Properties

Label 1840.2.a.r.1.1
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $1$
Dimension $3$
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] = [1840,2,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, 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("1840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(14.6924739719\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.68740\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.68740 q^{3} -1.00000 q^{5} +4.59692 q^{7} +4.22212 q^{9} +O(q^{10})\) \(q-2.68740 q^{3} -1.00000 q^{5} +4.59692 q^{7} +4.22212 q^{9} -5.13163 q^{11} -1.22212 q^{13} +2.68740 q^{15} -4.68740 q^{17} +4.59692 q^{19} -12.3537 q^{21} +1.00000 q^{23} +1.00000 q^{25} -3.28432 q^{27} +3.37480 q^{29} +0.777884 q^{31} +13.7907 q^{33} -4.59692 q^{35} +5.81903 q^{37} +3.28432 q^{39} -8.50643 q^{41} -8.00000 q^{43} -4.22212 q^{45} +6.44423 q^{47} +14.1316 q^{49} +12.5969 q^{51} -6.00000 q^{53} +5.13163 q^{55} -12.3537 q^{57} -9.37480 q^{59} +10.9507 q^{61} +19.4087 q^{63} +1.22212 q^{65} -15.6381 q^{67} -2.68740 q^{69} -1.31260 q^{71} -4.44423 q^{73} -2.68740 q^{75} -23.5897 q^{77} +4.88847 q^{79} -3.84008 q^{81} +3.81903 q^{83} +4.68740 q^{85} -9.06943 q^{87} +8.93057 q^{89} -5.61797 q^{91} -2.09048 q^{93} -4.59692 q^{95} -18.0622 q^{97} -21.6663 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - 3 q^{5} - 3 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} - 3 q^{5} - 3 q^{7} + 10 q^{9} - 3 q^{11} - q^{13} + q^{15} - 7 q^{17} - 3 q^{19} - 22 q^{21} + 3 q^{23} + 3 q^{25} + 14 q^{27} - 4 q^{29} + 5 q^{31} - 9 q^{33} + 3 q^{35} - 2 q^{37} - 14 q^{39} + q^{41} - 24 q^{43} - 10 q^{45} + 14 q^{47} + 30 q^{49} + 21 q^{51} - 18 q^{53} + 3 q^{55} - 22 q^{57} - 14 q^{59} + q^{61} - 8 q^{63} + q^{65} - 8 q^{67} - q^{69} - 11 q^{71} - 8 q^{73} - q^{75} - 24 q^{77} + 4 q^{79} + 7 q^{81} - 8 q^{83} + 7 q^{85} - 36 q^{87} + 18 q^{89} - q^{91} - 16 q^{93} + 3 q^{95} - 33 q^{97} - 57 q^{99}+O(q^{100}) \) 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
\(3\) −2.68740 −1.55157 −0.775785 0.630997i \(-0.782646\pi\)
−0.775785 + 0.630997i \(0.782646\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.59692 1.73747 0.868735 0.495277i \(-0.164933\pi\)
0.868735 + 0.495277i \(0.164933\pi\)
\(8\) 0 0
\(9\) 4.22212 1.40737
\(10\) 0 0
\(11\) −5.13163 −1.54725 −0.773623 0.633647i \(-0.781557\pi\)
−0.773623 + 0.633647i \(0.781557\pi\)
\(12\) 0 0
\(13\) −1.22212 −0.338954 −0.169477 0.985534i \(-0.554208\pi\)
−0.169477 + 0.985534i \(0.554208\pi\)
\(14\) 0 0
\(15\) 2.68740 0.693884
\(16\) 0 0
\(17\) −4.68740 −1.13686 −0.568431 0.822731i \(-0.692449\pi\)
−0.568431 + 0.822731i \(0.692449\pi\)
\(18\) 0 0
\(19\) 4.59692 1.05460 0.527302 0.849678i \(-0.323204\pi\)
0.527302 + 0.849678i \(0.323204\pi\)
\(20\) 0 0
\(21\) −12.3537 −2.69581
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.28432 −0.632067
\(28\) 0 0
\(29\) 3.37480 0.626684 0.313342 0.949640i \(-0.398551\pi\)
0.313342 + 0.949640i \(0.398551\pi\)
\(30\) 0 0
\(31\) 0.777884 0.139712 0.0698560 0.997557i \(-0.477746\pi\)
0.0698560 + 0.997557i \(0.477746\pi\)
\(32\) 0 0
\(33\) 13.7907 2.40066
\(34\) 0 0
\(35\) −4.59692 −0.777021
\(36\) 0 0
\(37\) 5.81903 0.956643 0.478321 0.878185i \(-0.341245\pi\)
0.478321 + 0.878185i \(0.341245\pi\)
\(38\) 0 0
\(39\) 3.28432 0.525911
\(40\) 0 0
\(41\) −8.50643 −1.32848 −0.664241 0.747519i \(-0.731245\pi\)
−0.664241 + 0.747519i \(0.731245\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) −4.22212 −0.629396
\(46\) 0 0
\(47\) 6.44423 0.939988 0.469994 0.882670i \(-0.344256\pi\)
0.469994 + 0.882670i \(0.344256\pi\)
\(48\) 0 0
\(49\) 14.1316 2.01880
\(50\) 0 0
\(51\) 12.5969 1.76392
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 5.13163 0.691949
\(56\) 0 0
\(57\) −12.3537 −1.63629
\(58\) 0 0
\(59\) −9.37480 −1.22049 −0.610247 0.792211i \(-0.708930\pi\)
−0.610247 + 0.792211i \(0.708930\pi\)
\(60\) 0 0
\(61\) 10.9507 1.40209 0.701044 0.713118i \(-0.252717\pi\)
0.701044 + 0.713118i \(0.252717\pi\)
\(62\) 0 0
\(63\) 19.4087 2.44527
\(64\) 0 0
\(65\) 1.22212 0.151585
\(66\) 0 0
\(67\) −15.6381 −1.91049 −0.955247 0.295810i \(-0.904410\pi\)
−0.955247 + 0.295810i \(0.904410\pi\)
\(68\) 0 0
\(69\) −2.68740 −0.323525
\(70\) 0 0
\(71\) −1.31260 −0.155777 −0.0778885 0.996962i \(-0.524818\pi\)
−0.0778885 + 0.996962i \(0.524818\pi\)
\(72\) 0 0
\(73\) −4.44423 −0.520158 −0.260079 0.965587i \(-0.583749\pi\)
−0.260079 + 0.965587i \(0.583749\pi\)
\(74\) 0 0
\(75\) −2.68740 −0.310314
\(76\) 0 0
\(77\) −23.5897 −2.68829
\(78\) 0 0
\(79\) 4.88847 0.549995 0.274998 0.961445i \(-0.411323\pi\)
0.274998 + 0.961445i \(0.411323\pi\)
\(80\) 0 0
\(81\) −3.84008 −0.426676
\(82\) 0 0
\(83\) 3.81903 0.419193 0.209597 0.977788i \(-0.432785\pi\)
0.209597 + 0.977788i \(0.432785\pi\)
\(84\) 0 0
\(85\) 4.68740 0.508420
\(86\) 0 0
\(87\) −9.06943 −0.972345
\(88\) 0 0
\(89\) 8.93057 0.946638 0.473319 0.880891i \(-0.343056\pi\)
0.473319 + 0.880891i \(0.343056\pi\)
\(90\) 0 0
\(91\) −5.61797 −0.588923
\(92\) 0 0
\(93\) −2.09048 −0.216773
\(94\) 0 0
\(95\) −4.59692 −0.471634
\(96\) 0 0
\(97\) −18.0622 −1.83394 −0.916969 0.398958i \(-0.869372\pi\)
−0.916969 + 0.398958i \(0.869372\pi\)
\(98\) 0 0
\(99\) −21.6663 −2.17755
\(100\) 0 0
\(101\) 3.37480 0.335805 0.167903 0.985804i \(-0.446301\pi\)
0.167903 + 0.985804i \(0.446301\pi\)
\(102\) 0 0
\(103\) −13.1316 −1.29390 −0.646949 0.762533i \(-0.723955\pi\)
−0.646949 + 0.762533i \(0.723955\pi\)
\(104\) 0 0
\(105\) 12.3537 1.20560
\(106\) 0 0
\(107\) −12.3054 −1.18960 −0.594802 0.803872i \(-0.702770\pi\)
−0.594802 + 0.803872i \(0.702770\pi\)
\(108\) 0 0
\(109\) −10.5969 −1.01500 −0.507500 0.861652i \(-0.669430\pi\)
−0.507500 + 0.861652i \(0.669430\pi\)
\(110\) 0 0
\(111\) −15.6381 −1.48430
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −5.15992 −0.477035
\(118\) 0 0
\(119\) −21.5476 −1.97526
\(120\) 0 0
\(121\) 15.3337 1.39397
\(122\) 0 0
\(123\) 22.8602 2.06123
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.26326 0.200832 0.100416 0.994946i \(-0.467983\pi\)
0.100416 + 0.994946i \(0.467983\pi\)
\(128\) 0 0
\(129\) 21.4992 1.89290
\(130\) 0 0
\(131\) −2.93057 −0.256045 −0.128022 0.991771i \(-0.540863\pi\)
−0.128022 + 0.991771i \(0.540863\pi\)
\(132\) 0 0
\(133\) 21.1316 1.83234
\(134\) 0 0
\(135\) 3.28432 0.282669
\(136\) 0 0
\(137\) −19.7907 −1.69084 −0.845419 0.534104i \(-0.820649\pi\)
−0.845419 + 0.534104i \(0.820649\pi\)
\(138\) 0 0
\(139\) −6.26326 −0.531243 −0.265622 0.964077i \(-0.585577\pi\)
−0.265622 + 0.964077i \(0.585577\pi\)
\(140\) 0 0
\(141\) −17.3182 −1.45846
\(142\) 0 0
\(143\) 6.27145 0.524445
\(144\) 0 0
\(145\) −3.37480 −0.280262
\(146\) 0 0
\(147\) −37.9773 −3.13232
\(148\) 0 0
\(149\) −19.7907 −1.62132 −0.810661 0.585516i \(-0.800892\pi\)
−0.810661 + 0.585516i \(0.800892\pi\)
\(150\) 0 0
\(151\) −4.06220 −0.330577 −0.165289 0.986245i \(-0.552856\pi\)
−0.165289 + 0.986245i \(0.552856\pi\)
\(152\) 0 0
\(153\) −19.7907 −1.59999
\(154\) 0 0
\(155\) −0.777884 −0.0624811
\(156\) 0 0
\(157\) 4.62520 0.369131 0.184566 0.982820i \(-0.440912\pi\)
0.184566 + 0.982820i \(0.440912\pi\)
\(158\) 0 0
\(159\) 16.1244 1.27875
\(160\) 0 0
\(161\) 4.59692 0.362288
\(162\) 0 0
\(163\) −12.4159 −0.972492 −0.486246 0.873822i \(-0.661634\pi\)
−0.486246 + 0.873822i \(0.661634\pi\)
\(164\) 0 0
\(165\) −13.7907 −1.07361
\(166\) 0 0
\(167\) −12.8885 −0.997339 −0.498670 0.866792i \(-0.666178\pi\)
−0.498670 + 0.866792i \(0.666178\pi\)
\(168\) 0 0
\(169\) −11.5064 −0.885110
\(170\) 0 0
\(171\) 19.4087 1.48422
\(172\) 0 0
\(173\) −10.2432 −0.778774 −0.389387 0.921074i \(-0.627313\pi\)
−0.389387 + 0.921074i \(0.627313\pi\)
\(174\) 0 0
\(175\) 4.59692 0.347494
\(176\) 0 0
\(177\) 25.1938 1.89368
\(178\) 0 0
\(179\) 13.1938 0.986153 0.493077 0.869986i \(-0.335872\pi\)
0.493077 + 0.869986i \(0.335872\pi\)
\(180\) 0 0
\(181\) 15.7907 1.17372 0.586858 0.809690i \(-0.300364\pi\)
0.586858 + 0.809690i \(0.300364\pi\)
\(182\) 0 0
\(183\) −29.4288 −2.17544
\(184\) 0 0
\(185\) −5.81903 −0.427824
\(186\) 0 0
\(187\) 24.0540 1.75900
\(188\) 0 0
\(189\) −15.0977 −1.09820
\(190\) 0 0
\(191\) −16.1244 −1.16672 −0.583360 0.812214i \(-0.698262\pi\)
−0.583360 + 0.812214i \(0.698262\pi\)
\(192\) 0 0
\(193\) −17.9434 −1.29160 −0.645798 0.763508i \(-0.723475\pi\)
−0.645798 + 0.763508i \(0.723475\pi\)
\(194\) 0 0
\(195\) −3.28432 −0.235195
\(196\) 0 0
\(197\) 5.88123 0.419020 0.209510 0.977806i \(-0.432813\pi\)
0.209510 + 0.977806i \(0.432813\pi\)
\(198\) 0 0
\(199\) −6.56863 −0.465638 −0.232819 0.972520i \(-0.574795\pi\)
−0.232819 + 0.972520i \(0.574795\pi\)
\(200\) 0 0
\(201\) 42.0257 2.96427
\(202\) 0 0
\(203\) 15.5137 1.08885
\(204\) 0 0
\(205\) 8.50643 0.594115
\(206\) 0 0
\(207\) 4.22212 0.293457
\(208\) 0 0
\(209\) −23.5897 −1.63173
\(210\) 0 0
\(211\) 15.4571 1.06411 0.532055 0.846710i \(-0.321420\pi\)
0.532055 + 0.846710i \(0.321420\pi\)
\(212\) 0 0
\(213\) 3.52748 0.241699
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 3.57587 0.242746
\(218\) 0 0
\(219\) 11.9434 0.807062
\(220\) 0 0
\(221\) 5.72855 0.385344
\(222\) 0 0
\(223\) 10.7496 0.719846 0.359923 0.932982i \(-0.382803\pi\)
0.359923 + 0.932982i \(0.382803\pi\)
\(224\) 0 0
\(225\) 4.22212 0.281474
\(226\) 0 0
\(227\) 12.3054 0.816736 0.408368 0.912817i \(-0.366098\pi\)
0.408368 + 0.912817i \(0.366098\pi\)
\(228\) 0 0
\(229\) 9.63806 0.636901 0.318451 0.947939i \(-0.396838\pi\)
0.318451 + 0.947939i \(0.396838\pi\)
\(230\) 0 0
\(231\) 63.3949 4.17108
\(232\) 0 0
\(233\) 13.9434 0.913464 0.456732 0.889604i \(-0.349020\pi\)
0.456732 + 0.889604i \(0.349020\pi\)
\(234\) 0 0
\(235\) −6.44423 −0.420375
\(236\) 0 0
\(237\) −13.1373 −0.853357
\(238\) 0 0
\(239\) 26.3877 1.70688 0.853438 0.521194i \(-0.174513\pi\)
0.853438 + 0.521194i \(0.174513\pi\)
\(240\) 0 0
\(241\) −7.06943 −0.455382 −0.227691 0.973733i \(-0.573118\pi\)
−0.227691 + 0.973733i \(0.573118\pi\)
\(242\) 0 0
\(243\) 20.1728 1.29408
\(244\) 0 0
\(245\) −14.1316 −0.902837
\(246\) 0 0
\(247\) −5.61797 −0.357463
\(248\) 0 0
\(249\) −10.2633 −0.650408
\(250\) 0 0
\(251\) 24.9023 1.57182 0.785909 0.618342i \(-0.212195\pi\)
0.785909 + 0.618342i \(0.212195\pi\)
\(252\) 0 0
\(253\) −5.13163 −0.322623
\(254\) 0 0
\(255\) −12.5969 −0.788849
\(256\) 0 0
\(257\) 0.444233 0.0277105 0.0138552 0.999904i \(-0.495590\pi\)
0.0138552 + 0.999904i \(0.495590\pi\)
\(258\) 0 0
\(259\) 26.7496 1.66214
\(260\) 0 0
\(261\) 14.2488 0.881978
\(262\) 0 0
\(263\) 23.8812 1.47258 0.736290 0.676666i \(-0.236576\pi\)
0.736290 + 0.676666i \(0.236576\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) −24.0000 −1.46878
\(268\) 0 0
\(269\) 16.2633 0.991589 0.495794 0.868440i \(-0.334877\pi\)
0.495794 + 0.868440i \(0.334877\pi\)
\(270\) 0 0
\(271\) −25.6098 −1.55568 −0.777842 0.628460i \(-0.783685\pi\)
−0.777842 + 0.628460i \(0.783685\pi\)
\(272\) 0 0
\(273\) 15.0977 0.913756
\(274\) 0 0
\(275\) −5.13163 −0.309449
\(276\) 0 0
\(277\) 2.88847 0.173551 0.0867755 0.996228i \(-0.472344\pi\)
0.0867755 + 0.996228i \(0.472344\pi\)
\(278\) 0 0
\(279\) 3.28432 0.196627
\(280\) 0 0
\(281\) 4.26326 0.254325 0.127163 0.991882i \(-0.459413\pi\)
0.127163 + 0.991882i \(0.459413\pi\)
\(282\) 0 0
\(283\) −30.5686 −1.81712 −0.908558 0.417758i \(-0.862816\pi\)
−0.908558 + 0.417758i \(0.862816\pi\)
\(284\) 0 0
\(285\) 12.3537 0.731773
\(286\) 0 0
\(287\) −39.1033 −2.30820
\(288\) 0 0
\(289\) 4.97171 0.292454
\(290\) 0 0
\(291\) 48.5403 2.84549
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 9.37480 0.545822
\(296\) 0 0
\(297\) 16.8539 0.977962
\(298\) 0 0
\(299\) −1.22212 −0.0706768
\(300\) 0 0
\(301\) −36.7753 −2.11969
\(302\) 0 0
\(303\) −9.06943 −0.521025
\(304\) 0 0
\(305\) −10.9507 −0.627033
\(306\) 0 0
\(307\) 8.54853 0.487891 0.243945 0.969789i \(-0.421558\pi\)
0.243945 + 0.969789i \(0.421558\pi\)
\(308\) 0 0
\(309\) 35.2899 2.00757
\(310\) 0 0
\(311\) 7.63806 0.433115 0.216557 0.976270i \(-0.430517\pi\)
0.216557 + 0.976270i \(0.430517\pi\)
\(312\) 0 0
\(313\) −18.2350 −1.03070 −0.515351 0.856979i \(-0.672338\pi\)
−0.515351 + 0.856979i \(0.672338\pi\)
\(314\) 0 0
\(315\) −19.4087 −1.09356
\(316\) 0 0
\(317\) −16.8602 −0.946962 −0.473481 0.880804i \(-0.657003\pi\)
−0.473481 + 0.880804i \(0.657003\pi\)
\(318\) 0 0
\(319\) −17.3182 −0.969635
\(320\) 0 0
\(321\) 33.0694 1.84576
\(322\) 0 0
\(323\) −21.5476 −1.19894
\(324\) 0 0
\(325\) −1.22212 −0.0677908
\(326\) 0 0
\(327\) 28.4781 1.57485
\(328\) 0 0
\(329\) 29.6236 1.63320
\(330\) 0 0
\(331\) −19.1517 −1.05267 −0.526337 0.850276i \(-0.676435\pi\)
−0.526337 + 0.850276i \(0.676435\pi\)
\(332\) 0 0
\(333\) 24.5686 1.34635
\(334\) 0 0
\(335\) 15.6381 0.854399
\(336\) 0 0
\(337\) 1.70845 0.0930652 0.0465326 0.998917i \(-0.485183\pi\)
0.0465326 + 0.998917i \(0.485183\pi\)
\(338\) 0 0
\(339\) 16.1244 0.875757
\(340\) 0 0
\(341\) −3.99181 −0.216169
\(342\) 0 0
\(343\) 32.7835 1.77014
\(344\) 0 0
\(345\) 2.68740 0.144685
\(346\) 0 0
\(347\) −10.6874 −0.573730 −0.286865 0.957971i \(-0.592613\pi\)
−0.286865 + 0.957971i \(0.592613\pi\)
\(348\) 0 0
\(349\) −12.3877 −0.663096 −0.331548 0.943438i \(-0.607571\pi\)
−0.331548 + 0.943438i \(0.607571\pi\)
\(350\) 0 0
\(351\) 4.01382 0.214242
\(352\) 0 0
\(353\) 5.45710 0.290452 0.145226 0.989399i \(-0.453609\pi\)
0.145226 + 0.989399i \(0.453609\pi\)
\(354\) 0 0
\(355\) 1.31260 0.0696656
\(356\) 0 0
\(357\) 57.9070 3.06476
\(358\) 0 0
\(359\) −20.1810 −1.06511 −0.532555 0.846395i \(-0.678768\pi\)
−0.532555 + 0.846395i \(0.678768\pi\)
\(360\) 0 0
\(361\) 2.13163 0.112191
\(362\) 0 0
\(363\) −41.2076 −2.16284
\(364\) 0 0
\(365\) 4.44423 0.232622
\(366\) 0 0
\(367\) −2.74960 −0.143528 −0.0717639 0.997422i \(-0.522863\pi\)
−0.0717639 + 0.997422i \(0.522863\pi\)
\(368\) 0 0
\(369\) −35.9151 −1.86967
\(370\) 0 0
\(371\) −27.5815 −1.43196
\(372\) 0 0
\(373\) −12.0823 −0.625598 −0.312799 0.949819i \(-0.601267\pi\)
−0.312799 + 0.949819i \(0.601267\pi\)
\(374\) 0 0
\(375\) 2.68740 0.138777
\(376\) 0 0
\(377\) −4.12440 −0.212417
\(378\) 0 0
\(379\) −5.25603 −0.269984 −0.134992 0.990847i \(-0.543101\pi\)
−0.134992 + 0.990847i \(0.543101\pi\)
\(380\) 0 0
\(381\) −6.08230 −0.311605
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 23.5897 1.20224
\(386\) 0 0
\(387\) −33.7769 −1.71698
\(388\) 0 0
\(389\) 0.325463 0.0165017 0.00825083 0.999966i \(-0.497374\pi\)
0.00825083 + 0.999966i \(0.497374\pi\)
\(390\) 0 0
\(391\) −4.68740 −0.237052
\(392\) 0 0
\(393\) 7.87560 0.397272
\(394\) 0 0
\(395\) −4.88847 −0.245965
\(396\) 0 0
\(397\) −17.7568 −0.891190 −0.445595 0.895235i \(-0.647008\pi\)
−0.445595 + 0.895235i \(0.647008\pi\)
\(398\) 0 0
\(399\) −56.7891 −2.84301
\(400\) 0 0
\(401\) 3.91770 0.195641 0.0978204 0.995204i \(-0.468813\pi\)
0.0978204 + 0.995204i \(0.468813\pi\)
\(402\) 0 0
\(403\) −0.950664 −0.0473560
\(404\) 0 0
\(405\) 3.84008 0.190815
\(406\) 0 0
\(407\) −29.8611 −1.48016
\(408\) 0 0
\(409\) 9.58405 0.473901 0.236950 0.971522i \(-0.423852\pi\)
0.236950 + 0.971522i \(0.423852\pi\)
\(410\) 0 0
\(411\) 53.1856 2.62345
\(412\) 0 0
\(413\) −43.0952 −2.12057
\(414\) 0 0
\(415\) −3.81903 −0.187469
\(416\) 0 0
\(417\) 16.8319 0.824261
\(418\) 0 0
\(419\) 20.5265 1.00279 0.501393 0.865219i \(-0.332821\pi\)
0.501393 + 0.865219i \(0.332821\pi\)
\(420\) 0 0
\(421\) 2.29155 0.111683 0.0558417 0.998440i \(-0.482216\pi\)
0.0558417 + 0.998440i \(0.482216\pi\)
\(422\) 0 0
\(423\) 27.2083 1.32291
\(424\) 0 0
\(425\) −4.68740 −0.227372
\(426\) 0 0
\(427\) 50.3393 2.43609
\(428\) 0 0
\(429\) −16.8539 −0.813714
\(430\) 0 0
\(431\) 8.83189 0.425417 0.212709 0.977116i \(-0.431771\pi\)
0.212709 + 0.977116i \(0.431771\pi\)
\(432\) 0 0
\(433\) 33.3465 1.60253 0.801266 0.598309i \(-0.204160\pi\)
0.801266 + 0.598309i \(0.204160\pi\)
\(434\) 0 0
\(435\) 9.06943 0.434846
\(436\) 0 0
\(437\) 4.59692 0.219900
\(438\) 0 0
\(439\) 6.02829 0.287714 0.143857 0.989598i \(-0.454049\pi\)
0.143857 + 0.989598i \(0.454049\pi\)
\(440\) 0 0
\(441\) 59.6654 2.84121
\(442\) 0 0
\(443\) 15.5275 0.737733 0.368866 0.929482i \(-0.379746\pi\)
0.368866 + 0.929482i \(0.379746\pi\)
\(444\) 0 0
\(445\) −8.93057 −0.423349
\(446\) 0 0
\(447\) 53.1856 2.51559
\(448\) 0 0
\(449\) −18.3594 −0.866433 −0.433216 0.901290i \(-0.642621\pi\)
−0.433216 + 0.901290i \(0.642621\pi\)
\(450\) 0 0
\(451\) 43.6519 2.05549
\(452\) 0 0
\(453\) 10.9168 0.512914
\(454\) 0 0
\(455\) 5.61797 0.263374
\(456\) 0 0
\(457\) −11.4992 −0.537910 −0.268955 0.963153i \(-0.586678\pi\)
−0.268955 + 0.963153i \(0.586678\pi\)
\(458\) 0 0
\(459\) 15.3949 0.718572
\(460\) 0 0
\(461\) 1.33270 0.0620700 0.0310350 0.999518i \(-0.490120\pi\)
0.0310350 + 0.999518i \(0.490120\pi\)
\(462\) 0 0
\(463\) −35.8190 −1.66465 −0.832326 0.554287i \(-0.812991\pi\)
−0.832326 + 0.554287i \(0.812991\pi\)
\(464\) 0 0
\(465\) 2.09048 0.0969439
\(466\) 0 0
\(467\) 23.7625 1.09960 0.549798 0.835298i \(-0.314705\pi\)
0.549798 + 0.835298i \(0.314705\pi\)
\(468\) 0 0
\(469\) −71.8869 −3.31943
\(470\) 0 0
\(471\) −12.4298 −0.572733
\(472\) 0 0
\(473\) 41.0531 1.88762
\(474\) 0 0
\(475\) 4.59692 0.210921
\(476\) 0 0
\(477\) −25.3327 −1.15990
\(478\) 0 0
\(479\) 2.04210 0.0933060 0.0466530 0.998911i \(-0.485145\pi\)
0.0466530 + 0.998911i \(0.485145\pi\)
\(480\) 0 0
\(481\) −7.11153 −0.324258
\(482\) 0 0
\(483\) −12.3537 −0.562115
\(484\) 0 0
\(485\) 18.0622 0.820162
\(486\) 0 0
\(487\) 18.6252 0.843988 0.421994 0.906599i \(-0.361330\pi\)
0.421994 + 0.906599i \(0.361330\pi\)
\(488\) 0 0
\(489\) 33.3666 1.50889
\(490\) 0 0
\(491\) −33.7204 −1.52178 −0.760889 0.648882i \(-0.775237\pi\)
−0.760889 + 0.648882i \(0.775237\pi\)
\(492\) 0 0
\(493\) −15.8190 −0.712453
\(494\) 0 0
\(495\) 21.6663 0.973830
\(496\) 0 0
\(497\) −6.03391 −0.270658
\(498\) 0 0
\(499\) 16.8885 0.756032 0.378016 0.925799i \(-0.376607\pi\)
0.378016 + 0.925799i \(0.376607\pi\)
\(500\) 0 0
\(501\) 34.6365 1.54744
\(502\) 0 0
\(503\) −11.4031 −0.508438 −0.254219 0.967147i \(-0.581818\pi\)
−0.254219 + 0.967147i \(0.581818\pi\)
\(504\) 0 0
\(505\) −3.37480 −0.150177
\(506\) 0 0
\(507\) 30.9224 1.37331
\(508\) 0 0
\(509\) −1.87560 −0.0831346 −0.0415673 0.999136i \(-0.513235\pi\)
−0.0415673 + 0.999136i \(0.513235\pi\)
\(510\) 0 0
\(511\) −20.4298 −0.903759
\(512\) 0 0
\(513\) −15.0977 −0.666581
\(514\) 0 0
\(515\) 13.1316 0.578649
\(516\) 0 0
\(517\) −33.0694 −1.45439
\(518\) 0 0
\(519\) 27.5275 1.20832
\(520\) 0 0
\(521\) 5.11153 0.223940 0.111970 0.993712i \(-0.464284\pi\)
0.111970 + 0.993712i \(0.464284\pi\)
\(522\) 0 0
\(523\) −19.4571 −0.850799 −0.425400 0.905006i \(-0.639866\pi\)
−0.425400 + 0.905006i \(0.639866\pi\)
\(524\) 0 0
\(525\) −12.3537 −0.539162
\(526\) 0 0
\(527\) −3.64625 −0.158833
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −39.5815 −1.71769
\(532\) 0 0
\(533\) 10.3958 0.450294
\(534\) 0 0
\(535\) 12.3054 0.532007
\(536\) 0 0
\(537\) −35.4571 −1.53009
\(538\) 0 0
\(539\) −72.5183 −3.12359
\(540\) 0 0
\(541\) −2.70750 −0.116404 −0.0582022 0.998305i \(-0.518537\pi\)
−0.0582022 + 0.998305i \(0.518537\pi\)
\(542\) 0 0
\(543\) −42.4360 −1.82111
\(544\) 0 0
\(545\) 10.5969 0.453922
\(546\) 0 0
\(547\) 1.66635 0.0712479 0.0356240 0.999365i \(-0.488658\pi\)
0.0356240 + 0.999365i \(0.488658\pi\)
\(548\) 0 0
\(549\) 46.2350 1.97326
\(550\) 0 0
\(551\) 15.5137 0.660904
\(552\) 0 0
\(553\) 22.4719 0.955601
\(554\) 0 0
\(555\) 15.6381 0.663799
\(556\) 0 0
\(557\) −20.9306 −0.886857 −0.443428 0.896310i \(-0.646238\pi\)
−0.443428 + 0.896310i \(0.646238\pi\)
\(558\) 0 0
\(559\) 9.77693 0.413520
\(560\) 0 0
\(561\) −64.6427 −2.72922
\(562\) 0 0
\(563\) 15.2761 0.643812 0.321906 0.946772i \(-0.395676\pi\)
0.321906 + 0.946772i \(0.395676\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) −17.6525 −0.741337
\(568\) 0 0
\(569\) 20.3877 0.854695 0.427348 0.904087i \(-0.359448\pi\)
0.427348 + 0.904087i \(0.359448\pi\)
\(570\) 0 0
\(571\) −5.49357 −0.229899 −0.114949 0.993371i \(-0.536671\pi\)
−0.114949 + 0.993371i \(0.536671\pi\)
\(572\) 0 0
\(573\) 43.3327 1.81025
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 17.2761 0.719215 0.359607 0.933104i \(-0.382911\pi\)
0.359607 + 0.933104i \(0.382911\pi\)
\(578\) 0 0
\(579\) 48.2212 2.00400
\(580\) 0 0
\(581\) 17.5558 0.728336
\(582\) 0 0
\(583\) 30.7898 1.27518
\(584\) 0 0
\(585\) 5.15992 0.213336
\(586\) 0 0
\(587\) −31.8247 −1.31354 −0.656772 0.754089i \(-0.728079\pi\)
−0.656772 + 0.754089i \(0.728079\pi\)
\(588\) 0 0
\(589\) 3.57587 0.147341
\(590\) 0 0
\(591\) −15.8052 −0.650140
\(592\) 0 0
\(593\) −12.4442 −0.511023 −0.255512 0.966806i \(-0.582244\pi\)
−0.255512 + 0.966806i \(0.582244\pi\)
\(594\) 0 0
\(595\) 21.5476 0.883365
\(596\) 0 0
\(597\) 17.6525 0.722470
\(598\) 0 0
\(599\) −17.4370 −0.712456 −0.356228 0.934399i \(-0.615937\pi\)
−0.356228 + 0.934399i \(0.615937\pi\)
\(600\) 0 0
\(601\) −0.916751 −0.0373951 −0.0186975 0.999825i \(-0.505952\pi\)
−0.0186975 + 0.999825i \(0.505952\pi\)
\(602\) 0 0
\(603\) −66.0257 −2.68878
\(604\) 0 0
\(605\) −15.3337 −0.623402
\(606\) 0 0
\(607\) −36.7753 −1.49266 −0.746332 0.665574i \(-0.768187\pi\)
−0.746332 + 0.665574i \(0.768187\pi\)
\(608\) 0 0
\(609\) −41.6914 −1.68942
\(610\) 0 0
\(611\) −7.87560 −0.318613
\(612\) 0 0
\(613\) 4.38766 0.177216 0.0886080 0.996067i \(-0.471758\pi\)
0.0886080 + 0.996067i \(0.471758\pi\)
\(614\) 0 0
\(615\) −22.8602 −0.921811
\(616\) 0 0
\(617\) 40.4499 1.62845 0.814225 0.580549i \(-0.197162\pi\)
0.814225 + 0.580549i \(0.197162\pi\)
\(618\) 0 0
\(619\) 39.8165 1.60036 0.800180 0.599760i \(-0.204737\pi\)
0.800180 + 0.599760i \(0.204737\pi\)
\(620\) 0 0
\(621\) −3.28432 −0.131795
\(622\) 0 0
\(623\) 41.0531 1.64476
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 63.3949 2.53175
\(628\) 0 0
\(629\) −27.2761 −1.08757
\(630\) 0 0
\(631\) −25.9013 −1.03112 −0.515558 0.856855i \(-0.672415\pi\)
−0.515558 + 0.856855i \(0.672415\pi\)
\(632\) 0 0
\(633\) −41.5394 −1.65104
\(634\) 0 0
\(635\) −2.26326 −0.0898149
\(636\) 0 0
\(637\) −17.2705 −0.684282
\(638\) 0 0
\(639\) −5.54195 −0.219236
\(640\) 0 0
\(641\) −43.8448 −1.73176 −0.865882 0.500248i \(-0.833242\pi\)
−0.865882 + 0.500248i \(0.833242\pi\)
\(642\) 0 0
\(643\) −3.94343 −0.155514 −0.0777568 0.996972i \(-0.524776\pi\)
−0.0777568 + 0.996972i \(0.524776\pi\)
\(644\) 0 0
\(645\) −21.4992 −0.846530
\(646\) 0 0
\(647\) −24.3456 −0.957123 −0.478561 0.878054i \(-0.658842\pi\)
−0.478561 + 0.878054i \(0.658842\pi\)
\(648\) 0 0
\(649\) 48.1080 1.88841
\(650\) 0 0
\(651\) −9.60978 −0.376637
\(652\) 0 0
\(653\) 37.6921 1.47500 0.737502 0.675344i \(-0.236005\pi\)
0.737502 + 0.675344i \(0.236005\pi\)
\(654\) 0 0
\(655\) 2.93057 0.114507
\(656\) 0 0
\(657\) −18.7641 −0.732056
\(658\) 0 0
\(659\) −24.6107 −0.958698 −0.479349 0.877624i \(-0.659127\pi\)
−0.479349 + 0.877624i \(0.659127\pi\)
\(660\) 0 0
\(661\) 27.3126 1.06234 0.531169 0.847266i \(-0.321753\pi\)
0.531169 + 0.847266i \(0.321753\pi\)
\(662\) 0 0
\(663\) −15.3949 −0.597888
\(664\) 0 0
\(665\) −21.1316 −0.819450
\(666\) 0 0
\(667\) 3.37480 0.130673
\(668\) 0 0
\(669\) −28.8885 −1.11689
\(670\) 0 0
\(671\) −56.1948 −2.16938
\(672\) 0 0
\(673\) 22.5265 0.868334 0.434167 0.900832i \(-0.357043\pi\)
0.434167 + 0.900832i \(0.357043\pi\)
\(674\) 0 0
\(675\) −3.28432 −0.126413
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) −83.0304 −3.18641
\(680\) 0 0
\(681\) −33.0694 −1.26722
\(682\) 0 0
\(683\) 31.9974 1.22435 0.612174 0.790723i \(-0.290295\pi\)
0.612174 + 0.790723i \(0.290295\pi\)
\(684\) 0 0
\(685\) 19.7907 0.756166
\(686\) 0 0
\(687\) −25.9013 −0.988197
\(688\) 0 0
\(689\) 7.33270 0.279354
\(690\) 0 0
\(691\) 2.80617 0.106752 0.0533758 0.998574i \(-0.483002\pi\)
0.0533758 + 0.998574i \(0.483002\pi\)
\(692\) 0 0
\(693\) −99.5984 −3.78343
\(694\) 0 0
\(695\) 6.26326 0.237579
\(696\) 0 0
\(697\) 39.8730 1.51030
\(698\) 0 0
\(699\) −37.4716 −1.41730
\(700\) 0 0
\(701\) 8.37385 0.316276 0.158138 0.987417i \(-0.449451\pi\)
0.158138 + 0.987417i \(0.449451\pi\)
\(702\) 0 0
\(703\) 26.7496 1.00888
\(704\) 0 0
\(705\) 17.3182 0.652242
\(706\) 0 0
\(707\) 15.5137 0.583451
\(708\) 0 0
\(709\) 21.2139 0.796706 0.398353 0.917232i \(-0.369582\pi\)
0.398353 + 0.917232i \(0.369582\pi\)
\(710\) 0 0
\(711\) 20.6397 0.774048
\(712\) 0 0
\(713\) 0.777884 0.0291320
\(714\) 0 0
\(715\) −6.27145 −0.234539
\(716\) 0 0
\(717\) −70.9142 −2.64834
\(718\) 0 0
\(719\) 28.1106 1.04835 0.524174 0.851611i \(-0.324374\pi\)
0.524174 + 0.851611i \(0.324374\pi\)
\(720\) 0 0
\(721\) −60.3650 −2.24811
\(722\) 0 0
\(723\) 18.9984 0.706558
\(724\) 0 0
\(725\) 3.37480 0.125337
\(726\) 0 0
\(727\) 39.0330 1.44765 0.723826 0.689982i \(-0.242382\pi\)
0.723826 + 0.689982i \(0.242382\pi\)
\(728\) 0 0
\(729\) −42.6921 −1.58119
\(730\) 0 0
\(731\) 37.4992 1.38696
\(732\) 0 0
\(733\) 47.1373 1.74105 0.870527 0.492120i \(-0.163778\pi\)
0.870527 + 0.492120i \(0.163778\pi\)
\(734\) 0 0
\(735\) 37.9773 1.40082
\(736\) 0 0
\(737\) 80.2488 2.95600
\(738\) 0 0
\(739\) 24.5265 0.902223 0.451111 0.892468i \(-0.351028\pi\)
0.451111 + 0.892468i \(0.351028\pi\)
\(740\) 0 0
\(741\) 15.0977 0.554629
\(742\) 0 0
\(743\) 7.34651 0.269517 0.134759 0.990878i \(-0.456974\pi\)
0.134759 + 0.990878i \(0.456974\pi\)
\(744\) 0 0
\(745\) 19.7907 0.725077
\(746\) 0 0
\(747\) 16.1244 0.589961
\(748\) 0 0
\(749\) −56.5667 −2.06690
\(750\) 0 0
\(751\) −11.2359 −0.410005 −0.205002 0.978761i \(-0.565720\pi\)
−0.205002 + 0.978761i \(0.565720\pi\)
\(752\) 0 0
\(753\) −66.9224 −2.43879
\(754\) 0 0
\(755\) 4.06220 0.147839
\(756\) 0 0
\(757\) 49.1794 1.78745 0.893727 0.448611i \(-0.148081\pi\)
0.893727 + 0.448611i \(0.148081\pi\)
\(758\) 0 0
\(759\) 13.7907 0.500572
\(760\) 0 0
\(761\) 43.2478 1.56773 0.783867 0.620929i \(-0.213245\pi\)
0.783867 + 0.620929i \(0.213245\pi\)
\(762\) 0 0
\(763\) −48.7131 −1.76353
\(764\) 0 0
\(765\) 19.7907 0.715536
\(766\) 0 0
\(767\) 11.4571 0.413692
\(768\) 0 0
\(769\) −39.6638 −1.43031 −0.715156 0.698964i \(-0.753645\pi\)
−0.715156 + 0.698964i \(0.753645\pi\)
\(770\) 0 0
\(771\) −1.19383 −0.0429948
\(772\) 0 0
\(773\) −2.18097 −0.0784440 −0.0392220 0.999231i \(-0.512488\pi\)
−0.0392220 + 0.999231i \(0.512488\pi\)
\(774\) 0 0
\(775\) 0.777884 0.0279424
\(776\) 0 0
\(777\) −71.8869 −2.57893
\(778\) 0 0
\(779\) −39.1033 −1.40102
\(780\) 0 0
\(781\) 6.73578 0.241025
\(782\) 0 0
\(783\) −11.0839 −0.396106
\(784\) 0 0
\(785\) −4.62520 −0.165080
\(786\) 0 0
\(787\) −4.76407 −0.169821 −0.0849103 0.996389i \(-0.527060\pi\)
−0.0849103 + 0.996389i \(0.527060\pi\)
\(788\) 0 0
\(789\) −64.1784 −2.28481
\(790\) 0 0
\(791\) −27.5815 −0.980685
\(792\) 0 0
\(793\) −13.3830 −0.475244
\(794\) 0 0
\(795\) −16.1244 −0.571873
\(796\) 0 0
\(797\) 39.7204 1.40697 0.703484 0.710711i \(-0.251627\pi\)
0.703484 + 0.710711i \(0.251627\pi\)
\(798\) 0 0
\(799\) −30.2067 −1.06864
\(800\) 0 0
\(801\) 37.7059 1.33227
\(802\) 0 0
\(803\) 22.8062 0.804812
\(804\) 0 0
\(805\) −4.59692 −0.162020
\(806\) 0 0
\(807\) −43.7059 −1.53852
\(808\) 0 0
\(809\) 46.8941 1.64871 0.824354 0.566074i \(-0.191538\pi\)
0.824354 + 0.566074i \(0.191538\pi\)
\(810\) 0 0
\(811\) 36.8319 1.29334 0.646671 0.762769i \(-0.276161\pi\)
0.646671 + 0.762769i \(0.276161\pi\)
\(812\) 0 0
\(813\) 68.8237 2.41375
\(814\) 0 0
\(815\) 12.4159 0.434912
\(816\) 0 0
\(817\) −36.7753 −1.28661
\(818\) 0 0
\(819\) −23.7197 −0.828834
\(820\) 0 0
\(821\) −18.6107 −0.649519 −0.324759 0.945797i \(-0.605283\pi\)
−0.324759 + 0.945797i \(0.605283\pi\)
\(822\) 0 0
\(823\) 9.31823 0.324813 0.162407 0.986724i \(-0.448074\pi\)
0.162407 + 0.986724i \(0.448074\pi\)
\(824\) 0 0
\(825\) 13.7907 0.480132
\(826\) 0 0
\(827\) −53.0129 −1.84344 −0.921719 0.387858i \(-0.873215\pi\)
−0.921719 + 0.387858i \(0.873215\pi\)
\(828\) 0 0
\(829\) −25.2761 −0.877876 −0.438938 0.898517i \(-0.644645\pi\)
−0.438938 + 0.898517i \(0.644645\pi\)
\(830\) 0 0
\(831\) −7.76246 −0.269277
\(832\) 0 0
\(833\) −66.2406 −2.29510
\(834\) 0 0
\(835\) 12.8885 0.446024
\(836\) 0 0
\(837\) −2.55481 −0.0883073
\(838\) 0 0
\(839\) −11.6946 −0.403744 −0.201872 0.979412i \(-0.564702\pi\)
−0.201872 + 0.979412i \(0.564702\pi\)
\(840\) 0 0
\(841\) −17.6107 −0.607267
\(842\) 0 0
\(843\) −11.4571 −0.394603
\(844\) 0 0
\(845\) 11.5064 0.395833
\(846\) 0 0
\(847\) 70.4875 2.42198
\(848\) 0 0
\(849\) 82.1501 2.81938
\(850\) 0 0
\(851\) 5.81903 0.199474
\(852\) 0 0
\(853\) 38.6371 1.32291 0.661455 0.749985i \(-0.269939\pi\)
0.661455 + 0.749985i \(0.269939\pi\)
\(854\) 0 0
\(855\) −19.4087 −0.663764
\(856\) 0 0
\(857\) 26.2488 0.896642 0.448321 0.893873i \(-0.352022\pi\)
0.448321 + 0.893873i \(0.352022\pi\)
\(858\) 0 0
\(859\) −37.5558 −1.28139 −0.640693 0.767797i \(-0.721353\pi\)
−0.640693 + 0.767797i \(0.721353\pi\)
\(860\) 0 0
\(861\) 105.086 3.58133
\(862\) 0 0
\(863\) 21.9855 0.748396 0.374198 0.927349i \(-0.377918\pi\)
0.374198 + 0.927349i \(0.377918\pi\)
\(864\) 0 0
\(865\) 10.2432 0.348278
\(866\) 0 0
\(867\) −13.3610 −0.453763
\(868\) 0 0
\(869\) −25.0858 −0.850978
\(870\) 0 0
\(871\) 19.1115 0.647570
\(872\) 0 0
\(873\) −76.2607 −2.58103
\(874\) 0 0
\(875\) −4.59692 −0.155404
\(876\) 0 0
\(877\) −36.5064 −1.23273 −0.616367 0.787459i \(-0.711396\pi\)
−0.616367 + 0.787459i \(0.711396\pi\)
\(878\) 0 0
\(879\) 16.1244 0.543862
\(880\) 0 0
\(881\) −17.4571 −0.588145 −0.294072 0.955783i \(-0.595011\pi\)
−0.294072 + 0.955783i \(0.595011\pi\)
\(882\) 0 0
\(883\) −22.9444 −0.772140 −0.386070 0.922470i \(-0.626168\pi\)
−0.386070 + 0.922470i \(0.626168\pi\)
\(884\) 0 0
\(885\) −25.1938 −0.846881
\(886\) 0 0
\(887\) 11.7223 0.393595 0.196798 0.980444i \(-0.436946\pi\)
0.196798 + 0.980444i \(0.436946\pi\)
\(888\) 0 0
\(889\) 10.4040 0.348940
\(890\) 0 0
\(891\) 19.7059 0.660172
\(892\) 0 0
\(893\) 29.6236 0.991316
\(894\) 0 0
\(895\) −13.1938 −0.441021
\(896\) 0 0
\(897\) 3.28432 0.109660
\(898\) 0 0
\(899\) 2.62520 0.0875554
\(900\) 0 0
\(901\) 28.1244 0.936960
\(902\) 0 0
\(903\) 98.8300 3.28886
\(904\) 0 0
\(905\) −15.7907 −0.524902
\(906\) 0 0
\(907\) 1.91770 0.0636763 0.0318381 0.999493i \(-0.489864\pi\)
0.0318381 + 0.999493i \(0.489864\pi\)
\(908\) 0 0
\(909\) 14.2488 0.472603
\(910\) 0 0
\(911\) 56.8319 1.88292 0.941462 0.337118i \(-0.109452\pi\)
0.941462 + 0.337118i \(0.109452\pi\)
\(912\) 0 0
\(913\) −19.5979 −0.648595
\(914\) 0 0
\(915\) 29.4288 0.972886
\(916\) 0 0
\(917\) −13.4716 −0.444870
\(918\) 0 0
\(919\) 26.0257 0.858509 0.429255 0.903183i \(-0.358776\pi\)
0.429255 + 0.903183i \(0.358776\pi\)
\(920\) 0 0
\(921\) −22.9733 −0.756997
\(922\) 0 0
\(923\) 1.60415 0.0528013
\(924\) 0 0
\(925\) 5.81903 0.191329
\(926\) 0 0
\(927\) −55.4433 −1.82100
\(928\) 0 0
\(929\) −17.1115 −0.561411 −0.280706 0.959794i \(-0.590568\pi\)
−0.280706 + 0.959794i \(0.590568\pi\)
\(930\) 0 0
\(931\) 64.9619 2.12904
\(932\) 0 0
\(933\) −20.5265 −0.672008
\(934\) 0 0
\(935\) −24.0540 −0.786650
\(936\) 0 0
\(937\) 16.3738 0.534910 0.267455 0.963570i \(-0.413817\pi\)
0.267455 + 0.963570i \(0.413817\pi\)
\(938\) 0 0
\(939\) 49.0047 1.59921
\(940\) 0 0
\(941\) −16.4097 −0.534940 −0.267470 0.963566i \(-0.586188\pi\)
−0.267470 + 0.963566i \(0.586188\pi\)
\(942\) 0 0
\(943\) −8.50643 −0.277008
\(944\) 0 0
\(945\) 15.0977 0.491129
\(946\) 0 0
\(947\) 21.3886 0.695037 0.347518 0.937673i \(-0.387024\pi\)
0.347518 + 0.937673i \(0.387024\pi\)
\(948\) 0 0
\(949\) 5.43137 0.176310
\(950\) 0 0
\(951\) 45.3100 1.46928
\(952\) 0 0
\(953\) −16.7276 −0.541860 −0.270930 0.962599i \(-0.587331\pi\)
−0.270930 + 0.962599i \(0.587331\pi\)
\(954\) 0 0
\(955\) 16.1244 0.521773
\(956\) 0 0
\(957\) 46.5410 1.50446
\(958\) 0 0
\(959\) −90.9764 −2.93778
\(960\) 0 0
\(961\) −30.3949 −0.980481
\(962\) 0 0
\(963\) −51.9547 −1.67422
\(964\) 0 0
\(965\) 17.9434 0.577619
\(966\) 0 0
\(967\) −45.2617 −1.45552 −0.727758 0.685834i \(-0.759438\pi\)
−0.727758 + 0.685834i \(0.759438\pi\)
\(968\) 0 0
\(969\) 57.9070 1.86024
\(970\) 0 0
\(971\) −45.2560 −1.45234 −0.726168 0.687518i \(-0.758700\pi\)
−0.726168 + 0.687518i \(0.758700\pi\)
\(972\) 0 0
\(973\) −28.7917 −0.923019
\(974\) 0 0
\(975\) 3.28432 0.105182
\(976\) 0 0
\(977\) −13.3465 −0.426993 −0.213496 0.976944i \(-0.568485\pi\)
−0.213496 + 0.976944i \(0.568485\pi\)
\(978\) 0 0
\(979\) −45.8284 −1.46468
\(980\) 0 0
\(981\) −44.7414 −1.42848
\(982\) 0 0
\(983\) −5.13163 −0.163674 −0.0818368 0.996646i \(-0.526079\pi\)
−0.0818368 + 0.996646i \(0.526079\pi\)
\(984\) 0 0
\(985\) −5.88123 −0.187392
\(986\) 0 0
\(987\) −79.6104 −2.53403
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −17.7989 −0.565402 −0.282701 0.959208i \(-0.591230\pi\)
−0.282701 + 0.959208i \(0.591230\pi\)
\(992\) 0 0
\(993\) 51.4684 1.63330
\(994\) 0 0
\(995\) 6.56863 0.208240
\(996\) 0 0
\(997\) 40.3877 1.27909 0.639545 0.768754i \(-0.279123\pi\)
0.639545 + 0.768754i \(0.279123\pi\)
\(998\) 0 0
\(999\) −19.1115 −0.604662
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 1840.2.a.r.1.1 3
4.3 odd 2 230.2.a.d.1.3 3
5.4 even 2 9200.2.a.cf.1.3 3
8.3 odd 2 7360.2.a.bz.1.1 3
8.5 even 2 7360.2.a.ce.1.3 3
12.11 even 2 2070.2.a.z.1.1 3
20.3 even 4 1150.2.b.j.599.3 6
20.7 even 4 1150.2.b.j.599.4 6
20.19 odd 2 1150.2.a.q.1.1 3
92.91 even 2 5290.2.a.r.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.3 3 4.3 odd 2
1150.2.a.q.1.1 3 20.19 odd 2
1150.2.b.j.599.3 6 20.3 even 4
1150.2.b.j.599.4 6 20.7 even 4
1840.2.a.r.1.1 3 1.1 even 1 trivial
2070.2.a.z.1.1 3 12.11 even 2
5290.2.a.r.1.3 3 92.91 even 2
7360.2.a.bz.1.1 3 8.3 odd 2
7360.2.a.ce.1.3 3 8.5 even 2
9200.2.a.cf.1.3 3 5.4 even 2