Properties

Label 4016.2.a.j.1.2
Level $4016$
Weight $2$
Character 4016.1
Self dual yes
Analytic conductor $32.068$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4016,2,Mod(1,4016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4016, 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("4016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4016.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: \(32.0679214517\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} - 27 x^{12} + 79 x^{11} + 274 x^{10} - 747 x^{9} - 1422 x^{8} + 3287 x^{7} + 4161 x^{6} - 6861 x^{5} - 6676 x^{4} + 5599 x^{3} + 4627 x^{2} - 359 x - 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1004)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.87232\) of defining polynomial
Character \(\chi\) \(=\) 4016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.87232 q^{3} -0.801100 q^{5} +4.32848 q^{7} +5.25020 q^{9} +O(q^{10})\) \(q-2.87232 q^{3} -0.801100 q^{5} +4.32848 q^{7} +5.25020 q^{9} -5.29679 q^{11} +0.355175 q^{13} +2.30101 q^{15} +6.75803 q^{17} -2.61157 q^{19} -12.4328 q^{21} -4.43653 q^{23} -4.35824 q^{25} -6.46328 q^{27} +6.80961 q^{29} -0.0652340 q^{31} +15.2141 q^{33} -3.46755 q^{35} -5.99433 q^{37} -1.02017 q^{39} -8.40285 q^{41} -4.52917 q^{43} -4.20593 q^{45} -4.41115 q^{47} +11.7358 q^{49} -19.4112 q^{51} +9.98971 q^{53} +4.24326 q^{55} +7.50126 q^{57} +12.9609 q^{59} -7.45173 q^{61} +22.7254 q^{63} -0.284531 q^{65} -3.80450 q^{67} +12.7431 q^{69} +15.0301 q^{71} -3.56498 q^{73} +12.5182 q^{75} -22.9271 q^{77} +11.7170 q^{79} +2.81398 q^{81} +5.25692 q^{83} -5.41386 q^{85} -19.5593 q^{87} +2.90166 q^{89} +1.53737 q^{91} +0.187373 q^{93} +2.09213 q^{95} -13.2836 q^{97} -27.8092 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) \) 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
\(3\) −2.87232 −1.65833 −0.829166 0.559002i \(-0.811184\pi\)
−0.829166 + 0.559002i \(0.811184\pi\)
\(4\) 0 0
\(5\) −0.801100 −0.358263 −0.179131 0.983825i \(-0.557329\pi\)
−0.179131 + 0.983825i \(0.557329\pi\)
\(6\) 0 0
\(7\) 4.32848 1.63601 0.818007 0.575209i \(-0.195079\pi\)
0.818007 + 0.575209i \(0.195079\pi\)
\(8\) 0 0
\(9\) 5.25020 1.75007
\(10\) 0 0
\(11\) −5.29679 −1.59704 −0.798521 0.601967i \(-0.794384\pi\)
−0.798521 + 0.601967i \(0.794384\pi\)
\(12\) 0 0
\(13\) 0.355175 0.0985078 0.0492539 0.998786i \(-0.484316\pi\)
0.0492539 + 0.998786i \(0.484316\pi\)
\(14\) 0 0
\(15\) 2.30101 0.594119
\(16\) 0 0
\(17\) 6.75803 1.63906 0.819531 0.573035i \(-0.194234\pi\)
0.819531 + 0.573035i \(0.194234\pi\)
\(18\) 0 0
\(19\) −2.61157 −0.599136 −0.299568 0.954075i \(-0.596843\pi\)
−0.299568 + 0.954075i \(0.596843\pi\)
\(20\) 0 0
\(21\) −12.4328 −2.71305
\(22\) 0 0
\(23\) −4.43653 −0.925080 −0.462540 0.886598i \(-0.653062\pi\)
−0.462540 + 0.886598i \(0.653062\pi\)
\(24\) 0 0
\(25\) −4.35824 −0.871648
\(26\) 0 0
\(27\) −6.46328 −1.24386
\(28\) 0 0
\(29\) 6.80961 1.26451 0.632256 0.774759i \(-0.282129\pi\)
0.632256 + 0.774759i \(0.282129\pi\)
\(30\) 0 0
\(31\) −0.0652340 −0.0117164 −0.00585819 0.999983i \(-0.501865\pi\)
−0.00585819 + 0.999983i \(0.501865\pi\)
\(32\) 0 0
\(33\) 15.2141 2.64843
\(34\) 0 0
\(35\) −3.46755 −0.586123
\(36\) 0 0
\(37\) −5.99433 −0.985462 −0.492731 0.870182i \(-0.664001\pi\)
−0.492731 + 0.870182i \(0.664001\pi\)
\(38\) 0 0
\(39\) −1.02017 −0.163359
\(40\) 0 0
\(41\) −8.40285 −1.31230 −0.656152 0.754628i \(-0.727817\pi\)
−0.656152 + 0.754628i \(0.727817\pi\)
\(42\) 0 0
\(43\) −4.52917 −0.690692 −0.345346 0.938475i \(-0.612239\pi\)
−0.345346 + 0.938475i \(0.612239\pi\)
\(44\) 0 0
\(45\) −4.20593 −0.626984
\(46\) 0 0
\(47\) −4.41115 −0.643432 −0.321716 0.946836i \(-0.604260\pi\)
−0.321716 + 0.946836i \(0.604260\pi\)
\(48\) 0 0
\(49\) 11.7358 1.67654
\(50\) 0 0
\(51\) −19.4112 −2.71811
\(52\) 0 0
\(53\) 9.98971 1.37219 0.686096 0.727511i \(-0.259323\pi\)
0.686096 + 0.727511i \(0.259323\pi\)
\(54\) 0 0
\(55\) 4.24326 0.572161
\(56\) 0 0
\(57\) 7.50126 0.993566
\(58\) 0 0
\(59\) 12.9609 1.68737 0.843683 0.536842i \(-0.180383\pi\)
0.843683 + 0.536842i \(0.180383\pi\)
\(60\) 0 0
\(61\) −7.45173 −0.954096 −0.477048 0.878877i \(-0.658293\pi\)
−0.477048 + 0.878877i \(0.658293\pi\)
\(62\) 0 0
\(63\) 22.7254 2.86313
\(64\) 0 0
\(65\) −0.284531 −0.0352917
\(66\) 0 0
\(67\) −3.80450 −0.464794 −0.232397 0.972621i \(-0.574657\pi\)
−0.232397 + 0.972621i \(0.574657\pi\)
\(68\) 0 0
\(69\) 12.7431 1.53409
\(70\) 0 0
\(71\) 15.0301 1.78374 0.891872 0.452288i \(-0.149392\pi\)
0.891872 + 0.452288i \(0.149392\pi\)
\(72\) 0 0
\(73\) −3.56498 −0.417249 −0.208625 0.977996i \(-0.566899\pi\)
−0.208625 + 0.977996i \(0.566899\pi\)
\(74\) 0 0
\(75\) 12.5182 1.44548
\(76\) 0 0
\(77\) −22.9271 −2.61278
\(78\) 0 0
\(79\) 11.7170 1.31826 0.659131 0.752028i \(-0.270924\pi\)
0.659131 + 0.752028i \(0.270924\pi\)
\(80\) 0 0
\(81\) 2.81398 0.312665
\(82\) 0 0
\(83\) 5.25692 0.577022 0.288511 0.957477i \(-0.406840\pi\)
0.288511 + 0.957477i \(0.406840\pi\)
\(84\) 0 0
\(85\) −5.41386 −0.587215
\(86\) 0 0
\(87\) −19.5593 −2.09698
\(88\) 0 0
\(89\) 2.90166 0.307576 0.153788 0.988104i \(-0.450853\pi\)
0.153788 + 0.988104i \(0.450853\pi\)
\(90\) 0 0
\(91\) 1.53737 0.161160
\(92\) 0 0
\(93\) 0.187373 0.0194296
\(94\) 0 0
\(95\) 2.09213 0.214648
\(96\) 0 0
\(97\) −13.2836 −1.34874 −0.674372 0.738392i \(-0.735586\pi\)
−0.674372 + 0.738392i \(0.735586\pi\)
\(98\) 0 0
\(99\) −27.8092 −2.79493
\(100\) 0 0
\(101\) −7.25334 −0.721735 −0.360867 0.932617i \(-0.617519\pi\)
−0.360867 + 0.932617i \(0.617519\pi\)
\(102\) 0 0
\(103\) −9.92994 −0.978426 −0.489213 0.872164i \(-0.662716\pi\)
−0.489213 + 0.872164i \(0.662716\pi\)
\(104\) 0 0
\(105\) 9.95990 0.971987
\(106\) 0 0
\(107\) 2.49608 0.241305 0.120653 0.992695i \(-0.461501\pi\)
0.120653 + 0.992695i \(0.461501\pi\)
\(108\) 0 0
\(109\) 9.89004 0.947294 0.473647 0.880715i \(-0.342937\pi\)
0.473647 + 0.880715i \(0.342937\pi\)
\(110\) 0 0
\(111\) 17.2176 1.63422
\(112\) 0 0
\(113\) −8.27604 −0.778544 −0.389272 0.921123i \(-0.627273\pi\)
−0.389272 + 0.921123i \(0.627273\pi\)
\(114\) 0 0
\(115\) 3.55410 0.331422
\(116\) 0 0
\(117\) 1.86474 0.172395
\(118\) 0 0
\(119\) 29.2520 2.68153
\(120\) 0 0
\(121\) 17.0560 1.55054
\(122\) 0 0
\(123\) 24.1356 2.17624
\(124\) 0 0
\(125\) 7.49689 0.670542
\(126\) 0 0
\(127\) 13.9646 1.23915 0.619577 0.784936i \(-0.287304\pi\)
0.619577 + 0.784936i \(0.287304\pi\)
\(128\) 0 0
\(129\) 13.0092 1.14540
\(130\) 0 0
\(131\) −15.7257 −1.37396 −0.686980 0.726676i \(-0.741064\pi\)
−0.686980 + 0.726676i \(0.741064\pi\)
\(132\) 0 0
\(133\) −11.3042 −0.980194
\(134\) 0 0
\(135\) 5.17773 0.445628
\(136\) 0 0
\(137\) −9.05086 −0.773267 −0.386634 0.922233i \(-0.626362\pi\)
−0.386634 + 0.922233i \(0.626362\pi\)
\(138\) 0 0
\(139\) −14.9944 −1.27181 −0.635903 0.771769i \(-0.719372\pi\)
−0.635903 + 0.771769i \(0.719372\pi\)
\(140\) 0 0
\(141\) 12.6702 1.06702
\(142\) 0 0
\(143\) −1.88129 −0.157321
\(144\) 0 0
\(145\) −5.45518 −0.453028
\(146\) 0 0
\(147\) −33.7089 −2.78026
\(148\) 0 0
\(149\) 11.5209 0.943830 0.471915 0.881644i \(-0.343563\pi\)
0.471915 + 0.881644i \(0.343563\pi\)
\(150\) 0 0
\(151\) −17.9441 −1.46027 −0.730133 0.683305i \(-0.760542\pi\)
−0.730133 + 0.683305i \(0.760542\pi\)
\(152\) 0 0
\(153\) 35.4810 2.86847
\(154\) 0 0
\(155\) 0.0522590 0.00419754
\(156\) 0 0
\(157\) −3.18254 −0.253994 −0.126997 0.991903i \(-0.540534\pi\)
−0.126997 + 0.991903i \(0.540534\pi\)
\(158\) 0 0
\(159\) −28.6936 −2.27555
\(160\) 0 0
\(161\) −19.2034 −1.51344
\(162\) 0 0
\(163\) −0.164487 −0.0128836 −0.00644180 0.999979i \(-0.502051\pi\)
−0.00644180 + 0.999979i \(0.502051\pi\)
\(164\) 0 0
\(165\) −12.1880 −0.948833
\(166\) 0 0
\(167\) 1.59809 0.123664 0.0618318 0.998087i \(-0.480306\pi\)
0.0618318 + 0.998087i \(0.480306\pi\)
\(168\) 0 0
\(169\) −12.8739 −0.990296
\(170\) 0 0
\(171\) −13.7113 −1.04853
\(172\) 0 0
\(173\) 0.673099 0.0511748 0.0255874 0.999673i \(-0.491854\pi\)
0.0255874 + 0.999673i \(0.491854\pi\)
\(174\) 0 0
\(175\) −18.8646 −1.42603
\(176\) 0 0
\(177\) −37.2278 −2.79821
\(178\) 0 0
\(179\) −10.6930 −0.799235 −0.399618 0.916682i \(-0.630857\pi\)
−0.399618 + 0.916682i \(0.630857\pi\)
\(180\) 0 0
\(181\) −4.73003 −0.351580 −0.175790 0.984428i \(-0.556248\pi\)
−0.175790 + 0.984428i \(0.556248\pi\)
\(182\) 0 0
\(183\) 21.4037 1.58221
\(184\) 0 0
\(185\) 4.80206 0.353054
\(186\) 0 0
\(187\) −35.7958 −2.61765
\(188\) 0 0
\(189\) −27.9762 −2.03497
\(190\) 0 0
\(191\) 6.50279 0.470525 0.235263 0.971932i \(-0.424405\pi\)
0.235263 + 0.971932i \(0.424405\pi\)
\(192\) 0 0
\(193\) −21.2434 −1.52914 −0.764568 0.644543i \(-0.777048\pi\)
−0.764568 + 0.644543i \(0.777048\pi\)
\(194\) 0 0
\(195\) 0.817262 0.0585254
\(196\) 0 0
\(197\) 11.1562 0.794845 0.397422 0.917636i \(-0.369905\pi\)
0.397422 + 0.917636i \(0.369905\pi\)
\(198\) 0 0
\(199\) 12.1091 0.858388 0.429194 0.903212i \(-0.358798\pi\)
0.429194 + 0.903212i \(0.358798\pi\)
\(200\) 0 0
\(201\) 10.9277 0.770783
\(202\) 0 0
\(203\) 29.4753 2.06876
\(204\) 0 0
\(205\) 6.73153 0.470150
\(206\) 0 0
\(207\) −23.2927 −1.61895
\(208\) 0 0
\(209\) 13.8330 0.956845
\(210\) 0 0
\(211\) −25.4749 −1.75377 −0.876883 0.480703i \(-0.840381\pi\)
−0.876883 + 0.480703i \(0.840381\pi\)
\(212\) 0 0
\(213\) −43.1712 −2.95804
\(214\) 0 0
\(215\) 3.62832 0.247450
\(216\) 0 0
\(217\) −0.282364 −0.0191681
\(218\) 0 0
\(219\) 10.2397 0.691938
\(220\) 0 0
\(221\) 2.40028 0.161460
\(222\) 0 0
\(223\) −5.59130 −0.374421 −0.187211 0.982320i \(-0.559945\pi\)
−0.187211 + 0.982320i \(0.559945\pi\)
\(224\) 0 0
\(225\) −22.8816 −1.52544
\(226\) 0 0
\(227\) 23.7847 1.57864 0.789321 0.613980i \(-0.210433\pi\)
0.789321 + 0.613980i \(0.210433\pi\)
\(228\) 0 0
\(229\) −3.73615 −0.246892 −0.123446 0.992351i \(-0.539395\pi\)
−0.123446 + 0.992351i \(0.539395\pi\)
\(230\) 0 0
\(231\) 65.8538 4.33286
\(232\) 0 0
\(233\) −28.3122 −1.85479 −0.927397 0.374078i \(-0.877959\pi\)
−0.927397 + 0.374078i \(0.877959\pi\)
\(234\) 0 0
\(235\) 3.53377 0.230518
\(236\) 0 0
\(237\) −33.6548 −2.18612
\(238\) 0 0
\(239\) −25.2022 −1.63020 −0.815098 0.579323i \(-0.803317\pi\)
−0.815098 + 0.579323i \(0.803317\pi\)
\(240\) 0 0
\(241\) 4.67062 0.300861 0.150431 0.988621i \(-0.451934\pi\)
0.150431 + 0.988621i \(0.451934\pi\)
\(242\) 0 0
\(243\) 11.3072 0.725357
\(244\) 0 0
\(245\) −9.40153 −0.600642
\(246\) 0 0
\(247\) −0.927566 −0.0590196
\(248\) 0 0
\(249\) −15.0995 −0.956895
\(250\) 0 0
\(251\) 1.00000 0.0631194
\(252\) 0 0
\(253\) 23.4994 1.47739
\(254\) 0 0
\(255\) 15.5503 0.973798
\(256\) 0 0
\(257\) 10.3679 0.646730 0.323365 0.946274i \(-0.395186\pi\)
0.323365 + 0.946274i \(0.395186\pi\)
\(258\) 0 0
\(259\) −25.9464 −1.61223
\(260\) 0 0
\(261\) 35.7518 2.21298
\(262\) 0 0
\(263\) −27.6365 −1.70414 −0.852070 0.523429i \(-0.824653\pi\)
−0.852070 + 0.523429i \(0.824653\pi\)
\(264\) 0 0
\(265\) −8.00276 −0.491606
\(266\) 0 0
\(267\) −8.33449 −0.510062
\(268\) 0 0
\(269\) −12.4810 −0.760980 −0.380490 0.924785i \(-0.624245\pi\)
−0.380490 + 0.924785i \(0.624245\pi\)
\(270\) 0 0
\(271\) −21.7491 −1.32116 −0.660581 0.750755i \(-0.729690\pi\)
−0.660581 + 0.750755i \(0.729690\pi\)
\(272\) 0 0
\(273\) −4.41581 −0.267257
\(274\) 0 0
\(275\) 23.0847 1.39206
\(276\) 0 0
\(277\) −4.24654 −0.255150 −0.127575 0.991829i \(-0.540719\pi\)
−0.127575 + 0.991829i \(0.540719\pi\)
\(278\) 0 0
\(279\) −0.342491 −0.0205044
\(280\) 0 0
\(281\) −9.93757 −0.592826 −0.296413 0.955060i \(-0.595790\pi\)
−0.296413 + 0.955060i \(0.595790\pi\)
\(282\) 0 0
\(283\) 30.5140 1.81387 0.906933 0.421274i \(-0.138417\pi\)
0.906933 + 0.421274i \(0.138417\pi\)
\(284\) 0 0
\(285\) −6.00926 −0.355958
\(286\) 0 0
\(287\) −36.3716 −2.14695
\(288\) 0 0
\(289\) 28.6709 1.68652
\(290\) 0 0
\(291\) 38.1547 2.23667
\(292\) 0 0
\(293\) −19.1769 −1.12033 −0.560165 0.828381i \(-0.689262\pi\)
−0.560165 + 0.828381i \(0.689262\pi\)
\(294\) 0 0
\(295\) −10.3830 −0.604521
\(296\) 0 0
\(297\) 34.2346 1.98649
\(298\) 0 0
\(299\) −1.57574 −0.0911277
\(300\) 0 0
\(301\) −19.6045 −1.12998
\(302\) 0 0
\(303\) 20.8339 1.19688
\(304\) 0 0
\(305\) 5.96958 0.341817
\(306\) 0 0
\(307\) −24.2689 −1.38510 −0.692550 0.721369i \(-0.743513\pi\)
−0.692550 + 0.721369i \(0.743513\pi\)
\(308\) 0 0
\(309\) 28.5219 1.62255
\(310\) 0 0
\(311\) 31.6530 1.79488 0.897439 0.441139i \(-0.145425\pi\)
0.897439 + 0.441139i \(0.145425\pi\)
\(312\) 0 0
\(313\) −32.4679 −1.83519 −0.917597 0.397512i \(-0.869874\pi\)
−0.917597 + 0.397512i \(0.869874\pi\)
\(314\) 0 0
\(315\) −18.2053 −1.02575
\(316\) 0 0
\(317\) 3.47733 0.195307 0.0976533 0.995220i \(-0.468866\pi\)
0.0976533 + 0.995220i \(0.468866\pi\)
\(318\) 0 0
\(319\) −36.0691 −2.01948
\(320\) 0 0
\(321\) −7.16953 −0.400164
\(322\) 0 0
\(323\) −17.6491 −0.982021
\(324\) 0 0
\(325\) −1.54794 −0.0858641
\(326\) 0 0
\(327\) −28.4073 −1.57093
\(328\) 0 0
\(329\) −19.0936 −1.05266
\(330\) 0 0
\(331\) −20.1784 −1.10911 −0.554554 0.832148i \(-0.687111\pi\)
−0.554554 + 0.832148i \(0.687111\pi\)
\(332\) 0 0
\(333\) −31.4714 −1.72462
\(334\) 0 0
\(335\) 3.04779 0.166519
\(336\) 0 0
\(337\) −15.5362 −0.846313 −0.423157 0.906057i \(-0.639078\pi\)
−0.423157 + 0.906057i \(0.639078\pi\)
\(338\) 0 0
\(339\) 23.7714 1.29108
\(340\) 0 0
\(341\) 0.345531 0.0187115
\(342\) 0 0
\(343\) 20.4987 1.10683
\(344\) 0 0
\(345\) −10.2085 −0.549608
\(346\) 0 0
\(347\) −3.38220 −0.181566 −0.0907831 0.995871i \(-0.528937\pi\)
−0.0907831 + 0.995871i \(0.528937\pi\)
\(348\) 0 0
\(349\) 14.7704 0.790640 0.395320 0.918543i \(-0.370634\pi\)
0.395320 + 0.918543i \(0.370634\pi\)
\(350\) 0 0
\(351\) −2.29559 −0.122530
\(352\) 0 0
\(353\) −11.4231 −0.607990 −0.303995 0.952674i \(-0.598321\pi\)
−0.303995 + 0.952674i \(0.598321\pi\)
\(354\) 0 0
\(355\) −12.0406 −0.639049
\(356\) 0 0
\(357\) −84.0210 −4.44686
\(358\) 0 0
\(359\) −22.7369 −1.20001 −0.600003 0.799998i \(-0.704834\pi\)
−0.600003 + 0.799998i \(0.704834\pi\)
\(360\) 0 0
\(361\) −12.1797 −0.641036
\(362\) 0 0
\(363\) −48.9902 −2.57132
\(364\) 0 0
\(365\) 2.85591 0.149485
\(366\) 0 0
\(367\) 4.85042 0.253190 0.126595 0.991955i \(-0.459595\pi\)
0.126595 + 0.991955i \(0.459595\pi\)
\(368\) 0 0
\(369\) −44.1166 −2.29662
\(370\) 0 0
\(371\) 43.2403 2.24492
\(372\) 0 0
\(373\) −18.5539 −0.960683 −0.480341 0.877082i \(-0.659487\pi\)
−0.480341 + 0.877082i \(0.659487\pi\)
\(374\) 0 0
\(375\) −21.5334 −1.11198
\(376\) 0 0
\(377\) 2.41860 0.124564
\(378\) 0 0
\(379\) −16.8723 −0.866673 −0.433337 0.901232i \(-0.642664\pi\)
−0.433337 + 0.901232i \(0.642664\pi\)
\(380\) 0 0
\(381\) −40.1106 −2.05493
\(382\) 0 0
\(383\) 20.2439 1.03442 0.517208 0.855860i \(-0.326971\pi\)
0.517208 + 0.855860i \(0.326971\pi\)
\(384\) 0 0
\(385\) 18.3669 0.936063
\(386\) 0 0
\(387\) −23.7791 −1.20876
\(388\) 0 0
\(389\) 4.30439 0.218241 0.109121 0.994029i \(-0.465196\pi\)
0.109121 + 0.994029i \(0.465196\pi\)
\(390\) 0 0
\(391\) −29.9822 −1.51626
\(392\) 0 0
\(393\) 45.1692 2.27848
\(394\) 0 0
\(395\) −9.38647 −0.472284
\(396\) 0 0
\(397\) 30.3077 1.52110 0.760551 0.649279i \(-0.224929\pi\)
0.760551 + 0.649279i \(0.224929\pi\)
\(398\) 0 0
\(399\) 32.4691 1.62549
\(400\) 0 0
\(401\) 15.7892 0.788475 0.394238 0.919009i \(-0.371009\pi\)
0.394238 + 0.919009i \(0.371009\pi\)
\(402\) 0 0
\(403\) −0.0231695 −0.00115415
\(404\) 0 0
\(405\) −2.25428 −0.112016
\(406\) 0 0
\(407\) 31.7507 1.57382
\(408\) 0 0
\(409\) −10.4833 −0.518364 −0.259182 0.965828i \(-0.583453\pi\)
−0.259182 + 0.965828i \(0.583453\pi\)
\(410\) 0 0
\(411\) 25.9969 1.28233
\(412\) 0 0
\(413\) 56.1011 2.76055
\(414\) 0 0
\(415\) −4.21132 −0.206726
\(416\) 0 0
\(417\) 43.0686 2.10908
\(418\) 0 0
\(419\) −12.1794 −0.595001 −0.297500 0.954722i \(-0.596153\pi\)
−0.297500 + 0.954722i \(0.596153\pi\)
\(420\) 0 0
\(421\) −26.5828 −1.29556 −0.647782 0.761826i \(-0.724303\pi\)
−0.647782 + 0.761826i \(0.724303\pi\)
\(422\) 0 0
\(423\) −23.1594 −1.12605
\(424\) 0 0
\(425\) −29.4531 −1.42868
\(426\) 0 0
\(427\) −32.2547 −1.56091
\(428\) 0 0
\(429\) 5.40365 0.260891
\(430\) 0 0
\(431\) 16.8284 0.810598 0.405299 0.914184i \(-0.367167\pi\)
0.405299 + 0.914184i \(0.367167\pi\)
\(432\) 0 0
\(433\) −5.73005 −0.275369 −0.137684 0.990476i \(-0.543966\pi\)
−0.137684 + 0.990476i \(0.543966\pi\)
\(434\) 0 0
\(435\) 15.6690 0.751271
\(436\) 0 0
\(437\) 11.5863 0.554249
\(438\) 0 0
\(439\) −13.2135 −0.630648 −0.315324 0.948984i \(-0.602113\pi\)
−0.315324 + 0.948984i \(0.602113\pi\)
\(440\) 0 0
\(441\) 61.6151 2.93405
\(442\) 0 0
\(443\) 9.21026 0.437593 0.218796 0.975771i \(-0.429787\pi\)
0.218796 + 0.975771i \(0.429787\pi\)
\(444\) 0 0
\(445\) −2.32452 −0.110193
\(446\) 0 0
\(447\) −33.0917 −1.56518
\(448\) 0 0
\(449\) 15.7721 0.744332 0.372166 0.928166i \(-0.378615\pi\)
0.372166 + 0.928166i \(0.378615\pi\)
\(450\) 0 0
\(451\) 44.5081 2.09581
\(452\) 0 0
\(453\) 51.5410 2.42161
\(454\) 0 0
\(455\) −1.23159 −0.0577377
\(456\) 0 0
\(457\) −36.7878 −1.72086 −0.860431 0.509567i \(-0.829806\pi\)
−0.860431 + 0.509567i \(0.829806\pi\)
\(458\) 0 0
\(459\) −43.6790 −2.03876
\(460\) 0 0
\(461\) −30.9537 −1.44166 −0.720828 0.693114i \(-0.756238\pi\)
−0.720828 + 0.693114i \(0.756238\pi\)
\(462\) 0 0
\(463\) 10.6966 0.497112 0.248556 0.968617i \(-0.420044\pi\)
0.248556 + 0.968617i \(0.420044\pi\)
\(464\) 0 0
\(465\) −0.150104 −0.00696092
\(466\) 0 0
\(467\) −24.9092 −1.15266 −0.576331 0.817216i \(-0.695516\pi\)
−0.576331 + 0.817216i \(0.695516\pi\)
\(468\) 0 0
\(469\) −16.4677 −0.760410
\(470\) 0 0
\(471\) 9.14125 0.421207
\(472\) 0 0
\(473\) 23.9901 1.10306
\(474\) 0 0
\(475\) 11.3819 0.522235
\(476\) 0 0
\(477\) 52.4479 2.40143
\(478\) 0 0
\(479\) −22.9085 −1.04672 −0.523359 0.852112i \(-0.675321\pi\)
−0.523359 + 0.852112i \(0.675321\pi\)
\(480\) 0 0
\(481\) −2.12904 −0.0970757
\(482\) 0 0
\(483\) 55.1584 2.50979
\(484\) 0 0
\(485\) 10.6415 0.483205
\(486\) 0 0
\(487\) 29.4106 1.33272 0.666360 0.745630i \(-0.267852\pi\)
0.666360 + 0.745630i \(0.267852\pi\)
\(488\) 0 0
\(489\) 0.472458 0.0213653
\(490\) 0 0
\(491\) 14.0114 0.632328 0.316164 0.948705i \(-0.397605\pi\)
0.316164 + 0.948705i \(0.397605\pi\)
\(492\) 0 0
\(493\) 46.0195 2.07261
\(494\) 0 0
\(495\) 22.2780 1.00132
\(496\) 0 0
\(497\) 65.0575 2.91823
\(498\) 0 0
\(499\) 3.66328 0.163991 0.0819954 0.996633i \(-0.473871\pi\)
0.0819954 + 0.996633i \(0.473871\pi\)
\(500\) 0 0
\(501\) −4.59021 −0.205075
\(502\) 0 0
\(503\) 7.92668 0.353433 0.176717 0.984262i \(-0.443452\pi\)
0.176717 + 0.984262i \(0.443452\pi\)
\(504\) 0 0
\(505\) 5.81066 0.258571
\(506\) 0 0
\(507\) 36.9778 1.64224
\(508\) 0 0
\(509\) −8.11458 −0.359672 −0.179836 0.983697i \(-0.557557\pi\)
−0.179836 + 0.983697i \(0.557557\pi\)
\(510\) 0 0
\(511\) −15.4310 −0.682626
\(512\) 0 0
\(513\) 16.8793 0.745240
\(514\) 0 0
\(515\) 7.95487 0.350534
\(516\) 0 0
\(517\) 23.3649 1.02759
\(518\) 0 0
\(519\) −1.93335 −0.0848648
\(520\) 0 0
\(521\) 28.8551 1.26416 0.632082 0.774902i \(-0.282201\pi\)
0.632082 + 0.774902i \(0.282201\pi\)
\(522\) 0 0
\(523\) −24.4433 −1.06883 −0.534415 0.845222i \(-0.679468\pi\)
−0.534415 + 0.845222i \(0.679468\pi\)
\(524\) 0 0
\(525\) 54.1850 2.36483
\(526\) 0 0
\(527\) −0.440853 −0.0192039
\(528\) 0 0
\(529\) −3.31721 −0.144226
\(530\) 0 0
\(531\) 68.0473 2.95300
\(532\) 0 0
\(533\) −2.98448 −0.129272
\(534\) 0 0
\(535\) −1.99961 −0.0864507
\(536\) 0 0
\(537\) 30.7138 1.32540
\(538\) 0 0
\(539\) −62.1619 −2.67750
\(540\) 0 0
\(541\) −7.60751 −0.327073 −0.163536 0.986537i \(-0.552290\pi\)
−0.163536 + 0.986537i \(0.552290\pi\)
\(542\) 0 0
\(543\) 13.5861 0.583037
\(544\) 0 0
\(545\) −7.92292 −0.339380
\(546\) 0 0
\(547\) 28.3469 1.21203 0.606013 0.795455i \(-0.292768\pi\)
0.606013 + 0.795455i \(0.292768\pi\)
\(548\) 0 0
\(549\) −39.1230 −1.66973
\(550\) 0 0
\(551\) −17.7838 −0.757615
\(552\) 0 0
\(553\) 50.7167 2.15669
\(554\) 0 0
\(555\) −13.7930 −0.585482
\(556\) 0 0
\(557\) 23.7882 1.00794 0.503969 0.863721i \(-0.331872\pi\)
0.503969 + 0.863721i \(0.331872\pi\)
\(558\) 0 0
\(559\) −1.60865 −0.0680386
\(560\) 0 0
\(561\) 102.817 4.34094
\(562\) 0 0
\(563\) 2.88451 0.121568 0.0607839 0.998151i \(-0.480640\pi\)
0.0607839 + 0.998151i \(0.480640\pi\)
\(564\) 0 0
\(565\) 6.62994 0.278924
\(566\) 0 0
\(567\) 12.1803 0.511523
\(568\) 0 0
\(569\) 8.34899 0.350008 0.175004 0.984568i \(-0.444006\pi\)
0.175004 + 0.984568i \(0.444006\pi\)
\(570\) 0 0
\(571\) −26.0203 −1.08892 −0.544458 0.838788i \(-0.683265\pi\)
−0.544458 + 0.838788i \(0.683265\pi\)
\(572\) 0 0
\(573\) −18.6781 −0.780287
\(574\) 0 0
\(575\) 19.3355 0.806344
\(576\) 0 0
\(577\) 31.4162 1.30787 0.653936 0.756550i \(-0.273117\pi\)
0.653936 + 0.756550i \(0.273117\pi\)
\(578\) 0 0
\(579\) 61.0179 2.53582
\(580\) 0 0
\(581\) 22.7545 0.944016
\(582\) 0 0
\(583\) −52.9134 −2.19145
\(584\) 0 0
\(585\) −1.49384 −0.0617628
\(586\) 0 0
\(587\) 22.3674 0.923202 0.461601 0.887088i \(-0.347275\pi\)
0.461601 + 0.887088i \(0.347275\pi\)
\(588\) 0 0
\(589\) 0.170363 0.00701970
\(590\) 0 0
\(591\) −32.0441 −1.31812
\(592\) 0 0
\(593\) −29.9886 −1.23149 −0.615743 0.787947i \(-0.711144\pi\)
−0.615743 + 0.787947i \(0.711144\pi\)
\(594\) 0 0
\(595\) −23.4338 −0.960692
\(596\) 0 0
\(597\) −34.7810 −1.42349
\(598\) 0 0
\(599\) 2.82921 0.115598 0.0577991 0.998328i \(-0.481592\pi\)
0.0577991 + 0.998328i \(0.481592\pi\)
\(600\) 0 0
\(601\) 35.4709 1.44689 0.723443 0.690384i \(-0.242558\pi\)
0.723443 + 0.690384i \(0.242558\pi\)
\(602\) 0 0
\(603\) −19.9744 −0.813420
\(604\) 0 0
\(605\) −13.6636 −0.555502
\(606\) 0 0
\(607\) −8.67595 −0.352146 −0.176073 0.984377i \(-0.556339\pi\)
−0.176073 + 0.984377i \(0.556339\pi\)
\(608\) 0 0
\(609\) −84.6623 −3.43069
\(610\) 0 0
\(611\) −1.56673 −0.0633831
\(612\) 0 0
\(613\) 6.42430 0.259475 0.129738 0.991548i \(-0.458587\pi\)
0.129738 + 0.991548i \(0.458587\pi\)
\(614\) 0 0
\(615\) −19.3351 −0.779665
\(616\) 0 0
\(617\) −3.38475 −0.136265 −0.0681324 0.997676i \(-0.521704\pi\)
−0.0681324 + 0.997676i \(0.521704\pi\)
\(618\) 0 0
\(619\) 37.5948 1.51106 0.755532 0.655112i \(-0.227378\pi\)
0.755532 + 0.655112i \(0.227378\pi\)
\(620\) 0 0
\(621\) 28.6745 1.15067
\(622\) 0 0
\(623\) 12.5598 0.503198
\(624\) 0 0
\(625\) 15.7854 0.631417
\(626\) 0 0
\(627\) −39.7326 −1.58677
\(628\) 0 0
\(629\) −40.5098 −1.61523
\(630\) 0 0
\(631\) −2.02277 −0.0805251 −0.0402626 0.999189i \(-0.512819\pi\)
−0.0402626 + 0.999189i \(0.512819\pi\)
\(632\) 0 0
\(633\) 73.1720 2.90833
\(634\) 0 0
\(635\) −11.1870 −0.443943
\(636\) 0 0
\(637\) 4.16825 0.165152
\(638\) 0 0
\(639\) 78.9110 3.12167
\(640\) 0 0
\(641\) −5.30542 −0.209552 −0.104776 0.994496i \(-0.533412\pi\)
−0.104776 + 0.994496i \(0.533412\pi\)
\(642\) 0 0
\(643\) 40.5167 1.59782 0.798910 0.601450i \(-0.205410\pi\)
0.798910 + 0.601450i \(0.205410\pi\)
\(644\) 0 0
\(645\) −10.4217 −0.410354
\(646\) 0 0
\(647\) 8.40204 0.330318 0.165159 0.986267i \(-0.447186\pi\)
0.165159 + 0.986267i \(0.447186\pi\)
\(648\) 0 0
\(649\) −68.6512 −2.69479
\(650\) 0 0
\(651\) 0.811040 0.0317872
\(652\) 0 0
\(653\) 7.00801 0.274245 0.137122 0.990554i \(-0.456215\pi\)
0.137122 + 0.990554i \(0.456215\pi\)
\(654\) 0 0
\(655\) 12.5979 0.492239
\(656\) 0 0
\(657\) −18.7169 −0.730214
\(658\) 0 0
\(659\) −48.3463 −1.88330 −0.941652 0.336589i \(-0.890727\pi\)
−0.941652 + 0.336589i \(0.890727\pi\)
\(660\) 0 0
\(661\) 26.1898 1.01866 0.509332 0.860570i \(-0.329892\pi\)
0.509332 + 0.860570i \(0.329892\pi\)
\(662\) 0 0
\(663\) −6.89437 −0.267755
\(664\) 0 0
\(665\) 9.05576 0.351167
\(666\) 0 0
\(667\) −30.2110 −1.16978
\(668\) 0 0
\(669\) 16.0600 0.620914
\(670\) 0 0
\(671\) 39.4702 1.52373
\(672\) 0 0
\(673\) 39.9702 1.54074 0.770369 0.637598i \(-0.220072\pi\)
0.770369 + 0.637598i \(0.220072\pi\)
\(674\) 0 0
\(675\) 28.1685 1.08421
\(676\) 0 0
\(677\) −10.3948 −0.399505 −0.199753 0.979846i \(-0.564014\pi\)
−0.199753 + 0.979846i \(0.564014\pi\)
\(678\) 0 0
\(679\) −57.4978 −2.20656
\(680\) 0 0
\(681\) −68.3170 −2.61791
\(682\) 0 0
\(683\) 0.414911 0.0158761 0.00793807 0.999968i \(-0.497473\pi\)
0.00793807 + 0.999968i \(0.497473\pi\)
\(684\) 0 0
\(685\) 7.25065 0.277033
\(686\) 0 0
\(687\) 10.7314 0.409428
\(688\) 0 0
\(689\) 3.54810 0.135172
\(690\) 0 0
\(691\) −4.96293 −0.188799 −0.0943994 0.995534i \(-0.530093\pi\)
−0.0943994 + 0.995534i \(0.530093\pi\)
\(692\) 0 0
\(693\) −120.372 −4.57254
\(694\) 0 0
\(695\) 12.0120 0.455641
\(696\) 0 0
\(697\) −56.7867 −2.15095
\(698\) 0 0
\(699\) 81.3216 3.07587
\(700\) 0 0
\(701\) −4.09222 −0.154561 −0.0772804 0.997009i \(-0.524624\pi\)
−0.0772804 + 0.997009i \(0.524624\pi\)
\(702\) 0 0
\(703\) 15.6546 0.590426
\(704\) 0 0
\(705\) −10.1501 −0.382275
\(706\) 0 0
\(707\) −31.3960 −1.18077
\(708\) 0 0
\(709\) −31.1864 −1.17123 −0.585615 0.810589i \(-0.699147\pi\)
−0.585615 + 0.810589i \(0.699147\pi\)
\(710\) 0 0
\(711\) 61.5164 2.30705
\(712\) 0 0
\(713\) 0.289413 0.0108386
\(714\) 0 0
\(715\) 1.50710 0.0563623
\(716\) 0 0
\(717\) 72.3887 2.70341
\(718\) 0 0
\(719\) 0.987492 0.0368272 0.0184136 0.999830i \(-0.494138\pi\)
0.0184136 + 0.999830i \(0.494138\pi\)
\(720\) 0 0
\(721\) −42.9816 −1.60072
\(722\) 0 0
\(723\) −13.4155 −0.498928
\(724\) 0 0
\(725\) −29.6779 −1.10221
\(726\) 0 0
\(727\) 46.5550 1.72663 0.863314 0.504667i \(-0.168385\pi\)
0.863314 + 0.504667i \(0.168385\pi\)
\(728\) 0 0
\(729\) −40.9198 −1.51555
\(730\) 0 0
\(731\) −30.6083 −1.13209
\(732\) 0 0
\(733\) 23.0763 0.852341 0.426171 0.904643i \(-0.359862\pi\)
0.426171 + 0.904643i \(0.359862\pi\)
\(734\) 0 0
\(735\) 27.0042 0.996064
\(736\) 0 0
\(737\) 20.1517 0.742296
\(738\) 0 0
\(739\) 21.9962 0.809145 0.404572 0.914506i \(-0.367420\pi\)
0.404572 + 0.914506i \(0.367420\pi\)
\(740\) 0 0
\(741\) 2.66426 0.0978741
\(742\) 0 0
\(743\) −1.69614 −0.0622254 −0.0311127 0.999516i \(-0.509905\pi\)
−0.0311127 + 0.999516i \(0.509905\pi\)
\(744\) 0 0
\(745\) −9.22941 −0.338139
\(746\) 0 0
\(747\) 27.5999 1.00983
\(748\) 0 0
\(749\) 10.8042 0.394778
\(750\) 0 0
\(751\) −27.4070 −1.00010 −0.500049 0.865997i \(-0.666685\pi\)
−0.500049 + 0.865997i \(0.666685\pi\)
\(752\) 0 0
\(753\) −2.87232 −0.104673
\(754\) 0 0
\(755\) 14.3750 0.523159
\(756\) 0 0
\(757\) 5.88790 0.213999 0.107000 0.994259i \(-0.465876\pi\)
0.107000 + 0.994259i \(0.465876\pi\)
\(758\) 0 0
\(759\) −67.4976 −2.45001
\(760\) 0 0
\(761\) 4.02614 0.145947 0.0729737 0.997334i \(-0.476751\pi\)
0.0729737 + 0.997334i \(0.476751\pi\)
\(762\) 0 0
\(763\) 42.8089 1.54979
\(764\) 0 0
\(765\) −28.4238 −1.02767
\(766\) 0 0
\(767\) 4.60339 0.166219
\(768\) 0 0
\(769\) 38.9600 1.40493 0.702467 0.711716i \(-0.252082\pi\)
0.702467 + 0.711716i \(0.252082\pi\)
\(770\) 0 0
\(771\) −29.7798 −1.07249
\(772\) 0 0
\(773\) −40.0509 −1.44053 −0.720265 0.693699i \(-0.755980\pi\)
−0.720265 + 0.693699i \(0.755980\pi\)
\(774\) 0 0
\(775\) 0.284305 0.0102126
\(776\) 0 0
\(777\) 74.5262 2.67361
\(778\) 0 0
\(779\) 21.9447 0.786249
\(780\) 0 0
\(781\) −79.6112 −2.84871
\(782\) 0 0
\(783\) −44.0124 −1.57287
\(784\) 0 0
\(785\) 2.54953 0.0909967
\(786\) 0 0
\(787\) −52.2706 −1.86324 −0.931622 0.363428i \(-0.881606\pi\)
−0.931622 + 0.363428i \(0.881606\pi\)
\(788\) 0 0
\(789\) 79.3807 2.82603
\(790\) 0 0
\(791\) −35.8227 −1.27371
\(792\) 0 0
\(793\) −2.64667 −0.0939859
\(794\) 0 0
\(795\) 22.9865 0.815245
\(796\) 0 0
\(797\) −44.0948 −1.56192 −0.780959 0.624582i \(-0.785269\pi\)
−0.780959 + 0.624582i \(0.785269\pi\)
\(798\) 0 0
\(799\) −29.8107 −1.05463
\(800\) 0 0
\(801\) 15.2343 0.538277
\(802\) 0 0
\(803\) 18.8830 0.666365
\(804\) 0 0
\(805\) 15.3839 0.542211
\(806\) 0 0
\(807\) 35.8494 1.26196
\(808\) 0 0
\(809\) 1.50361 0.0528640 0.0264320 0.999651i \(-0.491585\pi\)
0.0264320 + 0.999651i \(0.491585\pi\)
\(810\) 0 0
\(811\) −27.3113 −0.959031 −0.479515 0.877533i \(-0.659188\pi\)
−0.479515 + 0.877533i \(0.659188\pi\)
\(812\) 0 0
\(813\) 62.4702 2.19093
\(814\) 0 0
\(815\) 0.131770 0.00461572
\(816\) 0 0
\(817\) 11.8283 0.413819
\(818\) 0 0
\(819\) 8.07149 0.282041
\(820\) 0 0
\(821\) −53.0473 −1.85136 −0.925681 0.378305i \(-0.876507\pi\)
−0.925681 + 0.378305i \(0.876507\pi\)
\(822\) 0 0
\(823\) 49.1868 1.71454 0.857271 0.514865i \(-0.172158\pi\)
0.857271 + 0.514865i \(0.172158\pi\)
\(824\) 0 0
\(825\) −66.3065 −2.30849
\(826\) 0 0
\(827\) −9.26392 −0.322138 −0.161069 0.986943i \(-0.551494\pi\)
−0.161069 + 0.986943i \(0.551494\pi\)
\(828\) 0 0
\(829\) 11.0625 0.384215 0.192107 0.981374i \(-0.438468\pi\)
0.192107 + 0.981374i \(0.438468\pi\)
\(830\) 0 0
\(831\) 12.1974 0.423123
\(832\) 0 0
\(833\) 79.3107 2.74795
\(834\) 0 0
\(835\) −1.28023 −0.0443041
\(836\) 0 0
\(837\) 0.421626 0.0145735
\(838\) 0 0
\(839\) −40.3371 −1.39259 −0.696295 0.717756i \(-0.745170\pi\)
−0.696295 + 0.717756i \(0.745170\pi\)
\(840\) 0 0
\(841\) 17.3708 0.598992
\(842\) 0 0
\(843\) 28.5438 0.983102
\(844\) 0 0
\(845\) 10.3132 0.354786
\(846\) 0 0
\(847\) 73.8265 2.53671
\(848\) 0 0
\(849\) −87.6457 −3.00799
\(850\) 0 0
\(851\) 26.5940 0.911631
\(852\) 0 0
\(853\) 53.2519 1.82331 0.911655 0.410956i \(-0.134805\pi\)
0.911655 + 0.410956i \(0.134805\pi\)
\(854\) 0 0
\(855\) 10.9841 0.375648
\(856\) 0 0
\(857\) −4.24507 −0.145009 −0.0725043 0.997368i \(-0.523099\pi\)
−0.0725043 + 0.997368i \(0.523099\pi\)
\(858\) 0 0
\(859\) 32.5555 1.11078 0.555390 0.831590i \(-0.312569\pi\)
0.555390 + 0.831590i \(0.312569\pi\)
\(860\) 0 0
\(861\) 104.471 3.56035
\(862\) 0 0
\(863\) −52.5375 −1.78840 −0.894199 0.447669i \(-0.852254\pi\)
−0.894199 + 0.447669i \(0.852254\pi\)
\(864\) 0 0
\(865\) −0.539220 −0.0183340
\(866\) 0 0
\(867\) −82.3519 −2.79682
\(868\) 0 0
\(869\) −62.0623 −2.10532
\(870\) 0 0
\(871\) −1.35127 −0.0457859
\(872\) 0 0
\(873\) −69.7415 −2.36039
\(874\) 0 0
\(875\) 32.4502 1.09702
\(876\) 0 0
\(877\) −51.6449 −1.74392 −0.871962 0.489573i \(-0.837153\pi\)
−0.871962 + 0.489573i \(0.837153\pi\)
\(878\) 0 0
\(879\) 55.0822 1.85788
\(880\) 0 0
\(881\) −16.6645 −0.561441 −0.280721 0.959789i \(-0.590573\pi\)
−0.280721 + 0.959789i \(0.590573\pi\)
\(882\) 0 0
\(883\) −9.48960 −0.319350 −0.159675 0.987170i \(-0.551045\pi\)
−0.159675 + 0.987170i \(0.551045\pi\)
\(884\) 0 0
\(885\) 29.8232 1.00250
\(886\) 0 0
\(887\) −15.6151 −0.524305 −0.262152 0.965026i \(-0.584432\pi\)
−0.262152 + 0.965026i \(0.584432\pi\)
\(888\) 0 0
\(889\) 60.4454 2.02727
\(890\) 0 0
\(891\) −14.9051 −0.499338
\(892\) 0 0
\(893\) 11.5200 0.385503
\(894\) 0 0
\(895\) 8.56620 0.286336
\(896\) 0 0
\(897\) 4.52604 0.151120
\(898\) 0 0
\(899\) −0.444218 −0.0148155
\(900\) 0 0
\(901\) 67.5107 2.24911
\(902\) 0 0
\(903\) 56.3102 1.87389
\(904\) 0 0
\(905\) 3.78923 0.125958
\(906\) 0 0
\(907\) −11.4311 −0.379562 −0.189781 0.981826i \(-0.560778\pi\)
−0.189781 + 0.981826i \(0.560778\pi\)
\(908\) 0 0
\(909\) −38.0815 −1.26308
\(910\) 0 0
\(911\) −10.7068 −0.354733 −0.177367 0.984145i \(-0.556758\pi\)
−0.177367 + 0.984145i \(0.556758\pi\)
\(912\) 0 0
\(913\) −27.8448 −0.921529
\(914\) 0 0
\(915\) −17.1465 −0.566846
\(916\) 0 0
\(917\) −68.0684 −2.24782
\(918\) 0 0
\(919\) 9.67015 0.318989 0.159494 0.987199i \(-0.449014\pi\)
0.159494 + 0.987199i \(0.449014\pi\)
\(920\) 0 0
\(921\) 69.7080 2.29696
\(922\) 0 0
\(923\) 5.33831 0.175713
\(924\) 0 0
\(925\) 26.1247 0.858976
\(926\) 0 0
\(927\) −52.1341 −1.71231
\(928\) 0 0
\(929\) 29.9009 0.981018 0.490509 0.871436i \(-0.336811\pi\)
0.490509 + 0.871436i \(0.336811\pi\)
\(930\) 0 0
\(931\) −30.6488 −1.00447
\(932\) 0 0
\(933\) −90.9175 −2.97650
\(934\) 0 0
\(935\) 28.6761 0.937807
\(936\) 0 0
\(937\) 16.9955 0.555218 0.277609 0.960694i \(-0.410458\pi\)
0.277609 + 0.960694i \(0.410458\pi\)
\(938\) 0 0
\(939\) 93.2581 3.04336
\(940\) 0 0
\(941\) −24.8089 −0.808748 −0.404374 0.914594i \(-0.632511\pi\)
−0.404374 + 0.914594i \(0.632511\pi\)
\(942\) 0 0
\(943\) 37.2795 1.21399
\(944\) 0 0
\(945\) 22.4117 0.729054
\(946\) 0 0
\(947\) −4.82922 −0.156929 −0.0784643 0.996917i \(-0.525002\pi\)
−0.0784643 + 0.996917i \(0.525002\pi\)
\(948\) 0 0
\(949\) −1.26619 −0.0411023
\(950\) 0 0
\(951\) −9.98800 −0.323883
\(952\) 0 0
\(953\) 7.65459 0.247956 0.123978 0.992285i \(-0.460435\pi\)
0.123978 + 0.992285i \(0.460435\pi\)
\(954\) 0 0
\(955\) −5.20938 −0.168572
\(956\) 0 0
\(957\) 103.602 3.34897
\(958\) 0 0
\(959\) −39.1765 −1.26508
\(960\) 0 0
\(961\) −30.9957 −0.999863
\(962\) 0 0
\(963\) 13.1049 0.422300
\(964\) 0 0
\(965\) 17.0181 0.547833
\(966\) 0 0
\(967\) −17.3405 −0.557632 −0.278816 0.960345i \(-0.589942\pi\)
−0.278816 + 0.960345i \(0.589942\pi\)
\(968\) 0 0
\(969\) 50.6937 1.62852
\(970\) 0 0
\(971\) −17.6264 −0.565659 −0.282830 0.959170i \(-0.591273\pi\)
−0.282830 + 0.959170i \(0.591273\pi\)
\(972\) 0 0
\(973\) −64.9029 −2.08069
\(974\) 0 0
\(975\) 4.44617 0.142391
\(976\) 0 0
\(977\) 1.76153 0.0563564 0.0281782 0.999603i \(-0.491029\pi\)
0.0281782 + 0.999603i \(0.491029\pi\)
\(978\) 0 0
\(979\) −15.3695 −0.491211
\(980\) 0 0
\(981\) 51.9247 1.65783
\(982\) 0 0
\(983\) 27.6266 0.881152 0.440576 0.897715i \(-0.354774\pi\)
0.440576 + 0.897715i \(0.354774\pi\)
\(984\) 0 0
\(985\) −8.93722 −0.284763
\(986\) 0 0
\(987\) 54.8428 1.74567
\(988\) 0 0
\(989\) 20.0938 0.638946
\(990\) 0 0
\(991\) −46.7671 −1.48560 −0.742802 0.669511i \(-0.766504\pi\)
−0.742802 + 0.669511i \(0.766504\pi\)
\(992\) 0 0
\(993\) 57.9589 1.83927
\(994\) 0 0
\(995\) −9.70057 −0.307529
\(996\) 0 0
\(997\) 21.0053 0.665243 0.332622 0.943060i \(-0.392067\pi\)
0.332622 + 0.943060i \(0.392067\pi\)
\(998\) 0 0
\(999\) 38.7430 1.22577
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 4016.2.a.j.1.2 14
4.3 odd 2 1004.2.a.b.1.13 14
12.11 even 2 9036.2.a.m.1.9 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1004.2.a.b.1.13 14 4.3 odd 2
4016.2.a.j.1.2 14 1.1 even 1 trivial
9036.2.a.m.1.9 14 12.11 even 2