Properties

Label 4012.2.a.h.1.15
Level $4012$
Weight $2$
Character 4012.1
Self dual yes
Analytic conductor $32.036$
Analytic rank $1$
Dimension $15$
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] = [4012,2,Mod(1,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.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: \(32.0359812909\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} - 249 x^{6} + 2736 x^{5} - 801 x^{4} - 900 x^{3} + 429 x^{2} - 36 x - 2 \) 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 1.15
Root \(1.82373\) of defining polynomial
Character \(\chi\) \(=\) 4012.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.66529 q^{3} +1.22875 q^{5} -3.12758 q^{7} +4.10376 q^{9} +O(q^{10})\) \(q+2.66529 q^{3} +1.22875 q^{5} -3.12758 q^{7} +4.10376 q^{9} -1.80947 q^{11} -4.05098 q^{13} +3.27498 q^{15} +1.00000 q^{17} -5.41700 q^{19} -8.33589 q^{21} -9.32843 q^{23} -3.49017 q^{25} +2.94184 q^{27} +10.0200 q^{29} +0.429720 q^{31} -4.82277 q^{33} -3.84302 q^{35} +4.26089 q^{37} -10.7970 q^{39} +0.0793498 q^{41} -9.41857 q^{43} +5.04251 q^{45} -2.21591 q^{47} +2.78173 q^{49} +2.66529 q^{51} -8.34332 q^{53} -2.22339 q^{55} -14.4379 q^{57} -1.00000 q^{59} +0.967357 q^{61} -12.8348 q^{63} -4.97766 q^{65} -8.09029 q^{67} -24.8629 q^{69} +10.0748 q^{71} +3.10494 q^{73} -9.30230 q^{75} +5.65926 q^{77} +8.46026 q^{79} -4.47043 q^{81} -12.3705 q^{83} +1.22875 q^{85} +26.7062 q^{87} +12.7521 q^{89} +12.6698 q^{91} +1.14533 q^{93} -6.65615 q^{95} -5.26348 q^{97} -7.42565 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) \) 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
\(3\) 2.66529 1.53880 0.769402 0.638764i \(-0.220554\pi\)
0.769402 + 0.638764i \(0.220554\pi\)
\(4\) 0 0
\(5\) 1.22875 0.549515 0.274757 0.961514i \(-0.411402\pi\)
0.274757 + 0.961514i \(0.411402\pi\)
\(6\) 0 0
\(7\) −3.12758 −1.18211 −0.591056 0.806630i \(-0.701289\pi\)
−0.591056 + 0.806630i \(0.701289\pi\)
\(8\) 0 0
\(9\) 4.10376 1.36792
\(10\) 0 0
\(11\) −1.80947 −0.545577 −0.272788 0.962074i \(-0.587946\pi\)
−0.272788 + 0.962074i \(0.587946\pi\)
\(12\) 0 0
\(13\) −4.05098 −1.12354 −0.561770 0.827293i \(-0.689880\pi\)
−0.561770 + 0.827293i \(0.689880\pi\)
\(14\) 0 0
\(15\) 3.27498 0.845596
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) −5.41700 −1.24275 −0.621373 0.783515i \(-0.713425\pi\)
−0.621373 + 0.783515i \(0.713425\pi\)
\(20\) 0 0
\(21\) −8.33589 −1.81904
\(22\) 0 0
\(23\) −9.32843 −1.94511 −0.972556 0.232670i \(-0.925254\pi\)
−0.972556 + 0.232670i \(0.925254\pi\)
\(24\) 0 0
\(25\) −3.49017 −0.698034
\(26\) 0 0
\(27\) 2.94184 0.566158
\(28\) 0 0
\(29\) 10.0200 1.86067 0.930336 0.366709i \(-0.119516\pi\)
0.930336 + 0.366709i \(0.119516\pi\)
\(30\) 0 0
\(31\) 0.429720 0.0771800 0.0385900 0.999255i \(-0.487713\pi\)
0.0385900 + 0.999255i \(0.487713\pi\)
\(32\) 0 0
\(33\) −4.82277 −0.839536
\(34\) 0 0
\(35\) −3.84302 −0.649588
\(36\) 0 0
\(37\) 4.26089 0.700486 0.350243 0.936659i \(-0.386099\pi\)
0.350243 + 0.936659i \(0.386099\pi\)
\(38\) 0 0
\(39\) −10.7970 −1.72891
\(40\) 0 0
\(41\) 0.0793498 0.0123923 0.00619617 0.999981i \(-0.498028\pi\)
0.00619617 + 0.999981i \(0.498028\pi\)
\(42\) 0 0
\(43\) −9.41857 −1.43632 −0.718160 0.695878i \(-0.755015\pi\)
−0.718160 + 0.695878i \(0.755015\pi\)
\(44\) 0 0
\(45\) 5.04251 0.751693
\(46\) 0 0
\(47\) −2.21591 −0.323224 −0.161612 0.986854i \(-0.551669\pi\)
−0.161612 + 0.986854i \(0.551669\pi\)
\(48\) 0 0
\(49\) 2.78173 0.397390
\(50\) 0 0
\(51\) 2.66529 0.373215
\(52\) 0 0
\(53\) −8.34332 −1.14604 −0.573022 0.819540i \(-0.694229\pi\)
−0.573022 + 0.819540i \(0.694229\pi\)
\(54\) 0 0
\(55\) −2.22339 −0.299802
\(56\) 0 0
\(57\) −14.4379 −1.91234
\(58\) 0 0
\(59\) −1.00000 −0.130189
\(60\) 0 0
\(61\) 0.967357 0.123857 0.0619287 0.998081i \(-0.480275\pi\)
0.0619287 + 0.998081i \(0.480275\pi\)
\(62\) 0 0
\(63\) −12.8348 −1.61704
\(64\) 0 0
\(65\) −4.97766 −0.617402
\(66\) 0 0
\(67\) −8.09029 −0.988386 −0.494193 0.869352i \(-0.664536\pi\)
−0.494193 + 0.869352i \(0.664536\pi\)
\(68\) 0 0
\(69\) −24.8629 −2.99315
\(70\) 0 0
\(71\) 10.0748 1.19565 0.597827 0.801625i \(-0.296031\pi\)
0.597827 + 0.801625i \(0.296031\pi\)
\(72\) 0 0
\(73\) 3.10494 0.363406 0.181703 0.983353i \(-0.441839\pi\)
0.181703 + 0.983353i \(0.441839\pi\)
\(74\) 0 0
\(75\) −9.30230 −1.07414
\(76\) 0 0
\(77\) 5.65926 0.644933
\(78\) 0 0
\(79\) 8.46026 0.951854 0.475927 0.879485i \(-0.342113\pi\)
0.475927 + 0.879485i \(0.342113\pi\)
\(80\) 0 0
\(81\) −4.47043 −0.496714
\(82\) 0 0
\(83\) −12.3705 −1.35784 −0.678921 0.734211i \(-0.737552\pi\)
−0.678921 + 0.734211i \(0.737552\pi\)
\(84\) 0 0
\(85\) 1.22875 0.133277
\(86\) 0 0
\(87\) 26.7062 2.86321
\(88\) 0 0
\(89\) 12.7521 1.35171 0.675857 0.737032i \(-0.263774\pi\)
0.675857 + 0.737032i \(0.263774\pi\)
\(90\) 0 0
\(91\) 12.6698 1.32815
\(92\) 0 0
\(93\) 1.14533 0.118765
\(94\) 0 0
\(95\) −6.65615 −0.682907
\(96\) 0 0
\(97\) −5.26348 −0.534425 −0.267213 0.963638i \(-0.586103\pi\)
−0.267213 + 0.963638i \(0.586103\pi\)
\(98\) 0 0
\(99\) −7.42565 −0.746306
\(100\) 0 0
\(101\) 11.4756 1.14187 0.570935 0.820996i \(-0.306581\pi\)
0.570935 + 0.820996i \(0.306581\pi\)
\(102\) 0 0
\(103\) −16.3684 −1.61283 −0.806416 0.591349i \(-0.798596\pi\)
−0.806416 + 0.591349i \(0.798596\pi\)
\(104\) 0 0
\(105\) −10.2427 −0.999590
\(106\) 0 0
\(107\) 19.0759 1.84414 0.922069 0.387025i \(-0.126497\pi\)
0.922069 + 0.387025i \(0.126497\pi\)
\(108\) 0 0
\(109\) 13.5619 1.29899 0.649495 0.760365i \(-0.274980\pi\)
0.649495 + 0.760365i \(0.274980\pi\)
\(110\) 0 0
\(111\) 11.3565 1.07791
\(112\) 0 0
\(113\) 10.6150 0.998580 0.499290 0.866435i \(-0.333594\pi\)
0.499290 + 0.866435i \(0.333594\pi\)
\(114\) 0 0
\(115\) −11.4623 −1.06887
\(116\) 0 0
\(117\) −16.6243 −1.53691
\(118\) 0 0
\(119\) −3.12758 −0.286704
\(120\) 0 0
\(121\) −7.72581 −0.702346
\(122\) 0 0
\(123\) 0.211490 0.0190694
\(124\) 0 0
\(125\) −10.4323 −0.933095
\(126\) 0 0
\(127\) −10.0668 −0.893287 −0.446644 0.894712i \(-0.647381\pi\)
−0.446644 + 0.894712i \(0.647381\pi\)
\(128\) 0 0
\(129\) −25.1032 −2.21022
\(130\) 0 0
\(131\) −3.26722 −0.285458 −0.142729 0.989762i \(-0.545588\pi\)
−0.142729 + 0.989762i \(0.545588\pi\)
\(132\) 0 0
\(133\) 16.9421 1.46906
\(134\) 0 0
\(135\) 3.61480 0.311112
\(136\) 0 0
\(137\) −9.87323 −0.843527 −0.421764 0.906706i \(-0.638589\pi\)
−0.421764 + 0.906706i \(0.638589\pi\)
\(138\) 0 0
\(139\) −6.63396 −0.562685 −0.281343 0.959607i \(-0.590780\pi\)
−0.281343 + 0.959607i \(0.590780\pi\)
\(140\) 0 0
\(141\) −5.90605 −0.497379
\(142\) 0 0
\(143\) 7.33015 0.612978
\(144\) 0 0
\(145\) 12.3121 1.02247
\(146\) 0 0
\(147\) 7.41411 0.611506
\(148\) 0 0
\(149\) 5.51433 0.451751 0.225876 0.974156i \(-0.427476\pi\)
0.225876 + 0.974156i \(0.427476\pi\)
\(150\) 0 0
\(151\) −12.9784 −1.05617 −0.528083 0.849193i \(-0.677089\pi\)
−0.528083 + 0.849193i \(0.677089\pi\)
\(152\) 0 0
\(153\) 4.10376 0.331769
\(154\) 0 0
\(155\) 0.528019 0.0424115
\(156\) 0 0
\(157\) −22.9176 −1.82902 −0.914511 0.404562i \(-0.867424\pi\)
−0.914511 + 0.404562i \(0.867424\pi\)
\(158\) 0 0
\(159\) −22.2374 −1.76354
\(160\) 0 0
\(161\) 29.1754 2.29934
\(162\) 0 0
\(163\) −20.8633 −1.63414 −0.817069 0.576540i \(-0.804402\pi\)
−0.817069 + 0.576540i \(0.804402\pi\)
\(164\) 0 0
\(165\) −5.92599 −0.461337
\(166\) 0 0
\(167\) 20.0402 1.55076 0.775379 0.631496i \(-0.217559\pi\)
0.775379 + 0.631496i \(0.217559\pi\)
\(168\) 0 0
\(169\) 3.41046 0.262343
\(170\) 0 0
\(171\) −22.2301 −1.69998
\(172\) 0 0
\(173\) −3.65014 −0.277515 −0.138757 0.990326i \(-0.544311\pi\)
−0.138757 + 0.990326i \(0.544311\pi\)
\(174\) 0 0
\(175\) 10.9158 0.825154
\(176\) 0 0
\(177\) −2.66529 −0.200335
\(178\) 0 0
\(179\) 17.2813 1.29167 0.645833 0.763478i \(-0.276510\pi\)
0.645833 + 0.763478i \(0.276510\pi\)
\(180\) 0 0
\(181\) 13.4288 0.998156 0.499078 0.866557i \(-0.333672\pi\)
0.499078 + 0.866557i \(0.333672\pi\)
\(182\) 0 0
\(183\) 2.57828 0.190592
\(184\) 0 0
\(185\) 5.23558 0.384927
\(186\) 0 0
\(187\) −1.80947 −0.132322
\(188\) 0 0
\(189\) −9.20084 −0.669262
\(190\) 0 0
\(191\) 3.80814 0.275547 0.137774 0.990464i \(-0.456005\pi\)
0.137774 + 0.990464i \(0.456005\pi\)
\(192\) 0 0
\(193\) −12.5574 −0.903898 −0.451949 0.892044i \(-0.649271\pi\)
−0.451949 + 0.892044i \(0.649271\pi\)
\(194\) 0 0
\(195\) −13.2669 −0.950061
\(196\) 0 0
\(197\) 9.95161 0.709023 0.354512 0.935052i \(-0.384647\pi\)
0.354512 + 0.935052i \(0.384647\pi\)
\(198\) 0 0
\(199\) 21.4079 1.51757 0.758783 0.651343i \(-0.225794\pi\)
0.758783 + 0.651343i \(0.225794\pi\)
\(200\) 0 0
\(201\) −21.5630 −1.52093
\(202\) 0 0
\(203\) −31.3384 −2.19952
\(204\) 0 0
\(205\) 0.0975012 0.00680978
\(206\) 0 0
\(207\) −38.2816 −2.66076
\(208\) 0 0
\(209\) 9.80192 0.678013
\(210\) 0 0
\(211\) 12.1461 0.836169 0.418084 0.908408i \(-0.362702\pi\)
0.418084 + 0.908408i \(0.362702\pi\)
\(212\) 0 0
\(213\) 26.8521 1.83988
\(214\) 0 0
\(215\) −11.5731 −0.789279
\(216\) 0 0
\(217\) −1.34398 −0.0912354
\(218\) 0 0
\(219\) 8.27556 0.559211
\(220\) 0 0
\(221\) −4.05098 −0.272499
\(222\) 0 0
\(223\) 4.89663 0.327903 0.163951 0.986468i \(-0.447576\pi\)
0.163951 + 0.986468i \(0.447576\pi\)
\(224\) 0 0
\(225\) −14.3228 −0.954854
\(226\) 0 0
\(227\) −10.2524 −0.680472 −0.340236 0.940340i \(-0.610507\pi\)
−0.340236 + 0.940340i \(0.610507\pi\)
\(228\) 0 0
\(229\) 16.8844 1.11575 0.557876 0.829924i \(-0.311616\pi\)
0.557876 + 0.829924i \(0.311616\pi\)
\(230\) 0 0
\(231\) 15.0836 0.992426
\(232\) 0 0
\(233\) −18.2195 −1.19360 −0.596799 0.802391i \(-0.703561\pi\)
−0.596799 + 0.802391i \(0.703561\pi\)
\(234\) 0 0
\(235\) −2.72281 −0.177616
\(236\) 0 0
\(237\) 22.5490 1.46472
\(238\) 0 0
\(239\) −24.6352 −1.59352 −0.796758 0.604298i \(-0.793454\pi\)
−0.796758 + 0.604298i \(0.793454\pi\)
\(240\) 0 0
\(241\) 8.95060 0.576559 0.288279 0.957546i \(-0.406917\pi\)
0.288279 + 0.957546i \(0.406917\pi\)
\(242\) 0 0
\(243\) −20.7405 −1.33050
\(244\) 0 0
\(245\) 3.41806 0.218372
\(246\) 0 0
\(247\) 21.9442 1.39627
\(248\) 0 0
\(249\) −32.9710 −2.08945
\(250\) 0 0
\(251\) 5.74174 0.362416 0.181208 0.983445i \(-0.441999\pi\)
0.181208 + 0.983445i \(0.441999\pi\)
\(252\) 0 0
\(253\) 16.8795 1.06121
\(254\) 0 0
\(255\) 3.27498 0.205087
\(256\) 0 0
\(257\) −11.7469 −0.732749 −0.366375 0.930467i \(-0.619401\pi\)
−0.366375 + 0.930467i \(0.619401\pi\)
\(258\) 0 0
\(259\) −13.3262 −0.828053
\(260\) 0 0
\(261\) 41.1198 2.54525
\(262\) 0 0
\(263\) 25.1564 1.55121 0.775604 0.631219i \(-0.217445\pi\)
0.775604 + 0.631219i \(0.217445\pi\)
\(264\) 0 0
\(265\) −10.2519 −0.629768
\(266\) 0 0
\(267\) 33.9879 2.08003
\(268\) 0 0
\(269\) −11.3493 −0.691981 −0.345990 0.938238i \(-0.612457\pi\)
−0.345990 + 0.938238i \(0.612457\pi\)
\(270\) 0 0
\(271\) −20.8351 −1.26565 −0.632823 0.774297i \(-0.718104\pi\)
−0.632823 + 0.774297i \(0.718104\pi\)
\(272\) 0 0
\(273\) 33.7686 2.04377
\(274\) 0 0
\(275\) 6.31536 0.380831
\(276\) 0 0
\(277\) 15.7728 0.947694 0.473847 0.880607i \(-0.342865\pi\)
0.473847 + 0.880607i \(0.342865\pi\)
\(278\) 0 0
\(279\) 1.76347 0.105576
\(280\) 0 0
\(281\) 0.989841 0.0590490 0.0295245 0.999564i \(-0.490601\pi\)
0.0295245 + 0.999564i \(0.490601\pi\)
\(282\) 0 0
\(283\) 19.0067 1.12983 0.564916 0.825148i \(-0.308908\pi\)
0.564916 + 0.825148i \(0.308908\pi\)
\(284\) 0 0
\(285\) −17.7406 −1.05086
\(286\) 0 0
\(287\) −0.248172 −0.0146492
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −14.0287 −0.822376
\(292\) 0 0
\(293\) 24.5161 1.43224 0.716122 0.697975i \(-0.245915\pi\)
0.716122 + 0.697975i \(0.245915\pi\)
\(294\) 0 0
\(295\) −1.22875 −0.0715407
\(296\) 0 0
\(297\) −5.32319 −0.308883
\(298\) 0 0
\(299\) 37.7893 2.18541
\(300\) 0 0
\(301\) 29.4573 1.69789
\(302\) 0 0
\(303\) 30.5859 1.75711
\(304\) 0 0
\(305\) 1.18864 0.0680614
\(306\) 0 0
\(307\) −14.5879 −0.832577 −0.416289 0.909232i \(-0.636669\pi\)
−0.416289 + 0.909232i \(0.636669\pi\)
\(308\) 0 0
\(309\) −43.6266 −2.48183
\(310\) 0 0
\(311\) 20.0198 1.13522 0.567609 0.823298i \(-0.307869\pi\)
0.567609 + 0.823298i \(0.307869\pi\)
\(312\) 0 0
\(313\) 2.42363 0.136992 0.0684959 0.997651i \(-0.478180\pi\)
0.0684959 + 0.997651i \(0.478180\pi\)
\(314\) 0 0
\(315\) −15.7708 −0.888585
\(316\) 0 0
\(317\) −10.2630 −0.576427 −0.288214 0.957566i \(-0.593061\pi\)
−0.288214 + 0.957566i \(0.593061\pi\)
\(318\) 0 0
\(319\) −18.1310 −1.01514
\(320\) 0 0
\(321\) 50.8428 2.83777
\(322\) 0 0
\(323\) −5.41700 −0.301410
\(324\) 0 0
\(325\) 14.1386 0.784269
\(326\) 0 0
\(327\) 36.1463 1.99889
\(328\) 0 0
\(329\) 6.93043 0.382087
\(330\) 0 0
\(331\) 4.32736 0.237853 0.118927 0.992903i \(-0.462055\pi\)
0.118927 + 0.992903i \(0.462055\pi\)
\(332\) 0 0
\(333\) 17.4857 0.958209
\(334\) 0 0
\(335\) −9.94096 −0.543133
\(336\) 0 0
\(337\) −36.5048 −1.98854 −0.994270 0.106895i \(-0.965909\pi\)
−0.994270 + 0.106895i \(0.965909\pi\)
\(338\) 0 0
\(339\) 28.2922 1.53662
\(340\) 0 0
\(341\) −0.777567 −0.0421076
\(342\) 0 0
\(343\) 13.1930 0.712353
\(344\) 0 0
\(345\) −30.5504 −1.64478
\(346\) 0 0
\(347\) −5.63200 −0.302342 −0.151171 0.988508i \(-0.548304\pi\)
−0.151171 + 0.988508i \(0.548304\pi\)
\(348\) 0 0
\(349\) −34.6802 −1.85639 −0.928194 0.372096i \(-0.878639\pi\)
−0.928194 + 0.372096i \(0.878639\pi\)
\(350\) 0 0
\(351\) −11.9174 −0.636101
\(352\) 0 0
\(353\) −0.970983 −0.0516802 −0.0258401 0.999666i \(-0.508226\pi\)
−0.0258401 + 0.999666i \(0.508226\pi\)
\(354\) 0 0
\(355\) 12.3794 0.657029
\(356\) 0 0
\(357\) −8.33589 −0.441182
\(358\) 0 0
\(359\) 20.6141 1.08797 0.543985 0.839095i \(-0.316915\pi\)
0.543985 + 0.839095i \(0.316915\pi\)
\(360\) 0 0
\(361\) 10.3439 0.544416
\(362\) 0 0
\(363\) −20.5915 −1.08077
\(364\) 0 0
\(365\) 3.81520 0.199697
\(366\) 0 0
\(367\) −5.85365 −0.305558 −0.152779 0.988260i \(-0.548822\pi\)
−0.152779 + 0.988260i \(0.548822\pi\)
\(368\) 0 0
\(369\) 0.325632 0.0169517
\(370\) 0 0
\(371\) 26.0944 1.35475
\(372\) 0 0
\(373\) −3.59866 −0.186331 −0.0931657 0.995651i \(-0.529699\pi\)
−0.0931657 + 0.995651i \(0.529699\pi\)
\(374\) 0 0
\(375\) −27.8051 −1.43585
\(376\) 0 0
\(377\) −40.5909 −2.09054
\(378\) 0 0
\(379\) 14.9701 0.768960 0.384480 0.923133i \(-0.374381\pi\)
0.384480 + 0.923133i \(0.374381\pi\)
\(380\) 0 0
\(381\) −26.8310 −1.37459
\(382\) 0 0
\(383\) −2.43186 −0.124262 −0.0621311 0.998068i \(-0.519790\pi\)
−0.0621311 + 0.998068i \(0.519790\pi\)
\(384\) 0 0
\(385\) 6.95383 0.354400
\(386\) 0 0
\(387\) −38.6516 −1.96477
\(388\) 0 0
\(389\) −27.1743 −1.37779 −0.688895 0.724861i \(-0.741904\pi\)
−0.688895 + 0.724861i \(0.741904\pi\)
\(390\) 0 0
\(391\) −9.32843 −0.471759
\(392\) 0 0
\(393\) −8.70807 −0.439264
\(394\) 0 0
\(395\) 10.3956 0.523058
\(396\) 0 0
\(397\) −7.95557 −0.399279 −0.199639 0.979869i \(-0.563977\pi\)
−0.199639 + 0.979869i \(0.563977\pi\)
\(398\) 0 0
\(399\) 45.1555 2.26060
\(400\) 0 0
\(401\) −28.2089 −1.40868 −0.704342 0.709861i \(-0.748758\pi\)
−0.704342 + 0.709861i \(0.748758\pi\)
\(402\) 0 0
\(403\) −1.74079 −0.0867148
\(404\) 0 0
\(405\) −5.49305 −0.272952
\(406\) 0 0
\(407\) −7.70996 −0.382169
\(408\) 0 0
\(409\) −5.84972 −0.289250 −0.144625 0.989487i \(-0.546198\pi\)
−0.144625 + 0.989487i \(0.546198\pi\)
\(410\) 0 0
\(411\) −26.3150 −1.29802
\(412\) 0 0
\(413\) 3.12758 0.153898
\(414\) 0 0
\(415\) −15.2003 −0.746154
\(416\) 0 0
\(417\) −17.6814 −0.865863
\(418\) 0 0
\(419\) −28.6156 −1.39797 −0.698983 0.715139i \(-0.746364\pi\)
−0.698983 + 0.715139i \(0.746364\pi\)
\(420\) 0 0
\(421\) −9.28918 −0.452727 −0.226363 0.974043i \(-0.572684\pi\)
−0.226363 + 0.974043i \(0.572684\pi\)
\(422\) 0 0
\(423\) −9.09358 −0.442145
\(424\) 0 0
\(425\) −3.49017 −0.169298
\(426\) 0 0
\(427\) −3.02548 −0.146413
\(428\) 0 0
\(429\) 19.5370 0.943253
\(430\) 0 0
\(431\) −2.70573 −0.130330 −0.0651651 0.997874i \(-0.520757\pi\)
−0.0651651 + 0.997874i \(0.520757\pi\)
\(432\) 0 0
\(433\) 24.0687 1.15667 0.578335 0.815800i \(-0.303703\pi\)
0.578335 + 0.815800i \(0.303703\pi\)
\(434\) 0 0
\(435\) 32.8154 1.57338
\(436\) 0 0
\(437\) 50.5321 2.41728
\(438\) 0 0
\(439\) 34.3843 1.64107 0.820536 0.571594i \(-0.193675\pi\)
0.820536 + 0.571594i \(0.193675\pi\)
\(440\) 0 0
\(441\) 11.4156 0.543598
\(442\) 0 0
\(443\) 24.7999 1.17828 0.589140 0.808031i \(-0.299467\pi\)
0.589140 + 0.808031i \(0.299467\pi\)
\(444\) 0 0
\(445\) 15.6691 0.742787
\(446\) 0 0
\(447\) 14.6973 0.695157
\(448\) 0 0
\(449\) −29.6807 −1.40072 −0.700359 0.713791i \(-0.746977\pi\)
−0.700359 + 0.713791i \(0.746977\pi\)
\(450\) 0 0
\(451\) −0.143581 −0.00676098
\(452\) 0 0
\(453\) −34.5912 −1.62523
\(454\) 0 0
\(455\) 15.5680 0.729839
\(456\) 0 0
\(457\) 22.0512 1.03151 0.515757 0.856735i \(-0.327511\pi\)
0.515757 + 0.856735i \(0.327511\pi\)
\(458\) 0 0
\(459\) 2.94184 0.137313
\(460\) 0 0
\(461\) −34.3620 −1.60040 −0.800199 0.599735i \(-0.795273\pi\)
−0.800199 + 0.599735i \(0.795273\pi\)
\(462\) 0 0
\(463\) 23.5944 1.09653 0.548263 0.836306i \(-0.315289\pi\)
0.548263 + 0.836306i \(0.315289\pi\)
\(464\) 0 0
\(465\) 1.40732 0.0652631
\(466\) 0 0
\(467\) −26.4674 −1.22476 −0.612382 0.790562i \(-0.709789\pi\)
−0.612382 + 0.790562i \(0.709789\pi\)
\(468\) 0 0
\(469\) 25.3030 1.16838
\(470\) 0 0
\(471\) −61.0819 −2.81451
\(472\) 0 0
\(473\) 17.0427 0.783622
\(474\) 0 0
\(475\) 18.9062 0.867478
\(476\) 0 0
\(477\) −34.2390 −1.56770
\(478\) 0 0
\(479\) 2.29484 0.104854 0.0524269 0.998625i \(-0.483304\pi\)
0.0524269 + 0.998625i \(0.483304\pi\)
\(480\) 0 0
\(481\) −17.2608 −0.787024
\(482\) 0 0
\(483\) 77.7608 3.53824
\(484\) 0 0
\(485\) −6.46751 −0.293675
\(486\) 0 0
\(487\) −13.4074 −0.607546 −0.303773 0.952744i \(-0.598246\pi\)
−0.303773 + 0.952744i \(0.598246\pi\)
\(488\) 0 0
\(489\) −55.6066 −2.51462
\(490\) 0 0
\(491\) 29.1949 1.31755 0.658774 0.752341i \(-0.271075\pi\)
0.658774 + 0.752341i \(0.271075\pi\)
\(492\) 0 0
\(493\) 10.0200 0.451279
\(494\) 0 0
\(495\) −9.12428 −0.410106
\(496\) 0 0
\(497\) −31.5096 −1.41340
\(498\) 0 0
\(499\) −43.3041 −1.93856 −0.969278 0.245966i \(-0.920895\pi\)
−0.969278 + 0.245966i \(0.920895\pi\)
\(500\) 0 0
\(501\) 53.4129 2.38631
\(502\) 0 0
\(503\) −25.2316 −1.12502 −0.562510 0.826790i \(-0.690164\pi\)
−0.562510 + 0.826790i \(0.690164\pi\)
\(504\) 0 0
\(505\) 14.1007 0.627474
\(506\) 0 0
\(507\) 9.08987 0.403695
\(508\) 0 0
\(509\) 31.0390 1.37578 0.687890 0.725815i \(-0.258537\pi\)
0.687890 + 0.725815i \(0.258537\pi\)
\(510\) 0 0
\(511\) −9.71094 −0.429586
\(512\) 0 0
\(513\) −15.9360 −0.703590
\(514\) 0 0
\(515\) −20.1128 −0.886275
\(516\) 0 0
\(517\) 4.00963 0.176344
\(518\) 0 0
\(519\) −9.72867 −0.427041
\(520\) 0 0
\(521\) 20.4542 0.896115 0.448057 0.894005i \(-0.352116\pi\)
0.448057 + 0.894005i \(0.352116\pi\)
\(522\) 0 0
\(523\) −11.6091 −0.507629 −0.253815 0.967253i \(-0.581685\pi\)
−0.253815 + 0.967253i \(0.581685\pi\)
\(524\) 0 0
\(525\) 29.0937 1.26975
\(526\) 0 0
\(527\) 0.429720 0.0187189
\(528\) 0 0
\(529\) 64.0196 2.78346
\(530\) 0 0
\(531\) −4.10376 −0.178088
\(532\) 0 0
\(533\) −0.321445 −0.0139233
\(534\) 0 0
\(535\) 23.4396 1.01338
\(536\) 0 0
\(537\) 46.0597 1.98762
\(538\) 0 0
\(539\) −5.03346 −0.216807
\(540\) 0 0
\(541\) −24.7202 −1.06280 −0.531402 0.847120i \(-0.678334\pi\)
−0.531402 + 0.847120i \(0.678334\pi\)
\(542\) 0 0
\(543\) 35.7917 1.53597
\(544\) 0 0
\(545\) 16.6642 0.713815
\(546\) 0 0
\(547\) 14.4519 0.617918 0.308959 0.951075i \(-0.400019\pi\)
0.308959 + 0.951075i \(0.400019\pi\)
\(548\) 0 0
\(549\) 3.96980 0.169427
\(550\) 0 0
\(551\) −54.2785 −2.31234
\(552\) 0 0
\(553\) −26.4601 −1.12520
\(554\) 0 0
\(555\) 13.9543 0.592328
\(556\) 0 0
\(557\) 42.1816 1.78729 0.893645 0.448774i \(-0.148139\pi\)
0.893645 + 0.448774i \(0.148139\pi\)
\(558\) 0 0
\(559\) 38.1545 1.61376
\(560\) 0 0
\(561\) −4.82277 −0.203617
\(562\) 0 0
\(563\) −1.10583 −0.0466054 −0.0233027 0.999728i \(-0.507418\pi\)
−0.0233027 + 0.999728i \(0.507418\pi\)
\(564\) 0 0
\(565\) 13.0433 0.548734
\(566\) 0 0
\(567\) 13.9816 0.587172
\(568\) 0 0
\(569\) 15.6702 0.656929 0.328464 0.944516i \(-0.393469\pi\)
0.328464 + 0.944516i \(0.393469\pi\)
\(570\) 0 0
\(571\) −9.31950 −0.390009 −0.195005 0.980802i \(-0.562472\pi\)
−0.195005 + 0.980802i \(0.562472\pi\)
\(572\) 0 0
\(573\) 10.1498 0.424013
\(574\) 0 0
\(575\) 32.5578 1.35775
\(576\) 0 0
\(577\) −26.7254 −1.11259 −0.556297 0.830984i \(-0.687778\pi\)
−0.556297 + 0.830984i \(0.687778\pi\)
\(578\) 0 0
\(579\) −33.4690 −1.39092
\(580\) 0 0
\(581\) 38.6898 1.60512
\(582\) 0 0
\(583\) 15.0970 0.625255
\(584\) 0 0
\(585\) −20.4271 −0.844557
\(586\) 0 0
\(587\) −11.9112 −0.491627 −0.245813 0.969317i \(-0.579055\pi\)
−0.245813 + 0.969317i \(0.579055\pi\)
\(588\) 0 0
\(589\) −2.32779 −0.0959151
\(590\) 0 0
\(591\) 26.5239 1.09105
\(592\) 0 0
\(593\) −33.2850 −1.36685 −0.683426 0.730020i \(-0.739511\pi\)
−0.683426 + 0.730020i \(0.739511\pi\)
\(594\) 0 0
\(595\) −3.84302 −0.157548
\(596\) 0 0
\(597\) 57.0582 2.33524
\(598\) 0 0
\(599\) 31.2912 1.27852 0.639262 0.768989i \(-0.279240\pi\)
0.639262 + 0.768989i \(0.279240\pi\)
\(600\) 0 0
\(601\) −33.4347 −1.36383 −0.681915 0.731432i \(-0.738852\pi\)
−0.681915 + 0.731432i \(0.738852\pi\)
\(602\) 0 0
\(603\) −33.2006 −1.35203
\(604\) 0 0
\(605\) −9.49310 −0.385950
\(606\) 0 0
\(607\) −39.4291 −1.60038 −0.800189 0.599748i \(-0.795267\pi\)
−0.800189 + 0.599748i \(0.795267\pi\)
\(608\) 0 0
\(609\) −83.5258 −3.38464
\(610\) 0 0
\(611\) 8.97662 0.363155
\(612\) 0 0
\(613\) −29.4169 −1.18814 −0.594068 0.804415i \(-0.702479\pi\)
−0.594068 + 0.804415i \(0.702479\pi\)
\(614\) 0 0
\(615\) 0.259869 0.0104789
\(616\) 0 0
\(617\) −35.4580 −1.42748 −0.713742 0.700409i \(-0.753001\pi\)
−0.713742 + 0.700409i \(0.753001\pi\)
\(618\) 0 0
\(619\) 9.46718 0.380518 0.190259 0.981734i \(-0.439067\pi\)
0.190259 + 0.981734i \(0.439067\pi\)
\(620\) 0 0
\(621\) −27.4428 −1.10124
\(622\) 0 0
\(623\) −39.8830 −1.59788
\(624\) 0 0
\(625\) 4.63211 0.185284
\(626\) 0 0
\(627\) 26.1249 1.04333
\(628\) 0 0
\(629\) 4.26089 0.169893
\(630\) 0 0
\(631\) −19.2080 −0.764660 −0.382330 0.924026i \(-0.624878\pi\)
−0.382330 + 0.924026i \(0.624878\pi\)
\(632\) 0 0
\(633\) 32.3727 1.28670
\(634\) 0 0
\(635\) −12.3696 −0.490875
\(636\) 0 0
\(637\) −11.2687 −0.446484
\(638\) 0 0
\(639\) 41.3444 1.63556
\(640\) 0 0
\(641\) −13.5997 −0.537155 −0.268577 0.963258i \(-0.586553\pi\)
−0.268577 + 0.963258i \(0.586553\pi\)
\(642\) 0 0
\(643\) 1.19490 0.0471225 0.0235612 0.999722i \(-0.492500\pi\)
0.0235612 + 0.999722i \(0.492500\pi\)
\(644\) 0 0
\(645\) −30.8456 −1.21455
\(646\) 0 0
\(647\) −42.5758 −1.67383 −0.836914 0.547334i \(-0.815643\pi\)
−0.836914 + 0.547334i \(0.815643\pi\)
\(648\) 0 0
\(649\) 1.80947 0.0710280
\(650\) 0 0
\(651\) −3.58210 −0.140393
\(652\) 0 0
\(653\) −0.772012 −0.0302112 −0.0151056 0.999886i \(-0.504808\pi\)
−0.0151056 + 0.999886i \(0.504808\pi\)
\(654\) 0 0
\(655\) −4.01460 −0.156863
\(656\) 0 0
\(657\) 12.7419 0.497110
\(658\) 0 0
\(659\) −22.8835 −0.891416 −0.445708 0.895178i \(-0.647048\pi\)
−0.445708 + 0.895178i \(0.647048\pi\)
\(660\) 0 0
\(661\) −31.0916 −1.20932 −0.604662 0.796482i \(-0.706692\pi\)
−0.604662 + 0.796482i \(0.706692\pi\)
\(662\) 0 0
\(663\) −10.7970 −0.419322
\(664\) 0 0
\(665\) 20.8176 0.807273
\(666\) 0 0
\(667\) −93.4710 −3.61921
\(668\) 0 0
\(669\) 13.0509 0.504578
\(670\) 0 0
\(671\) −1.75041 −0.0675737
\(672\) 0 0
\(673\) 35.2250 1.35782 0.678912 0.734219i \(-0.262452\pi\)
0.678912 + 0.734219i \(0.262452\pi\)
\(674\) 0 0
\(675\) −10.2675 −0.395197
\(676\) 0 0
\(677\) 10.7097 0.411607 0.205804 0.978593i \(-0.434019\pi\)
0.205804 + 0.978593i \(0.434019\pi\)
\(678\) 0 0
\(679\) 16.4619 0.631751
\(680\) 0 0
\(681\) −27.3255 −1.04711
\(682\) 0 0
\(683\) −9.79463 −0.374781 −0.187391 0.982285i \(-0.560003\pi\)
−0.187391 + 0.982285i \(0.560003\pi\)
\(684\) 0 0
\(685\) −12.1318 −0.463531
\(686\) 0 0
\(687\) 45.0018 1.71692
\(688\) 0 0
\(689\) 33.7987 1.28763
\(690\) 0 0
\(691\) 6.38249 0.242801 0.121401 0.992604i \(-0.461261\pi\)
0.121401 + 0.992604i \(0.461261\pi\)
\(692\) 0 0
\(693\) 23.2243 0.882217
\(694\) 0 0
\(695\) −8.15149 −0.309204
\(696\) 0 0
\(697\) 0.0793498 0.00300559
\(698\) 0 0
\(699\) −48.5602 −1.83671
\(700\) 0 0
\(701\) −15.9980 −0.604235 −0.302117 0.953271i \(-0.597693\pi\)
−0.302117 + 0.953271i \(0.597693\pi\)
\(702\) 0 0
\(703\) −23.0812 −0.870525
\(704\) 0 0
\(705\) −7.25707 −0.273317
\(706\) 0 0
\(707\) −35.8909 −1.34982
\(708\) 0 0
\(709\) 10.1141 0.379843 0.189922 0.981799i \(-0.439177\pi\)
0.189922 + 0.981799i \(0.439177\pi\)
\(710\) 0 0
\(711\) 34.7189 1.30206
\(712\) 0 0
\(713\) −4.00861 −0.150124
\(714\) 0 0
\(715\) 9.00693 0.336840
\(716\) 0 0
\(717\) −65.6598 −2.45211
\(718\) 0 0
\(719\) −22.6802 −0.845829 −0.422915 0.906170i \(-0.638993\pi\)
−0.422915 + 0.906170i \(0.638993\pi\)
\(720\) 0 0
\(721\) 51.1936 1.90655
\(722\) 0 0
\(723\) 23.8559 0.887211
\(724\) 0 0
\(725\) −34.9715 −1.29881
\(726\) 0 0
\(727\) 15.9601 0.591929 0.295964 0.955199i \(-0.404359\pi\)
0.295964 + 0.955199i \(0.404359\pi\)
\(728\) 0 0
\(729\) −41.8681 −1.55067
\(730\) 0 0
\(731\) −9.41857 −0.348359
\(732\) 0 0
\(733\) 26.2409 0.969229 0.484615 0.874728i \(-0.338960\pi\)
0.484615 + 0.874728i \(0.338960\pi\)
\(734\) 0 0
\(735\) 9.11011 0.336031
\(736\) 0 0
\(737\) 14.6392 0.539240
\(738\) 0 0
\(739\) 27.5569 1.01370 0.506848 0.862036i \(-0.330811\pi\)
0.506848 + 0.862036i \(0.330811\pi\)
\(740\) 0 0
\(741\) 58.4876 2.14859
\(742\) 0 0
\(743\) 19.2489 0.706174 0.353087 0.935591i \(-0.385132\pi\)
0.353087 + 0.935591i \(0.385132\pi\)
\(744\) 0 0
\(745\) 6.77574 0.248244
\(746\) 0 0
\(747\) −50.7657 −1.85742
\(748\) 0 0
\(749\) −59.6614 −2.17998
\(750\) 0 0
\(751\) 2.78247 0.101534 0.0507668 0.998711i \(-0.483833\pi\)
0.0507668 + 0.998711i \(0.483833\pi\)
\(752\) 0 0
\(753\) 15.3034 0.557687
\(754\) 0 0
\(755\) −15.9472 −0.580379
\(756\) 0 0
\(757\) −4.55489 −0.165550 −0.0827751 0.996568i \(-0.526378\pi\)
−0.0827751 + 0.996568i \(0.526378\pi\)
\(758\) 0 0
\(759\) 44.9888 1.63299
\(760\) 0 0
\(761\) 48.0049 1.74017 0.870087 0.492898i \(-0.164062\pi\)
0.870087 + 0.492898i \(0.164062\pi\)
\(762\) 0 0
\(763\) −42.4158 −1.53555
\(764\) 0 0
\(765\) 5.04251 0.182312
\(766\) 0 0
\(767\) 4.05098 0.146273
\(768\) 0 0
\(769\) 41.6711 1.50270 0.751350 0.659904i \(-0.229403\pi\)
0.751350 + 0.659904i \(0.229403\pi\)
\(770\) 0 0
\(771\) −31.3088 −1.12756
\(772\) 0 0
\(773\) −17.4436 −0.627403 −0.313701 0.949522i \(-0.601569\pi\)
−0.313701 + 0.949522i \(0.601569\pi\)
\(774\) 0 0
\(775\) −1.49979 −0.0538742
\(776\) 0 0
\(777\) −35.5183 −1.27421
\(778\) 0 0
\(779\) −0.429838 −0.0154005
\(780\) 0 0
\(781\) −18.2300 −0.652321
\(782\) 0 0
\(783\) 29.4773 1.05343
\(784\) 0 0
\(785\) −28.1600 −1.00507
\(786\) 0 0
\(787\) −22.0109 −0.784604 −0.392302 0.919837i \(-0.628321\pi\)
−0.392302 + 0.919837i \(0.628321\pi\)
\(788\) 0 0
\(789\) 67.0490 2.38701
\(790\) 0 0
\(791\) −33.1994 −1.18043
\(792\) 0 0
\(793\) −3.91875 −0.139159
\(794\) 0 0
\(795\) −27.3242 −0.969090
\(796\) 0 0
\(797\) −30.6652 −1.08622 −0.543108 0.839663i \(-0.682753\pi\)
−0.543108 + 0.839663i \(0.682753\pi\)
\(798\) 0 0
\(799\) −2.21591 −0.0783933
\(800\) 0 0
\(801\) 52.3314 1.84904
\(802\) 0 0
\(803\) −5.61831 −0.198266
\(804\) 0 0
\(805\) 35.8493 1.26352
\(806\) 0 0
\(807\) −30.2492 −1.06482
\(808\) 0 0
\(809\) 27.4843 0.966298 0.483149 0.875538i \(-0.339493\pi\)
0.483149 + 0.875538i \(0.339493\pi\)
\(810\) 0 0
\(811\) −7.53676 −0.264651 −0.132326 0.991206i \(-0.542244\pi\)
−0.132326 + 0.991206i \(0.542244\pi\)
\(812\) 0 0
\(813\) −55.5317 −1.94758
\(814\) 0 0
\(815\) −25.6358 −0.897983
\(816\) 0 0
\(817\) 51.0204 1.78498
\(818\) 0 0
\(819\) 51.9937 1.81681
\(820\) 0 0
\(821\) 1.99702 0.0696964 0.0348482 0.999393i \(-0.488905\pi\)
0.0348482 + 0.999393i \(0.488905\pi\)
\(822\) 0 0
\(823\) −11.2296 −0.391439 −0.195720 0.980660i \(-0.562704\pi\)
−0.195720 + 0.980660i \(0.562704\pi\)
\(824\) 0 0
\(825\) 16.8323 0.586024
\(826\) 0 0
\(827\) −9.09139 −0.316139 −0.158069 0.987428i \(-0.550527\pi\)
−0.158069 + 0.987428i \(0.550527\pi\)
\(828\) 0 0
\(829\) −54.6984 −1.89975 −0.949877 0.312622i \(-0.898793\pi\)
−0.949877 + 0.312622i \(0.898793\pi\)
\(830\) 0 0
\(831\) 42.0390 1.45832
\(832\) 0 0
\(833\) 2.78173 0.0963812
\(834\) 0 0
\(835\) 24.6245 0.852164
\(836\) 0 0
\(837\) 1.26417 0.0436960
\(838\) 0 0
\(839\) 38.6343 1.33380 0.666902 0.745145i \(-0.267620\pi\)
0.666902 + 0.745145i \(0.267620\pi\)
\(840\) 0 0
\(841\) 71.4008 2.46210
\(842\) 0 0
\(843\) 2.63821 0.0908649
\(844\) 0 0
\(845\) 4.19062 0.144162
\(846\) 0 0
\(847\) 24.1630 0.830252
\(848\) 0 0
\(849\) 50.6584 1.73859
\(850\) 0 0
\(851\) −39.7474 −1.36252
\(852\) 0 0
\(853\) −13.7301 −0.470111 −0.235055 0.971982i \(-0.575527\pi\)
−0.235055 + 0.971982i \(0.575527\pi\)
\(854\) 0 0
\(855\) −27.3153 −0.934162
\(856\) 0 0
\(857\) 30.5782 1.04453 0.522266 0.852783i \(-0.325087\pi\)
0.522266 + 0.852783i \(0.325087\pi\)
\(858\) 0 0
\(859\) −23.2853 −0.794485 −0.397243 0.917714i \(-0.630033\pi\)
−0.397243 + 0.917714i \(0.630033\pi\)
\(860\) 0 0
\(861\) −0.661451 −0.0225422
\(862\) 0 0
\(863\) −1.15746 −0.0394005 −0.0197002 0.999806i \(-0.506271\pi\)
−0.0197002 + 0.999806i \(0.506271\pi\)
\(864\) 0 0
\(865\) −4.48511 −0.152498
\(866\) 0 0
\(867\) 2.66529 0.0905179
\(868\) 0 0
\(869\) −15.3086 −0.519309
\(870\) 0 0
\(871\) 32.7736 1.11049
\(872\) 0 0
\(873\) −21.6001 −0.731051
\(874\) 0 0
\(875\) 32.6279 1.10302
\(876\) 0 0
\(877\) −5.85805 −0.197812 −0.0989061 0.995097i \(-0.531534\pi\)
−0.0989061 + 0.995097i \(0.531534\pi\)
\(878\) 0 0
\(879\) 65.3424 2.20395
\(880\) 0 0
\(881\) −9.15450 −0.308423 −0.154212 0.988038i \(-0.549284\pi\)
−0.154212 + 0.988038i \(0.549284\pi\)
\(882\) 0 0
\(883\) 52.4140 1.76387 0.881935 0.471371i \(-0.156241\pi\)
0.881935 + 0.471371i \(0.156241\pi\)
\(884\) 0 0
\(885\) −3.27498 −0.110087
\(886\) 0 0
\(887\) −10.3427 −0.347275 −0.173637 0.984810i \(-0.555552\pi\)
−0.173637 + 0.984810i \(0.555552\pi\)
\(888\) 0 0
\(889\) 31.4848 1.05597
\(890\) 0 0
\(891\) 8.08912 0.270996
\(892\) 0 0
\(893\) 12.0036 0.401685
\(894\) 0 0
\(895\) 21.2345 0.709790
\(896\) 0 0
\(897\) 100.719 3.36292
\(898\) 0 0
\(899\) 4.30580 0.143607
\(900\) 0 0
\(901\) −8.34332 −0.277956
\(902\) 0 0
\(903\) 78.5122 2.61272
\(904\) 0 0
\(905\) 16.5007 0.548502
\(906\) 0 0
\(907\) 7.54200 0.250428 0.125214 0.992130i \(-0.460038\pi\)
0.125214 + 0.992130i \(0.460038\pi\)
\(908\) 0 0
\(909\) 47.0933 1.56199
\(910\) 0 0
\(911\) 12.5967 0.417349 0.208674 0.977985i \(-0.433085\pi\)
0.208674 + 0.977985i \(0.433085\pi\)
\(912\) 0 0
\(913\) 22.3841 0.740807
\(914\) 0 0
\(915\) 3.16807 0.104733
\(916\) 0 0
\(917\) 10.2185 0.337443
\(918\) 0 0
\(919\) 30.1436 0.994347 0.497174 0.867651i \(-0.334371\pi\)
0.497174 + 0.867651i \(0.334371\pi\)
\(920\) 0 0
\(921\) −38.8811 −1.28117
\(922\) 0 0
\(923\) −40.8127 −1.34337
\(924\) 0 0
\(925\) −14.8712 −0.488962
\(926\) 0 0
\(927\) −67.1722 −2.20622
\(928\) 0 0
\(929\) 21.2975 0.698750 0.349375 0.936983i \(-0.386394\pi\)
0.349375 + 0.936983i \(0.386394\pi\)
\(930\) 0 0
\(931\) −15.0686 −0.493855
\(932\) 0 0
\(933\) 53.3585 1.74688
\(934\) 0 0
\(935\) −2.22339 −0.0727128
\(936\) 0 0
\(937\) 29.5345 0.964851 0.482426 0.875937i \(-0.339756\pi\)
0.482426 + 0.875937i \(0.339756\pi\)
\(938\) 0 0
\(939\) 6.45968 0.210804
\(940\) 0 0
\(941\) 23.7209 0.773281 0.386640 0.922231i \(-0.373635\pi\)
0.386640 + 0.922231i \(0.373635\pi\)
\(942\) 0 0
\(943\) −0.740208 −0.0241045
\(944\) 0 0
\(945\) −11.3055 −0.367770
\(946\) 0 0
\(947\) −7.01806 −0.228056 −0.114028 0.993478i \(-0.536375\pi\)
−0.114028 + 0.993478i \(0.536375\pi\)
\(948\) 0 0
\(949\) −12.5781 −0.408301
\(950\) 0 0
\(951\) −27.3538 −0.887009
\(952\) 0 0
\(953\) 35.5749 1.15238 0.576192 0.817314i \(-0.304538\pi\)
0.576192 + 0.817314i \(0.304538\pi\)
\(954\) 0 0
\(955\) 4.67926 0.151417
\(956\) 0 0
\(957\) −48.3242 −1.56210
\(958\) 0 0
\(959\) 30.8793 0.997144
\(960\) 0 0
\(961\) −30.8153 −0.994043
\(962\) 0 0
\(963\) 78.2830 2.52263
\(964\) 0 0
\(965\) −15.4299 −0.496705
\(966\) 0 0
\(967\) −26.5148 −0.852658 −0.426329 0.904568i \(-0.640193\pi\)
−0.426329 + 0.904568i \(0.640193\pi\)
\(968\) 0 0
\(969\) −14.4379 −0.463811
\(970\) 0 0
\(971\) −40.7636 −1.30816 −0.654082 0.756423i \(-0.726945\pi\)
−0.654082 + 0.756423i \(0.726945\pi\)
\(972\) 0 0
\(973\) 20.7482 0.665157
\(974\) 0 0
\(975\) 37.6835 1.20684
\(976\) 0 0
\(977\) −46.3207 −1.48193 −0.740966 0.671543i \(-0.765632\pi\)
−0.740966 + 0.671543i \(0.765632\pi\)
\(978\) 0 0
\(979\) −23.0745 −0.737464
\(980\) 0 0
\(981\) 55.6547 1.77692
\(982\) 0 0
\(983\) −0.0236923 −0.000755668 0 −0.000377834 1.00000i \(-0.500120\pi\)
−0.000377834 1.00000i \(0.500120\pi\)
\(984\) 0 0
\(985\) 12.2281 0.389619
\(986\) 0 0
\(987\) 18.4716 0.587958
\(988\) 0 0
\(989\) 87.8605 2.79380
\(990\) 0 0
\(991\) 25.8366 0.820726 0.410363 0.911922i \(-0.365402\pi\)
0.410363 + 0.911922i \(0.365402\pi\)
\(992\) 0 0
\(993\) 11.5337 0.366010
\(994\) 0 0
\(995\) 26.3050 0.833925
\(996\) 0 0
\(997\) −3.76142 −0.119125 −0.0595627 0.998225i \(-0.518971\pi\)
−0.0595627 + 0.998225i \(0.518971\pi\)
\(998\) 0 0
\(999\) 12.5349 0.396585
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 4012.2.a.h.1.15 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4012.2.a.h.1.15 15 1.1 even 1 trivial