Properties

Label 1849.4.a.k.1.5
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $60$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1849,4,Mod(1,1849)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1849, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1849.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(109.094531601\)
Analytic rank: \(1\)
Dimension: \(60\)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) \(=\) 1849.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.24990 q^{2} -9.69865 q^{3} +19.5614 q^{4} -5.59920 q^{5} +50.9169 q^{6} +0.820208 q^{7} -60.6963 q^{8} +67.0638 q^{9} +O(q^{10})\) \(q-5.24990 q^{2} -9.69865 q^{3} +19.5614 q^{4} -5.59920 q^{5} +50.9169 q^{6} +0.820208 q^{7} -60.6963 q^{8} +67.0638 q^{9} +29.3952 q^{10} -48.4653 q^{11} -189.719 q^{12} +4.57016 q^{13} -4.30601 q^{14} +54.3046 q^{15} +162.158 q^{16} +108.420 q^{17} -352.078 q^{18} +24.1770 q^{19} -109.528 q^{20} -7.95490 q^{21} +254.438 q^{22} -1.18989 q^{23} +588.672 q^{24} -93.6490 q^{25} -23.9928 q^{26} -388.564 q^{27} +16.0444 q^{28} +32.2620 q^{29} -285.094 q^{30} -59.1739 q^{31} -365.742 q^{32} +470.048 q^{33} -569.192 q^{34} -4.59250 q^{35} +1311.86 q^{36} +353.501 q^{37} -126.927 q^{38} -44.3243 q^{39} +339.850 q^{40} -62.8277 q^{41} +41.7624 q^{42} -948.050 q^{44} -375.503 q^{45} +6.24678 q^{46} -213.247 q^{47} -1572.71 q^{48} -342.327 q^{49} +491.648 q^{50} -1051.52 q^{51} +89.3987 q^{52} -119.382 q^{53} +2039.92 q^{54} +271.367 q^{55} -49.7835 q^{56} -234.484 q^{57} -169.372 q^{58} -367.191 q^{59} +1062.28 q^{60} -36.8200 q^{61} +310.657 q^{62} +55.0062 q^{63} +622.845 q^{64} -25.5892 q^{65} -2467.70 q^{66} +173.580 q^{67} +2120.84 q^{68} +11.5403 q^{69} +24.1102 q^{70} -918.269 q^{71} -4070.52 q^{72} -33.8560 q^{73} -1855.84 q^{74} +908.269 q^{75} +472.936 q^{76} -39.7516 q^{77} +232.698 q^{78} -612.712 q^{79} -907.954 q^{80} +1957.83 q^{81} +329.839 q^{82} -416.023 q^{83} -155.609 q^{84} -607.063 q^{85} -312.898 q^{87} +2941.66 q^{88} -921.950 q^{89} +1971.35 q^{90} +3.74848 q^{91} -23.2759 q^{92} +573.907 q^{93} +1119.53 q^{94} -135.372 q^{95} +3547.20 q^{96} +431.946 q^{97} +1797.18 q^{98} -3250.27 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q - 15 q^{2} - 37 q^{3} + 213 q^{4} - 51 q^{5} + 22 q^{6} - 54 q^{7} - 204 q^{8} + 387 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 60 q - 15 q^{2} - 37 q^{3} + 213 q^{4} - 51 q^{5} + 22 q^{6} - 54 q^{7} - 204 q^{8} + 387 q^{9} + 27 q^{10} + 43 q^{11} - 432 q^{12} + 13 q^{13} + 96 q^{14} + 65 q^{15} + 717 q^{16} - 138 q^{17} - 625 q^{18} - 610 q^{19} - 345 q^{20} + 611 q^{21} - 118 q^{22} - 243 q^{23} + 258 q^{24} + 899 q^{25} - 1071 q^{26} - 1609 q^{27} - 46 q^{28} - 773 q^{29} - 375 q^{30} - 97 q^{31} - 1967 q^{32} - 500 q^{33} - 217 q^{34} + 247 q^{35} + 175 q^{36} - 228 q^{37} + 1253 q^{38} - 1493 q^{39} + 2220 q^{40} - 951 q^{41} - 2643 q^{42} - 1378 q^{44} - 1086 q^{45} + 565 q^{46} - 2 q^{47} - 2303 q^{48} + 1264 q^{49} - 3273 q^{50} - 3076 q^{51} - 2825 q^{52} - 39 q^{53} + 5201 q^{54} - 1306 q^{55} + 3683 q^{56} + 1342 q^{57} + 2588 q^{58} - 1065 q^{59} + 2803 q^{60} - 2999 q^{61} - 5569 q^{62} - 2377 q^{63} + 2082 q^{64} - 5578 q^{65} + 3338 q^{66} + 961 q^{67} - 3754 q^{68} - 1817 q^{69} - 2738 q^{70} - 8003 q^{71} - 1412 q^{72} + 1011 q^{73} - 1413 q^{74} - 7457 q^{75} - 5516 q^{76} - 4052 q^{77} + 1091 q^{78} - 4422 q^{79} - 1610 q^{80} + 2108 q^{81} - 4676 q^{82} - 297 q^{83} - 54 q^{84} - 4333 q^{85} + 1377 q^{87} - 3652 q^{88} - 2480 q^{89} - 1414 q^{90} - 4551 q^{91} - 3286 q^{92} - 4 q^{93} - 4609 q^{94} - 835 q^{95} + 5864 q^{96} + 3785 q^{97} - 753 q^{98} + 5072 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.24990 −1.85612 −0.928060 0.372432i \(-0.878524\pi\)
−0.928060 + 0.372432i \(0.878524\pi\)
\(3\) −9.69865 −1.86651 −0.933253 0.359220i \(-0.883043\pi\)
−0.933253 + 0.359220i \(0.883043\pi\)
\(4\) 19.5614 2.44518
\(5\) −5.59920 −0.500807 −0.250404 0.968142i \(-0.580563\pi\)
−0.250404 + 0.968142i \(0.580563\pi\)
\(6\) 50.9169 3.46446
\(7\) 0.820208 0.0442870 0.0221435 0.999755i \(-0.492951\pi\)
0.0221435 + 0.999755i \(0.492951\pi\)
\(8\) −60.6963 −2.68242
\(9\) 67.0638 2.48384
\(10\) 29.3952 0.929558
\(11\) −48.4653 −1.32844 −0.664220 0.747537i \(-0.731236\pi\)
−0.664220 + 0.747537i \(0.731236\pi\)
\(12\) −189.719 −4.56394
\(13\) 4.57016 0.0975025 0.0487513 0.998811i \(-0.484476\pi\)
0.0487513 + 0.998811i \(0.484476\pi\)
\(14\) −4.30601 −0.0822020
\(15\) 54.3046 0.934760
\(16\) 162.158 2.53372
\(17\) 108.420 1.54680 0.773400 0.633918i \(-0.218554\pi\)
0.773400 + 0.633918i \(0.218554\pi\)
\(18\) −352.078 −4.61031
\(19\) 24.1770 0.291925 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(20\) −109.528 −1.22456
\(21\) −7.95490 −0.0826620
\(22\) 254.438 2.46574
\(23\) −1.18989 −0.0107873 −0.00539366 0.999985i \(-0.501717\pi\)
−0.00539366 + 0.999985i \(0.501717\pi\)
\(24\) 588.672 5.00676
\(25\) −93.6490 −0.749192
\(26\) −23.9928 −0.180976
\(27\) −388.564 −2.76960
\(28\) 16.0444 0.108290
\(29\) 32.2620 0.206583 0.103291 0.994651i \(-0.467063\pi\)
0.103291 + 0.994651i \(0.467063\pi\)
\(30\) −285.094 −1.73503
\(31\) −59.1739 −0.342837 −0.171419 0.985198i \(-0.554835\pi\)
−0.171419 + 0.985198i \(0.554835\pi\)
\(32\) −365.742 −2.02046
\(33\) 470.048 2.47954
\(34\) −569.192 −2.87105
\(35\) −4.59250 −0.0221793
\(36\) 1311.86 6.07344
\(37\) 353.501 1.57068 0.785340 0.619065i \(-0.212488\pi\)
0.785340 + 0.619065i \(0.212488\pi\)
\(38\) −126.927 −0.541847
\(39\) −44.3243 −0.181989
\(40\) 339.850 1.34338
\(41\) −62.8277 −0.239318 −0.119659 0.992815i \(-0.538180\pi\)
−0.119659 + 0.992815i \(0.538180\pi\)
\(42\) 41.7624 0.153431
\(43\) 0 0
\(44\) −948.050 −3.24827
\(45\) −375.503 −1.24393
\(46\) 6.24678 0.0200226
\(47\) −213.247 −0.661816 −0.330908 0.943663i \(-0.607355\pi\)
−0.330908 + 0.943663i \(0.607355\pi\)
\(48\) −1572.71 −4.72920
\(49\) −342.327 −0.998039
\(50\) 491.648 1.39059
\(51\) −1051.52 −2.88711
\(52\) 89.3987 0.238411
\(53\) −119.382 −0.309402 −0.154701 0.987961i \(-0.549441\pi\)
−0.154701 + 0.987961i \(0.549441\pi\)
\(54\) 2039.92 5.14071
\(55\) 271.367 0.665292
\(56\) −49.7835 −0.118797
\(57\) −234.484 −0.544880
\(58\) −169.372 −0.383442
\(59\) −367.191 −0.810240 −0.405120 0.914264i \(-0.632770\pi\)
−0.405120 + 0.914264i \(0.632770\pi\)
\(60\) 1062.28 2.28565
\(61\) −36.8200 −0.0772839 −0.0386419 0.999253i \(-0.512303\pi\)
−0.0386419 + 0.999253i \(0.512303\pi\)
\(62\) 310.657 0.636347
\(63\) 55.0062 0.110002
\(64\) 622.845 1.21649
\(65\) −25.5892 −0.0488300
\(66\) −2467.70 −4.60232
\(67\) 173.580 0.316510 0.158255 0.987398i \(-0.449413\pi\)
0.158255 + 0.987398i \(0.449413\pi\)
\(68\) 2120.84 3.78220
\(69\) 11.5403 0.0201346
\(70\) 24.1102 0.0411674
\(71\) −918.269 −1.53491 −0.767454 0.641104i \(-0.778477\pi\)
−0.767454 + 0.641104i \(0.778477\pi\)
\(72\) −4070.52 −6.66272
\(73\) −33.8560 −0.0542814 −0.0271407 0.999632i \(-0.508640\pi\)
−0.0271407 + 0.999632i \(0.508640\pi\)
\(74\) −1855.84 −2.91537
\(75\) 908.269 1.39837
\(76\) 472.936 0.713808
\(77\) −39.7516 −0.0588327
\(78\) 232.698 0.337793
\(79\) −612.712 −0.872601 −0.436301 0.899801i \(-0.643712\pi\)
−0.436301 + 0.899801i \(0.643712\pi\)
\(80\) −907.954 −1.26890
\(81\) 1957.83 2.68564
\(82\) 329.839 0.444202
\(83\) −416.023 −0.550174 −0.275087 0.961419i \(-0.588707\pi\)
−0.275087 + 0.961419i \(0.588707\pi\)
\(84\) −155.609 −0.202123
\(85\) −607.063 −0.774649
\(86\) 0 0
\(87\) −312.898 −0.385588
\(88\) 2941.66 3.56344
\(89\) −921.950 −1.09805 −0.549025 0.835806i \(-0.685001\pi\)
−0.549025 + 0.835806i \(0.685001\pi\)
\(90\) 1971.35 2.30888
\(91\) 3.74848 0.00431810
\(92\) −23.2759 −0.0263769
\(93\) 573.907 0.639908
\(94\) 1119.53 1.22841
\(95\) −135.372 −0.146198
\(96\) 3547.20 3.77120
\(97\) 431.946 0.452139 0.226070 0.974111i \(-0.427412\pi\)
0.226070 + 0.974111i \(0.427412\pi\)
\(98\) 1797.18 1.85248
\(99\) −3250.27 −3.29964
\(100\) −1831.91 −1.83191
\(101\) 793.557 0.781801 0.390900 0.920433i \(-0.372164\pi\)
0.390900 + 0.920433i \(0.372164\pi\)
\(102\) 5520.39 5.35882
\(103\) 1475.90 1.41189 0.705945 0.708266i \(-0.250522\pi\)
0.705945 + 0.708266i \(0.250522\pi\)
\(104\) −277.391 −0.261543
\(105\) 44.5411 0.0413978
\(106\) 626.741 0.574288
\(107\) 524.938 0.474277 0.237139 0.971476i \(-0.423790\pi\)
0.237139 + 0.971476i \(0.423790\pi\)
\(108\) −7600.87 −6.77217
\(109\) 1931.34 1.69714 0.848572 0.529081i \(-0.177463\pi\)
0.848572 + 0.529081i \(0.177463\pi\)
\(110\) −1424.65 −1.23486
\(111\) −3428.48 −2.93168
\(112\) 133.003 0.112211
\(113\) −1163.28 −0.968424 −0.484212 0.874951i \(-0.660894\pi\)
−0.484212 + 0.874951i \(0.660894\pi\)
\(114\) 1231.02 1.01136
\(115\) 6.66241 0.00540237
\(116\) 631.091 0.505132
\(117\) 306.492 0.242181
\(118\) 1927.71 1.50390
\(119\) 88.9266 0.0685032
\(120\) −3296.09 −2.50742
\(121\) 1017.88 0.764752
\(122\) 193.301 0.143448
\(123\) 609.343 0.446688
\(124\) −1157.53 −0.838298
\(125\) 1224.26 0.876008
\(126\) −288.777 −0.204177
\(127\) 1801.72 1.25887 0.629437 0.777051i \(-0.283285\pi\)
0.629437 + 0.777051i \(0.283285\pi\)
\(128\) −343.935 −0.237499
\(129\) 0 0
\(130\) 134.341 0.0906343
\(131\) 670.848 0.447422 0.223711 0.974656i \(-0.428183\pi\)
0.223711 + 0.974656i \(0.428183\pi\)
\(132\) 9194.80 6.06292
\(133\) 19.8301 0.0129285
\(134\) −911.277 −0.587480
\(135\) 2175.65 1.38704
\(136\) −6580.67 −4.14917
\(137\) 1536.94 0.958462 0.479231 0.877689i \(-0.340916\pi\)
0.479231 + 0.877689i \(0.340916\pi\)
\(138\) −60.5853 −0.0373722
\(139\) 1697.23 1.03566 0.517831 0.855483i \(-0.326739\pi\)
0.517831 + 0.855483i \(0.326739\pi\)
\(140\) −89.8359 −0.0542323
\(141\) 2068.21 1.23528
\(142\) 4820.82 2.84897
\(143\) −221.494 −0.129526
\(144\) 10874.9 6.29336
\(145\) −180.641 −0.103458
\(146\) 177.740 0.100753
\(147\) 3320.11 1.86284
\(148\) 6914.98 3.84059
\(149\) 2923.14 1.60720 0.803600 0.595170i \(-0.202915\pi\)
0.803600 + 0.595170i \(0.202915\pi\)
\(150\) −4768.32 −2.59554
\(151\) 1455.71 0.784529 0.392265 0.919852i \(-0.371692\pi\)
0.392265 + 0.919852i \(0.371692\pi\)
\(152\) −1467.45 −0.783066
\(153\) 7271.03 3.84201
\(154\) 208.692 0.109200
\(155\) 331.327 0.171695
\(156\) −867.047 −0.444996
\(157\) −1258.86 −0.639921 −0.319961 0.947431i \(-0.603670\pi\)
−0.319961 + 0.947431i \(0.603670\pi\)
\(158\) 3216.68 1.61965
\(159\) 1157.84 0.577501
\(160\) 2047.86 1.01186
\(161\) −0.975954 −0.000477739 0
\(162\) −10278.4 −4.98486
\(163\) 2717.96 1.30606 0.653029 0.757333i \(-0.273498\pi\)
0.653029 + 0.757333i \(0.273498\pi\)
\(164\) −1229.00 −0.585175
\(165\) −2631.89 −1.24177
\(166\) 2184.08 1.02119
\(167\) −3875.29 −1.79568 −0.897841 0.440319i \(-0.854865\pi\)
−0.897841 + 0.440319i \(0.854865\pi\)
\(168\) 482.833 0.221734
\(169\) −2176.11 −0.990493
\(170\) 3187.02 1.43784
\(171\) 1621.40 0.725096
\(172\) 0 0
\(173\) −3701.24 −1.62659 −0.813296 0.581851i \(-0.802329\pi\)
−0.813296 + 0.581851i \(0.802329\pi\)
\(174\) 1642.68 0.715697
\(175\) −76.8116 −0.0331795
\(176\) −7859.03 −3.36589
\(177\) 3561.25 1.51232
\(178\) 4840.14 2.03811
\(179\) −1787.48 −0.746382 −0.373191 0.927755i \(-0.621736\pi\)
−0.373191 + 0.927755i \(0.621736\pi\)
\(180\) −7345.38 −3.04162
\(181\) 3722.70 1.52876 0.764381 0.644765i \(-0.223045\pi\)
0.764381 + 0.644765i \(0.223045\pi\)
\(182\) −19.6791 −0.00801491
\(183\) 357.104 0.144251
\(184\) 72.2217 0.0289362
\(185\) −1979.32 −0.786608
\(186\) −3012.95 −1.18774
\(187\) −5254.59 −2.05483
\(188\) −4171.42 −1.61826
\(189\) −318.704 −0.122658
\(190\) 710.687 0.271361
\(191\) −4238.79 −1.60580 −0.802900 0.596114i \(-0.796711\pi\)
−0.802900 + 0.596114i \(0.796711\pi\)
\(192\) −6040.75 −2.27059
\(193\) 2087.86 0.778691 0.389345 0.921092i \(-0.372701\pi\)
0.389345 + 0.921092i \(0.372701\pi\)
\(194\) −2267.67 −0.839224
\(195\) 248.181 0.0911414
\(196\) −6696.41 −2.44038
\(197\) 2825.59 1.02190 0.510952 0.859609i \(-0.329293\pi\)
0.510952 + 0.859609i \(0.329293\pi\)
\(198\) 17063.6 6.12452
\(199\) −44.6321 −0.0158989 −0.00794945 0.999968i \(-0.502530\pi\)
−0.00794945 + 0.999968i \(0.502530\pi\)
\(200\) 5684.15 2.00965
\(201\) −1683.49 −0.590768
\(202\) −4166.09 −1.45112
\(203\) 26.4615 0.00914894
\(204\) −20569.3 −7.05950
\(205\) 351.784 0.119852
\(206\) −7748.32 −2.62064
\(207\) −79.7983 −0.0267940
\(208\) 741.087 0.247044
\(209\) −1171.74 −0.387805
\(210\) −233.836 −0.0768392
\(211\) −4090.17 −1.33450 −0.667249 0.744835i \(-0.732528\pi\)
−0.667249 + 0.744835i \(0.732528\pi\)
\(212\) −2335.27 −0.756544
\(213\) 8905.97 2.86492
\(214\) −2755.87 −0.880315
\(215\) 0 0
\(216\) 23584.4 7.42924
\(217\) −48.5349 −0.0151832
\(218\) −10139.3 −3.15010
\(219\) 328.357 0.101317
\(220\) 5308.32 1.62676
\(221\) 495.494 0.150817
\(222\) 17999.2 5.44155
\(223\) 1439.56 0.432288 0.216144 0.976362i \(-0.430652\pi\)
0.216144 + 0.976362i \(0.430652\pi\)
\(224\) −299.984 −0.0894801
\(225\) −6280.46 −1.86088
\(226\) 6107.08 1.79751
\(227\) 3376.13 0.987143 0.493571 0.869705i \(-0.335691\pi\)
0.493571 + 0.869705i \(0.335691\pi\)
\(228\) −4586.84 −1.33233
\(229\) 3863.41 1.11485 0.557427 0.830226i \(-0.311789\pi\)
0.557427 + 0.830226i \(0.311789\pi\)
\(230\) −34.9770 −0.0100274
\(231\) 385.537 0.109812
\(232\) −1958.18 −0.554142
\(233\) −1421.57 −0.399701 −0.199850 0.979826i \(-0.564046\pi\)
−0.199850 + 0.979826i \(0.564046\pi\)
\(234\) −1609.05 −0.449517
\(235\) 1194.01 0.331442
\(236\) −7182.77 −1.98118
\(237\) 5942.48 1.62872
\(238\) −466.855 −0.127150
\(239\) 4388.76 1.18780 0.593902 0.804537i \(-0.297587\pi\)
0.593902 + 0.804537i \(0.297587\pi\)
\(240\) 8805.92 2.36842
\(241\) 2027.92 0.542031 0.271016 0.962575i \(-0.412641\pi\)
0.271016 + 0.962575i \(0.412641\pi\)
\(242\) −5343.79 −1.41947
\(243\) −8497.05 −2.24315
\(244\) −720.251 −0.188973
\(245\) 1916.76 0.499825
\(246\) −3198.99 −0.829106
\(247\) 110.492 0.0284634
\(248\) 3591.64 0.919634
\(249\) 4034.86 1.02690
\(250\) −6427.23 −1.62598
\(251\) −2680.07 −0.673962 −0.336981 0.941511i \(-0.609406\pi\)
−0.336981 + 0.941511i \(0.609406\pi\)
\(252\) 1076.00 0.268975
\(253\) 57.6682 0.0143303
\(254\) −9458.86 −2.33662
\(255\) 5887.69 1.44589
\(256\) −3177.13 −0.775668
\(257\) 5500.90 1.33516 0.667581 0.744537i \(-0.267330\pi\)
0.667581 + 0.744537i \(0.267330\pi\)
\(258\) 0 0
\(259\) 289.944 0.0695608
\(260\) −500.561 −0.119398
\(261\) 2163.61 0.513119
\(262\) −3521.88 −0.830468
\(263\) 842.845 0.197612 0.0988062 0.995107i \(-0.468498\pi\)
0.0988062 + 0.995107i \(0.468498\pi\)
\(264\) −28530.2 −6.65117
\(265\) 668.441 0.154951
\(266\) −104.106 −0.0239968
\(267\) 8941.67 2.04952
\(268\) 3395.47 0.773923
\(269\) −2415.99 −0.547605 −0.273802 0.961786i \(-0.588281\pi\)
−0.273802 + 0.961786i \(0.588281\pi\)
\(270\) −11421.9 −2.57451
\(271\) −4315.30 −0.967290 −0.483645 0.875264i \(-0.660687\pi\)
−0.483645 + 0.875264i \(0.660687\pi\)
\(272\) 17581.1 3.91916
\(273\) −36.3551 −0.00805976
\(274\) −8068.76 −1.77902
\(275\) 4538.73 0.995256
\(276\) 225.745 0.0492327
\(277\) 6373.76 1.38253 0.691267 0.722599i \(-0.257053\pi\)
0.691267 + 0.722599i \(0.257053\pi\)
\(278\) −8910.28 −1.92231
\(279\) −3968.43 −0.851554
\(280\) 278.748 0.0594942
\(281\) −6639.69 −1.40958 −0.704788 0.709418i \(-0.748958\pi\)
−0.704788 + 0.709418i \(0.748958\pi\)
\(282\) −10857.9 −2.29283
\(283\) −2035.59 −0.427573 −0.213787 0.976880i \(-0.568580\pi\)
−0.213787 + 0.976880i \(0.568580\pi\)
\(284\) −17962.7 −3.75312
\(285\) 1312.92 0.272880
\(286\) 1162.82 0.240416
\(287\) −51.5317 −0.0105987
\(288\) −24528.0 −5.01850
\(289\) 6841.81 1.39259
\(290\) 948.348 0.192031
\(291\) −4189.29 −0.843920
\(292\) −662.271 −0.132728
\(293\) −3279.18 −0.653829 −0.326915 0.945054i \(-0.606009\pi\)
−0.326915 + 0.945054i \(0.606009\pi\)
\(294\) −17430.2 −3.45766
\(295\) 2055.97 0.405774
\(296\) −21456.2 −4.21323
\(297\) 18831.9 3.67925
\(298\) −15346.2 −2.98315
\(299\) −5.43797 −0.00105179
\(300\) 17767.0 3.41927
\(301\) 0 0
\(302\) −7642.33 −1.45618
\(303\) −7696.43 −1.45924
\(304\) 3920.48 0.739655
\(305\) 206.162 0.0387043
\(306\) −38172.1 −7.13123
\(307\) 5716.06 1.06265 0.531324 0.847169i \(-0.321695\pi\)
0.531324 + 0.847169i \(0.321695\pi\)
\(308\) −777.598 −0.143856
\(309\) −14314.2 −2.63530
\(310\) −1739.43 −0.318687
\(311\) 5282.52 0.963165 0.481583 0.876401i \(-0.340062\pi\)
0.481583 + 0.876401i \(0.340062\pi\)
\(312\) 2690.32 0.488171
\(313\) 1343.26 0.242575 0.121287 0.992617i \(-0.461298\pi\)
0.121287 + 0.992617i \(0.461298\pi\)
\(314\) 6608.86 1.18777
\(315\) −307.991 −0.0550899
\(316\) −11985.5 −2.13367
\(317\) 1907.98 0.338054 0.169027 0.985611i \(-0.445937\pi\)
0.169027 + 0.985611i \(0.445937\pi\)
\(318\) −6078.54 −1.07191
\(319\) −1563.59 −0.274433
\(320\) −3487.43 −0.609229
\(321\) −5091.19 −0.885242
\(322\) 5.12366 0.000886740 0
\(323\) 2621.26 0.451550
\(324\) 38297.9 6.56686
\(325\) −427.990 −0.0730481
\(326\) −14269.0 −2.42420
\(327\) −18731.4 −3.16773
\(328\) 3813.41 0.641952
\(329\) −174.907 −0.0293099
\(330\) 13817.2 2.30488
\(331\) −9225.01 −1.53188 −0.765940 0.642912i \(-0.777726\pi\)
−0.765940 + 0.642912i \(0.777726\pi\)
\(332\) −8138.00 −1.34527
\(333\) 23707.1 3.90132
\(334\) 20344.9 3.33300
\(335\) −971.908 −0.158511
\(336\) −1289.95 −0.209442
\(337\) 2193.66 0.354589 0.177295 0.984158i \(-0.443265\pi\)
0.177295 + 0.984158i \(0.443265\pi\)
\(338\) 11424.4 1.83847
\(339\) 11282.2 1.80757
\(340\) −11875.0 −1.89416
\(341\) 2867.88 0.455439
\(342\) −8512.17 −1.34586
\(343\) −562.111 −0.0884872
\(344\) 0 0
\(345\) −64.6164 −0.0100836
\(346\) 19431.1 3.01915
\(347\) 7606.27 1.17673 0.588366 0.808595i \(-0.299771\pi\)
0.588366 + 0.808595i \(0.299771\pi\)
\(348\) −6120.72 −0.942831
\(349\) 11139.6 1.70857 0.854285 0.519805i \(-0.173995\pi\)
0.854285 + 0.519805i \(0.173995\pi\)
\(350\) 403.253 0.0615851
\(351\) −1775.80 −0.270043
\(352\) 17725.8 2.68406
\(353\) −6695.74 −1.00957 −0.504785 0.863245i \(-0.668428\pi\)
−0.504785 + 0.863245i \(0.668428\pi\)
\(354\) −18696.2 −2.80704
\(355\) 5141.57 0.768694
\(356\) −18034.7 −2.68493
\(357\) −862.468 −0.127862
\(358\) 9384.08 1.38537
\(359\) 1450.57 0.213253 0.106627 0.994299i \(-0.465995\pi\)
0.106627 + 0.994299i \(0.465995\pi\)
\(360\) 22791.7 3.33674
\(361\) −6274.47 −0.914780
\(362\) −19543.8 −2.83756
\(363\) −9872.10 −1.42741
\(364\) 73.3255 0.0105585
\(365\) 189.566 0.0271845
\(366\) −1874.76 −0.267747
\(367\) 852.755 0.121290 0.0606450 0.998159i \(-0.480684\pi\)
0.0606450 + 0.998159i \(0.480684\pi\)
\(368\) −192.949 −0.0273320
\(369\) −4213.46 −0.594428
\(370\) 10391.2 1.46004
\(371\) −97.9177 −0.0137025
\(372\) 11226.4 1.56469
\(373\) −5505.36 −0.764228 −0.382114 0.924115i \(-0.624804\pi\)
−0.382114 + 0.924115i \(0.624804\pi\)
\(374\) 27586.0 3.81401
\(375\) −11873.7 −1.63507
\(376\) 12943.3 1.77527
\(377\) 147.442 0.0201423
\(378\) 1673.16 0.227667
\(379\) −1565.31 −0.212149 −0.106074 0.994358i \(-0.533828\pi\)
−0.106074 + 0.994358i \(0.533828\pi\)
\(380\) −2648.06 −0.357480
\(381\) −17474.3 −2.34970
\(382\) 22253.2 2.98056
\(383\) 9163.31 1.22251 0.611257 0.791432i \(-0.290664\pi\)
0.611257 + 0.791432i \(0.290664\pi\)
\(384\) 3335.70 0.443292
\(385\) 222.577 0.0294638
\(386\) −10961.0 −1.44534
\(387\) 0 0
\(388\) 8449.48 1.10556
\(389\) −6892.14 −0.898318 −0.449159 0.893452i \(-0.648276\pi\)
−0.449159 + 0.893452i \(0.648276\pi\)
\(390\) −1302.92 −0.169169
\(391\) −129.007 −0.0166858
\(392\) 20778.0 2.67716
\(393\) −6506.32 −0.835116
\(394\) −14834.1 −1.89678
\(395\) 3430.70 0.437005
\(396\) −63579.8 −8.06820
\(397\) −9704.76 −1.22687 −0.613436 0.789744i \(-0.710213\pi\)
−0.613436 + 0.789744i \(0.710213\pi\)
\(398\) 234.314 0.0295103
\(399\) −192.325 −0.0241311
\(400\) −15185.9 −1.89824
\(401\) −6933.44 −0.863440 −0.431720 0.902008i \(-0.642093\pi\)
−0.431720 + 0.902008i \(0.642093\pi\)
\(402\) 8838.16 1.09654
\(403\) −270.434 −0.0334275
\(404\) 15523.1 1.91164
\(405\) −10962.3 −1.34499
\(406\) −138.920 −0.0169815
\(407\) −17132.5 −2.08655
\(408\) 63823.6 7.74445
\(409\) 3733.75 0.451399 0.225699 0.974197i \(-0.427533\pi\)
0.225699 + 0.974197i \(0.427533\pi\)
\(410\) −1846.83 −0.222460
\(411\) −14906.2 −1.78898
\(412\) 28870.7 3.45232
\(413\) −301.173 −0.0358831
\(414\) 418.933 0.0497329
\(415\) 2329.39 0.275531
\(416\) −1671.50 −0.197000
\(417\) −16460.8 −1.93307
\(418\) 6151.53 0.719812
\(419\) −7400.59 −0.862869 −0.431435 0.902144i \(-0.641992\pi\)
−0.431435 + 0.902144i \(0.641992\pi\)
\(420\) 871.287 0.101225
\(421\) −813.451 −0.0941691 −0.0470845 0.998891i \(-0.514993\pi\)
−0.0470845 + 0.998891i \(0.514993\pi\)
\(422\) 21473.0 2.47699
\(423\) −14301.2 −1.64385
\(424\) 7246.02 0.829948
\(425\) −10153.4 −1.15885
\(426\) −46755.4 −5.31763
\(427\) −30.2000 −0.00342267
\(428\) 10268.5 1.15969
\(429\) 2148.19 0.241761
\(430\) 0 0
\(431\) −6470.90 −0.723184 −0.361592 0.932337i \(-0.617767\pi\)
−0.361592 + 0.932337i \(0.617767\pi\)
\(432\) −63008.8 −7.01739
\(433\) −11614.5 −1.28905 −0.644523 0.764585i \(-0.722944\pi\)
−0.644523 + 0.764585i \(0.722944\pi\)
\(434\) 254.803 0.0281819
\(435\) 1751.98 0.193105
\(436\) 37779.7 4.14982
\(437\) −28.7678 −0.00314909
\(438\) −1723.84 −0.188056
\(439\) 7410.59 0.805668 0.402834 0.915273i \(-0.368025\pi\)
0.402834 + 0.915273i \(0.368025\pi\)
\(440\) −16470.9 −1.78459
\(441\) −22957.8 −2.47897
\(442\) −2601.29 −0.279934
\(443\) 10749.4 1.15287 0.576433 0.817145i \(-0.304444\pi\)
0.576433 + 0.817145i \(0.304444\pi\)
\(444\) −67065.9 −7.16849
\(445\) 5162.18 0.549912
\(446\) −7557.55 −0.802378
\(447\) −28350.5 −2.99985
\(448\) 510.862 0.0538749
\(449\) −2958.61 −0.310970 −0.155485 0.987838i \(-0.549694\pi\)
−0.155485 + 0.987838i \(0.549694\pi\)
\(450\) 32971.7 3.45401
\(451\) 3044.96 0.317919
\(452\) −22755.4 −2.36797
\(453\) −14118.4 −1.46433
\(454\) −17724.3 −1.83225
\(455\) −20.9885 −0.00216254
\(456\) 14232.3 1.46160
\(457\) 7447.46 0.762314 0.381157 0.924510i \(-0.375526\pi\)
0.381157 + 0.924510i \(0.375526\pi\)
\(458\) −20282.5 −2.06930
\(459\) −42128.0 −4.28402
\(460\) 130.326 0.0132098
\(461\) 13145.0 1.32803 0.664015 0.747719i \(-0.268851\pi\)
0.664015 + 0.747719i \(0.268851\pi\)
\(462\) −2024.03 −0.203823
\(463\) −8298.41 −0.832958 −0.416479 0.909145i \(-0.636736\pi\)
−0.416479 + 0.909145i \(0.636736\pi\)
\(464\) 5231.54 0.523422
\(465\) −3213.42 −0.320470
\(466\) 7463.10 0.741892
\(467\) 5139.31 0.509248 0.254624 0.967040i \(-0.418048\pi\)
0.254624 + 0.967040i \(0.418048\pi\)
\(468\) 5995.42 0.592176
\(469\) 142.372 0.0140173
\(470\) −6268.45 −0.615196
\(471\) 12209.2 1.19442
\(472\) 22287.1 2.17340
\(473\) 0 0
\(474\) −31197.4 −3.02309
\(475\) −2264.15 −0.218708
\(476\) 1739.53 0.167503
\(477\) −8006.18 −0.768507
\(478\) −23040.5 −2.20471
\(479\) 10403.5 0.992377 0.496188 0.868215i \(-0.334733\pi\)
0.496188 + 0.868215i \(0.334733\pi\)
\(480\) −19861.5 −1.88864
\(481\) 1615.55 0.153145
\(482\) −10646.3 −1.00607
\(483\) 9.46543 0.000891702 0
\(484\) 19911.3 1.86995
\(485\) −2418.55 −0.226435
\(486\) 44608.6 4.16356
\(487\) 130.371 0.0121308 0.00606538 0.999982i \(-0.498069\pi\)
0.00606538 + 0.999982i \(0.498069\pi\)
\(488\) 2234.84 0.207308
\(489\) −26360.6 −2.43776
\(490\) −10062.8 −0.927735
\(491\) 6549.97 0.602029 0.301014 0.953620i \(-0.402675\pi\)
0.301014 + 0.953620i \(0.402675\pi\)
\(492\) 11919.6 1.09223
\(493\) 3497.83 0.319543
\(494\) −580.074 −0.0528315
\(495\) 18198.9 1.65248
\(496\) −9595.52 −0.868652
\(497\) −753.171 −0.0679766
\(498\) −21182.6 −1.90606
\(499\) −12423.5 −1.11454 −0.557268 0.830333i \(-0.688150\pi\)
−0.557268 + 0.830333i \(0.688150\pi\)
\(500\) 23948.2 2.14200
\(501\) 37585.1 3.35165
\(502\) 14070.1 1.25095
\(503\) 16203.4 1.43633 0.718166 0.695871i \(-0.244982\pi\)
0.718166 + 0.695871i \(0.244982\pi\)
\(504\) −3338.67 −0.295072
\(505\) −4443.28 −0.391532
\(506\) −302.752 −0.0265988
\(507\) 21105.4 1.84876
\(508\) 35244.3 3.07817
\(509\) 21966.2 1.91283 0.956417 0.292004i \(-0.0943222\pi\)
0.956417 + 0.292004i \(0.0943222\pi\)
\(510\) −30909.8 −2.68374
\(511\) −27.7689 −0.00240396
\(512\) 19431.1 1.67723
\(513\) −9394.31 −0.808516
\(514\) −28879.2 −2.47822
\(515\) −8263.85 −0.707085
\(516\) 0 0
\(517\) 10335.1 0.879182
\(518\) −1522.18 −0.129113
\(519\) 35897.1 3.03604
\(520\) 1553.17 0.130983
\(521\) −15304.4 −1.28694 −0.643471 0.765470i \(-0.722506\pi\)
−0.643471 + 0.765470i \(0.722506\pi\)
\(522\) −11358.7 −0.952411
\(523\) −5101.60 −0.426534 −0.213267 0.976994i \(-0.568410\pi\)
−0.213267 + 0.976994i \(0.568410\pi\)
\(524\) 13122.7 1.09403
\(525\) 744.969 0.0619297
\(526\) −4424.85 −0.366792
\(527\) −6415.62 −0.530301
\(528\) 76222.0 6.28245
\(529\) −12165.6 −0.999884
\(530\) −3509.25 −0.287607
\(531\) −24625.2 −2.01251
\(532\) 387.905 0.0316125
\(533\) −287.132 −0.0233341
\(534\) −46942.9 −3.80415
\(535\) −2939.23 −0.237522
\(536\) −10535.7 −0.849013
\(537\) 17336.1 1.39313
\(538\) 12683.7 1.01642
\(539\) 16591.0 1.32583
\(540\) 42558.8 3.39155
\(541\) 3790.76 0.301253 0.150626 0.988591i \(-0.451871\pi\)
0.150626 + 0.988591i \(0.451871\pi\)
\(542\) 22654.9 1.79540
\(543\) −36105.1 −2.85344
\(544\) −39653.6 −3.12525
\(545\) −10813.9 −0.849942
\(546\) 190.861 0.0149599
\(547\) 3861.12 0.301809 0.150905 0.988548i \(-0.451781\pi\)
0.150905 + 0.988548i \(0.451781\pi\)
\(548\) 30064.7 2.34361
\(549\) −2469.29 −0.191961
\(550\) −23827.8 −1.84731
\(551\) 779.997 0.0603067
\(552\) −700.453 −0.0540095
\(553\) −502.551 −0.0386449
\(554\) −33461.6 −2.56615
\(555\) 19196.7 1.46821
\(556\) 33200.2 2.53238
\(557\) 14089.2 1.07177 0.535886 0.844290i \(-0.319978\pi\)
0.535886 + 0.844290i \(0.319978\pi\)
\(558\) 20833.8 1.58059
\(559\) 0 0
\(560\) −744.711 −0.0561960
\(561\) 50962.4 3.83535
\(562\) 34857.7 2.61634
\(563\) −4417.45 −0.330681 −0.165340 0.986237i \(-0.552872\pi\)
−0.165340 + 0.986237i \(0.552872\pi\)
\(564\) 40457.2 3.02049
\(565\) 6513.42 0.484994
\(566\) 10686.6 0.793627
\(567\) 1605.83 0.118939
\(568\) 55735.5 4.11727
\(569\) −2935.33 −0.216266 −0.108133 0.994136i \(-0.534487\pi\)
−0.108133 + 0.994136i \(0.534487\pi\)
\(570\) −6892.70 −0.506497
\(571\) −14735.7 −1.07998 −0.539990 0.841672i \(-0.681572\pi\)
−0.539990 + 0.841672i \(0.681572\pi\)
\(572\) −4332.74 −0.316715
\(573\) 41110.5 2.99723
\(574\) 270.536 0.0196724
\(575\) 111.432 0.00808178
\(576\) 41770.3 3.02158
\(577\) 12632.7 0.911450 0.455725 0.890121i \(-0.349380\pi\)
0.455725 + 0.890121i \(0.349380\pi\)
\(578\) −35918.8 −2.58482
\(579\) −20249.4 −1.45343
\(580\) −3533.60 −0.252974
\(581\) −341.225 −0.0243656
\(582\) 21993.4 1.56642
\(583\) 5785.86 0.411022
\(584\) 2054.93 0.145606
\(585\) −1716.11 −0.121286
\(586\) 17215.4 1.21358
\(587\) −11971.0 −0.841730 −0.420865 0.907123i \(-0.638273\pi\)
−0.420865 + 0.907123i \(0.638273\pi\)
\(588\) 64946.1 4.55499
\(589\) −1430.65 −0.100083
\(590\) −10793.6 −0.753165
\(591\) −27404.4 −1.90739
\(592\) 57322.9 3.97966
\(593\) −25205.0 −1.74544 −0.872719 0.488223i \(-0.837645\pi\)
−0.872719 + 0.488223i \(0.837645\pi\)
\(594\) −98865.5 −6.82913
\(595\) −497.917 −0.0343069
\(596\) 57180.7 3.92989
\(597\) 432.871 0.0296754
\(598\) 28.5488 0.00195225
\(599\) 10917.1 0.744674 0.372337 0.928098i \(-0.378557\pi\)
0.372337 + 0.928098i \(0.378557\pi\)
\(600\) −55128.5 −3.75102
\(601\) 7264.54 0.493056 0.246528 0.969136i \(-0.420710\pi\)
0.246528 + 0.969136i \(0.420710\pi\)
\(602\) 0 0
\(603\) 11640.9 0.786161
\(604\) 28475.7 1.91831
\(605\) −5699.34 −0.382993
\(606\) 40405.5 2.70852
\(607\) −21342.0 −1.42709 −0.713547 0.700607i \(-0.752913\pi\)
−0.713547 + 0.700607i \(0.752913\pi\)
\(608\) −8842.53 −0.589822
\(609\) −256.641 −0.0170766
\(610\) −1082.33 −0.0718398
\(611\) −974.574 −0.0645287
\(612\) 142232. 9.39440
\(613\) −11531.3 −0.759779 −0.379889 0.925032i \(-0.624038\pi\)
−0.379889 + 0.925032i \(0.624038\pi\)
\(614\) −30008.7 −1.97240
\(615\) −3411.83 −0.223705
\(616\) 2412.77 0.157814
\(617\) 9055.91 0.590887 0.295444 0.955360i \(-0.404533\pi\)
0.295444 + 0.955360i \(0.404533\pi\)
\(618\) 75148.3 4.89143
\(619\) −4260.01 −0.276615 −0.138307 0.990389i \(-0.544166\pi\)
−0.138307 + 0.990389i \(0.544166\pi\)
\(620\) 6481.22 0.419826
\(621\) 462.348 0.0298766
\(622\) −27732.7 −1.78775
\(623\) −756.191 −0.0486294
\(624\) −7187.54 −0.461109
\(625\) 4851.26 0.310481
\(626\) −7052.00 −0.450247
\(627\) 11364.3 0.723840
\(628\) −24625.0 −1.56472
\(629\) 38326.4 2.42953
\(630\) 1616.92 0.102253
\(631\) 5353.21 0.337731 0.168865 0.985639i \(-0.445990\pi\)
0.168865 + 0.985639i \(0.445990\pi\)
\(632\) 37189.4 2.34069
\(633\) 39669.1 2.49085
\(634\) −10016.7 −0.627468
\(635\) −10088.2 −0.630454
\(636\) 22649.0 1.41209
\(637\) −1564.49 −0.0973113
\(638\) 8208.67 0.509380
\(639\) −61582.6 −3.81247
\(640\) 1925.76 0.118941
\(641\) 18014.6 1.11004 0.555020 0.831837i \(-0.312711\pi\)
0.555020 + 0.831837i \(0.312711\pi\)
\(642\) 26728.2 1.64311
\(643\) 13215.0 0.810496 0.405248 0.914207i \(-0.367185\pi\)
0.405248 + 0.914207i \(0.367185\pi\)
\(644\) −19.0910 −0.00116816
\(645\) 0 0
\(646\) −13761.3 −0.838130
\(647\) 9446.10 0.573979 0.286989 0.957934i \(-0.407346\pi\)
0.286989 + 0.957934i \(0.407346\pi\)
\(648\) −118833. −7.20401
\(649\) 17796.0 1.07635
\(650\) 2246.91 0.135586
\(651\) 470.723 0.0283396
\(652\) 53167.2 3.19354
\(653\) −16070.1 −0.963048 −0.481524 0.876433i \(-0.659917\pi\)
−0.481524 + 0.876433i \(0.659917\pi\)
\(654\) 98337.7 5.87968
\(655\) −3756.21 −0.224072
\(656\) −10188.0 −0.606364
\(657\) −2270.51 −0.134826
\(658\) 918.245 0.0544026
\(659\) 24256.6 1.43385 0.716923 0.697152i \(-0.245550\pi\)
0.716923 + 0.697152i \(0.245550\pi\)
\(660\) −51483.5 −3.03635
\(661\) −14345.2 −0.844123 −0.422061 0.906567i \(-0.638693\pi\)
−0.422061 + 0.906567i \(0.638693\pi\)
\(662\) 48430.4 2.84335
\(663\) −4805.63 −0.281501
\(664\) 25251.1 1.47580
\(665\) −111.033 −0.00647468
\(666\) −124460. −7.24132
\(667\) −38.3881 −0.00222848
\(668\) −75806.2 −4.39076
\(669\) −13961.8 −0.806868
\(670\) 5102.42 0.294214
\(671\) 1784.49 0.102667
\(672\) 2909.44 0.167015
\(673\) −19972.6 −1.14397 −0.571983 0.820266i \(-0.693826\pi\)
−0.571983 + 0.820266i \(0.693826\pi\)
\(674\) −11516.5 −0.658160
\(675\) 36388.7 2.07496
\(676\) −42567.9 −2.42193
\(677\) −4701.00 −0.266874 −0.133437 0.991057i \(-0.542601\pi\)
−0.133437 + 0.991057i \(0.542601\pi\)
\(678\) −59230.5 −3.35506
\(679\) 354.286 0.0200239
\(680\) 36846.4 2.07794
\(681\) −32743.9 −1.84251
\(682\) −15056.1 −0.845348
\(683\) −9546.69 −0.534838 −0.267419 0.963580i \(-0.586171\pi\)
−0.267419 + 0.963580i \(0.586171\pi\)
\(684\) 31716.8 1.77299
\(685\) −8605.61 −0.480005
\(686\) 2951.02 0.164243
\(687\) −37469.9 −2.08088
\(688\) 0 0
\(689\) −545.592 −0.0301675
\(690\) 339.229 0.0187163
\(691\) 5642.15 0.310619 0.155309 0.987866i \(-0.450363\pi\)
0.155309 + 0.987866i \(0.450363\pi\)
\(692\) −72401.6 −3.97731
\(693\) −2665.89 −0.146131
\(694\) −39932.2 −2.18416
\(695\) −9503.12 −0.518668
\(696\) 18991.7 1.03431
\(697\) −6811.75 −0.370177
\(698\) −58481.9 −3.17131
\(699\) 13787.3 0.746043
\(700\) −1502.54 −0.0811298
\(701\) 4272.35 0.230192 0.115096 0.993354i \(-0.463282\pi\)
0.115096 + 0.993354i \(0.463282\pi\)
\(702\) 9322.77 0.501232
\(703\) 8546.57 0.458521
\(704\) −30186.3 −1.61604
\(705\) −11580.3 −0.618639
\(706\) 35151.9 1.87388
\(707\) 650.882 0.0346236
\(708\) 69663.2 3.69788
\(709\) −5618.45 −0.297610 −0.148805 0.988867i \(-0.547543\pi\)
−0.148805 + 0.988867i \(0.547543\pi\)
\(710\) −26992.7 −1.42679
\(711\) −41090.8 −2.16741
\(712\) 55959.0 2.94544
\(713\) 70.4103 0.00369830
\(714\) 4527.87 0.237327
\(715\) 1240.19 0.0648677
\(716\) −34965.6 −1.82504
\(717\) −42565.0 −2.21704
\(718\) −7615.32 −0.395824
\(719\) −20888.3 −1.08345 −0.541726 0.840555i \(-0.682229\pi\)
−0.541726 + 0.840555i \(0.682229\pi\)
\(720\) −60890.8 −3.15176
\(721\) 1210.54 0.0625285
\(722\) 32940.3 1.69794
\(723\) −19668.0 −1.01170
\(724\) 72821.3 3.73810
\(725\) −3021.30 −0.154770
\(726\) 51827.5 2.64945
\(727\) −22269.4 −1.13607 −0.568036 0.823004i \(-0.692297\pi\)
−0.568036 + 0.823004i \(0.692297\pi\)
\(728\) −227.519 −0.0115830
\(729\) 29548.5 1.50122
\(730\) −995.203 −0.0504577
\(731\) 0 0
\(732\) 6985.46 0.352719
\(733\) −3529.35 −0.177844 −0.0889219 0.996039i \(-0.528342\pi\)
−0.0889219 + 0.996039i \(0.528342\pi\)
\(734\) −4476.88 −0.225129
\(735\) −18590.0 −0.932926
\(736\) 435.191 0.0217953
\(737\) −8412.60 −0.420464
\(738\) 22120.2 1.10333
\(739\) −2076.61 −0.103369 −0.0516843 0.998663i \(-0.516459\pi\)
−0.0516843 + 0.998663i \(0.516459\pi\)
\(740\) −38718.3 −1.92340
\(741\) −1071.63 −0.0531271
\(742\) 514.058 0.0254335
\(743\) 13559.1 0.669496 0.334748 0.942308i \(-0.391349\pi\)
0.334748 + 0.942308i \(0.391349\pi\)
\(744\) −34834.0 −1.71650
\(745\) −16367.2 −0.804898
\(746\) 28902.6 1.41850
\(747\) −27900.1 −1.36655
\(748\) −102787. −5.02443
\(749\) 430.558 0.0210043
\(750\) 62335.5 3.03489
\(751\) −26360.2 −1.28082 −0.640410 0.768033i \(-0.721236\pi\)
−0.640410 + 0.768033i \(0.721236\pi\)
\(752\) −34579.8 −1.67685
\(753\) 25993.0 1.25795
\(754\) −774.057 −0.0373866
\(755\) −8150.80 −0.392898
\(756\) −6234.29 −0.299919
\(757\) −6025.87 −0.289318 −0.144659 0.989482i \(-0.546209\pi\)
−0.144659 + 0.989482i \(0.546209\pi\)
\(758\) 8217.70 0.393774
\(759\) −559.304 −0.0267476
\(760\) 8216.55 0.392165
\(761\) 11127.9 0.530073 0.265037 0.964238i \(-0.414616\pi\)
0.265037 + 0.964238i \(0.414616\pi\)
\(762\) 91738.2 4.36132
\(763\) 1584.10 0.0751615
\(764\) −82916.7 −3.92647
\(765\) −40711.9 −1.92411
\(766\) −48106.4 −2.26913
\(767\) −1678.12 −0.0790004
\(768\) 30813.9 1.44779
\(769\) 26446.7 1.24017 0.620085 0.784534i \(-0.287098\pi\)
0.620085 + 0.784534i \(0.287098\pi\)
\(770\) −1168.51 −0.0546884
\(771\) −53351.3 −2.49209
\(772\) 40841.5 1.90404
\(773\) −3081.60 −0.143386 −0.0716930 0.997427i \(-0.522840\pi\)
−0.0716930 + 0.997427i \(0.522840\pi\)
\(774\) 0 0
\(775\) 5541.58 0.256851
\(776\) −26217.5 −1.21283
\(777\) −2812.06 −0.129836
\(778\) 36183.1 1.66738
\(779\) −1518.98 −0.0698629
\(780\) 4854.77 0.222857
\(781\) 44504.2 2.03903
\(782\) 677.274 0.0309709
\(783\) −12535.9 −0.572152
\(784\) −55511.1 −2.52875
\(785\) 7048.58 0.320477
\(786\) 34157.5 1.55007
\(787\) −17984.9 −0.814605 −0.407302 0.913293i \(-0.633530\pi\)
−0.407302 + 0.913293i \(0.633530\pi\)
\(788\) 55272.6 2.49874
\(789\) −8174.46 −0.368845
\(790\) −18010.8 −0.811134
\(791\) −954.129 −0.0428886
\(792\) 197279. 8.85102
\(793\) −168.273 −0.00753537
\(794\) 50949.0 2.27722
\(795\) −6482.97 −0.289217
\(796\) −873.067 −0.0388757
\(797\) 35163.6 1.56281 0.781403 0.624026i \(-0.214504\pi\)
0.781403 + 0.624026i \(0.214504\pi\)
\(798\) 1009.69 0.0447902
\(799\) −23120.2 −1.02370
\(800\) 34251.4 1.51371
\(801\) −61829.5 −2.72739
\(802\) 36399.8 1.60265
\(803\) 1640.84 0.0721095
\(804\) −32931.5 −1.44453
\(805\) 5.46456 0.000239255 0
\(806\) 1419.75 0.0620454
\(807\) 23431.9 1.02211
\(808\) −48166.0 −2.09712
\(809\) 4623.40 0.200927 0.100464 0.994941i \(-0.467967\pi\)
0.100464 + 0.994941i \(0.467967\pi\)
\(810\) 57550.8 2.49645
\(811\) 20925.5 0.906034 0.453017 0.891502i \(-0.350348\pi\)
0.453017 + 0.891502i \(0.350348\pi\)
\(812\) 517.625 0.0223708
\(813\) 41852.5 1.80545
\(814\) 89944.0 3.87289
\(815\) −15218.4 −0.654083
\(816\) −170513. −7.31513
\(817\) 0 0
\(818\) −19601.8 −0.837850
\(819\) 251.387 0.0107255
\(820\) 6881.40 0.293060
\(821\) −10907.2 −0.463658 −0.231829 0.972757i \(-0.574471\pi\)
−0.231829 + 0.972757i \(0.574471\pi\)
\(822\) 78256.0 3.32055
\(823\) 15600.9 0.660769 0.330384 0.943846i \(-0.392822\pi\)
0.330384 + 0.943846i \(0.392822\pi\)
\(824\) −89581.6 −3.78729
\(825\) −44019.5 −1.85765
\(826\) 1581.12 0.0666033
\(827\) 15888.5 0.668075 0.334038 0.942560i \(-0.391589\pi\)
0.334038 + 0.942560i \(0.391589\pi\)
\(828\) −1560.97 −0.0655162
\(829\) 44098.8 1.84754 0.923772 0.382942i \(-0.125089\pi\)
0.923772 + 0.382942i \(0.125089\pi\)
\(830\) −12229.1 −0.511419
\(831\) −61816.9 −2.58051
\(832\) 2846.50 0.118611
\(833\) −37115.0 −1.54377
\(834\) 86417.7 3.58801
\(835\) 21698.5 0.899291
\(836\) −22921.0 −0.948251
\(837\) 22992.9 0.949523
\(838\) 38852.3 1.60159
\(839\) −36334.9 −1.49514 −0.747569 0.664184i \(-0.768779\pi\)
−0.747569 + 0.664184i \(0.768779\pi\)
\(840\) −2703.48 −0.111046
\(841\) −23348.2 −0.957324
\(842\) 4270.54 0.174789
\(843\) 64396.1 2.63098
\(844\) −80009.5 −3.26308
\(845\) 12184.5 0.496046
\(846\) 75079.7 3.05118
\(847\) 834.877 0.0338686
\(848\) −19358.7 −0.783938
\(849\) 19742.5 0.798068
\(850\) 53304.2 2.15097
\(851\) −420.626 −0.0169434
\(852\) 174213. 7.00523
\(853\) −35791.3 −1.43666 −0.718330 0.695703i \(-0.755093\pi\)
−0.718330 + 0.695703i \(0.755093\pi\)
\(854\) 158.547 0.00635289
\(855\) −9078.53 −0.363133
\(856\) −31861.8 −1.27221
\(857\) −21314.2 −0.849566 −0.424783 0.905295i \(-0.639650\pi\)
−0.424783 + 0.905295i \(0.639650\pi\)
\(858\) −11277.8 −0.448738
\(859\) 20018.3 0.795129 0.397565 0.917574i \(-0.369855\pi\)
0.397565 + 0.917574i \(0.369855\pi\)
\(860\) 0 0
\(861\) 499.788 0.0197825
\(862\) 33971.5 1.34231
\(863\) −19104.2 −0.753550 −0.376775 0.926305i \(-0.622967\pi\)
−0.376775 + 0.926305i \(0.622967\pi\)
\(864\) 142114. 5.59586
\(865\) 20724.0 0.814609
\(866\) 60974.9 2.39262
\(867\) −66356.3 −2.59928
\(868\) −949.412 −0.0371257
\(869\) 29695.3 1.15920
\(870\) −9197.69 −0.358426
\(871\) 793.287 0.0308605
\(872\) −117225. −4.55245
\(873\) 28967.9 1.12304
\(874\) 151.028 0.00584508
\(875\) 1004.15 0.0387958
\(876\) 6423.13 0.247737
\(877\) −2573.42 −0.0990859 −0.0495430 0.998772i \(-0.515776\pi\)
−0.0495430 + 0.998772i \(0.515776\pi\)
\(878\) −38904.8 −1.49542
\(879\) 31803.6 1.22038
\(880\) 44004.2 1.68566
\(881\) −2855.96 −0.109217 −0.0546083 0.998508i \(-0.517391\pi\)
−0.0546083 + 0.998508i \(0.517391\pi\)
\(882\) 120526. 4.60127
\(883\) −38336.8 −1.46108 −0.730541 0.682869i \(-0.760732\pi\)
−0.730541 + 0.682869i \(0.760732\pi\)
\(884\) 9692.57 0.368774
\(885\) −19940.2 −0.757379
\(886\) −56433.2 −2.13986
\(887\) −18543.8 −0.701960 −0.350980 0.936383i \(-0.614152\pi\)
−0.350980 + 0.936383i \(0.614152\pi\)
\(888\) 208096. 7.86401
\(889\) 1477.79 0.0557519
\(890\) −27100.9 −1.02070
\(891\) −94886.7 −3.56770
\(892\) 28159.9 1.05702
\(893\) −5155.67 −0.193201
\(894\) 148837. 5.56808
\(895\) 10008.4 0.373794
\(896\) −282.098 −0.0105181
\(897\) 52.7409 0.00196318
\(898\) 15532.4 0.577197
\(899\) −1909.07 −0.0708243
\(900\) −122855. −4.55017
\(901\) −12943.3 −0.478584
\(902\) −15985.7 −0.590096
\(903\) 0 0
\(904\) 70606.6 2.59772
\(905\) −20844.1 −0.765615
\(906\) 74120.2 2.71797
\(907\) 27448.6 1.00487 0.502434 0.864615i \(-0.332438\pi\)
0.502434 + 0.864615i \(0.332438\pi\)
\(908\) 66041.8 2.41374
\(909\) 53218.9 1.94187
\(910\) 110.187 0.00401392
\(911\) −21983.3 −0.799495 −0.399748 0.916625i \(-0.630902\pi\)
−0.399748 + 0.916625i \(0.630902\pi\)
\(912\) −38023.4 −1.38057
\(913\) 20162.7 0.730873
\(914\) −39098.4 −1.41495
\(915\) −1999.50 −0.0722418
\(916\) 75573.9 2.72602
\(917\) 550.235 0.0198150
\(918\) 221168. 7.95166
\(919\) 44001.0 1.57939 0.789695 0.613499i \(-0.210239\pi\)
0.789695 + 0.613499i \(0.210239\pi\)
\(920\) −404.383 −0.0144914
\(921\) −55438.1 −1.98344
\(922\) −69009.7 −2.46498
\(923\) −4196.63 −0.149657
\(924\) 7541.65 0.268509
\(925\) −33105.0 −1.17674
\(926\) 43565.8 1.54607
\(927\) 98979.4 3.50692
\(928\) −11799.6 −0.417392
\(929\) −7977.00 −0.281719 −0.140860 0.990030i \(-0.544987\pi\)
−0.140860 + 0.990030i \(0.544987\pi\)
\(930\) 16870.1 0.594831
\(931\) −8276.43 −0.291352
\(932\) −27808.0 −0.977339
\(933\) −51233.3 −1.79775
\(934\) −26980.8 −0.945225
\(935\) 29421.5 1.02907
\(936\) −18602.9 −0.649632
\(937\) −16116.5 −0.561904 −0.280952 0.959722i \(-0.590650\pi\)
−0.280952 + 0.959722i \(0.590650\pi\)
\(938\) −747.436 −0.0260178
\(939\) −13027.9 −0.452767
\(940\) 23356.6 0.810435
\(941\) 24389.1 0.844912 0.422456 0.906384i \(-0.361168\pi\)
0.422456 + 0.906384i \(0.361168\pi\)
\(942\) −64097.0 −2.21698
\(943\) 74.7578 0.00258160
\(944\) −59542.8 −2.05292
\(945\) 1784.48 0.0614278
\(946\) 0 0
\(947\) −6934.33 −0.237946 −0.118973 0.992897i \(-0.537960\pi\)
−0.118973 + 0.992897i \(0.537960\pi\)
\(948\) 116243. 3.98250
\(949\) −154.727 −0.00529257
\(950\) 11886.5 0.405948
\(951\) −18504.9 −0.630980
\(952\) −5397.51 −0.183755
\(953\) −507.536 −0.0172515 −0.00862577 0.999963i \(-0.502746\pi\)
−0.00862577 + 0.999963i \(0.502746\pi\)
\(954\) 42031.6 1.42644
\(955\) 23733.8 0.804196
\(956\) 85850.3 2.90439
\(957\) 15164.7 0.512230
\(958\) −54617.4 −1.84197
\(959\) 1260.61 0.0424475
\(960\) 33823.3 1.13713
\(961\) −26289.4 −0.882463
\(962\) −8481.49 −0.284256
\(963\) 35204.3 1.17803
\(964\) 39668.9 1.32536
\(965\) −11690.3 −0.389974
\(966\) −49.6926 −0.00165511
\(967\) −35811.1 −1.19091 −0.595454 0.803389i \(-0.703028\pi\)
−0.595454 + 0.803389i \(0.703028\pi\)
\(968\) −61781.8 −2.05139
\(969\) −25422.6 −0.842820
\(970\) 12697.1 0.420290
\(971\) −46854.8 −1.54855 −0.774275 0.632849i \(-0.781885\pi\)
−0.774275 + 0.632849i \(0.781885\pi\)
\(972\) −166214. −5.48490
\(973\) 1392.08 0.0458664
\(974\) −684.435 −0.0225161
\(975\) 4150.93 0.136345
\(976\) −5970.65 −0.195815
\(977\) 4116.58 0.134802 0.0674008 0.997726i \(-0.478529\pi\)
0.0674008 + 0.997726i \(0.478529\pi\)
\(978\) 138390. 4.52478
\(979\) 44682.6 1.45869
\(980\) 37494.5 1.22216
\(981\) 129523. 4.21544
\(982\) −34386.7 −1.11744
\(983\) 18015.4 0.584539 0.292270 0.956336i \(-0.405590\pi\)
0.292270 + 0.956336i \(0.405590\pi\)
\(984\) −36984.9 −1.19821
\(985\) −15821.0 −0.511777
\(986\) −18363.3 −0.593109
\(987\) 1696.36 0.0547070
\(988\) 2161.39 0.0695981
\(989\) 0 0
\(990\) −95542.2 −3.06720
\(991\) −51294.1 −1.64421 −0.822105 0.569336i \(-0.807200\pi\)
−0.822105 + 0.569336i \(0.807200\pi\)
\(992\) 21642.4 0.692688
\(993\) 89470.1 2.85926
\(994\) 3954.07 0.126173
\(995\) 249.904 0.00796229
\(996\) 78927.6 2.51096
\(997\) −17610.7 −0.559415 −0.279707 0.960085i \(-0.590237\pi\)
−0.279707 + 0.960085i \(0.590237\pi\)
\(998\) 65222.3 2.06871
\(999\) −137358. −4.35016
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 1849.4.a.k.1.5 60
43.3 odd 42 43.4.g.a.9.10 120
43.29 odd 42 43.4.g.a.24.10 yes 120
43.42 odd 2 1849.4.a.l.1.56 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.g.a.9.10 120 43.3 odd 42
43.4.g.a.24.10 yes 120 43.29 odd 42
1849.4.a.k.1.5 60 1.1 even 1 trivial
1849.4.a.l.1.56 60 43.42 odd 2