Properties

Label 1521.4.a.bf.1.4
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.100291\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.34727 q^{2} -2.49032 q^{4} -15.3991 q^{5} -10.1317 q^{7} +24.6236 q^{8} +O(q^{10})\) \(q-2.34727 q^{2} -2.49032 q^{4} -15.3991 q^{5} -10.1317 q^{7} +24.6236 q^{8} +36.1458 q^{10} -15.0669 q^{11} +23.7819 q^{14} -37.8757 q^{16} -90.8352 q^{17} +114.640 q^{19} +38.3486 q^{20} +35.3661 q^{22} -75.7635 q^{23} +112.132 q^{25} +25.2313 q^{28} -214.817 q^{29} -284.476 q^{31} -108.084 q^{32} +213.215 q^{34} +156.019 q^{35} +358.878 q^{37} -269.091 q^{38} -379.181 q^{40} -313.154 q^{41} -296.702 q^{43} +37.5214 q^{44} +177.837 q^{46} +316.691 q^{47} -240.348 q^{49} -263.203 q^{50} -163.911 q^{53} +232.016 q^{55} -249.480 q^{56} +504.233 q^{58} -254.149 q^{59} -935.247 q^{61} +667.742 q^{62} +556.709 q^{64} -240.494 q^{67} +226.209 q^{68} -366.220 q^{70} -947.455 q^{71} -430.712 q^{73} -842.384 q^{74} -285.490 q^{76} +152.654 q^{77} -496.620 q^{79} +583.251 q^{80} +735.058 q^{82} -392.527 q^{83} +1398.78 q^{85} +696.439 q^{86} -371.002 q^{88} -979.895 q^{89} +188.675 q^{92} -743.360 q^{94} -1765.35 q^{95} -553.356 q^{97} +564.162 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29} + 717 q^{31} - 707 q^{32} + 632 q^{34} + 418 q^{35} + 1108 q^{37} + 660 q^{38} - 1506 q^{40} - 334 q^{41} + 242 q^{43} + 307 q^{44} + 979 q^{46} + 184 q^{47} - 38 q^{49} + 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} + 1161 q^{58} - 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} + 2308 q^{67} - 2785 q^{68} - 1420 q^{70} - 96 q^{71} + 2505 q^{73} + 1191 q^{74} + 2409 q^{76} + 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 1517 q^{82} - 1539 q^{83} + 4296 q^{85} + 3763 q^{86} - 3716 q^{88} + 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} + 1445 q^{97} - 1457 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.34727 −0.829886 −0.414943 0.909848i \(-0.636198\pi\)
−0.414943 + 0.909848i \(0.636198\pi\)
\(3\) 0 0
\(4\) −2.49032 −0.311290
\(5\) −15.3991 −1.37734 −0.688668 0.725077i \(-0.741804\pi\)
−0.688668 + 0.725077i \(0.741804\pi\)
\(6\) 0 0
\(7\) −10.1317 −0.547062 −0.273531 0.961863i \(-0.588192\pi\)
−0.273531 + 0.961863i \(0.588192\pi\)
\(8\) 24.6236 1.08822
\(9\) 0 0
\(10\) 36.1458 1.14303
\(11\) −15.0669 −0.412985 −0.206493 0.978448i \(-0.566205\pi\)
−0.206493 + 0.978448i \(0.566205\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 23.7819 0.453999
\(15\) 0 0
\(16\) −37.8757 −0.591809
\(17\) −90.8352 −1.29593 −0.647964 0.761671i \(-0.724379\pi\)
−0.647964 + 0.761671i \(0.724379\pi\)
\(18\) 0 0
\(19\) 114.640 1.38422 0.692110 0.721792i \(-0.256681\pi\)
0.692110 + 0.721792i \(0.256681\pi\)
\(20\) 38.3486 0.428751
\(21\) 0 0
\(22\) 35.3661 0.342731
\(23\) −75.7635 −0.686860 −0.343430 0.939178i \(-0.611589\pi\)
−0.343430 + 0.939178i \(0.611589\pi\)
\(24\) 0 0
\(25\) 112.132 0.897052
\(26\) 0 0
\(27\) 0 0
\(28\) 25.2313 0.170295
\(29\) −214.817 −1.37553 −0.687767 0.725931i \(-0.741409\pi\)
−0.687767 + 0.725931i \(0.741409\pi\)
\(30\) 0 0
\(31\) −284.476 −1.64817 −0.824087 0.566463i \(-0.808311\pi\)
−0.824087 + 0.566463i \(0.808311\pi\)
\(32\) −108.084 −0.597087
\(33\) 0 0
\(34\) 213.215 1.07547
\(35\) 156.019 0.753488
\(36\) 0 0
\(37\) 358.878 1.59457 0.797286 0.603602i \(-0.206268\pi\)
0.797286 + 0.603602i \(0.206268\pi\)
\(38\) −269.091 −1.14874
\(39\) 0 0
\(40\) −379.181 −1.49884
\(41\) −313.154 −1.19284 −0.596421 0.802672i \(-0.703411\pi\)
−0.596421 + 0.802672i \(0.703411\pi\)
\(42\) 0 0
\(43\) −296.702 −1.05225 −0.526123 0.850409i \(-0.676355\pi\)
−0.526123 + 0.850409i \(0.676355\pi\)
\(44\) 37.5214 0.128558
\(45\) 0 0
\(46\) 177.837 0.570015
\(47\) 316.691 0.982854 0.491427 0.870919i \(-0.336475\pi\)
0.491427 + 0.870919i \(0.336475\pi\)
\(48\) 0 0
\(49\) −240.348 −0.700723
\(50\) −263.203 −0.744451
\(51\) 0 0
\(52\) 0 0
\(53\) −163.911 −0.424810 −0.212405 0.977182i \(-0.568130\pi\)
−0.212405 + 0.977182i \(0.568130\pi\)
\(54\) 0 0
\(55\) 232.016 0.568819
\(56\) −249.480 −0.595324
\(57\) 0 0
\(58\) 504.233 1.14154
\(59\) −254.149 −0.560803 −0.280401 0.959883i \(-0.590468\pi\)
−0.280401 + 0.959883i \(0.590468\pi\)
\(60\) 0 0
\(61\) −935.247 −1.96305 −0.981526 0.191330i \(-0.938720\pi\)
−0.981526 + 0.191330i \(0.938720\pi\)
\(62\) 667.742 1.36780
\(63\) 0 0
\(64\) 556.709 1.08732
\(65\) 0 0
\(66\) 0 0
\(67\) −240.494 −0.438522 −0.219261 0.975666i \(-0.570365\pi\)
−0.219261 + 0.975666i \(0.570365\pi\)
\(68\) 226.209 0.403409
\(69\) 0 0
\(70\) −366.220 −0.625309
\(71\) −947.455 −1.58369 −0.791847 0.610720i \(-0.790880\pi\)
−0.791847 + 0.610720i \(0.790880\pi\)
\(72\) 0 0
\(73\) −430.712 −0.690562 −0.345281 0.938499i \(-0.612216\pi\)
−0.345281 + 0.938499i \(0.612216\pi\)
\(74\) −842.384 −1.32331
\(75\) 0 0
\(76\) −285.490 −0.430894
\(77\) 152.654 0.225929
\(78\) 0 0
\(79\) −496.620 −0.707268 −0.353634 0.935384i \(-0.615054\pi\)
−0.353634 + 0.935384i \(0.615054\pi\)
\(80\) 583.251 0.815119
\(81\) 0 0
\(82\) 735.058 0.989922
\(83\) −392.527 −0.519102 −0.259551 0.965729i \(-0.583575\pi\)
−0.259551 + 0.965729i \(0.583575\pi\)
\(84\) 0 0
\(85\) 1398.78 1.78493
\(86\) 696.439 0.873244
\(87\) 0 0
\(88\) −371.002 −0.449419
\(89\) −979.895 −1.16706 −0.583532 0.812090i \(-0.698330\pi\)
−0.583532 + 0.812090i \(0.698330\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 188.675 0.213813
\(93\) 0 0
\(94\) −743.360 −0.815656
\(95\) −1765.35 −1.90654
\(96\) 0 0
\(97\) −553.356 −0.579225 −0.289613 0.957144i \(-0.593526\pi\)
−0.289613 + 0.957144i \(0.593526\pi\)
\(98\) 564.162 0.581520
\(99\) 0 0
\(100\) −279.243 −0.279243
\(101\) 763.951 0.752633 0.376317 0.926491i \(-0.377190\pi\)
0.376317 + 0.926491i \(0.377190\pi\)
\(102\) 0 0
\(103\) 182.518 0.174602 0.0873010 0.996182i \(-0.472176\pi\)
0.0873010 + 0.996182i \(0.472176\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 384.744 0.352544
\(107\) −183.699 −0.165971 −0.0829855 0.996551i \(-0.526446\pi\)
−0.0829855 + 0.996551i \(0.526446\pi\)
\(108\) 0 0
\(109\) 1774.42 1.55925 0.779627 0.626244i \(-0.215408\pi\)
0.779627 + 0.626244i \(0.215408\pi\)
\(110\) −544.605 −0.472055
\(111\) 0 0
\(112\) 383.747 0.323756
\(113\) −417.288 −0.347391 −0.173695 0.984799i \(-0.555571\pi\)
−0.173695 + 0.984799i \(0.555571\pi\)
\(114\) 0 0
\(115\) 1166.69 0.946037
\(116\) 534.963 0.428190
\(117\) 0 0
\(118\) 596.556 0.465402
\(119\) 920.318 0.708953
\(120\) 0 0
\(121\) −1103.99 −0.829443
\(122\) 2195.28 1.62911
\(123\) 0 0
\(124\) 708.436 0.513060
\(125\) 198.162 0.141793
\(126\) 0 0
\(127\) −1951.69 −1.36366 −0.681828 0.731513i \(-0.738815\pi\)
−0.681828 + 0.731513i \(0.738815\pi\)
\(128\) −442.072 −0.305266
\(129\) 0 0
\(130\) 0 0
\(131\) 1475.61 0.984159 0.492080 0.870550i \(-0.336237\pi\)
0.492080 + 0.870550i \(0.336237\pi\)
\(132\) 0 0
\(133\) −1161.50 −0.757255
\(134\) 564.503 0.363923
\(135\) 0 0
\(136\) −2236.69 −1.41026
\(137\) 1900.71 1.18532 0.592660 0.805453i \(-0.298078\pi\)
0.592660 + 0.805453i \(0.298078\pi\)
\(138\) 0 0
\(139\) −2326.76 −1.41981 −0.709903 0.704299i \(-0.751261\pi\)
−0.709903 + 0.704299i \(0.751261\pi\)
\(140\) −388.538 −0.234553
\(141\) 0 0
\(142\) 2223.93 1.31428
\(143\) 0 0
\(144\) 0 0
\(145\) 3307.98 1.89457
\(146\) 1011.00 0.573088
\(147\) 0 0
\(148\) −893.721 −0.496374
\(149\) 1370.92 0.753761 0.376881 0.926262i \(-0.376997\pi\)
0.376881 + 0.926262i \(0.376997\pi\)
\(150\) 0 0
\(151\) 1177.57 0.634628 0.317314 0.948320i \(-0.397219\pi\)
0.317314 + 0.948320i \(0.397219\pi\)
\(152\) 2822.85 1.50634
\(153\) 0 0
\(154\) −358.320 −0.187495
\(155\) 4380.67 2.27009
\(156\) 0 0
\(157\) 1621.57 0.824302 0.412151 0.911116i \(-0.364778\pi\)
0.412151 + 0.911116i \(0.364778\pi\)
\(158\) 1165.70 0.586951
\(159\) 0 0
\(160\) 1664.40 0.822389
\(161\) 767.616 0.375755
\(162\) 0 0
\(163\) −133.130 −0.0639727 −0.0319864 0.999488i \(-0.510183\pi\)
−0.0319864 + 0.999488i \(0.510183\pi\)
\(164\) 779.854 0.371320
\(165\) 0 0
\(166\) 921.368 0.430795
\(167\) 490.724 0.227385 0.113693 0.993516i \(-0.463732\pi\)
0.113693 + 0.993516i \(0.463732\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −3283.31 −1.48129
\(171\) 0 0
\(172\) 738.882 0.327554
\(173\) −2008.13 −0.882518 −0.441259 0.897380i \(-0.645468\pi\)
−0.441259 + 0.897380i \(0.645468\pi\)
\(174\) 0 0
\(175\) −1136.09 −0.490743
\(176\) 570.670 0.244408
\(177\) 0 0
\(178\) 2300.08 0.968529
\(179\) 2152.60 0.898843 0.449421 0.893320i \(-0.351630\pi\)
0.449421 + 0.893320i \(0.351630\pi\)
\(180\) 0 0
\(181\) 834.690 0.342774 0.171387 0.985204i \(-0.445175\pi\)
0.171387 + 0.985204i \(0.445175\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1865.57 −0.747455
\(185\) −5526.39 −2.19626
\(186\) 0 0
\(187\) 1368.60 0.535200
\(188\) −788.662 −0.305953
\(189\) 0 0
\(190\) 4143.75 1.58221
\(191\) −4464.71 −1.69139 −0.845695 0.533667i \(-0.820814\pi\)
−0.845695 + 0.533667i \(0.820814\pi\)
\(192\) 0 0
\(193\) −3299.84 −1.23071 −0.615357 0.788248i \(-0.710988\pi\)
−0.615357 + 0.788248i \(0.710988\pi\)
\(194\) 1298.88 0.480691
\(195\) 0 0
\(196\) 598.543 0.218128
\(197\) −2973.59 −1.07543 −0.537714 0.843127i \(-0.680712\pi\)
−0.537714 + 0.843127i \(0.680712\pi\)
\(198\) 0 0
\(199\) −5430.82 −1.93457 −0.967287 0.253684i \(-0.918358\pi\)
−0.967287 + 0.253684i \(0.918358\pi\)
\(200\) 2761.08 0.976191
\(201\) 0 0
\(202\) −1793.20 −0.624600
\(203\) 2176.47 0.752503
\(204\) 0 0
\(205\) 4822.29 1.64294
\(206\) −428.419 −0.144900
\(207\) 0 0
\(208\) 0 0
\(209\) −1727.27 −0.571663
\(210\) 0 0
\(211\) 1228.25 0.400742 0.200371 0.979720i \(-0.435785\pi\)
0.200371 + 0.979720i \(0.435785\pi\)
\(212\) 408.192 0.132239
\(213\) 0 0
\(214\) 431.192 0.137737
\(215\) 4568.93 1.44930
\(216\) 0 0
\(217\) 2882.24 0.901654
\(218\) −4165.05 −1.29400
\(219\) 0 0
\(220\) −577.795 −0.177068
\(221\) 0 0
\(222\) 0 0
\(223\) −685.256 −0.205776 −0.102888 0.994693i \(-0.532808\pi\)
−0.102888 + 0.994693i \(0.532808\pi\)
\(224\) 1095.08 0.326644
\(225\) 0 0
\(226\) 979.487 0.288294
\(227\) −287.067 −0.0839354 −0.0419677 0.999119i \(-0.513363\pi\)
−0.0419677 + 0.999119i \(0.513363\pi\)
\(228\) 0 0
\(229\) −2302.23 −0.664347 −0.332174 0.943218i \(-0.607782\pi\)
−0.332174 + 0.943218i \(0.607782\pi\)
\(230\) −2738.53 −0.785102
\(231\) 0 0
\(232\) −5289.57 −1.49688
\(233\) 970.620 0.272908 0.136454 0.990646i \(-0.456429\pi\)
0.136454 + 0.990646i \(0.456429\pi\)
\(234\) 0 0
\(235\) −4876.75 −1.35372
\(236\) 632.912 0.174572
\(237\) 0 0
\(238\) −2160.24 −0.588350
\(239\) 5007.40 1.35524 0.677619 0.735413i \(-0.263012\pi\)
0.677619 + 0.735413i \(0.263012\pi\)
\(240\) 0 0
\(241\) −540.092 −0.144358 −0.0721792 0.997392i \(-0.522995\pi\)
−0.0721792 + 0.997392i \(0.522995\pi\)
\(242\) 2591.36 0.688343
\(243\) 0 0
\(244\) 2329.07 0.611078
\(245\) 3701.14 0.965130
\(246\) 0 0
\(247\) 0 0
\(248\) −7004.83 −1.79358
\(249\) 0 0
\(250\) −465.141 −0.117672
\(251\) 6087.80 1.53091 0.765455 0.643489i \(-0.222514\pi\)
0.765455 + 0.643489i \(0.222514\pi\)
\(252\) 0 0
\(253\) 1141.52 0.283663
\(254\) 4581.14 1.13168
\(255\) 0 0
\(256\) −3416.01 −0.833987
\(257\) 5096.34 1.23697 0.618484 0.785797i \(-0.287747\pi\)
0.618484 + 0.785797i \(0.287747\pi\)
\(258\) 0 0
\(259\) −3636.06 −0.872330
\(260\) 0 0
\(261\) 0 0
\(262\) −3463.66 −0.816739
\(263\) −3405.50 −0.798450 −0.399225 0.916853i \(-0.630721\pi\)
−0.399225 + 0.916853i \(0.630721\pi\)
\(264\) 0 0
\(265\) 2524.08 0.585106
\(266\) 2726.36 0.628435
\(267\) 0 0
\(268\) 598.906 0.136507
\(269\) 2720.44 0.616611 0.308306 0.951287i \(-0.400238\pi\)
0.308306 + 0.951287i \(0.400238\pi\)
\(270\) 0 0
\(271\) 6954.27 1.55883 0.779413 0.626511i \(-0.215518\pi\)
0.779413 + 0.626511i \(0.215518\pi\)
\(272\) 3440.45 0.766941
\(273\) 0 0
\(274\) −4461.49 −0.983680
\(275\) −1689.47 −0.370470
\(276\) 0 0
\(277\) −6563.96 −1.42379 −0.711895 0.702286i \(-0.752163\pi\)
−0.711895 + 0.702286i \(0.752163\pi\)
\(278\) 5461.53 1.17828
\(279\) 0 0
\(280\) 3841.76 0.819961
\(281\) 652.800 0.138586 0.0692932 0.997596i \(-0.477926\pi\)
0.0692932 + 0.997596i \(0.477926\pi\)
\(282\) 0 0
\(283\) 6010.62 1.26252 0.631262 0.775570i \(-0.282537\pi\)
0.631262 + 0.775570i \(0.282537\pi\)
\(284\) 2359.47 0.492988
\(285\) 0 0
\(286\) 0 0
\(287\) 3172.80 0.652558
\(288\) 0 0
\(289\) 3338.04 0.679430
\(290\) −7764.73 −1.57228
\(291\) 0 0
\(292\) 1072.61 0.214965
\(293\) 1912.72 0.381372 0.190686 0.981651i \(-0.438929\pi\)
0.190686 + 0.981651i \(0.438929\pi\)
\(294\) 0 0
\(295\) 3913.66 0.772413
\(296\) 8836.87 1.73525
\(297\) 0 0
\(298\) −3217.93 −0.625535
\(299\) 0 0
\(300\) 0 0
\(301\) 3006.10 0.575644
\(302\) −2764.06 −0.526669
\(303\) 0 0
\(304\) −4342.07 −0.819194
\(305\) 14401.9 2.70378
\(306\) 0 0
\(307\) −1983.07 −0.368665 −0.184332 0.982864i \(-0.559012\pi\)
−0.184332 + 0.982864i \(0.559012\pi\)
\(308\) −380.157 −0.0703294
\(309\) 0 0
\(310\) −10282.6 −1.88391
\(311\) −1893.76 −0.345291 −0.172645 0.984984i \(-0.555231\pi\)
−0.172645 + 0.984984i \(0.555231\pi\)
\(312\) 0 0
\(313\) 4574.19 0.826034 0.413017 0.910723i \(-0.364475\pi\)
0.413017 + 0.910723i \(0.364475\pi\)
\(314\) −3806.26 −0.684076
\(315\) 0 0
\(316\) 1236.74 0.220165
\(317\) −7594.53 −1.34559 −0.672794 0.739830i \(-0.734906\pi\)
−0.672794 + 0.739830i \(0.734906\pi\)
\(318\) 0 0
\(319\) 3236.62 0.568076
\(320\) −8572.81 −1.49761
\(321\) 0 0
\(322\) −1801.80 −0.311834
\(323\) −10413.3 −1.79385
\(324\) 0 0
\(325\) 0 0
\(326\) 312.492 0.0530901
\(327\) 0 0
\(328\) −7710.99 −1.29807
\(329\) −3208.63 −0.537682
\(330\) 0 0
\(331\) −1738.58 −0.288705 −0.144352 0.989526i \(-0.546110\pi\)
−0.144352 + 0.989526i \(0.546110\pi\)
\(332\) 977.519 0.161591
\(333\) 0 0
\(334\) −1151.86 −0.188704
\(335\) 3703.38 0.603992
\(336\) 0 0
\(337\) −2710.61 −0.438149 −0.219074 0.975708i \(-0.570304\pi\)
−0.219074 + 0.975708i \(0.570304\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −3483.41 −0.555630
\(341\) 4286.17 0.680672
\(342\) 0 0
\(343\) 5910.33 0.930401
\(344\) −7305.87 −1.14508
\(345\) 0 0
\(346\) 4713.63 0.732389
\(347\) 2506.81 0.387817 0.193908 0.981020i \(-0.437884\pi\)
0.193908 + 0.981020i \(0.437884\pi\)
\(348\) 0 0
\(349\) 7536.48 1.15593 0.577963 0.816063i \(-0.303848\pi\)
0.577963 + 0.816063i \(0.303848\pi\)
\(350\) 2666.70 0.407261
\(351\) 0 0
\(352\) 1628.50 0.246588
\(353\) 8992.88 1.35593 0.677964 0.735095i \(-0.262862\pi\)
0.677964 + 0.735095i \(0.262862\pi\)
\(354\) 0 0
\(355\) 14589.9 2.18128
\(356\) 2440.25 0.363295
\(357\) 0 0
\(358\) −5052.74 −0.745937
\(359\) 4566.64 0.671359 0.335679 0.941976i \(-0.391034\pi\)
0.335679 + 0.941976i \(0.391034\pi\)
\(360\) 0 0
\(361\) 6283.31 0.916068
\(362\) −1959.24 −0.284463
\(363\) 0 0
\(364\) 0 0
\(365\) 6632.57 0.951136
\(366\) 0 0
\(367\) −6449.50 −0.917332 −0.458666 0.888609i \(-0.651673\pi\)
−0.458666 + 0.888609i \(0.651673\pi\)
\(368\) 2869.60 0.406490
\(369\) 0 0
\(370\) 12971.9 1.82264
\(371\) 1660.71 0.232398
\(372\) 0 0
\(373\) 7648.89 1.06178 0.530891 0.847440i \(-0.321857\pi\)
0.530891 + 0.847440i \(0.321857\pi\)
\(374\) −3212.49 −0.444154
\(375\) 0 0
\(376\) 7798.08 1.06956
\(377\) 0 0
\(378\) 0 0
\(379\) −10297.8 −1.39567 −0.697837 0.716256i \(-0.745854\pi\)
−0.697837 + 0.716256i \(0.745854\pi\)
\(380\) 4396.28 0.593486
\(381\) 0 0
\(382\) 10479.9 1.40366
\(383\) −10258.0 −1.36856 −0.684282 0.729217i \(-0.739884\pi\)
−0.684282 + 0.729217i \(0.739884\pi\)
\(384\) 0 0
\(385\) −2350.73 −0.311180
\(386\) 7745.63 1.02135
\(387\) 0 0
\(388\) 1378.03 0.180307
\(389\) 4771.57 0.621924 0.310962 0.950422i \(-0.399349\pi\)
0.310962 + 0.950422i \(0.399349\pi\)
\(390\) 0 0
\(391\) 6882.00 0.890122
\(392\) −5918.24 −0.762541
\(393\) 0 0
\(394\) 6979.82 0.892483
\(395\) 7647.50 0.974145
\(396\) 0 0
\(397\) 2291.22 0.289655 0.144827 0.989457i \(-0.453737\pi\)
0.144827 + 0.989457i \(0.453737\pi\)
\(398\) 12747.6 1.60548
\(399\) 0 0
\(400\) −4247.07 −0.530883
\(401\) −7534.63 −0.938308 −0.469154 0.883116i \(-0.655441\pi\)
−0.469154 + 0.883116i \(0.655441\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −1902.48 −0.234287
\(405\) 0 0
\(406\) −5108.76 −0.624491
\(407\) −5407.18 −0.658535
\(408\) 0 0
\(409\) −1517.97 −0.183517 −0.0917587 0.995781i \(-0.529249\pi\)
−0.0917587 + 0.995781i \(0.529249\pi\)
\(410\) −11319.2 −1.36345
\(411\) 0 0
\(412\) −454.528 −0.0543519
\(413\) 2574.97 0.306794
\(414\) 0 0
\(415\) 6044.56 0.714978
\(416\) 0 0
\(417\) 0 0
\(418\) 4054.36 0.474415
\(419\) −8080.97 −0.942199 −0.471100 0.882080i \(-0.656143\pi\)
−0.471100 + 0.882080i \(0.656143\pi\)
\(420\) 0 0
\(421\) 8073.13 0.934585 0.467293 0.884103i \(-0.345229\pi\)
0.467293 + 0.884103i \(0.345229\pi\)
\(422\) −2883.05 −0.332570
\(423\) 0 0
\(424\) −4036.09 −0.462287
\(425\) −10185.5 −1.16252
\(426\) 0 0
\(427\) 9475.68 1.07391
\(428\) 457.470 0.0516651
\(429\) 0 0
\(430\) −10724.5 −1.20275
\(431\) −2241.39 −0.250496 −0.125248 0.992125i \(-0.539973\pi\)
−0.125248 + 0.992125i \(0.539973\pi\)
\(432\) 0 0
\(433\) 10237.9 1.13626 0.568130 0.822939i \(-0.307667\pi\)
0.568130 + 0.822939i \(0.307667\pi\)
\(434\) −6765.39 −0.748270
\(435\) 0 0
\(436\) −4418.88 −0.485380
\(437\) −8685.52 −0.950766
\(438\) 0 0
\(439\) −5416.69 −0.588894 −0.294447 0.955668i \(-0.595135\pi\)
−0.294447 + 0.955668i \(0.595135\pi\)
\(440\) 5713.08 0.619001
\(441\) 0 0
\(442\) 0 0
\(443\) 2537.44 0.272138 0.136069 0.990699i \(-0.456553\pi\)
0.136069 + 0.990699i \(0.456553\pi\)
\(444\) 0 0
\(445\) 15089.5 1.60744
\(446\) 1608.48 0.170771
\(447\) 0 0
\(448\) −5640.43 −0.594833
\(449\) −7790.61 −0.818846 −0.409423 0.912345i \(-0.634270\pi\)
−0.409423 + 0.912345i \(0.634270\pi\)
\(450\) 0 0
\(451\) 4718.26 0.492626
\(452\) 1039.18 0.108139
\(453\) 0 0
\(454\) 673.825 0.0696568
\(455\) 0 0
\(456\) 0 0
\(457\) 6599.71 0.675539 0.337770 0.941229i \(-0.390328\pi\)
0.337770 + 0.941229i \(0.390328\pi\)
\(458\) 5403.95 0.551332
\(459\) 0 0
\(460\) −2905.43 −0.294492
\(461\) −4482.57 −0.452872 −0.226436 0.974026i \(-0.572707\pi\)
−0.226436 + 0.974026i \(0.572707\pi\)
\(462\) 0 0
\(463\) −3805.86 −0.382016 −0.191008 0.981589i \(-0.561176\pi\)
−0.191008 + 0.981589i \(0.561176\pi\)
\(464\) 8136.35 0.814053
\(465\) 0 0
\(466\) −2278.31 −0.226482
\(467\) −14778.8 −1.46441 −0.732207 0.681082i \(-0.761510\pi\)
−0.732207 + 0.681082i \(0.761510\pi\)
\(468\) 0 0
\(469\) 2436.62 0.239899
\(470\) 11447.0 1.12343
\(471\) 0 0
\(472\) −6258.06 −0.610277
\(473\) 4470.37 0.434562
\(474\) 0 0
\(475\) 12854.8 1.24172
\(476\) −2291.89 −0.220690
\(477\) 0 0
\(478\) −11753.7 −1.12469
\(479\) −11171.2 −1.06560 −0.532801 0.846240i \(-0.678861\pi\)
−0.532801 + 0.846240i \(0.678861\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 1267.74 0.119801
\(483\) 0 0
\(484\) 2749.29 0.258197
\(485\) 8521.18 0.797787
\(486\) 0 0
\(487\) 6046.57 0.562621 0.281310 0.959617i \(-0.409231\pi\)
0.281310 + 0.959617i \(0.409231\pi\)
\(488\) −23029.2 −2.13623
\(489\) 0 0
\(490\) −8687.57 −0.800948
\(491\) 1035.04 0.0951338 0.0475669 0.998868i \(-0.484853\pi\)
0.0475669 + 0.998868i \(0.484853\pi\)
\(492\) 0 0
\(493\) 19512.9 1.78259
\(494\) 0 0
\(495\) 0 0
\(496\) 10774.7 0.975404
\(497\) 9599.37 0.866379
\(498\) 0 0
\(499\) −11698.3 −1.04947 −0.524736 0.851265i \(-0.675836\pi\)
−0.524736 + 0.851265i \(0.675836\pi\)
\(500\) −493.488 −0.0441389
\(501\) 0 0
\(502\) −14289.7 −1.27048
\(503\) 13552.0 1.20130 0.600651 0.799511i \(-0.294908\pi\)
0.600651 + 0.799511i \(0.294908\pi\)
\(504\) 0 0
\(505\) −11764.1 −1.03663
\(506\) −2679.46 −0.235408
\(507\) 0 0
\(508\) 4860.33 0.424492
\(509\) −5076.46 −0.442064 −0.221032 0.975267i \(-0.570942\pi\)
−0.221032 + 0.975267i \(0.570942\pi\)
\(510\) 0 0
\(511\) 4363.86 0.377781
\(512\) 11554.9 0.997380
\(513\) 0 0
\(514\) −11962.5 −1.02654
\(515\) −2810.60 −0.240486
\(516\) 0 0
\(517\) −4771.55 −0.405904
\(518\) 8534.81 0.723934
\(519\) 0 0
\(520\) 0 0
\(521\) 8493.73 0.714236 0.357118 0.934059i \(-0.383759\pi\)
0.357118 + 0.934059i \(0.383759\pi\)
\(522\) 0 0
\(523\) −1384.53 −0.115758 −0.0578789 0.998324i \(-0.518434\pi\)
−0.0578789 + 0.998324i \(0.518434\pi\)
\(524\) −3674.75 −0.306359
\(525\) 0 0
\(526\) 7993.64 0.662622
\(527\) 25840.4 2.13592
\(528\) 0 0
\(529\) −6426.89 −0.528223
\(530\) −5924.71 −0.485571
\(531\) 0 0
\(532\) 2892.51 0.235726
\(533\) 0 0
\(534\) 0 0
\(535\) 2828.80 0.228598
\(536\) −5921.82 −0.477208
\(537\) 0 0
\(538\) −6385.62 −0.511717
\(539\) 3621.30 0.289388
\(540\) 0 0
\(541\) 5026.83 0.399483 0.199742 0.979849i \(-0.435990\pi\)
0.199742 + 0.979849i \(0.435990\pi\)
\(542\) −16323.6 −1.29365
\(543\) 0 0
\(544\) 9817.87 0.773782
\(545\) −27324.5 −2.14762
\(546\) 0 0
\(547\) 1540.36 0.120404 0.0602021 0.998186i \(-0.480825\pi\)
0.0602021 + 0.998186i \(0.480825\pi\)
\(548\) −4733.38 −0.368978
\(549\) 0 0
\(550\) 3965.65 0.307447
\(551\) −24626.6 −1.90404
\(552\) 0 0
\(553\) 5031.63 0.386920
\(554\) 15407.4 1.18158
\(555\) 0 0
\(556\) 5794.38 0.441972
\(557\) 8550.51 0.650443 0.325221 0.945638i \(-0.394561\pi\)
0.325221 + 0.945638i \(0.394561\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −5909.35 −0.445921
\(561\) 0 0
\(562\) −1532.30 −0.115011
\(563\) 6569.35 0.491768 0.245884 0.969299i \(-0.420922\pi\)
0.245884 + 0.969299i \(0.420922\pi\)
\(564\) 0 0
\(565\) 6425.85 0.478473
\(566\) −14108.5 −1.04775
\(567\) 0 0
\(568\) −23329.8 −1.72341
\(569\) −23766.1 −1.75102 −0.875508 0.483204i \(-0.839473\pi\)
−0.875508 + 0.483204i \(0.839473\pi\)
\(570\) 0 0
\(571\) −24971.7 −1.83018 −0.915091 0.403248i \(-0.867881\pi\)
−0.915091 + 0.403248i \(0.867881\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −7447.41 −0.541549
\(575\) −8495.48 −0.616150
\(576\) 0 0
\(577\) −16643.4 −1.20082 −0.600409 0.799693i \(-0.704996\pi\)
−0.600409 + 0.799693i \(0.704996\pi\)
\(578\) −7835.28 −0.563849
\(579\) 0 0
\(580\) −8237.93 −0.589761
\(581\) 3976.98 0.283981
\(582\) 0 0
\(583\) 2469.64 0.175441
\(584\) −10605.7 −0.751484
\(585\) 0 0
\(586\) −4489.66 −0.316495
\(587\) 6720.67 0.472558 0.236279 0.971685i \(-0.424072\pi\)
0.236279 + 0.971685i \(0.424072\pi\)
\(588\) 0 0
\(589\) −32612.3 −2.28144
\(590\) −9186.41 −0.641014
\(591\) 0 0
\(592\) −13592.8 −0.943681
\(593\) −26349.8 −1.82471 −0.912357 0.409395i \(-0.865740\pi\)
−0.912357 + 0.409395i \(0.865740\pi\)
\(594\) 0 0
\(595\) −14172.1 −0.976466
\(596\) −3414.04 −0.234638
\(597\) 0 0
\(598\) 0 0
\(599\) 6136.81 0.418603 0.209302 0.977851i \(-0.432881\pi\)
0.209302 + 0.977851i \(0.432881\pi\)
\(600\) 0 0
\(601\) 12493.7 0.847966 0.423983 0.905670i \(-0.360632\pi\)
0.423983 + 0.905670i \(0.360632\pi\)
\(602\) −7056.14 −0.477719
\(603\) 0 0
\(604\) −2932.51 −0.197553
\(605\) 17000.4 1.14242
\(606\) 0 0
\(607\) 18696.6 1.25020 0.625099 0.780545i \(-0.285059\pi\)
0.625099 + 0.780545i \(0.285059\pi\)
\(608\) −12390.8 −0.826501
\(609\) 0 0
\(610\) −33805.3 −2.24383
\(611\) 0 0
\(612\) 0 0
\(613\) 1238.64 0.0816123 0.0408062 0.999167i \(-0.487007\pi\)
0.0408062 + 0.999167i \(0.487007\pi\)
\(614\) 4654.81 0.305949
\(615\) 0 0
\(616\) 3758.89 0.245860
\(617\) −16549.5 −1.07983 −0.539917 0.841718i \(-0.681544\pi\)
−0.539917 + 0.841718i \(0.681544\pi\)
\(618\) 0 0
\(619\) 13945.4 0.905513 0.452756 0.891634i \(-0.350441\pi\)
0.452756 + 0.891634i \(0.350441\pi\)
\(620\) −10909.3 −0.706656
\(621\) 0 0
\(622\) 4445.17 0.286552
\(623\) 9928.03 0.638456
\(624\) 0 0
\(625\) −17068.0 −1.09235
\(626\) −10736.9 −0.685513
\(627\) 0 0
\(628\) −4038.23 −0.256597
\(629\) −32598.8 −2.06645
\(630\) 0 0
\(631\) 15343.1 0.967987 0.483994 0.875072i \(-0.339186\pi\)
0.483994 + 0.875072i \(0.339186\pi\)
\(632\) −12228.6 −0.769663
\(633\) 0 0
\(634\) 17826.4 1.11668
\(635\) 30054.2 1.87821
\(636\) 0 0
\(637\) 0 0
\(638\) −7597.23 −0.471438
\(639\) 0 0
\(640\) 6807.51 0.420454
\(641\) −11067.7 −0.681979 −0.340989 0.940067i \(-0.610762\pi\)
−0.340989 + 0.940067i \(0.610762\pi\)
\(642\) 0 0
\(643\) −25118.7 −1.54057 −0.770284 0.637701i \(-0.779885\pi\)
−0.770284 + 0.637701i \(0.779885\pi\)
\(644\) −1911.61 −0.116969
\(645\) 0 0
\(646\) 24442.9 1.48869
\(647\) 4447.03 0.270217 0.135109 0.990831i \(-0.456862\pi\)
0.135109 + 0.990831i \(0.456862\pi\)
\(648\) 0 0
\(649\) 3829.23 0.231603
\(650\) 0 0
\(651\) 0 0
\(652\) 331.537 0.0199141
\(653\) 19426.2 1.16418 0.582088 0.813126i \(-0.302236\pi\)
0.582088 + 0.813126i \(0.302236\pi\)
\(654\) 0 0
\(655\) −22723.1 −1.35552
\(656\) 11861.0 0.705934
\(657\) 0 0
\(658\) 7531.52 0.446215
\(659\) 14099.4 0.833438 0.416719 0.909035i \(-0.363180\pi\)
0.416719 + 0.909035i \(0.363180\pi\)
\(660\) 0 0
\(661\) 2754.25 0.162070 0.0810348 0.996711i \(-0.474177\pi\)
0.0810348 + 0.996711i \(0.474177\pi\)
\(662\) 4080.93 0.239592
\(663\) 0 0
\(664\) −9665.44 −0.564898
\(665\) 17886.0 1.04299
\(666\) 0 0
\(667\) 16275.3 0.944800
\(668\) −1222.06 −0.0707828
\(669\) 0 0
\(670\) −8692.83 −0.501244
\(671\) 14091.3 0.810712
\(672\) 0 0
\(673\) 11936.8 0.683700 0.341850 0.939754i \(-0.388946\pi\)
0.341850 + 0.939754i \(0.388946\pi\)
\(674\) 6362.53 0.363613
\(675\) 0 0
\(676\) 0 0
\(677\) 10255.2 0.582186 0.291093 0.956695i \(-0.405981\pi\)
0.291093 + 0.956695i \(0.405981\pi\)
\(678\) 0 0
\(679\) 5606.46 0.316872
\(680\) 34443.0 1.94239
\(681\) 0 0
\(682\) −10060.8 −0.564880
\(683\) −10605.7 −0.594169 −0.297085 0.954851i \(-0.596014\pi\)
−0.297085 + 0.954851i \(0.596014\pi\)
\(684\) 0 0
\(685\) −29269.2 −1.63258
\(686\) −13873.1 −0.772127
\(687\) 0 0
\(688\) 11237.8 0.622728
\(689\) 0 0
\(690\) 0 0
\(691\) 4660.14 0.256556 0.128278 0.991738i \(-0.459055\pi\)
0.128278 + 0.991738i \(0.459055\pi\)
\(692\) 5000.90 0.274719
\(693\) 0 0
\(694\) −5884.15 −0.321844
\(695\) 35829.9 1.95555
\(696\) 0 0
\(697\) 28445.4 1.54584
\(698\) −17690.2 −0.959287
\(699\) 0 0
\(700\) 2829.22 0.152764
\(701\) 24016.9 1.29402 0.647008 0.762483i \(-0.276020\pi\)
0.647008 + 0.762483i \(0.276020\pi\)
\(702\) 0 0
\(703\) 41141.7 2.20724
\(704\) −8387.88 −0.449048
\(705\) 0 0
\(706\) −21108.7 −1.12527
\(707\) −7740.15 −0.411737
\(708\) 0 0
\(709\) −7252.56 −0.384169 −0.192084 0.981378i \(-0.561525\pi\)
−0.192084 + 0.981378i \(0.561525\pi\)
\(710\) −34246.5 −1.81021
\(711\) 0 0
\(712\) −24128.6 −1.27002
\(713\) 21552.9 1.13207
\(714\) 0 0
\(715\) 0 0
\(716\) −5360.66 −0.279801
\(717\) 0 0
\(718\) −10719.1 −0.557151
\(719\) −10951.7 −0.568050 −0.284025 0.958817i \(-0.591670\pi\)
−0.284025 + 0.958817i \(0.591670\pi\)
\(720\) 0 0
\(721\) −1849.22 −0.0955182
\(722\) −14748.6 −0.760231
\(723\) 0 0
\(724\) −2078.65 −0.106702
\(725\) −24087.7 −1.23393
\(726\) 0 0
\(727\) −27856.0 −1.42108 −0.710538 0.703658i \(-0.751549\pi\)
−0.710538 + 0.703658i \(0.751549\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −15568.4 −0.789334
\(731\) 26951.0 1.36364
\(732\) 0 0
\(733\) −31213.8 −1.57286 −0.786431 0.617678i \(-0.788073\pi\)
−0.786431 + 0.617678i \(0.788073\pi\)
\(734\) 15138.7 0.761281
\(735\) 0 0
\(736\) 8188.85 0.410116
\(737\) 3623.49 0.181103
\(738\) 0 0
\(739\) 14423.1 0.717946 0.358973 0.933348i \(-0.383127\pi\)
0.358973 + 0.933348i \(0.383127\pi\)
\(740\) 13762.5 0.683674
\(741\) 0 0
\(742\) −3898.13 −0.192864
\(743\) −13469.7 −0.665079 −0.332539 0.943089i \(-0.607905\pi\)
−0.332539 + 0.943089i \(0.607905\pi\)
\(744\) 0 0
\(745\) −21111.0 −1.03818
\(746\) −17954.0 −0.881158
\(747\) 0 0
\(748\) −3408.26 −0.166602
\(749\) 1861.19 0.0907965
\(750\) 0 0
\(751\) −32033.6 −1.55649 −0.778245 0.627961i \(-0.783890\pi\)
−0.778245 + 0.627961i \(0.783890\pi\)
\(752\) −11994.9 −0.581661
\(753\) 0 0
\(754\) 0 0
\(755\) −18133.4 −0.874096
\(756\) 0 0
\(757\) −26097.2 −1.25300 −0.626498 0.779423i \(-0.715512\pi\)
−0.626498 + 0.779423i \(0.715512\pi\)
\(758\) 24171.6 1.15825
\(759\) 0 0
\(760\) −43469.3 −2.07473
\(761\) −18238.2 −0.868772 −0.434386 0.900727i \(-0.643035\pi\)
−0.434386 + 0.900727i \(0.643035\pi\)
\(762\) 0 0
\(763\) −17978.0 −0.853009
\(764\) 11118.6 0.526513
\(765\) 0 0
\(766\) 24078.4 1.13575
\(767\) 0 0
\(768\) 0 0
\(769\) −18817.0 −0.882390 −0.441195 0.897411i \(-0.645445\pi\)
−0.441195 + 0.897411i \(0.645445\pi\)
\(770\) 5517.79 0.258243
\(771\) 0 0
\(772\) 8217.67 0.383109
\(773\) 13795.6 0.641904 0.320952 0.947095i \(-0.395997\pi\)
0.320952 + 0.947095i \(0.395997\pi\)
\(774\) 0 0
\(775\) −31898.7 −1.47850
\(776\) −13625.6 −0.630325
\(777\) 0 0
\(778\) −11200.2 −0.516126
\(779\) −35900.0 −1.65116
\(780\) 0 0
\(781\) 14275.2 0.654043
\(782\) −16153.9 −0.738699
\(783\) 0 0
\(784\) 9103.36 0.414694
\(785\) −24970.7 −1.13534
\(786\) 0 0
\(787\) −40545.4 −1.83645 −0.918227 0.396055i \(-0.870379\pi\)
−0.918227 + 0.396055i \(0.870379\pi\)
\(788\) 7405.19 0.334770
\(789\) 0 0
\(790\) −17950.7 −0.808429
\(791\) 4227.85 0.190044
\(792\) 0 0
\(793\) 0 0
\(794\) −5378.11 −0.240380
\(795\) 0 0
\(796\) 13524.5 0.602214
\(797\) 31576.4 1.40338 0.701690 0.712482i \(-0.252429\pi\)
0.701690 + 0.712482i \(0.252429\pi\)
\(798\) 0 0
\(799\) −28766.7 −1.27371
\(800\) −12119.7 −0.535619
\(801\) 0 0
\(802\) 17685.8 0.778688
\(803\) 6489.50 0.285192
\(804\) 0 0
\(805\) −11820.6 −0.517541
\(806\) 0 0
\(807\) 0 0
\(808\) 18811.2 0.819031
\(809\) −31130.1 −1.35287 −0.676437 0.736501i \(-0.736477\pi\)
−0.676437 + 0.736501i \(0.736477\pi\)
\(810\) 0 0
\(811\) −8733.71 −0.378153 −0.189076 0.981962i \(-0.560549\pi\)
−0.189076 + 0.981962i \(0.560549\pi\)
\(812\) −5420.10 −0.234247
\(813\) 0 0
\(814\) 12692.1 0.546509
\(815\) 2050.08 0.0881119
\(816\) 0 0
\(817\) −34013.8 −1.45654
\(818\) 3563.08 0.152298
\(819\) 0 0
\(820\) −12009.0 −0.511431
\(821\) 36960.1 1.57115 0.785576 0.618765i \(-0.212367\pi\)
0.785576 + 0.618765i \(0.212367\pi\)
\(822\) 0 0
\(823\) 20509.7 0.868679 0.434340 0.900749i \(-0.356982\pi\)
0.434340 + 0.900749i \(0.356982\pi\)
\(824\) 4494.25 0.190006
\(825\) 0 0
\(826\) −6044.15 −0.254604
\(827\) 11533.4 0.484952 0.242476 0.970157i \(-0.422040\pi\)
0.242476 + 0.970157i \(0.422040\pi\)
\(828\) 0 0
\(829\) 34096.4 1.42849 0.714244 0.699897i \(-0.246771\pi\)
0.714244 + 0.699897i \(0.246771\pi\)
\(830\) −14188.2 −0.593350
\(831\) 0 0
\(832\) 0 0
\(833\) 21832.1 0.908087
\(834\) 0 0
\(835\) −7556.69 −0.313186
\(836\) 4301.45 0.177953
\(837\) 0 0
\(838\) 18968.2 0.781917
\(839\) 1855.42 0.0763484 0.0381742 0.999271i \(-0.487846\pi\)
0.0381742 + 0.999271i \(0.487846\pi\)
\(840\) 0 0
\(841\) 21757.3 0.892094
\(842\) −18949.8 −0.775599
\(843\) 0 0
\(844\) −3058.75 −0.124747
\(845\) 0 0
\(846\) 0 0
\(847\) 11185.3 0.453757
\(848\) 6208.26 0.251406
\(849\) 0 0
\(850\) 23908.1 0.964755
\(851\) −27189.9 −1.09525
\(852\) 0 0
\(853\) −28668.7 −1.15076 −0.575380 0.817886i \(-0.695146\pi\)
−0.575380 + 0.817886i \(0.695146\pi\)
\(854\) −22242.0 −0.891224
\(855\) 0 0
\(856\) −4523.35 −0.180613
\(857\) −449.310 −0.0179091 −0.00895457 0.999960i \(-0.502850\pi\)
−0.00895457 + 0.999960i \(0.502850\pi\)
\(858\) 0 0
\(859\) 33466.9 1.32931 0.664654 0.747151i \(-0.268579\pi\)
0.664654 + 0.747151i \(0.268579\pi\)
\(860\) −11378.1 −0.451151
\(861\) 0 0
\(862\) 5261.14 0.207883
\(863\) 25097.5 0.989953 0.494976 0.868906i \(-0.335177\pi\)
0.494976 + 0.868906i \(0.335177\pi\)
\(864\) 0 0
\(865\) 30923.4 1.21552
\(866\) −24031.0 −0.942965
\(867\) 0 0
\(868\) −7177.69 −0.280676
\(869\) 7482.53 0.292091
\(870\) 0 0
\(871\) 0 0
\(872\) 43692.7 1.69681
\(873\) 0 0
\(874\) 20387.3 0.789027
\(875\) −2007.73 −0.0775698
\(876\) 0 0
\(877\) 27015.2 1.04018 0.520090 0.854112i \(-0.325899\pi\)
0.520090 + 0.854112i \(0.325899\pi\)
\(878\) 12714.4 0.488715
\(879\) 0 0
\(880\) −8787.79 −0.336632
\(881\) −48638.7 −1.86002 −0.930010 0.367534i \(-0.880202\pi\)
−0.930010 + 0.367534i \(0.880202\pi\)
\(882\) 0 0
\(883\) −25479.0 −0.971048 −0.485524 0.874223i \(-0.661371\pi\)
−0.485524 + 0.874223i \(0.661371\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −5956.05 −0.225843
\(887\) −42359.7 −1.60350 −0.801748 0.597663i \(-0.796096\pi\)
−0.801748 + 0.597663i \(0.796096\pi\)
\(888\) 0 0
\(889\) 19774.0 0.746005
\(890\) −35419.1 −1.33399
\(891\) 0 0
\(892\) 1706.51 0.0640561
\(893\) 36305.4 1.36049
\(894\) 0 0
\(895\) −33148.1 −1.23801
\(896\) 4478.96 0.167000
\(897\) 0 0
\(898\) 18286.7 0.679548
\(899\) 61110.3 2.26712
\(900\) 0 0
\(901\) 14888.9 0.550524
\(902\) −11075.0 −0.408823
\(903\) 0 0
\(904\) −10275.1 −0.378038
\(905\) −12853.5 −0.472114
\(906\) 0 0
\(907\) −5314.91 −0.194574 −0.0972871 0.995256i \(-0.531016\pi\)
−0.0972871 + 0.995256i \(0.531016\pi\)
\(908\) 714.890 0.0261283
\(909\) 0 0
\(910\) 0 0
\(911\) −2471.71 −0.0898919 −0.0449459 0.998989i \(-0.514312\pi\)
−0.0449459 + 0.998989i \(0.514312\pi\)
\(912\) 0 0
\(913\) 5914.17 0.214382
\(914\) −15491.3 −0.560620
\(915\) 0 0
\(916\) 5733.29 0.206805
\(917\) −14950.5 −0.538396
\(918\) 0 0
\(919\) −9636.13 −0.345883 −0.172942 0.984932i \(-0.555327\pi\)
−0.172942 + 0.984932i \(0.555327\pi\)
\(920\) 28728.1 1.02950
\(921\) 0 0
\(922\) 10521.8 0.375832
\(923\) 0 0
\(924\) 0 0
\(925\) 40241.5 1.43041
\(926\) 8933.38 0.317029
\(927\) 0 0
\(928\) 23218.3 0.821314
\(929\) 53249.1 1.88057 0.940283 0.340394i \(-0.110561\pi\)
0.940283 + 0.340394i \(0.110561\pi\)
\(930\) 0 0
\(931\) −27553.5 −0.969955
\(932\) −2417.16 −0.0849534
\(933\) 0 0
\(934\) 34689.8 1.21530
\(935\) −21075.3 −0.737149
\(936\) 0 0
\(937\) −41122.6 −1.43374 −0.716871 0.697205i \(-0.754427\pi\)
−0.716871 + 0.697205i \(0.754427\pi\)
\(938\) −5719.40 −0.199088
\(939\) 0 0
\(940\) 12144.7 0.421399
\(941\) 8005.71 0.277342 0.138671 0.990339i \(-0.455717\pi\)
0.138671 + 0.990339i \(0.455717\pi\)
\(942\) 0 0
\(943\) 23725.7 0.819315
\(944\) 9626.07 0.331888
\(945\) 0 0
\(946\) −10493.2 −0.360637
\(947\) 17468.6 0.599424 0.299712 0.954030i \(-0.403109\pi\)
0.299712 + 0.954030i \(0.403109\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −30173.6 −1.03048
\(951\) 0 0
\(952\) 22661.6 0.771498
\(953\) −30681.4 −1.04288 −0.521442 0.853287i \(-0.674606\pi\)
−0.521442 + 0.853287i \(0.674606\pi\)
\(954\) 0 0
\(955\) 68752.5 2.32961
\(956\) −12470.0 −0.421872
\(957\) 0 0
\(958\) 26221.8 0.884328
\(959\) −19257.5 −0.648444
\(960\) 0 0
\(961\) 51135.6 1.71648
\(962\) 0 0
\(963\) 0 0
\(964\) 1345.00 0.0449373
\(965\) 50814.5 1.69511
\(966\) 0 0
\(967\) 15616.0 0.519313 0.259656 0.965701i \(-0.416391\pi\)
0.259656 + 0.965701i \(0.416391\pi\)
\(968\) −27184.2 −0.902617
\(969\) 0 0
\(970\) −20001.5 −0.662072
\(971\) −7185.98 −0.237496 −0.118748 0.992924i \(-0.537888\pi\)
−0.118748 + 0.992924i \(0.537888\pi\)
\(972\) 0 0
\(973\) 23574.1 0.776723
\(974\) −14192.9 −0.466911
\(975\) 0 0
\(976\) 35423.2 1.16175
\(977\) −33165.3 −1.08603 −0.543015 0.839723i \(-0.682717\pi\)
−0.543015 + 0.839723i \(0.682717\pi\)
\(978\) 0 0
\(979\) 14764.0 0.481980
\(980\) −9217.01 −0.300435
\(981\) 0 0
\(982\) −2429.52 −0.0789502
\(983\) 11658.7 0.378287 0.189143 0.981949i \(-0.439429\pi\)
0.189143 + 0.981949i \(0.439429\pi\)
\(984\) 0 0
\(985\) 45790.5 1.48123
\(986\) −45802.1 −1.47935
\(987\) 0 0
\(988\) 0 0
\(989\) 22479.2 0.722746
\(990\) 0 0
\(991\) −1957.32 −0.0627409 −0.0313704 0.999508i \(-0.509987\pi\)
−0.0313704 + 0.999508i \(0.509987\pi\)
\(992\) 30747.4 0.984104
\(993\) 0 0
\(994\) −22532.3 −0.718996
\(995\) 83629.5 2.66456
\(996\) 0 0
\(997\) −11434.8 −0.363233 −0.181617 0.983369i \(-0.558133\pi\)
−0.181617 + 0.983369i \(0.558133\pi\)
\(998\) 27459.0 0.870942
\(999\) 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.bf.1.4 9
3.2 odd 2 507.4.a.p.1.6 yes 9
13.12 even 2 1521.4.a.bi.1.6 9
39.5 even 4 507.4.b.k.337.7 18
39.8 even 4 507.4.b.k.337.12 18
39.38 odd 2 507.4.a.o.1.4 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.4 9 39.38 odd 2
507.4.a.p.1.6 yes 9 3.2 odd 2
507.4.b.k.337.7 18 39.5 even 4
507.4.b.k.337.12 18 39.8 even 4
1521.4.a.bf.1.4 9 1.1 even 1 trivial
1521.4.a.bi.1.6 9 13.12 even 2