Properties

Label 1045.4.a.c.1.9
Level $1045$
Weight $4$
Character 1045.1
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.6569959560\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + 1317797 x^{12} - 1160501 x^{11} - 9914845 x^{10} + 7570653 x^{9} + 42786958 x^{8} - 23777633 x^{7} - 102801526 x^{6} + 28436356 x^{5} + 122325928 x^{4} + 411232 x^{3} - 47350496 x^{2} - 4782848 x + 150528\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(0.783853\) of defining polynomial
Character \(\chi\) \(=\) 1045.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.783853 q^{2} +0.0603595 q^{3} -7.38557 q^{4} -5.00000 q^{5} -0.0473130 q^{6} -1.09978 q^{7} +12.0600 q^{8} -26.9964 q^{9} +O(q^{10})\) \(q-0.783853 q^{2} +0.0603595 q^{3} -7.38557 q^{4} -5.00000 q^{5} -0.0473130 q^{6} -1.09978 q^{7} +12.0600 q^{8} -26.9964 q^{9} +3.91926 q^{10} -11.0000 q^{11} -0.445790 q^{12} -22.1285 q^{13} +0.862068 q^{14} -0.301798 q^{15} +49.6313 q^{16} +88.1083 q^{17} +21.1612 q^{18} +19.0000 q^{19} +36.9279 q^{20} -0.0663823 q^{21} +8.62238 q^{22} +168.327 q^{23} +0.727938 q^{24} +25.0000 q^{25} +17.3455 q^{26} -3.25919 q^{27} +8.12252 q^{28} +214.076 q^{29} +0.236565 q^{30} -36.7173 q^{31} -135.384 q^{32} -0.663955 q^{33} -69.0640 q^{34} +5.49891 q^{35} +199.384 q^{36} -218.260 q^{37} -14.8932 q^{38} -1.33567 q^{39} -60.3001 q^{40} +28.0068 q^{41} +0.0520340 q^{42} -376.154 q^{43} +81.2413 q^{44} +134.982 q^{45} -131.944 q^{46} -249.449 q^{47} +2.99572 q^{48} -341.790 q^{49} -19.5963 q^{50} +5.31818 q^{51} +163.432 q^{52} +446.371 q^{53} +2.55473 q^{54} +55.0000 q^{55} -13.2634 q^{56} +1.14683 q^{57} -167.804 q^{58} +826.710 q^{59} +2.22895 q^{60} +349.824 q^{61} +28.7809 q^{62} +29.6901 q^{63} -290.929 q^{64} +110.643 q^{65} +0.520443 q^{66} -340.516 q^{67} -650.731 q^{68} +10.1602 q^{69} -4.31034 q^{70} -1068.42 q^{71} -325.577 q^{72} -99.5645 q^{73} +171.084 q^{74} +1.50899 q^{75} -140.326 q^{76} +12.0976 q^{77} +1.04697 q^{78} +1027.99 q^{79} -248.157 q^{80} +728.705 q^{81} -21.9532 q^{82} +609.027 q^{83} +0.490272 q^{84} -440.542 q^{85} +294.849 q^{86} +12.9215 q^{87} -132.660 q^{88} +175.527 q^{89} -105.806 q^{90} +24.3366 q^{91} -1243.19 q^{92} -2.21624 q^{93} +195.531 q^{94} -95.0000 q^{95} -8.17171 q^{96} +38.6274 q^{97} +267.913 q^{98} +296.960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + O(q^{10}) \) \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.783853 −0.277134 −0.138567 0.990353i \(-0.544250\pi\)
−0.138567 + 0.990353i \(0.544250\pi\)
\(3\) 0.0603595 0.0116162 0.00580810 0.999983i \(-0.498151\pi\)
0.00580810 + 0.999983i \(0.498151\pi\)
\(4\) −7.38557 −0.923197
\(5\) −5.00000 −0.447214
\(6\) −0.0473130 −0.00321924
\(7\) −1.09978 −0.0593827 −0.0296913 0.999559i \(-0.509452\pi\)
−0.0296913 + 0.999559i \(0.509452\pi\)
\(8\) 12.0600 0.532983
\(9\) −26.9964 −0.999865
\(10\) 3.91926 0.123938
\(11\) −11.0000 −0.301511
\(12\) −0.445790 −0.0107240
\(13\) −22.1285 −0.472104 −0.236052 0.971740i \(-0.575854\pi\)
−0.236052 + 0.971740i \(0.575854\pi\)
\(14\) 0.862068 0.0164569
\(15\) −0.301798 −0.00519492
\(16\) 49.6313 0.775489
\(17\) 88.1083 1.25702 0.628512 0.777800i \(-0.283664\pi\)
0.628512 + 0.777800i \(0.283664\pi\)
\(18\) 21.1612 0.277096
\(19\) 19.0000 0.229416
\(20\) 36.9279 0.412866
\(21\) −0.0663823 −0.000689801 0
\(22\) 8.62238 0.0835590
\(23\) 168.327 1.52603 0.763015 0.646381i \(-0.223718\pi\)
0.763015 + 0.646381i \(0.223718\pi\)
\(24\) 0.727938 0.00619123
\(25\) 25.0000 0.200000
\(26\) 17.3455 0.130836
\(27\) −3.25919 −0.0232308
\(28\) 8.12252 0.0548219
\(29\) 214.076 1.37079 0.685395 0.728171i \(-0.259629\pi\)
0.685395 + 0.728171i \(0.259629\pi\)
\(30\) 0.236565 0.00143969
\(31\) −36.7173 −0.212730 −0.106365 0.994327i \(-0.533921\pi\)
−0.106365 + 0.994327i \(0.533921\pi\)
\(32\) −135.384 −0.747897
\(33\) −0.663955 −0.00350241
\(34\) −69.0640 −0.348364
\(35\) 5.49891 0.0265567
\(36\) 199.384 0.923072
\(37\) −218.260 −0.969778 −0.484889 0.874576i \(-0.661140\pi\)
−0.484889 + 0.874576i \(0.661140\pi\)
\(38\) −14.8932 −0.0635789
\(39\) −1.33567 −0.00548405
\(40\) −60.3001 −0.238357
\(41\) 28.0068 0.106681 0.0533406 0.998576i \(-0.483013\pi\)
0.0533406 + 0.998576i \(0.483013\pi\)
\(42\) 0.0520340 0.000191167 0
\(43\) −376.154 −1.33402 −0.667011 0.745048i \(-0.732427\pi\)
−0.667011 + 0.745048i \(0.732427\pi\)
\(44\) 81.2413 0.278354
\(45\) 134.982 0.447153
\(46\) −131.944 −0.422915
\(47\) −249.449 −0.774168 −0.387084 0.922044i \(-0.626518\pi\)
−0.387084 + 0.922044i \(0.626518\pi\)
\(48\) 2.99572 0.00900823
\(49\) −341.790 −0.996474
\(50\) −19.5963 −0.0554268
\(51\) 5.31818 0.0146018
\(52\) 163.432 0.435845
\(53\) 446.371 1.15686 0.578432 0.815731i \(-0.303665\pi\)
0.578432 + 0.815731i \(0.303665\pi\)
\(54\) 2.55473 0.00643805
\(55\) 55.0000 0.134840
\(56\) −13.2634 −0.0316499
\(57\) 1.14683 0.00266494
\(58\) −167.804 −0.379893
\(59\) 826.710 1.82421 0.912106 0.409955i \(-0.134456\pi\)
0.912106 + 0.409955i \(0.134456\pi\)
\(60\) 2.22895 0.00479593
\(61\) 349.824 0.734269 0.367134 0.930168i \(-0.380339\pi\)
0.367134 + 0.930168i \(0.380339\pi\)
\(62\) 28.7809 0.0589546
\(63\) 29.6901 0.0593746
\(64\) −290.929 −0.568222
\(65\) 110.643 0.211131
\(66\) 0.520443 0.000970638 0
\(67\) −340.516 −0.620906 −0.310453 0.950589i \(-0.600481\pi\)
−0.310453 + 0.950589i \(0.600481\pi\)
\(68\) −650.731 −1.16048
\(69\) 10.1602 0.0177267
\(70\) −4.31034 −0.00735977
\(71\) −1068.42 −1.78588 −0.892942 0.450172i \(-0.851363\pi\)
−0.892942 + 0.450172i \(0.851363\pi\)
\(72\) −325.577 −0.532911
\(73\) −99.5645 −0.159632 −0.0798161 0.996810i \(-0.525433\pi\)
−0.0798161 + 0.996810i \(0.525433\pi\)
\(74\) 171.084 0.268758
\(75\) 1.50899 0.00232324
\(76\) −140.326 −0.211796
\(77\) 12.0976 0.0179045
\(78\) 1.04697 0.00151982
\(79\) 1027.99 1.46402 0.732012 0.681292i \(-0.238582\pi\)
0.732012 + 0.681292i \(0.238582\pi\)
\(80\) −248.157 −0.346809
\(81\) 728.705 0.999595
\(82\) −21.9532 −0.0295650
\(83\) 609.027 0.805414 0.402707 0.915329i \(-0.368069\pi\)
0.402707 + 0.915329i \(0.368069\pi\)
\(84\) 0.490272 0.000636822 0
\(85\) −440.542 −0.562158
\(86\) 294.849 0.369703
\(87\) 12.9215 0.0159234
\(88\) −132.660 −0.160700
\(89\) 175.527 0.209055 0.104527 0.994522i \(-0.466667\pi\)
0.104527 + 0.994522i \(0.466667\pi\)
\(90\) −105.806 −0.123921
\(91\) 24.3366 0.0280348
\(92\) −1243.19 −1.40883
\(93\) −2.21624 −0.00247111
\(94\) 195.531 0.214548
\(95\) −95.0000 −0.102598
\(96\) −8.17171 −0.00868772
\(97\) 38.6274 0.0404332 0.0202166 0.999796i \(-0.493564\pi\)
0.0202166 + 0.999796i \(0.493564\pi\)
\(98\) 267.913 0.276157
\(99\) 296.960 0.301471
\(100\) −184.639 −0.184639
\(101\) −592.060 −0.583289 −0.291644 0.956527i \(-0.594202\pi\)
−0.291644 + 0.956527i \(0.594202\pi\)
\(102\) −4.16867 −0.00404666
\(103\) 886.258 0.847821 0.423910 0.905704i \(-0.360657\pi\)
0.423910 + 0.905704i \(0.360657\pi\)
\(104\) −266.871 −0.251623
\(105\) 0.331912 0.000308488 0
\(106\) −349.889 −0.320606
\(107\) −1745.46 −1.57701 −0.788506 0.615027i \(-0.789145\pi\)
−0.788506 + 0.615027i \(0.789145\pi\)
\(108\) 24.0710 0.0214466
\(109\) −1923.93 −1.69063 −0.845316 0.534266i \(-0.820588\pi\)
−0.845316 + 0.534266i \(0.820588\pi\)
\(110\) −43.1119 −0.0373687
\(111\) −13.1741 −0.0112651
\(112\) −54.5836 −0.0460506
\(113\) −720.292 −0.599641 −0.299820 0.953996i \(-0.596927\pi\)
−0.299820 + 0.953996i \(0.596927\pi\)
\(114\) −0.898947 −0.000738545 0
\(115\) −841.637 −0.682461
\(116\) −1581.08 −1.26551
\(117\) 597.390 0.472040
\(118\) −648.019 −0.505551
\(119\) −96.9000 −0.0746454
\(120\) −3.63969 −0.00276880
\(121\) 121.000 0.0909091
\(122\) −274.211 −0.203491
\(123\) 1.69048 0.00123923
\(124\) 271.178 0.196391
\(125\) −125.000 −0.0894427
\(126\) −23.2727 −0.0164547
\(127\) −1879.77 −1.31341 −0.656704 0.754149i \(-0.728050\pi\)
−0.656704 + 0.754149i \(0.728050\pi\)
\(128\) 1311.12 0.905371
\(129\) −22.7045 −0.0154963
\(130\) −86.7276 −0.0585117
\(131\) −77.2212 −0.0515027 −0.0257513 0.999668i \(-0.508198\pi\)
−0.0257513 + 0.999668i \(0.508198\pi\)
\(132\) 4.90369 0.00323342
\(133\) −20.8959 −0.0136233
\(134\) 266.915 0.172074
\(135\) 16.2960 0.0103891
\(136\) 1062.59 0.669972
\(137\) −244.748 −0.152629 −0.0763147 0.997084i \(-0.524315\pi\)
−0.0763147 + 0.997084i \(0.524315\pi\)
\(138\) −7.96407 −0.00491266
\(139\) −1554.03 −0.948279 −0.474139 0.880450i \(-0.657241\pi\)
−0.474139 + 0.880450i \(0.657241\pi\)
\(140\) −40.6126 −0.0245171
\(141\) −15.0566 −0.00899289
\(142\) 837.482 0.494929
\(143\) 243.414 0.142345
\(144\) −1339.86 −0.775385
\(145\) −1070.38 −0.613036
\(146\) 78.0440 0.0442395
\(147\) −20.6303 −0.0115752
\(148\) 1611.98 0.895296
\(149\) −1846.74 −1.01538 −0.507688 0.861541i \(-0.669500\pi\)
−0.507688 + 0.861541i \(0.669500\pi\)
\(150\) −1.18282 −0.000643848 0
\(151\) −1013.96 −0.546458 −0.273229 0.961949i \(-0.588092\pi\)
−0.273229 + 0.961949i \(0.588092\pi\)
\(152\) 229.141 0.122275
\(153\) −2378.60 −1.25685
\(154\) −9.48274 −0.00496196
\(155\) 183.586 0.0951355
\(156\) 9.86468 0.00506286
\(157\) −3340.56 −1.69812 −0.849062 0.528293i \(-0.822832\pi\)
−0.849062 + 0.528293i \(0.822832\pi\)
\(158\) −805.793 −0.405731
\(159\) 26.9428 0.0134384
\(160\) 676.919 0.334470
\(161\) −185.123 −0.0906197
\(162\) −571.197 −0.277022
\(163\) 720.474 0.346208 0.173104 0.984904i \(-0.444620\pi\)
0.173104 + 0.984904i \(0.444620\pi\)
\(164\) −206.846 −0.0984878
\(165\) 3.31977 0.00156633
\(166\) −477.388 −0.223208
\(167\) −278.976 −0.129268 −0.0646342 0.997909i \(-0.520588\pi\)
−0.0646342 + 0.997909i \(0.520588\pi\)
\(168\) −0.800573 −0.000367652 0
\(169\) −1707.33 −0.777118
\(170\) 345.320 0.155793
\(171\) −512.931 −0.229385
\(172\) 2778.11 1.23157
\(173\) −228.531 −0.100433 −0.0502165 0.998738i \(-0.515991\pi\)
−0.0502165 + 0.998738i \(0.515991\pi\)
\(174\) −10.1286 −0.00441291
\(175\) −27.4946 −0.0118765
\(176\) −545.944 −0.233819
\(177\) 49.8998 0.0211904
\(178\) −137.588 −0.0579361
\(179\) 3443.47 1.43786 0.718930 0.695083i \(-0.244632\pi\)
0.718930 + 0.695083i \(0.244632\pi\)
\(180\) −996.918 −0.412810
\(181\) 3327.36 1.36641 0.683207 0.730225i \(-0.260585\pi\)
0.683207 + 0.730225i \(0.260585\pi\)
\(182\) −19.0763 −0.00776939
\(183\) 21.1152 0.00852941
\(184\) 2030.03 0.813348
\(185\) 1091.30 0.433698
\(186\) 1.73720 0.000684828 0
\(187\) −969.192 −0.379007
\(188\) 1842.32 0.714709
\(189\) 3.58440 0.00137951
\(190\) 74.4660 0.0284333
\(191\) 1998.23 0.757000 0.378500 0.925601i \(-0.376440\pi\)
0.378500 + 0.925601i \(0.376440\pi\)
\(192\) −17.5604 −0.00660057
\(193\) −1524.15 −0.568448 −0.284224 0.958758i \(-0.591736\pi\)
−0.284224 + 0.958758i \(0.591736\pi\)
\(194\) −30.2782 −0.0112054
\(195\) 6.67834 0.00245254
\(196\) 2524.32 0.919941
\(197\) −281.245 −0.101715 −0.0508575 0.998706i \(-0.516195\pi\)
−0.0508575 + 0.998706i \(0.516195\pi\)
\(198\) −232.773 −0.0835477
\(199\) −3911.67 −1.39342 −0.696711 0.717352i \(-0.745354\pi\)
−0.696711 + 0.717352i \(0.745354\pi\)
\(200\) 301.501 0.106597
\(201\) −20.5534 −0.00721256
\(202\) 464.088 0.161649
\(203\) −235.437 −0.0814012
\(204\) −39.2778 −0.0134804
\(205\) −140.034 −0.0477093
\(206\) −694.696 −0.234960
\(207\) −4544.23 −1.52582
\(208\) −1098.27 −0.366112
\(209\) −209.000 −0.0691714
\(210\) −0.260170 −8.54925e−5 0
\(211\) 297.386 0.0970279 0.0485139 0.998823i \(-0.484551\pi\)
0.0485139 + 0.998823i \(0.484551\pi\)
\(212\) −3296.71 −1.06801
\(213\) −64.4891 −0.0207452
\(214\) 1368.19 0.437043
\(215\) 1880.77 0.596593
\(216\) −39.3060 −0.0123816
\(217\) 40.3810 0.0126324
\(218\) 1508.08 0.468532
\(219\) −6.00967 −0.00185432
\(220\) −406.207 −0.124484
\(221\) −1949.71 −0.593446
\(222\) 10.3266 0.00312195
\(223\) 2419.03 0.726413 0.363206 0.931709i \(-0.381682\pi\)
0.363206 + 0.931709i \(0.381682\pi\)
\(224\) 148.893 0.0444121
\(225\) −674.909 −0.199973
\(226\) 564.603 0.166181
\(227\) −3083.26 −0.901512 −0.450756 0.892647i \(-0.648846\pi\)
−0.450756 + 0.892647i \(0.648846\pi\)
\(228\) −8.47001 −0.00246026
\(229\) 1552.11 0.447887 0.223944 0.974602i \(-0.428107\pi\)
0.223944 + 0.974602i \(0.428107\pi\)
\(230\) 659.720 0.189133
\(231\) 0.730206 0.000207983 0
\(232\) 2581.76 0.730608
\(233\) 1396.79 0.392733 0.196366 0.980531i \(-0.437086\pi\)
0.196366 + 0.980531i \(0.437086\pi\)
\(234\) −468.266 −0.130818
\(235\) 1247.25 0.346218
\(236\) −6105.73 −1.68411
\(237\) 62.0490 0.0170064
\(238\) 75.9553 0.0206868
\(239\) 2918.42 0.789861 0.394930 0.918711i \(-0.370769\pi\)
0.394930 + 0.918711i \(0.370769\pi\)
\(240\) −14.9786 −0.00402861
\(241\) −2402.93 −0.642267 −0.321133 0.947034i \(-0.604064\pi\)
−0.321133 + 0.947034i \(0.604064\pi\)
\(242\) −94.8462 −0.0251940
\(243\) 131.983 0.0348423
\(244\) −2583.65 −0.677875
\(245\) 1708.95 0.445637
\(246\) −1.32509 −0.000343433 0
\(247\) −420.442 −0.108308
\(248\) −442.811 −0.113381
\(249\) 36.7606 0.00935585
\(250\) 97.9816 0.0247876
\(251\) 4142.81 1.04180 0.520900 0.853617i \(-0.325596\pi\)
0.520900 + 0.853617i \(0.325596\pi\)
\(252\) −219.279 −0.0548145
\(253\) −1851.60 −0.460115
\(254\) 1473.46 0.363990
\(255\) −26.5909 −0.00653014
\(256\) 1299.71 0.317313
\(257\) −2926.38 −0.710283 −0.355142 0.934813i \(-0.615567\pi\)
−0.355142 + 0.934813i \(0.615567\pi\)
\(258\) 17.7970 0.00429454
\(259\) 240.039 0.0575880
\(260\) −817.160 −0.194916
\(261\) −5779.27 −1.37061
\(262\) 60.5301 0.0142731
\(263\) −2387.23 −0.559707 −0.279853 0.960043i \(-0.590286\pi\)
−0.279853 + 0.960043i \(0.590286\pi\)
\(264\) −8.00731 −0.00186673
\(265\) −2231.86 −0.517365
\(266\) 16.3793 0.00377548
\(267\) 10.5947 0.00242842
\(268\) 2514.91 0.573218
\(269\) −5830.08 −1.32144 −0.660718 0.750634i \(-0.729748\pi\)
−0.660718 + 0.750634i \(0.729748\pi\)
\(270\) −12.7736 −0.00287918
\(271\) −6575.21 −1.47386 −0.736930 0.675970i \(-0.763725\pi\)
−0.736930 + 0.675970i \(0.763725\pi\)
\(272\) 4372.93 0.974809
\(273\) 1.46894 0.000325658 0
\(274\) 191.846 0.0422988
\(275\) −275.000 −0.0603023
\(276\) −75.0386 −0.0163652
\(277\) 5770.55 1.25169 0.625846 0.779947i \(-0.284754\pi\)
0.625846 + 0.779947i \(0.284754\pi\)
\(278\) 1218.13 0.262800
\(279\) 991.232 0.212701
\(280\) 66.3170 0.0141543
\(281\) 372.824 0.0791488 0.0395744 0.999217i \(-0.487400\pi\)
0.0395744 + 0.999217i \(0.487400\pi\)
\(282\) 11.8022 0.00249223
\(283\) 1853.03 0.389226 0.194613 0.980880i \(-0.437655\pi\)
0.194613 + 0.980880i \(0.437655\pi\)
\(284\) 7890.87 1.64872
\(285\) −5.73416 −0.00119180
\(286\) −190.801 −0.0394486
\(287\) −30.8014 −0.00633501
\(288\) 3654.87 0.747796
\(289\) 2850.08 0.580109
\(290\) 839.021 0.169893
\(291\) 2.33153 0.000469680 0
\(292\) 735.341 0.147372
\(293\) −3682.91 −0.734328 −0.367164 0.930156i \(-0.619671\pi\)
−0.367164 + 0.930156i \(0.619671\pi\)
\(294\) 16.1711 0.00320789
\(295\) −4133.55 −0.815812
\(296\) −2632.23 −0.516875
\(297\) 35.8511 0.00700436
\(298\) 1447.57 0.281395
\(299\) −3724.84 −0.720445
\(300\) −11.1447 −0.00214481
\(301\) 413.688 0.0792178
\(302\) 794.798 0.151442
\(303\) −35.7364 −0.00677559
\(304\) 942.995 0.177909
\(305\) −1749.12 −0.328375
\(306\) 1864.48 0.348317
\(307\) −4358.02 −0.810180 −0.405090 0.914277i \(-0.632760\pi\)
−0.405090 + 0.914277i \(0.632760\pi\)
\(308\) −89.3478 −0.0165294
\(309\) 53.4941 0.00984845
\(310\) −143.905 −0.0263653
\(311\) −467.150 −0.0851756 −0.0425878 0.999093i \(-0.513560\pi\)
−0.0425878 + 0.999093i \(0.513560\pi\)
\(312\) −16.1082 −0.00292291
\(313\) 8790.72 1.58748 0.793740 0.608258i \(-0.208131\pi\)
0.793740 + 0.608258i \(0.208131\pi\)
\(314\) 2618.51 0.470608
\(315\) −148.451 −0.0265531
\(316\) −7592.29 −1.35158
\(317\) 636.066 0.112697 0.0563486 0.998411i \(-0.482054\pi\)
0.0563486 + 0.998411i \(0.482054\pi\)
\(318\) −21.1192 −0.00372422
\(319\) −2354.84 −0.413309
\(320\) 1454.65 0.254116
\(321\) −105.355 −0.0183189
\(322\) 145.110 0.0251138
\(323\) 1674.06 0.288381
\(324\) −5381.90 −0.922823
\(325\) −553.214 −0.0944208
\(326\) −564.746 −0.0959459
\(327\) −116.127 −0.0196387
\(328\) 337.763 0.0568593
\(329\) 274.340 0.0459721
\(330\) −2.60221 −0.000434082 0
\(331\) −8584.94 −1.42559 −0.712796 0.701371i \(-0.752572\pi\)
−0.712796 + 0.701371i \(0.752572\pi\)
\(332\) −4498.01 −0.743556
\(333\) 5892.24 0.969647
\(334\) 218.676 0.0358246
\(335\) 1702.58 0.277677
\(336\) −3.29464 −0.000534933 0
\(337\) −1382.13 −0.223411 −0.111706 0.993741i \(-0.535631\pi\)
−0.111706 + 0.993741i \(0.535631\pi\)
\(338\) 1338.29 0.215366
\(339\) −43.4765 −0.00696554
\(340\) 3253.65 0.518983
\(341\) 403.890 0.0641404
\(342\) 402.062 0.0635703
\(343\) 753.120 0.118556
\(344\) −4536.43 −0.711011
\(345\) −50.8008 −0.00792760
\(346\) 179.135 0.0278334
\(347\) 11869.6 1.83629 0.918143 0.396249i \(-0.129688\pi\)
0.918143 + 0.396249i \(0.129688\pi\)
\(348\) −95.4329 −0.0147004
\(349\) −413.020 −0.0633481 −0.0316740 0.999498i \(-0.510084\pi\)
−0.0316740 + 0.999498i \(0.510084\pi\)
\(350\) 21.5517 0.00329139
\(351\) 72.1212 0.0109674
\(352\) 1489.22 0.225500
\(353\) −3256.96 −0.491078 −0.245539 0.969387i \(-0.578965\pi\)
−0.245539 + 0.969387i \(0.578965\pi\)
\(354\) −39.1141 −0.00587258
\(355\) 5342.08 0.798672
\(356\) −1296.37 −0.192999
\(357\) −5.84884 −0.000867096 0
\(358\) −2699.17 −0.398479
\(359\) −5256.97 −0.772848 −0.386424 0.922321i \(-0.626290\pi\)
−0.386424 + 0.922321i \(0.626290\pi\)
\(360\) 1627.88 0.238325
\(361\) 361.000 0.0526316
\(362\) −2608.16 −0.378680
\(363\) 7.30350 0.00105602
\(364\) −179.740 −0.0258816
\(365\) 497.823 0.0713897
\(366\) −16.5512 −0.00236379
\(367\) 1958.55 0.278571 0.139286 0.990252i \(-0.455519\pi\)
0.139286 + 0.990252i \(0.455519\pi\)
\(368\) 8354.31 1.18342
\(369\) −756.082 −0.106667
\(370\) −855.420 −0.120192
\(371\) −490.911 −0.0686977
\(372\) 16.3682 0.00228132
\(373\) −5965.31 −0.828076 −0.414038 0.910260i \(-0.635882\pi\)
−0.414038 + 0.910260i \(0.635882\pi\)
\(374\) 759.704 0.105036
\(375\) −7.54494 −0.00103898
\(376\) −3008.36 −0.412618
\(377\) −4737.19 −0.647156
\(378\) −2.80965 −0.000382308 0
\(379\) 375.676 0.0509161 0.0254580 0.999676i \(-0.491896\pi\)
0.0254580 + 0.999676i \(0.491896\pi\)
\(380\) 701.630 0.0947180
\(381\) −113.462 −0.0152568
\(382\) −1566.32 −0.209790
\(383\) 292.289 0.0389954 0.0194977 0.999810i \(-0.493793\pi\)
0.0194977 + 0.999810i \(0.493793\pi\)
\(384\) 79.1384 0.0105170
\(385\) −60.4880 −0.00800716
\(386\) 1194.71 0.157536
\(387\) 10154.8 1.33384
\(388\) −285.286 −0.0373278
\(389\) −6104.86 −0.795704 −0.397852 0.917450i \(-0.630244\pi\)
−0.397852 + 0.917450i \(0.630244\pi\)
\(390\) −5.23484 −0.000679683 0
\(391\) 14831.0 1.91826
\(392\) −4122.00 −0.531104
\(393\) −4.66104 −0.000598265 0
\(394\) 220.455 0.0281887
\(395\) −5139.95 −0.654731
\(396\) −2193.22 −0.278317
\(397\) 2537.19 0.320751 0.160375 0.987056i \(-0.448730\pi\)
0.160375 + 0.987056i \(0.448730\pi\)
\(398\) 3066.17 0.386164
\(399\) −1.26126 −0.000158251 0
\(400\) 1240.78 0.155098
\(401\) −2019.82 −0.251534 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(402\) 16.1108 0.00199884
\(403\) 812.500 0.100430
\(404\) 4372.70 0.538490
\(405\) −3643.52 −0.447033
\(406\) 184.548 0.0225590
\(407\) 2400.86 0.292399
\(408\) 64.1374 0.00778253
\(409\) −5930.91 −0.717029 −0.358514 0.933524i \(-0.616717\pi\)
−0.358514 + 0.933524i \(0.616717\pi\)
\(410\) 109.766 0.0132219
\(411\) −14.7729 −0.00177297
\(412\) −6545.52 −0.782706
\(413\) −909.201 −0.108327
\(414\) 3562.00 0.422857
\(415\) −3045.14 −0.360192
\(416\) 2995.85 0.353085
\(417\) −93.8003 −0.0110154
\(418\) 163.825 0.0191697
\(419\) 13939.9 1.62532 0.812658 0.582741i \(-0.198020\pi\)
0.812658 + 0.582741i \(0.198020\pi\)
\(420\) −2.45136 −0.000284795 0
\(421\) −11207.9 −1.29748 −0.648738 0.761012i \(-0.724703\pi\)
−0.648738 + 0.761012i \(0.724703\pi\)
\(422\) −233.107 −0.0268897
\(423\) 6734.22 0.774063
\(424\) 5383.25 0.616589
\(425\) 2202.71 0.251405
\(426\) 50.5500 0.00574919
\(427\) −384.730 −0.0436028
\(428\) 12891.2 1.45589
\(429\) 14.6924 0.00165350
\(430\) −1474.25 −0.165336
\(431\) 2686.39 0.300230 0.150115 0.988669i \(-0.452036\pi\)
0.150115 + 0.988669i \(0.452036\pi\)
\(432\) −161.758 −0.0180153
\(433\) 5824.77 0.646468 0.323234 0.946319i \(-0.395230\pi\)
0.323234 + 0.946319i \(0.395230\pi\)
\(434\) −31.6528 −0.00350088
\(435\) −64.6077 −0.00712115
\(436\) 14209.3 1.56079
\(437\) 3198.22 0.350095
\(438\) 4.71070 0.000513894 0
\(439\) −16128.4 −1.75346 −0.876728 0.480986i \(-0.840279\pi\)
−0.876728 + 0.480986i \(0.840279\pi\)
\(440\) 663.302 0.0718674
\(441\) 9227.10 0.996339
\(442\) 1528.29 0.164464
\(443\) −16147.4 −1.73179 −0.865896 0.500224i \(-0.833251\pi\)
−0.865896 + 0.500224i \(0.833251\pi\)
\(444\) 97.2983 0.0103999
\(445\) −877.637 −0.0934921
\(446\) −1896.16 −0.201314
\(447\) −111.468 −0.0117948
\(448\) 319.959 0.0337425
\(449\) 10233.4 1.07560 0.537801 0.843072i \(-0.319255\pi\)
0.537801 + 0.843072i \(0.319255\pi\)
\(450\) 529.029 0.0554193
\(451\) −308.075 −0.0321656
\(452\) 5319.77 0.553586
\(453\) −61.2023 −0.00634776
\(454\) 2416.82 0.249840
\(455\) −121.683 −0.0125375
\(456\) 13.8308 0.00142037
\(457\) 6256.80 0.640439 0.320220 0.947343i \(-0.396243\pi\)
0.320220 + 0.947343i \(0.396243\pi\)
\(458\) −1216.62 −0.124125
\(459\) −287.162 −0.0292017
\(460\) 6215.97 0.630046
\(461\) 11010.4 1.11238 0.556189 0.831056i \(-0.312263\pi\)
0.556189 + 0.831056i \(0.312263\pi\)
\(462\) −0.572374 −5.76391e−5 0
\(463\) 16642.0 1.67046 0.835228 0.549904i \(-0.185336\pi\)
0.835228 + 0.549904i \(0.185336\pi\)
\(464\) 10624.9 1.06303
\(465\) 11.0812 0.00110511
\(466\) −1094.88 −0.108840
\(467\) −6140.06 −0.608411 −0.304206 0.952606i \(-0.598391\pi\)
−0.304206 + 0.952606i \(0.598391\pi\)
\(468\) −4412.07 −0.435786
\(469\) 374.494 0.0368710
\(470\) −977.657 −0.0959488
\(471\) −201.635 −0.0197258
\(472\) 9970.15 0.972274
\(473\) 4137.70 0.402223
\(474\) −48.6373 −0.00471305
\(475\) 475.000 0.0458831
\(476\) 715.662 0.0689124
\(477\) −12050.4 −1.15671
\(478\) −2287.61 −0.218897
\(479\) −2999.01 −0.286072 −0.143036 0.989718i \(-0.545686\pi\)
−0.143036 + 0.989718i \(0.545686\pi\)
\(480\) 40.8585 0.00388527
\(481\) 4829.79 0.457836
\(482\) 1883.54 0.177994
\(483\) −11.1740 −0.00105266
\(484\) −893.655 −0.0839270
\(485\) −193.137 −0.0180823
\(486\) −103.455 −0.00965599
\(487\) 13804.2 1.28445 0.642224 0.766517i \(-0.278012\pi\)
0.642224 + 0.766517i \(0.278012\pi\)
\(488\) 4218.89 0.391353
\(489\) 43.4875 0.00402162
\(490\) −1339.57 −0.123501
\(491\) 1097.44 0.100869 0.0504346 0.998727i \(-0.483939\pi\)
0.0504346 + 0.998727i \(0.483939\pi\)
\(492\) −12.4852 −0.00114405
\(493\) 18861.9 1.72312
\(494\) 329.565 0.0300158
\(495\) −1484.80 −0.134822
\(496\) −1822.33 −0.164969
\(497\) 1175.03 0.106051
\(498\) −28.8149 −0.00259282
\(499\) −11098.7 −0.995679 −0.497840 0.867269i \(-0.665873\pi\)
−0.497840 + 0.867269i \(0.665873\pi\)
\(500\) 923.197 0.0825732
\(501\) −16.8389 −0.00150161
\(502\) −3247.36 −0.288718
\(503\) 6753.57 0.598662 0.299331 0.954149i \(-0.403237\pi\)
0.299331 + 0.954149i \(0.403237\pi\)
\(504\) 358.064 0.0316457
\(505\) 2960.30 0.260855
\(506\) 1451.38 0.127514
\(507\) −103.053 −0.00902715
\(508\) 13883.2 1.21253
\(509\) −10002.2 −0.870997 −0.435499 0.900189i \(-0.643428\pi\)
−0.435499 + 0.900189i \(0.643428\pi\)
\(510\) 20.8433 0.00180972
\(511\) 109.499 0.00947938
\(512\) −11507.7 −0.993309
\(513\) −61.9247 −0.00532952
\(514\) 2293.85 0.196843
\(515\) −4431.29 −0.379157
\(516\) 167.686 0.0143061
\(517\) 2743.94 0.233420
\(518\) −188.155 −0.0159596
\(519\) −13.7940 −0.00116665
\(520\) 1334.35 0.112529
\(521\) −107.338 −0.00902602 −0.00451301 0.999990i \(-0.501437\pi\)
−0.00451301 + 0.999990i \(0.501437\pi\)
\(522\) 4530.10 0.379841
\(523\) 9371.01 0.783491 0.391745 0.920074i \(-0.371871\pi\)
0.391745 + 0.920074i \(0.371871\pi\)
\(524\) 570.323 0.0475471
\(525\) −1.65956 −0.000137960 0
\(526\) 1871.24 0.155114
\(527\) −3235.10 −0.267406
\(528\) −32.9529 −0.00271608
\(529\) 16167.1 1.32877
\(530\) 1749.45 0.143379
\(531\) −22318.2 −1.82397
\(532\) 154.328 0.0125770
\(533\) −619.750 −0.0503646
\(534\) −8.30472 −0.000672997 0
\(535\) 8727.31 0.705261
\(536\) −4106.63 −0.330932
\(537\) 207.846 0.0167025
\(538\) 4569.92 0.366215
\(539\) 3759.70 0.300448
\(540\) −120.355 −0.00959122
\(541\) −3851.36 −0.306068 −0.153034 0.988221i \(-0.548904\pi\)
−0.153034 + 0.988221i \(0.548904\pi\)
\(542\) 5154.00 0.408456
\(543\) 200.838 0.0158725
\(544\) −11928.4 −0.940125
\(545\) 9619.64 0.756074
\(546\) −1.15144 −9.02508e−5 0
\(547\) 572.812 0.0447746 0.0223873 0.999749i \(-0.492873\pi\)
0.0223873 + 0.999749i \(0.492873\pi\)
\(548\) 1807.60 0.140907
\(549\) −9443.98 −0.734170
\(550\) 215.560 0.0167118
\(551\) 4067.45 0.314481
\(552\) 122.532 0.00944801
\(553\) −1130.56 −0.0869376
\(554\) −4523.26 −0.346886
\(555\) 65.8705 0.00503792
\(556\) 11477.4 0.875448
\(557\) −21782.7 −1.65703 −0.828513 0.559971i \(-0.810812\pi\)
−0.828513 + 0.559971i \(0.810812\pi\)
\(558\) −776.980 −0.0589466
\(559\) 8323.74 0.629798
\(560\) 272.918 0.0205945
\(561\) −58.4999 −0.00440262
\(562\) −292.239 −0.0219348
\(563\) 7408.62 0.554593 0.277297 0.960784i \(-0.410562\pi\)
0.277297 + 0.960784i \(0.410562\pi\)
\(564\) 111.202 0.00830220
\(565\) 3601.46 0.268168
\(566\) −1452.50 −0.107868
\(567\) −801.417 −0.0593586
\(568\) −12885.1 −0.951846
\(569\) 2367.61 0.174438 0.0872190 0.996189i \(-0.472202\pi\)
0.0872190 + 0.996189i \(0.472202\pi\)
\(570\) 4.49473 0.000330287 0
\(571\) −22777.5 −1.66937 −0.834683 0.550731i \(-0.814349\pi\)
−0.834683 + 0.550731i \(0.814349\pi\)
\(572\) −1797.75 −0.131412
\(573\) 120.612 0.00879346
\(574\) 24.1438 0.00175565
\(575\) 4208.18 0.305206
\(576\) 7854.03 0.568145
\(577\) −16073.0 −1.15967 −0.579833 0.814735i \(-0.696882\pi\)
−0.579833 + 0.814735i \(0.696882\pi\)
\(578\) −2234.04 −0.160768
\(579\) −91.9967 −0.00660320
\(580\) 7905.38 0.565953
\(581\) −669.797 −0.0478276
\(582\) −1.82758 −0.000130164 0
\(583\) −4910.08 −0.348808
\(584\) −1200.75 −0.0850812
\(585\) −2986.95 −0.211103
\(586\) 2886.86 0.203507
\(587\) −14991.8 −1.05413 −0.527067 0.849823i \(-0.676708\pi\)
−0.527067 + 0.849823i \(0.676708\pi\)
\(588\) 152.367 0.0106862
\(589\) −697.628 −0.0488035
\(590\) 3240.10 0.226089
\(591\) −16.9758 −0.00118154
\(592\) −10832.6 −0.752053
\(593\) 8315.72 0.575861 0.287930 0.957651i \(-0.407033\pi\)
0.287930 + 0.957651i \(0.407033\pi\)
\(594\) −28.1020 −0.00194114
\(595\) 484.500 0.0333824
\(596\) 13639.2 0.937391
\(597\) −236.106 −0.0161863
\(598\) 2919.73 0.199660
\(599\) −18674.7 −1.27383 −0.636916 0.770933i \(-0.719790\pi\)
−0.636916 + 0.770933i \(0.719790\pi\)
\(600\) 18.1984 0.00123825
\(601\) −807.103 −0.0547794 −0.0273897 0.999625i \(-0.508719\pi\)
−0.0273897 + 0.999625i \(0.508719\pi\)
\(602\) −324.270 −0.0219539
\(603\) 9192.70 0.620822
\(604\) 7488.70 0.504488
\(605\) −605.000 −0.0406558
\(606\) 28.0121 0.00187775
\(607\) 16324.2 1.09156 0.545782 0.837927i \(-0.316233\pi\)
0.545782 + 0.837927i \(0.316233\pi\)
\(608\) −2572.29 −0.171579
\(609\) −14.2109 −0.000945572 0
\(610\) 1371.05 0.0910038
\(611\) 5519.95 0.365488
\(612\) 17567.4 1.16032
\(613\) −824.080 −0.0542973 −0.0271487 0.999631i \(-0.508643\pi\)
−0.0271487 + 0.999631i \(0.508643\pi\)
\(614\) 3416.04 0.224528
\(615\) −8.45239 −0.000554200 0
\(616\) 145.897 0.00954282
\(617\) −15112.5 −0.986070 −0.493035 0.870009i \(-0.664113\pi\)
−0.493035 + 0.870009i \(0.664113\pi\)
\(618\) −41.9315 −0.00272934
\(619\) 3574.05 0.232073 0.116036 0.993245i \(-0.462981\pi\)
0.116036 + 0.993245i \(0.462981\pi\)
\(620\) −1355.89 −0.0878288
\(621\) −548.612 −0.0354509
\(622\) 366.177 0.0236051
\(623\) −193.042 −0.0124142
\(624\) −66.2910 −0.00425283
\(625\) 625.000 0.0400000
\(626\) −6890.63 −0.439944
\(627\) −12.6151 −0.000803509 0
\(628\) 24671.9 1.56770
\(629\) −19230.6 −1.21903
\(630\) 116.363 0.00735878
\(631\) 1678.56 0.105899 0.0529495 0.998597i \(-0.483138\pi\)
0.0529495 + 0.998597i \(0.483138\pi\)
\(632\) 12397.6 0.780300
\(633\) 17.9501 0.00112709
\(634\) −498.582 −0.0312322
\(635\) 9398.86 0.587374
\(636\) −198.988 −0.0124063
\(637\) 7563.33 0.470439
\(638\) 1845.85 0.114542
\(639\) 28843.4 1.78564
\(640\) −6555.58 −0.404894
\(641\) 2186.08 0.134704 0.0673518 0.997729i \(-0.478545\pi\)
0.0673518 + 0.997729i \(0.478545\pi\)
\(642\) 82.5831 0.00507678
\(643\) −22353.0 −1.37094 −0.685472 0.728099i \(-0.740404\pi\)
−0.685472 + 0.728099i \(0.740404\pi\)
\(644\) 1367.24 0.0836598
\(645\) 113.522 0.00693014
\(646\) −1312.22 −0.0799202
\(647\) 848.196 0.0515394 0.0257697 0.999668i \(-0.491796\pi\)
0.0257697 + 0.999668i \(0.491796\pi\)
\(648\) 8788.20 0.532767
\(649\) −9093.81 −0.550021
\(650\) 433.638 0.0261672
\(651\) 2.43738 0.000146741 0
\(652\) −5321.12 −0.319618
\(653\) 20748.5 1.24341 0.621707 0.783250i \(-0.286439\pi\)
0.621707 + 0.783250i \(0.286439\pi\)
\(654\) 91.0268 0.00544255
\(655\) 386.106 0.0230327
\(656\) 1390.01 0.0827301
\(657\) 2687.88 0.159611
\(658\) −215.042 −0.0127404
\(659\) −5565.49 −0.328984 −0.164492 0.986378i \(-0.552599\pi\)
−0.164492 + 0.986378i \(0.552599\pi\)
\(660\) −24.5184 −0.00144603
\(661\) −6981.39 −0.410809 −0.205404 0.978677i \(-0.565851\pi\)
−0.205404 + 0.978677i \(0.565851\pi\)
\(662\) 6729.33 0.395080
\(663\) −117.684 −0.00689359
\(664\) 7344.88 0.429272
\(665\) 104.479 0.00609253
\(666\) −4618.65 −0.268722
\(667\) 36034.9 2.09187
\(668\) 2060.40 0.119340
\(669\) 146.011 0.00843816
\(670\) −1334.57 −0.0769538
\(671\) −3848.07 −0.221390
\(672\) 8.98710 0.000515900 0
\(673\) 15367.4 0.880192 0.440096 0.897951i \(-0.354944\pi\)
0.440096 + 0.897951i \(0.354944\pi\)
\(674\) 1083.39 0.0619148
\(675\) −81.4799 −0.00464617
\(676\) 12609.6 0.717433
\(677\) −20345.3 −1.15500 −0.577500 0.816391i \(-0.695972\pi\)
−0.577500 + 0.816391i \(0.695972\pi\)
\(678\) 34.0792 0.00193039
\(679\) −42.4818 −0.00240103
\(680\) −5312.94 −0.299621
\(681\) −186.104 −0.0104721
\(682\) −316.590 −0.0177755
\(683\) −17653.0 −0.988982 −0.494491 0.869183i \(-0.664645\pi\)
−0.494491 + 0.869183i \(0.664645\pi\)
\(684\) 3788.29 0.211767
\(685\) 1223.74 0.0682579
\(686\) −590.336 −0.0328559
\(687\) 93.6845 0.00520275
\(688\) −18669.0 −1.03452
\(689\) −9877.54 −0.546160
\(690\) 39.8204 0.00219701
\(691\) −27619.5 −1.52054 −0.760272 0.649605i \(-0.774934\pi\)
−0.760272 + 0.649605i \(0.774934\pi\)
\(692\) 1687.83 0.0927194
\(693\) −326.591 −0.0179021
\(694\) −9303.99 −0.508897
\(695\) 7770.13 0.424083
\(696\) 155.834 0.00848689
\(697\) 2467.63 0.134101
\(698\) 323.747 0.0175559
\(699\) 84.3095 0.00456206
\(700\) 203.063 0.0109644
\(701\) −10809.2 −0.582393 −0.291196 0.956663i \(-0.594053\pi\)
−0.291196 + 0.956663i \(0.594053\pi\)
\(702\) −56.5324 −0.00303943
\(703\) −4146.95 −0.222482
\(704\) 3200.22 0.171325
\(705\) 75.2831 0.00402174
\(706\) 2552.98 0.136094
\(707\) 651.137 0.0346372
\(708\) −368.539 −0.0195629
\(709\) −14360.2 −0.760663 −0.380331 0.924850i \(-0.624190\pi\)
−0.380331 + 0.924850i \(0.624190\pi\)
\(710\) −4187.41 −0.221339
\(711\) −27752.0 −1.46383
\(712\) 2116.86 0.111423
\(713\) −6180.52 −0.324632
\(714\) 4.58463 0.000240302 0
\(715\) −1217.07 −0.0636585
\(716\) −25432.0 −1.32743
\(717\) 176.154 0.00917518
\(718\) 4120.69 0.214182
\(719\) 19614.5 1.01738 0.508690 0.860950i \(-0.330130\pi\)
0.508690 + 0.860950i \(0.330130\pi\)
\(720\) 6699.32 0.346763
\(721\) −974.690 −0.0503459
\(722\) −282.971 −0.0145860
\(723\) −145.040 −0.00746070
\(724\) −24574.5 −1.26147
\(725\) 5351.90 0.274158
\(726\) −5.72487 −0.000292658 0
\(727\) 7497.02 0.382461 0.191231 0.981545i \(-0.438752\pi\)
0.191231 + 0.981545i \(0.438752\pi\)
\(728\) 293.500 0.0149421
\(729\) −19667.1 −0.999190
\(730\) −390.220 −0.0197845
\(731\) −33142.3 −1.67690
\(732\) −155.948 −0.00787433
\(733\) 21328.5 1.07474 0.537371 0.843346i \(-0.319418\pi\)
0.537371 + 0.843346i \(0.319418\pi\)
\(734\) −1535.22 −0.0772016
\(735\) 103.152 0.00517660
\(736\) −22788.8 −1.14131
\(737\) 3745.68 0.187210
\(738\) 592.657 0.0295610
\(739\) 13437.4 0.668879 0.334440 0.942417i \(-0.391453\pi\)
0.334440 + 0.942417i \(0.391453\pi\)
\(740\) −8059.89 −0.400389
\(741\) −25.3777 −0.00125813
\(742\) 384.802 0.0190384
\(743\) −3760.57 −0.185682 −0.0928411 0.995681i \(-0.529595\pi\)
−0.0928411 + 0.995681i \(0.529595\pi\)
\(744\) −26.7279 −0.00131706
\(745\) 9233.70 0.454090
\(746\) 4675.93 0.229488
\(747\) −16441.5 −0.805306
\(748\) 7158.04 0.349898
\(749\) 1919.63 0.0936471
\(750\) 5.91412 0.000287938 0
\(751\) −27004.2 −1.31211 −0.656056 0.754712i \(-0.727777\pi\)
−0.656056 + 0.754712i \(0.727777\pi\)
\(752\) −12380.5 −0.600359
\(753\) 250.058 0.0121018
\(754\) 3713.26 0.179349
\(755\) 5069.82 0.244383
\(756\) −26.4729 −0.00127356
\(757\) 4049.74 0.194439 0.0972195 0.995263i \(-0.469005\pi\)
0.0972195 + 0.995263i \(0.469005\pi\)
\(758\) −294.475 −0.0141106
\(759\) −111.762 −0.00534479
\(760\) −1145.70 −0.0546829
\(761\) −25094.8 −1.19538 −0.597692 0.801726i \(-0.703915\pi\)
−0.597692 + 0.801726i \(0.703915\pi\)
\(762\) 88.9376 0.00422818
\(763\) 2115.90 0.100394
\(764\) −14758.1 −0.698860
\(765\) 11893.0 0.562082
\(766\) −229.111 −0.0108070
\(767\) −18293.9 −0.861218
\(768\) 78.4500 0.00368597
\(769\) 12245.3 0.574224 0.287112 0.957897i \(-0.407305\pi\)
0.287112 + 0.957897i \(0.407305\pi\)
\(770\) 47.4137 0.00221905
\(771\) −176.635 −0.00825079
\(772\) 11256.7 0.524789
\(773\) 4554.29 0.211910 0.105955 0.994371i \(-0.466210\pi\)
0.105955 + 0.994371i \(0.466210\pi\)
\(774\) −7959.86 −0.369653
\(775\) −917.932 −0.0425459
\(776\) 465.848 0.0215502
\(777\) 14.4886 0.000668954 0
\(778\) 4785.32 0.220517
\(779\) 532.129 0.0244743
\(780\) −49.3234 −0.00226418
\(781\) 11752.6 0.538464
\(782\) −11625.4 −0.531614
\(783\) −697.716 −0.0318446
\(784\) −16963.5 −0.772755
\(785\) 16702.8 0.759424
\(786\) 3.65357 0.000165800 0
\(787\) 4110.23 0.186167 0.0930837 0.995658i \(-0.470328\pi\)
0.0930837 + 0.995658i \(0.470328\pi\)
\(788\) 2077.15 0.0939030
\(789\) −144.092 −0.00650167
\(790\) 4028.96 0.181448
\(791\) 792.165 0.0356083
\(792\) 3581.34 0.160679
\(793\) −7741.10 −0.346651
\(794\) −1988.79 −0.0888909
\(795\) −134.714 −0.00600982
\(796\) 28889.9 1.28640
\(797\) 28046.0 1.24647 0.623237 0.782033i \(-0.285817\pi\)
0.623237 + 0.782033i \(0.285817\pi\)
\(798\) 0.988646 4.38567e−5 0
\(799\) −21978.5 −0.973148
\(800\) −3384.60 −0.149579
\(801\) −4738.60 −0.209026
\(802\) 1583.25 0.0697087
\(803\) 1095.21 0.0481309
\(804\) 151.799 0.00665861
\(805\) 925.617 0.0405264
\(806\) −636.880 −0.0278327
\(807\) −351.901 −0.0153501
\(808\) −7140.26 −0.310883
\(809\) −12876.7 −0.559607 −0.279803 0.960057i \(-0.590269\pi\)
−0.279803 + 0.960057i \(0.590269\pi\)
\(810\) 2855.99 0.123888
\(811\) 2515.04 0.108896 0.0544482 0.998517i \(-0.482660\pi\)
0.0544482 + 0.998517i \(0.482660\pi\)
\(812\) 1738.84 0.0751493
\(813\) −396.877 −0.0171206
\(814\) −1881.92 −0.0810337
\(815\) −3602.37 −0.154829
\(816\) 263.948 0.0113236
\(817\) −7146.93 −0.306046
\(818\) 4648.96 0.198713
\(819\) −656.999 −0.0280310
\(820\) 1034.23 0.0440451
\(821\) −39018.8 −1.65867 −0.829333 0.558754i \(-0.811280\pi\)
−0.829333 + 0.558754i \(0.811280\pi\)
\(822\) 11.5798 0.000491351 0
\(823\) −3391.84 −0.143660 −0.0718299 0.997417i \(-0.522884\pi\)
−0.0718299 + 0.997417i \(0.522884\pi\)
\(824\) 10688.3 0.451874
\(825\) −16.5989 −0.000700483 0
\(826\) 712.680 0.0300210
\(827\) −36397.7 −1.53044 −0.765220 0.643769i \(-0.777370\pi\)
−0.765220 + 0.643769i \(0.777370\pi\)
\(828\) 33561.7 1.40864
\(829\) −25754.4 −1.07899 −0.539497 0.841987i \(-0.681386\pi\)
−0.539497 + 0.841987i \(0.681386\pi\)
\(830\) 2386.94 0.0998215
\(831\) 348.308 0.0145399
\(832\) 6437.85 0.268260
\(833\) −30114.6 −1.25259
\(834\) 73.5256 0.00305274
\(835\) 1394.88 0.0578105
\(836\) 1543.59 0.0638589
\(837\) 119.669 0.00494188
\(838\) −10926.8 −0.450430
\(839\) −26723.0 −1.09962 −0.549809 0.835290i \(-0.685300\pi\)
−0.549809 + 0.835290i \(0.685300\pi\)
\(840\) 4.00286 0.000164419 0
\(841\) 21439.6 0.879067
\(842\) 8785.31 0.359575
\(843\) 22.5035 0.000919408 0
\(844\) −2196.36 −0.0895758
\(845\) 8536.64 0.347538
\(846\) −5278.64 −0.214519
\(847\) −133.074 −0.00539842
\(848\) 22154.0 0.897136
\(849\) 111.848 0.00452133
\(850\) −1726.60 −0.0696728
\(851\) −36739.2 −1.47991
\(852\) 476.289 0.0191519
\(853\) 31404.1 1.26056 0.630279 0.776368i \(-0.282940\pi\)
0.630279 + 0.776368i \(0.282940\pi\)
\(854\) 301.572 0.0120838
\(855\) 2564.65 0.102584
\(856\) −21050.3 −0.840520
\(857\) −19490.7 −0.776884 −0.388442 0.921473i \(-0.626987\pi\)
−0.388442 + 0.921473i \(0.626987\pi\)
\(858\) −11.5166 −0.000458242 0
\(859\) −2031.11 −0.0806758 −0.0403379 0.999186i \(-0.512843\pi\)
−0.0403379 + 0.999186i \(0.512843\pi\)
\(860\) −13890.6 −0.550773
\(861\) −1.85916 −7.35888e−5 0
\(862\) −2105.74 −0.0832038
\(863\) −31486.2 −1.24195 −0.620975 0.783830i \(-0.713263\pi\)
−0.620975 + 0.783830i \(0.713263\pi\)
\(864\) 441.242 0.0173743
\(865\) 1142.66 0.0449150
\(866\) −4565.77 −0.179158
\(867\) 172.029 0.00673866
\(868\) −298.237 −0.0116622
\(869\) −11307.9 −0.441420
\(870\) 50.6429 0.00197351
\(871\) 7535.13 0.293132
\(872\) −23202.6 −0.901078
\(873\) −1042.80 −0.0404278
\(874\) −2506.93 −0.0970232
\(875\) 137.473 0.00531135
\(876\) 44.3849 0.00171190
\(877\) −16898.1 −0.650638 −0.325319 0.945604i \(-0.605472\pi\)
−0.325319 + 0.945604i \(0.605472\pi\)
\(878\) 12642.3 0.485942
\(879\) −222.299 −0.00853009
\(880\) 2729.72 0.104567
\(881\) 25394.8 0.971138 0.485569 0.874198i \(-0.338612\pi\)
0.485569 + 0.874198i \(0.338612\pi\)
\(882\) −7232.69 −0.276119
\(883\) −19200.4 −0.731762 −0.365881 0.930662i \(-0.619232\pi\)
−0.365881 + 0.930662i \(0.619232\pi\)
\(884\) 14399.7 0.547868
\(885\) −249.499 −0.00947664
\(886\) 12657.2 0.479938
\(887\) 41952.3 1.58807 0.794036 0.607871i \(-0.207976\pi\)
0.794036 + 0.607871i \(0.207976\pi\)
\(888\) −158.880 −0.00600412
\(889\) 2067.34 0.0779936
\(890\) 687.938 0.0259098
\(891\) −8015.75 −0.301389
\(892\) −17865.9 −0.670622
\(893\) −4739.53 −0.177606
\(894\) 87.3748 0.00326874
\(895\) −17217.3 −0.643030
\(896\) −1441.94 −0.0537633
\(897\) −224.830 −0.00836883
\(898\) −8021.50 −0.298086
\(899\) −7860.29 −0.291608
\(900\) 4984.59 0.184614
\(901\) 39329.0 1.45421
\(902\) 241.485 0.00891418
\(903\) 24.9700 0.000920209 0
\(904\) −8686.75 −0.319598
\(905\) −16636.8 −0.611079
\(906\) 47.9736 0.00175918
\(907\) 7143.32 0.261511 0.130755 0.991415i \(-0.458260\pi\)
0.130755 + 0.991415i \(0.458260\pi\)
\(908\) 22771.7 0.832273
\(909\) 15983.5 0.583210
\(910\) 95.3815 0.00347458
\(911\) −19674.4 −0.715522 −0.357761 0.933813i \(-0.616460\pi\)
−0.357761 + 0.933813i \(0.616460\pi\)
\(912\) 56.9187 0.00206663
\(913\) −6699.30 −0.242842
\(914\) −4904.41 −0.177487
\(915\) −105.576 −0.00381447
\(916\) −11463.2 −0.413488
\(917\) 84.9265 0.00305837
\(918\) 225.093 0.00809278
\(919\) 30418.3 1.09185 0.545924 0.837835i \(-0.316179\pi\)
0.545924 + 0.837835i \(0.316179\pi\)
\(920\) −10150.2 −0.363740
\(921\) −263.048 −0.00941121
\(922\) −8630.54 −0.308277
\(923\) 23642.5 0.843123
\(924\) −5.39299 −0.000192009 0
\(925\) −5456.51 −0.193956
\(926\) −13044.9 −0.462940
\(927\) −23925.7 −0.847707
\(928\) −28982.5 −1.02521
\(929\) 20882.8 0.737504 0.368752 0.929528i \(-0.379785\pi\)
0.368752 + 0.929528i \(0.379785\pi\)
\(930\) −8.68602 −0.000306264 0
\(931\) −6494.02 −0.228607
\(932\) −10316.1 −0.362570
\(933\) −28.1969 −0.000989417 0
\(934\) 4812.90 0.168611
\(935\) 4845.96 0.169497
\(936\) 7204.54 0.251590
\(937\) −49312.2 −1.71927 −0.859636 0.510907i \(-0.829310\pi\)
−0.859636 + 0.510907i \(0.829310\pi\)
\(938\) −293.548 −0.0102182
\(939\) 530.604 0.0184405
\(940\) −9211.62 −0.319628
\(941\) 20987.9 0.727083 0.363542 0.931578i \(-0.381567\pi\)
0.363542 + 0.931578i \(0.381567\pi\)
\(942\) 158.052 0.00546667
\(943\) 4714.31 0.162799
\(944\) 41030.7 1.41466
\(945\) −17.9220 −0.000616935 0
\(946\) −3243.34 −0.111470
\(947\) −50485.8 −1.73238 −0.866192 0.499711i \(-0.833440\pi\)
−0.866192 + 0.499711i \(0.833440\pi\)
\(948\) −458.267 −0.0157002
\(949\) 2203.22 0.0753630
\(950\) −372.330 −0.0127158
\(951\) 38.3926 0.00130911
\(952\) −1168.62 −0.0397847
\(953\) 35925.7 1.22114 0.610570 0.791962i \(-0.290940\pi\)
0.610570 + 0.791962i \(0.290940\pi\)
\(954\) 9445.74 0.320563
\(955\) −9991.16 −0.338541
\(956\) −21554.2 −0.729197
\(957\) −142.137 −0.00480108
\(958\) 2350.78 0.0792802
\(959\) 269.169 0.00906354
\(960\) 87.8018 0.00295187
\(961\) −28442.8 −0.954746
\(962\) −3785.84 −0.126882
\(963\) 47121.1 1.57680
\(964\) 17747.0 0.592939
\(965\) 7620.73 0.254218
\(966\) 8.75874 0.000291727 0
\(967\) −13309.2 −0.442600 −0.221300 0.975206i \(-0.571030\pi\)
−0.221300 + 0.975206i \(0.571030\pi\)
\(968\) 1459.26 0.0484530
\(969\) 101.045 0.00334989
\(970\) 151.391 0.00501121
\(971\) −15424.5 −0.509781 −0.254890 0.966970i \(-0.582039\pi\)
−0.254890 + 0.966970i \(0.582039\pi\)
\(972\) −974.767 −0.0321663
\(973\) 1709.09 0.0563113
\(974\) −10820.4 −0.355964
\(975\) −33.3917 −0.00109681
\(976\) 17362.2 0.569418
\(977\) 31101.5 1.01845 0.509224 0.860634i \(-0.329932\pi\)
0.509224 + 0.860634i \(0.329932\pi\)
\(978\) −34.0878 −0.00111453
\(979\) −1930.80 −0.0630323
\(980\) −12621.6 −0.411410
\(981\) 51939.1 1.69040
\(982\) −860.231 −0.0279543
\(983\) 6010.71 0.195027 0.0975136 0.995234i \(-0.468911\pi\)
0.0975136 + 0.995234i \(0.468911\pi\)
\(984\) 20.3872 0.000660488 0
\(985\) 1406.22 0.0454884
\(986\) −14784.9 −0.477534
\(987\) 16.5590 0.000534021 0
\(988\) 3105.21 0.0999897
\(989\) −63317.0 −2.03576
\(990\) 1163.86 0.0373637
\(991\) −19192.5 −0.615208 −0.307604 0.951515i \(-0.599527\pi\)
−0.307604 + 0.951515i \(0.599527\pi\)
\(992\) 4970.93 0.159100
\(993\) −518.183 −0.0165600
\(994\) −921.047 −0.0293902
\(995\) 19558.3 0.623157
\(996\) −271.498 −0.00863729
\(997\) 53233.1 1.69098 0.845490 0.533990i \(-0.179308\pi\)
0.845490 + 0.533990i \(0.179308\pi\)
\(998\) 8699.71 0.275936
\(999\) 711.353 0.0225287
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 1045.4.a.c.1.9 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.4.a.c.1.9 20 1.1 even 1 trivial