Properties

Label 1521.4.a.bm.1.10
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \( x^{18} - 100 x^{16} + 4066 x^{14} - 86197 x^{12} + 1016064 x^{10} - 6594119 x^{8} + 22251817 x^{6} - 37096332 x^{4} + 27824160 x^{2} - 6889792 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 7\cdot 13^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(0.685025\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.685025 q^{2} -7.53074 q^{4} +9.28431 q^{5} -6.04730 q^{7} -10.6389 q^{8} +O(q^{10})\) \(q+0.685025 q^{2} -7.53074 q^{4} +9.28431 q^{5} -6.04730 q^{7} -10.6389 q^{8} +6.35999 q^{10} +59.2146 q^{11} -4.14255 q^{14} +52.9580 q^{16} -117.407 q^{17} -49.2492 q^{19} -69.9177 q^{20} +40.5635 q^{22} +23.1522 q^{23} -38.8015 q^{25} +45.5406 q^{28} -145.028 q^{29} +94.9826 q^{31} +121.389 q^{32} -80.4268 q^{34} -56.1450 q^{35} +379.757 q^{37} -33.7370 q^{38} -98.7753 q^{40} -268.749 q^{41} -23.6647 q^{43} -445.930 q^{44} +15.8599 q^{46} +301.164 q^{47} -306.430 q^{49} -26.5800 q^{50} -391.713 q^{53} +549.767 q^{55} +64.3369 q^{56} -99.3480 q^{58} +510.800 q^{59} +520.734 q^{61} +65.0655 q^{62} -340.509 q^{64} +470.377 q^{67} +884.162 q^{68} -38.4607 q^{70} -466.583 q^{71} -314.174 q^{73} +260.143 q^{74} +370.883 q^{76} -358.088 q^{77} +47.9136 q^{79} +491.678 q^{80} -184.100 q^{82} +310.030 q^{83} -1090.04 q^{85} -16.2109 q^{86} -629.981 q^{88} +216.244 q^{89} -174.353 q^{92} +206.305 q^{94} -457.245 q^{95} -219.507 q^{97} -209.912 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 56 q^{4} - 94 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 56 q^{4} - 94 q^{7} + 156 q^{10} + 88 q^{16} - 448 q^{19} - 256 q^{22} - 16 q^{25} - 1300 q^{28} - 818 q^{31} - 900 q^{34} - 524 q^{37} + 800 q^{40} + 752 q^{43} - 1650 q^{46} + 2948 q^{49} - 464 q^{55} - 1626 q^{58} - 1784 q^{61} - 4274 q^{64} - 2872 q^{67} - 5772 q^{70} - 3078 q^{73} - 5726 q^{76} + 3326 q^{79} - 1038 q^{82} - 2340 q^{85} + 780 q^{88} + 1220 q^{94} - 4630 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.685025 0.242193 0.121096 0.992641i \(-0.461359\pi\)
0.121096 + 0.992641i \(0.461359\pi\)
\(3\) 0 0
\(4\) −7.53074 −0.941343
\(5\) 9.28431 0.830414 0.415207 0.909727i \(-0.363709\pi\)
0.415207 + 0.909727i \(0.363709\pi\)
\(6\) 0 0
\(7\) −6.04730 −0.326523 −0.163262 0.986583i \(-0.552201\pi\)
−0.163262 + 0.986583i \(0.552201\pi\)
\(8\) −10.6389 −0.470180
\(9\) 0 0
\(10\) 6.35999 0.201120
\(11\) 59.2146 1.62308 0.811539 0.584298i \(-0.198630\pi\)
0.811539 + 0.584298i \(0.198630\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −4.14255 −0.0790817
\(15\) 0 0
\(16\) 52.9580 0.827468
\(17\) −117.407 −1.67502 −0.837512 0.546420i \(-0.815990\pi\)
−0.837512 + 0.546420i \(0.815990\pi\)
\(18\) 0 0
\(19\) −49.2492 −0.594660 −0.297330 0.954775i \(-0.596096\pi\)
−0.297330 + 0.954775i \(0.596096\pi\)
\(20\) −69.9177 −0.781704
\(21\) 0 0
\(22\) 40.5635 0.393098
\(23\) 23.1522 0.209894 0.104947 0.994478i \(-0.466533\pi\)
0.104947 + 0.994478i \(0.466533\pi\)
\(24\) 0 0
\(25\) −38.8015 −0.310412
\(26\) 0 0
\(27\) 0 0
\(28\) 45.5406 0.307370
\(29\) −145.028 −0.928658 −0.464329 0.885663i \(-0.653704\pi\)
−0.464329 + 0.885663i \(0.653704\pi\)
\(30\) 0 0
\(31\) 94.9826 0.550303 0.275151 0.961401i \(-0.411272\pi\)
0.275151 + 0.961401i \(0.411272\pi\)
\(32\) 121.389 0.670587
\(33\) 0 0
\(34\) −80.4268 −0.405679
\(35\) −56.1450 −0.271150
\(36\) 0 0
\(37\) 379.757 1.68734 0.843672 0.536860i \(-0.180390\pi\)
0.843672 + 0.536860i \(0.180390\pi\)
\(38\) −33.7370 −0.144023
\(39\) 0 0
\(40\) −98.7753 −0.390444
\(41\) −268.749 −1.02370 −0.511848 0.859076i \(-0.671039\pi\)
−0.511848 + 0.859076i \(0.671039\pi\)
\(42\) 0 0
\(43\) −23.6647 −0.0839264 −0.0419632 0.999119i \(-0.513361\pi\)
−0.0419632 + 0.999119i \(0.513361\pi\)
\(44\) −445.930 −1.52787
\(45\) 0 0
\(46\) 15.8599 0.0508350
\(47\) 301.164 0.934666 0.467333 0.884081i \(-0.345215\pi\)
0.467333 + 0.884081i \(0.345215\pi\)
\(48\) 0 0
\(49\) −306.430 −0.893383
\(50\) −26.5800 −0.0751797
\(51\) 0 0
\(52\) 0 0
\(53\) −391.713 −1.01521 −0.507603 0.861591i \(-0.669468\pi\)
−0.507603 + 0.861591i \(0.669468\pi\)
\(54\) 0 0
\(55\) 549.767 1.34783
\(56\) 64.3369 0.153525
\(57\) 0 0
\(58\) −99.3480 −0.224914
\(59\) 510.800 1.12713 0.563563 0.826073i \(-0.309430\pi\)
0.563563 + 0.826073i \(0.309430\pi\)
\(60\) 0 0
\(61\) 520.734 1.09300 0.546501 0.837458i \(-0.315959\pi\)
0.546501 + 0.837458i \(0.315959\pi\)
\(62\) 65.0655 0.133279
\(63\) 0 0
\(64\) −340.509 −0.665057
\(65\) 0 0
\(66\) 0 0
\(67\) 470.377 0.857697 0.428849 0.903376i \(-0.358919\pi\)
0.428849 + 0.903376i \(0.358919\pi\)
\(68\) 884.162 1.57677
\(69\) 0 0
\(70\) −38.4607 −0.0656705
\(71\) −466.583 −0.779905 −0.389953 0.920835i \(-0.627509\pi\)
−0.389953 + 0.920835i \(0.627509\pi\)
\(72\) 0 0
\(73\) −314.174 −0.503716 −0.251858 0.967764i \(-0.581042\pi\)
−0.251858 + 0.967764i \(0.581042\pi\)
\(74\) 260.143 0.408663
\(75\) 0 0
\(76\) 370.883 0.559779
\(77\) −358.088 −0.529973
\(78\) 0 0
\(79\) 47.9136 0.0682368 0.0341184 0.999418i \(-0.489138\pi\)
0.0341184 + 0.999418i \(0.489138\pi\)
\(80\) 491.678 0.687141
\(81\) 0 0
\(82\) −184.100 −0.247932
\(83\) 310.030 0.410003 0.205001 0.978762i \(-0.434280\pi\)
0.205001 + 0.978762i \(0.434280\pi\)
\(84\) 0 0
\(85\) −1090.04 −1.39096
\(86\) −16.2109 −0.0203264
\(87\) 0 0
\(88\) −629.981 −0.763138
\(89\) 216.244 0.257549 0.128774 0.991674i \(-0.458896\pi\)
0.128774 + 0.991674i \(0.458896\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −174.353 −0.197583
\(93\) 0 0
\(94\) 206.305 0.226369
\(95\) −457.245 −0.493814
\(96\) 0 0
\(97\) −219.507 −0.229768 −0.114884 0.993379i \(-0.536650\pi\)
−0.114884 + 0.993379i \(0.536650\pi\)
\(98\) −209.912 −0.216371
\(99\) 0 0
\(100\) 292.204 0.292204
\(101\) −921.359 −0.907710 −0.453855 0.891076i \(-0.649952\pi\)
−0.453855 + 0.891076i \(0.649952\pi\)
\(102\) 0 0
\(103\) −1277.55 −1.22214 −0.611072 0.791575i \(-0.709262\pi\)
−0.611072 + 0.791575i \(0.709262\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −268.333 −0.245876
\(107\) −1626.78 −1.46978 −0.734892 0.678184i \(-0.762767\pi\)
−0.734892 + 0.678184i \(0.762767\pi\)
\(108\) 0 0
\(109\) −1626.49 −1.42926 −0.714629 0.699504i \(-0.753404\pi\)
−0.714629 + 0.699504i \(0.753404\pi\)
\(110\) 376.604 0.326434
\(111\) 0 0
\(112\) −320.253 −0.270188
\(113\) −2003.32 −1.66776 −0.833878 0.551949i \(-0.813884\pi\)
−0.833878 + 0.551949i \(0.813884\pi\)
\(114\) 0 0
\(115\) 214.952 0.174299
\(116\) 1092.17 0.874185
\(117\) 0 0
\(118\) 349.911 0.272982
\(119\) 709.995 0.546934
\(120\) 0 0
\(121\) 2175.36 1.63438
\(122\) 356.716 0.264717
\(123\) 0 0
\(124\) −715.290 −0.518023
\(125\) −1520.78 −1.08818
\(126\) 0 0
\(127\) −1966.75 −1.37418 −0.687091 0.726571i \(-0.741113\pi\)
−0.687091 + 0.726571i \(0.741113\pi\)
\(128\) −1204.37 −0.831659
\(129\) 0 0
\(130\) 0 0
\(131\) 2863.90 1.91008 0.955039 0.296480i \(-0.0958128\pi\)
0.955039 + 0.296480i \(0.0958128\pi\)
\(132\) 0 0
\(133\) 297.825 0.194171
\(134\) 322.220 0.207728
\(135\) 0 0
\(136\) 1249.09 0.787562
\(137\) −1774.52 −1.10663 −0.553313 0.832974i \(-0.686637\pi\)
−0.553313 + 0.832974i \(0.686637\pi\)
\(138\) 0 0
\(139\) −1349.35 −0.823382 −0.411691 0.911324i \(-0.635062\pi\)
−0.411691 + 0.911324i \(0.635062\pi\)
\(140\) 422.813 0.255245
\(141\) 0 0
\(142\) −319.621 −0.188888
\(143\) 0 0
\(144\) 0 0
\(145\) −1346.49 −0.771171
\(146\) −215.217 −0.121996
\(147\) 0 0
\(148\) −2859.85 −1.58837
\(149\) −1267.51 −0.696903 −0.348452 0.937327i \(-0.613292\pi\)
−0.348452 + 0.937327i \(0.613292\pi\)
\(150\) 0 0
\(151\) −2928.66 −1.57835 −0.789176 0.614168i \(-0.789492\pi\)
−0.789176 + 0.614168i \(0.789492\pi\)
\(152\) 523.960 0.279597
\(153\) 0 0
\(154\) −245.299 −0.128356
\(155\) 881.849 0.456979
\(156\) 0 0
\(157\) 2520.07 1.28104 0.640521 0.767940i \(-0.278718\pi\)
0.640521 + 0.767940i \(0.278718\pi\)
\(158\) 32.8220 0.0165265
\(159\) 0 0
\(160\) 1127.01 0.556865
\(161\) −140.008 −0.0685354
\(162\) 0 0
\(163\) −3596.35 −1.72815 −0.864073 0.503366i \(-0.832095\pi\)
−0.864073 + 0.503366i \(0.832095\pi\)
\(164\) 2023.88 0.963649
\(165\) 0 0
\(166\) 212.378 0.0992998
\(167\) 1988.55 0.921430 0.460715 0.887548i \(-0.347593\pi\)
0.460715 + 0.887548i \(0.347593\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −746.707 −0.336881
\(171\) 0 0
\(172\) 178.213 0.0790035
\(173\) 584.054 0.256675 0.128338 0.991731i \(-0.459036\pi\)
0.128338 + 0.991731i \(0.459036\pi\)
\(174\) 0 0
\(175\) 234.644 0.101357
\(176\) 3135.88 1.34305
\(177\) 0 0
\(178\) 148.133 0.0623765
\(179\) −1253.49 −0.523409 −0.261704 0.965148i \(-0.584285\pi\)
−0.261704 + 0.965148i \(0.584285\pi\)
\(180\) 0 0
\(181\) −3426.69 −1.40720 −0.703601 0.710595i \(-0.748426\pi\)
−0.703601 + 0.710595i \(0.748426\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −246.315 −0.0986881
\(185\) 3525.78 1.40119
\(186\) 0 0
\(187\) −6952.21 −2.71869
\(188\) −2267.99 −0.879840
\(189\) 0 0
\(190\) −313.225 −0.119598
\(191\) 3100.81 1.17470 0.587348 0.809335i \(-0.300172\pi\)
0.587348 + 0.809335i \(0.300172\pi\)
\(192\) 0 0
\(193\) −338.564 −0.126271 −0.0631357 0.998005i \(-0.520110\pi\)
−0.0631357 + 0.998005i \(0.520110\pi\)
\(194\) −150.368 −0.0556482
\(195\) 0 0
\(196\) 2307.65 0.840979
\(197\) 1317.22 0.476387 0.238193 0.971218i \(-0.423445\pi\)
0.238193 + 0.971218i \(0.423445\pi\)
\(198\) 0 0
\(199\) 3454.89 1.23071 0.615354 0.788251i \(-0.289013\pi\)
0.615354 + 0.788251i \(0.289013\pi\)
\(200\) 412.808 0.145950
\(201\) 0 0
\(202\) −631.154 −0.219841
\(203\) 877.029 0.303228
\(204\) 0 0
\(205\) −2495.15 −0.850092
\(206\) −875.155 −0.295995
\(207\) 0 0
\(208\) 0 0
\(209\) −2916.27 −0.965181
\(210\) 0 0
\(211\) 2391.57 0.780295 0.390148 0.920752i \(-0.372424\pi\)
0.390148 + 0.920752i \(0.372424\pi\)
\(212\) 2949.89 0.955657
\(213\) 0 0
\(214\) −1114.39 −0.355971
\(215\) −219.711 −0.0696937
\(216\) 0 0
\(217\) −574.388 −0.179687
\(218\) −1114.18 −0.346156
\(219\) 0 0
\(220\) −4140.15 −1.26877
\(221\) 0 0
\(222\) 0 0
\(223\) −1389.30 −0.417194 −0.208597 0.978002i \(-0.566890\pi\)
−0.208597 + 0.978002i \(0.566890\pi\)
\(224\) −734.076 −0.218962
\(225\) 0 0
\(226\) −1372.32 −0.403919
\(227\) 4465.41 1.30564 0.652818 0.757514i \(-0.273586\pi\)
0.652818 + 0.757514i \(0.273586\pi\)
\(228\) 0 0
\(229\) −3605.58 −1.04045 −0.520226 0.854029i \(-0.674152\pi\)
−0.520226 + 0.854029i \(0.674152\pi\)
\(230\) 147.248 0.0422141
\(231\) 0 0
\(232\) 1542.95 0.436636
\(233\) −559.829 −0.157406 −0.0787031 0.996898i \(-0.525078\pi\)
−0.0787031 + 0.996898i \(0.525078\pi\)
\(234\) 0 0
\(235\) 2796.10 0.776159
\(236\) −3846.70 −1.06101
\(237\) 0 0
\(238\) 486.365 0.132464
\(239\) −3411.14 −0.923215 −0.461607 0.887084i \(-0.652727\pi\)
−0.461607 + 0.887084i \(0.652727\pi\)
\(240\) 0 0
\(241\) −6188.58 −1.65411 −0.827057 0.562119i \(-0.809986\pi\)
−0.827057 + 0.562119i \(0.809986\pi\)
\(242\) 1490.18 0.395836
\(243\) 0 0
\(244\) −3921.51 −1.02889
\(245\) −2844.99 −0.741877
\(246\) 0 0
\(247\) 0 0
\(248\) −1010.52 −0.258741
\(249\) 0 0
\(250\) −1041.78 −0.263551
\(251\) −6493.27 −1.63288 −0.816438 0.577434i \(-0.804054\pi\)
−0.816438 + 0.577434i \(0.804054\pi\)
\(252\) 0 0
\(253\) 1370.95 0.340675
\(254\) −1347.28 −0.332817
\(255\) 0 0
\(256\) 1899.05 0.463635
\(257\) 7490.26 1.81801 0.909007 0.416780i \(-0.136841\pi\)
0.909007 + 0.416780i \(0.136841\pi\)
\(258\) 0 0
\(259\) −2296.50 −0.550957
\(260\) 0 0
\(261\) 0 0
\(262\) 1961.84 0.462608
\(263\) −4040.55 −0.947342 −0.473671 0.880702i \(-0.657071\pi\)
−0.473671 + 0.880702i \(0.657071\pi\)
\(264\) 0 0
\(265\) −3636.79 −0.843041
\(266\) 204.017 0.0470267
\(267\) 0 0
\(268\) −3542.29 −0.807387
\(269\) −7720.88 −1.75000 −0.875001 0.484121i \(-0.839140\pi\)
−0.875001 + 0.484121i \(0.839140\pi\)
\(270\) 0 0
\(271\) 6584.34 1.47590 0.737952 0.674853i \(-0.235793\pi\)
0.737952 + 0.674853i \(0.235793\pi\)
\(272\) −6217.64 −1.38603
\(273\) 0 0
\(274\) −1215.59 −0.268017
\(275\) −2297.62 −0.503824
\(276\) 0 0
\(277\) 3790.56 0.822213 0.411106 0.911587i \(-0.365142\pi\)
0.411106 + 0.911587i \(0.365142\pi\)
\(278\) −924.336 −0.199417
\(279\) 0 0
\(280\) 597.324 0.127489
\(281\) −6408.89 −1.36058 −0.680288 0.732945i \(-0.738145\pi\)
−0.680288 + 0.732945i \(0.738145\pi\)
\(282\) 0 0
\(283\) −7328.79 −1.53940 −0.769702 0.638403i \(-0.779595\pi\)
−0.769702 + 0.638403i \(0.779595\pi\)
\(284\) 3513.72 0.734158
\(285\) 0 0
\(286\) 0 0
\(287\) 1625.20 0.334261
\(288\) 0 0
\(289\) 8871.42 1.80570
\(290\) −922.378 −0.186772
\(291\) 0 0
\(292\) 2365.96 0.474169
\(293\) −527.824 −0.105242 −0.0526209 0.998615i \(-0.516757\pi\)
−0.0526209 + 0.998615i \(0.516757\pi\)
\(294\) 0 0
\(295\) 4742.42 0.935981
\(296\) −4040.22 −0.793354
\(297\) 0 0
\(298\) −868.277 −0.168785
\(299\) 0 0
\(300\) 0 0
\(301\) 143.108 0.0274039
\(302\) −2006.21 −0.382266
\(303\) 0 0
\(304\) −2608.14 −0.492063
\(305\) 4834.66 0.907645
\(306\) 0 0
\(307\) −827.031 −0.153750 −0.0768749 0.997041i \(-0.524494\pi\)
−0.0768749 + 0.997041i \(0.524494\pi\)
\(308\) 2696.67 0.498886
\(309\) 0 0
\(310\) 604.088 0.110677
\(311\) 2022.53 0.368769 0.184384 0.982854i \(-0.440971\pi\)
0.184384 + 0.982854i \(0.440971\pi\)
\(312\) 0 0
\(313\) −6839.50 −1.23512 −0.617559 0.786525i \(-0.711878\pi\)
−0.617559 + 0.786525i \(0.711878\pi\)
\(314\) 1726.31 0.310260
\(315\) 0 0
\(316\) −360.825 −0.0642342
\(317\) 7535.31 1.33510 0.667548 0.744567i \(-0.267344\pi\)
0.667548 + 0.744567i \(0.267344\pi\)
\(318\) 0 0
\(319\) −8587.79 −1.50728
\(320\) −3161.39 −0.552273
\(321\) 0 0
\(322\) −95.9092 −0.0165988
\(323\) 5782.21 0.996070
\(324\) 0 0
\(325\) 0 0
\(326\) −2463.59 −0.418545
\(327\) 0 0
\(328\) 2859.21 0.481321
\(329\) −1821.23 −0.305190
\(330\) 0 0
\(331\) −7616.11 −1.26471 −0.632355 0.774678i \(-0.717912\pi\)
−0.632355 + 0.774678i \(0.717912\pi\)
\(332\) −2334.76 −0.385953
\(333\) 0 0
\(334\) 1362.21 0.223164
\(335\) 4367.13 0.712244
\(336\) 0 0
\(337\) −5959.66 −0.963334 −0.481667 0.876354i \(-0.659968\pi\)
−0.481667 + 0.876354i \(0.659968\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 8208.84 1.30937
\(341\) 5624.36 0.893185
\(342\) 0 0
\(343\) 3927.30 0.618234
\(344\) 251.768 0.0394605
\(345\) 0 0
\(346\) 400.092 0.0621649
\(347\) 9976.27 1.54338 0.771692 0.635997i \(-0.219411\pi\)
0.771692 + 0.635997i \(0.219411\pi\)
\(348\) 0 0
\(349\) 5660.84 0.868246 0.434123 0.900854i \(-0.357058\pi\)
0.434123 + 0.900854i \(0.357058\pi\)
\(350\) 160.737 0.0245479
\(351\) 0 0
\(352\) 7188.01 1.08841
\(353\) 682.431 0.102896 0.0514478 0.998676i \(-0.483616\pi\)
0.0514478 + 0.998676i \(0.483616\pi\)
\(354\) 0 0
\(355\) −4331.91 −0.647644
\(356\) −1628.48 −0.242442
\(357\) 0 0
\(358\) −858.672 −0.126766
\(359\) 8328.79 1.22445 0.612224 0.790685i \(-0.290275\pi\)
0.612224 + 0.790685i \(0.290275\pi\)
\(360\) 0 0
\(361\) −4433.51 −0.646379
\(362\) −2347.37 −0.340815
\(363\) 0 0
\(364\) 0 0
\(365\) −2916.89 −0.418293
\(366\) 0 0
\(367\) 559.900 0.0796364 0.0398182 0.999207i \(-0.487322\pi\)
0.0398182 + 0.999207i \(0.487322\pi\)
\(368\) 1226.09 0.173681
\(369\) 0 0
\(370\) 2415.25 0.339359
\(371\) 2368.80 0.331488
\(372\) 0 0
\(373\) 548.079 0.0760816 0.0380408 0.999276i \(-0.487888\pi\)
0.0380408 + 0.999276i \(0.487888\pi\)
\(374\) −4762.44 −0.658449
\(375\) 0 0
\(376\) −3204.07 −0.439461
\(377\) 0 0
\(378\) 0 0
\(379\) −4680.27 −0.634325 −0.317163 0.948371i \(-0.602730\pi\)
−0.317163 + 0.948371i \(0.602730\pi\)
\(380\) 3443.40 0.464849
\(381\) 0 0
\(382\) 2124.13 0.284503
\(383\) −9174.13 −1.22396 −0.611979 0.790874i \(-0.709626\pi\)
−0.611979 + 0.790874i \(0.709626\pi\)
\(384\) 0 0
\(385\) −3324.60 −0.440097
\(386\) −231.925 −0.0305821
\(387\) 0 0
\(388\) 1653.05 0.216291
\(389\) 346.645 0.0451816 0.0225908 0.999745i \(-0.492809\pi\)
0.0225908 + 0.999745i \(0.492809\pi\)
\(390\) 0 0
\(391\) −2718.23 −0.351578
\(392\) 3260.10 0.420050
\(393\) 0 0
\(394\) 902.331 0.115378
\(395\) 444.845 0.0566648
\(396\) 0 0
\(397\) −513.327 −0.0648946 −0.0324473 0.999473i \(-0.510330\pi\)
−0.0324473 + 0.999473i \(0.510330\pi\)
\(398\) 2366.69 0.298069
\(399\) 0 0
\(400\) −2054.85 −0.256856
\(401\) −11168.4 −1.39083 −0.695413 0.718610i \(-0.744779\pi\)
−0.695413 + 0.718610i \(0.744779\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 6938.52 0.854466
\(405\) 0 0
\(406\) 600.787 0.0734398
\(407\) 22487.2 2.73869
\(408\) 0 0
\(409\) −4286.40 −0.518212 −0.259106 0.965849i \(-0.583428\pi\)
−0.259106 + 0.965849i \(0.583428\pi\)
\(410\) −1709.24 −0.205886
\(411\) 0 0
\(412\) 9620.91 1.15046
\(413\) −3088.96 −0.368033
\(414\) 0 0
\(415\) 2878.42 0.340472
\(416\) 0 0
\(417\) 0 0
\(418\) −1997.72 −0.233760
\(419\) −3822.03 −0.445629 −0.222814 0.974861i \(-0.571524\pi\)
−0.222814 + 0.974861i \(0.571524\pi\)
\(420\) 0 0
\(421\) −2751.33 −0.318508 −0.159254 0.987238i \(-0.550909\pi\)
−0.159254 + 0.987238i \(0.550909\pi\)
\(422\) 1638.28 0.188982
\(423\) 0 0
\(424\) 4167.41 0.477329
\(425\) 4555.58 0.519948
\(426\) 0 0
\(427\) −3149.03 −0.356891
\(428\) 12250.9 1.38357
\(429\) 0 0
\(430\) −150.507 −0.0168793
\(431\) −8079.52 −0.902962 −0.451481 0.892281i \(-0.649104\pi\)
−0.451481 + 0.892281i \(0.649104\pi\)
\(432\) 0 0
\(433\) 3793.63 0.421040 0.210520 0.977590i \(-0.432484\pi\)
0.210520 + 0.977590i \(0.432484\pi\)
\(434\) −393.470 −0.0435189
\(435\) 0 0
\(436\) 12248.6 1.34542
\(437\) −1140.23 −0.124816
\(438\) 0 0
\(439\) 11525.0 1.25298 0.626491 0.779428i \(-0.284490\pi\)
0.626491 + 0.779428i \(0.284490\pi\)
\(440\) −5848.94 −0.633721
\(441\) 0 0
\(442\) 0 0
\(443\) 13127.4 1.40791 0.703954 0.710245i \(-0.251416\pi\)
0.703954 + 0.710245i \(0.251416\pi\)
\(444\) 0 0
\(445\) 2007.68 0.213872
\(446\) −951.705 −0.101042
\(447\) 0 0
\(448\) 2059.16 0.217157
\(449\) 7763.80 0.816028 0.408014 0.912976i \(-0.366221\pi\)
0.408014 + 0.912976i \(0.366221\pi\)
\(450\) 0 0
\(451\) −15913.9 −1.66154
\(452\) 15086.5 1.56993
\(453\) 0 0
\(454\) 3058.92 0.316216
\(455\) 0 0
\(456\) 0 0
\(457\) −291.436 −0.0298311 −0.0149156 0.999889i \(-0.504748\pi\)
−0.0149156 + 0.999889i \(0.504748\pi\)
\(458\) −2469.91 −0.251990
\(459\) 0 0
\(460\) −1618.75 −0.164075
\(461\) −1612.08 −0.162868 −0.0814338 0.996679i \(-0.525950\pi\)
−0.0814338 + 0.996679i \(0.525950\pi\)
\(462\) 0 0
\(463\) −916.747 −0.0920191 −0.0460095 0.998941i \(-0.514650\pi\)
−0.0460095 + 0.998941i \(0.514650\pi\)
\(464\) −7680.40 −0.768435
\(465\) 0 0
\(466\) −383.497 −0.0381227
\(467\) −3524.88 −0.349277 −0.174638 0.984633i \(-0.555876\pi\)
−0.174638 + 0.984633i \(0.555876\pi\)
\(468\) 0 0
\(469\) −2844.51 −0.280058
\(470\) 1915.40 0.187980
\(471\) 0 0
\(472\) −5434.37 −0.529952
\(473\) −1401.30 −0.136219
\(474\) 0 0
\(475\) 1910.95 0.184590
\(476\) −5346.79 −0.514852
\(477\) 0 0
\(478\) −2336.72 −0.223596
\(479\) 12392.8 1.18213 0.591066 0.806623i \(-0.298707\pi\)
0.591066 + 0.806623i \(0.298707\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −4239.33 −0.400615
\(483\) 0 0
\(484\) −16382.1 −1.53851
\(485\) −2037.97 −0.190803
\(486\) 0 0
\(487\) 14186.4 1.32002 0.660009 0.751258i \(-0.270552\pi\)
0.660009 + 0.751258i \(0.270552\pi\)
\(488\) −5540.06 −0.513907
\(489\) 0 0
\(490\) −1948.89 −0.179678
\(491\) −14559.4 −1.33820 −0.669101 0.743172i \(-0.733320\pi\)
−0.669101 + 0.743172i \(0.733320\pi\)
\(492\) 0 0
\(493\) 17027.3 1.55552
\(494\) 0 0
\(495\) 0 0
\(496\) 5030.09 0.455358
\(497\) 2821.57 0.254657
\(498\) 0 0
\(499\) 8470.02 0.759860 0.379930 0.925015i \(-0.375948\pi\)
0.379930 + 0.925015i \(0.375948\pi\)
\(500\) 11452.6 1.02435
\(501\) 0 0
\(502\) −4448.05 −0.395471
\(503\) −7649.38 −0.678070 −0.339035 0.940774i \(-0.610100\pi\)
−0.339035 + 0.940774i \(0.610100\pi\)
\(504\) 0 0
\(505\) −8554.19 −0.753775
\(506\) 939.134 0.0825091
\(507\) 0 0
\(508\) 14811.1 1.29358
\(509\) 13019.8 1.13378 0.566889 0.823794i \(-0.308147\pi\)
0.566889 + 0.823794i \(0.308147\pi\)
\(510\) 0 0
\(511\) 1899.90 0.164475
\(512\) 10935.9 0.943948
\(513\) 0 0
\(514\) 5131.02 0.440310
\(515\) −11861.2 −1.01489
\(516\) 0 0
\(517\) 17833.3 1.51704
\(518\) −1573.16 −0.133438
\(519\) 0 0
\(520\) 0 0
\(521\) −1003.37 −0.0843734 −0.0421867 0.999110i \(-0.513432\pi\)
−0.0421867 + 0.999110i \(0.513432\pi\)
\(522\) 0 0
\(523\) −4216.27 −0.352514 −0.176257 0.984344i \(-0.556399\pi\)
−0.176257 + 0.984344i \(0.556399\pi\)
\(524\) −21567.3 −1.79804
\(525\) 0 0
\(526\) −2767.88 −0.229439
\(527\) −11151.6 −0.921770
\(528\) 0 0
\(529\) −11631.0 −0.955944
\(530\) −2491.29 −0.204179
\(531\) 0 0
\(532\) −2242.84 −0.182781
\(533\) 0 0
\(534\) 0 0
\(535\) −15103.5 −1.22053
\(536\) −5004.32 −0.403272
\(537\) 0 0
\(538\) −5289.00 −0.423838
\(539\) −18145.1 −1.45003
\(540\) 0 0
\(541\) 23139.4 1.83889 0.919447 0.393213i \(-0.128637\pi\)
0.919447 + 0.393213i \(0.128637\pi\)
\(542\) 4510.44 0.357454
\(543\) 0 0
\(544\) −14251.9 −1.12325
\(545\) −15100.8 −1.18688
\(546\) 0 0
\(547\) −12532.9 −0.979646 −0.489823 0.871822i \(-0.662939\pi\)
−0.489823 + 0.871822i \(0.662939\pi\)
\(548\) 13363.5 1.04171
\(549\) 0 0
\(550\) −1573.93 −0.122023
\(551\) 7142.53 0.552236
\(552\) 0 0
\(553\) −289.748 −0.0222809
\(554\) 2596.63 0.199134
\(555\) 0 0
\(556\) 10161.6 0.775084
\(557\) 6240.38 0.474710 0.237355 0.971423i \(-0.423720\pi\)
0.237355 + 0.971423i \(0.423720\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2973.32 −0.224368
\(561\) 0 0
\(562\) −4390.25 −0.329522
\(563\) −6742.87 −0.504756 −0.252378 0.967629i \(-0.581213\pi\)
−0.252378 + 0.967629i \(0.581213\pi\)
\(564\) 0 0
\(565\) −18599.4 −1.38493
\(566\) −5020.41 −0.372833
\(567\) 0 0
\(568\) 4963.96 0.366695
\(569\) 2935.57 0.216283 0.108142 0.994135i \(-0.465510\pi\)
0.108142 + 0.994135i \(0.465510\pi\)
\(570\) 0 0
\(571\) 9709.50 0.711611 0.355805 0.934560i \(-0.384207\pi\)
0.355805 + 0.934560i \(0.384207\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1113.31 0.0809556
\(575\) −898.342 −0.0651538
\(576\) 0 0
\(577\) 18892.5 1.36309 0.681546 0.731775i \(-0.261308\pi\)
0.681546 + 0.731775i \(0.261308\pi\)
\(578\) 6077.14 0.437329
\(579\) 0 0
\(580\) 10140.1 0.725936
\(581\) −1874.84 −0.133875
\(582\) 0 0
\(583\) −23195.1 −1.64776
\(584\) 3342.48 0.236837
\(585\) 0 0
\(586\) −361.573 −0.0254888
\(587\) −24642.6 −1.73272 −0.866362 0.499416i \(-0.833548\pi\)
−0.866362 + 0.499416i \(0.833548\pi\)
\(588\) 0 0
\(589\) −4677.82 −0.327243
\(590\) 3248.68 0.226688
\(591\) 0 0
\(592\) 20111.2 1.39622
\(593\) 6433.11 0.445491 0.222745 0.974877i \(-0.428498\pi\)
0.222745 + 0.974877i \(0.428498\pi\)
\(594\) 0 0
\(595\) 6591.82 0.454182
\(596\) 9545.30 0.656025
\(597\) 0 0
\(598\) 0 0
\(599\) −19055.2 −1.29979 −0.649897 0.760023i \(-0.725188\pi\)
−0.649897 + 0.760023i \(0.725188\pi\)
\(600\) 0 0
\(601\) −20141.7 −1.36705 −0.683524 0.729928i \(-0.739554\pi\)
−0.683524 + 0.729928i \(0.739554\pi\)
\(602\) 98.0323 0.00663704
\(603\) 0 0
\(604\) 22055.0 1.48577
\(605\) 20196.8 1.35722
\(606\) 0 0
\(607\) −10627.2 −0.710620 −0.355310 0.934749i \(-0.615625\pi\)
−0.355310 + 0.934749i \(0.615625\pi\)
\(608\) −5978.32 −0.398771
\(609\) 0 0
\(610\) 3311.86 0.219825
\(611\) 0 0
\(612\) 0 0
\(613\) −18460.9 −1.21636 −0.608181 0.793798i \(-0.708101\pi\)
−0.608181 + 0.793798i \(0.708101\pi\)
\(614\) −566.537 −0.0372371
\(615\) 0 0
\(616\) 3809.68 0.249182
\(617\) 21119.7 1.37804 0.689018 0.724744i \(-0.258042\pi\)
0.689018 + 0.724744i \(0.258042\pi\)
\(618\) 0 0
\(619\) −571.628 −0.0371174 −0.0185587 0.999828i \(-0.505908\pi\)
−0.0185587 + 0.999828i \(0.505908\pi\)
\(620\) −6640.97 −0.430174
\(621\) 0 0
\(622\) 1385.48 0.0893132
\(623\) −1307.69 −0.0840957
\(624\) 0 0
\(625\) −9269.25 −0.593232
\(626\) −4685.23 −0.299137
\(627\) 0 0
\(628\) −18978.0 −1.20590
\(629\) −44586.2 −2.82634
\(630\) 0 0
\(631\) 7543.32 0.475903 0.237952 0.971277i \(-0.423524\pi\)
0.237952 + 0.971277i \(0.423524\pi\)
\(632\) −509.751 −0.0320835
\(633\) 0 0
\(634\) 5161.88 0.323351
\(635\) −18260.0 −1.14114
\(636\) 0 0
\(637\) 0 0
\(638\) −5882.85 −0.365054
\(639\) 0 0
\(640\) −11181.8 −0.690621
\(641\) 6800.31 0.419027 0.209513 0.977806i \(-0.432812\pi\)
0.209513 + 0.977806i \(0.432812\pi\)
\(642\) 0 0
\(643\) −11713.1 −0.718380 −0.359190 0.933265i \(-0.616947\pi\)
−0.359190 + 0.933265i \(0.616947\pi\)
\(644\) 1054.37 0.0645153
\(645\) 0 0
\(646\) 3960.96 0.241241
\(647\) 12455.4 0.756835 0.378417 0.925635i \(-0.376468\pi\)
0.378417 + 0.925635i \(0.376468\pi\)
\(648\) 0 0
\(649\) 30246.8 1.82941
\(650\) 0 0
\(651\) 0 0
\(652\) 27083.2 1.62678
\(653\) 14073.0 0.843369 0.421685 0.906743i \(-0.361439\pi\)
0.421685 + 0.906743i \(0.361439\pi\)
\(654\) 0 0
\(655\) 26589.4 1.58616
\(656\) −14232.4 −0.847076
\(657\) 0 0
\(658\) −1247.59 −0.0739149
\(659\) −8037.27 −0.475095 −0.237548 0.971376i \(-0.576344\pi\)
−0.237548 + 0.971376i \(0.576344\pi\)
\(660\) 0 0
\(661\) −1994.71 −0.117375 −0.0586877 0.998276i \(-0.518692\pi\)
−0.0586877 + 0.998276i \(0.518692\pi\)
\(662\) −5217.23 −0.306304
\(663\) 0 0
\(664\) −3298.40 −0.192775
\(665\) 2765.10 0.161242
\(666\) 0 0
\(667\) −3357.73 −0.194920
\(668\) −14975.3 −0.867382
\(669\) 0 0
\(670\) 2991.59 0.172500
\(671\) 30835.0 1.77403
\(672\) 0 0
\(673\) 13886.1 0.795348 0.397674 0.917527i \(-0.369817\pi\)
0.397674 + 0.917527i \(0.369817\pi\)
\(674\) −4082.52 −0.233313
\(675\) 0 0
\(676\) 0 0
\(677\) 11024.0 0.625831 0.312915 0.949781i \(-0.398694\pi\)
0.312915 + 0.949781i \(0.398694\pi\)
\(678\) 0 0
\(679\) 1327.42 0.0750247
\(680\) 11596.9 0.654002
\(681\) 0 0
\(682\) 3852.83 0.216323
\(683\) 22696.5 1.27154 0.635768 0.771880i \(-0.280683\pi\)
0.635768 + 0.771880i \(0.280683\pi\)
\(684\) 0 0
\(685\) −16475.2 −0.918957
\(686\) 2690.30 0.149732
\(687\) 0 0
\(688\) −1253.24 −0.0694465
\(689\) 0 0
\(690\) 0 0
\(691\) 3516.06 0.193570 0.0967852 0.995305i \(-0.469144\pi\)
0.0967852 + 0.995305i \(0.469144\pi\)
\(692\) −4398.36 −0.241619
\(693\) 0 0
\(694\) 6834.00 0.373797
\(695\) −12527.8 −0.683748
\(696\) 0 0
\(697\) 31553.0 1.71471
\(698\) 3877.82 0.210283
\(699\) 0 0
\(700\) −1767.05 −0.0954115
\(701\) 26652.4 1.43602 0.718009 0.696034i \(-0.245054\pi\)
0.718009 + 0.696034i \(0.245054\pi\)
\(702\) 0 0
\(703\) −18702.7 −1.00340
\(704\) −20163.1 −1.07944
\(705\) 0 0
\(706\) 467.482 0.0249206
\(707\) 5571.73 0.296388
\(708\) 0 0
\(709\) 8938.91 0.473495 0.236747 0.971571i \(-0.423919\pi\)
0.236747 + 0.971571i \(0.423919\pi\)
\(710\) −2967.46 −0.156855
\(711\) 0 0
\(712\) −2300.61 −0.121094
\(713\) 2199.06 0.115505
\(714\) 0 0
\(715\) 0 0
\(716\) 9439.70 0.492707
\(717\) 0 0
\(718\) 5705.43 0.296553
\(719\) −5294.09 −0.274599 −0.137299 0.990530i \(-0.543842\pi\)
−0.137299 + 0.990530i \(0.543842\pi\)
\(720\) 0 0
\(721\) 7725.73 0.399059
\(722\) −3037.07 −0.156548
\(723\) 0 0
\(724\) 25805.5 1.32466
\(725\) 5627.32 0.288267
\(726\) 0 0
\(727\) 30010.0 1.53096 0.765480 0.643459i \(-0.222501\pi\)
0.765480 + 0.643459i \(0.222501\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1998.14 −0.101308
\(731\) 2778.40 0.140579
\(732\) 0 0
\(733\) 35185.8 1.77301 0.886505 0.462720i \(-0.153126\pi\)
0.886505 + 0.462720i \(0.153126\pi\)
\(734\) 383.546 0.0192874
\(735\) 0 0
\(736\) 2810.43 0.140752
\(737\) 27853.2 1.39211
\(738\) 0 0
\(739\) 16737.3 0.833140 0.416570 0.909104i \(-0.363232\pi\)
0.416570 + 0.909104i \(0.363232\pi\)
\(740\) −26551.8 −1.31900
\(741\) 0 0
\(742\) 1622.69 0.0802842
\(743\) 14564.0 0.719115 0.359558 0.933123i \(-0.382928\pi\)
0.359558 + 0.933123i \(0.382928\pi\)
\(744\) 0 0
\(745\) −11768.0 −0.578718
\(746\) 375.448 0.0184264
\(747\) 0 0
\(748\) 52355.3 2.55922
\(749\) 9837.63 0.479919
\(750\) 0 0
\(751\) 22935.6 1.11442 0.557212 0.830370i \(-0.311871\pi\)
0.557212 + 0.830370i \(0.311871\pi\)
\(752\) 15949.0 0.773406
\(753\) 0 0
\(754\) 0 0
\(755\) −27190.6 −1.31069
\(756\) 0 0
\(757\) 15356.3 0.737296 0.368648 0.929569i \(-0.379821\pi\)
0.368648 + 0.929569i \(0.379821\pi\)
\(758\) −3206.10 −0.153629
\(759\) 0 0
\(760\) 4864.61 0.232181
\(761\) −25838.1 −1.23079 −0.615394 0.788219i \(-0.711003\pi\)
−0.615394 + 0.788219i \(0.711003\pi\)
\(762\) 0 0
\(763\) 9835.84 0.466686
\(764\) −23351.4 −1.10579
\(765\) 0 0
\(766\) −6284.51 −0.296434
\(767\) 0 0
\(768\) 0 0
\(769\) 11315.0 0.530597 0.265299 0.964166i \(-0.414529\pi\)
0.265299 + 0.964166i \(0.414529\pi\)
\(770\) −2277.44 −0.106588
\(771\) 0 0
\(772\) 2549.64 0.118865
\(773\) 1991.25 0.0926524 0.0463262 0.998926i \(-0.485249\pi\)
0.0463262 + 0.998926i \(0.485249\pi\)
\(774\) 0 0
\(775\) −3685.47 −0.170821
\(776\) 2335.32 0.108032
\(777\) 0 0
\(778\) 237.461 0.0109427
\(779\) 13235.7 0.608752
\(780\) 0 0
\(781\) −27628.5 −1.26585
\(782\) −1862.06 −0.0851497
\(783\) 0 0
\(784\) −16227.9 −0.739246
\(785\) 23397.1 1.06380
\(786\) 0 0
\(787\) −13262.7 −0.600717 −0.300359 0.953826i \(-0.597106\pi\)
−0.300359 + 0.953826i \(0.597106\pi\)
\(788\) −9919.66 −0.448443
\(789\) 0 0
\(790\) 304.730 0.0137238
\(791\) 12114.7 0.544561
\(792\) 0 0
\(793\) 0 0
\(794\) −351.642 −0.0157170
\(795\) 0 0
\(796\) −26017.9 −1.15852
\(797\) −36462.7 −1.62055 −0.810273 0.586052i \(-0.800681\pi\)
−0.810273 + 0.586052i \(0.800681\pi\)
\(798\) 0 0
\(799\) −35358.8 −1.56559
\(800\) −4710.09 −0.208158
\(801\) 0 0
\(802\) −7650.61 −0.336848
\(803\) −18603.7 −0.817570
\(804\) 0 0
\(805\) −1299.88 −0.0569128
\(806\) 0 0
\(807\) 0 0
\(808\) 9802.29 0.426786
\(809\) 2469.45 0.107319 0.0536596 0.998559i \(-0.482911\pi\)
0.0536596 + 0.998559i \(0.482911\pi\)
\(810\) 0 0
\(811\) 42081.3 1.82204 0.911019 0.412363i \(-0.135297\pi\)
0.911019 + 0.412363i \(0.135297\pi\)
\(812\) −6604.68 −0.285442
\(813\) 0 0
\(814\) 15404.3 0.663292
\(815\) −33389.6 −1.43508
\(816\) 0 0
\(817\) 1165.47 0.0499077
\(818\) −2936.29 −0.125507
\(819\) 0 0
\(820\) 18790.3 0.800228
\(821\) 43282.1 1.83990 0.919949 0.392038i \(-0.128230\pi\)
0.919949 + 0.392038i \(0.128230\pi\)
\(822\) 0 0
\(823\) −38193.7 −1.61768 −0.808838 0.588031i \(-0.799903\pi\)
−0.808838 + 0.588031i \(0.799903\pi\)
\(824\) 13591.8 0.574628
\(825\) 0 0
\(826\) −2116.01 −0.0891350
\(827\) −35724.1 −1.50211 −0.751057 0.660238i \(-0.770456\pi\)
−0.751057 + 0.660238i \(0.770456\pi\)
\(828\) 0 0
\(829\) 7345.94 0.307762 0.153881 0.988089i \(-0.450823\pi\)
0.153881 + 0.988089i \(0.450823\pi\)
\(830\) 1971.79 0.0824600
\(831\) 0 0
\(832\) 0 0
\(833\) 35977.1 1.49644
\(834\) 0 0
\(835\) 18462.3 0.765169
\(836\) 21961.7 0.908566
\(837\) 0 0
\(838\) −2618.19 −0.107928
\(839\) −11691.7 −0.481100 −0.240550 0.970637i \(-0.577328\pi\)
−0.240550 + 0.970637i \(0.577328\pi\)
\(840\) 0 0
\(841\) −3355.80 −0.137595
\(842\) −1884.73 −0.0771404
\(843\) 0 0
\(844\) −18010.3 −0.734525
\(845\) 0 0
\(846\) 0 0
\(847\) −13155.1 −0.533664
\(848\) −20744.3 −0.840051
\(849\) 0 0
\(850\) 3120.68 0.125928
\(851\) 8792.22 0.354164
\(852\) 0 0
\(853\) −15321.4 −0.615000 −0.307500 0.951548i \(-0.599492\pi\)
−0.307500 + 0.951548i \(0.599492\pi\)
\(854\) −2157.17 −0.0864364
\(855\) 0 0
\(856\) 17307.2 0.691063
\(857\) 32949.6 1.31335 0.656673 0.754176i \(-0.271963\pi\)
0.656673 + 0.754176i \(0.271963\pi\)
\(858\) 0 0
\(859\) −8132.72 −0.323032 −0.161516 0.986870i \(-0.551638\pi\)
−0.161516 + 0.986870i \(0.551638\pi\)
\(860\) 1654.58 0.0656056
\(861\) 0 0
\(862\) −5534.67 −0.218691
\(863\) 25770.4 1.01649 0.508247 0.861212i \(-0.330294\pi\)
0.508247 + 0.861212i \(0.330294\pi\)
\(864\) 0 0
\(865\) 5422.54 0.213147
\(866\) 2598.73 0.101973
\(867\) 0 0
\(868\) 4325.57 0.169147
\(869\) 2837.19 0.110754
\(870\) 0 0
\(871\) 0 0
\(872\) 17304.1 0.672008
\(873\) 0 0
\(874\) −781.086 −0.0302295
\(875\) 9196.64 0.355318
\(876\) 0 0
\(877\) −31715.4 −1.22116 −0.610578 0.791956i \(-0.709063\pi\)
−0.610578 + 0.791956i \(0.709063\pi\)
\(878\) 7894.93 0.303464
\(879\) 0 0
\(880\) 29114.5 1.11528
\(881\) 29755.9 1.13791 0.568957 0.822367i \(-0.307347\pi\)
0.568957 + 0.822367i \(0.307347\pi\)
\(882\) 0 0
\(883\) 41818.4 1.59377 0.796886 0.604129i \(-0.206479\pi\)
0.796886 + 0.604129i \(0.206479\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 8992.63 0.340986
\(887\) −6514.49 −0.246601 −0.123301 0.992369i \(-0.539348\pi\)
−0.123301 + 0.992369i \(0.539348\pi\)
\(888\) 0 0
\(889\) 11893.5 0.448703
\(890\) 1375.31 0.0517984
\(891\) 0 0
\(892\) 10462.5 0.392723
\(893\) −14832.1 −0.555809
\(894\) 0 0
\(895\) −11637.8 −0.434646
\(896\) 7283.19 0.271556
\(897\) 0 0
\(898\) 5318.40 0.197636
\(899\) −13775.2 −0.511043
\(900\) 0 0
\(901\) 45989.9 1.70049
\(902\) −10901.4 −0.402413
\(903\) 0 0
\(904\) 21313.2 0.784145
\(905\) −31814.4 −1.16856
\(906\) 0 0
\(907\) −1750.33 −0.0640780 −0.0320390 0.999487i \(-0.510200\pi\)
−0.0320390 + 0.999487i \(0.510200\pi\)
\(908\) −33627.8 −1.22905
\(909\) 0 0
\(910\) 0 0
\(911\) 14268.9 0.518936 0.259468 0.965752i \(-0.416453\pi\)
0.259468 + 0.965752i \(0.416453\pi\)
\(912\) 0 0
\(913\) 18358.3 0.665467
\(914\) −199.641 −0.00722489
\(915\) 0 0
\(916\) 27152.7 0.979421
\(917\) −17318.9 −0.623685
\(918\) 0 0
\(919\) −19466.2 −0.698729 −0.349364 0.936987i \(-0.613602\pi\)
−0.349364 + 0.936987i \(0.613602\pi\)
\(920\) −2286.87 −0.0819520
\(921\) 0 0
\(922\) −1104.31 −0.0394454
\(923\) 0 0
\(924\) 0 0
\(925\) −14735.2 −0.523772
\(926\) −627.995 −0.0222864
\(927\) 0 0
\(928\) −17604.9 −0.622745
\(929\) −866.279 −0.0305939 −0.0152969 0.999883i \(-0.504869\pi\)
−0.0152969 + 0.999883i \(0.504869\pi\)
\(930\) 0 0
\(931\) 15091.5 0.531259
\(932\) 4215.93 0.148173
\(933\) 0 0
\(934\) −2414.63 −0.0845924
\(935\) −64546.5 −2.25764
\(936\) 0 0
\(937\) −33092.8 −1.15378 −0.576891 0.816821i \(-0.695734\pi\)
−0.576891 + 0.816821i \(0.695734\pi\)
\(938\) −1948.56 −0.0678281
\(939\) 0 0
\(940\) −21056.7 −0.730632
\(941\) −27089.5 −0.938461 −0.469231 0.883076i \(-0.655469\pi\)
−0.469231 + 0.883076i \(0.655469\pi\)
\(942\) 0 0
\(943\) −6222.14 −0.214868
\(944\) 27050.9 0.932661
\(945\) 0 0
\(946\) −959.923 −0.0329913
\(947\) −45146.5 −1.54917 −0.774586 0.632469i \(-0.782042\pi\)
−0.774586 + 0.632469i \(0.782042\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 1309.05 0.0447064
\(951\) 0 0
\(952\) −7553.60 −0.257157
\(953\) 32600.5 1.10812 0.554058 0.832478i \(-0.313079\pi\)
0.554058 + 0.832478i \(0.313079\pi\)
\(954\) 0 0
\(955\) 28788.9 0.975484
\(956\) 25688.4 0.869062
\(957\) 0 0
\(958\) 8489.38 0.286304
\(959\) 10731.1 0.361339
\(960\) 0 0
\(961\) −20769.3 −0.697167
\(962\) 0 0
\(963\) 0 0
\(964\) 46604.6 1.55709
\(965\) −3143.34 −0.104858
\(966\) 0 0
\(967\) −13124.6 −0.436462 −0.218231 0.975897i \(-0.570029\pi\)
−0.218231 + 0.975897i \(0.570029\pi\)
\(968\) −23143.6 −0.768454
\(969\) 0 0
\(970\) −1396.06 −0.0462111
\(971\) 7776.85 0.257025 0.128512 0.991708i \(-0.458980\pi\)
0.128512 + 0.991708i \(0.458980\pi\)
\(972\) 0 0
\(973\) 8159.90 0.268853
\(974\) 9718.07 0.319699
\(975\) 0 0
\(976\) 27577.0 0.904425
\(977\) −25662.3 −0.840339 −0.420169 0.907446i \(-0.638029\pi\)
−0.420169 + 0.907446i \(0.638029\pi\)
\(978\) 0 0
\(979\) 12804.8 0.418022
\(980\) 21424.9 0.698361
\(981\) 0 0
\(982\) −9973.56 −0.324103
\(983\) 1747.89 0.0567132 0.0283566 0.999598i \(-0.490973\pi\)
0.0283566 + 0.999598i \(0.490973\pi\)
\(984\) 0 0
\(985\) 12229.5 0.395598
\(986\) 11664.2 0.376737
\(987\) 0 0
\(988\) 0 0
\(989\) −547.891 −0.0176157
\(990\) 0 0
\(991\) −14983.4 −0.480286 −0.240143 0.970738i \(-0.577194\pi\)
−0.240143 + 0.970738i \(0.577194\pi\)
\(992\) 11529.9 0.369026
\(993\) 0 0
\(994\) 1932.85 0.0616762
\(995\) 32076.3 1.02200
\(996\) 0 0
\(997\) −51834.8 −1.64656 −0.823282 0.567632i \(-0.807860\pi\)
−0.823282 + 0.567632i \(0.807860\pi\)
\(998\) 5802.18 0.184033
\(999\) 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 1521.4.a.bm.1.10 yes 18
3.2 odd 2 inner 1521.4.a.bm.1.9 18
13.12 even 2 1521.4.a.bn.1.9 yes 18
39.38 odd 2 1521.4.a.bn.1.10 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1521.4.a.bm.1.9 18 3.2 odd 2 inner
1521.4.a.bm.1.10 yes 18 1.1 even 1 trivial
1521.4.a.bn.1.9 yes 18 13.12 even 2
1521.4.a.bn.1.10 yes 18 39.38 odd 2