Properties

Label 4140.3.d.a.2161.10
Level $4140$
Weight $3$
Character 4140.2161
Analytic conductor $112.807$
Analytic rank $0$
Dimension $16$
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] = [4140,3,Mod(2161,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4140.2161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 4140.d (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: \(112.806829445\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 64 x^{14} - 16 x^{13} + 2252 x^{12} + 648 x^{11} - 30106 x^{10} + 12360 x^{9} + 374528 x^{8} + 196544 x^{7} + 1261236 x^{6} - 4237944 x^{5} + \cdots + 1535848276 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 460)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2161.10
Root \(-5.14043 + 2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 4140.2161
Dual form 4140.3.d.a.2161.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.23607i q^{5} -7.88014i q^{7} +O(q^{10})\) \(q+2.23607i q^{5} -7.88014i q^{7} -3.98380i q^{11} +13.5122 q^{13} +8.06277i q^{17} +2.96508i q^{19} +(4.80075 + 22.4934i) q^{23} -5.00000 q^{25} -0.688711 q^{29} -36.6485 q^{31} +17.6205 q^{35} +3.63322i q^{37} +32.4636 q^{41} -42.5054i q^{43} +59.3870 q^{47} -13.0966 q^{49} +57.5176i q^{53} +8.90804 q^{55} +25.3686 q^{59} +31.8220i q^{61} +30.2143i q^{65} +108.405i q^{67} +67.6191 q^{71} -102.151 q^{73} -31.3929 q^{77} +48.5263i q^{79} +73.0389i q^{83} -18.0289 q^{85} -92.9465i q^{89} -106.478i q^{91} -6.63011 q^{95} +138.539i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{13} + 14 q^{23} - 80 q^{25} - 90 q^{29} + 10 q^{31} - 30 q^{35} - 186 q^{41} + 320 q^{47} + 2 q^{49} - 120 q^{55} + 90 q^{59} + 238 q^{71} - 280 q^{73} - 324 q^{77} - 30 q^{85} - 80 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) 2.23607i 0.447214i
\(6\) 0 0
\(7\) 7.88014i 1.12573i −0.826547 0.562867i \(-0.809698\pi\)
0.826547 0.562867i \(-0.190302\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.98380i 0.362163i −0.983468 0.181082i \(-0.942040\pi\)
0.983468 0.181082i \(-0.0579598\pi\)
\(12\) 0 0
\(13\) 13.5122 1.03940 0.519702 0.854348i \(-0.326043\pi\)
0.519702 + 0.854348i \(0.326043\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.06277i 0.474280i 0.971475 + 0.237140i \(0.0762101\pi\)
−0.971475 + 0.237140i \(0.923790\pi\)
\(18\) 0 0
\(19\) 2.96508i 0.156057i 0.996951 + 0.0780284i \(0.0248625\pi\)
−0.996951 + 0.0780284i \(0.975138\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.80075 + 22.4934i 0.208728 + 0.977974i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.688711 −0.0237487 −0.0118743 0.999929i \(-0.503780\pi\)
−0.0118743 + 0.999929i \(0.503780\pi\)
\(30\) 0 0
\(31\) −36.6485 −1.18221 −0.591105 0.806595i \(-0.701308\pi\)
−0.591105 + 0.806595i \(0.701308\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 17.6205 0.503444
\(36\) 0 0
\(37\) 3.63322i 0.0981952i 0.998794 + 0.0490976i \(0.0156345\pi\)
−0.998794 + 0.0490976i \(0.984365\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 32.4636 0.791796 0.395898 0.918294i \(-0.370433\pi\)
0.395898 + 0.918294i \(0.370433\pi\)
\(42\) 0 0
\(43\) 42.5054i 0.988497i −0.869321 0.494248i \(-0.835443\pi\)
0.869321 0.494248i \(-0.164557\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 59.3870 1.26355 0.631777 0.775151i \(-0.282326\pi\)
0.631777 + 0.775151i \(0.282326\pi\)
\(48\) 0 0
\(49\) −13.0966 −0.267279
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 57.5176i 1.08524i 0.839979 + 0.542618i \(0.182567\pi\)
−0.839979 + 0.542618i \(0.817433\pi\)
\(54\) 0 0
\(55\) 8.90804 0.161964
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 25.3686 0.429977 0.214989 0.976617i \(-0.431029\pi\)
0.214989 + 0.976617i \(0.431029\pi\)
\(60\) 0 0
\(61\) 31.8220i 0.521672i 0.965383 + 0.260836i \(0.0839981\pi\)
−0.965383 + 0.260836i \(0.916002\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 30.2143i 0.464835i
\(66\) 0 0
\(67\) 108.405i 1.61799i 0.587819 + 0.808993i \(0.299987\pi\)
−0.587819 + 0.808993i \(0.700013\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 67.6191 0.952382 0.476191 0.879342i \(-0.342017\pi\)
0.476191 + 0.879342i \(0.342017\pi\)
\(72\) 0 0
\(73\) −102.151 −1.39933 −0.699663 0.714473i \(-0.746667\pi\)
−0.699663 + 0.714473i \(0.746667\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −31.3929 −0.407700
\(78\) 0 0
\(79\) 48.5263i 0.614257i 0.951668 + 0.307128i \(0.0993681\pi\)
−0.951668 + 0.307128i \(0.900632\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 73.0389i 0.879987i 0.898001 + 0.439993i \(0.145019\pi\)
−0.898001 + 0.439993i \(0.854981\pi\)
\(84\) 0 0
\(85\) −18.0289 −0.212105
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 92.9465i 1.04434i −0.852840 0.522172i \(-0.825122\pi\)
0.852840 0.522172i \(-0.174878\pi\)
\(90\) 0 0
\(91\) 106.478i 1.17009i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.63011 −0.0697907
\(96\) 0 0
\(97\) 138.539i 1.42824i 0.700025 + 0.714119i \(0.253172\pi\)
−0.700025 + 0.714119i \(0.746828\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −71.2312 −0.705259 −0.352630 0.935763i \(-0.614712\pi\)
−0.352630 + 0.935763i \(0.614712\pi\)
\(102\) 0 0
\(103\) 173.582i 1.68526i −0.538492 0.842630i \(-0.681006\pi\)
0.538492 0.842630i \(-0.318994\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 88.3635i 0.825827i −0.910770 0.412914i \(-0.864511\pi\)
0.910770 0.412914i \(-0.135489\pi\)
\(108\) 0 0
\(109\) 142.645i 1.30867i −0.756203 0.654337i \(-0.772948\pi\)
0.756203 0.654337i \(-0.227052\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 91.5083i 0.809808i −0.914359 0.404904i \(-0.867305\pi\)
0.914359 0.404904i \(-0.132695\pi\)
\(114\) 0 0
\(115\) −50.2968 + 10.7348i −0.437363 + 0.0933462i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 63.5358 0.533914
\(120\) 0 0
\(121\) 105.129 0.868838
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) 101.492 0.799147 0.399574 0.916701i \(-0.369158\pi\)
0.399574 + 0.916701i \(0.369158\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 43.9095 0.335187 0.167594 0.985856i \(-0.446400\pi\)
0.167594 + 0.985856i \(0.446400\pi\)
\(132\) 0 0
\(133\) 23.3652 0.175678
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 215.195i 1.57077i 0.619009 + 0.785384i \(0.287535\pi\)
−0.619009 + 0.785384i \(0.712465\pi\)
\(138\) 0 0
\(139\) 63.6545 0.457946 0.228973 0.973433i \(-0.426463\pi\)
0.228973 + 0.973433i \(0.426463\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 53.8300i 0.376434i
\(144\) 0 0
\(145\) 1.54001i 0.0106207i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 97.7394i 0.655969i 0.944683 + 0.327985i \(0.106369\pi\)
−0.944683 + 0.327985i \(0.893631\pi\)
\(150\) 0 0
\(151\) −209.737 −1.38899 −0.694493 0.719500i \(-0.744371\pi\)
−0.694493 + 0.719500i \(0.744371\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 81.9485i 0.528700i
\(156\) 0 0
\(157\) 287.459i 1.83095i −0.402378 0.915474i \(-0.631816\pi\)
0.402378 0.915474i \(-0.368184\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 177.251 37.8306i 1.10094 0.234973i
\(162\) 0 0
\(163\) 112.054 0.687446 0.343723 0.939071i \(-0.388312\pi\)
0.343723 + 0.939071i \(0.388312\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 192.709 1.15394 0.576972 0.816764i \(-0.304234\pi\)
0.576972 + 0.816764i \(0.304234\pi\)
\(168\) 0 0
\(169\) 13.5807 0.0803593
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 303.454 1.75407 0.877034 0.480429i \(-0.159519\pi\)
0.877034 + 0.480429i \(0.159519\pi\)
\(174\) 0 0
\(175\) 39.4007i 0.225147i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 94.6527 0.528786 0.264393 0.964415i \(-0.414828\pi\)
0.264393 + 0.964415i \(0.414828\pi\)
\(180\) 0 0
\(181\) 251.014i 1.38682i 0.720545 + 0.693408i \(0.243892\pi\)
−0.720545 + 0.693408i \(0.756108\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.12413 −0.0439142
\(186\) 0 0
\(187\) 32.1204 0.171767
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 74.3125i 0.389071i 0.980895 + 0.194535i \(0.0623199\pi\)
−0.980895 + 0.194535i \(0.937680\pi\)
\(192\) 0 0
\(193\) −279.864 −1.45007 −0.725037 0.688710i \(-0.758177\pi\)
−0.725037 + 0.688710i \(0.758177\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 239.461 1.21554 0.607770 0.794113i \(-0.292064\pi\)
0.607770 + 0.794113i \(0.292064\pi\)
\(198\) 0 0
\(199\) 302.424i 1.51972i −0.650087 0.759860i \(-0.725267\pi\)
0.650087 0.759860i \(-0.274733\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5.42714i 0.0267347i
\(204\) 0 0
\(205\) 72.5909i 0.354102i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 11.8123 0.0565180
\(210\) 0 0
\(211\) 313.131 1.48403 0.742016 0.670382i \(-0.233870\pi\)
0.742016 + 0.670382i \(0.233870\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 95.0449 0.442069
\(216\) 0 0
\(217\) 288.795i 1.33085i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 108.946i 0.492969i
\(222\) 0 0
\(223\) −42.5933 −0.191001 −0.0955007 0.995429i \(-0.530445\pi\)
−0.0955007 + 0.995429i \(0.530445\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 246.716i 1.08686i −0.839456 0.543428i \(-0.817126\pi\)
0.839456 0.543428i \(-0.182874\pi\)
\(228\) 0 0
\(229\) 192.536i 0.840767i 0.907347 + 0.420383i \(0.138104\pi\)
−0.907347 + 0.420383i \(0.861896\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −246.508 −1.05797 −0.528986 0.848630i \(-0.677428\pi\)
−0.528986 + 0.848630i \(0.677428\pi\)
\(234\) 0 0
\(235\) 132.793i 0.565078i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 36.1050 0.151067 0.0755335 0.997143i \(-0.475934\pi\)
0.0755335 + 0.997143i \(0.475934\pi\)
\(240\) 0 0
\(241\) 137.672i 0.571253i −0.958341 0.285627i \(-0.907798\pi\)
0.958341 0.285627i \(-0.0922017\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 29.2850i 0.119531i
\(246\) 0 0
\(247\) 40.0648i 0.162206i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 60.0114i 0.239089i −0.992829 0.119545i \(-0.961857\pi\)
0.992829 0.119545i \(-0.0381435\pi\)
\(252\) 0 0
\(253\) 89.6091 19.1252i 0.354186 0.0755938i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 162.546 0.632475 0.316237 0.948680i \(-0.397580\pi\)
0.316237 + 0.948680i \(0.397580\pi\)
\(258\) 0 0
\(259\) 28.6303 0.110542
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 416.956i 1.58539i −0.609621 0.792693i \(-0.708678\pi\)
0.609621 0.792693i \(-0.291322\pi\)
\(264\) 0 0
\(265\) −128.613 −0.485333
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 354.383 1.31741 0.658704 0.752402i \(-0.271105\pi\)
0.658704 + 0.752402i \(0.271105\pi\)
\(270\) 0 0
\(271\) 430.921 1.59011 0.795056 0.606535i \(-0.207441\pi\)
0.795056 + 0.606535i \(0.207441\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 19.9190i 0.0724327i
\(276\) 0 0
\(277\) −17.8119 −0.0643029 −0.0321515 0.999483i \(-0.510236\pi\)
−0.0321515 + 0.999483i \(0.510236\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 537.350i 1.91228i 0.292915 + 0.956138i \(0.405375\pi\)
−0.292915 + 0.956138i \(0.594625\pi\)
\(282\) 0 0
\(283\) 227.251i 0.803007i 0.915858 + 0.401503i \(0.131512\pi\)
−0.915858 + 0.401503i \(0.868488\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 255.818i 0.891352i
\(288\) 0 0
\(289\) 223.992 0.775058
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 570.580i 1.94737i 0.227896 + 0.973685i \(0.426815\pi\)
−0.227896 + 0.973685i \(0.573185\pi\)
\(294\) 0 0
\(295\) 56.7260i 0.192292i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 64.8690 + 303.936i 0.216953 + 1.01651i
\(300\) 0 0
\(301\) −334.948 −1.11279
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −71.1561 −0.233299
\(306\) 0 0
\(307\) −17.4675 −0.0568973 −0.0284487 0.999595i \(-0.509057\pi\)
−0.0284487 + 0.999595i \(0.509057\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 41.5565 0.133622 0.0668111 0.997766i \(-0.478718\pi\)
0.0668111 + 0.997766i \(0.478718\pi\)
\(312\) 0 0
\(313\) 193.750i 0.619011i 0.950898 + 0.309505i \(0.100164\pi\)
−0.950898 + 0.309505i \(0.899836\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −292.088 −0.921413 −0.460707 0.887553i \(-0.652404\pi\)
−0.460707 + 0.887553i \(0.652404\pi\)
\(318\) 0 0
\(319\) 2.74369i 0.00860090i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −23.9067 −0.0740146
\(324\) 0 0
\(325\) −67.5612 −0.207881
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 467.978i 1.42243i
\(330\) 0 0
\(331\) 562.886 1.70056 0.850281 0.526329i \(-0.176432\pi\)
0.850281 + 0.526329i \(0.176432\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −242.401 −0.723585
\(336\) 0 0
\(337\) 178.192i 0.528759i −0.964419 0.264379i \(-0.914833\pi\)
0.964419 0.264379i \(-0.0851671\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 146.000i 0.428153i
\(342\) 0 0
\(343\) 282.924i 0.824850i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 462.453 1.33272 0.666359 0.745631i \(-0.267852\pi\)
0.666359 + 0.745631i \(0.267852\pi\)
\(348\) 0 0
\(349\) 159.441 0.456850 0.228425 0.973562i \(-0.426642\pi\)
0.228425 + 0.973562i \(0.426642\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 474.031 1.34286 0.671432 0.741067i \(-0.265680\pi\)
0.671432 + 0.741067i \(0.265680\pi\)
\(354\) 0 0
\(355\) 151.201i 0.425918i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 462.027i 1.28698i 0.765453 + 0.643492i \(0.222515\pi\)
−0.765453 + 0.643492i \(0.777485\pi\)
\(360\) 0 0
\(361\) 352.208 0.975646
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 228.416i 0.625798i
\(366\) 0 0
\(367\) 215.737i 0.587839i 0.955830 + 0.293919i \(0.0949597\pi\)
−0.955830 + 0.293919i \(0.905040\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 453.247 1.22169
\(372\) 0 0
\(373\) 137.933i 0.369795i −0.982758 0.184897i \(-0.940805\pi\)
0.982758 0.184897i \(-0.0591952\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9.30604 −0.0246844
\(378\) 0 0
\(379\) 15.4281i 0.0407073i −0.999793 0.0203536i \(-0.993521\pi\)
0.999793 0.0203536i \(-0.00647921\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 713.842i 1.86382i 0.362694 + 0.931908i \(0.381857\pi\)
−0.362694 + 0.931908i \(0.618143\pi\)
\(384\) 0 0
\(385\) 70.1966i 0.182329i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 165.475i 0.425385i 0.977119 + 0.212692i \(0.0682232\pi\)
−0.977119 + 0.212692i \(0.931777\pi\)
\(390\) 0 0
\(391\) −181.359 + 38.7074i −0.463834 + 0.0989958i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −108.508 −0.274704
\(396\) 0 0
\(397\) −92.3570 −0.232637 −0.116319 0.993212i \(-0.537109\pi\)
−0.116319 + 0.993212i \(0.537109\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 162.145i 0.404352i −0.979349 0.202176i \(-0.935199\pi\)
0.979349 0.202176i \(-0.0648013\pi\)
\(402\) 0 0
\(403\) −495.203 −1.22879
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 14.4740 0.0355627
\(408\) 0 0
\(409\) −91.8175 −0.224493 −0.112246 0.993680i \(-0.535805\pi\)
−0.112246 + 0.993680i \(0.535805\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 199.909i 0.484040i
\(414\) 0 0
\(415\) −163.320 −0.393542
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 210.857i 0.503238i −0.967826 0.251619i \(-0.919037\pi\)
0.967826 0.251619i \(-0.0809629\pi\)
\(420\) 0 0
\(421\) 557.635i 1.32455i 0.749261 + 0.662274i \(0.230409\pi\)
−0.749261 + 0.662274i \(0.769591\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 40.3138i 0.0948561i
\(426\) 0 0
\(427\) 250.762 0.587264
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 162.258i 0.376469i −0.982124 0.188234i \(-0.939724\pi\)
0.982124 0.188234i \(-0.0602765\pi\)
\(432\) 0 0
\(433\) 465.859i 1.07589i −0.842981 0.537944i \(-0.819201\pi\)
0.842981 0.537944i \(-0.180799\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −66.6947 + 14.2346i −0.152619 + 0.0325735i
\(438\) 0 0
\(439\) −112.072 −0.255289 −0.127644 0.991820i \(-0.540742\pi\)
−0.127644 + 0.991820i \(0.540742\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −765.565 −1.72814 −0.864069 0.503373i \(-0.832092\pi\)
−0.864069 + 0.503373i \(0.832092\pi\)
\(444\) 0 0
\(445\) 207.835 0.467044
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 167.225 0.372439 0.186220 0.982508i \(-0.440376\pi\)
0.186220 + 0.982508i \(0.440376\pi\)
\(450\) 0 0
\(451\) 129.329i 0.286759i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 238.093 0.523281
\(456\) 0 0
\(457\) 283.250i 0.619804i 0.950768 + 0.309902i \(0.100296\pi\)
−0.950768 + 0.309902i \(0.899704\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 98.1109 0.212822 0.106411 0.994322i \(-0.466064\pi\)
0.106411 + 0.994322i \(0.466064\pi\)
\(462\) 0 0
\(463\) 749.068 1.61786 0.808929 0.587907i \(-0.200048\pi\)
0.808929 + 0.587907i \(0.200048\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 71.8733i 0.153904i −0.997035 0.0769522i \(-0.975481\pi\)
0.997035 0.0769522i \(-0.0245189\pi\)
\(468\) 0 0
\(469\) 854.247 1.82142
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −169.333 −0.357997
\(474\) 0 0
\(475\) 14.8254i 0.0312113i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 624.966i 1.30473i 0.757905 + 0.652365i \(0.226223\pi\)
−0.757905 + 0.652365i \(0.773777\pi\)
\(480\) 0 0
\(481\) 49.0930i 0.102064i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −309.783 −0.638727
\(486\) 0 0
\(487\) 879.504 1.80596 0.902982 0.429679i \(-0.141373\pi\)
0.902982 + 0.429679i \(0.141373\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −716.334 −1.45893 −0.729464 0.684019i \(-0.760230\pi\)
−0.729464 + 0.684019i \(0.760230\pi\)
\(492\) 0 0
\(493\) 5.55292i 0.0112635i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 532.848i 1.07213i
\(498\) 0 0
\(499\) −34.6000 −0.0693387 −0.0346693 0.999399i \(-0.511038\pi\)
−0.0346693 + 0.999399i \(0.511038\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 287.726i 0.572020i 0.958227 + 0.286010i \(0.0923291\pi\)
−0.958227 + 0.286010i \(0.907671\pi\)
\(504\) 0 0
\(505\) 159.278i 0.315402i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −284.848 −0.559624 −0.279812 0.960055i \(-0.590272\pi\)
−0.279812 + 0.960055i \(0.590272\pi\)
\(510\) 0 0
\(511\) 804.963i 1.57527i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 388.141 0.753672
\(516\) 0 0
\(517\) 236.586i 0.457613i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 195.190i 0.374645i 0.982298 + 0.187323i \(0.0599810\pi\)
−0.982298 + 0.187323i \(0.940019\pi\)
\(522\) 0 0
\(523\) 0.505041i 0.000965661i −1.00000 0.000482831i \(-0.999846\pi\)
1.00000 0.000482831i \(-0.000153690\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 295.488i 0.560699i
\(528\) 0 0
\(529\) −482.906 + 215.970i −0.912865 + 0.408262i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 438.657 0.822995
\(534\) 0 0
\(535\) 197.587 0.369321
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 52.1744i 0.0967985i
\(540\) 0 0
\(541\) 579.191 1.07059 0.535296 0.844664i \(-0.320200\pi\)
0.535296 + 0.844664i \(0.320200\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 318.965 0.585257
\(546\) 0 0
\(547\) 623.934 1.14065 0.570324 0.821420i \(-0.306818\pi\)
0.570324 + 0.821420i \(0.306818\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.04208i 0.00370614i
\(552\) 0 0
\(553\) 382.394 0.691490
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 445.491i 0.799805i −0.916558 0.399903i \(-0.869044\pi\)
0.916558 0.399903i \(-0.130956\pi\)
\(558\) 0 0
\(559\) 574.343i 1.02745i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 315.518i 0.560422i −0.959938 0.280211i \(-0.909595\pi\)
0.959938 0.280211i \(-0.0904045\pi\)
\(564\) 0 0
\(565\) 204.619 0.362157
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 954.594i 1.67767i 0.544386 + 0.838835i \(0.316763\pi\)
−0.544386 + 0.838835i \(0.683237\pi\)
\(570\) 0 0
\(571\) 266.616i 0.466929i 0.972365 + 0.233464i \(0.0750062\pi\)
−0.972365 + 0.233464i \(0.924994\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −24.0038 112.467i −0.0417457 0.195595i
\(576\) 0 0
\(577\) −1069.07 −1.85280 −0.926401 0.376539i \(-0.877114\pi\)
−0.926401 + 0.376539i \(0.877114\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 575.557 0.990632
\(582\) 0 0
\(583\) 229.138 0.393033
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1065.52 1.81520 0.907600 0.419835i \(-0.137912\pi\)
0.907600 + 0.419835i \(0.137912\pi\)
\(588\) 0 0
\(589\) 108.666i 0.184492i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 337.936 0.569875 0.284938 0.958546i \(-0.408027\pi\)
0.284938 + 0.958546i \(0.408027\pi\)
\(594\) 0 0
\(595\) 142.070i 0.238774i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −549.708 −0.917709 −0.458854 0.888511i \(-0.651740\pi\)
−0.458854 + 0.888511i \(0.651740\pi\)
\(600\) 0 0
\(601\) −375.387 −0.624605 −0.312302 0.949983i \(-0.601100\pi\)
−0.312302 + 0.949983i \(0.601100\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 235.076i 0.388556i
\(606\) 0 0
\(607\) 440.263 0.725309 0.362654 0.931924i \(-0.381871\pi\)
0.362654 + 0.931924i \(0.381871\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 802.452 1.31334
\(612\) 0 0
\(613\) 49.5676i 0.0808607i −0.999182 0.0404303i \(-0.987127\pi\)
0.999182 0.0404303i \(-0.0128729\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 785.550i 1.27318i −0.771204 0.636588i \(-0.780345\pi\)
0.771204 0.636588i \(-0.219655\pi\)
\(618\) 0 0
\(619\) 693.014i 1.11957i −0.828638 0.559785i \(-0.810884\pi\)
0.828638 0.559785i \(-0.189116\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −732.432 −1.17565
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −29.2938 −0.0465721
\(630\) 0 0
\(631\) 9.68025i 0.0153411i −0.999971 0.00767056i \(-0.997558\pi\)
0.999971 0.00767056i \(-0.00244164\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 226.942i 0.357389i
\(636\) 0 0
\(637\) −176.965 −0.277810
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 174.812i 0.272718i −0.990659 0.136359i \(-0.956460\pi\)
0.990659 0.136359i \(-0.0435401\pi\)
\(642\) 0 0
\(643\) 285.997i 0.444785i −0.974957 0.222393i \(-0.928613\pi\)
0.974957 0.222393i \(-0.0713867\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −698.951 −1.08030 −0.540148 0.841570i \(-0.681632\pi\)
−0.540148 + 0.841570i \(0.681632\pi\)
\(648\) 0 0
\(649\) 101.064i 0.155722i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 639.144 0.978781 0.489390 0.872065i \(-0.337219\pi\)
0.489390 + 0.872065i \(0.337219\pi\)
\(654\) 0 0
\(655\) 98.1846i 0.149900i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1204.48i 1.82774i −0.406004 0.913871i \(-0.633078\pi\)
0.406004 0.913871i \(-0.366922\pi\)
\(660\) 0 0
\(661\) 442.012i 0.668702i −0.942449 0.334351i \(-0.891483\pi\)
0.942449 0.334351i \(-0.108517\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 52.2462i 0.0785658i
\(666\) 0 0
\(667\) −3.30633 15.4915i −0.00495702 0.0232256i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 126.772 0.188930
\(672\) 0 0
\(673\) 401.763 0.596974 0.298487 0.954414i \(-0.403518\pi\)
0.298487 + 0.954414i \(0.403518\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 575.340i 0.849838i −0.905231 0.424919i \(-0.860303\pi\)
0.905231 0.424919i \(-0.139697\pi\)
\(678\) 0 0
\(679\) 1091.71 1.60782
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 148.111 0.216854 0.108427 0.994104i \(-0.465419\pi\)
0.108427 + 0.994104i \(0.465419\pi\)
\(684\) 0 0
\(685\) −481.191 −0.702469
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 777.191i 1.12800i
\(690\) 0 0
\(691\) −176.640 −0.255629 −0.127815 0.991798i \(-0.540796\pi\)
−0.127815 + 0.991798i \(0.540796\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 142.336i 0.204800i
\(696\) 0 0
\(697\) 261.747i 0.375533i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 179.122i 0.255524i 0.991805 + 0.127762i \(0.0407793\pi\)
−0.991805 + 0.127762i \(0.959221\pi\)
\(702\) 0 0
\(703\) −10.7728 −0.0153240
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 561.312i 0.793935i
\(708\) 0 0
\(709\) 612.124i 0.863363i −0.902026 0.431681i \(-0.857920\pi\)
0.902026 0.431681i \(-0.142080\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −175.940 824.349i −0.246761 1.15617i
\(714\) 0 0
\(715\) 120.368 0.168346
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 332.970 0.463102 0.231551 0.972823i \(-0.425620\pi\)
0.231551 + 0.972823i \(0.425620\pi\)
\(720\) 0 0
\(721\) −1367.85 −1.89716
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.44356 0.00474973
\(726\) 0 0
\(727\) 363.921i 0.500579i 0.968171 + 0.250290i \(0.0805258\pi\)
−0.968171 + 0.250290i \(0.919474\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 342.711 0.468825
\(732\) 0 0
\(733\) 414.694i 0.565749i 0.959157 + 0.282875i \(0.0912880\pi\)
−0.959157 + 0.282875i \(0.908712\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 431.864 0.585975
\(738\) 0 0
\(739\) −750.501 −1.01556 −0.507782 0.861486i \(-0.669534\pi\)
−0.507782 + 0.861486i \(0.669534\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1301.85i 1.75215i 0.482171 + 0.876077i \(0.339848\pi\)
−0.482171 + 0.876077i \(0.660152\pi\)
\(744\) 0 0
\(745\) −218.552 −0.293358
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −696.317 −0.929662
\(750\) 0 0
\(751\) 860.771i 1.14617i 0.819497 + 0.573083i \(0.194253\pi\)
−0.819497 + 0.573083i \(0.805747\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 468.986i 0.621173i
\(756\) 0 0
\(757\) 281.079i 0.371306i −0.982615 0.185653i \(-0.940560\pi\)
0.982615 0.185653i \(-0.0594400\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −155.150 −0.203877 −0.101938 0.994791i \(-0.532504\pi\)
−0.101938 + 0.994791i \(0.532504\pi\)
\(762\) 0 0
\(763\) −1124.07 −1.47322
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 342.787 0.446920
\(768\) 0 0
\(769\) 1242.19i 1.61533i −0.589642 0.807665i \(-0.700731\pi\)
0.589642 0.807665i \(-0.299269\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1105.56i 1.43022i −0.699014 0.715108i \(-0.746377\pi\)
0.699014 0.715108i \(-0.253623\pi\)
\(774\) 0 0
\(775\) 183.242 0.236442
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 96.2572i 0.123565i
\(780\) 0 0
\(781\) 269.381i 0.344918i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 642.777 0.818825
\(786\) 0 0
\(787\) 166.440i 0.211486i −0.994393 0.105743i \(-0.966278\pi\)
0.994393 0.105743i \(-0.0337221\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −721.098 −0.911629
\(792\) 0 0
\(793\) 429.986i 0.542227i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 702.946i 0.881990i −0.897510 0.440995i \(-0.854626\pi\)
0.897510 0.440995i \(-0.145374\pi\)
\(798\) 0 0
\(799\) 478.824i 0.599279i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 406.948i 0.506785i
\(804\) 0 0
\(805\) 84.5919 + 396.346i 0.105083 + 0.492355i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −612.933 −0.757643 −0.378822 0.925470i \(-0.623671\pi\)
−0.378822 + 0.925470i \(0.623671\pi\)
\(810\) 0 0
\(811\) −664.035 −0.818785 −0.409393 0.912358i \(-0.634259\pi\)
−0.409393 + 0.912358i \(0.634259\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 250.560i 0.307435i
\(816\) 0 0
\(817\) 126.032 0.154262
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −598.546 −0.729046 −0.364523 0.931194i \(-0.618768\pi\)
−0.364523 + 0.931194i \(0.618768\pi\)
\(822\) 0 0
\(823\) 1216.57 1.47821 0.739105 0.673591i \(-0.235249\pi\)
0.739105 + 0.673591i \(0.235249\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 719.387i 0.869875i −0.900461 0.434937i \(-0.856770\pi\)
0.900461 0.434937i \(-0.143230\pi\)
\(828\) 0 0
\(829\) −340.645 −0.410911 −0.205455 0.978666i \(-0.565868\pi\)
−0.205455 + 0.978666i \(0.565868\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 105.595i 0.126765i
\(834\) 0 0
\(835\) 430.909i 0.516059i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9.11326i 0.0108621i 0.999985 + 0.00543103i \(0.00172876\pi\)
−0.999985 + 0.00543103i \(0.998271\pi\)
\(840\) 0 0
\(841\) −840.526 −0.999436
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 30.3674i 0.0359378i
\(846\) 0 0
\(847\) 828.434i 0.978081i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −81.7235 + 17.4422i −0.0960323 + 0.0204961i
\(852\) 0 0
\(853\) −1029.00 −1.20633 −0.603163 0.797618i \(-0.706093\pi\)
−0.603163 + 0.797618i \(0.706093\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −606.460 −0.707654 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(858\) 0 0
\(859\) 1147.62 1.33600 0.668001 0.744161i \(-0.267150\pi\)
0.668001 + 0.744161i \(0.267150\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1299.38 1.50566 0.752829 0.658216i \(-0.228689\pi\)
0.752829 + 0.658216i \(0.228689\pi\)
\(864\) 0 0
\(865\) 678.543i 0.784443i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 193.319 0.222461
\(870\) 0 0
\(871\) 1464.80i 1.68174i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −88.1027 −0.100689
\(876\) 0 0
\(877\) −869.529 −0.991482 −0.495741 0.868470i \(-0.665103\pi\)
−0.495741 + 0.868470i \(0.665103\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1210.10i 1.37355i 0.726869 + 0.686776i \(0.240975\pi\)
−0.726869 + 0.686776i \(0.759025\pi\)
\(882\) 0 0
\(883\) 275.674 0.312202 0.156101 0.987741i \(-0.450107\pi\)
0.156101 + 0.987741i \(0.450107\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −742.524 −0.837119 −0.418559 0.908189i \(-0.637465\pi\)
−0.418559 + 0.908189i \(0.637465\pi\)
\(888\) 0 0
\(889\) 799.769i 0.899628i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 176.087i 0.197186i
\(894\) 0 0
\(895\) 211.650i 0.236480i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 25.2402 0.0280759
\(900\) 0 0
\(901\) −463.751 −0.514707
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −561.284 −0.620203
\(906\) 0 0
\(907\) 1335.53i 1.47247i −0.676727 0.736234i \(-0.736602\pi\)
0.676727 0.736234i \(-0.263398\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 511.312i 0.561265i −0.959815 0.280632i \(-0.909456\pi\)
0.959815 0.280632i \(-0.0905442\pi\)
\(912\) 0 0
\(913\) 290.972 0.318699
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 346.013i 0.377332i
\(918\) 0 0
\(919\) 712.793i 0.775618i −0.921740 0.387809i \(-0.873232\pi\)
0.921740 0.387809i \(-0.126768\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 913.686 0.989909
\(924\) 0 0
\(925\) 18.1661i 0.0196390i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −86.3665 −0.0929672 −0.0464836 0.998919i \(-0.514802\pi\)
−0.0464836 + 0.998919i \(0.514802\pi\)
\(930\) 0 0
\(931\) 38.8326i 0.0417106i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 71.8235i 0.0768165i
\(936\) 0 0
\(937\) 994.559i 1.06143i −0.847551 0.530714i \(-0.821924\pi\)
0.847551 0.530714i \(-0.178076\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1456.29i 1.54760i 0.633429 + 0.773800i \(0.281647\pi\)
−0.633429 + 0.773800i \(0.718353\pi\)
\(942\) 0 0
\(943\) 155.850 + 730.217i 0.165270 + 0.774356i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 206.079 0.217613 0.108806 0.994063i \(-0.465297\pi\)
0.108806 + 0.994063i \(0.465297\pi\)
\(948\) 0 0
\(949\) −1380.29 −1.45446
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 457.750i 0.480325i 0.970733 + 0.240163i \(0.0772007\pi\)
−0.970733 + 0.240163i \(0.922799\pi\)
\(954\) 0 0
\(955\) −166.168 −0.173998
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1695.77 1.76827
\(960\) 0 0
\(961\) 382.112 0.397619
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 625.795i 0.648492i
\(966\) 0 0
\(967\) −942.336 −0.974495 −0.487247 0.873264i \(-0.661999\pi\)
−0.487247 + 0.873264i \(0.661999\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 218.442i 0.224966i 0.993654 + 0.112483i \(0.0358804\pi\)
−0.993654 + 0.112483i \(0.964120\pi\)
\(972\) 0 0
\(973\) 501.607i 0.515526i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 378.433i 0.387342i 0.981067 + 0.193671i \(0.0620394\pi\)
−0.981067 + 0.193671i \(0.937961\pi\)
\(978\) 0 0
\(979\) −370.280 −0.378223
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1359.48i 1.38299i −0.722381 0.691495i \(-0.756952\pi\)
0.722381 0.691495i \(-0.243048\pi\)
\(984\) 0 0
\(985\) 535.452i 0.543606i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 956.090 204.058i 0.966724 0.206327i
\(990\) 0 0
\(991\) −648.537 −0.654426 −0.327213 0.944951i \(-0.606110\pi\)
−0.327213 + 0.944951i \(0.606110\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 676.241 0.679640
\(996\) 0 0
\(997\) −155.843 −0.156312 −0.0781562 0.996941i \(-0.524903\pi\)
−0.0781562 + 0.996941i \(0.524903\pi\)
\(998\) 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 4140.3.d.a.2161.10 16
3.2 odd 2 460.3.f.a.321.1 16
12.11 even 2 1840.3.k.c.321.15 16
15.2 even 4 2300.3.d.b.1149.4 32
15.8 even 4 2300.3.d.b.1149.29 32
15.14 odd 2 2300.3.f.e.1701.16 16
23.22 odd 2 inner 4140.3.d.a.2161.7 16
69.68 even 2 460.3.f.a.321.2 yes 16
276.275 odd 2 1840.3.k.c.321.16 16
345.68 odd 4 2300.3.d.b.1149.3 32
345.137 odd 4 2300.3.d.b.1149.30 32
345.344 even 2 2300.3.f.e.1701.15 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
460.3.f.a.321.1 16 3.2 odd 2
460.3.f.a.321.2 yes 16 69.68 even 2
1840.3.k.c.321.15 16 12.11 even 2
1840.3.k.c.321.16 16 276.275 odd 2
2300.3.d.b.1149.3 32 345.68 odd 4
2300.3.d.b.1149.4 32 15.2 even 4
2300.3.d.b.1149.29 32 15.8 even 4
2300.3.d.b.1149.30 32 345.137 odd 4
2300.3.f.e.1701.15 16 345.344 even 2
2300.3.f.e.1701.16 16 15.14 odd 2
4140.3.d.a.2161.7 16 23.22 odd 2 inner
4140.3.d.a.2161.10 16 1.1 even 1 trivial