Properties

Label 1875.4.a.g.1.3
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 1.3
Root \(-4.01755\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.01755 q^{2} +3.00000 q^{3} +8.14069 q^{4} -12.0526 q^{6} +1.75849 q^{7} -0.565245 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.01755 q^{2} +3.00000 q^{3} +8.14069 q^{4} -12.0526 q^{6} +1.75849 q^{7} -0.565245 q^{8} +9.00000 q^{9} -23.3703 q^{11} +24.4221 q^{12} -53.2789 q^{13} -7.06482 q^{14} -62.8547 q^{16} -20.8945 q^{17} -36.1579 q^{18} +18.3753 q^{19} +5.27547 q^{21} +93.8914 q^{22} +43.0993 q^{23} -1.69574 q^{24} +214.051 q^{26} +27.0000 q^{27} +14.3153 q^{28} +133.522 q^{29} +242.234 q^{31} +257.044 q^{32} -70.1110 q^{33} +83.9447 q^{34} +73.2662 q^{36} +357.009 q^{37} -73.8238 q^{38} -159.837 q^{39} -424.487 q^{41} -21.1945 q^{42} -93.5407 q^{43} -190.251 q^{44} -173.153 q^{46} -233.643 q^{47} -188.564 q^{48} -339.908 q^{49} -62.6835 q^{51} -433.728 q^{52} -28.0378 q^{53} -108.474 q^{54} -0.993978 q^{56} +55.1260 q^{57} -536.430 q^{58} +451.380 q^{59} +861.337 q^{61} -973.189 q^{62} +15.8264 q^{63} -529.848 q^{64} +281.674 q^{66} -621.276 q^{67} -170.096 q^{68} +129.298 q^{69} +944.967 q^{71} -5.08721 q^{72} -687.417 q^{73} -1434.30 q^{74} +149.588 q^{76} -41.0965 q^{77} +642.152 q^{78} -560.019 q^{79} +81.0000 q^{81} +1705.40 q^{82} -941.309 q^{83} +42.9460 q^{84} +375.804 q^{86} +400.565 q^{87} +13.2100 q^{88} +1128.29 q^{89} -93.6905 q^{91} +350.858 q^{92} +726.703 q^{93} +938.671 q^{94} +771.131 q^{96} -1250.98 q^{97} +1365.60 q^{98} -210.333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.01755 −1.42042 −0.710209 0.703991i \(-0.751400\pi\)
−0.710209 + 0.703991i \(0.751400\pi\)
\(3\) 3.00000 0.577350
\(4\) 8.14069 1.01759
\(5\) 0 0
\(6\) −12.0526 −0.820079
\(7\) 1.75849 0.0949496 0.0474748 0.998872i \(-0.484883\pi\)
0.0474748 + 0.998872i \(0.484883\pi\)
\(8\) −0.565245 −0.0249805
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −23.3703 −0.640583 −0.320292 0.947319i \(-0.603781\pi\)
−0.320292 + 0.947319i \(0.603781\pi\)
\(12\) 24.4221 0.587504
\(13\) −53.2789 −1.13669 −0.568343 0.822792i \(-0.692415\pi\)
−0.568343 + 0.822792i \(0.692415\pi\)
\(14\) −7.06482 −0.134868
\(15\) 0 0
\(16\) −62.8547 −0.982104
\(17\) −20.8945 −0.298098 −0.149049 0.988830i \(-0.547621\pi\)
−0.149049 + 0.988830i \(0.547621\pi\)
\(18\) −36.1579 −0.473473
\(19\) 18.3753 0.221873 0.110937 0.993827i \(-0.464615\pi\)
0.110937 + 0.993827i \(0.464615\pi\)
\(20\) 0 0
\(21\) 5.27547 0.0548192
\(22\) 93.8914 0.909896
\(23\) 43.0993 0.390731 0.195366 0.980730i \(-0.437411\pi\)
0.195366 + 0.980730i \(0.437411\pi\)
\(24\) −1.69574 −0.0144225
\(25\) 0 0
\(26\) 214.051 1.61457
\(27\) 27.0000 0.192450
\(28\) 14.3153 0.0966194
\(29\) 133.522 0.854979 0.427489 0.904020i \(-0.359398\pi\)
0.427489 + 0.904020i \(0.359398\pi\)
\(30\) 0 0
\(31\) 242.234 1.40344 0.701719 0.712454i \(-0.252416\pi\)
0.701719 + 0.712454i \(0.252416\pi\)
\(32\) 257.044 1.41998
\(33\) −70.1110 −0.369841
\(34\) 83.9447 0.423423
\(35\) 0 0
\(36\) 73.2662 0.339196
\(37\) 357.009 1.58627 0.793133 0.609048i \(-0.208448\pi\)
0.793133 + 0.609048i \(0.208448\pi\)
\(38\) −73.8238 −0.315153
\(39\) −159.837 −0.656266
\(40\) 0 0
\(41\) −424.487 −1.61692 −0.808460 0.588552i \(-0.799698\pi\)
−0.808460 + 0.588552i \(0.799698\pi\)
\(42\) −21.1945 −0.0778661
\(43\) −93.5407 −0.331740 −0.165870 0.986148i \(-0.553043\pi\)
−0.165870 + 0.986148i \(0.553043\pi\)
\(44\) −190.251 −0.651849
\(45\) 0 0
\(46\) −173.153 −0.555002
\(47\) −233.643 −0.725113 −0.362556 0.931962i \(-0.618096\pi\)
−0.362556 + 0.931962i \(0.618096\pi\)
\(48\) −188.564 −0.567018
\(49\) −339.908 −0.990985
\(50\) 0 0
\(51\) −62.6835 −0.172107
\(52\) −433.728 −1.15668
\(53\) −28.0378 −0.0726659 −0.0363330 0.999340i \(-0.511568\pi\)
−0.0363330 + 0.999340i \(0.511568\pi\)
\(54\) −108.474 −0.273360
\(55\) 0 0
\(56\) −0.993978 −0.00237189
\(57\) 55.1260 0.128099
\(58\) −536.430 −1.21443
\(59\) 451.380 0.996012 0.498006 0.867174i \(-0.334066\pi\)
0.498006 + 0.867174i \(0.334066\pi\)
\(60\) 0 0
\(61\) 861.337 1.80792 0.903959 0.427620i \(-0.140648\pi\)
0.903959 + 0.427620i \(0.140648\pi\)
\(62\) −973.189 −1.99347
\(63\) 15.8264 0.0316499
\(64\) −529.848 −1.03486
\(65\) 0 0
\(66\) 281.674 0.525329
\(67\) −621.276 −1.13285 −0.566425 0.824113i \(-0.691674\pi\)
−0.566425 + 0.824113i \(0.691674\pi\)
\(68\) −170.096 −0.303340
\(69\) 129.298 0.225589
\(70\) 0 0
\(71\) 944.967 1.57953 0.789767 0.613407i \(-0.210201\pi\)
0.789767 + 0.613407i \(0.210201\pi\)
\(72\) −5.08721 −0.00832685
\(73\) −687.417 −1.10214 −0.551069 0.834460i \(-0.685780\pi\)
−0.551069 + 0.834460i \(0.685780\pi\)
\(74\) −1434.30 −2.25316
\(75\) 0 0
\(76\) 149.588 0.225775
\(77\) −41.0965 −0.0608231
\(78\) 642.152 0.932172
\(79\) −560.019 −0.797558 −0.398779 0.917047i \(-0.630566\pi\)
−0.398779 + 0.917047i \(0.630566\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 1705.40 2.29670
\(83\) −941.309 −1.24484 −0.622422 0.782682i \(-0.713851\pi\)
−0.622422 + 0.782682i \(0.713851\pi\)
\(84\) 42.9460 0.0557833
\(85\) 0 0
\(86\) 375.804 0.471209
\(87\) 400.565 0.493622
\(88\) 13.2100 0.0160021
\(89\) 1128.29 1.34381 0.671904 0.740638i \(-0.265477\pi\)
0.671904 + 0.740638i \(0.265477\pi\)
\(90\) 0 0
\(91\) −93.6905 −0.107928
\(92\) 350.858 0.397603
\(93\) 726.703 0.810276
\(94\) 938.671 1.02996
\(95\) 0 0
\(96\) 771.131 0.819825
\(97\) −1250.98 −1.30946 −0.654730 0.755863i \(-0.727218\pi\)
−0.654730 + 0.755863i \(0.727218\pi\)
\(98\) 1365.60 1.40761
\(99\) −210.333 −0.213528
\(100\) 0 0
\(101\) 665.647 0.655785 0.327893 0.944715i \(-0.393662\pi\)
0.327893 + 0.944715i \(0.393662\pi\)
\(102\) 251.834 0.244464
\(103\) 289.533 0.276976 0.138488 0.990364i \(-0.455776\pi\)
0.138488 + 0.990364i \(0.455776\pi\)
\(104\) 30.1157 0.0283950
\(105\) 0 0
\(106\) 112.643 0.103216
\(107\) 50.4077 0.0455430 0.0227715 0.999741i \(-0.492751\pi\)
0.0227715 + 0.999741i \(0.492751\pi\)
\(108\) 219.799 0.195835
\(109\) −1606.79 −1.41195 −0.705977 0.708235i \(-0.749492\pi\)
−0.705977 + 0.708235i \(0.749492\pi\)
\(110\) 0 0
\(111\) 1071.03 0.915831
\(112\) −110.529 −0.0932504
\(113\) −1209.22 −1.00668 −0.503338 0.864090i \(-0.667895\pi\)
−0.503338 + 0.864090i \(0.667895\pi\)
\(114\) −221.471 −0.181953
\(115\) 0 0
\(116\) 1086.96 0.870015
\(117\) −479.510 −0.378895
\(118\) −1813.44 −1.41475
\(119\) −36.7428 −0.0283043
\(120\) 0 0
\(121\) −784.828 −0.589653
\(122\) −3460.46 −2.56800
\(123\) −1273.46 −0.933529
\(124\) 1971.96 1.42812
\(125\) 0 0
\(126\) −63.5834 −0.0449560
\(127\) −688.004 −0.480713 −0.240356 0.970685i \(-0.577264\pi\)
−0.240356 + 0.970685i \(0.577264\pi\)
\(128\) 72.3402 0.0499534
\(129\) −280.622 −0.191530
\(130\) 0 0
\(131\) 1467.74 0.978911 0.489456 0.872028i \(-0.337195\pi\)
0.489456 + 0.872028i \(0.337195\pi\)
\(132\) −570.752 −0.376345
\(133\) 32.3129 0.0210668
\(134\) 2496.01 1.60912
\(135\) 0 0
\(136\) 11.8105 0.00744664
\(137\) 2626.32 1.63782 0.818910 0.573922i \(-0.194579\pi\)
0.818910 + 0.573922i \(0.194579\pi\)
\(138\) −519.460 −0.320430
\(139\) −3159.82 −1.92815 −0.964074 0.265635i \(-0.914418\pi\)
−0.964074 + 0.265635i \(0.914418\pi\)
\(140\) 0 0
\(141\) −700.928 −0.418644
\(142\) −3796.45 −2.24360
\(143\) 1245.15 0.728142
\(144\) −565.692 −0.327368
\(145\) 0 0
\(146\) 2761.73 1.56550
\(147\) −1019.72 −0.572145
\(148\) 2906.30 1.61416
\(149\) 1591.21 0.874879 0.437440 0.899248i \(-0.355885\pi\)
0.437440 + 0.899248i \(0.355885\pi\)
\(150\) 0 0
\(151\) 1150.25 0.619909 0.309955 0.950751i \(-0.399686\pi\)
0.309955 + 0.950751i \(0.399686\pi\)
\(152\) −10.3866 −0.00554251
\(153\) −188.051 −0.0993659
\(154\) 165.107 0.0863943
\(155\) 0 0
\(156\) −1301.18 −0.667808
\(157\) −629.792 −0.320146 −0.160073 0.987105i \(-0.551173\pi\)
−0.160073 + 0.987105i \(0.551173\pi\)
\(158\) 2249.90 1.13287
\(159\) −84.1135 −0.0419537
\(160\) 0 0
\(161\) 75.7897 0.0370998
\(162\) −325.421 −0.157824
\(163\) 645.868 0.310358 0.155179 0.987886i \(-0.450405\pi\)
0.155179 + 0.987886i \(0.450405\pi\)
\(164\) −3455.62 −1.64536
\(165\) 0 0
\(166\) 3781.75 1.76820
\(167\) −3427.01 −1.58796 −0.793981 0.607942i \(-0.791995\pi\)
−0.793981 + 0.607942i \(0.791995\pi\)
\(168\) −2.98194 −0.00136941
\(169\) 641.645 0.292055
\(170\) 0 0
\(171\) 165.378 0.0739577
\(172\) −761.486 −0.337574
\(173\) 1106.51 0.486281 0.243140 0.969991i \(-0.421822\pi\)
0.243140 + 0.969991i \(0.421822\pi\)
\(174\) −1609.29 −0.701150
\(175\) 0 0
\(176\) 1468.93 0.629120
\(177\) 1354.14 0.575048
\(178\) −4532.98 −1.90877
\(179\) −151.565 −0.0632878 −0.0316439 0.999499i \(-0.510074\pi\)
−0.0316439 + 0.999499i \(0.510074\pi\)
\(180\) 0 0
\(181\) 2768.13 1.13676 0.568379 0.822767i \(-0.307571\pi\)
0.568379 + 0.822767i \(0.307571\pi\)
\(182\) 376.406 0.153303
\(183\) 2584.01 1.04380
\(184\) −24.3617 −0.00976068
\(185\) 0 0
\(186\) −2919.57 −1.15093
\(187\) 488.311 0.190956
\(188\) −1902.01 −0.737865
\(189\) 47.4793 0.0182731
\(190\) 0 0
\(191\) 2375.05 0.899752 0.449876 0.893091i \(-0.351468\pi\)
0.449876 + 0.893091i \(0.351468\pi\)
\(192\) −1589.54 −0.597476
\(193\) −1876.65 −0.699917 −0.349958 0.936765i \(-0.613804\pi\)
−0.349958 + 0.936765i \(0.613804\pi\)
\(194\) 5025.86 1.85998
\(195\) 0 0
\(196\) −2767.08 −1.00841
\(197\) −956.430 −0.345903 −0.172951 0.984930i \(-0.555330\pi\)
−0.172951 + 0.984930i \(0.555330\pi\)
\(198\) 845.023 0.303299
\(199\) −633.678 −0.225730 −0.112865 0.993610i \(-0.536003\pi\)
−0.112865 + 0.993610i \(0.536003\pi\)
\(200\) 0 0
\(201\) −1863.83 −0.654051
\(202\) −2674.27 −0.931489
\(203\) 234.797 0.0811799
\(204\) −510.287 −0.175134
\(205\) 0 0
\(206\) −1163.21 −0.393421
\(207\) 387.894 0.130244
\(208\) 3348.83 1.11634
\(209\) −429.438 −0.142128
\(210\) 0 0
\(211\) −818.815 −0.267154 −0.133577 0.991038i \(-0.542646\pi\)
−0.133577 + 0.991038i \(0.542646\pi\)
\(212\) −228.248 −0.0739439
\(213\) 2834.90 0.911944
\(214\) −202.515 −0.0646900
\(215\) 0 0
\(216\) −15.2616 −0.00480751
\(217\) 425.967 0.133256
\(218\) 6455.37 2.00556
\(219\) −2062.25 −0.636320
\(220\) 0 0
\(221\) 1113.24 0.338844
\(222\) −4302.90 −1.30086
\(223\) −3310.99 −0.994262 −0.497131 0.867676i \(-0.665613\pi\)
−0.497131 + 0.867676i \(0.665613\pi\)
\(224\) 452.009 0.134826
\(225\) 0 0
\(226\) 4858.12 1.42990
\(227\) −986.372 −0.288404 −0.144202 0.989548i \(-0.546062\pi\)
−0.144202 + 0.989548i \(0.546062\pi\)
\(228\) 448.764 0.130351
\(229\) −356.162 −0.102777 −0.0513883 0.998679i \(-0.516365\pi\)
−0.0513883 + 0.998679i \(0.516365\pi\)
\(230\) 0 0
\(231\) −123.290 −0.0351163
\(232\) −75.4726 −0.0213578
\(233\) −5095.94 −1.43281 −0.716407 0.697682i \(-0.754215\pi\)
−0.716407 + 0.697682i \(0.754215\pi\)
\(234\) 1926.46 0.538190
\(235\) 0 0
\(236\) 3674.55 1.01353
\(237\) −1680.06 −0.460470
\(238\) 147.616 0.0402039
\(239\) −2984.39 −0.807715 −0.403858 0.914822i \(-0.632331\pi\)
−0.403858 + 0.914822i \(0.632331\pi\)
\(240\) 0 0
\(241\) 4367.46 1.16736 0.583678 0.811985i \(-0.301613\pi\)
0.583678 + 0.811985i \(0.301613\pi\)
\(242\) 3153.08 0.837553
\(243\) 243.000 0.0641500
\(244\) 7011.88 1.83971
\(245\) 0 0
\(246\) 5116.19 1.32600
\(247\) −979.018 −0.252200
\(248\) −136.922 −0.0350587
\(249\) −2823.93 −0.718711
\(250\) 0 0
\(251\) −6267.54 −1.57611 −0.788055 0.615605i \(-0.788912\pi\)
−0.788055 + 0.615605i \(0.788912\pi\)
\(252\) 128.838 0.0322065
\(253\) −1007.24 −0.250296
\(254\) 2764.09 0.682813
\(255\) 0 0
\(256\) 3948.15 0.963904
\(257\) −6002.51 −1.45691 −0.728456 0.685093i \(-0.759762\pi\)
−0.728456 + 0.685093i \(0.759762\pi\)
\(258\) 1127.41 0.272053
\(259\) 627.796 0.150615
\(260\) 0 0
\(261\) 1201.70 0.284993
\(262\) −5896.73 −1.39046
\(263\) 6644.05 1.55776 0.778878 0.627176i \(-0.215789\pi\)
0.778878 + 0.627176i \(0.215789\pi\)
\(264\) 39.6299 0.00923883
\(265\) 0 0
\(266\) −129.818 −0.0299236
\(267\) 3384.88 0.775848
\(268\) −5057.62 −1.15277
\(269\) 8457.47 1.91695 0.958477 0.285168i \(-0.0920495\pi\)
0.958477 + 0.285168i \(0.0920495\pi\)
\(270\) 0 0
\(271\) −8399.42 −1.88276 −0.941380 0.337347i \(-0.890470\pi\)
−0.941380 + 0.337347i \(0.890470\pi\)
\(272\) 1313.32 0.292763
\(273\) −281.072 −0.0623122
\(274\) −10551.4 −2.32639
\(275\) 0 0
\(276\) 1052.57 0.229556
\(277\) 1569.09 0.340352 0.170176 0.985414i \(-0.445566\pi\)
0.170176 + 0.985414i \(0.445566\pi\)
\(278\) 12694.7 2.73877
\(279\) 2180.11 0.467813
\(280\) 0 0
\(281\) −1495.83 −0.317559 −0.158779 0.987314i \(-0.550756\pi\)
−0.158779 + 0.987314i \(0.550756\pi\)
\(282\) 2816.01 0.594649
\(283\) −5970.75 −1.25415 −0.627074 0.778959i \(-0.715748\pi\)
−0.627074 + 0.778959i \(0.715748\pi\)
\(284\) 7692.68 1.60731
\(285\) 0 0
\(286\) −5002.43 −1.03427
\(287\) −746.456 −0.153526
\(288\) 2313.39 0.473326
\(289\) −4476.42 −0.911138
\(290\) 0 0
\(291\) −3752.93 −0.756017
\(292\) −5596.05 −1.12152
\(293\) 1269.10 0.253043 0.126521 0.991964i \(-0.459619\pi\)
0.126521 + 0.991964i \(0.459619\pi\)
\(294\) 4096.79 0.812685
\(295\) 0 0
\(296\) −201.797 −0.0396258
\(297\) −630.999 −0.123280
\(298\) −6392.76 −1.24269
\(299\) −2296.28 −0.444139
\(300\) 0 0
\(301\) −164.490 −0.0314986
\(302\) −4621.20 −0.880530
\(303\) 1996.94 0.378618
\(304\) −1154.98 −0.217903
\(305\) 0 0
\(306\) 755.502 0.141141
\(307\) 5218.55 0.970157 0.485078 0.874471i \(-0.338791\pi\)
0.485078 + 0.874471i \(0.338791\pi\)
\(308\) −334.554 −0.0618928
\(309\) 868.598 0.159912
\(310\) 0 0
\(311\) 644.592 0.117529 0.0587644 0.998272i \(-0.481284\pi\)
0.0587644 + 0.998272i \(0.481284\pi\)
\(312\) 90.3470 0.0163939
\(313\) 922.833 0.166650 0.0833252 0.996522i \(-0.473446\pi\)
0.0833252 + 0.996522i \(0.473446\pi\)
\(314\) 2530.22 0.454741
\(315\) 0 0
\(316\) −4558.94 −0.811584
\(317\) −1224.32 −0.216924 −0.108462 0.994101i \(-0.534593\pi\)
−0.108462 + 0.994101i \(0.534593\pi\)
\(318\) 337.930 0.0595918
\(319\) −3120.45 −0.547685
\(320\) 0 0
\(321\) 151.223 0.0262942
\(322\) −304.489 −0.0526972
\(323\) −383.943 −0.0661399
\(324\) 659.396 0.113065
\(325\) 0 0
\(326\) −2594.81 −0.440838
\(327\) −4820.38 −0.815192
\(328\) 239.939 0.0403915
\(329\) −410.859 −0.0688491
\(330\) 0 0
\(331\) −10643.3 −1.76739 −0.883696 0.468062i \(-0.844953\pi\)
−0.883696 + 0.468062i \(0.844953\pi\)
\(332\) −7662.91 −1.26674
\(333\) 3213.08 0.528755
\(334\) 13768.2 2.25557
\(335\) 0 0
\(336\) −331.588 −0.0538381
\(337\) −3954.03 −0.639139 −0.319569 0.947563i \(-0.603538\pi\)
−0.319569 + 0.947563i \(0.603538\pi\)
\(338\) −2577.84 −0.414841
\(339\) −3627.67 −0.581204
\(340\) 0 0
\(341\) −5661.10 −0.899020
\(342\) −664.414 −0.105051
\(343\) −1200.89 −0.189043
\(344\) 52.8734 0.00828704
\(345\) 0 0
\(346\) −4445.47 −0.690722
\(347\) 9477.20 1.46617 0.733087 0.680135i \(-0.238079\pi\)
0.733087 + 0.680135i \(0.238079\pi\)
\(348\) 3260.88 0.502303
\(349\) 5550.12 0.851264 0.425632 0.904896i \(-0.360052\pi\)
0.425632 + 0.904896i \(0.360052\pi\)
\(350\) 0 0
\(351\) −1438.53 −0.218755
\(352\) −6007.19 −0.909615
\(353\) −12135.0 −1.82970 −0.914849 0.403796i \(-0.867691\pi\)
−0.914849 + 0.403796i \(0.867691\pi\)
\(354\) −5440.32 −0.816808
\(355\) 0 0
\(356\) 9185.10 1.36744
\(357\) −110.228 −0.0163415
\(358\) 608.921 0.0898951
\(359\) −1199.51 −0.176344 −0.0881722 0.996105i \(-0.528103\pi\)
−0.0881722 + 0.996105i \(0.528103\pi\)
\(360\) 0 0
\(361\) −6521.35 −0.950772
\(362\) −11121.1 −1.61467
\(363\) −2354.48 −0.340436
\(364\) −762.706 −0.109826
\(365\) 0 0
\(366\) −10381.4 −1.48263
\(367\) 8405.65 1.19556 0.597781 0.801659i \(-0.296049\pi\)
0.597781 + 0.801659i \(0.296049\pi\)
\(368\) −2708.99 −0.383739
\(369\) −3820.38 −0.538973
\(370\) 0 0
\(371\) −49.3043 −0.00689960
\(372\) 5915.87 0.824526
\(373\) −5200.17 −0.721863 −0.360931 0.932592i \(-0.617541\pi\)
−0.360931 + 0.932592i \(0.617541\pi\)
\(374\) −1961.81 −0.271238
\(375\) 0 0
\(376\) 132.065 0.0181137
\(377\) −7113.90 −0.971842
\(378\) −190.750 −0.0259554
\(379\) 418.100 0.0566658 0.0283329 0.999599i \(-0.490980\pi\)
0.0283329 + 0.999599i \(0.490980\pi\)
\(380\) 0 0
\(381\) −2064.01 −0.277540
\(382\) −9541.88 −1.27802
\(383\) 3290.25 0.438966 0.219483 0.975616i \(-0.429563\pi\)
0.219483 + 0.975616i \(0.429563\pi\)
\(384\) 217.021 0.0288406
\(385\) 0 0
\(386\) 7539.51 0.994174
\(387\) −841.866 −0.110580
\(388\) −10183.8 −1.33249
\(389\) −1339.04 −0.174530 −0.0872650 0.996185i \(-0.527813\pi\)
−0.0872650 + 0.996185i \(0.527813\pi\)
\(390\) 0 0
\(391\) −900.538 −0.116476
\(392\) 192.131 0.0247553
\(393\) 4403.23 0.565175
\(394\) 3842.50 0.491326
\(395\) 0 0
\(396\) −1712.26 −0.217283
\(397\) 929.457 0.117502 0.0587508 0.998273i \(-0.481288\pi\)
0.0587508 + 0.998273i \(0.481288\pi\)
\(398\) 2545.83 0.320631
\(399\) 96.9386 0.0121629
\(400\) 0 0
\(401\) 6924.87 0.862372 0.431186 0.902263i \(-0.358095\pi\)
0.431186 + 0.902263i \(0.358095\pi\)
\(402\) 7488.02 0.929026
\(403\) −12906.0 −1.59527
\(404\) 5418.83 0.667319
\(405\) 0 0
\(406\) −943.308 −0.115309
\(407\) −8343.41 −1.01614
\(408\) 35.4315 0.00429932
\(409\) −1786.31 −0.215959 −0.107980 0.994153i \(-0.534438\pi\)
−0.107980 + 0.994153i \(0.534438\pi\)
\(410\) 0 0
\(411\) 7878.95 0.945596
\(412\) 2357.00 0.281847
\(413\) 793.748 0.0945709
\(414\) −1558.38 −0.185001
\(415\) 0 0
\(416\) −13695.0 −1.61407
\(417\) −9479.46 −1.11322
\(418\) 1725.29 0.201882
\(419\) −10085.9 −1.17596 −0.587980 0.808875i \(-0.700077\pi\)
−0.587980 + 0.808875i \(0.700077\pi\)
\(420\) 0 0
\(421\) −4600.54 −0.532581 −0.266290 0.963893i \(-0.585798\pi\)
−0.266290 + 0.963893i \(0.585798\pi\)
\(422\) 3289.63 0.379470
\(423\) −2102.78 −0.241704
\(424\) 15.8483 0.00181523
\(425\) 0 0
\(426\) −11389.3 −1.29534
\(427\) 1514.65 0.171661
\(428\) 410.354 0.0463439
\(429\) 3735.44 0.420393
\(430\) 0 0
\(431\) −5687.96 −0.635683 −0.317841 0.948144i \(-0.602958\pi\)
−0.317841 + 0.948144i \(0.602958\pi\)
\(432\) −1697.08 −0.189006
\(433\) −14103.5 −1.56529 −0.782643 0.622471i \(-0.786129\pi\)
−0.782643 + 0.622471i \(0.786129\pi\)
\(434\) −1711.34 −0.189279
\(435\) 0 0
\(436\) −13080.4 −1.43679
\(437\) 791.964 0.0866928
\(438\) 8285.20 0.903840
\(439\) −13955.0 −1.51717 −0.758583 0.651576i \(-0.774108\pi\)
−0.758583 + 0.651576i \(0.774108\pi\)
\(440\) 0 0
\(441\) −3059.17 −0.330328
\(442\) −4472.48 −0.481299
\(443\) −6671.65 −0.715530 −0.357765 0.933812i \(-0.616461\pi\)
−0.357765 + 0.933812i \(0.616461\pi\)
\(444\) 8718.89 0.931938
\(445\) 0 0
\(446\) 13302.1 1.41227
\(447\) 4773.63 0.505112
\(448\) −931.732 −0.0982594
\(449\) −5676.67 −0.596656 −0.298328 0.954463i \(-0.596429\pi\)
−0.298328 + 0.954463i \(0.596429\pi\)
\(450\) 0 0
\(451\) 9920.39 1.03577
\(452\) −9843.93 −1.02438
\(453\) 3450.76 0.357905
\(454\) 3962.80 0.409655
\(455\) 0 0
\(456\) −31.1597 −0.00319997
\(457\) −16775.1 −1.71708 −0.858539 0.512749i \(-0.828627\pi\)
−0.858539 + 0.512749i \(0.828627\pi\)
\(458\) 1430.90 0.145986
\(459\) −564.152 −0.0573689
\(460\) 0 0
\(461\) −1308.27 −0.132174 −0.0660868 0.997814i \(-0.521051\pi\)
−0.0660868 + 0.997814i \(0.521051\pi\)
\(462\) 495.322 0.0498798
\(463\) −753.004 −0.0755833 −0.0377916 0.999286i \(-0.512032\pi\)
−0.0377916 + 0.999286i \(0.512032\pi\)
\(464\) −8392.47 −0.839678
\(465\) 0 0
\(466\) 20473.2 2.03520
\(467\) 15968.1 1.58226 0.791132 0.611645i \(-0.209492\pi\)
0.791132 + 0.611645i \(0.209492\pi\)
\(468\) −3903.55 −0.385559
\(469\) −1092.51 −0.107564
\(470\) 0 0
\(471\) −1889.38 −0.184836
\(472\) −255.140 −0.0248809
\(473\) 2186.08 0.212507
\(474\) 6749.71 0.654060
\(475\) 0 0
\(476\) −299.112 −0.0288020
\(477\) −252.341 −0.0242220
\(478\) 11989.9 1.14729
\(479\) −10985.5 −1.04789 −0.523945 0.851752i \(-0.675540\pi\)
−0.523945 + 0.851752i \(0.675540\pi\)
\(480\) 0 0
\(481\) −19021.0 −1.80309
\(482\) −17546.5 −1.65813
\(483\) 227.369 0.0214196
\(484\) −6389.04 −0.600023
\(485\) 0 0
\(486\) −976.264 −0.0911198
\(487\) −6290.00 −0.585271 −0.292636 0.956224i \(-0.594532\pi\)
−0.292636 + 0.956224i \(0.594532\pi\)
\(488\) −486.867 −0.0451628
\(489\) 1937.61 0.179185
\(490\) 0 0
\(491\) −12483.9 −1.14743 −0.573716 0.819054i \(-0.694499\pi\)
−0.573716 + 0.819054i \(0.694499\pi\)
\(492\) −10366.8 −0.949946
\(493\) −2789.87 −0.254867
\(494\) 3933.25 0.358230
\(495\) 0 0
\(496\) −15225.6 −1.37832
\(497\) 1661.72 0.149976
\(498\) 11345.3 1.02087
\(499\) −7074.89 −0.634700 −0.317350 0.948308i \(-0.602793\pi\)
−0.317350 + 0.948308i \(0.602793\pi\)
\(500\) 0 0
\(501\) −10281.0 −0.916811
\(502\) 25180.1 2.23873
\(503\) −8658.48 −0.767520 −0.383760 0.923433i \(-0.625371\pi\)
−0.383760 + 0.923433i \(0.625371\pi\)
\(504\) −8.94581 −0.000790631 0
\(505\) 0 0
\(506\) 4046.65 0.355525
\(507\) 1924.94 0.168618
\(508\) −5600.83 −0.489167
\(509\) −9061.62 −0.789094 −0.394547 0.918876i \(-0.629099\pi\)
−0.394547 + 0.918876i \(0.629099\pi\)
\(510\) 0 0
\(511\) −1208.82 −0.104648
\(512\) −16440.6 −1.41910
\(513\) 496.134 0.0426995
\(514\) 24115.4 2.06942
\(515\) 0 0
\(516\) −2284.46 −0.194899
\(517\) 5460.31 0.464495
\(518\) −2522.20 −0.213937
\(519\) 3319.54 0.280754
\(520\) 0 0
\(521\) 9821.93 0.825924 0.412962 0.910748i \(-0.364494\pi\)
0.412962 + 0.910748i \(0.364494\pi\)
\(522\) −4827.87 −0.404809
\(523\) −20861.7 −1.74420 −0.872101 0.489326i \(-0.837243\pi\)
−0.872101 + 0.489326i \(0.837243\pi\)
\(524\) 11948.5 0.996127
\(525\) 0 0
\(526\) −26692.8 −2.21266
\(527\) −5061.37 −0.418362
\(528\) 4406.80 0.363222
\(529\) −10309.5 −0.847329
\(530\) 0 0
\(531\) 4062.42 0.332004
\(532\) 263.049 0.0214373
\(533\) 22616.2 1.83793
\(534\) −13598.9 −1.10203
\(535\) 0 0
\(536\) 351.173 0.0282992
\(537\) −454.696 −0.0365392
\(538\) −33978.3 −2.72288
\(539\) 7943.75 0.634808
\(540\) 0 0
\(541\) 7917.14 0.629177 0.314588 0.949228i \(-0.398134\pi\)
0.314588 + 0.949228i \(0.398134\pi\)
\(542\) 33745.1 2.67431
\(543\) 8304.38 0.656307
\(544\) −5370.80 −0.423292
\(545\) 0 0
\(546\) 1129.22 0.0885093
\(547\) −13960.6 −1.09124 −0.545622 0.838031i \(-0.683707\pi\)
−0.545622 + 0.838031i \(0.683707\pi\)
\(548\) 21380.0 1.66662
\(549\) 7752.04 0.602639
\(550\) 0 0
\(551\) 2453.51 0.189697
\(552\) −73.0850 −0.00563533
\(553\) −984.788 −0.0757278
\(554\) −6303.91 −0.483443
\(555\) 0 0
\(556\) −25723.1 −1.96206
\(557\) 1925.41 0.146467 0.0732337 0.997315i \(-0.476668\pi\)
0.0732337 + 0.997315i \(0.476668\pi\)
\(558\) −8758.70 −0.664490
\(559\) 4983.75 0.377084
\(560\) 0 0
\(561\) 1464.93 0.110249
\(562\) 6009.59 0.451066
\(563\) 12581.0 0.941788 0.470894 0.882190i \(-0.343931\pi\)
0.470894 + 0.882190i \(0.343931\pi\)
\(564\) −5706.04 −0.426006
\(565\) 0 0
\(566\) 23987.8 1.78142
\(567\) 142.438 0.0105500
\(568\) −534.138 −0.0394576
\(569\) 917.143 0.0675723 0.0337861 0.999429i \(-0.489243\pi\)
0.0337861 + 0.999429i \(0.489243\pi\)
\(570\) 0 0
\(571\) −7954.65 −0.582998 −0.291499 0.956571i \(-0.594154\pi\)
−0.291499 + 0.956571i \(0.594154\pi\)
\(572\) 10136.4 0.740948
\(573\) 7125.15 0.519472
\(574\) 2998.92 0.218071
\(575\) 0 0
\(576\) −4768.63 −0.344953
\(577\) −458.160 −0.0330563 −0.0165281 0.999863i \(-0.505261\pi\)
−0.0165281 + 0.999863i \(0.505261\pi\)
\(578\) 17984.2 1.29420
\(579\) −5629.94 −0.404097
\(580\) 0 0
\(581\) −1655.28 −0.118197
\(582\) 15077.6 1.07386
\(583\) 655.254 0.0465486
\(584\) 388.559 0.0275320
\(585\) 0 0
\(586\) −5098.67 −0.359427
\(587\) 4149.10 0.291740 0.145870 0.989304i \(-0.453402\pi\)
0.145870 + 0.989304i \(0.453402\pi\)
\(588\) −8301.25 −0.582207
\(589\) 4451.14 0.311385
\(590\) 0 0
\(591\) −2869.29 −0.199707
\(592\) −22439.7 −1.55788
\(593\) 5267.14 0.364748 0.182374 0.983229i \(-0.441622\pi\)
0.182374 + 0.983229i \(0.441622\pi\)
\(594\) 2535.07 0.175110
\(595\) 0 0
\(596\) 12953.6 0.890266
\(597\) −1901.03 −0.130325
\(598\) 9225.43 0.630863
\(599\) −20272.2 −1.38281 −0.691403 0.722470i \(-0.743007\pi\)
−0.691403 + 0.722470i \(0.743007\pi\)
\(600\) 0 0
\(601\) 11957.9 0.811601 0.405801 0.913962i \(-0.366993\pi\)
0.405801 + 0.913962i \(0.366993\pi\)
\(602\) 660.848 0.0447411
\(603\) −5591.48 −0.377617
\(604\) 9363.86 0.630812
\(605\) 0 0
\(606\) −8022.80 −0.537796
\(607\) 11679.0 0.780949 0.390474 0.920614i \(-0.372311\pi\)
0.390474 + 0.920614i \(0.372311\pi\)
\(608\) 4723.26 0.315055
\(609\) 704.391 0.0468692
\(610\) 0 0
\(611\) 12448.2 0.824225
\(612\) −1530.86 −0.101113
\(613\) 1799.17 0.118544 0.0592722 0.998242i \(-0.481122\pi\)
0.0592722 + 0.998242i \(0.481122\pi\)
\(614\) −20965.8 −1.37803
\(615\) 0 0
\(616\) 23.2296 0.00151939
\(617\) 852.988 0.0556564 0.0278282 0.999613i \(-0.491141\pi\)
0.0278282 + 0.999613i \(0.491141\pi\)
\(618\) −3489.64 −0.227142
\(619\) 10187.7 0.661516 0.330758 0.943716i \(-0.392696\pi\)
0.330758 + 0.943716i \(0.392696\pi\)
\(620\) 0 0
\(621\) 1163.68 0.0751963
\(622\) −2589.68 −0.166940
\(623\) 1984.10 0.127594
\(624\) 10046.5 0.644521
\(625\) 0 0
\(626\) −3707.53 −0.236713
\(627\) −1288.31 −0.0820578
\(628\) −5126.94 −0.325776
\(629\) −7459.52 −0.472862
\(630\) 0 0
\(631\) 15476.9 0.976425 0.488213 0.872725i \(-0.337649\pi\)
0.488213 + 0.872725i \(0.337649\pi\)
\(632\) 316.548 0.0199234
\(633\) −2456.44 −0.154241
\(634\) 4918.78 0.308122
\(635\) 0 0
\(636\) −684.743 −0.0426915
\(637\) 18109.9 1.12644
\(638\) 12536.6 0.777942
\(639\) 8504.70 0.526511
\(640\) 0 0
\(641\) 16853.5 1.03849 0.519247 0.854624i \(-0.326212\pi\)
0.519247 + 0.854624i \(0.326212\pi\)
\(642\) −607.546 −0.0373488
\(643\) 25045.6 1.53608 0.768041 0.640401i \(-0.221232\pi\)
0.768041 + 0.640401i \(0.221232\pi\)
\(644\) 616.981 0.0377522
\(645\) 0 0
\(646\) 1542.51 0.0939463
\(647\) 6140.11 0.373095 0.186548 0.982446i \(-0.440270\pi\)
0.186548 + 0.982446i \(0.440270\pi\)
\(648\) −45.7849 −0.00277562
\(649\) −10548.9 −0.638029
\(650\) 0 0
\(651\) 1277.90 0.0769353
\(652\) 5257.82 0.315816
\(653\) −10637.8 −0.637502 −0.318751 0.947838i \(-0.603263\pi\)
−0.318751 + 0.947838i \(0.603263\pi\)
\(654\) 19366.1 1.15791
\(655\) 0 0
\(656\) 26681.0 1.58798
\(657\) −6186.75 −0.367379
\(658\) 1650.64 0.0977945
\(659\) 6344.68 0.375044 0.187522 0.982260i \(-0.439954\pi\)
0.187522 + 0.982260i \(0.439954\pi\)
\(660\) 0 0
\(661\) −31546.8 −1.85632 −0.928159 0.372183i \(-0.878609\pi\)
−0.928159 + 0.372183i \(0.878609\pi\)
\(662\) 42759.8 2.51043
\(663\) 3339.71 0.195631
\(664\) 532.070 0.0310969
\(665\) 0 0
\(666\) −12908.7 −0.751054
\(667\) 5754.69 0.334067
\(668\) −27898.2 −1.61589
\(669\) −9932.98 −0.574037
\(670\) 0 0
\(671\) −20129.7 −1.15812
\(672\) 1356.03 0.0778420
\(673\) 14986.0 0.858347 0.429173 0.903222i \(-0.358805\pi\)
0.429173 + 0.903222i \(0.358805\pi\)
\(674\) 15885.5 0.907844
\(675\) 0 0
\(676\) 5223.44 0.297192
\(677\) 551.325 0.0312986 0.0156493 0.999878i \(-0.495018\pi\)
0.0156493 + 0.999878i \(0.495018\pi\)
\(678\) 14574.4 0.825553
\(679\) −2199.83 −0.124333
\(680\) 0 0
\(681\) −2959.12 −0.166510
\(682\) 22743.7 1.27698
\(683\) −2372.30 −0.132904 −0.0664522 0.997790i \(-0.521168\pi\)
−0.0664522 + 0.997790i \(0.521168\pi\)
\(684\) 1346.29 0.0752584
\(685\) 0 0
\(686\) 4824.62 0.268520
\(687\) −1068.49 −0.0593381
\(688\) 5879.47 0.325803
\(689\) 1493.83 0.0825984
\(690\) 0 0
\(691\) 11255.4 0.619645 0.309822 0.950794i \(-0.399730\pi\)
0.309822 + 0.950794i \(0.399730\pi\)
\(692\) 9007.78 0.494833
\(693\) −369.869 −0.0202744
\(694\) −38075.1 −2.08258
\(695\) 0 0
\(696\) −226.418 −0.0123309
\(697\) 8869.44 0.482000
\(698\) −22297.9 −1.20915
\(699\) −15287.8 −0.827236
\(700\) 0 0
\(701\) 19255.2 1.03746 0.518729 0.854939i \(-0.326405\pi\)
0.518729 + 0.854939i \(0.326405\pi\)
\(702\) 5779.37 0.310724
\(703\) 6560.15 0.351950
\(704\) 12382.7 0.662913
\(705\) 0 0
\(706\) 48753.1 2.59894
\(707\) 1170.53 0.0622666
\(708\) 11023.6 0.585161
\(709\) −24897.9 −1.31884 −0.659421 0.751774i \(-0.729199\pi\)
−0.659421 + 0.751774i \(0.729199\pi\)
\(710\) 0 0
\(711\) −5040.17 −0.265853
\(712\) −637.763 −0.0335691
\(713\) 10440.1 0.548367
\(714\) 442.848 0.0232117
\(715\) 0 0
\(716\) −1233.85 −0.0644008
\(717\) −8953.16 −0.466335
\(718\) 4819.08 0.250483
\(719\) −37023.4 −1.92036 −0.960182 0.279377i \(-0.909872\pi\)
−0.960182 + 0.279377i \(0.909872\pi\)
\(720\) 0 0
\(721\) 509.141 0.0262987
\(722\) 26199.8 1.35049
\(723\) 13102.4 0.673973
\(724\) 22534.5 1.15675
\(725\) 0 0
\(726\) 9459.25 0.483562
\(727\) 26634.7 1.35877 0.679385 0.733782i \(-0.262246\pi\)
0.679385 + 0.733782i \(0.262246\pi\)
\(728\) 52.9581 0.00269610
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 1954.49 0.0988909
\(732\) 21035.7 1.06216
\(733\) −1452.15 −0.0731735 −0.0365868 0.999330i \(-0.511649\pi\)
−0.0365868 + 0.999330i \(0.511649\pi\)
\(734\) −33770.1 −1.69820
\(735\) 0 0
\(736\) 11078.4 0.554830
\(737\) 14519.4 0.725685
\(738\) 15348.6 0.765567
\(739\) 5831.78 0.290291 0.145146 0.989410i \(-0.453635\pi\)
0.145146 + 0.989410i \(0.453635\pi\)
\(740\) 0 0
\(741\) −2937.06 −0.145608
\(742\) 198.082 0.00980032
\(743\) 34683.6 1.71254 0.856271 0.516527i \(-0.172775\pi\)
0.856271 + 0.516527i \(0.172775\pi\)
\(744\) −410.766 −0.0202411
\(745\) 0 0
\(746\) 20891.9 1.02535
\(747\) −8471.78 −0.414948
\(748\) 3975.19 0.194315
\(749\) 88.6415 0.00432429
\(750\) 0 0
\(751\) 34364.2 1.66973 0.834865 0.550455i \(-0.185546\pi\)
0.834865 + 0.550455i \(0.185546\pi\)
\(752\) 14685.5 0.712136
\(753\) −18802.6 −0.909967
\(754\) 28580.4 1.38042
\(755\) 0 0
\(756\) 386.514 0.0185944
\(757\) 32402.5 1.55573 0.777865 0.628431i \(-0.216303\pi\)
0.777865 + 0.628431i \(0.216303\pi\)
\(758\) −1679.74 −0.0804891
\(759\) −3021.73 −0.144508
\(760\) 0 0
\(761\) 21787.1 1.03782 0.518910 0.854829i \(-0.326338\pi\)
0.518910 + 0.854829i \(0.326338\pi\)
\(762\) 8292.27 0.394222
\(763\) −2825.53 −0.134064
\(764\) 19334.6 0.915576
\(765\) 0 0
\(766\) −13218.8 −0.623516
\(767\) −24049.1 −1.13215
\(768\) 11844.5 0.556510
\(769\) 8475.81 0.397459 0.198729 0.980054i \(-0.436319\pi\)
0.198729 + 0.980054i \(0.436319\pi\)
\(770\) 0 0
\(771\) −18007.5 −0.841148
\(772\) −15277.2 −0.712226
\(773\) −39267.1 −1.82709 −0.913544 0.406740i \(-0.866666\pi\)
−0.913544 + 0.406740i \(0.866666\pi\)
\(774\) 3382.24 0.157070
\(775\) 0 0
\(776\) 707.109 0.0327110
\(777\) 1883.39 0.0869578
\(778\) 5379.67 0.247905
\(779\) −7800.09 −0.358751
\(780\) 0 0
\(781\) −22084.2 −1.01182
\(782\) 3617.95 0.165445
\(783\) 3605.09 0.164541
\(784\) 21364.8 0.973250
\(785\) 0 0
\(786\) −17690.2 −0.802784
\(787\) −22732.3 −1.02963 −0.514815 0.857301i \(-0.672139\pi\)
−0.514815 + 0.857301i \(0.672139\pi\)
\(788\) −7786.00 −0.351986
\(789\) 19932.1 0.899370
\(790\) 0 0
\(791\) −2126.41 −0.0955834
\(792\) 118.890 0.00533404
\(793\) −45891.1 −2.05503
\(794\) −3734.14 −0.166901
\(795\) 0 0
\(796\) −5158.58 −0.229700
\(797\) −3379.13 −0.150182 −0.0750909 0.997177i \(-0.523925\pi\)
−0.0750909 + 0.997177i \(0.523925\pi\)
\(798\) −389.455 −0.0172764
\(799\) 4881.85 0.216154
\(800\) 0 0
\(801\) 10154.6 0.447936
\(802\) −27821.0 −1.22493
\(803\) 16065.2 0.706012
\(804\) −15172.8 −0.665554
\(805\) 0 0
\(806\) 51850.5 2.26595
\(807\) 25372.4 1.10675
\(808\) −376.254 −0.0163819
\(809\) −18020.8 −0.783163 −0.391581 0.920143i \(-0.628072\pi\)
−0.391581 + 0.920143i \(0.628072\pi\)
\(810\) 0 0
\(811\) 15100.0 0.653800 0.326900 0.945059i \(-0.393996\pi\)
0.326900 + 0.945059i \(0.393996\pi\)
\(812\) 1911.41 0.0826076
\(813\) −25198.2 −1.08701
\(814\) 33520.0 1.44334
\(815\) 0 0
\(816\) 3939.95 0.169027
\(817\) −1718.84 −0.0736042
\(818\) 7176.59 0.306752
\(819\) −843.215 −0.0359760
\(820\) 0 0
\(821\) 6594.93 0.280347 0.140173 0.990127i \(-0.455234\pi\)
0.140173 + 0.990127i \(0.455234\pi\)
\(822\) −31654.1 −1.34314
\(823\) −2955.28 −0.125170 −0.0625848 0.998040i \(-0.519934\pi\)
−0.0625848 + 0.998040i \(0.519934\pi\)
\(824\) −163.657 −0.00691901
\(825\) 0 0
\(826\) −3188.92 −0.134330
\(827\) −28059.5 −1.17984 −0.589919 0.807463i \(-0.700840\pi\)
−0.589919 + 0.807463i \(0.700840\pi\)
\(828\) 3157.72 0.132534
\(829\) −17823.7 −0.746733 −0.373367 0.927684i \(-0.621797\pi\)
−0.373367 + 0.927684i \(0.621797\pi\)
\(830\) 0 0
\(831\) 4707.28 0.196503
\(832\) 28229.7 1.17631
\(833\) 7102.20 0.295410
\(834\) 38084.2 1.58123
\(835\) 0 0
\(836\) −3495.92 −0.144628
\(837\) 6540.33 0.270092
\(838\) 40520.5 1.67036
\(839\) 10573.5 0.435086 0.217543 0.976051i \(-0.430196\pi\)
0.217543 + 0.976051i \(0.430196\pi\)
\(840\) 0 0
\(841\) −6560.92 −0.269012
\(842\) 18482.9 0.756487
\(843\) −4487.50 −0.183343
\(844\) −6665.72 −0.271852
\(845\) 0 0
\(846\) 8448.04 0.343321
\(847\) −1380.11 −0.0559873
\(848\) 1762.31 0.0713655
\(849\) −17912.2 −0.724083
\(850\) 0 0
\(851\) 15386.8 0.619804
\(852\) 23078.1 0.927982
\(853\) −42034.9 −1.68728 −0.843639 0.536911i \(-0.819591\pi\)
−0.843639 + 0.536911i \(0.819591\pi\)
\(854\) −6085.20 −0.243830
\(855\) 0 0
\(856\) −28.4927 −0.00113769
\(857\) 13137.5 0.523649 0.261824 0.965116i \(-0.415676\pi\)
0.261824 + 0.965116i \(0.415676\pi\)
\(858\) −15007.3 −0.597134
\(859\) 19059.0 0.757023 0.378512 0.925597i \(-0.376436\pi\)
0.378512 + 0.925597i \(0.376436\pi\)
\(860\) 0 0
\(861\) −2239.37 −0.0886382
\(862\) 22851.6 0.902935
\(863\) −15663.3 −0.617827 −0.308914 0.951090i \(-0.599965\pi\)
−0.308914 + 0.951090i \(0.599965\pi\)
\(864\) 6940.18 0.273275
\(865\) 0 0
\(866\) 56661.3 2.22336
\(867\) −13429.3 −0.526046
\(868\) 3467.67 0.135599
\(869\) 13087.8 0.510902
\(870\) 0 0
\(871\) 33100.9 1.28769
\(872\) 908.233 0.0352714
\(873\) −11258.8 −0.436487
\(874\) −3181.75 −0.123140
\(875\) 0 0
\(876\) −16788.2 −0.647511
\(877\) 10218.6 0.393452 0.196726 0.980458i \(-0.436969\pi\)
0.196726 + 0.980458i \(0.436969\pi\)
\(878\) 56064.9 2.15501
\(879\) 3807.30 0.146094
\(880\) 0 0
\(881\) −13828.7 −0.528833 −0.264417 0.964409i \(-0.585179\pi\)
−0.264417 + 0.964409i \(0.585179\pi\)
\(882\) 12290.4 0.469204
\(883\) −41109.3 −1.56675 −0.783374 0.621551i \(-0.786503\pi\)
−0.783374 + 0.621551i \(0.786503\pi\)
\(884\) 9062.52 0.344803
\(885\) 0 0
\(886\) 26803.7 1.01635
\(887\) −34941.8 −1.32270 −0.661348 0.750080i \(-0.730015\pi\)
−0.661348 + 0.750080i \(0.730015\pi\)
\(888\) −605.392 −0.0228780
\(889\) −1209.85 −0.0456435
\(890\) 0 0
\(891\) −1893.00 −0.0711759
\(892\) −26953.8 −1.01175
\(893\) −4293.26 −0.160883
\(894\) −19178.3 −0.717470
\(895\) 0 0
\(896\) 127.210 0.00474305
\(897\) −6888.85 −0.256424
\(898\) 22806.3 0.847501
\(899\) 32343.6 1.19991
\(900\) 0 0
\(901\) 585.837 0.0216615
\(902\) −39855.7 −1.47123
\(903\) −493.471 −0.0181857
\(904\) 683.508 0.0251473
\(905\) 0 0
\(906\) −13863.6 −0.508374
\(907\) 32117.9 1.17581 0.587903 0.808932i \(-0.299954\pi\)
0.587903 + 0.808932i \(0.299954\pi\)
\(908\) −8029.75 −0.293477
\(909\) 5990.82 0.218595
\(910\) 0 0
\(911\) 39364.3 1.43161 0.715806 0.698300i \(-0.246060\pi\)
0.715806 + 0.698300i \(0.246060\pi\)
\(912\) −3464.93 −0.125806
\(913\) 21998.7 0.797426
\(914\) 67394.6 2.43897
\(915\) 0 0
\(916\) −2899.41 −0.104584
\(917\) 2581.01 0.0929472
\(918\) 2266.51 0.0814878
\(919\) −17762.7 −0.637580 −0.318790 0.947825i \(-0.603276\pi\)
−0.318790 + 0.947825i \(0.603276\pi\)
\(920\) 0 0
\(921\) 15655.6 0.560120
\(922\) 5256.02 0.187742
\(923\) −50346.8 −1.79543
\(924\) −1003.66 −0.0357338
\(925\) 0 0
\(926\) 3025.23 0.107360
\(927\) 2605.79 0.0923253
\(928\) 34320.9 1.21405
\(929\) 45096.1 1.59263 0.796315 0.604882i \(-0.206780\pi\)
0.796315 + 0.604882i \(0.206780\pi\)
\(930\) 0 0
\(931\) −6245.92 −0.219873
\(932\) −41484.4 −1.45801
\(933\) 1933.78 0.0678553
\(934\) −64152.8 −2.24748
\(935\) 0 0
\(936\) 271.041 0.00946501
\(937\) 12655.9 0.441249 0.220625 0.975359i \(-0.429190\pi\)
0.220625 + 0.975359i \(0.429190\pi\)
\(938\) 4389.20 0.152785
\(939\) 2768.50 0.0962157
\(940\) 0 0
\(941\) 5410.17 0.187424 0.0937122 0.995599i \(-0.470127\pi\)
0.0937122 + 0.995599i \(0.470127\pi\)
\(942\) 7590.66 0.262545
\(943\) −18295.1 −0.631781
\(944\) −28371.3 −0.978187
\(945\) 0 0
\(946\) −8782.66 −0.301849
\(947\) −25710.8 −0.882250 −0.441125 0.897446i \(-0.645420\pi\)
−0.441125 + 0.897446i \(0.645420\pi\)
\(948\) −13676.8 −0.468568
\(949\) 36624.9 1.25279
\(950\) 0 0
\(951\) −3672.97 −0.125241
\(952\) 20.7687 0.000707056 0
\(953\) −38228.0 −1.29940 −0.649699 0.760192i \(-0.725105\pi\)
−0.649699 + 0.760192i \(0.725105\pi\)
\(954\) 1013.79 0.0344053
\(955\) 0 0
\(956\) −24295.0 −0.821920
\(957\) −9361.35 −0.316206
\(958\) 44134.7 1.48844
\(959\) 4618.35 0.155510
\(960\) 0 0
\(961\) 28886.5 0.969640
\(962\) 76418.0 2.56114
\(963\) 453.669 0.0151810
\(964\) 35554.2 1.18789
\(965\) 0 0
\(966\) −913.466 −0.0304247
\(967\) 20971.8 0.697423 0.348712 0.937230i \(-0.386619\pi\)
0.348712 + 0.937230i \(0.386619\pi\)
\(968\) 443.620 0.0147298
\(969\) −1151.83 −0.0381859
\(970\) 0 0
\(971\) 3841.02 0.126946 0.0634728 0.997984i \(-0.479782\pi\)
0.0634728 + 0.997984i \(0.479782\pi\)
\(972\) 1978.19 0.0652782
\(973\) −5556.52 −0.183077
\(974\) 25270.4 0.831330
\(975\) 0 0
\(976\) −54139.1 −1.77556
\(977\) 17709.1 0.579901 0.289950 0.957042i \(-0.406361\pi\)
0.289950 + 0.957042i \(0.406361\pi\)
\(978\) −7784.42 −0.254518
\(979\) −26368.6 −0.860822
\(980\) 0 0
\(981\) −14461.1 −0.470651
\(982\) 50154.6 1.62983
\(983\) −57068.0 −1.85166 −0.925832 0.377935i \(-0.876634\pi\)
−0.925832 + 0.377935i \(0.876634\pi\)
\(984\) 719.817 0.0233201
\(985\) 0 0
\(986\) 11208.4 0.362018
\(987\) −1232.58 −0.0397501
\(988\) −7969.89 −0.256636
\(989\) −4031.54 −0.129621
\(990\) 0 0
\(991\) −45875.2 −1.47051 −0.735255 0.677791i \(-0.762938\pi\)
−0.735255 + 0.677791i \(0.762938\pi\)
\(992\) 62264.8 1.99285
\(993\) −31929.8 −1.02040
\(994\) −6676.02 −0.213029
\(995\) 0 0
\(996\) −22988.7 −0.731351
\(997\) −37421.7 −1.18872 −0.594362 0.804198i \(-0.702595\pi\)
−0.594362 + 0.804198i \(0.702595\pi\)
\(998\) 28423.7 0.901540
\(999\) 9639.23 0.305277
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.1.3 14
5.4 even 2 1875.4.a.f.1.12 14
25.6 even 5 75.4.g.b.61.6 yes 28
25.21 even 5 75.4.g.b.16.6 28
75.56 odd 10 225.4.h.a.136.2 28
75.71 odd 10 225.4.h.a.91.2 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.4.g.b.16.6 28 25.21 even 5
75.4.g.b.61.6 yes 28 25.6 even 5
225.4.h.a.91.2 28 75.71 odd 10
225.4.h.a.136.2 28 75.56 odd 10
1875.4.a.f.1.12 14 5.4 even 2
1875.4.a.g.1.3 14 1.1 even 1 trivial