Properties

Label 8004.2.a.k.1.9
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $18$
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] = [8004,2,Mod(1,8004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8004, 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("8004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.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: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + 76571 x^{10} - 655793 x^{9} - 114554 x^{8} + 2789438 x^{7} + \cdots - 115488 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(0.235801\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.235801 q^{5} +1.32081 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.235801 q^{5} +1.32081 q^{7} +1.00000 q^{9} +1.55746 q^{11} -3.66795 q^{13} +0.235801 q^{15} -2.58826 q^{17} -1.45355 q^{19} +1.32081 q^{21} -1.00000 q^{23} -4.94440 q^{25} +1.00000 q^{27} +1.00000 q^{29} +6.25177 q^{31} +1.55746 q^{33} +0.311448 q^{35} -9.15202 q^{37} -3.66795 q^{39} +11.1000 q^{41} +9.13805 q^{43} +0.235801 q^{45} +7.96289 q^{47} -5.25546 q^{49} -2.58826 q^{51} -5.80551 q^{53} +0.367252 q^{55} -1.45355 q^{57} +5.27926 q^{59} +12.9066 q^{61} +1.32081 q^{63} -0.864906 q^{65} -0.0629591 q^{67} -1.00000 q^{69} +1.10433 q^{71} +12.3233 q^{73} -4.94440 q^{75} +2.05711 q^{77} +12.5754 q^{79} +1.00000 q^{81} +11.0990 q^{83} -0.610314 q^{85} +1.00000 q^{87} +9.93825 q^{89} -4.84466 q^{91} +6.25177 q^{93} -0.342749 q^{95} -0.659173 q^{97} +1.55746 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) \) 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
\(5\) 0.235801 0.105453 0.0527267 0.998609i \(-0.483209\pi\)
0.0527267 + 0.998609i \(0.483209\pi\)
\(6\) 0 0
\(7\) 1.32081 0.499219 0.249609 0.968347i \(-0.419698\pi\)
0.249609 + 0.968347i \(0.419698\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.55746 0.469593 0.234797 0.972045i \(-0.424558\pi\)
0.234797 + 0.972045i \(0.424558\pi\)
\(12\) 0 0
\(13\) −3.66795 −1.01731 −0.508653 0.860972i \(-0.669856\pi\)
−0.508653 + 0.860972i \(0.669856\pi\)
\(14\) 0 0
\(15\) 0.235801 0.0608835
\(16\) 0 0
\(17\) −2.58826 −0.627745 −0.313872 0.949465i \(-0.601626\pi\)
−0.313872 + 0.949465i \(0.601626\pi\)
\(18\) 0 0
\(19\) −1.45355 −0.333468 −0.166734 0.986002i \(-0.553322\pi\)
−0.166734 + 0.986002i \(0.553322\pi\)
\(20\) 0 0
\(21\) 1.32081 0.288224
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.94440 −0.988880
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 6.25177 1.12285 0.561425 0.827527i \(-0.310253\pi\)
0.561425 + 0.827527i \(0.310253\pi\)
\(32\) 0 0
\(33\) 1.55746 0.271120
\(34\) 0 0
\(35\) 0.311448 0.0526443
\(36\) 0 0
\(37\) −9.15202 −1.50458 −0.752291 0.658831i \(-0.771051\pi\)
−0.752291 + 0.658831i \(0.771051\pi\)
\(38\) 0 0
\(39\) −3.66795 −0.587342
\(40\) 0 0
\(41\) 11.1000 1.73353 0.866764 0.498719i \(-0.166196\pi\)
0.866764 + 0.498719i \(0.166196\pi\)
\(42\) 0 0
\(43\) 9.13805 1.39354 0.696770 0.717295i \(-0.254620\pi\)
0.696770 + 0.717295i \(0.254620\pi\)
\(44\) 0 0
\(45\) 0.235801 0.0351511
\(46\) 0 0
\(47\) 7.96289 1.16151 0.580753 0.814080i \(-0.302758\pi\)
0.580753 + 0.814080i \(0.302758\pi\)
\(48\) 0 0
\(49\) −5.25546 −0.750781
\(50\) 0 0
\(51\) −2.58826 −0.362429
\(52\) 0 0
\(53\) −5.80551 −0.797448 −0.398724 0.917071i \(-0.630547\pi\)
−0.398724 + 0.917071i \(0.630547\pi\)
\(54\) 0 0
\(55\) 0.367252 0.0495202
\(56\) 0 0
\(57\) −1.45355 −0.192528
\(58\) 0 0
\(59\) 5.27926 0.687301 0.343650 0.939098i \(-0.388337\pi\)
0.343650 + 0.939098i \(0.388337\pi\)
\(60\) 0 0
\(61\) 12.9066 1.65252 0.826260 0.563289i \(-0.190464\pi\)
0.826260 + 0.563289i \(0.190464\pi\)
\(62\) 0 0
\(63\) 1.32081 0.166406
\(64\) 0 0
\(65\) −0.864906 −0.107278
\(66\) 0 0
\(67\) −0.0629591 −0.00769167 −0.00384584 0.999993i \(-0.501224\pi\)
−0.00384584 + 0.999993i \(0.501224\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 1.10433 0.131060 0.0655298 0.997851i \(-0.479126\pi\)
0.0655298 + 0.997851i \(0.479126\pi\)
\(72\) 0 0
\(73\) 12.3233 1.44233 0.721166 0.692763i \(-0.243607\pi\)
0.721166 + 0.692763i \(0.243607\pi\)
\(74\) 0 0
\(75\) −4.94440 −0.570930
\(76\) 0 0
\(77\) 2.05711 0.234430
\(78\) 0 0
\(79\) 12.5754 1.41484 0.707422 0.706792i \(-0.249858\pi\)
0.707422 + 0.706792i \(0.249858\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.0990 1.21827 0.609137 0.793065i \(-0.291516\pi\)
0.609137 + 0.793065i \(0.291516\pi\)
\(84\) 0 0
\(85\) −0.610314 −0.0661978
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 9.93825 1.05345 0.526726 0.850035i \(-0.323419\pi\)
0.526726 + 0.850035i \(0.323419\pi\)
\(90\) 0 0
\(91\) −4.84466 −0.507858
\(92\) 0 0
\(93\) 6.25177 0.648278
\(94\) 0 0
\(95\) −0.342749 −0.0351653
\(96\) 0 0
\(97\) −0.659173 −0.0669289 −0.0334645 0.999440i \(-0.510654\pi\)
−0.0334645 + 0.999440i \(0.510654\pi\)
\(98\) 0 0
\(99\) 1.55746 0.156531
\(100\) 0 0
\(101\) −16.8646 −1.67809 −0.839043 0.544065i \(-0.816884\pi\)
−0.839043 + 0.544065i \(0.816884\pi\)
\(102\) 0 0
\(103\) −12.4751 −1.22921 −0.614604 0.788836i \(-0.710684\pi\)
−0.614604 + 0.788836i \(0.710684\pi\)
\(104\) 0 0
\(105\) 0.311448 0.0303942
\(106\) 0 0
\(107\) 17.7008 1.71120 0.855598 0.517640i \(-0.173189\pi\)
0.855598 + 0.517640i \(0.173189\pi\)
\(108\) 0 0
\(109\) 18.0712 1.73090 0.865452 0.500992i \(-0.167032\pi\)
0.865452 + 0.500992i \(0.167032\pi\)
\(110\) 0 0
\(111\) −9.15202 −0.868671
\(112\) 0 0
\(113\) −5.31581 −0.500069 −0.250035 0.968237i \(-0.580442\pi\)
−0.250035 + 0.968237i \(0.580442\pi\)
\(114\) 0 0
\(115\) −0.235801 −0.0219886
\(116\) 0 0
\(117\) −3.66795 −0.339102
\(118\) 0 0
\(119\) −3.41859 −0.313382
\(120\) 0 0
\(121\) −8.57430 −0.779482
\(122\) 0 0
\(123\) 11.1000 1.00085
\(124\) 0 0
\(125\) −2.34490 −0.209734
\(126\) 0 0
\(127\) 18.1025 1.60634 0.803170 0.595750i \(-0.203145\pi\)
0.803170 + 0.595750i \(0.203145\pi\)
\(128\) 0 0
\(129\) 9.13805 0.804561
\(130\) 0 0
\(131\) 17.1033 1.49432 0.747162 0.664642i \(-0.231416\pi\)
0.747162 + 0.664642i \(0.231416\pi\)
\(132\) 0 0
\(133\) −1.91986 −0.166473
\(134\) 0 0
\(135\) 0.235801 0.0202945
\(136\) 0 0
\(137\) −3.30297 −0.282192 −0.141096 0.989996i \(-0.545063\pi\)
−0.141096 + 0.989996i \(0.545063\pi\)
\(138\) 0 0
\(139\) 5.61502 0.476260 0.238130 0.971233i \(-0.423466\pi\)
0.238130 + 0.971233i \(0.423466\pi\)
\(140\) 0 0
\(141\) 7.96289 0.670596
\(142\) 0 0
\(143\) −5.71270 −0.477720
\(144\) 0 0
\(145\) 0.235801 0.0195822
\(146\) 0 0
\(147\) −5.25546 −0.433463
\(148\) 0 0
\(149\) −8.57546 −0.702529 −0.351265 0.936276i \(-0.614248\pi\)
−0.351265 + 0.936276i \(0.614248\pi\)
\(150\) 0 0
\(151\) −1.96597 −0.159988 −0.0799942 0.996795i \(-0.525490\pi\)
−0.0799942 + 0.996795i \(0.525490\pi\)
\(152\) 0 0
\(153\) −2.58826 −0.209248
\(154\) 0 0
\(155\) 1.47417 0.118408
\(156\) 0 0
\(157\) 21.8057 1.74029 0.870143 0.492799i \(-0.164026\pi\)
0.870143 + 0.492799i \(0.164026\pi\)
\(158\) 0 0
\(159\) −5.80551 −0.460407
\(160\) 0 0
\(161\) −1.32081 −0.104094
\(162\) 0 0
\(163\) −0.957364 −0.0749865 −0.0374933 0.999297i \(-0.511937\pi\)
−0.0374933 + 0.999297i \(0.511937\pi\)
\(164\) 0 0
\(165\) 0.367252 0.0285905
\(166\) 0 0
\(167\) 9.43303 0.729950 0.364975 0.931017i \(-0.381078\pi\)
0.364975 + 0.931017i \(0.381078\pi\)
\(168\) 0 0
\(169\) 0.453852 0.0349117
\(170\) 0 0
\(171\) −1.45355 −0.111156
\(172\) 0 0
\(173\) −9.93746 −0.755531 −0.377765 0.925901i \(-0.623307\pi\)
−0.377765 + 0.925901i \(0.623307\pi\)
\(174\) 0 0
\(175\) −6.53060 −0.493667
\(176\) 0 0
\(177\) 5.27926 0.396813
\(178\) 0 0
\(179\) 7.78781 0.582088 0.291044 0.956710i \(-0.405997\pi\)
0.291044 + 0.956710i \(0.405997\pi\)
\(180\) 0 0
\(181\) 17.6621 1.31282 0.656408 0.754406i \(-0.272075\pi\)
0.656408 + 0.754406i \(0.272075\pi\)
\(182\) 0 0
\(183\) 12.9066 0.954083
\(184\) 0 0
\(185\) −2.15805 −0.158663
\(186\) 0 0
\(187\) −4.03112 −0.294785
\(188\) 0 0
\(189\) 1.32081 0.0960747
\(190\) 0 0
\(191\) −16.3446 −1.18265 −0.591326 0.806433i \(-0.701395\pi\)
−0.591326 + 0.806433i \(0.701395\pi\)
\(192\) 0 0
\(193\) −2.95073 −0.212398 −0.106199 0.994345i \(-0.533868\pi\)
−0.106199 + 0.994345i \(0.533868\pi\)
\(194\) 0 0
\(195\) −0.864906 −0.0619372
\(196\) 0 0
\(197\) −7.66843 −0.546353 −0.273177 0.961964i \(-0.588074\pi\)
−0.273177 + 0.961964i \(0.588074\pi\)
\(198\) 0 0
\(199\) −1.25025 −0.0886278 −0.0443139 0.999018i \(-0.514110\pi\)
−0.0443139 + 0.999018i \(0.514110\pi\)
\(200\) 0 0
\(201\) −0.0629591 −0.00444079
\(202\) 0 0
\(203\) 1.32081 0.0927026
\(204\) 0 0
\(205\) 2.61739 0.182806
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −2.26386 −0.156594
\(210\) 0 0
\(211\) −18.0844 −1.24498 −0.622489 0.782628i \(-0.713878\pi\)
−0.622489 + 0.782628i \(0.713878\pi\)
\(212\) 0 0
\(213\) 1.10433 0.0756673
\(214\) 0 0
\(215\) 2.15476 0.146953
\(216\) 0 0
\(217\) 8.25739 0.560548
\(218\) 0 0
\(219\) 12.3233 0.832730
\(220\) 0 0
\(221\) 9.49360 0.638609
\(222\) 0 0
\(223\) −21.9030 −1.46674 −0.733368 0.679832i \(-0.762053\pi\)
−0.733368 + 0.679832i \(0.762053\pi\)
\(224\) 0 0
\(225\) −4.94440 −0.329627
\(226\) 0 0
\(227\) −15.7643 −1.04631 −0.523155 0.852237i \(-0.675245\pi\)
−0.523155 + 0.852237i \(0.675245\pi\)
\(228\) 0 0
\(229\) −0.580742 −0.0383765 −0.0191883 0.999816i \(-0.506108\pi\)
−0.0191883 + 0.999816i \(0.506108\pi\)
\(230\) 0 0
\(231\) 2.05711 0.135348
\(232\) 0 0
\(233\) −5.02325 −0.329084 −0.164542 0.986370i \(-0.552615\pi\)
−0.164542 + 0.986370i \(0.552615\pi\)
\(234\) 0 0
\(235\) 1.87766 0.122485
\(236\) 0 0
\(237\) 12.5754 0.816860
\(238\) 0 0
\(239\) −17.5493 −1.13517 −0.567585 0.823315i \(-0.692122\pi\)
−0.567585 + 0.823315i \(0.692122\pi\)
\(240\) 0 0
\(241\) 7.66197 0.493551 0.246776 0.969073i \(-0.420629\pi\)
0.246776 + 0.969073i \(0.420629\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −1.23924 −0.0791723
\(246\) 0 0
\(247\) 5.33155 0.339239
\(248\) 0 0
\(249\) 11.0990 0.703371
\(250\) 0 0
\(251\) −7.51257 −0.474189 −0.237095 0.971487i \(-0.576195\pi\)
−0.237095 + 0.971487i \(0.576195\pi\)
\(252\) 0 0
\(253\) −1.55746 −0.0979169
\(254\) 0 0
\(255\) −0.610314 −0.0382193
\(256\) 0 0
\(257\) −28.3787 −1.77022 −0.885108 0.465386i \(-0.845916\pi\)
−0.885108 + 0.465386i \(0.845916\pi\)
\(258\) 0 0
\(259\) −12.0881 −0.751116
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −10.3465 −0.637990 −0.318995 0.947756i \(-0.603345\pi\)
−0.318995 + 0.947756i \(0.603345\pi\)
\(264\) 0 0
\(265\) −1.36894 −0.0840936
\(266\) 0 0
\(267\) 9.93825 0.608211
\(268\) 0 0
\(269\) 27.0699 1.65048 0.825241 0.564781i \(-0.191039\pi\)
0.825241 + 0.564781i \(0.191039\pi\)
\(270\) 0 0
\(271\) 31.5819 1.91847 0.959233 0.282615i \(-0.0912019\pi\)
0.959233 + 0.282615i \(0.0912019\pi\)
\(272\) 0 0
\(273\) −4.84466 −0.293212
\(274\) 0 0
\(275\) −7.70072 −0.464371
\(276\) 0 0
\(277\) −0.186204 −0.0111879 −0.00559395 0.999984i \(-0.501781\pi\)
−0.00559395 + 0.999984i \(0.501781\pi\)
\(278\) 0 0
\(279\) 6.25177 0.374284
\(280\) 0 0
\(281\) 20.3180 1.21207 0.606035 0.795438i \(-0.292759\pi\)
0.606035 + 0.795438i \(0.292759\pi\)
\(282\) 0 0
\(283\) −2.11388 −0.125657 −0.0628284 0.998024i \(-0.520012\pi\)
−0.0628284 + 0.998024i \(0.520012\pi\)
\(284\) 0 0
\(285\) −0.342749 −0.0203027
\(286\) 0 0
\(287\) 14.6610 0.865410
\(288\) 0 0
\(289\) −10.3009 −0.605937
\(290\) 0 0
\(291\) −0.659173 −0.0386414
\(292\) 0 0
\(293\) −15.9170 −0.929881 −0.464940 0.885342i \(-0.653924\pi\)
−0.464940 + 0.885342i \(0.653924\pi\)
\(294\) 0 0
\(295\) 1.24485 0.0724782
\(296\) 0 0
\(297\) 1.55746 0.0903732
\(298\) 0 0
\(299\) 3.66795 0.212123
\(300\) 0 0
\(301\) 12.0696 0.695681
\(302\) 0 0
\(303\) −16.8646 −0.968843
\(304\) 0 0
\(305\) 3.04339 0.174264
\(306\) 0 0
\(307\) 21.5322 1.22891 0.614454 0.788953i \(-0.289376\pi\)
0.614454 + 0.788953i \(0.289376\pi\)
\(308\) 0 0
\(309\) −12.4751 −0.709684
\(310\) 0 0
\(311\) 22.6621 1.28505 0.642526 0.766264i \(-0.277887\pi\)
0.642526 + 0.766264i \(0.277887\pi\)
\(312\) 0 0
\(313\) −7.15436 −0.404389 −0.202194 0.979345i \(-0.564807\pi\)
−0.202194 + 0.979345i \(0.564807\pi\)
\(314\) 0 0
\(315\) 0.311448 0.0175481
\(316\) 0 0
\(317\) −3.26223 −0.183225 −0.0916125 0.995795i \(-0.529202\pi\)
−0.0916125 + 0.995795i \(0.529202\pi\)
\(318\) 0 0
\(319\) 1.55746 0.0872013
\(320\) 0 0
\(321\) 17.7008 0.987960
\(322\) 0 0
\(323\) 3.76217 0.209333
\(324\) 0 0
\(325\) 18.1358 1.00599
\(326\) 0 0
\(327\) 18.0712 0.999337
\(328\) 0 0
\(329\) 10.5175 0.579846
\(330\) 0 0
\(331\) 2.35489 0.129437 0.0647183 0.997904i \(-0.479385\pi\)
0.0647183 + 0.997904i \(0.479385\pi\)
\(332\) 0 0
\(333\) −9.15202 −0.501527
\(334\) 0 0
\(335\) −0.0148458 −0.000811113 0
\(336\) 0 0
\(337\) 2.47977 0.135081 0.0675407 0.997717i \(-0.478485\pi\)
0.0675407 + 0.997717i \(0.478485\pi\)
\(338\) 0 0
\(339\) −5.31581 −0.288715
\(340\) 0 0
\(341\) 9.73690 0.527283
\(342\) 0 0
\(343\) −16.1871 −0.874023
\(344\) 0 0
\(345\) −0.235801 −0.0126951
\(346\) 0 0
\(347\) 14.2901 0.767132 0.383566 0.923514i \(-0.374696\pi\)
0.383566 + 0.923514i \(0.374696\pi\)
\(348\) 0 0
\(349\) −11.3138 −0.605614 −0.302807 0.953052i \(-0.597924\pi\)
−0.302807 + 0.953052i \(0.597924\pi\)
\(350\) 0 0
\(351\) −3.66795 −0.195781
\(352\) 0 0
\(353\) −8.17924 −0.435337 −0.217669 0.976023i \(-0.569845\pi\)
−0.217669 + 0.976023i \(0.569845\pi\)
\(354\) 0 0
\(355\) 0.260402 0.0138207
\(356\) 0 0
\(357\) −3.41859 −0.180931
\(358\) 0 0
\(359\) 13.6446 0.720134 0.360067 0.932926i \(-0.382754\pi\)
0.360067 + 0.932926i \(0.382754\pi\)
\(360\) 0 0
\(361\) −16.8872 −0.888799
\(362\) 0 0
\(363\) −8.57430 −0.450034
\(364\) 0 0
\(365\) 2.90584 0.152099
\(366\) 0 0
\(367\) 4.14678 0.216460 0.108230 0.994126i \(-0.465482\pi\)
0.108230 + 0.994126i \(0.465482\pi\)
\(368\) 0 0
\(369\) 11.1000 0.577843
\(370\) 0 0
\(371\) −7.66797 −0.398101
\(372\) 0 0
\(373\) −19.5238 −1.01090 −0.505451 0.862855i \(-0.668674\pi\)
−0.505451 + 0.862855i \(0.668674\pi\)
\(374\) 0 0
\(375\) −2.34490 −0.121090
\(376\) 0 0
\(377\) −3.66795 −0.188909
\(378\) 0 0
\(379\) 22.2083 1.14076 0.570381 0.821380i \(-0.306796\pi\)
0.570381 + 0.821380i \(0.306796\pi\)
\(380\) 0 0
\(381\) 18.1025 0.927421
\(382\) 0 0
\(383\) −16.3785 −0.836902 −0.418451 0.908239i \(-0.637427\pi\)
−0.418451 + 0.908239i \(0.637427\pi\)
\(384\) 0 0
\(385\) 0.485069 0.0247214
\(386\) 0 0
\(387\) 9.13805 0.464513
\(388\) 0 0
\(389\) −8.61686 −0.436892 −0.218446 0.975849i \(-0.570099\pi\)
−0.218446 + 0.975849i \(0.570099\pi\)
\(390\) 0 0
\(391\) 2.58826 0.130894
\(392\) 0 0
\(393\) 17.1033 0.862749
\(394\) 0 0
\(395\) 2.96529 0.149200
\(396\) 0 0
\(397\) −22.7894 −1.14377 −0.571885 0.820334i \(-0.693787\pi\)
−0.571885 + 0.820334i \(0.693787\pi\)
\(398\) 0 0
\(399\) −1.91986 −0.0961134
\(400\) 0 0
\(401\) −12.6941 −0.633916 −0.316958 0.948440i \(-0.602661\pi\)
−0.316958 + 0.948440i \(0.602661\pi\)
\(402\) 0 0
\(403\) −22.9312 −1.14228
\(404\) 0 0
\(405\) 0.235801 0.0117170
\(406\) 0 0
\(407\) −14.2539 −0.706541
\(408\) 0 0
\(409\) 24.3119 1.20215 0.601073 0.799194i \(-0.294740\pi\)
0.601073 + 0.799194i \(0.294740\pi\)
\(410\) 0 0
\(411\) −3.30297 −0.162924
\(412\) 0 0
\(413\) 6.97289 0.343113
\(414\) 0 0
\(415\) 2.61716 0.128471
\(416\) 0 0
\(417\) 5.61502 0.274969
\(418\) 0 0
\(419\) −21.9676 −1.07319 −0.536594 0.843840i \(-0.680289\pi\)
−0.536594 + 0.843840i \(0.680289\pi\)
\(420\) 0 0
\(421\) 20.4023 0.994348 0.497174 0.867651i \(-0.334371\pi\)
0.497174 + 0.867651i \(0.334371\pi\)
\(422\) 0 0
\(423\) 7.96289 0.387169
\(424\) 0 0
\(425\) 12.7974 0.620764
\(426\) 0 0
\(427\) 17.0471 0.824969
\(428\) 0 0
\(429\) −5.71270 −0.275812
\(430\) 0 0
\(431\) −13.8642 −0.667817 −0.333909 0.942605i \(-0.608368\pi\)
−0.333909 + 0.942605i \(0.608368\pi\)
\(432\) 0 0
\(433\) −7.17070 −0.344602 −0.172301 0.985044i \(-0.555120\pi\)
−0.172301 + 0.985044i \(0.555120\pi\)
\(434\) 0 0
\(435\) 0.235801 0.0113058
\(436\) 0 0
\(437\) 1.45355 0.0695328
\(438\) 0 0
\(439\) 19.7238 0.941365 0.470683 0.882303i \(-0.344008\pi\)
0.470683 + 0.882303i \(0.344008\pi\)
\(440\) 0 0
\(441\) −5.25546 −0.250260
\(442\) 0 0
\(443\) 9.56265 0.454335 0.227167 0.973856i \(-0.427054\pi\)
0.227167 + 0.973856i \(0.427054\pi\)
\(444\) 0 0
\(445\) 2.34345 0.111090
\(446\) 0 0
\(447\) −8.57546 −0.405605
\(448\) 0 0
\(449\) −8.47592 −0.400003 −0.200002 0.979796i \(-0.564095\pi\)
−0.200002 + 0.979796i \(0.564095\pi\)
\(450\) 0 0
\(451\) 17.2878 0.814053
\(452\) 0 0
\(453\) −1.96597 −0.0923694
\(454\) 0 0
\(455\) −1.14238 −0.0535554
\(456\) 0 0
\(457\) 9.53016 0.445802 0.222901 0.974841i \(-0.428447\pi\)
0.222901 + 0.974841i \(0.428447\pi\)
\(458\) 0 0
\(459\) −2.58826 −0.120810
\(460\) 0 0
\(461\) 2.60871 0.121500 0.0607498 0.998153i \(-0.480651\pi\)
0.0607498 + 0.998153i \(0.480651\pi\)
\(462\) 0 0
\(463\) 18.0415 0.838461 0.419230 0.907880i \(-0.362300\pi\)
0.419230 + 0.907880i \(0.362300\pi\)
\(464\) 0 0
\(465\) 1.47417 0.0683631
\(466\) 0 0
\(467\) 3.51586 0.162695 0.0813473 0.996686i \(-0.474078\pi\)
0.0813473 + 0.996686i \(0.474078\pi\)
\(468\) 0 0
\(469\) −0.0831569 −0.00383983
\(470\) 0 0
\(471\) 21.8057 1.00475
\(472\) 0 0
\(473\) 14.2322 0.654397
\(474\) 0 0
\(475\) 7.18694 0.329759
\(476\) 0 0
\(477\) −5.80551 −0.265816
\(478\) 0 0
\(479\) −0.603449 −0.0275723 −0.0137861 0.999905i \(-0.504388\pi\)
−0.0137861 + 0.999905i \(0.504388\pi\)
\(480\) 0 0
\(481\) 33.5691 1.53062
\(482\) 0 0
\(483\) −1.32081 −0.0600989
\(484\) 0 0
\(485\) −0.155434 −0.00705788
\(486\) 0 0
\(487\) 6.61982 0.299973 0.149986 0.988688i \(-0.452077\pi\)
0.149986 + 0.988688i \(0.452077\pi\)
\(488\) 0 0
\(489\) −0.957364 −0.0432935
\(490\) 0 0
\(491\) −15.6882 −0.707998 −0.353999 0.935246i \(-0.615178\pi\)
−0.353999 + 0.935246i \(0.615178\pi\)
\(492\) 0 0
\(493\) −2.58826 −0.116569
\(494\) 0 0
\(495\) 0.367252 0.0165067
\(496\) 0 0
\(497\) 1.45861 0.0654274
\(498\) 0 0
\(499\) −19.0720 −0.853779 −0.426890 0.904304i \(-0.640391\pi\)
−0.426890 + 0.904304i \(0.640391\pi\)
\(500\) 0 0
\(501\) 9.43303 0.421437
\(502\) 0 0
\(503\) −7.50381 −0.334578 −0.167289 0.985908i \(-0.553501\pi\)
−0.167289 + 0.985908i \(0.553501\pi\)
\(504\) 0 0
\(505\) −3.97668 −0.176960
\(506\) 0 0
\(507\) 0.453852 0.0201563
\(508\) 0 0
\(509\) 29.7328 1.31789 0.658943 0.752193i \(-0.271004\pi\)
0.658943 + 0.752193i \(0.271004\pi\)
\(510\) 0 0
\(511\) 16.2767 0.720039
\(512\) 0 0
\(513\) −1.45355 −0.0641759
\(514\) 0 0
\(515\) −2.94164 −0.129624
\(516\) 0 0
\(517\) 12.4019 0.545436
\(518\) 0 0
\(519\) −9.93746 −0.436206
\(520\) 0 0
\(521\) 1.14188 0.0500268 0.0250134 0.999687i \(-0.492037\pi\)
0.0250134 + 0.999687i \(0.492037\pi\)
\(522\) 0 0
\(523\) 15.9260 0.696395 0.348197 0.937421i \(-0.386794\pi\)
0.348197 + 0.937421i \(0.386794\pi\)
\(524\) 0 0
\(525\) −6.53060 −0.285019
\(526\) 0 0
\(527\) −16.1812 −0.704864
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 5.27926 0.229100
\(532\) 0 0
\(533\) −40.7142 −1.76353
\(534\) 0 0
\(535\) 4.17386 0.180452
\(536\) 0 0
\(537\) 7.78781 0.336069
\(538\) 0 0
\(539\) −8.18520 −0.352561
\(540\) 0 0
\(541\) −9.79768 −0.421235 −0.210618 0.977569i \(-0.567547\pi\)
−0.210618 + 0.977569i \(0.567547\pi\)
\(542\) 0 0
\(543\) 17.6621 0.757955
\(544\) 0 0
\(545\) 4.26120 0.182530
\(546\) 0 0
\(547\) 12.9607 0.554161 0.277081 0.960847i \(-0.410633\pi\)
0.277081 + 0.960847i \(0.410633\pi\)
\(548\) 0 0
\(549\) 12.9066 0.550840
\(550\) 0 0
\(551\) −1.45355 −0.0619234
\(552\) 0 0
\(553\) 16.6097 0.706316
\(554\) 0 0
\(555\) −2.15805 −0.0916043
\(556\) 0 0
\(557\) 13.0755 0.554026 0.277013 0.960866i \(-0.410656\pi\)
0.277013 + 0.960866i \(0.410656\pi\)
\(558\) 0 0
\(559\) −33.5179 −1.41766
\(560\) 0 0
\(561\) −4.03112 −0.170194
\(562\) 0 0
\(563\) −26.8969 −1.13357 −0.566785 0.823865i \(-0.691813\pi\)
−0.566785 + 0.823865i \(0.691813\pi\)
\(564\) 0 0
\(565\) −1.25347 −0.0527340
\(566\) 0 0
\(567\) 1.32081 0.0554688
\(568\) 0 0
\(569\) −15.4961 −0.649631 −0.324816 0.945777i \(-0.605302\pi\)
−0.324816 + 0.945777i \(0.605302\pi\)
\(570\) 0 0
\(571\) 22.7800 0.953312 0.476656 0.879090i \(-0.341849\pi\)
0.476656 + 0.879090i \(0.341849\pi\)
\(572\) 0 0
\(573\) −16.3446 −0.682804
\(574\) 0 0
\(575\) 4.94440 0.206196
\(576\) 0 0
\(577\) 38.6151 1.60757 0.803785 0.594920i \(-0.202816\pi\)
0.803785 + 0.594920i \(0.202816\pi\)
\(578\) 0 0
\(579\) −2.95073 −0.122628
\(580\) 0 0
\(581\) 14.6597 0.608185
\(582\) 0 0
\(583\) −9.04187 −0.374476
\(584\) 0 0
\(585\) −0.864906 −0.0357595
\(586\) 0 0
\(587\) 9.48452 0.391468 0.195734 0.980657i \(-0.437291\pi\)
0.195734 + 0.980657i \(0.437291\pi\)
\(588\) 0 0
\(589\) −9.08727 −0.374434
\(590\) 0 0
\(591\) −7.66843 −0.315437
\(592\) 0 0
\(593\) 23.4436 0.962714 0.481357 0.876525i \(-0.340144\pi\)
0.481357 + 0.876525i \(0.340144\pi\)
\(594\) 0 0
\(595\) −0.806108 −0.0330472
\(596\) 0 0
\(597\) −1.25025 −0.0511693
\(598\) 0 0
\(599\) −44.2702 −1.80883 −0.904416 0.426651i \(-0.859693\pi\)
−0.904416 + 0.426651i \(0.859693\pi\)
\(600\) 0 0
\(601\) −8.15420 −0.332617 −0.166308 0.986074i \(-0.553185\pi\)
−0.166308 + 0.986074i \(0.553185\pi\)
\(602\) 0 0
\(603\) −0.0629591 −0.00256389
\(604\) 0 0
\(605\) −2.02183 −0.0821990
\(606\) 0 0
\(607\) −23.4638 −0.952366 −0.476183 0.879346i \(-0.657980\pi\)
−0.476183 + 0.879346i \(0.657980\pi\)
\(608\) 0 0
\(609\) 1.32081 0.0535219
\(610\) 0 0
\(611\) −29.2075 −1.18161
\(612\) 0 0
\(613\) −24.1828 −0.976735 −0.488367 0.872638i \(-0.662407\pi\)
−0.488367 + 0.872638i \(0.662407\pi\)
\(614\) 0 0
\(615\) 2.61739 0.105543
\(616\) 0 0
\(617\) 14.9245 0.600839 0.300420 0.953807i \(-0.402873\pi\)
0.300420 + 0.953807i \(0.402873\pi\)
\(618\) 0 0
\(619\) 36.0985 1.45092 0.725460 0.688264i \(-0.241627\pi\)
0.725460 + 0.688264i \(0.241627\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 13.1265 0.525903
\(624\) 0 0
\(625\) 24.1691 0.966762
\(626\) 0 0
\(627\) −2.26386 −0.0904097
\(628\) 0 0
\(629\) 23.6878 0.944493
\(630\) 0 0
\(631\) 20.8935 0.831758 0.415879 0.909420i \(-0.363474\pi\)
0.415879 + 0.909420i \(0.363474\pi\)
\(632\) 0 0
\(633\) −18.0844 −0.718788
\(634\) 0 0
\(635\) 4.26859 0.169394
\(636\) 0 0
\(637\) 19.2768 0.763774
\(638\) 0 0
\(639\) 1.10433 0.0436865
\(640\) 0 0
\(641\) −22.4373 −0.886218 −0.443109 0.896468i \(-0.646125\pi\)
−0.443109 + 0.896468i \(0.646125\pi\)
\(642\) 0 0
\(643\) −16.1046 −0.635103 −0.317552 0.948241i \(-0.602861\pi\)
−0.317552 + 0.948241i \(0.602861\pi\)
\(644\) 0 0
\(645\) 2.15476 0.0848436
\(646\) 0 0
\(647\) −39.9830 −1.57189 −0.785947 0.618294i \(-0.787824\pi\)
−0.785947 + 0.618294i \(0.787824\pi\)
\(648\) 0 0
\(649\) 8.22225 0.322752
\(650\) 0 0
\(651\) 8.25739 0.323633
\(652\) 0 0
\(653\) 33.3654 1.30569 0.652845 0.757492i \(-0.273575\pi\)
0.652845 + 0.757492i \(0.273575\pi\)
\(654\) 0 0
\(655\) 4.03298 0.157582
\(656\) 0 0
\(657\) 12.3233 0.480777
\(658\) 0 0
\(659\) 19.8932 0.774930 0.387465 0.921884i \(-0.373351\pi\)
0.387465 + 0.921884i \(0.373351\pi\)
\(660\) 0 0
\(661\) 29.4777 1.14655 0.573274 0.819363i \(-0.305673\pi\)
0.573274 + 0.819363i \(0.305673\pi\)
\(662\) 0 0
\(663\) 9.49360 0.368701
\(664\) 0 0
\(665\) −0.452706 −0.0175552
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −21.9030 −0.846821
\(670\) 0 0
\(671\) 20.1016 0.776012
\(672\) 0 0
\(673\) 29.1505 1.12367 0.561835 0.827249i \(-0.310095\pi\)
0.561835 + 0.827249i \(0.310095\pi\)
\(674\) 0 0
\(675\) −4.94440 −0.190310
\(676\) 0 0
\(677\) −30.9708 −1.19031 −0.595153 0.803612i \(-0.702909\pi\)
−0.595153 + 0.803612i \(0.702909\pi\)
\(678\) 0 0
\(679\) −0.870642 −0.0334122
\(680\) 0 0
\(681\) −15.7643 −0.604088
\(682\) 0 0
\(683\) −14.0933 −0.539266 −0.269633 0.962963i \(-0.586902\pi\)
−0.269633 + 0.962963i \(0.586902\pi\)
\(684\) 0 0
\(685\) −0.778844 −0.0297581
\(686\) 0 0
\(687\) −0.580742 −0.0221567
\(688\) 0 0
\(689\) 21.2943 0.811249
\(690\) 0 0
\(691\) −40.2834 −1.53245 −0.766227 0.642570i \(-0.777868\pi\)
−0.766227 + 0.642570i \(0.777868\pi\)
\(692\) 0 0
\(693\) 2.05711 0.0781433
\(694\) 0 0
\(695\) 1.32403 0.0502232
\(696\) 0 0
\(697\) −28.7296 −1.08821
\(698\) 0 0
\(699\) −5.02325 −0.189997
\(700\) 0 0
\(701\) 30.9405 1.16861 0.584303 0.811536i \(-0.301368\pi\)
0.584303 + 0.811536i \(0.301368\pi\)
\(702\) 0 0
\(703\) 13.3029 0.501729
\(704\) 0 0
\(705\) 1.87766 0.0707167
\(706\) 0 0
\(707\) −22.2749 −0.837732
\(708\) 0 0
\(709\) −52.0684 −1.95547 −0.977735 0.209842i \(-0.932705\pi\)
−0.977735 + 0.209842i \(0.932705\pi\)
\(710\) 0 0
\(711\) 12.5754 0.471614
\(712\) 0 0
\(713\) −6.25177 −0.234131
\(714\) 0 0
\(715\) −1.34706 −0.0503772
\(716\) 0 0
\(717\) −17.5493 −0.655390
\(718\) 0 0
\(719\) −5.53680 −0.206488 −0.103244 0.994656i \(-0.532922\pi\)
−0.103244 + 0.994656i \(0.532922\pi\)
\(720\) 0 0
\(721\) −16.4772 −0.613644
\(722\) 0 0
\(723\) 7.66197 0.284952
\(724\) 0 0
\(725\) −4.94440 −0.183630
\(726\) 0 0
\(727\) 12.0694 0.447628 0.223814 0.974632i \(-0.428149\pi\)
0.223814 + 0.974632i \(0.428149\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −23.6516 −0.874787
\(732\) 0 0
\(733\) 21.7033 0.801629 0.400815 0.916159i \(-0.368727\pi\)
0.400815 + 0.916159i \(0.368727\pi\)
\(734\) 0 0
\(735\) −1.23924 −0.0457102
\(736\) 0 0
\(737\) −0.0980565 −0.00361196
\(738\) 0 0
\(739\) 9.73830 0.358229 0.179115 0.983828i \(-0.442677\pi\)
0.179115 + 0.983828i \(0.442677\pi\)
\(740\) 0 0
\(741\) 5.33155 0.195860
\(742\) 0 0
\(743\) −10.6942 −0.392333 −0.196166 0.980571i \(-0.562849\pi\)
−0.196166 + 0.980571i \(0.562849\pi\)
\(744\) 0 0
\(745\) −2.02210 −0.0740841
\(746\) 0 0
\(747\) 11.0990 0.406091
\(748\) 0 0
\(749\) 23.3793 0.854262
\(750\) 0 0
\(751\) −44.4957 −1.62367 −0.811836 0.583886i \(-0.801532\pi\)
−0.811836 + 0.583886i \(0.801532\pi\)
\(752\) 0 0
\(753\) −7.51257 −0.273773
\(754\) 0 0
\(755\) −0.463578 −0.0168713
\(756\) 0 0
\(757\) −13.9527 −0.507118 −0.253559 0.967320i \(-0.581601\pi\)
−0.253559 + 0.967320i \(0.581601\pi\)
\(758\) 0 0
\(759\) −1.55746 −0.0565324
\(760\) 0 0
\(761\) 7.28775 0.264181 0.132090 0.991238i \(-0.457831\pi\)
0.132090 + 0.991238i \(0.457831\pi\)
\(762\) 0 0
\(763\) 23.8686 0.864100
\(764\) 0 0
\(765\) −0.610314 −0.0220659
\(766\) 0 0
\(767\) −19.3640 −0.699195
\(768\) 0 0
\(769\) 24.5824 0.886463 0.443232 0.896407i \(-0.353832\pi\)
0.443232 + 0.896407i \(0.353832\pi\)
\(770\) 0 0
\(771\) −28.3787 −1.02203
\(772\) 0 0
\(773\) 7.30187 0.262630 0.131315 0.991341i \(-0.458080\pi\)
0.131315 + 0.991341i \(0.458080\pi\)
\(774\) 0 0
\(775\) −30.9112 −1.11036
\(776\) 0 0
\(777\) −12.0881 −0.433657
\(778\) 0 0
\(779\) −16.1344 −0.578075
\(780\) 0 0
\(781\) 1.71995 0.0615447
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) 5.14181 0.183519
\(786\) 0 0
\(787\) 47.6152 1.69730 0.848650 0.528955i \(-0.177416\pi\)
0.848650 + 0.528955i \(0.177416\pi\)
\(788\) 0 0
\(789\) −10.3465 −0.368344
\(790\) 0 0
\(791\) −7.02117 −0.249644
\(792\) 0 0
\(793\) −47.3407 −1.68112
\(794\) 0 0
\(795\) −1.36894 −0.0485515
\(796\) 0 0
\(797\) −22.7331 −0.805246 −0.402623 0.915366i \(-0.631901\pi\)
−0.402623 + 0.915366i \(0.631901\pi\)
\(798\) 0 0
\(799\) −20.6100 −0.729130
\(800\) 0 0
\(801\) 9.93825 0.351151
\(802\) 0 0
\(803\) 19.1931 0.677309
\(804\) 0 0
\(805\) −0.311448 −0.0109771
\(806\) 0 0
\(807\) 27.0699 0.952906
\(808\) 0 0
\(809\) 14.7281 0.517812 0.258906 0.965903i \(-0.416638\pi\)
0.258906 + 0.965903i \(0.416638\pi\)
\(810\) 0 0
\(811\) 11.2811 0.396133 0.198067 0.980189i \(-0.436534\pi\)
0.198067 + 0.980189i \(0.436534\pi\)
\(812\) 0 0
\(813\) 31.5819 1.10763
\(814\) 0 0
\(815\) −0.225747 −0.00790758
\(816\) 0 0
\(817\) −13.2826 −0.464700
\(818\) 0 0
\(819\) −4.84466 −0.169286
\(820\) 0 0
\(821\) 40.1945 1.40280 0.701399 0.712769i \(-0.252559\pi\)
0.701399 + 0.712769i \(0.252559\pi\)
\(822\) 0 0
\(823\) 37.7569 1.31612 0.658061 0.752964i \(-0.271377\pi\)
0.658061 + 0.752964i \(0.271377\pi\)
\(824\) 0 0
\(825\) −7.70072 −0.268105
\(826\) 0 0
\(827\) −20.9301 −0.727810 −0.363905 0.931436i \(-0.618557\pi\)
−0.363905 + 0.931436i \(0.618557\pi\)
\(828\) 0 0
\(829\) −38.4034 −1.33380 −0.666902 0.745145i \(-0.732380\pi\)
−0.666902 + 0.745145i \(0.732380\pi\)
\(830\) 0 0
\(831\) −0.186204 −0.00645934
\(832\) 0 0
\(833\) 13.6025 0.471299
\(834\) 0 0
\(835\) 2.22432 0.0769757
\(836\) 0 0
\(837\) 6.25177 0.216093
\(838\) 0 0
\(839\) 33.5390 1.15790 0.578948 0.815365i \(-0.303464\pi\)
0.578948 + 0.815365i \(0.303464\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 20.3180 0.699789
\(844\) 0 0
\(845\) 0.107019 0.00368155
\(846\) 0 0
\(847\) −11.3250 −0.389132
\(848\) 0 0
\(849\) −2.11388 −0.0725480
\(850\) 0 0
\(851\) 9.15202 0.313727
\(852\) 0 0
\(853\) 9.62422 0.329527 0.164764 0.986333i \(-0.447314\pi\)
0.164764 + 0.986333i \(0.447314\pi\)
\(854\) 0 0
\(855\) −0.342749 −0.0117218
\(856\) 0 0
\(857\) −36.5360 −1.24804 −0.624022 0.781406i \(-0.714503\pi\)
−0.624022 + 0.781406i \(0.714503\pi\)
\(858\) 0 0
\(859\) −35.0505 −1.19591 −0.597954 0.801530i \(-0.704020\pi\)
−0.597954 + 0.801530i \(0.704020\pi\)
\(860\) 0 0
\(861\) 14.6610 0.499645
\(862\) 0 0
\(863\) −5.83740 −0.198708 −0.0993538 0.995052i \(-0.531678\pi\)
−0.0993538 + 0.995052i \(0.531678\pi\)
\(864\) 0 0
\(865\) −2.34326 −0.0796733
\(866\) 0 0
\(867\) −10.3009 −0.349838
\(868\) 0 0
\(869\) 19.5857 0.664401
\(870\) 0 0
\(871\) 0.230931 0.00782479
\(872\) 0 0
\(873\) −0.659173 −0.0223096
\(874\) 0 0
\(875\) −3.09716 −0.104703
\(876\) 0 0
\(877\) −48.6139 −1.64157 −0.820787 0.571234i \(-0.806465\pi\)
−0.820787 + 0.571234i \(0.806465\pi\)
\(878\) 0 0
\(879\) −15.9170 −0.536867
\(880\) 0 0
\(881\) −29.6969 −1.00051 −0.500256 0.865877i \(-0.666761\pi\)
−0.500256 + 0.865877i \(0.666761\pi\)
\(882\) 0 0
\(883\) −19.2019 −0.646196 −0.323098 0.946366i \(-0.604724\pi\)
−0.323098 + 0.946366i \(0.604724\pi\)
\(884\) 0 0
\(885\) 1.24485 0.0418453
\(886\) 0 0
\(887\) −32.9751 −1.10720 −0.553598 0.832784i \(-0.686746\pi\)
−0.553598 + 0.832784i \(0.686746\pi\)
\(888\) 0 0
\(889\) 23.9100 0.801915
\(890\) 0 0
\(891\) 1.55746 0.0521770
\(892\) 0 0
\(893\) −11.5745 −0.387325
\(894\) 0 0
\(895\) 1.83637 0.0613832
\(896\) 0 0
\(897\) 3.66795 0.122469
\(898\) 0 0
\(899\) 6.25177 0.208508
\(900\) 0 0
\(901\) 15.0262 0.500594
\(902\) 0 0
\(903\) 12.0696 0.401652
\(904\) 0 0
\(905\) 4.16475 0.138441
\(906\) 0 0
\(907\) −1.11525 −0.0370312 −0.0185156 0.999829i \(-0.505894\pi\)
−0.0185156 + 0.999829i \(0.505894\pi\)
\(908\) 0 0
\(909\) −16.8646 −0.559362
\(910\) 0 0
\(911\) 11.7434 0.389076 0.194538 0.980895i \(-0.437679\pi\)
0.194538 + 0.980895i \(0.437679\pi\)
\(912\) 0 0
\(913\) 17.2863 0.572093
\(914\) 0 0
\(915\) 3.04339 0.100611
\(916\) 0 0
\(917\) 22.5902 0.745995
\(918\) 0 0
\(919\) −0.976902 −0.0322250 −0.0161125 0.999870i \(-0.505129\pi\)
−0.0161125 + 0.999870i \(0.505129\pi\)
\(920\) 0 0
\(921\) 21.5322 0.709511
\(922\) 0 0
\(923\) −4.05062 −0.133328
\(924\) 0 0
\(925\) 45.2512 1.48785
\(926\) 0 0
\(927\) −12.4751 −0.409736
\(928\) 0 0
\(929\) 23.8871 0.783710 0.391855 0.920027i \(-0.371833\pi\)
0.391855 + 0.920027i \(0.371833\pi\)
\(930\) 0 0
\(931\) 7.63909 0.250361
\(932\) 0 0
\(933\) 22.6621 0.741925
\(934\) 0 0
\(935\) −0.950542 −0.0310860
\(936\) 0 0
\(937\) −5.19928 −0.169853 −0.0849265 0.996387i \(-0.527066\pi\)
−0.0849265 + 0.996387i \(0.527066\pi\)
\(938\) 0 0
\(939\) −7.15436 −0.233474
\(940\) 0 0
\(941\) −31.5399 −1.02817 −0.514085 0.857739i \(-0.671869\pi\)
−0.514085 + 0.857739i \(0.671869\pi\)
\(942\) 0 0
\(943\) −11.1000 −0.361465
\(944\) 0 0
\(945\) 0.311448 0.0101314
\(946\) 0 0
\(947\) 38.9817 1.26673 0.633367 0.773851i \(-0.281672\pi\)
0.633367 + 0.773851i \(0.281672\pi\)
\(948\) 0 0
\(949\) −45.2012 −1.46729
\(950\) 0 0
\(951\) −3.26223 −0.105785
\(952\) 0 0
\(953\) −26.4546 −0.856948 −0.428474 0.903554i \(-0.640949\pi\)
−0.428474 + 0.903554i \(0.640949\pi\)
\(954\) 0 0
\(955\) −3.85406 −0.124715
\(956\) 0 0
\(957\) 1.55746 0.0503457
\(958\) 0 0
\(959\) −4.36260 −0.140876
\(960\) 0 0
\(961\) 8.08459 0.260793
\(962\) 0 0
\(963\) 17.7008 0.570399
\(964\) 0 0
\(965\) −0.695785 −0.0223981
\(966\) 0 0
\(967\) −46.6669 −1.50071 −0.750353 0.661037i \(-0.770117\pi\)
−0.750353 + 0.661037i \(0.770117\pi\)
\(968\) 0 0
\(969\) 3.76217 0.120858
\(970\) 0 0
\(971\) −8.19172 −0.262885 −0.131442 0.991324i \(-0.541961\pi\)
−0.131442 + 0.991324i \(0.541961\pi\)
\(972\) 0 0
\(973\) 7.41637 0.237758
\(974\) 0 0
\(975\) 18.1358 0.580810
\(976\) 0 0
\(977\) 6.54176 0.209289 0.104645 0.994510i \(-0.466629\pi\)
0.104645 + 0.994510i \(0.466629\pi\)
\(978\) 0 0
\(979\) 15.4785 0.494694
\(980\) 0 0
\(981\) 18.0712 0.576968
\(982\) 0 0
\(983\) −21.2203 −0.676823 −0.338411 0.940998i \(-0.609890\pi\)
−0.338411 + 0.940998i \(0.609890\pi\)
\(984\) 0 0
\(985\) −1.80822 −0.0576148
\(986\) 0 0
\(987\) 10.5175 0.334774
\(988\) 0 0
\(989\) −9.13805 −0.290573
\(990\) 0 0
\(991\) 18.0972 0.574878 0.287439 0.957799i \(-0.407196\pi\)
0.287439 + 0.957799i \(0.407196\pi\)
\(992\) 0 0
\(993\) 2.35489 0.0747302
\(994\) 0 0
\(995\) −0.294810 −0.00934610
\(996\) 0 0
\(997\) 0.0807050 0.00255595 0.00127798 0.999999i \(-0.499593\pi\)
0.00127798 + 0.999999i \(0.499593\pi\)
\(998\) 0 0
\(999\) −9.15202 −0.289557
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 8004.2.a.k.1.9 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.k.1.9 18 1.1 even 1 trivial