Properties

Label 1216.4.a.b.1.1
Level $1216$
Weight $4$
Character 1216.1
Self dual yes
Analytic conductor $71.746$
Analytic rank $0$
Dimension $1$
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] = [1216,4,Mod(1,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.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: \(71.7463225670\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1216.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} +9.00000 q^{5} +31.0000 q^{7} -23.0000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +9.00000 q^{5} +31.0000 q^{7} -23.0000 q^{9} +57.0000 q^{11} +52.0000 q^{13} -18.0000 q^{15} +69.0000 q^{17} +19.0000 q^{19} -62.0000 q^{21} +72.0000 q^{23} -44.0000 q^{25} +100.000 q^{27} +150.000 q^{29} -32.0000 q^{31} -114.000 q^{33} +279.000 q^{35} +226.000 q^{37} -104.000 q^{39} -258.000 q^{41} -67.0000 q^{43} -207.000 q^{45} -579.000 q^{47} +618.000 q^{49} -138.000 q^{51} +432.000 q^{53} +513.000 q^{55} -38.0000 q^{57} -330.000 q^{59} +13.0000 q^{61} -713.000 q^{63} +468.000 q^{65} -856.000 q^{67} -144.000 q^{69} -642.000 q^{71} -487.000 q^{73} +88.0000 q^{75} +1767.00 q^{77} +700.000 q^{79} +421.000 q^{81} -12.0000 q^{83} +621.000 q^{85} -300.000 q^{87} -600.000 q^{89} +1612.00 q^{91} +64.0000 q^{93} +171.000 q^{95} +1424.00 q^{97} -1311.00 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.00000 −0.384900 −0.192450 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(4\) 0 0
\(5\) 9.00000 0.804984 0.402492 0.915423i \(-0.368144\pi\)
0.402492 + 0.915423i \(0.368144\pi\)
\(6\) 0 0
\(7\) 31.0000 1.67384 0.836921 0.547323i \(-0.184353\pi\)
0.836921 + 0.547323i \(0.184353\pi\)
\(8\) 0 0
\(9\) −23.0000 −0.851852
\(10\) 0 0
\(11\) 57.0000 1.56238 0.781188 0.624295i \(-0.214614\pi\)
0.781188 + 0.624295i \(0.214614\pi\)
\(12\) 0 0
\(13\) 52.0000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) −18.0000 −0.309839
\(16\) 0 0
\(17\) 69.0000 0.984409 0.492205 0.870480i \(-0.336191\pi\)
0.492205 + 0.870480i \(0.336191\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416
\(20\) 0 0
\(21\) −62.0000 −0.644262
\(22\) 0 0
\(23\) 72.0000 0.652741 0.326370 0.945242i \(-0.394174\pi\)
0.326370 + 0.945242i \(0.394174\pi\)
\(24\) 0 0
\(25\) −44.0000 −0.352000
\(26\) 0 0
\(27\) 100.000 0.712778
\(28\) 0 0
\(29\) 150.000 0.960493 0.480247 0.877134i \(-0.340547\pi\)
0.480247 + 0.877134i \(0.340547\pi\)
\(30\) 0 0
\(31\) −32.0000 −0.185399 −0.0926995 0.995694i \(-0.529550\pi\)
−0.0926995 + 0.995694i \(0.529550\pi\)
\(32\) 0 0
\(33\) −114.000 −0.601359
\(34\) 0 0
\(35\) 279.000 1.34742
\(36\) 0 0
\(37\) 226.000 1.00417 0.502083 0.864819i \(-0.332567\pi\)
0.502083 + 0.864819i \(0.332567\pi\)
\(38\) 0 0
\(39\) −104.000 −0.427008
\(40\) 0 0
\(41\) −258.000 −0.982752 −0.491376 0.870948i \(-0.663506\pi\)
−0.491376 + 0.870948i \(0.663506\pi\)
\(42\) 0 0
\(43\) −67.0000 −0.237614 −0.118807 0.992917i \(-0.537907\pi\)
−0.118807 + 0.992917i \(0.537907\pi\)
\(44\) 0 0
\(45\) −207.000 −0.685728
\(46\) 0 0
\(47\) −579.000 −1.79693 −0.898466 0.439043i \(-0.855318\pi\)
−0.898466 + 0.439043i \(0.855318\pi\)
\(48\) 0 0
\(49\) 618.000 1.80175
\(50\) 0 0
\(51\) −138.000 −0.378899
\(52\) 0 0
\(53\) 432.000 1.11962 0.559809 0.828622i \(-0.310874\pi\)
0.559809 + 0.828622i \(0.310874\pi\)
\(54\) 0 0
\(55\) 513.000 1.25769
\(56\) 0 0
\(57\) −38.0000 −0.0883022
\(58\) 0 0
\(59\) −330.000 −0.728175 −0.364088 0.931365i \(-0.618619\pi\)
−0.364088 + 0.931365i \(0.618619\pi\)
\(60\) 0 0
\(61\) 13.0000 0.0272865 0.0136433 0.999907i \(-0.495657\pi\)
0.0136433 + 0.999907i \(0.495657\pi\)
\(62\) 0 0
\(63\) −713.000 −1.42587
\(64\) 0 0
\(65\) 468.000 0.893050
\(66\) 0 0
\(67\) −856.000 −1.56085 −0.780426 0.625249i \(-0.784998\pi\)
−0.780426 + 0.625249i \(0.784998\pi\)
\(68\) 0 0
\(69\) −144.000 −0.251240
\(70\) 0 0
\(71\) −642.000 −1.07312 −0.536559 0.843863i \(-0.680276\pi\)
−0.536559 + 0.843863i \(0.680276\pi\)
\(72\) 0 0
\(73\) −487.000 −0.780809 −0.390404 0.920643i \(-0.627665\pi\)
−0.390404 + 0.920643i \(0.627665\pi\)
\(74\) 0 0
\(75\) 88.0000 0.135485
\(76\) 0 0
\(77\) 1767.00 2.61517
\(78\) 0 0
\(79\) 700.000 0.996913 0.498457 0.866915i \(-0.333900\pi\)
0.498457 + 0.866915i \(0.333900\pi\)
\(80\) 0 0
\(81\) 421.000 0.577503
\(82\) 0 0
\(83\) −12.0000 −0.0158695 −0.00793477 0.999969i \(-0.502526\pi\)
−0.00793477 + 0.999969i \(0.502526\pi\)
\(84\) 0 0
\(85\) 621.000 0.792434
\(86\) 0 0
\(87\) −300.000 −0.369694
\(88\) 0 0
\(89\) −600.000 −0.714605 −0.357303 0.933989i \(-0.616304\pi\)
−0.357303 + 0.933989i \(0.616304\pi\)
\(90\) 0 0
\(91\) 1612.00 1.85696
\(92\) 0 0
\(93\) 64.0000 0.0713601
\(94\) 0 0
\(95\) 171.000 0.184676
\(96\) 0 0
\(97\) 1424.00 1.49057 0.745285 0.666746i \(-0.232313\pi\)
0.745285 + 0.666746i \(0.232313\pi\)
\(98\) 0 0
\(99\) −1311.00 −1.33091
\(100\) 0 0
\(101\) −1062.00 −1.04627 −0.523133 0.852251i \(-0.675237\pi\)
−0.523133 + 0.852251i \(0.675237\pi\)
\(102\) 0 0
\(103\) −1178.00 −1.12691 −0.563455 0.826147i \(-0.690528\pi\)
−0.563455 + 0.826147i \(0.690528\pi\)
\(104\) 0 0
\(105\) −558.000 −0.518621
\(106\) 0 0
\(107\) 114.000 0.102998 0.0514990 0.998673i \(-0.483600\pi\)
0.0514990 + 0.998673i \(0.483600\pi\)
\(108\) 0 0
\(109\) −1460.00 −1.28296 −0.641480 0.767140i \(-0.721679\pi\)
−0.641480 + 0.767140i \(0.721679\pi\)
\(110\) 0 0
\(111\) −452.000 −0.386504
\(112\) 0 0
\(113\) −822.000 −0.684312 −0.342156 0.939643i \(-0.611157\pi\)
−0.342156 + 0.939643i \(0.611157\pi\)
\(114\) 0 0
\(115\) 648.000 0.525446
\(116\) 0 0
\(117\) −1196.00 −0.945045
\(118\) 0 0
\(119\) 2139.00 1.64775
\(120\) 0 0
\(121\) 1918.00 1.44102
\(122\) 0 0
\(123\) 516.000 0.378261
\(124\) 0 0
\(125\) −1521.00 −1.08834
\(126\) 0 0
\(127\) 2086.00 1.45750 0.728750 0.684780i \(-0.240102\pi\)
0.728750 + 0.684780i \(0.240102\pi\)
\(128\) 0 0
\(129\) 134.000 0.0914577
\(130\) 0 0
\(131\) −93.0000 −0.0620263 −0.0310132 0.999519i \(-0.509873\pi\)
−0.0310132 + 0.999519i \(0.509873\pi\)
\(132\) 0 0
\(133\) 589.000 0.384006
\(134\) 0 0
\(135\) 900.000 0.573775
\(136\) 0 0
\(137\) 1269.00 0.791372 0.395686 0.918386i \(-0.370507\pi\)
0.395686 + 0.918386i \(0.370507\pi\)
\(138\) 0 0
\(139\) −1975.00 −1.20516 −0.602580 0.798058i \(-0.705861\pi\)
−0.602580 + 0.798058i \(0.705861\pi\)
\(140\) 0 0
\(141\) 1158.00 0.691640
\(142\) 0 0
\(143\) 2964.00 1.73330
\(144\) 0 0
\(145\) 1350.00 0.773182
\(146\) 0 0
\(147\) −1236.00 −0.693494
\(148\) 0 0
\(149\) 1695.00 0.931945 0.465973 0.884799i \(-0.345705\pi\)
0.465973 + 0.884799i \(0.345705\pi\)
\(150\) 0 0
\(151\) −1802.00 −0.971157 −0.485578 0.874193i \(-0.661391\pi\)
−0.485578 + 0.874193i \(0.661391\pi\)
\(152\) 0 0
\(153\) −1587.00 −0.838571
\(154\) 0 0
\(155\) −288.000 −0.149243
\(156\) 0 0
\(157\) 3226.00 1.63989 0.819945 0.572442i \(-0.194004\pi\)
0.819945 + 0.572442i \(0.194004\pi\)
\(158\) 0 0
\(159\) −864.000 −0.430941
\(160\) 0 0
\(161\) 2232.00 1.09259
\(162\) 0 0
\(163\) 1268.00 0.609309 0.304655 0.952463i \(-0.401459\pi\)
0.304655 + 0.952463i \(0.401459\pi\)
\(164\) 0 0
\(165\) −1026.00 −0.484085
\(166\) 0 0
\(167\) −654.000 −0.303042 −0.151521 0.988454i \(-0.548417\pi\)
−0.151521 + 0.988454i \(0.548417\pi\)
\(168\) 0 0
\(169\) 507.000 0.230769
\(170\) 0 0
\(171\) −437.000 −0.195428
\(172\) 0 0
\(173\) 1362.00 0.598560 0.299280 0.954165i \(-0.403253\pi\)
0.299280 + 0.954165i \(0.403253\pi\)
\(174\) 0 0
\(175\) −1364.00 −0.589193
\(176\) 0 0
\(177\) 660.000 0.280275
\(178\) 0 0
\(179\) −210.000 −0.0876879 −0.0438440 0.999038i \(-0.513960\pi\)
−0.0438440 + 0.999038i \(0.513960\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.000821319 0 −0.000410660 1.00000i \(-0.500131\pi\)
−0.000410660 1.00000i \(0.500131\pi\)
\(182\) 0 0
\(183\) −26.0000 −0.0105026
\(184\) 0 0
\(185\) 2034.00 0.808339
\(186\) 0 0
\(187\) 3933.00 1.53802
\(188\) 0 0
\(189\) 3100.00 1.19308
\(190\) 0 0
\(191\) 2643.00 1.00126 0.500630 0.865661i \(-0.333102\pi\)
0.500630 + 0.865661i \(0.333102\pi\)
\(192\) 0 0
\(193\) 3248.00 1.21138 0.605690 0.795701i \(-0.292897\pi\)
0.605690 + 0.795701i \(0.292897\pi\)
\(194\) 0 0
\(195\) −936.000 −0.343735
\(196\) 0 0
\(197\) 3126.00 1.13055 0.565275 0.824903i \(-0.308770\pi\)
0.565275 + 0.824903i \(0.308770\pi\)
\(198\) 0 0
\(199\) 2995.00 1.06688 0.533442 0.845837i \(-0.320898\pi\)
0.533442 + 0.845837i \(0.320898\pi\)
\(200\) 0 0
\(201\) 1712.00 0.600772
\(202\) 0 0
\(203\) 4650.00 1.60771
\(204\) 0 0
\(205\) −2322.00 −0.791100
\(206\) 0 0
\(207\) −1656.00 −0.556038
\(208\) 0 0
\(209\) 1083.00 0.358434
\(210\) 0 0
\(211\) −4318.00 −1.40883 −0.704416 0.709788i \(-0.748791\pi\)
−0.704416 + 0.709788i \(0.748791\pi\)
\(212\) 0 0
\(213\) 1284.00 0.413043
\(214\) 0 0
\(215\) −603.000 −0.191276
\(216\) 0 0
\(217\) −992.000 −0.310329
\(218\) 0 0
\(219\) 974.000 0.300533
\(220\) 0 0
\(221\) 3588.00 1.09210
\(222\) 0 0
\(223\) −518.000 −0.155551 −0.0777754 0.996971i \(-0.524782\pi\)
−0.0777754 + 0.996971i \(0.524782\pi\)
\(224\) 0 0
\(225\) 1012.00 0.299852
\(226\) 0 0
\(227\) 2844.00 0.831555 0.415777 0.909466i \(-0.363510\pi\)
0.415777 + 0.909466i \(0.363510\pi\)
\(228\) 0 0
\(229\) −1745.00 −0.503550 −0.251775 0.967786i \(-0.581014\pi\)
−0.251775 + 0.967786i \(0.581014\pi\)
\(230\) 0 0
\(231\) −3534.00 −1.00658
\(232\) 0 0
\(233\) 5283.00 1.48541 0.742706 0.669618i \(-0.233542\pi\)
0.742706 + 0.669618i \(0.233542\pi\)
\(234\) 0 0
\(235\) −5211.00 −1.44650
\(236\) 0 0
\(237\) −1400.00 −0.383712
\(238\) 0 0
\(239\) −465.000 −0.125851 −0.0629254 0.998018i \(-0.520043\pi\)
−0.0629254 + 0.998018i \(0.520043\pi\)
\(240\) 0 0
\(241\) −7078.00 −1.89184 −0.945921 0.324396i \(-0.894839\pi\)
−0.945921 + 0.324396i \(0.894839\pi\)
\(242\) 0 0
\(243\) −3542.00 −0.935059
\(244\) 0 0
\(245\) 5562.00 1.45038
\(246\) 0 0
\(247\) 988.000 0.254514
\(248\) 0 0
\(249\) 24.0000 0.00610819
\(250\) 0 0
\(251\) 3567.00 0.897000 0.448500 0.893783i \(-0.351958\pi\)
0.448500 + 0.893783i \(0.351958\pi\)
\(252\) 0 0
\(253\) 4104.00 1.01983
\(254\) 0 0
\(255\) −1242.00 −0.305008
\(256\) 0 0
\(257\) −1896.00 −0.460192 −0.230096 0.973168i \(-0.573904\pi\)
−0.230096 + 0.973168i \(0.573904\pi\)
\(258\) 0 0
\(259\) 7006.00 1.68082
\(260\) 0 0
\(261\) −3450.00 −0.818198
\(262\) 0 0
\(263\) 57.0000 0.0133641 0.00668207 0.999978i \(-0.497873\pi\)
0.00668207 + 0.999978i \(0.497873\pi\)
\(264\) 0 0
\(265\) 3888.00 0.901275
\(266\) 0 0
\(267\) 1200.00 0.275052
\(268\) 0 0
\(269\) −2700.00 −0.611977 −0.305989 0.952035i \(-0.598987\pi\)
−0.305989 + 0.952035i \(0.598987\pi\)
\(270\) 0 0
\(271\) −3872.00 −0.867923 −0.433962 0.900931i \(-0.642885\pi\)
−0.433962 + 0.900931i \(0.642885\pi\)
\(272\) 0 0
\(273\) −3224.00 −0.714745
\(274\) 0 0
\(275\) −2508.00 −0.549957
\(276\) 0 0
\(277\) 7711.00 1.67260 0.836298 0.548275i \(-0.184715\pi\)
0.836298 + 0.548275i \(0.184715\pi\)
\(278\) 0 0
\(279\) 736.000 0.157932
\(280\) 0 0
\(281\) −6858.00 −1.45592 −0.727961 0.685619i \(-0.759532\pi\)
−0.727961 + 0.685619i \(0.759532\pi\)
\(282\) 0 0
\(283\) −1807.00 −0.379558 −0.189779 0.981827i \(-0.560777\pi\)
−0.189779 + 0.981827i \(0.560777\pi\)
\(284\) 0 0
\(285\) −342.000 −0.0710819
\(286\) 0 0
\(287\) −7998.00 −1.64497
\(288\) 0 0
\(289\) −152.000 −0.0309383
\(290\) 0 0
\(291\) −2848.00 −0.573721
\(292\) 0 0
\(293\) 3012.00 0.600556 0.300278 0.953852i \(-0.402921\pi\)
0.300278 + 0.953852i \(0.402921\pi\)
\(294\) 0 0
\(295\) −2970.00 −0.586170
\(296\) 0 0
\(297\) 5700.00 1.11363
\(298\) 0 0
\(299\) 3744.00 0.724151
\(300\) 0 0
\(301\) −2077.00 −0.397729
\(302\) 0 0
\(303\) 2124.00 0.402708
\(304\) 0 0
\(305\) 117.000 0.0219652
\(306\) 0 0
\(307\) −1096.00 −0.203753 −0.101876 0.994797i \(-0.532485\pi\)
−0.101876 + 0.994797i \(0.532485\pi\)
\(308\) 0 0
\(309\) 2356.00 0.433748
\(310\) 0 0
\(311\) −1947.00 −0.354998 −0.177499 0.984121i \(-0.556801\pi\)
−0.177499 + 0.984121i \(0.556801\pi\)
\(312\) 0 0
\(313\) 7598.00 1.37209 0.686045 0.727559i \(-0.259345\pi\)
0.686045 + 0.727559i \(0.259345\pi\)
\(314\) 0 0
\(315\) −6417.00 −1.14780
\(316\) 0 0
\(317\) −8334.00 −1.47661 −0.738303 0.674469i \(-0.764373\pi\)
−0.738303 + 0.674469i \(0.764373\pi\)
\(318\) 0 0
\(319\) 8550.00 1.50065
\(320\) 0 0
\(321\) −228.000 −0.0396440
\(322\) 0 0
\(323\) 1311.00 0.225839
\(324\) 0 0
\(325\) −2288.00 −0.390509
\(326\) 0 0
\(327\) 2920.00 0.493812
\(328\) 0 0
\(329\) −17949.0 −3.00778
\(330\) 0 0
\(331\) −8368.00 −1.38957 −0.694784 0.719219i \(-0.744500\pi\)
−0.694784 + 0.719219i \(0.744500\pi\)
\(332\) 0 0
\(333\) −5198.00 −0.855401
\(334\) 0 0
\(335\) −7704.00 −1.25646
\(336\) 0 0
\(337\) −10336.0 −1.67074 −0.835368 0.549692i \(-0.814745\pi\)
−0.835368 + 0.549692i \(0.814745\pi\)
\(338\) 0 0
\(339\) 1644.00 0.263392
\(340\) 0 0
\(341\) −1824.00 −0.289663
\(342\) 0 0
\(343\) 8525.00 1.34200
\(344\) 0 0
\(345\) −1296.00 −0.202244
\(346\) 0 0
\(347\) 6879.00 1.06422 0.532110 0.846675i \(-0.321399\pi\)
0.532110 + 0.846675i \(0.321399\pi\)
\(348\) 0 0
\(349\) 6355.00 0.974714 0.487357 0.873203i \(-0.337961\pi\)
0.487357 + 0.873203i \(0.337961\pi\)
\(350\) 0 0
\(351\) 5200.00 0.790756
\(352\) 0 0
\(353\) 7218.00 1.08832 0.544158 0.838983i \(-0.316849\pi\)
0.544158 + 0.838983i \(0.316849\pi\)
\(354\) 0 0
\(355\) −5778.00 −0.863843
\(356\) 0 0
\(357\) −4278.00 −0.634218
\(358\) 0 0
\(359\) −1665.00 −0.244778 −0.122389 0.992482i \(-0.539056\pi\)
−0.122389 + 0.992482i \(0.539056\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) −3836.00 −0.554650
\(364\) 0 0
\(365\) −4383.00 −0.628539
\(366\) 0 0
\(367\) −13064.0 −1.85813 −0.929067 0.369911i \(-0.879388\pi\)
−0.929067 + 0.369911i \(0.879388\pi\)
\(368\) 0 0
\(369\) 5934.00 0.837159
\(370\) 0 0
\(371\) 13392.0 1.87406
\(372\) 0 0
\(373\) 10492.0 1.45645 0.728224 0.685339i \(-0.240346\pi\)
0.728224 + 0.685339i \(0.240346\pi\)
\(374\) 0 0
\(375\) 3042.00 0.418902
\(376\) 0 0
\(377\) 7800.00 1.06557
\(378\) 0 0
\(379\) 7610.00 1.03140 0.515698 0.856770i \(-0.327532\pi\)
0.515698 + 0.856770i \(0.327532\pi\)
\(380\) 0 0
\(381\) −4172.00 −0.560992
\(382\) 0 0
\(383\) −4008.00 −0.534724 −0.267362 0.963596i \(-0.586152\pi\)
−0.267362 + 0.963596i \(0.586152\pi\)
\(384\) 0 0
\(385\) 15903.0 2.10517
\(386\) 0 0
\(387\) 1541.00 0.202412
\(388\) 0 0
\(389\) 3525.00 0.459446 0.229723 0.973256i \(-0.426218\pi\)
0.229723 + 0.973256i \(0.426218\pi\)
\(390\) 0 0
\(391\) 4968.00 0.642564
\(392\) 0 0
\(393\) 186.000 0.0238739
\(394\) 0 0
\(395\) 6300.00 0.802500
\(396\) 0 0
\(397\) −6629.00 −0.838035 −0.419018 0.907978i \(-0.637625\pi\)
−0.419018 + 0.907978i \(0.637625\pi\)
\(398\) 0 0
\(399\) −1178.00 −0.147804
\(400\) 0 0
\(401\) −10848.0 −1.35093 −0.675465 0.737392i \(-0.736057\pi\)
−0.675465 + 0.737392i \(0.736057\pi\)
\(402\) 0 0
\(403\) −1664.00 −0.205682
\(404\) 0 0
\(405\) 3789.00 0.464881
\(406\) 0 0
\(407\) 12882.0 1.56889
\(408\) 0 0
\(409\) −3040.00 −0.367526 −0.183763 0.982971i \(-0.558828\pi\)
−0.183763 + 0.982971i \(0.558828\pi\)
\(410\) 0 0
\(411\) −2538.00 −0.304599
\(412\) 0 0
\(413\) −10230.0 −1.21885
\(414\) 0 0
\(415\) −108.000 −0.0127747
\(416\) 0 0
\(417\) 3950.00 0.463867
\(418\) 0 0
\(419\) −3900.00 −0.454719 −0.227360 0.973811i \(-0.573009\pi\)
−0.227360 + 0.973811i \(0.573009\pi\)
\(420\) 0 0
\(421\) −4412.00 −0.510755 −0.255377 0.966841i \(-0.582200\pi\)
−0.255377 + 0.966841i \(0.582200\pi\)
\(422\) 0 0
\(423\) 13317.0 1.53072
\(424\) 0 0
\(425\) −3036.00 −0.346512
\(426\) 0 0
\(427\) 403.000 0.0456734
\(428\) 0 0
\(429\) −5928.00 −0.667148
\(430\) 0 0
\(431\) −432.000 −0.0482801 −0.0241400 0.999709i \(-0.507685\pi\)
−0.0241400 + 0.999709i \(0.507685\pi\)
\(432\) 0 0
\(433\) −2002.00 −0.222194 −0.111097 0.993810i \(-0.535436\pi\)
−0.111097 + 0.993810i \(0.535436\pi\)
\(434\) 0 0
\(435\) −2700.00 −0.297598
\(436\) 0 0
\(437\) 1368.00 0.149749
\(438\) 0 0
\(439\) 1690.00 0.183734 0.0918671 0.995771i \(-0.470717\pi\)
0.0918671 + 0.995771i \(0.470717\pi\)
\(440\) 0 0
\(441\) −14214.0 −1.53482
\(442\) 0 0
\(443\) −1977.00 −0.212032 −0.106016 0.994364i \(-0.533809\pi\)
−0.106016 + 0.994364i \(0.533809\pi\)
\(444\) 0 0
\(445\) −5400.00 −0.575246
\(446\) 0 0
\(447\) −3390.00 −0.358706
\(448\) 0 0
\(449\) −2760.00 −0.290095 −0.145047 0.989425i \(-0.546333\pi\)
−0.145047 + 0.989425i \(0.546333\pi\)
\(450\) 0 0
\(451\) −14706.0 −1.53543
\(452\) 0 0
\(453\) 3604.00 0.373798
\(454\) 0 0
\(455\) 14508.0 1.49483
\(456\) 0 0
\(457\) 4499.00 0.460513 0.230256 0.973130i \(-0.426044\pi\)
0.230256 + 0.973130i \(0.426044\pi\)
\(458\) 0 0
\(459\) 6900.00 0.701665
\(460\) 0 0
\(461\) 11643.0 1.17629 0.588144 0.808756i \(-0.299859\pi\)
0.588144 + 0.808756i \(0.299859\pi\)
\(462\) 0 0
\(463\) 1537.00 0.154277 0.0771387 0.997020i \(-0.475422\pi\)
0.0771387 + 0.997020i \(0.475422\pi\)
\(464\) 0 0
\(465\) 576.000 0.0574438
\(466\) 0 0
\(467\) −7641.00 −0.757138 −0.378569 0.925573i \(-0.623584\pi\)
−0.378569 + 0.925573i \(0.623584\pi\)
\(468\) 0 0
\(469\) −26536.0 −2.61262
\(470\) 0 0
\(471\) −6452.00 −0.631194
\(472\) 0 0
\(473\) −3819.00 −0.371243
\(474\) 0 0
\(475\) −836.000 −0.0807543
\(476\) 0 0
\(477\) −9936.00 −0.953749
\(478\) 0 0
\(479\) 8580.00 0.818435 0.409217 0.912437i \(-0.365802\pi\)
0.409217 + 0.912437i \(0.365802\pi\)
\(480\) 0 0
\(481\) 11752.0 1.11402
\(482\) 0 0
\(483\) −4464.00 −0.420536
\(484\) 0 0
\(485\) 12816.0 1.19989
\(486\) 0 0
\(487\) −12134.0 −1.12904 −0.564522 0.825418i \(-0.690939\pi\)
−0.564522 + 0.825418i \(0.690939\pi\)
\(488\) 0 0
\(489\) −2536.00 −0.234523
\(490\) 0 0
\(491\) −5508.00 −0.506258 −0.253129 0.967433i \(-0.581460\pi\)
−0.253129 + 0.967433i \(0.581460\pi\)
\(492\) 0 0
\(493\) 10350.0 0.945518
\(494\) 0 0
\(495\) −11799.0 −1.07136
\(496\) 0 0
\(497\) −19902.0 −1.79623
\(498\) 0 0
\(499\) −11905.0 −1.06802 −0.534009 0.845479i \(-0.679315\pi\)
−0.534009 + 0.845479i \(0.679315\pi\)
\(500\) 0 0
\(501\) 1308.00 0.116641
\(502\) 0 0
\(503\) −9108.00 −0.807367 −0.403684 0.914899i \(-0.632270\pi\)
−0.403684 + 0.914899i \(0.632270\pi\)
\(504\) 0 0
\(505\) −9558.00 −0.842229
\(506\) 0 0
\(507\) −1014.00 −0.0888231
\(508\) 0 0
\(509\) 2520.00 0.219444 0.109722 0.993962i \(-0.465004\pi\)
0.109722 + 0.993962i \(0.465004\pi\)
\(510\) 0 0
\(511\) −15097.0 −1.30695
\(512\) 0 0
\(513\) 1900.00 0.163523
\(514\) 0 0
\(515\) −10602.0 −0.907146
\(516\) 0 0
\(517\) −33003.0 −2.80749
\(518\) 0 0
\(519\) −2724.00 −0.230386
\(520\) 0 0
\(521\) 21612.0 1.81735 0.908675 0.417505i \(-0.137095\pi\)
0.908675 + 0.417505i \(0.137095\pi\)
\(522\) 0 0
\(523\) −9022.00 −0.754311 −0.377155 0.926150i \(-0.623098\pi\)
−0.377155 + 0.926150i \(0.623098\pi\)
\(524\) 0 0
\(525\) 2728.00 0.226780
\(526\) 0 0
\(527\) −2208.00 −0.182509
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) 7590.00 0.620297
\(532\) 0 0
\(533\) −13416.0 −1.09027
\(534\) 0 0
\(535\) 1026.00 0.0829119
\(536\) 0 0
\(537\) 420.000 0.0337511
\(538\) 0 0
\(539\) 35226.0 2.81501
\(540\) 0 0
\(541\) 9253.00 0.735337 0.367669 0.929957i \(-0.380156\pi\)
0.367669 + 0.929957i \(0.380156\pi\)
\(542\) 0 0
\(543\) 4.00000 0.000316126 0
\(544\) 0 0
\(545\) −13140.0 −1.03276
\(546\) 0 0
\(547\) 13244.0 1.03523 0.517617 0.855613i \(-0.326819\pi\)
0.517617 + 0.855613i \(0.326819\pi\)
\(548\) 0 0
\(549\) −299.000 −0.0232441
\(550\) 0 0
\(551\) 2850.00 0.220352
\(552\) 0 0
\(553\) 21700.0 1.66868
\(554\) 0 0
\(555\) −4068.00 −0.311130
\(556\) 0 0
\(557\) −1569.00 −0.119355 −0.0596774 0.998218i \(-0.519007\pi\)
−0.0596774 + 0.998218i \(0.519007\pi\)
\(558\) 0 0
\(559\) −3484.00 −0.263609
\(560\) 0 0
\(561\) −7866.00 −0.591984
\(562\) 0 0
\(563\) −15762.0 −1.17991 −0.589955 0.807436i \(-0.700854\pi\)
−0.589955 + 0.807436i \(0.700854\pi\)
\(564\) 0 0
\(565\) −7398.00 −0.550861
\(566\) 0 0
\(567\) 13051.0 0.966650
\(568\) 0 0
\(569\) −13800.0 −1.01674 −0.508371 0.861138i \(-0.669752\pi\)
−0.508371 + 0.861138i \(0.669752\pi\)
\(570\) 0 0
\(571\) −4348.00 −0.318666 −0.159333 0.987225i \(-0.550934\pi\)
−0.159333 + 0.987225i \(0.550934\pi\)
\(572\) 0 0
\(573\) −5286.00 −0.385385
\(574\) 0 0
\(575\) −3168.00 −0.229765
\(576\) 0 0
\(577\) 3539.00 0.255339 0.127669 0.991817i \(-0.459250\pi\)
0.127669 + 0.991817i \(0.459250\pi\)
\(578\) 0 0
\(579\) −6496.00 −0.466260
\(580\) 0 0
\(581\) −372.000 −0.0265631
\(582\) 0 0
\(583\) 24624.0 1.74927
\(584\) 0 0
\(585\) −10764.0 −0.760746
\(586\) 0 0
\(587\) −6321.00 −0.444456 −0.222228 0.974995i \(-0.571333\pi\)
−0.222228 + 0.974995i \(0.571333\pi\)
\(588\) 0 0
\(589\) −608.000 −0.0425335
\(590\) 0 0
\(591\) −6252.00 −0.435149
\(592\) 0 0
\(593\) 13278.0 0.919498 0.459749 0.888049i \(-0.347939\pi\)
0.459749 + 0.888049i \(0.347939\pi\)
\(594\) 0 0
\(595\) 19251.0 1.32641
\(596\) 0 0
\(597\) −5990.00 −0.410644
\(598\) 0 0
\(599\) −20400.0 −1.39152 −0.695761 0.718274i \(-0.744933\pi\)
−0.695761 + 0.718274i \(0.744933\pi\)
\(600\) 0 0
\(601\) −22198.0 −1.50661 −0.753307 0.657669i \(-0.771543\pi\)
−0.753307 + 0.657669i \(0.771543\pi\)
\(602\) 0 0
\(603\) 19688.0 1.32961
\(604\) 0 0
\(605\) 17262.0 1.16000
\(606\) 0 0
\(607\) −9824.00 −0.656909 −0.328455 0.944520i \(-0.606528\pi\)
−0.328455 + 0.944520i \(0.606528\pi\)
\(608\) 0 0
\(609\) −9300.00 −0.618810
\(610\) 0 0
\(611\) −30108.0 −1.99352
\(612\) 0 0
\(613\) 4327.00 0.285099 0.142550 0.989788i \(-0.454470\pi\)
0.142550 + 0.989788i \(0.454470\pi\)
\(614\) 0 0
\(615\) 4644.00 0.304495
\(616\) 0 0
\(617\) −14151.0 −0.923335 −0.461668 0.887053i \(-0.652749\pi\)
−0.461668 + 0.887053i \(0.652749\pi\)
\(618\) 0 0
\(619\) 22460.0 1.45839 0.729195 0.684306i \(-0.239895\pi\)
0.729195 + 0.684306i \(0.239895\pi\)
\(620\) 0 0
\(621\) 7200.00 0.465259
\(622\) 0 0
\(623\) −18600.0 −1.19614
\(624\) 0 0
\(625\) −8189.00 −0.524096
\(626\) 0 0
\(627\) −2166.00 −0.137961
\(628\) 0 0
\(629\) 15594.0 0.988511
\(630\) 0 0
\(631\) 16363.0 1.03233 0.516165 0.856489i \(-0.327359\pi\)
0.516165 + 0.856489i \(0.327359\pi\)
\(632\) 0 0
\(633\) 8636.00 0.542259
\(634\) 0 0
\(635\) 18774.0 1.17327
\(636\) 0 0
\(637\) 32136.0 1.99886
\(638\) 0 0
\(639\) 14766.0 0.914138
\(640\) 0 0
\(641\) 5592.00 0.344572 0.172286 0.985047i \(-0.444885\pi\)
0.172286 + 0.985047i \(0.444885\pi\)
\(642\) 0 0
\(643\) 16553.0 1.01522 0.507610 0.861587i \(-0.330529\pi\)
0.507610 + 0.861587i \(0.330529\pi\)
\(644\) 0 0
\(645\) 1206.00 0.0736220
\(646\) 0 0
\(647\) 4611.00 0.280181 0.140091 0.990139i \(-0.455261\pi\)
0.140091 + 0.990139i \(0.455261\pi\)
\(648\) 0 0
\(649\) −18810.0 −1.13768
\(650\) 0 0
\(651\) 1984.00 0.119446
\(652\) 0 0
\(653\) −16413.0 −0.983599 −0.491800 0.870708i \(-0.663661\pi\)
−0.491800 + 0.870708i \(0.663661\pi\)
\(654\) 0 0
\(655\) −837.000 −0.0499302
\(656\) 0 0
\(657\) 11201.0 0.665133
\(658\) 0 0
\(659\) 27390.0 1.61906 0.809532 0.587076i \(-0.199721\pi\)
0.809532 + 0.587076i \(0.199721\pi\)
\(660\) 0 0
\(661\) −26912.0 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(662\) 0 0
\(663\) −7176.00 −0.420351
\(664\) 0 0
\(665\) 5301.00 0.309119
\(666\) 0 0
\(667\) 10800.0 0.626953
\(668\) 0 0
\(669\) 1036.00 0.0598716
\(670\) 0 0
\(671\) 741.000 0.0426319
\(672\) 0 0
\(673\) −21562.0 −1.23500 −0.617499 0.786571i \(-0.711854\pi\)
−0.617499 + 0.786571i \(0.711854\pi\)
\(674\) 0 0
\(675\) −4400.00 −0.250898
\(676\) 0 0
\(677\) 21966.0 1.24700 0.623502 0.781822i \(-0.285709\pi\)
0.623502 + 0.781822i \(0.285709\pi\)
\(678\) 0 0
\(679\) 44144.0 2.49498
\(680\) 0 0
\(681\) −5688.00 −0.320066
\(682\) 0 0
\(683\) 15348.0 0.859846 0.429923 0.902866i \(-0.358541\pi\)
0.429923 + 0.902866i \(0.358541\pi\)
\(684\) 0 0
\(685\) 11421.0 0.637042
\(686\) 0 0
\(687\) 3490.00 0.193816
\(688\) 0 0
\(689\) 22464.0 1.24210
\(690\) 0 0
\(691\) 8147.00 0.448519 0.224259 0.974529i \(-0.428004\pi\)
0.224259 + 0.974529i \(0.428004\pi\)
\(692\) 0 0
\(693\) −40641.0 −2.22774
\(694\) 0 0
\(695\) −17775.0 −0.970136
\(696\) 0 0
\(697\) −17802.0 −0.967430
\(698\) 0 0
\(699\) −10566.0 −0.571735
\(700\) 0 0
\(701\) −14982.0 −0.807222 −0.403611 0.914931i \(-0.632245\pi\)
−0.403611 + 0.914931i \(0.632245\pi\)
\(702\) 0 0
\(703\) 4294.00 0.230372
\(704\) 0 0
\(705\) 10422.0 0.556759
\(706\) 0 0
\(707\) −32922.0 −1.75129
\(708\) 0 0
\(709\) −21890.0 −1.15952 −0.579758 0.814789i \(-0.696853\pi\)
−0.579758 + 0.814789i \(0.696853\pi\)
\(710\) 0 0
\(711\) −16100.0 −0.849222
\(712\) 0 0
\(713\) −2304.00 −0.121018
\(714\) 0 0
\(715\) 26676.0 1.39528
\(716\) 0 0
\(717\) 930.000 0.0484400
\(718\) 0 0
\(719\) 27015.0 1.40124 0.700619 0.713536i \(-0.252907\pi\)
0.700619 + 0.713536i \(0.252907\pi\)
\(720\) 0 0
\(721\) −36518.0 −1.88627
\(722\) 0 0
\(723\) 14156.0 0.728171
\(724\) 0 0
\(725\) −6600.00 −0.338094
\(726\) 0 0
\(727\) 13021.0 0.664267 0.332134 0.943232i \(-0.392232\pi\)
0.332134 + 0.943232i \(0.392232\pi\)
\(728\) 0 0
\(729\) −4283.00 −0.217599
\(730\) 0 0
\(731\) −4623.00 −0.233909
\(732\) 0 0
\(733\) 6262.00 0.315542 0.157771 0.987476i \(-0.449569\pi\)
0.157771 + 0.987476i \(0.449569\pi\)
\(734\) 0 0
\(735\) −11124.0 −0.558252
\(736\) 0 0
\(737\) −48792.0 −2.43864
\(738\) 0 0
\(739\) −10855.0 −0.540335 −0.270168 0.962813i \(-0.587079\pi\)
−0.270168 + 0.962813i \(0.587079\pi\)
\(740\) 0 0
\(741\) −1976.00 −0.0979624
\(742\) 0 0
\(743\) 14892.0 0.735309 0.367654 0.929962i \(-0.380161\pi\)
0.367654 + 0.929962i \(0.380161\pi\)
\(744\) 0 0
\(745\) 15255.0 0.750201
\(746\) 0 0
\(747\) 276.000 0.0135185
\(748\) 0 0
\(749\) 3534.00 0.172403
\(750\) 0 0
\(751\) −28952.0 −1.40676 −0.703378 0.710816i \(-0.748326\pi\)
−0.703378 + 0.710816i \(0.748326\pi\)
\(752\) 0 0
\(753\) −7134.00 −0.345256
\(754\) 0 0
\(755\) −16218.0 −0.781766
\(756\) 0 0
\(757\) 3541.00 0.170013 0.0850065 0.996380i \(-0.472909\pi\)
0.0850065 + 0.996380i \(0.472909\pi\)
\(758\) 0 0
\(759\) −8208.00 −0.392532
\(760\) 0 0
\(761\) 22617.0 1.07735 0.538676 0.842513i \(-0.318925\pi\)
0.538676 + 0.842513i \(0.318925\pi\)
\(762\) 0 0
\(763\) −45260.0 −2.14747
\(764\) 0 0
\(765\) −14283.0 −0.675037
\(766\) 0 0
\(767\) −17160.0 −0.807838
\(768\) 0 0
\(769\) 11495.0 0.539038 0.269519 0.962995i \(-0.413135\pi\)
0.269519 + 0.962995i \(0.413135\pi\)
\(770\) 0 0
\(771\) 3792.00 0.177128
\(772\) 0 0
\(773\) 14622.0 0.680358 0.340179 0.940361i \(-0.389512\pi\)
0.340179 + 0.940361i \(0.389512\pi\)
\(774\) 0 0
\(775\) 1408.00 0.0652605
\(776\) 0 0
\(777\) −14012.0 −0.646947
\(778\) 0 0
\(779\) −4902.00 −0.225459
\(780\) 0 0
\(781\) −36594.0 −1.67661
\(782\) 0 0
\(783\) 15000.0 0.684618
\(784\) 0 0
\(785\) 29034.0 1.32009
\(786\) 0 0
\(787\) 7124.00 0.322672 0.161336 0.986900i \(-0.448420\pi\)
0.161336 + 0.986900i \(0.448420\pi\)
\(788\) 0 0
\(789\) −114.000 −0.00514386
\(790\) 0 0
\(791\) −25482.0 −1.14543
\(792\) 0 0
\(793\) 676.000 0.0302717
\(794\) 0 0
\(795\) −7776.00 −0.346901
\(796\) 0 0
\(797\) 3576.00 0.158932 0.0794658 0.996838i \(-0.474679\pi\)
0.0794658 + 0.996838i \(0.474679\pi\)
\(798\) 0 0
\(799\) −39951.0 −1.76892
\(800\) 0 0
\(801\) 13800.0 0.608738
\(802\) 0 0
\(803\) −27759.0 −1.21992
\(804\) 0 0
\(805\) 20088.0 0.879514
\(806\) 0 0
\(807\) 5400.00 0.235550
\(808\) 0 0
\(809\) 42855.0 1.86242 0.931212 0.364477i \(-0.118752\pi\)
0.931212 + 0.364477i \(0.118752\pi\)
\(810\) 0 0
\(811\) −15568.0 −0.674065 −0.337032 0.941493i \(-0.609423\pi\)
−0.337032 + 0.941493i \(0.609423\pi\)
\(812\) 0 0
\(813\) 7744.00 0.334064
\(814\) 0 0
\(815\) 11412.0 0.490485
\(816\) 0 0
\(817\) −1273.00 −0.0545124
\(818\) 0 0
\(819\) −37076.0 −1.58186
\(820\) 0 0
\(821\) −2517.00 −0.106996 −0.0534981 0.998568i \(-0.517037\pi\)
−0.0534981 + 0.998568i \(0.517037\pi\)
\(822\) 0 0
\(823\) 9727.00 0.411983 0.205991 0.978554i \(-0.433958\pi\)
0.205991 + 0.978554i \(0.433958\pi\)
\(824\) 0 0
\(825\) 5016.00 0.211678
\(826\) 0 0
\(827\) 28224.0 1.18675 0.593376 0.804925i \(-0.297795\pi\)
0.593376 + 0.804925i \(0.297795\pi\)
\(828\) 0 0
\(829\) −3080.00 −0.129038 −0.0645192 0.997916i \(-0.520551\pi\)
−0.0645192 + 0.997916i \(0.520551\pi\)
\(830\) 0 0
\(831\) −15422.0 −0.643782
\(832\) 0 0
\(833\) 42642.0 1.77366
\(834\) 0 0
\(835\) −5886.00 −0.243944
\(836\) 0 0
\(837\) −3200.00 −0.132148
\(838\) 0 0
\(839\) −26790.0 −1.10238 −0.551188 0.834381i \(-0.685825\pi\)
−0.551188 + 0.834381i \(0.685825\pi\)
\(840\) 0 0
\(841\) −1889.00 −0.0774530
\(842\) 0 0
\(843\) 13716.0 0.560385
\(844\) 0 0
\(845\) 4563.00 0.185766
\(846\) 0 0
\(847\) 59458.0 2.41204
\(848\) 0 0
\(849\) 3614.00 0.146092
\(850\) 0 0
\(851\) 16272.0 0.655461
\(852\) 0 0
\(853\) −19178.0 −0.769803 −0.384902 0.922958i \(-0.625765\pi\)
−0.384902 + 0.922958i \(0.625765\pi\)
\(854\) 0 0
\(855\) −3933.00 −0.157317
\(856\) 0 0
\(857\) −2406.00 −0.0959013 −0.0479506 0.998850i \(-0.515269\pi\)
−0.0479506 + 0.998850i \(0.515269\pi\)
\(858\) 0 0
\(859\) 9125.00 0.362446 0.181223 0.983442i \(-0.441994\pi\)
0.181223 + 0.983442i \(0.441994\pi\)
\(860\) 0 0
\(861\) 15996.0 0.633150
\(862\) 0 0
\(863\) −8898.00 −0.350975 −0.175488 0.984482i \(-0.556150\pi\)
−0.175488 + 0.984482i \(0.556150\pi\)
\(864\) 0 0
\(865\) 12258.0 0.481832
\(866\) 0 0
\(867\) 304.000 0.0119082
\(868\) 0 0
\(869\) 39900.0 1.55755
\(870\) 0 0
\(871\) −44512.0 −1.73161
\(872\) 0 0
\(873\) −32752.0 −1.26974
\(874\) 0 0
\(875\) −47151.0 −1.82171
\(876\) 0 0
\(877\) 15886.0 0.611667 0.305834 0.952085i \(-0.401065\pi\)
0.305834 + 0.952085i \(0.401065\pi\)
\(878\) 0 0
\(879\) −6024.00 −0.231154
\(880\) 0 0
\(881\) −25683.0 −0.982159 −0.491080 0.871115i \(-0.663398\pi\)
−0.491080 + 0.871115i \(0.663398\pi\)
\(882\) 0 0
\(883\) −28267.0 −1.07730 −0.538652 0.842528i \(-0.681066\pi\)
−0.538652 + 0.842528i \(0.681066\pi\)
\(884\) 0 0
\(885\) 5940.00 0.225617
\(886\) 0 0
\(887\) 2466.00 0.0933486 0.0466743 0.998910i \(-0.485138\pi\)
0.0466743 + 0.998910i \(0.485138\pi\)
\(888\) 0 0
\(889\) 64666.0 2.43963
\(890\) 0 0
\(891\) 23997.0 0.902278
\(892\) 0 0
\(893\) −11001.0 −0.412245
\(894\) 0 0
\(895\) −1890.00 −0.0705874
\(896\) 0 0
\(897\) −7488.00 −0.278726
\(898\) 0 0
\(899\) −4800.00 −0.178074
\(900\) 0 0
\(901\) 29808.0 1.10216
\(902\) 0 0
\(903\) 4154.00 0.153086
\(904\) 0 0
\(905\) −18.0000 −0.000661149 0
\(906\) 0 0
\(907\) 29324.0 1.07353 0.536763 0.843733i \(-0.319647\pi\)
0.536763 + 0.843733i \(0.319647\pi\)
\(908\) 0 0
\(909\) 24426.0 0.891264
\(910\) 0 0
\(911\) −47142.0 −1.71447 −0.857236 0.514924i \(-0.827820\pi\)
−0.857236 + 0.514924i \(0.827820\pi\)
\(912\) 0 0
\(913\) −684.000 −0.0247942
\(914\) 0 0
\(915\) −234.000 −0.00845443
\(916\) 0 0
\(917\) −2883.00 −0.103822
\(918\) 0 0
\(919\) 39940.0 1.43362 0.716811 0.697267i \(-0.245601\pi\)
0.716811 + 0.697267i \(0.245601\pi\)
\(920\) 0 0
\(921\) 2192.00 0.0784244
\(922\) 0 0
\(923\) −33384.0 −1.19052
\(924\) 0 0
\(925\) −9944.00 −0.353467
\(926\) 0 0
\(927\) 27094.0 0.959961
\(928\) 0 0
\(929\) −4410.00 −0.155745 −0.0778727 0.996963i \(-0.524813\pi\)
−0.0778727 + 0.996963i \(0.524813\pi\)
\(930\) 0 0
\(931\) 11742.0 0.413350
\(932\) 0 0
\(933\) 3894.00 0.136639
\(934\) 0 0
\(935\) 35397.0 1.23808
\(936\) 0 0
\(937\) −41671.0 −1.45286 −0.726431 0.687239i \(-0.758822\pi\)
−0.726431 + 0.687239i \(0.758822\pi\)
\(938\) 0 0
\(939\) −15196.0 −0.528118
\(940\) 0 0
\(941\) −4062.00 −0.140720 −0.0703599 0.997522i \(-0.522415\pi\)
−0.0703599 + 0.997522i \(0.522415\pi\)
\(942\) 0 0
\(943\) −18576.0 −0.641482
\(944\) 0 0
\(945\) 27900.0 0.960410
\(946\) 0 0
\(947\) −45036.0 −1.54538 −0.772689 0.634785i \(-0.781089\pi\)
−0.772689 + 0.634785i \(0.781089\pi\)
\(948\) 0 0
\(949\) −25324.0 −0.866230
\(950\) 0 0
\(951\) 16668.0 0.568346
\(952\) 0 0
\(953\) 26508.0 0.901027 0.450513 0.892770i \(-0.351241\pi\)
0.450513 + 0.892770i \(0.351241\pi\)
\(954\) 0 0
\(955\) 23787.0 0.805999
\(956\) 0 0
\(957\) −17100.0 −0.577601
\(958\) 0 0
\(959\) 39339.0 1.32463
\(960\) 0 0
\(961\) −28767.0 −0.965627
\(962\) 0 0
\(963\) −2622.00 −0.0877391
\(964\) 0 0
\(965\) 29232.0 0.975141
\(966\) 0 0
\(967\) 15976.0 0.531286 0.265643 0.964071i \(-0.414416\pi\)
0.265643 + 0.964071i \(0.414416\pi\)
\(968\) 0 0
\(969\) −2622.00 −0.0869255
\(970\) 0 0
\(971\) −39468.0 −1.30442 −0.652208 0.758040i \(-0.726157\pi\)
−0.652208 + 0.758040i \(0.726157\pi\)
\(972\) 0 0
\(973\) −61225.0 −2.01725
\(974\) 0 0
\(975\) 4576.00 0.150307
\(976\) 0 0
\(977\) 21804.0 0.713994 0.356997 0.934106i \(-0.383801\pi\)
0.356997 + 0.934106i \(0.383801\pi\)
\(978\) 0 0
\(979\) −34200.0 −1.11648
\(980\) 0 0
\(981\) 33580.0 1.09289
\(982\) 0 0
\(983\) −11268.0 −0.365609 −0.182804 0.983149i \(-0.558517\pi\)
−0.182804 + 0.983149i \(0.558517\pi\)
\(984\) 0 0
\(985\) 28134.0 0.910075
\(986\) 0 0
\(987\) 35898.0 1.15770
\(988\) 0 0
\(989\) −4824.00 −0.155100
\(990\) 0 0
\(991\) 778.000 0.0249384 0.0124692 0.999922i \(-0.496031\pi\)
0.0124692 + 0.999922i \(0.496031\pi\)
\(992\) 0 0
\(993\) 16736.0 0.534845
\(994\) 0 0
\(995\) 26955.0 0.858825
\(996\) 0 0
\(997\) −389.000 −0.0123568 −0.00617841 0.999981i \(-0.501967\pi\)
−0.00617841 + 0.999981i \(0.501967\pi\)
\(998\) 0 0
\(999\) 22600.0 0.715748
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 1216.4.a.b.1.1 1
4.3 odd 2 1216.4.a.e.1.1 1
8.3 odd 2 38.4.a.a.1.1 1
8.5 even 2 304.4.a.a.1.1 1
24.11 even 2 342.4.a.d.1.1 1
40.3 even 4 950.4.b.d.799.2 2
40.19 odd 2 950.4.a.d.1.1 1
40.27 even 4 950.4.b.d.799.1 2
56.27 even 2 1862.4.a.a.1.1 1
152.75 even 2 722.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.4.a.a.1.1 1 8.3 odd 2
304.4.a.a.1.1 1 8.5 even 2
342.4.a.d.1.1 1 24.11 even 2
722.4.a.d.1.1 1 152.75 even 2
950.4.a.d.1.1 1 40.19 odd 2
950.4.b.d.799.1 2 40.27 even 4
950.4.b.d.799.2 2 40.3 even 4
1216.4.a.b.1.1 1 1.1 even 1 trivial
1216.4.a.e.1.1 1 4.3 odd 2
1862.4.a.a.1.1 1 56.27 even 2