Properties

Label 1875.4.a.g.1.6
Level $1875$
Weight $4$
Character 1875.1
Self dual yes
Analytic conductor $110.629$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1875.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(110.628581261\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \( x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + \cdots + 42064 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.955230\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.955230 q^{2} +3.00000 q^{3} -7.08754 q^{4} -2.86569 q^{6} +12.4836 q^{7} +14.4121 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-0.955230 q^{2} +3.00000 q^{3} -7.08754 q^{4} -2.86569 q^{6} +12.4836 q^{7} +14.4121 q^{8} +9.00000 q^{9} -44.7266 q^{11} -21.2626 q^{12} -7.13628 q^{13} -11.9247 q^{14} +42.9334 q^{16} +26.5116 q^{17} -8.59707 q^{18} -46.3901 q^{19} +37.4507 q^{21} +42.7241 q^{22} +145.454 q^{23} +43.2362 q^{24} +6.81679 q^{26} +27.0000 q^{27} -88.4777 q^{28} -213.354 q^{29} -148.082 q^{31} -156.308 q^{32} -134.180 q^{33} -25.3247 q^{34} -63.7878 q^{36} +402.312 q^{37} +44.3132 q^{38} -21.4089 q^{39} +233.656 q^{41} -35.7740 q^{42} +87.5080 q^{43} +317.001 q^{44} -138.942 q^{46} -75.9842 q^{47} +128.800 q^{48} -187.161 q^{49} +79.5348 q^{51} +50.5787 q^{52} +167.029 q^{53} -25.7912 q^{54} +179.914 q^{56} -139.170 q^{57} +203.802 q^{58} -595.082 q^{59} -865.794 q^{61} +141.452 q^{62} +112.352 q^{63} -194.158 q^{64} +128.172 q^{66} -406.302 q^{67} -187.902 q^{68} +436.361 q^{69} +526.158 q^{71} +129.709 q^{72} +1001.69 q^{73} -384.301 q^{74} +328.792 q^{76} -558.347 q^{77} +20.4504 q^{78} -133.956 q^{79} +81.0000 q^{81} -223.195 q^{82} +516.642 q^{83} -265.433 q^{84} -83.5903 q^{86} -640.061 q^{87} -644.602 q^{88} -610.698 q^{89} -89.0862 q^{91} -1030.91 q^{92} -444.246 q^{93} +72.5824 q^{94} -468.924 q^{96} -1639.57 q^{97} +178.782 q^{98} -402.539 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9} - 33 q^{11} + 150 q^{12} - 188 q^{13} - 287 q^{14} + 82 q^{16} - 146 q^{17} - 184 q^{19} - 81 q^{21} - 277 q^{22} - 164 q^{23} + 63 q^{24} + 103 q^{26} + 378 q^{27} + 224 q^{28} + 252 q^{29} - 889 q^{31} - 422 q^{32} - 99 q^{33} - 230 q^{34} + 450 q^{36} - 642 q^{37} - 1833 q^{38} - 564 q^{39} - 164 q^{41} - 861 q^{42} - 696 q^{43} - 6 q^{44} - 419 q^{46} + 92 q^{47} + 246 q^{48} + 81 q^{49} - 438 q^{51} - 1956 q^{52} - 949 q^{53} - 2380 q^{56} - 552 q^{57} - 1041 q^{58} - 81 q^{59} - 496 q^{61} - 1454 q^{62} - 243 q^{63} - 3903 q^{64} - 831 q^{66} - 1926 q^{67} - 685 q^{68} - 492 q^{69} + 2498 q^{71} + 189 q^{72} + 1026 q^{73} - 707 q^{74} - 5704 q^{76} - 5434 q^{77} + 309 q^{78} - 695 q^{79} + 1134 q^{81} + 886 q^{82} - 5315 q^{83} + 672 q^{84} + 3997 q^{86} + 756 q^{87} - 2969 q^{88} - 1424 q^{89} - 1194 q^{91} - 1607 q^{92} - 2667 q^{93} - 629 q^{94} - 1266 q^{96} - 291 q^{97} + 1018 q^{98} - 297 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.955230 −0.337725 −0.168862 0.985640i \(-0.554009\pi\)
−0.168862 + 0.985640i \(0.554009\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.08754 −0.885942
\(5\) 0 0
\(6\) −2.86569 −0.194986
\(7\) 12.4836 0.674049 0.337024 0.941496i \(-0.390580\pi\)
0.337024 + 0.941496i \(0.390580\pi\)
\(8\) 14.4121 0.636929
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −44.7266 −1.22596 −0.612980 0.790098i \(-0.710029\pi\)
−0.612980 + 0.790098i \(0.710029\pi\)
\(12\) −21.2626 −0.511499
\(13\) −7.13628 −0.152250 −0.0761250 0.997098i \(-0.524255\pi\)
−0.0761250 + 0.997098i \(0.524255\pi\)
\(14\) −11.9247 −0.227643
\(15\) 0 0
\(16\) 42.9334 0.670835
\(17\) 26.5116 0.378236 0.189118 0.981954i \(-0.439437\pi\)
0.189118 + 0.981954i \(0.439437\pi\)
\(18\) −8.59707 −0.112575
\(19\) −46.3901 −0.560138 −0.280069 0.959980i \(-0.590357\pi\)
−0.280069 + 0.959980i \(0.590357\pi\)
\(20\) 0 0
\(21\) 37.4507 0.389162
\(22\) 42.7241 0.414037
\(23\) 145.454 1.31866 0.659330 0.751854i \(-0.270840\pi\)
0.659330 + 0.751854i \(0.270840\pi\)
\(24\) 43.2362 0.367731
\(25\) 0 0
\(26\) 6.81679 0.0514186
\(27\) 27.0000 0.192450
\(28\) −88.4777 −0.597168
\(29\) −213.354 −1.36616 −0.683082 0.730341i \(-0.739361\pi\)
−0.683082 + 0.730341i \(0.739361\pi\)
\(30\) 0 0
\(31\) −148.082 −0.857945 −0.428972 0.903318i \(-0.641124\pi\)
−0.428972 + 0.903318i \(0.641124\pi\)
\(32\) −156.308 −0.863487
\(33\) −134.180 −0.707809
\(34\) −25.3247 −0.127740
\(35\) 0 0
\(36\) −63.7878 −0.295314
\(37\) 402.312 1.78756 0.893781 0.448505i \(-0.148043\pi\)
0.893781 + 0.448505i \(0.148043\pi\)
\(38\) 44.3132 0.189173
\(39\) −21.4089 −0.0879015
\(40\) 0 0
\(41\) 233.656 0.890022 0.445011 0.895525i \(-0.353200\pi\)
0.445011 + 0.895525i \(0.353200\pi\)
\(42\) −35.7740 −0.131430
\(43\) 87.5080 0.310345 0.155173 0.987887i \(-0.450407\pi\)
0.155173 + 0.987887i \(0.450407\pi\)
\(44\) 317.001 1.08613
\(45\) 0 0
\(46\) −138.942 −0.445344
\(47\) −75.9842 −0.235818 −0.117909 0.993024i \(-0.537619\pi\)
−0.117909 + 0.993024i \(0.537619\pi\)
\(48\) 128.800 0.387307
\(49\) −187.161 −0.545658
\(50\) 0 0
\(51\) 79.5348 0.218375
\(52\) 50.5787 0.134885
\(53\) 167.029 0.432891 0.216446 0.976295i \(-0.430554\pi\)
0.216446 + 0.976295i \(0.430554\pi\)
\(54\) −25.7912 −0.0649952
\(55\) 0 0
\(56\) 179.914 0.429321
\(57\) −139.170 −0.323396
\(58\) 203.802 0.461388
\(59\) −595.082 −1.31310 −0.656551 0.754282i \(-0.727985\pi\)
−0.656551 + 0.754282i \(0.727985\pi\)
\(60\) 0 0
\(61\) −865.794 −1.81727 −0.908636 0.417589i \(-0.862875\pi\)
−0.908636 + 0.417589i \(0.862875\pi\)
\(62\) 141.452 0.289749
\(63\) 112.352 0.224683
\(64\) −194.158 −0.379214
\(65\) 0 0
\(66\) 128.172 0.239044
\(67\) −406.302 −0.740862 −0.370431 0.928860i \(-0.620790\pi\)
−0.370431 + 0.928860i \(0.620790\pi\)
\(68\) −187.902 −0.335095
\(69\) 436.361 0.761329
\(70\) 0 0
\(71\) 526.158 0.879486 0.439743 0.898124i \(-0.355070\pi\)
0.439743 + 0.898124i \(0.355070\pi\)
\(72\) 129.709 0.212310
\(73\) 1001.69 1.60602 0.803009 0.595967i \(-0.203231\pi\)
0.803009 + 0.595967i \(0.203231\pi\)
\(74\) −384.301 −0.603704
\(75\) 0 0
\(76\) 328.792 0.496250
\(77\) −558.347 −0.826357
\(78\) 20.4504 0.0296865
\(79\) −133.956 −0.190774 −0.0953872 0.995440i \(-0.530409\pi\)
−0.0953872 + 0.995440i \(0.530409\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −223.195 −0.300582
\(83\) 516.642 0.683238 0.341619 0.939838i \(-0.389025\pi\)
0.341619 + 0.939838i \(0.389025\pi\)
\(84\) −265.433 −0.344775
\(85\) 0 0
\(86\) −83.5903 −0.104811
\(87\) −640.061 −0.788756
\(88\) −644.602 −0.780850
\(89\) −610.698 −0.727347 −0.363673 0.931527i \(-0.618478\pi\)
−0.363673 + 0.931527i \(0.618478\pi\)
\(90\) 0 0
\(91\) −89.0862 −0.102624
\(92\) −1030.91 −1.16826
\(93\) −444.246 −0.495335
\(94\) 72.5824 0.0796415
\(95\) 0 0
\(96\) −468.924 −0.498534
\(97\) −1639.57 −1.71621 −0.858107 0.513471i \(-0.828359\pi\)
−0.858107 + 0.513471i \(0.828359\pi\)
\(98\) 178.782 0.184282
\(99\) −402.539 −0.408653
\(100\) 0 0
\(101\) −433.248 −0.426830 −0.213415 0.976962i \(-0.568459\pi\)
−0.213415 + 0.976962i \(0.568459\pi\)
\(102\) −75.9741 −0.0737505
\(103\) 81.3244 0.0777974 0.0388987 0.999243i \(-0.487615\pi\)
0.0388987 + 0.999243i \(0.487615\pi\)
\(104\) −102.849 −0.0969725
\(105\) 0 0
\(106\) −159.551 −0.146198
\(107\) 391.773 0.353964 0.176982 0.984214i \(-0.443367\pi\)
0.176982 + 0.984214i \(0.443367\pi\)
\(108\) −191.363 −0.170500
\(109\) 1815.92 1.59572 0.797862 0.602840i \(-0.205964\pi\)
0.797862 + 0.602840i \(0.205964\pi\)
\(110\) 0 0
\(111\) 1206.94 1.03205
\(112\) 535.962 0.452176
\(113\) 1087.20 0.905094 0.452547 0.891741i \(-0.350515\pi\)
0.452547 + 0.891741i \(0.350515\pi\)
\(114\) 132.940 0.109219
\(115\) 0 0
\(116\) 1512.15 1.21034
\(117\) −64.2266 −0.0507500
\(118\) 568.440 0.443467
\(119\) 330.959 0.254949
\(120\) 0 0
\(121\) 669.465 0.502979
\(122\) 827.033 0.613738
\(123\) 700.967 0.513854
\(124\) 1049.54 0.760089
\(125\) 0 0
\(126\) −107.322 −0.0758810
\(127\) 871.466 0.608898 0.304449 0.952529i \(-0.401528\pi\)
0.304449 + 0.952529i \(0.401528\pi\)
\(128\) 1435.93 0.991557
\(129\) 262.524 0.179178
\(130\) 0 0
\(131\) 1708.16 1.13925 0.569627 0.821904i \(-0.307088\pi\)
0.569627 + 0.821904i \(0.307088\pi\)
\(132\) 951.003 0.627077
\(133\) −579.114 −0.377560
\(134\) 388.112 0.250207
\(135\) 0 0
\(136\) 382.087 0.240910
\(137\) −2401.86 −1.49784 −0.748921 0.662659i \(-0.769428\pi\)
−0.748921 + 0.662659i \(0.769428\pi\)
\(138\) −416.825 −0.257120
\(139\) −2161.94 −1.31923 −0.659617 0.751602i \(-0.729282\pi\)
−0.659617 + 0.751602i \(0.729282\pi\)
\(140\) 0 0
\(141\) −227.953 −0.136150
\(142\) −502.602 −0.297024
\(143\) 319.181 0.186652
\(144\) 386.401 0.223612
\(145\) 0 0
\(146\) −956.847 −0.542392
\(147\) −561.482 −0.315036
\(148\) −2851.40 −1.58368
\(149\) −1527.70 −0.839959 −0.419980 0.907534i \(-0.637963\pi\)
−0.419980 + 0.907534i \(0.637963\pi\)
\(150\) 0 0
\(151\) 2584.71 1.39299 0.696493 0.717564i \(-0.254743\pi\)
0.696493 + 0.717564i \(0.254743\pi\)
\(152\) −668.578 −0.356768
\(153\) 238.605 0.126079
\(154\) 533.349 0.279081
\(155\) 0 0
\(156\) 151.736 0.0778757
\(157\) −692.965 −0.352259 −0.176129 0.984367i \(-0.556358\pi\)
−0.176129 + 0.984367i \(0.556358\pi\)
\(158\) 127.958 0.0644293
\(159\) 501.088 0.249930
\(160\) 0 0
\(161\) 1815.78 0.888841
\(162\) −77.3736 −0.0375250
\(163\) −3620.52 −1.73976 −0.869881 0.493262i \(-0.835805\pi\)
−0.869881 + 0.493262i \(0.835805\pi\)
\(164\) −1656.04 −0.788508
\(165\) 0 0
\(166\) −493.511 −0.230746
\(167\) 1555.61 0.720817 0.360409 0.932794i \(-0.382637\pi\)
0.360409 + 0.932794i \(0.382637\pi\)
\(168\) 539.742 0.247869
\(169\) −2146.07 −0.976820
\(170\) 0 0
\(171\) −417.511 −0.186713
\(172\) −620.216 −0.274948
\(173\) −2175.91 −0.956250 −0.478125 0.878292i \(-0.658683\pi\)
−0.478125 + 0.878292i \(0.658683\pi\)
\(174\) 611.405 0.266382
\(175\) 0 0
\(176\) −1920.27 −0.822417
\(177\) −1785.24 −0.758120
\(178\) 583.357 0.245643
\(179\) −1865.44 −0.778937 −0.389468 0.921040i \(-0.627341\pi\)
−0.389468 + 0.921040i \(0.627341\pi\)
\(180\) 0 0
\(181\) −257.830 −0.105881 −0.0529403 0.998598i \(-0.516859\pi\)
−0.0529403 + 0.998598i \(0.516859\pi\)
\(182\) 85.0978 0.0346586
\(183\) −2597.38 −1.04920
\(184\) 2096.29 0.839893
\(185\) 0 0
\(186\) 424.357 0.167287
\(187\) −1185.77 −0.463702
\(188\) 538.541 0.208921
\(189\) 337.056 0.129721
\(190\) 0 0
\(191\) −3960.87 −1.50052 −0.750258 0.661145i \(-0.770071\pi\)
−0.750258 + 0.661145i \(0.770071\pi\)
\(192\) −582.473 −0.218939
\(193\) 1253.72 0.467590 0.233795 0.972286i \(-0.424886\pi\)
0.233795 + 0.972286i \(0.424886\pi\)
\(194\) 1566.16 0.579608
\(195\) 0 0
\(196\) 1326.51 0.483422
\(197\) −5059.84 −1.82994 −0.914971 0.403519i \(-0.867787\pi\)
−0.914971 + 0.403519i \(0.867787\pi\)
\(198\) 384.517 0.138012
\(199\) −5323.91 −1.89649 −0.948246 0.317536i \(-0.897145\pi\)
−0.948246 + 0.317536i \(0.897145\pi\)
\(200\) 0 0
\(201\) −1218.91 −0.427737
\(202\) 413.852 0.144151
\(203\) −2663.41 −0.920862
\(204\) −563.706 −0.193467
\(205\) 0 0
\(206\) −77.6835 −0.0262741
\(207\) 1309.08 0.439553
\(208\) −306.385 −0.102135
\(209\) 2074.87 0.686707
\(210\) 0 0
\(211\) −1029.61 −0.335931 −0.167965 0.985793i \(-0.553720\pi\)
−0.167965 + 0.985793i \(0.553720\pi\)
\(212\) −1183.83 −0.383517
\(213\) 1578.47 0.507771
\(214\) −374.233 −0.119542
\(215\) 0 0
\(216\) 389.126 0.122577
\(217\) −1848.59 −0.578297
\(218\) −1734.62 −0.538915
\(219\) 3005.08 0.927235
\(220\) 0 0
\(221\) −189.194 −0.0575864
\(222\) −1152.90 −0.348548
\(223\) 1245.08 0.373886 0.186943 0.982371i \(-0.440142\pi\)
0.186943 + 0.982371i \(0.440142\pi\)
\(224\) −1951.28 −0.582032
\(225\) 0 0
\(226\) −1038.53 −0.305673
\(227\) −4598.29 −1.34449 −0.672245 0.740329i \(-0.734670\pi\)
−0.672245 + 0.740329i \(0.734670\pi\)
\(228\) 986.375 0.286510
\(229\) 6354.76 1.83378 0.916888 0.399145i \(-0.130693\pi\)
0.916888 + 0.399145i \(0.130693\pi\)
\(230\) 0 0
\(231\) −1675.04 −0.477097
\(232\) −3074.87 −0.870150
\(233\) 635.967 0.178814 0.0894068 0.995995i \(-0.471503\pi\)
0.0894068 + 0.995995i \(0.471503\pi\)
\(234\) 61.3511 0.0171395
\(235\) 0 0
\(236\) 4217.66 1.16333
\(237\) −401.867 −0.110144
\(238\) −316.142 −0.0861028
\(239\) −6458.06 −1.74786 −0.873928 0.486056i \(-0.838435\pi\)
−0.873928 + 0.486056i \(0.838435\pi\)
\(240\) 0 0
\(241\) 509.747 0.136248 0.0681239 0.997677i \(-0.478299\pi\)
0.0681239 + 0.997677i \(0.478299\pi\)
\(242\) −639.493 −0.169868
\(243\) 243.000 0.0641500
\(244\) 6136.35 1.61000
\(245\) 0 0
\(246\) −669.585 −0.173541
\(247\) 331.053 0.0852810
\(248\) −2134.17 −0.546450
\(249\) 1549.92 0.394468
\(250\) 0 0
\(251\) 2645.82 0.665349 0.332674 0.943042i \(-0.392049\pi\)
0.332674 + 0.943042i \(0.392049\pi\)
\(252\) −796.299 −0.199056
\(253\) −6505.64 −1.61662
\(254\) −832.450 −0.205640
\(255\) 0 0
\(256\) 181.620 0.0443408
\(257\) −3975.12 −0.964829 −0.482415 0.875943i \(-0.660240\pi\)
−0.482415 + 0.875943i \(0.660240\pi\)
\(258\) −250.771 −0.0605128
\(259\) 5022.29 1.20490
\(260\) 0 0
\(261\) −1920.18 −0.455388
\(262\) −1631.68 −0.384754
\(263\) 4820.25 1.13015 0.565075 0.825040i \(-0.308847\pi\)
0.565075 + 0.825040i \(0.308847\pi\)
\(264\) −1933.81 −0.450824
\(265\) 0 0
\(266\) 553.187 0.127512
\(267\) −1832.09 −0.419934
\(268\) 2879.68 0.656360
\(269\) 4196.29 0.951123 0.475562 0.879682i \(-0.342245\pi\)
0.475562 + 0.879682i \(0.342245\pi\)
\(270\) 0 0
\(271\) −2039.98 −0.457269 −0.228634 0.973512i \(-0.573426\pi\)
−0.228634 + 0.973512i \(0.573426\pi\)
\(272\) 1138.24 0.253734
\(273\) −267.259 −0.0592499
\(274\) 2294.32 0.505858
\(275\) 0 0
\(276\) −3092.72 −0.674493
\(277\) −4613.40 −1.00069 −0.500347 0.865825i \(-0.666794\pi\)
−0.500347 + 0.865825i \(0.666794\pi\)
\(278\) 2065.15 0.445538
\(279\) −1332.74 −0.285982
\(280\) 0 0
\(281\) −2512.14 −0.533317 −0.266658 0.963791i \(-0.585919\pi\)
−0.266658 + 0.963791i \(0.585919\pi\)
\(282\) 217.747 0.0459811
\(283\) 1061.91 0.223054 0.111527 0.993761i \(-0.464426\pi\)
0.111527 + 0.993761i \(0.464426\pi\)
\(284\) −3729.17 −0.779174
\(285\) 0 0
\(286\) −304.892 −0.0630371
\(287\) 2916.85 0.599918
\(288\) −1406.77 −0.287829
\(289\) −4210.13 −0.856938
\(290\) 0 0
\(291\) −4918.70 −0.990856
\(292\) −7099.54 −1.42284
\(293\) −7941.52 −1.58344 −0.791721 0.610883i \(-0.790814\pi\)
−0.791721 + 0.610883i \(0.790814\pi\)
\(294\) 536.345 0.106395
\(295\) 0 0
\(296\) 5798.15 1.13855
\(297\) −1207.62 −0.235936
\(298\) 1459.30 0.283675
\(299\) −1038.00 −0.200766
\(300\) 0 0
\(301\) 1092.41 0.209188
\(302\) −2468.99 −0.470446
\(303\) −1299.75 −0.246430
\(304\) −1991.69 −0.375760
\(305\) 0 0
\(306\) −227.922 −0.0425799
\(307\) −5405.88 −1.00498 −0.502492 0.864582i \(-0.667583\pi\)
−0.502492 + 0.864582i \(0.667583\pi\)
\(308\) 3957.30 0.732104
\(309\) 243.973 0.0449164
\(310\) 0 0
\(311\) −4625.83 −0.843431 −0.421715 0.906728i \(-0.638572\pi\)
−0.421715 + 0.906728i \(0.638572\pi\)
\(312\) −308.546 −0.0559871
\(313\) −1053.32 −0.190214 −0.0951069 0.995467i \(-0.530319\pi\)
−0.0951069 + 0.995467i \(0.530319\pi\)
\(314\) 661.941 0.118967
\(315\) 0 0
\(316\) 949.415 0.169015
\(317\) 2710.00 0.480154 0.240077 0.970754i \(-0.422827\pi\)
0.240077 + 0.970754i \(0.422827\pi\)
\(318\) −478.654 −0.0844075
\(319\) 9542.57 1.67486
\(320\) 0 0
\(321\) 1175.32 0.204361
\(322\) −1734.49 −0.300184
\(323\) −1229.88 −0.211864
\(324\) −574.090 −0.0984380
\(325\) 0 0
\(326\) 3458.43 0.587561
\(327\) 5447.77 0.921292
\(328\) 3367.46 0.566881
\(329\) −948.554 −0.158953
\(330\) 0 0
\(331\) −4387.70 −0.728609 −0.364304 0.931280i \(-0.618693\pi\)
−0.364304 + 0.931280i \(0.618693\pi\)
\(332\) −3661.72 −0.605309
\(333\) 3620.81 0.595854
\(334\) −1485.96 −0.243438
\(335\) 0 0
\(336\) 1607.89 0.261064
\(337\) −3872.64 −0.625983 −0.312992 0.949756i \(-0.601331\pi\)
−0.312992 + 0.949756i \(0.601331\pi\)
\(338\) 2049.99 0.329896
\(339\) 3261.61 0.522556
\(340\) 0 0
\(341\) 6623.19 1.05181
\(342\) 398.819 0.0630575
\(343\) −6618.29 −1.04185
\(344\) 1261.17 0.197668
\(345\) 0 0
\(346\) 2078.49 0.322949
\(347\) 5782.93 0.894651 0.447326 0.894371i \(-0.352377\pi\)
0.447326 + 0.894371i \(0.352377\pi\)
\(348\) 4536.45 0.698792
\(349\) −2916.63 −0.447345 −0.223673 0.974664i \(-0.571805\pi\)
−0.223673 + 0.974664i \(0.571805\pi\)
\(350\) 0 0
\(351\) −192.680 −0.0293005
\(352\) 6991.11 1.05860
\(353\) −3487.56 −0.525847 −0.262924 0.964817i \(-0.584687\pi\)
−0.262924 + 0.964817i \(0.584687\pi\)
\(354\) 1705.32 0.256036
\(355\) 0 0
\(356\) 4328.34 0.644387
\(357\) 992.878 0.147195
\(358\) 1781.93 0.263066
\(359\) 10606.5 1.55931 0.779653 0.626212i \(-0.215396\pi\)
0.779653 + 0.626212i \(0.215396\pi\)
\(360\) 0 0
\(361\) −4706.96 −0.686245
\(362\) 246.287 0.0357585
\(363\) 2008.39 0.290395
\(364\) 631.402 0.0909188
\(365\) 0 0
\(366\) 2481.10 0.354342
\(367\) −2685.12 −0.381913 −0.190957 0.981598i \(-0.561159\pi\)
−0.190957 + 0.981598i \(0.561159\pi\)
\(368\) 6244.83 0.884604
\(369\) 2102.90 0.296674
\(370\) 0 0
\(371\) 2085.12 0.291790
\(372\) 3148.61 0.438838
\(373\) −688.479 −0.0955712 −0.0477856 0.998858i \(-0.515216\pi\)
−0.0477856 + 0.998858i \(0.515216\pi\)
\(374\) 1132.69 0.156604
\(375\) 0 0
\(376\) −1095.09 −0.150199
\(377\) 1522.55 0.207999
\(378\) −321.966 −0.0438099
\(379\) −12235.4 −1.65829 −0.829146 0.559033i \(-0.811173\pi\)
−0.829146 + 0.559033i \(0.811173\pi\)
\(380\) 0 0
\(381\) 2614.40 0.351548
\(382\) 3783.54 0.506762
\(383\) 1902.17 0.253777 0.126888 0.991917i \(-0.459501\pi\)
0.126888 + 0.991917i \(0.459501\pi\)
\(384\) 4307.78 0.572476
\(385\) 0 0
\(386\) −1197.59 −0.157917
\(387\) 787.572 0.103448
\(388\) 11620.5 1.52047
\(389\) 187.985 0.0245019 0.0122509 0.999925i \(-0.496100\pi\)
0.0122509 + 0.999925i \(0.496100\pi\)
\(390\) 0 0
\(391\) 3856.21 0.498765
\(392\) −2697.37 −0.347546
\(393\) 5124.47 0.657748
\(394\) 4833.31 0.618017
\(395\) 0 0
\(396\) 2853.01 0.362043
\(397\) −7547.41 −0.954140 −0.477070 0.878865i \(-0.658301\pi\)
−0.477070 + 0.878865i \(0.658301\pi\)
\(398\) 5085.56 0.640493
\(399\) −1737.34 −0.217985
\(400\) 0 0
\(401\) 5396.95 0.672097 0.336049 0.941845i \(-0.390909\pi\)
0.336049 + 0.941845i \(0.390909\pi\)
\(402\) 1164.34 0.144457
\(403\) 1056.75 0.130622
\(404\) 3070.66 0.378147
\(405\) 0 0
\(406\) 2544.17 0.310998
\(407\) −17994.1 −2.19148
\(408\) 1146.26 0.139089
\(409\) −4090.41 −0.494518 −0.247259 0.968949i \(-0.579530\pi\)
−0.247259 + 0.968949i \(0.579530\pi\)
\(410\) 0 0
\(411\) −7205.57 −0.864780
\(412\) −576.390 −0.0689240
\(413\) −7428.73 −0.885095
\(414\) −1250.48 −0.148448
\(415\) 0 0
\(416\) 1115.46 0.131466
\(417\) −6485.83 −0.761661
\(418\) −1981.98 −0.231918
\(419\) 2841.80 0.331339 0.165670 0.986181i \(-0.447021\pi\)
0.165670 + 0.986181i \(0.447021\pi\)
\(420\) 0 0
\(421\) −16539.8 −1.91473 −0.957363 0.288889i \(-0.906714\pi\)
−0.957363 + 0.288889i \(0.906714\pi\)
\(422\) 983.516 0.113452
\(423\) −683.858 −0.0786060
\(424\) 2407.24 0.275721
\(425\) 0 0
\(426\) −1507.81 −0.171487
\(427\) −10808.2 −1.22493
\(428\) −2776.71 −0.313592
\(429\) 957.544 0.107764
\(430\) 0 0
\(431\) −2301.49 −0.257214 −0.128607 0.991696i \(-0.541050\pi\)
−0.128607 + 0.991696i \(0.541050\pi\)
\(432\) 1159.20 0.129102
\(433\) −7556.52 −0.838668 −0.419334 0.907832i \(-0.637736\pi\)
−0.419334 + 0.907832i \(0.637736\pi\)
\(434\) 1765.83 0.195305
\(435\) 0 0
\(436\) −12870.4 −1.41372
\(437\) −6747.61 −0.738632
\(438\) −2870.54 −0.313150
\(439\) −8328.02 −0.905409 −0.452705 0.891660i \(-0.649541\pi\)
−0.452705 + 0.891660i \(0.649541\pi\)
\(440\) 0 0
\(441\) −1684.45 −0.181886
\(442\) 180.724 0.0194484
\(443\) 11675.9 1.25223 0.626114 0.779731i \(-0.284644\pi\)
0.626114 + 0.779731i \(0.284644\pi\)
\(444\) −8554.21 −0.914335
\(445\) 0 0
\(446\) −1189.34 −0.126271
\(447\) −4583.10 −0.484951
\(448\) −2423.78 −0.255609
\(449\) −1034.94 −0.108779 −0.0543893 0.998520i \(-0.517321\pi\)
−0.0543893 + 0.998520i \(0.517321\pi\)
\(450\) 0 0
\(451\) −10450.6 −1.09113
\(452\) −7705.60 −0.801861
\(453\) 7754.13 0.804241
\(454\) 4392.43 0.454068
\(455\) 0 0
\(456\) −2005.73 −0.205980
\(457\) 15279.9 1.56403 0.782016 0.623259i \(-0.214192\pi\)
0.782016 + 0.623259i \(0.214192\pi\)
\(458\) −6070.26 −0.619311
\(459\) 715.814 0.0727915
\(460\) 0 0
\(461\) 15795.3 1.59580 0.797898 0.602793i \(-0.205946\pi\)
0.797898 + 0.602793i \(0.205946\pi\)
\(462\) 1600.05 0.161128
\(463\) 1773.18 0.177985 0.0889923 0.996032i \(-0.471635\pi\)
0.0889923 + 0.996032i \(0.471635\pi\)
\(464\) −9160.01 −0.916471
\(465\) 0 0
\(466\) −607.494 −0.0603898
\(467\) 13051.2 1.29323 0.646615 0.762817i \(-0.276184\pi\)
0.646615 + 0.762817i \(0.276184\pi\)
\(468\) 455.208 0.0449615
\(469\) −5072.10 −0.499377
\(470\) 0 0
\(471\) −2078.90 −0.203377
\(472\) −8576.35 −0.836353
\(473\) −3913.93 −0.380471
\(474\) 383.875 0.0371982
\(475\) 0 0
\(476\) −2345.69 −0.225870
\(477\) 1503.26 0.144297
\(478\) 6168.94 0.590294
\(479\) −633.119 −0.0603924 −0.0301962 0.999544i \(-0.509613\pi\)
−0.0301962 + 0.999544i \(0.509613\pi\)
\(480\) 0 0
\(481\) −2871.02 −0.272156
\(482\) −486.926 −0.0460142
\(483\) 5447.34 0.513173
\(484\) −4744.85 −0.445610
\(485\) 0 0
\(486\) −232.121 −0.0216651
\(487\) −6437.29 −0.598976 −0.299488 0.954100i \(-0.596816\pi\)
−0.299488 + 0.954100i \(0.596816\pi\)
\(488\) −12477.9 −1.15747
\(489\) −10861.6 −1.00445
\(490\) 0 0
\(491\) 3114.41 0.286256 0.143128 0.989704i \(-0.454284\pi\)
0.143128 + 0.989704i \(0.454284\pi\)
\(492\) −4968.13 −0.455245
\(493\) −5656.35 −0.516733
\(494\) −316.232 −0.0288015
\(495\) 0 0
\(496\) −6357.67 −0.575540
\(497\) 6568.33 0.592816
\(498\) −1480.53 −0.133222
\(499\) −7664.80 −0.687623 −0.343811 0.939039i \(-0.611718\pi\)
−0.343811 + 0.939039i \(0.611718\pi\)
\(500\) 0 0
\(501\) 4666.82 0.416164
\(502\) −2527.37 −0.224705
\(503\) 8278.58 0.733844 0.366922 0.930252i \(-0.380412\pi\)
0.366922 + 0.930252i \(0.380412\pi\)
\(504\) 1619.22 0.143107
\(505\) 0 0
\(506\) 6214.38 0.545974
\(507\) −6438.22 −0.563967
\(508\) −6176.55 −0.539449
\(509\) −7111.22 −0.619252 −0.309626 0.950858i \(-0.600204\pi\)
−0.309626 + 0.950858i \(0.600204\pi\)
\(510\) 0 0
\(511\) 12504.7 1.08253
\(512\) −11660.9 −1.00653
\(513\) −1252.53 −0.107799
\(514\) 3797.15 0.325847
\(515\) 0 0
\(516\) −1860.65 −0.158741
\(517\) 3398.51 0.289103
\(518\) −4797.44 −0.406926
\(519\) −6527.72 −0.552091
\(520\) 0 0
\(521\) −812.539 −0.0683262 −0.0341631 0.999416i \(-0.510877\pi\)
−0.0341631 + 0.999416i \(0.510877\pi\)
\(522\) 1834.22 0.153796
\(523\) 6446.12 0.538946 0.269473 0.963008i \(-0.413150\pi\)
0.269473 + 0.963008i \(0.413150\pi\)
\(524\) −12106.6 −1.00931
\(525\) 0 0
\(526\) −4604.45 −0.381680
\(527\) −3925.89 −0.324506
\(528\) −5760.80 −0.474823
\(529\) 8989.76 0.738865
\(530\) 0 0
\(531\) −5355.73 −0.437701
\(532\) 4104.49 0.334497
\(533\) −1667.43 −0.135506
\(534\) 1750.07 0.141822
\(535\) 0 0
\(536\) −5855.66 −0.471877
\(537\) −5596.33 −0.449719
\(538\) −4008.42 −0.321218
\(539\) 8371.06 0.668955
\(540\) 0 0
\(541\) −4552.44 −0.361783 −0.180892 0.983503i \(-0.557898\pi\)
−0.180892 + 0.983503i \(0.557898\pi\)
\(542\) 1948.65 0.154431
\(543\) −773.491 −0.0611302
\(544\) −4143.97 −0.326602
\(545\) 0 0
\(546\) 255.293 0.0200102
\(547\) 14318.6 1.11923 0.559615 0.828753i \(-0.310949\pi\)
0.559615 + 0.828753i \(0.310949\pi\)
\(548\) 17023.2 1.32700
\(549\) −7792.15 −0.605757
\(550\) 0 0
\(551\) 9897.50 0.765241
\(552\) 6288.86 0.484913
\(553\) −1672.24 −0.128591
\(554\) 4406.86 0.337959
\(555\) 0 0
\(556\) 15322.8 1.16877
\(557\) −14882.2 −1.13210 −0.566050 0.824371i \(-0.691529\pi\)
−0.566050 + 0.824371i \(0.691529\pi\)
\(558\) 1273.07 0.0965831
\(559\) −624.482 −0.0472500
\(560\) 0 0
\(561\) −3557.32 −0.267719
\(562\) 2399.68 0.180114
\(563\) 5655.46 0.423356 0.211678 0.977339i \(-0.432107\pi\)
0.211678 + 0.977339i \(0.432107\pi\)
\(564\) 1615.62 0.120621
\(565\) 0 0
\(566\) −1014.37 −0.0753308
\(567\) 1011.17 0.0748943
\(568\) 7583.03 0.560170
\(569\) −10016.7 −0.738001 −0.369001 0.929429i \(-0.620300\pi\)
−0.369001 + 0.929429i \(0.620300\pi\)
\(570\) 0 0
\(571\) −17386.3 −1.27424 −0.637121 0.770764i \(-0.719875\pi\)
−0.637121 + 0.770764i \(0.719875\pi\)
\(572\) −2262.21 −0.165363
\(573\) −11882.6 −0.866324
\(574\) −2786.27 −0.202607
\(575\) 0 0
\(576\) −1747.42 −0.126405
\(577\) −8886.10 −0.641132 −0.320566 0.947226i \(-0.603873\pi\)
−0.320566 + 0.947226i \(0.603873\pi\)
\(578\) 4021.65 0.289409
\(579\) 3761.16 0.269963
\(580\) 0 0
\(581\) 6449.52 0.460536
\(582\) 4698.49 0.334637
\(583\) −7470.65 −0.530708
\(584\) 14436.5 1.02292
\(585\) 0 0
\(586\) 7585.97 0.534767
\(587\) 20817.3 1.46375 0.731876 0.681438i \(-0.238645\pi\)
0.731876 + 0.681438i \(0.238645\pi\)
\(588\) 3979.53 0.279104
\(589\) 6869.54 0.480568
\(590\) 0 0
\(591\) −15179.5 −1.05652
\(592\) 17272.7 1.19916
\(593\) 24320.9 1.68422 0.842108 0.539309i \(-0.181314\pi\)
0.842108 + 0.539309i \(0.181314\pi\)
\(594\) 1153.55 0.0796815
\(595\) 0 0
\(596\) 10827.6 0.744155
\(597\) −15971.7 −1.09494
\(598\) 991.527 0.0678036
\(599\) 18650.6 1.27219 0.636096 0.771610i \(-0.280548\pi\)
0.636096 + 0.771610i \(0.280548\pi\)
\(600\) 0 0
\(601\) −15558.8 −1.05600 −0.528000 0.849245i \(-0.677058\pi\)
−0.528000 + 0.849245i \(0.677058\pi\)
\(602\) −1043.50 −0.0706479
\(603\) −3656.72 −0.246954
\(604\) −18319.2 −1.23410
\(605\) 0 0
\(606\) 1241.56 0.0832257
\(607\) 27620.1 1.84690 0.923448 0.383722i \(-0.125358\pi\)
0.923448 + 0.383722i \(0.125358\pi\)
\(608\) 7251.14 0.483672
\(609\) −7990.24 −0.531660
\(610\) 0 0
\(611\) 542.245 0.0359033
\(612\) −1691.12 −0.111698
\(613\) −10581.1 −0.697173 −0.348587 0.937277i \(-0.613338\pi\)
−0.348587 + 0.937277i \(0.613338\pi\)
\(614\) 5163.86 0.339408
\(615\) 0 0
\(616\) −8046.93 −0.526331
\(617\) 26415.2 1.72356 0.861780 0.507283i \(-0.169350\pi\)
0.861780 + 0.507283i \(0.169350\pi\)
\(618\) −233.051 −0.0151694
\(619\) −4747.92 −0.308296 −0.154148 0.988048i \(-0.549263\pi\)
−0.154148 + 0.988048i \(0.549263\pi\)
\(620\) 0 0
\(621\) 3927.25 0.253776
\(622\) 4418.73 0.284847
\(623\) −7623.68 −0.490267
\(624\) −919.156 −0.0589674
\(625\) 0 0
\(626\) 1006.16 0.0642399
\(627\) 6224.61 0.396471
\(628\) 4911.42 0.312081
\(629\) 10666.0 0.676120
\(630\) 0 0
\(631\) −24505.3 −1.54602 −0.773012 0.634391i \(-0.781251\pi\)
−0.773012 + 0.634391i \(0.781251\pi\)
\(632\) −1930.58 −0.121510
\(633\) −3088.84 −0.193950
\(634\) −2588.68 −0.162160
\(635\) 0 0
\(636\) −3551.48 −0.221423
\(637\) 1335.63 0.0830764
\(638\) −9115.35 −0.565643
\(639\) 4735.42 0.293162
\(640\) 0 0
\(641\) −6380.17 −0.393138 −0.196569 0.980490i \(-0.562980\pi\)
−0.196569 + 0.980490i \(0.562980\pi\)
\(642\) −1122.70 −0.0690178
\(643\) −21527.0 −1.32029 −0.660143 0.751140i \(-0.729504\pi\)
−0.660143 + 0.751140i \(0.729504\pi\)
\(644\) −12869.4 −0.787462
\(645\) 0 0
\(646\) 1174.82 0.0715519
\(647\) 23849.2 1.44916 0.724581 0.689189i \(-0.242033\pi\)
0.724581 + 0.689189i \(0.242033\pi\)
\(648\) 1167.38 0.0707699
\(649\) 26615.9 1.60981
\(650\) 0 0
\(651\) −5545.77 −0.333880
\(652\) 25660.6 1.54133
\(653\) 11845.8 0.709898 0.354949 0.934886i \(-0.384498\pi\)
0.354949 + 0.934886i \(0.384498\pi\)
\(654\) −5203.87 −0.311143
\(655\) 0 0
\(656\) 10031.6 0.597058
\(657\) 9015.24 0.535339
\(658\) 906.087 0.0536823
\(659\) −13126.8 −0.775944 −0.387972 0.921671i \(-0.626824\pi\)
−0.387972 + 0.921671i \(0.626824\pi\)
\(660\) 0 0
\(661\) 24494.4 1.44133 0.720667 0.693281i \(-0.243836\pi\)
0.720667 + 0.693281i \(0.243836\pi\)
\(662\) 4191.26 0.246069
\(663\) −567.583 −0.0332475
\(664\) 7445.87 0.435174
\(665\) 0 0
\(666\) −3458.71 −0.201235
\(667\) −31033.1 −1.80151
\(668\) −11025.4 −0.638602
\(669\) 3735.24 0.215863
\(670\) 0 0
\(671\) 38724.0 2.22790
\(672\) −5853.83 −0.336037
\(673\) −28081.1 −1.60839 −0.804196 0.594364i \(-0.797404\pi\)
−0.804196 + 0.594364i \(0.797404\pi\)
\(674\) 3699.27 0.211410
\(675\) 0 0
\(676\) 15210.4 0.865406
\(677\) −6825.24 −0.387467 −0.193733 0.981054i \(-0.562060\pi\)
−0.193733 + 0.981054i \(0.562060\pi\)
\(678\) −3115.59 −0.176480
\(679\) −20467.6 −1.15681
\(680\) 0 0
\(681\) −13794.9 −0.776242
\(682\) −6326.67 −0.355221
\(683\) −31450.7 −1.76197 −0.880987 0.473140i \(-0.843121\pi\)
−0.880987 + 0.473140i \(0.843121\pi\)
\(684\) 2959.12 0.165417
\(685\) 0 0
\(686\) 6321.99 0.351858
\(687\) 19064.3 1.05873
\(688\) 3757.02 0.208190
\(689\) −1191.97 −0.0659077
\(690\) 0 0
\(691\) 6136.45 0.337832 0.168916 0.985630i \(-0.445973\pi\)
0.168916 + 0.985630i \(0.445973\pi\)
\(692\) 15421.8 0.847182
\(693\) −5025.12 −0.275452
\(694\) −5524.03 −0.302146
\(695\) 0 0
\(696\) −9224.60 −0.502382
\(697\) 6194.59 0.336638
\(698\) 2786.05 0.151080
\(699\) 1907.90 0.103238
\(700\) 0 0
\(701\) 5291.43 0.285099 0.142550 0.989788i \(-0.454470\pi\)
0.142550 + 0.989788i \(0.454470\pi\)
\(702\) 184.053 0.00989551
\(703\) −18663.3 −1.00128
\(704\) 8684.00 0.464902
\(705\) 0 0
\(706\) 3331.42 0.177592
\(707\) −5408.48 −0.287704
\(708\) 12653.0 0.671650
\(709\) −4385.62 −0.232307 −0.116153 0.993231i \(-0.537056\pi\)
−0.116153 + 0.993231i \(0.537056\pi\)
\(710\) 0 0
\(711\) −1205.60 −0.0635915
\(712\) −8801.42 −0.463268
\(713\) −21539.1 −1.13134
\(714\) −948.427 −0.0497115
\(715\) 0 0
\(716\) 13221.4 0.690093
\(717\) −19374.2 −1.00912
\(718\) −10131.7 −0.526616
\(719\) 6410.26 0.332493 0.166246 0.986084i \(-0.446835\pi\)
0.166246 + 0.986084i \(0.446835\pi\)
\(720\) 0 0
\(721\) 1015.22 0.0524393
\(722\) 4496.23 0.231762
\(723\) 1529.24 0.0786627
\(724\) 1827.38 0.0938040
\(725\) 0 0
\(726\) −1918.48 −0.0980736
\(727\) 31663.6 1.61532 0.807661 0.589647i \(-0.200733\pi\)
0.807661 + 0.589647i \(0.200733\pi\)
\(728\) −1283.92 −0.0653642
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 2319.98 0.117384
\(732\) 18409.0 0.929532
\(733\) 25767.7 1.29843 0.649215 0.760605i \(-0.275097\pi\)
0.649215 + 0.760605i \(0.275097\pi\)
\(734\) 2564.91 0.128982
\(735\) 0 0
\(736\) −22735.5 −1.13865
\(737\) 18172.5 0.908267
\(738\) −2008.75 −0.100194
\(739\) 3389.01 0.168697 0.0843484 0.996436i \(-0.473119\pi\)
0.0843484 + 0.996436i \(0.473119\pi\)
\(740\) 0 0
\(741\) 993.159 0.0492370
\(742\) −1991.77 −0.0985447
\(743\) −17203.7 −0.849453 −0.424727 0.905322i \(-0.639630\pi\)
−0.424727 + 0.905322i \(0.639630\pi\)
\(744\) −6402.50 −0.315493
\(745\) 0 0
\(746\) 657.655 0.0322768
\(747\) 4649.77 0.227746
\(748\) 8404.21 0.410813
\(749\) 4890.72 0.238589
\(750\) 0 0
\(751\) 17081.3 0.829970 0.414985 0.909828i \(-0.363787\pi\)
0.414985 + 0.909828i \(0.363787\pi\)
\(752\) −3262.27 −0.158195
\(753\) 7937.46 0.384139
\(754\) −1454.39 −0.0702463
\(755\) 0 0
\(756\) −2388.90 −0.114925
\(757\) 4684.65 0.224923 0.112461 0.993656i \(-0.464127\pi\)
0.112461 + 0.993656i \(0.464127\pi\)
\(758\) 11687.7 0.560046
\(759\) −19516.9 −0.933359
\(760\) 0 0
\(761\) 25366.8 1.20834 0.604169 0.796856i \(-0.293505\pi\)
0.604169 + 0.796856i \(0.293505\pi\)
\(762\) −2497.35 −0.118726
\(763\) 22669.2 1.07560
\(764\) 28072.8 1.32937
\(765\) 0 0
\(766\) −1817.01 −0.0857068
\(767\) 4246.67 0.199920
\(768\) 544.859 0.0256001
\(769\) −21287.7 −0.998250 −0.499125 0.866530i \(-0.666345\pi\)
−0.499125 + 0.866530i \(0.666345\pi\)
\(770\) 0 0
\(771\) −11925.4 −0.557044
\(772\) −8885.79 −0.414257
\(773\) 23748.4 1.10501 0.552504 0.833511i \(-0.313673\pi\)
0.552504 + 0.833511i \(0.313673\pi\)
\(774\) −752.312 −0.0349371
\(775\) 0 0
\(776\) −23629.5 −1.09311
\(777\) 15066.9 0.695651
\(778\) −179.569 −0.00827490
\(779\) −10839.3 −0.498535
\(780\) 0 0
\(781\) −23533.2 −1.07821
\(782\) −3683.57 −0.168445
\(783\) −5760.55 −0.262919
\(784\) −8035.46 −0.366047
\(785\) 0 0
\(786\) −4895.04 −0.222138
\(787\) 19191.0 0.869231 0.434615 0.900616i \(-0.356884\pi\)
0.434615 + 0.900616i \(0.356884\pi\)
\(788\) 35861.8 1.62122
\(789\) 14460.8 0.652492
\(790\) 0 0
\(791\) 13572.2 0.610077
\(792\) −5801.42 −0.260283
\(793\) 6178.55 0.276680
\(794\) 7209.51 0.322237
\(795\) 0 0
\(796\) 37733.4 1.68018
\(797\) −20205.5 −0.898012 −0.449006 0.893529i \(-0.648222\pi\)
−0.449006 + 0.893529i \(0.648222\pi\)
\(798\) 1659.56 0.0736188
\(799\) −2014.46 −0.0891948
\(800\) 0 0
\(801\) −5496.28 −0.242449
\(802\) −5155.33 −0.226984
\(803\) −44802.3 −1.96891
\(804\) 8639.05 0.378950
\(805\) 0 0
\(806\) −1009.44 −0.0441143
\(807\) 12588.9 0.549131
\(808\) −6244.00 −0.271861
\(809\) −18217.3 −0.791700 −0.395850 0.918315i \(-0.629550\pi\)
−0.395850 + 0.918315i \(0.629550\pi\)
\(810\) 0 0
\(811\) −25637.1 −1.11004 −0.555019 0.831837i \(-0.687289\pi\)
−0.555019 + 0.831837i \(0.687289\pi\)
\(812\) 18877.0 0.815830
\(813\) −6119.93 −0.264004
\(814\) 17188.5 0.740117
\(815\) 0 0
\(816\) 3414.71 0.146493
\(817\) −4059.51 −0.173836
\(818\) 3907.29 0.167011
\(819\) −801.776 −0.0342080
\(820\) 0 0
\(821\) −3325.18 −0.141352 −0.0706758 0.997499i \(-0.522516\pi\)
−0.0706758 + 0.997499i \(0.522516\pi\)
\(822\) 6882.97 0.292057
\(823\) 4282.17 0.181370 0.0906848 0.995880i \(-0.471094\pi\)
0.0906848 + 0.995880i \(0.471094\pi\)
\(824\) 1172.05 0.0495515
\(825\) 0 0
\(826\) 7096.15 0.298918
\(827\) −15503.1 −0.651869 −0.325935 0.945392i \(-0.605679\pi\)
−0.325935 + 0.945392i \(0.605679\pi\)
\(828\) −9278.17 −0.389419
\(829\) 26065.5 1.09203 0.546015 0.837775i \(-0.316144\pi\)
0.546015 + 0.837775i \(0.316144\pi\)
\(830\) 0 0
\(831\) −13840.2 −0.577751
\(832\) 1385.56 0.0577353
\(833\) −4961.93 −0.206388
\(834\) 6195.46 0.257232
\(835\) 0 0
\(836\) −14705.7 −0.608383
\(837\) −3998.21 −0.165112
\(838\) −2714.57 −0.111901
\(839\) 44675.9 1.83836 0.919180 0.393838i \(-0.128853\pi\)
0.919180 + 0.393838i \(0.128853\pi\)
\(840\) 0 0
\(841\) 21130.8 0.866406
\(842\) 15799.3 0.646650
\(843\) −7536.43 −0.307910
\(844\) 7297.41 0.297615
\(845\) 0 0
\(846\) 653.242 0.0265472
\(847\) 8357.30 0.339032
\(848\) 7171.14 0.290399
\(849\) 3185.74 0.128780
\(850\) 0 0
\(851\) 58517.8 2.35719
\(852\) −11187.5 −0.449856
\(853\) −16623.4 −0.667261 −0.333631 0.942704i \(-0.608274\pi\)
−0.333631 + 0.942704i \(0.608274\pi\)
\(854\) 10324.3 0.413689
\(855\) 0 0
\(856\) 5646.26 0.225450
\(857\) −5804.93 −0.231380 −0.115690 0.993285i \(-0.536908\pi\)
−0.115690 + 0.993285i \(0.536908\pi\)
\(858\) −914.675 −0.0363945
\(859\) −30984.8 −1.23072 −0.615360 0.788246i \(-0.710989\pi\)
−0.615360 + 0.788246i \(0.710989\pi\)
\(860\) 0 0
\(861\) 8750.56 0.346363
\(862\) 2198.46 0.0868674
\(863\) 3511.89 0.138524 0.0692620 0.997599i \(-0.477936\pi\)
0.0692620 + 0.997599i \(0.477936\pi\)
\(864\) −4220.31 −0.166178
\(865\) 0 0
\(866\) 7218.22 0.283239
\(867\) −12630.4 −0.494753
\(868\) 13101.9 0.512337
\(869\) 5991.37 0.233882
\(870\) 0 0
\(871\) 2899.49 0.112796
\(872\) 26171.2 1.01636
\(873\) −14756.1 −0.572071
\(874\) 6445.52 0.249454
\(875\) 0 0
\(876\) −21298.6 −0.821476
\(877\) 36724.0 1.41401 0.707003 0.707211i \(-0.250047\pi\)
0.707003 + 0.707211i \(0.250047\pi\)
\(878\) 7955.18 0.305779
\(879\) −23824.5 −0.914200
\(880\) 0 0
\(881\) 41245.8 1.57731 0.788654 0.614838i \(-0.210779\pi\)
0.788654 + 0.614838i \(0.210779\pi\)
\(882\) 1609.03 0.0614274
\(883\) 2065.87 0.0787340 0.0393670 0.999225i \(-0.487466\pi\)
0.0393670 + 0.999225i \(0.487466\pi\)
\(884\) 1340.92 0.0510182
\(885\) 0 0
\(886\) −11153.1 −0.422909
\(887\) −10866.5 −0.411342 −0.205671 0.978621i \(-0.565938\pi\)
−0.205671 + 0.978621i \(0.565938\pi\)
\(888\) 17394.5 0.657342
\(889\) 10879.0 0.410427
\(890\) 0 0
\(891\) −3622.85 −0.136218
\(892\) −8824.54 −0.331242
\(893\) 3524.92 0.132091
\(894\) 4377.91 0.163780
\(895\) 0 0
\(896\) 17925.5 0.668358
\(897\) −3114.00 −0.115912
\(898\) 988.601 0.0367372
\(899\) 31593.8 1.17209
\(900\) 0 0
\(901\) 4428.22 0.163735
\(902\) 9982.74 0.368502
\(903\) 3277.23 0.120775
\(904\) 15668.9 0.576481
\(905\) 0 0
\(906\) −7406.98 −0.271612
\(907\) 12436.5 0.455290 0.227645 0.973744i \(-0.426897\pi\)
0.227645 + 0.973744i \(0.426897\pi\)
\(908\) 32590.6 1.19114
\(909\) −3899.24 −0.142277
\(910\) 0 0
\(911\) 19230.5 0.699380 0.349690 0.936866i \(-0.386287\pi\)
0.349690 + 0.936866i \(0.386287\pi\)
\(912\) −5975.06 −0.216945
\(913\) −23107.6 −0.837623
\(914\) −14595.8 −0.528212
\(915\) 0 0
\(916\) −45039.6 −1.62462
\(917\) 21323.9 0.767912
\(918\) −683.767 −0.0245835
\(919\) −24215.5 −0.869201 −0.434600 0.900623i \(-0.643110\pi\)
−0.434600 + 0.900623i \(0.643110\pi\)
\(920\) 0 0
\(921\) −16217.7 −0.580228
\(922\) −15088.2 −0.538940
\(923\) −3754.82 −0.133902
\(924\) 11871.9 0.422681
\(925\) 0 0
\(926\) −1693.80 −0.0601098
\(927\) 731.920 0.0259325
\(928\) 33348.8 1.17967
\(929\) −19194.4 −0.677878 −0.338939 0.940808i \(-0.610068\pi\)
−0.338939 + 0.940808i \(0.610068\pi\)
\(930\) 0 0
\(931\) 8682.41 0.305644
\(932\) −4507.44 −0.158418
\(933\) −13877.5 −0.486955
\(934\) −12466.9 −0.436756
\(935\) 0 0
\(936\) −925.637 −0.0323242
\(937\) 37196.1 1.29685 0.648423 0.761281i \(-0.275429\pi\)
0.648423 + 0.761281i \(0.275429\pi\)
\(938\) 4845.02 0.168652
\(939\) −3159.95 −0.109820
\(940\) 0 0
\(941\) 33682.5 1.16686 0.583431 0.812163i \(-0.301710\pi\)
0.583431 + 0.812163i \(0.301710\pi\)
\(942\) 1985.82 0.0686854
\(943\) 33986.1 1.17364
\(944\) −25548.9 −0.880875
\(945\) 0 0
\(946\) 3738.70 0.128494
\(947\) 44386.5 1.52309 0.761546 0.648110i \(-0.224440\pi\)
0.761546 + 0.648110i \(0.224440\pi\)
\(948\) 2848.24 0.0975809
\(949\) −7148.37 −0.244516
\(950\) 0 0
\(951\) 8130.01 0.277217
\(952\) 4769.81 0.162385
\(953\) 7655.67 0.260222 0.130111 0.991499i \(-0.458467\pi\)
0.130111 + 0.991499i \(0.458467\pi\)
\(954\) −1435.96 −0.0487327
\(955\) 0 0
\(956\) 45771.8 1.54850
\(957\) 28627.7 0.966983
\(958\) 604.774 0.0203960
\(959\) −29983.7 −1.00962
\(960\) 0 0
\(961\) −7862.76 −0.263931
\(962\) 2742.48 0.0919139
\(963\) 3525.96 0.117988
\(964\) −3612.85 −0.120708
\(965\) 0 0
\(966\) −5203.46 −0.173311
\(967\) −14646.4 −0.487070 −0.243535 0.969892i \(-0.578307\pi\)
−0.243535 + 0.969892i \(0.578307\pi\)
\(968\) 9648.37 0.320362
\(969\) −3689.63 −0.122320
\(970\) 0 0
\(971\) −44429.7 −1.46840 −0.734200 0.678933i \(-0.762443\pi\)
−0.734200 + 0.678933i \(0.762443\pi\)
\(972\) −1722.27 −0.0568332
\(973\) −26988.7 −0.889229
\(974\) 6149.09 0.202289
\(975\) 0 0
\(976\) −37171.5 −1.21909
\(977\) 4615.83 0.151150 0.0755749 0.997140i \(-0.475921\pi\)
0.0755749 + 0.997140i \(0.475921\pi\)
\(978\) 10375.3 0.339228
\(979\) 27314.4 0.891698
\(980\) 0 0
\(981\) 16343.3 0.531908
\(982\) −2974.98 −0.0966756
\(983\) −20086.5 −0.651738 −0.325869 0.945415i \(-0.605657\pi\)
−0.325869 + 0.945415i \(0.605657\pi\)
\(984\) 10102.4 0.327289
\(985\) 0 0
\(986\) 5403.12 0.174513
\(987\) −2845.66 −0.0917714
\(988\) −2346.35 −0.0755540
\(989\) 12728.4 0.409240
\(990\) 0 0
\(991\) 983.787 0.0315348 0.0157674 0.999876i \(-0.494981\pi\)
0.0157674 + 0.999876i \(0.494981\pi\)
\(992\) 23146.4 0.740824
\(993\) −13163.1 −0.420663
\(994\) −6274.26 −0.200209
\(995\) 0 0
\(996\) −10985.1 −0.349476
\(997\) 884.133 0.0280850 0.0140425 0.999901i \(-0.495530\pi\)
0.0140425 + 0.999901i \(0.495530\pi\)
\(998\) 7321.65 0.232227
\(999\) 10862.4 0.344016
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 1875.4.a.g.1.6 14
5.4 even 2 1875.4.a.f.1.9 14
25.6 even 5 75.4.g.b.61.5 yes 28
25.21 even 5 75.4.g.b.16.5 28
75.56 odd 10 225.4.h.a.136.3 28
75.71 odd 10 225.4.h.a.91.3 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.4.g.b.16.5 28 25.21 even 5
75.4.g.b.61.5 yes 28 25.6 even 5
225.4.h.a.91.3 28 75.71 odd 10
225.4.h.a.136.3 28 75.56 odd 10
1875.4.a.f.1.9 14 5.4 even 2
1875.4.a.g.1.6 14 1.1 even 1 trivial