Properties

Label 1575.4.a.u.1.1
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 315)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155 q^{2} -5.56155 q^{4} +7.00000 q^{7} +21.1771 q^{8} +O(q^{10})\) \(q-1.56155 q^{2} -5.56155 q^{4} +7.00000 q^{7} +21.1771 q^{8} +10.2462 q^{11} -34.3542 q^{13} -10.9309 q^{14} +11.4233 q^{16} -82.6004 q^{17} +90.7083 q^{19} -16.0000 q^{22} -12.1383 q^{23} +53.6458 q^{26} -38.9309 q^{28} +105.153 q^{29} -142.108 q^{31} -187.255 q^{32} +128.985 q^{34} -64.8466 q^{37} -141.646 q^{38} +195.201 q^{41} +319.218 q^{43} -56.9848 q^{44} +18.9545 q^{46} -318.847 q^{47} +49.0000 q^{49} +191.062 q^{52} -296.799 q^{53} +148.240 q^{56} -164.203 q^{58} +284.000 q^{59} -494.540 q^{61} +221.909 q^{62} +201.022 q^{64} -549.032 q^{67} +459.386 q^{68} +740.972 q^{71} +556.850 q^{73} +101.261 q^{74} -504.479 q^{76} +71.7235 q^{77} -376.189 q^{79} -304.816 q^{82} +752.466 q^{83} -498.475 q^{86} +216.985 q^{88} +945.299 q^{89} -240.479 q^{91} +67.5076 q^{92} +497.896 q^{94} +180.668 q^{97} -76.5161 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 7 q^{4} + 14 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 7 q^{4} + 14 q^{7} - 3 q^{8} + 4 q^{11} + 22 q^{13} + 7 q^{14} - 39 q^{16} - 58 q^{17} - 32 q^{22} - 82 q^{23} + 198 q^{26} - 49 q^{28} + 334 q^{29} - 210 q^{31} - 123 q^{32} + 192 q^{34} - 6 q^{37} - 374 q^{38} + 176 q^{41} - 46 q^{43} - 48 q^{44} - 160 q^{46} - 514 q^{47} + 98 q^{49} + 110 q^{52} - 808 q^{53} - 21 q^{56} + 422 q^{58} + 568 q^{59} - 618 q^{61} + 48 q^{62} + 769 q^{64} - 694 q^{67} + 424 q^{68} + 814 q^{71} - 82 q^{73} + 252 q^{74} - 374 q^{76} + 28 q^{77} + 600 q^{79} - 354 q^{82} + 268 q^{83} - 1434 q^{86} + 368 q^{88} - 72 q^{89} + 154 q^{91} + 168 q^{92} - 2 q^{94} - 1626 q^{97} + 49 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.56155 −0.552092 −0.276046 0.961144i \(-0.589024\pi\)
−0.276046 + 0.961144i \(0.589024\pi\)
\(3\) 0 0
\(4\) −5.56155 −0.695194
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 21.1771 0.935904
\(9\) 0 0
\(10\) 0 0
\(11\) 10.2462 0.280850 0.140425 0.990091i \(-0.455153\pi\)
0.140425 + 0.990091i \(0.455153\pi\)
\(12\) 0 0
\(13\) −34.3542 −0.732933 −0.366467 0.930431i \(-0.619433\pi\)
−0.366467 + 0.930431i \(0.619433\pi\)
\(14\) −10.9309 −0.208671
\(15\) 0 0
\(16\) 11.4233 0.178489
\(17\) −82.6004 −1.17844 −0.589222 0.807972i \(-0.700565\pi\)
−0.589222 + 0.807972i \(0.700565\pi\)
\(18\) 0 0
\(19\) 90.7083 1.09526 0.547629 0.836721i \(-0.315530\pi\)
0.547629 + 0.836721i \(0.315530\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −16.0000 −0.155055
\(23\) −12.1383 −0.110044 −0.0550218 0.998485i \(-0.517523\pi\)
−0.0550218 + 0.998485i \(0.517523\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 53.6458 0.404647
\(27\) 0 0
\(28\) −38.9309 −0.262759
\(29\) 105.153 0.673328 0.336664 0.941625i \(-0.390701\pi\)
0.336664 + 0.941625i \(0.390701\pi\)
\(30\) 0 0
\(31\) −142.108 −0.823334 −0.411667 0.911334i \(-0.635053\pi\)
−0.411667 + 0.911334i \(0.635053\pi\)
\(32\) −187.255 −1.03445
\(33\) 0 0
\(34\) 128.985 0.650609
\(35\) 0 0
\(36\) 0 0
\(37\) −64.8466 −0.288127 −0.144064 0.989568i \(-0.546017\pi\)
−0.144064 + 0.989568i \(0.546017\pi\)
\(38\) −141.646 −0.604684
\(39\) 0 0
\(40\) 0 0
\(41\) 195.201 0.743542 0.371771 0.928324i \(-0.378751\pi\)
0.371771 + 0.928324i \(0.378751\pi\)
\(42\) 0 0
\(43\) 319.218 1.13210 0.566049 0.824371i \(-0.308471\pi\)
0.566049 + 0.824371i \(0.308471\pi\)
\(44\) −56.9848 −0.195245
\(45\) 0 0
\(46\) 18.9545 0.0607542
\(47\) −318.847 −0.989544 −0.494772 0.869023i \(-0.664748\pi\)
−0.494772 + 0.869023i \(0.664748\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 191.062 0.509531
\(53\) −296.799 −0.769217 −0.384609 0.923080i \(-0.625664\pi\)
−0.384609 + 0.923080i \(0.625664\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 148.240 0.353738
\(57\) 0 0
\(58\) −164.203 −0.371739
\(59\) 284.000 0.626672 0.313336 0.949642i \(-0.398553\pi\)
0.313336 + 0.949642i \(0.398553\pi\)
\(60\) 0 0
\(61\) −494.540 −1.03802 −0.519011 0.854768i \(-0.673700\pi\)
−0.519011 + 0.854768i \(0.673700\pi\)
\(62\) 221.909 0.454556
\(63\) 0 0
\(64\) 201.022 0.392621
\(65\) 0 0
\(66\) 0 0
\(67\) −549.032 −1.00112 −0.500559 0.865702i \(-0.666872\pi\)
−0.500559 + 0.865702i \(0.666872\pi\)
\(68\) 459.386 0.819247
\(69\) 0 0
\(70\) 0 0
\(71\) 740.972 1.23855 0.619276 0.785174i \(-0.287426\pi\)
0.619276 + 0.785174i \(0.287426\pi\)
\(72\) 0 0
\(73\) 556.850 0.892800 0.446400 0.894834i \(-0.352706\pi\)
0.446400 + 0.894834i \(0.352706\pi\)
\(74\) 101.261 0.159073
\(75\) 0 0
\(76\) −504.479 −0.761417
\(77\) 71.7235 0.106151
\(78\) 0 0
\(79\) −376.189 −0.535755 −0.267877 0.963453i \(-0.586322\pi\)
−0.267877 + 0.963453i \(0.586322\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −304.816 −0.410504
\(83\) 752.466 0.995107 0.497553 0.867433i \(-0.334232\pi\)
0.497553 + 0.867433i \(0.334232\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −498.475 −0.625023
\(87\) 0 0
\(88\) 216.985 0.262848
\(89\) 945.299 1.12586 0.562930 0.826505i \(-0.309674\pi\)
0.562930 + 0.826505i \(0.309674\pi\)
\(90\) 0 0
\(91\) −240.479 −0.277023
\(92\) 67.5076 0.0765016
\(93\) 0 0
\(94\) 497.896 0.546319
\(95\) 0 0
\(96\) 0 0
\(97\) 180.668 0.189114 0.0945572 0.995519i \(-0.469856\pi\)
0.0945572 + 0.995519i \(0.469856\pi\)
\(98\) −76.5161 −0.0788703
\(99\) 0 0
\(100\) 0 0
\(101\) 1463.05 1.44138 0.720689 0.693258i \(-0.243826\pi\)
0.720689 + 0.693258i \(0.243826\pi\)
\(102\) 0 0
\(103\) −535.049 −0.511844 −0.255922 0.966697i \(-0.582379\pi\)
−0.255922 + 0.966697i \(0.582379\pi\)
\(104\) −727.521 −0.685955
\(105\) 0 0
\(106\) 463.468 0.424679
\(107\) 327.153 0.295581 0.147790 0.989019i \(-0.452784\pi\)
0.147790 + 0.989019i \(0.452784\pi\)
\(108\) 0 0
\(109\) 1621.70 1.42506 0.712528 0.701644i \(-0.247550\pi\)
0.712528 + 0.701644i \(0.247550\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 79.9630 0.0674625
\(113\) 450.716 0.375219 0.187610 0.982244i \(-0.439926\pi\)
0.187610 + 0.982244i \(0.439926\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −584.816 −0.468093
\(117\) 0 0
\(118\) −443.481 −0.345981
\(119\) −578.203 −0.445410
\(120\) 0 0
\(121\) −1226.02 −0.921123
\(122\) 772.250 0.573084
\(123\) 0 0
\(124\) 790.341 0.572377
\(125\) 0 0
\(126\) 0 0
\(127\) −1710.10 −1.19486 −0.597428 0.801923i \(-0.703811\pi\)
−0.597428 + 0.801923i \(0.703811\pi\)
\(128\) 1184.13 0.817683
\(129\) 0 0
\(130\) 0 0
\(131\) −1584.49 −1.05678 −0.528388 0.849003i \(-0.677203\pi\)
−0.528388 + 0.849003i \(0.677203\pi\)
\(132\) 0 0
\(133\) 634.958 0.413969
\(134\) 857.343 0.552710
\(135\) 0 0
\(136\) −1749.23 −1.10291
\(137\) −2095.24 −1.30663 −0.653316 0.757085i \(-0.726623\pi\)
−0.653316 + 0.757085i \(0.726623\pi\)
\(138\) 0 0
\(139\) −1385.00 −0.845135 −0.422568 0.906331i \(-0.638871\pi\)
−0.422568 + 0.906331i \(0.638871\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1157.07 −0.683795
\(143\) −352.000 −0.205844
\(144\) 0 0
\(145\) 0 0
\(146\) −869.551 −0.492908
\(147\) 0 0
\(148\) 360.648 0.200304
\(149\) −2502.89 −1.37614 −0.688069 0.725645i \(-0.741541\pi\)
−0.688069 + 0.725645i \(0.741541\pi\)
\(150\) 0 0
\(151\) 395.212 0.212993 0.106496 0.994313i \(-0.466037\pi\)
0.106496 + 0.994313i \(0.466037\pi\)
\(152\) 1920.94 1.02506
\(153\) 0 0
\(154\) −112.000 −0.0586053
\(155\) 0 0
\(156\) 0 0
\(157\) −1732.35 −0.880615 −0.440308 0.897847i \(-0.645131\pi\)
−0.440308 + 0.897847i \(0.645131\pi\)
\(158\) 587.439 0.295786
\(159\) 0 0
\(160\) 0 0
\(161\) −84.9678 −0.0415926
\(162\) 0 0
\(163\) −3303.26 −1.58731 −0.793654 0.608369i \(-0.791824\pi\)
−0.793654 + 0.608369i \(0.791824\pi\)
\(164\) −1085.62 −0.516906
\(165\) 0 0
\(166\) −1175.02 −0.549391
\(167\) 601.941 0.278920 0.139460 0.990228i \(-0.455463\pi\)
0.139460 + 0.990228i \(0.455463\pi\)
\(168\) 0 0
\(169\) −1016.79 −0.462809
\(170\) 0 0
\(171\) 0 0
\(172\) −1775.35 −0.787028
\(173\) 3741.17 1.64414 0.822068 0.569389i \(-0.192820\pi\)
0.822068 + 0.569389i \(0.192820\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 117.045 0.0501286
\(177\) 0 0
\(178\) −1476.13 −0.621578
\(179\) −968.352 −0.404347 −0.202173 0.979350i \(-0.564800\pi\)
−0.202173 + 0.979350i \(0.564800\pi\)
\(180\) 0 0
\(181\) −1895.65 −0.778465 −0.389233 0.921139i \(-0.627260\pi\)
−0.389233 + 0.921139i \(0.627260\pi\)
\(182\) 375.521 0.152942
\(183\) 0 0
\(184\) −257.053 −0.102990
\(185\) 0 0
\(186\) 0 0
\(187\) −846.341 −0.330966
\(188\) 1773.28 0.687925
\(189\) 0 0
\(190\) 0 0
\(191\) 2729.88 1.03417 0.517086 0.855933i \(-0.327017\pi\)
0.517086 + 0.855933i \(0.327017\pi\)
\(192\) 0 0
\(193\) −1460.44 −0.544689 −0.272345 0.962200i \(-0.587799\pi\)
−0.272345 + 0.962200i \(0.587799\pi\)
\(194\) −282.123 −0.104409
\(195\) 0 0
\(196\) −272.516 −0.0993134
\(197\) −4831.42 −1.74733 −0.873665 0.486527i \(-0.838263\pi\)
−0.873665 + 0.486527i \(0.838263\pi\)
\(198\) 0 0
\(199\) −873.070 −0.311006 −0.155503 0.987835i \(-0.549700\pi\)
−0.155503 + 0.987835i \(0.549700\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −2284.63 −0.795774
\(203\) 736.074 0.254494
\(204\) 0 0
\(205\) 0 0
\(206\) 835.508 0.282585
\(207\) 0 0
\(208\) −392.438 −0.130820
\(209\) 929.417 0.307603
\(210\) 0 0
\(211\) −5148.02 −1.67964 −0.839821 0.542863i \(-0.817340\pi\)
−0.839821 + 0.542863i \(0.817340\pi\)
\(212\) 1650.66 0.534755
\(213\) 0 0
\(214\) −510.867 −0.163188
\(215\) 0 0
\(216\) 0 0
\(217\) −994.756 −0.311191
\(218\) −2532.38 −0.786762
\(219\) 0 0
\(220\) 0 0
\(221\) 2837.67 0.863720
\(222\) 0 0
\(223\) 2435.31 0.731304 0.365652 0.930752i \(-0.380846\pi\)
0.365652 + 0.930752i \(0.380846\pi\)
\(224\) −1310.78 −0.390984
\(225\) 0 0
\(226\) −703.817 −0.207156
\(227\) −3508.47 −1.02584 −0.512919 0.858437i \(-0.671436\pi\)
−0.512919 + 0.858437i \(0.671436\pi\)
\(228\) 0 0
\(229\) 3429.54 0.989653 0.494827 0.868992i \(-0.335232\pi\)
0.494827 + 0.868992i \(0.335232\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2226.84 0.630170
\(233\) −3989.71 −1.12178 −0.560890 0.827890i \(-0.689541\pi\)
−0.560890 + 0.827890i \(0.689541\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1579.48 −0.435659
\(237\) 0 0
\(238\) 902.894 0.245907
\(239\) 5934.71 1.60621 0.803105 0.595837i \(-0.203180\pi\)
0.803105 + 0.595837i \(0.203180\pi\)
\(240\) 0 0
\(241\) −6481.90 −1.73251 −0.866257 0.499598i \(-0.833481\pi\)
−0.866257 + 0.499598i \(0.833481\pi\)
\(242\) 1914.49 0.508545
\(243\) 0 0
\(244\) 2750.41 0.721627
\(245\) 0 0
\(246\) 0 0
\(247\) −3116.21 −0.802751
\(248\) −3009.43 −0.770561
\(249\) 0 0
\(250\) 0 0
\(251\) −5257.70 −1.32216 −0.661081 0.750314i \(-0.729902\pi\)
−0.661081 + 0.750314i \(0.729902\pi\)
\(252\) 0 0
\(253\) −124.371 −0.0309057
\(254\) 2670.41 0.659671
\(255\) 0 0
\(256\) −3457.26 −0.844057
\(257\) 4108.52 0.997209 0.498604 0.866830i \(-0.333846\pi\)
0.498604 + 0.866830i \(0.333846\pi\)
\(258\) 0 0
\(259\) −453.926 −0.108902
\(260\) 0 0
\(261\) 0 0
\(262\) 2474.27 0.583438
\(263\) −6236.89 −1.46229 −0.731146 0.682221i \(-0.761014\pi\)
−0.731146 + 0.682221i \(0.761014\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −991.521 −0.228549
\(267\) 0 0
\(268\) 3053.47 0.695972
\(269\) 4356.67 0.987475 0.493738 0.869611i \(-0.335630\pi\)
0.493738 + 0.869611i \(0.335630\pi\)
\(270\) 0 0
\(271\) −63.0970 −0.0141434 −0.00707171 0.999975i \(-0.502251\pi\)
−0.00707171 + 0.999975i \(0.502251\pi\)
\(272\) −943.568 −0.210339
\(273\) 0 0
\(274\) 3271.83 0.721382
\(275\) 0 0
\(276\) 0 0
\(277\) −5804.64 −1.25909 −0.629543 0.776966i \(-0.716758\pi\)
−0.629543 + 0.776966i \(0.716758\pi\)
\(278\) 2162.74 0.466593
\(279\) 0 0
\(280\) 0 0
\(281\) 8430.89 1.78984 0.894920 0.446227i \(-0.147233\pi\)
0.894920 + 0.446227i \(0.147233\pi\)
\(282\) 0 0
\(283\) −7415.73 −1.55767 −0.778833 0.627232i \(-0.784188\pi\)
−0.778833 + 0.627232i \(0.784188\pi\)
\(284\) −4120.95 −0.861034
\(285\) 0 0
\(286\) 549.667 0.113645
\(287\) 1366.41 0.281033
\(288\) 0 0
\(289\) 1909.82 0.388728
\(290\) 0 0
\(291\) 0 0
\(292\) −3096.95 −0.620669
\(293\) −482.022 −0.0961094 −0.0480547 0.998845i \(-0.515302\pi\)
−0.0480547 + 0.998845i \(0.515302\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1373.26 −0.269659
\(297\) 0 0
\(298\) 3908.39 0.759755
\(299\) 417.000 0.0806546
\(300\) 0 0
\(301\) 2234.52 0.427893
\(302\) −617.144 −0.117592
\(303\) 0 0
\(304\) 1036.19 0.195492
\(305\) 0 0
\(306\) 0 0
\(307\) −2757.90 −0.512709 −0.256354 0.966583i \(-0.582521\pi\)
−0.256354 + 0.966583i \(0.582521\pi\)
\(308\) −398.894 −0.0737957
\(309\) 0 0
\(310\) 0 0
\(311\) −5817.25 −1.06066 −0.530331 0.847791i \(-0.677932\pi\)
−0.530331 + 0.847791i \(0.677932\pi\)
\(312\) 0 0
\(313\) −2232.83 −0.403218 −0.201609 0.979466i \(-0.564617\pi\)
−0.201609 + 0.979466i \(0.564617\pi\)
\(314\) 2705.16 0.486181
\(315\) 0 0
\(316\) 2092.20 0.372453
\(317\) 81.4013 0.0144226 0.00721128 0.999974i \(-0.497705\pi\)
0.00721128 + 0.999974i \(0.497705\pi\)
\(318\) 0 0
\(319\) 1077.42 0.189104
\(320\) 0 0
\(321\) 0 0
\(322\) 132.682 0.0229629
\(323\) −7492.54 −1.29070
\(324\) 0 0
\(325\) 0 0
\(326\) 5158.21 0.876341
\(327\) 0 0
\(328\) 4133.78 0.695884
\(329\) −2231.93 −0.374012
\(330\) 0 0
\(331\) −6550.18 −1.08771 −0.543853 0.839181i \(-0.683035\pi\)
−0.543853 + 0.839181i \(0.683035\pi\)
\(332\) −4184.88 −0.691792
\(333\) 0 0
\(334\) −939.963 −0.153990
\(335\) 0 0
\(336\) 0 0
\(337\) −4124.63 −0.666715 −0.333358 0.942800i \(-0.608182\pi\)
−0.333358 + 0.942800i \(0.608182\pi\)
\(338\) 1587.77 0.255513
\(339\) 0 0
\(340\) 0 0
\(341\) −1456.07 −0.231233
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 6760.10 1.05954
\(345\) 0 0
\(346\) −5842.03 −0.907715
\(347\) 1228.44 0.190047 0.0950233 0.995475i \(-0.469707\pi\)
0.0950233 + 0.995475i \(0.469707\pi\)
\(348\) 0 0
\(349\) 3320.93 0.509357 0.254678 0.967026i \(-0.418030\pi\)
0.254678 + 0.967026i \(0.418030\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1918.65 −0.290524
\(353\) −4589.25 −0.691958 −0.345979 0.938242i \(-0.612453\pi\)
−0.345979 + 0.938242i \(0.612453\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −5257.33 −0.782691
\(357\) 0 0
\(358\) 1512.13 0.223237
\(359\) −12381.6 −1.82026 −0.910130 0.414323i \(-0.864018\pi\)
−0.910130 + 0.414323i \(0.864018\pi\)
\(360\) 0 0
\(361\) 1369.00 0.199592
\(362\) 2960.15 0.429785
\(363\) 0 0
\(364\) 1337.44 0.192585
\(365\) 0 0
\(366\) 0 0
\(367\) −5612.33 −0.798259 −0.399130 0.916894i \(-0.630688\pi\)
−0.399130 + 0.916894i \(0.630688\pi\)
\(368\) −138.659 −0.0196416
\(369\) 0 0
\(370\) 0 0
\(371\) −2077.59 −0.290737
\(372\) 0 0
\(373\) 3691.05 0.512374 0.256187 0.966627i \(-0.417534\pi\)
0.256187 + 0.966627i \(0.417534\pi\)
\(374\) 1321.61 0.182724
\(375\) 0 0
\(376\) −6752.24 −0.926118
\(377\) −3612.46 −0.493504
\(378\) 0 0
\(379\) 5350.16 0.725117 0.362559 0.931961i \(-0.381903\pi\)
0.362559 + 0.931961i \(0.381903\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4262.85 −0.570959
\(383\) 7300.09 0.973936 0.486968 0.873420i \(-0.338103\pi\)
0.486968 + 0.873420i \(0.338103\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2280.56 0.300719
\(387\) 0 0
\(388\) −1004.80 −0.131471
\(389\) 9102.27 1.18638 0.593192 0.805061i \(-0.297867\pi\)
0.593192 + 0.805061i \(0.297867\pi\)
\(390\) 0 0
\(391\) 1002.62 0.129680
\(392\) 1037.68 0.133701
\(393\) 0 0
\(394\) 7544.51 0.964688
\(395\) 0 0
\(396\) 0 0
\(397\) 11520.1 1.45637 0.728185 0.685381i \(-0.240364\pi\)
0.728185 + 0.685381i \(0.240364\pi\)
\(398\) 1363.34 0.171704
\(399\) 0 0
\(400\) 0 0
\(401\) −3812.93 −0.474834 −0.237417 0.971408i \(-0.576301\pi\)
−0.237417 + 0.971408i \(0.576301\pi\)
\(402\) 0 0
\(403\) 4882.00 0.603448
\(404\) −8136.85 −1.00204
\(405\) 0 0
\(406\) −1149.42 −0.140504
\(407\) −664.432 −0.0809205
\(408\) 0 0
\(409\) −4098.82 −0.495535 −0.247767 0.968820i \(-0.579697\pi\)
−0.247767 + 0.968820i \(0.579697\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2975.70 0.355831
\(413\) 1988.00 0.236860
\(414\) 0 0
\(415\) 0 0
\(416\) 6432.98 0.758180
\(417\) 0 0
\(418\) −1451.33 −0.169825
\(419\) −14312.4 −1.66876 −0.834378 0.551193i \(-0.814173\pi\)
−0.834378 + 0.551193i \(0.814173\pi\)
\(420\) 0 0
\(421\) 6679.57 0.773259 0.386630 0.922235i \(-0.373639\pi\)
0.386630 + 0.922235i \(0.373639\pi\)
\(422\) 8038.91 0.927317
\(423\) 0 0
\(424\) −6285.34 −0.719913
\(425\) 0 0
\(426\) 0 0
\(427\) −3461.78 −0.392335
\(428\) −1819.48 −0.205486
\(429\) 0 0
\(430\) 0 0
\(431\) −11503.8 −1.28566 −0.642828 0.766010i \(-0.722239\pi\)
−0.642828 + 0.766010i \(0.722239\pi\)
\(432\) 0 0
\(433\) −3243.99 −0.360037 −0.180019 0.983663i \(-0.557616\pi\)
−0.180019 + 0.983663i \(0.557616\pi\)
\(434\) 1553.36 0.171806
\(435\) 0 0
\(436\) −9019.19 −0.990691
\(437\) −1101.04 −0.120526
\(438\) 0 0
\(439\) −2744.21 −0.298346 −0.149173 0.988811i \(-0.547661\pi\)
−0.149173 + 0.988811i \(0.547661\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4431.17 −0.476853
\(443\) 8180.27 0.877328 0.438664 0.898651i \(-0.355452\pi\)
0.438664 + 0.898651i \(0.355452\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −3802.87 −0.403747
\(447\) 0 0
\(448\) 1407.15 0.148397
\(449\) −6725.70 −0.706916 −0.353458 0.935450i \(-0.614994\pi\)
−0.353458 + 0.935450i \(0.614994\pi\)
\(450\) 0 0
\(451\) 2000.07 0.208824
\(452\) −2506.68 −0.260850
\(453\) 0 0
\(454\) 5478.65 0.566357
\(455\) 0 0
\(456\) 0 0
\(457\) 10807.0 1.10619 0.553095 0.833118i \(-0.313447\pi\)
0.553095 + 0.833118i \(0.313447\pi\)
\(458\) −5355.41 −0.546380
\(459\) 0 0
\(460\) 0 0
\(461\) −17775.8 −1.79588 −0.897942 0.440113i \(-0.854938\pi\)
−0.897942 + 0.440113i \(0.854938\pi\)
\(462\) 0 0
\(463\) 4418.93 0.443554 0.221777 0.975097i \(-0.428814\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(464\) 1201.20 0.120182
\(465\) 0 0
\(466\) 6230.15 0.619326
\(467\) 3373.08 0.334235 0.167117 0.985937i \(-0.446554\pi\)
0.167117 + 0.985937i \(0.446554\pi\)
\(468\) 0 0
\(469\) −3843.23 −0.378387
\(470\) 0 0
\(471\) 0 0
\(472\) 6014.29 0.586505
\(473\) 3270.77 0.317950
\(474\) 0 0
\(475\) 0 0
\(476\) 3215.70 0.309646
\(477\) 0 0
\(478\) −9267.36 −0.886776
\(479\) −4620.02 −0.440697 −0.220349 0.975421i \(-0.570720\pi\)
−0.220349 + 0.975421i \(0.570720\pi\)
\(480\) 0 0
\(481\) 2227.75 0.211178
\(482\) 10121.8 0.956508
\(483\) 0 0
\(484\) 6818.55 0.640360
\(485\) 0 0
\(486\) 0 0
\(487\) 9108.62 0.847538 0.423769 0.905770i \(-0.360707\pi\)
0.423769 + 0.905770i \(0.360707\pi\)
\(488\) −10472.9 −0.971488
\(489\) 0 0
\(490\) 0 0
\(491\) 2696.84 0.247875 0.123938 0.992290i \(-0.460448\pi\)
0.123938 + 0.992290i \(0.460448\pi\)
\(492\) 0 0
\(493\) −8685.71 −0.793478
\(494\) 4866.12 0.443193
\(495\) 0 0
\(496\) −1623.34 −0.146956
\(497\) 5186.80 0.468128
\(498\) 0 0
\(499\) −20054.2 −1.79910 −0.899549 0.436820i \(-0.856104\pi\)
−0.899549 + 0.436820i \(0.856104\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 8210.17 0.729956
\(503\) 8106.49 0.718590 0.359295 0.933224i \(-0.383017\pi\)
0.359295 + 0.933224i \(0.383017\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 194.212 0.0170628
\(507\) 0 0
\(508\) 9510.80 0.830657
\(509\) −12455.8 −1.08466 −0.542332 0.840164i \(-0.682458\pi\)
−0.542332 + 0.840164i \(0.682458\pi\)
\(510\) 0 0
\(511\) 3897.95 0.337447
\(512\) −4074.36 −0.351686
\(513\) 0 0
\(514\) −6415.68 −0.550551
\(515\) 0 0
\(516\) 0 0
\(517\) −3266.97 −0.277913
\(518\) 708.830 0.0601239
\(519\) 0 0
\(520\) 0 0
\(521\) 5539.61 0.465825 0.232913 0.972498i \(-0.425174\pi\)
0.232913 + 0.972498i \(0.425174\pi\)
\(522\) 0 0
\(523\) −15504.2 −1.29628 −0.648138 0.761523i \(-0.724452\pi\)
−0.648138 + 0.761523i \(0.724452\pi\)
\(524\) 8812.24 0.734665
\(525\) 0 0
\(526\) 9739.23 0.807321
\(527\) 11738.2 0.970252
\(528\) 0 0
\(529\) −12019.7 −0.987890
\(530\) 0 0
\(531\) 0 0
\(532\) −3531.35 −0.287789
\(533\) −6705.96 −0.544967
\(534\) 0 0
\(535\) 0 0
\(536\) −11626.9 −0.936951
\(537\) 0 0
\(538\) −6803.17 −0.545178
\(539\) 502.064 0.0401214
\(540\) 0 0
\(541\) 7853.16 0.624092 0.312046 0.950067i \(-0.398986\pi\)
0.312046 + 0.950067i \(0.398986\pi\)
\(542\) 98.5293 0.00780848
\(543\) 0 0
\(544\) 15467.3 1.21904
\(545\) 0 0
\(546\) 0 0
\(547\) 9177.75 0.717390 0.358695 0.933455i \(-0.383222\pi\)
0.358695 + 0.933455i \(0.383222\pi\)
\(548\) 11652.8 0.908363
\(549\) 0 0
\(550\) 0 0
\(551\) 9538.29 0.737468
\(552\) 0 0
\(553\) −2633.33 −0.202496
\(554\) 9064.25 0.695132
\(555\) 0 0
\(556\) 7702.73 0.587533
\(557\) −25030.2 −1.90407 −0.952033 0.305996i \(-0.901010\pi\)
−0.952033 + 0.305996i \(0.901010\pi\)
\(558\) 0 0
\(559\) −10966.5 −0.829753
\(560\) 0 0
\(561\) 0 0
\(562\) −13165.3 −0.988157
\(563\) 19486.1 1.45869 0.729345 0.684146i \(-0.239825\pi\)
0.729345 + 0.684146i \(0.239825\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 11580.0 0.859975
\(567\) 0 0
\(568\) 15691.6 1.15916
\(569\) 5813.63 0.428331 0.214165 0.976797i \(-0.431297\pi\)
0.214165 + 0.976797i \(0.431297\pi\)
\(570\) 0 0
\(571\) 268.770 0.0196982 0.00984910 0.999951i \(-0.496865\pi\)
0.00984910 + 0.999951i \(0.496865\pi\)
\(572\) 1957.67 0.143102
\(573\) 0 0
\(574\) −2133.71 −0.155156
\(575\) 0 0
\(576\) 0 0
\(577\) −8283.57 −0.597660 −0.298830 0.954306i \(-0.596596\pi\)
−0.298830 + 0.954306i \(0.596596\pi\)
\(578\) −2982.29 −0.214614
\(579\) 0 0
\(580\) 0 0
\(581\) 5267.26 0.376115
\(582\) 0 0
\(583\) −3041.07 −0.216035
\(584\) 11792.5 0.835575
\(585\) 0 0
\(586\) 752.703 0.0530612
\(587\) −6671.87 −0.469127 −0.234563 0.972101i \(-0.575366\pi\)
−0.234563 + 0.972101i \(0.575366\pi\)
\(588\) 0 0
\(589\) −12890.4 −0.901763
\(590\) 0 0
\(591\) 0 0
\(592\) −740.761 −0.0514275
\(593\) −11409.0 −0.790071 −0.395035 0.918666i \(-0.629268\pi\)
−0.395035 + 0.918666i \(0.629268\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 13919.9 0.956683
\(597\) 0 0
\(598\) −651.167 −0.0445288
\(599\) 13118.3 0.894826 0.447413 0.894327i \(-0.352345\pi\)
0.447413 + 0.894327i \(0.352345\pi\)
\(600\) 0 0
\(601\) 21970.2 1.49115 0.745577 0.666420i \(-0.232174\pi\)
0.745577 + 0.666420i \(0.232174\pi\)
\(602\) −3489.33 −0.236237
\(603\) 0 0
\(604\) −2197.99 −0.148071
\(605\) 0 0
\(606\) 0 0
\(607\) 21115.1 1.41192 0.705959 0.708253i \(-0.250516\pi\)
0.705959 + 0.708253i \(0.250516\pi\)
\(608\) −16985.6 −1.13299
\(609\) 0 0
\(610\) 0 0
\(611\) 10953.7 0.725269
\(612\) 0 0
\(613\) 1963.61 0.129379 0.0646895 0.997905i \(-0.479394\pi\)
0.0646895 + 0.997905i \(0.479394\pi\)
\(614\) 4306.60 0.283062
\(615\) 0 0
\(616\) 1518.89 0.0993474
\(617\) 17352.1 1.13221 0.566103 0.824335i \(-0.308450\pi\)
0.566103 + 0.824335i \(0.308450\pi\)
\(618\) 0 0
\(619\) −3614.68 −0.234711 −0.117356 0.993090i \(-0.537442\pi\)
−0.117356 + 0.993090i \(0.537442\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 9083.94 0.585583
\(623\) 6617.09 0.425535
\(624\) 0 0
\(625\) 0 0
\(626\) 3486.68 0.222613
\(627\) 0 0
\(628\) 9634.56 0.612199
\(629\) 5356.35 0.339542
\(630\) 0 0
\(631\) 16368.8 1.03269 0.516347 0.856379i \(-0.327291\pi\)
0.516347 + 0.856379i \(0.327291\pi\)
\(632\) −7966.59 −0.501415
\(633\) 0 0
\(634\) −127.112 −0.00796258
\(635\) 0 0
\(636\) 0 0
\(637\) −1683.35 −0.104705
\(638\) −1682.45 −0.104403
\(639\) 0 0
\(640\) 0 0
\(641\) 849.649 0.0523543 0.0261772 0.999657i \(-0.491667\pi\)
0.0261772 + 0.999657i \(0.491667\pi\)
\(642\) 0 0
\(643\) 1335.12 0.0818850 0.0409425 0.999162i \(-0.486964\pi\)
0.0409425 + 0.999162i \(0.486964\pi\)
\(644\) 472.553 0.0289149
\(645\) 0 0
\(646\) 11700.0 0.712586
\(647\) −5944.39 −0.361202 −0.180601 0.983556i \(-0.557804\pi\)
−0.180601 + 0.983556i \(0.557804\pi\)
\(648\) 0 0
\(649\) 2909.92 0.176001
\(650\) 0 0
\(651\) 0 0
\(652\) 18371.3 1.10349
\(653\) −5561.29 −0.333277 −0.166639 0.986018i \(-0.553291\pi\)
−0.166639 + 0.986018i \(0.553291\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2229.84 0.132714
\(657\) 0 0
\(658\) 3485.27 0.206489
\(659\) −27399.5 −1.61963 −0.809813 0.586688i \(-0.800431\pi\)
−0.809813 + 0.586688i \(0.800431\pi\)
\(660\) 0 0
\(661\) 3931.52 0.231344 0.115672 0.993287i \(-0.463098\pi\)
0.115672 + 0.993287i \(0.463098\pi\)
\(662\) 10228.5 0.600514
\(663\) 0 0
\(664\) 15935.0 0.931324
\(665\) 0 0
\(666\) 0 0
\(667\) −1276.38 −0.0740954
\(668\) −3347.73 −0.193903
\(669\) 0 0
\(670\) 0 0
\(671\) −5067.16 −0.291528
\(672\) 0 0
\(673\) −11427.3 −0.654515 −0.327257 0.944935i \(-0.606124\pi\)
−0.327257 + 0.944935i \(0.606124\pi\)
\(674\) 6440.83 0.368088
\(675\) 0 0
\(676\) 5654.94 0.321742
\(677\) −16259.3 −0.923035 −0.461518 0.887131i \(-0.652695\pi\)
−0.461518 + 0.887131i \(0.652695\pi\)
\(678\) 0 0
\(679\) 1264.68 0.0714785
\(680\) 0 0
\(681\) 0 0
\(682\) 2273.73 0.127662
\(683\) 10873.9 0.609192 0.304596 0.952482i \(-0.401479\pi\)
0.304596 + 0.952482i \(0.401479\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −535.613 −0.0298102
\(687\) 0 0
\(688\) 3646.52 0.202067
\(689\) 10196.3 0.563785
\(690\) 0 0
\(691\) 21613.9 1.18991 0.594957 0.803758i \(-0.297169\pi\)
0.594957 + 0.803758i \(0.297169\pi\)
\(692\) −20806.7 −1.14299
\(693\) 0 0
\(694\) −1918.28 −0.104923
\(695\) 0 0
\(696\) 0 0
\(697\) −16123.7 −0.876222
\(698\) −5185.81 −0.281212
\(699\) 0 0
\(700\) 0 0
\(701\) 20667.7 1.11357 0.556783 0.830658i \(-0.312036\pi\)
0.556783 + 0.830658i \(0.312036\pi\)
\(702\) 0 0
\(703\) −5882.12 −0.315574
\(704\) 2059.71 0.110267
\(705\) 0 0
\(706\) 7166.36 0.382025
\(707\) 10241.4 0.544790
\(708\) 0 0
\(709\) −22813.3 −1.20842 −0.604212 0.796824i \(-0.706512\pi\)
−0.604212 + 0.796824i \(0.706512\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 20018.7 1.05370
\(713\) 1724.94 0.0906026
\(714\) 0 0
\(715\) 0 0
\(716\) 5385.54 0.281099
\(717\) 0 0
\(718\) 19334.4 1.00495
\(719\) −9551.14 −0.495407 −0.247703 0.968836i \(-0.579676\pi\)
−0.247703 + 0.968836i \(0.579676\pi\)
\(720\) 0 0
\(721\) −3745.34 −0.193459
\(722\) −2137.77 −0.110193
\(723\) 0 0
\(724\) 10542.7 0.541185
\(725\) 0 0
\(726\) 0 0
\(727\) 30003.0 1.53060 0.765302 0.643672i \(-0.222590\pi\)
0.765302 + 0.643672i \(0.222590\pi\)
\(728\) −5092.65 −0.259267
\(729\) 0 0
\(730\) 0 0
\(731\) −26367.5 −1.33411
\(732\) 0 0
\(733\) −16863.7 −0.849762 −0.424881 0.905249i \(-0.639684\pi\)
−0.424881 + 0.905249i \(0.639684\pi\)
\(734\) 8763.95 0.440713
\(735\) 0 0
\(736\) 2272.95 0.113834
\(737\) −5625.50 −0.281164
\(738\) 0 0
\(739\) −30778.9 −1.53210 −0.766048 0.642783i \(-0.777780\pi\)
−0.766048 + 0.642783i \(0.777780\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 3244.27 0.160514
\(743\) 28510.9 1.40776 0.703879 0.710320i \(-0.251450\pi\)
0.703879 + 0.710320i \(0.251450\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −5763.77 −0.282878
\(747\) 0 0
\(748\) 4706.97 0.230085
\(749\) 2290.07 0.111719
\(750\) 0 0
\(751\) 12480.0 0.606395 0.303197 0.952928i \(-0.401946\pi\)
0.303197 + 0.952928i \(0.401946\pi\)
\(752\) −3642.28 −0.176623
\(753\) 0 0
\(754\) 5641.04 0.272460
\(755\) 0 0
\(756\) 0 0
\(757\) 11272.1 0.541203 0.270601 0.962691i \(-0.412777\pi\)
0.270601 + 0.962691i \(0.412777\pi\)
\(758\) −8354.56 −0.400332
\(759\) 0 0
\(760\) 0 0
\(761\) 28775.4 1.37071 0.685354 0.728210i \(-0.259647\pi\)
0.685354 + 0.728210i \(0.259647\pi\)
\(762\) 0 0
\(763\) 11351.9 0.538621
\(764\) −15182.4 −0.718951
\(765\) 0 0
\(766\) −11399.5 −0.537702
\(767\) −9756.58 −0.459309
\(768\) 0 0
\(769\) −17298.3 −0.811175 −0.405587 0.914056i \(-0.632933\pi\)
−0.405587 + 0.914056i \(0.632933\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8122.33 0.378665
\(773\) 5348.18 0.248849 0.124425 0.992229i \(-0.460291\pi\)
0.124425 + 0.992229i \(0.460291\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 3826.03 0.176993
\(777\) 0 0
\(778\) −14213.7 −0.654994
\(779\) 17706.3 0.814371
\(780\) 0 0
\(781\) 7592.15 0.347847
\(782\) −1565.65 −0.0715954
\(783\) 0 0
\(784\) 559.741 0.0254984
\(785\) 0 0
\(786\) 0 0
\(787\) 21615.1 0.979030 0.489515 0.871995i \(-0.337174\pi\)
0.489515 + 0.871995i \(0.337174\pi\)
\(788\) 26870.2 1.21473
\(789\) 0 0
\(790\) 0 0
\(791\) 3155.01 0.141820
\(792\) 0 0
\(793\) 16989.5 0.760800
\(794\) −17989.3 −0.804050
\(795\) 0 0
\(796\) 4855.62 0.216210
\(797\) −38266.0 −1.70069 −0.850345 0.526225i \(-0.823607\pi\)
−0.850345 + 0.526225i \(0.823607\pi\)
\(798\) 0 0
\(799\) 26336.8 1.16612
\(800\) 0 0
\(801\) 0 0
\(802\) 5954.09 0.262152
\(803\) 5705.61 0.250743
\(804\) 0 0
\(805\) 0 0
\(806\) −7623.50 −0.333159
\(807\) 0 0
\(808\) 30983.2 1.34899
\(809\) −4724.05 −0.205301 −0.102651 0.994717i \(-0.532732\pi\)
−0.102651 + 0.994717i \(0.532732\pi\)
\(810\) 0 0
\(811\) 9687.39 0.419446 0.209723 0.977761i \(-0.432744\pi\)
0.209723 + 0.977761i \(0.432744\pi\)
\(812\) −4093.71 −0.176923
\(813\) 0 0
\(814\) 1037.55 0.0446756
\(815\) 0 0
\(816\) 0 0
\(817\) 28955.7 1.23994
\(818\) 6400.53 0.273581
\(819\) 0 0
\(820\) 0 0
\(821\) 32250.8 1.37096 0.685482 0.728090i \(-0.259592\pi\)
0.685482 + 0.728090i \(0.259592\pi\)
\(822\) 0 0
\(823\) −5858.93 −0.248153 −0.124076 0.992273i \(-0.539597\pi\)
−0.124076 + 0.992273i \(0.539597\pi\)
\(824\) −11330.8 −0.479037
\(825\) 0 0
\(826\) −3104.37 −0.130768
\(827\) 16375.7 0.688560 0.344280 0.938867i \(-0.388123\pi\)
0.344280 + 0.938867i \(0.388123\pi\)
\(828\) 0 0
\(829\) −15152.9 −0.634841 −0.317421 0.948285i \(-0.602817\pi\)
−0.317421 + 0.948285i \(0.602817\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −6905.94 −0.287765
\(833\) −4047.42 −0.168349
\(834\) 0 0
\(835\) 0 0
\(836\) −5169.00 −0.213844
\(837\) 0 0
\(838\) 22349.6 0.921307
\(839\) 8829.87 0.363338 0.181669 0.983360i \(-0.441850\pi\)
0.181669 + 0.983360i \(0.441850\pi\)
\(840\) 0 0
\(841\) −13331.8 −0.546630
\(842\) −10430.5 −0.426911
\(843\) 0 0
\(844\) 28631.0 1.16768
\(845\) 0 0
\(846\) 0 0
\(847\) −8582.11 −0.348152
\(848\) −3390.42 −0.137297
\(849\) 0 0
\(850\) 0 0
\(851\) 787.125 0.0317066
\(852\) 0 0
\(853\) 4227.64 0.169697 0.0848486 0.996394i \(-0.472959\pi\)
0.0848486 + 0.996394i \(0.472959\pi\)
\(854\) 5405.75 0.216605
\(855\) 0 0
\(856\) 6928.15 0.276635
\(857\) 32233.1 1.28479 0.642394 0.766375i \(-0.277941\pi\)
0.642394 + 0.766375i \(0.277941\pi\)
\(858\) 0 0
\(859\) 37392.3 1.48523 0.742613 0.669720i \(-0.233586\pi\)
0.742613 + 0.669720i \(0.233586\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 17963.8 0.709801
\(863\) 8372.73 0.330256 0.165128 0.986272i \(-0.447196\pi\)
0.165128 + 0.986272i \(0.447196\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 5065.66 0.198774
\(867\) 0 0
\(868\) 5532.39 0.216338
\(869\) −3854.52 −0.150467
\(870\) 0 0
\(871\) 18861.5 0.733753
\(872\) 34343.0 1.33372
\(873\) 0 0
\(874\) 1719.33 0.0665416
\(875\) 0 0
\(876\) 0 0
\(877\) −38583.0 −1.48558 −0.742790 0.669524i \(-0.766498\pi\)
−0.742790 + 0.669524i \(0.766498\pi\)
\(878\) 4285.23 0.164715
\(879\) 0 0
\(880\) 0 0
\(881\) −41611.7 −1.59130 −0.795650 0.605757i \(-0.792870\pi\)
−0.795650 + 0.605757i \(0.792870\pi\)
\(882\) 0 0
\(883\) −43408.1 −1.65436 −0.827180 0.561937i \(-0.810056\pi\)
−0.827180 + 0.561937i \(0.810056\pi\)
\(884\) −15781.8 −0.600453
\(885\) 0 0
\(886\) −12773.9 −0.484366
\(887\) 19523.6 0.739050 0.369525 0.929221i \(-0.379520\pi\)
0.369525 + 0.929221i \(0.379520\pi\)
\(888\) 0 0
\(889\) −11970.7 −0.451613
\(890\) 0 0
\(891\) 0 0
\(892\) −13544.1 −0.508398
\(893\) −28922.0 −1.08381
\(894\) 0 0
\(895\) 0 0
\(896\) 8288.92 0.309055
\(897\) 0 0
\(898\) 10502.5 0.390283
\(899\) −14943.1 −0.554373
\(900\) 0 0
\(901\) 24515.7 0.906479
\(902\) −3123.21 −0.115290
\(903\) 0 0
\(904\) 9544.84 0.351169
\(905\) 0 0
\(906\) 0 0
\(907\) −44114.7 −1.61500 −0.807499 0.589869i \(-0.799180\pi\)
−0.807499 + 0.589869i \(0.799180\pi\)
\(908\) 19512.5 0.713156
\(909\) 0 0
\(910\) 0 0
\(911\) −49231.9 −1.79048 −0.895240 0.445585i \(-0.852996\pi\)
−0.895240 + 0.445585i \(0.852996\pi\)
\(912\) 0 0
\(913\) 7709.92 0.279476
\(914\) −16875.7 −0.610719
\(915\) 0 0
\(916\) −19073.6 −0.688001
\(917\) −11091.4 −0.399424
\(918\) 0 0
\(919\) −46993.6 −1.68681 −0.843403 0.537282i \(-0.819451\pi\)
−0.843403 + 0.537282i \(0.819451\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 27757.9 0.991494
\(923\) −25455.5 −0.907775
\(924\) 0 0
\(925\) 0 0
\(926\) −6900.40 −0.244883
\(927\) 0 0
\(928\) −19690.5 −0.696521
\(929\) 33023.5 1.16627 0.583135 0.812375i \(-0.301826\pi\)
0.583135 + 0.812375i \(0.301826\pi\)
\(930\) 0 0
\(931\) 4444.71 0.156466
\(932\) 22189.0 0.779855
\(933\) 0 0
\(934\) −5267.25 −0.184529
\(935\) 0 0
\(936\) 0 0
\(937\) 23431.0 0.816924 0.408462 0.912775i \(-0.366065\pi\)
0.408462 + 0.912775i \(0.366065\pi\)
\(938\) 6001.40 0.208905
\(939\) 0 0
\(940\) 0 0
\(941\) −43688.0 −1.51348 −0.756742 0.653714i \(-0.773210\pi\)
−0.756742 + 0.653714i \(0.773210\pi\)
\(942\) 0 0
\(943\) −2369.40 −0.0818221
\(944\) 3244.21 0.111854
\(945\) 0 0
\(946\) −5107.48 −0.175538
\(947\) 34884.0 1.19702 0.598511 0.801115i \(-0.295759\pi\)
0.598511 + 0.801115i \(0.295759\pi\)
\(948\) 0 0
\(949\) −19130.1 −0.654363
\(950\) 0 0
\(951\) 0 0
\(952\) −12244.6 −0.416860
\(953\) 34486.4 1.17222 0.586108 0.810233i \(-0.300659\pi\)
0.586108 + 0.810233i \(0.300659\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −33006.2 −1.11663
\(957\) 0 0
\(958\) 7214.40 0.243306
\(959\) −14666.7 −0.493861
\(960\) 0 0
\(961\) −9596.33 −0.322122
\(962\) −3478.75 −0.116590
\(963\) 0 0
\(964\) 36049.4 1.20443
\(965\) 0 0
\(966\) 0 0
\(967\) 46341.6 1.54110 0.770550 0.637379i \(-0.219981\pi\)
0.770550 + 0.637379i \(0.219981\pi\)
\(968\) −25963.4 −0.862083
\(969\) 0 0
\(970\) 0 0
\(971\) 34551.4 1.14192 0.570962 0.820977i \(-0.306571\pi\)
0.570962 + 0.820977i \(0.306571\pi\)
\(972\) 0 0
\(973\) −9694.97 −0.319431
\(974\) −14223.6 −0.467919
\(975\) 0 0
\(976\) −5649.27 −0.185275
\(977\) 970.127 0.0317678 0.0158839 0.999874i \(-0.494944\pi\)
0.0158839 + 0.999874i \(0.494944\pi\)
\(978\) 0 0
\(979\) 9685.73 0.316198
\(980\) 0 0
\(981\) 0 0
\(982\) −4211.26 −0.136850
\(983\) 18342.2 0.595142 0.297571 0.954700i \(-0.403824\pi\)
0.297571 + 0.954700i \(0.403824\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 13563.2 0.438073
\(987\) 0 0
\(988\) 17331.0 0.558068
\(989\) −3874.75 −0.124580
\(990\) 0 0
\(991\) 48979.6 1.57002 0.785010 0.619484i \(-0.212658\pi\)
0.785010 + 0.619484i \(0.212658\pi\)
\(992\) 26610.4 0.851694
\(993\) 0 0
\(994\) −8099.46 −0.258450
\(995\) 0 0
\(996\) 0 0
\(997\) −31492.9 −1.00039 −0.500196 0.865912i \(-0.666739\pi\)
−0.500196 + 0.865912i \(0.666739\pi\)
\(998\) 31315.7 0.993268
\(999\) 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 1575.4.a.u.1.1 2
3.2 odd 2 1575.4.a.r.1.2 2
5.4 even 2 315.4.a.h.1.2 2
15.14 odd 2 315.4.a.j.1.1 yes 2
35.34 odd 2 2205.4.a.y.1.2 2
105.104 even 2 2205.4.a.ba.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
315.4.a.h.1.2 2 5.4 even 2
315.4.a.j.1.1 yes 2 15.14 odd 2
1575.4.a.r.1.2 2 3.2 odd 2
1575.4.a.u.1.1 2 1.1 even 1 trivial
2205.4.a.y.1.2 2 35.34 odd 2
2205.4.a.ba.1.1 2 105.104 even 2