Properties

Label 5994.2.a.bb.1.2
Level $5994$
Weight $2$
Character 5994.1
Self dual yes
Analytic conductor $47.862$
Analytic rank $0$
Dimension $10$
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] = [5994,2,Mod(1,5994)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5994, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5994.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5994 = 2 \cdot 3^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5994.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: \(47.8623309716\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 26x^{8} + 49x^{7} + 236x^{6} - 420x^{5} - 860x^{4} + 1461x^{3} + 993x^{2} - 1638x + 99 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 666)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.15778\) of defining polynomial
Character \(\chi\) \(=\) 5994.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.33861 q^{5} +0.195962 q^{7} +1.00000 q^{8} -3.33861 q^{10} -5.08601 q^{11} -5.32840 q^{13} +0.195962 q^{14} +1.00000 q^{16} +0.739743 q^{17} +5.26526 q^{19} -3.33861 q^{20} -5.08601 q^{22} +2.50863 q^{23} +6.14633 q^{25} -5.32840 q^{26} +0.195962 q^{28} -6.13351 q^{29} -1.59981 q^{31} +1.00000 q^{32} +0.739743 q^{34} -0.654240 q^{35} -1.00000 q^{37} +5.26526 q^{38} -3.33861 q^{40} -5.25874 q^{41} +4.47515 q^{43} -5.08601 q^{44} +2.50863 q^{46} +13.4814 q^{47} -6.96160 q^{49} +6.14633 q^{50} -5.32840 q^{52} -9.76490 q^{53} +16.9802 q^{55} +0.195962 q^{56} -6.13351 q^{58} +0.821294 q^{59} +6.12609 q^{61} -1.59981 q^{62} +1.00000 q^{64} +17.7895 q^{65} +7.02877 q^{67} +0.739743 q^{68} -0.654240 q^{70} +10.4556 q^{71} +7.73574 q^{73} -1.00000 q^{74} +5.26526 q^{76} -0.996662 q^{77} -0.927623 q^{79} -3.33861 q^{80} -5.25874 q^{82} -8.85942 q^{83} -2.46971 q^{85} +4.47515 q^{86} -5.08601 q^{88} -10.6441 q^{89} -1.04416 q^{91} +2.50863 q^{92} +13.4814 q^{94} -17.5787 q^{95} +10.4087 q^{97} -6.96160 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 10 q^{4} + q^{5} + q^{7} + 10 q^{8} + q^{10} + 3 q^{11} + 12 q^{13} + q^{14} + 10 q^{16} + 12 q^{17} + 24 q^{19} + q^{20} + 3 q^{22} + 3 q^{23} + 21 q^{25} + 12 q^{26} + q^{28} + 4 q^{29}+ \cdots + 35 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.33861 −1.49307 −0.746536 0.665344i \(-0.768285\pi\)
−0.746536 + 0.665344i \(0.768285\pi\)
\(6\) 0 0
\(7\) 0.195962 0.0740665 0.0370333 0.999314i \(-0.488209\pi\)
0.0370333 + 0.999314i \(0.488209\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −3.33861 −1.05576
\(11\) −5.08601 −1.53349 −0.766745 0.641952i \(-0.778125\pi\)
−0.766745 + 0.641952i \(0.778125\pi\)
\(12\) 0 0
\(13\) −5.32840 −1.47783 −0.738917 0.673797i \(-0.764662\pi\)
−0.738917 + 0.673797i \(0.764662\pi\)
\(14\) 0.195962 0.0523729
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.739743 0.179414 0.0897070 0.995968i \(-0.471407\pi\)
0.0897070 + 0.995968i \(0.471407\pi\)
\(18\) 0 0
\(19\) 5.26526 1.20793 0.603966 0.797010i \(-0.293586\pi\)
0.603966 + 0.797010i \(0.293586\pi\)
\(20\) −3.33861 −0.746536
\(21\) 0 0
\(22\) −5.08601 −1.08434
\(23\) 2.50863 0.523085 0.261543 0.965192i \(-0.415769\pi\)
0.261543 + 0.965192i \(0.415769\pi\)
\(24\) 0 0
\(25\) 6.14633 1.22927
\(26\) −5.32840 −1.04499
\(27\) 0 0
\(28\) 0.195962 0.0370333
\(29\) −6.13351 −1.13896 −0.569482 0.822004i \(-0.692856\pi\)
−0.569482 + 0.822004i \(0.692856\pi\)
\(30\) 0 0
\(31\) −1.59981 −0.287334 −0.143667 0.989626i \(-0.545889\pi\)
−0.143667 + 0.989626i \(0.545889\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.739743 0.126865
\(35\) −0.654240 −0.110587
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 5.26526 0.854137
\(39\) 0 0
\(40\) −3.33861 −0.527881
\(41\) −5.25874 −0.821277 −0.410638 0.911798i \(-0.634694\pi\)
−0.410638 + 0.911798i \(0.634694\pi\)
\(42\) 0 0
\(43\) 4.47515 0.682453 0.341227 0.939981i \(-0.389158\pi\)
0.341227 + 0.939981i \(0.389158\pi\)
\(44\) −5.08601 −0.766745
\(45\) 0 0
\(46\) 2.50863 0.369877
\(47\) 13.4814 1.96647 0.983233 0.182352i \(-0.0583709\pi\)
0.983233 + 0.182352i \(0.0583709\pi\)
\(48\) 0 0
\(49\) −6.96160 −0.994514
\(50\) 6.14633 0.869223
\(51\) 0 0
\(52\) −5.32840 −0.738917
\(53\) −9.76490 −1.34131 −0.670656 0.741768i \(-0.733987\pi\)
−0.670656 + 0.741768i \(0.733987\pi\)
\(54\) 0 0
\(55\) 16.9802 2.28961
\(56\) 0.195962 0.0261865
\(57\) 0 0
\(58\) −6.13351 −0.805369
\(59\) 0.821294 0.106923 0.0534617 0.998570i \(-0.482974\pi\)
0.0534617 + 0.998570i \(0.482974\pi\)
\(60\) 0 0
\(61\) 6.12609 0.784366 0.392183 0.919887i \(-0.371720\pi\)
0.392183 + 0.919887i \(0.371720\pi\)
\(62\) −1.59981 −0.203176
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 17.7895 2.20651
\(66\) 0 0
\(67\) 7.02877 0.858700 0.429350 0.903138i \(-0.358743\pi\)
0.429350 + 0.903138i \(0.358743\pi\)
\(68\) 0.739743 0.0897070
\(69\) 0 0
\(70\) −0.654240 −0.0781966
\(71\) 10.4556 1.24085 0.620426 0.784265i \(-0.286960\pi\)
0.620426 + 0.784265i \(0.286960\pi\)
\(72\) 0 0
\(73\) 7.73574 0.905400 0.452700 0.891663i \(-0.350461\pi\)
0.452700 + 0.891663i \(0.350461\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 5.26526 0.603966
\(77\) −0.996662 −0.113580
\(78\) 0 0
\(79\) −0.927623 −0.104366 −0.0521829 0.998638i \(-0.516618\pi\)
−0.0521829 + 0.998638i \(0.516618\pi\)
\(80\) −3.33861 −0.373268
\(81\) 0 0
\(82\) −5.25874 −0.580730
\(83\) −8.85942 −0.972448 −0.486224 0.873834i \(-0.661626\pi\)
−0.486224 + 0.873834i \(0.661626\pi\)
\(84\) 0 0
\(85\) −2.46971 −0.267878
\(86\) 4.47515 0.482567
\(87\) 0 0
\(88\) −5.08601 −0.542170
\(89\) −10.6441 −1.12827 −0.564137 0.825682i \(-0.690791\pi\)
−0.564137 + 0.825682i \(0.690791\pi\)
\(90\) 0 0
\(91\) −1.04416 −0.109458
\(92\) 2.50863 0.261543
\(93\) 0 0
\(94\) 13.4814 1.39050
\(95\) −17.5787 −1.80353
\(96\) 0 0
\(97\) 10.4087 1.05684 0.528422 0.848982i \(-0.322784\pi\)
0.528422 + 0.848982i \(0.322784\pi\)
\(98\) −6.96160 −0.703228
\(99\) 0 0
\(100\) 6.14633 0.614633
\(101\) −2.91641 −0.290193 −0.145097 0.989417i \(-0.546349\pi\)
−0.145097 + 0.989417i \(0.546349\pi\)
\(102\) 0 0
\(103\) 12.6134 1.24284 0.621419 0.783478i \(-0.286557\pi\)
0.621419 + 0.783478i \(0.286557\pi\)
\(104\) −5.32840 −0.522493
\(105\) 0 0
\(106\) −9.76490 −0.948451
\(107\) 3.87889 0.374987 0.187493 0.982266i \(-0.439964\pi\)
0.187493 + 0.982266i \(0.439964\pi\)
\(108\) 0 0
\(109\) 0.536436 0.0513812 0.0256906 0.999670i \(-0.491822\pi\)
0.0256906 + 0.999670i \(0.491822\pi\)
\(110\) 16.9802 1.61900
\(111\) 0 0
\(112\) 0.195962 0.0185166
\(113\) 19.0344 1.79061 0.895304 0.445456i \(-0.146959\pi\)
0.895304 + 0.445456i \(0.146959\pi\)
\(114\) 0 0
\(115\) −8.37534 −0.781005
\(116\) −6.13351 −0.569482
\(117\) 0 0
\(118\) 0.821294 0.0756063
\(119\) 0.144961 0.0132886
\(120\) 0 0
\(121\) 14.8675 1.35159
\(122\) 6.12609 0.554630
\(123\) 0 0
\(124\) −1.59981 −0.143667
\(125\) −3.82717 −0.342312
\(126\) 0 0
\(127\) 20.0651 1.78049 0.890247 0.455479i \(-0.150532\pi\)
0.890247 + 0.455479i \(0.150532\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 17.7895 1.56024
\(131\) 8.42646 0.736223 0.368112 0.929782i \(-0.380004\pi\)
0.368112 + 0.929782i \(0.380004\pi\)
\(132\) 0 0
\(133\) 1.03179 0.0894674
\(134\) 7.02877 0.607193
\(135\) 0 0
\(136\) 0.739743 0.0634324
\(137\) −20.8102 −1.77794 −0.888969 0.457968i \(-0.848577\pi\)
−0.888969 + 0.457968i \(0.848577\pi\)
\(138\) 0 0
\(139\) −0.122437 −0.0103849 −0.00519246 0.999987i \(-0.501653\pi\)
−0.00519246 + 0.999987i \(0.501653\pi\)
\(140\) −0.654240 −0.0552934
\(141\) 0 0
\(142\) 10.4556 0.877414
\(143\) 27.1003 2.26624
\(144\) 0 0
\(145\) 20.4774 1.70056
\(146\) 7.73574 0.640214
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) 6.84115 0.560449 0.280224 0.959935i \(-0.409591\pi\)
0.280224 + 0.959935i \(0.409591\pi\)
\(150\) 0 0
\(151\) 21.8371 1.77708 0.888538 0.458803i \(-0.151722\pi\)
0.888538 + 0.458803i \(0.151722\pi\)
\(152\) 5.26526 0.427069
\(153\) 0 0
\(154\) −0.996662 −0.0803133
\(155\) 5.34115 0.429011
\(156\) 0 0
\(157\) −19.8674 −1.58559 −0.792797 0.609486i \(-0.791376\pi\)
−0.792797 + 0.609486i \(0.791376\pi\)
\(158\) −0.927623 −0.0737977
\(159\) 0 0
\(160\) −3.33861 −0.263941
\(161\) 0.491595 0.0387431
\(162\) 0 0
\(163\) −4.27443 −0.334799 −0.167399 0.985889i \(-0.553537\pi\)
−0.167399 + 0.985889i \(0.553537\pi\)
\(164\) −5.25874 −0.410638
\(165\) 0 0
\(166\) −8.85942 −0.687624
\(167\) 0.615260 0.0476102 0.0238051 0.999717i \(-0.492422\pi\)
0.0238051 + 0.999717i \(0.492422\pi\)
\(168\) 0 0
\(169\) 15.3919 1.18399
\(170\) −2.46971 −0.189418
\(171\) 0 0
\(172\) 4.47515 0.341227
\(173\) 20.9888 1.59575 0.797873 0.602826i \(-0.205959\pi\)
0.797873 + 0.602826i \(0.205959\pi\)
\(174\) 0 0
\(175\) 1.20445 0.0910475
\(176\) −5.08601 −0.383372
\(177\) 0 0
\(178\) −10.6441 −0.797810
\(179\) −22.4414 −1.67735 −0.838673 0.544635i \(-0.816668\pi\)
−0.838673 + 0.544635i \(0.816668\pi\)
\(180\) 0 0
\(181\) 5.60462 0.416588 0.208294 0.978066i \(-0.433209\pi\)
0.208294 + 0.978066i \(0.433209\pi\)
\(182\) −1.04416 −0.0773985
\(183\) 0 0
\(184\) 2.50863 0.184939
\(185\) 3.33861 0.245460
\(186\) 0 0
\(187\) −3.76234 −0.275129
\(188\) 13.4814 0.983233
\(189\) 0 0
\(190\) −17.5787 −1.27529
\(191\) −4.12388 −0.298393 −0.149197 0.988808i \(-0.547669\pi\)
−0.149197 + 0.988808i \(0.547669\pi\)
\(192\) 0 0
\(193\) −8.55099 −0.615513 −0.307757 0.951465i \(-0.599578\pi\)
−0.307757 + 0.951465i \(0.599578\pi\)
\(194\) 10.4087 0.747302
\(195\) 0 0
\(196\) −6.96160 −0.497257
\(197\) 13.1755 0.938714 0.469357 0.883009i \(-0.344486\pi\)
0.469357 + 0.883009i \(0.344486\pi\)
\(198\) 0 0
\(199\) 18.5811 1.31718 0.658589 0.752503i \(-0.271154\pi\)
0.658589 + 0.752503i \(0.271154\pi\)
\(200\) 6.14633 0.434611
\(201\) 0 0
\(202\) −2.91641 −0.205198
\(203\) −1.20193 −0.0843591
\(204\) 0 0
\(205\) 17.5569 1.22623
\(206\) 12.6134 0.878819
\(207\) 0 0
\(208\) −5.32840 −0.369458
\(209\) −26.7791 −1.85235
\(210\) 0 0
\(211\) −4.80875 −0.331048 −0.165524 0.986206i \(-0.552932\pi\)
−0.165524 + 0.986206i \(0.552932\pi\)
\(212\) −9.76490 −0.670656
\(213\) 0 0
\(214\) 3.87889 0.265156
\(215\) −14.9408 −1.01895
\(216\) 0 0
\(217\) −0.313501 −0.0212819
\(218\) 0.536436 0.0363320
\(219\) 0 0
\(220\) 16.9802 1.14481
\(221\) −3.94165 −0.265144
\(222\) 0 0
\(223\) 5.28621 0.353991 0.176995 0.984212i \(-0.443362\pi\)
0.176995 + 0.984212i \(0.443362\pi\)
\(224\) 0.195962 0.0130932
\(225\) 0 0
\(226\) 19.0344 1.26615
\(227\) 6.52175 0.432864 0.216432 0.976298i \(-0.430558\pi\)
0.216432 + 0.976298i \(0.430558\pi\)
\(228\) 0 0
\(229\) 5.93818 0.392406 0.196203 0.980563i \(-0.437139\pi\)
0.196203 + 0.980563i \(0.437139\pi\)
\(230\) −8.37534 −0.552254
\(231\) 0 0
\(232\) −6.13351 −0.402685
\(233\) 18.4107 1.20613 0.603063 0.797693i \(-0.293947\pi\)
0.603063 + 0.797693i \(0.293947\pi\)
\(234\) 0 0
\(235\) −45.0092 −2.93608
\(236\) 0.821294 0.0534617
\(237\) 0 0
\(238\) 0.144961 0.00939644
\(239\) 24.0032 1.55264 0.776320 0.630339i \(-0.217084\pi\)
0.776320 + 0.630339i \(0.217084\pi\)
\(240\) 0 0
\(241\) −17.2454 −1.11087 −0.555437 0.831559i \(-0.687449\pi\)
−0.555437 + 0.831559i \(0.687449\pi\)
\(242\) 14.8675 0.955717
\(243\) 0 0
\(244\) 6.12609 0.392183
\(245\) 23.2421 1.48488
\(246\) 0 0
\(247\) −28.0554 −1.78512
\(248\) −1.59981 −0.101588
\(249\) 0 0
\(250\) −3.82717 −0.242051
\(251\) 5.56820 0.351462 0.175731 0.984438i \(-0.443771\pi\)
0.175731 + 0.984438i \(0.443771\pi\)
\(252\) 0 0
\(253\) −12.7589 −0.802146
\(254\) 20.0651 1.25900
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.8042 −0.798705 −0.399352 0.916798i \(-0.630765\pi\)
−0.399352 + 0.916798i \(0.630765\pi\)
\(258\) 0 0
\(259\) −0.195962 −0.0121765
\(260\) 17.7895 1.10326
\(261\) 0 0
\(262\) 8.42646 0.520588
\(263\) 3.24586 0.200149 0.100074 0.994980i \(-0.468092\pi\)
0.100074 + 0.994980i \(0.468092\pi\)
\(264\) 0 0
\(265\) 32.6012 2.00268
\(266\) 1.03179 0.0632630
\(267\) 0 0
\(268\) 7.02877 0.429350
\(269\) −20.4388 −1.24618 −0.623089 0.782151i \(-0.714122\pi\)
−0.623089 + 0.782151i \(0.714122\pi\)
\(270\) 0 0
\(271\) −18.6519 −1.13303 −0.566513 0.824053i \(-0.691708\pi\)
−0.566513 + 0.824053i \(0.691708\pi\)
\(272\) 0.739743 0.0448535
\(273\) 0 0
\(274\) −20.8102 −1.25719
\(275\) −31.2603 −1.88507
\(276\) 0 0
\(277\) −6.39476 −0.384224 −0.192112 0.981373i \(-0.561534\pi\)
−0.192112 + 0.981373i \(0.561534\pi\)
\(278\) −0.122437 −0.00734325
\(279\) 0 0
\(280\) −0.654240 −0.0390983
\(281\) −20.2860 −1.21016 −0.605081 0.796164i \(-0.706859\pi\)
−0.605081 + 0.796164i \(0.706859\pi\)
\(282\) 0 0
\(283\) 21.7594 1.29346 0.646730 0.762719i \(-0.276136\pi\)
0.646730 + 0.762719i \(0.276136\pi\)
\(284\) 10.4556 0.620426
\(285\) 0 0
\(286\) 27.1003 1.60247
\(287\) −1.03051 −0.0608291
\(288\) 0 0
\(289\) −16.4528 −0.967811
\(290\) 20.4774 1.20248
\(291\) 0 0
\(292\) 7.73574 0.452700
\(293\) 29.9252 1.74825 0.874125 0.485700i \(-0.161435\pi\)
0.874125 + 0.485700i \(0.161435\pi\)
\(294\) 0 0
\(295\) −2.74198 −0.159644
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) 6.84115 0.396297
\(299\) −13.3670 −0.773033
\(300\) 0 0
\(301\) 0.876957 0.0505469
\(302\) 21.8371 1.25658
\(303\) 0 0
\(304\) 5.26526 0.301983
\(305\) −20.4527 −1.17112
\(306\) 0 0
\(307\) −22.7730 −1.29972 −0.649861 0.760053i \(-0.725173\pi\)
−0.649861 + 0.760053i \(0.725173\pi\)
\(308\) −0.996662 −0.0567901
\(309\) 0 0
\(310\) 5.34115 0.303357
\(311\) −31.3806 −1.77943 −0.889716 0.456513i \(-0.849098\pi\)
−0.889716 + 0.456513i \(0.849098\pi\)
\(312\) 0 0
\(313\) 4.18723 0.236676 0.118338 0.992973i \(-0.462243\pi\)
0.118338 + 0.992973i \(0.462243\pi\)
\(314\) −19.8674 −1.12118
\(315\) 0 0
\(316\) −0.927623 −0.0521829
\(317\) −9.71753 −0.545791 −0.272895 0.962044i \(-0.587981\pi\)
−0.272895 + 0.962044i \(0.587981\pi\)
\(318\) 0 0
\(319\) 31.1951 1.74659
\(320\) −3.33861 −0.186634
\(321\) 0 0
\(322\) 0.491595 0.0273955
\(323\) 3.89493 0.216720
\(324\) 0 0
\(325\) −32.7502 −1.81665
\(326\) −4.27443 −0.236739
\(327\) 0 0
\(328\) −5.25874 −0.290365
\(329\) 2.64184 0.145649
\(330\) 0 0
\(331\) 4.19467 0.230560 0.115280 0.993333i \(-0.463223\pi\)
0.115280 + 0.993333i \(0.463223\pi\)
\(332\) −8.85942 −0.486224
\(333\) 0 0
\(334\) 0.615260 0.0336655
\(335\) −23.4663 −1.28210
\(336\) 0 0
\(337\) 17.3297 0.944010 0.472005 0.881596i \(-0.343530\pi\)
0.472005 + 0.881596i \(0.343530\pi\)
\(338\) 15.3919 0.837208
\(339\) 0 0
\(340\) −2.46971 −0.133939
\(341\) 8.13665 0.440624
\(342\) 0 0
\(343\) −2.73594 −0.147727
\(344\) 4.47515 0.241284
\(345\) 0 0
\(346\) 20.9888 1.12836
\(347\) −0.604429 −0.0324474 −0.0162237 0.999868i \(-0.505164\pi\)
−0.0162237 + 0.999868i \(0.505164\pi\)
\(348\) 0 0
\(349\) 15.8975 0.850971 0.425486 0.904965i \(-0.360103\pi\)
0.425486 + 0.904965i \(0.360103\pi\)
\(350\) 1.20445 0.0643803
\(351\) 0 0
\(352\) −5.08601 −0.271085
\(353\) −5.98441 −0.318518 −0.159259 0.987237i \(-0.550911\pi\)
−0.159259 + 0.987237i \(0.550911\pi\)
\(354\) 0 0
\(355\) −34.9072 −1.85268
\(356\) −10.6441 −0.564137
\(357\) 0 0
\(358\) −22.4414 −1.18606
\(359\) 7.52687 0.397253 0.198627 0.980075i \(-0.436352\pi\)
0.198627 + 0.980075i \(0.436352\pi\)
\(360\) 0 0
\(361\) 8.72292 0.459101
\(362\) 5.60462 0.294572
\(363\) 0 0
\(364\) −1.04416 −0.0547290
\(365\) −25.8266 −1.35183
\(366\) 0 0
\(367\) −18.5266 −0.967080 −0.483540 0.875322i \(-0.660649\pi\)
−0.483540 + 0.875322i \(0.660649\pi\)
\(368\) 2.50863 0.130771
\(369\) 0 0
\(370\) 3.33861 0.173566
\(371\) −1.91355 −0.0993464
\(372\) 0 0
\(373\) −29.4102 −1.52280 −0.761401 0.648282i \(-0.775488\pi\)
−0.761401 + 0.648282i \(0.775488\pi\)
\(374\) −3.76234 −0.194546
\(375\) 0 0
\(376\) 13.4814 0.695251
\(377\) 32.6818 1.68320
\(378\) 0 0
\(379\) 10.0742 0.517475 0.258738 0.965948i \(-0.416694\pi\)
0.258738 + 0.965948i \(0.416694\pi\)
\(380\) −17.5787 −0.901766
\(381\) 0 0
\(382\) −4.12388 −0.210996
\(383\) 15.6220 0.798248 0.399124 0.916897i \(-0.369314\pi\)
0.399124 + 0.916897i \(0.369314\pi\)
\(384\) 0 0
\(385\) 3.32747 0.169584
\(386\) −8.55099 −0.435234
\(387\) 0 0
\(388\) 10.4087 0.528422
\(389\) 16.6098 0.842151 0.421076 0.907026i \(-0.361653\pi\)
0.421076 + 0.907026i \(0.361653\pi\)
\(390\) 0 0
\(391\) 1.85574 0.0938488
\(392\) −6.96160 −0.351614
\(393\) 0 0
\(394\) 13.1755 0.663771
\(395\) 3.09697 0.155826
\(396\) 0 0
\(397\) 23.9512 1.20207 0.601037 0.799221i \(-0.294754\pi\)
0.601037 + 0.799221i \(0.294754\pi\)
\(398\) 18.5811 0.931385
\(399\) 0 0
\(400\) 6.14633 0.307317
\(401\) −28.2179 −1.40913 −0.704566 0.709638i \(-0.748858\pi\)
−0.704566 + 0.709638i \(0.748858\pi\)
\(402\) 0 0
\(403\) 8.52443 0.424632
\(404\) −2.91641 −0.145097
\(405\) 0 0
\(406\) −1.20193 −0.0596509
\(407\) 5.08601 0.252104
\(408\) 0 0
\(409\) 20.8364 1.03029 0.515147 0.857102i \(-0.327737\pi\)
0.515147 + 0.857102i \(0.327737\pi\)
\(410\) 17.5569 0.867073
\(411\) 0 0
\(412\) 12.6134 0.621419
\(413\) 0.160942 0.00791945
\(414\) 0 0
\(415\) 29.5782 1.45194
\(416\) −5.32840 −0.261246
\(417\) 0 0
\(418\) −26.7791 −1.30981
\(419\) 23.4768 1.14691 0.573457 0.819235i \(-0.305602\pi\)
0.573457 + 0.819235i \(0.305602\pi\)
\(420\) 0 0
\(421\) −26.6544 −1.29905 −0.649527 0.760338i \(-0.725033\pi\)
−0.649527 + 0.760338i \(0.725033\pi\)
\(422\) −4.80875 −0.234086
\(423\) 0 0
\(424\) −9.76490 −0.474226
\(425\) 4.54671 0.220548
\(426\) 0 0
\(427\) 1.20048 0.0580952
\(428\) 3.87889 0.187493
\(429\) 0 0
\(430\) −14.9408 −0.720508
\(431\) 27.9472 1.34617 0.673085 0.739565i \(-0.264969\pi\)
0.673085 + 0.739565i \(0.264969\pi\)
\(432\) 0 0
\(433\) 14.4980 0.696730 0.348365 0.937359i \(-0.386737\pi\)
0.348365 + 0.937359i \(0.386737\pi\)
\(434\) −0.313501 −0.0150485
\(435\) 0 0
\(436\) 0.536436 0.0256906
\(437\) 13.2086 0.631852
\(438\) 0 0
\(439\) 10.2482 0.489122 0.244561 0.969634i \(-0.421356\pi\)
0.244561 + 0.969634i \(0.421356\pi\)
\(440\) 16.9802 0.809500
\(441\) 0 0
\(442\) −3.94165 −0.187485
\(443\) 8.13881 0.386687 0.193343 0.981131i \(-0.438067\pi\)
0.193343 + 0.981131i \(0.438067\pi\)
\(444\) 0 0
\(445\) 35.5365 1.68459
\(446\) 5.28621 0.250309
\(447\) 0 0
\(448\) 0.195962 0.00925832
\(449\) 1.81755 0.0857754 0.0428877 0.999080i \(-0.486344\pi\)
0.0428877 + 0.999080i \(0.486344\pi\)
\(450\) 0 0
\(451\) 26.7460 1.25942
\(452\) 19.0344 0.895304
\(453\) 0 0
\(454\) 6.52175 0.306081
\(455\) 3.48605 0.163429
\(456\) 0 0
\(457\) −38.9876 −1.82376 −0.911880 0.410456i \(-0.865370\pi\)
−0.911880 + 0.410456i \(0.865370\pi\)
\(458\) 5.93818 0.277473
\(459\) 0 0
\(460\) −8.37534 −0.390502
\(461\) −14.6527 −0.682445 −0.341223 0.939983i \(-0.610841\pi\)
−0.341223 + 0.939983i \(0.610841\pi\)
\(462\) 0 0
\(463\) 25.3342 1.17738 0.588689 0.808359i \(-0.299644\pi\)
0.588689 + 0.808359i \(0.299644\pi\)
\(464\) −6.13351 −0.284741
\(465\) 0 0
\(466\) 18.4107 0.852861
\(467\) −6.52767 −0.302065 −0.151032 0.988529i \(-0.548260\pi\)
−0.151032 + 0.988529i \(0.548260\pi\)
\(468\) 0 0
\(469\) 1.37737 0.0636010
\(470\) −45.0092 −2.07612
\(471\) 0 0
\(472\) 0.821294 0.0378031
\(473\) −22.7606 −1.04653
\(474\) 0 0
\(475\) 32.3620 1.48487
\(476\) 0.144961 0.00664428
\(477\) 0 0
\(478\) 24.0032 1.09788
\(479\) 19.4006 0.886438 0.443219 0.896413i \(-0.353836\pi\)
0.443219 + 0.896413i \(0.353836\pi\)
\(480\) 0 0
\(481\) 5.32840 0.242954
\(482\) −17.2454 −0.785506
\(483\) 0 0
\(484\) 14.8675 0.675794
\(485\) −34.7507 −1.57795
\(486\) 0 0
\(487\) −11.5204 −0.522040 −0.261020 0.965333i \(-0.584059\pi\)
−0.261020 + 0.965333i \(0.584059\pi\)
\(488\) 6.12609 0.277315
\(489\) 0 0
\(490\) 23.2421 1.04997
\(491\) −30.0958 −1.35821 −0.679103 0.734043i \(-0.737631\pi\)
−0.679103 + 0.734043i \(0.737631\pi\)
\(492\) 0 0
\(493\) −4.53722 −0.204346
\(494\) −28.0554 −1.26227
\(495\) 0 0
\(496\) −1.59981 −0.0718336
\(497\) 2.04890 0.0919055
\(498\) 0 0
\(499\) 19.6116 0.877934 0.438967 0.898503i \(-0.355345\pi\)
0.438967 + 0.898503i \(0.355345\pi\)
\(500\) −3.82717 −0.171156
\(501\) 0 0
\(502\) 5.56820 0.248521
\(503\) −19.1959 −0.855903 −0.427951 0.903802i \(-0.640765\pi\)
−0.427951 + 0.903802i \(0.640765\pi\)
\(504\) 0 0
\(505\) 9.73675 0.433280
\(506\) −12.7589 −0.567203
\(507\) 0 0
\(508\) 20.0651 0.890247
\(509\) 29.4055 1.30337 0.651687 0.758488i \(-0.274062\pi\)
0.651687 + 0.758488i \(0.274062\pi\)
\(510\) 0 0
\(511\) 1.51591 0.0670598
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −12.8042 −0.564769
\(515\) −42.1113 −1.85565
\(516\) 0 0
\(517\) −68.5666 −3.01556
\(518\) −0.195962 −0.00861006
\(519\) 0 0
\(520\) 17.7895 0.780120
\(521\) −1.28738 −0.0564013 −0.0282006 0.999602i \(-0.508978\pi\)
−0.0282006 + 0.999602i \(0.508978\pi\)
\(522\) 0 0
\(523\) −4.27752 −0.187043 −0.0935215 0.995617i \(-0.529812\pi\)
−0.0935215 + 0.995617i \(0.529812\pi\)
\(524\) 8.42646 0.368112
\(525\) 0 0
\(526\) 3.24586 0.141526
\(527\) −1.18345 −0.0515518
\(528\) 0 0
\(529\) −16.7068 −0.726382
\(530\) 32.6012 1.41611
\(531\) 0 0
\(532\) 1.03179 0.0447337
\(533\) 28.0207 1.21371
\(534\) 0 0
\(535\) −12.9501 −0.559882
\(536\) 7.02877 0.303596
\(537\) 0 0
\(538\) −20.4388 −0.881180
\(539\) 35.4067 1.52508
\(540\) 0 0
\(541\) 5.96148 0.256304 0.128152 0.991755i \(-0.459095\pi\)
0.128152 + 0.991755i \(0.459095\pi\)
\(542\) −18.6519 −0.801170
\(543\) 0 0
\(544\) 0.739743 0.0317162
\(545\) −1.79095 −0.0767159
\(546\) 0 0
\(547\) 28.8625 1.23407 0.617036 0.786935i \(-0.288333\pi\)
0.617036 + 0.786935i \(0.288333\pi\)
\(548\) −20.8102 −0.888969
\(549\) 0 0
\(550\) −31.2603 −1.33294
\(551\) −32.2945 −1.37579
\(552\) 0 0
\(553\) −0.181778 −0.00773001
\(554\) −6.39476 −0.271687
\(555\) 0 0
\(556\) −0.122437 −0.00519246
\(557\) 27.3289 1.15796 0.578981 0.815341i \(-0.303451\pi\)
0.578981 + 0.815341i \(0.303451\pi\)
\(558\) 0 0
\(559\) −23.8454 −1.00855
\(560\) −0.654240 −0.0276467
\(561\) 0 0
\(562\) −20.2860 −0.855714
\(563\) 31.1305 1.31199 0.655996 0.754764i \(-0.272249\pi\)
0.655996 + 0.754764i \(0.272249\pi\)
\(564\) 0 0
\(565\) −63.5485 −2.67351
\(566\) 21.7594 0.914615
\(567\) 0 0
\(568\) 10.4556 0.438707
\(569\) −20.9407 −0.877879 −0.438939 0.898517i \(-0.644646\pi\)
−0.438939 + 0.898517i \(0.644646\pi\)
\(570\) 0 0
\(571\) 37.5254 1.57039 0.785194 0.619250i \(-0.212563\pi\)
0.785194 + 0.619250i \(0.212563\pi\)
\(572\) 27.1003 1.13312
\(573\) 0 0
\(574\) −1.03051 −0.0430127
\(575\) 15.4189 0.643012
\(576\) 0 0
\(577\) −17.9434 −0.746993 −0.373496 0.927632i \(-0.621841\pi\)
−0.373496 + 0.927632i \(0.621841\pi\)
\(578\) −16.4528 −0.684345
\(579\) 0 0
\(580\) 20.4774 0.850278
\(581\) −1.73611 −0.0720258
\(582\) 0 0
\(583\) 49.6644 2.05689
\(584\) 7.73574 0.320107
\(585\) 0 0
\(586\) 29.9252 1.23620
\(587\) −37.8461 −1.56207 −0.781037 0.624484i \(-0.785309\pi\)
−0.781037 + 0.624484i \(0.785309\pi\)
\(588\) 0 0
\(589\) −8.42341 −0.347081
\(590\) −2.74198 −0.112886
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) −14.5823 −0.598823 −0.299412 0.954124i \(-0.596790\pi\)
−0.299412 + 0.954124i \(0.596790\pi\)
\(594\) 0 0
\(595\) −0.483969 −0.0198408
\(596\) 6.84115 0.280224
\(597\) 0 0
\(598\) −13.3670 −0.546617
\(599\) 0.156940 0.00641241 0.00320620 0.999995i \(-0.498979\pi\)
0.00320620 + 0.999995i \(0.498979\pi\)
\(600\) 0 0
\(601\) 36.9757 1.50827 0.754135 0.656720i \(-0.228056\pi\)
0.754135 + 0.656720i \(0.228056\pi\)
\(602\) 0.876957 0.0357421
\(603\) 0 0
\(604\) 21.8371 0.888538
\(605\) −49.6367 −2.01802
\(606\) 0 0
\(607\) 40.9855 1.66355 0.831776 0.555112i \(-0.187324\pi\)
0.831776 + 0.555112i \(0.187324\pi\)
\(608\) 5.26526 0.213534
\(609\) 0 0
\(610\) −20.4527 −0.828104
\(611\) −71.8344 −2.90611
\(612\) 0 0
\(613\) 3.72829 0.150584 0.0752921 0.997162i \(-0.476011\pi\)
0.0752921 + 0.997162i \(0.476011\pi\)
\(614\) −22.7730 −0.919042
\(615\) 0 0
\(616\) −0.996662 −0.0401567
\(617\) −16.5328 −0.665584 −0.332792 0.943000i \(-0.607991\pi\)
−0.332792 + 0.943000i \(0.607991\pi\)
\(618\) 0 0
\(619\) 24.5417 0.986415 0.493208 0.869912i \(-0.335824\pi\)
0.493208 + 0.869912i \(0.335824\pi\)
\(620\) 5.34115 0.214506
\(621\) 0 0
\(622\) −31.3806 −1.25825
\(623\) −2.08584 −0.0835673
\(624\) 0 0
\(625\) −17.9542 −0.718170
\(626\) 4.18723 0.167355
\(627\) 0 0
\(628\) −19.8674 −0.792797
\(629\) −0.739743 −0.0294955
\(630\) 0 0
\(631\) 6.74272 0.268423 0.134212 0.990953i \(-0.457150\pi\)
0.134212 + 0.990953i \(0.457150\pi\)
\(632\) −0.927623 −0.0368989
\(633\) 0 0
\(634\) −9.71753 −0.385932
\(635\) −66.9897 −2.65841
\(636\) 0 0
\(637\) 37.0942 1.46973
\(638\) 31.1951 1.23502
\(639\) 0 0
\(640\) −3.33861 −0.131970
\(641\) 33.4562 1.32144 0.660719 0.750633i \(-0.270251\pi\)
0.660719 + 0.750633i \(0.270251\pi\)
\(642\) 0 0
\(643\) 44.9939 1.77439 0.887193 0.461398i \(-0.152652\pi\)
0.887193 + 0.461398i \(0.152652\pi\)
\(644\) 0.491595 0.0193716
\(645\) 0 0
\(646\) 3.89493 0.153244
\(647\) 4.93692 0.194090 0.0970451 0.995280i \(-0.469061\pi\)
0.0970451 + 0.995280i \(0.469061\pi\)
\(648\) 0 0
\(649\) −4.17711 −0.163966
\(650\) −32.7502 −1.28457
\(651\) 0 0
\(652\) −4.27443 −0.167399
\(653\) 36.7104 1.43659 0.718294 0.695740i \(-0.244923\pi\)
0.718294 + 0.695740i \(0.244923\pi\)
\(654\) 0 0
\(655\) −28.1327 −1.09923
\(656\) −5.25874 −0.205319
\(657\) 0 0
\(658\) 2.64184 0.102990
\(659\) −43.6251 −1.69939 −0.849696 0.527273i \(-0.823215\pi\)
−0.849696 + 0.527273i \(0.823215\pi\)
\(660\) 0 0
\(661\) −13.6589 −0.531268 −0.265634 0.964074i \(-0.585581\pi\)
−0.265634 + 0.964074i \(0.585581\pi\)
\(662\) 4.19467 0.163031
\(663\) 0 0
\(664\) −8.85942 −0.343812
\(665\) −3.44474 −0.133581
\(666\) 0 0
\(667\) −15.3867 −0.595776
\(668\) 0.615260 0.0238051
\(669\) 0 0
\(670\) −23.4663 −0.906583
\(671\) −31.1574 −1.20282
\(672\) 0 0
\(673\) −33.0580 −1.27429 −0.637146 0.770743i \(-0.719885\pi\)
−0.637146 + 0.770743i \(0.719885\pi\)
\(674\) 17.3297 0.667516
\(675\) 0 0
\(676\) 15.3919 0.591996
\(677\) −17.1146 −0.657767 −0.328884 0.944370i \(-0.606672\pi\)
−0.328884 + 0.944370i \(0.606672\pi\)
\(678\) 0 0
\(679\) 2.03971 0.0782768
\(680\) −2.46971 −0.0947092
\(681\) 0 0
\(682\) 8.13665 0.311568
\(683\) −0.957620 −0.0366423 −0.0183212 0.999832i \(-0.505832\pi\)
−0.0183212 + 0.999832i \(0.505832\pi\)
\(684\) 0 0
\(685\) 69.4773 2.65459
\(686\) −2.73594 −0.104459
\(687\) 0 0
\(688\) 4.47515 0.170613
\(689\) 52.0314 1.98224
\(690\) 0 0
\(691\) −9.99795 −0.380340 −0.190170 0.981751i \(-0.560904\pi\)
−0.190170 + 0.981751i \(0.560904\pi\)
\(692\) 20.9888 0.797873
\(693\) 0 0
\(694\) −0.604429 −0.0229438
\(695\) 0.408768 0.0155055
\(696\) 0 0
\(697\) −3.89011 −0.147349
\(698\) 15.8975 0.601728
\(699\) 0 0
\(700\) 1.20445 0.0455238
\(701\) 3.05328 0.115321 0.0576604 0.998336i \(-0.481636\pi\)
0.0576604 + 0.998336i \(0.481636\pi\)
\(702\) 0 0
\(703\) −5.26526 −0.198583
\(704\) −5.08601 −0.191686
\(705\) 0 0
\(706\) −5.98441 −0.225226
\(707\) −0.571504 −0.0214936
\(708\) 0 0
\(709\) −47.9650 −1.80136 −0.900682 0.434479i \(-0.856933\pi\)
−0.900682 + 0.434479i \(0.856933\pi\)
\(710\) −34.9072 −1.31004
\(711\) 0 0
\(712\) −10.6441 −0.398905
\(713\) −4.01333 −0.150300
\(714\) 0 0
\(715\) −90.4774 −3.38366
\(716\) −22.4414 −0.838673
\(717\) 0 0
\(718\) 7.52687 0.280900
\(719\) 32.0932 1.19687 0.598437 0.801170i \(-0.295789\pi\)
0.598437 + 0.801170i \(0.295789\pi\)
\(720\) 0 0
\(721\) 2.47175 0.0920527
\(722\) 8.72292 0.324633
\(723\) 0 0
\(724\) 5.60462 0.208294
\(725\) −37.6986 −1.40009
\(726\) 0 0
\(727\) 23.7998 0.882686 0.441343 0.897339i \(-0.354502\pi\)
0.441343 + 0.897339i \(0.354502\pi\)
\(728\) −1.04416 −0.0386992
\(729\) 0 0
\(730\) −25.8266 −0.955887
\(731\) 3.31046 0.122442
\(732\) 0 0
\(733\) 6.63216 0.244964 0.122482 0.992471i \(-0.460915\pi\)
0.122482 + 0.992471i \(0.460915\pi\)
\(734\) −18.5266 −0.683829
\(735\) 0 0
\(736\) 2.50863 0.0924693
\(737\) −35.7484 −1.31681
\(738\) 0 0
\(739\) −42.4986 −1.56334 −0.781669 0.623694i \(-0.785631\pi\)
−0.781669 + 0.623694i \(0.785631\pi\)
\(740\) 3.33861 0.122730
\(741\) 0 0
\(742\) −1.91355 −0.0702485
\(743\) −43.9188 −1.61122 −0.805612 0.592443i \(-0.798164\pi\)
−0.805612 + 0.592443i \(0.798164\pi\)
\(744\) 0 0
\(745\) −22.8399 −0.836791
\(746\) −29.4102 −1.07678
\(747\) 0 0
\(748\) −3.76234 −0.137565
\(749\) 0.760114 0.0277740
\(750\) 0 0
\(751\) 10.6148 0.387340 0.193670 0.981067i \(-0.437961\pi\)
0.193670 + 0.981067i \(0.437961\pi\)
\(752\) 13.4814 0.491617
\(753\) 0 0
\(754\) 32.6818 1.19020
\(755\) −72.9055 −2.65330
\(756\) 0 0
\(757\) 40.1639 1.45978 0.729892 0.683563i \(-0.239570\pi\)
0.729892 + 0.683563i \(0.239570\pi\)
\(758\) 10.0742 0.365910
\(759\) 0 0
\(760\) −17.5787 −0.637645
\(761\) 17.1467 0.621568 0.310784 0.950481i \(-0.399408\pi\)
0.310784 + 0.950481i \(0.399408\pi\)
\(762\) 0 0
\(763\) 0.105121 0.00380563
\(764\) −4.12388 −0.149197
\(765\) 0 0
\(766\) 15.6220 0.564446
\(767\) −4.37619 −0.158015
\(768\) 0 0
\(769\) 11.9471 0.430824 0.215412 0.976523i \(-0.430891\pi\)
0.215412 + 0.976523i \(0.430891\pi\)
\(770\) 3.32747 0.119914
\(771\) 0 0
\(772\) −8.55099 −0.307757
\(773\) −50.8932 −1.83050 −0.915250 0.402886i \(-0.868007\pi\)
−0.915250 + 0.402886i \(0.868007\pi\)
\(774\) 0 0
\(775\) −9.83297 −0.353211
\(776\) 10.4087 0.373651
\(777\) 0 0
\(778\) 16.6098 0.595491
\(779\) −27.6886 −0.992047
\(780\) 0 0
\(781\) −53.1773 −1.90283
\(782\) 1.85574 0.0663611
\(783\) 0 0
\(784\) −6.96160 −0.248629
\(785\) 66.3297 2.36741
\(786\) 0 0
\(787\) −12.0089 −0.428071 −0.214036 0.976826i \(-0.568661\pi\)
−0.214036 + 0.976826i \(0.568661\pi\)
\(788\) 13.1755 0.469357
\(789\) 0 0
\(790\) 3.09697 0.110185
\(791\) 3.73001 0.132624
\(792\) 0 0
\(793\) −32.6423 −1.15916
\(794\) 23.9512 0.849995
\(795\) 0 0
\(796\) 18.5811 0.658589
\(797\) −1.21781 −0.0431372 −0.0215686 0.999767i \(-0.506866\pi\)
−0.0215686 + 0.999767i \(0.506866\pi\)
\(798\) 0 0
\(799\) 9.97278 0.352812
\(800\) 6.14633 0.217306
\(801\) 0 0
\(802\) −28.2179 −0.996407
\(803\) −39.3440 −1.38842
\(804\) 0 0
\(805\) −1.64125 −0.0578463
\(806\) 8.52443 0.300260
\(807\) 0 0
\(808\) −2.91641 −0.102599
\(809\) −31.5509 −1.10927 −0.554635 0.832094i \(-0.687142\pi\)
−0.554635 + 0.832094i \(0.687142\pi\)
\(810\) 0 0
\(811\) −11.0562 −0.388235 −0.194118 0.980978i \(-0.562184\pi\)
−0.194118 + 0.980978i \(0.562184\pi\)
\(812\) −1.20193 −0.0421796
\(813\) 0 0
\(814\) 5.08601 0.178264
\(815\) 14.2707 0.499879
\(816\) 0 0
\(817\) 23.5628 0.824358
\(818\) 20.8364 0.728527
\(819\) 0 0
\(820\) 17.5569 0.613113
\(821\) 19.6171 0.684642 0.342321 0.939583i \(-0.388787\pi\)
0.342321 + 0.939583i \(0.388787\pi\)
\(822\) 0 0
\(823\) −40.3719 −1.40728 −0.703638 0.710559i \(-0.748442\pi\)
−0.703638 + 0.710559i \(0.748442\pi\)
\(824\) 12.6134 0.439410
\(825\) 0 0
\(826\) 0.160942 0.00559989
\(827\) −44.3183 −1.54110 −0.770549 0.637381i \(-0.780018\pi\)
−0.770549 + 0.637381i \(0.780018\pi\)
\(828\) 0 0
\(829\) −14.1163 −0.490279 −0.245139 0.969488i \(-0.578834\pi\)
−0.245139 + 0.969488i \(0.578834\pi\)
\(830\) 29.5782 1.02667
\(831\) 0 0
\(832\) −5.32840 −0.184729
\(833\) −5.14979 −0.178430
\(834\) 0 0
\(835\) −2.05411 −0.0710855
\(836\) −26.7791 −0.926176
\(837\) 0 0
\(838\) 23.4768 0.810991
\(839\) −4.20041 −0.145014 −0.0725071 0.997368i \(-0.523100\pi\)
−0.0725071 + 0.997368i \(0.523100\pi\)
\(840\) 0 0
\(841\) 8.61994 0.297239
\(842\) −26.6544 −0.918570
\(843\) 0 0
\(844\) −4.80875 −0.165524
\(845\) −51.3876 −1.76779
\(846\) 0 0
\(847\) 2.91345 0.100107
\(848\) −9.76490 −0.335328
\(849\) 0 0
\(850\) 4.54671 0.155951
\(851\) −2.50863 −0.0859947
\(852\) 0 0
\(853\) −15.9537 −0.546243 −0.273121 0.961980i \(-0.588056\pi\)
−0.273121 + 0.961980i \(0.588056\pi\)
\(854\) 1.20048 0.0410795
\(855\) 0 0
\(856\) 3.87889 0.132578
\(857\) −0.986693 −0.0337048 −0.0168524 0.999858i \(-0.505365\pi\)
−0.0168524 + 0.999858i \(0.505365\pi\)
\(858\) 0 0
\(859\) 6.41493 0.218874 0.109437 0.993994i \(-0.465095\pi\)
0.109437 + 0.993994i \(0.465095\pi\)
\(860\) −14.9408 −0.509476
\(861\) 0 0
\(862\) 27.9472 0.951886
\(863\) −19.6447 −0.668714 −0.334357 0.942446i \(-0.608519\pi\)
−0.334357 + 0.942446i \(0.608519\pi\)
\(864\) 0 0
\(865\) −70.0733 −2.38257
\(866\) 14.4980 0.492663
\(867\) 0 0
\(868\) −0.313501 −0.0106409
\(869\) 4.71790 0.160044
\(870\) 0 0
\(871\) −37.4521 −1.26902
\(872\) 0.536436 0.0181660
\(873\) 0 0
\(874\) 13.2086 0.446787
\(875\) −0.749978 −0.0253539
\(876\) 0 0
\(877\) −25.0212 −0.844905 −0.422452 0.906385i \(-0.638831\pi\)
−0.422452 + 0.906385i \(0.638831\pi\)
\(878\) 10.2482 0.345861
\(879\) 0 0
\(880\) 16.9802 0.572403
\(881\) 20.0574 0.675751 0.337875 0.941191i \(-0.390292\pi\)
0.337875 + 0.941191i \(0.390292\pi\)
\(882\) 0 0
\(883\) −10.4184 −0.350608 −0.175304 0.984514i \(-0.556091\pi\)
−0.175304 + 0.984514i \(0.556091\pi\)
\(884\) −3.94165 −0.132572
\(885\) 0 0
\(886\) 8.13881 0.273429
\(887\) −28.0688 −0.942459 −0.471229 0.882011i \(-0.656190\pi\)
−0.471229 + 0.882011i \(0.656190\pi\)
\(888\) 0 0
\(889\) 3.93200 0.131875
\(890\) 35.5365 1.19119
\(891\) 0 0
\(892\) 5.28621 0.176995
\(893\) 70.9831 2.37536
\(894\) 0 0
\(895\) 74.9230 2.50440
\(896\) 0.195962 0.00654662
\(897\) 0 0
\(898\) 1.81755 0.0606524
\(899\) 9.81245 0.327264
\(900\) 0 0
\(901\) −7.22352 −0.240650
\(902\) 26.7460 0.890544
\(903\) 0 0
\(904\) 19.0344 0.633075
\(905\) −18.7116 −0.621996
\(906\) 0 0
\(907\) −6.81634 −0.226333 −0.113166 0.993576i \(-0.536099\pi\)
−0.113166 + 0.993576i \(0.536099\pi\)
\(908\) 6.52175 0.216432
\(909\) 0 0
\(910\) 3.48605 0.115562
\(911\) 46.1659 1.52954 0.764772 0.644301i \(-0.222851\pi\)
0.764772 + 0.644301i \(0.222851\pi\)
\(912\) 0 0
\(913\) 45.0591 1.49124
\(914\) −38.9876 −1.28959
\(915\) 0 0
\(916\) 5.93818 0.196203
\(917\) 1.65126 0.0545295
\(918\) 0 0
\(919\) 26.8038 0.884175 0.442087 0.896972i \(-0.354238\pi\)
0.442087 + 0.896972i \(0.354238\pi\)
\(920\) −8.37534 −0.276127
\(921\) 0 0
\(922\) −14.6527 −0.482562
\(923\) −55.7117 −1.83377
\(924\) 0 0
\(925\) −6.14633 −0.202090
\(926\) 25.3342 0.832533
\(927\) 0 0
\(928\) −6.13351 −0.201342
\(929\) 25.9178 0.850334 0.425167 0.905115i \(-0.360215\pi\)
0.425167 + 0.905115i \(0.360215\pi\)
\(930\) 0 0
\(931\) −36.6546 −1.20131
\(932\) 18.4107 0.603063
\(933\) 0 0
\(934\) −6.52767 −0.213592
\(935\) 12.5610 0.410788
\(936\) 0 0
\(937\) 8.04348 0.262769 0.131384 0.991331i \(-0.458058\pi\)
0.131384 + 0.991331i \(0.458058\pi\)
\(938\) 1.37737 0.0449727
\(939\) 0 0
\(940\) −45.0092 −1.46804
\(941\) 57.0300 1.85913 0.929563 0.368664i \(-0.120185\pi\)
0.929563 + 0.368664i \(0.120185\pi\)
\(942\) 0 0
\(943\) −13.1922 −0.429598
\(944\) 0.821294 0.0267309
\(945\) 0 0
\(946\) −22.7606 −0.740012
\(947\) −11.9823 −0.389372 −0.194686 0.980866i \(-0.562369\pi\)
−0.194686 + 0.980866i \(0.562369\pi\)
\(948\) 0 0
\(949\) −41.2191 −1.33803
\(950\) 32.3620 1.04996
\(951\) 0 0
\(952\) 0.144961 0.00469822
\(953\) 53.7478 1.74106 0.870532 0.492112i \(-0.163775\pi\)
0.870532 + 0.492112i \(0.163775\pi\)
\(954\) 0 0
\(955\) 13.7680 0.445523
\(956\) 24.0032 0.776320
\(957\) 0 0
\(958\) 19.4006 0.626806
\(959\) −4.07800 −0.131686
\(960\) 0 0
\(961\) −28.4406 −0.917439
\(962\) 5.32840 0.171795
\(963\) 0 0
\(964\) −17.2454 −0.555437
\(965\) 28.5484 0.919006
\(966\) 0 0
\(967\) −54.0756 −1.73895 −0.869476 0.493975i \(-0.835544\pi\)
−0.869476 + 0.493975i \(0.835544\pi\)
\(968\) 14.8675 0.477859
\(969\) 0 0
\(970\) −34.7507 −1.11578
\(971\) −16.9965 −0.545442 −0.272721 0.962093i \(-0.587924\pi\)
−0.272721 + 0.962093i \(0.587924\pi\)
\(972\) 0 0
\(973\) −0.0239929 −0.000769176 0
\(974\) −11.5204 −0.369138
\(975\) 0 0
\(976\) 6.12609 0.196091
\(977\) −14.3965 −0.460585 −0.230293 0.973121i \(-0.573968\pi\)
−0.230293 + 0.973121i \(0.573968\pi\)
\(978\) 0 0
\(979\) 54.1360 1.73019
\(980\) 23.2421 0.742441
\(981\) 0 0
\(982\) −30.0958 −0.960397
\(983\) −46.2710 −1.47582 −0.737908 0.674901i \(-0.764186\pi\)
−0.737908 + 0.674901i \(0.764186\pi\)
\(984\) 0 0
\(985\) −43.9878 −1.40157
\(986\) −4.53722 −0.144494
\(987\) 0 0
\(988\) −28.0554 −0.892561
\(989\) 11.2265 0.356981
\(990\) 0 0
\(991\) 50.4576 1.60284 0.801419 0.598103i \(-0.204079\pi\)
0.801419 + 0.598103i \(0.204079\pi\)
\(992\) −1.59981 −0.0507940
\(993\) 0 0
\(994\) 2.04890 0.0649870
\(995\) −62.0350 −1.96664
\(996\) 0 0
\(997\) 51.1368 1.61952 0.809759 0.586762i \(-0.199598\pi\)
0.809759 + 0.586762i \(0.199598\pi\)
\(998\) 19.6116 0.620793
\(999\) 0 0
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 5994.2.a.bb.1.2 10
3.2 odd 2 5994.2.a.ba.1.9 10
9.2 odd 6 1998.2.e.e.1333.2 20
9.4 even 3 666.2.e.e.223.1 20
9.5 odd 6 1998.2.e.e.667.2 20
9.7 even 3 666.2.e.e.445.1 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
666.2.e.e.223.1 20 9.4 even 3
666.2.e.e.445.1 yes 20 9.7 even 3
1998.2.e.e.667.2 20 9.5 odd 6
1998.2.e.e.1333.2 20 9.2 odd 6
5994.2.a.ba.1.9 10 3.2 odd 2
5994.2.a.bb.1.2 10 1.1 even 1 trivial