Properties

Label 448.2.p.d.255.2
Level $448$
Weight $2$
Character 448.255
Analytic conductor $3.577$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,2,Mod(255,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.255");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.p (of order \(6\), 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: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 255.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 448.255
Dual form 448.2.p.d.383.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 - 1.50000i) q^{3} +(1.50000 - 0.866025i) q^{5} +(1.73205 - 2.00000i) q^{7} +O(q^{10})\) \(q+(0.866025 - 1.50000i) q^{3} +(1.50000 - 0.866025i) q^{5} +(1.73205 - 2.00000i) q^{7} +(0.866025 + 0.500000i) q^{11} +3.46410i q^{13} -3.00000i q^{15} +(-1.50000 - 0.866025i) q^{17} +(-2.59808 - 4.50000i) q^{19} +(-1.50000 - 4.33013i) q^{21} +(-0.866025 + 0.500000i) q^{23} +(-1.00000 + 1.73205i) q^{25} +5.19615 q^{27} -4.00000 q^{29} +(0.866025 - 1.50000i) q^{31} +(1.50000 - 0.866025i) q^{33} +(0.866025 - 4.50000i) q^{35} +(1.50000 + 2.59808i) q^{37} +(5.19615 + 3.00000i) q^{39} +3.46410i q^{41} +2.00000i q^{43} +(-4.33013 - 7.50000i) q^{47} +(-1.00000 - 6.92820i) q^{49} +(-2.59808 + 1.50000i) q^{51} +(-0.500000 + 0.866025i) q^{53} +1.73205 q^{55} -9.00000 q^{57} +(-2.59808 + 4.50000i) q^{59} +(4.50000 - 2.59808i) q^{61} +(3.00000 + 5.19615i) q^{65} +(2.59808 + 1.50000i) q^{67} +1.73205i q^{69} +14.0000i q^{71} +(7.50000 + 4.33013i) q^{73} +(1.73205 + 3.00000i) q^{75} +(2.50000 - 0.866025i) q^{77} +(-7.79423 + 4.50000i) q^{79} +(4.50000 - 7.79423i) q^{81} -13.8564 q^{83} -3.00000 q^{85} +(-3.46410 + 6.00000i) q^{87} +(13.5000 - 7.79423i) q^{89} +(6.92820 + 6.00000i) q^{91} +(-1.50000 - 2.59808i) q^{93} +(-7.79423 - 4.50000i) q^{95} +17.3205i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{5} - 6 q^{17} - 6 q^{21} - 4 q^{25} - 16 q^{29} + 6 q^{33} + 6 q^{37} - 4 q^{49} - 2 q^{53} - 36 q^{57} + 18 q^{61} + 12 q^{65} + 30 q^{73} + 10 q^{77} + 18 q^{81} - 12 q^{85} + 54 q^{89} - 6 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(1\)

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
\(3\) 0.866025 1.50000i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(4\) 0 0
\(5\) 1.50000 0.866025i 0.670820 0.387298i −0.125567 0.992085i \(-0.540075\pi\)
0.796387 + 0.604787i \(0.206742\pi\)
\(6\) 0 0
\(7\) 1.73205 2.00000i 0.654654 0.755929i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.866025 + 0.500000i 0.261116 + 0.150756i 0.624844 0.780750i \(-0.285163\pi\)
−0.363727 + 0.931505i \(0.618496\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 3.00000i 0.774597i
\(16\) 0 0
\(17\) −1.50000 0.866025i −0.363803 0.210042i 0.306944 0.951727i \(-0.400693\pi\)
−0.670748 + 0.741685i \(0.734027\pi\)
\(18\) 0 0
\(19\) −2.59808 4.50000i −0.596040 1.03237i −0.993399 0.114708i \(-0.963407\pi\)
0.397360 0.917663i \(-0.369927\pi\)
\(20\) 0 0
\(21\) −1.50000 4.33013i −0.327327 0.944911i
\(22\) 0 0
\(23\) −0.866025 + 0.500000i −0.180579 + 0.104257i −0.587565 0.809177i \(-0.699913\pi\)
0.406986 + 0.913434i \(0.366580\pi\)
\(24\) 0 0
\(25\) −1.00000 + 1.73205i −0.200000 + 0.346410i
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 0.866025 1.50000i 0.155543 0.269408i −0.777714 0.628619i \(-0.783621\pi\)
0.933257 + 0.359211i \(0.116954\pi\)
\(32\) 0 0
\(33\) 1.50000 0.866025i 0.261116 0.150756i
\(34\) 0 0
\(35\) 0.866025 4.50000i 0.146385 0.760639i
\(36\) 0 0
\(37\) 1.50000 + 2.59808i 0.246598 + 0.427121i 0.962580 0.270998i \(-0.0873538\pi\)
−0.715981 + 0.698119i \(0.754020\pi\)
\(38\) 0 0
\(39\) 5.19615 + 3.00000i 0.832050 + 0.480384i
\(40\) 0 0
\(41\) 3.46410i 0.541002i 0.962720 + 0.270501i \(0.0871893\pi\)
−0.962720 + 0.270501i \(0.912811\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.33013 7.50000i −0.631614 1.09399i −0.987222 0.159352i \(-0.949059\pi\)
0.355608 0.934635i \(-0.384274\pi\)
\(48\) 0 0
\(49\) −1.00000 6.92820i −0.142857 0.989743i
\(50\) 0 0
\(51\) −2.59808 + 1.50000i −0.363803 + 0.210042i
\(52\) 0 0
\(53\) −0.500000 + 0.866025i −0.0686803 + 0.118958i −0.898321 0.439340i \(-0.855212\pi\)
0.829640 + 0.558298i \(0.188546\pi\)
\(54\) 0 0
\(55\) 1.73205 0.233550
\(56\) 0 0
\(57\) −9.00000 −1.19208
\(58\) 0 0
\(59\) −2.59808 + 4.50000i −0.338241 + 0.585850i −0.984102 0.177605i \(-0.943165\pi\)
0.645861 + 0.763455i \(0.276498\pi\)
\(60\) 0 0
\(61\) 4.50000 2.59808i 0.576166 0.332650i −0.183442 0.983030i \(-0.558724\pi\)
0.759608 + 0.650381i \(0.225391\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) 0 0
\(67\) 2.59808 + 1.50000i 0.317406 + 0.183254i 0.650236 0.759733i \(-0.274670\pi\)
−0.332830 + 0.942987i \(0.608004\pi\)
\(68\) 0 0
\(69\) 1.73205i 0.208514i
\(70\) 0 0
\(71\) 14.0000i 1.66149i 0.556650 + 0.830747i \(0.312086\pi\)
−0.556650 + 0.830747i \(0.687914\pi\)
\(72\) 0 0
\(73\) 7.50000 + 4.33013i 0.877809 + 0.506803i 0.869935 0.493166i \(-0.164160\pi\)
0.00787336 + 0.999969i \(0.497494\pi\)
\(74\) 0 0
\(75\) 1.73205 + 3.00000i 0.200000 + 0.346410i
\(76\) 0 0
\(77\) 2.50000 0.866025i 0.284901 0.0986928i
\(78\) 0 0
\(79\) −7.79423 + 4.50000i −0.876919 + 0.506290i −0.869641 0.493684i \(-0.835650\pi\)
−0.00727784 + 0.999974i \(0.502317\pi\)
\(80\) 0 0
\(81\) 4.50000 7.79423i 0.500000 0.866025i
\(82\) 0 0
\(83\) −13.8564 −1.52094 −0.760469 0.649374i \(-0.775031\pi\)
−0.760469 + 0.649374i \(0.775031\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) −3.46410 + 6.00000i −0.371391 + 0.643268i
\(88\) 0 0
\(89\) 13.5000 7.79423i 1.43100 0.826187i 0.433800 0.901009i \(-0.357172\pi\)
0.997197 + 0.0748225i \(0.0238390\pi\)
\(90\) 0 0
\(91\) 6.92820 + 6.00000i 0.726273 + 0.628971i
\(92\) 0 0
\(93\) −1.50000 2.59808i −0.155543 0.269408i
\(94\) 0 0
\(95\) −7.79423 4.50000i −0.799671 0.461690i
\(96\) 0 0
\(97\) 17.3205i 1.75863i 0.476240 + 0.879316i \(0.342000\pi\)
−0.476240 + 0.879316i \(0.658000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.50000 + 4.33013i 0.746278 + 0.430864i 0.824347 0.566084i \(-0.191542\pi\)
−0.0780696 + 0.996948i \(0.524876\pi\)
\(102\) 0 0
\(103\) 4.33013 + 7.50000i 0.426660 + 0.738997i 0.996574 0.0827075i \(-0.0263567\pi\)
−0.569914 + 0.821705i \(0.693023\pi\)
\(104\) 0 0
\(105\) −6.00000 5.19615i −0.585540 0.507093i
\(106\) 0 0
\(107\) −11.2583 + 6.50000i −1.08838 + 0.628379i −0.933146 0.359498i \(-0.882948\pi\)
−0.155238 + 0.987877i \(0.549614\pi\)
\(108\) 0 0
\(109\) 4.50000 7.79423i 0.431022 0.746552i −0.565940 0.824447i \(-0.691487\pi\)
0.996962 + 0.0778949i \(0.0248199\pi\)
\(110\) 0 0
\(111\) 5.19615 0.493197
\(112\) 0 0
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) −0.866025 + 1.50000i −0.0807573 + 0.139876i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4.33013 + 1.50000i −0.396942 + 0.137505i
\(120\) 0 0
\(121\) −5.00000 8.66025i −0.454545 0.787296i
\(122\) 0 0
\(123\) 5.19615 + 3.00000i 0.468521 + 0.270501i
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) 6.00000i 0.532414i −0.963916 0.266207i \(-0.914230\pi\)
0.963916 0.266207i \(-0.0857705\pi\)
\(128\) 0 0
\(129\) 3.00000 + 1.73205i 0.264135 + 0.152499i
\(130\) 0 0
\(131\) 2.59808 + 4.50000i 0.226995 + 0.393167i 0.956916 0.290365i \(-0.0937766\pi\)
−0.729921 + 0.683531i \(0.760443\pi\)
\(132\) 0 0
\(133\) −13.5000 2.59808i −1.17060 0.225282i
\(134\) 0 0
\(135\) 7.79423 4.50000i 0.670820 0.387298i
\(136\) 0 0
\(137\) −0.500000 + 0.866025i −0.0427179 + 0.0739895i −0.886594 0.462549i \(-0.846935\pi\)
0.843876 + 0.536538i \(0.180268\pi\)
\(138\) 0 0
\(139\) 6.92820 0.587643 0.293821 0.955860i \(-0.405073\pi\)
0.293821 + 0.955860i \(0.405073\pi\)
\(140\) 0 0
\(141\) −15.0000 −1.26323
\(142\) 0 0
\(143\) −1.73205 + 3.00000i −0.144841 + 0.250873i
\(144\) 0 0
\(145\) −6.00000 + 3.46410i −0.498273 + 0.287678i
\(146\) 0 0
\(147\) −11.2583 4.50000i −0.928571 0.371154i
\(148\) 0 0
\(149\) 0.500000 + 0.866025i 0.0409616 + 0.0709476i 0.885779 0.464107i \(-0.153625\pi\)
−0.844818 + 0.535054i \(0.820291\pi\)
\(150\) 0 0
\(151\) 6.06218 + 3.50000i 0.493333 + 0.284826i 0.725956 0.687741i \(-0.241398\pi\)
−0.232623 + 0.972567i \(0.574731\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.00000i 0.240966i
\(156\) 0 0
\(157\) −1.50000 0.866025i −0.119713 0.0691164i 0.438948 0.898513i \(-0.355351\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 0.866025 + 1.50000i 0.0686803 + 0.118958i
\(160\) 0 0
\(161\) −0.500000 + 2.59808i −0.0394055 + 0.204757i
\(162\) 0 0
\(163\) 18.1865 10.5000i 1.42448 0.822423i 0.427802 0.903873i \(-0.359288\pi\)
0.996678 + 0.0814491i \(0.0259548\pi\)
\(164\) 0 0
\(165\) 1.50000 2.59808i 0.116775 0.202260i
\(166\) 0 0
\(167\) −17.3205 −1.34030 −0.670151 0.742225i \(-0.733770\pi\)
−0.670151 + 0.742225i \(0.733770\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.5000 6.06218i 0.798300 0.460899i −0.0445762 0.999006i \(-0.514194\pi\)
0.842876 + 0.538107i \(0.180860\pi\)
\(174\) 0 0
\(175\) 1.73205 + 5.00000i 0.130931 + 0.377964i
\(176\) 0 0
\(177\) 4.50000 + 7.79423i 0.338241 + 0.585850i
\(178\) 0 0
\(179\) −16.4545 9.50000i −1.22987 0.710063i −0.262864 0.964833i \(-0.584667\pi\)
−0.967002 + 0.254770i \(0.918000\pi\)
\(180\) 0 0
\(181\) 6.92820i 0.514969i 0.966282 + 0.257485i \(0.0828937\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 0 0
\(183\) 9.00000i 0.665299i
\(184\) 0 0
\(185\) 4.50000 + 2.59808i 0.330847 + 0.191014i
\(186\) 0 0
\(187\) −0.866025 1.50000i −0.0633300 0.109691i
\(188\) 0 0
\(189\) 9.00000 10.3923i 0.654654 0.755929i
\(190\) 0 0
\(191\) 0.866025 0.500000i 0.0626634 0.0361787i −0.468341 0.883548i \(-0.655148\pi\)
0.531004 + 0.847369i \(0.321815\pi\)
\(192\) 0 0
\(193\) 7.50000 12.9904i 0.539862 0.935068i −0.459049 0.888411i \(-0.651810\pi\)
0.998911 0.0466572i \(-0.0148568\pi\)
\(194\) 0 0
\(195\) 10.3923 0.744208
\(196\) 0 0
\(197\) −16.0000 −1.13995 −0.569976 0.821661i \(-0.693048\pi\)
−0.569976 + 0.821661i \(0.693048\pi\)
\(198\) 0 0
\(199\) 11.2583 19.5000i 0.798082 1.38232i −0.122782 0.992434i \(-0.539182\pi\)
0.920864 0.389885i \(-0.127485\pi\)
\(200\) 0 0
\(201\) 4.50000 2.59808i 0.317406 0.183254i
\(202\) 0 0
\(203\) −6.92820 + 8.00000i −0.486265 + 0.561490i
\(204\) 0 0
\(205\) 3.00000 + 5.19615i 0.209529 + 0.362915i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.19615i 0.359425i
\(210\) 0 0
\(211\) 10.0000i 0.688428i 0.938891 + 0.344214i \(0.111855\pi\)
−0.938891 + 0.344214i \(0.888145\pi\)
\(212\) 0 0
\(213\) 21.0000 + 12.1244i 1.43890 + 0.830747i
\(214\) 0 0
\(215\) 1.73205 + 3.00000i 0.118125 + 0.204598i
\(216\) 0 0
\(217\) −1.50000 4.33013i −0.101827 0.293948i
\(218\) 0 0
\(219\) 12.9904 7.50000i 0.877809 0.506803i
\(220\) 0 0
\(221\) 3.00000 5.19615i 0.201802 0.349531i
\(222\) 0 0
\(223\) 6.92820 0.463947 0.231973 0.972722i \(-0.425482\pi\)
0.231973 + 0.972722i \(0.425482\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.52628 16.5000i 0.632281 1.09514i −0.354803 0.934941i \(-0.615452\pi\)
0.987084 0.160202i \(-0.0512147\pi\)
\(228\) 0 0
\(229\) 13.5000 7.79423i 0.892105 0.515057i 0.0174746 0.999847i \(-0.494437\pi\)
0.874630 + 0.484790i \(0.161104\pi\)
\(230\) 0 0
\(231\) 0.866025 4.50000i 0.0569803 0.296078i
\(232\) 0 0
\(233\) 3.50000 + 6.06218i 0.229293 + 0.397146i 0.957599 0.288106i \(-0.0930254\pi\)
−0.728306 + 0.685252i \(0.759692\pi\)
\(234\) 0 0
\(235\) −12.9904 7.50000i −0.847399 0.489246i
\(236\) 0 0
\(237\) 15.5885i 1.01258i
\(238\) 0 0
\(239\) 20.0000i 1.29369i −0.762620 0.646846i \(-0.776088\pi\)
0.762620 0.646846i \(-0.223912\pi\)
\(240\) 0 0
\(241\) −4.50000 2.59808i −0.289870 0.167357i 0.348013 0.937490i \(-0.386857\pi\)
−0.637883 + 0.770133i \(0.720190\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −7.50000 9.52628i −0.479157 0.608612i
\(246\) 0 0
\(247\) 15.5885 9.00000i 0.991870 0.572656i
\(248\) 0 0
\(249\) −12.0000 + 20.7846i −0.760469 + 1.31717i
\(250\) 0 0
\(251\) 3.46410 0.218652 0.109326 0.994006i \(-0.465131\pi\)
0.109326 + 0.994006i \(0.465131\pi\)
\(252\) 0 0
\(253\) −1.00000 −0.0628695
\(254\) 0 0
\(255\) −2.59808 + 4.50000i −0.162698 + 0.281801i
\(256\) 0 0
\(257\) 4.50000 2.59808i 0.280702 0.162064i −0.353039 0.935609i \(-0.614852\pi\)
0.633741 + 0.773545i \(0.281518\pi\)
\(258\) 0 0
\(259\) 7.79423 + 1.50000i 0.484310 + 0.0932055i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −19.9186 11.5000i −1.22823 0.709120i −0.261573 0.965184i \(-0.584241\pi\)
−0.966660 + 0.256063i \(0.917574\pi\)
\(264\) 0 0
\(265\) 1.73205i 0.106399i
\(266\) 0 0
\(267\) 27.0000i 1.65237i
\(268\) 0 0
\(269\) −19.5000 11.2583i −1.18894 0.686433i −0.230871 0.972984i \(-0.574158\pi\)
−0.958065 + 0.286552i \(0.907491\pi\)
\(270\) 0 0
\(271\) 7.79423 + 13.5000i 0.473466 + 0.820067i 0.999539 0.0303728i \(-0.00966946\pi\)
−0.526073 + 0.850439i \(0.676336\pi\)
\(272\) 0 0
\(273\) 15.0000 5.19615i 0.907841 0.314485i
\(274\) 0 0
\(275\) −1.73205 + 1.00000i −0.104447 + 0.0603023i
\(276\) 0 0
\(277\) −6.50000 + 11.2583i −0.390547 + 0.676448i −0.992522 0.122068i \(-0.961047\pi\)
0.601975 + 0.798515i \(0.294381\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.00000 −0.238620 −0.119310 0.992857i \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) 0 0
\(283\) 6.06218 10.5000i 0.360359 0.624160i −0.627661 0.778487i \(-0.715988\pi\)
0.988020 + 0.154327i \(0.0493208\pi\)
\(284\) 0 0
\(285\) −13.5000 + 7.79423i −0.799671 + 0.461690i
\(286\) 0 0
\(287\) 6.92820 + 6.00000i 0.408959 + 0.354169i
\(288\) 0 0
\(289\) −7.00000 12.1244i −0.411765 0.713197i
\(290\) 0 0
\(291\) 25.9808 + 15.0000i 1.52302 + 0.879316i
\(292\) 0 0
\(293\) 20.7846i 1.21425i −0.794606 0.607125i \(-0.792323\pi\)
0.794606 0.607125i \(-0.207677\pi\)
\(294\) 0 0
\(295\) 9.00000i 0.524000i
\(296\) 0 0
\(297\) 4.50000 + 2.59808i 0.261116 + 0.150756i
\(298\) 0 0
\(299\) −1.73205 3.00000i −0.100167 0.173494i
\(300\) 0 0
\(301\) 4.00000 + 3.46410i 0.230556 + 0.199667i
\(302\) 0 0
\(303\) 12.9904 7.50000i 0.746278 0.430864i
\(304\) 0 0
\(305\) 4.50000 7.79423i 0.257669 0.446296i
\(306\) 0 0
\(307\) −20.7846 −1.18624 −0.593120 0.805114i \(-0.702104\pi\)
−0.593120 + 0.805114i \(0.702104\pi\)
\(308\) 0 0
\(309\) 15.0000 0.853320
\(310\) 0 0
\(311\) −4.33013 + 7.50000i −0.245539 + 0.425286i −0.962283 0.272050i \(-0.912298\pi\)
0.716744 + 0.697336i \(0.245632\pi\)
\(312\) 0 0
\(313\) 1.50000 0.866025i 0.0847850 0.0489506i −0.457008 0.889463i \(-0.651079\pi\)
0.541793 + 0.840512i \(0.317746\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.50000 + 9.52628i 0.308911 + 0.535049i 0.978124 0.208021i \(-0.0667022\pi\)
−0.669214 + 0.743070i \(0.733369\pi\)
\(318\) 0 0
\(319\) −3.46410 2.00000i −0.193952 0.111979i
\(320\) 0 0
\(321\) 22.5167i 1.25676i
\(322\) 0 0
\(323\) 9.00000i 0.500773i
\(324\) 0 0
\(325\) −6.00000 3.46410i −0.332820 0.192154i
\(326\) 0 0
\(327\) −7.79423 13.5000i −0.431022 0.746552i
\(328\) 0 0
\(329\) −22.5000 4.33013i −1.24047 0.238728i
\(330\) 0 0
\(331\) −6.06218 + 3.50000i −0.333207 + 0.192377i −0.657264 0.753660i \(-0.728286\pi\)
0.324057 + 0.946038i \(0.394953\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.19615 0.283896
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) −13.8564 + 24.0000i −0.752577 + 1.30350i
\(340\) 0 0
\(341\) 1.50000 0.866025i 0.0812296 0.0468979i
\(342\) 0 0
\(343\) −15.5885 10.0000i −0.841698 0.539949i
\(344\) 0 0
\(345\) 1.50000 + 2.59808i 0.0807573 + 0.139876i
\(346\) 0 0
\(347\) −11.2583 6.50000i −0.604379 0.348938i 0.166383 0.986061i \(-0.446791\pi\)
−0.770762 + 0.637123i \(0.780124\pi\)
\(348\) 0 0
\(349\) 10.3923i 0.556287i 0.960539 + 0.278144i \(0.0897191\pi\)
−0.960539 + 0.278144i \(0.910281\pi\)
\(350\) 0 0
\(351\) 18.0000i 0.960769i
\(352\) 0 0
\(353\) −25.5000 14.7224i −1.35723 0.783596i −0.367979 0.929834i \(-0.619950\pi\)
−0.989249 + 0.146238i \(0.953283\pi\)
\(354\) 0 0
\(355\) 12.1244 + 21.0000i 0.643494 + 1.11456i
\(356\) 0 0
\(357\) −1.50000 + 7.79423i −0.0793884 + 0.412514i
\(358\) 0 0
\(359\) −19.9186 + 11.5000i −1.05126 + 0.606947i −0.923003 0.384794i \(-0.874273\pi\)
−0.128260 + 0.991741i \(0.540939\pi\)
\(360\) 0 0
\(361\) −4.00000 + 6.92820i −0.210526 + 0.364642i
\(362\) 0 0
\(363\) −17.3205 −0.909091
\(364\) 0 0
\(365\) 15.0000 0.785136
\(366\) 0 0
\(367\) −0.866025 + 1.50000i −0.0452062 + 0.0782994i −0.887743 0.460339i \(-0.847728\pi\)
0.842537 + 0.538639i \(0.181061\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.866025 + 2.50000i 0.0449618 + 0.129794i
\(372\) 0 0
\(373\) −14.5000 25.1147i −0.750782 1.30039i −0.947444 0.319921i \(-0.896344\pi\)
0.196663 0.980471i \(-0.436990\pi\)
\(374\) 0 0
\(375\) 18.1865 + 10.5000i 0.939149 + 0.542218i
\(376\) 0 0
\(377\) 13.8564i 0.713641i
\(378\) 0 0
\(379\) 8.00000i 0.410932i −0.978664 0.205466i \(-0.934129\pi\)
0.978664 0.205466i \(-0.0658711\pi\)
\(380\) 0 0
\(381\) −9.00000 5.19615i −0.461084 0.266207i
\(382\) 0 0
\(383\) 2.59808 + 4.50000i 0.132755 + 0.229939i 0.924738 0.380605i \(-0.124284\pi\)
−0.791982 + 0.610544i \(0.790951\pi\)
\(384\) 0 0
\(385\) 3.00000 3.46410i 0.152894 0.176547i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9.50000 + 16.4545i −0.481669 + 0.834275i −0.999779 0.0210389i \(-0.993303\pi\)
0.518110 + 0.855314i \(0.326636\pi\)
\(390\) 0 0
\(391\) 1.73205 0.0875936
\(392\) 0 0
\(393\) 9.00000 0.453990
\(394\) 0 0
\(395\) −7.79423 + 13.5000i −0.392170 + 0.679259i
\(396\) 0 0
\(397\) −16.5000 + 9.52628i −0.828111 + 0.478110i −0.853206 0.521575i \(-0.825345\pi\)
0.0250943 + 0.999685i \(0.492011\pi\)
\(398\) 0 0
\(399\) −15.5885 + 18.0000i −0.780399 + 0.901127i
\(400\) 0 0
\(401\) 11.5000 + 19.9186i 0.574283 + 0.994687i 0.996119 + 0.0880147i \(0.0280523\pi\)
−0.421837 + 0.906672i \(0.638614\pi\)
\(402\) 0 0
\(403\) 5.19615 + 3.00000i 0.258839 + 0.149441i
\(404\) 0 0
\(405\) 15.5885i 0.774597i
\(406\) 0 0
\(407\) 3.00000i 0.148704i
\(408\) 0 0
\(409\) 22.5000 + 12.9904i 1.11255 + 0.642333i 0.939490 0.342578i \(-0.111300\pi\)
0.173064 + 0.984911i \(0.444633\pi\)
\(410\) 0 0
\(411\) 0.866025 + 1.50000i 0.0427179 + 0.0739895i
\(412\) 0 0
\(413\) 4.50000 + 12.9904i 0.221431 + 0.639215i
\(414\) 0 0
\(415\) −20.7846 + 12.0000i −1.02028 + 0.589057i
\(416\) 0 0
\(417\) 6.00000 10.3923i 0.293821 0.508913i
\(418\) 0 0
\(419\) 20.7846 1.01539 0.507697 0.861536i \(-0.330497\pi\)
0.507697 + 0.861536i \(0.330497\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.00000 1.73205i 0.145521 0.0840168i
\(426\) 0 0
\(427\) 2.59808 13.5000i 0.125730 0.653311i
\(428\) 0 0
\(429\) 3.00000 + 5.19615i 0.144841 + 0.250873i
\(430\) 0 0
\(431\) 19.9186 + 11.5000i 0.959444 + 0.553936i 0.896002 0.444050i \(-0.146459\pi\)
0.0634424 + 0.997985i \(0.479792\pi\)
\(432\) 0 0
\(433\) 10.3923i 0.499422i 0.968320 + 0.249711i \(0.0803357\pi\)
−0.968320 + 0.249711i \(0.919664\pi\)
\(434\) 0 0
\(435\) 12.0000i 0.575356i
\(436\) 0 0
\(437\) 4.50000 + 2.59808i 0.215264 + 0.124283i
\(438\) 0 0
\(439\) −11.2583 19.5000i −0.537331 0.930684i −0.999047 0.0436563i \(-0.986099\pi\)
0.461716 0.887028i \(-0.347234\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −14.7224 + 8.50000i −0.699484 + 0.403847i −0.807155 0.590339i \(-0.798994\pi\)
0.107671 + 0.994187i \(0.465661\pi\)
\(444\) 0 0
\(445\) 13.5000 23.3827i 0.639961 1.10845i
\(446\) 0 0
\(447\) 1.73205 0.0819232
\(448\) 0 0
\(449\) 8.00000 0.377543 0.188772 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) 0 0
\(451\) −1.73205 + 3.00000i −0.0815591 + 0.141264i
\(452\) 0 0
\(453\) 10.5000 6.06218i 0.493333 0.284826i
\(454\) 0 0
\(455\) 15.5885 + 3.00000i 0.730798 + 0.140642i
\(456\) 0 0
\(457\) −7.50000 12.9904i −0.350835 0.607664i 0.635561 0.772051i \(-0.280769\pi\)
−0.986396 + 0.164386i \(0.947436\pi\)
\(458\) 0 0
\(459\) −7.79423 4.50000i −0.363803 0.210042i
\(460\) 0 0
\(461\) 17.3205i 0.806696i −0.915047 0.403348i \(-0.867846\pi\)
0.915047 0.403348i \(-0.132154\pi\)
\(462\) 0 0
\(463\) 30.0000i 1.39422i 0.716965 + 0.697109i \(0.245531\pi\)
−0.716965 + 0.697109i \(0.754469\pi\)
\(464\) 0 0
\(465\) −4.50000 2.59808i −0.208683 0.120483i
\(466\) 0 0
\(467\) −4.33013 7.50000i −0.200374 0.347059i 0.748275 0.663389i \(-0.230883\pi\)
−0.948649 + 0.316330i \(0.897549\pi\)
\(468\) 0 0
\(469\) 7.50000 2.59808i 0.346318 0.119968i
\(470\) 0 0
\(471\) −2.59808 + 1.50000i −0.119713 + 0.0691164i
\(472\) 0 0
\(473\) −1.00000 + 1.73205i −0.0459800 + 0.0796398i
\(474\) 0 0
\(475\) 10.3923 0.476832
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6.06218 10.5000i 0.276988 0.479757i −0.693647 0.720315i \(-0.743997\pi\)
0.970635 + 0.240558i \(0.0773304\pi\)
\(480\) 0 0
\(481\) −9.00000 + 5.19615i −0.410365 + 0.236924i
\(482\) 0 0
\(483\) 3.46410 + 3.00000i 0.157622 + 0.136505i
\(484\) 0 0
\(485\) 15.0000 + 25.9808i 0.681115 + 1.17973i
\(486\) 0 0
\(487\) −26.8468 15.5000i −1.21654 0.702372i −0.252367 0.967632i \(-0.581209\pi\)
−0.964177 + 0.265260i \(0.914542\pi\)
\(488\) 0 0
\(489\) 36.3731i 1.64485i
\(490\) 0 0
\(491\) 32.0000i 1.44414i 0.691820 + 0.722070i \(0.256809\pi\)
−0.691820 + 0.722070i \(0.743191\pi\)
\(492\) 0 0
\(493\) 6.00000 + 3.46410i 0.270226 + 0.156015i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 28.0000 + 24.2487i 1.25597 + 1.08770i
\(498\) 0 0
\(499\) −30.3109 + 17.5000i −1.35690 + 0.783408i −0.989205 0.146538i \(-0.953187\pi\)
−0.367697 + 0.929946i \(0.619854\pi\)
\(500\) 0 0
\(501\) −15.0000 + 25.9808i −0.670151 + 1.16073i
\(502\) 0 0
\(503\) −6.92820 −0.308913 −0.154457 0.988000i \(-0.549363\pi\)
−0.154457 + 0.988000i \(0.549363\pi\)
\(504\) 0 0
\(505\) 15.0000 0.667491
\(506\) 0 0
\(507\) 0.866025 1.50000i 0.0384615 0.0666173i
\(508\) 0 0
\(509\) −10.5000 + 6.06218i −0.465404 + 0.268701i −0.714314 0.699825i \(-0.753261\pi\)
0.248910 + 0.968527i \(0.419928\pi\)
\(510\) 0 0
\(511\) 21.6506 7.50000i 0.957768 0.331780i
\(512\) 0 0
\(513\) −13.5000 23.3827i −0.596040 1.03237i
\(514\) 0 0
\(515\) 12.9904 + 7.50000i 0.572425 + 0.330489i
\(516\) 0 0
\(517\) 8.66025i 0.380878i
\(518\) 0 0
\(519\) 21.0000i 0.921798i
\(520\) 0 0
\(521\) 1.50000 + 0.866025i 0.0657162 + 0.0379413i 0.532498 0.846431i \(-0.321253\pi\)
−0.466782 + 0.884372i \(0.654587\pi\)
\(522\) 0 0
\(523\) −12.9904 22.5000i −0.568030 0.983856i −0.996761 0.0804241i \(-0.974373\pi\)
0.428731 0.903432i \(-0.358961\pi\)
\(524\) 0 0
\(525\) 9.00000 + 1.73205i 0.392792 + 0.0755929i
\(526\) 0 0
\(527\) −2.59808 + 1.50000i −0.113174 + 0.0653410i
\(528\) 0 0
\(529\) −11.0000 + 19.0526i −0.478261 + 0.828372i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) −11.2583 + 19.5000i −0.486740 + 0.843059i
\(536\) 0 0
\(537\) −28.5000 + 16.4545i −1.22987 + 0.710063i
\(538\) 0 0
\(539\) 2.59808 6.50000i 0.111907 0.279975i
\(540\) 0 0
\(541\) −9.50000 16.4545i −0.408437 0.707433i 0.586278 0.810110i \(-0.300593\pi\)
−0.994715 + 0.102677i \(0.967259\pi\)
\(542\) 0 0
\(543\) 10.3923 + 6.00000i 0.445976 + 0.257485i
\(544\) 0 0
\(545\) 15.5885i 0.667736i
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.3923 + 18.0000i 0.442727 + 0.766826i
\(552\) 0 0
\(553\) −4.50000 + 23.3827i −0.191359 + 0.994333i
\(554\) 0 0
\(555\) 7.79423 4.50000i 0.330847 0.191014i
\(556\) 0 0
\(557\) 18.5000 32.0429i 0.783870 1.35770i −0.145802 0.989314i \(-0.546576\pi\)
0.929672 0.368389i \(-0.120091\pi\)
\(558\) 0 0
\(559\) −6.92820 −0.293032
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) 0 0
\(563\) 11.2583 19.5000i 0.474482 0.821827i −0.525091 0.851046i \(-0.675969\pi\)
0.999573 + 0.0292191i \(0.00930205\pi\)
\(564\) 0 0
\(565\) −24.0000 + 13.8564i −1.00969 + 0.582943i
\(566\) 0 0
\(567\) −7.79423 22.5000i −0.327327 0.944911i
\(568\) 0 0
\(569\) 6.50000 + 11.2583i 0.272494 + 0.471974i 0.969500 0.245092i \(-0.0788181\pi\)
−0.697006 + 0.717066i \(0.745485\pi\)
\(570\) 0 0
\(571\) 18.1865 + 10.5000i 0.761083 + 0.439411i 0.829684 0.558233i \(-0.188520\pi\)
−0.0686016 + 0.997644i \(0.521854\pi\)
\(572\) 0 0
\(573\) 1.73205i 0.0723575i
\(574\) 0 0
\(575\) 2.00000i 0.0834058i
\(576\) 0 0
\(577\) −28.5000 16.4545i −1.18647 0.685009i −0.228968 0.973434i \(-0.573535\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) 0 0
\(579\) −12.9904 22.5000i −0.539862 0.935068i
\(580\) 0 0
\(581\) −24.0000 + 27.7128i −0.995688 + 1.14972i
\(582\) 0 0
\(583\) −0.866025 + 0.500000i −0.0358671 + 0.0207079i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −6.92820 −0.285958 −0.142979 0.989726i \(-0.545668\pi\)
−0.142979 + 0.989726i \(0.545668\pi\)
\(588\) 0 0
\(589\) −9.00000 −0.370839
\(590\) 0 0
\(591\) −13.8564 + 24.0000i −0.569976 + 0.987228i
\(592\) 0 0
\(593\) −13.5000 + 7.79423i −0.554379 + 0.320071i −0.750886 0.660432i \(-0.770373\pi\)
0.196508 + 0.980502i \(0.437040\pi\)
\(594\) 0 0
\(595\) −5.19615 + 6.00000i −0.213021 + 0.245976i
\(596\) 0 0
\(597\) −19.5000 33.7750i −0.798082 1.38232i
\(598\) 0 0
\(599\) 14.7224 + 8.50000i 0.601542 + 0.347301i 0.769648 0.638468i \(-0.220432\pi\)
−0.168106 + 0.985769i \(0.553765\pi\)
\(600\) 0 0
\(601\) 38.1051i 1.55434i −0.629291 0.777170i \(-0.716654\pi\)
0.629291 0.777170i \(-0.283346\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −15.0000 8.66025i −0.609837 0.352089i
\(606\) 0 0
\(607\) −7.79423 13.5000i −0.316358 0.547948i 0.663367 0.748294i \(-0.269127\pi\)
−0.979725 + 0.200346i \(0.935793\pi\)
\(608\) 0 0
\(609\) 6.00000 + 17.3205i 0.243132 + 0.701862i
\(610\) 0 0
\(611\) 25.9808 15.0000i 1.05107 0.606835i
\(612\) 0 0
\(613\) −15.5000 + 26.8468i −0.626039 + 1.08433i 0.362300 + 0.932062i \(0.381992\pi\)
−0.988339 + 0.152270i \(0.951342\pi\)
\(614\) 0 0
\(615\) 10.3923 0.419058
\(616\) 0 0
\(617\) 20.0000 0.805170 0.402585 0.915383i \(-0.368112\pi\)
0.402585 + 0.915383i \(0.368112\pi\)
\(618\) 0 0
\(619\) 7.79423 13.5000i 0.313276 0.542611i −0.665793 0.746136i \(-0.731907\pi\)
0.979070 + 0.203526i \(0.0652400\pi\)
\(620\) 0 0
\(621\) −4.50000 + 2.59808i −0.180579 + 0.104257i
\(622\) 0 0
\(623\) 7.79423 40.5000i 0.312269 1.62260i
\(624\) 0 0
\(625\) 5.50000 + 9.52628i 0.220000 + 0.381051i
\(626\) 0 0
\(627\) −7.79423 4.50000i −0.311272 0.179713i
\(628\) 0 0
\(629\) 5.19615i 0.207184i
\(630\) 0 0
\(631\) 30.0000i 1.19428i −0.802137 0.597141i \(-0.796303\pi\)
0.802137 0.597141i \(-0.203697\pi\)
\(632\) 0 0
\(633\) 15.0000 + 8.66025i 0.596196 + 0.344214i
\(634\) 0 0
\(635\) −5.19615 9.00000i −0.206203 0.357154i
\(636\) 0 0
\(637\) 24.0000 3.46410i 0.950915 0.137253i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.50000 11.2583i 0.256735 0.444677i −0.708631 0.705580i \(-0.750687\pi\)
0.965365 + 0.260902i \(0.0840201\pi\)
\(642\) 0 0
\(643\) 13.8564 0.546443 0.273222 0.961951i \(-0.411911\pi\)
0.273222 + 0.961951i \(0.411911\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) 16.4545 28.5000i 0.646892 1.12045i −0.336968 0.941516i \(-0.609402\pi\)
0.983861 0.178935i \(-0.0572651\pi\)
\(648\) 0 0
\(649\) −4.50000 + 2.59808i −0.176640 + 0.101983i
\(650\) 0 0
\(651\) −7.79423 1.50000i −0.305480 0.0587896i
\(652\) 0 0
\(653\) 15.5000 + 26.8468i 0.606562 + 1.05060i 0.991803 + 0.127780i \(0.0407851\pi\)
−0.385241 + 0.922816i \(0.625882\pi\)
\(654\) 0 0
\(655\) 7.79423 + 4.50000i 0.304546 + 0.175830i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 38.0000i 1.48027i −0.672458 0.740135i \(-0.734762\pi\)
0.672458 0.740135i \(-0.265238\pi\)
\(660\) 0 0
\(661\) 34.5000 + 19.9186i 1.34189 + 0.774743i 0.987085 0.160196i \(-0.0512125\pi\)
0.354809 + 0.934939i \(0.384546\pi\)
\(662\) 0 0
\(663\) −5.19615 9.00000i −0.201802 0.349531i
\(664\) 0 0
\(665\) −22.5000 + 7.79423i −0.872513 + 0.302247i
\(666\) 0 0
\(667\) 3.46410 2.00000i 0.134131 0.0774403i
\(668\) 0 0
\(669\) 6.00000 10.3923i 0.231973 0.401790i
\(670\) 0 0
\(671\) 5.19615 0.200595
\(672\) 0 0
\(673\) 24.0000 0.925132 0.462566 0.886585i \(-0.346929\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) 0 0
\(675\) −5.19615 + 9.00000i −0.200000 + 0.346410i
\(676\) 0 0
\(677\) 37.5000 21.6506i 1.44124 0.832102i 0.443309 0.896369i \(-0.353804\pi\)
0.997933 + 0.0642672i \(0.0204710\pi\)
\(678\) 0 0
\(679\) 34.6410 + 30.0000i 1.32940 + 1.15129i
\(680\) 0 0
\(681\) −16.5000 28.5788i −0.632281 1.09514i
\(682\) 0 0
\(683\) 21.6506 + 12.5000i 0.828439 + 0.478299i 0.853318 0.521391i \(-0.174587\pi\)
−0.0248792 + 0.999690i \(0.507920\pi\)
\(684\) 0 0
\(685\) 1.73205i 0.0661783i
\(686\) 0 0
\(687\) 27.0000i 1.03011i
\(688\) 0 0
\(689\) −3.00000 1.73205i −0.114291 0.0659859i
\(690\) 0 0
\(691\) 6.06218 + 10.5000i 0.230616 + 0.399439i 0.957990 0.286803i \(-0.0925925\pi\)
−0.727373 + 0.686242i \(0.759259\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.3923 6.00000i 0.394203 0.227593i
\(696\) 0 0
\(697\) 3.00000 5.19615i 0.113633 0.196818i
\(698\) 0 0
\(699\) 12.1244 0.458585
\(700\) 0 0
\(701\) 26.0000 0.982006 0.491003 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 0 0
\(703\) 7.79423 13.5000i 0.293965 0.509162i
\(704\) 0 0
\(705\) −22.5000 + 12.9904i −0.847399 + 0.489246i
\(706\) 0 0
\(707\) 21.6506 7.50000i 0.814256 0.282067i
\(708\) 0 0
\(709\) −4.50000 7.79423i −0.169001 0.292718i 0.769068 0.639167i \(-0.220721\pi\)
−0.938069 + 0.346449i \(0.887387\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.73205i 0.0648658i
\(714\) 0 0
\(715\) 6.00000i 0.224387i
\(716\) 0 0
\(717\) −30.0000 17.3205i −1.12037 0.646846i
\(718\) 0 0
\(719\) −12.9904 22.5000i −0.484459 0.839108i 0.515381 0.856961i \(-0.327650\pi\)
−0.999841 + 0.0178527i \(0.994317\pi\)
\(720\) 0 0
\(721\) 22.5000 + 4.33013i 0.837944 + 0.161262i
\(722\) 0 0
\(723\) −7.79423 + 4.50000i −0.289870 + 0.167357i
\(724\) 0 0
\(725\) 4.00000 6.92820i 0.148556 0.257307i
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 1.73205 3.00000i 0.0640622 0.110959i
\(732\) 0 0
\(733\) −37.5000 + 21.6506i −1.38509 + 0.799684i −0.992757 0.120137i \(-0.961667\pi\)
−0.392337 + 0.919822i \(0.628333\pi\)
\(734\) 0 0
\(735\) −20.7846 + 3.00000i −0.766652 + 0.110657i
\(736\) 0 0
\(737\) 1.50000 + 2.59808i 0.0552532 + 0.0957014i
\(738\) 0 0
\(739\) −44.1673 25.5000i −1.62472 0.938033i −0.985634 0.168898i \(-0.945979\pi\)
−0.639087 0.769135i \(-0.720687\pi\)
\(740\) 0 0
\(741\) 31.1769i 1.14531i
\(742\) 0 0
\(743\) 34.0000i 1.24734i 0.781688 + 0.623670i \(0.214359\pi\)
−0.781688 + 0.623670i \(0.785641\pi\)
\(744\) 0 0
\(745\) 1.50000 + 0.866025i 0.0549557 + 0.0317287i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −6.50000 + 33.7750i −0.237505 + 1.23411i
\(750\) 0 0
\(751\) 21.6506 12.5000i 0.790043 0.456131i −0.0499348 0.998752i \(-0.515901\pi\)
0.839978 + 0.542621i \(0.182568\pi\)
\(752\) 0 0
\(753\) 3.00000 5.19615i 0.109326 0.189358i
\(754\) 0 0
\(755\) 12.1244 0.441250
\(756\) 0 0
\(757\) 48.0000 1.74459 0.872295 0.488980i \(-0.162631\pi\)
0.872295 + 0.488980i \(0.162631\pi\)
\(758\) 0 0
\(759\) −0.866025 + 1.50000i −0.0314347 + 0.0544466i
\(760\) 0 0
\(761\) 16.5000 9.52628i 0.598125 0.345327i −0.170179 0.985413i \(-0.554435\pi\)
0.768303 + 0.640086i \(0.221101\pi\)
\(762\) 0 0
\(763\) −7.79423 22.5000i −0.282170 0.814555i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −15.5885 9.00000i −0.562867 0.324971i
\(768\) 0 0
\(769\) 3.46410i 0.124919i 0.998048 + 0.0624593i \(0.0198944\pi\)
−0.998048 + 0.0624593i \(0.980106\pi\)
\(770\) 0 0
\(771\) 9.00000i 0.324127i
\(772\) 0 0
\(773\) −22.5000 12.9904i −0.809269 0.467232i 0.0374331 0.999299i \(-0.488082\pi\)
−0.846702 + 0.532068i \(0.821415\pi\)
\(774\) 0 0
\(775\) 1.73205 + 3.00000i 0.0622171 + 0.107763i
\(776\) 0 0
\(777\) 9.00000 10.3923i 0.322873 0.372822i
\(778\) 0 0
\(779\) 15.5885 9.00000i 0.558514 0.322458i
\(780\) 0 0
\(781\) −7.00000 + 12.1244i −0.250480 + 0.433844i
\(782\) 0 0
\(783\) −20.7846 −0.742781
\(784\) 0 0
\(785\) −3.00000 −0.107075
\(786\) 0 0
\(787\) −2.59808 + 4.50000i −0.0926114 + 0.160408i −0.908609 0.417647i \(-0.862855\pi\)
0.815998 + 0.578055i \(0.196188\pi\)
\(788\) 0 0
\(789\) −34.5000 + 19.9186i −1.22823 + 0.709120i
\(790\) 0 0
\(791\) −27.7128 + 32.0000i −0.985354 + 1.13779i
\(792\) 0 0
\(793\) 9.00000 + 15.5885i 0.319599 + 0.553562i
\(794\) 0 0
\(795\) 2.59808 + 1.50000i 0.0921443 + 0.0531995i
\(796\) 0 0
\(797\) 10.3923i 0.368114i −0.982916 0.184057i \(-0.941077\pi\)
0.982916 0.184057i \(-0.0589232\pi\)
\(798\) 0 0
\(799\) 15.0000i 0.530662i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.33013 + 7.50000i 0.152807 + 0.264669i
\(804\) 0 0
\(805\) 1.50000 + 4.33013i 0.0528681 + 0.152617i
\(806\) 0 0
\(807\) −33.7750 + 19.5000i −1.18894 + 0.686433i
\(808\) 0 0
\(809\) −21.5000 + 37.2391i −0.755900 + 1.30926i 0.189026 + 0.981972i \(0.439467\pi\)
−0.944926 + 0.327285i \(0.893866\pi\)
\(810\) 0 0
\(811\) −13.8564 −0.486564 −0.243282 0.969956i \(-0.578224\pi\)
−0.243282 + 0.969956i \(0.578224\pi\)
\(812\) 0 0
\(813\) 27.0000 0.946931
\(814\) 0 0
\(815\) 18.1865 31.5000i 0.637046 1.10340i
\(816\) 0 0
\(817\) 9.00000 5.19615i 0.314870 0.181790i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5.50000 9.52628i −0.191951 0.332469i 0.753946 0.656937i \(-0.228148\pi\)
−0.945897 + 0.324468i \(0.894815\pi\)
\(822\) 0 0
\(823\) 7.79423 + 4.50000i 0.271690 + 0.156860i 0.629655 0.776875i \(-0.283196\pi\)
−0.357966 + 0.933735i \(0.616529\pi\)
\(824\) 0 0
\(825\) 3.46410i 0.120605i
\(826\) 0 0
\(827\) 22.0000i 0.765015i 0.923952 + 0.382507i \(0.124939\pi\)
−0.923952 + 0.382507i \(0.875061\pi\)
\(828\) 0 0
\(829\) 7.50000 + 4.33013i 0.260486 + 0.150392i 0.624556 0.780980i \(-0.285280\pi\)
−0.364070 + 0.931371i \(0.618613\pi\)
\(830\) 0 0
\(831\) 11.2583 + 19.5000i 0.390547 + 0.676448i
\(832\) 0 0
\(833\) −4.50000 + 11.2583i −0.155916 + 0.390078i
\(834\) 0 0
\(835\) −25.9808 + 15.0000i −0.899101 + 0.519096i
\(836\) 0 0
\(837\) 4.50000 7.79423i 0.155543 0.269408i
\(838\) 0 0
\(839\) 48.4974 1.67432 0.837158 0.546960i \(-0.184215\pi\)
0.837158 + 0.546960i \(0.184215\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) −3.46410 + 6.00000i −0.119310 + 0.206651i
\(844\) 0 0
\(845\) 1.50000 0.866025i 0.0516016 0.0297922i
\(846\) 0 0
\(847\) −25.9808 5.00000i −0.892710 0.171802i
\(848\) 0 0
\(849\) −10.5000 18.1865i −0.360359 0.624160i
\(850\) 0 0
\(851\) −2.59808 1.50000i −0.0890609 0.0514193i
\(852\) 0 0
\(853\) 24.2487i 0.830260i −0.909762 0.415130i \(-0.863736\pi\)
0.909762 0.415130i \(-0.136264\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −22.5000 12.9904i −0.768585 0.443743i 0.0637844 0.997964i \(-0.479683\pi\)
−0.832370 + 0.554221i \(0.813016\pi\)
\(858\) 0 0
\(859\) 25.1147 + 43.5000i 0.856904 + 1.48420i 0.874868 + 0.484362i \(0.160948\pi\)
−0.0179638 + 0.999839i \(0.505718\pi\)
\(860\) 0 0
\(861\) 15.0000 5.19615i 0.511199 0.177084i
\(862\) 0 0
\(863\) 30.3109 17.5000i 1.03179 0.595707i 0.114296 0.993447i \(-0.463539\pi\)
0.917498 + 0.397740i \(0.130205\pi\)
\(864\) 0 0
\(865\) 10.5000 18.1865i 0.357011 0.618361i
\(866\) 0 0
\(867\) −24.2487 −0.823529
\(868\) 0 0
\(869\) −9.00000 −0.305304
\(870\) 0 0
\(871\) −5.19615 + 9.00000i −0.176065 + 0.304953i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 24.2487 + 21.0000i 0.819756 + 0.709930i
\(876\) 0 0
\(877\) 0.500000 + 0.866025i 0.0168838 + 0.0292436i 0.874344 0.485307i \(-0.161292\pi\)
−0.857460 + 0.514551i \(0.827959\pi\)
\(878\) 0 0
\(879\) −31.1769 18.0000i −1.05157 0.607125i
\(880\) 0 0
\(881\) 13.8564i 0.466834i −0.972377 0.233417i \(-0.925009\pi\)
0.972377 0.233417i \(-0.0749907\pi\)
\(882\) 0 0
\(883\) 10.0000i 0.336527i −0.985742 0.168263i \(-0.946184\pi\)
0.985742 0.168263i \(-0.0538159\pi\)
\(884\) 0 0
\(885\) 13.5000 + 7.79423i 0.453798 + 0.262000i
\(886\) 0 0
\(887\) 12.9904 + 22.5000i 0.436174 + 0.755476i 0.997391 0.0721931i \(-0.0229998\pi\)
−0.561216 + 0.827669i \(0.689666\pi\)
\(888\) 0 0
\(889\) −12.0000 10.3923i −0.402467 0.348547i
\(890\) 0 0
\(891\) 7.79423 4.50000i 0.261116 0.150756i
\(892\) 0 0
\(893\) −22.5000 + 38.9711i −0.752934 + 1.30412i
\(894\) 0 0
\(895\) −32.9090 −1.10003
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) 0 0
\(899\) −3.46410 + 6.00000i −0.115534 + 0.200111i
\(900\) 0 0
\(901\) 1.50000 0.866025i 0.0499722 0.0288515i
\(902\) 0 0
\(903\) 8.66025 3.00000i 0.288195 0.0998337i
\(904\) 0 0
\(905\) 6.00000 + 10.3923i 0.199447 + 0.345452i
\(906\) 0 0
\(907\) 6.06218 + 3.50000i 0.201291 + 0.116216i 0.597258 0.802049i \(-0.296257\pi\)
−0.395966 + 0.918265i \(0.629590\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 26.0000i 0.861418i 0.902491 + 0.430709i \(0.141737\pi\)
−0.902491 + 0.430709i \(0.858263\pi\)
\(912\) 0 0
\(913\) −12.0000 6.92820i −0.397142 0.229290i
\(914\) 0 0
\(915\) −7.79423 13.5000i −0.257669 0.446296i
\(916\) 0 0
\(917\) 13.5000 + 2.59808i 0.445809 + 0.0857960i
\(918\) 0 0
\(919\) 0.866025 0.500000i 0.0285675 0.0164935i −0.485648 0.874154i \(-0.661416\pi\)
0.514216 + 0.857661i \(0.328083\pi\)
\(920\) 0 0
\(921\) −18.0000 + 31.1769i −0.593120 + 1.02731i
\(922\) 0 0
\(923\) −48.4974 −1.59631
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.50000 4.33013i 0.246067 0.142067i −0.371895 0.928275i \(-0.621292\pi\)
0.617962 + 0.786208i \(0.287959\pi\)
\(930\) 0 0
\(931\) −28.5788 + 22.5000i −0.936634 + 0.737408i
\(932\) 0 0
\(933\) 7.50000 + 12.9904i 0.245539 + 0.425286i
\(934\) 0 0
\(935\) −2.59808 1.50000i −0.0849662 0.0490552i
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 3.00000i 0.0979013i
\(940\) 0 0
\(941\) −49.5000 28.5788i −1.61365 0.931644i −0.988514 0.151131i \(-0.951708\pi\)
−0.625140 0.780513i \(-0.714958\pi\)
\(942\) 0 0
\(943\) −1.73205 3.00000i −0.0564033 0.0976934i
\(944\) 0 0
\(945\) 4.50000 23.3827i 0.146385 0.760639i
\(946\) 0 0
\(947\) 25.1147 14.5000i 0.816119 0.471187i −0.0329571 0.999457i \(-0.510492\pi\)
0.849076 + 0.528270i \(0.177159\pi\)
\(948\) 0 0
\(949\) −15.0000 + 25.9808i −0.486921 + 0.843371i
\(950\) 0 0
\(951\) 19.0526 0.617822
\(952\) 0 0
\(953\) −8.00000 −0.259145 −0.129573 0.991570i \(-0.541361\pi\)
−0.129573 + 0.991570i \(0.541361\pi\)
\(954\) 0 0
\(955\) 0.866025 1.50000i 0.0280239 0.0485389i
\(956\) 0 0
\(957\) −6.00000 + 3.46410i −0.193952 + 0.111979i
\(958\) 0 0
\(959\) 0.866025 + 2.50000i 0.0279654 + 0.0807292i
\(960\) 0 0
\(961\) 14.0000 + 24.2487i 0.451613 + 0.782216i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 25.9808i 0.836350i
\(966\) 0 0
\(967\) 6.00000i 0.192947i 0.995336 + 0.0964735i \(0.0307563\pi\)
−0.995336 + 0.0964735i \(0.969244\pi\)
\(968\) 0 0
\(969\) 13.5000 + 7.79423i 0.433682 + 0.250387i
\(970\) 0 0
\(971\) 30.3109 + 52.5000i 0.972723 + 1.68481i 0.687254 + 0.726417i \(0.258816\pi\)
0.285469 + 0.958388i \(0.407851\pi\)
\(972\) 0 0
\(973\) 12.0000 13.8564i 0.384702 0.444216i
\(974\) 0 0
\(975\) −10.3923 + 6.00000i −0.332820 + 0.192154i
\(976\) 0 0
\(977\) 15.5000 26.8468i 0.495889 0.858905i −0.504100 0.863645i \(-0.668176\pi\)
0.999989 + 0.00474056i \(0.00150897\pi\)
\(978\) 0 0
\(979\) 15.5885 0.498209
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −30.3109 + 52.5000i −0.966767 + 1.67449i −0.261977 + 0.965074i \(0.584374\pi\)
−0.704790 + 0.709416i \(0.748959\pi\)
\(984\) 0 0
\(985\) −24.0000 + 13.8564i −0.764704 + 0.441502i
\(986\) 0 0
\(987\) −25.9808 + 30.0000i −0.826977 + 0.954911i
\(988\) 0 0
\(989\) −1.00000 1.73205i −0.0317982 0.0550760i
\(990\) 0 0
\(991\) −19.9186 11.5000i −0.632735 0.365310i 0.149076 0.988826i \(-0.452370\pi\)
−0.781810 + 0.623516i \(0.785704\pi\)
\(992\) 0 0
\(993\) 12.1244i 0.384755i
\(994\) 0 0
\(995\) 39.0000i 1.23638i
\(996\) 0 0
\(997\) −19.5000 11.2583i −0.617571 0.356555i 0.158352 0.987383i \(-0.449382\pi\)
−0.775923 + 0.630828i \(0.782715\pi\)
\(998\) 0 0
\(999\) 7.79423 + 13.5000i 0.246598 + 0.427121i
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 448.2.p.d.255.2 4
4.3 odd 2 inner 448.2.p.d.255.1 4
7.3 odd 6 3136.2.f.e.3135.3 4
7.4 even 3 3136.2.f.e.3135.2 4
7.5 odd 6 inner 448.2.p.d.383.1 4
8.3 odd 2 28.2.f.a.3.1 4
8.5 even 2 28.2.f.a.3.2 yes 4
24.5 odd 2 252.2.bf.e.199.1 4
24.11 even 2 252.2.bf.e.199.2 4
28.3 even 6 3136.2.f.e.3135.1 4
28.11 odd 6 3136.2.f.e.3135.4 4
28.19 even 6 inner 448.2.p.d.383.2 4
40.3 even 4 700.2.t.b.199.1 4
40.13 odd 4 700.2.t.a.199.1 4
40.19 odd 2 700.2.p.a.451.2 4
40.27 even 4 700.2.t.a.199.2 4
40.29 even 2 700.2.p.a.451.1 4
40.37 odd 4 700.2.t.b.199.2 4
56.3 even 6 196.2.d.b.195.2 4
56.5 odd 6 28.2.f.a.19.1 yes 4
56.11 odd 6 196.2.d.b.195.1 4
56.13 odd 2 196.2.f.a.31.2 4
56.19 even 6 28.2.f.a.19.2 yes 4
56.27 even 2 196.2.f.a.31.1 4
56.37 even 6 196.2.f.a.19.1 4
56.45 odd 6 196.2.d.b.195.3 4
56.51 odd 6 196.2.f.a.19.2 4
56.53 even 6 196.2.d.b.195.4 4
168.5 even 6 252.2.bf.e.19.2 4
168.11 even 6 1764.2.b.a.1567.4 4
168.53 odd 6 1764.2.b.a.1567.2 4
168.59 odd 6 1764.2.b.a.1567.3 4
168.101 even 6 1764.2.b.a.1567.1 4
168.131 odd 6 252.2.bf.e.19.1 4
280.19 even 6 700.2.p.a.551.1 4
280.117 even 12 700.2.t.b.299.1 4
280.173 even 12 700.2.t.a.299.2 4
280.187 odd 12 700.2.t.a.299.1 4
280.229 odd 6 700.2.p.a.551.2 4
280.243 odd 12 700.2.t.b.299.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.2.f.a.3.1 4 8.3 odd 2
28.2.f.a.3.2 yes 4 8.5 even 2
28.2.f.a.19.1 yes 4 56.5 odd 6
28.2.f.a.19.2 yes 4 56.19 even 6
196.2.d.b.195.1 4 56.11 odd 6
196.2.d.b.195.2 4 56.3 even 6
196.2.d.b.195.3 4 56.45 odd 6
196.2.d.b.195.4 4 56.53 even 6
196.2.f.a.19.1 4 56.37 even 6
196.2.f.a.19.2 4 56.51 odd 6
196.2.f.a.31.1 4 56.27 even 2
196.2.f.a.31.2 4 56.13 odd 2
252.2.bf.e.19.1 4 168.131 odd 6
252.2.bf.e.19.2 4 168.5 even 6
252.2.bf.e.199.1 4 24.5 odd 2
252.2.bf.e.199.2 4 24.11 even 2
448.2.p.d.255.1 4 4.3 odd 2 inner
448.2.p.d.255.2 4 1.1 even 1 trivial
448.2.p.d.383.1 4 7.5 odd 6 inner
448.2.p.d.383.2 4 28.19 even 6 inner
700.2.p.a.451.1 4 40.29 even 2
700.2.p.a.451.2 4 40.19 odd 2
700.2.p.a.551.1 4 280.19 even 6
700.2.p.a.551.2 4 280.229 odd 6
700.2.t.a.199.1 4 40.13 odd 4
700.2.t.a.199.2 4 40.27 even 4
700.2.t.a.299.1 4 280.187 odd 12
700.2.t.a.299.2 4 280.173 even 12
700.2.t.b.199.1 4 40.3 even 4
700.2.t.b.199.2 4 40.37 odd 4
700.2.t.b.299.1 4 280.117 even 12
700.2.t.b.299.2 4 280.243 odd 12
1764.2.b.a.1567.1 4 168.101 even 6
1764.2.b.a.1567.2 4 168.53 odd 6
1764.2.b.a.1567.3 4 168.59 odd 6
1764.2.b.a.1567.4 4 168.11 even 6
3136.2.f.e.3135.1 4 28.3 even 6
3136.2.f.e.3135.2 4 7.4 even 3
3136.2.f.e.3135.3 4 7.3 odd 6
3136.2.f.e.3135.4 4 28.11 odd 6