Properties

Label 441.4.a.s.1.3
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.57516.1
Defining polynomial: \(x^{3} - x^{2} - 24 x + 6\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-4.55637\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.55637 q^{2} +12.7605 q^{4} -17.8732 q^{5} +21.6905 q^{8} +O(q^{10})\) \(q+4.55637 q^{2} +12.7605 q^{4} -17.8732 q^{5} +21.6905 q^{8} -81.4369 q^{10} +11.3942 q^{11} -13.0987 q^{13} -3.25412 q^{16} -53.2674 q^{17} -42.4223 q^{19} -228.071 q^{20} +51.9159 q^{22} -152.085 q^{23} +194.451 q^{25} -59.6823 q^{26} -186.493 q^{29} -157.874 q^{31} -188.351 q^{32} -242.706 q^{34} +3.74588 q^{37} -193.291 q^{38} -387.678 q^{40} +39.3230 q^{41} +429.439 q^{43} +145.395 q^{44} -692.957 q^{46} -21.1869 q^{47} +885.992 q^{50} -167.145 q^{52} -365.904 q^{53} -203.650 q^{55} -849.732 q^{58} +226.578 q^{59} +651.973 q^{61} -719.331 q^{62} -832.161 q^{64} +234.115 q^{65} +145.433 q^{67} -679.717 q^{68} +368.962 q^{71} +608.906 q^{73} +17.0676 q^{74} -541.328 q^{76} +910.237 q^{79} +58.1615 q^{80} +179.170 q^{82} +327.929 q^{83} +952.058 q^{85} +1956.68 q^{86} +247.144 q^{88} +37.6118 q^{89} -1940.68 q^{92} -96.5352 q^{94} +758.222 q^{95} +722.013 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 25q^{4} - 11q^{5} - 39q^{8} + O(q^{10}) \) \( 3q - q^{2} + 25q^{4} - 11q^{5} - 39q^{8} - 55q^{10} - 35q^{11} + 62q^{13} + 241q^{16} - 48q^{17} - 202q^{19} - 439q^{20} - 7q^{22} - 216q^{23} + 130q^{25} - 274q^{26} - 53q^{29} - 95q^{31} - 683q^{32} - 24q^{34} + 262q^{37} + 398q^{38} + 21q^{40} - 244q^{41} + 360q^{43} + 905q^{44} - 1056q^{46} + 210q^{47} + 1378q^{50} + 324q^{52} - 393q^{53} - 1031q^{55} - 1249q^{58} - 1143q^{59} - 70q^{61} - 1059q^{62} - 399q^{64} + 472q^{65} - 628q^{67} - 1944q^{68} - 318q^{71} + 988q^{73} - 1002q^{74} - 2340q^{76} + 861q^{79} - 175q^{80} + 124q^{82} - 519q^{83} + 1800q^{85} + 3208q^{86} - 891q^{88} - 1766q^{89} + 672q^{92} - 3294q^{94} + 736q^{95} + 19q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.55637 1.61092 0.805459 0.592651i \(-0.201919\pi\)
0.805459 + 0.592651i \(0.201919\pi\)
\(3\) 0 0
\(4\) 12.7605 1.59506
\(5\) −17.8732 −1.59863 −0.799314 0.600914i \(-0.794804\pi\)
−0.799314 + 0.600914i \(0.794804\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 21.6905 0.958592
\(9\) 0 0
\(10\) −81.4369 −2.57526
\(11\) 11.3942 0.312315 0.156158 0.987732i \(-0.450089\pi\)
0.156158 + 0.987732i \(0.450089\pi\)
\(12\) 0 0
\(13\) −13.0987 −0.279455 −0.139728 0.990190i \(-0.544623\pi\)
−0.139728 + 0.990190i \(0.544623\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.25412 −0.0508456
\(17\) −53.2674 −0.759955 −0.379977 0.924996i \(-0.624068\pi\)
−0.379977 + 0.924996i \(0.624068\pi\)
\(18\) 0 0
\(19\) −42.4223 −0.512228 −0.256114 0.966647i \(-0.582442\pi\)
−0.256114 + 0.966647i \(0.582442\pi\)
\(20\) −228.071 −2.54991
\(21\) 0 0
\(22\) 51.9159 0.503114
\(23\) −152.085 −1.37878 −0.689391 0.724389i \(-0.742122\pi\)
−0.689391 + 0.724389i \(0.742122\pi\)
\(24\) 0 0
\(25\) 194.451 1.55561
\(26\) −59.6823 −0.450180
\(27\) 0 0
\(28\) 0 0
\(29\) −186.493 −1.19417 −0.597085 0.802178i \(-0.703675\pi\)
−0.597085 + 0.802178i \(0.703675\pi\)
\(30\) 0 0
\(31\) −157.874 −0.914676 −0.457338 0.889293i \(-0.651197\pi\)
−0.457338 + 0.889293i \(0.651197\pi\)
\(32\) −188.351 −1.04050
\(33\) 0 0
\(34\) −242.706 −1.22423
\(35\) 0 0
\(36\) 0 0
\(37\) 3.74588 0.0166438 0.00832188 0.999965i \(-0.497351\pi\)
0.00832188 + 0.999965i \(0.497351\pi\)
\(38\) −193.291 −0.825158
\(39\) 0 0
\(40\) −387.678 −1.53243
\(41\) 39.3230 0.149786 0.0748930 0.997192i \(-0.476138\pi\)
0.0748930 + 0.997192i \(0.476138\pi\)
\(42\) 0 0
\(43\) 429.439 1.52300 0.761498 0.648168i \(-0.224464\pi\)
0.761498 + 0.648168i \(0.224464\pi\)
\(44\) 145.395 0.498161
\(45\) 0 0
\(46\) −692.957 −2.22111
\(47\) −21.1869 −0.0657537 −0.0328768 0.999459i \(-0.510467\pi\)
−0.0328768 + 0.999459i \(0.510467\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 885.992 2.50596
\(51\) 0 0
\(52\) −167.145 −0.445748
\(53\) −365.904 −0.948317 −0.474158 0.880440i \(-0.657248\pi\)
−0.474158 + 0.880440i \(0.657248\pi\)
\(54\) 0 0
\(55\) −203.650 −0.499276
\(56\) 0 0
\(57\) 0 0
\(58\) −849.732 −1.92371
\(59\) 226.578 0.499964 0.249982 0.968250i \(-0.419575\pi\)
0.249982 + 0.968250i \(0.419575\pi\)
\(60\) 0 0
\(61\) 651.973 1.36847 0.684235 0.729262i \(-0.260136\pi\)
0.684235 + 0.729262i \(0.260136\pi\)
\(62\) −719.331 −1.47347
\(63\) 0 0
\(64\) −832.161 −1.62532
\(65\) 234.115 0.446745
\(66\) 0 0
\(67\) 145.433 0.265186 0.132593 0.991171i \(-0.457670\pi\)
0.132593 + 0.991171i \(0.457670\pi\)
\(68\) −679.717 −1.21217
\(69\) 0 0
\(70\) 0 0
\(71\) 368.962 0.616728 0.308364 0.951268i \(-0.400218\pi\)
0.308364 + 0.951268i \(0.400218\pi\)
\(72\) 0 0
\(73\) 608.906 0.976261 0.488130 0.872771i \(-0.337679\pi\)
0.488130 + 0.872771i \(0.337679\pi\)
\(74\) 17.0676 0.0268117
\(75\) 0 0
\(76\) −541.328 −0.817034
\(77\) 0 0
\(78\) 0 0
\(79\) 910.237 1.29633 0.648163 0.761502i \(-0.275538\pi\)
0.648163 + 0.761502i \(0.275538\pi\)
\(80\) 58.1615 0.0812832
\(81\) 0 0
\(82\) 179.170 0.241293
\(83\) 327.929 0.433674 0.216837 0.976208i \(-0.430426\pi\)
0.216837 + 0.976208i \(0.430426\pi\)
\(84\) 0 0
\(85\) 952.058 1.21489
\(86\) 1956.68 2.45342
\(87\) 0 0
\(88\) 247.144 0.299383
\(89\) 37.6118 0.0447960 0.0223980 0.999749i \(-0.492870\pi\)
0.0223980 + 0.999749i \(0.492870\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1940.68 −2.19924
\(93\) 0 0
\(94\) −96.5352 −0.105924
\(95\) 758.222 0.818863
\(96\) 0 0
\(97\) 722.013 0.755766 0.377883 0.925853i \(-0.376652\pi\)
0.377883 + 0.925853i \(0.376652\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 2481.29 2.48129
\(101\) −1518.67 −1.49617 −0.748087 0.663601i \(-0.769027\pi\)
−0.748087 + 0.663601i \(0.769027\pi\)
\(102\) 0 0
\(103\) −1051.88 −1.00626 −0.503132 0.864210i \(-0.667819\pi\)
−0.503132 + 0.864210i \(0.667819\pi\)
\(104\) −284.116 −0.267883
\(105\) 0 0
\(106\) −1667.19 −1.52766
\(107\) −766.520 −0.692545 −0.346273 0.938134i \(-0.612553\pi\)
−0.346273 + 0.938134i \(0.612553\pi\)
\(108\) 0 0
\(109\) −1427.05 −1.25400 −0.627002 0.779018i \(-0.715718\pi\)
−0.627002 + 0.779018i \(0.715718\pi\)
\(110\) −927.904 −0.804292
\(111\) 0 0
\(112\) 0 0
\(113\) −362.564 −0.301833 −0.150917 0.988546i \(-0.548222\pi\)
−0.150917 + 0.988546i \(0.548222\pi\)
\(114\) 0 0
\(115\) 2718.25 2.20416
\(116\) −2379.74 −1.90477
\(117\) 0 0
\(118\) 1032.37 0.805402
\(119\) 0 0
\(120\) 0 0
\(121\) −1201.17 −0.902459
\(122\) 2970.63 2.20449
\(123\) 0 0
\(124\) −2014.54 −1.45896
\(125\) −1241.32 −0.888216
\(126\) 0 0
\(127\) 974.777 0.681082 0.340541 0.940230i \(-0.389390\pi\)
0.340541 + 0.940230i \(0.389390\pi\)
\(128\) −2284.83 −1.57775
\(129\) 0 0
\(130\) 1066.71 0.719670
\(131\) 1792.70 1.19564 0.597821 0.801629i \(-0.296033\pi\)
0.597821 + 0.801629i \(0.296033\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 662.647 0.427194
\(135\) 0 0
\(136\) −1155.39 −0.728486
\(137\) −1684.42 −1.05043 −0.525217 0.850969i \(-0.676016\pi\)
−0.525217 + 0.850969i \(0.676016\pi\)
\(138\) 0 0
\(139\) 315.089 0.192270 0.0961350 0.995368i \(-0.469352\pi\)
0.0961350 + 0.995368i \(0.469352\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1681.13 0.993499
\(143\) −149.248 −0.0872781
\(144\) 0 0
\(145\) 3333.23 1.90903
\(146\) 2774.40 1.57268
\(147\) 0 0
\(148\) 47.7992 0.0265478
\(149\) −1893.77 −1.04123 −0.520617 0.853790i \(-0.674298\pi\)
−0.520617 + 0.853790i \(0.674298\pi\)
\(150\) 0 0
\(151\) −2011.84 −1.08425 −0.542124 0.840299i \(-0.682380\pi\)
−0.542124 + 0.840299i \(0.682380\pi\)
\(152\) −920.159 −0.491018
\(153\) 0 0
\(154\) 0 0
\(155\) 2821.71 1.46223
\(156\) 0 0
\(157\) −3828.50 −1.94616 −0.973082 0.230460i \(-0.925977\pi\)
−0.973082 + 0.230460i \(0.925977\pi\)
\(158\) 4147.38 2.08827
\(159\) 0 0
\(160\) 3366.43 1.66337
\(161\) 0 0
\(162\) 0 0
\(163\) −3509.26 −1.68630 −0.843148 0.537682i \(-0.819300\pi\)
−0.843148 + 0.537682i \(0.819300\pi\)
\(164\) 501.780 0.238917
\(165\) 0 0
\(166\) 1494.17 0.698613
\(167\) 343.008 0.158939 0.0794694 0.996837i \(-0.474677\pi\)
0.0794694 + 0.996837i \(0.474677\pi\)
\(168\) 0 0
\(169\) −2025.42 −0.921905
\(170\) 4337.93 1.95708
\(171\) 0 0
\(172\) 5479.84 2.42927
\(173\) 4187.21 1.84016 0.920081 0.391729i \(-0.128123\pi\)
0.920081 + 0.391729i \(0.128123\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −37.0779 −0.0158798
\(177\) 0 0
\(178\) 171.373 0.0721627
\(179\) 1970.29 0.822716 0.411358 0.911474i \(-0.365055\pi\)
0.411358 + 0.911474i \(0.365055\pi\)
\(180\) 0 0
\(181\) −3613.10 −1.48376 −0.741878 0.670535i \(-0.766065\pi\)
−0.741878 + 0.670535i \(0.766065\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −3298.80 −1.32169
\(185\) −66.9509 −0.0266072
\(186\) 0 0
\(187\) −606.936 −0.237345
\(188\) −270.355 −0.104881
\(189\) 0 0
\(190\) 3454.74 1.31912
\(191\) −1907.77 −0.722729 −0.361365 0.932425i \(-0.617689\pi\)
−0.361365 + 0.932425i \(0.617689\pi\)
\(192\) 0 0
\(193\) 2399.93 0.895080 0.447540 0.894264i \(-0.352300\pi\)
0.447540 + 0.894264i \(0.352300\pi\)
\(194\) 3289.75 1.21748
\(195\) 0 0
\(196\) 0 0
\(197\) −1514.32 −0.547668 −0.273834 0.961777i \(-0.588292\pi\)
−0.273834 + 0.961777i \(0.588292\pi\)
\(198\) 0 0
\(199\) 1367.78 0.487232 0.243616 0.969872i \(-0.421666\pi\)
0.243616 + 0.969872i \(0.421666\pi\)
\(200\) 4217.74 1.49120
\(201\) 0 0
\(202\) −6919.63 −2.41021
\(203\) 0 0
\(204\) 0 0
\(205\) −702.828 −0.239452
\(206\) −4792.76 −1.62101
\(207\) 0 0
\(208\) 42.6246 0.0142091
\(209\) −483.366 −0.159977
\(210\) 0 0
\(211\) 4302.52 1.40378 0.701891 0.712285i \(-0.252339\pi\)
0.701891 + 0.712285i \(0.252339\pi\)
\(212\) −4669.11 −1.51262
\(213\) 0 0
\(214\) −3492.55 −1.11563
\(215\) −7675.45 −2.43470
\(216\) 0 0
\(217\) 0 0
\(218\) −6502.16 −2.02010
\(219\) 0 0
\(220\) −2598.67 −0.796374
\(221\) 697.731 0.212373
\(222\) 0 0
\(223\) −1497.19 −0.449592 −0.224796 0.974406i \(-0.572172\pi\)
−0.224796 + 0.974406i \(0.572172\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1651.97 −0.486229
\(227\) −1603.32 −0.468795 −0.234397 0.972141i \(-0.575312\pi\)
−0.234397 + 0.972141i \(0.575312\pi\)
\(228\) 0 0
\(229\) 1010.52 0.291603 0.145802 0.989314i \(-0.453424\pi\)
0.145802 + 0.989314i \(0.453424\pi\)
\(230\) 12385.4 3.55072
\(231\) 0 0
\(232\) −4045.13 −1.14472
\(233\) −198.217 −0.0557323 −0.0278661 0.999612i \(-0.508871\pi\)
−0.0278661 + 0.999612i \(0.508871\pi\)
\(234\) 0 0
\(235\) 378.677 0.105116
\(236\) 2891.24 0.797472
\(237\) 0 0
\(238\) 0 0
\(239\) 1201.19 0.325098 0.162549 0.986700i \(-0.448028\pi\)
0.162549 + 0.986700i \(0.448028\pi\)
\(240\) 0 0
\(241\) −2732.69 −0.730407 −0.365204 0.930928i \(-0.619001\pi\)
−0.365204 + 0.930928i \(0.619001\pi\)
\(242\) −5472.99 −1.45379
\(243\) 0 0
\(244\) 8319.49 2.18279
\(245\) 0 0
\(246\) 0 0
\(247\) 555.675 0.143145
\(248\) −3424.35 −0.876801
\(249\) 0 0
\(250\) −5655.91 −1.43084
\(251\) −7565.82 −1.90259 −0.951295 0.308281i \(-0.900246\pi\)
−0.951295 + 0.308281i \(0.900246\pi\)
\(252\) 0 0
\(253\) −1732.88 −0.430615
\(254\) 4441.44 1.09717
\(255\) 0 0
\(256\) −3753.22 −0.916313
\(257\) −5008.68 −1.21569 −0.607846 0.794055i \(-0.707966\pi\)
−0.607846 + 0.794055i \(0.707966\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 2987.42 0.712584
\(261\) 0 0
\(262\) 8168.21 1.92608
\(263\) 6248.81 1.46509 0.732544 0.680720i \(-0.238333\pi\)
0.732544 + 0.680720i \(0.238333\pi\)
\(264\) 0 0
\(265\) 6539.88 1.51601
\(266\) 0 0
\(267\) 0 0
\(268\) 1855.80 0.422988
\(269\) 3588.45 0.813351 0.406676 0.913573i \(-0.366688\pi\)
0.406676 + 0.913573i \(0.366688\pi\)
\(270\) 0 0
\(271\) −1983.14 −0.444529 −0.222264 0.974986i \(-0.571345\pi\)
−0.222264 + 0.974986i \(0.571345\pi\)
\(272\) 173.338 0.0386404
\(273\) 0 0
\(274\) −7674.81 −1.69216
\(275\) 2215.61 0.485841
\(276\) 0 0
\(277\) 7363.91 1.59731 0.798654 0.601790i \(-0.205546\pi\)
0.798654 + 0.601790i \(0.205546\pi\)
\(278\) 1435.66 0.309731
\(279\) 0 0
\(280\) 0 0
\(281\) 5312.05 1.12772 0.563861 0.825869i \(-0.309315\pi\)
0.563861 + 0.825869i \(0.309315\pi\)
\(282\) 0 0
\(283\) −1091.76 −0.229324 −0.114662 0.993405i \(-0.536578\pi\)
−0.114662 + 0.993405i \(0.536578\pi\)
\(284\) 4708.13 0.983718
\(285\) 0 0
\(286\) −680.030 −0.140598
\(287\) 0 0
\(288\) 0 0
\(289\) −2075.59 −0.422469
\(290\) 15187.4 3.07530
\(291\) 0 0
\(292\) 7769.92 1.55719
\(293\) 7191.86 1.43397 0.716985 0.697089i \(-0.245522\pi\)
0.716985 + 0.697089i \(0.245522\pi\)
\(294\) 0 0
\(295\) −4049.67 −0.799257
\(296\) 81.2499 0.0159546
\(297\) 0 0
\(298\) −8628.73 −1.67734
\(299\) 1992.12 0.385308
\(300\) 0 0
\(301\) 0 0
\(302\) −9166.69 −1.74663
\(303\) 0 0
\(304\) 138.047 0.0260446
\(305\) −11652.9 −2.18767
\(306\) 0 0
\(307\) 541.355 0.100641 0.0503204 0.998733i \(-0.483976\pi\)
0.0503204 + 0.998733i \(0.483976\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 12856.7 2.35553
\(311\) −54.0168 −0.00984892 −0.00492446 0.999988i \(-0.501568\pi\)
−0.00492446 + 0.999988i \(0.501568\pi\)
\(312\) 0 0
\(313\) 3772.94 0.681340 0.340670 0.940183i \(-0.389346\pi\)
0.340670 + 0.940183i \(0.389346\pi\)
\(314\) −17444.1 −3.13511
\(315\) 0 0
\(316\) 11615.1 2.06772
\(317\) −1719.24 −0.304612 −0.152306 0.988333i \(-0.548670\pi\)
−0.152306 + 0.988333i \(0.548670\pi\)
\(318\) 0 0
\(319\) −2124.93 −0.372957
\(320\) 14873.4 2.59827
\(321\) 0 0
\(322\) 0 0
\(323\) 2259.72 0.389270
\(324\) 0 0
\(325\) −2547.06 −0.434724
\(326\) −15989.5 −2.71649
\(327\) 0 0
\(328\) 852.934 0.143584
\(329\) 0 0
\(330\) 0 0
\(331\) 8408.21 1.39625 0.698123 0.715978i \(-0.254019\pi\)
0.698123 + 0.715978i \(0.254019\pi\)
\(332\) 4184.53 0.691735
\(333\) 0 0
\(334\) 1562.87 0.256037
\(335\) −2599.36 −0.423935
\(336\) 0 0
\(337\) 2789.46 0.450894 0.225447 0.974255i \(-0.427616\pi\)
0.225447 + 0.974255i \(0.427616\pi\)
\(338\) −9228.58 −1.48511
\(339\) 0 0
\(340\) 12148.7 1.93781
\(341\) −1798.84 −0.285667
\(342\) 0 0
\(343\) 0 0
\(344\) 9314.72 1.45993
\(345\) 0 0
\(346\) 19078.5 2.96435
\(347\) 3471.96 0.537132 0.268566 0.963261i \(-0.413450\pi\)
0.268566 + 0.963261i \(0.413450\pi\)
\(348\) 0 0
\(349\) −6626.12 −1.01630 −0.508149 0.861269i \(-0.669670\pi\)
−0.508149 + 0.861269i \(0.669670\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2146.10 −0.324964
\(353\) −9468.40 −1.42763 −0.713813 0.700337i \(-0.753033\pi\)
−0.713813 + 0.700337i \(0.753033\pi\)
\(354\) 0 0
\(355\) −6594.53 −0.985919
\(356\) 479.944 0.0714522
\(357\) 0 0
\(358\) 8977.35 1.32533
\(359\) 6279.55 0.923182 0.461591 0.887093i \(-0.347279\pi\)
0.461591 + 0.887093i \(0.347279\pi\)
\(360\) 0 0
\(361\) −5059.35 −0.737622
\(362\) −16462.6 −2.39021
\(363\) 0 0
\(364\) 0 0
\(365\) −10883.1 −1.56068
\(366\) 0 0
\(367\) −10827.8 −1.54008 −0.770038 0.637998i \(-0.779763\pi\)
−0.770038 + 0.637998i \(0.779763\pi\)
\(368\) 494.904 0.0701050
\(369\) 0 0
\(370\) −305.053 −0.0428620
\(371\) 0 0
\(372\) 0 0
\(373\) 5239.23 0.727284 0.363642 0.931539i \(-0.381533\pi\)
0.363642 + 0.931539i \(0.381533\pi\)
\(374\) −2765.42 −0.382344
\(375\) 0 0
\(376\) −459.553 −0.0630310
\(377\) 2442.81 0.333717
\(378\) 0 0
\(379\) −11050.4 −1.49768 −0.748839 0.662751i \(-0.769389\pi\)
−0.748839 + 0.662751i \(0.769389\pi\)
\(380\) 9675.27 1.30613
\(381\) 0 0
\(382\) −8692.49 −1.16426
\(383\) 10468.0 1.39658 0.698292 0.715813i \(-0.253944\pi\)
0.698292 + 0.715813i \(0.253944\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10934.9 1.44190
\(387\) 0 0
\(388\) 9213.22 1.20549
\(389\) 11614.0 1.51377 0.756884 0.653550i \(-0.226721\pi\)
0.756884 + 0.653550i \(0.226721\pi\)
\(390\) 0 0
\(391\) 8101.19 1.04781
\(392\) 0 0
\(393\) 0 0
\(394\) −6899.78 −0.882248
\(395\) −16268.9 −2.07234
\(396\) 0 0
\(397\) −6707.30 −0.847934 −0.423967 0.905678i \(-0.639363\pi\)
−0.423967 + 0.905678i \(0.639363\pi\)
\(398\) 6232.10 0.784892
\(399\) 0 0
\(400\) −632.768 −0.0790960
\(401\) −5526.38 −0.688215 −0.344107 0.938930i \(-0.611818\pi\)
−0.344107 + 0.938930i \(0.611818\pi\)
\(402\) 0 0
\(403\) 2067.94 0.255611
\(404\) −19379.0 −2.38649
\(405\) 0 0
\(406\) 0 0
\(407\) 42.6811 0.00519810
\(408\) 0 0
\(409\) −1318.91 −0.159452 −0.0797258 0.996817i \(-0.525404\pi\)
−0.0797258 + 0.996817i \(0.525404\pi\)
\(410\) −3202.34 −0.385738
\(411\) 0 0
\(412\) −13422.5 −1.60505
\(413\) 0 0
\(414\) 0 0
\(415\) −5861.15 −0.693283
\(416\) 2467.14 0.290773
\(417\) 0 0
\(418\) −2202.39 −0.257709
\(419\) −3656.13 −0.426286 −0.213143 0.977021i \(-0.568370\pi\)
−0.213143 + 0.977021i \(0.568370\pi\)
\(420\) 0 0
\(421\) −135.389 −0.0156733 −0.00783663 0.999969i \(-0.502495\pi\)
−0.00783663 + 0.999969i \(0.502495\pi\)
\(422\) 19603.9 2.26138
\(423\) 0 0
\(424\) −7936.63 −0.909049
\(425\) −10357.9 −1.18219
\(426\) 0 0
\(427\) 0 0
\(428\) −9781.16 −1.10465
\(429\) 0 0
\(430\) −34972.1 −3.92211
\(431\) −8389.16 −0.937568 −0.468784 0.883313i \(-0.655308\pi\)
−0.468784 + 0.883313i \(0.655308\pi\)
\(432\) 0 0
\(433\) −8243.02 −0.914859 −0.457430 0.889246i \(-0.651230\pi\)
−0.457430 + 0.889246i \(0.651230\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −18209.8 −2.00021
\(437\) 6451.81 0.706252
\(438\) 0 0
\(439\) −18283.2 −1.98772 −0.993859 0.110649i \(-0.964707\pi\)
−0.993859 + 0.110649i \(0.964707\pi\)
\(440\) −4417.26 −0.478602
\(441\) 0 0
\(442\) 3179.12 0.342116
\(443\) −1210.44 −0.129818 −0.0649092 0.997891i \(-0.520676\pi\)
−0.0649092 + 0.997891i \(0.520676\pi\)
\(444\) 0 0
\(445\) −672.243 −0.0716121
\(446\) −6821.72 −0.724256
\(447\) 0 0
\(448\) 0 0
\(449\) 8301.16 0.872508 0.436254 0.899824i \(-0.356305\pi\)
0.436254 + 0.899824i \(0.356305\pi\)
\(450\) 0 0
\(451\) 448.052 0.0467804
\(452\) −4626.49 −0.481442
\(453\) 0 0
\(454\) −7305.34 −0.755190
\(455\) 0 0
\(456\) 0 0
\(457\) −12293.8 −1.25838 −0.629188 0.777253i \(-0.716612\pi\)
−0.629188 + 0.777253i \(0.716612\pi\)
\(458\) 4604.31 0.469749
\(459\) 0 0
\(460\) 34686.2 3.51577
\(461\) −19434.2 −1.96343 −0.981717 0.190346i \(-0.939039\pi\)
−0.981717 + 0.190346i \(0.939039\pi\)
\(462\) 0 0
\(463\) −12491.1 −1.25380 −0.626902 0.779098i \(-0.715678\pi\)
−0.626902 + 0.779098i \(0.715678\pi\)
\(464\) 606.871 0.0607183
\(465\) 0 0
\(466\) −903.149 −0.0897802
\(467\) −3385.17 −0.335433 −0.167716 0.985835i \(-0.553639\pi\)
−0.167716 + 0.985835i \(0.553639\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1725.39 0.169333
\(471\) 0 0
\(472\) 4914.57 0.479262
\(473\) 4893.09 0.475654
\(474\) 0 0
\(475\) −8249.07 −0.796828
\(476\) 0 0
\(477\) 0 0
\(478\) 5473.05 0.523706
\(479\) −5979.41 −0.570368 −0.285184 0.958473i \(-0.592055\pi\)
−0.285184 + 0.958473i \(0.592055\pi\)
\(480\) 0 0
\(481\) −49.0661 −0.00465119
\(482\) −12451.1 −1.17663
\(483\) 0 0
\(484\) −15327.5 −1.43948
\(485\) −12904.7 −1.20819
\(486\) 0 0
\(487\) −1114.96 −0.103745 −0.0518725 0.998654i \(-0.516519\pi\)
−0.0518725 + 0.998654i \(0.516519\pi\)
\(488\) 14141.6 1.31180
\(489\) 0 0
\(490\) 0 0
\(491\) −1086.23 −0.0998387 −0.0499194 0.998753i \(-0.515896\pi\)
−0.0499194 + 0.998753i \(0.515896\pi\)
\(492\) 0 0
\(493\) 9934.01 0.907516
\(494\) 2531.86 0.230595
\(495\) 0 0
\(496\) 513.740 0.0465073
\(497\) 0 0
\(498\) 0 0
\(499\) −2213.50 −0.198577 −0.0992884 0.995059i \(-0.531657\pi\)
−0.0992884 + 0.995059i \(0.531657\pi\)
\(500\) −15839.8 −1.41676
\(501\) 0 0
\(502\) −34472.6 −3.06492
\(503\) 2643.32 0.234314 0.117157 0.993113i \(-0.462622\pi\)
0.117157 + 0.993113i \(0.462622\pi\)
\(504\) 0 0
\(505\) 27143.5 2.39183
\(506\) −7895.66 −0.693685
\(507\) 0 0
\(508\) 12438.6 1.08637
\(509\) 665.169 0.0579236 0.0289618 0.999581i \(-0.490780\pi\)
0.0289618 + 0.999581i \(0.490780\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1177.58 0.101645
\(513\) 0 0
\(514\) −22821.4 −1.95838
\(515\) 18800.5 1.60864
\(516\) 0 0
\(517\) −241.406 −0.0205359
\(518\) 0 0
\(519\) 0 0
\(520\) 5078.07 0.428246
\(521\) 11762.0 0.989063 0.494531 0.869160i \(-0.335340\pi\)
0.494531 + 0.869160i \(0.335340\pi\)
\(522\) 0 0
\(523\) 10122.6 0.846330 0.423165 0.906053i \(-0.360919\pi\)
0.423165 + 0.906053i \(0.360919\pi\)
\(524\) 22875.7 1.90712
\(525\) 0 0
\(526\) 28471.9 2.36014
\(527\) 8409.52 0.695112
\(528\) 0 0
\(529\) 10963.0 0.901042
\(530\) 29798.1 2.44216
\(531\) 0 0
\(532\) 0 0
\(533\) −515.079 −0.0418585
\(534\) 0 0
\(535\) 13700.2 1.10712
\(536\) 3154.51 0.254206
\(537\) 0 0
\(538\) 16350.3 1.31024
\(539\) 0 0
\(540\) 0 0
\(541\) 16118.0 1.28090 0.640449 0.768001i \(-0.278748\pi\)
0.640449 + 0.768001i \(0.278748\pi\)
\(542\) −9035.92 −0.716100
\(543\) 0 0
\(544\) 10032.9 0.790733
\(545\) 25505.9 2.00469
\(546\) 0 0
\(547\) −626.100 −0.0489399 −0.0244699 0.999701i \(-0.507790\pi\)
−0.0244699 + 0.999701i \(0.507790\pi\)
\(548\) −21493.9 −1.67550
\(549\) 0 0
\(550\) 10095.1 0.782650
\(551\) 7911.47 0.611688
\(552\) 0 0
\(553\) 0 0
\(554\) 33552.7 2.57313
\(555\) 0 0
\(556\) 4020.69 0.306682
\(557\) −20771.2 −1.58008 −0.790039 0.613057i \(-0.789940\pi\)
−0.790039 + 0.613057i \(0.789940\pi\)
\(558\) 0 0
\(559\) −5625.08 −0.425609
\(560\) 0 0
\(561\) 0 0
\(562\) 24203.6 1.81667
\(563\) 5521.72 0.413344 0.206672 0.978410i \(-0.433737\pi\)
0.206672 + 0.978410i \(0.433737\pi\)
\(564\) 0 0
\(565\) 6480.18 0.482519
\(566\) −4974.48 −0.369422
\(567\) 0 0
\(568\) 8002.95 0.591191
\(569\) −7574.81 −0.558089 −0.279044 0.960278i \(-0.590018\pi\)
−0.279044 + 0.960278i \(0.590018\pi\)
\(570\) 0 0
\(571\) −331.248 −0.0242772 −0.0121386 0.999926i \(-0.503864\pi\)
−0.0121386 + 0.999926i \(0.503864\pi\)
\(572\) −1904.48 −0.139214
\(573\) 0 0
\(574\) 0 0
\(575\) −29573.2 −2.14485
\(576\) 0 0
\(577\) 2038.11 0.147050 0.0735248 0.997293i \(-0.476575\pi\)
0.0735248 + 0.997293i \(0.476575\pi\)
\(578\) −9457.14 −0.680563
\(579\) 0 0
\(580\) 42533.6 3.04502
\(581\) 0 0
\(582\) 0 0
\(583\) −4169.17 −0.296174
\(584\) 13207.4 0.935835
\(585\) 0 0
\(586\) 32768.8 2.31001
\(587\) −5232.90 −0.367947 −0.183973 0.982931i \(-0.558896\pi\)
−0.183973 + 0.982931i \(0.558896\pi\)
\(588\) 0 0
\(589\) 6697.36 0.468523
\(590\) −18451.8 −1.28754
\(591\) 0 0
\(592\) −12.1895 −0.000846262 0
\(593\) 5720.24 0.396125 0.198062 0.980189i \(-0.436535\pi\)
0.198062 + 0.980189i \(0.436535\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −24165.5 −1.66083
\(597\) 0 0
\(598\) 9076.81 0.620700
\(599\) −18088.4 −1.23384 −0.616922 0.787024i \(-0.711621\pi\)
−0.616922 + 0.787024i \(0.711621\pi\)
\(600\) 0 0
\(601\) −1821.43 −0.123623 −0.0618117 0.998088i \(-0.519688\pi\)
−0.0618117 + 0.998088i \(0.519688\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −25672.0 −1.72944
\(605\) 21468.8 1.44270
\(606\) 0 0
\(607\) 2372.20 0.158624 0.0793120 0.996850i \(-0.474728\pi\)
0.0793120 + 0.996850i \(0.474728\pi\)
\(608\) 7990.26 0.532974
\(609\) 0 0
\(610\) −53094.7 −3.52416
\(611\) 277.520 0.0183752
\(612\) 0 0
\(613\) 9725.08 0.640770 0.320385 0.947287i \(-0.396188\pi\)
0.320385 + 0.947287i \(0.396188\pi\)
\(614\) 2466.61 0.162124
\(615\) 0 0
\(616\) 0 0
\(617\) 5329.51 0.347744 0.173872 0.984768i \(-0.444372\pi\)
0.173872 + 0.984768i \(0.444372\pi\)
\(618\) 0 0
\(619\) −15976.6 −1.03740 −0.518702 0.854955i \(-0.673585\pi\)
−0.518702 + 0.854955i \(0.673585\pi\)
\(620\) 36006.3 2.33234
\(621\) 0 0
\(622\) −246.120 −0.0158658
\(623\) 0 0
\(624\) 0 0
\(625\) −2120.06 −0.135684
\(626\) 17190.9 1.09758
\(627\) 0 0
\(628\) −48853.5 −3.10425
\(629\) −199.533 −0.0126485
\(630\) 0 0
\(631\) −4199.98 −0.264974 −0.132487 0.991185i \(-0.542296\pi\)
−0.132487 + 0.991185i \(0.542296\pi\)
\(632\) 19743.5 1.24265
\(633\) 0 0
\(634\) −7833.47 −0.490705
\(635\) −17422.4 −1.08880
\(636\) 0 0
\(637\) 0 0
\(638\) −9681.97 −0.600804
\(639\) 0 0
\(640\) 40837.2 2.52224
\(641\) −2648.51 −0.163198 −0.0815988 0.996665i \(-0.526003\pi\)
−0.0815988 + 0.996665i \(0.526003\pi\)
\(642\) 0 0
\(643\) 13.4305 0.000823715 0 0.000411857 1.00000i \(-0.499869\pi\)
0.000411857 1.00000i \(0.499869\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 10296.1 0.627083
\(647\) −11624.1 −0.706324 −0.353162 0.935562i \(-0.614894\pi\)
−0.353162 + 0.935562i \(0.614894\pi\)
\(648\) 0 0
\(649\) 2581.66 0.156146
\(650\) −11605.3 −0.700305
\(651\) 0 0
\(652\) −44779.8 −2.68974
\(653\) 28516.6 1.70894 0.854471 0.519499i \(-0.173881\pi\)
0.854471 + 0.519499i \(0.173881\pi\)
\(654\) 0 0
\(655\) −32041.3 −1.91139
\(656\) −127.962 −0.00761595
\(657\) 0 0
\(658\) 0 0
\(659\) −18048.6 −1.06688 −0.533440 0.845838i \(-0.679101\pi\)
−0.533440 + 0.845838i \(0.679101\pi\)
\(660\) 0 0
\(661\) 17841.4 1.04985 0.524926 0.851148i \(-0.324093\pi\)
0.524926 + 0.851148i \(0.324093\pi\)
\(662\) 38310.9 2.24924
\(663\) 0 0
\(664\) 7112.94 0.415716
\(665\) 0 0
\(666\) 0 0
\(667\) 28362.9 1.64650
\(668\) 4376.95 0.253517
\(669\) 0 0
\(670\) −11843.6 −0.682924
\(671\) 7428.68 0.427394
\(672\) 0 0
\(673\) −6826.13 −0.390978 −0.195489 0.980706i \(-0.562629\pi\)
−0.195489 + 0.980706i \(0.562629\pi\)
\(674\) 12709.8 0.726354
\(675\) 0 0
\(676\) −25845.4 −1.47049
\(677\) 21286.9 1.20845 0.604225 0.796814i \(-0.293483\pi\)
0.604225 + 0.796814i \(0.293483\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 20650.6 1.16458
\(681\) 0 0
\(682\) −8196.16 −0.460187
\(683\) 20696.8 1.15951 0.579753 0.814793i \(-0.303149\pi\)
0.579753 + 0.814793i \(0.303149\pi\)
\(684\) 0 0
\(685\) 30105.9 1.67925
\(686\) 0 0
\(687\) 0 0
\(688\) −1397.44 −0.0774376
\(689\) 4792.86 0.265012
\(690\) 0 0
\(691\) 31341.9 1.72548 0.862738 0.505652i \(-0.168748\pi\)
0.862738 + 0.505652i \(0.168748\pi\)
\(692\) 53430.8 2.93517
\(693\) 0 0
\(694\) 15819.5 0.865276
\(695\) −5631.66 −0.307368
\(696\) 0 0
\(697\) −2094.63 −0.113831
\(698\) −30191.0 −1.63717
\(699\) 0 0
\(700\) 0 0
\(701\) 9213.32 0.496408 0.248204 0.968708i \(-0.420160\pi\)
0.248204 + 0.968708i \(0.420160\pi\)
\(702\) 0 0
\(703\) −158.909 −0.00852541
\(704\) −9481.77 −0.507610
\(705\) 0 0
\(706\) −43141.5 −2.29979
\(707\) 0 0
\(708\) 0 0
\(709\) 14516.5 0.768942 0.384471 0.923137i \(-0.374384\pi\)
0.384471 + 0.923137i \(0.374384\pi\)
\(710\) −30047.1 −1.58824
\(711\) 0 0
\(712\) 815.817 0.0429411
\(713\) 24010.3 1.26114
\(714\) 0 0
\(715\) 2667.54 0.139525
\(716\) 25141.8 1.31228
\(717\) 0 0
\(718\) 28611.9 1.48717
\(719\) −25882.4 −1.34249 −0.671246 0.741235i \(-0.734240\pi\)
−0.671246 + 0.741235i \(0.734240\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −23052.3 −1.18825
\(723\) 0 0
\(724\) −46104.9 −2.36668
\(725\) −36263.9 −1.85767
\(726\) 0 0
\(727\) 32181.2 1.64172 0.820862 0.571127i \(-0.193494\pi\)
0.820862 + 0.571127i \(0.193494\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −49587.4 −2.51412
\(731\) −22875.1 −1.15741
\(732\) 0 0
\(733\) 20836.1 1.04993 0.524966 0.851123i \(-0.324078\pi\)
0.524966 + 0.851123i \(0.324078\pi\)
\(734\) −49335.5 −2.48094
\(735\) 0 0
\(736\) 28645.4 1.43462
\(737\) 1657.09 0.0828217
\(738\) 0 0
\(739\) 26434.9 1.31586 0.657931 0.753078i \(-0.271432\pi\)
0.657931 + 0.753078i \(0.271432\pi\)
\(740\) −854.325 −0.0424400
\(741\) 0 0
\(742\) 0 0
\(743\) −9954.69 −0.491524 −0.245762 0.969330i \(-0.579038\pi\)
−0.245762 + 0.969330i \(0.579038\pi\)
\(744\) 0 0
\(745\) 33847.8 1.66455
\(746\) 23871.8 1.17160
\(747\) 0 0
\(748\) −7744.79 −0.378580
\(749\) 0 0
\(750\) 0 0
\(751\) −33204.5 −1.61338 −0.806692 0.590973i \(-0.798744\pi\)
−0.806692 + 0.590973i \(0.798744\pi\)
\(752\) 68.9446 0.00334329
\(753\) 0 0
\(754\) 11130.4 0.537591
\(755\) 35958.1 1.73331
\(756\) 0 0
\(757\) 1964.06 0.0942998 0.0471499 0.998888i \(-0.484986\pi\)
0.0471499 + 0.998888i \(0.484986\pi\)
\(758\) −50349.6 −2.41264
\(759\) 0 0
\(760\) 16446.2 0.784955
\(761\) 38553.8 1.83650 0.918248 0.396005i \(-0.129604\pi\)
0.918248 + 0.396005i \(0.129604\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −24344.0 −1.15280
\(765\) 0 0
\(766\) 47696.2 2.24978
\(767\) −2967.87 −0.139718
\(768\) 0 0
\(769\) −19715.0 −0.924501 −0.462251 0.886749i \(-0.652958\pi\)
−0.462251 + 0.886749i \(0.652958\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 30624.2 1.42771
\(773\) 14700.7 0.684019 0.342010 0.939696i \(-0.388892\pi\)
0.342010 + 0.939696i \(0.388892\pi\)
\(774\) 0 0
\(775\) −30698.8 −1.42288
\(776\) 15660.8 0.724471
\(777\) 0 0
\(778\) 52917.8 2.43856
\(779\) −1668.17 −0.0767246
\(780\) 0 0
\(781\) 4204.01 0.192614
\(782\) 36912.0 1.68794
\(783\) 0 0
\(784\) 0 0
\(785\) 68427.6 3.11119
\(786\) 0 0
\(787\) −23918.6 −1.08336 −0.541681 0.840584i \(-0.682212\pi\)
−0.541681 + 0.840584i \(0.682212\pi\)
\(788\) −19323.4 −0.873562
\(789\) 0 0
\(790\) −74126.9 −3.33837
\(791\) 0 0
\(792\) 0 0
\(793\) −8539.98 −0.382426
\(794\) −30560.9 −1.36595
\(795\) 0 0
\(796\) 17453.5 0.777164
\(797\) −38252.7 −1.70010 −0.850051 0.526700i \(-0.823429\pi\)
−0.850051 + 0.526700i \(0.823429\pi\)
\(798\) 0 0
\(799\) 1128.57 0.0499698
\(800\) −36625.1 −1.61861
\(801\) 0 0
\(802\) −25180.2 −1.10866
\(803\) 6937.96 0.304901
\(804\) 0 0
\(805\) 0 0
\(806\) 9422.27 0.411769
\(807\) 0 0
\(808\) −32940.7 −1.43422
\(809\) 31435.6 1.36615 0.683075 0.730348i \(-0.260642\pi\)
0.683075 + 0.730348i \(0.260642\pi\)
\(810\) 0 0
\(811\) 11467.0 0.496501 0.248250 0.968696i \(-0.420144\pi\)
0.248250 + 0.968696i \(0.420144\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 194.471 0.00837371
\(815\) 62721.7 2.69576
\(816\) 0 0
\(817\) −18217.8 −0.780121
\(818\) −6009.42 −0.256864
\(819\) 0 0
\(820\) −8968.42 −0.381940
\(821\) 5030.57 0.213847 0.106923 0.994267i \(-0.465900\pi\)
0.106923 + 0.994267i \(0.465900\pi\)
\(822\) 0 0
\(823\) −13985.2 −0.592336 −0.296168 0.955136i \(-0.595709\pi\)
−0.296168 + 0.955136i \(0.595709\pi\)
\(824\) −22815.8 −0.964596
\(825\) 0 0
\(826\) 0 0
\(827\) 13939.5 0.586125 0.293063 0.956093i \(-0.405326\pi\)
0.293063 + 0.956093i \(0.405326\pi\)
\(828\) 0 0
\(829\) −20104.4 −0.842286 −0.421143 0.906994i \(-0.638371\pi\)
−0.421143 + 0.906994i \(0.638371\pi\)
\(830\) −26705.5 −1.11682
\(831\) 0 0
\(832\) 10900.2 0.454203
\(833\) 0 0
\(834\) 0 0
\(835\) −6130.66 −0.254084
\(836\) −6167.98 −0.255172
\(837\) 0 0
\(838\) −16658.7 −0.686711
\(839\) −15949.5 −0.656302 −0.328151 0.944625i \(-0.606425\pi\)
−0.328151 + 0.944625i \(0.606425\pi\)
\(840\) 0 0
\(841\) 10390.8 0.426043
\(842\) −616.881 −0.0252484
\(843\) 0 0
\(844\) 54902.2 2.23911
\(845\) 36200.8 1.47378
\(846\) 0 0
\(847\) 0 0
\(848\) 1190.70 0.0482177
\(849\) 0 0
\(850\) −47194.5 −1.90442
\(851\) −569.694 −0.0229481
\(852\) 0 0
\(853\) 11802.0 0.473730 0.236865 0.971543i \(-0.423880\pi\)
0.236865 + 0.971543i \(0.423880\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −16626.2 −0.663868
\(857\) −9595.28 −0.382460 −0.191230 0.981545i \(-0.561248\pi\)
−0.191230 + 0.981545i \(0.561248\pi\)
\(858\) 0 0
\(859\) 21840.9 0.867521 0.433760 0.901028i \(-0.357186\pi\)
0.433760 + 0.901028i \(0.357186\pi\)
\(860\) −97942.3 −3.88350
\(861\) 0 0
\(862\) −38224.1 −1.51035
\(863\) 26531.6 1.04652 0.523260 0.852173i \(-0.324716\pi\)
0.523260 + 0.852173i \(0.324716\pi\)
\(864\) 0 0
\(865\) −74838.9 −2.94173
\(866\) −37558.2 −1.47376
\(867\) 0 0
\(868\) 0 0
\(869\) 10371.4 0.404862
\(870\) 0 0
\(871\) −1904.98 −0.0741077
\(872\) −30953.3 −1.20208
\(873\) 0 0
\(874\) 29396.8 1.13771
\(875\) 0 0
\(876\) 0 0
\(877\) −6832.46 −0.263074 −0.131537 0.991311i \(-0.541991\pi\)
−0.131537 + 0.991311i \(0.541991\pi\)
\(878\) −83304.9 −3.20205
\(879\) 0 0
\(880\) 662.701 0.0253860
\(881\) 3994.77 0.152766 0.0763832 0.997079i \(-0.475663\pi\)
0.0763832 + 0.997079i \(0.475663\pi\)
\(882\) 0 0
\(883\) 13727.0 0.523161 0.261580 0.965182i \(-0.415756\pi\)
0.261580 + 0.965182i \(0.415756\pi\)
\(884\) 8903.38 0.338748
\(885\) 0 0
\(886\) −5515.19 −0.209127
\(887\) −44119.4 −1.67011 −0.835054 0.550169i \(-0.814563\pi\)
−0.835054 + 0.550169i \(0.814563\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −3062.99 −0.115361
\(891\) 0 0
\(892\) −19104.8 −0.717125
\(893\) 898.796 0.0336809
\(894\) 0 0
\(895\) −35215.3 −1.31522
\(896\) 0 0
\(897\) 0 0
\(898\) 37823.1 1.40554
\(899\) 29442.4 1.09228
\(900\) 0 0
\(901\) 19490.7 0.720678
\(902\) 2041.49 0.0753594
\(903\) 0 0
\(904\) −7864.18 −0.289335
\(905\) 64577.7 2.37197
\(906\) 0 0
\(907\) 36905.8 1.35109 0.675545 0.737319i \(-0.263908\pi\)
0.675545 + 0.737319i \(0.263908\pi\)
\(908\) −20459.2 −0.747755
\(909\) 0 0
\(910\) 0 0
\(911\) 3169.56 0.115271 0.0576356 0.998338i \(-0.481644\pi\)
0.0576356 + 0.998338i \(0.481644\pi\)
\(912\) 0 0
\(913\) 3736.48 0.135443
\(914\) −56014.9 −2.02714
\(915\) 0 0
\(916\) 12894.7 0.465124
\(917\) 0 0
\(918\) 0 0
\(919\) 8727.00 0.313250 0.156625 0.987658i \(-0.449939\pi\)
0.156625 + 0.987658i \(0.449939\pi\)
\(920\) 58960.2 2.11289
\(921\) 0 0
\(922\) −88549.5 −3.16293
\(923\) −4832.91 −0.172348
\(924\) 0 0
\(925\) 728.392 0.0258912
\(926\) −56914.1 −2.01978
\(927\) 0 0
\(928\) 35126.1 1.24253
\(929\) −19405.1 −0.685317 −0.342659 0.939460i \(-0.611327\pi\)
−0.342659 + 0.939460i \(0.611327\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2529.34 −0.0888963
\(933\) 0 0
\(934\) −15424.1 −0.540355
\(935\) 10847.9 0.379427
\(936\) 0 0
\(937\) −615.692 −0.0214662 −0.0107331 0.999942i \(-0.503417\pi\)
−0.0107331 + 0.999942i \(0.503417\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 4832.10 0.167666
\(941\) 29602.0 1.02550 0.512751 0.858537i \(-0.328626\pi\)
0.512751 + 0.858537i \(0.328626\pi\)
\(942\) 0 0
\(943\) −5980.46 −0.206522
\(944\) −737.310 −0.0254210
\(945\) 0 0
\(946\) 22294.7 0.766240
\(947\) −13537.4 −0.464527 −0.232264 0.972653i \(-0.574613\pi\)
−0.232264 + 0.972653i \(0.574613\pi\)
\(948\) 0 0
\(949\) −7975.85 −0.272821
\(950\) −37585.8 −1.28363
\(951\) 0 0
\(952\) 0 0
\(953\) −33468.5 −1.13762 −0.568810 0.822469i \(-0.692596\pi\)
−0.568810 + 0.822469i \(0.692596\pi\)
\(954\) 0 0
\(955\) 34097.9 1.15538
\(956\) 15327.7 0.518550
\(957\) 0 0
\(958\) −27244.4 −0.918817
\(959\) 0 0
\(960\) 0 0
\(961\) −4866.88 −0.163368
\(962\) −223.563 −0.00749268
\(963\) 0 0
\(964\) −34870.4 −1.16504
\(965\) −42894.4 −1.43090
\(966\) 0 0
\(967\) −55733.5 −1.85343 −0.926715 0.375764i \(-0.877380\pi\)
−0.926715 + 0.375764i \(0.877380\pi\)
\(968\) −26054.0 −0.865090
\(969\) 0 0
\(970\) −58798.4 −1.94629
\(971\) 19491.3 0.644187 0.322094 0.946708i \(-0.395613\pi\)
0.322094 + 0.946708i \(0.395613\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −5080.18 −0.167125
\(975\) 0 0
\(976\) −2121.60 −0.0695807
\(977\) 6241.56 0.204386 0.102193 0.994765i \(-0.467414\pi\)
0.102193 + 0.994765i \(0.467414\pi\)
\(978\) 0 0
\(979\) 428.554 0.0139905
\(980\) 0 0
\(981\) 0 0
\(982\) −4949.25 −0.160832
\(983\) −59694.6 −1.93689 −0.968444 0.249231i \(-0.919822\pi\)
−0.968444 + 0.249231i \(0.919822\pi\)
\(984\) 0 0
\(985\) 27065.7 0.875517
\(986\) 45263.0 1.46193
\(987\) 0 0
\(988\) 7090.68 0.228324
\(989\) −65311.4 −2.09988
\(990\) 0 0
\(991\) 15561.6 0.498821 0.249411 0.968398i \(-0.419763\pi\)
0.249411 + 0.968398i \(0.419763\pi\)
\(992\) 29735.6 0.951720
\(993\) 0 0
\(994\) 0 0
\(995\) −24446.6 −0.778903
\(996\) 0 0
\(997\) 19884.9 0.631657 0.315829 0.948816i \(-0.397718\pi\)
0.315829 + 0.948816i \(0.397718\pi\)
\(998\) −10085.5 −0.319891
\(999\) 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 441.4.a.s.1.3 3
3.2 odd 2 147.4.a.l.1.1 3
7.2 even 3 63.4.e.c.46.1 6
7.3 odd 6 441.4.e.w.226.1 6
7.4 even 3 63.4.e.c.37.1 6
7.5 odd 6 441.4.e.w.361.1 6
7.6 odd 2 441.4.a.t.1.3 3
12.11 even 2 2352.4.a.ci.1.3 3
21.2 odd 6 21.4.e.b.4.3 6
21.5 even 6 147.4.e.n.67.3 6
21.11 odd 6 21.4.e.b.16.3 yes 6
21.17 even 6 147.4.e.n.79.3 6
21.20 even 2 147.4.a.m.1.1 3
84.11 even 6 336.4.q.k.289.1 6
84.23 even 6 336.4.q.k.193.1 6
84.83 odd 2 2352.4.a.cg.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.b.4.3 6 21.2 odd 6
21.4.e.b.16.3 yes 6 21.11 odd 6
63.4.e.c.37.1 6 7.4 even 3
63.4.e.c.46.1 6 7.2 even 3
147.4.a.l.1.1 3 3.2 odd 2
147.4.a.m.1.1 3 21.20 even 2
147.4.e.n.67.3 6 21.5 even 6
147.4.e.n.79.3 6 21.17 even 6
336.4.q.k.193.1 6 84.23 even 6
336.4.q.k.289.1 6 84.11 even 6
441.4.a.s.1.3 3 1.1 even 1 trivial
441.4.a.t.1.3 3 7.6 odd 2
441.4.e.w.226.1 6 7.3 odd 6
441.4.e.w.361.1 6 7.5 odd 6
2352.4.a.cg.1.1 3 84.83 odd 2
2352.4.a.ci.1.3 3 12.11 even 2