Properties

Label 1521.4.a.bk.1.7
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1521,4,Mod(1,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1521.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.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: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(2.04224\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.04224 q^{2} -3.82924 q^{4} +12.0825 q^{5} -29.7373 q^{7} -24.1582 q^{8} +O(q^{10})\) \(q+2.04224 q^{2} -3.82924 q^{4} +12.0825 q^{5} -29.7373 q^{7} -24.1582 q^{8} +24.6753 q^{10} +28.0636 q^{11} -60.7308 q^{14} -18.7029 q^{16} +50.6556 q^{17} -105.148 q^{19} -46.2667 q^{20} +57.3126 q^{22} +160.592 q^{23} +20.9857 q^{25} +113.871 q^{28} -140.105 q^{29} -223.593 q^{31} +155.070 q^{32} +103.451 q^{34} -359.300 q^{35} +228.352 q^{37} -214.739 q^{38} -291.890 q^{40} -295.902 q^{41} +192.103 q^{43} -107.462 q^{44} +327.968 q^{46} +36.9300 q^{47} +541.307 q^{49} +42.8579 q^{50} -149.102 q^{53} +339.077 q^{55} +718.399 q^{56} -286.128 q^{58} +438.867 q^{59} +286.146 q^{61} -456.631 q^{62} +466.313 q^{64} +537.128 q^{67} -193.973 q^{68} -733.777 q^{70} +102.729 q^{71} +75.5209 q^{73} +466.350 q^{74} +402.639 q^{76} -834.535 q^{77} +17.5526 q^{79} -225.977 q^{80} -604.304 q^{82} +1463.08 q^{83} +612.044 q^{85} +392.322 q^{86} -677.965 q^{88} -334.905 q^{89} -614.946 q^{92} +75.4200 q^{94} -1270.45 q^{95} +748.756 q^{97} +1105.48 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 60 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} + 3140 q^{64} - 1200 q^{68} + 4380 q^{74} - 4320 q^{77} + 1100 q^{79} - 4780 q^{82} + 6340 q^{88} + 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.04224 0.722042 0.361021 0.932558i \(-0.382428\pi\)
0.361021 + 0.932558i \(0.382428\pi\)
\(3\) 0 0
\(4\) −3.82924 −0.478656
\(5\) 12.0825 1.08069 0.540344 0.841444i \(-0.318294\pi\)
0.540344 + 0.841444i \(0.318294\pi\)
\(6\) 0 0
\(7\) −29.7373 −1.60566 −0.802832 0.596206i \(-0.796674\pi\)
−0.802832 + 0.596206i \(0.796674\pi\)
\(8\) −24.1582 −1.06765
\(9\) 0 0
\(10\) 24.6753 0.780302
\(11\) 28.0636 0.769226 0.384613 0.923078i \(-0.374335\pi\)
0.384613 + 0.923078i \(0.374335\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −60.7308 −1.15936
\(15\) 0 0
\(16\) −18.7029 −0.292233
\(17\) 50.6556 0.722693 0.361347 0.932432i \(-0.382317\pi\)
0.361347 + 0.932432i \(0.382317\pi\)
\(18\) 0 0
\(19\) −105.148 −1.26962 −0.634808 0.772670i \(-0.718921\pi\)
−0.634808 + 0.772670i \(0.718921\pi\)
\(20\) −46.2667 −0.517277
\(21\) 0 0
\(22\) 57.3126 0.555413
\(23\) 160.592 1.45590 0.727951 0.685629i \(-0.240473\pi\)
0.727951 + 0.685629i \(0.240473\pi\)
\(24\) 0 0
\(25\) 20.9857 0.167886
\(26\) 0 0
\(27\) 0 0
\(28\) 113.871 0.768560
\(29\) −140.105 −0.897132 −0.448566 0.893750i \(-0.648065\pi\)
−0.448566 + 0.893750i \(0.648065\pi\)
\(30\) 0 0
\(31\) −223.593 −1.29544 −0.647718 0.761880i \(-0.724276\pi\)
−0.647718 + 0.761880i \(0.724276\pi\)
\(32\) 155.070 0.856647
\(33\) 0 0
\(34\) 103.451 0.521815
\(35\) −359.300 −1.73522
\(36\) 0 0
\(37\) 228.352 1.01462 0.507308 0.861765i \(-0.330641\pi\)
0.507308 + 0.861765i \(0.330641\pi\)
\(38\) −214.739 −0.916716
\(39\) 0 0
\(40\) −291.890 −1.15380
\(41\) −295.902 −1.12713 −0.563563 0.826073i \(-0.690570\pi\)
−0.563563 + 0.826073i \(0.690570\pi\)
\(42\) 0 0
\(43\) 192.103 0.681291 0.340645 0.940192i \(-0.389354\pi\)
0.340645 + 0.940192i \(0.389354\pi\)
\(44\) −107.462 −0.368194
\(45\) 0 0
\(46\) 327.968 1.05122
\(47\) 36.9300 0.114613 0.0573063 0.998357i \(-0.481749\pi\)
0.0573063 + 0.998357i \(0.481749\pi\)
\(48\) 0 0
\(49\) 541.307 1.57815
\(50\) 42.8579 0.121220
\(51\) 0 0
\(52\) 0 0
\(53\) −149.102 −0.386429 −0.193214 0.981157i \(-0.561891\pi\)
−0.193214 + 0.981157i \(0.561891\pi\)
\(54\) 0 0
\(55\) 339.077 0.831293
\(56\) 718.399 1.71429
\(57\) 0 0
\(58\) −286.128 −0.647767
\(59\) 438.867 0.968400 0.484200 0.874957i \(-0.339111\pi\)
0.484200 + 0.874957i \(0.339111\pi\)
\(60\) 0 0
\(61\) 286.146 0.600610 0.300305 0.953843i \(-0.402912\pi\)
0.300305 + 0.953843i \(0.402912\pi\)
\(62\) −456.631 −0.935358
\(63\) 0 0
\(64\) 466.313 0.910768
\(65\) 0 0
\(66\) 0 0
\(67\) 537.128 0.979412 0.489706 0.871888i \(-0.337104\pi\)
0.489706 + 0.871888i \(0.337104\pi\)
\(68\) −193.973 −0.345921
\(69\) 0 0
\(70\) −733.777 −1.25290
\(71\) 102.729 0.171713 0.0858567 0.996307i \(-0.472637\pi\)
0.0858567 + 0.996307i \(0.472637\pi\)
\(72\) 0 0
\(73\) 75.5209 0.121083 0.0605414 0.998166i \(-0.480717\pi\)
0.0605414 + 0.998166i \(0.480717\pi\)
\(74\) 466.350 0.732596
\(75\) 0 0
\(76\) 402.639 0.607709
\(77\) −834.535 −1.23512
\(78\) 0 0
\(79\) 17.5526 0.0249978 0.0124989 0.999922i \(-0.496021\pi\)
0.0124989 + 0.999922i \(0.496021\pi\)
\(80\) −225.977 −0.315813
\(81\) 0 0
\(82\) −604.304 −0.813831
\(83\) 1463.08 1.93487 0.967434 0.253122i \(-0.0814573\pi\)
0.967434 + 0.253122i \(0.0814573\pi\)
\(84\) 0 0
\(85\) 612.044 0.781005
\(86\) 392.322 0.491920
\(87\) 0 0
\(88\) −677.965 −0.821265
\(89\) −334.905 −0.398875 −0.199438 0.979911i \(-0.563911\pi\)
−0.199438 + 0.979911i \(0.563911\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −614.946 −0.696876
\(93\) 0 0
\(94\) 75.4200 0.0827551
\(95\) −1270.45 −1.37206
\(96\) 0 0
\(97\) 748.756 0.783760 0.391880 0.920016i \(-0.371825\pi\)
0.391880 + 0.920016i \(0.371825\pi\)
\(98\) 1105.48 1.13949
\(99\) 0 0
\(100\) −80.3594 −0.0803594
\(101\) 784.002 0.772387 0.386194 0.922418i \(-0.373790\pi\)
0.386194 + 0.922418i \(0.373790\pi\)
\(102\) 0 0
\(103\) 396.040 0.378864 0.189432 0.981894i \(-0.439335\pi\)
0.189432 + 0.981894i \(0.439335\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −304.502 −0.279018
\(107\) 1436.59 1.29795 0.648974 0.760810i \(-0.275198\pi\)
0.648974 + 0.760810i \(0.275198\pi\)
\(108\) 0 0
\(109\) 1977.92 1.73807 0.869037 0.494746i \(-0.164739\pi\)
0.869037 + 0.494746i \(0.164739\pi\)
\(110\) 692.477 0.600228
\(111\) 0 0
\(112\) 556.175 0.469228
\(113\) −122.405 −0.101902 −0.0509509 0.998701i \(-0.516225\pi\)
−0.0509509 + 0.998701i \(0.516225\pi\)
\(114\) 0 0
\(115\) 1940.35 1.57338
\(116\) 536.496 0.429417
\(117\) 0 0
\(118\) 896.273 0.699225
\(119\) −1506.36 −1.16040
\(120\) 0 0
\(121\) −543.436 −0.408291
\(122\) 584.379 0.433665
\(123\) 0 0
\(124\) 856.192 0.620067
\(125\) −1256.75 −0.899256
\(126\) 0 0
\(127\) 2309.61 1.61374 0.806868 0.590731i \(-0.201161\pi\)
0.806868 + 0.590731i \(0.201161\pi\)
\(128\) −288.232 −0.199034
\(129\) 0 0
\(130\) 0 0
\(131\) 1444.26 0.963250 0.481625 0.876377i \(-0.340047\pi\)
0.481625 + 0.876377i \(0.340047\pi\)
\(132\) 0 0
\(133\) 3126.83 2.03858
\(134\) 1096.94 0.707176
\(135\) 0 0
\(136\) −1223.75 −0.771584
\(137\) −735.918 −0.458932 −0.229466 0.973317i \(-0.573698\pi\)
−0.229466 + 0.973317i \(0.573698\pi\)
\(138\) 0 0
\(139\) −1505.14 −0.918449 −0.459225 0.888320i \(-0.651873\pi\)
−0.459225 + 0.888320i \(0.651873\pi\)
\(140\) 1375.85 0.830573
\(141\) 0 0
\(142\) 209.797 0.123984
\(143\) 0 0
\(144\) 0 0
\(145\) −1692.81 −0.969520
\(146\) 154.232 0.0874269
\(147\) 0 0
\(148\) −874.415 −0.485652
\(149\) 427.843 0.235237 0.117618 0.993059i \(-0.462474\pi\)
0.117618 + 0.993059i \(0.462474\pi\)
\(150\) 0 0
\(151\) 1601.83 0.863278 0.431639 0.902046i \(-0.357935\pi\)
0.431639 + 0.902046i \(0.357935\pi\)
\(152\) 2540.20 1.35551
\(153\) 0 0
\(154\) −1704.32 −0.891807
\(155\) −2701.55 −1.39996
\(156\) 0 0
\(157\) −730.346 −0.371261 −0.185631 0.982620i \(-0.559433\pi\)
−0.185631 + 0.982620i \(0.559433\pi\)
\(158\) 35.8467 0.0180495
\(159\) 0 0
\(160\) 1873.62 0.925767
\(161\) −4775.58 −2.33769
\(162\) 0 0
\(163\) −1898.36 −0.912215 −0.456107 0.889925i \(-0.650757\pi\)
−0.456107 + 0.889925i \(0.650757\pi\)
\(164\) 1133.08 0.539505
\(165\) 0 0
\(166\) 2987.97 1.39706
\(167\) 1427.50 0.661457 0.330729 0.943726i \(-0.392706\pi\)
0.330729 + 0.943726i \(0.392706\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 1249.94 0.563919
\(171\) 0 0
\(172\) −735.611 −0.326104
\(173\) 2044.40 0.898454 0.449227 0.893418i \(-0.351699\pi\)
0.449227 + 0.893418i \(0.351699\pi\)
\(174\) 0 0
\(175\) −624.058 −0.269568
\(176\) −524.871 −0.224793
\(177\) 0 0
\(178\) −683.958 −0.288004
\(179\) 3889.72 1.62420 0.812098 0.583521i \(-0.198325\pi\)
0.812098 + 0.583521i \(0.198325\pi\)
\(180\) 0 0
\(181\) −2477.02 −1.01721 −0.508606 0.861000i \(-0.669839\pi\)
−0.508606 + 0.861000i \(0.669839\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −3879.61 −1.55440
\(185\) 2759.05 1.09648
\(186\) 0 0
\(187\) 1421.58 0.555914
\(188\) −141.414 −0.0548600
\(189\) 0 0
\(190\) −2594.57 −0.990683
\(191\) −2276.81 −0.862535 −0.431267 0.902224i \(-0.641933\pi\)
−0.431267 + 0.902224i \(0.641933\pi\)
\(192\) 0 0
\(193\) −3922.42 −1.46291 −0.731456 0.681888i \(-0.761159\pi\)
−0.731456 + 0.681888i \(0.761159\pi\)
\(194\) 1529.14 0.565907
\(195\) 0 0
\(196\) −2072.80 −0.755392
\(197\) −5063.23 −1.83117 −0.915584 0.402128i \(-0.868271\pi\)
−0.915584 + 0.402128i \(0.868271\pi\)
\(198\) 0 0
\(199\) 3270.06 1.16487 0.582433 0.812879i \(-0.302101\pi\)
0.582433 + 0.812879i \(0.302101\pi\)
\(200\) −506.977 −0.179243
\(201\) 0 0
\(202\) 1601.12 0.557696
\(203\) 4166.34 1.44049
\(204\) 0 0
\(205\) −3575.22 −1.21807
\(206\) 808.810 0.273555
\(207\) 0 0
\(208\) 0 0
\(209\) −2950.84 −0.976622
\(210\) 0 0
\(211\) −2812.18 −0.917527 −0.458764 0.888558i \(-0.651708\pi\)
−0.458764 + 0.888558i \(0.651708\pi\)
\(212\) 570.948 0.184966
\(213\) 0 0
\(214\) 2933.87 0.937173
\(215\) 2321.08 0.736262
\(216\) 0 0
\(217\) 6649.05 2.08003
\(218\) 4039.39 1.25496
\(219\) 0 0
\(220\) −1298.41 −0.397903
\(221\) 0 0
\(222\) 0 0
\(223\) 917.736 0.275588 0.137794 0.990461i \(-0.455999\pi\)
0.137794 + 0.990461i \(0.455999\pi\)
\(224\) −4611.35 −1.37549
\(225\) 0 0
\(226\) −249.981 −0.0735774
\(227\) −1336.39 −0.390746 −0.195373 0.980729i \(-0.562592\pi\)
−0.195373 + 0.980729i \(0.562592\pi\)
\(228\) 0 0
\(229\) −164.820 −0.0475617 −0.0237808 0.999717i \(-0.507570\pi\)
−0.0237808 + 0.999717i \(0.507570\pi\)
\(230\) 3962.66 1.13604
\(231\) 0 0
\(232\) 3384.68 0.957824
\(233\) 4243.42 1.19312 0.596558 0.802570i \(-0.296535\pi\)
0.596558 + 0.802570i \(0.296535\pi\)
\(234\) 0 0
\(235\) 446.205 0.123860
\(236\) −1680.53 −0.463530
\(237\) 0 0
\(238\) −3076.35 −0.837859
\(239\) −2491.07 −0.674200 −0.337100 0.941469i \(-0.609446\pi\)
−0.337100 + 0.941469i \(0.609446\pi\)
\(240\) 0 0
\(241\) −2917.40 −0.779776 −0.389888 0.920862i \(-0.627486\pi\)
−0.389888 + 0.920862i \(0.627486\pi\)
\(242\) −1109.83 −0.294803
\(243\) 0 0
\(244\) −1095.72 −0.287485
\(245\) 6540.32 1.70549
\(246\) 0 0
\(247\) 0 0
\(248\) 5401.60 1.38307
\(249\) 0 0
\(250\) −2566.58 −0.649300
\(251\) 1313.88 0.330403 0.165202 0.986260i \(-0.447173\pi\)
0.165202 + 0.986260i \(0.447173\pi\)
\(252\) 0 0
\(253\) 4506.79 1.11992
\(254\) 4716.78 1.16519
\(255\) 0 0
\(256\) −4319.15 −1.05448
\(257\) 987.582 0.239703 0.119851 0.992792i \(-0.461758\pi\)
0.119851 + 0.992792i \(0.461758\pi\)
\(258\) 0 0
\(259\) −6790.57 −1.62913
\(260\) 0 0
\(261\) 0 0
\(262\) 2949.53 0.695507
\(263\) 6986.45 1.63803 0.819017 0.573769i \(-0.194519\pi\)
0.819017 + 0.573769i \(0.194519\pi\)
\(264\) 0 0
\(265\) −1801.52 −0.417609
\(266\) 6385.74 1.47194
\(267\) 0 0
\(268\) −2056.79 −0.468801
\(269\) 5904.34 1.33827 0.669134 0.743142i \(-0.266665\pi\)
0.669134 + 0.743142i \(0.266665\pi\)
\(270\) 0 0
\(271\) 2131.54 0.477793 0.238897 0.971045i \(-0.423214\pi\)
0.238897 + 0.971045i \(0.423214\pi\)
\(272\) −947.408 −0.211195
\(273\) 0 0
\(274\) −1502.92 −0.331368
\(275\) 588.934 0.129142
\(276\) 0 0
\(277\) 4032.41 0.874673 0.437336 0.899298i \(-0.355922\pi\)
0.437336 + 0.899298i \(0.355922\pi\)
\(278\) −3073.86 −0.663159
\(279\) 0 0
\(280\) 8680.03 1.85261
\(281\) −2298.29 −0.487916 −0.243958 0.969786i \(-0.578446\pi\)
−0.243958 + 0.969786i \(0.578446\pi\)
\(282\) 0 0
\(283\) 6656.80 1.39825 0.699127 0.714998i \(-0.253572\pi\)
0.699127 + 0.714998i \(0.253572\pi\)
\(284\) −393.373 −0.0821916
\(285\) 0 0
\(286\) 0 0
\(287\) 8799.33 1.80978
\(288\) 0 0
\(289\) −2347.01 −0.477715
\(290\) −3457.13 −0.700034
\(291\) 0 0
\(292\) −289.188 −0.0579570
\(293\) 7466.99 1.48883 0.744413 0.667719i \(-0.232729\pi\)
0.744413 + 0.667719i \(0.232729\pi\)
\(294\) 0 0
\(295\) 5302.59 1.04654
\(296\) −5516.57 −1.08326
\(297\) 0 0
\(298\) 873.759 0.169851
\(299\) 0 0
\(300\) 0 0
\(301\) −5712.64 −1.09392
\(302\) 3271.32 0.623323
\(303\) 0 0
\(304\) 1966.58 0.371024
\(305\) 3457.34 0.649071
\(306\) 0 0
\(307\) 3965.99 0.737299 0.368650 0.929568i \(-0.379820\pi\)
0.368650 + 0.929568i \(0.379820\pi\)
\(308\) 3195.64 0.591196
\(309\) 0 0
\(310\) −5517.23 −1.01083
\(311\) −7372.29 −1.34419 −0.672097 0.740463i \(-0.734606\pi\)
−0.672097 + 0.740463i \(0.734606\pi\)
\(312\) 0 0
\(313\) 8249.55 1.48975 0.744875 0.667204i \(-0.232509\pi\)
0.744875 + 0.667204i \(0.232509\pi\)
\(314\) −1491.54 −0.268066
\(315\) 0 0
\(316\) −67.2133 −0.0119653
\(317\) −5575.26 −0.987817 −0.493909 0.869514i \(-0.664432\pi\)
−0.493909 + 0.869514i \(0.664432\pi\)
\(318\) 0 0
\(319\) −3931.85 −0.690098
\(320\) 5634.21 0.984256
\(321\) 0 0
\(322\) −9752.88 −1.68791
\(323\) −5326.35 −0.917543
\(324\) 0 0
\(325\) 0 0
\(326\) −3876.91 −0.658657
\(327\) 0 0
\(328\) 7148.46 1.20338
\(329\) −1098.20 −0.184029
\(330\) 0 0
\(331\) 4157.36 0.690361 0.345180 0.938536i \(-0.387818\pi\)
0.345180 + 0.938536i \(0.387818\pi\)
\(332\) −5602.50 −0.926136
\(333\) 0 0
\(334\) 2915.30 0.477600
\(335\) 6489.82 1.05844
\(336\) 0 0
\(337\) 3225.18 0.521326 0.260663 0.965430i \(-0.416059\pi\)
0.260663 + 0.965430i \(0.416059\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −2343.67 −0.373833
\(341\) −6274.82 −0.996483
\(342\) 0 0
\(343\) −5897.11 −0.928321
\(344\) −4640.87 −0.727381
\(345\) 0 0
\(346\) 4175.15 0.648721
\(347\) −3290.49 −0.509057 −0.254529 0.967065i \(-0.581920\pi\)
−0.254529 + 0.967065i \(0.581920\pi\)
\(348\) 0 0
\(349\) 4491.52 0.688899 0.344449 0.938805i \(-0.388066\pi\)
0.344449 + 0.938805i \(0.388066\pi\)
\(350\) −1274.48 −0.194639
\(351\) 0 0
\(352\) 4351.81 0.658955
\(353\) 5897.88 0.889270 0.444635 0.895712i \(-0.353333\pi\)
0.444635 + 0.895712i \(0.353333\pi\)
\(354\) 0 0
\(355\) 1241.21 0.185569
\(356\) 1282.43 0.190924
\(357\) 0 0
\(358\) 7943.75 1.17274
\(359\) −9277.20 −1.36388 −0.681938 0.731410i \(-0.738863\pi\)
−0.681938 + 0.731410i \(0.738863\pi\)
\(360\) 0 0
\(361\) 4197.19 0.611924
\(362\) −5058.67 −0.734469
\(363\) 0 0
\(364\) 0 0
\(365\) 912.477 0.130853
\(366\) 0 0
\(367\) −6574.36 −0.935092 −0.467546 0.883969i \(-0.654862\pi\)
−0.467546 + 0.883969i \(0.654862\pi\)
\(368\) −3003.54 −0.425463
\(369\) 0 0
\(370\) 5634.65 0.791707
\(371\) 4433.89 0.620474
\(372\) 0 0
\(373\) −5345.55 −0.742043 −0.371021 0.928624i \(-0.620992\pi\)
−0.371021 + 0.928624i \(0.620992\pi\)
\(374\) 2903.20 0.401393
\(375\) 0 0
\(376\) −892.162 −0.122366
\(377\) 0 0
\(378\) 0 0
\(379\) 1038.51 0.140751 0.0703757 0.997521i \(-0.477580\pi\)
0.0703757 + 0.997521i \(0.477580\pi\)
\(380\) 4864.87 0.656743
\(381\) 0 0
\(382\) −4649.80 −0.622786
\(383\) 6749.19 0.900437 0.450219 0.892918i \(-0.351346\pi\)
0.450219 + 0.892918i \(0.351346\pi\)
\(384\) 0 0
\(385\) −10083.2 −1.33478
\(386\) −8010.54 −1.05628
\(387\) 0 0
\(388\) −2867.17 −0.375151
\(389\) 1246.11 0.162417 0.0812083 0.996697i \(-0.474122\pi\)
0.0812083 + 0.996697i \(0.474122\pi\)
\(390\) 0 0
\(391\) 8134.89 1.05217
\(392\) −13077.0 −1.68492
\(393\) 0 0
\(394\) −10340.3 −1.32218
\(395\) 212.079 0.0270148
\(396\) 0 0
\(397\) −8355.69 −1.05632 −0.528161 0.849144i \(-0.677118\pi\)
−0.528161 + 0.849144i \(0.677118\pi\)
\(398\) 6678.25 0.841082
\(399\) 0 0
\(400\) −392.494 −0.0490618
\(401\) 3283.66 0.408923 0.204461 0.978875i \(-0.434456\pi\)
0.204461 + 0.978875i \(0.434456\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −3002.14 −0.369708
\(405\) 0 0
\(406\) 8508.68 1.04010
\(407\) 6408.37 0.780470
\(408\) 0 0
\(409\) −10928.9 −1.32127 −0.660636 0.750706i \(-0.729713\pi\)
−0.660636 + 0.750706i \(0.729713\pi\)
\(410\) −7301.47 −0.879497
\(411\) 0 0
\(412\) −1516.53 −0.181345
\(413\) −13050.7 −1.55492
\(414\) 0 0
\(415\) 17677.6 2.09099
\(416\) 0 0
\(417\) 0 0
\(418\) −6026.33 −0.705162
\(419\) −7302.94 −0.851485 −0.425742 0.904844i \(-0.639987\pi\)
−0.425742 + 0.904844i \(0.639987\pi\)
\(420\) 0 0
\(421\) −7580.99 −0.877612 −0.438806 0.898582i \(-0.644599\pi\)
−0.438806 + 0.898582i \(0.644599\pi\)
\(422\) −5743.15 −0.662493
\(423\) 0 0
\(424\) 3602.03 0.412571
\(425\) 1063.04 0.121330
\(426\) 0 0
\(427\) −8509.19 −0.964377
\(428\) −5501.06 −0.621270
\(429\) 0 0
\(430\) 4740.21 0.531612
\(431\) −10056.7 −1.12394 −0.561968 0.827159i \(-0.689956\pi\)
−0.561968 + 0.827159i \(0.689956\pi\)
\(432\) 0 0
\(433\) 2733.38 0.303367 0.151683 0.988429i \(-0.451531\pi\)
0.151683 + 0.988429i \(0.451531\pi\)
\(434\) 13579.0 1.50187
\(435\) 0 0
\(436\) −7573.93 −0.831939
\(437\) −16886.0 −1.84844
\(438\) 0 0
\(439\) 6744.23 0.733223 0.366611 0.930374i \(-0.380518\pi\)
0.366611 + 0.930374i \(0.380518\pi\)
\(440\) −8191.48 −0.887531
\(441\) 0 0
\(442\) 0 0
\(443\) −8655.69 −0.928317 −0.464158 0.885752i \(-0.653643\pi\)
−0.464158 + 0.885752i \(0.653643\pi\)
\(444\) 0 0
\(445\) −4046.48 −0.431059
\(446\) 1874.24 0.198986
\(447\) 0 0
\(448\) −13866.9 −1.46239
\(449\) −6522.46 −0.685555 −0.342777 0.939417i \(-0.611368\pi\)
−0.342777 + 0.939417i \(0.611368\pi\)
\(450\) 0 0
\(451\) −8304.07 −0.867014
\(452\) 468.719 0.0487759
\(453\) 0 0
\(454\) −2729.23 −0.282135
\(455\) 0 0
\(456\) 0 0
\(457\) −1551.23 −0.158782 −0.0793909 0.996844i \(-0.525298\pi\)
−0.0793909 + 0.996844i \(0.525298\pi\)
\(458\) −336.603 −0.0343415
\(459\) 0 0
\(460\) −7430.06 −0.753105
\(461\) −7766.25 −0.784621 −0.392310 0.919833i \(-0.628324\pi\)
−0.392310 + 0.919833i \(0.628324\pi\)
\(462\) 0 0
\(463\) −2004.52 −0.201205 −0.100603 0.994927i \(-0.532077\pi\)
−0.100603 + 0.994927i \(0.532077\pi\)
\(464\) 2620.37 0.262172
\(465\) 0 0
\(466\) 8666.10 0.861479
\(467\) −18674.3 −1.85042 −0.925209 0.379458i \(-0.876111\pi\)
−0.925209 + 0.379458i \(0.876111\pi\)
\(468\) 0 0
\(469\) −15972.7 −1.57261
\(470\) 911.259 0.0894324
\(471\) 0 0
\(472\) −10602.2 −1.03391
\(473\) 5391.11 0.524067
\(474\) 0 0
\(475\) −2206.61 −0.213150
\(476\) 5768.22 0.555433
\(477\) 0 0
\(478\) −5087.36 −0.486800
\(479\) 9313.02 0.888357 0.444178 0.895938i \(-0.353496\pi\)
0.444178 + 0.895938i \(0.353496\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −5958.03 −0.563031
\(483\) 0 0
\(484\) 2080.95 0.195431
\(485\) 9046.81 0.846999
\(486\) 0 0
\(487\) −3536.80 −0.329092 −0.164546 0.986369i \(-0.552616\pi\)
−0.164546 + 0.986369i \(0.552616\pi\)
\(488\) −6912.76 −0.641241
\(489\) 0 0
\(490\) 13356.9 1.23144
\(491\) 3361.78 0.308992 0.154496 0.987993i \(-0.450625\pi\)
0.154496 + 0.987993i \(0.450625\pi\)
\(492\) 0 0
\(493\) −7097.10 −0.648351
\(494\) 0 0
\(495\) 0 0
\(496\) 4181.84 0.378569
\(497\) −3054.87 −0.275714
\(498\) 0 0
\(499\) 4027.43 0.361308 0.180654 0.983547i \(-0.442179\pi\)
0.180654 + 0.983547i \(0.442179\pi\)
\(500\) 4812.40 0.430434
\(501\) 0 0
\(502\) 2683.26 0.238565
\(503\) −1766.67 −0.156604 −0.0783022 0.996930i \(-0.524950\pi\)
−0.0783022 + 0.996930i \(0.524950\pi\)
\(504\) 0 0
\(505\) 9472.67 0.834709
\(506\) 9203.96 0.808628
\(507\) 0 0
\(508\) −8844.05 −0.772424
\(509\) 6816.27 0.593567 0.296784 0.954945i \(-0.404086\pi\)
0.296784 + 0.954945i \(0.404086\pi\)
\(510\) 0 0
\(511\) −2245.79 −0.194418
\(512\) −6514.89 −0.562344
\(513\) 0 0
\(514\) 2016.88 0.173076
\(515\) 4785.13 0.409433
\(516\) 0 0
\(517\) 1036.39 0.0881630
\(518\) −13868.0 −1.17630
\(519\) 0 0
\(520\) 0 0
\(521\) 5442.27 0.457640 0.228820 0.973469i \(-0.426513\pi\)
0.228820 + 0.973469i \(0.426513\pi\)
\(522\) 0 0
\(523\) 20728.5 1.73307 0.866535 0.499117i \(-0.166342\pi\)
0.866535 + 0.499117i \(0.166342\pi\)
\(524\) −5530.43 −0.461065
\(525\) 0 0
\(526\) 14268.0 1.18273
\(527\) −11326.2 −0.936202
\(528\) 0 0
\(529\) 13622.8 1.11965
\(530\) −3679.13 −0.301531
\(531\) 0 0
\(532\) −11973.4 −0.975775
\(533\) 0 0
\(534\) 0 0
\(535\) 17357.5 1.40268
\(536\) −12976.0 −1.04567
\(537\) 0 0
\(538\) 12058.1 0.966285
\(539\) 15191.0 1.21396
\(540\) 0 0
\(541\) −8577.44 −0.681651 −0.340825 0.940127i \(-0.610706\pi\)
−0.340825 + 0.940127i \(0.610706\pi\)
\(542\) 4353.13 0.344987
\(543\) 0 0
\(544\) 7855.14 0.619093
\(545\) 23898.1 1.87832
\(546\) 0 0
\(547\) 8723.99 0.681921 0.340961 0.940078i \(-0.389248\pi\)
0.340961 + 0.940078i \(0.389248\pi\)
\(548\) 2818.01 0.219671
\(549\) 0 0
\(550\) 1202.75 0.0932459
\(551\) 14731.8 1.13901
\(552\) 0 0
\(553\) −521.968 −0.0401380
\(554\) 8235.17 0.631550
\(555\) 0 0
\(556\) 5763.56 0.439621
\(557\) −965.006 −0.0734087 −0.0367043 0.999326i \(-0.511686\pi\)
−0.0367043 + 0.999326i \(0.511686\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 6719.95 0.507089
\(561\) 0 0
\(562\) −4693.66 −0.352296
\(563\) 14605.2 1.09331 0.546657 0.837357i \(-0.315900\pi\)
0.546657 + 0.837357i \(0.315900\pi\)
\(564\) 0 0
\(565\) −1478.95 −0.110124
\(566\) 13594.8 1.00960
\(567\) 0 0
\(568\) −2481.74 −0.183330
\(569\) −7802.48 −0.574863 −0.287432 0.957801i \(-0.592801\pi\)
−0.287432 + 0.957801i \(0.592801\pi\)
\(570\) 0 0
\(571\) 11988.2 0.878618 0.439309 0.898336i \(-0.355223\pi\)
0.439309 + 0.898336i \(0.355223\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 17970.4 1.30674
\(575\) 3370.14 0.244425
\(576\) 0 0
\(577\) −5576.90 −0.402374 −0.201187 0.979553i \(-0.564480\pi\)
−0.201187 + 0.979553i \(0.564480\pi\)
\(578\) −4793.17 −0.344930
\(579\) 0 0
\(580\) 6482.19 0.464066
\(581\) −43508.1 −3.10675
\(582\) 0 0
\(583\) −4184.33 −0.297251
\(584\) −1824.45 −0.129274
\(585\) 0 0
\(586\) 15249.4 1.07499
\(587\) 26754.0 1.88119 0.940593 0.339535i \(-0.110270\pi\)
0.940593 + 0.339535i \(0.110270\pi\)
\(588\) 0 0
\(589\) 23510.5 1.64471
\(590\) 10829.2 0.755644
\(591\) 0 0
\(592\) −4270.85 −0.296505
\(593\) −3589.40 −0.248565 −0.124283 0.992247i \(-0.539663\pi\)
−0.124283 + 0.992247i \(0.539663\pi\)
\(594\) 0 0
\(595\) −18200.5 −1.25403
\(596\) −1638.32 −0.112597
\(597\) 0 0
\(598\) 0 0
\(599\) 7462.78 0.509050 0.254525 0.967066i \(-0.418081\pi\)
0.254525 + 0.967066i \(0.418081\pi\)
\(600\) 0 0
\(601\) −16511.0 −1.12063 −0.560316 0.828279i \(-0.689320\pi\)
−0.560316 + 0.828279i \(0.689320\pi\)
\(602\) −11666.6 −0.789858
\(603\) 0 0
\(604\) −6133.80 −0.413213
\(605\) −6566.04 −0.441235
\(606\) 0 0
\(607\) −11953.6 −0.799309 −0.399654 0.916666i \(-0.630870\pi\)
−0.399654 + 0.916666i \(0.630870\pi\)
\(608\) −16305.3 −1.08761
\(609\) 0 0
\(610\) 7060.73 0.468657
\(611\) 0 0
\(612\) 0 0
\(613\) 4575.65 0.301482 0.150741 0.988573i \(-0.451834\pi\)
0.150741 + 0.988573i \(0.451834\pi\)
\(614\) 8099.51 0.532361
\(615\) 0 0
\(616\) 20160.9 1.31868
\(617\) −19231.0 −1.25480 −0.627400 0.778697i \(-0.715881\pi\)
−0.627400 + 0.778697i \(0.715881\pi\)
\(618\) 0 0
\(619\) 11715.6 0.760727 0.380363 0.924837i \(-0.375799\pi\)
0.380363 + 0.924837i \(0.375799\pi\)
\(620\) 10344.9 0.670099
\(621\) 0 0
\(622\) −15056.0 −0.970564
\(623\) 9959.17 0.640459
\(624\) 0 0
\(625\) −17807.8 −1.13970
\(626\) 16847.6 1.07566
\(627\) 0 0
\(628\) 2796.68 0.177706
\(629\) 11567.3 0.733256
\(630\) 0 0
\(631\) 8780.09 0.553930 0.276965 0.960880i \(-0.410671\pi\)
0.276965 + 0.960880i \(0.410671\pi\)
\(632\) −424.040 −0.0266889
\(633\) 0 0
\(634\) −11386.0 −0.713245
\(635\) 27905.7 1.74395
\(636\) 0 0
\(637\) 0 0
\(638\) −8029.78 −0.498279
\(639\) 0 0
\(640\) −3482.55 −0.215094
\(641\) −24991.7 −1.53996 −0.769980 0.638068i \(-0.779734\pi\)
−0.769980 + 0.638068i \(0.779734\pi\)
\(642\) 0 0
\(643\) 2353.86 0.144365 0.0721827 0.997391i \(-0.477004\pi\)
0.0721827 + 0.997391i \(0.477004\pi\)
\(644\) 18286.8 1.11895
\(645\) 0 0
\(646\) −10877.7 −0.662504
\(647\) 5910.80 0.359162 0.179581 0.983743i \(-0.442526\pi\)
0.179581 + 0.983743i \(0.442526\pi\)
\(648\) 0 0
\(649\) 12316.2 0.744918
\(650\) 0 0
\(651\) 0 0
\(652\) 7269.28 0.436637
\(653\) −5924.34 −0.355034 −0.177517 0.984118i \(-0.556806\pi\)
−0.177517 + 0.984118i \(0.556806\pi\)
\(654\) 0 0
\(655\) 17450.2 1.04097
\(656\) 5534.23 0.329383
\(657\) 0 0
\(658\) −2242.79 −0.132877
\(659\) 12839.5 0.758964 0.379482 0.925199i \(-0.376102\pi\)
0.379482 + 0.925199i \(0.376102\pi\)
\(660\) 0 0
\(661\) −10265.5 −0.604057 −0.302028 0.953299i \(-0.597664\pi\)
−0.302028 + 0.953299i \(0.597664\pi\)
\(662\) 8490.35 0.498469
\(663\) 0 0
\(664\) −35345.4 −2.06577
\(665\) 37779.8 2.20306
\(666\) 0 0
\(667\) −22499.7 −1.30614
\(668\) −5466.25 −0.316610
\(669\) 0 0
\(670\) 13253.8 0.764236
\(671\) 8030.27 0.462004
\(672\) 0 0
\(673\) 9862.82 0.564909 0.282454 0.959281i \(-0.408851\pi\)
0.282454 + 0.959281i \(0.408851\pi\)
\(674\) 6586.61 0.376419
\(675\) 0 0
\(676\) 0 0
\(677\) −32615.5 −1.85158 −0.925788 0.378043i \(-0.876597\pi\)
−0.925788 + 0.378043i \(0.876597\pi\)
\(678\) 0 0
\(679\) −22266.0 −1.25845
\(680\) −14785.9 −0.833841
\(681\) 0 0
\(682\) −12814.7 −0.719502
\(683\) −21627.4 −1.21164 −0.605820 0.795602i \(-0.707155\pi\)
−0.605820 + 0.795602i \(0.707155\pi\)
\(684\) 0 0
\(685\) −8891.70 −0.495962
\(686\) −12043.3 −0.670286
\(687\) 0 0
\(688\) −3592.90 −0.199096
\(689\) 0 0
\(690\) 0 0
\(691\) −14233.1 −0.783580 −0.391790 0.920055i \(-0.628144\pi\)
−0.391790 + 0.920055i \(0.628144\pi\)
\(692\) −7828.49 −0.430050
\(693\) 0 0
\(694\) −6719.98 −0.367561
\(695\) −18185.8 −0.992557
\(696\) 0 0
\(697\) −14989.1 −0.814566
\(698\) 9172.78 0.497414
\(699\) 0 0
\(700\) 2389.67 0.129030
\(701\) 28747.0 1.54887 0.774437 0.632651i \(-0.218033\pi\)
0.774437 + 0.632651i \(0.218033\pi\)
\(702\) 0 0
\(703\) −24010.8 −1.28817
\(704\) 13086.4 0.700586
\(705\) 0 0
\(706\) 12044.9 0.642090
\(707\) −23314.1 −1.24019
\(708\) 0 0
\(709\) 1818.65 0.0963339 0.0481670 0.998839i \(-0.484662\pi\)
0.0481670 + 0.998839i \(0.484662\pi\)
\(710\) 2534.86 0.133988
\(711\) 0 0
\(712\) 8090.70 0.425859
\(713\) −35907.3 −1.88603
\(714\) 0 0
\(715\) 0 0
\(716\) −14894.7 −0.777431
\(717\) 0 0
\(718\) −18946.3 −0.984776
\(719\) −26141.8 −1.35595 −0.677973 0.735087i \(-0.737141\pi\)
−0.677973 + 0.735087i \(0.737141\pi\)
\(720\) 0 0
\(721\) −11777.2 −0.608328
\(722\) 8571.68 0.441835
\(723\) 0 0
\(724\) 9485.10 0.486894
\(725\) −2940.20 −0.150616
\(726\) 0 0
\(727\) −1340.10 −0.0683652 −0.0341826 0.999416i \(-0.510883\pi\)
−0.0341826 + 0.999416i \(0.510883\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1863.50 0.0944811
\(731\) 9731.11 0.492364
\(732\) 0 0
\(733\) 32517.1 1.63854 0.819269 0.573409i \(-0.194380\pi\)
0.819269 + 0.573409i \(0.194380\pi\)
\(734\) −13426.4 −0.675176
\(735\) 0 0
\(736\) 24903.0 1.24719
\(737\) 15073.7 0.753389
\(738\) 0 0
\(739\) 22170.5 1.10359 0.551797 0.833978i \(-0.313942\pi\)
0.551797 + 0.833978i \(0.313942\pi\)
\(740\) −10565.1 −0.524838
\(741\) 0 0
\(742\) 9055.07 0.448008
\(743\) −30387.8 −1.50043 −0.750216 0.661193i \(-0.770051\pi\)
−0.750216 + 0.661193i \(0.770051\pi\)
\(744\) 0 0
\(745\) 5169.39 0.254217
\(746\) −10916.9 −0.535786
\(747\) 0 0
\(748\) −5443.57 −0.266092
\(749\) −42720.3 −2.08407
\(750\) 0 0
\(751\) −16898.3 −0.821077 −0.410539 0.911843i \(-0.634659\pi\)
−0.410539 + 0.911843i \(0.634659\pi\)
\(752\) −690.699 −0.0334936
\(753\) 0 0
\(754\) 0 0
\(755\) 19354.0 0.932934
\(756\) 0 0
\(757\) 32925.8 1.58086 0.790428 0.612555i \(-0.209858\pi\)
0.790428 + 0.612555i \(0.209858\pi\)
\(758\) 2120.89 0.101628
\(759\) 0 0
\(760\) 30691.8 1.46488
\(761\) −16792.0 −0.799882 −0.399941 0.916541i \(-0.630969\pi\)
−0.399941 + 0.916541i \(0.630969\pi\)
\(762\) 0 0
\(763\) −58817.9 −2.79076
\(764\) 8718.46 0.412857
\(765\) 0 0
\(766\) 13783.5 0.650153
\(767\) 0 0
\(768\) 0 0
\(769\) 1541.57 0.0722891 0.0361445 0.999347i \(-0.488492\pi\)
0.0361445 + 0.999347i \(0.488492\pi\)
\(770\) −20592.4 −0.963765
\(771\) 0 0
\(772\) 15019.9 0.700231
\(773\) −38071.2 −1.77144 −0.885721 0.464218i \(-0.846335\pi\)
−0.885721 + 0.464218i \(0.846335\pi\)
\(774\) 0 0
\(775\) −4692.26 −0.217485
\(776\) −18088.6 −0.836782
\(777\) 0 0
\(778\) 2544.85 0.117272
\(779\) 31113.6 1.43102
\(780\) 0 0
\(781\) 2882.93 0.132086
\(782\) 16613.4 0.759712
\(783\) 0 0
\(784\) −10124.0 −0.461189
\(785\) −8824.38 −0.401217
\(786\) 0 0
\(787\) −20049.8 −0.908129 −0.454065 0.890969i \(-0.650027\pi\)
−0.454065 + 0.890969i \(0.650027\pi\)
\(788\) 19388.3 0.876498
\(789\) 0 0
\(790\) 433.117 0.0195058
\(791\) 3640.00 0.163620
\(792\) 0 0
\(793\) 0 0
\(794\) −17064.4 −0.762709
\(795\) 0 0
\(796\) −12521.9 −0.557570
\(797\) 22401.9 0.995627 0.497813 0.867284i \(-0.334136\pi\)
0.497813 + 0.867284i \(0.334136\pi\)
\(798\) 0 0
\(799\) 1870.71 0.0828298
\(800\) 3254.24 0.143819
\(801\) 0 0
\(802\) 6706.02 0.295259
\(803\) 2119.39 0.0931401
\(804\) 0 0
\(805\) −57700.7 −2.52631
\(806\) 0 0
\(807\) 0 0
\(808\) −18940.1 −0.824640
\(809\) 41966.4 1.82381 0.911903 0.410406i \(-0.134613\pi\)
0.911903 + 0.410406i \(0.134613\pi\)
\(810\) 0 0
\(811\) 13029.1 0.564133 0.282067 0.959395i \(-0.408980\pi\)
0.282067 + 0.959395i \(0.408980\pi\)
\(812\) −15953.9 −0.689500
\(813\) 0 0
\(814\) 13087.4 0.563532
\(815\) −22936.8 −0.985819
\(816\) 0 0
\(817\) −20199.4 −0.864977
\(818\) −22319.5 −0.954014
\(819\) 0 0
\(820\) 13690.4 0.583036
\(821\) 8898.14 0.378255 0.189127 0.981953i \(-0.439434\pi\)
0.189127 + 0.981953i \(0.439434\pi\)
\(822\) 0 0
\(823\) 22723.6 0.962448 0.481224 0.876598i \(-0.340192\pi\)
0.481224 + 0.876598i \(0.340192\pi\)
\(824\) −9567.61 −0.404494
\(825\) 0 0
\(826\) −26652.7 −1.12272
\(827\) 19073.3 0.801989 0.400994 0.916081i \(-0.368665\pi\)
0.400994 + 0.916081i \(0.368665\pi\)
\(828\) 0 0
\(829\) 42503.8 1.78072 0.890361 0.455255i \(-0.150452\pi\)
0.890361 + 0.455255i \(0.150452\pi\)
\(830\) 36102.0 1.50978
\(831\) 0 0
\(832\) 0 0
\(833\) 27420.2 1.14052
\(834\) 0 0
\(835\) 17247.7 0.714829
\(836\) 11299.5 0.467465
\(837\) 0 0
\(838\) −14914.4 −0.614808
\(839\) −19427.2 −0.799408 −0.399704 0.916644i \(-0.630887\pi\)
−0.399704 + 0.916644i \(0.630887\pi\)
\(840\) 0 0
\(841\) −4759.60 −0.195154
\(842\) −15482.2 −0.633673
\(843\) 0 0
\(844\) 10768.5 0.439180
\(845\) 0 0
\(846\) 0 0
\(847\) 16160.3 0.655578
\(848\) 2788.64 0.112927
\(849\) 0 0
\(850\) 2170.99 0.0876052
\(851\) 36671.5 1.47718
\(852\) 0 0
\(853\) −26851.8 −1.07783 −0.538914 0.842361i \(-0.681165\pi\)
−0.538914 + 0.842361i \(0.681165\pi\)
\(854\) −17377.8 −0.696320
\(855\) 0 0
\(856\) −34705.4 −1.38576
\(857\) −41539.4 −1.65573 −0.827864 0.560929i \(-0.810444\pi\)
−0.827864 + 0.560929i \(0.810444\pi\)
\(858\) 0 0
\(859\) −11936.2 −0.474107 −0.237054 0.971497i \(-0.576182\pi\)
−0.237054 + 0.971497i \(0.576182\pi\)
\(860\) −8887.99 −0.352416
\(861\) 0 0
\(862\) −20538.3 −0.811529
\(863\) 41128.6 1.62229 0.811143 0.584848i \(-0.198846\pi\)
0.811143 + 0.584848i \(0.198846\pi\)
\(864\) 0 0
\(865\) 24701.3 0.970948
\(866\) 5582.22 0.219043
\(867\) 0 0
\(868\) −25460.8 −0.995619
\(869\) 492.590 0.0192290
\(870\) 0 0
\(871\) 0 0
\(872\) −47782.9 −1.85566
\(873\) 0 0
\(874\) −34485.3 −1.33465
\(875\) 37372.3 1.44390
\(876\) 0 0
\(877\) 6406.86 0.246687 0.123343 0.992364i \(-0.460638\pi\)
0.123343 + 0.992364i \(0.460638\pi\)
\(878\) 13773.4 0.529417
\(879\) 0 0
\(880\) −6341.73 −0.242931
\(881\) 2938.07 0.112357 0.0561783 0.998421i \(-0.482108\pi\)
0.0561783 + 0.998421i \(0.482108\pi\)
\(882\) 0 0
\(883\) 3022.06 0.115176 0.0575881 0.998340i \(-0.481659\pi\)
0.0575881 + 0.998340i \(0.481659\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −17677.0 −0.670284
\(887\) 10060.4 0.380830 0.190415 0.981704i \(-0.439017\pi\)
0.190415 + 0.981704i \(0.439017\pi\)
\(888\) 0 0
\(889\) −68681.5 −2.59112
\(890\) −8263.89 −0.311243
\(891\) 0 0
\(892\) −3514.24 −0.131912
\(893\) −3883.13 −0.145514
\(894\) 0 0
\(895\) 46997.3 1.75525
\(896\) 8571.24 0.319582
\(897\) 0 0
\(898\) −13320.5 −0.494999
\(899\) 31326.5 1.16218
\(900\) 0 0
\(901\) −7552.84 −0.279269
\(902\) −16958.9 −0.626020
\(903\) 0 0
\(904\) 2957.09 0.108796
\(905\) −29928.4 −1.09929
\(906\) 0 0
\(907\) −43158.6 −1.58000 −0.789999 0.613108i \(-0.789919\pi\)
−0.789999 + 0.613108i \(0.789919\pi\)
\(908\) 5117.36 0.187033
\(909\) 0 0
\(910\) 0 0
\(911\) 32665.9 1.18800 0.594001 0.804464i \(-0.297547\pi\)
0.594001 + 0.804464i \(0.297547\pi\)
\(912\) 0 0
\(913\) 41059.3 1.48835
\(914\) −3167.98 −0.114647
\(915\) 0 0
\(916\) 631.137 0.0227657
\(917\) −42948.4 −1.54665
\(918\) 0 0
\(919\) 18989.9 0.681633 0.340816 0.940130i \(-0.389297\pi\)
0.340816 + 0.940130i \(0.389297\pi\)
\(920\) −46875.3 −1.67982
\(921\) 0 0
\(922\) −15860.6 −0.566529
\(923\) 0 0
\(924\) 0 0
\(925\) 4792.12 0.170340
\(926\) −4093.72 −0.145279
\(927\) 0 0
\(928\) −21726.0 −0.768525
\(929\) −5596.81 −0.197659 −0.0988295 0.995104i \(-0.531510\pi\)
−0.0988295 + 0.995104i \(0.531510\pi\)
\(930\) 0 0
\(931\) −56917.6 −2.00365
\(932\) −16249.1 −0.571091
\(933\) 0 0
\(934\) −38137.5 −1.33608
\(935\) 17176.1 0.600770
\(936\) 0 0
\(937\) 40294.4 1.40487 0.702433 0.711750i \(-0.252097\pi\)
0.702433 + 0.711750i \(0.252097\pi\)
\(938\) −32620.2 −1.13549
\(939\) 0 0
\(940\) −1708.63 −0.0592865
\(941\) 43648.8 1.51213 0.756063 0.654499i \(-0.227120\pi\)
0.756063 + 0.654499i \(0.227120\pi\)
\(942\) 0 0
\(943\) −47519.5 −1.64098
\(944\) −8208.10 −0.282999
\(945\) 0 0
\(946\) 11010.0 0.378398
\(947\) −10486.6 −0.359839 −0.179919 0.983681i \(-0.557584\pi\)
−0.179919 + 0.983681i \(0.557584\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −4506.44 −0.153903
\(951\) 0 0
\(952\) 36390.9 1.23890
\(953\) −33058.8 −1.12369 −0.561846 0.827242i \(-0.689909\pi\)
−0.561846 + 0.827242i \(0.689909\pi\)
\(954\) 0 0
\(955\) −27509.5 −0.932131
\(956\) 9538.90 0.322709
\(957\) 0 0
\(958\) 19019.5 0.641431
\(959\) 21884.2 0.736891
\(960\) 0 0
\(961\) 20202.8 0.678152
\(962\) 0 0
\(963\) 0 0
\(964\) 11171.4 0.373244
\(965\) −47392.5 −1.58095
\(966\) 0 0
\(967\) 53634.9 1.78364 0.891821 0.452389i \(-0.149428\pi\)
0.891821 + 0.452389i \(0.149428\pi\)
\(968\) 13128.4 0.435913
\(969\) 0 0
\(970\) 18475.8 0.611569
\(971\) 4086.80 0.135069 0.0675344 0.997717i \(-0.478487\pi\)
0.0675344 + 0.997717i \(0.478487\pi\)
\(972\) 0 0
\(973\) 44758.8 1.47472
\(974\) −7223.00 −0.237618
\(975\) 0 0
\(976\) −5351.76 −0.175518
\(977\) 14381.0 0.470919 0.235459 0.971884i \(-0.424341\pi\)
0.235459 + 0.971884i \(0.424341\pi\)
\(978\) 0 0
\(979\) −9398.64 −0.306825
\(980\) −25044.5 −0.816343
\(981\) 0 0
\(982\) 6865.57 0.223105
\(983\) −12916.5 −0.419099 −0.209549 0.977798i \(-0.567200\pi\)
−0.209549 + 0.977798i \(0.567200\pi\)
\(984\) 0 0
\(985\) −61176.2 −1.97892
\(986\) −14494.0 −0.468137
\(987\) 0 0
\(988\) 0 0
\(989\) 30850.3 0.991893
\(990\) 0 0
\(991\) −5838.98 −0.187166 −0.0935829 0.995611i \(-0.529832\pi\)
−0.0935829 + 0.995611i \(0.529832\pi\)
\(992\) −34672.5 −1.10973
\(993\) 0 0
\(994\) −6238.79 −0.199077
\(995\) 39510.3 1.25886
\(996\) 0 0
\(997\) 44290.1 1.40690 0.703452 0.710743i \(-0.251641\pi\)
0.703452 + 0.710743i \(0.251641\pi\)
\(998\) 8224.99 0.260879
\(999\) 0 0
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 1521.4.a.bk.1.7 10
3.2 odd 2 507.4.a.r.1.4 10
13.2 odd 12 117.4.q.e.82.4 10
13.7 odd 12 117.4.q.e.10.4 10
13.12 even 2 inner 1521.4.a.bk.1.4 10
39.2 even 12 39.4.j.c.4.2 10
39.5 even 4 507.4.b.i.337.7 10
39.8 even 4 507.4.b.i.337.4 10
39.20 even 12 39.4.j.c.10.2 yes 10
39.38 odd 2 507.4.a.r.1.7 10
156.59 odd 12 624.4.bv.h.49.2 10
156.119 odd 12 624.4.bv.h.433.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.2 10 39.2 even 12
39.4.j.c.10.2 yes 10 39.20 even 12
117.4.q.e.10.4 10 13.7 odd 12
117.4.q.e.82.4 10 13.2 odd 12
507.4.a.r.1.4 10 3.2 odd 2
507.4.a.r.1.7 10 39.38 odd 2
507.4.b.i.337.4 10 39.8 even 4
507.4.b.i.337.7 10 39.5 even 4
624.4.bv.h.49.2 10 156.59 odd 12
624.4.bv.h.433.4 10 156.119 odd 12
1521.4.a.bk.1.4 10 13.12 even 2 inner
1521.4.a.bk.1.7 10 1.1 even 1 trivial