Properties

Label 1521.4.a.bi.1.7
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
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: \(1\)
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 1.7
Root \(-4.83218\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.03025 q^{2} +8.24289 q^{4} +8.08864 q^{5} -5.95078 q^{7} +0.978887 q^{8} +O(q^{10})\) \(q+4.03025 q^{2} +8.24289 q^{4} +8.08864 q^{5} -5.95078 q^{7} +0.978887 q^{8} +32.5992 q^{10} +17.2359 q^{11} -23.9831 q^{14} -61.9979 q^{16} -92.9299 q^{17} -13.3832 q^{19} +66.6738 q^{20} +69.4648 q^{22} -219.710 q^{23} -59.5738 q^{25} -49.0516 q^{28} +199.485 q^{29} -307.777 q^{31} -257.698 q^{32} -374.531 q^{34} -48.1337 q^{35} +333.777 q^{37} -53.9376 q^{38} +7.91787 q^{40} -200.689 q^{41} +116.806 q^{43} +142.073 q^{44} -885.487 q^{46} +338.610 q^{47} -307.588 q^{49} -240.097 q^{50} +26.6215 q^{53} +139.415 q^{55} -5.82514 q^{56} +803.976 q^{58} -280.058 q^{59} -207.084 q^{61} -1240.42 q^{62} -542.603 q^{64} +285.981 q^{67} -766.011 q^{68} -193.991 q^{70} -317.673 q^{71} +63.0668 q^{73} +1345.20 q^{74} -110.316 q^{76} -102.567 q^{77} -623.835 q^{79} -501.479 q^{80} -808.824 q^{82} +659.874 q^{83} -751.677 q^{85} +470.755 q^{86} +16.8720 q^{88} +1273.19 q^{89} -1811.05 q^{92} +1364.68 q^{94} -108.252 q^{95} -603.746 q^{97} -1239.66 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.03025 1.42491 0.712454 0.701719i \(-0.247584\pi\)
0.712454 + 0.701719i \(0.247584\pi\)
\(3\) 0 0
\(4\) 8.24289 1.03036
\(5\) 8.08864 0.723470 0.361735 0.932281i \(-0.382184\pi\)
0.361735 + 0.932281i \(0.382184\pi\)
\(6\) 0 0
\(7\) −5.95078 −0.321312 −0.160656 0.987010i \(-0.551361\pi\)
−0.160656 + 0.987010i \(0.551361\pi\)
\(8\) 0.978887 0.0432611
\(9\) 0 0
\(10\) 32.5992 1.03088
\(11\) 17.2359 0.472437 0.236219 0.971700i \(-0.424092\pi\)
0.236219 + 0.971700i \(0.424092\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −23.9831 −0.457840
\(15\) 0 0
\(16\) −61.9979 −0.968718
\(17\) −92.9299 −1.32581 −0.662907 0.748702i \(-0.730677\pi\)
−0.662907 + 0.748702i \(0.730677\pi\)
\(18\) 0 0
\(19\) −13.3832 −0.161596 −0.0807979 0.996731i \(-0.525747\pi\)
−0.0807979 + 0.996731i \(0.525747\pi\)
\(20\) 66.6738 0.745435
\(21\) 0 0
\(22\) 69.4648 0.673179
\(23\) −219.710 −1.99186 −0.995930 0.0901293i \(-0.971272\pi\)
−0.995930 + 0.0901293i \(0.971272\pi\)
\(24\) 0 0
\(25\) −59.5738 −0.476591
\(26\) 0 0
\(27\) 0 0
\(28\) −49.0516 −0.331067
\(29\) 199.485 1.27736 0.638681 0.769471i \(-0.279480\pi\)
0.638681 + 0.769471i \(0.279480\pi\)
\(30\) 0 0
\(31\) −307.777 −1.78317 −0.891586 0.452851i \(-0.850407\pi\)
−0.891586 + 0.452851i \(0.850407\pi\)
\(32\) −257.698 −1.42359
\(33\) 0 0
\(34\) −374.531 −1.88916
\(35\) −48.1337 −0.232460
\(36\) 0 0
\(37\) 333.777 1.48304 0.741521 0.670930i \(-0.234105\pi\)
0.741521 + 0.670930i \(0.234105\pi\)
\(38\) −53.9376 −0.230259
\(39\) 0 0
\(40\) 7.91787 0.0312981
\(41\) −200.689 −0.764446 −0.382223 0.924070i \(-0.624841\pi\)
−0.382223 + 0.924070i \(0.624841\pi\)
\(42\) 0 0
\(43\) 116.806 0.414248 0.207124 0.978315i \(-0.433590\pi\)
0.207124 + 0.978315i \(0.433590\pi\)
\(44\) 142.073 0.486781
\(45\) 0 0
\(46\) −885.487 −2.83822
\(47\) 338.610 1.05088 0.525440 0.850831i \(-0.323901\pi\)
0.525440 + 0.850831i \(0.323901\pi\)
\(48\) 0 0
\(49\) −307.588 −0.896759
\(50\) −240.097 −0.679097
\(51\) 0 0
\(52\) 0 0
\(53\) 26.6215 0.0689951 0.0344975 0.999405i \(-0.489017\pi\)
0.0344975 + 0.999405i \(0.489017\pi\)
\(54\) 0 0
\(55\) 139.415 0.341794
\(56\) −5.82514 −0.0139003
\(57\) 0 0
\(58\) 803.976 1.82012
\(59\) −280.058 −0.617973 −0.308987 0.951066i \(-0.599990\pi\)
−0.308987 + 0.951066i \(0.599990\pi\)
\(60\) 0 0
\(61\) −207.084 −0.434663 −0.217332 0.976098i \(-0.569735\pi\)
−0.217332 + 0.976098i \(0.569735\pi\)
\(62\) −1240.42 −2.54086
\(63\) 0 0
\(64\) −542.603 −1.05977
\(65\) 0 0
\(66\) 0 0
\(67\) 285.981 0.521465 0.260732 0.965411i \(-0.416036\pi\)
0.260732 + 0.965411i \(0.416036\pi\)
\(68\) −766.011 −1.36607
\(69\) 0 0
\(70\) −193.991 −0.331233
\(71\) −317.673 −0.530998 −0.265499 0.964111i \(-0.585537\pi\)
−0.265499 + 0.964111i \(0.585537\pi\)
\(72\) 0 0
\(73\) 63.0668 0.101115 0.0505576 0.998721i \(-0.483900\pi\)
0.0505576 + 0.998721i \(0.483900\pi\)
\(74\) 1345.20 2.11320
\(75\) 0 0
\(76\) −110.316 −0.166502
\(77\) −102.567 −0.151800
\(78\) 0 0
\(79\) −623.835 −0.888443 −0.444221 0.895917i \(-0.646520\pi\)
−0.444221 + 0.895917i \(0.646520\pi\)
\(80\) −501.479 −0.700838
\(81\) 0 0
\(82\) −808.824 −1.08926
\(83\) 659.874 0.872658 0.436329 0.899787i \(-0.356278\pi\)
0.436329 + 0.899787i \(0.356278\pi\)
\(84\) 0 0
\(85\) −751.677 −0.959186
\(86\) 470.755 0.590265
\(87\) 0 0
\(88\) 16.8720 0.0204382
\(89\) 1273.19 1.51638 0.758191 0.652032i \(-0.226083\pi\)
0.758191 + 0.652032i \(0.226083\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1811.05 −2.05233
\(93\) 0 0
\(94\) 1364.68 1.49741
\(95\) −108.252 −0.116910
\(96\) 0 0
\(97\) −603.746 −0.631970 −0.315985 0.948764i \(-0.602335\pi\)
−0.315985 + 0.948764i \(0.602335\pi\)
\(98\) −1239.66 −1.27780
\(99\) 0 0
\(100\) −491.060 −0.491060
\(101\) 740.588 0.729616 0.364808 0.931083i \(-0.381135\pi\)
0.364808 + 0.931083i \(0.381135\pi\)
\(102\) 0 0
\(103\) −1888.57 −1.80666 −0.903332 0.428942i \(-0.858887\pi\)
−0.903332 + 0.428942i \(0.858887\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 107.291 0.0983116
\(107\) 919.463 0.830728 0.415364 0.909655i \(-0.363654\pi\)
0.415364 + 0.909655i \(0.363654\pi\)
\(108\) 0 0
\(109\) −1570.40 −1.37998 −0.689988 0.723821i \(-0.742384\pi\)
−0.689988 + 0.723821i \(0.742384\pi\)
\(110\) 561.876 0.487025
\(111\) 0 0
\(112\) 368.936 0.311260
\(113\) 1324.66 1.10277 0.551386 0.834250i \(-0.314099\pi\)
0.551386 + 0.834250i \(0.314099\pi\)
\(114\) 0 0
\(115\) −1777.16 −1.44105
\(116\) 1644.34 1.31614
\(117\) 0 0
\(118\) −1128.70 −0.880555
\(119\) 553.006 0.426000
\(120\) 0 0
\(121\) −1033.92 −0.776803
\(122\) −834.601 −0.619354
\(123\) 0 0
\(124\) −2536.97 −1.83731
\(125\) −1492.95 −1.06827
\(126\) 0 0
\(127\) −2350.39 −1.64223 −0.821115 0.570763i \(-0.806648\pi\)
−0.821115 + 0.570763i \(0.806648\pi\)
\(128\) −125.240 −0.0864824
\(129\) 0 0
\(130\) 0 0
\(131\) −2308.69 −1.53978 −0.769890 0.638177i \(-0.779689\pi\)
−0.769890 + 0.638177i \(0.779689\pi\)
\(132\) 0 0
\(133\) 79.6405 0.0519226
\(134\) 1152.57 0.743039
\(135\) 0 0
\(136\) −90.9680 −0.0573562
\(137\) −374.912 −0.233802 −0.116901 0.993144i \(-0.537296\pi\)
−0.116901 + 0.993144i \(0.537296\pi\)
\(138\) 0 0
\(139\) 487.711 0.297605 0.148802 0.988867i \(-0.452458\pi\)
0.148802 + 0.988867i \(0.452458\pi\)
\(140\) −396.761 −0.239517
\(141\) 0 0
\(142\) −1280.30 −0.756623
\(143\) 0 0
\(144\) 0 0
\(145\) 1613.57 0.924134
\(146\) 254.175 0.144080
\(147\) 0 0
\(148\) 2751.28 1.52807
\(149\) 1055.91 0.580558 0.290279 0.956942i \(-0.406252\pi\)
0.290279 + 0.956942i \(0.406252\pi\)
\(150\) 0 0
\(151\) 888.560 0.478874 0.239437 0.970912i \(-0.423037\pi\)
0.239437 + 0.970912i \(0.423037\pi\)
\(152\) −13.1007 −0.00699081
\(153\) 0 0
\(154\) −413.370 −0.216301
\(155\) −2489.50 −1.29007
\(156\) 0 0
\(157\) −3648.56 −1.85469 −0.927346 0.374206i \(-0.877915\pi\)
−0.927346 + 0.374206i \(0.877915\pi\)
\(158\) −2514.21 −1.26595
\(159\) 0 0
\(160\) −2084.43 −1.02993
\(161\) 1307.45 0.640008
\(162\) 0 0
\(163\) −1386.49 −0.666249 −0.333125 0.942883i \(-0.608103\pi\)
−0.333125 + 0.942883i \(0.608103\pi\)
\(164\) −1654.25 −0.787655
\(165\) 0 0
\(166\) 2659.46 1.24346
\(167\) −3376.19 −1.56442 −0.782209 0.623017i \(-0.785907\pi\)
−0.782209 + 0.623017i \(0.785907\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −3029.44 −1.36675
\(171\) 0 0
\(172\) 962.815 0.426825
\(173\) −341.817 −0.150219 −0.0751095 0.997175i \(-0.523931\pi\)
−0.0751095 + 0.997175i \(0.523931\pi\)
\(174\) 0 0
\(175\) 354.511 0.153134
\(176\) −1068.59 −0.457658
\(177\) 0 0
\(178\) 5131.28 2.16070
\(179\) 2885.55 1.20489 0.602446 0.798159i \(-0.294193\pi\)
0.602446 + 0.798159i \(0.294193\pi\)
\(180\) 0 0
\(181\) 3795.62 1.55871 0.779354 0.626584i \(-0.215547\pi\)
0.779354 + 0.626584i \(0.215547\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −215.072 −0.0861701
\(185\) 2699.80 1.07294
\(186\) 0 0
\(187\) −1601.73 −0.626363
\(188\) 2791.12 1.08278
\(189\) 0 0
\(190\) −436.282 −0.166586
\(191\) 2805.90 1.06297 0.531486 0.847067i \(-0.321634\pi\)
0.531486 + 0.847067i \(0.321634\pi\)
\(192\) 0 0
\(193\) 2485.64 0.927048 0.463524 0.886084i \(-0.346585\pi\)
0.463524 + 0.886084i \(0.346585\pi\)
\(194\) −2433.24 −0.900499
\(195\) 0 0
\(196\) −2535.41 −0.923985
\(197\) 2750.70 0.994818 0.497409 0.867516i \(-0.334285\pi\)
0.497409 + 0.867516i \(0.334285\pi\)
\(198\) 0 0
\(199\) 3998.53 1.42436 0.712182 0.701994i \(-0.247707\pi\)
0.712182 + 0.701994i \(0.247707\pi\)
\(200\) −58.3161 −0.0206178
\(201\) 0 0
\(202\) 2984.75 1.03964
\(203\) −1187.09 −0.410432
\(204\) 0 0
\(205\) −1623.30 −0.553054
\(206\) −7611.41 −2.57433
\(207\) 0 0
\(208\) 0 0
\(209\) −230.671 −0.0763438
\(210\) 0 0
\(211\) −1375.80 −0.448882 −0.224441 0.974488i \(-0.572056\pi\)
−0.224441 + 0.974488i \(0.572056\pi\)
\(212\) 219.438 0.0710898
\(213\) 0 0
\(214\) 3705.66 1.18371
\(215\) 944.799 0.299696
\(216\) 0 0
\(217\) 1831.51 0.572955
\(218\) −6329.12 −1.96634
\(219\) 0 0
\(220\) 1149.18 0.352171
\(221\) 0 0
\(222\) 0 0
\(223\) −1694.37 −0.508805 −0.254403 0.967098i \(-0.581879\pi\)
−0.254403 + 0.967098i \(0.581879\pi\)
\(224\) 1533.50 0.457418
\(225\) 0 0
\(226\) 5338.70 1.57135
\(227\) −405.400 −0.118534 −0.0592672 0.998242i \(-0.518876\pi\)
−0.0592672 + 0.998242i \(0.518876\pi\)
\(228\) 0 0
\(229\) −2359.54 −0.680884 −0.340442 0.940265i \(-0.610577\pi\)
−0.340442 + 0.940265i \(0.610577\pi\)
\(230\) −7162.39 −2.05337
\(231\) 0 0
\(232\) 195.274 0.0552601
\(233\) −938.507 −0.263878 −0.131939 0.991258i \(-0.542120\pi\)
−0.131939 + 0.991258i \(0.542120\pi\)
\(234\) 0 0
\(235\) 2738.90 0.760280
\(236\) −2308.49 −0.636736
\(237\) 0 0
\(238\) 2228.75 0.607010
\(239\) 4664.25 1.26237 0.631183 0.775634i \(-0.282570\pi\)
0.631183 + 0.775634i \(0.282570\pi\)
\(240\) 0 0
\(241\) 3774.94 1.00898 0.504492 0.863416i \(-0.331680\pi\)
0.504492 + 0.863416i \(0.331680\pi\)
\(242\) −4166.97 −1.10687
\(243\) 0 0
\(244\) −1706.97 −0.447860
\(245\) −2487.97 −0.648778
\(246\) 0 0
\(247\) 0 0
\(248\) −301.279 −0.0771421
\(249\) 0 0
\(250\) −6016.96 −1.52219
\(251\) 960.695 0.241588 0.120794 0.992678i \(-0.461456\pi\)
0.120794 + 0.992678i \(0.461456\pi\)
\(252\) 0 0
\(253\) −3786.90 −0.941029
\(254\) −9472.64 −2.34003
\(255\) 0 0
\(256\) 3836.08 0.936542
\(257\) 16.7302 0.00406070 0.00203035 0.999998i \(-0.499354\pi\)
0.00203035 + 0.999998i \(0.499354\pi\)
\(258\) 0 0
\(259\) −1986.23 −0.476519
\(260\) 0 0
\(261\) 0 0
\(262\) −9304.58 −2.19404
\(263\) 400.736 0.0939560 0.0469780 0.998896i \(-0.485041\pi\)
0.0469780 + 0.998896i \(0.485041\pi\)
\(264\) 0 0
\(265\) 215.332 0.0499159
\(266\) 320.971 0.0739849
\(267\) 0 0
\(268\) 2357.31 0.537297
\(269\) −2236.97 −0.507027 −0.253514 0.967332i \(-0.581586\pi\)
−0.253514 + 0.967332i \(0.581586\pi\)
\(270\) 0 0
\(271\) 4018.89 0.900850 0.450425 0.892814i \(-0.351273\pi\)
0.450425 + 0.892814i \(0.351273\pi\)
\(272\) 5761.46 1.28434
\(273\) 0 0
\(274\) −1510.99 −0.333147
\(275\) −1026.81 −0.225159
\(276\) 0 0
\(277\) 6792.95 1.47346 0.736731 0.676186i \(-0.236368\pi\)
0.736731 + 0.676186i \(0.236368\pi\)
\(278\) 1965.59 0.424059
\(279\) 0 0
\(280\) −47.1175 −0.0100565
\(281\) −7286.80 −1.54695 −0.773477 0.633824i \(-0.781484\pi\)
−0.773477 + 0.633824i \(0.781484\pi\)
\(282\) 0 0
\(283\) −2429.77 −0.510369 −0.255185 0.966892i \(-0.582136\pi\)
−0.255185 + 0.966892i \(0.582136\pi\)
\(284\) −2618.54 −0.547120
\(285\) 0 0
\(286\) 0 0
\(287\) 1194.25 0.245626
\(288\) 0 0
\(289\) 3722.97 0.757780
\(290\) 6503.07 1.31681
\(291\) 0 0
\(292\) 519.853 0.104185
\(293\) 4211.25 0.839672 0.419836 0.907600i \(-0.362088\pi\)
0.419836 + 0.907600i \(0.362088\pi\)
\(294\) 0 0
\(295\) −2265.29 −0.447085
\(296\) 326.730 0.0641580
\(297\) 0 0
\(298\) 4255.56 0.827242
\(299\) 0 0
\(300\) 0 0
\(301\) −695.084 −0.133103
\(302\) 3581.12 0.682351
\(303\) 0 0
\(304\) 829.731 0.156541
\(305\) −1675.03 −0.314466
\(306\) 0 0
\(307\) −8212.27 −1.52671 −0.763353 0.645982i \(-0.776448\pi\)
−0.763353 + 0.645982i \(0.776448\pi\)
\(308\) −845.447 −0.156408
\(309\) 0 0
\(310\) −10033.3 −1.83823
\(311\) 4238.55 0.772818 0.386409 0.922328i \(-0.373715\pi\)
0.386409 + 0.922328i \(0.373715\pi\)
\(312\) 0 0
\(313\) 3807.82 0.687638 0.343819 0.939036i \(-0.388279\pi\)
0.343819 + 0.939036i \(0.388279\pi\)
\(314\) −14704.6 −2.64276
\(315\) 0 0
\(316\) −5142.20 −0.915416
\(317\) −10497.7 −1.85996 −0.929981 0.367608i \(-0.880177\pi\)
−0.929981 + 0.367608i \(0.880177\pi\)
\(318\) 0 0
\(319\) 3438.31 0.603474
\(320\) −4388.92 −0.766713
\(321\) 0 0
\(322\) 5269.34 0.911953
\(323\) 1243.70 0.214246
\(324\) 0 0
\(325\) 0 0
\(326\) −5587.91 −0.949343
\(327\) 0 0
\(328\) −196.452 −0.0330708
\(329\) −2014.99 −0.337660
\(330\) 0 0
\(331\) −11905.2 −1.97695 −0.988474 0.151389i \(-0.951625\pi\)
−0.988474 + 0.151389i \(0.951625\pi\)
\(332\) 5439.27 0.899152
\(333\) 0 0
\(334\) −13606.9 −2.22915
\(335\) 2313.20 0.377264
\(336\) 0 0
\(337\) 8981.18 1.45174 0.725869 0.687833i \(-0.241438\pi\)
0.725869 + 0.687833i \(0.241438\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −6195.99 −0.988308
\(341\) −5304.80 −0.842437
\(342\) 0 0
\(343\) 3871.51 0.609451
\(344\) 114.339 0.0179208
\(345\) 0 0
\(346\) −1377.61 −0.214048
\(347\) 5712.87 0.883813 0.441906 0.897061i \(-0.354302\pi\)
0.441906 + 0.897061i \(0.354302\pi\)
\(348\) 0 0
\(349\) −9507.28 −1.45820 −0.729102 0.684406i \(-0.760062\pi\)
−0.729102 + 0.684406i \(0.760062\pi\)
\(350\) 1428.77 0.218202
\(351\) 0 0
\(352\) −4441.65 −0.672559
\(353\) −3972.96 −0.599035 −0.299518 0.954091i \(-0.596826\pi\)
−0.299518 + 0.954091i \(0.596826\pi\)
\(354\) 0 0
\(355\) −2569.55 −0.384162
\(356\) 10494.8 1.56242
\(357\) 0 0
\(358\) 11629.5 1.71686
\(359\) −109.866 −0.0161519 −0.00807594 0.999967i \(-0.502571\pi\)
−0.00807594 + 0.999967i \(0.502571\pi\)
\(360\) 0 0
\(361\) −6679.89 −0.973887
\(362\) 15297.3 2.22101
\(363\) 0 0
\(364\) 0 0
\(365\) 510.125 0.0731539
\(366\) 0 0
\(367\) 7020.02 0.998480 0.499240 0.866464i \(-0.333613\pi\)
0.499240 + 0.866464i \(0.333613\pi\)
\(368\) 13621.6 1.92955
\(369\) 0 0
\(370\) 10880.9 1.52884
\(371\) −158.418 −0.0221689
\(372\) 0 0
\(373\) 2227.18 0.309166 0.154583 0.987980i \(-0.450597\pi\)
0.154583 + 0.987980i \(0.450597\pi\)
\(374\) −6455.36 −0.892510
\(375\) 0 0
\(376\) 331.461 0.0454622
\(377\) 0 0
\(378\) 0 0
\(379\) 11004.6 1.49147 0.745734 0.666244i \(-0.232099\pi\)
0.745734 + 0.666244i \(0.232099\pi\)
\(380\) −892.309 −0.120459
\(381\) 0 0
\(382\) 11308.5 1.51464
\(383\) −4436.85 −0.591938 −0.295969 0.955198i \(-0.595643\pi\)
−0.295969 + 0.955198i \(0.595643\pi\)
\(384\) 0 0
\(385\) −829.627 −0.109823
\(386\) 10017.7 1.32096
\(387\) 0 0
\(388\) −4976.61 −0.651157
\(389\) 2561.58 0.333874 0.166937 0.985968i \(-0.446612\pi\)
0.166937 + 0.985968i \(0.446612\pi\)
\(390\) 0 0
\(391\) 20417.7 2.64083
\(392\) −301.094 −0.0387948
\(393\) 0 0
\(394\) 11086.0 1.41752
\(395\) −5045.98 −0.642762
\(396\) 0 0
\(397\) −2206.90 −0.278996 −0.139498 0.990222i \(-0.544549\pi\)
−0.139498 + 0.990222i \(0.544549\pi\)
\(398\) 16115.1 2.02959
\(399\) 0 0
\(400\) 3693.45 0.461682
\(401\) −8071.02 −1.00511 −0.502553 0.864546i \(-0.667606\pi\)
−0.502553 + 0.864546i \(0.667606\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 6104.58 0.751768
\(405\) 0 0
\(406\) −4784.28 −0.584827
\(407\) 5752.93 0.700644
\(408\) 0 0
\(409\) −6298.36 −0.761452 −0.380726 0.924688i \(-0.624326\pi\)
−0.380726 + 0.924688i \(0.624326\pi\)
\(410\) −6542.29 −0.788051
\(411\) 0 0
\(412\) −15567.3 −1.86152
\(413\) 1666.56 0.198562
\(414\) 0 0
\(415\) 5337.49 0.631342
\(416\) 0 0
\(417\) 0 0
\(418\) −929.662 −0.108783
\(419\) 1432.01 0.166965 0.0834824 0.996509i \(-0.473396\pi\)
0.0834824 + 0.996509i \(0.473396\pi\)
\(420\) 0 0
\(421\) −9641.43 −1.11614 −0.558069 0.829794i \(-0.688458\pi\)
−0.558069 + 0.829794i \(0.688458\pi\)
\(422\) −5544.82 −0.639615
\(423\) 0 0
\(424\) 26.0594 0.00298480
\(425\) 5536.19 0.631870
\(426\) 0 0
\(427\) 1232.31 0.139662
\(428\) 7579.03 0.855949
\(429\) 0 0
\(430\) 3807.77 0.427040
\(431\) 3370.61 0.376698 0.188349 0.982102i \(-0.439686\pi\)
0.188349 + 0.982102i \(0.439686\pi\)
\(432\) 0 0
\(433\) 6248.91 0.693541 0.346770 0.937950i \(-0.387278\pi\)
0.346770 + 0.937950i \(0.387278\pi\)
\(434\) 7381.45 0.816407
\(435\) 0 0
\(436\) −12944.7 −1.42187
\(437\) 2940.43 0.321876
\(438\) 0 0
\(439\) 2866.16 0.311604 0.155802 0.987788i \(-0.450204\pi\)
0.155802 + 0.987788i \(0.450204\pi\)
\(440\) 136.471 0.0147864
\(441\) 0 0
\(442\) 0 0
\(443\) −10776.3 −1.15575 −0.577877 0.816124i \(-0.696119\pi\)
−0.577877 + 0.816124i \(0.696119\pi\)
\(444\) 0 0
\(445\) 10298.4 1.09706
\(446\) −6828.74 −0.725000
\(447\) 0 0
\(448\) 3228.91 0.340517
\(449\) 13834.9 1.45414 0.727069 0.686564i \(-0.240882\pi\)
0.727069 + 0.686564i \(0.240882\pi\)
\(450\) 0 0
\(451\) −3459.04 −0.361153
\(452\) 10919.0 1.13625
\(453\) 0 0
\(454\) −1633.86 −0.168901
\(455\) 0 0
\(456\) 0 0
\(457\) −17233.0 −1.76395 −0.881976 0.471295i \(-0.843787\pi\)
−0.881976 + 0.471295i \(0.843787\pi\)
\(458\) −9509.51 −0.970197
\(459\) 0 0
\(460\) −14648.9 −1.48480
\(461\) 11485.0 1.16033 0.580164 0.814500i \(-0.302988\pi\)
0.580164 + 0.814500i \(0.302988\pi\)
\(462\) 0 0
\(463\) −1613.61 −0.161967 −0.0809837 0.996715i \(-0.525806\pi\)
−0.0809837 + 0.996715i \(0.525806\pi\)
\(464\) −12367.7 −1.23740
\(465\) 0 0
\(466\) −3782.41 −0.376002
\(467\) 8149.20 0.807495 0.403747 0.914871i \(-0.367707\pi\)
0.403747 + 0.914871i \(0.367707\pi\)
\(468\) 0 0
\(469\) −1701.81 −0.167553
\(470\) 11038.4 1.08333
\(471\) 0 0
\(472\) −274.145 −0.0267342
\(473\) 2013.24 0.195706
\(474\) 0 0
\(475\) 797.289 0.0770150
\(476\) 4558.36 0.438933
\(477\) 0 0
\(478\) 18798.1 1.79875
\(479\) 1827.62 0.174334 0.0871670 0.996194i \(-0.472219\pi\)
0.0871670 + 0.996194i \(0.472219\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 15213.9 1.43771
\(483\) 0 0
\(484\) −8522.52 −0.800387
\(485\) −4883.49 −0.457212
\(486\) 0 0
\(487\) 7838.95 0.729398 0.364699 0.931125i \(-0.381172\pi\)
0.364699 + 0.931125i \(0.381172\pi\)
\(488\) −202.712 −0.0188040
\(489\) 0 0
\(490\) −10027.1 −0.924449
\(491\) −17196.1 −1.58055 −0.790273 0.612755i \(-0.790061\pi\)
−0.790273 + 0.612755i \(0.790061\pi\)
\(492\) 0 0
\(493\) −18538.2 −1.69354
\(494\) 0 0
\(495\) 0 0
\(496\) 19081.5 1.72739
\(497\) 1890.40 0.170616
\(498\) 0 0
\(499\) −5355.28 −0.480431 −0.240216 0.970720i \(-0.577218\pi\)
−0.240216 + 0.970720i \(0.577218\pi\)
\(500\) −12306.2 −1.10070
\(501\) 0 0
\(502\) 3871.84 0.344240
\(503\) −2979.80 −0.264141 −0.132071 0.991240i \(-0.542163\pi\)
−0.132071 + 0.991240i \(0.542163\pi\)
\(504\) 0 0
\(505\) 5990.35 0.527856
\(506\) −15262.1 −1.34088
\(507\) 0 0
\(508\) −19374.0 −1.69209
\(509\) 15353.7 1.33701 0.668507 0.743706i \(-0.266934\pi\)
0.668507 + 0.743706i \(0.266934\pi\)
\(510\) 0 0
\(511\) −375.297 −0.0324895
\(512\) 16462.3 1.42097
\(513\) 0 0
\(514\) 67.4267 0.00578611
\(515\) −15276.0 −1.30707
\(516\) 0 0
\(517\) 5836.24 0.496475
\(518\) −8005.00 −0.678995
\(519\) 0 0
\(520\) 0 0
\(521\) −433.801 −0.0364783 −0.0182391 0.999834i \(-0.505806\pi\)
−0.0182391 + 0.999834i \(0.505806\pi\)
\(522\) 0 0
\(523\) 18900.3 1.58021 0.790106 0.612970i \(-0.210026\pi\)
0.790106 + 0.612970i \(0.210026\pi\)
\(524\) −19030.3 −1.58653
\(525\) 0 0
\(526\) 1615.06 0.133879
\(527\) 28601.7 2.36415
\(528\) 0 0
\(529\) 36105.7 2.96751
\(530\) 867.839 0.0711255
\(531\) 0 0
\(532\) 656.468 0.0534990
\(533\) 0 0
\(534\) 0 0
\(535\) 7437.21 0.601007
\(536\) 279.943 0.0225592
\(537\) 0 0
\(538\) −9015.53 −0.722467
\(539\) −5301.55 −0.423662
\(540\) 0 0
\(541\) −13247.5 −1.05278 −0.526391 0.850243i \(-0.676455\pi\)
−0.526391 + 0.850243i \(0.676455\pi\)
\(542\) 16197.1 1.28363
\(543\) 0 0
\(544\) 23947.9 1.88742
\(545\) −12702.4 −0.998372
\(546\) 0 0
\(547\) −5543.52 −0.433316 −0.216658 0.976248i \(-0.569516\pi\)
−0.216658 + 0.976248i \(0.569516\pi\)
\(548\) −3090.36 −0.240901
\(549\) 0 0
\(550\) −4138.28 −0.320831
\(551\) −2669.76 −0.206416
\(552\) 0 0
\(553\) 3712.31 0.285467
\(554\) 27377.3 2.09955
\(555\) 0 0
\(556\) 4020.14 0.306640
\(557\) 6733.19 0.512198 0.256099 0.966651i \(-0.417563\pi\)
0.256099 + 0.966651i \(0.417563\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 2984.19 0.225188
\(561\) 0 0
\(562\) −29367.6 −2.20427
\(563\) −24282.4 −1.81773 −0.908863 0.417096i \(-0.863048\pi\)
−0.908863 + 0.417096i \(0.863048\pi\)
\(564\) 0 0
\(565\) 10714.7 0.797823
\(566\) −9792.55 −0.727229
\(567\) 0 0
\(568\) −310.966 −0.0229716
\(569\) 19876.2 1.46441 0.732207 0.681082i \(-0.238490\pi\)
0.732207 + 0.681082i \(0.238490\pi\)
\(570\) 0 0
\(571\) 225.014 0.0164913 0.00824567 0.999966i \(-0.497375\pi\)
0.00824567 + 0.999966i \(0.497375\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 4813.14 0.349994
\(575\) 13089.0 0.949302
\(576\) 0 0
\(577\) 12258.1 0.884424 0.442212 0.896910i \(-0.354194\pi\)
0.442212 + 0.896910i \(0.354194\pi\)
\(578\) 15004.5 1.07977
\(579\) 0 0
\(580\) 13300.4 0.952191
\(581\) −3926.77 −0.280395
\(582\) 0 0
\(583\) 458.844 0.0325958
\(584\) 61.7353 0.00437436
\(585\) 0 0
\(586\) 16972.4 1.19645
\(587\) 10282.3 0.722992 0.361496 0.932374i \(-0.382266\pi\)
0.361496 + 0.932374i \(0.382266\pi\)
\(588\) 0 0
\(589\) 4119.04 0.288153
\(590\) −9129.67 −0.637055
\(591\) 0 0
\(592\) −20693.5 −1.43665
\(593\) 12355.1 0.855586 0.427793 0.903877i \(-0.359291\pi\)
0.427793 + 0.903877i \(0.359291\pi\)
\(594\) 0 0
\(595\) 4473.07 0.308198
\(596\) 8703.71 0.598184
\(597\) 0 0
\(598\) 0 0
\(599\) −13035.2 −0.889157 −0.444578 0.895740i \(-0.646646\pi\)
−0.444578 + 0.895740i \(0.646646\pi\)
\(600\) 0 0
\(601\) −5153.36 −0.349767 −0.174884 0.984589i \(-0.555955\pi\)
−0.174884 + 0.984589i \(0.555955\pi\)
\(602\) −2801.36 −0.189659
\(603\) 0 0
\(604\) 7324.30 0.493413
\(605\) −8363.05 −0.561994
\(606\) 0 0
\(607\) −1406.55 −0.0940530 −0.0470265 0.998894i \(-0.514975\pi\)
−0.0470265 + 0.998894i \(0.514975\pi\)
\(608\) 3448.83 0.230047
\(609\) 0 0
\(610\) −6750.79 −0.448085
\(611\) 0 0
\(612\) 0 0
\(613\) 15984.3 1.05318 0.526590 0.850120i \(-0.323470\pi\)
0.526590 + 0.850120i \(0.323470\pi\)
\(614\) −33097.5 −2.17541
\(615\) 0 0
\(616\) −100.401 −0.00656703
\(617\) −10324.4 −0.673652 −0.336826 0.941567i \(-0.609353\pi\)
−0.336826 + 0.941567i \(0.609353\pi\)
\(618\) 0 0
\(619\) 7423.00 0.481996 0.240998 0.970526i \(-0.422525\pi\)
0.240998 + 0.970526i \(0.422525\pi\)
\(620\) −20520.6 −1.32924
\(621\) 0 0
\(622\) 17082.4 1.10119
\(623\) −7576.48 −0.487232
\(624\) 0 0
\(625\) −4629.23 −0.296271
\(626\) 15346.4 0.979820
\(627\) 0 0
\(628\) −30074.6 −1.91100
\(629\) −31017.8 −1.96624
\(630\) 0 0
\(631\) −1190.30 −0.0750954 −0.0375477 0.999295i \(-0.511955\pi\)
−0.0375477 + 0.999295i \(0.511955\pi\)
\(632\) −610.665 −0.0384350
\(633\) 0 0
\(634\) −42308.2 −2.65027
\(635\) −19011.5 −1.18810
\(636\) 0 0
\(637\) 0 0
\(638\) 13857.2 0.859894
\(639\) 0 0
\(640\) −1013.02 −0.0625674
\(641\) −705.064 −0.0434451 −0.0217226 0.999764i \(-0.506915\pi\)
−0.0217226 + 0.999764i \(0.506915\pi\)
\(642\) 0 0
\(643\) −14641.4 −0.897976 −0.448988 0.893538i \(-0.648215\pi\)
−0.448988 + 0.893538i \(0.648215\pi\)
\(644\) 10777.1 0.659439
\(645\) 0 0
\(646\) 5012.42 0.305280
\(647\) −1520.83 −0.0924111 −0.0462056 0.998932i \(-0.514713\pi\)
−0.0462056 + 0.998932i \(0.514713\pi\)
\(648\) 0 0
\(649\) −4827.04 −0.291954
\(650\) 0 0
\(651\) 0 0
\(652\) −11428.7 −0.686477
\(653\) 22054.4 1.32168 0.660838 0.750529i \(-0.270201\pi\)
0.660838 + 0.750529i \(0.270201\pi\)
\(654\) 0 0
\(655\) −18674.2 −1.11398
\(656\) 12442.3 0.740532
\(657\) 0 0
\(658\) −8120.92 −0.481134
\(659\) 12652.3 0.747895 0.373947 0.927450i \(-0.378004\pi\)
0.373947 + 0.927450i \(0.378004\pi\)
\(660\) 0 0
\(661\) 10893.2 0.640994 0.320497 0.947250i \(-0.396150\pi\)
0.320497 + 0.947250i \(0.396150\pi\)
\(662\) −47981.0 −2.81697
\(663\) 0 0
\(664\) 645.942 0.0377522
\(665\) 644.184 0.0375645
\(666\) 0 0
\(667\) −43829.0 −2.54433
\(668\) −27829.6 −1.61191
\(669\) 0 0
\(670\) 9322.76 0.537567
\(671\) −3569.28 −0.205351
\(672\) 0 0
\(673\) −9019.75 −0.516621 −0.258310 0.966062i \(-0.583166\pi\)
−0.258310 + 0.966062i \(0.583166\pi\)
\(674\) 36196.4 2.06859
\(675\) 0 0
\(676\) 0 0
\(677\) 24923.6 1.41491 0.707454 0.706760i \(-0.249844\pi\)
0.707454 + 0.706760i \(0.249844\pi\)
\(678\) 0 0
\(679\) 3592.76 0.203060
\(680\) −735.807 −0.0414955
\(681\) 0 0
\(682\) −21379.7 −1.20039
\(683\) −24634.2 −1.38009 −0.690044 0.723767i \(-0.742409\pi\)
−0.690044 + 0.723767i \(0.742409\pi\)
\(684\) 0 0
\(685\) −3032.53 −0.169149
\(686\) 15603.1 0.868411
\(687\) 0 0
\(688\) −7241.70 −0.401290
\(689\) 0 0
\(690\) 0 0
\(691\) 10340.2 0.569264 0.284632 0.958637i \(-0.408129\pi\)
0.284632 + 0.958637i \(0.408129\pi\)
\(692\) −2817.56 −0.154780
\(693\) 0 0
\(694\) 23024.3 1.25935
\(695\) 3944.92 0.215308
\(696\) 0 0
\(697\) 18650.0 1.01351
\(698\) −38316.7 −2.07780
\(699\) 0 0
\(700\) 2922.19 0.157784
\(701\) 20833.9 1.12252 0.561258 0.827641i \(-0.310317\pi\)
0.561258 + 0.827641i \(0.310317\pi\)
\(702\) 0 0
\(703\) −4467.00 −0.239653
\(704\) −9352.23 −0.500676
\(705\) 0 0
\(706\) −16012.0 −0.853569
\(707\) −4407.07 −0.234434
\(708\) 0 0
\(709\) −9189.56 −0.486772 −0.243386 0.969930i \(-0.578258\pi\)
−0.243386 + 0.969930i \(0.578258\pi\)
\(710\) −10355.9 −0.547395
\(711\) 0 0
\(712\) 1246.31 0.0656004
\(713\) 67621.8 3.55183
\(714\) 0 0
\(715\) 0 0
\(716\) 23785.2 1.24147
\(717\) 0 0
\(718\) −442.788 −0.0230149
\(719\) −7503.22 −0.389183 −0.194592 0.980884i \(-0.562338\pi\)
−0.194592 + 0.980884i \(0.562338\pi\)
\(720\) 0 0
\(721\) 11238.5 0.580503
\(722\) −26921.6 −1.38770
\(723\) 0 0
\(724\) 31286.8 1.60603
\(725\) −11884.1 −0.608779
\(726\) 0 0
\(727\) −20727.2 −1.05740 −0.528700 0.848809i \(-0.677320\pi\)
−0.528700 + 0.848809i \(0.677320\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2055.93 0.104237
\(731\) −10854.7 −0.549216
\(732\) 0 0
\(733\) −17361.3 −0.874836 −0.437418 0.899258i \(-0.644107\pi\)
−0.437418 + 0.899258i \(0.644107\pi\)
\(734\) 28292.4 1.42274
\(735\) 0 0
\(736\) 56618.9 2.83560
\(737\) 4929.13 0.246359
\(738\) 0 0
\(739\) −18093.8 −0.900663 −0.450331 0.892861i \(-0.648694\pi\)
−0.450331 + 0.892861i \(0.648694\pi\)
\(740\) 22254.1 1.10551
\(741\) 0 0
\(742\) −638.466 −0.0315887
\(743\) −14875.3 −0.734482 −0.367241 0.930126i \(-0.619698\pi\)
−0.367241 + 0.930126i \(0.619698\pi\)
\(744\) 0 0
\(745\) 8540.85 0.420017
\(746\) 8976.07 0.440533
\(747\) 0 0
\(748\) −13202.9 −0.645380
\(749\) −5471.52 −0.266923
\(750\) 0 0
\(751\) −17750.9 −0.862502 −0.431251 0.902232i \(-0.641928\pi\)
−0.431251 + 0.902232i \(0.641928\pi\)
\(752\) −20993.1 −1.01801
\(753\) 0 0
\(754\) 0 0
\(755\) 7187.25 0.346451
\(756\) 0 0
\(757\) −10220.3 −0.490702 −0.245351 0.969434i \(-0.578903\pi\)
−0.245351 + 0.969434i \(0.578903\pi\)
\(758\) 44351.1 2.12520
\(759\) 0 0
\(760\) −105.967 −0.00505765
\(761\) −14112.0 −0.672221 −0.336110 0.941823i \(-0.609112\pi\)
−0.336110 + 0.941823i \(0.609112\pi\)
\(762\) 0 0
\(763\) 9345.13 0.443403
\(764\) 23128.7 1.09525
\(765\) 0 0
\(766\) −17881.6 −0.843457
\(767\) 0 0
\(768\) 0 0
\(769\) 22736.7 1.06620 0.533100 0.846052i \(-0.321027\pi\)
0.533100 + 0.846052i \(0.321027\pi\)
\(770\) −3343.60 −0.156487
\(771\) 0 0
\(772\) 20488.8 0.955194
\(773\) −343.173 −0.0159678 −0.00798388 0.999968i \(-0.502541\pi\)
−0.00798388 + 0.999968i \(0.502541\pi\)
\(774\) 0 0
\(775\) 18335.4 0.849844
\(776\) −590.999 −0.0273397
\(777\) 0 0
\(778\) 10323.8 0.475740
\(779\) 2685.86 0.123531
\(780\) 0 0
\(781\) −5475.37 −0.250863
\(782\) 82288.3 3.76294
\(783\) 0 0
\(784\) 19069.8 0.868706
\(785\) −29511.9 −1.34181
\(786\) 0 0
\(787\) 19086.4 0.864496 0.432248 0.901755i \(-0.357721\pi\)
0.432248 + 0.901755i \(0.357721\pi\)
\(788\) 22673.7 1.02502
\(789\) 0 0
\(790\) −20336.6 −0.915876
\(791\) −7882.74 −0.354334
\(792\) 0 0
\(793\) 0 0
\(794\) −8894.36 −0.397543
\(795\) 0 0
\(796\) 32959.5 1.46761
\(797\) −12031.1 −0.534709 −0.267354 0.963598i \(-0.586149\pi\)
−0.267354 + 0.963598i \(0.586149\pi\)
\(798\) 0 0
\(799\) −31467.0 −1.39327
\(800\) 15352.1 0.678471
\(801\) 0 0
\(802\) −32528.2 −1.43218
\(803\) 1087.01 0.0477706
\(804\) 0 0
\(805\) 10575.5 0.463027
\(806\) 0 0
\(807\) 0 0
\(808\) 724.952 0.0315640
\(809\) −10175.4 −0.442210 −0.221105 0.975250i \(-0.570966\pi\)
−0.221105 + 0.975250i \(0.570966\pi\)
\(810\) 0 0
\(811\) −26754.1 −1.15840 −0.579201 0.815185i \(-0.696635\pi\)
−0.579201 + 0.815185i \(0.696635\pi\)
\(812\) −9785.08 −0.422893
\(813\) 0 0
\(814\) 23185.7 0.998353
\(815\) −11214.9 −0.482011
\(816\) 0 0
\(817\) −1563.23 −0.0669408
\(818\) −25383.9 −1.08500
\(819\) 0 0
\(820\) −13380.7 −0.569845
\(821\) −15777.5 −0.670693 −0.335347 0.942095i \(-0.608853\pi\)
−0.335347 + 0.942095i \(0.608853\pi\)
\(822\) 0 0
\(823\) −13863.9 −0.587197 −0.293599 0.955929i \(-0.594853\pi\)
−0.293599 + 0.955929i \(0.594853\pi\)
\(824\) −1848.70 −0.0781583
\(825\) 0 0
\(826\) 6716.66 0.282933
\(827\) −26835.0 −1.12835 −0.564175 0.825655i \(-0.690806\pi\)
−0.564175 + 0.825655i \(0.690806\pi\)
\(828\) 0 0
\(829\) 625.251 0.0261953 0.0130976 0.999914i \(-0.495831\pi\)
0.0130976 + 0.999914i \(0.495831\pi\)
\(830\) 21511.4 0.899604
\(831\) 0 0
\(832\) 0 0
\(833\) 28584.2 1.18893
\(834\) 0 0
\(835\) −27308.8 −1.13181
\(836\) −1901.40 −0.0786617
\(837\) 0 0
\(838\) 5771.35 0.237909
\(839\) −27307.4 −1.12367 −0.561833 0.827251i \(-0.689904\pi\)
−0.561833 + 0.827251i \(0.689904\pi\)
\(840\) 0 0
\(841\) 15405.4 0.631656
\(842\) −38857.3 −1.59039
\(843\) 0 0
\(844\) −11340.6 −0.462510
\(845\) 0 0
\(846\) 0 0
\(847\) 6152.66 0.249596
\(848\) −1650.48 −0.0668368
\(849\) 0 0
\(850\) 22312.2 0.900356
\(851\) −73334.2 −2.95401
\(852\) 0 0
\(853\) −44801.9 −1.79834 −0.899172 0.437595i \(-0.855830\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(854\) 4966.53 0.199006
\(855\) 0 0
\(856\) 900.051 0.0359382
\(857\) 25167.1 1.00314 0.501571 0.865116i \(-0.332756\pi\)
0.501571 + 0.865116i \(0.332756\pi\)
\(858\) 0 0
\(859\) 4059.10 0.161228 0.0806138 0.996745i \(-0.474312\pi\)
0.0806138 + 0.996745i \(0.474312\pi\)
\(860\) 7787.87 0.308795
\(861\) 0 0
\(862\) 13584.4 0.536759
\(863\) 818.924 0.0323019 0.0161509 0.999870i \(-0.494859\pi\)
0.0161509 + 0.999870i \(0.494859\pi\)
\(864\) 0 0
\(865\) −2764.84 −0.108679
\(866\) 25184.6 0.988231
\(867\) 0 0
\(868\) 15096.9 0.590350
\(869\) −10752.3 −0.419733
\(870\) 0 0
\(871\) 0 0
\(872\) −1537.25 −0.0596993
\(873\) 0 0
\(874\) 11850.7 0.458644
\(875\) 8884.23 0.343248
\(876\) 0 0
\(877\) −190.251 −0.00732533 −0.00366267 0.999993i \(-0.501166\pi\)
−0.00366267 + 0.999993i \(0.501166\pi\)
\(878\) 11551.3 0.444007
\(879\) 0 0
\(880\) −8643.43 −0.331102
\(881\) 19803.4 0.757315 0.378657 0.925537i \(-0.376386\pi\)
0.378657 + 0.925537i \(0.376386\pi\)
\(882\) 0 0
\(883\) 19652.1 0.748974 0.374487 0.927232i \(-0.377819\pi\)
0.374487 + 0.927232i \(0.377819\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −43431.2 −1.64684
\(887\) 26295.0 0.995379 0.497689 0.867355i \(-0.334182\pi\)
0.497689 + 0.867355i \(0.334182\pi\)
\(888\) 0 0
\(889\) 13986.6 0.527668
\(890\) 41505.1 1.56321
\(891\) 0 0
\(892\) −13966.5 −0.524253
\(893\) −4531.69 −0.169818
\(894\) 0 0
\(895\) 23340.2 0.871704
\(896\) 745.275 0.0277878
\(897\) 0 0
\(898\) 55758.0 2.07201
\(899\) −61397.0 −2.27776
\(900\) 0 0
\(901\) −2473.93 −0.0914746
\(902\) −13940.8 −0.514609
\(903\) 0 0
\(904\) 1296.69 0.0477072
\(905\) 30701.4 1.12768
\(906\) 0 0
\(907\) 42417.3 1.55286 0.776429 0.630204i \(-0.217029\pi\)
0.776429 + 0.630204i \(0.217029\pi\)
\(908\) −3341.66 −0.122133
\(909\) 0 0
\(910\) 0 0
\(911\) 872.245 0.0317220 0.0158610 0.999874i \(-0.494951\pi\)
0.0158610 + 0.999874i \(0.494951\pi\)
\(912\) 0 0
\(913\) 11373.5 0.412276
\(914\) −69453.2 −2.51347
\(915\) 0 0
\(916\) −19449.4 −0.701557
\(917\) 13738.5 0.494749
\(918\) 0 0
\(919\) −17181.4 −0.616716 −0.308358 0.951270i \(-0.599779\pi\)
−0.308358 + 0.951270i \(0.599779\pi\)
\(920\) −1739.64 −0.0623415
\(921\) 0 0
\(922\) 46287.5 1.65336
\(923\) 0 0
\(924\) 0 0
\(925\) −19884.4 −0.706804
\(926\) −6503.25 −0.230788
\(927\) 0 0
\(928\) −51407.0 −1.81845
\(929\) −56042.9 −1.97923 −0.989617 0.143728i \(-0.954091\pi\)
−0.989617 + 0.143728i \(0.954091\pi\)
\(930\) 0 0
\(931\) 4116.52 0.144912
\(932\) −7736.00 −0.271890
\(933\) 0 0
\(934\) 32843.3 1.15060
\(935\) −12955.8 −0.453155
\(936\) 0 0
\(937\) −36672.5 −1.27859 −0.639295 0.768961i \(-0.720774\pi\)
−0.639295 + 0.768961i \(0.720774\pi\)
\(938\) −6858.71 −0.238747
\(939\) 0 0
\(940\) 22576.4 0.783363
\(941\) 21069.0 0.729895 0.364947 0.931028i \(-0.381087\pi\)
0.364947 + 0.931028i \(0.381087\pi\)
\(942\) 0 0
\(943\) 44093.4 1.52267
\(944\) 17363.0 0.598642
\(945\) 0 0
\(946\) 8113.87 0.278863
\(947\) 1838.70 0.0630938 0.0315469 0.999502i \(-0.489957\pi\)
0.0315469 + 0.999502i \(0.489957\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 3213.27 0.109739
\(951\) 0 0
\(952\) 541.330 0.0184292
\(953\) −13599.8 −0.462266 −0.231133 0.972922i \(-0.574243\pi\)
−0.231133 + 0.972922i \(0.574243\pi\)
\(954\) 0 0
\(955\) 22695.9 0.769029
\(956\) 38446.9 1.30069
\(957\) 0 0
\(958\) 7365.75 0.248410
\(959\) 2231.02 0.0751235
\(960\) 0 0
\(961\) 64935.6 2.17971
\(962\) 0 0
\(963\) 0 0
\(964\) 31116.4 1.03962
\(965\) 20105.5 0.670692
\(966\) 0 0
\(967\) 2081.30 0.0692141 0.0346070 0.999401i \(-0.488982\pi\)
0.0346070 + 0.999401i \(0.488982\pi\)
\(968\) −1012.10 −0.0336054
\(969\) 0 0
\(970\) −19681.6 −0.651484
\(971\) −1636.62 −0.0540904 −0.0270452 0.999634i \(-0.508610\pi\)
−0.0270452 + 0.999634i \(0.508610\pi\)
\(972\) 0 0
\(973\) −2902.26 −0.0956240
\(974\) 31592.9 1.03932
\(975\) 0 0
\(976\) 12838.8 0.421066
\(977\) 29387.1 0.962311 0.481156 0.876635i \(-0.340217\pi\)
0.481156 + 0.876635i \(0.340217\pi\)
\(978\) 0 0
\(979\) 21944.6 0.716396
\(980\) −20508.1 −0.668476
\(981\) 0 0
\(982\) −69304.4 −2.25213
\(983\) −24084.4 −0.781457 −0.390728 0.920506i \(-0.627777\pi\)
−0.390728 + 0.920506i \(0.627777\pi\)
\(984\) 0 0
\(985\) 22249.4 0.719722
\(986\) −74713.4 −2.41314
\(987\) 0 0
\(988\) 0 0
\(989\) −25663.4 −0.825125
\(990\) 0 0
\(991\) 1413.43 0.0453068 0.0226534 0.999743i \(-0.492789\pi\)
0.0226534 + 0.999743i \(0.492789\pi\)
\(992\) 79313.5 2.53851
\(993\) 0 0
\(994\) 7618.79 0.243112
\(995\) 32342.7 1.03049
\(996\) 0 0
\(997\) −33357.4 −1.05962 −0.529809 0.848117i \(-0.677737\pi\)
−0.529809 + 0.848117i \(0.677737\pi\)
\(998\) −21583.1 −0.684570
\(999\) 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.bi.1.7 9
3.2 odd 2 507.4.a.o.1.3 9
13.12 even 2 1521.4.a.bf.1.3 9
39.5 even 4 507.4.b.k.337.15 18
39.8 even 4 507.4.b.k.337.4 18
39.38 odd 2 507.4.a.p.1.7 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.3 9 3.2 odd 2
507.4.a.p.1.7 yes 9 39.38 odd 2
507.4.b.k.337.4 18 39.8 even 4
507.4.b.k.337.15 18 39.5 even 4
1521.4.a.bf.1.3 9 13.12 even 2
1521.4.a.bi.1.7 9 1.1 even 1 trivial