Properties

Label 4140.2.f.b.829.8
Level $4140$
Weight $2$
Character 4140.829
Analytic conductor $33.058$
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] = [4140,2,Mod(829,4140)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4140.829");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.f (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: \(33.0580664368\)
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: \( x^{12} + 24x^{10} + 188x^{8} + 530x^{6} + 508x^{4} + 80x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 460)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 829.8
Root \(1.26443i\) of defining polynomial
Character \(\chi\) \(=\) 4140.829
Dual form 4140.2.f.b.829.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.817027 + 2.08146i) q^{5} -4.41307i q^{7} +O(q^{10})\) \(q+(0.817027 + 2.08146i) q^{5} -4.41307i q^{7} -2.29289 q^{11} +6.92936i q^{13} +1.51387i q^{17} -2.89920 q^{19} -1.00000i q^{23} +(-3.66493 + 3.40122i) q^{25} -7.68764 q^{29} +3.85746 q^{31} +(9.18562 - 3.60560i) q^{35} -8.62830i q^{37} +6.44324 q^{41} -3.48497i q^{43} -6.19747i q^{47} -12.4752 q^{49} -2.17710i q^{53} +(-1.87335 - 4.77255i) q^{55} -11.7637 q^{59} -5.11443 q^{61} +(-14.4232 + 5.66148i) q^{65} -9.94597i q^{67} -3.41407 q^{71} -8.95307i q^{73} +10.1187i q^{77} +1.92694 q^{79} -8.04131i q^{83} +(-3.15106 + 1.23687i) q^{85} -1.09273 q^{89} +30.5798 q^{91} +(-2.36872 - 6.03456i) q^{95} +16.9208i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{11} - 8 q^{19} + 8 q^{25} + 10 q^{29} + 18 q^{31} + 10 q^{35} + 2 q^{41} - 38 q^{49} + 16 q^{55} - 22 q^{59} - 8 q^{61} - 38 q^{65} + 34 q^{71} - 20 q^{79} + 6 q^{85} - 48 q^{89} - 8 q^{91} - 12 q^{95}+O(q^{100}) \) 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
\(5\) 0.817027 + 2.08146i 0.365386 + 0.930856i
\(6\) 0 0
\(7\) 4.41307i 1.66798i −0.551777 0.833992i \(-0.686050\pi\)
0.551777 0.833992i \(-0.313950\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.29289 −0.691332 −0.345666 0.938358i \(-0.612347\pi\)
−0.345666 + 0.938358i \(0.612347\pi\)
\(12\) 0 0
\(13\) 6.92936i 1.92186i 0.276792 + 0.960930i \(0.410729\pi\)
−0.276792 + 0.960930i \(0.589271\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.51387i 0.367168i 0.983004 + 0.183584i \(0.0587699\pi\)
−0.983004 + 0.183584i \(0.941230\pi\)
\(18\) 0 0
\(19\) −2.89920 −0.665121 −0.332561 0.943082i \(-0.607913\pi\)
−0.332561 + 0.943082i \(0.607913\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) −3.66493 + 3.40122i −0.732987 + 0.680243i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.68764 −1.42756 −0.713779 0.700371i \(-0.753018\pi\)
−0.713779 + 0.700371i \(0.753018\pi\)
\(30\) 0 0
\(31\) 3.85746 0.692821 0.346410 0.938083i \(-0.387401\pi\)
0.346410 + 0.938083i \(0.387401\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 9.18562 3.60560i 1.55265 0.609457i
\(36\) 0 0
\(37\) 8.62830i 1.41848i −0.704965 0.709242i \(-0.749037\pi\)
0.704965 0.709242i \(-0.250963\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.44324 1.00626 0.503132 0.864209i \(-0.332181\pi\)
0.503132 + 0.864209i \(0.332181\pi\)
\(42\) 0 0
\(43\) 3.48497i 0.531453i −0.964048 0.265727i \(-0.914388\pi\)
0.964048 0.265727i \(-0.0856119\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.19747i 0.903994i −0.892019 0.451997i \(-0.850712\pi\)
0.892019 0.451997i \(-0.149288\pi\)
\(48\) 0 0
\(49\) −12.4752 −1.78217
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.17710i 0.299047i −0.988758 0.149524i \(-0.952226\pi\)
0.988758 0.149524i \(-0.0477740\pi\)
\(54\) 0 0
\(55\) −1.87335 4.77255i −0.252603 0.643530i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.7637 −1.53150 −0.765750 0.643138i \(-0.777632\pi\)
−0.765750 + 0.643138i \(0.777632\pi\)
\(60\) 0 0
\(61\) −5.11443 −0.654835 −0.327418 0.944880i \(-0.606178\pi\)
−0.327418 + 0.944880i \(0.606178\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −14.4232 + 5.66148i −1.78898 + 0.702220i
\(66\) 0 0
\(67\) 9.94597i 1.21509i −0.794284 0.607547i \(-0.792154\pi\)
0.794284 0.607547i \(-0.207846\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.41407 −0.405175 −0.202588 0.979264i \(-0.564935\pi\)
−0.202588 + 0.979264i \(0.564935\pi\)
\(72\) 0 0
\(73\) 8.95307i 1.04788i −0.851756 0.523939i \(-0.824462\pi\)
0.851756 0.523939i \(-0.175538\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.1187i 1.15313i
\(78\) 0 0
\(79\) 1.92694 0.216798 0.108399 0.994107i \(-0.465428\pi\)
0.108399 + 0.994107i \(0.465428\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.04131i 0.882648i −0.897348 0.441324i \(-0.854509\pi\)
0.897348 0.441324i \(-0.145491\pi\)
\(84\) 0 0
\(85\) −3.15106 + 1.23687i −0.341781 + 0.134158i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.09273 −0.115829 −0.0579147 0.998322i \(-0.518445\pi\)
−0.0579147 + 0.998322i \(0.518445\pi\)
\(90\) 0 0
\(91\) 30.5798 3.20563
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.36872 6.03456i −0.243026 0.619132i
\(96\) 0 0
\(97\) 16.9208i 1.71805i 0.511933 + 0.859026i \(0.328930\pi\)
−0.511933 + 0.859026i \(0.671070\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.9497 −1.28855 −0.644274 0.764795i \(-0.722840\pi\)
−0.644274 + 0.764795i \(0.722840\pi\)
\(102\) 0 0
\(103\) 10.9754i 1.08144i 0.841202 + 0.540721i \(0.181848\pi\)
−0.841202 + 0.540721i \(0.818152\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 19.8821i 1.92208i 0.276411 + 0.961040i \(0.410855\pi\)
−0.276411 + 0.961040i \(0.589145\pi\)
\(108\) 0 0
\(109\) −0.427249 −0.0409231 −0.0204615 0.999791i \(-0.506514\pi\)
−0.0204615 + 0.999791i \(0.506514\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.38533i 0.506609i 0.967387 + 0.253304i \(0.0815174\pi\)
−0.967387 + 0.253304i \(0.918483\pi\)
\(114\) 0 0
\(115\) 2.08146 0.817027i 0.194097 0.0761882i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.68082 0.612430
\(120\) 0 0
\(121\) −5.74267 −0.522061
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.0738 4.84952i −0.901031 0.433754i
\(126\) 0 0
\(127\) 6.16014i 0.546624i −0.961925 0.273312i \(-0.911881\pi\)
0.961925 0.273312i \(-0.0881191\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −14.6325 −1.27845 −0.639225 0.769020i \(-0.720745\pi\)
−0.639225 + 0.769020i \(0.720745\pi\)
\(132\) 0 0
\(133\) 12.7944i 1.10941i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.972256i 0.0830654i 0.999137 + 0.0415327i \(0.0132241\pi\)
−0.999137 + 0.0415327i \(0.986776\pi\)
\(138\) 0 0
\(139\) −7.74312 −0.656763 −0.328382 0.944545i \(-0.606503\pi\)
−0.328382 + 0.944545i \(0.606503\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 15.8883i 1.32864i
\(144\) 0 0
\(145\) −6.28101 16.0015i −0.521609 1.32885i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −17.6823 −1.44859 −0.724293 0.689492i \(-0.757834\pi\)
−0.724293 + 0.689492i \(0.757834\pi\)
\(150\) 0 0
\(151\) −2.01286 −0.163804 −0.0819022 0.996640i \(-0.526100\pi\)
−0.0819022 + 0.996640i \(0.526100\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.15165 + 8.02914i 0.253147 + 0.644916i
\(156\) 0 0
\(157\) 5.82703i 0.465048i 0.972591 + 0.232524i \(0.0746984\pi\)
−0.972591 + 0.232524i \(0.925302\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.41307 −0.347799
\(162\) 0 0
\(163\) 6.75147i 0.528816i −0.964411 0.264408i \(-0.914823\pi\)
0.964411 0.264408i \(-0.0851765\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 23.4541i 1.81493i −0.420125 0.907466i \(-0.638014\pi\)
0.420125 0.907466i \(-0.361986\pi\)
\(168\) 0 0
\(169\) −35.0161 −2.69355
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.28850i 0.630163i 0.949065 + 0.315081i \(0.102032\pi\)
−0.949065 + 0.315081i \(0.897968\pi\)
\(174\) 0 0
\(175\) 15.0098 + 16.1736i 1.13463 + 1.22261i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.04002 0.0777345 0.0388673 0.999244i \(-0.487625\pi\)
0.0388673 + 0.999244i \(0.487625\pi\)
\(180\) 0 0
\(181\) 2.71850 0.202065 0.101032 0.994883i \(-0.467785\pi\)
0.101032 + 0.994883i \(0.467785\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 17.9594 7.04956i 1.32040 0.518294i
\(186\) 0 0
\(187\) 3.47114i 0.253835i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.04858 −0.582375 −0.291187 0.956666i \(-0.594050\pi\)
−0.291187 + 0.956666i \(0.594050\pi\)
\(192\) 0 0
\(193\) 16.0276i 1.15370i −0.816852 0.576848i \(-0.804283\pi\)
0.816852 0.576848i \(-0.195717\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.06316i 0.289488i 0.989469 + 0.144744i \(0.0462359\pi\)
−0.989469 + 0.144744i \(0.953764\pi\)
\(198\) 0 0
\(199\) −18.7042 −1.32590 −0.662952 0.748662i \(-0.730697\pi\)
−0.662952 + 0.748662i \(0.730697\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 33.9261i 2.38114i
\(204\) 0 0
\(205\) 5.26430 + 13.4113i 0.367675 + 0.936688i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.64753 0.459819
\(210\) 0 0
\(211\) 7.87839 0.542371 0.271185 0.962527i \(-0.412584\pi\)
0.271185 + 0.962527i \(0.412584\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 7.25382 2.84732i 0.494707 0.194185i
\(216\) 0 0
\(217\) 17.0232i 1.15561i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −10.4902 −0.705645
\(222\) 0 0
\(223\) 2.29263i 0.153526i −0.997049 0.0767628i \(-0.975542\pi\)
0.997049 0.0767628i \(-0.0244584\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.8325i 0.851725i 0.904788 + 0.425862i \(0.140029\pi\)
−0.904788 + 0.425862i \(0.859971\pi\)
\(228\) 0 0
\(229\) 16.2528 1.07401 0.537007 0.843578i \(-0.319555\pi\)
0.537007 + 0.843578i \(0.319555\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.5410i 1.41120i −0.708613 0.705598i \(-0.750679\pi\)
0.708613 0.705598i \(-0.249321\pi\)
\(234\) 0 0
\(235\) 12.8998 5.06350i 0.841489 0.330307i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −11.0622 −0.715555 −0.357778 0.933807i \(-0.616465\pi\)
−0.357778 + 0.933807i \(0.616465\pi\)
\(240\) 0 0
\(241\) 18.2164 1.17342 0.586710 0.809797i \(-0.300423\pi\)
0.586710 + 0.809797i \(0.300423\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −10.1926 25.9666i −0.651179 1.65894i
\(246\) 0 0
\(247\) 20.0896i 1.27827i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 10.0182 0.632345 0.316172 0.948702i \(-0.397602\pi\)
0.316172 + 0.948702i \(0.397602\pi\)
\(252\) 0 0
\(253\) 2.29289i 0.144153i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.3021i 0.892141i −0.894998 0.446070i \(-0.852823\pi\)
0.894998 0.446070i \(-0.147177\pi\)
\(258\) 0 0
\(259\) −38.0773 −2.36601
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.44361i 0.582318i 0.956675 + 0.291159i \(0.0940409\pi\)
−0.956675 + 0.291159i \(0.905959\pi\)
\(264\) 0 0
\(265\) 4.53153 1.77875i 0.278370 0.109268i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 23.7630 1.44886 0.724428 0.689350i \(-0.242104\pi\)
0.724428 + 0.689350i \(0.242104\pi\)
\(270\) 0 0
\(271\) −17.5939 −1.06875 −0.534375 0.845247i \(-0.679453\pi\)
−0.534375 + 0.845247i \(0.679453\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.40328 7.79860i 0.506737 0.470273i
\(276\) 0 0
\(277\) 29.6582i 1.78199i 0.454014 + 0.890995i \(0.349992\pi\)
−0.454014 + 0.890995i \(0.650008\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.25484 0.492442 0.246221 0.969214i \(-0.420811\pi\)
0.246221 + 0.969214i \(0.420811\pi\)
\(282\) 0 0
\(283\) 9.21317i 0.547666i 0.961777 + 0.273833i \(0.0882916\pi\)
−0.961777 + 0.273833i \(0.911708\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 28.4344i 1.67843i
\(288\) 0 0
\(289\) 14.7082 0.865188
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.1138i 0.649277i −0.945838 0.324639i \(-0.894757\pi\)
0.945838 0.324639i \(-0.105243\pi\)
\(294\) 0 0
\(295\) −9.61125 24.4856i −0.559588 1.42561i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.92936 0.400735
\(300\) 0 0
\(301\) −15.3794 −0.886455
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −4.17863 10.6455i −0.239267 0.609558i
\(306\) 0 0
\(307\) 18.7800i 1.07183i −0.844272 0.535916i \(-0.819967\pi\)
0.844272 0.535916i \(-0.180033\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2.51258 −0.142475 −0.0712377 0.997459i \(-0.522695\pi\)
−0.0712377 + 0.997459i \(0.522695\pi\)
\(312\) 0 0
\(313\) 19.9278i 1.12638i 0.826326 + 0.563192i \(0.190427\pi\)
−0.826326 + 0.563192i \(0.809573\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 27.8849i 1.56617i −0.621912 0.783087i \(-0.713644\pi\)
0.621912 0.783087i \(-0.286356\pi\)
\(318\) 0 0
\(319\) 17.6269 0.986916
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.38901i 0.244211i
\(324\) 0 0
\(325\) −23.5683 25.3957i −1.30733 1.40870i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −27.3499 −1.50785
\(330\) 0 0
\(331\) −28.5409 −1.56875 −0.784375 0.620287i \(-0.787016\pi\)
−0.784375 + 0.620287i \(0.787016\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 20.7021 8.12612i 1.13108 0.443978i
\(336\) 0 0
\(337\) 3.71371i 0.202298i −0.994871 0.101149i \(-0.967748\pi\)
0.994871 0.101149i \(-0.0322520\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −8.84472 −0.478969
\(342\) 0 0
\(343\) 24.1624i 1.30464i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 14.1459i 0.759393i 0.925111 + 0.379696i \(0.123972\pi\)
−0.925111 + 0.379696i \(0.876028\pi\)
\(348\) 0 0
\(349\) 21.2802 1.13910 0.569550 0.821957i \(-0.307117\pi\)
0.569550 + 0.821957i \(0.307117\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 24.4641i 1.30209i −0.759038 0.651046i \(-0.774331\pi\)
0.759038 0.651046i \(-0.225669\pi\)
\(354\) 0 0
\(355\) −2.78939 7.10624i −0.148045 0.377160i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −23.0252 −1.21522 −0.607612 0.794234i \(-0.707872\pi\)
−0.607612 + 0.794234i \(0.707872\pi\)
\(360\) 0 0
\(361\) −10.5947 −0.557613
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 18.6354 7.31490i 0.975424 0.382880i
\(366\) 0 0
\(367\) 0.478057i 0.0249544i −0.999922 0.0124772i \(-0.996028\pi\)
0.999922 0.0124772i \(-0.00397171\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −9.60767 −0.498806
\(372\) 0 0
\(373\) 27.3508i 1.41617i 0.706127 + 0.708085i \(0.250441\pi\)
−0.706127 + 0.708085i \(0.749559\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 53.2704i 2.74357i
\(378\) 0 0
\(379\) 7.35358 0.377728 0.188864 0.982003i \(-0.439519\pi\)
0.188864 + 0.982003i \(0.439519\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.34108i 0.170721i 0.996350 + 0.0853606i \(0.0272042\pi\)
−0.996350 + 0.0853606i \(0.972796\pi\)
\(384\) 0 0
\(385\) −21.0616 + 8.26723i −1.07340 + 0.421337i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0.366568 0.0185858 0.00929288 0.999957i \(-0.497042\pi\)
0.00929288 + 0.999957i \(0.497042\pi\)
\(390\) 0 0
\(391\) 1.51387 0.0765598
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.57436 + 4.01085i 0.0792148 + 0.201808i
\(396\) 0 0
\(397\) 11.3222i 0.568245i 0.958788 + 0.284122i \(0.0917022\pi\)
−0.958788 + 0.284122i \(0.908298\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −37.1673 −1.85605 −0.928024 0.372520i \(-0.878494\pi\)
−0.928024 + 0.372520i \(0.878494\pi\)
\(402\) 0 0
\(403\) 26.7298i 1.33150i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 19.7837i 0.980643i
\(408\) 0 0
\(409\) 13.7745 0.681107 0.340553 0.940225i \(-0.389386\pi\)
0.340553 + 0.940225i \(0.389386\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 51.9139i 2.55452i
\(414\) 0 0
\(415\) 16.7376 6.56997i 0.821619 0.322507i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 7.50726 0.366753 0.183377 0.983043i \(-0.441297\pi\)
0.183377 + 0.983043i \(0.441297\pi\)
\(420\) 0 0
\(421\) −7.65685 −0.373172 −0.186586 0.982439i \(-0.559742\pi\)
−0.186586 + 0.982439i \(0.559742\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.14900 5.54824i −0.249763 0.269129i
\(426\) 0 0
\(427\) 22.5703i 1.09225i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.22764 −0.299975 −0.149987 0.988688i \(-0.547923\pi\)
−0.149987 + 0.988688i \(0.547923\pi\)
\(432\) 0 0
\(433\) 0.704087i 0.0338362i 0.999857 + 0.0169181i \(0.00538546\pi\)
−0.999857 + 0.0169181i \(0.994615\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.89920i 0.138687i
\(438\) 0 0
\(439\) 34.8570 1.66363 0.831816 0.555052i \(-0.187301\pi\)
0.831816 + 0.555052i \(0.187301\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.8165i 1.17907i 0.807745 + 0.589533i \(0.200688\pi\)
−0.807745 + 0.589533i \(0.799312\pi\)
\(444\) 0 0
\(445\) −0.892792 2.27448i −0.0423224 0.107820i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.77738 0.319844 0.159922 0.987130i \(-0.448876\pi\)
0.159922 + 0.987130i \(0.448876\pi\)
\(450\) 0 0
\(451\) −14.7736 −0.695662
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 24.9845 + 63.6505i 1.17129 + 2.98398i
\(456\) 0 0
\(457\) 1.47169i 0.0688429i −0.999407 0.0344214i \(-0.989041\pi\)
0.999407 0.0344214i \(-0.0109588\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −25.8037 −1.20180 −0.600899 0.799325i \(-0.705191\pi\)
−0.600899 + 0.799325i \(0.705191\pi\)
\(462\) 0 0
\(463\) 15.3469i 0.713231i −0.934251 0.356616i \(-0.883931\pi\)
0.934251 0.356616i \(-0.116069\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.2933i 0.476315i 0.971227 + 0.238157i \(0.0765434\pi\)
−0.971227 + 0.238157i \(0.923457\pi\)
\(468\) 0 0
\(469\) −43.8922 −2.02676
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.99065i 0.367410i
\(474\) 0 0
\(475\) 10.6254 9.86079i 0.487525 0.452444i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.50453 0.205817 0.102909 0.994691i \(-0.467185\pi\)
0.102909 + 0.994691i \(0.467185\pi\)
\(480\) 0 0
\(481\) 59.7886 2.72613
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −35.2200 + 13.8248i −1.59926 + 0.627751i
\(486\) 0 0
\(487\) 27.2669i 1.23558i 0.786343 + 0.617790i \(0.211972\pi\)
−0.786343 + 0.617790i \(0.788028\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −24.4866 −1.10507 −0.552533 0.833491i \(-0.686339\pi\)
−0.552533 + 0.833491i \(0.686339\pi\)
\(492\) 0 0
\(493\) 11.6381i 0.524154i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15.0665i 0.675826i
\(498\) 0 0
\(499\) 24.1892 1.08286 0.541429 0.840746i \(-0.317883\pi\)
0.541429 + 0.840746i \(0.317883\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 30.6230i 1.36541i −0.730693 0.682706i \(-0.760803\pi\)
0.730693 0.682706i \(-0.239197\pi\)
\(504\) 0 0
\(505\) −10.5803 26.9543i −0.470817 1.19945i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.56826 −0.113836 −0.0569181 0.998379i \(-0.518127\pi\)
−0.0569181 + 0.998379i \(0.518127\pi\)
\(510\) 0 0
\(511\) −39.5105 −1.74784
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −22.8449 + 8.96722i −1.00667 + 0.395143i
\(516\) 0 0
\(517\) 14.2101i 0.624960i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 44.1355 1.93361 0.966807 0.255509i \(-0.0822432\pi\)
0.966807 + 0.255509i \(0.0822432\pi\)
\(522\) 0 0
\(523\) 42.2884i 1.84914i 0.381008 + 0.924572i \(0.375577\pi\)
−0.381008 + 0.924572i \(0.624423\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.83970i 0.254381i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 44.6475i 1.93390i
\(534\) 0 0
\(535\) −41.3838 + 16.2443i −1.78918 + 0.702300i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 28.6042 1.23207
\(540\) 0 0
\(541\) 10.2776 0.441871 0.220935 0.975288i \(-0.429089\pi\)
0.220935 + 0.975288i \(0.429089\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −0.349074 0.889302i −0.0149527 0.0380935i
\(546\) 0 0
\(547\) 3.87305i 0.165600i 0.996566 + 0.0827998i \(0.0263862\pi\)
−0.996566 + 0.0827998i \(0.973614\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 22.2880 0.949500
\(552\) 0 0
\(553\) 8.50373i 0.361615i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.62364i 0.407767i −0.978995 0.203883i \(-0.934644\pi\)
0.978995 0.203883i \(-0.0653563\pi\)
\(558\) 0 0
\(559\) 24.1486 1.02138
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.35865i 0.141550i −0.997492 0.0707751i \(-0.977453\pi\)
0.997492 0.0707751i \(-0.0225473\pi\)
\(564\) 0 0
\(565\) −11.2093 + 4.39996i −0.471580 + 0.185108i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.26939 −0.178982 −0.0894910 0.995988i \(-0.528524\pi\)
−0.0894910 + 0.995988i \(0.528524\pi\)
\(570\) 0 0
\(571\) 16.4567 0.688689 0.344345 0.938843i \(-0.388101\pi\)
0.344345 + 0.938843i \(0.388101\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.40122 + 3.66493i 0.141840 + 0.152838i
\(576\) 0 0
\(577\) 1.98780i 0.0827530i −0.999144 0.0413765i \(-0.986826\pi\)
0.999144 0.0413765i \(-0.0131743\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −35.4869 −1.47224
\(582\) 0 0
\(583\) 4.99184i 0.206741i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.9720i 1.19580i −0.801570 0.597900i \(-0.796002\pi\)
0.801570 0.597900i \(-0.203998\pi\)
\(588\) 0 0
\(589\) −11.1835 −0.460810
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 19.4799i 0.799944i 0.916527 + 0.399972i \(0.130980\pi\)
−0.916527 + 0.399972i \(0.869020\pi\)
\(594\) 0 0
\(595\) 5.45841 + 13.9059i 0.223773 + 0.570084i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 37.8442 1.54627 0.773136 0.634240i \(-0.218687\pi\)
0.773136 + 0.634240i \(0.218687\pi\)
\(600\) 0 0
\(601\) −20.2347 −0.825389 −0.412695 0.910869i \(-0.635412\pi\)
−0.412695 + 0.910869i \(0.635412\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.69191 11.9531i −0.190753 0.485963i
\(606\) 0 0
\(607\) 24.4075i 0.990670i −0.868702 0.495335i \(-0.835045\pi\)
0.868702 0.495335i \(-0.164955\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 42.9445 1.73735
\(612\) 0 0
\(613\) 10.6964i 0.432025i 0.976391 + 0.216012i \(0.0693051\pi\)
−0.976391 + 0.216012i \(0.930695\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 27.4738i 1.10605i −0.833164 0.553026i \(-0.813473\pi\)
0.833164 0.553026i \(-0.186527\pi\)
\(618\) 0 0
\(619\) 0.389330 0.0156485 0.00782425 0.999969i \(-0.497509\pi\)
0.00782425 + 0.999969i \(0.497509\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.82230i 0.193201i
\(624\) 0 0
\(625\) 1.86347 24.9305i 0.0745389 0.997218i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 13.0621 0.520822
\(630\) 0 0
\(631\) −11.9330 −0.475047 −0.237523 0.971382i \(-0.576336\pi\)
−0.237523 + 0.971382i \(0.576336\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.8221 5.03300i 0.508828 0.199729i
\(636\) 0 0
\(637\) 86.4451i 3.42508i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −22.8525 −0.902621 −0.451311 0.892367i \(-0.649043\pi\)
−0.451311 + 0.892367i \(0.649043\pi\)
\(642\) 0 0
\(643\) 34.9279i 1.37742i 0.725036 + 0.688711i \(0.241823\pi\)
−0.725036 + 0.688711i \(0.758177\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 26.3738i 1.03686i 0.855120 + 0.518430i \(0.173483\pi\)
−0.855120 + 0.518430i \(0.826517\pi\)
\(648\) 0 0
\(649\) 26.9728 1.05877
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.17022i 0.0849274i 0.999098 + 0.0424637i \(0.0135207\pi\)
−0.999098 + 0.0424637i \(0.986479\pi\)
\(654\) 0 0
\(655\) −11.9552 30.4570i −0.467127 1.19005i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 19.7820 0.770596 0.385298 0.922792i \(-0.374099\pi\)
0.385298 + 0.922792i \(0.374099\pi\)
\(660\) 0 0
\(661\) 45.7505 1.77949 0.889743 0.456461i \(-0.150883\pi\)
0.889743 + 0.456461i \(0.150883\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −26.6309 + 10.4533i −1.03270 + 0.405363i
\(666\) 0 0
\(667\) 7.68764i 0.297666i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 11.7268 0.452708
\(672\) 0 0
\(673\) 6.99940i 0.269807i 0.990859 + 0.134904i \(0.0430725\pi\)
−0.990859 + 0.134904i \(0.956928\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 26.7178i 1.02685i 0.858134 + 0.513425i \(0.171624\pi\)
−0.858134 + 0.513425i \(0.828376\pi\)
\(678\) 0 0
\(679\) 74.6728 2.86568
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 19.8947i 0.761248i 0.924730 + 0.380624i \(0.124291\pi\)
−0.924730 + 0.380624i \(0.875709\pi\)
\(684\) 0 0
\(685\) −2.02371 + 0.794359i −0.0773219 + 0.0303509i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 15.0859 0.574727
\(690\) 0 0
\(691\) 1.24438 0.0473385 0.0236693 0.999720i \(-0.492465\pi\)
0.0236693 + 0.999720i \(0.492465\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.32634 16.1170i −0.239972 0.611352i
\(696\) 0 0
\(697\) 9.75424i 0.369468i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 29.8454 1.12724 0.563622 0.826033i \(-0.309408\pi\)
0.563622 + 0.826033i \(0.309408\pi\)
\(702\) 0 0
\(703\) 25.0151i 0.943464i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 57.1481i 2.14928i
\(708\) 0 0
\(709\) −25.2255 −0.947362 −0.473681 0.880697i \(-0.657075\pi\)
−0.473681 + 0.880697i \(0.657075\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.85746i 0.144463i
\(714\) 0 0
\(715\) 33.0707 12.9811i 1.23678 0.485467i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −23.7787 −0.886797 −0.443398 0.896325i \(-0.646227\pi\)
−0.443398 + 0.896325i \(0.646227\pi\)
\(720\) 0 0
\(721\) 48.4353 1.80383
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 28.1747 26.1473i 1.04638 0.971086i
\(726\) 0 0
\(727\) 36.5849i 1.35686i −0.734666 0.678429i \(-0.762661\pi\)
0.734666 0.678429i \(-0.237339\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 5.27580 0.195133
\(732\) 0 0
\(733\) 21.7593i 0.803698i −0.915706 0.401849i \(-0.868368\pi\)
0.915706 0.401849i \(-0.131632\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 22.8050i 0.840032i
\(738\) 0 0
\(739\) −2.72303 −0.100168 −0.0500842 0.998745i \(-0.515949\pi\)
−0.0500842 + 0.998745i \(0.515949\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 32.0161i 1.17456i 0.809385 + 0.587278i \(0.199800\pi\)
−0.809385 + 0.587278i \(0.800200\pi\)
\(744\) 0 0
\(745\) −14.4469 36.8049i −0.529293 1.34843i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 87.7413 3.20600
\(750\) 0 0
\(751\) −36.0949 −1.31712 −0.658561 0.752527i \(-0.728835\pi\)
−0.658561 + 0.752527i \(0.728835\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.64456 4.18969i −0.0598517 0.152478i
\(756\) 0 0
\(757\) 46.8188i 1.70166i 0.525442 + 0.850830i \(0.323900\pi\)
−0.525442 + 0.850830i \(0.676100\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 28.8182 1.04466 0.522330 0.852744i \(-0.325063\pi\)
0.522330 + 0.852744i \(0.325063\pi\)
\(762\) 0 0
\(763\) 1.88548i 0.0682590i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 81.5148i 2.94333i
\(768\) 0 0
\(769\) −51.6243 −1.86162 −0.930811 0.365501i \(-0.880898\pi\)
−0.930811 + 0.365501i \(0.880898\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25.9610i 0.933753i −0.884322 0.466877i \(-0.845379\pi\)
0.884322 0.466877i \(-0.154621\pi\)
\(774\) 0 0
\(775\) −14.1373 + 13.1201i −0.507828 + 0.471286i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −18.6802 −0.669288
\(780\) 0 0
\(781\) 7.82807 0.280110
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −12.1287 + 4.76084i −0.432893 + 0.169922i
\(786\) 0 0
\(787\) 53.2258i 1.89730i −0.316333 0.948648i \(-0.602452\pi\)
0.316333 0.948648i \(-0.397548\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 23.7658 0.845015
\(792\) 0 0
\(793\) 35.4397i 1.25850i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 31.1874i 1.10471i 0.833608 + 0.552357i \(0.186271\pi\)
−0.833608 + 0.552357i \(0.813729\pi\)
\(798\) 0 0
\(799\) 9.38218 0.331918
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 20.5284i 0.724431i
\(804\) 0 0
\(805\) −3.60560 9.18562i −0.127081 0.323750i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.60251 −0.0563414 −0.0281707 0.999603i \(-0.508968\pi\)
−0.0281707 + 0.999603i \(0.508968\pi\)
\(810\) 0 0
\(811\) −36.9545 −1.29765 −0.648824 0.760938i \(-0.724739\pi\)
−0.648824 + 0.760938i \(0.724739\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 14.0529 5.51613i 0.492251 0.193222i
\(816\) 0 0
\(817\) 10.1036i 0.353481i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 16.5583 0.577890 0.288945 0.957346i \(-0.406696\pi\)
0.288945 + 0.957346i \(0.406696\pi\)
\(822\) 0 0
\(823\) 7.63201i 0.266035i −0.991114 0.133018i \(-0.957533\pi\)
0.991114 0.133018i \(-0.0424667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 19.1969i 0.667543i −0.942654 0.333772i \(-0.891679\pi\)
0.942654 0.333772i \(-0.108321\pi\)
\(828\) 0 0
\(829\) 6.32413 0.219646 0.109823 0.993951i \(-0.464972\pi\)
0.109823 + 0.993951i \(0.464972\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 18.8858i 0.654355i
\(834\) 0 0
\(835\) 48.8187 19.1626i 1.68944 0.663150i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 23.1158 0.798046 0.399023 0.916941i \(-0.369349\pi\)
0.399023 + 0.916941i \(0.369349\pi\)
\(840\) 0 0
\(841\) 30.0997 1.03792
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −28.6091 72.8845i −0.984183 2.50730i
\(846\) 0 0
\(847\) 25.3428i 0.870789i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −8.62830 −0.295774
\(852\) 0 0
\(853\) 37.7049i 1.29099i −0.763764 0.645496i \(-0.776651\pi\)
0.763764 0.645496i \(-0.223349\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.7133i 0.673395i 0.941613 + 0.336697i \(0.109310\pi\)
−0.941613 + 0.336697i \(0.890690\pi\)
\(858\) 0 0
\(859\) −2.00609 −0.0684471 −0.0342235 0.999414i \(-0.510896\pi\)
−0.0342235 + 0.999414i \(0.510896\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16.8127i 0.572312i −0.958183 0.286156i \(-0.907623\pi\)
0.958183 0.286156i \(-0.0923775\pi\)
\(864\) 0 0
\(865\) −17.2522 + 6.77193i −0.586591 + 0.230252i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −4.41826 −0.149879
\(870\) 0 0
\(871\) 68.9192 2.33524
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −21.4013 + 44.4565i −0.723495 + 1.50291i
\(876\) 0 0
\(877\) 14.8922i 0.502874i −0.967874 0.251437i \(-0.919097\pi\)
0.967874 0.251437i \(-0.0809031\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 16.5218 0.556632 0.278316 0.960490i \(-0.410224\pi\)
0.278316 + 0.960490i \(0.410224\pi\)
\(882\) 0 0
\(883\) 38.7254i 1.30321i 0.758557 + 0.651607i \(0.225905\pi\)
−0.758557 + 0.651607i \(0.774095\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.16364i 0.0390711i 0.999809 + 0.0195355i \(0.00621875\pi\)
−0.999809 + 0.0195355i \(0.993781\pi\)
\(888\) 0 0
\(889\) −27.1851 −0.911760
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 17.9677i 0.601266i
\(894\) 0 0
\(895\) 0.849722 + 2.16475i 0.0284031 + 0.0723597i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −29.6548 −0.989042
\(900\) 0 0
\(901\) 3.29584 0.109800
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.22109 + 5.65845i 0.0738316 + 0.188093i
\(906\) 0 0
\(907\) 30.2796i 1.00542i −0.864456 0.502709i \(-0.832337\pi\)
0.864456 0.502709i \(-0.167663\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 28.2296 0.935290 0.467645 0.883916i \(-0.345103\pi\)
0.467645 + 0.883916i \(0.345103\pi\)
\(912\) 0 0
\(913\) 18.4378i 0.610203i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 64.5744i 2.13243i
\(918\) 0 0
\(919\) −3.28150 −0.108247 −0.0541233 0.998534i \(-0.517236\pi\)
−0.0541233 + 0.998534i \(0.517236\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 23.6573i 0.778690i
\(924\) 0 0
\(925\) 29.3467 + 31.6221i 0.964914 + 1.03973i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.64745 0.250905 0.125452 0.992100i \(-0.459962\pi\)
0.125452 + 0.992100i \(0.459962\pi\)
\(930\) 0 0
\(931\) 36.1680 1.18536
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 7.22503 2.83601i 0.236284 0.0927476i
\(936\) 0 0
\(937\) 4.95226i 0.161783i 0.996723 + 0.0808917i \(0.0257768\pi\)
−0.996723 + 0.0808917i \(0.974223\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 8.62677 0.281225 0.140612 0.990065i \(-0.455093\pi\)
0.140612 + 0.990065i \(0.455093\pi\)
\(942\) 0 0
\(943\) 6.44324i 0.209821i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 46.6807i 1.51692i −0.651720 0.758460i \(-0.725952\pi\)
0.651720 0.758460i \(-0.274048\pi\)
\(948\) 0 0
\(949\) 62.0391 2.01387
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 12.2343i 0.396307i 0.980171 + 0.198154i \(0.0634945\pi\)
−0.980171 + 0.198154i \(0.936505\pi\)
\(954\) 0 0
\(955\) −6.57591 16.7528i −0.212791 0.542107i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.29063 0.138552
\(960\) 0 0
\(961\) −16.1200 −0.520000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 33.3609 13.0950i 1.07392 0.421544i
\(966\) 0 0
\(967\) 54.8318i 1.76327i 0.471932 + 0.881635i \(0.343557\pi\)
−0.471932 + 0.881635i \(0.656443\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −40.3804 −1.29587 −0.647935 0.761696i \(-0.724367\pi\)
−0.647935 + 0.761696i \(0.724367\pi\)
\(972\) 0 0
\(973\) 34.1709i 1.09547i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10.6700i 0.341363i −0.985326 0.170681i \(-0.945403\pi\)
0.985326 0.170681i \(-0.0545968\pi\)
\(978\) 0 0
\(979\) 2.50551 0.0800765
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 6.27812i 0.200241i −0.994975 0.100120i \(-0.968077\pi\)
0.994975 0.100120i \(-0.0319228\pi\)
\(984\) 0 0
\(985\) −8.45730 + 3.31971i −0.269472 + 0.105775i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −3.48497 −0.110816
\(990\) 0 0
\(991\) 61.2864 1.94683 0.973413 0.229057i \(-0.0735643\pi\)
0.973413 + 0.229057i \(0.0735643\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −15.2818 38.9320i −0.484466 1.23423i
\(996\) 0 0
\(997\) 39.9799i 1.26618i −0.774080 0.633088i \(-0.781787\pi\)
0.774080 0.633088i \(-0.218213\pi\)
\(998\) 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 4140.2.f.b.829.8 12
3.2 odd 2 460.2.c.a.369.3 12
5.4 even 2 inner 4140.2.f.b.829.7 12
12.11 even 2 1840.2.e.f.369.10 12
15.2 even 4 2300.2.a.o.1.1 6
15.8 even 4 2300.2.a.n.1.6 6
15.14 odd 2 460.2.c.a.369.10 yes 12
60.23 odd 4 9200.2.a.cy.1.1 6
60.47 odd 4 9200.2.a.cx.1.6 6
60.59 even 2 1840.2.e.f.369.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.c.a.369.3 12 3.2 odd 2
460.2.c.a.369.10 yes 12 15.14 odd 2
1840.2.e.f.369.3 12 60.59 even 2
1840.2.e.f.369.10 12 12.11 even 2
2300.2.a.n.1.6 6 15.8 even 4
2300.2.a.o.1.1 6 15.2 even 4
4140.2.f.b.829.7 12 5.4 even 2 inner
4140.2.f.b.829.8 12 1.1 even 1 trivial
9200.2.a.cx.1.6 6 60.47 odd 4
9200.2.a.cy.1.1 6 60.23 odd 4