Properties

Label 1856.4.a.bj.1.2
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 1.2
Root \(8.22268\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.22268 q^{3} -9.91795 q^{5} -18.1458 q^{7} +58.0578 q^{9} +O(q^{10})\) \(q-9.22268 q^{3} -9.91795 q^{5} -18.1458 q^{7} +58.0578 q^{9} -30.3346 q^{11} +38.5792 q^{13} +91.4701 q^{15} +59.0959 q^{17} -50.7598 q^{19} +167.353 q^{21} +155.842 q^{23} -26.6342 q^{25} -286.437 q^{27} -29.0000 q^{29} -241.283 q^{31} +279.767 q^{33} +179.969 q^{35} -371.714 q^{37} -355.804 q^{39} +61.4490 q^{41} -369.530 q^{43} -575.815 q^{45} -281.699 q^{47} -13.7311 q^{49} -545.023 q^{51} +374.175 q^{53} +300.857 q^{55} +468.141 q^{57} +129.028 q^{59} -463.061 q^{61} -1053.50 q^{63} -382.627 q^{65} -970.507 q^{67} -1437.28 q^{69} +241.521 q^{71} +565.154 q^{73} +245.639 q^{75} +550.445 q^{77} +424.422 q^{79} +1074.15 q^{81} +161.180 q^{83} -586.110 q^{85} +267.458 q^{87} -1138.20 q^{89} -700.050 q^{91} +2225.28 q^{93} +503.433 q^{95} -1449.36 q^{97} -1761.16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9}+O(q^{10}) \) 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
\(3\) −9.22268 −1.77491 −0.887453 0.460898i \(-0.847527\pi\)
−0.887453 + 0.460898i \(0.847527\pi\)
\(4\) 0 0
\(5\) −9.91795 −0.887089 −0.443544 0.896252i \(-0.646279\pi\)
−0.443544 + 0.896252i \(0.646279\pi\)
\(6\) 0 0
\(7\) −18.1458 −0.979779 −0.489890 0.871784i \(-0.662963\pi\)
−0.489890 + 0.871784i \(0.662963\pi\)
\(8\) 0 0
\(9\) 58.0578 2.15029
\(10\) 0 0
\(11\) −30.3346 −0.831476 −0.415738 0.909485i \(-0.636477\pi\)
−0.415738 + 0.909485i \(0.636477\pi\)
\(12\) 0 0
\(13\) 38.5792 0.823073 0.411536 0.911393i \(-0.364992\pi\)
0.411536 + 0.911393i \(0.364992\pi\)
\(14\) 0 0
\(15\) 91.4701 1.57450
\(16\) 0 0
\(17\) 59.0959 0.843109 0.421555 0.906803i \(-0.361485\pi\)
0.421555 + 0.906803i \(0.361485\pi\)
\(18\) 0 0
\(19\) −50.7598 −0.612900 −0.306450 0.951887i \(-0.599141\pi\)
−0.306450 + 0.951887i \(0.599141\pi\)
\(20\) 0 0
\(21\) 167.353 1.73902
\(22\) 0 0
\(23\) 155.842 1.41284 0.706418 0.707795i \(-0.250310\pi\)
0.706418 + 0.707795i \(0.250310\pi\)
\(24\) 0 0
\(25\) −26.6342 −0.213073
\(26\) 0 0
\(27\) −286.437 −2.04166
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) −241.283 −1.39793 −0.698964 0.715157i \(-0.746355\pi\)
−0.698964 + 0.715157i \(0.746355\pi\)
\(32\) 0 0
\(33\) 279.767 1.47579
\(34\) 0 0
\(35\) 179.969 0.869151
\(36\) 0 0
\(37\) −371.714 −1.65160 −0.825802 0.563960i \(-0.809277\pi\)
−0.825802 + 0.563960i \(0.809277\pi\)
\(38\) 0 0
\(39\) −355.804 −1.46088
\(40\) 0 0
\(41\) 61.4490 0.234066 0.117033 0.993128i \(-0.462662\pi\)
0.117033 + 0.993128i \(0.462662\pi\)
\(42\) 0 0
\(43\) −369.530 −1.31053 −0.655265 0.755399i \(-0.727443\pi\)
−0.655265 + 0.755399i \(0.727443\pi\)
\(44\) 0 0
\(45\) −575.815 −1.90750
\(46\) 0 0
\(47\) −281.699 −0.874255 −0.437127 0.899400i \(-0.644004\pi\)
−0.437127 + 0.899400i \(0.644004\pi\)
\(48\) 0 0
\(49\) −13.7311 −0.0400322
\(50\) 0 0
\(51\) −545.023 −1.49644
\(52\) 0 0
\(53\) 374.175 0.969753 0.484876 0.874583i \(-0.338865\pi\)
0.484876 + 0.874583i \(0.338865\pi\)
\(54\) 0 0
\(55\) 300.857 0.737593
\(56\) 0 0
\(57\) 468.141 1.08784
\(58\) 0 0
\(59\) 129.028 0.284713 0.142357 0.989815i \(-0.454532\pi\)
0.142357 + 0.989815i \(0.454532\pi\)
\(60\) 0 0
\(61\) −463.061 −0.971948 −0.485974 0.873973i \(-0.661535\pi\)
−0.485974 + 0.873973i \(0.661535\pi\)
\(62\) 0 0
\(63\) −1053.50 −2.10681
\(64\) 0 0
\(65\) −382.627 −0.730139
\(66\) 0 0
\(67\) −970.507 −1.76965 −0.884823 0.465927i \(-0.845721\pi\)
−0.884823 + 0.465927i \(0.845721\pi\)
\(68\) 0 0
\(69\) −1437.28 −2.50765
\(70\) 0 0
\(71\) 241.521 0.403708 0.201854 0.979416i \(-0.435303\pi\)
0.201854 + 0.979416i \(0.435303\pi\)
\(72\) 0 0
\(73\) 565.154 0.906113 0.453057 0.891482i \(-0.350334\pi\)
0.453057 + 0.891482i \(0.350334\pi\)
\(74\) 0 0
\(75\) 245.639 0.378185
\(76\) 0 0
\(77\) 550.445 0.814663
\(78\) 0 0
\(79\) 424.422 0.604446 0.302223 0.953237i \(-0.402271\pi\)
0.302223 + 0.953237i \(0.402271\pi\)
\(80\) 0 0
\(81\) 1074.15 1.47346
\(82\) 0 0
\(83\) 161.180 0.213154 0.106577 0.994304i \(-0.466011\pi\)
0.106577 + 0.994304i \(0.466011\pi\)
\(84\) 0 0
\(85\) −586.110 −0.747913
\(86\) 0 0
\(87\) 267.458 0.329592
\(88\) 0 0
\(89\) −1138.20 −1.35561 −0.677806 0.735241i \(-0.737069\pi\)
−0.677806 + 0.735241i \(0.737069\pi\)
\(90\) 0 0
\(91\) −700.050 −0.806430
\(92\) 0 0
\(93\) 2225.28 2.48119
\(94\) 0 0
\(95\) 503.433 0.543696
\(96\) 0 0
\(97\) −1449.36 −1.51712 −0.758558 0.651606i \(-0.774096\pi\)
−0.758558 + 0.651606i \(0.774096\pi\)
\(98\) 0 0
\(99\) −1761.16 −1.78791
\(100\) 0 0
\(101\) −979.900 −0.965383 −0.482692 0.875790i \(-0.660341\pi\)
−0.482692 + 0.875790i \(0.660341\pi\)
\(102\) 0 0
\(103\) −375.174 −0.358903 −0.179451 0.983767i \(-0.557432\pi\)
−0.179451 + 0.983767i \(0.557432\pi\)
\(104\) 0 0
\(105\) −1659.80 −1.54266
\(106\) 0 0
\(107\) 779.197 0.703999 0.351999 0.936000i \(-0.385502\pi\)
0.351999 + 0.936000i \(0.385502\pi\)
\(108\) 0 0
\(109\) 146.498 0.128734 0.0643669 0.997926i \(-0.479497\pi\)
0.0643669 + 0.997926i \(0.479497\pi\)
\(110\) 0 0
\(111\) 3428.20 2.93144
\(112\) 0 0
\(113\) −209.441 −0.174359 −0.0871795 0.996193i \(-0.527785\pi\)
−0.0871795 + 0.996193i \(0.527785\pi\)
\(114\) 0 0
\(115\) −1545.63 −1.25331
\(116\) 0 0
\(117\) 2239.83 1.76985
\(118\) 0 0
\(119\) −1072.34 −0.826061
\(120\) 0 0
\(121\) −410.811 −0.308648
\(122\) 0 0
\(123\) −566.725 −0.415446
\(124\) 0 0
\(125\) 1503.90 1.07610
\(126\) 0 0
\(127\) −2328.39 −1.62686 −0.813429 0.581664i \(-0.802402\pi\)
−0.813429 + 0.581664i \(0.802402\pi\)
\(128\) 0 0
\(129\) 3408.06 2.32607
\(130\) 0 0
\(131\) −2293.89 −1.52991 −0.764955 0.644084i \(-0.777239\pi\)
−0.764955 + 0.644084i \(0.777239\pi\)
\(132\) 0 0
\(133\) 921.075 0.600506
\(134\) 0 0
\(135\) 2840.87 1.81113
\(136\) 0 0
\(137\) −1579.37 −0.984927 −0.492463 0.870333i \(-0.663903\pi\)
−0.492463 + 0.870333i \(0.663903\pi\)
\(138\) 0 0
\(139\) 1220.98 0.745054 0.372527 0.928021i \(-0.378491\pi\)
0.372527 + 0.928021i \(0.378491\pi\)
\(140\) 0 0
\(141\) 2598.02 1.55172
\(142\) 0 0
\(143\) −1170.29 −0.684365
\(144\) 0 0
\(145\) 287.621 0.164728
\(146\) 0 0
\(147\) 126.637 0.0710534
\(148\) 0 0
\(149\) −400.388 −0.220141 −0.110071 0.993924i \(-0.535108\pi\)
−0.110071 + 0.993924i \(0.535108\pi\)
\(150\) 0 0
\(151\) 611.484 0.329549 0.164774 0.986331i \(-0.447310\pi\)
0.164774 + 0.986331i \(0.447310\pi\)
\(152\) 0 0
\(153\) 3430.98 1.81293
\(154\) 0 0
\(155\) 2393.04 1.24009
\(156\) 0 0
\(157\) 2219.20 1.12810 0.564048 0.825742i \(-0.309243\pi\)
0.564048 + 0.825742i \(0.309243\pi\)
\(158\) 0 0
\(159\) −3450.90 −1.72122
\(160\) 0 0
\(161\) −2827.87 −1.38427
\(162\) 0 0
\(163\) −3384.80 −1.62649 −0.813245 0.581922i \(-0.802301\pi\)
−0.813245 + 0.581922i \(0.802301\pi\)
\(164\) 0 0
\(165\) −2774.71 −1.30916
\(166\) 0 0
\(167\) −1469.18 −0.680769 −0.340384 0.940286i \(-0.610557\pi\)
−0.340384 + 0.940286i \(0.610557\pi\)
\(168\) 0 0
\(169\) −708.644 −0.322551
\(170\) 0 0
\(171\) −2947.00 −1.31791
\(172\) 0 0
\(173\) 2494.19 1.09613 0.548063 0.836437i \(-0.315365\pi\)
0.548063 + 0.836437i \(0.315365\pi\)
\(174\) 0 0
\(175\) 483.298 0.208765
\(176\) 0 0
\(177\) −1189.99 −0.505339
\(178\) 0 0
\(179\) −1681.66 −0.702197 −0.351099 0.936339i \(-0.614192\pi\)
−0.351099 + 0.936339i \(0.614192\pi\)
\(180\) 0 0
\(181\) 3463.28 1.42223 0.711115 0.703076i \(-0.248191\pi\)
0.711115 + 0.703076i \(0.248191\pi\)
\(182\) 0 0
\(183\) 4270.66 1.72512
\(184\) 0 0
\(185\) 3686.64 1.46512
\(186\) 0 0
\(187\) −1792.65 −0.701025
\(188\) 0 0
\(189\) 5197.61 2.00037
\(190\) 0 0
\(191\) 2962.78 1.12240 0.561202 0.827679i \(-0.310339\pi\)
0.561202 + 0.827679i \(0.310339\pi\)
\(192\) 0 0
\(193\) 3212.73 1.19822 0.599112 0.800665i \(-0.295520\pi\)
0.599112 + 0.800665i \(0.295520\pi\)
\(194\) 0 0
\(195\) 3528.85 1.29593
\(196\) 0 0
\(197\) 40.3205 0.0145823 0.00729115 0.999973i \(-0.497679\pi\)
0.00729115 + 0.999973i \(0.497679\pi\)
\(198\) 0 0
\(199\) −492.569 −0.175464 −0.0877319 0.996144i \(-0.527962\pi\)
−0.0877319 + 0.996144i \(0.527962\pi\)
\(200\) 0 0
\(201\) 8950.68 3.14096
\(202\) 0 0
\(203\) 526.227 0.181940
\(204\) 0 0
\(205\) −609.449 −0.207638
\(206\) 0 0
\(207\) 9047.83 3.03801
\(208\) 0 0
\(209\) 1539.78 0.509611
\(210\) 0 0
\(211\) 2004.02 0.653850 0.326925 0.945050i \(-0.393988\pi\)
0.326925 + 0.945050i \(0.393988\pi\)
\(212\) 0 0
\(213\) −2227.47 −0.716544
\(214\) 0 0
\(215\) 3664.98 1.16256
\(216\) 0 0
\(217\) 4378.27 1.36966
\(218\) 0 0
\(219\) −5212.23 −1.60827
\(220\) 0 0
\(221\) 2279.87 0.693941
\(222\) 0 0
\(223\) 1975.53 0.593234 0.296617 0.954996i \(-0.404141\pi\)
0.296617 + 0.954996i \(0.404141\pi\)
\(224\) 0 0
\(225\) −1546.32 −0.458170
\(226\) 0 0
\(227\) −5304.20 −1.55089 −0.775446 0.631414i \(-0.782475\pi\)
−0.775446 + 0.631414i \(0.782475\pi\)
\(228\) 0 0
\(229\) 2463.70 0.710944 0.355472 0.934687i \(-0.384320\pi\)
0.355472 + 0.934687i \(0.384320\pi\)
\(230\) 0 0
\(231\) −5076.58 −1.44595
\(232\) 0 0
\(233\) −3230.57 −0.908332 −0.454166 0.890917i \(-0.650063\pi\)
−0.454166 + 0.890917i \(0.650063\pi\)
\(234\) 0 0
\(235\) 2793.87 0.775541
\(236\) 0 0
\(237\) −3914.31 −1.07283
\(238\) 0 0
\(239\) −4280.39 −1.15848 −0.579238 0.815159i \(-0.696650\pi\)
−0.579238 + 0.815159i \(0.696650\pi\)
\(240\) 0 0
\(241\) 6488.41 1.73425 0.867127 0.498087i \(-0.165964\pi\)
0.867127 + 0.498087i \(0.165964\pi\)
\(242\) 0 0
\(243\) −2172.77 −0.573594
\(244\) 0 0
\(245\) 136.184 0.0355121
\(246\) 0 0
\(247\) −1958.27 −0.504461
\(248\) 0 0
\(249\) −1486.51 −0.378328
\(250\) 0 0
\(251\) 407.291 0.102422 0.0512111 0.998688i \(-0.483692\pi\)
0.0512111 + 0.998688i \(0.483692\pi\)
\(252\) 0 0
\(253\) −4727.40 −1.17474
\(254\) 0 0
\(255\) 5405.51 1.32748
\(256\) 0 0
\(257\) −5519.50 −1.33968 −0.669839 0.742507i \(-0.733637\pi\)
−0.669839 + 0.742507i \(0.733637\pi\)
\(258\) 0 0
\(259\) 6745.03 1.61821
\(260\) 0 0
\(261\) −1683.68 −0.399299
\(262\) 0 0
\(263\) −4159.14 −0.975146 −0.487573 0.873082i \(-0.662118\pi\)
−0.487573 + 0.873082i \(0.662118\pi\)
\(264\) 0 0
\(265\) −3711.05 −0.860257
\(266\) 0 0
\(267\) 10497.3 2.40608
\(268\) 0 0
\(269\) −3408.03 −0.772459 −0.386230 0.922403i \(-0.626223\pi\)
−0.386230 + 0.922403i \(0.626223\pi\)
\(270\) 0 0
\(271\) 7638.65 1.71223 0.856116 0.516783i \(-0.172871\pi\)
0.856116 + 0.516783i \(0.172871\pi\)
\(272\) 0 0
\(273\) 6456.33 1.43134
\(274\) 0 0
\(275\) 807.938 0.177165
\(276\) 0 0
\(277\) −254.643 −0.0552347 −0.0276173 0.999619i \(-0.508792\pi\)
−0.0276173 + 0.999619i \(0.508792\pi\)
\(278\) 0 0
\(279\) −14008.4 −3.00595
\(280\) 0 0
\(281\) −3750.69 −0.796255 −0.398128 0.917330i \(-0.630340\pi\)
−0.398128 + 0.917330i \(0.630340\pi\)
\(282\) 0 0
\(283\) 7621.54 1.60090 0.800448 0.599402i \(-0.204595\pi\)
0.800448 + 0.599402i \(0.204595\pi\)
\(284\) 0 0
\(285\) −4643.00 −0.965010
\(286\) 0 0
\(287\) −1115.04 −0.229334
\(288\) 0 0
\(289\) −1420.67 −0.289166
\(290\) 0 0
\(291\) 13367.0 2.69274
\(292\) 0 0
\(293\) 3420.15 0.681935 0.340968 0.940075i \(-0.389245\pi\)
0.340968 + 0.940075i \(0.389245\pi\)
\(294\) 0 0
\(295\) −1279.70 −0.252566
\(296\) 0 0
\(297\) 8688.95 1.69759
\(298\) 0 0
\(299\) 6012.25 1.16287
\(300\) 0 0
\(301\) 6705.40 1.28403
\(302\) 0 0
\(303\) 9037.31 1.71346
\(304\) 0 0
\(305\) 4592.61 0.862204
\(306\) 0 0
\(307\) −6813.99 −1.26676 −0.633380 0.773841i \(-0.718333\pi\)
−0.633380 + 0.773841i \(0.718333\pi\)
\(308\) 0 0
\(309\) 3460.11 0.637018
\(310\) 0 0
\(311\) 6160.82 1.12331 0.561653 0.827373i \(-0.310166\pi\)
0.561653 + 0.827373i \(0.310166\pi\)
\(312\) 0 0
\(313\) −5355.56 −0.967138 −0.483569 0.875306i \(-0.660660\pi\)
−0.483569 + 0.875306i \(0.660660\pi\)
\(314\) 0 0
\(315\) 10448.6 1.86893
\(316\) 0 0
\(317\) −1812.38 −0.321115 −0.160557 0.987027i \(-0.551329\pi\)
−0.160557 + 0.987027i \(0.551329\pi\)
\(318\) 0 0
\(319\) 879.704 0.154401
\(320\) 0 0
\(321\) −7186.29 −1.24953
\(322\) 0 0
\(323\) −2999.69 −0.516741
\(324\) 0 0
\(325\) −1027.53 −0.175375
\(326\) 0 0
\(327\) −1351.11 −0.228490
\(328\) 0 0
\(329\) 5111.64 0.856577
\(330\) 0 0
\(331\) −10192.2 −1.69249 −0.846243 0.532797i \(-0.821141\pi\)
−0.846243 + 0.532797i \(0.821141\pi\)
\(332\) 0 0
\(333\) −21580.9 −3.55143
\(334\) 0 0
\(335\) 9625.45 1.56983
\(336\) 0 0
\(337\) 4600.14 0.743578 0.371789 0.928317i \(-0.378745\pi\)
0.371789 + 0.928317i \(0.378745\pi\)
\(338\) 0 0
\(339\) 1931.61 0.309471
\(340\) 0 0
\(341\) 7319.24 1.16234
\(342\) 0 0
\(343\) 6473.16 1.01900
\(344\) 0 0
\(345\) 14254.8 2.22451
\(346\) 0 0
\(347\) 3833.04 0.592992 0.296496 0.955034i \(-0.404182\pi\)
0.296496 + 0.955034i \(0.404182\pi\)
\(348\) 0 0
\(349\) −3398.43 −0.521243 −0.260622 0.965441i \(-0.583928\pi\)
−0.260622 + 0.965441i \(0.583928\pi\)
\(350\) 0 0
\(351\) −11050.5 −1.68043
\(352\) 0 0
\(353\) −8663.33 −1.30624 −0.653119 0.757255i \(-0.726540\pi\)
−0.653119 + 0.757255i \(0.726540\pi\)
\(354\) 0 0
\(355\) −2395.39 −0.358125
\(356\) 0 0
\(357\) 9889.85 1.46618
\(358\) 0 0
\(359\) −12373.5 −1.81908 −0.909540 0.415616i \(-0.863566\pi\)
−0.909540 + 0.415616i \(0.863566\pi\)
\(360\) 0 0
\(361\) −4282.44 −0.624354
\(362\) 0 0
\(363\) 3788.78 0.547822
\(364\) 0 0
\(365\) −5605.17 −0.803803
\(366\) 0 0
\(367\) −3507.48 −0.498880 −0.249440 0.968390i \(-0.580246\pi\)
−0.249440 + 0.968390i \(0.580246\pi\)
\(368\) 0 0
\(369\) 3567.60 0.503311
\(370\) 0 0
\(371\) −6789.69 −0.950144
\(372\) 0 0
\(373\) 12460.9 1.72976 0.864878 0.501982i \(-0.167396\pi\)
0.864878 + 0.501982i \(0.167396\pi\)
\(374\) 0 0
\(375\) −13870.0 −1.90998
\(376\) 0 0
\(377\) −1118.80 −0.152841
\(378\) 0 0
\(379\) 11480.8 1.55601 0.778003 0.628260i \(-0.216233\pi\)
0.778003 + 0.628260i \(0.216233\pi\)
\(380\) 0 0
\(381\) 21474.0 2.88752
\(382\) 0 0
\(383\) 11746.7 1.56718 0.783591 0.621278i \(-0.213386\pi\)
0.783591 + 0.621278i \(0.213386\pi\)
\(384\) 0 0
\(385\) −5459.29 −0.722678
\(386\) 0 0
\(387\) −21454.1 −2.81802
\(388\) 0 0
\(389\) −2142.34 −0.279232 −0.139616 0.990206i \(-0.544587\pi\)
−0.139616 + 0.990206i \(0.544587\pi\)
\(390\) 0 0
\(391\) 9209.60 1.19117
\(392\) 0 0
\(393\) 21155.8 2.71544
\(394\) 0 0
\(395\) −4209.40 −0.536197
\(396\) 0 0
\(397\) −3330.09 −0.420989 −0.210494 0.977595i \(-0.567507\pi\)
−0.210494 + 0.977595i \(0.567507\pi\)
\(398\) 0 0
\(399\) −8494.78 −1.06584
\(400\) 0 0
\(401\) −2449.26 −0.305013 −0.152507 0.988302i \(-0.548735\pi\)
−0.152507 + 0.988302i \(0.548735\pi\)
\(402\) 0 0
\(403\) −9308.52 −1.15060
\(404\) 0 0
\(405\) −10653.4 −1.30709
\(406\) 0 0
\(407\) 11275.8 1.37327
\(408\) 0 0
\(409\) 1978.69 0.239217 0.119608 0.992821i \(-0.461836\pi\)
0.119608 + 0.992821i \(0.461836\pi\)
\(410\) 0 0
\(411\) 14566.1 1.74815
\(412\) 0 0
\(413\) −2341.32 −0.278956
\(414\) 0 0
\(415\) −1598.57 −0.189087
\(416\) 0 0
\(417\) −11260.7 −1.32240
\(418\) 0 0
\(419\) 3715.93 0.433258 0.216629 0.976254i \(-0.430494\pi\)
0.216629 + 0.976254i \(0.430494\pi\)
\(420\) 0 0
\(421\) 16124.9 1.86670 0.933350 0.358968i \(-0.116871\pi\)
0.933350 + 0.358968i \(0.116871\pi\)
\(422\) 0 0
\(423\) −16354.8 −1.87990
\(424\) 0 0
\(425\) −1573.97 −0.179644
\(426\) 0 0
\(427\) 8402.59 0.952295
\(428\) 0 0
\(429\) 10793.2 1.21468
\(430\) 0 0
\(431\) −6764.83 −0.756034 −0.378017 0.925799i \(-0.623394\pi\)
−0.378017 + 0.925799i \(0.623394\pi\)
\(432\) 0 0
\(433\) −1794.00 −0.199109 −0.0995544 0.995032i \(-0.531742\pi\)
−0.0995544 + 0.995032i \(0.531742\pi\)
\(434\) 0 0
\(435\) −2652.63 −0.292377
\(436\) 0 0
\(437\) −7910.48 −0.865926
\(438\) 0 0
\(439\) −5513.07 −0.599372 −0.299686 0.954038i \(-0.596882\pi\)
−0.299686 + 0.954038i \(0.596882\pi\)
\(440\) 0 0
\(441\) −797.195 −0.0860809
\(442\) 0 0
\(443\) 11355.7 1.21789 0.608945 0.793213i \(-0.291593\pi\)
0.608945 + 0.793213i \(0.291593\pi\)
\(444\) 0 0
\(445\) 11288.7 1.20255
\(446\) 0 0
\(447\) 3692.65 0.390730
\(448\) 0 0
\(449\) −3049.12 −0.320483 −0.160242 0.987078i \(-0.551227\pi\)
−0.160242 + 0.987078i \(0.551227\pi\)
\(450\) 0 0
\(451\) −1864.03 −0.194621
\(452\) 0 0
\(453\) −5639.52 −0.584918
\(454\) 0 0
\(455\) 6943.06 0.715375
\(456\) 0 0
\(457\) −12033.7 −1.23176 −0.615879 0.787841i \(-0.711199\pi\)
−0.615879 + 0.787841i \(0.711199\pi\)
\(458\) 0 0
\(459\) −16927.2 −1.72134
\(460\) 0 0
\(461\) 629.093 0.0635570 0.0317785 0.999495i \(-0.489883\pi\)
0.0317785 + 0.999495i \(0.489883\pi\)
\(462\) 0 0
\(463\) 4.68659 0.000470420 0 0.000235210 1.00000i \(-0.499925\pi\)
0.000235210 1.00000i \(0.499925\pi\)
\(464\) 0 0
\(465\) −22070.2 −2.20104
\(466\) 0 0
\(467\) −694.076 −0.0687752 −0.0343876 0.999409i \(-0.510948\pi\)
−0.0343876 + 0.999409i \(0.510948\pi\)
\(468\) 0 0
\(469\) 17610.6 1.73386
\(470\) 0 0
\(471\) −20466.9 −2.00227
\(472\) 0 0
\(473\) 11209.5 1.08967
\(474\) 0 0
\(475\) 1351.95 0.130593
\(476\) 0 0
\(477\) 21723.8 2.08525
\(478\) 0 0
\(479\) 10669.7 1.01777 0.508883 0.860836i \(-0.330059\pi\)
0.508883 + 0.860836i \(0.330059\pi\)
\(480\) 0 0
\(481\) −14340.4 −1.35939
\(482\) 0 0
\(483\) 26080.5 2.45694
\(484\) 0 0
\(485\) 14374.7 1.34582
\(486\) 0 0
\(487\) −1565.26 −0.145644 −0.0728221 0.997345i \(-0.523201\pi\)
−0.0728221 + 0.997345i \(0.523201\pi\)
\(488\) 0 0
\(489\) 31216.9 2.88687
\(490\) 0 0
\(491\) 15547.9 1.42906 0.714529 0.699606i \(-0.246641\pi\)
0.714529 + 0.699606i \(0.246641\pi\)
\(492\) 0 0
\(493\) −1713.78 −0.156561
\(494\) 0 0
\(495\) 17467.1 1.58604
\(496\) 0 0
\(497\) −4382.58 −0.395545
\(498\) 0 0
\(499\) 15465.1 1.38740 0.693700 0.720264i \(-0.255979\pi\)
0.693700 + 0.720264i \(0.255979\pi\)
\(500\) 0 0
\(501\) 13549.8 1.20830
\(502\) 0 0
\(503\) 6096.03 0.540375 0.270187 0.962808i \(-0.412914\pi\)
0.270187 + 0.962808i \(0.412914\pi\)
\(504\) 0 0
\(505\) 9718.60 0.856381
\(506\) 0 0
\(507\) 6535.60 0.572497
\(508\) 0 0
\(509\) −18928.0 −1.64827 −0.824133 0.566397i \(-0.808337\pi\)
−0.824133 + 0.566397i \(0.808337\pi\)
\(510\) 0 0
\(511\) −10255.2 −0.887791
\(512\) 0 0
\(513\) 14539.5 1.25133
\(514\) 0 0
\(515\) 3720.96 0.318379
\(516\) 0 0
\(517\) 8545.22 0.726921
\(518\) 0 0
\(519\) −23003.1 −1.94552
\(520\) 0 0
\(521\) −20910.7 −1.75838 −0.879189 0.476474i \(-0.841915\pi\)
−0.879189 + 0.476474i \(0.841915\pi\)
\(522\) 0 0
\(523\) −16510.9 −1.38044 −0.690220 0.723600i \(-0.742486\pi\)
−0.690220 + 0.723600i \(0.742486\pi\)
\(524\) 0 0
\(525\) −4457.30 −0.370538
\(526\) 0 0
\(527\) −14258.9 −1.17861
\(528\) 0 0
\(529\) 12119.6 0.996104
\(530\) 0 0
\(531\) 7491.12 0.612216
\(532\) 0 0
\(533\) 2370.66 0.192654
\(534\) 0 0
\(535\) −7728.04 −0.624509
\(536\) 0 0
\(537\) 15509.4 1.24633
\(538\) 0 0
\(539\) 416.526 0.0332858
\(540\) 0 0
\(541\) −24189.4 −1.92234 −0.961169 0.275960i \(-0.911004\pi\)
−0.961169 + 0.275960i \(0.911004\pi\)
\(542\) 0 0
\(543\) −31940.7 −2.52432
\(544\) 0 0
\(545\) −1452.96 −0.114198
\(546\) 0 0
\(547\) −8110.28 −0.633950 −0.316975 0.948434i \(-0.602667\pi\)
−0.316975 + 0.948434i \(0.602667\pi\)
\(548\) 0 0
\(549\) −26884.3 −2.08997
\(550\) 0 0
\(551\) 1472.03 0.113813
\(552\) 0 0
\(553\) −7701.47 −0.592224
\(554\) 0 0
\(555\) −34000.7 −2.60045
\(556\) 0 0
\(557\) 22133.0 1.68367 0.841835 0.539735i \(-0.181475\pi\)
0.841835 + 0.539735i \(0.181475\pi\)
\(558\) 0 0
\(559\) −14256.2 −1.07866
\(560\) 0 0
\(561\) 16533.1 1.24425
\(562\) 0 0
\(563\) −18464.4 −1.38220 −0.691102 0.722757i \(-0.742874\pi\)
−0.691102 + 0.722757i \(0.742874\pi\)
\(564\) 0 0
\(565\) 2077.23 0.154672
\(566\) 0 0
\(567\) −19491.3 −1.44366
\(568\) 0 0
\(569\) −5538.92 −0.408091 −0.204045 0.978961i \(-0.565409\pi\)
−0.204045 + 0.978961i \(0.565409\pi\)
\(570\) 0 0
\(571\) 11219.3 0.822262 0.411131 0.911576i \(-0.365134\pi\)
0.411131 + 0.911576i \(0.365134\pi\)
\(572\) 0 0
\(573\) −27324.8 −1.99216
\(574\) 0 0
\(575\) −4150.71 −0.301038
\(576\) 0 0
\(577\) −8239.06 −0.594448 −0.297224 0.954808i \(-0.596061\pi\)
−0.297224 + 0.954808i \(0.596061\pi\)
\(578\) 0 0
\(579\) −29630.0 −2.12674
\(580\) 0 0
\(581\) −2924.73 −0.208844
\(582\) 0 0
\(583\) −11350.5 −0.806326
\(584\) 0 0
\(585\) −22214.5 −1.57001
\(586\) 0 0
\(587\) 6452.46 0.453699 0.226850 0.973930i \(-0.427157\pi\)
0.226850 + 0.973930i \(0.427157\pi\)
\(588\) 0 0
\(589\) 12247.5 0.856790
\(590\) 0 0
\(591\) −371.863 −0.0258822
\(592\) 0 0
\(593\) 9937.52 0.688171 0.344085 0.938938i \(-0.388189\pi\)
0.344085 + 0.938938i \(0.388189\pi\)
\(594\) 0 0
\(595\) 10635.4 0.732790
\(596\) 0 0
\(597\) 4542.81 0.311432
\(598\) 0 0
\(599\) 5754.72 0.392540 0.196270 0.980550i \(-0.437117\pi\)
0.196270 + 0.980550i \(0.437117\pi\)
\(600\) 0 0
\(601\) 17242.3 1.17026 0.585132 0.810938i \(-0.301043\pi\)
0.585132 + 0.810938i \(0.301043\pi\)
\(602\) 0 0
\(603\) −56345.6 −3.80525
\(604\) 0 0
\(605\) 4074.40 0.273798
\(606\) 0 0
\(607\) −11857.0 −0.792854 −0.396427 0.918066i \(-0.629750\pi\)
−0.396427 + 0.918066i \(0.629750\pi\)
\(608\) 0 0
\(609\) −4853.23 −0.322927
\(610\) 0 0
\(611\) −10867.7 −0.719575
\(612\) 0 0
\(613\) 6818.16 0.449238 0.224619 0.974447i \(-0.427886\pi\)
0.224619 + 0.974447i \(0.427886\pi\)
\(614\) 0 0
\(615\) 5620.75 0.368537
\(616\) 0 0
\(617\) −30340.0 −1.97965 −0.989825 0.142289i \(-0.954554\pi\)
−0.989825 + 0.142289i \(0.954554\pi\)
\(618\) 0 0
\(619\) −8090.29 −0.525325 −0.262663 0.964888i \(-0.584601\pi\)
−0.262663 + 0.964888i \(0.584601\pi\)
\(620\) 0 0
\(621\) −44638.7 −2.88453
\(622\) 0 0
\(623\) 20653.6 1.32820
\(624\) 0 0
\(625\) −11586.3 −0.741526
\(626\) 0 0
\(627\) −14200.9 −0.904512
\(628\) 0 0
\(629\) −21966.8 −1.39248
\(630\) 0 0
\(631\) −12014.9 −0.758010 −0.379005 0.925395i \(-0.623734\pi\)
−0.379005 + 0.925395i \(0.623734\pi\)
\(632\) 0 0
\(633\) −18482.4 −1.16052
\(634\) 0 0
\(635\) 23092.8 1.44317
\(636\) 0 0
\(637\) −529.733 −0.0329494
\(638\) 0 0
\(639\) 14022.2 0.868089
\(640\) 0 0
\(641\) 2106.54 0.129802 0.0649011 0.997892i \(-0.479327\pi\)
0.0649011 + 0.997892i \(0.479327\pi\)
\(642\) 0 0
\(643\) 9074.16 0.556532 0.278266 0.960504i \(-0.410240\pi\)
0.278266 + 0.960504i \(0.410240\pi\)
\(644\) 0 0
\(645\) −33800.9 −2.06343
\(646\) 0 0
\(647\) 25416.1 1.54437 0.772187 0.635395i \(-0.219163\pi\)
0.772187 + 0.635395i \(0.219163\pi\)
\(648\) 0 0
\(649\) −3914.03 −0.236732
\(650\) 0 0
\(651\) −40379.4 −2.43102
\(652\) 0 0
\(653\) −18241.8 −1.09319 −0.546597 0.837396i \(-0.684077\pi\)
−0.546597 + 0.837396i \(0.684077\pi\)
\(654\) 0 0
\(655\) 22750.7 1.35717
\(656\) 0 0
\(657\) 32811.6 1.94841
\(658\) 0 0
\(659\) 6878.96 0.406626 0.203313 0.979114i \(-0.434829\pi\)
0.203313 + 0.979114i \(0.434829\pi\)
\(660\) 0 0
\(661\) −26498.4 −1.55925 −0.779627 0.626244i \(-0.784592\pi\)
−0.779627 + 0.626244i \(0.784592\pi\)
\(662\) 0 0
\(663\) −21026.5 −1.23168
\(664\) 0 0
\(665\) −9135.18 −0.532703
\(666\) 0 0
\(667\) −4519.41 −0.262357
\(668\) 0 0
\(669\) −18219.7 −1.05293
\(670\) 0 0
\(671\) 14046.8 0.808151
\(672\) 0 0
\(673\) 1232.11 0.0705710 0.0352855 0.999377i \(-0.488766\pi\)
0.0352855 + 0.999377i \(0.488766\pi\)
\(674\) 0 0
\(675\) 7629.00 0.435023
\(676\) 0 0
\(677\) −6769.54 −0.384305 −0.192153 0.981365i \(-0.561547\pi\)
−0.192153 + 0.981365i \(0.561547\pi\)
\(678\) 0 0
\(679\) 26299.8 1.48644
\(680\) 0 0
\(681\) 48919.0 2.75269
\(682\) 0 0
\(683\) −14965.1 −0.838393 −0.419196 0.907896i \(-0.637688\pi\)
−0.419196 + 0.907896i \(0.637688\pi\)
\(684\) 0 0
\(685\) 15664.2 0.873717
\(686\) 0 0
\(687\) −22722.0 −1.26186
\(688\) 0 0
\(689\) 14435.4 0.798177
\(690\) 0 0
\(691\) 26158.4 1.44011 0.720053 0.693919i \(-0.244117\pi\)
0.720053 + 0.693919i \(0.244117\pi\)
\(692\) 0 0
\(693\) 31957.7 1.75176
\(694\) 0 0
\(695\) −12109.7 −0.660929
\(696\) 0 0
\(697\) 3631.39 0.197344
\(698\) 0 0
\(699\) 29794.5 1.61220
\(700\) 0 0
\(701\) −1249.14 −0.0673031 −0.0336515 0.999434i \(-0.510714\pi\)
−0.0336515 + 0.999434i \(0.510714\pi\)
\(702\) 0 0
\(703\) 18868.1 1.01227
\(704\) 0 0
\(705\) −25767.0 −1.37651
\(706\) 0 0
\(707\) 17781.0 0.945863
\(708\) 0 0
\(709\) −16036.0 −0.849427 −0.424713 0.905328i \(-0.639625\pi\)
−0.424713 + 0.905328i \(0.639625\pi\)
\(710\) 0 0
\(711\) 24641.0 1.29973
\(712\) 0 0
\(713\) −37602.0 −1.97504
\(714\) 0 0
\(715\) 11606.8 0.607093
\(716\) 0 0
\(717\) 39476.7 2.05618
\(718\) 0 0
\(719\) 17662.1 0.916111 0.458056 0.888923i \(-0.348546\pi\)
0.458056 + 0.888923i \(0.348546\pi\)
\(720\) 0 0
\(721\) 6807.82 0.351645
\(722\) 0 0
\(723\) −59840.5 −3.07814
\(724\) 0 0
\(725\) 772.391 0.0395667
\(726\) 0 0
\(727\) −700.740 −0.0357483 −0.0178741 0.999840i \(-0.505690\pi\)
−0.0178741 + 0.999840i \(0.505690\pi\)
\(728\) 0 0
\(729\) −8963.33 −0.455385
\(730\) 0 0
\(731\) −21837.7 −1.10492
\(732\) 0 0
\(733\) −31338.0 −1.57912 −0.789561 0.613672i \(-0.789692\pi\)
−0.789561 + 0.613672i \(0.789692\pi\)
\(734\) 0 0
\(735\) −1255.98 −0.0630307
\(736\) 0 0
\(737\) 29440.0 1.47142
\(738\) 0 0
\(739\) 34066.8 1.69576 0.847880 0.530188i \(-0.177879\pi\)
0.847880 + 0.530188i \(0.177879\pi\)
\(740\) 0 0
\(741\) 18060.5 0.895371
\(742\) 0 0
\(743\) 6009.39 0.296720 0.148360 0.988933i \(-0.452601\pi\)
0.148360 + 0.988933i \(0.452601\pi\)
\(744\) 0 0
\(745\) 3971.03 0.195285
\(746\) 0 0
\(747\) 9357.75 0.458343
\(748\) 0 0
\(749\) −14139.1 −0.689763
\(750\) 0 0
\(751\) 4761.05 0.231336 0.115668 0.993288i \(-0.463099\pi\)
0.115668 + 0.993288i \(0.463099\pi\)
\(752\) 0 0
\(753\) −3756.32 −0.181790
\(754\) 0 0
\(755\) −6064.67 −0.292339
\(756\) 0 0
\(757\) −28880.1 −1.38661 −0.693307 0.720643i \(-0.743847\pi\)
−0.693307 + 0.720643i \(0.743847\pi\)
\(758\) 0 0
\(759\) 43599.3 2.08505
\(760\) 0 0
\(761\) 14423.5 0.687059 0.343530 0.939142i \(-0.388377\pi\)
0.343530 + 0.939142i \(0.388377\pi\)
\(762\) 0 0
\(763\) −2658.32 −0.126131
\(764\) 0 0
\(765\) −34028.3 −1.60823
\(766\) 0 0
\(767\) 4977.82 0.234340
\(768\) 0 0
\(769\) 38830.7 1.82090 0.910449 0.413621i \(-0.135736\pi\)
0.910449 + 0.413621i \(0.135736\pi\)
\(770\) 0 0
\(771\) 50904.6 2.37780
\(772\) 0 0
\(773\) −15417.7 −0.717380 −0.358690 0.933457i \(-0.616776\pi\)
−0.358690 + 0.933457i \(0.616776\pi\)
\(774\) 0 0
\(775\) 6426.39 0.297861
\(776\) 0 0
\(777\) −62207.3 −2.87217
\(778\) 0 0
\(779\) −3119.14 −0.143459
\(780\) 0 0
\(781\) −7326.45 −0.335673
\(782\) 0 0
\(783\) 8306.66 0.379126
\(784\) 0 0
\(785\) −22009.9 −1.00072
\(786\) 0 0
\(787\) 1122.93 0.0508618 0.0254309 0.999677i \(-0.491904\pi\)
0.0254309 + 0.999677i \(0.491904\pi\)
\(788\) 0 0
\(789\) 38358.4 1.73079
\(790\) 0 0
\(791\) 3800.47 0.170833
\(792\) 0 0
\(793\) −17864.5 −0.799984
\(794\) 0 0
\(795\) 34225.8 1.52687
\(796\) 0 0
\(797\) 17273.1 0.767684 0.383842 0.923399i \(-0.374601\pi\)
0.383842 + 0.923399i \(0.374601\pi\)
\(798\) 0 0
\(799\) −16647.2 −0.737092
\(800\) 0 0
\(801\) −66081.7 −2.91496
\(802\) 0 0
\(803\) −17143.7 −0.753411
\(804\) 0 0
\(805\) 28046.6 1.22797
\(806\) 0 0
\(807\) 31431.2 1.37104
\(808\) 0 0
\(809\) 31287.2 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(810\) 0 0
\(811\) 20399.1 0.883244 0.441622 0.897201i \(-0.354403\pi\)
0.441622 + 0.897201i \(0.354403\pi\)
\(812\) 0 0
\(813\) −70448.8 −3.03905
\(814\) 0 0
\(815\) 33570.3 1.44284
\(816\) 0 0
\(817\) 18757.3 0.803223
\(818\) 0 0
\(819\) −40643.4 −1.73406
\(820\) 0 0
\(821\) 9158.81 0.389336 0.194668 0.980869i \(-0.437637\pi\)
0.194668 + 0.980869i \(0.437637\pi\)
\(822\) 0 0
\(823\) −20147.5 −0.853340 −0.426670 0.904407i \(-0.640313\pi\)
−0.426670 + 0.904407i \(0.640313\pi\)
\(824\) 0 0
\(825\) −7451.35 −0.314452
\(826\) 0 0
\(827\) 15491.3 0.651372 0.325686 0.945478i \(-0.394405\pi\)
0.325686 + 0.945478i \(0.394405\pi\)
\(828\) 0 0
\(829\) 36020.1 1.50908 0.754542 0.656252i \(-0.227859\pi\)
0.754542 + 0.656252i \(0.227859\pi\)
\(830\) 0 0
\(831\) 2348.49 0.0980363
\(832\) 0 0
\(833\) −811.449 −0.0337515
\(834\) 0 0
\(835\) 14571.2 0.603902
\(836\) 0 0
\(837\) 69112.4 2.85409
\(838\) 0 0
\(839\) 32919.5 1.35460 0.677298 0.735709i \(-0.263151\pi\)
0.677298 + 0.735709i \(0.263151\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) 34591.5 1.41328
\(844\) 0 0
\(845\) 7028.30 0.286131
\(846\) 0 0
\(847\) 7454.48 0.302407
\(848\) 0 0
\(849\) −70291.0 −2.84144
\(850\) 0 0
\(851\) −57928.5 −2.33345
\(852\) 0 0
\(853\) −15790.8 −0.633842 −0.316921 0.948452i \(-0.602649\pi\)
−0.316921 + 0.948452i \(0.602649\pi\)
\(854\) 0 0
\(855\) 29228.2 1.16911
\(856\) 0 0
\(857\) 31142.7 1.24132 0.620661 0.784079i \(-0.286864\pi\)
0.620661 + 0.784079i \(0.286864\pi\)
\(858\) 0 0
\(859\) −15038.5 −0.597331 −0.298665 0.954358i \(-0.596541\pi\)
−0.298665 + 0.954358i \(0.596541\pi\)
\(860\) 0 0
\(861\) 10283.7 0.407045
\(862\) 0 0
\(863\) 28642.7 1.12979 0.564894 0.825163i \(-0.308917\pi\)
0.564894 + 0.825163i \(0.308917\pi\)
\(864\) 0 0
\(865\) −24737.3 −0.972361
\(866\) 0 0
\(867\) 13102.4 0.513243
\(868\) 0 0
\(869\) −12874.7 −0.502582
\(870\) 0 0
\(871\) −37441.4 −1.45655
\(872\) 0 0
\(873\) −84146.7 −3.26224
\(874\) 0 0
\(875\) −27289.4 −1.05434
\(876\) 0 0
\(877\) 23442.1 0.902603 0.451302 0.892371i \(-0.350960\pi\)
0.451302 + 0.892371i \(0.350960\pi\)
\(878\) 0 0
\(879\) −31542.9 −1.21037
\(880\) 0 0
\(881\) 48730.1 1.86352 0.931758 0.363079i \(-0.118275\pi\)
0.931758 + 0.363079i \(0.118275\pi\)
\(882\) 0 0
\(883\) 37501.4 1.42924 0.714622 0.699511i \(-0.246599\pi\)
0.714622 + 0.699511i \(0.246599\pi\)
\(884\) 0 0
\(885\) 11802.3 0.448281
\(886\) 0 0
\(887\) 23885.6 0.904170 0.452085 0.891975i \(-0.350680\pi\)
0.452085 + 0.891975i \(0.350680\pi\)
\(888\) 0 0
\(889\) 42250.4 1.59396
\(890\) 0 0
\(891\) −32584.0 −1.22515
\(892\) 0 0
\(893\) 14299.0 0.535830
\(894\) 0 0
\(895\) 16678.6 0.622911
\(896\) 0 0
\(897\) −55449.0 −2.06398
\(898\) 0 0
\(899\) 6997.22 0.259589
\(900\) 0 0
\(901\) 22112.2 0.817608
\(902\) 0 0
\(903\) −61841.8 −2.27903
\(904\) 0 0
\(905\) −34348.7 −1.26164
\(906\) 0 0
\(907\) −10117.1 −0.370380 −0.185190 0.982703i \(-0.559290\pi\)
−0.185190 + 0.982703i \(0.559290\pi\)
\(908\) 0 0
\(909\) −56890.9 −2.07585
\(910\) 0 0
\(911\) 7387.19 0.268659 0.134330 0.990937i \(-0.457112\pi\)
0.134330 + 0.990937i \(0.457112\pi\)
\(912\) 0 0
\(913\) −4889.33 −0.177232
\(914\) 0 0
\(915\) −42356.2 −1.53033
\(916\) 0 0
\(917\) 41624.4 1.49897
\(918\) 0 0
\(919\) 41196.3 1.47872 0.739359 0.673312i \(-0.235129\pi\)
0.739359 + 0.673312i \(0.235129\pi\)
\(920\) 0 0
\(921\) 62843.3 2.24838
\(922\) 0 0
\(923\) 9317.69 0.332281
\(924\) 0 0
\(925\) 9900.29 0.351913
\(926\) 0 0
\(927\) −21781.8 −0.771745
\(928\) 0 0
\(929\) −42590.3 −1.50414 −0.752068 0.659085i \(-0.770944\pi\)
−0.752068 + 0.659085i \(0.770944\pi\)
\(930\) 0 0
\(931\) 696.985 0.0245357
\(932\) 0 0
\(933\) −56819.3 −1.99376
\(934\) 0 0
\(935\) 17779.4 0.621871
\(936\) 0 0
\(937\) −19764.2 −0.689079 −0.344539 0.938772i \(-0.611965\pi\)
−0.344539 + 0.938772i \(0.611965\pi\)
\(938\) 0 0
\(939\) 49392.6 1.71658
\(940\) 0 0
\(941\) 19163.0 0.663866 0.331933 0.943303i \(-0.392299\pi\)
0.331933 + 0.943303i \(0.392299\pi\)
\(942\) 0 0
\(943\) 9576.31 0.330697
\(944\) 0 0
\(945\) −51549.7 −1.77451
\(946\) 0 0
\(947\) −29334.5 −1.00659 −0.503297 0.864114i \(-0.667880\pi\)
−0.503297 + 0.864114i \(0.667880\pi\)
\(948\) 0 0
\(949\) 21803.2 0.745797
\(950\) 0 0
\(951\) 16715.0 0.569948
\(952\) 0 0
\(953\) −26975.1 −0.916904 −0.458452 0.888719i \(-0.651596\pi\)
−0.458452 + 0.888719i \(0.651596\pi\)
\(954\) 0 0
\(955\) −29384.7 −0.995672
\(956\) 0 0
\(957\) −8113.23 −0.274048
\(958\) 0 0
\(959\) 28658.9 0.965011
\(960\) 0 0
\(961\) 28426.7 0.954204
\(962\) 0 0
\(963\) 45238.5 1.51380
\(964\) 0 0
\(965\) −31863.7 −1.06293
\(966\) 0 0
\(967\) −12036.6 −0.400280 −0.200140 0.979767i \(-0.564140\pi\)
−0.200140 + 0.979767i \(0.564140\pi\)
\(968\) 0 0
\(969\) 27665.2 0.917167
\(970\) 0 0
\(971\) 43608.2 1.44125 0.720625 0.693325i \(-0.243855\pi\)
0.720625 + 0.693325i \(0.243855\pi\)
\(972\) 0 0
\(973\) −22155.7 −0.729989
\(974\) 0 0
\(975\) 9476.54 0.311274
\(976\) 0 0
\(977\) −48826.4 −1.59887 −0.799434 0.600754i \(-0.794867\pi\)
−0.799434 + 0.600754i \(0.794867\pi\)
\(978\) 0 0
\(979\) 34527.0 1.12716
\(980\) 0 0
\(981\) 8505.37 0.276815
\(982\) 0 0
\(983\) 23279.6 0.755345 0.377672 0.925939i \(-0.376725\pi\)
0.377672 + 0.925939i \(0.376725\pi\)
\(984\) 0 0
\(985\) −399.897 −0.0129358
\(986\) 0 0
\(987\) −47143.0 −1.52034
\(988\) 0 0
\(989\) −57588.1 −1.85156
\(990\) 0 0
\(991\) −39511.7 −1.26653 −0.633264 0.773936i \(-0.718285\pi\)
−0.633264 + 0.773936i \(0.718285\pi\)
\(992\) 0 0
\(993\) 93999.3 3.00400
\(994\) 0 0
\(995\) 4885.28 0.155652
\(996\) 0 0
\(997\) −38690.6 −1.22903 −0.614515 0.788905i \(-0.710648\pi\)
−0.614515 + 0.788905i \(0.710648\pi\)
\(998\) 0 0
\(999\) 106472. 3.37201
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.1.2 12
4.3 odd 2 1856.4.a.bl.1.11 12
8.3 odd 2 928.4.a.h.1.2 12
8.5 even 2 928.4.a.j.1.11 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
928.4.a.h.1.2 12 8.3 odd 2
928.4.a.j.1.11 yes 12 8.5 even 2
1856.4.a.bj.1.2 12 1.1 even 1 trivial
1856.4.a.bl.1.11 12 4.3 odd 2