Properties

Label 1521.4.a.u.1.2
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.526440\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.52644 q^{2} -5.66998 q^{4} +19.3400 q^{5} -4.84136 q^{7} -20.8664 q^{8} +O(q^{10})\) \(q+1.52644 q^{2} -5.66998 q^{4} +19.3400 q^{5} -4.84136 q^{7} -20.8664 q^{8} +29.5213 q^{10} -61.0728 q^{11} -7.39005 q^{14} +13.5085 q^{16} +41.7885 q^{17} +107.561 q^{19} -109.657 q^{20} -93.2239 q^{22} -28.5138 q^{23} +249.034 q^{25} +27.4504 q^{28} +89.8886 q^{29} -183.108 q^{31} +187.551 q^{32} +63.7876 q^{34} -93.6318 q^{35} -418.029 q^{37} +164.185 q^{38} -403.555 q^{40} -142.674 q^{41} -71.0935 q^{43} +346.281 q^{44} -43.5246 q^{46} +323.711 q^{47} -319.561 q^{49} +380.136 q^{50} +25.1047 q^{53} -1181.15 q^{55} +101.022 q^{56} +137.210 q^{58} -684.508 q^{59} +308.125 q^{61} -279.503 q^{62} +178.217 q^{64} -672.808 q^{67} -236.940 q^{68} -142.923 q^{70} -326.837 q^{71} -24.3058 q^{73} -638.095 q^{74} -609.869 q^{76} +295.675 q^{77} +166.810 q^{79} +261.255 q^{80} -217.783 q^{82} -201.093 q^{83} +808.188 q^{85} -108.520 q^{86} +1274.37 q^{88} +108.834 q^{89} +161.673 q^{92} +494.126 q^{94} +2080.23 q^{95} -1157.95 q^{97} -487.791 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 10 q^{4} + 4 q^{5} - 30 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 10 q^{4} + 4 q^{5} - 30 q^{7} - 6 q^{8} - 4 q^{10} - 16 q^{11} + 176 q^{14} - 110 q^{16} + 146 q^{17} - 94 q^{19} - 244 q^{20} - 56 q^{22} + 48 q^{23} + 145 q^{25} - 80 q^{28} + 2 q^{29} - 302 q^{31} + 154 q^{32} - 164 q^{34} - 80 q^{35} - 374 q^{37} - 312 q^{38} - 516 q^{40} + 480 q^{41} - 260 q^{43} + 712 q^{44} + 1104 q^{46} - 24 q^{47} + 447 q^{49} + 814 q^{50} + 678 q^{53} - 1552 q^{55} - 96 q^{56} + 628 q^{58} - 1788 q^{59} + 230 q^{61} - 1952 q^{62} - 750 q^{64} - 74 q^{67} + 460 q^{68} - 1216 q^{70} - 948 q^{71} + 222 q^{73} - 1724 q^{74} - 2392 q^{76} - 112 q^{77} - 24 q^{79} + 1100 q^{80} + 564 q^{82} - 796 q^{83} + 248 q^{85} + 1800 q^{86} + 1608 q^{88} + 1436 q^{89} + 1296 q^{92} - 1920 q^{94} + 4032 q^{95} - 3242 q^{97} - 5070 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.52644 0.539678 0.269839 0.962905i \(-0.413030\pi\)
0.269839 + 0.962905i \(0.413030\pi\)
\(3\) 0 0
\(4\) −5.66998 −0.708748
\(5\) 19.3400 1.72982 0.864909 0.501928i \(-0.167376\pi\)
0.864909 + 0.501928i \(0.167376\pi\)
\(6\) 0 0
\(7\) −4.84136 −0.261409 −0.130704 0.991421i \(-0.541724\pi\)
−0.130704 + 0.991421i \(0.541724\pi\)
\(8\) −20.8664 −0.922173
\(9\) 0 0
\(10\) 29.5213 0.933545
\(11\) −61.0728 −1.67401 −0.837006 0.547194i \(-0.815696\pi\)
−0.837006 + 0.547194i \(0.815696\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −7.39005 −0.141077
\(15\) 0 0
\(16\) 13.5085 0.211071
\(17\) 41.7885 0.596188 0.298094 0.954537i \(-0.403649\pi\)
0.298094 + 0.954537i \(0.403649\pi\)
\(18\) 0 0
\(19\) 107.561 1.29875 0.649374 0.760469i \(-0.275031\pi\)
0.649374 + 0.760469i \(0.275031\pi\)
\(20\) −109.657 −1.22601
\(21\) 0 0
\(22\) −93.2239 −0.903427
\(23\) −28.5138 −0.258502 −0.129251 0.991612i \(-0.541257\pi\)
−0.129251 + 0.991612i \(0.541257\pi\)
\(24\) 0 0
\(25\) 249.034 1.99227
\(26\) 0 0
\(27\) 0 0
\(28\) 27.4504 0.185273
\(29\) 89.8886 0.575583 0.287791 0.957693i \(-0.407079\pi\)
0.287791 + 0.957693i \(0.407079\pi\)
\(30\) 0 0
\(31\) −183.108 −1.06087 −0.530437 0.847724i \(-0.677972\pi\)
−0.530437 + 0.847724i \(0.677972\pi\)
\(32\) 187.551 1.03608
\(33\) 0 0
\(34\) 63.7876 0.321750
\(35\) −93.6318 −0.452190
\(36\) 0 0
\(37\) −418.029 −1.85739 −0.928696 0.370843i \(-0.879069\pi\)
−0.928696 + 0.370843i \(0.879069\pi\)
\(38\) 164.185 0.700905
\(39\) 0 0
\(40\) −403.555 −1.59519
\(41\) −142.674 −0.543460 −0.271730 0.962373i \(-0.587596\pi\)
−0.271730 + 0.962373i \(0.587596\pi\)
\(42\) 0 0
\(43\) −71.0935 −0.252132 −0.126066 0.992022i \(-0.540235\pi\)
−0.126066 + 0.992022i \(0.540235\pi\)
\(44\) 346.281 1.18645
\(45\) 0 0
\(46\) −43.5246 −0.139508
\(47\) 323.711 1.00464 0.502321 0.864681i \(-0.332480\pi\)
0.502321 + 0.864681i \(0.332480\pi\)
\(48\) 0 0
\(49\) −319.561 −0.931665
\(50\) 380.136 1.07519
\(51\) 0 0
\(52\) 0 0
\(53\) 25.1047 0.0650641 0.0325321 0.999471i \(-0.489643\pi\)
0.0325321 + 0.999471i \(0.489643\pi\)
\(54\) 0 0
\(55\) −1181.15 −2.89574
\(56\) 101.022 0.241064
\(57\) 0 0
\(58\) 137.210 0.310629
\(59\) −684.508 −1.51043 −0.755215 0.655477i \(-0.772467\pi\)
−0.755215 + 0.655477i \(0.772467\pi\)
\(60\) 0 0
\(61\) 308.125 0.646744 0.323372 0.946272i \(-0.395184\pi\)
0.323372 + 0.946272i \(0.395184\pi\)
\(62\) −279.503 −0.572531
\(63\) 0 0
\(64\) 178.217 0.348081
\(65\) 0 0
\(66\) 0 0
\(67\) −672.808 −1.22681 −0.613407 0.789767i \(-0.710202\pi\)
−0.613407 + 0.789767i \(0.710202\pi\)
\(68\) −236.940 −0.422547
\(69\) 0 0
\(70\) −142.923 −0.244037
\(71\) −326.837 −0.546315 −0.273158 0.961969i \(-0.588068\pi\)
−0.273158 + 0.961969i \(0.588068\pi\)
\(72\) 0 0
\(73\) −24.3058 −0.0389695 −0.0194847 0.999810i \(-0.506203\pi\)
−0.0194847 + 0.999810i \(0.506203\pi\)
\(74\) −638.095 −1.00239
\(75\) 0 0
\(76\) −609.869 −0.920484
\(77\) 295.675 0.437602
\(78\) 0 0
\(79\) 166.810 0.237565 0.118783 0.992920i \(-0.462101\pi\)
0.118783 + 0.992920i \(0.462101\pi\)
\(80\) 261.255 0.365115
\(81\) 0 0
\(82\) −217.783 −0.293294
\(83\) −201.093 −0.265938 −0.132969 0.991120i \(-0.542451\pi\)
−0.132969 + 0.991120i \(0.542451\pi\)
\(84\) 0 0
\(85\) 808.188 1.03130
\(86\) −108.520 −0.136070
\(87\) 0 0
\(88\) 1274.37 1.54373
\(89\) 108.834 0.129622 0.0648109 0.997898i \(-0.479356\pi\)
0.0648109 + 0.997898i \(0.479356\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 161.673 0.183212
\(93\) 0 0
\(94\) 494.126 0.542183
\(95\) 2080.23 2.24660
\(96\) 0 0
\(97\) −1157.95 −1.21208 −0.606041 0.795434i \(-0.707243\pi\)
−0.606041 + 0.795434i \(0.707243\pi\)
\(98\) −487.791 −0.502799
\(99\) 0 0
\(100\) −1412.02 −1.41202
\(101\) −1702.75 −1.67752 −0.838761 0.544500i \(-0.816719\pi\)
−0.838761 + 0.544500i \(0.816719\pi\)
\(102\) 0 0
\(103\) −1455.14 −1.39203 −0.696015 0.718027i \(-0.745045\pi\)
−0.696015 + 0.718027i \(0.745045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 38.3209 0.0351137
\(107\) 822.762 0.743359 0.371679 0.928361i \(-0.378782\pi\)
0.371679 + 0.928361i \(0.378782\pi\)
\(108\) 0 0
\(109\) −457.264 −0.401816 −0.200908 0.979610i \(-0.564389\pi\)
−0.200908 + 0.979610i \(0.564389\pi\)
\(110\) −1802.95 −1.56277
\(111\) 0 0
\(112\) −65.3998 −0.0551759
\(113\) 381.693 0.317758 0.158879 0.987298i \(-0.449212\pi\)
0.158879 + 0.987298i \(0.449212\pi\)
\(114\) 0 0
\(115\) −551.456 −0.447161
\(116\) −509.667 −0.407943
\(117\) 0 0
\(118\) −1044.86 −0.815146
\(119\) −202.313 −0.155849
\(120\) 0 0
\(121\) 2398.88 1.80232
\(122\) 470.334 0.349033
\(123\) 0 0
\(124\) 1038.22 0.751892
\(125\) 2398.82 1.71645
\(126\) 0 0
\(127\) −1129.09 −0.788905 −0.394452 0.918916i \(-0.629066\pi\)
−0.394452 + 0.918916i \(0.629066\pi\)
\(128\) −1228.37 −0.848232
\(129\) 0 0
\(130\) 0 0
\(131\) 852.761 0.568749 0.284374 0.958713i \(-0.408214\pi\)
0.284374 + 0.958713i \(0.408214\pi\)
\(132\) 0 0
\(133\) −520.742 −0.339504
\(134\) −1027.00 −0.662085
\(135\) 0 0
\(136\) −871.975 −0.549789
\(137\) −488.903 −0.304889 −0.152445 0.988312i \(-0.548715\pi\)
−0.152445 + 0.988312i \(0.548715\pi\)
\(138\) 0 0
\(139\) 407.123 0.248430 0.124215 0.992255i \(-0.460359\pi\)
0.124215 + 0.992255i \(0.460359\pi\)
\(140\) 530.890 0.320489
\(141\) 0 0
\(142\) −498.897 −0.294834
\(143\) 0 0
\(144\) 0 0
\(145\) 1738.44 0.995654
\(146\) −37.1013 −0.0210310
\(147\) 0 0
\(148\) 2370.21 1.31642
\(149\) 1717.63 0.944388 0.472194 0.881495i \(-0.343462\pi\)
0.472194 + 0.881495i \(0.343462\pi\)
\(150\) 0 0
\(151\) −1341.79 −0.723133 −0.361567 0.932346i \(-0.617758\pi\)
−0.361567 + 0.932346i \(0.617758\pi\)
\(152\) −2244.41 −1.19767
\(153\) 0 0
\(154\) 451.331 0.236164
\(155\) −3541.30 −1.83512
\(156\) 0 0
\(157\) −760.546 −0.386612 −0.193306 0.981138i \(-0.561921\pi\)
−0.193306 + 0.981138i \(0.561921\pi\)
\(158\) 254.626 0.128209
\(159\) 0 0
\(160\) 3627.23 1.79224
\(161\) 138.046 0.0675746
\(162\) 0 0
\(163\) −2712.09 −1.30323 −0.651616 0.758549i \(-0.725909\pi\)
−0.651616 + 0.758549i \(0.725909\pi\)
\(164\) 808.957 0.385176
\(165\) 0 0
\(166\) −306.957 −0.143521
\(167\) 1551.69 0.719004 0.359502 0.933144i \(-0.382947\pi\)
0.359502 + 0.933144i \(0.382947\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 1233.65 0.556568
\(171\) 0 0
\(172\) 403.099 0.178698
\(173\) 3970.26 1.74482 0.872409 0.488777i \(-0.162557\pi\)
0.872409 + 0.488777i \(0.162557\pi\)
\(174\) 0 0
\(175\) −1205.66 −0.520798
\(176\) −825.004 −0.353335
\(177\) 0 0
\(178\) 166.128 0.0699540
\(179\) 2690.95 1.12364 0.561818 0.827261i \(-0.310102\pi\)
0.561818 + 0.827261i \(0.310102\pi\)
\(180\) 0 0
\(181\) −4371.10 −1.79503 −0.897517 0.440980i \(-0.854631\pi\)
−0.897517 + 0.440980i \(0.854631\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 594.980 0.238383
\(185\) −8084.66 −3.21295
\(186\) 0 0
\(187\) −2552.14 −0.998026
\(188\) −1835.44 −0.712038
\(189\) 0 0
\(190\) 3175.34 1.21244
\(191\) −1408.47 −0.533578 −0.266789 0.963755i \(-0.585963\pi\)
−0.266789 + 0.963755i \(0.585963\pi\)
\(192\) 0 0
\(193\) 4131.69 1.54096 0.770481 0.637463i \(-0.220016\pi\)
0.770481 + 0.637463i \(0.220016\pi\)
\(194\) −1767.54 −0.654134
\(195\) 0 0
\(196\) 1811.91 0.660316
\(197\) −3401.23 −1.23009 −0.615045 0.788492i \(-0.710862\pi\)
−0.615045 + 0.788492i \(0.710862\pi\)
\(198\) 0 0
\(199\) −3520.74 −1.25416 −0.627081 0.778954i \(-0.715751\pi\)
−0.627081 + 0.778954i \(0.715751\pi\)
\(200\) −5196.45 −1.83722
\(201\) 0 0
\(202\) −2599.14 −0.905321
\(203\) −435.183 −0.150463
\(204\) 0 0
\(205\) −2759.30 −0.940088
\(206\) −2221.18 −0.751248
\(207\) 0 0
\(208\) 0 0
\(209\) −6569.05 −2.17412
\(210\) 0 0
\(211\) −2245.22 −0.732545 −0.366272 0.930508i \(-0.619366\pi\)
−0.366272 + 0.930508i \(0.619366\pi\)
\(212\) −142.343 −0.0461141
\(213\) 0 0
\(214\) 1255.90 0.401174
\(215\) −1374.95 −0.436142
\(216\) 0 0
\(217\) 886.490 0.277322
\(218\) −697.986 −0.216851
\(219\) 0 0
\(220\) 6697.07 2.05235
\(221\) 0 0
\(222\) 0 0
\(223\) −3431.26 −1.03038 −0.515188 0.857077i \(-0.672278\pi\)
−0.515188 + 0.857077i \(0.672278\pi\)
\(224\) −908.003 −0.270842
\(225\) 0 0
\(226\) 582.631 0.171487
\(227\) −4757.91 −1.39116 −0.695581 0.718448i \(-0.744853\pi\)
−0.695581 + 0.718448i \(0.744853\pi\)
\(228\) 0 0
\(229\) 4368.93 1.26073 0.630364 0.776300i \(-0.282906\pi\)
0.630364 + 0.776300i \(0.282906\pi\)
\(230\) −841.764 −0.241323
\(231\) 0 0
\(232\) −1875.65 −0.530787
\(233\) 3642.00 1.02401 0.512007 0.858981i \(-0.328902\pi\)
0.512007 + 0.858981i \(0.328902\pi\)
\(234\) 0 0
\(235\) 6260.57 1.73785
\(236\) 3881.15 1.07051
\(237\) 0 0
\(238\) −308.819 −0.0841082
\(239\) 2236.17 0.605213 0.302606 0.953116i \(-0.402143\pi\)
0.302606 + 0.953116i \(0.402143\pi\)
\(240\) 0 0
\(241\) −6538.78 −1.74772 −0.873858 0.486181i \(-0.838390\pi\)
−0.873858 + 0.486181i \(0.838390\pi\)
\(242\) 3661.75 0.972670
\(243\) 0 0
\(244\) −1747.06 −0.458378
\(245\) −6180.30 −1.61161
\(246\) 0 0
\(247\) 0 0
\(248\) 3820.80 0.978310
\(249\) 0 0
\(250\) 3661.65 0.926332
\(251\) −2507.12 −0.630470 −0.315235 0.949014i \(-0.602083\pi\)
−0.315235 + 0.949014i \(0.602083\pi\)
\(252\) 0 0
\(253\) 1741.42 0.432735
\(254\) −1723.49 −0.425755
\(255\) 0 0
\(256\) −3300.77 −0.805853
\(257\) 808.131 0.196147 0.0980735 0.995179i \(-0.468732\pi\)
0.0980735 + 0.995179i \(0.468732\pi\)
\(258\) 0 0
\(259\) 2023.83 0.485539
\(260\) 0 0
\(261\) 0 0
\(262\) 1301.69 0.306941
\(263\) −2940.70 −0.689472 −0.344736 0.938700i \(-0.612032\pi\)
−0.344736 + 0.938700i \(0.612032\pi\)
\(264\) 0 0
\(265\) 485.525 0.112549
\(266\) −794.881 −0.183223
\(267\) 0 0
\(268\) 3814.81 0.869502
\(269\) −7111.50 −1.61188 −0.805940 0.591997i \(-0.798340\pi\)
−0.805940 + 0.591997i \(0.798340\pi\)
\(270\) 0 0
\(271\) −2034.96 −0.456145 −0.228072 0.973644i \(-0.573242\pi\)
−0.228072 + 0.973644i \(0.573242\pi\)
\(272\) 564.502 0.125838
\(273\) 0 0
\(274\) −746.281 −0.164542
\(275\) −15209.2 −3.33509
\(276\) 0 0
\(277\) 2723.20 0.590689 0.295345 0.955391i \(-0.404566\pi\)
0.295345 + 0.955391i \(0.404566\pi\)
\(278\) 621.449 0.134072
\(279\) 0 0
\(280\) 1953.76 0.416998
\(281\) 3265.56 0.693263 0.346632 0.938001i \(-0.387325\pi\)
0.346632 + 0.938001i \(0.387325\pi\)
\(282\) 0 0
\(283\) 1144.02 0.240299 0.120150 0.992756i \(-0.461663\pi\)
0.120150 + 0.992756i \(0.461663\pi\)
\(284\) 1853.16 0.387200
\(285\) 0 0
\(286\) 0 0
\(287\) 690.735 0.142065
\(288\) 0 0
\(289\) −3166.72 −0.644560
\(290\) 2653.63 0.537333
\(291\) 0 0
\(292\) 137.813 0.0276195
\(293\) −1677.35 −0.334444 −0.167222 0.985919i \(-0.553480\pi\)
−0.167222 + 0.985919i \(0.553480\pi\)
\(294\) 0 0
\(295\) −13238.4 −2.61277
\(296\) 8722.75 1.71284
\(297\) 0 0
\(298\) 2621.86 0.509666
\(299\) 0 0
\(300\) 0 0
\(301\) 344.190 0.0659095
\(302\) −2048.16 −0.390259
\(303\) 0 0
\(304\) 1452.99 0.274128
\(305\) 5959.13 1.11875
\(306\) 0 0
\(307\) −7207.70 −1.33995 −0.669975 0.742383i \(-0.733695\pi\)
−0.669975 + 0.742383i \(0.733695\pi\)
\(308\) −1676.47 −0.310149
\(309\) 0 0
\(310\) −5405.57 −0.990374
\(311\) −412.963 −0.0752958 −0.0376479 0.999291i \(-0.511987\pi\)
−0.0376479 + 0.999291i \(0.511987\pi\)
\(312\) 0 0
\(313\) 2936.39 0.530270 0.265135 0.964211i \(-0.414583\pi\)
0.265135 + 0.964211i \(0.414583\pi\)
\(314\) −1160.93 −0.208646
\(315\) 0 0
\(316\) −945.812 −0.168374
\(317\) 377.956 0.0669657 0.0334828 0.999439i \(-0.489340\pi\)
0.0334828 + 0.999439i \(0.489340\pi\)
\(318\) 0 0
\(319\) −5489.75 −0.963533
\(320\) 3446.71 0.602116
\(321\) 0 0
\(322\) 210.718 0.0364685
\(323\) 4494.81 0.774298
\(324\) 0 0
\(325\) 0 0
\(326\) −4139.83 −0.703326
\(327\) 0 0
\(328\) 2977.09 0.501165
\(329\) −1567.20 −0.262622
\(330\) 0 0
\(331\) 4428.17 0.735330 0.367665 0.929958i \(-0.380157\pi\)
0.367665 + 0.929958i \(0.380157\pi\)
\(332\) 1140.20 0.188483
\(333\) 0 0
\(334\) 2368.57 0.388031
\(335\) −13012.1 −2.12217
\(336\) 0 0
\(337\) −1768.76 −0.285907 −0.142953 0.989729i \(-0.545660\pi\)
−0.142953 + 0.989729i \(0.545660\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −4582.41 −0.730930
\(341\) 11182.9 1.77592
\(342\) 0 0
\(343\) 3207.70 0.504955
\(344\) 1483.47 0.232509
\(345\) 0 0
\(346\) 6060.36 0.941639
\(347\) −2412.97 −0.373300 −0.186650 0.982426i \(-0.559763\pi\)
−0.186650 + 0.982426i \(0.559763\pi\)
\(348\) 0 0
\(349\) 9967.45 1.52878 0.764392 0.644752i \(-0.223039\pi\)
0.764392 + 0.644752i \(0.223039\pi\)
\(350\) −1840.37 −0.281063
\(351\) 0 0
\(352\) −11454.3 −1.73442
\(353\) 4516.30 0.680959 0.340479 0.940252i \(-0.389411\pi\)
0.340479 + 0.940252i \(0.389411\pi\)
\(354\) 0 0
\(355\) −6321.01 −0.945027
\(356\) −617.084 −0.0918691
\(357\) 0 0
\(358\) 4107.57 0.606401
\(359\) 12159.8 1.78767 0.893833 0.448400i \(-0.148006\pi\)
0.893833 + 0.448400i \(0.148006\pi\)
\(360\) 0 0
\(361\) 4710.38 0.686745
\(362\) −6672.22 −0.968740
\(363\) 0 0
\(364\) 0 0
\(365\) −470.072 −0.0674102
\(366\) 0 0
\(367\) −2674.25 −0.380367 −0.190183 0.981749i \(-0.560908\pi\)
−0.190183 + 0.981749i \(0.560908\pi\)
\(368\) −385.180 −0.0545622
\(369\) 0 0
\(370\) −12340.7 −1.73396
\(371\) −121.541 −0.0170083
\(372\) 0 0
\(373\) 9601.74 1.33287 0.666433 0.745564i \(-0.267820\pi\)
0.666433 + 0.745564i \(0.267820\pi\)
\(374\) −3895.69 −0.538613
\(375\) 0 0
\(376\) −6754.69 −0.926454
\(377\) 0 0
\(378\) 0 0
\(379\) −9019.65 −1.22245 −0.611225 0.791457i \(-0.709323\pi\)
−0.611225 + 0.791457i \(0.709323\pi\)
\(380\) −11794.9 −1.59227
\(381\) 0 0
\(382\) −2149.95 −0.287960
\(383\) −4015.34 −0.535703 −0.267852 0.963460i \(-0.586314\pi\)
−0.267852 + 0.963460i \(0.586314\pi\)
\(384\) 0 0
\(385\) 5718.35 0.756972
\(386\) 6306.78 0.831623
\(387\) 0 0
\(388\) 6565.55 0.859060
\(389\) 2725.35 0.355221 0.177610 0.984101i \(-0.443163\pi\)
0.177610 + 0.984101i \(0.443163\pi\)
\(390\) 0 0
\(391\) −1191.55 −0.154115
\(392\) 6668.09 0.859157
\(393\) 0 0
\(394\) −5191.78 −0.663853
\(395\) 3226.11 0.410945
\(396\) 0 0
\(397\) 4391.59 0.555182 0.277591 0.960699i \(-0.410464\pi\)
0.277591 + 0.960699i \(0.410464\pi\)
\(398\) −5374.19 −0.676844
\(399\) 0 0
\(400\) 3364.09 0.420511
\(401\) 3762.48 0.468552 0.234276 0.972170i \(-0.424728\pi\)
0.234276 + 0.972170i \(0.424728\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 9654.55 1.18894
\(405\) 0 0
\(406\) −664.281 −0.0812013
\(407\) 25530.2 3.10930
\(408\) 0 0
\(409\) 6797.81 0.821833 0.410917 0.911673i \(-0.365209\pi\)
0.410917 + 0.911673i \(0.365209\pi\)
\(410\) −4211.91 −0.507345
\(411\) 0 0
\(412\) 8250.61 0.986599
\(413\) 3313.95 0.394840
\(414\) 0 0
\(415\) −3889.14 −0.460025
\(416\) 0 0
\(417\) 0 0
\(418\) −10027.3 −1.17332
\(419\) 12594.2 1.46841 0.734207 0.678925i \(-0.237554\pi\)
0.734207 + 0.678925i \(0.237554\pi\)
\(420\) 0 0
\(421\) −6888.04 −0.797393 −0.398697 0.917083i \(-0.630537\pi\)
−0.398697 + 0.917083i \(0.630537\pi\)
\(422\) −3427.19 −0.395338
\(423\) 0 0
\(424\) −523.845 −0.0600004
\(425\) 10406.8 1.18777
\(426\) 0 0
\(427\) −1491.74 −0.169065
\(428\) −4665.04 −0.526854
\(429\) 0 0
\(430\) −2098.77 −0.235376
\(431\) −7384.53 −0.825291 −0.412645 0.910892i \(-0.635395\pi\)
−0.412645 + 0.910892i \(0.635395\pi\)
\(432\) 0 0
\(433\) 9068.33 1.00646 0.503229 0.864153i \(-0.332145\pi\)
0.503229 + 0.864153i \(0.332145\pi\)
\(434\) 1353.17 0.149665
\(435\) 0 0
\(436\) 2592.68 0.284786
\(437\) −3066.97 −0.335728
\(438\) 0 0
\(439\) 16875.4 1.83466 0.917331 0.398125i \(-0.130339\pi\)
0.917331 + 0.398125i \(0.130339\pi\)
\(440\) 24646.3 2.67037
\(441\) 0 0
\(442\) 0 0
\(443\) 6766.18 0.725668 0.362834 0.931854i \(-0.381809\pi\)
0.362834 + 0.931854i \(0.381809\pi\)
\(444\) 0 0
\(445\) 2104.84 0.224222
\(446\) −5237.61 −0.556072
\(447\) 0 0
\(448\) −862.814 −0.0909914
\(449\) 140.944 0.0148141 0.00740706 0.999973i \(-0.497642\pi\)
0.00740706 + 0.999973i \(0.497642\pi\)
\(450\) 0 0
\(451\) 8713.47 0.909759
\(452\) −2164.19 −0.225210
\(453\) 0 0
\(454\) −7262.67 −0.750779
\(455\) 0 0
\(456\) 0 0
\(457\) 17733.1 1.81514 0.907571 0.419898i \(-0.137934\pi\)
0.907571 + 0.419898i \(0.137934\pi\)
\(458\) 6668.90 0.680387
\(459\) 0 0
\(460\) 3126.74 0.316924
\(461\) 2293.37 0.231699 0.115849 0.993267i \(-0.463041\pi\)
0.115849 + 0.993267i \(0.463041\pi\)
\(462\) 0 0
\(463\) −13770.9 −1.38226 −0.691129 0.722731i \(-0.742887\pi\)
−0.691129 + 0.722731i \(0.742887\pi\)
\(464\) 1214.27 0.121489
\(465\) 0 0
\(466\) 5559.29 0.552638
\(467\) 3477.37 0.344568 0.172284 0.985047i \(-0.444885\pi\)
0.172284 + 0.985047i \(0.444885\pi\)
\(468\) 0 0
\(469\) 3257.31 0.320700
\(470\) 9556.38 0.937879
\(471\) 0 0
\(472\) 14283.2 1.39288
\(473\) 4341.88 0.422072
\(474\) 0 0
\(475\) 26786.4 2.58746
\(476\) 1147.11 0.110458
\(477\) 0 0
\(478\) 3413.38 0.326620
\(479\) −3137.39 −0.299271 −0.149636 0.988741i \(-0.547810\pi\)
−0.149636 + 0.988741i \(0.547810\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −9981.05 −0.943204
\(483\) 0 0
\(484\) −13601.6 −1.27739
\(485\) −22394.7 −2.09668
\(486\) 0 0
\(487\) −5996.52 −0.557964 −0.278982 0.960296i \(-0.589997\pi\)
−0.278982 + 0.960296i \(0.589997\pi\)
\(488\) −6429.46 −0.596410
\(489\) 0 0
\(490\) −9433.86 −0.869752
\(491\) 9401.49 0.864121 0.432060 0.901845i \(-0.357787\pi\)
0.432060 + 0.901845i \(0.357787\pi\)
\(492\) 0 0
\(493\) 3756.31 0.343156
\(494\) 0 0
\(495\) 0 0
\(496\) −2473.52 −0.223920
\(497\) 1582.34 0.142812
\(498\) 0 0
\(499\) 5052.33 0.453253 0.226626 0.973982i \(-0.427230\pi\)
0.226626 + 0.973982i \(0.427230\pi\)
\(500\) −13601.2 −1.21653
\(501\) 0 0
\(502\) −3826.96 −0.340251
\(503\) −8184.02 −0.725462 −0.362731 0.931894i \(-0.618156\pi\)
−0.362731 + 0.931894i \(0.618156\pi\)
\(504\) 0 0
\(505\) −32931.1 −2.90181
\(506\) 2658.17 0.233537
\(507\) 0 0
\(508\) 6401.94 0.559134
\(509\) 6039.12 0.525892 0.262946 0.964811i \(-0.415306\pi\)
0.262946 + 0.964811i \(0.415306\pi\)
\(510\) 0 0
\(511\) 117.673 0.0101870
\(512\) 4788.54 0.413331
\(513\) 0 0
\(514\) 1233.56 0.105856
\(515\) −28142.3 −2.40796
\(516\) 0 0
\(517\) −19770.0 −1.68178
\(518\) 3089.25 0.262035
\(519\) 0 0
\(520\) 0 0
\(521\) 14602.5 1.22792 0.613960 0.789337i \(-0.289576\pi\)
0.613960 + 0.789337i \(0.289576\pi\)
\(522\) 0 0
\(523\) 8910.70 0.745005 0.372502 0.928031i \(-0.378500\pi\)
0.372502 + 0.928031i \(0.378500\pi\)
\(524\) −4835.14 −0.403099
\(525\) 0 0
\(526\) −4488.80 −0.372093
\(527\) −7651.79 −0.632481
\(528\) 0 0
\(529\) −11354.0 −0.933177
\(530\) 741.124 0.0607403
\(531\) 0 0
\(532\) 2952.60 0.240623
\(533\) 0 0
\(534\) 0 0
\(535\) 15912.2 1.28588
\(536\) 14039.1 1.13134
\(537\) 0 0
\(538\) −10855.3 −0.869897
\(539\) 19516.5 1.55962
\(540\) 0 0
\(541\) 13313.6 1.05803 0.529017 0.848611i \(-0.322561\pi\)
0.529017 + 0.848611i \(0.322561\pi\)
\(542\) −3106.25 −0.246171
\(543\) 0 0
\(544\) 7837.48 0.617701
\(545\) −8843.47 −0.695069
\(546\) 0 0
\(547\) −4116.94 −0.321806 −0.160903 0.986970i \(-0.551441\pi\)
−0.160903 + 0.986970i \(0.551441\pi\)
\(548\) 2772.07 0.216090
\(549\) 0 0
\(550\) −23215.9 −1.79987
\(551\) 9668.52 0.747537
\(552\) 0 0
\(553\) −807.590 −0.0621016
\(554\) 4156.79 0.318782
\(555\) 0 0
\(556\) −2308.38 −0.176074
\(557\) −6888.37 −0.524003 −0.262002 0.965067i \(-0.584383\pi\)
−0.262002 + 0.965067i \(0.584383\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −1264.83 −0.0954443
\(561\) 0 0
\(562\) 4984.68 0.374139
\(563\) −10537.1 −0.788782 −0.394391 0.918943i \(-0.629045\pi\)
−0.394391 + 0.918943i \(0.629045\pi\)
\(564\) 0 0
\(565\) 7381.92 0.549664
\(566\) 1746.27 0.129684
\(567\) 0 0
\(568\) 6819.91 0.503798
\(569\) −26930.1 −1.98413 −0.992065 0.125722i \(-0.959875\pi\)
−0.992065 + 0.125722i \(0.959875\pi\)
\(570\) 0 0
\(571\) −3125.60 −0.229076 −0.114538 0.993419i \(-0.536539\pi\)
−0.114538 + 0.993419i \(0.536539\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1054.36 0.0766696
\(575\) −7100.91 −0.515006
\(576\) 0 0
\(577\) −4787.13 −0.345391 −0.172696 0.984975i \(-0.555248\pi\)
−0.172696 + 0.984975i \(0.555248\pi\)
\(578\) −4833.81 −0.347855
\(579\) 0 0
\(580\) −9856.94 −0.705668
\(581\) 973.566 0.0695186
\(582\) 0 0
\(583\) −1533.22 −0.108918
\(584\) 507.174 0.0359366
\(585\) 0 0
\(586\) −2560.38 −0.180492
\(587\) 18380.8 1.29243 0.646214 0.763156i \(-0.276351\pi\)
0.646214 + 0.763156i \(0.276351\pi\)
\(588\) 0 0
\(589\) −19695.3 −1.37781
\(590\) −20207.6 −1.41005
\(591\) 0 0
\(592\) −5646.96 −0.392042
\(593\) −13831.7 −0.957843 −0.478922 0.877858i \(-0.658972\pi\)
−0.478922 + 0.877858i \(0.658972\pi\)
\(594\) 0 0
\(595\) −3912.73 −0.269590
\(596\) −9738.94 −0.669333
\(597\) 0 0
\(598\) 0 0
\(599\) −12248.6 −0.835502 −0.417751 0.908562i \(-0.637182\pi\)
−0.417751 + 0.908562i \(0.637182\pi\)
\(600\) 0 0
\(601\) 9719.56 0.659682 0.329841 0.944036i \(-0.393005\pi\)
0.329841 + 0.944036i \(0.393005\pi\)
\(602\) 525.385 0.0355699
\(603\) 0 0
\(604\) 7607.91 0.512519
\(605\) 46394.3 3.11768
\(606\) 0 0
\(607\) 1607.83 0.107512 0.0537560 0.998554i \(-0.482881\pi\)
0.0537560 + 0.998554i \(0.482881\pi\)
\(608\) 20173.2 1.34561
\(609\) 0 0
\(610\) 9096.25 0.603764
\(611\) 0 0
\(612\) 0 0
\(613\) −14731.1 −0.970610 −0.485305 0.874345i \(-0.661291\pi\)
−0.485305 + 0.874345i \(0.661291\pi\)
\(614\) −11002.1 −0.723142
\(615\) 0 0
\(616\) −6169.68 −0.403545
\(617\) 27951.8 1.82382 0.911909 0.410392i \(-0.134608\pi\)
0.911909 + 0.410392i \(0.134608\pi\)
\(618\) 0 0
\(619\) −16200.2 −1.05192 −0.525961 0.850509i \(-0.676294\pi\)
−0.525961 + 0.850509i \(0.676294\pi\)
\(620\) 20079.1 1.30064
\(621\) 0 0
\(622\) −630.364 −0.0406355
\(623\) −526.903 −0.0338843
\(624\) 0 0
\(625\) 15263.8 0.976880
\(626\) 4482.22 0.286175
\(627\) 0 0
\(628\) 4312.28 0.274011
\(629\) −17468.8 −1.10735
\(630\) 0 0
\(631\) −12731.8 −0.803239 −0.401619 0.915807i \(-0.631553\pi\)
−0.401619 + 0.915807i \(0.631553\pi\)
\(632\) −3480.73 −0.219076
\(633\) 0 0
\(634\) 576.927 0.0361399
\(635\) −21836.6 −1.36466
\(636\) 0 0
\(637\) 0 0
\(638\) −8379.77 −0.519997
\(639\) 0 0
\(640\) −23756.7 −1.46729
\(641\) 11556.9 0.712119 0.356059 0.934463i \(-0.384120\pi\)
0.356059 + 0.934463i \(0.384120\pi\)
\(642\) 0 0
\(643\) 9181.25 0.563100 0.281550 0.959547i \(-0.409152\pi\)
0.281550 + 0.959547i \(0.409152\pi\)
\(644\) −782.716 −0.0478934
\(645\) 0 0
\(646\) 6861.06 0.417871
\(647\) −5244.11 −0.318651 −0.159326 0.987226i \(-0.550932\pi\)
−0.159326 + 0.987226i \(0.550932\pi\)
\(648\) 0 0
\(649\) 41804.8 2.52848
\(650\) 0 0
\(651\) 0 0
\(652\) 15377.5 0.923663
\(653\) 16421.4 0.984106 0.492053 0.870565i \(-0.336247\pi\)
0.492053 + 0.870565i \(0.336247\pi\)
\(654\) 0 0
\(655\) 16492.4 0.983832
\(656\) −1927.31 −0.114709
\(657\) 0 0
\(658\) −2392.24 −0.141732
\(659\) −1838.11 −0.108653 −0.0543266 0.998523i \(-0.517301\pi\)
−0.0543266 + 0.998523i \(0.517301\pi\)
\(660\) 0 0
\(661\) 5500.93 0.323694 0.161847 0.986816i \(-0.448255\pi\)
0.161847 + 0.986816i \(0.448255\pi\)
\(662\) 6759.34 0.396841
\(663\) 0 0
\(664\) 4196.10 0.245241
\(665\) −10071.1 −0.587281
\(666\) 0 0
\(667\) −2563.07 −0.148789
\(668\) −8798.08 −0.509593
\(669\) 0 0
\(670\) −19862.2 −1.14529
\(671\) −18818.0 −1.08266
\(672\) 0 0
\(673\) −25986.7 −1.48843 −0.744216 0.667939i \(-0.767177\pi\)
−0.744216 + 0.667939i \(0.767177\pi\)
\(674\) −2699.91 −0.154298
\(675\) 0 0
\(676\) 0 0
\(677\) 11691.3 0.663714 0.331857 0.943330i \(-0.392325\pi\)
0.331857 + 0.943330i \(0.392325\pi\)
\(678\) 0 0
\(679\) 5606.05 0.316849
\(680\) −16864.0 −0.951035
\(681\) 0 0
\(682\) 17070.0 0.958423
\(683\) −11111.5 −0.622501 −0.311251 0.950328i \(-0.600748\pi\)
−0.311251 + 0.950328i \(0.600748\pi\)
\(684\) 0 0
\(685\) −9455.37 −0.527403
\(686\) 4896.36 0.272513
\(687\) 0 0
\(688\) −960.370 −0.0532177
\(689\) 0 0
\(690\) 0 0
\(691\) 7542.55 0.415242 0.207621 0.978209i \(-0.433428\pi\)
0.207621 + 0.978209i \(0.433428\pi\)
\(692\) −22511.3 −1.23664
\(693\) 0 0
\(694\) −3683.26 −0.201462
\(695\) 7873.75 0.429738
\(696\) 0 0
\(697\) −5962.11 −0.324005
\(698\) 15214.7 0.825051
\(699\) 0 0
\(700\) 6836.10 0.369114
\(701\) 8231.17 0.443491 0.221745 0.975105i \(-0.428825\pi\)
0.221745 + 0.975105i \(0.428825\pi\)
\(702\) 0 0
\(703\) −44963.6 −2.41228
\(704\) −10884.2 −0.582691
\(705\) 0 0
\(706\) 6893.86 0.367498
\(707\) 8243.61 0.438519
\(708\) 0 0
\(709\) −28044.6 −1.48553 −0.742764 0.669554i \(-0.766485\pi\)
−0.742764 + 0.669554i \(0.766485\pi\)
\(710\) −9648.64 −0.510010
\(711\) 0 0
\(712\) −2270.96 −0.119534
\(713\) 5221.09 0.274238
\(714\) 0 0
\(715\) 0 0
\(716\) −15257.6 −0.796374
\(717\) 0 0
\(718\) 18561.3 0.964764
\(719\) −29686.4 −1.53980 −0.769901 0.638163i \(-0.779694\pi\)
−0.769901 + 0.638163i \(0.779694\pi\)
\(720\) 0 0
\(721\) 7044.86 0.363889
\(722\) 7190.12 0.370621
\(723\) 0 0
\(724\) 24784.0 1.27223
\(725\) 22385.3 1.14672
\(726\) 0 0
\(727\) −27654.5 −1.41080 −0.705398 0.708812i \(-0.749232\pi\)
−0.705398 + 0.708812i \(0.749232\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −717.537 −0.0363798
\(731\) −2970.89 −0.150318
\(732\) 0 0
\(733\) 13077.5 0.658975 0.329488 0.944160i \(-0.393124\pi\)
0.329488 + 0.944160i \(0.393124\pi\)
\(734\) −4082.08 −0.205276
\(735\) 0 0
\(736\) −5347.79 −0.267829
\(737\) 41090.3 2.05370
\(738\) 0 0
\(739\) 4218.33 0.209978 0.104989 0.994473i \(-0.466519\pi\)
0.104989 + 0.994473i \(0.466519\pi\)
\(740\) 45839.9 2.27717
\(741\) 0 0
\(742\) −185.525 −0.00917903
\(743\) 7725.54 0.381457 0.190728 0.981643i \(-0.438915\pi\)
0.190728 + 0.981643i \(0.438915\pi\)
\(744\) 0 0
\(745\) 33218.9 1.63362
\(746\) 14656.5 0.719319
\(747\) 0 0
\(748\) 14470.6 0.707349
\(749\) −3983.29 −0.194321
\(750\) 0 0
\(751\) 7506.12 0.364717 0.182358 0.983232i \(-0.441627\pi\)
0.182358 + 0.983232i \(0.441627\pi\)
\(752\) 4372.87 0.212051
\(753\) 0 0
\(754\) 0 0
\(755\) −25950.1 −1.25089
\(756\) 0 0
\(757\) −2741.62 −0.131632 −0.0658162 0.997832i \(-0.520965\pi\)
−0.0658162 + 0.997832i \(0.520965\pi\)
\(758\) −13768.0 −0.659729
\(759\) 0 0
\(760\) −43406.9 −2.07175
\(761\) 29740.0 1.41666 0.708328 0.705883i \(-0.249450\pi\)
0.708328 + 0.705883i \(0.249450\pi\)
\(762\) 0 0
\(763\) 2213.78 0.105038
\(764\) 7986.01 0.378172
\(765\) 0 0
\(766\) −6129.18 −0.289107
\(767\) 0 0
\(768\) 0 0
\(769\) 19896.3 0.933004 0.466502 0.884520i \(-0.345514\pi\)
0.466502 + 0.884520i \(0.345514\pi\)
\(770\) 8728.72 0.408521
\(771\) 0 0
\(772\) −23426.6 −1.09215
\(773\) −13601.3 −0.632866 −0.316433 0.948615i \(-0.602485\pi\)
−0.316433 + 0.948615i \(0.602485\pi\)
\(774\) 0 0
\(775\) −45600.1 −2.11355
\(776\) 24162.2 1.11775
\(777\) 0 0
\(778\) 4160.09 0.191705
\(779\) −15346.1 −0.705818
\(780\) 0 0
\(781\) 19960.8 0.914539
\(782\) −1818.83 −0.0831727
\(783\) 0 0
\(784\) −4316.81 −0.196648
\(785\) −14708.9 −0.668770
\(786\) 0 0
\(787\) 1498.29 0.0678631 0.0339315 0.999424i \(-0.489197\pi\)
0.0339315 + 0.999424i \(0.489197\pi\)
\(788\) 19284.9 0.871824
\(789\) 0 0
\(790\) 4924.46 0.221778
\(791\) −1847.91 −0.0830647
\(792\) 0 0
\(793\) 0 0
\(794\) 6703.49 0.299620
\(795\) 0 0
\(796\) 19962.5 0.888885
\(797\) 3713.30 0.165034 0.0825168 0.996590i \(-0.473704\pi\)
0.0825168 + 0.996590i \(0.473704\pi\)
\(798\) 0 0
\(799\) 13527.4 0.598955
\(800\) 46706.7 2.06416
\(801\) 0 0
\(802\) 5743.20 0.252867
\(803\) 1484.42 0.0652354
\(804\) 0 0
\(805\) 2669.80 0.116892
\(806\) 0 0
\(807\) 0 0
\(808\) 35530.2 1.54697
\(809\) 34527.6 1.50053 0.750263 0.661139i \(-0.229927\pi\)
0.750263 + 0.661139i \(0.229927\pi\)
\(810\) 0 0
\(811\) 37279.2 1.61412 0.807059 0.590471i \(-0.201058\pi\)
0.807059 + 0.590471i \(0.201058\pi\)
\(812\) 2467.48 0.106640
\(813\) 0 0
\(814\) 38970.3 1.67802
\(815\) −52451.6 −2.25436
\(816\) 0 0
\(817\) −7646.90 −0.327455
\(818\) 10376.4 0.443525
\(819\) 0 0
\(820\) 15645.2 0.666285
\(821\) 13877.9 0.589943 0.294972 0.955506i \(-0.404690\pi\)
0.294972 + 0.955506i \(0.404690\pi\)
\(822\) 0 0
\(823\) 18945.1 0.802410 0.401205 0.915988i \(-0.368592\pi\)
0.401205 + 0.915988i \(0.368592\pi\)
\(824\) 30363.5 1.28369
\(825\) 0 0
\(826\) 5058.55 0.213086
\(827\) −7804.75 −0.328171 −0.164086 0.986446i \(-0.552467\pi\)
−0.164086 + 0.986446i \(0.552467\pi\)
\(828\) 0 0
\(829\) 5784.85 0.242360 0.121180 0.992631i \(-0.461332\pi\)
0.121180 + 0.992631i \(0.461332\pi\)
\(830\) −5936.54 −0.248265
\(831\) 0 0
\(832\) 0 0
\(833\) −13354.0 −0.555448
\(834\) 0 0
\(835\) 30009.7 1.24375
\(836\) 37246.4 1.54090
\(837\) 0 0
\(838\) 19224.3 0.792471
\(839\) −5011.42 −0.206214 −0.103107 0.994670i \(-0.532878\pi\)
−0.103107 + 0.994670i \(0.532878\pi\)
\(840\) 0 0
\(841\) −16309.0 −0.668704
\(842\) −10514.2 −0.430336
\(843\) 0 0
\(844\) 12730.3 0.519190
\(845\) 0 0
\(846\) 0 0
\(847\) −11613.9 −0.471142
\(848\) 339.128 0.0137332
\(849\) 0 0
\(850\) 15885.3 0.641013
\(851\) 11919.6 0.480138
\(852\) 0 0
\(853\) 22059.0 0.885446 0.442723 0.896659i \(-0.354013\pi\)
0.442723 + 0.896659i \(0.354013\pi\)
\(854\) −2277.06 −0.0912404
\(855\) 0 0
\(856\) −17168.1 −0.685506
\(857\) −13956.2 −0.556283 −0.278141 0.960540i \(-0.589718\pi\)
−0.278141 + 0.960540i \(0.589718\pi\)
\(858\) 0 0
\(859\) 12498.5 0.496442 0.248221 0.968703i \(-0.420154\pi\)
0.248221 + 0.968703i \(0.420154\pi\)
\(860\) 7795.92 0.309115
\(861\) 0 0
\(862\) −11272.0 −0.445391
\(863\) −38631.2 −1.52378 −0.761890 0.647707i \(-0.775728\pi\)
−0.761890 + 0.647707i \(0.775728\pi\)
\(864\) 0 0
\(865\) 76784.7 3.01822
\(866\) 13842.3 0.543163
\(867\) 0 0
\(868\) −5026.38 −0.196551
\(869\) −10187.6 −0.397687
\(870\) 0 0
\(871\) 0 0
\(872\) 9541.46 0.370544
\(873\) 0 0
\(874\) −4681.55 −0.181185
\(875\) −11613.5 −0.448696
\(876\) 0 0
\(877\) 856.756 0.0329881 0.0164941 0.999864i \(-0.494750\pi\)
0.0164941 + 0.999864i \(0.494750\pi\)
\(878\) 25759.2 0.990127
\(879\) 0 0
\(880\) −15955.6 −0.611206
\(881\) −33638.6 −1.28640 −0.643198 0.765700i \(-0.722393\pi\)
−0.643198 + 0.765700i \(0.722393\pi\)
\(882\) 0 0
\(883\) −31109.1 −1.18562 −0.592811 0.805342i \(-0.701982\pi\)
−0.592811 + 0.805342i \(0.701982\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 10328.2 0.391627
\(887\) −26080.3 −0.987248 −0.493624 0.869675i \(-0.664328\pi\)
−0.493624 + 0.869675i \(0.664328\pi\)
\(888\) 0 0
\(889\) 5466.35 0.206227
\(890\) 3212.91 0.121008
\(891\) 0 0
\(892\) 19455.2 0.730277
\(893\) 34818.8 1.30478
\(894\) 0 0
\(895\) 52042.8 1.94369
\(896\) 5946.99 0.221736
\(897\) 0 0
\(898\) 215.142 0.00799486
\(899\) −16459.3 −0.610621
\(900\) 0 0
\(901\) 1049.09 0.0387905
\(902\) 13300.6 0.490977
\(903\) 0 0
\(904\) −7964.56 −0.293028
\(905\) −84536.9 −3.10508
\(906\) 0 0
\(907\) 20169.0 0.738369 0.369184 0.929356i \(-0.379637\pi\)
0.369184 + 0.929356i \(0.379637\pi\)
\(908\) 26977.3 0.985983
\(909\) 0 0
\(910\) 0 0
\(911\) −19982.2 −0.726716 −0.363358 0.931650i \(-0.618370\pi\)
−0.363358 + 0.931650i \(0.618370\pi\)
\(912\) 0 0
\(913\) 12281.3 0.445184
\(914\) 27068.5 0.979593
\(915\) 0 0
\(916\) −24771.7 −0.893538
\(917\) −4128.52 −0.148676
\(918\) 0 0
\(919\) 38513.1 1.38241 0.691203 0.722661i \(-0.257081\pi\)
0.691203 + 0.722661i \(0.257081\pi\)
\(920\) 11506.9 0.412360
\(921\) 0 0
\(922\) 3500.70 0.125043
\(923\) 0 0
\(924\) 0 0
\(925\) −104103. −3.70043
\(926\) −21020.4 −0.745975
\(927\) 0 0
\(928\) 16858.7 0.596352
\(929\) 23218.9 0.820009 0.410005 0.912083i \(-0.365527\pi\)
0.410005 + 0.912083i \(0.365527\pi\)
\(930\) 0 0
\(931\) −34372.3 −1.21000
\(932\) −20650.1 −0.725768
\(933\) 0 0
\(934\) 5307.99 0.185956
\(935\) −49358.3 −1.72640
\(936\) 0 0
\(937\) −11112.9 −0.387452 −0.193726 0.981056i \(-0.562057\pi\)
−0.193726 + 0.981056i \(0.562057\pi\)
\(938\) 4972.08 0.173075
\(939\) 0 0
\(940\) −35497.3 −1.23170
\(941\) 45570.4 1.57869 0.789347 0.613947i \(-0.210419\pi\)
0.789347 + 0.613947i \(0.210419\pi\)
\(942\) 0 0
\(943\) 4068.16 0.140485
\(944\) −9246.71 −0.318808
\(945\) 0 0
\(946\) 6627.62 0.227783
\(947\) 34903.9 1.19770 0.598852 0.800860i \(-0.295624\pi\)
0.598852 + 0.800860i \(0.295624\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 40887.8 1.39640
\(951\) 0 0
\(952\) 4221.55 0.143720
\(953\) 2886.52 0.0981151 0.0490575 0.998796i \(-0.484378\pi\)
0.0490575 + 0.998796i \(0.484378\pi\)
\(954\) 0 0
\(955\) −27239.8 −0.922994
\(956\) −12679.0 −0.428943
\(957\) 0 0
\(958\) −4789.03 −0.161510
\(959\) 2366.96 0.0797008
\(960\) 0 0
\(961\) 3737.42 0.125455
\(962\) 0 0
\(963\) 0 0
\(964\) 37074.8 1.23869
\(965\) 79906.8 2.66559
\(966\) 0 0
\(967\) −11593.8 −0.385556 −0.192778 0.981242i \(-0.561750\pi\)
−0.192778 + 0.981242i \(0.561750\pi\)
\(968\) −50056.1 −1.66205
\(969\) 0 0
\(970\) −34184.1 −1.13153
\(971\) 4952.12 0.163667 0.0818337 0.996646i \(-0.473922\pi\)
0.0818337 + 0.996646i \(0.473922\pi\)
\(972\) 0 0
\(973\) −1971.03 −0.0649418
\(974\) −9153.33 −0.301121
\(975\) 0 0
\(976\) 4162.32 0.136509
\(977\) −19650.1 −0.643462 −0.321731 0.946831i \(-0.604265\pi\)
−0.321731 + 0.946831i \(0.604265\pi\)
\(978\) 0 0
\(979\) −6646.77 −0.216988
\(980\) 35042.2 1.14223
\(981\) 0 0
\(982\) 14350.8 0.466347
\(983\) 56818.4 1.84357 0.921783 0.387707i \(-0.126733\pi\)
0.921783 + 0.387707i \(0.126733\pi\)
\(984\) 0 0
\(985\) −65779.7 −2.12783
\(986\) 5733.78 0.185193
\(987\) 0 0
\(988\) 0 0
\(989\) 2027.15 0.0651764
\(990\) 0 0
\(991\) −19120.4 −0.612897 −0.306448 0.951887i \(-0.599141\pi\)
−0.306448 + 0.951887i \(0.599141\pi\)
\(992\) −34342.1 −1.09915
\(993\) 0 0
\(994\) 2415.34 0.0770723
\(995\) −68090.9 −2.16947
\(996\) 0 0
\(997\) 38887.9 1.23530 0.617650 0.786453i \(-0.288085\pi\)
0.617650 + 0.786453i \(0.288085\pi\)
\(998\) 7712.07 0.244611
\(999\) 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.u.1.2 3
3.2 odd 2 507.4.a.h.1.2 3
13.12 even 2 117.4.a.f.1.2 3
39.5 even 4 507.4.b.g.337.4 6
39.8 even 4 507.4.b.g.337.3 6
39.38 odd 2 39.4.a.c.1.2 3
52.51 odd 2 1872.4.a.bk.1.1 3
156.155 even 2 624.4.a.t.1.3 3
195.194 odd 2 975.4.a.l.1.2 3
273.272 even 2 1911.4.a.k.1.2 3
312.77 odd 2 2496.4.a.bl.1.1 3
312.155 even 2 2496.4.a.bp.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.2 3 39.38 odd 2
117.4.a.f.1.2 3 13.12 even 2
507.4.a.h.1.2 3 3.2 odd 2
507.4.b.g.337.3 6 39.8 even 4
507.4.b.g.337.4 6 39.5 even 4
624.4.a.t.1.3 3 156.155 even 2
975.4.a.l.1.2 3 195.194 odd 2
1521.4.a.u.1.2 3 1.1 even 1 trivial
1872.4.a.bk.1.1 3 52.51 odd 2
1911.4.a.k.1.2 3 273.272 even 2
2496.4.a.bl.1.1 3 312.77 odd 2
2496.4.a.bp.1.1 3 312.155 even 2