Properties

Label 2401.4.a.c.1.6
Level $2401$
Weight $4$
Character 2401.1
Self dual yes
Analytic conductor $141.664$
Analytic rank $1$
Dimension $39$
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] = [2401,4,Mod(1,2401)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2401, 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("2401.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2401 = 7^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2401.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: \(141.663585924\)
Analytic rank: \(1\)
Dimension: \(39\)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Character \(\chi\) \(=\) 2401.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.34611 q^{2} +2.32971 q^{3} +10.8887 q^{4} +2.24811 q^{5} -10.1252 q^{6} -12.5546 q^{8} -21.5725 q^{9} +O(q^{10})\) \(q-4.34611 q^{2} +2.32971 q^{3} +10.8887 q^{4} +2.24811 q^{5} -10.1252 q^{6} -12.5546 q^{8} -21.5725 q^{9} -9.77055 q^{10} -17.9888 q^{11} +25.3675 q^{12} -16.6689 q^{13} +5.23744 q^{15} -32.5457 q^{16} +21.0084 q^{17} +93.7564 q^{18} +74.5252 q^{19} +24.4790 q^{20} +78.1814 q^{22} -9.45614 q^{23} -29.2486 q^{24} -119.946 q^{25} +72.4448 q^{26} -113.160 q^{27} -146.445 q^{29} -22.7625 q^{30} +249.890 q^{31} +241.885 q^{32} -41.9086 q^{33} -91.3048 q^{34} -234.896 q^{36} -111.969 q^{37} -323.895 q^{38} -38.8336 q^{39} -28.2243 q^{40} +121.866 q^{41} +325.998 q^{43} -195.875 q^{44} -48.4973 q^{45} +41.0974 q^{46} +434.775 q^{47} -75.8220 q^{48} +521.299 q^{50} +48.9434 q^{51} -181.502 q^{52} +86.4205 q^{53} +491.804 q^{54} -40.4409 q^{55} +173.622 q^{57} +636.468 q^{58} +572.886 q^{59} +57.0290 q^{60} +84.8298 q^{61} -1086.05 q^{62} -790.892 q^{64} -37.4735 q^{65} +182.140 q^{66} -25.3988 q^{67} +228.754 q^{68} -22.0300 q^{69} -1172.76 q^{71} +270.835 q^{72} +957.095 q^{73} +486.629 q^{74} -279.439 q^{75} +811.483 q^{76} +168.775 q^{78} +352.495 q^{79} -73.1665 q^{80} +318.828 q^{81} -529.641 q^{82} -1307.72 q^{83} +47.2292 q^{85} -1416.83 q^{86} -341.175 q^{87} +225.843 q^{88} +1201.77 q^{89} +210.775 q^{90} -102.965 q^{92} +582.170 q^{93} -1889.58 q^{94} +167.541 q^{95} +563.520 q^{96} -1434.34 q^{97} +388.063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 39 q + q^{2} - q^{3} + 145 q^{4} - 27 q^{5} - 41 q^{6} - 12 q^{8} + 312 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 39 q + q^{2} - q^{3} + 145 q^{4} - 27 q^{5} - 41 q^{6} - 12 q^{8} + 312 q^{9} - 78 q^{10} + q^{11} + 91 q^{12} - 77 q^{13} - 161 q^{15} + 461 q^{16} - 211 q^{17} + 8 q^{18} - 314 q^{19} - 476 q^{20} - 61 q^{22} - 69 q^{23} - 330 q^{24} + 606 q^{25} - 504 q^{26} + 50 q^{27} + 57 q^{29} + 42 q^{30} - 638 q^{31} - 1600 q^{32} - 1574 q^{33} - 1343 q^{34} + 782 q^{36} + 71 q^{37} - 1359 q^{38} - 84 q^{39} + 155 q^{40} - 1393 q^{41} - 125 q^{43} + 52 q^{44} - 1129 q^{45} - 1454 q^{46} - 1483 q^{47} + 974 q^{48} + 3074 q^{50} - 2044 q^{51} - 3899 q^{52} + 2213 q^{53} - 1142 q^{54} - 1604 q^{55} + 98 q^{57} + 2403 q^{58} - 2073 q^{59} - 1519 q^{60} - 2575 q^{61} - 1742 q^{62} + 1358 q^{64} + 1876 q^{65} + 48 q^{66} + 176 q^{67} - 3038 q^{68} + 638 q^{69} - 1259 q^{71} - 3799 q^{72} + 307 q^{73} + 3845 q^{74} - 131 q^{75} - 1974 q^{76} - 6041 q^{78} + 22 q^{79} + 804 q^{80} + 795 q^{81} - 8043 q^{82} - 6349 q^{83} + 3094 q^{85} + 1745 q^{86} - 9508 q^{87} + 1299 q^{88} - 2253 q^{89} - 11156 q^{90} + 1284 q^{92} - 3430 q^{93} - 2738 q^{94} - 3290 q^{95} - 3031 q^{96} + 770 q^{97} + 5384 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.34611 −1.53658 −0.768292 0.640100i \(-0.778893\pi\)
−0.768292 + 0.640100i \(0.778893\pi\)
\(3\) 2.32971 0.448352 0.224176 0.974549i \(-0.428031\pi\)
0.224176 + 0.974549i \(0.428031\pi\)
\(4\) 10.8887 1.36109
\(5\) 2.24811 0.201077 0.100539 0.994933i \(-0.467943\pi\)
0.100539 + 0.994933i \(0.467943\pi\)
\(6\) −10.1252 −0.688931
\(7\) 0 0
\(8\) −12.5546 −0.554842
\(9\) −21.5725 −0.798980
\(10\) −9.77055 −0.308972
\(11\) −17.9888 −0.493075 −0.246538 0.969133i \(-0.579293\pi\)
−0.246538 + 0.969133i \(0.579293\pi\)
\(12\) 25.3675 0.610247
\(13\) −16.6689 −0.355624 −0.177812 0.984064i \(-0.556902\pi\)
−0.177812 + 0.984064i \(0.556902\pi\)
\(14\) 0 0
\(15\) 5.23744 0.0901535
\(16\) −32.5457 −0.508527
\(17\) 21.0084 0.299722 0.149861 0.988707i \(-0.452117\pi\)
0.149861 + 0.988707i \(0.452117\pi\)
\(18\) 93.7564 1.22770
\(19\) 74.5252 0.899856 0.449928 0.893065i \(-0.351450\pi\)
0.449928 + 0.893065i \(0.351450\pi\)
\(20\) 24.4790 0.273684
\(21\) 0 0
\(22\) 78.1814 0.757651
\(23\) −9.45614 −0.0857278 −0.0428639 0.999081i \(-0.513648\pi\)
−0.0428639 + 0.999081i \(0.513648\pi\)
\(24\) −29.2486 −0.248765
\(25\) −119.946 −0.959568
\(26\) 72.4448 0.546446
\(27\) −113.160 −0.806577
\(28\) 0 0
\(29\) −146.445 −0.937731 −0.468866 0.883269i \(-0.655337\pi\)
−0.468866 + 0.883269i \(0.655337\pi\)
\(30\) −22.7625 −0.138528
\(31\) 249.890 1.44779 0.723896 0.689909i \(-0.242350\pi\)
0.723896 + 0.689909i \(0.242350\pi\)
\(32\) 241.885 1.33624
\(33\) −41.9086 −0.221071
\(34\) −91.3048 −0.460548
\(35\) 0 0
\(36\) −234.896 −1.08748
\(37\) −111.969 −0.497501 −0.248750 0.968568i \(-0.580020\pi\)
−0.248750 + 0.968568i \(0.580020\pi\)
\(38\) −323.895 −1.38270
\(39\) −38.8336 −0.159445
\(40\) −28.2243 −0.111566
\(41\) 121.866 0.464200 0.232100 0.972692i \(-0.425440\pi\)
0.232100 + 0.972692i \(0.425440\pi\)
\(42\) 0 0
\(43\) 325.998 1.15615 0.578073 0.815985i \(-0.303805\pi\)
0.578073 + 0.815985i \(0.303805\pi\)
\(44\) −195.875 −0.671119
\(45\) −48.4973 −0.160657
\(46\) 41.0974 0.131728
\(47\) 434.775 1.34933 0.674664 0.738125i \(-0.264288\pi\)
0.674664 + 0.738125i \(0.264288\pi\)
\(48\) −75.8220 −0.227999
\(49\) 0 0
\(50\) 521.299 1.47446
\(51\) 48.9434 0.134381
\(52\) −181.502 −0.484036
\(53\) 86.4205 0.223977 0.111988 0.993710i \(-0.464278\pi\)
0.111988 + 0.993710i \(0.464278\pi\)
\(54\) 491.804 1.23937
\(55\) −40.4409 −0.0991463
\(56\) 0 0
\(57\) 173.622 0.403452
\(58\) 636.468 1.44090
\(59\) 572.886 1.26412 0.632062 0.774917i \(-0.282209\pi\)
0.632062 + 0.774917i \(0.282209\pi\)
\(60\) 57.0290 0.122707
\(61\) 84.8298 0.178055 0.0890274 0.996029i \(-0.471624\pi\)
0.0890274 + 0.996029i \(0.471624\pi\)
\(62\) −1086.05 −2.22465
\(63\) 0 0
\(64\) −790.892 −1.54471
\(65\) −37.4735 −0.0715080
\(66\) 182.140 0.339695
\(67\) −25.3988 −0.0463127 −0.0231564 0.999732i \(-0.507372\pi\)
−0.0231564 + 0.999732i \(0.507372\pi\)
\(68\) 228.754 0.407949
\(69\) −22.0300 −0.0384363
\(70\) 0 0
\(71\) −1172.76 −1.96030 −0.980150 0.198255i \(-0.936472\pi\)
−0.980150 + 0.198255i \(0.936472\pi\)
\(72\) 270.835 0.443308
\(73\) 957.095 1.53451 0.767257 0.641340i \(-0.221621\pi\)
0.767257 + 0.641340i \(0.221621\pi\)
\(74\) 486.629 0.764452
\(75\) −279.439 −0.430224
\(76\) 811.483 1.22478
\(77\) 0 0
\(78\) 168.775 0.245000
\(79\) 352.495 0.502010 0.251005 0.967986i \(-0.419239\pi\)
0.251005 + 0.967986i \(0.419239\pi\)
\(80\) −73.1665 −0.102253
\(81\) 318.828 0.437350
\(82\) −529.641 −0.713282
\(83\) −1307.72 −1.72941 −0.864707 0.502277i \(-0.832496\pi\)
−0.864707 + 0.502277i \(0.832496\pi\)
\(84\) 0 0
\(85\) 47.2292 0.0602674
\(86\) −1416.83 −1.77652
\(87\) −341.175 −0.420434
\(88\) 225.843 0.273579
\(89\) 1201.77 1.43132 0.715660 0.698449i \(-0.246126\pi\)
0.715660 + 0.698449i \(0.246126\pi\)
\(90\) 210.775 0.246863
\(91\) 0 0
\(92\) −102.965 −0.116683
\(93\) 582.170 0.649121
\(94\) −1889.58 −2.07336
\(95\) 167.541 0.180941
\(96\) 563.520 0.599105
\(97\) −1434.34 −1.50139 −0.750697 0.660647i \(-0.770282\pi\)
−0.750697 + 0.660647i \(0.770282\pi\)
\(98\) 0 0
\(99\) 388.063 0.393957
\(100\) −1306.06 −1.30606
\(101\) −1609.75 −1.58590 −0.792949 0.609288i \(-0.791455\pi\)
−0.792949 + 0.609288i \(0.791455\pi\)
\(102\) −212.713 −0.206488
\(103\) 41.4991 0.0396993 0.0198496 0.999803i \(-0.493681\pi\)
0.0198496 + 0.999803i \(0.493681\pi\)
\(104\) 209.272 0.197315
\(105\) 0 0
\(106\) −375.593 −0.344159
\(107\) 1025.00 0.926083 0.463041 0.886337i \(-0.346758\pi\)
0.463041 + 0.886337i \(0.346758\pi\)
\(108\) −1232.16 −1.09782
\(109\) −1031.80 −0.906685 −0.453342 0.891336i \(-0.649769\pi\)
−0.453342 + 0.891336i \(0.649769\pi\)
\(110\) 175.761 0.152346
\(111\) −260.854 −0.223056
\(112\) 0 0
\(113\) 764.600 0.636527 0.318264 0.948002i \(-0.396900\pi\)
0.318264 + 0.948002i \(0.396900\pi\)
\(114\) −754.580 −0.619938
\(115\) −21.2585 −0.0172379
\(116\) −1594.60 −1.27634
\(117\) 359.589 0.284137
\(118\) −2489.83 −1.94243
\(119\) 0 0
\(120\) −65.7542 −0.0500210
\(121\) −1007.40 −0.756877
\(122\) −368.680 −0.273596
\(123\) 283.911 0.208125
\(124\) 2720.98 1.97057
\(125\) −550.666 −0.394025
\(126\) 0 0
\(127\) −2381.49 −1.66396 −0.831980 0.554806i \(-0.812792\pi\)
−0.831980 + 0.554806i \(0.812792\pi\)
\(128\) 1502.23 1.03734
\(129\) 759.481 0.518361
\(130\) 162.864 0.109878
\(131\) 1771.25 1.18134 0.590668 0.806915i \(-0.298864\pi\)
0.590668 + 0.806915i \(0.298864\pi\)
\(132\) −456.331 −0.300898
\(133\) 0 0
\(134\) 110.386 0.0711633
\(135\) −254.396 −0.162184
\(136\) −263.753 −0.166299
\(137\) −744.084 −0.464025 −0.232012 0.972713i \(-0.574531\pi\)
−0.232012 + 0.972713i \(0.574531\pi\)
\(138\) 95.7450 0.0590605
\(139\) 2853.92 1.74148 0.870742 0.491741i \(-0.163639\pi\)
0.870742 + 0.491741i \(0.163639\pi\)
\(140\) 0 0
\(141\) 1012.90 0.604975
\(142\) 5096.96 3.01217
\(143\) 299.853 0.175349
\(144\) 702.092 0.406303
\(145\) −329.226 −0.188557
\(146\) −4159.64 −2.35791
\(147\) 0 0
\(148\) −1219.19 −0.677143
\(149\) −3344.16 −1.83869 −0.919343 0.393457i \(-0.871279\pi\)
−0.919343 + 0.393457i \(0.871279\pi\)
\(150\) 1214.47 0.661076
\(151\) 1872.86 1.00934 0.504672 0.863311i \(-0.331613\pi\)
0.504672 + 0.863311i \(0.331613\pi\)
\(152\) −935.638 −0.499278
\(153\) −453.203 −0.239472
\(154\) 0 0
\(155\) 561.781 0.291118
\(156\) −422.847 −0.217019
\(157\) −1498.57 −0.761779 −0.380889 0.924621i \(-0.624382\pi\)
−0.380889 + 0.924621i \(0.624382\pi\)
\(158\) −1531.98 −0.771380
\(159\) 201.334 0.100420
\(160\) 543.784 0.268687
\(161\) 0 0
\(162\) −1385.66 −0.672024
\(163\) −2429.98 −1.16767 −0.583835 0.811872i \(-0.698449\pi\)
−0.583835 + 0.811872i \(0.698449\pi\)
\(164\) 1326.96 0.631817
\(165\) −94.2153 −0.0444524
\(166\) 5683.52 2.65739
\(167\) 1097.74 0.508658 0.254329 0.967118i \(-0.418145\pi\)
0.254329 + 0.967118i \(0.418145\pi\)
\(168\) 0 0
\(169\) −1919.15 −0.873531
\(170\) −205.264 −0.0926059
\(171\) −1607.69 −0.718967
\(172\) 3549.70 1.57362
\(173\) 893.115 0.392499 0.196249 0.980554i \(-0.437124\pi\)
0.196249 + 0.980554i \(0.437124\pi\)
\(174\) 1482.78 0.646032
\(175\) 0 0
\(176\) 585.459 0.250742
\(177\) 1334.66 0.566773
\(178\) −5223.04 −2.19934
\(179\) −196.938 −0.0822337 −0.0411169 0.999154i \(-0.513092\pi\)
−0.0411169 + 0.999154i \(0.513092\pi\)
\(180\) −528.073 −0.218668
\(181\) −798.131 −0.327760 −0.163880 0.986480i \(-0.552401\pi\)
−0.163880 + 0.986480i \(0.552401\pi\)
\(182\) 0 0
\(183\) 197.629 0.0798313
\(184\) 118.718 0.0475654
\(185\) −251.718 −0.100036
\(186\) −2530.18 −0.997428
\(187\) −377.916 −0.147786
\(188\) 4734.14 1.83656
\(189\) 0 0
\(190\) −728.153 −0.278030
\(191\) 1707.50 0.646861 0.323431 0.946252i \(-0.395164\pi\)
0.323431 + 0.946252i \(0.395164\pi\)
\(192\) −1842.55 −0.692575
\(193\) −3539.06 −1.31993 −0.659967 0.751294i \(-0.729430\pi\)
−0.659967 + 0.751294i \(0.729430\pi\)
\(194\) 6233.80 2.30702
\(195\) −87.3023 −0.0320608
\(196\) 0 0
\(197\) −805.236 −0.291222 −0.145611 0.989342i \(-0.546515\pi\)
−0.145611 + 0.989342i \(0.546515\pi\)
\(198\) −1686.57 −0.605348
\(199\) −143.693 −0.0511867 −0.0255933 0.999672i \(-0.508148\pi\)
−0.0255933 + 0.999672i \(0.508148\pi\)
\(200\) 1505.88 0.532409
\(201\) −59.1716 −0.0207644
\(202\) 6996.14 2.43686
\(203\) 0 0
\(204\) 532.930 0.182905
\(205\) 273.967 0.0933401
\(206\) −180.360 −0.0610012
\(207\) 203.992 0.0684949
\(208\) 542.501 0.180844
\(209\) −1340.62 −0.443697
\(210\) 0 0
\(211\) 2762.56 0.901340 0.450670 0.892691i \(-0.351185\pi\)
0.450670 + 0.892691i \(0.351185\pi\)
\(212\) 941.007 0.304852
\(213\) −2732.19 −0.878905
\(214\) −4454.78 −1.42300
\(215\) 732.881 0.232475
\(216\) 1420.68 0.447523
\(217\) 0 0
\(218\) 4484.33 1.39320
\(219\) 2229.75 0.688003
\(220\) −440.349 −0.134947
\(221\) −350.186 −0.106589
\(222\) 1133.70 0.342744
\(223\) 4232.25 1.27091 0.635454 0.772138i \(-0.280813\pi\)
0.635454 + 0.772138i \(0.280813\pi\)
\(224\) 0 0
\(225\) 2587.53 0.766676
\(226\) −3323.04 −0.978077
\(227\) −880.898 −0.257565 −0.128783 0.991673i \(-0.541107\pi\)
−0.128783 + 0.991673i \(0.541107\pi\)
\(228\) 1890.52 0.549134
\(229\) 1622.26 0.468131 0.234065 0.972221i \(-0.424797\pi\)
0.234065 + 0.972221i \(0.424797\pi\)
\(230\) 92.3917 0.0264875
\(231\) 0 0
\(232\) 1838.57 0.520293
\(233\) −3926.05 −1.10388 −0.551940 0.833884i \(-0.686112\pi\)
−0.551940 + 0.833884i \(0.686112\pi\)
\(234\) −1562.81 −0.436600
\(235\) 977.423 0.271319
\(236\) 6237.98 1.72059
\(237\) 821.209 0.225077
\(238\) 0 0
\(239\) −3458.66 −0.936077 −0.468038 0.883708i \(-0.655039\pi\)
−0.468038 + 0.883708i \(0.655039\pi\)
\(240\) −170.456 −0.0458455
\(241\) 4466.02 1.19370 0.596850 0.802353i \(-0.296419\pi\)
0.596850 + 0.802353i \(0.296419\pi\)
\(242\) 4378.29 1.16300
\(243\) 3798.08 1.00266
\(244\) 923.687 0.242348
\(245\) 0 0
\(246\) −1233.91 −0.319802
\(247\) −1242.25 −0.320010
\(248\) −3137.28 −0.803296
\(249\) −3046.61 −0.775387
\(250\) 2393.26 0.605452
\(251\) 2978.25 0.748947 0.374473 0.927238i \(-0.377824\pi\)
0.374473 + 0.927238i \(0.377824\pi\)
\(252\) 0 0
\(253\) 170.105 0.0422703
\(254\) 10350.2 2.55681
\(255\) 110.030 0.0270210
\(256\) −201.729 −0.0492502
\(257\) −4278.07 −1.03836 −0.519180 0.854665i \(-0.673763\pi\)
−0.519180 + 0.854665i \(0.673763\pi\)
\(258\) −3300.79 −0.796505
\(259\) 0 0
\(260\) −408.038 −0.0973286
\(261\) 3159.19 0.749229
\(262\) −7698.06 −1.81522
\(263\) −601.979 −0.141139 −0.0705696 0.997507i \(-0.522482\pi\)
−0.0705696 + 0.997507i \(0.522482\pi\)
\(264\) 526.148 0.122660
\(265\) 194.283 0.0450366
\(266\) 0 0
\(267\) 2799.77 0.641736
\(268\) −276.560 −0.0630357
\(269\) −7920.85 −1.79533 −0.897663 0.440682i \(-0.854737\pi\)
−0.897663 + 0.440682i \(0.854737\pi\)
\(270\) 1105.63 0.249210
\(271\) 4473.92 1.00285 0.501423 0.865202i \(-0.332810\pi\)
0.501423 + 0.865202i \(0.332810\pi\)
\(272\) −683.733 −0.152417
\(273\) 0 0
\(274\) 3233.87 0.713013
\(275\) 2157.68 0.473139
\(276\) −239.878 −0.0523152
\(277\) −1656.22 −0.359251 −0.179625 0.983735i \(-0.557489\pi\)
−0.179625 + 0.983735i \(0.557489\pi\)
\(278\) −12403.5 −2.67593
\(279\) −5390.74 −1.15676
\(280\) 0 0
\(281\) −3481.30 −0.739063 −0.369532 0.929218i \(-0.620482\pi\)
−0.369532 + 0.929218i \(0.620482\pi\)
\(282\) −4402.17 −0.929594
\(283\) 7363.19 1.54663 0.773314 0.634023i \(-0.218597\pi\)
0.773314 + 0.634023i \(0.218597\pi\)
\(284\) −12769.9 −2.66814
\(285\) 390.322 0.0811251
\(286\) −1303.20 −0.269439
\(287\) 0 0
\(288\) −5218.05 −1.06763
\(289\) −4471.65 −0.910166
\(290\) 1430.85 0.289733
\(291\) −3341.59 −0.673153
\(292\) 10421.5 2.08861
\(293\) −1440.22 −0.287162 −0.143581 0.989639i \(-0.545862\pi\)
−0.143581 + 0.989639i \(0.545862\pi\)
\(294\) 0 0
\(295\) 1287.91 0.254187
\(296\) 1405.73 0.276035
\(297\) 2035.61 0.397703
\(298\) 14534.1 2.82529
\(299\) 157.623 0.0304869
\(300\) −3042.73 −0.585573
\(301\) 0 0
\(302\) −8139.65 −1.55094
\(303\) −3750.24 −0.711041
\(304\) −2425.48 −0.457601
\(305\) 190.707 0.0358028
\(306\) 1969.67 0.367969
\(307\) 561.058 0.104304 0.0521519 0.998639i \(-0.483392\pi\)
0.0521519 + 0.998639i \(0.483392\pi\)
\(308\) 0 0
\(309\) 96.6806 0.0177993
\(310\) −2441.56 −0.447327
\(311\) −3641.23 −0.663908 −0.331954 0.943296i \(-0.607708\pi\)
−0.331954 + 0.943296i \(0.607708\pi\)
\(312\) 487.542 0.0884668
\(313\) 8474.44 1.53036 0.765182 0.643815i \(-0.222649\pi\)
0.765182 + 0.643815i \(0.222649\pi\)
\(314\) 6512.98 1.17054
\(315\) 0 0
\(316\) 3838.21 0.683279
\(317\) −661.990 −0.117290 −0.0586452 0.998279i \(-0.518678\pi\)
−0.0586452 + 0.998279i \(0.518678\pi\)
\(318\) −875.022 −0.154304
\(319\) 2634.38 0.462372
\(320\) −1778.02 −0.310606
\(321\) 2387.96 0.415211
\(322\) 0 0
\(323\) 1565.65 0.269707
\(324\) 3471.62 0.595272
\(325\) 1999.36 0.341246
\(326\) 10560.9 1.79422
\(327\) −2403.79 −0.406514
\(328\) −1529.98 −0.257558
\(329\) 0 0
\(330\) 409.471 0.0683049
\(331\) 2857.55 0.474517 0.237258 0.971447i \(-0.423751\pi\)
0.237258 + 0.971447i \(0.423751\pi\)
\(332\) −14239.4 −2.35389
\(333\) 2415.44 0.397493
\(334\) −4770.91 −0.781595
\(335\) −57.0993 −0.00931244
\(336\) 0 0
\(337\) 2053.91 0.331998 0.165999 0.986126i \(-0.446915\pi\)
0.165999 + 0.986126i \(0.446915\pi\)
\(338\) 8340.84 1.34225
\(339\) 1781.29 0.285388
\(340\) 514.265 0.0820292
\(341\) −4495.22 −0.713870
\(342\) 6987.22 1.10475
\(343\) 0 0
\(344\) −4092.79 −0.641479
\(345\) −49.5260 −0.00772866
\(346\) −3881.58 −0.603107
\(347\) 4806.99 0.743669 0.371834 0.928299i \(-0.378729\pi\)
0.371834 + 0.928299i \(0.378729\pi\)
\(348\) −3714.95 −0.572248
\(349\) −785.561 −0.120487 −0.0602437 0.998184i \(-0.519188\pi\)
−0.0602437 + 0.998184i \(0.519188\pi\)
\(350\) 0 0
\(351\) 1886.24 0.286838
\(352\) −4351.21 −0.658865
\(353\) −4340.98 −0.654524 −0.327262 0.944934i \(-0.606126\pi\)
−0.327262 + 0.944934i \(0.606126\pi\)
\(354\) −5800.57 −0.870894
\(355\) −2636.50 −0.394172
\(356\) 13085.7 1.94815
\(357\) 0 0
\(358\) 855.915 0.126359
\(359\) −3640.84 −0.535254 −0.267627 0.963523i \(-0.586239\pi\)
−0.267627 + 0.963523i \(0.586239\pi\)
\(360\) 608.867 0.0891392
\(361\) −1304.99 −0.190260
\(362\) 3468.77 0.503631
\(363\) −2346.95 −0.339347
\(364\) 0 0
\(365\) 2151.66 0.308556
\(366\) −858.916 −0.122667
\(367\) −4305.91 −0.612444 −0.306222 0.951960i \(-0.599065\pi\)
−0.306222 + 0.951960i \(0.599065\pi\)
\(368\) 307.757 0.0435949
\(369\) −2628.94 −0.370887
\(370\) 1094.00 0.153714
\(371\) 0 0
\(372\) 6339.08 0.883510
\(373\) 4904.40 0.680805 0.340403 0.940280i \(-0.389437\pi\)
0.340403 + 0.940280i \(0.389437\pi\)
\(374\) 1642.46 0.227085
\(375\) −1282.89 −0.176662
\(376\) −5458.45 −0.748665
\(377\) 2441.08 0.333480
\(378\) 0 0
\(379\) −4748.66 −0.643594 −0.321797 0.946809i \(-0.604287\pi\)
−0.321797 + 0.946809i \(0.604287\pi\)
\(380\) 1824.31 0.246276
\(381\) −5548.17 −0.746040
\(382\) −7421.00 −0.993956
\(383\) −5520.26 −0.736481 −0.368240 0.929731i \(-0.620040\pi\)
−0.368240 + 0.929731i \(0.620040\pi\)
\(384\) 3499.76 0.465094
\(385\) 0 0
\(386\) 15381.2 2.02819
\(387\) −7032.59 −0.923738
\(388\) −15618.1 −2.04353
\(389\) 11831.2 1.54207 0.771034 0.636794i \(-0.219740\pi\)
0.771034 + 0.636794i \(0.219740\pi\)
\(390\) 379.426 0.0492640
\(391\) −198.658 −0.0256946
\(392\) 0 0
\(393\) 4126.50 0.529654
\(394\) 3499.65 0.447487
\(395\) 792.448 0.100943
\(396\) 4225.50 0.536211
\(397\) −5923.53 −0.748849 −0.374425 0.927257i \(-0.622160\pi\)
−0.374425 + 0.927257i \(0.622160\pi\)
\(398\) 624.508 0.0786526
\(399\) 0 0
\(400\) 3903.73 0.487966
\(401\) 3689.81 0.459502 0.229751 0.973249i \(-0.426209\pi\)
0.229751 + 0.973249i \(0.426209\pi\)
\(402\) 257.167 0.0319062
\(403\) −4165.38 −0.514870
\(404\) −17528.1 −2.15855
\(405\) 716.761 0.0879411
\(406\) 0 0
\(407\) 2014.18 0.245305
\(408\) −614.467 −0.0745604
\(409\) 551.712 0.0667002 0.0333501 0.999444i \(-0.489382\pi\)
0.0333501 + 0.999444i \(0.489382\pi\)
\(410\) −1190.69 −0.143425
\(411\) −1733.50 −0.208047
\(412\) 451.871 0.0540342
\(413\) 0 0
\(414\) −886.573 −0.105248
\(415\) −2939.91 −0.347746
\(416\) −4031.94 −0.475198
\(417\) 6648.79 0.780798
\(418\) 5826.48 0.681777
\(419\) −13584.8 −1.58391 −0.791955 0.610579i \(-0.790937\pi\)
−0.791955 + 0.610579i \(0.790937\pi\)
\(420\) 0 0
\(421\) −14750.9 −1.70764 −0.853820 0.520569i \(-0.825720\pi\)
−0.853820 + 0.520569i \(0.825720\pi\)
\(422\) −12006.4 −1.38498
\(423\) −9379.17 −1.07809
\(424\) −1084.98 −0.124272
\(425\) −2519.87 −0.287604
\(426\) 11874.4 1.35051
\(427\) 0 0
\(428\) 11161.0 1.26048
\(429\) 698.570 0.0786183
\(430\) −3185.19 −0.357217
\(431\) −8917.00 −0.996559 −0.498280 0.867016i \(-0.666035\pi\)
−0.498280 + 0.867016i \(0.666035\pi\)
\(432\) 3682.86 0.410166
\(433\) −13038.4 −1.44708 −0.723541 0.690281i \(-0.757487\pi\)
−0.723541 + 0.690281i \(0.757487\pi\)
\(434\) 0 0
\(435\) −766.999 −0.0845397
\(436\) −11235.0 −1.23408
\(437\) −704.721 −0.0771427
\(438\) −9690.75 −1.05717
\(439\) −407.776 −0.0443328 −0.0221664 0.999754i \(-0.507056\pi\)
−0.0221664 + 0.999754i \(0.507056\pi\)
\(440\) 507.721 0.0550105
\(441\) 0 0
\(442\) 1521.95 0.163782
\(443\) 6603.65 0.708237 0.354119 0.935201i \(-0.384781\pi\)
0.354119 + 0.935201i \(0.384781\pi\)
\(444\) −2840.36 −0.303598
\(445\) 2701.72 0.287806
\(446\) −18393.9 −1.95286
\(447\) −7790.91 −0.824379
\(448\) 0 0
\(449\) 3047.11 0.320272 0.160136 0.987095i \(-0.448807\pi\)
0.160136 + 0.987095i \(0.448807\pi\)
\(450\) −11245.7 −1.17806
\(451\) −2192.21 −0.228886
\(452\) 8325.51 0.866369
\(453\) 4363.21 0.452542
\(454\) 3828.48 0.395770
\(455\) 0 0
\(456\) −2179.76 −0.223852
\(457\) 8839.77 0.904829 0.452415 0.891808i \(-0.350563\pi\)
0.452415 + 0.891808i \(0.350563\pi\)
\(458\) −7050.53 −0.719322
\(459\) −2377.30 −0.241749
\(460\) −231.477 −0.0234623
\(461\) −9929.15 −1.00314 −0.501569 0.865118i \(-0.667244\pi\)
−0.501569 + 0.865118i \(0.667244\pi\)
\(462\) 0 0
\(463\) −15165.6 −1.52225 −0.761127 0.648603i \(-0.775354\pi\)
−0.761127 + 0.648603i \(0.775354\pi\)
\(464\) 4766.17 0.476862
\(465\) 1308.78 0.130523
\(466\) 17063.1 1.69620
\(467\) −13652.8 −1.35284 −0.676421 0.736515i \(-0.736470\pi\)
−0.676421 + 0.736515i \(0.736470\pi\)
\(468\) 3915.46 0.386735
\(469\) 0 0
\(470\) −4247.99 −0.416905
\(471\) −3491.24 −0.341545
\(472\) −7192.38 −0.701390
\(473\) −5864.32 −0.570067
\(474\) −3569.07 −0.345850
\(475\) −8939.00 −0.863473
\(476\) 0 0
\(477\) −1864.30 −0.178953
\(478\) 15031.7 1.43836
\(479\) 9267.83 0.884046 0.442023 0.897004i \(-0.354261\pi\)
0.442023 + 0.897004i \(0.354261\pi\)
\(480\) 1266.86 0.120466
\(481\) 1866.39 0.176923
\(482\) −19409.8 −1.83422
\(483\) 0 0
\(484\) −10969.3 −1.03018
\(485\) −3224.56 −0.301896
\(486\) −16506.9 −1.54068
\(487\) −4135.09 −0.384762 −0.192381 0.981320i \(-0.561621\pi\)
−0.192381 + 0.981320i \(0.561621\pi\)
\(488\) −1065.01 −0.0987924
\(489\) −5661.13 −0.523528
\(490\) 0 0
\(491\) 4511.68 0.414683 0.207342 0.978269i \(-0.433519\pi\)
0.207342 + 0.978269i \(0.433519\pi\)
\(492\) 3091.42 0.283277
\(493\) −3076.58 −0.281059
\(494\) 5398.97 0.491723
\(495\) 872.409 0.0792159
\(496\) −8132.85 −0.736241
\(497\) 0 0
\(498\) 13240.9 1.19145
\(499\) −14416.5 −1.29332 −0.646662 0.762776i \(-0.723836\pi\)
−0.646662 + 0.762776i \(0.723836\pi\)
\(500\) −5996.04 −0.536302
\(501\) 2557.42 0.228058
\(502\) −12943.8 −1.15082
\(503\) −16181.4 −1.43438 −0.717188 0.696880i \(-0.754571\pi\)
−0.717188 + 0.696880i \(0.754571\pi\)
\(504\) 0 0
\(505\) −3618.89 −0.318888
\(506\) −739.294 −0.0649518
\(507\) −4471.05 −0.391650
\(508\) −25931.3 −2.26480
\(509\) −15301.3 −1.33245 −0.666227 0.745749i \(-0.732092\pi\)
−0.666227 + 0.745749i \(0.732092\pi\)
\(510\) −478.204 −0.0415200
\(511\) 0 0
\(512\) −11141.1 −0.961664
\(513\) −8433.24 −0.725803
\(514\) 18593.0 1.59553
\(515\) 93.2946 0.00798262
\(516\) 8269.76 0.705535
\(517\) −7821.08 −0.665321
\(518\) 0 0
\(519\) 2080.70 0.175978
\(520\) 470.467 0.0396756
\(521\) −3639.78 −0.306069 −0.153034 0.988221i \(-0.548905\pi\)
−0.153034 + 0.988221i \(0.548905\pi\)
\(522\) −13730.2 −1.15125
\(523\) −21163.1 −1.76940 −0.884700 0.466160i \(-0.845637\pi\)
−0.884700 + 0.466160i \(0.845637\pi\)
\(524\) 19286.6 1.60790
\(525\) 0 0
\(526\) 2616.27 0.216872
\(527\) 5249.78 0.433936
\(528\) 1363.95 0.112421
\(529\) −12077.6 −0.992651
\(530\) −844.376 −0.0692026
\(531\) −12358.6 −1.01001
\(532\) 0 0
\(533\) −2031.36 −0.165081
\(534\) −12168.1 −0.986080
\(535\) 2304.32 0.186214
\(536\) 318.872 0.0256962
\(537\) −458.808 −0.0368697
\(538\) 34424.9 2.75867
\(539\) 0 0
\(540\) −2770.04 −0.220747
\(541\) 20129.7 1.59971 0.799855 0.600193i \(-0.204910\pi\)
0.799855 + 0.600193i \(0.204910\pi\)
\(542\) −19444.2 −1.54096
\(543\) −1859.41 −0.146952
\(544\) 5081.60 0.400500
\(545\) −2319.61 −0.182314
\(546\) 0 0
\(547\) 15914.9 1.24401 0.622003 0.783015i \(-0.286319\pi\)
0.622003 + 0.783015i \(0.286319\pi\)
\(548\) −8102.11 −0.631579
\(549\) −1829.99 −0.142262
\(550\) −9377.54 −0.727018
\(551\) −10913.9 −0.843823
\(552\) 276.579 0.0213261
\(553\) 0 0
\(554\) 7198.11 0.552019
\(555\) −586.430 −0.0448514
\(556\) 31075.5 2.37031
\(557\) −5577.43 −0.424279 −0.212139 0.977239i \(-0.568043\pi\)
−0.212139 + 0.977239i \(0.568043\pi\)
\(558\) 23428.8 1.77745
\(559\) −5434.03 −0.411154
\(560\) 0 0
\(561\) −880.433 −0.0662601
\(562\) 15130.1 1.13563
\(563\) 11272.1 0.843809 0.421904 0.906640i \(-0.361362\pi\)
0.421904 + 0.906640i \(0.361362\pi\)
\(564\) 11029.2 0.823424
\(565\) 1718.91 0.127991
\(566\) −32001.2 −2.37652
\(567\) 0 0
\(568\) 14723.6 1.08766
\(569\) −15667.9 −1.15437 −0.577183 0.816615i \(-0.695848\pi\)
−0.577183 + 0.816615i \(0.695848\pi\)
\(570\) −1696.38 −0.124655
\(571\) 15747.6 1.15415 0.577073 0.816692i \(-0.304195\pi\)
0.577073 + 0.816692i \(0.304195\pi\)
\(572\) 3265.01 0.238666
\(573\) 3977.98 0.290022
\(574\) 0 0
\(575\) 1134.23 0.0822617
\(576\) 17061.5 1.23419
\(577\) 5393.16 0.389116 0.194558 0.980891i \(-0.437673\pi\)
0.194558 + 0.980891i \(0.437673\pi\)
\(578\) 19434.3 1.39855
\(579\) −8244.98 −0.591796
\(580\) −3584.84 −0.256642
\(581\) 0 0
\(582\) 14522.9 1.03436
\(583\) −1554.60 −0.110437
\(584\) −12016.0 −0.851413
\(585\) 808.396 0.0571334
\(586\) 6259.35 0.441248
\(587\) −12745.0 −0.896157 −0.448078 0.893994i \(-0.647891\pi\)
−0.448078 + 0.893994i \(0.647891\pi\)
\(588\) 0 0
\(589\) 18623.1 1.30280
\(590\) −5597.41 −0.390579
\(591\) −1875.96 −0.130570
\(592\) 3644.10 0.252993
\(593\) −12151.4 −0.841483 −0.420741 0.907181i \(-0.638230\pi\)
−0.420741 + 0.907181i \(0.638230\pi\)
\(594\) −8846.97 −0.611104
\(595\) 0 0
\(596\) −36413.6 −2.50261
\(597\) −334.763 −0.0229497
\(598\) −685.048 −0.0468457
\(599\) −4620.70 −0.315186 −0.157593 0.987504i \(-0.550373\pi\)
−0.157593 + 0.987504i \(0.550373\pi\)
\(600\) 3508.26 0.238707
\(601\) 19257.6 1.30704 0.653522 0.756908i \(-0.273291\pi\)
0.653522 + 0.756908i \(0.273291\pi\)
\(602\) 0 0
\(603\) 547.914 0.0370029
\(604\) 20393.0 1.37381
\(605\) −2264.76 −0.152191
\(606\) 16299.0 1.09257
\(607\) −10802.5 −0.722342 −0.361171 0.932500i \(-0.617623\pi\)
−0.361171 + 0.932500i \(0.617623\pi\)
\(608\) 18026.5 1.20242
\(609\) 0 0
\(610\) −828.834 −0.0550140
\(611\) −7247.21 −0.479854
\(612\) −4934.79 −0.325943
\(613\) −23434.5 −1.54406 −0.772030 0.635586i \(-0.780759\pi\)
−0.772030 + 0.635586i \(0.780759\pi\)
\(614\) −2438.42 −0.160271
\(615\) 638.264 0.0418492
\(616\) 0 0
\(617\) −25636.7 −1.67276 −0.836380 0.548149i \(-0.815333\pi\)
−0.836380 + 0.548149i \(0.815333\pi\)
\(618\) −420.185 −0.0273500
\(619\) 9673.75 0.628144 0.314072 0.949399i \(-0.398307\pi\)
0.314072 + 0.949399i \(0.398307\pi\)
\(620\) 6117.06 0.396237
\(621\) 1070.05 0.0691461
\(622\) 15825.2 1.02015
\(623\) 0 0
\(624\) 1263.87 0.0810820
\(625\) 13755.3 0.880338
\(626\) −36830.9 −2.35153
\(627\) −3123.25 −0.198932
\(628\) −16317.5 −1.03685
\(629\) −2352.28 −0.149112
\(630\) 0 0
\(631\) 14878.8 0.938694 0.469347 0.883014i \(-0.344489\pi\)
0.469347 + 0.883014i \(0.344489\pi\)
\(632\) −4425.45 −0.278536
\(633\) 6435.96 0.404118
\(634\) 2877.08 0.180227
\(635\) −5353.85 −0.334585
\(636\) 2192.27 0.136681
\(637\) 0 0
\(638\) −11449.3 −0.710473
\(639\) 25299.4 1.56624
\(640\) 3377.18 0.208586
\(641\) −21274.3 −1.31089 −0.655447 0.755241i \(-0.727520\pi\)
−0.655447 + 0.755241i \(0.727520\pi\)
\(642\) −10378.3 −0.638007
\(643\) −30347.4 −1.86125 −0.930627 0.365969i \(-0.880738\pi\)
−0.930627 + 0.365969i \(0.880738\pi\)
\(644\) 0 0
\(645\) 1707.40 0.104231
\(646\) −6804.51 −0.414427
\(647\) 3556.80 0.216124 0.108062 0.994144i \(-0.465536\pi\)
0.108062 + 0.994144i \(0.465536\pi\)
\(648\) −4002.77 −0.242660
\(649\) −10305.5 −0.623309
\(650\) −8689.47 −0.524352
\(651\) 0 0
\(652\) −26459.3 −1.58930
\(653\) 3633.28 0.217736 0.108868 0.994056i \(-0.465277\pi\)
0.108868 + 0.994056i \(0.465277\pi\)
\(654\) 10447.2 0.624643
\(655\) 3981.97 0.237540
\(656\) −3966.20 −0.236058
\(657\) −20646.9 −1.22605
\(658\) 0 0
\(659\) −5559.27 −0.328617 −0.164308 0.986409i \(-0.552539\pi\)
−0.164308 + 0.986409i \(0.552539\pi\)
\(660\) −1025.88 −0.0605037
\(661\) −29787.5 −1.75280 −0.876399 0.481586i \(-0.840061\pi\)
−0.876399 + 0.481586i \(0.840061\pi\)
\(662\) −12419.2 −0.729135
\(663\) −815.831 −0.0477892
\(664\) 16418.0 0.959552
\(665\) 0 0
\(666\) −10497.8 −0.610782
\(667\) 1384.81 0.0803897
\(668\) 11953.0 0.692328
\(669\) 9859.91 0.569815
\(670\) 248.160 0.0143093
\(671\) −1525.99 −0.0877944
\(672\) 0 0
\(673\) −19124.8 −1.09540 −0.547702 0.836673i \(-0.684497\pi\)
−0.547702 + 0.836673i \(0.684497\pi\)
\(674\) −8926.51 −0.510143
\(675\) 13573.0 0.773965
\(676\) −20897.0 −1.18895
\(677\) 2966.75 0.168421 0.0842107 0.996448i \(-0.473163\pi\)
0.0842107 + 0.996448i \(0.473163\pi\)
\(678\) −7741.71 −0.438523
\(679\) 0 0
\(680\) −592.946 −0.0334389
\(681\) −2052.23 −0.115480
\(682\) 19536.7 1.09692
\(683\) −11936.7 −0.668736 −0.334368 0.942443i \(-0.608523\pi\)
−0.334368 + 0.942443i \(0.608523\pi\)
\(684\) −17505.7 −0.978577
\(685\) −1672.78 −0.0933049
\(686\) 0 0
\(687\) 3779.39 0.209887
\(688\) −10609.9 −0.587932
\(689\) −1440.53 −0.0796515
\(690\) 215.246 0.0118757
\(691\) 10447.6 0.575174 0.287587 0.957754i \(-0.407147\pi\)
0.287587 + 0.957754i \(0.407147\pi\)
\(692\) 9724.86 0.534225
\(693\) 0 0
\(694\) −20891.7 −1.14271
\(695\) 6415.93 0.350173
\(696\) 4283.33 0.233275
\(697\) 2560.20 0.139131
\(698\) 3414.14 0.185139
\(699\) −9146.55 −0.494927
\(700\) 0 0
\(701\) 1573.42 0.0847748 0.0423874 0.999101i \(-0.486504\pi\)
0.0423874 + 0.999101i \(0.486504\pi\)
\(702\) −8197.83 −0.440751
\(703\) −8344.49 −0.447679
\(704\) 14227.2 0.761659
\(705\) 2277.11 0.121647
\(706\) 18866.4 1.00573
\(707\) 0 0
\(708\) 14532.7 0.771428
\(709\) 595.641 0.0315511 0.0157756 0.999876i \(-0.494978\pi\)
0.0157756 + 0.999876i \(0.494978\pi\)
\(710\) 11458.5 0.605678
\(711\) −7604.18 −0.401096
\(712\) −15087.8 −0.794157
\(713\) −2362.99 −0.124116
\(714\) 0 0
\(715\) 674.104 0.0352588
\(716\) −2144.40 −0.111927
\(717\) −8057.67 −0.419692
\(718\) 15823.5 0.822462
\(719\) −21498.1 −1.11508 −0.557542 0.830149i \(-0.688255\pi\)
−0.557542 + 0.830149i \(0.688255\pi\)
\(720\) 1578.38 0.0816983
\(721\) 0 0
\(722\) 5671.64 0.292350
\(723\) 10404.5 0.535198
\(724\) −8690.61 −0.446111
\(725\) 17565.5 0.899817
\(726\) 10200.1 0.521436
\(727\) 15331.0 0.782110 0.391055 0.920367i \(-0.372110\pi\)
0.391055 + 0.920367i \(0.372110\pi\)
\(728\) 0 0
\(729\) 240.067 0.0121967
\(730\) −9351.35 −0.474122
\(731\) 6848.70 0.346523
\(732\) 2151.92 0.108657
\(733\) 12319.8 0.620794 0.310397 0.950607i \(-0.399538\pi\)
0.310397 + 0.950607i \(0.399538\pi\)
\(734\) 18714.0 0.941071
\(735\) 0 0
\(736\) −2287.29 −0.114553
\(737\) 456.893 0.0228357
\(738\) 11425.7 0.569898
\(739\) 19796.5 0.985421 0.492711 0.870193i \(-0.336006\pi\)
0.492711 + 0.870193i \(0.336006\pi\)
\(740\) −2740.89 −0.136158
\(741\) −2894.08 −0.143477
\(742\) 0 0
\(743\) 25309.1 1.24967 0.624833 0.780759i \(-0.285167\pi\)
0.624833 + 0.780759i \(0.285167\pi\)
\(744\) −7308.94 −0.360159
\(745\) −7518.05 −0.369718
\(746\) −21315.1 −1.04611
\(747\) 28210.8 1.38177
\(748\) −4115.01 −0.201149
\(749\) 0 0
\(750\) 5575.59 0.271456
\(751\) 15125.9 0.734957 0.367479 0.930032i \(-0.380221\pi\)
0.367479 + 0.930032i \(0.380221\pi\)
\(752\) −14150.1 −0.686170
\(753\) 6938.45 0.335792
\(754\) −10609.2 −0.512420
\(755\) 4210.40 0.202956
\(756\) 0 0
\(757\) 23598.6 1.13303 0.566517 0.824050i \(-0.308290\pi\)
0.566517 + 0.824050i \(0.308290\pi\)
\(758\) 20638.2 0.988937
\(759\) 396.294 0.0189520
\(760\) −2103.42 −0.100393
\(761\) −19827.3 −0.944468 −0.472234 0.881473i \(-0.656552\pi\)
−0.472234 + 0.881473i \(0.656552\pi\)
\(762\) 24113.0 1.14635
\(763\) 0 0
\(764\) 18592.5 0.880435
\(765\) −1018.85 −0.0481525
\(766\) 23991.7 1.13166
\(767\) −9549.36 −0.449553
\(768\) −469.969 −0.0220814
\(769\) −14670.9 −0.687968 −0.343984 0.938976i \(-0.611777\pi\)
−0.343984 + 0.938976i \(0.611777\pi\)
\(770\) 0 0
\(771\) −9966.64 −0.465551
\(772\) −38535.8 −1.79655
\(773\) 32441.5 1.50949 0.754747 0.656016i \(-0.227760\pi\)
0.754747 + 0.656016i \(0.227760\pi\)
\(774\) 30564.4 1.41940
\(775\) −29973.3 −1.38925
\(776\) 18007.6 0.833036
\(777\) 0 0
\(778\) −51419.7 −2.36952
\(779\) 9082.05 0.417713
\(780\) −950.609 −0.0436375
\(781\) 21096.6 0.966576
\(782\) 863.391 0.0394818
\(783\) 16571.7 0.756352
\(784\) 0 0
\(785\) −3368.97 −0.153176
\(786\) −17934.2 −0.813858
\(787\) 26331.4 1.19265 0.596324 0.802744i \(-0.296627\pi\)
0.596324 + 0.802744i \(0.296627\pi\)
\(788\) −8767.98 −0.396379
\(789\) −1402.43 −0.0632801
\(790\) −3444.07 −0.155107
\(791\) 0 0
\(792\) −4871.99 −0.218584
\(793\) −1414.02 −0.0633206
\(794\) 25744.3 1.15067
\(795\) 452.622 0.0201923
\(796\) −1564.63 −0.0696696
\(797\) −18317.9 −0.814121 −0.407061 0.913401i \(-0.633446\pi\)
−0.407061 + 0.913401i \(0.633446\pi\)
\(798\) 0 0
\(799\) 9133.92 0.404424
\(800\) −29013.1 −1.28221
\(801\) −25925.2 −1.14360
\(802\) −16036.3 −0.706064
\(803\) −17217.0 −0.756631
\(804\) −644.303 −0.0282622
\(805\) 0 0
\(806\) 18103.2 0.791140
\(807\) −18453.3 −0.804939
\(808\) 20209.8 0.879923
\(809\) 19252.0 0.836666 0.418333 0.908294i \(-0.362615\pi\)
0.418333 + 0.908294i \(0.362615\pi\)
\(810\) −3115.13 −0.135129
\(811\) 15983.0 0.692033 0.346017 0.938228i \(-0.387534\pi\)
0.346017 + 0.938228i \(0.387534\pi\)
\(812\) 0 0
\(813\) 10422.9 0.449628
\(814\) −8753.86 −0.376932
\(815\) −5462.86 −0.234792
\(816\) −1592.90 −0.0683365
\(817\) 24295.1 1.04036
\(818\) −2397.80 −0.102490
\(819\) 0 0
\(820\) 2983.15 0.127044
\(821\) −37248.2 −1.58340 −0.791700 0.610910i \(-0.790804\pi\)
−0.791700 + 0.610910i \(0.790804\pi\)
\(822\) 7533.98 0.319681
\(823\) −1197.09 −0.0507023 −0.0253512 0.999679i \(-0.508070\pi\)
−0.0253512 + 0.999679i \(0.508070\pi\)
\(824\) −521.006 −0.0220268
\(825\) 5026.77 0.212133
\(826\) 0 0
\(827\) −36095.2 −1.51772 −0.758859 0.651255i \(-0.774243\pi\)
−0.758859 + 0.651255i \(0.774243\pi\)
\(828\) 2221.21 0.0932276
\(829\) 39715.1 1.66389 0.831943 0.554861i \(-0.187229\pi\)
0.831943 + 0.554861i \(0.187229\pi\)
\(830\) 12777.2 0.534341
\(831\) −3858.50 −0.161071
\(832\) 13183.3 0.549337
\(833\) 0 0
\(834\) −28896.4 −1.19976
\(835\) 2467.85 0.102280
\(836\) −14597.6 −0.603910
\(837\) −28277.4 −1.16776
\(838\) 59040.9 2.43381
\(839\) 44254.4 1.82102 0.910508 0.413492i \(-0.135691\pi\)
0.910508 + 0.413492i \(0.135691\pi\)
\(840\) 0 0
\(841\) −2942.77 −0.120660
\(842\) 64109.2 2.62393
\(843\) −8110.40 −0.331361
\(844\) 30080.7 1.22680
\(845\) −4314.46 −0.175647
\(846\) 40762.9 1.65657
\(847\) 0 0
\(848\) −2812.62 −0.113898
\(849\) 17154.1 0.693434
\(850\) 10951.6 0.441928
\(851\) 1058.79 0.0426497
\(852\) −29750.1 −1.19627
\(853\) −12595.8 −0.505593 −0.252797 0.967519i \(-0.581350\pi\)
−0.252797 + 0.967519i \(0.581350\pi\)
\(854\) 0 0
\(855\) −3614.27 −0.144568
\(856\) −12868.6 −0.513830
\(857\) −31318.6 −1.24833 −0.624167 0.781291i \(-0.714561\pi\)
−0.624167 + 0.781291i \(0.714561\pi\)
\(858\) −3036.06 −0.120804
\(859\) 12280.8 0.487795 0.243898 0.969801i \(-0.421574\pi\)
0.243898 + 0.969801i \(0.421574\pi\)
\(860\) 7980.13 0.316419
\(861\) 0 0
\(862\) 38754.3 1.53130
\(863\) 43670.3 1.72254 0.861272 0.508144i \(-0.169668\pi\)
0.861272 + 0.508144i \(0.169668\pi\)
\(864\) −27371.6 −1.07778
\(865\) 2007.82 0.0789226
\(866\) 56666.5 2.22356
\(867\) −10417.6 −0.408075
\(868\) 0 0
\(869\) −6340.96 −0.247529
\(870\) 3333.47 0.129902
\(871\) 423.369 0.0164699
\(872\) 12953.9 0.503067
\(873\) 30942.2 1.19958
\(874\) 3062.80 0.118536
\(875\) 0 0
\(876\) 24279.1 0.936432
\(877\) −26037.3 −1.00253 −0.501265 0.865294i \(-0.667132\pi\)
−0.501265 + 0.865294i \(0.667132\pi\)
\(878\) 1772.24 0.0681210
\(879\) −3355.29 −0.128750
\(880\) 1316.18 0.0504185
\(881\) −45357.9 −1.73456 −0.867280 0.497820i \(-0.834134\pi\)
−0.867280 + 0.497820i \(0.834134\pi\)
\(882\) 0 0
\(883\) 3211.59 0.122399 0.0611996 0.998126i \(-0.480507\pi\)
0.0611996 + 0.998126i \(0.480507\pi\)
\(884\) −3813.07 −0.145076
\(885\) 3000.46 0.113965
\(886\) −28700.2 −1.08827
\(887\) −4760.50 −0.180205 −0.0901025 0.995932i \(-0.528719\pi\)
−0.0901025 + 0.995932i \(0.528719\pi\)
\(888\) 3274.93 0.123761
\(889\) 0 0
\(890\) −11742.0 −0.442238
\(891\) −5735.33 −0.215646
\(892\) 46083.8 1.72982
\(893\) 32401.7 1.21420
\(894\) 33860.2 1.26673
\(895\) −442.739 −0.0165353
\(896\) 0 0
\(897\) 367.216 0.0136689
\(898\) −13243.1 −0.492125
\(899\) −36595.2 −1.35764
\(900\) 28174.9 1.04351
\(901\) 1815.55 0.0671309
\(902\) 9527.62 0.351702
\(903\) 0 0
\(904\) −9599.29 −0.353172
\(905\) −1794.29 −0.0659052
\(906\) −18963.0 −0.695368
\(907\) 10177.3 0.372582 0.186291 0.982495i \(-0.440353\pi\)
0.186291 + 0.982495i \(0.440353\pi\)
\(908\) −9591.84 −0.350569
\(909\) 34726.2 1.26710
\(910\) 0 0
\(911\) −15584.8 −0.566792 −0.283396 0.959003i \(-0.591461\pi\)
−0.283396 + 0.959003i \(0.591461\pi\)
\(912\) −5650.65 −0.205166
\(913\) 23524.4 0.852731
\(914\) −38418.7 −1.39035
\(915\) 444.291 0.0160523
\(916\) 17664.3 0.637167
\(917\) 0 0
\(918\) 10332.0 0.371468
\(919\) 10782.4 0.387026 0.193513 0.981098i \(-0.438012\pi\)
0.193513 + 0.981098i \(0.438012\pi\)
\(920\) 266.892 0.00956433
\(921\) 1307.10 0.0467648
\(922\) 43153.2 1.54140
\(923\) 19548.6 0.697130
\(924\) 0 0
\(925\) 13430.2 0.477386
\(926\) 65911.3 2.33907
\(927\) −895.237 −0.0317189
\(928\) −35422.9 −1.25303
\(929\) −53482.3 −1.88880 −0.944400 0.328798i \(-0.893357\pi\)
−0.944400 + 0.328798i \(0.893357\pi\)
\(930\) −5688.12 −0.200560
\(931\) 0 0
\(932\) −42749.6 −1.50248
\(933\) −8483.01 −0.297665
\(934\) 59336.7 2.07876
\(935\) −849.597 −0.0297164
\(936\) −4514.51 −0.157651
\(937\) 30490.5 1.06305 0.531526 0.847042i \(-0.321619\pi\)
0.531526 + 0.847042i \(0.321619\pi\)
\(938\) 0 0
\(939\) 19743.0 0.686142
\(940\) 10642.9 0.369290
\(941\) −36848.7 −1.27655 −0.638276 0.769808i \(-0.720352\pi\)
−0.638276 + 0.769808i \(0.720352\pi\)
\(942\) 15173.3 0.524813
\(943\) −1152.38 −0.0397949
\(944\) −18645.0 −0.642842
\(945\) 0 0
\(946\) 25487.0 0.875956
\(947\) −14328.9 −0.491687 −0.245844 0.969309i \(-0.579065\pi\)
−0.245844 + 0.969309i \(0.579065\pi\)
\(948\) 8941.91 0.306350
\(949\) −15953.7 −0.545710
\(950\) 38849.9 1.32680
\(951\) −1542.24 −0.0525874
\(952\) 0 0
\(953\) 1760.96 0.0598563 0.0299281 0.999552i \(-0.490472\pi\)
0.0299281 + 0.999552i \(0.490472\pi\)
\(954\) 8102.47 0.274976
\(955\) 3838.66 0.130069
\(956\) −37660.4 −1.27408
\(957\) 6137.32 0.207306
\(958\) −40279.1 −1.35841
\(959\) 0 0
\(960\) −4142.25 −0.139261
\(961\) 32653.9 1.09610
\(962\) −8111.55 −0.271857
\(963\) −22111.9 −0.739922
\(964\) 48629.2 1.62473
\(965\) −7956.21 −0.265409
\(966\) 0 0
\(967\) −909.752 −0.0302540 −0.0151270 0.999886i \(-0.504815\pi\)
−0.0151270 + 0.999886i \(0.504815\pi\)
\(968\) 12647.6 0.419947
\(969\) 3647.51 0.120924
\(970\) 14014.3 0.463889
\(971\) −4900.51 −0.161962 −0.0809809 0.996716i \(-0.525805\pi\)
−0.0809809 + 0.996716i \(0.525805\pi\)
\(972\) 41356.2 1.36471
\(973\) 0 0
\(974\) 17971.6 0.591218
\(975\) 4657.93 0.152998
\(976\) −2760.85 −0.0905457
\(977\) 8173.98 0.267665 0.133833 0.991004i \(-0.457272\pi\)
0.133833 + 0.991004i \(0.457272\pi\)
\(978\) 24603.9 0.804444
\(979\) −21618.4 −0.705749
\(980\) 0 0
\(981\) 22258.5 0.724423
\(982\) −19608.3 −0.637195
\(983\) 30729.2 0.997059 0.498530 0.866873i \(-0.333874\pi\)
0.498530 + 0.866873i \(0.333874\pi\)
\(984\) −3564.40 −0.115477
\(985\) −1810.26 −0.0585581
\(986\) 13371.2 0.431871
\(987\) 0 0
\(988\) −13526.5 −0.435562
\(989\) −3082.69 −0.0991139
\(990\) −3791.59 −0.121722
\(991\) −42375.8 −1.35834 −0.679169 0.733982i \(-0.737660\pi\)
−0.679169 + 0.733982i \(0.737660\pi\)
\(992\) 60444.5 1.93459
\(993\) 6657.25 0.212751
\(994\) 0 0
\(995\) −323.039 −0.0102925
\(996\) −33173.7 −1.05537
\(997\) 13594.3 0.431831 0.215916 0.976412i \(-0.430726\pi\)
0.215916 + 0.976412i \(0.430726\pi\)
\(998\) 62655.6 1.98730
\(999\) 12670.3 0.401273
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 2401.4.a.c.1.6 39
7.6 odd 2 2401.4.a.d.1.6 39
49.13 odd 14 49.4.e.a.22.12 78
49.34 odd 14 49.4.e.a.29.12 yes 78
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.4.e.a.22.12 78 49.13 odd 14
49.4.e.a.29.12 yes 78 49.34 odd 14
2401.4.a.c.1.6 39 1.1 even 1 trivial
2401.4.a.d.1.6 39 7.6 odd 2