Properties

Label 2112.4.a.bg.1.1
Level $2112$
Weight $4$
Character 2112.1
Self dual yes
Analytic conductor $124.612$
Analytic rank $1$
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] = [2112,4,Mod(1,2112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2112, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2112.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2112.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: \(124.612033932\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.42443\) of defining polynomial
Character \(\chi\) \(=\) 2112.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -2.84886 q^{5} -31.6977 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -2.84886 q^{5} -31.6977 q^{7} +9.00000 q^{9} -11.0000 q^{11} -5.15114 q^{13} +8.54657 q^{15} +121.942 q^{17} +34.8489 q^{19} +95.0931 q^{21} -116.244 q^{23} -116.884 q^{25} -27.0000 q^{27} +69.4534 q^{29} -140.605 q^{31} +33.0000 q^{33} +90.3023 q^{35} +420.070 q^{37} +15.4534 q^{39} -322.058 q^{41} +321.035 q^{43} -25.6397 q^{45} +231.408 q^{47} +661.745 q^{49} -365.826 q^{51} -4.91916 q^{53} +31.3374 q^{55} -104.547 q^{57} +406.443 q^{59} +556.431 q^{61} -285.279 q^{63} +14.6749 q^{65} +84.7452 q^{67} +348.733 q^{69} -49.0808 q^{71} +785.884 q^{73} +350.652 q^{75} +348.675 q^{77} +383.118 q^{79} +81.0000 q^{81} -930.211 q^{83} -347.395 q^{85} -208.360 q^{87} -732.559 q^{89} +163.279 q^{91} +421.814 q^{93} -99.2794 q^{95} -1171.49 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 14 q^{5} - 24 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 14 q^{5} - 24 q^{7} + 18 q^{9} - 22 q^{11} - 30 q^{13} - 42 q^{15} + 106 q^{17} + 50 q^{19} + 72 q^{21} - 134 q^{23} + 42 q^{25} - 54 q^{27} + 198 q^{29} - 360 q^{31} + 66 q^{33} + 220 q^{35} + 328 q^{37} + 90 q^{39} - 782 q^{41} + 386 q^{43} + 126 q^{45} - 266 q^{47} + 378 q^{49} - 318 q^{51} + 522 q^{53} - 154 q^{55} - 150 q^{57} - 172 q^{59} + 778 q^{61} - 216 q^{63} - 404 q^{65} - 776 q^{67} + 402 q^{69} - 630 q^{71} + 1296 q^{73} - 126 q^{75} + 264 q^{77} - 652 q^{79} + 162 q^{81} - 324 q^{83} - 616 q^{85} - 594 q^{87} - 756 q^{89} - 28 q^{91} + 1080 q^{93} + 156 q^{95} - 452 q^{97} - 198 q^{99}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −2.84886 −0.254810 −0.127405 0.991851i \(-0.540665\pi\)
−0.127405 + 0.991851i \(0.540665\pi\)
\(6\) 0 0
\(7\) −31.6977 −1.71152 −0.855758 0.517377i \(-0.826909\pi\)
−0.855758 + 0.517377i \(0.826909\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) −5.15114 −0.109898 −0.0549488 0.998489i \(-0.517500\pi\)
−0.0549488 + 0.998489i \(0.517500\pi\)
\(14\) 0 0
\(15\) 8.54657 0.147114
\(16\) 0 0
\(17\) 121.942 1.73972 0.869861 0.493297i \(-0.164208\pi\)
0.869861 + 0.493297i \(0.164208\pi\)
\(18\) 0 0
\(19\) 34.8489 0.420783 0.210391 0.977617i \(-0.432526\pi\)
0.210391 + 0.977617i \(0.432526\pi\)
\(20\) 0 0
\(21\) 95.0931 0.988144
\(22\) 0 0
\(23\) −116.244 −1.05385 −0.526926 0.849911i \(-0.676656\pi\)
−0.526926 + 0.849911i \(0.676656\pi\)
\(24\) 0 0
\(25\) −116.884 −0.935072
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 69.4534 0.444730 0.222365 0.974963i \(-0.428622\pi\)
0.222365 + 0.974963i \(0.428622\pi\)
\(30\) 0 0
\(31\) −140.605 −0.814623 −0.407312 0.913289i \(-0.633534\pi\)
−0.407312 + 0.913289i \(0.633534\pi\)
\(32\) 0 0
\(33\) 33.0000 0.174078
\(34\) 0 0
\(35\) 90.3023 0.436111
\(36\) 0 0
\(37\) 420.070 1.86646 0.933232 0.359276i \(-0.116976\pi\)
0.933232 + 0.359276i \(0.116976\pi\)
\(38\) 0 0
\(39\) 15.4534 0.0634495
\(40\) 0 0
\(41\) −322.058 −1.22676 −0.613378 0.789789i \(-0.710190\pi\)
−0.613378 + 0.789789i \(0.710190\pi\)
\(42\) 0 0
\(43\) 321.035 1.13854 0.569272 0.822149i \(-0.307225\pi\)
0.569272 + 0.822149i \(0.307225\pi\)
\(44\) 0 0
\(45\) −25.6397 −0.0849365
\(46\) 0 0
\(47\) 231.408 0.718176 0.359088 0.933304i \(-0.383088\pi\)
0.359088 + 0.933304i \(0.383088\pi\)
\(48\) 0 0
\(49\) 661.745 1.92929
\(50\) 0 0
\(51\) −365.826 −1.00443
\(52\) 0 0
\(53\) −4.91916 −0.0127490 −0.00637452 0.999980i \(-0.502029\pi\)
−0.00637452 + 0.999980i \(0.502029\pi\)
\(54\) 0 0
\(55\) 31.3374 0.0768280
\(56\) 0 0
\(57\) −104.547 −0.242939
\(58\) 0 0
\(59\) 406.443 0.896854 0.448427 0.893820i \(-0.351984\pi\)
0.448427 + 0.893820i \(0.351984\pi\)
\(60\) 0 0
\(61\) 556.431 1.16793 0.583964 0.811779i \(-0.301501\pi\)
0.583964 + 0.811779i \(0.301501\pi\)
\(62\) 0 0
\(63\) −285.279 −0.570505
\(64\) 0 0
\(65\) 14.6749 0.0280030
\(66\) 0 0
\(67\) 84.7452 0.154526 0.0772632 0.997011i \(-0.475382\pi\)
0.0772632 + 0.997011i \(0.475382\pi\)
\(68\) 0 0
\(69\) 348.733 0.608442
\(70\) 0 0
\(71\) −49.0808 −0.0820398 −0.0410199 0.999158i \(-0.513061\pi\)
−0.0410199 + 0.999158i \(0.513061\pi\)
\(72\) 0 0
\(73\) 785.884 1.26001 0.630005 0.776591i \(-0.283053\pi\)
0.630005 + 0.776591i \(0.283053\pi\)
\(74\) 0 0
\(75\) 350.652 0.539864
\(76\) 0 0
\(77\) 348.675 0.516041
\(78\) 0 0
\(79\) 383.118 0.545622 0.272811 0.962068i \(-0.412047\pi\)
0.272811 + 0.962068i \(0.412047\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −930.211 −1.23017 −0.615084 0.788462i \(-0.710878\pi\)
−0.615084 + 0.788462i \(0.710878\pi\)
\(84\) 0 0
\(85\) −347.395 −0.443298
\(86\) 0 0
\(87\) −208.360 −0.256765
\(88\) 0 0
\(89\) −732.559 −0.872484 −0.436242 0.899829i \(-0.643691\pi\)
−0.436242 + 0.899829i \(0.643691\pi\)
\(90\) 0 0
\(91\) 163.279 0.188092
\(92\) 0 0
\(93\) 421.814 0.470323
\(94\) 0 0
\(95\) −99.2794 −0.107220
\(96\) 0 0
\(97\) −1171.49 −1.22626 −0.613128 0.789984i \(-0.710089\pi\)
−0.613128 + 0.789984i \(0.710089\pi\)
\(98\) 0 0
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) 1221.27 1.20318 0.601589 0.798806i \(-0.294535\pi\)
0.601589 + 0.798806i \(0.294535\pi\)
\(102\) 0 0
\(103\) −516.745 −0.494334 −0.247167 0.968973i \(-0.579500\pi\)
−0.247167 + 0.968973i \(0.579500\pi\)
\(104\) 0 0
\(105\) −270.907 −0.251789
\(106\) 0 0
\(107\) −152.025 −0.137353 −0.0686765 0.997639i \(-0.521878\pi\)
−0.0686765 + 0.997639i \(0.521878\pi\)
\(108\) 0 0
\(109\) −2170.32 −1.90714 −0.953572 0.301164i \(-0.902625\pi\)
−0.953572 + 0.301164i \(0.902625\pi\)
\(110\) 0 0
\(111\) −1260.21 −1.07760
\(112\) 0 0
\(113\) −646.397 −0.538123 −0.269062 0.963123i \(-0.586714\pi\)
−0.269062 + 0.963123i \(0.586714\pi\)
\(114\) 0 0
\(115\) 331.163 0.268532
\(116\) 0 0
\(117\) −46.3603 −0.0366326
\(118\) 0 0
\(119\) −3865.28 −2.97756
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) 966.174 0.708268
\(124\) 0 0
\(125\) 689.093 0.493075
\(126\) 0 0
\(127\) 993.304 0.694027 0.347014 0.937860i \(-0.387196\pi\)
0.347014 + 0.937860i \(0.387196\pi\)
\(128\) 0 0
\(129\) −963.105 −0.657339
\(130\) 0 0
\(131\) 385.814 0.257318 0.128659 0.991689i \(-0.458933\pi\)
0.128659 + 0.991689i \(0.458933\pi\)
\(132\) 0 0
\(133\) −1104.63 −0.720177
\(134\) 0 0
\(135\) 76.9192 0.0490381
\(136\) 0 0
\(137\) 884.840 0.551803 0.275901 0.961186i \(-0.411024\pi\)
0.275901 + 0.961186i \(0.411024\pi\)
\(138\) 0 0
\(139\) −1091.94 −0.666312 −0.333156 0.942872i \(-0.608114\pi\)
−0.333156 + 0.942872i \(0.608114\pi\)
\(140\) 0 0
\(141\) −694.223 −0.414639
\(142\) 0 0
\(143\) 56.6626 0.0331354
\(144\) 0 0
\(145\) −197.863 −0.113322
\(146\) 0 0
\(147\) −1985.24 −1.11387
\(148\) 0 0
\(149\) −297.014 −0.163304 −0.0816522 0.996661i \(-0.526020\pi\)
−0.0816522 + 0.996661i \(0.526020\pi\)
\(150\) 0 0
\(151\) 1887.86 1.01743 0.508716 0.860935i \(-0.330120\pi\)
0.508716 + 0.860935i \(0.330120\pi\)
\(152\) 0 0
\(153\) 1097.48 0.579907
\(154\) 0 0
\(155\) 400.562 0.207574
\(156\) 0 0
\(157\) 56.5343 0.0287384 0.0143692 0.999897i \(-0.495426\pi\)
0.0143692 + 0.999897i \(0.495426\pi\)
\(158\) 0 0
\(159\) 14.7575 0.00736066
\(160\) 0 0
\(161\) 3684.68 1.80369
\(162\) 0 0
\(163\) −49.2338 −0.0236582 −0.0118291 0.999930i \(-0.503765\pi\)
−0.0118291 + 0.999930i \(0.503765\pi\)
\(164\) 0 0
\(165\) −94.0123 −0.0443567
\(166\) 0 0
\(167\) −2068.75 −0.958589 −0.479294 0.877654i \(-0.659107\pi\)
−0.479294 + 0.877654i \(0.659107\pi\)
\(168\) 0 0
\(169\) −2170.47 −0.987923
\(170\) 0 0
\(171\) 313.640 0.140261
\(172\) 0 0
\(173\) 604.012 0.265446 0.132723 0.991153i \(-0.457628\pi\)
0.132723 + 0.991153i \(0.457628\pi\)
\(174\) 0 0
\(175\) 3704.96 1.60039
\(176\) 0 0
\(177\) −1219.33 −0.517799
\(178\) 0 0
\(179\) −2132.02 −0.890251 −0.445126 0.895468i \(-0.646841\pi\)
−0.445126 + 0.895468i \(0.646841\pi\)
\(180\) 0 0
\(181\) 589.371 0.242031 0.121015 0.992651i \(-0.461385\pi\)
0.121015 + 0.992651i \(0.461385\pi\)
\(182\) 0 0
\(183\) −1669.29 −0.674304
\(184\) 0 0
\(185\) −1196.72 −0.475593
\(186\) 0 0
\(187\) −1341.36 −0.524546
\(188\) 0 0
\(189\) 855.838 0.329381
\(190\) 0 0
\(191\) 2160.90 0.818624 0.409312 0.912395i \(-0.365769\pi\)
0.409312 + 0.912395i \(0.365769\pi\)
\(192\) 0 0
\(193\) −1490.91 −0.556052 −0.278026 0.960574i \(-0.589680\pi\)
−0.278026 + 0.960574i \(0.589680\pi\)
\(194\) 0 0
\(195\) −44.0246 −0.0161675
\(196\) 0 0
\(197\) 230.529 0.0833732 0.0416866 0.999131i \(-0.486727\pi\)
0.0416866 + 0.999131i \(0.486727\pi\)
\(198\) 0 0
\(199\) −22.4007 −0.00797963 −0.00398982 0.999992i \(-0.501270\pi\)
−0.00398982 + 0.999992i \(0.501270\pi\)
\(200\) 0 0
\(201\) −254.236 −0.0892159
\(202\) 0 0
\(203\) −2201.51 −0.761163
\(204\) 0 0
\(205\) 917.497 0.312589
\(206\) 0 0
\(207\) −1046.20 −0.351284
\(208\) 0 0
\(209\) −383.337 −0.126871
\(210\) 0 0
\(211\) −1051.64 −0.343117 −0.171558 0.985174i \(-0.554880\pi\)
−0.171558 + 0.985174i \(0.554880\pi\)
\(212\) 0 0
\(213\) 147.243 0.0473657
\(214\) 0 0
\(215\) −914.583 −0.290112
\(216\) 0 0
\(217\) 4456.84 1.39424
\(218\) 0 0
\(219\) −2357.65 −0.727467
\(220\) 0 0
\(221\) −628.141 −0.191191
\(222\) 0 0
\(223\) −3861.80 −1.15966 −0.579832 0.814736i \(-0.696882\pi\)
−0.579832 + 0.814736i \(0.696882\pi\)
\(224\) 0 0
\(225\) −1051.96 −0.311691
\(226\) 0 0
\(227\) −872.721 −0.255174 −0.127587 0.991827i \(-0.540723\pi\)
−0.127587 + 0.991827i \(0.540723\pi\)
\(228\) 0 0
\(229\) −1841.72 −0.531459 −0.265730 0.964048i \(-0.585613\pi\)
−0.265730 + 0.964048i \(0.585613\pi\)
\(230\) 0 0
\(231\) −1046.02 −0.297937
\(232\) 0 0
\(233\) 3932.14 1.10559 0.552796 0.833317i \(-0.313561\pi\)
0.552796 + 0.833317i \(0.313561\pi\)
\(234\) 0 0
\(235\) −659.248 −0.182998
\(236\) 0 0
\(237\) −1149.35 −0.315015
\(238\) 0 0
\(239\) −4772.10 −1.29155 −0.645777 0.763526i \(-0.723466\pi\)
−0.645777 + 0.763526i \(0.723466\pi\)
\(240\) 0 0
\(241\) 3988.84 1.06616 0.533078 0.846066i \(-0.321035\pi\)
0.533078 + 0.846066i \(0.321035\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −1885.22 −0.491601
\(246\) 0 0
\(247\) −179.511 −0.0462431
\(248\) 0 0
\(249\) 2790.63 0.710238
\(250\) 0 0
\(251\) −5474.22 −1.37661 −0.688306 0.725421i \(-0.741645\pi\)
−0.688306 + 0.725421i \(0.741645\pi\)
\(252\) 0 0
\(253\) 1278.69 0.317749
\(254\) 0 0
\(255\) 1042.19 0.255938
\(256\) 0 0
\(257\) −6434.01 −1.56164 −0.780822 0.624754i \(-0.785199\pi\)
−0.780822 + 0.624754i \(0.785199\pi\)
\(258\) 0 0
\(259\) −13315.3 −3.19448
\(260\) 0 0
\(261\) 625.081 0.148243
\(262\) 0 0
\(263\) −7589.00 −1.77931 −0.889654 0.456636i \(-0.849054\pi\)
−0.889654 + 0.456636i \(0.849054\pi\)
\(264\) 0 0
\(265\) 14.0140 0.00324858
\(266\) 0 0
\(267\) 2197.68 0.503729
\(268\) 0 0
\(269\) −478.178 −0.108383 −0.0541914 0.998531i \(-0.517258\pi\)
−0.0541914 + 0.998531i \(0.517258\pi\)
\(270\) 0 0
\(271\) 122.323 0.0274192 0.0137096 0.999906i \(-0.495636\pi\)
0.0137096 + 0.999906i \(0.495636\pi\)
\(272\) 0 0
\(273\) −489.838 −0.108595
\(274\) 0 0
\(275\) 1285.72 0.281935
\(276\) 0 0
\(277\) −8199.41 −1.77854 −0.889269 0.457385i \(-0.848786\pi\)
−0.889269 + 0.457385i \(0.848786\pi\)
\(278\) 0 0
\(279\) −1265.44 −0.271541
\(280\) 0 0
\(281\) 6943.79 1.47413 0.737067 0.675820i \(-0.236210\pi\)
0.737067 + 0.675820i \(0.236210\pi\)
\(282\) 0 0
\(283\) 1035.14 0.217429 0.108715 0.994073i \(-0.465327\pi\)
0.108715 + 0.994073i \(0.465327\pi\)
\(284\) 0 0
\(285\) 297.838 0.0619032
\(286\) 0 0
\(287\) 10208.5 2.09961
\(288\) 0 0
\(289\) 9956.85 2.02663
\(290\) 0 0
\(291\) 3514.47 0.707979
\(292\) 0 0
\(293\) 6144.81 1.22520 0.612600 0.790393i \(-0.290124\pi\)
0.612600 + 0.790393i \(0.290124\pi\)
\(294\) 0 0
\(295\) −1157.90 −0.228527
\(296\) 0 0
\(297\) 297.000 0.0580259
\(298\) 0 0
\(299\) 598.791 0.115816
\(300\) 0 0
\(301\) −10176.1 −1.94864
\(302\) 0 0
\(303\) −3663.81 −0.694655
\(304\) 0 0
\(305\) −1585.19 −0.297599
\(306\) 0 0
\(307\) −2186.09 −0.406406 −0.203203 0.979137i \(-0.565135\pi\)
−0.203203 + 0.979137i \(0.565135\pi\)
\(308\) 0 0
\(309\) 1550.24 0.285404
\(310\) 0 0
\(311\) 7484.83 1.36471 0.682357 0.731019i \(-0.260955\pi\)
0.682357 + 0.731019i \(0.260955\pi\)
\(312\) 0 0
\(313\) −6833.33 −1.23400 −0.617001 0.786962i \(-0.711653\pi\)
−0.617001 + 0.786962i \(0.711653\pi\)
\(314\) 0 0
\(315\) 812.721 0.145370
\(316\) 0 0
\(317\) −924.265 −0.163760 −0.0818800 0.996642i \(-0.526092\pi\)
−0.0818800 + 0.996642i \(0.526092\pi\)
\(318\) 0 0
\(319\) −763.988 −0.134091
\(320\) 0 0
\(321\) 456.074 0.0793008
\(322\) 0 0
\(323\) 4249.54 0.732046
\(324\) 0 0
\(325\) 602.086 0.102762
\(326\) 0 0
\(327\) 6510.95 1.10109
\(328\) 0 0
\(329\) −7335.10 −1.22917
\(330\) 0 0
\(331\) −9820.46 −1.63076 −0.815380 0.578927i \(-0.803472\pi\)
−0.815380 + 0.578927i \(0.803472\pi\)
\(332\) 0 0
\(333\) 3780.63 0.622154
\(334\) 0 0
\(335\) −241.427 −0.0393748
\(336\) 0 0
\(337\) 600.808 0.0971161 0.0485580 0.998820i \(-0.484537\pi\)
0.0485580 + 0.998820i \(0.484537\pi\)
\(338\) 0 0
\(339\) 1939.19 0.310686
\(340\) 0 0
\(341\) 1546.65 0.245618
\(342\) 0 0
\(343\) −10103.5 −1.59049
\(344\) 0 0
\(345\) −993.490 −0.155037
\(346\) 0 0
\(347\) −3143.41 −0.486303 −0.243152 0.969988i \(-0.578181\pi\)
−0.243152 + 0.969988i \(0.578181\pi\)
\(348\) 0 0
\(349\) −720.663 −0.110533 −0.0552667 0.998472i \(-0.517601\pi\)
−0.0552667 + 0.998472i \(0.517601\pi\)
\(350\) 0 0
\(351\) 139.081 0.0211498
\(352\) 0 0
\(353\) 1207.12 0.182007 0.0910034 0.995851i \(-0.470993\pi\)
0.0910034 + 0.995851i \(0.470993\pi\)
\(354\) 0 0
\(355\) 139.824 0.0209045
\(356\) 0 0
\(357\) 11595.8 1.71910
\(358\) 0 0
\(359\) −8748.31 −1.28612 −0.643062 0.765814i \(-0.722336\pi\)
−0.643062 + 0.765814i \(0.722336\pi\)
\(360\) 0 0
\(361\) −5644.56 −0.822942
\(362\) 0 0
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) −2238.87 −0.321063
\(366\) 0 0
\(367\) 6730.45 0.957293 0.478647 0.878008i \(-0.341128\pi\)
0.478647 + 0.878008i \(0.341128\pi\)
\(368\) 0 0
\(369\) −2898.52 −0.408919
\(370\) 0 0
\(371\) 155.926 0.0218202
\(372\) 0 0
\(373\) 227.394 0.0315657 0.0157828 0.999875i \(-0.494976\pi\)
0.0157828 + 0.999875i \(0.494976\pi\)
\(374\) 0 0
\(375\) −2067.28 −0.284677
\(376\) 0 0
\(377\) −357.764 −0.0488748
\(378\) 0 0
\(379\) 11356.2 1.53913 0.769565 0.638568i \(-0.220473\pi\)
0.769565 + 0.638568i \(0.220473\pi\)
\(380\) 0 0
\(381\) −2979.91 −0.400697
\(382\) 0 0
\(383\) −10753.6 −1.43468 −0.717338 0.696725i \(-0.754640\pi\)
−0.717338 + 0.696725i \(0.754640\pi\)
\(384\) 0 0
\(385\) −993.325 −0.131492
\(386\) 0 0
\(387\) 2889.32 0.379515
\(388\) 0 0
\(389\) 11727.1 1.52850 0.764252 0.644918i \(-0.223109\pi\)
0.764252 + 0.644918i \(0.223109\pi\)
\(390\) 0 0
\(391\) −14175.1 −1.83341
\(392\) 0 0
\(393\) −1157.44 −0.148563
\(394\) 0 0
\(395\) −1091.45 −0.139030
\(396\) 0 0
\(397\) 359.905 0.0454990 0.0227495 0.999741i \(-0.492758\pi\)
0.0227495 + 0.999741i \(0.492758\pi\)
\(398\) 0 0
\(399\) 3313.89 0.415794
\(400\) 0 0
\(401\) −4066.71 −0.506438 −0.253219 0.967409i \(-0.581489\pi\)
−0.253219 + 0.967409i \(0.581489\pi\)
\(402\) 0 0
\(403\) 724.274 0.0895252
\(404\) 0 0
\(405\) −230.757 −0.0283122
\(406\) 0 0
\(407\) −4620.77 −0.562760
\(408\) 0 0
\(409\) −13488.8 −1.63076 −0.815379 0.578927i \(-0.803472\pi\)
−0.815379 + 0.578927i \(0.803472\pi\)
\(410\) 0 0
\(411\) −2654.52 −0.318584
\(412\) 0 0
\(413\) −12883.3 −1.53498
\(414\) 0 0
\(415\) 2650.04 0.313459
\(416\) 0 0
\(417\) 3275.83 0.384695
\(418\) 0 0
\(419\) −7040.12 −0.820841 −0.410420 0.911896i \(-0.634618\pi\)
−0.410420 + 0.911896i \(0.634618\pi\)
\(420\) 0 0
\(421\) −9171.74 −1.06177 −0.530883 0.847445i \(-0.678140\pi\)
−0.530883 + 0.847445i \(0.678140\pi\)
\(422\) 0 0
\(423\) 2082.67 0.239392
\(424\) 0 0
\(425\) −14253.1 −1.62677
\(426\) 0 0
\(427\) −17637.6 −1.99893
\(428\) 0 0
\(429\) −169.988 −0.0191307
\(430\) 0 0
\(431\) −992.995 −0.110976 −0.0554882 0.998459i \(-0.517672\pi\)
−0.0554882 + 0.998459i \(0.517672\pi\)
\(432\) 0 0
\(433\) 3790.21 0.420660 0.210330 0.977630i \(-0.432546\pi\)
0.210330 + 0.977630i \(0.432546\pi\)
\(434\) 0 0
\(435\) 593.589 0.0654262
\(436\) 0 0
\(437\) −4050.98 −0.443443
\(438\) 0 0
\(439\) 5136.97 0.558483 0.279242 0.960221i \(-0.409917\pi\)
0.279242 + 0.960221i \(0.409917\pi\)
\(440\) 0 0
\(441\) 5955.71 0.643095
\(442\) 0 0
\(443\) 10676.8 1.14508 0.572541 0.819876i \(-0.305958\pi\)
0.572541 + 0.819876i \(0.305958\pi\)
\(444\) 0 0
\(445\) 2086.96 0.222317
\(446\) 0 0
\(447\) 891.042 0.0942838
\(448\) 0 0
\(449\) 10529.9 1.10676 0.553379 0.832929i \(-0.313338\pi\)
0.553379 + 0.832929i \(0.313338\pi\)
\(450\) 0 0
\(451\) 3542.64 0.369881
\(452\) 0 0
\(453\) −5663.59 −0.587414
\(454\) 0 0
\(455\) −465.160 −0.0479275
\(456\) 0 0
\(457\) −14072.5 −1.44045 −0.720225 0.693741i \(-0.755961\pi\)
−0.720225 + 0.693741i \(0.755961\pi\)
\(458\) 0 0
\(459\) −3292.43 −0.334810
\(460\) 0 0
\(461\) 30.8173 0.00311346 0.00155673 0.999999i \(-0.499504\pi\)
0.00155673 + 0.999999i \(0.499504\pi\)
\(462\) 0 0
\(463\) −17591.3 −1.76573 −0.882867 0.469622i \(-0.844390\pi\)
−0.882867 + 0.469622i \(0.844390\pi\)
\(464\) 0 0
\(465\) −1201.69 −0.119843
\(466\) 0 0
\(467\) 13273.1 1.31522 0.657609 0.753360i \(-0.271568\pi\)
0.657609 + 0.753360i \(0.271568\pi\)
\(468\) 0 0
\(469\) −2686.23 −0.264474
\(470\) 0 0
\(471\) −169.603 −0.0165921
\(472\) 0 0
\(473\) −3531.39 −0.343284
\(474\) 0 0
\(475\) −4073.27 −0.393462
\(476\) 0 0
\(477\) −44.2724 −0.00424968
\(478\) 0 0
\(479\) −2496.68 −0.238155 −0.119077 0.992885i \(-0.537994\pi\)
−0.119077 + 0.992885i \(0.537994\pi\)
\(480\) 0 0
\(481\) −2163.84 −0.205120
\(482\) 0 0
\(483\) −11054.0 −1.04136
\(484\) 0 0
\(485\) 3337.41 0.312462
\(486\) 0 0
\(487\) 3464.42 0.322357 0.161178 0.986925i \(-0.448471\pi\)
0.161178 + 0.986925i \(0.448471\pi\)
\(488\) 0 0
\(489\) 147.701 0.0136591
\(490\) 0 0
\(491\) −16224.6 −1.49125 −0.745625 0.666366i \(-0.767849\pi\)
−0.745625 + 0.666366i \(0.767849\pi\)
\(492\) 0 0
\(493\) 8469.29 0.773707
\(494\) 0 0
\(495\) 282.037 0.0256093
\(496\) 0 0
\(497\) 1555.75 0.140412
\(498\) 0 0
\(499\) 9993.81 0.896562 0.448281 0.893893i \(-0.352036\pi\)
0.448281 + 0.893893i \(0.352036\pi\)
\(500\) 0 0
\(501\) 6206.24 0.553441
\(502\) 0 0
\(503\) 15334.8 1.35933 0.679667 0.733520i \(-0.262124\pi\)
0.679667 + 0.733520i \(0.262124\pi\)
\(504\) 0 0
\(505\) −3479.23 −0.306581
\(506\) 0 0
\(507\) 6511.40 0.570377
\(508\) 0 0
\(509\) 7291.23 0.634927 0.317464 0.948270i \(-0.397169\pi\)
0.317464 + 0.948270i \(0.397169\pi\)
\(510\) 0 0
\(511\) −24910.7 −2.15653
\(512\) 0 0
\(513\) −940.919 −0.0809797
\(514\) 0 0
\(515\) 1472.13 0.125961
\(516\) 0 0
\(517\) −2545.49 −0.216538
\(518\) 0 0
\(519\) −1812.04 −0.153255
\(520\) 0 0
\(521\) 16794.3 1.41223 0.706114 0.708098i \(-0.250447\pi\)
0.706114 + 0.708098i \(0.250447\pi\)
\(522\) 0 0
\(523\) −21009.4 −1.75655 −0.878275 0.478157i \(-0.841305\pi\)
−0.878275 + 0.478157i \(0.841305\pi\)
\(524\) 0 0
\(525\) −11114.9 −0.923986
\(526\) 0 0
\(527\) −17145.6 −1.41722
\(528\) 0 0
\(529\) 1345.73 0.110605
\(530\) 0 0
\(531\) 3657.99 0.298951
\(532\) 0 0
\(533\) 1658.97 0.134818
\(534\) 0 0
\(535\) 433.097 0.0349989
\(536\) 0 0
\(537\) 6396.07 0.513987
\(538\) 0 0
\(539\) −7279.20 −0.581702
\(540\) 0 0
\(541\) 16802.8 1.33532 0.667662 0.744464i \(-0.267295\pi\)
0.667662 + 0.744464i \(0.267295\pi\)
\(542\) 0 0
\(543\) −1768.11 −0.139737
\(544\) 0 0
\(545\) 6182.93 0.485959
\(546\) 0 0
\(547\) 16784.5 1.31198 0.655990 0.754770i \(-0.272251\pi\)
0.655990 + 0.754770i \(0.272251\pi\)
\(548\) 0 0
\(549\) 5007.88 0.389309
\(550\) 0 0
\(551\) 2420.37 0.187135
\(552\) 0 0
\(553\) −12144.0 −0.933840
\(554\) 0 0
\(555\) 3590.16 0.274584
\(556\) 0 0
\(557\) −18127.0 −1.37893 −0.689467 0.724317i \(-0.742155\pi\)
−0.689467 + 0.724317i \(0.742155\pi\)
\(558\) 0 0
\(559\) −1653.70 −0.125123
\(560\) 0 0
\(561\) 4024.09 0.302847
\(562\) 0 0
\(563\) 2090.88 0.156518 0.0782592 0.996933i \(-0.475064\pi\)
0.0782592 + 0.996933i \(0.475064\pi\)
\(564\) 0 0
\(565\) 1841.49 0.137119
\(566\) 0 0
\(567\) −2567.51 −0.190168
\(568\) 0 0
\(569\) 6249.23 0.460424 0.230212 0.973140i \(-0.426058\pi\)
0.230212 + 0.973140i \(0.426058\pi\)
\(570\) 0 0
\(571\) 6048.79 0.443317 0.221659 0.975124i \(-0.428853\pi\)
0.221659 + 0.975124i \(0.428853\pi\)
\(572\) 0 0
\(573\) −6482.69 −0.472633
\(574\) 0 0
\(575\) 13587.1 0.985428
\(576\) 0 0
\(577\) −15729.1 −1.13486 −0.567429 0.823423i \(-0.692062\pi\)
−0.567429 + 0.823423i \(0.692062\pi\)
\(578\) 0 0
\(579\) 4472.73 0.321037
\(580\) 0 0
\(581\) 29485.6 2.10545
\(582\) 0 0
\(583\) 54.1108 0.00384398
\(584\) 0 0
\(585\) 132.074 0.00933433
\(586\) 0 0
\(587\) 15620.5 1.09835 0.549173 0.835709i \(-0.314943\pi\)
0.549173 + 0.835709i \(0.314943\pi\)
\(588\) 0 0
\(589\) −4899.91 −0.342780
\(590\) 0 0
\(591\) −691.587 −0.0481355
\(592\) 0 0
\(593\) −493.541 −0.0341776 −0.0170888 0.999854i \(-0.505440\pi\)
−0.0170888 + 0.999854i \(0.505440\pi\)
\(594\) 0 0
\(595\) 11011.6 0.758711
\(596\) 0 0
\(597\) 67.2022 0.00460704
\(598\) 0 0
\(599\) 12455.1 0.849585 0.424793 0.905291i \(-0.360347\pi\)
0.424793 + 0.905291i \(0.360347\pi\)
\(600\) 0 0
\(601\) 12454.8 0.845329 0.422664 0.906286i \(-0.361095\pi\)
0.422664 + 0.906286i \(0.361095\pi\)
\(602\) 0 0
\(603\) 762.707 0.0515088
\(604\) 0 0
\(605\) −344.712 −0.0231645
\(606\) 0 0
\(607\) 4243.19 0.283733 0.141867 0.989886i \(-0.454690\pi\)
0.141867 + 0.989886i \(0.454690\pi\)
\(608\) 0 0
\(609\) 6604.54 0.439458
\(610\) 0 0
\(611\) −1192.01 −0.0789259
\(612\) 0 0
\(613\) −5733.14 −0.377748 −0.188874 0.982001i \(-0.560484\pi\)
−0.188874 + 0.982001i \(0.560484\pi\)
\(614\) 0 0
\(615\) −2752.49 −0.180473
\(616\) 0 0
\(617\) 15642.1 1.02063 0.510314 0.859988i \(-0.329529\pi\)
0.510314 + 0.859988i \(0.329529\pi\)
\(618\) 0 0
\(619\) −7467.40 −0.484879 −0.242440 0.970167i \(-0.577948\pi\)
−0.242440 + 0.970167i \(0.577948\pi\)
\(620\) 0 0
\(621\) 3138.60 0.202814
\(622\) 0 0
\(623\) 23220.4 1.49327
\(624\) 0 0
\(625\) 12647.4 0.809432
\(626\) 0 0
\(627\) 1150.01 0.0732489
\(628\) 0 0
\(629\) 51224.2 3.24713
\(630\) 0 0
\(631\) 1486.38 0.0937745 0.0468872 0.998900i \(-0.485070\pi\)
0.0468872 + 0.998900i \(0.485070\pi\)
\(632\) 0 0
\(633\) 3154.91 0.198099
\(634\) 0 0
\(635\) −2829.78 −0.176845
\(636\) 0 0
\(637\) −3408.74 −0.212024
\(638\) 0 0
\(639\) −441.728 −0.0273466
\(640\) 0 0
\(641\) 12386.0 0.763211 0.381606 0.924325i \(-0.375371\pi\)
0.381606 + 0.924325i \(0.375371\pi\)
\(642\) 0 0
\(643\) −14458.1 −0.886737 −0.443369 0.896339i \(-0.646217\pi\)
−0.443369 + 0.896339i \(0.646217\pi\)
\(644\) 0 0
\(645\) 2743.75 0.167496
\(646\) 0 0
\(647\) −15792.8 −0.959625 −0.479813 0.877371i \(-0.659295\pi\)
−0.479813 + 0.877371i \(0.659295\pi\)
\(648\) 0 0
\(649\) −4470.87 −0.270412
\(650\) 0 0
\(651\) −13370.5 −0.804965
\(652\) 0 0
\(653\) 3179.93 0.190567 0.0952837 0.995450i \(-0.469624\pi\)
0.0952837 + 0.995450i \(0.469624\pi\)
\(654\) 0 0
\(655\) −1099.13 −0.0655672
\(656\) 0 0
\(657\) 7072.96 0.420003
\(658\) 0 0
\(659\) 11593.5 0.685308 0.342654 0.939462i \(-0.388674\pi\)
0.342654 + 0.939462i \(0.388674\pi\)
\(660\) 0 0
\(661\) −3233.88 −0.190293 −0.0951464 0.995463i \(-0.530332\pi\)
−0.0951464 + 0.995463i \(0.530332\pi\)
\(662\) 0 0
\(663\) 1884.42 0.110384
\(664\) 0 0
\(665\) 3146.93 0.183508
\(666\) 0 0
\(667\) −8073.56 −0.468680
\(668\) 0 0
\(669\) 11585.4 0.669532
\(670\) 0 0
\(671\) −6120.74 −0.352144
\(672\) 0 0
\(673\) −5495.72 −0.314776 −0.157388 0.987537i \(-0.550307\pi\)
−0.157388 + 0.987537i \(0.550307\pi\)
\(674\) 0 0
\(675\) 3155.87 0.179955
\(676\) 0 0
\(677\) −33836.7 −1.92090 −0.960451 0.278448i \(-0.910180\pi\)
−0.960451 + 0.278448i \(0.910180\pi\)
\(678\) 0 0
\(679\) 37133.6 2.09876
\(680\) 0 0
\(681\) 2618.16 0.147325
\(682\) 0 0
\(683\) −21080.3 −1.18099 −0.590493 0.807043i \(-0.701067\pi\)
−0.590493 + 0.807043i \(0.701067\pi\)
\(684\) 0 0
\(685\) −2520.78 −0.140605
\(686\) 0 0
\(687\) 5525.16 0.306838
\(688\) 0 0
\(689\) 25.3393 0.00140109
\(690\) 0 0
\(691\) 11811.3 0.650253 0.325127 0.945671i \(-0.394593\pi\)
0.325127 + 0.945671i \(0.394593\pi\)
\(692\) 0 0
\(693\) 3138.07 0.172014
\(694\) 0 0
\(695\) 3110.79 0.169783
\(696\) 0 0
\(697\) −39272.4 −2.13422
\(698\) 0 0
\(699\) −11796.4 −0.638313
\(700\) 0 0
\(701\) −4244.99 −0.228718 −0.114359 0.993440i \(-0.536481\pi\)
−0.114359 + 0.993440i \(0.536481\pi\)
\(702\) 0 0
\(703\) 14639.0 0.785376
\(704\) 0 0
\(705\) 1977.74 0.105654
\(706\) 0 0
\(707\) −38711.5 −2.05926
\(708\) 0 0
\(709\) 898.822 0.0476107 0.0238053 0.999717i \(-0.492422\pi\)
0.0238053 + 0.999717i \(0.492422\pi\)
\(710\) 0 0
\(711\) 3448.06 0.181874
\(712\) 0 0
\(713\) 16344.5 0.858493
\(714\) 0 0
\(715\) −161.424 −0.00844322
\(716\) 0 0
\(717\) 14316.3 0.745679
\(718\) 0 0
\(719\) −10741.8 −0.557165 −0.278582 0.960412i \(-0.589865\pi\)
−0.278582 + 0.960412i \(0.589865\pi\)
\(720\) 0 0
\(721\) 16379.6 0.846061
\(722\) 0 0
\(723\) −11966.5 −0.615546
\(724\) 0 0
\(725\) −8117.99 −0.415855
\(726\) 0 0
\(727\) −16794.2 −0.856758 −0.428379 0.903599i \(-0.640915\pi\)
−0.428379 + 0.903599i \(0.640915\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 39147.7 1.98075
\(732\) 0 0
\(733\) −8659.40 −0.436347 −0.218173 0.975910i \(-0.570010\pi\)
−0.218173 + 0.975910i \(0.570010\pi\)
\(734\) 0 0
\(735\) 5655.65 0.283826
\(736\) 0 0
\(737\) −932.197 −0.0465915
\(738\) 0 0
\(739\) 16705.7 0.831567 0.415783 0.909464i \(-0.363507\pi\)
0.415783 + 0.909464i \(0.363507\pi\)
\(740\) 0 0
\(741\) 538.534 0.0266984
\(742\) 0 0
\(743\) −1292.12 −0.0637996 −0.0318998 0.999491i \(-0.510156\pi\)
−0.0318998 + 0.999491i \(0.510156\pi\)
\(744\) 0 0
\(745\) 846.151 0.0416115
\(746\) 0 0
\(747\) −8371.90 −0.410056
\(748\) 0 0
\(749\) 4818.83 0.235082
\(750\) 0 0
\(751\) 14980.4 0.727886 0.363943 0.931421i \(-0.381430\pi\)
0.363943 + 0.931421i \(0.381430\pi\)
\(752\) 0 0
\(753\) 16422.7 0.794787
\(754\) 0 0
\(755\) −5378.25 −0.259251
\(756\) 0 0
\(757\) −3003.41 −0.144202 −0.0721010 0.997397i \(-0.522970\pi\)
−0.0721010 + 0.997397i \(0.522970\pi\)
\(758\) 0 0
\(759\) −3836.06 −0.183452
\(760\) 0 0
\(761\) −20375.0 −0.970555 −0.485277 0.874360i \(-0.661281\pi\)
−0.485277 + 0.874360i \(0.661281\pi\)
\(762\) 0 0
\(763\) 68794.1 3.26411
\(764\) 0 0
\(765\) −3126.56 −0.147766
\(766\) 0 0
\(767\) −2093.65 −0.0985621
\(768\) 0 0
\(769\) −12372.4 −0.580184 −0.290092 0.956999i \(-0.593686\pi\)
−0.290092 + 0.956999i \(0.593686\pi\)
\(770\) 0 0
\(771\) 19302.0 0.901615
\(772\) 0 0
\(773\) −21023.6 −0.978225 −0.489113 0.872221i \(-0.662679\pi\)
−0.489113 + 0.872221i \(0.662679\pi\)
\(774\) 0 0
\(775\) 16434.4 0.761732
\(776\) 0 0
\(777\) 39945.8 1.84433
\(778\) 0 0
\(779\) −11223.4 −0.516198
\(780\) 0 0
\(781\) 539.889 0.0247359
\(782\) 0 0
\(783\) −1875.24 −0.0855884
\(784\) 0 0
\(785\) −161.058 −0.00732281
\(786\) 0 0
\(787\) 30286.2 1.37177 0.685886 0.727709i \(-0.259415\pi\)
0.685886 + 0.727709i \(0.259415\pi\)
\(788\) 0 0
\(789\) 22767.0 1.02728
\(790\) 0 0
\(791\) 20489.3 0.921007
\(792\) 0 0
\(793\) −2866.25 −0.128353
\(794\) 0 0
\(795\) −42.0420 −0.00187557
\(796\) 0 0
\(797\) −32337.8 −1.43722 −0.718610 0.695413i \(-0.755221\pi\)
−0.718610 + 0.695413i \(0.755221\pi\)
\(798\) 0 0
\(799\) 28218.3 1.24943
\(800\) 0 0
\(801\) −6593.03 −0.290828
\(802\) 0 0
\(803\) −8644.72 −0.379907
\(804\) 0 0
\(805\) −10497.1 −0.459596
\(806\) 0 0
\(807\) 1434.53 0.0625749
\(808\) 0 0
\(809\) −891.707 −0.0387525 −0.0193762 0.999812i \(-0.506168\pi\)
−0.0193762 + 0.999812i \(0.506168\pi\)
\(810\) 0 0
\(811\) −10114.9 −0.437957 −0.218978 0.975730i \(-0.570272\pi\)
−0.218978 + 0.975730i \(0.570272\pi\)
\(812\) 0 0
\(813\) −366.970 −0.0158305
\(814\) 0 0
\(815\) 140.260 0.00602833
\(816\) 0 0
\(817\) 11187.7 0.479080
\(818\) 0 0
\(819\) 1469.51 0.0626972
\(820\) 0 0
\(821\) −10833.5 −0.460525 −0.230262 0.973129i \(-0.573958\pi\)
−0.230262 + 0.973129i \(0.573958\pi\)
\(822\) 0 0
\(823\) −31958.5 −1.35359 −0.676794 0.736173i \(-0.736631\pi\)
−0.676794 + 0.736173i \(0.736631\pi\)
\(824\) 0 0
\(825\) −3857.17 −0.162775
\(826\) 0 0
\(827\) 34847.3 1.46525 0.732624 0.680634i \(-0.238296\pi\)
0.732624 + 0.680634i \(0.238296\pi\)
\(828\) 0 0
\(829\) −6537.91 −0.273910 −0.136955 0.990577i \(-0.543732\pi\)
−0.136955 + 0.990577i \(0.543732\pi\)
\(830\) 0 0
\(831\) 24598.2 1.02684
\(832\) 0 0
\(833\) 80694.5 3.35642
\(834\) 0 0
\(835\) 5893.56 0.244258
\(836\) 0 0
\(837\) 3796.32 0.156774
\(838\) 0 0
\(839\) −2710.34 −0.111527 −0.0557635 0.998444i \(-0.517759\pi\)
−0.0557635 + 0.998444i \(0.517759\pi\)
\(840\) 0 0
\(841\) −19565.2 −0.802215
\(842\) 0 0
\(843\) −20831.4 −0.851092
\(844\) 0 0
\(845\) 6183.35 0.251732
\(846\) 0 0
\(847\) −3835.42 −0.155592
\(848\) 0 0
\(849\) −3105.41 −0.125533
\(850\) 0 0
\(851\) −48830.8 −1.96698
\(852\) 0 0
\(853\) −9759.32 −0.391738 −0.195869 0.980630i \(-0.562753\pi\)
−0.195869 + 0.980630i \(0.562753\pi\)
\(854\) 0 0
\(855\) −893.515 −0.0357398
\(856\) 0 0
\(857\) −13649.8 −0.544072 −0.272036 0.962287i \(-0.587697\pi\)
−0.272036 + 0.962287i \(0.587697\pi\)
\(858\) 0 0
\(859\) 7796.42 0.309674 0.154837 0.987940i \(-0.450515\pi\)
0.154837 + 0.987940i \(0.450515\pi\)
\(860\) 0 0
\(861\) −30625.5 −1.21221
\(862\) 0 0
\(863\) −7183.57 −0.283350 −0.141675 0.989913i \(-0.545249\pi\)
−0.141675 + 0.989913i \(0.545249\pi\)
\(864\) 0 0
\(865\) −1720.75 −0.0676383
\(866\) 0 0
\(867\) −29870.6 −1.17008
\(868\) 0 0
\(869\) −4214.30 −0.164511
\(870\) 0 0
\(871\) −436.534 −0.0169821
\(872\) 0 0
\(873\) −10543.4 −0.408752
\(874\) 0 0
\(875\) −21842.7 −0.843905
\(876\) 0 0
\(877\) −17063.1 −0.656991 −0.328495 0.944506i \(-0.606542\pi\)
−0.328495 + 0.944506i \(0.606542\pi\)
\(878\) 0 0
\(879\) −18434.4 −0.707369
\(880\) 0 0
\(881\) −32174.9 −1.23042 −0.615210 0.788363i \(-0.710929\pi\)
−0.615210 + 0.788363i \(0.710929\pi\)
\(882\) 0 0
\(883\) 2843.68 0.108378 0.0541889 0.998531i \(-0.482743\pi\)
0.0541889 + 0.998531i \(0.482743\pi\)
\(884\) 0 0
\(885\) 3473.69 0.131940
\(886\) 0 0
\(887\) 31417.8 1.18930 0.594649 0.803985i \(-0.297291\pi\)
0.594649 + 0.803985i \(0.297291\pi\)
\(888\) 0 0
\(889\) −31485.5 −1.18784
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) 0 0
\(893\) 8064.30 0.302196
\(894\) 0 0
\(895\) 6073.83 0.226845
\(896\) 0 0
\(897\) −1796.37 −0.0668664
\(898\) 0 0
\(899\) −9765.47 −0.362288
\(900\) 0 0
\(901\) −599.852 −0.0221798
\(902\) 0 0
\(903\) 30528.2 1.12505
\(904\) 0 0
\(905\) −1679.03 −0.0616718
\(906\) 0 0
\(907\) 12253.1 0.448573 0.224287 0.974523i \(-0.427995\pi\)
0.224287 + 0.974523i \(0.427995\pi\)
\(908\) 0 0
\(909\) 10991.4 0.401059
\(910\) 0 0
\(911\) 48422.4 1.76104 0.880518 0.474012i \(-0.157195\pi\)
0.880518 + 0.474012i \(0.157195\pi\)
\(912\) 0 0
\(913\) 10232.3 0.370909
\(914\) 0 0
\(915\) 4755.57 0.171819
\(916\) 0 0
\(917\) −12229.4 −0.440404
\(918\) 0 0
\(919\) −5546.18 −0.199077 −0.0995385 0.995034i \(-0.531737\pi\)
−0.0995385 + 0.995034i \(0.531737\pi\)
\(920\) 0 0
\(921\) 6558.26 0.234638
\(922\) 0 0
\(923\) 252.822 0.00901598
\(924\) 0 0
\(925\) −49099.5 −1.74528
\(926\) 0 0
\(927\) −4650.71 −0.164778
\(928\) 0 0
\(929\) −35684.5 −1.26025 −0.630125 0.776494i \(-0.716996\pi\)
−0.630125 + 0.776494i \(0.716996\pi\)
\(930\) 0 0
\(931\) 23061.1 0.811811
\(932\) 0 0
\(933\) −22454.5 −0.787918
\(934\) 0 0
\(935\) 3821.35 0.133659
\(936\) 0 0
\(937\) −48903.6 −1.70503 −0.852514 0.522705i \(-0.824923\pi\)
−0.852514 + 0.522705i \(0.824923\pi\)
\(938\) 0 0
\(939\) 20500.0 0.712451
\(940\) 0 0
\(941\) 23741.9 0.822490 0.411245 0.911525i \(-0.365094\pi\)
0.411245 + 0.911525i \(0.365094\pi\)
\(942\) 0 0
\(943\) 37437.4 1.29282
\(944\) 0 0
\(945\) −2438.16 −0.0839295
\(946\) 0 0
\(947\) 37612.4 1.29064 0.645321 0.763911i \(-0.276724\pi\)
0.645321 + 0.763911i \(0.276724\pi\)
\(948\) 0 0
\(949\) −4048.20 −0.138472
\(950\) 0 0
\(951\) 2772.80 0.0945469
\(952\) 0 0
\(953\) −48294.3 −1.64156 −0.820779 0.571246i \(-0.806460\pi\)
−0.820779 + 0.571246i \(0.806460\pi\)
\(954\) 0 0
\(955\) −6156.09 −0.208593
\(956\) 0 0
\(957\) 2291.96 0.0774176
\(958\) 0 0
\(959\) −28047.4 −0.944419
\(960\) 0 0
\(961\) −10021.4 −0.336389
\(962\) 0 0
\(963\) −1368.22 −0.0457843
\(964\) 0 0
\(965\) 4247.39 0.141687
\(966\) 0 0
\(967\) −1840.92 −0.0612204 −0.0306102 0.999531i \(-0.509745\pi\)
−0.0306102 + 0.999531i \(0.509745\pi\)
\(968\) 0 0
\(969\) −12748.6 −0.422647
\(970\) 0 0
\(971\) 31461.8 1.03981 0.519906 0.854223i \(-0.325967\pi\)
0.519906 + 0.854223i \(0.325967\pi\)
\(972\) 0 0
\(973\) 34612.1 1.14040
\(974\) 0 0
\(975\) −1806.26 −0.0593298
\(976\) 0 0
\(977\) −7040.11 −0.230535 −0.115268 0.993334i \(-0.536773\pi\)
−0.115268 + 0.993334i \(0.536773\pi\)
\(978\) 0 0
\(979\) 8058.15 0.263064
\(980\) 0 0
\(981\) −19532.9 −0.635715
\(982\) 0 0
\(983\) 24610.9 0.798541 0.399270 0.916833i \(-0.369263\pi\)
0.399270 + 0.916833i \(0.369263\pi\)
\(984\) 0 0
\(985\) −656.744 −0.0212443
\(986\) 0 0
\(987\) 22005.3 0.709662
\(988\) 0 0
\(989\) −37318.5 −1.19986
\(990\) 0 0
\(991\) 40003.3 1.28229 0.641144 0.767421i \(-0.278460\pi\)
0.641144 + 0.767421i \(0.278460\pi\)
\(992\) 0 0
\(993\) 29461.4 0.941519
\(994\) 0 0
\(995\) 63.8165 0.00203329
\(996\) 0 0
\(997\) 7342.61 0.233242 0.116621 0.993176i \(-0.462794\pi\)
0.116621 + 0.993176i \(0.462794\pi\)
\(998\) 0 0
\(999\) −11341.9 −0.359201
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 2112.4.a.bg.1.1 2
4.3 odd 2 2112.4.a.bn.1.1 2
8.3 odd 2 33.4.a.c.1.1 2
8.5 even 2 528.4.a.p.1.2 2
24.5 odd 2 1584.4.a.bj.1.1 2
24.11 even 2 99.4.a.f.1.2 2
40.3 even 4 825.4.c.h.199.3 4
40.19 odd 2 825.4.a.l.1.2 2
40.27 even 4 825.4.c.h.199.2 4
56.27 even 2 1617.4.a.k.1.1 2
88.43 even 2 363.4.a.i.1.2 2
120.59 even 2 2475.4.a.p.1.1 2
264.131 odd 2 1089.4.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.1 2 8.3 odd 2
99.4.a.f.1.2 2 24.11 even 2
363.4.a.i.1.2 2 88.43 even 2
528.4.a.p.1.2 2 8.5 even 2
825.4.a.l.1.2 2 40.19 odd 2
825.4.c.h.199.2 4 40.27 even 4
825.4.c.h.199.3 4 40.3 even 4
1089.4.a.u.1.1 2 264.131 odd 2
1584.4.a.bj.1.1 2 24.5 odd 2
1617.4.a.k.1.1 2 56.27 even 2
2112.4.a.bg.1.1 2 1.1 even 1 trivial
2112.4.a.bn.1.1 2 4.3 odd 2
2475.4.a.p.1.1 2 120.59 even 2