Properties

Label 1350.4.c.k.649.2
Level $1350$
Weight $4$
Character 1350.649
Analytic conductor $79.653$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.6525785077\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 270)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1350.649
Dual form 1350.4.c.k.649.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} -4.00000 q^{4} -14.0000i q^{7} -8.00000i q^{8} +O(q^{10})\) \(q+2.00000i q^{2} -4.00000 q^{4} -14.0000i q^{7} -8.00000i q^{8} +3.00000 q^{11} +47.0000i q^{13} +28.0000 q^{14} +16.0000 q^{16} +39.0000i q^{17} -32.0000 q^{19} +6.00000i q^{22} -99.0000i q^{23} -94.0000 q^{26} +56.0000i q^{28} -51.0000 q^{29} +83.0000 q^{31} +32.0000i q^{32} -78.0000 q^{34} -314.000i q^{37} -64.0000i q^{38} -108.000 q^{41} +299.000i q^{43} -12.0000 q^{44} +198.000 q^{46} -531.000i q^{47} +147.000 q^{49} -188.000i q^{52} +564.000i q^{53} -112.000 q^{56} -102.000i q^{58} -12.0000 q^{59} +230.000 q^{61} +166.000i q^{62} -64.0000 q^{64} +268.000i q^{67} -156.000i q^{68} +120.000 q^{71} +1106.00i q^{73} +628.000 q^{74} +128.000 q^{76} -42.0000i q^{77} +739.000 q^{79} -216.000i q^{82} +1086.00i q^{83} -598.000 q^{86} -24.0000i q^{88} +120.000 q^{89} +658.000 q^{91} +396.000i q^{92} +1062.00 q^{94} +1642.00i q^{97} +294.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 6 q^{11} + 56 q^{14} + 32 q^{16} - 64 q^{19} - 188 q^{26} - 102 q^{29} + 166 q^{31} - 156 q^{34} - 216 q^{41} - 24 q^{44} + 396 q^{46} + 294 q^{49} - 224 q^{56} - 24 q^{59} + 460 q^{61} - 128 q^{64} + 240 q^{71} + 1256 q^{74} + 256 q^{76} + 1478 q^{79} - 1196 q^{86} + 240 q^{89} + 1316 q^{91} + 2124 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) 0 0
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 14.0000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 0.0822304 0.0411152 0.999154i \(-0.486909\pi\)
0.0411152 + 0.999154i \(0.486909\pi\)
\(12\) 0 0
\(13\) 47.0000i 1.00273i 0.865237 + 0.501364i \(0.167168\pi\)
−0.865237 + 0.501364i \(0.832832\pi\)
\(14\) 28.0000 0.534522
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 39.0000i 0.556405i 0.960522 + 0.278203i \(0.0897387\pi\)
−0.960522 + 0.278203i \(0.910261\pi\)
\(18\) 0 0
\(19\) −32.0000 −0.386384 −0.193192 0.981161i \(-0.561884\pi\)
−0.193192 + 0.981161i \(0.561884\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.00000i 0.0581456i
\(23\) − 99.0000i − 0.897519i −0.893653 0.448759i \(-0.851866\pi\)
0.893653 0.448759i \(-0.148134\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −94.0000 −0.709035
\(27\) 0 0
\(28\) 56.0000i 0.377964i
\(29\) −51.0000 −0.326568 −0.163284 0.986579i \(-0.552209\pi\)
−0.163284 + 0.986579i \(0.552209\pi\)
\(30\) 0 0
\(31\) 83.0000 0.480879 0.240439 0.970664i \(-0.422708\pi\)
0.240439 + 0.970664i \(0.422708\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 0 0
\(34\) −78.0000 −0.393438
\(35\) 0 0
\(36\) 0 0
\(37\) − 314.000i − 1.39517i −0.716502 0.697585i \(-0.754258\pi\)
0.716502 0.697585i \(-0.245742\pi\)
\(38\) − 64.0000i − 0.273215i
\(39\) 0 0
\(40\) 0 0
\(41\) −108.000 −0.411385 −0.205692 0.978617i \(-0.565945\pi\)
−0.205692 + 0.978617i \(0.565945\pi\)
\(42\) 0 0
\(43\) 299.000i 1.06040i 0.847874 + 0.530199i \(0.177883\pi\)
−0.847874 + 0.530199i \(0.822117\pi\)
\(44\) −12.0000 −0.0411152
\(45\) 0 0
\(46\) 198.000 0.634641
\(47\) − 531.000i − 1.64796i −0.566616 0.823982i \(-0.691748\pi\)
0.566616 0.823982i \(-0.308252\pi\)
\(48\) 0 0
\(49\) 147.000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) − 188.000i − 0.501364i
\(53\) 564.000i 1.46172i 0.682525 + 0.730862i \(0.260882\pi\)
−0.682525 + 0.730862i \(0.739118\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −112.000 −0.267261
\(57\) 0 0
\(58\) − 102.000i − 0.230918i
\(59\) −12.0000 −0.0264791 −0.0132396 0.999912i \(-0.504214\pi\)
−0.0132396 + 0.999912i \(0.504214\pi\)
\(60\) 0 0
\(61\) 230.000 0.482762 0.241381 0.970430i \(-0.422400\pi\)
0.241381 + 0.970430i \(0.422400\pi\)
\(62\) 166.000i 0.340033i
\(63\) 0 0
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 268.000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) − 156.000i − 0.278203i
\(69\) 0 0
\(70\) 0 0
\(71\) 120.000 0.200583 0.100291 0.994958i \(-0.468022\pi\)
0.100291 + 0.994958i \(0.468022\pi\)
\(72\) 0 0
\(73\) 1106.00i 1.77325i 0.462486 + 0.886627i \(0.346958\pi\)
−0.462486 + 0.886627i \(0.653042\pi\)
\(74\) 628.000 0.986534
\(75\) 0 0
\(76\) 128.000 0.193192
\(77\) − 42.0000i − 0.0621603i
\(78\) 0 0
\(79\) 739.000 1.05246 0.526228 0.850344i \(-0.323606\pi\)
0.526228 + 0.850344i \(0.323606\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 216.000i − 0.290893i
\(83\) 1086.00i 1.43619i 0.695944 + 0.718096i \(0.254986\pi\)
−0.695944 + 0.718096i \(0.745014\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −598.000 −0.749814
\(87\) 0 0
\(88\) − 24.0000i − 0.0290728i
\(89\) 120.000 0.142921 0.0714605 0.997443i \(-0.477234\pi\)
0.0714605 + 0.997443i \(0.477234\pi\)
\(90\) 0 0
\(91\) 658.000 0.757991
\(92\) 396.000i 0.448759i
\(93\) 0 0
\(94\) 1062.00 1.16529
\(95\) 0 0
\(96\) 0 0
\(97\) 1642.00i 1.71876i 0.511336 + 0.859381i \(0.329151\pi\)
−0.511336 + 0.859381i \(0.670849\pi\)
\(98\) 294.000i 0.303046i
\(99\) 0 0
\(100\) 0 0
\(101\) 33.0000 0.0325111 0.0162556 0.999868i \(-0.494825\pi\)
0.0162556 + 0.999868i \(0.494825\pi\)
\(102\) 0 0
\(103\) − 1198.00i − 1.14604i −0.819540 0.573022i \(-0.805771\pi\)
0.819540 0.573022i \(-0.194229\pi\)
\(104\) 376.000 0.354518
\(105\) 0 0
\(106\) −1128.00 −1.03359
\(107\) 1542.00i 1.39318i 0.717467 + 0.696592i \(0.245301\pi\)
−0.717467 + 0.696592i \(0.754699\pi\)
\(108\) 0 0
\(109\) 556.000 0.488579 0.244290 0.969702i \(-0.421445\pi\)
0.244290 + 0.969702i \(0.421445\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 224.000i − 0.188982i
\(113\) 1605.00i 1.33616i 0.744091 + 0.668078i \(0.232883\pi\)
−0.744091 + 0.668078i \(0.767117\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 204.000 0.163284
\(117\) 0 0
\(118\) − 24.0000i − 0.0187236i
\(119\) 546.000 0.420603
\(120\) 0 0
\(121\) −1322.00 −0.993238
\(122\) 460.000i 0.341364i
\(123\) 0 0
\(124\) −332.000 −0.240439
\(125\) 0 0
\(126\) 0 0
\(127\) − 1334.00i − 0.932074i −0.884765 0.466037i \(-0.845681\pi\)
0.884765 0.466037i \(-0.154319\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −2883.00 −1.92282 −0.961408 0.275127i \(-0.911280\pi\)
−0.961408 + 0.275127i \(0.911280\pi\)
\(132\) 0 0
\(133\) 448.000i 0.292079i
\(134\) −536.000 −0.345547
\(135\) 0 0
\(136\) 312.000 0.196719
\(137\) − 282.000i − 0.175860i −0.996127 0.0879302i \(-0.971975\pi\)
0.996127 0.0879302i \(-0.0280253\pi\)
\(138\) 0 0
\(139\) 2494.00 1.52186 0.760929 0.648835i \(-0.224743\pi\)
0.760929 + 0.648835i \(0.224743\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 240.000i 0.141833i
\(143\) 141.000i 0.0824546i
\(144\) 0 0
\(145\) 0 0
\(146\) −2212.00 −1.25388
\(147\) 0 0
\(148\) 1256.00i 0.697585i
\(149\) −2595.00 −1.42678 −0.713392 0.700766i \(-0.752842\pi\)
−0.713392 + 0.700766i \(0.752842\pi\)
\(150\) 0 0
\(151\) 1229.00 0.662348 0.331174 0.943570i \(-0.392555\pi\)
0.331174 + 0.943570i \(0.392555\pi\)
\(152\) 256.000i 0.136608i
\(153\) 0 0
\(154\) 84.0000 0.0439540
\(155\) 0 0
\(156\) 0 0
\(157\) 1591.00i 0.808762i 0.914591 + 0.404381i \(0.132513\pi\)
−0.914591 + 0.404381i \(0.867487\pi\)
\(158\) 1478.00i 0.744199i
\(159\) 0 0
\(160\) 0 0
\(161\) −1386.00 −0.678460
\(162\) 0 0
\(163\) − 457.000i − 0.219601i −0.993954 0.109801i \(-0.964979\pi\)
0.993954 0.109801i \(-0.0350212\pi\)
\(164\) 432.000 0.205692
\(165\) 0 0
\(166\) −2172.00 −1.01554
\(167\) 1164.00i 0.539359i 0.962950 + 0.269680i \(0.0869178\pi\)
−0.962950 + 0.269680i \(0.913082\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.00546199
\(170\) 0 0
\(171\) 0 0
\(172\) − 1196.00i − 0.530199i
\(173\) 3942.00i 1.73240i 0.499700 + 0.866199i \(0.333444\pi\)
−0.499700 + 0.866199i \(0.666556\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 48.0000 0.0205576
\(177\) 0 0
\(178\) 240.000i 0.101060i
\(179\) 1212.00 0.506085 0.253042 0.967455i \(-0.418569\pi\)
0.253042 + 0.967455i \(0.418569\pi\)
\(180\) 0 0
\(181\) 2288.00 0.939590 0.469795 0.882776i \(-0.344328\pi\)
0.469795 + 0.882776i \(0.344328\pi\)
\(182\) 1316.00i 0.535980i
\(183\) 0 0
\(184\) −792.000 −0.317321
\(185\) 0 0
\(186\) 0 0
\(187\) 117.000i 0.0457534i
\(188\) 2124.00i 0.823982i
\(189\) 0 0
\(190\) 0 0
\(191\) −1938.00 −0.734182 −0.367091 0.930185i \(-0.619646\pi\)
−0.367091 + 0.930185i \(0.619646\pi\)
\(192\) 0 0
\(193\) − 1498.00i − 0.558696i −0.960190 0.279348i \(-0.909882\pi\)
0.960190 0.279348i \(-0.0901184\pi\)
\(194\) −3284.00 −1.21535
\(195\) 0 0
\(196\) −588.000 −0.214286
\(197\) 2124.00i 0.768166i 0.923299 + 0.384083i \(0.125482\pi\)
−0.923299 + 0.384083i \(0.874518\pi\)
\(198\) 0 0
\(199\) 385.000 0.137145 0.0685727 0.997646i \(-0.478155\pi\)
0.0685727 + 0.997646i \(0.478155\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 66.0000i 0.0229888i
\(203\) 714.000i 0.246862i
\(204\) 0 0
\(205\) 0 0
\(206\) 2396.00 0.810375
\(207\) 0 0
\(208\) 752.000i 0.250682i
\(209\) −96.0000 −0.0317725
\(210\) 0 0
\(211\) 3170.00 1.03427 0.517137 0.855903i \(-0.326998\pi\)
0.517137 + 0.855903i \(0.326998\pi\)
\(212\) − 2256.00i − 0.730862i
\(213\) 0 0
\(214\) −3084.00 −0.985130
\(215\) 0 0
\(216\) 0 0
\(217\) − 1162.00i − 0.363510i
\(218\) 1112.00i 0.345478i
\(219\) 0 0
\(220\) 0 0
\(221\) −1833.00 −0.557923
\(222\) 0 0
\(223\) 1388.00i 0.416804i 0.978043 + 0.208402i \(0.0668263\pi\)
−0.978043 + 0.208402i \(0.933174\pi\)
\(224\) 448.000 0.133631
\(225\) 0 0
\(226\) −3210.00 −0.944805
\(227\) 4644.00i 1.35786i 0.734205 + 0.678928i \(0.237555\pi\)
−0.734205 + 0.678928i \(0.762445\pi\)
\(228\) 0 0
\(229\) −4736.00 −1.36665 −0.683327 0.730113i \(-0.739468\pi\)
−0.683327 + 0.730113i \(0.739468\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 408.000i 0.115459i
\(233\) 2814.00i 0.791207i 0.918421 + 0.395604i \(0.129465\pi\)
−0.918421 + 0.395604i \(0.870535\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 48.0000 0.0132396
\(237\) 0 0
\(238\) 1092.00i 0.297411i
\(239\) 2202.00 0.595965 0.297982 0.954571i \(-0.403686\pi\)
0.297982 + 0.954571i \(0.403686\pi\)
\(240\) 0 0
\(241\) 3485.00 0.931488 0.465744 0.884920i \(-0.345787\pi\)
0.465744 + 0.884920i \(0.345787\pi\)
\(242\) − 2644.00i − 0.702325i
\(243\) 0 0
\(244\) −920.000 −0.241381
\(245\) 0 0
\(246\) 0 0
\(247\) − 1504.00i − 0.387438i
\(248\) − 664.000i − 0.170016i
\(249\) 0 0
\(250\) 0 0
\(251\) −6345.00 −1.59559 −0.797795 0.602929i \(-0.794000\pi\)
−0.797795 + 0.602929i \(0.794000\pi\)
\(252\) 0 0
\(253\) − 297.000i − 0.0738033i
\(254\) 2668.00 0.659076
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 525.000i − 0.127426i −0.997968 0.0637132i \(-0.979706\pi\)
0.997968 0.0637132i \(-0.0202943\pi\)
\(258\) 0 0
\(259\) −4396.00 −1.05465
\(260\) 0 0
\(261\) 0 0
\(262\) − 5766.00i − 1.35964i
\(263\) 5196.00i 1.21825i 0.793075 + 0.609124i \(0.208479\pi\)
−0.793075 + 0.609124i \(0.791521\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −896.000 −0.206531
\(267\) 0 0
\(268\) − 1072.00i − 0.244339i
\(269\) 7479.00 1.69518 0.847589 0.530654i \(-0.178054\pi\)
0.847589 + 0.530654i \(0.178054\pi\)
\(270\) 0 0
\(271\) −856.000 −0.191876 −0.0959378 0.995387i \(-0.530585\pi\)
−0.0959378 + 0.995387i \(0.530585\pi\)
\(272\) 624.000i 0.139101i
\(273\) 0 0
\(274\) 564.000 0.124352
\(275\) 0 0
\(276\) 0 0
\(277\) 7054.00i 1.53009i 0.643979 + 0.765043i \(0.277282\pi\)
−0.643979 + 0.765043i \(0.722718\pi\)
\(278\) 4988.00i 1.07612i
\(279\) 0 0
\(280\) 0 0
\(281\) 1014.00 0.215268 0.107634 0.994191i \(-0.465673\pi\)
0.107634 + 0.994191i \(0.465673\pi\)
\(282\) 0 0
\(283\) 992.000i 0.208368i 0.994558 + 0.104184i \(0.0332232\pi\)
−0.994558 + 0.104184i \(0.966777\pi\)
\(284\) −480.000 −0.100291
\(285\) 0 0
\(286\) −282.000 −0.0583042
\(287\) 1512.00i 0.310977i
\(288\) 0 0
\(289\) 3392.00 0.690413
\(290\) 0 0
\(291\) 0 0
\(292\) − 4424.00i − 0.886627i
\(293\) − 4950.00i − 0.986970i −0.869754 0.493485i \(-0.835723\pi\)
0.869754 0.493485i \(-0.164277\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2512.00 −0.493267
\(297\) 0 0
\(298\) − 5190.00i − 1.00889i
\(299\) 4653.00 0.899966
\(300\) 0 0
\(301\) 4186.00 0.801585
\(302\) 2458.00i 0.468351i
\(303\) 0 0
\(304\) −512.000 −0.0965961
\(305\) 0 0
\(306\) 0 0
\(307\) 4777.00i 0.888071i 0.896009 + 0.444035i \(0.146454\pi\)
−0.896009 + 0.444035i \(0.853546\pi\)
\(308\) 168.000i 0.0310802i
\(309\) 0 0
\(310\) 0 0
\(311\) −7692.00 −1.40249 −0.701243 0.712922i \(-0.747371\pi\)
−0.701243 + 0.712922i \(0.747371\pi\)
\(312\) 0 0
\(313\) − 2932.00i − 0.529477i −0.964320 0.264739i \(-0.914714\pi\)
0.964320 0.264739i \(-0.0852857\pi\)
\(314\) −3182.00 −0.571881
\(315\) 0 0
\(316\) −2956.00 −0.526228
\(317\) − 8352.00i − 1.47980i −0.672720 0.739898i \(-0.734874\pi\)
0.672720 0.739898i \(-0.265126\pi\)
\(318\) 0 0
\(319\) −153.000 −0.0268538
\(320\) 0 0
\(321\) 0 0
\(322\) − 2772.00i − 0.479744i
\(323\) − 1248.00i − 0.214986i
\(324\) 0 0
\(325\) 0 0
\(326\) 914.000 0.155282
\(327\) 0 0
\(328\) 864.000i 0.145446i
\(329\) −7434.00 −1.24574
\(330\) 0 0
\(331\) −3070.00 −0.509796 −0.254898 0.966968i \(-0.582042\pi\)
−0.254898 + 0.966968i \(0.582042\pi\)
\(332\) − 4344.00i − 0.718096i
\(333\) 0 0
\(334\) −2328.00 −0.381385
\(335\) 0 0
\(336\) 0 0
\(337\) 1672.00i 0.270266i 0.990827 + 0.135133i \(0.0431462\pi\)
−0.990827 + 0.135133i \(0.956854\pi\)
\(338\) − 24.0000i − 0.00386221i
\(339\) 0 0
\(340\) 0 0
\(341\) 249.000 0.0395428
\(342\) 0 0
\(343\) − 6860.00i − 1.07990i
\(344\) 2392.00 0.374907
\(345\) 0 0
\(346\) −7884.00 −1.22499
\(347\) 5076.00i 0.785285i 0.919691 + 0.392643i \(0.128439\pi\)
−0.919691 + 0.392643i \(0.871561\pi\)
\(348\) 0 0
\(349\) −8594.00 −1.31813 −0.659063 0.752087i \(-0.729047\pi\)
−0.659063 + 0.752087i \(0.729047\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 96.0000i 0.0145364i
\(353\) 12711.0i 1.91654i 0.285866 + 0.958269i \(0.407719\pi\)
−0.285866 + 0.958269i \(0.592281\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −480.000 −0.0714605
\(357\) 0 0
\(358\) 2424.00i 0.357856i
\(359\) 1464.00 0.215228 0.107614 0.994193i \(-0.465679\pi\)
0.107614 + 0.994193i \(0.465679\pi\)
\(360\) 0 0
\(361\) −5835.00 −0.850707
\(362\) 4576.00i 0.664390i
\(363\) 0 0
\(364\) −2632.00 −0.378995
\(365\) 0 0
\(366\) 0 0
\(367\) 7630.00i 1.08524i 0.839979 + 0.542620i \(0.182567\pi\)
−0.839979 + 0.542620i \(0.817433\pi\)
\(368\) − 1584.00i − 0.224380i
\(369\) 0 0
\(370\) 0 0
\(371\) 7896.00 1.10496
\(372\) 0 0
\(373\) − 3883.00i − 0.539019i −0.962998 0.269510i \(-0.913138\pi\)
0.962998 0.269510i \(-0.0868616\pi\)
\(374\) −234.000 −0.0323525
\(375\) 0 0
\(376\) −4248.00 −0.582643
\(377\) − 2397.00i − 0.327458i
\(378\) 0 0
\(379\) 13768.0 1.86600 0.933001 0.359874i \(-0.117180\pi\)
0.933001 + 0.359874i \(0.117180\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 3876.00i − 0.519145i
\(383\) − 14139.0i − 1.88634i −0.332307 0.943171i \(-0.607827\pi\)
0.332307 0.943171i \(-0.392173\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2996.00 0.395058
\(387\) 0 0
\(388\) − 6568.00i − 0.859381i
\(389\) −567.000 −0.0739024 −0.0369512 0.999317i \(-0.511765\pi\)
−0.0369512 + 0.999317i \(0.511765\pi\)
\(390\) 0 0
\(391\) 3861.00 0.499384
\(392\) − 1176.00i − 0.151523i
\(393\) 0 0
\(394\) −4248.00 −0.543176
\(395\) 0 0
\(396\) 0 0
\(397\) 6685.00i 0.845115i 0.906336 + 0.422557i \(0.138867\pi\)
−0.906336 + 0.422557i \(0.861133\pi\)
\(398\) 770.000i 0.0969764i
\(399\) 0 0
\(400\) 0 0
\(401\) 4572.00 0.569364 0.284682 0.958622i \(-0.408112\pi\)
0.284682 + 0.958622i \(0.408112\pi\)
\(402\) 0 0
\(403\) 3901.00i 0.482190i
\(404\) −132.000 −0.0162556
\(405\) 0 0
\(406\) −1428.00 −0.174558
\(407\) − 942.000i − 0.114725i
\(408\) 0 0
\(409\) 25.0000 0.00302242 0.00151121 0.999999i \(-0.499519\pi\)
0.00151121 + 0.999999i \(0.499519\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4792.00i 0.573022i
\(413\) 168.000i 0.0200163i
\(414\) 0 0
\(415\) 0 0
\(416\) −1504.00 −0.177259
\(417\) 0 0
\(418\) − 192.000i − 0.0224666i
\(419\) −12453.0 −1.45195 −0.725977 0.687719i \(-0.758612\pi\)
−0.725977 + 0.687719i \(0.758612\pi\)
\(420\) 0 0
\(421\) 5048.00 0.584381 0.292191 0.956360i \(-0.405616\pi\)
0.292191 + 0.956360i \(0.405616\pi\)
\(422\) 6340.00i 0.731342i
\(423\) 0 0
\(424\) 4512.00 0.516797
\(425\) 0 0
\(426\) 0 0
\(427\) − 3220.00i − 0.364934i
\(428\) − 6168.00i − 0.696592i
\(429\) 0 0
\(430\) 0 0
\(431\) 5400.00 0.603501 0.301750 0.953387i \(-0.402429\pi\)
0.301750 + 0.953387i \(0.402429\pi\)
\(432\) 0 0
\(433\) − 6298.00i − 0.698990i −0.936938 0.349495i \(-0.886353\pi\)
0.936938 0.349495i \(-0.113647\pi\)
\(434\) 2324.00 0.257040
\(435\) 0 0
\(436\) −2224.00 −0.244290
\(437\) 3168.00i 0.346787i
\(438\) 0 0
\(439\) 6208.00 0.674924 0.337462 0.941339i \(-0.390432\pi\)
0.337462 + 0.941339i \(0.390432\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 3666.00i − 0.394511i
\(443\) − 3360.00i − 0.360358i −0.983634 0.180179i \(-0.942332\pi\)
0.983634 0.180179i \(-0.0576676\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −2776.00 −0.294725
\(447\) 0 0
\(448\) 896.000i 0.0944911i
\(449\) −14394.0 −1.51291 −0.756453 0.654048i \(-0.773069\pi\)
−0.756453 + 0.654048i \(0.773069\pi\)
\(450\) 0 0
\(451\) −324.000 −0.0338283
\(452\) − 6420.00i − 0.668078i
\(453\) 0 0
\(454\) −9288.00 −0.960149
\(455\) 0 0
\(456\) 0 0
\(457\) 916.000i 0.0937608i 0.998901 + 0.0468804i \(0.0149280\pi\)
−0.998901 + 0.0468804i \(0.985072\pi\)
\(458\) − 9472.00i − 0.966370i
\(459\) 0 0
\(460\) 0 0
\(461\) 8550.00 0.863803 0.431902 0.901921i \(-0.357843\pi\)
0.431902 + 0.901921i \(0.357843\pi\)
\(462\) 0 0
\(463\) 3734.00i 0.374803i 0.982283 + 0.187401i \(0.0600065\pi\)
−0.982283 + 0.187401i \(0.939993\pi\)
\(464\) −816.000 −0.0816419
\(465\) 0 0
\(466\) −5628.00 −0.559468
\(467\) − 9840.00i − 0.975034i −0.873113 0.487517i \(-0.837903\pi\)
0.873113 0.487517i \(-0.162097\pi\)
\(468\) 0 0
\(469\) 3752.00 0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 96.0000i 0.00936178i
\(473\) 897.000i 0.0871968i
\(474\) 0 0
\(475\) 0 0
\(476\) −2184.00 −0.210301
\(477\) 0 0
\(478\) 4404.00i 0.421411i
\(479\) 17280.0 1.64832 0.824158 0.566360i \(-0.191649\pi\)
0.824158 + 0.566360i \(0.191649\pi\)
\(480\) 0 0
\(481\) 14758.0 1.39897
\(482\) 6970.00i 0.658661i
\(483\) 0 0
\(484\) 5288.00 0.496619
\(485\) 0 0
\(486\) 0 0
\(487\) 4588.00i 0.426904i 0.976954 + 0.213452i \(0.0684707\pi\)
−0.976954 + 0.213452i \(0.931529\pi\)
\(488\) − 1840.00i − 0.170682i
\(489\) 0 0
\(490\) 0 0
\(491\) 636.000 0.0584568 0.0292284 0.999573i \(-0.490695\pi\)
0.0292284 + 0.999573i \(0.490695\pi\)
\(492\) 0 0
\(493\) − 1989.00i − 0.181704i
\(494\) 3008.00 0.273960
\(495\) 0 0
\(496\) 1328.00 0.120220
\(497\) − 1680.00i − 0.151626i
\(498\) 0 0
\(499\) 11716.0 1.05106 0.525531 0.850774i \(-0.323867\pi\)
0.525531 + 0.850774i \(0.323867\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 12690.0i − 1.12825i
\(503\) 4653.00i 0.412459i 0.978504 + 0.206230i \(0.0661194\pi\)
−0.978504 + 0.206230i \(0.933881\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 594.000 0.0521868
\(507\) 0 0
\(508\) 5336.00i 0.466037i
\(509\) −16479.0 −1.43501 −0.717504 0.696555i \(-0.754715\pi\)
−0.717504 + 0.696555i \(0.754715\pi\)
\(510\) 0 0
\(511\) 15484.0 1.34045
\(512\) 512.000i 0.0441942i
\(513\) 0 0
\(514\) 1050.00 0.0901041
\(515\) 0 0
\(516\) 0 0
\(517\) − 1593.00i − 0.135513i
\(518\) − 8792.00i − 0.745750i
\(519\) 0 0
\(520\) 0 0
\(521\) 3120.00 0.262360 0.131180 0.991359i \(-0.458123\pi\)
0.131180 + 0.991359i \(0.458123\pi\)
\(522\) 0 0
\(523\) 17645.0i 1.47526i 0.675204 + 0.737631i \(0.264056\pi\)
−0.675204 + 0.737631i \(0.735944\pi\)
\(524\) 11532.0 0.961408
\(525\) 0 0
\(526\) −10392.0 −0.861431
\(527\) 3237.00i 0.267563i
\(528\) 0 0
\(529\) 2366.00 0.194460
\(530\) 0 0
\(531\) 0 0
\(532\) − 1792.00i − 0.146040i
\(533\) − 5076.00i − 0.412507i
\(534\) 0 0
\(535\) 0 0
\(536\) 2144.00 0.172774
\(537\) 0 0
\(538\) 14958.0i 1.19867i
\(539\) 441.000 0.0352416
\(540\) 0 0
\(541\) −2182.00 −0.173404 −0.0867019 0.996234i \(-0.527633\pi\)
−0.0867019 + 0.996234i \(0.527633\pi\)
\(542\) − 1712.00i − 0.135677i
\(543\) 0 0
\(544\) −1248.00 −0.0983595
\(545\) 0 0
\(546\) 0 0
\(547\) 4033.00i 0.315244i 0.987499 + 0.157622i \(0.0503828\pi\)
−0.987499 + 0.157622i \(0.949617\pi\)
\(548\) 1128.00i 0.0879302i
\(549\) 0 0
\(550\) 0 0
\(551\) 1632.00 0.126181
\(552\) 0 0
\(553\) − 10346.0i − 0.795582i
\(554\) −14108.0 −1.08193
\(555\) 0 0
\(556\) −9976.00 −0.760929
\(557\) 960.000i 0.0730278i 0.999333 + 0.0365139i \(0.0116253\pi\)
−0.999333 + 0.0365139i \(0.988375\pi\)
\(558\) 0 0
\(559\) −14053.0 −1.06329
\(560\) 0 0
\(561\) 0 0
\(562\) 2028.00i 0.152217i
\(563\) − 23754.0i − 1.77817i −0.457739 0.889087i \(-0.651340\pi\)
0.457739 0.889087i \(-0.348660\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1984.00 −0.147339
\(567\) 0 0
\(568\) − 960.000i − 0.0709167i
\(569\) 22536.0 1.66038 0.830192 0.557478i \(-0.188231\pi\)
0.830192 + 0.557478i \(0.188231\pi\)
\(570\) 0 0
\(571\) 17726.0 1.29914 0.649571 0.760301i \(-0.274949\pi\)
0.649571 + 0.760301i \(0.274949\pi\)
\(572\) − 564.000i − 0.0412273i
\(573\) 0 0
\(574\) −3024.00 −0.219894
\(575\) 0 0
\(576\) 0 0
\(577\) − 17168.0i − 1.23867i −0.785126 0.619336i \(-0.787402\pi\)
0.785126 0.619336i \(-0.212598\pi\)
\(578\) 6784.00i 0.488196i
\(579\) 0 0
\(580\) 0 0
\(581\) 15204.0 1.08566
\(582\) 0 0
\(583\) 1692.00i 0.120198i
\(584\) 8848.00 0.626940
\(585\) 0 0
\(586\) 9900.00 0.697893
\(587\) − 7542.00i − 0.530309i −0.964206 0.265155i \(-0.914577\pi\)
0.964206 0.265155i \(-0.0854230\pi\)
\(588\) 0 0
\(589\) −2656.00 −0.185804
\(590\) 0 0
\(591\) 0 0
\(592\) − 5024.00i − 0.348792i
\(593\) − 15543.0i − 1.07635i −0.842834 0.538174i \(-0.819114\pi\)
0.842834 0.538174i \(-0.180886\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10380.0 0.713392
\(597\) 0 0
\(598\) 9306.00i 0.636372i
\(599\) −16026.0 −1.09316 −0.546581 0.837406i \(-0.684071\pi\)
−0.546581 + 0.837406i \(0.684071\pi\)
\(600\) 0 0
\(601\) 10469.0 0.710548 0.355274 0.934762i \(-0.384388\pi\)
0.355274 + 0.934762i \(0.384388\pi\)
\(602\) 8372.00i 0.566806i
\(603\) 0 0
\(604\) −4916.00 −0.331174
\(605\) 0 0
\(606\) 0 0
\(607\) 8074.00i 0.539891i 0.962876 + 0.269945i \(0.0870056\pi\)
−0.962876 + 0.269945i \(0.912994\pi\)
\(608\) − 1024.00i − 0.0683038i
\(609\) 0 0
\(610\) 0 0
\(611\) 24957.0 1.65246
\(612\) 0 0
\(613\) 26855.0i 1.76943i 0.466128 + 0.884717i \(0.345649\pi\)
−0.466128 + 0.884717i \(0.654351\pi\)
\(614\) −9554.00 −0.627961
\(615\) 0 0
\(616\) −336.000 −0.0219770
\(617\) − 24447.0i − 1.59514i −0.603229 0.797568i \(-0.706119\pi\)
0.603229 0.797568i \(-0.293881\pi\)
\(618\) 0 0
\(619\) −1850.00 −0.120126 −0.0600628 0.998195i \(-0.519130\pi\)
−0.0600628 + 0.998195i \(0.519130\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 15384.0i − 0.991708i
\(623\) − 1680.00i − 0.108038i
\(624\) 0 0
\(625\) 0 0
\(626\) 5864.00 0.374397
\(627\) 0 0
\(628\) − 6364.00i − 0.404381i
\(629\) 12246.0 0.776280
\(630\) 0 0
\(631\) 21728.0 1.37081 0.685403 0.728164i \(-0.259626\pi\)
0.685403 + 0.728164i \(0.259626\pi\)
\(632\) − 5912.00i − 0.372099i
\(633\) 0 0
\(634\) 16704.0 1.04637
\(635\) 0 0
\(636\) 0 0
\(637\) 6909.00i 0.429740i
\(638\) − 306.000i − 0.0189885i
\(639\) 0 0
\(640\) 0 0
\(641\) 23862.0 1.47035 0.735173 0.677879i \(-0.237101\pi\)
0.735173 + 0.677879i \(0.237101\pi\)
\(642\) 0 0
\(643\) 10523.0i 0.645391i 0.946503 + 0.322696i \(0.104589\pi\)
−0.946503 + 0.322696i \(0.895411\pi\)
\(644\) 5544.00 0.339230
\(645\) 0 0
\(646\) 2496.00 0.152018
\(647\) − 5484.00i − 0.333228i −0.986022 0.166614i \(-0.946717\pi\)
0.986022 0.166614i \(-0.0532833\pi\)
\(648\) 0 0
\(649\) −36.0000 −0.00217739
\(650\) 0 0
\(651\) 0 0
\(652\) 1828.00i 0.109801i
\(653\) − 26784.0i − 1.60511i −0.596576 0.802557i \(-0.703473\pi\)
0.596576 0.802557i \(-0.296527\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1728.00 −0.102846
\(657\) 0 0
\(658\) − 14868.0i − 0.880874i
\(659\) 12120.0 0.716431 0.358216 0.933639i \(-0.383385\pi\)
0.358216 + 0.933639i \(0.383385\pi\)
\(660\) 0 0
\(661\) −18226.0 −1.07248 −0.536240 0.844066i \(-0.680156\pi\)
−0.536240 + 0.844066i \(0.680156\pi\)
\(662\) − 6140.00i − 0.360480i
\(663\) 0 0
\(664\) 8688.00 0.507771
\(665\) 0 0
\(666\) 0 0
\(667\) 5049.00i 0.293101i
\(668\) − 4656.00i − 0.269680i
\(669\) 0 0
\(670\) 0 0
\(671\) 690.000 0.0396977
\(672\) 0 0
\(673\) − 11062.0i − 0.633594i −0.948493 0.316797i \(-0.897393\pi\)
0.948493 0.316797i \(-0.102607\pi\)
\(674\) −3344.00 −0.191107
\(675\) 0 0
\(676\) 48.0000 0.00273100
\(677\) 9348.00i 0.530684i 0.964154 + 0.265342i \(0.0854848\pi\)
−0.964154 + 0.265342i \(0.914515\pi\)
\(678\) 0 0
\(679\) 22988.0 1.29926
\(680\) 0 0
\(681\) 0 0
\(682\) 498.000i 0.0279610i
\(683\) 19248.0i 1.07834i 0.842198 + 0.539169i \(0.181261\pi\)
−0.842198 + 0.539169i \(0.818739\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 13720.0 0.763604
\(687\) 0 0
\(688\) 4784.00i 0.265099i
\(689\) −26508.0 −1.46571
\(690\) 0 0
\(691\) −17710.0 −0.974993 −0.487496 0.873125i \(-0.662090\pi\)
−0.487496 + 0.873125i \(0.662090\pi\)
\(692\) − 15768.0i − 0.866199i
\(693\) 0 0
\(694\) −10152.0 −0.555280
\(695\) 0 0
\(696\) 0 0
\(697\) − 4212.00i − 0.228897i
\(698\) − 17188.0i − 0.932056i
\(699\) 0 0
\(700\) 0 0
\(701\) −19437.0 −1.04725 −0.523627 0.851947i \(-0.675422\pi\)
−0.523627 + 0.851947i \(0.675422\pi\)
\(702\) 0 0
\(703\) 10048.0i 0.539072i
\(704\) −192.000 −0.0102788
\(705\) 0 0
\(706\) −25422.0 −1.35520
\(707\) − 462.000i − 0.0245761i
\(708\) 0 0
\(709\) 19516.0 1.03376 0.516882 0.856057i \(-0.327093\pi\)
0.516882 + 0.856057i \(0.327093\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 960.000i − 0.0505302i
\(713\) − 8217.00i − 0.431598i
\(714\) 0 0
\(715\) 0 0
\(716\) −4848.00 −0.253042
\(717\) 0 0
\(718\) 2928.00i 0.152189i
\(719\) −17358.0 −0.900340 −0.450170 0.892943i \(-0.648637\pi\)
−0.450170 + 0.892943i \(0.648637\pi\)
\(720\) 0 0
\(721\) −16772.0 −0.866327
\(722\) − 11670.0i − 0.601541i
\(723\) 0 0
\(724\) −9152.00 −0.469795
\(725\) 0 0
\(726\) 0 0
\(727\) − 24428.0i − 1.24620i −0.782144 0.623098i \(-0.785874\pi\)
0.782144 0.623098i \(-0.214126\pi\)
\(728\) − 5264.00i − 0.267990i
\(729\) 0 0
\(730\) 0 0
\(731\) −11661.0 −0.590010
\(732\) 0 0
\(733\) − 21418.0i − 1.07925i −0.841905 0.539626i \(-0.818566\pi\)
0.841905 0.539626i \(-0.181434\pi\)
\(734\) −15260.0 −0.767380
\(735\) 0 0
\(736\) 3168.00 0.158660
\(737\) 804.000i 0.0401842i
\(738\) 0 0
\(739\) 664.000 0.0330523 0.0165261 0.999863i \(-0.494739\pi\)
0.0165261 + 0.999863i \(0.494739\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 15792.0i 0.781324i
\(743\) − 34209.0i − 1.68911i −0.535471 0.844553i \(-0.679866\pi\)
0.535471 0.844553i \(-0.320134\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 7766.00 0.381144
\(747\) 0 0
\(748\) − 468.000i − 0.0228767i
\(749\) 21588.0 1.05315
\(750\) 0 0
\(751\) 6857.00 0.333176 0.166588 0.986027i \(-0.446725\pi\)
0.166588 + 0.986027i \(0.446725\pi\)
\(752\) − 8496.00i − 0.411991i
\(753\) 0 0
\(754\) 4794.00 0.231548
\(755\) 0 0
\(756\) 0 0
\(757\) 23719.0i 1.13881i 0.822056 + 0.569407i \(0.192827\pi\)
−0.822056 + 0.569407i \(0.807173\pi\)
\(758\) 27536.0i 1.31946i
\(759\) 0 0
\(760\) 0 0
\(761\) −14418.0 −0.686796 −0.343398 0.939190i \(-0.611578\pi\)
−0.343398 + 0.939190i \(0.611578\pi\)
\(762\) 0 0
\(763\) − 7784.00i − 0.369331i
\(764\) 7752.00 0.367091
\(765\) 0 0
\(766\) 28278.0 1.33385
\(767\) − 564.000i − 0.0265513i
\(768\) 0 0
\(769\) 4849.00 0.227385 0.113693 0.993516i \(-0.463732\pi\)
0.113693 + 0.993516i \(0.463732\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5992.00i 0.279348i
\(773\) − 36258.0i − 1.68708i −0.537070 0.843538i \(-0.680469\pi\)
0.537070 0.843538i \(-0.319531\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 13136.0 0.607674
\(777\) 0 0
\(778\) − 1134.00i − 0.0522569i
\(779\) 3456.00 0.158953
\(780\) 0 0
\(781\) 360.000 0.0164940
\(782\) 7722.00i 0.353118i
\(783\) 0 0
\(784\) 2352.00 0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) 18877.0i 0.855009i 0.904013 + 0.427505i \(0.140607\pi\)
−0.904013 + 0.427505i \(0.859393\pi\)
\(788\) − 8496.00i − 0.384083i
\(789\) 0 0
\(790\) 0 0
\(791\) 22470.0 1.01004
\(792\) 0 0
\(793\) 10810.0i 0.484079i
\(794\) −13370.0 −0.597586
\(795\) 0 0
\(796\) −1540.00 −0.0685727
\(797\) 16200.0i 0.719992i 0.932954 + 0.359996i \(0.117222\pi\)
−0.932954 + 0.359996i \(0.882778\pi\)
\(798\) 0 0
\(799\) 20709.0 0.916936
\(800\) 0 0
\(801\) 0 0
\(802\) 9144.00i 0.402601i
\(803\) 3318.00i 0.145815i
\(804\) 0 0
\(805\) 0 0
\(806\) −7802.00 −0.340960
\(807\) 0 0
\(808\) − 264.000i − 0.0114944i
\(809\) 26760.0 1.16296 0.581478 0.813562i \(-0.302475\pi\)
0.581478 + 0.813562i \(0.302475\pi\)
\(810\) 0 0
\(811\) −10510.0 −0.455063 −0.227531 0.973771i \(-0.573065\pi\)
−0.227531 + 0.973771i \(0.573065\pi\)
\(812\) − 2856.00i − 0.123431i
\(813\) 0 0
\(814\) 1884.00 0.0811231
\(815\) 0 0
\(816\) 0 0
\(817\) − 9568.00i − 0.409721i
\(818\) 50.0000i 0.00213717i
\(819\) 0 0
\(820\) 0 0
\(821\) −28230.0 −1.20004 −0.600021 0.799985i \(-0.704841\pi\)
−0.600021 + 0.799985i \(0.704841\pi\)
\(822\) 0 0
\(823\) − 39868.0i − 1.68859i −0.535877 0.844296i \(-0.680019\pi\)
0.535877 0.844296i \(-0.319981\pi\)
\(824\) −9584.00 −0.405187
\(825\) 0 0
\(826\) −336.000 −0.0141537
\(827\) − 32394.0i − 1.36209i −0.732241 0.681046i \(-0.761525\pi\)
0.732241 0.681046i \(-0.238475\pi\)
\(828\) 0 0
\(829\) −34820.0 −1.45880 −0.729402 0.684085i \(-0.760202\pi\)
−0.729402 + 0.684085i \(0.760202\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 3008.00i − 0.125341i
\(833\) 5733.00i 0.238459i
\(834\) 0 0
\(835\) 0 0
\(836\) 384.000 0.0158863
\(837\) 0 0
\(838\) − 24906.0i − 1.02669i
\(839\) 1146.00 0.0471565 0.0235783 0.999722i \(-0.492494\pi\)
0.0235783 + 0.999722i \(0.492494\pi\)
\(840\) 0 0
\(841\) −21788.0 −0.893354
\(842\) 10096.0i 0.413220i
\(843\) 0 0
\(844\) −12680.0 −0.517137
\(845\) 0 0
\(846\) 0 0
\(847\) 18508.0i 0.750817i
\(848\) 9024.00i 0.365431i
\(849\) 0 0
\(850\) 0 0
\(851\) −31086.0 −1.25219
\(852\) 0 0
\(853\) − 19393.0i − 0.778433i −0.921146 0.389217i \(-0.872746\pi\)
0.921146 0.389217i \(-0.127254\pi\)
\(854\) 6440.00 0.258047
\(855\) 0 0
\(856\) 12336.0 0.492565
\(857\) 8430.00i 0.336013i 0.985786 + 0.168007i \(0.0537330\pi\)
−0.985786 + 0.168007i \(0.946267\pi\)
\(858\) 0 0
\(859\) −15470.0 −0.614470 −0.307235 0.951634i \(-0.599404\pi\)
−0.307235 + 0.951634i \(0.599404\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 10800.0i 0.426740i
\(863\) 5871.00i 0.231577i 0.993274 + 0.115789i \(0.0369395\pi\)
−0.993274 + 0.115789i \(0.963060\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 12596.0 0.494260
\(867\) 0 0
\(868\) 4648.00i 0.181755i
\(869\) 2217.00 0.0865438
\(870\) 0 0
\(871\) −12596.0 −0.490011
\(872\) − 4448.00i − 0.172739i
\(873\) 0 0
\(874\) −6336.00 −0.245216
\(875\) 0 0
\(876\) 0 0
\(877\) 11299.0i 0.435051i 0.976055 + 0.217526i \(0.0697986\pi\)
−0.976055 + 0.217526i \(0.930201\pi\)
\(878\) 12416.0i 0.477243i
\(879\) 0 0
\(880\) 0 0
\(881\) 29682.0 1.13509 0.567544 0.823343i \(-0.307894\pi\)
0.567544 + 0.823343i \(0.307894\pi\)
\(882\) 0 0
\(883\) 40316.0i 1.53651i 0.640142 + 0.768257i \(0.278876\pi\)
−0.640142 + 0.768257i \(0.721124\pi\)
\(884\) 7332.00 0.278961
\(885\) 0 0
\(886\) 6720.00 0.254811
\(887\) 21945.0i 0.830711i 0.909659 + 0.415356i \(0.136343\pi\)
−0.909659 + 0.415356i \(0.863657\pi\)
\(888\) 0 0
\(889\) −18676.0 −0.704581
\(890\) 0 0
\(891\) 0 0
\(892\) − 5552.00i − 0.208402i
\(893\) 16992.0i 0.636748i
\(894\) 0 0
\(895\) 0 0
\(896\) −1792.00 −0.0668153
\(897\) 0 0
\(898\) − 28788.0i − 1.06979i
\(899\) −4233.00 −0.157039
\(900\) 0 0
\(901\) −21996.0 −0.813311
\(902\) − 648.000i − 0.0239202i
\(903\) 0 0
\(904\) 12840.0 0.472403
\(905\) 0 0
\(906\) 0 0
\(907\) − 24911.0i − 0.911969i −0.889988 0.455985i \(-0.849287\pi\)
0.889988 0.455985i \(-0.150713\pi\)
\(908\) − 18576.0i − 0.678928i
\(909\) 0 0
\(910\) 0 0
\(911\) 33264.0 1.20975 0.604877 0.796319i \(-0.293222\pi\)
0.604877 + 0.796319i \(0.293222\pi\)
\(912\) 0 0
\(913\) 3258.00i 0.118099i
\(914\) −1832.00 −0.0662989
\(915\) 0 0
\(916\) 18944.0 0.683327
\(917\) 40362.0i 1.45351i
\(918\) 0 0
\(919\) 23191.0 0.832427 0.416214 0.909267i \(-0.363357\pi\)
0.416214 + 0.909267i \(0.363357\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 17100.0i 0.610801i
\(923\) 5640.00i 0.201130i
\(924\) 0 0
\(925\) 0 0
\(926\) −7468.00 −0.265026
\(927\) 0 0
\(928\) − 1632.00i − 0.0577296i
\(929\) −2160.00 −0.0762834 −0.0381417 0.999272i \(-0.512144\pi\)
−0.0381417 + 0.999272i \(0.512144\pi\)
\(930\) 0 0
\(931\) −4704.00 −0.165593
\(932\) − 11256.0i − 0.395604i
\(933\) 0 0
\(934\) 19680.0 0.689453
\(935\) 0 0
\(936\) 0 0
\(937\) − 2066.00i − 0.0720312i −0.999351 0.0360156i \(-0.988533\pi\)
0.999351 0.0360156i \(-0.0114666\pi\)
\(938\) 7504.00i 0.261209i
\(939\) 0 0
\(940\) 0 0
\(941\) −22233.0 −0.770218 −0.385109 0.922871i \(-0.625836\pi\)
−0.385109 + 0.922871i \(0.625836\pi\)
\(942\) 0 0
\(943\) 10692.0i 0.369225i
\(944\) −192.000 −0.00661978
\(945\) 0 0
\(946\) −1794.00 −0.0616575
\(947\) 17754.0i 0.609216i 0.952478 + 0.304608i \(0.0985254\pi\)
−0.952478 + 0.304608i \(0.901475\pi\)
\(948\) 0 0
\(949\) −51982.0 −1.77809
\(950\) 0 0
\(951\) 0 0
\(952\) − 4368.00i − 0.148706i
\(953\) 33891.0i 1.15198i 0.817457 + 0.575990i \(0.195383\pi\)
−0.817457 + 0.575990i \(0.804617\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −8808.00 −0.297982
\(957\) 0 0
\(958\) 34560.0i 1.16554i
\(959\) −3948.00 −0.132938
\(960\) 0 0
\(961\) −22902.0 −0.768756
\(962\) 29516.0i 0.989225i
\(963\) 0 0
\(964\) −13940.0 −0.465744
\(965\) 0 0
\(966\) 0 0
\(967\) − 51074.0i − 1.69848i −0.528008 0.849239i \(-0.677061\pi\)
0.528008 0.849239i \(-0.322939\pi\)
\(968\) 10576.0i 0.351163i
\(969\) 0 0
\(970\) 0 0
\(971\) 20967.0 0.692959 0.346479 0.938058i \(-0.387377\pi\)
0.346479 + 0.938058i \(0.387377\pi\)
\(972\) 0 0
\(973\) − 34916.0i − 1.15042i
\(974\) −9176.00 −0.301867
\(975\) 0 0
\(976\) 3680.00 0.120691
\(977\) − 31749.0i − 1.03965i −0.854272 0.519826i \(-0.825997\pi\)
0.854272 0.519826i \(-0.174003\pi\)
\(978\) 0 0
\(979\) 360.000 0.0117525
\(980\) 0 0
\(981\) 0 0
\(982\) 1272.00i 0.0413352i
\(983\) 47325.0i 1.53554i 0.640727 + 0.767769i \(0.278633\pi\)
−0.640727 + 0.767769i \(0.721367\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 3978.00 0.128484
\(987\) 0 0
\(988\) 6016.00i 0.193719i
\(989\) 29601.0 0.951726
\(990\) 0 0
\(991\) 2363.00 0.0757449 0.0378724 0.999283i \(-0.487942\pi\)
0.0378724 + 0.999283i \(0.487942\pi\)
\(992\) 2656.00i 0.0850081i
\(993\) 0 0
\(994\) 3360.00 0.107216
\(995\) 0 0
\(996\) 0 0
\(997\) − 45569.0i − 1.44753i −0.690048 0.723764i \(-0.742411\pi\)
0.690048 0.723764i \(-0.257589\pi\)
\(998\) 23432.0i 0.743213i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1350.4.c.k.649.2 2
3.2 odd 2 1350.4.c.j.649.1 2
5.2 odd 4 270.4.a.f.1.1 1
5.3 odd 4 1350.4.a.r.1.1 1
5.4 even 2 inner 1350.4.c.k.649.1 2
15.2 even 4 270.4.a.j.1.1 yes 1
15.8 even 4 1350.4.a.e.1.1 1
15.14 odd 2 1350.4.c.j.649.2 2
20.7 even 4 2160.4.a.l.1.1 1
45.2 even 12 810.4.e.f.271.1 2
45.7 odd 12 810.4.e.n.271.1 2
45.22 odd 12 810.4.e.n.541.1 2
45.32 even 12 810.4.e.f.541.1 2
60.47 odd 4 2160.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.4.a.f.1.1 1 5.2 odd 4
270.4.a.j.1.1 yes 1 15.2 even 4
810.4.e.f.271.1 2 45.2 even 12
810.4.e.f.541.1 2 45.32 even 12
810.4.e.n.271.1 2 45.7 odd 12
810.4.e.n.541.1 2 45.22 odd 12
1350.4.a.e.1.1 1 15.8 even 4
1350.4.a.r.1.1 1 5.3 odd 4
1350.4.c.j.649.1 2 3.2 odd 2
1350.4.c.j.649.2 2 15.14 odd 2
1350.4.c.k.649.1 2 5.4 even 2 inner
1350.4.c.k.649.2 2 1.1 even 1 trivial
2160.4.a.b.1.1 1 60.47 odd 4
2160.4.a.l.1.1 1 20.7 even 4