Properties

Label 4020.2.a.e.1.2
Level $4020$
Weight $2$
Character 4020.1
Self dual yes
Analytic conductor $32.100$
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] = [4020,2,Mod(1,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, 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("4020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.0998616126\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.1257629.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 9x^{3} + 9x^{2} + 17x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.49431\) of defining polynomial
Character \(\chi\) \(=\) 4020.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -1.20813 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -1.20813 q^{7} +1.00000 q^{9} +5.26856 q^{11} +0.221575 q^{13} -1.00000 q^{15} -4.28618 q^{17} -5.28201 q^{19} +1.20813 q^{21} -2.67771 q^{23} +1.00000 q^{25} -1.00000 q^{27} -6.52207 q^{29} +2.38432 q^{31} -5.26856 q^{33} -1.20813 q^{35} +8.84802 q^{37} -0.221575 q^{39} -7.02055 q^{41} -9.44939 q^{43} +1.00000 q^{45} +1.74782 q^{47} -5.54042 q^{49} +4.28618 q^{51} -5.05419 q^{53} +5.26856 q^{55} +5.28201 q^{57} +10.1910 q^{59} -2.62352 q^{61} -1.20813 q^{63} +0.221575 q^{65} -1.00000 q^{67} +2.67771 q^{69} +6.47669 q^{71} -6.83165 q^{73} -1.00000 q^{75} -6.36510 q^{77} +16.6612 q^{79} +1.00000 q^{81} -10.2039 q^{83} -4.28618 q^{85} +6.52207 q^{87} -4.85648 q^{89} -0.267691 q^{91} -2.38432 q^{93} -5.28201 q^{95} +13.8518 q^{97} +5.26856 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} + 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{3} + 5 q^{5} - 3 q^{7} + 5 q^{9} + 3 q^{11} - 11 q^{13} - 5 q^{15} - 11 q^{17} + 6 q^{19} + 3 q^{21} + 3 q^{23} + 5 q^{25} - 5 q^{27} - 2 q^{29} + q^{31} - 3 q^{33} - 3 q^{35} - 13 q^{37} + 11 q^{39} - 13 q^{41} - 9 q^{43} + 5 q^{45} - 12 q^{47} - 2 q^{49} + 11 q^{51} - 19 q^{53} + 3 q^{55} - 6 q^{57} + 6 q^{59} - 3 q^{61} - 3 q^{63} - 11 q^{65} - 5 q^{67} - 3 q^{69} + 6 q^{71} - 21 q^{73} - 5 q^{75} - 15 q^{77} - q^{79} + 5 q^{81} - 8 q^{83} - 11 q^{85} + 2 q^{87} - 29 q^{89} + 23 q^{91} - q^{93} + 6 q^{95} - 13 q^{97} + 3 q^{99}+O(q^{100}) \) 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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.20813 −0.456630 −0.228315 0.973587i \(-0.573322\pi\)
−0.228315 + 0.973587i \(0.573322\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.26856 1.58853 0.794265 0.607571i \(-0.207856\pi\)
0.794265 + 0.607571i \(0.207856\pi\)
\(12\) 0 0
\(13\) 0.221575 0.0614538 0.0307269 0.999528i \(-0.490218\pi\)
0.0307269 + 0.999528i \(0.490218\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −4.28618 −1.03955 −0.519776 0.854303i \(-0.673984\pi\)
−0.519776 + 0.854303i \(0.673984\pi\)
\(18\) 0 0
\(19\) −5.28201 −1.21178 −0.605888 0.795550i \(-0.707182\pi\)
−0.605888 + 0.795550i \(0.707182\pi\)
\(20\) 0 0
\(21\) 1.20813 0.263635
\(22\) 0 0
\(23\) −2.67771 −0.558342 −0.279171 0.960241i \(-0.590060\pi\)
−0.279171 + 0.960241i \(0.590060\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.52207 −1.21112 −0.605559 0.795800i \(-0.707050\pi\)
−0.605559 + 0.795800i \(0.707050\pi\)
\(30\) 0 0
\(31\) 2.38432 0.428237 0.214119 0.976808i \(-0.431312\pi\)
0.214119 + 0.976808i \(0.431312\pi\)
\(32\) 0 0
\(33\) −5.26856 −0.917139
\(34\) 0 0
\(35\) −1.20813 −0.204211
\(36\) 0 0
\(37\) 8.84802 1.45461 0.727303 0.686316i \(-0.240773\pi\)
0.727303 + 0.686316i \(0.240773\pi\)
\(38\) 0 0
\(39\) −0.221575 −0.0354804
\(40\) 0 0
\(41\) −7.02055 −1.09643 −0.548213 0.836339i \(-0.684692\pi\)
−0.548213 + 0.836339i \(0.684692\pi\)
\(42\) 0 0
\(43\) −9.44939 −1.44102 −0.720509 0.693446i \(-0.756092\pi\)
−0.720509 + 0.693446i \(0.756092\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 1.74782 0.254945 0.127473 0.991842i \(-0.459313\pi\)
0.127473 + 0.991842i \(0.459313\pi\)
\(48\) 0 0
\(49\) −5.54042 −0.791489
\(50\) 0 0
\(51\) 4.28618 0.600185
\(52\) 0 0
\(53\) −5.05419 −0.694247 −0.347123 0.937819i \(-0.612842\pi\)
−0.347123 + 0.937819i \(0.612842\pi\)
\(54\) 0 0
\(55\) 5.26856 0.710412
\(56\) 0 0
\(57\) 5.28201 0.699619
\(58\) 0 0
\(59\) 10.1910 1.32675 0.663376 0.748287i \(-0.269123\pi\)
0.663376 + 0.748287i \(0.269123\pi\)
\(60\) 0 0
\(61\) −2.62352 −0.335907 −0.167954 0.985795i \(-0.553716\pi\)
−0.167954 + 0.985795i \(0.553716\pi\)
\(62\) 0 0
\(63\) −1.20813 −0.152210
\(64\) 0 0
\(65\) 0.221575 0.0274830
\(66\) 0 0
\(67\) −1.00000 −0.122169
\(68\) 0 0
\(69\) 2.67771 0.322359
\(70\) 0 0
\(71\) 6.47669 0.768642 0.384321 0.923200i \(-0.374436\pi\)
0.384321 + 0.923200i \(0.374436\pi\)
\(72\) 0 0
\(73\) −6.83165 −0.799584 −0.399792 0.916606i \(-0.630918\pi\)
−0.399792 + 0.916606i \(0.630918\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −6.36510 −0.725370
\(78\) 0 0
\(79\) 16.6612 1.87453 0.937265 0.348618i \(-0.113349\pi\)
0.937265 + 0.348618i \(0.113349\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.2039 −1.12002 −0.560009 0.828486i \(-0.689202\pi\)
−0.560009 + 0.828486i \(0.689202\pi\)
\(84\) 0 0
\(85\) −4.28618 −0.464901
\(86\) 0 0
\(87\) 6.52207 0.699239
\(88\) 0 0
\(89\) −4.85648 −0.514785 −0.257393 0.966307i \(-0.582863\pi\)
−0.257393 + 0.966307i \(0.582863\pi\)
\(90\) 0 0
\(91\) −0.267691 −0.0280617
\(92\) 0 0
\(93\) −2.38432 −0.247243
\(94\) 0 0
\(95\) −5.28201 −0.541922
\(96\) 0 0
\(97\) 13.8518 1.40644 0.703218 0.710974i \(-0.251746\pi\)
0.703218 + 0.710974i \(0.251746\pi\)
\(98\) 0 0
\(99\) 5.26856 0.529510
\(100\) 0 0
\(101\) −3.48797 −0.347066 −0.173533 0.984828i \(-0.555518\pi\)
−0.173533 + 0.984828i \(0.555518\pi\)
\(102\) 0 0
\(103\) −2.08149 −0.205095 −0.102548 0.994728i \(-0.532699\pi\)
−0.102548 + 0.994728i \(0.532699\pi\)
\(104\) 0 0
\(105\) 1.20813 0.117901
\(106\) 0 0
\(107\) −6.20298 −0.599665 −0.299832 0.953992i \(-0.596931\pi\)
−0.299832 + 0.953992i \(0.596931\pi\)
\(108\) 0 0
\(109\) 1.50055 0.143726 0.0718631 0.997415i \(-0.477106\pi\)
0.0718631 + 0.997415i \(0.477106\pi\)
\(110\) 0 0
\(111\) −8.84802 −0.839817
\(112\) 0 0
\(113\) −7.71515 −0.725780 −0.362890 0.931832i \(-0.618210\pi\)
−0.362890 + 0.931832i \(0.618210\pi\)
\(114\) 0 0
\(115\) −2.67771 −0.249698
\(116\) 0 0
\(117\) 0.221575 0.0204846
\(118\) 0 0
\(119\) 5.17826 0.474690
\(120\) 0 0
\(121\) 16.7577 1.52343
\(122\) 0 0
\(123\) 7.02055 0.633022
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −18.2425 −1.61876 −0.809380 0.587286i \(-0.800196\pi\)
−0.809380 + 0.587286i \(0.800196\pi\)
\(128\) 0 0
\(129\) 9.44939 0.831972
\(130\) 0 0
\(131\) 7.94163 0.693864 0.346932 0.937890i \(-0.387224\pi\)
0.346932 + 0.937890i \(0.387224\pi\)
\(132\) 0 0
\(133\) 6.38134 0.553333
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −14.4964 −1.23851 −0.619254 0.785190i \(-0.712565\pi\)
−0.619254 + 0.785190i \(0.712565\pi\)
\(138\) 0 0
\(139\) −12.8983 −1.09402 −0.547010 0.837126i \(-0.684234\pi\)
−0.547010 + 0.837126i \(0.684234\pi\)
\(140\) 0 0
\(141\) −1.74782 −0.147193
\(142\) 0 0
\(143\) 1.16738 0.0976213
\(144\) 0 0
\(145\) −6.52207 −0.541628
\(146\) 0 0
\(147\) 5.54042 0.456967
\(148\) 0 0
\(149\) 2.64431 0.216630 0.108315 0.994117i \(-0.465454\pi\)
0.108315 + 0.994117i \(0.465454\pi\)
\(150\) 0 0
\(151\) 1.00344 0.0816587 0.0408293 0.999166i \(-0.487000\pi\)
0.0408293 + 0.999166i \(0.487000\pi\)
\(152\) 0 0
\(153\) −4.28618 −0.346517
\(154\) 0 0
\(155\) 2.38432 0.191513
\(156\) 0 0
\(157\) −10.0151 −0.799288 −0.399644 0.916670i \(-0.630866\pi\)
−0.399644 + 0.916670i \(0.630866\pi\)
\(158\) 0 0
\(159\) 5.05419 0.400824
\(160\) 0 0
\(161\) 3.23502 0.254955
\(162\) 0 0
\(163\) −13.5123 −1.05836 −0.529182 0.848508i \(-0.677501\pi\)
−0.529182 + 0.848508i \(0.677501\pi\)
\(164\) 0 0
\(165\) −5.26856 −0.410157
\(166\) 0 0
\(167\) 3.12997 0.242205 0.121102 0.992640i \(-0.461357\pi\)
0.121102 + 0.992640i \(0.461357\pi\)
\(168\) 0 0
\(169\) −12.9509 −0.996223
\(170\) 0 0
\(171\) −5.28201 −0.403925
\(172\) 0 0
\(173\) −14.7522 −1.12159 −0.560795 0.827954i \(-0.689505\pi\)
−0.560795 + 0.827954i \(0.689505\pi\)
\(174\) 0 0
\(175\) −1.20813 −0.0913260
\(176\) 0 0
\(177\) −10.1910 −0.766000
\(178\) 0 0
\(179\) 12.2500 0.915606 0.457803 0.889054i \(-0.348637\pi\)
0.457803 + 0.889054i \(0.348637\pi\)
\(180\) 0 0
\(181\) −12.9768 −0.964556 −0.482278 0.876018i \(-0.660191\pi\)
−0.482278 + 0.876018i \(0.660191\pi\)
\(182\) 0 0
\(183\) 2.62352 0.193936
\(184\) 0 0
\(185\) 8.84802 0.650520
\(186\) 0 0
\(187\) −22.5820 −1.65136
\(188\) 0 0
\(189\) 1.20813 0.0878784
\(190\) 0 0
\(191\) 12.6718 0.916901 0.458451 0.888720i \(-0.348405\pi\)
0.458451 + 0.888720i \(0.348405\pi\)
\(192\) 0 0
\(193\) −20.9701 −1.50946 −0.754731 0.656034i \(-0.772233\pi\)
−0.754731 + 0.656034i \(0.772233\pi\)
\(194\) 0 0
\(195\) −0.221575 −0.0158673
\(196\) 0 0
\(197\) −0.0298715 −0.00212826 −0.00106413 0.999999i \(-0.500339\pi\)
−0.00106413 + 0.999999i \(0.500339\pi\)
\(198\) 0 0
\(199\) −8.79971 −0.623795 −0.311898 0.950116i \(-0.600965\pi\)
−0.311898 + 0.950116i \(0.600965\pi\)
\(200\) 0 0
\(201\) 1.00000 0.0705346
\(202\) 0 0
\(203\) 7.87950 0.553033
\(204\) 0 0
\(205\) −7.02055 −0.490337
\(206\) 0 0
\(207\) −2.67771 −0.186114
\(208\) 0 0
\(209\) −27.8286 −1.92494
\(210\) 0 0
\(211\) −15.3630 −1.05763 −0.528816 0.848737i \(-0.677364\pi\)
−0.528816 + 0.848737i \(0.677364\pi\)
\(212\) 0 0
\(213\) −6.47669 −0.443775
\(214\) 0 0
\(215\) −9.44939 −0.644443
\(216\) 0 0
\(217\) −2.88057 −0.195546
\(218\) 0 0
\(219\) 6.83165 0.461640
\(220\) 0 0
\(221\) −0.949710 −0.0638844
\(222\) 0 0
\(223\) 16.4518 1.10169 0.550847 0.834606i \(-0.314305\pi\)
0.550847 + 0.834606i \(0.314305\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −4.69592 −0.311679 −0.155840 0.987782i \(-0.549808\pi\)
−0.155840 + 0.987782i \(0.549808\pi\)
\(228\) 0 0
\(229\) −19.8217 −1.30985 −0.654927 0.755692i \(-0.727301\pi\)
−0.654927 + 0.755692i \(0.727301\pi\)
\(230\) 0 0
\(231\) 6.36510 0.418793
\(232\) 0 0
\(233\) 8.11745 0.531792 0.265896 0.964002i \(-0.414332\pi\)
0.265896 + 0.964002i \(0.414332\pi\)
\(234\) 0 0
\(235\) 1.74782 0.114015
\(236\) 0 0
\(237\) −16.6612 −1.08226
\(238\) 0 0
\(239\) −2.19458 −0.141956 −0.0709778 0.997478i \(-0.522612\pi\)
−0.0709778 + 0.997478i \(0.522612\pi\)
\(240\) 0 0
\(241\) −5.89615 −0.379804 −0.189902 0.981803i \(-0.560817\pi\)
−0.189902 + 0.981803i \(0.560817\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −5.54042 −0.353965
\(246\) 0 0
\(247\) −1.17036 −0.0744682
\(248\) 0 0
\(249\) 10.2039 0.646643
\(250\) 0 0
\(251\) 20.5661 1.29812 0.649060 0.760738i \(-0.275163\pi\)
0.649060 + 0.760738i \(0.275163\pi\)
\(252\) 0 0
\(253\) −14.1077 −0.886943
\(254\) 0 0
\(255\) 4.28618 0.268411
\(256\) 0 0
\(257\) 22.0885 1.37784 0.688920 0.724838i \(-0.258085\pi\)
0.688920 + 0.724838i \(0.258085\pi\)
\(258\) 0 0
\(259\) −10.6896 −0.664217
\(260\) 0 0
\(261\) −6.52207 −0.403706
\(262\) 0 0
\(263\) −4.52774 −0.279193 −0.139596 0.990208i \(-0.544580\pi\)
−0.139596 + 0.990208i \(0.544580\pi\)
\(264\) 0 0
\(265\) −5.05419 −0.310477
\(266\) 0 0
\(267\) 4.85648 0.297211
\(268\) 0 0
\(269\) −14.7766 −0.900942 −0.450471 0.892791i \(-0.648744\pi\)
−0.450471 + 0.892791i \(0.648744\pi\)
\(270\) 0 0
\(271\) −3.43852 −0.208875 −0.104438 0.994531i \(-0.533304\pi\)
−0.104438 + 0.994531i \(0.533304\pi\)
\(272\) 0 0
\(273\) 0.267691 0.0162014
\(274\) 0 0
\(275\) 5.26856 0.317706
\(276\) 0 0
\(277\) 3.39446 0.203954 0.101977 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(278\) 0 0
\(279\) 2.38432 0.142746
\(280\) 0 0
\(281\) 25.3562 1.51263 0.756313 0.654210i \(-0.226999\pi\)
0.756313 + 0.654210i \(0.226999\pi\)
\(282\) 0 0
\(283\) 16.2710 0.967209 0.483605 0.875287i \(-0.339327\pi\)
0.483605 + 0.875287i \(0.339327\pi\)
\(284\) 0 0
\(285\) 5.28201 0.312879
\(286\) 0 0
\(287\) 8.48173 0.500661
\(288\) 0 0
\(289\) 1.37134 0.0806668
\(290\) 0 0
\(291\) −13.8518 −0.812007
\(292\) 0 0
\(293\) −10.5037 −0.613634 −0.306817 0.951769i \(-0.599264\pi\)
−0.306817 + 0.951769i \(0.599264\pi\)
\(294\) 0 0
\(295\) 10.1910 0.593341
\(296\) 0 0
\(297\) −5.26856 −0.305713
\(298\) 0 0
\(299\) −0.593314 −0.0343122
\(300\) 0 0
\(301\) 11.4161 0.658012
\(302\) 0 0
\(303\) 3.48797 0.200379
\(304\) 0 0
\(305\) −2.62352 −0.150222
\(306\) 0 0
\(307\) 6.75566 0.385566 0.192783 0.981241i \(-0.438249\pi\)
0.192783 + 0.981241i \(0.438249\pi\)
\(308\) 0 0
\(309\) 2.08149 0.118412
\(310\) 0 0
\(311\) −27.7675 −1.57455 −0.787274 0.616603i \(-0.788508\pi\)
−0.787274 + 0.616603i \(0.788508\pi\)
\(312\) 0 0
\(313\) −1.69703 −0.0959220 −0.0479610 0.998849i \(-0.515272\pi\)
−0.0479610 + 0.998849i \(0.515272\pi\)
\(314\) 0 0
\(315\) −1.20813 −0.0680703
\(316\) 0 0
\(317\) −15.6249 −0.877579 −0.438790 0.898590i \(-0.644593\pi\)
−0.438790 + 0.898590i \(0.644593\pi\)
\(318\) 0 0
\(319\) −34.3619 −1.92390
\(320\) 0 0
\(321\) 6.20298 0.346217
\(322\) 0 0
\(323\) 22.6396 1.25970
\(324\) 0 0
\(325\) 0.221575 0.0122908
\(326\) 0 0
\(327\) −1.50055 −0.0829804
\(328\) 0 0
\(329\) −2.11159 −0.116416
\(330\) 0 0
\(331\) 21.8890 1.20313 0.601564 0.798824i \(-0.294544\pi\)
0.601564 + 0.798824i \(0.294544\pi\)
\(332\) 0 0
\(333\) 8.84802 0.484869
\(334\) 0 0
\(335\) −1.00000 −0.0546358
\(336\) 0 0
\(337\) −20.5155 −1.11755 −0.558775 0.829319i \(-0.688728\pi\)
−0.558775 + 0.829319i \(0.688728\pi\)
\(338\) 0 0
\(339\) 7.71515 0.419029
\(340\) 0 0
\(341\) 12.5619 0.680268
\(342\) 0 0
\(343\) 15.1504 0.818047
\(344\) 0 0
\(345\) 2.67771 0.144163
\(346\) 0 0
\(347\) 5.81881 0.312370 0.156185 0.987728i \(-0.450080\pi\)
0.156185 + 0.987728i \(0.450080\pi\)
\(348\) 0 0
\(349\) −31.1985 −1.67002 −0.835010 0.550235i \(-0.814538\pi\)
−0.835010 + 0.550235i \(0.814538\pi\)
\(350\) 0 0
\(351\) −0.221575 −0.0118268
\(352\) 0 0
\(353\) 10.0053 0.532527 0.266263 0.963900i \(-0.414211\pi\)
0.266263 + 0.963900i \(0.414211\pi\)
\(354\) 0 0
\(355\) 6.47669 0.343747
\(356\) 0 0
\(357\) −5.17826 −0.274062
\(358\) 0 0
\(359\) −2.08282 −0.109927 −0.0549634 0.998488i \(-0.517504\pi\)
−0.0549634 + 0.998488i \(0.517504\pi\)
\(360\) 0 0
\(361\) 8.89959 0.468399
\(362\) 0 0
\(363\) −16.7577 −0.879552
\(364\) 0 0
\(365\) −6.83165 −0.357585
\(366\) 0 0
\(367\) 5.23892 0.273469 0.136735 0.990608i \(-0.456339\pi\)
0.136735 + 0.990608i \(0.456339\pi\)
\(368\) 0 0
\(369\) −7.02055 −0.365475
\(370\) 0 0
\(371\) 6.10612 0.317014
\(372\) 0 0
\(373\) 1.55057 0.0802852 0.0401426 0.999194i \(-0.487219\pi\)
0.0401426 + 0.999194i \(0.487219\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −1.44513 −0.0744278
\(378\) 0 0
\(379\) 19.2300 0.987778 0.493889 0.869525i \(-0.335575\pi\)
0.493889 + 0.869525i \(0.335575\pi\)
\(380\) 0 0
\(381\) 18.2425 0.934591
\(382\) 0 0
\(383\) −30.7083 −1.56912 −0.784559 0.620054i \(-0.787111\pi\)
−0.784559 + 0.620054i \(0.787111\pi\)
\(384\) 0 0
\(385\) −6.36510 −0.324395
\(386\) 0 0
\(387\) −9.44939 −0.480339
\(388\) 0 0
\(389\) 31.0267 1.57312 0.786559 0.617515i \(-0.211861\pi\)
0.786559 + 0.617515i \(0.211861\pi\)
\(390\) 0 0
\(391\) 11.4772 0.580425
\(392\) 0 0
\(393\) −7.94163 −0.400602
\(394\) 0 0
\(395\) 16.6612 0.838315
\(396\) 0 0
\(397\) −22.0597 −1.10714 −0.553572 0.832801i \(-0.686736\pi\)
−0.553572 + 0.832801i \(0.686736\pi\)
\(398\) 0 0
\(399\) −6.38134 −0.319467
\(400\) 0 0
\(401\) 6.53299 0.326242 0.163121 0.986606i \(-0.447844\pi\)
0.163121 + 0.986606i \(0.447844\pi\)
\(402\) 0 0
\(403\) 0.528306 0.0263168
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 46.6163 2.31069
\(408\) 0 0
\(409\) −13.9387 −0.689222 −0.344611 0.938746i \(-0.611989\pi\)
−0.344611 + 0.938746i \(0.611989\pi\)
\(410\) 0 0
\(411\) 14.4964 0.715053
\(412\) 0 0
\(413\) −12.3120 −0.605834
\(414\) 0 0
\(415\) −10.2039 −0.500887
\(416\) 0 0
\(417\) 12.8983 0.631633
\(418\) 0 0
\(419\) 13.0939 0.639677 0.319839 0.947472i \(-0.396371\pi\)
0.319839 + 0.947472i \(0.396371\pi\)
\(420\) 0 0
\(421\) −3.70216 −0.180432 −0.0902161 0.995922i \(-0.528756\pi\)
−0.0902161 + 0.995922i \(0.528756\pi\)
\(422\) 0 0
\(423\) 1.74782 0.0849818
\(424\) 0 0
\(425\) −4.28618 −0.207910
\(426\) 0 0
\(427\) 3.16955 0.153385
\(428\) 0 0
\(429\) −1.16738 −0.0563617
\(430\) 0 0
\(431\) −11.8212 −0.569407 −0.284703 0.958616i \(-0.591895\pi\)
−0.284703 + 0.958616i \(0.591895\pi\)
\(432\) 0 0
\(433\) −30.2560 −1.45401 −0.727006 0.686631i \(-0.759089\pi\)
−0.727006 + 0.686631i \(0.759089\pi\)
\(434\) 0 0
\(435\) 6.52207 0.312709
\(436\) 0 0
\(437\) 14.1437 0.676584
\(438\) 0 0
\(439\) 26.7871 1.27848 0.639240 0.769007i \(-0.279249\pi\)
0.639240 + 0.769007i \(0.279249\pi\)
\(440\) 0 0
\(441\) −5.54042 −0.263830
\(442\) 0 0
\(443\) 22.3894 1.06375 0.531876 0.846822i \(-0.321487\pi\)
0.531876 + 0.846822i \(0.321487\pi\)
\(444\) 0 0
\(445\) −4.85648 −0.230219
\(446\) 0 0
\(447\) −2.64431 −0.125071
\(448\) 0 0
\(449\) 15.7716 0.744309 0.372155 0.928171i \(-0.378619\pi\)
0.372155 + 0.928171i \(0.378619\pi\)
\(450\) 0 0
\(451\) −36.9882 −1.74171
\(452\) 0 0
\(453\) −1.00344 −0.0471456
\(454\) 0 0
\(455\) −0.267691 −0.0125496
\(456\) 0 0
\(457\) 26.3749 1.23377 0.616883 0.787055i \(-0.288395\pi\)
0.616883 + 0.787055i \(0.288395\pi\)
\(458\) 0 0
\(459\) 4.28618 0.200062
\(460\) 0 0
\(461\) −37.2864 −1.73660 −0.868300 0.496039i \(-0.834787\pi\)
−0.868300 + 0.496039i \(0.834787\pi\)
\(462\) 0 0
\(463\) −30.1519 −1.40128 −0.700639 0.713516i \(-0.747102\pi\)
−0.700639 + 0.713516i \(0.747102\pi\)
\(464\) 0 0
\(465\) −2.38432 −0.110570
\(466\) 0 0
\(467\) 1.98512 0.0918605 0.0459303 0.998945i \(-0.485375\pi\)
0.0459303 + 0.998945i \(0.485375\pi\)
\(468\) 0 0
\(469\) 1.20813 0.0557862
\(470\) 0 0
\(471\) 10.0151 0.461469
\(472\) 0 0
\(473\) −49.7847 −2.28910
\(474\) 0 0
\(475\) −5.28201 −0.242355
\(476\) 0 0
\(477\) −5.05419 −0.231416
\(478\) 0 0
\(479\) −35.2629 −1.61120 −0.805602 0.592457i \(-0.798158\pi\)
−0.805602 + 0.592457i \(0.798158\pi\)
\(480\) 0 0
\(481\) 1.96050 0.0893911
\(482\) 0 0
\(483\) −3.23502 −0.147199
\(484\) 0 0
\(485\) 13.8518 0.628978
\(486\) 0 0
\(487\) −16.7469 −0.758875 −0.379437 0.925217i \(-0.623882\pi\)
−0.379437 + 0.925217i \(0.623882\pi\)
\(488\) 0 0
\(489\) 13.5123 0.611047
\(490\) 0 0
\(491\) 24.1645 1.09053 0.545265 0.838264i \(-0.316429\pi\)
0.545265 + 0.838264i \(0.316429\pi\)
\(492\) 0 0
\(493\) 27.9548 1.25902
\(494\) 0 0
\(495\) 5.26856 0.236804
\(496\) 0 0
\(497\) −7.82467 −0.350985
\(498\) 0 0
\(499\) 9.75145 0.436535 0.218267 0.975889i \(-0.429960\pi\)
0.218267 + 0.975889i \(0.429960\pi\)
\(500\) 0 0
\(501\) −3.12997 −0.139837
\(502\) 0 0
\(503\) −4.65830 −0.207703 −0.103852 0.994593i \(-0.533117\pi\)
−0.103852 + 0.994593i \(0.533117\pi\)
\(504\) 0 0
\(505\) −3.48797 −0.155213
\(506\) 0 0
\(507\) 12.9509 0.575170
\(508\) 0 0
\(509\) 5.20120 0.230539 0.115269 0.993334i \(-0.463227\pi\)
0.115269 + 0.993334i \(0.463227\pi\)
\(510\) 0 0
\(511\) 8.25351 0.365114
\(512\) 0 0
\(513\) 5.28201 0.233206
\(514\) 0 0
\(515\) −2.08149 −0.0917214
\(516\) 0 0
\(517\) 9.20848 0.404989
\(518\) 0 0
\(519\) 14.7522 0.647551
\(520\) 0 0
\(521\) −15.7719 −0.690980 −0.345490 0.938422i \(-0.612287\pi\)
−0.345490 + 0.938422i \(0.612287\pi\)
\(522\) 0 0
\(523\) −40.4599 −1.76919 −0.884595 0.466360i \(-0.845565\pi\)
−0.884595 + 0.466360i \(0.845565\pi\)
\(524\) 0 0
\(525\) 1.20813 0.0527271
\(526\) 0 0
\(527\) −10.2196 −0.445174
\(528\) 0 0
\(529\) −15.8299 −0.688255
\(530\) 0 0
\(531\) 10.1910 0.442250
\(532\) 0 0
\(533\) −1.55558 −0.0673796
\(534\) 0 0
\(535\) −6.20298 −0.268178
\(536\) 0 0
\(537\) −12.2500 −0.528625
\(538\) 0 0
\(539\) −29.1901 −1.25730
\(540\) 0 0
\(541\) −19.2953 −0.829571 −0.414785 0.909919i \(-0.636143\pi\)
−0.414785 + 0.909919i \(0.636143\pi\)
\(542\) 0 0
\(543\) 12.9768 0.556887
\(544\) 0 0
\(545\) 1.50055 0.0642763
\(546\) 0 0
\(547\) −29.6679 −1.26851 −0.634254 0.773125i \(-0.718693\pi\)
−0.634254 + 0.773125i \(0.718693\pi\)
\(548\) 0 0
\(549\) −2.62352 −0.111969
\(550\) 0 0
\(551\) 34.4496 1.46760
\(552\) 0 0
\(553\) −20.1289 −0.855966
\(554\) 0 0
\(555\) −8.84802 −0.375578
\(556\) 0 0
\(557\) 2.85195 0.120841 0.0604204 0.998173i \(-0.480756\pi\)
0.0604204 + 0.998173i \(0.480756\pi\)
\(558\) 0 0
\(559\) −2.09375 −0.0885561
\(560\) 0 0
\(561\) 22.5820 0.953413
\(562\) 0 0
\(563\) 6.76040 0.284917 0.142458 0.989801i \(-0.454499\pi\)
0.142458 + 0.989801i \(0.454499\pi\)
\(564\) 0 0
\(565\) −7.71515 −0.324579
\(566\) 0 0
\(567\) −1.20813 −0.0507366
\(568\) 0 0
\(569\) −22.0963 −0.926326 −0.463163 0.886273i \(-0.653286\pi\)
−0.463163 + 0.886273i \(0.653286\pi\)
\(570\) 0 0
\(571\) −28.2440 −1.18197 −0.590987 0.806681i \(-0.701262\pi\)
−0.590987 + 0.806681i \(0.701262\pi\)
\(572\) 0 0
\(573\) −12.6718 −0.529373
\(574\) 0 0
\(575\) −2.67771 −0.111668
\(576\) 0 0
\(577\) 7.97207 0.331882 0.165941 0.986136i \(-0.446934\pi\)
0.165941 + 0.986136i \(0.446934\pi\)
\(578\) 0 0
\(579\) 20.9701 0.871489
\(580\) 0 0
\(581\) 12.3276 0.511434
\(582\) 0 0
\(583\) −26.6283 −1.10283
\(584\) 0 0
\(585\) 0.221575 0.00916100
\(586\) 0 0
\(587\) 9.72439 0.401369 0.200684 0.979656i \(-0.435683\pi\)
0.200684 + 0.979656i \(0.435683\pi\)
\(588\) 0 0
\(589\) −12.5940 −0.518927
\(590\) 0 0
\(591\) 0.0298715 0.00122875
\(592\) 0 0
\(593\) 24.1802 0.992961 0.496480 0.868048i \(-0.334626\pi\)
0.496480 + 0.868048i \(0.334626\pi\)
\(594\) 0 0
\(595\) 5.17826 0.212288
\(596\) 0 0
\(597\) 8.79971 0.360148
\(598\) 0 0
\(599\) −6.26002 −0.255778 −0.127889 0.991789i \(-0.540820\pi\)
−0.127889 + 0.991789i \(0.540820\pi\)
\(600\) 0 0
\(601\) −2.78039 −0.113415 −0.0567073 0.998391i \(-0.518060\pi\)
−0.0567073 + 0.998391i \(0.518060\pi\)
\(602\) 0 0
\(603\) −1.00000 −0.0407231
\(604\) 0 0
\(605\) 16.7577 0.681298
\(606\) 0 0
\(607\) 12.3197 0.500040 0.250020 0.968241i \(-0.419563\pi\)
0.250020 + 0.968241i \(0.419563\pi\)
\(608\) 0 0
\(609\) −7.87950 −0.319293
\(610\) 0 0
\(611\) 0.387273 0.0156674
\(612\) 0 0
\(613\) 11.3931 0.460164 0.230082 0.973171i \(-0.426101\pi\)
0.230082 + 0.973171i \(0.426101\pi\)
\(614\) 0 0
\(615\) 7.02055 0.283096
\(616\) 0 0
\(617\) −11.9748 −0.482086 −0.241043 0.970514i \(-0.577489\pi\)
−0.241043 + 0.970514i \(0.577489\pi\)
\(618\) 0 0
\(619\) 8.25911 0.331962 0.165981 0.986129i \(-0.446921\pi\)
0.165981 + 0.986129i \(0.446921\pi\)
\(620\) 0 0
\(621\) 2.67771 0.107453
\(622\) 0 0
\(623\) 5.86725 0.235066
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 27.8286 1.11137
\(628\) 0 0
\(629\) −37.9242 −1.51214
\(630\) 0 0
\(631\) 47.5428 1.89265 0.946325 0.323218i \(-0.104765\pi\)
0.946325 + 0.323218i \(0.104765\pi\)
\(632\) 0 0
\(633\) 15.3630 0.610624
\(634\) 0 0
\(635\) −18.2425 −0.723931
\(636\) 0 0
\(637\) −1.22762 −0.0486401
\(638\) 0 0
\(639\) 6.47669 0.256214
\(640\) 0 0
\(641\) 29.4053 1.16144 0.580720 0.814103i \(-0.302771\pi\)
0.580720 + 0.814103i \(0.302771\pi\)
\(642\) 0 0
\(643\) 17.8999 0.705903 0.352952 0.935642i \(-0.385178\pi\)
0.352952 + 0.935642i \(0.385178\pi\)
\(644\) 0 0
\(645\) 9.44939 0.372069
\(646\) 0 0
\(647\) 34.8635 1.37063 0.685313 0.728248i \(-0.259665\pi\)
0.685313 + 0.728248i \(0.259665\pi\)
\(648\) 0 0
\(649\) 53.6917 2.10758
\(650\) 0 0
\(651\) 2.88057 0.112898
\(652\) 0 0
\(653\) 35.7869 1.40045 0.700225 0.713922i \(-0.253083\pi\)
0.700225 + 0.713922i \(0.253083\pi\)
\(654\) 0 0
\(655\) 7.94163 0.310305
\(656\) 0 0
\(657\) −6.83165 −0.266528
\(658\) 0 0
\(659\) 38.7234 1.50845 0.754225 0.656616i \(-0.228013\pi\)
0.754225 + 0.656616i \(0.228013\pi\)
\(660\) 0 0
\(661\) 3.65942 0.142335 0.0711675 0.997464i \(-0.477328\pi\)
0.0711675 + 0.997464i \(0.477328\pi\)
\(662\) 0 0
\(663\) 0.949710 0.0368837
\(664\) 0 0
\(665\) 6.38134 0.247458
\(666\) 0 0
\(667\) 17.4642 0.676217
\(668\) 0 0
\(669\) −16.4518 −0.636064
\(670\) 0 0
\(671\) −13.8222 −0.533599
\(672\) 0 0
\(673\) 38.6162 1.48855 0.744273 0.667875i \(-0.232796\pi\)
0.744273 + 0.667875i \(0.232796\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 40.1414 1.54276 0.771379 0.636376i \(-0.219567\pi\)
0.771379 + 0.636376i \(0.219567\pi\)
\(678\) 0 0
\(679\) −16.7348 −0.642221
\(680\) 0 0
\(681\) 4.69592 0.179948
\(682\) 0 0
\(683\) −12.7462 −0.487720 −0.243860 0.969810i \(-0.578414\pi\)
−0.243860 + 0.969810i \(0.578414\pi\)
\(684\) 0 0
\(685\) −14.4964 −0.553878
\(686\) 0 0
\(687\) 19.8217 0.756245
\(688\) 0 0
\(689\) −1.11988 −0.0426641
\(690\) 0 0
\(691\) −10.8852 −0.414094 −0.207047 0.978331i \(-0.566385\pi\)
−0.207047 + 0.978331i \(0.566385\pi\)
\(692\) 0 0
\(693\) −6.36510 −0.241790
\(694\) 0 0
\(695\) −12.8983 −0.489261
\(696\) 0 0
\(697\) 30.0913 1.13979
\(698\) 0 0
\(699\) −8.11745 −0.307030
\(700\) 0 0
\(701\) 27.2030 1.02744 0.513721 0.857957i \(-0.328267\pi\)
0.513721 + 0.857957i \(0.328267\pi\)
\(702\) 0 0
\(703\) −46.7353 −1.76266
\(704\) 0 0
\(705\) −1.74782 −0.0658266
\(706\) 0 0
\(707\) 4.21391 0.158481
\(708\) 0 0
\(709\) 41.2293 1.54840 0.774200 0.632941i \(-0.218152\pi\)
0.774200 + 0.632941i \(0.218152\pi\)
\(710\) 0 0
\(711\) 16.6612 0.624843
\(712\) 0 0
\(713\) −6.38453 −0.239103
\(714\) 0 0
\(715\) 1.16738 0.0436576
\(716\) 0 0
\(717\) 2.19458 0.0819581
\(718\) 0 0
\(719\) −28.7367 −1.07170 −0.535848 0.844314i \(-0.680008\pi\)
−0.535848 + 0.844314i \(0.680008\pi\)
\(720\) 0 0
\(721\) 2.51471 0.0936526
\(722\) 0 0
\(723\) 5.89615 0.219280
\(724\) 0 0
\(725\) −6.52207 −0.242224
\(726\) 0 0
\(727\) −16.2623 −0.603136 −0.301568 0.953445i \(-0.597510\pi\)
−0.301568 + 0.953445i \(0.597510\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 40.5018 1.49801
\(732\) 0 0
\(733\) −46.5411 −1.71904 −0.859518 0.511105i \(-0.829236\pi\)
−0.859518 + 0.511105i \(0.829236\pi\)
\(734\) 0 0
\(735\) 5.54042 0.204362
\(736\) 0 0
\(737\) −5.26856 −0.194070
\(738\) 0 0
\(739\) 36.1832 1.33102 0.665511 0.746388i \(-0.268214\pi\)
0.665511 + 0.746388i \(0.268214\pi\)
\(740\) 0 0
\(741\) 1.17036 0.0429943
\(742\) 0 0
\(743\) 31.3825 1.15131 0.575657 0.817691i \(-0.304746\pi\)
0.575657 + 0.817691i \(0.304746\pi\)
\(744\) 0 0
\(745\) 2.64431 0.0968799
\(746\) 0 0
\(747\) −10.2039 −0.373339
\(748\) 0 0
\(749\) 7.49400 0.273825
\(750\) 0 0
\(751\) 17.0310 0.621472 0.310736 0.950496i \(-0.399425\pi\)
0.310736 + 0.950496i \(0.399425\pi\)
\(752\) 0 0
\(753\) −20.5661 −0.749469
\(754\) 0 0
\(755\) 1.00344 0.0365189
\(756\) 0 0
\(757\) −20.0230 −0.727749 −0.363875 0.931448i \(-0.618546\pi\)
−0.363875 + 0.931448i \(0.618546\pi\)
\(758\) 0 0
\(759\) 14.1077 0.512077
\(760\) 0 0
\(761\) 26.0826 0.945493 0.472747 0.881198i \(-0.343263\pi\)
0.472747 + 0.881198i \(0.343263\pi\)
\(762\) 0 0
\(763\) −1.81285 −0.0656297
\(764\) 0 0
\(765\) −4.28618 −0.154967
\(766\) 0 0
\(767\) 2.25806 0.0815339
\(768\) 0 0
\(769\) 18.8245 0.678829 0.339414 0.940637i \(-0.389771\pi\)
0.339414 + 0.940637i \(0.389771\pi\)
\(770\) 0 0
\(771\) −22.0885 −0.795496
\(772\) 0 0
\(773\) 25.5855 0.920248 0.460124 0.887855i \(-0.347805\pi\)
0.460124 + 0.887855i \(0.347805\pi\)
\(774\) 0 0
\(775\) 2.38432 0.0856474
\(776\) 0 0
\(777\) 10.6896 0.383486
\(778\) 0 0
\(779\) 37.0826 1.32862
\(780\) 0 0
\(781\) 34.1228 1.22101
\(782\) 0 0
\(783\) 6.52207 0.233080
\(784\) 0 0
\(785\) −10.0151 −0.357452
\(786\) 0 0
\(787\) −8.06377 −0.287442 −0.143721 0.989618i \(-0.545907\pi\)
−0.143721 + 0.989618i \(0.545907\pi\)
\(788\) 0 0
\(789\) 4.52774 0.161192
\(790\) 0 0
\(791\) 9.32089 0.331413
\(792\) 0 0
\(793\) −0.581306 −0.0206428
\(794\) 0 0
\(795\) 5.05419 0.179254
\(796\) 0 0
\(797\) 25.8730 0.916468 0.458234 0.888832i \(-0.348482\pi\)
0.458234 + 0.888832i \(0.348482\pi\)
\(798\) 0 0
\(799\) −7.49146 −0.265029
\(800\) 0 0
\(801\) −4.85648 −0.171595
\(802\) 0 0
\(803\) −35.9929 −1.27016
\(804\) 0 0
\(805\) 3.23502 0.114020
\(806\) 0 0
\(807\) 14.7766 0.520159
\(808\) 0 0
\(809\) −16.2083 −0.569854 −0.284927 0.958549i \(-0.591969\pi\)
−0.284927 + 0.958549i \(0.591969\pi\)
\(810\) 0 0
\(811\) 40.9205 1.43691 0.718457 0.695571i \(-0.244849\pi\)
0.718457 + 0.695571i \(0.244849\pi\)
\(812\) 0 0
\(813\) 3.43852 0.120594
\(814\) 0 0
\(815\) −13.5123 −0.473315
\(816\) 0 0
\(817\) 49.9117 1.74619
\(818\) 0 0
\(819\) −0.267691 −0.00935388
\(820\) 0 0
\(821\) 8.74373 0.305158 0.152579 0.988291i \(-0.451242\pi\)
0.152579 + 0.988291i \(0.451242\pi\)
\(822\) 0 0
\(823\) 4.33914 0.151253 0.0756265 0.997136i \(-0.475904\pi\)
0.0756265 + 0.997136i \(0.475904\pi\)
\(824\) 0 0
\(825\) −5.26856 −0.183428
\(826\) 0 0
\(827\) −13.8600 −0.481959 −0.240979 0.970530i \(-0.577469\pi\)
−0.240979 + 0.970530i \(0.577469\pi\)
\(828\) 0 0
\(829\) 34.5706 1.20069 0.600343 0.799742i \(-0.295031\pi\)
0.600343 + 0.799742i \(0.295031\pi\)
\(830\) 0 0
\(831\) −3.39446 −0.117753
\(832\) 0 0
\(833\) 23.7473 0.822794
\(834\) 0 0
\(835\) 3.12997 0.108317
\(836\) 0 0
\(837\) −2.38432 −0.0824143
\(838\) 0 0
\(839\) −37.9355 −1.30968 −0.654840 0.755767i \(-0.727264\pi\)
−0.654840 + 0.755767i \(0.727264\pi\)
\(840\) 0 0
\(841\) 13.5374 0.466807
\(842\) 0 0
\(843\) −25.3562 −0.873315
\(844\) 0 0
\(845\) −12.9509 −0.445525
\(846\) 0 0
\(847\) −20.2455 −0.695643
\(848\) 0 0
\(849\) −16.2710 −0.558419
\(850\) 0 0
\(851\) −23.6925 −0.812167
\(852\) 0 0
\(853\) −42.2892 −1.44795 −0.723977 0.689824i \(-0.757688\pi\)
−0.723977 + 0.689824i \(0.757688\pi\)
\(854\) 0 0
\(855\) −5.28201 −0.180641
\(856\) 0 0
\(857\) −13.8774 −0.474044 −0.237022 0.971504i \(-0.576171\pi\)
−0.237022 + 0.971504i \(0.576171\pi\)
\(858\) 0 0
\(859\) 49.3256 1.68297 0.841483 0.540283i \(-0.181683\pi\)
0.841483 + 0.540283i \(0.181683\pi\)
\(860\) 0 0
\(861\) −8.48173 −0.289057
\(862\) 0 0
\(863\) 22.5871 0.768874 0.384437 0.923151i \(-0.374396\pi\)
0.384437 + 0.923151i \(0.374396\pi\)
\(864\) 0 0
\(865\) −14.7522 −0.501591
\(866\) 0 0
\(867\) −1.37134 −0.0465730
\(868\) 0 0
\(869\) 87.7804 2.97775
\(870\) 0 0
\(871\) −0.221575 −0.00750778
\(872\) 0 0
\(873\) 13.8518 0.468812
\(874\) 0 0
\(875\) −1.20813 −0.0408422
\(876\) 0 0
\(877\) −23.9314 −0.808105 −0.404053 0.914736i \(-0.632399\pi\)
−0.404053 + 0.914736i \(0.632399\pi\)
\(878\) 0 0
\(879\) 10.5037 0.354281
\(880\) 0 0
\(881\) 27.9624 0.942079 0.471039 0.882112i \(-0.343879\pi\)
0.471039 + 0.882112i \(0.343879\pi\)
\(882\) 0 0
\(883\) −4.42710 −0.148984 −0.0744920 0.997222i \(-0.523734\pi\)
−0.0744920 + 0.997222i \(0.523734\pi\)
\(884\) 0 0
\(885\) −10.1910 −0.342566
\(886\) 0 0
\(887\) 29.7480 0.998842 0.499421 0.866360i \(-0.333546\pi\)
0.499421 + 0.866360i \(0.333546\pi\)
\(888\) 0 0
\(889\) 22.0393 0.739174
\(890\) 0 0
\(891\) 5.26856 0.176503
\(892\) 0 0
\(893\) −9.23199 −0.308937
\(894\) 0 0
\(895\) 12.2500 0.409471
\(896\) 0 0
\(897\) 0.593314 0.0198102
\(898\) 0 0
\(899\) −15.5507 −0.518646
\(900\) 0 0
\(901\) 21.6632 0.721705
\(902\) 0 0
\(903\) −11.4161 −0.379903
\(904\) 0 0
\(905\) −12.9768 −0.431363
\(906\) 0 0
\(907\) 12.3392 0.409716 0.204858 0.978792i \(-0.434327\pi\)
0.204858 + 0.978792i \(0.434327\pi\)
\(908\) 0 0
\(909\) −3.48797 −0.115689
\(910\) 0 0
\(911\) −2.77581 −0.0919666 −0.0459833 0.998942i \(-0.514642\pi\)
−0.0459833 + 0.998942i \(0.514642\pi\)
\(912\) 0 0
\(913\) −53.7596 −1.77918
\(914\) 0 0
\(915\) 2.62352 0.0867308
\(916\) 0 0
\(917\) −9.59452 −0.316839
\(918\) 0 0
\(919\) −25.2264 −0.832141 −0.416070 0.909332i \(-0.636593\pi\)
−0.416070 + 0.909332i \(0.636593\pi\)
\(920\) 0 0
\(921\) −6.75566 −0.222607
\(922\) 0 0
\(923\) 1.43507 0.0472360
\(924\) 0 0
\(925\) 8.84802 0.290921
\(926\) 0 0
\(927\) −2.08149 −0.0683651
\(928\) 0 0
\(929\) −53.4132 −1.75243 −0.876216 0.481919i \(-0.839940\pi\)
−0.876216 + 0.481919i \(0.839940\pi\)
\(930\) 0 0
\(931\) 29.2646 0.959107
\(932\) 0 0
\(933\) 27.7675 0.909066
\(934\) 0 0
\(935\) −22.5820 −0.738510
\(936\) 0 0
\(937\) −1.30085 −0.0424969 −0.0212485 0.999774i \(-0.506764\pi\)
−0.0212485 + 0.999774i \(0.506764\pi\)
\(938\) 0 0
\(939\) 1.69703 0.0553806
\(940\) 0 0
\(941\) 0.872072 0.0284287 0.0142144 0.999899i \(-0.495475\pi\)
0.0142144 + 0.999899i \(0.495475\pi\)
\(942\) 0 0
\(943\) 18.7990 0.612180
\(944\) 0 0
\(945\) 1.20813 0.0393004
\(946\) 0 0
\(947\) −31.3246 −1.01791 −0.508957 0.860792i \(-0.669969\pi\)
−0.508957 + 0.860792i \(0.669969\pi\)
\(948\) 0 0
\(949\) −1.51372 −0.0491375
\(950\) 0 0
\(951\) 15.6249 0.506671
\(952\) 0 0
\(953\) 39.9757 1.29494 0.647470 0.762091i \(-0.275827\pi\)
0.647470 + 0.762091i \(0.275827\pi\)
\(954\) 0 0
\(955\) 12.6718 0.410051
\(956\) 0 0
\(957\) 34.3619 1.11076
\(958\) 0 0
\(959\) 17.5135 0.565540
\(960\) 0 0
\(961\) −25.3150 −0.816613
\(962\) 0 0
\(963\) −6.20298 −0.199888
\(964\) 0 0
\(965\) −20.9701 −0.675052
\(966\) 0 0
\(967\) 21.8166 0.701576 0.350788 0.936455i \(-0.385914\pi\)
0.350788 + 0.936455i \(0.385914\pi\)
\(968\) 0 0
\(969\) −22.6396 −0.727290
\(970\) 0 0
\(971\) 33.1306 1.06321 0.531606 0.846992i \(-0.321589\pi\)
0.531606 + 0.846992i \(0.321589\pi\)
\(972\) 0 0
\(973\) 15.5828 0.499562
\(974\) 0 0
\(975\) −0.221575 −0.00709608
\(976\) 0 0
\(977\) 60.5558 1.93735 0.968675 0.248331i \(-0.0798820\pi\)
0.968675 + 0.248331i \(0.0798820\pi\)
\(978\) 0 0
\(979\) −25.5866 −0.817752
\(980\) 0 0
\(981\) 1.50055 0.0479087
\(982\) 0 0
\(983\) 25.4291 0.811064 0.405532 0.914081i \(-0.367086\pi\)
0.405532 + 0.914081i \(0.367086\pi\)
\(984\) 0 0
\(985\) −0.0298715 −0.000951786 0
\(986\) 0 0
\(987\) 2.11159 0.0672126
\(988\) 0 0
\(989\) 25.3027 0.804580
\(990\) 0 0
\(991\) 3.77341 0.119866 0.0599332 0.998202i \(-0.480911\pi\)
0.0599332 + 0.998202i \(0.480911\pi\)
\(992\) 0 0
\(993\) −21.8890 −0.694626
\(994\) 0 0
\(995\) −8.79971 −0.278970
\(996\) 0 0
\(997\) −25.0286 −0.792662 −0.396331 0.918108i \(-0.629717\pi\)
−0.396331 + 0.918108i \(0.629717\pi\)
\(998\) 0 0
\(999\) −8.84802 −0.279939
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 4020.2.a.e.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.a.e.1.2 5 1.1 even 1 trivial