Properties

Label 1152.3.e.c.1025.3
Level $1152$
Weight $3$
Character 1152.1025
Analytic conductor $31.390$
Analytic rank $0$
Dimension $4$
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] = [1152,3,Mod(1025,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.1025");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.e (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.3
Root \(0.500000 - 0.244099i\) of defining polynomial
Character \(\chi\) \(=\) 1152.1025
Dual form 1152.3.e.c.1025.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.21904i q^{5} -2.00000 q^{7} +O(q^{10})\) \(q+5.21904i q^{5} -2.00000 q^{7} +4.78122i q^{11} -1.38083 q^{13} -14.6807i q^{17} -26.7617 q^{19} +18.0477i q^{23} -2.23834 q^{25} +25.0180i q^{29} -39.5233 q^{31} -10.4381i q^{35} +26.0000 q^{37} -28.8228i q^{41} +9.52333 q^{43} -80.2731i q^{47} -45.0000 q^{49} +9.79874i q^{53} -24.9533 q^{55} -73.5391i q^{59} +67.5233 q^{61} -7.20661i q^{65} -102.762 q^{67} -21.9533i q^{71} -140.570 q^{73} -9.56244i q^{77} -0.476674 q^{79} -31.3142i q^{83} +76.6192 q^{85} +13.1310i q^{89} +2.76166 q^{91} -139.670i q^{95} -69.2383 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{7} + 32 q^{13} - 32 q^{19} - 84 q^{25} - 8 q^{31} + 104 q^{37} - 112 q^{43} - 180 q^{49} - 400 q^{55} + 120 q^{61} - 336 q^{67} - 112 q^{73} - 152 q^{79} + 344 q^{85} - 64 q^{91} - 352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 5.21904i 1.04381i 0.853004 + 0.521904i \(0.174778\pi\)
−0.853004 + 0.521904i \(0.825222\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.285714 −0.142857 0.989743i \(-0.545629\pi\)
−0.142857 + 0.989743i \(0.545629\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.78122i 0.434656i 0.976099 + 0.217328i \(0.0697341\pi\)
−0.976099 + 0.217328i \(0.930266\pi\)
\(12\) 0 0
\(13\) −1.38083 −0.106218 −0.0531089 0.998589i \(-0.516913\pi\)
−0.0531089 + 0.998589i \(0.516913\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 14.6807i − 0.863571i −0.901976 0.431786i \(-0.857884\pi\)
0.901976 0.431786i \(-0.142116\pi\)
\(18\) 0 0
\(19\) −26.7617 −1.40851 −0.704254 0.709948i \(-0.748719\pi\)
−0.704254 + 0.709948i \(0.748719\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 18.0477i 0.784683i 0.919820 + 0.392342i \(0.128335\pi\)
−0.919820 + 0.392342i \(0.871665\pi\)
\(24\) 0 0
\(25\) −2.23834 −0.0895335
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 25.0180i 0.862691i 0.902187 + 0.431345i \(0.141961\pi\)
−0.902187 + 0.431345i \(0.858039\pi\)
\(30\) 0 0
\(31\) −39.5233 −1.27495 −0.637473 0.770473i \(-0.720020\pi\)
−0.637473 + 0.770473i \(0.720020\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 10.4381i − 0.298231i
\(36\) 0 0
\(37\) 26.0000 0.702703 0.351351 0.936244i \(-0.385722\pi\)
0.351351 + 0.936244i \(0.385722\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 28.8228i − 0.702996i −0.936189 0.351498i \(-0.885672\pi\)
0.936189 0.351498i \(-0.114328\pi\)
\(42\) 0 0
\(43\) 9.52333 0.221473 0.110736 0.993850i \(-0.464679\pi\)
0.110736 + 0.993850i \(0.464679\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 80.2731i − 1.70794i −0.520323 0.853969i \(-0.674189\pi\)
0.520323 0.853969i \(-0.325811\pi\)
\(48\) 0 0
\(49\) −45.0000 −0.918367
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.79874i 0.184882i 0.995718 + 0.0924409i \(0.0294669\pi\)
−0.995718 + 0.0924409i \(0.970533\pi\)
\(54\) 0 0
\(55\) −24.9533 −0.453697
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 73.5391i − 1.24643i −0.782052 0.623213i \(-0.785827\pi\)
0.782052 0.623213i \(-0.214173\pi\)
\(60\) 0 0
\(61\) 67.5233 1.10694 0.553470 0.832869i \(-0.313303\pi\)
0.553470 + 0.832869i \(0.313303\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 7.20661i − 0.110871i
\(66\) 0 0
\(67\) −102.762 −1.53376 −0.766878 0.641793i \(-0.778191\pi\)
−0.766878 + 0.641793i \(0.778191\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 21.9533i − 0.309201i −0.987977 0.154601i \(-0.950591\pi\)
0.987977 0.154601i \(-0.0494091\pi\)
\(72\) 0 0
\(73\) −140.570 −1.92562 −0.962808 0.270186i \(-0.912915\pi\)
−0.962808 + 0.270186i \(0.912915\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 9.56244i − 0.124187i
\(78\) 0 0
\(79\) −0.476674 −0.00603385 −0.00301692 0.999995i \(-0.500960\pi\)
−0.00301692 + 0.999995i \(0.500960\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 31.3142i − 0.377280i −0.982046 0.188640i \(-0.939592\pi\)
0.982046 0.188640i \(-0.0604079\pi\)
\(84\) 0 0
\(85\) 76.6192 0.901402
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.1310i 0.147539i 0.997275 + 0.0737694i \(0.0235029\pi\)
−0.997275 + 0.0737694i \(0.976497\pi\)
\(90\) 0 0
\(91\) 2.76166 0.0303479
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 139.670i − 1.47021i
\(96\) 0 0
\(97\) −69.2383 −0.713797 −0.356899 0.934143i \(-0.616166\pi\)
−0.356899 + 0.934143i \(0.616166\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 78.5566i 0.777788i 0.921282 + 0.388894i \(0.127143\pi\)
−0.921282 + 0.388894i \(0.872857\pi\)
\(102\) 0 0
\(103\) −198.570 −1.92786 −0.963932 0.266149i \(-0.914249\pi\)
−0.963932 + 0.266149i \(0.914249\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 177.517i − 1.65904i −0.558480 0.829518i \(-0.688615\pi\)
0.558480 0.829518i \(-0.311385\pi\)
\(108\) 0 0
\(109\) 137.381 1.26037 0.630187 0.776443i \(-0.282978\pi\)
0.630187 + 0.776443i \(0.282978\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 72.5279i − 0.641840i −0.947106 0.320920i \(-0.896008\pi\)
0.947106 0.320920i \(-0.103992\pi\)
\(114\) 0 0
\(115\) −94.1917 −0.819058
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 29.3614i 0.246735i
\(120\) 0 0
\(121\) 98.1400 0.811074
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 118.794i 0.950352i
\(126\) 0 0
\(127\) 81.0467 0.638163 0.319081 0.947727i \(-0.396626\pi\)
0.319081 + 0.947727i \(0.396626\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 164.452i 1.25536i 0.778473 + 0.627679i \(0.215995\pi\)
−0.778473 + 0.627679i \(0.784005\pi\)
\(132\) 0 0
\(133\) 53.5233 0.402431
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 224.790i − 1.64081i −0.571786 0.820403i \(-0.693749\pi\)
0.571786 0.820403i \(-0.306251\pi\)
\(138\) 0 0
\(139\) −170.378 −1.22574 −0.612872 0.790183i \(-0.709986\pi\)
−0.612872 + 0.790183i \(0.709986\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 6.60206i − 0.0461682i
\(144\) 0 0
\(145\) −130.570 −0.900483
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 59.5637i 0.399756i 0.979821 + 0.199878i \(0.0640546\pi\)
−0.979821 + 0.199878i \(0.935945\pi\)
\(150\) 0 0
\(151\) −242.570 −1.60642 −0.803212 0.595693i \(-0.796877\pi\)
−0.803212 + 0.595693i \(0.796877\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 206.274i − 1.33080i
\(156\) 0 0
\(157\) −161.617 −1.02941 −0.514703 0.857369i \(-0.672098\pi\)
−0.514703 + 0.857369i \(0.672098\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 36.0954i − 0.224195i
\(162\) 0 0
\(163\) −80.9533 −0.496646 −0.248323 0.968677i \(-0.579879\pi\)
−0.248323 + 0.968677i \(0.579879\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 179.268i 1.07346i 0.843754 + 0.536731i \(0.180341\pi\)
−0.843754 + 0.536731i \(0.819659\pi\)
\(168\) 0 0
\(169\) −167.093 −0.988718
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 26.4982i 0.153169i 0.997063 + 0.0765844i \(0.0244015\pi\)
−0.997063 + 0.0765844i \(0.975599\pi\)
\(174\) 0 0
\(175\) 4.47667 0.0255810
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 306.679i 1.71329i 0.515905 + 0.856646i \(0.327456\pi\)
−0.515905 + 0.856646i \(0.672544\pi\)
\(180\) 0 0
\(181\) 57.3808 0.317021 0.158511 0.987357i \(-0.449331\pi\)
0.158511 + 0.987357i \(0.449331\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 135.695i 0.733486i
\(186\) 0 0
\(187\) 70.1917 0.375357
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 216.712i − 1.13462i −0.823506 0.567308i \(-0.807985\pi\)
0.823506 0.567308i \(-0.192015\pi\)
\(192\) 0 0
\(193\) 187.140 0.969637 0.484819 0.874615i \(-0.338886\pi\)
0.484819 + 0.874615i \(0.338886\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 65.8947i 0.334491i 0.985915 + 0.167245i \(0.0534872\pi\)
−0.985915 + 0.167245i \(0.946513\pi\)
\(198\) 0 0
\(199\) −114.953 −0.577655 −0.288828 0.957381i \(-0.593265\pi\)
−0.288828 + 0.957381i \(0.593265\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 50.0361i − 0.246483i
\(204\) 0 0
\(205\) 150.427 0.733793
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 127.953i − 0.612217i
\(210\) 0 0
\(211\) −103.332 −0.489723 −0.244862 0.969558i \(-0.578743\pi\)
−0.244862 + 0.969558i \(0.578743\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 49.7026i 0.231175i
\(216\) 0 0
\(217\) 79.0467 0.364270
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 20.2716i 0.0917267i
\(222\) 0 0
\(223\) 241.047 1.08093 0.540463 0.841368i \(-0.318249\pi\)
0.540463 + 0.841368i \(0.318249\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 22.5579i − 0.0993738i −0.998765 0.0496869i \(-0.984178\pi\)
0.998765 0.0496869i \(-0.0158223\pi\)
\(228\) 0 0
\(229\) −212.997 −0.930120 −0.465060 0.885279i \(-0.653967\pi\)
−0.465060 + 0.885279i \(0.653967\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 63.9107i 0.274295i 0.990551 + 0.137147i \(0.0437934\pi\)
−0.990551 + 0.137147i \(0.956207\pi\)
\(234\) 0 0
\(235\) 418.948 1.78276
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 174.417i − 0.729779i −0.931051 0.364890i \(-0.881107\pi\)
0.931051 0.364890i \(-0.118893\pi\)
\(240\) 0 0
\(241\) −47.3316 −0.196397 −0.0981984 0.995167i \(-0.531308\pi\)
−0.0981984 + 0.995167i \(0.531308\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 234.857i − 0.958598i
\(246\) 0 0
\(247\) 36.9533 0.149609
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 205.731i − 0.819647i −0.912165 0.409824i \(-0.865590\pi\)
0.912165 0.409824i \(-0.134410\pi\)
\(252\) 0 0
\(253\) −86.2901 −0.341067
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 251.796i 0.979751i 0.871792 + 0.489875i \(0.162958\pi\)
−0.871792 + 0.489875i \(0.837042\pi\)
\(258\) 0 0
\(259\) −52.0000 −0.200772
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 206.607i 0.785578i 0.919629 + 0.392789i \(0.128490\pi\)
−0.919629 + 0.392789i \(0.871510\pi\)
\(264\) 0 0
\(265\) −51.1400 −0.192981
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 430.963i − 1.60209i −0.598601 0.801047i \(-0.704277\pi\)
0.598601 0.801047i \(-0.295723\pi\)
\(270\) 0 0
\(271\) 290.953 1.07363 0.536814 0.843700i \(-0.319628\pi\)
0.536814 + 0.843700i \(0.319628\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 10.7020i − 0.0389163i
\(276\) 0 0
\(277\) 57.9508 0.209209 0.104604 0.994514i \(-0.466642\pi\)
0.104604 + 0.994514i \(0.466642\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 124.517i 0.443120i 0.975147 + 0.221560i \(0.0711149\pi\)
−0.975147 + 0.221560i \(0.928885\pi\)
\(282\) 0 0
\(283\) 254.093 0.897856 0.448928 0.893568i \(-0.351806\pi\)
0.448928 + 0.893568i \(0.351806\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 57.6457i 0.200856i
\(288\) 0 0
\(289\) 73.4767 0.254245
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 239.909i − 0.818802i −0.912355 0.409401i \(-0.865738\pi\)
0.912355 0.409401i \(-0.134262\pi\)
\(294\) 0 0
\(295\) 383.803 1.30103
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 24.9209i − 0.0833473i
\(300\) 0 0
\(301\) −19.0467 −0.0632779
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 352.407i 1.15543i
\(306\) 0 0
\(307\) −3.71501 −0.0121010 −0.00605051 0.999982i \(-0.501926\pi\)
−0.00605051 + 0.999982i \(0.501926\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 473.828i 1.52356i 0.647835 + 0.761781i \(0.275675\pi\)
−0.647835 + 0.761781i \(0.724325\pi\)
\(312\) 0 0
\(313\) 310.000 0.990415 0.495208 0.868775i \(-0.335092\pi\)
0.495208 + 0.868775i \(0.335092\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 475.342i 1.49950i 0.661719 + 0.749752i \(0.269827\pi\)
−0.661719 + 0.749752i \(0.730173\pi\)
\(318\) 0 0
\(319\) −119.617 −0.374974
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 392.880i 1.21635i
\(324\) 0 0
\(325\) 3.09077 0.00951005
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 160.546i 0.487982i
\(330\) 0 0
\(331\) 390.762 1.18055 0.590274 0.807203i \(-0.299020\pi\)
0.590274 + 0.807203i \(0.299020\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 536.317i − 1.60095i
\(336\) 0 0
\(337\) 316.000 0.937685 0.468843 0.883282i \(-0.344671\pi\)
0.468843 + 0.883282i \(0.344671\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 188.970i − 0.554163i
\(342\) 0 0
\(343\) 188.000 0.548105
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 485.475i 1.39906i 0.714602 + 0.699531i \(0.246608\pi\)
−0.714602 + 0.699531i \(0.753392\pi\)
\(348\) 0 0
\(349\) −561.233 −1.60812 −0.804059 0.594549i \(-0.797330\pi\)
−0.804059 + 0.594549i \(0.797330\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 114.280i − 0.323740i −0.986812 0.161870i \(-0.948247\pi\)
0.986812 0.161870i \(-0.0517525\pi\)
\(354\) 0 0
\(355\) 114.575 0.322747
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 217.789i 0.606654i 0.952886 + 0.303327i \(0.0980975\pi\)
−0.952886 + 0.303327i \(0.901903\pi\)
\(360\) 0 0
\(361\) 355.187 0.983896
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 733.640i − 2.00997i
\(366\) 0 0
\(367\) 143.907 0.392116 0.196058 0.980592i \(-0.437186\pi\)
0.196058 + 0.980592i \(0.437186\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 19.5975i − 0.0528234i
\(372\) 0 0
\(373\) −400.663 −1.07416 −0.537082 0.843530i \(-0.680473\pi\)
−0.537082 + 0.843530i \(0.680473\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 34.5457i − 0.0916331i
\(378\) 0 0
\(379\) 156.668 0.413373 0.206686 0.978407i \(-0.433732\pi\)
0.206686 + 0.978407i \(0.433732\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 483.390i 1.26211i 0.775736 + 0.631057i \(0.217379\pi\)
−0.775736 + 0.631057i \(0.782621\pi\)
\(384\) 0 0
\(385\) 49.9067 0.129628
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 726.933i 1.86872i 0.356326 + 0.934362i \(0.384029\pi\)
−0.356326 + 0.934362i \(0.615971\pi\)
\(390\) 0 0
\(391\) 264.953 0.677630
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 2.48778i − 0.00629817i
\(396\) 0 0
\(397\) −94.1866 −0.237246 −0.118623 0.992939i \(-0.537848\pi\)
−0.118623 + 0.992939i \(0.537848\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.42898i 0.00605731i 0.999995 + 0.00302865i \(0.000964052\pi\)
−0.999995 + 0.00302865i \(0.999036\pi\)
\(402\) 0 0
\(403\) 54.5751 0.135422
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 124.312i 0.305434i
\(408\) 0 0
\(409\) −66.7617 −0.163231 −0.0816157 0.996664i \(-0.526008\pi\)
−0.0816157 + 0.996664i \(0.526008\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 147.078i 0.356122i
\(414\) 0 0
\(415\) 163.430 0.393807
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 241.021i 0.575229i 0.957746 + 0.287614i \(0.0928621\pi\)
−0.957746 + 0.287614i \(0.907138\pi\)
\(420\) 0 0
\(421\) −698.806 −1.65987 −0.829936 0.557859i \(-0.811623\pi\)
−0.829936 + 0.557859i \(0.811623\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 32.8604i 0.0773185i
\(426\) 0 0
\(427\) −135.047 −0.316269
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 783.599i 1.81810i 0.416692 + 0.909048i \(0.363189\pi\)
−0.416692 + 0.909048i \(0.636811\pi\)
\(432\) 0 0
\(433\) 317.233 0.732640 0.366320 0.930489i \(-0.380617\pi\)
0.366320 + 0.930489i \(0.380617\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 482.987i − 1.10523i
\(438\) 0 0
\(439\) −219.140 −0.499180 −0.249590 0.968352i \(-0.580296\pi\)
−0.249590 + 0.968352i \(0.580296\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 229.707i 0.518526i 0.965807 + 0.259263i \(0.0834797\pi\)
−0.965807 + 0.259263i \(0.916520\pi\)
\(444\) 0 0
\(445\) −68.5309 −0.154002
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 107.074i − 0.238471i −0.992866 0.119236i \(-0.961956\pi\)
0.992866 0.119236i \(-0.0380444\pi\)
\(450\) 0 0
\(451\) 137.808 0.305562
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 14.4132i 0.0316774i
\(456\) 0 0
\(457\) 605.042 1.32394 0.661971 0.749529i \(-0.269720\pi\)
0.661971 + 0.749529i \(0.269720\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 305.435i − 0.662550i −0.943534 0.331275i \(-0.892521\pi\)
0.943534 0.331275i \(-0.107479\pi\)
\(462\) 0 0
\(463\) 59.1400 0.127732 0.0638660 0.997958i \(-0.479657\pi\)
0.0638660 + 0.997958i \(0.479657\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 565.477i 1.21087i 0.795894 + 0.605435i \(0.207001\pi\)
−0.795894 + 0.605435i \(0.792999\pi\)
\(468\) 0 0
\(469\) 205.523 0.438216
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 45.5331i 0.0962645i
\(474\) 0 0
\(475\) 59.9016 0.126109
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 249.576i − 0.521035i −0.965469 0.260517i \(-0.916107\pi\)
0.965469 0.260517i \(-0.0838932\pi\)
\(480\) 0 0
\(481\) −35.9016 −0.0746395
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 361.357i − 0.745067i
\(486\) 0 0
\(487\) −435.897 −0.895065 −0.447532 0.894268i \(-0.647697\pi\)
−0.447532 + 0.894268i \(0.647697\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 335.506i 0.683311i 0.939825 + 0.341655i \(0.110988\pi\)
−0.939825 + 0.341655i \(0.889012\pi\)
\(492\) 0 0
\(493\) 367.282 0.744995
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 43.9066i 0.0883433i
\(498\) 0 0
\(499\) 712.373 1.42760 0.713801 0.700349i \(-0.246972\pi\)
0.713801 + 0.700349i \(0.246972\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 527.832i − 1.04937i −0.851297 0.524683i \(-0.824184\pi\)
0.851297 0.524683i \(-0.175816\pi\)
\(504\) 0 0
\(505\) −409.990 −0.811861
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 736.294i 1.44655i 0.690560 + 0.723275i \(0.257364\pi\)
−0.690560 + 0.723275i \(0.742636\pi\)
\(510\) 0 0
\(511\) 281.140 0.550176
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 1036.34i − 2.01232i
\(516\) 0 0
\(517\) 383.803 0.742366
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 372.001i 0.714013i 0.934102 + 0.357006i \(0.116203\pi\)
−0.934102 + 0.357006i \(0.883797\pi\)
\(522\) 0 0
\(523\) −251.135 −0.480181 −0.240091 0.970750i \(-0.577177\pi\)
−0.240091 + 0.970750i \(0.577177\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 580.231i 1.10101i
\(528\) 0 0
\(529\) 203.280 0.384272
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 39.7995i 0.0746707i
\(534\) 0 0
\(535\) 926.467 1.73171
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 215.155i − 0.399174i
\(540\) 0 0
\(541\) −966.992 −1.78742 −0.893708 0.448648i \(-0.851906\pi\)
−0.893708 + 0.448648i \(0.851906\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 716.996i 1.31559i
\(546\) 0 0
\(547\) 293.710 0.536947 0.268473 0.963287i \(-0.413481\pi\)
0.268473 + 0.963287i \(0.413481\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 669.524i − 1.21511i
\(552\) 0 0
\(553\) 0.953348 0.00172396
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 566.193i − 1.01650i −0.861208 0.508252i \(-0.830292\pi\)
0.861208 0.508252i \(-0.169708\pi\)
\(558\) 0 0
\(559\) −13.1501 −0.0235243
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 150.108i − 0.266622i −0.991074 0.133311i \(-0.957439\pi\)
0.991074 0.133311i \(-0.0425609\pi\)
\(564\) 0 0
\(565\) 378.526 0.669957
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 899.707i − 1.58121i −0.612328 0.790604i \(-0.709767\pi\)
0.612328 0.790604i \(-0.290233\pi\)
\(570\) 0 0
\(571\) −680.570 −1.19189 −0.595946 0.803025i \(-0.703223\pi\)
−0.595946 + 0.803025i \(0.703223\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 40.3969i − 0.0702554i
\(576\) 0 0
\(577\) −249.233 −0.431947 −0.215973 0.976399i \(-0.569292\pi\)
−0.215973 + 0.976399i \(0.569292\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 62.6284i 0.107794i
\(582\) 0 0
\(583\) −46.8499 −0.0803600
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 602.587i − 1.02655i −0.858223 0.513277i \(-0.828431\pi\)
0.858223 0.513277i \(-0.171569\pi\)
\(588\) 0 0
\(589\) 1057.71 1.79577
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 154.142i 0.259936i 0.991518 + 0.129968i \(0.0414875\pi\)
−0.991518 + 0.129968i \(0.958513\pi\)
\(594\) 0 0
\(595\) −153.238 −0.257543
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 309.244i − 0.516267i −0.966109 0.258133i \(-0.916893\pi\)
0.966109 0.258133i \(-0.0831074\pi\)
\(600\) 0 0
\(601\) −811.327 −1.34996 −0.674981 0.737836i \(-0.735848\pi\)
−0.674981 + 0.737836i \(0.735848\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 512.196i 0.846605i
\(606\) 0 0
\(607\) −202.197 −0.333108 −0.166554 0.986032i \(-0.553264\pi\)
−0.166554 + 0.986032i \(0.553264\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 110.844i 0.181413i
\(612\) 0 0
\(613\) −694.560 −1.13305 −0.566525 0.824044i \(-0.691713\pi\)
−0.566525 + 0.824044i \(0.691713\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 896.267i 1.45262i 0.687367 + 0.726311i \(0.258767\pi\)
−0.687367 + 0.726311i \(0.741233\pi\)
\(618\) 0 0
\(619\) −672.570 −1.08654 −0.543271 0.839557i \(-0.682815\pi\)
−0.543271 + 0.839557i \(0.682815\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 26.2619i − 0.0421540i
\(624\) 0 0
\(625\) −675.948 −1.08152
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 381.699i − 0.606834i
\(630\) 0 0
\(631\) −218.953 −0.346994 −0.173497 0.984834i \(-0.555507\pi\)
−0.173497 + 0.984834i \(0.555507\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 422.985i 0.666119i
\(636\) 0 0
\(637\) 62.1374 0.0975470
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 1193.18i − 1.86144i −0.365734 0.930720i \(-0.619182\pi\)
0.365734 0.930720i \(-0.380818\pi\)
\(642\) 0 0
\(643\) 1009.33 1.56971 0.784857 0.619676i \(-0.212736\pi\)
0.784857 + 0.619676i \(0.212736\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 637.605i − 0.985479i −0.870177 0.492740i \(-0.835995\pi\)
0.870177 0.492740i \(-0.164005\pi\)
\(648\) 0 0
\(649\) 351.606 0.541767
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 939.073i − 1.43809i −0.694964 0.719045i \(-0.744580\pi\)
0.694964 0.719045i \(-0.255420\pi\)
\(654\) 0 0
\(655\) −858.280 −1.31035
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 1224.84i − 1.85864i −0.369281 0.929318i \(-0.620396\pi\)
0.369281 0.929318i \(-0.379604\pi\)
\(660\) 0 0
\(661\) 1034.18 1.56456 0.782282 0.622925i \(-0.214056\pi\)
0.782282 + 0.622925i \(0.214056\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 279.340i 0.420060i
\(666\) 0 0
\(667\) −451.518 −0.676939
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 322.844i 0.481138i
\(672\) 0 0
\(673\) −18.9533 −0.0281625 −0.0140812 0.999901i \(-0.504482\pi\)
−0.0140812 + 0.999901i \(0.504482\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1024.66i 1.51353i 0.653685 + 0.756767i \(0.273222\pi\)
−0.653685 + 0.756767i \(0.726778\pi\)
\(678\) 0 0
\(679\) 138.477 0.203942
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 965.904i 1.41421i 0.707109 + 0.707104i \(0.249999\pi\)
−0.707109 + 0.707104i \(0.750001\pi\)
\(684\) 0 0
\(685\) 1173.19 1.71268
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 13.5304i − 0.0196377i
\(690\) 0 0
\(691\) −419.233 −0.606705 −0.303353 0.952878i \(-0.598106\pi\)
−0.303353 + 0.952878i \(0.598106\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 889.210i − 1.27944i
\(696\) 0 0
\(697\) −423.140 −0.607087
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1085.27i 1.54817i 0.633081 + 0.774086i \(0.281790\pi\)
−0.633081 + 0.774086i \(0.718210\pi\)
\(702\) 0 0
\(703\) −695.803 −0.989763
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 157.113i − 0.222225i
\(708\) 0 0
\(709\) 369.469 0.521113 0.260556 0.965459i \(-0.416094\pi\)
0.260556 + 0.965459i \(0.416094\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 713.306i − 1.00043i
\(714\) 0 0
\(715\) 34.4564 0.0481907
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 468.567i 0.651692i 0.945423 + 0.325846i \(0.105649\pi\)
−0.945423 + 0.325846i \(0.894351\pi\)
\(720\) 0 0
\(721\) 397.140 0.550818
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 55.9988i − 0.0772397i
\(726\) 0 0
\(727\) −502.570 −0.691293 −0.345646 0.938365i \(-0.612340\pi\)
−0.345646 + 0.938365i \(0.612340\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 139.809i − 0.191257i
\(732\) 0 0
\(733\) −1189.37 −1.62261 −0.811303 0.584625i \(-0.801241\pi\)
−0.811303 + 0.584625i \(0.801241\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 491.326i − 0.666657i
\(738\) 0 0
\(739\) −1431.04 −1.93645 −0.968225 0.250081i \(-0.919543\pi\)
−0.968225 + 0.250081i \(0.919543\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 203.508i 0.273900i 0.990578 + 0.136950i \(0.0437300\pi\)
−0.990578 + 0.136950i \(0.956270\pi\)
\(744\) 0 0
\(745\) −310.865 −0.417269
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 355.034i 0.474010i
\(750\) 0 0
\(751\) −930.570 −1.23911 −0.619554 0.784954i \(-0.712686\pi\)
−0.619554 + 0.784954i \(0.712686\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 1265.98i − 1.67680i
\(756\) 0 0
\(757\) −306.894 −0.405408 −0.202704 0.979240i \(-0.564973\pi\)
−0.202704 + 0.979240i \(0.564973\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 939.701i − 1.23482i −0.786640 0.617412i \(-0.788181\pi\)
0.786640 0.617412i \(-0.211819\pi\)
\(762\) 0 0
\(763\) −274.762 −0.360107
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 101.545i 0.132393i
\(768\) 0 0
\(769\) −1238.19 −1.61013 −0.805063 0.593190i \(-0.797868\pi\)
−0.805063 + 0.593190i \(0.797868\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 697.502i 0.902332i 0.892440 + 0.451166i \(0.148992\pi\)
−0.892440 + 0.451166i \(0.851008\pi\)
\(774\) 0 0
\(775\) 88.4665 0.114150
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 771.347i 0.990176i
\(780\) 0 0
\(781\) 104.964 0.134396
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 843.483i − 1.07450i
\(786\) 0 0
\(787\) 206.575 0.262484 0.131242 0.991350i \(-0.458103\pi\)
0.131242 + 0.991350i \(0.458103\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 145.056i 0.183383i
\(792\) 0 0
\(793\) −93.2383 −0.117577
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 566.596i 0.710911i 0.934693 + 0.355455i \(0.115674\pi\)
−0.934693 + 0.355455i \(0.884326\pi\)
\(798\) 0 0
\(799\) −1178.47 −1.47493
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 672.096i − 0.836981i
\(804\) 0 0
\(805\) 188.383 0.234017
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 806.091i − 0.996404i −0.867061 0.498202i \(-0.833994\pi\)
0.867061 0.498202i \(-0.166006\pi\)
\(810\) 0 0
\(811\) −420.472 −0.518461 −0.259230 0.965816i \(-0.583469\pi\)
−0.259230 + 0.965816i \(0.583469\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 422.498i − 0.518403i
\(816\) 0 0
\(817\) −254.860 −0.311946
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 386.042i 0.470209i 0.971970 + 0.235105i \(0.0755433\pi\)
−0.971970 + 0.235105i \(0.924457\pi\)
\(822\) 0 0
\(823\) 998.570 1.21333 0.606665 0.794958i \(-0.292507\pi\)
0.606665 + 0.794958i \(0.292507\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 684.604i 0.827816i 0.910319 + 0.413908i \(0.135837\pi\)
−0.910319 + 0.413908i \(0.864163\pi\)
\(828\) 0 0
\(829\) 533.567 0.643628 0.321814 0.946803i \(-0.395707\pi\)
0.321814 + 0.946803i \(0.395707\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 660.632i 0.793076i
\(834\) 0 0
\(835\) −935.606 −1.12049
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 1441.02i − 1.71754i −0.512360 0.858771i \(-0.671229\pi\)
0.512360 0.858771i \(-0.328771\pi\)
\(840\) 0 0
\(841\) 215.098 0.255765
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 872.066i − 1.03203i
\(846\) 0 0
\(847\) −196.280 −0.231735
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 469.241i 0.551399i
\(852\) 0 0
\(853\) 670.383 0.785913 0.392956 0.919557i \(-0.371452\pi\)
0.392956 + 0.919557i \(0.371452\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 223.046i − 0.260264i −0.991497 0.130132i \(-0.958460\pi\)
0.991497 0.130132i \(-0.0415401\pi\)
\(858\) 0 0
\(859\) −1174.48 −1.36726 −0.683630 0.729829i \(-0.739600\pi\)
−0.683630 + 0.729829i \(0.739600\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 780.785i 0.904734i 0.891832 + 0.452367i \(0.149420\pi\)
−0.891832 + 0.452367i \(0.850580\pi\)
\(864\) 0 0
\(865\) −138.295 −0.159879
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 2.27908i − 0.00262265i
\(870\) 0 0
\(871\) 141.897 0.162912
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 237.588i − 0.271529i
\(876\) 0 0
\(877\) 1096.28 1.25003 0.625017 0.780611i \(-0.285092\pi\)
0.625017 + 0.780611i \(0.285092\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1164.43i 1.32171i 0.750514 + 0.660854i \(0.229806\pi\)
−0.750514 + 0.660854i \(0.770194\pi\)
\(882\) 0 0
\(883\) −11.8032 −0.0133672 −0.00668360 0.999978i \(-0.502127\pi\)
−0.00668360 + 0.999978i \(0.502127\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 295.505i 0.333151i 0.986029 + 0.166575i \(0.0532709\pi\)
−0.986029 + 0.166575i \(0.946729\pi\)
\(888\) 0 0
\(889\) −162.093 −0.182332
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2148.24i 2.40565i
\(894\) 0 0
\(895\) −1600.57 −1.78835
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 988.796i − 1.09988i
\(900\) 0 0
\(901\) 143.852 0.159659
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 299.473i 0.330909i
\(906\) 0 0
\(907\) −423.617 −0.467053 −0.233526 0.972350i \(-0.575026\pi\)
−0.233526 + 0.972350i \(0.575026\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 1008.39i − 1.10691i −0.832880 0.553454i \(-0.813309\pi\)
0.832880 0.553454i \(-0.186691\pi\)
\(912\) 0 0
\(913\) 149.720 0.163987
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 328.904i − 0.358674i
\(918\) 0 0
\(919\) −1067.51 −1.16160 −0.580802 0.814045i \(-0.697261\pi\)
−0.580802 + 0.814045i \(0.697261\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 30.3138i 0.0328427i
\(924\) 0 0
\(925\) −58.1968 −0.0629154
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 720.842i − 0.775934i −0.921673 0.387967i \(-0.873178\pi\)
0.921673 0.387967i \(-0.126822\pi\)
\(930\) 0 0
\(931\) 1204.27 1.29353
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 366.333i 0.391800i
\(936\) 0 0
\(937\) −1032.47 −1.10189 −0.550943 0.834543i \(-0.685732\pi\)
−0.550943 + 0.834543i \(0.685732\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 1597.62i − 1.69779i −0.528565 0.848893i \(-0.677270\pi\)
0.528565 0.848893i \(-0.322730\pi\)
\(942\) 0 0
\(943\) 520.187 0.551629
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 625.214i − 0.660205i −0.943945 0.330103i \(-0.892917\pi\)
0.943945 0.330103i \(-0.107083\pi\)
\(948\) 0 0
\(949\) 194.103 0.204535
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 807.036i 0.846838i 0.905934 + 0.423419i \(0.139170\pi\)
−0.905934 + 0.423419i \(0.860830\pi\)
\(954\) 0 0
\(955\) 1131.03 1.18432
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 449.581i 0.468802i
\(960\) 0 0
\(961\) 601.093 0.625487
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 976.690i 1.01211i
\(966\) 0 0
\(967\) 1296.47 1.34071 0.670355 0.742041i \(-0.266142\pi\)
0.670355 + 0.742041i \(0.266142\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 200.075i 0.206050i 0.994679 + 0.103025i \(0.0328522\pi\)
−0.994679 + 0.103025i \(0.967148\pi\)
\(972\) 0 0
\(973\) 340.757 0.350212
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1309.95i − 1.34079i −0.742003 0.670396i \(-0.766124\pi\)
0.742003 0.670396i \(-0.233876\pi\)
\(978\) 0 0
\(979\) −62.7820 −0.0641287
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1553.47i − 1.58033i −0.612893 0.790166i \(-0.709994\pi\)
0.612893 0.790166i \(-0.290006\pi\)
\(984\) 0 0
\(985\) −343.907 −0.349144
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 171.874i 0.173786i
\(990\) 0 0
\(991\) 922.757 0.931137 0.465568 0.885012i \(-0.345850\pi\)
0.465568 + 0.885012i \(0.345850\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 599.946i − 0.602960i
\(996\) 0 0
\(997\) −787.710 −0.790080 −0.395040 0.918664i \(-0.629269\pi\)
−0.395040 + 0.918664i \(0.629269\pi\)
\(998\) 0 0
\(999\) 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 1152.3.e.c.1025.3 yes 4
3.2 odd 2 inner 1152.3.e.c.1025.2 yes 4
4.3 odd 2 1152.3.e.g.1025.3 yes 4
8.3 odd 2 1152.3.e.e.1025.2 yes 4
8.5 even 2 1152.3.e.a.1025.2 4
12.11 even 2 1152.3.e.g.1025.2 yes 4
16.3 odd 4 2304.3.h.j.2177.5 8
16.5 even 4 2304.3.h.l.2177.4 8
16.11 odd 4 2304.3.h.j.2177.4 8
16.13 even 4 2304.3.h.l.2177.5 8
24.5 odd 2 1152.3.e.a.1025.3 yes 4
24.11 even 2 1152.3.e.e.1025.3 yes 4
48.5 odd 4 2304.3.h.l.2177.6 8
48.11 even 4 2304.3.h.j.2177.6 8
48.29 odd 4 2304.3.h.l.2177.3 8
48.35 even 4 2304.3.h.j.2177.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.3.e.a.1025.2 4 8.5 even 2
1152.3.e.a.1025.3 yes 4 24.5 odd 2
1152.3.e.c.1025.2 yes 4 3.2 odd 2 inner
1152.3.e.c.1025.3 yes 4 1.1 even 1 trivial
1152.3.e.e.1025.2 yes 4 8.3 odd 2
1152.3.e.e.1025.3 yes 4 24.11 even 2
1152.3.e.g.1025.2 yes 4 12.11 even 2
1152.3.e.g.1025.3 yes 4 4.3 odd 2
2304.3.h.j.2177.3 8 48.35 even 4
2304.3.h.j.2177.4 8 16.11 odd 4
2304.3.h.j.2177.5 8 16.3 odd 4
2304.3.h.j.2177.6 8 48.11 even 4
2304.3.h.l.2177.3 8 48.29 odd 4
2304.3.h.l.2177.4 8 16.5 even 4
2304.3.h.l.2177.5 8 16.13 even 4
2304.3.h.l.2177.6 8 48.5 odd 4