Properties

Label 8016.2.a.s.1.5
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $5$
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] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, 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("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.284897.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 5x^{2} + 10x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(3.04177\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.26608 q^{5} +4.21058 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.26608 q^{5} +4.21058 q^{7} +1.00000 q^{9} -1.64992 q^{11} -5.09727 q^{13} -3.26608 q^{15} -7.91600 q^{17} +2.28918 q^{19} -4.21058 q^{21} -2.37812 q^{23} +5.66730 q^{25} -1.00000 q^{27} +3.30523 q^{29} -6.02804 q^{31} +1.64992 q^{33} +13.7521 q^{35} -3.25775 q^{37} +5.09727 q^{39} -6.76188 q^{41} -0.995567 q^{43} +3.26608 q^{45} +7.03765 q^{47} +10.7290 q^{49} +7.91600 q^{51} -7.78190 q^{53} -5.38877 q^{55} -2.28918 q^{57} +2.76900 q^{59} -13.5958 q^{61} +4.21058 q^{63} -16.6481 q^{65} -10.5454 q^{67} +2.37812 q^{69} +0.104019 q^{71} -6.12397 q^{73} -5.66730 q^{75} -6.94712 q^{77} -3.17407 q^{79} +1.00000 q^{81} +15.1125 q^{83} -25.8543 q^{85} -3.30523 q^{87} -11.4567 q^{89} -21.4625 q^{91} +6.02804 q^{93} +7.47667 q^{95} -0.758299 q^{97} -1.64992 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9} + 3 q^{11} - 14 q^{13} - q^{15} - 13 q^{17} + 2 q^{19} - 4 q^{21} + 5 q^{23} + 2 q^{25} - 5 q^{27} + 13 q^{29} - 2 q^{31} - 3 q^{33} + 12 q^{35} - 5 q^{37} + 14 q^{39} - 20 q^{41} + 20 q^{43} + q^{45} - q^{47} - 9 q^{49} + 13 q^{51} - 3 q^{53} + 3 q^{55} - 2 q^{57} + q^{59} - 34 q^{61} + 4 q^{63} - 22 q^{65} + 16 q^{67} - 5 q^{69} + 5 q^{71} - 12 q^{73} - 2 q^{75} - 8 q^{77} + 20 q^{79} + 5 q^{81} + 15 q^{83} - 27 q^{85} - 13 q^{87} - 48 q^{89} - 7 q^{91} + 2 q^{93} + 5 q^{95} - 21 q^{97} + 3 q^{99}+O(q^{100}) \) 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
\(5\) 3.26608 1.46064 0.730319 0.683107i \(-0.239372\pi\)
0.730319 + 0.683107i \(0.239372\pi\)
\(6\) 0 0
\(7\) 4.21058 1.59145 0.795726 0.605657i \(-0.207090\pi\)
0.795726 + 0.605657i \(0.207090\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.64992 −0.497469 −0.248734 0.968572i \(-0.580015\pi\)
−0.248734 + 0.968572i \(0.580015\pi\)
\(12\) 0 0
\(13\) −5.09727 −1.41373 −0.706864 0.707350i \(-0.749891\pi\)
−0.706864 + 0.707350i \(0.749891\pi\)
\(14\) 0 0
\(15\) −3.26608 −0.843299
\(16\) 0 0
\(17\) −7.91600 −1.91991 −0.959956 0.280150i \(-0.909616\pi\)
−0.959956 + 0.280150i \(0.909616\pi\)
\(18\) 0 0
\(19\) 2.28918 0.525175 0.262587 0.964908i \(-0.415424\pi\)
0.262587 + 0.964908i \(0.415424\pi\)
\(20\) 0 0
\(21\) −4.21058 −0.918825
\(22\) 0 0
\(23\) −2.37812 −0.495872 −0.247936 0.968776i \(-0.579752\pi\)
−0.247936 + 0.968776i \(0.579752\pi\)
\(24\) 0 0
\(25\) 5.66730 1.13346
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.30523 0.613767 0.306883 0.951747i \(-0.400714\pi\)
0.306883 + 0.951747i \(0.400714\pi\)
\(30\) 0 0
\(31\) −6.02804 −1.08267 −0.541334 0.840808i \(-0.682080\pi\)
−0.541334 + 0.840808i \(0.682080\pi\)
\(32\) 0 0
\(33\) 1.64992 0.287214
\(34\) 0 0
\(35\) 13.7521 2.32453
\(36\) 0 0
\(37\) −3.25775 −0.535571 −0.267785 0.963479i \(-0.586292\pi\)
−0.267785 + 0.963479i \(0.586292\pi\)
\(38\) 0 0
\(39\) 5.09727 0.816216
\(40\) 0 0
\(41\) −6.76188 −1.05603 −0.528014 0.849236i \(-0.677063\pi\)
−0.528014 + 0.849236i \(0.677063\pi\)
\(42\) 0 0
\(43\) −0.995567 −0.151822 −0.0759112 0.997115i \(-0.524187\pi\)
−0.0759112 + 0.997115i \(0.524187\pi\)
\(44\) 0 0
\(45\) 3.26608 0.486879
\(46\) 0 0
\(47\) 7.03765 1.02655 0.513274 0.858225i \(-0.328433\pi\)
0.513274 + 0.858225i \(0.328433\pi\)
\(48\) 0 0
\(49\) 10.7290 1.53272
\(50\) 0 0
\(51\) 7.91600 1.10846
\(52\) 0 0
\(53\) −7.78190 −1.06893 −0.534463 0.845192i \(-0.679486\pi\)
−0.534463 + 0.845192i \(0.679486\pi\)
\(54\) 0 0
\(55\) −5.38877 −0.726622
\(56\) 0 0
\(57\) −2.28918 −0.303210
\(58\) 0 0
\(59\) 2.76900 0.360494 0.180247 0.983621i \(-0.442310\pi\)
0.180247 + 0.983621i \(0.442310\pi\)
\(60\) 0 0
\(61\) −13.5958 −1.74076 −0.870379 0.492382i \(-0.836126\pi\)
−0.870379 + 0.492382i \(0.836126\pi\)
\(62\) 0 0
\(63\) 4.21058 0.530484
\(64\) 0 0
\(65\) −16.6481 −2.06494
\(66\) 0 0
\(67\) −10.5454 −1.28833 −0.644165 0.764887i \(-0.722795\pi\)
−0.644165 + 0.764887i \(0.722795\pi\)
\(68\) 0 0
\(69\) 2.37812 0.286292
\(70\) 0 0
\(71\) 0.104019 0.0123448 0.00617239 0.999981i \(-0.498035\pi\)
0.00617239 + 0.999981i \(0.498035\pi\)
\(72\) 0 0
\(73\) −6.12397 −0.716756 −0.358378 0.933577i \(-0.616670\pi\)
−0.358378 + 0.933577i \(0.616670\pi\)
\(74\) 0 0
\(75\) −5.66730 −0.654404
\(76\) 0 0
\(77\) −6.94712 −0.791698
\(78\) 0 0
\(79\) −3.17407 −0.357111 −0.178555 0.983930i \(-0.557142\pi\)
−0.178555 + 0.983930i \(0.557142\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 15.1125 1.65882 0.829408 0.558643i \(-0.188678\pi\)
0.829408 + 0.558643i \(0.188678\pi\)
\(84\) 0 0
\(85\) −25.8543 −2.80430
\(86\) 0 0
\(87\) −3.30523 −0.354358
\(88\) 0 0
\(89\) −11.4567 −1.21441 −0.607205 0.794545i \(-0.707709\pi\)
−0.607205 + 0.794545i \(0.707709\pi\)
\(90\) 0 0
\(91\) −21.4625 −2.24988
\(92\) 0 0
\(93\) 6.02804 0.625078
\(94\) 0 0
\(95\) 7.47667 0.767090
\(96\) 0 0
\(97\) −0.758299 −0.0769936 −0.0384968 0.999259i \(-0.512257\pi\)
−0.0384968 + 0.999259i \(0.512257\pi\)
\(98\) 0 0
\(99\) −1.64992 −0.165823
\(100\) 0 0
\(101\) 15.7478 1.56696 0.783480 0.621417i \(-0.213443\pi\)
0.783480 + 0.621417i \(0.213443\pi\)
\(102\) 0 0
\(103\) 0.604753 0.0595880 0.0297940 0.999556i \(-0.490515\pi\)
0.0297940 + 0.999556i \(0.490515\pi\)
\(104\) 0 0
\(105\) −13.7521 −1.34207
\(106\) 0 0
\(107\) −12.7791 −1.23540 −0.617702 0.786412i \(-0.711936\pi\)
−0.617702 + 0.786412i \(0.711936\pi\)
\(108\) 0 0
\(109\) −11.0076 −1.05433 −0.527166 0.849762i \(-0.676745\pi\)
−0.527166 + 0.849762i \(0.676745\pi\)
\(110\) 0 0
\(111\) 3.25775 0.309212
\(112\) 0 0
\(113\) −17.2772 −1.62531 −0.812653 0.582748i \(-0.801977\pi\)
−0.812653 + 0.582748i \(0.801977\pi\)
\(114\) 0 0
\(115\) −7.76713 −0.724289
\(116\) 0 0
\(117\) −5.09727 −0.471243
\(118\) 0 0
\(119\) −33.3310 −3.05545
\(120\) 0 0
\(121\) −8.27777 −0.752525
\(122\) 0 0
\(123\) 6.76188 0.609698
\(124\) 0 0
\(125\) 2.17947 0.194938
\(126\) 0 0
\(127\) 5.31722 0.471827 0.235914 0.971774i \(-0.424192\pi\)
0.235914 + 0.971774i \(0.424192\pi\)
\(128\) 0 0
\(129\) 0.995567 0.0876548
\(130\) 0 0
\(131\) 5.15770 0.450631 0.225315 0.974286i \(-0.427659\pi\)
0.225315 + 0.974286i \(0.427659\pi\)
\(132\) 0 0
\(133\) 9.63880 0.835790
\(134\) 0 0
\(135\) −3.26608 −0.281100
\(136\) 0 0
\(137\) 9.21320 0.787137 0.393568 0.919295i \(-0.371240\pi\)
0.393568 + 0.919295i \(0.371240\pi\)
\(138\) 0 0
\(139\) 12.5130 1.06134 0.530671 0.847578i \(-0.321940\pi\)
0.530671 + 0.847578i \(0.321940\pi\)
\(140\) 0 0
\(141\) −7.03765 −0.592677
\(142\) 0 0
\(143\) 8.41007 0.703286
\(144\) 0 0
\(145\) 10.7952 0.896490
\(146\) 0 0
\(147\) −10.7290 −0.884915
\(148\) 0 0
\(149\) −6.57887 −0.538962 −0.269481 0.963006i \(-0.586852\pi\)
−0.269481 + 0.963006i \(0.586852\pi\)
\(150\) 0 0
\(151\) −22.0126 −1.79136 −0.895678 0.444702i \(-0.853309\pi\)
−0.895678 + 0.444702i \(0.853309\pi\)
\(152\) 0 0
\(153\) −7.91600 −0.639971
\(154\) 0 0
\(155\) −19.6881 −1.58138
\(156\) 0 0
\(157\) 12.3950 0.989232 0.494616 0.869112i \(-0.335309\pi\)
0.494616 + 0.869112i \(0.335309\pi\)
\(158\) 0 0
\(159\) 7.78190 0.617145
\(160\) 0 0
\(161\) −10.0133 −0.789156
\(162\) 0 0
\(163\) −0.575434 −0.0450715 −0.0225357 0.999746i \(-0.507174\pi\)
−0.0225357 + 0.999746i \(0.507174\pi\)
\(164\) 0 0
\(165\) 5.38877 0.419515
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 12.9821 0.998626
\(170\) 0 0
\(171\) 2.28918 0.175058
\(172\) 0 0
\(173\) 17.0978 1.29992 0.649960 0.759969i \(-0.274786\pi\)
0.649960 + 0.759969i \(0.274786\pi\)
\(174\) 0 0
\(175\) 23.8627 1.80385
\(176\) 0 0
\(177\) −2.76900 −0.208131
\(178\) 0 0
\(179\) −3.74199 −0.279689 −0.139845 0.990173i \(-0.544660\pi\)
−0.139845 + 0.990173i \(0.544660\pi\)
\(180\) 0 0
\(181\) 7.68447 0.571182 0.285591 0.958352i \(-0.407810\pi\)
0.285591 + 0.958352i \(0.407810\pi\)
\(182\) 0 0
\(183\) 13.5958 1.00503
\(184\) 0 0
\(185\) −10.6401 −0.782275
\(186\) 0 0
\(187\) 13.0608 0.955097
\(188\) 0 0
\(189\) −4.21058 −0.306275
\(190\) 0 0
\(191\) 11.8665 0.858628 0.429314 0.903155i \(-0.358755\pi\)
0.429314 + 0.903155i \(0.358755\pi\)
\(192\) 0 0
\(193\) −7.56592 −0.544607 −0.272303 0.962211i \(-0.587785\pi\)
−0.272303 + 0.962211i \(0.587785\pi\)
\(194\) 0 0
\(195\) 16.6481 1.19220
\(196\) 0 0
\(197\) 13.2456 0.943710 0.471855 0.881676i \(-0.343585\pi\)
0.471855 + 0.881676i \(0.343585\pi\)
\(198\) 0 0
\(199\) −2.82206 −0.200051 −0.100025 0.994985i \(-0.531892\pi\)
−0.100025 + 0.994985i \(0.531892\pi\)
\(200\) 0 0
\(201\) 10.5454 0.743818
\(202\) 0 0
\(203\) 13.9170 0.976780
\(204\) 0 0
\(205\) −22.0849 −1.54247
\(206\) 0 0
\(207\) −2.37812 −0.165291
\(208\) 0 0
\(209\) −3.77697 −0.261258
\(210\) 0 0
\(211\) −9.97232 −0.686523 −0.343261 0.939240i \(-0.611532\pi\)
−0.343261 + 0.939240i \(0.611532\pi\)
\(212\) 0 0
\(213\) −0.104019 −0.00712726
\(214\) 0 0
\(215\) −3.25160 −0.221758
\(216\) 0 0
\(217\) −25.3816 −1.72301
\(218\) 0 0
\(219\) 6.12397 0.413819
\(220\) 0 0
\(221\) 40.3500 2.71423
\(222\) 0 0
\(223\) 22.9399 1.53617 0.768085 0.640347i \(-0.221210\pi\)
0.768085 + 0.640347i \(0.221210\pi\)
\(224\) 0 0
\(225\) 5.66730 0.377820
\(226\) 0 0
\(227\) −25.7876 −1.71159 −0.855793 0.517319i \(-0.826930\pi\)
−0.855793 + 0.517319i \(0.826930\pi\)
\(228\) 0 0
\(229\) 12.1888 0.805460 0.402730 0.915319i \(-0.368061\pi\)
0.402730 + 0.915319i \(0.368061\pi\)
\(230\) 0 0
\(231\) 6.94712 0.457087
\(232\) 0 0
\(233\) −8.47149 −0.554986 −0.277493 0.960728i \(-0.589503\pi\)
−0.277493 + 0.960728i \(0.589503\pi\)
\(234\) 0 0
\(235\) 22.9856 1.49941
\(236\) 0 0
\(237\) 3.17407 0.206178
\(238\) 0 0
\(239\) 17.5250 1.13360 0.566798 0.823857i \(-0.308182\pi\)
0.566798 + 0.823857i \(0.308182\pi\)
\(240\) 0 0
\(241\) −25.1764 −1.62176 −0.810878 0.585215i \(-0.801010\pi\)
−0.810878 + 0.585215i \(0.801010\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 35.0419 2.23874
\(246\) 0 0
\(247\) −11.6686 −0.742454
\(248\) 0 0
\(249\) −15.1125 −0.957718
\(250\) 0 0
\(251\) 2.14668 0.135497 0.0677486 0.997702i \(-0.478418\pi\)
0.0677486 + 0.997702i \(0.478418\pi\)
\(252\) 0 0
\(253\) 3.92370 0.246681
\(254\) 0 0
\(255\) 25.8543 1.61906
\(256\) 0 0
\(257\) −21.7250 −1.35517 −0.677585 0.735444i \(-0.736973\pi\)
−0.677585 + 0.735444i \(0.736973\pi\)
\(258\) 0 0
\(259\) −13.7170 −0.852335
\(260\) 0 0
\(261\) 3.30523 0.204589
\(262\) 0 0
\(263\) 18.3272 1.13011 0.565053 0.825055i \(-0.308856\pi\)
0.565053 + 0.825055i \(0.308856\pi\)
\(264\) 0 0
\(265\) −25.4163 −1.56131
\(266\) 0 0
\(267\) 11.4567 0.701140
\(268\) 0 0
\(269\) −18.7577 −1.14368 −0.571838 0.820367i \(-0.693769\pi\)
−0.571838 + 0.820367i \(0.693769\pi\)
\(270\) 0 0
\(271\) −30.6404 −1.86127 −0.930635 0.365950i \(-0.880744\pi\)
−0.930635 + 0.365950i \(0.880744\pi\)
\(272\) 0 0
\(273\) 21.4625 1.29897
\(274\) 0 0
\(275\) −9.35058 −0.563861
\(276\) 0 0
\(277\) 14.6795 0.882005 0.441003 0.897506i \(-0.354623\pi\)
0.441003 + 0.897506i \(0.354623\pi\)
\(278\) 0 0
\(279\) −6.02804 −0.360889
\(280\) 0 0
\(281\) 30.5620 1.82318 0.911588 0.411106i \(-0.134857\pi\)
0.911588 + 0.411106i \(0.134857\pi\)
\(282\) 0 0
\(283\) 23.0703 1.37139 0.685694 0.727890i \(-0.259499\pi\)
0.685694 + 0.727890i \(0.259499\pi\)
\(284\) 0 0
\(285\) −7.47667 −0.442880
\(286\) 0 0
\(287\) −28.4715 −1.68062
\(288\) 0 0
\(289\) 45.6631 2.68606
\(290\) 0 0
\(291\) 0.758299 0.0444523
\(292\) 0 0
\(293\) 19.9252 1.16404 0.582020 0.813174i \(-0.302263\pi\)
0.582020 + 0.813174i \(0.302263\pi\)
\(294\) 0 0
\(295\) 9.04380 0.526550
\(296\) 0 0
\(297\) 1.64992 0.0957379
\(298\) 0 0
\(299\) 12.1219 0.701028
\(300\) 0 0
\(301\) −4.19192 −0.241618
\(302\) 0 0
\(303\) −15.7478 −0.904685
\(304\) 0 0
\(305\) −44.4049 −2.54262
\(306\) 0 0
\(307\) 3.67492 0.209739 0.104869 0.994486i \(-0.466558\pi\)
0.104869 + 0.994486i \(0.466558\pi\)
\(308\) 0 0
\(309\) −0.604753 −0.0344032
\(310\) 0 0
\(311\) −33.1759 −1.88123 −0.940615 0.339475i \(-0.889751\pi\)
−0.940615 + 0.339475i \(0.889751\pi\)
\(312\) 0 0
\(313\) 10.2804 0.581082 0.290541 0.956863i \(-0.406165\pi\)
0.290541 + 0.956863i \(0.406165\pi\)
\(314\) 0 0
\(315\) 13.7521 0.774844
\(316\) 0 0
\(317\) 16.4030 0.921281 0.460641 0.887587i \(-0.347620\pi\)
0.460641 + 0.887587i \(0.347620\pi\)
\(318\) 0 0
\(319\) −5.45336 −0.305330
\(320\) 0 0
\(321\) 12.7791 0.713261
\(322\) 0 0
\(323\) −18.1212 −1.00829
\(324\) 0 0
\(325\) −28.8878 −1.60240
\(326\) 0 0
\(327\) 11.0076 0.608719
\(328\) 0 0
\(329\) 29.6326 1.63370
\(330\) 0 0
\(331\) −3.20238 −0.176018 −0.0880092 0.996120i \(-0.528051\pi\)
−0.0880092 + 0.996120i \(0.528051\pi\)
\(332\) 0 0
\(333\) −3.25775 −0.178524
\(334\) 0 0
\(335\) −34.4423 −1.88178
\(336\) 0 0
\(337\) 16.0805 0.875960 0.437980 0.898985i \(-0.355694\pi\)
0.437980 + 0.898985i \(0.355694\pi\)
\(338\) 0 0
\(339\) 17.2772 0.938371
\(340\) 0 0
\(341\) 9.94577 0.538593
\(342\) 0 0
\(343\) 15.7014 0.847793
\(344\) 0 0
\(345\) 7.76713 0.418168
\(346\) 0 0
\(347\) −0.541237 −0.0290551 −0.0145276 0.999894i \(-0.504624\pi\)
−0.0145276 + 0.999894i \(0.504624\pi\)
\(348\) 0 0
\(349\) 16.6118 0.889212 0.444606 0.895726i \(-0.353344\pi\)
0.444606 + 0.895726i \(0.353344\pi\)
\(350\) 0 0
\(351\) 5.09727 0.272072
\(352\) 0 0
\(353\) 29.8202 1.58717 0.793584 0.608461i \(-0.208213\pi\)
0.793584 + 0.608461i \(0.208213\pi\)
\(354\) 0 0
\(355\) 0.339734 0.0180312
\(356\) 0 0
\(357\) 33.3310 1.76406
\(358\) 0 0
\(359\) −6.74911 −0.356205 −0.178102 0.984012i \(-0.556996\pi\)
−0.178102 + 0.984012i \(0.556996\pi\)
\(360\) 0 0
\(361\) −13.7596 −0.724191
\(362\) 0 0
\(363\) 8.27777 0.434470
\(364\) 0 0
\(365\) −20.0014 −1.04692
\(366\) 0 0
\(367\) 19.1555 0.999910 0.499955 0.866051i \(-0.333350\pi\)
0.499955 + 0.866051i \(0.333350\pi\)
\(368\) 0 0
\(369\) −6.76188 −0.352009
\(370\) 0 0
\(371\) −32.7664 −1.70114
\(372\) 0 0
\(373\) 17.7032 0.916637 0.458319 0.888788i \(-0.348452\pi\)
0.458319 + 0.888788i \(0.348452\pi\)
\(374\) 0 0
\(375\) −2.17947 −0.112547
\(376\) 0 0
\(377\) −16.8477 −0.867699
\(378\) 0 0
\(379\) −11.4738 −0.589370 −0.294685 0.955594i \(-0.595215\pi\)
−0.294685 + 0.955594i \(0.595215\pi\)
\(380\) 0 0
\(381\) −5.31722 −0.272409
\(382\) 0 0
\(383\) 30.9550 1.58173 0.790864 0.611991i \(-0.209631\pi\)
0.790864 + 0.611991i \(0.209631\pi\)
\(384\) 0 0
\(385\) −22.6899 −1.15638
\(386\) 0 0
\(387\) −0.995567 −0.0506075
\(388\) 0 0
\(389\) −4.92327 −0.249620 −0.124810 0.992181i \(-0.539832\pi\)
−0.124810 + 0.992181i \(0.539832\pi\)
\(390\) 0 0
\(391\) 18.8252 0.952031
\(392\) 0 0
\(393\) −5.15770 −0.260172
\(394\) 0 0
\(395\) −10.3668 −0.521609
\(396\) 0 0
\(397\) −22.7967 −1.14413 −0.572066 0.820208i \(-0.693858\pi\)
−0.572066 + 0.820208i \(0.693858\pi\)
\(398\) 0 0
\(399\) −9.63880 −0.482544
\(400\) 0 0
\(401\) 8.15323 0.407153 0.203576 0.979059i \(-0.434743\pi\)
0.203576 + 0.979059i \(0.434743\pi\)
\(402\) 0 0
\(403\) 30.7265 1.53060
\(404\) 0 0
\(405\) 3.26608 0.162293
\(406\) 0 0
\(407\) 5.37502 0.266430
\(408\) 0 0
\(409\) −28.8407 −1.42608 −0.713041 0.701122i \(-0.752683\pi\)
−0.713041 + 0.701122i \(0.752683\pi\)
\(410\) 0 0
\(411\) −9.21320 −0.454454
\(412\) 0 0
\(413\) 11.6591 0.573708
\(414\) 0 0
\(415\) 49.3588 2.42293
\(416\) 0 0
\(417\) −12.5130 −0.612766
\(418\) 0 0
\(419\) −6.28566 −0.307075 −0.153537 0.988143i \(-0.549067\pi\)
−0.153537 + 0.988143i \(0.549067\pi\)
\(420\) 0 0
\(421\) −7.48032 −0.364569 −0.182284 0.983246i \(-0.558349\pi\)
−0.182284 + 0.983246i \(0.558349\pi\)
\(422\) 0 0
\(423\) 7.03765 0.342182
\(424\) 0 0
\(425\) −44.8624 −2.17615
\(426\) 0 0
\(427\) −57.2461 −2.77033
\(428\) 0 0
\(429\) −8.41007 −0.406042
\(430\) 0 0
\(431\) −1.40836 −0.0678384 −0.0339192 0.999425i \(-0.510799\pi\)
−0.0339192 + 0.999425i \(0.510799\pi\)
\(432\) 0 0
\(433\) −34.2424 −1.64558 −0.822791 0.568344i \(-0.807584\pi\)
−0.822791 + 0.568344i \(0.807584\pi\)
\(434\) 0 0
\(435\) −10.7952 −0.517589
\(436\) 0 0
\(437\) −5.44395 −0.260420
\(438\) 0 0
\(439\) 34.5156 1.64734 0.823669 0.567071i \(-0.191924\pi\)
0.823669 + 0.567071i \(0.191924\pi\)
\(440\) 0 0
\(441\) 10.7290 0.510906
\(442\) 0 0
\(443\) 5.46501 0.259650 0.129825 0.991537i \(-0.458558\pi\)
0.129825 + 0.991537i \(0.458558\pi\)
\(444\) 0 0
\(445\) −37.4186 −1.77381
\(446\) 0 0
\(447\) 6.57887 0.311170
\(448\) 0 0
\(449\) −29.4100 −1.38794 −0.693971 0.720003i \(-0.744141\pi\)
−0.693971 + 0.720003i \(0.744141\pi\)
\(450\) 0 0
\(451\) 11.1565 0.525341
\(452\) 0 0
\(453\) 22.0126 1.03424
\(454\) 0 0
\(455\) −70.0982 −3.28626
\(456\) 0 0
\(457\) −23.1605 −1.08340 −0.541700 0.840572i \(-0.682219\pi\)
−0.541700 + 0.840572i \(0.682219\pi\)
\(458\) 0 0
\(459\) 7.91600 0.369487
\(460\) 0 0
\(461\) −31.6357 −1.47342 −0.736711 0.676208i \(-0.763622\pi\)
−0.736711 + 0.676208i \(0.763622\pi\)
\(462\) 0 0
\(463\) −19.0045 −0.883216 −0.441608 0.897208i \(-0.645592\pi\)
−0.441608 + 0.897208i \(0.645592\pi\)
\(464\) 0 0
\(465\) 19.6881 0.913012
\(466\) 0 0
\(467\) 35.6469 1.64954 0.824772 0.565466i \(-0.191304\pi\)
0.824772 + 0.565466i \(0.191304\pi\)
\(468\) 0 0
\(469\) −44.4024 −2.05031
\(470\) 0 0
\(471\) −12.3950 −0.571133
\(472\) 0 0
\(473\) 1.64260 0.0755270
\(474\) 0 0
\(475\) 12.9735 0.595265
\(476\) 0 0
\(477\) −7.78190 −0.356309
\(478\) 0 0
\(479\) −28.3089 −1.29347 −0.646733 0.762717i \(-0.723865\pi\)
−0.646733 + 0.762717i \(0.723865\pi\)
\(480\) 0 0
\(481\) 16.6056 0.757151
\(482\) 0 0
\(483\) 10.0133 0.455620
\(484\) 0 0
\(485\) −2.47667 −0.112460
\(486\) 0 0
\(487\) −12.7102 −0.575955 −0.287978 0.957637i \(-0.592983\pi\)
−0.287978 + 0.957637i \(0.592983\pi\)
\(488\) 0 0
\(489\) 0.575434 0.0260220
\(490\) 0 0
\(491\) −34.2050 −1.54365 −0.771826 0.635834i \(-0.780656\pi\)
−0.771826 + 0.635834i \(0.780656\pi\)
\(492\) 0 0
\(493\) −26.1642 −1.17838
\(494\) 0 0
\(495\) −5.38877 −0.242207
\(496\) 0 0
\(497\) 0.437980 0.0196461
\(498\) 0 0
\(499\) −40.9349 −1.83250 −0.916248 0.400612i \(-0.868798\pi\)
−0.916248 + 0.400612i \(0.868798\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) 12.1601 0.542192 0.271096 0.962552i \(-0.412614\pi\)
0.271096 + 0.962552i \(0.412614\pi\)
\(504\) 0 0
\(505\) 51.4335 2.28876
\(506\) 0 0
\(507\) −12.9821 −0.576557
\(508\) 0 0
\(509\) 8.63676 0.382818 0.191409 0.981510i \(-0.438694\pi\)
0.191409 + 0.981510i \(0.438694\pi\)
\(510\) 0 0
\(511\) −25.7855 −1.14068
\(512\) 0 0
\(513\) −2.28918 −0.101070
\(514\) 0 0
\(515\) 1.97517 0.0870365
\(516\) 0 0
\(517\) −11.6115 −0.510675
\(518\) 0 0
\(519\) −17.0978 −0.750509
\(520\) 0 0
\(521\) 0.450632 0.0197426 0.00987128 0.999951i \(-0.496858\pi\)
0.00987128 + 0.999951i \(0.496858\pi\)
\(522\) 0 0
\(523\) 21.6091 0.944901 0.472451 0.881357i \(-0.343370\pi\)
0.472451 + 0.881357i \(0.343370\pi\)
\(524\) 0 0
\(525\) −23.8627 −1.04145
\(526\) 0 0
\(527\) 47.7179 2.07863
\(528\) 0 0
\(529\) −17.3446 −0.754111
\(530\) 0 0
\(531\) 2.76900 0.120165
\(532\) 0 0
\(533\) 34.4671 1.49294
\(534\) 0 0
\(535\) −41.7377 −1.80448
\(536\) 0 0
\(537\) 3.74199 0.161479
\(538\) 0 0
\(539\) −17.7020 −0.762479
\(540\) 0 0
\(541\) 42.0014 1.80578 0.902890 0.429872i \(-0.141441\pi\)
0.902890 + 0.429872i \(0.141441\pi\)
\(542\) 0 0
\(543\) −7.68447 −0.329772
\(544\) 0 0
\(545\) −35.9516 −1.54000
\(546\) 0 0
\(547\) 11.4897 0.491266 0.245633 0.969363i \(-0.421004\pi\)
0.245633 + 0.969363i \(0.421004\pi\)
\(548\) 0 0
\(549\) −13.5958 −0.580253
\(550\) 0 0
\(551\) 7.56629 0.322335
\(552\) 0 0
\(553\) −13.3647 −0.568324
\(554\) 0 0
\(555\) 10.6401 0.451647
\(556\) 0 0
\(557\) 1.64653 0.0697659 0.0348829 0.999391i \(-0.488894\pi\)
0.0348829 + 0.999391i \(0.488894\pi\)
\(558\) 0 0
\(559\) 5.07467 0.214636
\(560\) 0 0
\(561\) −13.0608 −0.551425
\(562\) 0 0
\(563\) 9.99544 0.421258 0.210629 0.977566i \(-0.432449\pi\)
0.210629 + 0.977566i \(0.432449\pi\)
\(564\) 0 0
\(565\) −56.4289 −2.37398
\(566\) 0 0
\(567\) 4.21058 0.176828
\(568\) 0 0
\(569\) 36.9807 1.55031 0.775156 0.631770i \(-0.217671\pi\)
0.775156 + 0.631770i \(0.217671\pi\)
\(570\) 0 0
\(571\) 28.5674 1.19551 0.597755 0.801679i \(-0.296060\pi\)
0.597755 + 0.801679i \(0.296060\pi\)
\(572\) 0 0
\(573\) −11.8665 −0.495729
\(574\) 0 0
\(575\) −13.4775 −0.562051
\(576\) 0 0
\(577\) 7.46548 0.310792 0.155396 0.987852i \(-0.450335\pi\)
0.155396 + 0.987852i \(0.450335\pi\)
\(578\) 0 0
\(579\) 7.56592 0.314429
\(580\) 0 0
\(581\) 63.6326 2.63993
\(582\) 0 0
\(583\) 12.8395 0.531758
\(584\) 0 0
\(585\) −16.6481 −0.688314
\(586\) 0 0
\(587\) 27.1153 1.11917 0.559585 0.828773i \(-0.310961\pi\)
0.559585 + 0.828773i \(0.310961\pi\)
\(588\) 0 0
\(589\) −13.7993 −0.568590
\(590\) 0 0
\(591\) −13.2456 −0.544851
\(592\) 0 0
\(593\) −32.8939 −1.35079 −0.675395 0.737456i \(-0.736027\pi\)
−0.675395 + 0.737456i \(0.736027\pi\)
\(594\) 0 0
\(595\) −108.862 −4.46290
\(596\) 0 0
\(597\) 2.82206 0.115499
\(598\) 0 0
\(599\) 20.4006 0.833545 0.416772 0.909011i \(-0.363161\pi\)
0.416772 + 0.909011i \(0.363161\pi\)
\(600\) 0 0
\(601\) −34.4674 −1.40596 −0.702978 0.711212i \(-0.748147\pi\)
−0.702978 + 0.711212i \(0.748147\pi\)
\(602\) 0 0
\(603\) −10.5454 −0.429443
\(604\) 0 0
\(605\) −27.0359 −1.09917
\(606\) 0 0
\(607\) −7.27037 −0.295095 −0.147548 0.989055i \(-0.547138\pi\)
−0.147548 + 0.989055i \(0.547138\pi\)
\(608\) 0 0
\(609\) −13.9170 −0.563944
\(610\) 0 0
\(611\) −35.8728 −1.45126
\(612\) 0 0
\(613\) −11.7901 −0.476199 −0.238100 0.971241i \(-0.576525\pi\)
−0.238100 + 0.971241i \(0.576525\pi\)
\(614\) 0 0
\(615\) 22.0849 0.890548
\(616\) 0 0
\(617\) −28.2396 −1.13688 −0.568442 0.822723i \(-0.692454\pi\)
−0.568442 + 0.822723i \(0.692454\pi\)
\(618\) 0 0
\(619\) −11.3137 −0.454735 −0.227368 0.973809i \(-0.573012\pi\)
−0.227368 + 0.973809i \(0.573012\pi\)
\(620\) 0 0
\(621\) 2.37812 0.0954306
\(622\) 0 0
\(623\) −48.2395 −1.93267
\(624\) 0 0
\(625\) −21.2182 −0.848728
\(626\) 0 0
\(627\) 3.77697 0.150838
\(628\) 0 0
\(629\) 25.7884 1.02825
\(630\) 0 0
\(631\) −18.4312 −0.733733 −0.366866 0.930274i \(-0.619569\pi\)
−0.366866 + 0.930274i \(0.619569\pi\)
\(632\) 0 0
\(633\) 9.97232 0.396364
\(634\) 0 0
\(635\) 17.3665 0.689168
\(636\) 0 0
\(637\) −54.6887 −2.16684
\(638\) 0 0
\(639\) 0.104019 0.00411492
\(640\) 0 0
\(641\) 38.3952 1.51652 0.758260 0.651952i \(-0.226050\pi\)
0.758260 + 0.651952i \(0.226050\pi\)
\(642\) 0 0
\(643\) −11.5338 −0.454849 −0.227425 0.973796i \(-0.573031\pi\)
−0.227425 + 0.973796i \(0.573031\pi\)
\(644\) 0 0
\(645\) 3.25160 0.128032
\(646\) 0 0
\(647\) −49.8032 −1.95797 −0.978983 0.203942i \(-0.934625\pi\)
−0.978983 + 0.203942i \(0.934625\pi\)
\(648\) 0 0
\(649\) −4.56863 −0.179334
\(650\) 0 0
\(651\) 25.3816 0.994782
\(652\) 0 0
\(653\) 16.6308 0.650814 0.325407 0.945574i \(-0.394499\pi\)
0.325407 + 0.945574i \(0.394499\pi\)
\(654\) 0 0
\(655\) 16.8455 0.658208
\(656\) 0 0
\(657\) −6.12397 −0.238919
\(658\) 0 0
\(659\) −5.03596 −0.196173 −0.0980866 0.995178i \(-0.531272\pi\)
−0.0980866 + 0.995178i \(0.531272\pi\)
\(660\) 0 0
\(661\) 6.59712 0.256598 0.128299 0.991736i \(-0.459048\pi\)
0.128299 + 0.991736i \(0.459048\pi\)
\(662\) 0 0
\(663\) −40.3500 −1.56706
\(664\) 0 0
\(665\) 31.4811 1.22079
\(666\) 0 0
\(667\) −7.86024 −0.304350
\(668\) 0 0
\(669\) −22.9399 −0.886909
\(670\) 0 0
\(671\) 22.4319 0.865973
\(672\) 0 0
\(673\) 11.7616 0.453377 0.226688 0.973967i \(-0.427210\pi\)
0.226688 + 0.973967i \(0.427210\pi\)
\(674\) 0 0
\(675\) −5.66730 −0.218135
\(676\) 0 0
\(677\) −0.322074 −0.0123783 −0.00618915 0.999981i \(-0.501970\pi\)
−0.00618915 + 0.999981i \(0.501970\pi\)
\(678\) 0 0
\(679\) −3.19288 −0.122532
\(680\) 0 0
\(681\) 25.7876 0.988185
\(682\) 0 0
\(683\) 11.3210 0.433185 0.216593 0.976262i \(-0.430506\pi\)
0.216593 + 0.976262i \(0.430506\pi\)
\(684\) 0 0
\(685\) 30.0911 1.14972
\(686\) 0 0
\(687\) −12.1888 −0.465032
\(688\) 0 0
\(689\) 39.6664 1.51117
\(690\) 0 0
\(691\) −29.2991 −1.11459 −0.557295 0.830314i \(-0.688161\pi\)
−0.557295 + 0.830314i \(0.688161\pi\)
\(692\) 0 0
\(693\) −6.94712 −0.263899
\(694\) 0 0
\(695\) 40.8686 1.55024
\(696\) 0 0
\(697\) 53.5271 2.02748
\(698\) 0 0
\(699\) 8.47149 0.320421
\(700\) 0 0
\(701\) 19.6166 0.740910 0.370455 0.928850i \(-0.379202\pi\)
0.370455 + 0.928850i \(0.379202\pi\)
\(702\) 0 0
\(703\) −7.45759 −0.281268
\(704\) 0 0
\(705\) −22.9856 −0.865686
\(706\) 0 0
\(707\) 66.3073 2.49374
\(708\) 0 0
\(709\) 21.2453 0.797884 0.398942 0.916976i \(-0.369378\pi\)
0.398942 + 0.916976i \(0.369378\pi\)
\(710\) 0 0
\(711\) −3.17407 −0.119037
\(712\) 0 0
\(713\) 14.3354 0.536864
\(714\) 0 0
\(715\) 27.4680 1.02724
\(716\) 0 0
\(717\) −17.5250 −0.654481
\(718\) 0 0
\(719\) −0.176660 −0.00658832 −0.00329416 0.999995i \(-0.501049\pi\)
−0.00329416 + 0.999995i \(0.501049\pi\)
\(720\) 0 0
\(721\) 2.54636 0.0948315
\(722\) 0 0
\(723\) 25.1764 0.936321
\(724\) 0 0
\(725\) 18.7318 0.695680
\(726\) 0 0
\(727\) −32.4701 −1.20425 −0.602125 0.798402i \(-0.705679\pi\)
−0.602125 + 0.798402i \(0.705679\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 7.88091 0.291486
\(732\) 0 0
\(733\) −22.7271 −0.839445 −0.419723 0.907652i \(-0.637873\pi\)
−0.419723 + 0.907652i \(0.637873\pi\)
\(734\) 0 0
\(735\) −35.0419 −1.29254
\(736\) 0 0
\(737\) 17.3991 0.640904
\(738\) 0 0
\(739\) −6.53177 −0.240275 −0.120137 0.992757i \(-0.538334\pi\)
−0.120137 + 0.992757i \(0.538334\pi\)
\(740\) 0 0
\(741\) 11.6686 0.428656
\(742\) 0 0
\(743\) 13.2769 0.487084 0.243542 0.969890i \(-0.421691\pi\)
0.243542 + 0.969890i \(0.421691\pi\)
\(744\) 0 0
\(745\) −21.4871 −0.787228
\(746\) 0 0
\(747\) 15.1125 0.552939
\(748\) 0 0
\(749\) −53.8076 −1.96609
\(750\) 0 0
\(751\) 47.5336 1.73453 0.867263 0.497851i \(-0.165877\pi\)
0.867263 + 0.497851i \(0.165877\pi\)
\(752\) 0 0
\(753\) −2.14668 −0.0782294
\(754\) 0 0
\(755\) −71.8949 −2.61652
\(756\) 0 0
\(757\) −1.99499 −0.0725093 −0.0362546 0.999343i \(-0.511543\pi\)
−0.0362546 + 0.999343i \(0.511543\pi\)
\(758\) 0 0
\(759\) −3.92370 −0.142421
\(760\) 0 0
\(761\) −23.3966 −0.848126 −0.424063 0.905633i \(-0.639397\pi\)
−0.424063 + 0.905633i \(0.639397\pi\)
\(762\) 0 0
\(763\) −46.3482 −1.67792
\(764\) 0 0
\(765\) −25.8543 −0.934765
\(766\) 0 0
\(767\) −14.1144 −0.509640
\(768\) 0 0
\(769\) −4.40764 −0.158944 −0.0794718 0.996837i \(-0.525323\pi\)
−0.0794718 + 0.996837i \(0.525323\pi\)
\(770\) 0 0
\(771\) 21.7250 0.782408
\(772\) 0 0
\(773\) −43.4847 −1.56404 −0.782018 0.623256i \(-0.785810\pi\)
−0.782018 + 0.623256i \(0.785810\pi\)
\(774\) 0 0
\(775\) −34.1627 −1.22716
\(776\) 0 0
\(777\) 13.7170 0.492096
\(778\) 0 0
\(779\) −15.4792 −0.554600
\(780\) 0 0
\(781\) −0.171623 −0.00614114
\(782\) 0 0
\(783\) −3.30523 −0.118119
\(784\) 0 0
\(785\) 40.4832 1.44491
\(786\) 0 0
\(787\) −7.63599 −0.272194 −0.136097 0.990696i \(-0.543456\pi\)
−0.136097 + 0.990696i \(0.543456\pi\)
\(788\) 0 0
\(789\) −18.3272 −0.652467
\(790\) 0 0
\(791\) −72.7473 −2.58659
\(792\) 0 0
\(793\) 69.3012 2.46096
\(794\) 0 0
\(795\) 25.4163 0.901425
\(796\) 0 0
\(797\) −1.17701 −0.0416920 −0.0208460 0.999783i \(-0.506636\pi\)
−0.0208460 + 0.999783i \(0.506636\pi\)
\(798\) 0 0
\(799\) −55.7101 −1.97088
\(800\) 0 0
\(801\) −11.4567 −0.404803
\(802\) 0 0
\(803\) 10.1040 0.356564
\(804\) 0 0
\(805\) −32.7042 −1.15267
\(806\) 0 0
\(807\) 18.7577 0.660301
\(808\) 0 0
\(809\) 50.7537 1.78441 0.892203 0.451634i \(-0.149159\pi\)
0.892203 + 0.451634i \(0.149159\pi\)
\(810\) 0 0
\(811\) −4.42078 −0.155235 −0.0776174 0.996983i \(-0.524731\pi\)
−0.0776174 + 0.996983i \(0.524731\pi\)
\(812\) 0 0
\(813\) 30.6404 1.07460
\(814\) 0 0
\(815\) −1.87942 −0.0658331
\(816\) 0 0
\(817\) −2.27904 −0.0797334
\(818\) 0 0
\(819\) −21.4625 −0.749960
\(820\) 0 0
\(821\) 24.7260 0.862943 0.431472 0.902127i \(-0.357995\pi\)
0.431472 + 0.902127i \(0.357995\pi\)
\(822\) 0 0
\(823\) −5.09032 −0.177437 −0.0887187 0.996057i \(-0.528277\pi\)
−0.0887187 + 0.996057i \(0.528277\pi\)
\(824\) 0 0
\(825\) 9.35058 0.325546
\(826\) 0 0
\(827\) −32.2760 −1.12234 −0.561172 0.827699i \(-0.689650\pi\)
−0.561172 + 0.827699i \(0.689650\pi\)
\(828\) 0 0
\(829\) 9.44996 0.328211 0.164105 0.986443i \(-0.447526\pi\)
0.164105 + 0.986443i \(0.447526\pi\)
\(830\) 0 0
\(831\) −14.6795 −0.509226
\(832\) 0 0
\(833\) −84.9309 −2.94268
\(834\) 0 0
\(835\) −3.26608 −0.113027
\(836\) 0 0
\(837\) 6.02804 0.208359
\(838\) 0 0
\(839\) −1.01604 −0.0350776 −0.0175388 0.999846i \(-0.505583\pi\)
−0.0175388 + 0.999846i \(0.505583\pi\)
\(840\) 0 0
\(841\) −18.0754 −0.623291
\(842\) 0 0
\(843\) −30.5620 −1.05261
\(844\) 0 0
\(845\) 42.4007 1.45863
\(846\) 0 0
\(847\) −34.8543 −1.19761
\(848\) 0 0
\(849\) −23.0703 −0.791771
\(850\) 0 0
\(851\) 7.74732 0.265575
\(852\) 0 0
\(853\) 10.2405 0.350629 0.175315 0.984512i \(-0.443906\pi\)
0.175315 + 0.984512i \(0.443906\pi\)
\(854\) 0 0
\(855\) 7.47667 0.255697
\(856\) 0 0
\(857\) 16.5117 0.564030 0.282015 0.959410i \(-0.408997\pi\)
0.282015 + 0.959410i \(0.408997\pi\)
\(858\) 0 0
\(859\) 10.4220 0.355592 0.177796 0.984067i \(-0.443103\pi\)
0.177796 + 0.984067i \(0.443103\pi\)
\(860\) 0 0
\(861\) 28.4715 0.970305
\(862\) 0 0
\(863\) 17.3376 0.590179 0.295089 0.955470i \(-0.404651\pi\)
0.295089 + 0.955470i \(0.404651\pi\)
\(864\) 0 0
\(865\) 55.8427 1.89871
\(866\) 0 0
\(867\) −45.6631 −1.55080
\(868\) 0 0
\(869\) 5.23695 0.177651
\(870\) 0 0
\(871\) 53.7529 1.82135
\(872\) 0 0
\(873\) −0.758299 −0.0256645
\(874\) 0 0
\(875\) 9.17683 0.310234
\(876\) 0 0
\(877\) −52.7758 −1.78211 −0.891056 0.453893i \(-0.850035\pi\)
−0.891056 + 0.453893i \(0.850035\pi\)
\(878\) 0 0
\(879\) −19.9252 −0.672059
\(880\) 0 0
\(881\) 18.7239 0.630825 0.315412 0.948955i \(-0.397857\pi\)
0.315412 + 0.948955i \(0.397857\pi\)
\(882\) 0 0
\(883\) −1.18388 −0.0398408 −0.0199204 0.999802i \(-0.506341\pi\)
−0.0199204 + 0.999802i \(0.506341\pi\)
\(884\) 0 0
\(885\) −9.04380 −0.304004
\(886\) 0 0
\(887\) −49.6215 −1.66613 −0.833063 0.553178i \(-0.813415\pi\)
−0.833063 + 0.553178i \(0.813415\pi\)
\(888\) 0 0
\(889\) 22.3886 0.750890
\(890\) 0 0
\(891\) −1.64992 −0.0552743
\(892\) 0 0
\(893\) 16.1105 0.539117
\(894\) 0 0
\(895\) −12.2216 −0.408524
\(896\) 0 0
\(897\) −12.1219 −0.404739
\(898\) 0 0
\(899\) −19.9241 −0.664505
\(900\) 0 0
\(901\) 61.6015 2.05225
\(902\) 0 0
\(903\) 4.19192 0.139498
\(904\) 0 0
\(905\) 25.0981 0.834290
\(906\) 0 0
\(907\) 39.7086 1.31850 0.659251 0.751923i \(-0.270873\pi\)
0.659251 + 0.751923i \(0.270873\pi\)
\(908\) 0 0
\(909\) 15.7478 0.522320
\(910\) 0 0
\(911\) −13.0082 −0.430980 −0.215490 0.976506i \(-0.569135\pi\)
−0.215490 + 0.976506i \(0.569135\pi\)
\(912\) 0 0
\(913\) −24.9344 −0.825210
\(914\) 0 0
\(915\) 44.4049 1.46798
\(916\) 0 0
\(917\) 21.7169 0.717157
\(918\) 0 0
\(919\) −20.7415 −0.684198 −0.342099 0.939664i \(-0.611138\pi\)
−0.342099 + 0.939664i \(0.611138\pi\)
\(920\) 0 0
\(921\) −3.67492 −0.121093
\(922\) 0 0
\(923\) −0.530212 −0.0174521
\(924\) 0 0
\(925\) −18.4627 −0.607049
\(926\) 0 0
\(927\) 0.604753 0.0198627
\(928\) 0 0
\(929\) −16.5351 −0.542500 −0.271250 0.962509i \(-0.587437\pi\)
−0.271250 + 0.962509i \(0.587437\pi\)
\(930\) 0 0
\(931\) 24.5607 0.804945
\(932\) 0 0
\(933\) 33.1759 1.08613
\(934\) 0 0
\(935\) 42.6575 1.39505
\(936\) 0 0
\(937\) 32.6845 1.06776 0.533879 0.845561i \(-0.320734\pi\)
0.533879 + 0.845561i \(0.320734\pi\)
\(938\) 0 0
\(939\) −10.2804 −0.335488
\(940\) 0 0
\(941\) −24.2334 −0.789987 −0.394994 0.918684i \(-0.629253\pi\)
−0.394994 + 0.918684i \(0.629253\pi\)
\(942\) 0 0
\(943\) 16.0806 0.523655
\(944\) 0 0
\(945\) −13.7521 −0.447357
\(946\) 0 0
\(947\) 47.7988 1.55325 0.776626 0.629962i \(-0.216930\pi\)
0.776626 + 0.629962i \(0.216930\pi\)
\(948\) 0 0
\(949\) 31.2155 1.01330
\(950\) 0 0
\(951\) −16.4030 −0.531902
\(952\) 0 0
\(953\) −11.6792 −0.378325 −0.189163 0.981946i \(-0.560577\pi\)
−0.189163 + 0.981946i \(0.560577\pi\)
\(954\) 0 0
\(955\) 38.7569 1.25414
\(956\) 0 0
\(957\) 5.45336 0.176282
\(958\) 0 0
\(959\) 38.7930 1.25269
\(960\) 0 0
\(961\) 5.33722 0.172169
\(962\) 0 0
\(963\) −12.7791 −0.411801
\(964\) 0 0
\(965\) −24.7109 −0.795473
\(966\) 0 0
\(967\) −53.5374 −1.72165 −0.860823 0.508905i \(-0.830051\pi\)
−0.860823 + 0.508905i \(0.830051\pi\)
\(968\) 0 0
\(969\) 18.1212 0.582136
\(970\) 0 0
\(971\) 25.8894 0.830830 0.415415 0.909632i \(-0.363636\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(972\) 0 0
\(973\) 52.6872 1.68907
\(974\) 0 0
\(975\) 28.8878 0.925149
\(976\) 0 0
\(977\) 54.3200 1.73785 0.868925 0.494943i \(-0.164811\pi\)
0.868925 + 0.494943i \(0.164811\pi\)
\(978\) 0 0
\(979\) 18.9026 0.604131
\(980\) 0 0
\(981\) −11.0076 −0.351444
\(982\) 0 0
\(983\) 42.7846 1.36462 0.682309 0.731064i \(-0.260976\pi\)
0.682309 + 0.731064i \(0.260976\pi\)
\(984\) 0 0
\(985\) 43.2612 1.37842
\(986\) 0 0
\(987\) −29.6326 −0.943217
\(988\) 0 0
\(989\) 2.36758 0.0752845
\(990\) 0 0
\(991\) 42.2570 1.34234 0.671169 0.741305i \(-0.265793\pi\)
0.671169 + 0.741305i \(0.265793\pi\)
\(992\) 0 0
\(993\) 3.20238 0.101624
\(994\) 0 0
\(995\) −9.21709 −0.292201
\(996\) 0 0
\(997\) 34.6191 1.09640 0.548198 0.836348i \(-0.315314\pi\)
0.548198 + 0.836348i \(0.315314\pi\)
\(998\) 0 0
\(999\) 3.25775 0.103071
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 8016.2.a.s.1.5 5
4.3 odd 2 4008.2.a.f.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.f.1.5 5 4.3 odd 2
8016.2.a.s.1.5 5 1.1 even 1 trivial