Properties

Label 1856.4.a.bj.1.12
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 1.12
Root \(-10.7228\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.72277 q^{3} +6.27199 q^{5} +31.0130 q^{7} +67.5323 q^{9} +O(q^{10})\) \(q+9.72277 q^{3} +6.27199 q^{5} +31.0130 q^{7} +67.5323 q^{9} -8.24716 q^{11} -32.0907 q^{13} +60.9811 q^{15} +51.1369 q^{17} -158.789 q^{19} +301.533 q^{21} +129.494 q^{23} -85.6622 q^{25} +394.086 q^{27} -29.0000 q^{29} -14.6426 q^{31} -80.1853 q^{33} +194.513 q^{35} +257.398 q^{37} -312.011 q^{39} -26.0208 q^{41} -3.40346 q^{43} +423.562 q^{45} +482.228 q^{47} +618.808 q^{49} +497.192 q^{51} -169.104 q^{53} -51.7261 q^{55} -1543.87 q^{57} +569.745 q^{59} +500.616 q^{61} +2094.38 q^{63} -201.273 q^{65} -335.555 q^{67} +1259.04 q^{69} +958.137 q^{71} -946.043 q^{73} -832.874 q^{75} -255.769 q^{77} -179.326 q^{79} +2008.24 q^{81} +285.090 q^{83} +320.730 q^{85} -281.960 q^{87} +832.078 q^{89} -995.231 q^{91} -142.367 q^{93} -995.921 q^{95} -908.370 q^{97} -556.950 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 14 q^{3} + 10 q^{5} + 44 q^{7} + 130 q^{9}+O(q^{10}) \) 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
\(3\) 9.72277 1.87115 0.935574 0.353130i \(-0.114883\pi\)
0.935574 + 0.353130i \(0.114883\pi\)
\(4\) 0 0
\(5\) 6.27199 0.560984 0.280492 0.959856i \(-0.409502\pi\)
0.280492 + 0.959856i \(0.409502\pi\)
\(6\) 0 0
\(7\) 31.0130 1.67455 0.837273 0.546785i \(-0.184149\pi\)
0.837273 + 0.546785i \(0.184149\pi\)
\(8\) 0 0
\(9\) 67.5323 2.50120
\(10\) 0 0
\(11\) −8.24716 −0.226056 −0.113028 0.993592i \(-0.536055\pi\)
−0.113028 + 0.993592i \(0.536055\pi\)
\(12\) 0 0
\(13\) −32.0907 −0.684644 −0.342322 0.939583i \(-0.611213\pi\)
−0.342322 + 0.939583i \(0.611213\pi\)
\(14\) 0 0
\(15\) 60.9811 1.04968
\(16\) 0 0
\(17\) 51.1369 0.729560 0.364780 0.931094i \(-0.381144\pi\)
0.364780 + 0.931094i \(0.381144\pi\)
\(18\) 0 0
\(19\) −158.789 −1.91730 −0.958648 0.284593i \(-0.908141\pi\)
−0.958648 + 0.284593i \(0.908141\pi\)
\(20\) 0 0
\(21\) 301.533 3.13332
\(22\) 0 0
\(23\) 129.494 1.17398 0.586988 0.809596i \(-0.300314\pi\)
0.586988 + 0.809596i \(0.300314\pi\)
\(24\) 0 0
\(25\) −85.6622 −0.685297
\(26\) 0 0
\(27\) 394.086 2.80896
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) −14.6426 −0.0848353 −0.0424177 0.999100i \(-0.513506\pi\)
−0.0424177 + 0.999100i \(0.513506\pi\)
\(32\) 0 0
\(33\) −80.1853 −0.422984
\(34\) 0 0
\(35\) 194.513 0.939393
\(36\) 0 0
\(37\) 257.398 1.14367 0.571837 0.820367i \(-0.306231\pi\)
0.571837 + 0.820367i \(0.306231\pi\)
\(38\) 0 0
\(39\) −312.011 −1.28107
\(40\) 0 0
\(41\) −26.0208 −0.0991163 −0.0495582 0.998771i \(-0.515781\pi\)
−0.0495582 + 0.998771i \(0.515781\pi\)
\(42\) 0 0
\(43\) −3.40346 −0.0120703 −0.00603516 0.999982i \(-0.501921\pi\)
−0.00603516 + 0.999982i \(0.501921\pi\)
\(44\) 0 0
\(45\) 423.562 1.40313
\(46\) 0 0
\(47\) 482.228 1.49660 0.748300 0.663361i \(-0.230871\pi\)
0.748300 + 0.663361i \(0.230871\pi\)
\(48\) 0 0
\(49\) 618.808 1.80410
\(50\) 0 0
\(51\) 497.192 1.36511
\(52\) 0 0
\(53\) −169.104 −0.438269 −0.219135 0.975695i \(-0.570323\pi\)
−0.219135 + 0.975695i \(0.570323\pi\)
\(54\) 0 0
\(55\) −51.7261 −0.126814
\(56\) 0 0
\(57\) −1543.87 −3.58755
\(58\) 0 0
\(59\) 569.745 1.25719 0.628597 0.777731i \(-0.283630\pi\)
0.628597 + 0.777731i \(0.283630\pi\)
\(60\) 0 0
\(61\) 500.616 1.05078 0.525388 0.850863i \(-0.323920\pi\)
0.525388 + 0.850863i \(0.323920\pi\)
\(62\) 0 0
\(63\) 2094.38 4.18837
\(64\) 0 0
\(65\) −201.273 −0.384074
\(66\) 0 0
\(67\) −335.555 −0.611859 −0.305930 0.952054i \(-0.598967\pi\)
−0.305930 + 0.952054i \(0.598967\pi\)
\(68\) 0 0
\(69\) 1259.04 2.19668
\(70\) 0 0
\(71\) 958.137 1.60155 0.800774 0.598966i \(-0.204422\pi\)
0.800774 + 0.598966i \(0.204422\pi\)
\(72\) 0 0
\(73\) −946.043 −1.51679 −0.758397 0.651793i \(-0.774017\pi\)
−0.758397 + 0.651793i \(0.774017\pi\)
\(74\) 0 0
\(75\) −832.874 −1.28229
\(76\) 0 0
\(77\) −255.769 −0.378541
\(78\) 0 0
\(79\) −179.326 −0.255389 −0.127694 0.991814i \(-0.540758\pi\)
−0.127694 + 0.991814i \(0.540758\pi\)
\(80\) 0 0
\(81\) 2008.24 2.75479
\(82\) 0 0
\(83\) 285.090 0.377020 0.188510 0.982071i \(-0.439634\pi\)
0.188510 + 0.982071i \(0.439634\pi\)
\(84\) 0 0
\(85\) 320.730 0.409271
\(86\) 0 0
\(87\) −281.960 −0.347464
\(88\) 0 0
\(89\) 832.078 0.991013 0.495506 0.868604i \(-0.334983\pi\)
0.495506 + 0.868604i \(0.334983\pi\)
\(90\) 0 0
\(91\) −995.231 −1.14647
\(92\) 0 0
\(93\) −142.367 −0.158740
\(94\) 0 0
\(95\) −995.921 −1.07557
\(96\) 0 0
\(97\) −908.370 −0.950835 −0.475417 0.879760i \(-0.657703\pi\)
−0.475417 + 0.879760i \(0.657703\pi\)
\(98\) 0 0
\(99\) −556.950 −0.565410
\(100\) 0 0
\(101\) −996.317 −0.981557 −0.490778 0.871284i \(-0.663288\pi\)
−0.490778 + 0.871284i \(0.663288\pi\)
\(102\) 0 0
\(103\) 1295.46 1.23928 0.619639 0.784887i \(-0.287279\pi\)
0.619639 + 0.784887i \(0.287279\pi\)
\(104\) 0 0
\(105\) 1891.21 1.75774
\(106\) 0 0
\(107\) −1276.23 −1.15307 −0.576533 0.817074i \(-0.695595\pi\)
−0.576533 + 0.817074i \(0.695595\pi\)
\(108\) 0 0
\(109\) −1481.99 −1.30228 −0.651141 0.758957i \(-0.725709\pi\)
−0.651141 + 0.758957i \(0.725709\pi\)
\(110\) 0 0
\(111\) 2502.62 2.13998
\(112\) 0 0
\(113\) 565.572 0.470837 0.235418 0.971894i \(-0.424354\pi\)
0.235418 + 0.971894i \(0.424354\pi\)
\(114\) 0 0
\(115\) 812.187 0.658581
\(116\) 0 0
\(117\) −2167.16 −1.71243
\(118\) 0 0
\(119\) 1585.91 1.22168
\(120\) 0 0
\(121\) −1262.98 −0.948899
\(122\) 0 0
\(123\) −252.995 −0.185461
\(124\) 0 0
\(125\) −1321.27 −0.945424
\(126\) 0 0
\(127\) −2132.64 −1.49009 −0.745045 0.667014i \(-0.767572\pi\)
−0.745045 + 0.667014i \(0.767572\pi\)
\(128\) 0 0
\(129\) −33.0911 −0.0225853
\(130\) 0 0
\(131\) −2518.91 −1.67999 −0.839993 0.542598i \(-0.817441\pi\)
−0.839993 + 0.542598i \(0.817441\pi\)
\(132\) 0 0
\(133\) −4924.52 −3.21060
\(134\) 0 0
\(135\) 2471.71 1.57578
\(136\) 0 0
\(137\) −57.5792 −0.0359075 −0.0179537 0.999839i \(-0.505715\pi\)
−0.0179537 + 0.999839i \(0.505715\pi\)
\(138\) 0 0
\(139\) 1250.64 0.763148 0.381574 0.924338i \(-0.375382\pi\)
0.381574 + 0.924338i \(0.375382\pi\)
\(140\) 0 0
\(141\) 4688.59 2.80036
\(142\) 0 0
\(143\) 264.658 0.154768
\(144\) 0 0
\(145\) −181.888 −0.104172
\(146\) 0 0
\(147\) 6016.52 3.37575
\(148\) 0 0
\(149\) 1811.93 0.996235 0.498118 0.867109i \(-0.334025\pi\)
0.498118 + 0.867109i \(0.334025\pi\)
\(150\) 0 0
\(151\) 680.525 0.366757 0.183379 0.983042i \(-0.441297\pi\)
0.183379 + 0.983042i \(0.441297\pi\)
\(152\) 0 0
\(153\) 3453.39 1.82477
\(154\) 0 0
\(155\) −91.8385 −0.0475912
\(156\) 0 0
\(157\) 1422.01 0.722858 0.361429 0.932400i \(-0.382289\pi\)
0.361429 + 0.932400i \(0.382289\pi\)
\(158\) 0 0
\(159\) −1644.16 −0.820066
\(160\) 0 0
\(161\) 4016.01 1.96588
\(162\) 0 0
\(163\) 3154.01 1.51559 0.757794 0.652493i \(-0.226277\pi\)
0.757794 + 0.652493i \(0.226277\pi\)
\(164\) 0 0
\(165\) −502.921 −0.237287
\(166\) 0 0
\(167\) −1539.64 −0.713417 −0.356708 0.934216i \(-0.616101\pi\)
−0.356708 + 0.934216i \(0.616101\pi\)
\(168\) 0 0
\(169\) −1167.18 −0.531263
\(170\) 0 0
\(171\) −10723.4 −4.79554
\(172\) 0 0
\(173\) −2816.06 −1.23758 −0.618789 0.785557i \(-0.712376\pi\)
−0.618789 + 0.785557i \(0.712376\pi\)
\(174\) 0 0
\(175\) −2656.64 −1.14756
\(176\) 0 0
\(177\) 5539.50 2.35240
\(178\) 0 0
\(179\) 3564.28 1.48830 0.744152 0.668010i \(-0.232854\pi\)
0.744152 + 0.668010i \(0.232854\pi\)
\(180\) 0 0
\(181\) 261.021 0.107191 0.0535953 0.998563i \(-0.482932\pi\)
0.0535953 + 0.998563i \(0.482932\pi\)
\(182\) 0 0
\(183\) 4867.38 1.96616
\(184\) 0 0
\(185\) 1614.40 0.641583
\(186\) 0 0
\(187\) −421.734 −0.164921
\(188\) 0 0
\(189\) 12221.8 4.70373
\(190\) 0 0
\(191\) −2242.54 −0.849551 −0.424776 0.905299i \(-0.639647\pi\)
−0.424776 + 0.905299i \(0.639647\pi\)
\(192\) 0 0
\(193\) 120.858 0.0450752 0.0225376 0.999746i \(-0.492825\pi\)
0.0225376 + 0.999746i \(0.492825\pi\)
\(194\) 0 0
\(195\) −1956.93 −0.718660
\(196\) 0 0
\(197\) 864.591 0.312688 0.156344 0.987703i \(-0.450029\pi\)
0.156344 + 0.987703i \(0.450029\pi\)
\(198\) 0 0
\(199\) −4885.39 −1.74028 −0.870140 0.492804i \(-0.835972\pi\)
−0.870140 + 0.492804i \(0.835972\pi\)
\(200\) 0 0
\(201\) −3262.52 −1.14488
\(202\) 0 0
\(203\) −899.378 −0.310955
\(204\) 0 0
\(205\) −163.202 −0.0556027
\(206\) 0 0
\(207\) 8745.06 2.93634
\(208\) 0 0
\(209\) 1309.56 0.433416
\(210\) 0 0
\(211\) −5261.82 −1.71677 −0.858386 0.513005i \(-0.828532\pi\)
−0.858386 + 0.513005i \(0.828532\pi\)
\(212\) 0 0
\(213\) 9315.75 2.99673
\(214\) 0 0
\(215\) −21.3465 −0.00677125
\(216\) 0 0
\(217\) −454.112 −0.142061
\(218\) 0 0
\(219\) −9198.16 −2.83815
\(220\) 0 0
\(221\) −1641.02 −0.499489
\(222\) 0 0
\(223\) 2726.49 0.818740 0.409370 0.912368i \(-0.365748\pi\)
0.409370 + 0.912368i \(0.365748\pi\)
\(224\) 0 0
\(225\) −5784.96 −1.71406
\(226\) 0 0
\(227\) −6079.66 −1.77763 −0.888813 0.458270i \(-0.848469\pi\)
−0.888813 + 0.458270i \(0.848469\pi\)
\(228\) 0 0
\(229\) −538.146 −0.155291 −0.0776456 0.996981i \(-0.524740\pi\)
−0.0776456 + 0.996981i \(0.524740\pi\)
\(230\) 0 0
\(231\) −2486.79 −0.708306
\(232\) 0 0
\(233\) 2834.78 0.797051 0.398525 0.917157i \(-0.369522\pi\)
0.398525 + 0.917157i \(0.369522\pi\)
\(234\) 0 0
\(235\) 3024.53 0.839568
\(236\) 0 0
\(237\) −1743.54 −0.477870
\(238\) 0 0
\(239\) −354.937 −0.0960627 −0.0480313 0.998846i \(-0.515295\pi\)
−0.0480313 + 0.998846i \(0.515295\pi\)
\(240\) 0 0
\(241\) 4887.24 1.30628 0.653142 0.757235i \(-0.273450\pi\)
0.653142 + 0.757235i \(0.273450\pi\)
\(242\) 0 0
\(243\) 8885.33 2.34565
\(244\) 0 0
\(245\) 3881.15 1.01207
\(246\) 0 0
\(247\) 5095.65 1.31267
\(248\) 0 0
\(249\) 2771.86 0.705461
\(250\) 0 0
\(251\) 250.836 0.0630783 0.0315391 0.999503i \(-0.489959\pi\)
0.0315391 + 0.999503i \(0.489959\pi\)
\(252\) 0 0
\(253\) −1067.96 −0.265384
\(254\) 0 0
\(255\) 3118.38 0.765807
\(256\) 0 0
\(257\) 721.700 0.175169 0.0875845 0.996157i \(-0.472085\pi\)
0.0875845 + 0.996157i \(0.472085\pi\)
\(258\) 0 0
\(259\) 7982.68 1.91513
\(260\) 0 0
\(261\) −1958.44 −0.464461
\(262\) 0 0
\(263\) −3765.00 −0.882737 −0.441368 0.897326i \(-0.645507\pi\)
−0.441368 + 0.897326i \(0.645507\pi\)
\(264\) 0 0
\(265\) −1060.62 −0.245862
\(266\) 0 0
\(267\) 8090.11 1.85433
\(268\) 0 0
\(269\) −552.384 −0.125202 −0.0626012 0.998039i \(-0.519940\pi\)
−0.0626012 + 0.998039i \(0.519940\pi\)
\(270\) 0 0
\(271\) −5370.32 −1.20378 −0.601888 0.798580i \(-0.705585\pi\)
−0.601888 + 0.798580i \(0.705585\pi\)
\(272\) 0 0
\(273\) −9676.40 −2.14521
\(274\) 0 0
\(275\) 706.470 0.154915
\(276\) 0 0
\(277\) 3413.04 0.740324 0.370162 0.928967i \(-0.379302\pi\)
0.370162 + 0.928967i \(0.379302\pi\)
\(278\) 0 0
\(279\) −988.851 −0.212190
\(280\) 0 0
\(281\) −6727.56 −1.42823 −0.714115 0.700028i \(-0.753171\pi\)
−0.714115 + 0.700028i \(0.753171\pi\)
\(282\) 0 0
\(283\) −8718.25 −1.83126 −0.915630 0.402023i \(-0.868307\pi\)
−0.915630 + 0.402023i \(0.868307\pi\)
\(284\) 0 0
\(285\) −9683.11 −2.01256
\(286\) 0 0
\(287\) −806.984 −0.165975
\(288\) 0 0
\(289\) −2298.02 −0.467742
\(290\) 0 0
\(291\) −8831.87 −1.77915
\(292\) 0 0
\(293\) −1279.00 −0.255018 −0.127509 0.991837i \(-0.540698\pi\)
−0.127509 + 0.991837i \(0.540698\pi\)
\(294\) 0 0
\(295\) 3573.43 0.705265
\(296\) 0 0
\(297\) −3250.09 −0.634982
\(298\) 0 0
\(299\) −4155.57 −0.803755
\(300\) 0 0
\(301\) −105.552 −0.0202123
\(302\) 0 0
\(303\) −9686.96 −1.83664
\(304\) 0 0
\(305\) 3139.86 0.589468
\(306\) 0 0
\(307\) 8483.52 1.57713 0.788567 0.614949i \(-0.210823\pi\)
0.788567 + 0.614949i \(0.210823\pi\)
\(308\) 0 0
\(309\) 12595.5 2.31887
\(310\) 0 0
\(311\) −7130.60 −1.30013 −0.650063 0.759880i \(-0.725257\pi\)
−0.650063 + 0.759880i \(0.725257\pi\)
\(312\) 0 0
\(313\) 3430.86 0.619565 0.309782 0.950808i \(-0.399744\pi\)
0.309782 + 0.950808i \(0.399744\pi\)
\(314\) 0 0
\(315\) 13135.9 2.34961
\(316\) 0 0
\(317\) −7352.21 −1.30265 −0.651327 0.758797i \(-0.725787\pi\)
−0.651327 + 0.758797i \(0.725787\pi\)
\(318\) 0 0
\(319\) 239.168 0.0419775
\(320\) 0 0
\(321\) −12408.5 −2.15756
\(322\) 0 0
\(323\) −8119.96 −1.39878
\(324\) 0 0
\(325\) 2748.96 0.469185
\(326\) 0 0
\(327\) −14409.0 −2.43676
\(328\) 0 0
\(329\) 14955.3 2.50612
\(330\) 0 0
\(331\) −5975.85 −0.992333 −0.496166 0.868227i \(-0.665259\pi\)
−0.496166 + 0.868227i \(0.665259\pi\)
\(332\) 0 0
\(333\) 17382.7 2.86055
\(334\) 0 0
\(335\) −2104.60 −0.343243
\(336\) 0 0
\(337\) 6455.18 1.04343 0.521715 0.853120i \(-0.325292\pi\)
0.521715 + 0.853120i \(0.325292\pi\)
\(338\) 0 0
\(339\) 5498.93 0.881005
\(340\) 0 0
\(341\) 120.760 0.0191775
\(342\) 0 0
\(343\) 8553.62 1.34651
\(344\) 0 0
\(345\) 7896.71 1.23230
\(346\) 0 0
\(347\) −12624.0 −1.95300 −0.976502 0.215509i \(-0.930859\pi\)
−0.976502 + 0.215509i \(0.930859\pi\)
\(348\) 0 0
\(349\) 35.6888 0.00547386 0.00273693 0.999996i \(-0.499129\pi\)
0.00273693 + 0.999996i \(0.499129\pi\)
\(350\) 0 0
\(351\) −12646.5 −1.92314
\(352\) 0 0
\(353\) −639.322 −0.0963957 −0.0481978 0.998838i \(-0.515348\pi\)
−0.0481978 + 0.998838i \(0.515348\pi\)
\(354\) 0 0
\(355\) 6009.42 0.898443
\(356\) 0 0
\(357\) 15419.4 2.28595
\(358\) 0 0
\(359\) 7152.10 1.05146 0.525729 0.850652i \(-0.323793\pi\)
0.525729 + 0.850652i \(0.323793\pi\)
\(360\) 0 0
\(361\) 18354.9 2.67603
\(362\) 0 0
\(363\) −12279.7 −1.77553
\(364\) 0 0
\(365\) −5933.57 −0.850897
\(366\) 0 0
\(367\) −6894.06 −0.980564 −0.490282 0.871564i \(-0.663106\pi\)
−0.490282 + 0.871564i \(0.663106\pi\)
\(368\) 0 0
\(369\) −1757.25 −0.247909
\(370\) 0 0
\(371\) −5244.43 −0.733902
\(372\) 0 0
\(373\) 8479.89 1.17714 0.588569 0.808447i \(-0.299692\pi\)
0.588569 + 0.808447i \(0.299692\pi\)
\(374\) 0 0
\(375\) −12846.4 −1.76903
\(376\) 0 0
\(377\) 930.631 0.127135
\(378\) 0 0
\(379\) 4935.89 0.668970 0.334485 0.942401i \(-0.391438\pi\)
0.334485 + 0.942401i \(0.391438\pi\)
\(380\) 0 0
\(381\) −20735.2 −2.78818
\(382\) 0 0
\(383\) 1790.95 0.238938 0.119469 0.992838i \(-0.461881\pi\)
0.119469 + 0.992838i \(0.461881\pi\)
\(384\) 0 0
\(385\) −1604.18 −0.212355
\(386\) 0 0
\(387\) −229.844 −0.0301902
\(388\) 0 0
\(389\) 6332.31 0.825350 0.412675 0.910878i \(-0.364595\pi\)
0.412675 + 0.910878i \(0.364595\pi\)
\(390\) 0 0
\(391\) 6621.94 0.856486
\(392\) 0 0
\(393\) −24490.8 −3.14350
\(394\) 0 0
\(395\) −1124.73 −0.143269
\(396\) 0 0
\(397\) −14324.5 −1.81089 −0.905446 0.424462i \(-0.860463\pi\)
−0.905446 + 0.424462i \(0.860463\pi\)
\(398\) 0 0
\(399\) −47880.0 −6.00751
\(400\) 0 0
\(401\) 6430.30 0.800783 0.400392 0.916344i \(-0.368874\pi\)
0.400392 + 0.916344i \(0.368874\pi\)
\(402\) 0 0
\(403\) 469.893 0.0580820
\(404\) 0 0
\(405\) 12595.7 1.54539
\(406\) 0 0
\(407\) −2122.80 −0.258534
\(408\) 0 0
\(409\) 44.4605 0.00537513 0.00268757 0.999996i \(-0.499145\pi\)
0.00268757 + 0.999996i \(0.499145\pi\)
\(410\) 0 0
\(411\) −559.830 −0.0671882
\(412\) 0 0
\(413\) 17669.5 2.10523
\(414\) 0 0
\(415\) 1788.08 0.211502
\(416\) 0 0
\(417\) 12159.7 1.42796
\(418\) 0 0
\(419\) 23.4862 0.00273837 0.00136918 0.999999i \(-0.499564\pi\)
0.00136918 + 0.999999i \(0.499564\pi\)
\(420\) 0 0
\(421\) −3365.39 −0.389594 −0.194797 0.980844i \(-0.562405\pi\)
−0.194797 + 0.980844i \(0.562405\pi\)
\(422\) 0 0
\(423\) 32566.0 3.74329
\(424\) 0 0
\(425\) −4380.50 −0.499965
\(426\) 0 0
\(427\) 15525.6 1.75957
\(428\) 0 0
\(429\) 2573.21 0.289593
\(430\) 0 0
\(431\) −9557.80 −1.06817 −0.534087 0.845429i \(-0.679345\pi\)
−0.534087 + 0.845429i \(0.679345\pi\)
\(432\) 0 0
\(433\) 11660.3 1.29413 0.647065 0.762435i \(-0.275996\pi\)
0.647065 + 0.762435i \(0.275996\pi\)
\(434\) 0 0
\(435\) −1768.45 −0.194921
\(436\) 0 0
\(437\) −20562.3 −2.25086
\(438\) 0 0
\(439\) 16716.4 1.81738 0.908689 0.417474i \(-0.137084\pi\)
0.908689 + 0.417474i \(0.137084\pi\)
\(440\) 0 0
\(441\) 41789.5 4.51242
\(442\) 0 0
\(443\) 367.759 0.0394419 0.0197209 0.999806i \(-0.493722\pi\)
0.0197209 + 0.999806i \(0.493722\pi\)
\(444\) 0 0
\(445\) 5218.79 0.555942
\(446\) 0 0
\(447\) 17617.0 1.86410
\(448\) 0 0
\(449\) −4823.82 −0.507015 −0.253508 0.967333i \(-0.581584\pi\)
−0.253508 + 0.967333i \(0.581584\pi\)
\(450\) 0 0
\(451\) 214.598 0.0224058
\(452\) 0 0
\(453\) 6616.59 0.686257
\(454\) 0 0
\(455\) −6242.08 −0.643150
\(456\) 0 0
\(457\) −2785.87 −0.285159 −0.142579 0.989783i \(-0.545540\pi\)
−0.142579 + 0.989783i \(0.545540\pi\)
\(458\) 0 0
\(459\) 20152.4 2.04931
\(460\) 0 0
\(461\) 9052.88 0.914608 0.457304 0.889310i \(-0.348815\pi\)
0.457304 + 0.889310i \(0.348815\pi\)
\(462\) 0 0
\(463\) 3825.06 0.383943 0.191972 0.981400i \(-0.438512\pi\)
0.191972 + 0.981400i \(0.438512\pi\)
\(464\) 0 0
\(465\) −892.924 −0.0890503
\(466\) 0 0
\(467\) −3943.34 −0.390741 −0.195371 0.980729i \(-0.562591\pi\)
−0.195371 + 0.980729i \(0.562591\pi\)
\(468\) 0 0
\(469\) −10406.6 −1.02459
\(470\) 0 0
\(471\) 13825.9 1.35258
\(472\) 0 0
\(473\) 28.0689 0.00272856
\(474\) 0 0
\(475\) 13602.2 1.31392
\(476\) 0 0
\(477\) −11420.0 −1.09620
\(478\) 0 0
\(479\) 7621.40 0.726995 0.363497 0.931595i \(-0.381583\pi\)
0.363497 + 0.931595i \(0.381583\pi\)
\(480\) 0 0
\(481\) −8260.09 −0.783009
\(482\) 0 0
\(483\) 39046.8 3.67845
\(484\) 0 0
\(485\) −5697.28 −0.533403
\(486\) 0 0
\(487\) 2939.21 0.273488 0.136744 0.990606i \(-0.456336\pi\)
0.136744 + 0.990606i \(0.456336\pi\)
\(488\) 0 0
\(489\) 30665.7 2.83589
\(490\) 0 0
\(491\) 21365.5 1.96377 0.981886 0.189473i \(-0.0606780\pi\)
0.981886 + 0.189473i \(0.0606780\pi\)
\(492\) 0 0
\(493\) −1482.97 −0.135476
\(494\) 0 0
\(495\) −3493.18 −0.317186
\(496\) 0 0
\(497\) 29714.7 2.68187
\(498\) 0 0
\(499\) −5450.33 −0.488958 −0.244479 0.969655i \(-0.578617\pi\)
−0.244479 + 0.969655i \(0.578617\pi\)
\(500\) 0 0
\(501\) −14969.5 −1.33491
\(502\) 0 0
\(503\) 19531.8 1.73137 0.865686 0.500587i \(-0.166882\pi\)
0.865686 + 0.500587i \(0.166882\pi\)
\(504\) 0 0
\(505\) −6248.89 −0.550637
\(506\) 0 0
\(507\) −11348.3 −0.994072
\(508\) 0 0
\(509\) −7726.61 −0.672841 −0.336420 0.941712i \(-0.609216\pi\)
−0.336420 + 0.941712i \(0.609216\pi\)
\(510\) 0 0
\(511\) −29339.7 −2.53994
\(512\) 0 0
\(513\) −62576.5 −5.38561
\(514\) 0 0
\(515\) 8125.12 0.695215
\(516\) 0 0
\(517\) −3977.01 −0.338315
\(518\) 0 0
\(519\) −27379.9 −2.31569
\(520\) 0 0
\(521\) −7915.18 −0.665586 −0.332793 0.943000i \(-0.607991\pi\)
−0.332793 + 0.943000i \(0.607991\pi\)
\(522\) 0 0
\(523\) −8078.88 −0.675459 −0.337729 0.941243i \(-0.609659\pi\)
−0.337729 + 0.941243i \(0.609659\pi\)
\(524\) 0 0
\(525\) −25829.9 −2.14726
\(526\) 0 0
\(527\) −748.779 −0.0618925
\(528\) 0 0
\(529\) 4601.80 0.378220
\(530\) 0 0
\(531\) 38476.2 3.14449
\(532\) 0 0
\(533\) 835.028 0.0678594
\(534\) 0 0
\(535\) −8004.51 −0.646851
\(536\) 0 0
\(537\) 34654.6 2.78484
\(538\) 0 0
\(539\) −5103.41 −0.407828
\(540\) 0 0
\(541\) 12220.5 0.971163 0.485582 0.874191i \(-0.338608\pi\)
0.485582 + 0.874191i \(0.338608\pi\)
\(542\) 0 0
\(543\) 2537.84 0.200570
\(544\) 0 0
\(545\) −9295.01 −0.730559
\(546\) 0 0
\(547\) −6172.25 −0.482462 −0.241231 0.970468i \(-0.577551\pi\)
−0.241231 + 0.970468i \(0.577551\pi\)
\(548\) 0 0
\(549\) 33807.8 2.62820
\(550\) 0 0
\(551\) 4604.87 0.356033
\(552\) 0 0
\(553\) −5561.43 −0.427660
\(554\) 0 0
\(555\) 15696.4 1.20050
\(556\) 0 0
\(557\) 3103.59 0.236092 0.118046 0.993008i \(-0.462337\pi\)
0.118046 + 0.993008i \(0.462337\pi\)
\(558\) 0 0
\(559\) 109.220 0.00826386
\(560\) 0 0
\(561\) −4100.43 −0.308592
\(562\) 0 0
\(563\) −21390.9 −1.60128 −0.800639 0.599147i \(-0.795506\pi\)
−0.800639 + 0.599147i \(0.795506\pi\)
\(564\) 0 0
\(565\) 3547.26 0.264132
\(566\) 0 0
\(567\) 62281.6 4.61302
\(568\) 0 0
\(569\) −23212.7 −1.71024 −0.855120 0.518431i \(-0.826516\pi\)
−0.855120 + 0.518431i \(0.826516\pi\)
\(570\) 0 0
\(571\) 11439.8 0.838424 0.419212 0.907888i \(-0.362306\pi\)
0.419212 + 0.907888i \(0.362306\pi\)
\(572\) 0 0
\(573\) −21803.7 −1.58964
\(574\) 0 0
\(575\) −11092.8 −0.804523
\(576\) 0 0
\(577\) −8568.11 −0.618189 −0.309095 0.951031i \(-0.600026\pi\)
−0.309095 + 0.951031i \(0.600026\pi\)
\(578\) 0 0
\(579\) 1175.07 0.0843425
\(580\) 0 0
\(581\) 8841.50 0.631338
\(582\) 0 0
\(583\) 1394.63 0.0990732
\(584\) 0 0
\(585\) −13592.4 −0.960645
\(586\) 0 0
\(587\) 2184.34 0.153590 0.0767950 0.997047i \(-0.475531\pi\)
0.0767950 + 0.997047i \(0.475531\pi\)
\(588\) 0 0
\(589\) 2325.09 0.162655
\(590\) 0 0
\(591\) 8406.22 0.585086
\(592\) 0 0
\(593\) −8518.37 −0.589895 −0.294947 0.955513i \(-0.595302\pi\)
−0.294947 + 0.955513i \(0.595302\pi\)
\(594\) 0 0
\(595\) 9946.81 0.685343
\(596\) 0 0
\(597\) −47499.5 −3.25632
\(598\) 0 0
\(599\) −17697.5 −1.20718 −0.603591 0.797294i \(-0.706264\pi\)
−0.603591 + 0.797294i \(0.706264\pi\)
\(600\) 0 0
\(601\) 25751.4 1.74779 0.873894 0.486116i \(-0.161587\pi\)
0.873894 + 0.486116i \(0.161587\pi\)
\(602\) 0 0
\(603\) −22660.8 −1.53038
\(604\) 0 0
\(605\) −7921.42 −0.532317
\(606\) 0 0
\(607\) 12644.9 0.845538 0.422769 0.906238i \(-0.361058\pi\)
0.422769 + 0.906238i \(0.361058\pi\)
\(608\) 0 0
\(609\) −8744.44 −0.581844
\(610\) 0 0
\(611\) −15475.1 −1.02464
\(612\) 0 0
\(613\) −18792.6 −1.23821 −0.619107 0.785307i \(-0.712505\pi\)
−0.619107 + 0.785307i \(0.712505\pi\)
\(614\) 0 0
\(615\) −1586.78 −0.104041
\(616\) 0 0
\(617\) −8630.30 −0.563116 −0.281558 0.959544i \(-0.590851\pi\)
−0.281558 + 0.959544i \(0.590851\pi\)
\(618\) 0 0
\(619\) −26553.4 −1.72419 −0.862094 0.506748i \(-0.830847\pi\)
−0.862094 + 0.506748i \(0.830847\pi\)
\(620\) 0 0
\(621\) 51032.0 3.29765
\(622\) 0 0
\(623\) 25805.3 1.65950
\(624\) 0 0
\(625\) 2420.78 0.154930
\(626\) 0 0
\(627\) 12732.5 0.810985
\(628\) 0 0
\(629\) 13162.5 0.834379
\(630\) 0 0
\(631\) 18310.6 1.15521 0.577603 0.816318i \(-0.303988\pi\)
0.577603 + 0.816318i \(0.303988\pi\)
\(632\) 0 0
\(633\) −51159.5 −3.21233
\(634\) 0 0
\(635\) −13375.9 −0.835916
\(636\) 0 0
\(637\) −19858.0 −1.23517
\(638\) 0 0
\(639\) 64705.2 4.00579
\(640\) 0 0
\(641\) −7753.98 −0.477790 −0.238895 0.971045i \(-0.576785\pi\)
−0.238895 + 0.971045i \(0.576785\pi\)
\(642\) 0 0
\(643\) 24720.7 1.51616 0.758078 0.652163i \(-0.226138\pi\)
0.758078 + 0.652163i \(0.226138\pi\)
\(644\) 0 0
\(645\) −207.547 −0.0126700
\(646\) 0 0
\(647\) 10318.0 0.626957 0.313478 0.949595i \(-0.398506\pi\)
0.313478 + 0.949595i \(0.398506\pi\)
\(648\) 0 0
\(649\) −4698.78 −0.284196
\(650\) 0 0
\(651\) −4415.23 −0.265817
\(652\) 0 0
\(653\) 15277.3 0.915542 0.457771 0.889070i \(-0.348648\pi\)
0.457771 + 0.889070i \(0.348648\pi\)
\(654\) 0 0
\(655\) −15798.6 −0.942444
\(656\) 0 0
\(657\) −63888.5 −3.79380
\(658\) 0 0
\(659\) −9600.60 −0.567506 −0.283753 0.958897i \(-0.591580\pi\)
−0.283753 + 0.958897i \(0.591580\pi\)
\(660\) 0 0
\(661\) −6812.43 −0.400866 −0.200433 0.979707i \(-0.564235\pi\)
−0.200433 + 0.979707i \(0.564235\pi\)
\(662\) 0 0
\(663\) −15955.3 −0.934617
\(664\) 0 0
\(665\) −30886.5 −1.80109
\(666\) 0 0
\(667\) −3755.34 −0.218002
\(668\) 0 0
\(669\) 26509.0 1.53198
\(670\) 0 0
\(671\) −4128.66 −0.237534
\(672\) 0 0
\(673\) 9459.01 0.541780 0.270890 0.962610i \(-0.412682\pi\)
0.270890 + 0.962610i \(0.412682\pi\)
\(674\) 0 0
\(675\) −33758.3 −1.92497
\(676\) 0 0
\(677\) 25231.8 1.43241 0.716203 0.697892i \(-0.245879\pi\)
0.716203 + 0.697892i \(0.245879\pi\)
\(678\) 0 0
\(679\) −28171.3 −1.59222
\(680\) 0 0
\(681\) −59111.1 −3.32620
\(682\) 0 0
\(683\) 16443.6 0.921226 0.460613 0.887601i \(-0.347630\pi\)
0.460613 + 0.887601i \(0.347630\pi\)
\(684\) 0 0
\(685\) −361.136 −0.0201435
\(686\) 0 0
\(687\) −5232.27 −0.290573
\(688\) 0 0
\(689\) 5426.68 0.300058
\(690\) 0 0
\(691\) −8693.86 −0.478625 −0.239313 0.970943i \(-0.576922\pi\)
−0.239313 + 0.970943i \(0.576922\pi\)
\(692\) 0 0
\(693\) −17272.7 −0.946805
\(694\) 0 0
\(695\) 7843.98 0.428114
\(696\) 0 0
\(697\) −1330.62 −0.0723113
\(698\) 0 0
\(699\) 27562.0 1.49140
\(700\) 0 0
\(701\) 2006.88 0.108130 0.0540648 0.998537i \(-0.482782\pi\)
0.0540648 + 0.998537i \(0.482782\pi\)
\(702\) 0 0
\(703\) −40871.9 −2.19276
\(704\) 0 0
\(705\) 29406.8 1.57096
\(706\) 0 0
\(707\) −30898.8 −1.64366
\(708\) 0 0
\(709\) 10597.0 0.561322 0.280661 0.959807i \(-0.409446\pi\)
0.280661 + 0.959807i \(0.409446\pi\)
\(710\) 0 0
\(711\) −12110.3 −0.638778
\(712\) 0 0
\(713\) −1896.14 −0.0995947
\(714\) 0 0
\(715\) 1659.93 0.0868221
\(716\) 0 0
\(717\) −3450.97 −0.179748
\(718\) 0 0
\(719\) −2248.05 −0.116604 −0.0583020 0.998299i \(-0.518569\pi\)
−0.0583020 + 0.998299i \(0.518569\pi\)
\(720\) 0 0
\(721\) 40176.2 2.07523
\(722\) 0 0
\(723\) 47517.5 2.44425
\(724\) 0 0
\(725\) 2484.20 0.127257
\(726\) 0 0
\(727\) −26960.8 −1.37541 −0.687704 0.725991i \(-0.741381\pi\)
−0.687704 + 0.725991i \(0.741381\pi\)
\(728\) 0 0
\(729\) 32167.5 1.63428
\(730\) 0 0
\(731\) −174.043 −0.00880601
\(732\) 0 0
\(733\) −10345.6 −0.521317 −0.260658 0.965431i \(-0.583940\pi\)
−0.260658 + 0.965431i \(0.583940\pi\)
\(734\) 0 0
\(735\) 37735.6 1.89374
\(736\) 0 0
\(737\) 2767.38 0.138314
\(738\) 0 0
\(739\) −11942.7 −0.594479 −0.297239 0.954803i \(-0.596066\pi\)
−0.297239 + 0.954803i \(0.596066\pi\)
\(740\) 0 0
\(741\) 49543.8 2.45619
\(742\) 0 0
\(743\) 1334.16 0.0658754 0.0329377 0.999457i \(-0.489514\pi\)
0.0329377 + 0.999457i \(0.489514\pi\)
\(744\) 0 0
\(745\) 11364.4 0.558872
\(746\) 0 0
\(747\) 19252.8 0.943002
\(748\) 0 0
\(749\) −39579.8 −1.93086
\(750\) 0 0
\(751\) −17347.0 −0.842876 −0.421438 0.906857i \(-0.638474\pi\)
−0.421438 + 0.906857i \(0.638474\pi\)
\(752\) 0 0
\(753\) 2438.82 0.118029
\(754\) 0 0
\(755\) 4268.25 0.205745
\(756\) 0 0
\(757\) −5359.59 −0.257328 −0.128664 0.991688i \(-0.541069\pi\)
−0.128664 + 0.991688i \(0.541069\pi\)
\(758\) 0 0
\(759\) −10383.5 −0.496573
\(760\) 0 0
\(761\) 10355.3 0.493272 0.246636 0.969108i \(-0.420675\pi\)
0.246636 + 0.969108i \(0.420675\pi\)
\(762\) 0 0
\(763\) −45960.9 −2.18073
\(764\) 0 0
\(765\) 21659.6 1.02367
\(766\) 0 0
\(767\) −18283.5 −0.860730
\(768\) 0 0
\(769\) −9339.42 −0.437956 −0.218978 0.975730i \(-0.570272\pi\)
−0.218978 + 0.975730i \(0.570272\pi\)
\(770\) 0 0
\(771\) 7016.93 0.327767
\(772\) 0 0
\(773\) −21743.9 −1.01174 −0.505869 0.862610i \(-0.668828\pi\)
−0.505869 + 0.862610i \(0.668828\pi\)
\(774\) 0 0
\(775\) 1254.32 0.0581374
\(776\) 0 0
\(777\) 77613.8 3.58350
\(778\) 0 0
\(779\) 4131.81 0.190035
\(780\) 0 0
\(781\) −7901.91 −0.362039
\(782\) 0 0
\(783\) −11428.5 −0.521611
\(784\) 0 0
\(785\) 8918.83 0.405512
\(786\) 0 0
\(787\) −26378.6 −1.19478 −0.597391 0.801950i \(-0.703796\pi\)
−0.597391 + 0.801950i \(0.703796\pi\)
\(788\) 0 0
\(789\) −36606.2 −1.65173
\(790\) 0 0
\(791\) 17540.1 0.788438
\(792\) 0 0
\(793\) −16065.1 −0.719407
\(794\) 0 0
\(795\) −10312.2 −0.460044
\(796\) 0 0
\(797\) 36134.8 1.60597 0.802986 0.595999i \(-0.203244\pi\)
0.802986 + 0.595999i \(0.203244\pi\)
\(798\) 0 0
\(799\) 24659.6 1.09186
\(800\) 0 0
\(801\) 56192.2 2.47872
\(802\) 0 0
\(803\) 7802.17 0.342880
\(804\) 0 0
\(805\) 25188.4 1.10282
\(806\) 0 0
\(807\) −5370.70 −0.234272
\(808\) 0 0
\(809\) 368.546 0.0160166 0.00800828 0.999968i \(-0.497451\pi\)
0.00800828 + 0.999968i \(0.497451\pi\)
\(810\) 0 0
\(811\) −24252.1 −1.05007 −0.525034 0.851081i \(-0.675948\pi\)
−0.525034 + 0.851081i \(0.675948\pi\)
\(812\) 0 0
\(813\) −52214.4 −2.25244
\(814\) 0 0
\(815\) 19781.9 0.850221
\(816\) 0 0
\(817\) 540.432 0.0231424
\(818\) 0 0
\(819\) −67210.2 −2.86754
\(820\) 0 0
\(821\) −8693.57 −0.369559 −0.184779 0.982780i \(-0.559157\pi\)
−0.184779 + 0.982780i \(0.559157\pi\)
\(822\) 0 0
\(823\) 4674.45 0.197984 0.0989922 0.995088i \(-0.468438\pi\)
0.0989922 + 0.995088i \(0.468438\pi\)
\(824\) 0 0
\(825\) 6868.84 0.289870
\(826\) 0 0
\(827\) −18877.2 −0.793740 −0.396870 0.917875i \(-0.629904\pi\)
−0.396870 + 0.917875i \(0.629904\pi\)
\(828\) 0 0
\(829\) −30674.6 −1.28513 −0.642565 0.766231i \(-0.722130\pi\)
−0.642565 + 0.766231i \(0.722130\pi\)
\(830\) 0 0
\(831\) 33184.2 1.38526
\(832\) 0 0
\(833\) 31643.9 1.31620
\(834\) 0 0
\(835\) −9656.58 −0.400215
\(836\) 0 0
\(837\) −5770.46 −0.238299
\(838\) 0 0
\(839\) −6238.05 −0.256688 −0.128344 0.991730i \(-0.540966\pi\)
−0.128344 + 0.991730i \(0.540966\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −65410.5 −2.67243
\(844\) 0 0
\(845\) −7320.57 −0.298030
\(846\) 0 0
\(847\) −39169.0 −1.58897
\(848\) 0 0
\(849\) −84765.6 −3.42656
\(850\) 0 0
\(851\) 33331.6 1.34265
\(852\) 0 0
\(853\) 16138.1 0.647780 0.323890 0.946095i \(-0.395009\pi\)
0.323890 + 0.946095i \(0.395009\pi\)
\(854\) 0 0
\(855\) −67256.9 −2.69022
\(856\) 0 0
\(857\) 47596.5 1.89716 0.948579 0.316541i \(-0.102522\pi\)
0.948579 + 0.316541i \(0.102522\pi\)
\(858\) 0 0
\(859\) −572.644 −0.0227455 −0.0113727 0.999935i \(-0.503620\pi\)
−0.0113727 + 0.999935i \(0.503620\pi\)
\(860\) 0 0
\(861\) −7846.13 −0.310564
\(862\) 0 0
\(863\) 23203.3 0.915237 0.457619 0.889149i \(-0.348703\pi\)
0.457619 + 0.889149i \(0.348703\pi\)
\(864\) 0 0
\(865\) −17662.3 −0.694261
\(866\) 0 0
\(867\) −22343.1 −0.875215
\(868\) 0 0
\(869\) 1478.93 0.0577321
\(870\) 0 0
\(871\) 10768.2 0.418906
\(872\) 0 0
\(873\) −61344.3 −2.37822
\(874\) 0 0
\(875\) −40976.6 −1.58316
\(876\) 0 0
\(877\) 41334.1 1.59151 0.795755 0.605619i \(-0.207074\pi\)
0.795755 + 0.605619i \(0.207074\pi\)
\(878\) 0 0
\(879\) −12435.5 −0.477176
\(880\) 0 0
\(881\) −30870.9 −1.18055 −0.590276 0.807202i \(-0.700981\pi\)
−0.590276 + 0.807202i \(0.700981\pi\)
\(882\) 0 0
\(883\) 11615.6 0.442691 0.221346 0.975195i \(-0.428955\pi\)
0.221346 + 0.975195i \(0.428955\pi\)
\(884\) 0 0
\(885\) 34743.7 1.31966
\(886\) 0 0
\(887\) −9853.82 −0.373009 −0.186505 0.982454i \(-0.559716\pi\)
−0.186505 + 0.982454i \(0.559716\pi\)
\(888\) 0 0
\(889\) −66139.7 −2.49522
\(890\) 0 0
\(891\) −16562.3 −0.622735
\(892\) 0 0
\(893\) −76572.4 −2.86942
\(894\) 0 0
\(895\) 22355.1 0.834914
\(896\) 0 0
\(897\) −40403.7 −1.50395
\(898\) 0 0
\(899\) 424.637 0.0157535
\(900\) 0 0
\(901\) −8647.47 −0.319744
\(902\) 0 0
\(903\) −1026.26 −0.0378202
\(904\) 0 0
\(905\) 1637.12 0.0601322
\(906\) 0 0
\(907\) 10244.0 0.375024 0.187512 0.982262i \(-0.439958\pi\)
0.187512 + 0.982262i \(0.439958\pi\)
\(908\) 0 0
\(909\) −67283.6 −2.45507
\(910\) 0 0
\(911\) 20813.9 0.756966 0.378483 0.925608i \(-0.376446\pi\)
0.378483 + 0.925608i \(0.376446\pi\)
\(912\) 0 0
\(913\) −2351.18 −0.0852276
\(914\) 0 0
\(915\) 30528.1 1.10298
\(916\) 0 0
\(917\) −78119.0 −2.81321
\(918\) 0 0
\(919\) −9794.71 −0.351575 −0.175788 0.984428i \(-0.556247\pi\)
−0.175788 + 0.984428i \(0.556247\pi\)
\(920\) 0 0
\(921\) 82483.3 2.95105
\(922\) 0 0
\(923\) −30747.3 −1.09649
\(924\) 0 0
\(925\) −22049.3 −0.783757
\(926\) 0 0
\(927\) 87485.5 3.09968
\(928\) 0 0
\(929\) −29395.4 −1.03814 −0.519070 0.854732i \(-0.673722\pi\)
−0.519070 + 0.854732i \(0.673722\pi\)
\(930\) 0 0
\(931\) −98259.7 −3.45900
\(932\) 0 0
\(933\) −69329.2 −2.43273
\(934\) 0 0
\(935\) −2645.11 −0.0925181
\(936\) 0 0
\(937\) 50174.1 1.74932 0.874661 0.484735i \(-0.161084\pi\)
0.874661 + 0.484735i \(0.161084\pi\)
\(938\) 0 0
\(939\) 33357.5 1.15930
\(940\) 0 0
\(941\) 37984.8 1.31591 0.657955 0.753058i \(-0.271422\pi\)
0.657955 + 0.753058i \(0.271422\pi\)
\(942\) 0 0
\(943\) −3369.55 −0.116360
\(944\) 0 0
\(945\) 76655.0 2.63872
\(946\) 0 0
\(947\) −15217.3 −0.522171 −0.261085 0.965316i \(-0.584080\pi\)
−0.261085 + 0.965316i \(0.584080\pi\)
\(948\) 0 0
\(949\) 30359.2 1.03846
\(950\) 0 0
\(951\) −71483.9 −2.43746
\(952\) 0 0
\(953\) 17012.1 0.578253 0.289126 0.957291i \(-0.406635\pi\)
0.289126 + 0.957291i \(0.406635\pi\)
\(954\) 0 0
\(955\) −14065.2 −0.476584
\(956\) 0 0
\(957\) 2325.37 0.0785461
\(958\) 0 0
\(959\) −1785.71 −0.0601287
\(960\) 0 0
\(961\) −29576.6 −0.992803
\(962\) 0 0
\(963\) −86186.9 −2.88404
\(964\) 0 0
\(965\) 758.018 0.0252865
\(966\) 0 0
\(967\) −37192.8 −1.23686 −0.618428 0.785841i \(-0.712230\pi\)
−0.618428 + 0.785841i \(0.712230\pi\)
\(968\) 0 0
\(969\) −78948.5 −2.61733
\(970\) 0 0
\(971\) −55741.1 −1.84224 −0.921121 0.389276i \(-0.872725\pi\)
−0.921121 + 0.389276i \(0.872725\pi\)
\(972\) 0 0
\(973\) 38786.0 1.27793
\(974\) 0 0
\(975\) 26727.5 0.877914
\(976\) 0 0
\(977\) 7356.33 0.240890 0.120445 0.992720i \(-0.461568\pi\)
0.120445 + 0.992720i \(0.461568\pi\)
\(978\) 0 0
\(979\) −6862.29 −0.224024
\(980\) 0 0
\(981\) −100082. −3.25726
\(982\) 0 0
\(983\) 1951.93 0.0633337 0.0316668 0.999498i \(-0.489918\pi\)
0.0316668 + 0.999498i \(0.489918\pi\)
\(984\) 0 0
\(985\) 5422.70 0.175413
\(986\) 0 0
\(987\) 145407. 4.68933
\(988\) 0 0
\(989\) −440.729 −0.0141703
\(990\) 0 0
\(991\) −31606.2 −1.01312 −0.506561 0.862204i \(-0.669083\pi\)
−0.506561 + 0.862204i \(0.669083\pi\)
\(992\) 0 0
\(993\) −58101.8 −1.85680
\(994\) 0 0
\(995\) −30641.1 −0.976269
\(996\) 0 0
\(997\) 6443.52 0.204682 0.102341 0.994749i \(-0.467367\pi\)
0.102341 + 0.994749i \(0.467367\pi\)
\(998\) 0 0
\(999\) 101437. 3.21254
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.1.12 12
4.3 odd 2 1856.4.a.bl.1.1 12
8.3 odd 2 928.4.a.h.1.12 12
8.5 even 2 928.4.a.j.1.1 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
928.4.a.h.1.12 12 8.3 odd 2
928.4.a.j.1.1 yes 12 8.5 even 2
1856.4.a.bj.1.12 12 1.1 even 1 trivial
1856.4.a.bl.1.1 12 4.3 odd 2