Properties

Label 624.4.a.j.1.2
Level $624$
Weight $4$
Character 624.1
Self dual yes
Analytic conductor $36.817$
Analytic rank $1$
Dimension $2$
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] = [624,4,Mod(1,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.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: \(36.8171918436\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -4.87689 q^{5} +25.6155 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -4.87689 q^{5} +25.6155 q^{7} +9.00000 q^{9} -59.7235 q^{11} +13.0000 q^{13} +14.6307 q^{15} -75.4773 q^{17} +116.354 q^{19} -76.8466 q^{21} +90.5227 q^{23} -101.216 q^{25} -27.0000 q^{27} -187.261 q^{29} +225.062 q^{31} +179.170 q^{33} -124.924 q^{35} +290.648 q^{37} -39.0000 q^{39} -191.339 q^{41} -326.833 q^{43} -43.8920 q^{45} +406.773 q^{47} +313.155 q^{49} +226.432 q^{51} -426.985 q^{53} +291.265 q^{55} -349.062 q^{57} -331.015 q^{59} -524.678 q^{61} +230.540 q^{63} -63.3996 q^{65} -968.172 q^{67} -271.568 q^{69} +8.39776 q^{71} -903.329 q^{73} +303.648 q^{75} -1529.85 q^{77} -1157.98 q^{79} +81.0000 q^{81} -952.371 q^{83} +368.095 q^{85} +561.784 q^{87} +1059.96 q^{89} +333.002 q^{91} -675.187 q^{93} -567.447 q^{95} -90.7689 q^{97} -537.511 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 18 q^{5} + 10 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 18 q^{5} + 10 q^{7} + 18 q^{9} - 4 q^{11} + 26 q^{13} + 54 q^{15} - 52 q^{17} + 142 q^{19} - 30 q^{21} + 280 q^{23} - 54 q^{25} - 54 q^{27} - 424 q^{29} + 178 q^{31} + 12 q^{33} + 80 q^{35} + 136 q^{37} - 78 q^{39} - 226 q^{41} + 72 q^{43} - 162 q^{45} - 176 q^{47} + 214 q^{49} + 156 q^{51} - 788 q^{53} - 440 q^{55} - 426 q^{57} - 728 q^{59} - 736 q^{61} + 90 q^{63} - 234 q^{65} - 1054 q^{67} - 840 q^{69} + 660 q^{71} + 24 q^{73} + 162 q^{75} - 2400 q^{77} - 40 q^{79} + 162 q^{81} - 1344 q^{83} + 60 q^{85} + 1272 q^{87} + 314 q^{89} + 130 q^{91} - 534 q^{93} - 904 q^{95} - 264 q^{97} - 36 q^{99}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −4.87689 −0.436203 −0.218101 0.975926i \(-0.569986\pi\)
−0.218101 + 0.975926i \(0.569986\pi\)
\(6\) 0 0
\(7\) 25.6155 1.38311 0.691554 0.722325i \(-0.256926\pi\)
0.691554 + 0.722325i \(0.256926\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −59.7235 −1.63703 −0.818514 0.574487i \(-0.805202\pi\)
−0.818514 + 0.574487i \(0.805202\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) 14.6307 0.251842
\(16\) 0 0
\(17\) −75.4773 −1.07682 −0.538410 0.842683i \(-0.680975\pi\)
−0.538410 + 0.842683i \(0.680975\pi\)
\(18\) 0 0
\(19\) 116.354 1.40492 0.702460 0.711723i \(-0.252085\pi\)
0.702460 + 0.711723i \(0.252085\pi\)
\(20\) 0 0
\(21\) −76.8466 −0.798538
\(22\) 0 0
\(23\) 90.5227 0.820665 0.410332 0.911936i \(-0.365413\pi\)
0.410332 + 0.911936i \(0.365413\pi\)
\(24\) 0 0
\(25\) −101.216 −0.809727
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −187.261 −1.19909 −0.599544 0.800342i \(-0.704651\pi\)
−0.599544 + 0.800342i \(0.704651\pi\)
\(30\) 0 0
\(31\) 225.062 1.30395 0.651974 0.758241i \(-0.273941\pi\)
0.651974 + 0.758241i \(0.273941\pi\)
\(32\) 0 0
\(33\) 179.170 0.945138
\(34\) 0 0
\(35\) −124.924 −0.603316
\(36\) 0 0
\(37\) 290.648 1.29141 0.645705 0.763587i \(-0.276563\pi\)
0.645705 + 0.763587i \(0.276563\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) −191.339 −0.728833 −0.364416 0.931236i \(-0.618731\pi\)
−0.364416 + 0.931236i \(0.618731\pi\)
\(42\) 0 0
\(43\) −326.833 −1.15911 −0.579554 0.814934i \(-0.696773\pi\)
−0.579554 + 0.814934i \(0.696773\pi\)
\(44\) 0 0
\(45\) −43.8920 −0.145401
\(46\) 0 0
\(47\) 406.773 1.26242 0.631212 0.775611i \(-0.282558\pi\)
0.631212 + 0.775611i \(0.282558\pi\)
\(48\) 0 0
\(49\) 313.155 0.912989
\(50\) 0 0
\(51\) 226.432 0.621702
\(52\) 0 0
\(53\) −426.985 −1.10662 −0.553310 0.832975i \(-0.686636\pi\)
−0.553310 + 0.832975i \(0.686636\pi\)
\(54\) 0 0
\(55\) 291.265 0.714076
\(56\) 0 0
\(57\) −349.062 −0.811131
\(58\) 0 0
\(59\) −331.015 −0.730415 −0.365208 0.930926i \(-0.619002\pi\)
−0.365208 + 0.930926i \(0.619002\pi\)
\(60\) 0 0
\(61\) −524.678 −1.10128 −0.550640 0.834743i \(-0.685616\pi\)
−0.550640 + 0.834743i \(0.685616\pi\)
\(62\) 0 0
\(63\) 230.540 0.461036
\(64\) 0 0
\(65\) −63.3996 −0.120981
\(66\) 0 0
\(67\) −968.172 −1.76539 −0.882695 0.469947i \(-0.844273\pi\)
−0.882695 + 0.469947i \(0.844273\pi\)
\(68\) 0 0
\(69\) −271.568 −0.473811
\(70\) 0 0
\(71\) 8.39776 0.0140371 0.00701853 0.999975i \(-0.497766\pi\)
0.00701853 + 0.999975i \(0.497766\pi\)
\(72\) 0 0
\(73\) −903.329 −1.44831 −0.724156 0.689637i \(-0.757770\pi\)
−0.724156 + 0.689637i \(0.757770\pi\)
\(74\) 0 0
\(75\) 303.648 0.467496
\(76\) 0 0
\(77\) −1529.85 −2.26419
\(78\) 0 0
\(79\) −1157.98 −1.64915 −0.824573 0.565755i \(-0.808585\pi\)
−0.824573 + 0.565755i \(0.808585\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −952.371 −1.25947 −0.629737 0.776809i \(-0.716837\pi\)
−0.629737 + 0.776809i \(0.716837\pi\)
\(84\) 0 0
\(85\) 368.095 0.469711
\(86\) 0 0
\(87\) 561.784 0.692294
\(88\) 0 0
\(89\) 1059.96 1.26242 0.631211 0.775611i \(-0.282558\pi\)
0.631211 + 0.775611i \(0.282558\pi\)
\(90\) 0 0
\(91\) 333.002 0.383605
\(92\) 0 0
\(93\) −675.187 −0.752835
\(94\) 0 0
\(95\) −567.447 −0.612830
\(96\) 0 0
\(97\) −90.7689 −0.0950123 −0.0475061 0.998871i \(-0.515127\pi\)
−0.0475061 + 0.998871i \(0.515127\pi\)
\(98\) 0 0
\(99\) −537.511 −0.545676
\(100\) 0 0
\(101\) 39.4507 0.0388662 0.0194331 0.999811i \(-0.493814\pi\)
0.0194331 + 0.999811i \(0.493814\pi\)
\(102\) 0 0
\(103\) −445.599 −0.426273 −0.213137 0.977022i \(-0.568368\pi\)
−0.213137 + 0.977022i \(0.568368\pi\)
\(104\) 0 0
\(105\) 374.773 0.348324
\(106\) 0 0
\(107\) 1079.89 0.975669 0.487834 0.872936i \(-0.337787\pi\)
0.487834 + 0.872936i \(0.337787\pi\)
\(108\) 0 0
\(109\) −551.788 −0.484878 −0.242439 0.970167i \(-0.577947\pi\)
−0.242439 + 0.970167i \(0.577947\pi\)
\(110\) 0 0
\(111\) −871.943 −0.745596
\(112\) 0 0
\(113\) −2173.98 −1.80983 −0.904916 0.425591i \(-0.860066\pi\)
−0.904916 + 0.425591i \(0.860066\pi\)
\(114\) 0 0
\(115\) −441.470 −0.357976
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) −1933.39 −1.48936
\(120\) 0 0
\(121\) 2235.89 1.67986
\(122\) 0 0
\(123\) 574.017 0.420792
\(124\) 0 0
\(125\) 1103.23 0.789408
\(126\) 0 0
\(127\) 959.360 0.670310 0.335155 0.942163i \(-0.391211\pi\)
0.335155 + 0.942163i \(0.391211\pi\)
\(128\) 0 0
\(129\) 980.500 0.669211
\(130\) 0 0
\(131\) 1859.58 1.24025 0.620124 0.784504i \(-0.287082\pi\)
0.620124 + 0.784504i \(0.287082\pi\)
\(132\) 0 0
\(133\) 2980.47 1.94316
\(134\) 0 0
\(135\) 131.676 0.0839472
\(136\) 0 0
\(137\) −1601.07 −0.998455 −0.499227 0.866471i \(-0.666383\pi\)
−0.499227 + 0.866471i \(0.666383\pi\)
\(138\) 0 0
\(139\) −1278.56 −0.780185 −0.390093 0.920776i \(-0.627557\pi\)
−0.390093 + 0.920776i \(0.627557\pi\)
\(140\) 0 0
\(141\) −1220.32 −0.728860
\(142\) 0 0
\(143\) −776.405 −0.454030
\(144\) 0 0
\(145\) 913.254 0.523046
\(146\) 0 0
\(147\) −939.466 −0.527115
\(148\) 0 0
\(149\) 1928.65 1.06041 0.530205 0.847869i \(-0.322115\pi\)
0.530205 + 0.847869i \(0.322115\pi\)
\(150\) 0 0
\(151\) −2822.23 −1.52099 −0.760496 0.649343i \(-0.775044\pi\)
−0.760496 + 0.649343i \(0.775044\pi\)
\(152\) 0 0
\(153\) −679.295 −0.358940
\(154\) 0 0
\(155\) −1097.61 −0.568786
\(156\) 0 0
\(157\) −690.345 −0.350927 −0.175463 0.984486i \(-0.556142\pi\)
−0.175463 + 0.984486i \(0.556142\pi\)
\(158\) 0 0
\(159\) 1280.95 0.638908
\(160\) 0 0
\(161\) 2318.79 1.13507
\(162\) 0 0
\(163\) −3606.51 −1.73303 −0.866515 0.499151i \(-0.833645\pi\)
−0.866515 + 0.499151i \(0.833645\pi\)
\(164\) 0 0
\(165\) −873.795 −0.412272
\(166\) 0 0
\(167\) 1019.70 0.472494 0.236247 0.971693i \(-0.424083\pi\)
0.236247 + 0.971693i \(0.424083\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 1047.19 0.468307
\(172\) 0 0
\(173\) −2395.16 −1.05260 −0.526301 0.850298i \(-0.676422\pi\)
−0.526301 + 0.850298i \(0.676422\pi\)
\(174\) 0 0
\(175\) −2592.70 −1.11994
\(176\) 0 0
\(177\) 993.045 0.421705
\(178\) 0 0
\(179\) −3609.61 −1.50724 −0.753618 0.657313i \(-0.771693\pi\)
−0.753618 + 0.657313i \(0.771693\pi\)
\(180\) 0 0
\(181\) −1848.55 −0.759125 −0.379562 0.925166i \(-0.623925\pi\)
−0.379562 + 0.925166i \(0.623925\pi\)
\(182\) 0 0
\(183\) 1574.03 0.635825
\(184\) 0 0
\(185\) −1417.46 −0.563317
\(186\) 0 0
\(187\) 4507.76 1.76278
\(188\) 0 0
\(189\) −691.619 −0.266179
\(190\) 0 0
\(191\) 2364.55 0.895772 0.447886 0.894091i \(-0.352177\pi\)
0.447886 + 0.894091i \(0.352177\pi\)
\(192\) 0 0
\(193\) 4199.18 1.56613 0.783067 0.621938i \(-0.213654\pi\)
0.783067 + 0.621938i \(0.213654\pi\)
\(194\) 0 0
\(195\) 190.199 0.0698483
\(196\) 0 0
\(197\) −848.869 −0.307002 −0.153501 0.988148i \(-0.549055\pi\)
−0.153501 + 0.988148i \(0.549055\pi\)
\(198\) 0 0
\(199\) −620.958 −0.221199 −0.110599 0.993865i \(-0.535277\pi\)
−0.110599 + 0.993865i \(0.535277\pi\)
\(200\) 0 0
\(201\) 2904.52 1.01925
\(202\) 0 0
\(203\) −4796.80 −1.65847
\(204\) 0 0
\(205\) 933.140 0.317919
\(206\) 0 0
\(207\) 814.705 0.273555
\(208\) 0 0
\(209\) −6949.08 −2.29989
\(210\) 0 0
\(211\) 2911.89 0.950059 0.475030 0.879970i \(-0.342437\pi\)
0.475030 + 0.879970i \(0.342437\pi\)
\(212\) 0 0
\(213\) −25.1933 −0.00810430
\(214\) 0 0
\(215\) 1593.93 0.505606
\(216\) 0 0
\(217\) 5765.09 1.80350
\(218\) 0 0
\(219\) 2709.99 0.836183
\(220\) 0 0
\(221\) −981.204 −0.298656
\(222\) 0 0
\(223\) 5433.63 1.63167 0.815836 0.578283i \(-0.196277\pi\)
0.815836 + 0.578283i \(0.196277\pi\)
\(224\) 0 0
\(225\) −910.943 −0.269909
\(226\) 0 0
\(227\) 4797.03 1.40260 0.701300 0.712866i \(-0.252603\pi\)
0.701300 + 0.712866i \(0.252603\pi\)
\(228\) 0 0
\(229\) 932.947 0.269218 0.134609 0.990899i \(-0.457022\pi\)
0.134609 + 0.990899i \(0.457022\pi\)
\(230\) 0 0
\(231\) 4589.55 1.30723
\(232\) 0 0
\(233\) −169.095 −0.0475441 −0.0237720 0.999717i \(-0.507568\pi\)
−0.0237720 + 0.999717i \(0.507568\pi\)
\(234\) 0 0
\(235\) −1983.79 −0.550672
\(236\) 0 0
\(237\) 3473.93 0.952135
\(238\) 0 0
\(239\) 1467.25 0.397106 0.198553 0.980090i \(-0.436376\pi\)
0.198553 + 0.980090i \(0.436376\pi\)
\(240\) 0 0
\(241\) 361.064 0.0965070 0.0482535 0.998835i \(-0.484634\pi\)
0.0482535 + 0.998835i \(0.484634\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −1527.23 −0.398248
\(246\) 0 0
\(247\) 1512.60 0.389655
\(248\) 0 0
\(249\) 2857.11 0.727157
\(250\) 0 0
\(251\) −1271.80 −0.319823 −0.159911 0.987131i \(-0.551121\pi\)
−0.159911 + 0.987131i \(0.551121\pi\)
\(252\) 0 0
\(253\) −5406.33 −1.34345
\(254\) 0 0
\(255\) −1104.28 −0.271188
\(256\) 0 0
\(257\) −842.178 −0.204411 −0.102205 0.994763i \(-0.532590\pi\)
−0.102205 + 0.994763i \(0.532590\pi\)
\(258\) 0 0
\(259\) 7445.09 1.78616
\(260\) 0 0
\(261\) −1685.35 −0.399696
\(262\) 0 0
\(263\) −3988.33 −0.935099 −0.467550 0.883967i \(-0.654863\pi\)
−0.467550 + 0.883967i \(0.654863\pi\)
\(264\) 0 0
\(265\) 2082.36 0.482711
\(266\) 0 0
\(267\) −3179.88 −0.728860
\(268\) 0 0
\(269\) 4850.14 1.09932 0.549662 0.835387i \(-0.314756\pi\)
0.549662 + 0.835387i \(0.314756\pi\)
\(270\) 0 0
\(271\) −1997.17 −0.447674 −0.223837 0.974627i \(-0.571858\pi\)
−0.223837 + 0.974627i \(0.571858\pi\)
\(272\) 0 0
\(273\) −999.006 −0.221475
\(274\) 0 0
\(275\) 6044.97 1.32555
\(276\) 0 0
\(277\) −615.909 −0.133597 −0.0667985 0.997766i \(-0.521278\pi\)
−0.0667985 + 0.997766i \(0.521278\pi\)
\(278\) 0 0
\(279\) 2025.56 0.434650
\(280\) 0 0
\(281\) 4322.77 0.917703 0.458852 0.888513i \(-0.348261\pi\)
0.458852 + 0.888513i \(0.348261\pi\)
\(282\) 0 0
\(283\) 1033.91 0.217172 0.108586 0.994087i \(-0.465368\pi\)
0.108586 + 0.994087i \(0.465368\pi\)
\(284\) 0 0
\(285\) 1702.34 0.353817
\(286\) 0 0
\(287\) −4901.25 −1.00805
\(288\) 0 0
\(289\) 783.818 0.159540
\(290\) 0 0
\(291\) 272.307 0.0548554
\(292\) 0 0
\(293\) −8391.93 −1.67325 −0.836625 0.547777i \(-0.815474\pi\)
−0.836625 + 0.547777i \(0.815474\pi\)
\(294\) 0 0
\(295\) 1614.33 0.318609
\(296\) 0 0
\(297\) 1612.53 0.315046
\(298\) 0 0
\(299\) 1176.80 0.227612
\(300\) 0 0
\(301\) −8372.01 −1.60317
\(302\) 0 0
\(303\) −118.352 −0.0224394
\(304\) 0 0
\(305\) 2558.80 0.480382
\(306\) 0 0
\(307\) 4374.08 0.813166 0.406583 0.913614i \(-0.366720\pi\)
0.406583 + 0.913614i \(0.366720\pi\)
\(308\) 0 0
\(309\) 1336.80 0.246109
\(310\) 0 0
\(311\) −5981.42 −1.09060 −0.545298 0.838242i \(-0.683584\pi\)
−0.545298 + 0.838242i \(0.683584\pi\)
\(312\) 0 0
\(313\) 1159.92 0.209466 0.104733 0.994500i \(-0.466601\pi\)
0.104733 + 0.994500i \(0.466601\pi\)
\(314\) 0 0
\(315\) −1124.32 −0.201105
\(316\) 0 0
\(317\) −1502.60 −0.266229 −0.133115 0.991101i \(-0.542498\pi\)
−0.133115 + 0.991101i \(0.542498\pi\)
\(318\) 0 0
\(319\) 11183.9 1.96294
\(320\) 0 0
\(321\) −3239.66 −0.563303
\(322\) 0 0
\(323\) −8782.09 −1.51284
\(324\) 0 0
\(325\) −1315.81 −0.224578
\(326\) 0 0
\(327\) 1655.36 0.279944
\(328\) 0 0
\(329\) 10419.7 1.74607
\(330\) 0 0
\(331\) 472.260 0.0784221 0.0392111 0.999231i \(-0.487516\pi\)
0.0392111 + 0.999231i \(0.487516\pi\)
\(332\) 0 0
\(333\) 2615.83 0.430470
\(334\) 0 0
\(335\) 4721.67 0.770067
\(336\) 0 0
\(337\) −8318.44 −1.34461 −0.672306 0.740273i \(-0.734696\pi\)
−0.672306 + 0.740273i \(0.734696\pi\)
\(338\) 0 0
\(339\) 6521.94 1.04491
\(340\) 0 0
\(341\) −13441.5 −2.13460
\(342\) 0 0
\(343\) −764.488 −0.120345
\(344\) 0 0
\(345\) 1324.41 0.206678
\(346\) 0 0
\(347\) 11115.7 1.71967 0.859834 0.510574i \(-0.170567\pi\)
0.859834 + 0.510574i \(0.170567\pi\)
\(348\) 0 0
\(349\) 6057.81 0.929133 0.464566 0.885538i \(-0.346210\pi\)
0.464566 + 0.885538i \(0.346210\pi\)
\(350\) 0 0
\(351\) −351.000 −0.0533761
\(352\) 0 0
\(353\) 675.403 0.101836 0.0509179 0.998703i \(-0.483785\pi\)
0.0509179 + 0.998703i \(0.483785\pi\)
\(354\) 0 0
\(355\) −40.9550 −0.00612300
\(356\) 0 0
\(357\) 5800.17 0.859881
\(358\) 0 0
\(359\) 351.424 0.0516642 0.0258321 0.999666i \(-0.491776\pi\)
0.0258321 + 0.999666i \(0.491776\pi\)
\(360\) 0 0
\(361\) 6679.29 0.973800
\(362\) 0 0
\(363\) −6707.68 −0.969868
\(364\) 0 0
\(365\) 4405.44 0.631757
\(366\) 0 0
\(367\) 6707.82 0.954075 0.477037 0.878883i \(-0.341711\pi\)
0.477037 + 0.878883i \(0.341711\pi\)
\(368\) 0 0
\(369\) −1722.05 −0.242944
\(370\) 0 0
\(371\) −10937.4 −1.53058
\(372\) 0 0
\(373\) 7236.36 1.00452 0.502258 0.864718i \(-0.332503\pi\)
0.502258 + 0.864718i \(0.332503\pi\)
\(374\) 0 0
\(375\) −3309.69 −0.455765
\(376\) 0 0
\(377\) −2434.40 −0.332567
\(378\) 0 0
\(379\) −4084.92 −0.553636 −0.276818 0.960922i \(-0.589280\pi\)
−0.276818 + 0.960922i \(0.589280\pi\)
\(380\) 0 0
\(381\) −2878.08 −0.387004
\(382\) 0 0
\(383\) −7764.76 −1.03593 −0.517965 0.855402i \(-0.673310\pi\)
−0.517965 + 0.855402i \(0.673310\pi\)
\(384\) 0 0
\(385\) 7460.91 0.987645
\(386\) 0 0
\(387\) −2941.50 −0.386369
\(388\) 0 0
\(389\) −1927.25 −0.251197 −0.125598 0.992081i \(-0.540085\pi\)
−0.125598 + 0.992081i \(0.540085\pi\)
\(390\) 0 0
\(391\) −6832.41 −0.883708
\(392\) 0 0
\(393\) −5578.75 −0.716058
\(394\) 0 0
\(395\) 5647.33 0.719362
\(396\) 0 0
\(397\) −2607.17 −0.329597 −0.164799 0.986327i \(-0.552697\pi\)
−0.164799 + 0.986327i \(0.552697\pi\)
\(398\) 0 0
\(399\) −8941.42 −1.12188
\(400\) 0 0
\(401\) −3890.24 −0.484462 −0.242231 0.970219i \(-0.577879\pi\)
−0.242231 + 0.970219i \(0.577879\pi\)
\(402\) 0 0
\(403\) 2925.81 0.361650
\(404\) 0 0
\(405\) −395.028 −0.0484670
\(406\) 0 0
\(407\) −17358.5 −2.11407
\(408\) 0 0
\(409\) 5916.54 0.715291 0.357646 0.933857i \(-0.383580\pi\)
0.357646 + 0.933857i \(0.383580\pi\)
\(410\) 0 0
\(411\) 4803.20 0.576458
\(412\) 0 0
\(413\) −8479.13 −1.01024
\(414\) 0 0
\(415\) 4644.61 0.549386
\(416\) 0 0
\(417\) 3835.67 0.450440
\(418\) 0 0
\(419\) −1729.83 −0.201689 −0.100845 0.994902i \(-0.532155\pi\)
−0.100845 + 0.994902i \(0.532155\pi\)
\(420\) 0 0
\(421\) −5546.89 −0.642135 −0.321067 0.947056i \(-0.604042\pi\)
−0.321067 + 0.947056i \(0.604042\pi\)
\(422\) 0 0
\(423\) 3660.95 0.420808
\(424\) 0 0
\(425\) 7639.50 0.871930
\(426\) 0 0
\(427\) −13439.9 −1.52319
\(428\) 0 0
\(429\) 2329.22 0.262134
\(430\) 0 0
\(431\) 11184.4 1.24996 0.624979 0.780641i \(-0.285107\pi\)
0.624979 + 0.780641i \(0.285107\pi\)
\(432\) 0 0
\(433\) 4298.00 0.477017 0.238509 0.971140i \(-0.423341\pi\)
0.238509 + 0.971140i \(0.423341\pi\)
\(434\) 0 0
\(435\) −2739.76 −0.301981
\(436\) 0 0
\(437\) 10532.7 1.15297
\(438\) 0 0
\(439\) 12481.1 1.35692 0.678461 0.734636i \(-0.262647\pi\)
0.678461 + 0.734636i \(0.262647\pi\)
\(440\) 0 0
\(441\) 2818.40 0.304330
\(442\) 0 0
\(443\) 12999.3 1.39417 0.697084 0.716989i \(-0.254480\pi\)
0.697084 + 0.716989i \(0.254480\pi\)
\(444\) 0 0
\(445\) −5169.31 −0.550672
\(446\) 0 0
\(447\) −5785.95 −0.612228
\(448\) 0 0
\(449\) 14636.6 1.53840 0.769201 0.639007i \(-0.220654\pi\)
0.769201 + 0.639007i \(0.220654\pi\)
\(450\) 0 0
\(451\) 11427.4 1.19312
\(452\) 0 0
\(453\) 8466.69 0.878145
\(454\) 0 0
\(455\) −1624.01 −0.167330
\(456\) 0 0
\(457\) −8803.22 −0.901088 −0.450544 0.892754i \(-0.648770\pi\)
−0.450544 + 0.892754i \(0.648770\pi\)
\(458\) 0 0
\(459\) 2037.89 0.207234
\(460\) 0 0
\(461\) −13223.9 −1.33601 −0.668004 0.744158i \(-0.732851\pi\)
−0.668004 + 0.744158i \(0.732851\pi\)
\(462\) 0 0
\(463\) 14136.4 1.41895 0.709476 0.704729i \(-0.248932\pi\)
0.709476 + 0.704729i \(0.248932\pi\)
\(464\) 0 0
\(465\) 3292.82 0.328389
\(466\) 0 0
\(467\) −8654.11 −0.857525 −0.428763 0.903417i \(-0.641050\pi\)
−0.428763 + 0.903417i \(0.641050\pi\)
\(468\) 0 0
\(469\) −24800.2 −2.44172
\(470\) 0 0
\(471\) 2071.03 0.202608
\(472\) 0 0
\(473\) 19519.6 1.89749
\(474\) 0 0
\(475\) −11776.9 −1.13760
\(476\) 0 0
\(477\) −3842.86 −0.368873
\(478\) 0 0
\(479\) −3622.59 −0.345554 −0.172777 0.984961i \(-0.555274\pi\)
−0.172777 + 0.984961i \(0.555274\pi\)
\(480\) 0 0
\(481\) 3778.42 0.358173
\(482\) 0 0
\(483\) −6956.36 −0.655332
\(484\) 0 0
\(485\) 442.671 0.0414446
\(486\) 0 0
\(487\) −20857.4 −1.94074 −0.970368 0.241632i \(-0.922317\pi\)
−0.970368 + 0.241632i \(0.922317\pi\)
\(488\) 0 0
\(489\) 10819.5 1.00057
\(490\) 0 0
\(491\) 5060.09 0.465089 0.232545 0.972586i \(-0.425295\pi\)
0.232545 + 0.972586i \(0.425295\pi\)
\(492\) 0 0
\(493\) 14134.0 1.29120
\(494\) 0 0
\(495\) 2621.39 0.238025
\(496\) 0 0
\(497\) 215.113 0.0194148
\(498\) 0 0
\(499\) 1150.05 0.103173 0.0515864 0.998669i \(-0.483572\pi\)
0.0515864 + 0.998669i \(0.483572\pi\)
\(500\) 0 0
\(501\) −3059.09 −0.272795
\(502\) 0 0
\(503\) −7145.59 −0.633412 −0.316706 0.948524i \(-0.602577\pi\)
−0.316706 + 0.948524i \(0.602577\pi\)
\(504\) 0 0
\(505\) −192.397 −0.0169536
\(506\) 0 0
\(507\) −507.000 −0.0444116
\(508\) 0 0
\(509\) 7240.54 0.630513 0.315257 0.949006i \(-0.397909\pi\)
0.315257 + 0.949006i \(0.397909\pi\)
\(510\) 0 0
\(511\) −23139.3 −2.00317
\(512\) 0 0
\(513\) −3141.56 −0.270377
\(514\) 0 0
\(515\) 2173.14 0.185941
\(516\) 0 0
\(517\) −24293.9 −2.06662
\(518\) 0 0
\(519\) 7185.47 0.607720
\(520\) 0 0
\(521\) 2690.90 0.226277 0.113139 0.993579i \(-0.463910\pi\)
0.113139 + 0.993579i \(0.463910\pi\)
\(522\) 0 0
\(523\) −507.043 −0.0423928 −0.0211964 0.999775i \(-0.506748\pi\)
−0.0211964 + 0.999775i \(0.506748\pi\)
\(524\) 0 0
\(525\) 7778.10 0.646598
\(526\) 0 0
\(527\) −16987.1 −1.40412
\(528\) 0 0
\(529\) −3972.63 −0.326509
\(530\) 0 0
\(531\) −2979.14 −0.243472
\(532\) 0 0
\(533\) −2487.41 −0.202142
\(534\) 0 0
\(535\) −5266.49 −0.425589
\(536\) 0 0
\(537\) 10828.8 0.870203
\(538\) 0 0
\(539\) −18702.7 −1.49459
\(540\) 0 0
\(541\) 997.477 0.0792696 0.0396348 0.999214i \(-0.487381\pi\)
0.0396348 + 0.999214i \(0.487381\pi\)
\(542\) 0 0
\(543\) 5545.65 0.438281
\(544\) 0 0
\(545\) 2691.01 0.211505
\(546\) 0 0
\(547\) 15191.1 1.18743 0.593716 0.804675i \(-0.297660\pi\)
0.593716 + 0.804675i \(0.297660\pi\)
\(548\) 0 0
\(549\) −4722.10 −0.367094
\(550\) 0 0
\(551\) −21788.6 −1.68462
\(552\) 0 0
\(553\) −29662.2 −2.28095
\(554\) 0 0
\(555\) 4252.37 0.325231
\(556\) 0 0
\(557\) −1426.17 −0.108489 −0.0542447 0.998528i \(-0.517275\pi\)
−0.0542447 + 0.998528i \(0.517275\pi\)
\(558\) 0 0
\(559\) −4248.83 −0.321478
\(560\) 0 0
\(561\) −13523.3 −1.01774
\(562\) 0 0
\(563\) −14066.7 −1.05300 −0.526502 0.850174i \(-0.676497\pi\)
−0.526502 + 0.850174i \(0.676497\pi\)
\(564\) 0 0
\(565\) 10602.3 0.789453
\(566\) 0 0
\(567\) 2074.86 0.153679
\(568\) 0 0
\(569\) −20020.8 −1.47507 −0.737537 0.675307i \(-0.764011\pi\)
−0.737537 + 0.675307i \(0.764011\pi\)
\(570\) 0 0
\(571\) 6023.04 0.441430 0.220715 0.975338i \(-0.429161\pi\)
0.220715 + 0.975338i \(0.429161\pi\)
\(572\) 0 0
\(573\) −7093.64 −0.517174
\(574\) 0 0
\(575\) −9162.34 −0.664515
\(576\) 0 0
\(577\) −6825.67 −0.492472 −0.246236 0.969210i \(-0.579194\pi\)
−0.246236 + 0.969210i \(0.579194\pi\)
\(578\) 0 0
\(579\) −12597.5 −0.904207
\(580\) 0 0
\(581\) −24395.5 −1.74199
\(582\) 0 0
\(583\) 25501.0 1.81157
\(584\) 0 0
\(585\) −570.597 −0.0403270
\(586\) 0 0
\(587\) 24638.4 1.73243 0.866213 0.499675i \(-0.166547\pi\)
0.866213 + 0.499675i \(0.166547\pi\)
\(588\) 0 0
\(589\) 26187.0 1.83194
\(590\) 0 0
\(591\) 2546.61 0.177248
\(592\) 0 0
\(593\) −4694.14 −0.325068 −0.162534 0.986703i \(-0.551967\pi\)
−0.162534 + 0.986703i \(0.551967\pi\)
\(594\) 0 0
\(595\) 9428.94 0.649662
\(596\) 0 0
\(597\) 1862.87 0.127709
\(598\) 0 0
\(599\) −10043.5 −0.685087 −0.342544 0.939502i \(-0.611288\pi\)
−0.342544 + 0.939502i \(0.611288\pi\)
\(600\) 0 0
\(601\) −20677.8 −1.40343 −0.701717 0.712456i \(-0.747583\pi\)
−0.701717 + 0.712456i \(0.747583\pi\)
\(602\) 0 0
\(603\) −8713.55 −0.588463
\(604\) 0 0
\(605\) −10904.2 −0.732760
\(606\) 0 0
\(607\) −4990.49 −0.333703 −0.166851 0.985982i \(-0.553360\pi\)
−0.166851 + 0.985982i \(0.553360\pi\)
\(608\) 0 0
\(609\) 14390.4 0.957518
\(610\) 0 0
\(611\) 5288.04 0.350133
\(612\) 0 0
\(613\) −27970.0 −1.84290 −0.921450 0.388496i \(-0.872995\pi\)
−0.921450 + 0.388496i \(0.872995\pi\)
\(614\) 0 0
\(615\) −2799.42 −0.183550
\(616\) 0 0
\(617\) −2978.15 −0.194321 −0.0971603 0.995269i \(-0.530976\pi\)
−0.0971603 + 0.995269i \(0.530976\pi\)
\(618\) 0 0
\(619\) −29605.9 −1.92239 −0.961196 0.275865i \(-0.911036\pi\)
−0.961196 + 0.275865i \(0.911036\pi\)
\(620\) 0 0
\(621\) −2444.11 −0.157937
\(622\) 0 0
\(623\) 27151.4 1.74607
\(624\) 0 0
\(625\) 7271.65 0.465385
\(626\) 0 0
\(627\) 20847.2 1.32784
\(628\) 0 0
\(629\) −21937.3 −1.39062
\(630\) 0 0
\(631\) 937.256 0.0591309 0.0295654 0.999563i \(-0.490588\pi\)
0.0295654 + 0.999563i \(0.490588\pi\)
\(632\) 0 0
\(633\) −8735.66 −0.548517
\(634\) 0 0
\(635\) −4678.70 −0.292391
\(636\) 0 0
\(637\) 4071.02 0.253218
\(638\) 0 0
\(639\) 75.5799 0.00467902
\(640\) 0 0
\(641\) −8879.08 −0.547118 −0.273559 0.961855i \(-0.588201\pi\)
−0.273559 + 0.961855i \(0.588201\pi\)
\(642\) 0 0
\(643\) 8098.91 0.496718 0.248359 0.968668i \(-0.420109\pi\)
0.248359 + 0.968668i \(0.420109\pi\)
\(644\) 0 0
\(645\) −4781.79 −0.291912
\(646\) 0 0
\(647\) 16069.3 0.976429 0.488214 0.872724i \(-0.337648\pi\)
0.488214 + 0.872724i \(0.337648\pi\)
\(648\) 0 0
\(649\) 19769.4 1.19571
\(650\) 0 0
\(651\) −17295.3 −1.04125
\(652\) 0 0
\(653\) 4610.46 0.276296 0.138148 0.990412i \(-0.455885\pi\)
0.138148 + 0.990412i \(0.455885\pi\)
\(654\) 0 0
\(655\) −9068.99 −0.541000
\(656\) 0 0
\(657\) −8129.97 −0.482770
\(658\) 0 0
\(659\) 24960.1 1.47543 0.737714 0.675113i \(-0.235905\pi\)
0.737714 + 0.675113i \(0.235905\pi\)
\(660\) 0 0
\(661\) 24437.3 1.43798 0.718988 0.695023i \(-0.244606\pi\)
0.718988 + 0.695023i \(0.244606\pi\)
\(662\) 0 0
\(663\) 2943.61 0.172429
\(664\) 0 0
\(665\) −14535.5 −0.847610
\(666\) 0 0
\(667\) −16951.4 −0.984050
\(668\) 0 0
\(669\) −16300.9 −0.942046
\(670\) 0 0
\(671\) 31335.6 1.80283
\(672\) 0 0
\(673\) 2702.96 0.154816 0.0774081 0.996999i \(-0.475336\pi\)
0.0774081 + 0.996999i \(0.475336\pi\)
\(674\) 0 0
\(675\) 2732.83 0.155832
\(676\) 0 0
\(677\) 11071.8 0.628544 0.314272 0.949333i \(-0.398240\pi\)
0.314272 + 0.949333i \(0.398240\pi\)
\(678\) 0 0
\(679\) −2325.09 −0.131412
\(680\) 0 0
\(681\) −14391.1 −0.809792
\(682\) 0 0
\(683\) 11778.4 0.659864 0.329932 0.944005i \(-0.392974\pi\)
0.329932 + 0.944005i \(0.392974\pi\)
\(684\) 0 0
\(685\) 7808.23 0.435529
\(686\) 0 0
\(687\) −2798.84 −0.155433
\(688\) 0 0
\(689\) −5550.80 −0.306921
\(690\) 0 0
\(691\) −30715.1 −1.69097 −0.845483 0.534002i \(-0.820687\pi\)
−0.845483 + 0.534002i \(0.820687\pi\)
\(692\) 0 0
\(693\) −13768.6 −0.754729
\(694\) 0 0
\(695\) 6235.39 0.340319
\(696\) 0 0
\(697\) 14441.7 0.784821
\(698\) 0 0
\(699\) 507.285 0.0274496
\(700\) 0 0
\(701\) 741.115 0.0399308 0.0199654 0.999801i \(-0.493644\pi\)
0.0199654 + 0.999801i \(0.493644\pi\)
\(702\) 0 0
\(703\) 33818.1 1.81433
\(704\) 0 0
\(705\) 5951.36 0.317931
\(706\) 0 0
\(707\) 1010.55 0.0537562
\(708\) 0 0
\(709\) −24269.6 −1.28556 −0.642781 0.766050i \(-0.722219\pi\)
−0.642781 + 0.766050i \(0.722219\pi\)
\(710\) 0 0
\(711\) −10421.8 −0.549716
\(712\) 0 0
\(713\) 20373.3 1.07011
\(714\) 0 0
\(715\) 3786.45 0.198049
\(716\) 0 0
\(717\) −4401.74 −0.229269
\(718\) 0 0
\(719\) −15598.6 −0.809083 −0.404541 0.914520i \(-0.632569\pi\)
−0.404541 + 0.914520i \(0.632569\pi\)
\(720\) 0 0
\(721\) −11414.2 −0.589582
\(722\) 0 0
\(723\) −1083.19 −0.0557183
\(724\) 0 0
\(725\) 18953.8 0.970934
\(726\) 0 0
\(727\) 32399.2 1.65285 0.826423 0.563050i \(-0.190372\pi\)
0.826423 + 0.563050i \(0.190372\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 24668.5 1.24815
\(732\) 0 0
\(733\) 7851.92 0.395658 0.197829 0.980237i \(-0.436611\pi\)
0.197829 + 0.980237i \(0.436611\pi\)
\(734\) 0 0
\(735\) 4581.68 0.229929
\(736\) 0 0
\(737\) 57822.6 2.88999
\(738\) 0 0
\(739\) −13348.4 −0.664451 −0.332226 0.943200i \(-0.607800\pi\)
−0.332226 + 0.943200i \(0.607800\pi\)
\(740\) 0 0
\(741\) −4537.81 −0.224967
\(742\) 0 0
\(743\) 9303.61 0.459376 0.229688 0.973264i \(-0.426229\pi\)
0.229688 + 0.973264i \(0.426229\pi\)
\(744\) 0 0
\(745\) −9405.82 −0.462554
\(746\) 0 0
\(747\) −8571.34 −0.419825
\(748\) 0 0
\(749\) 27661.9 1.34946
\(750\) 0 0
\(751\) 9774.38 0.474930 0.237465 0.971396i \(-0.423684\pi\)
0.237465 + 0.971396i \(0.423684\pi\)
\(752\) 0 0
\(753\) 3815.41 0.184650
\(754\) 0 0
\(755\) 13763.7 0.663461
\(756\) 0 0
\(757\) 24541.6 1.17831 0.589154 0.808021i \(-0.299461\pi\)
0.589154 + 0.808021i \(0.299461\pi\)
\(758\) 0 0
\(759\) 16219.0 0.775642
\(760\) 0 0
\(761\) −14409.4 −0.686389 −0.343194 0.939264i \(-0.611509\pi\)
−0.343194 + 0.939264i \(0.611509\pi\)
\(762\) 0 0
\(763\) −14134.3 −0.670639
\(764\) 0 0
\(765\) 3312.85 0.156570
\(766\) 0 0
\(767\) −4303.20 −0.202581
\(768\) 0 0
\(769\) 4426.67 0.207581 0.103791 0.994599i \(-0.466903\pi\)
0.103791 + 0.994599i \(0.466903\pi\)
\(770\) 0 0
\(771\) 2526.53 0.118017
\(772\) 0 0
\(773\) −4608.64 −0.214439 −0.107219 0.994235i \(-0.534195\pi\)
−0.107219 + 0.994235i \(0.534195\pi\)
\(774\) 0 0
\(775\) −22779.9 −1.05584
\(776\) 0 0
\(777\) −22335.3 −1.03124
\(778\) 0 0
\(779\) −22263.1 −1.02395
\(780\) 0 0
\(781\) −501.544 −0.0229790
\(782\) 0 0
\(783\) 5056.06 0.230765
\(784\) 0 0
\(785\) 3366.74 0.153075
\(786\) 0 0
\(787\) −2743.83 −0.124278 −0.0621392 0.998067i \(-0.519792\pi\)
−0.0621392 + 0.998067i \(0.519792\pi\)
\(788\) 0 0
\(789\) 11965.0 0.539880
\(790\) 0 0
\(791\) −55687.7 −2.50319
\(792\) 0 0
\(793\) −6820.81 −0.305440
\(794\) 0 0
\(795\) −6247.08 −0.278693
\(796\) 0 0
\(797\) −2474.62 −0.109982 −0.0549910 0.998487i \(-0.517513\pi\)
−0.0549910 + 0.998487i \(0.517513\pi\)
\(798\) 0 0
\(799\) −30702.1 −1.35940
\(800\) 0 0
\(801\) 9539.64 0.420807
\(802\) 0 0
\(803\) 53950.0 2.37093
\(804\) 0 0
\(805\) −11308.5 −0.495120
\(806\) 0 0
\(807\) −14550.4 −0.634695
\(808\) 0 0
\(809\) 13410.2 0.582790 0.291395 0.956603i \(-0.405881\pi\)
0.291395 + 0.956603i \(0.405881\pi\)
\(810\) 0 0
\(811\) −24145.8 −1.04547 −0.522733 0.852497i \(-0.675087\pi\)
−0.522733 + 0.852497i \(0.675087\pi\)
\(812\) 0 0
\(813\) 5991.52 0.258465
\(814\) 0 0
\(815\) 17588.6 0.755952
\(816\) 0 0
\(817\) −38028.4 −1.62845
\(818\) 0 0
\(819\) 2997.02 0.127868
\(820\) 0 0
\(821\) −5569.07 −0.236738 −0.118369 0.992970i \(-0.537767\pi\)
−0.118369 + 0.992970i \(0.537767\pi\)
\(822\) 0 0
\(823\) −17139.2 −0.725921 −0.362961 0.931804i \(-0.618234\pi\)
−0.362961 + 0.931804i \(0.618234\pi\)
\(824\) 0 0
\(825\) −18134.9 −0.765304
\(826\) 0 0
\(827\) −3756.54 −0.157954 −0.0789769 0.996876i \(-0.525165\pi\)
−0.0789769 + 0.996876i \(0.525165\pi\)
\(828\) 0 0
\(829\) −15534.3 −0.650819 −0.325410 0.945573i \(-0.605502\pi\)
−0.325410 + 0.945573i \(0.605502\pi\)
\(830\) 0 0
\(831\) 1847.73 0.0771323
\(832\) 0 0
\(833\) −23636.1 −0.983124
\(834\) 0 0
\(835\) −4972.95 −0.206103
\(836\) 0 0
\(837\) −6076.69 −0.250945
\(838\) 0 0
\(839\) 31953.0 1.31483 0.657413 0.753531i \(-0.271651\pi\)
0.657413 + 0.753531i \(0.271651\pi\)
\(840\) 0 0
\(841\) 10677.8 0.437813
\(842\) 0 0
\(843\) −12968.3 −0.529836
\(844\) 0 0
\(845\) −824.195 −0.0335541
\(846\) 0 0
\(847\) 57273.6 2.32343
\(848\) 0 0
\(849\) −3101.74 −0.125384
\(850\) 0 0
\(851\) 26310.2 1.05982
\(852\) 0 0
\(853\) 41587.2 1.66931 0.834654 0.550775i \(-0.185668\pi\)
0.834654 + 0.550775i \(0.185668\pi\)
\(854\) 0 0
\(855\) −5107.02 −0.204277
\(856\) 0 0
\(857\) −8080.58 −0.322086 −0.161043 0.986947i \(-0.551486\pi\)
−0.161043 + 0.986947i \(0.551486\pi\)
\(858\) 0 0
\(859\) −33848.0 −1.34444 −0.672222 0.740349i \(-0.734660\pi\)
−0.672222 + 0.740349i \(0.734660\pi\)
\(860\) 0 0
\(861\) 14703.7 0.582000
\(862\) 0 0
\(863\) 27002.6 1.06510 0.532548 0.846400i \(-0.321234\pi\)
0.532548 + 0.846400i \(0.321234\pi\)
\(864\) 0 0
\(865\) 11680.9 0.459148
\(866\) 0 0
\(867\) −2351.45 −0.0921102
\(868\) 0 0
\(869\) 69158.4 2.69970
\(870\) 0 0
\(871\) −12586.2 −0.489631
\(872\) 0 0
\(873\) −816.920 −0.0316708
\(874\) 0 0
\(875\) 28259.8 1.09184
\(876\) 0 0
\(877\) 45535.6 1.75328 0.876641 0.481145i \(-0.159779\pi\)
0.876641 + 0.481145i \(0.159779\pi\)
\(878\) 0 0
\(879\) 25175.8 0.966051
\(880\) 0 0
\(881\) 864.335 0.0330536 0.0165268 0.999863i \(-0.494739\pi\)
0.0165268 + 0.999863i \(0.494739\pi\)
\(882\) 0 0
\(883\) 1075.10 0.0409741 0.0204871 0.999790i \(-0.493478\pi\)
0.0204871 + 0.999790i \(0.493478\pi\)
\(884\) 0 0
\(885\) −4842.98 −0.183949
\(886\) 0 0
\(887\) 36604.8 1.38565 0.692824 0.721107i \(-0.256366\pi\)
0.692824 + 0.721107i \(0.256366\pi\)
\(888\) 0 0
\(889\) 24574.5 0.927112
\(890\) 0 0
\(891\) −4837.60 −0.181892
\(892\) 0 0
\(893\) 47329.7 1.77360
\(894\) 0 0
\(895\) 17603.7 0.657460
\(896\) 0 0
\(897\) −3530.39 −0.131412
\(898\) 0 0
\(899\) −42145.5 −1.56355
\(900\) 0 0
\(901\) 32227.6 1.19163
\(902\) 0 0
\(903\) 25116.0 0.925591
\(904\) 0 0
\(905\) 9015.18 0.331132
\(906\) 0 0
\(907\) −19657.7 −0.719649 −0.359825 0.933020i \(-0.617163\pi\)
−0.359825 + 0.933020i \(0.617163\pi\)
\(908\) 0 0
\(909\) 355.056 0.0129554
\(910\) 0 0
\(911\) 8278.09 0.301060 0.150530 0.988605i \(-0.451902\pi\)
0.150530 + 0.988605i \(0.451902\pi\)
\(912\) 0 0
\(913\) 56878.9 2.06179
\(914\) 0 0
\(915\) −7676.40 −0.277349
\(916\) 0 0
\(917\) 47634.2 1.71540
\(918\) 0 0
\(919\) −20750.6 −0.744829 −0.372414 0.928067i \(-0.621470\pi\)
−0.372414 + 0.928067i \(0.621470\pi\)
\(920\) 0 0
\(921\) −13122.2 −0.469482
\(922\) 0 0
\(923\) 109.171 0.00389318
\(924\) 0 0
\(925\) −29418.2 −1.04569
\(926\) 0 0
\(927\) −4010.39 −0.142091
\(928\) 0 0
\(929\) −27905.6 −0.985526 −0.492763 0.870164i \(-0.664013\pi\)
−0.492763 + 0.870164i \(0.664013\pi\)
\(930\) 0 0
\(931\) 36436.9 1.28268
\(932\) 0 0
\(933\) 17944.3 0.629656
\(934\) 0 0
\(935\) −21983.9 −0.768931
\(936\) 0 0
\(937\) 1734.90 0.0604873 0.0302437 0.999543i \(-0.490372\pi\)
0.0302437 + 0.999543i \(0.490372\pi\)
\(938\) 0 0
\(939\) −3479.77 −0.120935
\(940\) 0 0
\(941\) 18761.2 0.649943 0.324972 0.945724i \(-0.394645\pi\)
0.324972 + 0.945724i \(0.394645\pi\)
\(942\) 0 0
\(943\) −17320.5 −0.598127
\(944\) 0 0
\(945\) 3372.95 0.116108
\(946\) 0 0
\(947\) 20003.3 0.686400 0.343200 0.939262i \(-0.388489\pi\)
0.343200 + 0.939262i \(0.388489\pi\)
\(948\) 0 0
\(949\) −11743.3 −0.401689
\(950\) 0 0
\(951\) 4507.81 0.153707
\(952\) 0 0
\(953\) −14639.9 −0.497620 −0.248810 0.968552i \(-0.580039\pi\)
−0.248810 + 0.968552i \(0.580039\pi\)
\(954\) 0 0
\(955\) −11531.6 −0.390738
\(956\) 0 0
\(957\) −33551.7 −1.13330
\(958\) 0 0
\(959\) −41012.2 −1.38097
\(960\) 0 0
\(961\) 20862.1 0.700283
\(962\) 0 0
\(963\) 9718.98 0.325223
\(964\) 0 0
\(965\) −20479.0 −0.683152
\(966\) 0 0
\(967\) 29052.1 0.966136 0.483068 0.875583i \(-0.339522\pi\)
0.483068 + 0.875583i \(0.339522\pi\)
\(968\) 0 0
\(969\) 26346.3 0.873441
\(970\) 0 0
\(971\) 30911.6 1.02163 0.510814 0.859691i \(-0.329344\pi\)
0.510814 + 0.859691i \(0.329344\pi\)
\(972\) 0 0
\(973\) −32750.9 −1.07908
\(974\) 0 0
\(975\) 3947.42 0.129660
\(976\) 0 0
\(977\) −34458.8 −1.12839 −0.564194 0.825642i \(-0.690813\pi\)
−0.564194 + 0.825642i \(0.690813\pi\)
\(978\) 0 0
\(979\) −63304.5 −2.06662
\(980\) 0 0
\(981\) −4966.09 −0.161626
\(982\) 0 0
\(983\) −41353.9 −1.34180 −0.670898 0.741550i \(-0.734091\pi\)
−0.670898 + 0.741550i \(0.734091\pi\)
\(984\) 0 0
\(985\) 4139.85 0.133915
\(986\) 0 0
\(987\) −31259.1 −1.00809
\(988\) 0 0
\(989\) −29585.8 −0.951239
\(990\) 0 0
\(991\) −4649.74 −0.149045 −0.0745227 0.997219i \(-0.523743\pi\)
−0.0745227 + 0.997219i \(0.523743\pi\)
\(992\) 0 0
\(993\) −1416.78 −0.0452770
\(994\) 0 0
\(995\) 3028.35 0.0964875
\(996\) 0 0
\(997\) −45125.0 −1.43342 −0.716712 0.697369i \(-0.754354\pi\)
−0.716712 + 0.697369i \(0.754354\pi\)
\(998\) 0 0
\(999\) −7847.49 −0.248532
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 624.4.a.j.1.2 2
3.2 odd 2 1872.4.a.bi.1.1 2
4.3 odd 2 312.4.a.d.1.2 2
8.3 odd 2 2496.4.a.ba.1.1 2
8.5 even 2 2496.4.a.bj.1.1 2
12.11 even 2 936.4.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.d.1.2 2 4.3 odd 2
624.4.a.j.1.2 2 1.1 even 1 trivial
936.4.a.j.1.1 2 12.11 even 2
1872.4.a.bi.1.1 2 3.2 odd 2
2496.4.a.ba.1.1 2 8.3 odd 2
2496.4.a.bj.1.1 2 8.5 even 2