Properties

Label 1856.4.a.bj.1.6
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 2 x^{11} - 217 x^{10} + 520 x^{9} + 16022 x^{8} - 37368 x^{7} - 509640 x^{6} + 989168 x^{5} + 7106592 x^{4} - 7979328 x^{3} - 38912400 x^{2} + \cdots + 14751072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: no (minimal twist has level 928)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.613335\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.61334 q^{3} +10.5322 q^{5} +13.7418 q^{7} -24.3972 q^{9} +O(q^{10})\) \(q-1.61334 q^{3} +10.5322 q^{5} +13.7418 q^{7} -24.3972 q^{9} +30.2948 q^{11} -19.9286 q^{13} -16.9919 q^{15} +105.498 q^{17} +79.5429 q^{19} -22.1701 q^{21} -14.8819 q^{23} -14.0732 q^{25} +82.9208 q^{27} -29.0000 q^{29} -61.9562 q^{31} -48.8756 q^{33} +144.731 q^{35} -6.43644 q^{37} +32.1516 q^{39} +351.682 q^{41} -473.240 q^{43} -256.955 q^{45} +492.253 q^{47} -154.163 q^{49} -170.203 q^{51} +668.911 q^{53} +319.070 q^{55} -128.329 q^{57} -12.7826 q^{59} -80.8937 q^{61} -335.261 q^{63} -209.892 q^{65} -529.153 q^{67} +24.0095 q^{69} -211.869 q^{71} +627.334 q^{73} +22.7048 q^{75} +416.305 q^{77} +915.349 q^{79} +524.944 q^{81} -411.330 q^{83} +1111.12 q^{85} +46.7867 q^{87} +1377.88 q^{89} -273.856 q^{91} +99.9561 q^{93} +837.761 q^{95} -649.398 q^{97} -739.106 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9} - 46 q^{11} + 34 q^{13} - 50 q^{15} + 36 q^{17} - 148 q^{19} + 92 q^{21} + 328 q^{23} + 486 q^{25} - 326 q^{27} - 348 q^{29} + 374 q^{31} + 710 q^{33} - 204 q^{35} + 340 q^{37} - 122 q^{39} + 32 q^{41} - 462 q^{43} + 1132 q^{45} + 434 q^{47} + 1508 q^{49} - 440 q^{51} - 610 q^{53} + 46 q^{55} - 932 q^{57} - 1240 q^{59} + 1228 q^{61} + 4240 q^{63} + 730 q^{65} - 1672 q^{67} + 528 q^{69} + 3220 q^{71} + 564 q^{73} - 6032 q^{75} - 644 q^{77} + 1862 q^{79} + 3040 q^{81} - 3736 q^{83} + 808 q^{85} + 406 q^{87} + 584 q^{89} - 4844 q^{91} + 3226 q^{93} + 2844 q^{95} + 904 q^{97} - 6832 q^{99}+O(q^{100}) \) 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 0
\(3\) −1.61334 −0.310486 −0.155243 0.987876i \(-0.549616\pi\)
−0.155243 + 0.987876i \(0.549616\pi\)
\(4\) 0 0
\(5\) 10.5322 0.942027 0.471013 0.882126i \(-0.343888\pi\)
0.471013 + 0.882126i \(0.343888\pi\)
\(6\) 0 0
\(7\) 13.7418 0.741988 0.370994 0.928635i \(-0.379017\pi\)
0.370994 + 0.928635i \(0.379017\pi\)
\(8\) 0 0
\(9\) −24.3972 −0.903598
\(10\) 0 0
\(11\) 30.2948 0.830384 0.415192 0.909734i \(-0.363714\pi\)
0.415192 + 0.909734i \(0.363714\pi\)
\(12\) 0 0
\(13\) −19.9286 −0.425170 −0.212585 0.977143i \(-0.568188\pi\)
−0.212585 + 0.977143i \(0.568188\pi\)
\(14\) 0 0
\(15\) −16.9919 −0.292487
\(16\) 0 0
\(17\) 105.498 1.50512 0.752558 0.658526i \(-0.228820\pi\)
0.752558 + 0.658526i \(0.228820\pi\)
\(18\) 0 0
\(19\) 79.5429 0.960442 0.480221 0.877147i \(-0.340556\pi\)
0.480221 + 0.877147i \(0.340556\pi\)
\(20\) 0 0
\(21\) −22.1701 −0.230377
\(22\) 0 0
\(23\) −14.8819 −0.134917 −0.0674586 0.997722i \(-0.521489\pi\)
−0.0674586 + 0.997722i \(0.521489\pi\)
\(24\) 0 0
\(25\) −14.0732 −0.112586
\(26\) 0 0
\(27\) 82.9208 0.591041
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) −61.9562 −0.358957 −0.179478 0.983762i \(-0.557441\pi\)
−0.179478 + 0.983762i \(0.557441\pi\)
\(32\) 0 0
\(33\) −48.8756 −0.257823
\(34\) 0 0
\(35\) 144.731 0.698972
\(36\) 0 0
\(37\) −6.43644 −0.0285985 −0.0142993 0.999898i \(-0.504552\pi\)
−0.0142993 + 0.999898i \(0.504552\pi\)
\(38\) 0 0
\(39\) 32.1516 0.132010
\(40\) 0 0
\(41\) 351.682 1.33960 0.669799 0.742543i \(-0.266380\pi\)
0.669799 + 0.742543i \(0.266380\pi\)
\(42\) 0 0
\(43\) −473.240 −1.67834 −0.839169 0.543871i \(-0.816958\pi\)
−0.839169 + 0.543871i \(0.816958\pi\)
\(44\) 0 0
\(45\) −256.955 −0.851214
\(46\) 0 0
\(47\) 492.253 1.52771 0.763856 0.645387i \(-0.223304\pi\)
0.763856 + 0.645387i \(0.223304\pi\)
\(48\) 0 0
\(49\) −154.163 −0.449454
\(50\) 0 0
\(51\) −170.203 −0.467318
\(52\) 0 0
\(53\) 668.911 1.73362 0.866811 0.498637i \(-0.166166\pi\)
0.866811 + 0.498637i \(0.166166\pi\)
\(54\) 0 0
\(55\) 319.070 0.782244
\(56\) 0 0
\(57\) −128.329 −0.298204
\(58\) 0 0
\(59\) −12.7826 −0.0282060 −0.0141030 0.999901i \(-0.504489\pi\)
−0.0141030 + 0.999901i \(0.504489\pi\)
\(60\) 0 0
\(61\) −80.8937 −0.169793 −0.0848965 0.996390i \(-0.527056\pi\)
−0.0848965 + 0.996390i \(0.527056\pi\)
\(62\) 0 0
\(63\) −335.261 −0.670459
\(64\) 0 0
\(65\) −209.892 −0.400521
\(66\) 0 0
\(67\) −529.153 −0.964870 −0.482435 0.875932i \(-0.660248\pi\)
−0.482435 + 0.875932i \(0.660248\pi\)
\(68\) 0 0
\(69\) 24.0095 0.0418899
\(70\) 0 0
\(71\) −211.869 −0.354144 −0.177072 0.984198i \(-0.556663\pi\)
−0.177072 + 0.984198i \(0.556663\pi\)
\(72\) 0 0
\(73\) 627.334 1.00581 0.502903 0.864343i \(-0.332265\pi\)
0.502903 + 0.864343i \(0.332265\pi\)
\(74\) 0 0
\(75\) 22.7048 0.0349563
\(76\) 0 0
\(77\) 416.305 0.616135
\(78\) 0 0
\(79\) 915.349 1.30361 0.651803 0.758389i \(-0.274013\pi\)
0.651803 + 0.758389i \(0.274013\pi\)
\(80\) 0 0
\(81\) 524.944 0.720088
\(82\) 0 0
\(83\) −411.330 −0.543967 −0.271984 0.962302i \(-0.587680\pi\)
−0.271984 + 0.962302i \(0.587680\pi\)
\(84\) 0 0
\(85\) 1111.12 1.41786
\(86\) 0 0
\(87\) 46.7867 0.0576559
\(88\) 0 0
\(89\) 1377.88 1.64107 0.820536 0.571595i \(-0.193675\pi\)
0.820536 + 0.571595i \(0.193675\pi\)
\(90\) 0 0
\(91\) −273.856 −0.315471
\(92\) 0 0
\(93\) 99.9561 0.111451
\(94\) 0 0
\(95\) 837.761 0.904762
\(96\) 0 0
\(97\) −649.398 −0.679756 −0.339878 0.940470i \(-0.610386\pi\)
−0.339878 + 0.940470i \(0.610386\pi\)
\(98\) 0 0
\(99\) −739.106 −0.750333
\(100\) 0 0
\(101\) 851.483 0.838869 0.419434 0.907786i \(-0.362228\pi\)
0.419434 + 0.907786i \(0.362228\pi\)
\(102\) 0 0
\(103\) −1923.60 −1.84017 −0.920085 0.391719i \(-0.871881\pi\)
−0.920085 + 0.391719i \(0.871881\pi\)
\(104\) 0 0
\(105\) −233.500 −0.217021
\(106\) 0 0
\(107\) −1648.55 −1.48945 −0.744726 0.667370i \(-0.767420\pi\)
−0.744726 + 0.667370i \(0.767420\pi\)
\(108\) 0 0
\(109\) −770.267 −0.676865 −0.338432 0.940991i \(-0.609897\pi\)
−0.338432 + 0.940991i \(0.609897\pi\)
\(110\) 0 0
\(111\) 10.3841 0.00887945
\(112\) 0 0
\(113\) 1429.70 1.19022 0.595111 0.803643i \(-0.297108\pi\)
0.595111 + 0.803643i \(0.297108\pi\)
\(114\) 0 0
\(115\) −156.739 −0.127096
\(116\) 0 0
\(117\) 486.202 0.384183
\(118\) 0 0
\(119\) 1449.73 1.11678
\(120\) 0 0
\(121\) −413.226 −0.310463
\(122\) 0 0
\(123\) −567.381 −0.415927
\(124\) 0 0
\(125\) −1464.74 −1.04809
\(126\) 0 0
\(127\) 2102.32 1.46890 0.734451 0.678662i \(-0.237440\pi\)
0.734451 + 0.678662i \(0.237440\pi\)
\(128\) 0 0
\(129\) 763.495 0.521101
\(130\) 0 0
\(131\) −1376.03 −0.917742 −0.458871 0.888503i \(-0.651746\pi\)
−0.458871 + 0.888503i \(0.651746\pi\)
\(132\) 0 0
\(133\) 1093.06 0.712637
\(134\) 0 0
\(135\) 873.337 0.556777
\(136\) 0 0
\(137\) 304.947 0.190171 0.0950853 0.995469i \(-0.469688\pi\)
0.0950853 + 0.995469i \(0.469688\pi\)
\(138\) 0 0
\(139\) 364.348 0.222328 0.111164 0.993802i \(-0.464542\pi\)
0.111164 + 0.993802i \(0.464542\pi\)
\(140\) 0 0
\(141\) −794.169 −0.474334
\(142\) 0 0
\(143\) −603.734 −0.353054
\(144\) 0 0
\(145\) −305.433 −0.174930
\(146\) 0 0
\(147\) 248.716 0.139549
\(148\) 0 0
\(149\) −200.018 −0.109974 −0.0549869 0.998487i \(-0.517512\pi\)
−0.0549869 + 0.998487i \(0.517512\pi\)
\(150\) 0 0
\(151\) −35.5937 −0.0191826 −0.00959130 0.999954i \(-0.503053\pi\)
−0.00959130 + 0.999954i \(0.503053\pi\)
\(152\) 0 0
\(153\) −2573.85 −1.36002
\(154\) 0 0
\(155\) −652.534 −0.338147
\(156\) 0 0
\(157\) −137.397 −0.0698437 −0.0349218 0.999390i \(-0.511118\pi\)
−0.0349218 + 0.999390i \(0.511118\pi\)
\(158\) 0 0
\(159\) −1079.18 −0.538266
\(160\) 0 0
\(161\) −204.504 −0.100107
\(162\) 0 0
\(163\) 867.825 0.417014 0.208507 0.978021i \(-0.433140\pi\)
0.208507 + 0.978021i \(0.433140\pi\)
\(164\) 0 0
\(165\) −514.767 −0.242876
\(166\) 0 0
\(167\) −66.2847 −0.0307142 −0.0153571 0.999882i \(-0.504889\pi\)
−0.0153571 + 0.999882i \(0.504889\pi\)
\(168\) 0 0
\(169\) −1799.85 −0.819230
\(170\) 0 0
\(171\) −1940.62 −0.867854
\(172\) 0 0
\(173\) 1798.76 0.790505 0.395253 0.918572i \(-0.370657\pi\)
0.395253 + 0.918572i \(0.370657\pi\)
\(174\) 0 0
\(175\) −193.391 −0.0835372
\(176\) 0 0
\(177\) 20.6226 0.00875759
\(178\) 0 0
\(179\) −1214.69 −0.507208 −0.253604 0.967308i \(-0.581616\pi\)
−0.253604 + 0.967308i \(0.581616\pi\)
\(180\) 0 0
\(181\) −643.623 −0.264310 −0.132155 0.991229i \(-0.542190\pi\)
−0.132155 + 0.991229i \(0.542190\pi\)
\(182\) 0 0
\(183\) 130.509 0.0527184
\(184\) 0 0
\(185\) −67.7898 −0.0269406
\(186\) 0 0
\(187\) 3196.03 1.24982
\(188\) 0 0
\(189\) 1139.48 0.438546
\(190\) 0 0
\(191\) 4085.97 1.54791 0.773953 0.633243i \(-0.218277\pi\)
0.773953 + 0.633243i \(0.218277\pi\)
\(192\) 0 0
\(193\) 864.326 0.322360 0.161180 0.986925i \(-0.448470\pi\)
0.161180 + 0.986925i \(0.448470\pi\)
\(194\) 0 0
\(195\) 338.626 0.124357
\(196\) 0 0
\(197\) 1935.81 0.700106 0.350053 0.936730i \(-0.386164\pi\)
0.350053 + 0.936730i \(0.386164\pi\)
\(198\) 0 0
\(199\) 3593.39 1.28005 0.640023 0.768356i \(-0.278925\pi\)
0.640023 + 0.768356i \(0.278925\pi\)
\(200\) 0 0
\(201\) 853.701 0.299579
\(202\) 0 0
\(203\) −398.512 −0.137784
\(204\) 0 0
\(205\) 3703.98 1.26194
\(206\) 0 0
\(207\) 363.076 0.121911
\(208\) 0 0
\(209\) 2409.74 0.797536
\(210\) 0 0
\(211\) −692.347 −0.225892 −0.112946 0.993601i \(-0.536029\pi\)
−0.112946 + 0.993601i \(0.536029\pi\)
\(212\) 0 0
\(213\) 341.816 0.109957
\(214\) 0 0
\(215\) −4984.25 −1.58104
\(216\) 0 0
\(217\) −851.390 −0.266342
\(218\) 0 0
\(219\) −1012.10 −0.312289
\(220\) 0 0
\(221\) −2102.43 −0.639930
\(222\) 0 0
\(223\) 1770.55 0.531680 0.265840 0.964017i \(-0.414351\pi\)
0.265840 + 0.964017i \(0.414351\pi\)
\(224\) 0 0
\(225\) 343.346 0.101732
\(226\) 0 0
\(227\) 2996.07 0.876017 0.438008 0.898971i \(-0.355684\pi\)
0.438008 + 0.898971i \(0.355684\pi\)
\(228\) 0 0
\(229\) 3790.17 1.09372 0.546860 0.837224i \(-0.315823\pi\)
0.546860 + 0.837224i \(0.315823\pi\)
\(230\) 0 0
\(231\) −671.640 −0.191301
\(232\) 0 0
\(233\) 6257.35 1.75937 0.879684 0.475558i \(-0.157754\pi\)
0.879684 + 0.475558i \(0.157754\pi\)
\(234\) 0 0
\(235\) 5184.50 1.43915
\(236\) 0 0
\(237\) −1476.77 −0.404752
\(238\) 0 0
\(239\) 5893.50 1.59506 0.797529 0.603280i \(-0.206140\pi\)
0.797529 + 0.603280i \(0.206140\pi\)
\(240\) 0 0
\(241\) −418.978 −0.111986 −0.0559932 0.998431i \(-0.517833\pi\)
−0.0559932 + 0.998431i \(0.517833\pi\)
\(242\) 0 0
\(243\) −3085.77 −0.814619
\(244\) 0 0
\(245\) −1623.67 −0.423398
\(246\) 0 0
\(247\) −1585.18 −0.408351
\(248\) 0 0
\(249\) 663.612 0.168894
\(250\) 0 0
\(251\) −7444.38 −1.87205 −0.936026 0.351932i \(-0.885525\pi\)
−0.936026 + 0.351932i \(0.885525\pi\)
\(252\) 0 0
\(253\) −450.844 −0.112033
\(254\) 0 0
\(255\) −1792.61 −0.440226
\(256\) 0 0
\(257\) 5332.46 1.29428 0.647140 0.762371i \(-0.275965\pi\)
0.647140 + 0.762371i \(0.275965\pi\)
\(258\) 0 0
\(259\) −88.4484 −0.0212197
\(260\) 0 0
\(261\) 707.517 0.167794
\(262\) 0 0
\(263\) 832.925 0.195286 0.0976432 0.995221i \(-0.468870\pi\)
0.0976432 + 0.995221i \(0.468870\pi\)
\(264\) 0 0
\(265\) 7045.09 1.63312
\(266\) 0 0
\(267\) −2222.99 −0.509531
\(268\) 0 0
\(269\) 4291.22 0.972640 0.486320 0.873781i \(-0.338339\pi\)
0.486320 + 0.873781i \(0.338339\pi\)
\(270\) 0 0
\(271\) −5967.70 −1.33768 −0.668841 0.743405i \(-0.733209\pi\)
−0.668841 + 0.743405i \(0.733209\pi\)
\(272\) 0 0
\(273\) 441.821 0.0979495
\(274\) 0 0
\(275\) −426.345 −0.0934893
\(276\) 0 0
\(277\) 2234.76 0.484742 0.242371 0.970184i \(-0.422075\pi\)
0.242371 + 0.970184i \(0.422075\pi\)
\(278\) 0 0
\(279\) 1511.55 0.324353
\(280\) 0 0
\(281\) −1672.59 −0.355083 −0.177542 0.984113i \(-0.556814\pi\)
−0.177542 + 0.984113i \(0.556814\pi\)
\(282\) 0 0
\(283\) 4837.14 1.01604 0.508018 0.861347i \(-0.330379\pi\)
0.508018 + 0.861347i \(0.330379\pi\)
\(284\) 0 0
\(285\) −1351.59 −0.280916
\(286\) 0 0
\(287\) 4832.75 0.993965
\(288\) 0 0
\(289\) 6216.79 1.26538
\(290\) 0 0
\(291\) 1047.70 0.211055
\(292\) 0 0
\(293\) 8937.62 1.78205 0.891027 0.453951i \(-0.149986\pi\)
0.891027 + 0.453951i \(0.149986\pi\)
\(294\) 0 0
\(295\) −134.629 −0.0265708
\(296\) 0 0
\(297\) 2512.07 0.490791
\(298\) 0 0
\(299\) 296.576 0.0573627
\(300\) 0 0
\(301\) −6503.18 −1.24531
\(302\) 0 0
\(303\) −1373.73 −0.260457
\(304\) 0 0
\(305\) −851.987 −0.159950
\(306\) 0 0
\(307\) −4175.09 −0.776173 −0.388086 0.921623i \(-0.626864\pi\)
−0.388086 + 0.921623i \(0.626864\pi\)
\(308\) 0 0
\(309\) 3103.40 0.571348
\(310\) 0 0
\(311\) 6986.98 1.27394 0.636970 0.770889i \(-0.280187\pi\)
0.636970 + 0.770889i \(0.280187\pi\)
\(312\) 0 0
\(313\) −7458.22 −1.34685 −0.673425 0.739256i \(-0.735177\pi\)
−0.673425 + 0.739256i \(0.735177\pi\)
\(314\) 0 0
\(315\) −3531.03 −0.631590
\(316\) 0 0
\(317\) 9362.43 1.65882 0.829411 0.558640i \(-0.188676\pi\)
0.829411 + 0.558640i \(0.188676\pi\)
\(318\) 0 0
\(319\) −878.549 −0.154198
\(320\) 0 0
\(321\) 2659.67 0.462455
\(322\) 0 0
\(323\) 8391.61 1.44558
\(324\) 0 0
\(325\) 280.460 0.0478680
\(326\) 0 0
\(327\) 1242.70 0.210157
\(328\) 0 0
\(329\) 6764.45 1.13354
\(330\) 0 0
\(331\) 7034.25 1.16809 0.584044 0.811722i \(-0.301469\pi\)
0.584044 + 0.811722i \(0.301469\pi\)
\(332\) 0 0
\(333\) 157.031 0.0258416
\(334\) 0 0
\(335\) −5573.13 −0.908933
\(336\) 0 0
\(337\) 3071.54 0.496490 0.248245 0.968697i \(-0.420146\pi\)
0.248245 + 0.968697i \(0.420146\pi\)
\(338\) 0 0
\(339\) −2306.59 −0.369548
\(340\) 0 0
\(341\) −1876.95 −0.298072
\(342\) 0 0
\(343\) −6831.91 −1.07548
\(344\) 0 0
\(345\) 252.872 0.0394614
\(346\) 0 0
\(347\) 3918.32 0.606186 0.303093 0.952961i \(-0.401981\pi\)
0.303093 + 0.952961i \(0.401981\pi\)
\(348\) 0 0
\(349\) −4880.00 −0.748482 −0.374241 0.927331i \(-0.622097\pi\)
−0.374241 + 0.927331i \(0.622097\pi\)
\(350\) 0 0
\(351\) −1652.50 −0.251293
\(352\) 0 0
\(353\) −1386.41 −0.209041 −0.104520 0.994523i \(-0.533331\pi\)
−0.104520 + 0.994523i \(0.533331\pi\)
\(354\) 0 0
\(355\) −2231.44 −0.333613
\(356\) 0 0
\(357\) −2338.90 −0.346745
\(358\) 0 0
\(359\) 1798.75 0.264441 0.132220 0.991220i \(-0.457789\pi\)
0.132220 + 0.991220i \(0.457789\pi\)
\(360\) 0 0
\(361\) −531.919 −0.0775506
\(362\) 0 0
\(363\) 666.673 0.0963946
\(364\) 0 0
\(365\) 6607.19 0.947497
\(366\) 0 0
\(367\) −8826.24 −1.25538 −0.627692 0.778462i \(-0.716000\pi\)
−0.627692 + 0.778462i \(0.716000\pi\)
\(368\) 0 0
\(369\) −8580.04 −1.21046
\(370\) 0 0
\(371\) 9192.04 1.28633
\(372\) 0 0
\(373\) 5324.16 0.739074 0.369537 0.929216i \(-0.379516\pi\)
0.369537 + 0.929216i \(0.379516\pi\)
\(374\) 0 0
\(375\) 2363.12 0.325416
\(376\) 0 0
\(377\) 577.930 0.0789521
\(378\) 0 0
\(379\) 486.170 0.0658915 0.0329458 0.999457i \(-0.489511\pi\)
0.0329458 + 0.999457i \(0.489511\pi\)
\(380\) 0 0
\(381\) −3391.74 −0.456074
\(382\) 0 0
\(383\) −10274.8 −1.37081 −0.685405 0.728162i \(-0.740375\pi\)
−0.685405 + 0.728162i \(0.740375\pi\)
\(384\) 0 0
\(385\) 4384.60 0.580415
\(386\) 0 0
\(387\) 11545.7 1.51654
\(388\) 0 0
\(389\) 8823.81 1.15009 0.575045 0.818122i \(-0.304985\pi\)
0.575045 + 0.818122i \(0.304985\pi\)
\(390\) 0 0
\(391\) −1570.01 −0.203066
\(392\) 0 0
\(393\) 2220.00 0.284946
\(394\) 0 0
\(395\) 9640.62 1.22803
\(396\) 0 0
\(397\) 7624.66 0.963906 0.481953 0.876197i \(-0.339928\pi\)
0.481953 + 0.876197i \(0.339928\pi\)
\(398\) 0 0
\(399\) −1763.48 −0.221264
\(400\) 0 0
\(401\) 5192.75 0.646667 0.323333 0.946285i \(-0.395196\pi\)
0.323333 + 0.946285i \(0.395196\pi\)
\(402\) 0 0
\(403\) 1234.70 0.152618
\(404\) 0 0
\(405\) 5528.80 0.678342
\(406\) 0 0
\(407\) −194.991 −0.0237477
\(408\) 0 0
\(409\) 15423.3 1.86463 0.932316 0.361645i \(-0.117785\pi\)
0.932316 + 0.361645i \(0.117785\pi\)
\(410\) 0 0
\(411\) −491.981 −0.0590454
\(412\) 0 0
\(413\) −175.656 −0.0209285
\(414\) 0 0
\(415\) −4332.20 −0.512432
\(416\) 0 0
\(417\) −587.815 −0.0690298
\(418\) 0 0
\(419\) 2112.52 0.246308 0.123154 0.992388i \(-0.460699\pi\)
0.123154 + 0.992388i \(0.460699\pi\)
\(420\) 0 0
\(421\) −2010.09 −0.232698 −0.116349 0.993208i \(-0.537119\pi\)
−0.116349 + 0.993208i \(0.537119\pi\)
\(422\) 0 0
\(423\) −12009.6 −1.38044
\(424\) 0 0
\(425\) −1484.69 −0.169454
\(426\) 0 0
\(427\) −1111.63 −0.125984
\(428\) 0 0
\(429\) 974.025 0.109619
\(430\) 0 0
\(431\) −10433.5 −1.16604 −0.583021 0.812457i \(-0.698129\pi\)
−0.583021 + 0.812457i \(0.698129\pi\)
\(432\) 0 0
\(433\) −6973.11 −0.773917 −0.386959 0.922097i \(-0.626474\pi\)
−0.386959 + 0.922097i \(0.626474\pi\)
\(434\) 0 0
\(435\) 492.766 0.0543134
\(436\) 0 0
\(437\) −1183.75 −0.129580
\(438\) 0 0
\(439\) −3247.81 −0.353096 −0.176548 0.984292i \(-0.556493\pi\)
−0.176548 + 0.984292i \(0.556493\pi\)
\(440\) 0 0
\(441\) 3761.13 0.406126
\(442\) 0 0
\(443\) −8227.84 −0.882430 −0.441215 0.897401i \(-0.645452\pi\)
−0.441215 + 0.897401i \(0.645452\pi\)
\(444\) 0 0
\(445\) 14512.1 1.54593
\(446\) 0 0
\(447\) 322.696 0.0341454
\(448\) 0 0
\(449\) −8823.99 −0.927460 −0.463730 0.885976i \(-0.653489\pi\)
−0.463730 + 0.885976i \(0.653489\pi\)
\(450\) 0 0
\(451\) 10654.1 1.11238
\(452\) 0 0
\(453\) 57.4245 0.00595594
\(454\) 0 0
\(455\) −2884.30 −0.297182
\(456\) 0 0
\(457\) −1162.68 −0.119010 −0.0595052 0.998228i \(-0.518952\pi\)
−0.0595052 + 0.998228i \(0.518952\pi\)
\(458\) 0 0
\(459\) 8747.97 0.889586
\(460\) 0 0
\(461\) −13275.8 −1.34124 −0.670622 0.741799i \(-0.733973\pi\)
−0.670622 + 0.741799i \(0.733973\pi\)
\(462\) 0 0
\(463\) 17666.1 1.77325 0.886625 0.462488i \(-0.153043\pi\)
0.886625 + 0.462488i \(0.153043\pi\)
\(464\) 0 0
\(465\) 1052.76 0.104990
\(466\) 0 0
\(467\) −19505.3 −1.93276 −0.966380 0.257117i \(-0.917227\pi\)
−0.966380 + 0.257117i \(0.917227\pi\)
\(468\) 0 0
\(469\) −7271.52 −0.715922
\(470\) 0 0
\(471\) 221.667 0.0216855
\(472\) 0 0
\(473\) −14336.7 −1.39366
\(474\) 0 0
\(475\) −1119.42 −0.108132
\(476\) 0 0
\(477\) −16319.5 −1.56650
\(478\) 0 0
\(479\) 9224.14 0.879878 0.439939 0.898028i \(-0.355000\pi\)
0.439939 + 0.898028i \(0.355000\pi\)
\(480\) 0 0
\(481\) 128.270 0.0121592
\(482\) 0 0
\(483\) 329.934 0.0310818
\(484\) 0 0
\(485\) −6839.57 −0.640348
\(486\) 0 0
\(487\) −9009.75 −0.838338 −0.419169 0.907908i \(-0.637679\pi\)
−0.419169 + 0.907908i \(0.637679\pi\)
\(488\) 0 0
\(489\) −1400.09 −0.129477
\(490\) 0 0
\(491\) 2822.52 0.259426 0.129713 0.991552i \(-0.458594\pi\)
0.129713 + 0.991552i \(0.458594\pi\)
\(492\) 0 0
\(493\) −3059.44 −0.279493
\(494\) 0 0
\(495\) −7784.40 −0.706834
\(496\) 0 0
\(497\) −2911.46 −0.262771
\(498\) 0 0
\(499\) −5407.39 −0.485106 −0.242553 0.970138i \(-0.577985\pi\)
−0.242553 + 0.970138i \(0.577985\pi\)
\(500\) 0 0
\(501\) 106.939 0.00953633
\(502\) 0 0
\(503\) −757.352 −0.0671345 −0.0335673 0.999436i \(-0.510687\pi\)
−0.0335673 + 0.999436i \(0.510687\pi\)
\(504\) 0 0
\(505\) 8967.97 0.790237
\(506\) 0 0
\(507\) 2903.76 0.254360
\(508\) 0 0
\(509\) −2637.36 −0.229664 −0.114832 0.993385i \(-0.536633\pi\)
−0.114832 + 0.993385i \(0.536633\pi\)
\(510\) 0 0
\(511\) 8620.70 0.746296
\(512\) 0 0
\(513\) 6595.77 0.567661
\(514\) 0 0
\(515\) −20259.7 −1.73349
\(516\) 0 0
\(517\) 14912.7 1.26859
\(518\) 0 0
\(519\) −2902.01 −0.245441
\(520\) 0 0
\(521\) −16495.4 −1.38709 −0.693546 0.720413i \(-0.743952\pi\)
−0.693546 + 0.720413i \(0.743952\pi\)
\(522\) 0 0
\(523\) 713.955 0.0596923 0.0298462 0.999555i \(-0.490498\pi\)
0.0298462 + 0.999555i \(0.490498\pi\)
\(524\) 0 0
\(525\) 312.005 0.0259372
\(526\) 0 0
\(527\) −6536.24 −0.540272
\(528\) 0 0
\(529\) −11945.5 −0.981797
\(530\) 0 0
\(531\) 311.859 0.0254869
\(532\) 0 0
\(533\) −7008.54 −0.569557
\(534\) 0 0
\(535\) −17362.8 −1.40310
\(536\) 0 0
\(537\) 1959.70 0.157481
\(538\) 0 0
\(539\) −4670.32 −0.373219
\(540\) 0 0
\(541\) −21514.0 −1.70972 −0.854861 0.518857i \(-0.826358\pi\)
−0.854861 + 0.518857i \(0.826358\pi\)
\(542\) 0 0
\(543\) 1038.38 0.0820648
\(544\) 0 0
\(545\) −8112.60 −0.637625
\(546\) 0 0
\(547\) −6782.38 −0.530153 −0.265077 0.964227i \(-0.585397\pi\)
−0.265077 + 0.964227i \(0.585397\pi\)
\(548\) 0 0
\(549\) 1973.58 0.153425
\(550\) 0 0
\(551\) −2306.75 −0.178350
\(552\) 0 0
\(553\) 12578.6 0.967260
\(554\) 0 0
\(555\) 109.368 0.00836468
\(556\) 0 0
\(557\) 11559.5 0.879337 0.439668 0.898160i \(-0.355096\pi\)
0.439668 + 0.898160i \(0.355096\pi\)
\(558\) 0 0
\(559\) 9431.04 0.713579
\(560\) 0 0
\(561\) −5156.27 −0.388053
\(562\) 0 0
\(563\) 11151.0 0.834736 0.417368 0.908737i \(-0.362953\pi\)
0.417368 + 0.908737i \(0.362953\pi\)
\(564\) 0 0
\(565\) 15057.9 1.12122
\(566\) 0 0
\(567\) 7213.68 0.534296
\(568\) 0 0
\(569\) 11955.9 0.880873 0.440436 0.897784i \(-0.354824\pi\)
0.440436 + 0.897784i \(0.354824\pi\)
\(570\) 0 0
\(571\) 18691.6 1.36991 0.684956 0.728585i \(-0.259822\pi\)
0.684956 + 0.728585i \(0.259822\pi\)
\(572\) 0 0
\(573\) −6592.03 −0.480604
\(574\) 0 0
\(575\) 209.436 0.0151897
\(576\) 0 0
\(577\) −11793.2 −0.850876 −0.425438 0.904987i \(-0.639880\pi\)
−0.425438 + 0.904987i \(0.639880\pi\)
\(578\) 0 0
\(579\) −1394.45 −0.100089
\(580\) 0 0
\(581\) −5652.41 −0.403617
\(582\) 0 0
\(583\) 20264.5 1.43957
\(584\) 0 0
\(585\) 5120.77 0.361910
\(586\) 0 0
\(587\) 759.837 0.0534273 0.0267137 0.999643i \(-0.491496\pi\)
0.0267137 + 0.999643i \(0.491496\pi\)
\(588\) 0 0
\(589\) −4928.18 −0.344757
\(590\) 0 0
\(591\) −3123.11 −0.217373
\(592\) 0 0
\(593\) −1286.39 −0.0890825 −0.0445412 0.999008i \(-0.514183\pi\)
−0.0445412 + 0.999008i \(0.514183\pi\)
\(594\) 0 0
\(595\) 15268.8 1.05203
\(596\) 0 0
\(597\) −5797.35 −0.397437
\(598\) 0 0
\(599\) 4463.66 0.304474 0.152237 0.988344i \(-0.451352\pi\)
0.152237 + 0.988344i \(0.451352\pi\)
\(600\) 0 0
\(601\) −13078.7 −0.887673 −0.443837 0.896108i \(-0.646383\pi\)
−0.443837 + 0.896108i \(0.646383\pi\)
\(602\) 0 0
\(603\) 12909.8 0.871855
\(604\) 0 0
\(605\) −4352.17 −0.292465
\(606\) 0 0
\(607\) −475.770 −0.0318137 −0.0159069 0.999873i \(-0.505064\pi\)
−0.0159069 + 0.999873i \(0.505064\pi\)
\(608\) 0 0
\(609\) 642.934 0.0427800
\(610\) 0 0
\(611\) −9809.93 −0.649537
\(612\) 0 0
\(613\) −12137.1 −0.799697 −0.399849 0.916581i \(-0.630937\pi\)
−0.399849 + 0.916581i \(0.630937\pi\)
\(614\) 0 0
\(615\) −5975.76 −0.391814
\(616\) 0 0
\(617\) 3751.56 0.244785 0.122392 0.992482i \(-0.460943\pi\)
0.122392 + 0.992482i \(0.460943\pi\)
\(618\) 0 0
\(619\) 14965.7 0.971763 0.485882 0.874025i \(-0.338499\pi\)
0.485882 + 0.874025i \(0.338499\pi\)
\(620\) 0 0
\(621\) −1234.02 −0.0797416
\(622\) 0 0
\(623\) 18934.6 1.21766
\(624\) 0 0
\(625\) −13667.8 −0.874739
\(626\) 0 0
\(627\) −3887.71 −0.247624
\(628\) 0 0
\(629\) −679.031 −0.0430441
\(630\) 0 0
\(631\) 4591.19 0.289655 0.144828 0.989457i \(-0.453737\pi\)
0.144828 + 0.989457i \(0.453737\pi\)
\(632\) 0 0
\(633\) 1116.99 0.0701363
\(634\) 0 0
\(635\) 22142.0 1.38375
\(636\) 0 0
\(637\) 3072.25 0.191094
\(638\) 0 0
\(639\) 5169.00 0.320004
\(640\) 0 0
\(641\) 25387.9 1.56437 0.782187 0.623044i \(-0.214104\pi\)
0.782187 + 0.623044i \(0.214104\pi\)
\(642\) 0 0
\(643\) −25270.1 −1.54986 −0.774928 0.632050i \(-0.782214\pi\)
−0.774928 + 0.632050i \(0.782214\pi\)
\(644\) 0 0
\(645\) 8041.27 0.490891
\(646\) 0 0
\(647\) 18802.7 1.14252 0.571259 0.820770i \(-0.306455\pi\)
0.571259 + 0.820770i \(0.306455\pi\)
\(648\) 0 0
\(649\) −387.246 −0.0234218
\(650\) 0 0
\(651\) 1373.58 0.0826955
\(652\) 0 0
\(653\) 16345.7 0.979564 0.489782 0.871845i \(-0.337076\pi\)
0.489782 + 0.871845i \(0.337076\pi\)
\(654\) 0 0
\(655\) −14492.6 −0.864537
\(656\) 0 0
\(657\) −15305.2 −0.908845
\(658\) 0 0
\(659\) −25185.3 −1.48874 −0.744370 0.667768i \(-0.767250\pi\)
−0.744370 + 0.667768i \(0.767250\pi\)
\(660\) 0 0
\(661\) −17987.6 −1.05845 −0.529227 0.848480i \(-0.677518\pi\)
−0.529227 + 0.848480i \(0.677518\pi\)
\(662\) 0 0
\(663\) 3391.92 0.198690
\(664\) 0 0
\(665\) 11512.3 0.671323
\(666\) 0 0
\(667\) 431.576 0.0250535
\(668\) 0 0
\(669\) −2856.49 −0.165079
\(670\) 0 0
\(671\) −2450.66 −0.140993
\(672\) 0 0
\(673\) 10249.1 0.587033 0.293516 0.955954i \(-0.405174\pi\)
0.293516 + 0.955954i \(0.405174\pi\)
\(674\) 0 0
\(675\) −1166.96 −0.0665428
\(676\) 0 0
\(677\) 4149.01 0.235538 0.117769 0.993041i \(-0.462426\pi\)
0.117769 + 0.993041i \(0.462426\pi\)
\(678\) 0 0
\(679\) −8923.90 −0.504371
\(680\) 0 0
\(681\) −4833.66 −0.271991
\(682\) 0 0
\(683\) −33571.1 −1.88076 −0.940382 0.340121i \(-0.889532\pi\)
−0.940382 + 0.340121i \(0.889532\pi\)
\(684\) 0 0
\(685\) 3211.75 0.179146
\(686\) 0 0
\(687\) −6114.82 −0.339585
\(688\) 0 0
\(689\) −13330.5 −0.737084
\(690\) 0 0
\(691\) 27656.6 1.52258 0.761292 0.648409i \(-0.224565\pi\)
0.761292 + 0.648409i \(0.224565\pi\)
\(692\) 0 0
\(693\) −10156.7 −0.556738
\(694\) 0 0
\(695\) 3837.37 0.209439
\(696\) 0 0
\(697\) 37101.7 2.01625
\(698\) 0 0
\(699\) −10095.2 −0.546260
\(700\) 0 0
\(701\) 14493.7 0.780912 0.390456 0.920622i \(-0.372317\pi\)
0.390456 + 0.920622i \(0.372317\pi\)
\(702\) 0 0
\(703\) −511.974 −0.0274672
\(704\) 0 0
\(705\) −8364.33 −0.446835
\(706\) 0 0
\(707\) 11700.9 0.622431
\(708\) 0 0
\(709\) 8279.81 0.438582 0.219291 0.975659i \(-0.429626\pi\)
0.219291 + 0.975659i \(0.429626\pi\)
\(710\) 0 0
\(711\) −22331.9 −1.17794
\(712\) 0 0
\(713\) 922.027 0.0484294
\(714\) 0 0
\(715\) −6358.63 −0.332586
\(716\) 0 0
\(717\) −9508.19 −0.495244
\(718\) 0 0
\(719\) −3180.45 −0.164966 −0.0824832 0.996592i \(-0.526285\pi\)
−0.0824832 + 0.996592i \(0.526285\pi\)
\(720\) 0 0
\(721\) −26433.7 −1.36538
\(722\) 0 0
\(723\) 675.952 0.0347703
\(724\) 0 0
\(725\) 408.123 0.0209066
\(726\) 0 0
\(727\) 11749.6 0.599409 0.299704 0.954032i \(-0.403112\pi\)
0.299704 + 0.954032i \(0.403112\pi\)
\(728\) 0 0
\(729\) −9195.10 −0.467160
\(730\) 0 0
\(731\) −49925.8 −2.52609
\(732\) 0 0
\(733\) 15005.6 0.756130 0.378065 0.925779i \(-0.376590\pi\)
0.378065 + 0.925779i \(0.376590\pi\)
\(734\) 0 0
\(735\) 2619.52 0.131459
\(736\) 0 0
\(737\) −16030.6 −0.801212
\(738\) 0 0
\(739\) −5555.46 −0.276537 −0.138269 0.990395i \(-0.544154\pi\)
−0.138269 + 0.990395i \(0.544154\pi\)
\(740\) 0 0
\(741\) 2557.43 0.126788
\(742\) 0 0
\(743\) −18358.9 −0.906493 −0.453246 0.891385i \(-0.649734\pi\)
−0.453246 + 0.891385i \(0.649734\pi\)
\(744\) 0 0
\(745\) −2106.63 −0.103598
\(746\) 0 0
\(747\) 10035.3 0.491528
\(748\) 0 0
\(749\) −22654.1 −1.10516
\(750\) 0 0
\(751\) −8818.75 −0.428496 −0.214248 0.976779i \(-0.568730\pi\)
−0.214248 + 0.976779i \(0.568730\pi\)
\(752\) 0 0
\(753\) 12010.3 0.581247
\(754\) 0 0
\(755\) −374.879 −0.0180705
\(756\) 0 0
\(757\) 1462.27 0.0702073 0.0351037 0.999384i \(-0.488824\pi\)
0.0351037 + 0.999384i \(0.488824\pi\)
\(758\) 0 0
\(759\) 727.363 0.0347847
\(760\) 0 0
\(761\) 15232.2 0.725581 0.362790 0.931871i \(-0.381824\pi\)
0.362790 + 0.931871i \(0.381824\pi\)
\(762\) 0 0
\(763\) −10584.9 −0.502225
\(764\) 0 0
\(765\) −27108.2 −1.28118
\(766\) 0 0
\(767\) 254.740 0.0119923
\(768\) 0 0
\(769\) −32110.5 −1.50577 −0.752883 0.658155i \(-0.771337\pi\)
−0.752883 + 0.658155i \(0.771337\pi\)
\(770\) 0 0
\(771\) −8603.05 −0.401856
\(772\) 0 0
\(773\) 13012.4 0.605465 0.302733 0.953076i \(-0.402101\pi\)
0.302733 + 0.953076i \(0.402101\pi\)
\(774\) 0 0
\(775\) 871.922 0.0404134
\(776\) 0 0
\(777\) 142.697 0.00658844
\(778\) 0 0
\(779\) 27973.8 1.28661
\(780\) 0 0
\(781\) −6418.52 −0.294075
\(782\) 0 0
\(783\) −2404.70 −0.109754
\(784\) 0 0
\(785\) −1447.09 −0.0657946
\(786\) 0 0
\(787\) 2738.98 0.124059 0.0620293 0.998074i \(-0.480243\pi\)
0.0620293 + 0.998074i \(0.480243\pi\)
\(788\) 0 0
\(789\) −1343.79 −0.0606338
\(790\) 0 0
\(791\) 19646.7 0.883131
\(792\) 0 0
\(793\) 1612.10 0.0721909
\(794\) 0 0
\(795\) −11366.1 −0.507061
\(796\) 0 0
\(797\) −43359.0 −1.92704 −0.963522 0.267628i \(-0.913760\pi\)
−0.963522 + 0.267628i \(0.913760\pi\)
\(798\) 0 0
\(799\) 51931.6 2.29938
\(800\) 0 0
\(801\) −33616.4 −1.48287
\(802\) 0 0
\(803\) 19004.9 0.835205
\(804\) 0 0
\(805\) −2153.88 −0.0943033
\(806\) 0 0
\(807\) −6923.17 −0.301992
\(808\) 0 0
\(809\) 400.518 0.0174060 0.00870300 0.999962i \(-0.497230\pi\)
0.00870300 + 0.999962i \(0.497230\pi\)
\(810\) 0 0
\(811\) −37589.7 −1.62756 −0.813782 0.581170i \(-0.802595\pi\)
−0.813782 + 0.581170i \(0.802595\pi\)
\(812\) 0 0
\(813\) 9627.90 0.415332
\(814\) 0 0
\(815\) 9140.09 0.392839
\(816\) 0 0
\(817\) −37642.9 −1.61195
\(818\) 0 0
\(819\) 6681.29 0.285059
\(820\) 0 0
\(821\) −17756.6 −0.754821 −0.377411 0.926046i \(-0.623185\pi\)
−0.377411 + 0.926046i \(0.623185\pi\)
\(822\) 0 0
\(823\) 38533.6 1.63207 0.816036 0.578001i \(-0.196167\pi\)
0.816036 + 0.578001i \(0.196167\pi\)
\(824\) 0 0
\(825\) 687.837 0.0290272
\(826\) 0 0
\(827\) −45875.2 −1.92895 −0.964473 0.264182i \(-0.914898\pi\)
−0.964473 + 0.264182i \(0.914898\pi\)
\(828\) 0 0
\(829\) −22725.1 −0.952082 −0.476041 0.879423i \(-0.657929\pi\)
−0.476041 + 0.879423i \(0.657929\pi\)
\(830\) 0 0
\(831\) −3605.41 −0.150506
\(832\) 0 0
\(833\) −16263.8 −0.676480
\(834\) 0 0
\(835\) −698.123 −0.0289336
\(836\) 0 0
\(837\) −5137.46 −0.212158
\(838\) 0 0
\(839\) 31480.4 1.29538 0.647690 0.761904i \(-0.275735\pi\)
0.647690 + 0.761904i \(0.275735\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) 2698.45 0.110249
\(844\) 0 0
\(845\) −18956.3 −0.771737
\(846\) 0 0
\(847\) −5678.48 −0.230360
\(848\) 0 0
\(849\) −7803.92 −0.315465
\(850\) 0 0
\(851\) 95.7866 0.00385843
\(852\) 0 0
\(853\) −14863.4 −0.596617 −0.298309 0.954470i \(-0.596422\pi\)
−0.298309 + 0.954470i \(0.596422\pi\)
\(854\) 0 0
\(855\) −20439.0 −0.817542
\(856\) 0 0
\(857\) 30070.7 1.19859 0.599297 0.800527i \(-0.295447\pi\)
0.599297 + 0.800527i \(0.295447\pi\)
\(858\) 0 0
\(859\) 16015.7 0.636146 0.318073 0.948066i \(-0.396964\pi\)
0.318073 + 0.948066i \(0.396964\pi\)
\(860\) 0 0
\(861\) −7796.84 −0.308613
\(862\) 0 0
\(863\) 4726.12 0.186418 0.0932092 0.995647i \(-0.470287\pi\)
0.0932092 + 0.995647i \(0.470287\pi\)
\(864\) 0 0
\(865\) 18944.9 0.744677
\(866\) 0 0
\(867\) −10029.8 −0.392882
\(868\) 0 0
\(869\) 27730.3 1.08249
\(870\) 0 0
\(871\) 10545.3 0.410234
\(872\) 0 0
\(873\) 15843.5 0.614226
\(874\) 0 0
\(875\) −20128.2 −0.777667
\(876\) 0 0
\(877\) −35681.6 −1.37387 −0.686933 0.726720i \(-0.741044\pi\)
−0.686933 + 0.726720i \(0.741044\pi\)
\(878\) 0 0
\(879\) −14419.4 −0.553303
\(880\) 0 0
\(881\) −14945.7 −0.571548 −0.285774 0.958297i \(-0.592251\pi\)
−0.285774 + 0.958297i \(0.592251\pi\)
\(882\) 0 0
\(883\) −39264.7 −1.49645 −0.748223 0.663447i \(-0.769093\pi\)
−0.748223 + 0.663447i \(0.769093\pi\)
\(884\) 0 0
\(885\) 217.201 0.00824988
\(886\) 0 0
\(887\) 19020.1 0.719991 0.359995 0.932954i \(-0.382778\pi\)
0.359995 + 0.932954i \(0.382778\pi\)
\(888\) 0 0
\(889\) 28889.7 1.08991
\(890\) 0 0
\(891\) 15903.1 0.597949
\(892\) 0 0
\(893\) 39155.3 1.46728
\(894\) 0 0
\(895\) −12793.3 −0.477803
\(896\) 0 0
\(897\) −478.477 −0.0178103
\(898\) 0 0
\(899\) 1796.73 0.0666566
\(900\) 0 0
\(901\) 70568.6 2.60930
\(902\) 0 0
\(903\) 10491.8 0.386651
\(904\) 0 0
\(905\) −6778.76 −0.248987
\(906\) 0 0
\(907\) −24945.4 −0.913228 −0.456614 0.889665i \(-0.650938\pi\)
−0.456614 + 0.889665i \(0.650938\pi\)
\(908\) 0 0
\(909\) −20773.8 −0.758000
\(910\) 0 0
\(911\) 2868.98 0.104340 0.0521698 0.998638i \(-0.483386\pi\)
0.0521698 + 0.998638i \(0.483386\pi\)
\(912\) 0 0
\(913\) −12461.1 −0.451702
\(914\) 0 0
\(915\) 1374.54 0.0496622
\(916\) 0 0
\(917\) −18909.1 −0.680953
\(918\) 0 0
\(919\) −20331.1 −0.729774 −0.364887 0.931052i \(-0.618892\pi\)
−0.364887 + 0.931052i \(0.618892\pi\)
\(920\) 0 0
\(921\) 6735.82 0.240991
\(922\) 0 0
\(923\) 4222.26 0.150571
\(924\) 0 0
\(925\) 90.5814 0.00321978
\(926\) 0 0
\(927\) 46930.2 1.66277
\(928\) 0 0
\(929\) 2069.37 0.0730827 0.0365413 0.999332i \(-0.488366\pi\)
0.0365413 + 0.999332i \(0.488366\pi\)
\(930\) 0 0
\(931\) −12262.6 −0.431675
\(932\) 0 0
\(933\) −11272.3 −0.395541
\(934\) 0 0
\(935\) 33661.2 1.17737
\(936\) 0 0
\(937\) −27162.9 −0.947038 −0.473519 0.880784i \(-0.657016\pi\)
−0.473519 + 0.880784i \(0.657016\pi\)
\(938\) 0 0
\(939\) 12032.6 0.418178
\(940\) 0 0
\(941\) −8127.05 −0.281546 −0.140773 0.990042i \(-0.544959\pi\)
−0.140773 + 0.990042i \(0.544959\pi\)
\(942\) 0 0
\(943\) −5233.70 −0.180735
\(944\) 0 0
\(945\) 12001.2 0.413122
\(946\) 0 0
\(947\) −9170.78 −0.314689 −0.157344 0.987544i \(-0.550293\pi\)
−0.157344 + 0.987544i \(0.550293\pi\)
\(948\) 0 0
\(949\) −12501.9 −0.427639
\(950\) 0 0
\(951\) −15104.7 −0.515042
\(952\) 0 0
\(953\) −25938.9 −0.881683 −0.440841 0.897585i \(-0.645320\pi\)
−0.440841 + 0.897585i \(0.645320\pi\)
\(954\) 0 0
\(955\) 43034.1 1.45817
\(956\) 0 0
\(957\) 1417.39 0.0478765
\(958\) 0 0
\(959\) 4190.52 0.141104
\(960\) 0 0
\(961\) −25952.4 −0.871150
\(962\) 0 0
\(963\) 40220.0 1.34587
\(964\) 0 0
\(965\) 9103.24 0.303672
\(966\) 0 0
\(967\) −6987.35 −0.232366 −0.116183 0.993228i \(-0.537066\pi\)
−0.116183 + 0.993228i \(0.537066\pi\)
\(968\) 0 0
\(969\) −13538.5 −0.448832
\(970\) 0 0
\(971\) −37865.2 −1.25144 −0.625722 0.780046i \(-0.715196\pi\)
−0.625722 + 0.780046i \(0.715196\pi\)
\(972\) 0 0
\(973\) 5006.80 0.164965
\(974\) 0 0
\(975\) −452.476 −0.0148624
\(976\) 0 0
\(977\) −47682.8 −1.56142 −0.780711 0.624893i \(-0.785143\pi\)
−0.780711 + 0.624893i \(0.785143\pi\)
\(978\) 0 0
\(979\) 41742.7 1.36272
\(980\) 0 0
\(981\) 18792.3 0.611614
\(982\) 0 0
\(983\) 10728.5 0.348103 0.174052 0.984737i \(-0.444314\pi\)
0.174052 + 0.984737i \(0.444314\pi\)
\(984\) 0 0
\(985\) 20388.3 0.659518
\(986\) 0 0
\(987\) −10913.3 −0.351950
\(988\) 0 0
\(989\) 7042.72 0.226436
\(990\) 0 0
\(991\) −33130.6 −1.06198 −0.530992 0.847377i \(-0.678181\pi\)
−0.530992 + 0.847377i \(0.678181\pi\)
\(992\) 0 0
\(993\) −11348.6 −0.362676
\(994\) 0 0
\(995\) 37846.3 1.20584
\(996\) 0 0
\(997\) 19283.2 0.612543 0.306272 0.951944i \(-0.400918\pi\)
0.306272 + 0.951944i \(0.400918\pi\)
\(998\) 0 0
\(999\) −533.715 −0.0169029
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 1856.4.a.bj.1.6 12
4.3 odd 2 1856.4.a.bl.1.7 12
8.3 odd 2 928.4.a.h.1.6 12
8.5 even 2 928.4.a.j.1.7 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
928.4.a.h.1.6 12 8.3 odd 2
928.4.a.j.1.7 yes 12 8.5 even 2
1856.4.a.bj.1.6 12 1.1 even 1 trivial
1856.4.a.bl.1.7 12 4.3 odd 2