Properties

Label 1575.4.a.p.1.2
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
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 1.2
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.27492 q^{2} -2.82475 q^{4} -7.00000 q^{7} -24.6254 q^{8} +O(q^{10})\) \(q+2.27492 q^{2} -2.82475 q^{4} -7.00000 q^{7} -24.6254 q^{8} +40.7492 q^{11} -53.2990 q^{13} -15.9244 q^{14} -33.4228 q^{16} +4.54983 q^{17} +122.598 q^{19} +92.7010 q^{22} +131.347 q^{23} -121.251 q^{26} +19.7733 q^{28} +216.598 q^{29} -251.794 q^{31} +120.969 q^{32} +10.3505 q^{34} -11.8970 q^{37} +278.900 q^{38} +111.752 q^{41} -369.196 q^{43} -115.106 q^{44} +298.804 q^{46} -262.694 q^{47} +49.0000 q^{49} +150.556 q^{52} -567.100 q^{53} +172.378 q^{56} +492.743 q^{58} -839.890 q^{59} -485.794 q^{61} -572.811 q^{62} +542.577 q^{64} +333.691 q^{67} -12.8522 q^{68} -590.248 q^{71} -490.701 q^{73} -27.0647 q^{74} -346.309 q^{76} -285.244 q^{77} +121.691 q^{79} +254.228 q^{82} +609.608 q^{83} -839.890 q^{86} -1003.47 q^{88} -719.038 q^{89} +373.093 q^{91} -371.023 q^{92} -597.608 q^{94} +637.877 q^{97} +111.471 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8} + O(q^{10}) \) \( 2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8} + 6 q^{11} - 16 q^{13} + 21 q^{14} + 137 q^{16} - 6 q^{17} + 64 q^{19} + 276 q^{22} + 6 q^{23} - 318 q^{26} - 119 q^{28} + 252 q^{29} + 40 q^{31} - 279 q^{32} + 66 q^{34} + 248 q^{37} + 588 q^{38} + 450 q^{41} - 376 q^{43} - 804 q^{44} + 960 q^{46} - 12 q^{47} + 98 q^{49} + 890 q^{52} - 1104 q^{53} + 609 q^{56} + 306 q^{58} - 804 q^{59} - 428 q^{61} - 2112 q^{62} + 1289 q^{64} - 148 q^{67} - 222 q^{68} - 954 q^{71} - 1072 q^{73} - 1398 q^{74} - 1508 q^{76} - 42 q^{77} - 572 q^{79} - 1530 q^{82} + 1944 q^{83} - 804 q^{86} + 1164 q^{88} - 366 q^{89} + 112 q^{91} - 2856 q^{92} - 1920 q^{94} - 808 q^{97} - 147 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.27492 0.804305 0.402152 0.915573i \(-0.368262\pi\)
0.402152 + 0.915573i \(0.368262\pi\)
\(3\) 0 0
\(4\) −2.82475 −0.353094
\(5\) 0 0
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) −24.6254 −1.08830
\(9\) 0 0
\(10\) 0 0
\(11\) 40.7492 1.11694 0.558470 0.829525i \(-0.311389\pi\)
0.558470 + 0.829525i \(0.311389\pi\)
\(12\) 0 0
\(13\) −53.2990 −1.13711 −0.568557 0.822644i \(-0.692498\pi\)
−0.568557 + 0.822644i \(0.692498\pi\)
\(14\) −15.9244 −0.303999
\(15\) 0 0
\(16\) −33.4228 −0.522231
\(17\) 4.54983 0.0649116 0.0324558 0.999473i \(-0.489667\pi\)
0.0324558 + 0.999473i \(0.489667\pi\)
\(18\) 0 0
\(19\) 122.598 1.48031 0.740156 0.672436i \(-0.234752\pi\)
0.740156 + 0.672436i \(0.234752\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 92.7010 0.898360
\(23\) 131.347 1.19077 0.595387 0.803439i \(-0.296999\pi\)
0.595387 + 0.803439i \(0.296999\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −121.251 −0.914586
\(27\) 0 0
\(28\) 19.7733 0.133457
\(29\) 216.598 1.38694 0.693470 0.720486i \(-0.256081\pi\)
0.693470 + 0.720486i \(0.256081\pi\)
\(30\) 0 0
\(31\) −251.794 −1.45882 −0.729412 0.684075i \(-0.760206\pi\)
−0.729412 + 0.684075i \(0.760206\pi\)
\(32\) 120.969 0.668267
\(33\) 0 0
\(34\) 10.3505 0.0522087
\(35\) 0 0
\(36\) 0 0
\(37\) −11.8970 −0.0528610 −0.0264305 0.999651i \(-0.508414\pi\)
−0.0264305 + 0.999651i \(0.508414\pi\)
\(38\) 278.900 1.19062
\(39\) 0 0
\(40\) 0 0
\(41\) 111.752 0.425678 0.212839 0.977087i \(-0.431729\pi\)
0.212839 + 0.977087i \(0.431729\pi\)
\(42\) 0 0
\(43\) −369.196 −1.30935 −0.654673 0.755912i \(-0.727194\pi\)
−0.654673 + 0.755912i \(0.727194\pi\)
\(44\) −115.106 −0.394385
\(45\) 0 0
\(46\) 298.804 0.957744
\(47\) −262.694 −0.815275 −0.407637 0.913144i \(-0.633647\pi\)
−0.407637 + 0.913144i \(0.633647\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 150.556 0.401508
\(53\) −567.100 −1.46976 −0.734879 0.678199i \(-0.762761\pi\)
−0.734879 + 0.678199i \(0.762761\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 172.378 0.411339
\(57\) 0 0
\(58\) 492.743 1.11552
\(59\) −839.890 −1.85330 −0.926648 0.375931i \(-0.877323\pi\)
−0.926648 + 0.375931i \(0.877323\pi\)
\(60\) 0 0
\(61\) −485.794 −1.01966 −0.509832 0.860274i \(-0.670293\pi\)
−0.509832 + 0.860274i \(0.670293\pi\)
\(62\) −572.811 −1.17334
\(63\) 0 0
\(64\) 542.577 1.05972
\(65\) 0 0
\(66\) 0 0
\(67\) 333.691 0.608460 0.304230 0.952599i \(-0.401601\pi\)
0.304230 + 0.952599i \(0.401601\pi\)
\(68\) −12.8522 −0.0229199
\(69\) 0 0
\(70\) 0 0
\(71\) −590.248 −0.986613 −0.493306 0.869856i \(-0.664212\pi\)
−0.493306 + 0.869856i \(0.664212\pi\)
\(72\) 0 0
\(73\) −490.701 −0.786743 −0.393371 0.919380i \(-0.628691\pi\)
−0.393371 + 0.919380i \(0.628691\pi\)
\(74\) −27.0647 −0.0425164
\(75\) 0 0
\(76\) −346.309 −0.522689
\(77\) −285.244 −0.422164
\(78\) 0 0
\(79\) 121.691 0.173308 0.0866539 0.996238i \(-0.472383\pi\)
0.0866539 + 0.996238i \(0.472383\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 254.228 0.342375
\(83\) 609.608 0.806183 0.403091 0.915160i \(-0.367936\pi\)
0.403091 + 0.915160i \(0.367936\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −839.890 −1.05311
\(87\) 0 0
\(88\) −1003.47 −1.21557
\(89\) −719.038 −0.856381 −0.428190 0.903689i \(-0.640849\pi\)
−0.428190 + 0.903689i \(0.640849\pi\)
\(90\) 0 0
\(91\) 373.093 0.429789
\(92\) −371.023 −0.420455
\(93\) 0 0
\(94\) −597.608 −0.655729
\(95\) 0 0
\(96\) 0 0
\(97\) 637.877 0.667697 0.333849 0.942627i \(-0.391653\pi\)
0.333849 + 0.942627i \(0.391653\pi\)
\(98\) 111.471 0.114901
\(99\) 0 0
\(100\) 0 0
\(101\) −671.148 −0.661205 −0.330603 0.943770i \(-0.607252\pi\)
−0.330603 + 0.943770i \(0.607252\pi\)
\(102\) 0 0
\(103\) 912.412 0.872841 0.436420 0.899743i \(-0.356246\pi\)
0.436420 + 0.899743i \(0.356246\pi\)
\(104\) 1312.51 1.23752
\(105\) 0 0
\(106\) −1290.10 −1.18213
\(107\) −116.736 −0.105470 −0.0527350 0.998609i \(-0.516794\pi\)
−0.0527350 + 0.998609i \(0.516794\pi\)
\(108\) 0 0
\(109\) 837.176 0.735660 0.367830 0.929893i \(-0.380101\pi\)
0.367830 + 0.929893i \(0.380101\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 233.959 0.197385
\(113\) −1086.58 −0.904572 −0.452286 0.891873i \(-0.649391\pi\)
−0.452286 + 0.891873i \(0.649391\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −611.836 −0.489720
\(117\) 0 0
\(118\) −1910.68 −1.49061
\(119\) −31.8488 −0.0245343
\(120\) 0 0
\(121\) 329.495 0.247554
\(122\) −1105.14 −0.820121
\(123\) 0 0
\(124\) 711.256 0.515102
\(125\) 0 0
\(126\) 0 0
\(127\) 537.113 0.375284 0.187642 0.982237i \(-0.439916\pi\)
0.187642 + 0.982237i \(0.439916\pi\)
\(128\) 266.564 0.184071
\(129\) 0 0
\(130\) 0 0
\(131\) −1497.39 −0.998683 −0.499341 0.866405i \(-0.666425\pi\)
−0.499341 + 0.866405i \(0.666425\pi\)
\(132\) 0 0
\(133\) −858.186 −0.559505
\(134\) 759.120 0.489388
\(135\) 0 0
\(136\) −112.042 −0.0706433
\(137\) −1380.09 −0.860650 −0.430325 0.902674i \(-0.641601\pi\)
−0.430325 + 0.902674i \(0.641601\pi\)
\(138\) 0 0
\(139\) −141.980 −0.0866374 −0.0433187 0.999061i \(-0.513793\pi\)
−0.0433187 + 0.999061i \(0.513793\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1342.76 −0.793537
\(143\) −2171.89 −1.27009
\(144\) 0 0
\(145\) 0 0
\(146\) −1116.30 −0.632781
\(147\) 0 0
\(148\) 33.6061 0.0186649
\(149\) 1943.87 1.06878 0.534390 0.845238i \(-0.320542\pi\)
0.534390 + 0.845238i \(0.320542\pi\)
\(150\) 0 0
\(151\) −2654.76 −1.43074 −0.715370 0.698746i \(-0.753742\pi\)
−0.715370 + 0.698746i \(0.753742\pi\)
\(152\) −3019.03 −1.61102
\(153\) 0 0
\(154\) −648.907 −0.339548
\(155\) 0 0
\(156\) 0 0
\(157\) −1665.22 −0.846489 −0.423244 0.906016i \(-0.639109\pi\)
−0.423244 + 0.906016i \(0.639109\pi\)
\(158\) 276.837 0.139392
\(159\) 0 0
\(160\) 0 0
\(161\) −919.430 −0.450070
\(162\) 0 0
\(163\) 33.0732 0.0158926 0.00794629 0.999968i \(-0.497471\pi\)
0.00794629 + 0.999968i \(0.497471\pi\)
\(164\) −315.673 −0.150304
\(165\) 0 0
\(166\) 1386.81 0.648417
\(167\) 1654.48 0.766630 0.383315 0.923618i \(-0.374782\pi\)
0.383315 + 0.923618i \(0.374782\pi\)
\(168\) 0 0
\(169\) 643.784 0.293029
\(170\) 0 0
\(171\) 0 0
\(172\) 1042.89 0.462322
\(173\) 64.1909 0.0282101 0.0141050 0.999901i \(-0.495510\pi\)
0.0141050 + 0.999901i \(0.495510\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1361.95 −0.583300
\(177\) 0 0
\(178\) −1635.75 −0.688791
\(179\) −3914.68 −1.63462 −0.817309 0.576200i \(-0.804535\pi\)
−0.817309 + 0.576200i \(0.804535\pi\)
\(180\) 0 0
\(181\) −2058.04 −0.845156 −0.422578 0.906327i \(-0.638875\pi\)
−0.422578 + 0.906327i \(0.638875\pi\)
\(182\) 848.756 0.345681
\(183\) 0 0
\(184\) −3234.48 −1.29592
\(185\) 0 0
\(186\) 0 0
\(187\) 185.402 0.0725023
\(188\) 742.046 0.287869
\(189\) 0 0
\(190\) 0 0
\(191\) −428.048 −0.162160 −0.0810798 0.996708i \(-0.525837\pi\)
−0.0810798 + 0.996708i \(0.525837\pi\)
\(192\) 0 0
\(193\) −1604.93 −0.598576 −0.299288 0.954163i \(-0.596749\pi\)
−0.299288 + 0.954163i \(0.596749\pi\)
\(194\) 1451.12 0.537032
\(195\) 0 0
\(196\) −138.413 −0.0504420
\(197\) 3738.83 1.35218 0.676092 0.736817i \(-0.263672\pi\)
0.676092 + 0.736817i \(0.263672\pi\)
\(198\) 0 0
\(199\) −349.030 −0.124332 −0.0621660 0.998066i \(-0.519801\pi\)
−0.0621660 + 0.998066i \(0.519801\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −1526.81 −0.531810
\(203\) −1516.19 −0.524214
\(204\) 0 0
\(205\) 0 0
\(206\) 2075.66 0.702030
\(207\) 0 0
\(208\) 1781.40 0.593836
\(209\) 4995.77 1.65342
\(210\) 0 0
\(211\) 2588.58 0.844574 0.422287 0.906462i \(-0.361227\pi\)
0.422287 + 0.906462i \(0.361227\pi\)
\(212\) 1601.92 0.518962
\(213\) 0 0
\(214\) −265.565 −0.0848300
\(215\) 0 0
\(216\) 0 0
\(217\) 1762.56 0.551384
\(218\) 1904.51 0.591695
\(219\) 0 0
\(220\) 0 0
\(221\) −242.502 −0.0738119
\(222\) 0 0
\(223\) 3236.21 0.971804 0.485902 0.874013i \(-0.338491\pi\)
0.485902 + 0.874013i \(0.338491\pi\)
\(224\) −846.785 −0.252581
\(225\) 0 0
\(226\) −2471.88 −0.727552
\(227\) −5631.62 −1.64662 −0.823312 0.567589i \(-0.807876\pi\)
−0.823312 + 0.567589i \(0.807876\pi\)
\(228\) 0 0
\(229\) 3770.25 1.08797 0.543985 0.839095i \(-0.316915\pi\)
0.543985 + 0.839095i \(0.316915\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −5333.82 −1.50941
\(233\) −6560.90 −1.84472 −0.922358 0.386336i \(-0.873741\pi\)
−0.922358 + 0.386336i \(0.873741\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 2372.48 0.654387
\(237\) 0 0
\(238\) −72.4535 −0.0197330
\(239\) 771.444 0.208789 0.104394 0.994536i \(-0.466710\pi\)
0.104394 + 0.994536i \(0.466710\pi\)
\(240\) 0 0
\(241\) 1252.10 0.334668 0.167334 0.985900i \(-0.446484\pi\)
0.167334 + 0.985900i \(0.446484\pi\)
\(242\) 749.574 0.199109
\(243\) 0 0
\(244\) 1372.25 0.360037
\(245\) 0 0
\(246\) 0 0
\(247\) −6534.35 −1.68328
\(248\) 6200.53 1.58764
\(249\) 0 0
\(250\) 0 0
\(251\) −5166.27 −1.29917 −0.649586 0.760288i \(-0.725058\pi\)
−0.649586 + 0.760288i \(0.725058\pi\)
\(252\) 0 0
\(253\) 5352.29 1.33002
\(254\) 1221.89 0.301843
\(255\) 0 0
\(256\) −3734.21 −0.911672
\(257\) 2767.45 0.671707 0.335854 0.941914i \(-0.390975\pi\)
0.335854 + 0.941914i \(0.390975\pi\)
\(258\) 0 0
\(259\) 83.2791 0.0199796
\(260\) 0 0
\(261\) 0 0
\(262\) −3406.44 −0.803245
\(263\) −4101.78 −0.961699 −0.480849 0.876803i \(-0.659672\pi\)
−0.480849 + 0.876803i \(0.659672\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1952.30 −0.450013
\(267\) 0 0
\(268\) −942.594 −0.214844
\(269\) 6950.84 1.57546 0.787732 0.616018i \(-0.211255\pi\)
0.787732 + 0.616018i \(0.211255\pi\)
\(270\) 0 0
\(271\) −7140.29 −1.60052 −0.800262 0.599651i \(-0.795306\pi\)
−0.800262 + 0.599651i \(0.795306\pi\)
\(272\) −152.068 −0.0338988
\(273\) 0 0
\(274\) −3139.59 −0.692225
\(275\) 0 0
\(276\) 0 0
\(277\) −1320.51 −0.286433 −0.143217 0.989691i \(-0.545745\pi\)
−0.143217 + 0.989691i \(0.545745\pi\)
\(278\) −322.993 −0.0696829
\(279\) 0 0
\(280\) 0 0
\(281\) 204.309 0.0433738 0.0216869 0.999765i \(-0.493096\pi\)
0.0216869 + 0.999765i \(0.493096\pi\)
\(282\) 0 0
\(283\) −975.794 −0.204964 −0.102482 0.994735i \(-0.532678\pi\)
−0.102482 + 0.994735i \(0.532678\pi\)
\(284\) 1667.30 0.348367
\(285\) 0 0
\(286\) −4940.87 −1.02154
\(287\) −782.267 −0.160891
\(288\) 0 0
\(289\) −4892.30 −0.995786
\(290\) 0 0
\(291\) 0 0
\(292\) 1386.11 0.277794
\(293\) 607.919 0.121212 0.0606058 0.998162i \(-0.480697\pi\)
0.0606058 + 0.998162i \(0.480697\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 292.969 0.0575286
\(297\) 0 0
\(298\) 4422.14 0.859624
\(299\) −7000.67 −1.35405
\(300\) 0 0
\(301\) 2584.37 0.494886
\(302\) −6039.37 −1.15075
\(303\) 0 0
\(304\) −4097.56 −0.773064
\(305\) 0 0
\(306\) 0 0
\(307\) −8037.08 −1.49414 −0.747069 0.664747i \(-0.768539\pi\)
−0.747069 + 0.664747i \(0.768539\pi\)
\(308\) 805.744 0.149063
\(309\) 0 0
\(310\) 0 0
\(311\) −5311.60 −0.968468 −0.484234 0.874939i \(-0.660902\pi\)
−0.484234 + 0.874939i \(0.660902\pi\)
\(312\) 0 0
\(313\) 1531.61 0.276587 0.138293 0.990391i \(-0.455838\pi\)
0.138293 + 0.990391i \(0.455838\pi\)
\(314\) −3788.23 −0.680835
\(315\) 0 0
\(316\) −343.747 −0.0611939
\(317\) 4219.19 0.747549 0.373775 0.927520i \(-0.378063\pi\)
0.373775 + 0.927520i \(0.378063\pi\)
\(318\) 0 0
\(319\) 8826.19 1.54913
\(320\) 0 0
\(321\) 0 0
\(322\) −2091.63 −0.361993
\(323\) 557.801 0.0960893
\(324\) 0 0
\(325\) 0 0
\(326\) 75.2387 0.0127825
\(327\) 0 0
\(328\) −2751.95 −0.463265
\(329\) 1838.86 0.308145
\(330\) 0 0
\(331\) 8298.19 1.37797 0.688987 0.724773i \(-0.258056\pi\)
0.688987 + 0.724773i \(0.258056\pi\)
\(332\) −1721.99 −0.284658
\(333\) 0 0
\(334\) 3763.79 0.616604
\(335\) 0 0
\(336\) 0 0
\(337\) 4348.44 0.702892 0.351446 0.936208i \(-0.385690\pi\)
0.351446 + 0.936208i \(0.385690\pi\)
\(338\) 1464.56 0.235684
\(339\) 0 0
\(340\) 0 0
\(341\) −10260.4 −1.62942
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) 9091.60 1.42496
\(345\) 0 0
\(346\) 146.029 0.0226895
\(347\) −8345.54 −1.29110 −0.645550 0.763718i \(-0.723372\pi\)
−0.645550 + 0.763718i \(0.723372\pi\)
\(348\) 0 0
\(349\) −9982.54 −1.53110 −0.765549 0.643378i \(-0.777532\pi\)
−0.765549 + 0.643378i \(0.777532\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4929.40 0.746414
\(353\) 8801.59 1.32709 0.663543 0.748138i \(-0.269052\pi\)
0.663543 + 0.748138i \(0.269052\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2031.10 0.302383
\(357\) 0 0
\(358\) −8905.56 −1.31473
\(359\) 524.039 0.0770409 0.0385205 0.999258i \(-0.487736\pi\)
0.0385205 + 0.999258i \(0.487736\pi\)
\(360\) 0 0
\(361\) 8171.27 1.19132
\(362\) −4681.88 −0.679763
\(363\) 0 0
\(364\) −1053.90 −0.151756
\(365\) 0 0
\(366\) 0 0
\(367\) 6362.72 0.904991 0.452495 0.891767i \(-0.350534\pi\)
0.452495 + 0.891767i \(0.350534\pi\)
\(368\) −4389.99 −0.621858
\(369\) 0 0
\(370\) 0 0
\(371\) 3969.70 0.555516
\(372\) 0 0
\(373\) 11265.8 1.56387 0.781935 0.623361i \(-0.214233\pi\)
0.781935 + 0.623361i \(0.214233\pi\)
\(374\) 421.774 0.0583140
\(375\) 0 0
\(376\) 6468.96 0.887263
\(377\) −11544.5 −1.57711
\(378\) 0 0
\(379\) −1151.71 −0.156094 −0.0780470 0.996950i \(-0.524868\pi\)
−0.0780470 + 0.996950i \(0.524868\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −973.774 −0.130426
\(383\) −151.554 −0.0202195 −0.0101097 0.999949i \(-0.503218\pi\)
−0.0101097 + 0.999949i \(0.503218\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3651.08 −0.481437
\(387\) 0 0
\(388\) −1801.84 −0.235760
\(389\) −4794.18 −0.624870 −0.312435 0.949939i \(-0.601145\pi\)
−0.312435 + 0.949939i \(0.601145\pi\)
\(390\) 0 0
\(391\) 597.608 0.0772950
\(392\) −1206.65 −0.155471
\(393\) 0 0
\(394\) 8505.52 1.08757
\(395\) 0 0
\(396\) 0 0
\(397\) 4623.94 0.584556 0.292278 0.956333i \(-0.405587\pi\)
0.292278 + 0.956333i \(0.405587\pi\)
\(398\) −794.014 −0.100001
\(399\) 0 0
\(400\) 0 0
\(401\) 3610.63 0.449642 0.224821 0.974400i \(-0.427820\pi\)
0.224821 + 0.974400i \(0.427820\pi\)
\(402\) 0 0
\(403\) 13420.4 1.65885
\(404\) 1895.83 0.233467
\(405\) 0 0
\(406\) −3449.20 −0.421628
\(407\) −484.794 −0.0590426
\(408\) 0 0
\(409\) 8959.57 1.08318 0.541592 0.840641i \(-0.317822\pi\)
0.541592 + 0.840641i \(0.317822\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2577.34 −0.308195
\(413\) 5879.23 0.700480
\(414\) 0 0
\(415\) 0 0
\(416\) −6447.54 −0.759896
\(417\) 0 0
\(418\) 11365.0 1.32985
\(419\) 7078.28 0.825290 0.412645 0.910892i \(-0.364605\pi\)
0.412645 + 0.910892i \(0.364605\pi\)
\(420\) 0 0
\(421\) 11551.5 1.33725 0.668626 0.743599i \(-0.266883\pi\)
0.668626 + 0.743599i \(0.266883\pi\)
\(422\) 5888.80 0.679295
\(423\) 0 0
\(424\) 13965.1 1.59954
\(425\) 0 0
\(426\) 0 0
\(427\) 3400.56 0.385397
\(428\) 329.750 0.0372408
\(429\) 0 0
\(430\) 0 0
\(431\) 4064.38 0.454232 0.227116 0.973868i \(-0.427070\pi\)
0.227116 + 0.973868i \(0.427070\pi\)
\(432\) 0 0
\(433\) −17456.3 −1.93740 −0.968701 0.248229i \(-0.920151\pi\)
−0.968701 + 0.248229i \(0.920151\pi\)
\(434\) 4009.67 0.443480
\(435\) 0 0
\(436\) −2364.81 −0.259757
\(437\) 16102.9 1.76271
\(438\) 0 0
\(439\) 4595.39 0.499604 0.249802 0.968297i \(-0.419635\pi\)
0.249802 + 0.968297i \(0.419635\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −551.671 −0.0593672
\(443\) −306.214 −0.0328412 −0.0164206 0.999865i \(-0.505227\pi\)
−0.0164206 + 0.999865i \(0.505227\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 7362.10 0.781627
\(447\) 0 0
\(448\) −3798.04 −0.400537
\(449\) −9229.22 −0.970053 −0.485026 0.874500i \(-0.661190\pi\)
−0.485026 + 0.874500i \(0.661190\pi\)
\(450\) 0 0
\(451\) 4553.82 0.475457
\(452\) 3069.31 0.319399
\(453\) 0 0
\(454\) −12811.5 −1.32439
\(455\) 0 0
\(456\) 0 0
\(457\) 10992.2 1.12515 0.562577 0.826745i \(-0.309810\pi\)
0.562577 + 0.826745i \(0.309810\pi\)
\(458\) 8577.01 0.875059
\(459\) 0 0
\(460\) 0 0
\(461\) −7387.88 −0.746394 −0.373197 0.927752i \(-0.621739\pi\)
−0.373197 + 0.927752i \(0.621739\pi\)
\(462\) 0 0
\(463\) −10163.8 −1.02020 −0.510101 0.860114i \(-0.670392\pi\)
−0.510101 + 0.860114i \(0.670392\pi\)
\(464\) −7239.30 −0.724302
\(465\) 0 0
\(466\) −14925.5 −1.48371
\(467\) −15814.6 −1.56705 −0.783524 0.621362i \(-0.786580\pi\)
−0.783524 + 0.621362i \(0.786580\pi\)
\(468\) 0 0
\(469\) −2335.84 −0.229976
\(470\) 0 0
\(471\) 0 0
\(472\) 20682.6 2.01694
\(473\) −15044.4 −1.46246
\(474\) 0 0
\(475\) 0 0
\(476\) 89.9651 0.00866290
\(477\) 0 0
\(478\) 1754.97 0.167930
\(479\) 1444.85 0.137823 0.0689113 0.997623i \(-0.478047\pi\)
0.0689113 + 0.997623i \(0.478047\pi\)
\(480\) 0 0
\(481\) 634.099 0.0601090
\(482\) 2848.43 0.269175
\(483\) 0 0
\(484\) −930.742 −0.0874100
\(485\) 0 0
\(486\) 0 0
\(487\) 489.402 0.0455378 0.0227689 0.999741i \(-0.492752\pi\)
0.0227689 + 0.999741i \(0.492752\pi\)
\(488\) 11962.9 1.10970
\(489\) 0 0
\(490\) 0 0
\(491\) 3941.30 0.362257 0.181129 0.983459i \(-0.442025\pi\)
0.181129 + 0.983459i \(0.442025\pi\)
\(492\) 0 0
\(493\) 985.485 0.0900284
\(494\) −14865.1 −1.35387
\(495\) 0 0
\(496\) 8415.65 0.761843
\(497\) 4131.73 0.372905
\(498\) 0 0
\(499\) 11.0894 0.000994850 0 0.000497425 1.00000i \(-0.499842\pi\)
0.000497425 1.00000i \(0.499842\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −11752.8 −1.04493
\(503\) 7088.41 0.628343 0.314172 0.949366i \(-0.398273\pi\)
0.314172 + 0.949366i \(0.398273\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 12176.0 1.06974
\(507\) 0 0
\(508\) −1517.21 −0.132511
\(509\) −17588.4 −1.53162 −0.765810 0.643067i \(-0.777662\pi\)
−0.765810 + 0.643067i \(0.777662\pi\)
\(510\) 0 0
\(511\) 3434.91 0.297361
\(512\) −10627.5 −0.917333
\(513\) 0 0
\(514\) 6295.72 0.540257
\(515\) 0 0
\(516\) 0 0
\(517\) −10704.6 −0.910613
\(518\) 189.453 0.0160697
\(519\) 0 0
\(520\) 0 0
\(521\) 11646.6 0.979360 0.489680 0.871902i \(-0.337114\pi\)
0.489680 + 0.871902i \(0.337114\pi\)
\(522\) 0 0
\(523\) −8965.82 −0.749614 −0.374807 0.927103i \(-0.622291\pi\)
−0.374807 + 0.927103i \(0.622291\pi\)
\(524\) 4229.75 0.352629
\(525\) 0 0
\(526\) −9331.22 −0.773499
\(527\) −1145.62 −0.0946946
\(528\) 0 0
\(529\) 5085.08 0.417941
\(530\) 0 0
\(531\) 0 0
\(532\) 2424.16 0.197558
\(533\) −5956.30 −0.484045
\(534\) 0 0
\(535\) 0 0
\(536\) −8217.28 −0.662187
\(537\) 0 0
\(538\) 15812.6 1.26715
\(539\) 1996.71 0.159563
\(540\) 0 0
\(541\) −195.272 −0.0155183 −0.00775914 0.999970i \(-0.502470\pi\)
−0.00775914 + 0.999970i \(0.502470\pi\)
\(542\) −16243.6 −1.28731
\(543\) 0 0
\(544\) 550.390 0.0433783
\(545\) 0 0
\(546\) 0 0
\(547\) 1399.26 0.109375 0.0546874 0.998504i \(-0.482584\pi\)
0.0546874 + 0.998504i \(0.482584\pi\)
\(548\) 3898.41 0.303890
\(549\) 0 0
\(550\) 0 0
\(551\) 26554.5 2.05310
\(552\) 0 0
\(553\) −851.837 −0.0655042
\(554\) −3004.06 −0.230380
\(555\) 0 0
\(556\) 401.059 0.0305911
\(557\) 43.0467 0.00327459 0.00163730 0.999999i \(-0.499479\pi\)
0.00163730 + 0.999999i \(0.499479\pi\)
\(558\) 0 0
\(559\) 19677.8 1.48888
\(560\) 0 0
\(561\) 0 0
\(562\) 464.786 0.0348858
\(563\) −19232.9 −1.43973 −0.719865 0.694114i \(-0.755797\pi\)
−0.719865 + 0.694114i \(0.755797\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −2219.85 −0.164854
\(567\) 0 0
\(568\) 14535.1 1.07373
\(569\) −5163.98 −0.380466 −0.190233 0.981739i \(-0.560924\pi\)
−0.190233 + 0.981739i \(0.560924\pi\)
\(570\) 0 0
\(571\) −10231.9 −0.749899 −0.374950 0.927045i \(-0.622340\pi\)
−0.374950 + 0.927045i \(0.622340\pi\)
\(572\) 6135.05 0.448460
\(573\) 0 0
\(574\) −1779.59 −0.129406
\(575\) 0 0
\(576\) 0 0
\(577\) −16563.7 −1.19507 −0.597537 0.801842i \(-0.703854\pi\)
−0.597537 + 0.801842i \(0.703854\pi\)
\(578\) −11129.6 −0.800916
\(579\) 0 0
\(580\) 0 0
\(581\) −4267.26 −0.304708
\(582\) 0 0
\(583\) −23108.8 −1.64163
\(584\) 12083.7 0.856212
\(585\) 0 0
\(586\) 1382.96 0.0974910
\(587\) 16020.6 1.12648 0.563239 0.826294i \(-0.309555\pi\)
0.563239 + 0.826294i \(0.309555\pi\)
\(588\) 0 0
\(589\) −30869.4 −2.15951
\(590\) 0 0
\(591\) 0 0
\(592\) 397.631 0.0276057
\(593\) −6771.14 −0.468900 −0.234450 0.972128i \(-0.575329\pi\)
−0.234450 + 0.972128i \(0.575329\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −5490.95 −0.377379
\(597\) 0 0
\(598\) −15926.0 −1.08906
\(599\) −11070.2 −0.755120 −0.377560 0.925985i \(-0.623237\pi\)
−0.377560 + 0.925985i \(0.623237\pi\)
\(600\) 0 0
\(601\) −24187.7 −1.64166 −0.820830 0.571173i \(-0.806489\pi\)
−0.820830 + 0.571173i \(0.806489\pi\)
\(602\) 5879.23 0.398039
\(603\) 0 0
\(604\) 7499.05 0.505185
\(605\) 0 0
\(606\) 0 0
\(607\) 10074.1 0.673631 0.336816 0.941571i \(-0.390650\pi\)
0.336816 + 0.941571i \(0.390650\pi\)
\(608\) 14830.6 0.989243
\(609\) 0 0
\(610\) 0 0
\(611\) 14001.3 0.927060
\(612\) 0 0
\(613\) 11114.6 0.732323 0.366161 0.930551i \(-0.380672\pi\)
0.366161 + 0.930551i \(0.380672\pi\)
\(614\) −18283.7 −1.20174
\(615\) 0 0
\(616\) 7024.26 0.459441
\(617\) 20496.4 1.33737 0.668683 0.743548i \(-0.266858\pi\)
0.668683 + 0.743548i \(0.266858\pi\)
\(618\) 0 0
\(619\) −16714.4 −1.08532 −0.542658 0.839954i \(-0.682582\pi\)
−0.542658 + 0.839954i \(0.682582\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −12083.5 −0.778943
\(623\) 5033.27 0.323682
\(624\) 0 0
\(625\) 0 0
\(626\) 3484.28 0.222460
\(627\) 0 0
\(628\) 4703.82 0.298890
\(629\) −54.1295 −0.00343129
\(630\) 0 0
\(631\) 9168.53 0.578437 0.289218 0.957263i \(-0.406605\pi\)
0.289218 + 0.957263i \(0.406605\pi\)
\(632\) −2996.69 −0.188611
\(633\) 0 0
\(634\) 9598.30 0.601257
\(635\) 0 0
\(636\) 0 0
\(637\) −2611.65 −0.162445
\(638\) 20078.9 1.24597
\(639\) 0 0
\(640\) 0 0
\(641\) 4273.37 0.263319 0.131660 0.991295i \(-0.457969\pi\)
0.131660 + 0.991295i \(0.457969\pi\)
\(642\) 0 0
\(643\) −2955.75 −0.181281 −0.0906404 0.995884i \(-0.528891\pi\)
−0.0906404 + 0.995884i \(0.528891\pi\)
\(644\) 2597.16 0.158917
\(645\) 0 0
\(646\) 1268.95 0.0772851
\(647\) 22701.2 1.37941 0.689704 0.724091i \(-0.257741\pi\)
0.689704 + 0.724091i \(0.257741\pi\)
\(648\) 0 0
\(649\) −34224.8 −2.07002
\(650\) 0 0
\(651\) 0 0
\(652\) −93.4235 −0.00561158
\(653\) 1537.81 0.0921582 0.0460791 0.998938i \(-0.485327\pi\)
0.0460791 + 0.998938i \(0.485327\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3735.08 −0.222302
\(657\) 0 0
\(658\) 4183.26 0.247842
\(659\) −12338.1 −0.729323 −0.364661 0.931140i \(-0.618815\pi\)
−0.364661 + 0.931140i \(0.618815\pi\)
\(660\) 0 0
\(661\) 1845.10 0.108572 0.0542859 0.998525i \(-0.482712\pi\)
0.0542859 + 0.998525i \(0.482712\pi\)
\(662\) 18877.7 1.10831
\(663\) 0 0
\(664\) −15011.8 −0.877369
\(665\) 0 0
\(666\) 0 0
\(667\) 28449.5 1.65153
\(668\) −4673.48 −0.270692
\(669\) 0 0
\(670\) 0 0
\(671\) −19795.7 −1.13890
\(672\) 0 0
\(673\) −23955.4 −1.37208 −0.686041 0.727563i \(-0.740653\pi\)
−0.686041 + 0.727563i \(0.740653\pi\)
\(674\) 9892.34 0.565339
\(675\) 0 0
\(676\) −1818.53 −0.103467
\(677\) −3678.26 −0.208814 −0.104407 0.994535i \(-0.533294\pi\)
−0.104407 + 0.994535i \(0.533294\pi\)
\(678\) 0 0
\(679\) −4465.14 −0.252366
\(680\) 0 0
\(681\) 0 0
\(682\) −23341.6 −1.31055
\(683\) 4390.87 0.245991 0.122996 0.992407i \(-0.460750\pi\)
0.122996 + 0.992407i \(0.460750\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −780.297 −0.0434284
\(687\) 0 0
\(688\) 12339.6 0.683781
\(689\) 30225.8 1.67128
\(690\) 0 0
\(691\) 10371.7 0.570994 0.285497 0.958380i \(-0.407841\pi\)
0.285497 + 0.958380i \(0.407841\pi\)
\(692\) −181.323 −0.00996081
\(693\) 0 0
\(694\) −18985.4 −1.03844
\(695\) 0 0
\(696\) 0 0
\(697\) 508.455 0.0276314
\(698\) −22709.4 −1.23147
\(699\) 0 0
\(700\) 0 0
\(701\) −109.675 −0.00590922 −0.00295461 0.999996i \(-0.500940\pi\)
−0.00295461 + 0.999996i \(0.500940\pi\)
\(702\) 0 0
\(703\) −1458.55 −0.0782508
\(704\) 22109.6 1.18364
\(705\) 0 0
\(706\) 20022.9 1.06738
\(707\) 4698.03 0.249912
\(708\) 0 0
\(709\) 26918.8 1.42589 0.712944 0.701221i \(-0.247361\pi\)
0.712944 + 0.701221i \(0.247361\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 17706.6 0.931999
\(713\) −33072.4 −1.73713
\(714\) 0 0
\(715\) 0 0
\(716\) 11058.0 0.577174
\(717\) 0 0
\(718\) 1192.14 0.0619644
\(719\) −15170.8 −0.786889 −0.393445 0.919348i \(-0.628717\pi\)
−0.393445 + 0.919348i \(0.628717\pi\)
\(720\) 0 0
\(721\) −6386.88 −0.329903
\(722\) 18589.0 0.958185
\(723\) 0 0
\(724\) 5813.46 0.298419
\(725\) 0 0
\(726\) 0 0
\(727\) 33286.9 1.69813 0.849066 0.528288i \(-0.177166\pi\)
0.849066 + 0.528288i \(0.177166\pi\)
\(728\) −9187.57 −0.467739
\(729\) 0 0
\(730\) 0 0
\(731\) −1679.78 −0.0849917
\(732\) 0 0
\(733\) −20544.0 −1.03521 −0.517607 0.855619i \(-0.673177\pi\)
−0.517607 + 0.855619i \(0.673177\pi\)
\(734\) 14474.7 0.727888
\(735\) 0 0
\(736\) 15889.0 0.795755
\(737\) 13597.6 0.679614
\(738\) 0 0
\(739\) 34357.2 1.71022 0.855109 0.518449i \(-0.173490\pi\)
0.855109 + 0.518449i \(0.173490\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 9030.73 0.446804
\(743\) 8166.99 0.403254 0.201627 0.979462i \(-0.435377\pi\)
0.201627 + 0.979462i \(0.435377\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 25628.9 1.25783
\(747\) 0 0
\(748\) −523.715 −0.0256001
\(749\) 817.151 0.0398639
\(750\) 0 0
\(751\) 17080.1 0.829909 0.414954 0.909842i \(-0.363798\pi\)
0.414954 + 0.909842i \(0.363798\pi\)
\(752\) 8779.97 0.425761
\(753\) 0 0
\(754\) −26262.7 −1.26848
\(755\) 0 0
\(756\) 0 0
\(757\) 16324.0 0.783758 0.391879 0.920017i \(-0.371825\pi\)
0.391879 + 0.920017i \(0.371825\pi\)
\(758\) −2620.06 −0.125547
\(759\) 0 0
\(760\) 0 0
\(761\) 32366.2 1.54175 0.770875 0.636986i \(-0.219819\pi\)
0.770875 + 0.636986i \(0.219819\pi\)
\(762\) 0 0
\(763\) −5860.23 −0.278053
\(764\) 1209.13 0.0572576
\(765\) 0 0
\(766\) −344.774 −0.0162626
\(767\) 44765.3 2.10741
\(768\) 0 0
\(769\) −7948.44 −0.372728 −0.186364 0.982481i \(-0.559670\pi\)
−0.186364 + 0.982481i \(0.559670\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4533.52 0.211354
\(773\) 17819.3 0.829127 0.414564 0.910020i \(-0.363934\pi\)
0.414564 + 0.910020i \(0.363934\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −15708.0 −0.726655
\(777\) 0 0
\(778\) −10906.4 −0.502586
\(779\) 13700.6 0.630136
\(780\) 0 0
\(781\) −24052.1 −1.10199
\(782\) 1359.51 0.0621687
\(783\) 0 0
\(784\) −1637.72 −0.0746044
\(785\) 0 0
\(786\) 0 0
\(787\) −2912.38 −0.131912 −0.0659562 0.997823i \(-0.521010\pi\)
−0.0659562 + 0.997823i \(0.521010\pi\)
\(788\) −10561.3 −0.477448
\(789\) 0 0
\(790\) 0 0
\(791\) 7606.05 0.341896
\(792\) 0 0
\(793\) 25892.3 1.15948
\(794\) 10519.1 0.470162
\(795\) 0 0
\(796\) 985.923 0.0439009
\(797\) −33789.1 −1.50172 −0.750861 0.660460i \(-0.770361\pi\)
−0.750861 + 0.660460i \(0.770361\pi\)
\(798\) 0 0
\(799\) −1195.22 −0.0529208
\(800\) 0 0
\(801\) 0 0
\(802\) 8213.90 0.361649
\(803\) −19995.7 −0.878744
\(804\) 0 0
\(805\) 0 0
\(806\) 30530.2 1.33422
\(807\) 0 0
\(808\) 16527.3 0.719589
\(809\) −1252.13 −0.0544159 −0.0272079 0.999630i \(-0.508662\pi\)
−0.0272079 + 0.999630i \(0.508662\pi\)
\(810\) 0 0
\(811\) −31913.1 −1.38178 −0.690889 0.722961i \(-0.742781\pi\)
−0.690889 + 0.722961i \(0.742781\pi\)
\(812\) 4282.85 0.185097
\(813\) 0 0
\(814\) −1102.87 −0.0474882
\(815\) 0 0
\(816\) 0 0
\(817\) −45262.7 −1.93824
\(818\) 20382.3 0.871210
\(819\) 0 0
\(820\) 0 0
\(821\) −30742.4 −1.30684 −0.653421 0.756995i \(-0.726667\pi\)
−0.653421 + 0.756995i \(0.726667\pi\)
\(822\) 0 0
\(823\) 13822.6 0.585449 0.292724 0.956197i \(-0.405438\pi\)
0.292724 + 0.956197i \(0.405438\pi\)
\(824\) −22468.5 −0.949913
\(825\) 0 0
\(826\) 13374.8 0.563399
\(827\) −42107.1 −1.77051 −0.885253 0.465110i \(-0.846015\pi\)
−0.885253 + 0.465110i \(0.846015\pi\)
\(828\) 0 0
\(829\) −38763.8 −1.62403 −0.812015 0.583636i \(-0.801629\pi\)
−0.812015 + 0.583636i \(0.801629\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −28918.8 −1.20502
\(833\) 222.942 0.00927308
\(834\) 0 0
\(835\) 0 0
\(836\) −14111.8 −0.583812
\(837\) 0 0
\(838\) 16102.5 0.663784
\(839\) −16896.3 −0.695262 −0.347631 0.937631i \(-0.613014\pi\)
−0.347631 + 0.937631i \(0.613014\pi\)
\(840\) 0 0
\(841\) 22525.7 0.923601
\(842\) 26278.6 1.07556
\(843\) 0 0
\(844\) −7312.09 −0.298214
\(845\) 0 0
\(846\) 0 0
\(847\) −2306.47 −0.0935668
\(848\) 18954.0 0.767552
\(849\) 0 0
\(850\) 0 0
\(851\) −1562.64 −0.0629455
\(852\) 0 0
\(853\) −46429.3 −1.86367 −0.931833 0.362887i \(-0.881791\pi\)
−0.931833 + 0.362887i \(0.881791\pi\)
\(854\) 7735.99 0.309977
\(855\) 0 0
\(856\) 2874.67 0.114783
\(857\) 21206.4 0.845272 0.422636 0.906300i \(-0.361105\pi\)
0.422636 + 0.906300i \(0.361105\pi\)
\(858\) 0 0
\(859\) 13876.2 0.551163 0.275581 0.961278i \(-0.411130\pi\)
0.275581 + 0.961278i \(0.411130\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 9246.12 0.365341
\(863\) 14337.1 0.565515 0.282757 0.959191i \(-0.408751\pi\)
0.282757 + 0.959191i \(0.408751\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −39711.6 −1.55826
\(867\) 0 0
\(868\) −4978.79 −0.194690
\(869\) 4958.81 0.193574
\(870\) 0 0
\(871\) −17785.4 −0.691889
\(872\) −20615.8 −0.800619
\(873\) 0 0
\(874\) 36632.8 1.41776
\(875\) 0 0
\(876\) 0 0
\(877\) 24369.3 0.938304 0.469152 0.883118i \(-0.344560\pi\)
0.469152 + 0.883118i \(0.344560\pi\)
\(878\) 10454.1 0.401834
\(879\) 0 0
\(880\) 0 0
\(881\) 26127.0 0.999140 0.499570 0.866273i \(-0.333491\pi\)
0.499570 + 0.866273i \(0.333491\pi\)
\(882\) 0 0
\(883\) 15713.1 0.598855 0.299428 0.954119i \(-0.403204\pi\)
0.299428 + 0.954119i \(0.403204\pi\)
\(884\) 685.007 0.0260625
\(885\) 0 0
\(886\) −696.611 −0.0264143
\(887\) 13139.5 0.497385 0.248692 0.968583i \(-0.419999\pi\)
0.248692 + 0.968583i \(0.419999\pi\)
\(888\) 0 0
\(889\) −3759.79 −0.141844
\(890\) 0 0
\(891\) 0 0
\(892\) −9141.48 −0.343138
\(893\) −32205.8 −1.20686
\(894\) 0 0
\(895\) 0 0
\(896\) −1865.95 −0.0695725
\(897\) 0 0
\(898\) −20995.7 −0.780218
\(899\) −54538.1 −2.02330
\(900\) 0 0
\(901\) −2580.21 −0.0954043
\(902\) 10359.6 0.382412
\(903\) 0 0
\(904\) 26757.4 0.984446
\(905\) 0 0
\(906\) 0 0
\(907\) 3799.71 0.139104 0.0695519 0.997578i \(-0.477843\pi\)
0.0695519 + 0.997578i \(0.477843\pi\)
\(908\) 15907.9 0.581413
\(909\) 0 0
\(910\) 0 0
\(911\) 51528.4 1.87400 0.936998 0.349334i \(-0.113592\pi\)
0.936998 + 0.349334i \(0.113592\pi\)
\(912\) 0 0
\(913\) 24841.0 0.900458
\(914\) 25006.4 0.904966
\(915\) 0 0
\(916\) −10650.0 −0.384156
\(917\) 10481.7 0.377467
\(918\) 0 0
\(919\) −16984.7 −0.609657 −0.304828 0.952407i \(-0.598599\pi\)
−0.304828 + 0.952407i \(0.598599\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −16806.8 −0.600329
\(923\) 31459.6 1.12189
\(924\) 0 0
\(925\) 0 0
\(926\) −23121.9 −0.820553
\(927\) 0 0
\(928\) 26201.7 0.926846
\(929\) 5451.85 0.192540 0.0962699 0.995355i \(-0.469309\pi\)
0.0962699 + 0.995355i \(0.469309\pi\)
\(930\) 0 0
\(931\) 6007.30 0.211473
\(932\) 18532.9 0.651358
\(933\) 0 0
\(934\) −35976.8 −1.26038
\(935\) 0 0
\(936\) 0 0
\(937\) −42429.4 −1.47930 −0.739652 0.672989i \(-0.765010\pi\)
−0.739652 + 0.672989i \(0.765010\pi\)
\(938\) −5313.84 −0.184971
\(939\) 0 0
\(940\) 0 0
\(941\) 32977.9 1.14245 0.571226 0.820793i \(-0.306468\pi\)
0.571226 + 0.820793i \(0.306468\pi\)
\(942\) 0 0
\(943\) 14678.4 0.506886
\(944\) 28071.5 0.967848
\(945\) 0 0
\(946\) −34224.8 −1.17626
\(947\) −23753.4 −0.815082 −0.407541 0.913187i \(-0.633614\pi\)
−0.407541 + 0.913187i \(0.633614\pi\)
\(948\) 0 0
\(949\) 26153.9 0.894616
\(950\) 0 0
\(951\) 0 0
\(952\) 784.291 0.0267006
\(953\) −28074.3 −0.954267 −0.477134 0.878831i \(-0.658324\pi\)
−0.477134 + 0.878831i \(0.658324\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −2179.14 −0.0737221
\(957\) 0 0
\(958\) 3286.92 0.110851
\(959\) 9660.63 0.325295
\(960\) 0 0
\(961\) 33609.2 1.12817
\(962\) 1442.52 0.0483460
\(963\) 0 0
\(964\) −3536.88 −0.118169
\(965\) 0 0
\(966\) 0 0
\(967\) 11150.3 0.370806 0.185403 0.982663i \(-0.440641\pi\)
0.185403 + 0.982663i \(0.440641\pi\)
\(968\) −8113.95 −0.269414
\(969\) 0 0
\(970\) 0 0
\(971\) −6059.04 −0.200251 −0.100126 0.994975i \(-0.531924\pi\)
−0.100126 + 0.994975i \(0.531924\pi\)
\(972\) 0 0
\(973\) 993.861 0.0327459
\(974\) 1113.35 0.0366263
\(975\) 0 0
\(976\) 16236.6 0.532500
\(977\) 5700.49 0.186668 0.0933341 0.995635i \(-0.470248\pi\)
0.0933341 + 0.995635i \(0.470248\pi\)
\(978\) 0 0
\(979\) −29300.2 −0.956526
\(980\) 0 0
\(981\) 0 0
\(982\) 8966.12 0.291365
\(983\) 197.480 0.00640757 0.00320378 0.999995i \(-0.498980\pi\)
0.00320378 + 0.999995i \(0.498980\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 2241.90 0.0724103
\(987\) 0 0
\(988\) 18457.9 0.594357
\(989\) −48492.9 −1.55913
\(990\) 0 0
\(991\) 20620.8 0.660990 0.330495 0.943808i \(-0.392784\pi\)
0.330495 + 0.943808i \(0.392784\pi\)
\(992\) −30459.3 −0.974884
\(993\) 0 0
\(994\) 9399.35 0.299929
\(995\) 0 0
\(996\) 0 0
\(997\) −19326.8 −0.613928 −0.306964 0.951721i \(-0.599313\pi\)
−0.306964 + 0.951721i \(0.599313\pi\)
\(998\) 25.2275 0.000800162 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1575.4.a.p.1.2 2
3.2 odd 2 525.4.a.n.1.1 2
5.4 even 2 63.4.a.e.1.1 2
15.2 even 4 525.4.d.g.274.2 4
15.8 even 4 525.4.d.g.274.3 4
15.14 odd 2 21.4.a.c.1.2 2
20.19 odd 2 1008.4.a.ba.1.2 2
35.4 even 6 441.4.e.q.226.2 4
35.9 even 6 441.4.e.q.361.2 4
35.19 odd 6 441.4.e.p.361.2 4
35.24 odd 6 441.4.e.p.226.2 4
35.34 odd 2 441.4.a.r.1.1 2
60.59 even 2 336.4.a.m.1.1 2
105.44 odd 6 147.4.e.l.67.1 4
105.59 even 6 147.4.e.m.79.1 4
105.74 odd 6 147.4.e.l.79.1 4
105.89 even 6 147.4.e.m.67.1 4
105.104 even 2 147.4.a.i.1.2 2
120.29 odd 2 1344.4.a.bg.1.2 2
120.59 even 2 1344.4.a.bo.1.2 2
420.419 odd 2 2352.4.a.bz.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.c.1.2 2 15.14 odd 2
63.4.a.e.1.1 2 5.4 even 2
147.4.a.i.1.2 2 105.104 even 2
147.4.e.l.67.1 4 105.44 odd 6
147.4.e.l.79.1 4 105.74 odd 6
147.4.e.m.67.1 4 105.89 even 6
147.4.e.m.79.1 4 105.59 even 6
336.4.a.m.1.1 2 60.59 even 2
441.4.a.r.1.1 2 35.34 odd 2
441.4.e.p.226.2 4 35.24 odd 6
441.4.e.p.361.2 4 35.19 odd 6
441.4.e.q.226.2 4 35.4 even 6
441.4.e.q.361.2 4 35.9 even 6
525.4.a.n.1.1 2 3.2 odd 2
525.4.d.g.274.2 4 15.2 even 4
525.4.d.g.274.3 4 15.8 even 4
1008.4.a.ba.1.2 2 20.19 odd 2
1344.4.a.bg.1.2 2 120.29 odd 2
1344.4.a.bo.1.2 2 120.59 even 2
1575.4.a.p.1.2 2 1.1 even 1 trivial
2352.4.a.bz.1.2 2 420.419 odd 2