Properties

Label 3040.2.a.s.1.1
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $1$
Dimension $4$
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] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.78292.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 8x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.78678\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.78678 q^{3} -1.00000 q^{5} -3.76616 q^{7} +4.76616 q^{9} +O(q^{10})\) \(q-2.78678 q^{3} -1.00000 q^{5} -3.76616 q^{7} +4.76616 q^{9} +1.15575 q^{11} +1.63103 q^{13} +2.78678 q^{15} -1.38959 q^{17} +1.00000 q^{19} +10.4955 q^{21} -7.76616 q^{23} +1.00000 q^{25} -4.92191 q^{27} +0.651655 q^{29} +4.31150 q^{31} -3.22082 q^{33} +3.76616 q^{35} +3.02062 q^{37} -4.54534 q^{39} +9.84382 q^{41} +9.95014 q^{43} -4.76616 q^{45} -3.62343 q^{47} +7.18398 q^{49} +3.87248 q^{51} +5.20460 q^{53} -1.15575 q^{55} -2.78678 q^{57} +1.65985 q^{59} +1.15575 q^{61} -17.9501 q^{63} -1.63103 q^{65} +1.99239 q^{67} +21.6426 q^{69} -5.53232 q^{71} +1.87248 q^{73} -2.78678 q^{75} -4.35274 q^{77} -0.0412422 q^{79} -0.582183 q^{81} +5.95014 q^{83} +1.38959 q^{85} -1.81602 q^{87} -0.794392 q^{89} -6.14274 q^{91} -12.0152 q^{93} -1.00000 q^{95} -18.9209 q^{97} +5.50849 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} - 4 q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} - 4 q^{5} - 5 q^{7} + 9 q^{9} - 6 q^{11} + 5 q^{13} - q^{15} - 5 q^{17} + 4 q^{19} - 3 q^{21} - 21 q^{23} + 4 q^{25} + q^{27} - q^{29} - 4 q^{31} - 14 q^{33} + 5 q^{35} + 10 q^{37} - 7 q^{39} - 2 q^{41} + 6 q^{43} - 9 q^{45} - 24 q^{47} + 5 q^{49} + 13 q^{51} - 5 q^{53} + 6 q^{55} + q^{57} - 11 q^{59} - 6 q^{61} - 38 q^{63} - 5 q^{65} + 19 q^{67} - 7 q^{69} - 2 q^{71} + 5 q^{73} + q^{75} + 8 q^{77} + 4 q^{79} - 16 q^{81} - 10 q^{83} + 5 q^{85} - 31 q^{87} + 20 q^{89} - 5 q^{91} - 26 q^{93} - 4 q^{95} - 6 q^{97} - 14 q^{99}+O(q^{100}) \) 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
\(3\) −2.78678 −1.60895 −0.804475 0.593986i \(-0.797553\pi\)
−0.804475 + 0.593986i \(0.797553\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.76616 −1.42348 −0.711738 0.702445i \(-0.752092\pi\)
−0.711738 + 0.702445i \(0.752092\pi\)
\(8\) 0 0
\(9\) 4.76616 1.58872
\(10\) 0 0
\(11\) 1.15575 0.348472 0.174236 0.984704i \(-0.444254\pi\)
0.174236 + 0.984704i \(0.444254\pi\)
\(12\) 0 0
\(13\) 1.63103 0.452367 0.226184 0.974085i \(-0.427375\pi\)
0.226184 + 0.974085i \(0.427375\pi\)
\(14\) 0 0
\(15\) 2.78678 0.719544
\(16\) 0 0
\(17\) −1.38959 −0.337024 −0.168512 0.985700i \(-0.553896\pi\)
−0.168512 + 0.985700i \(0.553896\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 10.4955 2.29030
\(22\) 0 0
\(23\) −7.76616 −1.61936 −0.809678 0.586874i \(-0.800358\pi\)
−0.809678 + 0.586874i \(0.800358\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.92191 −0.947222
\(28\) 0 0
\(29\) 0.651655 0.121009 0.0605046 0.998168i \(-0.480729\pi\)
0.0605046 + 0.998168i \(0.480729\pi\)
\(30\) 0 0
\(31\) 4.31150 0.774368 0.387184 0.922002i \(-0.373448\pi\)
0.387184 + 0.922002i \(0.373448\pi\)
\(32\) 0 0
\(33\) −3.22082 −0.560674
\(34\) 0 0
\(35\) 3.76616 0.636598
\(36\) 0 0
\(37\) 3.02062 0.496587 0.248294 0.968685i \(-0.420130\pi\)
0.248294 + 0.968685i \(0.420130\pi\)
\(38\) 0 0
\(39\) −4.54534 −0.727836
\(40\) 0 0
\(41\) 9.84382 1.53735 0.768674 0.639641i \(-0.220917\pi\)
0.768674 + 0.639641i \(0.220917\pi\)
\(42\) 0 0
\(43\) 9.95014 1.51738 0.758691 0.651450i \(-0.225839\pi\)
0.758691 + 0.651450i \(0.225839\pi\)
\(44\) 0 0
\(45\) −4.76616 −0.710498
\(46\) 0 0
\(47\) −3.62343 −0.528531 −0.264265 0.964450i \(-0.585129\pi\)
−0.264265 + 0.964450i \(0.585129\pi\)
\(48\) 0 0
\(49\) 7.18398 1.02628
\(50\) 0 0
\(51\) 3.87248 0.542256
\(52\) 0 0
\(53\) 5.20460 0.714907 0.357453 0.933931i \(-0.383645\pi\)
0.357453 + 0.933931i \(0.383645\pi\)
\(54\) 0 0
\(55\) −1.15575 −0.155841
\(56\) 0 0
\(57\) −2.78678 −0.369118
\(58\) 0 0
\(59\) 1.65985 0.216093 0.108047 0.994146i \(-0.465540\pi\)
0.108047 + 0.994146i \(0.465540\pi\)
\(60\) 0 0
\(61\) 1.15575 0.147979 0.0739893 0.997259i \(-0.476427\pi\)
0.0739893 + 0.997259i \(0.476427\pi\)
\(62\) 0 0
\(63\) −17.9501 −2.26151
\(64\) 0 0
\(65\) −1.63103 −0.202305
\(66\) 0 0
\(67\) 1.99239 0.243409 0.121705 0.992566i \(-0.461164\pi\)
0.121705 + 0.992566i \(0.461164\pi\)
\(68\) 0 0
\(69\) 21.6426 2.60546
\(70\) 0 0
\(71\) −5.53232 −0.656566 −0.328283 0.944579i \(-0.606470\pi\)
−0.328283 + 0.944579i \(0.606470\pi\)
\(72\) 0 0
\(73\) 1.87248 0.219157 0.109579 0.993978i \(-0.465050\pi\)
0.109579 + 0.993978i \(0.465050\pi\)
\(74\) 0 0
\(75\) −2.78678 −0.321790
\(76\) 0 0
\(77\) −4.35274 −0.496041
\(78\) 0 0
\(79\) −0.0412422 −0.00464011 −0.00232005 0.999997i \(-0.500738\pi\)
−0.00232005 + 0.999997i \(0.500738\pi\)
\(80\) 0 0
\(81\) −0.582183 −0.0646870
\(82\) 0 0
\(83\) 5.95014 0.653113 0.326556 0.945178i \(-0.394112\pi\)
0.326556 + 0.945178i \(0.394112\pi\)
\(84\) 0 0
\(85\) 1.38959 0.150722
\(86\) 0 0
\(87\) −1.81602 −0.194698
\(88\) 0 0
\(89\) −0.794392 −0.0842054 −0.0421027 0.999113i \(-0.513406\pi\)
−0.0421027 + 0.999113i \(0.513406\pi\)
\(90\) 0 0
\(91\) −6.14274 −0.643934
\(92\) 0 0
\(93\) −12.0152 −1.24592
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −18.9209 −1.92113 −0.960563 0.278061i \(-0.910308\pi\)
−0.960563 + 0.278061i \(0.910308\pi\)
\(98\) 0 0
\(99\) 5.50849 0.553624
\(100\) 0 0
\(101\) −7.95014 −0.791069 −0.395534 0.918451i \(-0.629441\pi\)
−0.395534 + 0.918451i \(0.629441\pi\)
\(102\) 0 0
\(103\) −15.7087 −1.54782 −0.773912 0.633293i \(-0.781703\pi\)
−0.773912 + 0.633293i \(0.781703\pi\)
\(104\) 0 0
\(105\) −10.4955 −1.02425
\(106\) 0 0
\(107\) −1.99239 −0.192612 −0.0963059 0.995352i \(-0.530703\pi\)
−0.0963059 + 0.995352i \(0.530703\pi\)
\(108\) 0 0
\(109\) 0.963155 0.0922535 0.0461267 0.998936i \(-0.485312\pi\)
0.0461267 + 0.998936i \(0.485312\pi\)
\(110\) 0 0
\(111\) −8.41782 −0.798984
\(112\) 0 0
\(113\) −0.938137 −0.0882525 −0.0441262 0.999026i \(-0.514050\pi\)
−0.0441262 + 0.999026i \(0.514050\pi\)
\(114\) 0 0
\(115\) 7.76616 0.724198
\(116\) 0 0
\(117\) 7.77377 0.718685
\(118\) 0 0
\(119\) 5.23341 0.479746
\(120\) 0 0
\(121\) −9.66424 −0.878567
\(122\) 0 0
\(123\) −27.4326 −2.47352
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 10.8146 0.959639 0.479820 0.877367i \(-0.340702\pi\)
0.479820 + 0.877367i \(0.340702\pi\)
\(128\) 0 0
\(129\) −27.7289 −2.44139
\(130\) 0 0
\(131\) −19.4174 −1.69651 −0.848253 0.529592i \(-0.822345\pi\)
−0.848253 + 0.529592i \(0.822345\pi\)
\(132\) 0 0
\(133\) −3.76616 −0.326568
\(134\) 0 0
\(135\) 4.92191 0.423611
\(136\) 0 0
\(137\) −2.22522 −0.190114 −0.0950568 0.995472i \(-0.530303\pi\)
−0.0950568 + 0.995472i \(0.530303\pi\)
\(138\) 0 0
\(139\) 1.90890 0.161911 0.0809554 0.996718i \(-0.474203\pi\)
0.0809554 + 0.996718i \(0.474203\pi\)
\(140\) 0 0
\(141\) 10.0977 0.850380
\(142\) 0 0
\(143\) 1.88507 0.157637
\(144\) 0 0
\(145\) −0.651655 −0.0541170
\(146\) 0 0
\(147\) −20.0202 −1.65124
\(148\) 0 0
\(149\) 19.3675 1.58665 0.793325 0.608798i \(-0.208348\pi\)
0.793325 + 0.608798i \(0.208348\pi\)
\(150\) 0 0
\(151\) 14.3267 1.16589 0.582946 0.812511i \(-0.301900\pi\)
0.582946 + 0.812511i \(0.301900\pi\)
\(152\) 0 0
\(153\) −6.62300 −0.535438
\(154\) 0 0
\(155\) −4.31150 −0.346308
\(156\) 0 0
\(157\) −20.2943 −1.61966 −0.809829 0.586665i \(-0.800440\pi\)
−0.809829 + 0.586665i \(0.800440\pi\)
\(158\) 0 0
\(159\) −14.5041 −1.15025
\(160\) 0 0
\(161\) 29.2486 2.30511
\(162\) 0 0
\(163\) 16.2616 1.27371 0.636855 0.770984i \(-0.280235\pi\)
0.636855 + 0.770984i \(0.280235\pi\)
\(164\) 0 0
\(165\) 3.22082 0.250741
\(166\) 0 0
\(167\) −11.1269 −0.861028 −0.430514 0.902584i \(-0.641668\pi\)
−0.430514 + 0.902584i \(0.641668\pi\)
\(168\) 0 0
\(169\) −10.3397 −0.795364
\(170\) 0 0
\(171\) 4.76616 0.364478
\(172\) 0 0
\(173\) −2.32393 −0.176685 −0.0883426 0.996090i \(-0.528157\pi\)
−0.0883426 + 0.996090i \(0.528157\pi\)
\(174\) 0 0
\(175\) −3.76616 −0.284695
\(176\) 0 0
\(177\) −4.62563 −0.347684
\(178\) 0 0
\(179\) −17.8851 −1.33679 −0.668396 0.743805i \(-0.733019\pi\)
−0.668396 + 0.743805i \(0.733019\pi\)
\(180\) 0 0
\(181\) −24.2118 −1.79965 −0.899824 0.436253i \(-0.856305\pi\)
−0.899824 + 0.436253i \(0.856305\pi\)
\(182\) 0 0
\(183\) −3.22082 −0.238090
\(184\) 0 0
\(185\) −3.02062 −0.222080
\(186\) 0 0
\(187\) −1.60602 −0.117444
\(188\) 0 0
\(189\) 18.5367 1.34835
\(190\) 0 0
\(191\) −15.7575 −1.14018 −0.570088 0.821584i \(-0.693091\pi\)
−0.570088 + 0.821584i \(0.693091\pi\)
\(192\) 0 0
\(193\) 14.4380 1.03927 0.519635 0.854388i \(-0.326068\pi\)
0.519635 + 0.854388i \(0.326068\pi\)
\(194\) 0 0
\(195\) 4.54534 0.325498
\(196\) 0 0
\(197\) 19.0234 1.35536 0.677681 0.735356i \(-0.262985\pi\)
0.677681 + 0.735356i \(0.262985\pi\)
\(198\) 0 0
\(199\) −13.9866 −0.991481 −0.495741 0.868471i \(-0.665103\pi\)
−0.495741 + 0.868471i \(0.665103\pi\)
\(200\) 0 0
\(201\) −5.55236 −0.391634
\(202\) 0 0
\(203\) −2.45424 −0.172254
\(204\) 0 0
\(205\) −9.84382 −0.687523
\(206\) 0 0
\(207\) −37.0148 −2.57271
\(208\) 0 0
\(209\) 1.15575 0.0799449
\(210\) 0 0
\(211\) −17.3311 −1.19312 −0.596562 0.802567i \(-0.703467\pi\)
−0.596562 + 0.802567i \(0.703467\pi\)
\(212\) 0 0
\(213\) 15.4174 1.05638
\(214\) 0 0
\(215\) −9.95014 −0.678594
\(216\) 0 0
\(217\) −16.2378 −1.10229
\(218\) 0 0
\(219\) −5.21820 −0.352613
\(220\) 0 0
\(221\) −2.26646 −0.152459
\(222\) 0 0
\(223\) 3.60983 0.241732 0.120866 0.992669i \(-0.461433\pi\)
0.120866 + 0.992669i \(0.461433\pi\)
\(224\) 0 0
\(225\) 4.76616 0.317744
\(226\) 0 0
\(227\) 10.1629 0.674538 0.337269 0.941408i \(-0.390497\pi\)
0.337269 + 0.941408i \(0.390497\pi\)
\(228\) 0 0
\(229\) 13.7940 0.911531 0.455765 0.890100i \(-0.349366\pi\)
0.455765 + 0.890100i \(0.349366\pi\)
\(230\) 0 0
\(231\) 12.1301 0.798105
\(232\) 0 0
\(233\) −19.1059 −1.25167 −0.625834 0.779956i \(-0.715241\pi\)
−0.625834 + 0.779956i \(0.715241\pi\)
\(234\) 0 0
\(235\) 3.62343 0.236366
\(236\) 0 0
\(237\) 0.114933 0.00746570
\(238\) 0 0
\(239\) 8.70811 0.563281 0.281641 0.959520i \(-0.409121\pi\)
0.281641 + 0.959520i \(0.409121\pi\)
\(240\) 0 0
\(241\) 25.4586 1.63993 0.819967 0.572410i \(-0.193992\pi\)
0.819967 + 0.572410i \(0.193992\pi\)
\(242\) 0 0
\(243\) 16.3882 1.05130
\(244\) 0 0
\(245\) −7.18398 −0.458968
\(246\) 0 0
\(247\) 1.63103 0.103780
\(248\) 0 0
\(249\) −16.5818 −1.05083
\(250\) 0 0
\(251\) 23.9828 1.51378 0.756889 0.653543i \(-0.226718\pi\)
0.756889 + 0.653543i \(0.226718\pi\)
\(252\) 0 0
\(253\) −8.97574 −0.564300
\(254\) 0 0
\(255\) −3.87248 −0.242504
\(256\) 0 0
\(257\) −9.06186 −0.565263 −0.282632 0.959228i \(-0.591207\pi\)
−0.282632 + 0.959228i \(0.591207\pi\)
\(258\) 0 0
\(259\) −11.3761 −0.706880
\(260\) 0 0
\(261\) 3.10589 0.192250
\(262\) 0 0
\(263\) −0.459059 −0.0283068 −0.0141534 0.999900i \(-0.504505\pi\)
−0.0141534 + 0.999900i \(0.504505\pi\)
\(264\) 0 0
\(265\) −5.20460 −0.319716
\(266\) 0 0
\(267\) 2.21380 0.135482
\(268\) 0 0
\(269\) −8.63822 −0.526681 −0.263341 0.964703i \(-0.584824\pi\)
−0.263341 + 0.964703i \(0.584824\pi\)
\(270\) 0 0
\(271\) 5.29849 0.321860 0.160930 0.986966i \(-0.448551\pi\)
0.160930 + 0.986966i \(0.448551\pi\)
\(272\) 0 0
\(273\) 17.1185 1.03606
\(274\) 0 0
\(275\) 1.15575 0.0696943
\(276\) 0 0
\(277\) −26.3680 −1.58430 −0.792149 0.610328i \(-0.791038\pi\)
−0.792149 + 0.610328i \(0.791038\pi\)
\(278\) 0 0
\(279\) 20.5493 1.23025
\(280\) 0 0
\(281\) −9.22082 −0.550068 −0.275034 0.961434i \(-0.588689\pi\)
−0.275034 + 0.961434i \(0.588689\pi\)
\(282\) 0 0
\(283\) 1.96737 0.116948 0.0584741 0.998289i \(-0.481376\pi\)
0.0584741 + 0.998289i \(0.481376\pi\)
\(284\) 0 0
\(285\) 2.78678 0.165075
\(286\) 0 0
\(287\) −37.0734 −2.18838
\(288\) 0 0
\(289\) −15.0690 −0.886415
\(290\) 0 0
\(291\) 52.7285 3.09100
\(292\) 0 0
\(293\) −23.6866 −1.38379 −0.691894 0.721999i \(-0.743224\pi\)
−0.691894 + 0.721999i \(0.743224\pi\)
\(294\) 0 0
\(295\) −1.65985 −0.0966399
\(296\) 0 0
\(297\) −5.68850 −0.330080
\(298\) 0 0
\(299\) −12.6669 −0.732544
\(300\) 0 0
\(301\) −37.4739 −2.15996
\(302\) 0 0
\(303\) 22.1553 1.27279
\(304\) 0 0
\(305\) −1.15575 −0.0661781
\(306\) 0 0
\(307\) 4.47908 0.255634 0.127817 0.991798i \(-0.459203\pi\)
0.127817 + 0.991798i \(0.459203\pi\)
\(308\) 0 0
\(309\) 43.7767 2.49037
\(310\) 0 0
\(311\) 28.5281 1.61768 0.808840 0.588028i \(-0.200096\pi\)
0.808840 + 0.588028i \(0.200096\pi\)
\(312\) 0 0
\(313\) 28.8070 1.62827 0.814133 0.580678i \(-0.197212\pi\)
0.814133 + 0.580678i \(0.197212\pi\)
\(314\) 0 0
\(315\) 17.9501 1.01138
\(316\) 0 0
\(317\) −23.6550 −1.32860 −0.664300 0.747466i \(-0.731270\pi\)
−0.664300 + 0.747466i \(0.731270\pi\)
\(318\) 0 0
\(319\) 0.753150 0.0421683
\(320\) 0 0
\(321\) 5.55236 0.309903
\(322\) 0 0
\(323\) −1.38959 −0.0773187
\(324\) 0 0
\(325\) 1.63103 0.0904735
\(326\) 0 0
\(327\) −2.68410 −0.148431
\(328\) 0 0
\(329\) 13.6464 0.752351
\(330\) 0 0
\(331\) 25.3483 1.39327 0.696636 0.717425i \(-0.254679\pi\)
0.696636 + 0.717425i \(0.254679\pi\)
\(332\) 0 0
\(333\) 14.3968 0.788938
\(334\) 0 0
\(335\) −1.99239 −0.108856
\(336\) 0 0
\(337\) −29.0935 −1.58482 −0.792411 0.609988i \(-0.791174\pi\)
−0.792411 + 0.609988i \(0.791174\pi\)
\(338\) 0 0
\(339\) 2.61438 0.141994
\(340\) 0 0
\(341\) 4.98302 0.269845
\(342\) 0 0
\(343\) −0.692897 −0.0374129
\(344\) 0 0
\(345\) −21.6426 −1.16520
\(346\) 0 0
\(347\) 7.23823 0.388569 0.194284 0.980945i \(-0.437762\pi\)
0.194284 + 0.980945i \(0.437762\pi\)
\(348\) 0 0
\(349\) 19.0234 1.01830 0.509150 0.860678i \(-0.329960\pi\)
0.509150 + 0.860678i \(0.329960\pi\)
\(350\) 0 0
\(351\) −8.02780 −0.428492
\(352\) 0 0
\(353\) 10.4379 0.555551 0.277776 0.960646i \(-0.410403\pi\)
0.277776 + 0.960646i \(0.410403\pi\)
\(354\) 0 0
\(355\) 5.53232 0.293625
\(356\) 0 0
\(357\) −14.5844 −0.771888
\(358\) 0 0
\(359\) −28.4456 −1.50130 −0.750651 0.660699i \(-0.770260\pi\)
−0.750651 + 0.660699i \(0.770260\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 26.9322 1.41357
\(364\) 0 0
\(365\) −1.87248 −0.0980101
\(366\) 0 0
\(367\) −29.3523 −1.53218 −0.766089 0.642734i \(-0.777800\pi\)
−0.766089 + 0.642734i \(0.777800\pi\)
\(368\) 0 0
\(369\) 46.9173 2.44242
\(370\) 0 0
\(371\) −19.6014 −1.01765
\(372\) 0 0
\(373\) −9.47486 −0.490590 −0.245295 0.969449i \(-0.578885\pi\)
−0.245295 + 0.969449i \(0.578885\pi\)
\(374\) 0 0
\(375\) 2.78678 0.143909
\(376\) 0 0
\(377\) 1.06287 0.0547406
\(378\) 0 0
\(379\) −19.8304 −1.01862 −0.509309 0.860584i \(-0.670099\pi\)
−0.509309 + 0.860584i \(0.670099\pi\)
\(380\) 0 0
\(381\) −30.1379 −1.54401
\(382\) 0 0
\(383\) −17.9626 −0.917845 −0.458922 0.888476i \(-0.651764\pi\)
−0.458922 + 0.888476i \(0.651764\pi\)
\(384\) 0 0
\(385\) 4.35274 0.221836
\(386\) 0 0
\(387\) 47.4240 2.41070
\(388\) 0 0
\(389\) 9.34455 0.473788 0.236894 0.971536i \(-0.423871\pi\)
0.236894 + 0.971536i \(0.423871\pi\)
\(390\) 0 0
\(391\) 10.7918 0.545763
\(392\) 0 0
\(393\) 54.1121 2.72959
\(394\) 0 0
\(395\) 0.0412422 0.00207512
\(396\) 0 0
\(397\) −31.4739 −1.57963 −0.789814 0.613347i \(-0.789823\pi\)
−0.789814 + 0.613347i \(0.789823\pi\)
\(398\) 0 0
\(399\) 10.4955 0.525431
\(400\) 0 0
\(401\) 5.35977 0.267654 0.133827 0.991005i \(-0.457273\pi\)
0.133827 + 0.991005i \(0.457273\pi\)
\(402\) 0 0
\(403\) 7.03220 0.350299
\(404\) 0 0
\(405\) 0.582183 0.0289289
\(406\) 0 0
\(407\) 3.49108 0.173047
\(408\) 0 0
\(409\) 11.1319 0.550438 0.275219 0.961382i \(-0.411250\pi\)
0.275219 + 0.961382i \(0.411250\pi\)
\(410\) 0 0
\(411\) 6.20121 0.305883
\(412\) 0 0
\(413\) −6.25125 −0.307604
\(414\) 0 0
\(415\) −5.95014 −0.292081
\(416\) 0 0
\(417\) −5.31969 −0.260506
\(418\) 0 0
\(419\) −35.5727 −1.73784 −0.868920 0.494952i \(-0.835186\pi\)
−0.868920 + 0.494952i \(0.835186\pi\)
\(420\) 0 0
\(421\) −26.8222 −1.30723 −0.653617 0.756826i \(-0.726749\pi\)
−0.653617 + 0.756826i \(0.726749\pi\)
\(422\) 0 0
\(423\) −17.2698 −0.839688
\(424\) 0 0
\(425\) −1.38959 −0.0674049
\(426\) 0 0
\(427\) −4.35274 −0.210644
\(428\) 0 0
\(429\) −5.25327 −0.253630
\(430\) 0 0
\(431\) 33.0887 1.59382 0.796912 0.604095i \(-0.206465\pi\)
0.796912 + 0.604095i \(0.206465\pi\)
\(432\) 0 0
\(433\) 32.9613 1.58402 0.792009 0.610509i \(-0.209035\pi\)
0.792009 + 0.610509i \(0.209035\pi\)
\(434\) 0 0
\(435\) 1.81602 0.0870715
\(436\) 0 0
\(437\) −7.76616 −0.371506
\(438\) 0 0
\(439\) −22.4657 −1.07223 −0.536114 0.844146i \(-0.680108\pi\)
−0.536114 + 0.844146i \(0.680108\pi\)
\(440\) 0 0
\(441\) 34.2400 1.63048
\(442\) 0 0
\(443\) 14.4754 0.687749 0.343874 0.939016i \(-0.388261\pi\)
0.343874 + 0.939016i \(0.388261\pi\)
\(444\) 0 0
\(445\) 0.794392 0.0376578
\(446\) 0 0
\(447\) −53.9731 −2.55284
\(448\) 0 0
\(449\) −11.2621 −0.531490 −0.265745 0.964043i \(-0.585618\pi\)
−0.265745 + 0.964043i \(0.585618\pi\)
\(450\) 0 0
\(451\) 11.3770 0.535722
\(452\) 0 0
\(453\) −39.9255 −1.87586
\(454\) 0 0
\(455\) 6.14274 0.287976
\(456\) 0 0
\(457\) 19.3887 0.906967 0.453483 0.891265i \(-0.350181\pi\)
0.453483 + 0.891265i \(0.350181\pi\)
\(458\) 0 0
\(459\) 6.83943 0.319237
\(460\) 0 0
\(461\) −22.3106 −1.03911 −0.519555 0.854437i \(-0.673902\pi\)
−0.519555 + 0.854437i \(0.673902\pi\)
\(462\) 0 0
\(463\) 23.6062 1.09707 0.548536 0.836127i \(-0.315185\pi\)
0.548536 + 0.836127i \(0.315185\pi\)
\(464\) 0 0
\(465\) 12.0152 0.557192
\(466\) 0 0
\(467\) −31.9341 −1.47773 −0.738866 0.673852i \(-0.764639\pi\)
−0.738866 + 0.673852i \(0.764639\pi\)
\(468\) 0 0
\(469\) −7.50367 −0.346487
\(470\) 0 0
\(471\) 56.5557 2.60595
\(472\) 0 0
\(473\) 11.4999 0.528765
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 24.8060 1.13579
\(478\) 0 0
\(479\) −4.53275 −0.207107 −0.103553 0.994624i \(-0.533021\pi\)
−0.103553 + 0.994624i \(0.533021\pi\)
\(480\) 0 0
\(481\) 4.92673 0.224640
\(482\) 0 0
\(483\) −81.5096 −3.70882
\(484\) 0 0
\(485\) 18.9209 0.859154
\(486\) 0 0
\(487\) −2.70051 −0.122372 −0.0611858 0.998126i \(-0.519488\pi\)
−0.0611858 + 0.998126i \(0.519488\pi\)
\(488\) 0 0
\(489\) −45.3177 −2.04934
\(490\) 0 0
\(491\) 37.9091 1.71081 0.855406 0.517957i \(-0.173307\pi\)
0.855406 + 0.517957i \(0.173307\pi\)
\(492\) 0 0
\(493\) −0.905531 −0.0407831
\(494\) 0 0
\(495\) −5.50849 −0.247588
\(496\) 0 0
\(497\) 20.8356 0.934606
\(498\) 0 0
\(499\) −5.54094 −0.248047 −0.124023 0.992279i \(-0.539580\pi\)
−0.124023 + 0.992279i \(0.539580\pi\)
\(500\) 0 0
\(501\) 31.0084 1.38535
\(502\) 0 0
\(503\) 1.28211 0.0571663 0.0285831 0.999591i \(-0.490900\pi\)
0.0285831 + 0.999591i \(0.490900\pi\)
\(504\) 0 0
\(505\) 7.95014 0.353777
\(506\) 0 0
\(507\) 28.8146 1.27970
\(508\) 0 0
\(509\) −42.7935 −1.89679 −0.948395 0.317091i \(-0.897294\pi\)
−0.948395 + 0.317091i \(0.897294\pi\)
\(510\) 0 0
\(511\) −7.05206 −0.311965
\(512\) 0 0
\(513\) −4.92191 −0.217308
\(514\) 0 0
\(515\) 15.7087 0.692208
\(516\) 0 0
\(517\) −4.18777 −0.184178
\(518\) 0 0
\(519\) 6.47629 0.284278
\(520\) 0 0
\(521\) −38.8088 −1.70024 −0.850121 0.526587i \(-0.823471\pi\)
−0.850121 + 0.526587i \(0.823471\pi\)
\(522\) 0 0
\(523\) 15.4923 0.677430 0.338715 0.940889i \(-0.390008\pi\)
0.338715 + 0.940889i \(0.390008\pi\)
\(524\) 0 0
\(525\) 10.4955 0.458060
\(526\) 0 0
\(527\) −5.99121 −0.260981
\(528\) 0 0
\(529\) 37.3133 1.62232
\(530\) 0 0
\(531\) 7.91109 0.343312
\(532\) 0 0
\(533\) 16.0556 0.695446
\(534\) 0 0
\(535\) 1.99239 0.0861386
\(536\) 0 0
\(537\) 49.8418 2.15083
\(538\) 0 0
\(539\) 8.30288 0.357631
\(540\) 0 0
\(541\) 6.90187 0.296735 0.148367 0.988932i \(-0.452598\pi\)
0.148367 + 0.988932i \(0.452598\pi\)
\(542\) 0 0
\(543\) 67.4730 2.89554
\(544\) 0 0
\(545\) −0.963155 −0.0412570
\(546\) 0 0
\(547\) 27.3647 1.17003 0.585016 0.811022i \(-0.301088\pi\)
0.585016 + 0.811022i \(0.301088\pi\)
\(548\) 0 0
\(549\) 5.50849 0.235097
\(550\) 0 0
\(551\) 0.651655 0.0277614
\(552\) 0 0
\(553\) 0.155325 0.00660508
\(554\) 0 0
\(555\) 8.41782 0.357316
\(556\) 0 0
\(557\) 31.8438 1.34927 0.674633 0.738153i \(-0.264302\pi\)
0.674633 + 0.738153i \(0.264302\pi\)
\(558\) 0 0
\(559\) 16.2290 0.686414
\(560\) 0 0
\(561\) 4.47562 0.188961
\(562\) 0 0
\(563\) −11.5121 −0.485178 −0.242589 0.970129i \(-0.577997\pi\)
−0.242589 + 0.970129i \(0.577997\pi\)
\(564\) 0 0
\(565\) 0.938137 0.0394677
\(566\) 0 0
\(567\) 2.19260 0.0920804
\(568\) 0 0
\(569\) −21.8286 −0.915103 −0.457551 0.889183i \(-0.651273\pi\)
−0.457551 + 0.889183i \(0.651273\pi\)
\(570\) 0 0
\(571\) 25.5237 1.06813 0.534067 0.845442i \(-0.320663\pi\)
0.534067 + 0.845442i \(0.320663\pi\)
\(572\) 0 0
\(573\) 43.9129 1.83449
\(574\) 0 0
\(575\) −7.76616 −0.323871
\(576\) 0 0
\(577\) 1.32232 0.0550489 0.0275245 0.999621i \(-0.491238\pi\)
0.0275245 + 0.999621i \(0.491238\pi\)
\(578\) 0 0
\(579\) −40.2356 −1.67214
\(580\) 0 0
\(581\) −22.4092 −0.929690
\(582\) 0 0
\(583\) 6.01522 0.249125
\(584\) 0 0
\(585\) −7.77377 −0.321406
\(586\) 0 0
\(587\) −44.7849 −1.84847 −0.924236 0.381822i \(-0.875297\pi\)
−0.924236 + 0.381822i \(0.875297\pi\)
\(588\) 0 0
\(589\) 4.31150 0.177652
\(590\) 0 0
\(591\) −53.0141 −2.18071
\(592\) 0 0
\(593\) 1.22962 0.0504944 0.0252472 0.999681i \(-0.491963\pi\)
0.0252472 + 0.999681i \(0.491963\pi\)
\(594\) 0 0
\(595\) −5.23341 −0.214549
\(596\) 0 0
\(597\) 38.9775 1.59524
\(598\) 0 0
\(599\) −11.8590 −0.484547 −0.242274 0.970208i \(-0.577893\pi\)
−0.242274 + 0.970208i \(0.577893\pi\)
\(600\) 0 0
\(601\) −2.10851 −0.0860079 −0.0430039 0.999075i \(-0.513693\pi\)
−0.0430039 + 0.999075i \(0.513693\pi\)
\(602\) 0 0
\(603\) 9.49606 0.386710
\(604\) 0 0
\(605\) 9.66424 0.392907
\(606\) 0 0
\(607\) 15.4940 0.628884 0.314442 0.949277i \(-0.398183\pi\)
0.314442 + 0.949277i \(0.398183\pi\)
\(608\) 0 0
\(609\) 6.83943 0.277148
\(610\) 0 0
\(611\) −5.90993 −0.239090
\(612\) 0 0
\(613\) −4.53935 −0.183343 −0.0916713 0.995789i \(-0.529221\pi\)
−0.0916713 + 0.995789i \(0.529221\pi\)
\(614\) 0 0
\(615\) 27.4326 1.10619
\(616\) 0 0
\(617\) −0.711908 −0.0286603 −0.0143302 0.999897i \(-0.504562\pi\)
−0.0143302 + 0.999897i \(0.504562\pi\)
\(618\) 0 0
\(619\) 44.0382 1.77004 0.885022 0.465549i \(-0.154143\pi\)
0.885022 + 0.465549i \(0.154143\pi\)
\(620\) 0 0
\(621\) 38.2244 1.53389
\(622\) 0 0
\(623\) 2.99181 0.119864
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −3.22082 −0.128627
\(628\) 0 0
\(629\) −4.19742 −0.167362
\(630\) 0 0
\(631\) −22.6448 −0.901476 −0.450738 0.892656i \(-0.648839\pi\)
−0.450738 + 0.892656i \(0.648839\pi\)
\(632\) 0 0
\(633\) 48.2981 1.91968
\(634\) 0 0
\(635\) −10.8146 −0.429164
\(636\) 0 0
\(637\) 11.7173 0.464257
\(638\) 0 0
\(639\) −26.3680 −1.04310
\(640\) 0 0
\(641\) −33.4066 −1.31948 −0.659740 0.751494i \(-0.729334\pi\)
−0.659740 + 0.751494i \(0.729334\pi\)
\(642\) 0 0
\(643\) −6.22040 −0.245309 −0.122654 0.992449i \(-0.539141\pi\)
−0.122654 + 0.992449i \(0.539141\pi\)
\(644\) 0 0
\(645\) 27.7289 1.09182
\(646\) 0 0
\(647\) 39.3693 1.54777 0.773884 0.633328i \(-0.218311\pi\)
0.773884 + 0.633328i \(0.218311\pi\)
\(648\) 0 0
\(649\) 1.91837 0.0753025
\(650\) 0 0
\(651\) 45.2513 1.77354
\(652\) 0 0
\(653\) 21.0494 0.823728 0.411864 0.911245i \(-0.364878\pi\)
0.411864 + 0.911245i \(0.364878\pi\)
\(654\) 0 0
\(655\) 19.4174 0.758700
\(656\) 0 0
\(657\) 8.92454 0.348180
\(658\) 0 0
\(659\) 5.17494 0.201587 0.100793 0.994907i \(-0.467862\pi\)
0.100793 + 0.994907i \(0.467862\pi\)
\(660\) 0 0
\(661\) 29.8868 1.16246 0.581232 0.813738i \(-0.302571\pi\)
0.581232 + 0.813738i \(0.302571\pi\)
\(662\) 0 0
\(663\) 6.31614 0.245299
\(664\) 0 0
\(665\) 3.76616 0.146046
\(666\) 0 0
\(667\) −5.06086 −0.195957
\(668\) 0 0
\(669\) −10.0598 −0.388935
\(670\) 0 0
\(671\) 1.33576 0.0515664
\(672\) 0 0
\(673\) 41.5439 1.60140 0.800700 0.599066i \(-0.204461\pi\)
0.800700 + 0.599066i \(0.204461\pi\)
\(674\) 0 0
\(675\) −4.92191 −0.189444
\(676\) 0 0
\(677\) −38.6384 −1.48499 −0.742497 0.669850i \(-0.766358\pi\)
−0.742497 + 0.669850i \(0.766358\pi\)
\(678\) 0 0
\(679\) 71.2592 2.73468
\(680\) 0 0
\(681\) −28.3219 −1.08530
\(682\) 0 0
\(683\) 10.9295 0.418206 0.209103 0.977894i \(-0.432946\pi\)
0.209103 + 0.977894i \(0.432946\pi\)
\(684\) 0 0
\(685\) 2.22522 0.0850214
\(686\) 0 0
\(687\) −38.4408 −1.46661
\(688\) 0 0
\(689\) 8.48888 0.323401
\(690\) 0 0
\(691\) 11.9349 0.454026 0.227013 0.973892i \(-0.427104\pi\)
0.227013 + 0.973892i \(0.427104\pi\)
\(692\) 0 0
\(693\) −20.7459 −0.788071
\(694\) 0 0
\(695\) −1.90890 −0.0724087
\(696\) 0 0
\(697\) −13.6789 −0.518124
\(698\) 0 0
\(699\) 53.2440 2.01387
\(700\) 0 0
\(701\) 18.4494 0.696825 0.348412 0.937341i \(-0.386721\pi\)
0.348412 + 0.937341i \(0.386721\pi\)
\(702\) 0 0
\(703\) 3.02062 0.113925
\(704\) 0 0
\(705\) −10.0977 −0.380301
\(706\) 0 0
\(707\) 29.9415 1.12607
\(708\) 0 0
\(709\) 13.7125 0.514984 0.257492 0.966280i \(-0.417104\pi\)
0.257492 + 0.966280i \(0.417104\pi\)
\(710\) 0 0
\(711\) −0.196567 −0.00737184
\(712\) 0 0
\(713\) −33.4838 −1.25398
\(714\) 0 0
\(715\) −1.88507 −0.0704975
\(716\) 0 0
\(717\) −24.2676 −0.906291
\(718\) 0 0
\(719\) −25.5363 −0.952343 −0.476172 0.879352i \(-0.657976\pi\)
−0.476172 + 0.879352i \(0.657976\pi\)
\(720\) 0 0
\(721\) 59.1615 2.20329
\(722\) 0 0
\(723\) −70.9477 −2.63857
\(724\) 0 0
\(725\) 0.651655 0.0242018
\(726\) 0 0
\(727\) 5.33973 0.198040 0.0990198 0.995085i \(-0.468429\pi\)
0.0990198 + 0.995085i \(0.468429\pi\)
\(728\) 0 0
\(729\) −43.9237 −1.62680
\(730\) 0 0
\(731\) −13.8266 −0.511395
\(732\) 0 0
\(733\) −48.6201 −1.79583 −0.897913 0.440173i \(-0.854917\pi\)
−0.897913 + 0.440173i \(0.854917\pi\)
\(734\) 0 0
\(735\) 20.0202 0.738456
\(736\) 0 0
\(737\) 2.30271 0.0848213
\(738\) 0 0
\(739\) 22.3019 0.820387 0.410194 0.911999i \(-0.365461\pi\)
0.410194 + 0.911999i \(0.365461\pi\)
\(740\) 0 0
\(741\) −4.54534 −0.166977
\(742\) 0 0
\(743\) 17.1433 0.628927 0.314464 0.949269i \(-0.398175\pi\)
0.314464 + 0.949269i \(0.398175\pi\)
\(744\) 0 0
\(745\) −19.3675 −0.709572
\(746\) 0 0
\(747\) 28.3593 1.03761
\(748\) 0 0
\(749\) 7.50367 0.274178
\(750\) 0 0
\(751\) −5.65605 −0.206392 −0.103196 0.994661i \(-0.532907\pi\)
−0.103196 + 0.994661i \(0.532907\pi\)
\(752\) 0 0
\(753\) −66.8348 −2.43559
\(754\) 0 0
\(755\) −14.3267 −0.521403
\(756\) 0 0
\(757\) 39.8114 1.44697 0.723485 0.690341i \(-0.242539\pi\)
0.723485 + 0.690341i \(0.242539\pi\)
\(758\) 0 0
\(759\) 25.0134 0.907931
\(760\) 0 0
\(761\) 16.8712 0.611581 0.305790 0.952099i \(-0.401079\pi\)
0.305790 + 0.952099i \(0.401079\pi\)
\(762\) 0 0
\(763\) −3.62740 −0.131321
\(764\) 0 0
\(765\) 6.62300 0.239455
\(766\) 0 0
\(767\) 2.70726 0.0977536
\(768\) 0 0
\(769\) −14.4140 −0.519783 −0.259891 0.965638i \(-0.583687\pi\)
−0.259891 + 0.965638i \(0.583687\pi\)
\(770\) 0 0
\(771\) 25.2535 0.909481
\(772\) 0 0
\(773\) 9.24584 0.332550 0.166275 0.986079i \(-0.446826\pi\)
0.166275 + 0.986079i \(0.446826\pi\)
\(774\) 0 0
\(775\) 4.31150 0.154874
\(776\) 0 0
\(777\) 31.7029 1.13733
\(778\) 0 0
\(779\) 9.84382 0.352692
\(780\) 0 0
\(781\) −6.39398 −0.228795
\(782\) 0 0
\(783\) −3.20739 −0.114623
\(784\) 0 0
\(785\) 20.2943 0.724333
\(786\) 0 0
\(787\) −53.7345 −1.91543 −0.957714 0.287723i \(-0.907102\pi\)
−0.957714 + 0.287723i \(0.907102\pi\)
\(788\) 0 0
\(789\) 1.27930 0.0455442
\(790\) 0 0
\(791\) 3.53318 0.125625
\(792\) 0 0
\(793\) 1.88507 0.0669407
\(794\) 0 0
\(795\) 14.5041 0.514407
\(796\) 0 0
\(797\) −46.5960 −1.65051 −0.825257 0.564757i \(-0.808970\pi\)
−0.825257 + 0.564757i \(0.808970\pi\)
\(798\) 0 0
\(799\) 5.03507 0.178128
\(800\) 0 0
\(801\) −3.78620 −0.133779
\(802\) 0 0
\(803\) 2.16412 0.0763701
\(804\) 0 0
\(805\) −29.2486 −1.03088
\(806\) 0 0
\(807\) 24.0728 0.847404
\(808\) 0 0
\(809\) 8.43902 0.296700 0.148350 0.988935i \(-0.452604\pi\)
0.148350 + 0.988935i \(0.452604\pi\)
\(810\) 0 0
\(811\) −26.6660 −0.936371 −0.468185 0.883630i \(-0.655092\pi\)
−0.468185 + 0.883630i \(0.655092\pi\)
\(812\) 0 0
\(813\) −14.7657 −0.517857
\(814\) 0 0
\(815\) −16.2616 −0.569620
\(816\) 0 0
\(817\) 9.95014 0.348111
\(818\) 0 0
\(819\) −29.2773 −1.02303
\(820\) 0 0
\(821\) −51.3265 −1.79131 −0.895653 0.444753i \(-0.853291\pi\)
−0.895653 + 0.444753i \(0.853291\pi\)
\(822\) 0 0
\(823\) −31.6340 −1.10269 −0.551346 0.834277i \(-0.685886\pi\)
−0.551346 + 0.834277i \(0.685886\pi\)
\(824\) 0 0
\(825\) −3.22082 −0.112135
\(826\) 0 0
\(827\) −10.5165 −0.365695 −0.182848 0.983141i \(-0.558532\pi\)
−0.182848 + 0.983141i \(0.558532\pi\)
\(828\) 0 0
\(829\) −12.3653 −0.429466 −0.214733 0.976673i \(-0.568888\pi\)
−0.214733 + 0.976673i \(0.568888\pi\)
\(830\) 0 0
\(831\) 73.4818 2.54906
\(832\) 0 0
\(833\) −9.98277 −0.345882
\(834\) 0 0
\(835\) 11.1269 0.385064
\(836\) 0 0
\(837\) −21.2208 −0.733499
\(838\) 0 0
\(839\) −31.2612 −1.07926 −0.539629 0.841903i \(-0.681435\pi\)
−0.539629 + 0.841903i \(0.681435\pi\)
\(840\) 0 0
\(841\) −28.5753 −0.985357
\(842\) 0 0
\(843\) 25.6964 0.885033
\(844\) 0 0
\(845\) 10.3397 0.355698
\(846\) 0 0
\(847\) 36.3971 1.25062
\(848\) 0 0
\(849\) −5.48264 −0.188164
\(850\) 0 0
\(851\) −23.4586 −0.804152
\(852\) 0 0
\(853\) 38.0132 1.30155 0.650774 0.759272i \(-0.274445\pi\)
0.650774 + 0.759272i \(0.274445\pi\)
\(854\) 0 0
\(855\) −4.76616 −0.162999
\(856\) 0 0
\(857\) −19.6761 −0.672122 −0.336061 0.941840i \(-0.609095\pi\)
−0.336061 + 0.941840i \(0.609095\pi\)
\(858\) 0 0
\(859\) 14.4665 0.493591 0.246795 0.969068i \(-0.420622\pi\)
0.246795 + 0.969068i \(0.420622\pi\)
\(860\) 0 0
\(861\) 103.316 3.52099
\(862\) 0 0
\(863\) 22.1515 0.754046 0.377023 0.926204i \(-0.376948\pi\)
0.377023 + 0.926204i \(0.376948\pi\)
\(864\) 0 0
\(865\) 2.32393 0.0790160
\(866\) 0 0
\(867\) 41.9942 1.42620
\(868\) 0 0
\(869\) −0.0476656 −0.00161695
\(870\) 0 0
\(871\) 3.24966 0.110110
\(872\) 0 0
\(873\) −90.1801 −3.05213
\(874\) 0 0
\(875\) 3.76616 0.127320
\(876\) 0 0
\(877\) −3.99899 −0.135036 −0.0675182 0.997718i \(-0.521508\pi\)
−0.0675182 + 0.997718i \(0.521508\pi\)
\(878\) 0 0
\(879\) 66.0095 2.22645
\(880\) 0 0
\(881\) −54.5223 −1.83690 −0.918451 0.395535i \(-0.870559\pi\)
−0.918451 + 0.395535i \(0.870559\pi\)
\(882\) 0 0
\(883\) 7.22100 0.243006 0.121503 0.992591i \(-0.461229\pi\)
0.121503 + 0.992591i \(0.461229\pi\)
\(884\) 0 0
\(885\) 4.62563 0.155489
\(886\) 0 0
\(887\) −19.2258 −0.645539 −0.322770 0.946478i \(-0.604614\pi\)
−0.322770 + 0.946478i \(0.604614\pi\)
\(888\) 0 0
\(889\) −40.7295 −1.36602
\(890\) 0 0
\(891\) −0.672858 −0.0225416
\(892\) 0 0
\(893\) −3.62343 −0.121253
\(894\) 0 0
\(895\) 17.8851 0.597832
\(896\) 0 0
\(897\) 35.2998 1.17863
\(898\) 0 0
\(899\) 2.80961 0.0937057
\(900\) 0 0
\(901\) −7.23225 −0.240941
\(902\) 0 0
\(903\) 104.432 3.47526
\(904\) 0 0
\(905\) 24.2118 0.804827
\(906\) 0 0
\(907\) 12.4188 0.412360 0.206180 0.978514i \(-0.433897\pi\)
0.206180 + 0.978514i \(0.433897\pi\)
\(908\) 0 0
\(909\) −37.8917 −1.25679
\(910\) 0 0
\(911\) 19.4999 0.646060 0.323030 0.946389i \(-0.395299\pi\)
0.323030 + 0.946389i \(0.395299\pi\)
\(912\) 0 0
\(913\) 6.87688 0.227591
\(914\) 0 0
\(915\) 3.22082 0.106477
\(916\) 0 0
\(917\) 73.1291 2.41493
\(918\) 0 0
\(919\) 58.4870 1.92931 0.964655 0.263517i \(-0.0848825\pi\)
0.964655 + 0.263517i \(0.0848825\pi\)
\(920\) 0 0
\(921\) −12.4822 −0.411303
\(922\) 0 0
\(923\) −9.02341 −0.297009
\(924\) 0 0
\(925\) 3.02062 0.0993174
\(926\) 0 0
\(927\) −74.8702 −2.45906
\(928\) 0 0
\(929\) −48.2071 −1.58162 −0.790812 0.612059i \(-0.790342\pi\)
−0.790812 + 0.612059i \(0.790342\pi\)
\(930\) 0 0
\(931\) 7.18398 0.235445
\(932\) 0 0
\(933\) −79.5017 −2.60277
\(934\) 0 0
\(935\) 1.60602 0.0525223
\(936\) 0 0
\(937\) −25.5028 −0.833141 −0.416570 0.909103i \(-0.636768\pi\)
−0.416570 + 0.909103i \(0.636768\pi\)
\(938\) 0 0
\(939\) −80.2788 −2.61980
\(940\) 0 0
\(941\) 3.54576 0.115589 0.0577943 0.998329i \(-0.481593\pi\)
0.0577943 + 0.998329i \(0.481593\pi\)
\(942\) 0 0
\(943\) −76.4487 −2.48951
\(944\) 0 0
\(945\) −18.5367 −0.603000
\(946\) 0 0
\(947\) 19.5065 0.633875 0.316938 0.948446i \(-0.397345\pi\)
0.316938 + 0.948446i \(0.397345\pi\)
\(948\) 0 0
\(949\) 3.05408 0.0991395
\(950\) 0 0
\(951\) 65.9215 2.13765
\(952\) 0 0
\(953\) −5.13470 −0.166329 −0.0831647 0.996536i \(-0.526503\pi\)
−0.0831647 + 0.996536i \(0.526503\pi\)
\(954\) 0 0
\(955\) 15.7575 0.509902
\(956\) 0 0
\(957\) −2.09887 −0.0678467
\(958\) 0 0
\(959\) 8.38055 0.270622
\(960\) 0 0
\(961\) −12.4110 −0.400354
\(962\) 0 0
\(963\) −9.49606 −0.306006
\(964\) 0 0
\(965\) −14.4380 −0.464776
\(966\) 0 0
\(967\) −37.9753 −1.22120 −0.610602 0.791938i \(-0.709072\pi\)
−0.610602 + 0.791938i \(0.709072\pi\)
\(968\) 0 0
\(969\) 3.87248 0.124402
\(970\) 0 0
\(971\) −17.0571 −0.547387 −0.273694 0.961817i \(-0.588245\pi\)
−0.273694 + 0.961817i \(0.588245\pi\)
\(972\) 0 0
\(973\) −7.18923 −0.230476
\(974\) 0 0
\(975\) −4.54534 −0.145567
\(976\) 0 0
\(977\) 28.4705 0.910851 0.455425 0.890274i \(-0.349487\pi\)
0.455425 + 0.890274i \(0.349487\pi\)
\(978\) 0 0
\(979\) −0.918119 −0.0293432
\(980\) 0 0
\(981\) 4.59055 0.146565
\(982\) 0 0
\(983\) −49.9529 −1.59325 −0.796625 0.604473i \(-0.793384\pi\)
−0.796625 + 0.604473i \(0.793384\pi\)
\(984\) 0 0
\(985\) −19.0234 −0.606136
\(986\) 0 0
\(987\) −38.0296 −1.21049
\(988\) 0 0
\(989\) −77.2744 −2.45718
\(990\) 0 0
\(991\) −24.6210 −0.782111 −0.391056 0.920367i \(-0.627890\pi\)
−0.391056 + 0.920367i \(0.627890\pi\)
\(992\) 0 0
\(993\) −70.6404 −2.24170
\(994\) 0 0
\(995\) 13.9866 0.443404
\(996\) 0 0
\(997\) 9.04742 0.286535 0.143267 0.989684i \(-0.454239\pi\)
0.143267 + 0.989684i \(0.454239\pi\)
\(998\) 0 0
\(999\) −14.8672 −0.470378
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 3040.2.a.s.1.1 yes 4
4.3 odd 2 3040.2.a.q.1.4 4
8.3 odd 2 6080.2.a.cg.1.1 4
8.5 even 2 6080.2.a.ce.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.q.1.4 4 4.3 odd 2
3040.2.a.s.1.1 yes 4 1.1 even 1 trivial
6080.2.a.ce.1.4 4 8.5 even 2
6080.2.a.cg.1.1 4 8.3 odd 2