Properties

Label 320.6.d.c.161.10
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 161.10
Root \(10.2435 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 320.161
Dual form 320.6.d.c.161.3

$q$-expansion

\(f(q)\) \(=\) \(q+16.5588i q^{3} -25.0000i q^{5} -55.4690 q^{7} -31.1927 q^{9} +O(q^{10})\) \(q+16.5588i q^{3} -25.0000i q^{5} -55.4690 q^{7} -31.1927 q^{9} -153.628i q^{11} -57.3312i q^{13} +413.969 q^{15} +1783.25 q^{17} +1595.98i q^{19} -918.498i q^{21} -4115.40 q^{23} -625.000 q^{25} +3507.27i q^{27} +5003.52i q^{29} -1823.88 q^{31} +2543.90 q^{33} +1386.72i q^{35} -1800.07i q^{37} +949.334 q^{39} +5455.32 q^{41} +8258.91i q^{43} +779.817i q^{45} -8409.69 q^{47} -13730.2 q^{49} +29528.5i q^{51} +6743.29i q^{53} -3840.71 q^{55} -26427.5 q^{57} +17.5455i q^{59} -18829.3i q^{61} +1730.23 q^{63} -1433.28 q^{65} -5972.99i q^{67} -68145.9i q^{69} +16857.0 q^{71} -56634.5 q^{73} -10349.2i q^{75} +8521.61i q^{77} -37295.3 q^{79} -65655.8 q^{81} +57832.9i q^{83} -44581.3i q^{85} -82852.2 q^{87} -93163.3 q^{89} +3180.10i q^{91} -30201.2i q^{93} +39899.6 q^{95} -91299.9 q^{97} +4792.08i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 268q^{7} - 428q^{9} + O(q^{10}) \) \( 12q - 268q^{7} - 428q^{9} + 900q^{15} + 2400q^{17} - 8108q^{23} - 7500q^{25} - 7976q^{31} - 20776q^{33} - 40984q^{39} - 56408q^{41} - 21172q^{47} - 5540q^{49} - 18400q^{55} + 39992q^{57} - 179516q^{63} + 44000q^{65} - 367704q^{71} + 58736q^{73} - 26192q^{79} + 411692q^{81} - 183200q^{87} + 87672q^{89} - 121000q^{95} - 172336q^{97} + O(q^{100}) \)

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
\(3\) 16.5588i 1.06225i 0.847295 + 0.531123i \(0.178230\pi\)
−0.847295 + 0.531123i \(0.821770\pi\)
\(4\) 0 0
\(5\) − 25.0000i − 0.447214i
\(6\) 0 0
\(7\) −55.4690 −0.427863 −0.213932 0.976849i \(-0.568627\pi\)
−0.213932 + 0.976849i \(0.568627\pi\)
\(8\) 0 0
\(9\) −31.1927 −0.128365
\(10\) 0 0
\(11\) − 153.628i − 0.382816i −0.981511 0.191408i \(-0.938695\pi\)
0.981511 0.191408i \(-0.0613053\pi\)
\(12\) 0 0
\(13\) − 57.3312i − 0.0940877i −0.998893 0.0470438i \(-0.985020\pi\)
0.998893 0.0470438i \(-0.0149800\pi\)
\(14\) 0 0
\(15\) 413.969 0.475051
\(16\) 0 0
\(17\) 1783.25 1.49655 0.748274 0.663389i \(-0.230883\pi\)
0.748274 + 0.663389i \(0.230883\pi\)
\(18\) 0 0
\(19\) 1595.98i 1.01425i 0.861873 + 0.507124i \(0.169291\pi\)
−0.861873 + 0.507124i \(0.830709\pi\)
\(20\) 0 0
\(21\) − 918.498i − 0.454496i
\(22\) 0 0
\(23\) −4115.40 −1.62215 −0.811077 0.584939i \(-0.801118\pi\)
−0.811077 + 0.584939i \(0.801118\pi\)
\(24\) 0 0
\(25\) −625.000 −0.200000
\(26\) 0 0
\(27\) 3507.27i 0.925890i
\(28\) 0 0
\(29\) 5003.52i 1.10479i 0.833581 + 0.552397i \(0.186287\pi\)
−0.833581 + 0.552397i \(0.813713\pi\)
\(30\) 0 0
\(31\) −1823.88 −0.340873 −0.170436 0.985369i \(-0.554518\pi\)
−0.170436 + 0.985369i \(0.554518\pi\)
\(32\) 0 0
\(33\) 2543.90 0.406644
\(34\) 0 0
\(35\) 1386.72i 0.191346i
\(36\) 0 0
\(37\) − 1800.07i − 0.216165i −0.994142 0.108083i \(-0.965529\pi\)
0.994142 0.108083i \(-0.0344711\pi\)
\(38\) 0 0
\(39\) 949.334 0.0999442
\(40\) 0 0
\(41\) 5455.32 0.506828 0.253414 0.967358i \(-0.418447\pi\)
0.253414 + 0.967358i \(0.418447\pi\)
\(42\) 0 0
\(43\) 8258.91i 0.681164i 0.940215 + 0.340582i \(0.110624\pi\)
−0.940215 + 0.340582i \(0.889376\pi\)
\(44\) 0 0
\(45\) 779.817i 0.0574066i
\(46\) 0 0
\(47\) −8409.69 −0.555310 −0.277655 0.960681i \(-0.589557\pi\)
−0.277655 + 0.960681i \(0.589557\pi\)
\(48\) 0 0
\(49\) −13730.2 −0.816933
\(50\) 0 0
\(51\) 29528.5i 1.58970i
\(52\) 0 0
\(53\) 6743.29i 0.329748i 0.986315 + 0.164874i \(0.0527217\pi\)
−0.986315 + 0.164874i \(0.947278\pi\)
\(54\) 0 0
\(55\) −3840.71 −0.171200
\(56\) 0 0
\(57\) −26427.5 −1.07738
\(58\) 0 0
\(59\) 17.5455i 0 0.000656201i 1.00000 0.000328100i \(0.000104438\pi\)
−1.00000 0.000328100i \(0.999896\pi\)
\(60\) 0 0
\(61\) − 18829.3i − 0.647901i −0.946074 0.323951i \(-0.894989\pi\)
0.946074 0.323951i \(-0.105011\pi\)
\(62\) 0 0
\(63\) 1730.23 0.0549227
\(64\) 0 0
\(65\) −1433.28 −0.0420773
\(66\) 0 0
\(67\) − 5972.99i − 0.162557i −0.996691 0.0812783i \(-0.974100\pi\)
0.996691 0.0812783i \(-0.0259003\pi\)
\(68\) 0 0
\(69\) − 68145.9i − 1.72313i
\(70\) 0 0
\(71\) 16857.0 0.396857 0.198428 0.980115i \(-0.436416\pi\)
0.198428 + 0.980115i \(0.436416\pi\)
\(72\) 0 0
\(73\) −56634.5 −1.24387 −0.621933 0.783070i \(-0.713653\pi\)
−0.621933 + 0.783070i \(0.713653\pi\)
\(74\) 0 0
\(75\) − 10349.2i − 0.212449i
\(76\) 0 0
\(77\) 8521.61i 0.163793i
\(78\) 0 0
\(79\) −37295.3 −0.672336 −0.336168 0.941802i \(-0.609131\pi\)
−0.336168 + 0.941802i \(0.609131\pi\)
\(80\) 0 0
\(81\) −65655.8 −1.11189
\(82\) 0 0
\(83\) 57832.9i 0.921466i 0.887539 + 0.460733i \(0.152413\pi\)
−0.887539 + 0.460733i \(0.847587\pi\)
\(84\) 0 0
\(85\) − 44581.3i − 0.669277i
\(86\) 0 0
\(87\) −82852.2 −1.17356
\(88\) 0 0
\(89\) −93163.3 −1.24672 −0.623361 0.781934i \(-0.714233\pi\)
−0.623361 + 0.781934i \(0.714233\pi\)
\(90\) 0 0
\(91\) 3180.10i 0.0402567i
\(92\) 0 0
\(93\) − 30201.2i − 0.362090i
\(94\) 0 0
\(95\) 39899.6 0.453585
\(96\) 0 0
\(97\) −91299.9 −0.985238 −0.492619 0.870245i \(-0.663960\pi\)
−0.492619 + 0.870245i \(0.663960\pi\)
\(98\) 0 0
\(99\) 4792.08i 0.0491402i
\(100\) 0 0
\(101\) − 167055.i − 1.62951i −0.579807 0.814754i \(-0.696872\pi\)
0.579807 0.814754i \(-0.303128\pi\)
\(102\) 0 0
\(103\) −66663.7 −0.619151 −0.309575 0.950875i \(-0.600187\pi\)
−0.309575 + 0.950875i \(0.600187\pi\)
\(104\) 0 0
\(105\) −22962.4 −0.203257
\(106\) 0 0
\(107\) 102568.i 0.866072i 0.901377 + 0.433036i \(0.142558\pi\)
−0.901377 + 0.433036i \(0.857442\pi\)
\(108\) 0 0
\(109\) 151278.i 1.21958i 0.792563 + 0.609790i \(0.208746\pi\)
−0.792563 + 0.609790i \(0.791254\pi\)
\(110\) 0 0
\(111\) 29807.0 0.229621
\(112\) 0 0
\(113\) −27795.5 −0.204776 −0.102388 0.994745i \(-0.532648\pi\)
−0.102388 + 0.994745i \(0.532648\pi\)
\(114\) 0 0
\(115\) 102885.i 0.725449i
\(116\) 0 0
\(117\) 1788.31i 0.0120776i
\(118\) 0 0
\(119\) −98915.3 −0.640318
\(120\) 0 0
\(121\) 137449. 0.853452
\(122\) 0 0
\(123\) 90333.4i 0.538376i
\(124\) 0 0
\(125\) 15625.0i 0.0894427i
\(126\) 0 0
\(127\) −134073. −0.737620 −0.368810 0.929505i \(-0.620235\pi\)
−0.368810 + 0.929505i \(0.620235\pi\)
\(128\) 0 0
\(129\) −136757. −0.723563
\(130\) 0 0
\(131\) 372871.i 1.89837i 0.314723 + 0.949184i \(0.398088\pi\)
−0.314723 + 0.949184i \(0.601912\pi\)
\(132\) 0 0
\(133\) − 88527.5i − 0.433959i
\(134\) 0 0
\(135\) 87681.7 0.414071
\(136\) 0 0
\(137\) −146243. −0.665690 −0.332845 0.942981i \(-0.608009\pi\)
−0.332845 + 0.942981i \(0.608009\pi\)
\(138\) 0 0
\(139\) 226750.i 0.995428i 0.867341 + 0.497714i \(0.165827\pi\)
−0.867341 + 0.497714i \(0.834173\pi\)
\(140\) 0 0
\(141\) − 139254.i − 0.589875i
\(142\) 0 0
\(143\) −8807.71 −0.0360183
\(144\) 0 0
\(145\) 125088. 0.494079
\(146\) 0 0
\(147\) − 227355.i − 0.867783i
\(148\) 0 0
\(149\) 4820.91i 0.0177895i 0.999960 + 0.00889474i \(0.00283132\pi\)
−0.999960 + 0.00889474i \(0.997169\pi\)
\(150\) 0 0
\(151\) 7260.42 0.0259131 0.0129566 0.999916i \(-0.495876\pi\)
0.0129566 + 0.999916i \(0.495876\pi\)
\(152\) 0 0
\(153\) −55624.5 −0.192104
\(154\) 0 0
\(155\) 45597.0i 0.152443i
\(156\) 0 0
\(157\) 420038.i 1.36000i 0.733211 + 0.680001i \(0.238021\pi\)
−0.733211 + 0.680001i \(0.761979\pi\)
\(158\) 0 0
\(159\) −111661. −0.350273
\(160\) 0 0
\(161\) 228277. 0.694060
\(162\) 0 0
\(163\) − 34691.5i − 0.102271i −0.998692 0.0511356i \(-0.983716\pi\)
0.998692 0.0511356i \(-0.0162841\pi\)
\(164\) 0 0
\(165\) − 63597.4i − 0.181857i
\(166\) 0 0
\(167\) −519139. −1.44043 −0.720216 0.693750i \(-0.755957\pi\)
−0.720216 + 0.693750i \(0.755957\pi\)
\(168\) 0 0
\(169\) 368006. 0.991148
\(170\) 0 0
\(171\) − 49783.0i − 0.130194i
\(172\) 0 0
\(173\) 579939.i 1.47322i 0.676319 + 0.736609i \(0.263574\pi\)
−0.676319 + 0.736609i \(0.736426\pi\)
\(174\) 0 0
\(175\) 34668.1 0.0855727
\(176\) 0 0
\(177\) −290.532 −0.000697046 0
\(178\) 0 0
\(179\) − 280387.i − 0.654072i −0.945012 0.327036i \(-0.893950\pi\)
0.945012 0.327036i \(-0.106050\pi\)
\(180\) 0 0
\(181\) − 877810.i − 1.99161i −0.0915109 0.995804i \(-0.529170\pi\)
0.0915109 0.995804i \(-0.470830\pi\)
\(182\) 0 0
\(183\) 311789. 0.688230
\(184\) 0 0
\(185\) −45001.9 −0.0966721
\(186\) 0 0
\(187\) − 273959.i − 0.572903i
\(188\) 0 0
\(189\) − 194545.i − 0.396154i
\(190\) 0 0
\(191\) 998587. 1.98063 0.990313 0.138852i \(-0.0443413\pi\)
0.990313 + 0.138852i \(0.0443413\pi\)
\(192\) 0 0
\(193\) −483429. −0.934199 −0.467100 0.884205i \(-0.654701\pi\)
−0.467100 + 0.884205i \(0.654701\pi\)
\(194\) 0 0
\(195\) − 23733.3i − 0.0446964i
\(196\) 0 0
\(197\) 219120.i 0.402269i 0.979564 + 0.201135i \(0.0644629\pi\)
−0.979564 + 0.201135i \(0.935537\pi\)
\(198\) 0 0
\(199\) −492419. −0.881460 −0.440730 0.897640i \(-0.645280\pi\)
−0.440730 + 0.897640i \(0.645280\pi\)
\(200\) 0 0
\(201\) 98905.3 0.172675
\(202\) 0 0
\(203\) − 277540.i − 0.472700i
\(204\) 0 0
\(205\) − 136383.i − 0.226660i
\(206\) 0 0
\(207\) 128370. 0.208228
\(208\) 0 0
\(209\) 245188. 0.388270
\(210\) 0 0
\(211\) − 940463.i − 1.45424i −0.686511 0.727119i \(-0.740859\pi\)
0.686511 0.727119i \(-0.259141\pi\)
\(212\) 0 0
\(213\) 279131.i 0.421559i
\(214\) 0 0
\(215\) 206473. 0.304626
\(216\) 0 0
\(217\) 101169. 0.145847
\(218\) 0 0
\(219\) − 937797.i − 1.32129i
\(220\) 0 0
\(221\) − 102236.i − 0.140807i
\(222\) 0 0
\(223\) 1.11376e6 1.49979 0.749895 0.661557i \(-0.230104\pi\)
0.749895 + 0.661557i \(0.230104\pi\)
\(224\) 0 0
\(225\) 19495.4 0.0256730
\(226\) 0 0
\(227\) − 605351.i − 0.779728i −0.920873 0.389864i \(-0.872522\pi\)
0.920873 0.389864i \(-0.127478\pi\)
\(228\) 0 0
\(229\) 1.24526e6i 1.56917i 0.620020 + 0.784586i \(0.287125\pi\)
−0.620020 + 0.784586i \(0.712875\pi\)
\(230\) 0 0
\(231\) −141107. −0.173988
\(232\) 0 0
\(233\) 457782. 0.552420 0.276210 0.961097i \(-0.410921\pi\)
0.276210 + 0.961097i \(0.410921\pi\)
\(234\) 0 0
\(235\) 210242.i 0.248342i
\(236\) 0 0
\(237\) − 617565.i − 0.714186i
\(238\) 0 0
\(239\) 984849. 1.11526 0.557629 0.830091i \(-0.311711\pi\)
0.557629 + 0.830091i \(0.311711\pi\)
\(240\) 0 0
\(241\) 1.38553e6 1.53665 0.768324 0.640061i \(-0.221091\pi\)
0.768324 + 0.640061i \(0.221091\pi\)
\(242\) 0 0
\(243\) − 234914.i − 0.255207i
\(244\) 0 0
\(245\) 343255.i 0.365344i
\(246\) 0 0
\(247\) 91499.6 0.0954282
\(248\) 0 0
\(249\) −957641. −0.978823
\(250\) 0 0
\(251\) 534745.i 0.535750i 0.963454 + 0.267875i \(0.0863214\pi\)
−0.963454 + 0.267875i \(0.913679\pi\)
\(252\) 0 0
\(253\) 632242.i 0.620986i
\(254\) 0 0
\(255\) 738212. 0.710936
\(256\) 0 0
\(257\) 855941. 0.808372 0.404186 0.914677i \(-0.367555\pi\)
0.404186 + 0.914677i \(0.367555\pi\)
\(258\) 0 0
\(259\) 99848.3i 0.0924892i
\(260\) 0 0
\(261\) − 156073.i − 0.141817i
\(262\) 0 0
\(263\) −666515. −0.594183 −0.297092 0.954849i \(-0.596017\pi\)
−0.297092 + 0.954849i \(0.596017\pi\)
\(264\) 0 0
\(265\) 168582. 0.147468
\(266\) 0 0
\(267\) − 1.54267e6i − 1.32432i
\(268\) 0 0
\(269\) − 189848.i − 0.159965i −0.996796 0.0799826i \(-0.974514\pi\)
0.996796 0.0799826i \(-0.0254865\pi\)
\(270\) 0 0
\(271\) −1.45698e6 −1.20512 −0.602559 0.798075i \(-0.705852\pi\)
−0.602559 + 0.798075i \(0.705852\pi\)
\(272\) 0 0
\(273\) −52658.6 −0.0427624
\(274\) 0 0
\(275\) 96017.8i 0.0765632i
\(276\) 0 0
\(277\) 1.08051e6i 0.846115i 0.906103 + 0.423057i \(0.139043\pi\)
−0.906103 + 0.423057i \(0.860957\pi\)
\(278\) 0 0
\(279\) 56891.7 0.0437561
\(280\) 0 0
\(281\) 1.48051e6 1.11852 0.559261 0.828991i \(-0.311085\pi\)
0.559261 + 0.828991i \(0.311085\pi\)
\(282\) 0 0
\(283\) 479977.i 0.356250i 0.984008 + 0.178125i \(0.0570031\pi\)
−0.984008 + 0.178125i \(0.942997\pi\)
\(284\) 0 0
\(285\) 660687.i 0.481819i
\(286\) 0 0
\(287\) −302601. −0.216853
\(288\) 0 0
\(289\) 1.76014e6 1.23966
\(290\) 0 0
\(291\) − 1.51181e6i − 1.04656i
\(292\) 0 0
\(293\) 2.07194e6i 1.40996i 0.709226 + 0.704981i \(0.249044\pi\)
−0.709226 + 0.704981i \(0.750956\pi\)
\(294\) 0 0
\(295\) 438.639 0.000293462 0
\(296\) 0 0
\(297\) 538816. 0.354445
\(298\) 0 0
\(299\) 235941.i 0.152625i
\(300\) 0 0
\(301\) − 458113.i − 0.291445i
\(302\) 0 0
\(303\) 2.76623e6 1.73094
\(304\) 0 0
\(305\) −470732. −0.289750
\(306\) 0 0
\(307\) 2.04780e6i 1.24006i 0.784579 + 0.620029i \(0.212879\pi\)
−0.784579 + 0.620029i \(0.787121\pi\)
\(308\) 0 0
\(309\) − 1.10387e6i − 0.657690i
\(310\) 0 0
\(311\) −2.77842e6 −1.62891 −0.814455 0.580227i \(-0.802964\pi\)
−0.814455 + 0.580227i \(0.802964\pi\)
\(312\) 0 0
\(313\) 1.78088e6 1.02748 0.513741 0.857945i \(-0.328259\pi\)
0.513741 + 0.857945i \(0.328259\pi\)
\(314\) 0 0
\(315\) − 43255.7i − 0.0245622i
\(316\) 0 0
\(317\) − 1.67386e6i − 0.935560i −0.883845 0.467780i \(-0.845054\pi\)
0.883845 0.467780i \(-0.154946\pi\)
\(318\) 0 0
\(319\) 768684. 0.422932
\(320\) 0 0
\(321\) −1.69841e6 −0.919981
\(322\) 0 0
\(323\) 2.84604e6i 1.51787i
\(324\) 0 0
\(325\) 35832.0i 0.0188175i
\(326\) 0 0
\(327\) −2.50498e6 −1.29549
\(328\) 0 0
\(329\) 466477. 0.237597
\(330\) 0 0
\(331\) − 1.33416e6i − 0.669325i −0.942338 0.334663i \(-0.891378\pi\)
0.942338 0.334663i \(-0.108622\pi\)
\(332\) 0 0
\(333\) 56149.1i 0.0277481i
\(334\) 0 0
\(335\) −149325. −0.0726975
\(336\) 0 0
\(337\) 1.18056e6 0.566257 0.283128 0.959082i \(-0.408628\pi\)
0.283128 + 0.959082i \(0.408628\pi\)
\(338\) 0 0
\(339\) − 460260.i − 0.217522i
\(340\) 0 0
\(341\) 280200.i 0.130491i
\(342\) 0 0
\(343\) 1.69387e6 0.777399
\(344\) 0 0
\(345\) −1.70365e6 −0.770605
\(346\) 0 0
\(347\) 669989.i 0.298706i 0.988784 + 0.149353i \(0.0477191\pi\)
−0.988784 + 0.149353i \(0.952281\pi\)
\(348\) 0 0
\(349\) − 1.64640e6i − 0.723554i −0.932265 0.361777i \(-0.882170\pi\)
0.932265 0.361777i \(-0.117830\pi\)
\(350\) 0 0
\(351\) 201076. 0.0871148
\(352\) 0 0
\(353\) −4.33852e6 −1.85312 −0.926562 0.376143i \(-0.877250\pi\)
−0.926562 + 0.376143i \(0.877250\pi\)
\(354\) 0 0
\(355\) − 421425.i − 0.177480i
\(356\) 0 0
\(357\) − 1.63791e6i − 0.680175i
\(358\) 0 0
\(359\) −3.33513e6 −1.36577 −0.682883 0.730527i \(-0.739274\pi\)
−0.682883 + 0.730527i \(0.739274\pi\)
\(360\) 0 0
\(361\) −71060.5 −0.0286986
\(362\) 0 0
\(363\) 2.27599e6i 0.906575i
\(364\) 0 0
\(365\) 1.41586e6i 0.556274i
\(366\) 0 0
\(367\) 4.53771e6 1.75862 0.879309 0.476252i \(-0.158005\pi\)
0.879309 + 0.476252i \(0.158005\pi\)
\(368\) 0 0
\(369\) −170166. −0.0650590
\(370\) 0 0
\(371\) − 374043.i − 0.141087i
\(372\) 0 0
\(373\) 4.47262e6i 1.66452i 0.554383 + 0.832262i \(0.312954\pi\)
−0.554383 + 0.832262i \(0.687046\pi\)
\(374\) 0 0
\(375\) −258731. −0.0950101
\(376\) 0 0
\(377\) 286858. 0.103947
\(378\) 0 0
\(379\) − 189146.i − 0.0676391i −0.999428 0.0338196i \(-0.989233\pi\)
0.999428 0.0338196i \(-0.0107672\pi\)
\(380\) 0 0
\(381\) − 2.22009e6i − 0.783534i
\(382\) 0 0
\(383\) 3.16389e6 1.10211 0.551054 0.834469i \(-0.314226\pi\)
0.551054 + 0.834469i \(0.314226\pi\)
\(384\) 0 0
\(385\) 213040. 0.0732504
\(386\) 0 0
\(387\) − 257618.i − 0.0874376i
\(388\) 0 0
\(389\) − 2.39411e6i − 0.802178i −0.916039 0.401089i \(-0.868632\pi\)
0.916039 0.401089i \(-0.131368\pi\)
\(390\) 0 0
\(391\) −7.33880e6 −2.42763
\(392\) 0 0
\(393\) −6.17428e6 −2.01653
\(394\) 0 0
\(395\) 932383.i 0.300678i
\(396\) 0 0
\(397\) − 5.57288e6i − 1.77461i −0.461182 0.887306i \(-0.652574\pi\)
0.461182 0.887306i \(-0.347426\pi\)
\(398\) 0 0
\(399\) 1.46591e6 0.460971
\(400\) 0 0
\(401\) −1.95214e6 −0.606248 −0.303124 0.952951i \(-0.598030\pi\)
−0.303124 + 0.952951i \(0.598030\pi\)
\(402\) 0 0
\(403\) 104565.i 0.0320719i
\(404\) 0 0
\(405\) 1.64140e6i 0.497251i
\(406\) 0 0
\(407\) −276543. −0.0827515
\(408\) 0 0
\(409\) −5.80819e6 −1.71685 −0.858425 0.512939i \(-0.828557\pi\)
−0.858425 + 0.512939i \(0.828557\pi\)
\(410\) 0 0
\(411\) − 2.42160e6i − 0.707127i
\(412\) 0 0
\(413\) − 973.233i 0 0.000280764i
\(414\) 0 0
\(415\) 1.44582e6 0.412092
\(416\) 0 0
\(417\) −3.75470e6 −1.05739
\(418\) 0 0
\(419\) 3.04430e6i 0.847133i 0.905865 + 0.423567i \(0.139222\pi\)
−0.905865 + 0.423567i \(0.860778\pi\)
\(420\) 0 0
\(421\) − 1.07953e6i − 0.296846i −0.988924 0.148423i \(-0.952580\pi\)
0.988924 0.148423i \(-0.0474197\pi\)
\(422\) 0 0
\(423\) 262321. 0.0712823
\(424\) 0 0
\(425\) −1.11453e6 −0.299310
\(426\) 0 0
\(427\) 1.04444e6i 0.277213i
\(428\) 0 0
\(429\) − 145845.i − 0.0382602i
\(430\) 0 0
\(431\) −6.45588e6 −1.67403 −0.837013 0.547183i \(-0.815700\pi\)
−0.837013 + 0.547183i \(0.815700\pi\)
\(432\) 0 0
\(433\) −1.74868e6 −0.448221 −0.224110 0.974564i \(-0.571948\pi\)
−0.224110 + 0.974564i \(0.571948\pi\)
\(434\) 0 0
\(435\) 2.07130e6i 0.524833i
\(436\) 0 0
\(437\) − 6.56810e6i − 1.64527i
\(438\) 0 0
\(439\) 1.58712e6 0.393049 0.196525 0.980499i \(-0.437034\pi\)
0.196525 + 0.980499i \(0.437034\pi\)
\(440\) 0 0
\(441\) 428282. 0.104866
\(442\) 0 0
\(443\) 2.17901e6i 0.527534i 0.964586 + 0.263767i \(0.0849650\pi\)
−0.964586 + 0.263767i \(0.915035\pi\)
\(444\) 0 0
\(445\) 2.32908e6i 0.557551i
\(446\) 0 0
\(447\) −79828.3 −0.0188968
\(448\) 0 0
\(449\) 3.99854e6 0.936021 0.468010 0.883723i \(-0.344971\pi\)
0.468010 + 0.883723i \(0.344971\pi\)
\(450\) 0 0
\(451\) − 838092.i − 0.194022i
\(452\) 0 0
\(453\) 120224.i 0.0275261i
\(454\) 0 0
\(455\) 79502.6 0.0180033
\(456\) 0 0
\(457\) 5.36373e6 1.20137 0.600685 0.799486i \(-0.294895\pi\)
0.600685 + 0.799486i \(0.294895\pi\)
\(458\) 0 0
\(459\) 6.25435e6i 1.38564i
\(460\) 0 0
\(461\) − 2.44852e6i − 0.536601i −0.963335 0.268300i \(-0.913538\pi\)
0.963335 0.268300i \(-0.0864620\pi\)
\(462\) 0 0
\(463\) 2.81552e6 0.610389 0.305195 0.952290i \(-0.401278\pi\)
0.305195 + 0.952290i \(0.401278\pi\)
\(464\) 0 0
\(465\) −755030. −0.161932
\(466\) 0 0
\(467\) 1.24614e6i 0.264408i 0.991223 + 0.132204i \(0.0422053\pi\)
−0.991223 + 0.132204i \(0.957795\pi\)
\(468\) 0 0
\(469\) 331316.i 0.0695520i
\(470\) 0 0
\(471\) −6.95531e6 −1.44466
\(472\) 0 0
\(473\) 1.26880e6 0.260760
\(474\) 0 0
\(475\) − 997489.i − 0.202850i
\(476\) 0 0
\(477\) − 210341.i − 0.0423281i
\(478\) 0 0
\(479\) 2.01494e6 0.401258 0.200629 0.979667i \(-0.435701\pi\)
0.200629 + 0.979667i \(0.435701\pi\)
\(480\) 0 0
\(481\) −103200. −0.0203385
\(482\) 0 0
\(483\) 3.77998e6i 0.737262i
\(484\) 0 0
\(485\) 2.28250e6i 0.440612i
\(486\) 0 0
\(487\) −8.40493e6 −1.60587 −0.802937 0.596064i \(-0.796730\pi\)
−0.802937 + 0.596064i \(0.796730\pi\)
\(488\) 0 0
\(489\) 574448. 0.108637
\(490\) 0 0
\(491\) − 4.54738e6i − 0.851250i −0.904900 0.425625i \(-0.860054\pi\)
0.904900 0.425625i \(-0.139946\pi\)
\(492\) 0 0
\(493\) 8.92255e6i 1.65338i
\(494\) 0 0
\(495\) 119802. 0.0219761
\(496\) 0 0
\(497\) −935040. −0.169801
\(498\) 0 0
\(499\) − 3.00176e6i − 0.539665i −0.962907 0.269833i \(-0.913032\pi\)
0.962907 0.269833i \(-0.0869684\pi\)
\(500\) 0 0
\(501\) − 8.59630e6i − 1.53009i
\(502\) 0 0
\(503\) 6.17709e6 1.08859 0.544294 0.838894i \(-0.316798\pi\)
0.544294 + 0.838894i \(0.316798\pi\)
\(504\) 0 0
\(505\) −4.17638e6 −0.728738
\(506\) 0 0
\(507\) 6.09373e6i 1.05284i
\(508\) 0 0
\(509\) − 628764.i − 0.107571i −0.998553 0.0537853i \(-0.982871\pi\)
0.998553 0.0537853i \(-0.0171287\pi\)
\(510\) 0 0
\(511\) 3.14146e6 0.532205
\(512\) 0 0
\(513\) −5.59754e6 −0.939082
\(514\) 0 0
\(515\) 1.66659e6i 0.276893i
\(516\) 0 0
\(517\) 1.29197e6i 0.212581i
\(518\) 0 0
\(519\) −9.60307e6 −1.56492
\(520\) 0 0
\(521\) 3.46406e6 0.559102 0.279551 0.960131i \(-0.409814\pi\)
0.279551 + 0.960131i \(0.409814\pi\)
\(522\) 0 0
\(523\) 7.86718e6i 1.25766i 0.777541 + 0.628832i \(0.216467\pi\)
−0.777541 + 0.628832i \(0.783533\pi\)
\(524\) 0 0
\(525\) 574061.i 0.0908992i
\(526\) 0 0
\(527\) −3.25244e6 −0.510133
\(528\) 0 0
\(529\) 1.05001e7 1.63138
\(530\) 0 0
\(531\) − 547.293i 0 8.42332e-5i
\(532\) 0 0
\(533\) − 312760.i − 0.0476863i
\(534\) 0 0
\(535\) 2.56421e6 0.387319
\(536\) 0 0
\(537\) 4.64286e6 0.694785
\(538\) 0 0
\(539\) 2.10935e6i 0.312735i
\(540\) 0 0
\(541\) 5.36006e6i 0.787366i 0.919246 + 0.393683i \(0.128799\pi\)
−0.919246 + 0.393683i \(0.871201\pi\)
\(542\) 0 0
\(543\) 1.45354e7 2.11558
\(544\) 0 0
\(545\) 3.78196e6 0.545413
\(546\) 0 0
\(547\) − 9.71604e6i − 1.38842i −0.719773 0.694210i \(-0.755754\pi\)
0.719773 0.694210i \(-0.244246\pi\)
\(548\) 0 0
\(549\) 587335.i 0.0831678i
\(550\) 0 0
\(551\) −7.98554e6 −1.12053
\(552\) 0 0
\(553\) 2.06873e6 0.287668
\(554\) 0 0
\(555\) − 745175.i − 0.102689i
\(556\) 0 0
\(557\) 6.34509e6i 0.866562i 0.901259 + 0.433281i \(0.142644\pi\)
−0.901259 + 0.433281i \(0.857356\pi\)
\(558\) 0 0
\(559\) 473493. 0.0640891
\(560\) 0 0
\(561\) 4.53642e6 0.608563
\(562\) 0 0
\(563\) 2.79024e6i 0.370997i 0.982645 + 0.185498i \(0.0593899\pi\)
−0.982645 + 0.185498i \(0.940610\pi\)
\(564\) 0 0
\(565\) 694889.i 0.0915786i
\(566\) 0 0
\(567\) 3.64186e6 0.475736
\(568\) 0 0
\(569\) −1.13522e7 −1.46994 −0.734972 0.678098i \(-0.762805\pi\)
−0.734972 + 0.678098i \(0.762805\pi\)
\(570\) 0 0
\(571\) − 6.20109e6i − 0.795935i −0.917400 0.397967i \(-0.869716\pi\)
0.917400 0.397967i \(-0.130284\pi\)
\(572\) 0 0
\(573\) 1.65354e7i 2.10391i
\(574\) 0 0
\(575\) 2.57212e6 0.324431
\(576\) 0 0
\(577\) 7.88135e6 0.985510 0.492755 0.870168i \(-0.335990\pi\)
0.492755 + 0.870168i \(0.335990\pi\)
\(578\) 0 0
\(579\) − 8.00499e6i − 0.992349i
\(580\) 0 0
\(581\) − 3.20793e6i − 0.394262i
\(582\) 0 0
\(583\) 1.03596e6 0.126233
\(584\) 0 0
\(585\) 44707.9 0.00540125
\(586\) 0 0
\(587\) − 1.32264e7i − 1.58433i −0.610306 0.792166i \(-0.708953\pi\)
0.610306 0.792166i \(-0.291047\pi\)
\(588\) 0 0
\(589\) − 2.91088e6i − 0.345729i
\(590\) 0 0
\(591\) −3.62836e6 −0.427309
\(592\) 0 0
\(593\) −1.27301e7 −1.48660 −0.743301 0.668957i \(-0.766741\pi\)
−0.743301 + 0.668957i \(0.766741\pi\)
\(594\) 0 0
\(595\) 2.47288e6i 0.286359i
\(596\) 0 0
\(597\) − 8.15386e6i − 0.936326i
\(598\) 0 0
\(599\) 4.85153e6 0.552474 0.276237 0.961090i \(-0.410913\pi\)
0.276237 + 0.961090i \(0.410913\pi\)
\(600\) 0 0
\(601\) 1.36513e7 1.54166 0.770829 0.637042i \(-0.219842\pi\)
0.770829 + 0.637042i \(0.219842\pi\)
\(602\) 0 0
\(603\) 186314.i 0.0208666i
\(604\) 0 0
\(605\) − 3.43623e6i − 0.381675i
\(606\) 0 0
\(607\) 3.82670e6 0.421554 0.210777 0.977534i \(-0.432401\pi\)
0.210777 + 0.977534i \(0.432401\pi\)
\(608\) 0 0
\(609\) 4.59573e6 0.502124
\(610\) 0 0
\(611\) 482138.i 0.0522478i
\(612\) 0 0
\(613\) 8.97723e6i 0.964920i 0.875918 + 0.482460i \(0.160257\pi\)
−0.875918 + 0.482460i \(0.839743\pi\)
\(614\) 0 0
\(615\) 2.25833e6 0.240769
\(616\) 0 0
\(617\) −1.21136e7 −1.28103 −0.640514 0.767946i \(-0.721279\pi\)
−0.640514 + 0.767946i \(0.721279\pi\)
\(618\) 0 0
\(619\) − 1.72513e7i − 1.80965i −0.425779 0.904827i \(-0.640000\pi\)
0.425779 0.904827i \(-0.360000\pi\)
\(620\) 0 0
\(621\) − 1.44338e7i − 1.50194i
\(622\) 0 0
\(623\) 5.16767e6 0.533427
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) 4.06002e6i 0.412438i
\(628\) 0 0
\(629\) − 3.20999e6i − 0.323502i
\(630\) 0 0
\(631\) 1.60671e7 1.60644 0.803220 0.595683i \(-0.203119\pi\)
0.803220 + 0.595683i \(0.203119\pi\)
\(632\) 0 0
\(633\) 1.55729e7 1.54476
\(634\) 0 0
\(635\) 3.35183e6i 0.329874i
\(636\) 0 0
\(637\) 787169.i 0.0768633i
\(638\) 0 0
\(639\) −525815. −0.0509425
\(640\) 0 0
\(641\) 4.51328e6 0.433857 0.216929 0.976187i \(-0.430396\pi\)
0.216929 + 0.976187i \(0.430396\pi\)
\(642\) 0 0
\(643\) − 1.72356e7i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(644\) 0 0
\(645\) 3.41893e6i 0.323587i
\(646\) 0 0
\(647\) −4.25296e6 −0.399421 −0.199711 0.979855i \(-0.564000\pi\)
−0.199711 + 0.979855i \(0.564000\pi\)
\(648\) 0 0
\(649\) 2695.49 0.000251204 0
\(650\) 0 0
\(651\) 1.67523e6i 0.154925i
\(652\) 0 0
\(653\) 2.28808e6i 0.209985i 0.994473 + 0.104992i \(0.0334818\pi\)
−0.994473 + 0.104992i \(0.966518\pi\)
\(654\) 0 0
\(655\) 9.32177e6 0.848976
\(656\) 0 0
\(657\) 1.76658e6 0.159669
\(658\) 0 0
\(659\) − 6.31745e6i − 0.566668i −0.959021 0.283334i \(-0.908560\pi\)
0.959021 0.283334i \(-0.0914404\pi\)
\(660\) 0 0
\(661\) − 9.62603e6i − 0.856926i −0.903559 0.428463i \(-0.859055\pi\)
0.903559 0.428463i \(-0.140945\pi\)
\(662\) 0 0
\(663\) 1.69290e6 0.149571
\(664\) 0 0
\(665\) −2.21319e6 −0.194073
\(666\) 0 0
\(667\) − 2.05915e7i − 1.79214i
\(668\) 0 0
\(669\) 1.84425e7i 1.59314i
\(670\) 0 0
\(671\) −2.89271e6 −0.248027
\(672\) 0 0
\(673\) −5.75251e6 −0.489575 −0.244788 0.969577i \(-0.578718\pi\)
−0.244788 + 0.969577i \(0.578718\pi\)
\(674\) 0 0
\(675\) − 2.19204e6i − 0.185178i
\(676\) 0 0
\(677\) 6.21796e6i 0.521406i 0.965419 + 0.260703i \(0.0839543\pi\)
−0.965419 + 0.260703i \(0.916046\pi\)
\(678\) 0 0
\(679\) 5.06431e6 0.421547
\(680\) 0 0
\(681\) 1.00239e7 0.828262
\(682\) 0 0
\(683\) − 1.93186e6i − 0.158461i −0.996856 0.0792306i \(-0.974754\pi\)
0.996856 0.0792306i \(-0.0252463\pi\)
\(684\) 0 0
\(685\) 3.65606e6i 0.297706i
\(686\) 0 0
\(687\) −2.06199e7 −1.66685
\(688\) 0 0
\(689\) 386601. 0.0310252
\(690\) 0 0
\(691\) − 2.04595e7i − 1.63005i −0.579429 0.815023i \(-0.696724\pi\)
0.579429 0.815023i \(-0.303276\pi\)
\(692\) 0 0
\(693\) − 265812.i − 0.0210253i
\(694\) 0 0
\(695\) 5.66874e6 0.445169
\(696\) 0 0
\(697\) 9.72822e6 0.758493
\(698\) 0 0
\(699\) 7.58031e6i 0.586805i
\(700\) 0 0
\(701\) − 2.02793e7i − 1.55868i −0.626599 0.779342i \(-0.715553\pi\)
0.626599 0.779342i \(-0.284447\pi\)
\(702\) 0 0
\(703\) 2.87289e6 0.219245
\(704\) 0 0
\(705\) −3.48135e6 −0.263800
\(706\) 0 0
\(707\) 9.26637e6i 0.697206i
\(708\) 0 0
\(709\) 7.36145e6i 0.549981i 0.961447 + 0.274991i \(0.0886747\pi\)
−0.961447 + 0.274991i \(0.911325\pi\)
\(710\) 0 0
\(711\) 1.16334e6 0.0863044
\(712\) 0 0
\(713\) 7.50599e6 0.552948
\(714\) 0 0
\(715\) 220193.i 0.0161079i
\(716\) 0 0
\(717\) 1.63079e7i 1.18468i
\(718\) 0 0
\(719\) −325405. −0.0234748 −0.0117374 0.999931i \(-0.503736\pi\)
−0.0117374 + 0.999931i \(0.503736\pi\)
\(720\) 0 0
\(721\) 3.69777e6 0.264912
\(722\) 0 0
\(723\) 2.29427e7i 1.63230i
\(724\) 0 0
\(725\) − 3.12720e6i − 0.220959i
\(726\) 0 0
\(727\) −7.04688e6 −0.494494 −0.247247 0.968953i \(-0.579526\pi\)
−0.247247 + 0.968953i \(0.579526\pi\)
\(728\) 0 0
\(729\) −1.20645e7 −0.840795
\(730\) 0 0
\(731\) 1.47277e7i 1.01939i
\(732\) 0 0
\(733\) − 705266.i − 0.0484834i −0.999706 0.0242417i \(-0.992283\pi\)
0.999706 0.0242417i \(-0.00771713\pi\)
\(734\) 0 0
\(735\) −5.68388e6 −0.388084
\(736\) 0 0
\(737\) −917621. −0.0622293
\(738\) 0 0
\(739\) 4.61861e6i 0.311100i 0.987828 + 0.155550i \(0.0497150\pi\)
−0.987828 + 0.155550i \(0.950285\pi\)
\(740\) 0 0
\(741\) 1.51512e6i 0.101368i
\(742\) 0 0
\(743\) 4.68905e6 0.311611 0.155805 0.987788i \(-0.450203\pi\)
0.155805 + 0.987788i \(0.450203\pi\)
\(744\) 0 0
\(745\) 120523. 0.00795570
\(746\) 0 0
\(747\) − 1.80396e6i − 0.118284i
\(748\) 0 0
\(749\) − 5.68936e6i − 0.370560i
\(750\) 0 0
\(751\) 1.71512e7 1.10967 0.554836 0.831960i \(-0.312781\pi\)
0.554836 + 0.831960i \(0.312781\pi\)
\(752\) 0 0
\(753\) −8.85471e6 −0.569098
\(754\) 0 0
\(755\) − 181511.i − 0.0115887i
\(756\) 0 0
\(757\) 1.45909e7i 0.925427i 0.886508 + 0.462713i \(0.153124\pi\)
−0.886508 + 0.462713i \(0.846876\pi\)
\(758\) 0 0
\(759\) −1.04691e7 −0.659640
\(760\) 0 0
\(761\) 6.25243e6 0.391370 0.195685 0.980667i \(-0.437307\pi\)
0.195685 + 0.980667i \(0.437307\pi\)
\(762\) 0 0
\(763\) − 8.39126e6i − 0.521814i
\(764\) 0 0
\(765\) 1.39061e6i 0.0859117i
\(766\) 0 0
\(767\) 1005.91 6.17404e−5 0
\(768\) 0 0
\(769\) 2.81919e7 1.71913 0.859566 0.511025i \(-0.170734\pi\)
0.859566 + 0.511025i \(0.170734\pi\)
\(770\) 0 0
\(771\) 1.41733e7i 0.858689i
\(772\) 0 0
\(773\) 2.92638e7i 1.76150i 0.473583 + 0.880749i \(0.342960\pi\)
−0.473583 + 0.880749i \(0.657040\pi\)
\(774\) 0 0
\(775\) 1.13993e6 0.0681745
\(776\) 0 0
\(777\) −1.65336e6 −0.0982462
\(778\) 0 0
\(779\) 8.70659e6i 0.514049i
\(780\) 0 0
\(781\) − 2.58971e6i − 0.151923i
\(782\) 0 0
\(783\) −1.75487e7 −1.02292
\(784\) 0 0
\(785\) 1.05010e7 0.608211
\(786\) 0 0
\(787\) − 765622.i − 0.0440634i −0.999757 0.0220317i \(-0.992987\pi\)
0.999757 0.0220317i \(-0.00701347\pi\)
\(788\) 0 0
\(789\) − 1.10367e7i − 0.631168i
\(790\) 0 0
\(791\) 1.54179e6 0.0876161
\(792\) 0 0
\(793\) −1.07950e6 −0.0609595
\(794\) 0 0
\(795\) 2.79151e6i 0.156647i
\(796\) 0 0
\(797\) − 1.13923e7i − 0.635281i −0.948211 0.317641i \(-0.897109\pi\)
0.948211 0.317641i \(-0.102891\pi\)
\(798\) 0 0
\(799\) −1.49966e7 −0.831048
\(800\) 0 0
\(801\) 2.90601e6 0.160035
\(802\) 0 0
\(803\) 8.70067e6i 0.476172i
\(804\) 0 0
\(805\) − 5.70692e6i − 0.310393i
\(806\) 0 0
\(807\) 3.14365e6 0.169922
\(808\) 0 0
\(809\) −1.46651e7 −0.787798 −0.393899 0.919154i \(-0.628874\pi\)
−0.393899 + 0.919154i \(0.628874\pi\)
\(810\) 0 0
\(811\) 8.79076e6i 0.469326i 0.972077 + 0.234663i \(0.0753986\pi\)
−0.972077 + 0.234663i \(0.924601\pi\)
\(812\) 0 0
\(813\) − 2.41257e7i − 1.28013i
\(814\) 0 0
\(815\) −867286. −0.0457371
\(816\) 0 0
\(817\) −1.31811e7 −0.690869
\(818\) 0 0
\(819\) − 99196.0i − 0.00516754i
\(820\) 0 0
\(821\) 1.25031e7i 0.647383i 0.946163 + 0.323691i \(0.104924\pi\)
−0.946163 + 0.323691i \(0.895076\pi\)
\(822\) 0 0
\(823\) 2.36472e7 1.21697 0.608486 0.793565i \(-0.291777\pi\)
0.608486 + 0.793565i \(0.291777\pi\)
\(824\) 0 0
\(825\) −1.58994e6 −0.0813289
\(826\) 0 0
\(827\) − 4.91285e6i − 0.249787i −0.992170 0.124893i \(-0.960141\pi\)
0.992170 0.124893i \(-0.0398589\pi\)
\(828\) 0 0
\(829\) − 4.52433e6i − 0.228648i −0.993443 0.114324i \(-0.963530\pi\)
0.993443 0.114324i \(-0.0364703\pi\)
\(830\) 0 0
\(831\) −1.78919e7 −0.898781
\(832\) 0 0
\(833\) −2.44844e7 −1.22258
\(834\) 0 0
\(835\) 1.29785e7i 0.644181i
\(836\) 0 0
\(837\) − 6.39684e6i − 0.315611i
\(838\) 0 0
\(839\) 4.83585e6 0.237175 0.118587 0.992944i \(-0.462163\pi\)
0.118587 + 0.992944i \(0.462163\pi\)
\(840\) 0 0
\(841\) −4.52410e6 −0.220568
\(842\) 0 0
\(843\) 2.45154e7i 1.18815i
\(844\) 0 0
\(845\) − 9.20015e6i − 0.443255i
\(846\) 0 0
\(847\) −7.62417e6 −0.365161
\(848\) 0 0
\(849\) −7.94783e6 −0.378424
\(850\) 0 0
\(851\) 7.40802e6i 0.350654i
\(852\) 0 0
\(853\) − 4.52210e6i − 0.212798i −0.994324 0.106399i \(-0.966068\pi\)
0.994324 0.106399i \(-0.0339321\pi\)
\(854\) 0 0
\(855\) −1.24457e6 −0.0582245
\(856\) 0 0
\(857\) −2.18075e7 −1.01427 −0.507135 0.861867i \(-0.669295\pi\)
−0.507135 + 0.861867i \(0.669295\pi\)
\(858\) 0 0
\(859\) 3.47558e6i 0.160711i 0.996766 + 0.0803553i \(0.0256055\pi\)
−0.996766 + 0.0803553i \(0.974395\pi\)
\(860\) 0 0
\(861\) − 5.01070e6i − 0.230351i
\(862\) 0 0
\(863\) −2.20281e7 −1.00682 −0.503409 0.864048i \(-0.667921\pi\)
−0.503409 + 0.864048i \(0.667921\pi\)
\(864\) 0 0
\(865\) 1.44985e7 0.658843
\(866\) 0 0
\(867\) 2.91457e7i 1.31682i
\(868\) 0 0
\(869\) 5.72962e6i 0.257381i
\(870\) 0 0
\(871\) −342439. −0.0152946
\(872\) 0 0
\(873\) 2.84789e6 0.126470
\(874\) 0 0
\(875\) − 866703.i − 0.0382693i
\(876\) 0 0
\(877\) − 2.22181e7i − 0.975456i −0.872996 0.487728i \(-0.837826\pi\)
0.872996 0.487728i \(-0.162174\pi\)
\(878\) 0 0
\(879\) −3.43087e7 −1.49773
\(880\) 0 0
\(881\) −3.40974e7 −1.48007 −0.740033 0.672570i \(-0.765190\pi\)
−0.740033 + 0.672570i \(0.765190\pi\)
\(882\) 0 0
\(883\) − 3.76822e7i − 1.62642i −0.581967 0.813212i \(-0.697717\pi\)
0.581967 0.813212i \(-0.302283\pi\)
\(884\) 0 0
\(885\) 7263.31i 0 0.000311729i
\(886\) 0 0
\(887\) 2.33279e6 0.0995558 0.0497779 0.998760i \(-0.484149\pi\)
0.0497779 + 0.998760i \(0.484149\pi\)
\(888\) 0 0
\(889\) 7.43691e6 0.315601
\(890\) 0 0
\(891\) 1.00866e7i 0.425648i
\(892\) 0 0
\(893\) − 1.34217e7i − 0.563222i
\(894\) 0 0
\(895\) −7.00968e6 −0.292510
\(896\) 0 0
\(897\) −3.90689e6 −0.162125
\(898\) 0 0
\(899\) − 9.12583e6i − 0.376594i
\(900\) 0 0
\(901\) 1.20250e7i 0.493484i
\(902\) 0 0
\(903\) 7.58579e6 0.309586
\(904\) 0 0
\(905\) −2.19452e7 −0.890674
\(906\) 0 0
\(907\) − 3.00605e6i − 0.121333i −0.998158 0.0606664i \(-0.980677\pi\)
0.998158 0.0606664i \(-0.0193226\pi\)
\(908\) 0 0
\(909\) 5.21090e6i 0.209172i
\(910\) 0 0
\(911\) −9.65671e6 −0.385508 −0.192754 0.981247i \(-0.561742\pi\)
−0.192754 + 0.981247i \(0.561742\pi\)
\(912\) 0 0
\(913\) 8.88477e6 0.352752
\(914\) 0 0
\(915\) − 7.79474e6i − 0.307786i
\(916\) 0 0
\(917\) − 2.06828e7i − 0.812242i
\(918\) 0 0
\(919\) 3.59667e7 1.40479 0.702396 0.711786i \(-0.252114\pi\)
0.702396 + 0.711786i \(0.252114\pi\)
\(920\) 0 0
\(921\) −3.39091e7 −1.31725
\(922\) 0 0
\(923\) − 966431.i − 0.0373393i
\(924\) 0 0
\(925\) 1.12505e6i 0.0432331i
\(926\) 0 0
\(927\) 2.07942e6 0.0794773
\(928\) 0 0
\(929\) −2.66167e7 −1.01185 −0.505923 0.862579i \(-0.668848\pi\)
−0.505923 + 0.862579i \(0.668848\pi\)
\(930\) 0 0
\(931\) − 2.19131e7i − 0.828572i
\(932\) 0 0
\(933\) − 4.60072e7i − 1.73030i
\(934\) 0 0
\(935\) −6.84896e6 −0.256210
\(936\) 0 0
\(937\) 1.41039e7 0.524797 0.262399 0.964960i \(-0.415486\pi\)
0.262399 + 0.964960i \(0.415486\pi\)
\(938\) 0 0
\(939\) 2.94892e7i 1.09144i
\(940\) 0 0
\(941\) − 2.94208e6i − 0.108313i −0.998532 0.0541565i \(-0.982753\pi\)
0.998532 0.0541565i \(-0.0172470\pi\)
\(942\) 0 0
\(943\) −2.24508e7 −0.822153
\(944\) 0 0
\(945\) −4.86361e6 −0.177166
\(946\) 0 0
\(947\) 1.97465e7i 0.715511i 0.933815 + 0.357755i \(0.116458\pi\)
−0.933815 + 0.357755i \(0.883542\pi\)
\(948\) 0 0
\(949\) 3.24692e6i 0.117033i
\(950\) 0 0
\(951\) 2.77171e7 0.993795
\(952\) 0 0
\(953\) −1.51163e7 −0.539155 −0.269577 0.962979i \(-0.586884\pi\)
−0.269577 + 0.962979i \(0.586884\pi\)
\(954\) 0 0
\(955\) − 2.49647e7i − 0.885763i
\(956\) 0 0
\(957\) 1.27285e7i 0.449258i
\(958\) 0 0
\(959\) 8.11192e6 0.284825
\(960\) 0 0
\(961\) −2.53026e7 −0.883806
\(962\) 0 0
\(963\) − 3.19938e6i − 0.111173i
\(964\) 0 0
\(965\) 1.20857e7i 0.417787i
\(966\) 0 0
\(967\) 4.34383e7 1.49385 0.746925 0.664908i \(-0.231529\pi\)
0.746925 + 0.664908i \(0.231529\pi\)
\(968\) 0 0
\(969\) −4.71269e7 −1.61235
\(970\) 0 0
\(971\) − 9.27361e6i − 0.315646i −0.987467 0.157823i \(-0.949552\pi\)
0.987467 0.157823i \(-0.0504476\pi\)
\(972\) 0 0
\(973\) − 1.25776e7i − 0.425907i
\(974\) 0 0
\(975\) −593334. −0.0199888
\(976\) 0 0
\(977\) −6.80505e6 −0.228084 −0.114042 0.993476i \(-0.536380\pi\)
−0.114042 + 0.993476i \(0.536380\pi\)
\(978\) 0 0
\(979\) 1.43125e7i 0.477265i
\(980\) 0 0
\(981\) − 4.71878e6i − 0.156551i
\(982\) 0 0
\(983\) −4.94290e7 −1.63154 −0.815771 0.578375i \(-0.803687\pi\)
−0.815771 + 0.578375i \(0.803687\pi\)
\(984\) 0 0
\(985\) 5.47800e6 0.179900
\(986\) 0 0
\(987\) 7.72428e6i 0.252386i
\(988\) 0 0
\(989\) − 3.39887e7i − 1.10495i
\(990\) 0 0
\(991\) −3.41846e7 −1.10572 −0.552862 0.833273i \(-0.686464\pi\)
−0.552862 + 0.833273i \(0.686464\pi\)
\(992\) 0 0
\(993\) 2.20920e7 0.710988
\(994\) 0 0
\(995\) 1.23105e7i 0.394201i
\(996\) 0 0
\(997\) 4.99930e7i 1.59284i 0.604746 + 0.796418i \(0.293274\pi\)
−0.604746 + 0.796418i \(0.706726\pi\)
\(998\) 0 0
\(999\) 6.31334e6 0.200145
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.161.10 yes 12
4.3 odd 2 320.6.d.d.161.3 yes 12
8.3 odd 2 320.6.d.d.161.10 yes 12
8.5 even 2 inner 320.6.d.c.161.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.6.d.c.161.3 12 8.5 even 2 inner
320.6.d.c.161.10 yes 12 1.1 even 1 trivial
320.6.d.d.161.3 yes 12 4.3 odd 2
320.6.d.d.161.10 yes 12 8.3 odd 2