Properties

Label 2736.3.m.e.1711.8
Level $2736$
Weight $3$
Character 2736.1711
Analytic conductor $74.551$
Analytic rank $0$
Dimension $12$
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] = [2736,3,Mod(1711,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.1711");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.m (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: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 2 x^{10} - 28 x^{9} - 400 x^{8} - 520 x^{7} + 17067 x^{6} - 3250 x^{5} - 195494 x^{4} + 302996 x^{3} + 602332 x^{2} - 2263536 x + 2052928 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 912)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1711.8
Root \(4.06962 + 0.767506i\) of defining polynomial
Character \(\chi\) \(=\) 2736.1711
Dual form 2736.3.m.e.1711.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.65777 q^{5} +4.49506i q^{7} +O(q^{10})\) \(q+4.65777 q^{5} +4.49506i q^{7} +10.3003i q^{11} +10.9629 q^{13} +16.3920 q^{17} +4.35890i q^{19} -22.7559i q^{23} -3.30515 q^{25} +55.9066 q^{29} -61.5568i q^{31} +20.9370i q^{35} +38.6211 q^{37} -43.7169 q^{41} -39.1687i q^{43} +30.2344i q^{47} +28.7944 q^{49} -77.4119 q^{53} +47.9763i q^{55} +90.9651i q^{59} +115.683 q^{61} +51.0628 q^{65} +37.2160i q^{67} +103.972i q^{71} -74.5741 q^{73} -46.3003 q^{77} -120.050i q^{79} -29.4019i q^{83} +76.3503 q^{85} +110.359 q^{89} +49.2790i q^{91} +20.3028i q^{95} -13.1839 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 10 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 10 q^{5} + 36 q^{13} - 14 q^{17} + 10 q^{25} + 108 q^{29} - 16 q^{37} - 16 q^{41} - 118 q^{49} - 220 q^{53} + 366 q^{61} - 140 q^{65} - 158 q^{73} + 286 q^{77} + 98 q^{85} + 396 q^{89} + 224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(5\) 4.65777 0.931555 0.465777 0.884902i \(-0.345775\pi\)
0.465777 + 0.884902i \(0.345775\pi\)
\(6\) 0 0
\(7\) 4.49506i 0.642151i 0.947054 + 0.321076i \(0.104044\pi\)
−0.947054 + 0.321076i \(0.895956\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 10.3003i 0.936388i 0.883626 + 0.468194i \(0.155095\pi\)
−0.883626 + 0.468194i \(0.844905\pi\)
\(12\) 0 0
\(13\) 10.9629 0.843302 0.421651 0.906758i \(-0.361451\pi\)
0.421651 + 0.906758i \(0.361451\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 16.3920 0.964237 0.482118 0.876106i \(-0.339867\pi\)
0.482118 + 0.876106i \(0.339867\pi\)
\(18\) 0 0
\(19\) 4.35890i 0.229416i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 22.7559i − 0.989386i −0.869068 0.494693i \(-0.835280\pi\)
0.869068 0.494693i \(-0.164720\pi\)
\(24\) 0 0
\(25\) −3.30515 −0.132206
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 55.9066 1.92781 0.963907 0.266238i \(-0.0857806\pi\)
0.963907 + 0.266238i \(0.0857806\pi\)
\(30\) 0 0
\(31\) − 61.5568i − 1.98570i −0.119354 0.992852i \(-0.538082\pi\)
0.119354 0.992852i \(-0.461918\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 20.9370i 0.598199i
\(36\) 0 0
\(37\) 38.6211 1.04381 0.521907 0.853002i \(-0.325221\pi\)
0.521907 + 0.853002i \(0.325221\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −43.7169 −1.06627 −0.533133 0.846031i \(-0.678986\pi\)
−0.533133 + 0.846031i \(0.678986\pi\)
\(42\) 0 0
\(43\) − 39.1687i − 0.910900i −0.890261 0.455450i \(-0.849478\pi\)
0.890261 0.455450i \(-0.150522\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 30.2344i 0.643285i 0.946861 + 0.321642i \(0.104235\pi\)
−0.946861 + 0.321642i \(0.895765\pi\)
\(48\) 0 0
\(49\) 28.7944 0.587642
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −77.4119 −1.46060 −0.730301 0.683126i \(-0.760620\pi\)
−0.730301 + 0.683126i \(0.760620\pi\)
\(54\) 0 0
\(55\) 47.9763i 0.872297i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 90.9651i 1.54178i 0.636968 + 0.770891i \(0.280189\pi\)
−0.636968 + 0.770891i \(0.719811\pi\)
\(60\) 0 0
\(61\) 115.683 1.89644 0.948221 0.317611i \(-0.102881\pi\)
0.948221 + 0.317611i \(0.102881\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 51.0628 0.785582
\(66\) 0 0
\(67\) 37.2160i 0.555462i 0.960659 + 0.277731i \(0.0895824\pi\)
−0.960659 + 0.277731i \(0.910418\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 103.972i 1.46439i 0.681096 + 0.732194i \(0.261503\pi\)
−0.681096 + 0.732194i \(0.738497\pi\)
\(72\) 0 0
\(73\) −74.5741 −1.02156 −0.510782 0.859710i \(-0.670644\pi\)
−0.510782 + 0.859710i \(0.670644\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −46.3003 −0.601303
\(78\) 0 0
\(79\) − 120.050i − 1.51962i −0.650147 0.759809i \(-0.725293\pi\)
0.650147 0.759809i \(-0.274707\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 29.4019i − 0.354240i −0.984189 0.177120i \(-0.943322\pi\)
0.984189 0.177120i \(-0.0566781\pi\)
\(84\) 0 0
\(85\) 76.3503 0.898239
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 110.359 1.23998 0.619992 0.784608i \(-0.287136\pi\)
0.619992 + 0.784608i \(0.287136\pi\)
\(90\) 0 0
\(91\) 49.2790i 0.541527i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 20.3028i 0.213713i
\(96\) 0 0
\(97\) −13.1839 −0.135916 −0.0679581 0.997688i \(-0.521648\pi\)
−0.0679581 + 0.997688i \(0.521648\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −25.4992 −0.252468 −0.126234 0.992001i \(-0.540289\pi\)
−0.126234 + 0.992001i \(0.540289\pi\)
\(102\) 0 0
\(103\) 164.210i 1.59427i 0.603802 + 0.797134i \(0.293652\pi\)
−0.603802 + 0.797134i \(0.706348\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 3.78316i − 0.0353566i −0.999844 0.0176783i \(-0.994373\pi\)
0.999844 0.0176783i \(-0.00562747\pi\)
\(108\) 0 0
\(109\) 7.83296 0.0718620 0.0359310 0.999354i \(-0.488560\pi\)
0.0359310 + 0.999354i \(0.488560\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −7.22373 −0.0639268 −0.0319634 0.999489i \(-0.510176\pi\)
−0.0319634 + 0.999489i \(0.510176\pi\)
\(114\) 0 0
\(115\) − 105.992i − 0.921667i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 73.6831i 0.619186i
\(120\) 0 0
\(121\) 14.9044 0.123177
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −131.839 −1.05471
\(126\) 0 0
\(127\) − 167.160i − 1.31622i −0.752920 0.658112i \(-0.771355\pi\)
0.752920 0.658112i \(-0.228645\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 211.318i 1.61311i 0.591157 + 0.806557i \(0.298672\pi\)
−0.591157 + 0.806557i \(0.701328\pi\)
\(132\) 0 0
\(133\) −19.5935 −0.147320
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 195.425 1.42646 0.713229 0.700931i \(-0.247232\pi\)
0.713229 + 0.700931i \(0.247232\pi\)
\(138\) 0 0
\(139\) 62.5804i 0.450219i 0.974334 + 0.225109i \(0.0722739\pi\)
−0.974334 + 0.225109i \(0.927726\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 112.921i 0.789658i
\(144\) 0 0
\(145\) 260.400 1.79586
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 166.917 1.12025 0.560125 0.828408i \(-0.310753\pi\)
0.560125 + 0.828408i \(0.310753\pi\)
\(150\) 0 0
\(151\) − 2.43568i − 0.0161303i −0.999967 0.00806516i \(-0.997433\pi\)
0.999967 0.00806516i \(-0.00256725\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 286.718i − 1.84979i
\(156\) 0 0
\(157\) 186.584 1.18843 0.594216 0.804306i \(-0.297463\pi\)
0.594216 + 0.804306i \(0.297463\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 102.289 0.635336
\(162\) 0 0
\(163\) 155.612i 0.954673i 0.878721 + 0.477336i \(0.158398\pi\)
−0.878721 + 0.477336i \(0.841602\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 112.843i 0.675708i 0.941199 + 0.337854i \(0.109701\pi\)
−0.941199 + 0.337854i \(0.890299\pi\)
\(168\) 0 0
\(169\) −48.8142 −0.288842
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −127.897 −0.739290 −0.369645 0.929173i \(-0.620521\pi\)
−0.369645 + 0.929173i \(0.620521\pi\)
\(174\) 0 0
\(175\) − 14.8569i − 0.0848964i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 16.0432i − 0.0896267i −0.998995 0.0448134i \(-0.985731\pi\)
0.998995 0.0448134i \(-0.0142693\pi\)
\(180\) 0 0
\(181\) −219.288 −1.21153 −0.605767 0.795642i \(-0.707134\pi\)
−0.605767 + 0.795642i \(0.707134\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 179.888 0.972370
\(186\) 0 0
\(187\) 168.842i 0.902900i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 49.0084i 0.256588i 0.991736 + 0.128294i \(0.0409502\pi\)
−0.991736 + 0.128294i \(0.959050\pi\)
\(192\) 0 0
\(193\) −336.473 −1.74338 −0.871691 0.490056i \(-0.836976\pi\)
−0.871691 + 0.490056i \(0.836976\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 50.7872 0.257803 0.128902 0.991657i \(-0.458855\pi\)
0.128902 + 0.991657i \(0.458855\pi\)
\(198\) 0 0
\(199\) − 262.657i − 1.31989i −0.751316 0.659943i \(-0.770580\pi\)
0.751316 0.659943i \(-0.229420\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 251.304i 1.23795i
\(204\) 0 0
\(205\) −203.623 −0.993285
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −44.8978 −0.214822
\(210\) 0 0
\(211\) − 131.345i − 0.622488i −0.950330 0.311244i \(-0.899254\pi\)
0.950330 0.311244i \(-0.100746\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 182.439i − 0.848553i
\(216\) 0 0
\(217\) 276.701 1.27512
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 179.705 0.813143
\(222\) 0 0
\(223\) − 240.108i − 1.07672i −0.842715 0.538360i \(-0.819044\pi\)
0.842715 0.538360i \(-0.180956\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 129.300i − 0.569605i −0.958586 0.284802i \(-0.908072\pi\)
0.958586 0.284802i \(-0.0919280\pi\)
\(228\) 0 0
\(229\) −23.2842 −0.101678 −0.0508389 0.998707i \(-0.516189\pi\)
−0.0508389 + 0.998707i \(0.516189\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 141.563 0.607568 0.303784 0.952741i \(-0.401750\pi\)
0.303784 + 0.952741i \(0.401750\pi\)
\(234\) 0 0
\(235\) 140.825i 0.599255i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 47.6515i − 0.199379i −0.995019 0.0996893i \(-0.968215\pi\)
0.995019 0.0996893i \(-0.0317849\pi\)
\(240\) 0 0
\(241\) −164.728 −0.683519 −0.341759 0.939788i \(-0.611023\pi\)
−0.341759 + 0.939788i \(0.611023\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 134.118 0.547420
\(246\) 0 0
\(247\) 47.7863i 0.193467i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 158.643i 0.632045i 0.948752 + 0.316022i \(0.102347\pi\)
−0.948752 + 0.316022i \(0.897653\pi\)
\(252\) 0 0
\(253\) 234.392 0.926450
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 399.158 1.55314 0.776571 0.630029i \(-0.216957\pi\)
0.776571 + 0.630029i \(0.216957\pi\)
\(258\) 0 0
\(259\) 173.604i 0.670287i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.2067i 0.0464134i 0.999731 + 0.0232067i \(0.00738758\pi\)
−0.999731 + 0.0232067i \(0.992612\pi\)
\(264\) 0 0
\(265\) −360.567 −1.36063
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 348.721 1.29636 0.648180 0.761487i \(-0.275530\pi\)
0.648180 + 0.761487i \(0.275530\pi\)
\(270\) 0 0
\(271\) 314.514i 1.16057i 0.814413 + 0.580285i \(0.197059\pi\)
−0.814413 + 0.580285i \(0.802941\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 34.0440i − 0.123796i
\(276\) 0 0
\(277\) 360.287 1.30068 0.650338 0.759645i \(-0.274627\pi\)
0.650338 + 0.759645i \(0.274627\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −404.603 −1.43987 −0.719935 0.694042i \(-0.755828\pi\)
−0.719935 + 0.694042i \(0.755828\pi\)
\(282\) 0 0
\(283\) 109.783i 0.387925i 0.981009 + 0.193962i \(0.0621339\pi\)
−0.981009 + 0.193962i \(0.937866\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 196.510i − 0.684704i
\(288\) 0 0
\(289\) −20.3014 −0.0702472
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 51.7076 0.176476 0.0882382 0.996099i \(-0.471876\pi\)
0.0882382 + 0.996099i \(0.471876\pi\)
\(294\) 0 0
\(295\) 423.695i 1.43625i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 249.471i − 0.834352i
\(300\) 0 0
\(301\) 176.066 0.584935
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 538.825 1.76664
\(306\) 0 0
\(307\) − 43.7081i − 0.142372i −0.997463 0.0711858i \(-0.977322\pi\)
0.997463 0.0711858i \(-0.0226783\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 350.883i − 1.12824i −0.825692 0.564121i \(-0.809215\pi\)
0.825692 0.564121i \(-0.190785\pi\)
\(312\) 0 0
\(313\) 130.501 0.416936 0.208468 0.978029i \(-0.433152\pi\)
0.208468 + 0.978029i \(0.433152\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −214.564 −0.676858 −0.338429 0.940992i \(-0.609895\pi\)
−0.338429 + 0.940992i \(0.609895\pi\)
\(318\) 0 0
\(319\) 575.853i 1.80518i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 71.4512i 0.221211i
\(324\) 0 0
\(325\) −36.2342 −0.111490
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −135.905 −0.413086
\(330\) 0 0
\(331\) 261.592i 0.790308i 0.918615 + 0.395154i \(0.129309\pi\)
−0.918615 + 0.395154i \(0.870691\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 173.344i 0.517443i
\(336\) 0 0
\(337\) 485.821 1.44160 0.720802 0.693141i \(-0.243774\pi\)
0.720802 + 0.693141i \(0.243774\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 634.052 1.85939
\(342\) 0 0
\(343\) 349.691i 1.01951i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 106.505i 0.306931i 0.988154 + 0.153465i \(0.0490434\pi\)
−0.988154 + 0.153465i \(0.950957\pi\)
\(348\) 0 0
\(349\) 167.989 0.481345 0.240673 0.970606i \(-0.422632\pi\)
0.240673 + 0.970606i \(0.422632\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 596.666 1.69027 0.845136 0.534551i \(-0.179519\pi\)
0.845136 + 0.534551i \(0.179519\pi\)
\(354\) 0 0
\(355\) 484.276i 1.36416i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 212.379i 0.591584i 0.955252 + 0.295792i \(0.0955835\pi\)
−0.955252 + 0.295792i \(0.904416\pi\)
\(360\) 0 0
\(361\) −19.0000 −0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −347.349 −0.951642
\(366\) 0 0
\(367\) 429.659i 1.17073i 0.810768 + 0.585367i \(0.199050\pi\)
−0.810768 + 0.585367i \(0.800950\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 347.971i − 0.937927i
\(372\) 0 0
\(373\) 207.539 0.556405 0.278202 0.960523i \(-0.410261\pi\)
0.278202 + 0.960523i \(0.410261\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 612.900 1.62573
\(378\) 0 0
\(379\) − 647.875i − 1.70943i −0.519096 0.854716i \(-0.673731\pi\)
0.519096 0.854716i \(-0.326269\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 485.111i 1.26661i 0.773903 + 0.633304i \(0.218302\pi\)
−0.773903 + 0.633304i \(0.781698\pi\)
\(384\) 0 0
\(385\) −215.656 −0.560146
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −419.923 −1.07949 −0.539747 0.841828i \(-0.681480\pi\)
−0.539747 + 0.841828i \(0.681480\pi\)
\(390\) 0 0
\(391\) − 373.015i − 0.954003i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 559.165i − 1.41561i
\(396\) 0 0
\(397\) −544.430 −1.37136 −0.685680 0.727903i \(-0.740495\pi\)
−0.685680 + 0.727903i \(0.740495\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 103.777 0.258796 0.129398 0.991593i \(-0.458695\pi\)
0.129398 + 0.991593i \(0.458695\pi\)
\(402\) 0 0
\(403\) − 674.843i − 1.67455i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 397.808i 0.977416i
\(408\) 0 0
\(409\) −291.937 −0.713782 −0.356891 0.934146i \(-0.616163\pi\)
−0.356891 + 0.934146i \(0.616163\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −408.893 −0.990057
\(414\) 0 0
\(415\) − 136.947i − 0.329994i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 153.052i − 0.365280i −0.983180 0.182640i \(-0.941536\pi\)
0.983180 0.182640i \(-0.0584643\pi\)
\(420\) 0 0
\(421\) 438.962 1.04266 0.521332 0.853354i \(-0.325435\pi\)
0.521332 + 0.853354i \(0.325435\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −54.1782 −0.127478
\(426\) 0 0
\(427\) 520.002i 1.21780i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 407.423i 0.945296i 0.881251 + 0.472648i \(0.156702\pi\)
−0.881251 + 0.472648i \(0.843298\pi\)
\(432\) 0 0
\(433\) 139.799 0.322861 0.161430 0.986884i \(-0.448389\pi\)
0.161430 + 0.986884i \(0.448389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 99.1906 0.226981
\(438\) 0 0
\(439\) − 178.585i − 0.406799i −0.979096 0.203400i \(-0.934801\pi\)
0.979096 0.203400i \(-0.0651990\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 354.944i − 0.801227i −0.916247 0.400614i \(-0.868797\pi\)
0.916247 0.400614i \(-0.131203\pi\)
\(444\) 0 0
\(445\) 514.025 1.15511
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 28.7188 0.0639618 0.0319809 0.999488i \(-0.489818\pi\)
0.0319809 + 0.999488i \(0.489818\pi\)
\(450\) 0 0
\(451\) − 450.296i − 0.998439i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 229.530i 0.504462i
\(456\) 0 0
\(457\) −344.238 −0.753256 −0.376628 0.926365i \(-0.622917\pi\)
−0.376628 + 0.926365i \(0.622917\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 173.337 0.376002 0.188001 0.982169i \(-0.439799\pi\)
0.188001 + 0.982169i \(0.439799\pi\)
\(462\) 0 0
\(463\) 277.141i 0.598577i 0.954163 + 0.299289i \(0.0967493\pi\)
−0.954163 + 0.299289i \(0.903251\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 191.671i 0.410430i 0.978717 + 0.205215i \(0.0657893\pi\)
−0.978717 + 0.205215i \(0.934211\pi\)
\(468\) 0 0
\(469\) −167.288 −0.356691
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 403.448 0.852956
\(474\) 0 0
\(475\) − 14.4068i − 0.0303302i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 245.009i − 0.511501i −0.966743 0.255751i \(-0.917677\pi\)
0.966743 0.255751i \(-0.0823226\pi\)
\(480\) 0 0
\(481\) 423.401 0.880251
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −61.4075 −0.126613
\(486\) 0 0
\(487\) 944.321i 1.93906i 0.244978 + 0.969529i \(0.421219\pi\)
−0.244978 + 0.969529i \(0.578781\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 844.187i 1.71932i 0.510866 + 0.859660i \(0.329325\pi\)
−0.510866 + 0.859660i \(0.670675\pi\)
\(492\) 0 0
\(493\) 916.423 1.85887
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −467.358 −0.940358
\(498\) 0 0
\(499\) − 662.375i − 1.32741i −0.747997 0.663703i \(-0.768984\pi\)
0.747997 0.663703i \(-0.231016\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 772.361i 1.53551i 0.640744 + 0.767755i \(0.278626\pi\)
−0.640744 + 0.767755i \(0.721374\pi\)
\(504\) 0 0
\(505\) −118.770 −0.235187
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −278.829 −0.547797 −0.273899 0.961759i \(-0.588313\pi\)
−0.273899 + 0.961759i \(0.588313\pi\)
\(510\) 0 0
\(511\) − 335.215i − 0.655998i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 764.851i 1.48515i
\(516\) 0 0
\(517\) −311.422 −0.602364
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −746.210 −1.43226 −0.716132 0.697965i \(-0.754089\pi\)
−0.716132 + 0.697965i \(0.754089\pi\)
\(522\) 0 0
\(523\) − 523.892i − 1.00171i −0.865532 0.500853i \(-0.833020\pi\)
0.865532 0.500853i \(-0.166980\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 1009.04i − 1.91469i
\(528\) 0 0
\(529\) 11.1696 0.0211145
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −479.265 −0.899185
\(534\) 0 0
\(535\) − 17.6211i − 0.0329366i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 296.591i 0.550261i
\(540\) 0 0
\(541\) −738.463 −1.36500 −0.682498 0.730888i \(-0.739106\pi\)
−0.682498 + 0.730888i \(0.739106\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 36.4841 0.0669433
\(546\) 0 0
\(547\) − 715.442i − 1.30794i −0.756521 0.653969i \(-0.773103\pi\)
0.756521 0.653969i \(-0.226897\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 243.691i 0.442271i
\(552\) 0 0
\(553\) 539.631 0.975824
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −142.896 −0.256546 −0.128273 0.991739i \(-0.540943\pi\)
−0.128273 + 0.991739i \(0.540943\pi\)
\(558\) 0 0
\(559\) − 429.403i − 0.768164i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 1004.23i − 1.78372i −0.452315 0.891858i \(-0.649402\pi\)
0.452315 0.891858i \(-0.350598\pi\)
\(564\) 0 0
\(565\) −33.6465 −0.0595513
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 672.894 1.18259 0.591295 0.806455i \(-0.298617\pi\)
0.591295 + 0.806455i \(0.298617\pi\)
\(570\) 0 0
\(571\) 401.764i 0.703614i 0.936073 + 0.351807i \(0.114433\pi\)
−0.936073 + 0.351807i \(0.885567\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 75.2117i 0.130803i
\(576\) 0 0
\(577\) −1128.20 −1.95528 −0.977640 0.210284i \(-0.932561\pi\)
−0.977640 + 0.210284i \(0.932561\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 132.163 0.227476
\(582\) 0 0
\(583\) − 797.363i − 1.36769i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 118.129i − 0.201241i −0.994925 0.100621i \(-0.967917\pi\)
0.994925 0.100621i \(-0.0320828\pi\)
\(588\) 0 0
\(589\) 268.320 0.455552
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −64.5109 −0.108787 −0.0543937 0.998520i \(-0.517323\pi\)
−0.0543937 + 0.998520i \(0.517323\pi\)
\(594\) 0 0
\(595\) 343.199i 0.576805i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 759.072i − 1.26723i −0.773647 0.633616i \(-0.781570\pi\)
0.773647 0.633616i \(-0.218430\pi\)
\(600\) 0 0
\(601\) −609.382 −1.01395 −0.506974 0.861962i \(-0.669236\pi\)
−0.506974 + 0.861962i \(0.669236\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 69.4214 0.114746
\(606\) 0 0
\(607\) 106.361i 0.175224i 0.996155 + 0.0876122i \(0.0279236\pi\)
−0.996155 + 0.0876122i \(0.972076\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 331.457i 0.542484i
\(612\) 0 0
\(613\) 470.099 0.766882 0.383441 0.923565i \(-0.374739\pi\)
0.383441 + 0.923565i \(0.374739\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1173.48 −1.90191 −0.950955 0.309330i \(-0.899895\pi\)
−0.950955 + 0.309330i \(0.899895\pi\)
\(618\) 0 0
\(619\) 655.252i 1.05857i 0.848446 + 0.529283i \(0.177539\pi\)
−0.848446 + 0.529283i \(0.822461\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 496.068i 0.796257i
\(624\) 0 0
\(625\) −531.447 −0.850315
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 633.079 1.00648
\(630\) 0 0
\(631\) − 488.854i − 0.774729i −0.921927 0.387364i \(-0.873386\pi\)
0.921927 0.387364i \(-0.126614\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 778.596i − 1.22613i
\(636\) 0 0
\(637\) 315.671 0.495560
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −181.853 −0.283702 −0.141851 0.989888i \(-0.545305\pi\)
−0.141851 + 0.989888i \(0.545305\pi\)
\(642\) 0 0
\(643\) − 340.140i − 0.528990i −0.964387 0.264495i \(-0.914795\pi\)
0.964387 0.264495i \(-0.0852052\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 638.906i 0.987490i 0.869607 + 0.493745i \(0.164372\pi\)
−0.869607 + 0.493745i \(0.835628\pi\)
\(648\) 0 0
\(649\) −936.965 −1.44371
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −420.515 −0.643975 −0.321987 0.946744i \(-0.604351\pi\)
−0.321987 + 0.946744i \(0.604351\pi\)
\(654\) 0 0
\(655\) 984.271i 1.50270i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 549.294i − 0.833527i −0.909015 0.416763i \(-0.863164\pi\)
0.909015 0.416763i \(-0.136836\pi\)
\(660\) 0 0
\(661\) −621.536 −0.940297 −0.470148 0.882587i \(-0.655800\pi\)
−0.470148 + 0.882587i \(0.655800\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −91.2621 −0.137236
\(666\) 0 0
\(667\) − 1272.20i − 1.90735i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1191.57i 1.77581i
\(672\) 0 0
\(673\) 782.350 1.16248 0.581241 0.813732i \(-0.302568\pi\)
0.581241 + 0.813732i \(0.302568\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1007.37 −1.48798 −0.743992 0.668189i \(-0.767070\pi\)
−0.743992 + 0.668189i \(0.767070\pi\)
\(678\) 0 0
\(679\) − 59.2623i − 0.0872788i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 217.854i − 0.318966i −0.987201 0.159483i \(-0.949017\pi\)
0.987201 0.159483i \(-0.0509827\pi\)
\(684\) 0 0
\(685\) 910.244 1.32882
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −848.661 −1.23173
\(690\) 0 0
\(691\) − 569.606i − 0.824321i −0.911111 0.412160i \(-0.864774\pi\)
0.911111 0.412160i \(-0.135226\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 291.485i 0.419403i
\(696\) 0 0
\(697\) −716.609 −1.02813
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 98.9249 0.141120 0.0705598 0.997508i \(-0.477521\pi\)
0.0705598 + 0.997508i \(0.477521\pi\)
\(702\) 0 0
\(703\) 168.346i 0.239467i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 114.621i − 0.162122i
\(708\) 0 0
\(709\) 878.241 1.23870 0.619352 0.785113i \(-0.287395\pi\)
0.619352 + 0.785113i \(0.287395\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1400.78 −1.96463
\(714\) 0 0
\(715\) 525.961i 0.735610i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 295.797i − 0.411401i −0.978615 0.205701i \(-0.934053\pi\)
0.978615 0.205701i \(-0.0659473\pi\)
\(720\) 0 0
\(721\) −738.132 −1.02376
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −184.780 −0.254869
\(726\) 0 0
\(727\) − 702.264i − 0.965975i −0.875627 0.482988i \(-0.839552\pi\)
0.875627 0.482988i \(-0.160448\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 642.054i − 0.878323i
\(732\) 0 0
\(733\) −1118.98 −1.52657 −0.763285 0.646062i \(-0.776415\pi\)
−0.763285 + 0.646062i \(0.776415\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −383.335 −0.520128
\(738\) 0 0
\(739\) − 437.441i − 0.591936i −0.955198 0.295968i \(-0.904358\pi\)
0.955198 0.295968i \(-0.0956422\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 296.960i − 0.399677i −0.979829 0.199839i \(-0.935958\pi\)
0.979829 0.199839i \(-0.0640418\pi\)
\(744\) 0 0
\(745\) 777.463 1.04357
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 17.0055 0.0227043
\(750\) 0 0
\(751\) − 978.482i − 1.30291i −0.758689 0.651453i \(-0.774160\pi\)
0.758689 0.651453i \(-0.225840\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 11.3448i − 0.0150263i
\(756\) 0 0
\(757\) −1303.36 −1.72175 −0.860874 0.508819i \(-0.830082\pi\)
−0.860874 + 0.508819i \(0.830082\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 368.703 0.484498 0.242249 0.970214i \(-0.422115\pi\)
0.242249 + 0.970214i \(0.422115\pi\)
\(762\) 0 0
\(763\) 35.2096i 0.0461463i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 997.244i 1.30019i
\(768\) 0 0
\(769\) 723.932 0.941393 0.470697 0.882295i \(-0.344003\pi\)
0.470697 + 0.882295i \(0.344003\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1172.19 1.51642 0.758208 0.652013i \(-0.226075\pi\)
0.758208 + 0.652013i \(0.226075\pi\)
\(774\) 0 0
\(775\) 203.455i 0.262522i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 190.558i − 0.244618i
\(780\) 0 0
\(781\) −1070.93 −1.37124
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 869.064 1.10709
\(786\) 0 0
\(787\) − 896.417i − 1.13903i −0.821981 0.569515i \(-0.807131\pi\)
0.821981 0.569515i \(-0.192869\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 32.4711i − 0.0410507i
\(792\) 0 0
\(793\) 1268.22 1.59927
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −200.885 −0.252052 −0.126026 0.992027i \(-0.540222\pi\)
−0.126026 + 0.992027i \(0.540222\pi\)
\(798\) 0 0
\(799\) 495.603i 0.620279i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 768.134i − 0.956580i
\(804\) 0 0
\(805\) 476.439 0.591850
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1204.46 −1.48883 −0.744415 0.667717i \(-0.767272\pi\)
−0.744415 + 0.667717i \(0.767272\pi\)
\(810\) 0 0
\(811\) − 324.831i − 0.400531i −0.979742 0.200265i \(-0.935820\pi\)
0.979742 0.200265i \(-0.0641805\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 724.804i 0.889330i
\(816\) 0 0
\(817\) 170.732 0.208975
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1525.76 −1.85841 −0.929206 0.369562i \(-0.879508\pi\)
−0.929206 + 0.369562i \(0.879508\pi\)
\(822\) 0 0
\(823\) − 632.977i − 0.769110i −0.923102 0.384555i \(-0.874355\pi\)
0.923102 0.384555i \(-0.125645\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 75.0303i − 0.0907258i −0.998971 0.0453629i \(-0.985556\pi\)
0.998971 0.0453629i \(-0.0144444\pi\)
\(828\) 0 0
\(829\) 912.657 1.10091 0.550457 0.834864i \(-0.314454\pi\)
0.550457 + 0.834864i \(0.314454\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 471.999 0.566626
\(834\) 0 0
\(835\) 525.598i 0.629458i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 927.103i − 1.10501i −0.833510 0.552505i \(-0.813672\pi\)
0.833510 0.552505i \(-0.186328\pi\)
\(840\) 0 0
\(841\) 2284.55 2.71647
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −227.366 −0.269072
\(846\) 0 0
\(847\) 66.9963i 0.0790983i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 878.858i − 1.03274i
\(852\) 0 0
\(853\) −1383.56 −1.62199 −0.810997 0.585051i \(-0.801074\pi\)
−0.810997 + 0.585051i \(0.801074\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −405.494 −0.473155 −0.236577 0.971613i \(-0.576026\pi\)
−0.236577 + 0.971613i \(0.576026\pi\)
\(858\) 0 0
\(859\) 442.450i 0.515076i 0.966268 + 0.257538i \(0.0829113\pi\)
−0.966268 + 0.257538i \(0.917089\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 703.125i 0.814745i 0.913262 + 0.407373i \(0.133555\pi\)
−0.913262 + 0.407373i \(0.866445\pi\)
\(864\) 0 0
\(865\) −595.716 −0.688689
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1236.55 1.42295
\(870\) 0 0
\(871\) 407.996i 0.468422i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 592.624i − 0.677284i
\(876\) 0 0
\(877\) 1034.99 1.18015 0.590074 0.807349i \(-0.299099\pi\)
0.590074 + 0.807349i \(0.299099\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 222.517 0.252573 0.126287 0.991994i \(-0.459694\pi\)
0.126287 + 0.991994i \(0.459694\pi\)
\(882\) 0 0
\(883\) − 616.520i − 0.698211i −0.937084 0.349105i \(-0.886486\pi\)
0.937084 0.349105i \(-0.113514\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 603.814i − 0.680737i −0.940292 0.340368i \(-0.889448\pi\)
0.940292 0.340368i \(-0.110552\pi\)
\(888\) 0 0
\(889\) 751.396 0.845215
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −131.789 −0.147580
\(894\) 0 0
\(895\) − 74.7255i − 0.0834922i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 3441.43i − 3.82807i
\(900\) 0 0
\(901\) −1268.94 −1.40837
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1021.39 −1.12861
\(906\) 0 0
\(907\) − 685.479i − 0.755765i −0.925854 0.377882i \(-0.876652\pi\)
0.925854 0.377882i \(-0.123348\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 937.378i − 1.02895i −0.857504 0.514477i \(-0.827986\pi\)
0.857504 0.514477i \(-0.172014\pi\)
\(912\) 0 0
\(913\) 302.848 0.331706
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −949.886 −1.03586
\(918\) 0 0
\(919\) 251.118i 0.273252i 0.990623 + 0.136626i \(0.0436258\pi\)
−0.990623 + 0.136626i \(0.956374\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1139.83i 1.23492i
\(924\) 0 0
\(925\) −127.649 −0.137999
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 119.382 0.128506 0.0642531 0.997934i \(-0.479534\pi\)
0.0642531 + 0.997934i \(0.479534\pi\)
\(930\) 0 0
\(931\) 125.512i 0.134814i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 786.429i 0.841101i
\(936\) 0 0
\(937\) −1028.78 −1.09795 −0.548973 0.835840i \(-0.684981\pi\)
−0.548973 + 0.835840i \(0.684981\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 789.816 0.839337 0.419669 0.907677i \(-0.362146\pi\)
0.419669 + 0.907677i \(0.362146\pi\)
\(942\) 0 0
\(943\) 994.817i 1.05495i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1386.53i 1.46413i 0.681237 + 0.732063i \(0.261442\pi\)
−0.681237 + 0.732063i \(0.738558\pi\)
\(948\) 0 0
\(949\) −817.551 −0.861487
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1433.00 −1.50368 −0.751838 0.659348i \(-0.770832\pi\)
−0.751838 + 0.659348i \(0.770832\pi\)
\(954\) 0 0
\(955\) 228.270i 0.239026i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 878.446i 0.916002i
\(960\) 0 0
\(961\) −2828.24 −2.94302
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1567.21 −1.62406
\(966\) 0 0
\(967\) 987.717i 1.02142i 0.859752 + 0.510712i \(0.170618\pi\)
−0.859752 + 0.510712i \(0.829382\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 871.164i − 0.897182i −0.893737 0.448591i \(-0.851926\pi\)
0.893737 0.448591i \(-0.148074\pi\)
\(972\) 0 0
\(973\) −281.302 −0.289108
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 395.790 0.405108 0.202554 0.979271i \(-0.435076\pi\)
0.202554 + 0.979271i \(0.435076\pi\)
\(978\) 0 0
\(979\) 1136.72i 1.16111i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 189.551i 0.192830i 0.995341 + 0.0964148i \(0.0307375\pi\)
−0.995341 + 0.0964148i \(0.969262\pi\)
\(984\) 0 0
\(985\) 236.555 0.240158
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −891.318 −0.901232
\(990\) 0 0
\(991\) 1028.23i 1.03757i 0.854905 + 0.518784i \(0.173615\pi\)
−0.854905 + 0.518784i \(0.826385\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 1223.40i − 1.22955i
\(996\) 0 0
\(997\) −788.638 −0.791011 −0.395505 0.918464i \(-0.629431\pi\)
−0.395505 + 0.918464i \(0.629431\pi\)
\(998\) 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 2736.3.m.e.1711.8 12
3.2 odd 2 912.3.m.b.799.9 yes 12
4.3 odd 2 inner 2736.3.m.e.1711.7 12
12.11 even 2 912.3.m.b.799.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.3.m.b.799.3 12 12.11 even 2
912.3.m.b.799.9 yes 12 3.2 odd 2
2736.3.m.e.1711.7 12 4.3 odd 2 inner
2736.3.m.e.1711.8 12 1.1 even 1 trivial