Properties

Label 165.4.a.d.1.3
Level $165$
Weight $4$
Character 165.1
Self dual yes
Analytic conductor $9.735$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.73531515095\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.23612.1
Defining polynomial: \(x^{3} - x^{2} - 20 x + 26\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-4.59056\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.59056 q^{2} -3.00000 q^{3} +4.89212 q^{4} -5.00000 q^{5} -10.7717 q^{6} -16.1465 q^{7} -11.1590 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.59056 q^{2} -3.00000 q^{3} +4.89212 q^{4} -5.00000 q^{5} -10.7717 q^{6} -16.1465 q^{7} -11.1590 q^{8} +9.00000 q^{9} -17.9528 q^{10} +11.0000 q^{11} -14.6764 q^{12} -54.1214 q^{13} -57.9749 q^{14} +15.0000 q^{15} -79.2041 q^{16} -107.010 q^{17} +32.3150 q^{18} +48.7496 q^{19} -24.4606 q^{20} +48.4394 q^{21} +39.4962 q^{22} +11.9498 q^{23} +33.4771 q^{24} +25.0000 q^{25} -194.326 q^{26} -27.0000 q^{27} -78.9905 q^{28} +239.733 q^{29} +53.8584 q^{30} -82.0851 q^{31} -195.115 q^{32} -33.0000 q^{33} -384.224 q^{34} +80.7324 q^{35} +44.0291 q^{36} -21.7573 q^{37} +175.038 q^{38} +162.364 q^{39} +55.7952 q^{40} -124.835 q^{41} +173.925 q^{42} +224.459 q^{43} +53.8133 q^{44} -45.0000 q^{45} +42.9064 q^{46} -186.832 q^{47} +237.612 q^{48} -82.2913 q^{49} +89.7640 q^{50} +321.029 q^{51} -264.768 q^{52} +233.997 q^{53} -96.9451 q^{54} -55.0000 q^{55} +180.179 q^{56} -146.249 q^{57} +860.774 q^{58} +232.936 q^{59} +73.3818 q^{60} +163.849 q^{61} -294.731 q^{62} -145.318 q^{63} -66.9386 q^{64} +270.607 q^{65} -118.488 q^{66} -876.918 q^{67} -523.503 q^{68} -35.8493 q^{69} +289.874 q^{70} -733.141 q^{71} -100.431 q^{72} +1161.97 q^{73} -78.1208 q^{74} -75.0000 q^{75} +238.489 q^{76} -177.611 q^{77} +582.978 q^{78} -588.831 q^{79} +396.021 q^{80} +81.0000 q^{81} -448.226 q^{82} -1161.06 q^{83} +236.971 q^{84} +535.048 q^{85} +805.933 q^{86} -719.198 q^{87} -122.749 q^{88} -1042.16 q^{89} -161.575 q^{90} +873.869 q^{91} +58.4597 q^{92} +246.255 q^{93} -670.831 q^{94} -243.748 q^{95} +585.345 q^{96} +1546.63 q^{97} -295.472 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} + 12 q^{6} - 4 q^{7} - 48 q^{8} + 27 q^{9} + O(q^{10}) \) \( 3 q - 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} + 12 q^{6} - 4 q^{7} - 48 q^{8} + 27 q^{9} + 20 q^{10} + 33 q^{11} - 66 q^{12} - 56 q^{14} + 45 q^{15} + 50 q^{16} - 218 q^{17} - 36 q^{18} + 146 q^{19} - 110 q^{20} + 12 q^{21} - 44 q^{22} - 200 q^{23} + 144 q^{24} + 75 q^{25} - 508 q^{26} - 81 q^{27} - 340 q^{28} + 68 q^{29} - 60 q^{30} - 68 q^{31} - 688 q^{32} - 99 q^{33} - 176 q^{34} + 20 q^{35} + 198 q^{36} - 390 q^{37} - 316 q^{38} + 240 q^{40} - 196 q^{41} + 168 q^{42} - 524 q^{43} + 242 q^{44} - 135 q^{45} + 1160 q^{46} - 60 q^{47} - 150 q^{48} - 157 q^{49} - 100 q^{50} + 654 q^{51} + 1020 q^{52} - 158 q^{53} + 108 q^{54} - 165 q^{55} + 1368 q^{56} - 438 q^{57} + 1092 q^{58} - 1044 q^{59} + 330 q^{60} + 642 q^{61} + 88 q^{62} - 36 q^{63} + 1166 q^{64} + 132 q^{66} - 236 q^{67} + 144 q^{68} + 600 q^{69} + 280 q^{70} - 544 q^{71} - 432 q^{72} + 900 q^{73} + 1536 q^{74} - 225 q^{75} + 1996 q^{76} - 44 q^{77} + 1524 q^{78} - 1586 q^{79} - 250 q^{80} + 243 q^{81} + 380 q^{82} - 1582 q^{83} + 1020 q^{84} + 1090 q^{85} + 3568 q^{86} - 204 q^{87} - 528 q^{88} - 2122 q^{89} + 180 q^{90} - 8 q^{91} - 4128 q^{92} + 204 q^{93} - 2152 q^{94} - 730 q^{95} + 2064 q^{96} + 618 q^{97} + 572 q^{98} + 297 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.59056 1.26945 0.634727 0.772736i \(-0.281112\pi\)
0.634727 + 0.772736i \(0.281112\pi\)
\(3\) −3.00000 −0.577350
\(4\) 4.89212 0.611515
\(5\) −5.00000 −0.447214
\(6\) −10.7717 −0.732920
\(7\) −16.1465 −0.871828 −0.435914 0.899988i \(-0.643575\pi\)
−0.435914 + 0.899988i \(0.643575\pi\)
\(8\) −11.1590 −0.493164
\(9\) 9.00000 0.333333
\(10\) −17.9528 −0.567717
\(11\) 11.0000 0.301511
\(12\) −14.6764 −0.353058
\(13\) −54.1214 −1.15466 −0.577329 0.816511i \(-0.695905\pi\)
−0.577329 + 0.816511i \(0.695905\pi\)
\(14\) −57.9749 −1.10675
\(15\) 15.0000 0.258199
\(16\) −79.2041 −1.23756
\(17\) −107.010 −1.52668 −0.763342 0.645995i \(-0.776443\pi\)
−0.763342 + 0.645995i \(0.776443\pi\)
\(18\) 32.3150 0.423152
\(19\) 48.7496 0.588628 0.294314 0.955709i \(-0.404909\pi\)
0.294314 + 0.955709i \(0.404909\pi\)
\(20\) −24.4606 −0.273478
\(21\) 48.4394 0.503350
\(22\) 39.4962 0.382755
\(23\) 11.9498 0.108335 0.0541674 0.998532i \(-0.482750\pi\)
0.0541674 + 0.998532i \(0.482750\pi\)
\(24\) 33.4771 0.284729
\(25\) 25.0000 0.200000
\(26\) −194.326 −1.46579
\(27\) −27.0000 −0.192450
\(28\) −78.9905 −0.533136
\(29\) 239.733 1.53508 0.767538 0.641003i \(-0.221481\pi\)
0.767538 + 0.641003i \(0.221481\pi\)
\(30\) 53.8584 0.327772
\(31\) −82.0851 −0.475578 −0.237789 0.971317i \(-0.576423\pi\)
−0.237789 + 0.971317i \(0.576423\pi\)
\(32\) −195.115 −1.07787
\(33\) −33.0000 −0.174078
\(34\) −384.224 −1.93806
\(35\) 80.7324 0.389893
\(36\) 44.0291 0.203838
\(37\) −21.7573 −0.0966723 −0.0483361 0.998831i \(-0.515392\pi\)
−0.0483361 + 0.998831i \(0.515392\pi\)
\(38\) 175.038 0.747236
\(39\) 162.364 0.666643
\(40\) 55.7952 0.220550
\(41\) −124.835 −0.475510 −0.237755 0.971325i \(-0.576412\pi\)
−0.237755 + 0.971325i \(0.576412\pi\)
\(42\) 173.925 0.638980
\(43\) 224.459 0.796039 0.398019 0.917377i \(-0.369698\pi\)
0.398019 + 0.917377i \(0.369698\pi\)
\(44\) 53.8133 0.184379
\(45\) −45.0000 −0.149071
\(46\) 42.9064 0.137526
\(47\) −186.832 −0.579835 −0.289917 0.957052i \(-0.593628\pi\)
−0.289917 + 0.957052i \(0.593628\pi\)
\(48\) 237.612 0.714508
\(49\) −82.2913 −0.239916
\(50\) 89.7640 0.253891
\(51\) 321.029 0.881431
\(52\) −264.768 −0.706091
\(53\) 233.997 0.606453 0.303226 0.952919i \(-0.401936\pi\)
0.303226 + 0.952919i \(0.401936\pi\)
\(54\) −96.9451 −0.244307
\(55\) −55.0000 −0.134840
\(56\) 180.179 0.429954
\(57\) −146.249 −0.339844
\(58\) 860.774 1.94871
\(59\) 232.936 0.513996 0.256998 0.966412i \(-0.417267\pi\)
0.256998 + 0.966412i \(0.417267\pi\)
\(60\) 73.3818 0.157892
\(61\) 163.849 0.343913 0.171957 0.985105i \(-0.444991\pi\)
0.171957 + 0.985105i \(0.444991\pi\)
\(62\) −294.731 −0.603725
\(63\) −145.318 −0.290609
\(64\) −66.9386 −0.130739
\(65\) 270.607 0.516379
\(66\) −118.488 −0.220984
\(67\) −876.918 −1.59899 −0.799497 0.600670i \(-0.794900\pi\)
−0.799497 + 0.600670i \(0.794900\pi\)
\(68\) −523.503 −0.933590
\(69\) −35.8493 −0.0625471
\(70\) 289.874 0.494952
\(71\) −733.141 −1.22546 −0.612731 0.790291i \(-0.709929\pi\)
−0.612731 + 0.790291i \(0.709929\pi\)
\(72\) −100.431 −0.164388
\(73\) 1161.97 1.86299 0.931496 0.363750i \(-0.118504\pi\)
0.931496 + 0.363750i \(0.118504\pi\)
\(74\) −78.1208 −0.122721
\(75\) −75.0000 −0.115470
\(76\) 238.489 0.359954
\(77\) −177.611 −0.262866
\(78\) 582.978 0.846272
\(79\) −588.831 −0.838591 −0.419296 0.907850i \(-0.637723\pi\)
−0.419296 + 0.907850i \(0.637723\pi\)
\(80\) 396.021 0.553456
\(81\) 81.0000 0.111111
\(82\) −448.226 −0.603638
\(83\) −1161.06 −1.53546 −0.767731 0.640772i \(-0.778614\pi\)
−0.767731 + 0.640772i \(0.778614\pi\)
\(84\) 236.971 0.307806
\(85\) 535.048 0.682754
\(86\) 805.933 1.01053
\(87\) −719.198 −0.886277
\(88\) −122.749 −0.148695
\(89\) −1042.16 −1.24122 −0.620610 0.784120i \(-0.713115\pi\)
−0.620610 + 0.784120i \(0.713115\pi\)
\(90\) −161.575 −0.189239
\(91\) 873.869 1.00666
\(92\) 58.4597 0.0662483
\(93\) 246.255 0.274575
\(94\) −670.831 −0.736074
\(95\) −243.748 −0.263242
\(96\) 585.345 0.622307
\(97\) 1546.63 1.61893 0.809464 0.587169i \(-0.199758\pi\)
0.809464 + 0.587169i \(0.199758\pi\)
\(98\) −295.472 −0.304563
\(99\) 99.0000 0.100504
\(100\) 122.303 0.122303
\(101\) −662.282 −0.652470 −0.326235 0.945289i \(-0.605780\pi\)
−0.326235 + 0.945289i \(0.605780\pi\)
\(102\) 1152.67 1.11894
\(103\) 399.592 0.382262 0.191131 0.981565i \(-0.438784\pi\)
0.191131 + 0.981565i \(0.438784\pi\)
\(104\) 603.942 0.569436
\(105\) −242.197 −0.225105
\(106\) 840.181 0.769864
\(107\) −1591.22 −1.43765 −0.718827 0.695189i \(-0.755321\pi\)
−0.718827 + 0.695189i \(0.755321\pi\)
\(108\) −132.087 −0.117686
\(109\) 755.128 0.663561 0.331780 0.943357i \(-0.392351\pi\)
0.331780 + 0.943357i \(0.392351\pi\)
\(110\) −197.481 −0.171173
\(111\) 65.2718 0.0558138
\(112\) 1278.87 1.07894
\(113\) −1145.65 −0.953753 −0.476876 0.878970i \(-0.658231\pi\)
−0.476876 + 0.878970i \(0.658231\pi\)
\(114\) −525.115 −0.431417
\(115\) −59.7488 −0.0484488
\(116\) 1172.80 0.938722
\(117\) −487.092 −0.384886
\(118\) 836.372 0.652494
\(119\) 1727.83 1.33101
\(120\) −167.386 −0.127334
\(121\) 121.000 0.0909091
\(122\) 588.309 0.436582
\(123\) 374.504 0.274536
\(124\) −401.570 −0.290823
\(125\) −125.000 −0.0894427
\(126\) −521.774 −0.368915
\(127\) 1461.29 1.02101 0.510505 0.859875i \(-0.329458\pi\)
0.510505 + 0.859875i \(0.329458\pi\)
\(128\) 1320.57 0.911900
\(129\) −673.377 −0.459593
\(130\) 971.630 0.655520
\(131\) −1524.94 −1.01706 −0.508528 0.861045i \(-0.669810\pi\)
−0.508528 + 0.861045i \(0.669810\pi\)
\(132\) −161.440 −0.106451
\(133\) −787.134 −0.513182
\(134\) −3148.63 −2.02985
\(135\) 135.000 0.0860663
\(136\) 1194.12 0.752906
\(137\) −2125.68 −1.32561 −0.662805 0.748792i \(-0.730634\pi\)
−0.662805 + 0.748792i \(0.730634\pi\)
\(138\) −128.719 −0.0794007
\(139\) −1774.28 −1.08268 −0.541339 0.840805i \(-0.682082\pi\)
−0.541339 + 0.840805i \(0.682082\pi\)
\(140\) 394.952 0.238425
\(141\) 560.496 0.334768
\(142\) −2632.39 −1.55567
\(143\) −595.335 −0.348143
\(144\) −712.837 −0.412521
\(145\) −1198.66 −0.686507
\(146\) 4172.13 2.36498
\(147\) 246.874 0.138516
\(148\) −106.439 −0.0591165
\(149\) 1575.78 0.866393 0.433197 0.901299i \(-0.357386\pi\)
0.433197 + 0.901299i \(0.357386\pi\)
\(150\) −269.292 −0.146584
\(151\) 420.978 0.226879 0.113439 0.993545i \(-0.463813\pi\)
0.113439 + 0.993545i \(0.463813\pi\)
\(152\) −543.998 −0.290290
\(153\) −963.086 −0.508895
\(154\) −637.724 −0.333696
\(155\) 410.425 0.212685
\(156\) 794.304 0.407662
\(157\) −2224.30 −1.13069 −0.565345 0.824854i \(-0.691257\pi\)
−0.565345 + 0.824854i \(0.691257\pi\)
\(158\) −2114.23 −1.06455
\(159\) −701.992 −0.350136
\(160\) 975.574 0.482037
\(161\) −192.947 −0.0944492
\(162\) 290.835 0.141051
\(163\) −3093.37 −1.48645 −0.743226 0.669040i \(-0.766705\pi\)
−0.743226 + 0.669040i \(0.766705\pi\)
\(164\) −610.706 −0.290781
\(165\) 165.000 0.0778499
\(166\) −4168.87 −1.94920
\(167\) −2416.43 −1.11970 −0.559848 0.828595i \(-0.689140\pi\)
−0.559848 + 0.828595i \(0.689140\pi\)
\(168\) −540.537 −0.248234
\(169\) 732.122 0.333237
\(170\) 1921.12 0.866725
\(171\) 438.746 0.196209
\(172\) 1098.08 0.486789
\(173\) −3758.02 −1.65154 −0.825771 0.564005i \(-0.809260\pi\)
−0.825771 + 0.564005i \(0.809260\pi\)
\(174\) −2582.32 −1.12509
\(175\) −403.662 −0.174366
\(176\) −871.245 −0.373140
\(177\) −698.809 −0.296756
\(178\) −3741.93 −1.57567
\(179\) 2533.99 1.05810 0.529049 0.848591i \(-0.322549\pi\)
0.529049 + 0.848591i \(0.322549\pi\)
\(180\) −220.145 −0.0911592
\(181\) −13.8995 −0.00570798 −0.00285399 0.999996i \(-0.500908\pi\)
−0.00285399 + 0.999996i \(0.500908\pi\)
\(182\) 3137.68 1.27791
\(183\) −491.547 −0.198558
\(184\) −133.348 −0.0534268
\(185\) 108.786 0.0432332
\(186\) 884.194 0.348561
\(187\) −1177.10 −0.460312
\(188\) −914.004 −0.354577
\(189\) 435.955 0.167783
\(190\) −875.192 −0.334174
\(191\) 3495.39 1.32417 0.662087 0.749427i \(-0.269671\pi\)
0.662087 + 0.749427i \(0.269671\pi\)
\(192\) 200.816 0.0754824
\(193\) 3469.33 1.29393 0.646963 0.762522i \(-0.276039\pi\)
0.646963 + 0.762522i \(0.276039\pi\)
\(194\) 5553.25 2.05516
\(195\) −811.820 −0.298132
\(196\) −402.579 −0.146712
\(197\) 3638.39 1.31586 0.657930 0.753079i \(-0.271432\pi\)
0.657930 + 0.753079i \(0.271432\pi\)
\(198\) 355.465 0.127585
\(199\) 51.6049 0.0183828 0.00919140 0.999958i \(-0.497074\pi\)
0.00919140 + 0.999958i \(0.497074\pi\)
\(200\) −278.976 −0.0986329
\(201\) 2630.75 0.923179
\(202\) −2377.96 −0.828281
\(203\) −3870.84 −1.33832
\(204\) 1570.51 0.539008
\(205\) 624.173 0.212654
\(206\) 1434.76 0.485265
\(207\) 107.548 0.0361116
\(208\) 4286.63 1.42896
\(209\) 536.246 0.177478
\(210\) −869.623 −0.285761
\(211\) 2084.57 0.680131 0.340065 0.940402i \(-0.389551\pi\)
0.340065 + 0.940402i \(0.389551\pi\)
\(212\) 1144.74 0.370855
\(213\) 2199.42 0.707521
\(214\) −5713.37 −1.82504
\(215\) −1122.29 −0.355999
\(216\) 301.294 0.0949095
\(217\) 1325.38 0.414622
\(218\) 2711.33 0.842361
\(219\) −3485.91 −1.07560
\(220\) −269.067 −0.0824566
\(221\) 5791.50 1.76280
\(222\) 234.362 0.0708531
\(223\) 1887.36 0.566757 0.283378 0.959008i \(-0.408545\pi\)
0.283378 + 0.959008i \(0.408545\pi\)
\(224\) 3150.42 0.939715
\(225\) 225.000 0.0666667
\(226\) −4113.54 −1.21075
\(227\) −1150.73 −0.336462 −0.168231 0.985748i \(-0.553806\pi\)
−0.168231 + 0.985748i \(0.553806\pi\)
\(228\) −715.466 −0.207820
\(229\) 4106.79 1.18508 0.592542 0.805540i \(-0.298125\pi\)
0.592542 + 0.805540i \(0.298125\pi\)
\(230\) −214.532 −0.0615035
\(231\) 532.834 0.151766
\(232\) −2675.18 −0.757045
\(233\) −5733.58 −1.61210 −0.806050 0.591848i \(-0.798399\pi\)
−0.806050 + 0.591848i \(0.798399\pi\)
\(234\) −1748.93 −0.488596
\(235\) 934.159 0.259310
\(236\) 1139.55 0.314316
\(237\) 1766.49 0.484161
\(238\) 6203.86 1.68965
\(239\) 6036.18 1.63367 0.816837 0.576868i \(-0.195725\pi\)
0.816837 + 0.576868i \(0.195725\pi\)
\(240\) −1188.06 −0.319538
\(241\) 3720.90 0.994540 0.497270 0.867596i \(-0.334336\pi\)
0.497270 + 0.867596i \(0.334336\pi\)
\(242\) 434.458 0.115405
\(243\) −243.000 −0.0641500
\(244\) 801.568 0.210308
\(245\) 411.457 0.107294
\(246\) 1344.68 0.348511
\(247\) −2638.39 −0.679664
\(248\) 915.990 0.234538
\(249\) 3483.19 0.886500
\(250\) −448.820 −0.113543
\(251\) −3809.88 −0.958077 −0.479038 0.877794i \(-0.659015\pi\)
−0.479038 + 0.877794i \(0.659015\pi\)
\(252\) −710.914 −0.177712
\(253\) 131.447 0.0326641
\(254\) 5246.84 1.29613
\(255\) −1605.14 −0.394188
\(256\) 5277.10 1.28835
\(257\) 1225.95 0.297559 0.148780 0.988870i \(-0.452465\pi\)
0.148780 + 0.988870i \(0.452465\pi\)
\(258\) −2417.80 −0.583433
\(259\) 351.303 0.0842816
\(260\) 1323.84 0.315773
\(261\) 2157.59 0.511692
\(262\) −5475.38 −1.29111
\(263\) 7397.68 1.73445 0.867225 0.497916i \(-0.165901\pi\)
0.867225 + 0.497916i \(0.165901\pi\)
\(264\) 368.248 0.0858489
\(265\) −1169.99 −0.271214
\(266\) −2826.25 −0.651461
\(267\) 3126.47 0.716618
\(268\) −4289.99 −0.977808
\(269\) −3214.48 −0.728589 −0.364295 0.931284i \(-0.618690\pi\)
−0.364295 + 0.931284i \(0.618690\pi\)
\(270\) 484.726 0.109257
\(271\) −7377.37 −1.65367 −0.826833 0.562448i \(-0.809860\pi\)
−0.826833 + 0.562448i \(0.809860\pi\)
\(272\) 8475.60 1.88937
\(273\) −2621.61 −0.581197
\(274\) −7632.36 −1.68280
\(275\) 275.000 0.0603023
\(276\) −175.379 −0.0382485
\(277\) −810.606 −0.175829 −0.0879144 0.996128i \(-0.528020\pi\)
−0.0879144 + 0.996128i \(0.528020\pi\)
\(278\) −6370.65 −1.37441
\(279\) −738.766 −0.158526
\(280\) −900.895 −0.192281
\(281\) 1114.72 0.236651 0.118325 0.992975i \(-0.462247\pi\)
0.118325 + 0.992975i \(0.462247\pi\)
\(282\) 2012.49 0.424972
\(283\) 2265.40 0.475844 0.237922 0.971284i \(-0.423534\pi\)
0.237922 + 0.971284i \(0.423534\pi\)
\(284\) −3586.61 −0.749388
\(285\) 731.244 0.151983
\(286\) −2137.59 −0.441951
\(287\) 2015.64 0.414563
\(288\) −1756.03 −0.359289
\(289\) 6538.04 1.33076
\(290\) −4303.87 −0.871490
\(291\) −4639.88 −0.934689
\(292\) 5684.50 1.13925
\(293\) 3802.06 0.758084 0.379042 0.925380i \(-0.376254\pi\)
0.379042 + 0.925380i \(0.376254\pi\)
\(294\) 886.416 0.175840
\(295\) −1164.68 −0.229866
\(296\) 242.790 0.0476753
\(297\) −297.000 −0.0580259
\(298\) 5657.92 1.09985
\(299\) −646.738 −0.125090
\(300\) −366.909 −0.0706116
\(301\) −3624.22 −0.694009
\(302\) 1511.55 0.288013
\(303\) 1986.85 0.376704
\(304\) −3861.17 −0.728465
\(305\) −819.244 −0.153803
\(306\) −3458.02 −0.646019
\(307\) −1356.35 −0.252153 −0.126077 0.992021i \(-0.540239\pi\)
−0.126077 + 0.992021i \(0.540239\pi\)
\(308\) −868.895 −0.160746
\(309\) −1198.78 −0.220699
\(310\) 1473.66 0.269994
\(311\) −8078.07 −1.47288 −0.736440 0.676503i \(-0.763495\pi\)
−0.736440 + 0.676503i \(0.763495\pi\)
\(312\) −1811.83 −0.328764
\(313\) 5761.54 1.04045 0.520226 0.854029i \(-0.325848\pi\)
0.520226 + 0.854029i \(0.325848\pi\)
\(314\) −7986.48 −1.43536
\(315\) 726.591 0.129964
\(316\) −2880.63 −0.512811
\(317\) 5107.00 0.904851 0.452426 0.891802i \(-0.350559\pi\)
0.452426 + 0.891802i \(0.350559\pi\)
\(318\) −2520.54 −0.444481
\(319\) 2637.06 0.462843
\(320\) 334.693 0.0584684
\(321\) 4773.66 0.830030
\(322\) −692.786 −0.119899
\(323\) −5216.67 −0.898648
\(324\) 396.262 0.0679461
\(325\) −1353.03 −0.230932
\(326\) −11106.9 −1.88698
\(327\) −2265.38 −0.383107
\(328\) 1393.03 0.234504
\(329\) 3016.68 0.505516
\(330\) 592.442 0.0988269
\(331\) −2780.94 −0.461796 −0.230898 0.972978i \(-0.574166\pi\)
−0.230898 + 0.972978i \(0.574166\pi\)
\(332\) −5680.06 −0.938958
\(333\) −195.816 −0.0322241
\(334\) −8676.35 −1.42140
\(335\) 4384.59 0.715092
\(336\) −3836.60 −0.622928
\(337\) 4939.28 0.798397 0.399198 0.916865i \(-0.369288\pi\)
0.399198 + 0.916865i \(0.369288\pi\)
\(338\) 2628.73 0.423029
\(339\) 3436.96 0.550649
\(340\) 2617.52 0.417514
\(341\) −902.936 −0.143392
\(342\) 1575.34 0.249079
\(343\) 6866.96 1.08099
\(344\) −2504.74 −0.392578
\(345\) 179.247 0.0279719
\(346\) −13493.4 −2.09656
\(347\) −2711.58 −0.419496 −0.209748 0.977755i \(-0.567264\pi\)
−0.209748 + 0.977755i \(0.567264\pi\)
\(348\) −3518.40 −0.541971
\(349\) 5496.03 0.842967 0.421484 0.906836i \(-0.361510\pi\)
0.421484 + 0.906836i \(0.361510\pi\)
\(350\) −1449.37 −0.221349
\(351\) 1461.28 0.222214
\(352\) −2146.26 −0.324989
\(353\) 6372.23 0.960792 0.480396 0.877052i \(-0.340493\pi\)
0.480396 + 0.877052i \(0.340493\pi\)
\(354\) −2509.12 −0.376718
\(355\) 3665.71 0.548043
\(356\) −5098.36 −0.759024
\(357\) −5183.48 −0.768456
\(358\) 9098.46 1.34321
\(359\) 630.622 0.0927101 0.0463551 0.998925i \(-0.485239\pi\)
0.0463551 + 0.998925i \(0.485239\pi\)
\(360\) 502.157 0.0735166
\(361\) −4482.48 −0.653518
\(362\) −49.9071 −0.00724602
\(363\) −363.000 −0.0524864
\(364\) 4275.07 0.615590
\(365\) −5809.86 −0.833156
\(366\) −1764.93 −0.252061
\(367\) −5374.49 −0.764431 −0.382216 0.924073i \(-0.624839\pi\)
−0.382216 + 0.924073i \(0.624839\pi\)
\(368\) −946.471 −0.134071
\(369\) −1123.51 −0.158503
\(370\) 390.604 0.0548825
\(371\) −3778.23 −0.528722
\(372\) 1204.71 0.167907
\(373\) 7520.12 1.04391 0.521953 0.852974i \(-0.325204\pi\)
0.521953 + 0.852974i \(0.325204\pi\)
\(374\) −4226.46 −0.584346
\(375\) 375.000 0.0516398
\(376\) 2084.86 0.285954
\(377\) −12974.7 −1.77249
\(378\) 1565.32 0.212993
\(379\) −12509.1 −1.69538 −0.847690 0.530492i \(-0.822007\pi\)
−0.847690 + 0.530492i \(0.822007\pi\)
\(380\) −1192.44 −0.160977
\(381\) −4383.86 −0.589481
\(382\) 12550.4 1.68098
\(383\) −11149.9 −1.48755 −0.743775 0.668430i \(-0.766967\pi\)
−0.743775 + 0.668430i \(0.766967\pi\)
\(384\) −3961.72 −0.526486
\(385\) 888.056 0.117557
\(386\) 12456.8 1.64258
\(387\) 2020.13 0.265346
\(388\) 7566.28 0.989999
\(389\) 3194.22 0.416333 0.208166 0.978093i \(-0.433250\pi\)
0.208166 + 0.978093i \(0.433250\pi\)
\(390\) −2914.89 −0.378465
\(391\) −1278.74 −0.165393
\(392\) 918.292 0.118318
\(393\) 4574.81 0.587198
\(394\) 13063.8 1.67042
\(395\) 2944.16 0.375029
\(396\) 484.320 0.0614596
\(397\) −584.410 −0.0738809 −0.0369404 0.999317i \(-0.511761\pi\)
−0.0369404 + 0.999317i \(0.511761\pi\)
\(398\) 185.291 0.0233361
\(399\) 2361.40 0.296286
\(400\) −1980.10 −0.247513
\(401\) −6951.24 −0.865657 −0.432829 0.901476i \(-0.642484\pi\)
−0.432829 + 0.901476i \(0.642484\pi\)
\(402\) 9445.88 1.17193
\(403\) 4442.56 0.549130
\(404\) −3239.96 −0.398995
\(405\) −405.000 −0.0496904
\(406\) −13898.5 −1.69894
\(407\) −239.330 −0.0291478
\(408\) −3582.37 −0.434690
\(409\) 11754.1 1.42104 0.710519 0.703678i \(-0.248460\pi\)
0.710519 + 0.703678i \(0.248460\pi\)
\(410\) 2241.13 0.269955
\(411\) 6377.03 0.765342
\(412\) 1954.85 0.233759
\(413\) −3761.10 −0.448116
\(414\) 386.157 0.0458420
\(415\) 5805.32 0.686680
\(416\) 10559.9 1.24457
\(417\) 5322.83 0.625084
\(418\) 1925.42 0.225300
\(419\) 11829.8 1.37929 0.689646 0.724146i \(-0.257766\pi\)
0.689646 + 0.724146i \(0.257766\pi\)
\(420\) −1184.86 −0.137655
\(421\) −2000.07 −0.231538 −0.115769 0.993276i \(-0.536933\pi\)
−0.115769 + 0.993276i \(0.536933\pi\)
\(422\) 7484.76 0.863395
\(423\) −1681.49 −0.193278
\(424\) −2611.18 −0.299081
\(425\) −2675.24 −0.305337
\(426\) 7897.16 0.898166
\(427\) −2645.58 −0.299833
\(428\) −7784.43 −0.879147
\(429\) 1786.00 0.201000
\(430\) −4029.67 −0.451925
\(431\) −8064.11 −0.901240 −0.450620 0.892716i \(-0.648797\pi\)
−0.450620 + 0.892716i \(0.648797\pi\)
\(432\) 2138.51 0.238169
\(433\) −10710.3 −1.18869 −0.594345 0.804210i \(-0.702589\pi\)
−0.594345 + 0.804210i \(0.702589\pi\)
\(434\) 4758.87 0.526344
\(435\) 3595.99 0.396355
\(436\) 3694.18 0.405777
\(437\) 582.546 0.0637688
\(438\) −12516.4 −1.36542
\(439\) 4658.08 0.506419 0.253210 0.967411i \(-0.418514\pi\)
0.253210 + 0.967411i \(0.418514\pi\)
\(440\) 613.747 0.0664983
\(441\) −740.622 −0.0799721
\(442\) 20794.7 2.23779
\(443\) 2094.73 0.224658 0.112329 0.993671i \(-0.464169\pi\)
0.112329 + 0.993671i \(0.464169\pi\)
\(444\) 319.318 0.0341310
\(445\) 5210.79 0.555090
\(446\) 6776.67 0.719472
\(447\) −4727.33 −0.500212
\(448\) 1080.82 0.113982
\(449\) −1402.21 −0.147382 −0.0736909 0.997281i \(-0.523478\pi\)
−0.0736909 + 0.997281i \(0.523478\pi\)
\(450\) 807.876 0.0846303
\(451\) −1373.18 −0.143372
\(452\) −5604.67 −0.583234
\(453\) −1262.93 −0.130989
\(454\) −4131.78 −0.427124
\(455\) −4369.35 −0.450194
\(456\) 1632.00 0.167599
\(457\) −4914.19 −0.503011 −0.251506 0.967856i \(-0.580926\pi\)
−0.251506 + 0.967856i \(0.580926\pi\)
\(458\) 14745.7 1.50441
\(459\) 2889.26 0.293810
\(460\) −292.298 −0.0296271
\(461\) −2214.08 −0.223688 −0.111844 0.993726i \(-0.535676\pi\)
−0.111844 + 0.993726i \(0.535676\pi\)
\(462\) 1913.17 0.192660
\(463\) −5567.02 −0.558793 −0.279396 0.960176i \(-0.590134\pi\)
−0.279396 + 0.960176i \(0.590134\pi\)
\(464\) −18987.8 −1.89976
\(465\) −1231.28 −0.122794
\(466\) −20586.8 −2.04649
\(467\) 497.054 0.0492525 0.0246263 0.999697i \(-0.492160\pi\)
0.0246263 + 0.999697i \(0.492160\pi\)
\(468\) −2382.91 −0.235364
\(469\) 14159.1 1.39405
\(470\) 3354.15 0.329182
\(471\) 6672.90 0.652804
\(472\) −2599.35 −0.253484
\(473\) 2469.05 0.240015
\(474\) 6342.70 0.614620
\(475\) 1218.74 0.117726
\(476\) 8452.73 0.813929
\(477\) 2105.97 0.202151
\(478\) 21673.3 2.07388
\(479\) −9349.28 −0.891815 −0.445908 0.895079i \(-0.647119\pi\)
−0.445908 + 0.895079i \(0.647119\pi\)
\(480\) −2926.72 −0.278304
\(481\) 1177.53 0.111624
\(482\) 13360.1 1.26252
\(483\) 578.840 0.0545303
\(484\) 591.946 0.0555923
\(485\) −7733.13 −0.724007
\(486\) −872.506 −0.0814355
\(487\) 197.750 0.0184003 0.00920013 0.999958i \(-0.497071\pi\)
0.00920013 + 0.999958i \(0.497071\pi\)
\(488\) −1828.40 −0.169606
\(489\) 9280.12 0.858204
\(490\) 1477.36 0.136205
\(491\) −9997.05 −0.918861 −0.459430 0.888214i \(-0.651946\pi\)
−0.459430 + 0.888214i \(0.651946\pi\)
\(492\) 1832.12 0.167883
\(493\) −25653.7 −2.34358
\(494\) −9473.31 −0.862803
\(495\) −495.000 −0.0449467
\(496\) 6501.48 0.588558
\(497\) 11837.6 1.06839
\(498\) 12506.6 1.12537
\(499\) −8714.73 −0.781813 −0.390907 0.920430i \(-0.627838\pi\)
−0.390907 + 0.920430i \(0.627838\pi\)
\(500\) −611.515 −0.0546955
\(501\) 7249.30 0.646457
\(502\) −13679.6 −1.21623
\(503\) 5978.53 0.529959 0.264979 0.964254i \(-0.414635\pi\)
0.264979 + 0.964254i \(0.414635\pi\)
\(504\) 1621.61 0.143318
\(505\) 3311.41 0.291794
\(506\) 471.970 0.0414657
\(507\) −2196.36 −0.192394
\(508\) 7148.79 0.624363
\(509\) −8205.79 −0.714569 −0.357284 0.933996i \(-0.616297\pi\)
−0.357284 + 0.933996i \(0.616297\pi\)
\(510\) −5763.36 −0.500404
\(511\) −18761.7 −1.62421
\(512\) 8383.17 0.723608
\(513\) −1316.24 −0.113281
\(514\) 4401.85 0.377738
\(515\) −1997.96 −0.170953
\(516\) −3294.24 −0.281048
\(517\) −2055.15 −0.174827
\(518\) 1261.38 0.106992
\(519\) 11274.1 0.953519
\(520\) −3019.71 −0.254660
\(521\) −5266.06 −0.442822 −0.221411 0.975181i \(-0.571066\pi\)
−0.221411 + 0.975181i \(0.571066\pi\)
\(522\) 7746.97 0.649570
\(523\) −22398.6 −1.87270 −0.936350 0.351068i \(-0.885819\pi\)
−0.936350 + 0.351068i \(0.885819\pi\)
\(524\) −7460.18 −0.621945
\(525\) 1210.99 0.100670
\(526\) 26561.8 2.20181
\(527\) 8783.89 0.726057
\(528\) 2613.74 0.215432
\(529\) −12024.2 −0.988264
\(530\) −4200.90 −0.344294
\(531\) 2096.43 0.171332
\(532\) −3850.75 −0.313818
\(533\) 6756.22 0.549052
\(534\) 11225.8 0.909714
\(535\) 7956.10 0.642938
\(536\) 9785.56 0.788566
\(537\) −7601.98 −0.610893
\(538\) −11541.8 −0.924911
\(539\) −905.205 −0.0723375
\(540\) 660.436 0.0526308
\(541\) −13030.2 −1.03551 −0.517757 0.855528i \(-0.673233\pi\)
−0.517757 + 0.855528i \(0.673233\pi\)
\(542\) −26488.9 −2.09925
\(543\) 41.6986 0.00329550
\(544\) 20879.1 1.64556
\(545\) −3775.64 −0.296753
\(546\) −9413.04 −0.737804
\(547\) 10448.9 0.816747 0.408374 0.912815i \(-0.366096\pi\)
0.408374 + 0.912815i \(0.366096\pi\)
\(548\) −10399.1 −0.810631
\(549\) 1474.64 0.114638
\(550\) 987.404 0.0765510
\(551\) 11686.9 0.903588
\(552\) 400.044 0.0308460
\(553\) 9507.55 0.731107
\(554\) −2910.53 −0.223207
\(555\) −326.359 −0.0249607
\(556\) −8679.97 −0.662073
\(557\) −11448.7 −0.870911 −0.435456 0.900210i \(-0.643413\pi\)
−0.435456 + 0.900210i \(0.643413\pi\)
\(558\) −2652.58 −0.201242
\(559\) −12148.0 −0.919153
\(560\) −6394.34 −0.482518
\(561\) 3531.31 0.265762
\(562\) 4002.49 0.300418
\(563\) −26035.8 −1.94898 −0.974492 0.224422i \(-0.927951\pi\)
−0.974492 + 0.224422i \(0.927951\pi\)
\(564\) 2742.01 0.204715
\(565\) 5728.27 0.426531
\(566\) 8134.05 0.604063
\(567\) −1307.86 −0.0968697
\(568\) 8181.15 0.604354
\(569\) 25075.0 1.84745 0.923725 0.383057i \(-0.125129\pi\)
0.923725 + 0.383057i \(0.125129\pi\)
\(570\) 2625.57 0.192935
\(571\) 19056.6 1.39666 0.698330 0.715776i \(-0.253927\pi\)
0.698330 + 0.715776i \(0.253927\pi\)
\(572\) −2912.45 −0.212894
\(573\) −10486.2 −0.764512
\(574\) 7237.28 0.526268
\(575\) 298.744 0.0216669
\(576\) −602.447 −0.0435798
\(577\) −6482.55 −0.467716 −0.233858 0.972271i \(-0.575135\pi\)
−0.233858 + 0.972271i \(0.575135\pi\)
\(578\) 23475.2 1.68934
\(579\) −10408.0 −0.747048
\(580\) −5864.00 −0.419809
\(581\) 18747.1 1.33866
\(582\) −16659.8 −1.18654
\(583\) 2573.97 0.182852
\(584\) −12966.5 −0.918762
\(585\) 2435.46 0.172126
\(586\) 13651.5 0.962353
\(587\) 24650.3 1.73327 0.866633 0.498946i \(-0.166280\pi\)
0.866633 + 0.498946i \(0.166280\pi\)
\(588\) 1207.74 0.0847045
\(589\) −4001.61 −0.279938
\(590\) −4181.86 −0.291804
\(591\) −10915.2 −0.759712
\(592\) 1723.27 0.119638
\(593\) 23163.2 1.60405 0.802024 0.597292i \(-0.203757\pi\)
0.802024 + 0.597292i \(0.203757\pi\)
\(594\) −1066.40 −0.0736612
\(595\) −8639.13 −0.595244
\(596\) 7708.88 0.529812
\(597\) −154.815 −0.0106133
\(598\) −2322.15 −0.158796
\(599\) 5355.44 0.365304 0.182652 0.983178i \(-0.441532\pi\)
0.182652 + 0.983178i \(0.441532\pi\)
\(600\) 836.928 0.0569457
\(601\) −20417.6 −1.38578 −0.692889 0.721045i \(-0.743662\pi\)
−0.692889 + 0.721045i \(0.743662\pi\)
\(602\) −13013.0 −0.881012
\(603\) −7892.26 −0.532998
\(604\) 2059.48 0.138740
\(605\) −605.000 −0.0406558
\(606\) 7133.89 0.478208
\(607\) −6749.81 −0.451345 −0.225673 0.974203i \(-0.572458\pi\)
−0.225673 + 0.974203i \(0.572458\pi\)
\(608\) −9511.77 −0.634463
\(609\) 11612.5 0.772681
\(610\) −2941.55 −0.195245
\(611\) 10111.6 0.669511
\(612\) −4711.53 −0.311197
\(613\) −30321.0 −1.99780 −0.998901 0.0468613i \(-0.985078\pi\)
−0.998901 + 0.0468613i \(0.985078\pi\)
\(614\) −4870.06 −0.320097
\(615\) −1872.52 −0.122776
\(616\) 1981.97 0.129636
\(617\) 15236.8 0.994182 0.497091 0.867699i \(-0.334402\pi\)
0.497091 + 0.867699i \(0.334402\pi\)
\(618\) −4304.28 −0.280168
\(619\) −20875.4 −1.35550 −0.677749 0.735293i \(-0.737044\pi\)
−0.677749 + 0.735293i \(0.737044\pi\)
\(620\) 2007.85 0.130060
\(621\) −322.644 −0.0208490
\(622\) −29004.8 −1.86975
\(623\) 16827.2 1.08213
\(624\) −12859.9 −0.825013
\(625\) 625.000 0.0400000
\(626\) 20687.2 1.32081
\(627\) −1608.74 −0.102467
\(628\) −10881.5 −0.691434
\(629\) 2328.24 0.147588
\(630\) 2608.87 0.164984
\(631\) 27966.1 1.76437 0.882183 0.470907i \(-0.156073\pi\)
0.882183 + 0.470907i \(0.156073\pi\)
\(632\) 6570.79 0.413563
\(633\) −6253.70 −0.392674
\(634\) 18337.0 1.14867
\(635\) −7306.44 −0.456610
\(636\) −3434.23 −0.214113
\(637\) 4453.72 0.277022
\(638\) 9468.52 0.587558
\(639\) −6598.27 −0.408487
\(640\) −6602.86 −0.407814
\(641\) −17992.0 −1.10865 −0.554323 0.832301i \(-0.687023\pi\)
−0.554323 + 0.832301i \(0.687023\pi\)
\(642\) 17140.1 1.05369
\(643\) −9448.64 −0.579499 −0.289750 0.957102i \(-0.593572\pi\)
−0.289750 + 0.957102i \(0.593572\pi\)
\(644\) −943.918 −0.0577571
\(645\) 3366.88 0.205536
\(646\) −18730.8 −1.14079
\(647\) −7429.22 −0.451426 −0.225713 0.974194i \(-0.572471\pi\)
−0.225713 + 0.974194i \(0.572471\pi\)
\(648\) −903.882 −0.0547960
\(649\) 2562.30 0.154976
\(650\) −4858.15 −0.293157
\(651\) −3976.15 −0.239382
\(652\) −15133.2 −0.908988
\(653\) −4488.63 −0.268995 −0.134497 0.990914i \(-0.542942\pi\)
−0.134497 + 0.990914i \(0.542942\pi\)
\(654\) −8134.00 −0.486337
\(655\) 7624.69 0.454842
\(656\) 9887.42 0.588474
\(657\) 10457.7 0.620998
\(658\) 10831.6 0.641730
\(659\) −25326.7 −1.49710 −0.748550 0.663078i \(-0.769250\pi\)
−0.748550 + 0.663078i \(0.769250\pi\)
\(660\) 807.200 0.0476064
\(661\) 15192.3 0.893969 0.446984 0.894542i \(-0.352498\pi\)
0.446984 + 0.894542i \(0.352498\pi\)
\(662\) −9985.15 −0.586229
\(663\) −17374.5 −1.01775
\(664\) 12956.4 0.757235
\(665\) 3935.67 0.229502
\(666\) −703.087 −0.0409070
\(667\) 2864.75 0.166302
\(668\) −11821.5 −0.684711
\(669\) −5662.07 −0.327217
\(670\) 15743.1 0.907776
\(671\) 1802.34 0.103694
\(672\) −9451.25 −0.542545
\(673\) 11718.2 0.671180 0.335590 0.942008i \(-0.391064\pi\)
0.335590 + 0.942008i \(0.391064\pi\)
\(674\) 17734.8 1.01353
\(675\) −675.000 −0.0384900
\(676\) 3581.63 0.203779
\(677\) −15649.1 −0.888397 −0.444198 0.895928i \(-0.646511\pi\)
−0.444198 + 0.895928i \(0.646511\pi\)
\(678\) 12340.6 0.699024
\(679\) −24972.6 −1.41143
\(680\) −5970.61 −0.336710
\(681\) 3452.20 0.194257
\(682\) −3242.05 −0.182030
\(683\) −18162.8 −1.01754 −0.508770 0.860903i \(-0.669899\pi\)
−0.508770 + 0.860903i \(0.669899\pi\)
\(684\) 2146.40 0.119985
\(685\) 10628.4 0.592831
\(686\) 24656.2 1.37227
\(687\) −12320.4 −0.684208
\(688\) −17778.1 −0.985149
\(689\) −12664.2 −0.700246
\(690\) 643.595 0.0355091
\(691\) 29606.7 1.62995 0.814973 0.579500i \(-0.196752\pi\)
0.814973 + 0.579500i \(0.196752\pi\)
\(692\) −18384.7 −1.00994
\(693\) −1598.50 −0.0876220
\(694\) −9736.08 −0.532531
\(695\) 8871.38 0.484188
\(696\) 8025.55 0.437080
\(697\) 13358.5 0.725953
\(698\) 19733.8 1.07011
\(699\) 17200.7 0.930746
\(700\) −1974.76 −0.106627
\(701\) −26164.7 −1.40974 −0.704870 0.709336i \(-0.748995\pi\)
−0.704870 + 0.709336i \(0.748995\pi\)
\(702\) 5246.80 0.282091
\(703\) −1060.66 −0.0569040
\(704\) −736.324 −0.0394194
\(705\) −2802.48 −0.149713
\(706\) 22879.9 1.21968
\(707\) 10693.5 0.568842
\(708\) −3418.66 −0.181470
\(709\) −14508.9 −0.768535 −0.384268 0.923222i \(-0.625546\pi\)
−0.384268 + 0.923222i \(0.625546\pi\)
\(710\) 13161.9 0.695716
\(711\) −5299.48 −0.279530
\(712\) 11629.5 0.612125
\(713\) −980.898 −0.0515216
\(714\) −18611.6 −0.975520
\(715\) 2976.67 0.155694
\(716\) 12396.6 0.647043
\(717\) −18108.5 −0.943202
\(718\) 2264.28 0.117691
\(719\) −2545.80 −0.132048 −0.0660239 0.997818i \(-0.521031\pi\)
−0.0660239 + 0.997818i \(0.521031\pi\)
\(720\) 3564.19 0.184485
\(721\) −6452.01 −0.333267
\(722\) −16094.6 −0.829611
\(723\) −11162.7 −0.574198
\(724\) −67.9982 −0.00349051
\(725\) 5993.31 0.307015
\(726\) −1303.37 −0.0666291
\(727\) −18984.8 −0.968511 −0.484255 0.874927i \(-0.660909\pi\)
−0.484255 + 0.874927i \(0.660909\pi\)
\(728\) −9751.54 −0.496451
\(729\) 729.000 0.0370370
\(730\) −20860.6 −1.05765
\(731\) −24019.2 −1.21530
\(732\) −2404.70 −0.121421
\(733\) 36740.4 1.85134 0.925672 0.378326i \(-0.123500\pi\)
0.925672 + 0.378326i \(0.123500\pi\)
\(734\) −19297.4 −0.970411
\(735\) −1234.37 −0.0619461
\(736\) −2331.58 −0.116770
\(737\) −9646.10 −0.482115
\(738\) −4034.04 −0.201213
\(739\) 1755.17 0.0873682 0.0436841 0.999045i \(-0.486090\pi\)
0.0436841 + 0.999045i \(0.486090\pi\)
\(740\) 532.196 0.0264377
\(741\) 7915.18 0.392404
\(742\) −13566.0 −0.671189
\(743\) 17972.6 0.887419 0.443709 0.896171i \(-0.353662\pi\)
0.443709 + 0.896171i \(0.353662\pi\)
\(744\) −2747.97 −0.135411
\(745\) −7878.88 −0.387463
\(746\) 27001.4 1.32519
\(747\) −10449.6 −0.511821
\(748\) −5758.54 −0.281488
\(749\) 25692.6 1.25339
\(750\) 1346.46 0.0655544
\(751\) 22704.9 1.10321 0.551606 0.834105i \(-0.314015\pi\)
0.551606 + 0.834105i \(0.314015\pi\)
\(752\) 14797.9 0.717583
\(753\) 11429.6 0.553146
\(754\) −46586.3 −2.25010
\(755\) −2104.89 −0.101463
\(756\) 2132.74 0.102602
\(757\) 39860.3 1.91380 0.956901 0.290414i \(-0.0937929\pi\)
0.956901 + 0.290414i \(0.0937929\pi\)
\(758\) −44914.6 −2.15221
\(759\) −394.342 −0.0188587
\(760\) 2719.99 0.129822
\(761\) 19631.4 0.935135 0.467568 0.883957i \(-0.345130\pi\)
0.467568 + 0.883957i \(0.345130\pi\)
\(762\) −15740.5 −0.748319
\(763\) −12192.7 −0.578511
\(764\) 17099.8 0.809752
\(765\) 4815.43 0.227585
\(766\) −40034.3 −1.88838
\(767\) −12606.8 −0.593490
\(768\) −15831.3 −0.743832
\(769\) 35029.9 1.64267 0.821333 0.570449i \(-0.193231\pi\)
0.821333 + 0.570449i \(0.193231\pi\)
\(770\) 3188.62 0.149234
\(771\) −3677.86 −0.171796
\(772\) 16972.4 0.791255
\(773\) −26736.9 −1.24406 −0.622032 0.782992i \(-0.713693\pi\)
−0.622032 + 0.782992i \(0.713693\pi\)
\(774\) 7253.40 0.336845
\(775\) −2052.13 −0.0951156
\(776\) −17258.8 −0.798398
\(777\) −1053.91 −0.0486600
\(778\) 11469.0 0.528515
\(779\) −6085.64 −0.279898
\(780\) −3971.52 −0.182312
\(781\) −8064.55 −0.369491
\(782\) −4591.39 −0.209959
\(783\) −6472.78 −0.295426
\(784\) 6517.81 0.296912
\(785\) 11121.5 0.505660
\(786\) 16426.1 0.745421
\(787\) 4799.45 0.217385 0.108693 0.994075i \(-0.465334\pi\)
0.108693 + 0.994075i \(0.465334\pi\)
\(788\) 17799.4 0.804668
\(789\) −22193.0 −1.00139
\(790\) 10571.2 0.476083
\(791\) 18498.3 0.831508
\(792\) −1104.74 −0.0495649
\(793\) −8867.72 −0.397102
\(794\) −2098.36 −0.0937884
\(795\) 3509.96 0.156585
\(796\) 252.457 0.0112413
\(797\) 38438.9 1.70838 0.854189 0.519963i \(-0.174054\pi\)
0.854189 + 0.519963i \(0.174054\pi\)
\(798\) 8478.76 0.376121
\(799\) 19992.8 0.885224
\(800\) −4877.87 −0.215574
\(801\) −9379.42 −0.413740
\(802\) −24958.9 −1.09891
\(803\) 12781.7 0.561713
\(804\) 12870.0 0.564538
\(805\) 964.733 0.0422390
\(806\) 15951.3 0.697096
\(807\) 9643.45 0.420651
\(808\) 7390.42 0.321775
\(809\) 9960.91 0.432889 0.216444 0.976295i \(-0.430554\pi\)
0.216444 + 0.976295i \(0.430554\pi\)
\(810\) −1454.18 −0.0630797
\(811\) −34199.9 −1.48079 −0.740396 0.672171i \(-0.765362\pi\)
−0.740396 + 0.672171i \(0.765362\pi\)
\(812\) −18936.6 −0.818404
\(813\) 22132.1 0.954744
\(814\) −859.329 −0.0370018
\(815\) 15466.9 0.664762
\(816\) −25426.8 −1.09083
\(817\) 10942.3 0.468570
\(818\) 42203.9 1.80394
\(819\) 7864.82 0.335555
\(820\) 3053.53 0.130041
\(821\) −31796.4 −1.35165 −0.675823 0.737064i \(-0.736212\pi\)
−0.675823 + 0.737064i \(0.736212\pi\)
\(822\) 22897.1 0.971567
\(823\) −10783.6 −0.456733 −0.228366 0.973575i \(-0.573338\pi\)
−0.228366 + 0.973575i \(0.573338\pi\)
\(824\) −4459.07 −0.188518
\(825\) −825.000 −0.0348155
\(826\) −13504.5 −0.568862
\(827\) 44197.5 1.85840 0.929201 0.369576i \(-0.120497\pi\)
0.929201 + 0.369576i \(0.120497\pi\)
\(828\) 526.137 0.0220828
\(829\) 22487.7 0.942137 0.471069 0.882097i \(-0.343868\pi\)
0.471069 + 0.882097i \(0.343868\pi\)
\(830\) 20844.4 0.871709
\(831\) 2431.82 0.101515
\(832\) 3622.81 0.150959
\(833\) 8805.96 0.366276
\(834\) 19111.9 0.793516
\(835\) 12082.2 0.500743
\(836\) 2623.38 0.108530
\(837\) 2216.30 0.0915250
\(838\) 42475.6 1.75095
\(839\) 22491.0 0.925477 0.462738 0.886495i \(-0.346867\pi\)
0.462738 + 0.886495i \(0.346867\pi\)
\(840\) 2702.69 0.111014
\(841\) 33082.7 1.35646
\(842\) −7181.37 −0.293927
\(843\) −3344.17 −0.136630
\(844\) 10198.0 0.415910
\(845\) −3660.61 −0.149028
\(846\) −6037.48 −0.245358
\(847\) −1953.72 −0.0792571
\(848\) −18533.5 −0.750524
\(849\) −6796.19 −0.274729
\(850\) −9605.60 −0.387611
\(851\) −259.994 −0.0104730
\(852\) 10759.8 0.432660
\(853\) 22327.9 0.896241 0.448120 0.893973i \(-0.352094\pi\)
0.448120 + 0.893973i \(0.352094\pi\)
\(854\) −9499.12 −0.380624
\(855\) −2193.73 −0.0877474
\(856\) 17756.5 0.708999
\(857\) −14505.9 −0.578193 −0.289096 0.957300i \(-0.593355\pi\)
−0.289096 + 0.957300i \(0.593355\pi\)
\(858\) 6412.76 0.255161
\(859\) −8411.45 −0.334104 −0.167052 0.985948i \(-0.553425\pi\)
−0.167052 + 0.985948i \(0.553425\pi\)
\(860\) −5490.40 −0.217699
\(861\) −6046.92 −0.239348
\(862\) −28954.7 −1.14408
\(863\) 32499.6 1.28192 0.640961 0.767573i \(-0.278536\pi\)
0.640961 + 0.767573i \(0.278536\pi\)
\(864\) 5268.10 0.207436
\(865\) 18790.1 0.738592
\(866\) −38455.9 −1.50899
\(867\) −19614.1 −0.768316
\(868\) 6483.94 0.253548
\(869\) −6477.15 −0.252845
\(870\) 12911.6 0.503155
\(871\) 47460.0 1.84629
\(872\) −8426.50 −0.327245
\(873\) 13919.6 0.539643
\(874\) 2091.67 0.0809516
\(875\) 2018.31 0.0779786
\(876\) −17053.5 −0.657745
\(877\) −32183.1 −1.23916 −0.619581 0.784933i \(-0.712697\pi\)
−0.619581 + 0.784933i \(0.712697\pi\)
\(878\) 16725.1 0.642876
\(879\) −11406.2 −0.437680
\(880\) 4356.23 0.166873
\(881\) −6246.34 −0.238870 −0.119435 0.992842i \(-0.538108\pi\)
−0.119435 + 0.992842i \(0.538108\pi\)
\(882\) −2659.25 −0.101521
\(883\) −11801.1 −0.449762 −0.224881 0.974386i \(-0.572199\pi\)
−0.224881 + 0.974386i \(0.572199\pi\)
\(884\) 28332.7 1.07798
\(885\) 3494.05 0.132713
\(886\) 7521.24 0.285193
\(887\) 32375.1 1.22553 0.612767 0.790264i \(-0.290056\pi\)
0.612767 + 0.790264i \(0.290056\pi\)
\(888\) −728.371 −0.0275254
\(889\) −23594.6 −0.890145
\(890\) 18709.7 0.704662
\(891\) 891.000 0.0335013
\(892\) 9233.17 0.346580
\(893\) −9107.98 −0.341307
\(894\) −16973.8 −0.634997
\(895\) −12670.0 −0.473196
\(896\) −21322.6 −0.795020
\(897\) 1940.21 0.0722205
\(898\) −5034.72 −0.187095
\(899\) −19678.5 −0.730049
\(900\) 1100.73 0.0407677
\(901\) −25039.9 −0.925861
\(902\) −4930.49 −0.182004
\(903\) 10872.7 0.400686
\(904\) 12784.4 0.470357
\(905\) 69.4977 0.00255269
\(906\) −4534.64 −0.166284
\(907\) −19592.8 −0.717277 −0.358638 0.933477i \(-0.616759\pi\)
−0.358638 + 0.933477i \(0.616759\pi\)
\(908\) −5629.53 −0.205752
\(909\) −5960.54 −0.217490
\(910\) −15688.4 −0.571500
\(911\) 18673.8 0.679133 0.339567 0.940582i \(-0.389720\pi\)
0.339567 + 0.940582i \(0.389720\pi\)
\(912\) 11583.5 0.420579
\(913\) −12771.7 −0.462959
\(914\) −17644.7 −0.638550
\(915\) 2457.73 0.0887980
\(916\) 20090.9 0.724696
\(917\) 24622.4 0.886698
\(918\) 10374.1 0.372979
\(919\) −4572.90 −0.164142 −0.0820708 0.996627i \(-0.526153\pi\)
−0.0820708 + 0.996627i \(0.526153\pi\)
\(920\) 666.739 0.0238932
\(921\) 4069.05 0.145581
\(922\) −7949.78 −0.283961
\(923\) 39678.6 1.41499
\(924\) 2606.69 0.0928070
\(925\) −543.932 −0.0193345
\(926\) −19988.7 −0.709362
\(927\) 3596.33 0.127421
\(928\) −46775.4 −1.65461
\(929\) −44222.1 −1.56176 −0.780882 0.624679i \(-0.785230\pi\)
−0.780882 + 0.624679i \(0.785230\pi\)
\(930\) −4420.97 −0.155881
\(931\) −4011.67 −0.141221
\(932\) −28049.3 −0.985823
\(933\) 24234.2 0.850367
\(934\) 1784.70 0.0625238
\(935\) 5885.52 0.205858
\(936\) 5435.48 0.189812
\(937\) 9218.62 0.321408 0.160704 0.987003i \(-0.448624\pi\)
0.160704 + 0.987003i \(0.448624\pi\)
\(938\) 50839.2 1.76968
\(939\) −17284.6 −0.600705
\(940\) 4570.02 0.158572
\(941\) −26484.9 −0.917516 −0.458758 0.888561i \(-0.651706\pi\)
−0.458758 + 0.888561i \(0.651706\pi\)
\(942\) 23959.4 0.828705
\(943\) −1491.75 −0.0515142
\(944\) −18449.5 −0.636103
\(945\) −2179.77 −0.0750350
\(946\) 8865.26 0.304688
\(947\) −44972.4 −1.54320 −0.771599 0.636110i \(-0.780543\pi\)
−0.771599 + 0.636110i \(0.780543\pi\)
\(948\) 8641.90 0.296072
\(949\) −62887.5 −2.15112
\(950\) 4375.96 0.149447
\(951\) −15321.0 −0.522416
\(952\) −19280.9 −0.656404
\(953\) −2052.50 −0.0697659 −0.0348829 0.999391i \(-0.511106\pi\)
−0.0348829 + 0.999391i \(0.511106\pi\)
\(954\) 7561.63 0.256621
\(955\) −17476.9 −0.592189
\(956\) 29529.7 0.999016
\(957\) −7911.18 −0.267223
\(958\) −33569.2 −1.13212
\(959\) 34322.2 1.15570
\(960\) −1004.08 −0.0337568
\(961\) −23053.0 −0.773826
\(962\) 4228.00 0.141701
\(963\) −14321.0 −0.479218
\(964\) 18203.1 0.608176
\(965\) −17346.6 −0.578661
\(966\) 2078.36 0.0692237
\(967\) 14950.9 0.497195 0.248598 0.968607i \(-0.420030\pi\)
0.248598 + 0.968607i \(0.420030\pi\)
\(968\) −1350.24 −0.0448331
\(969\) 15650.0 0.518835
\(970\) −27766.3 −0.919094
\(971\) 26751.9 0.884151 0.442076 0.896978i \(-0.354242\pi\)
0.442076 + 0.896978i \(0.354242\pi\)
\(972\) −1188.78 −0.0392287
\(973\) 28648.3 0.943908
\(974\) 710.034 0.0233583
\(975\) 4059.10 0.133329
\(976\) −12977.5 −0.425615
\(977\) −56893.7 −1.86304 −0.931520 0.363690i \(-0.881517\pi\)
−0.931520 + 0.363690i \(0.881517\pi\)
\(978\) 33320.8 1.08945
\(979\) −11463.7 −0.374242
\(980\) 2012.89 0.0656118
\(981\) 6796.15 0.221187
\(982\) −35895.0 −1.16645
\(983\) −34633.1 −1.12373 −0.561863 0.827230i \(-0.689915\pi\)
−0.561863 + 0.827230i \(0.689915\pi\)
\(984\) −4179.10 −0.135391
\(985\) −18191.9 −0.588470
\(986\) −92111.0 −2.97506
\(987\) −9050.03 −0.291860
\(988\) −12907.3 −0.415625
\(989\) 2682.23 0.0862386
\(990\) −1777.33 −0.0570577
\(991\) 1961.79 0.0628843 0.0314422 0.999506i \(-0.489990\pi\)
0.0314422 + 0.999506i \(0.489990\pi\)
\(992\) 16016.0 0.512610
\(993\) 8342.83 0.266618
\(994\) 42503.8 1.35628
\(995\) −258.025 −0.00822103
\(996\) 17040.2 0.542108
\(997\) 13096.5 0.416017 0.208009 0.978127i \(-0.433302\pi\)
0.208009 + 0.978127i \(0.433302\pi\)
\(998\) −31290.8 −0.992476
\(999\) 587.447 0.0186046
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 165.4.a.d.1.3 3
3.2 odd 2 495.4.a.l.1.1 3
5.2 odd 4 825.4.c.l.199.5 6
5.3 odd 4 825.4.c.l.199.2 6
5.4 even 2 825.4.a.s.1.1 3
11.10 odd 2 1815.4.a.s.1.1 3
15.14 odd 2 2475.4.a.s.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.d.1.3 3 1.1 even 1 trivial
495.4.a.l.1.1 3 3.2 odd 2
825.4.a.s.1.1 3 5.4 even 2
825.4.c.l.199.2 6 5.3 odd 4
825.4.c.l.199.5 6 5.2 odd 4
1815.4.a.s.1.1 3 11.10 odd 2
2475.4.a.s.1.3 3 15.14 odd 2