Properties

Label 234.4.a.c.1.1
Level $234$
Weight $4$
Character 234.1
Self dual yes
Analytic conductor $13.806$
Analytic rank $1$
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] = [234,4,Mod(1,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, 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("234.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 234.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: \(13.8064469413\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 234.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -4.00000 q^{5} +4.00000 q^{7} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} -4.00000 q^{5} +4.00000 q^{7} -8.00000 q^{8} +8.00000 q^{10} -2.00000 q^{11} -13.0000 q^{13} -8.00000 q^{14} +16.0000 q^{16} +6.00000 q^{17} -36.0000 q^{19} -16.0000 q^{20} +4.00000 q^{22} +20.0000 q^{23} -109.000 q^{25} +26.0000 q^{26} +16.0000 q^{28} +14.0000 q^{29} -152.000 q^{31} -32.0000 q^{32} -12.0000 q^{34} -16.0000 q^{35} -258.000 q^{37} +72.0000 q^{38} +32.0000 q^{40} -84.0000 q^{41} -188.000 q^{43} -8.00000 q^{44} -40.0000 q^{46} -254.000 q^{47} -327.000 q^{49} +218.000 q^{50} -52.0000 q^{52} -366.000 q^{53} +8.00000 q^{55} -32.0000 q^{56} -28.0000 q^{58} -550.000 q^{59} -14.0000 q^{61} +304.000 q^{62} +64.0000 q^{64} +52.0000 q^{65} +448.000 q^{67} +24.0000 q^{68} +32.0000 q^{70} -926.000 q^{71} +254.000 q^{73} +516.000 q^{74} -144.000 q^{76} -8.00000 q^{77} +1328.00 q^{79} -64.0000 q^{80} +168.000 q^{82} -186.000 q^{83} -24.0000 q^{85} +376.000 q^{86} +16.0000 q^{88} +336.000 q^{89} -52.0000 q^{91} +80.0000 q^{92} +508.000 q^{94} +144.000 q^{95} +614.000 q^{97} +654.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −4.00000 −0.357771 −0.178885 0.983870i \(-0.557249\pi\)
−0.178885 + 0.983870i \(0.557249\pi\)
\(6\) 0 0
\(7\) 4.00000 0.215980 0.107990 0.994152i \(-0.465559\pi\)
0.107990 + 0.994152i \(0.465559\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 8.00000 0.252982
\(11\) −2.00000 −0.0548202 −0.0274101 0.999624i \(-0.508726\pi\)
−0.0274101 + 0.999624i \(0.508726\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) −8.00000 −0.152721
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 6.00000 0.0856008 0.0428004 0.999084i \(-0.486372\pi\)
0.0428004 + 0.999084i \(0.486372\pi\)
\(18\) 0 0
\(19\) −36.0000 −0.434682 −0.217341 0.976096i \(-0.569738\pi\)
−0.217341 + 0.976096i \(0.569738\pi\)
\(20\) −16.0000 −0.178885
\(21\) 0 0
\(22\) 4.00000 0.0387638
\(23\) 20.0000 0.181317 0.0906584 0.995882i \(-0.471103\pi\)
0.0906584 + 0.995882i \(0.471103\pi\)
\(24\) 0 0
\(25\) −109.000 −0.872000
\(26\) 26.0000 0.196116
\(27\) 0 0
\(28\) 16.0000 0.107990
\(29\) 14.0000 0.0896460 0.0448230 0.998995i \(-0.485728\pi\)
0.0448230 + 0.998995i \(0.485728\pi\)
\(30\) 0 0
\(31\) −152.000 −0.880645 −0.440323 0.897840i \(-0.645136\pi\)
−0.440323 + 0.897840i \(0.645136\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −12.0000 −0.0605289
\(35\) −16.0000 −0.0772712
\(36\) 0 0
\(37\) −258.000 −1.14635 −0.573175 0.819433i \(-0.694288\pi\)
−0.573175 + 0.819433i \(0.694288\pi\)
\(38\) 72.0000 0.307367
\(39\) 0 0
\(40\) 32.0000 0.126491
\(41\) −84.0000 −0.319966 −0.159983 0.987120i \(-0.551144\pi\)
−0.159983 + 0.987120i \(0.551144\pi\)
\(42\) 0 0
\(43\) −188.000 −0.666738 −0.333369 0.942796i \(-0.608185\pi\)
−0.333369 + 0.942796i \(0.608185\pi\)
\(44\) −8.00000 −0.0274101
\(45\) 0 0
\(46\) −40.0000 −0.128210
\(47\) −254.000 −0.788292 −0.394146 0.919048i \(-0.628960\pi\)
−0.394146 + 0.919048i \(0.628960\pi\)
\(48\) 0 0
\(49\) −327.000 −0.953353
\(50\) 218.000 0.616597
\(51\) 0 0
\(52\) −52.0000 −0.138675
\(53\) −366.000 −0.948565 −0.474283 0.880373i \(-0.657293\pi\)
−0.474283 + 0.880373i \(0.657293\pi\)
\(54\) 0 0
\(55\) 8.00000 0.0196131
\(56\) −32.0000 −0.0763604
\(57\) 0 0
\(58\) −28.0000 −0.0633893
\(59\) −550.000 −1.21363 −0.606813 0.794845i \(-0.707552\pi\)
−0.606813 + 0.794845i \(0.707552\pi\)
\(60\) 0 0
\(61\) −14.0000 −0.0293855 −0.0146928 0.999892i \(-0.504677\pi\)
−0.0146928 + 0.999892i \(0.504677\pi\)
\(62\) 304.000 0.622710
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 52.0000 0.0992278
\(66\) 0 0
\(67\) 448.000 0.816894 0.408447 0.912782i \(-0.366070\pi\)
0.408447 + 0.912782i \(0.366070\pi\)
\(68\) 24.0000 0.0428004
\(69\) 0 0
\(70\) 32.0000 0.0546390
\(71\) −926.000 −1.54783 −0.773915 0.633289i \(-0.781704\pi\)
−0.773915 + 0.633289i \(0.781704\pi\)
\(72\) 0 0
\(73\) 254.000 0.407239 0.203620 0.979050i \(-0.434729\pi\)
0.203620 + 0.979050i \(0.434729\pi\)
\(74\) 516.000 0.810592
\(75\) 0 0
\(76\) −144.000 −0.217341
\(77\) −8.00000 −0.0118401
\(78\) 0 0
\(79\) 1328.00 1.89129 0.945644 0.325205i \(-0.105433\pi\)
0.945644 + 0.325205i \(0.105433\pi\)
\(80\) −64.0000 −0.0894427
\(81\) 0 0
\(82\) 168.000 0.226250
\(83\) −186.000 −0.245978 −0.122989 0.992408i \(-0.539248\pi\)
−0.122989 + 0.992408i \(0.539248\pi\)
\(84\) 0 0
\(85\) −24.0000 −0.0306255
\(86\) 376.000 0.471455
\(87\) 0 0
\(88\) 16.0000 0.0193819
\(89\) 336.000 0.400179 0.200089 0.979778i \(-0.435877\pi\)
0.200089 + 0.979778i \(0.435877\pi\)
\(90\) 0 0
\(91\) −52.0000 −0.0599020
\(92\) 80.0000 0.0906584
\(93\) 0 0
\(94\) 508.000 0.557406
\(95\) 144.000 0.155517
\(96\) 0 0
\(97\) 614.000 0.642704 0.321352 0.946960i \(-0.395863\pi\)
0.321352 + 0.946960i \(0.395863\pi\)
\(98\) 654.000 0.674122
\(99\) 0 0
\(100\) −436.000 −0.436000
\(101\) 1606.00 1.58221 0.791104 0.611682i \(-0.209507\pi\)
0.791104 + 0.611682i \(0.209507\pi\)
\(102\) 0 0
\(103\) 208.000 0.198979 0.0994896 0.995039i \(-0.468279\pi\)
0.0994896 + 0.995039i \(0.468279\pi\)
\(104\) 104.000 0.0980581
\(105\) 0 0
\(106\) 732.000 0.670737
\(107\) 248.000 0.224066 0.112033 0.993704i \(-0.464264\pi\)
0.112033 + 0.993704i \(0.464264\pi\)
\(108\) 0 0
\(109\) −542.000 −0.476277 −0.238138 0.971231i \(-0.576537\pi\)
−0.238138 + 0.971231i \(0.576537\pi\)
\(110\) −16.0000 −0.0138685
\(111\) 0 0
\(112\) 64.0000 0.0539949
\(113\) 2042.00 1.69996 0.849979 0.526817i \(-0.176615\pi\)
0.849979 + 0.526817i \(0.176615\pi\)
\(114\) 0 0
\(115\) −80.0000 −0.0648699
\(116\) 56.0000 0.0448230
\(117\) 0 0
\(118\) 1100.00 0.858163
\(119\) 24.0000 0.0184880
\(120\) 0 0
\(121\) −1327.00 −0.996995
\(122\) 28.0000 0.0207787
\(123\) 0 0
\(124\) −608.000 −0.440323
\(125\) 936.000 0.669747
\(126\) 0 0
\(127\) −488.000 −0.340968 −0.170484 0.985360i \(-0.554533\pi\)
−0.170484 + 0.985360i \(0.554533\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) −104.000 −0.0701646
\(131\) −1744.00 −1.16316 −0.581580 0.813489i \(-0.697565\pi\)
−0.581580 + 0.813489i \(0.697565\pi\)
\(132\) 0 0
\(133\) −144.000 −0.0938826
\(134\) −896.000 −0.577631
\(135\) 0 0
\(136\) −48.0000 −0.0302645
\(137\) 828.000 0.516356 0.258178 0.966097i \(-0.416878\pi\)
0.258178 + 0.966097i \(0.416878\pi\)
\(138\) 0 0
\(139\) −404.000 −0.246524 −0.123262 0.992374i \(-0.539336\pi\)
−0.123262 + 0.992374i \(0.539336\pi\)
\(140\) −64.0000 −0.0386356
\(141\) 0 0
\(142\) 1852.00 1.09448
\(143\) 26.0000 0.0152044
\(144\) 0 0
\(145\) −56.0000 −0.0320727
\(146\) −508.000 −0.287962
\(147\) 0 0
\(148\) −1032.00 −0.573175
\(149\) −2928.00 −1.60987 −0.804937 0.593361i \(-0.797801\pi\)
−0.804937 + 0.593361i \(0.797801\pi\)
\(150\) 0 0
\(151\) 1944.00 1.04769 0.523843 0.851815i \(-0.324498\pi\)
0.523843 + 0.851815i \(0.324498\pi\)
\(152\) 288.000 0.153683
\(153\) 0 0
\(154\) 16.0000 0.00837219
\(155\) 608.000 0.315069
\(156\) 0 0
\(157\) 3590.00 1.82492 0.912462 0.409161i \(-0.134178\pi\)
0.912462 + 0.409161i \(0.134178\pi\)
\(158\) −2656.00 −1.33734
\(159\) 0 0
\(160\) 128.000 0.0632456
\(161\) 80.0000 0.0391608
\(162\) 0 0
\(163\) −2284.00 −1.09753 −0.548763 0.835978i \(-0.684901\pi\)
−0.548763 + 0.835978i \(0.684901\pi\)
\(164\) −336.000 −0.159983
\(165\) 0 0
\(166\) 372.000 0.173933
\(167\) −3174.00 −1.47073 −0.735364 0.677673i \(-0.762989\pi\)
−0.735364 + 0.677673i \(0.762989\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 48.0000 0.0216555
\(171\) 0 0
\(172\) −752.000 −0.333369
\(173\) 1358.00 0.596802 0.298401 0.954441i \(-0.403547\pi\)
0.298401 + 0.954441i \(0.403547\pi\)
\(174\) 0 0
\(175\) −436.000 −0.188334
\(176\) −32.0000 −0.0137051
\(177\) 0 0
\(178\) −672.000 −0.282969
\(179\) −708.000 −0.295634 −0.147817 0.989015i \(-0.547225\pi\)
−0.147817 + 0.989015i \(0.547225\pi\)
\(180\) 0 0
\(181\) −546.000 −0.224220 −0.112110 0.993696i \(-0.535761\pi\)
−0.112110 + 0.993696i \(0.535761\pi\)
\(182\) 104.000 0.0423571
\(183\) 0 0
\(184\) −160.000 −0.0641052
\(185\) 1032.00 0.410131
\(186\) 0 0
\(187\) −12.0000 −0.00469266
\(188\) −1016.00 −0.394146
\(189\) 0 0
\(190\) −288.000 −0.109967
\(191\) 3472.00 1.31531 0.657657 0.753317i \(-0.271547\pi\)
0.657657 + 0.753317i \(0.271547\pi\)
\(192\) 0 0
\(193\) −310.000 −0.115618 −0.0578090 0.998328i \(-0.518411\pi\)
−0.0578090 + 0.998328i \(0.518411\pi\)
\(194\) −1228.00 −0.454460
\(195\) 0 0
\(196\) −1308.00 −0.476676
\(197\) −1020.00 −0.368893 −0.184447 0.982843i \(-0.559049\pi\)
−0.184447 + 0.982843i \(0.559049\pi\)
\(198\) 0 0
\(199\) −3256.00 −1.15986 −0.579929 0.814667i \(-0.696920\pi\)
−0.579929 + 0.814667i \(0.696920\pi\)
\(200\) 872.000 0.308299
\(201\) 0 0
\(202\) −3212.00 −1.11879
\(203\) 56.0000 0.0193617
\(204\) 0 0
\(205\) 336.000 0.114474
\(206\) −416.000 −0.140699
\(207\) 0 0
\(208\) −208.000 −0.0693375
\(209\) 72.0000 0.0238294
\(210\) 0 0
\(211\) −4564.00 −1.48909 −0.744547 0.667570i \(-0.767334\pi\)
−0.744547 + 0.667570i \(0.767334\pi\)
\(212\) −1464.00 −0.474283
\(213\) 0 0
\(214\) −496.000 −0.158439
\(215\) 752.000 0.238539
\(216\) 0 0
\(217\) −608.000 −0.190202
\(218\) 1084.00 0.336779
\(219\) 0 0
\(220\) 32.0000 0.00980654
\(221\) −78.0000 −0.0237414
\(222\) 0 0
\(223\) −72.0000 −0.0216210 −0.0108105 0.999942i \(-0.503441\pi\)
−0.0108105 + 0.999942i \(0.503441\pi\)
\(224\) −128.000 −0.0381802
\(225\) 0 0
\(226\) −4084.00 −1.20205
\(227\) −2694.00 −0.787696 −0.393848 0.919176i \(-0.628856\pi\)
−0.393848 + 0.919176i \(0.628856\pi\)
\(228\) 0 0
\(229\) 5922.00 1.70889 0.854447 0.519538i \(-0.173896\pi\)
0.854447 + 0.519538i \(0.173896\pi\)
\(230\) 160.000 0.0458699
\(231\) 0 0
\(232\) −112.000 −0.0316947
\(233\) 5122.00 1.44014 0.720072 0.693900i \(-0.244109\pi\)
0.720072 + 0.693900i \(0.244109\pi\)
\(234\) 0 0
\(235\) 1016.00 0.282028
\(236\) −2200.00 −0.606813
\(237\) 0 0
\(238\) −48.0000 −0.0130730
\(239\) −5022.00 −1.35919 −0.679595 0.733588i \(-0.737844\pi\)
−0.679595 + 0.733588i \(0.737844\pi\)
\(240\) 0 0
\(241\) −1218.00 −0.325553 −0.162777 0.986663i \(-0.552045\pi\)
−0.162777 + 0.986663i \(0.552045\pi\)
\(242\) 2654.00 0.704982
\(243\) 0 0
\(244\) −56.0000 −0.0146928
\(245\) 1308.00 0.341082
\(246\) 0 0
\(247\) 468.000 0.120559
\(248\) 1216.00 0.311355
\(249\) 0 0
\(250\) −1872.00 −0.473583
\(251\) 2112.00 0.531109 0.265554 0.964096i \(-0.414445\pi\)
0.265554 + 0.964096i \(0.414445\pi\)
\(252\) 0 0
\(253\) −40.0000 −0.00993984
\(254\) 976.000 0.241101
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −2814.00 −0.683006 −0.341503 0.939881i \(-0.610936\pi\)
−0.341503 + 0.939881i \(0.610936\pi\)
\(258\) 0 0
\(259\) −1032.00 −0.247588
\(260\) 208.000 0.0496139
\(261\) 0 0
\(262\) 3488.00 0.822478
\(263\) 4044.00 0.948151 0.474076 0.880484i \(-0.342782\pi\)
0.474076 + 0.880484i \(0.342782\pi\)
\(264\) 0 0
\(265\) 1464.00 0.339369
\(266\) 288.000 0.0663850
\(267\) 0 0
\(268\) 1792.00 0.408447
\(269\) 1470.00 0.333188 0.166594 0.986026i \(-0.446723\pi\)
0.166594 + 0.986026i \(0.446723\pi\)
\(270\) 0 0
\(271\) −1844.00 −0.413340 −0.206670 0.978411i \(-0.566263\pi\)
−0.206670 + 0.978411i \(0.566263\pi\)
\(272\) 96.0000 0.0214002
\(273\) 0 0
\(274\) −1656.00 −0.365119
\(275\) 218.000 0.0478033
\(276\) 0 0
\(277\) 5766.00 1.25071 0.625353 0.780342i \(-0.284955\pi\)
0.625353 + 0.780342i \(0.284955\pi\)
\(278\) 808.000 0.174319
\(279\) 0 0
\(280\) 128.000 0.0273195
\(281\) 7468.00 1.58542 0.792711 0.609598i \(-0.208669\pi\)
0.792711 + 0.609598i \(0.208669\pi\)
\(282\) 0 0
\(283\) 1228.00 0.257940 0.128970 0.991648i \(-0.458833\pi\)
0.128970 + 0.991648i \(0.458833\pi\)
\(284\) −3704.00 −0.773915
\(285\) 0 0
\(286\) −52.0000 −0.0107511
\(287\) −336.000 −0.0691061
\(288\) 0 0
\(289\) −4877.00 −0.992673
\(290\) 112.000 0.0226788
\(291\) 0 0
\(292\) 1016.00 0.203620
\(293\) −6608.00 −1.31755 −0.658777 0.752338i \(-0.728926\pi\)
−0.658777 + 0.752338i \(0.728926\pi\)
\(294\) 0 0
\(295\) 2200.00 0.434200
\(296\) 2064.00 0.405296
\(297\) 0 0
\(298\) 5856.00 1.13835
\(299\) −260.000 −0.0502883
\(300\) 0 0
\(301\) −752.000 −0.144002
\(302\) −3888.00 −0.740825
\(303\) 0 0
\(304\) −576.000 −0.108671
\(305\) 56.0000 0.0105133
\(306\) 0 0
\(307\) 7664.00 1.42478 0.712390 0.701784i \(-0.247613\pi\)
0.712390 + 0.701784i \(0.247613\pi\)
\(308\) −32.0000 −0.00592003
\(309\) 0 0
\(310\) −1216.00 −0.222788
\(311\) 2340.00 0.426653 0.213327 0.976981i \(-0.431570\pi\)
0.213327 + 0.976981i \(0.431570\pi\)
\(312\) 0 0
\(313\) 6710.00 1.21173 0.605865 0.795567i \(-0.292827\pi\)
0.605865 + 0.795567i \(0.292827\pi\)
\(314\) −7180.00 −1.29042
\(315\) 0 0
\(316\) 5312.00 0.945644
\(317\) −4164.00 −0.737771 −0.368886 0.929475i \(-0.620261\pi\)
−0.368886 + 0.929475i \(0.620261\pi\)
\(318\) 0 0
\(319\) −28.0000 −0.00491442
\(320\) −256.000 −0.0447214
\(321\) 0 0
\(322\) −160.000 −0.0276908
\(323\) −216.000 −0.0372092
\(324\) 0 0
\(325\) 1417.00 0.241849
\(326\) 4568.00 0.776068
\(327\) 0 0
\(328\) 672.000 0.113125
\(329\) −1016.00 −0.170255
\(330\) 0 0
\(331\) −10072.0 −1.67253 −0.836265 0.548326i \(-0.815265\pi\)
−0.836265 + 0.548326i \(0.815265\pi\)
\(332\) −744.000 −0.122989
\(333\) 0 0
\(334\) 6348.00 1.03996
\(335\) −1792.00 −0.292261
\(336\) 0 0
\(337\) 2990.00 0.483311 0.241655 0.970362i \(-0.422310\pi\)
0.241655 + 0.970362i \(0.422310\pi\)
\(338\) −338.000 −0.0543928
\(339\) 0 0
\(340\) −96.0000 −0.0153127
\(341\) 304.000 0.0482772
\(342\) 0 0
\(343\) −2680.00 −0.421885
\(344\) 1504.00 0.235727
\(345\) 0 0
\(346\) −2716.00 −0.422003
\(347\) −6564.00 −1.01549 −0.507743 0.861508i \(-0.669520\pi\)
−0.507743 + 0.861508i \(0.669520\pi\)
\(348\) 0 0
\(349\) −674.000 −0.103376 −0.0516882 0.998663i \(-0.516460\pi\)
−0.0516882 + 0.998663i \(0.516460\pi\)
\(350\) 872.000 0.133172
\(351\) 0 0
\(352\) 64.0000 0.00969094
\(353\) 10732.0 1.61815 0.809075 0.587706i \(-0.199969\pi\)
0.809075 + 0.587706i \(0.199969\pi\)
\(354\) 0 0
\(355\) 3704.00 0.553769
\(356\) 1344.00 0.200089
\(357\) 0 0
\(358\) 1416.00 0.209044
\(359\) 4842.00 0.711841 0.355921 0.934516i \(-0.384167\pi\)
0.355921 + 0.934516i \(0.384167\pi\)
\(360\) 0 0
\(361\) −5563.00 −0.811051
\(362\) 1092.00 0.158548
\(363\) 0 0
\(364\) −208.000 −0.0299510
\(365\) −1016.00 −0.145698
\(366\) 0 0
\(367\) −6280.00 −0.893224 −0.446612 0.894728i \(-0.647370\pi\)
−0.446612 + 0.894728i \(0.647370\pi\)
\(368\) 320.000 0.0453292
\(369\) 0 0
\(370\) −2064.00 −0.290006
\(371\) −1464.00 −0.204871
\(372\) 0 0
\(373\) 6434.00 0.893136 0.446568 0.894750i \(-0.352646\pi\)
0.446568 + 0.894750i \(0.352646\pi\)
\(374\) 24.0000 0.00331821
\(375\) 0 0
\(376\) 2032.00 0.278703
\(377\) −182.000 −0.0248633
\(378\) 0 0
\(379\) −9068.00 −1.22900 −0.614501 0.788916i \(-0.710643\pi\)
−0.614501 + 0.788916i \(0.710643\pi\)
\(380\) 576.000 0.0777584
\(381\) 0 0
\(382\) −6944.00 −0.930068
\(383\) −3162.00 −0.421855 −0.210928 0.977502i \(-0.567648\pi\)
−0.210928 + 0.977502i \(0.567648\pi\)
\(384\) 0 0
\(385\) 32.0000 0.00423603
\(386\) 620.000 0.0817543
\(387\) 0 0
\(388\) 2456.00 0.321352
\(389\) 3666.00 0.477824 0.238912 0.971041i \(-0.423209\pi\)
0.238912 + 0.971041i \(0.423209\pi\)
\(390\) 0 0
\(391\) 120.000 0.0155209
\(392\) 2616.00 0.337061
\(393\) 0 0
\(394\) 2040.00 0.260847
\(395\) −5312.00 −0.676647
\(396\) 0 0
\(397\) 11054.0 1.39744 0.698721 0.715394i \(-0.253753\pi\)
0.698721 + 0.715394i \(0.253753\pi\)
\(398\) 6512.00 0.820143
\(399\) 0 0
\(400\) −1744.00 −0.218000
\(401\) 5328.00 0.663510 0.331755 0.943366i \(-0.392359\pi\)
0.331755 + 0.943366i \(0.392359\pi\)
\(402\) 0 0
\(403\) 1976.00 0.244247
\(404\) 6424.00 0.791104
\(405\) 0 0
\(406\) −112.000 −0.0136908
\(407\) 516.000 0.0628432
\(408\) 0 0
\(409\) −12074.0 −1.45971 −0.729854 0.683603i \(-0.760412\pi\)
−0.729854 + 0.683603i \(0.760412\pi\)
\(410\) −672.000 −0.0809456
\(411\) 0 0
\(412\) 832.000 0.0994896
\(413\) −2200.00 −0.262118
\(414\) 0 0
\(415\) 744.000 0.0880037
\(416\) 416.000 0.0490290
\(417\) 0 0
\(418\) −144.000 −0.0168499
\(419\) −13584.0 −1.58382 −0.791911 0.610636i \(-0.790914\pi\)
−0.791911 + 0.610636i \(0.790914\pi\)
\(420\) 0 0
\(421\) −7406.00 −0.857355 −0.428677 0.903458i \(-0.641020\pi\)
−0.428677 + 0.903458i \(0.641020\pi\)
\(422\) 9128.00 1.05295
\(423\) 0 0
\(424\) 2928.00 0.335369
\(425\) −654.000 −0.0746439
\(426\) 0 0
\(427\) −56.0000 −0.00634667
\(428\) 992.000 0.112033
\(429\) 0 0
\(430\) −1504.00 −0.168673
\(431\) 10134.0 1.13257 0.566285 0.824210i \(-0.308380\pi\)
0.566285 + 0.824210i \(0.308380\pi\)
\(432\) 0 0
\(433\) 9406.00 1.04393 0.521967 0.852966i \(-0.325198\pi\)
0.521967 + 0.852966i \(0.325198\pi\)
\(434\) 1216.00 0.134493
\(435\) 0 0
\(436\) −2168.00 −0.238138
\(437\) −720.000 −0.0788153
\(438\) 0 0
\(439\) 4088.00 0.444441 0.222220 0.974996i \(-0.428670\pi\)
0.222220 + 0.974996i \(0.428670\pi\)
\(440\) −64.0000 −0.00693427
\(441\) 0 0
\(442\) 156.000 0.0167877
\(443\) 5328.00 0.571424 0.285712 0.958315i \(-0.407770\pi\)
0.285712 + 0.958315i \(0.407770\pi\)
\(444\) 0 0
\(445\) −1344.00 −0.143172
\(446\) 144.000 0.0152883
\(447\) 0 0
\(448\) 256.000 0.0269975
\(449\) −13160.0 −1.38320 −0.691602 0.722279i \(-0.743095\pi\)
−0.691602 + 0.722279i \(0.743095\pi\)
\(450\) 0 0
\(451\) 168.000 0.0175406
\(452\) 8168.00 0.849979
\(453\) 0 0
\(454\) 5388.00 0.556985
\(455\) 208.000 0.0214312
\(456\) 0 0
\(457\) −9146.00 −0.936175 −0.468087 0.883682i \(-0.655057\pi\)
−0.468087 + 0.883682i \(0.655057\pi\)
\(458\) −11844.0 −1.20837
\(459\) 0 0
\(460\) −320.000 −0.0324349
\(461\) −5580.00 −0.563745 −0.281873 0.959452i \(-0.590956\pi\)
−0.281873 + 0.959452i \(0.590956\pi\)
\(462\) 0 0
\(463\) 14788.0 1.48436 0.742178 0.670203i \(-0.233793\pi\)
0.742178 + 0.670203i \(0.233793\pi\)
\(464\) 224.000 0.0224115
\(465\) 0 0
\(466\) −10244.0 −1.01834
\(467\) −12376.0 −1.22632 −0.613162 0.789957i \(-0.710103\pi\)
−0.613162 + 0.789957i \(0.710103\pi\)
\(468\) 0 0
\(469\) 1792.00 0.176433
\(470\) −2032.00 −0.199424
\(471\) 0 0
\(472\) 4400.00 0.429081
\(473\) 376.000 0.0365507
\(474\) 0 0
\(475\) 3924.00 0.379043
\(476\) 96.0000 0.00924402
\(477\) 0 0
\(478\) 10044.0 0.961092
\(479\) −834.000 −0.0795541 −0.0397771 0.999209i \(-0.512665\pi\)
−0.0397771 + 0.999209i \(0.512665\pi\)
\(480\) 0 0
\(481\) 3354.00 0.317940
\(482\) 2436.00 0.230201
\(483\) 0 0
\(484\) −5308.00 −0.498497
\(485\) −2456.00 −0.229941
\(486\) 0 0
\(487\) −13192.0 −1.22749 −0.613744 0.789505i \(-0.710337\pi\)
−0.613744 + 0.789505i \(0.710337\pi\)
\(488\) 112.000 0.0103893
\(489\) 0 0
\(490\) −2616.00 −0.241181
\(491\) −16568.0 −1.52282 −0.761409 0.648272i \(-0.775492\pi\)
−0.761409 + 0.648272i \(0.775492\pi\)
\(492\) 0 0
\(493\) 84.0000 0.00767377
\(494\) −936.000 −0.0852482
\(495\) 0 0
\(496\) −2432.00 −0.220161
\(497\) −3704.00 −0.334300
\(498\) 0 0
\(499\) −10136.0 −0.909318 −0.454659 0.890666i \(-0.650239\pi\)
−0.454659 + 0.890666i \(0.650239\pi\)
\(500\) 3744.00 0.334874
\(501\) 0 0
\(502\) −4224.00 −0.375550
\(503\) −10412.0 −0.922959 −0.461479 0.887151i \(-0.652681\pi\)
−0.461479 + 0.887151i \(0.652681\pi\)
\(504\) 0 0
\(505\) −6424.00 −0.566068
\(506\) 80.0000 0.00702853
\(507\) 0 0
\(508\) −1952.00 −0.170484
\(509\) 4180.00 0.363999 0.181999 0.983299i \(-0.441743\pi\)
0.181999 + 0.983299i \(0.441743\pi\)
\(510\) 0 0
\(511\) 1016.00 0.0879554
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) 5628.00 0.482958
\(515\) −832.000 −0.0711889
\(516\) 0 0
\(517\) 508.000 0.0432143
\(518\) 2064.00 0.175071
\(519\) 0 0
\(520\) −416.000 −0.0350823
\(521\) 14610.0 1.22855 0.614276 0.789091i \(-0.289448\pi\)
0.614276 + 0.789091i \(0.289448\pi\)
\(522\) 0 0
\(523\) −2172.00 −0.181596 −0.0907982 0.995869i \(-0.528942\pi\)
−0.0907982 + 0.995869i \(0.528942\pi\)
\(524\) −6976.00 −0.581580
\(525\) 0 0
\(526\) −8088.00 −0.670444
\(527\) −912.000 −0.0753840
\(528\) 0 0
\(529\) −11767.0 −0.967124
\(530\) −2928.00 −0.239970
\(531\) 0 0
\(532\) −576.000 −0.0469413
\(533\) 1092.00 0.0887425
\(534\) 0 0
\(535\) −992.000 −0.0801643
\(536\) −3584.00 −0.288816
\(537\) 0 0
\(538\) −2940.00 −0.235599
\(539\) 654.000 0.0522630
\(540\) 0 0
\(541\) −11758.0 −0.934410 −0.467205 0.884149i \(-0.654739\pi\)
−0.467205 + 0.884149i \(0.654739\pi\)
\(542\) 3688.00 0.292275
\(543\) 0 0
\(544\) −192.000 −0.0151322
\(545\) 2168.00 0.170398
\(546\) 0 0
\(547\) 340.000 0.0265765 0.0132883 0.999912i \(-0.495770\pi\)
0.0132883 + 0.999912i \(0.495770\pi\)
\(548\) 3312.00 0.258178
\(549\) 0 0
\(550\) −436.000 −0.0338020
\(551\) −504.000 −0.0389676
\(552\) 0 0
\(553\) 5312.00 0.408480
\(554\) −11532.0 −0.884382
\(555\) 0 0
\(556\) −1616.00 −0.123262
\(557\) 3768.00 0.286634 0.143317 0.989677i \(-0.454223\pi\)
0.143317 + 0.989677i \(0.454223\pi\)
\(558\) 0 0
\(559\) 2444.00 0.184920
\(560\) −256.000 −0.0193178
\(561\) 0 0
\(562\) −14936.0 −1.12106
\(563\) 10172.0 0.761454 0.380727 0.924687i \(-0.375674\pi\)
0.380727 + 0.924687i \(0.375674\pi\)
\(564\) 0 0
\(565\) −8168.00 −0.608195
\(566\) −2456.00 −0.182391
\(567\) 0 0
\(568\) 7408.00 0.547241
\(569\) 5506.00 0.405665 0.202833 0.979213i \(-0.434985\pi\)
0.202833 + 0.979213i \(0.434985\pi\)
\(570\) 0 0
\(571\) 2340.00 0.171499 0.0857495 0.996317i \(-0.472672\pi\)
0.0857495 + 0.996317i \(0.472672\pi\)
\(572\) 104.000 0.00760220
\(573\) 0 0
\(574\) 672.000 0.0488654
\(575\) −2180.00 −0.158108
\(576\) 0 0
\(577\) −20094.0 −1.44978 −0.724891 0.688864i \(-0.758110\pi\)
−0.724891 + 0.688864i \(0.758110\pi\)
\(578\) 9754.00 0.701925
\(579\) 0 0
\(580\) −224.000 −0.0160364
\(581\) −744.000 −0.0531262
\(582\) 0 0
\(583\) 732.000 0.0520006
\(584\) −2032.00 −0.143981
\(585\) 0 0
\(586\) 13216.0 0.931652
\(587\) 7118.00 0.500496 0.250248 0.968182i \(-0.419488\pi\)
0.250248 + 0.968182i \(0.419488\pi\)
\(588\) 0 0
\(589\) 5472.00 0.382801
\(590\) −4400.00 −0.307026
\(591\) 0 0
\(592\) −4128.00 −0.286587
\(593\) 10328.0 0.715211 0.357606 0.933873i \(-0.383593\pi\)
0.357606 + 0.933873i \(0.383593\pi\)
\(594\) 0 0
\(595\) −96.0000 −0.00661448
\(596\) −11712.0 −0.804937
\(597\) 0 0
\(598\) 520.000 0.0355592
\(599\) 19732.0 1.34596 0.672978 0.739662i \(-0.265015\pi\)
0.672978 + 0.739662i \(0.265015\pi\)
\(600\) 0 0
\(601\) −12026.0 −0.816224 −0.408112 0.912932i \(-0.633813\pi\)
−0.408112 + 0.912932i \(0.633813\pi\)
\(602\) 1504.00 0.101825
\(603\) 0 0
\(604\) 7776.00 0.523843
\(605\) 5308.00 0.356696
\(606\) 0 0
\(607\) 17016.0 1.13782 0.568911 0.822399i \(-0.307365\pi\)
0.568911 + 0.822399i \(0.307365\pi\)
\(608\) 1152.00 0.0768417
\(609\) 0 0
\(610\) −112.000 −0.00743401
\(611\) 3302.00 0.218633
\(612\) 0 0
\(613\) 11654.0 0.767864 0.383932 0.923361i \(-0.374570\pi\)
0.383932 + 0.923361i \(0.374570\pi\)
\(614\) −15328.0 −1.00747
\(615\) 0 0
\(616\) 64.0000 0.00418609
\(617\) −11612.0 −0.757669 −0.378834 0.925465i \(-0.623675\pi\)
−0.378834 + 0.925465i \(0.623675\pi\)
\(618\) 0 0
\(619\) 4024.00 0.261290 0.130645 0.991429i \(-0.458295\pi\)
0.130645 + 0.991429i \(0.458295\pi\)
\(620\) 2432.00 0.157535
\(621\) 0 0
\(622\) −4680.00 −0.301690
\(623\) 1344.00 0.0864305
\(624\) 0 0
\(625\) 9881.00 0.632384
\(626\) −13420.0 −0.856823
\(627\) 0 0
\(628\) 14360.0 0.912462
\(629\) −1548.00 −0.0981285
\(630\) 0 0
\(631\) −1088.00 −0.0686412 −0.0343206 0.999411i \(-0.510927\pi\)
−0.0343206 + 0.999411i \(0.510927\pi\)
\(632\) −10624.0 −0.668671
\(633\) 0 0
\(634\) 8328.00 0.521683
\(635\) 1952.00 0.121989
\(636\) 0 0
\(637\) 4251.00 0.264412
\(638\) 56.0000 0.00347502
\(639\) 0 0
\(640\) 512.000 0.0316228
\(641\) 7078.00 0.436138 0.218069 0.975933i \(-0.430024\pi\)
0.218069 + 0.975933i \(0.430024\pi\)
\(642\) 0 0
\(643\) 8336.00 0.511259 0.255630 0.966775i \(-0.417717\pi\)
0.255630 + 0.966775i \(0.417717\pi\)
\(644\) 320.000 0.0195804
\(645\) 0 0
\(646\) 432.000 0.0263109
\(647\) −32.0000 −0.00194444 −0.000972218 1.00000i \(-0.500309\pi\)
−0.000972218 1.00000i \(0.500309\pi\)
\(648\) 0 0
\(649\) 1100.00 0.0665312
\(650\) −2834.00 −0.171013
\(651\) 0 0
\(652\) −9136.00 −0.548763
\(653\) 15822.0 0.948182 0.474091 0.880476i \(-0.342777\pi\)
0.474091 + 0.880476i \(0.342777\pi\)
\(654\) 0 0
\(655\) 6976.00 0.416145
\(656\) −1344.00 −0.0799914
\(657\) 0 0
\(658\) 2032.00 0.120388
\(659\) −21540.0 −1.27326 −0.636631 0.771169i \(-0.719672\pi\)
−0.636631 + 0.771169i \(0.719672\pi\)
\(660\) 0 0
\(661\) 8270.00 0.486635 0.243317 0.969947i \(-0.421764\pi\)
0.243317 + 0.969947i \(0.421764\pi\)
\(662\) 20144.0 1.18266
\(663\) 0 0
\(664\) 1488.00 0.0869663
\(665\) 576.000 0.0335885
\(666\) 0 0
\(667\) 280.000 0.0162543
\(668\) −12696.0 −0.735364
\(669\) 0 0
\(670\) 3584.00 0.206660
\(671\) 28.0000 0.00161092
\(672\) 0 0
\(673\) 8482.00 0.485820 0.242910 0.970049i \(-0.421898\pi\)
0.242910 + 0.970049i \(0.421898\pi\)
\(674\) −5980.00 −0.341752
\(675\) 0 0
\(676\) 676.000 0.0384615
\(677\) −2550.00 −0.144763 −0.0723814 0.997377i \(-0.523060\pi\)
−0.0723814 + 0.997377i \(0.523060\pi\)
\(678\) 0 0
\(679\) 2456.00 0.138811
\(680\) 192.000 0.0108277
\(681\) 0 0
\(682\) −608.000 −0.0341371
\(683\) 31534.0 1.76664 0.883320 0.468771i \(-0.155303\pi\)
0.883320 + 0.468771i \(0.155303\pi\)
\(684\) 0 0
\(685\) −3312.00 −0.184737
\(686\) 5360.00 0.298317
\(687\) 0 0
\(688\) −3008.00 −0.166684
\(689\) 4758.00 0.263085
\(690\) 0 0
\(691\) 33832.0 1.86256 0.931281 0.364302i \(-0.118693\pi\)
0.931281 + 0.364302i \(0.118693\pi\)
\(692\) 5432.00 0.298401
\(693\) 0 0
\(694\) 13128.0 0.718058
\(695\) 1616.00 0.0881991
\(696\) 0 0
\(697\) −504.000 −0.0273893
\(698\) 1348.00 0.0730982
\(699\) 0 0
\(700\) −1744.00 −0.0941671
\(701\) −19422.0 −1.04645 −0.523223 0.852196i \(-0.675271\pi\)
−0.523223 + 0.852196i \(0.675271\pi\)
\(702\) 0 0
\(703\) 9288.00 0.498298
\(704\) −128.000 −0.00685253
\(705\) 0 0
\(706\) −21464.0 −1.14420
\(707\) 6424.00 0.341725
\(708\) 0 0
\(709\) −1894.00 −0.100325 −0.0501627 0.998741i \(-0.515974\pi\)
−0.0501627 + 0.998741i \(0.515974\pi\)
\(710\) −7408.00 −0.391574
\(711\) 0 0
\(712\) −2688.00 −0.141485
\(713\) −3040.00 −0.159676
\(714\) 0 0
\(715\) −104.000 −0.00543969
\(716\) −2832.00 −0.147817
\(717\) 0 0
\(718\) −9684.00 −0.503348
\(719\) 20156.0 1.04547 0.522734 0.852496i \(-0.324912\pi\)
0.522734 + 0.852496i \(0.324912\pi\)
\(720\) 0 0
\(721\) 832.000 0.0429754
\(722\) 11126.0 0.573500
\(723\) 0 0
\(724\) −2184.00 −0.112110
\(725\) −1526.00 −0.0781713
\(726\) 0 0
\(727\) 11128.0 0.567696 0.283848 0.958869i \(-0.408389\pi\)
0.283848 + 0.958869i \(0.408389\pi\)
\(728\) 416.000 0.0211786
\(729\) 0 0
\(730\) 2032.00 0.103024
\(731\) −1128.00 −0.0570733
\(732\) 0 0
\(733\) 16202.0 0.816418 0.408209 0.912888i \(-0.366153\pi\)
0.408209 + 0.912888i \(0.366153\pi\)
\(734\) 12560.0 0.631605
\(735\) 0 0
\(736\) −640.000 −0.0320526
\(737\) −896.000 −0.0447823
\(738\) 0 0
\(739\) −5328.00 −0.265215 −0.132607 0.991169i \(-0.542335\pi\)
−0.132607 + 0.991169i \(0.542335\pi\)
\(740\) 4128.00 0.205065
\(741\) 0 0
\(742\) 2928.00 0.144866
\(743\) 20482.0 1.01132 0.505661 0.862732i \(-0.331249\pi\)
0.505661 + 0.862732i \(0.331249\pi\)
\(744\) 0 0
\(745\) 11712.0 0.575966
\(746\) −12868.0 −0.631543
\(747\) 0 0
\(748\) −48.0000 −0.00234633
\(749\) 992.000 0.0483937
\(750\) 0 0
\(751\) 8040.00 0.390657 0.195329 0.980738i \(-0.437423\pi\)
0.195329 + 0.980738i \(0.437423\pi\)
\(752\) −4064.00 −0.197073
\(753\) 0 0
\(754\) 364.000 0.0175810
\(755\) −7776.00 −0.374831
\(756\) 0 0
\(757\) −15822.0 −0.759657 −0.379829 0.925057i \(-0.624017\pi\)
−0.379829 + 0.925057i \(0.624017\pi\)
\(758\) 18136.0 0.869036
\(759\) 0 0
\(760\) −1152.00 −0.0549835
\(761\) 1452.00 0.0691655 0.0345828 0.999402i \(-0.488990\pi\)
0.0345828 + 0.999402i \(0.488990\pi\)
\(762\) 0 0
\(763\) −2168.00 −0.102866
\(764\) 13888.0 0.657657
\(765\) 0 0
\(766\) 6324.00 0.298297
\(767\) 7150.00 0.336599
\(768\) 0 0
\(769\) 32298.0 1.51456 0.757279 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) −64.0000 −0.00299532
\(771\) 0 0
\(772\) −1240.00 −0.0578090
\(773\) −18736.0 −0.871781 −0.435891 0.900000i \(-0.643567\pi\)
−0.435891 + 0.900000i \(0.643567\pi\)
\(774\) 0 0
\(775\) 16568.0 0.767923
\(776\) −4912.00 −0.227230
\(777\) 0 0
\(778\) −7332.00 −0.337873
\(779\) 3024.00 0.139083
\(780\) 0 0
\(781\) 1852.00 0.0848525
\(782\) −240.000 −0.0109749
\(783\) 0 0
\(784\) −5232.00 −0.238338
\(785\) −14360.0 −0.652905
\(786\) 0 0
\(787\) −40816.0 −1.84871 −0.924354 0.381536i \(-0.875395\pi\)
−0.924354 + 0.381536i \(0.875395\pi\)
\(788\) −4080.00 −0.184447
\(789\) 0 0
\(790\) 10624.0 0.478462
\(791\) 8168.00 0.367156
\(792\) 0 0
\(793\) 182.000 0.00815008
\(794\) −22108.0 −0.988141
\(795\) 0 0
\(796\) −13024.0 −0.579929
\(797\) −4518.00 −0.200798 −0.100399 0.994947i \(-0.532012\pi\)
−0.100399 + 0.994947i \(0.532012\pi\)
\(798\) 0 0
\(799\) −1524.00 −0.0674784
\(800\) 3488.00 0.154149
\(801\) 0 0
\(802\) −10656.0 −0.469173
\(803\) −508.000 −0.0223249
\(804\) 0 0
\(805\) −320.000 −0.0140106
\(806\) −3952.00 −0.172709
\(807\) 0 0
\(808\) −12848.0 −0.559395
\(809\) 5058.00 0.219814 0.109907 0.993942i \(-0.464945\pi\)
0.109907 + 0.993942i \(0.464945\pi\)
\(810\) 0 0
\(811\) −22564.0 −0.976978 −0.488489 0.872570i \(-0.662452\pi\)
−0.488489 + 0.872570i \(0.662452\pi\)
\(812\) 224.000 0.00968086
\(813\) 0 0
\(814\) −1032.00 −0.0444368
\(815\) 9136.00 0.392663
\(816\) 0 0
\(817\) 6768.00 0.289819
\(818\) 24148.0 1.03217
\(819\) 0 0
\(820\) 1344.00 0.0572372
\(821\) −32584.0 −1.38513 −0.692564 0.721357i \(-0.743519\pi\)
−0.692564 + 0.721357i \(0.743519\pi\)
\(822\) 0 0
\(823\) −9288.00 −0.393389 −0.196695 0.980465i \(-0.563021\pi\)
−0.196695 + 0.980465i \(0.563021\pi\)
\(824\) −1664.00 −0.0703497
\(825\) 0 0
\(826\) 4400.00 0.185346
\(827\) −20586.0 −0.865593 −0.432796 0.901492i \(-0.642473\pi\)
−0.432796 + 0.901492i \(0.642473\pi\)
\(828\) 0 0
\(829\) −46118.0 −1.93214 −0.966070 0.258280i \(-0.916844\pi\)
−0.966070 + 0.258280i \(0.916844\pi\)
\(830\) −1488.00 −0.0622280
\(831\) 0 0
\(832\) −832.000 −0.0346688
\(833\) −1962.00 −0.0816078
\(834\) 0 0
\(835\) 12696.0 0.526183
\(836\) 288.000 0.0119147
\(837\) 0 0
\(838\) 27168.0 1.11993
\(839\) 39230.0 1.61427 0.807133 0.590369i \(-0.201018\pi\)
0.807133 + 0.590369i \(0.201018\pi\)
\(840\) 0 0
\(841\) −24193.0 −0.991964
\(842\) 14812.0 0.606241
\(843\) 0 0
\(844\) −18256.0 −0.744547
\(845\) −676.000 −0.0275208
\(846\) 0 0
\(847\) −5308.00 −0.215331
\(848\) −5856.00 −0.237141
\(849\) 0 0
\(850\) 1308.00 0.0527812
\(851\) −5160.00 −0.207853
\(852\) 0 0
\(853\) −18674.0 −0.749573 −0.374786 0.927111i \(-0.622284\pi\)
−0.374786 + 0.927111i \(0.622284\pi\)
\(854\) 112.000 0.00448778
\(855\) 0 0
\(856\) −1984.00 −0.0792193
\(857\) −41678.0 −1.66125 −0.830626 0.556830i \(-0.812017\pi\)
−0.830626 + 0.556830i \(0.812017\pi\)
\(858\) 0 0
\(859\) −14740.0 −0.585474 −0.292737 0.956193i \(-0.594566\pi\)
−0.292737 + 0.956193i \(0.594566\pi\)
\(860\) 3008.00 0.119270
\(861\) 0 0
\(862\) −20268.0 −0.800848
\(863\) 24982.0 0.985396 0.492698 0.870200i \(-0.336011\pi\)
0.492698 + 0.870200i \(0.336011\pi\)
\(864\) 0 0
\(865\) −5432.00 −0.213519
\(866\) −18812.0 −0.738173
\(867\) 0 0
\(868\) −2432.00 −0.0951008
\(869\) −2656.00 −0.103681
\(870\) 0 0
\(871\) −5824.00 −0.226566
\(872\) 4336.00 0.168389
\(873\) 0 0
\(874\) 1440.00 0.0557308
\(875\) 3744.00 0.144652
\(876\) 0 0
\(877\) 1134.00 0.0436630 0.0218315 0.999762i \(-0.493050\pi\)
0.0218315 + 0.999762i \(0.493050\pi\)
\(878\) −8176.00 −0.314267
\(879\) 0 0
\(880\) 128.000 0.00490327
\(881\) −34950.0 −1.33654 −0.668272 0.743917i \(-0.732966\pi\)
−0.668272 + 0.743917i \(0.732966\pi\)
\(882\) 0 0
\(883\) −3068.00 −0.116927 −0.0584634 0.998290i \(-0.518620\pi\)
−0.0584634 + 0.998290i \(0.518620\pi\)
\(884\) −312.000 −0.0118707
\(885\) 0 0
\(886\) −10656.0 −0.404058
\(887\) 14080.0 0.532988 0.266494 0.963837i \(-0.414135\pi\)
0.266494 + 0.963837i \(0.414135\pi\)
\(888\) 0 0
\(889\) −1952.00 −0.0736423
\(890\) 2688.00 0.101238
\(891\) 0 0
\(892\) −288.000 −0.0108105
\(893\) 9144.00 0.342657
\(894\) 0 0
\(895\) 2832.00 0.105769
\(896\) −512.000 −0.0190901
\(897\) 0 0
\(898\) 26320.0 0.978073
\(899\) −2128.00 −0.0789464
\(900\) 0 0
\(901\) −2196.00 −0.0811980
\(902\) −336.000 −0.0124031
\(903\) 0 0
\(904\) −16336.0 −0.601026
\(905\) 2184.00 0.0802195
\(906\) 0 0
\(907\) −24876.0 −0.910688 −0.455344 0.890316i \(-0.650484\pi\)
−0.455344 + 0.890316i \(0.650484\pi\)
\(908\) −10776.0 −0.393848
\(909\) 0 0
\(910\) −416.000 −0.0151541
\(911\) −51456.0 −1.87136 −0.935682 0.352843i \(-0.885215\pi\)
−0.935682 + 0.352843i \(0.885215\pi\)
\(912\) 0 0
\(913\) 372.000 0.0134846
\(914\) 18292.0 0.661975
\(915\) 0 0
\(916\) 23688.0 0.854447
\(917\) −6976.00 −0.251219
\(918\) 0 0
\(919\) −31032.0 −1.11388 −0.556938 0.830554i \(-0.688024\pi\)
−0.556938 + 0.830554i \(0.688024\pi\)
\(920\) 640.000 0.0229350
\(921\) 0 0
\(922\) 11160.0 0.398628
\(923\) 12038.0 0.429291
\(924\) 0 0
\(925\) 28122.0 0.999617
\(926\) −29576.0 −1.04960
\(927\) 0 0
\(928\) −448.000 −0.0158473
\(929\) −50820.0 −1.79478 −0.897390 0.441239i \(-0.854539\pi\)
−0.897390 + 0.441239i \(0.854539\pi\)
\(930\) 0 0
\(931\) 11772.0 0.414406
\(932\) 20488.0 0.720072
\(933\) 0 0
\(934\) 24752.0 0.867142
\(935\) 48.0000 0.00167890
\(936\) 0 0
\(937\) 5982.00 0.208563 0.104281 0.994548i \(-0.466746\pi\)
0.104281 + 0.994548i \(0.466746\pi\)
\(938\) −3584.00 −0.124757
\(939\) 0 0
\(940\) 4064.00 0.141014
\(941\) 20224.0 0.700620 0.350310 0.936634i \(-0.386076\pi\)
0.350310 + 0.936634i \(0.386076\pi\)
\(942\) 0 0
\(943\) −1680.00 −0.0580152
\(944\) −8800.00 −0.303406
\(945\) 0 0
\(946\) −752.000 −0.0258453
\(947\) −8478.00 −0.290917 −0.145458 0.989364i \(-0.546466\pi\)
−0.145458 + 0.989364i \(0.546466\pi\)
\(948\) 0 0
\(949\) −3302.00 −0.112948
\(950\) −7848.00 −0.268024
\(951\) 0 0
\(952\) −192.000 −0.00653651
\(953\) −40918.0 −1.39083 −0.695417 0.718607i \(-0.744780\pi\)
−0.695417 + 0.718607i \(0.744780\pi\)
\(954\) 0 0
\(955\) −13888.0 −0.470581
\(956\) −20088.0 −0.679595
\(957\) 0 0
\(958\) 1668.00 0.0562533
\(959\) 3312.00 0.111522
\(960\) 0 0
\(961\) −6687.00 −0.224464
\(962\) −6708.00 −0.224818
\(963\) 0 0
\(964\) −4872.00 −0.162777
\(965\) 1240.00 0.0413648
\(966\) 0 0
\(967\) −4624.00 −0.153772 −0.0768862 0.997040i \(-0.524498\pi\)
−0.0768862 + 0.997040i \(0.524498\pi\)
\(968\) 10616.0 0.352491
\(969\) 0 0
\(970\) 4912.00 0.162593
\(971\) −15300.0 −0.505665 −0.252832 0.967510i \(-0.581362\pi\)
−0.252832 + 0.967510i \(0.581362\pi\)
\(972\) 0 0
\(973\) −1616.00 −0.0532442
\(974\) 26384.0 0.867965
\(975\) 0 0
\(976\) −224.000 −0.00734638
\(977\) −19584.0 −0.641298 −0.320649 0.947198i \(-0.603901\pi\)
−0.320649 + 0.947198i \(0.603901\pi\)
\(978\) 0 0
\(979\) −672.000 −0.0219379
\(980\) 5232.00 0.170541
\(981\) 0 0
\(982\) 33136.0 1.07679
\(983\) 17582.0 0.570477 0.285238 0.958457i \(-0.407927\pi\)
0.285238 + 0.958457i \(0.407927\pi\)
\(984\) 0 0
\(985\) 4080.00 0.131979
\(986\) −168.000 −0.00542618
\(987\) 0 0
\(988\) 1872.00 0.0602796
\(989\) −3760.00 −0.120891
\(990\) 0 0
\(991\) 47904.0 1.53554 0.767770 0.640725i \(-0.221366\pi\)
0.767770 + 0.640725i \(0.221366\pi\)
\(992\) 4864.00 0.155678
\(993\) 0 0
\(994\) 7408.00 0.236386
\(995\) 13024.0 0.414963
\(996\) 0 0
\(997\) −44578.0 −1.41605 −0.708024 0.706189i \(-0.750413\pi\)
−0.708024 + 0.706189i \(0.750413\pi\)
\(998\) 20272.0 0.642985
\(999\) 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 234.4.a.c.1.1 1
3.2 odd 2 78.4.a.f.1.1 1
4.3 odd 2 1872.4.a.f.1.1 1
12.11 even 2 624.4.a.c.1.1 1
15.14 odd 2 1950.4.a.a.1.1 1
24.5 odd 2 2496.4.a.c.1.1 1
24.11 even 2 2496.4.a.l.1.1 1
39.5 even 4 1014.4.b.g.337.1 2
39.8 even 4 1014.4.b.g.337.2 2
39.38 odd 2 1014.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.f.1.1 1 3.2 odd 2
234.4.a.c.1.1 1 1.1 even 1 trivial
624.4.a.c.1.1 1 12.11 even 2
1014.4.a.e.1.1 1 39.38 odd 2
1014.4.b.g.337.1 2 39.5 even 4
1014.4.b.g.337.2 2 39.8 even 4
1872.4.a.f.1.1 1 4.3 odd 2
1950.4.a.a.1.1 1 15.14 odd 2
2496.4.a.c.1.1 1 24.5 odd 2
2496.4.a.l.1.1 1 24.11 even 2