Properties

Label 1521.4.a.ba.1.4
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: 4.4.1520092.1
Defining polynomial: \( x^{4} - x^{3} - 40x^{2} - 9x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 117)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(6.81072\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.48928 q^{2} +22.1322 q^{4} -14.0859 q^{5} -24.2643 q^{7} +77.5754 q^{8} +O(q^{10})\) \(q+5.48928 q^{2} +22.1322 q^{4} -14.0859 q^{5} -24.2643 q^{7} +77.5754 q^{8} -77.3217 q^{10} -3.10739 q^{11} -133.194 q^{14} +248.776 q^{16} +43.9142 q^{17} +85.8504 q^{19} -311.753 q^{20} -17.0574 q^{22} +203.829 q^{23} +73.4140 q^{25} -537.022 q^{28} -31.0739 q^{29} +135.736 q^{31} +744.995 q^{32} +241.057 q^{34} +341.786 q^{35} -290.229 q^{37} +471.256 q^{38} -1092.72 q^{40} +148.731 q^{41} +281.057 q^{43} -68.7734 q^{44} +1118.87 q^{46} +225.991 q^{47} +245.758 q^{49} +402.990 q^{50} +172.755 q^{53} +43.7706 q^{55} -1882.32 q^{56} -170.574 q^{58} +41.2175 q^{59} +499.815 q^{61} +745.091 q^{62} +2099.28 q^{64} -503.506 q^{67} +971.917 q^{68} +1876.16 q^{70} -946.442 q^{71} +1115.97 q^{73} -1593.15 q^{74} +1900.05 q^{76} +75.3989 q^{77} +674.803 q^{79} -3504.24 q^{80} +816.424 q^{82} -59.4512 q^{83} -618.574 q^{85} +1542.80 q^{86} -241.057 q^{88} +1218.41 q^{89} +4511.17 q^{92} +1240.53 q^{94} -1209.28 q^{95} +879.746 q^{97} +1349.03 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 58 q^{4} - 36 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 58 q^{4} - 36 q^{7} - 4 q^{10} + 354 q^{16} - 84 q^{19} + 176 q^{22} + 660 q^{25} - 988 q^{28} + 604 q^{31} + 720 q^{34} - 184 q^{37} - 2356 q^{40} + 880 q^{43} + 2888 q^{46} - 116 q^{49} + 1152 q^{55} + 1760 q^{58} + 656 q^{61} + 3482 q^{64} - 3052 q^{67} + 4696 q^{70} + 312 q^{73} + 2044 q^{76} - 720 q^{79} + 396 q^{82} - 32 q^{85} - 720 q^{88} + 4840 q^{94} + 344 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.48928 1.94075 0.970376 0.241598i \(-0.0776716\pi\)
0.970376 + 0.241598i \(0.0776716\pi\)
\(3\) 0 0
\(4\) 22.1322 2.76652
\(5\) −14.0859 −1.25989 −0.629943 0.776642i \(-0.716922\pi\)
−0.629943 + 0.776642i \(0.716922\pi\)
\(6\) 0 0
\(7\) −24.2643 −1.31015 −0.655076 0.755563i \(-0.727363\pi\)
−0.655076 + 0.755563i \(0.727363\pi\)
\(8\) 77.5754 3.42838
\(9\) 0 0
\(10\) −77.3217 −2.44513
\(11\) −3.10739 −0.0851741 −0.0425870 0.999093i \(-0.513560\pi\)
−0.0425870 + 0.999093i \(0.513560\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −133.194 −2.54268
\(15\) 0 0
\(16\) 248.776 3.88712
\(17\) 43.9142 0.626515 0.313258 0.949668i \(-0.398580\pi\)
0.313258 + 0.949668i \(0.398580\pi\)
\(18\) 0 0
\(19\) 85.8504 1.03660 0.518301 0.855198i \(-0.326565\pi\)
0.518301 + 0.855198i \(0.326565\pi\)
\(20\) −311.753 −3.48550
\(21\) 0 0
\(22\) −17.0574 −0.165302
\(23\) 203.829 1.84788 0.923940 0.382537i \(-0.124950\pi\)
0.923940 + 0.382537i \(0.124950\pi\)
\(24\) 0 0
\(25\) 73.4140 0.587312
\(26\) 0 0
\(27\) 0 0
\(28\) −537.022 −3.62456
\(29\) −31.0739 −0.198975 −0.0994877 0.995039i \(-0.531720\pi\)
−0.0994877 + 0.995039i \(0.531720\pi\)
\(30\) 0 0
\(31\) 135.736 0.786414 0.393207 0.919450i \(-0.371366\pi\)
0.393207 + 0.919450i \(0.371366\pi\)
\(32\) 744.995 4.11555
\(33\) 0 0
\(34\) 241.057 1.21591
\(35\) 341.786 1.65064
\(36\) 0 0
\(37\) −290.229 −1.28955 −0.644776 0.764372i \(-0.723049\pi\)
−0.644776 + 0.764372i \(0.723049\pi\)
\(38\) 471.256 2.01179
\(39\) 0 0
\(40\) −1092.72 −4.31937
\(41\) 148.731 0.566532 0.283266 0.959041i \(-0.408582\pi\)
0.283266 + 0.959041i \(0.408582\pi\)
\(42\) 0 0
\(43\) 281.057 0.996764 0.498382 0.866958i \(-0.333928\pi\)
0.498382 + 0.866958i \(0.333928\pi\)
\(44\) −68.7734 −0.235636
\(45\) 0 0
\(46\) 1118.87 3.58628
\(47\) 225.991 0.701366 0.350683 0.936494i \(-0.385949\pi\)
0.350683 + 0.936494i \(0.385949\pi\)
\(48\) 0 0
\(49\) 245.758 0.716496
\(50\) 402.990 1.13983
\(51\) 0 0
\(52\) 0 0
\(53\) 172.755 0.447730 0.223865 0.974620i \(-0.428132\pi\)
0.223865 + 0.974620i \(0.428132\pi\)
\(54\) 0 0
\(55\) 43.7706 0.107310
\(56\) −1882.32 −4.49170
\(57\) 0 0
\(58\) −170.574 −0.386162
\(59\) 41.2175 0.0909503 0.0454751 0.998965i \(-0.485520\pi\)
0.0454751 + 0.998965i \(0.485520\pi\)
\(60\) 0 0
\(61\) 499.815 1.04910 0.524548 0.851381i \(-0.324234\pi\)
0.524548 + 0.851381i \(0.324234\pi\)
\(62\) 745.091 1.52624
\(63\) 0 0
\(64\) 2099.28 4.10015
\(65\) 0 0
\(66\) 0 0
\(67\) −503.506 −0.918106 −0.459053 0.888409i \(-0.651811\pi\)
−0.459053 + 0.888409i \(0.651811\pi\)
\(68\) 971.917 1.73327
\(69\) 0 0
\(70\) 1876.16 3.20349
\(71\) −946.442 −1.58200 −0.791000 0.611816i \(-0.790439\pi\)
−0.791000 + 0.611816i \(0.790439\pi\)
\(72\) 0 0
\(73\) 1115.97 1.78925 0.894623 0.446821i \(-0.147444\pi\)
0.894623 + 0.446821i \(0.147444\pi\)
\(74\) −1593.15 −2.50270
\(75\) 0 0
\(76\) 1900.05 2.86778
\(77\) 75.3989 0.111591
\(78\) 0 0
\(79\) 674.803 0.961029 0.480514 0.876987i \(-0.340450\pi\)
0.480514 + 0.876987i \(0.340450\pi\)
\(80\) −3504.24 −4.89732
\(81\) 0 0
\(82\) 816.424 1.09950
\(83\) −59.4512 −0.0786219 −0.0393109 0.999227i \(-0.512516\pi\)
−0.0393109 + 0.999227i \(0.512516\pi\)
\(84\) 0 0
\(85\) −618.574 −0.789338
\(86\) 1542.80 1.93447
\(87\) 0 0
\(88\) −241.057 −0.292009
\(89\) 1218.41 1.45114 0.725571 0.688147i \(-0.241576\pi\)
0.725571 + 0.688147i \(0.241576\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4511.17 5.11220
\(93\) 0 0
\(94\) 1240.53 1.36118
\(95\) −1209.28 −1.30600
\(96\) 0 0
\(97\) 879.746 0.920872 0.460436 0.887693i \(-0.347693\pi\)
0.460436 + 0.887693i \(0.347693\pi\)
\(98\) 1349.03 1.39054
\(99\) 0 0
\(100\) 1624.81 1.62481
\(101\) −255.628 −0.251841 −0.125920 0.992040i \(-0.540188\pi\)
−0.125920 + 0.992040i \(0.540188\pi\)
\(102\) 0 0
\(103\) 176.369 0.168720 0.0843600 0.996435i \(-0.473115\pi\)
0.0843600 + 0.996435i \(0.473115\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 948.299 0.868934
\(107\) −816.164 −0.737397 −0.368699 0.929549i \(-0.620197\pi\)
−0.368699 + 0.929549i \(0.620197\pi\)
\(108\) 0 0
\(109\) −174.369 −0.153225 −0.0766125 0.997061i \(-0.524410\pi\)
−0.0766125 + 0.997061i \(0.524410\pi\)
\(110\) 240.269 0.208261
\(111\) 0 0
\(112\) −6036.37 −5.09271
\(113\) −199.667 −0.166222 −0.0831112 0.996540i \(-0.526486\pi\)
−0.0831112 + 0.996540i \(0.526486\pi\)
\(114\) 0 0
\(115\) −2871.12 −2.32812
\(116\) −687.734 −0.550470
\(117\) 0 0
\(118\) 226.254 0.176512
\(119\) −1065.55 −0.820830
\(120\) 0 0
\(121\) −1321.34 −0.992745
\(122\) 2743.63 2.03603
\(123\) 0 0
\(124\) 3004.12 2.17563
\(125\) 726.638 0.519940
\(126\) 0 0
\(127\) 407.840 0.284961 0.142480 0.989798i \(-0.454492\pi\)
0.142480 + 0.989798i \(0.454492\pi\)
\(128\) 5563.57 3.84183
\(129\) 0 0
\(130\) 0 0
\(131\) −2177.50 −1.45229 −0.726143 0.687544i \(-0.758689\pi\)
−0.726143 + 0.687544i \(0.758689\pi\)
\(132\) 0 0
\(133\) −2083.10 −1.35810
\(134\) −2763.89 −1.78182
\(135\) 0 0
\(136\) 3406.66 2.14793
\(137\) −723.750 −0.451344 −0.225672 0.974203i \(-0.572458\pi\)
−0.225672 + 0.974203i \(0.572458\pi\)
\(138\) 0 0
\(139\) −1520.85 −0.928033 −0.464017 0.885826i \(-0.653592\pi\)
−0.464017 + 0.885826i \(0.653592\pi\)
\(140\) 7564.47 4.56653
\(141\) 0 0
\(142\) −5195.28 −3.07027
\(143\) 0 0
\(144\) 0 0
\(145\) 437.706 0.250686
\(146\) 6125.90 3.47248
\(147\) 0 0
\(148\) −6423.41 −3.56757
\(149\) 2027.51 1.11477 0.557384 0.830255i \(-0.311805\pi\)
0.557384 + 0.830255i \(0.311805\pi\)
\(150\) 0 0
\(151\) 11.8952 0.00641071 0.00320535 0.999995i \(-0.498980\pi\)
0.00320535 + 0.999995i \(0.498980\pi\)
\(152\) 6659.88 3.55386
\(153\) 0 0
\(154\) 413.885 0.216570
\(155\) −1911.97 −0.990792
\(156\) 0 0
\(157\) 1369.10 0.695963 0.347982 0.937501i \(-0.386867\pi\)
0.347982 + 0.937501i \(0.386867\pi\)
\(158\) 3704.18 1.86512
\(159\) 0 0
\(160\) −10494.0 −5.18513
\(161\) −4945.77 −2.42100
\(162\) 0 0
\(163\) −1650.54 −0.793130 −0.396565 0.918007i \(-0.629798\pi\)
−0.396565 + 0.918007i \(0.629798\pi\)
\(164\) 3291.73 1.56732
\(165\) 0 0
\(166\) −326.344 −0.152586
\(167\) −872.686 −0.404374 −0.202187 0.979347i \(-0.564805\pi\)
−0.202187 + 0.979347i \(0.564805\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −3395.52 −1.53191
\(171\) 0 0
\(172\) 6220.41 2.75757
\(173\) 732.880 0.322080 0.161040 0.986948i \(-0.448515\pi\)
0.161040 + 0.986948i \(0.448515\pi\)
\(174\) 0 0
\(175\) −1781.34 −0.769467
\(176\) −773.044 −0.331082
\(177\) 0 0
\(178\) 6688.21 2.81631
\(179\) −1530.81 −0.639207 −0.319604 0.947551i \(-0.603550\pi\)
−0.319604 + 0.947551i \(0.603550\pi\)
\(180\) 0 0
\(181\) −1198.21 −0.492056 −0.246028 0.969263i \(-0.579126\pi\)
−0.246028 + 0.969263i \(0.579126\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 15812.1 6.33524
\(185\) 4088.16 1.62469
\(186\) 0 0
\(187\) −136.459 −0.0533629
\(188\) 5001.68 1.94034
\(189\) 0 0
\(190\) −6638.09 −2.53462
\(191\) 2465.44 0.933994 0.466997 0.884259i \(-0.345336\pi\)
0.466997 + 0.884259i \(0.345336\pi\)
\(192\) 0 0
\(193\) 4588.52 1.71134 0.855671 0.517520i \(-0.173145\pi\)
0.855671 + 0.517520i \(0.173145\pi\)
\(194\) 4829.17 1.78719
\(195\) 0 0
\(196\) 5439.16 1.98220
\(197\) −3862.82 −1.39703 −0.698514 0.715596i \(-0.746155\pi\)
−0.698514 + 0.715596i \(0.746155\pi\)
\(198\) 0 0
\(199\) 1434.59 0.511033 0.255517 0.966805i \(-0.417754\pi\)
0.255517 + 0.966805i \(0.417754\pi\)
\(200\) 5695.12 2.01353
\(201\) 0 0
\(202\) −1403.21 −0.488761
\(203\) 753.989 0.260688
\(204\) 0 0
\(205\) −2095.01 −0.713766
\(206\) 968.139 0.327444
\(207\) 0 0
\(208\) 0 0
\(209\) −266.771 −0.0882915
\(210\) 0 0
\(211\) −1577.24 −0.514604 −0.257302 0.966331i \(-0.582834\pi\)
−0.257302 + 0.966331i \(0.582834\pi\)
\(212\) 3823.44 1.23866
\(213\) 0 0
\(214\) −4480.15 −1.43111
\(215\) −3958.96 −1.25581
\(216\) 0 0
\(217\) −3293.54 −1.03032
\(218\) −957.161 −0.297372
\(219\) 0 0
\(220\) 968.738 0.296874
\(221\) 0 0
\(222\) 0 0
\(223\) 3123.69 0.938016 0.469008 0.883194i \(-0.344612\pi\)
0.469008 + 0.883194i \(0.344612\pi\)
\(224\) −18076.8 −5.39200
\(225\) 0 0
\(226\) −1096.03 −0.322597
\(227\) −3006.22 −0.878987 −0.439493 0.898246i \(-0.644842\pi\)
−0.439493 + 0.898246i \(0.644842\pi\)
\(228\) 0 0
\(229\) 2344.43 0.676526 0.338263 0.941052i \(-0.390161\pi\)
0.338263 + 0.941052i \(0.390161\pi\)
\(230\) −15760.4 −4.51830
\(231\) 0 0
\(232\) −2410.57 −0.682163
\(233\) −5913.50 −1.66269 −0.831344 0.555759i \(-0.812428\pi\)
−0.831344 + 0.555759i \(0.812428\pi\)
\(234\) 0 0
\(235\) −3183.30 −0.883641
\(236\) 912.233 0.251616
\(237\) 0 0
\(238\) −5849.10 −1.59303
\(239\) −250.440 −0.0677807 −0.0338904 0.999426i \(-0.510790\pi\)
−0.0338904 + 0.999426i \(0.510790\pi\)
\(240\) 0 0
\(241\) −1112.43 −0.297337 −0.148668 0.988887i \(-0.547499\pi\)
−0.148668 + 0.988887i \(0.547499\pi\)
\(242\) −7253.22 −1.92667
\(243\) 0 0
\(244\) 11062.0 2.90234
\(245\) −3461.74 −0.902703
\(246\) 0 0
\(247\) 0 0
\(248\) 10529.7 2.69613
\(249\) 0 0
\(250\) 3988.72 1.00907
\(251\) −3496.19 −0.879194 −0.439597 0.898195i \(-0.644879\pi\)
−0.439597 + 0.898195i \(0.644879\pi\)
\(252\) 0 0
\(253\) −633.376 −0.157391
\(254\) 2238.75 0.553038
\(255\) 0 0
\(256\) 13745.7 3.35589
\(257\) −4644.20 −1.12723 −0.563614 0.826039i \(-0.690589\pi\)
−0.563614 + 0.826039i \(0.690589\pi\)
\(258\) 0 0
\(259\) 7042.22 1.68951
\(260\) 0 0
\(261\) 0 0
\(262\) −11952.9 −2.81853
\(263\) 4930.41 1.15598 0.577989 0.816045i \(-0.303838\pi\)
0.577989 + 0.816045i \(0.303838\pi\)
\(264\) 0 0
\(265\) −2433.42 −0.564089
\(266\) −11434.7 −2.63574
\(267\) 0 0
\(268\) −11143.7 −2.53996
\(269\) 1276.44 0.289316 0.144658 0.989482i \(-0.453792\pi\)
0.144658 + 0.989482i \(0.453792\pi\)
\(270\) 0 0
\(271\) 2519.91 0.564848 0.282424 0.959290i \(-0.408861\pi\)
0.282424 + 0.959290i \(0.408861\pi\)
\(272\) 10924.8 2.43534
\(273\) 0 0
\(274\) −3972.86 −0.875947
\(275\) −228.126 −0.0500237
\(276\) 0 0
\(277\) −2462.00 −0.534033 −0.267017 0.963692i \(-0.586038\pi\)
−0.267017 + 0.963692i \(0.586038\pi\)
\(278\) −8348.36 −1.80108
\(279\) 0 0
\(280\) 26514.2 5.65902
\(281\) 5058.09 1.07381 0.536904 0.843643i \(-0.319594\pi\)
0.536904 + 0.843643i \(0.319594\pi\)
\(282\) 0 0
\(283\) 3078.55 0.646647 0.323323 0.946289i \(-0.395200\pi\)
0.323323 + 0.946289i \(0.395200\pi\)
\(284\) −20946.8 −4.37664
\(285\) 0 0
\(286\) 0 0
\(287\) −3608.85 −0.742243
\(288\) 0 0
\(289\) −2984.54 −0.607478
\(290\) 2402.69 0.486520
\(291\) 0 0
\(292\) 24698.9 4.94999
\(293\) 8626.31 1.71998 0.859990 0.510310i \(-0.170469\pi\)
0.859990 + 0.510310i \(0.170469\pi\)
\(294\) 0 0
\(295\) −580.588 −0.114587
\(296\) −22514.7 −4.42107
\(297\) 0 0
\(298\) 11129.6 2.16349
\(299\) 0 0
\(300\) 0 0
\(301\) −6819.67 −1.30591
\(302\) 65.2960 0.0124416
\(303\) 0 0
\(304\) 21357.5 4.02939
\(305\) −7040.37 −1.32174
\(306\) 0 0
\(307\) −545.591 −0.101428 −0.0507142 0.998713i \(-0.516150\pi\)
−0.0507142 + 0.998713i \(0.516150\pi\)
\(308\) 1668.74 0.308719
\(309\) 0 0
\(310\) −10495.3 −1.92288
\(311\) −3024.25 −0.551413 −0.275707 0.961242i \(-0.588912\pi\)
−0.275707 + 0.961242i \(0.588912\pi\)
\(312\) 0 0
\(313\) −10082.4 −1.82074 −0.910370 0.413796i \(-0.864203\pi\)
−0.910370 + 0.413796i \(0.864203\pi\)
\(314\) 7515.38 1.35069
\(315\) 0 0
\(316\) 14934.9 2.65871
\(317\) −8498.67 −1.50578 −0.752891 0.658145i \(-0.771341\pi\)
−0.752891 + 0.658145i \(0.771341\pi\)
\(318\) 0 0
\(319\) 96.5590 0.0169475
\(320\) −29570.3 −5.16573
\(321\) 0 0
\(322\) −27148.7 −4.69857
\(323\) 3770.05 0.649447
\(324\) 0 0
\(325\) 0 0
\(326\) −9060.26 −1.53927
\(327\) 0 0
\(328\) 11537.8 1.94229
\(329\) −5483.53 −0.918896
\(330\) 0 0
\(331\) −9969.94 −1.65558 −0.827791 0.561037i \(-0.810403\pi\)
−0.827791 + 0.561037i \(0.810403\pi\)
\(332\) −1315.78 −0.217509
\(333\) 0 0
\(334\) −4790.41 −0.784790
\(335\) 7092.36 1.15671
\(336\) 0 0
\(337\) −3231.96 −0.522422 −0.261211 0.965282i \(-0.584122\pi\)
−0.261211 + 0.965282i \(0.584122\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −13690.4 −2.18372
\(341\) −421.784 −0.0669821
\(342\) 0 0
\(343\) 2359.51 0.371433
\(344\) 21803.1 3.41729
\(345\) 0 0
\(346\) 4022.98 0.625078
\(347\) −5989.75 −0.926647 −0.463323 0.886189i \(-0.653343\pi\)
−0.463323 + 0.886189i \(0.653343\pi\)
\(348\) 0 0
\(349\) 9974.91 1.52993 0.764964 0.644074i \(-0.222757\pi\)
0.764964 + 0.644074i \(0.222757\pi\)
\(350\) −9778.28 −1.49335
\(351\) 0 0
\(352\) −2314.99 −0.350538
\(353\) 10834.0 1.63352 0.816761 0.576976i \(-0.195767\pi\)
0.816761 + 0.576976i \(0.195767\pi\)
\(354\) 0 0
\(355\) 13331.5 1.99314
\(356\) 26966.2 4.01462
\(357\) 0 0
\(358\) −8403.04 −1.24054
\(359\) 4315.76 0.634477 0.317239 0.948346i \(-0.397244\pi\)
0.317239 + 0.948346i \(0.397244\pi\)
\(360\) 0 0
\(361\) 511.285 0.0745422
\(362\) −6577.31 −0.954960
\(363\) 0 0
\(364\) 0 0
\(365\) −15719.6 −2.25425
\(366\) 0 0
\(367\) 10505.3 1.49420 0.747101 0.664711i \(-0.231445\pi\)
0.747101 + 0.664711i \(0.231445\pi\)
\(368\) 50707.6 7.18293
\(369\) 0 0
\(370\) 22441.0 3.15312
\(371\) −4191.78 −0.586594
\(372\) 0 0
\(373\) 5869.44 0.814767 0.407384 0.913257i \(-0.366441\pi\)
0.407384 + 0.913257i \(0.366441\pi\)
\(374\) −749.060 −0.103564
\(375\) 0 0
\(376\) 17531.4 2.40455
\(377\) 0 0
\(378\) 0 0
\(379\) 6525.75 0.884447 0.442223 0.896905i \(-0.354190\pi\)
0.442223 + 0.896905i \(0.354190\pi\)
\(380\) −26764.1 −3.61307
\(381\) 0 0
\(382\) 13533.5 1.81265
\(383\) −7042.66 −0.939590 −0.469795 0.882775i \(-0.655672\pi\)
−0.469795 + 0.882775i \(0.655672\pi\)
\(384\) 0 0
\(385\) −1062.06 −0.140592
\(386\) 25187.7 3.32129
\(387\) 0 0
\(388\) 19470.7 2.54761
\(389\) 10339.4 1.34763 0.673815 0.738900i \(-0.264655\pi\)
0.673815 + 0.738900i \(0.264655\pi\)
\(390\) 0 0
\(391\) 8950.98 1.15773
\(392\) 19064.8 2.45642
\(393\) 0 0
\(394\) −21204.1 −2.71129
\(395\) −9505.24 −1.21079
\(396\) 0 0
\(397\) 7465.86 0.943831 0.471915 0.881644i \(-0.343563\pi\)
0.471915 + 0.881644i \(0.343563\pi\)
\(398\) 7874.88 0.991789
\(399\) 0 0
\(400\) 18263.6 2.28295
\(401\) −3224.37 −0.401539 −0.200770 0.979638i \(-0.564344\pi\)
−0.200770 + 0.979638i \(0.564344\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −5657.60 −0.696723
\(405\) 0 0
\(406\) 4138.85 0.505931
\(407\) 901.857 0.109836
\(408\) 0 0
\(409\) −12391.9 −1.49814 −0.749072 0.662489i \(-0.769500\pi\)
−0.749072 + 0.662489i \(0.769500\pi\)
\(410\) −11500.1 −1.38524
\(411\) 0 0
\(412\) 3903.43 0.466768
\(413\) −1000.12 −0.119159
\(414\) 0 0
\(415\) 837.426 0.0990546
\(416\) 0 0
\(417\) 0 0
\(418\) −1464.38 −0.171352
\(419\) 13464.4 1.56988 0.784938 0.619574i \(-0.212694\pi\)
0.784938 + 0.619574i \(0.212694\pi\)
\(420\) 0 0
\(421\) 6265.49 0.725324 0.362662 0.931921i \(-0.381868\pi\)
0.362662 + 0.931921i \(0.381868\pi\)
\(422\) −8657.89 −0.998719
\(423\) 0 0
\(424\) 13401.5 1.53499
\(425\) 3223.92 0.367960
\(426\) 0 0
\(427\) −12127.7 −1.37447
\(428\) −18063.5 −2.04003
\(429\) 0 0
\(430\) −21731.8 −2.43721
\(431\) 9499.09 1.06161 0.530806 0.847493i \(-0.321889\pi\)
0.530806 + 0.847493i \(0.321889\pi\)
\(432\) 0 0
\(433\) 5002.67 0.555226 0.277613 0.960693i \(-0.410457\pi\)
0.277613 + 0.960693i \(0.410457\pi\)
\(434\) −18079.1 −1.99960
\(435\) 0 0
\(436\) −3859.17 −0.423900
\(437\) 17498.8 1.91551
\(438\) 0 0
\(439\) 13664.9 1.48563 0.742816 0.669496i \(-0.233490\pi\)
0.742816 + 0.669496i \(0.233490\pi\)
\(440\) 3395.52 0.367898
\(441\) 0 0
\(442\) 0 0
\(443\) −10780.6 −1.15621 −0.578105 0.815962i \(-0.696208\pi\)
−0.578105 + 0.815962i \(0.696208\pi\)
\(444\) 0 0
\(445\) −17162.5 −1.82827
\(446\) 17146.8 1.82046
\(447\) 0 0
\(448\) −50937.6 −5.37182
\(449\) −11959.1 −1.25699 −0.628493 0.777815i \(-0.716328\pi\)
−0.628493 + 0.777815i \(0.716328\pi\)
\(450\) 0 0
\(451\) −462.165 −0.0482539
\(452\) −4419.07 −0.459858
\(453\) 0 0
\(454\) −16502.0 −1.70590
\(455\) 0 0
\(456\) 0 0
\(457\) 2120.66 0.217069 0.108534 0.994093i \(-0.465384\pi\)
0.108534 + 0.994093i \(0.465384\pi\)
\(458\) 12869.2 1.31297
\(459\) 0 0
\(460\) −63544.2 −6.44079
\(461\) −15462.9 −1.56221 −0.781103 0.624402i \(-0.785343\pi\)
−0.781103 + 0.624402i \(0.785343\pi\)
\(462\) 0 0
\(463\) 10395.4 1.04344 0.521722 0.853116i \(-0.325290\pi\)
0.521722 + 0.853116i \(0.325290\pi\)
\(464\) −7730.44 −0.773441
\(465\) 0 0
\(466\) −32460.8 −3.22687
\(467\) −11845.8 −1.17379 −0.586893 0.809664i \(-0.699649\pi\)
−0.586893 + 0.809664i \(0.699649\pi\)
\(468\) 0 0
\(469\) 12217.2 1.20286
\(470\) −17474.0 −1.71493
\(471\) 0 0
\(472\) 3197.47 0.311812
\(473\) −873.356 −0.0848984
\(474\) 0 0
\(475\) 6302.62 0.608808
\(476\) −23582.9 −2.27084
\(477\) 0 0
\(478\) −1374.73 −0.131546
\(479\) −12879.9 −1.22860 −0.614299 0.789074i \(-0.710561\pi\)
−0.614299 + 0.789074i \(0.710561\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −6106.46 −0.577057
\(483\) 0 0
\(484\) −29244.2 −2.74645
\(485\) −12392.1 −1.16019
\(486\) 0 0
\(487\) 4946.81 0.460290 0.230145 0.973156i \(-0.426080\pi\)
0.230145 + 0.973156i \(0.426080\pi\)
\(488\) 38773.4 3.59670
\(489\) 0 0
\(490\) −19002.4 −1.75192
\(491\) 14390.3 1.32266 0.661330 0.750095i \(-0.269992\pi\)
0.661330 + 0.750095i \(0.269992\pi\)
\(492\) 0 0
\(493\) −1364.59 −0.124661
\(494\) 0 0
\(495\) 0 0
\(496\) 33767.7 3.05689
\(497\) 22964.8 2.07266
\(498\) 0 0
\(499\) 1427.01 0.128019 0.0640096 0.997949i \(-0.479611\pi\)
0.0640096 + 0.997949i \(0.479611\pi\)
\(500\) 16082.1 1.43842
\(501\) 0 0
\(502\) −19191.6 −1.70630
\(503\) −18033.6 −1.59857 −0.799283 0.600955i \(-0.794787\pi\)
−0.799283 + 0.600955i \(0.794787\pi\)
\(504\) 0 0
\(505\) 3600.76 0.317290
\(506\) −3476.78 −0.305458
\(507\) 0 0
\(508\) 9026.39 0.788349
\(509\) −7512.52 −0.654198 −0.327099 0.944990i \(-0.606071\pi\)
−0.327099 + 0.944990i \(0.606071\pi\)
\(510\) 0 0
\(511\) −27078.4 −2.34418
\(512\) 30945.6 2.67112
\(513\) 0 0
\(514\) −25493.3 −2.18767
\(515\) −2484.33 −0.212568
\(516\) 0 0
\(517\) −702.244 −0.0597382
\(518\) 38656.7 3.27892
\(519\) 0 0
\(520\) 0 0
\(521\) 21355.8 1.79580 0.897901 0.440198i \(-0.145092\pi\)
0.897901 + 0.440198i \(0.145092\pi\)
\(522\) 0 0
\(523\) 3086.57 0.258062 0.129031 0.991641i \(-0.458813\pi\)
0.129031 + 0.991641i \(0.458813\pi\)
\(524\) −48192.9 −4.01778
\(525\) 0 0
\(526\) 27064.4 2.24347
\(527\) 5960.73 0.492701
\(528\) 0 0
\(529\) 29379.2 2.41466
\(530\) −13357.7 −1.09476
\(531\) 0 0
\(532\) −46103.6 −3.75722
\(533\) 0 0
\(534\) 0 0
\(535\) 11496.4 0.929036
\(536\) −39059.7 −3.14762
\(537\) 0 0
\(538\) 7006.74 0.561491
\(539\) −763.667 −0.0610269
\(540\) 0 0
\(541\) 2029.46 0.161282 0.0806408 0.996743i \(-0.474303\pi\)
0.0806408 + 0.996743i \(0.474303\pi\)
\(542\) 13832.5 1.09623
\(543\) 0 0
\(544\) 32715.9 2.57846
\(545\) 2456.16 0.193046
\(546\) 0 0
\(547\) 22144.8 1.73098 0.865488 0.500929i \(-0.167008\pi\)
0.865488 + 0.500929i \(0.167008\pi\)
\(548\) −16018.1 −1.24865
\(549\) 0 0
\(550\) −1252.25 −0.0970837
\(551\) −2667.71 −0.206258
\(552\) 0 0
\(553\) −16373.6 −1.25909
\(554\) −13514.6 −1.03643
\(555\) 0 0
\(556\) −33659.7 −2.56742
\(557\) −23136.5 −1.76001 −0.880004 0.474967i \(-0.842460\pi\)
−0.880004 + 0.474967i \(0.842460\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 85028.1 6.41623
\(561\) 0 0
\(562\) 27765.2 2.08400
\(563\) −3978.29 −0.297806 −0.148903 0.988852i \(-0.547574\pi\)
−0.148903 + 0.988852i \(0.547574\pi\)
\(564\) 0 0
\(565\) 2812.51 0.209421
\(566\) 16899.0 1.25498
\(567\) 0 0
\(568\) −73420.6 −5.42370
\(569\) −25871.5 −1.90613 −0.953065 0.302764i \(-0.902090\pi\)
−0.953065 + 0.302764i \(0.902090\pi\)
\(570\) 0 0
\(571\) −7241.31 −0.530717 −0.265358 0.964150i \(-0.585490\pi\)
−0.265358 + 0.964150i \(0.585490\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −19810.0 −1.44051
\(575\) 14963.9 1.08528
\(576\) 0 0
\(577\) −11537.4 −0.832426 −0.416213 0.909267i \(-0.636643\pi\)
−0.416213 + 0.909267i \(0.636643\pi\)
\(578\) −16383.0 −1.17897
\(579\) 0 0
\(580\) 9687.38 0.693529
\(581\) 1442.54 0.103007
\(582\) 0 0
\(583\) −536.817 −0.0381350
\(584\) 86572.2 6.13422
\(585\) 0 0
\(586\) 47352.2 3.33806
\(587\) −16195.2 −1.13875 −0.569376 0.822078i \(-0.692815\pi\)
−0.569376 + 0.822078i \(0.692815\pi\)
\(588\) 0 0
\(589\) 11653.0 0.815198
\(590\) −3187.01 −0.222385
\(591\) 0 0
\(592\) −72202.0 −5.01264
\(593\) 14885.6 1.03083 0.515413 0.856942i \(-0.327638\pi\)
0.515413 + 0.856942i \(0.327638\pi\)
\(594\) 0 0
\(595\) 15009.3 1.03415
\(596\) 44873.3 3.08403
\(597\) 0 0
\(598\) 0 0
\(599\) −21516.2 −1.46766 −0.733829 0.679334i \(-0.762269\pi\)
−0.733829 + 0.679334i \(0.762269\pi\)
\(600\) 0 0
\(601\) −675.727 −0.0458627 −0.0229313 0.999737i \(-0.507300\pi\)
−0.0229313 + 0.999737i \(0.507300\pi\)
\(602\) −37435.1 −2.53445
\(603\) 0 0
\(604\) 263.266 0.0177354
\(605\) 18612.4 1.25075
\(606\) 0 0
\(607\) −12166.0 −0.813512 −0.406756 0.913537i \(-0.633340\pi\)
−0.406756 + 0.913537i \(0.633340\pi\)
\(608\) 63958.1 4.26619
\(609\) 0 0
\(610\) −38646.6 −2.56517
\(611\) 0 0
\(612\) 0 0
\(613\) 22799.0 1.50219 0.751095 0.660194i \(-0.229526\pi\)
0.751095 + 0.660194i \(0.229526\pi\)
\(614\) −2994.90 −0.196847
\(615\) 0 0
\(616\) 5849.10 0.382576
\(617\) 1635.30 0.106701 0.0533506 0.998576i \(-0.483010\pi\)
0.0533506 + 0.998576i \(0.483010\pi\)
\(618\) 0 0
\(619\) 4435.20 0.287990 0.143995 0.989578i \(-0.454005\pi\)
0.143995 + 0.989578i \(0.454005\pi\)
\(620\) −42315.9 −2.74105
\(621\) 0 0
\(622\) −16600.9 −1.07016
\(623\) −29564.0 −1.90122
\(624\) 0 0
\(625\) −19412.1 −1.24238
\(626\) −55345.1 −3.53360
\(627\) 0 0
\(628\) 30301.2 1.92540
\(629\) −12745.2 −0.807924
\(630\) 0 0
\(631\) 17645.7 1.11325 0.556626 0.830763i \(-0.312096\pi\)
0.556626 + 0.830763i \(0.312096\pi\)
\(632\) 52348.1 3.29477
\(633\) 0 0
\(634\) −46651.6 −2.92235
\(635\) −5744.82 −0.359018
\(636\) 0 0
\(637\) 0 0
\(638\) 530.039 0.0328910
\(639\) 0 0
\(640\) −78368.1 −4.84027
\(641\) 1482.41 0.0913440 0.0456720 0.998956i \(-0.485457\pi\)
0.0456720 + 0.998956i \(0.485457\pi\)
\(642\) 0 0
\(643\) −9629.03 −0.590563 −0.295281 0.955410i \(-0.595413\pi\)
−0.295281 + 0.955410i \(0.595413\pi\)
\(644\) −109461. −6.69775
\(645\) 0 0
\(646\) 20694.9 1.26042
\(647\) 15991.6 0.971709 0.485854 0.874040i \(-0.338509\pi\)
0.485854 + 0.874040i \(0.338509\pi\)
\(648\) 0 0
\(649\) −128.079 −0.00774660
\(650\) 0 0
\(651\) 0 0
\(652\) −36530.0 −2.19421
\(653\) 20321.2 1.21781 0.608907 0.793242i \(-0.291608\pi\)
0.608907 + 0.793242i \(0.291608\pi\)
\(654\) 0 0
\(655\) 30672.2 1.82971
\(656\) 37000.5 2.20218
\(657\) 0 0
\(658\) −30100.6 −1.78335
\(659\) −21597.1 −1.27664 −0.638318 0.769773i \(-0.720369\pi\)
−0.638318 + 0.769773i \(0.720369\pi\)
\(660\) 0 0
\(661\) −513.172 −0.0301968 −0.0150984 0.999886i \(-0.504806\pi\)
−0.0150984 + 0.999886i \(0.504806\pi\)
\(662\) −54727.8 −3.21307
\(663\) 0 0
\(664\) −4611.95 −0.269546
\(665\) 29342.5 1.71106
\(666\) 0 0
\(667\) −6333.76 −0.367683
\(668\) −19314.4 −1.11871
\(669\) 0 0
\(670\) 38932.0 2.24488
\(671\) −1553.12 −0.0893557
\(672\) 0 0
\(673\) 11983.8 0.686389 0.343195 0.939264i \(-0.388491\pi\)
0.343195 + 0.939264i \(0.388491\pi\)
\(674\) −17741.1 −1.01389
\(675\) 0 0
\(676\) 0 0
\(677\) −21876.5 −1.24192 −0.620960 0.783842i \(-0.713257\pi\)
−0.620960 + 0.783842i \(0.713257\pi\)
\(678\) 0 0
\(679\) −21346.4 −1.20648
\(680\) −47986.1 −2.70615
\(681\) 0 0
\(682\) −2315.29 −0.129996
\(683\) −2551.17 −0.142925 −0.0714625 0.997443i \(-0.522767\pi\)
−0.0714625 + 0.997443i \(0.522767\pi\)
\(684\) 0 0
\(685\) 10194.7 0.568642
\(686\) 12952.0 0.720860
\(687\) 0 0
\(688\) 69920.2 3.87454
\(689\) 0 0
\(690\) 0 0
\(691\) 15321.2 0.843481 0.421741 0.906717i \(-0.361419\pi\)
0.421741 + 0.906717i \(0.361419\pi\)
\(692\) 16220.2 0.891041
\(693\) 0 0
\(694\) −32879.4 −1.79839
\(695\) 21422.6 1.16922
\(696\) 0 0
\(697\) 6531.39 0.354941
\(698\) 54755.0 2.96921
\(699\) 0 0
\(700\) −39424.9 −2.12875
\(701\) 558.289 0.0300803 0.0150401 0.999887i \(-0.495212\pi\)
0.0150401 + 0.999887i \(0.495212\pi\)
\(702\) 0 0
\(703\) −24916.3 −1.33675
\(704\) −6523.29 −0.349227
\(705\) 0 0
\(706\) 59470.6 3.17026
\(707\) 6202.64 0.329949
\(708\) 0 0
\(709\) 7058.31 0.373879 0.186940 0.982371i \(-0.440143\pi\)
0.186940 + 0.982371i \(0.440143\pi\)
\(710\) 73180.5 3.86819
\(711\) 0 0
\(712\) 94519.0 4.97507
\(713\) 27666.8 1.45320
\(714\) 0 0
\(715\) 0 0
\(716\) −33880.1 −1.76838
\(717\) 0 0
\(718\) 23690.4 1.23136
\(719\) −14673.0 −0.761073 −0.380536 0.924766i \(-0.624261\pi\)
−0.380536 + 0.924766i \(0.624261\pi\)
\(720\) 0 0
\(721\) −4279.48 −0.221049
\(722\) 2806.58 0.144668
\(723\) 0 0
\(724\) −26519.0 −1.36128
\(725\) −2281.26 −0.116861
\(726\) 0 0
\(727\) 35867.8 1.82980 0.914899 0.403684i \(-0.132270\pi\)
0.914899 + 0.403684i \(0.132270\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −86289.1 −4.37493
\(731\) 12342.4 0.624488
\(732\) 0 0
\(733\) −27817.7 −1.40173 −0.700865 0.713294i \(-0.747203\pi\)
−0.700865 + 0.713294i \(0.747203\pi\)
\(734\) 57666.5 2.89988
\(735\) 0 0
\(736\) 151851. 7.60505
\(737\) 1564.59 0.0781988
\(738\) 0 0
\(739\) 7162.20 0.356517 0.178258 0.983984i \(-0.442954\pi\)
0.178258 + 0.983984i \(0.442954\pi\)
\(740\) 90479.8 4.49473
\(741\) 0 0
\(742\) −23009.9 −1.13843
\(743\) −12529.8 −0.618671 −0.309336 0.950953i \(-0.600107\pi\)
−0.309336 + 0.950953i \(0.600107\pi\)
\(744\) 0 0
\(745\) −28559.5 −1.40448
\(746\) 32219.0 1.58126
\(747\) 0 0
\(748\) −3020.13 −0.147629
\(749\) 19803.7 0.966102
\(750\) 0 0
\(751\) 8282.62 0.402446 0.201223 0.979545i \(-0.435508\pi\)
0.201223 + 0.979545i \(0.435508\pi\)
\(752\) 56221.1 2.72629
\(753\) 0 0
\(754\) 0 0
\(755\) −167.555 −0.00807676
\(756\) 0 0
\(757\) −22044.3 −1.05840 −0.529202 0.848496i \(-0.677509\pi\)
−0.529202 + 0.848496i \(0.677509\pi\)
\(758\) 35821.7 1.71649
\(759\) 0 0
\(760\) −93810.7 −4.47746
\(761\) −5710.64 −0.272025 −0.136012 0.990707i \(-0.543429\pi\)
−0.136012 + 0.990707i \(0.543429\pi\)
\(762\) 0 0
\(763\) 4230.95 0.200748
\(764\) 54565.5 2.58391
\(765\) 0 0
\(766\) −38659.1 −1.82351
\(767\) 0 0
\(768\) 0 0
\(769\) −16851.6 −0.790228 −0.395114 0.918632i \(-0.629295\pi\)
−0.395114 + 0.918632i \(0.629295\pi\)
\(770\) −5829.97 −0.272854
\(771\) 0 0
\(772\) 101554. 4.73447
\(773\) −593.103 −0.0275970 −0.0137985 0.999905i \(-0.504392\pi\)
−0.0137985 + 0.999905i \(0.504392\pi\)
\(774\) 0 0
\(775\) 9964.89 0.461870
\(776\) 68246.6 3.15710
\(777\) 0 0
\(778\) 56755.8 2.61542
\(779\) 12768.6 0.587268
\(780\) 0 0
\(781\) 2940.97 0.134745
\(782\) 49134.4 2.24686
\(783\) 0 0
\(784\) 61138.6 2.78510
\(785\) −19285.1 −0.876834
\(786\) 0 0
\(787\) −18564.0 −0.840834 −0.420417 0.907331i \(-0.638116\pi\)
−0.420417 + 0.907331i \(0.638116\pi\)
\(788\) −85492.6 −3.86491
\(789\) 0 0
\(790\) −52176.9 −2.34984
\(791\) 4844.80 0.217777
\(792\) 0 0
\(793\) 0 0
\(794\) 40982.2 1.83174
\(795\) 0 0
\(796\) 31750.7 1.41378
\(797\) 5227.16 0.232316 0.116158 0.993231i \(-0.462942\pi\)
0.116158 + 0.993231i \(0.462942\pi\)
\(798\) 0 0
\(799\) 9924.23 0.439417
\(800\) 54693.0 2.41711
\(801\) 0 0
\(802\) −17699.5 −0.779289
\(803\) −3467.77 −0.152397
\(804\) 0 0
\(805\) 69665.9 3.05019
\(806\) 0 0
\(807\) 0 0
\(808\) −19830.4 −0.863406
\(809\) −8907.33 −0.387101 −0.193551 0.981090i \(-0.562000\pi\)
−0.193551 + 0.981090i \(0.562000\pi\)
\(810\) 0 0
\(811\) 15352.3 0.664725 0.332363 0.943152i \(-0.392154\pi\)
0.332363 + 0.943152i \(0.392154\pi\)
\(812\) 16687.4 0.721198
\(813\) 0 0
\(814\) 4950.54 0.213165
\(815\) 23249.4 0.999253
\(816\) 0 0
\(817\) 24128.9 1.03325
\(818\) −68022.7 −2.90753
\(819\) 0 0
\(820\) −46367.2 −1.97465
\(821\) 12869.6 0.547080 0.273540 0.961861i \(-0.411805\pi\)
0.273540 + 0.961861i \(0.411805\pi\)
\(822\) 0 0
\(823\) −402.065 −0.0170293 −0.00851464 0.999964i \(-0.502710\pi\)
−0.00851464 + 0.999964i \(0.502710\pi\)
\(824\) 13681.9 0.578437
\(825\) 0 0
\(826\) −5489.91 −0.231257
\(827\) −42772.7 −1.79849 −0.899245 0.437446i \(-0.855883\pi\)
−0.899245 + 0.437446i \(0.855883\pi\)
\(828\) 0 0
\(829\) −22933.5 −0.960814 −0.480407 0.877046i \(-0.659511\pi\)
−0.480407 + 0.877046i \(0.659511\pi\)
\(830\) 4596.87 0.192240
\(831\) 0 0
\(832\) 0 0
\(833\) 10792.3 0.448896
\(834\) 0 0
\(835\) 12292.6 0.509465
\(836\) −5904.22 −0.244260
\(837\) 0 0
\(838\) 73909.8 3.04674
\(839\) −42510.1 −1.74924 −0.874619 0.484811i \(-0.838888\pi\)
−0.874619 + 0.484811i \(0.838888\pi\)
\(840\) 0 0
\(841\) −23423.4 −0.960409
\(842\) 34393.0 1.40767
\(843\) 0 0
\(844\) −34907.7 −1.42366
\(845\) 0 0
\(846\) 0 0
\(847\) 32061.5 1.30065
\(848\) 42977.2 1.74038
\(849\) 0 0
\(850\) 17697.0 0.714119
\(851\) −59157.1 −2.38294
\(852\) 0 0
\(853\) −40464.6 −1.62425 −0.812123 0.583487i \(-0.801688\pi\)
−0.812123 + 0.583487i \(0.801688\pi\)
\(854\) −66572.3 −2.66751
\(855\) 0 0
\(856\) −63314.2 −2.52808
\(857\) 38964.3 1.55308 0.776542 0.630065i \(-0.216972\pi\)
0.776542 + 0.630065i \(0.216972\pi\)
\(858\) 0 0
\(859\) −6379.47 −0.253393 −0.126697 0.991942i \(-0.540437\pi\)
−0.126697 + 0.991942i \(0.540437\pi\)
\(860\) −87620.4 −3.47422
\(861\) 0 0
\(862\) 52143.1 2.06033
\(863\) 33744.0 1.33101 0.665504 0.746395i \(-0.268217\pi\)
0.665504 + 0.746395i \(0.268217\pi\)
\(864\) 0 0
\(865\) −10323.3 −0.405784
\(866\) 27461.0 1.07756
\(867\) 0 0
\(868\) −72893.1 −2.85041
\(869\) −2096.88 −0.0818547
\(870\) 0 0
\(871\) 0 0
\(872\) −13526.8 −0.525314
\(873\) 0 0
\(874\) 96055.6 3.71754
\(875\) −17631.4 −0.681200
\(876\) 0 0
\(877\) −42898.6 −1.65175 −0.825874 0.563854i \(-0.809318\pi\)
−0.825874 + 0.563854i \(0.809318\pi\)
\(878\) 75010.7 2.88324
\(879\) 0 0
\(880\) 10889.1 0.417125
\(881\) 1750.90 0.0669572 0.0334786 0.999439i \(-0.489341\pi\)
0.0334786 + 0.999439i \(0.489341\pi\)
\(882\) 0 0
\(883\) 33196.7 1.26518 0.632591 0.774486i \(-0.281991\pi\)
0.632591 + 0.774486i \(0.281991\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −59177.6 −2.24392
\(887\) −29998.9 −1.13559 −0.567794 0.823171i \(-0.692203\pi\)
−0.567794 + 0.823171i \(0.692203\pi\)
\(888\) 0 0
\(889\) −9895.98 −0.373341
\(890\) −94209.8 −3.54823
\(891\) 0 0
\(892\) 69133.9 2.59504
\(893\) 19401.4 0.727037
\(894\) 0 0
\(895\) 21562.9 0.805328
\(896\) −134996. −5.03338
\(897\) 0 0
\(898\) −65647.0 −2.43950
\(899\) −4217.84 −0.156477
\(900\) 0 0
\(901\) 7586.39 0.280510
\(902\) −2536.95 −0.0936488
\(903\) 0 0
\(904\) −15489.3 −0.569874
\(905\) 16877.9 0.619935
\(906\) 0 0
\(907\) 24629.3 0.901657 0.450829 0.892611i \(-0.351129\pi\)
0.450829 + 0.892611i \(0.351129\pi\)
\(908\) −66534.2 −2.43173
\(909\) 0 0
\(910\) 0 0
\(911\) 7338.72 0.266896 0.133448 0.991056i \(-0.457395\pi\)
0.133448 + 0.991056i \(0.457395\pi\)
\(912\) 0 0
\(913\) 184.738 0.00669654
\(914\) 11640.9 0.421277
\(915\) 0 0
\(916\) 51887.4 1.87162
\(917\) 52835.7 1.90271
\(918\) 0 0
\(919\) −1449.13 −0.0520155 −0.0260078 0.999662i \(-0.508279\pi\)
−0.0260078 + 0.999662i \(0.508279\pi\)
\(920\) −222728. −7.98167
\(921\) 0 0
\(922\) −84879.9 −3.03186
\(923\) 0 0
\(924\) 0 0
\(925\) −21306.9 −0.757369
\(926\) 57063.2 2.02507
\(927\) 0 0
\(928\) −23149.9 −0.818894
\(929\) 6478.07 0.228782 0.114391 0.993436i \(-0.463508\pi\)
0.114391 + 0.993436i \(0.463508\pi\)
\(930\) 0 0
\(931\) 21098.4 0.742720
\(932\) −130879. −4.59986
\(933\) 0 0
\(934\) −65024.9 −2.27803
\(935\) 1922.15 0.0672311
\(936\) 0 0
\(937\) 4679.24 0.163142 0.0815710 0.996668i \(-0.474006\pi\)
0.0815710 + 0.996668i \(0.474006\pi\)
\(938\) 67063.9 2.33445
\(939\) 0 0
\(940\) −70453.4 −2.44461
\(941\) −47081.7 −1.63105 −0.815525 0.578722i \(-0.803552\pi\)
−0.815525 + 0.578722i \(0.803552\pi\)
\(942\) 0 0
\(943\) 30315.6 1.04688
\(944\) 10253.9 0.353534
\(945\) 0 0
\(946\) −4794.09 −0.164767
\(947\) −4264.62 −0.146338 −0.0731688 0.997320i \(-0.523311\pi\)
−0.0731688 + 0.997320i \(0.523311\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 34596.8 1.18155
\(951\) 0 0
\(952\) −82660.4 −2.81412
\(953\) 24368.1 0.828292 0.414146 0.910211i \(-0.364080\pi\)
0.414146 + 0.910211i \(0.364080\pi\)
\(954\) 0 0
\(955\) −34728.0 −1.17673
\(956\) −5542.77 −0.187517
\(957\) 0 0
\(958\) −70701.4 −2.38440
\(959\) 17561.3 0.591329
\(960\) 0 0
\(961\) −11366.8 −0.381552
\(962\) 0 0
\(963\) 0 0
\(964\) −24620.6 −0.822588
\(965\) −64633.7 −2.15610
\(966\) 0 0
\(967\) −11355.1 −0.377616 −0.188808 0.982014i \(-0.560462\pi\)
−0.188808 + 0.982014i \(0.560462\pi\)
\(968\) −102504. −3.40351
\(969\) 0 0
\(970\) −68023.4 −2.25165
\(971\) 3024.06 0.0999451 0.0499726 0.998751i \(-0.484087\pi\)
0.0499726 + 0.998751i \(0.484087\pi\)
\(972\) 0 0
\(973\) 36902.4 1.21586
\(974\) 27154.4 0.893309
\(975\) 0 0
\(976\) 124342. 4.07796
\(977\) −41218.9 −1.34975 −0.674877 0.737930i \(-0.735803\pi\)
−0.674877 + 0.737930i \(0.735803\pi\)
\(978\) 0 0
\(979\) −3786.09 −0.123600
\(980\) −76615.7 −2.49735
\(981\) 0 0
\(982\) 78992.4 2.56695
\(983\) 21579.3 0.700177 0.350088 0.936717i \(-0.386152\pi\)
0.350088 + 0.936717i \(0.386152\pi\)
\(984\) 0 0
\(985\) 54411.5 1.76010
\(986\) −7490.60 −0.241936
\(987\) 0 0
\(988\) 0 0
\(989\) 57287.6 1.84190
\(990\) 0 0
\(991\) −34613.5 −1.10952 −0.554760 0.832010i \(-0.687190\pi\)
−0.554760 + 0.832010i \(0.687190\pi\)
\(992\) 101122. 3.23653
\(993\) 0 0
\(994\) 126060. 4.02252
\(995\) −20207.6 −0.643843
\(996\) 0 0
\(997\) 20841.5 0.662043 0.331021 0.943623i \(-0.392607\pi\)
0.331021 + 0.943623i \(0.392607\pi\)
\(998\) 7833.24 0.248454
\(999\) 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.ba.1.4 4
3.2 odd 2 inner 1521.4.a.ba.1.1 4
13.12 even 2 117.4.a.g.1.1 4
39.38 odd 2 117.4.a.g.1.4 yes 4
52.51 odd 2 1872.4.a.bo.1.3 4
156.155 even 2 1872.4.a.bo.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.a.g.1.1 4 13.12 even 2
117.4.a.g.1.4 yes 4 39.38 odd 2
1521.4.a.ba.1.1 4 3.2 odd 2 inner
1521.4.a.ba.1.4 4 1.1 even 1 trivial
1872.4.a.bo.1.2 4 156.155 even 2
1872.4.a.bo.1.3 4 52.51 odd 2