Properties

Label 637.4.a.d.1.2
Level $637$
Weight $4$
Character 637.1
Self dual yes
Analytic conductor $37.584$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.5842166737\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5364412.1
Defining polynomial: \( x^{4} - 27x^{2} - 24x + 76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.63459\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.63459 q^{2} +5.33748 q^{3} +5.21021 q^{4} -11.0031 q^{5} -19.3995 q^{6} +10.1397 q^{8} +1.48874 q^{9} +O(q^{10})\) \(q-3.63459 q^{2} +5.33748 q^{3} +5.21021 q^{4} -11.0031 q^{5} -19.3995 q^{6} +10.1397 q^{8} +1.48874 q^{9} +39.9917 q^{10} -11.7038 q^{11} +27.8094 q^{12} +13.0000 q^{13} -58.7289 q^{15} -78.5354 q^{16} +102.636 q^{17} -5.41094 q^{18} -43.2825 q^{19} -57.3285 q^{20} +42.5383 q^{22} -25.9576 q^{23} +54.1206 q^{24} -3.93182 q^{25} -47.2496 q^{26} -136.166 q^{27} -272.081 q^{29} +213.455 q^{30} +121.190 q^{31} +204.326 q^{32} -62.4686 q^{33} -373.039 q^{34} +7.75663 q^{36} +168.192 q^{37} +157.314 q^{38} +69.3873 q^{39} -111.568 q^{40} +451.218 q^{41} +94.6309 q^{43} -60.9790 q^{44} -16.3807 q^{45} +94.3450 q^{46} -50.6431 q^{47} -419.181 q^{48} +14.2905 q^{50} +547.817 q^{51} +67.7327 q^{52} -398.509 q^{53} +494.907 q^{54} +128.778 q^{55} -231.020 q^{57} +988.901 q^{58} +686.474 q^{59} -305.990 q^{60} +75.3794 q^{61} -440.476 q^{62} -114.356 q^{64} -143.040 q^{65} +227.047 q^{66} -336.720 q^{67} +534.754 q^{68} -138.548 q^{69} +427.161 q^{71} +15.0954 q^{72} +134.191 q^{73} -611.308 q^{74} -20.9860 q^{75} -225.511 q^{76} -252.194 q^{78} +253.005 q^{79} +864.133 q^{80} -766.980 q^{81} -1639.99 q^{82} +193.536 q^{83} -1129.31 q^{85} -343.944 q^{86} -1452.23 q^{87} -118.673 q^{88} +996.000 q^{89} +59.5371 q^{90} -135.244 q^{92} +646.851 q^{93} +184.067 q^{94} +476.242 q^{95} +1090.59 q^{96} -761.982 q^{97} -17.4238 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 5 q^{3} + 26 q^{4} + 36 q^{5} + 45 q^{6} - 30 q^{8} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 5 q^{3} + 26 q^{4} + 36 q^{5} + 45 q^{6} - 30 q^{8} + 21 q^{9} + 44 q^{10} - 95 q^{11} + 17 q^{12} + 52 q^{13} - 16 q^{15} + 58 q^{16} + 146 q^{17} + 65 q^{18} + 48 q^{19} + 474 q^{20} - 143 q^{22} - 121 q^{23} + 469 q^{24} + 506 q^{25} - 52 q^{26} + 83 q^{27} - 440 q^{29} + 1548 q^{30} + 283 q^{31} - 114 q^{32} - 227 q^{33} - 1234 q^{34} + 755 q^{36} - 209 q^{37} - 440 q^{38} + 65 q^{39} - 754 q^{40} + 93 q^{41} + 526 q^{43} + 217 q^{44} + 768 q^{45} - 841 q^{46} + 783 q^{47} - 1407 q^{48} + 446 q^{50} - 672 q^{51} + 338 q^{52} - 340 q^{53} + 199 q^{54} - 756 q^{55} - 1014 q^{57} + 1916 q^{58} + 922 q^{59} - 396 q^{60} + 141 q^{61} - 1745 q^{62} - 1510 q^{64} + 468 q^{65} - 503 q^{66} - 523 q^{67} + 1710 q^{68} - 1595 q^{69} + 1468 q^{71} - 9 q^{72} + 47 q^{73} - 2249 q^{74} + 1547 q^{75} + 1382 q^{76} + 585 q^{78} + 1025 q^{79} + 2538 q^{80} - 1772 q^{81} + 1561 q^{82} + 1190 q^{83} - 568 q^{85} + 738 q^{86} - 720 q^{87} - 555 q^{88} + 2962 q^{89} + 1960 q^{90} - 599 q^{92} - 763 q^{93} - 317 q^{94} + 2082 q^{95} + 45 q^{96} - 2715 q^{97} + 586 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.63459 −1.28502 −0.642510 0.766277i \(-0.722107\pi\)
−0.642510 + 0.766277i \(0.722107\pi\)
\(3\) 5.33748 1.02720 0.513600 0.858030i \(-0.328312\pi\)
0.513600 + 0.858030i \(0.328312\pi\)
\(4\) 5.21021 0.651276
\(5\) −11.0031 −0.984147 −0.492074 0.870554i \(-0.663761\pi\)
−0.492074 + 0.870554i \(0.663761\pi\)
\(6\) −19.3995 −1.31997
\(7\) 0 0
\(8\) 10.1397 0.448117
\(9\) 1.48874 0.0551384
\(10\) 39.9917 1.26465
\(11\) −11.7038 −0.320801 −0.160401 0.987052i \(-0.551279\pi\)
−0.160401 + 0.987052i \(0.551279\pi\)
\(12\) 27.8094 0.668991
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) −58.7289 −1.01092
\(16\) −78.5354 −1.22712
\(17\) 102.636 1.46429 0.732143 0.681151i \(-0.238520\pi\)
0.732143 + 0.681151i \(0.238520\pi\)
\(18\) −5.41094 −0.0708539
\(19\) −43.2825 −0.522616 −0.261308 0.965256i \(-0.584154\pi\)
−0.261308 + 0.965256i \(0.584154\pi\)
\(20\) −57.3285 −0.640952
\(21\) 0 0
\(22\) 42.5383 0.412236
\(23\) −25.9576 −0.235327 −0.117664 0.993054i \(-0.537540\pi\)
−0.117664 + 0.993054i \(0.537540\pi\)
\(24\) 54.1206 0.460305
\(25\) −3.93182 −0.0314546
\(26\) −47.2496 −0.356400
\(27\) −136.166 −0.970561
\(28\) 0 0
\(29\) −272.081 −1.74221 −0.871106 0.491095i \(-0.836597\pi\)
−0.871106 + 0.491095i \(0.836597\pi\)
\(30\) 213.455 1.29905
\(31\) 121.190 0.702142 0.351071 0.936349i \(-0.385818\pi\)
0.351071 + 0.936349i \(0.385818\pi\)
\(32\) 204.326 1.12875
\(33\) −62.4686 −0.329527
\(34\) −373.039 −1.88164
\(35\) 0 0
\(36\) 7.75663 0.0359103
\(37\) 168.192 0.747314 0.373657 0.927567i \(-0.378104\pi\)
0.373657 + 0.927567i \(0.378104\pi\)
\(38\) 157.314 0.671572
\(39\) 69.3873 0.284894
\(40\) −111.568 −0.441013
\(41\) 451.218 1.71874 0.859372 0.511351i \(-0.170855\pi\)
0.859372 + 0.511351i \(0.170855\pi\)
\(42\) 0 0
\(43\) 94.6309 0.335606 0.167803 0.985821i \(-0.446333\pi\)
0.167803 + 0.985821i \(0.446333\pi\)
\(44\) −60.9790 −0.208930
\(45\) −16.3807 −0.0542643
\(46\) 94.3450 0.302400
\(47\) −50.6431 −0.157171 −0.0785857 0.996907i \(-0.525040\pi\)
−0.0785857 + 0.996907i \(0.525040\pi\)
\(48\) −419.181 −1.26049
\(49\) 0 0
\(50\) 14.2905 0.0404198
\(51\) 547.817 1.50411
\(52\) 67.7327 0.180632
\(53\) −398.509 −1.03282 −0.516410 0.856341i \(-0.672732\pi\)
−0.516410 + 0.856341i \(0.672732\pi\)
\(54\) 494.907 1.24719
\(55\) 128.778 0.315716
\(56\) 0 0
\(57\) −231.020 −0.536830
\(58\) 988.901 2.23878
\(59\) 686.474 1.51477 0.757384 0.652970i \(-0.226477\pi\)
0.757384 + 0.652970i \(0.226477\pi\)
\(60\) −305.990 −0.658385
\(61\) 75.3794 0.158219 0.0791094 0.996866i \(-0.474792\pi\)
0.0791094 + 0.996866i \(0.474792\pi\)
\(62\) −440.476 −0.902267
\(63\) 0 0
\(64\) −114.356 −0.223352
\(65\) −143.040 −0.272953
\(66\) 227.047 0.423449
\(67\) −336.720 −0.613983 −0.306991 0.951712i \(-0.599322\pi\)
−0.306991 + 0.951712i \(0.599322\pi\)
\(68\) 534.754 0.953654
\(69\) −138.548 −0.241728
\(70\) 0 0
\(71\) 427.161 0.714009 0.357005 0.934103i \(-0.383798\pi\)
0.357005 + 0.934103i \(0.383798\pi\)
\(72\) 15.0954 0.0247084
\(73\) 134.191 0.215148 0.107574 0.994197i \(-0.465692\pi\)
0.107574 + 0.994197i \(0.465692\pi\)
\(74\) −611.308 −0.960313
\(75\) −20.9860 −0.0323101
\(76\) −225.511 −0.340367
\(77\) 0 0
\(78\) −252.194 −0.366094
\(79\) 253.005 0.360319 0.180160 0.983637i \(-0.442339\pi\)
0.180160 + 0.983637i \(0.442339\pi\)
\(80\) 864.133 1.20766
\(81\) −766.980 −1.05210
\(82\) −1639.99 −2.20862
\(83\) 193.536 0.255944 0.127972 0.991778i \(-0.459153\pi\)
0.127972 + 0.991778i \(0.459153\pi\)
\(84\) 0 0
\(85\) −1129.31 −1.44107
\(86\) −343.944 −0.431261
\(87\) −1452.23 −1.78960
\(88\) −118.673 −0.143756
\(89\) 996.000 1.18624 0.593122 0.805112i \(-0.297895\pi\)
0.593122 + 0.805112i \(0.297895\pi\)
\(90\) 59.5371 0.0697307
\(91\) 0 0
\(92\) −135.244 −0.153263
\(93\) 646.851 0.721240
\(94\) 184.067 0.201968
\(95\) 476.242 0.514331
\(96\) 1090.59 1.15945
\(97\) −761.982 −0.797604 −0.398802 0.917037i \(-0.630574\pi\)
−0.398802 + 0.917037i \(0.630574\pi\)
\(98\) 0 0
\(99\) −17.4238 −0.0176885
\(100\) −20.4856 −0.0204856
\(101\) −822.058 −0.809879 −0.404940 0.914343i \(-0.632707\pi\)
−0.404940 + 0.914343i \(0.632707\pi\)
\(102\) −1991.09 −1.93282
\(103\) −794.608 −0.760146 −0.380073 0.924956i \(-0.624101\pi\)
−0.380073 + 0.924956i \(0.624101\pi\)
\(104\) 131.816 0.124285
\(105\) 0 0
\(106\) 1448.42 1.32719
\(107\) 2115.28 1.91114 0.955571 0.294761i \(-0.0952402\pi\)
0.955571 + 0.294761i \(0.0952402\pi\)
\(108\) −709.453 −0.632104
\(109\) 1364.17 1.19875 0.599377 0.800467i \(-0.295415\pi\)
0.599377 + 0.800467i \(0.295415\pi\)
\(110\) −468.053 −0.405701
\(111\) 897.722 0.767640
\(112\) 0 0
\(113\) 1277.27 1.06332 0.531660 0.846958i \(-0.321568\pi\)
0.531660 + 0.846958i \(0.321568\pi\)
\(114\) 839.662 0.689838
\(115\) 285.614 0.231597
\(116\) −1417.60 −1.13466
\(117\) 19.3536 0.0152926
\(118\) −2495.05 −1.94651
\(119\) 0 0
\(120\) −595.495 −0.453008
\(121\) −1194.02 −0.897087
\(122\) −273.973 −0.203314
\(123\) 2408.37 1.76549
\(124\) 631.427 0.457289
\(125\) 1418.65 1.01510
\(126\) 0 0
\(127\) 1278.83 0.893526 0.446763 0.894652i \(-0.352577\pi\)
0.446763 + 0.894652i \(0.352577\pi\)
\(128\) −1218.97 −0.841739
\(129\) 505.091 0.344735
\(130\) 519.892 0.350750
\(131\) 2865.00 1.91081 0.955405 0.295297i \(-0.0954187\pi\)
0.955405 + 0.295297i \(0.0954187\pi\)
\(132\) −325.475 −0.214613
\(133\) 0 0
\(134\) 1223.84 0.788980
\(135\) 1498.25 0.955175
\(136\) 1040.70 0.656171
\(137\) 1494.96 0.932283 0.466141 0.884710i \(-0.345644\pi\)
0.466141 + 0.884710i \(0.345644\pi\)
\(138\) 503.565 0.310625
\(139\) 1783.85 1.08852 0.544260 0.838917i \(-0.316811\pi\)
0.544260 + 0.838917i \(0.316811\pi\)
\(140\) 0 0
\(141\) −270.307 −0.161446
\(142\) −1552.55 −0.917516
\(143\) −152.149 −0.0889743
\(144\) −116.918 −0.0676612
\(145\) 2993.73 1.71459
\(146\) −487.728 −0.276470
\(147\) 0 0
\(148\) 876.316 0.486708
\(149\) 1727.01 0.949547 0.474774 0.880108i \(-0.342530\pi\)
0.474774 + 0.880108i \(0.342530\pi\)
\(150\) 76.2756 0.0415192
\(151\) 1499.77 0.808277 0.404139 0.914698i \(-0.367571\pi\)
0.404139 + 0.914698i \(0.367571\pi\)
\(152\) −438.873 −0.234193
\(153\) 152.798 0.0807383
\(154\) 0 0
\(155\) −1333.47 −0.691011
\(156\) 361.522 0.185545
\(157\) 1021.82 0.519430 0.259715 0.965685i \(-0.416371\pi\)
0.259715 + 0.965685i \(0.416371\pi\)
\(158\) −919.567 −0.463018
\(159\) −2127.04 −1.06091
\(160\) −2248.22 −1.11086
\(161\) 0 0
\(162\) 2787.65 1.35197
\(163\) −3847.87 −1.84901 −0.924504 0.381172i \(-0.875520\pi\)
−0.924504 + 0.381172i \(0.875520\pi\)
\(164\) 2350.94 1.11938
\(165\) 687.348 0.324303
\(166\) −703.423 −0.328893
\(167\) 424.130 0.196528 0.0982639 0.995160i \(-0.468671\pi\)
0.0982639 + 0.995160i \(0.468671\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 4104.58 1.85181
\(171\) −64.4363 −0.0288162
\(172\) 493.047 0.218573
\(173\) 3117.51 1.37006 0.685029 0.728516i \(-0.259790\pi\)
0.685029 + 0.728516i \(0.259790\pi\)
\(174\) 5278.24 2.29967
\(175\) 0 0
\(176\) 919.159 0.393660
\(177\) 3664.05 1.55597
\(178\) −3620.05 −1.52435
\(179\) 2962.21 1.23690 0.618452 0.785823i \(-0.287760\pi\)
0.618452 + 0.785823i \(0.287760\pi\)
\(180\) −85.3470 −0.0353410
\(181\) 1198.27 0.492081 0.246041 0.969259i \(-0.420870\pi\)
0.246041 + 0.969259i \(0.420870\pi\)
\(182\) 0 0
\(183\) 402.336 0.162522
\(184\) −263.203 −0.105454
\(185\) −1850.63 −0.735466
\(186\) −2351.04 −0.926808
\(187\) −1201.22 −0.469745
\(188\) −263.861 −0.102362
\(189\) 0 0
\(190\) −1730.94 −0.660925
\(191\) −3940.99 −1.49298 −0.746492 0.665394i \(-0.768263\pi\)
−0.746492 + 0.665394i \(0.768263\pi\)
\(192\) −610.376 −0.229427
\(193\) −1974.13 −0.736276 −0.368138 0.929771i \(-0.620005\pi\)
−0.368138 + 0.929771i \(0.620005\pi\)
\(194\) 2769.49 1.02494
\(195\) −763.475 −0.280377
\(196\) 0 0
\(197\) 2772.61 1.00274 0.501372 0.865232i \(-0.332829\pi\)
0.501372 + 0.865232i \(0.332829\pi\)
\(198\) 63.3283 0.0227300
\(199\) 919.870 0.327678 0.163839 0.986487i \(-0.447612\pi\)
0.163839 + 0.986487i \(0.447612\pi\)
\(200\) −39.8676 −0.0140953
\(201\) −1797.24 −0.630683
\(202\) 2987.84 1.04071
\(203\) 0 0
\(204\) 2854.24 0.979593
\(205\) −4964.80 −1.69150
\(206\) 2888.07 0.976803
\(207\) −38.6440 −0.0129756
\(208\) −1020.96 −0.340341
\(209\) 506.568 0.167656
\(210\) 0 0
\(211\) 936.550 0.305568 0.152784 0.988260i \(-0.451176\pi\)
0.152784 + 0.988260i \(0.451176\pi\)
\(212\) −2076.32 −0.672651
\(213\) 2279.96 0.733430
\(214\) −7688.18 −2.45586
\(215\) −1041.23 −0.330286
\(216\) −1380.69 −0.434925
\(217\) 0 0
\(218\) −4958.20 −1.54042
\(219\) 716.241 0.221000
\(220\) 670.958 0.205618
\(221\) 1334.27 0.406120
\(222\) −3262.85 −0.986433
\(223\) −5243.34 −1.57453 −0.787264 0.616616i \(-0.788503\pi\)
−0.787264 + 0.616616i \(0.788503\pi\)
\(224\) 0 0
\(225\) −5.85345 −0.00173435
\(226\) −4642.34 −1.36639
\(227\) 3821.36 1.11732 0.558662 0.829396i \(-0.311315\pi\)
0.558662 + 0.829396i \(0.311315\pi\)
\(228\) −1203.66 −0.349625
\(229\) −2144.86 −0.618936 −0.309468 0.950910i \(-0.600151\pi\)
−0.309468 + 0.950910i \(0.600151\pi\)
\(230\) −1038.09 −0.297606
\(231\) 0 0
\(232\) −2758.83 −0.780714
\(233\) −3473.06 −0.976513 −0.488257 0.872700i \(-0.662367\pi\)
−0.488257 + 0.872700i \(0.662367\pi\)
\(234\) −70.3422 −0.0196513
\(235\) 557.231 0.154680
\(236\) 3576.68 0.986533
\(237\) 1350.41 0.370120
\(238\) 0 0
\(239\) 3691.00 0.998959 0.499479 0.866326i \(-0.333525\pi\)
0.499479 + 0.866326i \(0.333525\pi\)
\(240\) 4612.29 1.24051
\(241\) 914.807 0.244514 0.122257 0.992498i \(-0.460987\pi\)
0.122257 + 0.992498i \(0.460987\pi\)
\(242\) 4339.78 1.15277
\(243\) −417.260 −0.110153
\(244\) 392.743 0.103044
\(245\) 0 0
\(246\) −8753.43 −2.26869
\(247\) −562.673 −0.144948
\(248\) 1228.84 0.314642
\(249\) 1032.99 0.262905
\(250\) −5156.20 −1.30443
\(251\) −5817.09 −1.46283 −0.731417 0.681930i \(-0.761141\pi\)
−0.731417 + 0.681930i \(0.761141\pi\)
\(252\) 0 0
\(253\) 303.801 0.0754933
\(254\) −4648.02 −1.14820
\(255\) −6027.69 −1.48027
\(256\) 5345.30 1.30500
\(257\) −3066.16 −0.744210 −0.372105 0.928191i \(-0.621364\pi\)
−0.372105 + 0.928191i \(0.621364\pi\)
\(258\) −1835.80 −0.442991
\(259\) 0 0
\(260\) −745.270 −0.177768
\(261\) −405.057 −0.0960627
\(262\) −10413.1 −2.45543
\(263\) 1215.01 0.284869 0.142435 0.989804i \(-0.454507\pi\)
0.142435 + 0.989804i \(0.454507\pi\)
\(264\) −633.414 −0.147666
\(265\) 4384.84 1.01645
\(266\) 0 0
\(267\) 5316.13 1.21851
\(268\) −1754.38 −0.399873
\(269\) −3689.75 −0.836311 −0.418156 0.908375i \(-0.637323\pi\)
−0.418156 + 0.908375i \(0.637323\pi\)
\(270\) −5445.51 −1.22742
\(271\) −5672.09 −1.27142 −0.635710 0.771928i \(-0.719292\pi\)
−0.635710 + 0.771928i \(0.719292\pi\)
\(272\) −8060.55 −1.79685
\(273\) 0 0
\(274\) −5433.55 −1.19800
\(275\) 46.0171 0.0100907
\(276\) −721.865 −0.157432
\(277\) 6266.61 1.35929 0.679647 0.733539i \(-0.262133\pi\)
0.679647 + 0.733539i \(0.262133\pi\)
\(278\) −6483.56 −1.39877
\(279\) 180.420 0.0387150
\(280\) 0 0
\(281\) −7122.92 −1.51216 −0.756082 0.654477i \(-0.772889\pi\)
−0.756082 + 0.654477i \(0.772889\pi\)
\(282\) 982.453 0.207462
\(283\) 2576.18 0.541123 0.270561 0.962703i \(-0.412791\pi\)
0.270561 + 0.962703i \(0.412791\pi\)
\(284\) 2225.60 0.465017
\(285\) 2541.93 0.528320
\(286\) 552.998 0.114334
\(287\) 0 0
\(288\) 304.187 0.0622375
\(289\) 5621.12 1.14413
\(290\) −10881.0 −2.20329
\(291\) −4067.07 −0.819298
\(292\) 699.162 0.140121
\(293\) 6955.17 1.38678 0.693388 0.720565i \(-0.256117\pi\)
0.693388 + 0.720565i \(0.256117\pi\)
\(294\) 0 0
\(295\) −7553.34 −1.49075
\(296\) 1705.42 0.334884
\(297\) 1593.65 0.311357
\(298\) −6276.98 −1.22019
\(299\) −337.448 −0.0652681
\(300\) −109.342 −0.0210428
\(301\) 0 0
\(302\) −5451.06 −1.03865
\(303\) −4387.72 −0.831908
\(304\) 3399.21 0.641310
\(305\) −829.407 −0.155710
\(306\) −555.356 −0.103750
\(307\) −1690.34 −0.314244 −0.157122 0.987579i \(-0.550222\pi\)
−0.157122 + 0.987579i \(0.550222\pi\)
\(308\) 0 0
\(309\) −4241.21 −0.780822
\(310\) 4846.61 0.887963
\(311\) −7291.22 −1.32941 −0.664706 0.747105i \(-0.731443\pi\)
−0.664706 + 0.747105i \(0.731443\pi\)
\(312\) 703.568 0.127666
\(313\) 4730.37 0.854237 0.427118 0.904196i \(-0.359529\pi\)
0.427118 + 0.904196i \(0.359529\pi\)
\(314\) −3713.91 −0.667477
\(315\) 0 0
\(316\) 1318.21 0.234668
\(317\) 8225.90 1.45745 0.728726 0.684805i \(-0.240113\pi\)
0.728726 + 0.684805i \(0.240113\pi\)
\(318\) 7730.90 1.36329
\(319\) 3184.37 0.558904
\(320\) 1258.27 0.219812
\(321\) 11290.3 1.96312
\(322\) 0 0
\(323\) −4442.34 −0.765258
\(324\) −3996.13 −0.685207
\(325\) −51.1137 −0.00872393
\(326\) 13985.4 2.37601
\(327\) 7281.25 1.23136
\(328\) 4575.23 0.770198
\(329\) 0 0
\(330\) −2498.23 −0.416736
\(331\) −7168.03 −1.19030 −0.595152 0.803613i \(-0.702908\pi\)
−0.595152 + 0.803613i \(0.702908\pi\)
\(332\) 1008.36 0.166690
\(333\) 250.394 0.0412057
\(334\) −1541.54 −0.252542
\(335\) 3704.96 0.604249
\(336\) 0 0
\(337\) 4566.95 0.738213 0.369107 0.929387i \(-0.379664\pi\)
0.369107 + 0.929387i \(0.379664\pi\)
\(338\) −614.245 −0.0988477
\(339\) 6817.39 1.09224
\(340\) −5883.96 −0.938536
\(341\) −1418.38 −0.225248
\(342\) 234.199 0.0370294
\(343\) 0 0
\(344\) 959.531 0.150391
\(345\) 1524.46 0.237896
\(346\) −11330.9 −1.76055
\(347\) −4616.95 −0.714267 −0.357134 0.934053i \(-0.616246\pi\)
−0.357134 + 0.934053i \(0.616246\pi\)
\(348\) −7566.41 −1.16552
\(349\) 7688.68 1.17927 0.589636 0.807669i \(-0.299271\pi\)
0.589636 + 0.807669i \(0.299271\pi\)
\(350\) 0 0
\(351\) −1770.16 −0.269185
\(352\) −2391.38 −0.362105
\(353\) 8037.67 1.21190 0.605952 0.795501i \(-0.292792\pi\)
0.605952 + 0.795501i \(0.292792\pi\)
\(354\) −13317.3 −1.99945
\(355\) −4700.09 −0.702690
\(356\) 5189.37 0.772573
\(357\) 0 0
\(358\) −10766.4 −1.58945
\(359\) 170.112 0.0250089 0.0125044 0.999922i \(-0.496020\pi\)
0.0125044 + 0.999922i \(0.496020\pi\)
\(360\) −166.096 −0.0243167
\(361\) −4985.62 −0.726873
\(362\) −4355.22 −0.632335
\(363\) −6373.07 −0.921487
\(364\) 0 0
\(365\) −1476.51 −0.211738
\(366\) −1462.33 −0.208844
\(367\) −6547.73 −0.931304 −0.465652 0.884968i \(-0.654180\pi\)
−0.465652 + 0.884968i \(0.654180\pi\)
\(368\) 2038.59 0.288774
\(369\) 671.745 0.0947687
\(370\) 6726.29 0.945089
\(371\) 0 0
\(372\) 3370.23 0.469727
\(373\) −12845.2 −1.78311 −0.891554 0.452915i \(-0.850384\pi\)
−0.891554 + 0.452915i \(0.850384\pi\)
\(374\) 4365.95 0.603631
\(375\) 7572.02 1.04271
\(376\) −513.507 −0.0704312
\(377\) −3537.05 −0.483203
\(378\) 0 0
\(379\) 9494.88 1.28686 0.643429 0.765505i \(-0.277511\pi\)
0.643429 + 0.765505i \(0.277511\pi\)
\(380\) 2481.32 0.334971
\(381\) 6825.74 0.917829
\(382\) 14323.9 1.91851
\(383\) −6344.95 −0.846506 −0.423253 0.906012i \(-0.639112\pi\)
−0.423253 + 0.906012i \(0.639112\pi\)
\(384\) −6506.22 −0.864634
\(385\) 0 0
\(386\) 7175.16 0.946130
\(387\) 140.880 0.0185048
\(388\) −3970.09 −0.519461
\(389\) −13196.7 −1.72005 −0.860026 0.510250i \(-0.829553\pi\)
−0.860026 + 0.510250i \(0.829553\pi\)
\(390\) 2774.92 0.360291
\(391\) −2664.18 −0.344586
\(392\) 0 0
\(393\) 15291.9 1.96278
\(394\) −10077.3 −1.28855
\(395\) −2783.83 −0.354607
\(396\) −90.7817 −0.0115201
\(397\) 32.4042 0.00409653 0.00204826 0.999998i \(-0.499348\pi\)
0.00204826 + 0.999998i \(0.499348\pi\)
\(398\) −3343.35 −0.421072
\(399\) 0 0
\(400\) 308.787 0.0385984
\(401\) 3853.07 0.479834 0.239917 0.970793i \(-0.422880\pi\)
0.239917 + 0.970793i \(0.422880\pi\)
\(402\) 6532.21 0.810440
\(403\) 1575.47 0.194739
\(404\) −4283.10 −0.527455
\(405\) 8439.15 1.03542
\(406\) 0 0
\(407\) −1968.48 −0.239739
\(408\) 5554.72 0.674018
\(409\) −10221.7 −1.23577 −0.617886 0.786268i \(-0.712011\pi\)
−0.617886 + 0.786268i \(0.712011\pi\)
\(410\) 18045.0 2.17361
\(411\) 7979.31 0.957640
\(412\) −4140.08 −0.495065
\(413\) 0 0
\(414\) 140.455 0.0166739
\(415\) −2129.49 −0.251886
\(416\) 2656.24 0.313059
\(417\) 9521.27 1.11813
\(418\) −1841.17 −0.215441
\(419\) 12153.6 1.41704 0.708521 0.705689i \(-0.249363\pi\)
0.708521 + 0.705689i \(0.249363\pi\)
\(420\) 0 0
\(421\) 13630.9 1.57798 0.788990 0.614405i \(-0.210604\pi\)
0.788990 + 0.614405i \(0.210604\pi\)
\(422\) −3403.97 −0.392660
\(423\) −75.3942 −0.00866618
\(424\) −4040.77 −0.462824
\(425\) −403.546 −0.0460585
\(426\) −8286.72 −0.942472
\(427\) 0 0
\(428\) 11021.1 1.24468
\(429\) −812.092 −0.0913943
\(430\) 3784.45 0.424424
\(431\) −5959.31 −0.666009 −0.333004 0.942925i \(-0.608062\pi\)
−0.333004 + 0.942925i \(0.608062\pi\)
\(432\) 10693.8 1.19099
\(433\) −13019.1 −1.44494 −0.722470 0.691402i \(-0.756993\pi\)
−0.722470 + 0.691402i \(0.756993\pi\)
\(434\) 0 0
\(435\) 15979.0 1.76123
\(436\) 7107.63 0.780720
\(437\) 1123.51 0.122986
\(438\) −2603.24 −0.283990
\(439\) 892.739 0.0970572 0.0485286 0.998822i \(-0.484547\pi\)
0.0485286 + 0.998822i \(0.484547\pi\)
\(440\) 1305.77 0.141477
\(441\) 0 0
\(442\) −4849.50 −0.521872
\(443\) −8779.74 −0.941621 −0.470810 0.882234i \(-0.656038\pi\)
−0.470810 + 0.882234i \(0.656038\pi\)
\(444\) 4677.32 0.499946
\(445\) −10959.1 −1.16744
\(446\) 19057.4 2.02330
\(447\) 9217.91 0.975374
\(448\) 0 0
\(449\) 13122.2 1.37923 0.689617 0.724174i \(-0.257779\pi\)
0.689617 + 0.724174i \(0.257779\pi\)
\(450\) 21.2749 0.00222868
\(451\) −5280.95 −0.551375
\(452\) 6654.83 0.692515
\(453\) 8005.02 0.830262
\(454\) −13889.1 −1.43578
\(455\) 0 0
\(456\) −2342.48 −0.240563
\(457\) −9100.56 −0.931524 −0.465762 0.884910i \(-0.654220\pi\)
−0.465762 + 0.884910i \(0.654220\pi\)
\(458\) 7795.67 0.795345
\(459\) −13975.5 −1.42118
\(460\) 1488.11 0.150833
\(461\) 11153.6 1.12685 0.563423 0.826169i \(-0.309484\pi\)
0.563423 + 0.826169i \(0.309484\pi\)
\(462\) 0 0
\(463\) −10006.3 −1.00439 −0.502197 0.864753i \(-0.667475\pi\)
−0.502197 + 0.864753i \(0.667475\pi\)
\(464\) 21368.0 2.13790
\(465\) −7117.37 −0.709806
\(466\) 12623.1 1.25484
\(467\) −10455.5 −1.03603 −0.518014 0.855372i \(-0.673328\pi\)
−0.518014 + 0.855372i \(0.673328\pi\)
\(468\) 100.836 0.00995973
\(469\) 0 0
\(470\) −2025.30 −0.198767
\(471\) 5453.97 0.533558
\(472\) 6960.66 0.678793
\(473\) −1107.54 −0.107663
\(474\) −4908.17 −0.475611
\(475\) 170.179 0.0164387
\(476\) 0 0
\(477\) −593.275 −0.0569480
\(478\) −13415.3 −1.28368
\(479\) 11161.0 1.06463 0.532314 0.846547i \(-0.321322\pi\)
0.532314 + 0.846547i \(0.321322\pi\)
\(480\) −11999.8 −1.14107
\(481\) 2186.50 0.207267
\(482\) −3324.94 −0.314205
\(483\) 0 0
\(484\) −6221.11 −0.584251
\(485\) 8384.17 0.784959
\(486\) 1516.57 0.141549
\(487\) 3941.17 0.366717 0.183359 0.983046i \(-0.441303\pi\)
0.183359 + 0.983046i \(0.441303\pi\)
\(488\) 764.326 0.0709005
\(489\) −20537.9 −1.89930
\(490\) 0 0
\(491\) 11636.5 1.06955 0.534775 0.844995i \(-0.320397\pi\)
0.534775 + 0.844995i \(0.320397\pi\)
\(492\) 12548.1 1.14982
\(493\) −27925.3 −2.55110
\(494\) 2045.08 0.186260
\(495\) 191.716 0.0174080
\(496\) −9517.72 −0.861610
\(497\) 0 0
\(498\) −3754.51 −0.337838
\(499\) 2370.55 0.212666 0.106333 0.994331i \(-0.466089\pi\)
0.106333 + 0.994331i \(0.466089\pi\)
\(500\) 7391.46 0.661113
\(501\) 2263.79 0.201873
\(502\) 21142.7 1.87977
\(503\) −16697.1 −1.48010 −0.740048 0.672554i \(-0.765197\pi\)
−0.740048 + 0.672554i \(0.765197\pi\)
\(504\) 0 0
\(505\) 9045.18 0.797040
\(506\) −1104.19 −0.0970104
\(507\) 902.035 0.0790153
\(508\) 6662.98 0.581933
\(509\) 12297.2 1.07085 0.535426 0.844582i \(-0.320151\pi\)
0.535426 + 0.844582i \(0.320151\pi\)
\(510\) 21908.1 1.90217
\(511\) 0 0
\(512\) −9676.19 −0.835217
\(513\) 5893.61 0.507230
\(514\) 11144.2 0.956324
\(515\) 8743.15 0.748096
\(516\) 2631.63 0.224518
\(517\) 592.714 0.0504208
\(518\) 0 0
\(519\) 16639.7 1.40732
\(520\) −1450.39 −0.122315
\(521\) −18304.4 −1.53922 −0.769609 0.638516i \(-0.779549\pi\)
−0.769609 + 0.638516i \(0.779549\pi\)
\(522\) 1472.21 0.123443
\(523\) 12673.2 1.05958 0.529788 0.848130i \(-0.322271\pi\)
0.529788 + 0.848130i \(0.322271\pi\)
\(524\) 14927.3 1.24447
\(525\) 0 0
\(526\) −4416.05 −0.366063
\(527\) 12438.5 1.02814
\(528\) 4905.99 0.404367
\(529\) −11493.2 −0.944621
\(530\) −15937.1 −1.30615
\(531\) 1021.98 0.0835219
\(532\) 0 0
\(533\) 5865.84 0.476694
\(534\) −19321.9 −1.56581
\(535\) −23274.7 −1.88084
\(536\) −3414.24 −0.275136
\(537\) 15810.7 1.27055
\(538\) 13410.7 1.07468
\(539\) 0 0
\(540\) 7806.19 0.622083
\(541\) −21283.1 −1.69137 −0.845684 0.533684i \(-0.820807\pi\)
−0.845684 + 0.533684i \(0.820807\pi\)
\(542\) 20615.7 1.63380
\(543\) 6395.75 0.505466
\(544\) 20971.1 1.65281
\(545\) −15010.1 −1.17975
\(546\) 0 0
\(547\) −19081.2 −1.49150 −0.745751 0.666225i \(-0.767909\pi\)
−0.745751 + 0.666225i \(0.767909\pi\)
\(548\) 7789.04 0.607174
\(549\) 112.220 0.00872392
\(550\) −167.253 −0.0129667
\(551\) 11776.4 0.910507
\(552\) −1404.84 −0.108322
\(553\) 0 0
\(554\) −22776.5 −1.74672
\(555\) −9877.73 −0.755471
\(556\) 9294.24 0.708927
\(557\) −13693.3 −1.04166 −0.520828 0.853661i \(-0.674377\pi\)
−0.520828 + 0.853661i \(0.674377\pi\)
\(558\) −655.753 −0.0497495
\(559\) 1230.20 0.0930805
\(560\) 0 0
\(561\) −6411.52 −0.482521
\(562\) 25888.9 1.94316
\(563\) 15762.2 1.17993 0.589964 0.807430i \(-0.299142\pi\)
0.589964 + 0.807430i \(0.299142\pi\)
\(564\) −1408.36 −0.105146
\(565\) −14053.9 −1.04646
\(566\) −9363.33 −0.695354
\(567\) 0 0
\(568\) 4331.29 0.319960
\(569\) 22202.1 1.63578 0.817892 0.575372i \(-0.195143\pi\)
0.817892 + 0.575372i \(0.195143\pi\)
\(570\) −9238.88 −0.678902
\(571\) 21989.8 1.61164 0.805819 0.592162i \(-0.201725\pi\)
0.805819 + 0.592162i \(0.201725\pi\)
\(572\) −792.727 −0.0579468
\(573\) −21035.0 −1.53359
\(574\) 0 0
\(575\) 102.061 0.00740212
\(576\) −170.247 −0.0123153
\(577\) 18405.1 1.32793 0.663965 0.747764i \(-0.268872\pi\)
0.663965 + 0.747764i \(0.268872\pi\)
\(578\) −20430.4 −1.47023
\(579\) −10536.9 −0.756303
\(580\) 15598.0 1.11667
\(581\) 0 0
\(582\) 14782.1 1.05281
\(583\) 4664.05 0.331330
\(584\) 1360.66 0.0964116
\(585\) −212.949 −0.0150502
\(586\) −25279.1 −1.78203
\(587\) 13059.4 0.918260 0.459130 0.888369i \(-0.348161\pi\)
0.459130 + 0.888369i \(0.348161\pi\)
\(588\) 0 0
\(589\) −5245.42 −0.366951
\(590\) 27453.3 1.91565
\(591\) 14798.8 1.03002
\(592\) −13209.0 −0.917040
\(593\) −11511.8 −0.797189 −0.398595 0.917127i \(-0.630502\pi\)
−0.398595 + 0.917127i \(0.630502\pi\)
\(594\) −5792.27 −0.400100
\(595\) 0 0
\(596\) 8998.11 0.618418
\(597\) 4909.79 0.336590
\(598\) 1226.49 0.0838708
\(599\) 7113.35 0.485215 0.242607 0.970125i \(-0.421997\pi\)
0.242607 + 0.970125i \(0.421997\pi\)
\(600\) −212.793 −0.0144787
\(601\) 7802.91 0.529596 0.264798 0.964304i \(-0.414695\pi\)
0.264798 + 0.964304i \(0.414695\pi\)
\(602\) 0 0
\(603\) −501.287 −0.0338540
\(604\) 7814.14 0.526412
\(605\) 13137.9 0.882865
\(606\) 15947.5 1.06902
\(607\) 13290.9 0.888732 0.444366 0.895845i \(-0.353429\pi\)
0.444366 + 0.895845i \(0.353429\pi\)
\(608\) −8843.74 −0.589903
\(609\) 0 0
\(610\) 3014.55 0.200091
\(611\) −658.360 −0.0435915
\(612\) 796.108 0.0525830
\(613\) −3133.55 −0.206465 −0.103232 0.994657i \(-0.532919\pi\)
−0.103232 + 0.994657i \(0.532919\pi\)
\(614\) 6143.69 0.403810
\(615\) −26499.5 −1.73750
\(616\) 0 0
\(617\) −126.313 −0.00824175 −0.00412088 0.999992i \(-0.501312\pi\)
−0.00412088 + 0.999992i \(0.501312\pi\)
\(618\) 15415.0 1.00337
\(619\) −5888.40 −0.382350 −0.191175 0.981556i \(-0.561230\pi\)
−0.191175 + 0.981556i \(0.561230\pi\)
\(620\) −6947.65 −0.450039
\(621\) 3534.54 0.228400
\(622\) 26500.6 1.70832
\(623\) 0 0
\(624\) −5449.36 −0.349598
\(625\) −15118.1 −0.967556
\(626\) −17192.9 −1.09771
\(627\) 2703.80 0.172216
\(628\) 5323.92 0.338292
\(629\) 17262.5 1.09428
\(630\) 0 0
\(631\) −21316.2 −1.34483 −0.672413 0.740176i \(-0.734742\pi\)
−0.672413 + 0.740176i \(0.734742\pi\)
\(632\) 2565.40 0.161465
\(633\) 4998.82 0.313879
\(634\) −29897.7 −1.87286
\(635\) −14071.1 −0.879361
\(636\) −11082.3 −0.690947
\(637\) 0 0
\(638\) −11573.9 −0.718203
\(639\) 635.930 0.0393693
\(640\) 13412.4 0.828395
\(641\) −11687.9 −0.720193 −0.360097 0.932915i \(-0.617256\pi\)
−0.360097 + 0.932915i \(0.617256\pi\)
\(642\) −41035.5 −2.52265
\(643\) 20771.5 1.27395 0.636974 0.770885i \(-0.280186\pi\)
0.636974 + 0.770885i \(0.280186\pi\)
\(644\) 0 0
\(645\) −5557.56 −0.339270
\(646\) 16146.1 0.983372
\(647\) 15685.1 0.953086 0.476543 0.879151i \(-0.341890\pi\)
0.476543 + 0.879151i \(0.341890\pi\)
\(648\) −7776.96 −0.471463
\(649\) −8034.32 −0.485940
\(650\) 185.777 0.0112104
\(651\) 0 0
\(652\) −20048.2 −1.20422
\(653\) 5161.91 0.309343 0.154672 0.987966i \(-0.450568\pi\)
0.154672 + 0.987966i \(0.450568\pi\)
\(654\) −26464.3 −1.58232
\(655\) −31523.9 −1.88052
\(656\) −35436.6 −2.10910
\(657\) 199.775 0.0118629
\(658\) 0 0
\(659\) −19647.1 −1.16137 −0.580683 0.814130i \(-0.697214\pi\)
−0.580683 + 0.814130i \(0.697214\pi\)
\(660\) 3581.23 0.211211
\(661\) −25197.0 −1.48268 −0.741340 0.671130i \(-0.765809\pi\)
−0.741340 + 0.671130i \(0.765809\pi\)
\(662\) 26052.8 1.52956
\(663\) 7121.62 0.417166
\(664\) 1962.40 0.114693
\(665\) 0 0
\(666\) −910.077 −0.0529501
\(667\) 7062.56 0.409990
\(668\) 2209.81 0.127994
\(669\) −27986.2 −1.61735
\(670\) −13466.0 −0.776473
\(671\) −882.222 −0.0507568
\(672\) 0 0
\(673\) 31227.3 1.78860 0.894298 0.447472i \(-0.147676\pi\)
0.894298 + 0.447472i \(0.147676\pi\)
\(674\) −16599.0 −0.948619
\(675\) 535.381 0.0305286
\(676\) 880.526 0.0500982
\(677\) 1846.13 0.104804 0.0524022 0.998626i \(-0.483312\pi\)
0.0524022 + 0.998626i \(0.483312\pi\)
\(678\) −24778.4 −1.40355
\(679\) 0 0
\(680\) −11450.9 −0.645769
\(681\) 20396.4 1.14771
\(682\) 5155.23 0.289448
\(683\) −5829.61 −0.326594 −0.163297 0.986577i \(-0.552213\pi\)
−0.163297 + 0.986577i \(0.552213\pi\)
\(684\) −335.727 −0.0187673
\(685\) −16449.2 −0.917503
\(686\) 0 0
\(687\) −11448.1 −0.635770
\(688\) −7431.87 −0.411828
\(689\) −5180.62 −0.286453
\(690\) −5540.78 −0.305701
\(691\) 12829.3 0.706294 0.353147 0.935568i \(-0.385112\pi\)
0.353147 + 0.935568i \(0.385112\pi\)
\(692\) 16242.9 0.892286
\(693\) 0 0
\(694\) 16780.7 0.917847
\(695\) −19627.9 −1.07126
\(696\) −14725.2 −0.801949
\(697\) 46311.2 2.51673
\(698\) −27945.2 −1.51539
\(699\) −18537.4 −1.00307
\(700\) 0 0
\(701\) −30321.8 −1.63372 −0.816862 0.576834i \(-0.804288\pi\)
−0.816862 + 0.576834i \(0.804288\pi\)
\(702\) 6433.79 0.345908
\(703\) −7279.78 −0.390558
\(704\) 1338.40 0.0716517
\(705\) 2974.21 0.158887
\(706\) −29213.6 −1.55732
\(707\) 0 0
\(708\) 19090.5 1.01337
\(709\) 13767.1 0.729246 0.364623 0.931155i \(-0.381198\pi\)
0.364623 + 0.931155i \(0.381198\pi\)
\(710\) 17082.9 0.902971
\(711\) 376.657 0.0198674
\(712\) 10099.2 0.531576
\(713\) −3145.81 −0.165233
\(714\) 0 0
\(715\) 1674.11 0.0875638
\(716\) 15433.7 0.805566
\(717\) 19700.7 1.02613
\(718\) −618.288 −0.0321369
\(719\) 25769.7 1.33664 0.668322 0.743872i \(-0.267013\pi\)
0.668322 + 0.743872i \(0.267013\pi\)
\(720\) 1286.47 0.0665885
\(721\) 0 0
\(722\) 18120.7 0.934046
\(723\) 4882.77 0.251165
\(724\) 6243.24 0.320481
\(725\) 1069.77 0.0548006
\(726\) 23163.5 1.18413
\(727\) −20584.7 −1.05013 −0.525064 0.851063i \(-0.675959\pi\)
−0.525064 + 0.851063i \(0.675959\pi\)
\(728\) 0 0
\(729\) 18481.3 0.938949
\(730\) 5366.52 0.272087
\(731\) 9712.52 0.491424
\(732\) 2096.26 0.105847
\(733\) −12560.0 −0.632900 −0.316450 0.948609i \(-0.602491\pi\)
−0.316450 + 0.948609i \(0.602491\pi\)
\(734\) 23798.3 1.19674
\(735\) 0 0
\(736\) −5303.80 −0.265626
\(737\) 3940.88 0.196966
\(738\) −2441.52 −0.121780
\(739\) 2247.01 0.111850 0.0559252 0.998435i \(-0.482189\pi\)
0.0559252 + 0.998435i \(0.482189\pi\)
\(740\) −9642.19 −0.478992
\(741\) −3003.26 −0.148890
\(742\) 0 0
\(743\) −17437.7 −0.861006 −0.430503 0.902589i \(-0.641664\pi\)
−0.430503 + 0.902589i \(0.641664\pi\)
\(744\) 6558.89 0.323200
\(745\) −19002.5 −0.934494
\(746\) 46687.0 2.29133
\(747\) 288.124 0.0141123
\(748\) −6258.63 −0.305934
\(749\) 0 0
\(750\) −27521.2 −1.33991
\(751\) −13096.0 −0.636323 −0.318162 0.948036i \(-0.603065\pi\)
−0.318162 + 0.948036i \(0.603065\pi\)
\(752\) 3977.28 0.192867
\(753\) −31048.6 −1.50262
\(754\) 12855.7 0.620925
\(755\) −16502.2 −0.795464
\(756\) 0 0
\(757\) 29343.6 1.40887 0.704433 0.709771i \(-0.251202\pi\)
0.704433 + 0.709771i \(0.251202\pi\)
\(758\) −34510.0 −1.65364
\(759\) 1621.53 0.0775467
\(760\) 4828.96 0.230480
\(761\) 23788.7 1.13316 0.566582 0.824005i \(-0.308265\pi\)
0.566582 + 0.824005i \(0.308265\pi\)
\(762\) −24808.7 −1.17943
\(763\) 0 0
\(764\) −20533.4 −0.972345
\(765\) −1681.25 −0.0794584
\(766\) 23061.3 1.08778
\(767\) 8924.16 0.420121
\(768\) 28530.4 1.34050
\(769\) −11235.6 −0.526875 −0.263437 0.964676i \(-0.584856\pi\)
−0.263437 + 0.964676i \(0.584856\pi\)
\(770\) 0 0
\(771\) −16365.6 −0.764452
\(772\) −10285.7 −0.479519
\(773\) 7742.41 0.360252 0.180126 0.983644i \(-0.442349\pi\)
0.180126 + 0.983644i \(0.442349\pi\)
\(774\) −512.042 −0.0237790
\(775\) −476.499 −0.0220856
\(776\) −7726.29 −0.357420
\(777\) 0 0
\(778\) 47964.6 2.21030
\(779\) −19529.9 −0.898242
\(780\) −3977.87 −0.182603
\(781\) −4999.38 −0.229055
\(782\) 9683.18 0.442800
\(783\) 37048.2 1.69092
\(784\) 0 0
\(785\) −11243.2 −0.511195
\(786\) −55579.7 −2.52222
\(787\) −73.6706 −0.00333681 −0.00166841 0.999999i \(-0.500531\pi\)
−0.00166841 + 0.999999i \(0.500531\pi\)
\(788\) 14445.9 0.653063
\(789\) 6485.09 0.292618
\(790\) 10118.1 0.455678
\(791\) 0 0
\(792\) −176.673 −0.00792650
\(793\) 979.932 0.0438820
\(794\) −117.776 −0.00526412
\(795\) 23404.0 1.04409
\(796\) 4792.72 0.213409
\(797\) −5805.43 −0.258016 −0.129008 0.991644i \(-0.541179\pi\)
−0.129008 + 0.991644i \(0.541179\pi\)
\(798\) 0 0
\(799\) −5197.80 −0.230144
\(800\) −803.373 −0.0355044
\(801\) 1482.78 0.0654076
\(802\) −14004.3 −0.616596
\(803\) −1570.53 −0.0690199
\(804\) −9363.98 −0.410749
\(805\) 0 0
\(806\) −5726.19 −0.250244
\(807\) −19694.0 −0.859059
\(808\) −8335.44 −0.362921
\(809\) 27095.9 1.17755 0.588776 0.808296i \(-0.299610\pi\)
0.588776 + 0.808296i \(0.299610\pi\)
\(810\) −30672.8 −1.33053
\(811\) 28006.3 1.21262 0.606309 0.795229i \(-0.292650\pi\)
0.606309 + 0.795229i \(0.292650\pi\)
\(812\) 0 0
\(813\) −30274.7 −1.30600
\(814\) 7154.60 0.308070
\(815\) 42338.5 1.81970
\(816\) −43023.0 −1.84572
\(817\) −4095.87 −0.175393
\(818\) 37151.7 1.58799
\(819\) 0 0
\(820\) −25867.7 −1.10163
\(821\) −16408.0 −0.697493 −0.348747 0.937217i \(-0.613393\pi\)
−0.348747 + 0.937217i \(0.613393\pi\)
\(822\) −29001.5 −1.23059
\(823\) 7614.47 0.322507 0.161254 0.986913i \(-0.448446\pi\)
0.161254 + 0.986913i \(0.448446\pi\)
\(824\) −8057.11 −0.340634
\(825\) 245.615 0.0103651
\(826\) 0 0
\(827\) −17850.7 −0.750581 −0.375290 0.926907i \(-0.622457\pi\)
−0.375290 + 0.926907i \(0.622457\pi\)
\(828\) −201.343 −0.00845068
\(829\) 3802.11 0.159292 0.0796459 0.996823i \(-0.474621\pi\)
0.0796459 + 0.996823i \(0.474621\pi\)
\(830\) 7739.83 0.323679
\(831\) 33448.0 1.39627
\(832\) −1486.63 −0.0619468
\(833\) 0 0
\(834\) −34605.9 −1.43681
\(835\) −4666.74 −0.193412
\(836\) 2639.33 0.109190
\(837\) −16502.0 −0.681472
\(838\) −44173.2 −1.82093
\(839\) 3538.68 0.145613 0.0728063 0.997346i \(-0.476805\pi\)
0.0728063 + 0.997346i \(0.476805\pi\)
\(840\) 0 0
\(841\) 49639.0 2.03530
\(842\) −49542.7 −2.02774
\(843\) −38018.5 −1.55329
\(844\) 4879.62 0.199009
\(845\) −1859.52 −0.0757036
\(846\) 274.027 0.0111362
\(847\) 0 0
\(848\) 31297.1 1.26739
\(849\) 13750.3 0.555841
\(850\) 1466.72 0.0591861
\(851\) −4365.86 −0.175863
\(852\) 11879.1 0.477666
\(853\) 21190.1 0.850570 0.425285 0.905059i \(-0.360174\pi\)
0.425285 + 0.905059i \(0.360174\pi\)
\(854\) 0 0
\(855\) 708.999 0.0283594
\(856\) 21448.4 0.856415
\(857\) 19184.4 0.764673 0.382337 0.924023i \(-0.375119\pi\)
0.382337 + 0.924023i \(0.375119\pi\)
\(858\) 2951.62 0.117443
\(859\) 37876.3 1.50445 0.752226 0.658905i \(-0.228980\pi\)
0.752226 + 0.658905i \(0.228980\pi\)
\(860\) −5425.04 −0.215108
\(861\) 0 0
\(862\) 21659.6 0.855834
\(863\) 9543.13 0.376422 0.188211 0.982129i \(-0.439731\pi\)
0.188211 + 0.982129i \(0.439731\pi\)
\(864\) −27822.2 −1.09552
\(865\) −34302.3 −1.34834
\(866\) 47319.1 1.85678
\(867\) 30002.6 1.17525
\(868\) 0 0
\(869\) −2961.10 −0.115591
\(870\) −58077.0 −2.26321
\(871\) −4377.35 −0.170288
\(872\) 13832.3 0.537181
\(873\) −1134.39 −0.0439786
\(874\) −4083.49 −0.158039
\(875\) 0 0
\(876\) 3731.77 0.143932
\(877\) 27017.4 1.04027 0.520133 0.854085i \(-0.325882\pi\)
0.520133 + 0.854085i \(0.325882\pi\)
\(878\) −3244.74 −0.124720
\(879\) 37123.1 1.42449
\(880\) −10113.6 −0.387419
\(881\) 31954.5 1.22199 0.610996 0.791634i \(-0.290769\pi\)
0.610996 + 0.791634i \(0.290769\pi\)
\(882\) 0 0
\(883\) 42055.4 1.60280 0.801402 0.598126i \(-0.204088\pi\)
0.801402 + 0.598126i \(0.204088\pi\)
\(884\) 6951.81 0.264496
\(885\) −40315.8 −1.53130
\(886\) 31910.7 1.21000
\(887\) −34830.5 −1.31848 −0.659242 0.751931i \(-0.729123\pi\)
−0.659242 + 0.751931i \(0.729123\pi\)
\(888\) 9102.66 0.343992
\(889\) 0 0
\(890\) 39831.7 1.50018
\(891\) 8976.54 0.337514
\(892\) −27318.9 −1.02545
\(893\) 2191.96 0.0821403
\(894\) −33503.3 −1.25338
\(895\) −32593.4 −1.21729
\(896\) 0 0
\(897\) −1801.13 −0.0670433
\(898\) −47693.9 −1.77234
\(899\) −32973.6 −1.22328
\(900\) −30.4977 −0.00112954
\(901\) −40901.3 −1.51234
\(902\) 19194.1 0.708528
\(903\) 0 0
\(904\) 12951.1 0.476492
\(905\) −13184.7 −0.484281
\(906\) −29094.9 −1.06690
\(907\) 28035.7 1.02636 0.513180 0.858281i \(-0.328467\pi\)
0.513180 + 0.858281i \(0.328467\pi\)
\(908\) 19910.1 0.727686
\(909\) −1223.83 −0.0446554
\(910\) 0 0
\(911\) 19163.8 0.696954 0.348477 0.937317i \(-0.386699\pi\)
0.348477 + 0.937317i \(0.386699\pi\)
\(912\) 18143.2 0.658753
\(913\) −2265.10 −0.0821070
\(914\) 33076.8 1.19703
\(915\) −4426.95 −0.159946
\(916\) −11175.2 −0.403098
\(917\) 0 0
\(918\) 50795.2 1.82624
\(919\) −34654.1 −1.24389 −0.621945 0.783061i \(-0.713657\pi\)
−0.621945 + 0.783061i \(0.713657\pi\)
\(920\) 2896.04 0.103782
\(921\) −9022.18 −0.322791
\(922\) −40538.8 −1.44802
\(923\) 5553.09 0.198031
\(924\) 0 0
\(925\) −661.301 −0.0235064
\(926\) 36368.9 1.29067
\(927\) −1182.96 −0.0419132
\(928\) −55593.1 −1.96652
\(929\) −42110.2 −1.48718 −0.743591 0.668635i \(-0.766879\pi\)
−0.743591 + 0.668635i \(0.766879\pi\)
\(930\) 25868.7 0.912115
\(931\) 0 0
\(932\) −18095.4 −0.635980
\(933\) −38916.8 −1.36557
\(934\) 38001.6 1.33132
\(935\) 13217.2 0.462298
\(936\) 196.240 0.00685289
\(937\) 18514.6 0.645514 0.322757 0.946482i \(-0.395390\pi\)
0.322757 + 0.946482i \(0.395390\pi\)
\(938\) 0 0
\(939\) 25248.3 0.877471
\(940\) 2903.29 0.100739
\(941\) 8119.45 0.281282 0.140641 0.990061i \(-0.455084\pi\)
0.140641 + 0.990061i \(0.455084\pi\)
\(942\) −19822.9 −0.685632
\(943\) −11712.5 −0.404467
\(944\) −53912.5 −1.85880
\(945\) 0 0
\(946\) 4025.44 0.138349
\(947\) 49354.7 1.69357 0.846786 0.531934i \(-0.178534\pi\)
0.846786 + 0.531934i \(0.178534\pi\)
\(948\) 7035.91 0.241050
\(949\) 1744.48 0.0596714
\(950\) −618.531 −0.0211240
\(951\) 43905.6 1.49709
\(952\) 0 0
\(953\) −300.006 −0.0101974 −0.00509872 0.999987i \(-0.501623\pi\)
−0.00509872 + 0.999987i \(0.501623\pi\)
\(954\) 2156.31 0.0731793
\(955\) 43363.1 1.46932
\(956\) 19230.9 0.650598
\(957\) 16996.5 0.574106
\(958\) −40565.4 −1.36807
\(959\) 0 0
\(960\) 6716.02 0.225790
\(961\) −15103.9 −0.506996
\(962\) −7947.01 −0.266343
\(963\) 3149.10 0.105377
\(964\) 4766.34 0.159246
\(965\) 21721.6 0.724604
\(966\) 0 0
\(967\) −20103.3 −0.668539 −0.334270 0.942477i \(-0.608490\pi\)
−0.334270 + 0.942477i \(0.608490\pi\)
\(968\) −12107.1 −0.402000
\(969\) −23710.9 −0.786073
\(970\) −30473.0 −1.00869
\(971\) −39115.5 −1.29277 −0.646383 0.763013i \(-0.723719\pi\)
−0.646383 + 0.763013i \(0.723719\pi\)
\(972\) −2174.01 −0.0717402
\(973\) 0 0
\(974\) −14324.5 −0.471239
\(975\) −272.819 −0.00896122
\(976\) −5919.95 −0.194153
\(977\) 21590.6 0.707005 0.353502 0.935434i \(-0.384991\pi\)
0.353502 + 0.935434i \(0.384991\pi\)
\(978\) 74646.9 2.44064
\(979\) −11656.9 −0.380549
\(980\) 0 0
\(981\) 2030.89 0.0660973
\(982\) −42293.9 −1.37439
\(983\) 7689.51 0.249499 0.124749 0.992188i \(-0.460187\pi\)
0.124749 + 0.992188i \(0.460187\pi\)
\(984\) 24420.2 0.791147
\(985\) −30507.3 −0.986847
\(986\) 101497. 3.27821
\(987\) 0 0
\(988\) −2931.65 −0.0944009
\(989\) −2456.39 −0.0789774
\(990\) −696.807 −0.0223697
\(991\) 23913.5 0.766537 0.383269 0.923637i \(-0.374798\pi\)
0.383269 + 0.923637i \(0.374798\pi\)
\(992\) 24762.3 0.792544
\(993\) −38259.3 −1.22268
\(994\) 0 0
\(995\) −10121.4 −0.322483
\(996\) 5382.12 0.171224
\(997\) 33.8007 0.00107370 0.000536851 1.00000i \(-0.499829\pi\)
0.000536851 1.00000i \(0.499829\pi\)
\(998\) −8615.97 −0.273280
\(999\) −22902.0 −0.725314
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 637.4.a.d.1.2 4
7.6 odd 2 91.4.a.b.1.2 4
21.20 even 2 819.4.a.h.1.3 4
28.27 even 2 1456.4.a.s.1.3 4
35.34 odd 2 2275.4.a.h.1.3 4
91.90 odd 2 1183.4.a.e.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.4.a.b.1.2 4 7.6 odd 2
637.4.a.d.1.2 4 1.1 even 1 trivial
819.4.a.h.1.3 4 21.20 even 2
1183.4.a.e.1.3 4 91.90 odd 2
1456.4.a.s.1.3 4 28.27 even 2
2275.4.a.h.1.3 4 35.34 odd 2