Properties

Label 1152.3.m.f.415.1
Level $1152$
Weight $3$
Character 1152.415
Analytic conductor $31.390$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,3,Mod(415,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.415");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.m (of order \(4\), degree \(2\), 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: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 415.1
Root \(-1.87459 + 0.697079i\) of defining polynomial
Character \(\chi\) \(=\) 1152.415
Dual form 1152.3.m.f.991.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-5.24354 - 5.24354i) q^{5} -5.32796 q^{7} +O(q^{10})\) \(q+(-5.24354 - 5.24354i) q^{5} -5.32796 q^{7} +(12.2863 - 12.2863i) q^{11} +(5.73657 - 5.73657i) q^{13} +23.3997 q^{17} +(-11.7492 - 11.7492i) q^{19} -5.80841 q^{23} +29.9894i q^{25} +(18.3914 - 18.3914i) q^{29} +16.9053i q^{31} +(27.9374 + 27.9374i) q^{35} +(-15.3391 - 15.3391i) q^{37} -29.2351i q^{41} +(-33.4099 + 33.4099i) q^{43} -18.2125i q^{47} -20.6128 q^{49} +(-66.9856 - 66.9856i) q^{53} -128.847 q^{55} +(-27.1523 + 27.1523i) q^{59} +(-65.2399 + 65.2399i) q^{61} -60.1599 q^{65} +(37.6951 + 37.6951i) q^{67} -42.6559 q^{71} -106.391i q^{73} +(-65.4607 + 65.4607i) q^{77} +21.2821i q^{79} +(24.1638 + 24.1638i) q^{83} +(-122.697 - 122.697i) q^{85} +52.8029i q^{89} +(-30.5643 + 30.5643i) q^{91} +123.215i q^{95} -21.0222 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{11} + 32 q^{19} + 128 q^{23} + 32 q^{29} + 96 q^{35} + 96 q^{37} - 160 q^{43} + 112 q^{49} - 160 q^{53} - 256 q^{55} - 128 q^{59} + 32 q^{61} + 32 q^{65} - 320 q^{67} - 512 q^{71} + 224 q^{77} - 160 q^{83} - 160 q^{85} + 480 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −5.24354 5.24354i −1.04871 1.04871i −0.998751 0.0499563i \(-0.984092\pi\)
−0.0499563 0.998751i \(-0.515908\pi\)
\(6\) 0 0
\(7\) −5.32796 −0.761138 −0.380569 0.924753i \(-0.624272\pi\)
−0.380569 + 0.924753i \(0.624272\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 12.2863 12.2863i 1.11693 1.11693i 0.124743 0.992189i \(-0.460189\pi\)
0.992189 0.124743i \(-0.0398107\pi\)
\(12\) 0 0
\(13\) 5.73657 5.73657i 0.441275 0.441275i −0.451165 0.892440i \(-0.648992\pi\)
0.892440 + 0.451165i \(0.148992\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 23.3997 1.37645 0.688226 0.725496i \(-0.258390\pi\)
0.688226 + 0.725496i \(0.258390\pi\)
\(18\) 0 0
\(19\) −11.7492 11.7492i −0.618380 0.618380i 0.326736 0.945116i \(-0.394051\pi\)
−0.945116 + 0.326736i \(0.894051\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.80841 −0.252540 −0.126270 0.991996i \(-0.540301\pi\)
−0.126270 + 0.991996i \(0.540301\pi\)
\(24\) 0 0
\(25\) 29.9894i 1.19958i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 18.3914 18.3914i 0.634185 0.634185i −0.314930 0.949115i \(-0.601981\pi\)
0.949115 + 0.314930i \(0.101981\pi\)
\(30\) 0 0
\(31\) 16.9053i 0.545332i 0.962109 + 0.272666i \(0.0879053\pi\)
−0.962109 + 0.272666i \(0.912095\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 27.9374 + 27.9374i 0.798211 + 0.798211i
\(36\) 0 0
\(37\) −15.3391 15.3391i −0.414571 0.414571i 0.468756 0.883327i \(-0.344702\pi\)
−0.883327 + 0.468756i \(0.844702\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 29.2351i 0.713051i −0.934286 0.356526i \(-0.883961\pi\)
0.934286 0.356526i \(-0.116039\pi\)
\(42\) 0 0
\(43\) −33.4099 + 33.4099i −0.776975 + 0.776975i −0.979315 0.202340i \(-0.935145\pi\)
0.202340 + 0.979315i \(0.435145\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 18.2125i 0.387500i −0.981051 0.193750i \(-0.937935\pi\)
0.981051 0.193750i \(-0.0620650\pi\)
\(48\) 0 0
\(49\) −20.6128 −0.420670
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −66.9856 66.9856i −1.26388 1.26388i −0.949197 0.314681i \(-0.898102\pi\)
−0.314681 0.949197i \(-0.601898\pi\)
\(54\) 0 0
\(55\) −128.847 −2.34267
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −27.1523 + 27.1523i −0.460209 + 0.460209i −0.898724 0.438515i \(-0.855505\pi\)
0.438515 + 0.898724i \(0.355505\pi\)
\(60\) 0 0
\(61\) −65.2399 + 65.2399i −1.06951 + 1.06951i −0.0721103 + 0.997397i \(0.522973\pi\)
−0.997397 + 0.0721103i \(0.977027\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −60.1599 −0.925537
\(66\) 0 0
\(67\) 37.6951 + 37.6951i 0.562614 + 0.562614i 0.930049 0.367435i \(-0.119764\pi\)
−0.367435 + 0.930049i \(0.619764\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −42.6559 −0.600788 −0.300394 0.953815i \(-0.597118\pi\)
−0.300394 + 0.953815i \(0.597118\pi\)
\(72\) 0 0
\(73\) 106.391i 1.45742i −0.684825 0.728708i \(-0.740121\pi\)
0.684825 0.728708i \(-0.259879\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −65.4607 + 65.4607i −0.850139 + 0.850139i
\(78\) 0 0
\(79\) 21.2821i 0.269394i 0.990887 + 0.134697i \(0.0430061\pi\)
−0.990887 + 0.134697i \(0.956994\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 24.1638 + 24.1638i 0.291130 + 0.291130i 0.837527 0.546396i \(-0.184001\pi\)
−0.546396 + 0.837527i \(0.684001\pi\)
\(84\) 0 0
\(85\) −122.697 122.697i −1.44350 1.44350i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 52.8029i 0.593291i 0.954988 + 0.296645i \(0.0958679\pi\)
−0.954988 + 0.296645i \(0.904132\pi\)
\(90\) 0 0
\(91\) −30.5643 + 30.5643i −0.335871 + 0.335871i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 123.215i 1.29700i
\(96\) 0 0
\(97\) −21.0222 −0.216724 −0.108362 0.994112i \(-0.534560\pi\)
−0.108362 + 0.994112i \(0.534560\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.24960 3.24960i −0.0321743 0.0321743i 0.690837 0.723011i \(-0.257242\pi\)
−0.723011 + 0.690837i \(0.757242\pi\)
\(102\) 0 0
\(103\) 105.112 1.02050 0.510252 0.860025i \(-0.329552\pi\)
0.510252 + 0.860025i \(0.329552\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −99.6160 + 99.6160i −0.930991 + 0.930991i −0.997768 0.0667770i \(-0.978728\pi\)
0.0667770 + 0.997768i \(0.478728\pi\)
\(108\) 0 0
\(109\) 108.050 108.050i 0.991282 0.991282i −0.00868078 0.999962i \(-0.502763\pi\)
0.999962 + 0.00868078i \(0.00276321\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 23.2835 0.206048 0.103024 0.994679i \(-0.467148\pi\)
0.103024 + 0.994679i \(0.467148\pi\)
\(114\) 0 0
\(115\) 30.4566 + 30.4566i 0.264840 + 0.264840i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −124.673 −1.04767
\(120\) 0 0
\(121\) 180.904i 1.49508i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 26.1621 26.1621i 0.209297 0.209297i
\(126\) 0 0
\(127\) 118.180i 0.930550i 0.885166 + 0.465275i \(0.154045\pi\)
−0.885166 + 0.465275i \(0.845955\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −69.2067 69.2067i −0.528296 0.528296i 0.391768 0.920064i \(-0.371863\pi\)
−0.920064 + 0.391768i \(0.871863\pi\)
\(132\) 0 0
\(133\) 62.5994 + 62.5994i 0.470672 + 0.470672i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 124.474i 0.908572i −0.890856 0.454286i \(-0.849894\pi\)
0.890856 0.454286i \(-0.150106\pi\)
\(138\) 0 0
\(139\) −169.014 + 169.014i −1.21593 + 1.21593i −0.246881 + 0.969046i \(0.579406\pi\)
−0.969046 + 0.246881i \(0.920594\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 140.962i 0.985749i
\(144\) 0 0
\(145\) −192.872 −1.33015
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 146.988 + 146.988i 0.986495 + 0.986495i 0.999910 0.0134145i \(-0.00427011\pi\)
−0.0134145 + 0.999910i \(0.504270\pi\)
\(150\) 0 0
\(151\) 75.5456 0.500302 0.250151 0.968207i \(-0.419520\pi\)
0.250151 + 0.968207i \(0.419520\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 88.6435 88.6435i 0.571893 0.571893i
\(156\) 0 0
\(157\) 81.5356 81.5356i 0.519335 0.519335i −0.398035 0.917370i \(-0.630308\pi\)
0.917370 + 0.398035i \(0.130308\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 30.9470 0.192217
\(162\) 0 0
\(163\) −55.8065 55.8065i −0.342371 0.342371i 0.514887 0.857258i \(-0.327834\pi\)
−0.857258 + 0.514887i \(0.827834\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 24.6339 0.147508 0.0737540 0.997276i \(-0.476502\pi\)
0.0737540 + 0.997276i \(0.476502\pi\)
\(168\) 0 0
\(169\) 103.183i 0.610553i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.88551 4.88551i 0.0282399 0.0282399i −0.692846 0.721086i \(-0.743643\pi\)
0.721086 + 0.692846i \(0.243643\pi\)
\(174\) 0 0
\(175\) 159.782i 0.913042i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −229.504 229.504i −1.28215 1.28215i −0.939444 0.342702i \(-0.888658\pi\)
−0.342702 0.939444i \(-0.611342\pi\)
\(180\) 0 0
\(181\) −116.607 116.607i −0.644238 0.644238i 0.307356 0.951595i \(-0.400556\pi\)
−0.951595 + 0.307356i \(0.900556\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 160.863i 0.869528i
\(186\) 0 0
\(187\) 287.495 287.495i 1.53740 1.53740i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 94.2316i 0.493359i 0.969097 + 0.246680i \(0.0793395\pi\)
−0.969097 + 0.246680i \(0.920660\pi\)
\(192\) 0 0
\(193\) 84.2667 0.436615 0.218308 0.975880i \(-0.429946\pi\)
0.218308 + 0.975880i \(0.429946\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −56.9578 56.9578i −0.289126 0.289126i 0.547609 0.836734i \(-0.315538\pi\)
−0.836734 + 0.547609i \(0.815538\pi\)
\(198\) 0 0
\(199\) −196.179 −0.985827 −0.492913 0.870078i \(-0.664068\pi\)
−0.492913 + 0.870078i \(0.664068\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −97.9886 + 97.9886i −0.482702 + 0.482702i
\(204\) 0 0
\(205\) −153.295 + 153.295i −0.747782 + 0.747782i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −288.708 −1.38138
\(210\) 0 0
\(211\) −177.340 177.340i −0.840475 0.840475i 0.148445 0.988921i \(-0.452573\pi\)
−0.988921 + 0.148445i \(0.952573\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 350.373 1.62964
\(216\) 0 0
\(217\) 90.0707i 0.415072i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 134.234 134.234i 0.607394 0.607394i
\(222\) 0 0
\(223\) 377.924i 1.69473i 0.531012 + 0.847364i \(0.321812\pi\)
−0.531012 + 0.847364i \(0.678188\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 103.909 + 103.909i 0.457750 + 0.457750i 0.897916 0.440166i \(-0.145080\pi\)
−0.440166 + 0.897916i \(0.645080\pi\)
\(228\) 0 0
\(229\) 101.055 + 101.055i 0.441290 + 0.441290i 0.892445 0.451156i \(-0.148988\pi\)
−0.451156 + 0.892445i \(0.648988\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 287.259i 1.23287i −0.787405 0.616436i \(-0.788576\pi\)
0.787405 0.616436i \(-0.211424\pi\)
\(234\) 0 0
\(235\) −95.4979 + 95.4979i −0.406374 + 0.406374i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 150.941i 0.631554i 0.948833 + 0.315777i \(0.102265\pi\)
−0.948833 + 0.315777i \(0.897735\pi\)
\(240\) 0 0
\(241\) 37.7817 0.156771 0.0783853 0.996923i \(-0.475024\pi\)
0.0783853 + 0.996923i \(0.475024\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 108.084 + 108.084i 0.441159 + 0.441159i
\(246\) 0 0
\(247\) −134.800 −0.545751
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 100.915 100.915i 0.402050 0.402050i −0.476905 0.878955i \(-0.658241\pi\)
0.878955 + 0.476905i \(0.158241\pi\)
\(252\) 0 0
\(253\) −71.3637 + 71.3637i −0.282070 + 0.282070i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −241.295 −0.938891 −0.469446 0.882961i \(-0.655546\pi\)
−0.469446 + 0.882961i \(0.655546\pi\)
\(258\) 0 0
\(259\) 81.7263 + 81.7263i 0.315546 + 0.315546i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 118.747 0.451509 0.225754 0.974184i \(-0.427515\pi\)
0.225754 + 0.974184i \(0.427515\pi\)
\(264\) 0 0
\(265\) 702.483i 2.65088i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.74853 7.74853i 0.0288050 0.0288050i −0.692558 0.721363i \(-0.743516\pi\)
0.721363 + 0.692558i \(0.243516\pi\)
\(270\) 0 0
\(271\) 131.899i 0.486712i −0.969937 0.243356i \(-0.921752\pi\)
0.969937 0.243356i \(-0.0782484\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 368.457 + 368.457i 1.33984 + 1.33984i
\(276\) 0 0
\(277\) 202.352 + 202.352i 0.730513 + 0.730513i 0.970721 0.240208i \(-0.0772157\pi\)
−0.240208 + 0.970721i \(0.577216\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 68.8493i 0.245015i 0.992468 + 0.122508i \(0.0390936\pi\)
−0.992468 + 0.122508i \(0.960906\pi\)
\(282\) 0 0
\(283\) −206.773 + 206.773i −0.730646 + 0.730646i −0.970748 0.240102i \(-0.922819\pi\)
0.240102 + 0.970748i \(0.422819\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 155.764i 0.542730i
\(288\) 0 0
\(289\) 258.545 0.894620
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −361.237 361.237i −1.23289 1.23289i −0.962848 0.270043i \(-0.912962\pi\)
−0.270043 0.962848i \(-0.587038\pi\)
\(294\) 0 0
\(295\) 284.749 0.965250
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −33.3204 + 33.3204i −0.111439 + 0.111439i
\(300\) 0 0
\(301\) 178.007 178.007i 0.591385 0.591385i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 684.176 2.24320
\(306\) 0 0
\(307\) 10.9073 + 10.9073i 0.0355286 + 0.0355286i 0.724648 0.689119i \(-0.242002\pi\)
−0.689119 + 0.724648i \(0.742002\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 160.251 0.515278 0.257639 0.966241i \(-0.417055\pi\)
0.257639 + 0.966241i \(0.417055\pi\)
\(312\) 0 0
\(313\) 355.500i 1.13578i −0.823103 0.567892i \(-0.807759\pi\)
0.823103 0.567892i \(-0.192241\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 72.5192 72.5192i 0.228767 0.228767i −0.583410 0.812178i \(-0.698282\pi\)
0.812178 + 0.583410i \(0.198282\pi\)
\(318\) 0 0
\(319\) 451.922i 1.41668i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −274.928 274.928i −0.851170 0.851170i
\(324\) 0 0
\(325\) 172.036 + 172.036i 0.529343 + 0.529343i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 97.0355i 0.294941i
\(330\) 0 0
\(331\) 248.096 248.096i 0.749536 0.749536i −0.224856 0.974392i \(-0.572191\pi\)
0.974392 + 0.224856i \(0.0721912\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 395.312i 1.18003i
\(336\) 0 0
\(337\) −467.271 −1.38656 −0.693280 0.720668i \(-0.743835\pi\)
−0.693280 + 0.720668i \(0.743835\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 207.703 + 207.703i 0.609098 + 0.609098i
\(342\) 0 0
\(343\) 370.894 1.08133
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 292.821 292.821i 0.843863 0.843863i −0.145496 0.989359i \(-0.546478\pi\)
0.989359 + 0.145496i \(0.0464776\pi\)
\(348\) 0 0
\(349\) −346.260 + 346.260i −0.992150 + 0.992150i −0.999969 0.00781941i \(-0.997511\pi\)
0.00781941 + 0.999969i \(0.497511\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8.01816 −0.0227143 −0.0113572 0.999936i \(-0.503615\pi\)
−0.0113572 + 0.999936i \(0.503615\pi\)
\(354\) 0 0
\(355\) 223.668 + 223.668i 0.630051 + 0.630051i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −590.403 −1.64458 −0.822289 0.569071i \(-0.807303\pi\)
−0.822289 + 0.569071i \(0.807303\pi\)
\(360\) 0 0
\(361\) 84.9121i 0.235213i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −557.867 + 557.867i −1.52840 + 1.52840i
\(366\) 0 0
\(367\) 397.100i 1.08202i −0.841017 0.541008i \(-0.818043\pi\)
0.841017 0.541008i \(-0.181957\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 356.897 + 356.897i 0.961986 + 0.961986i
\(372\) 0 0
\(373\) 165.010 + 165.010i 0.442387 + 0.442387i 0.892814 0.450427i \(-0.148728\pi\)
−0.450427 + 0.892814i \(0.648728\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 211.007i 0.559700i
\(378\) 0 0
\(379\) 206.669 206.669i 0.545300 0.545300i −0.379778 0.925078i \(-0.624000\pi\)
0.925078 + 0.379778i \(0.124000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 598.414i 1.56244i −0.624257 0.781219i \(-0.714598\pi\)
0.624257 0.781219i \(-0.285402\pi\)
\(384\) 0 0
\(385\) 686.492 1.78310
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 186.696 + 186.696i 0.479939 + 0.479939i 0.905112 0.425173i \(-0.139787\pi\)
−0.425173 + 0.905112i \(0.639787\pi\)
\(390\) 0 0
\(391\) −135.915 −0.347609
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 111.594 111.594i 0.282515 0.282515i
\(396\) 0 0
\(397\) 57.3727 57.3727i 0.144516 0.144516i −0.631147 0.775663i \(-0.717416\pi\)
0.775663 + 0.631147i \(0.217416\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 466.082 1.16230 0.581149 0.813797i \(-0.302603\pi\)
0.581149 + 0.813797i \(0.302603\pi\)
\(402\) 0 0
\(403\) 96.9784 + 96.9784i 0.240641 + 0.240641i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −376.921 −0.926096
\(408\) 0 0
\(409\) 597.952i 1.46198i 0.682386 + 0.730992i \(0.260942\pi\)
−0.682386 + 0.730992i \(0.739058\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 144.667 144.667i 0.350282 0.350282i
\(414\) 0 0
\(415\) 253.408i 0.610621i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.65301 4.65301i −0.0111050 0.0111050i 0.701532 0.712638i \(-0.252500\pi\)
−0.712638 + 0.701532i \(0.752500\pi\)
\(420\) 0 0
\(421\) −34.3754 34.3754i −0.0816519 0.0816519i 0.665101 0.746753i \(-0.268388\pi\)
−0.746753 + 0.665101i \(0.768388\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 701.742i 1.65116i
\(426\) 0 0
\(427\) 347.596 347.596i 0.814042 0.814042i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 423.823i 0.983347i −0.870780 0.491674i \(-0.836385\pi\)
0.870780 0.491674i \(-0.163615\pi\)
\(432\) 0 0
\(433\) 833.377 1.92466 0.962330 0.271885i \(-0.0876472\pi\)
0.962330 + 0.271885i \(0.0876472\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 68.2443 + 68.2443i 0.156165 + 0.156165i
\(438\) 0 0
\(439\) 32.3193 0.0736203 0.0368102 0.999322i \(-0.488280\pi\)
0.0368102 + 0.999322i \(0.488280\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 119.527 119.527i 0.269813 0.269813i −0.559212 0.829025i \(-0.688896\pi\)
0.829025 + 0.559212i \(0.188896\pi\)
\(444\) 0 0
\(445\) 276.874 276.874i 0.622189 0.622189i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 182.359 0.406146 0.203073 0.979164i \(-0.434907\pi\)
0.203073 + 0.979164i \(0.434907\pi\)
\(450\) 0 0
\(451\) −359.190 359.190i −0.796430 0.796430i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 320.530 0.704461
\(456\) 0 0
\(457\) 272.942i 0.597246i −0.954371 0.298623i \(-0.903473\pi\)
0.954371 0.298623i \(-0.0965274\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 188.323 188.323i 0.408510 0.408510i −0.472709 0.881219i \(-0.656724\pi\)
0.881219 + 0.472709i \(0.156724\pi\)
\(462\) 0 0
\(463\) 116.023i 0.250590i 0.992120 + 0.125295i \(0.0399877\pi\)
−0.992120 + 0.125295i \(0.960012\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −271.914 271.914i −0.582257 0.582257i 0.353266 0.935523i \(-0.385071\pi\)
−0.935523 + 0.353266i \(0.885071\pi\)
\(468\) 0 0
\(469\) −200.838 200.838i −0.428227 0.428227i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 820.966i 1.73566i
\(474\) 0 0
\(475\) 352.352 352.352i 0.741793 0.741793i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 775.808i 1.61964i 0.586678 + 0.809820i \(0.300435\pi\)
−0.586678 + 0.809820i \(0.699565\pi\)
\(480\) 0 0
\(481\) −175.988 −0.365880
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 110.231 + 110.231i 0.227280 + 0.227280i
\(486\) 0 0
\(487\) 174.891 0.359118 0.179559 0.983747i \(-0.442533\pi\)
0.179559 + 0.983747i \(0.442533\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −348.578 + 348.578i −0.709934 + 0.709934i −0.966521 0.256587i \(-0.917402\pi\)
0.256587 + 0.966521i \(0.417402\pi\)
\(492\) 0 0
\(493\) 430.352 430.352i 0.872926 0.872926i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 227.269 0.457282
\(498\) 0 0
\(499\) 607.544 + 607.544i 1.21752 + 1.21752i 0.968496 + 0.249027i \(0.0801109\pi\)
0.249027 + 0.968496i \(0.419889\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 130.935 0.260309 0.130154 0.991494i \(-0.458453\pi\)
0.130154 + 0.991494i \(0.458453\pi\)
\(504\) 0 0
\(505\) 34.0789i 0.0674829i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −61.5539 + 61.5539i −0.120931 + 0.120931i −0.764982 0.644051i \(-0.777252\pi\)
0.644051 + 0.764982i \(0.277252\pi\)
\(510\) 0 0
\(511\) 566.849i 1.10929i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −551.159 551.159i −1.07021 1.07021i
\(516\) 0 0
\(517\) −223.763 223.763i −0.432811 0.432811i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 32.5929i 0.0625584i 0.999511 + 0.0312792i \(0.00995810\pi\)
−0.999511 + 0.0312792i \(0.990042\pi\)
\(522\) 0 0
\(523\) 226.407 226.407i 0.432900 0.432900i −0.456713 0.889614i \(-0.650974\pi\)
0.889614 + 0.456713i \(0.150974\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 395.578i 0.750623i
\(528\) 0 0
\(529\) −495.262 −0.936224
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −167.709 167.709i −0.314652 0.314652i
\(534\) 0 0
\(535\) 1044.68 1.95267
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −253.254 + 253.254i −0.469859 + 0.469859i
\(540\) 0 0
\(541\) −510.912 + 510.912i −0.944385 + 0.944385i −0.998533 0.0541480i \(-0.982756\pi\)
0.0541480 + 0.998533i \(0.482756\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1133.13 −2.07913
\(546\) 0 0
\(547\) −512.889 512.889i −0.937639 0.937639i 0.0605271 0.998167i \(-0.480722\pi\)
−0.998167 + 0.0605271i \(0.980722\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −432.168 −0.784334
\(552\) 0 0
\(553\) 113.390i 0.205046i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 566.691 566.691i 1.01740 1.01740i 0.0175529 0.999846i \(-0.494412\pi\)
0.999846 0.0175529i \(-0.00558754\pi\)
\(558\) 0 0
\(559\) 383.317i 0.685720i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 548.653 + 548.653i 0.974517 + 0.974517i 0.999683 0.0251665i \(-0.00801159\pi\)
−0.0251665 + 0.999683i \(0.508012\pi\)
\(564\) 0 0
\(565\) −122.088 122.088i −0.216085 0.216085i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 551.224i 0.968760i −0.874858 0.484380i \(-0.839045\pi\)
0.874858 0.484380i \(-0.160955\pi\)
\(570\) 0 0
\(571\) −458.387 + 458.387i −0.802780 + 0.802780i −0.983529 0.180749i \(-0.942148\pi\)
0.180749 + 0.983529i \(0.442148\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 174.191i 0.302941i
\(576\) 0 0
\(577\) −718.488 −1.24521 −0.622607 0.782535i \(-0.713926\pi\)
−0.622607 + 0.782535i \(0.713926\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −128.744 128.744i −0.221590 0.221590i
\(582\) 0 0
\(583\) −1646.00 −2.82333
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.02450 3.02450i 0.00515247 0.00515247i −0.704526 0.709678i \(-0.748840\pi\)
0.709678 + 0.704526i \(0.248840\pi\)
\(588\) 0 0
\(589\) 198.624 198.624i 0.337222 0.337222i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −576.193 −0.971657 −0.485829 0.874054i \(-0.661482\pi\)
−0.485829 + 0.874054i \(0.661482\pi\)
\(594\) 0 0
\(595\) 653.726 + 653.726i 1.09870 + 1.09870i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1101.40 1.83873 0.919365 0.393406i \(-0.128703\pi\)
0.919365 + 0.393406i \(0.128703\pi\)
\(600\) 0 0
\(601\) 7.11053i 0.0118312i 0.999983 + 0.00591558i \(0.00188300\pi\)
−0.999983 + 0.00591558i \(0.998117\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −948.578 + 948.578i −1.56790 + 1.56790i
\(606\) 0 0
\(607\) 528.384i 0.870485i −0.900313 0.435242i \(-0.856663\pi\)
0.900313 0.435242i \(-0.143337\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −104.477 104.477i −0.170994 0.170994i
\(612\) 0 0
\(613\) 642.364 + 642.364i 1.04790 + 1.04790i 0.998793 + 0.0491093i \(0.0156383\pi\)
0.0491093 + 0.998793i \(0.484362\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1068.16i 1.73122i −0.500717 0.865611i \(-0.666930\pi\)
0.500717 0.865611i \(-0.333070\pi\)
\(618\) 0 0
\(619\) −691.136 + 691.136i −1.11654 + 1.11654i −0.124290 + 0.992246i \(0.539665\pi\)
−0.992246 + 0.124290i \(0.960335\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 281.332i 0.451576i
\(624\) 0 0
\(625\) 475.371 0.760594
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −358.931 358.931i −0.570637 0.570637i
\(630\) 0 0
\(631\) 486.622 0.771191 0.385596 0.922668i \(-0.373996\pi\)
0.385596 + 0.922668i \(0.373996\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 619.681 619.681i 0.975875 0.975875i
\(636\) 0 0
\(637\) −118.247 + 118.247i −0.185631 + 0.185631i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 691.017 1.07803 0.539015 0.842296i \(-0.318797\pi\)
0.539015 + 0.842296i \(0.318797\pi\)
\(642\) 0 0
\(643\) −652.605 652.605i −1.01494 1.01494i −0.999887 0.0150512i \(-0.995209\pi\)
−0.0150512 0.999887i \(-0.504791\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1156.72 −1.78782 −0.893911 0.448245i \(-0.852049\pi\)
−0.893911 + 0.448245i \(0.852049\pi\)
\(648\) 0 0
\(649\) 667.201i 1.02804i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −209.105 + 209.105i −0.320222 + 0.320222i −0.848852 0.528630i \(-0.822706\pi\)
0.528630 + 0.848852i \(0.322706\pi\)
\(654\) 0 0
\(655\) 725.776i 1.10806i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 533.902 + 533.902i 0.810170 + 0.810170i 0.984659 0.174489i \(-0.0558274\pi\)
−0.174489 + 0.984659i \(0.555827\pi\)
\(660\) 0 0
\(661\) −283.120 283.120i −0.428320 0.428320i 0.459736 0.888056i \(-0.347944\pi\)
−0.888056 + 0.459736i \(0.847944\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 656.484i 0.987195i
\(666\) 0 0
\(667\) −106.825 + 106.825i −0.160157 + 0.160157i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1603.11i 2.38913i
\(672\) 0 0
\(673\) −397.854 −0.591164 −0.295582 0.955317i \(-0.595514\pi\)
−0.295582 + 0.955317i \(0.595514\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 289.959 + 289.959i 0.428299 + 0.428299i 0.888049 0.459749i \(-0.152061\pi\)
−0.459749 + 0.888049i \(0.652061\pi\)
\(678\) 0 0
\(679\) 112.005 0.164956
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −150.197 + 150.197i −0.219908 + 0.219908i −0.808460 0.588551i \(-0.799698\pi\)
0.588551 + 0.808460i \(0.299698\pi\)
\(684\) 0 0
\(685\) −652.686 + 652.686i −0.952826 + 0.952826i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −768.535 −1.11544
\(690\) 0 0
\(691\) −791.212 791.212i −1.14502 1.14502i −0.987518 0.157506i \(-0.949655\pi\)
−0.157506 0.987518i \(-0.550345\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1772.46 2.55030
\(696\) 0 0
\(697\) 684.092i 0.981481i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 900.201 900.201i 1.28417 1.28417i 0.345893 0.938274i \(-0.387576\pi\)
0.938274 0.345893i \(-0.112424\pi\)
\(702\) 0 0
\(703\) 360.445i 0.512724i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 17.3138 + 17.3138i 0.0244891 + 0.0244891i
\(708\) 0 0
\(709\) −128.490 128.490i −0.181227 0.181227i 0.610663 0.791891i \(-0.290903\pi\)
−0.791891 + 0.610663i \(0.790903\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 98.1928i 0.137718i
\(714\) 0 0
\(715\) −739.140 + 739.140i −1.03376 + 1.03376i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1246.14i 1.73315i −0.499045 0.866576i \(-0.666316\pi\)
0.499045 0.866576i \(-0.333684\pi\)
\(720\) 0 0
\(721\) −560.033 −0.776745
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 551.546 + 551.546i 0.760753 + 0.760753i
\(726\) 0 0
\(727\) −1130.07 −1.55443 −0.777216 0.629234i \(-0.783369\pi\)
−0.777216 + 0.629234i \(0.783369\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −781.782 + 781.782i −1.06947 + 1.06947i
\(732\) 0 0
\(733\) 708.087 708.087i 0.966012 0.966012i −0.0334292 0.999441i \(-0.510643\pi\)
0.999441 + 0.0334292i \(0.0106428\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 926.264 1.25680
\(738\) 0 0
\(739\) 32.7516 + 32.7516i 0.0443188 + 0.0443188i 0.728919 0.684600i \(-0.240023\pi\)
−0.684600 + 0.728919i \(0.740023\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −708.128 −0.953066 −0.476533 0.879157i \(-0.658107\pi\)
−0.476533 + 0.879157i \(0.658107\pi\)
\(744\) 0 0
\(745\) 1541.47i 2.06909i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 530.751 530.751i 0.708612 0.708612i
\(750\) 0 0
\(751\) 1242.37i 1.65429i −0.561990 0.827144i \(-0.689964\pi\)
0.561990 0.827144i \(-0.310036\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −396.127 396.127i −0.524671 0.524671i
\(756\) 0 0
\(757\) 311.304 + 311.304i 0.411233 + 0.411233i 0.882168 0.470935i \(-0.156083\pi\)
−0.470935 + 0.882168i \(0.656083\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 179.137i 0.235397i 0.993049 + 0.117699i \(0.0375517\pi\)
−0.993049 + 0.117699i \(0.962448\pi\)
\(762\) 0 0
\(763\) −575.685 + 575.685i −0.754502 + 0.754502i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 311.523i 0.406158i
\(768\) 0 0
\(769\) −967.409 −1.25801 −0.629005 0.777402i \(-0.716537\pi\)
−0.629005 + 0.777402i \(0.716537\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −96.7342 96.7342i −0.125141 0.125141i 0.641762 0.766904i \(-0.278204\pi\)
−0.766904 + 0.641762i \(0.778204\pi\)
\(774\) 0 0
\(775\) −506.979 −0.654166
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −343.489 + 343.489i −0.440936 + 0.440936i
\(780\) 0 0
\(781\) −524.082 + 524.082i −0.671039 + 0.671039i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −855.070 −1.08926
\(786\) 0 0
\(787\) 381.038 + 381.038i 0.484166 + 0.484166i 0.906459 0.422293i \(-0.138775\pi\)
−0.422293 + 0.906459i \(0.638775\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −124.054 −0.156831
\(792\) 0 0
\(793\) 748.507i 0.943893i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −371.148 + 371.148i −0.465681 + 0.465681i −0.900512 0.434831i \(-0.856808\pi\)
0.434831 + 0.900512i \(0.356808\pi\)
\(798\) 0 0
\(799\) 426.167i 0.533375i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1307.15 1307.15i −1.62783 1.62783i
\(804\) 0 0
\(805\) −162.272 162.272i −0.201580 0.201580i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 309.566i 0.382653i −0.981526 0.191326i \(-0.938721\pi\)
0.981526 0.191326i \(-0.0612789\pi\)
\(810\) 0 0
\(811\) 27.2916 27.2916i 0.0336517 0.0336517i −0.690081 0.723732i \(-0.742425\pi\)
0.723732 + 0.690081i \(0.242425\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 585.247i 0.718095i
\(816\) 0 0
\(817\) 785.081 0.960931
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −879.903 879.903i −1.07175 1.07175i −0.997219 0.0745264i \(-0.976255\pi\)
−0.0745264 0.997219i \(-0.523745\pi\)
\(822\) 0 0
\(823\) 68.6842 0.0834559 0.0417280 0.999129i \(-0.486714\pi\)
0.0417280 + 0.999129i \(0.486714\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 942.097 942.097i 1.13917 1.13917i 0.150575 0.988599i \(-0.451887\pi\)
0.988599 0.150575i \(-0.0481126\pi\)
\(828\) 0 0
\(829\) 568.532 568.532i 0.685805 0.685805i −0.275497 0.961302i \(-0.588842\pi\)
0.961302 + 0.275497i \(0.0888424\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −482.333 −0.579031
\(834\) 0 0
\(835\) −129.169 129.169i −0.154693 0.154693i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1346.87 1.60533 0.802666 0.596429i \(-0.203414\pi\)
0.802666 + 0.596429i \(0.203414\pi\)
\(840\) 0 0
\(841\) 164.515i 0.195618i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 541.046 541.046i 0.640291 0.640291i
\(846\) 0 0
\(847\) 963.851i 1.13796i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 89.0960 + 89.0960i 0.104696 + 0.104696i
\(852\) 0 0
\(853\) −74.4816 74.4816i −0.0873172 0.0873172i 0.662099 0.749416i \(-0.269666\pi\)
−0.749416 + 0.662099i \(0.769666\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 53.7221i 0.0626862i 0.999509 + 0.0313431i \(0.00997845\pi\)
−0.999509 + 0.0313431i \(0.990022\pi\)
\(858\) 0 0
\(859\) 537.704 537.704i 0.625965 0.625965i −0.321085 0.947050i \(-0.604048\pi\)
0.947050 + 0.321085i \(0.104048\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1390.97i 1.61178i −0.592064 0.805891i \(-0.701687\pi\)
0.592064 0.805891i \(-0.298313\pi\)
\(864\) 0 0
\(865\) −51.2347 −0.0592309
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 261.477 + 261.477i 0.300895 + 0.300895i
\(870\) 0 0
\(871\) 432.482 0.496535
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −139.391 + 139.391i −0.159303 + 0.159303i
\(876\) 0 0
\(877\) −940.115 + 940.115i −1.07197 + 1.07197i −0.0747652 + 0.997201i \(0.523821\pi\)
−0.997201 + 0.0747652i \(0.976179\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 140.985 0.160029 0.0800143 0.996794i \(-0.474503\pi\)
0.0800143 + 0.996794i \(0.474503\pi\)
\(882\) 0 0
\(883\) 482.231 + 482.231i 0.546127 + 0.546127i 0.925318 0.379191i \(-0.123798\pi\)
−0.379191 + 0.925318i \(0.623798\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −266.180 −0.300091 −0.150045 0.988679i \(-0.547942\pi\)
−0.150045 + 0.988679i \(0.547942\pi\)
\(888\) 0 0
\(889\) 629.658i 0.708277i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −213.982 + 213.982i −0.239622 + 0.239622i
\(894\) 0 0
\(895\) 2406.83i 2.68919i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 310.911 + 310.911i 0.345841 + 0.345841i
\(900\) 0 0
\(901\) −1567.44 1567.44i −1.73967 1.73967i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1222.87i 1.35124i
\(906\) 0 0
\(907\) 303.117 303.117i 0.334197 0.334197i −0.519981 0.854178i \(-0.674061\pi\)
0.854178 + 0.519981i \(0.174061\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 296.228i 0.325168i −0.986695 0.162584i \(-0.948017\pi\)
0.986695 0.162584i \(-0.0519829\pi\)
\(912\) 0 0
\(913\) 593.765 0.650346
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 368.731 + 368.731i 0.402106 + 0.402106i
\(918\) 0 0
\(919\) −228.052 −0.248153 −0.124076 0.992273i \(-0.539597\pi\)
−0.124076 + 0.992273i \(0.539597\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −244.699 + 244.699i −0.265113 + 0.265113i
\(924\) 0 0
\(925\) 460.011 460.011i 0.497309 0.497309i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 574.026 0.617897 0.308948 0.951079i \(-0.400023\pi\)
0.308948 + 0.951079i \(0.400023\pi\)
\(930\) 0 0
\(931\) 242.184 + 242.184i 0.260133 + 0.260133i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3014.98 −3.22457
\(936\) 0 0
\(937\) 1098.22i 1.17206i −0.810291 0.586028i \(-0.800691\pi\)
0.810291 0.586028i \(-0.199309\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −857.669 + 857.669i −0.911444 + 0.911444i −0.996386 0.0849418i \(-0.972930\pi\)
0.0849418 + 0.996386i \(0.472930\pi\)
\(942\) 0 0
\(943\) 169.810i 0.180074i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1041.67 + 1041.67i 1.09997 + 1.09997i 0.994413 + 0.105556i \(0.0336622\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(948\) 0 0
\(949\) −610.322 610.322i −0.643121 0.643121i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 910.089i 0.954973i −0.878639 0.477486i \(-0.841548\pi\)
0.878639 0.477486i \(-0.158452\pi\)
\(954\) 0 0
\(955\) 494.107 494.107i 0.517389 0.517389i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 663.195i 0.691548i
\(960\) 0 0
\(961\) 675.212 0.702614
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −441.856 441.856i −0.457882 0.457882i
\(966\) 0 0
\(967\) −695.071 −0.718791 −0.359396 0.933185i \(-0.617017\pi\)
−0.359396 + 0.933185i \(0.617017\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1208.40 + 1208.40i −1.24449 + 1.24449i −0.286366 + 0.958120i \(0.592447\pi\)
−0.958120 + 0.286366i \(0.907553\pi\)
\(972\) 0 0
\(973\) 900.500 900.500i 0.925488 0.925488i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −141.036 −0.144356 −0.0721780 0.997392i \(-0.522995\pi\)
−0.0721780 + 0.997392i \(0.522995\pi\)
\(978\) 0 0
\(979\) 648.750 + 648.750i 0.662666 + 0.662666i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1692.71 −1.72199 −0.860994 0.508616i \(-0.830157\pi\)
−0.860994 + 0.508616i \(0.830157\pi\)
\(984\) 0 0
\(985\) 597.320i 0.606417i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 194.059 194.059i 0.196217 0.196217i
\(990\) 0 0
\(991\) 1532.62i 1.54654i −0.634079 0.773268i \(-0.718621\pi\)
0.634079 0.773268i \(-0.281379\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1028.67 + 1028.67i 1.03384 + 1.03384i
\(996\) 0 0
\(997\) −1131.91 1131.91i −1.13532 1.13532i −0.989279 0.146039i \(-0.953348\pi\)
−0.146039 0.989279i \(-0.546652\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1152.3.m.f.415.1 16
3.2 odd 2 384.3.l.a.31.4 16
4.3 odd 2 1152.3.m.c.415.1 16
8.3 odd 2 576.3.m.c.271.8 16
8.5 even 2 144.3.m.c.91.2 16
12.11 even 2 384.3.l.b.31.8 16
16.3 odd 4 inner 1152.3.m.f.991.1 16
16.5 even 4 576.3.m.c.559.8 16
16.11 odd 4 144.3.m.c.19.2 16
16.13 even 4 1152.3.m.c.991.1 16
24.5 odd 2 48.3.l.a.43.7 yes 16
24.11 even 2 192.3.l.a.79.1 16
48.5 odd 4 192.3.l.a.175.1 16
48.11 even 4 48.3.l.a.19.7 16
48.29 odd 4 384.3.l.b.223.8 16
48.35 even 4 384.3.l.a.223.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.3.l.a.19.7 16 48.11 even 4
48.3.l.a.43.7 yes 16 24.5 odd 2
144.3.m.c.19.2 16 16.11 odd 4
144.3.m.c.91.2 16 8.5 even 2
192.3.l.a.79.1 16 24.11 even 2
192.3.l.a.175.1 16 48.5 odd 4
384.3.l.a.31.4 16 3.2 odd 2
384.3.l.a.223.4 16 48.35 even 4
384.3.l.b.31.8 16 12.11 even 2
384.3.l.b.223.8 16 48.29 odd 4
576.3.m.c.271.8 16 8.3 odd 2
576.3.m.c.559.8 16 16.5 even 4
1152.3.m.c.415.1 16 4.3 odd 2
1152.3.m.c.991.1 16 16.13 even 4
1152.3.m.f.415.1 16 1.1 even 1 trivial
1152.3.m.f.991.1 16 16.3 odd 4 inner