Properties

Label 8016.2.a.bd.1.7
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
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] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, 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("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.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: \(64.0080822603\)
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: \( x^{10} - 26x^{8} - 3x^{7} + 220x^{6} + 42x^{5} - 675x^{4} - 67x^{3} + 628x^{2} - 48x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.39234\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.392343 q^{5} -0.0476950 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.392343 q^{5} -0.0476950 q^{7} +1.00000 q^{9} -1.67730 q^{11} -4.70371 q^{13} +0.392343 q^{15} +0.835685 q^{17} -0.998285 q^{19} -0.0476950 q^{21} -1.31016 q^{23} -4.84607 q^{25} +1.00000 q^{27} +6.86363 q^{29} +9.48974 q^{31} -1.67730 q^{33} -0.0187128 q^{35} +2.44944 q^{37} -4.70371 q^{39} +0.448099 q^{41} -9.51987 q^{43} +0.392343 q^{45} +3.80613 q^{47} -6.99773 q^{49} +0.835685 q^{51} -9.39965 q^{53} -0.658077 q^{55} -0.998285 q^{57} +3.24192 q^{59} +0.335380 q^{61} -0.0476950 q^{63} -1.84547 q^{65} +14.2570 q^{67} -1.31016 q^{69} -5.78446 q^{71} -2.80343 q^{73} -4.84607 q^{75} +0.0799990 q^{77} -15.3705 q^{79} +1.00000 q^{81} +7.43473 q^{83} +0.327875 q^{85} +6.86363 q^{87} -5.92823 q^{89} +0.224344 q^{91} +9.48974 q^{93} -0.391670 q^{95} +4.70012 q^{97} -1.67730 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9} + q^{11} - 6 q^{13} - 10 q^{15} - 9 q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 12 q^{25} + 10 q^{27} - 13 q^{29} - 23 q^{31} + q^{33} - q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 10 q^{45} - 10 q^{47} + 7 q^{49} - 9 q^{51} - 26 q^{53} - 11 q^{55} - 2 q^{57} + 10 q^{59} - 10 q^{61} - q^{63} - 22 q^{65} + 5 q^{67} - 7 q^{69} - 25 q^{71} - 8 q^{73} + 12 q^{75} - 46 q^{77} - 26 q^{79} + 10 q^{81} + 14 q^{83} + 9 q^{85} - 13 q^{87} - 31 q^{89} + 3 q^{91} - 23 q^{93} + 5 q^{95} - 32 q^{97} + q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0.392343 0.175461 0.0877305 0.996144i \(-0.472039\pi\)
0.0877305 + 0.996144i \(0.472039\pi\)
\(6\) 0 0
\(7\) −0.0476950 −0.0180270 −0.00901352 0.999959i \(-0.502869\pi\)
−0.00901352 + 0.999959i \(0.502869\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.67730 −0.505726 −0.252863 0.967502i \(-0.581372\pi\)
−0.252863 + 0.967502i \(0.581372\pi\)
\(12\) 0 0
\(13\) −4.70371 −1.30458 −0.652288 0.757971i \(-0.726191\pi\)
−0.652288 + 0.757971i \(0.726191\pi\)
\(14\) 0 0
\(15\) 0.392343 0.101302
\(16\) 0 0
\(17\) 0.835685 0.202683 0.101342 0.994852i \(-0.467686\pi\)
0.101342 + 0.994852i \(0.467686\pi\)
\(18\) 0 0
\(19\) −0.998285 −0.229022 −0.114511 0.993422i \(-0.536530\pi\)
−0.114511 + 0.993422i \(0.536530\pi\)
\(20\) 0 0
\(21\) −0.0476950 −0.0104079
\(22\) 0 0
\(23\) −1.31016 −0.273187 −0.136593 0.990627i \(-0.543615\pi\)
−0.136593 + 0.990627i \(0.543615\pi\)
\(24\) 0 0
\(25\) −4.84607 −0.969213
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.86363 1.27454 0.637272 0.770639i \(-0.280063\pi\)
0.637272 + 0.770639i \(0.280063\pi\)
\(30\) 0 0
\(31\) 9.48974 1.70441 0.852204 0.523210i \(-0.175266\pi\)
0.852204 + 0.523210i \(0.175266\pi\)
\(32\) 0 0
\(33\) −1.67730 −0.291981
\(34\) 0 0
\(35\) −0.0187128 −0.00316304
\(36\) 0 0
\(37\) 2.44944 0.402685 0.201343 0.979521i \(-0.435470\pi\)
0.201343 + 0.979521i \(0.435470\pi\)
\(38\) 0 0
\(39\) −4.70371 −0.753197
\(40\) 0 0
\(41\) 0.448099 0.0699813 0.0349907 0.999388i \(-0.488860\pi\)
0.0349907 + 0.999388i \(0.488860\pi\)
\(42\) 0 0
\(43\) −9.51987 −1.45177 −0.725883 0.687818i \(-0.758569\pi\)
−0.725883 + 0.687818i \(0.758569\pi\)
\(44\) 0 0
\(45\) 0.392343 0.0584870
\(46\) 0 0
\(47\) 3.80613 0.555182 0.277591 0.960699i \(-0.410464\pi\)
0.277591 + 0.960699i \(0.410464\pi\)
\(48\) 0 0
\(49\) −6.99773 −0.999675
\(50\) 0 0
\(51\) 0.835685 0.117019
\(52\) 0 0
\(53\) −9.39965 −1.29114 −0.645571 0.763700i \(-0.723380\pi\)
−0.645571 + 0.763700i \(0.723380\pi\)
\(54\) 0 0
\(55\) −0.658077 −0.0887351
\(56\) 0 0
\(57\) −0.998285 −0.132226
\(58\) 0 0
\(59\) 3.24192 0.422062 0.211031 0.977479i \(-0.432318\pi\)
0.211031 + 0.977479i \(0.432318\pi\)
\(60\) 0 0
\(61\) 0.335380 0.0429411 0.0214705 0.999769i \(-0.493165\pi\)
0.0214705 + 0.999769i \(0.493165\pi\)
\(62\) 0 0
\(63\) −0.0476950 −0.00600901
\(64\) 0 0
\(65\) −1.84547 −0.228902
\(66\) 0 0
\(67\) 14.2570 1.74177 0.870883 0.491491i \(-0.163548\pi\)
0.870883 + 0.491491i \(0.163548\pi\)
\(68\) 0 0
\(69\) −1.31016 −0.157724
\(70\) 0 0
\(71\) −5.78446 −0.686489 −0.343244 0.939246i \(-0.611526\pi\)
−0.343244 + 0.939246i \(0.611526\pi\)
\(72\) 0 0
\(73\) −2.80343 −0.328116 −0.164058 0.986451i \(-0.552458\pi\)
−0.164058 + 0.986451i \(0.552458\pi\)
\(74\) 0 0
\(75\) −4.84607 −0.559576
\(76\) 0 0
\(77\) 0.0799990 0.00911674
\(78\) 0 0
\(79\) −15.3705 −1.72932 −0.864658 0.502362i \(-0.832465\pi\)
−0.864658 + 0.502362i \(0.832465\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.43473 0.816068 0.408034 0.912967i \(-0.366215\pi\)
0.408034 + 0.912967i \(0.366215\pi\)
\(84\) 0 0
\(85\) 0.327875 0.0355630
\(86\) 0 0
\(87\) 6.86363 0.735859
\(88\) 0 0
\(89\) −5.92823 −0.628391 −0.314195 0.949358i \(-0.601735\pi\)
−0.314195 + 0.949358i \(0.601735\pi\)
\(90\) 0 0
\(91\) 0.224344 0.0235176
\(92\) 0 0
\(93\) 9.48974 0.984040
\(94\) 0 0
\(95\) −0.391670 −0.0401845
\(96\) 0 0
\(97\) 4.70012 0.477225 0.238612 0.971115i \(-0.423308\pi\)
0.238612 + 0.971115i \(0.423308\pi\)
\(98\) 0 0
\(99\) −1.67730 −0.168575
\(100\) 0 0
\(101\) −3.08547 −0.307015 −0.153508 0.988147i \(-0.549057\pi\)
−0.153508 + 0.988147i \(0.549057\pi\)
\(102\) 0 0
\(103\) −6.58913 −0.649246 −0.324623 0.945844i \(-0.605237\pi\)
−0.324623 + 0.945844i \(0.605237\pi\)
\(104\) 0 0
\(105\) −0.0187128 −0.00182618
\(106\) 0 0
\(107\) −10.7446 −1.03872 −0.519362 0.854555i \(-0.673830\pi\)
−0.519362 + 0.854555i \(0.673830\pi\)
\(108\) 0 0
\(109\) −5.37085 −0.514434 −0.257217 0.966354i \(-0.582806\pi\)
−0.257217 + 0.966354i \(0.582806\pi\)
\(110\) 0 0
\(111\) 2.44944 0.232491
\(112\) 0 0
\(113\) −8.97790 −0.844570 −0.422285 0.906463i \(-0.638772\pi\)
−0.422285 + 0.906463i \(0.638772\pi\)
\(114\) 0 0
\(115\) −0.514030 −0.0479336
\(116\) 0 0
\(117\) −4.70371 −0.434859
\(118\) 0 0
\(119\) −0.0398580 −0.00365378
\(120\) 0 0
\(121\) −8.18665 −0.744241
\(122\) 0 0
\(123\) 0.448099 0.0404037
\(124\) 0 0
\(125\) −3.86303 −0.345520
\(126\) 0 0
\(127\) −20.6485 −1.83225 −0.916127 0.400887i \(-0.868702\pi\)
−0.916127 + 0.400887i \(0.868702\pi\)
\(128\) 0 0
\(129\) −9.51987 −0.838178
\(130\) 0 0
\(131\) 7.40239 0.646750 0.323375 0.946271i \(-0.395182\pi\)
0.323375 + 0.946271i \(0.395182\pi\)
\(132\) 0 0
\(133\) 0.0476132 0.00412859
\(134\) 0 0
\(135\) 0.392343 0.0337675
\(136\) 0 0
\(137\) −18.9040 −1.61508 −0.807539 0.589814i \(-0.799201\pi\)
−0.807539 + 0.589814i \(0.799201\pi\)
\(138\) 0 0
\(139\) −4.58198 −0.388638 −0.194319 0.980938i \(-0.562250\pi\)
−0.194319 + 0.980938i \(0.562250\pi\)
\(140\) 0 0
\(141\) 3.80613 0.320534
\(142\) 0 0
\(143\) 7.88955 0.659758
\(144\) 0 0
\(145\) 2.69290 0.223633
\(146\) 0 0
\(147\) −6.99773 −0.577163
\(148\) 0 0
\(149\) −20.8271 −1.70622 −0.853112 0.521728i \(-0.825287\pi\)
−0.853112 + 0.521728i \(0.825287\pi\)
\(150\) 0 0
\(151\) −5.79277 −0.471409 −0.235704 0.971825i \(-0.575740\pi\)
−0.235704 + 0.971825i \(0.575740\pi\)
\(152\) 0 0
\(153\) 0.835685 0.0675612
\(154\) 0 0
\(155\) 3.72323 0.299057
\(156\) 0 0
\(157\) −12.6017 −1.00573 −0.502864 0.864366i \(-0.667720\pi\)
−0.502864 + 0.864366i \(0.667720\pi\)
\(158\) 0 0
\(159\) −9.39965 −0.745441
\(160\) 0 0
\(161\) 0.0624880 0.00492474
\(162\) 0 0
\(163\) 20.8695 1.63463 0.817313 0.576194i \(-0.195463\pi\)
0.817313 + 0.576194i \(0.195463\pi\)
\(164\) 0 0
\(165\) −0.658077 −0.0512313
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) 9.12493 0.701918
\(170\) 0 0
\(171\) −0.998285 −0.0763408
\(172\) 0 0
\(173\) −16.6941 −1.26923 −0.634614 0.772829i \(-0.718841\pi\)
−0.634614 + 0.772829i \(0.718841\pi\)
\(174\) 0 0
\(175\) 0.231133 0.0174720
\(176\) 0 0
\(177\) 3.24192 0.243678
\(178\) 0 0
\(179\) −18.5138 −1.38378 −0.691892 0.722001i \(-0.743223\pi\)
−0.691892 + 0.722001i \(0.743223\pi\)
\(180\) 0 0
\(181\) −11.1554 −0.829177 −0.414588 0.910009i \(-0.636074\pi\)
−0.414588 + 0.910009i \(0.636074\pi\)
\(182\) 0 0
\(183\) 0.335380 0.0247920
\(184\) 0 0
\(185\) 0.961020 0.0706556
\(186\) 0 0
\(187\) −1.40170 −0.102502
\(188\) 0 0
\(189\) −0.0476950 −0.00346930
\(190\) 0 0
\(191\) −1.00647 −0.0728259 −0.0364129 0.999337i \(-0.511593\pi\)
−0.0364129 + 0.999337i \(0.511593\pi\)
\(192\) 0 0
\(193\) −0.00263902 −0.000189961 0 −9.49805e−5 1.00000i \(-0.500030\pi\)
−9.49805e−5 1.00000i \(0.500030\pi\)
\(194\) 0 0
\(195\) −1.84547 −0.132157
\(196\) 0 0
\(197\) 3.29685 0.234891 0.117445 0.993079i \(-0.462529\pi\)
0.117445 + 0.993079i \(0.462529\pi\)
\(198\) 0 0
\(199\) 2.00744 0.142304 0.0711519 0.997465i \(-0.477332\pi\)
0.0711519 + 0.997465i \(0.477332\pi\)
\(200\) 0 0
\(201\) 14.2570 1.00561
\(202\) 0 0
\(203\) −0.327361 −0.0229763
\(204\) 0 0
\(205\) 0.175808 0.0122790
\(206\) 0 0
\(207\) −1.31016 −0.0910622
\(208\) 0 0
\(209\) 1.67443 0.115823
\(210\) 0 0
\(211\) 23.1716 1.59520 0.797598 0.603189i \(-0.206103\pi\)
0.797598 + 0.603189i \(0.206103\pi\)
\(212\) 0 0
\(213\) −5.78446 −0.396344
\(214\) 0 0
\(215\) −3.73505 −0.254728
\(216\) 0 0
\(217\) −0.452614 −0.0307254
\(218\) 0 0
\(219\) −2.80343 −0.189438
\(220\) 0 0
\(221\) −3.93083 −0.264416
\(222\) 0 0
\(223\) −11.6448 −0.779791 −0.389895 0.920859i \(-0.627489\pi\)
−0.389895 + 0.920859i \(0.627489\pi\)
\(224\) 0 0
\(225\) −4.84607 −0.323071
\(226\) 0 0
\(227\) 22.8199 1.51461 0.757306 0.653060i \(-0.226515\pi\)
0.757306 + 0.653060i \(0.226515\pi\)
\(228\) 0 0
\(229\) 13.7218 0.906759 0.453380 0.891318i \(-0.350218\pi\)
0.453380 + 0.891318i \(0.350218\pi\)
\(230\) 0 0
\(231\) 0.0799990 0.00526355
\(232\) 0 0
\(233\) 1.63184 0.106905 0.0534525 0.998570i \(-0.482977\pi\)
0.0534525 + 0.998570i \(0.482977\pi\)
\(234\) 0 0
\(235\) 1.49331 0.0974127
\(236\) 0 0
\(237\) −15.3705 −0.998420
\(238\) 0 0
\(239\) −15.6884 −1.01480 −0.507400 0.861711i \(-0.669393\pi\)
−0.507400 + 0.861711i \(0.669393\pi\)
\(240\) 0 0
\(241\) 1.42558 0.0918297 0.0459149 0.998945i \(-0.485380\pi\)
0.0459149 + 0.998945i \(0.485380\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −2.74551 −0.175404
\(246\) 0 0
\(247\) 4.69565 0.298777
\(248\) 0 0
\(249\) 7.43473 0.471157
\(250\) 0 0
\(251\) 20.7834 1.31183 0.655917 0.754833i \(-0.272282\pi\)
0.655917 + 0.754833i \(0.272282\pi\)
\(252\) 0 0
\(253\) 2.19753 0.138158
\(254\) 0 0
\(255\) 0.327875 0.0205323
\(256\) 0 0
\(257\) −19.0163 −1.18620 −0.593101 0.805128i \(-0.702097\pi\)
−0.593101 + 0.805128i \(0.702097\pi\)
\(258\) 0 0
\(259\) −0.116826 −0.00725922
\(260\) 0 0
\(261\) 6.86363 0.424848
\(262\) 0 0
\(263\) 7.63989 0.471096 0.235548 0.971863i \(-0.424312\pi\)
0.235548 + 0.971863i \(0.424312\pi\)
\(264\) 0 0
\(265\) −3.68788 −0.226545
\(266\) 0 0
\(267\) −5.92823 −0.362802
\(268\) 0 0
\(269\) −27.2370 −1.66067 −0.830336 0.557264i \(-0.811851\pi\)
−0.830336 + 0.557264i \(0.811851\pi\)
\(270\) 0 0
\(271\) 6.54938 0.397847 0.198923 0.980015i \(-0.436256\pi\)
0.198923 + 0.980015i \(0.436256\pi\)
\(272\) 0 0
\(273\) 0.224344 0.0135779
\(274\) 0 0
\(275\) 8.12832 0.490156
\(276\) 0 0
\(277\) −14.1616 −0.850889 −0.425445 0.904984i \(-0.639882\pi\)
−0.425445 + 0.904984i \(0.639882\pi\)
\(278\) 0 0
\(279\) 9.48974 0.568136
\(280\) 0 0
\(281\) 20.7817 1.23973 0.619867 0.784707i \(-0.287187\pi\)
0.619867 + 0.784707i \(0.287187\pi\)
\(282\) 0 0
\(283\) 16.9536 1.00779 0.503894 0.863766i \(-0.331900\pi\)
0.503894 + 0.863766i \(0.331900\pi\)
\(284\) 0 0
\(285\) −0.391670 −0.0232005
\(286\) 0 0
\(287\) −0.0213721 −0.00126156
\(288\) 0 0
\(289\) −16.3016 −0.958919
\(290\) 0 0
\(291\) 4.70012 0.275526
\(292\) 0 0
\(293\) −2.31142 −0.135035 −0.0675173 0.997718i \(-0.521508\pi\)
−0.0675173 + 0.997718i \(0.521508\pi\)
\(294\) 0 0
\(295\) 1.27194 0.0740554
\(296\) 0 0
\(297\) −1.67730 −0.0973270
\(298\) 0 0
\(299\) 6.16261 0.356393
\(300\) 0 0
\(301\) 0.454051 0.0261710
\(302\) 0 0
\(303\) −3.08547 −0.177255
\(304\) 0 0
\(305\) 0.131584 0.00753448
\(306\) 0 0
\(307\) 26.6433 1.52062 0.760308 0.649563i \(-0.225048\pi\)
0.760308 + 0.649563i \(0.225048\pi\)
\(308\) 0 0
\(309\) −6.58913 −0.374842
\(310\) 0 0
\(311\) 2.71532 0.153972 0.0769859 0.997032i \(-0.475470\pi\)
0.0769859 + 0.997032i \(0.475470\pi\)
\(312\) 0 0
\(313\) −6.54601 −0.370002 −0.185001 0.982738i \(-0.559229\pi\)
−0.185001 + 0.982738i \(0.559229\pi\)
\(314\) 0 0
\(315\) −0.0187128 −0.00105435
\(316\) 0 0
\(317\) −19.4676 −1.09341 −0.546704 0.837326i \(-0.684118\pi\)
−0.546704 + 0.837326i \(0.684118\pi\)
\(318\) 0 0
\(319\) −11.5124 −0.644570
\(320\) 0 0
\(321\) −10.7446 −0.599707
\(322\) 0 0
\(323\) −0.834252 −0.0464190
\(324\) 0 0
\(325\) 22.7945 1.26441
\(326\) 0 0
\(327\) −5.37085 −0.297009
\(328\) 0 0
\(329\) −0.181534 −0.0100083
\(330\) 0 0
\(331\) −11.3987 −0.626530 −0.313265 0.949666i \(-0.601423\pi\)
−0.313265 + 0.949666i \(0.601423\pi\)
\(332\) 0 0
\(333\) 2.44944 0.134228
\(334\) 0 0
\(335\) 5.59361 0.305612
\(336\) 0 0
\(337\) 9.46084 0.515365 0.257682 0.966230i \(-0.417041\pi\)
0.257682 + 0.966230i \(0.417041\pi\)
\(338\) 0 0
\(339\) −8.97790 −0.487613
\(340\) 0 0
\(341\) −15.9172 −0.861963
\(342\) 0 0
\(343\) 0.667622 0.0360482
\(344\) 0 0
\(345\) −0.514030 −0.0276745
\(346\) 0 0
\(347\) 18.1189 0.972674 0.486337 0.873771i \(-0.338333\pi\)
0.486337 + 0.873771i \(0.338333\pi\)
\(348\) 0 0
\(349\) 7.17579 0.384111 0.192056 0.981384i \(-0.438485\pi\)
0.192056 + 0.981384i \(0.438485\pi\)
\(350\) 0 0
\(351\) −4.70371 −0.251066
\(352\) 0 0
\(353\) 32.7788 1.74464 0.872319 0.488938i \(-0.162616\pi\)
0.872319 + 0.488938i \(0.162616\pi\)
\(354\) 0 0
\(355\) −2.26949 −0.120452
\(356\) 0 0
\(357\) −0.0398580 −0.00210951
\(358\) 0 0
\(359\) −25.9833 −1.37135 −0.685674 0.727909i \(-0.740492\pi\)
−0.685674 + 0.727909i \(0.740492\pi\)
\(360\) 0 0
\(361\) −18.0034 −0.947549
\(362\) 0 0
\(363\) −8.18665 −0.429688
\(364\) 0 0
\(365\) −1.09990 −0.0575716
\(366\) 0 0
\(367\) −0.622748 −0.0325072 −0.0162536 0.999868i \(-0.505174\pi\)
−0.0162536 + 0.999868i \(0.505174\pi\)
\(368\) 0 0
\(369\) 0.448099 0.0233271
\(370\) 0 0
\(371\) 0.448317 0.0232754
\(372\) 0 0
\(373\) 25.3579 1.31298 0.656492 0.754333i \(-0.272040\pi\)
0.656492 + 0.754333i \(0.272040\pi\)
\(374\) 0 0
\(375\) −3.86303 −0.199486
\(376\) 0 0
\(377\) −32.2846 −1.66274
\(378\) 0 0
\(379\) −21.1584 −1.08684 −0.543418 0.839462i \(-0.682870\pi\)
−0.543418 + 0.839462i \(0.682870\pi\)
\(380\) 0 0
\(381\) −20.6485 −1.05785
\(382\) 0 0
\(383\) 25.0508 1.28004 0.640019 0.768359i \(-0.278927\pi\)
0.640019 + 0.768359i \(0.278927\pi\)
\(384\) 0 0
\(385\) 0.0313870 0.00159963
\(386\) 0 0
\(387\) −9.51987 −0.483922
\(388\) 0 0
\(389\) −25.1811 −1.27673 −0.638367 0.769732i \(-0.720390\pi\)
−0.638367 + 0.769732i \(0.720390\pi\)
\(390\) 0 0
\(391\) −1.09488 −0.0553704
\(392\) 0 0
\(393\) 7.40239 0.373401
\(394\) 0 0
\(395\) −6.03050 −0.303427
\(396\) 0 0
\(397\) 16.5383 0.830035 0.415017 0.909813i \(-0.363775\pi\)
0.415017 + 0.909813i \(0.363775\pi\)
\(398\) 0 0
\(399\) 0.0476132 0.00238364
\(400\) 0 0
\(401\) −29.5026 −1.47329 −0.736645 0.676280i \(-0.763591\pi\)
−0.736645 + 0.676280i \(0.763591\pi\)
\(402\) 0 0
\(403\) −44.6370 −2.22353
\(404\) 0 0
\(405\) 0.392343 0.0194957
\(406\) 0 0
\(407\) −4.10845 −0.203648
\(408\) 0 0
\(409\) 10.9493 0.541410 0.270705 0.962662i \(-0.412743\pi\)
0.270705 + 0.962662i \(0.412743\pi\)
\(410\) 0 0
\(411\) −18.9040 −0.932466
\(412\) 0 0
\(413\) −0.154624 −0.00760853
\(414\) 0 0
\(415\) 2.91696 0.143188
\(416\) 0 0
\(417\) −4.58198 −0.224380
\(418\) 0 0
\(419\) −19.1069 −0.933433 −0.466716 0.884407i \(-0.654563\pi\)
−0.466716 + 0.884407i \(0.654563\pi\)
\(420\) 0 0
\(421\) 18.8853 0.920414 0.460207 0.887812i \(-0.347775\pi\)
0.460207 + 0.887812i \(0.347775\pi\)
\(422\) 0 0
\(423\) 3.80613 0.185061
\(424\) 0 0
\(425\) −4.04979 −0.196444
\(426\) 0 0
\(427\) −0.0159960 −0.000774100 0
\(428\) 0 0
\(429\) 7.88955 0.380911
\(430\) 0 0
\(431\) −1.54122 −0.0742381 −0.0371191 0.999311i \(-0.511818\pi\)
−0.0371191 + 0.999311i \(0.511818\pi\)
\(432\) 0 0
\(433\) −1.63875 −0.0787535 −0.0393768 0.999224i \(-0.512537\pi\)
−0.0393768 + 0.999224i \(0.512537\pi\)
\(434\) 0 0
\(435\) 2.69290 0.129114
\(436\) 0 0
\(437\) 1.30791 0.0625658
\(438\) 0 0
\(439\) 27.9964 1.33620 0.668098 0.744073i \(-0.267109\pi\)
0.668098 + 0.744073i \(0.267109\pi\)
\(440\) 0 0
\(441\) −6.99773 −0.333225
\(442\) 0 0
\(443\) 17.8603 0.848568 0.424284 0.905529i \(-0.360526\pi\)
0.424284 + 0.905529i \(0.360526\pi\)
\(444\) 0 0
\(445\) −2.32590 −0.110258
\(446\) 0 0
\(447\) −20.8271 −0.985088
\(448\) 0 0
\(449\) −23.9248 −1.12908 −0.564540 0.825406i \(-0.690946\pi\)
−0.564540 + 0.825406i \(0.690946\pi\)
\(450\) 0 0
\(451\) −0.751598 −0.0353914
\(452\) 0 0
\(453\) −5.79277 −0.272168
\(454\) 0 0
\(455\) 0.0880196 0.00412642
\(456\) 0 0
\(457\) 30.8005 1.44079 0.720393 0.693566i \(-0.243962\pi\)
0.720393 + 0.693566i \(0.243962\pi\)
\(458\) 0 0
\(459\) 0.835685 0.0390065
\(460\) 0 0
\(461\) 28.1069 1.30907 0.654534 0.756032i \(-0.272865\pi\)
0.654534 + 0.756032i \(0.272865\pi\)
\(462\) 0 0
\(463\) −9.82630 −0.456667 −0.228333 0.973583i \(-0.573328\pi\)
−0.228333 + 0.973583i \(0.573328\pi\)
\(464\) 0 0
\(465\) 3.72323 0.172661
\(466\) 0 0
\(467\) 3.50796 0.162329 0.0811646 0.996701i \(-0.474136\pi\)
0.0811646 + 0.996701i \(0.474136\pi\)
\(468\) 0 0
\(469\) −0.679986 −0.0313989
\(470\) 0 0
\(471\) −12.6017 −0.580657
\(472\) 0 0
\(473\) 15.9677 0.734196
\(474\) 0 0
\(475\) 4.83776 0.221971
\(476\) 0 0
\(477\) −9.39965 −0.430380
\(478\) 0 0
\(479\) −15.9493 −0.728742 −0.364371 0.931254i \(-0.618716\pi\)
−0.364371 + 0.931254i \(0.618716\pi\)
\(480\) 0 0
\(481\) −11.5215 −0.525334
\(482\) 0 0
\(483\) 0.0624880 0.00284330
\(484\) 0 0
\(485\) 1.84406 0.0837343
\(486\) 0 0
\(487\) 34.8701 1.58012 0.790058 0.613032i \(-0.210050\pi\)
0.790058 + 0.613032i \(0.210050\pi\)
\(488\) 0 0
\(489\) 20.8695 0.943751
\(490\) 0 0
\(491\) −3.03356 −0.136902 −0.0684512 0.997654i \(-0.521806\pi\)
−0.0684512 + 0.997654i \(0.521806\pi\)
\(492\) 0 0
\(493\) 5.73584 0.258329
\(494\) 0 0
\(495\) −0.658077 −0.0295784
\(496\) 0 0
\(497\) 0.275890 0.0123754
\(498\) 0 0
\(499\) −8.84557 −0.395982 −0.197991 0.980204i \(-0.563442\pi\)
−0.197991 + 0.980204i \(0.563442\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) 10.9142 0.486642 0.243321 0.969946i \(-0.421763\pi\)
0.243321 + 0.969946i \(0.421763\pi\)
\(504\) 0 0
\(505\) −1.21056 −0.0538692
\(506\) 0 0
\(507\) 9.12493 0.405252
\(508\) 0 0
\(509\) −26.6881 −1.18293 −0.591466 0.806330i \(-0.701450\pi\)
−0.591466 + 0.806330i \(0.701450\pi\)
\(510\) 0 0
\(511\) 0.133710 0.00591496
\(512\) 0 0
\(513\) −0.998285 −0.0440754
\(514\) 0 0
\(515\) −2.58519 −0.113917
\(516\) 0 0
\(517\) −6.38404 −0.280770
\(518\) 0 0
\(519\) −16.6941 −0.732790
\(520\) 0 0
\(521\) 42.5402 1.86372 0.931860 0.362818i \(-0.118185\pi\)
0.931860 + 0.362818i \(0.118185\pi\)
\(522\) 0 0
\(523\) 3.07635 0.134520 0.0672598 0.997735i \(-0.478574\pi\)
0.0672598 + 0.997735i \(0.478574\pi\)
\(524\) 0 0
\(525\) 0.231133 0.0100875
\(526\) 0 0
\(527\) 7.93044 0.345455
\(528\) 0 0
\(529\) −21.2835 −0.925369
\(530\) 0 0
\(531\) 3.24192 0.140687
\(532\) 0 0
\(533\) −2.10773 −0.0912960
\(534\) 0 0
\(535\) −4.21558 −0.182255
\(536\) 0 0
\(537\) −18.5138 −0.798928
\(538\) 0 0
\(539\) 11.7373 0.505562
\(540\) 0 0
\(541\) 40.0473 1.72177 0.860883 0.508803i \(-0.169912\pi\)
0.860883 + 0.508803i \(0.169912\pi\)
\(542\) 0 0
\(543\) −11.1554 −0.478725
\(544\) 0 0
\(545\) −2.10721 −0.0902631
\(546\) 0 0
\(547\) 45.1866 1.93204 0.966019 0.258471i \(-0.0832185\pi\)
0.966019 + 0.258471i \(0.0832185\pi\)
\(548\) 0 0
\(549\) 0.335380 0.0143137
\(550\) 0 0
\(551\) −6.85186 −0.291899
\(552\) 0 0
\(553\) 0.733096 0.0311744
\(554\) 0 0
\(555\) 0.961020 0.0407930
\(556\) 0 0
\(557\) 23.6465 1.00193 0.500967 0.865466i \(-0.332978\pi\)
0.500967 + 0.865466i \(0.332978\pi\)
\(558\) 0 0
\(559\) 44.7788 1.89394
\(560\) 0 0
\(561\) −1.40170 −0.0591797
\(562\) 0 0
\(563\) −33.9807 −1.43211 −0.716057 0.698041i \(-0.754055\pi\)
−0.716057 + 0.698041i \(0.754055\pi\)
\(564\) 0 0
\(565\) −3.52241 −0.148189
\(566\) 0 0
\(567\) −0.0476950 −0.00200300
\(568\) 0 0
\(569\) 10.1043 0.423593 0.211797 0.977314i \(-0.432069\pi\)
0.211797 + 0.977314i \(0.432069\pi\)
\(570\) 0 0
\(571\) −45.2277 −1.89272 −0.946359 0.323116i \(-0.895270\pi\)
−0.946359 + 0.323116i \(0.895270\pi\)
\(572\) 0 0
\(573\) −1.00647 −0.0420460
\(574\) 0 0
\(575\) 6.34911 0.264776
\(576\) 0 0
\(577\) −21.0191 −0.875035 −0.437517 0.899210i \(-0.644142\pi\)
−0.437517 + 0.899210i \(0.644142\pi\)
\(578\) 0 0
\(579\) −0.00263902 −0.000109674 0
\(580\) 0 0
\(581\) −0.354600 −0.0147113
\(582\) 0 0
\(583\) 15.7661 0.652964
\(584\) 0 0
\(585\) −1.84547 −0.0763007
\(586\) 0 0
\(587\) −12.5907 −0.519673 −0.259837 0.965653i \(-0.583669\pi\)
−0.259837 + 0.965653i \(0.583669\pi\)
\(588\) 0 0
\(589\) −9.47347 −0.390347
\(590\) 0 0
\(591\) 3.29685 0.135614
\(592\) 0 0
\(593\) 18.6770 0.766973 0.383486 0.923547i \(-0.374723\pi\)
0.383486 + 0.923547i \(0.374723\pi\)
\(594\) 0 0
\(595\) −0.0156380 −0.000641096 0
\(596\) 0 0
\(597\) 2.00744 0.0821591
\(598\) 0 0
\(599\) −43.2696 −1.76795 −0.883975 0.467535i \(-0.845142\pi\)
−0.883975 + 0.467535i \(0.845142\pi\)
\(600\) 0 0
\(601\) 18.4237 0.751520 0.375760 0.926717i \(-0.377382\pi\)
0.375760 + 0.926717i \(0.377382\pi\)
\(602\) 0 0
\(603\) 14.2570 0.580588
\(604\) 0 0
\(605\) −3.21197 −0.130585
\(606\) 0 0
\(607\) −12.4290 −0.504478 −0.252239 0.967665i \(-0.581167\pi\)
−0.252239 + 0.967665i \(0.581167\pi\)
\(608\) 0 0
\(609\) −0.327361 −0.0132653
\(610\) 0 0
\(611\) −17.9030 −0.724276
\(612\) 0 0
\(613\) −3.08597 −0.124641 −0.0623205 0.998056i \(-0.519850\pi\)
−0.0623205 + 0.998056i \(0.519850\pi\)
\(614\) 0 0
\(615\) 0.175808 0.00708928
\(616\) 0 0
\(617\) 5.89338 0.237259 0.118629 0.992939i \(-0.462150\pi\)
0.118629 + 0.992939i \(0.462150\pi\)
\(618\) 0 0
\(619\) −11.0381 −0.443657 −0.221829 0.975086i \(-0.571203\pi\)
−0.221829 + 0.975086i \(0.571203\pi\)
\(620\) 0 0
\(621\) −1.31016 −0.0525748
\(622\) 0 0
\(623\) 0.282747 0.0113280
\(624\) 0 0
\(625\) 22.7147 0.908588
\(626\) 0 0
\(627\) 1.67443 0.0668702
\(628\) 0 0
\(629\) 2.04696 0.0816177
\(630\) 0 0
\(631\) −44.0298 −1.75280 −0.876400 0.481584i \(-0.840062\pi\)
−0.876400 + 0.481584i \(0.840062\pi\)
\(632\) 0 0
\(633\) 23.1716 0.920987
\(634\) 0 0
\(635\) −8.10127 −0.321489
\(636\) 0 0
\(637\) 32.9153 1.30415
\(638\) 0 0
\(639\) −5.78446 −0.228830
\(640\) 0 0
\(641\) −30.4393 −1.20228 −0.601140 0.799144i \(-0.705286\pi\)
−0.601140 + 0.799144i \(0.705286\pi\)
\(642\) 0 0
\(643\) −11.1979 −0.441601 −0.220801 0.975319i \(-0.570867\pi\)
−0.220801 + 0.975319i \(0.570867\pi\)
\(644\) 0 0
\(645\) −3.73505 −0.147067
\(646\) 0 0
\(647\) 3.77009 0.148218 0.0741089 0.997250i \(-0.476389\pi\)
0.0741089 + 0.997250i \(0.476389\pi\)
\(648\) 0 0
\(649\) −5.43769 −0.213448
\(650\) 0 0
\(651\) −0.452614 −0.0177393
\(652\) 0 0
\(653\) −44.9124 −1.75756 −0.878780 0.477227i \(-0.841642\pi\)
−0.878780 + 0.477227i \(0.841642\pi\)
\(654\) 0 0
\(655\) 2.90427 0.113479
\(656\) 0 0
\(657\) −2.80343 −0.109372
\(658\) 0 0
\(659\) −0.530432 −0.0206627 −0.0103313 0.999947i \(-0.503289\pi\)
−0.0103313 + 0.999947i \(0.503289\pi\)
\(660\) 0 0
\(661\) −37.7146 −1.46693 −0.733464 0.679728i \(-0.762098\pi\)
−0.733464 + 0.679728i \(0.762098\pi\)
\(662\) 0 0
\(663\) −3.93083 −0.152661
\(664\) 0 0
\(665\) 0.0186807 0.000724407 0
\(666\) 0 0
\(667\) −8.99244 −0.348189
\(668\) 0 0
\(669\) −11.6448 −0.450212
\(670\) 0 0
\(671\) −0.562535 −0.0217164
\(672\) 0 0
\(673\) 14.2368 0.548787 0.274393 0.961618i \(-0.411523\pi\)
0.274393 + 0.961618i \(0.411523\pi\)
\(674\) 0 0
\(675\) −4.84607 −0.186525
\(676\) 0 0
\(677\) −27.2571 −1.04758 −0.523789 0.851848i \(-0.675482\pi\)
−0.523789 + 0.851848i \(0.675482\pi\)
\(678\) 0 0
\(679\) −0.224172 −0.00860294
\(680\) 0 0
\(681\) 22.8199 0.874461
\(682\) 0 0
\(683\) 28.3603 1.08518 0.542589 0.839998i \(-0.317444\pi\)
0.542589 + 0.839998i \(0.317444\pi\)
\(684\) 0 0
\(685\) −7.41685 −0.283383
\(686\) 0 0
\(687\) 13.7218 0.523518
\(688\) 0 0
\(689\) 44.2133 1.68439
\(690\) 0 0
\(691\) −32.5658 −1.23886 −0.619430 0.785052i \(-0.712636\pi\)
−0.619430 + 0.785052i \(0.712636\pi\)
\(692\) 0 0
\(693\) 0.0799990 0.00303891
\(694\) 0 0
\(695\) −1.79770 −0.0681908
\(696\) 0 0
\(697\) 0.374470 0.0141841
\(698\) 0 0
\(699\) 1.63184 0.0617217
\(700\) 0 0
\(701\) −34.2700 −1.29436 −0.647179 0.762338i \(-0.724051\pi\)
−0.647179 + 0.762338i \(0.724051\pi\)
\(702\) 0 0
\(703\) −2.44524 −0.0922240
\(704\) 0 0
\(705\) 1.49331 0.0562412
\(706\) 0 0
\(707\) 0.147161 0.00553457
\(708\) 0 0
\(709\) 17.4855 0.656680 0.328340 0.944560i \(-0.393511\pi\)
0.328340 + 0.944560i \(0.393511\pi\)
\(710\) 0 0
\(711\) −15.3705 −0.576438
\(712\) 0 0
\(713\) −12.4331 −0.465622
\(714\) 0 0
\(715\) 3.09541 0.115762
\(716\) 0 0
\(717\) −15.6884 −0.585895
\(718\) 0 0
\(719\) 15.3333 0.571836 0.285918 0.958254i \(-0.407702\pi\)
0.285918 + 0.958254i \(0.407702\pi\)
\(720\) 0 0
\(721\) 0.314269 0.0117040
\(722\) 0 0
\(723\) 1.42558 0.0530179
\(724\) 0 0
\(725\) −33.2616 −1.23531
\(726\) 0 0
\(727\) 20.8931 0.774881 0.387440 0.921895i \(-0.373359\pi\)
0.387440 + 0.921895i \(0.373359\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −7.95562 −0.294249
\(732\) 0 0
\(733\) −40.9819 −1.51370 −0.756851 0.653587i \(-0.773263\pi\)
−0.756851 + 0.653587i \(0.773263\pi\)
\(734\) 0 0
\(735\) −2.74551 −0.101269
\(736\) 0 0
\(737\) −23.9132 −0.880856
\(738\) 0 0
\(739\) 4.74579 0.174577 0.0872883 0.996183i \(-0.472180\pi\)
0.0872883 + 0.996183i \(0.472180\pi\)
\(740\) 0 0
\(741\) 4.69565 0.172499
\(742\) 0 0
\(743\) −10.2902 −0.377509 −0.188755 0.982024i \(-0.560445\pi\)
−0.188755 + 0.982024i \(0.560445\pi\)
\(744\) 0 0
\(745\) −8.17136 −0.299376
\(746\) 0 0
\(747\) 7.43473 0.272023
\(748\) 0 0
\(749\) 0.512466 0.0187251
\(750\) 0 0
\(751\) 34.7969 1.26976 0.634878 0.772612i \(-0.281050\pi\)
0.634878 + 0.772612i \(0.281050\pi\)
\(752\) 0 0
\(753\) 20.7834 0.757387
\(754\) 0 0
\(755\) −2.27275 −0.0827138
\(756\) 0 0
\(757\) 24.1181 0.876588 0.438294 0.898832i \(-0.355583\pi\)
0.438294 + 0.898832i \(0.355583\pi\)
\(758\) 0 0
\(759\) 2.19753 0.0797653
\(760\) 0 0
\(761\) 22.7164 0.823470 0.411735 0.911304i \(-0.364923\pi\)
0.411735 + 0.911304i \(0.364923\pi\)
\(762\) 0 0
\(763\) 0.256163 0.00927372
\(764\) 0 0
\(765\) 0.327875 0.0118543
\(766\) 0 0
\(767\) −15.2491 −0.550612
\(768\) 0 0
\(769\) 42.8534 1.54533 0.772667 0.634811i \(-0.218922\pi\)
0.772667 + 0.634811i \(0.218922\pi\)
\(770\) 0 0
\(771\) −19.0163 −0.684855
\(772\) 0 0
\(773\) 51.2076 1.84181 0.920905 0.389786i \(-0.127451\pi\)
0.920905 + 0.389786i \(0.127451\pi\)
\(774\) 0 0
\(775\) −45.9879 −1.65194
\(776\) 0 0
\(777\) −0.116826 −0.00419111
\(778\) 0 0
\(779\) −0.447331 −0.0160273
\(780\) 0 0
\(781\) 9.70229 0.347175
\(782\) 0 0
\(783\) 6.86363 0.245286
\(784\) 0 0
\(785\) −4.94420 −0.176466
\(786\) 0 0
\(787\) 4.78898 0.170709 0.0853543 0.996351i \(-0.472798\pi\)
0.0853543 + 0.996351i \(0.472798\pi\)
\(788\) 0 0
\(789\) 7.63989 0.271987
\(790\) 0 0
\(791\) 0.428201 0.0152251
\(792\) 0 0
\(793\) −1.57753 −0.0560199
\(794\) 0 0
\(795\) −3.68788 −0.130796
\(796\) 0 0
\(797\) −9.75603 −0.345576 −0.172788 0.984959i \(-0.555278\pi\)
−0.172788 + 0.984959i \(0.555278\pi\)
\(798\) 0 0
\(799\) 3.18073 0.112526
\(800\) 0 0
\(801\) −5.92823 −0.209464
\(802\) 0 0
\(803\) 4.70220 0.165937
\(804\) 0 0
\(805\) 0.0245167 0.000864100 0
\(806\) 0 0
\(807\) −27.2370 −0.958789
\(808\) 0 0
\(809\) 24.1859 0.850332 0.425166 0.905115i \(-0.360216\pi\)
0.425166 + 0.905115i \(0.360216\pi\)
\(810\) 0 0
\(811\) −46.5662 −1.63516 −0.817581 0.575814i \(-0.804685\pi\)
−0.817581 + 0.575814i \(0.804685\pi\)
\(812\) 0 0
\(813\) 6.54938 0.229697
\(814\) 0 0
\(815\) 8.18799 0.286813
\(816\) 0 0
\(817\) 9.50355 0.332487
\(818\) 0 0
\(819\) 0.224344 0.00783921
\(820\) 0 0
\(821\) 41.0430 1.43241 0.716205 0.697890i \(-0.245878\pi\)
0.716205 + 0.697890i \(0.245878\pi\)
\(822\) 0 0
\(823\) −37.0327 −1.29088 −0.645439 0.763811i \(-0.723326\pi\)
−0.645439 + 0.763811i \(0.723326\pi\)
\(824\) 0 0
\(825\) 8.12832 0.282992
\(826\) 0 0
\(827\) 15.3486 0.533724 0.266862 0.963735i \(-0.414013\pi\)
0.266862 + 0.963735i \(0.414013\pi\)
\(828\) 0 0
\(829\) 49.5804 1.72200 0.861000 0.508605i \(-0.169839\pi\)
0.861000 + 0.508605i \(0.169839\pi\)
\(830\) 0 0
\(831\) −14.1616 −0.491261
\(832\) 0 0
\(833\) −5.84790 −0.202618
\(834\) 0 0
\(835\) 0.392343 0.0135776
\(836\) 0 0
\(837\) 9.48974 0.328013
\(838\) 0 0
\(839\) −6.18421 −0.213503 −0.106751 0.994286i \(-0.534045\pi\)
−0.106751 + 0.994286i \(0.534045\pi\)
\(840\) 0 0
\(841\) 18.1095 0.624464
\(842\) 0 0
\(843\) 20.7817 0.715760
\(844\) 0 0
\(845\) 3.58010 0.123159
\(846\) 0 0
\(847\) 0.390463 0.0134165
\(848\) 0 0
\(849\) 16.9536 0.581846
\(850\) 0 0
\(851\) −3.20915 −0.110008
\(852\) 0 0
\(853\) −5.43706 −0.186161 −0.0930807 0.995659i \(-0.529671\pi\)
−0.0930807 + 0.995659i \(0.529671\pi\)
\(854\) 0 0
\(855\) −0.391670 −0.0133948
\(856\) 0 0
\(857\) −27.2410 −0.930533 −0.465267 0.885171i \(-0.654042\pi\)
−0.465267 + 0.885171i \(0.654042\pi\)
\(858\) 0 0
\(859\) 14.1977 0.484419 0.242210 0.970224i \(-0.422128\pi\)
0.242210 + 0.970224i \(0.422128\pi\)
\(860\) 0 0
\(861\) −0.0213721 −0.000728360 0
\(862\) 0 0
\(863\) −3.59454 −0.122359 −0.0611797 0.998127i \(-0.519486\pi\)
−0.0611797 + 0.998127i \(0.519486\pi\)
\(864\) 0 0
\(865\) −6.54980 −0.222700
\(866\) 0 0
\(867\) −16.3016 −0.553632
\(868\) 0 0
\(869\) 25.7810 0.874559
\(870\) 0 0
\(871\) −67.0607 −2.27226
\(872\) 0 0
\(873\) 4.70012 0.159075
\(874\) 0 0
\(875\) 0.184247 0.00622870
\(876\) 0 0
\(877\) 15.7663 0.532389 0.266195 0.963919i \(-0.414234\pi\)
0.266195 + 0.963919i \(0.414234\pi\)
\(878\) 0 0
\(879\) −2.31142 −0.0779623
\(880\) 0 0
\(881\) −43.1684 −1.45438 −0.727190 0.686436i \(-0.759174\pi\)
−0.727190 + 0.686436i \(0.759174\pi\)
\(882\) 0 0
\(883\) −57.9427 −1.94993 −0.974963 0.222365i \(-0.928622\pi\)
−0.974963 + 0.222365i \(0.928622\pi\)
\(884\) 0 0
\(885\) 1.27194 0.0427559
\(886\) 0 0
\(887\) −57.4395 −1.92863 −0.964314 0.264760i \(-0.914707\pi\)
−0.964314 + 0.264760i \(0.914707\pi\)
\(888\) 0 0
\(889\) 0.984829 0.0330301
\(890\) 0 0
\(891\) −1.67730 −0.0561918
\(892\) 0 0
\(893\) −3.79961 −0.127149
\(894\) 0 0
\(895\) −7.26374 −0.242800
\(896\) 0 0
\(897\) 6.16261 0.205763
\(898\) 0 0
\(899\) 65.1341 2.17234
\(900\) 0 0
\(901\) −7.85515 −0.261693
\(902\) 0 0
\(903\) 0.454051 0.0151099
\(904\) 0 0
\(905\) −4.37675 −0.145488
\(906\) 0 0
\(907\) 49.5615 1.64566 0.822831 0.568286i \(-0.192393\pi\)
0.822831 + 0.568286i \(0.192393\pi\)
\(908\) 0 0
\(909\) −3.08547 −0.102338
\(910\) 0 0
\(911\) −30.6908 −1.01683 −0.508415 0.861112i \(-0.669768\pi\)
−0.508415 + 0.861112i \(0.669768\pi\)
\(912\) 0 0
\(913\) −12.4703 −0.412707
\(914\) 0 0
\(915\) 0.131584 0.00435003
\(916\) 0 0
\(917\) −0.353057 −0.0116590
\(918\) 0 0
\(919\) 29.4304 0.970819 0.485409 0.874287i \(-0.338671\pi\)
0.485409 + 0.874287i \(0.338671\pi\)
\(920\) 0 0
\(921\) 26.6433 0.877928
\(922\) 0 0
\(923\) 27.2084 0.895577
\(924\) 0 0
\(925\) −11.8702 −0.390288
\(926\) 0 0
\(927\) −6.58913 −0.216415
\(928\) 0 0
\(929\) −3.45739 −0.113433 −0.0567166 0.998390i \(-0.518063\pi\)
−0.0567166 + 0.998390i \(0.518063\pi\)
\(930\) 0 0
\(931\) 6.98572 0.228948
\(932\) 0 0
\(933\) 2.71532 0.0888957
\(934\) 0 0
\(935\) −0.549946 −0.0179851
\(936\) 0 0
\(937\) −2.19174 −0.0716010 −0.0358005 0.999359i \(-0.511398\pi\)
−0.0358005 + 0.999359i \(0.511398\pi\)
\(938\) 0 0
\(939\) −6.54601 −0.213621
\(940\) 0 0
\(941\) 37.7444 1.23043 0.615216 0.788358i \(-0.289069\pi\)
0.615216 + 0.788358i \(0.289069\pi\)
\(942\) 0 0
\(943\) −0.587081 −0.0191180
\(944\) 0 0
\(945\) −0.0187128 −0.000608727 0
\(946\) 0 0
\(947\) 49.2053 1.59896 0.799479 0.600694i \(-0.205109\pi\)
0.799479 + 0.600694i \(0.205109\pi\)
\(948\) 0 0
\(949\) 13.1865 0.428053
\(950\) 0 0
\(951\) −19.4676 −0.631280
\(952\) 0 0
\(953\) 23.3028 0.754851 0.377426 0.926040i \(-0.376809\pi\)
0.377426 + 0.926040i \(0.376809\pi\)
\(954\) 0 0
\(955\) −0.394882 −0.0127781
\(956\) 0 0
\(957\) −11.5124 −0.372143
\(958\) 0 0
\(959\) 0.901628 0.0291151
\(960\) 0 0
\(961\) 59.0552 1.90501
\(962\) 0 0
\(963\) −10.7446 −0.346241
\(964\) 0 0
\(965\) −0.00103540 −3.33307e−5 0
\(966\) 0 0
\(967\) −1.90814 −0.0613616 −0.0306808 0.999529i \(-0.509768\pi\)
−0.0306808 + 0.999529i \(0.509768\pi\)
\(968\) 0 0
\(969\) −0.834252 −0.0268000
\(970\) 0 0
\(971\) 25.9471 0.832682 0.416341 0.909209i \(-0.363312\pi\)
0.416341 + 0.909209i \(0.363312\pi\)
\(972\) 0 0
\(973\) 0.218538 0.00700599
\(974\) 0 0
\(975\) 22.7945 0.730009
\(976\) 0 0
\(977\) −27.6096 −0.883311 −0.441655 0.897185i \(-0.645609\pi\)
−0.441655 + 0.897185i \(0.645609\pi\)
\(978\) 0 0
\(979\) 9.94343 0.317793
\(980\) 0 0
\(981\) −5.37085 −0.171478
\(982\) 0 0
\(983\) −11.9349 −0.380665 −0.190332 0.981720i \(-0.560957\pi\)
−0.190332 + 0.981720i \(0.560957\pi\)
\(984\) 0 0
\(985\) 1.29350 0.0412142
\(986\) 0 0
\(987\) −0.181534 −0.00577828
\(988\) 0 0
\(989\) 12.4725 0.396603
\(990\) 0 0
\(991\) −26.4396 −0.839881 −0.419940 0.907552i \(-0.637949\pi\)
−0.419940 + 0.907552i \(0.637949\pi\)
\(992\) 0 0
\(993\) −11.3987 −0.361727
\(994\) 0 0
\(995\) 0.787605 0.0249688
\(996\) 0 0
\(997\) 44.4393 1.40741 0.703704 0.710494i \(-0.251528\pi\)
0.703704 + 0.710494i \(0.251528\pi\)
\(998\) 0 0
\(999\) 2.44944 0.0774969
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 8016.2.a.bd.1.7 10
4.3 odd 2 4008.2.a.j.1.7 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.j.1.7 10 4.3 odd 2
8016.2.a.bd.1.7 10 1.1 even 1 trivial