Properties

Label 6016.2.a.n.1.5
Level $6016$
Weight $2$
Character 6016.1
Self dual yes
Analytic conductor $48.038$
Analytic rank $0$
Dimension $13$
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] = [6016,2,Mod(1,6016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6016, 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("6016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6016 = 2^{7} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6016.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: \(48.0380018560\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} - 6230 x^{4} + 1488 x^{3} + 4720 x^{2} - 1440 x - 832 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.73608\) of defining polynomial
Character \(\chi\) \(=\) 6016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73608 q^{3} -3.07388 q^{5} +4.60716 q^{7} +0.0139658 q^{9} +O(q^{10})\) \(q-1.73608 q^{3} -3.07388 q^{5} +4.60716 q^{7} +0.0139658 q^{9} +3.25201 q^{11} -3.13174 q^{13} +5.33650 q^{15} +1.98403 q^{17} -3.15860 q^{19} -7.99840 q^{21} +8.21953 q^{23} +4.44876 q^{25} +5.18399 q^{27} -1.32121 q^{29} -0.155177 q^{31} -5.64575 q^{33} -14.1619 q^{35} -0.122812 q^{37} +5.43694 q^{39} -6.73192 q^{41} +10.2888 q^{43} -0.0429292 q^{45} -1.00000 q^{47} +14.2260 q^{49} -3.44443 q^{51} +9.44846 q^{53} -9.99631 q^{55} +5.48358 q^{57} -14.5361 q^{59} -0.165968 q^{61} +0.0643427 q^{63} +9.62660 q^{65} -0.709327 q^{67} -14.2697 q^{69} -6.43089 q^{71} -13.5046 q^{73} -7.72340 q^{75} +14.9826 q^{77} -2.71846 q^{79} -9.04170 q^{81} +15.2501 q^{83} -6.09867 q^{85} +2.29372 q^{87} +11.6474 q^{89} -14.4284 q^{91} +0.269400 q^{93} +9.70918 q^{95} +2.82193 q^{97} +0.0454170 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 4 q^{3} + 6 q^{5} - 2 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 4 q^{3} + 6 q^{5} - 2 q^{7} + 21 q^{9} - 10 q^{11} + 4 q^{13} + 14 q^{15} + 10 q^{17} - 8 q^{19} + 10 q^{21} + 18 q^{23} + 23 q^{25} - 16 q^{27} + 14 q^{29} + 4 q^{31} + 14 q^{33} - 14 q^{35} + 16 q^{37} + 12 q^{39} + 10 q^{41} - 12 q^{43} + 10 q^{45} - 13 q^{47} + 9 q^{49} - 22 q^{51} + 26 q^{53} - 2 q^{55} + 20 q^{57} - 30 q^{59} + 18 q^{61} + 12 q^{63} - 4 q^{65} - 4 q^{67} + 2 q^{69} + 36 q^{71} + 10 q^{73} - 38 q^{75} + 42 q^{77} + 21 q^{81} - 12 q^{83} + 4 q^{85} + 6 q^{87} + 50 q^{89} + 4 q^{91} + 52 q^{93} + 8 q^{95} - 10 q^{97} - 34 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.73608 −1.00232 −0.501162 0.865353i \(-0.667094\pi\)
−0.501162 + 0.865353i \(0.667094\pi\)
\(4\) 0 0
\(5\) −3.07388 −1.37468 −0.687341 0.726335i \(-0.741222\pi\)
−0.687341 + 0.726335i \(0.741222\pi\)
\(6\) 0 0
\(7\) 4.60716 1.74134 0.870672 0.491864i \(-0.163684\pi\)
0.870672 + 0.491864i \(0.163684\pi\)
\(8\) 0 0
\(9\) 0.0139658 0.00465526
\(10\) 0 0
\(11\) 3.25201 0.980519 0.490260 0.871576i \(-0.336902\pi\)
0.490260 + 0.871576i \(0.336902\pi\)
\(12\) 0 0
\(13\) −3.13174 −0.868588 −0.434294 0.900771i \(-0.643002\pi\)
−0.434294 + 0.900771i \(0.643002\pi\)
\(14\) 0 0
\(15\) 5.33650 1.37788
\(16\) 0 0
\(17\) 1.98403 0.481197 0.240599 0.970625i \(-0.422656\pi\)
0.240599 + 0.970625i \(0.422656\pi\)
\(18\) 0 0
\(19\) −3.15860 −0.724634 −0.362317 0.932055i \(-0.618014\pi\)
−0.362317 + 0.932055i \(0.618014\pi\)
\(20\) 0 0
\(21\) −7.99840 −1.74539
\(22\) 0 0
\(23\) 8.21953 1.71389 0.856946 0.515407i \(-0.172359\pi\)
0.856946 + 0.515407i \(0.172359\pi\)
\(24\) 0 0
\(25\) 4.44876 0.889753
\(26\) 0 0
\(27\) 5.18399 0.997659
\(28\) 0 0
\(29\) −1.32121 −0.245343 −0.122671 0.992447i \(-0.539146\pi\)
−0.122671 + 0.992447i \(0.539146\pi\)
\(30\) 0 0
\(31\) −0.155177 −0.0278706 −0.0139353 0.999903i \(-0.504436\pi\)
−0.0139353 + 0.999903i \(0.504436\pi\)
\(32\) 0 0
\(33\) −5.64575 −0.982799
\(34\) 0 0
\(35\) −14.1619 −2.39380
\(36\) 0 0
\(37\) −0.122812 −0.0201902 −0.0100951 0.999949i \(-0.503213\pi\)
−0.0100951 + 0.999949i \(0.503213\pi\)
\(38\) 0 0
\(39\) 5.43694 0.870607
\(40\) 0 0
\(41\) −6.73192 −1.05135 −0.525675 0.850686i \(-0.676187\pi\)
−0.525675 + 0.850686i \(0.676187\pi\)
\(42\) 0 0
\(43\) 10.2888 1.56903 0.784514 0.620111i \(-0.212912\pi\)
0.784514 + 0.620111i \(0.212912\pi\)
\(44\) 0 0
\(45\) −0.0429292 −0.00639951
\(46\) 0 0
\(47\) −1.00000 −0.145865
\(48\) 0 0
\(49\) 14.2260 2.03228
\(50\) 0 0
\(51\) −3.44443 −0.482316
\(52\) 0 0
\(53\) 9.44846 1.29785 0.648923 0.760854i \(-0.275220\pi\)
0.648923 + 0.760854i \(0.275220\pi\)
\(54\) 0 0
\(55\) −9.99631 −1.34790
\(56\) 0 0
\(57\) 5.48358 0.726318
\(58\) 0 0
\(59\) −14.5361 −1.89244 −0.946222 0.323517i \(-0.895135\pi\)
−0.946222 + 0.323517i \(0.895135\pi\)
\(60\) 0 0
\(61\) −0.165968 −0.0212500 −0.0106250 0.999944i \(-0.503382\pi\)
−0.0106250 + 0.999944i \(0.503382\pi\)
\(62\) 0 0
\(63\) 0.0643427 0.00810642
\(64\) 0 0
\(65\) 9.62660 1.19403
\(66\) 0 0
\(67\) −0.709327 −0.0866581 −0.0433291 0.999061i \(-0.513796\pi\)
−0.0433291 + 0.999061i \(0.513796\pi\)
\(68\) 0 0
\(69\) −14.2697 −1.71788
\(70\) 0 0
\(71\) −6.43089 −0.763206 −0.381603 0.924326i \(-0.624628\pi\)
−0.381603 + 0.924326i \(0.624628\pi\)
\(72\) 0 0
\(73\) −13.5046 −1.58060 −0.790299 0.612722i \(-0.790075\pi\)
−0.790299 + 0.612722i \(0.790075\pi\)
\(74\) 0 0
\(75\) −7.72340 −0.891821
\(76\) 0 0
\(77\) 14.9826 1.70742
\(78\) 0 0
\(79\) −2.71846 −0.305851 −0.152925 0.988238i \(-0.548869\pi\)
−0.152925 + 0.988238i \(0.548869\pi\)
\(80\) 0 0
\(81\) −9.04170 −1.00463
\(82\) 0 0
\(83\) 15.2501 1.67391 0.836957 0.547268i \(-0.184332\pi\)
0.836957 + 0.547268i \(0.184332\pi\)
\(84\) 0 0
\(85\) −6.09867 −0.661494
\(86\) 0 0
\(87\) 2.29372 0.245913
\(88\) 0 0
\(89\) 11.6474 1.23462 0.617311 0.786719i \(-0.288222\pi\)
0.617311 + 0.786719i \(0.288222\pi\)
\(90\) 0 0
\(91\) −14.4284 −1.51251
\(92\) 0 0
\(93\) 0.269400 0.0279354
\(94\) 0 0
\(95\) 9.70918 0.996141
\(96\) 0 0
\(97\) 2.82193 0.286524 0.143262 0.989685i \(-0.454241\pi\)
0.143262 + 0.989685i \(0.454241\pi\)
\(98\) 0 0
\(99\) 0.0454170 0.00456458
\(100\) 0 0
\(101\) −8.97861 −0.893405 −0.446703 0.894683i \(-0.647402\pi\)
−0.446703 + 0.894683i \(0.647402\pi\)
\(102\) 0 0
\(103\) −10.0164 −0.986947 −0.493474 0.869761i \(-0.664273\pi\)
−0.493474 + 0.869761i \(0.664273\pi\)
\(104\) 0 0
\(105\) 24.5861 2.39936
\(106\) 0 0
\(107\) 9.24291 0.893546 0.446773 0.894647i \(-0.352573\pi\)
0.446773 + 0.894647i \(0.352573\pi\)
\(108\) 0 0
\(109\) 3.46112 0.331515 0.165758 0.986167i \(-0.446993\pi\)
0.165758 + 0.986167i \(0.446993\pi\)
\(110\) 0 0
\(111\) 0.213211 0.0202371
\(112\) 0 0
\(113\) −9.51981 −0.895548 −0.447774 0.894147i \(-0.647783\pi\)
−0.447774 + 0.894147i \(0.647783\pi\)
\(114\) 0 0
\(115\) −25.2659 −2.35606
\(116\) 0 0
\(117\) −0.0437372 −0.00404351
\(118\) 0 0
\(119\) 9.14074 0.837930
\(120\) 0 0
\(121\) −0.424405 −0.0385823
\(122\) 0 0
\(123\) 11.6871 1.05379
\(124\) 0 0
\(125\) 1.69444 0.151555
\(126\) 0 0
\(127\) −4.15758 −0.368926 −0.184463 0.982839i \(-0.559055\pi\)
−0.184463 + 0.982839i \(0.559055\pi\)
\(128\) 0 0
\(129\) −17.8622 −1.57268
\(130\) 0 0
\(131\) −19.3061 −1.68678 −0.843391 0.537300i \(-0.819444\pi\)
−0.843391 + 0.537300i \(0.819444\pi\)
\(132\) 0 0
\(133\) −14.5522 −1.26184
\(134\) 0 0
\(135\) −15.9350 −1.37146
\(136\) 0 0
\(137\) 7.76257 0.663201 0.331601 0.943420i \(-0.392411\pi\)
0.331601 + 0.943420i \(0.392411\pi\)
\(138\) 0 0
\(139\) −13.1398 −1.11450 −0.557251 0.830344i \(-0.688144\pi\)
−0.557251 + 0.830344i \(0.688144\pi\)
\(140\) 0 0
\(141\) 1.73608 0.146204
\(142\) 0 0
\(143\) −10.1845 −0.851667
\(144\) 0 0
\(145\) 4.06125 0.337268
\(146\) 0 0
\(147\) −24.6974 −2.03701
\(148\) 0 0
\(149\) 16.9653 1.38985 0.694924 0.719083i \(-0.255438\pi\)
0.694924 + 0.719083i \(0.255438\pi\)
\(150\) 0 0
\(151\) 24.2794 1.97583 0.987914 0.155006i \(-0.0495397\pi\)
0.987914 + 0.155006i \(0.0495397\pi\)
\(152\) 0 0
\(153\) 0.0277085 0.00224010
\(154\) 0 0
\(155\) 0.476996 0.0383133
\(156\) 0 0
\(157\) 10.4430 0.833439 0.416720 0.909035i \(-0.363180\pi\)
0.416720 + 0.909035i \(0.363180\pi\)
\(158\) 0 0
\(159\) −16.4033 −1.30086
\(160\) 0 0
\(161\) 37.8687 2.98447
\(162\) 0 0
\(163\) 3.46132 0.271112 0.135556 0.990770i \(-0.456718\pi\)
0.135556 + 0.990770i \(0.456718\pi\)
\(164\) 0 0
\(165\) 17.3544 1.35104
\(166\) 0 0
\(167\) 22.3535 1.72977 0.864883 0.501973i \(-0.167392\pi\)
0.864883 + 0.501973i \(0.167392\pi\)
\(168\) 0 0
\(169\) −3.19222 −0.245555
\(170\) 0 0
\(171\) −0.0441124 −0.00337336
\(172\) 0 0
\(173\) −7.68171 −0.584030 −0.292015 0.956414i \(-0.594326\pi\)
−0.292015 + 0.956414i \(0.594326\pi\)
\(174\) 0 0
\(175\) 20.4962 1.54937
\(176\) 0 0
\(177\) 25.2359 1.89684
\(178\) 0 0
\(179\) −17.1009 −1.27818 −0.639091 0.769131i \(-0.720689\pi\)
−0.639091 + 0.769131i \(0.720689\pi\)
\(180\) 0 0
\(181\) 7.53867 0.560345 0.280173 0.959950i \(-0.409608\pi\)
0.280173 + 0.959950i \(0.409608\pi\)
\(182\) 0 0
\(183\) 0.288133 0.0212994
\(184\) 0 0
\(185\) 0.377510 0.0277551
\(186\) 0 0
\(187\) 6.45208 0.471823
\(188\) 0 0
\(189\) 23.8835 1.73727
\(190\) 0 0
\(191\) 6.47173 0.468278 0.234139 0.972203i \(-0.424773\pi\)
0.234139 + 0.972203i \(0.424773\pi\)
\(192\) 0 0
\(193\) −14.0571 −1.01185 −0.505927 0.862576i \(-0.668849\pi\)
−0.505927 + 0.862576i \(0.668849\pi\)
\(194\) 0 0
\(195\) −16.7125 −1.19681
\(196\) 0 0
\(197\) −4.06861 −0.289876 −0.144938 0.989441i \(-0.546298\pi\)
−0.144938 + 0.989441i \(0.546298\pi\)
\(198\) 0 0
\(199\) 0.747244 0.0529707 0.0264854 0.999649i \(-0.491568\pi\)
0.0264854 + 0.999649i \(0.491568\pi\)
\(200\) 0 0
\(201\) 1.23145 0.0868596
\(202\) 0 0
\(203\) −6.08703 −0.427226
\(204\) 0 0
\(205\) 20.6931 1.44527
\(206\) 0 0
\(207\) 0.114792 0.00797862
\(208\) 0 0
\(209\) −10.2718 −0.710517
\(210\) 0 0
\(211\) 13.4493 0.925890 0.462945 0.886387i \(-0.346793\pi\)
0.462945 + 0.886387i \(0.346793\pi\)
\(212\) 0 0
\(213\) 11.1645 0.764980
\(214\) 0 0
\(215\) −31.6266 −2.15692
\(216\) 0 0
\(217\) −0.714927 −0.0485324
\(218\) 0 0
\(219\) 23.4451 1.58427
\(220\) 0 0
\(221\) −6.21345 −0.417962
\(222\) 0 0
\(223\) −4.82629 −0.323193 −0.161596 0.986857i \(-0.551664\pi\)
−0.161596 + 0.986857i \(0.551664\pi\)
\(224\) 0 0
\(225\) 0.0621305 0.00414203
\(226\) 0 0
\(227\) 20.9131 1.38805 0.694026 0.719950i \(-0.255835\pi\)
0.694026 + 0.719950i \(0.255835\pi\)
\(228\) 0 0
\(229\) −10.5871 −0.699615 −0.349808 0.936822i \(-0.613753\pi\)
−0.349808 + 0.936822i \(0.613753\pi\)
\(230\) 0 0
\(231\) −26.0109 −1.71139
\(232\) 0 0
\(233\) 10.8038 0.707781 0.353891 0.935287i \(-0.384859\pi\)
0.353891 + 0.935287i \(0.384859\pi\)
\(234\) 0 0
\(235\) 3.07388 0.200518
\(236\) 0 0
\(237\) 4.71946 0.306562
\(238\) 0 0
\(239\) 3.23907 0.209518 0.104759 0.994498i \(-0.466593\pi\)
0.104759 + 0.994498i \(0.466593\pi\)
\(240\) 0 0
\(241\) −13.1864 −0.849411 −0.424706 0.905332i \(-0.639622\pi\)
−0.424706 + 0.905332i \(0.639622\pi\)
\(242\) 0 0
\(243\) 0.145136 0.00931045
\(244\) 0 0
\(245\) −43.7290 −2.79374
\(246\) 0 0
\(247\) 9.89192 0.629408
\(248\) 0 0
\(249\) −26.4753 −1.67781
\(250\) 0 0
\(251\) −6.22110 −0.392672 −0.196336 0.980537i \(-0.562904\pi\)
−0.196336 + 0.980537i \(0.562904\pi\)
\(252\) 0 0
\(253\) 26.7300 1.68050
\(254\) 0 0
\(255\) 10.5878 0.663031
\(256\) 0 0
\(257\) 20.8976 1.30356 0.651778 0.758410i \(-0.274023\pi\)
0.651778 + 0.758410i \(0.274023\pi\)
\(258\) 0 0
\(259\) −0.565815 −0.0351580
\(260\) 0 0
\(261\) −0.0184517 −0.00114213
\(262\) 0 0
\(263\) 6.26030 0.386026 0.193013 0.981196i \(-0.438174\pi\)
0.193013 + 0.981196i \(0.438174\pi\)
\(264\) 0 0
\(265\) −29.0435 −1.78413
\(266\) 0 0
\(267\) −20.2208 −1.23749
\(268\) 0 0
\(269\) 14.9502 0.911527 0.455764 0.890101i \(-0.349366\pi\)
0.455764 + 0.890101i \(0.349366\pi\)
\(270\) 0 0
\(271\) −8.82152 −0.535869 −0.267935 0.963437i \(-0.586341\pi\)
−0.267935 + 0.963437i \(0.586341\pi\)
\(272\) 0 0
\(273\) 25.0489 1.51603
\(274\) 0 0
\(275\) 14.4674 0.872419
\(276\) 0 0
\(277\) 27.1518 1.63139 0.815697 0.578479i \(-0.196353\pi\)
0.815697 + 0.578479i \(0.196353\pi\)
\(278\) 0 0
\(279\) −0.00216717 −0.000129745 0
\(280\) 0 0
\(281\) 30.7428 1.83396 0.916981 0.398930i \(-0.130618\pi\)
0.916981 + 0.398930i \(0.130618\pi\)
\(282\) 0 0
\(283\) 17.7217 1.05345 0.526723 0.850037i \(-0.323421\pi\)
0.526723 + 0.850037i \(0.323421\pi\)
\(284\) 0 0
\(285\) −16.8559 −0.998457
\(286\) 0 0
\(287\) −31.0151 −1.83076
\(288\) 0 0
\(289\) −13.0636 −0.768449
\(290\) 0 0
\(291\) −4.89909 −0.287190
\(292\) 0 0
\(293\) 33.4734 1.95554 0.977768 0.209690i \(-0.0672454\pi\)
0.977768 + 0.209690i \(0.0672454\pi\)
\(294\) 0 0
\(295\) 44.6824 2.60151
\(296\) 0 0
\(297\) 16.8584 0.978224
\(298\) 0 0
\(299\) −25.7414 −1.48866
\(300\) 0 0
\(301\) 47.4022 2.73222
\(302\) 0 0
\(303\) 15.5876 0.895482
\(304\) 0 0
\(305\) 0.510166 0.0292120
\(306\) 0 0
\(307\) 1.04732 0.0597735 0.0298867 0.999553i \(-0.490485\pi\)
0.0298867 + 0.999553i \(0.490485\pi\)
\(308\) 0 0
\(309\) 17.3893 0.989242
\(310\) 0 0
\(311\) −12.4806 −0.707712 −0.353856 0.935300i \(-0.615130\pi\)
−0.353856 + 0.935300i \(0.615130\pi\)
\(312\) 0 0
\(313\) 19.1869 1.08451 0.542255 0.840214i \(-0.317571\pi\)
0.542255 + 0.840214i \(0.317571\pi\)
\(314\) 0 0
\(315\) −0.197782 −0.0111438
\(316\) 0 0
\(317\) −18.2453 −1.02476 −0.512379 0.858760i \(-0.671236\pi\)
−0.512379 + 0.858760i \(0.671236\pi\)
\(318\) 0 0
\(319\) −4.29659 −0.240563
\(320\) 0 0
\(321\) −16.0464 −0.895623
\(322\) 0 0
\(323\) −6.26676 −0.348692
\(324\) 0 0
\(325\) −13.9324 −0.772828
\(326\) 0 0
\(327\) −6.00877 −0.332286
\(328\) 0 0
\(329\) −4.60716 −0.254001
\(330\) 0 0
\(331\) −21.3888 −1.17563 −0.587817 0.808994i \(-0.700013\pi\)
−0.587817 + 0.808994i \(0.700013\pi\)
\(332\) 0 0
\(333\) −0.00171517 −9.39905e−5 0
\(334\) 0 0
\(335\) 2.18039 0.119127
\(336\) 0 0
\(337\) −5.73773 −0.312554 −0.156277 0.987713i \(-0.549949\pi\)
−0.156277 + 0.987713i \(0.549949\pi\)
\(338\) 0 0
\(339\) 16.5271 0.897630
\(340\) 0 0
\(341\) −0.504638 −0.0273277
\(342\) 0 0
\(343\) 33.2912 1.79756
\(344\) 0 0
\(345\) 43.8635 2.36153
\(346\) 0 0
\(347\) 22.0471 1.18355 0.591776 0.806103i \(-0.298427\pi\)
0.591776 + 0.806103i \(0.298427\pi\)
\(348\) 0 0
\(349\) 32.7353 1.75228 0.876139 0.482059i \(-0.160111\pi\)
0.876139 + 0.482059i \(0.160111\pi\)
\(350\) 0 0
\(351\) −16.2349 −0.866554
\(352\) 0 0
\(353\) −19.5157 −1.03872 −0.519359 0.854556i \(-0.673829\pi\)
−0.519359 + 0.854556i \(0.673829\pi\)
\(354\) 0 0
\(355\) 19.7678 1.04917
\(356\) 0 0
\(357\) −15.8690 −0.839878
\(358\) 0 0
\(359\) 35.1527 1.85529 0.927644 0.373465i \(-0.121830\pi\)
0.927644 + 0.373465i \(0.121830\pi\)
\(360\) 0 0
\(361\) −9.02322 −0.474906
\(362\) 0 0
\(363\) 0.736800 0.0386720
\(364\) 0 0
\(365\) 41.5117 2.17282
\(366\) 0 0
\(367\) 29.3033 1.52962 0.764809 0.644257i \(-0.222833\pi\)
0.764809 + 0.644257i \(0.222833\pi\)
\(368\) 0 0
\(369\) −0.0940166 −0.00489431
\(370\) 0 0
\(371\) 43.5306 2.26000
\(372\) 0 0
\(373\) 16.7829 0.868986 0.434493 0.900675i \(-0.356928\pi\)
0.434493 + 0.900675i \(0.356928\pi\)
\(374\) 0 0
\(375\) −2.94168 −0.151908
\(376\) 0 0
\(377\) 4.13768 0.213102
\(378\) 0 0
\(379\) 2.81712 0.144706 0.0723529 0.997379i \(-0.476949\pi\)
0.0723529 + 0.997379i \(0.476949\pi\)
\(380\) 0 0
\(381\) 7.21788 0.369783
\(382\) 0 0
\(383\) 24.5324 1.25355 0.626774 0.779201i \(-0.284375\pi\)
0.626774 + 0.779201i \(0.284375\pi\)
\(384\) 0 0
\(385\) −46.0547 −2.34716
\(386\) 0 0
\(387\) 0.143691 0.00730424
\(388\) 0 0
\(389\) 18.7162 0.948948 0.474474 0.880269i \(-0.342638\pi\)
0.474474 + 0.880269i \(0.342638\pi\)
\(390\) 0 0
\(391\) 16.3078 0.824720
\(392\) 0 0
\(393\) 33.5169 1.69070
\(394\) 0 0
\(395\) 8.35624 0.420448
\(396\) 0 0
\(397\) 2.38286 0.119592 0.0597962 0.998211i \(-0.480955\pi\)
0.0597962 + 0.998211i \(0.480955\pi\)
\(398\) 0 0
\(399\) 25.2638 1.26477
\(400\) 0 0
\(401\) −19.4749 −0.972531 −0.486265 0.873811i \(-0.661641\pi\)
−0.486265 + 0.873811i \(0.661641\pi\)
\(402\) 0 0
\(403\) 0.485974 0.0242081
\(404\) 0 0
\(405\) 27.7931 1.38105
\(406\) 0 0
\(407\) −0.399386 −0.0197968
\(408\) 0 0
\(409\) 8.04740 0.397919 0.198959 0.980008i \(-0.436244\pi\)
0.198959 + 0.980008i \(0.436244\pi\)
\(410\) 0 0
\(411\) −13.4764 −0.664743
\(412\) 0 0
\(413\) −66.9704 −3.29540
\(414\) 0 0
\(415\) −46.8770 −2.30110
\(416\) 0 0
\(417\) 22.8117 1.11709
\(418\) 0 0
\(419\) 29.2752 1.43019 0.715094 0.699028i \(-0.246384\pi\)
0.715094 + 0.699028i \(0.246384\pi\)
\(420\) 0 0
\(421\) −14.6616 −0.714564 −0.357282 0.933997i \(-0.616296\pi\)
−0.357282 + 0.933997i \(0.616296\pi\)
\(422\) 0 0
\(423\) −0.0139658 −0.000679040 0
\(424\) 0 0
\(425\) 8.82647 0.428147
\(426\) 0 0
\(427\) −0.764642 −0.0370036
\(428\) 0 0
\(429\) 17.6810 0.853647
\(430\) 0 0
\(431\) 25.3220 1.21972 0.609859 0.792510i \(-0.291226\pi\)
0.609859 + 0.792510i \(0.291226\pi\)
\(432\) 0 0
\(433\) −9.52001 −0.457502 −0.228751 0.973485i \(-0.573464\pi\)
−0.228751 + 0.973485i \(0.573464\pi\)
\(434\) 0 0
\(435\) −7.05064 −0.338052
\(436\) 0 0
\(437\) −25.9623 −1.24194
\(438\) 0 0
\(439\) 38.7419 1.84905 0.924524 0.381123i \(-0.124463\pi\)
0.924524 + 0.381123i \(0.124463\pi\)
\(440\) 0 0
\(441\) 0.198677 0.00946081
\(442\) 0 0
\(443\) −22.1440 −1.05209 −0.526046 0.850456i \(-0.676326\pi\)
−0.526046 + 0.850456i \(0.676326\pi\)
\(444\) 0 0
\(445\) −35.8028 −1.69721
\(446\) 0 0
\(447\) −29.4530 −1.39308
\(448\) 0 0
\(449\) 7.16829 0.338293 0.169146 0.985591i \(-0.445899\pi\)
0.169146 + 0.985591i \(0.445899\pi\)
\(450\) 0 0
\(451\) −21.8923 −1.03087
\(452\) 0 0
\(453\) −42.1509 −1.98042
\(454\) 0 0
\(455\) 44.3513 2.07922
\(456\) 0 0
\(457\) −9.81999 −0.459360 −0.229680 0.973266i \(-0.573768\pi\)
−0.229680 + 0.973266i \(0.573768\pi\)
\(458\) 0 0
\(459\) 10.2852 0.480071
\(460\) 0 0
\(461\) 22.4375 1.04502 0.522510 0.852633i \(-0.324996\pi\)
0.522510 + 0.852633i \(0.324996\pi\)
\(462\) 0 0
\(463\) 12.6027 0.585698 0.292849 0.956159i \(-0.405397\pi\)
0.292849 + 0.956159i \(0.405397\pi\)
\(464\) 0 0
\(465\) −0.828103 −0.0384024
\(466\) 0 0
\(467\) 6.03412 0.279226 0.139613 0.990206i \(-0.455414\pi\)
0.139613 + 0.990206i \(0.455414\pi\)
\(468\) 0 0
\(469\) −3.26799 −0.150902
\(470\) 0 0
\(471\) −18.1298 −0.835377
\(472\) 0 0
\(473\) 33.4593 1.53846
\(474\) 0 0
\(475\) −14.0519 −0.644745
\(476\) 0 0
\(477\) 0.131955 0.00604181
\(478\) 0 0
\(479\) 15.6740 0.716163 0.358082 0.933690i \(-0.383431\pi\)
0.358082 + 0.933690i \(0.383431\pi\)
\(480\) 0 0
\(481\) 0.384615 0.0175369
\(482\) 0 0
\(483\) −65.7431 −2.99141
\(484\) 0 0
\(485\) −8.67428 −0.393879
\(486\) 0 0
\(487\) −10.6721 −0.483600 −0.241800 0.970326i \(-0.577738\pi\)
−0.241800 + 0.970326i \(0.577738\pi\)
\(488\) 0 0
\(489\) −6.00913 −0.271742
\(490\) 0 0
\(491\) 6.25802 0.282421 0.141210 0.989980i \(-0.454901\pi\)
0.141210 + 0.989980i \(0.454901\pi\)
\(492\) 0 0
\(493\) −2.62132 −0.118058
\(494\) 0 0
\(495\) −0.139606 −0.00627484
\(496\) 0 0
\(497\) −29.6282 −1.32900
\(498\) 0 0
\(499\) 9.25077 0.414121 0.207061 0.978328i \(-0.433610\pi\)
0.207061 + 0.978328i \(0.433610\pi\)
\(500\) 0 0
\(501\) −38.8074 −1.73379
\(502\) 0 0
\(503\) 30.5842 1.36368 0.681840 0.731501i \(-0.261180\pi\)
0.681840 + 0.731501i \(0.261180\pi\)
\(504\) 0 0
\(505\) 27.5992 1.22815
\(506\) 0 0
\(507\) 5.54194 0.246126
\(508\) 0 0
\(509\) 8.64623 0.383237 0.191619 0.981469i \(-0.438626\pi\)
0.191619 + 0.981469i \(0.438626\pi\)
\(510\) 0 0
\(511\) −62.2180 −2.75236
\(512\) 0 0
\(513\) −16.3742 −0.722937
\(514\) 0 0
\(515\) 30.7893 1.35674
\(516\) 0 0
\(517\) −3.25201 −0.143023
\(518\) 0 0
\(519\) 13.3360 0.585387
\(520\) 0 0
\(521\) 29.6556 1.29924 0.649618 0.760261i \(-0.274929\pi\)
0.649618 + 0.760261i \(0.274929\pi\)
\(522\) 0 0
\(523\) −31.8295 −1.39181 −0.695903 0.718135i \(-0.744996\pi\)
−0.695903 + 0.718135i \(0.744996\pi\)
\(524\) 0 0
\(525\) −35.5830 −1.55297
\(526\) 0 0
\(527\) −0.307876 −0.0134113
\(528\) 0 0
\(529\) 44.5607 1.93742
\(530\) 0 0
\(531\) −0.203009 −0.00880983
\(532\) 0 0
\(533\) 21.0826 0.913189
\(534\) 0 0
\(535\) −28.4116 −1.22834
\(536\) 0 0
\(537\) 29.6885 1.28115
\(538\) 0 0
\(539\) 46.2630 1.99269
\(540\) 0 0
\(541\) −23.6460 −1.01662 −0.508311 0.861174i \(-0.669730\pi\)
−0.508311 + 0.861174i \(0.669730\pi\)
\(542\) 0 0
\(543\) −13.0877 −0.561648
\(544\) 0 0
\(545\) −10.6391 −0.455728
\(546\) 0 0
\(547\) 24.2872 1.03844 0.519222 0.854639i \(-0.326222\pi\)
0.519222 + 0.854639i \(0.326222\pi\)
\(548\) 0 0
\(549\) −0.00231787 −9.89244e−5 0
\(550\) 0 0
\(551\) 4.17318 0.177783
\(552\) 0 0
\(553\) −12.5244 −0.532592
\(554\) 0 0
\(555\) −0.655386 −0.0278196
\(556\) 0 0
\(557\) −9.78198 −0.414476 −0.207238 0.978291i \(-0.566447\pi\)
−0.207238 + 0.978291i \(0.566447\pi\)
\(558\) 0 0
\(559\) −32.2218 −1.36284
\(560\) 0 0
\(561\) −11.2013 −0.472920
\(562\) 0 0
\(563\) −41.7247 −1.75849 −0.879243 0.476374i \(-0.841951\pi\)
−0.879243 + 0.476374i \(0.841951\pi\)
\(564\) 0 0
\(565\) 29.2628 1.23109
\(566\) 0 0
\(567\) −41.6566 −1.74941
\(568\) 0 0
\(569\) 24.0759 1.00931 0.504656 0.863320i \(-0.331619\pi\)
0.504656 + 0.863320i \(0.331619\pi\)
\(570\) 0 0
\(571\) −34.2010 −1.43127 −0.715634 0.698475i \(-0.753862\pi\)
−0.715634 + 0.698475i \(0.753862\pi\)
\(572\) 0 0
\(573\) −11.2354 −0.469366
\(574\) 0 0
\(575\) 36.5668 1.52494
\(576\) 0 0
\(577\) −24.3536 −1.01385 −0.506927 0.861989i \(-0.669219\pi\)
−0.506927 + 0.861989i \(0.669219\pi\)
\(578\) 0 0
\(579\) 24.4042 1.01421
\(580\) 0 0
\(581\) 70.2597 2.91486
\(582\) 0 0
\(583\) 30.7265 1.27256
\(584\) 0 0
\(585\) 0.134443 0.00555854
\(586\) 0 0
\(587\) −9.03895 −0.373078 −0.186539 0.982448i \(-0.559727\pi\)
−0.186539 + 0.982448i \(0.559727\pi\)
\(588\) 0 0
\(589\) 0.490143 0.0201960
\(590\) 0 0
\(591\) 7.06342 0.290550
\(592\) 0 0
\(593\) −33.7598 −1.38635 −0.693175 0.720769i \(-0.743789\pi\)
−0.693175 + 0.720769i \(0.743789\pi\)
\(594\) 0 0
\(595\) −28.0976 −1.15189
\(596\) 0 0
\(597\) −1.29727 −0.0530939
\(598\) 0 0
\(599\) 4.43195 0.181085 0.0905423 0.995893i \(-0.471140\pi\)
0.0905423 + 0.995893i \(0.471140\pi\)
\(600\) 0 0
\(601\) −29.6734 −1.21040 −0.605202 0.796072i \(-0.706908\pi\)
−0.605202 + 0.796072i \(0.706908\pi\)
\(602\) 0 0
\(603\) −0.00990632 −0.000403417 0
\(604\) 0 0
\(605\) 1.30457 0.0530384
\(606\) 0 0
\(607\) 10.6283 0.431390 0.215695 0.976461i \(-0.430798\pi\)
0.215695 + 0.976461i \(0.430798\pi\)
\(608\) 0 0
\(609\) 10.5676 0.428219
\(610\) 0 0
\(611\) 3.13174 0.126697
\(612\) 0 0
\(613\) −22.3184 −0.901432 −0.450716 0.892667i \(-0.648831\pi\)
−0.450716 + 0.892667i \(0.648831\pi\)
\(614\) 0 0
\(615\) −35.9249 −1.44863
\(616\) 0 0
\(617\) 9.57779 0.385587 0.192794 0.981239i \(-0.438245\pi\)
0.192794 + 0.981239i \(0.438245\pi\)
\(618\) 0 0
\(619\) −28.3889 −1.14105 −0.570523 0.821282i \(-0.693259\pi\)
−0.570523 + 0.821282i \(0.693259\pi\)
\(620\) 0 0
\(621\) 42.6100 1.70988
\(622\) 0 0
\(623\) 53.6615 2.14990
\(624\) 0 0
\(625\) −27.4523 −1.09809
\(626\) 0 0
\(627\) 17.8327 0.712169
\(628\) 0 0
\(629\) −0.243662 −0.00971545
\(630\) 0 0
\(631\) 6.87974 0.273878 0.136939 0.990579i \(-0.456274\pi\)
0.136939 + 0.990579i \(0.456274\pi\)
\(632\) 0 0
\(633\) −23.3491 −0.928043
\(634\) 0 0
\(635\) 12.7799 0.507156
\(636\) 0 0
\(637\) −44.5520 −1.76521
\(638\) 0 0
\(639\) −0.0898125 −0.00355293
\(640\) 0 0
\(641\) −24.4052 −0.963946 −0.481973 0.876186i \(-0.660080\pi\)
−0.481973 + 0.876186i \(0.660080\pi\)
\(642\) 0 0
\(643\) 16.5483 0.652601 0.326301 0.945266i \(-0.394198\pi\)
0.326301 + 0.945266i \(0.394198\pi\)
\(644\) 0 0
\(645\) 54.9062 2.16193
\(646\) 0 0
\(647\) 8.07755 0.317561 0.158781 0.987314i \(-0.449244\pi\)
0.158781 + 0.987314i \(0.449244\pi\)
\(648\) 0 0
\(649\) −47.2717 −1.85558
\(650\) 0 0
\(651\) 1.24117 0.0486452
\(652\) 0 0
\(653\) 1.66715 0.0652405 0.0326202 0.999468i \(-0.489615\pi\)
0.0326202 + 0.999468i \(0.489615\pi\)
\(654\) 0 0
\(655\) 59.3447 2.31879
\(656\) 0 0
\(657\) −0.188603 −0.00735810
\(658\) 0 0
\(659\) 0.748190 0.0291453 0.0145727 0.999894i \(-0.495361\pi\)
0.0145727 + 0.999894i \(0.495361\pi\)
\(660\) 0 0
\(661\) −31.3464 −1.21923 −0.609616 0.792697i \(-0.708676\pi\)
−0.609616 + 0.792697i \(0.708676\pi\)
\(662\) 0 0
\(663\) 10.7870 0.418934
\(664\) 0 0
\(665\) 44.7318 1.73463
\(666\) 0 0
\(667\) −10.8597 −0.420490
\(668\) 0 0
\(669\) 8.37882 0.323944
\(670\) 0 0
\(671\) −0.539730 −0.0208360
\(672\) 0 0
\(673\) 39.4025 1.51886 0.759428 0.650591i \(-0.225479\pi\)
0.759428 + 0.650591i \(0.225479\pi\)
\(674\) 0 0
\(675\) 23.0623 0.887670
\(676\) 0 0
\(677\) 12.0347 0.462532 0.231266 0.972891i \(-0.425713\pi\)
0.231266 + 0.972891i \(0.425713\pi\)
\(678\) 0 0
\(679\) 13.0011 0.498936
\(680\) 0 0
\(681\) −36.3068 −1.39128
\(682\) 0 0
\(683\) 6.61906 0.253271 0.126636 0.991949i \(-0.459582\pi\)
0.126636 + 0.991949i \(0.459582\pi\)
\(684\) 0 0
\(685\) −23.8612 −0.911691
\(686\) 0 0
\(687\) 18.3800 0.701242
\(688\) 0 0
\(689\) −29.5901 −1.12729
\(690\) 0 0
\(691\) 20.6843 0.786868 0.393434 0.919353i \(-0.371287\pi\)
0.393434 + 0.919353i \(0.371287\pi\)
\(692\) 0 0
\(693\) 0.209243 0.00794850
\(694\) 0 0
\(695\) 40.3902 1.53209
\(696\) 0 0
\(697\) −13.3563 −0.505906
\(698\) 0 0
\(699\) −18.7563 −0.709427
\(700\) 0 0
\(701\) 2.10692 0.0795773 0.0397887 0.999208i \(-0.487332\pi\)
0.0397887 + 0.999208i \(0.487332\pi\)
\(702\) 0 0
\(703\) 0.387914 0.0146305
\(704\) 0 0
\(705\) −5.33650 −0.200984
\(706\) 0 0
\(707\) −41.3659 −1.55573
\(708\) 0 0
\(709\) −18.2299 −0.684637 −0.342318 0.939584i \(-0.611212\pi\)
−0.342318 + 0.939584i \(0.611212\pi\)
\(710\) 0 0
\(711\) −0.0379655 −0.00142382
\(712\) 0 0
\(713\) −1.27548 −0.0477672
\(714\) 0 0
\(715\) 31.3058 1.17077
\(716\) 0 0
\(717\) −5.62328 −0.210005
\(718\) 0 0
\(719\) −3.52201 −0.131349 −0.0656744 0.997841i \(-0.520920\pi\)
−0.0656744 + 0.997841i \(0.520920\pi\)
\(720\) 0 0
\(721\) −46.1473 −1.71862
\(722\) 0 0
\(723\) 22.8926 0.851386
\(724\) 0 0
\(725\) −5.87775 −0.218294
\(726\) 0 0
\(727\) −45.9002 −1.70234 −0.851172 0.524886i \(-0.824108\pi\)
−0.851172 + 0.524886i \(0.824108\pi\)
\(728\) 0 0
\(729\) 26.8731 0.995301
\(730\) 0 0
\(731\) 20.4133 0.755012
\(732\) 0 0
\(733\) 8.94933 0.330551 0.165275 0.986247i \(-0.447149\pi\)
0.165275 + 0.986247i \(0.447149\pi\)
\(734\) 0 0
\(735\) 75.9169 2.80024
\(736\) 0 0
\(737\) −2.30674 −0.0849700
\(738\) 0 0
\(739\) −0.905173 −0.0332973 −0.0166487 0.999861i \(-0.505300\pi\)
−0.0166487 + 0.999861i \(0.505300\pi\)
\(740\) 0 0
\(741\) −17.1731 −0.630871
\(742\) 0 0
\(743\) −46.2792 −1.69782 −0.848909 0.528539i \(-0.822740\pi\)
−0.848909 + 0.528539i \(0.822740\pi\)
\(744\) 0 0
\(745\) −52.1493 −1.91060
\(746\) 0 0
\(747\) 0.212980 0.00779252
\(748\) 0 0
\(749\) 42.5836 1.55597
\(750\) 0 0
\(751\) −25.5173 −0.931140 −0.465570 0.885011i \(-0.654151\pi\)
−0.465570 + 0.885011i \(0.654151\pi\)
\(752\) 0 0
\(753\) 10.8003 0.393585
\(754\) 0 0
\(755\) −74.6320 −2.71614
\(756\) 0 0
\(757\) −24.1051 −0.876115 −0.438058 0.898947i \(-0.644333\pi\)
−0.438058 + 0.898947i \(0.644333\pi\)
\(758\) 0 0
\(759\) −46.4054 −1.68441
\(760\) 0 0
\(761\) 45.5094 1.64972 0.824858 0.565341i \(-0.191255\pi\)
0.824858 + 0.565341i \(0.191255\pi\)
\(762\) 0 0
\(763\) 15.9460 0.577282
\(764\) 0 0
\(765\) −0.0851727 −0.00307943
\(766\) 0 0
\(767\) 45.5234 1.64375
\(768\) 0 0
\(769\) −1.88574 −0.0680017 −0.0340008 0.999422i \(-0.510825\pi\)
−0.0340008 + 0.999422i \(0.510825\pi\)
\(770\) 0 0
\(771\) −36.2799 −1.30659
\(772\) 0 0
\(773\) 30.9069 1.11164 0.555821 0.831302i \(-0.312404\pi\)
0.555821 + 0.831302i \(0.312404\pi\)
\(774\) 0 0
\(775\) −0.690346 −0.0247980
\(776\) 0 0
\(777\) 0.982299 0.0352398
\(778\) 0 0
\(779\) 21.2635 0.761843
\(780\) 0 0
\(781\) −20.9133 −0.748338
\(782\) 0 0
\(783\) −6.84914 −0.244768
\(784\) 0 0
\(785\) −32.1005 −1.14571
\(786\) 0 0
\(787\) 32.9261 1.17369 0.586844 0.809700i \(-0.300370\pi\)
0.586844 + 0.809700i \(0.300370\pi\)
\(788\) 0 0
\(789\) −10.8684 −0.386924
\(790\) 0 0
\(791\) −43.8593 −1.55946
\(792\) 0 0
\(793\) 0.519768 0.0184575
\(794\) 0 0
\(795\) 50.4217 1.78827
\(796\) 0 0
\(797\) 41.1351 1.45708 0.728540 0.685004i \(-0.240199\pi\)
0.728540 + 0.685004i \(0.240199\pi\)
\(798\) 0 0
\(799\) −1.98403 −0.0701898
\(800\) 0 0
\(801\) 0.162665 0.00574749
\(802\) 0 0
\(803\) −43.9172 −1.54981
\(804\) 0 0
\(805\) −116.404 −4.10271
\(806\) 0 0
\(807\) −25.9546 −0.913646
\(808\) 0 0
\(809\) −1.47110 −0.0517210 −0.0258605 0.999666i \(-0.508233\pi\)
−0.0258605 + 0.999666i \(0.508233\pi\)
\(810\) 0 0
\(811\) −42.7445 −1.50096 −0.750480 0.660893i \(-0.770178\pi\)
−0.750480 + 0.660893i \(0.770178\pi\)
\(812\) 0 0
\(813\) 15.3148 0.537115
\(814\) 0 0
\(815\) −10.6397 −0.372693
\(816\) 0 0
\(817\) −32.4983 −1.13697
\(818\) 0 0
\(819\) −0.201504 −0.00704114
\(820\) 0 0
\(821\) 3.12827 0.109177 0.0545887 0.998509i \(-0.482615\pi\)
0.0545887 + 0.998509i \(0.482615\pi\)
\(822\) 0 0
\(823\) −53.9811 −1.88166 −0.940832 0.338873i \(-0.889954\pi\)
−0.940832 + 0.338873i \(0.889954\pi\)
\(824\) 0 0
\(825\) −25.1166 −0.874448
\(826\) 0 0
\(827\) 23.6763 0.823306 0.411653 0.911341i \(-0.364952\pi\)
0.411653 + 0.911341i \(0.364952\pi\)
\(828\) 0 0
\(829\) −4.39129 −0.152516 −0.0762580 0.997088i \(-0.524297\pi\)
−0.0762580 + 0.997088i \(0.524297\pi\)
\(830\) 0 0
\(831\) −47.1377 −1.63519
\(832\) 0 0
\(833\) 28.2247 0.977928
\(834\) 0 0
\(835\) −68.7121 −2.37788
\(836\) 0 0
\(837\) −0.804436 −0.0278054
\(838\) 0 0
\(839\) 10.4724 0.361549 0.180774 0.983525i \(-0.442140\pi\)
0.180774 + 0.983525i \(0.442140\pi\)
\(840\) 0 0
\(841\) −27.2544 −0.939807
\(842\) 0 0
\(843\) −53.3719 −1.83823
\(844\) 0 0
\(845\) 9.81251 0.337561
\(846\) 0 0
\(847\) −1.95530 −0.0671850
\(848\) 0 0
\(849\) −30.7662 −1.05589
\(850\) 0 0
\(851\) −1.00946 −0.0346037
\(852\) 0 0
\(853\) −12.5901 −0.431076 −0.215538 0.976495i \(-0.569150\pi\)
−0.215538 + 0.976495i \(0.569150\pi\)
\(854\) 0 0
\(855\) 0.135596 0.00463730
\(856\) 0 0
\(857\) −9.23699 −0.315529 −0.157765 0.987477i \(-0.550429\pi\)
−0.157765 + 0.987477i \(0.550429\pi\)
\(858\) 0 0
\(859\) 9.28932 0.316948 0.158474 0.987363i \(-0.449343\pi\)
0.158474 + 0.987363i \(0.449343\pi\)
\(860\) 0 0
\(861\) 53.8446 1.83502
\(862\) 0 0
\(863\) −25.5775 −0.870667 −0.435333 0.900269i \(-0.643369\pi\)
−0.435333 + 0.900269i \(0.643369\pi\)
\(864\) 0 0
\(865\) 23.6127 0.802855
\(866\) 0 0
\(867\) 22.6795 0.770236
\(868\) 0 0
\(869\) −8.84048 −0.299893
\(870\) 0 0
\(871\) 2.22143 0.0752702
\(872\) 0 0
\(873\) 0.0394105 0.00133384
\(874\) 0 0
\(875\) 7.80656 0.263910
\(876\) 0 0
\(877\) 21.9782 0.742151 0.371076 0.928603i \(-0.378989\pi\)
0.371076 + 0.928603i \(0.378989\pi\)
\(878\) 0 0
\(879\) −58.1124 −1.96008
\(880\) 0 0
\(881\) 36.8279 1.24076 0.620381 0.784301i \(-0.286978\pi\)
0.620381 + 0.784301i \(0.286978\pi\)
\(882\) 0 0
\(883\) −4.67060 −0.157178 −0.0785890 0.996907i \(-0.525041\pi\)
−0.0785890 + 0.996907i \(0.525041\pi\)
\(884\) 0 0
\(885\) −77.5721 −2.60756
\(886\) 0 0
\(887\) −46.3977 −1.55788 −0.778941 0.627097i \(-0.784243\pi\)
−0.778941 + 0.627097i \(0.784243\pi\)
\(888\) 0 0
\(889\) −19.1547 −0.642427
\(890\) 0 0
\(891\) −29.4037 −0.985062
\(892\) 0 0
\(893\) 3.15860 0.105699
\(894\) 0 0
\(895\) 52.5662 1.75709
\(896\) 0 0
\(897\) 44.6891 1.49213
\(898\) 0 0
\(899\) 0.205022 0.00683785
\(900\) 0 0
\(901\) 18.7460 0.624520
\(902\) 0 0
\(903\) −82.2939 −2.73857
\(904\) 0 0
\(905\) −23.1730 −0.770297
\(906\) 0 0
\(907\) 17.5707 0.583426 0.291713 0.956506i \(-0.405775\pi\)
0.291713 + 0.956506i \(0.405775\pi\)
\(908\) 0 0
\(909\) −0.125393 −0.00415904
\(910\) 0 0
\(911\) 20.6650 0.684660 0.342330 0.939580i \(-0.388784\pi\)
0.342330 + 0.939580i \(0.388784\pi\)
\(912\) 0 0
\(913\) 49.5935 1.64131
\(914\) 0 0
\(915\) −0.885688 −0.0292799
\(916\) 0 0
\(917\) −88.9464 −2.93727
\(918\) 0 0
\(919\) 7.64800 0.252284 0.126142 0.992012i \(-0.459740\pi\)
0.126142 + 0.992012i \(0.459740\pi\)
\(920\) 0 0
\(921\) −1.81822 −0.0599125
\(922\) 0 0
\(923\) 20.1399 0.662911
\(924\) 0 0
\(925\) −0.546361 −0.0179643
\(926\) 0 0
\(927\) −0.139887 −0.00459450
\(928\) 0 0
\(929\) 19.5411 0.641121 0.320561 0.947228i \(-0.396129\pi\)
0.320561 + 0.947228i \(0.396129\pi\)
\(930\) 0 0
\(931\) −44.9342 −1.47266
\(932\) 0 0
\(933\) 21.6674 0.709358
\(934\) 0 0
\(935\) −19.8330 −0.648607
\(936\) 0 0
\(937\) 37.1182 1.21260 0.606299 0.795236i \(-0.292653\pi\)
0.606299 + 0.795236i \(0.292653\pi\)
\(938\) 0 0
\(939\) −33.3100 −1.08703
\(940\) 0 0
\(941\) −52.4125 −1.70860 −0.854300 0.519781i \(-0.826014\pi\)
−0.854300 + 0.519781i \(0.826014\pi\)
\(942\) 0 0
\(943\) −55.3332 −1.80190
\(944\) 0 0
\(945\) −73.4151 −2.38819
\(946\) 0 0
\(947\) 28.2704 0.918665 0.459333 0.888264i \(-0.348089\pi\)
0.459333 + 0.888264i \(0.348089\pi\)
\(948\) 0 0
\(949\) 42.2930 1.37289
\(950\) 0 0
\(951\) 31.6752 1.02714
\(952\) 0 0
\(953\) −15.9896 −0.517955 −0.258977 0.965883i \(-0.583385\pi\)
−0.258977 + 0.965883i \(0.583385\pi\)
\(954\) 0 0
\(955\) −19.8933 −0.643733
\(956\) 0 0
\(957\) 7.45922 0.241122
\(958\) 0 0
\(959\) 35.7634 1.15486
\(960\) 0 0
\(961\) −30.9759 −0.999223
\(962\) 0 0
\(963\) 0.129085 0.00415969
\(964\) 0 0
\(965\) 43.2099 1.39098
\(966\) 0 0
\(967\) −9.13852 −0.293875 −0.146938 0.989146i \(-0.546942\pi\)
−0.146938 + 0.989146i \(0.546942\pi\)
\(968\) 0 0
\(969\) 10.8796 0.349502
\(970\) 0 0
\(971\) −48.8938 −1.56908 −0.784539 0.620080i \(-0.787100\pi\)
−0.784539 + 0.620080i \(0.787100\pi\)
\(972\) 0 0
\(973\) −60.5372 −1.94073
\(974\) 0 0
\(975\) 24.1877 0.774625
\(976\) 0 0
\(977\) −14.5766 −0.466346 −0.233173 0.972435i \(-0.574911\pi\)
−0.233173 + 0.972435i \(0.574911\pi\)
\(978\) 0 0
\(979\) 37.8775 1.21057
\(980\) 0 0
\(981\) 0.0483373 0.00154329
\(982\) 0 0
\(983\) −23.4721 −0.748645 −0.374322 0.927299i \(-0.622125\pi\)
−0.374322 + 0.927299i \(0.622125\pi\)
\(984\) 0 0
\(985\) 12.5064 0.398488
\(986\) 0 0
\(987\) 7.99840 0.254592
\(988\) 0 0
\(989\) 84.5692 2.68914
\(990\) 0 0
\(991\) −26.2885 −0.835083 −0.417541 0.908658i \(-0.637108\pi\)
−0.417541 + 0.908658i \(0.637108\pi\)
\(992\) 0 0
\(993\) 37.1326 1.17837
\(994\) 0 0
\(995\) −2.29694 −0.0728179
\(996\) 0 0
\(997\) 19.4651 0.616465 0.308232 0.951311i \(-0.400263\pi\)
0.308232 + 0.951311i \(0.400263\pi\)
\(998\) 0 0
\(999\) −0.636656 −0.0201429
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 6016.2.a.n.1.5 yes 13
4.3 odd 2 6016.2.a.p.1.9 yes 13
8.3 odd 2 6016.2.a.m.1.5 13
8.5 even 2 6016.2.a.o.1.9 yes 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6016.2.a.m.1.5 13 8.3 odd 2
6016.2.a.n.1.5 yes 13 1.1 even 1 trivial
6016.2.a.o.1.9 yes 13 8.5 even 2
6016.2.a.p.1.9 yes 13 4.3 odd 2