Properties

Label 2166.4.a.u.1.2
Level $2166$
Weight $4$
Character 2166.1
Self dual yes
Analytic conductor $127.798$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(127.798137072\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.14457.1
Defining polynomial: \(x^{3} - x^{2} - 32 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.0940524\) of defining polynomial
Character \(\chi\) \(=\) 2166.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +4.18810 q^{5} -6.00000 q^{6} +3.18810 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +4.18810 q^{5} -6.00000 q^{6} +3.18810 q^{7} +8.00000 q^{8} +9.00000 q^{9} +8.37621 q^{10} +69.4003 q^{11} -12.0000 q^{12} -8.12863 q^{13} +6.37621 q^{14} -12.5643 q^{15} +16.0000 q^{16} -106.084 q^{17} +18.0000 q^{18} +16.7524 q^{20} -9.56431 q^{21} +138.801 q^{22} -176.494 q^{23} -24.0000 q^{24} -107.460 q^{25} -16.2573 q^{26} -27.0000 q^{27} +12.7524 q^{28} -66.2219 q^{29} -25.1286 q^{30} -140.915 q^{31} +32.0000 q^{32} -208.201 q^{33} -212.167 q^{34} +13.3521 q^{35} +36.0000 q^{36} -156.003 q^{37} +24.3859 q^{39} +33.5048 q^{40} -414.563 q^{41} -19.1286 q^{42} +115.850 q^{43} +277.601 q^{44} +37.6929 q^{45} -352.987 q^{46} +620.283 q^{47} -48.0000 q^{48} -332.836 q^{49} -214.920 q^{50} +318.251 q^{51} -32.5145 q^{52} -371.986 q^{53} -54.0000 q^{54} +290.656 q^{55} +25.5048 q^{56} -132.444 q^{58} +91.6929 q^{59} -50.2573 q^{60} +218.621 q^{61} -281.830 q^{62} +28.6929 q^{63} +64.0000 q^{64} -34.0436 q^{65} -416.402 q^{66} -145.342 q^{67} -424.334 q^{68} +529.481 q^{69} +26.7042 q^{70} +887.829 q^{71} +72.0000 q^{72} +199.016 q^{73} -312.006 q^{74} +322.379 q^{75} +221.255 q^{77} +48.7718 q^{78} -389.558 q^{79} +67.0097 q^{80} +81.0000 q^{81} -829.125 q^{82} +380.039 q^{83} -38.2573 q^{84} -444.289 q^{85} +231.701 q^{86} +198.666 q^{87} +555.202 q^{88} -425.799 q^{89} +75.3859 q^{90} -25.9149 q^{91} -705.974 q^{92} +422.744 q^{93} +1240.57 q^{94} -96.0000 q^{96} +419.846 q^{97} -665.672 q^{98} +624.603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} + 24 q^{8} + 27 q^{9} + O(q^{10}) \) \( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} + 24 q^{8} + 27 q^{9} + 20 q^{10} - 44 q^{11} - 36 q^{12} - 9 q^{13} + 14 q^{14} - 30 q^{15} + 48 q^{16} - 84 q^{17} + 54 q^{18} + 40 q^{20} - 21 q^{21} - 88 q^{22} - 2 q^{23} - 72 q^{24} - 83 q^{25} - 18 q^{26} - 81 q^{27} + 28 q^{28} + 92 q^{29} - 60 q^{30} - 109 q^{31} + 96 q^{32} + 132 q^{33} - 168 q^{34} + 282 q^{35} + 108 q^{36} + 245 q^{37} + 27 q^{39} + 80 q^{40} - 688 q^{41} - 42 q^{42} - 103 q^{43} - 176 q^{44} + 90 q^{45} - 4 q^{46} + 322 q^{47} - 144 q^{48} - 754 q^{49} - 166 q^{50} + 252 q^{51} - 36 q^{52} - 1322 q^{53} - 162 q^{54} - 248 q^{55} + 56 q^{56} + 184 q^{58} + 252 q^{59} - 120 q^{60} - 435 q^{61} - 218 q^{62} + 63 q^{63} + 192 q^{64} - 1582 q^{65} + 264 q^{66} - 719 q^{67} - 336 q^{68} + 6 q^{69} + 564 q^{70} - 62 q^{71} + 216 q^{72} - 581 q^{73} + 490 q^{74} + 249 q^{75} - 204 q^{77} + 54 q^{78} - 489 q^{79} + 160 q^{80} + 243 q^{81} - 1376 q^{82} + 2496 q^{83} - 84 q^{84} + 1632 q^{85} - 206 q^{86} - 276 q^{87} - 352 q^{88} + 1584 q^{89} + 180 q^{90} - 1573 q^{91} - 8 q^{92} + 327 q^{93} + 644 q^{94} - 288 q^{96} + 974 q^{97} - 1508 q^{98} - 396 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) 4.18810 0.374595 0.187298 0.982303i \(-0.440027\pi\)
0.187298 + 0.982303i \(0.440027\pi\)
\(6\) −6.00000 −0.408248
\(7\) 3.18810 0.172141 0.0860707 0.996289i \(-0.472569\pi\)
0.0860707 + 0.996289i \(0.472569\pi\)
\(8\) 8.00000 0.353553
\(9\) 9.00000 0.333333
\(10\) 8.37621 0.264879
\(11\) 69.4003 1.90227 0.951135 0.308774i \(-0.0999187\pi\)
0.951135 + 0.308774i \(0.0999187\pi\)
\(12\) −12.0000 −0.288675
\(13\) −8.12863 −0.173421 −0.0867106 0.996234i \(-0.527636\pi\)
−0.0867106 + 0.996234i \(0.527636\pi\)
\(14\) 6.37621 0.121722
\(15\) −12.5643 −0.216273
\(16\) 16.0000 0.250000
\(17\) −106.084 −1.51347 −0.756737 0.653720i \(-0.773207\pi\)
−0.756737 + 0.653720i \(0.773207\pi\)
\(18\) 18.0000 0.235702
\(19\) 0 0
\(20\) 16.7524 0.187298
\(21\) −9.56431 −0.0993859
\(22\) 138.801 1.34511
\(23\) −176.494 −1.60006 −0.800031 0.599958i \(-0.795184\pi\)
−0.800031 + 0.599958i \(0.795184\pi\)
\(24\) −24.0000 −0.204124
\(25\) −107.460 −0.859678
\(26\) −16.2573 −0.122627
\(27\) −27.0000 −0.192450
\(28\) 12.7524 0.0860707
\(29\) −66.2219 −0.424038 −0.212019 0.977266i \(-0.568004\pi\)
−0.212019 + 0.977266i \(0.568004\pi\)
\(30\) −25.1286 −0.152928
\(31\) −140.915 −0.816421 −0.408210 0.912888i \(-0.633847\pi\)
−0.408210 + 0.912888i \(0.633847\pi\)
\(32\) 32.0000 0.176777
\(33\) −208.201 −1.09828
\(34\) −212.167 −1.07019
\(35\) 13.3521 0.0644834
\(36\) 36.0000 0.166667
\(37\) −156.003 −0.693156 −0.346578 0.938021i \(-0.612656\pi\)
−0.346578 + 0.938021i \(0.612656\pi\)
\(38\) 0 0
\(39\) 24.3859 0.100125
\(40\) 33.5048 0.132440
\(41\) −414.563 −1.57912 −0.789559 0.613675i \(-0.789691\pi\)
−0.789559 + 0.613675i \(0.789691\pi\)
\(42\) −19.1286 −0.0702765
\(43\) 115.850 0.410861 0.205430 0.978672i \(-0.434141\pi\)
0.205430 + 0.978672i \(0.434141\pi\)
\(44\) 277.601 0.951135
\(45\) 37.6929 0.124865
\(46\) −352.987 −1.13142
\(47\) 620.283 1.92505 0.962527 0.271185i \(-0.0874156\pi\)
0.962527 + 0.271185i \(0.0874156\pi\)
\(48\) −48.0000 −0.144338
\(49\) −332.836 −0.970367
\(50\) −214.920 −0.607884
\(51\) 318.251 0.873804
\(52\) −32.5145 −0.0867106
\(53\) −371.986 −0.964078 −0.482039 0.876150i \(-0.660104\pi\)
−0.482039 + 0.876150i \(0.660104\pi\)
\(54\) −54.0000 −0.136083
\(55\) 290.656 0.712582
\(56\) 25.5048 0.0608612
\(57\) 0 0
\(58\) −132.444 −0.299840
\(59\) 91.6929 0.202329 0.101164 0.994870i \(-0.467743\pi\)
0.101164 + 0.994870i \(0.467743\pi\)
\(60\) −50.2573 −0.108136
\(61\) 218.621 0.458877 0.229438 0.973323i \(-0.426311\pi\)
0.229438 + 0.973323i \(0.426311\pi\)
\(62\) −281.830 −0.577297
\(63\) 28.6929 0.0573805
\(64\) 64.0000 0.125000
\(65\) −34.0436 −0.0649628
\(66\) −416.402 −0.776599
\(67\) −145.342 −0.265021 −0.132510 0.991182i \(-0.542304\pi\)
−0.132510 + 0.991182i \(0.542304\pi\)
\(68\) −424.334 −0.756737
\(69\) 529.481 0.923797
\(70\) 26.7042 0.0455967
\(71\) 887.829 1.48403 0.742014 0.670385i \(-0.233871\pi\)
0.742014 + 0.670385i \(0.233871\pi\)
\(72\) 72.0000 0.117851
\(73\) 199.016 0.319083 0.159542 0.987191i \(-0.448998\pi\)
0.159542 + 0.987191i \(0.448998\pi\)
\(74\) −312.006 −0.490135
\(75\) 322.379 0.496335
\(76\) 0 0
\(77\) 221.255 0.327460
\(78\) 48.7718 0.0707989
\(79\) −389.558 −0.554793 −0.277397 0.960755i \(-0.589472\pi\)
−0.277397 + 0.960755i \(0.589472\pi\)
\(80\) 67.0097 0.0936489
\(81\) 81.0000 0.111111
\(82\) −829.125 −1.11660
\(83\) 380.039 0.502586 0.251293 0.967911i \(-0.419144\pi\)
0.251293 + 0.967911i \(0.419144\pi\)
\(84\) −38.2573 −0.0496930
\(85\) −444.289 −0.566940
\(86\) 231.701 0.290523
\(87\) 198.666 0.244818
\(88\) 555.202 0.672554
\(89\) −425.799 −0.507130 −0.253565 0.967318i \(-0.581603\pi\)
−0.253565 + 0.967318i \(0.581603\pi\)
\(90\) 75.3859 0.0882930
\(91\) −25.9149 −0.0298530
\(92\) −705.974 −0.800031
\(93\) 422.744 0.471361
\(94\) 1240.57 1.36122
\(95\) 0 0
\(96\) −96.0000 −0.102062
\(97\) 419.846 0.439473 0.219736 0.975559i \(-0.429480\pi\)
0.219736 + 0.975559i \(0.429480\pi\)
\(98\) −665.672 −0.686153
\(99\) 624.603 0.634090
\(100\) −429.839 −0.429839
\(101\) −1241.71 −1.22331 −0.611657 0.791123i \(-0.709497\pi\)
−0.611657 + 0.791123i \(0.709497\pi\)
\(102\) 636.501 0.617873
\(103\) −593.606 −0.567862 −0.283931 0.958845i \(-0.591639\pi\)
−0.283931 + 0.958845i \(0.591639\pi\)
\(104\) −65.0290 −0.0613137
\(105\) −40.0564 −0.0372295
\(106\) −743.971 −0.681706
\(107\) −1778.56 −1.60691 −0.803457 0.595363i \(-0.797008\pi\)
−0.803457 + 0.595363i \(0.797008\pi\)
\(108\) −108.000 −0.0962250
\(109\) −1069.99 −0.940243 −0.470121 0.882602i \(-0.655790\pi\)
−0.470121 + 0.882602i \(0.655790\pi\)
\(110\) 581.311 0.503872
\(111\) 468.009 0.400194
\(112\) 51.0097 0.0430354
\(113\) 583.197 0.485510 0.242755 0.970088i \(-0.421949\pi\)
0.242755 + 0.970088i \(0.421949\pi\)
\(114\) 0 0
\(115\) −739.173 −0.599376
\(116\) −264.887 −0.212019
\(117\) −73.1577 −0.0578071
\(118\) 183.386 0.143068
\(119\) −338.206 −0.260532
\(120\) −100.515 −0.0764640
\(121\) 3485.40 2.61863
\(122\) 437.241 0.324475
\(123\) 1243.69 0.911704
\(124\) −563.659 −0.408210
\(125\) −973.566 −0.696627
\(126\) 57.3859 0.0405741
\(127\) 1659.35 1.15940 0.579700 0.814830i \(-0.303170\pi\)
0.579700 + 0.814830i \(0.303170\pi\)
\(128\) 128.000 0.0883883
\(129\) −347.551 −0.237211
\(130\) −68.0871 −0.0459356
\(131\) 597.546 0.398533 0.199267 0.979945i \(-0.436144\pi\)
0.199267 + 0.979945i \(0.436144\pi\)
\(132\) −832.804 −0.549138
\(133\) 0 0
\(134\) −290.685 −0.187398
\(135\) −113.079 −0.0720909
\(136\) −848.669 −0.535094
\(137\) −1777.31 −1.10836 −0.554182 0.832396i \(-0.686969\pi\)
−0.554182 + 0.832396i \(0.686969\pi\)
\(138\) 1058.96 0.653223
\(139\) 163.567 0.0998099 0.0499050 0.998754i \(-0.484108\pi\)
0.0499050 + 0.998754i \(0.484108\pi\)
\(140\) 53.4085 0.0322417
\(141\) −1860.85 −1.11143
\(142\) 1775.66 1.04937
\(143\) −564.129 −0.329894
\(144\) 144.000 0.0833333
\(145\) −277.344 −0.158843
\(146\) 398.032 0.225626
\(147\) 998.508 0.560242
\(148\) −624.012 −0.346578
\(149\) −1119.80 −0.615688 −0.307844 0.951437i \(-0.599607\pi\)
−0.307844 + 0.951437i \(0.599607\pi\)
\(150\) 644.759 0.350962
\(151\) −2807.15 −1.51286 −0.756432 0.654073i \(-0.773059\pi\)
−0.756432 + 0.654073i \(0.773059\pi\)
\(152\) 0 0
\(153\) −954.752 −0.504491
\(154\) 442.511 0.231549
\(155\) −590.166 −0.305828
\(156\) 97.5436 0.0500624
\(157\) 1003.94 0.510339 0.255170 0.966896i \(-0.417869\pi\)
0.255170 + 0.966896i \(0.417869\pi\)
\(158\) −779.116 −0.392298
\(159\) 1115.96 0.556611
\(160\) 134.019 0.0662198
\(161\) −562.680 −0.275437
\(162\) 162.000 0.0785674
\(163\) 1271.99 0.611225 0.305612 0.952156i \(-0.401139\pi\)
0.305612 + 0.952156i \(0.401139\pi\)
\(164\) −1658.25 −0.789559
\(165\) −871.967 −0.411409
\(166\) 760.077 0.355382
\(167\) −2984.98 −1.38314 −0.691572 0.722307i \(-0.743082\pi\)
−0.691572 + 0.722307i \(0.743082\pi\)
\(168\) −76.5145 −0.0351382
\(169\) −2130.93 −0.969925
\(170\) −888.578 −0.400887
\(171\) 0 0
\(172\) 463.402 0.205430
\(173\) 56.7747 0.0249509 0.0124754 0.999922i \(-0.496029\pi\)
0.0124754 + 0.999922i \(0.496029\pi\)
\(174\) 397.331 0.173113
\(175\) −342.593 −0.147986
\(176\) 1110.40 0.475568
\(177\) −275.079 −0.116815
\(178\) −851.598 −0.358595
\(179\) −4640.98 −1.93790 −0.968948 0.247266i \(-0.920468\pi\)
−0.968948 + 0.247266i \(0.920468\pi\)
\(180\) 150.772 0.0624326
\(181\) 3775.41 1.55041 0.775204 0.631711i \(-0.217647\pi\)
0.775204 + 0.631711i \(0.217647\pi\)
\(182\) −51.8298 −0.0211093
\(183\) −655.862 −0.264933
\(184\) −1411.95 −0.565708
\(185\) −653.357 −0.259653
\(186\) 845.489 0.333302
\(187\) −7362.23 −2.87904
\(188\) 2481.13 0.962527
\(189\) −86.0788 −0.0331286
\(190\) 0 0
\(191\) −2762.53 −1.04654 −0.523271 0.852166i \(-0.675288\pi\)
−0.523271 + 0.852166i \(0.675288\pi\)
\(192\) −192.000 −0.0721688
\(193\) 2061.51 0.768866 0.384433 0.923153i \(-0.374397\pi\)
0.384433 + 0.923153i \(0.374397\pi\)
\(194\) 839.691 0.310754
\(195\) 102.131 0.0375063
\(196\) −1331.34 −0.485184
\(197\) 2094.82 0.757614 0.378807 0.925476i \(-0.376334\pi\)
0.378807 + 0.925476i \(0.376334\pi\)
\(198\) 1249.21 0.448370
\(199\) −1717.36 −0.611762 −0.305881 0.952070i \(-0.598951\pi\)
−0.305881 + 0.952070i \(0.598951\pi\)
\(200\) −859.678 −0.303942
\(201\) 436.027 0.153010
\(202\) −2483.42 −0.865014
\(203\) −211.122 −0.0729945
\(204\) 1273.00 0.436902
\(205\) −1736.23 −0.591530
\(206\) −1187.21 −0.401539
\(207\) −1588.44 −0.533354
\(208\) −130.058 −0.0433553
\(209\) 0 0
\(210\) −80.1127 −0.0263252
\(211\) −2291.05 −0.747500 −0.373750 0.927529i \(-0.621928\pi\)
−0.373750 + 0.927529i \(0.621928\pi\)
\(212\) −1487.94 −0.482039
\(213\) −2663.49 −0.856804
\(214\) −3557.12 −1.13626
\(215\) 485.194 0.153907
\(216\) −216.000 −0.0680414
\(217\) −449.251 −0.140540
\(218\) −2139.98 −0.664852
\(219\) −597.048 −0.184223
\(220\) 1162.62 0.356291
\(221\) 862.314 0.262468
\(222\) 936.019 0.282980
\(223\) 3256.19 0.977805 0.488902 0.872338i \(-0.337397\pi\)
0.488902 + 0.872338i \(0.337397\pi\)
\(224\) 102.019 0.0304306
\(225\) −967.138 −0.286559
\(226\) 1166.39 0.343307
\(227\) −998.044 −0.291817 −0.145909 0.989298i \(-0.546611\pi\)
−0.145909 + 0.989298i \(0.546611\pi\)
\(228\) 0 0
\(229\) −1028.59 −0.296816 −0.148408 0.988926i \(-0.547415\pi\)
−0.148408 + 0.988926i \(0.547415\pi\)
\(230\) −1478.35 −0.423823
\(231\) −663.766 −0.189059
\(232\) −529.775 −0.149920
\(233\) −125.486 −0.0352827 −0.0176414 0.999844i \(-0.505616\pi\)
−0.0176414 + 0.999844i \(0.505616\pi\)
\(234\) −146.315 −0.0408758
\(235\) 2597.81 0.721117
\(236\) 366.772 0.101164
\(237\) 1168.67 0.320310
\(238\) −676.411 −0.184224
\(239\) 3591.03 0.971901 0.485950 0.873986i \(-0.338474\pi\)
0.485950 + 0.873986i \(0.338474\pi\)
\(240\) −201.029 −0.0540682
\(241\) −3691.84 −0.986773 −0.493386 0.869810i \(-0.664241\pi\)
−0.493386 + 0.869810i \(0.664241\pi\)
\(242\) 6970.80 1.85165
\(243\) −243.000 −0.0641500
\(244\) 874.482 0.229438
\(245\) −1393.95 −0.363495
\(246\) 2487.38 0.644672
\(247\) 0 0
\(248\) −1127.32 −0.288648
\(249\) −1140.12 −0.290168
\(250\) −1947.13 −0.492590
\(251\) 7294.88 1.83446 0.917228 0.398362i \(-0.130421\pi\)
0.917228 + 0.398362i \(0.130421\pi\)
\(252\) 114.772 0.0286902
\(253\) −12248.7 −3.04375
\(254\) 3318.71 0.819820
\(255\) 1332.87 0.327323
\(256\) 256.000 0.0625000
\(257\) 2303.20 0.559026 0.279513 0.960142i \(-0.409827\pi\)
0.279513 + 0.960142i \(0.409827\pi\)
\(258\) −695.102 −0.167733
\(259\) −497.354 −0.119321
\(260\) −136.174 −0.0324814
\(261\) −595.997 −0.141346
\(262\) 1195.09 0.281806
\(263\) 7138.38 1.67365 0.836827 0.547467i \(-0.184408\pi\)
0.836827 + 0.547467i \(0.184408\pi\)
\(264\) −1665.61 −0.388299
\(265\) −1557.91 −0.361139
\(266\) 0 0
\(267\) 1277.40 0.292792
\(268\) −581.370 −0.132510
\(269\) −5446.66 −1.23453 −0.617265 0.786755i \(-0.711759\pi\)
−0.617265 + 0.786755i \(0.711759\pi\)
\(270\) −226.158 −0.0509760
\(271\) −3403.68 −0.762947 −0.381474 0.924380i \(-0.624583\pi\)
−0.381474 + 0.924380i \(0.624583\pi\)
\(272\) −1697.34 −0.378368
\(273\) 77.7448 0.0172356
\(274\) −3554.62 −0.783731
\(275\) −7457.74 −1.63534
\(276\) 2117.92 0.461898
\(277\) −5131.93 −1.11317 −0.556584 0.830791i \(-0.687888\pi\)
−0.556584 + 0.830791i \(0.687888\pi\)
\(278\) 327.134 0.0705763
\(279\) −1268.23 −0.272140
\(280\) 106.817 0.0227983
\(281\) 3366.68 0.714731 0.357365 0.933965i \(-0.383675\pi\)
0.357365 + 0.933965i \(0.383675\pi\)
\(282\) −3721.70 −0.785900
\(283\) 6685.76 1.40434 0.702168 0.712011i \(-0.252215\pi\)
0.702168 + 0.712011i \(0.252215\pi\)
\(284\) 3551.32 0.742014
\(285\) 0 0
\(286\) −1128.26 −0.233270
\(287\) −1321.67 −0.271832
\(288\) 288.000 0.0589256
\(289\) 6340.72 1.29060
\(290\) −554.688 −0.112319
\(291\) −1259.54 −0.253730
\(292\) 796.065 0.159542
\(293\) −5625.93 −1.12174 −0.560871 0.827903i \(-0.689534\pi\)
−0.560871 + 0.827903i \(0.689534\pi\)
\(294\) 1997.02 0.396151
\(295\) 384.020 0.0757915
\(296\) −1248.02 −0.245067
\(297\) −1873.81 −0.366092
\(298\) −2239.60 −0.435357
\(299\) 1434.65 0.277485
\(300\) 1289.52 0.248168
\(301\) 369.343 0.0707262
\(302\) −5614.29 −1.06976
\(303\) 3725.13 0.706281
\(304\) 0 0
\(305\) 915.606 0.171893
\(306\) −1909.50 −0.356729
\(307\) 5603.54 1.04173 0.520865 0.853639i \(-0.325610\pi\)
0.520865 + 0.853639i \(0.325610\pi\)
\(308\) 885.022 0.163730
\(309\) 1780.82 0.327855
\(310\) −1180.33 −0.216253
\(311\) 6668.20 1.21582 0.607908 0.794008i \(-0.292009\pi\)
0.607908 + 0.794008i \(0.292009\pi\)
\(312\) 195.087 0.0353995
\(313\) 4491.40 0.811083 0.405541 0.914077i \(-0.367083\pi\)
0.405541 + 0.914077i \(0.367083\pi\)
\(314\) 2007.88 0.360865
\(315\) 120.169 0.0214945
\(316\) −1558.23 −0.277397
\(317\) −1266.05 −0.224316 −0.112158 0.993690i \(-0.535776\pi\)
−0.112158 + 0.993690i \(0.535776\pi\)
\(318\) 2231.91 0.393583
\(319\) −4595.82 −0.806635
\(320\) 268.039 0.0468244
\(321\) 5335.68 0.927752
\(322\) −1125.36 −0.194764
\(323\) 0 0
\(324\) 324.000 0.0555556
\(325\) 873.501 0.149086
\(326\) 2543.97 0.432201
\(327\) 3209.97 0.542849
\(328\) −3316.50 −0.558302
\(329\) 1977.53 0.331382
\(330\) −1743.93 −0.290910
\(331\) −5068.09 −0.841594 −0.420797 0.907155i \(-0.638249\pi\)
−0.420797 + 0.907155i \(0.638249\pi\)
\(332\) 1520.15 0.251293
\(333\) −1404.03 −0.231052
\(334\) −5969.97 −0.978031
\(335\) −608.709 −0.0992757
\(336\) −153.029 −0.0248465
\(337\) 10744.1 1.73670 0.868352 0.495948i \(-0.165179\pi\)
0.868352 + 0.495948i \(0.165179\pi\)
\(338\) −4261.85 −0.685841
\(339\) −1749.59 −0.280309
\(340\) −1777.16 −0.283470
\(341\) −9779.53 −1.55305
\(342\) 0 0
\(343\) −2154.64 −0.339182
\(344\) 926.803 0.145261
\(345\) 2217.52 0.346050
\(346\) 113.549 0.0176429
\(347\) −4851.34 −0.750528 −0.375264 0.926918i \(-0.622448\pi\)
−0.375264 + 0.926918i \(0.622448\pi\)
\(348\) 794.662 0.122409
\(349\) −7611.35 −1.16741 −0.583705 0.811966i \(-0.698398\pi\)
−0.583705 + 0.811966i \(0.698398\pi\)
\(350\) −685.186 −0.104642
\(351\) 219.473 0.0333749
\(352\) 2220.81 0.336277
\(353\) 5567.64 0.839477 0.419739 0.907645i \(-0.362122\pi\)
0.419739 + 0.907645i \(0.362122\pi\)
\(354\) −550.158 −0.0826004
\(355\) 3718.32 0.555910
\(356\) −1703.20 −0.253565
\(357\) 1014.62 0.150418
\(358\) −9281.96 −1.37030
\(359\) 4341.31 0.638233 0.319116 0.947716i \(-0.396614\pi\)
0.319116 + 0.947716i \(0.396614\pi\)
\(360\) 301.544 0.0441465
\(361\) 0 0
\(362\) 7550.82 1.09630
\(363\) −10456.2 −1.51187
\(364\) −103.660 −0.0149265
\(365\) 833.500 0.119527
\(366\) −1311.72 −0.187336
\(367\) −8964.48 −1.27505 −0.637524 0.770431i \(-0.720041\pi\)
−0.637524 + 0.770431i \(0.720041\pi\)
\(368\) −2823.90 −0.400016
\(369\) −3731.06 −0.526372
\(370\) −1306.71 −0.183602
\(371\) −1185.93 −0.165958
\(372\) 1690.98 0.235680
\(373\) 5048.11 0.700755 0.350377 0.936609i \(-0.386053\pi\)
0.350377 + 0.936609i \(0.386053\pi\)
\(374\) −14724.5 −2.03579
\(375\) 2920.70 0.402198
\(376\) 4962.26 0.680609
\(377\) 538.293 0.0735371
\(378\) −172.158 −0.0234255
\(379\) −9290.41 −1.25915 −0.629573 0.776941i \(-0.716770\pi\)
−0.629573 + 0.776941i \(0.716770\pi\)
\(380\) 0 0
\(381\) −4978.06 −0.669380
\(382\) −5525.06 −0.740017
\(383\) 4162.26 0.555304 0.277652 0.960682i \(-0.410444\pi\)
0.277652 + 0.960682i \(0.410444\pi\)
\(384\) −384.000 −0.0510310
\(385\) 926.641 0.122665
\(386\) 4123.03 0.543670
\(387\) 1042.65 0.136954
\(388\) 1679.38 0.219736
\(389\) 4185.23 0.545500 0.272750 0.962085i \(-0.412067\pi\)
0.272750 + 0.962085i \(0.412067\pi\)
\(390\) 204.261 0.0265210
\(391\) 18723.1 2.42165
\(392\) −2662.69 −0.343077
\(393\) −1792.64 −0.230093
\(394\) 4189.64 0.535714
\(395\) −1631.51 −0.207823
\(396\) 2498.41 0.317045
\(397\) −4529.76 −0.572650 −0.286325 0.958133i \(-0.592434\pi\)
−0.286325 + 0.958133i \(0.592434\pi\)
\(398\) −3434.72 −0.432581
\(399\) 0 0
\(400\) −1719.36 −0.214920
\(401\) −9497.05 −1.18269 −0.591347 0.806417i \(-0.701404\pi\)
−0.591347 + 0.806417i \(0.701404\pi\)
\(402\) 872.055 0.108194
\(403\) 1145.44 0.141585
\(404\) −4966.84 −0.611657
\(405\) 339.236 0.0416217
\(406\) −422.245 −0.0516149
\(407\) −10826.7 −1.31857
\(408\) 2546.01 0.308936
\(409\) −5354.44 −0.647335 −0.323667 0.946171i \(-0.604916\pi\)
−0.323667 + 0.946171i \(0.604916\pi\)
\(410\) −3472.46 −0.418275
\(411\) 5331.93 0.639914
\(412\) −2374.43 −0.283931
\(413\) 292.327 0.0348292
\(414\) −3176.88 −0.377138
\(415\) 1591.64 0.188267
\(416\) −260.116 −0.0306568
\(417\) −490.701 −0.0576253
\(418\) 0 0
\(419\) 14506.1 1.69134 0.845668 0.533709i \(-0.179202\pi\)
0.845668 + 0.533709i \(0.179202\pi\)
\(420\) −160.225 −0.0186148
\(421\) −1263.45 −0.146263 −0.0731315 0.997322i \(-0.523299\pi\)
−0.0731315 + 0.997322i \(0.523299\pi\)
\(422\) −4582.10 −0.528562
\(423\) 5582.55 0.641685
\(424\) −2975.88 −0.340853
\(425\) 11399.7 1.30110
\(426\) −5326.98 −0.605852
\(427\) 696.985 0.0789918
\(428\) −7114.24 −0.803457
\(429\) 1692.39 0.190464
\(430\) 970.387 0.108828
\(431\) −8187.62 −0.915043 −0.457522 0.889198i \(-0.651263\pi\)
−0.457522 + 0.889198i \(0.651263\pi\)
\(432\) −432.000 −0.0481125
\(433\) −9242.94 −1.02584 −0.512919 0.858437i \(-0.671436\pi\)
−0.512919 + 0.858437i \(0.671436\pi\)
\(434\) −898.502 −0.0993767
\(435\) 832.032 0.0917078
\(436\) −4279.96 −0.470121
\(437\) 0 0
\(438\) −1194.10 −0.130265
\(439\) −2043.05 −0.222117 −0.111059 0.993814i \(-0.535424\pi\)
−0.111059 + 0.993814i \(0.535424\pi\)
\(440\) 2325.25 0.251936
\(441\) −2995.52 −0.323456
\(442\) 1724.63 0.185593
\(443\) −9669.13 −1.03701 −0.518504 0.855075i \(-0.673511\pi\)
−0.518504 + 0.855075i \(0.673511\pi\)
\(444\) 1872.04 0.200097
\(445\) −1783.29 −0.189969
\(446\) 6512.38 0.691412
\(447\) 3359.40 0.355467
\(448\) 204.039 0.0215177
\(449\) −13227.2 −1.39027 −0.695134 0.718881i \(-0.744655\pi\)
−0.695134 + 0.718881i \(0.744655\pi\)
\(450\) −1934.28 −0.202628
\(451\) −28770.8 −3.00391
\(452\) 2332.79 0.242755
\(453\) 8421.44 0.873452
\(454\) −1996.09 −0.206346
\(455\) −108.534 −0.0111828
\(456\) 0 0
\(457\) −5380.98 −0.550791 −0.275396 0.961331i \(-0.588809\pi\)
−0.275396 + 0.961331i \(0.588809\pi\)
\(458\) −2057.17 −0.209880
\(459\) 2864.26 0.291268
\(460\) −2956.69 −0.299688
\(461\) −2844.67 −0.287396 −0.143698 0.989622i \(-0.545899\pi\)
−0.143698 + 0.989622i \(0.545899\pi\)
\(462\) −1327.53 −0.133685
\(463\) 6625.39 0.665028 0.332514 0.943098i \(-0.392103\pi\)
0.332514 + 0.943098i \(0.392103\pi\)
\(464\) −1059.55 −0.106009
\(465\) 1770.50 0.176570
\(466\) −250.972 −0.0249487
\(467\) −18635.1 −1.84653 −0.923264 0.384167i \(-0.874489\pi\)
−0.923264 + 0.384167i \(0.874489\pi\)
\(468\) −292.631 −0.0289035
\(469\) −463.367 −0.0456211
\(470\) 5195.62 0.509906
\(471\) −3011.83 −0.294645
\(472\) 733.544 0.0715341
\(473\) 8040.05 0.781569
\(474\) 2337.35 0.226493
\(475\) 0 0
\(476\) −1352.82 −0.130266
\(477\) −3347.87 −0.321359
\(478\) 7182.05 0.687237
\(479\) 1556.62 0.148484 0.0742420 0.997240i \(-0.476346\pi\)
0.0742420 + 0.997240i \(0.476346\pi\)
\(480\) −402.058 −0.0382320
\(481\) 1268.09 0.120208
\(482\) −7383.68 −0.697754
\(483\) 1688.04 0.159024
\(484\) 13941.6 1.30932
\(485\) 1758.36 0.164625
\(486\) −486.000 −0.0453609
\(487\) 19339.5 1.79950 0.899751 0.436405i \(-0.143748\pi\)
0.899751 + 0.436405i \(0.143748\pi\)
\(488\) 1748.96 0.162238
\(489\) −3815.96 −0.352891
\(490\) −2787.90 −0.257030
\(491\) −6838.25 −0.628525 −0.314262 0.949336i \(-0.601757\pi\)
−0.314262 + 0.949336i \(0.601757\pi\)
\(492\) 4974.75 0.455852
\(493\) 7025.05 0.641770
\(494\) 0 0
\(495\) 2615.90 0.237527
\(496\) −2254.64 −0.204105
\(497\) 2830.49 0.255463
\(498\) −2280.23 −0.205180
\(499\) −18115.0 −1.62513 −0.812563 0.582874i \(-0.801928\pi\)
−0.812563 + 0.582874i \(0.801928\pi\)
\(500\) −3894.26 −0.348314
\(501\) 8954.95 0.798559
\(502\) 14589.8 1.29716
\(503\) −5831.85 −0.516957 −0.258478 0.966017i \(-0.583221\pi\)
−0.258478 + 0.966017i \(0.583221\pi\)
\(504\) 229.544 0.0202871
\(505\) −5200.41 −0.458248
\(506\) −24497.4 −2.15226
\(507\) 6392.78 0.559987
\(508\) 6637.41 0.579700
\(509\) 9914.65 0.863377 0.431689 0.902023i \(-0.357918\pi\)
0.431689 + 0.902023i \(0.357918\pi\)
\(510\) 2665.73 0.231452
\(511\) 634.484 0.0549275
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 4606.40 0.395291
\(515\) −2486.09 −0.212718
\(516\) −1390.20 −0.118605
\(517\) 43047.8 3.66197
\(518\) −994.709 −0.0843726
\(519\) −170.324 −0.0144054
\(520\) −272.348 −0.0229678
\(521\) −5851.91 −0.492086 −0.246043 0.969259i \(-0.579131\pi\)
−0.246043 + 0.969259i \(0.579131\pi\)
\(522\) −1191.99 −0.0999466
\(523\) 6894.09 0.576401 0.288200 0.957570i \(-0.406943\pi\)
0.288200 + 0.957570i \(0.406943\pi\)
\(524\) 2390.19 0.199267
\(525\) 1027.78 0.0854399
\(526\) 14276.8 1.18345
\(527\) 14948.7 1.23563
\(528\) −3331.21 −0.274569
\(529\) 18983.0 1.56020
\(530\) −3115.83 −0.255364
\(531\) 825.236 0.0674430
\(532\) 0 0
\(533\) 3369.83 0.273853
\(534\) 2554.79 0.207035
\(535\) −7448.79 −0.601943
\(536\) −1162.74 −0.0936991
\(537\) 13922.9 1.11884
\(538\) −10893.3 −0.872945
\(539\) −23098.9 −1.84590
\(540\) −452.315 −0.0360455
\(541\) −5624.99 −0.447019 −0.223509 0.974702i \(-0.571751\pi\)
−0.223509 + 0.974702i \(0.571751\pi\)
\(542\) −6807.36 −0.539485
\(543\) −11326.2 −0.895129
\(544\) −3394.67 −0.267547
\(545\) −4481.23 −0.352211
\(546\) 155.490 0.0121874
\(547\) 3596.14 0.281096 0.140548 0.990074i \(-0.455114\pi\)
0.140548 + 0.990074i \(0.455114\pi\)
\(548\) −7109.24 −0.554182
\(549\) 1967.59 0.152959
\(550\) −14915.5 −1.15636
\(551\) 0 0
\(552\) 4235.85 0.326611
\(553\) −1241.95 −0.0955029
\(554\) −10263.9 −0.787129
\(555\) 1960.07 0.149911
\(556\) 654.269 0.0499050
\(557\) 18381.5 1.39829 0.699145 0.714980i \(-0.253564\pi\)
0.699145 + 0.714980i \(0.253564\pi\)
\(558\) −2536.47 −0.192432
\(559\) −941.705 −0.0712520
\(560\) 213.634 0.0161209
\(561\) 22086.7 1.66221
\(562\) 6733.36 0.505391
\(563\) 5578.33 0.417582 0.208791 0.977960i \(-0.433047\pi\)
0.208791 + 0.977960i \(0.433047\pi\)
\(564\) −7443.39 −0.555715
\(565\) 2442.49 0.181870
\(566\) 13371.5 0.993016
\(567\) 258.236 0.0191268
\(568\) 7102.63 0.524683
\(569\) 16981.3 1.25113 0.625564 0.780173i \(-0.284869\pi\)
0.625564 + 0.780173i \(0.284869\pi\)
\(570\) 0 0
\(571\) −19520.5 −1.43066 −0.715332 0.698785i \(-0.753724\pi\)
−0.715332 + 0.698785i \(0.753724\pi\)
\(572\) −2256.52 −0.164947
\(573\) 8287.59 0.604221
\(574\) −2643.34 −0.192214
\(575\) 18966.0 1.37554
\(576\) 576.000 0.0416667
\(577\) 6371.35 0.459693 0.229846 0.973227i \(-0.426178\pi\)
0.229846 + 0.973227i \(0.426178\pi\)
\(578\) 12681.4 0.912593
\(579\) −6184.54 −0.443905
\(580\) −1109.38 −0.0794213
\(581\) 1211.60 0.0865159
\(582\) −2519.07 −0.179414
\(583\) −25815.9 −1.83394
\(584\) 1592.13 0.112813
\(585\) −306.392 −0.0216543
\(586\) −11251.9 −0.793191
\(587\) 10759.0 0.756513 0.378257 0.925701i \(-0.376524\pi\)
0.378257 + 0.925701i \(0.376524\pi\)
\(588\) 3994.03 0.280121
\(589\) 0 0
\(590\) 768.039 0.0535927
\(591\) −6284.47 −0.437408
\(592\) −2496.05 −0.173289
\(593\) 3611.76 0.250113 0.125057 0.992150i \(-0.460089\pi\)
0.125057 + 0.992150i \(0.460089\pi\)
\(594\) −3747.62 −0.258866
\(595\) −1416.44 −0.0975939
\(596\) −4479.19 −0.307844
\(597\) 5152.09 0.353201
\(598\) 2869.30 0.196211
\(599\) −9180.23 −0.626201 −0.313100 0.949720i \(-0.601368\pi\)
−0.313100 + 0.949720i \(0.601368\pi\)
\(600\) 2579.03 0.175481
\(601\) 12700.4 0.862000 0.431000 0.902352i \(-0.358161\pi\)
0.431000 + 0.902352i \(0.358161\pi\)
\(602\) 738.686 0.0500110
\(603\) −1308.08 −0.0883403
\(604\) −11228.6 −0.756432
\(605\) 14597.2 0.980928
\(606\) 7450.26 0.499416
\(607\) −24227.5 −1.62004 −0.810019 0.586404i \(-0.800543\pi\)
−0.810019 + 0.586404i \(0.800543\pi\)
\(608\) 0 0
\(609\) 633.367 0.0421434
\(610\) 1831.21 0.121547
\(611\) −5042.05 −0.333845
\(612\) −3819.01 −0.252246
\(613\) −16213.5 −1.06828 −0.534141 0.845396i \(-0.679365\pi\)
−0.534141 + 0.845396i \(0.679365\pi\)
\(614\) 11207.1 0.736614
\(615\) 5208.70 0.341520
\(616\) 1770.04 0.115774
\(617\) −21515.3 −1.40385 −0.701924 0.712252i \(-0.747675\pi\)
−0.701924 + 0.712252i \(0.747675\pi\)
\(618\) 3561.64 0.231829
\(619\) −15878.5 −1.03103 −0.515516 0.856880i \(-0.672400\pi\)
−0.515516 + 0.856880i \(0.672400\pi\)
\(620\) −2360.66 −0.152914
\(621\) 4765.33 0.307932
\(622\) 13336.4 0.859712
\(623\) −1357.49 −0.0872982
\(624\) 390.174 0.0250312
\(625\) 9355.08 0.598725
\(626\) 8982.80 0.573522
\(627\) 0 0
\(628\) 4015.77 0.255170
\(629\) 16549.4 1.04907
\(630\) 240.338 0.0151989
\(631\) 5559.75 0.350761 0.175380 0.984501i \(-0.443884\pi\)
0.175380 + 0.984501i \(0.443884\pi\)
\(632\) −3116.46 −0.196149
\(633\) 6873.16 0.431569
\(634\) −2532.09 −0.158616
\(635\) 6949.55 0.434306
\(636\) 4463.83 0.278305
\(637\) 2705.50 0.168282
\(638\) −9191.64 −0.570377
\(639\) 7990.46 0.494676
\(640\) 536.077 0.0331099
\(641\) −20486.9 −1.26238 −0.631190 0.775628i \(-0.717433\pi\)
−0.631190 + 0.775628i \(0.717433\pi\)
\(642\) 10671.4 0.656020
\(643\) −4102.30 −0.251600 −0.125800 0.992056i \(-0.540150\pi\)
−0.125800 + 0.992056i \(0.540150\pi\)
\(644\) −2250.72 −0.137719
\(645\) −1455.58 −0.0888580
\(646\) 0 0
\(647\) 22860.4 1.38908 0.694539 0.719455i \(-0.255608\pi\)
0.694539 + 0.719455i \(0.255608\pi\)
\(648\) 648.000 0.0392837
\(649\) 6363.52 0.384884
\(650\) 1747.00 0.105420
\(651\) 1347.75 0.0811408
\(652\) 5087.94 0.305612
\(653\) 26388.5 1.58141 0.790706 0.612197i \(-0.209714\pi\)
0.790706 + 0.612197i \(0.209714\pi\)
\(654\) 6419.94 0.383853
\(655\) 2502.59 0.149289
\(656\) −6633.00 −0.394779
\(657\) 1791.15 0.106361
\(658\) 3955.05 0.234322
\(659\) 3223.29 0.190534 0.0952668 0.995452i \(-0.469630\pi\)
0.0952668 + 0.995452i \(0.469630\pi\)
\(660\) −3487.87 −0.205705
\(661\) −27480.2 −1.61703 −0.808513 0.588478i \(-0.799727\pi\)
−0.808513 + 0.588478i \(0.799727\pi\)
\(662\) −10136.2 −0.595097
\(663\) −2586.94 −0.151536
\(664\) 3040.31 0.177691
\(665\) 0 0
\(666\) −2808.06 −0.163378
\(667\) 11687.7 0.678487
\(668\) −11939.9 −0.691572
\(669\) −9768.56 −0.564536
\(670\) −1217.42 −0.0701985
\(671\) 15172.3 0.872908
\(672\) −306.058 −0.0175691
\(673\) −2165.00 −0.124004 −0.0620020 0.998076i \(-0.519749\pi\)
−0.0620020 + 0.998076i \(0.519749\pi\)
\(674\) 21488.2 1.22804
\(675\) 2901.41 0.165445
\(676\) −8523.70 −0.484963
\(677\) 11999.0 0.681179 0.340589 0.940212i \(-0.389373\pi\)
0.340589 + 0.940212i \(0.389373\pi\)
\(678\) −3499.18 −0.198209
\(679\) 1338.51 0.0756515
\(680\) −3554.31 −0.200444
\(681\) 2994.13 0.168481
\(682\) −19559.1 −1.09817
\(683\) 5234.35 0.293246 0.146623 0.989192i \(-0.453160\pi\)
0.146623 + 0.989192i \(0.453160\pi\)
\(684\) 0 0
\(685\) −7443.56 −0.415188
\(686\) −4309.27 −0.239838
\(687\) 3085.76 0.171367
\(688\) 1853.61 0.102715
\(689\) 3023.73 0.167192
\(690\) 4435.04 0.244694
\(691\) −5160.63 −0.284109 −0.142055 0.989859i \(-0.545371\pi\)
−0.142055 + 0.989859i \(0.545371\pi\)
\(692\) 227.099 0.0124754
\(693\) 1991.30 0.109153
\(694\) −9702.67 −0.530704
\(695\) 685.036 0.0373884
\(696\) 1589.32 0.0865563
\(697\) 43978.3 2.38995
\(698\) −15222.7 −0.825484
\(699\) 376.459 0.0203705
\(700\) −1370.37 −0.0739931
\(701\) −3315.03 −0.178612 −0.0893059 0.996004i \(-0.528465\pi\)
−0.0893059 + 0.996004i \(0.528465\pi\)
\(702\) 438.946 0.0235996
\(703\) 0 0
\(704\) 4441.62 0.237784
\(705\) −7793.43 −0.416337
\(706\) 11135.3 0.593600
\(707\) −3958.70 −0.210583
\(708\) −1100.32 −0.0584073
\(709\) −18333.2 −0.971111 −0.485555 0.874206i \(-0.661383\pi\)
−0.485555 + 0.874206i \(0.661383\pi\)
\(710\) 7436.64 0.393088
\(711\) −3506.02 −0.184931
\(712\) −3406.39 −0.179298
\(713\) 24870.6 1.30632
\(714\) 2029.23 0.106362
\(715\) −2362.63 −0.123577
\(716\) −18563.9 −0.968948
\(717\) −10773.1 −0.561127
\(718\) 8682.62 0.451299
\(719\) 9747.00 0.505566 0.252783 0.967523i \(-0.418654\pi\)
0.252783 + 0.967523i \(0.418654\pi\)
\(720\) 603.087 0.0312163
\(721\) −1892.48 −0.0977526
\(722\) 0 0
\(723\) 11075.5 0.569713
\(724\) 15101.6 0.775204
\(725\) 7116.19 0.364536
\(726\) −20912.4 −1.06905
\(727\) 32723.6 1.66939 0.834697 0.550709i \(-0.185643\pi\)
0.834697 + 0.550709i \(0.185643\pi\)
\(728\) −207.319 −0.0105546
\(729\) 729.000 0.0370370
\(730\) 1667.00 0.0845185
\(731\) −12289.8 −0.621827
\(732\) −2623.45 −0.132466
\(733\) 36938.2 1.86131 0.930657 0.365894i \(-0.119237\pi\)
0.930657 + 0.365894i \(0.119237\pi\)
\(734\) −17929.0 −0.901594
\(735\) 4181.86 0.209864
\(736\) −5647.79 −0.282854
\(737\) −10086.8 −0.504142
\(738\) −7462.13 −0.372202
\(739\) −9987.19 −0.497138 −0.248569 0.968614i \(-0.579960\pi\)
−0.248569 + 0.968614i \(0.579960\pi\)
\(740\) −2613.43 −0.129826
\(741\) 0 0
\(742\) −2371.86 −0.117350
\(743\) 21739.8 1.07343 0.536713 0.843765i \(-0.319666\pi\)
0.536713 + 0.843765i \(0.319666\pi\)
\(744\) 3381.96 0.166651
\(745\) −4689.83 −0.230634
\(746\) 10096.2 0.495508
\(747\) 3420.35 0.167529
\(748\) −29448.9 −1.43952
\(749\) −5670.23 −0.276617
\(750\) 5841.40 0.284397
\(751\) −38067.4 −1.84967 −0.924833 0.380374i \(-0.875795\pi\)
−0.924833 + 0.380374i \(0.875795\pi\)
\(752\) 9924.53 0.481264
\(753\) −21884.6 −1.05912
\(754\) 1076.59 0.0519986
\(755\) −11756.6 −0.566712
\(756\) −344.315 −0.0165643
\(757\) −11365.2 −0.545675 −0.272838 0.962060i \(-0.587962\pi\)
−0.272838 + 0.962060i \(0.587962\pi\)
\(758\) −18580.8 −0.890351
\(759\) 36746.1 1.75731
\(760\) 0 0
\(761\) 24549.3 1.16940 0.584699 0.811250i \(-0.301213\pi\)
0.584699 + 0.811250i \(0.301213\pi\)
\(762\) −9956.12 −0.473323
\(763\) −3411.24 −0.161855
\(764\) −11050.1 −0.523271
\(765\) −3998.60 −0.188980
\(766\) 8324.51 0.392659
\(767\) −745.338 −0.0350881
\(768\) −768.000 −0.0360844
\(769\) 14222.5 0.666941 0.333471 0.942760i \(-0.391780\pi\)
0.333471 + 0.942760i \(0.391780\pi\)
\(770\) 1853.28 0.0867372
\(771\) −6909.60 −0.322754
\(772\) 8246.06 0.384433
\(773\) 16082.6 0.748319 0.374159 0.927364i \(-0.377931\pi\)
0.374159 + 0.927364i \(0.377931\pi\)
\(774\) 2085.31 0.0968409
\(775\) 15142.7 0.701859
\(776\) 3358.77 0.155377
\(777\) 1492.06 0.0688899
\(778\) 8370.46 0.385727
\(779\) 0 0
\(780\) 408.523 0.0187531
\(781\) 61615.6 2.82302
\(782\) 37446.1 1.71237
\(783\) 1787.99 0.0816061
\(784\) −5325.38 −0.242592
\(785\) 4204.61 0.191171
\(786\) −3585.28 −0.162701
\(787\) 30058.0 1.36144 0.680720 0.732544i \(-0.261667\pi\)
0.680720 + 0.732544i \(0.261667\pi\)
\(788\) 8379.29 0.378807
\(789\) −21415.1 −0.966285
\(790\) −3263.02 −0.146953
\(791\) 1859.29 0.0835764
\(792\) 4996.82 0.224185
\(793\) −1777.09 −0.0795790
\(794\) −9059.52 −0.404925
\(795\) 4673.74 0.208504
\(796\) −6869.45 −0.305881
\(797\) −20008.2 −0.889243 −0.444621 0.895719i \(-0.646662\pi\)
−0.444621 + 0.895719i \(0.646662\pi\)
\(798\) 0 0
\(799\) −65801.8 −2.91352
\(800\) −3438.71 −0.151971
\(801\) −3832.19 −0.169043
\(802\) −18994.1 −0.836291
\(803\) 13811.8 0.606983
\(804\) 1744.11 0.0765050
\(805\) −2356.56 −0.103178
\(806\) 2290.89 0.100116
\(807\) 16340.0 0.712757
\(808\) −9933.68 −0.432507
\(809\) 35302.3 1.53419 0.767097 0.641532i \(-0.221701\pi\)
0.767097 + 0.641532i \(0.221701\pi\)
\(810\) 678.473 0.0294310
\(811\) 10980.8 0.475446 0.237723 0.971333i \(-0.423599\pi\)
0.237723 + 0.971333i \(0.423599\pi\)
\(812\) −844.489 −0.0364972
\(813\) 10211.0 0.440488
\(814\) −21653.3 −0.932369
\(815\) 5327.21 0.228962
\(816\) 5092.01 0.218451
\(817\) 0 0
\(818\) −10708.9 −0.457735
\(819\) −233.234 −0.00995100
\(820\) −6944.93 −0.295765
\(821\) −5626.68 −0.239187 −0.119594 0.992823i \(-0.538159\pi\)
−0.119594 + 0.992823i \(0.538159\pi\)
\(822\) 10663.9 0.452487
\(823\) 28766.3 1.21838 0.609191 0.793023i \(-0.291494\pi\)
0.609191 + 0.793023i \(0.291494\pi\)
\(824\) −4748.85 −0.200769
\(825\) 22373.2 0.944164
\(826\) 584.653 0.0246280
\(827\) 44482.4 1.87038 0.935191 0.354145i \(-0.115228\pi\)
0.935191 + 0.354145i \(0.115228\pi\)
\(828\) −6353.77 −0.266677
\(829\) 23302.5 0.976271 0.488136 0.872768i \(-0.337677\pi\)
0.488136 + 0.872768i \(0.337677\pi\)
\(830\) 3183.28 0.133125
\(831\) 15395.8 0.642688
\(832\) −520.232 −0.0216777
\(833\) 35308.4 1.46862
\(834\) −981.403 −0.0407472
\(835\) −12501.4 −0.518120
\(836\) 0 0
\(837\) 3804.70 0.157120
\(838\) 29012.2 1.19596
\(839\) −27683.7 −1.13915 −0.569575 0.821939i \(-0.692892\pi\)
−0.569575 + 0.821939i \(0.692892\pi\)
\(840\) −320.451 −0.0131626
\(841\) −20003.7 −0.820192
\(842\) −2526.90 −0.103424
\(843\) −10100.0 −0.412650
\(844\) −9164.21 −0.373750
\(845\) −8924.54 −0.363330
\(846\) 11165.1 0.453740
\(847\) 11111.8 0.450776
\(848\) −5951.77 −0.241020
\(849\) −20057.3 −0.810794
\(850\) 22799.4 0.920017
\(851\) 27533.5 1.10909
\(852\) −10654.0 −0.428402
\(853\) −30609.5 −1.22866 −0.614331 0.789049i \(-0.710574\pi\)
−0.614331 + 0.789049i \(0.710574\pi\)
\(854\) 1393.97 0.0558556
\(855\) 0 0
\(856\) −14228.5 −0.568130
\(857\) 10918.3 0.435194 0.217597 0.976039i \(-0.430178\pi\)
0.217597 + 0.976039i \(0.430178\pi\)
\(858\) 3384.78 0.134679
\(859\) 14639.5 0.581482 0.290741 0.956802i \(-0.406098\pi\)
0.290741 + 0.956802i \(0.406098\pi\)
\(860\) 1940.77 0.0769533
\(861\) 3965.01 0.156942
\(862\) −16375.2 −0.647033
\(863\) −22775.2 −0.898353 −0.449176 0.893443i \(-0.648283\pi\)
−0.449176 + 0.893443i \(0.648283\pi\)
\(864\) −864.000 −0.0340207
\(865\) 237.778 0.00934648
\(866\) −18485.9 −0.725377
\(867\) −19022.2 −0.745129
\(868\) −1797.00 −0.0702700
\(869\) −27035.4 −1.05537
\(870\) 1664.06 0.0648472
\(871\) 1181.43 0.0459603
\(872\) −8559.92 −0.332426
\(873\) 3778.61 0.146491
\(874\) 0 0
\(875\) −3103.83 −0.119918
\(876\) −2388.19 −0.0921114
\(877\) 805.908 0.0310303 0.0155152 0.999880i \(-0.495061\pi\)
0.0155152 + 0.999880i \(0.495061\pi\)
\(878\) −4086.10 −0.157061
\(879\) 16877.8 0.647638
\(880\) 4650.49 0.178146
\(881\) −901.690 −0.0344821 −0.0172410 0.999851i \(-0.505488\pi\)
−0.0172410 + 0.999851i \(0.505488\pi\)
\(882\) −5991.05 −0.228718
\(883\) −25406.3 −0.968279 −0.484140 0.874991i \(-0.660867\pi\)
−0.484140 + 0.874991i \(0.660867\pi\)
\(884\) 3449.26 0.131234
\(885\) −1152.06 −0.0437582
\(886\) −19338.3 −0.733275
\(887\) 28560.5 1.08114 0.540568 0.841300i \(-0.318209\pi\)
0.540568 + 0.841300i \(0.318209\pi\)
\(888\) 3744.07 0.141490
\(889\) 5290.19 0.199581
\(890\) −3566.58 −0.134328
\(891\) 5621.42 0.211363
\(892\) 13024.8 0.488902
\(893\) 0 0
\(894\) 6718.79 0.251353
\(895\) −19436.9 −0.725927
\(896\) 408.077 0.0152153
\(897\) −4303.95 −0.160206
\(898\) −26454.4 −0.983067
\(899\) 9331.64 0.346193
\(900\) −3868.55 −0.143280
\(901\) 39461.6 1.45911
\(902\) −57541.6 −2.12408
\(903\) −1108.03 −0.0408338
\(904\) 4665.58 0.171654
\(905\) 15811.8 0.580776
\(906\) 16842.9 0.617624
\(907\) −40905.4 −1.49751 −0.748755 0.662847i \(-0.769348\pi\)
−0.748755 + 0.662847i \(0.769348\pi\)
\(908\) −3992.18 −0.145909
\(909\) −11175.4 −0.407772
\(910\) −217.069 −0.00790743
\(911\) 30047.9 1.09279 0.546395 0.837527i \(-0.316000\pi\)
0.546395 + 0.837527i \(0.316000\pi\)
\(912\) 0 0
\(913\) 26374.8 0.956055
\(914\) −10762.0 −0.389468
\(915\) −2746.82 −0.0992426
\(916\) −4114.34 −0.148408
\(917\) 1905.04 0.0686041
\(918\) 5728.51 0.205958
\(919\) −49600.1 −1.78036 −0.890182 0.455605i \(-0.849423\pi\)
−0.890182 + 0.455605i \(0.849423\pi\)
\(920\) −5913.39 −0.211912
\(921\) −16810.6 −0.601443
\(922\) −5689.33 −0.203219
\(923\) −7216.84 −0.257362
\(924\) −2655.07 −0.0945295
\(925\) 16764.1 0.595891
\(926\) 13250.8 0.470246
\(927\) −5342.46 −0.189287
\(928\) −2119.10 −0.0749600
\(929\) 27556.8 0.973205 0.486603 0.873623i \(-0.338236\pi\)
0.486603 + 0.873623i \(0.338236\pi\)
\(930\) 3541.00 0.124854
\(931\) 0 0
\(932\) −501.945 −0.0176414
\(933\) −20004.6 −0.701952
\(934\) −37270.1 −1.30569
\(935\) −30833.8 −1.07847
\(936\) −585.261 −0.0204379
\(937\) −26880.8 −0.937202 −0.468601 0.883410i \(-0.655242\pi\)
−0.468601 + 0.883410i \(0.655242\pi\)
\(938\) −926.734 −0.0322590
\(939\) −13474.2 −0.468279
\(940\) 10391.2 0.360558
\(941\) −6411.34 −0.222108 −0.111054 0.993814i \(-0.535423\pi\)
−0.111054 + 0.993814i \(0.535423\pi\)
\(942\) −6023.65 −0.208345
\(943\) 73167.6 2.52669
\(944\) 1467.09 0.0505822
\(945\) −360.507 −0.0124098
\(946\) 16080.1 0.552653
\(947\) −1989.83 −0.0682796 −0.0341398 0.999417i \(-0.510869\pi\)
−0.0341398 + 0.999417i \(0.510869\pi\)
\(948\) 4674.69 0.160155
\(949\) −1617.73 −0.0553358
\(950\) 0 0
\(951\) 3798.14 0.129509
\(952\) −2705.64 −0.0921118
\(953\) −38266.4 −1.30070 −0.650351 0.759633i \(-0.725378\pi\)
−0.650351 + 0.759633i \(0.725378\pi\)
\(954\) −6695.74 −0.227235
\(955\) −11569.8 −0.392030
\(956\) 14364.1 0.485950
\(957\) 13787.5 0.465711
\(958\) 3113.24 0.104994
\(959\) −5666.25 −0.190795
\(960\) −804.116 −0.0270341
\(961\) −9934.01 −0.333457
\(962\) 2536.18 0.0849998
\(963\) −16007.0 −0.535638
\(964\) −14767.4 −0.493386
\(965\) 8633.84 0.288014
\(966\) 3376.08 0.112447
\(967\) 220.116 0.00732002 0.00366001 0.999993i \(-0.498835\pi\)
0.00366001 + 0.999993i \(0.498835\pi\)
\(968\) 27883.2 0.925827
\(969\) 0 0
\(970\) 3516.72 0.116407
\(971\) 2175.36 0.0718955 0.0359478 0.999354i \(-0.488555\pi\)
0.0359478 + 0.999354i \(0.488555\pi\)
\(972\) −972.000 −0.0320750
\(973\) 521.469 0.0171814
\(974\) 38679.0 1.27244
\(975\) −2620.50 −0.0860751
\(976\) 3497.93 0.114719
\(977\) 13707.7 0.448872 0.224436 0.974489i \(-0.427946\pi\)
0.224436 + 0.974489i \(0.427946\pi\)
\(978\) −7631.91 −0.249531
\(979\) −29550.6 −0.964699
\(980\) −5575.81 −0.181748
\(981\) −9629.91 −0.313414
\(982\) −13676.5 −0.444434
\(983\) −18885.7 −0.612779 −0.306389 0.951906i \(-0.599121\pi\)
−0.306389 + 0.951906i \(0.599121\pi\)
\(984\) 9949.50 0.322336
\(985\) 8773.33 0.283799
\(986\) 14050.1 0.453800
\(987\) −5932.58 −0.191323
\(988\) 0 0
\(989\) −20446.8 −0.657403
\(990\) 5231.80 0.167957
\(991\) −40655.0 −1.30318 −0.651589 0.758572i \(-0.725897\pi\)
−0.651589 + 0.758572i \(0.725897\pi\)
\(992\) −4509.27 −0.144324
\(993\) 15204.3 0.485894
\(994\) 5660.99 0.180639
\(995\) −7192.49 −0.229163
\(996\) −4560.46 −0.145084
\(997\) −16433.3 −0.522014 −0.261007 0.965337i \(-0.584055\pi\)
−0.261007 + 0.965337i \(0.584055\pi\)
\(998\) −36229.9 −1.14914
\(999\) 4212.08 0.133398
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 2166.4.a.u.1.2 3
19.7 even 3 114.4.e.d.49.2 yes 6
19.11 even 3 114.4.e.d.7.2 6
19.18 odd 2 2166.4.a.t.1.2 3
57.11 odd 6 342.4.g.h.235.2 6
57.26 odd 6 342.4.g.h.163.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.4.e.d.7.2 6 19.11 even 3
114.4.e.d.49.2 yes 6 19.7 even 3
342.4.g.h.163.2 6 57.26 odd 6
342.4.g.h.235.2 6 57.11 odd 6
2166.4.a.t.1.2 3 19.18 odd 2
2166.4.a.u.1.2 3 1.1 even 1 trivial