Properties

Label 8043.2.a.u.1.7
Level $8043$
Weight $2$
Character 8043.1
Self dual yes
Analytic conductor $64.224$
Analytic rank $0$
Dimension $53$
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] = [8043,2,Mod(1,8043)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8043, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8043.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8043 = 3 \cdot 7 \cdot 383 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8043.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: \(64.2236783457\)
Analytic rank: \(0\)
Dimension: \(53\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Character \(\chi\) \(=\) 8043.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.15559 q^{2} +1.00000 q^{3} +2.64657 q^{4} +3.65015 q^{5} -2.15559 q^{6} +1.00000 q^{7} -1.39373 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.15559 q^{2} +1.00000 q^{3} +2.64657 q^{4} +3.65015 q^{5} -2.15559 q^{6} +1.00000 q^{7} -1.39373 q^{8} +1.00000 q^{9} -7.86823 q^{10} -2.40764 q^{11} +2.64657 q^{12} +5.41813 q^{13} -2.15559 q^{14} +3.65015 q^{15} -2.28882 q^{16} +2.19164 q^{17} -2.15559 q^{18} +3.14459 q^{19} +9.66037 q^{20} +1.00000 q^{21} +5.18988 q^{22} -3.57444 q^{23} -1.39373 q^{24} +8.32361 q^{25} -11.6793 q^{26} +1.00000 q^{27} +2.64657 q^{28} +6.48940 q^{29} -7.86823 q^{30} -9.56205 q^{31} +7.72122 q^{32} -2.40764 q^{33} -4.72427 q^{34} +3.65015 q^{35} +2.64657 q^{36} -6.05844 q^{37} -6.77845 q^{38} +5.41813 q^{39} -5.08734 q^{40} +10.1134 q^{41} -2.15559 q^{42} +5.55063 q^{43} -6.37197 q^{44} +3.65015 q^{45} +7.70503 q^{46} +4.65440 q^{47} -2.28882 q^{48} +1.00000 q^{49} -17.9423 q^{50} +2.19164 q^{51} +14.3394 q^{52} -4.77275 q^{53} -2.15559 q^{54} -8.78824 q^{55} -1.39373 q^{56} +3.14459 q^{57} -13.9885 q^{58} +2.92751 q^{59} +9.66037 q^{60} -5.72176 q^{61} +20.6119 q^{62} +1.00000 q^{63} -12.0661 q^{64} +19.7770 q^{65} +5.18988 q^{66} +5.54279 q^{67} +5.80032 q^{68} -3.57444 q^{69} -7.86823 q^{70} +0.0521291 q^{71} -1.39373 q^{72} +13.9762 q^{73} +13.0595 q^{74} +8.32361 q^{75} +8.32237 q^{76} -2.40764 q^{77} -11.6793 q^{78} -7.64587 q^{79} -8.35453 q^{80} +1.00000 q^{81} -21.8004 q^{82} -8.88061 q^{83} +2.64657 q^{84} +7.99981 q^{85} -11.9649 q^{86} +6.48940 q^{87} +3.35560 q^{88} -5.52135 q^{89} -7.86823 q^{90} +5.41813 q^{91} -9.46000 q^{92} -9.56205 q^{93} -10.0330 q^{94} +11.4782 q^{95} +7.72122 q^{96} +0.287463 q^{97} -2.15559 q^{98} -2.40764 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 53 q + 11 q^{2} + 53 q^{3} + 63 q^{4} + 24 q^{5} + 11 q^{6} + 53 q^{7} + 30 q^{8} + 53 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 53 q + 11 q^{2} + 53 q^{3} + 63 q^{4} + 24 q^{5} + 11 q^{6} + 53 q^{7} + 30 q^{8} + 53 q^{9} + 2 q^{10} + 46 q^{11} + 63 q^{12} + 32 q^{13} + 11 q^{14} + 24 q^{15} + 67 q^{16} + 46 q^{17} + 11 q^{18} + 14 q^{19} + 53 q^{20} + 53 q^{21} + 13 q^{22} + 68 q^{23} + 30 q^{24} + 71 q^{25} + 11 q^{26} + 53 q^{27} + 63 q^{28} + 55 q^{29} + 2 q^{30} - 2 q^{31} + 51 q^{32} + 46 q^{33} - 7 q^{34} + 24 q^{35} + 63 q^{36} + 53 q^{37} + 16 q^{38} + 32 q^{39} - 20 q^{40} + 38 q^{41} + 11 q^{42} + 36 q^{43} + 70 q^{44} + 24 q^{45} + 4 q^{46} + 51 q^{47} + 67 q^{48} + 53 q^{49} + 32 q^{50} + 46 q^{51} + 10 q^{52} + 104 q^{53} + 11 q^{54} + 11 q^{55} + 30 q^{56} + 14 q^{57} + 4 q^{58} + 36 q^{59} + 53 q^{60} + 3 q^{61} + 25 q^{62} + 53 q^{63} + 82 q^{64} + 46 q^{65} + 13 q^{66} + 54 q^{67} + 88 q^{68} + 68 q^{69} + 2 q^{70} + 101 q^{71} + 30 q^{72} + q^{73} + 32 q^{74} + 71 q^{75} - 35 q^{76} + 46 q^{77} + 11 q^{78} + 14 q^{79} + 39 q^{80} + 53 q^{81} - 29 q^{82} + 38 q^{83} + 63 q^{84} + 16 q^{85} + 23 q^{86} + 55 q^{87} - 8 q^{88} + 52 q^{89} + 2 q^{90} + 32 q^{91} + 76 q^{92} - 2 q^{93} - 53 q^{94} + 46 q^{95} + 51 q^{96} - 3 q^{97} + 11 q^{98} + 46 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.15559 −1.52423 −0.762116 0.647440i \(-0.775839\pi\)
−0.762116 + 0.647440i \(0.775839\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.64657 1.32328
\(5\) 3.65015 1.63240 0.816199 0.577771i \(-0.196077\pi\)
0.816199 + 0.577771i \(0.196077\pi\)
\(6\) −2.15559 −0.880016
\(7\) 1.00000 0.377964
\(8\) −1.39373 −0.492759
\(9\) 1.00000 0.333333
\(10\) −7.86823 −2.48815
\(11\) −2.40764 −0.725930 −0.362965 0.931803i \(-0.618236\pi\)
−0.362965 + 0.931803i \(0.618236\pi\)
\(12\) 2.64657 0.763998
\(13\) 5.41813 1.50272 0.751359 0.659893i \(-0.229399\pi\)
0.751359 + 0.659893i \(0.229399\pi\)
\(14\) −2.15559 −0.576106
\(15\) 3.65015 0.942465
\(16\) −2.28882 −0.572204
\(17\) 2.19164 0.531550 0.265775 0.964035i \(-0.414372\pi\)
0.265775 + 0.964035i \(0.414372\pi\)
\(18\) −2.15559 −0.508077
\(19\) 3.14459 0.721419 0.360709 0.932678i \(-0.382535\pi\)
0.360709 + 0.932678i \(0.382535\pi\)
\(20\) 9.66037 2.16012
\(21\) 1.00000 0.218218
\(22\) 5.18988 1.10649
\(23\) −3.57444 −0.745323 −0.372661 0.927967i \(-0.621555\pi\)
−0.372661 + 0.927967i \(0.621555\pi\)
\(24\) −1.39373 −0.284495
\(25\) 8.32361 1.66472
\(26\) −11.6793 −2.29049
\(27\) 1.00000 0.192450
\(28\) 2.64657 0.500154
\(29\) 6.48940 1.20505 0.602526 0.798099i \(-0.294161\pi\)
0.602526 + 0.798099i \(0.294161\pi\)
\(30\) −7.86823 −1.43654
\(31\) −9.56205 −1.71740 −0.858698 0.512482i \(-0.828726\pi\)
−0.858698 + 0.512482i \(0.828726\pi\)
\(32\) 7.72122 1.36493
\(33\) −2.40764 −0.419116
\(34\) −4.72427 −0.810206
\(35\) 3.65015 0.616988
\(36\) 2.64657 0.441094
\(37\) −6.05844 −0.996002 −0.498001 0.867177i \(-0.665932\pi\)
−0.498001 + 0.867177i \(0.665932\pi\)
\(38\) −6.77845 −1.09961
\(39\) 5.41813 0.867595
\(40\) −5.08734 −0.804379
\(41\) 10.1134 1.57945 0.789726 0.613460i \(-0.210223\pi\)
0.789726 + 0.613460i \(0.210223\pi\)
\(42\) −2.15559 −0.332615
\(43\) 5.55063 0.846464 0.423232 0.906021i \(-0.360896\pi\)
0.423232 + 0.906021i \(0.360896\pi\)
\(44\) −6.37197 −0.960611
\(45\) 3.65015 0.544133
\(46\) 7.70503 1.13604
\(47\) 4.65440 0.678913 0.339457 0.940622i \(-0.389757\pi\)
0.339457 + 0.940622i \(0.389757\pi\)
\(48\) −2.28882 −0.330362
\(49\) 1.00000 0.142857
\(50\) −17.9423 −2.53742
\(51\) 2.19164 0.306891
\(52\) 14.3394 1.98852
\(53\) −4.77275 −0.655587 −0.327793 0.944749i \(-0.606305\pi\)
−0.327793 + 0.944749i \(0.606305\pi\)
\(54\) −2.15559 −0.293339
\(55\) −8.78824 −1.18501
\(56\) −1.39373 −0.186245
\(57\) 3.14459 0.416511
\(58\) −13.9885 −1.83678
\(59\) 2.92751 0.381129 0.190564 0.981675i \(-0.438968\pi\)
0.190564 + 0.981675i \(0.438968\pi\)
\(60\) 9.66037 1.24715
\(61\) −5.72176 −0.732597 −0.366298 0.930497i \(-0.619375\pi\)
−0.366298 + 0.930497i \(0.619375\pi\)
\(62\) 20.6119 2.61771
\(63\) 1.00000 0.125988
\(64\) −12.0661 −1.50827
\(65\) 19.7770 2.45303
\(66\) 5.18988 0.638830
\(67\) 5.54279 0.677160 0.338580 0.940938i \(-0.390053\pi\)
0.338580 + 0.940938i \(0.390053\pi\)
\(68\) 5.80032 0.703392
\(69\) −3.57444 −0.430312
\(70\) −7.86823 −0.940433
\(71\) 0.0521291 0.00618658 0.00309329 0.999995i \(-0.499015\pi\)
0.00309329 + 0.999995i \(0.499015\pi\)
\(72\) −1.39373 −0.164253
\(73\) 13.9762 1.63579 0.817895 0.575367i \(-0.195141\pi\)
0.817895 + 0.575367i \(0.195141\pi\)
\(74\) 13.0595 1.51814
\(75\) 8.32361 0.961128
\(76\) 8.32237 0.954641
\(77\) −2.40764 −0.274376
\(78\) −11.6793 −1.32242
\(79\) −7.64587 −0.860228 −0.430114 0.902775i \(-0.641527\pi\)
−0.430114 + 0.902775i \(0.641527\pi\)
\(80\) −8.35453 −0.934065
\(81\) 1.00000 0.111111
\(82\) −21.8004 −2.40745
\(83\) −8.88061 −0.974774 −0.487387 0.873186i \(-0.662050\pi\)
−0.487387 + 0.873186i \(0.662050\pi\)
\(84\) 2.64657 0.288764
\(85\) 7.99981 0.867702
\(86\) −11.9649 −1.29021
\(87\) 6.48940 0.695737
\(88\) 3.35560 0.357708
\(89\) −5.52135 −0.585262 −0.292631 0.956225i \(-0.594531\pi\)
−0.292631 + 0.956225i \(0.594531\pi\)
\(90\) −7.86823 −0.829384
\(91\) 5.41813 0.567974
\(92\) −9.46000 −0.986273
\(93\) −9.56205 −0.991539
\(94\) −10.0330 −1.03482
\(95\) 11.4782 1.17764
\(96\) 7.72122 0.788043
\(97\) 0.287463 0.0291875 0.0145937 0.999894i \(-0.495355\pi\)
0.0145937 + 0.999894i \(0.495355\pi\)
\(98\) −2.15559 −0.217747
\(99\) −2.40764 −0.241977
\(100\) 22.0290 2.20290
\(101\) 4.34648 0.432491 0.216246 0.976339i \(-0.430619\pi\)
0.216246 + 0.976339i \(0.430619\pi\)
\(102\) −4.72427 −0.467773
\(103\) −11.5843 −1.14143 −0.570716 0.821147i \(-0.693334\pi\)
−0.570716 + 0.821147i \(0.693334\pi\)
\(104\) −7.55142 −0.740478
\(105\) 3.65015 0.356218
\(106\) 10.2881 0.999267
\(107\) −13.8875 −1.34255 −0.671276 0.741207i \(-0.734254\pi\)
−0.671276 + 0.741207i \(0.734254\pi\)
\(108\) 2.64657 0.254666
\(109\) 13.7979 1.32160 0.660802 0.750561i \(-0.270216\pi\)
0.660802 + 0.750561i \(0.270216\pi\)
\(110\) 18.9438 1.80622
\(111\) −6.05844 −0.575042
\(112\) −2.28882 −0.216273
\(113\) −3.49061 −0.328369 −0.164184 0.986430i \(-0.552499\pi\)
−0.164184 + 0.986430i \(0.552499\pi\)
\(114\) −6.77845 −0.634860
\(115\) −13.0473 −1.21666
\(116\) 17.1746 1.59463
\(117\) 5.41813 0.500906
\(118\) −6.31050 −0.580929
\(119\) 2.19164 0.200907
\(120\) −5.08734 −0.464408
\(121\) −5.20329 −0.473026
\(122\) 12.3338 1.11665
\(123\) 10.1134 0.911897
\(124\) −25.3066 −2.27260
\(125\) 12.1317 1.08509
\(126\) −2.15559 −0.192035
\(127\) 1.66509 0.147753 0.0738764 0.997267i \(-0.476463\pi\)
0.0738764 + 0.997267i \(0.476463\pi\)
\(128\) 10.5672 0.934019
\(129\) 5.55063 0.488706
\(130\) −42.6311 −3.73899
\(131\) −7.50074 −0.655343 −0.327672 0.944792i \(-0.606264\pi\)
−0.327672 + 0.944792i \(0.606264\pi\)
\(132\) −6.37197 −0.554609
\(133\) 3.14459 0.272671
\(134\) −11.9480 −1.03215
\(135\) 3.65015 0.314155
\(136\) −3.05456 −0.261926
\(137\) 13.7101 1.17133 0.585664 0.810554i \(-0.300834\pi\)
0.585664 + 0.810554i \(0.300834\pi\)
\(138\) 7.70503 0.655896
\(139\) 16.8845 1.43213 0.716063 0.698036i \(-0.245942\pi\)
0.716063 + 0.698036i \(0.245942\pi\)
\(140\) 9.66037 0.816450
\(141\) 4.65440 0.391971
\(142\) −0.112369 −0.00942979
\(143\) −13.0449 −1.09087
\(144\) −2.28882 −0.190735
\(145\) 23.6873 1.96712
\(146\) −30.1270 −2.49332
\(147\) 1.00000 0.0824786
\(148\) −16.0341 −1.31799
\(149\) 16.3997 1.34352 0.671759 0.740770i \(-0.265539\pi\)
0.671759 + 0.740770i \(0.265539\pi\)
\(150\) −17.9423 −1.46498
\(151\) 18.5869 1.51258 0.756289 0.654238i \(-0.227010\pi\)
0.756289 + 0.654238i \(0.227010\pi\)
\(152\) −4.38272 −0.355486
\(153\) 2.19164 0.177183
\(154\) 5.18988 0.418212
\(155\) −34.9030 −2.80347
\(156\) 14.3394 1.14807
\(157\) −4.65606 −0.371594 −0.185797 0.982588i \(-0.559487\pi\)
−0.185797 + 0.982588i \(0.559487\pi\)
\(158\) 16.4814 1.31119
\(159\) −4.77275 −0.378503
\(160\) 28.1836 2.22811
\(161\) −3.57444 −0.281706
\(162\) −2.15559 −0.169359
\(163\) −14.5124 −1.13670 −0.568351 0.822786i \(-0.692418\pi\)
−0.568351 + 0.822786i \(0.692418\pi\)
\(164\) 26.7659 2.09006
\(165\) −8.78824 −0.684164
\(166\) 19.1430 1.48578
\(167\) 18.3195 1.41760 0.708801 0.705408i \(-0.249236\pi\)
0.708801 + 0.705408i \(0.249236\pi\)
\(168\) −1.39373 −0.107529
\(169\) 16.3561 1.25816
\(170\) −17.2443 −1.32258
\(171\) 3.14459 0.240473
\(172\) 14.6901 1.12011
\(173\) 19.0336 1.44710 0.723549 0.690273i \(-0.242510\pi\)
0.723549 + 0.690273i \(0.242510\pi\)
\(174\) −13.9885 −1.06046
\(175\) 8.32361 0.629206
\(176\) 5.51064 0.415380
\(177\) 2.92751 0.220045
\(178\) 11.9018 0.892075
\(179\) 9.03796 0.675529 0.337764 0.941231i \(-0.390329\pi\)
0.337764 + 0.941231i \(0.390329\pi\)
\(180\) 9.66037 0.720042
\(181\) −26.3548 −1.95894 −0.979468 0.201599i \(-0.935386\pi\)
−0.979468 + 0.201599i \(0.935386\pi\)
\(182\) −11.6793 −0.865724
\(183\) −5.72176 −0.422965
\(184\) 4.98182 0.367264
\(185\) −22.1142 −1.62587
\(186\) 20.6119 1.51134
\(187\) −5.27667 −0.385868
\(188\) 12.3182 0.898395
\(189\) 1.00000 0.0727393
\(190\) −24.7424 −1.79500
\(191\) 23.2967 1.68569 0.842844 0.538158i \(-0.180880\pi\)
0.842844 + 0.538158i \(0.180880\pi\)
\(192\) −12.0661 −0.870799
\(193\) −7.71605 −0.555413 −0.277707 0.960666i \(-0.589574\pi\)
−0.277707 + 0.960666i \(0.589574\pi\)
\(194\) −0.619653 −0.0444885
\(195\) 19.7770 1.41626
\(196\) 2.64657 0.189040
\(197\) 3.04912 0.217241 0.108620 0.994083i \(-0.465357\pi\)
0.108620 + 0.994083i \(0.465357\pi\)
\(198\) 5.18988 0.368828
\(199\) −7.35446 −0.521344 −0.260672 0.965427i \(-0.583944\pi\)
−0.260672 + 0.965427i \(0.583944\pi\)
\(200\) −11.6009 −0.820307
\(201\) 5.54279 0.390958
\(202\) −9.36923 −0.659217
\(203\) 6.48940 0.455467
\(204\) 5.80032 0.406103
\(205\) 36.9156 2.57829
\(206\) 24.9709 1.73981
\(207\) −3.57444 −0.248441
\(208\) −12.4011 −0.859862
\(209\) −7.57103 −0.523699
\(210\) −7.86823 −0.542959
\(211\) 3.36809 0.231869 0.115935 0.993257i \(-0.463014\pi\)
0.115935 + 0.993257i \(0.463014\pi\)
\(212\) −12.6314 −0.867527
\(213\) 0.0521291 0.00357183
\(214\) 29.9357 2.04636
\(215\) 20.2607 1.38177
\(216\) −1.39373 −0.0948315
\(217\) −9.56205 −0.649115
\(218\) −29.7427 −2.01443
\(219\) 13.9762 0.944424
\(220\) −23.2587 −1.56810
\(221\) 11.8746 0.798770
\(222\) 13.0595 0.876497
\(223\) −2.37933 −0.159332 −0.0796660 0.996822i \(-0.525385\pi\)
−0.0796660 + 0.996822i \(0.525385\pi\)
\(224\) 7.72122 0.515896
\(225\) 8.32361 0.554907
\(226\) 7.52431 0.500510
\(227\) 14.8099 0.982967 0.491483 0.870887i \(-0.336455\pi\)
0.491483 + 0.870887i \(0.336455\pi\)
\(228\) 8.32237 0.551162
\(229\) 11.8892 0.785660 0.392830 0.919611i \(-0.371496\pi\)
0.392830 + 0.919611i \(0.371496\pi\)
\(230\) 28.1245 1.85448
\(231\) −2.40764 −0.158411
\(232\) −9.04450 −0.593800
\(233\) −21.2250 −1.39050 −0.695249 0.718769i \(-0.744706\pi\)
−0.695249 + 0.718769i \(0.744706\pi\)
\(234\) −11.6793 −0.763497
\(235\) 16.9893 1.10826
\(236\) 7.74784 0.504342
\(237\) −7.64587 −0.496653
\(238\) −4.72427 −0.306229
\(239\) −25.1699 −1.62810 −0.814052 0.580792i \(-0.802743\pi\)
−0.814052 + 0.580792i \(0.802743\pi\)
\(240\) −8.35453 −0.539283
\(241\) 13.8128 0.889761 0.444880 0.895590i \(-0.353246\pi\)
0.444880 + 0.895590i \(0.353246\pi\)
\(242\) 11.2161 0.721001
\(243\) 1.00000 0.0641500
\(244\) −15.1430 −0.969433
\(245\) 3.65015 0.233200
\(246\) −21.8004 −1.38994
\(247\) 17.0378 1.08409
\(248\) 13.3269 0.846262
\(249\) −8.88061 −0.562786
\(250\) −26.1509 −1.65393
\(251\) 20.4634 1.29164 0.645820 0.763490i \(-0.276516\pi\)
0.645820 + 0.763490i \(0.276516\pi\)
\(252\) 2.64657 0.166718
\(253\) 8.60596 0.541052
\(254\) −3.58925 −0.225210
\(255\) 7.99981 0.500968
\(256\) 1.35370 0.0846065
\(257\) −0.0952126 −0.00593920 −0.00296960 0.999996i \(-0.500945\pi\)
−0.00296960 + 0.999996i \(0.500945\pi\)
\(258\) −11.9649 −0.744901
\(259\) −6.05844 −0.376453
\(260\) 52.3411 3.24606
\(261\) 6.48940 0.401684
\(262\) 16.1685 0.998895
\(263\) −8.14349 −0.502149 −0.251075 0.967968i \(-0.580784\pi\)
−0.251075 + 0.967968i \(0.580784\pi\)
\(264\) 3.35560 0.206523
\(265\) −17.4212 −1.07018
\(266\) −6.77845 −0.415613
\(267\) −5.52135 −0.337901
\(268\) 14.6694 0.896074
\(269\) 16.7336 1.02027 0.510134 0.860095i \(-0.329596\pi\)
0.510134 + 0.860095i \(0.329596\pi\)
\(270\) −7.86823 −0.478845
\(271\) −8.33824 −0.506512 −0.253256 0.967399i \(-0.581502\pi\)
−0.253256 + 0.967399i \(0.581502\pi\)
\(272\) −5.01626 −0.304155
\(273\) 5.41813 0.327920
\(274\) −29.5532 −1.78538
\(275\) −20.0402 −1.20847
\(276\) −9.46000 −0.569425
\(277\) −28.1966 −1.69417 −0.847083 0.531460i \(-0.821644\pi\)
−0.847083 + 0.531460i \(0.821644\pi\)
\(278\) −36.3961 −2.18289
\(279\) −9.56205 −0.572465
\(280\) −5.08734 −0.304027
\(281\) 9.23440 0.550878 0.275439 0.961319i \(-0.411177\pi\)
0.275439 + 0.961319i \(0.411177\pi\)
\(282\) −10.0330 −0.597454
\(283\) −17.9477 −1.06688 −0.533441 0.845837i \(-0.679102\pi\)
−0.533441 + 0.845837i \(0.679102\pi\)
\(284\) 0.137963 0.00818660
\(285\) 11.4782 0.679912
\(286\) 28.1194 1.66274
\(287\) 10.1134 0.596977
\(288\) 7.72122 0.454977
\(289\) −12.1967 −0.717454
\(290\) −51.0601 −2.99835
\(291\) 0.287463 0.0168514
\(292\) 36.9889 2.16461
\(293\) 5.88333 0.343708 0.171854 0.985122i \(-0.445024\pi\)
0.171854 + 0.985122i \(0.445024\pi\)
\(294\) −2.15559 −0.125717
\(295\) 10.6858 0.622154
\(296\) 8.44385 0.490789
\(297\) −2.40764 −0.139705
\(298\) −35.3511 −2.04783
\(299\) −19.3668 −1.12001
\(300\) 22.0290 1.27184
\(301\) 5.55063 0.319933
\(302\) −40.0657 −2.30552
\(303\) 4.34648 0.249699
\(304\) −7.19740 −0.412799
\(305\) −20.8853 −1.19589
\(306\) −4.72427 −0.270069
\(307\) −15.0865 −0.861032 −0.430516 0.902583i \(-0.641668\pi\)
−0.430516 + 0.902583i \(0.641668\pi\)
\(308\) −6.37197 −0.363077
\(309\) −11.5843 −0.659006
\(310\) 75.2364 4.27314
\(311\) 16.1821 0.917605 0.458803 0.888538i \(-0.348278\pi\)
0.458803 + 0.888538i \(0.348278\pi\)
\(312\) −7.55142 −0.427515
\(313\) 2.62515 0.148382 0.0741912 0.997244i \(-0.476363\pi\)
0.0741912 + 0.997244i \(0.476363\pi\)
\(314\) 10.0366 0.566395
\(315\) 3.65015 0.205663
\(316\) −20.2353 −1.13833
\(317\) 15.6924 0.881374 0.440687 0.897661i \(-0.354735\pi\)
0.440687 + 0.897661i \(0.354735\pi\)
\(318\) 10.2881 0.576927
\(319\) −15.6241 −0.874783
\(320\) −44.0433 −2.46209
\(321\) −13.8875 −0.775123
\(322\) 7.70503 0.429385
\(323\) 6.89181 0.383470
\(324\) 2.64657 0.147031
\(325\) 45.0984 2.50161
\(326\) 31.2828 1.73260
\(327\) 13.7979 0.763028
\(328\) −14.0954 −0.778289
\(329\) 4.65440 0.256605
\(330\) 18.9438 1.04282
\(331\) −14.3093 −0.786510 −0.393255 0.919430i \(-0.628651\pi\)
−0.393255 + 0.919430i \(0.628651\pi\)
\(332\) −23.5031 −1.28990
\(333\) −6.05844 −0.332001
\(334\) −39.4892 −2.16075
\(335\) 20.2320 1.10539
\(336\) −2.28882 −0.124865
\(337\) −10.5620 −0.575350 −0.287675 0.957728i \(-0.592882\pi\)
−0.287675 + 0.957728i \(0.592882\pi\)
\(338\) −35.2571 −1.91773
\(339\) −3.49061 −0.189584
\(340\) 21.1720 1.14822
\(341\) 23.0220 1.24671
\(342\) −6.77845 −0.366536
\(343\) 1.00000 0.0539949
\(344\) −7.73610 −0.417103
\(345\) −13.0473 −0.702441
\(346\) −41.0286 −2.20571
\(347\) 16.5244 0.887075 0.443537 0.896256i \(-0.353723\pi\)
0.443537 + 0.896256i \(0.353723\pi\)
\(348\) 17.1746 0.920658
\(349\) −10.8420 −0.580360 −0.290180 0.956972i \(-0.593715\pi\)
−0.290180 + 0.956972i \(0.593715\pi\)
\(350\) −17.9423 −0.959056
\(351\) 5.41813 0.289198
\(352\) −18.5899 −0.990844
\(353\) −22.4569 −1.19526 −0.597631 0.801771i \(-0.703891\pi\)
−0.597631 + 0.801771i \(0.703891\pi\)
\(354\) −6.31050 −0.335399
\(355\) 0.190279 0.0100990
\(356\) −14.6126 −0.774468
\(357\) 2.19164 0.115994
\(358\) −19.4821 −1.02966
\(359\) −18.4578 −0.974168 −0.487084 0.873355i \(-0.661939\pi\)
−0.487084 + 0.873355i \(0.661939\pi\)
\(360\) −5.08734 −0.268126
\(361\) −9.11155 −0.479555
\(362\) 56.8101 2.98587
\(363\) −5.20329 −0.273102
\(364\) 14.3394 0.751591
\(365\) 51.0153 2.67026
\(366\) 12.3338 0.644697
\(367\) 1.80127 0.0940257 0.0470128 0.998894i \(-0.485030\pi\)
0.0470128 + 0.998894i \(0.485030\pi\)
\(368\) 8.18125 0.426477
\(369\) 10.1134 0.526484
\(370\) 47.6692 2.47820
\(371\) −4.77275 −0.247789
\(372\) −25.3066 −1.31209
\(373\) 12.0176 0.622249 0.311125 0.950369i \(-0.399294\pi\)
0.311125 + 0.950369i \(0.399294\pi\)
\(374\) 11.3743 0.588153
\(375\) 12.1317 0.626478
\(376\) −6.48698 −0.334541
\(377\) 35.1604 1.81085
\(378\) −2.15559 −0.110872
\(379\) 2.38003 0.122254 0.0611270 0.998130i \(-0.480531\pi\)
0.0611270 + 0.998130i \(0.480531\pi\)
\(380\) 30.3779 1.55835
\(381\) 1.66509 0.0853052
\(382\) −50.2181 −2.56938
\(383\) 1.00000 0.0510976
\(384\) 10.5672 0.539256
\(385\) −8.78824 −0.447890
\(386\) 16.6326 0.846579
\(387\) 5.55063 0.282155
\(388\) 0.760791 0.0386233
\(389\) 19.4132 0.984288 0.492144 0.870514i \(-0.336213\pi\)
0.492144 + 0.870514i \(0.336213\pi\)
\(390\) −42.6311 −2.15871
\(391\) −7.83388 −0.396177
\(392\) −1.39373 −0.0703941
\(393\) −7.50074 −0.378362
\(394\) −6.57265 −0.331125
\(395\) −27.9086 −1.40423
\(396\) −6.37197 −0.320204
\(397\) −21.1059 −1.05928 −0.529638 0.848224i \(-0.677672\pi\)
−0.529638 + 0.848224i \(0.677672\pi\)
\(398\) 15.8532 0.794649
\(399\) 3.14459 0.157426
\(400\) −19.0512 −0.952561
\(401\) −25.6464 −1.28072 −0.640360 0.768075i \(-0.721215\pi\)
−0.640360 + 0.768075i \(0.721215\pi\)
\(402\) −11.9480 −0.595911
\(403\) −51.8084 −2.58076
\(404\) 11.5033 0.572308
\(405\) 3.65015 0.181378
\(406\) −13.9885 −0.694237
\(407\) 14.5865 0.723027
\(408\) −3.05456 −0.151223
\(409\) −11.3471 −0.561080 −0.280540 0.959842i \(-0.590513\pi\)
−0.280540 + 0.959842i \(0.590513\pi\)
\(410\) −79.5748 −3.92992
\(411\) 13.7101 0.676267
\(412\) −30.6586 −1.51044
\(413\) 2.92751 0.144053
\(414\) 7.70503 0.378682
\(415\) −32.4156 −1.59122
\(416\) 41.8345 2.05111
\(417\) 16.8845 0.826838
\(418\) 16.3200 0.798239
\(419\) −14.5182 −0.709259 −0.354630 0.935007i \(-0.615393\pi\)
−0.354630 + 0.935007i \(0.615393\pi\)
\(420\) 9.66037 0.471378
\(421\) −21.7400 −1.05954 −0.529771 0.848141i \(-0.677722\pi\)
−0.529771 + 0.848141i \(0.677722\pi\)
\(422\) −7.26023 −0.353422
\(423\) 4.65440 0.226304
\(424\) 6.65193 0.323046
\(425\) 18.2423 0.884884
\(426\) −0.112369 −0.00544429
\(427\) −5.72176 −0.276896
\(428\) −36.7541 −1.77658
\(429\) −13.0449 −0.629813
\(430\) −43.6737 −2.10613
\(431\) −21.8362 −1.05181 −0.525907 0.850542i \(-0.676274\pi\)
−0.525907 + 0.850542i \(0.676274\pi\)
\(432\) −2.28882 −0.110121
\(433\) 14.5073 0.697176 0.348588 0.937276i \(-0.386661\pi\)
0.348588 + 0.937276i \(0.386661\pi\)
\(434\) 20.6119 0.989401
\(435\) 23.6873 1.13572
\(436\) 36.5172 1.74886
\(437\) −11.2402 −0.537690
\(438\) −30.1270 −1.43952
\(439\) −24.3720 −1.16321 −0.581606 0.813471i \(-0.697576\pi\)
−0.581606 + 0.813471i \(0.697576\pi\)
\(440\) 12.2485 0.583922
\(441\) 1.00000 0.0476190
\(442\) −25.5967 −1.21751
\(443\) −38.6261 −1.83518 −0.917591 0.397525i \(-0.869869\pi\)
−0.917591 + 0.397525i \(0.869869\pi\)
\(444\) −16.0341 −0.760943
\(445\) −20.1538 −0.955381
\(446\) 5.12887 0.242859
\(447\) 16.3997 0.775680
\(448\) −12.0661 −0.570072
\(449\) −6.20044 −0.292617 −0.146308 0.989239i \(-0.546739\pi\)
−0.146308 + 0.989239i \(0.546739\pi\)
\(450\) −17.9423 −0.845808
\(451\) −24.3495 −1.14657
\(452\) −9.23812 −0.434525
\(453\) 18.5869 0.873287
\(454\) −31.9240 −1.49827
\(455\) 19.7770 0.927160
\(456\) −4.38272 −0.205240
\(457\) 5.18910 0.242736 0.121368 0.992608i \(-0.461272\pi\)
0.121368 + 0.992608i \(0.461272\pi\)
\(458\) −25.6282 −1.19753
\(459\) 2.19164 0.102297
\(460\) −34.5304 −1.60999
\(461\) 34.6598 1.61427 0.807133 0.590370i \(-0.201018\pi\)
0.807133 + 0.590370i \(0.201018\pi\)
\(462\) 5.18988 0.241455
\(463\) 6.38523 0.296747 0.148373 0.988931i \(-0.452596\pi\)
0.148373 + 0.988931i \(0.452596\pi\)
\(464\) −14.8531 −0.689536
\(465\) −34.9030 −1.61859
\(466\) 45.7524 2.11944
\(467\) 34.4507 1.59419 0.797094 0.603855i \(-0.206369\pi\)
0.797094 + 0.603855i \(0.206369\pi\)
\(468\) 14.3394 0.662841
\(469\) 5.54279 0.255942
\(470\) −36.6219 −1.68924
\(471\) −4.65606 −0.214540
\(472\) −4.08016 −0.187805
\(473\) −13.3639 −0.614473
\(474\) 16.4814 0.757014
\(475\) 26.1744 1.20096
\(476\) 5.80032 0.265857
\(477\) −4.77275 −0.218529
\(478\) 54.2559 2.48161
\(479\) −10.2961 −0.470440 −0.235220 0.971942i \(-0.575581\pi\)
−0.235220 + 0.971942i \(0.575581\pi\)
\(480\) 28.1836 1.28640
\(481\) −32.8254 −1.49671
\(482\) −29.7747 −1.35620
\(483\) −3.57444 −0.162643
\(484\) −13.7708 −0.625947
\(485\) 1.04929 0.0476456
\(486\) −2.15559 −0.0977795
\(487\) −12.3741 −0.560724 −0.280362 0.959894i \(-0.590455\pi\)
−0.280362 + 0.959894i \(0.590455\pi\)
\(488\) 7.97461 0.360994
\(489\) −14.5124 −0.656275
\(490\) −7.86823 −0.355450
\(491\) −36.2982 −1.63811 −0.819057 0.573712i \(-0.805503\pi\)
−0.819057 + 0.573712i \(0.805503\pi\)
\(492\) 26.7659 1.20670
\(493\) 14.2224 0.640546
\(494\) −36.7265 −1.65240
\(495\) −8.78824 −0.395002
\(496\) 21.8858 0.982701
\(497\) 0.0521291 0.00233831
\(498\) 19.1430 0.857816
\(499\) −0.699596 −0.0313182 −0.0156591 0.999877i \(-0.504985\pi\)
−0.0156591 + 0.999877i \(0.504985\pi\)
\(500\) 32.1073 1.43588
\(501\) 18.3195 0.818453
\(502\) −44.1107 −1.96876
\(503\) 6.22672 0.277636 0.138818 0.990318i \(-0.455670\pi\)
0.138818 + 0.990318i \(0.455670\pi\)
\(504\) −1.39373 −0.0620818
\(505\) 15.8653 0.705998
\(506\) −18.5509 −0.824689
\(507\) 16.3561 0.726401
\(508\) 4.40677 0.195519
\(509\) 20.8313 0.923332 0.461666 0.887054i \(-0.347252\pi\)
0.461666 + 0.887054i \(0.347252\pi\)
\(510\) −17.2443 −0.763591
\(511\) 13.9762 0.618271
\(512\) −24.0525 −1.06298
\(513\) 3.14459 0.138837
\(514\) 0.205239 0.00905272
\(515\) −42.2844 −1.86327
\(516\) 14.6901 0.646697
\(517\) −11.2061 −0.492843
\(518\) 13.0595 0.573802
\(519\) 19.0336 0.835483
\(520\) −27.5638 −1.20875
\(521\) −4.12511 −0.180724 −0.0903622 0.995909i \(-0.528802\pi\)
−0.0903622 + 0.995909i \(0.528802\pi\)
\(522\) −13.9885 −0.612260
\(523\) 15.9217 0.696206 0.348103 0.937456i \(-0.386826\pi\)
0.348103 + 0.937456i \(0.386826\pi\)
\(524\) −19.8512 −0.867205
\(525\) 8.32361 0.363272
\(526\) 17.5540 0.765392
\(527\) −20.9566 −0.912882
\(528\) 5.51064 0.239820
\(529\) −10.2234 −0.444494
\(530\) 37.5531 1.63120
\(531\) 2.92751 0.127043
\(532\) 8.32237 0.360821
\(533\) 54.7959 2.37347
\(534\) 11.9018 0.515040
\(535\) −50.6914 −2.19158
\(536\) −7.72517 −0.333676
\(537\) 9.03796 0.390017
\(538\) −36.0709 −1.55513
\(539\) −2.40764 −0.103704
\(540\) 9.66037 0.415716
\(541\) 12.4358 0.534657 0.267328 0.963605i \(-0.413859\pi\)
0.267328 + 0.963605i \(0.413859\pi\)
\(542\) 17.9738 0.772042
\(543\) −26.3548 −1.13099
\(544\) 16.9221 0.725530
\(545\) 50.3646 2.15738
\(546\) −11.6793 −0.499826
\(547\) −29.2955 −1.25258 −0.626292 0.779589i \(-0.715428\pi\)
−0.626292 + 0.779589i \(0.715428\pi\)
\(548\) 36.2846 1.55000
\(549\) −5.72176 −0.244199
\(550\) 43.1985 1.84199
\(551\) 20.4065 0.869347
\(552\) 4.98182 0.212040
\(553\) −7.64587 −0.325136
\(554\) 60.7802 2.58230
\(555\) −22.1142 −0.938697
\(556\) 44.6860 1.89511
\(557\) 10.9887 0.465606 0.232803 0.972524i \(-0.425210\pi\)
0.232803 + 0.972524i \(0.425210\pi\)
\(558\) 20.6119 0.872570
\(559\) 30.0740 1.27200
\(560\) −8.35453 −0.353043
\(561\) −5.27667 −0.222781
\(562\) −19.9056 −0.839666
\(563\) −28.9692 −1.22091 −0.610454 0.792052i \(-0.709013\pi\)
−0.610454 + 0.792052i \(0.709013\pi\)
\(564\) 12.3182 0.518688
\(565\) −12.7412 −0.536028
\(566\) 38.6880 1.62618
\(567\) 1.00000 0.0419961
\(568\) −0.0726540 −0.00304849
\(569\) 22.0370 0.923838 0.461919 0.886922i \(-0.347161\pi\)
0.461919 + 0.886922i \(0.347161\pi\)
\(570\) −24.7424 −1.03634
\(571\) 15.8466 0.663160 0.331580 0.943427i \(-0.392418\pi\)
0.331580 + 0.943427i \(0.392418\pi\)
\(572\) −34.5242 −1.44353
\(573\) 23.2967 0.973233
\(574\) −21.8004 −0.909931
\(575\) −29.7523 −1.24076
\(576\) −12.0661 −0.502756
\(577\) 17.8690 0.743898 0.371949 0.928253i \(-0.378690\pi\)
0.371949 + 0.928253i \(0.378690\pi\)
\(578\) 26.2911 1.09357
\(579\) −7.71605 −0.320668
\(580\) 62.6901 2.60306
\(581\) −8.88061 −0.368430
\(582\) −0.619653 −0.0256855
\(583\) 11.4910 0.475910
\(584\) −19.4791 −0.806050
\(585\) 19.7770 0.817678
\(586\) −12.6821 −0.523891
\(587\) 22.4986 0.928617 0.464309 0.885674i \(-0.346303\pi\)
0.464309 + 0.885674i \(0.346303\pi\)
\(588\) 2.64657 0.109143
\(589\) −30.0687 −1.23896
\(590\) −23.0343 −0.948307
\(591\) 3.04912 0.125424
\(592\) 13.8667 0.569917
\(593\) 35.7329 1.46737 0.733687 0.679488i \(-0.237798\pi\)
0.733687 + 0.679488i \(0.237798\pi\)
\(594\) 5.18988 0.212943
\(595\) 7.99981 0.327960
\(596\) 43.4030 1.77785
\(597\) −7.35446 −0.300998
\(598\) 41.7468 1.70716
\(599\) −18.8601 −0.770602 −0.385301 0.922791i \(-0.625902\pi\)
−0.385301 + 0.922791i \(0.625902\pi\)
\(600\) −11.6009 −0.473604
\(601\) 4.74151 0.193410 0.0967051 0.995313i \(-0.469170\pi\)
0.0967051 + 0.995313i \(0.469170\pi\)
\(602\) −11.9649 −0.487652
\(603\) 5.54279 0.225720
\(604\) 49.1914 2.00157
\(605\) −18.9928 −0.772167
\(606\) −9.36923 −0.380599
\(607\) 7.51614 0.305071 0.152535 0.988298i \(-0.451256\pi\)
0.152535 + 0.988298i \(0.451256\pi\)
\(608\) 24.2801 0.984687
\(609\) 6.48940 0.262964
\(610\) 45.0202 1.82281
\(611\) 25.2181 1.02022
\(612\) 5.80032 0.234464
\(613\) 26.3762 1.06533 0.532663 0.846327i \(-0.321191\pi\)
0.532663 + 0.846327i \(0.321191\pi\)
\(614\) 32.5203 1.31241
\(615\) 36.9156 1.48858
\(616\) 3.35560 0.135201
\(617\) −22.1997 −0.893725 −0.446862 0.894603i \(-0.647459\pi\)
−0.446862 + 0.894603i \(0.647459\pi\)
\(618\) 24.9709 1.00448
\(619\) 1.31859 0.0529984 0.0264992 0.999649i \(-0.491564\pi\)
0.0264992 + 0.999649i \(0.491564\pi\)
\(620\) −92.3730 −3.70979
\(621\) −3.57444 −0.143437
\(622\) −34.8821 −1.39864
\(623\) −5.52135 −0.221208
\(624\) −12.4011 −0.496442
\(625\) 2.66445 0.106578
\(626\) −5.65875 −0.226169
\(627\) −7.57103 −0.302358
\(628\) −12.3226 −0.491724
\(629\) −13.2779 −0.529425
\(630\) −7.86823 −0.313478
\(631\) −9.62948 −0.383344 −0.191672 0.981459i \(-0.561391\pi\)
−0.191672 + 0.981459i \(0.561391\pi\)
\(632\) 10.6563 0.423885
\(633\) 3.36809 0.133870
\(634\) −33.8264 −1.34342
\(635\) 6.07783 0.241191
\(636\) −12.6314 −0.500867
\(637\) 5.41813 0.214674
\(638\) 33.6792 1.33337
\(639\) 0.0521291 0.00206219
\(640\) 38.5719 1.52469
\(641\) 0.154604 0.00610648 0.00305324 0.999995i \(-0.499028\pi\)
0.00305324 + 0.999995i \(0.499028\pi\)
\(642\) 29.9357 1.18147
\(643\) −10.6917 −0.421641 −0.210821 0.977525i \(-0.567614\pi\)
−0.210821 + 0.977525i \(0.567614\pi\)
\(644\) −9.46000 −0.372776
\(645\) 20.2607 0.797763
\(646\) −14.8559 −0.584498
\(647\) 40.7766 1.60309 0.801547 0.597932i \(-0.204011\pi\)
0.801547 + 0.597932i \(0.204011\pi\)
\(648\) −1.39373 −0.0547510
\(649\) −7.04837 −0.276673
\(650\) −97.2136 −3.81303
\(651\) −9.56205 −0.374766
\(652\) −38.4081 −1.50418
\(653\) 27.7113 1.08443 0.542214 0.840240i \(-0.317586\pi\)
0.542214 + 0.840240i \(0.317586\pi\)
\(654\) −29.7427 −1.16303
\(655\) −27.3789 −1.06978
\(656\) −23.1478 −0.903770
\(657\) 13.9762 0.545263
\(658\) −10.0330 −0.391126
\(659\) 22.9421 0.893696 0.446848 0.894610i \(-0.352547\pi\)
0.446848 + 0.894610i \(0.352547\pi\)
\(660\) −23.2587 −0.905342
\(661\) 24.4034 0.949184 0.474592 0.880206i \(-0.342596\pi\)
0.474592 + 0.880206i \(0.342596\pi\)
\(662\) 30.8450 1.19882
\(663\) 11.8746 0.461170
\(664\) 12.3772 0.480328
\(665\) 11.4782 0.445107
\(666\) 13.0595 0.506046
\(667\) −23.1960 −0.898153
\(668\) 48.4837 1.87589
\(669\) −2.37933 −0.0919904
\(670\) −43.6120 −1.68488
\(671\) 13.7759 0.531814
\(672\) 7.72122 0.297852
\(673\) 8.37386 0.322789 0.161394 0.986890i \(-0.448401\pi\)
0.161394 + 0.986890i \(0.448401\pi\)
\(674\) 22.7674 0.876966
\(675\) 8.32361 0.320376
\(676\) 43.2875 1.66491
\(677\) −40.2952 −1.54867 −0.774335 0.632775i \(-0.781916\pi\)
−0.774335 + 0.632775i \(0.781916\pi\)
\(678\) 7.52431 0.288970
\(679\) 0.287463 0.0110318
\(680\) −11.1496 −0.427568
\(681\) 14.8099 0.567516
\(682\) −49.6259 −1.90027
\(683\) −0.957326 −0.0366311 −0.0183155 0.999832i \(-0.505830\pi\)
−0.0183155 + 0.999832i \(0.505830\pi\)
\(684\) 8.32237 0.318214
\(685\) 50.0438 1.91207
\(686\) −2.15559 −0.0823008
\(687\) 11.8892 0.453601
\(688\) −12.7044 −0.484350
\(689\) −25.8593 −0.985163
\(690\) 28.1245 1.07068
\(691\) −48.7993 −1.85641 −0.928206 0.372065i \(-0.878650\pi\)
−0.928206 + 0.372065i \(0.878650\pi\)
\(692\) 50.3737 1.91492
\(693\) −2.40764 −0.0914586
\(694\) −35.6198 −1.35211
\(695\) 61.6311 2.33780
\(696\) −9.04450 −0.342831
\(697\) 22.1650 0.839558
\(698\) 23.3709 0.884603
\(699\) −21.2250 −0.802804
\(700\) 22.0290 0.832618
\(701\) 15.4261 0.582637 0.291319 0.956626i \(-0.405906\pi\)
0.291319 + 0.956626i \(0.405906\pi\)
\(702\) −11.6793 −0.440805
\(703\) −19.0513 −0.718534
\(704\) 29.0509 1.09490
\(705\) 16.9893 0.639852
\(706\) 48.4079 1.82186
\(707\) 4.34648 0.163466
\(708\) 7.74784 0.291182
\(709\) −22.3176 −0.838155 −0.419077 0.907951i \(-0.637646\pi\)
−0.419077 + 0.907951i \(0.637646\pi\)
\(710\) −0.410164 −0.0153932
\(711\) −7.64587 −0.286743
\(712\) 7.69529 0.288393
\(713\) 34.1790 1.28001
\(714\) −4.72427 −0.176801
\(715\) −47.6158 −1.78073
\(716\) 23.9196 0.893916
\(717\) −25.1699 −0.939987
\(718\) 39.7875 1.48486
\(719\) 21.2848 0.793791 0.396895 0.917864i \(-0.370088\pi\)
0.396895 + 0.917864i \(0.370088\pi\)
\(720\) −8.35453 −0.311355
\(721\) −11.5843 −0.431421
\(722\) 19.6408 0.730953
\(723\) 13.8128 0.513704
\(724\) −69.7497 −2.59223
\(725\) 54.0153 2.00608
\(726\) 11.2161 0.416270
\(727\) 8.94927 0.331910 0.165955 0.986133i \(-0.446929\pi\)
0.165955 + 0.986133i \(0.446929\pi\)
\(728\) −7.55142 −0.279874
\(729\) 1.00000 0.0370370
\(730\) −109.968 −4.07010
\(731\) 12.1650 0.449938
\(732\) −15.1430 −0.559703
\(733\) −36.1281 −1.33442 −0.667211 0.744869i \(-0.732512\pi\)
−0.667211 + 0.744869i \(0.732512\pi\)
\(734\) −3.88281 −0.143317
\(735\) 3.65015 0.134638
\(736\) −27.5990 −1.01731
\(737\) −13.3450 −0.491570
\(738\) −21.8004 −0.802484
\(739\) 12.4900 0.459452 0.229726 0.973255i \(-0.426217\pi\)
0.229726 + 0.973255i \(0.426217\pi\)
\(740\) −58.5268 −2.15149
\(741\) 17.0378 0.625899
\(742\) 10.2881 0.377687
\(743\) −35.2226 −1.29219 −0.646095 0.763257i \(-0.723599\pi\)
−0.646095 + 0.763257i \(0.723599\pi\)
\(744\) 13.3269 0.488590
\(745\) 59.8615 2.19315
\(746\) −25.9051 −0.948453
\(747\) −8.88061 −0.324925
\(748\) −13.9651 −0.510613
\(749\) −13.8875 −0.507437
\(750\) −26.1509 −0.954897
\(751\) 34.9681 1.27600 0.638002 0.770034i \(-0.279761\pi\)
0.638002 + 0.770034i \(0.279761\pi\)
\(752\) −10.6531 −0.388477
\(753\) 20.4634 0.745728
\(754\) −75.7915 −2.76016
\(755\) 67.8449 2.46913
\(756\) 2.64657 0.0962547
\(757\) −29.2029 −1.06140 −0.530698 0.847561i \(-0.678070\pi\)
−0.530698 + 0.847561i \(0.678070\pi\)
\(758\) −5.13037 −0.186344
\(759\) 8.60596 0.312376
\(760\) −15.9976 −0.580294
\(761\) 3.63818 0.131884 0.0659419 0.997823i \(-0.478995\pi\)
0.0659419 + 0.997823i \(0.478995\pi\)
\(762\) −3.58925 −0.130025
\(763\) 13.7979 0.499519
\(764\) 61.6562 2.23064
\(765\) 7.99981 0.289234
\(766\) −2.15559 −0.0778846
\(767\) 15.8616 0.572729
\(768\) 1.35370 0.0488476
\(769\) −0.544551 −0.0196370 −0.00981851 0.999952i \(-0.503125\pi\)
−0.00981851 + 0.999952i \(0.503125\pi\)
\(770\) 18.9438 0.682689
\(771\) −0.0952126 −0.00342900
\(772\) −20.4210 −0.734969
\(773\) −24.3661 −0.876389 −0.438195 0.898880i \(-0.644382\pi\)
−0.438195 + 0.898880i \(0.644382\pi\)
\(774\) −11.9649 −0.430069
\(775\) −79.5908 −2.85899
\(776\) −0.400647 −0.0143824
\(777\) −6.05844 −0.217345
\(778\) −41.8469 −1.50028
\(779\) 31.8026 1.13945
\(780\) 52.3411 1.87411
\(781\) −0.125508 −0.00449102
\(782\) 16.8866 0.603865
\(783\) 6.48940 0.231912
\(784\) −2.28882 −0.0817435
\(785\) −16.9953 −0.606589
\(786\) 16.1685 0.576712
\(787\) −36.0913 −1.28651 −0.643257 0.765650i \(-0.722417\pi\)
−0.643257 + 0.765650i \(0.722417\pi\)
\(788\) 8.06970 0.287471
\(789\) −8.14349 −0.289916
\(790\) 60.1595 2.14038
\(791\) −3.49061 −0.124112
\(792\) 3.35560 0.119236
\(793\) −31.0013 −1.10089
\(794\) 45.4957 1.61458
\(795\) −17.4212 −0.617868
\(796\) −19.4641 −0.689885
\(797\) −0.105685 −0.00374354 −0.00187177 0.999998i \(-0.500596\pi\)
−0.00187177 + 0.999998i \(0.500596\pi\)
\(798\) −6.77845 −0.239954
\(799\) 10.2008 0.360877
\(800\) 64.2684 2.27223
\(801\) −5.52135 −0.195087
\(802\) 55.2831 1.95211
\(803\) −33.6496 −1.18747
\(804\) 14.6694 0.517349
\(805\) −13.0473 −0.459855
\(806\) 111.678 3.93368
\(807\) 16.7336 0.589052
\(808\) −6.05783 −0.213114
\(809\) 10.2779 0.361351 0.180675 0.983543i \(-0.442172\pi\)
0.180675 + 0.983543i \(0.442172\pi\)
\(810\) −7.86823 −0.276461
\(811\) 28.8261 1.01222 0.506110 0.862469i \(-0.331083\pi\)
0.506110 + 0.862469i \(0.331083\pi\)
\(812\) 17.1746 0.602712
\(813\) −8.33824 −0.292435
\(814\) −31.4426 −1.10206
\(815\) −52.9726 −1.85555
\(816\) −5.01626 −0.175604
\(817\) 17.4545 0.610655
\(818\) 24.4598 0.855216
\(819\) 5.41813 0.189325
\(820\) 97.6995 3.41181
\(821\) −8.70278 −0.303729 −0.151865 0.988401i \(-0.548528\pi\)
−0.151865 + 0.988401i \(0.548528\pi\)
\(822\) −29.5532 −1.03079
\(823\) 3.23833 0.112881 0.0564405 0.998406i \(-0.482025\pi\)
0.0564405 + 0.998406i \(0.482025\pi\)
\(824\) 16.1454 0.562451
\(825\) −20.0402 −0.697711
\(826\) −6.31050 −0.219571
\(827\) 34.5219 1.20044 0.600222 0.799834i \(-0.295079\pi\)
0.600222 + 0.799834i \(0.295079\pi\)
\(828\) −9.46000 −0.328758
\(829\) −30.5484 −1.06099 −0.530495 0.847688i \(-0.677994\pi\)
−0.530495 + 0.847688i \(0.677994\pi\)
\(830\) 69.8747 2.42539
\(831\) −28.1966 −0.978128
\(832\) −65.3759 −2.26650
\(833\) 2.19164 0.0759358
\(834\) −36.3961 −1.26029
\(835\) 66.8688 2.31409
\(836\) −20.0372 −0.693003
\(837\) −9.56205 −0.330513
\(838\) 31.2952 1.08108
\(839\) 15.9327 0.550058 0.275029 0.961436i \(-0.411313\pi\)
0.275029 + 0.961436i \(0.411313\pi\)
\(840\) −5.08734 −0.175530
\(841\) 13.1124 0.452151
\(842\) 46.8624 1.61499
\(843\) 9.23440 0.318049
\(844\) 8.91389 0.306829
\(845\) 59.7023 2.05382
\(846\) −10.0330 −0.344940
\(847\) −5.20329 −0.178787
\(848\) 10.9239 0.375130
\(849\) −17.9477 −0.615965
\(850\) −39.3230 −1.34877
\(851\) 21.6556 0.742343
\(852\) 0.137963 0.00472654
\(853\) −32.2332 −1.10364 −0.551821 0.833962i \(-0.686067\pi\)
−0.551821 + 0.833962i \(0.686067\pi\)
\(854\) 12.3338 0.422053
\(855\) 11.4782 0.392547
\(856\) 19.3554 0.661555
\(857\) 25.4124 0.868072 0.434036 0.900896i \(-0.357089\pi\)
0.434036 + 0.900896i \(0.357089\pi\)
\(858\) 28.1194 0.959981
\(859\) 47.3247 1.61470 0.807348 0.590075i \(-0.200902\pi\)
0.807348 + 0.590075i \(0.200902\pi\)
\(860\) 53.6212 1.82847
\(861\) 10.1134 0.344665
\(862\) 47.0699 1.60321
\(863\) 37.9162 1.29068 0.645341 0.763894i \(-0.276715\pi\)
0.645341 + 0.763894i \(0.276715\pi\)
\(864\) 7.72122 0.262681
\(865\) 69.4756 2.36224
\(866\) −31.2718 −1.06266
\(867\) −12.1967 −0.414222
\(868\) −25.3066 −0.858962
\(869\) 18.4085 0.624465
\(870\) −51.0601 −1.73110
\(871\) 30.0315 1.01758
\(872\) −19.2306 −0.651232
\(873\) 0.287463 0.00972917
\(874\) 24.2292 0.819564
\(875\) 12.1317 0.410126
\(876\) 36.9889 1.24974
\(877\) −34.5553 −1.16685 −0.583424 0.812168i \(-0.698288\pi\)
−0.583424 + 0.812168i \(0.698288\pi\)
\(878\) 52.5360 1.77301
\(879\) 5.88333 0.198440
\(880\) 20.1147 0.678066
\(881\) −24.9627 −0.841015 −0.420508 0.907289i \(-0.638148\pi\)
−0.420508 + 0.907289i \(0.638148\pi\)
\(882\) −2.15559 −0.0725825
\(883\) −23.6177 −0.794800 −0.397400 0.917646i \(-0.630087\pi\)
−0.397400 + 0.917646i \(0.630087\pi\)
\(884\) 31.4269 1.05700
\(885\) 10.6858 0.359201
\(886\) 83.2621 2.79724
\(887\) −16.4447 −0.552158 −0.276079 0.961135i \(-0.589035\pi\)
−0.276079 + 0.961135i \(0.589035\pi\)
\(888\) 8.44385 0.283357
\(889\) 1.66509 0.0558453
\(890\) 43.4433 1.45622
\(891\) −2.40764 −0.0806589
\(892\) −6.29707 −0.210841
\(893\) 14.6362 0.489781
\(894\) −35.3511 −1.18232
\(895\) 32.9899 1.10273
\(896\) 10.5672 0.353026
\(897\) −19.3668 −0.646638
\(898\) 13.3656 0.446016
\(899\) −62.0520 −2.06955
\(900\) 22.0290 0.734300
\(901\) −10.4601 −0.348477
\(902\) 52.4875 1.74764
\(903\) 5.55063 0.184714
\(904\) 4.86497 0.161807
\(905\) −96.1990 −3.19776
\(906\) −40.0657 −1.33109
\(907\) −29.7229 −0.986932 −0.493466 0.869765i \(-0.664270\pi\)
−0.493466 + 0.869765i \(0.664270\pi\)
\(908\) 39.1954 1.30074
\(909\) 4.34648 0.144164
\(910\) −42.6311 −1.41321
\(911\) −30.4629 −1.00928 −0.504641 0.863329i \(-0.668375\pi\)
−0.504641 + 0.863329i \(0.668375\pi\)
\(912\) −7.19740 −0.238330
\(913\) 21.3813 0.707617
\(914\) −11.1856 −0.369985
\(915\) −20.8853 −0.690447
\(916\) 31.4655 1.03965
\(917\) −7.50074 −0.247696
\(918\) −4.72427 −0.155924
\(919\) 32.8179 1.08256 0.541282 0.840841i \(-0.317939\pi\)
0.541282 + 0.840841i \(0.317939\pi\)
\(920\) 18.1844 0.599522
\(921\) −15.0865 −0.497117
\(922\) −74.7122 −2.46052
\(923\) 0.282442 0.00929669
\(924\) −6.37197 −0.209622
\(925\) −50.4281 −1.65807
\(926\) −13.7639 −0.452311
\(927\) −11.5843 −0.380477
\(928\) 50.1061 1.64481
\(929\) −31.5384 −1.03474 −0.517371 0.855761i \(-0.673089\pi\)
−0.517371 + 0.855761i \(0.673089\pi\)
\(930\) 75.2364 2.46710
\(931\) 3.14459 0.103060
\(932\) −56.1735 −1.84002
\(933\) 16.1821 0.529780
\(934\) −74.2616 −2.42991
\(935\) −19.2606 −0.629890
\(936\) −7.55142 −0.246826
\(937\) 8.79735 0.287397 0.143698 0.989622i \(-0.454100\pi\)
0.143698 + 0.989622i \(0.454100\pi\)
\(938\) −11.9480 −0.390115
\(939\) 2.62515 0.0856686
\(940\) 44.9632 1.46654
\(941\) −23.5639 −0.768160 −0.384080 0.923300i \(-0.625481\pi\)
−0.384080 + 0.923300i \(0.625481\pi\)
\(942\) 10.0366 0.327008
\(943\) −36.1499 −1.17720
\(944\) −6.70053 −0.218084
\(945\) 3.65015 0.118739
\(946\) 28.8071 0.936600
\(947\) −14.3560 −0.466507 −0.233253 0.972416i \(-0.574937\pi\)
−0.233253 + 0.972416i \(0.574937\pi\)
\(948\) −20.2353 −0.657213
\(949\) 75.7248 2.45813
\(950\) −56.4212 −1.83054
\(951\) 15.6924 0.508862
\(952\) −3.05456 −0.0989988
\(953\) −40.7720 −1.32073 −0.660367 0.750943i \(-0.729599\pi\)
−0.660367 + 0.750943i \(0.729599\pi\)
\(954\) 10.2881 0.333089
\(955\) 85.0364 2.75171
\(956\) −66.6138 −2.15444
\(957\) −15.6241 −0.505056
\(958\) 22.1941 0.717059
\(959\) 13.7101 0.442721
\(960\) −44.0433 −1.42149
\(961\) 60.4329 1.94945
\(962\) 70.7581 2.28133
\(963\) −13.8875 −0.447518
\(964\) 36.5565 1.17741
\(965\) −28.1647 −0.906655
\(966\) 7.70503 0.247905
\(967\) 28.5024 0.916576 0.458288 0.888804i \(-0.348463\pi\)
0.458288 + 0.888804i \(0.348463\pi\)
\(968\) 7.25199 0.233088
\(969\) 6.89181 0.221397
\(970\) −2.26183 −0.0726230
\(971\) −25.6857 −0.824292 −0.412146 0.911118i \(-0.635221\pi\)
−0.412146 + 0.911118i \(0.635221\pi\)
\(972\) 2.64657 0.0848887
\(973\) 16.8845 0.541293
\(974\) 26.6735 0.854674
\(975\) 45.0984 1.44430
\(976\) 13.0961 0.419195
\(977\) 33.7704 1.08041 0.540205 0.841533i \(-0.318347\pi\)
0.540205 + 0.841533i \(0.318347\pi\)
\(978\) 31.2828 1.00032
\(979\) 13.2934 0.424859
\(980\) 9.66037 0.308589
\(981\) 13.7979 0.440534
\(982\) 78.2440 2.49687
\(983\) 53.2854 1.69954 0.849770 0.527154i \(-0.176741\pi\)
0.849770 + 0.527154i \(0.176741\pi\)
\(984\) −14.0954 −0.449346
\(985\) 11.1298 0.354623
\(986\) −30.6577 −0.976341
\(987\) 4.65440 0.148151
\(988\) 45.0917 1.43456
\(989\) −19.8404 −0.630889
\(990\) 18.9438 0.602075
\(991\) 29.4983 0.937045 0.468523 0.883452i \(-0.344786\pi\)
0.468523 + 0.883452i \(0.344786\pi\)
\(992\) −73.8307 −2.34413
\(993\) −14.3093 −0.454091
\(994\) −0.112369 −0.00356412
\(995\) −26.8449 −0.851040
\(996\) −23.5031 −0.744725
\(997\) −11.1138 −0.351979 −0.175990 0.984392i \(-0.556313\pi\)
−0.175990 + 0.984392i \(0.556313\pi\)
\(998\) 1.50804 0.0477362
\(999\) −6.05844 −0.191681
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 8043.2.a.u.1.7 53
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8043.2.a.u.1.7 53 1.1 even 1 trivial