Properties

Label 5225.2.a.k.1.5
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.131947641.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.04201\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.04201 q^{2} +1.09835 q^{3} +2.16980 q^{4} +2.24284 q^{6} +0.469142 q^{7} +0.346728 q^{8} -1.79363 q^{9} +O(q^{10})\) \(q+2.04201 q^{2} +1.09835 q^{3} +2.16980 q^{4} +2.24284 q^{6} +0.469142 q^{7} +0.346728 q^{8} -1.79363 q^{9} -1.00000 q^{11} +2.38320 q^{12} -2.14036 q^{13} +0.957992 q^{14} -3.63157 q^{16} -4.47995 q^{17} -3.66261 q^{18} -1.00000 q^{19} +0.515282 q^{21} -2.04201 q^{22} -0.200995 q^{23} +0.380828 q^{24} -4.37063 q^{26} -5.26508 q^{27} +1.01794 q^{28} -6.92379 q^{29} +4.65688 q^{31} -8.10916 q^{32} -1.09835 q^{33} -9.14810 q^{34} -3.89181 q^{36} +7.22421 q^{37} -2.04201 q^{38} -2.35086 q^{39} -3.34355 q^{41} +1.05221 q^{42} -3.75162 q^{43} -2.16980 q^{44} -0.410434 q^{46} +2.69748 q^{47} -3.98874 q^{48} -6.77991 q^{49} -4.92055 q^{51} -4.64414 q^{52} -10.4652 q^{53} -10.7513 q^{54} +0.162665 q^{56} -1.09835 q^{57} -14.1384 q^{58} +6.05634 q^{59} -5.04325 q^{61} +9.50939 q^{62} -0.841467 q^{63} -9.29582 q^{64} -2.24284 q^{66} -7.47221 q^{67} -9.72059 q^{68} -0.220763 q^{69} +15.9494 q^{71} -0.621901 q^{72} +15.5740 q^{73} +14.7519 q^{74} -2.16980 q^{76} -0.469142 q^{77} -4.80047 q^{78} +5.88178 q^{79} -0.402006 q^{81} -6.82756 q^{82} -12.9799 q^{83} +1.11806 q^{84} -7.66084 q^{86} -7.60473 q^{87} -0.346728 q^{88} -0.794873 q^{89} -1.00413 q^{91} -0.436119 q^{92} +5.11488 q^{93} +5.50828 q^{94} -8.90669 q^{96} -17.5493 q^{97} -13.8446 q^{98} +1.79363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28} - 5 q^{29} - q^{31} - 15 q^{32} + 3 q^{33} - 22 q^{34} + 12 q^{36} - 9 q^{37} - 32 q^{39} + 25 q^{41} - 11 q^{42} - 15 q^{43} - 8 q^{44} - 16 q^{46} - 24 q^{47} + 4 q^{48} + 13 q^{49} + 27 q^{52} - 5 q^{53} - 11 q^{54} - 12 q^{56} + 3 q^{57} - 13 q^{58} + 39 q^{59} - 11 q^{61} + 42 q^{62} - 38 q^{63} - 14 q^{64} - 2 q^{66} - 24 q^{67} - 45 q^{68} + 14 q^{69} - 24 q^{71} + 61 q^{72} + 26 q^{73} + q^{74} - 8 q^{76} + 5 q^{77} + 29 q^{78} + 11 q^{79} + 30 q^{81} - 8 q^{82} - 39 q^{83} + 25 q^{84} + 18 q^{86} + 16 q^{87} + 22 q^{89} - 26 q^{91} + 11 q^{92} + 6 q^{93} - 30 q^{94} - 15 q^{96} - 22 q^{97} - 33 q^{98} - 9 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.04201 1.44392 0.721959 0.691936i \(-0.243242\pi\)
0.721959 + 0.691936i \(0.243242\pi\)
\(3\) 1.09835 0.634132 0.317066 0.948403i \(-0.397302\pi\)
0.317066 + 0.948403i \(0.397302\pi\)
\(4\) 2.16980 1.08490
\(5\) 0 0
\(6\) 2.24284 0.915635
\(7\) 0.469142 0.177319 0.0886595 0.996062i \(-0.471742\pi\)
0.0886595 + 0.996062i \(0.471742\pi\)
\(8\) 0.346728 0.122587
\(9\) −1.79363 −0.597876
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 2.38320 0.687969
\(13\) −2.14036 −0.593628 −0.296814 0.954935i \(-0.595924\pi\)
−0.296814 + 0.954935i \(0.595924\pi\)
\(14\) 0.957992 0.256034
\(15\) 0 0
\(16\) −3.63157 −0.907893
\(17\) −4.47995 −1.08655 −0.543274 0.839555i \(-0.682816\pi\)
−0.543274 + 0.839555i \(0.682816\pi\)
\(18\) −3.66261 −0.863284
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.515282 0.112444
\(22\) −2.04201 −0.435358
\(23\) −0.200995 −0.0419105 −0.0209552 0.999780i \(-0.506671\pi\)
−0.0209552 + 0.999780i \(0.506671\pi\)
\(24\) 0.380828 0.0777362
\(25\) 0 0
\(26\) −4.37063 −0.857150
\(27\) −5.26508 −1.01326
\(28\) 1.01794 0.192373
\(29\) −6.92379 −1.28571 −0.642857 0.765986i \(-0.722251\pi\)
−0.642857 + 0.765986i \(0.722251\pi\)
\(30\) 0 0
\(31\) 4.65688 0.836401 0.418200 0.908355i \(-0.362661\pi\)
0.418200 + 0.908355i \(0.362661\pi\)
\(32\) −8.10916 −1.43351
\(33\) −1.09835 −0.191198
\(34\) −9.14810 −1.56889
\(35\) 0 0
\(36\) −3.89181 −0.648635
\(37\) 7.22421 1.18765 0.593826 0.804593i \(-0.297617\pi\)
0.593826 + 0.804593i \(0.297617\pi\)
\(38\) −2.04201 −0.331257
\(39\) −2.35086 −0.376439
\(40\) 0 0
\(41\) −3.34355 −0.522175 −0.261088 0.965315i \(-0.584081\pi\)
−0.261088 + 0.965315i \(0.584081\pi\)
\(42\) 1.05221 0.162359
\(43\) −3.75162 −0.572117 −0.286058 0.958212i \(-0.592345\pi\)
−0.286058 + 0.958212i \(0.592345\pi\)
\(44\) −2.16980 −0.327109
\(45\) 0 0
\(46\) −0.410434 −0.0605152
\(47\) 2.69748 0.393468 0.196734 0.980457i \(-0.436966\pi\)
0.196734 + 0.980457i \(0.436966\pi\)
\(48\) −3.98874 −0.575724
\(49\) −6.77991 −0.968558
\(50\) 0 0
\(51\) −4.92055 −0.689015
\(52\) −4.64414 −0.644027
\(53\) −10.4652 −1.43751 −0.718754 0.695264i \(-0.755287\pi\)
−0.718754 + 0.695264i \(0.755287\pi\)
\(54\) −10.7513 −1.46307
\(55\) 0 0
\(56\) 0.162665 0.0217370
\(57\) −1.09835 −0.145480
\(58\) −14.1384 −1.85647
\(59\) 6.05634 0.788468 0.394234 0.919010i \(-0.371010\pi\)
0.394234 + 0.919010i \(0.371010\pi\)
\(60\) 0 0
\(61\) −5.04325 −0.645722 −0.322861 0.946446i \(-0.604645\pi\)
−0.322861 + 0.946446i \(0.604645\pi\)
\(62\) 9.50939 1.20769
\(63\) −0.841467 −0.106015
\(64\) −9.29582 −1.16198
\(65\) 0 0
\(66\) −2.24284 −0.276074
\(67\) −7.47221 −0.912876 −0.456438 0.889755i \(-0.650875\pi\)
−0.456438 + 0.889755i \(0.650875\pi\)
\(68\) −9.72059 −1.17879
\(69\) −0.220763 −0.0265768
\(70\) 0 0
\(71\) 15.9494 1.89284 0.946421 0.322936i \(-0.104670\pi\)
0.946421 + 0.322936i \(0.104670\pi\)
\(72\) −0.621901 −0.0732918
\(73\) 15.5740 1.82280 0.911400 0.411522i \(-0.135003\pi\)
0.911400 + 0.411522i \(0.135003\pi\)
\(74\) 14.7519 1.71487
\(75\) 0 0
\(76\) −2.16980 −0.248893
\(77\) −0.469142 −0.0534637
\(78\) −4.80047 −0.543547
\(79\) 5.88178 0.661752 0.330876 0.943674i \(-0.392656\pi\)
0.330876 + 0.943674i \(0.392656\pi\)
\(80\) 0 0
\(81\) −0.402006 −0.0446673
\(82\) −6.82756 −0.753978
\(83\) −12.9799 −1.42473 −0.712364 0.701810i \(-0.752376\pi\)
−0.712364 + 0.701810i \(0.752376\pi\)
\(84\) 1.11806 0.121990
\(85\) 0 0
\(86\) −7.66084 −0.826090
\(87\) −7.60473 −0.815313
\(88\) −0.346728 −0.0369613
\(89\) −0.794873 −0.0842564 −0.0421282 0.999112i \(-0.513414\pi\)
−0.0421282 + 0.999112i \(0.513414\pi\)
\(90\) 0 0
\(91\) −1.00413 −0.105262
\(92\) −0.436119 −0.0454686
\(93\) 5.11488 0.530389
\(94\) 5.50828 0.568136
\(95\) 0 0
\(96\) −8.90669 −0.909035
\(97\) −17.5493 −1.78186 −0.890931 0.454139i \(-0.849947\pi\)
−0.890931 + 0.454139i \(0.849947\pi\)
\(98\) −13.8446 −1.39852
\(99\) 1.79363 0.180267
\(100\) 0 0
\(101\) 14.0995 1.40295 0.701476 0.712693i \(-0.252525\pi\)
0.701476 + 0.712693i \(0.252525\pi\)
\(102\) −10.0478 −0.994881
\(103\) −14.7905 −1.45735 −0.728673 0.684861i \(-0.759863\pi\)
−0.728673 + 0.684861i \(0.759863\pi\)
\(104\) −0.742122 −0.0727710
\(105\) 0 0
\(106\) −21.3701 −2.07564
\(107\) −3.73310 −0.360892 −0.180446 0.983585i \(-0.557754\pi\)
−0.180446 + 0.983585i \(0.557754\pi\)
\(108\) −11.4242 −1.09929
\(109\) 0.525483 0.0503321 0.0251661 0.999683i \(-0.491989\pi\)
0.0251661 + 0.999683i \(0.491989\pi\)
\(110\) 0 0
\(111\) 7.93470 0.753129
\(112\) −1.70372 −0.160987
\(113\) −9.63228 −0.906128 −0.453064 0.891478i \(-0.649669\pi\)
−0.453064 + 0.891478i \(0.649669\pi\)
\(114\) −2.24284 −0.210061
\(115\) 0 0
\(116\) −15.0232 −1.39487
\(117\) 3.83901 0.354916
\(118\) 12.3671 1.13848
\(119\) −2.10173 −0.192666
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −10.2984 −0.932370
\(123\) −3.67239 −0.331128
\(124\) 10.1045 0.907410
\(125\) 0 0
\(126\) −1.71828 −0.153077
\(127\) 17.6992 1.57055 0.785276 0.619146i \(-0.212521\pi\)
0.785276 + 0.619146i \(0.212521\pi\)
\(128\) −2.76383 −0.244290
\(129\) −4.12059 −0.362798
\(130\) 0 0
\(131\) 19.8018 1.73009 0.865046 0.501693i \(-0.167289\pi\)
0.865046 + 0.501693i \(0.167289\pi\)
\(132\) −2.38320 −0.207430
\(133\) −0.469142 −0.0406798
\(134\) −15.2583 −1.31812
\(135\) 0 0
\(136\) −1.55332 −0.133196
\(137\) −9.38019 −0.801403 −0.400702 0.916209i \(-0.631234\pi\)
−0.400702 + 0.916209i \(0.631234\pi\)
\(138\) −0.450800 −0.0383747
\(139\) −13.6770 −1.16007 −0.580034 0.814592i \(-0.696961\pi\)
−0.580034 + 0.814592i \(0.696961\pi\)
\(140\) 0 0
\(141\) 2.96278 0.249511
\(142\) 32.5687 2.73311
\(143\) 2.14036 0.178986
\(144\) 6.51370 0.542808
\(145\) 0 0
\(146\) 31.8022 2.63197
\(147\) −7.44670 −0.614194
\(148\) 15.6751 1.28848
\(149\) 9.34304 0.765412 0.382706 0.923870i \(-0.374992\pi\)
0.382706 + 0.923870i \(0.374992\pi\)
\(150\) 0 0
\(151\) 7.72133 0.628353 0.314176 0.949365i \(-0.398272\pi\)
0.314176 + 0.949365i \(0.398272\pi\)
\(152\) −0.346728 −0.0281233
\(153\) 8.03537 0.649621
\(154\) −0.957992 −0.0771972
\(155\) 0 0
\(156\) −5.10089 −0.408398
\(157\) −2.86254 −0.228455 −0.114228 0.993455i \(-0.536439\pi\)
−0.114228 + 0.993455i \(0.536439\pi\)
\(158\) 12.0106 0.955515
\(159\) −11.4945 −0.911570
\(160\) 0 0
\(161\) −0.0942954 −0.00743152
\(162\) −0.820899 −0.0644959
\(163\) −10.9042 −0.854083 −0.427041 0.904232i \(-0.640444\pi\)
−0.427041 + 0.904232i \(0.640444\pi\)
\(164\) −7.25483 −0.566507
\(165\) 0 0
\(166\) −26.5050 −2.05719
\(167\) 15.7161 1.21615 0.608074 0.793880i \(-0.291942\pi\)
0.608074 + 0.793880i \(0.291942\pi\)
\(168\) 0.178663 0.0137841
\(169\) −8.41887 −0.647605
\(170\) 0 0
\(171\) 1.79363 0.137162
\(172\) −8.14026 −0.620689
\(173\) −3.84278 −0.292161 −0.146080 0.989273i \(-0.546666\pi\)
−0.146080 + 0.989273i \(0.546666\pi\)
\(174\) −15.5289 −1.17724
\(175\) 0 0
\(176\) 3.63157 0.273740
\(177\) 6.65198 0.499993
\(178\) −1.62314 −0.121659
\(179\) 19.8584 1.48429 0.742143 0.670242i \(-0.233810\pi\)
0.742143 + 0.670242i \(0.233810\pi\)
\(180\) 0 0
\(181\) −15.0067 −1.11544 −0.557719 0.830030i \(-0.688323\pi\)
−0.557719 + 0.830030i \(0.688323\pi\)
\(182\) −2.05044 −0.151989
\(183\) −5.53925 −0.409473
\(184\) −0.0696907 −0.00513767
\(185\) 0 0
\(186\) 10.4446 0.765838
\(187\) 4.47995 0.327607
\(188\) 5.85299 0.426873
\(189\) −2.47007 −0.179671
\(190\) 0 0
\(191\) 14.4281 1.04398 0.521990 0.852951i \(-0.325190\pi\)
0.521990 + 0.852951i \(0.325190\pi\)
\(192\) −10.2101 −0.736847
\(193\) −23.3554 −1.68116 −0.840580 0.541688i \(-0.817785\pi\)
−0.840580 + 0.541688i \(0.817785\pi\)
\(194\) −35.8358 −2.57286
\(195\) 0 0
\(196\) −14.7110 −1.05079
\(197\) 10.3230 0.735484 0.367742 0.929928i \(-0.380131\pi\)
0.367742 + 0.929928i \(0.380131\pi\)
\(198\) 3.66261 0.260290
\(199\) −10.5459 −0.747579 −0.373789 0.927514i \(-0.621942\pi\)
−0.373789 + 0.927514i \(0.621942\pi\)
\(200\) 0 0
\(201\) −8.20710 −0.578884
\(202\) 28.7913 2.02575
\(203\) −3.24824 −0.227982
\(204\) −10.6766 −0.747512
\(205\) 0 0
\(206\) −30.2022 −2.10429
\(207\) 0.360511 0.0250573
\(208\) 7.77287 0.538951
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 7.35896 0.506612 0.253306 0.967386i \(-0.418482\pi\)
0.253306 + 0.967386i \(0.418482\pi\)
\(212\) −22.7074 −1.55955
\(213\) 17.5180 1.20031
\(214\) −7.62302 −0.521099
\(215\) 0 0
\(216\) −1.82555 −0.124213
\(217\) 2.18474 0.148310
\(218\) 1.07304 0.0726755
\(219\) 17.1057 1.15590
\(220\) 0 0
\(221\) 9.58870 0.645006
\(222\) 16.2027 1.08746
\(223\) −14.4057 −0.964674 −0.482337 0.875986i \(-0.660212\pi\)
−0.482337 + 0.875986i \(0.660212\pi\)
\(224\) −3.80435 −0.254189
\(225\) 0 0
\(226\) −19.6692 −1.30837
\(227\) 29.7139 1.97218 0.986091 0.166204i \(-0.0531510\pi\)
0.986091 + 0.166204i \(0.0531510\pi\)
\(228\) −2.38320 −0.157831
\(229\) −11.0142 −0.727836 −0.363918 0.931431i \(-0.618561\pi\)
−0.363918 + 0.931431i \(0.618561\pi\)
\(230\) 0 0
\(231\) −0.515282 −0.0339030
\(232\) −2.40067 −0.157612
\(233\) −9.53690 −0.624783 −0.312391 0.949953i \(-0.601130\pi\)
−0.312391 + 0.949953i \(0.601130\pi\)
\(234\) 7.83929 0.512470
\(235\) 0 0
\(236\) 13.1410 0.855408
\(237\) 6.46024 0.419638
\(238\) −4.29176 −0.278193
\(239\) 4.02979 0.260665 0.130333 0.991470i \(-0.458395\pi\)
0.130333 + 0.991470i \(0.458395\pi\)
\(240\) 0 0
\(241\) 6.06607 0.390750 0.195375 0.980729i \(-0.437408\pi\)
0.195375 + 0.980729i \(0.437408\pi\)
\(242\) 2.04201 0.131265
\(243\) 15.3537 0.984940
\(244\) −10.9428 −0.700543
\(245\) 0 0
\(246\) −7.49905 −0.478122
\(247\) 2.14036 0.136188
\(248\) 1.61467 0.102532
\(249\) −14.2565 −0.903466
\(250\) 0 0
\(251\) −20.7498 −1.30971 −0.654857 0.755753i \(-0.727271\pi\)
−0.654857 + 0.755753i \(0.727271\pi\)
\(252\) −1.82581 −0.115015
\(253\) 0.200995 0.0126365
\(254\) 36.1420 2.26775
\(255\) 0 0
\(256\) 12.9479 0.809243
\(257\) −21.3702 −1.33304 −0.666519 0.745488i \(-0.732216\pi\)
−0.666519 + 0.745488i \(0.732216\pi\)
\(258\) −8.41428 −0.523850
\(259\) 3.38918 0.210593
\(260\) 0 0
\(261\) 12.4187 0.768699
\(262\) 40.4355 2.49811
\(263\) 2.10893 0.130042 0.0650212 0.997884i \(-0.479288\pi\)
0.0650212 + 0.997884i \(0.479288\pi\)
\(264\) −0.380828 −0.0234384
\(265\) 0 0
\(266\) −0.957992 −0.0587382
\(267\) −0.873048 −0.0534297
\(268\) −16.2132 −0.990378
\(269\) 20.3100 1.23833 0.619163 0.785263i \(-0.287472\pi\)
0.619163 + 0.785263i \(0.287472\pi\)
\(270\) 0 0
\(271\) −2.81864 −0.171220 −0.0856100 0.996329i \(-0.527284\pi\)
−0.0856100 + 0.996329i \(0.527284\pi\)
\(272\) 16.2693 0.986470
\(273\) −1.10289 −0.0667497
\(274\) −19.1544 −1.15716
\(275\) 0 0
\(276\) −0.479011 −0.0288331
\(277\) 16.6723 1.00174 0.500872 0.865522i \(-0.333013\pi\)
0.500872 + 0.865522i \(0.333013\pi\)
\(278\) −27.9285 −1.67504
\(279\) −8.35272 −0.500064
\(280\) 0 0
\(281\) −27.9281 −1.66605 −0.833026 0.553234i \(-0.813393\pi\)
−0.833026 + 0.553234i \(0.813393\pi\)
\(282\) 6.05001 0.360273
\(283\) 4.88494 0.290380 0.145190 0.989404i \(-0.453621\pi\)
0.145190 + 0.989404i \(0.453621\pi\)
\(284\) 34.6069 2.05354
\(285\) 0 0
\(286\) 4.37063 0.258441
\(287\) −1.56860 −0.0925916
\(288\) 14.5448 0.857062
\(289\) 3.06997 0.180587
\(290\) 0 0
\(291\) −19.2753 −1.12994
\(292\) 33.7924 1.97755
\(293\) −0.648185 −0.0378674 −0.0189337 0.999821i \(-0.506027\pi\)
−0.0189337 + 0.999821i \(0.506027\pi\)
\(294\) −15.2062 −0.886845
\(295\) 0 0
\(296\) 2.50483 0.145591
\(297\) 5.26508 0.305511
\(298\) 19.0786 1.10519
\(299\) 0.430202 0.0248792
\(300\) 0 0
\(301\) −1.76004 −0.101447
\(302\) 15.7670 0.907290
\(303\) 15.4862 0.889657
\(304\) 3.63157 0.208285
\(305\) 0 0
\(306\) 16.4083 0.938000
\(307\) −11.8075 −0.673891 −0.336945 0.941524i \(-0.609394\pi\)
−0.336945 + 0.941524i \(0.609394\pi\)
\(308\) −1.01794 −0.0580027
\(309\) −16.2451 −0.924150
\(310\) 0 0
\(311\) −14.4213 −0.817756 −0.408878 0.912589i \(-0.634080\pi\)
−0.408878 + 0.912589i \(0.634080\pi\)
\(312\) −0.815108 −0.0461464
\(313\) 5.49337 0.310504 0.155252 0.987875i \(-0.450381\pi\)
0.155252 + 0.987875i \(0.450381\pi\)
\(314\) −5.84533 −0.329871
\(315\) 0 0
\(316\) 12.7623 0.717934
\(317\) −13.4742 −0.756789 −0.378395 0.925644i \(-0.623524\pi\)
−0.378395 + 0.925644i \(0.623524\pi\)
\(318\) −23.4718 −1.31623
\(319\) 6.92379 0.387658
\(320\) 0 0
\(321\) −4.10024 −0.228853
\(322\) −0.192552 −0.0107305
\(323\) 4.47995 0.249271
\(324\) −0.872271 −0.0484595
\(325\) 0 0
\(326\) −22.2665 −1.23323
\(327\) 0.577164 0.0319172
\(328\) −1.15930 −0.0640118
\(329\) 1.26550 0.0697694
\(330\) 0 0
\(331\) 10.6406 0.584859 0.292430 0.956287i \(-0.405536\pi\)
0.292430 + 0.956287i \(0.405536\pi\)
\(332\) −28.1637 −1.54569
\(333\) −12.9576 −0.710069
\(334\) 32.0924 1.75602
\(335\) 0 0
\(336\) −1.87128 −0.102087
\(337\) 2.99281 0.163029 0.0815145 0.996672i \(-0.474024\pi\)
0.0815145 + 0.996672i \(0.474024\pi\)
\(338\) −17.1914 −0.935089
\(339\) −10.5796 −0.574605
\(340\) 0 0
\(341\) −4.65688 −0.252184
\(342\) 3.66261 0.198051
\(343\) −6.46473 −0.349063
\(344\) −1.30079 −0.0701340
\(345\) 0 0
\(346\) −7.84698 −0.421856
\(347\) −1.57779 −0.0847001 −0.0423500 0.999103i \(-0.513484\pi\)
−0.0423500 + 0.999103i \(0.513484\pi\)
\(348\) −16.5007 −0.884532
\(349\) 12.9750 0.694534 0.347267 0.937766i \(-0.387110\pi\)
0.347267 + 0.937766i \(0.387110\pi\)
\(350\) 0 0
\(351\) 11.2691 0.601503
\(352\) 8.10916 0.432220
\(353\) 13.5044 0.718768 0.359384 0.933190i \(-0.382987\pi\)
0.359384 + 0.933190i \(0.382987\pi\)
\(354\) 13.5834 0.721949
\(355\) 0 0
\(356\) −1.72471 −0.0914096
\(357\) −2.30844 −0.122175
\(358\) 40.5510 2.14319
\(359\) −22.9543 −1.21148 −0.605741 0.795662i \(-0.707123\pi\)
−0.605741 + 0.795662i \(0.707123\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −30.6438 −1.61060
\(363\) 1.09835 0.0576484
\(364\) −2.17876 −0.114198
\(365\) 0 0
\(366\) −11.3112 −0.591246
\(367\) −5.77896 −0.301660 −0.150830 0.988560i \(-0.548195\pi\)
−0.150830 + 0.988560i \(0.548195\pi\)
\(368\) 0.729930 0.0380502
\(369\) 5.99710 0.312196
\(370\) 0 0
\(371\) −4.90967 −0.254898
\(372\) 11.0983 0.575418
\(373\) −32.2936 −1.67210 −0.836051 0.548652i \(-0.815141\pi\)
−0.836051 + 0.548652i \(0.815141\pi\)
\(374\) 9.14810 0.473037
\(375\) 0 0
\(376\) 0.935292 0.0482340
\(377\) 14.8194 0.763237
\(378\) −5.04390 −0.259430
\(379\) −12.2031 −0.626831 −0.313415 0.949616i \(-0.601473\pi\)
−0.313415 + 0.949616i \(0.601473\pi\)
\(380\) 0 0
\(381\) 19.4399 0.995938
\(382\) 29.4623 1.50742
\(383\) −11.5424 −0.589787 −0.294894 0.955530i \(-0.595284\pi\)
−0.294894 + 0.955530i \(0.595284\pi\)
\(384\) −3.03565 −0.154912
\(385\) 0 0
\(386\) −47.6919 −2.42746
\(387\) 6.72902 0.342055
\(388\) −38.0784 −1.93314
\(389\) −4.65210 −0.235871 −0.117935 0.993021i \(-0.537628\pi\)
−0.117935 + 0.993021i \(0.537628\pi\)
\(390\) 0 0
\(391\) 0.900450 0.0455377
\(392\) −2.35078 −0.118732
\(393\) 21.7493 1.09711
\(394\) 21.0797 1.06198
\(395\) 0 0
\(396\) 3.89181 0.195571
\(397\) 23.8285 1.19592 0.597960 0.801526i \(-0.295978\pi\)
0.597960 + 0.801526i \(0.295978\pi\)
\(398\) −21.5348 −1.07944
\(399\) −0.515282 −0.0257963
\(400\) 0 0
\(401\) −5.11101 −0.255232 −0.127616 0.991824i \(-0.540732\pi\)
−0.127616 + 0.991824i \(0.540732\pi\)
\(402\) −16.7590 −0.835861
\(403\) −9.96739 −0.496511
\(404\) 30.5930 1.52206
\(405\) 0 0
\(406\) −6.63293 −0.329187
\(407\) −7.22421 −0.358091
\(408\) −1.70609 −0.0844642
\(409\) −20.0164 −0.989748 −0.494874 0.868965i \(-0.664786\pi\)
−0.494874 + 0.868965i \(0.664786\pi\)
\(410\) 0 0
\(411\) −10.3027 −0.508196
\(412\) −32.0923 −1.58107
\(413\) 2.84128 0.139810
\(414\) 0.736167 0.0361806
\(415\) 0 0
\(416\) 17.3565 0.850972
\(417\) −15.0221 −0.735636
\(418\) 2.04201 0.0998779
\(419\) 34.3463 1.67793 0.838964 0.544188i \(-0.183162\pi\)
0.838964 + 0.544188i \(0.183162\pi\)
\(420\) 0 0
\(421\) 31.9075 1.55508 0.777539 0.628835i \(-0.216468\pi\)
0.777539 + 0.628835i \(0.216468\pi\)
\(422\) 15.0271 0.731506
\(423\) −4.83828 −0.235245
\(424\) −3.62858 −0.176220
\(425\) 0 0
\(426\) 35.7718 1.73315
\(427\) −2.36600 −0.114499
\(428\) −8.10006 −0.391531
\(429\) 2.35086 0.113501
\(430\) 0 0
\(431\) 38.2858 1.84416 0.922080 0.386998i \(-0.126488\pi\)
0.922080 + 0.386998i \(0.126488\pi\)
\(432\) 19.1205 0.919936
\(433\) 20.3681 0.978827 0.489413 0.872052i \(-0.337211\pi\)
0.489413 + 0.872052i \(0.337211\pi\)
\(434\) 4.46126 0.214147
\(435\) 0 0
\(436\) 1.14019 0.0546053
\(437\) 0.200995 0.00961492
\(438\) 34.9300 1.66902
\(439\) −24.9107 −1.18892 −0.594462 0.804123i \(-0.702635\pi\)
−0.594462 + 0.804123i \(0.702635\pi\)
\(440\) 0 0
\(441\) 12.1606 0.579078
\(442\) 19.5802 0.931335
\(443\) 22.7243 1.07966 0.539832 0.841773i \(-0.318488\pi\)
0.539832 + 0.841773i \(0.318488\pi\)
\(444\) 17.2167 0.817068
\(445\) 0 0
\(446\) −29.4165 −1.39291
\(447\) 10.2619 0.485372
\(448\) −4.36106 −0.206041
\(449\) −41.1163 −1.94040 −0.970198 0.242313i \(-0.922094\pi\)
−0.970198 + 0.242313i \(0.922094\pi\)
\(450\) 0 0
\(451\) 3.34355 0.157442
\(452\) −20.9001 −0.983057
\(453\) 8.48071 0.398459
\(454\) 60.6761 2.84767
\(455\) 0 0
\(456\) −0.380828 −0.0178339
\(457\) −6.07340 −0.284101 −0.142051 0.989859i \(-0.545370\pi\)
−0.142051 + 0.989859i \(0.545370\pi\)
\(458\) −22.4910 −1.05094
\(459\) 23.5873 1.10096
\(460\) 0 0
\(461\) −6.74719 −0.314248 −0.157124 0.987579i \(-0.550222\pi\)
−0.157124 + 0.987579i \(0.550222\pi\)
\(462\) −1.05221 −0.0489532
\(463\) −37.3772 −1.73707 −0.868534 0.495630i \(-0.834937\pi\)
−0.868534 + 0.495630i \(0.834937\pi\)
\(464\) 25.1442 1.16729
\(465\) 0 0
\(466\) −19.4744 −0.902135
\(467\) −28.4403 −1.31606 −0.658030 0.752992i \(-0.728610\pi\)
−0.658030 + 0.752992i \(0.728610\pi\)
\(468\) 8.32987 0.385048
\(469\) −3.50553 −0.161870
\(470\) 0 0
\(471\) −3.14407 −0.144871
\(472\) 2.09990 0.0966558
\(473\) 3.75162 0.172500
\(474\) 13.1919 0.605923
\(475\) 0 0
\(476\) −4.56034 −0.209023
\(477\) 18.7707 0.859453
\(478\) 8.22886 0.376379
\(479\) 35.7235 1.63225 0.816124 0.577877i \(-0.196119\pi\)
0.816124 + 0.577877i \(0.196119\pi\)
\(480\) 0 0
\(481\) −15.4624 −0.705024
\(482\) 12.3870 0.564210
\(483\) −0.103569 −0.00471257
\(484\) 2.16980 0.0986272
\(485\) 0 0
\(486\) 31.3524 1.42217
\(487\) 9.68607 0.438918 0.219459 0.975622i \(-0.429571\pi\)
0.219459 + 0.975622i \(0.429571\pi\)
\(488\) −1.74864 −0.0791570
\(489\) −11.9766 −0.541601
\(490\) 0 0
\(491\) −8.75793 −0.395240 −0.197620 0.980279i \(-0.563321\pi\)
−0.197620 + 0.980279i \(0.563321\pi\)
\(492\) −7.96834 −0.359241
\(493\) 31.0182 1.39699
\(494\) 4.37063 0.196644
\(495\) 0 0
\(496\) −16.9118 −0.759363
\(497\) 7.48252 0.335637
\(498\) −29.1118 −1.30453
\(499\) 15.3711 0.688104 0.344052 0.938951i \(-0.388200\pi\)
0.344052 + 0.938951i \(0.388200\pi\)
\(500\) 0 0
\(501\) 17.2618 0.771198
\(502\) −42.3712 −1.89112
\(503\) −22.5117 −1.00375 −0.501874 0.864941i \(-0.667356\pi\)
−0.501874 + 0.864941i \(0.667356\pi\)
\(504\) −0.291760 −0.0129960
\(505\) 0 0
\(506\) 0.410434 0.0182460
\(507\) −9.24686 −0.410667
\(508\) 38.4037 1.70389
\(509\) −30.6457 −1.35835 −0.679173 0.733978i \(-0.737661\pi\)
−0.679173 + 0.733978i \(0.737661\pi\)
\(510\) 0 0
\(511\) 7.30642 0.323217
\(512\) 31.9673 1.41277
\(513\) 5.26508 0.232459
\(514\) −43.6381 −1.92480
\(515\) 0 0
\(516\) −8.94084 −0.393599
\(517\) −2.69748 −0.118635
\(518\) 6.92073 0.304079
\(519\) −4.22071 −0.185269
\(520\) 0 0
\(521\) 6.68929 0.293063 0.146532 0.989206i \(-0.453189\pi\)
0.146532 + 0.989206i \(0.453189\pi\)
\(522\) 25.3591 1.10994
\(523\) −5.06543 −0.221496 −0.110748 0.993849i \(-0.535325\pi\)
−0.110748 + 0.993849i \(0.535325\pi\)
\(524\) 42.9659 1.87697
\(525\) 0 0
\(526\) 4.30646 0.187771
\(527\) −20.8626 −0.908790
\(528\) 3.98874 0.173587
\(529\) −22.9596 −0.998244
\(530\) 0 0
\(531\) −10.8628 −0.471407
\(532\) −1.01794 −0.0441334
\(533\) 7.15640 0.309978
\(534\) −1.78277 −0.0771481
\(535\) 0 0
\(536\) −2.59082 −0.111907
\(537\) 21.8114 0.941233
\(538\) 41.4733 1.78804
\(539\) 6.77991 0.292031
\(540\) 0 0
\(541\) −45.1702 −1.94202 −0.971010 0.239038i \(-0.923168\pi\)
−0.971010 + 0.239038i \(0.923168\pi\)
\(542\) −5.75568 −0.247227
\(543\) −16.4826 −0.707335
\(544\) 36.3286 1.55758
\(545\) 0 0
\(546\) −2.25210 −0.0963812
\(547\) 34.5604 1.47769 0.738847 0.673873i \(-0.235371\pi\)
0.738847 + 0.673873i \(0.235371\pi\)
\(548\) −20.3531 −0.869442
\(549\) 9.04572 0.386062
\(550\) 0 0
\(551\) 6.92379 0.294963
\(552\) −0.0765447 −0.00325796
\(553\) 2.75939 0.117341
\(554\) 34.0450 1.44643
\(555\) 0 0
\(556\) −29.6763 −1.25856
\(557\) −26.7387 −1.13296 −0.566478 0.824077i \(-0.691695\pi\)
−0.566478 + 0.824077i \(0.691695\pi\)
\(558\) −17.0563 −0.722052
\(559\) 8.02981 0.339625
\(560\) 0 0
\(561\) 4.92055 0.207746
\(562\) −57.0294 −2.40564
\(563\) 3.43066 0.144585 0.0722925 0.997383i \(-0.476968\pi\)
0.0722925 + 0.997383i \(0.476968\pi\)
\(564\) 6.42862 0.270694
\(565\) 0 0
\(566\) 9.97510 0.419285
\(567\) −0.188598 −0.00792036
\(568\) 5.53009 0.232037
\(569\) 6.20817 0.260260 0.130130 0.991497i \(-0.458461\pi\)
0.130130 + 0.991497i \(0.458461\pi\)
\(570\) 0 0
\(571\) −4.29662 −0.179808 −0.0899040 0.995950i \(-0.528656\pi\)
−0.0899040 + 0.995950i \(0.528656\pi\)
\(572\) 4.64414 0.194181
\(573\) 15.8471 0.662022
\(574\) −3.20310 −0.133695
\(575\) 0 0
\(576\) 16.6733 0.694719
\(577\) −24.6963 −1.02812 −0.514060 0.857754i \(-0.671859\pi\)
−0.514060 + 0.857754i \(0.671859\pi\)
\(578\) 6.26891 0.260752
\(579\) −25.6524 −1.06608
\(580\) 0 0
\(581\) −6.08941 −0.252631
\(582\) −39.3602 −1.63153
\(583\) 10.4652 0.433425
\(584\) 5.39994 0.223451
\(585\) 0 0
\(586\) −1.32360 −0.0546774
\(587\) −9.31384 −0.384423 −0.192212 0.981353i \(-0.561566\pi\)
−0.192212 + 0.981353i \(0.561566\pi\)
\(588\) −16.1578 −0.666338
\(589\) −4.65688 −0.191884
\(590\) 0 0
\(591\) 11.3383 0.466394
\(592\) −26.2352 −1.07826
\(593\) 25.8934 1.06331 0.531656 0.846960i \(-0.321570\pi\)
0.531656 + 0.846960i \(0.321570\pi\)
\(594\) 10.7513 0.441133
\(595\) 0 0
\(596\) 20.2725 0.830394
\(597\) −11.5831 −0.474064
\(598\) 0.878476 0.0359236
\(599\) 24.8277 1.01443 0.507215 0.861819i \(-0.330675\pi\)
0.507215 + 0.861819i \(0.330675\pi\)
\(600\) 0 0
\(601\) 9.59884 0.391545 0.195772 0.980649i \(-0.437279\pi\)
0.195772 + 0.980649i \(0.437279\pi\)
\(602\) −3.59402 −0.146481
\(603\) 13.4024 0.545787
\(604\) 16.7537 0.681699
\(605\) 0 0
\(606\) 31.6229 1.28459
\(607\) 11.9356 0.484452 0.242226 0.970220i \(-0.422123\pi\)
0.242226 + 0.970220i \(0.422123\pi\)
\(608\) 8.10916 0.328870
\(609\) −3.56770 −0.144570
\(610\) 0 0
\(611\) −5.77357 −0.233574
\(612\) 17.4351 0.704774
\(613\) −9.27813 −0.374740 −0.187370 0.982289i \(-0.559996\pi\)
−0.187370 + 0.982289i \(0.559996\pi\)
\(614\) −24.1111 −0.973043
\(615\) 0 0
\(616\) −0.162665 −0.00655394
\(617\) −4.13222 −0.166357 −0.0831784 0.996535i \(-0.526507\pi\)
−0.0831784 + 0.996535i \(0.526507\pi\)
\(618\) −33.1726 −1.33440
\(619\) −27.0790 −1.08840 −0.544198 0.838957i \(-0.683166\pi\)
−0.544198 + 0.838957i \(0.683166\pi\)
\(620\) 0 0
\(621\) 1.05826 0.0424664
\(622\) −29.4484 −1.18077
\(623\) −0.372908 −0.0149403
\(624\) 8.53732 0.341766
\(625\) 0 0
\(626\) 11.2175 0.448342
\(627\) 1.09835 0.0438638
\(628\) −6.21113 −0.247851
\(629\) −32.3641 −1.29044
\(630\) 0 0
\(631\) −2.93731 −0.116933 −0.0584663 0.998289i \(-0.518621\pi\)
−0.0584663 + 0.998289i \(0.518621\pi\)
\(632\) 2.03938 0.0811220
\(633\) 8.08271 0.321259
\(634\) −27.5145 −1.09274
\(635\) 0 0
\(636\) −24.9407 −0.988962
\(637\) 14.5114 0.574963
\(638\) 14.1384 0.559746
\(639\) −28.6072 −1.13169
\(640\) 0 0
\(641\) −4.32999 −0.171024 −0.0855122 0.996337i \(-0.527253\pi\)
−0.0855122 + 0.996337i \(0.527253\pi\)
\(642\) −8.37273 −0.330445
\(643\) 5.02416 0.198134 0.0990668 0.995081i \(-0.468414\pi\)
0.0990668 + 0.995081i \(0.468414\pi\)
\(644\) −0.204602 −0.00806245
\(645\) 0 0
\(646\) 9.14810 0.359927
\(647\) −10.7629 −0.423133 −0.211566 0.977364i \(-0.567856\pi\)
−0.211566 + 0.977364i \(0.567856\pi\)
\(648\) −0.139387 −0.00547562
\(649\) −6.05634 −0.237732
\(650\) 0 0
\(651\) 2.39961 0.0940480
\(652\) −23.6599 −0.926593
\(653\) 35.3311 1.38261 0.691307 0.722561i \(-0.257035\pi\)
0.691307 + 0.722561i \(0.257035\pi\)
\(654\) 1.17857 0.0460858
\(655\) 0 0
\(656\) 12.1424 0.474080
\(657\) −27.9340 −1.08981
\(658\) 2.58416 0.100741
\(659\) −18.0616 −0.703580 −0.351790 0.936079i \(-0.614427\pi\)
−0.351790 + 0.936079i \(0.614427\pi\)
\(660\) 0 0
\(661\) 10.6516 0.414300 0.207150 0.978309i \(-0.433581\pi\)
0.207150 + 0.978309i \(0.433581\pi\)
\(662\) 21.7282 0.844489
\(663\) 10.5317 0.409019
\(664\) −4.50049 −0.174653
\(665\) 0 0
\(666\) −26.4594 −1.02528
\(667\) 1.39165 0.0538849
\(668\) 34.1007 1.31940
\(669\) −15.8224 −0.611731
\(670\) 0 0
\(671\) 5.04325 0.194693
\(672\) −4.17850 −0.161189
\(673\) −7.05682 −0.272020 −0.136010 0.990707i \(-0.543428\pi\)
−0.136010 + 0.990707i \(0.543428\pi\)
\(674\) 6.11135 0.235401
\(675\) 0 0
\(676\) −18.2672 −0.702586
\(677\) 17.0051 0.653560 0.326780 0.945100i \(-0.394036\pi\)
0.326780 + 0.945100i \(0.394036\pi\)
\(678\) −21.6036 −0.829682
\(679\) −8.23312 −0.315958
\(680\) 0 0
\(681\) 32.6363 1.25062
\(682\) −9.50939 −0.364133
\(683\) 32.7782 1.25422 0.627112 0.778929i \(-0.284237\pi\)
0.627112 + 0.778929i \(0.284237\pi\)
\(684\) 3.89181 0.148807
\(685\) 0 0
\(686\) −13.2010 −0.504018
\(687\) −12.0974 −0.461544
\(688\) 13.6243 0.519421
\(689\) 22.3993 0.853346
\(690\) 0 0
\(691\) 23.4551 0.892274 0.446137 0.894965i \(-0.352799\pi\)
0.446137 + 0.894965i \(0.352799\pi\)
\(692\) −8.33805 −0.316965
\(693\) 0.841467 0.0319647
\(694\) −3.22185 −0.122300
\(695\) 0 0
\(696\) −2.63677 −0.0999466
\(697\) 14.9790 0.567369
\(698\) 26.4950 1.00285
\(699\) −10.4748 −0.396195
\(700\) 0 0
\(701\) 6.40155 0.241783 0.120892 0.992666i \(-0.461425\pi\)
0.120892 + 0.992666i \(0.461425\pi\)
\(702\) 23.0117 0.868520
\(703\) −7.22421 −0.272466
\(704\) 9.29582 0.350349
\(705\) 0 0
\(706\) 27.5761 1.03784
\(707\) 6.61466 0.248770
\(708\) 14.4334 0.542442
\(709\) 16.2437 0.610044 0.305022 0.952345i \(-0.401336\pi\)
0.305022 + 0.952345i \(0.401336\pi\)
\(710\) 0 0
\(711\) −10.5497 −0.395646
\(712\) −0.275605 −0.0103287
\(713\) −0.936012 −0.0350539
\(714\) −4.71385 −0.176411
\(715\) 0 0
\(716\) 43.0887 1.61030
\(717\) 4.42611 0.165296
\(718\) −46.8729 −1.74928
\(719\) −2.10857 −0.0786363 −0.0393181 0.999227i \(-0.512519\pi\)
−0.0393181 + 0.999227i \(0.512519\pi\)
\(720\) 0 0
\(721\) −6.93882 −0.258415
\(722\) 2.04201 0.0759957
\(723\) 6.66266 0.247787
\(724\) −32.5615 −1.21014
\(725\) 0 0
\(726\) 2.24284 0.0832395
\(727\) −1.59364 −0.0591048 −0.0295524 0.999563i \(-0.509408\pi\)
−0.0295524 + 0.999563i \(0.509408\pi\)
\(728\) −0.348160 −0.0129037
\(729\) 18.0697 0.669249
\(730\) 0 0
\(731\) 16.8071 0.621632
\(732\) −12.0191 −0.444237
\(733\) −10.6588 −0.393692 −0.196846 0.980434i \(-0.563070\pi\)
−0.196846 + 0.980434i \(0.563070\pi\)
\(734\) −11.8007 −0.435572
\(735\) 0 0
\(736\) 1.62990 0.0600791
\(737\) 7.47221 0.275242
\(738\) 12.2461 0.450786
\(739\) 3.32501 0.122313 0.0611563 0.998128i \(-0.480521\pi\)
0.0611563 + 0.998128i \(0.480521\pi\)
\(740\) 0 0
\(741\) 2.35086 0.0863610
\(742\) −10.0256 −0.368051
\(743\) 51.3600 1.88421 0.942107 0.335311i \(-0.108841\pi\)
0.942107 + 0.335311i \(0.108841\pi\)
\(744\) 1.77347 0.0650187
\(745\) 0 0
\(746\) −65.9439 −2.41438
\(747\) 23.2811 0.851812
\(748\) 9.72059 0.355420
\(749\) −1.75135 −0.0639930
\(750\) 0 0
\(751\) −22.1565 −0.808502 −0.404251 0.914648i \(-0.632468\pi\)
−0.404251 + 0.914648i \(0.632468\pi\)
\(752\) −9.79610 −0.357227
\(753\) −22.7905 −0.830532
\(754\) 30.2613 1.10205
\(755\) 0 0
\(756\) −5.35955 −0.194925
\(757\) −3.94366 −0.143335 −0.0716674 0.997429i \(-0.522832\pi\)
−0.0716674 + 0.997429i \(0.522832\pi\)
\(758\) −24.9188 −0.905092
\(759\) 0.220763 0.00801320
\(760\) 0 0
\(761\) −48.5719 −1.76073 −0.880366 0.474296i \(-0.842703\pi\)
−0.880366 + 0.474296i \(0.842703\pi\)
\(762\) 39.6965 1.43805
\(763\) 0.246526 0.00892484
\(764\) 31.3061 1.13261
\(765\) 0 0
\(766\) −23.5696 −0.851604
\(767\) −12.9627 −0.468057
\(768\) 14.2213 0.513167
\(769\) −34.5865 −1.24722 −0.623611 0.781735i \(-0.714335\pi\)
−0.623611 + 0.781735i \(0.714335\pi\)
\(770\) 0 0
\(771\) −23.4720 −0.845322
\(772\) −50.6765 −1.82389
\(773\) −9.98101 −0.358992 −0.179496 0.983759i \(-0.557447\pi\)
−0.179496 + 0.983759i \(0.557447\pi\)
\(774\) 13.7407 0.493900
\(775\) 0 0
\(776\) −6.08483 −0.218433
\(777\) 3.72250 0.133544
\(778\) −9.49962 −0.340578
\(779\) 3.34355 0.119795
\(780\) 0 0
\(781\) −15.9494 −0.570713
\(782\) 1.83873 0.0657527
\(783\) 36.4543 1.30277
\(784\) 24.6217 0.879347
\(785\) 0 0
\(786\) 44.4122 1.58413
\(787\) −8.47966 −0.302267 −0.151134 0.988513i \(-0.548292\pi\)
−0.151134 + 0.988513i \(0.548292\pi\)
\(788\) 22.3988 0.797926
\(789\) 2.31635 0.0824641
\(790\) 0 0
\(791\) −4.51890 −0.160674
\(792\) 0.621901 0.0220983
\(793\) 10.7944 0.383319
\(794\) 48.6581 1.72681
\(795\) 0 0
\(796\) −22.8825 −0.811047
\(797\) −50.8181 −1.80007 −0.900034 0.435820i \(-0.856459\pi\)
−0.900034 + 0.435820i \(0.856459\pi\)
\(798\) −1.05221 −0.0372478
\(799\) −12.0846 −0.427522
\(800\) 0 0
\(801\) 1.42571 0.0503749
\(802\) −10.4367 −0.368534
\(803\) −15.5740 −0.549595
\(804\) −17.8077 −0.628030
\(805\) 0 0
\(806\) −20.3535 −0.716921
\(807\) 22.3075 0.785262
\(808\) 4.88869 0.171983
\(809\) 26.2792 0.923926 0.461963 0.886899i \(-0.347145\pi\)
0.461963 + 0.886899i \(0.347145\pi\)
\(810\) 0 0
\(811\) 44.2582 1.55412 0.777058 0.629429i \(-0.216711\pi\)
0.777058 + 0.629429i \(0.216711\pi\)
\(812\) −7.04802 −0.247337
\(813\) −3.09585 −0.108576
\(814\) −14.7519 −0.517054
\(815\) 0 0
\(816\) 17.8693 0.625552
\(817\) 3.75162 0.131253
\(818\) −40.8737 −1.42912
\(819\) 1.80104 0.0629334
\(820\) 0 0
\(821\) −45.4098 −1.58481 −0.792406 0.609994i \(-0.791172\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(822\) −21.0382 −0.733793
\(823\) −20.0801 −0.699950 −0.349975 0.936759i \(-0.613810\pi\)
−0.349975 + 0.936759i \(0.613810\pi\)
\(824\) −5.12826 −0.178652
\(825\) 0 0
\(826\) 5.80192 0.201875
\(827\) −2.70217 −0.0939637 −0.0469818 0.998896i \(-0.514960\pi\)
−0.0469818 + 0.998896i \(0.514960\pi\)
\(828\) 0.782237 0.0271846
\(829\) 43.8322 1.52236 0.761178 0.648543i \(-0.224621\pi\)
0.761178 + 0.648543i \(0.224621\pi\)
\(830\) 0 0
\(831\) 18.3120 0.635237
\(832\) 19.8964 0.689783
\(833\) 30.3737 1.05238
\(834\) −30.6753 −1.06220
\(835\) 0 0
\(836\) 2.16980 0.0750440
\(837\) −24.5189 −0.847495
\(838\) 70.1354 2.42279
\(839\) 12.7989 0.441866 0.220933 0.975289i \(-0.429090\pi\)
0.220933 + 0.975289i \(0.429090\pi\)
\(840\) 0 0
\(841\) 18.9388 0.653062
\(842\) 65.1554 2.24540
\(843\) −30.6748 −1.05650
\(844\) 15.9675 0.549623
\(845\) 0 0
\(846\) −9.87981 −0.339675
\(847\) 0.469142 0.0161199
\(848\) 38.0052 1.30510
\(849\) 5.36537 0.184139
\(850\) 0 0
\(851\) −1.45203 −0.0497750
\(852\) 38.0104 1.30222
\(853\) 30.8728 1.05707 0.528533 0.848913i \(-0.322742\pi\)
0.528533 + 0.848913i \(0.322742\pi\)
\(854\) −4.83139 −0.165327
\(855\) 0 0
\(856\) −1.29437 −0.0442406
\(857\) 18.5582 0.633935 0.316967 0.948436i \(-0.397335\pi\)
0.316967 + 0.948436i \(0.397335\pi\)
\(858\) 4.80047 0.163885
\(859\) −0.760628 −0.0259523 −0.0129762 0.999916i \(-0.504131\pi\)
−0.0129762 + 0.999916i \(0.504131\pi\)
\(860\) 0 0
\(861\) −1.72287 −0.0587153
\(862\) 78.1799 2.66282
\(863\) 46.7268 1.59060 0.795300 0.606216i \(-0.207313\pi\)
0.795300 + 0.606216i \(0.207313\pi\)
\(864\) 42.6954 1.45253
\(865\) 0 0
\(866\) 41.5918 1.41335
\(867\) 3.37190 0.114516
\(868\) 4.74044 0.160901
\(869\) −5.88178 −0.199526
\(870\) 0 0
\(871\) 15.9932 0.541909
\(872\) 0.182200 0.00617006
\(873\) 31.4769 1.06533
\(874\) 0.410434 0.0138831
\(875\) 0 0
\(876\) 37.1159 1.25403
\(877\) −38.0871 −1.28611 −0.643056 0.765819i \(-0.722334\pi\)
−0.643056 + 0.765819i \(0.722334\pi\)
\(878\) −50.8679 −1.71671
\(879\) −0.711934 −0.0240129
\(880\) 0 0
\(881\) −14.4600 −0.487170 −0.243585 0.969880i \(-0.578323\pi\)
−0.243585 + 0.969880i \(0.578323\pi\)
\(882\) 24.8321 0.836141
\(883\) −0.159279 −0.00536017 −0.00268008 0.999996i \(-0.500853\pi\)
−0.00268008 + 0.999996i \(0.500853\pi\)
\(884\) 20.8055 0.699766
\(885\) 0 0
\(886\) 46.4032 1.55895
\(887\) 9.95048 0.334104 0.167052 0.985948i \(-0.446575\pi\)
0.167052 + 0.985948i \(0.446575\pi\)
\(888\) 2.75118 0.0923236
\(889\) 8.30345 0.278489
\(890\) 0 0
\(891\) 0.402006 0.0134677
\(892\) −31.2574 −1.04657
\(893\) −2.69748 −0.0902678
\(894\) 20.9549 0.700837
\(895\) 0 0
\(896\) −1.29663 −0.0433173
\(897\) 0.472512 0.0157767
\(898\) −83.9597 −2.80177
\(899\) −32.2433 −1.07537
\(900\) 0 0
\(901\) 46.8837 1.56192
\(902\) 6.82756 0.227333
\(903\) −1.93314 −0.0643309
\(904\) −3.33978 −0.111079
\(905\) 0 0
\(906\) 17.3177 0.575342
\(907\) −51.7590 −1.71863 −0.859315 0.511448i \(-0.829109\pi\)
−0.859315 + 0.511448i \(0.829109\pi\)
\(908\) 64.4732 2.13962
\(909\) −25.2893 −0.838792
\(910\) 0 0
\(911\) −29.1062 −0.964332 −0.482166 0.876080i \(-0.660150\pi\)
−0.482166 + 0.876080i \(0.660150\pi\)
\(912\) 3.98874 0.132080
\(913\) 12.9799 0.429572
\(914\) −12.4019 −0.410219
\(915\) 0 0
\(916\) −23.8985 −0.789629
\(917\) 9.28986 0.306778
\(918\) 48.1655 1.58970
\(919\) −12.7499 −0.420581 −0.210290 0.977639i \(-0.567441\pi\)
−0.210290 + 0.977639i \(0.567441\pi\)
\(920\) 0 0
\(921\) −12.9688 −0.427336
\(922\) −13.7778 −0.453748
\(923\) −34.1373 −1.12364
\(924\) −1.11806 −0.0367814
\(925\) 0 0
\(926\) −76.3246 −2.50818
\(927\) 26.5286 0.871313
\(928\) 56.1461 1.84309
\(929\) 2.50821 0.0822916 0.0411458 0.999153i \(-0.486899\pi\)
0.0411458 + 0.999153i \(0.486899\pi\)
\(930\) 0 0
\(931\) 6.77991 0.222202
\(932\) −20.6931 −0.677826
\(933\) −15.8396 −0.518566
\(934\) −58.0753 −1.90028
\(935\) 0 0
\(936\) 1.33109 0.0435081
\(937\) −47.9802 −1.56744 −0.783722 0.621112i \(-0.786681\pi\)
−0.783722 + 0.621112i \(0.786681\pi\)
\(938\) −7.15832 −0.233727
\(939\) 6.03364 0.196900
\(940\) 0 0
\(941\) −44.4858 −1.45020 −0.725098 0.688645i \(-0.758206\pi\)
−0.725098 + 0.688645i \(0.758206\pi\)
\(942\) −6.42021 −0.209182
\(943\) 0.672039 0.0218846
\(944\) −21.9940 −0.715845
\(945\) 0 0
\(946\) 7.66084 0.249075
\(947\) −6.70041 −0.217734 −0.108867 0.994056i \(-0.534722\pi\)
−0.108867 + 0.994056i \(0.534722\pi\)
\(948\) 14.0174 0.455265
\(949\) −33.3339 −1.08207
\(950\) 0 0
\(951\) −14.7994 −0.479904
\(952\) −0.728730 −0.0236183
\(953\) 57.1810 1.85227 0.926136 0.377189i \(-0.123109\pi\)
0.926136 + 0.377189i \(0.123109\pi\)
\(954\) 38.3300 1.24098
\(955\) 0 0
\(956\) 8.74382 0.282796
\(957\) 7.60473 0.245826
\(958\) 72.9477 2.35683
\(959\) −4.40064 −0.142104
\(960\) 0 0
\(961\) −9.31344 −0.300434
\(962\) −31.5743 −1.01800
\(963\) 6.69579 0.215769
\(964\) 13.1621 0.423924
\(965\) 0 0
\(966\) −0.211489 −0.00680456
\(967\) 5.24741 0.168745 0.0843727 0.996434i \(-0.473111\pi\)
0.0843727 + 0.996434i \(0.473111\pi\)
\(968\) 0.346728 0.0111443
\(969\) 4.92055 0.158071
\(970\) 0 0
\(971\) −27.4776 −0.881799 −0.440899 0.897557i \(-0.645340\pi\)
−0.440899 + 0.897557i \(0.645340\pi\)
\(972\) 33.3144 1.06856
\(973\) −6.41645 −0.205702
\(974\) 19.7790 0.633761
\(975\) 0 0
\(976\) 18.3149 0.586247
\(977\) −42.6475 −1.36442 −0.682208 0.731158i \(-0.738980\pi\)
−0.682208 + 0.731158i \(0.738980\pi\)
\(978\) −24.4563 −0.782028
\(979\) 0.794873 0.0254043
\(980\) 0 0
\(981\) −0.942521 −0.0300924
\(982\) −17.8838 −0.570694
\(983\) 48.8786 1.55899 0.779493 0.626411i \(-0.215477\pi\)
0.779493 + 0.626411i \(0.215477\pi\)
\(984\) −1.27332 −0.0405919
\(985\) 0 0
\(986\) 63.3395 2.01714
\(987\) 1.38996 0.0442430
\(988\) 4.64414 0.147750
\(989\) 0.754059 0.0239777
\(990\) 0 0
\(991\) −60.2922 −1.91524 −0.957622 0.288027i \(-0.907001\pi\)
−0.957622 + 0.288027i \(0.907001\pi\)
\(992\) −37.7634 −1.19899
\(993\) 11.6871 0.370878
\(994\) 15.2794 0.484632
\(995\) 0 0
\(996\) −30.9336 −0.980169
\(997\) 6.50314 0.205957 0.102978 0.994684i \(-0.467163\pi\)
0.102978 + 0.994684i \(0.467163\pi\)
\(998\) 31.3879 0.993565
\(999\) −38.0360 −1.20341
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 5225.2.a.k.1.5 6
5.4 even 2 1045.2.a.g.1.2 6
15.14 odd 2 9405.2.a.w.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.g.1.2 6 5.4 even 2
5225.2.a.k.1.5 6 1.1 even 1 trivial
9405.2.a.w.1.5 6 15.14 odd 2