Properties

Label 825.4.a.s.1.1
Level $825$
Weight $4$
Character 825.1
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
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: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.59056\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

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

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 \(-0.718888\pi\)
−0.634727 + 0.772736i \(0.718888\pi\)
\(3\) 3.00000 0.577350
\(4\) 4.89212 0.611515
\(5\) 0 0
\(6\) −10.7717 −0.732920
\(7\) 16.1465 0.871828 0.435914 0.899988i \(-0.356425\pi\)
0.435914 + 0.899988i \(0.356425\pi\)
\(8\) 11.1590 0.493164
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 14.6764 0.353058
\(13\) 54.1214 1.15466 0.577329 0.816511i \(-0.304095\pi\)
0.577329 + 0.816511i \(0.304095\pi\)
\(14\) −57.9749 −1.10675
\(15\) 0 0
\(16\) −79.2041 −1.23756
\(17\) 107.010 1.52668 0.763342 0.645995i \(-0.223557\pi\)
0.763342 + 0.645995i \(0.223557\pi\)
\(18\) −32.3150 −0.423152
\(19\) 48.7496 0.588628 0.294314 0.955709i \(-0.404909\pi\)
0.294314 + 0.955709i \(0.404909\pi\)
\(20\) 0 0
\(21\) 48.4394 0.503350
\(22\) −39.4962 −0.382755
\(23\) −11.9498 −0.108335 −0.0541674 0.998532i \(-0.517250\pi\)
−0.0541674 + 0.998532i \(0.517250\pi\)
\(24\) 33.4771 0.284729
\(25\) 0 0
\(26\) −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 \(0.221481\pi\)
\(30\) 0 0
\(31\) −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
\(35\) 0 0
\(36\) 44.0291 0.203838
\(37\) 21.7573 0.0966723 0.0483361 0.998831i \(-0.484608\pi\)
0.0483361 + 0.998831i \(0.484608\pi\)
\(38\) −175.038 −0.747236
\(39\) 162.364 0.666643
\(40\) 0 0
\(41\) −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 \(-0.630302\pi\)
−0.398019 + 0.917377i \(0.630302\pi\)
\(44\) 53.8133 0.184379
\(45\) 0 0
\(46\) 42.9064 0.137526
\(47\) 186.832 0.579835 0.289917 0.957052i \(-0.406372\pi\)
0.289917 + 0.957052i \(0.406372\pi\)
\(48\) −237.612 −0.714508
\(49\) −82.2913 −0.239916
\(50\) 0 0
\(51\) 321.029 0.881431
\(52\) 264.768 0.706091
\(53\) −233.997 −0.606453 −0.303226 0.952919i \(-0.598064\pi\)
−0.303226 + 0.952919i \(0.598064\pi\)
\(54\) −96.9451 −0.244307
\(55\) 0 0
\(56\) 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 \(0.417267\pi\)
\(60\) 0 0
\(61\) 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
\(65\) 0 0
\(66\) −118.488 −0.220984
\(67\) 876.918 1.59899 0.799497 0.600670i \(-0.205100\pi\)
0.799497 + 0.600670i \(0.205100\pi\)
\(68\) 523.503 0.933590
\(69\) −35.8493 −0.0625471
\(70\) 0 0
\(71\) −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 \(-0.881496\pi\)
−0.931496 + 0.363750i \(0.881496\pi\)
\(74\) −78.1208 −0.122721
\(75\) 0 0
\(76\) 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 \(0.637723\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 448.226 0.603638
\(83\) 1161.06 1.53546 0.767731 0.640772i \(-0.221386\pi\)
0.767731 + 0.640772i \(0.221386\pi\)
\(84\) 236.971 0.307806
\(85\) 0 0
\(86\) 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 \(0.713115\pi\)
\(90\) 0 0
\(91\) 873.869 1.00666
\(92\) −58.4597 −0.0662483
\(93\) −246.255 −0.274575
\(94\) −670.831 −0.736074
\(95\) 0 0
\(96\) 585.345 0.622307
\(97\) −1546.63 −1.61893 −0.809464 0.587169i \(-0.800242\pi\)
−0.809464 + 0.587169i \(0.800242\pi\)
\(98\) 295.472 0.304563
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) −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 \(-0.561216\pi\)
−0.191131 + 0.981565i \(0.561216\pi\)
\(104\) 603.942 0.569436
\(105\) 0 0
\(106\) 840.181 0.769864
\(107\) 1591.22 1.43765 0.718827 0.695189i \(-0.244679\pi\)
0.718827 + 0.695189i \(0.244679\pi\)
\(108\) 132.087 0.117686
\(109\) 755.128 0.663561 0.331780 0.943357i \(-0.392351\pi\)
0.331780 + 0.943357i \(0.392351\pi\)
\(110\) 0 0
\(111\) 65.2718 0.0558138
\(112\) −1278.87 −1.07894
\(113\) 1145.65 0.953753 0.476876 0.878970i \(-0.341769\pi\)
0.476876 + 0.878970i \(0.341769\pi\)
\(114\) −525.115 −0.431417
\(115\) 0 0
\(116\) 1172.80 0.938722
\(117\) 487.092 0.384886
\(118\) −836.372 −0.652494
\(119\) 1727.83 1.33101
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −588.309 −0.436582
\(123\) −374.504 −0.274536
\(124\) −401.570 −0.290823
\(125\) 0 0
\(126\) −521.774 −0.368915
\(127\) −1461.29 −1.02101 −0.510505 0.859875i \(-0.670542\pi\)
−0.510505 + 0.859875i \(0.670542\pi\)
\(128\) −1320.57 −0.911900
\(129\) −673.377 −0.459593
\(130\) 0 0
\(131\) −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
\(135\) 0 0
\(136\) 1194.12 0.752906
\(137\) 2125.68 1.32561 0.662805 0.748792i \(-0.269366\pi\)
0.662805 + 0.748792i \(0.269366\pi\)
\(138\) 128.719 0.0794007
\(139\) −1774.28 −1.08268 −0.541339 0.840805i \(-0.682082\pi\)
−0.541339 + 0.840805i \(0.682082\pi\)
\(140\) 0 0
\(141\) 560.496 0.334768
\(142\) 2632.39 1.55567
\(143\) 595.335 0.348143
\(144\) −712.837 −0.412521
\(145\) 0 0
\(146\) 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 \(0.357386\pi\)
\(150\) 0 0
\(151\) 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
\(155\) 0 0
\(156\) 794.304 0.407662
\(157\) 2224.30 1.13069 0.565345 0.824854i \(-0.308743\pi\)
0.565345 + 0.824854i \(0.308743\pi\)
\(158\) 2114.23 1.06455
\(159\) −701.992 −0.350136
\(160\) 0 0
\(161\) −192.947 −0.0944492
\(162\) −290.835 −0.141051
\(163\) 3093.37 1.48645 0.743226 0.669040i \(-0.233295\pi\)
0.743226 + 0.669040i \(0.233295\pi\)
\(164\) −610.706 −0.290781
\(165\) 0 0
\(166\) −4168.87 −1.94920
\(167\) 2416.43 1.11970 0.559848 0.828595i \(-0.310860\pi\)
0.559848 + 0.828595i \(0.310860\pi\)
\(168\) 540.537 0.248234
\(169\) 732.122 0.333237
\(170\) 0 0
\(171\) 438.746 0.196209
\(172\) −1098.08 −0.486789
\(173\) 3758.02 1.65154 0.825771 0.564005i \(-0.190740\pi\)
0.825771 + 0.564005i \(0.190740\pi\)
\(174\) −2582.32 −1.12509
\(175\) 0 0
\(176\) −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 \(0.322549\pi\)
\(180\) 0 0
\(181\) −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
\(185\) 0 0
\(186\) 884.194 0.348561
\(187\) 1177.10 0.460312
\(188\) 914.004 0.354577
\(189\) 435.955 0.167783
\(190\) 0 0
\(191\) 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 \(-0.723961\pi\)
−0.646963 + 0.762522i \(0.723961\pi\)
\(194\) 5553.25 2.05516
\(195\) 0 0
\(196\) −402.579 −0.146712
\(197\) −3638.39 −1.31586 −0.657930 0.753079i \(-0.728568\pi\)
−0.657930 + 0.753079i \(0.728568\pi\)
\(198\) −355.465 −0.127585
\(199\) 51.6049 0.0183828 0.00919140 0.999958i \(-0.497074\pi\)
0.00919140 + 0.999958i \(0.497074\pi\)
\(200\) 0 0
\(201\) 2630.75 0.923179
\(202\) 2377.96 0.828281
\(203\) 3870.84 1.33832
\(204\) 1570.51 0.539008
\(205\) 0 0
\(206\) 1434.76 0.485265
\(207\) −107.548 −0.0361116
\(208\) −4286.63 −1.42896
\(209\) 536.246 0.177478
\(210\) 0 0
\(211\) 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
\(215\) 0 0
\(216\) 301.294 0.0949095
\(217\) −1325.38 −0.414622
\(218\) −2711.33 −0.842361
\(219\) −3485.91 −1.07560
\(220\) 0 0
\(221\) 5791.50 1.76280
\(222\) −234.362 −0.0708531
\(223\) −1887.36 −0.566757 −0.283378 0.959008i \(-0.591455\pi\)
−0.283378 + 0.959008i \(0.591455\pi\)
\(224\) 3150.42 0.939715
\(225\) 0 0
\(226\) −4113.54 −1.21075
\(227\) 1150.73 0.336462 0.168231 0.985748i \(-0.446194\pi\)
0.168231 + 0.985748i \(0.446194\pi\)
\(228\) 715.466 0.207820
\(229\) 4106.79 1.18508 0.592542 0.805540i \(-0.298125\pi\)
0.592542 + 0.805540i \(0.298125\pi\)
\(230\) 0 0
\(231\) 532.834 0.151766
\(232\) 2675.18 0.757045
\(233\) 5733.58 1.61210 0.806050 0.591848i \(-0.201601\pi\)
0.806050 + 0.591848i \(0.201601\pi\)
\(234\) −1748.93 −0.488596
\(235\) 0 0
\(236\) 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 \(0.195725\pi\)
\(240\) 0 0
\(241\) 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
\(245\) 0 0
\(246\) 1344.68 0.348511
\(247\) 2638.39 0.679664
\(248\) −915.990 −0.234538
\(249\) 3483.19 0.886500
\(250\) 0 0
\(251\) −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
\(255\) 0 0
\(256\) 5277.10 1.28835
\(257\) −1225.95 −0.297559 −0.148780 0.988870i \(-0.547535\pi\)
−0.148780 + 0.988870i \(0.547535\pi\)
\(258\) 2417.80 0.583433
\(259\) 351.303 0.0842816
\(260\) 0 0
\(261\) 2157.59 0.511692
\(262\) 5475.38 1.29111
\(263\) −7397.68 −1.73445 −0.867225 0.497916i \(-0.834099\pi\)
−0.867225 + 0.497916i \(0.834099\pi\)
\(264\) 368.248 0.0858489
\(265\) 0 0
\(266\) −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 \(0.618690\pi\)
\(270\) 0 0
\(271\) −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
\(275\) 0 0
\(276\) −175.379 −0.0382485
\(277\) 810.606 0.175829 0.0879144 0.996128i \(-0.471980\pi\)
0.0879144 + 0.996128i \(0.471980\pi\)
\(278\) 6370.65 1.37441
\(279\) −738.766 −0.158526
\(280\) 0 0
\(281\) 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 \(-0.576466\pi\)
−0.237922 + 0.971284i \(0.576466\pi\)
\(284\) −3586.61 −0.749388
\(285\) 0 0
\(286\) −2137.59 −0.441951
\(287\) −2015.64 −0.414563
\(288\) 1756.03 0.359289
\(289\) 6538.04 1.33076
\(290\) 0 0
\(291\) −4639.88 −0.934689
\(292\) −5684.50 −1.13925
\(293\) −3802.06 −0.758084 −0.379042 0.925380i \(-0.623746\pi\)
−0.379042 + 0.925380i \(0.623746\pi\)
\(294\) 886.416 0.175840
\(295\) 0 0
\(296\) 242.790 0.0476753
\(297\) 297.000 0.0580259
\(298\) −5657.92 −1.09985
\(299\) −646.738 −0.125090
\(300\) 0 0
\(301\) −3624.22 −0.694009
\(302\) −1511.55 −0.288013
\(303\) −1986.85 −0.376704
\(304\) −3861.17 −0.728465
\(305\) 0 0
\(306\) −3458.02 −0.646019
\(307\) 1356.35 0.252153 0.126077 0.992021i \(-0.459761\pi\)
0.126077 + 0.992021i \(0.459761\pi\)
\(308\) 868.895 0.160746
\(309\) −1198.78 −0.220699
\(310\) 0 0
\(311\) −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 \(-0.674152\pi\)
−0.520226 + 0.854029i \(0.674152\pi\)
\(314\) −7986.48 −1.43536
\(315\) 0 0
\(316\) −2880.63 −0.512811
\(317\) −5107.00 −0.904851 −0.452426 0.891802i \(-0.649441\pi\)
−0.452426 + 0.891802i \(0.649441\pi\)
\(318\) 2520.54 0.444481
\(319\) 2637.06 0.462843
\(320\) 0 0
\(321\) 4773.66 0.830030
\(322\) 692.786 0.119899
\(323\) 5216.67 0.898648
\(324\) 396.262 0.0679461
\(325\) 0 0
\(326\) −11106.9 −1.88698
\(327\) 2265.38 0.383107
\(328\) −1393.03 −0.234504
\(329\) 3016.68 0.505516
\(330\) 0 0
\(331\) −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
\(335\) 0 0
\(336\) −3836.60 −0.622928
\(337\) −4939.28 −0.798397 −0.399198 0.916865i \(-0.630712\pi\)
−0.399198 + 0.916865i \(0.630712\pi\)
\(338\) −2628.73 −0.423029
\(339\) 3436.96 0.550649
\(340\) 0 0
\(341\) −902.936 −0.143392
\(342\) −1575.34 −0.249079
\(343\) −6866.96 −1.08099
\(344\) −2504.74 −0.392578
\(345\) 0 0
\(346\) −13493.4 −2.09656
\(347\) 2711.58 0.419496 0.209748 0.977755i \(-0.432736\pi\)
0.209748 + 0.977755i \(0.432736\pi\)
\(348\) 3518.40 0.541971
\(349\) 5496.03 0.842967 0.421484 0.906836i \(-0.361510\pi\)
0.421484 + 0.906836i \(0.361510\pi\)
\(350\) 0 0
\(351\) 1461.28 0.222214
\(352\) 2146.26 0.324989
\(353\) −6372.23 −0.960792 −0.480396 0.877052i \(-0.659507\pi\)
−0.480396 + 0.877052i \(0.659507\pi\)
\(354\) −2509.12 −0.376718
\(355\) 0 0
\(356\) −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 \(0.485239\pi\)
\(360\) 0 0
\(361\) −4482.48 −0.653518
\(362\) 49.9071 0.00724602
\(363\) 363.000 0.0524864
\(364\) 4275.07 0.615590
\(365\) 0 0
\(366\) −1764.93 −0.252061
\(367\) 5374.49 0.764431 0.382216 0.924073i \(-0.375161\pi\)
0.382216 + 0.924073i \(0.375161\pi\)
\(368\) 946.471 0.134071
\(369\) −1123.51 −0.158503
\(370\) 0 0
\(371\) −3778.23 −0.528722
\(372\) −1204.71 −0.167907
\(373\) −7520.12 −1.04391 −0.521953 0.852974i \(-0.674796\pi\)
−0.521953 + 0.852974i \(0.674796\pi\)
\(374\) −4226.46 −0.584346
\(375\) 0 0
\(376\) 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 \(0.822007\pi\)
\(380\) 0 0
\(381\) −4383.86 −0.589481
\(382\) −12550.4 −1.68098
\(383\) 11149.9 1.48755 0.743775 0.668430i \(-0.233033\pi\)
0.743775 + 0.668430i \(0.233033\pi\)
\(384\) −3961.72 −0.526486
\(385\) 0 0
\(386\) 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 \(0.433250\pi\)
\(390\) 0 0
\(391\) −1278.74 −0.165393
\(392\) −918.292 −0.118318
\(393\) −4574.81 −0.587198
\(394\) 13063.8 1.67042
\(395\) 0 0
\(396\) 484.320 0.0614596
\(397\) 584.410 0.0738809 0.0369404 0.999317i \(-0.488239\pi\)
0.0369404 + 0.999317i \(0.488239\pi\)
\(398\) −185.291 −0.0233361
\(399\) 2361.40 0.296286
\(400\) 0 0
\(401\) −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
\(405\) 0 0
\(406\) −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 \(0.248460\pi\)
\(410\) 0 0
\(411\) 6377.03 0.765342
\(412\) −1954.85 −0.233759
\(413\) 3761.10 0.448116
\(414\) 386.157 0.0458420
\(415\) 0 0
\(416\) 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 \(0.257766\pi\)
\(420\) 0 0
\(421\) −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
\(425\) 0 0
\(426\) 7897.16 0.898166
\(427\) 2645.58 0.299833
\(428\) 7784.43 0.879147
\(429\) 1786.00 0.201000
\(430\) 0 0
\(431\) −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 \(-0.297411\pi\)
0.594345 + 0.804210i \(0.297411\pi\)
\(434\) 4758.87 0.526344
\(435\) 0 0
\(436\) 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 \(0.418514\pi\)
\(440\) 0 0
\(441\) −740.622 −0.0799721
\(442\) −20794.7 −2.23779
\(443\) −2094.73 −0.224658 −0.112329 0.993671i \(-0.535831\pi\)
−0.112329 + 0.993671i \(0.535831\pi\)
\(444\) 319.318 0.0341310
\(445\) 0 0
\(446\) 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 \(0.523478\pi\)
\(450\) 0 0
\(451\) −1373.18 −0.143372
\(452\) 5604.67 0.583234
\(453\) 1262.93 0.130989
\(454\) −4131.78 −0.427124
\(455\) 0 0
\(456\) 1632.00 0.167599
\(457\) 4914.19 0.503011 0.251506 0.967856i \(-0.419074\pi\)
0.251506 + 0.967856i \(0.419074\pi\)
\(458\) −14745.7 −1.50441
\(459\) 2889.26 0.293810
\(460\) 0 0
\(461\) −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 \(-0.409866\pi\)
0.279396 + 0.960176i \(0.409866\pi\)
\(464\) −18987.8 −1.89976
\(465\) 0 0
\(466\) −20586.8 −2.04649
\(467\) −497.054 −0.0492525 −0.0246263 0.999697i \(-0.507840\pi\)
−0.0246263 + 0.999697i \(0.507840\pi\)
\(468\) 2382.91 0.235364
\(469\) 14159.1 1.39405
\(470\) 0 0
\(471\) 6672.90 0.652804
\(472\) 2599.35 0.253484
\(473\) −2469.05 −0.240015
\(474\) 6342.70 0.614620
\(475\) 0 0
\(476\) 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 \(0.647119\pi\)
\(480\) 0 0
\(481\) 1177.53 0.111624
\(482\) −13360.1 −1.26252
\(483\) −578.840 −0.0545303
\(484\) 591.946 0.0555923
\(485\) 0 0
\(486\) −872.506 −0.0814355
\(487\) −197.750 −0.0184003 −0.00920013 0.999958i \(-0.502929\pi\)
−0.00920013 + 0.999958i \(0.502929\pi\)
\(488\) 1828.40 0.169606
\(489\) 9280.12 0.858204
\(490\) 0 0
\(491\) −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
\(495\) 0 0
\(496\) 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 \(0.627838\pi\)
\(500\) 0 0
\(501\) 7249.30 0.646457
\(502\) 13679.6 1.21623
\(503\) −5978.53 −0.529959 −0.264979 0.964254i \(-0.585365\pi\)
−0.264979 + 0.964254i \(0.585365\pi\)
\(504\) 1621.61 0.143318
\(505\) 0 0
\(506\) 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 \(0.616297\pi\)
\(510\) 0 0
\(511\) −18761.7 −1.62421
\(512\) −8383.17 −0.723608
\(513\) 1316.24 0.113281
\(514\) 4401.85 0.377738
\(515\) 0 0
\(516\) −3294.24 −0.281048
\(517\) 2055.15 0.174827
\(518\) −1261.38 −0.106992
\(519\) 11274.1 0.953519
\(520\) 0 0
\(521\) −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 \(-0.114181\pi\)
0.936350 + 0.351068i \(0.114181\pi\)
\(524\) −7460.18 −0.621945
\(525\) 0 0
\(526\) 26561.8 2.20181
\(527\) −8783.89 −0.726057
\(528\) −2613.74 −0.215432
\(529\) −12024.2 −0.988264
\(530\) 0 0
\(531\) 2096.43 0.171332
\(532\) 3850.75 0.313818
\(533\) −6756.22 −0.549052
\(534\) 11225.8 0.909714
\(535\) 0 0
\(536\) 9785.56 0.788566
\(537\) 7601.98 0.610893
\(538\) 11541.8 0.924911
\(539\) −905.205 −0.0723375
\(540\) 0 0
\(541\) −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
\(545\) 0 0
\(546\) −9413.04 −0.737804
\(547\) −10448.9 −0.816747 −0.408374 0.912815i \(-0.633904\pi\)
−0.408374 + 0.912815i \(0.633904\pi\)
\(548\) 10399.1 0.810631
\(549\) 1474.64 0.114638
\(550\) 0 0
\(551\) 11686.9 0.903588
\(552\) −400.044 −0.0308460
\(553\) −9507.55 −0.731107
\(554\) −2910.53 −0.223207
\(555\) 0 0
\(556\) −8679.97 −0.662073
\(557\) 11448.7 0.870911 0.435456 0.900210i \(-0.356587\pi\)
0.435456 + 0.900210i \(0.356587\pi\)
\(558\) 2652.58 0.201242
\(559\) −12148.0 −0.919153
\(560\) 0 0
\(561\) 3531.31 0.265762
\(562\) −4002.49 −0.300418
\(563\) 26035.8 1.94898 0.974492 0.224422i \(-0.0720494\pi\)
0.974492 + 0.224422i \(0.0720494\pi\)
\(564\) 2742.01 0.204715
\(565\) 0 0
\(566\) 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 \(0.125129\pi\)
\(570\) 0 0
\(571\) 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
\(575\) 0 0
\(576\) −602.447 −0.0435798
\(577\) 6482.55 0.467716 0.233858 0.972271i \(-0.424865\pi\)
0.233858 + 0.972271i \(0.424865\pi\)
\(578\) −23475.2 −1.68934
\(579\) −10408.0 −0.747048
\(580\) 0 0
\(581\) 18747.1 1.33866
\(582\) 16659.8 1.18654
\(583\) −2573.97 −0.182852
\(584\) −12966.5 −0.918762
\(585\) 0 0
\(586\) 13651.5 0.962353
\(587\) −24650.3 −1.73327 −0.866633 0.498946i \(-0.833720\pi\)
−0.866633 + 0.498946i \(0.833720\pi\)
\(588\) −1207.74 −0.0847045
\(589\) −4001.61 −0.279938
\(590\) 0 0
\(591\) −10915.2 −0.759712
\(592\) −1723.27 −0.119638
\(593\) −23163.2 −1.60405 −0.802024 0.597292i \(-0.796243\pi\)
−0.802024 + 0.597292i \(0.796243\pi\)
\(594\) −1066.40 −0.0736612
\(595\) 0 0
\(596\) 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 \(0.441532\pi\)
\(600\) 0 0
\(601\) −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
\(605\) 0 0
\(606\) 7133.89 0.478208
\(607\) 6749.81 0.451345 0.225673 0.974203i \(-0.427542\pi\)
0.225673 + 0.974203i \(0.427542\pi\)
\(608\) 9511.77 0.634463
\(609\) 11612.5 0.772681
\(610\) 0 0
\(611\) 10111.6 0.669511
\(612\) 4711.53 0.311197
\(613\) 30321.0 1.99780 0.998901 0.0468613i \(-0.0149219\pi\)
0.998901 + 0.0468613i \(0.0149219\pi\)
\(614\) −4870.06 −0.320097
\(615\) 0 0
\(616\) 1981.97 0.129636
\(617\) −15236.8 −0.994182 −0.497091 0.867699i \(-0.665598\pi\)
−0.497091 + 0.867699i \(0.665598\pi\)
\(618\) 4304.28 0.280168
\(619\) −20875.4 −1.35550 −0.677749 0.735293i \(-0.737044\pi\)
−0.677749 + 0.735293i \(0.737044\pi\)
\(620\) 0 0
\(621\) −322.644 −0.0208490
\(622\) 29004.8 1.86975
\(623\) −16827.2 −1.08213
\(624\) −12859.9 −0.825013
\(625\) 0 0
\(626\) 20687.2 1.32081
\(627\) 1608.74 0.102467
\(628\) 10881.5 0.691434
\(629\) 2328.24 0.147588
\(630\) 0 0
\(631\) 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
\(635\) 0 0
\(636\) −3434.23 −0.214113
\(637\) −4453.72 −0.277022
\(638\) −9468.52 −0.587558
\(639\) −6598.27 −0.408487
\(640\) 0 0
\(641\) −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 \(-0.406428\pi\)
0.289750 + 0.957102i \(0.406428\pi\)
\(644\) −943.918 −0.0577571
\(645\) 0 0
\(646\) −18730.8 −1.14079
\(647\) 7429.22 0.451426 0.225713 0.974194i \(-0.427529\pi\)
0.225713 + 0.974194i \(0.427529\pi\)
\(648\) 903.882 0.0547960
\(649\) 2562.30 0.154976
\(650\) 0 0
\(651\) −3976.15 −0.239382
\(652\) 15133.2 0.908988
\(653\) 4488.63 0.268995 0.134497 0.990914i \(-0.457058\pi\)
0.134497 + 0.990914i \(0.457058\pi\)
\(654\) −8134.00 −0.486337
\(655\) 0 0
\(656\) 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 \(0.769250\pi\)
\(660\) 0 0
\(661\) 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
\(665\) 0 0
\(666\) −703.087 −0.0409070
\(667\) −2864.75 −0.166302
\(668\) 11821.5 0.684711
\(669\) −5662.07 −0.327217
\(670\) 0 0
\(671\) 1802.34 0.103694
\(672\) 9451.25 0.542545
\(673\) −11718.2 −0.671180 −0.335590 0.942008i \(-0.608936\pi\)
−0.335590 + 0.942008i \(0.608936\pi\)
\(674\) 17734.8 1.01353
\(675\) 0 0
\(676\) 3581.63 0.203779
\(677\) 15649.1 0.888397 0.444198 0.895928i \(-0.353489\pi\)
0.444198 + 0.895928i \(0.353489\pi\)
\(678\) −12340.6 −0.699024
\(679\) −24972.6 −1.41143
\(680\) 0 0
\(681\) 3452.20 0.194257
\(682\) 3242.05 0.182030
\(683\) 18162.8 1.01754 0.508770 0.860903i \(-0.330101\pi\)
0.508770 + 0.860903i \(0.330101\pi\)
\(684\) 2146.40 0.119985
\(685\) 0 0
\(686\) 24656.2 1.37227
\(687\) 12320.4 0.684208
\(688\) 17778.1 0.985149
\(689\) −12664.2 −0.700246
\(690\) 0 0
\(691\) 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
\(695\) 0 0
\(696\) 8025.55 0.437080
\(697\) −13358.5 −0.725953
\(698\) −19733.8 −1.07011
\(699\) 17200.7 0.930746
\(700\) 0 0
\(701\) −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
\(705\) 0 0
\(706\) 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 \(0.625546\pi\)
\(710\) 0 0
\(711\) −5299.48 −0.279530
\(712\) −11629.5 −0.612125
\(713\) 980.898 0.0515216
\(714\) −18611.6 −0.975520
\(715\) 0 0
\(716\) 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 \(0.521031\pi\)
\(720\) 0 0
\(721\) −6452.01 −0.333267
\(722\) 16094.6 0.829611
\(723\) 11162.7 0.574198
\(724\) −67.9982 −0.00349051
\(725\) 0 0
\(726\) −1303.37 −0.0666291
\(727\) 18984.8 0.968511 0.484255 0.874927i \(-0.339091\pi\)
0.484255 + 0.874927i \(0.339091\pi\)
\(728\) 9751.54 0.496451
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −24019.2 −1.21530
\(732\) 2404.70 0.121421
\(733\) −36740.4 −1.85134 −0.925672 0.378326i \(-0.876500\pi\)
−0.925672 + 0.378326i \(0.876500\pi\)
\(734\) −19297.4 −0.970411
\(735\) 0 0
\(736\) −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 \(0.486090\pi\)
\(740\) 0 0
\(741\) 7915.18 0.392404
\(742\) 13566.0 0.671189
\(743\) −17972.6 −0.887419 −0.443709 0.896171i \(-0.646338\pi\)
−0.443709 + 0.896171i \(0.646338\pi\)
\(744\) −2747.97 −0.135411
\(745\) 0 0
\(746\) 27001.4 1.32519
\(747\) 10449.6 0.511821
\(748\) 5758.54 0.281488
\(749\) 25692.6 1.25339
\(750\) 0 0
\(751\) 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
\(755\) 0 0
\(756\) 2132.74 0.102602
\(757\) −39860.3 −1.91380 −0.956901 0.290414i \(-0.906207\pi\)
−0.956901 + 0.290414i \(0.906207\pi\)
\(758\) 44914.6 2.15221
\(759\) −394.342 −0.0188587
\(760\) 0 0
\(761\) 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
\(765\) 0 0
\(766\) −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 \(0.193231\pi\)
\(770\) 0 0
\(771\) −3677.86 −0.171796
\(772\) −16972.4 −0.791255
\(773\) 26736.9 1.24406 0.622032 0.782992i \(-0.286307\pi\)
0.622032 + 0.782992i \(0.286307\pi\)
\(774\) 7253.40 0.336845
\(775\) 0 0
\(776\) −17258.8 −0.798398
\(777\) 1053.91 0.0486600
\(778\) −11469.0 −0.528515
\(779\) −6085.64 −0.279898
\(780\) 0 0
\(781\) −8064.55 −0.369491
\(782\) 4591.39 0.209959
\(783\) 6472.78 0.295426
\(784\) 6517.81 0.296912
\(785\) 0 0
\(786\) 16426.1 0.745421
\(787\) −4799.45 −0.217385 −0.108693 0.994075i \(-0.534666\pi\)
−0.108693 + 0.994075i \(0.534666\pi\)
\(788\) −17799.4 −0.804668
\(789\) −22193.0 −1.00139
\(790\) 0 0
\(791\) 18498.3 0.831508
\(792\) 1104.74 0.0495649
\(793\) 8867.72 0.397102
\(794\) −2098.36 −0.0937884
\(795\) 0 0
\(796\) 252.457 0.0112413
\(797\) −38438.9 −1.70838 −0.854189 0.519963i \(-0.825946\pi\)
−0.854189 + 0.519963i \(0.825946\pi\)
\(798\) −8478.76 −0.376121
\(799\) 19992.8 0.885224
\(800\) 0 0
\(801\) −9379.42 −0.413740
\(802\) 24958.9 1.09891
\(803\) −12781.7 −0.561713
\(804\) 12870.0 0.564538
\(805\) 0 0
\(806\) 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 \(0.430554\pi\)
\(810\) 0 0
\(811\) −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
\(815\) 0 0
\(816\) −25426.8 −1.09083
\(817\) −10942.3 −0.468570
\(818\) −42203.9 −1.80394
\(819\) 7864.82 0.335555
\(820\) 0 0
\(821\) −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 \(-0.426662\pi\)
0.228366 + 0.973575i \(0.426662\pi\)
\(824\) −4459.07 −0.188518
\(825\) 0 0
\(826\) −13504.5 −0.568862
\(827\) −44197.5 −1.85840 −0.929201 0.369576i \(-0.879503\pi\)
−0.929201 + 0.369576i \(0.879503\pi\)
\(828\) −526.137 −0.0220828
\(829\) 22487.7 0.942137 0.471069 0.882097i \(-0.343868\pi\)
0.471069 + 0.882097i \(0.343868\pi\)
\(830\) 0 0
\(831\) 2431.82 0.101515
\(832\) −3622.81 −0.150959
\(833\) −8805.96 −0.366276
\(834\) 19111.9 0.793516
\(835\) 0 0
\(836\) 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 \(0.346867\pi\)
\(840\) 0 0
\(841\) 33082.7 1.35646
\(842\) 7181.37 0.293927
\(843\) 3344.17 0.136630
\(844\) 10198.0 0.415910
\(845\) 0 0
\(846\) −6037.48 −0.245358
\(847\) 1953.72 0.0792571
\(848\) 18533.5 0.750524
\(849\) −6796.19 −0.274729
\(850\) 0 0
\(851\) −259.994 −0.0104730
\(852\) −10759.8 −0.432660
\(853\) −22327.9 −0.896241 −0.448120 0.893973i \(-0.647906\pi\)
−0.448120 + 0.893973i \(0.647906\pi\)
\(854\) −9499.12 −0.380624
\(855\) 0 0
\(856\) 17756.5 0.708999
\(857\) 14505.9 0.578193 0.289096 0.957300i \(-0.406645\pi\)
0.289096 + 0.957300i \(0.406645\pi\)
\(858\) −6412.76 −0.255161
\(859\) −8411.45 −0.334104 −0.167052 0.985948i \(-0.553425\pi\)
−0.167052 + 0.985948i \(0.553425\pi\)
\(860\) 0 0
\(861\) −6046.92 −0.239348
\(862\) 28954.7 1.14408
\(863\) −32499.6 −1.28192 −0.640961 0.767573i \(-0.721464\pi\)
−0.640961 + 0.767573i \(0.721464\pi\)
\(864\) 5268.10 0.207436
\(865\) 0 0
\(866\) −38455.9 −1.50899
\(867\) 19614.1 0.768316
\(868\) −6483.94 −0.253548
\(869\) −6477.15 −0.252845
\(870\) 0 0
\(871\) 47460.0 1.84629
\(872\) 8426.50 0.327245
\(873\) −13919.6 −0.539643
\(874\) 2091.67 0.0809516
\(875\) 0 0
\(876\) −17053.5 −0.657745
\(877\) 32183.1 1.23916 0.619581 0.784933i \(-0.287303\pi\)
0.619581 + 0.784933i \(0.287303\pi\)
\(878\) −16725.1 −0.642876
\(879\) −11406.2 −0.437680
\(880\) 0 0
\(881\) −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 \(-0.427801\pi\)
0.224881 + 0.974386i \(0.427801\pi\)
\(884\) 28332.7 1.07798
\(885\) 0 0
\(886\) 7521.24 0.285193
\(887\) −32375.1 −1.22553 −0.612767 0.790264i \(-0.709944\pi\)
−0.612767 + 0.790264i \(0.709944\pi\)
\(888\) 728.371 0.0275254
\(889\) −23594.6 −0.890145
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) −9233.17 −0.346580
\(893\) 9107.98 0.341307
\(894\) −16973.8 −0.634997
\(895\) 0 0
\(896\) −21322.6 −0.795020
\(897\) −1940.21 −0.0722205
\(898\) 5034.72 0.187095
\(899\) −19678.5 −0.730049
\(900\) 0 0
\(901\) −25039.9 −0.925861
\(902\) 4930.49 0.182004
\(903\) −10872.7 −0.400686
\(904\) 12784.4 0.470357
\(905\) 0 0
\(906\) −4534.64 −0.166284
\(907\) 19592.8 0.717277 0.358638 0.933477i \(-0.383241\pi\)
0.358638 + 0.933477i \(0.383241\pi\)
\(908\) 5629.53 0.205752
\(909\) −5960.54 −0.217490
\(910\) 0 0
\(911\) 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
\(915\) 0 0
\(916\) 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 \(0.526153\pi\)
\(920\) 0 0
\(921\) 4069.05 0.145581
\(922\) 7949.78 0.283961
\(923\) −39678.6 −1.41499
\(924\) 2606.69 0.0928070
\(925\) 0 0
\(926\) −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 \(0.785230\pi\)
\(930\) 0 0
\(931\) −4011.67 −0.141221
\(932\) 28049.3 0.985823
\(933\) −24234.2 −0.850367
\(934\) 1784.70 0.0625238
\(935\) 0 0
\(936\) 5435.48 0.189812
\(937\) −9218.62 −0.321408 −0.160704 0.987003i \(-0.551376\pi\)
−0.160704 + 0.987003i \(0.551376\pi\)
\(938\) −50839.2 −1.76968
\(939\) −17284.6 −0.600705
\(940\) 0 0
\(941\) −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
\(945\) 0 0
\(946\) 8865.26 0.304688
\(947\) 44972.4 1.54320 0.771599 0.636110i \(-0.219457\pi\)
0.771599 + 0.636110i \(0.219457\pi\)
\(948\) −8641.90 −0.296072
\(949\) −62887.5 −2.15112
\(950\) 0 0
\(951\) −15321.0 −0.522416
\(952\) 19280.9 0.656404
\(953\) 2052.50 0.0697659 0.0348829 0.999391i \(-0.488894\pi\)
0.0348829 + 0.999391i \(0.488894\pi\)
\(954\) 7561.63 0.256621
\(955\) 0 0
\(956\) 29529.7 0.999016
\(957\) 7911.18 0.267223
\(958\) 33569.2 1.13212
\(959\) 34322.2 1.15570
\(960\) 0 0
\(961\) −23053.0 −0.773826
\(962\) −4228.00 −0.141701
\(963\) 14321.0 0.479218
\(964\) 18203.1 0.608176
\(965\) 0 0
\(966\) 2078.36 0.0692237
\(967\) −14950.9 −0.497195 −0.248598 0.968607i \(-0.579970\pi\)
−0.248598 + 0.968607i \(0.579970\pi\)
\(968\) 1350.24 0.0448331
\(969\) 15650.0 0.518835
\(970\) 0 0
\(971\) 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
\(975\) 0 0
\(976\) −12977.5 −0.425615
\(977\) 56893.7 1.86304 0.931520 0.363690i \(-0.118483\pi\)
0.931520 + 0.363690i \(0.118483\pi\)
\(978\) −33320.8 −1.08945
\(979\) −11463.7 −0.374242
\(980\) 0 0
\(981\) 6796.15 0.221187
\(982\) 35895.0 1.16645
\(983\) 34633.1 1.12373 0.561863 0.827230i \(-0.310085\pi\)
0.561863 + 0.827230i \(0.310085\pi\)
\(984\) −4179.10 −0.135391
\(985\) 0 0
\(986\) −92111.0 −2.97506
\(987\) 9050.03 0.291860
\(988\) 12907.3 0.415625
\(989\) 2682.23 0.0862386
\(990\) 0 0
\(991\) 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
\(995\) 0 0
\(996\) 17040.2 0.542108
\(997\) −13096.5 −0.416017 −0.208009 0.978127i \(-0.566698\pi\)
−0.208009 + 0.978127i \(0.566698\pi\)
\(998\) 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 825.4.a.s.1.1 3
3.2 odd 2 2475.4.a.s.1.3 3
5.2 odd 4 825.4.c.l.199.2 6
5.3 odd 4 825.4.c.l.199.5 6
5.4 even 2 165.4.a.d.1.3 3
15.14 odd 2 495.4.a.l.1.1 3
55.54 odd 2 1815.4.a.s.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.d.1.3 3 5.4 even 2
495.4.a.l.1.1 3 15.14 odd 2
825.4.a.s.1.1 3 1.1 even 1 trivial
825.4.c.l.199.2 6 5.2 odd 4
825.4.c.l.199.5 6 5.3 odd 4
1815.4.a.s.1.1 3 55.54 odd 2
2475.4.a.s.1.3 3 3.2 odd 2