Properties

Label 1521.4.a.bf.1.1
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.27560\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.52257 q^{2} +22.4988 q^{4} +6.08065 q^{5} +20.2718 q^{7} -80.0709 q^{8} +O(q^{10})\) \(q-5.52257 q^{2} +22.4988 q^{4} +6.08065 q^{5} +20.2718 q^{7} -80.0709 q^{8} -33.5808 q^{10} +48.8284 q^{11} -111.953 q^{14} +262.207 q^{16} +37.7513 q^{17} +120.837 q^{19} +136.807 q^{20} -269.658 q^{22} -74.8543 q^{23} -88.0257 q^{25} +456.092 q^{28} +112.710 q^{29} +113.134 q^{31} -807.490 q^{32} -208.485 q^{34} +123.266 q^{35} -85.7704 q^{37} -667.331 q^{38} -486.883 q^{40} +133.993 q^{41} -319.135 q^{43} +1098.58 q^{44} +413.388 q^{46} +401.982 q^{47} +67.9471 q^{49} +486.129 q^{50} +384.493 q^{53} +296.908 q^{55} -1623.18 q^{56} -622.450 q^{58} +121.629 q^{59} +220.043 q^{61} -624.790 q^{62} +2361.77 q^{64} +975.363 q^{67} +849.361 q^{68} -680.745 q^{70} -106.725 q^{71} +43.2566 q^{73} +473.673 q^{74} +2718.69 q^{76} +989.841 q^{77} +539.339 q^{79} +1594.39 q^{80} -739.986 q^{82} -811.183 q^{83} +229.553 q^{85} +1762.45 q^{86} -3909.73 q^{88} +1130.64 q^{89} -1684.13 q^{92} -2219.98 q^{94} +734.767 q^{95} -229.088 q^{97} -375.243 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29} + 717 q^{31} - 707 q^{32} + 632 q^{34} + 418 q^{35} + 1108 q^{37} + 660 q^{38} - 1506 q^{40} - 334 q^{41} + 242 q^{43} + 307 q^{44} + 979 q^{46} + 184 q^{47} - 38 q^{49} + 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} + 1161 q^{58} - 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} + 2308 q^{67} - 2785 q^{68} - 1420 q^{70} - 96 q^{71} + 2505 q^{73} + 1191 q^{74} + 2409 q^{76} + 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 1517 q^{82} - 1539 q^{83} + 4296 q^{85} + 3763 q^{86} - 3716 q^{88} + 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} + 1445 q^{97} - 1457 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.52257 −1.95253 −0.976263 0.216591i \(-0.930506\pi\)
−0.976263 + 0.216591i \(0.930506\pi\)
\(3\) 0 0
\(4\) 22.4988 2.81235
\(5\) 6.08065 0.543870 0.271935 0.962316i \(-0.412336\pi\)
0.271935 + 0.962316i \(0.412336\pi\)
\(6\) 0 0
\(7\) 20.2718 1.09458 0.547288 0.836944i \(-0.315660\pi\)
0.547288 + 0.836944i \(0.315660\pi\)
\(8\) −80.0709 −3.53867
\(9\) 0 0
\(10\) −33.5808 −1.06192
\(11\) 48.8284 1.33839 0.669196 0.743086i \(-0.266639\pi\)
0.669196 + 0.743086i \(0.266639\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −111.953 −2.13719
\(15\) 0 0
\(16\) 262.207 4.09698
\(17\) 37.7513 0.538591 0.269295 0.963058i \(-0.413209\pi\)
0.269295 + 0.963058i \(0.413209\pi\)
\(18\) 0 0
\(19\) 120.837 1.45905 0.729524 0.683955i \(-0.239742\pi\)
0.729524 + 0.683955i \(0.239742\pi\)
\(20\) 136.807 1.52955
\(21\) 0 0
\(22\) −269.658 −2.61324
\(23\) −74.8543 −0.678617 −0.339309 0.940675i \(-0.610193\pi\)
−0.339309 + 0.940675i \(0.610193\pi\)
\(24\) 0 0
\(25\) −88.0257 −0.704206
\(26\) 0 0
\(27\) 0 0
\(28\) 456.092 3.07834
\(29\) 112.710 0.721715 0.360858 0.932621i \(-0.382484\pi\)
0.360858 + 0.932621i \(0.382484\pi\)
\(30\) 0 0
\(31\) 113.134 0.655465 0.327733 0.944770i \(-0.393715\pi\)
0.327733 + 0.944770i \(0.393715\pi\)
\(32\) −807.490 −4.46079
\(33\) 0 0
\(34\) −208.485 −1.05161
\(35\) 123.266 0.595307
\(36\) 0 0
\(37\) −85.7704 −0.381096 −0.190548 0.981678i \(-0.561027\pi\)
−0.190548 + 0.981678i \(0.561027\pi\)
\(38\) −667.331 −2.84883
\(39\) 0 0
\(40\) −486.883 −1.92457
\(41\) 133.993 0.510395 0.255197 0.966889i \(-0.417860\pi\)
0.255197 + 0.966889i \(0.417860\pi\)
\(42\) 0 0
\(43\) −319.135 −1.13181 −0.565903 0.824472i \(-0.691472\pi\)
−0.565903 + 0.824472i \(0.691472\pi\)
\(44\) 1098.58 3.76403
\(45\) 0 0
\(46\) 413.388 1.32502
\(47\) 401.982 1.24755 0.623777 0.781602i \(-0.285597\pi\)
0.623777 + 0.781602i \(0.285597\pi\)
\(48\) 0 0
\(49\) 67.9471 0.198096
\(50\) 486.129 1.37498
\(51\) 0 0
\(52\) 0 0
\(53\) 384.493 0.996494 0.498247 0.867035i \(-0.333977\pi\)
0.498247 + 0.867035i \(0.333977\pi\)
\(54\) 0 0
\(55\) 296.908 0.727911
\(56\) −1623.18 −3.87334
\(57\) 0 0
\(58\) −622.450 −1.40917
\(59\) 121.629 0.268385 0.134192 0.990955i \(-0.457156\pi\)
0.134192 + 0.990955i \(0.457156\pi\)
\(60\) 0 0
\(61\) 220.043 0.461864 0.230932 0.972970i \(-0.425823\pi\)
0.230932 + 0.972970i \(0.425823\pi\)
\(62\) −624.790 −1.27981
\(63\) 0 0
\(64\) 2361.77 4.61283
\(65\) 0 0
\(66\) 0 0
\(67\) 975.363 1.77850 0.889250 0.457421i \(-0.151227\pi\)
0.889250 + 0.457421i \(0.151227\pi\)
\(68\) 849.361 1.51471
\(69\) 0 0
\(70\) −680.745 −1.16235
\(71\) −106.725 −0.178394 −0.0891969 0.996014i \(-0.528430\pi\)
−0.0891969 + 0.996014i \(0.528430\pi\)
\(72\) 0 0
\(73\) 43.2566 0.0693535 0.0346767 0.999399i \(-0.488960\pi\)
0.0346767 + 0.999399i \(0.488960\pi\)
\(74\) 473.673 0.744100
\(75\) 0 0
\(76\) 2718.69 4.10336
\(77\) 989.841 1.46497
\(78\) 0 0
\(79\) 539.339 0.768106 0.384053 0.923311i \(-0.374528\pi\)
0.384053 + 0.923311i \(0.374528\pi\)
\(80\) 1594.39 2.22822
\(81\) 0 0
\(82\) −739.986 −0.996558
\(83\) −811.183 −1.07276 −0.536379 0.843977i \(-0.680208\pi\)
−0.536379 + 0.843977i \(0.680208\pi\)
\(84\) 0 0
\(85\) 229.553 0.292923
\(86\) 1762.45 2.20988
\(87\) 0 0
\(88\) −3909.73 −4.73612
\(89\) 1130.64 1.34661 0.673304 0.739366i \(-0.264875\pi\)
0.673304 + 0.739366i \(0.264875\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1684.13 −1.90851
\(93\) 0 0
\(94\) −2219.98 −2.43588
\(95\) 734.767 0.793532
\(96\) 0 0
\(97\) −229.088 −0.239797 −0.119899 0.992786i \(-0.538257\pi\)
−0.119899 + 0.992786i \(0.538257\pi\)
\(98\) −375.243 −0.386788
\(99\) 0 0
\(100\) −1980.48 −1.98048
\(101\) 845.077 0.832557 0.416279 0.909237i \(-0.363334\pi\)
0.416279 + 0.909237i \(0.363334\pi\)
\(102\) 0 0
\(103\) −1095.09 −1.04760 −0.523800 0.851841i \(-0.675486\pi\)
−0.523800 + 0.851841i \(0.675486\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2123.39 −1.94568
\(107\) −1183.33 −1.06913 −0.534563 0.845129i \(-0.679524\pi\)
−0.534563 + 0.845129i \(0.679524\pi\)
\(108\) 0 0
\(109\) −239.968 −0.210870 −0.105435 0.994426i \(-0.533623\pi\)
−0.105435 + 0.994426i \(0.533623\pi\)
\(110\) −1639.70 −1.42126
\(111\) 0 0
\(112\) 5315.41 4.48446
\(113\) 1031.22 0.858490 0.429245 0.903188i \(-0.358780\pi\)
0.429245 + 0.903188i \(0.358780\pi\)
\(114\) 0 0
\(115\) −455.162 −0.369079
\(116\) 2535.85 2.02972
\(117\) 0 0
\(118\) −671.703 −0.524028
\(119\) 765.288 0.589528
\(120\) 0 0
\(121\) 1053.21 0.791294
\(122\) −1215.21 −0.901800
\(123\) 0 0
\(124\) 2545.38 1.84340
\(125\) −1295.33 −0.926866
\(126\) 0 0
\(127\) −18.0201 −0.0125908 −0.00629539 0.999980i \(-0.502004\pi\)
−0.00629539 + 0.999980i \(0.502004\pi\)
\(128\) −6583.12 −4.54587
\(129\) 0 0
\(130\) 0 0
\(131\) 1282.52 0.855378 0.427689 0.903926i \(-0.359328\pi\)
0.427689 + 0.903926i \(0.359328\pi\)
\(132\) 0 0
\(133\) 2449.59 1.59704
\(134\) −5386.51 −3.47257
\(135\) 0 0
\(136\) −3022.78 −1.90589
\(137\) −1547.36 −0.964965 −0.482482 0.875906i \(-0.660265\pi\)
−0.482482 + 0.875906i \(0.660265\pi\)
\(138\) 0 0
\(139\) −3096.87 −1.88973 −0.944867 0.327455i \(-0.893809\pi\)
−0.944867 + 0.327455i \(0.893809\pi\)
\(140\) 2773.34 1.67421
\(141\) 0 0
\(142\) 589.398 0.348318
\(143\) 0 0
\(144\) 0 0
\(145\) 685.351 0.392519
\(146\) −238.888 −0.135414
\(147\) 0 0
\(148\) −1929.73 −1.07178
\(149\) −420.879 −0.231408 −0.115704 0.993284i \(-0.536912\pi\)
−0.115704 + 0.993284i \(0.536912\pi\)
\(150\) 0 0
\(151\) −2801.17 −1.50964 −0.754821 0.655931i \(-0.772276\pi\)
−0.754821 + 0.655931i \(0.772276\pi\)
\(152\) −9675.52 −5.16308
\(153\) 0 0
\(154\) −5466.47 −2.86039
\(155\) 687.927 0.356488
\(156\) 0 0
\(157\) −2344.20 −1.19164 −0.595820 0.803118i \(-0.703173\pi\)
−0.595820 + 0.803118i \(0.703173\pi\)
\(158\) −2978.54 −1.49975
\(159\) 0 0
\(160\) −4910.06 −2.42609
\(161\) −1517.43 −0.742798
\(162\) 0 0
\(163\) 3658.41 1.75797 0.878984 0.476851i \(-0.158222\pi\)
0.878984 + 0.476851i \(0.158222\pi\)
\(164\) 3014.68 1.43541
\(165\) 0 0
\(166\) 4479.82 2.09459
\(167\) −1987.85 −0.921105 −0.460552 0.887632i \(-0.652349\pi\)
−0.460552 + 0.887632i \(0.652349\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −1267.72 −0.571940
\(171\) 0 0
\(172\) −7180.17 −3.18304
\(173\) 3577.11 1.57204 0.786019 0.618202i \(-0.212139\pi\)
0.786019 + 0.618202i \(0.212139\pi\)
\(174\) 0 0
\(175\) −1784.44 −0.770807
\(176\) 12803.1 5.48337
\(177\) 0 0
\(178\) −6244.07 −2.62928
\(179\) −770.144 −0.321582 −0.160791 0.986988i \(-0.551405\pi\)
−0.160791 + 0.986988i \(0.551405\pi\)
\(180\) 0 0
\(181\) −1895.69 −0.778482 −0.389241 0.921136i \(-0.627263\pi\)
−0.389241 + 0.921136i \(0.627263\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 5993.65 2.40140
\(185\) −521.539 −0.207267
\(186\) 0 0
\(187\) 1843.34 0.720846
\(188\) 9044.12 3.50857
\(189\) 0 0
\(190\) −4057.81 −1.54939
\(191\) −2203.83 −0.834887 −0.417443 0.908703i \(-0.637074\pi\)
−0.417443 + 0.908703i \(0.637074\pi\)
\(192\) 0 0
\(193\) 1016.38 0.379069 0.189534 0.981874i \(-0.439302\pi\)
0.189534 + 0.981874i \(0.439302\pi\)
\(194\) 1265.15 0.468210
\(195\) 0 0
\(196\) 1528.73 0.557117
\(197\) −4318.58 −1.56186 −0.780930 0.624619i \(-0.785254\pi\)
−0.780930 + 0.624619i \(0.785254\pi\)
\(198\) 0 0
\(199\) 1363.40 0.485672 0.242836 0.970067i \(-0.421922\pi\)
0.242836 + 0.970067i \(0.421922\pi\)
\(200\) 7048.30 2.49195
\(201\) 0 0
\(202\) −4667.00 −1.62559
\(203\) 2284.84 0.789972
\(204\) 0 0
\(205\) 814.764 0.277588
\(206\) 6047.74 2.04546
\(207\) 0 0
\(208\) 0 0
\(209\) 5900.28 1.95278
\(210\) 0 0
\(211\) 5288.34 1.72542 0.862711 0.505697i \(-0.168765\pi\)
0.862711 + 0.505697i \(0.168765\pi\)
\(212\) 8650.64 2.80249
\(213\) 0 0
\(214\) 6535.00 2.08749
\(215\) −1940.55 −0.615555
\(216\) 0 0
\(217\) 2293.43 0.717457
\(218\) 1325.24 0.411728
\(219\) 0 0
\(220\) 6680.09 2.04714
\(221\) 0 0
\(222\) 0 0
\(223\) −1258.01 −0.377769 −0.188884 0.981999i \(-0.560487\pi\)
−0.188884 + 0.981999i \(0.560487\pi\)
\(224\) −16369.3 −4.88268
\(225\) 0 0
\(226\) −5695.01 −1.67622
\(227\) −1135.21 −0.331922 −0.165961 0.986132i \(-0.553073\pi\)
−0.165961 + 0.986132i \(0.553073\pi\)
\(228\) 0 0
\(229\) −218.002 −0.0629083 −0.0314541 0.999505i \(-0.510014\pi\)
−0.0314541 + 0.999505i \(0.510014\pi\)
\(230\) 2513.67 0.720636
\(231\) 0 0
\(232\) −9024.80 −2.55391
\(233\) 2.56367 0.000720823 0 0.000360412 1.00000i \(-0.499885\pi\)
0.000360412 1.00000i \(0.499885\pi\)
\(234\) 0 0
\(235\) 2444.31 0.678507
\(236\) 2736.50 0.754793
\(237\) 0 0
\(238\) −4226.36 −1.15107
\(239\) −3971.13 −1.07477 −0.537387 0.843336i \(-0.680589\pi\)
−0.537387 + 0.843336i \(0.680589\pi\)
\(240\) 0 0
\(241\) −658.032 −0.175882 −0.0879411 0.996126i \(-0.528029\pi\)
−0.0879411 + 0.996126i \(0.528029\pi\)
\(242\) −5816.44 −1.54502
\(243\) 0 0
\(244\) 4950.72 1.29892
\(245\) 413.162 0.107739
\(246\) 0 0
\(247\) 0 0
\(248\) −9058.72 −2.31947
\(249\) 0 0
\(250\) 7153.58 1.80973
\(251\) −7091.01 −1.78319 −0.891595 0.452833i \(-0.850413\pi\)
−0.891595 + 0.452833i \(0.850413\pi\)
\(252\) 0 0
\(253\) −3655.01 −0.908256
\(254\) 99.5176 0.0245838
\(255\) 0 0
\(256\) 17461.6 4.26309
\(257\) −792.530 −0.192361 −0.0961803 0.995364i \(-0.530663\pi\)
−0.0961803 + 0.995364i \(0.530663\pi\)
\(258\) 0 0
\(259\) −1738.72 −0.417139
\(260\) 0 0
\(261\) 0 0
\(262\) −7082.83 −1.67015
\(263\) 337.467 0.0791221 0.0395610 0.999217i \(-0.487404\pi\)
0.0395610 + 0.999217i \(0.487404\pi\)
\(264\) 0 0
\(265\) 2337.97 0.541963
\(266\) −13528.0 −3.11826
\(267\) 0 0
\(268\) 21944.5 5.00177
\(269\) 3523.93 0.798728 0.399364 0.916792i \(-0.369231\pi\)
0.399364 + 0.916792i \(0.369231\pi\)
\(270\) 0 0
\(271\) 8313.62 1.86353 0.931765 0.363063i \(-0.118269\pi\)
0.931765 + 0.363063i \(0.118269\pi\)
\(272\) 9898.65 2.20660
\(273\) 0 0
\(274\) 8545.43 1.88412
\(275\) −4298.16 −0.942504
\(276\) 0 0
\(277\) −2500.33 −0.542347 −0.271174 0.962530i \(-0.587412\pi\)
−0.271174 + 0.962530i \(0.587412\pi\)
\(278\) 17102.7 3.68975
\(279\) 0 0
\(280\) −9870.01 −2.10659
\(281\) 3053.13 0.648165 0.324083 0.946029i \(-0.394944\pi\)
0.324083 + 0.946029i \(0.394944\pi\)
\(282\) 0 0
\(283\) 1158.80 0.243404 0.121702 0.992567i \(-0.461165\pi\)
0.121702 + 0.992567i \(0.461165\pi\)
\(284\) −2401.19 −0.501706
\(285\) 0 0
\(286\) 0 0
\(287\) 2716.28 0.558666
\(288\) 0 0
\(289\) −3487.84 −0.709920
\(290\) −3784.90 −0.766403
\(291\) 0 0
\(292\) 973.223 0.195046
\(293\) 8357.45 1.66637 0.833187 0.552991i \(-0.186514\pi\)
0.833187 + 0.552991i \(0.186514\pi\)
\(294\) 0 0
\(295\) 739.581 0.145966
\(296\) 6867.71 1.34857
\(297\) 0 0
\(298\) 2324.33 0.451829
\(299\) 0 0
\(300\) 0 0
\(301\) −6469.46 −1.23885
\(302\) 15469.7 2.94761
\(303\) 0 0
\(304\) 31684.3 5.97769
\(305\) 1338.01 0.251194
\(306\) 0 0
\(307\) −8429.30 −1.56705 −0.783527 0.621357i \(-0.786582\pi\)
−0.783527 + 0.621357i \(0.786582\pi\)
\(308\) 22270.3 4.12002
\(309\) 0 0
\(310\) −3799.13 −0.696051
\(311\) −3602.87 −0.656914 −0.328457 0.944519i \(-0.606529\pi\)
−0.328457 + 0.944519i \(0.606529\pi\)
\(312\) 0 0
\(313\) 6626.98 1.19674 0.598369 0.801220i \(-0.295816\pi\)
0.598369 + 0.801220i \(0.295816\pi\)
\(314\) 12946.0 2.32671
\(315\) 0 0
\(316\) 12134.5 2.16019
\(317\) −3983.87 −0.705857 −0.352928 0.935650i \(-0.614814\pi\)
−0.352928 + 0.935650i \(0.614814\pi\)
\(318\) 0 0
\(319\) 5503.46 0.965938
\(320\) 14361.1 2.50878
\(321\) 0 0
\(322\) 8380.14 1.45033
\(323\) 4561.76 0.785829
\(324\) 0 0
\(325\) 0 0
\(326\) −20203.8 −3.43248
\(327\) 0 0
\(328\) −10728.9 −1.80612
\(329\) 8148.91 1.36554
\(330\) 0 0
\(331\) 4172.82 0.692927 0.346464 0.938063i \(-0.387382\pi\)
0.346464 + 0.938063i \(0.387382\pi\)
\(332\) −18250.7 −3.01697
\(333\) 0 0
\(334\) 10978.1 1.79848
\(335\) 5930.84 0.967272
\(336\) 0 0
\(337\) 4157.51 0.672030 0.336015 0.941857i \(-0.390921\pi\)
0.336015 + 0.941857i \(0.390921\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 5164.66 0.823804
\(341\) 5524.14 0.877270
\(342\) 0 0
\(343\) −5575.83 −0.877744
\(344\) 25553.4 4.00509
\(345\) 0 0
\(346\) −19754.9 −3.06944
\(347\) −4667.98 −0.722162 −0.361081 0.932535i \(-0.617592\pi\)
−0.361081 + 0.932535i \(0.617592\pi\)
\(348\) 0 0
\(349\) 11905.6 1.82606 0.913028 0.407896i \(-0.133737\pi\)
0.913028 + 0.407896i \(0.133737\pi\)
\(350\) 9854.72 1.50502
\(351\) 0 0
\(352\) −39428.4 −5.97029
\(353\) 1662.12 0.250610 0.125305 0.992118i \(-0.460009\pi\)
0.125305 + 0.992118i \(0.460009\pi\)
\(354\) 0 0
\(355\) −648.959 −0.0970229
\(356\) 25438.2 3.78714
\(357\) 0 0
\(358\) 4253.18 0.627898
\(359\) −12754.4 −1.87507 −0.937535 0.347890i \(-0.886898\pi\)
−0.937535 + 0.347890i \(0.886898\pi\)
\(360\) 0 0
\(361\) 7742.58 1.12882
\(362\) 10469.1 1.52001
\(363\) 0 0
\(364\) 0 0
\(365\) 263.028 0.0377192
\(366\) 0 0
\(367\) 4463.81 0.634901 0.317451 0.948275i \(-0.397173\pi\)
0.317451 + 0.948275i \(0.397173\pi\)
\(368\) −19627.3 −2.78028
\(369\) 0 0
\(370\) 2880.24 0.404693
\(371\) 7794.38 1.09074
\(372\) 0 0
\(373\) 9454.46 1.31242 0.656211 0.754578i \(-0.272158\pi\)
0.656211 + 0.754578i \(0.272158\pi\)
\(374\) −10180.0 −1.40747
\(375\) 0 0
\(376\) −32187.0 −4.41468
\(377\) 0 0
\(378\) 0 0
\(379\) 14085.9 1.90908 0.954542 0.298077i \(-0.0963453\pi\)
0.954542 + 0.298077i \(0.0963453\pi\)
\(380\) 16531.4 2.23169
\(381\) 0 0
\(382\) 12170.8 1.63014
\(383\) 8663.84 1.15588 0.577939 0.816080i \(-0.303857\pi\)
0.577939 + 0.816080i \(0.303857\pi\)
\(384\) 0 0
\(385\) 6018.87 0.796754
\(386\) −5613.01 −0.740141
\(387\) 0 0
\(388\) −5154.20 −0.674395
\(389\) −5085.58 −0.662852 −0.331426 0.943481i \(-0.607530\pi\)
−0.331426 + 0.943481i \(0.607530\pi\)
\(390\) 0 0
\(391\) −2825.85 −0.365497
\(392\) −5440.58 −0.700997
\(393\) 0 0
\(394\) 23849.7 3.04957
\(395\) 3279.53 0.417750
\(396\) 0 0
\(397\) 12983.7 1.64140 0.820700 0.571360i \(-0.193584\pi\)
0.820700 + 0.571360i \(0.193584\pi\)
\(398\) −7529.48 −0.948288
\(399\) 0 0
\(400\) −23080.9 −2.88512
\(401\) −6493.66 −0.808673 −0.404337 0.914610i \(-0.632498\pi\)
−0.404337 + 0.914610i \(0.632498\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 19013.2 2.34145
\(405\) 0 0
\(406\) −12618.2 −1.54244
\(407\) −4188.03 −0.510056
\(408\) 0 0
\(409\) −12546.2 −1.51680 −0.758398 0.651792i \(-0.774017\pi\)
−0.758398 + 0.651792i \(0.774017\pi\)
\(410\) −4499.59 −0.541998
\(411\) 0 0
\(412\) −24638.3 −2.94622
\(413\) 2465.63 0.293767
\(414\) 0 0
\(415\) −4932.52 −0.583440
\(416\) 0 0
\(417\) 0 0
\(418\) −32584.7 −3.81285
\(419\) −7945.58 −0.926413 −0.463206 0.886250i \(-0.653301\pi\)
−0.463206 + 0.886250i \(0.653301\pi\)
\(420\) 0 0
\(421\) −284.758 −0.0329650 −0.0164825 0.999864i \(-0.505247\pi\)
−0.0164825 + 0.999864i \(0.505247\pi\)
\(422\) −29205.2 −3.36893
\(423\) 0 0
\(424\) −30786.7 −3.52626
\(425\) −3323.09 −0.379279
\(426\) 0 0
\(427\) 4460.68 0.505545
\(428\) −26623.4 −3.00676
\(429\) 0 0
\(430\) 10716.8 1.20189
\(431\) 12512.1 1.39835 0.699173 0.714952i \(-0.253552\pi\)
0.699173 + 0.714952i \(0.253552\pi\)
\(432\) 0 0
\(433\) −13217.2 −1.46692 −0.733460 0.679732i \(-0.762096\pi\)
−0.733460 + 0.679732i \(0.762096\pi\)
\(434\) −12665.6 −1.40085
\(435\) 0 0
\(436\) −5399.01 −0.593040
\(437\) −9045.16 −0.990135
\(438\) 0 0
\(439\) 4635.40 0.503953 0.251977 0.967733i \(-0.418919\pi\)
0.251977 + 0.967733i \(0.418919\pi\)
\(440\) −23773.7 −2.57583
\(441\) 0 0
\(442\) 0 0
\(443\) 2945.92 0.315947 0.157974 0.987443i \(-0.449504\pi\)
0.157974 + 0.987443i \(0.449504\pi\)
\(444\) 0 0
\(445\) 6875.05 0.732379
\(446\) 6947.45 0.737603
\(447\) 0 0
\(448\) 47877.3 5.04909
\(449\) 8668.29 0.911095 0.455548 0.890211i \(-0.349443\pi\)
0.455548 + 0.890211i \(0.349443\pi\)
\(450\) 0 0
\(451\) 6542.66 0.683108
\(452\) 23201.3 2.41438
\(453\) 0 0
\(454\) 6269.27 0.648087
\(455\) 0 0
\(456\) 0 0
\(457\) −5195.36 −0.531791 −0.265896 0.964002i \(-0.585668\pi\)
−0.265896 + 0.964002i \(0.585668\pi\)
\(458\) 1203.93 0.122830
\(459\) 0 0
\(460\) −10240.6 −1.03798
\(461\) 2931.19 0.296136 0.148068 0.988977i \(-0.452694\pi\)
0.148068 + 0.988977i \(0.452694\pi\)
\(462\) 0 0
\(463\) −396.589 −0.0398079 −0.0199039 0.999802i \(-0.506336\pi\)
−0.0199039 + 0.999802i \(0.506336\pi\)
\(464\) 29553.4 2.95685
\(465\) 0 0
\(466\) −14.1581 −0.00140743
\(467\) −16627.5 −1.64760 −0.823801 0.566880i \(-0.808150\pi\)
−0.823801 + 0.566880i \(0.808150\pi\)
\(468\) 0 0
\(469\) 19772.4 1.94670
\(470\) −13498.9 −1.32480
\(471\) 0 0
\(472\) −9738.91 −0.949724
\(473\) −15582.9 −1.51480
\(474\) 0 0
\(475\) −10636.8 −1.02747
\(476\) 17218.1 1.65796
\(477\) 0 0
\(478\) 21930.9 2.09852
\(479\) −8903.28 −0.849272 −0.424636 0.905364i \(-0.639598\pi\)
−0.424636 + 0.905364i \(0.639598\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 3634.03 0.343414
\(483\) 0 0
\(484\) 23696.1 2.22540
\(485\) −1393.00 −0.130418
\(486\) 0 0
\(487\) −469.526 −0.0436884 −0.0218442 0.999761i \(-0.506954\pi\)
−0.0218442 + 0.999761i \(0.506954\pi\)
\(488\) −17619.1 −1.63438
\(489\) 0 0
\(490\) −2281.72 −0.210362
\(491\) 18823.8 1.73015 0.865077 0.501639i \(-0.167269\pi\)
0.865077 + 0.501639i \(0.167269\pi\)
\(492\) 0 0
\(493\) 4254.96 0.388709
\(494\) 0 0
\(495\) 0 0
\(496\) 29664.4 2.68543
\(497\) −2163.52 −0.195265
\(498\) 0 0
\(499\) −5651.95 −0.507046 −0.253523 0.967329i \(-0.581589\pi\)
−0.253523 + 0.967329i \(0.581589\pi\)
\(500\) −29143.5 −2.60667
\(501\) 0 0
\(502\) 39160.7 3.48172
\(503\) −8022.35 −0.711131 −0.355565 0.934651i \(-0.615712\pi\)
−0.355565 + 0.934651i \(0.615712\pi\)
\(504\) 0 0
\(505\) 5138.62 0.452803
\(506\) 20185.1 1.77339
\(507\) 0 0
\(508\) −405.432 −0.0354097
\(509\) −3940.56 −0.343148 −0.171574 0.985171i \(-0.554885\pi\)
−0.171574 + 0.985171i \(0.554885\pi\)
\(510\) 0 0
\(511\) 876.890 0.0759126
\(512\) −43768.2 −3.77793
\(513\) 0 0
\(514\) 4376.81 0.375589
\(515\) −6658.88 −0.569758
\(516\) 0 0
\(517\) 19628.1 1.66972
\(518\) 9602.22 0.814474
\(519\) 0 0
\(520\) 0 0
\(521\) −8644.84 −0.726943 −0.363472 0.931605i \(-0.618409\pi\)
−0.363472 + 0.931605i \(0.618409\pi\)
\(522\) 0 0
\(523\) 1956.76 0.163600 0.0818001 0.996649i \(-0.473933\pi\)
0.0818001 + 0.996649i \(0.473933\pi\)
\(524\) 28855.3 2.40563
\(525\) 0 0
\(526\) −1863.69 −0.154488
\(527\) 4270.95 0.353028
\(528\) 0 0
\(529\) −6563.84 −0.539479
\(530\) −12911.6 −1.05820
\(531\) 0 0
\(532\) 55112.8 4.49144
\(533\) 0 0
\(534\) 0 0
\(535\) −7195.39 −0.581465
\(536\) −78098.2 −6.29352
\(537\) 0 0
\(538\) −19461.2 −1.55954
\(539\) 3317.75 0.265131
\(540\) 0 0
\(541\) 7005.37 0.556718 0.278359 0.960477i \(-0.410210\pi\)
0.278359 + 0.960477i \(0.410210\pi\)
\(542\) −45912.6 −3.63859
\(543\) 0 0
\(544\) −30483.8 −2.40254
\(545\) −1459.16 −0.114686
\(546\) 0 0
\(547\) 23216.7 1.81476 0.907382 0.420307i \(-0.138077\pi\)
0.907382 + 0.420307i \(0.138077\pi\)
\(548\) −34813.9 −2.71382
\(549\) 0 0
\(550\) 23736.9 1.84026
\(551\) 13619.6 1.05302
\(552\) 0 0
\(553\) 10933.4 0.840751
\(554\) 13808.2 1.05895
\(555\) 0 0
\(556\) −69675.9 −5.31460
\(557\) 14912.9 1.13443 0.567217 0.823568i \(-0.308020\pi\)
0.567217 + 0.823568i \(0.308020\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 32321.1 2.43896
\(561\) 0 0
\(562\) −16861.1 −1.26556
\(563\) −10854.3 −0.812531 −0.406266 0.913755i \(-0.633169\pi\)
−0.406266 + 0.913755i \(0.633169\pi\)
\(564\) 0 0
\(565\) 6270.50 0.466906
\(566\) −6399.55 −0.475253
\(567\) 0 0
\(568\) 8545.58 0.631276
\(569\) −17892.5 −1.31827 −0.659134 0.752026i \(-0.729077\pi\)
−0.659134 + 0.752026i \(0.729077\pi\)
\(570\) 0 0
\(571\) 17319.8 1.26937 0.634686 0.772770i \(-0.281130\pi\)
0.634686 + 0.772770i \(0.281130\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −15000.9 −1.09081
\(575\) 6589.10 0.477886
\(576\) 0 0
\(577\) −9738.48 −0.702632 −0.351316 0.936257i \(-0.614266\pi\)
−0.351316 + 0.936257i \(0.614266\pi\)
\(578\) 19261.8 1.38614
\(579\) 0 0
\(580\) 15419.6 1.10390
\(581\) −16444.2 −1.17421
\(582\) 0 0
\(583\) 18774.2 1.33370
\(584\) −3463.59 −0.245419
\(585\) 0 0
\(586\) −46154.7 −3.25364
\(587\) 5960.31 0.419095 0.209547 0.977799i \(-0.432801\pi\)
0.209547 + 0.977799i \(0.432801\pi\)
\(588\) 0 0
\(589\) 13670.7 0.956355
\(590\) −4084.39 −0.285003
\(591\) 0 0
\(592\) −22489.6 −1.56134
\(593\) 17870.2 1.23751 0.618753 0.785586i \(-0.287638\pi\)
0.618753 + 0.785586i \(0.287638\pi\)
\(594\) 0 0
\(595\) 4653.45 0.320627
\(596\) −9469.28 −0.650800
\(597\) 0 0
\(598\) 0 0
\(599\) 17943.7 1.22397 0.611987 0.790868i \(-0.290371\pi\)
0.611987 + 0.790868i \(0.290371\pi\)
\(600\) 0 0
\(601\) 6548.20 0.444437 0.222219 0.974997i \(-0.428670\pi\)
0.222219 + 0.974997i \(0.428670\pi\)
\(602\) 35728.1 2.41888
\(603\) 0 0
\(604\) −63023.0 −4.24565
\(605\) 6404.21 0.430361
\(606\) 0 0
\(607\) −14886.0 −0.995391 −0.497696 0.867352i \(-0.665820\pi\)
−0.497696 + 0.867352i \(0.665820\pi\)
\(608\) −97574.6 −6.50851
\(609\) 0 0
\(610\) −7389.24 −0.490462
\(611\) 0 0
\(612\) 0 0
\(613\) −8774.65 −0.578148 −0.289074 0.957307i \(-0.593347\pi\)
−0.289074 + 0.957307i \(0.593347\pi\)
\(614\) 46551.5 3.05971
\(615\) 0 0
\(616\) −79257.4 −5.18405
\(617\) 3810.41 0.248625 0.124312 0.992243i \(-0.460327\pi\)
0.124312 + 0.992243i \(0.460327\pi\)
\(618\) 0 0
\(619\) −16392.6 −1.06441 −0.532207 0.846614i \(-0.678637\pi\)
−0.532207 + 0.846614i \(0.678637\pi\)
\(620\) 15477.5 1.00257
\(621\) 0 0
\(622\) 19897.1 1.28264
\(623\) 22920.2 1.47396
\(624\) 0 0
\(625\) 3126.74 0.200112
\(626\) −36598.0 −2.33666
\(627\) 0 0
\(628\) −52741.8 −3.35131
\(629\) −3237.95 −0.205255
\(630\) 0 0
\(631\) −11077.3 −0.698858 −0.349429 0.936963i \(-0.613624\pi\)
−0.349429 + 0.936963i \(0.613624\pi\)
\(632\) −43185.4 −2.71807
\(633\) 0 0
\(634\) 22001.2 1.37820
\(635\) −109.574 −0.00684775
\(636\) 0 0
\(637\) 0 0
\(638\) −30393.2 −1.88602
\(639\) 0 0
\(640\) −40029.6 −2.47236
\(641\) 2143.45 0.132077 0.0660383 0.997817i \(-0.478964\pi\)
0.0660383 + 0.997817i \(0.478964\pi\)
\(642\) 0 0
\(643\) 9774.93 0.599511 0.299755 0.954016i \(-0.403095\pi\)
0.299755 + 0.954016i \(0.403095\pi\)
\(644\) −34140.5 −2.08901
\(645\) 0 0
\(646\) −25192.6 −1.53435
\(647\) 22447.6 1.36400 0.681999 0.731353i \(-0.261111\pi\)
0.681999 + 0.731353i \(0.261111\pi\)
\(648\) 0 0
\(649\) 5938.93 0.359204
\(650\) 0 0
\(651\) 0 0
\(652\) 82310.0 4.94403
\(653\) −10582.0 −0.634157 −0.317079 0.948399i \(-0.602702\pi\)
−0.317079 + 0.948399i \(0.602702\pi\)
\(654\) 0 0
\(655\) 7798.57 0.465214
\(656\) 35133.9 2.09108
\(657\) 0 0
\(658\) −45003.0 −2.66626
\(659\) 15114.6 0.893445 0.446722 0.894673i \(-0.352591\pi\)
0.446722 + 0.894673i \(0.352591\pi\)
\(660\) 0 0
\(661\) 2229.61 0.131198 0.0655990 0.997846i \(-0.479104\pi\)
0.0655990 + 0.997846i \(0.479104\pi\)
\(662\) −23044.7 −1.35296
\(663\) 0 0
\(664\) 64952.1 3.79613
\(665\) 14895.1 0.868581
\(666\) 0 0
\(667\) −8436.83 −0.489768
\(668\) −44724.3 −2.59047
\(669\) 0 0
\(670\) −32753.5 −1.88862
\(671\) 10744.4 0.618155
\(672\) 0 0
\(673\) −3588.22 −0.205521 −0.102761 0.994706i \(-0.532768\pi\)
−0.102761 + 0.994706i \(0.532768\pi\)
\(674\) −22960.2 −1.31216
\(675\) 0 0
\(676\) 0 0
\(677\) 26458.4 1.50204 0.751019 0.660280i \(-0.229562\pi\)
0.751019 + 0.660280i \(0.229562\pi\)
\(678\) 0 0
\(679\) −4644.03 −0.262476
\(680\) −18380.5 −1.03656
\(681\) 0 0
\(682\) −30507.5 −1.71289
\(683\) 23598.4 1.32206 0.661031 0.750358i \(-0.270119\pi\)
0.661031 + 0.750358i \(0.270119\pi\)
\(684\) 0 0
\(685\) −9408.97 −0.524815
\(686\) 30792.9 1.71382
\(687\) 0 0
\(688\) −83679.4 −4.63699
\(689\) 0 0
\(690\) 0 0
\(691\) −16433.6 −0.904722 −0.452361 0.891835i \(-0.649418\pi\)
−0.452361 + 0.891835i \(0.649418\pi\)
\(692\) 80480.8 4.42113
\(693\) 0 0
\(694\) 25779.2 1.41004
\(695\) −18831.0 −1.02777
\(696\) 0 0
\(697\) 5058.41 0.274894
\(698\) −65749.7 −3.56542
\(699\) 0 0
\(700\) −40147.9 −2.16778
\(701\) −13899.4 −0.748892 −0.374446 0.927249i \(-0.622167\pi\)
−0.374446 + 0.927249i \(0.622167\pi\)
\(702\) 0 0
\(703\) −10364.2 −0.556038
\(704\) 115321. 6.17377
\(705\) 0 0
\(706\) −9179.16 −0.489323
\(707\) 17131.3 0.911297
\(708\) 0 0
\(709\) 32829.6 1.73899 0.869494 0.493944i \(-0.164445\pi\)
0.869494 + 0.493944i \(0.164445\pi\)
\(710\) 3583.92 0.189440
\(711\) 0 0
\(712\) −90531.7 −4.76519
\(713\) −8468.55 −0.444810
\(714\) 0 0
\(715\) 0 0
\(716\) −17327.3 −0.904403
\(717\) 0 0
\(718\) 70437.0 3.66112
\(719\) 4683.94 0.242951 0.121475 0.992594i \(-0.461237\pi\)
0.121475 + 0.992594i \(0.461237\pi\)
\(720\) 0 0
\(721\) −22199.5 −1.14668
\(722\) −42759.0 −2.20405
\(723\) 0 0
\(724\) −42650.7 −2.18937
\(725\) −9921.39 −0.508236
\(726\) 0 0
\(727\) −35368.3 −1.80432 −0.902159 0.431405i \(-0.858018\pi\)
−0.902159 + 0.431405i \(0.858018\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1452.59 −0.0736478
\(731\) −12047.8 −0.609580
\(732\) 0 0
\(733\) 33988.4 1.71268 0.856338 0.516416i \(-0.172734\pi\)
0.856338 + 0.516416i \(0.172734\pi\)
\(734\) −24651.7 −1.23966
\(735\) 0 0
\(736\) 60444.0 3.02717
\(737\) 47625.4 2.38033
\(738\) 0 0
\(739\) −7777.31 −0.387135 −0.193568 0.981087i \(-0.562006\pi\)
−0.193568 + 0.981087i \(0.562006\pi\)
\(740\) −11734.0 −0.582907
\(741\) 0 0
\(742\) −43045.0 −2.12969
\(743\) 7022.08 0.346723 0.173361 0.984858i \(-0.444537\pi\)
0.173361 + 0.984858i \(0.444537\pi\)
\(744\) 0 0
\(745\) −2559.22 −0.125856
\(746\) −52212.9 −2.56254
\(747\) 0 0
\(748\) 41472.9 2.02727
\(749\) −23988.2 −1.17024
\(750\) 0 0
\(751\) 8214.69 0.399146 0.199573 0.979883i \(-0.436045\pi\)
0.199573 + 0.979883i \(0.436045\pi\)
\(752\) 105402. 5.11121
\(753\) 0 0
\(754\) 0 0
\(755\) −17032.9 −0.821048
\(756\) 0 0
\(757\) −20846.9 −1.00092 −0.500459 0.865760i \(-0.666835\pi\)
−0.500459 + 0.865760i \(0.666835\pi\)
\(758\) −77790.3 −3.72753
\(759\) 0 0
\(760\) −58833.5 −2.80804
\(761\) 3563.06 0.169725 0.0848627 0.996393i \(-0.472955\pi\)
0.0848627 + 0.996393i \(0.472955\pi\)
\(762\) 0 0
\(763\) −4864.60 −0.230813
\(764\) −49583.6 −2.34800
\(765\) 0 0
\(766\) −47846.7 −2.25688
\(767\) 0 0
\(768\) 0 0
\(769\) 32557.9 1.52675 0.763373 0.645958i \(-0.223542\pi\)
0.763373 + 0.645958i \(0.223542\pi\)
\(770\) −33239.7 −1.55568
\(771\) 0 0
\(772\) 22867.3 1.06608
\(773\) −20126.4 −0.936477 −0.468238 0.883602i \(-0.655111\pi\)
−0.468238 + 0.883602i \(0.655111\pi\)
\(774\) 0 0
\(775\) −9958.68 −0.461583
\(776\) 18343.2 0.848562
\(777\) 0 0
\(778\) 28085.5 1.29423
\(779\) 16191.3 0.744690
\(780\) 0 0
\(781\) −5211.22 −0.238761
\(782\) 15606.0 0.713642
\(783\) 0 0
\(784\) 17816.2 0.811597
\(785\) −14254.3 −0.648097
\(786\) 0 0
\(787\) 18828.4 0.852806 0.426403 0.904533i \(-0.359781\pi\)
0.426403 + 0.904533i \(0.359781\pi\)
\(788\) −97163.1 −4.39250
\(789\) 0 0
\(790\) −18111.5 −0.815667
\(791\) 20904.8 0.939682
\(792\) 0 0
\(793\) 0 0
\(794\) −71703.7 −3.20487
\(795\) 0 0
\(796\) 30674.9 1.36588
\(797\) −7288.52 −0.323930 −0.161965 0.986796i \(-0.551783\pi\)
−0.161965 + 0.986796i \(0.551783\pi\)
\(798\) 0 0
\(799\) 15175.3 0.671921
\(800\) 71079.9 3.14132
\(801\) 0 0
\(802\) 35861.7 1.57896
\(803\) 2112.15 0.0928221
\(804\) 0 0
\(805\) −9226.97 −0.403985
\(806\) 0 0
\(807\) 0 0
\(808\) −67666.1 −2.94614
\(809\) 33799.0 1.46886 0.734430 0.678684i \(-0.237449\pi\)
0.734430 + 0.678684i \(0.237449\pi\)
\(810\) 0 0
\(811\) −29055.3 −1.25804 −0.629019 0.777390i \(-0.716543\pi\)
−0.629019 + 0.777390i \(0.716543\pi\)
\(812\) 51406.2 2.22168
\(813\) 0 0
\(814\) 23128.7 0.995898
\(815\) 22245.5 0.956106
\(816\) 0 0
\(817\) −38563.3 −1.65136
\(818\) 69287.3 2.96158
\(819\) 0 0
\(820\) 18331.2 0.780676
\(821\) 8921.98 0.379268 0.189634 0.981855i \(-0.439270\pi\)
0.189634 + 0.981855i \(0.439270\pi\)
\(822\) 0 0
\(823\) −9148.48 −0.387480 −0.193740 0.981053i \(-0.562062\pi\)
−0.193740 + 0.981053i \(0.562062\pi\)
\(824\) 87685.1 3.70711
\(825\) 0 0
\(826\) −13616.6 −0.573588
\(827\) 20206.0 0.849615 0.424807 0.905284i \(-0.360342\pi\)
0.424807 + 0.905284i \(0.360342\pi\)
\(828\) 0 0
\(829\) −10832.4 −0.453830 −0.226915 0.973915i \(-0.572864\pi\)
−0.226915 + 0.973915i \(0.572864\pi\)
\(830\) 27240.2 1.13918
\(831\) 0 0
\(832\) 0 0
\(833\) 2565.09 0.106693
\(834\) 0 0
\(835\) −12087.4 −0.500961
\(836\) 132749. 5.49190
\(837\) 0 0
\(838\) 43880.1 1.80884
\(839\) −7047.44 −0.289994 −0.144997 0.989432i \(-0.546317\pi\)
−0.144997 + 0.989432i \(0.546317\pi\)
\(840\) 0 0
\(841\) −11685.4 −0.479127
\(842\) 1572.60 0.0643650
\(843\) 0 0
\(844\) 118981. 4.85250
\(845\) 0 0
\(846\) 0 0
\(847\) 21350.5 0.866132
\(848\) 100817. 4.08262
\(849\) 0 0
\(850\) 18352.0 0.740551
\(851\) 6420.28 0.258618
\(852\) 0 0
\(853\) −24085.5 −0.966790 −0.483395 0.875402i \(-0.660596\pi\)
−0.483395 + 0.875402i \(0.660596\pi\)
\(854\) −24634.5 −0.987089
\(855\) 0 0
\(856\) 94749.9 3.78328
\(857\) 4161.31 0.165866 0.0829332 0.996555i \(-0.473571\pi\)
0.0829332 + 0.996555i \(0.473571\pi\)
\(858\) 0 0
\(859\) 27387.5 1.08784 0.543918 0.839139i \(-0.316940\pi\)
0.543918 + 0.839139i \(0.316940\pi\)
\(860\) −43660.1 −1.73116
\(861\) 0 0
\(862\) −69099.1 −2.73031
\(863\) 19420.4 0.766022 0.383011 0.923744i \(-0.374887\pi\)
0.383011 + 0.923744i \(0.374887\pi\)
\(864\) 0 0
\(865\) 21751.1 0.854984
\(866\) 72992.8 2.86420
\(867\) 0 0
\(868\) 51599.5 2.01774
\(869\) 26335.1 1.02803
\(870\) 0 0
\(871\) 0 0
\(872\) 19214.5 0.746198
\(873\) 0 0
\(874\) 49952.6 1.93326
\(875\) −26258.8 −1.01453
\(876\) 0 0
\(877\) 9908.85 0.381526 0.190763 0.981636i \(-0.438904\pi\)
0.190763 + 0.981636i \(0.438904\pi\)
\(878\) −25599.3 −0.983981
\(879\) 0 0
\(880\) 77851.4 2.98224
\(881\) −20323.3 −0.777197 −0.388598 0.921407i \(-0.627041\pi\)
−0.388598 + 0.921407i \(0.627041\pi\)
\(882\) 0 0
\(883\) 12443.3 0.474238 0.237119 0.971481i \(-0.423797\pi\)
0.237119 + 0.971481i \(0.423797\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −16269.0 −0.616895
\(887\) −42313.5 −1.60174 −0.800872 0.598835i \(-0.795631\pi\)
−0.800872 + 0.598835i \(0.795631\pi\)
\(888\) 0 0
\(889\) −365.301 −0.0137816
\(890\) −37968.0 −1.42999
\(891\) 0 0
\(892\) −28303.7 −1.06242
\(893\) 48574.3 1.82024
\(894\) 0 0
\(895\) −4682.97 −0.174899
\(896\) −133452. −4.97580
\(897\) 0 0
\(898\) −47871.3 −1.77894
\(899\) 12751.3 0.473059
\(900\) 0 0
\(901\) 14515.1 0.536702
\(902\) −36132.3 −1.33379
\(903\) 0 0
\(904\) −82571.0 −3.03791
\(905\) −11527.0 −0.423393
\(906\) 0 0
\(907\) −11002.3 −0.402786 −0.201393 0.979511i \(-0.564547\pi\)
−0.201393 + 0.979511i \(0.564547\pi\)
\(908\) −25540.8 −0.933483
\(909\) 0 0
\(910\) 0 0
\(911\) 40803.4 1.48395 0.741974 0.670428i \(-0.233890\pi\)
0.741974 + 0.670428i \(0.233890\pi\)
\(912\) 0 0
\(913\) −39608.8 −1.43577
\(914\) 28691.8 1.03834
\(915\) 0 0
\(916\) −4904.80 −0.176920
\(917\) 25999.1 0.936276
\(918\) 0 0
\(919\) 2642.69 0.0948579 0.0474290 0.998875i \(-0.484897\pi\)
0.0474290 + 0.998875i \(0.484897\pi\)
\(920\) 36445.3 1.30605
\(921\) 0 0
\(922\) −16187.7 −0.578214
\(923\) 0 0
\(924\) 0 0
\(925\) 7550.00 0.268370
\(926\) 2190.19 0.0777258
\(927\) 0 0
\(928\) −91012.3 −3.21942
\(929\) 12687.7 0.448083 0.224042 0.974580i \(-0.428075\pi\)
0.224042 + 0.974580i \(0.428075\pi\)
\(930\) 0 0
\(931\) 8210.52 0.289032
\(932\) 57.6796 0.00202721
\(933\) 0 0
\(934\) 91826.7 3.21698
\(935\) 11208.7 0.392046
\(936\) 0 0
\(937\) −23066.2 −0.804205 −0.402103 0.915595i \(-0.631721\pi\)
−0.402103 + 0.915595i \(0.631721\pi\)
\(938\) −109195. −3.80099
\(939\) 0 0
\(940\) 54994.1 1.90820
\(941\) 12660.7 0.438605 0.219303 0.975657i \(-0.429622\pi\)
0.219303 + 0.975657i \(0.429622\pi\)
\(942\) 0 0
\(943\) −10029.9 −0.346362
\(944\) 31891.8 1.09957
\(945\) 0 0
\(946\) 86057.5 2.95769
\(947\) 21059.1 0.722629 0.361315 0.932444i \(-0.382328\pi\)
0.361315 + 0.932444i \(0.382328\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 58742.3 2.00616
\(951\) 0 0
\(952\) −61277.3 −2.08614
\(953\) −35646.2 −1.21164 −0.605820 0.795602i \(-0.707155\pi\)
−0.605820 + 0.795602i \(0.707155\pi\)
\(954\) 0 0
\(955\) −13400.7 −0.454070
\(956\) −89345.8 −3.02265
\(957\) 0 0
\(958\) 49169.0 1.65822
\(959\) −31367.9 −1.05623
\(960\) 0 0
\(961\) −16991.7 −0.570365
\(962\) 0 0
\(963\) 0 0
\(964\) −14805.0 −0.494643
\(965\) 6180.22 0.206164
\(966\) 0 0
\(967\) −53452.1 −1.77756 −0.888782 0.458329i \(-0.848448\pi\)
−0.888782 + 0.458329i \(0.848448\pi\)
\(968\) −84331.7 −2.80013
\(969\) 0 0
\(970\) 7692.95 0.254645
\(971\) −6502.60 −0.214911 −0.107455 0.994210i \(-0.534270\pi\)
−0.107455 + 0.994210i \(0.534270\pi\)
\(972\) 0 0
\(973\) −62779.2 −2.06846
\(974\) 2592.99 0.0853027
\(975\) 0 0
\(976\) 57696.9 1.89225
\(977\) −4351.84 −0.142505 −0.0712527 0.997458i \(-0.522700\pi\)
−0.0712527 + 0.997458i \(0.522700\pi\)
\(978\) 0 0
\(979\) 55207.5 1.80229
\(980\) 9295.67 0.302999
\(981\) 0 0
\(982\) −103956. −3.37817
\(983\) 48698.2 1.58009 0.790047 0.613046i \(-0.210056\pi\)
0.790047 + 0.613046i \(0.210056\pi\)
\(984\) 0 0
\(985\) −26259.8 −0.849448
\(986\) −23498.3 −0.758964
\(987\) 0 0
\(988\) 0 0
\(989\) 23888.6 0.768063
\(990\) 0 0
\(991\) 46630.0 1.49470 0.747351 0.664429i \(-0.231325\pi\)
0.747351 + 0.664429i \(0.231325\pi\)
\(992\) −91354.3 −2.92389
\(993\) 0 0
\(994\) 11948.2 0.381261
\(995\) 8290.35 0.264143
\(996\) 0 0
\(997\) 37175.5 1.18090 0.590452 0.807073i \(-0.298950\pi\)
0.590452 + 0.807073i \(0.298950\pi\)
\(998\) 31213.3 0.990021
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1521.4.a.bf.1.1 9
3.2 odd 2 507.4.a.p.1.9 yes 9
13.12 even 2 1521.4.a.bi.1.9 9
39.5 even 4 507.4.b.k.337.1 18
39.8 even 4 507.4.b.k.337.18 18
39.38 odd 2 507.4.a.o.1.1 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.1 9 39.38 odd 2
507.4.a.p.1.9 yes 9 3.2 odd 2
507.4.b.k.337.1 18 39.5 even 4
507.4.b.k.337.18 18 39.8 even 4
1521.4.a.bf.1.1 9 1.1 even 1 trivial
1521.4.a.bi.1.9 9 13.12 even 2