Properties

Label 1040.2.a.m.1.2
Level $1040$
Weight $2$
Character 1040.1
Self dual yes
Analytic conductor $8.304$
Analytic rank $0$
Dimension $2$
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] = [1040,2,Mod(1,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, 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("1040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.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: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 520)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.73205 q^{3} -1.00000 q^{5} -3.46410 q^{7} +4.46410 q^{9} +O(q^{10})\) \(q+2.73205 q^{3} -1.00000 q^{5} -3.46410 q^{7} +4.46410 q^{9} +4.73205 q^{11} +1.00000 q^{13} -2.73205 q^{15} +3.46410 q^{17} +3.26795 q^{19} -9.46410 q^{21} +8.19615 q^{23} +1.00000 q^{25} +4.00000 q^{27} -5.46410 q^{29} +4.73205 q^{31} +12.9282 q^{33} +3.46410 q^{35} -2.92820 q^{37} +2.73205 q^{39} -11.4641 q^{41} +2.73205 q^{43} -4.46410 q^{45} +11.4641 q^{47} +5.00000 q^{49} +9.46410 q^{51} +11.4641 q^{53} -4.73205 q^{55} +8.92820 q^{57} +1.80385 q^{59} -5.46410 q^{61} -15.4641 q^{63} -1.00000 q^{65} -15.8564 q^{67} +22.3923 q^{69} -11.6603 q^{71} +2.73205 q^{75} -16.3923 q^{77} -2.53590 q^{79} -2.46410 q^{81} -11.4641 q^{83} -3.46410 q^{85} -14.9282 q^{87} -4.92820 q^{89} -3.46410 q^{91} +12.9282 q^{93} -3.26795 q^{95} -11.8564 q^{97} +21.1244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} + 6 q^{11} + 2 q^{13} - 2 q^{15} + 10 q^{19} - 12 q^{21} + 6 q^{23} + 2 q^{25} + 8 q^{27} - 4 q^{29} + 6 q^{31} + 12 q^{33} + 8 q^{37} + 2 q^{39} - 16 q^{41} + 2 q^{43} - 2 q^{45} + 16 q^{47} + 10 q^{49} + 12 q^{51} + 16 q^{53} - 6 q^{55} + 4 q^{57} + 14 q^{59} - 4 q^{61} - 24 q^{63} - 2 q^{65} - 4 q^{67} + 24 q^{69} - 6 q^{71} + 2 q^{75} - 12 q^{77} - 12 q^{79} + 2 q^{81} - 16 q^{83} - 16 q^{87} + 4 q^{89} + 12 q^{93} - 10 q^{95} + 4 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.73205 1.57735 0.788675 0.614810i \(-0.210767\pi\)
0.788675 + 0.614810i \(0.210767\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.46410 −1.30931 −0.654654 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) 4.73205 1.42677 0.713384 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −2.73205 −0.705412
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 3.26795 0.749719 0.374859 0.927082i \(-0.377691\pi\)
0.374859 + 0.927082i \(0.377691\pi\)
\(20\) 0 0
\(21\) −9.46410 −2.06524
\(22\) 0 0
\(23\) 8.19615 1.70902 0.854508 0.519438i \(-0.173859\pi\)
0.854508 + 0.519438i \(0.173859\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −5.46410 −1.01466 −0.507329 0.861752i \(-0.669367\pi\)
−0.507329 + 0.861752i \(0.669367\pi\)
\(30\) 0 0
\(31\) 4.73205 0.849901 0.424951 0.905216i \(-0.360291\pi\)
0.424951 + 0.905216i \(0.360291\pi\)
\(32\) 0 0
\(33\) 12.9282 2.25051
\(34\) 0 0
\(35\) 3.46410 0.585540
\(36\) 0 0
\(37\) −2.92820 −0.481394 −0.240697 0.970600i \(-0.577376\pi\)
−0.240697 + 0.970600i \(0.577376\pi\)
\(38\) 0 0
\(39\) 2.73205 0.437478
\(40\) 0 0
\(41\) −11.4641 −1.79039 −0.895196 0.445673i \(-0.852964\pi\)
−0.895196 + 0.445673i \(0.852964\pi\)
\(42\) 0 0
\(43\) 2.73205 0.416634 0.208317 0.978061i \(-0.433201\pi\)
0.208317 + 0.978061i \(0.433201\pi\)
\(44\) 0 0
\(45\) −4.46410 −0.665469
\(46\) 0 0
\(47\) 11.4641 1.67221 0.836106 0.548569i \(-0.184827\pi\)
0.836106 + 0.548569i \(0.184827\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 9.46410 1.32524
\(52\) 0 0
\(53\) 11.4641 1.57472 0.787358 0.616496i \(-0.211449\pi\)
0.787358 + 0.616496i \(0.211449\pi\)
\(54\) 0 0
\(55\) −4.73205 −0.638070
\(56\) 0 0
\(57\) 8.92820 1.18257
\(58\) 0 0
\(59\) 1.80385 0.234841 0.117420 0.993082i \(-0.462537\pi\)
0.117420 + 0.993082i \(0.462537\pi\)
\(60\) 0 0
\(61\) −5.46410 −0.699607 −0.349803 0.936823i \(-0.613752\pi\)
−0.349803 + 0.936823i \(0.613752\pi\)
\(62\) 0 0
\(63\) −15.4641 −1.94829
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −15.8564 −1.93717 −0.968584 0.248686i \(-0.920001\pi\)
−0.968584 + 0.248686i \(0.920001\pi\)
\(68\) 0 0
\(69\) 22.3923 2.69572
\(70\) 0 0
\(71\) −11.6603 −1.38382 −0.691909 0.721985i \(-0.743230\pi\)
−0.691909 + 0.721985i \(0.743230\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 2.73205 0.315470
\(76\) 0 0
\(77\) −16.3923 −1.86808
\(78\) 0 0
\(79\) −2.53590 −0.285311 −0.142655 0.989772i \(-0.545564\pi\)
−0.142655 + 0.989772i \(0.545564\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 0 0
\(83\) −11.4641 −1.25835 −0.629174 0.777264i \(-0.716607\pi\)
−0.629174 + 0.777264i \(0.716607\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) 0 0
\(87\) −14.9282 −1.60047
\(88\) 0 0
\(89\) −4.92820 −0.522388 −0.261194 0.965286i \(-0.584116\pi\)
−0.261194 + 0.965286i \(0.584116\pi\)
\(90\) 0 0
\(91\) −3.46410 −0.363137
\(92\) 0 0
\(93\) 12.9282 1.34059
\(94\) 0 0
\(95\) −3.26795 −0.335285
\(96\) 0 0
\(97\) −11.8564 −1.20384 −0.601918 0.798558i \(-0.705597\pi\)
−0.601918 + 0.798558i \(0.705597\pi\)
\(98\) 0 0
\(99\) 21.1244 2.12308
\(100\) 0 0
\(101\) −3.07180 −0.305655 −0.152828 0.988253i \(-0.548838\pi\)
−0.152828 + 0.988253i \(0.548838\pi\)
\(102\) 0 0
\(103\) 6.73205 0.663329 0.331664 0.943397i \(-0.392390\pi\)
0.331664 + 0.943397i \(0.392390\pi\)
\(104\) 0 0
\(105\) 9.46410 0.923602
\(106\) 0 0
\(107\) 3.80385 0.367732 0.183866 0.982951i \(-0.441139\pi\)
0.183866 + 0.982951i \(0.441139\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) −6.39230 −0.601337 −0.300669 0.953729i \(-0.597210\pi\)
−0.300669 + 0.953729i \(0.597210\pi\)
\(114\) 0 0
\(115\) −8.19615 −0.764295
\(116\) 0 0
\(117\) 4.46410 0.412706
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) 11.3923 1.03566
\(122\) 0 0
\(123\) −31.3205 −2.82408
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 7.12436 0.632184 0.316092 0.948728i \(-0.397629\pi\)
0.316092 + 0.948728i \(0.397629\pi\)
\(128\) 0 0
\(129\) 7.46410 0.657178
\(130\) 0 0
\(131\) 2.92820 0.255838 0.127919 0.991785i \(-0.459170\pi\)
0.127919 + 0.991785i \(0.459170\pi\)
\(132\) 0 0
\(133\) −11.3205 −0.981613
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) 0.928203 0.0793018 0.0396509 0.999214i \(-0.487375\pi\)
0.0396509 + 0.999214i \(0.487375\pi\)
\(138\) 0 0
\(139\) 12.3923 1.05110 0.525551 0.850762i \(-0.323859\pi\)
0.525551 + 0.850762i \(0.323859\pi\)
\(140\) 0 0
\(141\) 31.3205 2.63766
\(142\) 0 0
\(143\) 4.73205 0.395714
\(144\) 0 0
\(145\) 5.46410 0.453769
\(146\) 0 0
\(147\) 13.6603 1.12668
\(148\) 0 0
\(149\) 3.85641 0.315929 0.157965 0.987445i \(-0.449507\pi\)
0.157965 + 0.987445i \(0.449507\pi\)
\(150\) 0 0
\(151\) 8.73205 0.710604 0.355302 0.934752i \(-0.384378\pi\)
0.355302 + 0.934752i \(0.384378\pi\)
\(152\) 0 0
\(153\) 15.4641 1.25020
\(154\) 0 0
\(155\) −4.73205 −0.380087
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 31.3205 2.48388
\(160\) 0 0
\(161\) −28.3923 −2.23763
\(162\) 0 0
\(163\) 15.8564 1.24197 0.620985 0.783823i \(-0.286733\pi\)
0.620985 + 0.783823i \(0.286733\pi\)
\(164\) 0 0
\(165\) −12.9282 −1.00646
\(166\) 0 0
\(167\) −14.3923 −1.11371 −0.556855 0.830610i \(-0.687992\pi\)
−0.556855 + 0.830610i \(0.687992\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 14.5885 1.11561
\(172\) 0 0
\(173\) −22.3923 −1.70246 −0.851228 0.524797i \(-0.824141\pi\)
−0.851228 + 0.524797i \(0.824141\pi\)
\(174\) 0 0
\(175\) −3.46410 −0.261861
\(176\) 0 0
\(177\) 4.92820 0.370426
\(178\) 0 0
\(179\) −13.8564 −1.03568 −0.517838 0.855479i \(-0.673263\pi\)
−0.517838 + 0.855479i \(0.673263\pi\)
\(180\) 0 0
\(181\) 11.3205 0.841447 0.420723 0.907189i \(-0.361776\pi\)
0.420723 + 0.907189i \(0.361776\pi\)
\(182\) 0 0
\(183\) −14.9282 −1.10352
\(184\) 0 0
\(185\) 2.92820 0.215286
\(186\) 0 0
\(187\) 16.3923 1.19872
\(188\) 0 0
\(189\) −13.8564 −1.00791
\(190\) 0 0
\(191\) −5.07180 −0.366982 −0.183491 0.983021i \(-0.558740\pi\)
−0.183491 + 0.983021i \(0.558740\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) −2.73205 −0.195646
\(196\) 0 0
\(197\) 0.928203 0.0661317 0.0330659 0.999453i \(-0.489473\pi\)
0.0330659 + 0.999453i \(0.489473\pi\)
\(198\) 0 0
\(199\) −14.9282 −1.05823 −0.529116 0.848549i \(-0.677476\pi\)
−0.529116 + 0.848549i \(0.677476\pi\)
\(200\) 0 0
\(201\) −43.3205 −3.05559
\(202\) 0 0
\(203\) 18.9282 1.32850
\(204\) 0 0
\(205\) 11.4641 0.800688
\(206\) 0 0
\(207\) 36.5885 2.54307
\(208\) 0 0
\(209\) 15.4641 1.06967
\(210\) 0 0
\(211\) −24.7846 −1.70624 −0.853121 0.521712i \(-0.825293\pi\)
−0.853121 + 0.521712i \(0.825293\pi\)
\(212\) 0 0
\(213\) −31.8564 −2.18277
\(214\) 0 0
\(215\) −2.73205 −0.186324
\(216\) 0 0
\(217\) −16.3923 −1.11278
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.46410 0.233021
\(222\) 0 0
\(223\) 2.39230 0.160201 0.0801003 0.996787i \(-0.474476\pi\)
0.0801003 + 0.996787i \(0.474476\pi\)
\(224\) 0 0
\(225\) 4.46410 0.297607
\(226\) 0 0
\(227\) 28.9282 1.92003 0.960016 0.279945i \(-0.0903161\pi\)
0.960016 + 0.279945i \(0.0903161\pi\)
\(228\) 0 0
\(229\) −0.535898 −0.0354132 −0.0177066 0.999843i \(-0.505636\pi\)
−0.0177066 + 0.999843i \(0.505636\pi\)
\(230\) 0 0
\(231\) −44.7846 −2.94661
\(232\) 0 0
\(233\) 7.85641 0.514690 0.257345 0.966320i \(-0.417152\pi\)
0.257345 + 0.966320i \(0.417152\pi\)
\(234\) 0 0
\(235\) −11.4641 −0.747836
\(236\) 0 0
\(237\) −6.92820 −0.450035
\(238\) 0 0
\(239\) −25.5167 −1.65054 −0.825268 0.564742i \(-0.808976\pi\)
−0.825268 + 0.564742i \(0.808976\pi\)
\(240\) 0 0
\(241\) 2.39230 0.154102 0.0770510 0.997027i \(-0.475450\pi\)
0.0770510 + 0.997027i \(0.475450\pi\)
\(242\) 0 0
\(243\) −18.7321 −1.20166
\(244\) 0 0
\(245\) −5.00000 −0.319438
\(246\) 0 0
\(247\) 3.26795 0.207935
\(248\) 0 0
\(249\) −31.3205 −1.98486
\(250\) 0 0
\(251\) 13.4641 0.849847 0.424923 0.905229i \(-0.360301\pi\)
0.424923 + 0.905229i \(0.360301\pi\)
\(252\) 0 0
\(253\) 38.7846 2.43837
\(254\) 0 0
\(255\) −9.46410 −0.592665
\(256\) 0 0
\(257\) −15.8564 −0.989095 −0.494548 0.869150i \(-0.664666\pi\)
−0.494548 + 0.869150i \(0.664666\pi\)
\(258\) 0 0
\(259\) 10.1436 0.630292
\(260\) 0 0
\(261\) −24.3923 −1.50985
\(262\) 0 0
\(263\) −24.5885 −1.51619 −0.758095 0.652145i \(-0.773869\pi\)
−0.758095 + 0.652145i \(0.773869\pi\)
\(264\) 0 0
\(265\) −11.4641 −0.704234
\(266\) 0 0
\(267\) −13.4641 −0.823990
\(268\) 0 0
\(269\) 15.8564 0.966782 0.483391 0.875405i \(-0.339405\pi\)
0.483391 + 0.875405i \(0.339405\pi\)
\(270\) 0 0
\(271\) −18.1962 −1.10534 −0.552669 0.833401i \(-0.686391\pi\)
−0.552669 + 0.833401i \(0.686391\pi\)
\(272\) 0 0
\(273\) −9.46410 −0.572793
\(274\) 0 0
\(275\) 4.73205 0.285353
\(276\) 0 0
\(277\) 26.3923 1.58576 0.792880 0.609378i \(-0.208581\pi\)
0.792880 + 0.609378i \(0.208581\pi\)
\(278\) 0 0
\(279\) 21.1244 1.26468
\(280\) 0 0
\(281\) −30.3923 −1.81305 −0.906526 0.422149i \(-0.861276\pi\)
−0.906526 + 0.422149i \(0.861276\pi\)
\(282\) 0 0
\(283\) 15.8038 0.939441 0.469721 0.882815i \(-0.344355\pi\)
0.469721 + 0.882815i \(0.344355\pi\)
\(284\) 0 0
\(285\) −8.92820 −0.528861
\(286\) 0 0
\(287\) 39.7128 2.34417
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) −32.3923 −1.89887
\(292\) 0 0
\(293\) −25.8564 −1.51055 −0.755274 0.655410i \(-0.772496\pi\)
−0.755274 + 0.655410i \(0.772496\pi\)
\(294\) 0 0
\(295\) −1.80385 −0.105024
\(296\) 0 0
\(297\) 18.9282 1.09833
\(298\) 0 0
\(299\) 8.19615 0.473996
\(300\) 0 0
\(301\) −9.46410 −0.545502
\(302\) 0 0
\(303\) −8.39230 −0.482125
\(304\) 0 0
\(305\) 5.46410 0.312874
\(306\) 0 0
\(307\) 8.53590 0.487169 0.243585 0.969880i \(-0.421677\pi\)
0.243585 + 0.969880i \(0.421677\pi\)
\(308\) 0 0
\(309\) 18.3923 1.04630
\(310\) 0 0
\(311\) 13.4641 0.763479 0.381740 0.924270i \(-0.375325\pi\)
0.381740 + 0.924270i \(0.375325\pi\)
\(312\) 0 0
\(313\) 20.2487 1.14452 0.572262 0.820071i \(-0.306066\pi\)
0.572262 + 0.820071i \(0.306066\pi\)
\(314\) 0 0
\(315\) 15.4641 0.871303
\(316\) 0 0
\(317\) 12.7846 0.718055 0.359028 0.933327i \(-0.383108\pi\)
0.359028 + 0.933327i \(0.383108\pi\)
\(318\) 0 0
\(319\) −25.8564 −1.44768
\(320\) 0 0
\(321\) 10.3923 0.580042
\(322\) 0 0
\(323\) 11.3205 0.629890
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) −16.3923 −0.906497
\(328\) 0 0
\(329\) −39.7128 −2.18944
\(330\) 0 0
\(331\) 22.9808 1.26314 0.631568 0.775320i \(-0.282411\pi\)
0.631568 + 0.775320i \(0.282411\pi\)
\(332\) 0 0
\(333\) −13.0718 −0.716330
\(334\) 0 0
\(335\) 15.8564 0.866328
\(336\) 0 0
\(337\) 26.3923 1.43768 0.718840 0.695175i \(-0.244673\pi\)
0.718840 + 0.695175i \(0.244673\pi\)
\(338\) 0 0
\(339\) −17.4641 −0.948520
\(340\) 0 0
\(341\) 22.3923 1.21261
\(342\) 0 0
\(343\) 6.92820 0.374088
\(344\) 0 0
\(345\) −22.3923 −1.20556
\(346\) 0 0
\(347\) −29.6603 −1.59225 −0.796123 0.605135i \(-0.793119\pi\)
−0.796123 + 0.605135i \(0.793119\pi\)
\(348\) 0 0
\(349\) −5.32051 −0.284800 −0.142400 0.989809i \(-0.545482\pi\)
−0.142400 + 0.989809i \(0.545482\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) 1.85641 0.0988065 0.0494033 0.998779i \(-0.484268\pi\)
0.0494033 + 0.998779i \(0.484268\pi\)
\(354\) 0 0
\(355\) 11.6603 0.618862
\(356\) 0 0
\(357\) −32.7846 −1.73515
\(358\) 0 0
\(359\) −25.8038 −1.36187 −0.680937 0.732342i \(-0.738427\pi\)
−0.680937 + 0.732342i \(0.738427\pi\)
\(360\) 0 0
\(361\) −8.32051 −0.437921
\(362\) 0 0
\(363\) 31.1244 1.63361
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −30.0526 −1.56873 −0.784365 0.620299i \(-0.787011\pi\)
−0.784365 + 0.620299i \(0.787011\pi\)
\(368\) 0 0
\(369\) −51.1769 −2.66416
\(370\) 0 0
\(371\) −39.7128 −2.06179
\(372\) 0 0
\(373\) −18.0000 −0.932005 −0.466002 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) 0 0
\(375\) −2.73205 −0.141082
\(376\) 0 0
\(377\) −5.46410 −0.281416
\(378\) 0 0
\(379\) 27.6603 1.42081 0.710406 0.703792i \(-0.248511\pi\)
0.710406 + 0.703792i \(0.248511\pi\)
\(380\) 0 0
\(381\) 19.4641 0.997176
\(382\) 0 0
\(383\) 9.60770 0.490930 0.245465 0.969405i \(-0.421059\pi\)
0.245465 + 0.969405i \(0.421059\pi\)
\(384\) 0 0
\(385\) 16.3923 0.835429
\(386\) 0 0
\(387\) 12.1962 0.619965
\(388\) 0 0
\(389\) −29.7128 −1.50650 −0.753250 0.657735i \(-0.771515\pi\)
−0.753250 + 0.657735i \(0.771515\pi\)
\(390\) 0 0
\(391\) 28.3923 1.43586
\(392\) 0 0
\(393\) 8.00000 0.403547
\(394\) 0 0
\(395\) 2.53590 0.127595
\(396\) 0 0
\(397\) 16.0000 0.803017 0.401508 0.915855i \(-0.368486\pi\)
0.401508 + 0.915855i \(0.368486\pi\)
\(398\) 0 0
\(399\) −30.9282 −1.54835
\(400\) 0 0
\(401\) −2.78461 −0.139057 −0.0695284 0.997580i \(-0.522149\pi\)
−0.0695284 + 0.997580i \(0.522149\pi\)
\(402\) 0 0
\(403\) 4.73205 0.235720
\(404\) 0 0
\(405\) 2.46410 0.122442
\(406\) 0 0
\(407\) −13.8564 −0.686837
\(408\) 0 0
\(409\) −2.67949 −0.132492 −0.0662462 0.997803i \(-0.521102\pi\)
−0.0662462 + 0.997803i \(0.521102\pi\)
\(410\) 0 0
\(411\) 2.53590 0.125087
\(412\) 0 0
\(413\) −6.24871 −0.307479
\(414\) 0 0
\(415\) 11.4641 0.562751
\(416\) 0 0
\(417\) 33.8564 1.65796
\(418\) 0 0
\(419\) −11.3205 −0.553043 −0.276522 0.961008i \(-0.589182\pi\)
−0.276522 + 0.961008i \(0.589182\pi\)
\(420\) 0 0
\(421\) −22.7846 −1.11045 −0.555227 0.831699i \(-0.687369\pi\)
−0.555227 + 0.831699i \(0.687369\pi\)
\(422\) 0 0
\(423\) 51.1769 2.48831
\(424\) 0 0
\(425\) 3.46410 0.168034
\(426\) 0 0
\(427\) 18.9282 0.916000
\(428\) 0 0
\(429\) 12.9282 0.624180
\(430\) 0 0
\(431\) 32.4449 1.56281 0.781407 0.624022i \(-0.214502\pi\)
0.781407 + 0.624022i \(0.214502\pi\)
\(432\) 0 0
\(433\) −14.7846 −0.710503 −0.355251 0.934771i \(-0.615605\pi\)
−0.355251 + 0.934771i \(0.615605\pi\)
\(434\) 0 0
\(435\) 14.9282 0.715753
\(436\) 0 0
\(437\) 26.7846 1.28128
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 22.3205 1.06288
\(442\) 0 0
\(443\) −19.1244 −0.908626 −0.454313 0.890842i \(-0.650115\pi\)
−0.454313 + 0.890842i \(0.650115\pi\)
\(444\) 0 0
\(445\) 4.92820 0.233619
\(446\) 0 0
\(447\) 10.5359 0.498331
\(448\) 0 0
\(449\) 25.3205 1.19495 0.597474 0.801888i \(-0.296171\pi\)
0.597474 + 0.801888i \(0.296171\pi\)
\(450\) 0 0
\(451\) −54.2487 −2.55447
\(452\) 0 0
\(453\) 23.8564 1.12087
\(454\) 0 0
\(455\) 3.46410 0.162400
\(456\) 0 0
\(457\) −22.7846 −1.06582 −0.532910 0.846172i \(-0.678901\pi\)
−0.532910 + 0.846172i \(0.678901\pi\)
\(458\) 0 0
\(459\) 13.8564 0.646762
\(460\) 0 0
\(461\) 11.4641 0.533936 0.266968 0.963705i \(-0.413978\pi\)
0.266968 + 0.963705i \(0.413978\pi\)
\(462\) 0 0
\(463\) −10.7846 −0.501203 −0.250602 0.968090i \(-0.580628\pi\)
−0.250602 + 0.968090i \(0.580628\pi\)
\(464\) 0 0
\(465\) −12.9282 −0.599531
\(466\) 0 0
\(467\) −23.1244 −1.07007 −0.535034 0.844831i \(-0.679701\pi\)
−0.535034 + 0.844831i \(0.679701\pi\)
\(468\) 0 0
\(469\) 54.9282 2.53635
\(470\) 0 0
\(471\) 38.2487 1.76241
\(472\) 0 0
\(473\) 12.9282 0.594439
\(474\) 0 0
\(475\) 3.26795 0.149944
\(476\) 0 0
\(477\) 51.1769 2.34323
\(478\) 0 0
\(479\) 22.1962 1.01417 0.507084 0.861897i \(-0.330723\pi\)
0.507084 + 0.861897i \(0.330723\pi\)
\(480\) 0 0
\(481\) −2.92820 −0.133515
\(482\) 0 0
\(483\) −77.5692 −3.52952
\(484\) 0 0
\(485\) 11.8564 0.538372
\(486\) 0 0
\(487\) −23.0718 −1.04548 −0.522741 0.852491i \(-0.675091\pi\)
−0.522741 + 0.852491i \(0.675091\pi\)
\(488\) 0 0
\(489\) 43.3205 1.95902
\(490\) 0 0
\(491\) −13.4641 −0.607626 −0.303813 0.952732i \(-0.598260\pi\)
−0.303813 + 0.952732i \(0.598260\pi\)
\(492\) 0 0
\(493\) −18.9282 −0.852483
\(494\) 0 0
\(495\) −21.1244 −0.949469
\(496\) 0 0
\(497\) 40.3923 1.81184
\(498\) 0 0
\(499\) 16.3397 0.731467 0.365734 0.930720i \(-0.380818\pi\)
0.365734 + 0.930720i \(0.380818\pi\)
\(500\) 0 0
\(501\) −39.3205 −1.75671
\(502\) 0 0
\(503\) 25.6603 1.14413 0.572067 0.820207i \(-0.306142\pi\)
0.572067 + 0.820207i \(0.306142\pi\)
\(504\) 0 0
\(505\) 3.07180 0.136693
\(506\) 0 0
\(507\) 2.73205 0.121335
\(508\) 0 0
\(509\) 6.39230 0.283334 0.141667 0.989914i \(-0.454754\pi\)
0.141667 + 0.989914i \(0.454754\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 13.0718 0.577134
\(514\) 0 0
\(515\) −6.73205 −0.296650
\(516\) 0 0
\(517\) 54.2487 2.38586
\(518\) 0 0
\(519\) −61.1769 −2.68537
\(520\) 0 0
\(521\) −18.2487 −0.799491 −0.399745 0.916626i \(-0.630901\pi\)
−0.399745 + 0.916626i \(0.630901\pi\)
\(522\) 0 0
\(523\) −10.3397 −0.452126 −0.226063 0.974113i \(-0.572585\pi\)
−0.226063 + 0.974113i \(0.572585\pi\)
\(524\) 0 0
\(525\) −9.46410 −0.413047
\(526\) 0 0
\(527\) 16.3923 0.714060
\(528\) 0 0
\(529\) 44.1769 1.92074
\(530\) 0 0
\(531\) 8.05256 0.349451
\(532\) 0 0
\(533\) −11.4641 −0.496565
\(534\) 0 0
\(535\) −3.80385 −0.164455
\(536\) 0 0
\(537\) −37.8564 −1.63362
\(538\) 0 0
\(539\) 23.6603 1.01912
\(540\) 0 0
\(541\) 4.53590 0.195014 0.0975068 0.995235i \(-0.468913\pi\)
0.0975068 + 0.995235i \(0.468913\pi\)
\(542\) 0 0
\(543\) 30.9282 1.32726
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) −6.33975 −0.271068 −0.135534 0.990773i \(-0.543275\pi\)
−0.135534 + 0.990773i \(0.543275\pi\)
\(548\) 0 0
\(549\) −24.3923 −1.04104
\(550\) 0 0
\(551\) −17.8564 −0.760708
\(552\) 0 0
\(553\) 8.78461 0.373560
\(554\) 0 0
\(555\) 8.00000 0.339581
\(556\) 0 0
\(557\) 34.9282 1.47996 0.739978 0.672631i \(-0.234836\pi\)
0.739978 + 0.672631i \(0.234836\pi\)
\(558\) 0 0
\(559\) 2.73205 0.115553
\(560\) 0 0
\(561\) 44.7846 1.89081
\(562\) 0 0
\(563\) 18.7321 0.789462 0.394731 0.918797i \(-0.370838\pi\)
0.394731 + 0.918797i \(0.370838\pi\)
\(564\) 0 0
\(565\) 6.39230 0.268926
\(566\) 0 0
\(567\) 8.53590 0.358474
\(568\) 0 0
\(569\) −16.3923 −0.687201 −0.343601 0.939116i \(-0.611647\pi\)
−0.343601 + 0.939116i \(0.611647\pi\)
\(570\) 0 0
\(571\) 26.2487 1.09847 0.549237 0.835667i \(-0.314918\pi\)
0.549237 + 0.835667i \(0.314918\pi\)
\(572\) 0 0
\(573\) −13.8564 −0.578860
\(574\) 0 0
\(575\) 8.19615 0.341803
\(576\) 0 0
\(577\) 16.0000 0.666089 0.333044 0.942911i \(-0.391924\pi\)
0.333044 + 0.942911i \(0.391924\pi\)
\(578\) 0 0
\(579\) 38.2487 1.58956
\(580\) 0 0
\(581\) 39.7128 1.64757
\(582\) 0 0
\(583\) 54.2487 2.24675
\(584\) 0 0
\(585\) −4.46410 −0.184568
\(586\) 0 0
\(587\) 37.7128 1.55657 0.778287 0.627908i \(-0.216089\pi\)
0.778287 + 0.627908i \(0.216089\pi\)
\(588\) 0 0
\(589\) 15.4641 0.637187
\(590\) 0 0
\(591\) 2.53590 0.104313
\(592\) 0 0
\(593\) 4.92820 0.202377 0.101189 0.994867i \(-0.467735\pi\)
0.101189 + 0.994867i \(0.467735\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) 0 0
\(597\) −40.7846 −1.66920
\(598\) 0 0
\(599\) 0.392305 0.0160291 0.00801457 0.999968i \(-0.497449\pi\)
0.00801457 + 0.999968i \(0.497449\pi\)
\(600\) 0 0
\(601\) 12.1436 0.495348 0.247674 0.968843i \(-0.420334\pi\)
0.247674 + 0.968843i \(0.420334\pi\)
\(602\) 0 0
\(603\) −70.7846 −2.88257
\(604\) 0 0
\(605\) −11.3923 −0.463163
\(606\) 0 0
\(607\) −6.73205 −0.273246 −0.136623 0.990623i \(-0.543625\pi\)
−0.136623 + 0.990623i \(0.543625\pi\)
\(608\) 0 0
\(609\) 51.7128 2.09551
\(610\) 0 0
\(611\) 11.4641 0.463788
\(612\) 0 0
\(613\) −43.8564 −1.77134 −0.885672 0.464312i \(-0.846302\pi\)
−0.885672 + 0.464312i \(0.846302\pi\)
\(614\) 0 0
\(615\) 31.3205 1.26296
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 25.8038 1.03714 0.518572 0.855034i \(-0.326464\pi\)
0.518572 + 0.855034i \(0.326464\pi\)
\(620\) 0 0
\(621\) 32.7846 1.31560
\(622\) 0 0
\(623\) 17.0718 0.683967
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 42.2487 1.68725
\(628\) 0 0
\(629\) −10.1436 −0.404452
\(630\) 0 0
\(631\) −6.87564 −0.273715 −0.136858 0.990591i \(-0.543700\pi\)
−0.136858 + 0.990591i \(0.543700\pi\)
\(632\) 0 0
\(633\) −67.7128 −2.69134
\(634\) 0 0
\(635\) −7.12436 −0.282721
\(636\) 0 0
\(637\) 5.00000 0.198107
\(638\) 0 0
\(639\) −52.0526 −2.05917
\(640\) 0 0
\(641\) −24.9282 −0.984605 −0.492302 0.870424i \(-0.663845\pi\)
−0.492302 + 0.870424i \(0.663845\pi\)
\(642\) 0 0
\(643\) 23.4641 0.925334 0.462667 0.886532i \(-0.346893\pi\)
0.462667 + 0.886532i \(0.346893\pi\)
\(644\) 0 0
\(645\) −7.46410 −0.293899
\(646\) 0 0
\(647\) −12.5885 −0.494903 −0.247452 0.968900i \(-0.579593\pi\)
−0.247452 + 0.968900i \(0.579593\pi\)
\(648\) 0 0
\(649\) 8.53590 0.335063
\(650\) 0 0
\(651\) −44.7846 −1.75525
\(652\) 0 0
\(653\) 31.8564 1.24664 0.623319 0.781968i \(-0.285784\pi\)
0.623319 + 0.781968i \(0.285784\pi\)
\(654\) 0 0
\(655\) −2.92820 −0.114414
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −34.5359 −1.34533 −0.672664 0.739948i \(-0.734850\pi\)
−0.672664 + 0.739948i \(0.734850\pi\)
\(660\) 0 0
\(661\) −30.7846 −1.19738 −0.598691 0.800980i \(-0.704312\pi\)
−0.598691 + 0.800980i \(0.704312\pi\)
\(662\) 0 0
\(663\) 9.46410 0.367555
\(664\) 0 0
\(665\) 11.3205 0.438990
\(666\) 0 0
\(667\) −44.7846 −1.73407
\(668\) 0 0
\(669\) 6.53590 0.252692
\(670\) 0 0
\(671\) −25.8564 −0.998176
\(672\) 0 0
\(673\) −15.4641 −0.596097 −0.298049 0.954551i \(-0.596336\pi\)
−0.298049 + 0.954551i \(0.596336\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) −42.3923 −1.62927 −0.814634 0.579975i \(-0.803062\pi\)
−0.814634 + 0.579975i \(0.803062\pi\)
\(678\) 0 0
\(679\) 41.0718 1.57619
\(680\) 0 0
\(681\) 79.0333 3.02856
\(682\) 0 0
\(683\) 33.7128 1.28998 0.644992 0.764189i \(-0.276860\pi\)
0.644992 + 0.764189i \(0.276860\pi\)
\(684\) 0 0
\(685\) −0.928203 −0.0354648
\(686\) 0 0
\(687\) −1.46410 −0.0558590
\(688\) 0 0
\(689\) 11.4641 0.436747
\(690\) 0 0
\(691\) 16.3397 0.621593 0.310797 0.950476i \(-0.399404\pi\)
0.310797 + 0.950476i \(0.399404\pi\)
\(692\) 0 0
\(693\) −73.1769 −2.77976
\(694\) 0 0
\(695\) −12.3923 −0.470067
\(696\) 0 0
\(697\) −39.7128 −1.50423
\(698\) 0 0
\(699\) 21.4641 0.811847
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) −9.56922 −0.360910
\(704\) 0 0
\(705\) −31.3205 −1.17960
\(706\) 0 0
\(707\) 10.6410 0.400197
\(708\) 0 0
\(709\) −46.3923 −1.74230 −0.871150 0.491017i \(-0.836625\pi\)
−0.871150 + 0.491017i \(0.836625\pi\)
\(710\) 0 0
\(711\) −11.3205 −0.424552
\(712\) 0 0
\(713\) 38.7846 1.45250
\(714\) 0 0
\(715\) −4.73205 −0.176969
\(716\) 0 0
\(717\) −69.7128 −2.60347
\(718\) 0 0
\(719\) −25.0718 −0.935020 −0.467510 0.883988i \(-0.654849\pi\)
−0.467510 + 0.883988i \(0.654849\pi\)
\(720\) 0 0
\(721\) −23.3205 −0.868501
\(722\) 0 0
\(723\) 6.53590 0.243073
\(724\) 0 0
\(725\) −5.46410 −0.202932
\(726\) 0 0
\(727\) −27.9090 −1.03509 −0.517543 0.855657i \(-0.673153\pi\)
−0.517543 + 0.855657i \(0.673153\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) 0 0
\(731\) 9.46410 0.350042
\(732\) 0 0
\(733\) 27.8564 1.02890 0.514450 0.857520i \(-0.327996\pi\)
0.514450 + 0.857520i \(0.327996\pi\)
\(734\) 0 0
\(735\) −13.6603 −0.503866
\(736\) 0 0
\(737\) −75.0333 −2.76389
\(738\) 0 0
\(739\) −6.58846 −0.242360 −0.121180 0.992631i \(-0.538668\pi\)
−0.121180 + 0.992631i \(0.538668\pi\)
\(740\) 0 0
\(741\) 8.92820 0.327986
\(742\) 0 0
\(743\) −3.46410 −0.127086 −0.0635428 0.997979i \(-0.520240\pi\)
−0.0635428 + 0.997979i \(0.520240\pi\)
\(744\) 0 0
\(745\) −3.85641 −0.141288
\(746\) 0 0
\(747\) −51.1769 −1.87247
\(748\) 0 0
\(749\) −13.1769 −0.481474
\(750\) 0 0
\(751\) 17.4641 0.637274 0.318637 0.947877i \(-0.396775\pi\)
0.318637 + 0.947877i \(0.396775\pi\)
\(752\) 0 0
\(753\) 36.7846 1.34051
\(754\) 0 0
\(755\) −8.73205 −0.317792
\(756\) 0 0
\(757\) 12.5359 0.455625 0.227812 0.973705i \(-0.426843\pi\)
0.227812 + 0.973705i \(0.426843\pi\)
\(758\) 0 0
\(759\) 105.962 3.84616
\(760\) 0 0
\(761\) 45.7128 1.65709 0.828544 0.559924i \(-0.189170\pi\)
0.828544 + 0.559924i \(0.189170\pi\)
\(762\) 0 0
\(763\) 20.7846 0.752453
\(764\) 0 0
\(765\) −15.4641 −0.559106
\(766\) 0 0
\(767\) 1.80385 0.0651332
\(768\) 0 0
\(769\) −14.7846 −0.533147 −0.266573 0.963815i \(-0.585891\pi\)
−0.266573 + 0.963815i \(0.585891\pi\)
\(770\) 0 0
\(771\) −43.3205 −1.56015
\(772\) 0 0
\(773\) 10.1436 0.364840 0.182420 0.983221i \(-0.441607\pi\)
0.182420 + 0.983221i \(0.441607\pi\)
\(774\) 0 0
\(775\) 4.73205 0.169980
\(776\) 0 0
\(777\) 27.7128 0.994192
\(778\) 0 0
\(779\) −37.4641 −1.34229
\(780\) 0 0
\(781\) −55.1769 −1.97439
\(782\) 0 0
\(783\) −21.8564 −0.781084
\(784\) 0 0
\(785\) −14.0000 −0.499681
\(786\) 0 0
\(787\) 19.4641 0.693820 0.346910 0.937898i \(-0.387231\pi\)
0.346910 + 0.937898i \(0.387231\pi\)
\(788\) 0 0
\(789\) −67.1769 −2.39156
\(790\) 0 0
\(791\) 22.1436 0.787336
\(792\) 0 0
\(793\) −5.46410 −0.194036
\(794\) 0 0
\(795\) −31.3205 −1.11082
\(796\) 0 0
\(797\) −4.92820 −0.174566 −0.0872830 0.996184i \(-0.527818\pi\)
−0.0872830 + 0.996184i \(0.527818\pi\)
\(798\) 0 0
\(799\) 39.7128 1.40494
\(800\) 0 0
\(801\) −22.0000 −0.777332
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 28.3923 1.00070
\(806\) 0 0
\(807\) 43.3205 1.52495
\(808\) 0 0
\(809\) 39.0333 1.37234 0.686169 0.727442i \(-0.259291\pi\)
0.686169 + 0.727442i \(0.259291\pi\)
\(810\) 0 0
\(811\) −17.8038 −0.625178 −0.312589 0.949889i \(-0.601196\pi\)
−0.312589 + 0.949889i \(0.601196\pi\)
\(812\) 0 0
\(813\) −49.7128 −1.74350
\(814\) 0 0
\(815\) −15.8564 −0.555426
\(816\) 0 0
\(817\) 8.92820 0.312358
\(818\) 0 0
\(819\) −15.4641 −0.540359
\(820\) 0 0
\(821\) 0.928203 0.0323945 0.0161973 0.999869i \(-0.494844\pi\)
0.0161973 + 0.999869i \(0.494844\pi\)
\(822\) 0 0
\(823\) 41.6603 1.45219 0.726093 0.687597i \(-0.241334\pi\)
0.726093 + 0.687597i \(0.241334\pi\)
\(824\) 0 0
\(825\) 12.9282 0.450102
\(826\) 0 0
\(827\) −29.6077 −1.02956 −0.514780 0.857322i \(-0.672126\pi\)
−0.514780 + 0.857322i \(0.672126\pi\)
\(828\) 0 0
\(829\) −32.3923 −1.12503 −0.562516 0.826787i \(-0.690166\pi\)
−0.562516 + 0.826787i \(0.690166\pi\)
\(830\) 0 0
\(831\) 72.1051 2.50130
\(832\) 0 0
\(833\) 17.3205 0.600120
\(834\) 0 0
\(835\) 14.3923 0.498066
\(836\) 0 0
\(837\) 18.9282 0.654254
\(838\) 0 0
\(839\) −12.3397 −0.426015 −0.213008 0.977051i \(-0.568326\pi\)
−0.213008 + 0.977051i \(0.568326\pi\)
\(840\) 0 0
\(841\) 0.856406 0.0295313
\(842\) 0 0
\(843\) −83.0333 −2.85982
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −39.4641 −1.35600
\(848\) 0 0
\(849\) 43.1769 1.48183
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 0 0
\(853\) 18.6410 0.638256 0.319128 0.947712i \(-0.396610\pi\)
0.319128 + 0.947712i \(0.396610\pi\)
\(854\) 0 0
\(855\) −14.5885 −0.498915
\(856\) 0 0
\(857\) −26.0000 −0.888143 −0.444072 0.895991i \(-0.646466\pi\)
−0.444072 + 0.895991i \(0.646466\pi\)
\(858\) 0 0
\(859\) 27.3205 0.932164 0.466082 0.884742i \(-0.345665\pi\)
0.466082 + 0.884742i \(0.345665\pi\)
\(860\) 0 0
\(861\) 108.497 3.69758
\(862\) 0 0
\(863\) 43.1769 1.46976 0.734880 0.678198i \(-0.237239\pi\)
0.734880 + 0.678198i \(0.237239\pi\)
\(864\) 0 0
\(865\) 22.3923 0.761361
\(866\) 0 0
\(867\) −13.6603 −0.463927
\(868\) 0 0
\(869\) −12.0000 −0.407072
\(870\) 0 0
\(871\) −15.8564 −0.537274
\(872\) 0 0
\(873\) −52.9282 −1.79135
\(874\) 0 0
\(875\) 3.46410 0.117108
\(876\) 0 0
\(877\) 13.7128 0.463049 0.231524 0.972829i \(-0.425629\pi\)
0.231524 + 0.972829i \(0.425629\pi\)
\(878\) 0 0
\(879\) −70.6410 −2.38266
\(880\) 0 0
\(881\) 20.3923 0.687034 0.343517 0.939146i \(-0.388382\pi\)
0.343517 + 0.939146i \(0.388382\pi\)
\(882\) 0 0
\(883\) 32.5885 1.09669 0.548344 0.836253i \(-0.315258\pi\)
0.548344 + 0.836253i \(0.315258\pi\)
\(884\) 0 0
\(885\) −4.92820 −0.165660
\(886\) 0 0
\(887\) −22.0526 −0.740452 −0.370226 0.928942i \(-0.620720\pi\)
−0.370226 + 0.928942i \(0.620720\pi\)
\(888\) 0 0
\(889\) −24.6795 −0.827724
\(890\) 0 0
\(891\) −11.6603 −0.390633
\(892\) 0 0
\(893\) 37.4641 1.25369
\(894\) 0 0
\(895\) 13.8564 0.463169
\(896\) 0 0
\(897\) 22.3923 0.747657
\(898\) 0 0
\(899\) −25.8564 −0.862359
\(900\) 0 0
\(901\) 39.7128 1.32303
\(902\) 0 0
\(903\) −25.8564 −0.860447
\(904\) 0 0
\(905\) −11.3205 −0.376306
\(906\) 0 0
\(907\) −7.12436 −0.236560 −0.118280 0.992980i \(-0.537738\pi\)
−0.118280 + 0.992980i \(0.537738\pi\)
\(908\) 0 0
\(909\) −13.7128 −0.454825
\(910\) 0 0
\(911\) −12.7846 −0.423573 −0.211787 0.977316i \(-0.567928\pi\)
−0.211787 + 0.977316i \(0.567928\pi\)
\(912\) 0 0
\(913\) −54.2487 −1.79537
\(914\) 0 0
\(915\) 14.9282 0.493511
\(916\) 0 0
\(917\) −10.1436 −0.334971
\(918\) 0 0
\(919\) 33.1769 1.09441 0.547203 0.837000i \(-0.315693\pi\)
0.547203 + 0.837000i \(0.315693\pi\)
\(920\) 0 0
\(921\) 23.3205 0.768437
\(922\) 0 0
\(923\) −11.6603 −0.383802
\(924\) 0 0
\(925\) −2.92820 −0.0962787
\(926\) 0 0
\(927\) 30.0526 0.987056
\(928\) 0 0
\(929\) −48.2487 −1.58299 −0.791494 0.611176i \(-0.790697\pi\)
−0.791494 + 0.611176i \(0.790697\pi\)
\(930\) 0 0
\(931\) 16.3397 0.535514
\(932\) 0 0
\(933\) 36.7846 1.20427
\(934\) 0 0
\(935\) −16.3923 −0.536086
\(936\) 0 0
\(937\) 18.7846 0.613666 0.306833 0.951763i \(-0.400731\pi\)
0.306833 + 0.951763i \(0.400731\pi\)
\(938\) 0 0
\(939\) 55.3205 1.80532
\(940\) 0 0
\(941\) −17.3205 −0.564632 −0.282316 0.959321i \(-0.591103\pi\)
−0.282316 + 0.959321i \(0.591103\pi\)
\(942\) 0 0
\(943\) −93.9615 −3.05981
\(944\) 0 0
\(945\) 13.8564 0.450749
\(946\) 0 0
\(947\) 29.3205 0.952788 0.476394 0.879232i \(-0.341944\pi\)
0.476394 + 0.879232i \(0.341944\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 34.9282 1.13262
\(952\) 0 0
\(953\) 44.6410 1.44606 0.723032 0.690814i \(-0.242748\pi\)
0.723032 + 0.690814i \(0.242748\pi\)
\(954\) 0 0
\(955\) 5.07180 0.164119
\(956\) 0 0
\(957\) −70.6410 −2.28350
\(958\) 0 0
\(959\) −3.21539 −0.103830
\(960\) 0 0
\(961\) −8.60770 −0.277668
\(962\) 0 0
\(963\) 16.9808 0.547197
\(964\) 0 0
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 20.6410 0.663770 0.331885 0.943320i \(-0.392315\pi\)
0.331885 + 0.943320i \(0.392315\pi\)
\(968\) 0 0
\(969\) 30.9282 0.993557
\(970\) 0 0
\(971\) −40.0000 −1.28366 −0.641831 0.766846i \(-0.721825\pi\)
−0.641831 + 0.766846i \(0.721825\pi\)
\(972\) 0 0
\(973\) −42.9282 −1.37622
\(974\) 0 0
\(975\) 2.73205 0.0874957
\(976\) 0 0
\(977\) −55.4256 −1.77322 −0.886611 0.462515i \(-0.846947\pi\)
−0.886611 + 0.462515i \(0.846947\pi\)
\(978\) 0 0
\(979\) −23.3205 −0.745327
\(980\) 0 0
\(981\) −26.7846 −0.855167
\(982\) 0 0
\(983\) 12.6410 0.403186 0.201593 0.979469i \(-0.435388\pi\)
0.201593 + 0.979469i \(0.435388\pi\)
\(984\) 0 0
\(985\) −0.928203 −0.0295750
\(986\) 0 0
\(987\) −108.497 −3.45351
\(988\) 0 0
\(989\) 22.3923 0.712034
\(990\) 0 0
\(991\) 43.7128 1.38858 0.694292 0.719694i \(-0.255718\pi\)
0.694292 + 0.719694i \(0.255718\pi\)
\(992\) 0 0
\(993\) 62.7846 1.99241
\(994\) 0 0
\(995\) 14.9282 0.473256
\(996\) 0 0
\(997\) −9.60770 −0.304279 −0.152139 0.988359i \(-0.548616\pi\)
−0.152139 + 0.988359i \(0.548616\pi\)
\(998\) 0 0
\(999\) −11.7128 −0.370577
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 1040.2.a.m.1.2 2
3.2 odd 2 9360.2.a.cq.1.1 2
4.3 odd 2 520.2.a.d.1.1 2
5.4 even 2 5200.2.a.bo.1.1 2
8.3 odd 2 4160.2.a.bl.1.2 2
8.5 even 2 4160.2.a.v.1.1 2
12.11 even 2 4680.2.a.be.1.2 2
20.3 even 4 2600.2.d.l.1249.1 4
20.7 even 4 2600.2.d.l.1249.4 4
20.19 odd 2 2600.2.a.u.1.2 2
52.51 odd 2 6760.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
520.2.a.d.1.1 2 4.3 odd 2
1040.2.a.m.1.2 2 1.1 even 1 trivial
2600.2.a.u.1.2 2 20.19 odd 2
2600.2.d.l.1249.1 4 20.3 even 4
2600.2.d.l.1249.4 4 20.7 even 4
4160.2.a.v.1.1 2 8.5 even 2
4160.2.a.bl.1.2 2 8.3 odd 2
4680.2.a.be.1.2 2 12.11 even 2
5200.2.a.bo.1.1 2 5.4 even 2
6760.2.a.p.1.1 2 52.51 odd 2
9360.2.a.cq.1.1 2 3.2 odd 2