Properties

Label 4008.2.a.g.1.5
Level $4008$
Weight $2$
Character 4008.1
Self dual yes
Analytic conductor $32.004$
Analytic rank $0$
Dimension $7$
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] = [4008,2,Mod(1,4008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4008.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: \(32.0040411301\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 7x^{5} + 11x^{4} + 12x^{3} - 14x^{2} - 6x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.31154\) of defining polynomial
Character \(\chi\) \(=\) 4008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.0621653 q^{5} +0.782308 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +0.0621653 q^{5} +0.782308 q^{7} +1.00000 q^{9} +2.68525 q^{11} +6.79339 q^{13} -0.0621653 q^{15} +4.20707 q^{17} +7.94188 q^{19} -0.782308 q^{21} +5.42476 q^{23} -4.99614 q^{25} -1.00000 q^{27} +1.95740 q^{29} +2.86384 q^{31} -2.68525 q^{33} +0.0486324 q^{35} -7.68582 q^{37} -6.79339 q^{39} -2.03431 q^{41} -5.17082 q^{43} +0.0621653 q^{45} -7.62796 q^{47} -6.38799 q^{49} -4.20707 q^{51} +3.26794 q^{53} +0.166929 q^{55} -7.94188 q^{57} +2.33656 q^{59} -12.7201 q^{61} +0.782308 q^{63} +0.422313 q^{65} -1.25024 q^{67} -5.42476 q^{69} +12.3346 q^{71} +10.0502 q^{73} +4.99614 q^{75} +2.10069 q^{77} +3.71570 q^{79} +1.00000 q^{81} +10.1647 q^{83} +0.261533 q^{85} -1.95740 q^{87} -2.74777 q^{89} +5.31452 q^{91} -2.86384 q^{93} +0.493709 q^{95} +1.54807 q^{97} +2.68525 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{3} - 3 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 7 q^{3} - 3 q^{5} + 8 q^{7} + 7 q^{9} + q^{11} - 2 q^{13} + 3 q^{15} + 11 q^{17} + 2 q^{19} - 8 q^{21} + 17 q^{23} + 4 q^{25} - 7 q^{27} - 7 q^{29} + 10 q^{31} - q^{33} + 10 q^{35} - 21 q^{37} + 2 q^{39} + 8 q^{41} - 12 q^{43} - 3 q^{45} + 25 q^{47} - 7 q^{49} - 11 q^{51} - 7 q^{53} + 15 q^{55} - 2 q^{57} + 3 q^{59} - 14 q^{61} + 8 q^{63} + 4 q^{65} + 4 q^{67} - 17 q^{69} + 27 q^{71} - 12 q^{73} - 4 q^{75} + 16 q^{77} + 8 q^{79} + 7 q^{81} + 15 q^{83} - 3 q^{85} + 7 q^{87} + 14 q^{89} - 3 q^{91} - 10 q^{93} + 37 q^{95} + 3 q^{97} + q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0.0621653 0.0278011 0.0139006 0.999903i \(-0.495575\pi\)
0.0139006 + 0.999903i \(0.495575\pi\)
\(6\) 0 0
\(7\) 0.782308 0.295685 0.147842 0.989011i \(-0.452767\pi\)
0.147842 + 0.989011i \(0.452767\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.68525 0.809633 0.404817 0.914398i \(-0.367335\pi\)
0.404817 + 0.914398i \(0.367335\pi\)
\(12\) 0 0
\(13\) 6.79339 1.88415 0.942073 0.335407i \(-0.108874\pi\)
0.942073 + 0.335407i \(0.108874\pi\)
\(14\) 0 0
\(15\) −0.0621653 −0.0160510
\(16\) 0 0
\(17\) 4.20707 1.02036 0.510182 0.860067i \(-0.329578\pi\)
0.510182 + 0.860067i \(0.329578\pi\)
\(18\) 0 0
\(19\) 7.94188 1.82199 0.910996 0.412416i \(-0.135315\pi\)
0.910996 + 0.412416i \(0.135315\pi\)
\(20\) 0 0
\(21\) −0.782308 −0.170714
\(22\) 0 0
\(23\) 5.42476 1.13114 0.565570 0.824700i \(-0.308656\pi\)
0.565570 + 0.824700i \(0.308656\pi\)
\(24\) 0 0
\(25\) −4.99614 −0.999227
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.95740 0.363480 0.181740 0.983347i \(-0.441827\pi\)
0.181740 + 0.983347i \(0.441827\pi\)
\(30\) 0 0
\(31\) 2.86384 0.514361 0.257180 0.966363i \(-0.417207\pi\)
0.257180 + 0.966363i \(0.417207\pi\)
\(32\) 0 0
\(33\) −2.68525 −0.467442
\(34\) 0 0
\(35\) 0.0486324 0.00822037
\(36\) 0 0
\(37\) −7.68582 −1.26354 −0.631771 0.775155i \(-0.717672\pi\)
−0.631771 + 0.775155i \(0.717672\pi\)
\(38\) 0 0
\(39\) −6.79339 −1.08781
\(40\) 0 0
\(41\) −2.03431 −0.317706 −0.158853 0.987302i \(-0.550780\pi\)
−0.158853 + 0.987302i \(0.550780\pi\)
\(42\) 0 0
\(43\) −5.17082 −0.788543 −0.394271 0.918994i \(-0.629003\pi\)
−0.394271 + 0.918994i \(0.629003\pi\)
\(44\) 0 0
\(45\) 0.0621653 0.00926705
\(46\) 0 0
\(47\) −7.62796 −1.11265 −0.556326 0.830964i \(-0.687790\pi\)
−0.556326 + 0.830964i \(0.687790\pi\)
\(48\) 0 0
\(49\) −6.38799 −0.912571
\(50\) 0 0
\(51\) −4.20707 −0.589107
\(52\) 0 0
\(53\) 3.26794 0.448886 0.224443 0.974487i \(-0.427944\pi\)
0.224443 + 0.974487i \(0.427944\pi\)
\(54\) 0 0
\(55\) 0.166929 0.0225087
\(56\) 0 0
\(57\) −7.94188 −1.05193
\(58\) 0 0
\(59\) 2.33656 0.304195 0.152097 0.988366i \(-0.451397\pi\)
0.152097 + 0.988366i \(0.451397\pi\)
\(60\) 0 0
\(61\) −12.7201 −1.62864 −0.814322 0.580413i \(-0.802891\pi\)
−0.814322 + 0.580413i \(0.802891\pi\)
\(62\) 0 0
\(63\) 0.782308 0.0985615
\(64\) 0 0
\(65\) 0.422313 0.0523814
\(66\) 0 0
\(67\) −1.25024 −0.152741 −0.0763706 0.997080i \(-0.524333\pi\)
−0.0763706 + 0.997080i \(0.524333\pi\)
\(68\) 0 0
\(69\) −5.42476 −0.653064
\(70\) 0 0
\(71\) 12.3346 1.46384 0.731922 0.681388i \(-0.238623\pi\)
0.731922 + 0.681388i \(0.238623\pi\)
\(72\) 0 0
\(73\) 10.0502 1.17628 0.588141 0.808758i \(-0.299860\pi\)
0.588141 + 0.808758i \(0.299860\pi\)
\(74\) 0 0
\(75\) 4.99614 0.576904
\(76\) 0 0
\(77\) 2.10069 0.239396
\(78\) 0 0
\(79\) 3.71570 0.418049 0.209025 0.977910i \(-0.432971\pi\)
0.209025 + 0.977910i \(0.432971\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.1647 1.11572 0.557861 0.829934i \(-0.311622\pi\)
0.557861 + 0.829934i \(0.311622\pi\)
\(84\) 0 0
\(85\) 0.261533 0.0283673
\(86\) 0 0
\(87\) −1.95740 −0.209855
\(88\) 0 0
\(89\) −2.74777 −0.291263 −0.145631 0.989339i \(-0.546521\pi\)
−0.145631 + 0.989339i \(0.546521\pi\)
\(90\) 0 0
\(91\) 5.31452 0.557113
\(92\) 0 0
\(93\) −2.86384 −0.296966
\(94\) 0 0
\(95\) 0.493709 0.0506534
\(96\) 0 0
\(97\) 1.54807 0.157183 0.0785914 0.996907i \(-0.474958\pi\)
0.0785914 + 0.996907i \(0.474958\pi\)
\(98\) 0 0
\(99\) 2.68525 0.269878
\(100\) 0 0
\(101\) −11.5022 −1.14451 −0.572254 0.820076i \(-0.693931\pi\)
−0.572254 + 0.820076i \(0.693931\pi\)
\(102\) 0 0
\(103\) −11.8029 −1.16298 −0.581489 0.813554i \(-0.697530\pi\)
−0.581489 + 0.813554i \(0.697530\pi\)
\(104\) 0 0
\(105\) −0.0486324 −0.00474603
\(106\) 0 0
\(107\) 15.5983 1.50795 0.753974 0.656904i \(-0.228134\pi\)
0.753974 + 0.656904i \(0.228134\pi\)
\(108\) 0 0
\(109\) 17.4215 1.66867 0.834337 0.551255i \(-0.185851\pi\)
0.834337 + 0.551255i \(0.185851\pi\)
\(110\) 0 0
\(111\) 7.68582 0.729506
\(112\) 0 0
\(113\) −4.23856 −0.398731 −0.199365 0.979925i \(-0.563888\pi\)
−0.199365 + 0.979925i \(0.563888\pi\)
\(114\) 0 0
\(115\) 0.337232 0.0314470
\(116\) 0 0
\(117\) 6.79339 0.628049
\(118\) 0 0
\(119\) 3.29122 0.301706
\(120\) 0 0
\(121\) −3.78944 −0.344494
\(122\) 0 0
\(123\) 2.03431 0.183428
\(124\) 0 0
\(125\) −0.621412 −0.0555808
\(126\) 0 0
\(127\) 6.23657 0.553406 0.276703 0.960955i \(-0.410758\pi\)
0.276703 + 0.960955i \(0.410758\pi\)
\(128\) 0 0
\(129\) 5.17082 0.455265
\(130\) 0 0
\(131\) −6.03122 −0.526950 −0.263475 0.964666i \(-0.584869\pi\)
−0.263475 + 0.964666i \(0.584869\pi\)
\(132\) 0 0
\(133\) 6.21299 0.538735
\(134\) 0 0
\(135\) −0.0621653 −0.00535033
\(136\) 0 0
\(137\) −10.9123 −0.932301 −0.466151 0.884705i \(-0.654359\pi\)
−0.466151 + 0.884705i \(0.654359\pi\)
\(138\) 0 0
\(139\) −17.5444 −1.48810 −0.744049 0.668125i \(-0.767097\pi\)
−0.744049 + 0.668125i \(0.767097\pi\)
\(140\) 0 0
\(141\) 7.62796 0.642390
\(142\) 0 0
\(143\) 18.2419 1.52547
\(144\) 0 0
\(145\) 0.121682 0.0101052
\(146\) 0 0
\(147\) 6.38799 0.526873
\(148\) 0 0
\(149\) 13.5444 1.10960 0.554802 0.831982i \(-0.312794\pi\)
0.554802 + 0.831982i \(0.312794\pi\)
\(150\) 0 0
\(151\) −4.98785 −0.405905 −0.202953 0.979189i \(-0.565054\pi\)
−0.202953 + 0.979189i \(0.565054\pi\)
\(152\) 0 0
\(153\) 4.20707 0.340121
\(154\) 0 0
\(155\) 0.178031 0.0142998
\(156\) 0 0
\(157\) −10.1442 −0.809593 −0.404797 0.914407i \(-0.632658\pi\)
−0.404797 + 0.914407i \(0.632658\pi\)
\(158\) 0 0
\(159\) −3.26794 −0.259164
\(160\) 0 0
\(161\) 4.24383 0.334461
\(162\) 0 0
\(163\) 13.5338 1.06005 0.530025 0.847982i \(-0.322183\pi\)
0.530025 + 0.847982i \(0.322183\pi\)
\(164\) 0 0
\(165\) −0.166929 −0.0129954
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 33.1501 2.55001
\(170\) 0 0
\(171\) 7.94188 0.607330
\(172\) 0 0
\(173\) 12.5805 0.956476 0.478238 0.878230i \(-0.341276\pi\)
0.478238 + 0.878230i \(0.341276\pi\)
\(174\) 0 0
\(175\) −3.90852 −0.295456
\(176\) 0 0
\(177\) −2.33656 −0.175627
\(178\) 0 0
\(179\) −2.52638 −0.188830 −0.0944151 0.995533i \(-0.530098\pi\)
−0.0944151 + 0.995533i \(0.530098\pi\)
\(180\) 0 0
\(181\) 9.84207 0.731555 0.365778 0.930702i \(-0.380803\pi\)
0.365778 + 0.930702i \(0.380803\pi\)
\(182\) 0 0
\(183\) 12.7201 0.940298
\(184\) 0 0
\(185\) −0.477791 −0.0351279
\(186\) 0 0
\(187\) 11.2970 0.826120
\(188\) 0 0
\(189\) −0.782308 −0.0569045
\(190\) 0 0
\(191\) 1.47483 0.106715 0.0533574 0.998575i \(-0.483008\pi\)
0.0533574 + 0.998575i \(0.483008\pi\)
\(192\) 0 0
\(193\) −15.2332 −1.09651 −0.548256 0.836311i \(-0.684708\pi\)
−0.548256 + 0.836311i \(0.684708\pi\)
\(194\) 0 0
\(195\) −0.422313 −0.0302424
\(196\) 0 0
\(197\) −26.2117 −1.86750 −0.933752 0.357920i \(-0.883486\pi\)
−0.933752 + 0.357920i \(0.883486\pi\)
\(198\) 0 0
\(199\) −17.1733 −1.21739 −0.608693 0.793406i \(-0.708306\pi\)
−0.608693 + 0.793406i \(0.708306\pi\)
\(200\) 0 0
\(201\) 1.25024 0.0881852
\(202\) 0 0
\(203\) 1.53129 0.107476
\(204\) 0 0
\(205\) −0.126463 −0.00883259
\(206\) 0 0
\(207\) 5.42476 0.377047
\(208\) 0 0
\(209\) 21.3259 1.47514
\(210\) 0 0
\(211\) −17.7930 −1.22492 −0.612461 0.790501i \(-0.709820\pi\)
−0.612461 + 0.790501i \(0.709820\pi\)
\(212\) 0 0
\(213\) −12.3346 −0.845151
\(214\) 0 0
\(215\) −0.321445 −0.0219224
\(216\) 0 0
\(217\) 2.24040 0.152089
\(218\) 0 0
\(219\) −10.0502 −0.679127
\(220\) 0 0
\(221\) 28.5802 1.92251
\(222\) 0 0
\(223\) 17.3886 1.16443 0.582214 0.813035i \(-0.302187\pi\)
0.582214 + 0.813035i \(0.302187\pi\)
\(224\) 0 0
\(225\) −4.99614 −0.333076
\(226\) 0 0
\(227\) 12.9318 0.858313 0.429157 0.903230i \(-0.358811\pi\)
0.429157 + 0.903230i \(0.358811\pi\)
\(228\) 0 0
\(229\) −23.6209 −1.56092 −0.780458 0.625209i \(-0.785014\pi\)
−0.780458 + 0.625209i \(0.785014\pi\)
\(230\) 0 0
\(231\) −2.10069 −0.138215
\(232\) 0 0
\(233\) −21.4494 −1.40520 −0.702598 0.711587i \(-0.747977\pi\)
−0.702598 + 0.711587i \(0.747977\pi\)
\(234\) 0 0
\(235\) −0.474194 −0.0309330
\(236\) 0 0
\(237\) −3.71570 −0.241361
\(238\) 0 0
\(239\) 24.3499 1.57506 0.787531 0.616275i \(-0.211359\pi\)
0.787531 + 0.616275i \(0.211359\pi\)
\(240\) 0 0
\(241\) 14.1838 0.913656 0.456828 0.889555i \(-0.348985\pi\)
0.456828 + 0.889555i \(0.348985\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −0.397111 −0.0253705
\(246\) 0 0
\(247\) 53.9522 3.43290
\(248\) 0 0
\(249\) −10.1647 −0.644162
\(250\) 0 0
\(251\) −8.79375 −0.555057 −0.277528 0.960717i \(-0.589515\pi\)
−0.277528 + 0.960717i \(0.589515\pi\)
\(252\) 0 0
\(253\) 14.5668 0.915809
\(254\) 0 0
\(255\) −0.261533 −0.0163779
\(256\) 0 0
\(257\) 0.693124 0.0432359 0.0216179 0.999766i \(-0.493118\pi\)
0.0216179 + 0.999766i \(0.493118\pi\)
\(258\) 0 0
\(259\) −6.01268 −0.373610
\(260\) 0 0
\(261\) 1.95740 0.121160
\(262\) 0 0
\(263\) 14.8980 0.918653 0.459326 0.888268i \(-0.348091\pi\)
0.459326 + 0.888268i \(0.348091\pi\)
\(264\) 0 0
\(265\) 0.203152 0.0124795
\(266\) 0 0
\(267\) 2.74777 0.168161
\(268\) 0 0
\(269\) −4.68185 −0.285458 −0.142729 0.989762i \(-0.545588\pi\)
−0.142729 + 0.989762i \(0.545588\pi\)
\(270\) 0 0
\(271\) 7.76324 0.471583 0.235792 0.971804i \(-0.424232\pi\)
0.235792 + 0.971804i \(0.424232\pi\)
\(272\) 0 0
\(273\) −5.31452 −0.321649
\(274\) 0 0
\(275\) −13.4159 −0.809007
\(276\) 0 0
\(277\) −15.5566 −0.934703 −0.467352 0.884072i \(-0.654792\pi\)
−0.467352 + 0.884072i \(0.654792\pi\)
\(278\) 0 0
\(279\) 2.86384 0.171454
\(280\) 0 0
\(281\) −9.94616 −0.593338 −0.296669 0.954980i \(-0.595876\pi\)
−0.296669 + 0.954980i \(0.595876\pi\)
\(282\) 0 0
\(283\) 10.1239 0.601802 0.300901 0.953655i \(-0.402713\pi\)
0.300901 + 0.953655i \(0.402713\pi\)
\(284\) 0 0
\(285\) −0.493709 −0.0292448
\(286\) 0 0
\(287\) −1.59146 −0.0939407
\(288\) 0 0
\(289\) 0.699411 0.0411418
\(290\) 0 0
\(291\) −1.54807 −0.0907495
\(292\) 0 0
\(293\) −18.1886 −1.06259 −0.531294 0.847187i \(-0.678294\pi\)
−0.531294 + 0.847187i \(0.678294\pi\)
\(294\) 0 0
\(295\) 0.145253 0.00845696
\(296\) 0 0
\(297\) −2.68525 −0.155814
\(298\) 0 0
\(299\) 36.8525 2.13123
\(300\) 0 0
\(301\) −4.04517 −0.233160
\(302\) 0 0
\(303\) 11.5022 0.660782
\(304\) 0 0
\(305\) −0.790749 −0.0452782
\(306\) 0 0
\(307\) −8.83095 −0.504009 −0.252004 0.967726i \(-0.581090\pi\)
−0.252004 + 0.967726i \(0.581090\pi\)
\(308\) 0 0
\(309\) 11.8029 0.671446
\(310\) 0 0
\(311\) −15.6371 −0.886699 −0.443349 0.896349i \(-0.646210\pi\)
−0.443349 + 0.896349i \(0.646210\pi\)
\(312\) 0 0
\(313\) −14.0920 −0.796526 −0.398263 0.917271i \(-0.630387\pi\)
−0.398263 + 0.917271i \(0.630387\pi\)
\(314\) 0 0
\(315\) 0.0486324 0.00274012
\(316\) 0 0
\(317\) 16.3138 0.916273 0.458136 0.888882i \(-0.348517\pi\)
0.458136 + 0.888882i \(0.348517\pi\)
\(318\) 0 0
\(319\) 5.25611 0.294286
\(320\) 0 0
\(321\) −15.5983 −0.870615
\(322\) 0 0
\(323\) 33.4120 1.85909
\(324\) 0 0
\(325\) −33.9407 −1.88269
\(326\) 0 0
\(327\) −17.4215 −0.963409
\(328\) 0 0
\(329\) −5.96741 −0.328994
\(330\) 0 0
\(331\) −8.18087 −0.449661 −0.224831 0.974398i \(-0.572183\pi\)
−0.224831 + 0.974398i \(0.572183\pi\)
\(332\) 0 0
\(333\) −7.68582 −0.421181
\(334\) 0 0
\(335\) −0.0777215 −0.00424638
\(336\) 0 0
\(337\) 13.5232 0.736654 0.368327 0.929696i \(-0.379931\pi\)
0.368327 + 0.929696i \(0.379931\pi\)
\(338\) 0 0
\(339\) 4.23856 0.230207
\(340\) 0 0
\(341\) 7.69012 0.416444
\(342\) 0 0
\(343\) −10.4735 −0.565518
\(344\) 0 0
\(345\) −0.337232 −0.0181559
\(346\) 0 0
\(347\) 11.4880 0.616706 0.308353 0.951272i \(-0.400222\pi\)
0.308353 + 0.951272i \(0.400222\pi\)
\(348\) 0 0
\(349\) −10.7858 −0.577351 −0.288676 0.957427i \(-0.593215\pi\)
−0.288676 + 0.957427i \(0.593215\pi\)
\(350\) 0 0
\(351\) −6.79339 −0.362604
\(352\) 0 0
\(353\) 27.6103 1.46955 0.734773 0.678313i \(-0.237289\pi\)
0.734773 + 0.678313i \(0.237289\pi\)
\(354\) 0 0
\(355\) 0.766782 0.0406965
\(356\) 0 0
\(357\) −3.29122 −0.174190
\(358\) 0 0
\(359\) 22.3864 1.18151 0.590755 0.806851i \(-0.298830\pi\)
0.590755 + 0.806851i \(0.298830\pi\)
\(360\) 0 0
\(361\) 44.0734 2.31965
\(362\) 0 0
\(363\) 3.78944 0.198894
\(364\) 0 0
\(365\) 0.624771 0.0327020
\(366\) 0 0
\(367\) 2.00665 0.104746 0.0523731 0.998628i \(-0.483322\pi\)
0.0523731 + 0.998628i \(0.483322\pi\)
\(368\) 0 0
\(369\) −2.03431 −0.105902
\(370\) 0 0
\(371\) 2.55653 0.132729
\(372\) 0 0
\(373\) −13.7031 −0.709519 −0.354759 0.934958i \(-0.615437\pi\)
−0.354759 + 0.934958i \(0.615437\pi\)
\(374\) 0 0
\(375\) 0.621412 0.0320896
\(376\) 0 0
\(377\) 13.2974 0.684850
\(378\) 0 0
\(379\) −16.3044 −0.837499 −0.418749 0.908102i \(-0.637531\pi\)
−0.418749 + 0.908102i \(0.637531\pi\)
\(380\) 0 0
\(381\) −6.23657 −0.319509
\(382\) 0 0
\(383\) 18.4410 0.942293 0.471146 0.882055i \(-0.343840\pi\)
0.471146 + 0.882055i \(0.343840\pi\)
\(384\) 0 0
\(385\) 0.130590 0.00665548
\(386\) 0 0
\(387\) −5.17082 −0.262848
\(388\) 0 0
\(389\) −20.0693 −1.01756 −0.508778 0.860898i \(-0.669902\pi\)
−0.508778 + 0.860898i \(0.669902\pi\)
\(390\) 0 0
\(391\) 22.8223 1.15417
\(392\) 0 0
\(393\) 6.03122 0.304235
\(394\) 0 0
\(395\) 0.230988 0.0116222
\(396\) 0 0
\(397\) −5.72441 −0.287300 −0.143650 0.989629i \(-0.545884\pi\)
−0.143650 + 0.989629i \(0.545884\pi\)
\(398\) 0 0
\(399\) −6.21299 −0.311039
\(400\) 0 0
\(401\) 10.6836 0.533512 0.266756 0.963764i \(-0.414048\pi\)
0.266756 + 0.963764i \(0.414048\pi\)
\(402\) 0 0
\(403\) 19.4552 0.969131
\(404\) 0 0
\(405\) 0.0621653 0.00308902
\(406\) 0 0
\(407\) −20.6384 −1.02301
\(408\) 0 0
\(409\) 7.95014 0.393109 0.196554 0.980493i \(-0.437025\pi\)
0.196554 + 0.980493i \(0.437025\pi\)
\(410\) 0 0
\(411\) 10.9123 0.538264
\(412\) 0 0
\(413\) 1.82791 0.0899457
\(414\) 0 0
\(415\) 0.631892 0.0310183
\(416\) 0 0
\(417\) 17.5444 0.859154
\(418\) 0 0
\(419\) 4.68925 0.229085 0.114542 0.993418i \(-0.463460\pi\)
0.114542 + 0.993418i \(0.463460\pi\)
\(420\) 0 0
\(421\) −9.84889 −0.480006 −0.240003 0.970772i \(-0.577148\pi\)
−0.240003 + 0.970772i \(0.577148\pi\)
\(422\) 0 0
\(423\) −7.62796 −0.370884
\(424\) 0 0
\(425\) −21.0191 −1.01957
\(426\) 0 0
\(427\) −9.95105 −0.481565
\(428\) 0 0
\(429\) −18.2419 −0.880729
\(430\) 0 0
\(431\) 31.8797 1.53559 0.767796 0.640694i \(-0.221353\pi\)
0.767796 + 0.640694i \(0.221353\pi\)
\(432\) 0 0
\(433\) −5.80718 −0.279075 −0.139538 0.990217i \(-0.544562\pi\)
−0.139538 + 0.990217i \(0.544562\pi\)
\(434\) 0 0
\(435\) −0.121682 −0.00583422
\(436\) 0 0
\(437\) 43.0828 2.06093
\(438\) 0 0
\(439\) −25.0785 −1.19693 −0.598465 0.801149i \(-0.704222\pi\)
−0.598465 + 0.801149i \(0.704222\pi\)
\(440\) 0 0
\(441\) −6.38799 −0.304190
\(442\) 0 0
\(443\) −31.2338 −1.48396 −0.741981 0.670421i \(-0.766114\pi\)
−0.741981 + 0.670421i \(0.766114\pi\)
\(444\) 0 0
\(445\) −0.170816 −0.00809743
\(446\) 0 0
\(447\) −13.5444 −0.640630
\(448\) 0 0
\(449\) 4.12427 0.194636 0.0973182 0.995253i \(-0.468974\pi\)
0.0973182 + 0.995253i \(0.468974\pi\)
\(450\) 0 0
\(451\) −5.46263 −0.257225
\(452\) 0 0
\(453\) 4.98785 0.234350
\(454\) 0 0
\(455\) 0.330378 0.0154884
\(456\) 0 0
\(457\) 0.0521369 0.00243886 0.00121943 0.999999i \(-0.499612\pi\)
0.00121943 + 0.999999i \(0.499612\pi\)
\(458\) 0 0
\(459\) −4.20707 −0.196369
\(460\) 0 0
\(461\) −5.15218 −0.239961 −0.119981 0.992776i \(-0.538283\pi\)
−0.119981 + 0.992776i \(0.538283\pi\)
\(462\) 0 0
\(463\) −30.4668 −1.41591 −0.707957 0.706256i \(-0.750383\pi\)
−0.707957 + 0.706256i \(0.750383\pi\)
\(464\) 0 0
\(465\) −0.178031 −0.00825601
\(466\) 0 0
\(467\) −14.8990 −0.689444 −0.344722 0.938705i \(-0.612027\pi\)
−0.344722 + 0.938705i \(0.612027\pi\)
\(468\) 0 0
\(469\) −0.978073 −0.0451632
\(470\) 0 0
\(471\) 10.1442 0.467419
\(472\) 0 0
\(473\) −13.8849 −0.638430
\(474\) 0 0
\(475\) −39.6787 −1.82058
\(476\) 0 0
\(477\) 3.26794 0.149629
\(478\) 0 0
\(479\) 10.7395 0.490702 0.245351 0.969434i \(-0.421097\pi\)
0.245351 + 0.969434i \(0.421097\pi\)
\(480\) 0 0
\(481\) −52.2128 −2.38070
\(482\) 0 0
\(483\) −4.24383 −0.193101
\(484\) 0 0
\(485\) 0.0962362 0.00436986
\(486\) 0 0
\(487\) 27.7875 1.25917 0.629587 0.776930i \(-0.283224\pi\)
0.629587 + 0.776930i \(0.283224\pi\)
\(488\) 0 0
\(489\) −13.5338 −0.612020
\(490\) 0 0
\(491\) 20.3242 0.917216 0.458608 0.888639i \(-0.348348\pi\)
0.458608 + 0.888639i \(0.348348\pi\)
\(492\) 0 0
\(493\) 8.23492 0.370882
\(494\) 0 0
\(495\) 0.166929 0.00750291
\(496\) 0 0
\(497\) 9.64943 0.432836
\(498\) 0 0
\(499\) 25.4727 1.14032 0.570158 0.821535i \(-0.306882\pi\)
0.570158 + 0.821535i \(0.306882\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) 16.3937 0.730959 0.365480 0.930819i \(-0.380905\pi\)
0.365480 + 0.930819i \(0.380905\pi\)
\(504\) 0 0
\(505\) −0.715035 −0.0318186
\(506\) 0 0
\(507\) −33.1501 −1.47225
\(508\) 0 0
\(509\) 27.3496 1.21225 0.606124 0.795370i \(-0.292723\pi\)
0.606124 + 0.795370i \(0.292723\pi\)
\(510\) 0 0
\(511\) 7.86232 0.347809
\(512\) 0 0
\(513\) −7.94188 −0.350642
\(514\) 0 0
\(515\) −0.733733 −0.0323321
\(516\) 0 0
\(517\) −20.4830 −0.900840
\(518\) 0 0
\(519\) −12.5805 −0.552222
\(520\) 0 0
\(521\) 18.2555 0.799788 0.399894 0.916562i \(-0.369047\pi\)
0.399894 + 0.916562i \(0.369047\pi\)
\(522\) 0 0
\(523\) 14.8011 0.647205 0.323603 0.946193i \(-0.395106\pi\)
0.323603 + 0.946193i \(0.395106\pi\)
\(524\) 0 0
\(525\) 3.90852 0.170582
\(526\) 0 0
\(527\) 12.0484 0.524835
\(528\) 0 0
\(529\) 6.42801 0.279479
\(530\) 0 0
\(531\) 2.33656 0.101398
\(532\) 0 0
\(533\) −13.8199 −0.598604
\(534\) 0 0
\(535\) 0.969675 0.0419227
\(536\) 0 0
\(537\) 2.52638 0.109021
\(538\) 0 0
\(539\) −17.1534 −0.738847
\(540\) 0 0
\(541\) 0.165065 0.00709671 0.00354835 0.999994i \(-0.498871\pi\)
0.00354835 + 0.999994i \(0.498871\pi\)
\(542\) 0 0
\(543\) −9.84207 −0.422363
\(544\) 0 0
\(545\) 1.08301 0.0463910
\(546\) 0 0
\(547\) −14.3907 −0.615300 −0.307650 0.951500i \(-0.599543\pi\)
−0.307650 + 0.951500i \(0.599543\pi\)
\(548\) 0 0
\(549\) −12.7201 −0.542881
\(550\) 0 0
\(551\) 15.5454 0.662258
\(552\) 0 0
\(553\) 2.90682 0.123611
\(554\) 0 0
\(555\) 0.477791 0.0202811
\(556\) 0 0
\(557\) 3.70371 0.156931 0.0784656 0.996917i \(-0.474998\pi\)
0.0784656 + 0.996917i \(0.474998\pi\)
\(558\) 0 0
\(559\) −35.1274 −1.48573
\(560\) 0 0
\(561\) −11.2970 −0.476961
\(562\) 0 0
\(563\) −39.3783 −1.65960 −0.829798 0.558064i \(-0.811544\pi\)
−0.829798 + 0.558064i \(0.811544\pi\)
\(564\) 0 0
\(565\) −0.263491 −0.0110852
\(566\) 0 0
\(567\) 0.782308 0.0328538
\(568\) 0 0
\(569\) −19.8473 −0.832041 −0.416020 0.909355i \(-0.636575\pi\)
−0.416020 + 0.909355i \(0.636575\pi\)
\(570\) 0 0
\(571\) −14.1013 −0.590120 −0.295060 0.955479i \(-0.595340\pi\)
−0.295060 + 0.955479i \(0.595340\pi\)
\(572\) 0 0
\(573\) −1.47483 −0.0616118
\(574\) 0 0
\(575\) −27.1028 −1.13027
\(576\) 0 0
\(577\) 25.5547 1.06385 0.531927 0.846790i \(-0.321468\pi\)
0.531927 + 0.846790i \(0.321468\pi\)
\(578\) 0 0
\(579\) 15.2332 0.633071
\(580\) 0 0
\(581\) 7.95193 0.329902
\(582\) 0 0
\(583\) 8.77522 0.363433
\(584\) 0 0
\(585\) 0.422313 0.0174605
\(586\) 0 0
\(587\) 24.2117 0.999323 0.499661 0.866221i \(-0.333458\pi\)
0.499661 + 0.866221i \(0.333458\pi\)
\(588\) 0 0
\(589\) 22.7443 0.937161
\(590\) 0 0
\(591\) 26.2117 1.07820
\(592\) 0 0
\(593\) −19.2622 −0.791005 −0.395503 0.918465i \(-0.629430\pi\)
−0.395503 + 0.918465i \(0.629430\pi\)
\(594\) 0 0
\(595\) 0.204600 0.00838777
\(596\) 0 0
\(597\) 17.1733 0.702858
\(598\) 0 0
\(599\) 3.33279 0.136174 0.0680870 0.997679i \(-0.478310\pi\)
0.0680870 + 0.997679i \(0.478310\pi\)
\(600\) 0 0
\(601\) 23.2581 0.948719 0.474359 0.880331i \(-0.342680\pi\)
0.474359 + 0.880331i \(0.342680\pi\)
\(602\) 0 0
\(603\) −1.25024 −0.0509137
\(604\) 0 0
\(605\) −0.235571 −0.00957733
\(606\) 0 0
\(607\) 9.74996 0.395739 0.197869 0.980228i \(-0.436598\pi\)
0.197869 + 0.980228i \(0.436598\pi\)
\(608\) 0 0
\(609\) −1.53129 −0.0620510
\(610\) 0 0
\(611\) −51.8197 −2.09640
\(612\) 0 0
\(613\) 5.41453 0.218691 0.109345 0.994004i \(-0.465125\pi\)
0.109345 + 0.994004i \(0.465125\pi\)
\(614\) 0 0
\(615\) 0.126463 0.00509950
\(616\) 0 0
\(617\) 37.5966 1.51358 0.756792 0.653656i \(-0.226766\pi\)
0.756792 + 0.653656i \(0.226766\pi\)
\(618\) 0 0
\(619\) −22.5427 −0.906066 −0.453033 0.891494i \(-0.649658\pi\)
−0.453033 + 0.891494i \(0.649658\pi\)
\(620\) 0 0
\(621\) −5.42476 −0.217688
\(622\) 0 0
\(623\) −2.14960 −0.0861219
\(624\) 0 0
\(625\) 24.9420 0.997682
\(626\) 0 0
\(627\) −21.3259 −0.851675
\(628\) 0 0
\(629\) −32.3348 −1.28927
\(630\) 0 0
\(631\) 34.8468 1.38723 0.693615 0.720346i \(-0.256017\pi\)
0.693615 + 0.720346i \(0.256017\pi\)
\(632\) 0 0
\(633\) 17.7930 0.707209
\(634\) 0 0
\(635\) 0.387698 0.0153853
\(636\) 0 0
\(637\) −43.3961 −1.71942
\(638\) 0 0
\(639\) 12.3346 0.487948
\(640\) 0 0
\(641\) 23.8042 0.940211 0.470106 0.882610i \(-0.344216\pi\)
0.470106 + 0.882610i \(0.344216\pi\)
\(642\) 0 0
\(643\) −43.8213 −1.72814 −0.864072 0.503369i \(-0.832094\pi\)
−0.864072 + 0.503369i \(0.832094\pi\)
\(644\) 0 0
\(645\) 0.321445 0.0126569
\(646\) 0 0
\(647\) 25.9087 1.01858 0.509288 0.860596i \(-0.329909\pi\)
0.509288 + 0.860596i \(0.329909\pi\)
\(648\) 0 0
\(649\) 6.27426 0.246286
\(650\) 0 0
\(651\) −2.24040 −0.0878084
\(652\) 0 0
\(653\) −1.32912 −0.0520125 −0.0260063 0.999662i \(-0.508279\pi\)
−0.0260063 + 0.999662i \(0.508279\pi\)
\(654\) 0 0
\(655\) −0.374932 −0.0146498
\(656\) 0 0
\(657\) 10.0502 0.392094
\(658\) 0 0
\(659\) 3.44424 0.134168 0.0670842 0.997747i \(-0.478630\pi\)
0.0670842 + 0.997747i \(0.478630\pi\)
\(660\) 0 0
\(661\) −43.5969 −1.69572 −0.847862 0.530217i \(-0.822111\pi\)
−0.847862 + 0.530217i \(0.822111\pi\)
\(662\) 0 0
\(663\) −28.5802 −1.10996
\(664\) 0 0
\(665\) 0.386232 0.0149774
\(666\) 0 0
\(667\) 10.6184 0.411147
\(668\) 0 0
\(669\) −17.3886 −0.672283
\(670\) 0 0
\(671\) −34.1567 −1.31860
\(672\) 0 0
\(673\) −15.8422 −0.610674 −0.305337 0.952244i \(-0.598769\pi\)
−0.305337 + 0.952244i \(0.598769\pi\)
\(674\) 0 0
\(675\) 4.99614 0.192301
\(676\) 0 0
\(677\) −33.0836 −1.27151 −0.635753 0.771893i \(-0.719310\pi\)
−0.635753 + 0.771893i \(0.719310\pi\)
\(678\) 0 0
\(679\) 1.21107 0.0464765
\(680\) 0 0
\(681\) −12.9318 −0.495547
\(682\) 0 0
\(683\) 7.85853 0.300698 0.150349 0.988633i \(-0.451960\pi\)
0.150349 + 0.988633i \(0.451960\pi\)
\(684\) 0 0
\(685\) −0.678366 −0.0259190
\(686\) 0 0
\(687\) 23.6209 0.901195
\(688\) 0 0
\(689\) 22.2004 0.845766
\(690\) 0 0
\(691\) −30.6961 −1.16773 −0.583867 0.811849i \(-0.698461\pi\)
−0.583867 + 0.811849i \(0.698461\pi\)
\(692\) 0 0
\(693\) 2.10069 0.0797987
\(694\) 0 0
\(695\) −1.09065 −0.0413708
\(696\) 0 0
\(697\) −8.55848 −0.324176
\(698\) 0 0
\(699\) 21.4494 0.811290
\(700\) 0 0
\(701\) −19.9235 −0.752498 −0.376249 0.926519i \(-0.622786\pi\)
−0.376249 + 0.926519i \(0.622786\pi\)
\(702\) 0 0
\(703\) −61.0399 −2.30216
\(704\) 0 0
\(705\) 0.474194 0.0178592
\(706\) 0 0
\(707\) −8.99823 −0.338413
\(708\) 0 0
\(709\) 16.7984 0.630879 0.315439 0.948946i \(-0.397848\pi\)
0.315439 + 0.948946i \(0.397848\pi\)
\(710\) 0 0
\(711\) 3.71570 0.139350
\(712\) 0 0
\(713\) 15.5356 0.581814
\(714\) 0 0
\(715\) 1.13401 0.0424097
\(716\) 0 0
\(717\) −24.3499 −0.909362
\(718\) 0 0
\(719\) 32.9584 1.22914 0.614571 0.788862i \(-0.289329\pi\)
0.614571 + 0.788862i \(0.289329\pi\)
\(720\) 0 0
\(721\) −9.23354 −0.343875
\(722\) 0 0
\(723\) −14.1838 −0.527500
\(724\) 0 0
\(725\) −9.77944 −0.363199
\(726\) 0 0
\(727\) −3.70734 −0.137498 −0.0687488 0.997634i \(-0.521901\pi\)
−0.0687488 + 0.997634i \(0.521901\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −21.7540 −0.804600
\(732\) 0 0
\(733\) 36.2151 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(734\) 0 0
\(735\) 0.397111 0.0146477
\(736\) 0 0
\(737\) −3.35721 −0.123664
\(738\) 0 0
\(739\) −10.7819 −0.396620 −0.198310 0.980139i \(-0.563545\pi\)
−0.198310 + 0.980139i \(0.563545\pi\)
\(740\) 0 0
\(741\) −53.9522 −1.98198
\(742\) 0 0
\(743\) −19.6242 −0.719941 −0.359970 0.932964i \(-0.617213\pi\)
−0.359970 + 0.932964i \(0.617213\pi\)
\(744\) 0 0
\(745\) 0.841994 0.0308483
\(746\) 0 0
\(747\) 10.1647 0.371907
\(748\) 0 0
\(749\) 12.2027 0.445877
\(750\) 0 0
\(751\) −4.30083 −0.156939 −0.0784697 0.996916i \(-0.525003\pi\)
−0.0784697 + 0.996916i \(0.525003\pi\)
\(752\) 0 0
\(753\) 8.79375 0.320462
\(754\) 0 0
\(755\) −0.310071 −0.0112846
\(756\) 0 0
\(757\) −7.62298 −0.277062 −0.138531 0.990358i \(-0.544238\pi\)
−0.138531 + 0.990358i \(0.544238\pi\)
\(758\) 0 0
\(759\) −14.5668 −0.528742
\(760\) 0 0
\(761\) −17.0907 −0.619538 −0.309769 0.950812i \(-0.600252\pi\)
−0.309769 + 0.950812i \(0.600252\pi\)
\(762\) 0 0
\(763\) 13.6289 0.493401
\(764\) 0 0
\(765\) 0.261533 0.00945576
\(766\) 0 0
\(767\) 15.8732 0.573147
\(768\) 0 0
\(769\) 2.19109 0.0790126 0.0395063 0.999219i \(-0.487421\pi\)
0.0395063 + 0.999219i \(0.487421\pi\)
\(770\) 0 0
\(771\) −0.693124 −0.0249623
\(772\) 0 0
\(773\) 9.07474 0.326396 0.163198 0.986593i \(-0.447819\pi\)
0.163198 + 0.986593i \(0.447819\pi\)
\(774\) 0 0
\(775\) −14.3081 −0.513963
\(776\) 0 0
\(777\) 6.01268 0.215704
\(778\) 0 0
\(779\) −16.1562 −0.578857
\(780\) 0 0
\(781\) 33.1214 1.18518
\(782\) 0 0
\(783\) −1.95740 −0.0699518
\(784\) 0 0
\(785\) −0.630615 −0.0225076
\(786\) 0 0
\(787\) 16.6175 0.592349 0.296174 0.955134i \(-0.404289\pi\)
0.296174 + 0.955134i \(0.404289\pi\)
\(788\) 0 0
\(789\) −14.8980 −0.530384
\(790\) 0 0
\(791\) −3.31586 −0.117898
\(792\) 0 0
\(793\) −86.4127 −3.06860
\(794\) 0 0
\(795\) −0.203152 −0.00720506
\(796\) 0 0
\(797\) −30.1950 −1.06956 −0.534781 0.844991i \(-0.679606\pi\)
−0.534781 + 0.844991i \(0.679606\pi\)
\(798\) 0 0
\(799\) −32.0913 −1.13531
\(800\) 0 0
\(801\) −2.74777 −0.0970875
\(802\) 0 0
\(803\) 26.9872 0.952358
\(804\) 0 0
\(805\) 0.263819 0.00929839
\(806\) 0 0
\(807\) 4.68185 0.164809
\(808\) 0 0
\(809\) −41.1345 −1.44621 −0.723106 0.690737i \(-0.757286\pi\)
−0.723106 + 0.690737i \(0.757286\pi\)
\(810\) 0 0
\(811\) 20.9200 0.734599 0.367299 0.930103i \(-0.380282\pi\)
0.367299 + 0.930103i \(0.380282\pi\)
\(812\) 0 0
\(813\) −7.76324 −0.272269
\(814\) 0 0
\(815\) 0.841332 0.0294706
\(816\) 0 0
\(817\) −41.0660 −1.43672
\(818\) 0 0
\(819\) 5.31452 0.185704
\(820\) 0 0
\(821\) −53.3825 −1.86306 −0.931531 0.363661i \(-0.881527\pi\)
−0.931531 + 0.363661i \(0.881527\pi\)
\(822\) 0 0
\(823\) 38.6317 1.34661 0.673307 0.739363i \(-0.264873\pi\)
0.673307 + 0.739363i \(0.264873\pi\)
\(824\) 0 0
\(825\) 13.4159 0.467081
\(826\) 0 0
\(827\) −29.9769 −1.04240 −0.521199 0.853435i \(-0.674515\pi\)
−0.521199 + 0.853435i \(0.674515\pi\)
\(828\) 0 0
\(829\) 35.8928 1.24661 0.623304 0.781980i \(-0.285790\pi\)
0.623304 + 0.781980i \(0.285790\pi\)
\(830\) 0 0
\(831\) 15.5566 0.539651
\(832\) 0 0
\(833\) −26.8747 −0.931154
\(834\) 0 0
\(835\) −0.0621653 −0.00215132
\(836\) 0 0
\(837\) −2.86384 −0.0989888
\(838\) 0 0
\(839\) 30.0957 1.03902 0.519510 0.854464i \(-0.326115\pi\)
0.519510 + 0.854464i \(0.326115\pi\)
\(840\) 0 0
\(841\) −25.1686 −0.867882
\(842\) 0 0
\(843\) 9.94616 0.342564
\(844\) 0 0
\(845\) 2.06078 0.0708931
\(846\) 0 0
\(847\) −2.96451 −0.101862
\(848\) 0 0
\(849\) −10.1239 −0.347450
\(850\) 0 0
\(851\) −41.6937 −1.42924
\(852\) 0 0
\(853\) −7.80634 −0.267284 −0.133642 0.991030i \(-0.542667\pi\)
−0.133642 + 0.991030i \(0.542667\pi\)
\(854\) 0 0
\(855\) 0.493709 0.0168845
\(856\) 0 0
\(857\) 47.4720 1.62161 0.810806 0.585315i \(-0.199029\pi\)
0.810806 + 0.585315i \(0.199029\pi\)
\(858\) 0 0
\(859\) −14.5484 −0.496385 −0.248192 0.968711i \(-0.579836\pi\)
−0.248192 + 0.968711i \(0.579836\pi\)
\(860\) 0 0
\(861\) 1.59146 0.0542367
\(862\) 0 0
\(863\) 43.9903 1.49745 0.748724 0.662882i \(-0.230667\pi\)
0.748724 + 0.662882i \(0.230667\pi\)
\(864\) 0 0
\(865\) 0.782069 0.0265911
\(866\) 0 0
\(867\) −0.699411 −0.0237532
\(868\) 0 0
\(869\) 9.97759 0.338467
\(870\) 0 0
\(871\) −8.49337 −0.287787
\(872\) 0 0
\(873\) 1.54807 0.0523943
\(874\) 0 0
\(875\) −0.486136 −0.0164344
\(876\) 0 0
\(877\) −18.2054 −0.614754 −0.307377 0.951588i \(-0.599451\pi\)
−0.307377 + 0.951588i \(0.599451\pi\)
\(878\) 0 0
\(879\) 18.1886 0.613486
\(880\) 0 0
\(881\) −54.0285 −1.82027 −0.910134 0.414315i \(-0.864021\pi\)
−0.910134 + 0.414315i \(0.864021\pi\)
\(882\) 0 0
\(883\) −7.34816 −0.247285 −0.123643 0.992327i \(-0.539458\pi\)
−0.123643 + 0.992327i \(0.539458\pi\)
\(884\) 0 0
\(885\) −0.145253 −0.00488263
\(886\) 0 0
\(887\) 9.37946 0.314931 0.157466 0.987524i \(-0.449668\pi\)
0.157466 + 0.987524i \(0.449668\pi\)
\(888\) 0 0
\(889\) 4.87892 0.163634
\(890\) 0 0
\(891\) 2.68525 0.0899592
\(892\) 0 0
\(893\) −60.5803 −2.02724
\(894\) 0 0
\(895\) −0.157053 −0.00524970
\(896\) 0 0
\(897\) −36.8525 −1.23047
\(898\) 0 0
\(899\) 5.60568 0.186960
\(900\) 0 0
\(901\) 13.7484 0.458026
\(902\) 0 0
\(903\) 4.04517 0.134615
\(904\) 0 0
\(905\) 0.611835 0.0203381
\(906\) 0 0
\(907\) −43.2416 −1.43582 −0.717908 0.696138i \(-0.754900\pi\)
−0.717908 + 0.696138i \(0.754900\pi\)
\(908\) 0 0
\(909\) −11.5022 −0.381503
\(910\) 0 0
\(911\) −32.3176 −1.07073 −0.535366 0.844620i \(-0.679826\pi\)
−0.535366 + 0.844620i \(0.679826\pi\)
\(912\) 0 0
\(913\) 27.2948 0.903325
\(914\) 0 0
\(915\) 0.790749 0.0261414
\(916\) 0 0
\(917\) −4.71827 −0.155811
\(918\) 0 0
\(919\) −27.5180 −0.907735 −0.453868 0.891069i \(-0.649956\pi\)
−0.453868 + 0.891069i \(0.649956\pi\)
\(920\) 0 0
\(921\) 8.83095 0.290990
\(922\) 0 0
\(923\) 83.7935 2.75810
\(924\) 0 0
\(925\) 38.3994 1.26257
\(926\) 0 0
\(927\) −11.8029 −0.387660
\(928\) 0 0
\(929\) 56.3400 1.84846 0.924228 0.381841i \(-0.124710\pi\)
0.924228 + 0.381841i \(0.124710\pi\)
\(930\) 0 0
\(931\) −50.7327 −1.66270
\(932\) 0 0
\(933\) 15.6371 0.511936
\(934\) 0 0
\(935\) 0.702282 0.0229671
\(936\) 0 0
\(937\) 21.2162 0.693104 0.346552 0.938031i \(-0.387352\pi\)
0.346552 + 0.938031i \(0.387352\pi\)
\(938\) 0 0
\(939\) 14.0920 0.459874
\(940\) 0 0
\(941\) 51.1429 1.66721 0.833606 0.552360i \(-0.186273\pi\)
0.833606 + 0.552360i \(0.186273\pi\)
\(942\) 0 0
\(943\) −11.0356 −0.359370
\(944\) 0 0
\(945\) −0.0486324 −0.00158201
\(946\) 0 0
\(947\) −30.3320 −0.985658 −0.492829 0.870126i \(-0.664037\pi\)
−0.492829 + 0.870126i \(0.664037\pi\)
\(948\) 0 0
\(949\) 68.2746 2.21629
\(950\) 0 0
\(951\) −16.3138 −0.529010
\(952\) 0 0
\(953\) −30.9277 −1.00184 −0.500922 0.865492i \(-0.667006\pi\)
−0.500922 + 0.865492i \(0.667006\pi\)
\(954\) 0 0
\(955\) 0.0916830 0.00296679
\(956\) 0 0
\(957\) −5.25611 −0.169906
\(958\) 0 0
\(959\) −8.53678 −0.275667
\(960\) 0 0
\(961\) −22.7984 −0.735433
\(962\) 0 0
\(963\) 15.5983 0.502650
\(964\) 0 0
\(965\) −0.946977 −0.0304843
\(966\) 0 0
\(967\) 5.89168 0.189464 0.0947318 0.995503i \(-0.469801\pi\)
0.0947318 + 0.995503i \(0.469801\pi\)
\(968\) 0 0
\(969\) −33.4120 −1.07335
\(970\) 0 0
\(971\) 27.9307 0.896340 0.448170 0.893948i \(-0.352076\pi\)
0.448170 + 0.893948i \(0.352076\pi\)
\(972\) 0 0
\(973\) −13.7251 −0.440008
\(974\) 0 0
\(975\) 33.9407 1.08697
\(976\) 0 0
\(977\) 17.6646 0.565141 0.282570 0.959247i \(-0.408813\pi\)
0.282570 + 0.959247i \(0.408813\pi\)
\(978\) 0 0
\(979\) −7.37844 −0.235816
\(980\) 0 0
\(981\) 17.4215 0.556224
\(982\) 0 0
\(983\) 26.1753 0.834863 0.417432 0.908708i \(-0.362930\pi\)
0.417432 + 0.908708i \(0.362930\pi\)
\(984\) 0 0
\(985\) −1.62946 −0.0519188
\(986\) 0 0
\(987\) 5.96741 0.189945
\(988\) 0 0
\(989\) −28.0505 −0.891953
\(990\) 0 0
\(991\) −0.799067 −0.0253832 −0.0126916 0.999919i \(-0.504040\pi\)
−0.0126916 + 0.999919i \(0.504040\pi\)
\(992\) 0 0
\(993\) 8.18087 0.259612
\(994\) 0 0
\(995\) −1.06758 −0.0338447
\(996\) 0 0
\(997\) 48.0295 1.52111 0.760555 0.649274i \(-0.224927\pi\)
0.760555 + 0.649274i \(0.224927\pi\)
\(998\) 0 0
\(999\) 7.68582 0.243169
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 4008.2.a.g.1.5 7
4.3 odd 2 8016.2.a.w.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.g.1.5 7 1.1 even 1 trivial
8016.2.a.w.1.5 7 4.3 odd 2