Properties

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

$q$-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 \(-0.513091\pi\)
−0.0411152 + 0.999154i \(0.513091\pi\)
\(12\) 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 \(-0.910261\pi\)
0.960522 0.278203i \(-0.0897387\pi\)
\(18\) 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 \(0.148134\pi\)
−0.893653 + 0.448759i \(0.851866\pi\)
\(24\) 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 \(-0.447791\pi\)
0.163284 + 0.986579i \(0.447791\pi\)
\(30\) 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 \(-0.434055\pi\)
0.205692 + 0.978617i \(0.434055\pi\)
\(42\) 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 \(0.308252\pi\)
−0.566616 + 0.823982i \(0.691748\pi\)
\(48\) 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 \(-0.739118\pi\)
0.682525 0.730862i \(-0.260882\pi\)
\(54\) 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 \(-0.495786\pi\)
0.0132396 + 0.999912i \(0.495786\pi\)
\(60\) 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 \(-0.531978\pi\)
−0.100291 + 0.994958i \(0.531978\pi\)
\(72\) 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 \(-0.745014\pi\)
0.695944 0.718096i \(-0.254986\pi\)
\(84\) 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 \(-0.522766\pi\)
−0.0714605 + 0.997443i \(0.522766\pi\)
\(90\) 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 \(-0.505175\pi\)
−0.0162556 + 0.999868i \(0.505175\pi\)
\(102\) 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 \(-0.754699\pi\)
0.717467 0.696592i \(-0.245301\pi\)
\(108\) 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 \(-0.767117\pi\)
0.744091 0.668078i \(-0.232883\pi\)
\(114\) 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 \(-0.0887199\pi\)
0.961408 + 0.275127i \(0.0887199\pi\)
\(132\) 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 \(0.0280253\pi\)
−0.996127 + 0.0879302i \(0.971975\pi\)
\(138\) 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 \(-0.247158\pi\)
0.713392 + 0.700766i \(0.247158\pi\)
\(150\) 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 \(-0.913082\pi\)
0.962950 0.269680i \(-0.0869178\pi\)
\(168\) 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 \(-0.666556\pi\)
0.499700 0.866199i \(-0.333444\pi\)
\(174\) 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 \(-0.581431\pi\)
−0.253042 + 0.967455i \(0.581431\pi\)
\(180\) 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 \(-0.380354\pi\)
0.367091 + 0.930185i \(0.380354\pi\)
\(192\) 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 \(-0.874518\pi\)
0.923299 0.384083i \(-0.125482\pi\)
\(198\) 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 \(-0.762445\pi\)
0.734205 0.678928i \(-0.237555\pi\)
\(228\) 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 \(-0.870535\pi\)
0.918421 0.395604i \(-0.129465\pi\)
\(234\) 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 \(-0.596314\pi\)
−0.297982 + 0.954571i \(0.596314\pi\)
\(240\) 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 \(-0.206000\pi\)
0.797795 + 0.602929i \(0.206000\pi\)
\(252\) 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 \(0.0202943\pi\)
−0.997968 + 0.0637132i \(0.979706\pi\)
\(258\) 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 \(-0.791521\pi\)
0.793075 0.609124i \(-0.208479\pi\)
\(264\) 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 \(-0.821946\pi\)
−0.847589 + 0.530654i \(0.821946\pi\)
\(270\) 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 \(-0.534327\pi\)
−0.107634 + 0.994191i \(0.534327\pi\)
\(282\) 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 \(0.164277\pi\)
−0.869754 + 0.493485i \(0.835723\pi\)
\(294\) 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 \(-0.252629\pi\)
0.701243 + 0.712922i \(0.252629\pi\)
\(312\) 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 \(0.265126\pi\)
−0.672720 + 0.739898i \(0.734874\pi\)
\(318\) 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 \(-0.871561\pi\)
0.919691 0.392643i \(-0.128439\pi\)
\(348\) 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 \(-0.592281\pi\)
0.285866 0.958269i \(-0.407719\pi\)
\(354\) 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 \(-0.534321\pi\)
−0.107614 + 0.994193i \(0.534321\pi\)
\(360\) 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 \(0.392173\pi\)
−0.332307 + 0.943171i \(0.607827\pi\)
\(384\) 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 \(-0.488235\pi\)
0.0369512 + 0.999317i \(0.488235\pi\)
\(390\) 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 \(-0.591888\pi\)
−0.284682 + 0.958622i \(0.591888\pi\)
\(402\) 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 \(-0.241388\pi\)
0.725977 + 0.687719i \(0.241388\pi\)
\(420\) 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 \(-0.597571\pi\)
−0.301750 + 0.953387i \(0.597571\pi\)
\(432\) 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 \(0.0576676\pi\)
−0.983634 + 0.180179i \(0.942332\pi\)
\(444\) 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 \(-0.226931\pi\)
0.756453 + 0.654048i \(0.226931\pi\)
\(450\) 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 \(-0.642157\pi\)
−0.431902 + 0.901921i \(0.642157\pi\)
\(462\) 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 \(0.162097\pi\)
−0.873113 + 0.487517i \(0.837903\pi\)
\(468\) 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 \(-0.808351\pi\)
−0.824158 + 0.566360i \(0.808351\pi\)
\(480\) 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 \(-0.509305\pi\)
−0.0292284 + 0.999573i \(0.509305\pi\)
\(492\) 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 \(-0.933881\pi\)
0.978504 0.206230i \(-0.0661194\pi\)
\(504\) 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 \(-0.245285\pi\)
0.717504 + 0.696555i \(0.245285\pi\)
\(510\) 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 \(-0.541877\pi\)
−0.131180 + 0.991359i \(0.541877\pi\)
\(522\) 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 \(-0.988375\pi\)
0.999333 0.0365139i \(-0.0116253\pi\)
\(558\) 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 \(0.348660\pi\)
−0.457739 + 0.889087i \(0.651340\pi\)
\(564\) 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 \(-0.811769\pi\)
−0.830192 + 0.557478i \(0.811769\pi\)
\(570\) 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 \(0.0854230\pi\)
−0.964206 + 0.265155i \(0.914577\pi\)
\(588\) 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 \(0.180886\pi\)
−0.842834 + 0.538174i \(0.819114\pi\)
\(594\) 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 \(-0.315929\pi\)
0.546581 + 0.837406i \(0.315929\pi\)
\(600\) 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 \(0.293881\pi\)
−0.603229 + 0.797568i \(0.706119\pi\)
\(618\) 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 \(-0.762899\pi\)
−0.735173 + 0.677879i \(0.762899\pi\)
\(642\) 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 \(0.0532833\pi\)
−0.986022 + 0.166614i \(0.946717\pi\)
\(648\) 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 \(0.296527\pi\)
−0.596576 + 0.802557i \(0.703473\pi\)
\(654\) 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 \(-0.616615\pi\)
−0.358216 + 0.933639i \(0.616615\pi\)
\(660\) 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 \(-0.914515\pi\)
0.964154 0.265342i \(-0.0854848\pi\)
\(678\) 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 \(-0.818739\pi\)
0.842198 0.539169i \(-0.181261\pi\)
\(684\) 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 \(-0.324578\pi\)
0.523627 + 0.851947i \(0.324578\pi\)
\(702\) 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 \(-0.351363\pi\)
0.450170 + 0.892943i \(0.351363\pi\)
\(720\) 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 \(0.320134\pi\)
−0.535471 + 0.844553i \(0.679866\pi\)
\(744\) 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 \(-0.388422\pi\)
0.343398 + 0.939190i \(0.388422\pi\)
\(762\) 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 \(0.319531\pi\)
−0.537070 + 0.843538i \(0.680469\pi\)
\(774\) 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 \(-0.882778\pi\)
0.932954 0.359996i \(-0.117222\pi\)
\(798\) 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 \(-0.697525\pi\)
−0.581478 + 0.813562i \(0.697525\pi\)
\(810\) 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 \(-0.295159\pi\)
0.600021 + 0.799985i \(0.295159\pi\)
\(822\) 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 \(0.238475\pi\)
−0.732241 + 0.681046i \(0.761525\pi\)
\(828\) 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 \(-0.507506\pi\)
−0.0235783 + 0.999722i \(0.507506\pi\)
\(840\) 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 \(-0.946267\pi\)
0.985786 0.168007i \(-0.0537330\pi\)
\(858\) 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 \(-0.963060\pi\)
0.993274 0.115789i \(-0.0369395\pi\)
\(864\) 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 \(-0.692106\pi\)
−0.567544 + 0.823343i \(0.692106\pi\)
\(882\) 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 \(-0.863657\pi\)
0.909659 0.415356i \(-0.136343\pi\)
\(888\) 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 \(-0.706778\pi\)
−0.604877 + 0.796319i \(0.706778\pi\)
\(912\) 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 \(-0.487856\pi\)
0.0381417 + 0.999272i \(0.487856\pi\)
\(930\) 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 \(-0.374164\pi\)
0.385109 + 0.922871i \(0.374164\pi\)
\(942\) 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 \(-0.901475\pi\)
0.952478 0.304608i \(-0.0985254\pi\)
\(948\) 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 \(-0.804617\pi\)
0.817457 0.575990i \(-0.195383\pi\)
\(954\) 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 \(-0.612623\pi\)
−0.346479 + 0.938058i \(0.612623\pi\)
\(972\) 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 \(0.174003\pi\)
−0.854272 + 0.519826i \(0.825997\pi\)
\(978\) 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 \(-0.721367\pi\)
0.640727 0.767769i \(-0.278633\pi\)
\(984\) 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.j.649.1 2
3.2 odd 2 1350.4.c.k.649.2 2
5.2 odd 4 270.4.a.j.1.1 yes 1
5.3 odd 4 1350.4.a.e.1.1 1
5.4 even 2 inner 1350.4.c.j.649.2 2
15.2 even 4 270.4.a.f.1.1 1
15.8 even 4 1350.4.a.r.1.1 1
15.14 odd 2 1350.4.c.k.649.1 2
20.7 even 4 2160.4.a.b.1.1 1
45.2 even 12 810.4.e.n.271.1 2
45.7 odd 12 810.4.e.f.271.1 2
45.22 odd 12 810.4.e.f.541.1 2
45.32 even 12 810.4.e.n.541.1 2
60.47 odd 4 2160.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.4.a.f.1.1 1 15.2 even 4
270.4.a.j.1.1 yes 1 5.2 odd 4
810.4.e.f.271.1 2 45.7 odd 12
810.4.e.f.541.1 2 45.22 odd 12
810.4.e.n.271.1 2 45.2 even 12
810.4.e.n.541.1 2 45.32 even 12
1350.4.a.e.1.1 1 5.3 odd 4
1350.4.a.r.1.1 1 15.8 even 4
1350.4.c.j.649.1 2 1.1 even 1 trivial
1350.4.c.j.649.2 2 5.4 even 2 inner
1350.4.c.k.649.1 2 15.14 odd 2
1350.4.c.k.649.2 2 3.2 odd 2
2160.4.a.b.1.1 1 20.7 even 4
2160.4.a.l.1.1 1 60.47 odd 4