Properties

Label 1472.3.f.c.321.1
Level $1472$
Weight $3$
Character 1472.321
Analytic conductor $40.109$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1472,3,Mod(321,1472)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1472, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1472.321");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1472 = 2^{6} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1472.f (of order \(2\), degree \(1\), not 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: \(40.1090949138\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.613376.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 58x^{2} + 599 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 46)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 321.1
Root \(-6.67505i\) of defining polynomial
Character \(\chi\) \(=\) 1472.321
Dual form 1472.3.f.c.321.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.41421 q^{3} -9.43995i q^{5} -3.91016i q^{7} -3.17157 q^{9} +O(q^{10})\) \(q-2.41421 q^{3} -9.43995i q^{5} -3.91016i q^{7} -3.17157 q^{9} -3.91016i q^{11} -10.3137 q^{13} +22.7901i q^{15} +17.2603i q^{17} -26.7002i q^{19} +9.43995i q^{21} +(-3.10051 - 22.7901i) q^{23} -64.1127 q^{25} +29.3848 q^{27} -35.1421 q^{29} -33.2426 q^{31} +9.43995i q^{33} -36.9117 q^{35} -39.3794i q^{37} +24.8995 q^{39} +13.4853 q^{41} -29.9395i q^{43} +29.9395i q^{45} +31.0416 q^{47} +33.7107 q^{49} -41.6700i q^{51} +61.2207i q^{53} -36.9117 q^{55} +64.4600i q^{57} -26.2010 q^{59} -15.6406i q^{61} +12.4013i q^{63} +97.3609i q^{65} +68.3702i q^{67} +(7.48528 + 55.0201i) q^{69} +111.267 q^{71} +47.8528 q^{73} +154.782 q^{75} -15.2893 q^{77} -117.860i q^{79} -42.3970 q^{81} -74.8487i q^{83} +162.936 q^{85} +84.8406 q^{87} -39.1016i q^{89} +40.3282i q^{91} +80.2548 q^{93} -252.049 q^{95} +66.0797i q^{97} +12.4013i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 24 q^{9} + 4 q^{13} - 52 q^{23} - 132 q^{25} + 44 q^{27} - 84 q^{29} - 116 q^{31} + 56 q^{35} + 60 q^{39} + 20 q^{41} + 28 q^{47} - 148 q^{49} + 56 q^{55} - 184 q^{59} - 4 q^{69} + 100 q^{71} - 148 q^{73} + 308 q^{75} - 344 q^{77} + 68 q^{81} + 120 q^{85} + 164 q^{87} + 140 q^{93} - 352 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1472\mathbb{Z}\right)^\times\).

\(n\) \(645\) \(833\) \(1151\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.41421 −0.804738 −0.402369 0.915478i \(-0.631813\pi\)
−0.402369 + 0.915478i \(0.631813\pi\)
\(4\) 0 0
\(5\) 9.43995i 1.88799i −0.329959 0.943995i \(-0.607035\pi\)
0.329959 0.943995i \(-0.392965\pi\)
\(6\) 0 0
\(7\) 3.91016i 0.558594i −0.960205 0.279297i \(-0.909899\pi\)
0.960205 0.279297i \(-0.0901014\pi\)
\(8\) 0 0
\(9\) −3.17157 −0.352397
\(10\) 0 0
\(11\) 3.91016i 0.355469i −0.984078 0.177734i \(-0.943123\pi\)
0.984078 0.177734i \(-0.0568768\pi\)
\(12\) 0 0
\(13\) −10.3137 −0.793362 −0.396681 0.917956i \(-0.629838\pi\)
−0.396681 + 0.917956i \(0.629838\pi\)
\(14\) 0 0
\(15\) 22.7901i 1.51934i
\(16\) 0 0
\(17\) 17.2603i 1.01531i 0.861561 + 0.507655i \(0.169487\pi\)
−0.861561 + 0.507655i \(0.830513\pi\)
\(18\) 0 0
\(19\) 26.7002i 1.40527i −0.711548 0.702637i \(-0.752006\pi\)
0.711548 0.702637i \(-0.247994\pi\)
\(20\) 0 0
\(21\) 9.43995i 0.449522i
\(22\) 0 0
\(23\) −3.10051 22.7901i −0.134805 0.990872i
\(24\) 0 0
\(25\) −64.1127 −2.56451
\(26\) 0 0
\(27\) 29.3848 1.08833
\(28\) 0 0
\(29\) −35.1421 −1.21180 −0.605899 0.795542i \(-0.707186\pi\)
−0.605899 + 0.795542i \(0.707186\pi\)
\(30\) 0 0
\(31\) −33.2426 −1.07234 −0.536172 0.844109i \(-0.680130\pi\)
−0.536172 + 0.844109i \(0.680130\pi\)
\(32\) 0 0
\(33\) 9.43995i 0.286059i
\(34\) 0 0
\(35\) −36.9117 −1.05462
\(36\) 0 0
\(37\) 39.3794i 1.06431i −0.846647 0.532155i \(-0.821382\pi\)
0.846647 0.532155i \(-0.178618\pi\)
\(38\) 0 0
\(39\) 24.8995 0.638449
\(40\) 0 0
\(41\) 13.4853 0.328909 0.164455 0.986385i \(-0.447414\pi\)
0.164455 + 0.986385i \(0.447414\pi\)
\(42\) 0 0
\(43\) 29.9395i 0.696267i −0.937445 0.348134i \(-0.886816\pi\)
0.937445 0.348134i \(-0.113184\pi\)
\(44\) 0 0
\(45\) 29.9395i 0.665322i
\(46\) 0 0
\(47\) 31.0416 0.660460 0.330230 0.943900i \(-0.392874\pi\)
0.330230 + 0.943900i \(0.392874\pi\)
\(48\) 0 0
\(49\) 33.7107 0.687973
\(50\) 0 0
\(51\) 41.6700i 0.817058i
\(52\) 0 0
\(53\) 61.2207i 1.15511i 0.816352 + 0.577554i \(0.195993\pi\)
−0.816352 + 0.577554i \(0.804007\pi\)
\(54\) 0 0
\(55\) −36.9117 −0.671122
\(56\) 0 0
\(57\) 64.4600i 1.13088i
\(58\) 0 0
\(59\) −26.2010 −0.444085 −0.222042 0.975037i \(-0.571272\pi\)
−0.222042 + 0.975037i \(0.571272\pi\)
\(60\) 0 0
\(61\) 15.6406i 0.256404i −0.991748 0.128202i \(-0.959079\pi\)
0.991748 0.128202i \(-0.0409205\pi\)
\(62\) 0 0
\(63\) 12.4013i 0.196847i
\(64\) 0 0
\(65\) 97.3609i 1.49786i
\(66\) 0 0
\(67\) 68.3702i 1.02045i 0.860041 + 0.510225i \(0.170438\pi\)
−0.860041 + 0.510225i \(0.829562\pi\)
\(68\) 0 0
\(69\) 7.48528 + 55.0201i 0.108482 + 0.797392i
\(70\) 0 0
\(71\) 111.267 1.56714 0.783571 0.621303i \(-0.213396\pi\)
0.783571 + 0.621303i \(0.213396\pi\)
\(72\) 0 0
\(73\) 47.8528 0.655518 0.327759 0.944761i \(-0.393707\pi\)
0.327759 + 0.944761i \(0.393707\pi\)
\(74\) 0 0
\(75\) 154.782 2.06376
\(76\) 0 0
\(77\) −15.2893 −0.198563
\(78\) 0 0
\(79\) 117.860i 1.49190i −0.665999 0.745952i \(-0.731995\pi\)
0.665999 0.745952i \(-0.268005\pi\)
\(80\) 0 0
\(81\) −42.3970 −0.523419
\(82\) 0 0
\(83\) 74.8487i 0.901792i −0.892576 0.450896i \(-0.851105\pi\)
0.892576 0.450896i \(-0.148895\pi\)
\(84\) 0 0
\(85\) 162.936 1.91689
\(86\) 0 0
\(87\) 84.8406 0.975180
\(88\) 0 0
\(89\) 39.1016i 0.439343i −0.975574 0.219672i \(-0.929501\pi\)
0.975574 0.219672i \(-0.0704986\pi\)
\(90\) 0 0
\(91\) 40.3282i 0.443167i
\(92\) 0 0
\(93\) 80.2548 0.862955
\(94\) 0 0
\(95\) −252.049 −2.65314
\(96\) 0 0
\(97\) 66.0797i 0.681234i 0.940202 + 0.340617i \(0.110636\pi\)
−0.940202 + 0.340617i \(0.889364\pi\)
\(98\) 0 0
\(99\) 12.4013i 0.125266i
\(100\) 0 0
\(101\) −101.338 −1.00335 −0.501674 0.865057i \(-0.667282\pi\)
−0.501674 + 0.865057i \(0.667282\pi\)
\(102\) 0 0
\(103\) 101.549i 0.985912i 0.870054 + 0.492956i \(0.164084\pi\)
−0.870054 + 0.492956i \(0.835916\pi\)
\(104\) 0 0
\(105\) 89.1127 0.848692
\(106\) 0 0
\(107\) 75.5196i 0.705791i 0.935663 + 0.352895i \(0.114803\pi\)
−0.935663 + 0.352895i \(0.885197\pi\)
\(108\) 0 0
\(109\) 138.638i 1.27191i 0.771727 + 0.635954i \(0.219393\pi\)
−0.771727 + 0.635954i \(0.780607\pi\)
\(110\) 0 0
\(111\) 95.0704i 0.856490i
\(112\) 0 0
\(113\) 180.701i 1.59912i −0.600585 0.799561i \(-0.705065\pi\)
0.600585 0.799561i \(-0.294935\pi\)
\(114\) 0 0
\(115\) −215.137 + 29.2686i −1.87076 + 0.254510i
\(116\) 0 0
\(117\) 32.7107 0.279578
\(118\) 0 0
\(119\) 67.4903 0.567146
\(120\) 0 0
\(121\) 105.711 0.873642
\(122\) 0 0
\(123\) −32.5563 −0.264686
\(124\) 0 0
\(125\) 369.222i 2.95378i
\(126\) 0 0
\(127\) −36.3898 −0.286534 −0.143267 0.989684i \(-0.545761\pi\)
−0.143267 + 0.989684i \(0.545761\pi\)
\(128\) 0 0
\(129\) 72.2803i 0.560313i
\(130\) 0 0
\(131\) −65.3848 −0.499120 −0.249560 0.968359i \(-0.580286\pi\)
−0.249560 + 0.968359i \(0.580286\pi\)
\(132\) 0 0
\(133\) −104.402 −0.784978
\(134\) 0 0
\(135\) 277.391i 2.05475i
\(136\) 0 0
\(137\) 65.8018i 0.480305i 0.970735 + 0.240152i \(0.0771974\pi\)
−0.970735 + 0.240152i \(0.922803\pi\)
\(138\) 0 0
\(139\) 29.4092 0.211577 0.105788 0.994389i \(-0.466263\pi\)
0.105788 + 0.994389i \(0.466263\pi\)
\(140\) 0 0
\(141\) −74.9411 −0.531497
\(142\) 0 0
\(143\) 40.3282i 0.282015i
\(144\) 0 0
\(145\) 331.740i 2.28786i
\(146\) 0 0
\(147\) −81.3848 −0.553638
\(148\) 0 0
\(149\) 3.23928i 0.0217401i −0.999941 0.0108701i \(-0.996540\pi\)
0.999941 0.0108701i \(-0.00346012\pi\)
\(150\) 0 0
\(151\) 2.41421 0.0159882 0.00799408 0.999968i \(-0.497455\pi\)
0.00799408 + 0.999968i \(0.497455\pi\)
\(152\) 0 0
\(153\) 54.7422i 0.357792i
\(154\) 0 0
\(155\) 313.809i 2.02457i
\(156\) 0 0
\(157\) 73.6221i 0.468931i 0.972124 + 0.234465i \(0.0753339\pi\)
−0.972124 + 0.234465i \(0.924666\pi\)
\(158\) 0 0
\(159\) 147.800i 0.929559i
\(160\) 0 0
\(161\) −89.1127 + 12.1235i −0.553495 + 0.0753010i
\(162\) 0 0
\(163\) 125.586 0.770465 0.385232 0.922820i \(-0.374121\pi\)
0.385232 + 0.922820i \(0.374121\pi\)
\(164\) 0 0
\(165\) 89.1127 0.540077
\(166\) 0 0
\(167\) 92.8284 0.555859 0.277929 0.960601i \(-0.410352\pi\)
0.277929 + 0.960601i \(0.410352\pi\)
\(168\) 0 0
\(169\) −62.6274 −0.370576
\(170\) 0 0
\(171\) 84.6817i 0.495215i
\(172\) 0 0
\(173\) 6.71068 0.0387900 0.0193950 0.999812i \(-0.493826\pi\)
0.0193950 + 0.999812i \(0.493826\pi\)
\(174\) 0 0
\(175\) 250.691i 1.43252i
\(176\) 0 0
\(177\) 63.2548 0.357372
\(178\) 0 0
\(179\) 278.262 1.55454 0.777268 0.629170i \(-0.216605\pi\)
0.777268 + 0.629170i \(0.216605\pi\)
\(180\) 0 0
\(181\) 53.6783i 0.296565i 0.988945 + 0.148283i \(0.0473745\pi\)
−0.988945 + 0.148283i \(0.952625\pi\)
\(182\) 0 0
\(183\) 37.7598i 0.206338i
\(184\) 0 0
\(185\) −371.740 −2.00941
\(186\) 0 0
\(187\) 67.4903 0.360911
\(188\) 0 0
\(189\) 114.899i 0.607932i
\(190\) 0 0
\(191\) 168.692i 0.883207i 0.897210 + 0.441603i \(0.145590\pi\)
−0.897210 + 0.441603i \(0.854410\pi\)
\(192\) 0 0
\(193\) −95.3188 −0.493880 −0.246940 0.969031i \(-0.579425\pi\)
−0.246940 + 0.969031i \(0.579425\pi\)
\(194\) 0 0
\(195\) 235.050i 1.20538i
\(196\) 0 0
\(197\) −249.083 −1.26438 −0.632191 0.774813i \(-0.717844\pi\)
−0.632191 + 0.774813i \(0.717844\pi\)
\(198\) 0 0
\(199\) 76.1905i 0.382867i 0.981506 + 0.191433i \(0.0613136\pi\)
−0.981506 + 0.191433i \(0.938686\pi\)
\(200\) 0 0
\(201\) 165.060i 0.821195i
\(202\) 0 0
\(203\) 137.411i 0.676903i
\(204\) 0 0
\(205\) 127.300i 0.620978i
\(206\) 0 0
\(207\) 9.83348 + 72.2803i 0.0475047 + 0.349180i
\(208\) 0 0
\(209\) −104.402 −0.499531
\(210\) 0 0
\(211\) −71.1716 −0.337306 −0.168653 0.985675i \(-0.553942\pi\)
−0.168653 + 0.985675i \(0.553942\pi\)
\(212\) 0 0
\(213\) −268.622 −1.26114
\(214\) 0 0
\(215\) −282.627 −1.31455
\(216\) 0 0
\(217\) 129.984i 0.599004i
\(218\) 0 0
\(219\) −115.527 −0.527520
\(220\) 0 0
\(221\) 178.017i 0.805508i
\(222\) 0 0
\(223\) 84.0833 0.377055 0.188527 0.982068i \(-0.439629\pi\)
0.188527 + 0.982068i \(0.439629\pi\)
\(224\) 0 0
\(225\) 203.338 0.903725
\(226\) 0 0
\(227\) 263.092i 1.15900i −0.814974 0.579498i \(-0.803249\pi\)
0.814974 0.579498i \(-0.196751\pi\)
\(228\) 0 0
\(229\) 141.321i 0.617124i 0.951204 + 0.308562i \(0.0998477\pi\)
−0.951204 + 0.308562i \(0.900152\pi\)
\(230\) 0 0
\(231\) 36.9117 0.159791
\(232\) 0 0
\(233\) −353.784 −1.51839 −0.759193 0.650866i \(-0.774406\pi\)
−0.759193 + 0.650866i \(0.774406\pi\)
\(234\) 0 0
\(235\) 293.032i 1.24694i
\(236\) 0 0
\(237\) 284.540i 1.20059i
\(238\) 0 0
\(239\) 272.286 1.13927 0.569637 0.821897i \(-0.307084\pi\)
0.569637 + 0.821897i \(0.307084\pi\)
\(240\) 0 0
\(241\) 443.678i 1.84099i 0.390758 + 0.920493i \(0.372213\pi\)
−0.390758 + 0.920493i \(0.627787\pi\)
\(242\) 0 0
\(243\) −162.108 −0.667110
\(244\) 0 0
\(245\) 318.227i 1.29889i
\(246\) 0 0
\(247\) 275.378i 1.11489i
\(248\) 0 0
\(249\) 180.701i 0.725706i
\(250\) 0 0
\(251\) 194.607i 0.775326i −0.921801 0.387663i \(-0.873282\pi\)
0.921801 0.387663i \(-0.126718\pi\)
\(252\) 0 0
\(253\) −89.1127 + 12.1235i −0.352224 + 0.0479188i
\(254\) 0 0
\(255\) −393.362 −1.54260
\(256\) 0 0
\(257\) −22.1371 −0.0861365 −0.0430683 0.999072i \(-0.513713\pi\)
−0.0430683 + 0.999072i \(0.513713\pi\)
\(258\) 0 0
\(259\) −153.980 −0.594517
\(260\) 0 0
\(261\) 111.456 0.427034
\(262\) 0 0
\(263\) 425.076i 1.61626i −0.589006 0.808129i \(-0.700481\pi\)
0.589006 0.808129i \(-0.299519\pi\)
\(264\) 0 0
\(265\) 577.921 2.18083
\(266\) 0 0
\(267\) 94.3995i 0.353556i
\(268\) 0 0
\(269\) −411.215 −1.52868 −0.764341 0.644813i \(-0.776935\pi\)
−0.764341 + 0.644813i \(0.776935\pi\)
\(270\) 0 0
\(271\) 98.8183 0.364643 0.182322 0.983239i \(-0.441639\pi\)
0.182322 + 0.983239i \(0.441639\pi\)
\(272\) 0 0
\(273\) 97.3609i 0.356633i
\(274\) 0 0
\(275\) 250.691i 0.911602i
\(276\) 0 0
\(277\) 207.319 0.748443 0.374222 0.927339i \(-0.377910\pi\)
0.374222 + 0.927339i \(0.377910\pi\)
\(278\) 0 0
\(279\) 105.431 0.377891
\(280\) 0 0
\(281\) 31.0034i 0.110332i −0.998477 0.0551661i \(-0.982431\pi\)
0.998477 0.0551661i \(-0.0175688\pi\)
\(282\) 0 0
\(283\) 225.888i 0.798191i −0.916909 0.399095i \(-0.869324\pi\)
0.916909 0.399095i \(-0.130676\pi\)
\(284\) 0 0
\(285\) 608.500 2.13509
\(286\) 0 0
\(287\) 52.7296i 0.183727i
\(288\) 0 0
\(289\) −8.91674 −0.0308538
\(290\) 0 0
\(291\) 159.530i 0.548215i
\(292\) 0 0
\(293\) 162.099i 0.553238i −0.960980 0.276619i \(-0.910786\pi\)
0.960980 0.276619i \(-0.0892140\pi\)
\(294\) 0 0
\(295\) 247.336i 0.838428i
\(296\) 0 0
\(297\) 114.899i 0.386866i
\(298\) 0 0
\(299\) 31.9777 + 235.050i 0.106949 + 0.786121i
\(300\) 0 0
\(301\) −117.068 −0.388931
\(302\) 0 0
\(303\) 244.652 0.807432
\(304\) 0 0
\(305\) −147.647 −0.484088
\(306\) 0 0
\(307\) 342.299 1.11498 0.557490 0.830184i \(-0.311765\pi\)
0.557490 + 0.830184i \(0.311765\pi\)
\(308\) 0 0
\(309\) 245.161i 0.793401i
\(310\) 0 0
\(311\) −464.654 −1.49406 −0.747032 0.664788i \(-0.768522\pi\)
−0.747032 + 0.664788i \(0.768522\pi\)
\(312\) 0 0
\(313\) 95.1855i 0.304107i 0.988372 + 0.152054i \(0.0485886\pi\)
−0.988372 + 0.152054i \(0.951411\pi\)
\(314\) 0 0
\(315\) 117.068 0.371645
\(316\) 0 0
\(317\) 210.142 0.662909 0.331454 0.943471i \(-0.392461\pi\)
0.331454 + 0.943471i \(0.392461\pi\)
\(318\) 0 0
\(319\) 137.411i 0.430756i
\(320\) 0 0
\(321\) 182.320i 0.567977i
\(322\) 0 0
\(323\) 460.853 1.42679
\(324\) 0 0
\(325\) 661.240 2.03458
\(326\) 0 0
\(327\) 334.701i 1.02355i
\(328\) 0 0
\(329\) 121.378i 0.368929i
\(330\) 0 0
\(331\) 66.2822 0.200248 0.100124 0.994975i \(-0.468076\pi\)
0.100124 + 0.994975i \(0.468076\pi\)
\(332\) 0 0
\(333\) 124.895i 0.375059i
\(334\) 0 0
\(335\) 645.411 1.92660
\(336\) 0 0
\(337\) 453.348i 1.34525i −0.739985 0.672623i \(-0.765168\pi\)
0.739985 0.672623i \(-0.234832\pi\)
\(338\) 0 0
\(339\) 436.250i 1.28687i
\(340\) 0 0
\(341\) 129.984i 0.381185i
\(342\) 0 0
\(343\) 323.412i 0.942891i
\(344\) 0 0
\(345\) 519.387 70.6607i 1.50547 0.204814i
\(346\) 0 0
\(347\) −291.750 −0.840779 −0.420389 0.907344i \(-0.638107\pi\)
−0.420389 + 0.907344i \(0.638107\pi\)
\(348\) 0 0
\(349\) −150.951 −0.432525 −0.216263 0.976335i \(-0.569387\pi\)
−0.216263 + 0.976335i \(0.569387\pi\)
\(350\) 0 0
\(351\) −303.066 −0.863436
\(352\) 0 0
\(353\) 377.059 1.06816 0.534078 0.845435i \(-0.320659\pi\)
0.534078 + 0.845435i \(0.320659\pi\)
\(354\) 0 0
\(355\) 1050.36i 2.95875i
\(356\) 0 0
\(357\) −162.936 −0.456404
\(358\) 0 0
\(359\) 352.240i 0.981169i −0.871394 0.490584i \(-0.836783\pi\)
0.871394 0.490584i \(-0.163217\pi\)
\(360\) 0 0
\(361\) −351.902 −0.974797
\(362\) 0 0
\(363\) −255.208 −0.703053
\(364\) 0 0
\(365\) 451.728i 1.23761i
\(366\) 0 0
\(367\) 320.843i 0.874232i 0.899405 + 0.437116i \(0.144000\pi\)
−0.899405 + 0.437116i \(0.856000\pi\)
\(368\) 0 0
\(369\) −42.7696 −0.115907
\(370\) 0 0
\(371\) 239.383 0.645236
\(372\) 0 0
\(373\) 156.128i 0.418575i −0.977854 0.209287i \(-0.932886\pi\)
0.977854 0.209287i \(-0.0671144\pi\)
\(374\) 0 0
\(375\) 891.381i 2.37702i
\(376\) 0 0
\(377\) 362.446 0.961395
\(378\) 0 0
\(379\) 451.220i 1.19055i 0.803520 + 0.595277i \(0.202958\pi\)
−0.803520 + 0.595277i \(0.797042\pi\)
\(380\) 0 0
\(381\) 87.8528 0.230585
\(382\) 0 0
\(383\) 423.178i 1.10490i 0.833545 + 0.552452i \(0.186308\pi\)
−0.833545 + 0.552452i \(0.813692\pi\)
\(384\) 0 0
\(385\) 144.330i 0.374884i
\(386\) 0 0
\(387\) 94.9553i 0.245363i
\(388\) 0 0
\(389\) 11.0596i 0.0284308i −0.999899 0.0142154i \(-0.995475\pi\)
0.999899 0.0142154i \(-0.00452506\pi\)
\(390\) 0 0
\(391\) 393.362 53.5155i 1.00604 0.136868i
\(392\) 0 0
\(393\) 157.853 0.401661
\(394\) 0 0
\(395\) −1112.60 −2.81670
\(396\) 0 0
\(397\) 332.578 0.837727 0.418864 0.908049i \(-0.362429\pi\)
0.418864 + 0.908049i \(0.362429\pi\)
\(398\) 0 0
\(399\) 252.049 0.631701
\(400\) 0 0
\(401\) 51.7808i 0.129129i 0.997914 + 0.0645646i \(0.0205658\pi\)
−0.997914 + 0.0645646i \(0.979434\pi\)
\(402\) 0 0
\(403\) 342.855 0.850757
\(404\) 0 0
\(405\) 400.225i 0.988211i
\(406\) 0 0
\(407\) −153.980 −0.378329
\(408\) 0 0
\(409\) 256.088 0.626133 0.313066 0.949731i \(-0.398644\pi\)
0.313066 + 0.949731i \(0.398644\pi\)
\(410\) 0 0
\(411\) 158.860i 0.386520i
\(412\) 0 0
\(413\) 102.450i 0.248063i
\(414\) 0 0
\(415\) −706.569 −1.70257
\(416\) 0 0
\(417\) −71.0000 −0.170264
\(418\) 0 0
\(419\) 241.644i 0.576715i 0.957523 + 0.288358i \(0.0931092\pi\)
−0.957523 + 0.288358i \(0.906891\pi\)
\(420\) 0 0
\(421\) 312.860i 0.743136i −0.928406 0.371568i \(-0.878820\pi\)
0.928406 0.371568i \(-0.121180\pi\)
\(422\) 0 0
\(423\) −98.4508 −0.232744
\(424\) 0 0
\(425\) 1106.60i 2.60377i
\(426\) 0 0
\(427\) −61.1573 −0.143225
\(428\) 0 0
\(429\) 97.3609i 0.226949i
\(430\) 0 0
\(431\) 672.412i 1.56012i 0.625704 + 0.780060i \(0.284812\pi\)
−0.625704 + 0.780060i \(0.715188\pi\)
\(432\) 0 0
\(433\) 395.414i 0.913197i −0.889673 0.456598i \(-0.849068\pi\)
0.889673 0.456598i \(-0.150932\pi\)
\(434\) 0 0
\(435\) 800.891i 1.84113i
\(436\) 0 0
\(437\) −608.500 + 82.7842i −1.39245 + 0.189437i
\(438\) 0 0
\(439\) −836.997 −1.90660 −0.953300 0.302026i \(-0.902337\pi\)
−0.953300 + 0.302026i \(0.902337\pi\)
\(440\) 0 0
\(441\) −106.916 −0.242440
\(442\) 0 0
\(443\) −227.905 −0.514457 −0.257229 0.966351i \(-0.582809\pi\)
−0.257229 + 0.966351i \(0.582809\pi\)
\(444\) 0 0
\(445\) −369.117 −0.829476
\(446\) 0 0
\(447\) 7.82031i 0.0174951i
\(448\) 0 0
\(449\) −439.019 −0.977771 −0.488886 0.872348i \(-0.662596\pi\)
−0.488886 + 0.872348i \(0.662596\pi\)
\(450\) 0 0
\(451\) 52.7296i 0.116917i
\(452\) 0 0
\(453\) −5.82843 −0.0128663
\(454\) 0 0
\(455\) 380.696 0.836695
\(456\) 0 0
\(457\) 40.1654i 0.0878893i −0.999034 0.0439447i \(-0.986007\pi\)
0.999034 0.0439447i \(-0.0139925\pi\)
\(458\) 0 0
\(459\) 507.189i 1.10499i
\(460\) 0 0
\(461\) −65.6762 −0.142465 −0.0712323 0.997460i \(-0.522693\pi\)
−0.0712323 + 0.997460i \(0.522693\pi\)
\(462\) 0 0
\(463\) 673.436 1.45450 0.727252 0.686370i \(-0.240797\pi\)
0.727252 + 0.686370i \(0.240797\pi\)
\(464\) 0 0
\(465\) 757.602i 1.62925i
\(466\) 0 0
\(467\) 310.685i 0.665278i −0.943054 0.332639i \(-0.892061\pi\)
0.943054 0.332639i \(-0.107939\pi\)
\(468\) 0 0
\(469\) 267.338 0.570017
\(470\) 0 0
\(471\) 177.739i 0.377366i
\(472\) 0 0
\(473\) −117.068 −0.247501
\(474\) 0 0
\(475\) 1711.82i 3.60384i
\(476\) 0 0
\(477\) 194.166i 0.407057i
\(478\) 0 0
\(479\) 230.469i 0.481146i −0.970631 0.240573i \(-0.922665\pi\)
0.970631 0.240573i \(-0.0773354\pi\)
\(480\) 0 0
\(481\) 406.148i 0.844383i
\(482\) 0 0
\(483\) 215.137 29.2686i 0.445418 0.0605976i
\(484\) 0 0
\(485\) 623.789 1.28616
\(486\) 0 0
\(487\) −372.360 −0.764599 −0.382299 0.924039i \(-0.624868\pi\)
−0.382299 + 0.924039i \(0.624868\pi\)
\(488\) 0 0
\(489\) −303.191 −0.620022
\(490\) 0 0
\(491\) 258.026 0.525512 0.262756 0.964862i \(-0.415369\pi\)
0.262756 + 0.964862i \(0.415369\pi\)
\(492\) 0 0
\(493\) 606.563i 1.23035i
\(494\) 0 0
\(495\) 117.068 0.236501
\(496\) 0 0
\(497\) 435.071i 0.875395i
\(498\) 0 0
\(499\) −372.747 −0.746988 −0.373494 0.927632i \(-0.621840\pi\)
−0.373494 + 0.927632i \(0.621840\pi\)
\(500\) 0 0
\(501\) −224.108 −0.447321
\(502\) 0 0
\(503\) 611.191i 1.21509i −0.794284 0.607546i \(-0.792154\pi\)
0.794284 0.607546i \(-0.207846\pi\)
\(504\) 0 0
\(505\) 956.627i 1.89431i
\(506\) 0 0
\(507\) 151.196 0.298217
\(508\) 0 0
\(509\) −357.985 −0.703310 −0.351655 0.936130i \(-0.614381\pi\)
−0.351655 + 0.936130i \(0.614381\pi\)
\(510\) 0 0
\(511\) 187.112i 0.366168i
\(512\) 0 0
\(513\) 784.580i 1.52940i
\(514\) 0 0
\(515\) 958.617 1.86139
\(516\) 0 0
\(517\) 121.378i 0.234773i
\(518\) 0 0
\(519\) −16.2010 −0.0312158
\(520\) 0 0
\(521\) 123.227i 0.236521i −0.992983 0.118261i \(-0.962268\pi\)
0.992983 0.118261i \(-0.0377318\pi\)
\(522\) 0 0
\(523\) 246.110i 0.470573i −0.971926 0.235286i \(-0.924397\pi\)
0.971926 0.235286i \(-0.0756028\pi\)
\(524\) 0 0
\(525\) 605.221i 1.15280i
\(526\) 0 0
\(527\) 573.777i 1.08876i
\(528\) 0 0
\(529\) −509.774 + 141.321i −0.963655 + 0.267148i
\(530\) 0 0
\(531\) 83.0984 0.156494
\(532\) 0 0
\(533\) −139.083 −0.260944
\(534\) 0 0
\(535\) 712.902 1.33253
\(536\) 0 0
\(537\) −671.784 −1.25099
\(538\) 0 0
\(539\) 131.814i 0.244553i
\(540\) 0 0
\(541\) 425.922 0.787286 0.393643 0.919263i \(-0.371215\pi\)
0.393643 + 0.919263i \(0.371215\pi\)
\(542\) 0 0
\(543\) 129.591i 0.238657i
\(544\) 0 0
\(545\) 1308.74 2.40135
\(546\) 0 0
\(547\) −715.070 −1.30726 −0.653629 0.756815i \(-0.726754\pi\)
−0.653629 + 0.756815i \(0.726754\pi\)
\(548\) 0 0
\(549\) 49.6054i 0.0903559i
\(550\) 0 0
\(551\) 938.303i 1.70291i
\(552\) 0 0
\(553\) −460.853 −0.833369
\(554\) 0 0
\(555\) 897.460 1.61705
\(556\) 0 0
\(557\) 325.539i 0.584451i 0.956349 + 0.292226i \(0.0943958\pi\)
−0.956349 + 0.292226i \(0.905604\pi\)
\(558\) 0 0
\(559\) 308.787i 0.552392i
\(560\) 0 0
\(561\) −162.936 −0.290439
\(562\) 0 0
\(563\) 636.780i 1.13105i 0.824732 + 0.565524i \(0.191326\pi\)
−0.824732 + 0.565524i \(0.808674\pi\)
\(564\) 0 0
\(565\) −1705.81 −3.01913
\(566\) 0 0
\(567\) 165.779i 0.292379i
\(568\) 0 0
\(569\) 359.226i 0.631329i −0.948871 0.315665i \(-0.897773\pi\)
0.948871 0.315665i \(-0.102227\pi\)
\(570\) 0 0
\(571\) 766.371i 1.34216i 0.741387 + 0.671078i \(0.234168\pi\)
−0.741387 + 0.671078i \(0.765832\pi\)
\(572\) 0 0
\(573\) 407.260i 0.710750i
\(574\) 0 0
\(575\) 198.782 + 1461.13i 0.345707 + 2.54110i
\(576\) 0 0
\(577\) −966.661 −1.67532 −0.837661 0.546190i \(-0.816078\pi\)
−0.837661 + 0.546190i \(0.816078\pi\)
\(578\) 0 0
\(579\) 230.120 0.397444
\(580\) 0 0
\(581\) −292.670 −0.503735
\(582\) 0 0
\(583\) 239.383 0.410605
\(584\) 0 0
\(585\) 308.787i 0.527841i
\(586\) 0 0
\(587\) −855.933 −1.45815 −0.729074 0.684435i \(-0.760049\pi\)
−0.729074 + 0.684435i \(0.760049\pi\)
\(588\) 0 0
\(589\) 887.586i 1.50694i
\(590\) 0 0
\(591\) 601.340 1.01750
\(592\) 0 0
\(593\) 159.563 0.269078 0.134539 0.990908i \(-0.457045\pi\)
0.134539 + 0.990908i \(0.457045\pi\)
\(594\) 0 0
\(595\) 637.106i 1.07077i
\(596\) 0 0
\(597\) 183.940i 0.308107i
\(598\) 0 0
\(599\) 46.1320 0.0770151 0.0385075 0.999258i \(-0.487740\pi\)
0.0385075 + 0.999258i \(0.487740\pi\)
\(600\) 0 0
\(601\) −283.010 −0.470899 −0.235449 0.971887i \(-0.575656\pi\)
−0.235449 + 0.971887i \(0.575656\pi\)
\(602\) 0 0
\(603\) 216.841i 0.359604i
\(604\) 0 0
\(605\) 997.904i 1.64943i
\(606\) 0 0
\(607\) −690.250 −1.13715 −0.568575 0.822632i \(-0.692505\pi\)
−0.568575 + 0.822632i \(0.692505\pi\)
\(608\) 0 0
\(609\) 331.740i 0.544729i
\(610\) 0 0
\(611\) −320.154 −0.523984
\(612\) 0 0
\(613\) 247.058i 0.403032i 0.979485 + 0.201516i \(0.0645867\pi\)
−0.979485 + 0.201516i \(0.935413\pi\)
\(614\) 0 0
\(615\) 307.330i 0.499724i
\(616\) 0 0
\(617\) 548.303i 0.888660i −0.895863 0.444330i \(-0.853442\pi\)
0.895863 0.444330i \(-0.146558\pi\)
\(618\) 0 0
\(619\) 1090.68i 1.76201i −0.473108 0.881005i \(-0.656868\pi\)
0.473108 0.881005i \(-0.343132\pi\)
\(620\) 0 0
\(621\) −91.1076 669.681i −0.146711 1.07839i
\(622\) 0 0
\(623\) −152.893 −0.245414
\(624\) 0 0
\(625\) 1882.62 3.01219
\(626\) 0 0
\(627\) 252.049 0.401992
\(628\) 0 0
\(629\) 679.700 1.08060
\(630\) 0 0
\(631\) 132.159i 0.209444i −0.994502 0.104722i \(-0.966605\pi\)
0.994502 0.104722i \(-0.0333953\pi\)
\(632\) 0 0
\(633\) 171.823 0.271443
\(634\) 0 0
\(635\) 343.518i 0.540974i
\(636\) 0 0
\(637\) −347.682 −0.545812
\(638\) 0 0
\(639\) −352.891 −0.552256
\(640\) 0 0
\(641\) 349.786i 0.545688i 0.962058 + 0.272844i \(0.0879644\pi\)
−0.962058 + 0.272844i \(0.912036\pi\)
\(642\) 0 0
\(643\) 147.129i 0.228817i 0.993434 + 0.114408i \(0.0364972\pi\)
−0.993434 + 0.114408i \(0.963503\pi\)
\(644\) 0 0
\(645\) 682.323 1.05787
\(646\) 0 0
\(647\) −966.252 −1.49343 −0.746717 0.665142i \(-0.768371\pi\)
−0.746717 + 0.665142i \(0.768371\pi\)
\(648\) 0 0
\(649\) 102.450i 0.157858i
\(650\) 0 0
\(651\) 313.809i 0.482041i
\(652\) 0 0
\(653\) 817.378 1.25173 0.625863 0.779933i \(-0.284747\pi\)
0.625863 + 0.779933i \(0.284747\pi\)
\(654\) 0 0
\(655\) 617.229i 0.942335i
\(656\) 0 0
\(657\) −151.769 −0.231003
\(658\) 0 0
\(659\) 1053.82i 1.59913i −0.600581 0.799564i \(-0.705064\pi\)
0.600581 0.799564i \(-0.294936\pi\)
\(660\) 0 0
\(661\) 1162.41i 1.75856i −0.476305 0.879280i \(-0.658024\pi\)
0.476305 0.879280i \(-0.341976\pi\)
\(662\) 0 0
\(663\) 429.772i 0.648223i
\(664\) 0 0
\(665\) 985.550i 1.48203i
\(666\) 0 0
\(667\) 108.958 + 800.891i 0.163356 + 1.20074i
\(668\) 0 0
\(669\) −202.995 −0.303430
\(670\) 0 0
\(671\) −61.1573 −0.0911435
\(672\) 0 0
\(673\) −647.461 −0.962052 −0.481026 0.876706i \(-0.659736\pi\)
−0.481026 + 0.876706i \(0.659736\pi\)
\(674\) 0 0
\(675\) −1883.94 −2.79102
\(676\) 0 0
\(677\) 544.278i 0.803956i −0.915649 0.401978i \(-0.868323\pi\)
0.915649 0.401978i \(-0.131677\pi\)
\(678\) 0 0
\(679\) 258.382 0.380533
\(680\) 0 0
\(681\) 635.160i 0.932688i
\(682\) 0 0
\(683\) 77.8314 0.113955 0.0569776 0.998375i \(-0.481854\pi\)
0.0569776 + 0.998375i \(0.481854\pi\)
\(684\) 0 0
\(685\) 621.166 0.906811
\(686\) 0 0
\(687\) 341.180i 0.496623i
\(688\) 0 0
\(689\) 631.413i 0.916419i
\(690\) 0 0
\(691\) −223.769 −0.323833 −0.161917 0.986804i \(-0.551768\pi\)
−0.161917 + 0.986804i \(0.551768\pi\)
\(692\) 0 0
\(693\) 48.4912 0.0699729
\(694\) 0 0
\(695\) 277.621i 0.399455i
\(696\) 0 0
\(697\) 232.760i 0.333945i
\(698\) 0 0
\(699\) 854.110 1.22190
\(700\) 0 0
\(701\) 376.995i 0.537795i −0.963169 0.268898i \(-0.913341\pi\)
0.963169 0.268898i \(-0.0866594\pi\)
\(702\) 0 0
\(703\) −1051.44 −1.49565
\(704\) 0 0
\(705\) 707.441i 1.00346i
\(706\) 0 0
\(707\) 396.248i 0.560464i
\(708\) 0 0
\(709\) 1182.12i 1.66731i 0.552287 + 0.833654i \(0.313755\pi\)
−0.552287 + 0.833654i \(0.686245\pi\)
\(710\) 0 0
\(711\) 373.803i 0.525743i
\(712\) 0 0
\(713\) 103.069 + 757.602i 0.144557 + 1.06256i
\(714\) 0 0
\(715\) 380.696 0.532443
\(716\) 0 0
\(717\) −657.357 −0.916817
\(718\) 0 0
\(719\) −93.9941 −0.130729 −0.0653645 0.997861i \(-0.520821\pi\)
−0.0653645 + 0.997861i \(0.520821\pi\)
\(720\) 0 0
\(721\) 397.072 0.550724
\(722\) 0 0
\(723\) 1071.13i 1.48151i
\(724\) 0 0
\(725\) 2253.06 3.10767
\(726\) 0 0
\(727\) 1303.61i 1.79314i −0.442900 0.896571i \(-0.646050\pi\)
0.442900 0.896571i \(-0.353950\pi\)
\(728\) 0 0
\(729\) 772.935 1.06027
\(730\) 0 0
\(731\) 516.764 0.706927
\(732\) 0 0
\(733\) 635.946i 0.867594i 0.901011 + 0.433797i \(0.142826\pi\)
−0.901011 + 0.433797i \(0.857174\pi\)
\(734\) 0 0
\(735\) 768.268i 1.04526i
\(736\) 0 0
\(737\) 267.338 0.362738
\(738\) 0 0
\(739\) 200.380 0.271150 0.135575 0.990767i \(-0.456712\pi\)
0.135575 + 0.990767i \(0.456712\pi\)
\(740\) 0 0
\(741\) 664.822i 0.897196i
\(742\) 0 0
\(743\) 1210.79i 1.62959i −0.579748 0.814796i \(-0.696849\pi\)
0.579748 0.814796i \(-0.303151\pi\)
\(744\) 0 0
\(745\) −30.5786 −0.0410452
\(746\) 0 0
\(747\) 237.388i 0.317789i
\(748\) 0 0
\(749\) 295.294 0.394250
\(750\) 0 0
\(751\) 983.883i 1.31010i 0.755587 + 0.655048i \(0.227352\pi\)
−0.755587 + 0.655048i \(0.772648\pi\)
\(752\) 0 0
\(753\) 469.822i 0.623934i
\(754\) 0 0
\(755\) 22.7901i 0.0301855i
\(756\) 0 0
\(757\) 1494.15i 1.97378i 0.161407 + 0.986888i \(0.448397\pi\)
−0.161407 + 0.986888i \(0.551603\pi\)
\(758\) 0 0
\(759\) 215.137 29.2686i 0.283448 0.0385621i
\(760\) 0 0
\(761\) −779.132 −1.02383 −0.511913 0.859037i \(-0.671063\pi\)
−0.511913 + 0.859037i \(0.671063\pi\)
\(762\) 0 0
\(763\) 542.096 0.710479
\(764\) 0 0
\(765\) −516.764 −0.675508
\(766\) 0 0
\(767\) 270.230 0.352320
\(768\) 0 0
\(769\) 155.017i 0.201582i −0.994908 0.100791i \(-0.967863\pi\)
0.994908 0.100791i \(-0.0321374\pi\)
\(770\) 0 0
\(771\) 53.4437 0.0693173
\(772\) 0 0
\(773\) 97.8690i 0.126609i −0.997994 0.0633047i \(-0.979836\pi\)
0.997994 0.0633047i \(-0.0201640\pi\)
\(774\) 0 0
\(775\) 2131.28 2.75003
\(776\) 0 0
\(777\) 371.740 0.478430
\(778\) 0 0
\(779\) 360.060i 0.462208i
\(780\) 0 0
\(781\) 435.071i 0.557070i
\(782\) 0 0
\(783\) −1032.64 −1.31883
\(784\) 0 0
\(785\) 694.989 0.885336
\(786\) 0 0
\(787\) 728.726i 0.925954i 0.886370 + 0.462977i \(0.153219\pi\)
−0.886370 + 0.462977i \(0.846781\pi\)
\(788\) 0 0
\(789\) 1026.22i 1.30066i
\(790\) 0 0
\(791\) −706.569 −0.893260
\(792\) 0 0
\(793\) 161.313i 0.203421i
\(794\) 0 0
\(795\) −1395.22 −1.75500
\(796\) 0 0
\(797\) 279.729i 0.350978i 0.984481 + 0.175489i \(0.0561506\pi\)
−0.984481 + 0.175489i \(0.943849\pi\)
\(798\) 0 0
\(799\) 535.787i 0.670572i
\(800\) 0 0
\(801\) 124.013i 0.154823i
\(802\) 0 0
\(803\) 187.112i 0.233016i
\(804\) 0 0
\(805\) 114.445 + 841.220i 0.142168 + 1.04499i
\(806\) 0 0
\(807\) 992.762 1.23019
\(808\) 0 0
\(809\) 1156.82 1.42994 0.714968 0.699157i \(-0.246441\pi\)
0.714968 + 0.699157i \(0.246441\pi\)
\(810\) 0 0
\(811\) 1178.54 1.45319 0.726594 0.687067i \(-0.241102\pi\)
0.726594 + 0.687067i \(0.241102\pi\)
\(812\) 0 0
\(813\) −238.569 −0.293442
\(814\) 0 0
\(815\) 1185.52i 1.45463i
\(816\) 0 0
\(817\) −799.391 −0.978447
\(818\) 0 0
\(819\) 127.904i 0.156171i
\(820\) 0 0
\(821\) 732.347 0.892019 0.446009 0.895028i \(-0.352845\pi\)
0.446009 + 0.895028i \(0.352845\pi\)
\(822\) 0 0
\(823\) 882.012 1.07170 0.535852 0.844312i \(-0.319991\pi\)
0.535852 + 0.844312i \(0.319991\pi\)
\(824\) 0 0
\(825\) 605.221i 0.733601i
\(826\) 0 0
\(827\) 1039.74i 1.25724i 0.777713 + 0.628619i \(0.216380\pi\)
−0.777713 + 0.628619i \(0.783620\pi\)
\(828\) 0 0
\(829\) −413.505 −0.498799 −0.249400 0.968401i \(-0.580233\pi\)
−0.249400 + 0.968401i \(0.580233\pi\)
\(830\) 0 0
\(831\) −500.512 −0.602301
\(832\) 0 0
\(833\) 581.855i 0.698506i
\(834\) 0 0
\(835\) 876.296i 1.04946i
\(836\) 0 0
\(837\) −976.828 −1.16706
\(838\) 0 0
\(839\) 254.486i 0.303320i −0.988433 0.151660i \(-0.951538\pi\)
0.988433 0.151660i \(-0.0484619\pi\)
\(840\) 0 0
\(841\) 393.970 0.468454
\(842\) 0 0
\(843\) 74.8487i 0.0887885i
\(844\) 0 0
\(845\) 591.200i 0.699645i
\(846\) 0 0
\(847\) 413.345i 0.488011i
\(848\) 0 0
\(849\) 545.342i 0.642334i
\(850\) 0 0
\(851\) −897.460 + 122.096i −1.05459 + 0.143474i
\(852\) 0 0
\(853\) −744.132 −0.872370 −0.436185 0.899857i \(-0.643671\pi\)
−0.436185 + 0.899857i \(0.643671\pi\)
\(854\) 0 0
\(855\) 799.391 0.934960
\(856\) 0 0
\(857\) −578.921 −0.675520 −0.337760 0.941232i \(-0.609669\pi\)
−0.337760 + 0.941232i \(0.609669\pi\)
\(858\) 0 0
\(859\) −1395.20 −1.62421 −0.812106 0.583510i \(-0.801679\pi\)
−0.812106 + 0.583510i \(0.801679\pi\)
\(860\) 0 0
\(861\) 127.300i 0.147852i
\(862\) 0 0
\(863\) −565.704 −0.655508 −0.327754 0.944763i \(-0.606292\pi\)
−0.327754 + 0.944763i \(0.606292\pi\)
\(864\) 0 0
\(865\) 63.3485i 0.0732352i
\(866\) 0 0
\(867\) 21.5269 0.0248292
\(868\) 0 0
\(869\) −460.853 −0.530325
\(870\) 0 0
\(871\) 705.150i 0.809587i
\(872\) 0 0
\(873\) 209.576i 0.240065i
\(874\) 0 0
\(875\) 1443.72 1.64996
\(876\) 0 0
\(877\) 468.112 0.533765 0.266882 0.963729i \(-0.414006\pi\)
0.266882 + 0.963729i \(0.414006\pi\)
\(878\) 0 0
\(879\) 391.341i 0.445212i
\(880\) 0 0
\(881\) 1317.52i 1.49548i −0.663990 0.747741i \(-0.731138\pi\)
0.663990 0.747741i \(-0.268862\pi\)
\(882\) 0 0
\(883\) −257.377 −0.291480 −0.145740 0.989323i \(-0.546556\pi\)
−0.145740 + 0.989323i \(0.546556\pi\)
\(884\) 0 0
\(885\) 597.123i 0.674715i
\(886\) 0 0
\(887\) −331.722 −0.373982 −0.186991 0.982362i \(-0.559874\pi\)
−0.186991 + 0.982362i \(0.559874\pi\)
\(888\) 0 0
\(889\) 142.290i 0.160056i
\(890\) 0 0
\(891\) 165.779i 0.186059i
\(892\) 0 0
\(893\) 828.818i 0.928128i
\(894\) 0 0
\(895\) 2626.78i 2.93495i
\(896\) 0 0
\(897\) −77.2010 567.461i −0.0860658 0.632621i
\(898\) 0 0
\(899\) 1168.22 1.29946
\(900\) 0 0
\(901\) −1056.69 −1.17279
\(902\) 0 0
\(903\) 282.627 0.312987
\(904\) 0 0
\(905\) 506.721 0.559912
\(906\) 0 0
\(907\) 436.596i 0.481362i 0.970604 + 0.240681i \(0.0773708\pi\)
−0.970604 + 0.240681i \(0.922629\pi\)
\(908\) 0 0
\(909\) 321.401 0.353577
\(910\) 0 0
\(911\) 1530.29i 1.67979i 0.542749 + 0.839895i \(0.317383\pi\)
−0.542749 + 0.839895i \(0.682617\pi\)
\(912\) 0 0
\(913\) −292.670 −0.320559
\(914\) 0 0
\(915\) 356.451 0.389564
\(916\) 0 0
\(917\) 255.665i 0.278806i
\(918\) 0 0
\(919\) 676.783i 0.736434i 0.929740 + 0.368217i \(0.120032\pi\)
−0.929740 + 0.368217i \(0.879968\pi\)
\(920\) 0 0
\(921\) −826.382 −0.897266
\(922\) 0 0
\(923\) −1147.58 −1.24331
\(924\) 0 0
\(925\) 2524.72i 2.72943i
\(926\) 0 0
\(927\) 322.070i 0.347432i
\(928\) 0 0
\(929\) −686.535 −0.739004 −0.369502 0.929230i \(-0.620472\pi\)
−0.369502 + 0.929230i \(0.620472\pi\)
\(930\) 0 0
\(931\) 900.082i 0.966791i
\(932\) 0 0
\(933\) 1121.77 1.20233
\(934\) 0 0
\(935\) 637.106i 0.681396i
\(936\) 0 0
\(937\) 774.354i 0.826418i −0.910636 0.413209i \(-0.864408\pi\)
0.910636 0.413209i \(-0.135592\pi\)
\(938\) 0 0
\(939\) 229.798i 0.244726i
\(940\) 0 0
\(941\) 774.814i 0.823395i 0.911321 + 0.411697i \(0.135064\pi\)
−0.911321 + 0.411697i \(0.864936\pi\)
\(942\) 0 0
\(943\) −41.8112 307.330i −0.0443385 0.325907i
\(944\) 0 0
\(945\) −1084.64 −1.14777
\(946\) 0 0
\(947\) −1090.69 −1.15173 −0.575865 0.817545i \(-0.695335\pi\)
−0.575865 + 0.817545i \(0.695335\pi\)
\(948\) 0 0
\(949\) −493.540 −0.520063
\(950\) 0 0
\(951\) −507.328 −0.533468
\(952\) 0 0
\(953\) 60.7603i 0.0637569i −0.999492 0.0318785i \(-0.989851\pi\)
0.999492 0.0318785i \(-0.0101489\pi\)
\(954\) 0 0
\(955\) 1592.45 1.66749
\(956\) 0 0
\(957\) 331.740i 0.346646i
\(958\) 0 0
\(959\) 257.295 0.268295
\(960\) 0 0
\(961\) 144.073 0.149920
\(962\) 0 0
\(963\) 239.516i 0.248719i
\(964\) 0 0
\(965\) 899.805i 0.932440i
\(966\) 0 0
\(967\) −1715.83 −1.77438 −0.887192 0.461400i \(-0.847347\pi\)
−0.887192 + 0.461400i \(0.847347\pi\)
\(968\) 0 0
\(969\) −1112.60 −1.14819
\(970\) 0 0
\(971\) 1071.57i 1.10358i −0.833984 0.551789i \(-0.813946\pi\)
0.833984 0.551789i \(-0.186054\pi\)
\(972\) 0 0
\(973\) 114.994i 0.118185i
\(974\) 0 0
\(975\) −1596.37 −1.63731
\(976\) 0 0
\(977\) 456.079i 0.466816i 0.972379 + 0.233408i \(0.0749877\pi\)
−0.972379 + 0.233408i \(0.925012\pi\)
\(978\) 0 0
\(979\) −152.893 −0.156173
\(980\) 0 0
\(981\) 439.700i 0.448216i
\(982\) 0 0
\(983\) 216.055i 0.219791i 0.993943 + 0.109896i \(0.0350517\pi\)
−0.993943 + 0.109896i \(0.964948\pi\)
\(984\) 0 0
\(985\) 2351.33i 2.38714i
\(986\) 0 0
\(987\) 293.032i 0.296891i
\(988\) 0 0
\(989\) −682.323 + 92.8276i −0.689912 + 0.0938600i
\(990\) 0 0
\(991\) 520.142 0.524866 0.262433 0.964950i \(-0.415475\pi\)
0.262433 + 0.964950i \(0.415475\pi\)
\(992\) 0 0
\(993\) −160.019 −0.161147
\(994\) 0 0
\(995\) 719.235 0.722849
\(996\) 0 0
\(997\) −466.877 −0.468282 −0.234141 0.972203i \(-0.575228\pi\)
−0.234141 + 0.972203i \(0.575228\pi\)
\(998\) 0 0
\(999\) 1157.16i 1.15831i
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 1472.3.f.c.321.1 4
4.3 odd 2 1472.3.f.f.321.3 4
8.3 odd 2 46.3.b.a.45.2 yes 4
8.5 even 2 368.3.f.c.321.4 4
23.22 odd 2 inner 1472.3.f.c.321.2 4
24.11 even 2 414.3.b.a.91.3 4
40.3 even 4 1150.3.c.a.1149.6 8
40.19 odd 2 1150.3.d.a.551.3 4
40.27 even 4 1150.3.c.a.1149.3 8
92.91 even 2 1472.3.f.f.321.4 4
184.45 odd 2 368.3.f.c.321.3 4
184.91 even 2 46.3.b.a.45.1 4
552.275 odd 2 414.3.b.a.91.4 4
920.459 even 2 1150.3.d.a.551.4 4
920.643 odd 4 1150.3.c.a.1149.5 8
920.827 odd 4 1150.3.c.a.1149.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
46.3.b.a.45.1 4 184.91 even 2
46.3.b.a.45.2 yes 4 8.3 odd 2
368.3.f.c.321.3 4 184.45 odd 2
368.3.f.c.321.4 4 8.5 even 2
414.3.b.a.91.3 4 24.11 even 2
414.3.b.a.91.4 4 552.275 odd 2
1150.3.c.a.1149.3 8 40.27 even 4
1150.3.c.a.1149.4 8 920.827 odd 4
1150.3.c.a.1149.5 8 920.643 odd 4
1150.3.c.a.1149.6 8 40.3 even 4
1150.3.d.a.551.3 4 40.19 odd 2
1150.3.d.a.551.4 4 920.459 even 2
1472.3.f.c.321.1 4 1.1 even 1 trivial
1472.3.f.c.321.2 4 23.22 odd 2 inner
1472.3.f.f.321.3 4 4.3 odd 2
1472.3.f.f.321.4 4 92.91 even 2