −1.28694
\(814\) 0 0
\(815\) −5.35341e6 −0.282317
\(816\) 0 0
\(817\) 192194. 0.0100736
\(818\) 0 0
\(819\) 9.07674e6 0.472847
\(820\) 0 0
\(821\) −1.88597e6 −0.0976508 −0.0488254 0.998807i \(-0.515548\pi\)
−0.0488254 + 0.998807i \(0.515548\pi\)
\(822\) 0 0
\(823\) 1.52832e7 0.786530 0.393265 0.919425i \(-0.371346\pi\)
0.393265 + 0.919425i \(0.371346\pi\)
\(824\) 0 0
\(825\) 5.05984e6 0.258822
\(826\) 0 0
\(827\) 2.78935e6 0.141821 0.0709103 0.997483i \(-0.477410\pi\)
0.0709103 + 0.997483i \(0.477410\pi\)
\(828\) 0 0
\(829\) 1.87077e6 0.0945440 0.0472720 0.998882i \(-0.484947\pi\)
0.0472720 + 0.998882i \(0.484947\pi\)
\(830\) 0 0
\(831\) −3.23641e6 −0.162578
\(832\) 0 0
\(833\) 2.98934e6 0.149267
\(834\) 0 0
\(835\) −1.51593e7 −0.752422
\(836\) 0 0
\(837\) −7.19866e6 −0.355171
\(838\) 0 0
\(839\) −2.01293e6 −0.0987244 −0.0493622 0.998781i \(-0.515719\pi\)
−0.0493622 + 0.998781i \(0.515719\pi\)
\(840\) 0 0
\(841\) 1.94703e7 0.949255
\(842\) 0 0
\(843\) −3.29527e7 −1.59706
\(844\) 0 0
\(845\) 5.09259e6 0.245356
\(846\) 0 0
\(847\) 3.78381e6 0.181226
\(848\) 0 0
\(849\) −3.66618e7 −1.74560
\(850\) 0 0
\(851\) 2.67915e7 1.26816
\(852\) 0 0
\(853\) −1.09443e7 −0.515008 −0.257504 0.966277i \(-0.582900\pi\)
−0.257504 + 0.966277i \(0.582900\pi\)
\(854\) 0 0
\(855\) −186210. −0.00871140
\(856\) 0 0
\(857\) −2.17692e7 −1.01249 −0.506245 0.862390i \(-0.668967\pi\)
−0.506245 + 0.862390i \(0.668967\pi\)
\(858\) 0 0
\(859\) 6.86741e6 0.317549 0.158774 0.987315i \(-0.449246\pi\)
0.158774 + 0.987315i \(0.449246\pi\)
\(860\) 0 0
\(861\) 1.10632e7 0.508595
\(862\) 0 0
\(863\) −1.75635e7 −0.802755 −0.401378 0.915913i \(-0.631468\pi\)
−0.401378 + 0.915913i \(0.631468\pi\)
\(864\) 0 0
\(865\) 1.06740e7 0.485049
\(866\) 0 0
\(867\) 1.72880e7 0.781080
\(868\) 0 0
\(869\) −1.29311e7 −0.580878
\(870\) 0 0
\(871\) −3.00058e7 −1.34017
\(872\) 0 0
\(873\) 1.29456e7 0.574893
\(874\) 0 0
\(875\) 1.90467e6 0.0841008
\(876\) 0 0
\(877\) 2.95517e6 0.129743 0.0648715 0.997894i \(-0.479336\pi\)
0.0648715 + 0.997894i \(0.479336\pi\)
\(878\) 0 0
\(879\) −4.50975e7 −1.96870
\(880\) 0 0
\(881\) 3.53773e7 1.53562 0.767811 0.640676i \(-0.221346\pi\)
0.767811 + 0.640676i \(0.221346\pi\)
\(882\) 0 0
\(883\) −2.26106e7 −0.975912 −0.487956 0.872868i \(-0.662257\pi\)
−0.487956 + 0.872868i \(0.662257\pi\)
\(884\) 0 0
\(885\) 2.62345e6 0.112594
\(886\) 0 0
\(887\) −8.57410e6 −0.365914 −0.182957 0.983121i \(-0.558567\pi\)
−0.182957 + 0.983121i \(0.558567\pi\)
\(888\) 0 0
\(889\) 1.07763e7 0.457314
\(890\) 0 0
\(891\) −3.21122e7 −1.35511
\(892\) 0 0
\(893\) −863241. −0.0362246
\(894\) 0 0
\(895\) −1.70691e7 −0.712284
\(896\) 0 0
\(897\) 5.17444e7 2.14725
\(898\) 0 0
\(899\) −1.70177e7 −0.702265
\(900\) 0 0
\(901\) 4.51896e7 1.85450
\(902\) 0 0
\(903\) −5.70528e6 −0.232840
\(904\) 0 0
\(905\) −1.81416e7 −0.736301
\(906\) 0 0
\(907\) −1.76363e7 −0.711852 −0.355926 0.934514i \(-0.615834\pi\)
−0.355926 + 0.934514i \(0.615834\pi\)
\(908\) 0 0
\(909\) 3.01349e6 0.120965
\(910\) 0 0
\(911\) −1.48438e7 −0.592585 −0.296292 0.955097i \(-0.595750\pi\)
−0.296292 + 0.955097i \(0.595750\pi\)
\(912\) 0 0
\(913\) −4.91792e7 −1.95256
\(914\) 0 0
\(915\) 2.21842e7 0.875973
\(916\) 0 0
\(917\) 3.34365e7 1.31310
\(918\) 0 0
\(919\) −448602. −0.0175215 −0.00876077 0.999962i \(-0.502789\pi\)
−0.00876077 + 0.999962i \(0.502789\pi\)
\(920\) 0 0
\(921\) 4.69071e7 1.82217
\(922\) 0 0
\(923\) 9.57129e6 0.369799
\(924\) 0 0
\(925\) 4.53262e6 0.174179
\(926\) 0 0
\(927\) −1.38206e7 −0.528235
\(928\) 0 0
\(929\) 2.11976e6 0.0805836 0.0402918 0.999188i \(-0.487171\pi\)
0.0402918 + 0.999188i \(0.487171\pi\)
\(930\) 0 0
\(931\) 147732. 0.00558598
\(932\) 0 0
\(933\) 3.44647e7 1.29619
\(934\) 0 0
\(935\) −1.68175e7 −0.629119
\(936\) 0 0
\(937\) 5.54945e6 0.206491 0.103245 0.994656i \(-0.467077\pi\)
0.103245 + 0.994656i \(0.467077\pi\)
\(938\) 0 0
\(939\) 4.01680e7 1.48667
\(940\) 0 0
\(941\) −6.00433e6 −0.221050 −0.110525 0.993873i \(-0.535253\pi\)
−0.110525 + 0.993873i \(0.535253\pi\)
\(942\) 0 0
\(943\) 1.81512e7 0.664701
\(944\) 0 0
\(945\) −8.15119e6 −0.296921
\(946\) 0 0
\(947\) 1.19597e7 0.433357 0.216679 0.976243i \(-0.430478\pi\)
0.216679 + 0.976243i \(0.430478\pi\)
\(948\) 0 0
\(949\) 4.39496e7 1.58413
\(950\) 0 0
\(951\) 2.74156e7 0.982982
\(952\) 0 0
\(953\) −2.84885e7 −1.01610 −0.508051 0.861327i \(-0.669634\pi\)
−0.508051 + 0.861327i \(0.669634\pi\)
\(954\) 0 0
\(955\) 1.65500e7 0.587205
\(956\) 0 0
\(957\) −5.11901e7 −1.80678
\(958\) 0 0
\(959\) −3.76891e7 −1.32333
\(960\) 0 0
\(961\) −2.13858e7 −0.746993
\(962\) 0 0
\(963\) −1.39409e6 −0.0484422
\(964\) 0 0
\(965\) −1.20389e7 −0.416169
\(966\) 0 0
\(967\) 2.44173e6 0.0839714 0.0419857 0.999118i \(-0.486632\pi\)
0.0419857 + 0.999118i \(0.486632\pi\)
\(968\) 0 0
\(969\) 2.15048e6 0.0735744
\(970\) 0 0
\(971\) −2.63888e7 −0.898198 −0.449099 0.893482i \(-0.648255\pi\)
−0.449099 + 0.893482i \(0.648255\pi\)
\(972\) 0 0
\(973\) 4.64898e7 1.57426
\(974\) 0 0
\(975\) 8.75417e6 0.294920
\(976\) 0 0
\(977\) −6.67324e6 −0.223666 −0.111833 0.993727i \(-0.535672\pi\)
−0.111833 + 0.993727i \(0.535672\pi\)
\(978\) 0 0
\(979\) 2.93516e7 0.978756
\(980\) 0 0
\(981\) −969506. −0.0321646
\(982\) 0 0
\(983\) −1.21646e7 −0.401528 −0.200764 0.979640i \(-0.564342\pi\)
−0.200764 + 0.979640i \(0.564342\pi\)
\(984\) 0 0
\(985\) −1.42316e7 −0.467372
\(986\) 0 0
\(987\) 2.56253e7 0.837289
\(988\) 0 0
\(989\) −9.36057e6 −0.304307
\(990\) 0 0
\(991\) −1.16204e7 −0.375869 −0.187935 0.982182i \(-0.560179\pi\)
−0.187935 + 0.982182i \(0.560179\pi\)
\(992\) 0 0
\(993\) −3.74750e7 −1.20606
\(994\) 0 0
\(995\) −2.12538e7 −0.680579
\(996\) 0 0
\(997\) 4.61789e7 1.47132 0.735658 0.677354i \(-0.236873\pi\)
0.735658 + 0.677354i \(0.236873\pi\)
\(998\) 0 0
\(999\) −1.93977e7 −0.614944
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 320.6.a.x.1.3 3
4.3 odd 2 320.6.a.y.1.1 3
8.3 odd 2 160.6.a.f.1.3 3
8.5 even 2 160.6.a.g.1.1 yes 3
40.3 even 4 800.6.c.k.449.5 6
40.13 odd 4 800.6.c.j.449.2 6
40.19 odd 2 800.6.a.o.1.1 3
40.27 even 4 800.6.c.k.449.2 6
40.29 even 2 800.6.a.n.1.3 3
40.37 odd 4 800.6.c.j.449.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.6.a.f.1.3 3 8.3 odd 2
160.6.a.g.1.1 yes 3 8.5 even 2
320.6.a.x.1.3 3 1.1 even 1 trivial
320.6.a.y.1.1 3 4.3 odd 2
800.6.a.n.1.3 3 40.29 even 2
800.6.a.o.1.1 3 40.19 odd 2
800.6.c.j.449.2 6 40.13 odd 4
800.6.c.j.449.5 6 40.37 odd 4
800.6.c.k.449.2 6 40.27 even 4
800.6.c.k.449.5 6 40.3 even 4