Properties

Label 2352.4.a.bo.1.2
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
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] = [2352,4,Mod(1,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, 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("2352.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.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: \(138.772492334\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{505}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 126 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-10.7361\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +6.73610 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +6.73610 q^{5} +9.00000 q^{9} +53.6805 q^{11} -1.73610 q^{13} -20.2083 q^{15} +46.9444 q^{17} -20.1527 q^{19} -118.944 q^{23} -79.6249 q^{25} -27.0000 q^{27} -103.681 q^{29} -157.528 q^{31} -161.042 q^{33} -37.7361 q^{37} +5.20831 q^{39} +287.250 q^{41} -504.875 q^{43} +60.6249 q^{45} -220.500 q^{47} -140.833 q^{51} +292.319 q^{53} +361.597 q^{55} +60.4582 q^{57} -595.875 q^{59} +265.389 q^{61} -11.6946 q^{65} -936.735 q^{67} +356.833 q^{69} +545.944 q^{71} -299.374 q^{73} +238.875 q^{75} +940.333 q^{79} +81.0000 q^{81} +611.319 q^{83} +316.222 q^{85} +311.042 q^{87} +1176.19 q^{89} +472.583 q^{93} -135.751 q^{95} -1482.49 q^{97} +483.125 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 9 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 9 q^{5} + 18 q^{9} - 5 q^{11} + 19 q^{13} + 27 q^{15} + 4 q^{17} + 117 q^{19} - 148 q^{23} + 43 q^{25} - 54 q^{27} - 95 q^{29} - 360 q^{31} + 15 q^{33} - 53 q^{37} - 57 q^{39} + 170 q^{41} - 403 q^{43} - 81 q^{45} + 368 q^{47} - 12 q^{51} + 697 q^{53} + 1285 q^{55} - 351 q^{57} - 585 q^{59} + 1160 q^{61} - 338 q^{65} - 233 q^{67} + 444 q^{69} - 616 q^{71} + 817 q^{73} - 129 q^{75} + 802 q^{79} + 162 q^{81} - 283 q^{83} + 992 q^{85} + 285 q^{87} + 1858 q^{89} + 1080 q^{93} - 2294 q^{95} - 1729 q^{97} - 45 q^{99}+O(q^{100}) \) 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
\(5\) 6.73610 0.602495 0.301248 0.953546i \(-0.402597\pi\)
0.301248 + 0.953546i \(0.402597\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 53.6805 1.47139 0.735695 0.677313i \(-0.236856\pi\)
0.735695 + 0.677313i \(0.236856\pi\)
\(12\) 0 0
\(13\) −1.73610 −0.0370391 −0.0185195 0.999828i \(-0.505895\pi\)
−0.0185195 + 0.999828i \(0.505895\pi\)
\(14\) 0 0
\(15\) −20.2083 −0.347851
\(16\) 0 0
\(17\) 46.9444 0.669747 0.334873 0.942263i \(-0.391306\pi\)
0.334873 + 0.942263i \(0.391306\pi\)
\(18\) 0 0
\(19\) −20.1527 −0.243334 −0.121667 0.992571i \(-0.538824\pi\)
−0.121667 + 0.992571i \(0.538824\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −118.944 −1.07833 −0.539166 0.842200i \(-0.681260\pi\)
−0.539166 + 0.842200i \(0.681260\pi\)
\(24\) 0 0
\(25\) −79.6249 −0.636999
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −103.681 −0.663896 −0.331948 0.943298i \(-0.607706\pi\)
−0.331948 + 0.943298i \(0.607706\pi\)
\(30\) 0 0
\(31\) −157.528 −0.912672 −0.456336 0.889808i \(-0.650838\pi\)
−0.456336 + 0.889808i \(0.650838\pi\)
\(32\) 0 0
\(33\) −161.042 −0.849507
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −37.7361 −0.167670 −0.0838348 0.996480i \(-0.526717\pi\)
−0.0838348 + 0.996480i \(0.526717\pi\)
\(38\) 0 0
\(39\) 5.20831 0.0213845
\(40\) 0 0
\(41\) 287.250 1.09417 0.547084 0.837078i \(-0.315738\pi\)
0.547084 + 0.837078i \(0.315738\pi\)
\(42\) 0 0
\(43\) −504.875 −1.79053 −0.895264 0.445537i \(-0.853013\pi\)
−0.895264 + 0.445537i \(0.853013\pi\)
\(44\) 0 0
\(45\) 60.6249 0.200832
\(46\) 0 0
\(47\) −220.500 −0.684323 −0.342162 0.939641i \(-0.611159\pi\)
−0.342162 + 0.939641i \(0.611159\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −140.833 −0.386678
\(52\) 0 0
\(53\) 292.319 0.757607 0.378803 0.925477i \(-0.376336\pi\)
0.378803 + 0.925477i \(0.376336\pi\)
\(54\) 0 0
\(55\) 361.597 0.886505
\(56\) 0 0
\(57\) 60.4582 0.140489
\(58\) 0 0
\(59\) −595.875 −1.31485 −0.657426 0.753519i \(-0.728355\pi\)
−0.657426 + 0.753519i \(0.728355\pi\)
\(60\) 0 0
\(61\) 265.389 0.557043 0.278521 0.960430i \(-0.410156\pi\)
0.278521 + 0.960430i \(0.410156\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −11.6946 −0.0223159
\(66\) 0 0
\(67\) −936.735 −1.70807 −0.854033 0.520218i \(-0.825851\pi\)
−0.854033 + 0.520218i \(0.825851\pi\)
\(68\) 0 0
\(69\) 356.833 0.622575
\(70\) 0 0
\(71\) 545.944 0.912558 0.456279 0.889837i \(-0.349182\pi\)
0.456279 + 0.889837i \(0.349182\pi\)
\(72\) 0 0
\(73\) −299.374 −0.479988 −0.239994 0.970774i \(-0.577145\pi\)
−0.239994 + 0.970774i \(0.577145\pi\)
\(74\) 0 0
\(75\) 238.875 0.367772
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 940.333 1.33919 0.669593 0.742728i \(-0.266468\pi\)
0.669593 + 0.742728i \(0.266468\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 611.319 0.808445 0.404223 0.914661i \(-0.367542\pi\)
0.404223 + 0.914661i \(0.367542\pi\)
\(84\) 0 0
\(85\) 316.222 0.403519
\(86\) 0 0
\(87\) 311.042 0.383301
\(88\) 0 0
\(89\) 1176.19 1.40086 0.700429 0.713722i \(-0.252992\pi\)
0.700429 + 0.713722i \(0.252992\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 472.583 0.526931
\(94\) 0 0
\(95\) −135.751 −0.146608
\(96\) 0 0
\(97\) −1482.49 −1.55179 −0.775895 0.630862i \(-0.782701\pi\)
−0.775895 + 0.630862i \(0.782701\pi\)
\(98\) 0 0
\(99\) 483.125 0.490463
\(100\) 0 0
\(101\) −617.944 −0.608789 −0.304395 0.952546i \(-0.598454\pi\)
−0.304395 + 0.952546i \(0.598454\pi\)
\(102\) 0 0
\(103\) −816.764 −0.781341 −0.390670 0.920531i \(-0.627757\pi\)
−0.390670 + 0.920531i \(0.627757\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1685.38 −1.52272 −0.761361 0.648328i \(-0.775469\pi\)
−0.761361 + 0.648328i \(0.775469\pi\)
\(108\) 0 0
\(109\) 91.5968 0.0804898 0.0402449 0.999190i \(-0.487186\pi\)
0.0402449 + 0.999190i \(0.487186\pi\)
\(110\) 0 0
\(111\) 113.208 0.0968041
\(112\) 0 0
\(113\) 614.194 0.511314 0.255657 0.966767i \(-0.417708\pi\)
0.255657 + 0.966767i \(0.417708\pi\)
\(114\) 0 0
\(115\) −801.222 −0.649690
\(116\) 0 0
\(117\) −15.6249 −0.0123464
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1550.60 1.16499
\(122\) 0 0
\(123\) −861.750 −0.631718
\(124\) 0 0
\(125\) −1378.37 −0.986284
\(126\) 0 0
\(127\) −234.278 −0.163691 −0.0818457 0.996645i \(-0.526081\pi\)
−0.0818457 + 0.996645i \(0.526081\pi\)
\(128\) 0 0
\(129\) 1514.62 1.03376
\(130\) 0 0
\(131\) 2664.76 1.77726 0.888631 0.458622i \(-0.151657\pi\)
0.888631 + 0.458622i \(0.151657\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −181.875 −0.115950
\(136\) 0 0
\(137\) −1046.31 −0.652496 −0.326248 0.945284i \(-0.605784\pi\)
−0.326248 + 0.945284i \(0.605784\pi\)
\(138\) 0 0
\(139\) 647.265 0.394966 0.197483 0.980306i \(-0.436723\pi\)
0.197483 + 0.980306i \(0.436723\pi\)
\(140\) 0 0
\(141\) 661.499 0.395094
\(142\) 0 0
\(143\) −93.1949 −0.0544989
\(144\) 0 0
\(145\) −698.403 −0.399994
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1326.81 0.729505 0.364752 0.931105i \(-0.381154\pi\)
0.364752 + 0.931105i \(0.381154\pi\)
\(150\) 0 0
\(151\) −3683.24 −1.98502 −0.992508 0.122178i \(-0.961012\pi\)
−0.992508 + 0.122178i \(0.961012\pi\)
\(152\) 0 0
\(153\) 422.500 0.223249
\(154\) 0 0
\(155\) −1061.12 −0.549881
\(156\) 0 0
\(157\) 2818.25 1.43262 0.716308 0.697784i \(-0.245831\pi\)
0.716308 + 0.697784i \(0.245831\pi\)
\(158\) 0 0
\(159\) −876.958 −0.437405
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4106.61 −1.97334 −0.986670 0.162733i \(-0.947969\pi\)
−0.986670 + 0.162733i \(0.947969\pi\)
\(164\) 0 0
\(165\) −1084.79 −0.511824
\(166\) 0 0
\(167\) 1045.56 0.484476 0.242238 0.970217i \(-0.422119\pi\)
0.242238 + 0.970217i \(0.422119\pi\)
\(168\) 0 0
\(169\) −2193.99 −0.998628
\(170\) 0 0
\(171\) −181.374 −0.0811114
\(172\) 0 0
\(173\) 1910.31 0.839525 0.419763 0.907634i \(-0.362113\pi\)
0.419763 + 0.907634i \(0.362113\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1787.62 0.759130
\(178\) 0 0
\(179\) −231.140 −0.0965151 −0.0482576 0.998835i \(-0.515367\pi\)
−0.0482576 + 0.998835i \(0.515367\pi\)
\(180\) 0 0
\(181\) −3308.40 −1.35863 −0.679314 0.733848i \(-0.737722\pi\)
−0.679314 + 0.733848i \(0.737722\pi\)
\(182\) 0 0
\(183\) −796.167 −0.321609
\(184\) 0 0
\(185\) −254.194 −0.101020
\(186\) 0 0
\(187\) 2520.00 0.985458
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1800.75 −0.682187 −0.341093 0.940029i \(-0.610797\pi\)
−0.341093 + 0.940029i \(0.610797\pi\)
\(192\) 0 0
\(193\) −1283.36 −0.478644 −0.239322 0.970940i \(-0.576925\pi\)
−0.239322 + 0.970940i \(0.576925\pi\)
\(194\) 0 0
\(195\) 35.0837 0.0128841
\(196\) 0 0
\(197\) 60.7514 0.0219714 0.0109857 0.999940i \(-0.496503\pi\)
0.0109857 + 0.999940i \(0.496503\pi\)
\(198\) 0 0
\(199\) 3860.39 1.37515 0.687577 0.726112i \(-0.258675\pi\)
0.687577 + 0.726112i \(0.258675\pi\)
\(200\) 0 0
\(201\) 2810.21 0.986153
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1934.94 0.659231
\(206\) 0 0
\(207\) −1070.50 −0.359444
\(208\) 0 0
\(209\) −1081.81 −0.358039
\(210\) 0 0
\(211\) −491.637 −0.160406 −0.0802031 0.996779i \(-0.525557\pi\)
−0.0802031 + 0.996779i \(0.525557\pi\)
\(212\) 0 0
\(213\) −1637.83 −0.526866
\(214\) 0 0
\(215\) −3400.89 −1.07878
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 898.123 0.277121
\(220\) 0 0
\(221\) −81.5003 −0.0248068
\(222\) 0 0
\(223\) 823.376 0.247253 0.123626 0.992329i \(-0.460548\pi\)
0.123626 + 0.992329i \(0.460548\pi\)
\(224\) 0 0
\(225\) −716.624 −0.212333
\(226\) 0 0
\(227\) −3758.87 −1.09905 −0.549527 0.835476i \(-0.685192\pi\)
−0.549527 + 0.835476i \(0.685192\pi\)
\(228\) 0 0
\(229\) 2440.10 0.704132 0.352066 0.935975i \(-0.385479\pi\)
0.352066 + 0.935975i \(0.385479\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1949.86 −0.548239 −0.274120 0.961696i \(-0.588386\pi\)
−0.274120 + 0.961696i \(0.588386\pi\)
\(234\) 0 0
\(235\) −1485.31 −0.412301
\(236\) 0 0
\(237\) −2821.00 −0.773180
\(238\) 0 0
\(239\) −2705.64 −0.732273 −0.366137 0.930561i \(-0.619320\pi\)
−0.366137 + 0.930561i \(0.619320\pi\)
\(240\) 0 0
\(241\) −604.211 −0.161496 −0.0807482 0.996735i \(-0.525731\pi\)
−0.0807482 + 0.996735i \(0.525731\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 34.9872 0.00901288
\(248\) 0 0
\(249\) −1833.96 −0.466756
\(250\) 0 0
\(251\) −1631.82 −0.410356 −0.205178 0.978725i \(-0.565777\pi\)
−0.205178 + 0.978725i \(0.565777\pi\)
\(252\) 0 0
\(253\) −6385.00 −1.58665
\(254\) 0 0
\(255\) −948.667 −0.232972
\(256\) 0 0
\(257\) 7106.55 1.72488 0.862441 0.506158i \(-0.168935\pi\)
0.862441 + 0.506158i \(0.168935\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −933.125 −0.221299
\(262\) 0 0
\(263\) −5570.25 −1.30599 −0.652996 0.757361i \(-0.726488\pi\)
−0.652996 + 0.757361i \(0.726488\pi\)
\(264\) 0 0
\(265\) 1969.09 0.456455
\(266\) 0 0
\(267\) −3528.58 −0.808786
\(268\) 0 0
\(269\) −3913.18 −0.886955 −0.443477 0.896286i \(-0.646255\pi\)
−0.443477 + 0.896286i \(0.646255\pi\)
\(270\) 0 0
\(271\) 6072.51 1.36118 0.680588 0.732666i \(-0.261724\pi\)
0.680588 + 0.732666i \(0.261724\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4274.31 −0.937274
\(276\) 0 0
\(277\) 6140.62 1.33196 0.665982 0.745968i \(-0.268013\pi\)
0.665982 + 0.745968i \(0.268013\pi\)
\(278\) 0 0
\(279\) −1417.75 −0.304224
\(280\) 0 0
\(281\) −365.751 −0.0776472 −0.0388236 0.999246i \(-0.512361\pi\)
−0.0388236 + 0.999246i \(0.512361\pi\)
\(282\) 0 0
\(283\) −710.348 −0.149208 −0.0746039 0.997213i \(-0.523769\pi\)
−0.0746039 + 0.997213i \(0.523769\pi\)
\(284\) 0 0
\(285\) 407.252 0.0846440
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2709.22 −0.551440
\(290\) 0 0
\(291\) 4447.46 0.895926
\(292\) 0 0
\(293\) −296.373 −0.0590932 −0.0295466 0.999563i \(-0.509406\pi\)
−0.0295466 + 0.999563i \(0.509406\pi\)
\(294\) 0 0
\(295\) −4013.87 −0.792192
\(296\) 0 0
\(297\) −1449.37 −0.283169
\(298\) 0 0
\(299\) 206.500 0.0399404
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 1853.83 0.351485
\(304\) 0 0
\(305\) 1787.69 0.335616
\(306\) 0 0
\(307\) 8596.87 1.59821 0.799103 0.601194i \(-0.205308\pi\)
0.799103 + 0.601194i \(0.205308\pi\)
\(308\) 0 0
\(309\) 2450.29 0.451107
\(310\) 0 0
\(311\) −10475.3 −1.90997 −0.954984 0.296658i \(-0.904128\pi\)
−0.954984 + 0.296658i \(0.904128\pi\)
\(312\) 0 0
\(313\) −5496.52 −0.992594 −0.496297 0.868153i \(-0.665307\pi\)
−0.496297 + 0.868153i \(0.665307\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6556.37 1.16165 0.580824 0.814029i \(-0.302730\pi\)
0.580824 + 0.814029i \(0.302730\pi\)
\(318\) 0 0
\(319\) −5565.62 −0.976850
\(320\) 0 0
\(321\) 5056.13 0.879145
\(322\) 0 0
\(323\) −946.057 −0.162972
\(324\) 0 0
\(325\) 138.237 0.0235939
\(326\) 0 0
\(327\) −274.790 −0.0464708
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −102.376 −0.0170003 −0.00850017 0.999964i \(-0.502706\pi\)
−0.00850017 + 0.999964i \(0.502706\pi\)
\(332\) 0 0
\(333\) −339.625 −0.0558899
\(334\) 0 0
\(335\) −6309.95 −1.02910
\(336\) 0 0
\(337\) 1333.50 0.215549 0.107775 0.994175i \(-0.465627\pi\)
0.107775 + 0.994175i \(0.465627\pi\)
\(338\) 0 0
\(339\) −1842.58 −0.295208
\(340\) 0 0
\(341\) −8456.17 −1.34290
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2403.67 0.375099
\(346\) 0 0
\(347\) 1101.00 0.170331 0.0851655 0.996367i \(-0.472858\pi\)
0.0851655 + 0.996367i \(0.472858\pi\)
\(348\) 0 0
\(349\) −9467.89 −1.45216 −0.726081 0.687609i \(-0.758660\pi\)
−0.726081 + 0.687609i \(0.758660\pi\)
\(350\) 0 0
\(351\) 46.8748 0.00712818
\(352\) 0 0
\(353\) 1684.69 0.254015 0.127007 0.991902i \(-0.459463\pi\)
0.127007 + 0.991902i \(0.459463\pi\)
\(354\) 0 0
\(355\) 3677.53 0.549812
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6635.25 0.975474 0.487737 0.872991i \(-0.337823\pi\)
0.487737 + 0.872991i \(0.337823\pi\)
\(360\) 0 0
\(361\) −6452.87 −0.940788
\(362\) 0 0
\(363\) −4651.79 −0.672605
\(364\) 0 0
\(365\) −2016.62 −0.289191
\(366\) 0 0
\(367\) −7121.97 −1.01298 −0.506490 0.862246i \(-0.669057\pi\)
−0.506490 + 0.862246i \(0.669057\pi\)
\(368\) 0 0
\(369\) 2585.25 0.364723
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −13621.3 −1.89085 −0.945424 0.325843i \(-0.894352\pi\)
−0.945424 + 0.325843i \(0.894352\pi\)
\(374\) 0 0
\(375\) 4135.12 0.569432
\(376\) 0 0
\(377\) 180.000 0.0245901
\(378\) 0 0
\(379\) 3331.51 0.451525 0.225763 0.974182i \(-0.427513\pi\)
0.225763 + 0.974182i \(0.427513\pi\)
\(380\) 0 0
\(381\) 702.834 0.0945073
\(382\) 0 0
\(383\) 3275.75 0.437031 0.218515 0.975833i \(-0.429879\pi\)
0.218515 + 0.975833i \(0.429879\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4543.87 −0.596842
\(388\) 0 0
\(389\) −14975.4 −1.95188 −0.975941 0.218035i \(-0.930035\pi\)
−0.975941 + 0.218035i \(0.930035\pi\)
\(390\) 0 0
\(391\) −5583.78 −0.722209
\(392\) 0 0
\(393\) −7994.29 −1.02610
\(394\) 0 0
\(395\) 6334.18 0.806853
\(396\) 0 0
\(397\) 10758.6 1.36010 0.680050 0.733166i \(-0.261958\pi\)
0.680050 + 0.733166i \(0.261958\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2607.00 −0.324657 −0.162328 0.986737i \(-0.551900\pi\)
−0.162328 + 0.986737i \(0.551900\pi\)
\(402\) 0 0
\(403\) 273.484 0.0338045
\(404\) 0 0
\(405\) 545.624 0.0669439
\(406\) 0 0
\(407\) −2025.69 −0.246707
\(408\) 0 0
\(409\) −6214.64 −0.751330 −0.375665 0.926756i \(-0.622586\pi\)
−0.375665 + 0.926756i \(0.622586\pi\)
\(410\) 0 0
\(411\) 3138.92 0.376719
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4117.91 0.487085
\(416\) 0 0
\(417\) −1941.79 −0.228034
\(418\) 0 0
\(419\) −13111.4 −1.52872 −0.764358 0.644792i \(-0.776944\pi\)
−0.764358 + 0.644792i \(0.776944\pi\)
\(420\) 0 0
\(421\) −8410.12 −0.973596 −0.486798 0.873514i \(-0.661835\pi\)
−0.486798 + 0.873514i \(0.661835\pi\)
\(422\) 0 0
\(423\) −1984.50 −0.228108
\(424\) 0 0
\(425\) −3737.95 −0.426628
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 279.585 0.0314650
\(430\) 0 0
\(431\) −7783.31 −0.869858 −0.434929 0.900465i \(-0.643226\pi\)
−0.434929 + 0.900465i \(0.643226\pi\)
\(432\) 0 0
\(433\) −13870.1 −1.53939 −0.769695 0.638412i \(-0.779591\pi\)
−0.769695 + 0.638412i \(0.779591\pi\)
\(434\) 0 0
\(435\) 2095.21 0.230937
\(436\) 0 0
\(437\) 2397.05 0.262395
\(438\) 0 0
\(439\) 5679.63 0.617480 0.308740 0.951146i \(-0.400093\pi\)
0.308740 + 0.951146i \(0.400093\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2316.93 0.248489 0.124245 0.992252i \(-0.460349\pi\)
0.124245 + 0.992252i \(0.460349\pi\)
\(444\) 0 0
\(445\) 7922.97 0.844010
\(446\) 0 0
\(447\) −3980.42 −0.421180
\(448\) 0 0
\(449\) 9526.86 1.00134 0.500669 0.865639i \(-0.333088\pi\)
0.500669 + 0.865639i \(0.333088\pi\)
\(450\) 0 0
\(451\) 15419.7 1.60995
\(452\) 0 0
\(453\) 11049.7 1.14605
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16627.3 −1.70195 −0.850974 0.525208i \(-0.823987\pi\)
−0.850974 + 0.525208i \(0.823987\pi\)
\(458\) 0 0
\(459\) −1267.50 −0.128893
\(460\) 0 0
\(461\) 11338.3 1.14550 0.572749 0.819730i \(-0.305877\pi\)
0.572749 + 0.819730i \(0.305877\pi\)
\(462\) 0 0
\(463\) −9207.87 −0.924246 −0.462123 0.886816i \(-0.652912\pi\)
−0.462123 + 0.886816i \(0.652912\pi\)
\(464\) 0 0
\(465\) 3183.37 0.317474
\(466\) 0 0
\(467\) −17215.7 −1.70588 −0.852941 0.522007i \(-0.825184\pi\)
−0.852941 + 0.522007i \(0.825184\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −8454.74 −0.827121
\(472\) 0 0
\(473\) −27101.9 −2.63456
\(474\) 0 0
\(475\) 1604.66 0.155004
\(476\) 0 0
\(477\) 2630.88 0.252536
\(478\) 0 0
\(479\) 10293.8 0.981911 0.490956 0.871185i \(-0.336648\pi\)
0.490956 + 0.871185i \(0.336648\pi\)
\(480\) 0 0
\(481\) 65.5137 0.00621033
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −9986.18 −0.934946
\(486\) 0 0
\(487\) −11109.0 −1.03367 −0.516833 0.856086i \(-0.672889\pi\)
−0.516833 + 0.856086i \(0.672889\pi\)
\(488\) 0 0
\(489\) 12319.8 1.13931
\(490\) 0 0
\(491\) 6573.88 0.604226 0.302113 0.953272i \(-0.402308\pi\)
0.302113 + 0.953272i \(0.402308\pi\)
\(492\) 0 0
\(493\) −4867.22 −0.444642
\(494\) 0 0
\(495\) 3254.38 0.295502
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 16841.4 1.51087 0.755435 0.655223i \(-0.227425\pi\)
0.755435 + 0.655223i \(0.227425\pi\)
\(500\) 0 0
\(501\) −3136.67 −0.279712
\(502\) 0 0
\(503\) −18436.6 −1.63429 −0.817146 0.576431i \(-0.804445\pi\)
−0.817146 + 0.576431i \(0.804445\pi\)
\(504\) 0 0
\(505\) −4162.53 −0.366793
\(506\) 0 0
\(507\) 6581.96 0.576558
\(508\) 0 0
\(509\) −20825.2 −1.81348 −0.906738 0.421694i \(-0.861436\pi\)
−0.906738 + 0.421694i \(0.861436\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 544.123 0.0468297
\(514\) 0 0
\(515\) −5501.80 −0.470754
\(516\) 0 0
\(517\) −11836.5 −1.00691
\(518\) 0 0
\(519\) −5730.92 −0.484700
\(520\) 0 0
\(521\) −14384.7 −1.20961 −0.604805 0.796373i \(-0.706749\pi\)
−0.604805 + 0.796373i \(0.706749\pi\)
\(522\) 0 0
\(523\) 6487.49 0.542405 0.271203 0.962522i \(-0.412579\pi\)
0.271203 + 0.962522i \(0.412579\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7395.05 −0.611259
\(528\) 0 0
\(529\) 1980.77 0.162799
\(530\) 0 0
\(531\) −5362.87 −0.438284
\(532\) 0 0
\(533\) −498.695 −0.0405270
\(534\) 0 0
\(535\) −11352.9 −0.917433
\(536\) 0 0
\(537\) 693.420 0.0557230
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 14416.6 1.14569 0.572844 0.819664i \(-0.305840\pi\)
0.572844 + 0.819664i \(0.305840\pi\)
\(542\) 0 0
\(543\) 9925.21 0.784404
\(544\) 0 0
\(545\) 617.006 0.0484947
\(546\) 0 0
\(547\) 4881.38 0.381559 0.190780 0.981633i \(-0.438898\pi\)
0.190780 + 0.981633i \(0.438898\pi\)
\(548\) 0 0
\(549\) 2388.50 0.185681
\(550\) 0 0
\(551\) 2089.44 0.161549
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 762.583 0.0583240
\(556\) 0 0
\(557\) −3153.13 −0.239860 −0.119930 0.992782i \(-0.538267\pi\)
−0.119930 + 0.992782i \(0.538267\pi\)
\(558\) 0 0
\(559\) 876.514 0.0663195
\(560\) 0 0
\(561\) −7560.00 −0.568954
\(562\) 0 0
\(563\) 6795.87 0.508724 0.254362 0.967109i \(-0.418134\pi\)
0.254362 + 0.967109i \(0.418134\pi\)
\(564\) 0 0
\(565\) 4137.28 0.308065
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10708.3 −0.788955 −0.394477 0.918906i \(-0.629074\pi\)
−0.394477 + 0.918906i \(0.629074\pi\)
\(570\) 0 0
\(571\) −6531.66 −0.478706 −0.239353 0.970933i \(-0.576935\pi\)
−0.239353 + 0.970933i \(0.576935\pi\)
\(572\) 0 0
\(573\) 5402.25 0.393861
\(574\) 0 0
\(575\) 9470.94 0.686896
\(576\) 0 0
\(577\) −17461.9 −1.25987 −0.629937 0.776646i \(-0.716919\pi\)
−0.629937 + 0.776646i \(0.716919\pi\)
\(578\) 0 0
\(579\) 3850.08 0.276345
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 15691.9 1.11473
\(584\) 0 0
\(585\) −105.251 −0.00743863
\(586\) 0 0
\(587\) 9254.42 0.650717 0.325358 0.945591i \(-0.394515\pi\)
0.325358 + 0.945591i \(0.394515\pi\)
\(588\) 0 0
\(589\) 3174.61 0.222084
\(590\) 0 0
\(591\) −182.254 −0.0126852
\(592\) 0 0
\(593\) 21019.2 1.45557 0.727786 0.685804i \(-0.240549\pi\)
0.727786 + 0.685804i \(0.240549\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −11581.2 −0.793945
\(598\) 0 0
\(599\) 26971.7 1.83979 0.919894 0.392167i \(-0.128275\pi\)
0.919894 + 0.392167i \(0.128275\pi\)
\(600\) 0 0
\(601\) 26510.3 1.79930 0.899648 0.436616i \(-0.143823\pi\)
0.899648 + 0.436616i \(0.143823\pi\)
\(602\) 0 0
\(603\) −8430.62 −0.569355
\(604\) 0 0
\(605\) 10445.0 0.701899
\(606\) 0 0
\(607\) 29412.0 1.96671 0.983356 0.181687i \(-0.0581556\pi\)
0.983356 + 0.181687i \(0.0581556\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 382.810 0.0253467
\(612\) 0 0
\(613\) −6068.64 −0.399853 −0.199926 0.979811i \(-0.564070\pi\)
−0.199926 + 0.979811i \(0.564070\pi\)
\(614\) 0 0
\(615\) −5804.83 −0.380607
\(616\) 0 0
\(617\) 1904.56 0.124270 0.0621352 0.998068i \(-0.480209\pi\)
0.0621352 + 0.998068i \(0.480209\pi\)
\(618\) 0 0
\(619\) −406.478 −0.0263937 −0.0131969 0.999913i \(-0.504201\pi\)
−0.0131969 + 0.999913i \(0.504201\pi\)
\(620\) 0 0
\(621\) 3211.50 0.207525
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 668.244 0.0427676
\(626\) 0 0
\(627\) 3245.42 0.206714
\(628\) 0 0
\(629\) −1771.50 −0.112296
\(630\) 0 0
\(631\) 15294.2 0.964903 0.482452 0.875923i \(-0.339746\pi\)
0.482452 + 0.875923i \(0.339746\pi\)
\(632\) 0 0
\(633\) 1474.91 0.0926105
\(634\) 0 0
\(635\) −1578.12 −0.0986233
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 4913.49 0.304186
\(640\) 0 0
\(641\) −15722.2 −0.968782 −0.484391 0.874852i \(-0.660959\pi\)
−0.484391 + 0.874852i \(0.660959\pi\)
\(642\) 0 0
\(643\) 18529.9 1.13646 0.568232 0.822868i \(-0.307628\pi\)
0.568232 + 0.822868i \(0.307628\pi\)
\(644\) 0 0
\(645\) 10202.7 0.622836
\(646\) 0 0
\(647\) −8638.95 −0.524934 −0.262467 0.964941i \(-0.584536\pi\)
−0.262467 + 0.964941i \(0.584536\pi\)
\(648\) 0 0
\(649\) −31986.9 −1.93466
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27884.4 1.67106 0.835528 0.549448i \(-0.185162\pi\)
0.835528 + 0.549448i \(0.185162\pi\)
\(654\) 0 0
\(655\) 17950.1 1.07079
\(656\) 0 0
\(657\) −2694.37 −0.159996
\(658\) 0 0
\(659\) 5767.55 0.340928 0.170464 0.985364i \(-0.445473\pi\)
0.170464 + 0.985364i \(0.445473\pi\)
\(660\) 0 0
\(661\) −4090.12 −0.240677 −0.120338 0.992733i \(-0.538398\pi\)
−0.120338 + 0.992733i \(0.538398\pi\)
\(662\) 0 0
\(663\) 244.501 0.0143222
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12332.2 0.715900
\(668\) 0 0
\(669\) −2470.13 −0.142751
\(670\) 0 0
\(671\) 14246.2 0.819627
\(672\) 0 0
\(673\) −21667.7 −1.24106 −0.620528 0.784185i \(-0.713082\pi\)
−0.620528 + 0.784185i \(0.713082\pi\)
\(674\) 0 0
\(675\) 2149.87 0.122591
\(676\) 0 0
\(677\) 15318.9 0.869648 0.434824 0.900515i \(-0.356811\pi\)
0.434824 + 0.900515i \(0.356811\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 11276.6 0.634539
\(682\) 0 0
\(683\) −9981.36 −0.559189 −0.279595 0.960118i \(-0.590200\pi\)
−0.279595 + 0.960118i \(0.590200\pi\)
\(684\) 0 0
\(685\) −7048.02 −0.393126
\(686\) 0 0
\(687\) −7320.29 −0.406531
\(688\) 0 0
\(689\) −507.497 −0.0280611
\(690\) 0 0
\(691\) 23243.1 1.27961 0.639805 0.768537i \(-0.279015\pi\)
0.639805 + 0.768537i \(0.279015\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4360.04 0.237965
\(696\) 0 0
\(697\) 13484.8 0.732815
\(698\) 0 0
\(699\) 5849.59 0.316526
\(700\) 0 0
\(701\) −11295.6 −0.608598 −0.304299 0.952577i \(-0.598422\pi\)
−0.304299 + 0.952577i \(0.598422\pi\)
\(702\) 0 0
\(703\) 760.485 0.0407998
\(704\) 0 0
\(705\) 4455.93 0.238042
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6868.11 0.363804 0.181902 0.983317i \(-0.441775\pi\)
0.181902 + 0.983317i \(0.441775\pi\)
\(710\) 0 0
\(711\) 8463.00 0.446395
\(712\) 0 0
\(713\) 18737.1 0.984163
\(714\) 0 0
\(715\) −627.770 −0.0328353
\(716\) 0 0
\(717\) 8116.92 0.422778
\(718\) 0 0
\(719\) −31479.5 −1.63281 −0.816404 0.577481i \(-0.804036\pi\)
−0.816404 + 0.577481i \(0.804036\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1812.63 0.0932400
\(724\) 0 0
\(725\) 8255.55 0.422901
\(726\) 0 0
\(727\) 26045.8 1.32873 0.664363 0.747410i \(-0.268703\pi\)
0.664363 + 0.747410i \(0.268703\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −23701.0 −1.19920
\(732\) 0 0
\(733\) 3538.40 0.178300 0.0891499 0.996018i \(-0.471585\pi\)
0.0891499 + 0.996018i \(0.471585\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −50284.4 −2.51323
\(738\) 0 0
\(739\) −10426.9 −0.519024 −0.259512 0.965740i \(-0.583562\pi\)
−0.259512 + 0.965740i \(0.583562\pi\)
\(740\) 0 0
\(741\) −104.962 −0.00520359
\(742\) 0 0
\(743\) −2518.88 −0.124372 −0.0621862 0.998065i \(-0.519807\pi\)
−0.0621862 + 0.998065i \(0.519807\pi\)
\(744\) 0 0
\(745\) 8937.50 0.439523
\(746\) 0 0
\(747\) 5501.87 0.269482
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −14820.3 −0.720105 −0.360053 0.932932i \(-0.617241\pi\)
−0.360053 + 0.932932i \(0.617241\pi\)
\(752\) 0 0
\(753\) 4895.46 0.236919
\(754\) 0 0
\(755\) −24810.7 −1.19596
\(756\) 0 0
\(757\) 8892.84 0.426969 0.213485 0.976946i \(-0.431519\pi\)
0.213485 + 0.976946i \(0.431519\pi\)
\(758\) 0 0
\(759\) 19155.0 0.916050
\(760\) 0 0
\(761\) −22795.0 −1.08583 −0.542916 0.839787i \(-0.682680\pi\)
−0.542916 + 0.839787i \(0.682680\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2846.00 0.134506
\(766\) 0 0
\(767\) 1034.50 0.0487009
\(768\) 0 0
\(769\) 13025.5 0.610809 0.305405 0.952223i \(-0.401208\pi\)
0.305405 + 0.952223i \(0.401208\pi\)
\(770\) 0 0
\(771\) −21319.7 −0.995861
\(772\) 0 0
\(773\) −26467.0 −1.23150 −0.615751 0.787941i \(-0.711147\pi\)
−0.615751 + 0.787941i \(0.711147\pi\)
\(774\) 0 0
\(775\) 12543.1 0.581371
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5788.87 −0.266249
\(780\) 0 0
\(781\) 29306.5 1.34273
\(782\) 0 0
\(783\) 2799.37 0.127767
\(784\) 0 0
\(785\) 18984.0 0.863144
\(786\) 0 0
\(787\) 33216.6 1.50450 0.752252 0.658876i \(-0.228968\pi\)
0.752252 + 0.658876i \(0.228968\pi\)
\(788\) 0 0
\(789\) 16710.7 0.754015
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −460.743 −0.0206324
\(794\) 0 0
\(795\) −5907.28 −0.263534
\(796\) 0 0
\(797\) 15363.9 0.682830 0.341415 0.939913i \(-0.389094\pi\)
0.341415 + 0.939913i \(0.389094\pi\)
\(798\) 0 0
\(799\) −10351.2 −0.458323
\(800\) 0 0
\(801\) 10585.7 0.466953
\(802\) 0 0
\(803\) −16070.6 −0.706249
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 11739.5 0.512083
\(808\) 0 0
\(809\) 26742.7 1.16220 0.581102 0.813831i \(-0.302622\pi\)
0.581102 + 0.813831i \(0.302622\pi\)
\(810\) 0 0
\(811\) −23651.0 −1.02404 −0.512022 0.858973i \(-0.671103\pi\)
−0.512022 + 0.858973i \(0.671103\pi\)
\(812\) 0 0
\(813\) −18217.5 −0.785876
\(814\) 0 0
\(815\) −27662.5 −1.18893
\(816\) 0 0
\(817\) 10174.6 0.435697
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 16974.1 0.721558 0.360779 0.932651i \(-0.382511\pi\)
0.360779 + 0.932651i \(0.382511\pi\)
\(822\) 0 0
\(823\) −22328.9 −0.945730 −0.472865 0.881135i \(-0.656780\pi\)
−0.472865 + 0.881135i \(0.656780\pi\)
\(824\) 0 0
\(825\) 12822.9 0.541135
\(826\) 0 0
\(827\) −15731.8 −0.661485 −0.330743 0.943721i \(-0.607299\pi\)
−0.330743 + 0.943721i \(0.607299\pi\)
\(828\) 0 0
\(829\) 38025.4 1.59309 0.796547 0.604576i \(-0.206658\pi\)
0.796547 + 0.604576i \(0.206658\pi\)
\(830\) 0 0
\(831\) −18421.9 −0.769010
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 7042.97 0.291895
\(836\) 0 0
\(837\) 4253.25 0.175644
\(838\) 0 0
\(839\) 16546.5 0.680868 0.340434 0.940268i \(-0.389426\pi\)
0.340434 + 0.940268i \(0.389426\pi\)
\(840\) 0 0
\(841\) −13639.4 −0.559242
\(842\) 0 0
\(843\) 1097.25 0.0448296
\(844\) 0 0
\(845\) −14778.9 −0.601669
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 2131.04 0.0861452
\(850\) 0 0
\(851\) 4488.50 0.180803
\(852\) 0 0
\(853\) −13511.9 −0.542365 −0.271182 0.962528i \(-0.587415\pi\)
−0.271182 + 0.962528i \(0.587415\pi\)
\(854\) 0 0
\(855\) −1221.76 −0.0488692
\(856\) 0 0
\(857\) −17178.2 −0.684711 −0.342355 0.939570i \(-0.611225\pi\)
−0.342355 + 0.939570i \(0.611225\pi\)
\(858\) 0 0
\(859\) −35914.3 −1.42652 −0.713260 0.700899i \(-0.752782\pi\)
−0.713260 + 0.700899i \(0.752782\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 882.508 0.0348099 0.0174049 0.999849i \(-0.494460\pi\)
0.0174049 + 0.999849i \(0.494460\pi\)
\(864\) 0 0
\(865\) 12868.0 0.505810
\(866\) 0 0
\(867\) 8127.67 0.318374
\(868\) 0 0
\(869\) 50477.6 1.97046
\(870\) 0 0
\(871\) 1626.27 0.0632652
\(872\) 0 0
\(873\) −13342.4 −0.517263
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 17420.1 0.670737 0.335368 0.942087i \(-0.391139\pi\)
0.335368 + 0.942087i \(0.391139\pi\)
\(878\) 0 0
\(879\) 889.120 0.0341175
\(880\) 0 0
\(881\) −38097.8 −1.45692 −0.728460 0.685088i \(-0.759764\pi\)
−0.728460 + 0.685088i \(0.759764\pi\)
\(882\) 0 0
\(883\) 13870.6 0.528633 0.264317 0.964436i \(-0.414854\pi\)
0.264317 + 0.964436i \(0.414854\pi\)
\(884\) 0 0
\(885\) 12041.6 0.457372
\(886\) 0 0
\(887\) −17467.8 −0.661230 −0.330615 0.943766i \(-0.607256\pi\)
−0.330615 + 0.943766i \(0.607256\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4348.12 0.163488
\(892\) 0 0
\(893\) 4443.67 0.166519
\(894\) 0 0
\(895\) −1556.98 −0.0581499
\(896\) 0 0
\(897\) −619.499 −0.0230596
\(898\) 0 0
\(899\) 16332.6 0.605919
\(900\) 0 0
\(901\) 13722.8 0.507405
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −22285.7 −0.818567
\(906\) 0 0
\(907\) 28536.3 1.04469 0.522344 0.852735i \(-0.325058\pi\)
0.522344 + 0.852735i \(0.325058\pi\)
\(908\) 0 0
\(909\) −5561.49 −0.202930
\(910\) 0 0
\(911\) 50235.7 1.82698 0.913492 0.406857i \(-0.133375\pi\)
0.913492 + 0.406857i \(0.133375\pi\)
\(912\) 0 0
\(913\) 32815.9 1.18954
\(914\) 0 0
\(915\) −5363.07 −0.193768
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −5317.08 −0.190854 −0.0954268 0.995436i \(-0.530422\pi\)
−0.0954268 + 0.995436i \(0.530422\pi\)
\(920\) 0 0
\(921\) −25790.6 −0.922725
\(922\) 0 0
\(923\) −947.814 −0.0338003
\(924\) 0 0
\(925\) 3004.73 0.106805
\(926\) 0 0
\(927\) −7350.87 −0.260447
\(928\) 0 0
\(929\) −25493.1 −0.900325 −0.450162 0.892947i \(-0.648634\pi\)
−0.450162 + 0.892947i \(0.648634\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 31425.9 1.10272
\(934\) 0 0
\(935\) 16975.0 0.593734
\(936\) 0 0
\(937\) −12691.4 −0.442486 −0.221243 0.975219i \(-0.571011\pi\)
−0.221243 + 0.975219i \(0.571011\pi\)
\(938\) 0 0
\(939\) 16489.6 0.573074
\(940\) 0 0
\(941\) 15378.4 0.532753 0.266377 0.963869i \(-0.414174\pi\)
0.266377 + 0.963869i \(0.414174\pi\)
\(942\) 0 0
\(943\) −34166.8 −1.17988
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −50442.9 −1.73091 −0.865455 0.500986i \(-0.832971\pi\)
−0.865455 + 0.500986i \(0.832971\pi\)
\(948\) 0 0
\(949\) 519.745 0.0177783
\(950\) 0 0
\(951\) −19669.1 −0.670678
\(952\) 0 0
\(953\) 31787.3 1.08047 0.540237 0.841513i \(-0.318335\pi\)
0.540237 + 0.841513i \(0.318335\pi\)
\(954\) 0 0
\(955\) −12130.0 −0.411014
\(956\) 0 0
\(957\) 16696.9 0.563984
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −4975.99 −0.167030
\(962\) 0 0
\(963\) −15168.4 −0.507574
\(964\) 0 0
\(965\) −8644.84 −0.288381
\(966\) 0 0
\(967\) −1005.61 −0.0334417 −0.0167209 0.999860i \(-0.505323\pi\)
−0.0167209 + 0.999860i \(0.505323\pi\)
\(968\) 0 0
\(969\) 2838.17 0.0940921
\(970\) 0 0
\(971\) 7070.91 0.233693 0.116847 0.993150i \(-0.462721\pi\)
0.116847 + 0.993150i \(0.462721\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −414.711 −0.0136219
\(976\) 0 0
\(977\) −11097.1 −0.363384 −0.181692 0.983355i \(-0.558157\pi\)
−0.181692 + 0.983355i \(0.558157\pi\)
\(978\) 0 0
\(979\) 63138.7 2.06121
\(980\) 0 0
\(981\) 824.371 0.0268299
\(982\) 0 0
\(983\) −17775.5 −0.576755 −0.288377 0.957517i \(-0.593116\pi\)
−0.288377 + 0.957517i \(0.593116\pi\)
\(984\) 0 0
\(985\) 409.228 0.0132376
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 60052.0 1.93078
\(990\) 0 0
\(991\) −4691.86 −0.150395 −0.0751977 0.997169i \(-0.523959\pi\)
−0.0751977 + 0.997169i \(0.523959\pi\)
\(992\) 0 0
\(993\) 307.129 0.00981515
\(994\) 0 0
\(995\) 26004.0 0.828523
\(996\) 0 0
\(997\) 8471.39 0.269099 0.134549 0.990907i \(-0.457041\pi\)
0.134549 + 0.990907i \(0.457041\pi\)
\(998\) 0 0
\(999\) 1018.87 0.0322680
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 2352.4.a.bo.1.2 2
4.3 odd 2 1176.4.a.u.1.2 2
7.3 odd 6 336.4.q.g.289.2 4
7.5 odd 6 336.4.q.g.193.2 4
7.6 odd 2 2352.4.a.cc.1.1 2
28.3 even 6 168.4.q.d.121.2 yes 4
28.19 even 6 168.4.q.d.25.2 4
28.27 even 2 1176.4.a.r.1.1 2
84.47 odd 6 504.4.s.f.361.1 4
84.59 odd 6 504.4.s.f.289.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.d.25.2 4 28.19 even 6
168.4.q.d.121.2 yes 4 28.3 even 6
336.4.q.g.193.2 4 7.5 odd 6
336.4.q.g.289.2 4 7.3 odd 6
504.4.s.f.289.1 4 84.59 odd 6
504.4.s.f.361.1 4 84.47 odd 6
1176.4.a.r.1.1 2 28.27 even 2
1176.4.a.u.1.2 2 4.3 odd 2
2352.4.a.bo.1.2 2 1.1 even 1 trivial
2352.4.a.cc.1.1 2 7.6 odd 2