Properties

Label 320.6.d.c.161.7
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 161.7
Root \(-3.41426 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 320.161
Dual form 320.6.d.c.161.6

$q$-expansion

\(f(q)\) \(=\) \(q+3.09968i q^{3} -25.0000i q^{5} +118.821 q^{7} +233.392 q^{9} +O(q^{10})\) \(q+3.09968i q^{3} -25.0000i q^{5} +118.821 q^{7} +233.392 q^{9} -355.467i q^{11} -370.735i q^{13} +77.4920 q^{15} -586.019 q^{17} -972.770i q^{19} +368.307i q^{21} +1298.93 q^{23} -625.000 q^{25} +1476.66i q^{27} +5226.75i q^{29} -4622.64 q^{31} +1101.83 q^{33} -2970.52i q^{35} -5296.01i q^{37} +1149.16 q^{39} +14042.0 q^{41} -4229.77i q^{43} -5834.80i q^{45} -18577.1 q^{47} -2688.58 q^{49} -1816.47i q^{51} -20963.9i q^{53} -8886.67 q^{55} +3015.28 q^{57} -48573.7i q^{59} -3620.88i q^{61} +27731.9 q^{63} -9268.37 q^{65} -1683.52i q^{67} +4026.27i q^{69} -5332.40 q^{71} +80560.7 q^{73} -1937.30i q^{75} -42236.9i q^{77} -39272.8 q^{79} +52137.1 q^{81} -18385.7i q^{83} +14650.5i q^{85} -16201.3 q^{87} +43137.3 q^{89} -44051.1i q^{91} -14328.7i q^{93} -24319.2 q^{95} +134744. q^{97} -82963.1i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 268q^{7} - 428q^{9} + O(q^{10}) \) \( 12q - 268q^{7} - 428q^{9} + 900q^{15} + 2400q^{17} - 8108q^{23} - 7500q^{25} - 7976q^{31} - 20776q^{33} - 40984q^{39} - 56408q^{41} - 21172q^{47} - 5540q^{49} - 18400q^{55} + 39992q^{57} - 179516q^{63} + 44000q^{65} - 367704q^{71} + 58736q^{73} - 26192q^{79} + 411692q^{81} - 183200q^{87} + 87672q^{89} - 121000q^{95} - 172336q^{97} + O(q^{100}) \)

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
\(3\) 3.09968i 0.198845i 0.995045 + 0.0994223i \(0.0316995\pi\)
−0.995045 + 0.0994223i \(0.968301\pi\)
\(4\) 0 0
\(5\) − 25.0000i − 0.447214i
\(6\) 0 0
\(7\) 118.821 0.916533 0.458266 0.888815i \(-0.348471\pi\)
0.458266 + 0.888815i \(0.348471\pi\)
\(8\) 0 0
\(9\) 233.392 0.960461
\(10\) 0 0
\(11\) − 355.467i − 0.885763i −0.896580 0.442881i \(-0.853956\pi\)
0.896580 0.442881i \(-0.146044\pi\)
\(12\) 0 0
\(13\) − 370.735i − 0.608422i −0.952605 0.304211i \(-0.901607\pi\)
0.952605 0.304211i \(-0.0983928\pi\)
\(14\) 0 0
\(15\) 77.4920 0.0889260
\(16\) 0 0
\(17\) −586.019 −0.491801 −0.245900 0.969295i \(-0.579084\pi\)
−0.245900 + 0.969295i \(0.579084\pi\)
\(18\) 0 0
\(19\) − 972.770i − 0.618196i −0.951030 0.309098i \(-0.899973\pi\)
0.951030 0.309098i \(-0.100027\pi\)
\(20\) 0 0
\(21\) 368.307i 0.182248i
\(22\) 0 0
\(23\) 1298.93 0.511996 0.255998 0.966677i \(-0.417596\pi\)
0.255998 + 0.966677i \(0.417596\pi\)
\(24\) 0 0
\(25\) −625.000 −0.200000
\(26\) 0 0
\(27\) 1476.66i 0.389827i
\(28\) 0 0
\(29\) 5226.75i 1.15408i 0.816715 + 0.577041i \(0.195793\pi\)
−0.816715 + 0.577041i \(0.804207\pi\)
\(30\) 0 0
\(31\) −4622.64 −0.863944 −0.431972 0.901887i \(-0.642182\pi\)
−0.431972 + 0.901887i \(0.642182\pi\)
\(32\) 0 0
\(33\) 1101.83 0.176129
\(34\) 0 0
\(35\) − 2970.52i − 0.409886i
\(36\) 0 0
\(37\) − 5296.01i − 0.635982i −0.948094 0.317991i \(-0.896992\pi\)
0.948094 0.317991i \(-0.103008\pi\)
\(38\) 0 0
\(39\) 1149.16 0.120981
\(40\) 0 0
\(41\) 14042.0 1.30457 0.652286 0.757973i \(-0.273810\pi\)
0.652286 + 0.757973i \(0.273810\pi\)
\(42\) 0 0
\(43\) − 4229.77i − 0.348856i −0.984670 0.174428i \(-0.944192\pi\)
0.984670 0.174428i \(-0.0558076\pi\)
\(44\) 0 0
\(45\) − 5834.80i − 0.429531i
\(46\) 0 0
\(47\) −18577.1 −1.22668 −0.613341 0.789818i \(-0.710175\pi\)
−0.613341 + 0.789818i \(0.710175\pi\)
\(48\) 0 0
\(49\) −2688.58 −0.159968
\(50\) 0 0
\(51\) − 1816.47i − 0.0977920i
\(52\) 0 0
\(53\) − 20963.9i − 1.02514i −0.858645 0.512570i \(-0.828694\pi\)
0.858645 0.512570i \(-0.171306\pi\)
\(54\) 0 0
\(55\) −8886.67 −0.396125
\(56\) 0 0
\(57\) 3015.28 0.122925
\(58\) 0 0
\(59\) − 48573.7i − 1.81665i −0.418265 0.908325i \(-0.637362\pi\)
0.418265 0.908325i \(-0.362638\pi\)
\(60\) 0 0
\(61\) − 3620.88i − 0.124592i −0.998058 0.0622958i \(-0.980158\pi\)
0.998058 0.0622958i \(-0.0198422\pi\)
\(62\) 0 0
\(63\) 27731.9 0.880294
\(64\) 0 0
\(65\) −9268.37 −0.272095
\(66\) 0 0
\(67\) − 1683.52i − 0.0458174i −0.999738 0.0229087i \(-0.992707\pi\)
0.999738 0.0229087i \(-0.00729270\pi\)
\(68\) 0 0
\(69\) 4026.27i 0.101808i
\(70\) 0 0
\(71\) −5332.40 −0.125538 −0.0627692 0.998028i \(-0.519993\pi\)
−0.0627692 + 0.998028i \(0.519993\pi\)
\(72\) 0 0
\(73\) 80560.7 1.76936 0.884680 0.466199i \(-0.154377\pi\)
0.884680 + 0.466199i \(0.154377\pi\)
\(74\) 0 0
\(75\) − 1937.30i − 0.0397689i
\(76\) 0 0
\(77\) − 42236.9i − 0.811831i
\(78\) 0 0
\(79\) −39272.8 −0.707985 −0.353992 0.935248i \(-0.615176\pi\)
−0.353992 + 0.935248i \(0.615176\pi\)
\(80\) 0 0
\(81\) 52137.1 0.882946
\(82\) 0 0
\(83\) − 18385.7i − 0.292944i −0.989215 0.146472i \(-0.953208\pi\)
0.989215 0.146472i \(-0.0467918\pi\)
\(84\) 0 0
\(85\) 14650.5i 0.219940i
\(86\) 0 0
\(87\) −16201.3 −0.229483
\(88\) 0 0
\(89\) 43137.3 0.577269 0.288635 0.957439i \(-0.406799\pi\)
0.288635 + 0.957439i \(0.406799\pi\)
\(90\) 0 0
\(91\) − 44051.1i − 0.557639i
\(92\) 0 0
\(93\) − 14328.7i − 0.171791i
\(94\) 0 0
\(95\) −24319.2 −0.276466
\(96\) 0 0
\(97\) 134744. 1.45405 0.727027 0.686609i \(-0.240902\pi\)
0.727027 + 0.686609i \(0.240902\pi\)
\(98\) 0 0
\(99\) − 82963.1i − 0.850740i
\(100\) 0 0
\(101\) − 39471.3i − 0.385015i −0.981296 0.192507i \(-0.938338\pi\)
0.981296 0.192507i \(-0.0616620\pi\)
\(102\) 0 0
\(103\) −164437. −1.52724 −0.763620 0.645666i \(-0.776580\pi\)
−0.763620 + 0.645666i \(0.776580\pi\)
\(104\) 0 0
\(105\) 9207.68 0.0815036
\(106\) 0 0
\(107\) − 203860.i − 1.72137i −0.509142 0.860683i \(-0.670037\pi\)
0.509142 0.860683i \(-0.329963\pi\)
\(108\) 0 0
\(109\) − 65561.7i − 0.528548i −0.964448 0.264274i \(-0.914868\pi\)
0.964448 0.264274i \(-0.0851322\pi\)
\(110\) 0 0
\(111\) 16415.9 0.126462
\(112\) 0 0
\(113\) 35057.0 0.258273 0.129136 0.991627i \(-0.458780\pi\)
0.129136 + 0.991627i \(0.458780\pi\)
\(114\) 0 0
\(115\) − 32473.3i − 0.228972i
\(116\) 0 0
\(117\) − 86526.6i − 0.584366i
\(118\) 0 0
\(119\) −69631.3 −0.450752
\(120\) 0 0
\(121\) 34694.3 0.215424
\(122\) 0 0
\(123\) 43525.6i 0.259407i
\(124\) 0 0
\(125\) 15625.0i 0.0894427i
\(126\) 0 0
\(127\) 157228. 0.865007 0.432503 0.901632i \(-0.357630\pi\)
0.432503 + 0.901632i \(0.357630\pi\)
\(128\) 0 0
\(129\) 13110.9 0.0693681
\(130\) 0 0
\(131\) − 85680.9i − 0.436220i −0.975924 0.218110i \(-0.930011\pi\)
0.975924 0.218110i \(-0.0699892\pi\)
\(132\) 0 0
\(133\) − 115585.i − 0.566597i
\(134\) 0 0
\(135\) 36916.6 0.174336
\(136\) 0 0
\(137\) −99340.8 −0.452196 −0.226098 0.974105i \(-0.572597\pi\)
−0.226098 + 0.974105i \(0.572597\pi\)
\(138\) 0 0
\(139\) 36370.0i 0.159664i 0.996808 + 0.0798319i \(0.0254384\pi\)
−0.996808 + 0.0798319i \(0.974562\pi\)
\(140\) 0 0
\(141\) − 57582.9i − 0.243919i
\(142\) 0 0
\(143\) −131784. −0.538918
\(144\) 0 0
\(145\) 130669. 0.516121
\(146\) 0 0
\(147\) − 8333.74i − 0.0318088i
\(148\) 0 0
\(149\) − 241962.i − 0.892857i −0.894819 0.446428i \(-0.852696\pi\)
0.894819 0.446428i \(-0.147304\pi\)
\(150\) 0 0
\(151\) −241079. −0.860433 −0.430217 0.902726i \(-0.641563\pi\)
−0.430217 + 0.902726i \(0.641563\pi\)
\(152\) 0 0
\(153\) −136772. −0.472356
\(154\) 0 0
\(155\) 115566.i 0.386367i
\(156\) 0 0
\(157\) 450362.i 1.45818i 0.684416 + 0.729092i \(0.260057\pi\)
−0.684416 + 0.729092i \(0.739943\pi\)
\(158\) 0 0
\(159\) 64981.5 0.203844
\(160\) 0 0
\(161\) 154340. 0.469261
\(162\) 0 0
\(163\) − 388586.i − 1.14556i −0.819709 0.572780i \(-0.805865\pi\)
0.819709 0.572780i \(-0.194135\pi\)
\(164\) 0 0
\(165\) − 27545.8i − 0.0787674i
\(166\) 0 0
\(167\) −16419.7 −0.0455589 −0.0227794 0.999741i \(-0.507252\pi\)
−0.0227794 + 0.999741i \(0.507252\pi\)
\(168\) 0 0
\(169\) 233849. 0.629822
\(170\) 0 0
\(171\) − 227037.i − 0.593753i
\(172\) 0 0
\(173\) − 111836.i − 0.284097i −0.989860 0.142048i \(-0.954631\pi\)
0.989860 0.142048i \(-0.0453689\pi\)
\(174\) 0 0
\(175\) −74263.1 −0.183307
\(176\) 0 0
\(177\) 150563. 0.361231
\(178\) 0 0
\(179\) 432673.i 1.00932i 0.863319 + 0.504658i \(0.168381\pi\)
−0.863319 + 0.504658i \(0.831619\pi\)
\(180\) 0 0
\(181\) 800005.i 1.81508i 0.419962 + 0.907542i \(0.362043\pi\)
−0.419962 + 0.907542i \(0.637957\pi\)
\(182\) 0 0
\(183\) 11223.6 0.0247744
\(184\) 0 0
\(185\) −132400. −0.284420
\(186\) 0 0
\(187\) 208310.i 0.435619i
\(188\) 0 0
\(189\) 175459.i 0.357289i
\(190\) 0 0
\(191\) 372676. 0.739177 0.369588 0.929196i \(-0.379499\pi\)
0.369588 + 0.929196i \(0.379499\pi\)
\(192\) 0 0
\(193\) −784772. −1.51653 −0.758263 0.651948i \(-0.773952\pi\)
−0.758263 + 0.651948i \(0.773952\pi\)
\(194\) 0 0
\(195\) − 28729.0i − 0.0541046i
\(196\) 0 0
\(197\) − 328945.i − 0.603889i −0.953325 0.301945i \(-0.902364\pi\)
0.953325 0.301945i \(-0.0976358\pi\)
\(198\) 0 0
\(199\) 473709. 0.847968 0.423984 0.905670i \(-0.360631\pi\)
0.423984 + 0.905670i \(0.360631\pi\)
\(200\) 0 0
\(201\) 5218.36 0.00911054
\(202\) 0 0
\(203\) 621047.i 1.05775i
\(204\) 0 0
\(205\) − 351049.i − 0.583423i
\(206\) 0 0
\(207\) 303160. 0.491752
\(208\) 0 0
\(209\) −345787. −0.547575
\(210\) 0 0
\(211\) 704656.i 1.08961i 0.838563 + 0.544805i \(0.183396\pi\)
−0.838563 + 0.544805i \(0.816604\pi\)
\(212\) 0 0
\(213\) − 16528.7i − 0.0249626i
\(214\) 0 0
\(215\) −105744. −0.156013
\(216\) 0 0
\(217\) −549266. −0.791833
\(218\) 0 0
\(219\) 249712.i 0.351828i
\(220\) 0 0
\(221\) 217258.i 0.299223i
\(222\) 0 0
\(223\) 487789. 0.656855 0.328427 0.944529i \(-0.393481\pi\)
0.328427 + 0.944529i \(0.393481\pi\)
\(224\) 0 0
\(225\) −145870. −0.192092
\(226\) 0 0
\(227\) 280630.i 0.361467i 0.983532 + 0.180734i \(0.0578472\pi\)
−0.983532 + 0.180734i \(0.942153\pi\)
\(228\) 0 0
\(229\) 336464.i 0.423984i 0.977271 + 0.211992i \(0.0679950\pi\)
−0.977271 + 0.211992i \(0.932005\pi\)
\(230\) 0 0
\(231\) 130921. 0.161428
\(232\) 0 0
\(233\) −248441. −0.299802 −0.149901 0.988701i \(-0.547895\pi\)
−0.149901 + 0.988701i \(0.547895\pi\)
\(234\) 0 0
\(235\) 464426.i 0.548589i
\(236\) 0 0
\(237\) − 121733.i − 0.140779i
\(238\) 0 0
\(239\) 685682. 0.776476 0.388238 0.921559i \(-0.373084\pi\)
0.388238 + 0.921559i \(0.373084\pi\)
\(240\) 0 0
\(241\) −762667. −0.845848 −0.422924 0.906165i \(-0.638996\pi\)
−0.422924 + 0.906165i \(0.638996\pi\)
\(242\) 0 0
\(243\) 520437.i 0.565396i
\(244\) 0 0
\(245\) 67214.5i 0.0715398i
\(246\) 0 0
\(247\) −360640. −0.376124
\(248\) 0 0
\(249\) 56989.8 0.0582504
\(250\) 0 0
\(251\) − 92462.4i − 0.0926362i −0.998927 0.0463181i \(-0.985251\pi\)
0.998927 0.0463181i \(-0.0147488\pi\)
\(252\) 0 0
\(253\) − 461727.i − 0.453507i
\(254\) 0 0
\(255\) −45411.8 −0.0437339
\(256\) 0 0
\(257\) −715466. −0.675704 −0.337852 0.941199i \(-0.609700\pi\)
−0.337852 + 0.941199i \(0.609700\pi\)
\(258\) 0 0
\(259\) − 629277.i − 0.582898i
\(260\) 0 0
\(261\) 1.21988e6i 1.10845i
\(262\) 0 0
\(263\) −2.07592e6 −1.85064 −0.925320 0.379187i \(-0.876204\pi\)
−0.925320 + 0.379187i \(0.876204\pi\)
\(264\) 0 0
\(265\) −524098. −0.458457
\(266\) 0 0
\(267\) 133712.i 0.114787i
\(268\) 0 0
\(269\) 1.61680e6i 1.36231i 0.732140 + 0.681154i \(0.238522\pi\)
−0.732140 + 0.681154i \(0.761478\pi\)
\(270\) 0 0
\(271\) 2.21414e6 1.83139 0.915695 0.401873i \(-0.131641\pi\)
0.915695 + 0.401873i \(0.131641\pi\)
\(272\) 0 0
\(273\) 136544. 0.110883
\(274\) 0 0
\(275\) 222167.i 0.177153i
\(276\) 0 0
\(277\) 568957.i 0.445533i 0.974872 + 0.222767i \(0.0715088\pi\)
−0.974872 + 0.222767i \(0.928491\pi\)
\(278\) 0 0
\(279\) −1.07889e6 −0.829784
\(280\) 0 0
\(281\) 786523. 0.594217 0.297109 0.954844i \(-0.403978\pi\)
0.297109 + 0.954844i \(0.403978\pi\)
\(282\) 0 0
\(283\) 2.33703e6i 1.73460i 0.497789 + 0.867298i \(0.334145\pi\)
−0.497789 + 0.867298i \(0.665855\pi\)
\(284\) 0 0
\(285\) − 75381.9i − 0.0549737i
\(286\) 0 0
\(287\) 1.66848e6 1.19568
\(288\) 0 0
\(289\) −1.07644e6 −0.758132
\(290\) 0 0
\(291\) 417664.i 0.289131i
\(292\) 0 0
\(293\) − 1.06744e6i − 0.726398i −0.931712 0.363199i \(-0.881685\pi\)
0.931712 0.363199i \(-0.118315\pi\)
\(294\) 0 0
\(295\) −1.21434e6 −0.812431
\(296\) 0 0
\(297\) 524905. 0.345294
\(298\) 0 0
\(299\) − 481559.i − 0.311510i
\(300\) 0 0
\(301\) − 502585.i − 0.319738i
\(302\) 0 0
\(303\) 122348. 0.0765582
\(304\) 0 0
\(305\) −90521.9 −0.0557191
\(306\) 0 0
\(307\) 345396.i 0.209156i 0.994517 + 0.104578i \(0.0333492\pi\)
−0.994517 + 0.104578i \(0.966651\pi\)
\(308\) 0 0
\(309\) − 509703.i − 0.303683i
\(310\) 0 0
\(311\) 1.38990e6 0.814860 0.407430 0.913236i \(-0.366425\pi\)
0.407430 + 0.913236i \(0.366425\pi\)
\(312\) 0 0
\(313\) −1.35897e6 −0.784061 −0.392031 0.919952i \(-0.628227\pi\)
−0.392031 + 0.919952i \(0.628227\pi\)
\(314\) 0 0
\(315\) − 693296.i − 0.393679i
\(316\) 0 0
\(317\) 2.04882e6i 1.14513i 0.819859 + 0.572565i \(0.194052\pi\)
−0.819859 + 0.572565i \(0.805948\pi\)
\(318\) 0 0
\(319\) 1.85794e6 1.02224
\(320\) 0 0
\(321\) 631902. 0.342284
\(322\) 0 0
\(323\) 570061.i 0.304029i
\(324\) 0 0
\(325\) 231709.i 0.121684i
\(326\) 0 0
\(327\) 203220. 0.105099
\(328\) 0 0
\(329\) −2.20734e6 −1.12429
\(330\) 0 0
\(331\) − 2.75605e6i − 1.38267i −0.722536 0.691333i \(-0.757024\pi\)
0.722536 0.691333i \(-0.242976\pi\)
\(332\) 0 0
\(333\) − 1.23605e6i − 0.610836i
\(334\) 0 0
\(335\) −42087.9 −0.0204902
\(336\) 0 0
\(337\) 1.96450e6 0.942275 0.471137 0.882060i \(-0.343844\pi\)
0.471137 + 0.882060i \(0.343844\pi\)
\(338\) 0 0
\(339\) 108665.i 0.0513561i
\(340\) 0 0
\(341\) 1.64319e6i 0.765249i
\(342\) 0 0
\(343\) −2.31648e6 −1.06315
\(344\) 0 0
\(345\) 100657. 0.0455298
\(346\) 0 0
\(347\) − 2.37477e6i − 1.05876i −0.848384 0.529381i \(-0.822424\pi\)
0.848384 0.529381i \(-0.177576\pi\)
\(348\) 0 0
\(349\) − 1.89577e6i − 0.833146i −0.909102 0.416573i \(-0.863231\pi\)
0.909102 0.416573i \(-0.136769\pi\)
\(350\) 0 0
\(351\) 547451. 0.237179
\(352\) 0 0
\(353\) 89473.2 0.0382170 0.0191085 0.999817i \(-0.493917\pi\)
0.0191085 + 0.999817i \(0.493917\pi\)
\(354\) 0 0
\(355\) 133310.i 0.0561425i
\(356\) 0 0
\(357\) − 215835.i − 0.0896295i
\(358\) 0 0
\(359\) −4.27153e6 −1.74923 −0.874615 0.484818i \(-0.838886\pi\)
−0.874615 + 0.484818i \(0.838886\pi\)
\(360\) 0 0
\(361\) 1.52982e6 0.617834
\(362\) 0 0
\(363\) 107541.i 0.0428360i
\(364\) 0 0
\(365\) − 2.01402e6i − 0.791282i
\(366\) 0 0
\(367\) 2.20466e6 0.854429 0.427214 0.904150i \(-0.359495\pi\)
0.427214 + 0.904150i \(0.359495\pi\)
\(368\) 0 0
\(369\) 3.27728e6 1.25299
\(370\) 0 0
\(371\) − 2.49095e6i − 0.939574i
\(372\) 0 0
\(373\) − 5.02519e6i − 1.87017i −0.354427 0.935084i \(-0.615324\pi\)
0.354427 0.935084i \(-0.384676\pi\)
\(374\) 0 0
\(375\) −48432.5 −0.0177852
\(376\) 0 0
\(377\) 1.93774e6 0.702169
\(378\) 0 0
\(379\) 128535.i 0.0459647i 0.999736 + 0.0229823i \(0.00731615\pi\)
−0.999736 + 0.0229823i \(0.992684\pi\)
\(380\) 0 0
\(381\) 487355.i 0.172002i
\(382\) 0 0
\(383\) −3.97607e6 −1.38502 −0.692512 0.721406i \(-0.743496\pi\)
−0.692512 + 0.721406i \(0.743496\pi\)
\(384\) 0 0
\(385\) −1.05592e6 −0.363062
\(386\) 0 0
\(387\) − 987195.i − 0.335062i
\(388\) 0 0
\(389\) 724285.i 0.242681i 0.992611 + 0.121340i \(0.0387192\pi\)
−0.992611 + 0.121340i \(0.961281\pi\)
\(390\) 0 0
\(391\) −761198. −0.251800
\(392\) 0 0
\(393\) 265584. 0.0867401
\(394\) 0 0
\(395\) 981819.i 0.316620i
\(396\) 0 0
\(397\) 3.92474e6i 1.24978i 0.780711 + 0.624892i \(0.214857\pi\)
−0.780711 + 0.624892i \(0.785143\pi\)
\(398\) 0 0
\(399\) 358278. 0.112665
\(400\) 0 0
\(401\) 5.85395e6 1.81798 0.908988 0.416822i \(-0.136856\pi\)
0.908988 + 0.416822i \(0.136856\pi\)
\(402\) 0 0
\(403\) 1.71377e6i 0.525643i
\(404\) 0 0
\(405\) − 1.30343e6i − 0.394865i
\(406\) 0 0
\(407\) −1.88256e6 −0.563329
\(408\) 0 0
\(409\) 2.05857e6 0.608496 0.304248 0.952593i \(-0.401595\pi\)
0.304248 + 0.952593i \(0.401595\pi\)
\(410\) 0 0
\(411\) − 307925.i − 0.0899167i
\(412\) 0 0
\(413\) − 5.77158e6i − 1.66502i
\(414\) 0 0
\(415\) −459642. −0.131009
\(416\) 0 0
\(417\) −112735. −0.0317483
\(418\) 0 0
\(419\) − 1.30873e6i − 0.364180i −0.983282 0.182090i \(-0.941714\pi\)
0.983282 0.182090i \(-0.0582863\pi\)
\(420\) 0 0
\(421\) 4.48649e6i 1.23368i 0.787089 + 0.616839i \(0.211587\pi\)
−0.787089 + 0.616839i \(0.788413\pi\)
\(422\) 0 0
\(423\) −4.33574e6 −1.17818
\(424\) 0 0
\(425\) 366262. 0.0983602
\(426\) 0 0
\(427\) − 430236.i − 0.114192i
\(428\) 0 0
\(429\) − 408488.i − 0.107161i
\(430\) 0 0
\(431\) 6.60957e6 1.71388 0.856940 0.515417i \(-0.172363\pi\)
0.856940 + 0.515417i \(0.172363\pi\)
\(432\) 0 0
\(433\) −6.04227e6 −1.54875 −0.774373 0.632729i \(-0.781935\pi\)
−0.774373 + 0.632729i \(0.781935\pi\)
\(434\) 0 0
\(435\) 405031.i 0.102628i
\(436\) 0 0
\(437\) − 1.26356e6i − 0.316514i
\(438\) 0 0
\(439\) 656087. 0.162480 0.0812401 0.996695i \(-0.474112\pi\)
0.0812401 + 0.996695i \(0.474112\pi\)
\(440\) 0 0
\(441\) −627493. −0.153643
\(442\) 0 0
\(443\) − 5.38448e6i − 1.30357i −0.758403 0.651786i \(-0.774020\pi\)
0.758403 0.651786i \(-0.225980\pi\)
\(444\) 0 0
\(445\) − 1.07843e6i − 0.258163i
\(446\) 0 0
\(447\) 750006. 0.177540
\(448\) 0 0
\(449\) 7.00212e6 1.63913 0.819565 0.572986i \(-0.194215\pi\)
0.819565 + 0.572986i \(0.194215\pi\)
\(450\) 0 0
\(451\) − 4.99145e6i − 1.15554i
\(452\) 0 0
\(453\) − 747268.i − 0.171093i
\(454\) 0 0
\(455\) −1.10128e6 −0.249384
\(456\) 0 0
\(457\) 5.95736e6 1.33433 0.667165 0.744910i \(-0.267508\pi\)
0.667165 + 0.744910i \(0.267508\pi\)
\(458\) 0 0
\(459\) − 865353.i − 0.191717i
\(460\) 0 0
\(461\) 2.60215e6i 0.570269i 0.958487 + 0.285135i \(0.0920383\pi\)
−0.958487 + 0.285135i \(0.907962\pi\)
\(462\) 0 0
\(463\) −2.68993e6 −0.583160 −0.291580 0.956546i \(-0.594181\pi\)
−0.291580 + 0.956546i \(0.594181\pi\)
\(464\) 0 0
\(465\) −358217. −0.0768271
\(466\) 0 0
\(467\) − 6.49920e6i − 1.37901i −0.724280 0.689506i \(-0.757828\pi\)
0.724280 0.689506i \(-0.242172\pi\)
\(468\) 0 0
\(469\) − 200037.i − 0.0419931i
\(470\) 0 0
\(471\) −1.39598e6 −0.289952
\(472\) 0 0
\(473\) −1.50354e6 −0.309003
\(474\) 0 0
\(475\) 607981.i 0.123639i
\(476\) 0 0
\(477\) − 4.89281e6i − 0.984607i
\(478\) 0 0
\(479\) 7.30319e6 1.45437 0.727183 0.686443i \(-0.240829\pi\)
0.727183 + 0.686443i \(0.240829\pi\)
\(480\) 0 0
\(481\) −1.96342e6 −0.386945
\(482\) 0 0
\(483\) 478406.i 0.0933100i
\(484\) 0 0
\(485\) − 3.36860e6i − 0.650272i
\(486\) 0 0
\(487\) −4.25953e6 −0.813841 −0.406921 0.913464i \(-0.633397\pi\)
−0.406921 + 0.913464i \(0.633397\pi\)
\(488\) 0 0
\(489\) 1.20449e6 0.227789
\(490\) 0 0
\(491\) − 888023.i − 0.166234i −0.996540 0.0831171i \(-0.973512\pi\)
0.996540 0.0831171i \(-0.0264876\pi\)
\(492\) 0 0
\(493\) − 3.06297e6i − 0.567579i
\(494\) 0 0
\(495\) −2.07408e6 −0.380463
\(496\) 0 0
\(497\) −633601. −0.115060
\(498\) 0 0
\(499\) − 597315.i − 0.107387i −0.998557 0.0536935i \(-0.982901\pi\)
0.998557 0.0536935i \(-0.0170994\pi\)
\(500\) 0 0
\(501\) − 50895.7i − 0.00905914i
\(502\) 0 0
\(503\) −9.67283e6 −1.70464 −0.852322 0.523018i \(-0.824806\pi\)
−0.852322 + 0.523018i \(0.824806\pi\)
\(504\) 0 0
\(505\) −986782. −0.172184
\(506\) 0 0
\(507\) 724856.i 0.125237i
\(508\) 0 0
\(509\) 2.66674e6i 0.456232i 0.973634 + 0.228116i \(0.0732566\pi\)
−0.973634 + 0.228116i \(0.926743\pi\)
\(510\) 0 0
\(511\) 9.57230e6 1.62168
\(512\) 0 0
\(513\) 1.43645e6 0.240989
\(514\) 0 0
\(515\) 4.11093e6i 0.683002i
\(516\) 0 0
\(517\) 6.60353e6i 1.08655i
\(518\) 0 0
\(519\) 346656. 0.0564911
\(520\) 0 0
\(521\) 5.96858e6 0.963333 0.481667 0.876355i \(-0.340032\pi\)
0.481667 + 0.876355i \(0.340032\pi\)
\(522\) 0 0
\(523\) 9.24654e6i 1.47817i 0.673611 + 0.739086i \(0.264742\pi\)
−0.673611 + 0.739086i \(0.735258\pi\)
\(524\) 0 0
\(525\) − 230192.i − 0.0364495i
\(526\) 0 0
\(527\) 2.70895e6 0.424888
\(528\) 0 0
\(529\) −4.74912e6 −0.737860
\(530\) 0 0
\(531\) − 1.13367e7i − 1.74482i
\(532\) 0 0
\(533\) − 5.20585e6i − 0.793731i
\(534\) 0 0
\(535\) −5.09651e6 −0.769818
\(536\) 0 0
\(537\) −1.34115e6 −0.200697
\(538\) 0 0
\(539\) 955702.i 0.141694i
\(540\) 0 0
\(541\) 1.19550e7i 1.75612i 0.478547 + 0.878062i \(0.341164\pi\)
−0.478547 + 0.878062i \(0.658836\pi\)
\(542\) 0 0
\(543\) −2.47976e6 −0.360919
\(544\) 0 0
\(545\) −1.63904e6 −0.236374
\(546\) 0 0
\(547\) 1.07515e7i 1.53638i 0.640221 + 0.768190i \(0.278843\pi\)
−0.640221 + 0.768190i \(0.721157\pi\)
\(548\) 0 0
\(549\) − 845083.i − 0.119665i
\(550\) 0 0
\(551\) 5.08442e6 0.713449
\(552\) 0 0
\(553\) −4.66643e6 −0.648891
\(554\) 0 0
\(555\) − 410399.i − 0.0565553i
\(556\) 0 0
\(557\) 3.09710e6i 0.422978i 0.977380 + 0.211489i \(0.0678313\pi\)
−0.977380 + 0.211489i \(0.932169\pi\)
\(558\) 0 0
\(559\) −1.56812e6 −0.212252
\(560\) 0 0
\(561\) −645696. −0.0866205
\(562\) 0 0
\(563\) 1.01607e7i 1.35100i 0.737361 + 0.675499i \(0.236072\pi\)
−0.737361 + 0.675499i \(0.763928\pi\)
\(564\) 0 0
\(565\) − 876424.i − 0.115503i
\(566\) 0 0
\(567\) 6.19498e6 0.809249
\(568\) 0 0
\(569\) −2.86628e6 −0.371141 −0.185570 0.982631i \(-0.559413\pi\)
−0.185570 + 0.982631i \(0.559413\pi\)
\(570\) 0 0
\(571\) 6.30651e6i 0.809466i 0.914435 + 0.404733i \(0.132636\pi\)
−0.914435 + 0.404733i \(0.867364\pi\)
\(572\) 0 0
\(573\) 1.15518e6i 0.146981i
\(574\) 0 0
\(575\) −811832. −0.102399
\(576\) 0 0
\(577\) −8.94056e6 −1.11796 −0.558979 0.829182i \(-0.688807\pi\)
−0.558979 + 0.829182i \(0.688807\pi\)
\(578\) 0 0
\(579\) − 2.43254e6i − 0.301553i
\(580\) 0 0
\(581\) − 2.18461e6i − 0.268493i
\(582\) 0 0
\(583\) −7.45199e6 −0.908031
\(584\) 0 0
\(585\) −2.16316e6 −0.261336
\(586\) 0 0
\(587\) − 7.46374e6i − 0.894049i −0.894522 0.447024i \(-0.852484\pi\)
0.894522 0.447024i \(-0.147516\pi\)
\(588\) 0 0
\(589\) 4.49676e6i 0.534086i
\(590\) 0 0
\(591\) 1.01962e6 0.120080
\(592\) 0 0
\(593\) −7.68119e6 −0.896999 −0.448500 0.893783i \(-0.648041\pi\)
−0.448500 + 0.893783i \(0.648041\pi\)
\(594\) 0 0
\(595\) 1.74078e6i 0.201582i
\(596\) 0 0
\(597\) 1.46835e6i 0.168614i
\(598\) 0 0
\(599\) −4.04401e6 −0.460517 −0.230258 0.973130i \(-0.573957\pi\)
−0.230258 + 0.973130i \(0.573957\pi\)
\(600\) 0 0
\(601\) 733125. 0.0827927 0.0413963 0.999143i \(-0.486819\pi\)
0.0413963 + 0.999143i \(0.486819\pi\)
\(602\) 0 0
\(603\) − 392919.i − 0.0440058i
\(604\) 0 0
\(605\) − 867357.i − 0.0963407i
\(606\) 0 0
\(607\) 1.38418e7 1.52483 0.762413 0.647091i \(-0.224015\pi\)
0.762413 + 0.647091i \(0.224015\pi\)
\(608\) 0 0
\(609\) −1.92505e6 −0.210329
\(610\) 0 0
\(611\) 6.88716e6i 0.746341i
\(612\) 0 0
\(613\) − 4.82596e6i − 0.518719i −0.965781 0.259360i \(-0.916489\pi\)
0.965781 0.259360i \(-0.0835115\pi\)
\(614\) 0 0
\(615\) 1.08814e6 0.116010
\(616\) 0 0
\(617\) −3.36275e6 −0.355616 −0.177808 0.984065i \(-0.556901\pi\)
−0.177808 + 0.984065i \(0.556901\pi\)
\(618\) 0 0
\(619\) 9.71613e6i 1.01922i 0.860406 + 0.509608i \(0.170210\pi\)
−0.860406 + 0.509608i \(0.829790\pi\)
\(620\) 0 0
\(621\) 1.91808e6i 0.199590i
\(622\) 0 0
\(623\) 5.12562e6 0.529086
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) − 1.07183e6i − 0.108882i
\(628\) 0 0
\(629\) 3.10356e6i 0.312776i
\(630\) 0 0
\(631\) 1.51112e7 1.51086 0.755432 0.655227i \(-0.227427\pi\)
0.755432 + 0.655227i \(0.227427\pi\)
\(632\) 0 0
\(633\) −2.18421e6 −0.216663
\(634\) 0 0
\(635\) − 3.93069e6i − 0.386843i
\(636\) 0 0
\(637\) 996751.i 0.0973281i
\(638\) 0 0
\(639\) −1.24454e6 −0.120575
\(640\) 0 0
\(641\) 9.92719e6 0.954292 0.477146 0.878824i \(-0.341671\pi\)
0.477146 + 0.878824i \(0.341671\pi\)
\(642\) 0 0
\(643\) 1.34898e7i 1.28670i 0.765573 + 0.643349i \(0.222456\pi\)
−0.765573 + 0.643349i \(0.777544\pi\)
\(644\) 0 0
\(645\) − 327773.i − 0.0310223i
\(646\) 0 0
\(647\) 6.20615e6 0.582857 0.291428 0.956593i \(-0.405869\pi\)
0.291428 + 0.956593i \(0.405869\pi\)
\(648\) 0 0
\(649\) −1.72664e7 −1.60912
\(650\) 0 0
\(651\) − 1.70255e6i − 0.157452i
\(652\) 0 0
\(653\) − 1.72472e7i − 1.58283i −0.611277 0.791417i \(-0.709344\pi\)
0.611277 0.791417i \(-0.290656\pi\)
\(654\) 0 0
\(655\) −2.14202e6 −0.195084
\(656\) 0 0
\(657\) 1.88022e7 1.69940
\(658\) 0 0
\(659\) − 4.84307e6i − 0.434418i −0.976125 0.217209i \(-0.930305\pi\)
0.976125 0.217209i \(-0.0696952\pi\)
\(660\) 0 0
\(661\) 1.29176e7i 1.14995i 0.818172 + 0.574973i \(0.194987\pi\)
−0.818172 + 0.574973i \(0.805013\pi\)
\(662\) 0 0
\(663\) −673429. −0.0594988
\(664\) 0 0
\(665\) −2.88964e6 −0.253390
\(666\) 0 0
\(667\) 6.78919e6i 0.590885i
\(668\) 0 0
\(669\) 1.51199e6i 0.130612i
\(670\) 0 0
\(671\) −1.28710e6 −0.110359
\(672\) 0 0
\(673\) −6.16137e6 −0.524372 −0.262186 0.965017i \(-0.584443\pi\)
−0.262186 + 0.965017i \(0.584443\pi\)
\(674\) 0 0
\(675\) − 922914.i − 0.0779654i
\(676\) 0 0
\(677\) 1.95790e7i 1.64179i 0.571077 + 0.820896i \(0.306526\pi\)
−0.571077 + 0.820896i \(0.693474\pi\)
\(678\) 0 0
\(679\) 1.60104e7 1.33269
\(680\) 0 0
\(681\) −869862. −0.0718758
\(682\) 0 0
\(683\) 1.33176e7i 1.09238i 0.837661 + 0.546190i \(0.183922\pi\)
−0.837661 + 0.546190i \(0.816078\pi\)
\(684\) 0 0
\(685\) 2.48352e6i 0.202228i
\(686\) 0 0
\(687\) −1.04293e6 −0.0843069
\(688\) 0 0
\(689\) −7.77206e6 −0.623718
\(690\) 0 0
\(691\) 8.17155e6i 0.651043i 0.945535 + 0.325521i \(0.105540\pi\)
−0.945535 + 0.325521i \(0.894460\pi\)
\(692\) 0 0
\(693\) − 9.85776e6i − 0.779731i
\(694\) 0 0
\(695\) 909250. 0.0714038
\(696\) 0 0
\(697\) −8.22886e6 −0.641590
\(698\) 0 0
\(699\) − 770089.i − 0.0596140i
\(700\) 0 0
\(701\) 1.90780e7i 1.46635i 0.680041 + 0.733174i \(0.261962\pi\)
−0.680041 + 0.733174i \(0.738038\pi\)
\(702\) 0 0
\(703\) −5.15180e6 −0.393161
\(704\) 0 0
\(705\) −1.43957e6 −0.109084
\(706\) 0 0
\(707\) − 4.69001e6i − 0.352879i
\(708\) 0 0
\(709\) 1.13159e7i 0.845422i 0.906265 + 0.422711i \(0.138921\pi\)
−0.906265 + 0.422711i \(0.861079\pi\)
\(710\) 0 0
\(711\) −9.16595e6 −0.679992
\(712\) 0 0
\(713\) −6.00449e6 −0.442336
\(714\) 0 0
\(715\) 3.29460e6i 0.241011i
\(716\) 0 0
\(717\) 2.12540e6i 0.154398i
\(718\) 0 0
\(719\) −1.63847e7 −1.18200 −0.590998 0.806673i \(-0.701266\pi\)
−0.590998 + 0.806673i \(0.701266\pi\)
\(720\) 0 0
\(721\) −1.95386e7 −1.39976
\(722\) 0 0
\(723\) − 2.36402e6i − 0.168192i
\(724\) 0 0
\(725\) − 3.26672e6i − 0.230816i
\(726\) 0 0
\(727\) 8.06838e6 0.566175 0.283087 0.959094i \(-0.408641\pi\)
0.283087 + 0.959094i \(0.408641\pi\)
\(728\) 0 0
\(729\) 1.10561e7 0.770520
\(730\) 0 0
\(731\) 2.47873e6i 0.171568i
\(732\) 0 0
\(733\) − 563282.i − 0.0387227i −0.999813 0.0193614i \(-0.993837\pi\)
0.999813 0.0193614i \(-0.00616330\pi\)
\(734\) 0 0
\(735\) −208344. −0.0142253
\(736\) 0 0
\(737\) −598434. −0.0405833
\(738\) 0 0
\(739\) 6.99287e6i 0.471026i 0.971871 + 0.235513i \(0.0756770\pi\)
−0.971871 + 0.235513i \(0.924323\pi\)
\(740\) 0 0
\(741\) − 1.11787e6i − 0.0747902i
\(742\) 0 0
\(743\) −690902. −0.0459139 −0.0229570 0.999736i \(-0.507308\pi\)
−0.0229570 + 0.999736i \(0.507308\pi\)
\(744\) 0 0
\(745\) −6.04906e6 −0.399298
\(746\) 0 0
\(747\) − 4.29107e6i − 0.281361i
\(748\) 0 0
\(749\) − 2.42229e7i − 1.57769i
\(750\) 0 0
\(751\) 1.61091e6 0.104225 0.0521123 0.998641i \(-0.483405\pi\)
0.0521123 + 0.998641i \(0.483405\pi\)
\(752\) 0 0
\(753\) 286604. 0.0184202
\(754\) 0 0
\(755\) 6.02698e6i 0.384797i
\(756\) 0 0
\(757\) − 9.75412e6i − 0.618655i −0.950956 0.309327i \(-0.899896\pi\)
0.950956 0.309327i \(-0.100104\pi\)
\(758\) 0 0
\(759\) 1.43121e6 0.0901774
\(760\) 0 0
\(761\) 1.42949e7 0.894786 0.447393 0.894337i \(-0.352352\pi\)
0.447393 + 0.894337i \(0.352352\pi\)
\(762\) 0 0
\(763\) − 7.79011e6i − 0.484431i
\(764\) 0 0
\(765\) 3.41930e6i 0.211244i
\(766\) 0 0
\(767\) −1.80080e7 −1.10529
\(768\) 0 0
\(769\) 1.02217e7 0.623313 0.311656 0.950195i \(-0.399116\pi\)
0.311656 + 0.950195i \(0.399116\pi\)
\(770\) 0 0
\(771\) − 2.21772e6i − 0.134360i
\(772\) 0 0
\(773\) 6.53435e6i 0.393327i 0.980471 + 0.196663i \(0.0630106\pi\)
−0.980471 + 0.196663i \(0.936989\pi\)
\(774\) 0 0
\(775\) 2.88915e6 0.172789
\(776\) 0 0
\(777\) 1.95056e6 0.115906
\(778\) 0 0
\(779\) − 1.36596e7i − 0.806481i
\(780\) 0 0
\(781\) 1.89549e6i 0.111197i
\(782\) 0 0
\(783\) −7.71815e6 −0.449892
\(784\) 0 0
\(785\) 1.12590e7 0.652120
\(786\) 0 0
\(787\) 1.32185e7i 0.760756i 0.924831 + 0.380378i \(0.124206\pi\)
−0.924831 + 0.380378i \(0.875794\pi\)
\(788\) 0 0
\(789\) − 6.43470e6i − 0.367990i
\(790\) 0 0
\(791\) 4.16550e6 0.236715
\(792\) 0 0
\(793\) −1.34238e6 −0.0758043
\(794\) 0 0
\(795\) − 1.62454e6i − 0.0911616i
\(796\) 0 0
\(797\) − 1.45286e7i − 0.810173i −0.914278 0.405087i \(-0.867241\pi\)
0.914278 0.405087i \(-0.132759\pi\)
\(798\) 0 0
\(799\) 1.08865e7 0.603284
\(800\) 0 0
\(801\) 1.00679e7 0.554444
\(802\) 0 0
\(803\) − 2.86367e7i − 1.56723i
\(804\) 0 0
\(805\) − 3.85851e6i − 0.209860i
\(806\) 0 0
\(807\) −5.01156e6 −0.270888
\(808\) 0 0
\(809\) 1.59141e7 0.854891 0.427445 0.904041i \(-0.359414\pi\)
0.427445 + 0.904041i \(0.359414\pi\)
\(810\) 0 0
\(811\) − 1.87197e7i − 0.999418i −0.866193 0.499709i \(-0.833440\pi\)
0.866193 0.499709i \(-0.166560\pi\)
\(812\) 0 0
\(813\) 6.86311e6i 0.364162i
\(814\) 0 0
\(815\) −9.71465e6 −0.512310
\(816\) 0 0
\(817\) −4.11459e6 −0.215661
\(818\) 0 0
\(819\) − 1.02812e7i − 0.535590i
\(820\) 0 0
\(821\) − 3.44309e7i − 1.78275i −0.453267 0.891375i \(-0.649742\pi\)
0.453267 0.891375i \(-0.350258\pi\)
\(822\) 0 0
\(823\) 1.98856e7 1.02338 0.511692 0.859169i \(-0.329019\pi\)
0.511692 + 0.859169i \(0.329019\pi\)
\(824\) 0 0
\(825\) −688646. −0.0352258
\(826\) 0 0
\(827\) − 2.39990e7i − 1.22019i −0.792327 0.610097i \(-0.791131\pi\)
0.792327 0.610097i \(-0.208869\pi\)
\(828\) 0 0
\(829\) 1.03074e7i 0.520908i 0.965486 + 0.260454i \(0.0838722\pi\)
−0.965486 + 0.260454i \(0.916128\pi\)
\(830\) 0 0
\(831\) −1.76359e6 −0.0885919
\(832\) 0 0
\(833\) 1.57556e6 0.0786724
\(834\) 0 0
\(835\) 410492.i 0.0203746i
\(836\) 0 0
\(837\) − 6.82608e6i − 0.336789i
\(838\) 0 0
\(839\) 808698. 0.0396626 0.0198313 0.999803i \(-0.493687\pi\)
0.0198313 + 0.999803i \(0.493687\pi\)
\(840\) 0 0
\(841\) −6.80776e6 −0.331905
\(842\) 0 0
\(843\) 2.43797e6i 0.118157i
\(844\) 0 0
\(845\) − 5.84622e6i − 0.281665i
\(846\) 0 0
\(847\) 4.12241e6 0.197443
\(848\) 0 0
\(849\) −7.24405e6 −0.344915
\(850\) 0 0
\(851\) − 6.87916e6i − 0.325620i
\(852\) 0 0
\(853\) 1.48576e7i 0.699161i 0.936906 + 0.349580i \(0.113676\pi\)
−0.936906 + 0.349580i \(0.886324\pi\)
\(854\) 0 0
\(855\) −5.67592e6 −0.265534
\(856\) 0 0
\(857\) 873459. 0.0406247 0.0203124 0.999794i \(-0.493534\pi\)
0.0203124 + 0.999794i \(0.493534\pi\)
\(858\) 0 0
\(859\) 3.79173e7i 1.75329i 0.481135 + 0.876647i \(0.340225\pi\)
−0.481135 + 0.876647i \(0.659775\pi\)
\(860\) 0 0
\(861\) 5.17175e6i 0.237755i
\(862\) 0 0
\(863\) −3.76129e7 −1.71914 −0.859568 0.511022i \(-0.829267\pi\)
−0.859568 + 0.511022i \(0.829267\pi\)
\(864\) 0 0
\(865\) −2.79590e6 −0.127052
\(866\) 0 0
\(867\) − 3.33662e6i − 0.150750i
\(868\) 0 0
\(869\) 1.39602e7i 0.627107i
\(870\) 0 0
\(871\) −624138. −0.0278763
\(872\) 0 0
\(873\) 3.14482e7 1.39656
\(874\) 0 0
\(875\) 1.85658e6i 0.0819772i
\(876\) 0 0
\(877\) − 1.68861e7i − 0.741360i −0.928761 0.370680i \(-0.879125\pi\)
0.928761 0.370680i \(-0.120875\pi\)
\(878\) 0 0
\(879\) 3.30872e6 0.144440
\(880\) 0 0
\(881\) 1.09397e7 0.474858 0.237429 0.971405i \(-0.423695\pi\)
0.237429 + 0.971405i \(0.423695\pi\)
\(882\) 0 0
\(883\) − 1.07162e7i − 0.462530i −0.972891 0.231265i \(-0.925714\pi\)
0.972891 0.231265i \(-0.0742864\pi\)
\(884\) 0 0
\(885\) − 3.76408e6i − 0.161547i
\(886\) 0 0
\(887\) −3.53953e6 −0.151056 −0.0755278 0.997144i \(-0.524064\pi\)
−0.0755278 + 0.997144i \(0.524064\pi\)
\(888\) 0 0
\(889\) 1.86819e7 0.792807
\(890\) 0 0
\(891\) − 1.85330e7i − 0.782081i
\(892\) 0 0
\(893\) 1.80712e7i 0.758330i
\(894\) 0 0
\(895\) 1.08168e7 0.451380
\(896\) 0 0
\(897\) 1.49268e6 0.0619420
\(898\) 0 0
\(899\) − 2.41614e7i − 0.997062i
\(900\) 0 0
\(901\) 1.22853e7i 0.504165i
\(902\) 0 0
\(903\) 1.55785e6 0.0635781
\(904\) 0 0
\(905\) 2.00001e7 0.811730
\(906\) 0 0
\(907\) − 1.70933e7i − 0.689935i −0.938615 0.344967i \(-0.887890\pi\)
0.938615 0.344967i \(-0.112110\pi\)
\(908\) 0 0
\(909\) − 9.21228e6i − 0.369792i
\(910\) 0 0
\(911\) −2.54600e7 −1.01639 −0.508196 0.861241i \(-0.669688\pi\)
−0.508196 + 0.861241i \(0.669688\pi\)
\(912\) 0 0
\(913\) −6.53550e6 −0.259479
\(914\) 0 0
\(915\) − 280589.i − 0.0110794i
\(916\) 0 0
\(917\) − 1.01807e7i − 0.399810i
\(918\) 0 0
\(919\) 3.02310e6 0.118077 0.0590383 0.998256i \(-0.481197\pi\)
0.0590383 + 0.998256i \(0.481197\pi\)
\(920\) 0 0
\(921\) −1.07062e6 −0.0415896
\(922\) 0 0
\(923\) 1.97691e6i 0.0763804i
\(924\) 0 0
\(925\) 3.31001e6i 0.127196i
\(926\) 0 0
\(927\) −3.83783e7 −1.46685
\(928\) 0 0
\(929\) 3.67045e7 1.39534 0.697669 0.716420i \(-0.254220\pi\)
0.697669 + 0.716420i \(0.254220\pi\)
\(930\) 0 0
\(931\) 2.61537e6i 0.0988915i
\(932\) 0 0
\(933\) 4.30825e6i 0.162031i
\(934\) 0 0
\(935\) 5.20776e6 0.194815
\(936\) 0 0
\(937\) −2.27722e6 −0.0847338 −0.0423669 0.999102i \(-0.513490\pi\)
−0.0423669 + 0.999102i \(0.513490\pi\)
\(938\) 0 0
\(939\) − 4.21238e6i − 0.155906i
\(940\) 0 0
\(941\) − 4.68229e7i − 1.72379i −0.507087 0.861895i \(-0.669278\pi\)
0.507087 0.861895i \(-0.330722\pi\)
\(942\) 0 0
\(943\) 1.82395e7 0.667936
\(944\) 0 0
\(945\) 4.38646e6 0.159785
\(946\) 0 0
\(947\) 4.27905e7i 1.55050i 0.631654 + 0.775250i \(0.282376\pi\)
−0.631654 + 0.775250i \(0.717624\pi\)
\(948\) 0 0
\(949\) − 2.98667e7i − 1.07652i
\(950\) 0 0
\(951\) −6.35067e6 −0.227703
\(952\) 0 0
\(953\) 3.95519e7 1.41070 0.705351 0.708858i \(-0.250789\pi\)
0.705351 + 0.708858i \(0.250789\pi\)
\(954\) 0 0
\(955\) − 9.31691e6i − 0.330570i
\(956\) 0 0
\(957\) 5.75901e6i 0.203267i
\(958\) 0 0
\(959\) −1.18038e7 −0.414452
\(960\) 0 0
\(961\) −7.26038e6 −0.253601
\(962\) 0 0
\(963\) − 4.75793e7i − 1.65330i
\(964\) 0 0
\(965\) 1.96193e7i 0.678211i
\(966\) 0 0
\(967\) 2.15339e7 0.740553 0.370277 0.928922i \(-0.379263\pi\)
0.370277 + 0.928922i \(0.379263\pi\)
\(968\) 0 0
\(969\) −1.76701e6 −0.0604546
\(970\) 0 0
\(971\) 559354.i 0.0190388i 0.999955 + 0.00951939i \(0.00303016\pi\)
−0.999955 + 0.00951939i \(0.996970\pi\)
\(972\) 0 0
\(973\) 4.32152e6i 0.146337i
\(974\) 0 0
\(975\) −718225. −0.0241963
\(976\) 0 0
\(977\) 2.11889e7 0.710187 0.355094 0.934831i \(-0.384449\pi\)
0.355094 + 0.934831i \(0.384449\pi\)
\(978\) 0 0
\(979\) − 1.53339e7i − 0.511324i
\(980\) 0 0
\(981\) − 1.53016e7i − 0.507649i
\(982\) 0 0
\(983\) 4.00893e7 1.32326 0.661630 0.749830i \(-0.269865\pi\)
0.661630 + 0.749830i \(0.269865\pi\)
\(984\) 0 0
\(985\) −8.22362e6 −0.270067
\(986\) 0 0
\(987\) − 6.84206e6i − 0.223560i
\(988\) 0 0
\(989\) − 5.49418e6i − 0.178613i
\(990\) 0 0
\(991\) −5.50729e7 −1.78137 −0.890684 0.454623i \(-0.849774\pi\)
−0.890684 + 0.454623i \(0.849774\pi\)
\(992\) 0 0
\(993\) 8.54288e6 0.274936
\(994\) 0 0
\(995\) − 1.18427e7i − 0.379223i
\(996\) 0 0
\(997\) 1.44969e7i 0.461889i 0.972967 + 0.230944i \(0.0741816\pi\)
−0.972967 + 0.230944i \(0.925818\pi\)
\(998\) 0 0
\(999\) 7.82043e6 0.247923
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.161.7 yes 12
4.3 odd 2 320.6.d.d.161.6 yes 12
8.3 odd 2 320.6.d.d.161.7 yes 12
8.5 even 2 inner 320.6.d.c.161.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.6.d.c.161.6 12 8.5 even 2 inner
320.6.d.c.161.7 yes 12 1.1 even 1 trivial
320.6.d.d.161.6 yes 12 4.3 odd 2
320.6.d.d.161.7 yes 12 8.3 odd 2