Properties

Label 1350.4.c.q.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: \(\Z[a_1, \ldots, a_{17}]\)
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.q.649.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} -4.00000 q^{4} -4.00000i q^{7} -8.00000i q^{8} +O(q^{10})\) \(q+2.00000i q^{2} -4.00000 q^{4} -4.00000i q^{7} -8.00000i q^{8} +42.0000 q^{11} -20.0000i q^{13} +8.00000 q^{14} +16.0000 q^{16} +93.0000i q^{17} -59.0000 q^{19} +84.0000i q^{22} -9.00000i q^{23} +40.0000 q^{26} +16.0000i q^{28} -120.000 q^{29} +47.0000 q^{31} +32.0000i q^{32} -186.000 q^{34} -262.000i q^{37} -118.000i q^{38} +126.000 q^{41} +178.000i q^{43} -168.000 q^{44} +18.0000 q^{46} +144.000i q^{47} +327.000 q^{49} +80.0000i q^{52} -741.000i q^{53} -32.0000 q^{56} -240.000i q^{58} +444.000 q^{59} +221.000 q^{61} +94.0000i q^{62} -64.0000 q^{64} -538.000i q^{67} -372.000i q^{68} +690.000 q^{71} +1126.00i q^{73} +524.000 q^{74} +236.000 q^{76} -168.000i q^{77} -665.000 q^{79} +252.000i q^{82} -75.0000i q^{83} -356.000 q^{86} -336.000i q^{88} +1086.00 q^{89} -80.0000 q^{91} +36.0000i q^{92} -288.000 q^{94} +1544.00i q^{97} +654.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 84 q^{11} + 16 q^{14} + 32 q^{16} - 118 q^{19} + 80 q^{26} - 240 q^{29} + 94 q^{31} - 372 q^{34} + 252 q^{41} - 336 q^{44} + 36 q^{46} + 654 q^{49} - 64 q^{56} + 888 q^{59} + 442 q^{61} - 128 q^{64} + 1380 q^{71} + 1048 q^{74} + 472 q^{76} - 1330 q^{79} - 712 q^{86} + 2172 q^{89} - 160 q^{91} - 576 q^{94}+O(q^{100}) \) 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
\(7\) − 4.00000i − 0.215980i −0.994152 0.107990i \(-0.965559\pi\)
0.994152 0.107990i \(-0.0344414\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 42.0000 1.15123 0.575613 0.817723i \(-0.304764\pi\)
0.575613 + 0.817723i \(0.304764\pi\)
\(12\) 0 0
\(13\) − 20.0000i − 0.426692i −0.976977 0.213346i \(-0.931564\pi\)
0.976977 0.213346i \(-0.0684362\pi\)
\(14\) 8.00000 0.152721
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 93.0000i 1.32681i 0.748259 + 0.663406i \(0.230890\pi\)
−0.748259 + 0.663406i \(0.769110\pi\)
\(18\) 0 0
\(19\) −59.0000 −0.712396 −0.356198 0.934410i \(-0.615927\pi\)
−0.356198 + 0.934410i \(0.615927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 84.0000i 0.814039i
\(23\) − 9.00000i − 0.0815926i −0.999167 0.0407963i \(-0.987011\pi\)
0.999167 0.0407963i \(-0.0129895\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 40.0000 0.301717
\(27\) 0 0
\(28\) 16.0000i 0.107990i
\(29\) −120.000 −0.768395 −0.384197 0.923251i \(-0.625522\pi\)
−0.384197 + 0.923251i \(0.625522\pi\)
\(30\) 0 0
\(31\) 47.0000 0.272305 0.136152 0.990688i \(-0.456526\pi\)
0.136152 + 0.990688i \(0.456526\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 0 0
\(34\) −186.000 −0.938198
\(35\) 0 0
\(36\) 0 0
\(37\) − 262.000i − 1.16412i −0.813145 0.582061i \(-0.802246\pi\)
0.813145 0.582061i \(-0.197754\pi\)
\(38\) − 118.000i − 0.503740i
\(39\) 0 0
\(40\) 0 0
\(41\) 126.000 0.479949 0.239974 0.970779i \(-0.422861\pi\)
0.239974 + 0.970779i \(0.422861\pi\)
\(42\) 0 0
\(43\) 178.000i 0.631273i 0.948880 + 0.315637i \(0.102218\pi\)
−0.948880 + 0.315637i \(0.897782\pi\)
\(44\) −168.000 −0.575613
\(45\) 0 0
\(46\) 18.0000 0.0576947
\(47\) 144.000i 0.446906i 0.974715 + 0.223453i \(0.0717328\pi\)
−0.974715 + 0.223453i \(0.928267\pi\)
\(48\) 0 0
\(49\) 327.000 0.953353
\(50\) 0 0
\(51\) 0 0
\(52\) 80.0000i 0.213346i
\(53\) − 741.000i − 1.92046i −0.279217 0.960228i \(-0.590075\pi\)
0.279217 0.960228i \(-0.409925\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −32.0000 −0.0763604
\(57\) 0 0
\(58\) − 240.000i − 0.543337i
\(59\) 444.000 0.979727 0.489863 0.871799i \(-0.337047\pi\)
0.489863 + 0.871799i \(0.337047\pi\)
\(60\) 0 0
\(61\) 221.000 0.463871 0.231936 0.972731i \(-0.425494\pi\)
0.231936 + 0.972731i \(0.425494\pi\)
\(62\) 94.0000i 0.192549i
\(63\) 0 0
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 538.000i − 0.981002i −0.871441 0.490501i \(-0.836814\pi\)
0.871441 0.490501i \(-0.163186\pi\)
\(68\) − 372.000i − 0.663406i
\(69\) 0 0
\(70\) 0 0
\(71\) 690.000 1.15335 0.576676 0.816973i \(-0.304350\pi\)
0.576676 + 0.816973i \(0.304350\pi\)
\(72\) 0 0
\(73\) 1126.00i 1.80532i 0.430355 + 0.902660i \(0.358388\pi\)
−0.430355 + 0.902660i \(0.641612\pi\)
\(74\) 524.000 0.823159
\(75\) 0 0
\(76\) 236.000 0.356198
\(77\) − 168.000i − 0.248641i
\(78\) 0 0
\(79\) −665.000 −0.947068 −0.473534 0.880776i \(-0.657022\pi\)
−0.473534 + 0.880776i \(0.657022\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 252.000i 0.339375i
\(83\) − 75.0000i − 0.0991846i −0.998770 0.0495923i \(-0.984208\pi\)
0.998770 0.0495923i \(-0.0157922\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −356.000 −0.446378
\(87\) 0 0
\(88\) − 336.000i − 0.407020i
\(89\) 1086.00 1.29344 0.646718 0.762729i \(-0.276141\pi\)
0.646718 + 0.762729i \(0.276141\pi\)
\(90\) 0 0
\(91\) −80.0000 −0.0921569
\(92\) 36.0000i 0.0407963i
\(93\) 0 0
\(94\) −288.000 −0.316010
\(95\) 0 0
\(96\) 0 0
\(97\) 1544.00i 1.61618i 0.589059 + 0.808090i \(0.299499\pi\)
−0.589059 + 0.808090i \(0.700501\pi\)
\(98\) 654.000i 0.674122i
\(99\) 0 0
\(100\) 0 0
\(101\) −132.000 −0.130044 −0.0650222 0.997884i \(-0.520712\pi\)
−0.0650222 + 0.997884i \(0.520712\pi\)
\(102\) 0 0
\(103\) 892.000i 0.853314i 0.904413 + 0.426657i \(0.140309\pi\)
−0.904413 + 0.426657i \(0.859691\pi\)
\(104\) −160.000 −0.150859
\(105\) 0 0
\(106\) 1482.00 1.35797
\(107\) − 1140.00i − 1.02998i −0.857196 0.514990i \(-0.827795\pi\)
0.857196 0.514990i \(-0.172205\pi\)
\(108\) 0 0
\(109\) 1735.00 1.52461 0.762307 0.647216i \(-0.224067\pi\)
0.762307 + 0.647216i \(0.224067\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 64.0000i − 0.0539949i
\(113\) 1434.00i 1.19380i 0.802316 + 0.596900i \(0.203601\pi\)
−0.802316 + 0.596900i \(0.796399\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 480.000 0.384197
\(117\) 0 0
\(118\) 888.000i 0.692771i
\(119\) 372.000 0.286565
\(120\) 0 0
\(121\) 433.000 0.325319
\(122\) 442.000i 0.328007i
\(123\) 0 0
\(124\) −188.000 −0.136152
\(125\) 0 0
\(126\) 0 0
\(127\) 686.000i 0.479312i 0.970858 + 0.239656i \(0.0770347\pi\)
−0.970858 + 0.239656i \(0.922965\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −114.000 −0.0760323 −0.0380161 0.999277i \(-0.512104\pi\)
−0.0380161 + 0.999277i \(0.512104\pi\)
\(132\) 0 0
\(133\) 236.000i 0.153863i
\(134\) 1076.00 0.693673
\(135\) 0 0
\(136\) 744.000 0.469099
\(137\) 159.000i 0.0991554i 0.998770 + 0.0495777i \(0.0157875\pi\)
−0.998770 + 0.0495777i \(0.984212\pi\)
\(138\) 0 0
\(139\) −2276.00 −1.38883 −0.694417 0.719573i \(-0.744337\pi\)
−0.694417 + 0.719573i \(0.744337\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1380.00i 0.815542i
\(143\) − 840.000i − 0.491219i
\(144\) 0 0
\(145\) 0 0
\(146\) −2252.00 −1.27655
\(147\) 0 0
\(148\) 1048.00i 0.582061i
\(149\) 1398.00 0.768648 0.384324 0.923198i \(-0.374434\pi\)
0.384324 + 0.923198i \(0.374434\pi\)
\(150\) 0 0
\(151\) 2624.00 1.41416 0.707080 0.707134i \(-0.250012\pi\)
0.707080 + 0.707134i \(0.250012\pi\)
\(152\) 472.000i 0.251870i
\(153\) 0 0
\(154\) 336.000 0.175816
\(155\) 0 0
\(156\) 0 0
\(157\) − 394.000i − 0.200284i −0.994973 0.100142i \(-0.968070\pi\)
0.994973 0.100142i \(-0.0319297\pi\)
\(158\) − 1330.00i − 0.669678i
\(159\) 0 0
\(160\) 0 0
\(161\) −36.0000 −0.0176223
\(162\) 0 0
\(163\) 3346.00i 1.60785i 0.594733 + 0.803923i \(0.297258\pi\)
−0.594733 + 0.803923i \(0.702742\pi\)
\(164\) −504.000 −0.239974
\(165\) 0 0
\(166\) 150.000 0.0701341
\(167\) − 1491.00i − 0.690881i −0.938441 0.345440i \(-0.887730\pi\)
0.938441 0.345440i \(-0.112270\pi\)
\(168\) 0 0
\(169\) 1797.00 0.817934
\(170\) 0 0
\(171\) 0 0
\(172\) − 712.000i − 0.315637i
\(173\) − 2403.00i − 1.05605i −0.849229 0.528025i \(-0.822933\pi\)
0.849229 0.528025i \(-0.177067\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 672.000 0.287806
\(177\) 0 0
\(178\) 2172.00i 0.914597i
\(179\) 2640.00 1.10236 0.551181 0.834386i \(-0.314177\pi\)
0.551181 + 0.834386i \(0.314177\pi\)
\(180\) 0 0
\(181\) 1073.00 0.440638 0.220319 0.975428i \(-0.429290\pi\)
0.220319 + 0.975428i \(0.429290\pi\)
\(182\) − 160.000i − 0.0651648i
\(183\) 0 0
\(184\) −72.0000 −0.0288473
\(185\) 0 0
\(186\) 0 0
\(187\) 3906.00i 1.52746i
\(188\) − 576.000i − 0.223453i
\(189\) 0 0
\(190\) 0 0
\(191\) 1470.00 0.556887 0.278444 0.960453i \(-0.410181\pi\)
0.278444 + 0.960453i \(0.410181\pi\)
\(192\) 0 0
\(193\) 4720.00i 1.76038i 0.474623 + 0.880189i \(0.342584\pi\)
−0.474623 + 0.880189i \(0.657416\pi\)
\(194\) −3088.00 −1.14281
\(195\) 0 0
\(196\) −1308.00 −0.476676
\(197\) − 765.000i − 0.276670i −0.990385 0.138335i \(-0.955825\pi\)
0.990385 0.138335i \(-0.0441751\pi\)
\(198\) 0 0
\(199\) −668.000 −0.237956 −0.118978 0.992897i \(-0.537962\pi\)
−0.118978 + 0.992897i \(0.537962\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 264.000i − 0.0919553i
\(203\) 480.000i 0.165958i
\(204\) 0 0
\(205\) 0 0
\(206\) −1784.00 −0.603384
\(207\) 0 0
\(208\) − 320.000i − 0.106673i
\(209\) −2478.00 −0.820128
\(210\) 0 0
\(211\) 4601.00 1.50117 0.750583 0.660777i \(-0.229773\pi\)
0.750583 + 0.660777i \(0.229773\pi\)
\(212\) 2964.00i 0.960228i
\(213\) 0 0
\(214\) 2280.00 0.728307
\(215\) 0 0
\(216\) 0 0
\(217\) − 188.000i − 0.0588123i
\(218\) 3470.00i 1.07806i
\(219\) 0 0
\(220\) 0 0
\(221\) 1860.00 0.566141
\(222\) 0 0
\(223\) 2158.00i 0.648029i 0.946052 + 0.324014i \(0.105033\pi\)
−0.946052 + 0.324014i \(0.894967\pi\)
\(224\) 128.000 0.0381802
\(225\) 0 0
\(226\) −2868.00 −0.844144
\(227\) 3123.00i 0.913131i 0.889690 + 0.456566i \(0.150921\pi\)
−0.889690 + 0.456566i \(0.849079\pi\)
\(228\) 0 0
\(229\) −2027.00 −0.584925 −0.292463 0.956277i \(-0.594475\pi\)
−0.292463 + 0.956277i \(0.594475\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 960.000i 0.271668i
\(233\) 438.000i 0.123152i 0.998102 + 0.0615758i \(0.0196126\pi\)
−0.998102 + 0.0615758i \(0.980387\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1776.00 −0.489863
\(237\) 0 0
\(238\) 744.000i 0.202632i
\(239\) −6414.00 −1.73593 −0.867965 0.496626i \(-0.834572\pi\)
−0.867965 + 0.496626i \(0.834572\pi\)
\(240\) 0 0
\(241\) 3431.00 0.917055 0.458527 0.888680i \(-0.348377\pi\)
0.458527 + 0.888680i \(0.348377\pi\)
\(242\) 866.000i 0.230035i
\(243\) 0 0
\(244\) −884.000 −0.231936
\(245\) 0 0
\(246\) 0 0
\(247\) 1180.00i 0.303974i
\(248\) − 376.000i − 0.0962743i
\(249\) 0 0
\(250\) 0 0
\(251\) 7308.00 1.83776 0.918878 0.394541i \(-0.129096\pi\)
0.918878 + 0.394541i \(0.129096\pi\)
\(252\) 0 0
\(253\) − 378.000i − 0.0939314i
\(254\) −1372.00 −0.338925
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 3729.00i − 0.905092i −0.891741 0.452546i \(-0.850516\pi\)
0.891741 0.452546i \(-0.149484\pi\)
\(258\) 0 0
\(259\) −1048.00 −0.251427
\(260\) 0 0
\(261\) 0 0
\(262\) − 228.000i − 0.0537629i
\(263\) 1956.00i 0.458601i 0.973356 + 0.229301i \(0.0736439\pi\)
−0.973356 + 0.229301i \(0.926356\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −472.000 −0.108798
\(267\) 0 0
\(268\) 2152.00i 0.490501i
\(269\) −990.000 −0.224392 −0.112196 0.993686i \(-0.535788\pi\)
−0.112196 + 0.993686i \(0.535788\pi\)
\(270\) 0 0
\(271\) 8495.00 1.90419 0.952093 0.305808i \(-0.0989266\pi\)
0.952093 + 0.305808i \(0.0989266\pi\)
\(272\) 1488.00i 0.331703i
\(273\) 0 0
\(274\) −318.000 −0.0701134
\(275\) 0 0
\(276\) 0 0
\(277\) − 1366.00i − 0.296300i −0.988965 0.148150i \(-0.952668\pi\)
0.988965 0.148150i \(-0.0473318\pi\)
\(278\) − 4552.00i − 0.982053i
\(279\) 0 0
\(280\) 0 0
\(281\) 5520.00 1.17187 0.585935 0.810358i \(-0.300727\pi\)
0.585935 + 0.810358i \(0.300727\pi\)
\(282\) 0 0
\(283\) − 5438.00i − 1.14225i −0.820865 0.571123i \(-0.806508\pi\)
0.820865 0.571123i \(-0.193492\pi\)
\(284\) −2760.00 −0.576676
\(285\) 0 0
\(286\) 1680.00 0.347344
\(287\) − 504.000i − 0.103659i
\(288\) 0 0
\(289\) −3736.00 −0.760432
\(290\) 0 0
\(291\) 0 0
\(292\) − 4504.00i − 0.902660i
\(293\) 8253.00i 1.64555i 0.568369 + 0.822774i \(0.307575\pi\)
−0.568369 + 0.822774i \(0.692425\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2096.00 −0.411579
\(297\) 0 0
\(298\) 2796.00i 0.543517i
\(299\) −180.000 −0.0348149
\(300\) 0 0
\(301\) 712.000 0.136342
\(302\) 5248.00i 0.999962i
\(303\) 0 0
\(304\) −944.000 −0.178099
\(305\) 0 0
\(306\) 0 0
\(307\) 9290.00i 1.72706i 0.504295 + 0.863531i \(0.331752\pi\)
−0.504295 + 0.863531i \(0.668248\pi\)
\(308\) 672.000i 0.124321i
\(309\) 0 0
\(310\) 0 0
\(311\) −8112.00 −1.47907 −0.739533 0.673121i \(-0.764953\pi\)
−0.739533 + 0.673121i \(0.764953\pi\)
\(312\) 0 0
\(313\) 7900.00i 1.42663i 0.700845 + 0.713314i \(0.252807\pi\)
−0.700845 + 0.713314i \(0.747193\pi\)
\(314\) 788.000 0.141622
\(315\) 0 0
\(316\) 2660.00 0.473534
\(317\) − 4419.00i − 0.782952i −0.920188 0.391476i \(-0.871965\pi\)
0.920188 0.391476i \(-0.128035\pi\)
\(318\) 0 0
\(319\) −5040.00 −0.884595
\(320\) 0 0
\(321\) 0 0
\(322\) − 72.0000i − 0.0124609i
\(323\) − 5487.00i − 0.945216i
\(324\) 0 0
\(325\) 0 0
\(326\) −6692.00 −1.13692
\(327\) 0 0
\(328\) − 1008.00i − 0.169687i
\(329\) 576.000 0.0965225
\(330\) 0 0
\(331\) −8200.00 −1.36167 −0.680835 0.732437i \(-0.738383\pi\)
−0.680835 + 0.732437i \(0.738383\pi\)
\(332\) 300.000i 0.0495923i
\(333\) 0 0
\(334\) 2982.00 0.488526
\(335\) 0 0
\(336\) 0 0
\(337\) − 9556.00i − 1.54465i −0.635225 0.772327i \(-0.719093\pi\)
0.635225 0.772327i \(-0.280907\pi\)
\(338\) 3594.00i 0.578366i
\(339\) 0 0
\(340\) 0 0
\(341\) 1974.00 0.313484
\(342\) 0 0
\(343\) − 2680.00i − 0.421885i
\(344\) 1424.00 0.223189
\(345\) 0 0
\(346\) 4806.00 0.746740
\(347\) − 10116.0i − 1.56500i −0.622650 0.782500i \(-0.713944\pi\)
0.622650 0.782500i \(-0.286056\pi\)
\(348\) 0 0
\(349\) 6751.00 1.03545 0.517726 0.855546i \(-0.326779\pi\)
0.517726 + 0.855546i \(0.326779\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1344.00i 0.203510i
\(353\) 4062.00i 0.612460i 0.951958 + 0.306230i \(0.0990677\pi\)
−0.951958 + 0.306230i \(0.900932\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −4344.00 −0.646718
\(357\) 0 0
\(358\) 5280.00i 0.779488i
\(359\) 8778.00 1.29049 0.645244 0.763977i \(-0.276756\pi\)
0.645244 + 0.763977i \(0.276756\pi\)
\(360\) 0 0
\(361\) −3378.00 −0.492492
\(362\) 2146.00i 0.311578i
\(363\) 0 0
\(364\) 320.000 0.0460785
\(365\) 0 0
\(366\) 0 0
\(367\) 956.000i 0.135975i 0.997686 + 0.0679875i \(0.0216578\pi\)
−0.997686 + 0.0679875i \(0.978342\pi\)
\(368\) − 144.000i − 0.0203981i
\(369\) 0 0
\(370\) 0 0
\(371\) −2964.00 −0.414780
\(372\) 0 0
\(373\) − 2300.00i − 0.319275i −0.987176 0.159637i \(-0.948968\pi\)
0.987176 0.159637i \(-0.0510325\pi\)
\(374\) −7812.00 −1.08008
\(375\) 0 0
\(376\) 1152.00 0.158005
\(377\) 2400.00i 0.327868i
\(378\) 0 0
\(379\) −29.0000 −0.00393042 −0.00196521 0.999998i \(-0.500626\pi\)
−0.00196521 + 0.999998i \(0.500626\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 2940.00i 0.393779i
\(383\) 8127.00i 1.08426i 0.840296 + 0.542128i \(0.182381\pi\)
−0.840296 + 0.542128i \(0.817619\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −9440.00 −1.24478
\(387\) 0 0
\(388\) − 6176.00i − 0.808090i
\(389\) −7938.00 −1.03463 −0.517317 0.855794i \(-0.673069\pi\)
−0.517317 + 0.855794i \(0.673069\pi\)
\(390\) 0 0
\(391\) 837.000 0.108258
\(392\) − 2616.00i − 0.337061i
\(393\) 0 0
\(394\) 1530.00 0.195635
\(395\) 0 0
\(396\) 0 0
\(397\) 272.000i 0.0343861i 0.999852 + 0.0171931i \(0.00547299\pi\)
−0.999852 + 0.0171931i \(0.994527\pi\)
\(398\) − 1336.00i − 0.168260i
\(399\) 0 0
\(400\) 0 0
\(401\) −4554.00 −0.567122 −0.283561 0.958954i \(-0.591516\pi\)
−0.283561 + 0.958954i \(0.591516\pi\)
\(402\) 0 0
\(403\) − 940.000i − 0.116190i
\(404\) 528.000 0.0650222
\(405\) 0 0
\(406\) −960.000 −0.117350
\(407\) − 11004.0i − 1.34017i
\(408\) 0 0
\(409\) −1001.00 −0.121018 −0.0605089 0.998168i \(-0.519272\pi\)
−0.0605089 + 0.998168i \(0.519272\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 3568.00i − 0.426657i
\(413\) − 1776.00i − 0.211601i
\(414\) 0 0
\(415\) 0 0
\(416\) 640.000 0.0754293
\(417\) 0 0
\(418\) − 4956.00i − 0.579918i
\(419\) −1794.00 −0.209171 −0.104585 0.994516i \(-0.533352\pi\)
−0.104585 + 0.994516i \(0.533352\pi\)
\(420\) 0 0
\(421\) −16129.0 −1.86717 −0.933586 0.358354i \(-0.883338\pi\)
−0.933586 + 0.358354i \(0.883338\pi\)
\(422\) 9202.00i 1.06148i
\(423\) 0 0
\(424\) −5928.00 −0.678984
\(425\) 0 0
\(426\) 0 0
\(427\) − 884.000i − 0.100187i
\(428\) 4560.00i 0.514990i
\(429\) 0 0
\(430\) 0 0
\(431\) 13356.0 1.49266 0.746329 0.665577i \(-0.231814\pi\)
0.746329 + 0.665577i \(0.231814\pi\)
\(432\) 0 0
\(433\) 11500.0i 1.27634i 0.769896 + 0.638169i \(0.220308\pi\)
−0.769896 + 0.638169i \(0.779692\pi\)
\(434\) 376.000 0.0415866
\(435\) 0 0
\(436\) −6940.00 −0.762307
\(437\) 531.000i 0.0581263i
\(438\) 0 0
\(439\) 11149.0 1.21210 0.606051 0.795426i \(-0.292753\pi\)
0.606051 + 0.795426i \(0.292753\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3720.00i 0.400322i
\(443\) 3849.00i 0.412803i 0.978467 + 0.206401i \(0.0661752\pi\)
−0.978467 + 0.206401i \(0.933825\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4316.00 −0.458225
\(447\) 0 0
\(448\) 256.000i 0.0269975i
\(449\) 18048.0 1.89697 0.948483 0.316828i \(-0.102618\pi\)
0.948483 + 0.316828i \(0.102618\pi\)
\(450\) 0 0
\(451\) 5292.00 0.552529
\(452\) − 5736.00i − 0.596900i
\(453\) 0 0
\(454\) −6246.00 −0.645681
\(455\) 0 0
\(456\) 0 0
\(457\) − 4264.00i − 0.436458i −0.975898 0.218229i \(-0.929972\pi\)
0.975898 0.218229i \(-0.0700280\pi\)
\(458\) − 4054.00i − 0.413605i
\(459\) 0 0
\(460\) 0 0
\(461\) 10242.0 1.03475 0.517373 0.855760i \(-0.326910\pi\)
0.517373 + 0.855760i \(0.326910\pi\)
\(462\) 0 0
\(463\) − 3302.00i − 0.331441i −0.986173 0.165720i \(-0.947005\pi\)
0.986173 0.165720i \(-0.0529949\pi\)
\(464\) −1920.00 −0.192099
\(465\) 0 0
\(466\) −876.000 −0.0870814
\(467\) 1923.00i 0.190548i 0.995451 + 0.0952739i \(0.0303727\pi\)
−0.995451 + 0.0952739i \(0.969627\pi\)
\(468\) 0 0
\(469\) −2152.00 −0.211877
\(470\) 0 0
\(471\) 0 0
\(472\) − 3552.00i − 0.346386i
\(473\) 7476.00i 0.726738i
\(474\) 0 0
\(475\) 0 0
\(476\) −1488.00 −0.143282
\(477\) 0 0
\(478\) − 12828.0i − 1.22749i
\(479\) 15246.0 1.45430 0.727148 0.686481i \(-0.240846\pi\)
0.727148 + 0.686481i \(0.240846\pi\)
\(480\) 0 0
\(481\) −5240.00 −0.496722
\(482\) 6862.00i 0.648455i
\(483\) 0 0
\(484\) −1732.00 −0.162660
\(485\) 0 0
\(486\) 0 0
\(487\) − 8206.00i − 0.763551i −0.924255 0.381776i \(-0.875313\pi\)
0.924255 0.381776i \(-0.124687\pi\)
\(488\) − 1768.00i − 0.164003i
\(489\) 0 0
\(490\) 0 0
\(491\) 16806.0 1.54469 0.772346 0.635202i \(-0.219083\pi\)
0.772346 + 0.635202i \(0.219083\pi\)
\(492\) 0 0
\(493\) − 11160.0i − 1.01952i
\(494\) −2360.00 −0.214942
\(495\) 0 0
\(496\) 752.000 0.0680762
\(497\) − 2760.00i − 0.249100i
\(498\) 0 0
\(499\) 5425.00 0.486686 0.243343 0.969940i \(-0.421756\pi\)
0.243343 + 0.969940i \(0.421756\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 14616.0i 1.29949i
\(503\) − 19665.0i − 1.74318i −0.490236 0.871589i \(-0.663090\pi\)
0.490236 0.871589i \(-0.336910\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 756.000 0.0664196
\(507\) 0 0
\(508\) − 2744.00i − 0.239656i
\(509\) −14724.0 −1.28218 −0.641090 0.767466i \(-0.721518\pi\)
−0.641090 + 0.767466i \(0.721518\pi\)
\(510\) 0 0
\(511\) 4504.00 0.389912
\(512\) 512.000i 0.0441942i
\(513\) 0 0
\(514\) 7458.00 0.639997
\(515\) 0 0
\(516\) 0 0
\(517\) 6048.00i 0.514489i
\(518\) − 2096.00i − 0.177786i
\(519\) 0 0
\(520\) 0 0
\(521\) −2058.00 −0.173057 −0.0865284 0.996249i \(-0.527577\pi\)
−0.0865284 + 0.996249i \(0.527577\pi\)
\(522\) 0 0
\(523\) − 11912.0i − 0.995938i −0.867195 0.497969i \(-0.834079\pi\)
0.867195 0.497969i \(-0.165921\pi\)
\(524\) 456.000 0.0380161
\(525\) 0 0
\(526\) −3912.00 −0.324280
\(527\) 4371.00i 0.361297i
\(528\) 0 0
\(529\) 12086.0 0.993343
\(530\) 0 0
\(531\) 0 0
\(532\) − 944.000i − 0.0769316i
\(533\) − 2520.00i − 0.204790i
\(534\) 0 0
\(535\) 0 0
\(536\) −4304.00 −0.346837
\(537\) 0 0
\(538\) − 1980.00i − 0.158669i
\(539\) 13734.0 1.09752
\(540\) 0 0
\(541\) −5170.00 −0.410861 −0.205430 0.978672i \(-0.565859\pi\)
−0.205430 + 0.978672i \(0.565859\pi\)
\(542\) 16990.0i 1.34646i
\(543\) 0 0
\(544\) −2976.00 −0.234550
\(545\) 0 0
\(546\) 0 0
\(547\) − 4186.00i − 0.327204i −0.986526 0.163602i \(-0.947689\pi\)
0.986526 0.163602i \(-0.0523112\pi\)
\(548\) − 636.000i − 0.0495777i
\(549\) 0 0
\(550\) 0 0
\(551\) 7080.00 0.547401
\(552\) 0 0
\(553\) 2660.00i 0.204547i
\(554\) 2732.00 0.209515
\(555\) 0 0
\(556\) 9104.00 0.694417
\(557\) − 13026.0i − 0.990896i −0.868637 0.495448i \(-0.835004\pi\)
0.868637 0.495448i \(-0.164996\pi\)
\(558\) 0 0
\(559\) 3560.00 0.269359
\(560\) 0 0
\(561\) 0 0
\(562\) 11040.0i 0.828638i
\(563\) − 10668.0i − 0.798584i −0.916824 0.399292i \(-0.869256\pi\)
0.916824 0.399292i \(-0.130744\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 10876.0 0.807690
\(567\) 0 0
\(568\) − 5520.00i − 0.407771i
\(569\) −15372.0 −1.13256 −0.566281 0.824212i \(-0.691618\pi\)
−0.566281 + 0.824212i \(0.691618\pi\)
\(570\) 0 0
\(571\) −14989.0 −1.09855 −0.549273 0.835643i \(-0.685095\pi\)
−0.549273 + 0.835643i \(0.685095\pi\)
\(572\) 3360.00i 0.245610i
\(573\) 0 0
\(574\) 1008.00 0.0732981
\(575\) 0 0
\(576\) 0 0
\(577\) − 1066.00i − 0.0769119i −0.999260 0.0384559i \(-0.987756\pi\)
0.999260 0.0384559i \(-0.0122439\pi\)
\(578\) − 7472.00i − 0.537706i
\(579\) 0 0
\(580\) 0 0
\(581\) −300.000 −0.0214219
\(582\) 0 0
\(583\) − 31122.0i − 2.21088i
\(584\) 9008.00 0.638277
\(585\) 0 0
\(586\) −16506.0 −1.16358
\(587\) 621.000i 0.0436651i 0.999762 + 0.0218325i \(0.00695007\pi\)
−0.999762 + 0.0218325i \(0.993050\pi\)
\(588\) 0 0
\(589\) −2773.00 −0.193989
\(590\) 0 0
\(591\) 0 0
\(592\) − 4192.00i − 0.291031i
\(593\) − 20187.0i − 1.39794i −0.715149 0.698972i \(-0.753641\pi\)
0.715149 0.698972i \(-0.246359\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −5592.00 −0.384324
\(597\) 0 0
\(598\) − 360.000i − 0.0246179i
\(599\) −18228.0 −1.24337 −0.621683 0.783269i \(-0.713551\pi\)
−0.621683 + 0.783269i \(0.713551\pi\)
\(600\) 0 0
\(601\) −11743.0 −0.797017 −0.398508 0.917165i \(-0.630472\pi\)
−0.398508 + 0.917165i \(0.630472\pi\)
\(602\) 1424.00i 0.0964085i
\(603\) 0 0
\(604\) −10496.0 −0.707080
\(605\) 0 0
\(606\) 0 0
\(607\) − 24418.0i − 1.63278i −0.577503 0.816389i \(-0.695973\pi\)
0.577503 0.816389i \(-0.304027\pi\)
\(608\) − 1888.00i − 0.125935i
\(609\) 0 0
\(610\) 0 0
\(611\) 2880.00 0.190691
\(612\) 0 0
\(613\) − 2672.00i − 0.176054i −0.996118 0.0880270i \(-0.971944\pi\)
0.996118 0.0880270i \(-0.0280562\pi\)
\(614\) −18580.0 −1.22122
\(615\) 0 0
\(616\) −1344.00 −0.0879080
\(617\) 8601.00i 0.561205i 0.959824 + 0.280602i \(0.0905342\pi\)
−0.959824 + 0.280602i \(0.909466\pi\)
\(618\) 0 0
\(619\) −21308.0 −1.38359 −0.691794 0.722095i \(-0.743179\pi\)
−0.691794 + 0.722095i \(0.743179\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 16224.0i − 1.04586i
\(623\) − 4344.00i − 0.279356i
\(624\) 0 0
\(625\) 0 0
\(626\) −15800.0 −1.00878
\(627\) 0 0
\(628\) 1576.00i 0.100142i
\(629\) 24366.0 1.54457
\(630\) 0 0
\(631\) −19015.0 −1.19964 −0.599822 0.800134i \(-0.704762\pi\)
−0.599822 + 0.800134i \(0.704762\pi\)
\(632\) 5320.00i 0.334839i
\(633\) 0 0
\(634\) 8838.00 0.553631
\(635\) 0 0
\(636\) 0 0
\(637\) − 6540.00i − 0.406788i
\(638\) − 10080.0i − 0.625503i
\(639\) 0 0
\(640\) 0 0
\(641\) 4416.00 0.272108 0.136054 0.990701i \(-0.456558\pi\)
0.136054 + 0.990701i \(0.456558\pi\)
\(642\) 0 0
\(643\) − 7580.00i − 0.464893i −0.972609 0.232446i \(-0.925327\pi\)
0.972609 0.232446i \(-0.0746730\pi\)
\(644\) 144.000 0.00881117
\(645\) 0 0
\(646\) 10974.0 0.668369
\(647\) 14901.0i 0.905439i 0.891653 + 0.452719i \(0.149546\pi\)
−0.891653 + 0.452719i \(0.850454\pi\)
\(648\) 0 0
\(649\) 18648.0 1.12789
\(650\) 0 0
\(651\) 0 0
\(652\) − 13384.0i − 0.803923i
\(653\) 12915.0i 0.773971i 0.922086 + 0.386985i \(0.126484\pi\)
−0.922086 + 0.386985i \(0.873516\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2016.00 0.119987
\(657\) 0 0
\(658\) 1152.00i 0.0682517i
\(659\) 28128.0 1.66269 0.831344 0.555758i \(-0.187572\pi\)
0.831344 + 0.555758i \(0.187572\pi\)
\(660\) 0 0
\(661\) −8362.00 −0.492049 −0.246024 0.969264i \(-0.579124\pi\)
−0.246024 + 0.969264i \(0.579124\pi\)
\(662\) − 16400.0i − 0.962846i
\(663\) 0 0
\(664\) −600.000 −0.0350670
\(665\) 0 0
\(666\) 0 0
\(667\) 1080.00i 0.0626953i
\(668\) 5964.00i 0.345440i
\(669\) 0 0
\(670\) 0 0
\(671\) 9282.00 0.534020
\(672\) 0 0
\(673\) − 29708.0i − 1.70157i −0.525511 0.850787i \(-0.676126\pi\)
0.525511 0.850787i \(-0.323874\pi\)
\(674\) 19112.0 1.09224
\(675\) 0 0
\(676\) −7188.00 −0.408967
\(677\) − 6762.00i − 0.383877i −0.981407 0.191939i \(-0.938523\pi\)
0.981407 0.191939i \(-0.0614774\pi\)
\(678\) 0 0
\(679\) 6176.00 0.349062
\(680\) 0 0
\(681\) 0 0
\(682\) 3948.00i 0.221667i
\(683\) − 19155.0i − 1.07313i −0.843860 0.536563i \(-0.819722\pi\)
0.843860 0.536563i \(-0.180278\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 5360.00 0.298317
\(687\) 0 0
\(688\) 2848.00i 0.157818i
\(689\) −14820.0 −0.819444
\(690\) 0 0
\(691\) −22975.0 −1.26485 −0.632424 0.774622i \(-0.717940\pi\)
−0.632424 + 0.774622i \(0.717940\pi\)
\(692\) 9612.00i 0.528025i
\(693\) 0 0
\(694\) 20232.0 1.10662
\(695\) 0 0
\(696\) 0 0
\(697\) 11718.0i 0.636802i
\(698\) 13502.0i 0.732175i
\(699\) 0 0
\(700\) 0 0
\(701\) 6450.00 0.347522 0.173761 0.984788i \(-0.444408\pi\)
0.173761 + 0.984788i \(0.444408\pi\)
\(702\) 0 0
\(703\) 15458.0i 0.829317i
\(704\) −2688.00 −0.143903
\(705\) 0 0
\(706\) −8124.00 −0.433075
\(707\) 528.000i 0.0280870i
\(708\) 0 0
\(709\) −34538.0 −1.82948 −0.914740 0.404042i \(-0.867605\pi\)
−0.914740 + 0.404042i \(0.867605\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 8688.00i − 0.457299i
\(713\) − 423.000i − 0.0222181i
\(714\) 0 0
\(715\) 0 0
\(716\) −10560.0 −0.551181
\(717\) 0 0
\(718\) 17556.0i 0.912513i
\(719\) 27114.0 1.40637 0.703186 0.711006i \(-0.251760\pi\)
0.703186 + 0.711006i \(0.251760\pi\)
\(720\) 0 0
\(721\) 3568.00 0.184299
\(722\) − 6756.00i − 0.348244i
\(723\) 0 0
\(724\) −4292.00 −0.220319
\(725\) 0 0
\(726\) 0 0
\(727\) 236.000i 0.0120396i 0.999982 + 0.00601978i \(0.00191617\pi\)
−0.999982 + 0.00601978i \(0.998084\pi\)
\(728\) 640.000i 0.0325824i
\(729\) 0 0
\(730\) 0 0
\(731\) −16554.0 −0.837581
\(732\) 0 0
\(733\) − 27128.0i − 1.36698i −0.729960 0.683489i \(-0.760462\pi\)
0.729960 0.683489i \(-0.239538\pi\)
\(734\) −1912.00 −0.0961488
\(735\) 0 0
\(736\) 288.000 0.0144237
\(737\) − 22596.0i − 1.12935i
\(738\) 0 0
\(739\) −5249.00 −0.261282 −0.130641 0.991430i \(-0.541704\pi\)
−0.130641 + 0.991430i \(0.541704\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 5928.00i − 0.293293i
\(743\) 13896.0i 0.686130i 0.939312 + 0.343065i \(0.111465\pi\)
−0.939312 + 0.343065i \(0.888535\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4600.00 0.225761
\(747\) 0 0
\(748\) − 15624.0i − 0.763730i
\(749\) −4560.00 −0.222455
\(750\) 0 0
\(751\) 27665.0 1.34422 0.672111 0.740451i \(-0.265388\pi\)
0.672111 + 0.740451i \(0.265388\pi\)
\(752\) 2304.00i 0.111726i
\(753\) 0 0
\(754\) −4800.00 −0.231838
\(755\) 0 0
\(756\) 0 0
\(757\) − 8122.00i − 0.389959i −0.980807 0.194980i \(-0.937536\pi\)
0.980807 0.194980i \(-0.0624641\pi\)
\(758\) − 58.0000i − 0.00277923i
\(759\) 0 0
\(760\) 0 0
\(761\) −10584.0 −0.504165 −0.252083 0.967706i \(-0.581115\pi\)
−0.252083 + 0.967706i \(0.581115\pi\)
\(762\) 0 0
\(763\) − 6940.00i − 0.329286i
\(764\) −5880.00 −0.278444
\(765\) 0 0
\(766\) −16254.0 −0.766685
\(767\) − 8880.00i − 0.418042i
\(768\) 0 0
\(769\) 18619.0 0.873106 0.436553 0.899679i \(-0.356199\pi\)
0.436553 + 0.899679i \(0.356199\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 18880.0i − 0.880189i
\(773\) 22251.0i 1.03533i 0.855582 + 0.517667i \(0.173199\pi\)
−0.855582 + 0.517667i \(0.826801\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 12352.0 0.571406
\(777\) 0 0
\(778\) − 15876.0i − 0.731597i
\(779\) −7434.00 −0.341914
\(780\) 0 0
\(781\) 28980.0 1.32777
\(782\) 1674.00i 0.0765500i
\(783\) 0 0
\(784\) 5232.00 0.238338
\(785\) 0 0
\(786\) 0 0
\(787\) 24854.0i 1.12573i 0.826549 + 0.562865i \(0.190301\pi\)
−0.826549 + 0.562865i \(0.809699\pi\)
\(788\) 3060.00i 0.138335i
\(789\) 0 0
\(790\) 0 0
\(791\) 5736.00 0.257837
\(792\) 0 0
\(793\) − 4420.00i − 0.197930i
\(794\) −544.000 −0.0243147
\(795\) 0 0
\(796\) 2672.00 0.118978
\(797\) − 3681.00i − 0.163598i −0.996649 0.0817991i \(-0.973933\pi\)
0.996649 0.0817991i \(-0.0260666\pi\)
\(798\) 0 0
\(799\) −13392.0 −0.592960
\(800\) 0 0
\(801\) 0 0
\(802\) − 9108.00i − 0.401016i
\(803\) 47292.0i 2.07833i
\(804\) 0 0
\(805\) 0 0
\(806\) 1880.00 0.0821590
\(807\) 0 0
\(808\) 1056.00i 0.0459777i
\(809\) −5142.00 −0.223465 −0.111732 0.993738i \(-0.535640\pi\)
−0.111732 + 0.993738i \(0.535640\pi\)
\(810\) 0 0
\(811\) −18484.0 −0.800322 −0.400161 0.916445i \(-0.631046\pi\)
−0.400161 + 0.916445i \(0.631046\pi\)
\(812\) − 1920.00i − 0.0829788i
\(813\) 0 0
\(814\) 22008.0 0.947641
\(815\) 0 0
\(816\) 0 0
\(817\) − 10502.0i − 0.449717i
\(818\) − 2002.00i − 0.0855725i
\(819\) 0 0
\(820\) 0 0
\(821\) −25014.0 −1.06333 −0.531665 0.846954i \(-0.678434\pi\)
−0.531665 + 0.846954i \(0.678434\pi\)
\(822\) 0 0
\(823\) 32146.0i 1.36153i 0.732502 + 0.680765i \(0.238352\pi\)
−0.732502 + 0.680765i \(0.761648\pi\)
\(824\) 7136.00 0.301692
\(825\) 0 0
\(826\) 3552.00 0.149625
\(827\) 10977.0i 0.461557i 0.973006 + 0.230779i \(0.0741273\pi\)
−0.973006 + 0.230779i \(0.925873\pi\)
\(828\) 0 0
\(829\) −36602.0 −1.53346 −0.766731 0.641969i \(-0.778118\pi\)
−0.766731 + 0.641969i \(0.778118\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1280.00i 0.0533366i
\(833\) 30411.0i 1.26492i
\(834\) 0 0
\(835\) 0 0
\(836\) 9912.00 0.410064
\(837\) 0 0
\(838\) − 3588.00i − 0.147906i
\(839\) 11076.0 0.455764 0.227882 0.973689i \(-0.426820\pi\)
0.227882 + 0.973689i \(0.426820\pi\)
\(840\) 0 0
\(841\) −9989.00 −0.409570
\(842\) − 32258.0i − 1.32029i
\(843\) 0 0
\(844\) −18404.0 −0.750583
\(845\) 0 0
\(846\) 0 0
\(847\) − 1732.00i − 0.0702624i
\(848\) − 11856.0i − 0.480114i
\(849\) 0 0
\(850\) 0 0
\(851\) −2358.00 −0.0949838
\(852\) 0 0
\(853\) − 36848.0i − 1.47908i −0.673115 0.739538i \(-0.735044\pi\)
0.673115 0.739538i \(-0.264956\pi\)
\(854\) 1768.00 0.0708428
\(855\) 0 0
\(856\) −9120.00 −0.364153
\(857\) 26961.0i 1.07464i 0.843377 + 0.537322i \(0.180564\pi\)
−0.843377 + 0.537322i \(0.819436\pi\)
\(858\) 0 0
\(859\) 415.000 0.0164838 0.00824192 0.999966i \(-0.497376\pi\)
0.00824192 + 0.999966i \(0.497376\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 26712.0i 1.05547i
\(863\) − 45501.0i − 1.79475i −0.441265 0.897377i \(-0.645470\pi\)
0.441265 0.897377i \(-0.354530\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −23000.0 −0.902508
\(867\) 0 0
\(868\) 752.000i 0.0294062i
\(869\) −27930.0 −1.09029
\(870\) 0 0
\(871\) −10760.0 −0.418586
\(872\) − 13880.0i − 0.539032i
\(873\) 0 0
\(874\) −1062.00 −0.0411015
\(875\) 0 0
\(876\) 0 0
\(877\) 35042.0i 1.34924i 0.738165 + 0.674620i \(0.235693\pi\)
−0.738165 + 0.674620i \(0.764307\pi\)
\(878\) 22298.0i 0.857085i
\(879\) 0 0
\(880\) 0 0
\(881\) −1080.00 −0.0413009 −0.0206505 0.999787i \(-0.506574\pi\)
−0.0206505 + 0.999787i \(0.506574\pi\)
\(882\) 0 0
\(883\) 20164.0i 0.768485i 0.923232 + 0.384243i \(0.125537\pi\)
−0.923232 + 0.384243i \(0.874463\pi\)
\(884\) −7440.00 −0.283070
\(885\) 0 0
\(886\) −7698.00 −0.291895
\(887\) − 20067.0i − 0.759621i −0.925064 0.379811i \(-0.875989\pi\)
0.925064 0.379811i \(-0.124011\pi\)
\(888\) 0 0
\(889\) 2744.00 0.103522
\(890\) 0 0
\(891\) 0 0
\(892\) − 8632.00i − 0.324014i
\(893\) − 8496.00i − 0.318374i
\(894\) 0 0
\(895\) 0 0
\(896\) −512.000 −0.0190901
\(897\) 0 0
\(898\) 36096.0i 1.34136i
\(899\) −5640.00 −0.209238
\(900\) 0 0
\(901\) 68913.0 2.54809
\(902\) 10584.0i 0.390697i
\(903\) 0 0
\(904\) 11472.0 0.422072
\(905\) 0 0
\(906\) 0 0
\(907\) − 26524.0i − 0.971020i −0.874231 0.485510i \(-0.838634\pi\)
0.874231 0.485510i \(-0.161366\pi\)
\(908\) − 12492.0i − 0.456566i
\(909\) 0 0
\(910\) 0 0
\(911\) −35568.0 −1.29355 −0.646773 0.762683i \(-0.723882\pi\)
−0.646773 + 0.762683i \(0.723882\pi\)
\(912\) 0 0
\(913\) − 3150.00i − 0.114184i
\(914\) 8528.00 0.308623
\(915\) 0 0
\(916\) 8108.00 0.292463
\(917\) 456.000i 0.0164214i
\(918\) 0 0
\(919\) 23704.0 0.850841 0.425420 0.904996i \(-0.360126\pi\)
0.425420 + 0.904996i \(0.360126\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 20484.0i 0.731675i
\(923\) − 13800.0i − 0.492126i
\(924\) 0 0
\(925\) 0 0
\(926\) 6604.00 0.234364
\(927\) 0 0
\(928\) − 3840.00i − 0.135834i
\(929\) −40590.0 −1.43349 −0.716746 0.697334i \(-0.754369\pi\)
−0.716746 + 0.697334i \(0.754369\pi\)
\(930\) 0 0
\(931\) −19293.0 −0.679165
\(932\) − 1752.00i − 0.0615758i
\(933\) 0 0
\(934\) −3846.00 −0.134738
\(935\) 0 0
\(936\) 0 0
\(937\) − 12964.0i − 0.451991i −0.974128 0.225995i \(-0.927437\pi\)
0.974128 0.225995i \(-0.0725634\pi\)
\(938\) − 4304.00i − 0.149819i
\(939\) 0 0
\(940\) 0 0
\(941\) −29922.0 −1.03659 −0.518294 0.855203i \(-0.673433\pi\)
−0.518294 + 0.855203i \(0.673433\pi\)
\(942\) 0 0
\(943\) − 1134.00i − 0.0391603i
\(944\) 7104.00 0.244932
\(945\) 0 0
\(946\) −14952.0 −0.513881
\(947\) − 5241.00i − 0.179841i −0.995949 0.0899206i \(-0.971339\pi\)
0.995949 0.0899206i \(-0.0286613\pi\)
\(948\) 0 0
\(949\) 22520.0 0.770316
\(950\) 0 0
\(951\) 0 0
\(952\) − 2976.00i − 0.101316i
\(953\) 26214.0i 0.891033i 0.895274 + 0.445517i \(0.146980\pi\)
−0.895274 + 0.445517i \(0.853020\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 25656.0 0.867965
\(957\) 0 0
\(958\) 30492.0i 1.02834i
\(959\) 636.000 0.0214155
\(960\) 0 0
\(961\) −27582.0 −0.925850
\(962\) − 10480.0i − 0.351236i
\(963\) 0 0
\(964\) −13724.0 −0.458527
\(965\) 0 0
\(966\) 0 0
\(967\) 18278.0i 0.607840i 0.952698 + 0.303920i \(0.0982955\pi\)
−0.952698 + 0.303920i \(0.901705\pi\)
\(968\) − 3464.00i − 0.115018i
\(969\) 0 0
\(970\) 0 0
\(971\) 24942.0 0.824333 0.412166 0.911109i \(-0.364772\pi\)
0.412166 + 0.911109i \(0.364772\pi\)
\(972\) 0 0
\(973\) 9104.00i 0.299960i
\(974\) 16412.0 0.539912
\(975\) 0 0
\(976\) 3536.00 0.115968
\(977\) 11226.0i 0.367607i 0.982963 + 0.183803i \(0.0588409\pi\)
−0.982963 + 0.183803i \(0.941159\pi\)
\(978\) 0 0
\(979\) 45612.0 1.48904
\(980\) 0 0
\(981\) 0 0
\(982\) 33612.0i 1.09226i
\(983\) − 23073.0i − 0.748641i −0.927299 0.374321i \(-0.877876\pi\)
0.927299 0.374321i \(-0.122124\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 22320.0 0.720906
\(987\) 0 0
\(988\) − 4720.00i − 0.151987i
\(989\) 1602.00 0.0515072
\(990\) 0 0
\(991\) 22037.0 0.706386 0.353193 0.935551i \(-0.385096\pi\)
0.353193 + 0.935551i \(0.385096\pi\)
\(992\) 1504.00i 0.0481371i
\(993\) 0 0
\(994\) 5520.00 0.176141
\(995\) 0 0
\(996\) 0 0
\(997\) 19082.0i 0.606151i 0.952966 + 0.303076i \(0.0980135\pi\)
−0.952966 + 0.303076i \(0.901986\pi\)
\(998\) 10850.0i 0.344139i
\(999\) 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.q.649.2 2
3.2 odd 2 1350.4.c.d.649.1 2
5.2 odd 4 1350.4.a.g.1.1 1
5.3 odd 4 270.4.a.i.1.1 yes 1
5.4 even 2 inner 1350.4.c.q.649.1 2
15.2 even 4 1350.4.a.u.1.1 1
15.8 even 4 270.4.a.e.1.1 1
15.14 odd 2 1350.4.c.d.649.2 2
20.3 even 4 2160.4.a.e.1.1 1
45.13 odd 12 810.4.e.h.541.1 2
45.23 even 12 810.4.e.q.541.1 2
45.38 even 12 810.4.e.q.271.1 2
45.43 odd 12 810.4.e.h.271.1 2
60.23 odd 4 2160.4.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.4.a.e.1.1 1 15.8 even 4
270.4.a.i.1.1 yes 1 5.3 odd 4
810.4.e.h.271.1 2 45.43 odd 12
810.4.e.h.541.1 2 45.13 odd 12
810.4.e.q.271.1 2 45.38 even 12
810.4.e.q.541.1 2 45.23 even 12
1350.4.a.g.1.1 1 5.2 odd 4
1350.4.a.u.1.1 1 15.2 even 4
1350.4.c.d.649.1 2 3.2 odd 2
1350.4.c.d.649.2 2 15.14 odd 2
1350.4.c.q.649.1 2 5.4 even 2 inner
1350.4.c.q.649.2 2 1.1 even 1 trivial
2160.4.a.e.1.1 1 20.3 even 4
2160.4.a.o.1.1 1 60.23 odd 4