Properties

Label 425.4.a.g.1.2
Level $425$
Weight $4$
Character 425.1
Self dual yes
Analytic conductor $25.076$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.0758117524\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.2636.1
Defining polynomial: \( x^{3} - 14x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.87707\) of defining polynomial
Character \(\chi\) \(=\) 425.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.36122 q^{2} -3.15463 q^{3} -6.14708 q^{4} +4.29415 q^{6} +7.94049 q^{7} +19.2573 q^{8} -17.0483 q^{9} +O(q^{10})\) \(q-1.36122 q^{2} -3.15463 q^{3} -6.14708 q^{4} +4.29415 q^{6} +7.94049 q^{7} +19.2573 q^{8} -17.0483 q^{9} +27.6161 q^{11} +19.3918 q^{12} -58.1117 q^{13} -10.8088 q^{14} +22.9632 q^{16} +17.0000 q^{17} +23.2065 q^{18} +89.1688 q^{19} -25.0493 q^{21} -37.5916 q^{22} +115.269 q^{23} -60.7497 q^{24} +79.1029 q^{26} +138.956 q^{27} -48.8108 q^{28} -128.558 q^{29} +273.460 q^{31} -185.316 q^{32} -87.1187 q^{33} -23.1408 q^{34} +104.797 q^{36} +132.351 q^{37} -121.379 q^{38} +183.321 q^{39} -470.559 q^{41} +34.0977 q^{42} -352.642 q^{43} -169.758 q^{44} -156.907 q^{46} -152.598 q^{47} -72.4403 q^{48} -279.949 q^{49} -53.6287 q^{51} +357.217 q^{52} -527.614 q^{53} -189.150 q^{54} +152.912 q^{56} -281.295 q^{57} +174.995 q^{58} -292.020 q^{59} -53.8962 q^{61} -372.239 q^{62} -135.372 q^{63} +68.5514 q^{64} +118.588 q^{66} -52.9572 q^{67} -104.500 q^{68} -363.632 q^{69} +788.400 q^{71} -328.304 q^{72} -295.780 q^{73} -180.159 q^{74} -548.127 q^{76} +219.285 q^{77} -249.541 q^{78} -720.325 q^{79} +21.9487 q^{81} +640.535 q^{82} +116.051 q^{83} +153.980 q^{84} +480.024 q^{86} +405.552 q^{87} +531.812 q^{88} -813.329 q^{89} -461.435 q^{91} -708.569 q^{92} -862.664 q^{93} +207.720 q^{94} +584.605 q^{96} -794.693 q^{97} +381.072 q^{98} -470.808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - 4 q^{3} + 25 q^{4} - 74 q^{6} - 22 q^{7} + 39 q^{8} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - 4 q^{3} + 25 q^{4} - 74 q^{6} - 22 q^{7} + 39 q^{8} + 59 q^{9} - 28 q^{11} - 22 q^{12} - 30 q^{13} + 92 q^{14} + 137 q^{16} + 51 q^{17} + 103 q^{18} + 80 q^{19} - 192 q^{21} - 286 q^{22} - 142 q^{23} - 666 q^{24} + 26 q^{26} + 20 q^{27} - 476 q^{28} - 456 q^{29} + 230 q^{31} + 71 q^{32} + 332 q^{33} - 17 q^{34} + 1313 q^{36} - 356 q^{37} - 724 q^{38} + 268 q^{39} - 294 q^{41} + 1128 q^{42} - 556 q^{43} - 1122 q^{44} - 704 q^{46} - 640 q^{47} - 774 q^{48} - 269 q^{49} - 68 q^{51} + 774 q^{52} - 302 q^{53} - 1100 q^{54} + 684 q^{56} + 720 q^{57} + 1304 q^{58} + 636 q^{59} - 84 q^{61} - 508 q^{62} - 1122 q^{63} - 919 q^{64} + 2468 q^{66} - 1008 q^{67} + 425 q^{68} + 576 q^{69} - 402 q^{71} + 927 q^{72} - 838 q^{73} + 836 q^{74} - 908 q^{76} + 504 q^{77} - 1308 q^{78} - 594 q^{79} - 505 q^{81} - 358 q^{82} + 2396 q^{83} - 2040 q^{84} - 1264 q^{86} - 1428 q^{87} - 1838 q^{88} - 170 q^{89} - 1016 q^{91} - 4896 q^{92} - 632 q^{93} - 2016 q^{94} + 678 q^{96} + 270 q^{97} - 2857 q^{98} - 2920 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.36122 −0.481264 −0.240632 0.970616i \(-0.577355\pi\)
−0.240632 + 0.970616i \(0.577355\pi\)
\(3\) −3.15463 −0.607109 −0.303555 0.952814i \(-0.598173\pi\)
−0.303555 + 0.952814i \(0.598173\pi\)
\(4\) −6.14708 −0.768385
\(5\) 0 0
\(6\) 4.29415 0.292180
\(7\) 7.94049 0.428746 0.214373 0.976752i \(-0.431229\pi\)
0.214373 + 0.976752i \(0.431229\pi\)
\(8\) 19.2573 0.851061
\(9\) −17.0483 −0.631419
\(10\) 0 0
\(11\) 27.6161 0.756961 0.378481 0.925609i \(-0.376447\pi\)
0.378481 + 0.925609i \(0.376447\pi\)
\(12\) 19.3918 0.466493
\(13\) −58.1117 −1.23979 −0.619896 0.784684i \(-0.712825\pi\)
−0.619896 + 0.784684i \(0.712825\pi\)
\(14\) −10.8088 −0.206340
\(15\) 0 0
\(16\) 22.9632 0.358799
\(17\) 17.0000 0.242536
\(18\) 23.2065 0.303879
\(19\) 89.1688 1.07667 0.538335 0.842731i \(-0.319054\pi\)
0.538335 + 0.842731i \(0.319054\pi\)
\(20\) 0 0
\(21\) −25.0493 −0.260296
\(22\) −37.5916 −0.364298
\(23\) 115.269 1.04501 0.522507 0.852635i \(-0.324997\pi\)
0.522507 + 0.852635i \(0.324997\pi\)
\(24\) −60.7497 −0.516687
\(25\) 0 0
\(26\) 79.1029 0.596668
\(27\) 138.956 0.990449
\(28\) −48.8108 −0.329442
\(29\) −128.558 −0.823191 −0.411596 0.911367i \(-0.635028\pi\)
−0.411596 + 0.911367i \(0.635028\pi\)
\(30\) 0 0
\(31\) 273.460 1.58435 0.792174 0.610295i \(-0.208949\pi\)
0.792174 + 0.610295i \(0.208949\pi\)
\(32\) −185.316 −1.02374
\(33\) −87.1187 −0.459558
\(34\) −23.1408 −0.116724
\(35\) 0 0
\(36\) 104.797 0.485172
\(37\) 132.351 0.588063 0.294031 0.955796i \(-0.405003\pi\)
0.294031 + 0.955796i \(0.405003\pi\)
\(38\) −121.379 −0.518163
\(39\) 183.321 0.752689
\(40\) 0 0
\(41\) −470.559 −1.79241 −0.896207 0.443636i \(-0.853688\pi\)
−0.896207 + 0.443636i \(0.853688\pi\)
\(42\) 34.0977 0.125271
\(43\) −352.642 −1.25064 −0.625318 0.780370i \(-0.715031\pi\)
−0.625318 + 0.780370i \(0.715031\pi\)
\(44\) −169.758 −0.581637
\(45\) 0 0
\(46\) −156.907 −0.502928
\(47\) −152.598 −0.473589 −0.236795 0.971560i \(-0.576097\pi\)
−0.236795 + 0.971560i \(0.576097\pi\)
\(48\) −72.4403 −0.217830
\(49\) −279.949 −0.816177
\(50\) 0 0
\(51\) −53.6287 −0.147246
\(52\) 357.217 0.952637
\(53\) −527.614 −1.36742 −0.683711 0.729753i \(-0.739635\pi\)
−0.683711 + 0.729753i \(0.739635\pi\)
\(54\) −189.150 −0.476668
\(55\) 0 0
\(56\) 152.912 0.364889
\(57\) −281.295 −0.653656
\(58\) 174.995 0.396173
\(59\) −292.020 −0.644368 −0.322184 0.946677i \(-0.604417\pi\)
−0.322184 + 0.946677i \(0.604417\pi\)
\(60\) 0 0
\(61\) −53.8962 −0.113126 −0.0565632 0.998399i \(-0.518014\pi\)
−0.0565632 + 0.998399i \(0.518014\pi\)
\(62\) −372.239 −0.762490
\(63\) −135.372 −0.270718
\(64\) 68.5514 0.133889
\(65\) 0 0
\(66\) 118.588 0.221169
\(67\) −52.9572 −0.0965635 −0.0482817 0.998834i \(-0.515375\pi\)
−0.0482817 + 0.998834i \(0.515375\pi\)
\(68\) −104.500 −0.186361
\(69\) −363.632 −0.634437
\(70\) 0 0
\(71\) 788.400 1.31783 0.658915 0.752218i \(-0.271016\pi\)
0.658915 + 0.752218i \(0.271016\pi\)
\(72\) −328.304 −0.537375
\(73\) −295.780 −0.474224 −0.237112 0.971482i \(-0.576201\pi\)
−0.237112 + 0.971482i \(0.576201\pi\)
\(74\) −180.159 −0.283014
\(75\) 0 0
\(76\) −548.127 −0.827296
\(77\) 219.285 0.324544
\(78\) −249.541 −0.362242
\(79\) −720.325 −1.02586 −0.512930 0.858430i \(-0.671440\pi\)
−0.512930 + 0.858430i \(0.671440\pi\)
\(80\) 0 0
\(81\) 21.9487 0.0301079
\(82\) 640.535 0.862625
\(83\) 116.051 0.153473 0.0767363 0.997051i \(-0.475550\pi\)
0.0767363 + 0.997051i \(0.475550\pi\)
\(84\) 153.980 0.200007
\(85\) 0 0
\(86\) 480.024 0.601887
\(87\) 405.552 0.499767
\(88\) 531.812 0.644220
\(89\) −813.329 −0.968682 −0.484341 0.874879i \(-0.660941\pi\)
−0.484341 + 0.874879i \(0.660941\pi\)
\(90\) 0 0
\(91\) −461.435 −0.531556
\(92\) −708.569 −0.802972
\(93\) −862.664 −0.961872
\(94\) 207.720 0.227922
\(95\) 0 0
\(96\) 584.605 0.621521
\(97\) −794.693 −0.831844 −0.415922 0.909400i \(-0.636541\pi\)
−0.415922 + 0.909400i \(0.636541\pi\)
\(98\) 381.072 0.392797
\(99\) −470.808 −0.477959
\(100\) 0 0
\(101\) 265.513 0.261579 0.130790 0.991410i \(-0.458249\pi\)
0.130790 + 0.991410i \(0.458249\pi\)
\(102\) 73.0006 0.0708641
\(103\) −523.107 −0.500420 −0.250210 0.968192i \(-0.580500\pi\)
−0.250210 + 0.968192i \(0.580500\pi\)
\(104\) −1119.07 −1.05514
\(105\) 0 0
\(106\) 718.199 0.658091
\(107\) 986.039 0.890878 0.445439 0.895312i \(-0.353048\pi\)
0.445439 + 0.895312i \(0.353048\pi\)
\(108\) −854.174 −0.761046
\(109\) 1814.39 1.59438 0.797188 0.603732i \(-0.206320\pi\)
0.797188 + 0.603732i \(0.206320\pi\)
\(110\) 0 0
\(111\) −417.518 −0.357018
\(112\) 182.339 0.153834
\(113\) 707.339 0.588857 0.294429 0.955673i \(-0.404871\pi\)
0.294429 + 0.955673i \(0.404871\pi\)
\(114\) 382.904 0.314581
\(115\) 0 0
\(116\) 790.253 0.632527
\(117\) 990.706 0.782827
\(118\) 397.503 0.310112
\(119\) 134.988 0.103986
\(120\) 0 0
\(121\) −568.350 −0.427010
\(122\) 73.3647 0.0544437
\(123\) 1484.44 1.08819
\(124\) −1680.98 −1.21739
\(125\) 0 0
\(126\) 184.271 0.130287
\(127\) −2648.18 −1.85030 −0.925151 0.379600i \(-0.876062\pi\)
−0.925151 + 0.379600i \(0.876062\pi\)
\(128\) 1389.22 0.959302
\(129\) 1112.46 0.759273
\(130\) 0 0
\(131\) −1979.08 −1.31995 −0.659974 0.751289i \(-0.729433\pi\)
−0.659974 + 0.751289i \(0.729433\pi\)
\(132\) 535.525 0.353117
\(133\) 708.044 0.461618
\(134\) 72.0865 0.0464726
\(135\) 0 0
\(136\) 327.374 0.206413
\(137\) −3141.92 −1.95936 −0.979679 0.200570i \(-0.935721\pi\)
−0.979679 + 0.200570i \(0.935721\pi\)
\(138\) 494.984 0.305332
\(139\) 1468.07 0.895830 0.447915 0.894076i \(-0.352167\pi\)
0.447915 + 0.894076i \(0.352167\pi\)
\(140\) 0 0
\(141\) 481.390 0.287520
\(142\) −1073.19 −0.634224
\(143\) −1604.82 −0.938474
\(144\) −391.483 −0.226553
\(145\) 0 0
\(146\) 402.621 0.228227
\(147\) 883.135 0.495508
\(148\) −813.570 −0.451858
\(149\) −286.027 −0.157263 −0.0786316 0.996904i \(-0.525055\pi\)
−0.0786316 + 0.996904i \(0.525055\pi\)
\(150\) 0 0
\(151\) −669.626 −0.360883 −0.180442 0.983586i \(-0.557753\pi\)
−0.180442 + 0.983586i \(0.557753\pi\)
\(152\) 1717.15 0.916311
\(153\) −289.821 −0.153141
\(154\) −298.496 −0.156191
\(155\) 0 0
\(156\) −1126.89 −0.578354
\(157\) −720.809 −0.366413 −0.183206 0.983074i \(-0.558648\pi\)
−0.183206 + 0.983074i \(0.558648\pi\)
\(158\) 980.522 0.493710
\(159\) 1664.43 0.830174
\(160\) 0 0
\(161\) 915.294 0.448045
\(162\) −29.8770 −0.0144899
\(163\) 676.599 0.325125 0.162562 0.986698i \(-0.448024\pi\)
0.162562 + 0.986698i \(0.448024\pi\)
\(164\) 2892.56 1.37726
\(165\) 0 0
\(166\) −157.971 −0.0738609
\(167\) 2835.67 1.31396 0.656979 0.753909i \(-0.271834\pi\)
0.656979 + 0.753909i \(0.271834\pi\)
\(168\) −482.382 −0.221527
\(169\) 1179.97 0.537083
\(170\) 0 0
\(171\) −1520.18 −0.679829
\(172\) 2167.72 0.960970
\(173\) 177.314 0.0779243 0.0389621 0.999241i \(-0.487595\pi\)
0.0389621 + 0.999241i \(0.487595\pi\)
\(174\) −552.046 −0.240520
\(175\) 0 0
\(176\) 634.153 0.271597
\(177\) 921.214 0.391202
\(178\) 1107.12 0.466192
\(179\) −1023.76 −0.427483 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(180\) 0 0
\(181\) −3450.21 −1.41686 −0.708432 0.705779i \(-0.750597\pi\)
−0.708432 + 0.705779i \(0.750597\pi\)
\(182\) 628.116 0.255819
\(183\) 170.023 0.0686800
\(184\) 2219.78 0.889370
\(185\) 0 0
\(186\) 1174.28 0.462915
\(187\) 469.474 0.183590
\(188\) 938.031 0.363899
\(189\) 1103.38 0.424651
\(190\) 0 0
\(191\) −490.894 −0.185968 −0.0929839 0.995668i \(-0.529640\pi\)
−0.0929839 + 0.995668i \(0.529640\pi\)
\(192\) −216.254 −0.0812855
\(193\) 3548.80 1.32357 0.661783 0.749696i \(-0.269800\pi\)
0.661783 + 0.749696i \(0.269800\pi\)
\(194\) 1081.75 0.400337
\(195\) 0 0
\(196\) 1720.87 0.627138
\(197\) −1363.15 −0.492996 −0.246498 0.969143i \(-0.579280\pi\)
−0.246498 + 0.969143i \(0.579280\pi\)
\(198\) 640.874 0.230025
\(199\) 3737.46 1.33137 0.665683 0.746235i \(-0.268140\pi\)
0.665683 + 0.746235i \(0.268140\pi\)
\(200\) 0 0
\(201\) 167.060 0.0586246
\(202\) −361.422 −0.125889
\(203\) −1020.81 −0.352940
\(204\) 329.660 0.113141
\(205\) 0 0
\(206\) 712.064 0.240834
\(207\) −1965.15 −0.659841
\(208\) −1334.43 −0.444836
\(209\) 2462.50 0.814997
\(210\) 0 0
\(211\) −5266.12 −1.71817 −0.859087 0.511829i \(-0.828968\pi\)
−0.859087 + 0.511829i \(0.828968\pi\)
\(212\) 3243.28 1.05071
\(213\) −2487.11 −0.800066
\(214\) −1342.22 −0.428748
\(215\) 0 0
\(216\) 2675.92 0.842932
\(217\) 2171.40 0.679283
\(218\) −2469.78 −0.767316
\(219\) 933.075 0.287906
\(220\) 0 0
\(221\) −987.899 −0.300694
\(222\) 568.334 0.171820
\(223\) −704.546 −0.211569 −0.105785 0.994389i \(-0.533735\pi\)
−0.105785 + 0.994389i \(0.533735\pi\)
\(224\) −1471.50 −0.438923
\(225\) 0 0
\(226\) −962.845 −0.283396
\(227\) 2151.26 0.629006 0.314503 0.949256i \(-0.398162\pi\)
0.314503 + 0.949256i \(0.398162\pi\)
\(228\) 1729.14 0.502259
\(229\) −3916.94 −1.13030 −0.565149 0.824989i \(-0.691181\pi\)
−0.565149 + 0.824989i \(0.691181\pi\)
\(230\) 0 0
\(231\) −691.764 −0.197034
\(232\) −2475.67 −0.700586
\(233\) 5192.74 1.46003 0.730017 0.683429i \(-0.239512\pi\)
0.730017 + 0.683429i \(0.239512\pi\)
\(234\) −1348.57 −0.376747
\(235\) 0 0
\(236\) 1795.07 0.495123
\(237\) 2272.36 0.622809
\(238\) −183.749 −0.0500448
\(239\) 334.305 0.0904786 0.0452393 0.998976i \(-0.485595\pi\)
0.0452393 + 0.998976i \(0.485595\pi\)
\(240\) 0 0
\(241\) −1918.45 −0.512773 −0.256386 0.966574i \(-0.582532\pi\)
−0.256386 + 0.966574i \(0.582532\pi\)
\(242\) 773.651 0.205505
\(243\) −3821.06 −1.00873
\(244\) 331.304 0.0869245
\(245\) 0 0
\(246\) −2020.65 −0.523708
\(247\) −5181.75 −1.33485
\(248\) 5266.09 1.34838
\(249\) −366.097 −0.0931746
\(250\) 0 0
\(251\) 7695.71 1.93525 0.967627 0.252385i \(-0.0812148\pi\)
0.967627 + 0.252385i \(0.0812148\pi\)
\(252\) 832.141 0.208016
\(253\) 3183.29 0.791035
\(254\) 3604.76 0.890484
\(255\) 0 0
\(256\) −2439.44 −0.595567
\(257\) −5335.10 −1.29492 −0.647460 0.762099i \(-0.724169\pi\)
−0.647460 + 0.762099i \(0.724169\pi\)
\(258\) −1514.30 −0.365411
\(259\) 1050.93 0.252130
\(260\) 0 0
\(261\) 2191.69 0.519778
\(262\) 2693.97 0.635244
\(263\) −3934.15 −0.922396 −0.461198 0.887297i \(-0.652580\pi\)
−0.461198 + 0.887297i \(0.652580\pi\)
\(264\) −1677.67 −0.391112
\(265\) 0 0
\(266\) −963.804 −0.222160
\(267\) 2565.75 0.588095
\(268\) 325.532 0.0741979
\(269\) 3424.04 0.776088 0.388044 0.921641i \(-0.373151\pi\)
0.388044 + 0.921641i \(0.373151\pi\)
\(270\) 0 0
\(271\) 549.034 0.123068 0.0615340 0.998105i \(-0.480401\pi\)
0.0615340 + 0.998105i \(0.480401\pi\)
\(272\) 390.374 0.0870216
\(273\) 1455.66 0.322712
\(274\) 4276.85 0.942970
\(275\) 0 0
\(276\) 2235.27 0.487492
\(277\) −5203.65 −1.12873 −0.564363 0.825527i \(-0.690878\pi\)
−0.564363 + 0.825527i \(0.690878\pi\)
\(278\) −1998.37 −0.431131
\(279\) −4662.02 −1.00039
\(280\) 0 0
\(281\) −1986.73 −0.421774 −0.210887 0.977510i \(-0.567635\pi\)
−0.210887 + 0.977510i \(0.567635\pi\)
\(282\) −655.279 −0.138373
\(283\) −753.696 −0.158313 −0.0791565 0.996862i \(-0.525223\pi\)
−0.0791565 + 0.996862i \(0.525223\pi\)
\(284\) −4846.36 −1.01260
\(285\) 0 0
\(286\) 2184.51 0.451654
\(287\) −3736.47 −0.768490
\(288\) 3159.33 0.646407
\(289\) 289.000 0.0588235
\(290\) 0 0
\(291\) 2506.96 0.505020
\(292\) 1818.18 0.364387
\(293\) 7202.22 1.43603 0.718017 0.696025i \(-0.245050\pi\)
0.718017 + 0.696025i \(0.245050\pi\)
\(294\) −1202.14 −0.238471
\(295\) 0 0
\(296\) 2548.72 0.500477
\(297\) 3837.43 0.749731
\(298\) 389.345 0.0756852
\(299\) −6698.50 −1.29560
\(300\) 0 0
\(301\) −2800.15 −0.536205
\(302\) 911.509 0.173680
\(303\) −837.595 −0.158807
\(304\) 2047.60 0.386308
\(305\) 0 0
\(306\) 394.511 0.0737016
\(307\) −2425.71 −0.450953 −0.225477 0.974249i \(-0.572394\pi\)
−0.225477 + 0.974249i \(0.572394\pi\)
\(308\) −1347.96 −0.249375
\(309\) 1650.21 0.303809
\(310\) 0 0
\(311\) −9544.94 −1.74033 −0.870167 0.492757i \(-0.835989\pi\)
−0.870167 + 0.492757i \(0.835989\pi\)
\(312\) 3530.27 0.640584
\(313\) −588.379 −0.106253 −0.0531264 0.998588i \(-0.516919\pi\)
−0.0531264 + 0.998588i \(0.516919\pi\)
\(314\) 981.180 0.176341
\(315\) 0 0
\(316\) 4427.89 0.788255
\(317\) −7653.31 −1.35600 −0.678001 0.735061i \(-0.737154\pi\)
−0.678001 + 0.735061i \(0.737154\pi\)
\(318\) −2265.65 −0.399533
\(319\) −3550.26 −0.623124
\(320\) 0 0
\(321\) −3110.59 −0.540860
\(322\) −1245.92 −0.215628
\(323\) 1515.87 0.261131
\(324\) −134.920 −0.0231345
\(325\) 0 0
\(326\) −921.001 −0.156471
\(327\) −5723.73 −0.967960
\(328\) −9061.70 −1.52545
\(329\) −1211.70 −0.203050
\(330\) 0 0
\(331\) 752.266 0.124919 0.0624597 0.998047i \(-0.480106\pi\)
0.0624597 + 0.998047i \(0.480106\pi\)
\(332\) −713.373 −0.117926
\(333\) −2256.36 −0.371314
\(334\) −3859.98 −0.632361
\(335\) 0 0
\(336\) −575.211 −0.0933939
\(337\) 1968.57 0.318204 0.159102 0.987262i \(-0.449140\pi\)
0.159102 + 0.987262i \(0.449140\pi\)
\(338\) −1606.20 −0.258479
\(339\) −2231.39 −0.357501
\(340\) 0 0
\(341\) 7551.89 1.19929
\(342\) 2069.30 0.327178
\(343\) −4946.52 −0.778678
\(344\) −6790.93 −1.06437
\(345\) 0 0
\(346\) −241.363 −0.0375022
\(347\) −3983.10 −0.616207 −0.308104 0.951353i \(-0.599694\pi\)
−0.308104 + 0.951353i \(0.599694\pi\)
\(348\) −2492.96 −0.384013
\(349\) 1495.61 0.229393 0.114697 0.993401i \(-0.463410\pi\)
0.114697 + 0.993401i \(0.463410\pi\)
\(350\) 0 0
\(351\) −8074.98 −1.22795
\(352\) −5117.72 −0.774930
\(353\) −6482.49 −0.977417 −0.488708 0.872447i \(-0.662532\pi\)
−0.488708 + 0.872447i \(0.662532\pi\)
\(354\) −1253.98 −0.188272
\(355\) 0 0
\(356\) 4999.59 0.744320
\(357\) −425.838 −0.0631309
\(358\) 1393.56 0.205732
\(359\) −4943.42 −0.726751 −0.363376 0.931643i \(-0.618376\pi\)
−0.363376 + 0.931643i \(0.618376\pi\)
\(360\) 0 0
\(361\) 1092.08 0.159218
\(362\) 4696.50 0.681886
\(363\) 1792.94 0.259242
\(364\) 2836.48 0.408439
\(365\) 0 0
\(366\) −231.439 −0.0330533
\(367\) 14.8871 0.00211743 0.00105872 0.999999i \(-0.499663\pi\)
0.00105872 + 0.999999i \(0.499663\pi\)
\(368\) 2646.95 0.374950
\(369\) 8022.23 1.13176
\(370\) 0 0
\(371\) −4189.51 −0.586276
\(372\) 5302.86 0.739088
\(373\) −1923.18 −0.266966 −0.133483 0.991051i \(-0.542616\pi\)
−0.133483 + 0.991051i \(0.542616\pi\)
\(374\) −639.058 −0.0883554
\(375\) 0 0
\(376\) −2938.63 −0.403053
\(377\) 7470.70 1.02059
\(378\) −1501.94 −0.204369
\(379\) −9592.87 −1.30014 −0.650069 0.759875i \(-0.725260\pi\)
−0.650069 + 0.759875i \(0.725260\pi\)
\(380\) 0 0
\(381\) 8354.04 1.12333
\(382\) 668.215 0.0894996
\(383\) 9083.77 1.21190 0.605951 0.795502i \(-0.292793\pi\)
0.605951 + 0.795502i \(0.292793\pi\)
\(384\) −4382.47 −0.582401
\(385\) 0 0
\(386\) −4830.70 −0.636985
\(387\) 6011.95 0.789675
\(388\) 4885.04 0.639176
\(389\) −1143.78 −0.149079 −0.0745396 0.997218i \(-0.523749\pi\)
−0.0745396 + 0.997218i \(0.523749\pi\)
\(390\) 0 0
\(391\) 1959.58 0.253453
\(392\) −5391.06 −0.694616
\(393\) 6243.27 0.801352
\(394\) 1855.55 0.237262
\(395\) 0 0
\(396\) 2894.09 0.367256
\(397\) −10604.5 −1.34061 −0.670307 0.742084i \(-0.733838\pi\)
−0.670307 + 0.742084i \(0.733838\pi\)
\(398\) −5087.51 −0.640739
\(399\) −2233.62 −0.280252
\(400\) 0 0
\(401\) 13785.4 1.71674 0.858368 0.513035i \(-0.171479\pi\)
0.858368 + 0.513035i \(0.171479\pi\)
\(402\) −227.406 −0.0282139
\(403\) −15891.2 −1.96426
\(404\) −1632.13 −0.200993
\(405\) 0 0
\(406\) 1389.55 0.169857
\(407\) 3655.01 0.445141
\(408\) −1032.74 −0.125315
\(409\) −9505.94 −1.14924 −0.574619 0.818421i \(-0.694850\pi\)
−0.574619 + 0.818421i \(0.694850\pi\)
\(410\) 0 0
\(411\) 9911.59 1.18954
\(412\) 3215.58 0.384515
\(413\) −2318.78 −0.276270
\(414\) 2675.00 0.317558
\(415\) 0 0
\(416\) 10769.1 1.26922
\(417\) −4631.23 −0.543866
\(418\) −3352.00 −0.392229
\(419\) 9680.86 1.12874 0.564369 0.825523i \(-0.309120\pi\)
0.564369 + 0.825523i \(0.309120\pi\)
\(420\) 0 0
\(421\) −12360.3 −1.43089 −0.715444 0.698671i \(-0.753775\pi\)
−0.715444 + 0.698671i \(0.753775\pi\)
\(422\) 7168.36 0.826897
\(423\) 2601.54 0.299033
\(424\) −10160.4 −1.16376
\(425\) 0 0
\(426\) 3385.51 0.385043
\(427\) −427.962 −0.0485025
\(428\) −6061.25 −0.684537
\(429\) 5062.61 0.569756
\(430\) 0 0
\(431\) 2970.58 0.331990 0.165995 0.986127i \(-0.446916\pi\)
0.165995 + 0.986127i \(0.446916\pi\)
\(432\) 3190.87 0.355372
\(433\) −6131.50 −0.680510 −0.340255 0.940333i \(-0.610513\pi\)
−0.340255 + 0.940333i \(0.610513\pi\)
\(434\) −2955.76 −0.326915
\(435\) 0 0
\(436\) −11153.2 −1.22509
\(437\) 10278.4 1.12513
\(438\) −1270.12 −0.138559
\(439\) −2544.91 −0.276679 −0.138339 0.990385i \(-0.544176\pi\)
−0.138339 + 0.990385i \(0.544176\pi\)
\(440\) 0 0
\(441\) 4772.65 0.515349
\(442\) 1344.75 0.144713
\(443\) −8529.82 −0.914817 −0.457408 0.889257i \(-0.651222\pi\)
−0.457408 + 0.889257i \(0.651222\pi\)
\(444\) 2566.51 0.274327
\(445\) 0 0
\(446\) 959.043 0.101821
\(447\) 902.308 0.0954759
\(448\) 544.331 0.0574046
\(449\) 8855.74 0.930798 0.465399 0.885101i \(-0.345911\pi\)
0.465399 + 0.885101i \(0.345911\pi\)
\(450\) 0 0
\(451\) −12995.0 −1.35679
\(452\) −4348.07 −0.452469
\(453\) 2112.42 0.219095
\(454\) −2928.35 −0.302718
\(455\) 0 0
\(456\) −5416.98 −0.556301
\(457\) 7154.78 0.732356 0.366178 0.930545i \(-0.380666\pi\)
0.366178 + 0.930545i \(0.380666\pi\)
\(458\) 5331.82 0.543973
\(459\) 2362.25 0.240219
\(460\) 0 0
\(461\) −7263.06 −0.733784 −0.366892 0.930264i \(-0.619578\pi\)
−0.366892 + 0.930264i \(0.619578\pi\)
\(462\) 941.645 0.0948253
\(463\) −352.898 −0.0354224 −0.0177112 0.999843i \(-0.505638\pi\)
−0.0177112 + 0.999843i \(0.505638\pi\)
\(464\) −2952.09 −0.295360
\(465\) 0 0
\(466\) −7068.47 −0.702662
\(467\) −1483.02 −0.146951 −0.0734753 0.997297i \(-0.523409\pi\)
−0.0734753 + 0.997297i \(0.523409\pi\)
\(468\) −6089.94 −0.601512
\(469\) −420.506 −0.0414012
\(470\) 0 0
\(471\) 2273.89 0.222452
\(472\) −5623.51 −0.548396
\(473\) −9738.60 −0.946683
\(474\) −3093.19 −0.299736
\(475\) 0 0
\(476\) −829.783 −0.0799014
\(477\) 8994.92 0.863415
\(478\) −455.063 −0.0435441
\(479\) −9990.10 −0.952942 −0.476471 0.879190i \(-0.658084\pi\)
−0.476471 + 0.879190i \(0.658084\pi\)
\(480\) 0 0
\(481\) −7691.13 −0.729075
\(482\) 2611.44 0.246779
\(483\) −2887.42 −0.272012
\(484\) 3493.69 0.328108
\(485\) 0 0
\(486\) 5201.30 0.485465
\(487\) 1129.88 0.105133 0.0525663 0.998617i \(-0.483260\pi\)
0.0525663 + 0.998617i \(0.483260\pi\)
\(488\) −1037.90 −0.0962774
\(489\) −2134.42 −0.197386
\(490\) 0 0
\(491\) 18774.9 1.72566 0.862832 0.505491i \(-0.168689\pi\)
0.862832 + 0.505491i \(0.168689\pi\)
\(492\) −9124.97 −0.836149
\(493\) −2185.48 −0.199653
\(494\) 7053.51 0.642414
\(495\) 0 0
\(496\) 6279.49 0.568463
\(497\) 6260.28 0.565014
\(498\) 498.339 0.0448416
\(499\) 17329.1 1.55462 0.777310 0.629118i \(-0.216584\pi\)
0.777310 + 0.629118i \(0.216584\pi\)
\(500\) 0 0
\(501\) −8945.50 −0.797716
\(502\) −10475.6 −0.931369
\(503\) 20837.0 1.84707 0.923533 0.383518i \(-0.125288\pi\)
0.923533 + 0.383518i \(0.125288\pi\)
\(504\) −2606.90 −0.230398
\(505\) 0 0
\(506\) −4333.16 −0.380697
\(507\) −3722.37 −0.326068
\(508\) 16278.6 1.42174
\(509\) 11835.0 1.03060 0.515301 0.857009i \(-0.327680\pi\)
0.515301 + 0.857009i \(0.327680\pi\)
\(510\) 0 0
\(511\) −2348.63 −0.203322
\(512\) −7793.12 −0.672676
\(513\) 12390.6 1.06639
\(514\) 7262.26 0.623199
\(515\) 0 0
\(516\) −6838.35 −0.583414
\(517\) −4214.16 −0.358489
\(518\) −1430.55 −0.121341
\(519\) −559.359 −0.0473086
\(520\) 0 0
\(521\) 7686.37 0.646346 0.323173 0.946340i \(-0.395250\pi\)
0.323173 + 0.946340i \(0.395250\pi\)
\(522\) −2983.37 −0.250151
\(523\) −11476.4 −0.959518 −0.479759 0.877400i \(-0.659276\pi\)
−0.479759 + 0.877400i \(0.659276\pi\)
\(524\) 12165.6 1.01423
\(525\) 0 0
\(526\) 5355.25 0.443916
\(527\) 4648.81 0.384261
\(528\) −2000.52 −0.164889
\(529\) 1120.01 0.0920535
\(530\) 0 0
\(531\) 4978.44 0.406866
\(532\) −4352.40 −0.354700
\(533\) 27345.0 2.22222
\(534\) −3492.56 −0.283029
\(535\) 0 0
\(536\) −1019.81 −0.0821814
\(537\) 3229.59 0.259529
\(538\) −4660.88 −0.373504
\(539\) −7731.09 −0.617814
\(540\) 0 0
\(541\) −546.481 −0.0434289 −0.0217145 0.999764i \(-0.506912\pi\)
−0.0217145 + 0.999764i \(0.506912\pi\)
\(542\) −747.357 −0.0592283
\(543\) 10884.1 0.860191
\(544\) −3150.38 −0.248293
\(545\) 0 0
\(546\) −1981.47 −0.155310
\(547\) −8397.33 −0.656388 −0.328194 0.944610i \(-0.606440\pi\)
−0.328194 + 0.944610i \(0.606440\pi\)
\(548\) 19313.6 1.50554
\(549\) 918.839 0.0714301
\(550\) 0 0
\(551\) −11463.3 −0.886305
\(552\) −7002.58 −0.539945
\(553\) −5719.73 −0.439833
\(554\) 7083.32 0.543215
\(555\) 0 0
\(556\) −9024.36 −0.688342
\(557\) 4881.65 0.371350 0.185675 0.982611i \(-0.440553\pi\)
0.185675 + 0.982611i \(0.440553\pi\)
\(558\) 6346.04 0.481451
\(559\) 20492.6 1.55053
\(560\) 0 0
\(561\) −1481.02 −0.111459
\(562\) 2704.38 0.202985
\(563\) −7198.57 −0.538870 −0.269435 0.963019i \(-0.586837\pi\)
−0.269435 + 0.963019i \(0.586837\pi\)
\(564\) −2959.14 −0.220926
\(565\) 0 0
\(566\) 1025.95 0.0761904
\(567\) 174.283 0.0129087
\(568\) 15182.5 1.12155
\(569\) −23946.9 −1.76433 −0.882167 0.470937i \(-0.843916\pi\)
−0.882167 + 0.470937i \(0.843916\pi\)
\(570\) 0 0
\(571\) 1593.15 0.116763 0.0583813 0.998294i \(-0.481406\pi\)
0.0583813 + 0.998294i \(0.481406\pi\)
\(572\) 9864.95 0.721109
\(573\) 1548.59 0.112903
\(574\) 5086.16 0.369847
\(575\) 0 0
\(576\) −1168.69 −0.0845403
\(577\) −12937.4 −0.933435 −0.466717 0.884406i \(-0.654564\pi\)
−0.466717 + 0.884406i \(0.654564\pi\)
\(578\) −393.393 −0.0283097
\(579\) −11195.2 −0.803549
\(580\) 0 0
\(581\) 921.499 0.0658007
\(582\) −3412.53 −0.243048
\(583\) −14570.6 −1.03508
\(584\) −5695.92 −0.403594
\(585\) 0 0
\(586\) −9803.82 −0.691112
\(587\) 12899.2 0.906998 0.453499 0.891257i \(-0.350176\pi\)
0.453499 + 0.891257i \(0.350176\pi\)
\(588\) −5428.70 −0.380741
\(589\) 24384.1 1.70582
\(590\) 0 0
\(591\) 4300.23 0.299302
\(592\) 3039.19 0.210997
\(593\) −4357.13 −0.301730 −0.150865 0.988554i \(-0.548206\pi\)
−0.150865 + 0.988554i \(0.548206\pi\)
\(594\) −5223.59 −0.360819
\(595\) 0 0
\(596\) 1758.23 0.120839
\(597\) −11790.3 −0.808284
\(598\) 9118.14 0.623526
\(599\) 13726.8 0.936328 0.468164 0.883642i \(-0.344916\pi\)
0.468164 + 0.883642i \(0.344916\pi\)
\(600\) 0 0
\(601\) 2531.41 0.171811 0.0859056 0.996303i \(-0.472622\pi\)
0.0859056 + 0.996303i \(0.472622\pi\)
\(602\) 3811.62 0.258057
\(603\) 902.830 0.0609720
\(604\) 4116.24 0.277297
\(605\) 0 0
\(606\) 1140.15 0.0764283
\(607\) −185.004 −0.0123708 −0.00618540 0.999981i \(-0.501969\pi\)
−0.00618540 + 0.999981i \(0.501969\pi\)
\(608\) −16524.4 −1.10223
\(609\) 3220.28 0.214273
\(610\) 0 0
\(611\) 8867.73 0.587152
\(612\) 1781.55 0.117672
\(613\) 17706.9 1.16668 0.583339 0.812228i \(-0.301746\pi\)
0.583339 + 0.812228i \(0.301746\pi\)
\(614\) 3301.93 0.217028
\(615\) 0 0
\(616\) 4222.84 0.276207
\(617\) 6183.89 0.403491 0.201746 0.979438i \(-0.435339\pi\)
0.201746 + 0.979438i \(0.435339\pi\)
\(618\) −2246.30 −0.146213
\(619\) −1247.51 −0.0810046 −0.0405023 0.999179i \(-0.512896\pi\)
−0.0405023 + 0.999179i \(0.512896\pi\)
\(620\) 0 0
\(621\) 16017.4 1.03503
\(622\) 12992.8 0.837561
\(623\) −6458.23 −0.415318
\(624\) 4209.63 0.270064
\(625\) 0 0
\(626\) 800.914 0.0511357
\(627\) −7768.27 −0.494792
\(628\) 4430.87 0.281546
\(629\) 2249.96 0.142626
\(630\) 0 0
\(631\) 24053.3 1.51750 0.758752 0.651379i \(-0.225809\pi\)
0.758752 + 0.651379i \(0.225809\pi\)
\(632\) −13871.5 −0.873069
\(633\) 16612.7 1.04312
\(634\) 10417.8 0.652596
\(635\) 0 0
\(636\) −10231.4 −0.637893
\(637\) 16268.3 1.01189
\(638\) 4832.69 0.299887
\(639\) −13440.9 −0.832102
\(640\) 0 0
\(641\) −21286.8 −1.31167 −0.655834 0.754905i \(-0.727683\pi\)
−0.655834 + 0.754905i \(0.727683\pi\)
\(642\) 4234.20 0.260297
\(643\) 1789.41 0.109747 0.0548736 0.998493i \(-0.482524\pi\)
0.0548736 + 0.998493i \(0.482524\pi\)
\(644\) −5626.38 −0.344271
\(645\) 0 0
\(646\) −2063.43 −0.125673
\(647\) 4378.61 0.266060 0.133030 0.991112i \(-0.457529\pi\)
0.133030 + 0.991112i \(0.457529\pi\)
\(648\) 422.672 0.0256237
\(649\) −8064.45 −0.487762
\(650\) 0 0
\(651\) −6849.97 −0.412399
\(652\) −4159.11 −0.249821
\(653\) 7665.15 0.459358 0.229679 0.973266i \(-0.426232\pi\)
0.229679 + 0.973266i \(0.426232\pi\)
\(654\) 7791.26 0.465845
\(655\) 0 0
\(656\) −10805.5 −0.643117
\(657\) 5042.54 0.299434
\(658\) 1649.39 0.0977205
\(659\) −4710.22 −0.278428 −0.139214 0.990262i \(-0.544458\pi\)
−0.139214 + 0.990262i \(0.544458\pi\)
\(660\) 0 0
\(661\) −31266.6 −1.83983 −0.919916 0.392116i \(-0.871743\pi\)
−0.919916 + 0.392116i \(0.871743\pi\)
\(662\) −1024.00 −0.0601192
\(663\) 3116.46 0.182554
\(664\) 2234.82 0.130614
\(665\) 0 0
\(666\) 3071.40 0.178700
\(667\) −14818.7 −0.860246
\(668\) −17431.1 −1.00962
\(669\) 2222.58 0.128446
\(670\) 0 0
\(671\) −1488.40 −0.0856322
\(672\) 4642.05 0.266474
\(673\) −11723.0 −0.671454 −0.335727 0.941959i \(-0.608982\pi\)
−0.335727 + 0.941959i \(0.608982\pi\)
\(674\) −2679.65 −0.153140
\(675\) 0 0
\(676\) −7253.37 −0.412686
\(677\) 289.531 0.0164366 0.00821829 0.999966i \(-0.497384\pi\)
0.00821829 + 0.999966i \(0.497384\pi\)
\(678\) 3037.42 0.172052
\(679\) −6310.25 −0.356650
\(680\) 0 0
\(681\) −6786.45 −0.381875
\(682\) −10279.8 −0.577175
\(683\) −1720.10 −0.0963660 −0.0481830 0.998839i \(-0.515343\pi\)
−0.0481830 + 0.998839i \(0.515343\pi\)
\(684\) 9344.64 0.522370
\(685\) 0 0
\(686\) 6733.30 0.374750
\(687\) 12356.5 0.686215
\(688\) −8097.77 −0.448728
\(689\) 30660.5 1.69532
\(690\) 0 0
\(691\) −16777.7 −0.923665 −0.461832 0.886967i \(-0.652808\pi\)
−0.461832 + 0.886967i \(0.652808\pi\)
\(692\) −1089.96 −0.0598758
\(693\) −3738.44 −0.204923
\(694\) 5421.88 0.296559
\(695\) 0 0
\(696\) 7809.84 0.425332
\(697\) −7999.50 −0.434724
\(698\) −2035.86 −0.110399
\(699\) −16381.2 −0.886400
\(700\) 0 0
\(701\) 23981.1 1.29209 0.646043 0.763301i \(-0.276423\pi\)
0.646043 + 0.763301i \(0.276423\pi\)
\(702\) 10991.8 0.590969
\(703\) 11801.6 0.633150
\(704\) 1893.12 0.101349
\(705\) 0 0
\(706\) 8824.10 0.470396
\(707\) 2108.30 0.112151
\(708\) −5662.78 −0.300593
\(709\) −7709.28 −0.408361 −0.204181 0.978933i \(-0.565453\pi\)
−0.204181 + 0.978933i \(0.565453\pi\)
\(710\) 0 0
\(711\) 12280.3 0.647747
\(712\) −15662.5 −0.824407
\(713\) 31521.5 1.65567
\(714\) 579.660 0.0303827
\(715\) 0 0
\(716\) 6293.13 0.328471
\(717\) −1054.61 −0.0549304
\(718\) 6729.09 0.349760
\(719\) −11976.5 −0.621209 −0.310605 0.950539i \(-0.600532\pi\)
−0.310605 + 0.950539i \(0.600532\pi\)
\(720\) 0 0
\(721\) −4153.72 −0.214553
\(722\) −1486.56 −0.0766260
\(723\) 6052.00 0.311309
\(724\) 21208.7 1.08870
\(725\) 0 0
\(726\) −2440.58 −0.124764
\(727\) 18597.3 0.948745 0.474372 0.880324i \(-0.342675\pi\)
0.474372 + 0.880324i \(0.342675\pi\)
\(728\) −8886.00 −0.452386
\(729\) 11461.4 0.582300
\(730\) 0 0
\(731\) −5994.91 −0.303324
\(732\) −1045.14 −0.0527727
\(733\) 23569.5 1.18767 0.593833 0.804588i \(-0.297614\pi\)
0.593833 + 0.804588i \(0.297614\pi\)
\(734\) −20.2646 −0.00101905
\(735\) 0 0
\(736\) −21361.3 −1.06982
\(737\) −1462.47 −0.0730948
\(738\) −10920.0 −0.544678
\(739\) 10149.1 0.505199 0.252599 0.967571i \(-0.418715\pi\)
0.252599 + 0.967571i \(0.418715\pi\)
\(740\) 0 0
\(741\) 16346.5 0.810397
\(742\) 5702.85 0.282154
\(743\) −27758.0 −1.37058 −0.685291 0.728269i \(-0.740325\pi\)
−0.685291 + 0.728269i \(0.740325\pi\)
\(744\) −16612.6 −0.818611
\(745\) 0 0
\(746\) 2617.87 0.128481
\(747\) −1978.47 −0.0969054
\(748\) −2885.89 −0.141068
\(749\) 7829.63 0.381960
\(750\) 0 0
\(751\) −815.225 −0.0396112 −0.0198056 0.999804i \(-0.506305\pi\)
−0.0198056 + 0.999804i \(0.506305\pi\)
\(752\) −3504.13 −0.169924
\(753\) −24277.1 −1.17491
\(754\) −10169.3 −0.491172
\(755\) 0 0
\(756\) −6782.56 −0.326295
\(757\) 13239.4 0.635659 0.317829 0.948148i \(-0.397046\pi\)
0.317829 + 0.948148i \(0.397046\pi\)
\(758\) 13058.0 0.625710
\(759\) −10042.1 −0.480244
\(760\) 0 0
\(761\) −11028.2 −0.525324 −0.262662 0.964888i \(-0.584600\pi\)
−0.262662 + 0.964888i \(0.584600\pi\)
\(762\) −11371.7 −0.540621
\(763\) 14407.1 0.683582
\(764\) 3017.56 0.142895
\(765\) 0 0
\(766\) −12365.0 −0.583246
\(767\) 16969.8 0.798882
\(768\) 7695.55 0.361574
\(769\) −18921.2 −0.887277 −0.443639 0.896206i \(-0.646313\pi\)
−0.443639 + 0.896206i \(0.646313\pi\)
\(770\) 0 0
\(771\) 16830.3 0.786158
\(772\) −21814.7 −1.01701
\(773\) −38728.6 −1.80203 −0.901016 0.433786i \(-0.857177\pi\)
−0.901016 + 0.433786i \(0.857177\pi\)
\(774\) −8183.59 −0.380043
\(775\) 0 0
\(776\) −15303.6 −0.707949
\(777\) −3315.29 −0.153070
\(778\) 1556.93 0.0717465
\(779\) −41959.2 −1.92984
\(780\) 0 0
\(781\) 21772.5 0.997545
\(782\) −2667.42 −0.121978
\(783\) −17863.9 −0.815329
\(784\) −6428.50 −0.292844
\(785\) 0 0
\(786\) −8498.47 −0.385662
\(787\) 20587.3 0.932477 0.466239 0.884659i \(-0.345609\pi\)
0.466239 + 0.884659i \(0.345609\pi\)
\(788\) 8379.37 0.378811
\(789\) 12410.8 0.559995
\(790\) 0 0
\(791\) 5616.62 0.252470
\(792\) −9066.49 −0.406772
\(793\) 3132.00 0.140253
\(794\) 14435.0 0.645190
\(795\) 0 0
\(796\) −22974.5 −1.02300
\(797\) 15871.4 0.705385 0.352693 0.935739i \(-0.385266\pi\)
0.352693 + 0.935739i \(0.385266\pi\)
\(798\) 3040.45 0.134876
\(799\) −2594.17 −0.114862
\(800\) 0 0
\(801\) 13865.9 0.611644
\(802\) −18765.0 −0.826204
\(803\) −8168.28 −0.358969
\(804\) −1026.93 −0.0450462
\(805\) 0 0
\(806\) 21631.4 0.945329
\(807\) −10801.6 −0.471170
\(808\) 5113.06 0.222620
\(809\) 39667.1 1.72388 0.861942 0.507007i \(-0.169248\pi\)
0.861942 + 0.507007i \(0.169248\pi\)
\(810\) 0 0
\(811\) 8003.87 0.346552 0.173276 0.984873i \(-0.444565\pi\)
0.173276 + 0.984873i \(0.444565\pi\)
\(812\) 6275.00 0.271194
\(813\) −1732.00 −0.0747157
\(814\) −4975.28 −0.214230
\(815\) 0 0
\(816\) −1231.48 −0.0528316
\(817\) −31444.7 −1.34652
\(818\) 12939.7 0.553088
\(819\) 7866.69 0.335634
\(820\) 0 0
\(821\) −13279.1 −0.564489 −0.282244 0.959343i \(-0.591079\pi\)
−0.282244 + 0.959343i \(0.591079\pi\)
\(822\) −13491.9 −0.572486
\(823\) −28934.0 −1.22549 −0.612745 0.790281i \(-0.709935\pi\)
−0.612745 + 0.790281i \(0.709935\pi\)
\(824\) −10073.6 −0.425887
\(825\) 0 0
\(826\) 3156.37 0.132959
\(827\) 13679.6 0.575193 0.287597 0.957752i \(-0.407144\pi\)
0.287597 + 0.957752i \(0.407144\pi\)
\(828\) 12079.9 0.507012
\(829\) 16514.5 0.691886 0.345943 0.938256i \(-0.387559\pi\)
0.345943 + 0.938256i \(0.387559\pi\)
\(830\) 0 0
\(831\) 16415.6 0.685260
\(832\) −3983.64 −0.165995
\(833\) −4759.13 −0.197952
\(834\) 6304.13 0.261744
\(835\) 0 0
\(836\) −15137.1 −0.626231
\(837\) 37998.9 1.56922
\(838\) −13177.8 −0.543221
\(839\) −87.9839 −0.00362043 −0.00181022 0.999998i \(-0.500576\pi\)
−0.00181022 + 0.999998i \(0.500576\pi\)
\(840\) 0 0
\(841\) −7861.94 −0.322356
\(842\) 16825.1 0.688635
\(843\) 6267.41 0.256063
\(844\) 32371.3 1.32022
\(845\) 0 0
\(846\) −3541.27 −0.143914
\(847\) −4512.98 −0.183079
\(848\) −12115.7 −0.490630
\(849\) 2377.63 0.0961133
\(850\) 0 0
\(851\) 15256.0 0.614534
\(852\) 15288.5 0.614758
\(853\) −8162.96 −0.327660 −0.163830 0.986489i \(-0.552385\pi\)
−0.163830 + 0.986489i \(0.552385\pi\)
\(854\) 582.551 0.0233425
\(855\) 0 0
\(856\) 18988.4 0.758191
\(857\) −18724.9 −0.746361 −0.373181 0.927759i \(-0.621733\pi\)
−0.373181 + 0.927759i \(0.621733\pi\)
\(858\) −6891.34 −0.274203
\(859\) −46422.5 −1.84391 −0.921953 0.387301i \(-0.873407\pi\)
−0.921953 + 0.387301i \(0.873407\pi\)
\(860\) 0 0
\(861\) 11787.2 0.466557
\(862\) −4043.61 −0.159775
\(863\) −29112.3 −1.14831 −0.574157 0.818746i \(-0.694670\pi\)
−0.574157 + 0.818746i \(0.694670\pi\)
\(864\) −25750.8 −1.01396
\(865\) 0 0
\(866\) 8346.33 0.327505
\(867\) −911.688 −0.0357123
\(868\) −13347.8 −0.521950
\(869\) −19892.6 −0.776536
\(870\) 0 0
\(871\) 3077.43 0.119719
\(872\) 34940.2 1.35691
\(873\) 13548.2 0.525241
\(874\) −13991.2 −0.541487
\(875\) 0 0
\(876\) −5735.69 −0.221222
\(877\) −39163.0 −1.50791 −0.753957 0.656924i \(-0.771857\pi\)
−0.753957 + 0.656924i \(0.771857\pi\)
\(878\) 3464.19 0.133156
\(879\) −22720.3 −0.871830
\(880\) 0 0
\(881\) −35073.2 −1.34125 −0.670627 0.741795i \(-0.733975\pi\)
−0.670627 + 0.741795i \(0.733975\pi\)
\(882\) −6496.63 −0.248019
\(883\) 48775.7 1.85893 0.929463 0.368915i \(-0.120271\pi\)
0.929463 + 0.368915i \(0.120271\pi\)
\(884\) 6072.69 0.231048
\(885\) 0 0
\(886\) 11611.0 0.440269
\(887\) −13296.0 −0.503309 −0.251654 0.967817i \(-0.580975\pi\)
−0.251654 + 0.967817i \(0.580975\pi\)
\(888\) −8040.27 −0.303844
\(889\) −21027.9 −0.793309
\(890\) 0 0
\(891\) 606.137 0.0227905
\(892\) 4330.90 0.162566
\(893\) −13607.0 −0.509899
\(894\) −1228.24 −0.0459492
\(895\) 0 0
\(896\) 11031.1 0.411297
\(897\) 21131.3 0.786570
\(898\) −12054.6 −0.447960
\(899\) −35155.3 −1.30422
\(900\) 0 0
\(901\) −8969.43 −0.331648
\(902\) 17689.1 0.652974
\(903\) 8833.44 0.325535
\(904\) 13621.4 0.501153
\(905\) 0 0
\(906\) −2875.47 −0.105443
\(907\) 11675.0 0.427410 0.213705 0.976898i \(-0.431447\pi\)
0.213705 + 0.976898i \(0.431447\pi\)
\(908\) −13224.0 −0.483319
\(909\) −4526.54 −0.165166
\(910\) 0 0
\(911\) 18552.9 0.674738 0.337369 0.941372i \(-0.390463\pi\)
0.337369 + 0.941372i \(0.390463\pi\)
\(912\) −6459.41 −0.234531
\(913\) 3204.87 0.116173
\(914\) −9739.24 −0.352457
\(915\) 0 0
\(916\) 24077.7 0.868504
\(917\) −15714.9 −0.565922
\(918\) −3215.55 −0.115609
\(919\) 33956.8 1.21886 0.609429 0.792841i \(-0.291399\pi\)
0.609429 + 0.792841i \(0.291399\pi\)
\(920\) 0 0
\(921\) 7652.23 0.273778
\(922\) 9886.63 0.353144
\(923\) −45815.3 −1.63383
\(924\) 4252.33 0.151398
\(925\) 0 0
\(926\) 480.372 0.0170475
\(927\) 8918.08 0.315974
\(928\) 23823.8 0.842732
\(929\) −23695.3 −0.836832 −0.418416 0.908256i \(-0.637415\pi\)
−0.418416 + 0.908256i \(0.637415\pi\)
\(930\) 0 0
\(931\) −24962.7 −0.878753
\(932\) −31920.2 −1.12187
\(933\) 30110.8 1.05657
\(934\) 2018.72 0.0707221
\(935\) 0 0
\(936\) 19078.3 0.666234
\(937\) −7990.62 −0.278593 −0.139297 0.990251i \(-0.544484\pi\)
−0.139297 + 0.990251i \(0.544484\pi\)
\(938\) 572.402 0.0199249
\(939\) 1856.12 0.0645071
\(940\) 0 0
\(941\) 24385.9 0.844799 0.422400 0.906410i \(-0.361188\pi\)
0.422400 + 0.906410i \(0.361188\pi\)
\(942\) −3095.26 −0.107058
\(943\) −54241.0 −1.87310
\(944\) −6705.69 −0.231199
\(945\) 0 0
\(946\) 13256.4 0.455605
\(947\) 1174.62 0.0403064 0.0201532 0.999797i \(-0.493585\pi\)
0.0201532 + 0.999797i \(0.493585\pi\)
\(948\) −13968.4 −0.478557
\(949\) 17188.3 0.587939
\(950\) 0 0
\(951\) 24143.4 0.823241
\(952\) 2599.51 0.0884985
\(953\) 33546.9 1.14029 0.570143 0.821546i \(-0.306888\pi\)
0.570143 + 0.821546i \(0.306888\pi\)
\(954\) −12244.1 −0.415531
\(955\) 0 0
\(956\) −2055.00 −0.0695224
\(957\) 11199.8 0.378304
\(958\) 13598.7 0.458617
\(959\) −24948.4 −0.840067
\(960\) 0 0
\(961\) 44989.1 1.51016
\(962\) 10469.3 0.350878
\(963\) −16810.3 −0.562517
\(964\) 11792.9 0.394007
\(965\) 0 0
\(966\) 3930.41 0.130910
\(967\) −24766.8 −0.823625 −0.411813 0.911269i \(-0.635104\pi\)
−0.411813 + 0.911269i \(0.635104\pi\)
\(968\) −10944.9 −0.363411
\(969\) −4782.01 −0.158535
\(970\) 0 0
\(971\) 42324.3 1.39882 0.699409 0.714721i \(-0.253447\pi\)
0.699409 + 0.714721i \(0.253447\pi\)
\(972\) 23488.3 0.775091
\(973\) 11657.2 0.384083
\(974\) −1538.01 −0.0505966
\(975\) 0 0
\(976\) −1237.63 −0.0405896
\(977\) 11320.4 0.370698 0.185349 0.982673i \(-0.440658\pi\)
0.185349 + 0.982673i \(0.440658\pi\)
\(978\) 2905.42 0.0949949
\(979\) −22461.0 −0.733254
\(980\) 0 0
\(981\) −30932.2 −1.00672
\(982\) −25556.8 −0.830500
\(983\) 11311.9 0.367032 0.183516 0.983017i \(-0.441252\pi\)
0.183516 + 0.983017i \(0.441252\pi\)
\(984\) 28586.3 0.926116
\(985\) 0 0
\(986\) 2974.92 0.0960860
\(987\) 3822.47 0.123273
\(988\) 31852.6 1.02568
\(989\) −40648.8 −1.30693
\(990\) 0 0
\(991\) −29405.5 −0.942580 −0.471290 0.881978i \(-0.656212\pi\)
−0.471290 + 0.881978i \(0.656212\pi\)
\(992\) −50676.5 −1.62196
\(993\) −2373.12 −0.0758397
\(994\) −8521.63 −0.271921
\(995\) 0 0
\(996\) 2250.43 0.0715939
\(997\) 54905.9 1.74412 0.872060 0.489398i \(-0.162784\pi\)
0.872060 + 0.489398i \(0.162784\pi\)
\(998\) −23588.7 −0.748183
\(999\) 18390.9 0.582446
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 425.4.a.g.1.2 3
5.2 odd 4 425.4.b.f.324.3 6
5.3 odd 4 425.4.b.f.324.4 6
5.4 even 2 17.4.a.b.1.2 3
15.14 odd 2 153.4.a.g.1.2 3
20.19 odd 2 272.4.a.h.1.2 3
35.34 odd 2 833.4.a.d.1.2 3
40.19 odd 2 1088.4.a.x.1.2 3
40.29 even 2 1088.4.a.v.1.2 3
55.54 odd 2 2057.4.a.e.1.2 3
60.59 even 2 2448.4.a.bi.1.1 3
85.4 even 4 289.4.b.b.288.3 6
85.64 even 4 289.4.b.b.288.4 6
85.84 even 2 289.4.a.b.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.b.1.2 3 5.4 even 2
153.4.a.g.1.2 3 15.14 odd 2
272.4.a.h.1.2 3 20.19 odd 2
289.4.a.b.1.2 3 85.84 even 2
289.4.b.b.288.3 6 85.4 even 4
289.4.b.b.288.4 6 85.64 even 4
425.4.a.g.1.2 3 1.1 even 1 trivial
425.4.b.f.324.3 6 5.2 odd 4
425.4.b.f.324.4 6 5.3 odd 4
833.4.a.d.1.2 3 35.34 odd 2
1088.4.a.v.1.2 3 40.29 even 2
1088.4.a.x.1.2 3 40.19 odd 2
2057.4.a.e.1.2 3 55.54 odd 2
2448.4.a.bi.1.1 3 60.59 even 2