Properties

Label 2268.2.i.i.2053.1
Level $2268$
Weight $2$
Character 2268.2053
Analytic conductor $18.110$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2268,2,Mod(865,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.865");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.i (of order \(3\), 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: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 10x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 756)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 2053.1
Root \(-1.58114 - 2.73861i\) of defining polynomial
Character \(\chi\) \(=\) 2268.2053
Dual form 2268.2.i.i.865.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.58114 - 2.73861i) q^{5} +(-2.50000 + 0.866025i) q^{7} +O(q^{10})\) \(q+(-1.58114 - 2.73861i) q^{5} +(-2.50000 + 0.866025i) q^{7} +(-3.16228 + 5.47723i) q^{11} +(1.58114 + 2.73861i) q^{17} +(3.50000 - 6.06218i) q^{19} +(1.58114 + 2.73861i) q^{23} +(-2.50000 + 4.33013i) q^{25} +(-1.58114 - 2.73861i) q^{29} +3.00000 q^{31} +(6.32456 + 5.47723i) q^{35} +(2.00000 - 3.46410i) q^{37} +(-4.74342 + 8.21584i) q^{41} +(-2.50000 - 4.33013i) q^{43} +9.48683 q^{47} +(5.50000 - 4.33013i) q^{49} +(-4.74342 - 8.21584i) q^{53} +20.0000 q^{55} +12.6491 q^{59} -3.00000 q^{61} +10.0000 q^{67} -12.6491 q^{71} +(2.50000 + 4.33013i) q^{73} +(3.16228 - 16.4317i) q^{77} +12.0000 q^{79} +(-3.16228 - 5.47723i) q^{83} +(5.00000 - 8.66025i) q^{85} +(4.74342 - 8.21584i) q^{89} -22.1359 q^{95} +(2.50000 + 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{7} + 14 q^{19} - 10 q^{25} + 12 q^{31} + 8 q^{37} - 10 q^{43} + 22 q^{49} + 80 q^{55} - 12 q^{61} + 40 q^{67} + 10 q^{73} + 48 q^{79} + 20 q^{85} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) −1.58114 2.73861i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(6\) 0 0
\(7\) −2.50000 + 0.866025i −0.944911 + 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.16228 + 5.47723i −0.953463 + 1.65145i −0.215615 + 0.976478i \(0.569176\pi\)
−0.737848 + 0.674967i \(0.764158\pi\)
\(12\) 0 0
\(13\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.58114 + 2.73861i 0.383482 + 0.664211i 0.991557 0.129668i \(-0.0413913\pi\)
−0.608075 + 0.793880i \(0.708058\pi\)
\(18\) 0 0
\(19\) 3.50000 6.06218i 0.802955 1.39076i −0.114708 0.993399i \(-0.536593\pi\)
0.917663 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.58114 + 2.73861i 0.329690 + 0.571040i 0.982450 0.186524i \(-0.0597224\pi\)
−0.652760 + 0.757565i \(0.726389\pi\)
\(24\) 0 0
\(25\) −2.50000 + 4.33013i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.58114 2.73861i −0.293610 0.508548i 0.681051 0.732236i \(-0.261523\pi\)
−0.974661 + 0.223689i \(0.928190\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.32456 + 5.47723i 1.06904 + 0.925820i
\(36\) 0 0
\(37\) 2.00000 3.46410i 0.328798 0.569495i −0.653476 0.756948i \(-0.726690\pi\)
0.982274 + 0.187453i \(0.0600231\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.74342 + 8.21584i −0.740797 + 1.28310i 0.211336 + 0.977414i \(0.432219\pi\)
−0.952133 + 0.305685i \(0.901115\pi\)
\(42\) 0 0
\(43\) −2.50000 4.33013i −0.381246 0.660338i 0.609994 0.792406i \(-0.291172\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.48683 1.38380 0.691898 0.721995i \(-0.256775\pi\)
0.691898 + 0.721995i \(0.256775\pi\)
\(48\) 0 0
\(49\) 5.50000 4.33013i 0.785714 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.74342 8.21584i −0.651558 1.12853i −0.982745 0.184967i \(-0.940782\pi\)
0.331186 0.943565i \(-0.392551\pi\)
\(54\) 0 0
\(55\) 20.0000 2.69680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.6491 1.64677 0.823387 0.567480i \(-0.192082\pi\)
0.823387 + 0.567480i \(0.192082\pi\)
\(60\) 0 0
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.6491 −1.50117 −0.750587 0.660772i \(-0.770229\pi\)
−0.750587 + 0.660772i \(0.770229\pi\)
\(72\) 0 0
\(73\) 2.50000 + 4.33013i 0.292603 + 0.506803i 0.974424 0.224716i \(-0.0721453\pi\)
−0.681822 + 0.731519i \(0.738812\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.16228 16.4317i 0.360375 1.87256i
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.16228 5.47723i −0.347105 0.601204i 0.638629 0.769515i \(-0.279502\pi\)
−0.985734 + 0.168311i \(0.946169\pi\)
\(84\) 0 0
\(85\) 5.00000 8.66025i 0.542326 0.939336i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.74342 8.21584i 0.502801 0.870877i −0.497194 0.867640i \(-0.665636\pi\)
0.999995 0.00323751i \(-0.00103053\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −22.1359 −2.27110
\(96\) 0 0
\(97\) 2.50000 + 4.33013i 0.253837 + 0.439658i 0.964579 0.263795i \(-0.0849741\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.16228 5.47723i 0.314658 0.545004i −0.664706 0.747105i \(-0.731443\pi\)
0.979365 + 0.202100i \(0.0647767\pi\)
\(102\) 0 0
\(103\) −1.00000 1.73205i −0.0985329 0.170664i 0.812545 0.582899i \(-0.198082\pi\)
−0.911078 + 0.412235i \(0.864748\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(108\) 0 0
\(109\) −3.50000 6.06218i −0.335239 0.580651i 0.648292 0.761392i \(-0.275484\pi\)
−0.983531 + 0.180741i \(0.942150\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.16228 5.47723i 0.297482 0.515254i −0.678077 0.734991i \(-0.737186\pi\)
0.975559 + 0.219737i \(0.0705198\pi\)
\(114\) 0 0
\(115\) 5.00000 8.66025i 0.466252 0.807573i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.32456 5.47723i −0.579771 0.502096i
\(120\) 0 0
\(121\) −14.5000 25.1147i −1.31818 2.28316i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 21.0000 1.86345 0.931724 0.363166i \(-0.118304\pi\)
0.931724 + 0.363166i \(0.118304\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.90569 + 13.6931i 0.690724 + 1.19637i 0.971601 + 0.236625i \(0.0760412\pi\)
−0.280877 + 0.959744i \(0.590625\pi\)
\(132\) 0 0
\(133\) −3.50000 + 18.1865i −0.303488 + 1.57697i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.58114 + 2.73861i −0.135086 + 0.233975i −0.925630 0.378429i \(-0.876464\pi\)
0.790544 + 0.612405i \(0.209798\pi\)
\(138\) 0 0
\(139\) 1.00000 1.73205i 0.0848189 0.146911i −0.820495 0.571654i \(-0.806302\pi\)
0.905314 + 0.424743i \(0.139635\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −5.00000 + 8.66025i −0.415227 + 0.719195i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.74342 + 8.21584i 0.388596 + 0.673068i 0.992261 0.124170i \(-0.0396269\pi\)
−0.603665 + 0.797238i \(0.706294\pi\)
\(150\) 0 0
\(151\) 8.50000 14.7224i 0.691720 1.19809i −0.279554 0.960130i \(-0.590186\pi\)
0.971274 0.237964i \(-0.0764802\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.74342 8.21584i −0.381000 0.659912i
\(156\) 0 0
\(157\) −16.0000 −1.27694 −0.638470 0.769647i \(-0.720432\pi\)
−0.638470 + 0.769647i \(0.720432\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.32456 5.47723i −0.498445 0.431666i
\(162\) 0 0
\(163\) −8.50000 + 14.7224i −0.665771 + 1.15315i 0.313304 + 0.949653i \(0.398564\pi\)
−0.979076 + 0.203497i \(0.934769\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.16228 5.47723i 0.244704 0.423840i −0.717344 0.696719i \(-0.754642\pi\)
0.962048 + 0.272879i \(0.0879758\pi\)
\(168\) 0 0
\(169\) 6.50000 + 11.2583i 0.500000 + 0.866025i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.16228 0.240424 0.120212 0.992748i \(-0.461643\pi\)
0.120212 + 0.992748i \(0.461643\pi\)
\(174\) 0 0
\(175\) 2.50000 12.9904i 0.188982 0.981981i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.74342 + 8.21584i 0.354540 + 0.614081i 0.987039 0.160480i \(-0.0513043\pi\)
−0.632499 + 0.774561i \(0.717971\pi\)
\(180\) 0 0
\(181\) 21.0000 1.56092 0.780459 0.625207i \(-0.214986\pi\)
0.780459 + 0.625207i \(0.214986\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −12.6491 −0.929981
\(186\) 0 0
\(187\) −20.0000 −1.46254
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.48683 0.686443 0.343222 0.939254i \(-0.388482\pi\)
0.343222 + 0.939254i \(0.388482\pi\)
\(192\) 0 0
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 25.2982 1.80242 0.901212 0.433379i \(-0.142679\pi\)
0.901212 + 0.433379i \(0.142679\pi\)
\(198\) 0 0
\(199\) 6.50000 + 11.2583i 0.460773 + 0.798082i 0.999000 0.0447181i \(-0.0142390\pi\)
−0.538227 + 0.842800i \(0.680906\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.32456 + 5.47723i 0.443897 + 0.384426i
\(204\) 0 0
\(205\) 30.0000 2.09529
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 22.1359 + 38.3406i 1.53118 + 2.65207i
\(210\) 0 0
\(211\) 0.500000 0.866025i 0.0344214 0.0596196i −0.848301 0.529514i \(-0.822374\pi\)
0.882723 + 0.469894i \(0.155708\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −7.90569 + 13.6931i −0.539164 + 0.933859i
\(216\) 0 0
\(217\) −7.50000 + 2.59808i −0.509133 + 0.176369i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.00000 1.73205i −0.0669650 0.115987i 0.830599 0.556871i \(-0.187998\pi\)
−0.897564 + 0.440884i \(0.854665\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.74342 + 8.21584i −0.314832 + 0.545304i −0.979402 0.201922i \(-0.935281\pi\)
0.664570 + 0.747226i \(0.268615\pi\)
\(228\) 0 0
\(229\) −6.50000 11.2583i −0.429532 0.743971i 0.567300 0.823511i \(-0.307988\pi\)
−0.996832 + 0.0795401i \(0.974655\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.90569 + 13.6931i −0.517919 + 0.897062i 0.481864 + 0.876246i \(0.339960\pi\)
−0.999783 + 0.0208165i \(0.993373\pi\)
\(234\) 0 0
\(235\) −15.0000 25.9808i −0.978492 1.69480i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.74342 8.21584i 0.306826 0.531438i −0.670840 0.741602i \(-0.734066\pi\)
0.977666 + 0.210164i \(0.0673997\pi\)
\(240\) 0 0
\(241\) 0.500000 0.866025i 0.0322078 0.0557856i −0.849472 0.527633i \(-0.823079\pi\)
0.881680 + 0.471848i \(0.156413\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −20.5548 8.21584i −1.31320 0.524891i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −9.48683 −0.598804 −0.299402 0.954127i \(-0.596787\pi\)
−0.299402 + 0.954127i \(0.596787\pi\)
\(252\) 0 0
\(253\) −20.0000 −1.25739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.48683 + 16.4317i 0.591772 + 1.02498i 0.993994 + 0.109437i \(0.0349049\pi\)
−0.402221 + 0.915542i \(0.631762\pi\)
\(258\) 0 0
\(259\) −2.00000 + 10.3923i −0.124274 + 0.645746i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −9.48683 + 16.4317i −0.584983 + 1.01322i 0.409894 + 0.912133i \(0.365566\pi\)
−0.994878 + 0.101088i \(0.967768\pi\)
\(264\) 0 0
\(265\) −15.0000 + 25.9808i −0.921443 + 1.59599i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.16228 + 5.47723i 0.192807 + 0.333952i 0.946180 0.323642i \(-0.104907\pi\)
−0.753372 + 0.657594i \(0.771574\pi\)
\(270\) 0 0
\(271\) −4.50000 + 7.79423i −0.273356 + 0.473466i −0.969719 0.244224i \(-0.921467\pi\)
0.696363 + 0.717689i \(0.254800\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −15.8114 27.3861i −0.953463 1.65145i
\(276\) 0 0
\(277\) 0.500000 0.866025i 0.0300421 0.0520344i −0.850613 0.525792i \(-0.823769\pi\)
0.880656 + 0.473757i \(0.157103\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.16228 5.47723i −0.188646 0.326744i 0.756153 0.654395i \(-0.227076\pi\)
−0.944799 + 0.327651i \(0.893743\pi\)
\(282\) 0 0
\(283\) 13.0000 0.772770 0.386385 0.922338i \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.74342 24.6475i 0.279995 1.45490i
\(288\) 0 0
\(289\) 3.50000 6.06218i 0.205882 0.356599i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(294\) 0 0
\(295\) −20.0000 34.6410i −1.16445 2.01688i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 10.0000 + 8.66025i 0.576390 + 0.499169i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.74342 + 8.21584i 0.271607 + 0.470438i
\(306\) 0 0
\(307\) 25.0000 1.42683 0.713413 0.700744i \(-0.247149\pi\)
0.713413 + 0.700744i \(0.247149\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −6.32456 −0.358633 −0.179316 0.983791i \(-0.557389\pi\)
−0.179316 + 0.983791i \(0.557389\pi\)
\(312\) 0 0
\(313\) −17.0000 −0.960897 −0.480448 0.877023i \(-0.659526\pi\)
−0.480448 + 0.877023i \(0.659526\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −18.9737 −1.06567 −0.532834 0.846220i \(-0.678873\pi\)
−0.532834 + 0.846220i \(0.678873\pi\)
\(318\) 0 0
\(319\) 20.0000 1.11979
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 22.1359 1.23168
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −23.7171 + 8.21584i −1.30757 + 0.452954i
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −15.8114 27.3861i −0.863868 1.49626i
\(336\) 0 0
\(337\) −10.5000 + 18.1865i −0.571971 + 0.990684i 0.424392 + 0.905479i \(0.360488\pi\)
−0.996363 + 0.0852050i \(0.972845\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −9.48683 + 16.4317i −0.513741 + 0.889825i
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 8.50000 + 14.7224i 0.454995 + 0.788074i 0.998688 0.0512103i \(-0.0163079\pi\)
−0.543693 + 0.839284i \(0.682975\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.48683 16.4317i 0.504933 0.874570i −0.495051 0.868864i \(-0.664850\pi\)
0.999984 0.00570573i \(-0.00181620\pi\)
\(354\) 0 0
\(355\) 20.0000 + 34.6410i 1.06149 + 1.83855i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.6491 21.9089i 0.667595 1.15631i −0.310980 0.950416i \(-0.600657\pi\)
0.978575 0.205891i \(-0.0660093\pi\)
\(360\) 0 0
\(361\) −15.0000 25.9808i −0.789474 1.36741i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7.90569 13.6931i 0.413803 0.716728i
\(366\) 0 0
\(367\) 9.50000 16.4545i 0.495896 0.858917i −0.504093 0.863649i \(-0.668173\pi\)
0.999989 + 0.00473247i \(0.00150640\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 18.9737 + 16.4317i 0.985064 + 0.853090i
\(372\) 0 0
\(373\) 8.50000 + 14.7224i 0.440113 + 0.762299i 0.997697 0.0678218i \(-0.0216049\pi\)
−0.557584 + 0.830120i \(0.688272\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −26.0000 −1.33553 −0.667765 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6.32456 + 10.9545i 0.323170 + 0.559746i 0.981140 0.193297i \(-0.0619182\pi\)
−0.657971 + 0.753044i \(0.728585\pi\)
\(384\) 0 0
\(385\) −50.0000 + 17.3205i −2.54824 + 0.882735i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 17.3925 30.1247i 0.881836 1.52738i 0.0325386 0.999470i \(-0.489641\pi\)
0.849298 0.527915i \(-0.177026\pi\)
\(390\) 0 0
\(391\) −5.00000 + 8.66025i −0.252861 + 0.437968i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −18.9737 32.8634i −0.954669 1.65353i
\(396\) 0 0
\(397\) 7.50000 12.9904i 0.376414 0.651969i −0.614123 0.789210i \(-0.710490\pi\)
0.990538 + 0.137241i \(0.0438236\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.48683 + 16.4317i 0.473750 + 0.820559i 0.999548 0.0300503i \(-0.00956676\pi\)
−0.525799 + 0.850609i \(0.676233\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.6491 + 21.9089i 0.626993 + 1.08598i
\(408\) 0 0
\(409\) −16.0000 −0.791149 −0.395575 0.918434i \(-0.629455\pi\)
−0.395575 + 0.918434i \(0.629455\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −31.6228 + 10.9545i −1.55606 + 0.539033i
\(414\) 0 0
\(415\) −10.0000 + 17.3205i −0.490881 + 0.850230i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.74342 8.21584i 0.231731 0.401370i −0.726587 0.687075i \(-0.758894\pi\)
0.958318 + 0.285705i \(0.0922277\pi\)
\(420\) 0 0
\(421\) −1.50000 2.59808i −0.0731055 0.126622i 0.827155 0.561973i \(-0.189958\pi\)
−0.900261 + 0.435351i \(0.856624\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −15.8114 −0.766965
\(426\) 0 0
\(427\) 7.50000 2.59808i 0.362950 0.125730i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(432\) 0 0
\(433\) 5.00000 0.240285 0.120142 0.992757i \(-0.461665\pi\)
0.120142 + 0.992757i \(0.461665\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 22.1359 1.05891
\(438\) 0 0
\(439\) −18.0000 −0.859093 −0.429547 0.903045i \(-0.641327\pi\)
−0.429547 + 0.903045i \(0.641327\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 34.7851 1.65269 0.826344 0.563166i \(-0.190417\pi\)
0.826344 + 0.563166i \(0.190417\pi\)
\(444\) 0 0
\(445\) −30.0000 −1.42214
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −34.7851 −1.64161 −0.820804 0.571210i \(-0.806474\pi\)
−0.820804 + 0.571210i \(0.806474\pi\)
\(450\) 0 0
\(451\) −30.0000 51.9615i −1.41264 2.44677i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −29.0000 −1.35656 −0.678281 0.734802i \(-0.737275\pi\)
−0.678281 + 0.734802i \(0.737275\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.32456 10.9545i −0.294564 0.510200i 0.680319 0.732916i \(-0.261841\pi\)
−0.974883 + 0.222716i \(0.928508\pi\)
\(462\) 0 0
\(463\) −12.5000 + 21.6506i −0.580924 + 1.00619i 0.414446 + 0.910074i \(0.363975\pi\)
−0.995370 + 0.0961164i \(0.969358\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.16228 5.47723i 0.146333 0.253456i −0.783537 0.621346i \(-0.786586\pi\)
0.929869 + 0.367890i \(0.119920\pi\)
\(468\) 0 0
\(469\) −25.0000 + 8.66025i −1.15439 + 0.399893i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 31.6228 1.45402
\(474\) 0 0
\(475\) 17.5000 + 30.3109i 0.802955 + 1.39076i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 11.0680 19.1703i 0.505709 0.875913i −0.494270 0.869309i \(-0.664564\pi\)
0.999978 0.00660425i \(-0.00210221\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.90569 13.6931i 0.358979 0.621770i
\(486\) 0 0
\(487\) 0.500000 + 0.866025i 0.0226572 + 0.0392434i 0.877132 0.480250i \(-0.159454\pi\)
−0.854475 + 0.519493i \(0.826121\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −9.48683 + 16.4317i −0.428135 + 0.741551i −0.996707 0.0810819i \(-0.974162\pi\)
0.568573 + 0.822633i \(0.307496\pi\)
\(492\) 0 0
\(493\) 5.00000 8.66025i 0.225189 0.390038i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 31.6228 10.9545i 1.41848 0.491374i
\(498\) 0 0
\(499\) −3.50000 6.06218i −0.156682 0.271380i 0.776989 0.629515i \(-0.216746\pi\)
−0.933670 + 0.358134i \(0.883413\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 28.4605 1.26899 0.634495 0.772927i \(-0.281208\pi\)
0.634495 + 0.772927i \(0.281208\pi\)
\(504\) 0 0
\(505\) −20.0000 −0.889988
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −18.9737 32.8634i −0.840993 1.45664i −0.889056 0.457798i \(-0.848638\pi\)
0.0480632 0.998844i \(-0.484695\pi\)
\(510\) 0 0
\(511\) −10.0000 8.66025i −0.442374 0.383107i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.16228 + 5.47723i −0.139347 + 0.241355i
\(516\) 0 0
\(517\) −30.0000 + 51.9615i −1.31940 + 2.28527i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(522\) 0 0
\(523\) −16.0000 + 27.7128i −0.699631 + 1.21180i 0.268963 + 0.963150i \(0.413319\pi\)
−0.968594 + 0.248646i \(0.920014\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.74342 + 8.21584i 0.206626 + 0.357888i
\(528\) 0 0
\(529\) 6.50000 11.2583i 0.282609 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.32456 + 43.8178i 0.272418 + 1.88737i
\(540\) 0 0
\(541\) 10.5000 18.1865i 0.451430 0.781900i −0.547045 0.837103i \(-0.684247\pi\)
0.998475 + 0.0552031i \(0.0175806\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −11.0680 + 19.1703i −0.474100 + 0.821165i
\(546\) 0 0
\(547\) −10.5000 18.1865i −0.448948 0.777600i 0.549370 0.835579i \(-0.314868\pi\)
−0.998318 + 0.0579790i \(0.981534\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −22.1359 −0.943023
\(552\) 0 0
\(553\) −30.0000 + 10.3923i −1.27573 + 0.441926i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.58114 2.73861i −0.0669950 0.116039i 0.830582 0.556896i \(-0.188008\pi\)
−0.897577 + 0.440857i \(0.854674\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −41.1096 −1.73256 −0.866282 0.499556i \(-0.833497\pi\)
−0.866282 + 0.499556i \(0.833497\pi\)
\(564\) 0 0
\(565\) −20.0000 −0.841406
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −37.9473 −1.59083 −0.795417 0.606062i \(-0.792748\pi\)
−0.795417 + 0.606062i \(0.792748\pi\)
\(570\) 0 0
\(571\) 39.0000 1.63210 0.816050 0.577982i \(-0.196160\pi\)
0.816050 + 0.577982i \(0.196160\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −15.8114 −0.659380
\(576\) 0 0
\(577\) 22.0000 + 38.1051i 0.915872 + 1.58634i 0.805620 + 0.592433i \(0.201833\pi\)
0.110252 + 0.993904i \(0.464834\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 12.6491 + 10.9545i 0.524774 + 0.454467i
\(582\) 0 0
\(583\) 60.0000 2.48495
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −17.3925 30.1247i −0.717866 1.24338i −0.961844 0.273600i \(-0.911786\pi\)
0.243977 0.969781i \(-0.421548\pi\)
\(588\) 0 0
\(589\) 10.5000 18.1865i 0.432645 0.749363i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9.48683 16.4317i 0.389578 0.674768i −0.602815 0.797881i \(-0.705954\pi\)
0.992393 + 0.123113i \(0.0392877\pi\)
\(594\) 0 0
\(595\) −5.00000 + 25.9808i −0.204980 + 1.06511i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 28.4605 1.16286 0.581432 0.813595i \(-0.302493\pi\)
0.581432 + 0.813595i \(0.302493\pi\)
\(600\) 0 0
\(601\) 9.50000 + 16.4545i 0.387513 + 0.671192i 0.992114 0.125336i \(-0.0400009\pi\)
−0.604601 + 0.796528i \(0.706668\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −45.8530 + 79.4198i −1.86419 + 3.22887i
\(606\) 0 0
\(607\) −7.50000 12.9904i −0.304416 0.527263i 0.672715 0.739901i \(-0.265128\pi\)
−0.977131 + 0.212638i \(0.931794\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −16.5000 28.5788i −0.666429 1.15429i −0.978896 0.204360i \(-0.934489\pi\)
0.312467 0.949929i \(-0.398845\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.6491 + 21.9089i −0.509234 + 0.882019i 0.490709 + 0.871324i \(0.336738\pi\)
−0.999943 + 0.0106956i \(0.996595\pi\)
\(618\) 0 0
\(619\) −11.0000 + 19.0526i −0.442127 + 0.765787i −0.997847 0.0655827i \(-0.979109\pi\)
0.555720 + 0.831370i \(0.312443\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.74342 + 24.6475i −0.190041 + 0.987482i
\(624\) 0 0
\(625\) 12.5000 + 21.6506i 0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.6491 0.504353
\(630\) 0 0
\(631\) −31.0000 −1.23409 −0.617045 0.786928i \(-0.711670\pi\)
−0.617045 + 0.786928i \(0.711670\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −33.2039 57.5109i −1.31766 2.28225i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.58114 2.73861i 0.0624512 0.108169i −0.833109 0.553108i \(-0.813442\pi\)
0.895561 + 0.444940i \(0.146775\pi\)
\(642\) 0 0
\(643\) 18.5000 32.0429i 0.729569 1.26365i −0.227497 0.973779i \(-0.573054\pi\)
0.957066 0.289871i \(-0.0936125\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.9737 + 32.8634i 0.745932 + 1.29199i 0.949758 + 0.312985i \(0.101329\pi\)
−0.203826 + 0.979007i \(0.565338\pi\)
\(648\) 0 0
\(649\) −40.0000 + 69.2820i −1.57014 + 2.71956i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −14.2302 24.6475i −0.556873 0.964532i −0.997755 0.0669670i \(-0.978668\pi\)
0.440882 0.897565i \(-0.354666\pi\)
\(654\) 0 0
\(655\) 25.0000 43.3013i 0.976831 1.69192i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.58114 + 2.73861i 0.0615924 + 0.106681i 0.895177 0.445710i \(-0.147049\pi\)
−0.833585 + 0.552391i \(0.813715\pi\)
\(660\) 0 0
\(661\) −17.0000 −0.661223 −0.330612 0.943767i \(-0.607255\pi\)
−0.330612 + 0.943767i \(0.607255\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 55.3399 19.1703i 2.14599 0.743392i
\(666\) 0 0
\(667\) 5.00000 8.66025i 0.193601 0.335326i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 9.48683 16.4317i 0.366235 0.634338i
\(672\) 0 0
\(673\) −2.50000 4.33013i −0.0963679 0.166914i 0.813811 0.581130i \(-0.197389\pi\)
−0.910179 + 0.414216i \(0.864056\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 37.9473 1.45843 0.729217 0.684282i \(-0.239884\pi\)
0.729217 + 0.684282i \(0.239884\pi\)
\(678\) 0 0
\(679\) −10.0000 8.66025i −0.383765 0.332350i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −9.48683 16.4317i −0.363004 0.628741i 0.625450 0.780264i \(-0.284915\pi\)
−0.988454 + 0.151524i \(0.951582\pi\)
\(684\) 0 0
\(685\) 10.0000 0.382080
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 11.0000 0.418460 0.209230 0.977866i \(-0.432904\pi\)
0.209230 + 0.977866i \(0.432904\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.32456 −0.239904
\(696\) 0 0
\(697\) −30.0000 −1.13633
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3.16228 0.119438 0.0597188 0.998215i \(-0.480980\pi\)
0.0597188 + 0.998215i \(0.480980\pi\)
\(702\) 0 0
\(703\) −14.0000 24.2487i −0.528020 0.914557i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −3.16228 + 16.4317i −0.118930 + 0.617977i
\(708\) 0 0
\(709\) −1.00000 −0.0375558 −0.0187779 0.999824i \(-0.505978\pi\)
−0.0187779 + 0.999824i \(0.505978\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.74342 + 8.21584i 0.177642 + 0.307686i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) 4.00000 + 3.46410i 0.148968 + 0.129010i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 15.8114 0.587220
\(726\) 0 0
\(727\) 14.5000 + 25.1147i 0.537775 + 0.931454i 0.999023 + 0.0441829i \(0.0140684\pi\)
−0.461248 + 0.887271i \(0.652598\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7.90569 13.6931i 0.292403 0.506456i
\(732\) 0 0
\(733\) −1.50000 2.59808i −0.0554038 0.0959621i 0.836993 0.547213i \(-0.184311\pi\)
−0.892397 + 0.451251i \(0.850978\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −31.6228 + 54.7723i −1.16484 + 2.01756i
\(738\) 0 0
\(739\) −1.50000 2.59808i −0.0551784 0.0955718i 0.837117 0.547024i \(-0.184239\pi\)
−0.892295 + 0.451452i \(0.850906\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −14.2302 + 24.6475i −0.522057 + 0.904230i 0.477614 + 0.878570i \(0.341502\pi\)
−0.999671 + 0.0256596i \(0.991831\pi\)
\(744\) 0 0
\(745\) 15.0000 25.9808i 0.549557 0.951861i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.50000 2.59808i −0.0547358 0.0948051i 0.837359 0.546653i \(-0.184098\pi\)
−0.892095 + 0.451848i \(0.850765\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −53.7587 −1.95648
\(756\) 0 0
\(757\) 15.0000 0.545184 0.272592 0.962130i \(-0.412119\pi\)
0.272592 + 0.962130i \(0.412119\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −12.6491 21.9089i −0.458530 0.794197i 0.540354 0.841438i \(-0.318290\pi\)
−0.998884 + 0.0472409i \(0.984957\pi\)
\(762\) 0 0
\(763\) 14.0000 + 12.1244i 0.506834 + 0.438931i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 11.5000 19.9186i 0.414701 0.718283i −0.580696 0.814120i \(-0.697220\pi\)
0.995397 + 0.0958377i \(0.0305530\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 17.3925 + 30.1247i 0.625566 + 1.08351i 0.988431 + 0.151670i \(0.0484651\pi\)
−0.362865 + 0.931842i \(0.618202\pi\)
\(774\) 0 0
\(775\) −7.50000 + 12.9904i −0.269408 + 0.466628i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 33.2039 + 57.5109i 1.18965 + 2.06054i
\(780\) 0 0
\(781\) 40.0000 69.2820i 1.43131 2.47911i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 25.2982 + 43.8178i 0.902932 + 1.56392i
\(786\) 0 0
\(787\) 15.0000 0.534692 0.267346 0.963601i \(-0.413853\pi\)
0.267346 + 0.963601i \(0.413853\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.16228 + 16.4317i −0.112438 + 0.584243i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9.48683 16.4317i 0.336041 0.582040i −0.647643 0.761944i \(-0.724245\pi\)
0.983684 + 0.179904i \(0.0575786\pi\)
\(798\) 0 0
\(799\) 15.0000 + 25.9808i 0.530662 + 0.919133i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −31.6228 −1.11594
\(804\) 0 0
\(805\) −5.00000 + 25.9808i −0.176227 + 0.915702i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −15.8114 27.3861i −0.555899 0.962845i −0.997833 0.0657975i \(-0.979041\pi\)
0.441934 0.897047i \(-0.354292\pi\)
\(810\) 0 0
\(811\) −14.0000 −0.491606 −0.245803 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 53.7587 1.88309
\(816\) 0 0
\(817\) −35.0000 −1.22449
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 25.2982 0.882914 0.441457 0.897282i \(-0.354462\pi\)
0.441457 + 0.897282i \(0.354462\pi\)
\(822\) 0 0
\(823\) −23.0000 −0.801730 −0.400865 0.916137i \(-0.631290\pi\)
−0.400865 + 0.916137i \(0.631290\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −18.9737 −0.659779 −0.329890 0.944020i \(-0.607011\pi\)
−0.329890 + 0.944020i \(0.607011\pi\)
\(828\) 0 0
\(829\) 5.50000 + 9.52628i 0.191023 + 0.330861i 0.945589 0.325362i \(-0.105486\pi\)
−0.754567 + 0.656223i \(0.772153\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 20.5548 + 8.21584i 0.712182 + 0.284662i
\(834\) 0 0
\(835\) −20.0000 −0.692129
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 15.8114 + 27.3861i 0.545870 + 0.945474i 0.998552 + 0.0538020i \(0.0171340\pi\)
−0.452682 + 0.891672i \(0.649533\pi\)
\(840\) 0 0
\(841\) 9.50000 16.4545i 0.327586 0.567396i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 20.5548 35.6020i 0.707107 1.22474i
\(846\) 0 0
\(847\) 58.0000 + 50.2295i 1.99290 + 1.72591i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 12.6491 0.433606
\(852\) 0 0
\(853\) 21.5000 + 37.2391i 0.736146 + 1.27504i 0.954219 + 0.299110i \(0.0966897\pi\)
−0.218073 + 0.975933i \(0.569977\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.9737 32.8634i 0.648128 1.12259i −0.335441 0.942061i \(-0.608885\pi\)
0.983570 0.180530i \(-0.0577812\pi\)
\(858\) 0 0
\(859\) −8.50000 14.7224i −0.290016 0.502323i 0.683797 0.729672i \(-0.260327\pi\)
−0.973813 + 0.227349i \(0.926994\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 7.90569 13.6931i 0.269113 0.466117i −0.699520 0.714613i \(-0.746603\pi\)
0.968633 + 0.248496i \(0.0799362\pi\)
\(864\) 0 0
\(865\) −5.00000 8.66025i −0.170005 0.294457i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −37.9473 + 65.7267i −1.28728 + 2.22963i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.50000 + 4.33013i 0.0844190 + 0.146218i 0.905143 0.425106i \(-0.139763\pi\)
−0.820724 + 0.571324i \(0.806430\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 47.4342 1.59810 0.799049 0.601266i \(-0.205337\pi\)
0.799049 + 0.601266i \(0.205337\pi\)
\(882\) 0 0
\(883\) 23.0000 0.774012 0.387006 0.922077i \(-0.373509\pi\)
0.387006 + 0.922077i \(0.373509\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(888\) 0 0
\(889\) −52.5000 + 18.1865i −1.76079 + 0.609957i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 33.2039 57.5109i 1.11113 1.92453i
\(894\) 0 0
\(895\) 15.0000 25.9808i 0.501395 0.868441i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4.74342 8.21584i −0.158202 0.274014i
\(900\) 0 0
\(901\) 15.0000 25.9808i 0.499722 0.865545i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −33.2039 57.5109i −1.10374 1.91173i
\(906\) 0 0
\(907\) 12.0000 20.7846i 0.398453 0.690142i −0.595082 0.803665i \(-0.702880\pi\)
0.993535 + 0.113523i \(0.0362137\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −23.7171 41.0792i −0.785782 1.36101i −0.928531 0.371255i \(-0.878927\pi\)
0.142749 0.989759i \(-0.454406\pi\)
\(912\) 0 0
\(913\) 40.0000 1.32381
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −31.6228 27.3861i −1.04428 0.904370i
\(918\) 0 0
\(919\) 5.50000 9.52628i 0.181428 0.314243i −0.760939 0.648824i \(-0.775261\pi\)
0.942367 + 0.334581i \(0.108595\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 10.0000 + 17.3205i 0.328798 + 0.569495i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −28.4605 −0.933759 −0.466879 0.884321i \(-0.654622\pi\)
−0.466879 + 0.884321i \(0.654622\pi\)
\(930\) 0 0
\(931\) −7.00000 48.4974i −0.229416 1.58944i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 31.6228 + 54.7723i 1.03418 + 1.79124i
\(936\) 0 0
\(937\) 44.0000 1.43742 0.718709 0.695311i \(-0.244734\pi\)
0.718709 + 0.695311i \(0.244734\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −25.2982 −0.824698 −0.412349 0.911026i \(-0.635292\pi\)
−0.412349 + 0.911026i \(0.635292\pi\)
\(942\) 0 0
\(943\) −30.0000 −0.976934
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −34.7851 −1.13036 −0.565181 0.824967i \(-0.691194\pi\)
−0.565181 + 0.824967i \(0.691194\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −3.16228 −0.102436 −0.0512181 0.998687i \(-0.516310\pi\)
−0.0512181 + 0.998687i \(0.516310\pi\)
\(954\) 0 0
\(955\) −15.0000 25.9808i −0.485389 0.840718i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.58114 8.21584i 0.0510576 0.265303i
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −12.6491 21.9089i −0.407189 0.705273i
\(966\) 0 0
\(967\) −13.0000 + 22.5167i −0.418052 + 0.724087i −0.995743 0.0921681i \(-0.970620\pi\)
0.577692 + 0.816255i \(0.303954\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 23.7171 41.0792i 0.761117 1.31829i −0.181158 0.983454i \(-0.557984\pi\)
0.942275 0.334840i \(-0.108682\pi\)
\(972\) 0 0
\(973\) −1.00000 + 5.19615i −0.0320585 + 0.166581i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −37.9473 −1.21404 −0.607021 0.794686i \(-0.707636\pi\)
−0.607021 + 0.794686i \(0.707636\pi\)
\(978\) 0 0
\(979\) 30.0000 + 51.9615i 0.958804 + 1.66070i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(984\) 0 0
\(985\) −40.0000 69.2820i −1.27451 2.20751i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.90569 13.6931i 0.251386 0.435414i
\(990\) 0 0
\(991\) −7.00000 12.1244i −0.222362 0.385143i 0.733163 0.680053i \(-0.238043\pi\)
−0.955525 + 0.294911i \(0.904710\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 20.5548 35.6020i 0.651631 1.12866i
\(996\) 0 0
\(997\) −20.5000 + 35.5070i −0.649242 + 1.12452i 0.334063 + 0.942551i \(0.391580\pi\)
−0.983304 + 0.181968i \(0.941753\pi\)
\(998\) 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 2268.2.i.i.2053.1 4
3.2 odd 2 inner 2268.2.i.i.2053.2 4
7.4 even 3 2268.2.l.i.109.2 4
9.2 odd 6 2268.2.l.i.541.1 4
9.4 even 3 756.2.k.d.541.1 yes 4
9.5 odd 6 756.2.k.d.541.2 yes 4
9.7 even 3 2268.2.l.i.541.2 4
21.11 odd 6 2268.2.l.i.109.1 4
63.4 even 3 756.2.k.d.109.1 4
63.5 even 6 5292.2.a.r.1.2 2
63.11 odd 6 inner 2268.2.i.i.865.2 4
63.23 odd 6 5292.2.a.q.1.1 2
63.25 even 3 inner 2268.2.i.i.865.1 4
63.32 odd 6 756.2.k.d.109.2 yes 4
63.40 odd 6 5292.2.a.r.1.1 2
63.58 even 3 5292.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
756.2.k.d.109.1 4 63.4 even 3
756.2.k.d.109.2 yes 4 63.32 odd 6
756.2.k.d.541.1 yes 4 9.4 even 3
756.2.k.d.541.2 yes 4 9.5 odd 6
2268.2.i.i.865.1 4 63.25 even 3 inner
2268.2.i.i.865.2 4 63.11 odd 6 inner
2268.2.i.i.2053.1 4 1.1 even 1 trivial
2268.2.i.i.2053.2 4 3.2 odd 2 inner
2268.2.l.i.109.1 4 21.11 odd 6
2268.2.l.i.109.2 4 7.4 even 3
2268.2.l.i.541.1 4 9.2 odd 6
2268.2.l.i.541.2 4 9.7 even 3
5292.2.a.q.1.1 2 63.23 odd 6
5292.2.a.q.1.2 2 63.58 even 3
5292.2.a.r.1.1 2 63.40 odd 6
5292.2.a.r.1.2 2 63.5 even 6