Properties

Label 75.4.a.d.1.1
Level $75$
Weight $4$
Character 75.1
Self dual yes
Analytic conductor $4.425$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.42514325043\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.35890\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.35890 q^{2} +3.00000 q^{3} +20.7178 q^{4} -16.0767 q^{6} -4.43560 q^{7} -68.1534 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-5.35890 q^{2} +3.00000 q^{3} +20.7178 q^{4} -16.0767 q^{6} -4.43560 q^{7} -68.1534 q^{8} +9.00000 q^{9} -3.43560 q^{11} +62.1534 q^{12} +78.7424 q^{13} +23.7699 q^{14} +199.485 q^{16} +53.1780 q^{17} -48.2301 q^{18} +20.4356 q^{19} -13.3068 q^{21} +18.4110 q^{22} +118.307 q^{23} -204.460 q^{24} -421.972 q^{26} +27.0000 q^{27} -91.8958 q^{28} +168.049 q^{29} -61.0492 q^{31} -523.792 q^{32} -10.3068 q^{33} -284.975 q^{34} +186.460 q^{36} -246.614 q^{37} -109.512 q^{38} +236.227 q^{39} +422.663 q^{41} +71.3097 q^{42} +362.436 q^{43} -71.1780 q^{44} -633.994 q^{46} -170.515 q^{47} +598.454 q^{48} -323.325 q^{49} +159.534 q^{51} +1631.37 q^{52} -546.049 q^{53} -144.690 q^{54} +302.301 q^{56} +61.3068 q^{57} -900.559 q^{58} -216.970 q^{59} +130.902 q^{61} +327.156 q^{62} -39.9204 q^{63} +1211.07 q^{64} +55.2330 q^{66} -614.890 q^{67} +1101.73 q^{68} +354.920 q^{69} +324.822 q^{71} -613.381 q^{72} -88.8712 q^{73} +1321.58 q^{74} +423.381 q^{76} +15.2389 q^{77} -1265.92 q^{78} -1137.42 q^{79} +81.0000 q^{81} -2265.01 q^{82} -758.909 q^{83} -275.687 q^{84} -1942.26 q^{86} +504.148 q^{87} +234.148 q^{88} +195.681 q^{89} -349.269 q^{91} +2451.06 q^{92} -183.148 q^{93} +913.774 q^{94} -1571.37 q^{96} +521.000 q^{97} +1732.67 q^{98} -30.9204 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 6 q^{3} + 24 q^{4} - 6 q^{6} + 26 q^{7} - 84 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 6 q^{3} + 24 q^{4} - 6 q^{6} + 26 q^{7} - 84 q^{8} + 18 q^{9} + 28 q^{11} + 72 q^{12} + 18 q^{13} + 126 q^{14} + 120 q^{16} - 68 q^{17} - 18 q^{18} + 6 q^{19} + 78 q^{21} + 124 q^{22} + 132 q^{23} - 252 q^{24} - 626 q^{26} + 54 q^{27} + 8 q^{28} + 92 q^{29} + 122 q^{31} - 664 q^{32} + 84 q^{33} - 692 q^{34} + 216 q^{36} - 284 q^{37} - 158 q^{38} + 54 q^{39} + 392 q^{41} + 378 q^{42} + 690 q^{43} + 32 q^{44} - 588 q^{46} - 620 q^{47} + 360 q^{48} + 260 q^{49} - 204 q^{51} + 1432 q^{52} - 848 q^{53} - 54 q^{54} - 180 q^{56} + 18 q^{57} - 1156 q^{58} + 124 q^{59} + 750 q^{61} + 942 q^{62} + 234 q^{63} + 1376 q^{64} + 372 q^{66} - 358 q^{67} + 704 q^{68} + 396 q^{69} + 824 q^{71} - 756 q^{72} - 108 q^{73} + 1196 q^{74} + 376 q^{76} + 972 q^{77} - 1878 q^{78} - 880 q^{79} + 162 q^{81} - 2368 q^{82} + 156 q^{83} + 24 q^{84} - 842 q^{86} + 276 q^{87} - 264 q^{88} - 864 q^{89} - 2198 q^{91} + 2496 q^{92} + 366 q^{93} - 596 q^{94} - 1992 q^{96} + 1042 q^{97} + 3692 q^{98} + 252 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.35890 −1.89466 −0.947328 0.320264i \(-0.896228\pi\)
−0.947328 + 0.320264i \(0.896228\pi\)
\(3\) 3.00000 0.577350
\(4\) 20.7178 2.58972
\(5\) 0 0
\(6\) −16.0767 −1.09388
\(7\) −4.43560 −0.239500 −0.119750 0.992804i \(-0.538209\pi\)
−0.119750 + 0.992804i \(0.538209\pi\)
\(8\) −68.1534 −3.01198
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −3.43560 −0.0941701 −0.0470851 0.998891i \(-0.514993\pi\)
−0.0470851 + 0.998891i \(0.514993\pi\)
\(12\) 62.1534 1.49518
\(13\) 78.7424 1.67994 0.839970 0.542634i \(-0.182573\pi\)
0.839970 + 0.542634i \(0.182573\pi\)
\(14\) 23.7699 0.453770
\(15\) 0 0
\(16\) 199.485 3.11695
\(17\) 53.1780 0.758680 0.379340 0.925257i \(-0.376151\pi\)
0.379340 + 0.925257i \(0.376151\pi\)
\(18\) −48.2301 −0.631552
\(19\) 20.4356 0.246750 0.123375 0.992360i \(-0.460628\pi\)
0.123375 + 0.992360i \(0.460628\pi\)
\(20\) 0 0
\(21\) −13.3068 −0.138275
\(22\) 18.4110 0.178420
\(23\) 118.307 1.07255 0.536275 0.844043i \(-0.319831\pi\)
0.536275 + 0.844043i \(0.319831\pi\)
\(24\) −204.460 −1.73897
\(25\) 0 0
\(26\) −421.972 −3.18291
\(27\) 27.0000 0.192450
\(28\) −91.8958 −0.620238
\(29\) 168.049 1.07607 0.538034 0.842923i \(-0.319167\pi\)
0.538034 + 0.842923i \(0.319167\pi\)
\(30\) 0 0
\(31\) −61.0492 −0.353702 −0.176851 0.984238i \(-0.556591\pi\)
−0.176851 + 0.984238i \(0.556591\pi\)
\(32\) −523.792 −2.89357
\(33\) −10.3068 −0.0543691
\(34\) −284.975 −1.43744
\(35\) 0 0
\(36\) 186.460 0.863242
\(37\) −246.614 −1.09576 −0.547879 0.836558i \(-0.684564\pi\)
−0.547879 + 0.836558i \(0.684564\pi\)
\(38\) −109.512 −0.467506
\(39\) 236.227 0.969913
\(40\) 0 0
\(41\) 422.663 1.60997 0.804986 0.593294i \(-0.202173\pi\)
0.804986 + 0.593294i \(0.202173\pi\)
\(42\) 71.3097 0.261984
\(43\) 362.436 1.28537 0.642685 0.766131i \(-0.277820\pi\)
0.642685 + 0.766131i \(0.277820\pi\)
\(44\) −71.1780 −0.243875
\(45\) 0 0
\(46\) −633.994 −2.03212
\(47\) −170.515 −0.529196 −0.264598 0.964359i \(-0.585239\pi\)
−0.264598 + 0.964359i \(0.585239\pi\)
\(48\) 598.454 1.79957
\(49\) −323.325 −0.942640
\(50\) 0 0
\(51\) 159.534 0.438024
\(52\) 1631.37 4.35058
\(53\) −546.049 −1.41520 −0.707600 0.706613i \(-0.750222\pi\)
−0.707600 + 0.706613i \(0.750222\pi\)
\(54\) −144.690 −0.364627
\(55\) 0 0
\(56\) 302.301 0.721369
\(57\) 61.3068 0.142461
\(58\) −900.559 −2.03878
\(59\) −216.970 −0.478763 −0.239382 0.970926i \(-0.576945\pi\)
−0.239382 + 0.970926i \(0.576945\pi\)
\(60\) 0 0
\(61\) 130.902 0.274758 0.137379 0.990519i \(-0.456132\pi\)
0.137379 + 0.990519i \(0.456132\pi\)
\(62\) 327.156 0.670143
\(63\) −39.9204 −0.0798332
\(64\) 1211.07 2.36537
\(65\) 0 0
\(66\) 55.2330 0.103011
\(67\) −614.890 −1.12121 −0.560603 0.828085i \(-0.689430\pi\)
−0.560603 + 0.828085i \(0.689430\pi\)
\(68\) 1101.73 1.96477
\(69\) 354.920 0.619238
\(70\) 0 0
\(71\) 324.822 0.542948 0.271474 0.962446i \(-0.412489\pi\)
0.271474 + 0.962446i \(0.412489\pi\)
\(72\) −613.381 −1.00399
\(73\) −88.8712 −0.142487 −0.0712437 0.997459i \(-0.522697\pi\)
−0.0712437 + 0.997459i \(0.522697\pi\)
\(74\) 1321.58 2.07608
\(75\) 0 0
\(76\) 423.381 0.639014
\(77\) 15.2389 0.0225537
\(78\) −1265.92 −1.83765
\(79\) −1137.42 −1.61988 −0.809938 0.586516i \(-0.800499\pi\)
−0.809938 + 0.586516i \(0.800499\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −2265.01 −3.05034
\(83\) −758.909 −1.00363 −0.501813 0.864976i \(-0.667334\pi\)
−0.501813 + 0.864976i \(0.667334\pi\)
\(84\) −275.687 −0.358095
\(85\) 0 0
\(86\) −1942.26 −2.43534
\(87\) 504.148 0.621268
\(88\) 234.148 0.283639
\(89\) 195.681 0.233058 0.116529 0.993187i \(-0.462823\pi\)
0.116529 + 0.993187i \(0.462823\pi\)
\(90\) 0 0
\(91\) −349.269 −0.402345
\(92\) 2451.06 2.77761
\(93\) −183.148 −0.204210
\(94\) 913.774 1.00264
\(95\) 0 0
\(96\) −1571.37 −1.67060
\(97\) 521.000 0.545356 0.272678 0.962105i \(-0.412091\pi\)
0.272678 + 0.962105i \(0.412091\pi\)
\(98\) 1732.67 1.78598
\(99\) −30.9204 −0.0313900
\(100\) 0 0
\(101\) 660.920 0.651129 0.325565 0.945520i \(-0.394446\pi\)
0.325565 + 0.945520i \(0.394446\pi\)
\(102\) −854.926 −0.829905
\(103\) 1530.75 1.46436 0.732181 0.681110i \(-0.238503\pi\)
0.732181 + 0.681110i \(0.238503\pi\)
\(104\) −5366.56 −5.05995
\(105\) 0 0
\(106\) 2926.22 2.68132
\(107\) −264.625 −0.239087 −0.119543 0.992829i \(-0.538143\pi\)
−0.119543 + 0.992829i \(0.538143\pi\)
\(108\) 559.381 0.498393
\(109\) 1117.61 0.982091 0.491046 0.871134i \(-0.336615\pi\)
0.491046 + 0.871134i \(0.336615\pi\)
\(110\) 0 0
\(111\) −739.841 −0.632636
\(112\) −884.834 −0.746508
\(113\) −934.061 −0.777602 −0.388801 0.921322i \(-0.627111\pi\)
−0.388801 + 0.921322i \(0.627111\pi\)
\(114\) −328.537 −0.269915
\(115\) 0 0
\(116\) 3481.61 2.78672
\(117\) 708.681 0.559980
\(118\) 1162.72 0.907092
\(119\) −235.876 −0.181704
\(120\) 0 0
\(121\) −1319.20 −0.991132
\(122\) −701.489 −0.520572
\(123\) 1267.99 0.929517
\(124\) −1264.80 −0.915990
\(125\) 0 0
\(126\) 213.929 0.151257
\(127\) 630.356 0.440433 0.220217 0.975451i \(-0.429324\pi\)
0.220217 + 0.975451i \(0.429324\pi\)
\(128\) −2299.66 −1.58799
\(129\) 1087.31 0.742109
\(130\) 0 0
\(131\) −2163.06 −1.44265 −0.721325 0.692597i \(-0.756466\pi\)
−0.721325 + 0.692597i \(0.756466\pi\)
\(132\) −213.534 −0.140801
\(133\) −90.6440 −0.0590965
\(134\) 3295.13 2.12430
\(135\) 0 0
\(136\) −3624.26 −2.28513
\(137\) 1118.61 0.697588 0.348794 0.937199i \(-0.386591\pi\)
0.348794 + 0.937199i \(0.386591\pi\)
\(138\) −1901.98 −1.17324
\(139\) −166.478 −0.101586 −0.0507930 0.998709i \(-0.516175\pi\)
−0.0507930 + 0.998709i \(0.516175\pi\)
\(140\) 0 0
\(141\) −511.546 −0.305531
\(142\) −1740.69 −1.02870
\(143\) −270.527 −0.158200
\(144\) 1795.36 1.03898
\(145\) 0 0
\(146\) 476.252 0.269965
\(147\) −969.976 −0.544233
\(148\) −5109.29 −2.83771
\(149\) −653.143 −0.359111 −0.179555 0.983748i \(-0.557466\pi\)
−0.179555 + 0.983748i \(0.557466\pi\)
\(150\) 0 0
\(151\) −1929.38 −1.03981 −0.519903 0.854225i \(-0.674032\pi\)
−0.519903 + 0.854225i \(0.674032\pi\)
\(152\) −1392.76 −0.743206
\(153\) 478.602 0.252893
\(154\) −81.6638 −0.0427315
\(155\) 0 0
\(156\) 4894.11 2.51181
\(157\) 2169.75 1.10296 0.551480 0.834188i \(-0.314063\pi\)
0.551480 + 0.834188i \(0.314063\pi\)
\(158\) 6095.34 3.06911
\(159\) −1638.15 −0.817066
\(160\) 0 0
\(161\) −524.761 −0.256876
\(162\) −434.071 −0.210517
\(163\) −763.738 −0.366997 −0.183499 0.983020i \(-0.558742\pi\)
−0.183499 + 0.983020i \(0.558742\pi\)
\(164\) 8756.64 4.16938
\(165\) 0 0
\(166\) 4066.91 1.90153
\(167\) −2564.28 −1.18820 −0.594102 0.804389i \(-0.702493\pi\)
−0.594102 + 0.804389i \(0.702493\pi\)
\(168\) 906.903 0.416483
\(169\) 4003.36 1.82220
\(170\) 0 0
\(171\) 183.920 0.0822500
\(172\) 7508.87 3.32875
\(173\) 51.8290 0.0227774 0.0113887 0.999935i \(-0.496375\pi\)
0.0113887 + 0.999935i \(0.496375\pi\)
\(174\) −2701.68 −1.17709
\(175\) 0 0
\(176\) −685.349 −0.293523
\(177\) −650.909 −0.276414
\(178\) −1048.64 −0.441566
\(179\) −3956.63 −1.65214 −0.826068 0.563571i \(-0.809427\pi\)
−0.826068 + 0.563571i \(0.809427\pi\)
\(180\) 0 0
\(181\) 1804.04 0.740848 0.370424 0.928863i \(-0.379212\pi\)
0.370424 + 0.928863i \(0.379212\pi\)
\(182\) 1871.70 0.762305
\(183\) 392.705 0.158632
\(184\) −8063.01 −3.23050
\(185\) 0 0
\(186\) 981.469 0.386908
\(187\) −182.698 −0.0714449
\(188\) −3532.70 −1.37047
\(189\) −119.761 −0.0460917
\(190\) 0 0
\(191\) 3666.75 1.38909 0.694547 0.719448i \(-0.255605\pi\)
0.694547 + 0.719448i \(0.255605\pi\)
\(192\) 3633.20 1.36565
\(193\) −2716.98 −1.01333 −0.506664 0.862144i \(-0.669121\pi\)
−0.506664 + 0.862144i \(0.669121\pi\)
\(194\) −2791.99 −1.03326
\(195\) 0 0
\(196\) −6698.59 −2.44118
\(197\) 2034.30 0.735723 0.367862 0.929881i \(-0.380090\pi\)
0.367862 + 0.929881i \(0.380090\pi\)
\(198\) 165.699 0.0594733
\(199\) −1551.27 −0.552596 −0.276298 0.961072i \(-0.589108\pi\)
−0.276298 + 0.961072i \(0.589108\pi\)
\(200\) 0 0
\(201\) −1844.67 −0.647328
\(202\) −3541.81 −1.23367
\(203\) −745.398 −0.257718
\(204\) 3305.19 1.13436
\(205\) 0 0
\(206\) −8203.13 −2.77446
\(207\) 1064.76 0.357517
\(208\) 15707.9 5.23629
\(209\) −70.2084 −0.0232365
\(210\) 0 0
\(211\) 3192.51 1.04162 0.520809 0.853673i \(-0.325630\pi\)
0.520809 + 0.853673i \(0.325630\pi\)
\(212\) −11312.9 −3.66498
\(213\) 974.466 0.313471
\(214\) 1418.10 0.452988
\(215\) 0 0
\(216\) −1840.14 −0.579656
\(217\) 270.789 0.0847115
\(218\) −5989.18 −1.86073
\(219\) −266.614 −0.0822652
\(220\) 0 0
\(221\) 4187.36 1.27454
\(222\) 3964.73 1.19863
\(223\) −1555.55 −0.467120 −0.233560 0.972342i \(-0.575037\pi\)
−0.233560 + 0.972342i \(0.575037\pi\)
\(224\) 2323.33 0.693008
\(225\) 0 0
\(226\) 5005.54 1.47329
\(227\) −6206.86 −1.81482 −0.907409 0.420248i \(-0.861943\pi\)
−0.907409 + 0.420248i \(0.861943\pi\)
\(228\) 1270.14 0.368935
\(229\) 4679.51 1.35035 0.675176 0.737657i \(-0.264068\pi\)
0.675176 + 0.737657i \(0.264068\pi\)
\(230\) 0 0
\(231\) 45.7167 0.0130214
\(232\) −11453.1 −3.24110
\(233\) −3244.53 −0.912259 −0.456129 0.889913i \(-0.650765\pi\)
−0.456129 + 0.889913i \(0.650765\pi\)
\(234\) −3797.75 −1.06097
\(235\) 0 0
\(236\) −4495.13 −1.23986
\(237\) −3412.27 −0.935236
\(238\) 1264.04 0.344266
\(239\) −3658.62 −0.990193 −0.495097 0.868838i \(-0.664867\pi\)
−0.495097 + 0.868838i \(0.664867\pi\)
\(240\) 0 0
\(241\) 1931.01 0.516131 0.258065 0.966127i \(-0.416915\pi\)
0.258065 + 0.966127i \(0.416915\pi\)
\(242\) 7069.44 1.87786
\(243\) 243.000 0.0641500
\(244\) 2711.99 0.711548
\(245\) 0 0
\(246\) −6795.02 −1.76112
\(247\) 1609.15 0.414525
\(248\) 4160.71 1.06534
\(249\) −2276.73 −0.579444
\(250\) 0 0
\(251\) −5843.34 −1.46944 −0.734718 0.678373i \(-0.762685\pi\)
−0.734718 + 0.678373i \(0.762685\pi\)
\(252\) −827.062 −0.206746
\(253\) −406.454 −0.101002
\(254\) −3378.01 −0.834470
\(255\) 0 0
\(256\) 2635.09 0.643333
\(257\) 4506.11 1.09371 0.546855 0.837227i \(-0.315825\pi\)
0.546855 + 0.837227i \(0.315825\pi\)
\(258\) −5826.77 −1.40604
\(259\) 1093.88 0.262434
\(260\) 0 0
\(261\) 1512.44 0.358689
\(262\) 11591.6 2.73333
\(263\) 5340.16 1.25205 0.626024 0.779804i \(-0.284681\pi\)
0.626024 + 0.779804i \(0.284681\pi\)
\(264\) 702.443 0.163759
\(265\) 0 0
\(266\) 485.752 0.111968
\(267\) 587.044 0.134556
\(268\) −12739.2 −2.90361
\(269\) 2809.79 0.636863 0.318431 0.947946i \(-0.396844\pi\)
0.318431 + 0.947946i \(0.396844\pi\)
\(270\) 0 0
\(271\) 3102.95 0.695537 0.347769 0.937580i \(-0.386940\pi\)
0.347769 + 0.937580i \(0.386940\pi\)
\(272\) 10608.2 2.36477
\(273\) −1047.81 −0.232294
\(274\) −5994.54 −1.32169
\(275\) 0 0
\(276\) 7353.17 1.60365
\(277\) −4598.93 −0.997555 −0.498777 0.866730i \(-0.666217\pi\)
−0.498777 + 0.866730i \(0.666217\pi\)
\(278\) 892.138 0.192471
\(279\) −549.443 −0.117901
\(280\) 0 0
\(281\) 2571.83 0.545987 0.272994 0.962016i \(-0.411986\pi\)
0.272994 + 0.962016i \(0.411986\pi\)
\(282\) 2741.32 0.578877
\(283\) 5575.31 1.17109 0.585544 0.810641i \(-0.300881\pi\)
0.585544 + 0.810641i \(0.300881\pi\)
\(284\) 6729.60 1.40608
\(285\) 0 0
\(286\) 1449.73 0.299735
\(287\) −1874.76 −0.385588
\(288\) −4714.12 −0.964522
\(289\) −2085.10 −0.424405
\(290\) 0 0
\(291\) 1563.00 0.314861
\(292\) −1841.22 −0.369003
\(293\) 5794.27 1.15531 0.577654 0.816282i \(-0.303968\pi\)
0.577654 + 0.816282i \(0.303968\pi\)
\(294\) 5198.01 1.03114
\(295\) 0 0
\(296\) 16807.6 3.30040
\(297\) −92.7611 −0.0181230
\(298\) 3500.13 0.680392
\(299\) 9315.76 1.80182
\(300\) 0 0
\(301\) −1607.62 −0.307846
\(302\) 10339.4 1.97008
\(303\) 1982.76 0.375930
\(304\) 4076.59 0.769107
\(305\) 0 0
\(306\) −2564.78 −0.479146
\(307\) −1404.47 −0.261099 −0.130550 0.991442i \(-0.541674\pi\)
−0.130550 + 0.991442i \(0.541674\pi\)
\(308\) 315.717 0.0584079
\(309\) 4592.25 0.845449
\(310\) 0 0
\(311\) −4096.75 −0.746963 −0.373481 0.927638i \(-0.621836\pi\)
−0.373481 + 0.927638i \(0.621836\pi\)
\(312\) −16099.7 −2.92136
\(313\) 974.611 0.176001 0.0880004 0.996120i \(-0.471952\pi\)
0.0880004 + 0.996120i \(0.471952\pi\)
\(314\) −11627.5 −2.08973
\(315\) 0 0
\(316\) −23564.9 −4.19503
\(317\) −2071.69 −0.367058 −0.183529 0.983014i \(-0.558752\pi\)
−0.183529 + 0.983014i \(0.558752\pi\)
\(318\) 8778.67 1.54806
\(319\) −577.349 −0.101333
\(320\) 0 0
\(321\) −793.876 −0.138037
\(322\) 2812.14 0.486691
\(323\) 1086.72 0.187204
\(324\) 1678.14 0.287747
\(325\) 0 0
\(326\) 4092.79 0.695334
\(327\) 3352.84 0.567011
\(328\) −28805.9 −4.84921
\(329\) 756.337 0.126742
\(330\) 0 0
\(331\) −6159.17 −1.02278 −0.511388 0.859350i \(-0.670868\pi\)
−0.511388 + 0.859350i \(0.670868\pi\)
\(332\) −15722.9 −2.59912
\(333\) −2219.52 −0.365252
\(334\) 13741.7 2.25124
\(335\) 0 0
\(336\) −2654.50 −0.430997
\(337\) 2791.26 0.451186 0.225593 0.974222i \(-0.427568\pi\)
0.225593 + 0.974222i \(0.427568\pi\)
\(338\) −21453.6 −3.45243
\(339\) −2802.18 −0.448949
\(340\) 0 0
\(341\) 209.740 0.0333081
\(342\) −985.611 −0.155835
\(343\) 2955.55 0.465262
\(344\) −24701.2 −3.87151
\(345\) 0 0
\(346\) −277.746 −0.0431553
\(347\) 940.848 0.145554 0.0727772 0.997348i \(-0.476814\pi\)
0.0727772 + 0.997348i \(0.476814\pi\)
\(348\) 10444.8 1.60891
\(349\) −3519.62 −0.539831 −0.269915 0.962884i \(-0.586996\pi\)
−0.269915 + 0.962884i \(0.586996\pi\)
\(350\) 0 0
\(351\) 2126.04 0.323304
\(352\) 1799.54 0.272487
\(353\) 5021.60 0.757147 0.378573 0.925571i \(-0.376415\pi\)
0.378573 + 0.925571i \(0.376415\pi\)
\(354\) 3488.15 0.523710
\(355\) 0 0
\(356\) 4054.09 0.603557
\(357\) −707.628 −0.104907
\(358\) 21203.2 3.13023
\(359\) 6811.99 1.00146 0.500728 0.865604i \(-0.333066\pi\)
0.500728 + 0.865604i \(0.333066\pi\)
\(360\) 0 0
\(361\) −6441.39 −0.939115
\(362\) −9667.69 −1.40365
\(363\) −3957.59 −0.572230
\(364\) −7236.09 −1.04196
\(365\) 0 0
\(366\) −2104.47 −0.300553
\(367\) 3748.07 0.533099 0.266550 0.963821i \(-0.414116\pi\)
0.266550 + 0.963821i \(0.414116\pi\)
\(368\) 23600.4 3.34309
\(369\) 3803.96 0.536657
\(370\) 0 0
\(371\) 2422.05 0.338940
\(372\) −3794.41 −0.528847
\(373\) −898.302 −0.124698 −0.0623489 0.998054i \(-0.519859\pi\)
−0.0623489 + 0.998054i \(0.519859\pi\)
\(374\) 979.060 0.135364
\(375\) 0 0
\(376\) 11621.2 1.59393
\(377\) 13232.6 1.80773
\(378\) 641.788 0.0873280
\(379\) −9378.99 −1.27115 −0.635576 0.772038i \(-0.719237\pi\)
−0.635576 + 0.772038i \(0.719237\pi\)
\(380\) 0 0
\(381\) 1891.07 0.254284
\(382\) −19649.8 −2.63186
\(383\) −9446.29 −1.26027 −0.630134 0.776486i \(-0.717000\pi\)
−0.630134 + 0.776486i \(0.717000\pi\)
\(384\) −6898.97 −0.916828
\(385\) 0 0
\(386\) 14560.0 1.91991
\(387\) 3261.92 0.428457
\(388\) 10794.0 1.41232
\(389\) −7643.23 −0.996214 −0.498107 0.867116i \(-0.665971\pi\)
−0.498107 + 0.867116i \(0.665971\pi\)
\(390\) 0 0
\(391\) 6291.32 0.813723
\(392\) 22035.7 2.83922
\(393\) −6489.17 −0.832914
\(394\) −10901.6 −1.39394
\(395\) 0 0
\(396\) −640.602 −0.0812915
\(397\) −12013.6 −1.51876 −0.759378 0.650650i \(-0.774497\pi\)
−0.759378 + 0.650650i \(0.774497\pi\)
\(398\) 8313.10 1.04698
\(399\) −271.932 −0.0341194
\(400\) 0 0
\(401\) −8538.51 −1.06332 −0.531662 0.846957i \(-0.678432\pi\)
−0.531662 + 0.846957i \(0.678432\pi\)
\(402\) 9885.40 1.22646
\(403\) −4807.16 −0.594197
\(404\) 13692.8 1.68625
\(405\) 0 0
\(406\) 3994.51 0.488287
\(407\) 847.265 0.103188
\(408\) −10872.8 −1.31932
\(409\) −12267.6 −1.48312 −0.741558 0.670889i \(-0.765913\pi\)
−0.741558 + 0.670889i \(0.765913\pi\)
\(410\) 0 0
\(411\) 3355.84 0.402753
\(412\) 31713.8 3.79229
\(413\) 962.389 0.114664
\(414\) −5705.95 −0.677372
\(415\) 0 0
\(416\) −41244.6 −4.86102
\(417\) −499.433 −0.0586508
\(418\) 376.240 0.0440251
\(419\) 15493.0 1.80641 0.903204 0.429212i \(-0.141209\pi\)
0.903204 + 0.429212i \(0.141209\pi\)
\(420\) 0 0
\(421\) 7510.67 0.869472 0.434736 0.900558i \(-0.356842\pi\)
0.434736 + 0.900558i \(0.356842\pi\)
\(422\) −17108.3 −1.97351
\(423\) −1534.64 −0.176399
\(424\) 37215.1 4.26256
\(425\) 0 0
\(426\) −5222.07 −0.593920
\(427\) −580.627 −0.0658045
\(428\) −5482.45 −0.619169
\(429\) −811.581 −0.0913368
\(430\) 0 0
\(431\) −15675.3 −1.75186 −0.875932 0.482434i \(-0.839753\pi\)
−0.875932 + 0.482434i \(0.839753\pi\)
\(432\) 5386.09 0.599857
\(433\) −9604.78 −1.06600 −0.532998 0.846117i \(-0.678935\pi\)
−0.532998 + 0.846117i \(0.678935\pi\)
\(434\) −1451.13 −0.160499
\(435\) 0 0
\(436\) 23154.5 2.54335
\(437\) 2417.67 0.264652
\(438\) 1428.76 0.155864
\(439\) −6362.06 −0.691673 −0.345837 0.938295i \(-0.612405\pi\)
−0.345837 + 0.938295i \(0.612405\pi\)
\(440\) 0 0
\(441\) −2909.93 −0.314213
\(442\) −22439.6 −2.41481
\(443\) 931.658 0.0999196 0.0499598 0.998751i \(-0.484091\pi\)
0.0499598 + 0.998751i \(0.484091\pi\)
\(444\) −15327.9 −1.63835
\(445\) 0 0
\(446\) 8336.06 0.885031
\(447\) −1959.43 −0.207333
\(448\) −5371.81 −0.566505
\(449\) −18684.1 −1.96383 −0.981914 0.189329i \(-0.939369\pi\)
−0.981914 + 0.189329i \(0.939369\pi\)
\(450\) 0 0
\(451\) −1452.10 −0.151611
\(452\) −19351.7 −2.01378
\(453\) −5788.14 −0.600333
\(454\) 33261.9 3.43846
\(455\) 0 0
\(456\) −4178.27 −0.429090
\(457\) −11565.9 −1.18387 −0.591936 0.805985i \(-0.701636\pi\)
−0.591936 + 0.805985i \(0.701636\pi\)
\(458\) −25077.0 −2.55845
\(459\) 1435.81 0.146008
\(460\) 0 0
\(461\) 19401.0 1.96008 0.980039 0.198806i \(-0.0637062\pi\)
0.980039 + 0.198806i \(0.0637062\pi\)
\(462\) −244.991 −0.0246711
\(463\) 1576.28 0.158220 0.0791099 0.996866i \(-0.474792\pi\)
0.0791099 + 0.996866i \(0.474792\pi\)
\(464\) 33523.2 3.35405
\(465\) 0 0
\(466\) 17387.1 1.72842
\(467\) 3256.55 0.322687 0.161344 0.986898i \(-0.448417\pi\)
0.161344 + 0.986898i \(0.448417\pi\)
\(468\) 14682.3 1.45019
\(469\) 2727.40 0.268528
\(470\) 0 0
\(471\) 6509.25 0.636795
\(472\) 14787.2 1.44203
\(473\) −1245.18 −0.121043
\(474\) 18286.0 1.77195
\(475\) 0 0
\(476\) −4886.83 −0.470562
\(477\) −4914.44 −0.471733
\(478\) 19606.2 1.87608
\(479\) 8291.59 0.790924 0.395462 0.918482i \(-0.370585\pi\)
0.395462 + 0.918482i \(0.370585\pi\)
\(480\) 0 0
\(481\) −19418.9 −1.84081
\(482\) −10348.1 −0.977891
\(483\) −1574.28 −0.148307
\(484\) −27330.9 −2.56676
\(485\) 0 0
\(486\) −1302.21 −0.121542
\(487\) 4758.55 0.442773 0.221387 0.975186i \(-0.428942\pi\)
0.221387 + 0.975186i \(0.428942\pi\)
\(488\) −8921.39 −0.827567
\(489\) −2291.21 −0.211886
\(490\) 0 0
\(491\) 3906.46 0.359055 0.179528 0.983753i \(-0.442543\pi\)
0.179528 + 0.983753i \(0.442543\pi\)
\(492\) 26269.9 2.40719
\(493\) 8936.52 0.816390
\(494\) −8623.26 −0.785382
\(495\) 0 0
\(496\) −12178.4 −1.10247
\(497\) −1440.78 −0.130036
\(498\) 12200.7 1.09785
\(499\) −3093.31 −0.277506 −0.138753 0.990327i \(-0.544309\pi\)
−0.138753 + 0.990327i \(0.544309\pi\)
\(500\) 0 0
\(501\) −7692.85 −0.686010
\(502\) 31313.9 2.78408
\(503\) −18153.9 −1.60923 −0.804616 0.593796i \(-0.797629\pi\)
−0.804616 + 0.593796i \(0.797629\pi\)
\(504\) 2720.71 0.240456
\(505\) 0 0
\(506\) 2178.15 0.191365
\(507\) 12010.1 1.05204
\(508\) 13059.6 1.14060
\(509\) 2281.32 0.198660 0.0993298 0.995055i \(-0.468330\pi\)
0.0993298 + 0.995055i \(0.468330\pi\)
\(510\) 0 0
\(511\) 394.197 0.0341257
\(512\) 4276.07 0.369097
\(513\) 551.761 0.0474870
\(514\) −24147.8 −2.07221
\(515\) 0 0
\(516\) 22526.6 1.92186
\(517\) 585.821 0.0498344
\(518\) −5861.98 −0.497221
\(519\) 155.487 0.0131505
\(520\) 0 0
\(521\) −16691.9 −1.40362 −0.701809 0.712366i \(-0.747624\pi\)
−0.701809 + 0.712366i \(0.747624\pi\)
\(522\) −8105.03 −0.679593
\(523\) −17090.4 −1.42889 −0.714446 0.699690i \(-0.753321\pi\)
−0.714446 + 0.699690i \(0.753321\pi\)
\(524\) −44813.8 −3.73607
\(525\) 0 0
\(526\) −28617.4 −2.37220
\(527\) −3246.47 −0.268346
\(528\) −2056.05 −0.169466
\(529\) 1829.50 0.150365
\(530\) 0 0
\(531\) −1952.73 −0.159588
\(532\) −1877.94 −0.153044
\(533\) 33281.5 2.70465
\(534\) −3145.91 −0.254938
\(535\) 0 0
\(536\) 41906.8 3.37705
\(537\) −11869.9 −0.953861
\(538\) −15057.4 −1.20664
\(539\) 1110.82 0.0887685
\(540\) 0 0
\(541\) 5271.40 0.418919 0.209459 0.977817i \(-0.432830\pi\)
0.209459 + 0.977817i \(0.432830\pi\)
\(542\) −16628.4 −1.31780
\(543\) 5412.13 0.427729
\(544\) −27854.2 −2.19529
\(545\) 0 0
\(546\) 5615.10 0.440117
\(547\) 15182.2 1.18673 0.593366 0.804933i \(-0.297799\pi\)
0.593366 + 0.804933i \(0.297799\pi\)
\(548\) 23175.2 1.80656
\(549\) 1178.11 0.0915860
\(550\) 0 0
\(551\) 3434.18 0.265519
\(552\) −24189.0 −1.86513
\(553\) 5045.15 0.387960
\(554\) 24645.2 1.89002
\(555\) 0 0
\(556\) −3449.05 −0.263080
\(557\) 12241.2 0.931198 0.465599 0.884996i \(-0.345839\pi\)
0.465599 + 0.884996i \(0.345839\pi\)
\(558\) 2944.41 0.223381
\(559\) 28539.0 2.15934
\(560\) 0 0
\(561\) −548.094 −0.0412488
\(562\) −13782.2 −1.03446
\(563\) 14196.4 1.06271 0.531355 0.847149i \(-0.321683\pi\)
0.531355 + 0.847149i \(0.321683\pi\)
\(564\) −10598.1 −0.791242
\(565\) 0 0
\(566\) −29877.5 −2.21881
\(567\) −359.283 −0.0266111
\(568\) −22137.7 −1.63535
\(569\) 9150.05 0.674148 0.337074 0.941478i \(-0.390563\pi\)
0.337074 + 0.941478i \(0.390563\pi\)
\(570\) 0 0
\(571\) 23582.1 1.72833 0.864167 0.503206i \(-0.167846\pi\)
0.864167 + 0.503206i \(0.167846\pi\)
\(572\) −5604.72 −0.409695
\(573\) 11000.3 0.801993
\(574\) 10046.7 0.730556
\(575\) 0 0
\(576\) 10899.6 0.788456
\(577\) 3906.22 0.281834 0.140917 0.990021i \(-0.454995\pi\)
0.140917 + 0.990021i \(0.454995\pi\)
\(578\) 11173.9 0.804102
\(579\) −8150.93 −0.585045
\(580\) 0 0
\(581\) 3366.21 0.240368
\(582\) −8375.96 −0.596554
\(583\) 1876.00 0.133270
\(584\) 6056.87 0.429170
\(585\) 0 0
\(586\) −31050.9 −2.18891
\(587\) −25938.0 −1.82381 −0.911905 0.410401i \(-0.865389\pi\)
−0.911905 + 0.410401i \(0.865389\pi\)
\(588\) −20095.8 −1.40941
\(589\) −1247.58 −0.0872759
\(590\) 0 0
\(591\) 6102.89 0.424770
\(592\) −49195.7 −3.41542
\(593\) 1908.23 0.132145 0.0660723 0.997815i \(-0.478953\pi\)
0.0660723 + 0.997815i \(0.478953\pi\)
\(594\) 497.097 0.0343370
\(595\) 0 0
\(596\) −13531.7 −0.929999
\(597\) −4653.81 −0.319041
\(598\) −49922.2 −3.41383
\(599\) −3495.41 −0.238429 −0.119214 0.992869i \(-0.538038\pi\)
−0.119214 + 0.992869i \(0.538038\pi\)
\(600\) 0 0
\(601\) −18267.2 −1.23983 −0.619913 0.784671i \(-0.712832\pi\)
−0.619913 + 0.784671i \(0.712832\pi\)
\(602\) 8615.06 0.583262
\(603\) −5534.01 −0.373735
\(604\) −39972.5 −2.69281
\(605\) 0 0
\(606\) −10625.4 −0.712257
\(607\) 11538.2 0.771534 0.385767 0.922596i \(-0.373937\pi\)
0.385767 + 0.922596i \(0.373937\pi\)
\(608\) −10704.0 −0.713987
\(609\) −2236.19 −0.148793
\(610\) 0 0
\(611\) −13426.8 −0.889017
\(612\) 9915.58 0.654924
\(613\) 21136.9 1.39268 0.696340 0.717713i \(-0.254811\pi\)
0.696340 + 0.717713i \(0.254811\pi\)
\(614\) 7526.43 0.494694
\(615\) 0 0
\(616\) −1038.58 −0.0679314
\(617\) 15673.2 1.02266 0.511329 0.859385i \(-0.329153\pi\)
0.511329 + 0.859385i \(0.329153\pi\)
\(618\) −24609.4 −1.60184
\(619\) 22923.7 1.48850 0.744249 0.667902i \(-0.232807\pi\)
0.744249 + 0.667902i \(0.232807\pi\)
\(620\) 0 0
\(621\) 3194.28 0.206413
\(622\) 21954.1 1.41524
\(623\) −867.964 −0.0558174
\(624\) 47123.7 3.02317
\(625\) 0 0
\(626\) −5222.84 −0.333461
\(627\) −210.625 −0.0134156
\(628\) 44952.4 2.85636
\(629\) −13114.4 −0.831329
\(630\) 0 0
\(631\) 9108.23 0.574632 0.287316 0.957836i \(-0.407237\pi\)
0.287316 + 0.957836i \(0.407237\pi\)
\(632\) 77519.3 4.87904
\(633\) 9577.53 0.601379
\(634\) 11102.0 0.695449
\(635\) 0 0
\(636\) −33938.8 −2.11598
\(637\) −25459.4 −1.58358
\(638\) 3093.96 0.191992
\(639\) 2923.40 0.180983
\(640\) 0 0
\(641\) 20103.5 1.23875 0.619375 0.785095i \(-0.287386\pi\)
0.619375 + 0.785095i \(0.287386\pi\)
\(642\) 4254.30 0.261533
\(643\) 5934.92 0.363997 0.181999 0.983299i \(-0.441743\pi\)
0.181999 + 0.983299i \(0.441743\pi\)
\(644\) −10871.9 −0.665237
\(645\) 0 0
\(646\) −5823.64 −0.354688
\(647\) −14193.7 −0.862460 −0.431230 0.902242i \(-0.641920\pi\)
−0.431230 + 0.902242i \(0.641920\pi\)
\(648\) −5520.42 −0.334665
\(649\) 745.420 0.0450852
\(650\) 0 0
\(651\) 812.368 0.0489082
\(652\) −15823.0 −0.950422
\(653\) −4795.80 −0.287403 −0.143701 0.989621i \(-0.545900\pi\)
−0.143701 + 0.989621i \(0.545900\pi\)
\(654\) −17967.5 −1.07429
\(655\) 0 0
\(656\) 84314.8 5.01820
\(657\) −799.841 −0.0474958
\(658\) −4053.13 −0.240133
\(659\) 4399.57 0.260065 0.130032 0.991510i \(-0.458492\pi\)
0.130032 + 0.991510i \(0.458492\pi\)
\(660\) 0 0
\(661\) 24096.0 1.41789 0.708945 0.705263i \(-0.249171\pi\)
0.708945 + 0.705263i \(0.249171\pi\)
\(662\) 33006.4 1.93781
\(663\) 12562.1 0.735853
\(664\) 51722.2 3.02291
\(665\) 0 0
\(666\) 11894.2 0.692028
\(667\) 19881.4 1.15414
\(668\) −53126.3 −3.07712
\(669\) −4666.66 −0.269692
\(670\) 0 0
\(671\) −449.725 −0.0258740
\(672\) 6969.98 0.400109
\(673\) −27648.3 −1.58360 −0.791800 0.610781i \(-0.790856\pi\)
−0.791800 + 0.610781i \(0.790856\pi\)
\(674\) −14958.1 −0.854843
\(675\) 0 0
\(676\) 82940.9 4.71898
\(677\) −27605.5 −1.56716 −0.783580 0.621292i \(-0.786608\pi\)
−0.783580 + 0.621292i \(0.786608\pi\)
\(678\) 15016.6 0.850604
\(679\) −2310.95 −0.130613
\(680\) 0 0
\(681\) −18620.6 −1.04779
\(682\) −1123.98 −0.0631075
\(683\) −14949.4 −0.837513 −0.418756 0.908099i \(-0.637534\pi\)
−0.418756 + 0.908099i \(0.637534\pi\)
\(684\) 3810.42 0.213005
\(685\) 0 0
\(686\) −15838.5 −0.881511
\(687\) 14038.5 0.779626
\(688\) 72300.4 4.00643
\(689\) −42997.2 −2.37745
\(690\) 0 0
\(691\) 8884.30 0.489110 0.244555 0.969635i \(-0.421358\pi\)
0.244555 + 0.969635i \(0.421358\pi\)
\(692\) 1073.78 0.0589871
\(693\) 137.150 0.00751790
\(694\) −5041.91 −0.275775
\(695\) 0 0
\(696\) −34359.4 −1.87125
\(697\) 22476.4 1.22145
\(698\) 18861.3 1.02279
\(699\) −9733.59 −0.526693
\(700\) 0 0
\(701\) 10556.9 0.568798 0.284399 0.958706i \(-0.408206\pi\)
0.284399 + 0.958706i \(0.408206\pi\)
\(702\) −11393.3 −0.612551
\(703\) −5039.70 −0.270378
\(704\) −4160.74 −0.222747
\(705\) 0 0
\(706\) −26910.2 −1.43453
\(707\) −2931.58 −0.155945
\(708\) −13485.4 −0.715836
\(709\) 25351.9 1.34289 0.671445 0.741055i \(-0.265674\pi\)
0.671445 + 0.741055i \(0.265674\pi\)
\(710\) 0 0
\(711\) −10236.8 −0.539959
\(712\) −13336.4 −0.701968
\(713\) −7222.53 −0.379363
\(714\) 3792.11 0.198762
\(715\) 0 0
\(716\) −81972.6 −4.27858
\(717\) −10975.8 −0.571688
\(718\) −36504.8 −1.89742
\(719\) 9719.94 0.504162 0.252081 0.967706i \(-0.418885\pi\)
0.252081 + 0.967706i \(0.418885\pi\)
\(720\) 0 0
\(721\) −6789.79 −0.350714
\(722\) 34518.7 1.77930
\(723\) 5793.04 0.297988
\(724\) 37375.8 1.91859
\(725\) 0 0
\(726\) 21208.3 1.08418
\(727\) −27509.3 −1.40339 −0.701694 0.712479i \(-0.747572\pi\)
−0.701694 + 0.712479i \(0.747572\pi\)
\(728\) 23803.9 1.21186
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 19273.6 0.975184
\(732\) 8135.98 0.410812
\(733\) −7240.49 −0.364848 −0.182424 0.983220i \(-0.558394\pi\)
−0.182424 + 0.983220i \(0.558394\pi\)
\(734\) −20085.5 −1.01004
\(735\) 0 0
\(736\) −61968.1 −3.10350
\(737\) 2112.51 0.105584
\(738\) −20385.1 −1.01678
\(739\) −15875.3 −0.790234 −0.395117 0.918631i \(-0.629296\pi\)
−0.395117 + 0.918631i \(0.629296\pi\)
\(740\) 0 0
\(741\) 4827.44 0.239326
\(742\) −12979.5 −0.642175
\(743\) −25714.3 −1.26967 −0.634836 0.772647i \(-0.718932\pi\)
−0.634836 + 0.772647i \(0.718932\pi\)
\(744\) 12482.1 0.615076
\(745\) 0 0
\(746\) 4813.91 0.236260
\(747\) −6830.18 −0.334542
\(748\) −3785.10 −0.185023
\(749\) 1173.77 0.0572612
\(750\) 0 0
\(751\) −9709.09 −0.471757 −0.235879 0.971783i \(-0.575797\pi\)
−0.235879 + 0.971783i \(0.575797\pi\)
\(752\) −34015.2 −1.64948
\(753\) −17530.0 −0.848379
\(754\) −70912.1 −3.42502
\(755\) 0 0
\(756\) −2481.19 −0.119365
\(757\) −9567.13 −0.459344 −0.229672 0.973268i \(-0.573765\pi\)
−0.229672 + 0.973268i \(0.573765\pi\)
\(758\) 50261.1 2.40840
\(759\) −1219.36 −0.0583137
\(760\) 0 0
\(761\) −12322.5 −0.586980 −0.293490 0.955962i \(-0.594817\pi\)
−0.293490 + 0.955962i \(0.594817\pi\)
\(762\) −10134.0 −0.481782
\(763\) −4957.28 −0.235211
\(764\) 75967.0 3.59737
\(765\) 0 0
\(766\) 50621.7 2.38778
\(767\) −17084.7 −0.804293
\(768\) 7905.27 0.371428
\(769\) 2575.56 0.120776 0.0603881 0.998175i \(-0.480766\pi\)
0.0603881 + 0.998175i \(0.480766\pi\)
\(770\) 0 0
\(771\) 13518.3 0.631454
\(772\) −56289.8 −2.62424
\(773\) −6606.23 −0.307386 −0.153693 0.988119i \(-0.549117\pi\)
−0.153693 + 0.988119i \(0.549117\pi\)
\(774\) −17480.3 −0.811778
\(775\) 0 0
\(776\) −35507.9 −1.64260
\(777\) 3281.63 0.151516
\(778\) 40959.3 1.88748
\(779\) 8637.37 0.397260
\(780\) 0 0
\(781\) −1115.96 −0.0511294
\(782\) −33714.5 −1.54173
\(783\) 4537.33 0.207089
\(784\) −64498.5 −2.93816
\(785\) 0 0
\(786\) 34774.8 1.57809
\(787\) 16417.0 0.743587 0.371793 0.928315i \(-0.378743\pi\)
0.371793 + 0.928315i \(0.378743\pi\)
\(788\) 42146.1 1.90532
\(789\) 16020.5 0.722870
\(790\) 0 0
\(791\) 4143.12 0.186235
\(792\) 2107.33 0.0945462
\(793\) 10307.5 0.461577
\(794\) 64379.8 2.87752
\(795\) 0 0
\(796\) −32138.9 −1.43107
\(797\) −3944.19 −0.175295 −0.0876477 0.996152i \(-0.527935\pi\)
−0.0876477 + 0.996152i \(0.527935\pi\)
\(798\) 1457.26 0.0646445
\(799\) −9067.66 −0.401490
\(800\) 0 0
\(801\) 1761.13 0.0776861
\(802\) 45757.0 2.01463
\(803\) 305.325 0.0134181
\(804\) −38217.5 −1.67640
\(805\) 0 0
\(806\) 25761.1 1.12580
\(807\) 8429.37 0.367693
\(808\) −45044.0 −1.96119
\(809\) −17960.7 −0.780549 −0.390275 0.920699i \(-0.627620\pi\)
−0.390275 + 0.920699i \(0.627620\pi\)
\(810\) 0 0
\(811\) −13162.5 −0.569912 −0.284956 0.958541i \(-0.591979\pi\)
−0.284956 + 0.958541i \(0.591979\pi\)
\(812\) −15443.0 −0.667418
\(813\) 9308.84 0.401569
\(814\) −4540.41 −0.195505
\(815\) 0 0
\(816\) 31824.6 1.36530
\(817\) 7406.59 0.317165
\(818\) 65740.9 2.81000
\(819\) −3143.42 −0.134115
\(820\) 0 0
\(821\) 26502.4 1.12660 0.563302 0.826251i \(-0.309531\pi\)
0.563302 + 0.826251i \(0.309531\pi\)
\(822\) −17983.6 −0.763078
\(823\) 6937.86 0.293850 0.146925 0.989148i \(-0.453062\pi\)
0.146925 + 0.989148i \(0.453062\pi\)
\(824\) −104326. −4.41063
\(825\) 0 0
\(826\) −5157.35 −0.217248
\(827\) 41197.9 1.73228 0.866138 0.499805i \(-0.166595\pi\)
0.866138 + 0.499805i \(0.166595\pi\)
\(828\) 22059.5 0.925871
\(829\) −693.324 −0.0290472 −0.0145236 0.999895i \(-0.504623\pi\)
−0.0145236 + 0.999895i \(0.504623\pi\)
\(830\) 0 0
\(831\) −13796.8 −0.575938
\(832\) 95362.4 3.97367
\(833\) −17193.8 −0.715162
\(834\) 2676.41 0.111123
\(835\) 0 0
\(836\) −1454.56 −0.0601760
\(837\) −1648.33 −0.0680699
\(838\) −83025.7 −3.42252
\(839\) −6491.28 −0.267108 −0.133554 0.991042i \(-0.542639\pi\)
−0.133554 + 0.991042i \(0.542639\pi\)
\(840\) 0 0
\(841\) 3851.52 0.157921
\(842\) −40248.9 −1.64735
\(843\) 7715.49 0.315226
\(844\) 66141.8 2.69750
\(845\) 0 0
\(846\) 8223.97 0.334215
\(847\) 5851.42 0.237376
\(848\) −108928. −4.41111
\(849\) 16725.9 0.676128
\(850\) 0 0
\(851\) −29176.1 −1.17526
\(852\) 20188.8 0.811803
\(853\) −1116.68 −0.0448233 −0.0224117 0.999749i \(-0.507134\pi\)
−0.0224117 + 0.999749i \(0.507134\pi\)
\(854\) 3111.52 0.124677
\(855\) 0 0
\(856\) 18035.1 0.720126
\(857\) 44383.9 1.76911 0.884554 0.466438i \(-0.154463\pi\)
0.884554 + 0.466438i \(0.154463\pi\)
\(858\) 4349.18 0.173052
\(859\) −25579.3 −1.01601 −0.508006 0.861354i \(-0.669617\pi\)
−0.508006 + 0.861354i \(0.669617\pi\)
\(860\) 0 0
\(861\) −5624.28 −0.222619
\(862\) 84002.5 3.31918
\(863\) −11194.8 −0.441570 −0.220785 0.975323i \(-0.570862\pi\)
−0.220785 + 0.975323i \(0.570862\pi\)
\(864\) −14142.4 −0.556867
\(865\) 0 0
\(866\) 51471.0 2.01970
\(867\) −6255.31 −0.245030
\(868\) 5610.16 0.219379
\(869\) 3907.73 0.152544
\(870\) 0 0
\(871\) −48417.9 −1.88356
\(872\) −76169.2 −2.95804
\(873\) 4689.00 0.181785
\(874\) −12956.0 −0.501424
\(875\) 0 0
\(876\) −5523.65 −0.213044
\(877\) −5721.75 −0.220308 −0.110154 0.993915i \(-0.535134\pi\)
−0.110154 + 0.993915i \(0.535134\pi\)
\(878\) 34093.6 1.31048
\(879\) 17382.8 0.667017
\(880\) 0 0
\(881\) −34682.8 −1.32633 −0.663163 0.748475i \(-0.730786\pi\)
−0.663163 + 0.748475i \(0.730786\pi\)
\(882\) 15594.0 0.595326
\(883\) 37990.4 1.44788 0.723941 0.689862i \(-0.242329\pi\)
0.723941 + 0.689862i \(0.242329\pi\)
\(884\) 86752.9 3.30070
\(885\) 0 0
\(886\) −4992.66 −0.189313
\(887\) −28299.0 −1.07124 −0.535618 0.844460i \(-0.679921\pi\)
−0.535618 + 0.844460i \(0.679921\pi\)
\(888\) 50422.7 1.90549
\(889\) −2796.00 −0.105484
\(890\) 0 0
\(891\) −278.283 −0.0104633
\(892\) −32227.7 −1.20971
\(893\) −3484.58 −0.130579
\(894\) 10500.4 0.392825
\(895\) 0 0
\(896\) 10200.4 0.380324
\(897\) 27947.3 1.04028
\(898\) 100126. 3.72078
\(899\) −10259.3 −0.380607
\(900\) 0 0
\(901\) −29037.8 −1.07368
\(902\) 7781.65 0.287251
\(903\) −4822.85 −0.177735
\(904\) 63659.4 2.34212
\(905\) 0 0
\(906\) 31018.1 1.13742
\(907\) −17388.0 −0.636559 −0.318280 0.947997i \(-0.603105\pi\)
−0.318280 + 0.947997i \(0.603105\pi\)
\(908\) −128592. −4.69988
\(909\) 5948.28 0.217043
\(910\) 0 0
\(911\) 23555.3 0.856663 0.428332 0.903622i \(-0.359101\pi\)
0.428332 + 0.903622i \(0.359101\pi\)
\(912\) 12229.8 0.444044
\(913\) 2607.30 0.0945117
\(914\) 61980.4 2.24303
\(915\) 0 0
\(916\) 96949.1 3.49704
\(917\) 9594.44 0.345514
\(918\) −7694.34 −0.276635
\(919\) −5983.09 −0.214760 −0.107380 0.994218i \(-0.534246\pi\)
−0.107380 + 0.994218i \(0.534246\pi\)
\(920\) 0 0
\(921\) −4213.42 −0.150746
\(922\) −103968. −3.71368
\(923\) 25577.3 0.912119
\(924\) 947.150 0.0337218
\(925\) 0 0
\(926\) −8447.10 −0.299772
\(927\) 13776.7 0.488120
\(928\) −88022.7 −3.11367
\(929\) 20576.7 0.726694 0.363347 0.931654i \(-0.381634\pi\)
0.363347 + 0.931654i \(0.381634\pi\)
\(930\) 0 0
\(931\) −6607.35 −0.232596
\(932\) −67219.5 −2.36250
\(933\) −12290.2 −0.431259
\(934\) −17451.5 −0.611382
\(935\) 0 0
\(936\) −48299.0 −1.68665
\(937\) −11228.6 −0.391485 −0.195743 0.980655i \(-0.562712\pi\)
−0.195743 + 0.980655i \(0.562712\pi\)
\(938\) −14615.9 −0.508769
\(939\) 2923.83 0.101614
\(940\) 0 0
\(941\) 38567.6 1.33610 0.668049 0.744118i \(-0.267130\pi\)
0.668049 + 0.744118i \(0.267130\pi\)
\(942\) −34882.4 −1.20651
\(943\) 50003.9 1.72678
\(944\) −43282.1 −1.49228
\(945\) 0 0
\(946\) 6672.81 0.229336
\(947\) 4606.17 0.158057 0.0790287 0.996872i \(-0.474818\pi\)
0.0790287 + 0.996872i \(0.474818\pi\)
\(948\) −70694.8 −2.42200
\(949\) −6997.93 −0.239370
\(950\) 0 0
\(951\) −6215.06 −0.211921
\(952\) 16075.8 0.547288
\(953\) −25559.7 −0.868795 −0.434397 0.900721i \(-0.643039\pi\)
−0.434397 + 0.900721i \(0.643039\pi\)
\(954\) 26336.0 0.893773
\(955\) 0 0
\(956\) −75798.5 −2.56433
\(957\) −1732.05 −0.0585048
\(958\) −44433.8 −1.49853
\(959\) −4961.72 −0.167072
\(960\) 0 0
\(961\) −26064.0 −0.874895
\(962\) 104064. 3.48769
\(963\) −2381.63 −0.0796956
\(964\) 40006.4 1.33664
\(965\) 0 0
\(966\) 8436.42 0.280991
\(967\) 37895.8 1.26023 0.630117 0.776500i \(-0.283007\pi\)
0.630117 + 0.776500i \(0.283007\pi\)
\(968\) 89907.7 2.98527
\(969\) 3260.17 0.108082
\(970\) 0 0
\(971\) −46761.0 −1.54545 −0.772726 0.634740i \(-0.781107\pi\)
−0.772726 + 0.634740i \(0.781107\pi\)
\(972\) 5034.42 0.166131
\(973\) 738.428 0.0243298
\(974\) −25500.6 −0.838903
\(975\) 0 0
\(976\) 26112.9 0.856407
\(977\) 3070.29 0.100540 0.0502698 0.998736i \(-0.483992\pi\)
0.0502698 + 0.998736i \(0.483992\pi\)
\(978\) 12278.4 0.401451
\(979\) −672.282 −0.0219471
\(980\) 0 0
\(981\) 10058.5 0.327364
\(982\) −20934.3 −0.680287
\(983\) 16319.0 0.529498 0.264749 0.964317i \(-0.414711\pi\)
0.264749 + 0.964317i \(0.414711\pi\)
\(984\) −86417.7 −2.79969
\(985\) 0 0
\(986\) −47889.9 −1.54678
\(987\) 2269.01 0.0731747
\(988\) 33338.0 1.07350
\(989\) 42878.6 1.37862
\(990\) 0 0
\(991\) 5105.79 0.163664 0.0818319 0.996646i \(-0.473923\pi\)
0.0818319 + 0.996646i \(0.473923\pi\)
\(992\) 31977.0 1.02346
\(993\) −18477.5 −0.590500
\(994\) 7720.99 0.246373
\(995\) 0 0
\(996\) −47168.7 −1.50060
\(997\) 7206.97 0.228934 0.114467 0.993427i \(-0.463484\pi\)
0.114467 + 0.993427i \(0.463484\pi\)
\(998\) 16576.7 0.525779
\(999\) −6658.57 −0.210879
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 75.4.a.d.1.1 2
3.2 odd 2 225.4.a.n.1.2 2
4.3 odd 2 1200.4.a.bl.1.2 2
5.2 odd 4 75.4.b.c.49.1 4
5.3 odd 4 75.4.b.c.49.4 4
5.4 even 2 75.4.a.e.1.2 yes 2
15.2 even 4 225.4.b.h.199.4 4
15.8 even 4 225.4.b.h.199.1 4
15.14 odd 2 225.4.a.j.1.1 2
20.3 even 4 1200.4.f.v.49.1 4
20.7 even 4 1200.4.f.v.49.4 4
20.19 odd 2 1200.4.a.bu.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.4.a.d.1.1 2 1.1 even 1 trivial
75.4.a.e.1.2 yes 2 5.4 even 2
75.4.b.c.49.1 4 5.2 odd 4
75.4.b.c.49.4 4 5.3 odd 4
225.4.a.j.1.1 2 15.14 odd 2
225.4.a.n.1.2 2 3.2 odd 2
225.4.b.h.199.1 4 15.8 even 4
225.4.b.h.199.4 4 15.2 even 4
1200.4.a.bl.1.2 2 4.3 odd 2
1200.4.a.bu.1.1 2 20.19 odd 2
1200.4.f.v.49.1 4 20.3 even 4
1200.4.f.v.49.4 4 20.7 even 4