Properties

Label 320.6.d.c.161.8
Level 320
Weight 6
Character 320.161
Analytic conductor 51.323
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 343 x^{10} - 696 x^{9} + 44406 x^{8} + 179640 x^{7} - 2401691 x^{6} - 15554592 x^{5} + 26901210 x^{4} + 434775816 x^{3} + 1271335685 x^{2} + 1475231592 x + 653157349\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{30} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.8
Root \(-7.49748 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 320.161
Dual form 320.6.d.c.161.5

$q$-expansion

\(f(q)\) \(=\) \(q+5.06676i q^{3} +25.0000i q^{5} -250.333 q^{7} +217.328 q^{9} +O(q^{10})\) \(q+5.06676i q^{3} +25.0000i q^{5} -250.333 q^{7} +217.328 q^{9} -94.5683i q^{11} -887.073i q^{13} -126.669 q^{15} +1448.19 q^{17} -611.677i q^{19} -1268.38i q^{21} +814.118 q^{23} -625.000 q^{25} +2332.37i q^{27} +5291.31i q^{29} +3614.36 q^{31} +479.155 q^{33} -6258.32i q^{35} +9480.98i q^{37} +4494.59 q^{39} -9910.48 q^{41} +20213.1i q^{43} +5433.20i q^{45} -20765.1 q^{47} +45859.5 q^{49} +7337.66i q^{51} -13009.9i q^{53} +2364.21 q^{55} +3099.22 q^{57} -16632.8i q^{59} -715.975i q^{61} -54404.3 q^{63} +22176.8 q^{65} +21955.0i q^{67} +4124.94i q^{69} -38473.5 q^{71} -40857.0 q^{73} -3166.73i q^{75} +23673.5i q^{77} +84846.9 q^{79} +40993.1 q^{81} +116115. i q^{83} +36204.9i q^{85} -26809.8 q^{87} +121182. q^{89} +222064. i q^{91} +18313.1i q^{93} +15291.9 q^{95} +21598.7 q^{97} -20552.3i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 268q^{7} - 428q^{9} + O(q^{10}) \) \( 12q - 268q^{7} - 428q^{9} + 900q^{15} + 2400q^{17} - 8108q^{23} - 7500q^{25} - 7976q^{31} - 20776q^{33} - 40984q^{39} - 56408q^{41} - 21172q^{47} - 5540q^{49} - 18400q^{55} + 39992q^{57} - 179516q^{63} + 44000q^{65} - 367704q^{71} + 58736q^{73} - 26192q^{79} + 411692q^{81} - 183200q^{87} + 87672q^{89} - 121000q^{95} - 172336q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 5.06676i 0.325033i 0.986706 + 0.162516i \(0.0519611\pi\)
−0.986706 + 0.162516i \(0.948039\pi\)
\(4\) 0 0
\(5\) 25.0000i 0.447214i
\(6\) 0 0
\(7\) −250.333 −1.93096 −0.965479 0.260481i \(-0.916119\pi\)
−0.965479 + 0.260481i \(0.916119\pi\)
\(8\) 0 0
\(9\) 217.328 0.894354
\(10\) 0 0
\(11\) − 94.5683i − 0.235648i −0.993034 0.117824i \(-0.962408\pi\)
0.993034 0.117824i \(-0.0375919\pi\)
\(12\) 0 0
\(13\) − 887.073i − 1.45580i −0.685684 0.727899i \(-0.740497\pi\)
0.685684 0.727899i \(-0.259503\pi\)
\(14\) 0 0
\(15\) −126.669 −0.145359
\(16\) 0 0
\(17\) 1448.19 1.21536 0.607679 0.794182i \(-0.292101\pi\)
0.607679 + 0.794182i \(0.292101\pi\)
\(18\) 0 0
\(19\) − 611.677i − 0.388721i −0.980930 0.194361i \(-0.937737\pi\)
0.980930 0.194361i \(-0.0622632\pi\)
\(20\) 0 0
\(21\) − 1268.38i − 0.627625i
\(22\) 0 0
\(23\) 814.118 0.320899 0.160449 0.987044i \(-0.448706\pi\)
0.160449 + 0.987044i \(0.448706\pi\)
\(24\) 0 0
\(25\) −625.000 −0.200000
\(26\) 0 0
\(27\) 2332.37i 0.615727i
\(28\) 0 0
\(29\) 5291.31i 1.16834i 0.811633 + 0.584168i \(0.198579\pi\)
−0.811633 + 0.584168i \(0.801421\pi\)
\(30\) 0 0
\(31\) 3614.36 0.675502 0.337751 0.941236i \(-0.390334\pi\)
0.337751 + 0.941236i \(0.390334\pi\)
\(32\) 0 0
\(33\) 479.155 0.0765934
\(34\) 0 0
\(35\) − 6258.32i − 0.863551i
\(36\) 0 0
\(37\) 9480.98i 1.13854i 0.822150 + 0.569271i \(0.192774\pi\)
−0.822150 + 0.569271i \(0.807226\pi\)
\(38\) 0 0
\(39\) 4494.59 0.473182
\(40\) 0 0
\(41\) −9910.48 −0.920736 −0.460368 0.887728i \(-0.652282\pi\)
−0.460368 + 0.887728i \(0.652282\pi\)
\(42\) 0 0
\(43\) 20213.1i 1.66710i 0.552445 + 0.833549i \(0.313695\pi\)
−0.552445 + 0.833549i \(0.686305\pi\)
\(44\) 0 0
\(45\) 5433.20i 0.399967i
\(46\) 0 0
\(47\) −20765.1 −1.37116 −0.685581 0.727996i \(-0.740452\pi\)
−0.685581 + 0.727996i \(0.740452\pi\)
\(48\) 0 0
\(49\) 45859.5 2.72860
\(50\) 0 0
\(51\) 7337.66i 0.395032i
\(52\) 0 0
\(53\) − 13009.9i − 0.636187i −0.948059 0.318094i \(-0.896957\pi\)
0.948059 0.318094i \(-0.103043\pi\)
\(54\) 0 0
\(55\) 2364.21 0.105385
\(56\) 0 0
\(57\) 3099.22 0.126347
\(58\) 0 0
\(59\) − 16632.8i − 0.622065i −0.950399 0.311032i \(-0.899325\pi\)
0.950399 0.311032i \(-0.100675\pi\)
\(60\) 0 0
\(61\) − 715.975i − 0.0246362i −0.999924 0.0123181i \(-0.996079\pi\)
0.999924 0.0123181i \(-0.00392107\pi\)
\(62\) 0 0
\(63\) −54404.3 −1.72696
\(64\) 0 0
\(65\) 22176.8 0.651053
\(66\) 0 0
\(67\) 21955.0i 0.597513i 0.954329 + 0.298756i \(0.0965718\pi\)
−0.954329 + 0.298756i \(0.903428\pi\)
\(68\) 0 0
\(69\) 4124.94i 0.104303i
\(70\) 0 0
\(71\) −38473.5 −0.905765 −0.452883 0.891570i \(-0.649604\pi\)
−0.452883 + 0.891570i \(0.649604\pi\)
\(72\) 0 0
\(73\) −40857.0 −0.897344 −0.448672 0.893696i \(-0.648103\pi\)
−0.448672 + 0.893696i \(0.648103\pi\)
\(74\) 0 0
\(75\) − 3166.73i − 0.0650066i
\(76\) 0 0
\(77\) 23673.5i 0.455026i
\(78\) 0 0
\(79\) 84846.9 1.52957 0.764783 0.644288i \(-0.222846\pi\)
0.764783 + 0.644288i \(0.222846\pi\)
\(80\) 0 0
\(81\) 40993.1 0.694222
\(82\) 0 0
\(83\) 116115.i 1.85010i 0.379851 + 0.925048i \(0.375975\pi\)
−0.379851 + 0.925048i \(0.624025\pi\)
\(84\) 0 0
\(85\) 36204.9i 0.543525i
\(86\) 0 0
\(87\) −26809.8 −0.379748
\(88\) 0 0
\(89\) 121182. 1.62168 0.810838 0.585271i \(-0.199012\pi\)
0.810838 + 0.585271i \(0.199012\pi\)
\(90\) 0 0
\(91\) 222064.i 2.81108i
\(92\) 0 0
\(93\) 18313.1i 0.219560i
\(94\) 0 0
\(95\) 15291.9 0.173841
\(96\) 0 0
\(97\) 21598.7 0.233077 0.116538 0.993186i \(-0.462820\pi\)
0.116538 + 0.993186i \(0.462820\pi\)
\(98\) 0 0
\(99\) − 20552.3i − 0.210753i
\(100\) 0 0
\(101\) 105401.i 1.02811i 0.857756 + 0.514057i \(0.171858\pi\)
−0.857756 + 0.514057i \(0.828142\pi\)
\(102\) 0 0
\(103\) −15362.8 −0.142684 −0.0713421 0.997452i \(-0.522728\pi\)
−0.0713421 + 0.997452i \(0.522728\pi\)
\(104\) 0 0
\(105\) 31709.4 0.280682
\(106\) 0 0
\(107\) 14774.1i 0.124750i 0.998053 + 0.0623750i \(0.0198675\pi\)
−0.998053 + 0.0623750i \(0.980133\pi\)
\(108\) 0 0
\(109\) 165494.i 1.33419i 0.744973 + 0.667094i \(0.232462\pi\)
−0.744973 + 0.667094i \(0.767538\pi\)
\(110\) 0 0
\(111\) −48037.9 −0.370064
\(112\) 0 0
\(113\) 72833.6 0.536582 0.268291 0.963338i \(-0.413541\pi\)
0.268291 + 0.963338i \(0.413541\pi\)
\(114\) 0 0
\(115\) 20352.9i 0.143510i
\(116\) 0 0
\(117\) − 192786.i − 1.30200i
\(118\) 0 0
\(119\) −362531. −2.34681
\(120\) 0 0
\(121\) 152108. 0.944470
\(122\) 0 0
\(123\) − 50214.0i − 0.299269i
\(124\) 0 0
\(125\) − 15625.0i − 0.0894427i
\(126\) 0 0
\(127\) 8287.17 0.0455929 0.0227964 0.999740i \(-0.492743\pi\)
0.0227964 + 0.999740i \(0.492743\pi\)
\(128\) 0 0
\(129\) −102415. −0.541862
\(130\) 0 0
\(131\) − 177213.i − 0.902231i −0.892466 0.451115i \(-0.851026\pi\)
0.892466 0.451115i \(-0.148974\pi\)
\(132\) 0 0
\(133\) 153123.i 0.750605i
\(134\) 0 0
\(135\) −58309.3 −0.275362
\(136\) 0 0
\(137\) 352108. 1.60278 0.801391 0.598141i \(-0.204094\pi\)
0.801391 + 0.598141i \(0.204094\pi\)
\(138\) 0 0
\(139\) 145465.i 0.638589i 0.947656 + 0.319294i \(0.103446\pi\)
−0.947656 + 0.319294i \(0.896554\pi\)
\(140\) 0 0
\(141\) − 105212.i − 0.445673i
\(142\) 0 0
\(143\) −83889.0 −0.343056
\(144\) 0 0
\(145\) −132283. −0.522496
\(146\) 0 0
\(147\) 232359.i 0.886884i
\(148\) 0 0
\(149\) − 162250.i − 0.598712i −0.954142 0.299356i \(-0.903228\pi\)
0.954142 0.299356i \(-0.0967718\pi\)
\(150\) 0 0
\(151\) 427740. 1.52664 0.763321 0.646019i \(-0.223567\pi\)
0.763321 + 0.646019i \(0.223567\pi\)
\(152\) 0 0
\(153\) 314733. 1.08696
\(154\) 0 0
\(155\) 90358.9i 0.302094i
\(156\) 0 0
\(157\) 278644.i 0.902194i 0.892475 + 0.451097i \(0.148967\pi\)
−0.892475 + 0.451097i \(0.851033\pi\)
\(158\) 0 0
\(159\) 65918.2 0.206782
\(160\) 0 0
\(161\) −203800. −0.619642
\(162\) 0 0
\(163\) − 276615.i − 0.815467i −0.913101 0.407733i \(-0.866319\pi\)
0.913101 0.407733i \(-0.133681\pi\)
\(164\) 0 0
\(165\) 11978.9i 0.0342536i
\(166\) 0 0
\(167\) −3008.18 −0.00834667 −0.00417334 0.999991i \(-0.501328\pi\)
−0.00417334 + 0.999991i \(0.501328\pi\)
\(168\) 0 0
\(169\) −415606. −1.11935
\(170\) 0 0
\(171\) − 132935.i − 0.347654i
\(172\) 0 0
\(173\) 473850.i 1.20372i 0.798602 + 0.601860i \(0.205573\pi\)
−0.798602 + 0.601860i \(0.794427\pi\)
\(174\) 0 0
\(175\) 156458. 0.386192
\(176\) 0 0
\(177\) 84274.5 0.202192
\(178\) 0 0
\(179\) 661760.i 1.54372i 0.635794 + 0.771859i \(0.280673\pi\)
−0.635794 + 0.771859i \(0.719327\pi\)
\(180\) 0 0
\(181\) 390057.i 0.884977i 0.896774 + 0.442488i \(0.145904\pi\)
−0.896774 + 0.442488i \(0.854096\pi\)
\(182\) 0 0
\(183\) 3627.67 0.00800757
\(184\) 0 0
\(185\) −237025. −0.509171
\(186\) 0 0
\(187\) − 136953.i − 0.286397i
\(188\) 0 0
\(189\) − 583869.i − 1.18894i
\(190\) 0 0
\(191\) 456202. 0.904845 0.452422 0.891804i \(-0.350560\pi\)
0.452422 + 0.891804i \(0.350560\pi\)
\(192\) 0 0
\(193\) −28265.7 −0.0546218 −0.0273109 0.999627i \(-0.508694\pi\)
−0.0273109 + 0.999627i \(0.508694\pi\)
\(194\) 0 0
\(195\) 112365.i 0.211614i
\(196\) 0 0
\(197\) − 746684.i − 1.37079i −0.728171 0.685396i \(-0.759629\pi\)
0.728171 0.685396i \(-0.240371\pi\)
\(198\) 0 0
\(199\) −271671. −0.486307 −0.243153 0.969988i \(-0.578182\pi\)
−0.243153 + 0.969988i \(0.578182\pi\)
\(200\) 0 0
\(201\) −111241. −0.194211
\(202\) 0 0
\(203\) − 1.32459e6i − 2.25601i
\(204\) 0 0
\(205\) − 247762.i − 0.411766i
\(206\) 0 0
\(207\) 176931. 0.286997
\(208\) 0 0
\(209\) −57845.3 −0.0916014
\(210\) 0 0
\(211\) 20283.2i 0.0313640i 0.999877 + 0.0156820i \(0.00499194\pi\)
−0.999877 + 0.0156820i \(0.995008\pi\)
\(212\) 0 0
\(213\) − 194936.i − 0.294404i
\(214\) 0 0
\(215\) −505327. −0.745549
\(216\) 0 0
\(217\) −904792. −1.30437
\(218\) 0 0
\(219\) − 207013.i − 0.291667i
\(220\) 0 0
\(221\) − 1.28465e6i − 1.76932i
\(222\) 0 0
\(223\) −30380.0 −0.0409096 −0.0204548 0.999791i \(-0.506511\pi\)
−0.0204548 + 0.999791i \(0.506511\pi\)
\(224\) 0 0
\(225\) −135830. −0.178871
\(226\) 0 0
\(227\) − 1.45723e6i − 1.87699i −0.345286 0.938497i \(-0.612218\pi\)
0.345286 0.938497i \(-0.387782\pi\)
\(228\) 0 0
\(229\) − 963464.i − 1.21408i −0.794672 0.607039i \(-0.792357\pi\)
0.794672 0.607039i \(-0.207643\pi\)
\(230\) 0 0
\(231\) −119948. −0.147899
\(232\) 0 0
\(233\) −1.31864e6 −1.59125 −0.795623 0.605792i \(-0.792856\pi\)
−0.795623 + 0.605792i \(0.792856\pi\)
\(234\) 0 0
\(235\) − 519127.i − 0.613202i
\(236\) 0 0
\(237\) 429899.i 0.497159i
\(238\) 0 0
\(239\) 1.17974e6 1.33595 0.667974 0.744184i \(-0.267162\pi\)
0.667974 + 0.744184i \(0.267162\pi\)
\(240\) 0 0
\(241\) 785926. 0.871644 0.435822 0.900033i \(-0.356458\pi\)
0.435822 + 0.900033i \(0.356458\pi\)
\(242\) 0 0
\(243\) 774469.i 0.841372i
\(244\) 0 0
\(245\) 1.14649e6i 1.22027i
\(246\) 0 0
\(247\) −542603. −0.565900
\(248\) 0 0
\(249\) −588329. −0.601342
\(250\) 0 0
\(251\) 923722.i 0.925459i 0.886500 + 0.462729i \(0.153130\pi\)
−0.886500 + 0.462729i \(0.846870\pi\)
\(252\) 0 0
\(253\) − 76989.7i − 0.0756191i
\(254\) 0 0
\(255\) −183441. −0.176664
\(256\) 0 0
\(257\) −497776. −0.470112 −0.235056 0.971982i \(-0.575527\pi\)
−0.235056 + 0.971982i \(0.575527\pi\)
\(258\) 0 0
\(259\) − 2.37340e6i − 2.19848i
\(260\) 0 0
\(261\) 1.14995e6i 1.04491i
\(262\) 0 0
\(263\) −745381. −0.664491 −0.332245 0.943193i \(-0.607806\pi\)
−0.332245 + 0.943193i \(0.607806\pi\)
\(264\) 0 0
\(265\) 325248. 0.284512
\(266\) 0 0
\(267\) 614002.i 0.527098i
\(268\) 0 0
\(269\) 1.80992e6i 1.52503i 0.646970 + 0.762516i \(0.276036\pi\)
−0.646970 + 0.762516i \(0.723964\pi\)
\(270\) 0 0
\(271\) 2.20319e6 1.82234 0.911169 0.412032i \(-0.135181\pi\)
0.911169 + 0.412032i \(0.135181\pi\)
\(272\) 0 0
\(273\) −1.12514e6 −0.913695
\(274\) 0 0
\(275\) 59105.2i 0.0471296i
\(276\) 0 0
\(277\) − 1.85372e6i − 1.45159i −0.687912 0.725795i \(-0.741472\pi\)
0.687912 0.725795i \(-0.258528\pi\)
\(278\) 0 0
\(279\) 785500. 0.604138
\(280\) 0 0
\(281\) −1.06542e6 −0.804926 −0.402463 0.915436i \(-0.631846\pi\)
−0.402463 + 0.915436i \(0.631846\pi\)
\(282\) 0 0
\(283\) − 38350.4i − 0.0284645i −0.999899 0.0142323i \(-0.995470\pi\)
0.999899 0.0142323i \(-0.00453042\pi\)
\(284\) 0 0
\(285\) 77480.6i 0.0565042i
\(286\) 0 0
\(287\) 2.48092e6 1.77790
\(288\) 0 0
\(289\) 677410. 0.477097
\(290\) 0 0
\(291\) 109436.i 0.0757576i
\(292\) 0 0
\(293\) 463895.i 0.315683i 0.987464 + 0.157841i \(0.0504534\pi\)
−0.987464 + 0.157841i \(0.949547\pi\)
\(294\) 0 0
\(295\) 415820. 0.278196
\(296\) 0 0
\(297\) 220568. 0.145095
\(298\) 0 0
\(299\) − 722182.i − 0.467164i
\(300\) 0 0
\(301\) − 5.06000e6i − 3.21910i
\(302\) 0 0
\(303\) −534042. −0.334171
\(304\) 0 0
\(305\) 17899.4 0.0110176
\(306\) 0 0
\(307\) 203618.i 0.123302i 0.998098 + 0.0616509i \(0.0196365\pi\)
−0.998098 + 0.0616509i \(0.980363\pi\)
\(308\) 0 0
\(309\) − 77839.4i − 0.0463771i
\(310\) 0 0
\(311\) −1.32625e6 −0.777541 −0.388770 0.921335i \(-0.627100\pi\)
−0.388770 + 0.921335i \(0.627100\pi\)
\(312\) 0 0
\(313\) −1.62292e6 −0.936344 −0.468172 0.883637i \(-0.655087\pi\)
−0.468172 + 0.883637i \(0.655087\pi\)
\(314\) 0 0
\(315\) − 1.36011e6i − 0.772320i
\(316\) 0 0
\(317\) 920943.i 0.514736i 0.966313 + 0.257368i \(0.0828553\pi\)
−0.966313 + 0.257368i \(0.917145\pi\)
\(318\) 0 0
\(319\) 500390. 0.275316
\(320\) 0 0
\(321\) −74856.7 −0.0405479
\(322\) 0 0
\(323\) − 885828.i − 0.472436i
\(324\) 0 0
\(325\) 554421.i 0.291160i
\(326\) 0 0
\(327\) −838521. −0.433655
\(328\) 0 0
\(329\) 5.19818e6 2.64766
\(330\) 0 0
\(331\) − 689680.i − 0.346001i −0.984922 0.173001i \(-0.944654\pi\)
0.984922 0.173001i \(-0.0553463\pi\)
\(332\) 0 0
\(333\) 2.06048e6i 1.01826i
\(334\) 0 0
\(335\) −548876. −0.267216
\(336\) 0 0
\(337\) 1.23442e6 0.592089 0.296045 0.955174i \(-0.404332\pi\)
0.296045 + 0.955174i \(0.404332\pi\)
\(338\) 0 0
\(339\) 369031.i 0.174407i
\(340\) 0 0
\(341\) − 341803.i − 0.159181i
\(342\) 0 0
\(343\) −7.27281e6 −3.33785
\(344\) 0 0
\(345\) −103124. −0.0466455
\(346\) 0 0
\(347\) 2.59951e6i 1.15896i 0.814987 + 0.579480i \(0.196744\pi\)
−0.814987 + 0.579480i \(0.803256\pi\)
\(348\) 0 0
\(349\) 277542.i 0.121974i 0.998139 + 0.0609868i \(0.0194247\pi\)
−0.998139 + 0.0609868i \(0.980575\pi\)
\(350\) 0 0
\(351\) 2.06899e6 0.896375
\(352\) 0 0
\(353\) −2.51736e6 −1.07525 −0.537624 0.843184i \(-0.680678\pi\)
−0.537624 + 0.843184i \(0.680678\pi\)
\(354\) 0 0
\(355\) − 961837.i − 0.405071i
\(356\) 0 0
\(357\) − 1.83686e6i − 0.762790i
\(358\) 0 0
\(359\) −899617. −0.368402 −0.184201 0.982889i \(-0.558970\pi\)
−0.184201 + 0.982889i \(0.558970\pi\)
\(360\) 0 0
\(361\) 2.10195e6 0.848896
\(362\) 0 0
\(363\) 770694.i 0.306984i
\(364\) 0 0
\(365\) − 1.02142e6i − 0.401305i
\(366\) 0 0
\(367\) 2.07519e6 0.804253 0.402126 0.915584i \(-0.368271\pi\)
0.402126 + 0.915584i \(0.368271\pi\)
\(368\) 0 0
\(369\) −2.15382e6 −0.823463
\(370\) 0 0
\(371\) 3.25681e6i 1.22845i
\(372\) 0 0
\(373\) 5.12372e6i 1.90684i 0.301651 + 0.953419i \(0.402462\pi\)
−0.301651 + 0.953419i \(0.597538\pi\)
\(374\) 0 0
\(375\) 79168.2 0.0290718
\(376\) 0 0
\(377\) 4.69378e6 1.70086
\(378\) 0 0
\(379\) 1.55330e6i 0.555465i 0.960659 + 0.277732i \(0.0895828\pi\)
−0.960659 + 0.277732i \(0.910417\pi\)
\(380\) 0 0
\(381\) 41989.1i 0.0148192i
\(382\) 0 0
\(383\) 1.17661e6 0.409861 0.204930 0.978777i \(-0.434303\pi\)
0.204930 + 0.978777i \(0.434303\pi\)
\(384\) 0 0
\(385\) −591839. −0.203494
\(386\) 0 0
\(387\) 4.39287e6i 1.49098i
\(388\) 0 0
\(389\) 1.56804e6i 0.525392i 0.964879 + 0.262696i \(0.0846116\pi\)
−0.964879 + 0.262696i \(0.915388\pi\)
\(390\) 0 0
\(391\) 1.17900e6 0.390007
\(392\) 0 0
\(393\) 897897. 0.293255
\(394\) 0 0
\(395\) 2.12117e6i 0.684043i
\(396\) 0 0
\(397\) 1.55070e6i 0.493799i 0.969041 + 0.246900i \(0.0794118\pi\)
−0.969041 + 0.246900i \(0.920588\pi\)
\(398\) 0 0
\(399\) −775838. −0.243971
\(400\) 0 0
\(401\) 1.78114e6 0.553144 0.276572 0.960993i \(-0.410802\pi\)
0.276572 + 0.960993i \(0.410802\pi\)
\(402\) 0 0
\(403\) − 3.20620e6i − 0.983394i
\(404\) 0 0
\(405\) 1.02483e6i 0.310465i
\(406\) 0 0
\(407\) 896600. 0.268295
\(408\) 0 0
\(409\) −5.19593e6 −1.53587 −0.767936 0.640527i \(-0.778716\pi\)
−0.767936 + 0.640527i \(0.778716\pi\)
\(410\) 0 0
\(411\) 1.78405e6i 0.520957i
\(412\) 0 0
\(413\) 4.16374e6i 1.20118i
\(414\) 0 0
\(415\) −2.90288e6 −0.827388
\(416\) 0 0
\(417\) −737036. −0.207562
\(418\) 0 0
\(419\) 6.45443e6i 1.79607i 0.439927 + 0.898033i \(0.355004\pi\)
−0.439927 + 0.898033i \(0.644996\pi\)
\(420\) 0 0
\(421\) − 1.41145e6i − 0.388114i −0.980990 0.194057i \(-0.937835\pi\)
0.980990 0.194057i \(-0.0621646\pi\)
\(422\) 0 0
\(423\) −4.51283e6 −1.22630
\(424\) 0 0
\(425\) −905121. −0.243072
\(426\) 0 0
\(427\) 179232.i 0.0475714i
\(428\) 0 0
\(429\) − 425046.i − 0.111504i
\(430\) 0 0
\(431\) 2.94825e6 0.764488 0.382244 0.924061i \(-0.375151\pi\)
0.382244 + 0.924061i \(0.375151\pi\)
\(432\) 0 0
\(433\) 4.63564e6 1.18820 0.594100 0.804391i \(-0.297508\pi\)
0.594100 + 0.804391i \(0.297508\pi\)
\(434\) 0 0
\(435\) − 670245.i − 0.169828i
\(436\) 0 0
\(437\) − 497978.i − 0.124740i
\(438\) 0 0
\(439\) −2.15825e6 −0.534491 −0.267245 0.963629i \(-0.586113\pi\)
−0.267245 + 0.963629i \(0.586113\pi\)
\(440\) 0 0
\(441\) 9.96656e6 2.44033
\(442\) 0 0
\(443\) − 4.27689e6i − 1.03543i −0.855554 0.517713i \(-0.826783\pi\)
0.855554 0.517713i \(-0.173217\pi\)
\(444\) 0 0
\(445\) 3.02956e6i 0.725236i
\(446\) 0 0
\(447\) 822080. 0.194601
\(448\) 0 0
\(449\) 1.31292e6 0.307342 0.153671 0.988122i \(-0.450890\pi\)
0.153671 + 0.988122i \(0.450890\pi\)
\(450\) 0 0
\(451\) 937217.i 0.216969i
\(452\) 0 0
\(453\) 2.16726e6i 0.496209i
\(454\) 0 0
\(455\) −5.55159e6 −1.25716
\(456\) 0 0
\(457\) −5.14752e6 −1.15294 −0.576471 0.817117i \(-0.695571\pi\)
−0.576471 + 0.817117i \(0.695571\pi\)
\(458\) 0 0
\(459\) 3.37773e6i 0.748330i
\(460\) 0 0
\(461\) − 7.94846e6i − 1.74193i −0.491346 0.870965i \(-0.663495\pi\)
0.491346 0.870965i \(-0.336505\pi\)
\(462\) 0 0
\(463\) 374404. 0.0811687 0.0405843 0.999176i \(-0.487078\pi\)
0.0405843 + 0.999176i \(0.487078\pi\)
\(464\) 0 0
\(465\) −457827. −0.0981904
\(466\) 0 0
\(467\) − 3.84308e6i − 0.815432i −0.913109 0.407716i \(-0.866325\pi\)
0.913109 0.407716i \(-0.133675\pi\)
\(468\) 0 0
\(469\) − 5.49607e6i − 1.15377i
\(470\) 0 0
\(471\) −1.41182e6 −0.293243
\(472\) 0 0
\(473\) 1.91152e6 0.392848
\(474\) 0 0
\(475\) 382298.i 0.0777443i
\(476\) 0 0
\(477\) − 2.82742e6i − 0.568977i
\(478\) 0 0
\(479\) −1.08933e6 −0.216930 −0.108465 0.994100i \(-0.534594\pi\)
−0.108465 + 0.994100i \(0.534594\pi\)
\(480\) 0 0
\(481\) 8.41033e6 1.65749
\(482\) 0 0
\(483\) − 1.03261e6i − 0.201404i
\(484\) 0 0
\(485\) 539968.i 0.104235i
\(486\) 0 0
\(487\) 2.02078e6 0.386098 0.193049 0.981189i \(-0.438162\pi\)
0.193049 + 0.981189i \(0.438162\pi\)
\(488\) 0 0
\(489\) 1.40154e6 0.265054
\(490\) 0 0
\(491\) 9.05818e6i 1.69565i 0.530273 + 0.847827i \(0.322089\pi\)
−0.530273 + 0.847827i \(0.677911\pi\)
\(492\) 0 0
\(493\) 7.66284e6i 1.41995i
\(494\) 0 0
\(495\) 513808. 0.0942514
\(496\) 0 0
\(497\) 9.63118e6 1.74899
\(498\) 0 0
\(499\) 2.83503e6i 0.509690i 0.966982 + 0.254845i \(0.0820245\pi\)
−0.966982 + 0.254845i \(0.917976\pi\)
\(500\) 0 0
\(501\) − 15241.8i − 0.00271294i
\(502\) 0 0
\(503\) −7.09022e6 −1.24951 −0.624755 0.780821i \(-0.714801\pi\)
−0.624755 + 0.780821i \(0.714801\pi\)
\(504\) 0 0
\(505\) −2.63503e6 −0.459787
\(506\) 0 0
\(507\) − 2.10578e6i − 0.363825i
\(508\) 0 0
\(509\) − 3.09922e6i − 0.530222i −0.964218 0.265111i \(-0.914591\pi\)
0.964218 0.265111i \(-0.0854086\pi\)
\(510\) 0 0
\(511\) 1.02278e7 1.73273
\(512\) 0 0
\(513\) 1.42666e6 0.239346
\(514\) 0 0
\(515\) − 384069.i − 0.0638103i
\(516\) 0 0
\(517\) 1.96372e6i 0.323112i
\(518\) 0 0
\(519\) −2.40088e6 −0.391248
\(520\) 0 0
\(521\) −6.38166e6 −1.03000 −0.515002 0.857189i \(-0.672209\pi\)
−0.515002 + 0.857189i \(0.672209\pi\)
\(522\) 0 0
\(523\) − 4.15218e6i − 0.663776i −0.943319 0.331888i \(-0.892314\pi\)
0.943319 0.331888i \(-0.107686\pi\)
\(524\) 0 0
\(525\) 792736.i 0.125525i
\(526\) 0 0
\(527\) 5.23429e6 0.820977
\(528\) 0 0
\(529\) −5.77355e6 −0.897024
\(530\) 0 0
\(531\) − 3.61477e6i − 0.556346i
\(532\) 0 0
\(533\) 8.79132e6i 1.34041i
\(534\) 0 0
\(535\) −369352. −0.0557899
\(536\) 0 0
\(537\) −3.35298e6 −0.501759
\(538\) 0 0
\(539\) − 4.33686e6i − 0.642989i
\(540\) 0 0
\(541\) 6.36079e6i 0.934367i 0.884160 + 0.467184i \(0.154731\pi\)
−0.884160 + 0.467184i \(0.845269\pi\)
\(542\) 0 0
\(543\) −1.97633e6 −0.287647
\(544\) 0 0
\(545\) −4.13736e6 −0.596667
\(546\) 0 0
\(547\) 9.40603e6i 1.34412i 0.740497 + 0.672060i \(0.234590\pi\)
−0.740497 + 0.672060i \(0.765410\pi\)
\(548\) 0 0
\(549\) − 155601.i − 0.0220334i
\(550\) 0 0
\(551\) 3.23657e6 0.454157
\(552\) 0 0
\(553\) −2.12400e7 −2.95353
\(554\) 0 0
\(555\) − 1.20095e6i − 0.165497i
\(556\) 0 0
\(557\) 3.24324e6i 0.442936i 0.975168 + 0.221468i \(0.0710848\pi\)
−0.975168 + 0.221468i \(0.928915\pi\)
\(558\) 0 0
\(559\) 1.79305e7 2.42696
\(560\) 0 0
\(561\) 693910. 0.0930884
\(562\) 0 0
\(563\) − 8.80932e6i − 1.17131i −0.810561 0.585654i \(-0.800838\pi\)
0.810561 0.585654i \(-0.199162\pi\)
\(564\) 0 0
\(565\) 1.82084e6i 0.239967i
\(566\) 0 0
\(567\) −1.02619e7 −1.34051
\(568\) 0 0
\(569\) 9.59867e6 1.24288 0.621441 0.783461i \(-0.286547\pi\)
0.621441 + 0.783461i \(0.286547\pi\)
\(570\) 0 0
\(571\) − 1.28215e7i − 1.64569i −0.568262 0.822847i \(-0.692384\pi\)
0.568262 0.822847i \(-0.307616\pi\)
\(572\) 0 0
\(573\) 2.31147e6i 0.294104i
\(574\) 0 0
\(575\) −508824. −0.0641797
\(576\) 0 0
\(577\) −5.39655e6 −0.674803 −0.337401 0.941361i \(-0.609548\pi\)
−0.337401 + 0.941361i \(0.609548\pi\)
\(578\) 0 0
\(579\) − 143216.i − 0.0177539i
\(580\) 0 0
\(581\) − 2.90675e7i − 3.57246i
\(582\) 0 0
\(583\) −1.23033e6 −0.149916
\(584\) 0 0
\(585\) 4.81965e6 0.582271
\(586\) 0 0
\(587\) − 2.42415e6i − 0.290379i −0.989404 0.145189i \(-0.953621\pi\)
0.989404 0.145189i \(-0.0463792\pi\)
\(588\) 0 0
\(589\) − 2.21082e6i − 0.262582i
\(590\) 0 0
\(591\) 3.78327e6 0.445552
\(592\) 0 0
\(593\) −9.84341e6 −1.14950 −0.574750 0.818329i \(-0.694901\pi\)
−0.574750 + 0.818329i \(0.694901\pi\)
\(594\) 0 0
\(595\) − 9.06327e6i − 1.04952i
\(596\) 0 0
\(597\) − 1.37649e6i − 0.158066i
\(598\) 0 0
\(599\) 1.63365e6 0.186034 0.0930169 0.995665i \(-0.470349\pi\)
0.0930169 + 0.995665i \(0.470349\pi\)
\(600\) 0 0
\(601\) −8.99906e6 −1.01627 −0.508137 0.861276i \(-0.669666\pi\)
−0.508137 + 0.861276i \(0.669666\pi\)
\(602\) 0 0
\(603\) 4.77144e6i 0.534388i
\(604\) 0 0
\(605\) 3.80270e6i 0.422380i
\(606\) 0 0
\(607\) 6.70178e6 0.738276 0.369138 0.929375i \(-0.379653\pi\)
0.369138 + 0.929375i \(0.379653\pi\)
\(608\) 0 0
\(609\) 6.71137e6 0.733277
\(610\) 0 0
\(611\) 1.84201e7i 1.99613i
\(612\) 0 0
\(613\) − 5.70585e6i − 0.613294i −0.951823 0.306647i \(-0.900793\pi\)
0.951823 0.306647i \(-0.0992072\pi\)
\(614\) 0 0
\(615\) 1.25535e6 0.133837
\(616\) 0 0
\(617\) 1.55652e7 1.64605 0.823023 0.568008i \(-0.192286\pi\)
0.823023 + 0.568008i \(0.192286\pi\)
\(618\) 0 0
\(619\) − 1.26590e7i − 1.32792i −0.747766 0.663962i \(-0.768874\pi\)
0.747766 0.663962i \(-0.231126\pi\)
\(620\) 0 0
\(621\) 1.89883e6i 0.197586i
\(622\) 0 0
\(623\) −3.03359e7 −3.13139
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) − 293088.i − 0.0297735i
\(628\) 0 0
\(629\) 1.37303e7i 1.38374i
\(630\) 0 0
\(631\) 1.23076e7 1.23055 0.615274 0.788313i \(-0.289045\pi\)
0.615274 + 0.788313i \(0.289045\pi\)
\(632\) 0 0
\(633\) −102770. −0.0101943
\(634\) 0 0
\(635\) 207179.i 0.0203897i
\(636\) 0 0
\(637\) − 4.06808e7i − 3.97229i
\(638\) 0 0
\(639\) −8.36136e6 −0.810075
\(640\) 0 0
\(641\) −1.25160e7 −1.20315 −0.601576 0.798816i \(-0.705460\pi\)
−0.601576 + 0.798816i \(0.705460\pi\)
\(642\) 0 0
\(643\) − 5.85603e6i − 0.558568i −0.960209 0.279284i \(-0.909903\pi\)
0.960209 0.279284i \(-0.0900970\pi\)
\(644\) 0 0
\(645\) − 2.56037e6i − 0.242328i
\(646\) 0 0
\(647\) 1.02686e7 0.964387 0.482194 0.876065i \(-0.339840\pi\)
0.482194 + 0.876065i \(0.339840\pi\)
\(648\) 0 0
\(649\) −1.57294e6 −0.146588
\(650\) 0 0
\(651\) − 4.58437e6i − 0.423962i
\(652\) 0 0
\(653\) − 6277.16i 0 0.000576076i −1.00000 0.000288038i \(-0.999908\pi\)
1.00000 0.000288038i \(-9.16854e-5\pi\)
\(654\) 0 0
\(655\) 4.43033e6 0.403490
\(656\) 0 0
\(657\) −8.87936e6 −0.802543
\(658\) 0 0
\(659\) − 1.90209e7i − 1.70615i −0.521784 0.853077i \(-0.674733\pi\)
0.521784 0.853077i \(-0.325267\pi\)
\(660\) 0 0
\(661\) − 9.84622e6i − 0.876528i −0.898846 0.438264i \(-0.855594\pi\)
0.898846 0.438264i \(-0.144406\pi\)
\(662\) 0 0
\(663\) 6.50904e6 0.575086
\(664\) 0 0
\(665\) −3.82807e6 −0.335681
\(666\) 0 0
\(667\) 4.30775e6i 0.374918i
\(668\) 0 0
\(669\) − 153928.i − 0.0132970i
\(670\) 0 0
\(671\) −67708.5 −0.00580546
\(672\) 0 0
\(673\) −1.82184e7 −1.55050 −0.775252 0.631652i \(-0.782377\pi\)
−0.775252 + 0.631652i \(0.782377\pi\)
\(674\) 0 0
\(675\) − 1.45773e6i − 0.123145i
\(676\) 0 0
\(677\) 8.10909e6i 0.679987i 0.940428 + 0.339993i \(0.110425\pi\)
−0.940428 + 0.339993i \(0.889575\pi\)
\(678\) 0 0
\(679\) −5.40687e6 −0.450061
\(680\) 0 0
\(681\) 7.38343e6 0.610085
\(682\) 0 0
\(683\) 1.11760e7i 0.916717i 0.888767 + 0.458358i \(0.151562\pi\)
−0.888767 + 0.458358i \(0.848438\pi\)
\(684\) 0 0
\(685\) 8.80269e6i 0.716785i
\(686\) 0 0
\(687\) 4.88164e6 0.394616
\(688\) 0 0
\(689\) −1.15408e7 −0.926161
\(690\) 0 0
\(691\) − 1.40751e6i − 0.112139i −0.998427 0.0560695i \(-0.982143\pi\)
0.998427 0.0560695i \(-0.0178568\pi\)
\(692\) 0 0
\(693\) 5.14492e6i 0.406954i
\(694\) 0 0
\(695\) −3.63662e6 −0.285586
\(696\) 0 0
\(697\) −1.43523e7 −1.11902
\(698\) 0 0
\(699\) − 6.68125e6i − 0.517208i
\(700\) 0 0
\(701\) 1.40860e7i 1.08266i 0.840809 + 0.541331i \(0.182079\pi\)
−0.840809 + 0.541331i \(0.817921\pi\)
\(702\) 0 0
\(703\) 5.79930e6 0.442576
\(704\) 0 0
\(705\) 2.63029e6 0.199311
\(706\) 0 0
\(707\) − 2.63854e7i − 1.98525i
\(708\) 0 0
\(709\) − 1.38968e6i − 0.103824i −0.998652 0.0519122i \(-0.983468\pi\)
0.998652 0.0519122i \(-0.0165316\pi\)
\(710\) 0 0
\(711\) 1.84396e7 1.36797
\(712\) 0 0
\(713\) 2.94251e6 0.216768
\(714\) 0 0
\(715\) − 2.09722e6i − 0.153419i
\(716\) 0 0
\(717\) 5.97744e6i 0.434227i
\(718\) 0 0
\(719\) 1.91426e7 1.38096 0.690478 0.723354i \(-0.257400\pi\)
0.690478 + 0.723354i \(0.257400\pi\)
\(720\) 0 0
\(721\) 3.84580e6 0.275517
\(722\) 0 0
\(723\) 3.98210e6i 0.283313i
\(724\) 0 0
\(725\) − 3.30707e6i − 0.233667i
\(726\) 0 0
\(727\) 1.38873e7 0.974497 0.487248 0.873263i \(-0.338001\pi\)
0.487248 + 0.873263i \(0.338001\pi\)
\(728\) 0 0
\(729\) 6.03727e6 0.420748
\(730\) 0 0
\(731\) 2.92725e7i 2.02612i
\(732\) 0 0
\(733\) − 9.41321e6i − 0.647110i −0.946209 0.323555i \(-0.895122\pi\)
0.946209 0.323555i \(-0.104878\pi\)
\(734\) 0 0
\(735\) −5.80899e6 −0.396627
\(736\) 0 0
\(737\) 2.07625e6 0.140803
\(738\) 0 0
\(739\) − 118264.i − 0.00796605i −0.999992 0.00398302i \(-0.998732\pi\)
0.999992 0.00398302i \(-0.00126784\pi\)
\(740\) 0 0
\(741\) − 2.74924e6i − 0.183936i
\(742\) 0 0
\(743\) 1.62763e6 0.108164 0.0540820 0.998537i \(-0.482777\pi\)
0.0540820 + 0.998537i \(0.482777\pi\)
\(744\) 0 0
\(745\) 4.05624e6 0.267752
\(746\) 0 0
\(747\) 2.52351e7i 1.65464i
\(748\) 0 0
\(749\) − 3.69843e6i − 0.240887i
\(750\) 0 0
\(751\) −9.02408e6 −0.583853 −0.291926 0.956441i \(-0.594296\pi\)
−0.291926 + 0.956441i \(0.594296\pi\)
\(752\) 0 0
\(753\) −4.68028e6 −0.300805
\(754\) 0 0
\(755\) 1.06935e7i 0.682735i
\(756\) 0 0
\(757\) 645135.i 0.0409177i 0.999791 + 0.0204588i \(0.00651271\pi\)
−0.999791 + 0.0204588i \(0.993487\pi\)
\(758\) 0 0
\(759\) 390089. 0.0245787
\(760\) 0 0
\(761\) 4.61414e6 0.288821 0.144411 0.989518i \(-0.453871\pi\)
0.144411 + 0.989518i \(0.453871\pi\)
\(762\) 0 0
\(763\) − 4.14287e7i − 2.57626i
\(764\) 0 0
\(765\) 7.86833e6i 0.486104i
\(766\) 0 0
\(767\) −1.47545e7 −0.905601
\(768\) 0 0
\(769\) 59875.3 0.00365117 0.00182559 0.999998i \(-0.499419\pi\)
0.00182559 + 0.999998i \(0.499419\pi\)
\(770\) 0 0
\(771\) − 2.52211e6i − 0.152802i
\(772\) 0 0
\(773\) 1.78468e7i 1.07426i 0.843499 + 0.537132i \(0.180492\pi\)
−0.843499 + 0.537132i \(0.819508\pi\)
\(774\) 0 0
\(775\) −2.25897e6 −0.135100
\(776\) 0 0
\(777\) 1.20255e7 0.714577
\(778\) 0 0
\(779\) 6.06202e6i 0.357910i
\(780\) 0 0
\(781\) 3.63837e6i 0.213442i
\(782\) 0 0
\(783\) −1.23413e7 −0.719377
\(784\) 0 0
\(785\) −6.96609e6 −0.403473
\(786\) 0 0
\(787\) − 1.87666e7i − 1.08006i −0.841645 0.540031i \(-0.818412\pi\)
0.841645 0.540031i \(-0.181588\pi\)
\(788\) 0 0
\(789\) − 3.77667e6i − 0.215981i
\(790\) 0 0
\(791\) −1.82326e7 −1.03612
\(792\) 0 0
\(793\) −635122. −0.0358653
\(794\) 0 0
\(795\) 1.64796e6i 0.0924757i
\(796\) 0 0
\(797\) − 3.95963e6i − 0.220805i −0.993887 0.110403i \(-0.964786\pi\)
0.993887 0.110403i \(-0.0352140\pi\)
\(798\) 0 0
\(799\) −3.00719e7 −1.66645
\(800\) 0 0
\(801\) 2.63363e7 1.45035
\(802\) 0 0
\(803\) 3.86377e6i 0.211457i
\(804\) 0 0
\(805\) − 5.09501e6i − 0.277112i
\(806\) 0 0
\(807\) −9.17044e6 −0.495685
\(808\) 0 0
\(809\) −8.74535e6 −0.469792 −0.234896 0.972020i \(-0.575475\pi\)
−0.234896 + 0.972020i \(0.575475\pi\)
\(810\) 0 0
\(811\) − 2.74240e7i − 1.46413i −0.681236 0.732064i \(-0.738557\pi\)
0.681236 0.732064i \(-0.261443\pi\)
\(812\) 0 0
\(813\) 1.11631e7i 0.592320i
\(814\) 0 0
\(815\) 6.91537e6 0.364688
\(816\) 0 0
\(817\) 1.23639e7 0.648037
\(818\) 0 0
\(819\) 4.82606e7i 2.51410i
\(820\) 0 0
\(821\) 1.48659e7i 0.769719i 0.922975 + 0.384859i \(0.125750\pi\)
−0.922975 + 0.384859i \(0.874250\pi\)
\(822\) 0 0
\(823\) −9.27591e6 −0.477372 −0.238686 0.971097i \(-0.576717\pi\)
−0.238686 + 0.971097i \(0.576717\pi\)
\(824\) 0 0
\(825\) −299472. −0.0153187
\(826\) 0 0
\(827\) 6.99433e6i 0.355617i 0.984065 + 0.177808i \(0.0569007\pi\)
−0.984065 + 0.177808i \(0.943099\pi\)
\(828\) 0 0
\(829\) − 3.29484e7i − 1.66513i −0.553929 0.832564i \(-0.686872\pi\)
0.553929 0.832564i \(-0.313128\pi\)
\(830\) 0 0
\(831\) 9.39234e6 0.471814
\(832\) 0 0
\(833\) 6.64135e7 3.31623
\(834\) 0 0
\(835\) − 75204.6i − 0.00373274i
\(836\) 0 0
\(837\) 8.43002e6i 0.415925i
\(838\) 0 0
\(839\) 6.29215e6 0.308599 0.154299 0.988024i \(-0.450688\pi\)
0.154299 + 0.988024i \(0.450688\pi\)
\(840\) 0 0
\(841\) −7.48678e6 −0.365010
\(842\) 0 0
\(843\) − 5.39824e6i − 0.261627i
\(844\) 0 0
\(845\) − 1.03902e7i − 0.500588i
\(846\) 0 0
\(847\) −3.80776e7 −1.82373
\(848\) 0 0
\(849\) 194312. 0.00925191
\(850\) 0 0
\(851\) 7.71864e6i 0.365356i
\(852\) 0 0
\(853\) − 2.28452e6i − 0.107504i −0.998554 0.0537518i \(-0.982882\pi\)
0.998554 0.0537518i \(-0.0171180\pi\)
\(854\) 0 0
\(855\) 3.32336e6 0.155476
\(856\) 0 0
\(857\) 1.33739e6 0.0622023 0.0311012 0.999516i \(-0.490099\pi\)
0.0311012 + 0.999516i \(0.490099\pi\)
\(858\) 0 0
\(859\) 3.12847e7i 1.44660i 0.690533 + 0.723301i \(0.257376\pi\)
−0.690533 + 0.723301i \(0.742624\pi\)
\(860\) 0 0
\(861\) 1.25702e7i 0.577877i
\(862\) 0 0
\(863\) 3.83963e7 1.75494 0.877469 0.479633i \(-0.159230\pi\)
0.877469 + 0.479633i \(0.159230\pi\)
\(864\) 0 0
\(865\) −1.18462e7 −0.538320
\(866\) 0 0
\(867\) 3.43228e6i 0.155072i
\(868\) 0 0
\(869\) − 8.02382e6i − 0.360439i
\(870\) 0 0
\(871\) 1.94757e7 0.869858
\(872\) 0 0
\(873\) 4.69401e6 0.208453
\(874\) 0 0
\(875\) 3.91145e6i 0.172710i
\(876\) 0 0
\(877\) 1.62074e7i 0.711565i 0.934569 + 0.355783i \(0.115786\pi\)
−0.934569 + 0.355783i \(0.884214\pi\)
\(878\) 0 0
\(879\) −2.35045e6 −0.102607
\(880\) 0 0
\(881\) −3.64879e6 −0.158383 −0.0791916 0.996859i \(-0.525234\pi\)
−0.0791916 + 0.996859i \(0.525234\pi\)
\(882\) 0 0
\(883\) 3.21031e7i 1.38562i 0.721119 + 0.692811i \(0.243628\pi\)
−0.721119 + 0.692811i \(0.756372\pi\)
\(884\) 0 0
\(885\) 2.10686e6i 0.0904228i
\(886\) 0 0
\(887\) 1.29249e7 0.551594 0.275797 0.961216i \(-0.411058\pi\)
0.275797 + 0.961216i \(0.411058\pi\)
\(888\) 0 0
\(889\) −2.07455e6 −0.0880379
\(890\) 0 0
\(891\) − 3.87665e6i − 0.163592i
\(892\) 0 0
\(893\) 1.27015e7i 0.533000i
\(894\) 0 0
\(895\) −1.65440e7 −0.690372
\(896\) 0 0
\(897\) 3.65913e6 0.151844
\(898\) 0 0
\(899\) 1.91247e7i 0.789214i
\(900\) 0 0
\(901\) − 1.88409e7i − 0.773196i
\(902\) 0 0
\(903\) 2.56378e7 1.04631
\(904\) 0 0
\(905\) −9.75143e6 −0.395774
\(906\) 0 0
\(907\) − 541993.i − 0.0218764i −0.999940 0.0109382i \(-0.996518\pi\)
0.999940 0.0109382i \(-0.00348181\pi\)
\(908\) 0 0
\(909\) 2.29066e7i 0.919498i
\(910\) 0 0
\(911\) −3.25168e7 −1.29811 −0.649055 0.760741i \(-0.724836\pi\)
−0.649055 + 0.760741i \(0.724836\pi\)
\(912\) 0 0
\(913\) 1.09808e7 0.435971
\(914\) 0 0
\(915\) 90691.8i 0.00358109i
\(916\) 0 0
\(917\) 4.43623e7i 1.74217i
\(918\) 0 0
\(919\) 3.14025e7 1.22652 0.613261 0.789880i \(-0.289857\pi\)
0.613261 + 0.789880i \(0.289857\pi\)
\(920\) 0 0
\(921\) −1.03168e6 −0.0400772
\(922\) 0 0
\(923\) 3.41288e7i 1.31861i
\(924\) 0 0
\(925\) − 5.92561e6i − 0.227708i
\(926\) 0 0
\(927\) −3.33876e6 −0.127610
\(928\) 0 0
\(929\) 1.06286e7 0.404052 0.202026 0.979380i \(-0.435247\pi\)
0.202026 + 0.979380i \(0.435247\pi\)
\(930\) 0 0
\(931\) − 2.80513e7i − 1.06066i
\(932\) 0 0
\(933\) − 6.71978e6i − 0.252726i
\(934\) 0 0
\(935\) 3.42383e6 0.128081
\(936\) 0 0
\(937\) −1.41673e7 −0.527154 −0.263577 0.964638i \(-0.584902\pi\)
−0.263577 + 0.964638i \(0.584902\pi\)
\(938\) 0 0
\(939\) − 8.22293e6i − 0.304343i
\(940\) 0 0
\(941\) 4.78895e6i 0.176306i 0.996107 + 0.0881528i \(0.0280964\pi\)
−0.996107 + 0.0881528i \(0.971904\pi\)
\(942\) 0 0
\(943\) −8.06830e6 −0.295463
\(944\) 0 0
\(945\) 1.45967e7 0.531712
\(946\) 0 0
\(947\) 4.35818e7i 1.57917i 0.613639 + 0.789587i \(0.289705\pi\)
−0.613639 + 0.789587i \(0.710295\pi\)
\(948\) 0 0
\(949\) 3.62431e7i 1.30635i
\(950\) 0 0
\(951\) −4.66620e6 −0.167306
\(952\) 0 0
\(953\) −639034. −0.0227925 −0.0113962 0.999935i \(-0.503628\pi\)
−0.0113962 + 0.999935i \(0.503628\pi\)
\(954\) 0 0
\(955\) 1.14051e7i 0.404659i
\(956\) 0 0
\(957\) 2.53536e6i 0.0894868i
\(958\) 0 0
\(959\) −8.81442e7 −3.09490
\(960\) 0 0
\(961\) −1.55656e7 −0.543697
\(962\) 0 0
\(963\) 3.21082e6i 0.111571i
\(964\) 0 0
\(965\) − 706642.i − 0.0244276i
\(966\) 0 0
\(967\) 1.89639e6 0.0652172 0.0326086 0.999468i \(-0.489619\pi\)
0.0326086 + 0.999468i \(0.489619\pi\)
\(968\) 0 0
\(969\) 4.48828e6 0.153557
\(970\) 0 0
\(971\) 2.30710e7i 0.785268i 0.919695 + 0.392634i \(0.128436\pi\)
−0.919695 + 0.392634i \(0.871564\pi\)
\(972\) 0 0
\(973\) − 3.64147e7i − 1.23309i
\(974\) 0 0
\(975\) −2.80912e6 −0.0946365
\(976\) 0 0
\(977\) 4.36091e7 1.46164 0.730820 0.682571i \(-0.239138\pi\)
0.730820 + 0.682571i \(0.239138\pi\)
\(978\) 0 0
\(979\) − 1.14600e7i − 0.382145i
\(980\) 0 0
\(981\) 3.59666e7i 1.19324i
\(982\) 0 0
\(983\) −5.07856e6 −0.167632 −0.0838160 0.996481i \(-0.526711\pi\)
−0.0838160 + 0.996481i \(0.526711\pi\)
\(984\) 0 0
\(985\) 1.86671e7 0.613036
\(986\) 0 0
\(987\) 2.63379e7i 0.860575i
\(988\) 0 0
\(989\) 1.64558e7i 0.534970i
\(990\) 0 0
\(991\) −1.28580e7 −0.415901 −0.207950 0.978139i \(-0.566679\pi\)
−0.207950 + 0.978139i \(0.566679\pi\)
\(992\) 0 0
\(993\) 3.49445e6 0.112462
\(994\) 0 0
\(995\) − 6.79177e6i − 0.217483i
\(996\) 0 0
\(997\) − 3.44037e7i − 1.09614i −0.836431 0.548072i \(-0.815362\pi\)
0.836431 0.548072i \(-0.184638\pi\)
\(998\) 0 0
\(999\) −2.21132e7 −0.701031
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.d.c.161.8 yes 12
4.3 odd 2 320.6.d.d.161.5 yes 12
8.3 odd 2 320.6.d.d.161.8 yes 12
8.5 even 2 inner 320.6.d.c.161.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.6.d.c.161.5 12 8.5 even 2 inner
320.6.d.c.161.8 yes 12 1.1 even 1 trivial
320.6.d.d.161.5 yes 12 4.3 odd 2
320.6.d.d.161.8 yes 12 8.3 odd 2