Properties

Label 8008.2.a.s.1.4
Level $8008$
Weight $2$
Character 8008.1
Self dual yes
Analytic conductor $63.944$
Analytic rank $1$
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] = [8008,2,Mod(1,8008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8008.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: \(63.9442019386\)
Analytic rank: \(1\)
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: \( x^{10} - 3x^{9} - 15x^{8} + 43x^{7} + 66x^{6} - 173x^{5} - 127x^{4} + 246x^{3} + 99x^{2} - 82x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.79314\) of defining polynomial
Character \(\chi\) \(=\) 8008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.79314 q^{3} -4.09989 q^{5} +1.00000 q^{7} +0.215350 q^{9} +O(q^{10})\) \(q-1.79314 q^{3} -4.09989 q^{5} +1.00000 q^{7} +0.215350 q^{9} +1.00000 q^{11} -1.00000 q^{13} +7.35167 q^{15} -2.85208 q^{17} -2.43171 q^{19} -1.79314 q^{21} -4.18782 q^{23} +11.8091 q^{25} +4.99327 q^{27} +4.21360 q^{29} -5.03853 q^{31} -1.79314 q^{33} -4.09989 q^{35} -1.01738 q^{37} +1.79314 q^{39} +3.98517 q^{41} +6.70767 q^{43} -0.882910 q^{45} -10.2043 q^{47} +1.00000 q^{49} +5.11418 q^{51} -1.40503 q^{53} -4.09989 q^{55} +4.36040 q^{57} +1.74222 q^{59} +2.66442 q^{61} +0.215350 q^{63} +4.09989 q^{65} +15.7459 q^{67} +7.50935 q^{69} -11.7880 q^{71} +3.81536 q^{73} -21.1753 q^{75} +1.00000 q^{77} +14.6244 q^{79} -9.59967 q^{81} +13.0350 q^{83} +11.6932 q^{85} -7.55557 q^{87} -5.08568 q^{89} -1.00000 q^{91} +9.03479 q^{93} +9.96975 q^{95} +12.1623 q^{97} +0.215350 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9} + 10 q^{11} - 10 q^{13} - 5 q^{15} - 11 q^{17} + 2 q^{19} - 3 q^{21} - 8 q^{23} + 2 q^{25} - 15 q^{27} - 8 q^{29} - 23 q^{31} - 3 q^{33} - 4 q^{35} + 10 q^{37} + 3 q^{39} - 18 q^{41} + 12 q^{43} - 10 q^{45} - 36 q^{47} + 10 q^{49} + 9 q^{51} - 21 q^{53} - 4 q^{55} - 30 q^{57} - 13 q^{59} - 2 q^{61} + 9 q^{63} + 4 q^{65} - 4 q^{67} - 26 q^{69} - 24 q^{71} - 23 q^{73} - 28 q^{75} + 10 q^{77} + 14 q^{79} + 30 q^{81} - 9 q^{83} - 17 q^{85} + 7 q^{87} - 18 q^{89} - 10 q^{91} + q^{93} - 4 q^{95} - 9 q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.79314 −1.03527 −0.517635 0.855602i \(-0.673187\pi\)
−0.517635 + 0.855602i \(0.673187\pi\)
\(4\) 0 0
\(5\) −4.09989 −1.83353 −0.916763 0.399432i \(-0.869207\pi\)
−0.916763 + 0.399432i \(0.869207\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0.215350 0.0717832
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 7.35167 1.89819
\(16\) 0 0
\(17\) −2.85208 −0.691732 −0.345866 0.938284i \(-0.612415\pi\)
−0.345866 + 0.938284i \(0.612415\pi\)
\(18\) 0 0
\(19\) −2.43171 −0.557873 −0.278936 0.960310i \(-0.589982\pi\)
−0.278936 + 0.960310i \(0.589982\pi\)
\(20\) 0 0
\(21\) −1.79314 −0.391295
\(22\) 0 0
\(23\) −4.18782 −0.873221 −0.436611 0.899651i \(-0.643821\pi\)
−0.436611 + 0.899651i \(0.643821\pi\)
\(24\) 0 0
\(25\) 11.8091 2.36182
\(26\) 0 0
\(27\) 4.99327 0.960955
\(28\) 0 0
\(29\) 4.21360 0.782445 0.391223 0.920296i \(-0.372052\pi\)
0.391223 + 0.920296i \(0.372052\pi\)
\(30\) 0 0
\(31\) −5.03853 −0.904947 −0.452473 0.891778i \(-0.649458\pi\)
−0.452473 + 0.891778i \(0.649458\pi\)
\(32\) 0 0
\(33\) −1.79314 −0.312146
\(34\) 0 0
\(35\) −4.09989 −0.693008
\(36\) 0 0
\(37\) −1.01738 −0.167256 −0.0836278 0.996497i \(-0.526651\pi\)
−0.0836278 + 0.996497i \(0.526651\pi\)
\(38\) 0 0
\(39\) 1.79314 0.287132
\(40\) 0 0
\(41\) 3.98517 0.622379 0.311190 0.950348i \(-0.399273\pi\)
0.311190 + 0.950348i \(0.399273\pi\)
\(42\) 0 0
\(43\) 6.70767 1.02291 0.511455 0.859310i \(-0.329107\pi\)
0.511455 + 0.859310i \(0.329107\pi\)
\(44\) 0 0
\(45\) −0.882910 −0.131616
\(46\) 0 0
\(47\) −10.2043 −1.48845 −0.744224 0.667930i \(-0.767180\pi\)
−0.744224 + 0.667930i \(0.767180\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 5.11418 0.716129
\(52\) 0 0
\(53\) −1.40503 −0.192996 −0.0964982 0.995333i \(-0.530764\pi\)
−0.0964982 + 0.995333i \(0.530764\pi\)
\(54\) 0 0
\(55\) −4.09989 −0.552829
\(56\) 0 0
\(57\) 4.36040 0.577549
\(58\) 0 0
\(59\) 1.74222 0.226818 0.113409 0.993548i \(-0.463823\pi\)
0.113409 + 0.993548i \(0.463823\pi\)
\(60\) 0 0
\(61\) 2.66442 0.341144 0.170572 0.985345i \(-0.445439\pi\)
0.170572 + 0.985345i \(0.445439\pi\)
\(62\) 0 0
\(63\) 0.215350 0.0271315
\(64\) 0 0
\(65\) 4.09989 0.508529
\(66\) 0 0
\(67\) 15.7459 1.92367 0.961834 0.273634i \(-0.0882256\pi\)
0.961834 + 0.273634i \(0.0882256\pi\)
\(68\) 0 0
\(69\) 7.50935 0.904019
\(70\) 0 0
\(71\) −11.7880 −1.39898 −0.699488 0.714645i \(-0.746588\pi\)
−0.699488 + 0.714645i \(0.746588\pi\)
\(72\) 0 0
\(73\) 3.81536 0.446554 0.223277 0.974755i \(-0.428325\pi\)
0.223277 + 0.974755i \(0.428325\pi\)
\(74\) 0 0
\(75\) −21.1753 −2.44512
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 14.6244 1.64537 0.822685 0.568497i \(-0.192475\pi\)
0.822685 + 0.568497i \(0.192475\pi\)
\(80\) 0 0
\(81\) −9.59967 −1.06663
\(82\) 0 0
\(83\) 13.0350 1.43078 0.715390 0.698725i \(-0.246249\pi\)
0.715390 + 0.698725i \(0.246249\pi\)
\(84\) 0 0
\(85\) 11.6932 1.26831
\(86\) 0 0
\(87\) −7.55557 −0.810042
\(88\) 0 0
\(89\) −5.08568 −0.539081 −0.269541 0.962989i \(-0.586872\pi\)
−0.269541 + 0.962989i \(0.586872\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 9.03479 0.936864
\(94\) 0 0
\(95\) 9.96975 1.02287
\(96\) 0 0
\(97\) 12.1623 1.23489 0.617447 0.786613i \(-0.288167\pi\)
0.617447 + 0.786613i \(0.288167\pi\)
\(98\) 0 0
\(99\) 0.215350 0.0216435
\(100\) 0 0
\(101\) 1.40779 0.140081 0.0700403 0.997544i \(-0.477687\pi\)
0.0700403 + 0.997544i \(0.477687\pi\)
\(102\) 0 0
\(103\) −9.23231 −0.909687 −0.454843 0.890571i \(-0.650305\pi\)
−0.454843 + 0.890571i \(0.650305\pi\)
\(104\) 0 0
\(105\) 7.35167 0.717450
\(106\) 0 0
\(107\) −10.5588 −1.02075 −0.510376 0.859951i \(-0.670494\pi\)
−0.510376 + 0.859951i \(0.670494\pi\)
\(108\) 0 0
\(109\) 4.07093 0.389924 0.194962 0.980811i \(-0.437542\pi\)
0.194962 + 0.980811i \(0.437542\pi\)
\(110\) 0 0
\(111\) 1.82430 0.173155
\(112\) 0 0
\(113\) 3.76726 0.354394 0.177197 0.984175i \(-0.443297\pi\)
0.177197 + 0.984175i \(0.443297\pi\)
\(114\) 0 0
\(115\) 17.1696 1.60107
\(116\) 0 0
\(117\) −0.215350 −0.0199091
\(118\) 0 0
\(119\) −2.85208 −0.261450
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −7.14597 −0.644330
\(124\) 0 0
\(125\) −27.9165 −2.49693
\(126\) 0 0
\(127\) 11.6398 1.03286 0.516431 0.856329i \(-0.327260\pi\)
0.516431 + 0.856329i \(0.327260\pi\)
\(128\) 0 0
\(129\) −12.0278 −1.05899
\(130\) 0 0
\(131\) 15.2068 1.32862 0.664311 0.747457i \(-0.268725\pi\)
0.664311 + 0.747457i \(0.268725\pi\)
\(132\) 0 0
\(133\) −2.43171 −0.210856
\(134\) 0 0
\(135\) −20.4718 −1.76194
\(136\) 0 0
\(137\) 13.9323 1.19032 0.595159 0.803608i \(-0.297089\pi\)
0.595159 + 0.803608i \(0.297089\pi\)
\(138\) 0 0
\(139\) −3.67440 −0.311659 −0.155829 0.987784i \(-0.549805\pi\)
−0.155829 + 0.987784i \(0.549805\pi\)
\(140\) 0 0
\(141\) 18.2977 1.54094
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) −17.2753 −1.43463
\(146\) 0 0
\(147\) −1.79314 −0.147896
\(148\) 0 0
\(149\) 11.6662 0.955736 0.477868 0.878432i \(-0.341410\pi\)
0.477868 + 0.878432i \(0.341410\pi\)
\(150\) 0 0
\(151\) −22.6065 −1.83969 −0.919846 0.392280i \(-0.871686\pi\)
−0.919846 + 0.392280i \(0.871686\pi\)
\(152\) 0 0
\(153\) −0.614195 −0.0496548
\(154\) 0 0
\(155\) 20.6574 1.65924
\(156\) 0 0
\(157\) −9.25830 −0.738893 −0.369446 0.929252i \(-0.620453\pi\)
−0.369446 + 0.929252i \(0.620453\pi\)
\(158\) 0 0
\(159\) 2.51942 0.199803
\(160\) 0 0
\(161\) −4.18782 −0.330047
\(162\) 0 0
\(163\) 8.49778 0.665598 0.332799 0.942998i \(-0.392007\pi\)
0.332799 + 0.942998i \(0.392007\pi\)
\(164\) 0 0
\(165\) 7.35167 0.572327
\(166\) 0 0
\(167\) −9.51364 −0.736188 −0.368094 0.929789i \(-0.619990\pi\)
−0.368094 + 0.929789i \(0.619990\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −0.523668 −0.0400459
\(172\) 0 0
\(173\) −11.5824 −0.880594 −0.440297 0.897852i \(-0.645127\pi\)
−0.440297 + 0.897852i \(0.645127\pi\)
\(174\) 0 0
\(175\) 11.8091 0.892683
\(176\) 0 0
\(177\) −3.12405 −0.234818
\(178\) 0 0
\(179\) −11.2718 −0.842494 −0.421247 0.906946i \(-0.638407\pi\)
−0.421247 + 0.906946i \(0.638407\pi\)
\(180\) 0 0
\(181\) 14.8861 1.10647 0.553236 0.833024i \(-0.313393\pi\)
0.553236 + 0.833024i \(0.313393\pi\)
\(182\) 0 0
\(183\) −4.77767 −0.353176
\(184\) 0 0
\(185\) 4.17113 0.306667
\(186\) 0 0
\(187\) −2.85208 −0.208565
\(188\) 0 0
\(189\) 4.99327 0.363207
\(190\) 0 0
\(191\) 11.4505 0.828531 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(192\) 0 0
\(193\) −8.04999 −0.579451 −0.289726 0.957110i \(-0.593564\pi\)
−0.289726 + 0.957110i \(0.593564\pi\)
\(194\) 0 0
\(195\) −7.35167 −0.526464
\(196\) 0 0
\(197\) 5.19564 0.370174 0.185087 0.982722i \(-0.440743\pi\)
0.185087 + 0.982722i \(0.440743\pi\)
\(198\) 0 0
\(199\) 26.1305 1.85235 0.926173 0.377100i \(-0.123079\pi\)
0.926173 + 0.377100i \(0.123079\pi\)
\(200\) 0 0
\(201\) −28.2346 −1.99152
\(202\) 0 0
\(203\) 4.21360 0.295737
\(204\) 0 0
\(205\) −16.3388 −1.14115
\(206\) 0 0
\(207\) −0.901846 −0.0626826
\(208\) 0 0
\(209\) −2.43171 −0.168205
\(210\) 0 0
\(211\) −6.02320 −0.414654 −0.207327 0.978272i \(-0.566476\pi\)
−0.207327 + 0.978272i \(0.566476\pi\)
\(212\) 0 0
\(213\) 21.1375 1.44832
\(214\) 0 0
\(215\) −27.5007 −1.87553
\(216\) 0 0
\(217\) −5.03853 −0.342038
\(218\) 0 0
\(219\) −6.84147 −0.462304
\(220\) 0 0
\(221\) 2.85208 0.191852
\(222\) 0 0
\(223\) −21.7386 −1.45572 −0.727862 0.685723i \(-0.759486\pi\)
−0.727862 + 0.685723i \(0.759486\pi\)
\(224\) 0 0
\(225\) 2.54308 0.169539
\(226\) 0 0
\(227\) −6.19258 −0.411016 −0.205508 0.978655i \(-0.565885\pi\)
−0.205508 + 0.978655i \(0.565885\pi\)
\(228\) 0 0
\(229\) 3.40410 0.224949 0.112475 0.993655i \(-0.464122\pi\)
0.112475 + 0.993655i \(0.464122\pi\)
\(230\) 0 0
\(231\) −1.79314 −0.117980
\(232\) 0 0
\(233\) −12.4067 −0.812792 −0.406396 0.913697i \(-0.633215\pi\)
−0.406396 + 0.913697i \(0.633215\pi\)
\(234\) 0 0
\(235\) 41.8364 2.72911
\(236\) 0 0
\(237\) −26.2235 −1.70340
\(238\) 0 0
\(239\) 8.45750 0.547070 0.273535 0.961862i \(-0.411807\pi\)
0.273535 + 0.961862i \(0.411807\pi\)
\(240\) 0 0
\(241\) −28.2861 −1.82207 −0.911033 0.412334i \(-0.864714\pi\)
−0.911033 + 0.412334i \(0.864714\pi\)
\(242\) 0 0
\(243\) 2.23375 0.143295
\(244\) 0 0
\(245\) −4.09989 −0.261932
\(246\) 0 0
\(247\) 2.43171 0.154726
\(248\) 0 0
\(249\) −23.3736 −1.48124
\(250\) 0 0
\(251\) 16.7992 1.06036 0.530179 0.847886i \(-0.322125\pi\)
0.530179 + 0.847886i \(0.322125\pi\)
\(252\) 0 0
\(253\) −4.18782 −0.263286
\(254\) 0 0
\(255\) −20.9676 −1.31304
\(256\) 0 0
\(257\) −3.02687 −0.188811 −0.0944053 0.995534i \(-0.530095\pi\)
−0.0944053 + 0.995534i \(0.530095\pi\)
\(258\) 0 0
\(259\) −1.01738 −0.0632167
\(260\) 0 0
\(261\) 0.907397 0.0561665
\(262\) 0 0
\(263\) 21.7871 1.34345 0.671726 0.740799i \(-0.265553\pi\)
0.671726 + 0.740799i \(0.265553\pi\)
\(264\) 0 0
\(265\) 5.76049 0.353864
\(266\) 0 0
\(267\) 9.11934 0.558094
\(268\) 0 0
\(269\) −4.03465 −0.245997 −0.122998 0.992407i \(-0.539251\pi\)
−0.122998 + 0.992407i \(0.539251\pi\)
\(270\) 0 0
\(271\) 17.5642 1.06695 0.533474 0.845816i \(-0.320886\pi\)
0.533474 + 0.845816i \(0.320886\pi\)
\(272\) 0 0
\(273\) 1.79314 0.108526
\(274\) 0 0
\(275\) 11.8091 0.712115
\(276\) 0 0
\(277\) −11.3895 −0.684330 −0.342165 0.939640i \(-0.611160\pi\)
−0.342165 + 0.939640i \(0.611160\pi\)
\(278\) 0 0
\(279\) −1.08505 −0.0649600
\(280\) 0 0
\(281\) −22.9580 −1.36956 −0.684781 0.728749i \(-0.740102\pi\)
−0.684781 + 0.728749i \(0.740102\pi\)
\(282\) 0 0
\(283\) 14.4918 0.861447 0.430723 0.902484i \(-0.358258\pi\)
0.430723 + 0.902484i \(0.358258\pi\)
\(284\) 0 0
\(285\) −17.8771 −1.05895
\(286\) 0 0
\(287\) 3.98517 0.235237
\(288\) 0 0
\(289\) −8.86562 −0.521507
\(290\) 0 0
\(291\) −21.8087 −1.27845
\(292\) 0 0
\(293\) 10.4556 0.610823 0.305412 0.952220i \(-0.401206\pi\)
0.305412 + 0.952220i \(0.401206\pi\)
\(294\) 0 0
\(295\) −7.14292 −0.415877
\(296\) 0 0
\(297\) 4.99327 0.289739
\(298\) 0 0
\(299\) 4.18782 0.242188
\(300\) 0 0
\(301\) 6.70767 0.386624
\(302\) 0 0
\(303\) −2.52437 −0.145021
\(304\) 0 0
\(305\) −10.9238 −0.625495
\(306\) 0 0
\(307\) −12.4302 −0.709426 −0.354713 0.934975i \(-0.615422\pi\)
−0.354713 + 0.934975i \(0.615422\pi\)
\(308\) 0 0
\(309\) 16.5548 0.941771
\(310\) 0 0
\(311\) 2.36867 0.134315 0.0671575 0.997742i \(-0.478607\pi\)
0.0671575 + 0.997742i \(0.478607\pi\)
\(312\) 0 0
\(313\) −23.1821 −1.31033 −0.655166 0.755485i \(-0.727401\pi\)
−0.655166 + 0.755485i \(0.727401\pi\)
\(314\) 0 0
\(315\) −0.882910 −0.0497463
\(316\) 0 0
\(317\) −16.0219 −0.899878 −0.449939 0.893059i \(-0.648554\pi\)
−0.449939 + 0.893059i \(0.648554\pi\)
\(318\) 0 0
\(319\) 4.21360 0.235916
\(320\) 0 0
\(321\) 18.9333 1.05675
\(322\) 0 0
\(323\) 6.93544 0.385898
\(324\) 0 0
\(325\) −11.8091 −0.655050
\(326\) 0 0
\(327\) −7.29975 −0.403677
\(328\) 0 0
\(329\) −10.2043 −0.562580
\(330\) 0 0
\(331\) −3.52695 −0.193859 −0.0969295 0.995291i \(-0.530902\pi\)
−0.0969295 + 0.995291i \(0.530902\pi\)
\(332\) 0 0
\(333\) −0.219092 −0.0120061
\(334\) 0 0
\(335\) −64.5564 −3.52710
\(336\) 0 0
\(337\) −1.07357 −0.0584808 −0.0292404 0.999572i \(-0.509309\pi\)
−0.0292404 + 0.999572i \(0.509309\pi\)
\(338\) 0 0
\(339\) −6.75522 −0.366893
\(340\) 0 0
\(341\) −5.03853 −0.272852
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −30.7875 −1.65754
\(346\) 0 0
\(347\) 33.9847 1.82439 0.912196 0.409754i \(-0.134385\pi\)
0.912196 + 0.409754i \(0.134385\pi\)
\(348\) 0 0
\(349\) 15.7599 0.843609 0.421805 0.906687i \(-0.361397\pi\)
0.421805 + 0.906687i \(0.361397\pi\)
\(350\) 0 0
\(351\) −4.99327 −0.266521
\(352\) 0 0
\(353\) −34.7330 −1.84865 −0.924326 0.381604i \(-0.875372\pi\)
−0.924326 + 0.381604i \(0.875372\pi\)
\(354\) 0 0
\(355\) 48.3294 2.56506
\(356\) 0 0
\(357\) 5.11418 0.270671
\(358\) 0 0
\(359\) 10.6306 0.561060 0.280530 0.959845i \(-0.409490\pi\)
0.280530 + 0.959845i \(0.409490\pi\)
\(360\) 0 0
\(361\) −13.0868 −0.688778
\(362\) 0 0
\(363\) −1.79314 −0.0941154
\(364\) 0 0
\(365\) −15.6425 −0.818768
\(366\) 0 0
\(367\) −15.7445 −0.821855 −0.410928 0.911668i \(-0.634795\pi\)
−0.410928 + 0.911668i \(0.634795\pi\)
\(368\) 0 0
\(369\) 0.858206 0.0446764
\(370\) 0 0
\(371\) −1.40503 −0.0729458
\(372\) 0 0
\(373\) −5.84884 −0.302841 −0.151421 0.988469i \(-0.548385\pi\)
−0.151421 + 0.988469i \(0.548385\pi\)
\(374\) 0 0
\(375\) 50.0582 2.58499
\(376\) 0 0
\(377\) −4.21360 −0.217011
\(378\) 0 0
\(379\) −12.7116 −0.652950 −0.326475 0.945206i \(-0.605861\pi\)
−0.326475 + 0.945206i \(0.605861\pi\)
\(380\) 0 0
\(381\) −20.8717 −1.06929
\(382\) 0 0
\(383\) −19.8627 −1.01493 −0.507467 0.861671i \(-0.669418\pi\)
−0.507467 + 0.861671i \(0.669418\pi\)
\(384\) 0 0
\(385\) −4.09989 −0.208950
\(386\) 0 0
\(387\) 1.44449 0.0734278
\(388\) 0 0
\(389\) −26.1385 −1.32527 −0.662636 0.748941i \(-0.730562\pi\)
−0.662636 + 0.748941i \(0.730562\pi\)
\(390\) 0 0
\(391\) 11.9440 0.604035
\(392\) 0 0
\(393\) −27.2679 −1.37548
\(394\) 0 0
\(395\) −59.9583 −3.01683
\(396\) 0 0
\(397\) −29.0511 −1.45803 −0.729017 0.684496i \(-0.760022\pi\)
−0.729017 + 0.684496i \(0.760022\pi\)
\(398\) 0 0
\(399\) 4.36040 0.218293
\(400\) 0 0
\(401\) −0.930773 −0.0464806 −0.0232403 0.999730i \(-0.507398\pi\)
−0.0232403 + 0.999730i \(0.507398\pi\)
\(402\) 0 0
\(403\) 5.03853 0.250987
\(404\) 0 0
\(405\) 39.3576 1.95569
\(406\) 0 0
\(407\) −1.01738 −0.0504295
\(408\) 0 0
\(409\) −17.9392 −0.887036 −0.443518 0.896266i \(-0.646270\pi\)
−0.443518 + 0.896266i \(0.646270\pi\)
\(410\) 0 0
\(411\) −24.9826 −1.23230
\(412\) 0 0
\(413\) 1.74222 0.0857291
\(414\) 0 0
\(415\) −53.4422 −2.62337
\(416\) 0 0
\(417\) 6.58872 0.322651
\(418\) 0 0
\(419\) −18.6006 −0.908699 −0.454350 0.890823i \(-0.650128\pi\)
−0.454350 + 0.890823i \(0.650128\pi\)
\(420\) 0 0
\(421\) −35.4998 −1.73015 −0.865076 0.501640i \(-0.832730\pi\)
−0.865076 + 0.501640i \(0.832730\pi\)
\(422\) 0 0
\(423\) −2.19749 −0.106846
\(424\) 0 0
\(425\) −33.6805 −1.63374
\(426\) 0 0
\(427\) 2.66442 0.128940
\(428\) 0 0
\(429\) 1.79314 0.0865736
\(430\) 0 0
\(431\) 7.64974 0.368475 0.184237 0.982882i \(-0.441018\pi\)
0.184237 + 0.982882i \(0.441018\pi\)
\(432\) 0 0
\(433\) −7.22294 −0.347112 −0.173556 0.984824i \(-0.555526\pi\)
−0.173556 + 0.984824i \(0.555526\pi\)
\(434\) 0 0
\(435\) 30.9770 1.48523
\(436\) 0 0
\(437\) 10.1836 0.487146
\(438\) 0 0
\(439\) 38.9496 1.85896 0.929482 0.368868i \(-0.120255\pi\)
0.929482 + 0.368868i \(0.120255\pi\)
\(440\) 0 0
\(441\) 0.215350 0.0102547
\(442\) 0 0
\(443\) −6.09099 −0.289392 −0.144696 0.989476i \(-0.546220\pi\)
−0.144696 + 0.989476i \(0.546220\pi\)
\(444\) 0 0
\(445\) 20.8507 0.988419
\(446\) 0 0
\(447\) −20.9192 −0.989444
\(448\) 0 0
\(449\) −14.5301 −0.685716 −0.342858 0.939387i \(-0.611395\pi\)
−0.342858 + 0.939387i \(0.611395\pi\)
\(450\) 0 0
\(451\) 3.98517 0.187654
\(452\) 0 0
\(453\) 40.5366 1.90458
\(454\) 0 0
\(455\) 4.09989 0.192206
\(456\) 0 0
\(457\) −1.88912 −0.0883692 −0.0441846 0.999023i \(-0.514069\pi\)
−0.0441846 + 0.999023i \(0.514069\pi\)
\(458\) 0 0
\(459\) −14.2412 −0.664723
\(460\) 0 0
\(461\) 24.0023 1.11790 0.558949 0.829202i \(-0.311205\pi\)
0.558949 + 0.829202i \(0.311205\pi\)
\(462\) 0 0
\(463\) 10.0569 0.467386 0.233693 0.972310i \(-0.424919\pi\)
0.233693 + 0.972310i \(0.424919\pi\)
\(464\) 0 0
\(465\) −37.0416 −1.71776
\(466\) 0 0
\(467\) −33.8616 −1.56693 −0.783463 0.621438i \(-0.786549\pi\)
−0.783463 + 0.621438i \(0.786549\pi\)
\(468\) 0 0
\(469\) 15.7459 0.727078
\(470\) 0 0
\(471\) 16.6014 0.764953
\(472\) 0 0
\(473\) 6.70767 0.308419
\(474\) 0 0
\(475\) −28.7163 −1.31759
\(476\) 0 0
\(477\) −0.302574 −0.0138539
\(478\) 0 0
\(479\) −15.6911 −0.716945 −0.358472 0.933540i \(-0.616702\pi\)
−0.358472 + 0.933540i \(0.616702\pi\)
\(480\) 0 0
\(481\) 1.01738 0.0463883
\(482\) 0 0
\(483\) 7.50935 0.341687
\(484\) 0 0
\(485\) −49.8641 −2.26421
\(486\) 0 0
\(487\) −23.8974 −1.08290 −0.541448 0.840735i \(-0.682124\pi\)
−0.541448 + 0.840735i \(0.682124\pi\)
\(488\) 0 0
\(489\) −15.2377 −0.689073
\(490\) 0 0
\(491\) 6.60891 0.298256 0.149128 0.988818i \(-0.452353\pi\)
0.149128 + 0.988818i \(0.452353\pi\)
\(492\) 0 0
\(493\) −12.0175 −0.541243
\(494\) 0 0
\(495\) −0.882910 −0.0396838
\(496\) 0 0
\(497\) −11.7880 −0.528763
\(498\) 0 0
\(499\) 12.2334 0.547643 0.273822 0.961781i \(-0.411712\pi\)
0.273822 + 0.961781i \(0.411712\pi\)
\(500\) 0 0
\(501\) 17.0593 0.762153
\(502\) 0 0
\(503\) −30.8269 −1.37450 −0.687252 0.726419i \(-0.741183\pi\)
−0.687252 + 0.726419i \(0.741183\pi\)
\(504\) 0 0
\(505\) −5.77179 −0.256841
\(506\) 0 0
\(507\) −1.79314 −0.0796361
\(508\) 0 0
\(509\) 5.27765 0.233928 0.116964 0.993136i \(-0.462684\pi\)
0.116964 + 0.993136i \(0.462684\pi\)
\(510\) 0 0
\(511\) 3.81536 0.168781
\(512\) 0 0
\(513\) −12.1422 −0.536090
\(514\) 0 0
\(515\) 37.8515 1.66793
\(516\) 0 0
\(517\) −10.2043 −0.448784
\(518\) 0 0
\(519\) 20.7689 0.911652
\(520\) 0 0
\(521\) 30.4010 1.33189 0.665946 0.746000i \(-0.268028\pi\)
0.665946 + 0.746000i \(0.268028\pi\)
\(522\) 0 0
\(523\) 11.0070 0.481301 0.240651 0.970612i \(-0.422639\pi\)
0.240651 + 0.970612i \(0.422639\pi\)
\(524\) 0 0
\(525\) −21.1753 −0.924168
\(526\) 0 0
\(527\) 14.3703 0.625981
\(528\) 0 0
\(529\) −5.46215 −0.237485
\(530\) 0 0
\(531\) 0.375187 0.0162817
\(532\) 0 0
\(533\) −3.98517 −0.172617
\(534\) 0 0
\(535\) 43.2897 1.87158
\(536\) 0 0
\(537\) 20.2119 0.872208
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −12.2639 −0.527266 −0.263633 0.964623i \(-0.584921\pi\)
−0.263633 + 0.964623i \(0.584921\pi\)
\(542\) 0 0
\(543\) −26.6928 −1.14550
\(544\) 0 0
\(545\) −16.6904 −0.714937
\(546\) 0 0
\(547\) 18.5691 0.793959 0.396979 0.917828i \(-0.370058\pi\)
0.396979 + 0.917828i \(0.370058\pi\)
\(548\) 0 0
\(549\) 0.573781 0.0244884
\(550\) 0 0
\(551\) −10.2463 −0.436505
\(552\) 0 0
\(553\) 14.6244 0.621892
\(554\) 0 0
\(555\) −7.47941 −0.317483
\(556\) 0 0
\(557\) 11.9013 0.504274 0.252137 0.967692i \(-0.418867\pi\)
0.252137 + 0.967692i \(0.418867\pi\)
\(558\) 0 0
\(559\) −6.70767 −0.283704
\(560\) 0 0
\(561\) 5.11418 0.215921
\(562\) 0 0
\(563\) −17.3475 −0.731110 −0.365555 0.930790i \(-0.619121\pi\)
−0.365555 + 0.930790i \(0.619121\pi\)
\(564\) 0 0
\(565\) −15.4453 −0.649790
\(566\) 0 0
\(567\) −9.59967 −0.403148
\(568\) 0 0
\(569\) 2.45530 0.102932 0.0514658 0.998675i \(-0.483611\pi\)
0.0514658 + 0.998675i \(0.483611\pi\)
\(570\) 0 0
\(571\) 29.9694 1.25418 0.627091 0.778946i \(-0.284246\pi\)
0.627091 + 0.778946i \(0.284246\pi\)
\(572\) 0 0
\(573\) −20.5324 −0.857753
\(574\) 0 0
\(575\) −49.4544 −2.06239
\(576\) 0 0
\(577\) 4.74606 0.197581 0.0987905 0.995108i \(-0.468503\pi\)
0.0987905 + 0.995108i \(0.468503\pi\)
\(578\) 0 0
\(579\) 14.4348 0.599888
\(580\) 0 0
\(581\) 13.0350 0.540784
\(582\) 0 0
\(583\) −1.40503 −0.0581906
\(584\) 0 0
\(585\) 0.882910 0.0365038
\(586\) 0 0
\(587\) −38.1637 −1.57518 −0.787592 0.616198i \(-0.788672\pi\)
−0.787592 + 0.616198i \(0.788672\pi\)
\(588\) 0 0
\(589\) 12.2523 0.504845
\(590\) 0 0
\(591\) −9.31651 −0.383230
\(592\) 0 0
\(593\) 35.4574 1.45606 0.728031 0.685544i \(-0.240436\pi\)
0.728031 + 0.685544i \(0.240436\pi\)
\(594\) 0 0
\(595\) 11.6932 0.479376
\(596\) 0 0
\(597\) −46.8557 −1.91768
\(598\) 0 0
\(599\) −21.5036 −0.878612 −0.439306 0.898337i \(-0.644776\pi\)
−0.439306 + 0.898337i \(0.644776\pi\)
\(600\) 0 0
\(601\) −15.3470 −0.626016 −0.313008 0.949750i \(-0.601337\pi\)
−0.313008 + 0.949750i \(0.601337\pi\)
\(602\) 0 0
\(603\) 3.39088 0.138087
\(604\) 0 0
\(605\) −4.09989 −0.166684
\(606\) 0 0
\(607\) −38.3757 −1.55762 −0.778811 0.627258i \(-0.784177\pi\)
−0.778811 + 0.627258i \(0.784177\pi\)
\(608\) 0 0
\(609\) −7.55557 −0.306167
\(610\) 0 0
\(611\) 10.2043 0.412821
\(612\) 0 0
\(613\) 12.4970 0.504749 0.252375 0.967630i \(-0.418788\pi\)
0.252375 + 0.967630i \(0.418788\pi\)
\(614\) 0 0
\(615\) 29.2977 1.18140
\(616\) 0 0
\(617\) −10.1442 −0.408390 −0.204195 0.978930i \(-0.565458\pi\)
−0.204195 + 0.978930i \(0.565458\pi\)
\(618\) 0 0
\(619\) 20.1922 0.811593 0.405796 0.913964i \(-0.366994\pi\)
0.405796 + 0.913964i \(0.366994\pi\)
\(620\) 0 0
\(621\) −20.9109 −0.839126
\(622\) 0 0
\(623\) −5.08568 −0.203754
\(624\) 0 0
\(625\) 55.4091 2.21636
\(626\) 0 0
\(627\) 4.36040 0.174138
\(628\) 0 0
\(629\) 2.90164 0.115696
\(630\) 0 0
\(631\) 48.0614 1.91330 0.956648 0.291248i \(-0.0940705\pi\)
0.956648 + 0.291248i \(0.0940705\pi\)
\(632\) 0 0
\(633\) 10.8004 0.429279
\(634\) 0 0
\(635\) −47.7218 −1.89378
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −2.53854 −0.100423
\(640\) 0 0
\(641\) 22.9390 0.906037 0.453019 0.891501i \(-0.350347\pi\)
0.453019 + 0.891501i \(0.350347\pi\)
\(642\) 0 0
\(643\) 25.3284 0.998853 0.499427 0.866356i \(-0.333544\pi\)
0.499427 + 0.866356i \(0.333544\pi\)
\(644\) 0 0
\(645\) 49.3126 1.94168
\(646\) 0 0
\(647\) 13.3832 0.526146 0.263073 0.964776i \(-0.415264\pi\)
0.263073 + 0.964776i \(0.415264\pi\)
\(648\) 0 0
\(649\) 1.74222 0.0683882
\(650\) 0 0
\(651\) 9.03479 0.354101
\(652\) 0 0
\(653\) −3.15602 −0.123505 −0.0617523 0.998092i \(-0.519669\pi\)
−0.0617523 + 0.998092i \(0.519669\pi\)
\(654\) 0 0
\(655\) −62.3461 −2.43606
\(656\) 0 0
\(657\) 0.821636 0.0320551
\(658\) 0 0
\(659\) −32.8595 −1.28003 −0.640013 0.768364i \(-0.721071\pi\)
−0.640013 + 0.768364i \(0.721071\pi\)
\(660\) 0 0
\(661\) −6.06631 −0.235952 −0.117976 0.993016i \(-0.537641\pi\)
−0.117976 + 0.993016i \(0.537641\pi\)
\(662\) 0 0
\(663\) −5.11418 −0.198618
\(664\) 0 0
\(665\) 9.96975 0.386610
\(666\) 0 0
\(667\) −17.6458 −0.683248
\(668\) 0 0
\(669\) 38.9804 1.50707
\(670\) 0 0
\(671\) 2.66442 0.102859
\(672\) 0 0
\(673\) 10.3601 0.399353 0.199676 0.979862i \(-0.436011\pi\)
0.199676 + 0.979862i \(0.436011\pi\)
\(674\) 0 0
\(675\) 58.9659 2.26960
\(676\) 0 0
\(677\) 11.7326 0.450919 0.225459 0.974253i \(-0.427612\pi\)
0.225459 + 0.974253i \(0.427612\pi\)
\(678\) 0 0
\(679\) 12.1623 0.466746
\(680\) 0 0
\(681\) 11.1042 0.425512
\(682\) 0 0
\(683\) −10.6615 −0.407950 −0.203975 0.978976i \(-0.565386\pi\)
−0.203975 + 0.978976i \(0.565386\pi\)
\(684\) 0 0
\(685\) −57.1209 −2.18248
\(686\) 0 0
\(687\) −6.10402 −0.232883
\(688\) 0 0
\(689\) 1.40503 0.0535276
\(690\) 0 0
\(691\) 31.1921 1.18660 0.593302 0.804980i \(-0.297824\pi\)
0.593302 + 0.804980i \(0.297824\pi\)
\(692\) 0 0
\(693\) 0.215350 0.00818046
\(694\) 0 0
\(695\) 15.0646 0.571434
\(696\) 0 0
\(697\) −11.3660 −0.430520
\(698\) 0 0
\(699\) 22.2470 0.841458
\(700\) 0 0
\(701\) −37.9873 −1.43476 −0.717379 0.696683i \(-0.754659\pi\)
−0.717379 + 0.696683i \(0.754659\pi\)
\(702\) 0 0
\(703\) 2.47396 0.0933073
\(704\) 0 0
\(705\) −75.0186 −2.82536
\(706\) 0 0
\(707\) 1.40779 0.0529455
\(708\) 0 0
\(709\) −16.8018 −0.631006 −0.315503 0.948925i \(-0.602173\pi\)
−0.315503 + 0.948925i \(0.602173\pi\)
\(710\) 0 0
\(711\) 3.14935 0.118110
\(712\) 0 0
\(713\) 21.1005 0.790219
\(714\) 0 0
\(715\) 4.09989 0.153327
\(716\) 0 0
\(717\) −15.1655 −0.566365
\(718\) 0 0
\(719\) 6.40347 0.238809 0.119405 0.992846i \(-0.461901\pi\)
0.119405 + 0.992846i \(0.461901\pi\)
\(720\) 0 0
\(721\) −9.23231 −0.343829
\(722\) 0 0
\(723\) 50.7209 1.88633
\(724\) 0 0
\(725\) 49.7587 1.84799
\(726\) 0 0
\(727\) −19.1475 −0.710141 −0.355071 0.934839i \(-0.615543\pi\)
−0.355071 + 0.934839i \(0.615543\pi\)
\(728\) 0 0
\(729\) 24.7936 0.918281
\(730\) 0 0
\(731\) −19.1308 −0.707580
\(732\) 0 0
\(733\) 44.1429 1.63045 0.815227 0.579142i \(-0.196612\pi\)
0.815227 + 0.579142i \(0.196612\pi\)
\(734\) 0 0
\(735\) 7.35167 0.271171
\(736\) 0 0
\(737\) 15.7459 0.580008
\(738\) 0 0
\(739\) −17.7229 −0.651947 −0.325973 0.945379i \(-0.605692\pi\)
−0.325973 + 0.945379i \(0.605692\pi\)
\(740\) 0 0
\(741\) −4.36040 −0.160183
\(742\) 0 0
\(743\) 36.0867 1.32389 0.661946 0.749552i \(-0.269731\pi\)
0.661946 + 0.749552i \(0.269731\pi\)
\(744\) 0 0
\(745\) −47.8303 −1.75237
\(746\) 0 0
\(747\) 2.80709 0.102706
\(748\) 0 0
\(749\) −10.5588 −0.385808
\(750\) 0 0
\(751\) 18.0251 0.657744 0.328872 0.944375i \(-0.393332\pi\)
0.328872 + 0.944375i \(0.393332\pi\)
\(752\) 0 0
\(753\) −30.1234 −1.09776
\(754\) 0 0
\(755\) 92.6842 3.37312
\(756\) 0 0
\(757\) −10.8674 −0.394981 −0.197491 0.980305i \(-0.563279\pi\)
−0.197491 + 0.980305i \(0.563279\pi\)
\(758\) 0 0
\(759\) 7.50935 0.272572
\(760\) 0 0
\(761\) 13.2258 0.479433 0.239717 0.970843i \(-0.422945\pi\)
0.239717 + 0.970843i \(0.422945\pi\)
\(762\) 0 0
\(763\) 4.07093 0.147378
\(764\) 0 0
\(765\) 2.51813 0.0910433
\(766\) 0 0
\(767\) −1.74222 −0.0629080
\(768\) 0 0
\(769\) 0.639598 0.0230645 0.0115322 0.999934i \(-0.496329\pi\)
0.0115322 + 0.999934i \(0.496329\pi\)
\(770\) 0 0
\(771\) 5.42759 0.195470
\(772\) 0 0
\(773\) 0.945183 0.0339959 0.0169979 0.999856i \(-0.494589\pi\)
0.0169979 + 0.999856i \(0.494589\pi\)
\(774\) 0 0
\(775\) −59.5004 −2.13732
\(776\) 0 0
\(777\) 1.82430 0.0654463
\(778\) 0 0
\(779\) −9.69079 −0.347208
\(780\) 0 0
\(781\) −11.7880 −0.421807
\(782\) 0 0
\(783\) 21.0396 0.751895
\(784\) 0 0
\(785\) 37.9580 1.35478
\(786\) 0 0
\(787\) 51.3127 1.82910 0.914550 0.404474i \(-0.132545\pi\)
0.914550 + 0.404474i \(0.132545\pi\)
\(788\) 0 0
\(789\) −39.0674 −1.39084
\(790\) 0 0
\(791\) 3.76726 0.133948
\(792\) 0 0
\(793\) −2.66442 −0.0946162
\(794\) 0 0
\(795\) −10.3294 −0.366345
\(796\) 0 0
\(797\) −24.3477 −0.862439 −0.431220 0.902247i \(-0.641917\pi\)
−0.431220 + 0.902247i \(0.641917\pi\)
\(798\) 0 0
\(799\) 29.1035 1.02961
\(800\) 0 0
\(801\) −1.09520 −0.0386970
\(802\) 0 0
\(803\) 3.81536 0.134641
\(804\) 0 0
\(805\) 17.1696 0.605149
\(806\) 0 0
\(807\) 7.23469 0.254673
\(808\) 0 0
\(809\) 55.0873 1.93677 0.968384 0.249466i \(-0.0802550\pi\)
0.968384 + 0.249466i \(0.0802550\pi\)
\(810\) 0 0
\(811\) 24.1956 0.849624 0.424812 0.905282i \(-0.360340\pi\)
0.424812 + 0.905282i \(0.360340\pi\)
\(812\) 0 0
\(813\) −31.4950 −1.10458
\(814\) 0 0
\(815\) −34.8400 −1.22039
\(816\) 0 0
\(817\) −16.3111 −0.570654
\(818\) 0 0
\(819\) −0.215350 −0.00752493
\(820\) 0 0
\(821\) −36.0287 −1.25741 −0.628706 0.777643i \(-0.716415\pi\)
−0.628706 + 0.777643i \(0.716415\pi\)
\(822\) 0 0
\(823\) −20.3856 −0.710597 −0.355298 0.934753i \(-0.615621\pi\)
−0.355298 + 0.934753i \(0.615621\pi\)
\(824\) 0 0
\(825\) −21.1753 −0.737231
\(826\) 0 0
\(827\) −9.60229 −0.333904 −0.166952 0.985965i \(-0.553393\pi\)
−0.166952 + 0.985965i \(0.553393\pi\)
\(828\) 0 0
\(829\) −31.2765 −1.08628 −0.543140 0.839642i \(-0.682765\pi\)
−0.543140 + 0.839642i \(0.682765\pi\)
\(830\) 0 0
\(831\) 20.4230 0.708466
\(832\) 0 0
\(833\) −2.85208 −0.0988188
\(834\) 0 0
\(835\) 39.0049 1.34982
\(836\) 0 0
\(837\) −25.1587 −0.869613
\(838\) 0 0
\(839\) −48.1484 −1.66227 −0.831133 0.556074i \(-0.812307\pi\)
−0.831133 + 0.556074i \(0.812307\pi\)
\(840\) 0 0
\(841\) −11.2456 −0.387779
\(842\) 0 0
\(843\) 41.1670 1.41787
\(844\) 0 0
\(845\) −4.09989 −0.141040
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) −25.9858 −0.891830
\(850\) 0 0
\(851\) 4.26059 0.146051
\(852\) 0 0
\(853\) 36.2899 1.24254 0.621272 0.783595i \(-0.286616\pi\)
0.621272 + 0.783595i \(0.286616\pi\)
\(854\) 0 0
\(855\) 2.14698 0.0734252
\(856\) 0 0
\(857\) 9.25191 0.316039 0.158020 0.987436i \(-0.449489\pi\)
0.158020 + 0.987436i \(0.449489\pi\)
\(858\) 0 0
\(859\) 51.2281 1.74788 0.873940 0.486034i \(-0.161557\pi\)
0.873940 + 0.486034i \(0.161557\pi\)
\(860\) 0 0
\(861\) −7.14597 −0.243534
\(862\) 0 0
\(863\) −36.9102 −1.25644 −0.628219 0.778036i \(-0.716216\pi\)
−0.628219 + 0.778036i \(0.716216\pi\)
\(864\) 0 0
\(865\) 47.4866 1.61459
\(866\) 0 0
\(867\) 15.8973 0.539900
\(868\) 0 0
\(869\) 14.6244 0.496098
\(870\) 0 0
\(871\) −15.7459 −0.533530
\(872\) 0 0
\(873\) 2.61915 0.0886447
\(874\) 0 0
\(875\) −27.9165 −0.943750
\(876\) 0 0
\(877\) −19.2717 −0.650758 −0.325379 0.945584i \(-0.605492\pi\)
−0.325379 + 0.945584i \(0.605492\pi\)
\(878\) 0 0
\(879\) −18.7484 −0.632367
\(880\) 0 0
\(881\) −25.9816 −0.875342 −0.437671 0.899135i \(-0.644197\pi\)
−0.437671 + 0.899135i \(0.644197\pi\)
\(882\) 0 0
\(883\) −15.7065 −0.528566 −0.264283 0.964445i \(-0.585135\pi\)
−0.264283 + 0.964445i \(0.585135\pi\)
\(884\) 0 0
\(885\) 12.8082 0.430544
\(886\) 0 0
\(887\) −21.1410 −0.709844 −0.354922 0.934896i \(-0.615493\pi\)
−0.354922 + 0.934896i \(0.615493\pi\)
\(888\) 0 0
\(889\) 11.6398 0.390385
\(890\) 0 0
\(891\) −9.59967 −0.321601
\(892\) 0 0
\(893\) 24.8139 0.830364
\(894\) 0 0
\(895\) 46.2131 1.54473
\(896\) 0 0
\(897\) −7.50935 −0.250730
\(898\) 0 0
\(899\) −21.2303 −0.708072
\(900\) 0 0
\(901\) 4.00728 0.133502
\(902\) 0 0
\(903\) −12.0278 −0.400260
\(904\) 0 0
\(905\) −61.0312 −2.02875
\(906\) 0 0
\(907\) −24.4835 −0.812961 −0.406480 0.913660i \(-0.633244\pi\)
−0.406480 + 0.913660i \(0.633244\pi\)
\(908\) 0 0
\(909\) 0.303168 0.0100554
\(910\) 0 0
\(911\) 18.5593 0.614896 0.307448 0.951565i \(-0.400525\pi\)
0.307448 + 0.951565i \(0.400525\pi\)
\(912\) 0 0
\(913\) 13.0350 0.431397
\(914\) 0 0
\(915\) 19.5879 0.647556
\(916\) 0 0
\(917\) 15.2068 0.502172
\(918\) 0 0
\(919\) −26.5489 −0.875767 −0.437884 0.899032i \(-0.644272\pi\)
−0.437884 + 0.899032i \(0.644272\pi\)
\(920\) 0 0
\(921\) 22.2890 0.734448
\(922\) 0 0
\(923\) 11.7880 0.388006
\(924\) 0 0
\(925\) −12.0143 −0.395027
\(926\) 0 0
\(927\) −1.98818 −0.0653003
\(928\) 0 0
\(929\) −55.1128 −1.80819 −0.904096 0.427328i \(-0.859455\pi\)
−0.904096 + 0.427328i \(0.859455\pi\)
\(930\) 0 0
\(931\) −2.43171 −0.0796961
\(932\) 0 0
\(933\) −4.24736 −0.139052
\(934\) 0 0
\(935\) 11.6932 0.382409
\(936\) 0 0
\(937\) −17.0844 −0.558122 −0.279061 0.960273i \(-0.590023\pi\)
−0.279061 + 0.960273i \(0.590023\pi\)
\(938\) 0 0
\(939\) 41.5688 1.35655
\(940\) 0 0
\(941\) −39.2418 −1.27924 −0.639622 0.768689i \(-0.720909\pi\)
−0.639622 + 0.768689i \(0.720909\pi\)
\(942\) 0 0
\(943\) −16.6892 −0.543475
\(944\) 0 0
\(945\) −20.4718 −0.665949
\(946\) 0 0
\(947\) −10.2766 −0.333945 −0.166973 0.985962i \(-0.553399\pi\)
−0.166973 + 0.985962i \(0.553399\pi\)
\(948\) 0 0
\(949\) −3.81536 −0.123852
\(950\) 0 0
\(951\) 28.7295 0.931617
\(952\) 0 0
\(953\) −46.4935 −1.50607 −0.753037 0.657979i \(-0.771412\pi\)
−0.753037 + 0.657979i \(0.771412\pi\)
\(954\) 0 0
\(955\) −46.9459 −1.51913
\(956\) 0 0
\(957\) −7.55557 −0.244237
\(958\) 0 0
\(959\) 13.9323 0.449898
\(960\) 0 0
\(961\) −5.61321 −0.181071
\(962\) 0 0
\(963\) −2.27382 −0.0732730
\(964\) 0 0
\(965\) 33.0041 1.06244
\(966\) 0 0
\(967\) −13.6936 −0.440356 −0.220178 0.975460i \(-0.570664\pi\)
−0.220178 + 0.975460i \(0.570664\pi\)
\(968\) 0 0
\(969\) −12.4362 −0.399509
\(970\) 0 0
\(971\) −0.675977 −0.0216931 −0.0108466 0.999941i \(-0.503453\pi\)
−0.0108466 + 0.999941i \(0.503453\pi\)
\(972\) 0 0
\(973\) −3.67440 −0.117796
\(974\) 0 0
\(975\) 21.1753 0.678154
\(976\) 0 0
\(977\) 45.2312 1.44707 0.723537 0.690286i \(-0.242515\pi\)
0.723537 + 0.690286i \(0.242515\pi\)
\(978\) 0 0
\(979\) −5.08568 −0.162539
\(980\) 0 0
\(981\) 0.876674 0.0279900
\(982\) 0 0
\(983\) 7.25413 0.231371 0.115685 0.993286i \(-0.463094\pi\)
0.115685 + 0.993286i \(0.463094\pi\)
\(984\) 0 0
\(985\) −21.3015 −0.678724
\(986\) 0 0
\(987\) 18.2977 0.582422
\(988\) 0 0
\(989\) −28.0905 −0.893227
\(990\) 0 0
\(991\) 5.17319 0.164332 0.0821660 0.996619i \(-0.473816\pi\)
0.0821660 + 0.996619i \(0.473816\pi\)
\(992\) 0 0
\(993\) 6.32432 0.200696
\(994\) 0 0
\(995\) −107.132 −3.39632
\(996\) 0 0
\(997\) 4.59884 0.145647 0.0728233 0.997345i \(-0.476799\pi\)
0.0728233 + 0.997345i \(0.476799\pi\)
\(998\) 0 0
\(999\) −5.08003 −0.160725
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 8008.2.a.s.1.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8008.2.a.s.1.4 10 1.1 even 1 trivial