Properties

Label 1521.4.a.bm.1.17
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

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

Embedding invariants

Embedding label 1.17
Root \(4.80856\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.80856 q^{2} +15.1223 q^{4} +1.59043 q^{5} -34.5272 q^{7} +34.2479 q^{8} +O(q^{10})\) \(q+4.80856 q^{2} +15.1223 q^{4} +1.59043 q^{5} -34.5272 q^{7} +34.2479 q^{8} +7.64769 q^{10} +42.6836 q^{11} -166.026 q^{14} +43.7049 q^{16} +45.4494 q^{17} -41.0910 q^{19} +24.0509 q^{20} +205.247 q^{22} -171.733 q^{23} -122.471 q^{25} -522.129 q^{28} -178.040 q^{29} -18.5736 q^{31} -63.8255 q^{32} +218.546 q^{34} -54.9131 q^{35} +331.269 q^{37} -197.588 q^{38} +54.4689 q^{40} -411.934 q^{41} +11.3363 q^{43} +645.473 q^{44} -825.787 q^{46} -494.592 q^{47} +849.125 q^{49} -588.907 q^{50} +217.808 q^{53} +67.8853 q^{55} -1182.48 q^{56} -856.115 q^{58} +775.414 q^{59} +340.831 q^{61} -89.3123 q^{62} -656.547 q^{64} -676.191 q^{67} +687.298 q^{68} -264.053 q^{70} -449.604 q^{71} -645.014 q^{73} +1592.93 q^{74} -621.389 q^{76} -1473.74 q^{77} -778.289 q^{79} +69.5095 q^{80} -1980.81 q^{82} -1056.50 q^{83} +72.2842 q^{85} +54.5114 q^{86} +1461.82 q^{88} +160.228 q^{89} -2596.99 q^{92} -2378.28 q^{94} -65.3523 q^{95} -7.23476 q^{97} +4083.07 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 56 q^{4} - 94 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 56 q^{4} - 94 q^{7} + 156 q^{10} + 88 q^{16} - 448 q^{19} - 256 q^{22} - 16 q^{25} - 1300 q^{28} - 818 q^{31} - 900 q^{34} - 524 q^{37} + 800 q^{40} + 752 q^{43} - 1650 q^{46} + 2948 q^{49} - 464 q^{55} - 1626 q^{58} - 1784 q^{61} - 4274 q^{64} - 2872 q^{67} - 5772 q^{70} - 3078 q^{73} - 5726 q^{76} + 3326 q^{79} - 1038 q^{82} - 2340 q^{85} + 780 q^{88} + 1220 q^{94} - 4630 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.80856 1.70008 0.850042 0.526715i \(-0.176577\pi\)
0.850042 + 0.526715i \(0.176577\pi\)
\(3\) 0 0
\(4\) 15.1223 1.89028
\(5\) 1.59043 0.142252 0.0711262 0.997467i \(-0.477341\pi\)
0.0711262 + 0.997467i \(0.477341\pi\)
\(6\) 0 0
\(7\) −34.5272 −1.86429 −0.932146 0.362083i \(-0.882066\pi\)
−0.932146 + 0.362083i \(0.882066\pi\)
\(8\) 34.2479 1.51356
\(9\) 0 0
\(10\) 7.64769 0.241841
\(11\) 42.6836 1.16996 0.584981 0.811047i \(-0.301102\pi\)
0.584981 + 0.811047i \(0.301102\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −166.026 −3.16945
\(15\) 0 0
\(16\) 43.7049 0.682888
\(17\) 45.4494 0.648418 0.324209 0.945985i \(-0.394902\pi\)
0.324209 + 0.945985i \(0.394902\pi\)
\(18\) 0 0
\(19\) −41.0910 −0.496153 −0.248077 0.968740i \(-0.579798\pi\)
−0.248077 + 0.968740i \(0.579798\pi\)
\(20\) 24.0509 0.268897
\(21\) 0 0
\(22\) 205.247 1.98903
\(23\) −171.733 −1.55690 −0.778450 0.627706i \(-0.783994\pi\)
−0.778450 + 0.627706i \(0.783994\pi\)
\(24\) 0 0
\(25\) −122.471 −0.979764
\(26\) 0 0
\(27\) 0 0
\(28\) −522.129 −3.52404
\(29\) −178.040 −1.14004 −0.570019 0.821631i \(-0.693064\pi\)
−0.570019 + 0.821631i \(0.693064\pi\)
\(30\) 0 0
\(31\) −18.5736 −0.107610 −0.0538051 0.998551i \(-0.517135\pi\)
−0.0538051 + 0.998551i \(0.517135\pi\)
\(32\) −63.8255 −0.352589
\(33\) 0 0
\(34\) 218.546 1.10236
\(35\) −54.9131 −0.265200
\(36\) 0 0
\(37\) 331.269 1.47190 0.735949 0.677037i \(-0.236736\pi\)
0.735949 + 0.677037i \(0.236736\pi\)
\(38\) −197.588 −0.843502
\(39\) 0 0
\(40\) 54.4689 0.215307
\(41\) −411.934 −1.56911 −0.784553 0.620062i \(-0.787107\pi\)
−0.784553 + 0.620062i \(0.787107\pi\)
\(42\) 0 0
\(43\) 11.3363 0.0402040 0.0201020 0.999798i \(-0.493601\pi\)
0.0201020 + 0.999798i \(0.493601\pi\)
\(44\) 645.473 2.21156
\(45\) 0 0
\(46\) −825.787 −2.64686
\(47\) −494.592 −1.53497 −0.767486 0.641065i \(-0.778493\pi\)
−0.767486 + 0.641065i \(0.778493\pi\)
\(48\) 0 0
\(49\) 849.125 2.47558
\(50\) −588.907 −1.66568
\(51\) 0 0
\(52\) 0 0
\(53\) 217.808 0.564495 0.282248 0.959342i \(-0.408920\pi\)
0.282248 + 0.959342i \(0.408920\pi\)
\(54\) 0 0
\(55\) 67.8853 0.166430
\(56\) −1182.48 −2.82171
\(57\) 0 0
\(58\) −856.115 −1.93816
\(59\) 775.414 1.71102 0.855511 0.517785i \(-0.173243\pi\)
0.855511 + 0.517785i \(0.173243\pi\)
\(60\) 0 0
\(61\) 340.831 0.715392 0.357696 0.933838i \(-0.383562\pi\)
0.357696 + 0.933838i \(0.383562\pi\)
\(62\) −89.3123 −0.182946
\(63\) 0 0
\(64\) −656.547 −1.28232
\(65\) 0 0
\(66\) 0 0
\(67\) −676.191 −1.23298 −0.616492 0.787361i \(-0.711447\pi\)
−0.616492 + 0.787361i \(0.711447\pi\)
\(68\) 687.298 1.22569
\(69\) 0 0
\(70\) −264.053 −0.450862
\(71\) −449.604 −0.751524 −0.375762 0.926716i \(-0.622619\pi\)
−0.375762 + 0.926716i \(0.622619\pi\)
\(72\) 0 0
\(73\) −645.014 −1.03415 −0.517076 0.855939i \(-0.672980\pi\)
−0.517076 + 0.855939i \(0.672980\pi\)
\(74\) 1592.93 2.50235
\(75\) 0 0
\(76\) −621.389 −0.937870
\(77\) −1473.74 −2.18115
\(78\) 0 0
\(79\) −778.289 −1.10841 −0.554205 0.832380i \(-0.686978\pi\)
−0.554205 + 0.832380i \(0.686978\pi\)
\(80\) 69.5095 0.0971425
\(81\) 0 0
\(82\) −1980.81 −2.66761
\(83\) −1056.50 −1.39718 −0.698591 0.715521i \(-0.746189\pi\)
−0.698591 + 0.715521i \(0.746189\pi\)
\(84\) 0 0
\(85\) 72.2842 0.0922391
\(86\) 54.5114 0.0683501
\(87\) 0 0
\(88\) 1461.82 1.77080
\(89\) 160.228 0.190833 0.0954166 0.995437i \(-0.469582\pi\)
0.0954166 + 0.995437i \(0.469582\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2596.99 −2.94298
\(93\) 0 0
\(94\) −2378.28 −2.60958
\(95\) −65.3523 −0.0705790
\(96\) 0 0
\(97\) −7.23476 −0.00757297 −0.00378649 0.999993i \(-0.501205\pi\)
−0.00378649 + 0.999993i \(0.501205\pi\)
\(98\) 4083.07 4.20870
\(99\) 0 0
\(100\) −1852.03 −1.85203
\(101\) −1125.70 −1.10902 −0.554511 0.832176i \(-0.687095\pi\)
−0.554511 + 0.832176i \(0.687095\pi\)
\(102\) 0 0
\(103\) 1204.03 1.15181 0.575907 0.817515i \(-0.304649\pi\)
0.575907 + 0.817515i \(0.304649\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1047.34 0.959689
\(107\) 1491.74 1.34778 0.673889 0.738833i \(-0.264623\pi\)
0.673889 + 0.738833i \(0.264623\pi\)
\(108\) 0 0
\(109\) −375.430 −0.329905 −0.164953 0.986301i \(-0.552747\pi\)
−0.164953 + 0.986301i \(0.552747\pi\)
\(110\) 326.431 0.282945
\(111\) 0 0
\(112\) −1509.00 −1.27310
\(113\) 810.999 0.675154 0.337577 0.941298i \(-0.390393\pi\)
0.337577 + 0.941298i \(0.390393\pi\)
\(114\) 0 0
\(115\) −273.129 −0.221473
\(116\) −2692.36 −2.15500
\(117\) 0 0
\(118\) 3728.62 2.90888
\(119\) −1569.24 −1.20884
\(120\) 0 0
\(121\) 490.889 0.368812
\(122\) 1638.91 1.21623
\(123\) 0 0
\(124\) −280.875 −0.203414
\(125\) −393.585 −0.281626
\(126\) 0 0
\(127\) −584.796 −0.408600 −0.204300 0.978908i \(-0.565492\pi\)
−0.204300 + 0.978908i \(0.565492\pi\)
\(128\) −2646.45 −1.82746
\(129\) 0 0
\(130\) 0 0
\(131\) −1521.40 −1.01470 −0.507350 0.861740i \(-0.669375\pi\)
−0.507350 + 0.861740i \(0.669375\pi\)
\(132\) 0 0
\(133\) 1418.75 0.924975
\(134\) −3251.51 −2.09618
\(135\) 0 0
\(136\) 1556.55 0.981417
\(137\) −583.044 −0.363597 −0.181799 0.983336i \(-0.558192\pi\)
−0.181799 + 0.983336i \(0.558192\pi\)
\(138\) 0 0
\(139\) −2008.49 −1.22560 −0.612798 0.790240i \(-0.709956\pi\)
−0.612798 + 0.790240i \(0.709956\pi\)
\(140\) −830.410 −0.501303
\(141\) 0 0
\(142\) −2161.95 −1.27765
\(143\) 0 0
\(144\) 0 0
\(145\) −283.160 −0.162173
\(146\) −3101.59 −1.75815
\(147\) 0 0
\(148\) 5009.54 2.78231
\(149\) 1357.74 0.746515 0.373257 0.927728i \(-0.378241\pi\)
0.373257 + 0.927728i \(0.378241\pi\)
\(150\) 0 0
\(151\) −2305.46 −1.24249 −0.621243 0.783618i \(-0.713372\pi\)
−0.621243 + 0.783618i \(0.713372\pi\)
\(152\) −1407.28 −0.750956
\(153\) 0 0
\(154\) −7086.59 −3.70814
\(155\) −29.5400 −0.0153078
\(156\) 0 0
\(157\) 1254.03 0.637470 0.318735 0.947844i \(-0.396742\pi\)
0.318735 + 0.947844i \(0.396742\pi\)
\(158\) −3742.45 −1.88439
\(159\) 0 0
\(160\) −101.510 −0.0501567
\(161\) 5929.44 2.90252
\(162\) 0 0
\(163\) 2792.83 1.34203 0.671017 0.741442i \(-0.265858\pi\)
0.671017 + 0.741442i \(0.265858\pi\)
\(164\) −6229.38 −2.96605
\(165\) 0 0
\(166\) −5080.25 −2.37533
\(167\) −1218.71 −0.564712 −0.282356 0.959310i \(-0.591116\pi\)
−0.282356 + 0.959310i \(0.591116\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 347.583 0.156814
\(171\) 0 0
\(172\) 171.431 0.0759969
\(173\) −1526.65 −0.670922 −0.335461 0.942054i \(-0.608892\pi\)
−0.335461 + 0.942054i \(0.608892\pi\)
\(174\) 0 0
\(175\) 4228.56 1.82657
\(176\) 1865.48 0.798954
\(177\) 0 0
\(178\) 770.468 0.324433
\(179\) −1067.26 −0.445647 −0.222823 0.974859i \(-0.571527\pi\)
−0.222823 + 0.974859i \(0.571527\pi\)
\(180\) 0 0
\(181\) 3579.98 1.47015 0.735076 0.677984i \(-0.237146\pi\)
0.735076 + 0.677984i \(0.237146\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −5881.47 −2.35646
\(185\) 526.860 0.209381
\(186\) 0 0
\(187\) 1939.94 0.758625
\(188\) −7479.36 −2.90153
\(189\) 0 0
\(190\) −314.251 −0.119990
\(191\) 3087.19 1.16953 0.584767 0.811201i \(-0.301186\pi\)
0.584767 + 0.811201i \(0.301186\pi\)
\(192\) 0 0
\(193\) 897.457 0.334717 0.167358 0.985896i \(-0.446476\pi\)
0.167358 + 0.985896i \(0.446476\pi\)
\(194\) −34.7888 −0.0128747
\(195\) 0 0
\(196\) 12840.7 4.67956
\(197\) −23.3043 −0.00842825 −0.00421412 0.999991i \(-0.501341\pi\)
−0.00421412 + 0.999991i \(0.501341\pi\)
\(198\) 0 0
\(199\) 3910.89 1.39314 0.696571 0.717487i \(-0.254708\pi\)
0.696571 + 0.717487i \(0.254708\pi\)
\(200\) −4194.35 −1.48293
\(201\) 0 0
\(202\) −5412.99 −1.88543
\(203\) 6147.20 2.12537
\(204\) 0 0
\(205\) −655.153 −0.223209
\(206\) 5789.67 1.95818
\(207\) 0 0
\(208\) 0 0
\(209\) −1753.91 −0.580481
\(210\) 0 0
\(211\) −191.920 −0.0626177 −0.0313088 0.999510i \(-0.509968\pi\)
−0.0313088 + 0.999510i \(0.509968\pi\)
\(212\) 3293.75 1.06706
\(213\) 0 0
\(214\) 7173.14 2.29133
\(215\) 18.0296 0.00571912
\(216\) 0 0
\(217\) 641.294 0.200617
\(218\) −1805.28 −0.560866
\(219\) 0 0
\(220\) 1026.58 0.314600
\(221\) 0 0
\(222\) 0 0
\(223\) 1444.23 0.433688 0.216844 0.976206i \(-0.430424\pi\)
0.216844 + 0.976206i \(0.430424\pi\)
\(224\) 2203.71 0.657329
\(225\) 0 0
\(226\) 3899.74 1.14782
\(227\) 4307.34 1.25942 0.629710 0.776830i \(-0.283174\pi\)
0.629710 + 0.776830i \(0.283174\pi\)
\(228\) 0 0
\(229\) −1864.12 −0.537924 −0.268962 0.963151i \(-0.586681\pi\)
−0.268962 + 0.963151i \(0.586681\pi\)
\(230\) −1313.36 −0.376522
\(231\) 0 0
\(232\) −6097.48 −1.72551
\(233\) 3541.27 0.995691 0.497846 0.867266i \(-0.334125\pi\)
0.497846 + 0.867266i \(0.334125\pi\)
\(234\) 0 0
\(235\) −786.615 −0.218354
\(236\) 11726.0 3.23432
\(237\) 0 0
\(238\) −7545.79 −2.05513
\(239\) −2043.01 −0.552935 −0.276467 0.961023i \(-0.589164\pi\)
−0.276467 + 0.961023i \(0.589164\pi\)
\(240\) 0 0
\(241\) 228.833 0.0611636 0.0305818 0.999532i \(-0.490264\pi\)
0.0305818 + 0.999532i \(0.490264\pi\)
\(242\) 2360.47 0.627011
\(243\) 0 0
\(244\) 5154.13 1.35229
\(245\) 1350.48 0.352158
\(246\) 0 0
\(247\) 0 0
\(248\) −636.106 −0.162874
\(249\) 0 0
\(250\) −1892.58 −0.478788
\(251\) 3409.06 0.857282 0.428641 0.903475i \(-0.358993\pi\)
0.428641 + 0.903475i \(0.358993\pi\)
\(252\) 0 0
\(253\) −7330.16 −1.82151
\(254\) −2812.03 −0.694654
\(255\) 0 0
\(256\) −7473.22 −1.82452
\(257\) 3963.28 0.961955 0.480978 0.876733i \(-0.340282\pi\)
0.480978 + 0.876733i \(0.340282\pi\)
\(258\) 0 0
\(259\) −11437.8 −2.74405
\(260\) 0 0
\(261\) 0 0
\(262\) −7315.76 −1.72507
\(263\) 4890.12 1.14653 0.573266 0.819370i \(-0.305676\pi\)
0.573266 + 0.819370i \(0.305676\pi\)
\(264\) 0 0
\(265\) 346.409 0.0803008
\(266\) 6822.17 1.57253
\(267\) 0 0
\(268\) −10225.5 −2.33069
\(269\) 3946.47 0.894500 0.447250 0.894409i \(-0.352403\pi\)
0.447250 + 0.894409i \(0.352403\pi\)
\(270\) 0 0
\(271\) −2692.66 −0.603570 −0.301785 0.953376i \(-0.597582\pi\)
−0.301785 + 0.953376i \(0.597582\pi\)
\(272\) 1986.36 0.442797
\(273\) 0 0
\(274\) −2803.60 −0.618146
\(275\) −5227.48 −1.14629
\(276\) 0 0
\(277\) −2638.62 −0.572344 −0.286172 0.958178i \(-0.592383\pi\)
−0.286172 + 0.958178i \(0.592383\pi\)
\(278\) −9657.94 −2.08361
\(279\) 0 0
\(280\) −1880.66 −0.401395
\(281\) 3345.23 0.710176 0.355088 0.934833i \(-0.384451\pi\)
0.355088 + 0.934833i \(0.384451\pi\)
\(282\) 0 0
\(283\) 2799.71 0.588076 0.294038 0.955794i \(-0.405001\pi\)
0.294038 + 0.955794i \(0.405001\pi\)
\(284\) −6799.04 −1.42059
\(285\) 0 0
\(286\) 0 0
\(287\) 14222.9 2.92527
\(288\) 0 0
\(289\) −2847.35 −0.579554
\(290\) −1361.59 −0.275708
\(291\) 0 0
\(292\) −9754.07 −1.95484
\(293\) −8050.62 −1.60520 −0.802598 0.596521i \(-0.796549\pi\)
−0.802598 + 0.596521i \(0.796549\pi\)
\(294\) 0 0
\(295\) 1233.24 0.243397
\(296\) 11345.2 2.22780
\(297\) 0 0
\(298\) 6528.79 1.26914
\(299\) 0 0
\(300\) 0 0
\(301\) −391.411 −0.0749520
\(302\) −11085.9 −2.11233
\(303\) 0 0
\(304\) −1795.87 −0.338817
\(305\) 542.068 0.101766
\(306\) 0 0
\(307\) 5378.68 0.999926 0.499963 0.866047i \(-0.333347\pi\)
0.499963 + 0.866047i \(0.333347\pi\)
\(308\) −22286.3 −4.12299
\(309\) 0 0
\(310\) −142.045 −0.0260246
\(311\) −3727.50 −0.679636 −0.339818 0.940491i \(-0.610366\pi\)
−0.339818 + 0.940491i \(0.610366\pi\)
\(312\) 0 0
\(313\) 5336.69 0.963730 0.481865 0.876245i \(-0.339960\pi\)
0.481865 + 0.876245i \(0.339960\pi\)
\(314\) 6030.10 1.08375
\(315\) 0 0
\(316\) −11769.5 −2.09521
\(317\) 338.871 0.0600408 0.0300204 0.999549i \(-0.490443\pi\)
0.0300204 + 0.999549i \(0.490443\pi\)
\(318\) 0 0
\(319\) −7599.37 −1.33380
\(320\) −1044.19 −0.182413
\(321\) 0 0
\(322\) 28512.1 4.93452
\(323\) −1867.56 −0.321715
\(324\) 0 0
\(325\) 0 0
\(326\) 13429.5 2.28157
\(327\) 0 0
\(328\) −14107.9 −2.37493
\(329\) 17076.9 2.86164
\(330\) 0 0
\(331\) −2076.44 −0.344807 −0.172404 0.985026i \(-0.555153\pi\)
−0.172404 + 0.985026i \(0.555153\pi\)
\(332\) −15976.7 −2.64107
\(333\) 0 0
\(334\) −5860.26 −0.960057
\(335\) −1075.44 −0.175395
\(336\) 0 0
\(337\) 2584.64 0.417787 0.208893 0.977938i \(-0.433014\pi\)
0.208893 + 0.977938i \(0.433014\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 1093.10 0.174358
\(341\) −792.788 −0.125900
\(342\) 0 0
\(343\) −17475.1 −2.75092
\(344\) 388.244 0.0608510
\(345\) 0 0
\(346\) −7341.02 −1.14062
\(347\) 4191.32 0.648420 0.324210 0.945985i \(-0.394902\pi\)
0.324210 + 0.945985i \(0.394902\pi\)
\(348\) 0 0
\(349\) 7431.29 1.13979 0.569897 0.821716i \(-0.306983\pi\)
0.569897 + 0.821716i \(0.306983\pi\)
\(350\) 20333.3 3.10532
\(351\) 0 0
\(352\) −2724.30 −0.412516
\(353\) −12249.5 −1.84695 −0.923474 0.383661i \(-0.874663\pi\)
−0.923474 + 0.383661i \(0.874663\pi\)
\(354\) 0 0
\(355\) −715.065 −0.106906
\(356\) 2423.01 0.360729
\(357\) 0 0
\(358\) −5131.98 −0.757636
\(359\) −5485.46 −0.806439 −0.403219 0.915103i \(-0.632109\pi\)
−0.403219 + 0.915103i \(0.632109\pi\)
\(360\) 0 0
\(361\) −5170.53 −0.753832
\(362\) 17214.5 2.49938
\(363\) 0 0
\(364\) 0 0
\(365\) −1025.85 −0.147111
\(366\) 0 0
\(367\) −8752.18 −1.24485 −0.622425 0.782680i \(-0.713852\pi\)
−0.622425 + 0.782680i \(0.713852\pi\)
\(368\) −7505.54 −1.06319
\(369\) 0 0
\(370\) 2533.44 0.355966
\(371\) −7520.30 −1.05238
\(372\) 0 0
\(373\) 5987.30 0.831128 0.415564 0.909564i \(-0.363584\pi\)
0.415564 + 0.909564i \(0.363584\pi\)
\(374\) 9328.34 1.28973
\(375\) 0 0
\(376\) −16938.7 −2.32327
\(377\) 0 0
\(378\) 0 0
\(379\) −13603.0 −1.84364 −0.921819 0.387620i \(-0.873297\pi\)
−0.921819 + 0.387620i \(0.873297\pi\)
\(380\) −988.276 −0.133414
\(381\) 0 0
\(382\) 14844.9 1.98831
\(383\) −6648.90 −0.887057 −0.443528 0.896260i \(-0.646273\pi\)
−0.443528 + 0.896260i \(0.646273\pi\)
\(384\) 0 0
\(385\) −2343.89 −0.310274
\(386\) 4315.48 0.569047
\(387\) 0 0
\(388\) −109.406 −0.0143151
\(389\) 4923.40 0.641713 0.320856 0.947128i \(-0.396029\pi\)
0.320856 + 0.947128i \(0.396029\pi\)
\(390\) 0 0
\(391\) −7805.15 −1.00952
\(392\) 29080.7 3.74694
\(393\) 0 0
\(394\) −112.060 −0.0143287
\(395\) −1237.82 −0.157674
\(396\) 0 0
\(397\) 576.103 0.0728307 0.0364153 0.999337i \(-0.488406\pi\)
0.0364153 + 0.999337i \(0.488406\pi\)
\(398\) 18805.7 2.36846
\(399\) 0 0
\(400\) −5352.56 −0.669070
\(401\) 9053.52 1.12746 0.563730 0.825959i \(-0.309366\pi\)
0.563730 + 0.825959i \(0.309366\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −17023.1 −2.09637
\(405\) 0 0
\(406\) 29559.2 3.61330
\(407\) 14139.7 1.72207
\(408\) 0 0
\(409\) −15789.9 −1.90895 −0.954474 0.298293i \(-0.903583\pi\)
−0.954474 + 0.298293i \(0.903583\pi\)
\(410\) −3150.34 −0.379474
\(411\) 0 0
\(412\) 18207.7 2.17726
\(413\) −26772.8 −3.18984
\(414\) 0 0
\(415\) −1680.29 −0.198753
\(416\) 0 0
\(417\) 0 0
\(418\) −8433.78 −0.986866
\(419\) 4091.45 0.477041 0.238521 0.971137i \(-0.423338\pi\)
0.238521 + 0.971137i \(0.423338\pi\)
\(420\) 0 0
\(421\) 5540.81 0.641431 0.320715 0.947176i \(-0.396077\pi\)
0.320715 + 0.947176i \(0.396077\pi\)
\(422\) −922.860 −0.106455
\(423\) 0 0
\(424\) 7459.46 0.854395
\(425\) −5566.22 −0.635297
\(426\) 0 0
\(427\) −11767.9 −1.33370
\(428\) 22558.5 2.54768
\(429\) 0 0
\(430\) 86.6965 0.00972297
\(431\) 40.7848 0.00455808 0.00227904 0.999997i \(-0.499275\pi\)
0.00227904 + 0.999997i \(0.499275\pi\)
\(432\) 0 0
\(433\) 766.330 0.0850519 0.0425259 0.999095i \(-0.486459\pi\)
0.0425259 + 0.999095i \(0.486459\pi\)
\(434\) 3083.70 0.341066
\(435\) 0 0
\(436\) −5677.35 −0.623614
\(437\) 7056.65 0.772461
\(438\) 0 0
\(439\) −6054.63 −0.658250 −0.329125 0.944286i \(-0.606754\pi\)
−0.329125 + 0.944286i \(0.606754\pi\)
\(440\) 2324.93 0.251901
\(441\) 0 0
\(442\) 0 0
\(443\) 7422.44 0.796051 0.398025 0.917374i \(-0.369696\pi\)
0.398025 + 0.917374i \(0.369696\pi\)
\(444\) 0 0
\(445\) 254.832 0.0271465
\(446\) 6944.65 0.737306
\(447\) 0 0
\(448\) 22668.7 2.39062
\(449\) −1050.84 −0.110451 −0.0552253 0.998474i \(-0.517588\pi\)
−0.0552253 + 0.998474i \(0.517588\pi\)
\(450\) 0 0
\(451\) −17582.8 −1.83579
\(452\) 12264.1 1.27623
\(453\) 0 0
\(454\) 20712.1 2.14112
\(455\) 0 0
\(456\) 0 0
\(457\) 14462.4 1.48036 0.740179 0.672409i \(-0.234741\pi\)
0.740179 + 0.672409i \(0.234741\pi\)
\(458\) −8963.74 −0.914516
\(459\) 0 0
\(460\) −4130.33 −0.418647
\(461\) 1400.23 0.141465 0.0707323 0.997495i \(-0.477466\pi\)
0.0707323 + 0.997495i \(0.477466\pi\)
\(462\) 0 0
\(463\) 11922.3 1.19671 0.598353 0.801232i \(-0.295822\pi\)
0.598353 + 0.801232i \(0.295822\pi\)
\(464\) −7781.20 −0.778519
\(465\) 0 0
\(466\) 17028.4 1.69276
\(467\) −8048.13 −0.797479 −0.398740 0.917064i \(-0.630552\pi\)
−0.398740 + 0.917064i \(0.630552\pi\)
\(468\) 0 0
\(469\) 23347.0 2.29864
\(470\) −3782.49 −0.371219
\(471\) 0 0
\(472\) 26556.3 2.58973
\(473\) 483.874 0.0470371
\(474\) 0 0
\(475\) 5032.43 0.486113
\(476\) −23730.5 −2.28505
\(477\) 0 0
\(478\) −9823.95 −0.940035
\(479\) 15262.5 1.45587 0.727933 0.685648i \(-0.240481\pi\)
0.727933 + 0.685648i \(0.240481\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 1100.36 0.103983
\(483\) 0 0
\(484\) 7423.35 0.697159
\(485\) −11.5064 −0.00107727
\(486\) 0 0
\(487\) 5277.64 0.491073 0.245537 0.969387i \(-0.421036\pi\)
0.245537 + 0.969387i \(0.421036\pi\)
\(488\) 11672.7 1.08279
\(489\) 0 0
\(490\) 6493.84 0.598698
\(491\) 11058.1 1.01638 0.508192 0.861244i \(-0.330314\pi\)
0.508192 + 0.861244i \(0.330314\pi\)
\(492\) 0 0
\(493\) −8091.80 −0.739222
\(494\) 0 0
\(495\) 0 0
\(496\) −811.757 −0.0734858
\(497\) 15523.6 1.40106
\(498\) 0 0
\(499\) −112.405 −0.0100840 −0.00504201 0.999987i \(-0.501605\pi\)
−0.00504201 + 0.999987i \(0.501605\pi\)
\(500\) −5951.89 −0.532354
\(501\) 0 0
\(502\) 16392.7 1.45745
\(503\) 10080.5 0.893570 0.446785 0.894641i \(-0.352569\pi\)
0.446785 + 0.894641i \(0.352569\pi\)
\(504\) 0 0
\(505\) −1790.35 −0.157761
\(506\) −35247.5 −3.09673
\(507\) 0 0
\(508\) −8843.44 −0.772370
\(509\) −19683.5 −1.71406 −0.857031 0.515265i \(-0.827694\pi\)
−0.857031 + 0.515265i \(0.827694\pi\)
\(510\) 0 0
\(511\) 22270.5 1.92796
\(512\) −14763.9 −1.27437
\(513\) 0 0
\(514\) 19057.7 1.63540
\(515\) 1914.93 0.163849
\(516\) 0 0
\(517\) −21111.0 −1.79586
\(518\) −54999.2 −4.66511
\(519\) 0 0
\(520\) 0 0
\(521\) 14722.3 1.23800 0.618998 0.785393i \(-0.287539\pi\)
0.618998 + 0.785393i \(0.287539\pi\)
\(522\) 0 0
\(523\) −11311.6 −0.945738 −0.472869 0.881133i \(-0.656782\pi\)
−0.472869 + 0.881133i \(0.656782\pi\)
\(524\) −23007.1 −1.91807
\(525\) 0 0
\(526\) 23514.4 1.94920
\(527\) −844.160 −0.0697764
\(528\) 0 0
\(529\) 17325.1 1.42394
\(530\) 1665.73 0.136518
\(531\) 0 0
\(532\) 21454.8 1.74846
\(533\) 0 0
\(534\) 0 0
\(535\) 2372.51 0.191725
\(536\) −23158.1 −1.86619
\(537\) 0 0
\(538\) 18976.8 1.52073
\(539\) 36243.7 2.89634
\(540\) 0 0
\(541\) −12705.1 −1.00967 −0.504837 0.863215i \(-0.668447\pi\)
−0.504837 + 0.863215i \(0.668447\pi\)
\(542\) −12947.8 −1.02612
\(543\) 0 0
\(544\) −2900.83 −0.228625
\(545\) −597.095 −0.0469298
\(546\) 0 0
\(547\) −6624.52 −0.517814 −0.258907 0.965902i \(-0.583362\pi\)
−0.258907 + 0.965902i \(0.583362\pi\)
\(548\) −8816.95 −0.687302
\(549\) 0 0
\(550\) −25136.7 −1.94878
\(551\) 7315.82 0.565634
\(552\) 0 0
\(553\) 26872.1 2.06640
\(554\) −12688.0 −0.973032
\(555\) 0 0
\(556\) −30372.9 −2.31672
\(557\) −14637.9 −1.11351 −0.556757 0.830676i \(-0.687954\pi\)
−0.556757 + 0.830676i \(0.687954\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2399.97 −0.181102
\(561\) 0 0
\(562\) 16085.7 1.20736
\(563\) −18465.4 −1.38228 −0.691139 0.722722i \(-0.742891\pi\)
−0.691139 + 0.722722i \(0.742891\pi\)
\(564\) 0 0
\(565\) 1289.84 0.0960423
\(566\) 13462.6 0.999778
\(567\) 0 0
\(568\) −15398.0 −1.13747
\(569\) 195.327 0.0143911 0.00719557 0.999974i \(-0.497710\pi\)
0.00719557 + 0.999974i \(0.497710\pi\)
\(570\) 0 0
\(571\) −12479.1 −0.914592 −0.457296 0.889314i \(-0.651182\pi\)
−0.457296 + 0.889314i \(0.651182\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 68391.8 4.97320
\(575\) 21032.2 1.52540
\(576\) 0 0
\(577\) −5511.41 −0.397648 −0.198824 0.980035i \(-0.563712\pi\)
−0.198824 + 0.980035i \(0.563712\pi\)
\(578\) −13691.7 −0.985290
\(579\) 0 0
\(580\) −4282.02 −0.306554
\(581\) 36478.0 2.60476
\(582\) 0 0
\(583\) 9296.83 0.660438
\(584\) −22090.4 −1.56525
\(585\) 0 0
\(586\) −38711.9 −2.72897
\(587\) −11511.5 −0.809420 −0.404710 0.914445i \(-0.632628\pi\)
−0.404710 + 0.914445i \(0.632628\pi\)
\(588\) 0 0
\(589\) 763.207 0.0533912
\(590\) 5930.12 0.413795
\(591\) 0 0
\(592\) 14478.1 1.00514
\(593\) 22211.3 1.53812 0.769062 0.639174i \(-0.220724\pi\)
0.769062 + 0.639174i \(0.220724\pi\)
\(594\) 0 0
\(595\) −2495.77 −0.171961
\(596\) 20532.2 1.41112
\(597\) 0 0
\(598\) 0 0
\(599\) −18477.3 −1.26037 −0.630184 0.776446i \(-0.717021\pi\)
−0.630184 + 0.776446i \(0.717021\pi\)
\(600\) 0 0
\(601\) 1530.40 0.103871 0.0519355 0.998650i \(-0.483461\pi\)
0.0519355 + 0.998650i \(0.483461\pi\)
\(602\) −1882.12 −0.127425
\(603\) 0 0
\(604\) −34863.7 −2.34865
\(605\) 780.725 0.0524644
\(606\) 0 0
\(607\) −18934.8 −1.26613 −0.633063 0.774101i \(-0.718202\pi\)
−0.633063 + 0.774101i \(0.718202\pi\)
\(608\) 2622.65 0.174938
\(609\) 0 0
\(610\) 2606.57 0.173011
\(611\) 0 0
\(612\) 0 0
\(613\) −3888.26 −0.256191 −0.128096 0.991762i \(-0.540886\pi\)
−0.128096 + 0.991762i \(0.540886\pi\)
\(614\) 25863.7 1.69996
\(615\) 0 0
\(616\) −50472.6 −3.30130
\(617\) −13927.8 −0.908769 −0.454385 0.890806i \(-0.650141\pi\)
−0.454385 + 0.890806i \(0.650141\pi\)
\(618\) 0 0
\(619\) −29627.9 −1.92382 −0.961910 0.273368i \(-0.911862\pi\)
−0.961910 + 0.273368i \(0.911862\pi\)
\(620\) −446.712 −0.0289361
\(621\) 0 0
\(622\) −17923.9 −1.15544
\(623\) −5532.23 −0.355769
\(624\) 0 0
\(625\) 14682.8 0.939702
\(626\) 25661.8 1.63842
\(627\) 0 0
\(628\) 18963.8 1.20500
\(629\) 15056.0 0.954405
\(630\) 0 0
\(631\) −6066.13 −0.382708 −0.191354 0.981521i \(-0.561288\pi\)
−0.191354 + 0.981521i \(0.561288\pi\)
\(632\) −26654.8 −1.67764
\(633\) 0 0
\(634\) 1629.48 0.102074
\(635\) −930.077 −0.0581244
\(636\) 0 0
\(637\) 0 0
\(638\) −36542.0 −2.26758
\(639\) 0 0
\(640\) −4208.99 −0.259961
\(641\) −16798.4 −1.03510 −0.517548 0.855654i \(-0.673155\pi\)
−0.517548 + 0.855654i \(0.673155\pi\)
\(642\) 0 0
\(643\) 20373.2 1.24952 0.624758 0.780818i \(-0.285198\pi\)
0.624758 + 0.780818i \(0.285198\pi\)
\(644\) 89666.6 5.48658
\(645\) 0 0
\(646\) −8980.28 −0.546942
\(647\) −23098.6 −1.40356 −0.701778 0.712396i \(-0.747610\pi\)
−0.701778 + 0.712396i \(0.747610\pi\)
\(648\) 0 0
\(649\) 33097.4 2.00183
\(650\) 0 0
\(651\) 0 0
\(652\) 42234.0 2.53683
\(653\) −1795.81 −0.107619 −0.0538097 0.998551i \(-0.517136\pi\)
−0.0538097 + 0.998551i \(0.517136\pi\)
\(654\) 0 0
\(655\) −2419.69 −0.144344
\(656\) −18003.5 −1.07152
\(657\) 0 0
\(658\) 82115.2 4.86502
\(659\) 14640.3 0.865412 0.432706 0.901535i \(-0.357559\pi\)
0.432706 + 0.901535i \(0.357559\pi\)
\(660\) 0 0
\(661\) 20490.3 1.20572 0.602860 0.797847i \(-0.294028\pi\)
0.602860 + 0.797847i \(0.294028\pi\)
\(662\) −9984.67 −0.586201
\(663\) 0 0
\(664\) −36182.9 −2.11471
\(665\) 2256.43 0.131580
\(666\) 0 0
\(667\) 30575.2 1.77493
\(668\) −18429.7 −1.06746
\(669\) 0 0
\(670\) −5171.30 −0.298186
\(671\) 14547.9 0.836981
\(672\) 0 0
\(673\) 21436.5 1.22781 0.613904 0.789380i \(-0.289598\pi\)
0.613904 + 0.789380i \(0.289598\pi\)
\(674\) 12428.4 0.710273
\(675\) 0 0
\(676\) 0 0
\(677\) −20158.0 −1.14436 −0.572181 0.820127i \(-0.693903\pi\)
−0.572181 + 0.820127i \(0.693903\pi\)
\(678\) 0 0
\(679\) 249.796 0.0141182
\(680\) 2475.58 0.139609
\(681\) 0 0
\(682\) −3812.17 −0.214040
\(683\) 4273.98 0.239443 0.119721 0.992808i \(-0.461800\pi\)
0.119721 + 0.992808i \(0.461800\pi\)
\(684\) 0 0
\(685\) −927.291 −0.0517226
\(686\) −84030.0 −4.67679
\(687\) 0 0
\(688\) 495.452 0.0274548
\(689\) 0 0
\(690\) 0 0
\(691\) −10287.2 −0.566346 −0.283173 0.959069i \(-0.591387\pi\)
−0.283173 + 0.959069i \(0.591387\pi\)
\(692\) −23086.5 −1.26823
\(693\) 0 0
\(694\) 20154.2 1.10237
\(695\) −3194.36 −0.174344
\(696\) 0 0
\(697\) −18722.2 −1.01744
\(698\) 35733.8 1.93774
\(699\) 0 0
\(700\) 63945.4 3.45273
\(701\) −11726.5 −0.631815 −0.315907 0.948790i \(-0.602309\pi\)
−0.315907 + 0.948790i \(0.602309\pi\)
\(702\) 0 0
\(703\) −13612.2 −0.730287
\(704\) −28023.8 −1.50027
\(705\) 0 0
\(706\) −58902.3 −3.13997
\(707\) 38867.2 2.06754
\(708\) 0 0
\(709\) 11833.0 0.626795 0.313398 0.949622i \(-0.398533\pi\)
0.313398 + 0.949622i \(0.398533\pi\)
\(710\) −3438.43 −0.181749
\(711\) 0 0
\(712\) 5487.48 0.288837
\(713\) 3189.69 0.167538
\(714\) 0 0
\(715\) 0 0
\(716\) −16139.4 −0.842398
\(717\) 0 0
\(718\) −26377.2 −1.37101
\(719\) 5964.80 0.309387 0.154694 0.987962i \(-0.450561\pi\)
0.154694 + 0.987962i \(0.450561\pi\)
\(720\) 0 0
\(721\) −41571.9 −2.14732
\(722\) −24862.8 −1.28158
\(723\) 0 0
\(724\) 54137.4 2.77901
\(725\) 21804.6 1.11697
\(726\) 0 0
\(727\) −25972.8 −1.32500 −0.662501 0.749061i \(-0.730505\pi\)
−0.662501 + 0.749061i \(0.730505\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −4932.86 −0.250101
\(731\) 515.229 0.0260690
\(732\) 0 0
\(733\) −33025.9 −1.66417 −0.832087 0.554646i \(-0.812854\pi\)
−0.832087 + 0.554646i \(0.812854\pi\)
\(734\) −42085.4 −2.11635
\(735\) 0 0
\(736\) 10960.9 0.548946
\(737\) −28862.3 −1.44255
\(738\) 0 0
\(739\) −7564.70 −0.376552 −0.188276 0.982116i \(-0.560290\pi\)
−0.188276 + 0.982116i \(0.560290\pi\)
\(740\) 7967.32 0.395790
\(741\) 0 0
\(742\) −36161.8 −1.78914
\(743\) −12623.9 −0.623319 −0.311660 0.950194i \(-0.600885\pi\)
−0.311660 + 0.950194i \(0.600885\pi\)
\(744\) 0 0
\(745\) 2159.40 0.106194
\(746\) 28790.3 1.41299
\(747\) 0 0
\(748\) 29336.4 1.43402
\(749\) −51505.6 −2.51265
\(750\) 0 0
\(751\) 15054.7 0.731497 0.365749 0.930714i \(-0.380813\pi\)
0.365749 + 0.930714i \(0.380813\pi\)
\(752\) −21616.1 −1.04822
\(753\) 0 0
\(754\) 0 0
\(755\) −3666.67 −0.176747
\(756\) 0 0
\(757\) 17024.5 0.817391 0.408695 0.912671i \(-0.365984\pi\)
0.408695 + 0.912671i \(0.365984\pi\)
\(758\) −65410.9 −3.13434
\(759\) 0 0
\(760\) −2238.18 −0.106825
\(761\) 1037.46 0.0494191 0.0247095 0.999695i \(-0.492134\pi\)
0.0247095 + 0.999695i \(0.492134\pi\)
\(762\) 0 0
\(763\) 12962.5 0.615040
\(764\) 46685.3 2.21075
\(765\) 0 0
\(766\) −31971.6 −1.50807
\(767\) 0 0
\(768\) 0 0
\(769\) −33909.7 −1.59014 −0.795068 0.606520i \(-0.792565\pi\)
−0.795068 + 0.606520i \(0.792565\pi\)
\(770\) −11270.7 −0.527492
\(771\) 0 0
\(772\) 13571.6 0.632710
\(773\) 13566.7 0.631253 0.315626 0.948884i \(-0.397785\pi\)
0.315626 + 0.948884i \(0.397785\pi\)
\(774\) 0 0
\(775\) 2274.72 0.105433
\(776\) −247.775 −0.0114621
\(777\) 0 0
\(778\) 23674.5 1.09097
\(779\) 16926.8 0.778517
\(780\) 0 0
\(781\) −19190.7 −0.879255
\(782\) −37531.5 −1.71627
\(783\) 0 0
\(784\) 37110.9 1.69055
\(785\) 1994.46 0.0906817
\(786\) 0 0
\(787\) −12545.8 −0.568246 −0.284123 0.958788i \(-0.591702\pi\)
−0.284123 + 0.958788i \(0.591702\pi\)
\(788\) −352.414 −0.0159318
\(789\) 0 0
\(790\) −5952.11 −0.268059
\(791\) −28001.5 −1.25868
\(792\) 0 0
\(793\) 0 0
\(794\) 2770.23 0.123818
\(795\) 0 0
\(796\) 59141.5 2.63344
\(797\) 26658.5 1.18481 0.592403 0.805642i \(-0.298179\pi\)
0.592403 + 0.805642i \(0.298179\pi\)
\(798\) 0 0
\(799\) −22478.9 −0.995304
\(800\) 7816.74 0.345454
\(801\) 0 0
\(802\) 43534.4 1.91677
\(803\) −27531.5 −1.20992
\(804\) 0 0
\(805\) 9430.36 0.412890
\(806\) 0 0
\(807\) 0 0
\(808\) −38552.8 −1.67857
\(809\) −3448.15 −0.149852 −0.0749260 0.997189i \(-0.523872\pi\)
−0.0749260 + 0.997189i \(0.523872\pi\)
\(810\) 0 0
\(811\) −33920.4 −1.46869 −0.734344 0.678778i \(-0.762510\pi\)
−0.734344 + 0.678778i \(0.762510\pi\)
\(812\) 92959.7 4.01754
\(813\) 0 0
\(814\) 67991.8 2.92766
\(815\) 4441.81 0.190908
\(816\) 0 0
\(817\) −465.820 −0.0199473
\(818\) −75926.7 −3.24537
\(819\) 0 0
\(820\) −9907.40 −0.421928
\(821\) −5461.33 −0.232158 −0.116079 0.993240i \(-0.537033\pi\)
−0.116079 + 0.993240i \(0.537033\pi\)
\(822\) 0 0
\(823\) 21155.0 0.896010 0.448005 0.894031i \(-0.352135\pi\)
0.448005 + 0.894031i \(0.352135\pi\)
\(824\) 41235.6 1.74334
\(825\) 0 0
\(826\) −128739. −5.42300
\(827\) 18974.8 0.797846 0.398923 0.916984i \(-0.369384\pi\)
0.398923 + 0.916984i \(0.369384\pi\)
\(828\) 0 0
\(829\) 9497.90 0.397920 0.198960 0.980008i \(-0.436244\pi\)
0.198960 + 0.980008i \(0.436244\pi\)
\(830\) −8079.79 −0.337896
\(831\) 0 0
\(832\) 0 0
\(833\) 38592.3 1.60521
\(834\) 0 0
\(835\) −1938.28 −0.0803316
\(836\) −26523.1 −1.09727
\(837\) 0 0
\(838\) 19674.0 0.811010
\(839\) −8225.56 −0.338472 −0.169236 0.985576i \(-0.554130\pi\)
−0.169236 + 0.985576i \(0.554130\pi\)
\(840\) 0 0
\(841\) 7309.11 0.299689
\(842\) 26643.3 1.09049
\(843\) 0 0
\(844\) −2902.27 −0.118365
\(845\) 0 0
\(846\) 0 0
\(847\) −16949.0 −0.687573
\(848\) 9519.27 0.385487
\(849\) 0 0
\(850\) −26765.5 −1.08006
\(851\) −56889.6 −2.29160
\(852\) 0 0
\(853\) −19140.4 −0.768294 −0.384147 0.923272i \(-0.625504\pi\)
−0.384147 + 0.923272i \(0.625504\pi\)
\(854\) −56586.8 −2.26740
\(855\) 0 0
\(856\) 51089.0 2.03994
\(857\) −20602.9 −0.821215 −0.410608 0.911812i \(-0.634683\pi\)
−0.410608 + 0.911812i \(0.634683\pi\)
\(858\) 0 0
\(859\) 5247.92 0.208448 0.104224 0.994554i \(-0.466764\pi\)
0.104224 + 0.994554i \(0.466764\pi\)
\(860\) 272.649 0.0108107
\(861\) 0 0
\(862\) 196.116 0.00774912
\(863\) −38622.5 −1.52343 −0.761717 0.647909i \(-0.775644\pi\)
−0.761717 + 0.647909i \(0.775644\pi\)
\(864\) 0 0
\(865\) −2428.04 −0.0954402
\(866\) 3684.95 0.144595
\(867\) 0 0
\(868\) 9697.82 0.379223
\(869\) −33220.2 −1.29680
\(870\) 0 0
\(871\) 0 0
\(872\) −12857.7 −0.499330
\(873\) 0 0
\(874\) 33932.4 1.31325
\(875\) 13589.4 0.525034
\(876\) 0 0
\(877\) −19963.9 −0.768682 −0.384341 0.923191i \(-0.625571\pi\)
−0.384341 + 0.923191i \(0.625571\pi\)
\(878\) −29114.1 −1.11908
\(879\) 0 0
\(880\) 2966.92 0.113653
\(881\) 13175.8 0.503865 0.251933 0.967745i \(-0.418934\pi\)
0.251933 + 0.967745i \(0.418934\pi\)
\(882\) 0 0
\(883\) 14434.0 0.550105 0.275052 0.961429i \(-0.411305\pi\)
0.275052 + 0.961429i \(0.411305\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 35691.2 1.35335
\(887\) −20962.3 −0.793514 −0.396757 0.917924i \(-0.629864\pi\)
−0.396757 + 0.917924i \(0.629864\pi\)
\(888\) 0 0
\(889\) 20191.3 0.761750
\(890\) 1225.38 0.0461513
\(891\) 0 0
\(892\) 21840.0 0.819794
\(893\) 20323.3 0.761582
\(894\) 0 0
\(895\) −1697.40 −0.0633943
\(896\) 91374.3 3.40692
\(897\) 0 0
\(898\) −5053.04 −0.187775
\(899\) 3306.84 0.122680
\(900\) 0 0
\(901\) 9899.25 0.366029
\(902\) −84548.1 −3.12100
\(903\) 0 0
\(904\) 27775.0 1.02188
\(905\) 5693.71 0.209133
\(906\) 0 0
\(907\) −14079.6 −0.515441 −0.257721 0.966219i \(-0.582971\pi\)
−0.257721 + 0.966219i \(0.582971\pi\)
\(908\) 65136.8 2.38066
\(909\) 0 0
\(910\) 0 0
\(911\) −622.987 −0.0226569 −0.0113285 0.999936i \(-0.503606\pi\)
−0.0113285 + 0.999936i \(0.503606\pi\)
\(912\) 0 0
\(913\) −45095.3 −1.63465
\(914\) 69543.5 2.51673
\(915\) 0 0
\(916\) −28189.7 −1.01683
\(917\) 52529.8 1.89170
\(918\) 0 0
\(919\) −30293.2 −1.08735 −0.543677 0.839294i \(-0.682968\pi\)
−0.543677 + 0.839294i \(0.682968\pi\)
\(920\) −9354.08 −0.335212
\(921\) 0 0
\(922\) 6733.09 0.240502
\(923\) 0 0
\(924\) 0 0
\(925\) −40570.7 −1.44211
\(926\) 57329.0 2.03450
\(927\) 0 0
\(928\) 11363.5 0.401965
\(929\) 25045.6 0.884520 0.442260 0.896887i \(-0.354177\pi\)
0.442260 + 0.896887i \(0.354177\pi\)
\(930\) 0 0
\(931\) −34891.4 −1.22827
\(932\) 53552.0 1.88214
\(933\) 0 0
\(934\) −38699.9 −1.35578
\(935\) 3085.35 0.107916
\(936\) 0 0
\(937\) 4025.83 0.140361 0.0701805 0.997534i \(-0.477642\pi\)
0.0701805 + 0.997534i \(0.477642\pi\)
\(938\) 112265. 3.90788
\(939\) 0 0
\(940\) −11895.4 −0.412750
\(941\) −50575.8 −1.75210 −0.876049 0.482222i \(-0.839830\pi\)
−0.876049 + 0.482222i \(0.839830\pi\)
\(942\) 0 0
\(943\) 70742.5 2.44294
\(944\) 33889.3 1.16844
\(945\) 0 0
\(946\) 2326.74 0.0799671
\(947\) 51825.9 1.77837 0.889185 0.457547i \(-0.151272\pi\)
0.889185 + 0.457547i \(0.151272\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 24198.8 0.826433
\(951\) 0 0
\(952\) −53743.1 −1.82965
\(953\) 40037.1 1.36089 0.680446 0.732798i \(-0.261786\pi\)
0.680446 + 0.732798i \(0.261786\pi\)
\(954\) 0 0
\(955\) 4909.96 0.166369
\(956\) −30895.0 −1.04520
\(957\) 0 0
\(958\) 73390.5 2.47509
\(959\) 20130.9 0.677851
\(960\) 0 0
\(961\) −29446.0 −0.988420
\(962\) 0 0
\(963\) 0 0
\(964\) 3460.47 0.115617
\(965\) 1427.34 0.0476143
\(966\) 0 0
\(967\) −5557.42 −0.184813 −0.0924067 0.995721i \(-0.529456\pi\)
−0.0924067 + 0.995721i \(0.529456\pi\)
\(968\) 16811.9 0.558218
\(969\) 0 0
\(970\) −55.3292 −0.00183146
\(971\) −25730.6 −0.850396 −0.425198 0.905100i \(-0.639796\pi\)
−0.425198 + 0.905100i \(0.639796\pi\)
\(972\) 0 0
\(973\) 69347.4 2.28487
\(974\) 25377.9 0.834865
\(975\) 0 0
\(976\) 14896.0 0.488533
\(977\) 39868.8 1.30554 0.652772 0.757554i \(-0.273606\pi\)
0.652772 + 0.757554i \(0.273606\pi\)
\(978\) 0 0
\(979\) 6839.12 0.223268
\(980\) 20422.2 0.665678
\(981\) 0 0
\(982\) 53173.5 1.72794
\(983\) −27728.9 −0.899709 −0.449854 0.893102i \(-0.648524\pi\)
−0.449854 + 0.893102i \(0.648524\pi\)
\(984\) 0 0
\(985\) −37.0639 −0.00119894
\(986\) −38909.9 −1.25674
\(987\) 0 0
\(988\) 0 0
\(989\) −1946.81 −0.0625936
\(990\) 0 0
\(991\) 38088.3 1.22090 0.610452 0.792053i \(-0.290988\pi\)
0.610452 + 0.792053i \(0.290988\pi\)
\(992\) 1185.47 0.0379422
\(993\) 0 0
\(994\) 74646.0 2.38192
\(995\) 6220.00 0.198178
\(996\) 0 0
\(997\) −18606.0 −0.591033 −0.295516 0.955338i \(-0.595492\pi\)
−0.295516 + 0.955338i \(0.595492\pi\)
\(998\) −540.505 −0.0171437
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1521.4.a.bm.1.17 yes 18
3.2 odd 2 inner 1521.4.a.bm.1.2 18
13.12 even 2 1521.4.a.bn.1.2 yes 18
39.38 odd 2 1521.4.a.bn.1.17 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1521.4.a.bm.1.2 18 3.2 odd 2 inner
1521.4.a.bm.1.17 yes 18 1.1 even 1 trivial
1521.4.a.bn.1.2 yes 18 13.12 even 2
1521.4.a.bn.1.17 yes 18 39.38 odd 2