Properties

Label 1452.4.a.t.1.2
Level $1452$
Weight $4$
Character 1452.1
Self dual yes
Analytic conductor $85.671$
Analytic rank $1$
Dimension $6$
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] = [1452,4,Mod(1,1452)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1452, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1452.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1452.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: \(85.6707733283\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 174x^{4} + 63x^{3} + 7614x^{2} + 1579x - 12611 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 11 \)
Twist minimal: no (minimal twist has level 132)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(10.2999\) of defining polynomial
Character \(\chi\) \(=\) 1452.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -3.88318 q^{5} -1.97618 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -3.88318 q^{5} -1.97618 q^{7} +9.00000 q^{9} -48.0600 q^{13} -11.6495 q^{15} -8.31618 q^{17} +102.009 q^{19} -5.92854 q^{21} +108.080 q^{23} -109.921 q^{25} +27.0000 q^{27} -88.1341 q^{29} +71.7828 q^{31} +7.67386 q^{35} +203.433 q^{37} -144.180 q^{39} -177.182 q^{41} -103.581 q^{43} -34.9486 q^{45} +185.170 q^{47} -339.095 q^{49} -24.9485 q^{51} -698.821 q^{53} +306.027 q^{57} -157.271 q^{59} -207.114 q^{61} -17.7856 q^{63} +186.625 q^{65} +404.836 q^{67} +324.241 q^{69} -22.5941 q^{71} -848.773 q^{73} -329.763 q^{75} -526.699 q^{79} +81.0000 q^{81} -657.045 q^{83} +32.2932 q^{85} -264.402 q^{87} -102.610 q^{89} +94.9752 q^{91} +215.348 q^{93} -396.119 q^{95} +321.520 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9} - 66 q^{13} + 39 q^{15} - 44 q^{17} - 270 q^{19} - 69 q^{21} - 124 q^{23} + 131 q^{25} + 162 q^{27} - 141 q^{29} - 253 q^{31} - 884 q^{35} + 288 q^{37} - 198 q^{39} - 428 q^{41} - 1006 q^{43} + 117 q^{45} - 674 q^{47} + 181 q^{49} - 132 q^{51} - 773 q^{53} - 810 q^{57} - 17 q^{59} - 1016 q^{61} - 207 q^{63} - 1220 q^{65} + 1836 q^{67} - 372 q^{69} - 208 q^{71} - 1521 q^{73} + 393 q^{75} - 1425 q^{79} + 486 q^{81} - 3065 q^{83} - 1304 q^{85} - 423 q^{87} + 1444 q^{89} + 1328 q^{91} - 759 q^{93} - 1760 q^{95} - 3887 q^{97}+O(q^{100}) \) 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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −3.88318 −0.347322 −0.173661 0.984806i \(-0.555560\pi\)
−0.173661 + 0.984806i \(0.555560\pi\)
\(6\) 0 0
\(7\) −1.97618 −0.106704 −0.0533519 0.998576i \(-0.516990\pi\)
−0.0533519 + 0.998576i \(0.516990\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −48.0600 −1.02534 −0.512670 0.858585i \(-0.671344\pi\)
−0.512670 + 0.858585i \(0.671344\pi\)
\(14\) 0 0
\(15\) −11.6495 −0.200526
\(16\) 0 0
\(17\) −8.31618 −0.118645 −0.0593226 0.998239i \(-0.518894\pi\)
−0.0593226 + 0.998239i \(0.518894\pi\)
\(18\) 0 0
\(19\) 102.009 1.23171 0.615854 0.787860i \(-0.288811\pi\)
0.615854 + 0.787860i \(0.288811\pi\)
\(20\) 0 0
\(21\) −5.92854 −0.0616054
\(22\) 0 0
\(23\) 108.080 0.979839 0.489920 0.871768i \(-0.337026\pi\)
0.489920 + 0.871768i \(0.337026\pi\)
\(24\) 0 0
\(25\) −109.921 −0.879368
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −88.1341 −0.564348 −0.282174 0.959363i \(-0.591056\pi\)
−0.282174 + 0.959363i \(0.591056\pi\)
\(30\) 0 0
\(31\) 71.7828 0.415889 0.207945 0.978141i \(-0.433323\pi\)
0.207945 + 0.978141i \(0.433323\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 7.67386 0.0370605
\(36\) 0 0
\(37\) 203.433 0.903897 0.451948 0.892044i \(-0.350729\pi\)
0.451948 + 0.892044i \(0.350729\pi\)
\(38\) 0 0
\(39\) −144.180 −0.591981
\(40\) 0 0
\(41\) −177.182 −0.674905 −0.337452 0.941343i \(-0.609565\pi\)
−0.337452 + 0.941343i \(0.609565\pi\)
\(42\) 0 0
\(43\) −103.581 −0.367349 −0.183674 0.982987i \(-0.558799\pi\)
−0.183674 + 0.982987i \(0.558799\pi\)
\(44\) 0 0
\(45\) −34.9486 −0.115774
\(46\) 0 0
\(47\) 185.170 0.574676 0.287338 0.957829i \(-0.407230\pi\)
0.287338 + 0.957829i \(0.407230\pi\)
\(48\) 0 0
\(49\) −339.095 −0.988614
\(50\) 0 0
\(51\) −24.9485 −0.0684999
\(52\) 0 0
\(53\) −698.821 −1.81114 −0.905570 0.424197i \(-0.860556\pi\)
−0.905570 + 0.424197i \(0.860556\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 306.027 0.711127
\(58\) 0 0
\(59\) −157.271 −0.347033 −0.173517 0.984831i \(-0.555513\pi\)
−0.173517 + 0.984831i \(0.555513\pi\)
\(60\) 0 0
\(61\) −207.114 −0.434726 −0.217363 0.976091i \(-0.569745\pi\)
−0.217363 + 0.976091i \(0.569745\pi\)
\(62\) 0 0
\(63\) −17.7856 −0.0355679
\(64\) 0 0
\(65\) 186.625 0.356123
\(66\) 0 0
\(67\) 404.836 0.738188 0.369094 0.929392i \(-0.379668\pi\)
0.369094 + 0.929392i \(0.379668\pi\)
\(68\) 0 0
\(69\) 324.241 0.565710
\(70\) 0 0
\(71\) −22.5941 −0.0377665 −0.0188832 0.999822i \(-0.506011\pi\)
−0.0188832 + 0.999822i \(0.506011\pi\)
\(72\) 0 0
\(73\) −848.773 −1.36084 −0.680421 0.732822i \(-0.738203\pi\)
−0.680421 + 0.732822i \(0.738203\pi\)
\(74\) 0 0
\(75\) −329.763 −0.507703
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −526.699 −0.750105 −0.375053 0.927004i \(-0.622375\pi\)
−0.375053 + 0.927004i \(0.622375\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −657.045 −0.868917 −0.434458 0.900692i \(-0.643060\pi\)
−0.434458 + 0.900692i \(0.643060\pi\)
\(84\) 0 0
\(85\) 32.2932 0.0412081
\(86\) 0 0
\(87\) −264.402 −0.325826
\(88\) 0 0
\(89\) −102.610 −0.122210 −0.0611048 0.998131i \(-0.519462\pi\)
−0.0611048 + 0.998131i \(0.519462\pi\)
\(90\) 0 0
\(91\) 94.9752 0.109408
\(92\) 0 0
\(93\) 215.348 0.240114
\(94\) 0 0
\(95\) −396.119 −0.427799
\(96\) 0 0
\(97\) 321.520 0.336551 0.168275 0.985740i \(-0.446180\pi\)
0.168275 + 0.985740i \(0.446180\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −175.925 −0.173319 −0.0866594 0.996238i \(-0.527619\pi\)
−0.0866594 + 0.996238i \(0.527619\pi\)
\(102\) 0 0
\(103\) 234.691 0.224513 0.112256 0.993679i \(-0.464192\pi\)
0.112256 + 0.993679i \(0.464192\pi\)
\(104\) 0 0
\(105\) 23.0216 0.0213969
\(106\) 0 0
\(107\) 1606.08 1.45108 0.725540 0.688180i \(-0.241590\pi\)
0.725540 + 0.688180i \(0.241590\pi\)
\(108\) 0 0
\(109\) −1512.59 −1.32917 −0.664587 0.747211i \(-0.731393\pi\)
−0.664587 + 0.747211i \(0.731393\pi\)
\(110\) 0 0
\(111\) 610.299 0.521865
\(112\) 0 0
\(113\) 573.791 0.477679 0.238840 0.971059i \(-0.423233\pi\)
0.238840 + 0.971059i \(0.423233\pi\)
\(114\) 0 0
\(115\) −419.695 −0.340320
\(116\) 0 0
\(117\) −432.540 −0.341780
\(118\) 0 0
\(119\) 16.4343 0.0126599
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −531.545 −0.389657
\(124\) 0 0
\(125\) 912.239 0.652745
\(126\) 0 0
\(127\) −1502.03 −1.04948 −0.524738 0.851264i \(-0.675837\pi\)
−0.524738 + 0.851264i \(0.675837\pi\)
\(128\) 0 0
\(129\) −310.744 −0.212089
\(130\) 0 0
\(131\) −2825.97 −1.88478 −0.942391 0.334513i \(-0.891428\pi\)
−0.942391 + 0.334513i \(0.891428\pi\)
\(132\) 0 0
\(133\) −201.588 −0.131428
\(134\) 0 0
\(135\) −104.846 −0.0668421
\(136\) 0 0
\(137\) −1754.35 −1.09405 −0.547024 0.837117i \(-0.684239\pi\)
−0.547024 + 0.837117i \(0.684239\pi\)
\(138\) 0 0
\(139\) −319.127 −0.194734 −0.0973670 0.995249i \(-0.531042\pi\)
−0.0973670 + 0.995249i \(0.531042\pi\)
\(140\) 0 0
\(141\) 555.509 0.331790
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 342.240 0.196010
\(146\) 0 0
\(147\) −1017.28 −0.570777
\(148\) 0 0
\(149\) −1461.09 −0.803337 −0.401668 0.915785i \(-0.631570\pi\)
−0.401668 + 0.915785i \(0.631570\pi\)
\(150\) 0 0
\(151\) −2008.90 −1.08266 −0.541331 0.840810i \(-0.682079\pi\)
−0.541331 + 0.840810i \(0.682079\pi\)
\(152\) 0 0
\(153\) −74.8456 −0.0395484
\(154\) 0 0
\(155\) −278.745 −0.144447
\(156\) 0 0
\(157\) 2064.11 1.04926 0.524631 0.851330i \(-0.324203\pi\)
0.524631 + 0.851330i \(0.324203\pi\)
\(158\) 0 0
\(159\) −2096.46 −1.04566
\(160\) 0 0
\(161\) −213.586 −0.104552
\(162\) 0 0
\(163\) −3817.75 −1.83454 −0.917268 0.398270i \(-0.869611\pi\)
−0.917268 + 0.398270i \(0.869611\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 987.900 0.457761 0.228880 0.973455i \(-0.426494\pi\)
0.228880 + 0.973455i \(0.426494\pi\)
\(168\) 0 0
\(169\) 112.759 0.0513241
\(170\) 0 0
\(171\) 918.080 0.410569
\(172\) 0 0
\(173\) −916.203 −0.402645 −0.201323 0.979525i \(-0.564524\pi\)
−0.201323 + 0.979525i \(0.564524\pi\)
\(174\) 0 0
\(175\) 217.224 0.0938318
\(176\) 0 0
\(177\) −471.814 −0.200360
\(178\) 0 0
\(179\) 2586.42 1.07999 0.539996 0.841668i \(-0.318426\pi\)
0.539996 + 0.841668i \(0.318426\pi\)
\(180\) 0 0
\(181\) −1243.57 −0.510684 −0.255342 0.966851i \(-0.582188\pi\)
−0.255342 + 0.966851i \(0.582188\pi\)
\(182\) 0 0
\(183\) −621.343 −0.250989
\(184\) 0 0
\(185\) −789.966 −0.313943
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −53.3569 −0.0205351
\(190\) 0 0
\(191\) 3600.18 1.36387 0.681936 0.731412i \(-0.261138\pi\)
0.681936 + 0.731412i \(0.261138\pi\)
\(192\) 0 0
\(193\) −3968.27 −1.48001 −0.740007 0.672599i \(-0.765178\pi\)
−0.740007 + 0.672599i \(0.765178\pi\)
\(194\) 0 0
\(195\) 559.876 0.205608
\(196\) 0 0
\(197\) −1808.07 −0.653907 −0.326953 0.945040i \(-0.606022\pi\)
−0.326953 + 0.945040i \(0.606022\pi\)
\(198\) 0 0
\(199\) −2188.11 −0.779453 −0.389727 0.920931i \(-0.627430\pi\)
−0.389727 + 0.920931i \(0.627430\pi\)
\(200\) 0 0
\(201\) 1214.51 0.426193
\(202\) 0 0
\(203\) 174.169 0.0602180
\(204\) 0 0
\(205\) 688.027 0.234409
\(206\) 0 0
\(207\) 972.723 0.326613
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4659.26 1.52017 0.760086 0.649822i \(-0.225157\pi\)
0.760086 + 0.649822i \(0.225157\pi\)
\(212\) 0 0
\(213\) −67.7822 −0.0218045
\(214\) 0 0
\(215\) 402.224 0.127588
\(216\) 0 0
\(217\) −141.856 −0.0443769
\(218\) 0 0
\(219\) −2546.32 −0.785682
\(220\) 0 0
\(221\) 399.675 0.121652
\(222\) 0 0
\(223\) 6376.94 1.91494 0.957470 0.288533i \(-0.0931677\pi\)
0.957470 + 0.288533i \(0.0931677\pi\)
\(224\) 0 0
\(225\) −989.288 −0.293123
\(226\) 0 0
\(227\) −1343.12 −0.392713 −0.196357 0.980533i \(-0.562911\pi\)
−0.196357 + 0.980533i \(0.562911\pi\)
\(228\) 0 0
\(229\) 3894.92 1.12395 0.561973 0.827156i \(-0.310043\pi\)
0.561973 + 0.827156i \(0.310043\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2289.21 0.643654 0.321827 0.946798i \(-0.395703\pi\)
0.321827 + 0.946798i \(0.395703\pi\)
\(234\) 0 0
\(235\) −719.047 −0.199598
\(236\) 0 0
\(237\) −1580.10 −0.433074
\(238\) 0 0
\(239\) 6002.89 1.62466 0.812332 0.583195i \(-0.198198\pi\)
0.812332 + 0.583195i \(0.198198\pi\)
\(240\) 0 0
\(241\) −6921.56 −1.85003 −0.925014 0.379933i \(-0.875947\pi\)
−0.925014 + 0.379933i \(0.875947\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 1316.76 0.343367
\(246\) 0 0
\(247\) −4902.54 −1.26292
\(248\) 0 0
\(249\) −1971.14 −0.501669
\(250\) 0 0
\(251\) −3691.49 −0.928307 −0.464153 0.885755i \(-0.653641\pi\)
−0.464153 + 0.885755i \(0.653641\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 96.8796 0.0237915
\(256\) 0 0
\(257\) 493.994 0.119901 0.0599504 0.998201i \(-0.480906\pi\)
0.0599504 + 0.998201i \(0.480906\pi\)
\(258\) 0 0
\(259\) −402.020 −0.0964492
\(260\) 0 0
\(261\) −793.207 −0.188116
\(262\) 0 0
\(263\) 3105.36 0.728079 0.364039 0.931384i \(-0.381397\pi\)
0.364039 + 0.931384i \(0.381397\pi\)
\(264\) 0 0
\(265\) 2713.64 0.629048
\(266\) 0 0
\(267\) −307.830 −0.0705577
\(268\) 0 0
\(269\) −4174.64 −0.946217 −0.473109 0.881004i \(-0.656868\pi\)
−0.473109 + 0.881004i \(0.656868\pi\)
\(270\) 0 0
\(271\) −7053.43 −1.58105 −0.790526 0.612428i \(-0.790193\pi\)
−0.790526 + 0.612428i \(0.790193\pi\)
\(272\) 0 0
\(273\) 284.925 0.0631666
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2816.73 −0.610978 −0.305489 0.952196i \(-0.598820\pi\)
−0.305489 + 0.952196i \(0.598820\pi\)
\(278\) 0 0
\(279\) 646.045 0.138630
\(280\) 0 0
\(281\) −3215.27 −0.682587 −0.341294 0.939957i \(-0.610865\pi\)
−0.341294 + 0.939957i \(0.610865\pi\)
\(282\) 0 0
\(283\) 139.720 0.0293480 0.0146740 0.999892i \(-0.495329\pi\)
0.0146740 + 0.999892i \(0.495329\pi\)
\(284\) 0 0
\(285\) −1188.36 −0.246990
\(286\) 0 0
\(287\) 350.143 0.0720149
\(288\) 0 0
\(289\) −4843.84 −0.985923
\(290\) 0 0
\(291\) 964.560 0.194308
\(292\) 0 0
\(293\) 8956.56 1.78583 0.892914 0.450227i \(-0.148657\pi\)
0.892914 + 0.450227i \(0.148657\pi\)
\(294\) 0 0
\(295\) 610.712 0.120532
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5194.33 −1.00467
\(300\) 0 0
\(301\) 204.695 0.0391975
\(302\) 0 0
\(303\) −527.775 −0.100066
\(304\) 0 0
\(305\) 804.261 0.150990
\(306\) 0 0
\(307\) −5258.22 −0.977532 −0.488766 0.872415i \(-0.662553\pi\)
−0.488766 + 0.872415i \(0.662553\pi\)
\(308\) 0 0
\(309\) 704.074 0.129623
\(310\) 0 0
\(311\) 1570.23 0.286301 0.143151 0.989701i \(-0.454277\pi\)
0.143151 + 0.989701i \(0.454277\pi\)
\(312\) 0 0
\(313\) −3836.52 −0.692822 −0.346411 0.938083i \(-0.612600\pi\)
−0.346411 + 0.938083i \(0.612600\pi\)
\(314\) 0 0
\(315\) 69.0647 0.0123535
\(316\) 0 0
\(317\) −4861.41 −0.861338 −0.430669 0.902510i \(-0.641722\pi\)
−0.430669 + 0.902510i \(0.641722\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 4818.24 0.837781
\(322\) 0 0
\(323\) −848.324 −0.146136
\(324\) 0 0
\(325\) 5282.80 0.901652
\(326\) 0 0
\(327\) −4537.78 −0.767399
\(328\) 0 0
\(329\) −365.929 −0.0613201
\(330\) 0 0
\(331\) 6335.94 1.05213 0.526065 0.850445i \(-0.323667\pi\)
0.526065 + 0.850445i \(0.323667\pi\)
\(332\) 0 0
\(333\) 1830.90 0.301299
\(334\) 0 0
\(335\) −1572.05 −0.256389
\(336\) 0 0
\(337\) −2540.13 −0.410593 −0.205296 0.978700i \(-0.565816\pi\)
−0.205296 + 0.978700i \(0.565816\pi\)
\(338\) 0 0
\(339\) 1721.37 0.275788
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1347.94 0.212193
\(344\) 0 0
\(345\) −1259.08 −0.196484
\(346\) 0 0
\(347\) −11636.8 −1.80028 −0.900142 0.435596i \(-0.856538\pi\)
−0.900142 + 0.435596i \(0.856538\pi\)
\(348\) 0 0
\(349\) −438.968 −0.0673279 −0.0336639 0.999433i \(-0.510718\pi\)
−0.0336639 + 0.999433i \(0.510718\pi\)
\(350\) 0 0
\(351\) −1297.62 −0.197327
\(352\) 0 0
\(353\) −11352.7 −1.71174 −0.855872 0.517188i \(-0.826979\pi\)
−0.855872 + 0.517188i \(0.826979\pi\)
\(354\) 0 0
\(355\) 87.7367 0.0131171
\(356\) 0 0
\(357\) 49.3028 0.00730919
\(358\) 0 0
\(359\) −2861.95 −0.420746 −0.210373 0.977621i \(-0.567468\pi\)
−0.210373 + 0.977621i \(0.567468\pi\)
\(360\) 0 0
\(361\) 3546.81 0.517104
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3295.94 0.472650
\(366\) 0 0
\(367\) 2766.45 0.393482 0.196741 0.980456i \(-0.436964\pi\)
0.196741 + 0.980456i \(0.436964\pi\)
\(368\) 0 0
\(369\) −1594.63 −0.224968
\(370\) 0 0
\(371\) 1381.00 0.193255
\(372\) 0 0
\(373\) 1705.25 0.236715 0.118358 0.992971i \(-0.462237\pi\)
0.118358 + 0.992971i \(0.462237\pi\)
\(374\) 0 0
\(375\) 2736.72 0.376863
\(376\) 0 0
\(377\) 4235.72 0.578649
\(378\) 0 0
\(379\) 1416.56 0.191989 0.0959947 0.995382i \(-0.469397\pi\)
0.0959947 + 0.995382i \(0.469397\pi\)
\(380\) 0 0
\(381\) −4506.08 −0.605915
\(382\) 0 0
\(383\) −13515.7 −1.80319 −0.901593 0.432585i \(-0.857602\pi\)
−0.901593 + 0.432585i \(0.857602\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −932.231 −0.122450
\(388\) 0 0
\(389\) 14064.8 1.83319 0.916597 0.399811i \(-0.130924\pi\)
0.916597 + 0.399811i \(0.130924\pi\)
\(390\) 0 0
\(391\) −898.815 −0.116253
\(392\) 0 0
\(393\) −8477.92 −1.08818
\(394\) 0 0
\(395\) 2045.27 0.260528
\(396\) 0 0
\(397\) −7916.44 −1.00079 −0.500397 0.865796i \(-0.666812\pi\)
−0.500397 + 0.865796i \(0.666812\pi\)
\(398\) 0 0
\(399\) −604.764 −0.0758799
\(400\) 0 0
\(401\) 12406.0 1.54495 0.772475 0.635045i \(-0.219019\pi\)
0.772475 + 0.635045i \(0.219019\pi\)
\(402\) 0 0
\(403\) −3449.88 −0.426428
\(404\) 0 0
\(405\) −314.537 −0.0385913
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −5008.18 −0.605473 −0.302737 0.953074i \(-0.597900\pi\)
−0.302737 + 0.953074i \(0.597900\pi\)
\(410\) 0 0
\(411\) −5263.06 −0.631649
\(412\) 0 0
\(413\) 310.796 0.0370298
\(414\) 0 0
\(415\) 2551.42 0.301794
\(416\) 0 0
\(417\) −957.382 −0.112430
\(418\) 0 0
\(419\) 10471.9 1.22097 0.610487 0.792026i \(-0.290974\pi\)
0.610487 + 0.792026i \(0.290974\pi\)
\(420\) 0 0
\(421\) 7648.95 0.885480 0.442740 0.896650i \(-0.354007\pi\)
0.442740 + 0.896650i \(0.354007\pi\)
\(422\) 0 0
\(423\) 1666.53 0.191559
\(424\) 0 0
\(425\) 914.122 0.104333
\(426\) 0 0
\(427\) 409.295 0.0463869
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4181.33 −0.467303 −0.233651 0.972320i \(-0.575067\pi\)
−0.233651 + 0.972320i \(0.575067\pi\)
\(432\) 0 0
\(433\) −11067.7 −1.22836 −0.614181 0.789165i \(-0.710514\pi\)
−0.614181 + 0.789165i \(0.710514\pi\)
\(434\) 0 0
\(435\) 1026.72 0.113167
\(436\) 0 0
\(437\) 11025.2 1.20688
\(438\) 0 0
\(439\) 1824.08 0.198311 0.0991555 0.995072i \(-0.468386\pi\)
0.0991555 + 0.995072i \(0.468386\pi\)
\(440\) 0 0
\(441\) −3051.85 −0.329538
\(442\) 0 0
\(443\) −7464.88 −0.800604 −0.400302 0.916383i \(-0.631095\pi\)
−0.400302 + 0.916383i \(0.631095\pi\)
\(444\) 0 0
\(445\) 398.453 0.0424461
\(446\) 0 0
\(447\) −4383.27 −0.463807
\(448\) 0 0
\(449\) 2526.10 0.265510 0.132755 0.991149i \(-0.457618\pi\)
0.132755 + 0.991149i \(0.457618\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −6026.70 −0.625075
\(454\) 0 0
\(455\) −368.805 −0.0379997
\(456\) 0 0
\(457\) 3594.81 0.367961 0.183980 0.982930i \(-0.441102\pi\)
0.183980 + 0.982930i \(0.441102\pi\)
\(458\) 0 0
\(459\) −224.537 −0.0228333
\(460\) 0 0
\(461\) 11811.9 1.19335 0.596675 0.802483i \(-0.296488\pi\)
0.596675 + 0.802483i \(0.296488\pi\)
\(462\) 0 0
\(463\) 12046.4 1.20917 0.604585 0.796541i \(-0.293339\pi\)
0.604585 + 0.796541i \(0.293339\pi\)
\(464\) 0 0
\(465\) −836.235 −0.0833968
\(466\) 0 0
\(467\) 10461.3 1.03659 0.518297 0.855201i \(-0.326566\pi\)
0.518297 + 0.855201i \(0.326566\pi\)
\(468\) 0 0
\(469\) −800.029 −0.0787674
\(470\) 0 0
\(471\) 6192.33 0.605791
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −11212.9 −1.08312
\(476\) 0 0
\(477\) −6289.39 −0.603713
\(478\) 0 0
\(479\) 14128.2 1.34767 0.673837 0.738880i \(-0.264645\pi\)
0.673837 + 0.738880i \(0.264645\pi\)
\(480\) 0 0
\(481\) −9776.98 −0.926802
\(482\) 0 0
\(483\) −640.759 −0.0603634
\(484\) 0 0
\(485\) −1248.52 −0.116891
\(486\) 0 0
\(487\) −10985.0 −1.02213 −0.511066 0.859541i \(-0.670749\pi\)
−0.511066 + 0.859541i \(0.670749\pi\)
\(488\) 0 0
\(489\) −11453.3 −1.05917
\(490\) 0 0
\(491\) 15542.4 1.42855 0.714274 0.699866i \(-0.246757\pi\)
0.714274 + 0.699866i \(0.246757\pi\)
\(492\) 0 0
\(493\) 732.939 0.0669572
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 44.6499 0.00402983
\(498\) 0 0
\(499\) −14846.9 −1.33194 −0.665969 0.745980i \(-0.731982\pi\)
−0.665969 + 0.745980i \(0.731982\pi\)
\(500\) 0 0
\(501\) 2963.70 0.264288
\(502\) 0 0
\(503\) 10239.4 0.907660 0.453830 0.891088i \(-0.350057\pi\)
0.453830 + 0.891088i \(0.350057\pi\)
\(504\) 0 0
\(505\) 683.148 0.0601974
\(506\) 0 0
\(507\) 338.277 0.0296320
\(508\) 0 0
\(509\) −17148.3 −1.49329 −0.746647 0.665220i \(-0.768338\pi\)
−0.746647 + 0.665220i \(0.768338\pi\)
\(510\) 0 0
\(511\) 1677.33 0.145207
\(512\) 0 0
\(513\) 2754.24 0.237042
\(514\) 0 0
\(515\) −911.348 −0.0779782
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −2748.61 −0.232467
\(520\) 0 0
\(521\) 19288.1 1.62193 0.810967 0.585092i \(-0.198942\pi\)
0.810967 + 0.585092i \(0.198942\pi\)
\(522\) 0 0
\(523\) 338.841 0.0283298 0.0141649 0.999900i \(-0.495491\pi\)
0.0141649 + 0.999900i \(0.495491\pi\)
\(524\) 0 0
\(525\) 651.671 0.0541738
\(526\) 0 0
\(527\) −596.958 −0.0493433
\(528\) 0 0
\(529\) −485.648 −0.0399152
\(530\) 0 0
\(531\) −1415.44 −0.115678
\(532\) 0 0
\(533\) 8515.33 0.692008
\(534\) 0 0
\(535\) −6236.69 −0.503992
\(536\) 0 0
\(537\) 7759.27 0.623533
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 23019.2 1.82934 0.914668 0.404205i \(-0.132452\pi\)
0.914668 + 0.404205i \(0.132452\pi\)
\(542\) 0 0
\(543\) −3730.71 −0.294844
\(544\) 0 0
\(545\) 5873.66 0.461652
\(546\) 0 0
\(547\) −8458.05 −0.661134 −0.330567 0.943783i \(-0.607240\pi\)
−0.330567 + 0.943783i \(0.607240\pi\)
\(548\) 0 0
\(549\) −1864.03 −0.144909
\(550\) 0 0
\(551\) −8990.46 −0.695112
\(552\) 0 0
\(553\) 1040.85 0.0800390
\(554\) 0 0
\(555\) −2369.90 −0.181255
\(556\) 0 0
\(557\) 21946.1 1.66945 0.834726 0.550665i \(-0.185626\pi\)
0.834726 + 0.550665i \(0.185626\pi\)
\(558\) 0 0
\(559\) 4978.11 0.376658
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6269.35 −0.469310 −0.234655 0.972079i \(-0.575396\pi\)
−0.234655 + 0.972079i \(0.575396\pi\)
\(564\) 0 0
\(565\) −2228.13 −0.165908
\(566\) 0 0
\(567\) −160.071 −0.0118560
\(568\) 0 0
\(569\) 16932.8 1.24756 0.623780 0.781600i \(-0.285596\pi\)
0.623780 + 0.781600i \(0.285596\pi\)
\(570\) 0 0
\(571\) 3759.85 0.275560 0.137780 0.990463i \(-0.456003\pi\)
0.137780 + 0.990463i \(0.456003\pi\)
\(572\) 0 0
\(573\) 10800.5 0.787432
\(574\) 0 0
\(575\) −11880.3 −0.861639
\(576\) 0 0
\(577\) 2371.51 0.171105 0.0855524 0.996334i \(-0.472735\pi\)
0.0855524 + 0.996334i \(0.472735\pi\)
\(578\) 0 0
\(579\) −11904.8 −0.854486
\(580\) 0 0
\(581\) 1298.44 0.0927167
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 1679.63 0.118708
\(586\) 0 0
\(587\) 14040.1 0.987221 0.493611 0.869683i \(-0.335677\pi\)
0.493611 + 0.869683i \(0.335677\pi\)
\(588\) 0 0
\(589\) 7322.48 0.512254
\(590\) 0 0
\(591\) −5424.21 −0.377533
\(592\) 0 0
\(593\) 25984.1 1.79939 0.899695 0.436518i \(-0.143789\pi\)
0.899695 + 0.436518i \(0.143789\pi\)
\(594\) 0 0
\(595\) −63.8172 −0.00439706
\(596\) 0 0
\(597\) −6564.34 −0.450018
\(598\) 0 0
\(599\) 7796.27 0.531798 0.265899 0.964001i \(-0.414331\pi\)
0.265899 + 0.964001i \(0.414331\pi\)
\(600\) 0 0
\(601\) 19476.1 1.32188 0.660938 0.750441i \(-0.270159\pi\)
0.660938 + 0.750441i \(0.270159\pi\)
\(602\) 0 0
\(603\) 3643.52 0.246063
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 860.774 0.0575581 0.0287790 0.999586i \(-0.490838\pi\)
0.0287790 + 0.999586i \(0.490838\pi\)
\(608\) 0 0
\(609\) 522.507 0.0347669
\(610\) 0 0
\(611\) −8899.25 −0.589239
\(612\) 0 0
\(613\) −26105.7 −1.72006 −0.860031 0.510242i \(-0.829556\pi\)
−0.860031 + 0.510242i \(0.829556\pi\)
\(614\) 0 0
\(615\) 2064.08 0.135336
\(616\) 0 0
\(617\) −6945.34 −0.453175 −0.226588 0.973991i \(-0.572757\pi\)
−0.226588 + 0.973991i \(0.572757\pi\)
\(618\) 0 0
\(619\) −9404.77 −0.610678 −0.305339 0.952244i \(-0.598770\pi\)
−0.305339 + 0.952244i \(0.598770\pi\)
\(620\) 0 0
\(621\) 2918.17 0.188570
\(622\) 0 0
\(623\) 202.776 0.0130402
\(624\) 0 0
\(625\) 10197.7 0.652655
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1691.79 −0.107243
\(630\) 0 0
\(631\) −6237.03 −0.393490 −0.196745 0.980455i \(-0.563037\pi\)
−0.196745 + 0.980455i \(0.563037\pi\)
\(632\) 0 0
\(633\) 13977.8 0.877672
\(634\) 0 0
\(635\) 5832.64 0.364506
\(636\) 0 0
\(637\) 16296.9 1.01367
\(638\) 0 0
\(639\) −203.347 −0.0125888
\(640\) 0 0
\(641\) 7455.08 0.459373 0.229686 0.973265i \(-0.426230\pi\)
0.229686 + 0.973265i \(0.426230\pi\)
\(642\) 0 0
\(643\) 23532.8 1.44330 0.721650 0.692258i \(-0.243384\pi\)
0.721650 + 0.692258i \(0.243384\pi\)
\(644\) 0 0
\(645\) 1206.67 0.0736631
\(646\) 0 0
\(647\) −25806.4 −1.56809 −0.784046 0.620703i \(-0.786847\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −425.567 −0.0256210
\(652\) 0 0
\(653\) −28699.7 −1.71992 −0.859959 0.510363i \(-0.829511\pi\)
−0.859959 + 0.510363i \(0.829511\pi\)
\(654\) 0 0
\(655\) 10973.8 0.654626
\(656\) 0 0
\(657\) −7638.96 −0.453614
\(658\) 0 0
\(659\) 31386.2 1.85528 0.927641 0.373472i \(-0.121833\pi\)
0.927641 + 0.373472i \(0.121833\pi\)
\(660\) 0 0
\(661\) −23697.4 −1.39443 −0.697217 0.716860i \(-0.745579\pi\)
−0.697217 + 0.716860i \(0.745579\pi\)
\(662\) 0 0
\(663\) 1199.03 0.0702357
\(664\) 0 0
\(665\) 782.802 0.0456477
\(666\) 0 0
\(667\) −9525.56 −0.552970
\(668\) 0 0
\(669\) 19130.8 1.10559
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 14520.3 0.831674 0.415837 0.909439i \(-0.363489\pi\)
0.415837 + 0.909439i \(0.363489\pi\)
\(674\) 0 0
\(675\) −2967.87 −0.169234
\(676\) 0 0
\(677\) 30118.8 1.70984 0.854918 0.518763i \(-0.173607\pi\)
0.854918 + 0.518763i \(0.173607\pi\)
\(678\) 0 0
\(679\) −635.382 −0.0359112
\(680\) 0 0
\(681\) −4029.35 −0.226733
\(682\) 0 0
\(683\) −17714.8 −0.992442 −0.496221 0.868196i \(-0.665280\pi\)
−0.496221 + 0.868196i \(0.665280\pi\)
\(684\) 0 0
\(685\) 6812.46 0.379987
\(686\) 0 0
\(687\) 11684.8 0.648910
\(688\) 0 0
\(689\) 33585.3 1.85704
\(690\) 0 0
\(691\) 12303.4 0.677340 0.338670 0.940905i \(-0.390023\pi\)
0.338670 + 0.940905i \(0.390023\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1239.23 0.0676354
\(696\) 0 0
\(697\) 1473.47 0.0800743
\(698\) 0 0
\(699\) 6867.64 0.371614
\(700\) 0 0
\(701\) 18584.2 1.00131 0.500653 0.865648i \(-0.333093\pi\)
0.500653 + 0.865648i \(0.333093\pi\)
\(702\) 0 0
\(703\) 20752.0 1.11334
\(704\) 0 0
\(705\) −2157.14 −0.115238
\(706\) 0 0
\(707\) 347.660 0.0184938
\(708\) 0 0
\(709\) 3983.07 0.210983 0.105492 0.994420i \(-0.466358\pi\)
0.105492 + 0.994420i \(0.466358\pi\)
\(710\) 0 0
\(711\) −4740.30 −0.250035
\(712\) 0 0
\(713\) 7758.30 0.407505
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 18008.7 0.938000
\(718\) 0 0
\(719\) 20957.6 1.08705 0.543523 0.839394i \(-0.317090\pi\)
0.543523 + 0.839394i \(0.317090\pi\)
\(720\) 0 0
\(721\) −463.792 −0.0239564
\(722\) 0 0
\(723\) −20764.7 −1.06811
\(724\) 0 0
\(725\) 9687.78 0.496269
\(726\) 0 0
\(727\) 20080.1 1.02439 0.512193 0.858870i \(-0.328833\pi\)
0.512193 + 0.858870i \(0.328833\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 861.400 0.0435842
\(732\) 0 0
\(733\) 8481.55 0.427385 0.213693 0.976901i \(-0.431451\pi\)
0.213693 + 0.976901i \(0.431451\pi\)
\(734\) 0 0
\(735\) 3950.29 0.198243
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −28685.7 −1.42790 −0.713952 0.700195i \(-0.753097\pi\)
−0.713952 + 0.700195i \(0.753097\pi\)
\(740\) 0 0
\(741\) −14707.6 −0.729147
\(742\) 0 0
\(743\) −16818.1 −0.830412 −0.415206 0.909727i \(-0.636290\pi\)
−0.415206 + 0.909727i \(0.636290\pi\)
\(744\) 0 0
\(745\) 5673.67 0.279016
\(746\) 0 0
\(747\) −5913.41 −0.289639
\(748\) 0 0
\(749\) −3173.90 −0.154836
\(750\) 0 0
\(751\) −30901.1 −1.50146 −0.750731 0.660608i \(-0.770299\pi\)
−0.750731 + 0.660608i \(0.770299\pi\)
\(752\) 0 0
\(753\) −11074.5 −0.535958
\(754\) 0 0
\(755\) 7800.91 0.376032
\(756\) 0 0
\(757\) 32931.1 1.58111 0.790556 0.612390i \(-0.209792\pi\)
0.790556 + 0.612390i \(0.209792\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1425.69 −0.0679122 −0.0339561 0.999423i \(-0.510811\pi\)
−0.0339561 + 0.999423i \(0.510811\pi\)
\(762\) 0 0
\(763\) 2989.16 0.141828
\(764\) 0 0
\(765\) 290.639 0.0137360
\(766\) 0 0
\(767\) 7558.45 0.355828
\(768\) 0 0
\(769\) −32734.4 −1.53502 −0.767511 0.641036i \(-0.778505\pi\)
−0.767511 + 0.641036i \(0.778505\pi\)
\(770\) 0 0
\(771\) 1481.98 0.0692248
\(772\) 0 0
\(773\) −13967.1 −0.649884 −0.324942 0.945734i \(-0.605345\pi\)
−0.324942 + 0.945734i \(0.605345\pi\)
\(774\) 0 0
\(775\) −7890.43 −0.365719
\(776\) 0 0
\(777\) −1206.06 −0.0556849
\(778\) 0 0
\(779\) −18074.1 −0.831286
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −2379.62 −0.108609
\(784\) 0 0
\(785\) −8015.31 −0.364431
\(786\) 0 0
\(787\) 24071.9 1.09031 0.545153 0.838337i \(-0.316472\pi\)
0.545153 + 0.838337i \(0.316472\pi\)
\(788\) 0 0
\(789\) 9316.08 0.420356
\(790\) 0 0
\(791\) −1133.91 −0.0509701
\(792\) 0 0
\(793\) 9953.90 0.445742
\(794\) 0 0
\(795\) 8140.93 0.363181
\(796\) 0 0
\(797\) 27332.1 1.21475 0.607373 0.794417i \(-0.292223\pi\)
0.607373 + 0.794417i \(0.292223\pi\)
\(798\) 0 0
\(799\) −1539.91 −0.0681826
\(800\) 0 0
\(801\) −923.491 −0.0407365
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 829.393 0.0363134
\(806\) 0 0
\(807\) −12523.9 −0.546299
\(808\) 0 0
\(809\) 23284.6 1.01192 0.505961 0.862557i \(-0.331138\pi\)
0.505961 + 0.862557i \(0.331138\pi\)
\(810\) 0 0
\(811\) 1970.62 0.0853242 0.0426621 0.999090i \(-0.486416\pi\)
0.0426621 + 0.999090i \(0.486416\pi\)
\(812\) 0 0
\(813\) −21160.3 −0.912821
\(814\) 0 0
\(815\) 14825.0 0.637175
\(816\) 0 0
\(817\) −10566.2 −0.452466
\(818\) 0 0
\(819\) 854.776 0.0364692
\(820\) 0 0
\(821\) 38252.6 1.62610 0.813048 0.582196i \(-0.197806\pi\)
0.813048 + 0.582196i \(0.197806\pi\)
\(822\) 0 0
\(823\) 15773.8 0.668094 0.334047 0.942556i \(-0.391586\pi\)
0.334047 + 0.942556i \(0.391586\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11049.4 −0.464601 −0.232301 0.972644i \(-0.574625\pi\)
−0.232301 + 0.972644i \(0.574625\pi\)
\(828\) 0 0
\(829\) −7576.94 −0.317440 −0.158720 0.987324i \(-0.550737\pi\)
−0.158720 + 0.987324i \(0.550737\pi\)
\(830\) 0 0
\(831\) −8450.19 −0.352748
\(832\) 0 0
\(833\) 2819.97 0.117294
\(834\) 0 0
\(835\) −3836.19 −0.158990
\(836\) 0 0
\(837\) 1938.13 0.0800379
\(838\) 0 0
\(839\) −10308.2 −0.424172 −0.212086 0.977251i \(-0.568026\pi\)
−0.212086 + 0.977251i \(0.568026\pi\)
\(840\) 0 0
\(841\) −16621.4 −0.681512
\(842\) 0 0
\(843\) −9645.82 −0.394092
\(844\) 0 0
\(845\) −437.863 −0.0178260
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 419.160 0.0169441
\(850\) 0 0
\(851\) 21987.1 0.885673
\(852\) 0 0
\(853\) 27769.5 1.11467 0.557333 0.830289i \(-0.311825\pi\)
0.557333 + 0.830289i \(0.311825\pi\)
\(854\) 0 0
\(855\) −3565.07 −0.142600
\(856\) 0 0
\(857\) 23246.5 0.926585 0.463293 0.886205i \(-0.346668\pi\)
0.463293 + 0.886205i \(0.346668\pi\)
\(858\) 0 0
\(859\) −12026.9 −0.477708 −0.238854 0.971056i \(-0.576772\pi\)
−0.238854 + 0.971056i \(0.576772\pi\)
\(860\) 0 0
\(861\) 1050.43 0.0415778
\(862\) 0 0
\(863\) 23768.0 0.937511 0.468756 0.883328i \(-0.344703\pi\)
0.468756 + 0.883328i \(0.344703\pi\)
\(864\) 0 0
\(865\) 3557.78 0.139847
\(866\) 0 0
\(867\) −14531.5 −0.569223
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −19456.4 −0.756894
\(872\) 0 0
\(873\) 2893.68 0.112184
\(874\) 0 0
\(875\) −1802.75 −0.0696504
\(876\) 0 0
\(877\) −3943.18 −0.151826 −0.0759131 0.997114i \(-0.524187\pi\)
−0.0759131 + 0.997114i \(0.524187\pi\)
\(878\) 0 0
\(879\) 26869.7 1.03105
\(880\) 0 0
\(881\) −15697.2 −0.600287 −0.300144 0.953894i \(-0.597035\pi\)
−0.300144 + 0.953894i \(0.597035\pi\)
\(882\) 0 0
\(883\) −20119.7 −0.766795 −0.383398 0.923583i \(-0.625246\pi\)
−0.383398 + 0.923583i \(0.625246\pi\)
\(884\) 0 0
\(885\) 1832.14 0.0695893
\(886\) 0 0
\(887\) −3226.96 −0.122154 −0.0610771 0.998133i \(-0.519454\pi\)
−0.0610771 + 0.998133i \(0.519454\pi\)
\(888\) 0 0
\(889\) 2968.28 0.111983
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 18889.0 0.707833
\(894\) 0 0
\(895\) −10043.5 −0.375105
\(896\) 0 0
\(897\) −15583.0 −0.580046
\(898\) 0 0
\(899\) −6326.51 −0.234706
\(900\) 0 0
\(901\) 5811.52 0.214883
\(902\) 0 0
\(903\) 614.086 0.0226307
\(904\) 0 0
\(905\) 4829.00 0.177372
\(906\) 0 0
\(907\) −20474.0 −0.749536 −0.374768 0.927119i \(-0.622278\pi\)
−0.374768 + 0.927119i \(0.622278\pi\)
\(908\) 0 0
\(909\) −1583.33 −0.0577730
\(910\) 0 0
\(911\) −26699.3 −0.971008 −0.485504 0.874235i \(-0.661364\pi\)
−0.485504 + 0.874235i \(0.661364\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 2412.78 0.0871740
\(916\) 0 0
\(917\) 5584.63 0.201113
\(918\) 0 0
\(919\) −47769.1 −1.71464 −0.857322 0.514780i \(-0.827874\pi\)
−0.857322 + 0.514780i \(0.827874\pi\)
\(920\) 0 0
\(921\) −15774.7 −0.564379
\(922\) 0 0
\(923\) 1085.87 0.0387235
\(924\) 0 0
\(925\) −22361.5 −0.794857
\(926\) 0 0
\(927\) 2112.22 0.0748376
\(928\) 0 0
\(929\) 24267.5 0.857042 0.428521 0.903532i \(-0.359035\pi\)
0.428521 + 0.903532i \(0.359035\pi\)
\(930\) 0 0
\(931\) −34590.7 −1.21768
\(932\) 0 0
\(933\) 4710.69 0.165296
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −18353.9 −0.639911 −0.319955 0.947433i \(-0.603668\pi\)
−0.319955 + 0.947433i \(0.603668\pi\)
\(938\) 0 0
\(939\) −11509.6 −0.400001
\(940\) 0 0
\(941\) 3836.36 0.132903 0.0664514 0.997790i \(-0.478832\pi\)
0.0664514 + 0.997790i \(0.478832\pi\)
\(942\) 0 0
\(943\) −19149.8 −0.661298
\(944\) 0 0
\(945\) 207.194 0.00713230
\(946\) 0 0
\(947\) 15609.8 0.535638 0.267819 0.963469i \(-0.413697\pi\)
0.267819 + 0.963469i \(0.413697\pi\)
\(948\) 0 0
\(949\) 40792.0 1.39533
\(950\) 0 0
\(951\) −14584.2 −0.497294
\(952\) 0 0
\(953\) −5756.37 −0.195663 −0.0978316 0.995203i \(-0.531191\pi\)
−0.0978316 + 0.995203i \(0.531191\pi\)
\(954\) 0 0
\(955\) −13980.1 −0.473703
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3466.92 0.116739
\(960\) 0 0
\(961\) −24638.2 −0.827036
\(962\) 0 0
\(963\) 14454.7 0.483693
\(964\) 0 0
\(965\) 15409.5 0.514041
\(966\) 0 0
\(967\) −767.534 −0.0255246 −0.0127623 0.999919i \(-0.504062\pi\)
−0.0127623 + 0.999919i \(0.504062\pi\)
\(968\) 0 0
\(969\) −2544.97 −0.0843718
\(970\) 0 0
\(971\) 27639.6 0.913488 0.456744 0.889598i \(-0.349015\pi\)
0.456744 + 0.889598i \(0.349015\pi\)
\(972\) 0 0
\(973\) 630.653 0.0207788
\(974\) 0 0
\(975\) 15848.4 0.520569
\(976\) 0 0
\(977\) −8695.98 −0.284759 −0.142379 0.989812i \(-0.545475\pi\)
−0.142379 + 0.989812i \(0.545475\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −13613.3 −0.443058
\(982\) 0 0
\(983\) 26361.5 0.855341 0.427671 0.903935i \(-0.359334\pi\)
0.427671 + 0.903935i \(0.359334\pi\)
\(984\) 0 0
\(985\) 7021.05 0.227116
\(986\) 0 0
\(987\) −1097.79 −0.0354032
\(988\) 0 0
\(989\) −11195.1 −0.359943
\(990\) 0 0
\(991\) 41119.0 1.31805 0.659025 0.752121i \(-0.270969\pi\)
0.659025 + 0.752121i \(0.270969\pi\)
\(992\) 0 0
\(993\) 19007.8 0.607447
\(994\) 0 0
\(995\) 8496.83 0.270721
\(996\) 0 0
\(997\) −49821.0 −1.58259 −0.791297 0.611431i \(-0.790594\pi\)
−0.791297 + 0.611431i \(0.790594\pi\)
\(998\) 0 0
\(999\) 5492.69 0.173955
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 1452.4.a.t.1.2 6
11.7 odd 10 132.4.i.c.49.1 12
11.8 odd 10 132.4.i.c.97.1 yes 12
11.10 odd 2 1452.4.a.u.1.2 6
33.8 even 10 396.4.j.c.361.3 12
33.29 even 10 396.4.j.c.181.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
132.4.i.c.49.1 12 11.7 odd 10
132.4.i.c.97.1 yes 12 11.8 odd 10
396.4.j.c.181.3 12 33.29 even 10
396.4.j.c.361.3 12 33.8 even 10
1452.4.a.t.1.2 6 1.1 even 1 trivial
1452.4.a.u.1.2 6 11.10 odd 2