Properties

Label 2880.2.bl.b.431.9
Level $2880$
Weight $2$
Character 2880.431
Analytic conductor $22.997$
Analytic rank $0$
Dimension $24$
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] = [2880,2,Mod(431,2880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2880.431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.bl (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: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 720)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 431.9
Character \(\chi\) \(=\) 2880.431
Dual form 2880.2.bl.b.1871.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 + 0.707107i) q^{5} +0.527405 q^{7} +O(q^{10})\) \(q+(0.707107 + 0.707107i) q^{5} +0.527405 q^{7} +(-1.36838 + 1.36838i) q^{11} +(-2.79215 - 2.79215i) q^{13} -6.68546i q^{17} +(-3.79610 + 3.79610i) q^{19} +5.00971i q^{23} +1.00000i q^{25} +(2.94130 - 2.94130i) q^{29} -6.21862i q^{31} +(0.372932 + 0.372932i) q^{35} +(-6.42437 + 6.42437i) q^{37} +2.16008 q^{41} +(-4.71629 - 4.71629i) q^{43} -11.3428 q^{47} -6.72184 q^{49} +(-4.23408 - 4.23408i) q^{53} -1.93518 q^{55} +(0.322982 - 0.322982i) q^{59} +(-0.887268 - 0.887268i) q^{61} -3.94870i q^{65} +(-1.24370 + 1.24370i) q^{67} -14.3113i q^{71} +7.97372i q^{73} +(-0.721691 + 0.721691i) q^{77} +5.37359i q^{79} +(2.43723 + 2.43723i) q^{83} +(4.72733 - 4.72733i) q^{85} +11.1796 q^{89} +(-1.47259 - 1.47259i) q^{91} -5.36850 q^{95} +2.81084 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 8 q^{7} + 24 q^{19} + 8 q^{37} - 48 q^{43} + 24 q^{49} - 24 q^{55} + 40 q^{61} + 40 q^{67} + 24 q^{85} - 40 q^{91} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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 0
\(4\) 0 0
\(5\) 0.707107 + 0.707107i 0.316228 + 0.316228i
\(6\) 0 0
\(7\) 0.527405 0.199340 0.0996702 0.995021i \(-0.468221\pi\)
0.0996702 + 0.995021i \(0.468221\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.36838 + 1.36838i −0.412582 + 0.412582i −0.882637 0.470055i \(-0.844234\pi\)
0.470055 + 0.882637i \(0.344234\pi\)
\(12\) 0 0
\(13\) −2.79215 2.79215i −0.774403 0.774403i 0.204470 0.978873i \(-0.434453\pi\)
−0.978873 + 0.204470i \(0.934453\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.68546i 1.62146i −0.585419 0.810731i \(-0.699070\pi\)
0.585419 0.810731i \(-0.300930\pi\)
\(18\) 0 0
\(19\) −3.79610 + 3.79610i −0.870886 + 0.870886i −0.992569 0.121683i \(-0.961171\pi\)
0.121683 + 0.992569i \(0.461171\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.00971i 1.04460i 0.852763 + 0.522298i \(0.174925\pi\)
−0.852763 + 0.522298i \(0.825075\pi\)
\(24\) 0 0
\(25\) 1.00000i 0.200000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.94130 2.94130i 0.546185 0.546185i −0.379150 0.925335i \(-0.623784\pi\)
0.925335 + 0.379150i \(0.123784\pi\)
\(30\) 0 0
\(31\) 6.21862i 1.11690i −0.829539 0.558448i \(-0.811397\pi\)
0.829539 0.558448i \(-0.188603\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.372932 + 0.372932i 0.0630370 + 0.0630370i
\(36\) 0 0
\(37\) −6.42437 + 6.42437i −1.05616 + 1.05616i −0.0578329 + 0.998326i \(0.518419\pi\)
−0.998326 + 0.0578329i \(0.981581\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.16008 0.337347 0.168674 0.985672i \(-0.446052\pi\)
0.168674 + 0.985672i \(0.446052\pi\)
\(42\) 0 0
\(43\) −4.71629 4.71629i −0.719228 0.719228i 0.249219 0.968447i \(-0.419826\pi\)
−0.968447 + 0.249219i \(0.919826\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.3428 −1.65451 −0.827256 0.561826i \(-0.810099\pi\)
−0.827256 + 0.561826i \(0.810099\pi\)
\(48\) 0 0
\(49\) −6.72184 −0.960263
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.23408 4.23408i −0.581595 0.581595i 0.353746 0.935342i \(-0.384908\pi\)
−0.935342 + 0.353746i \(0.884908\pi\)
\(54\) 0 0
\(55\) −1.93518 −0.260940
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.322982 0.322982i 0.0420487 0.0420487i −0.685770 0.727818i \(-0.740534\pi\)
0.727818 + 0.685770i \(0.240534\pi\)
\(60\) 0 0
\(61\) −0.887268 0.887268i −0.113603 0.113603i 0.648020 0.761623i \(-0.275597\pi\)
−0.761623 + 0.648020i \(0.775597\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.94870i 0.489776i
\(66\) 0 0
\(67\) −1.24370 + 1.24370i −0.151942 + 0.151942i −0.778985 0.627043i \(-0.784265\pi\)
0.627043 + 0.778985i \(0.284265\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.3113i 1.69844i −0.528041 0.849219i \(-0.677073\pi\)
0.528041 0.849219i \(-0.322927\pi\)
\(72\) 0 0
\(73\) 7.97372i 0.933254i 0.884454 + 0.466627i \(0.154531\pi\)
−0.884454 + 0.466627i \(0.845469\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.721691 + 0.721691i −0.0822443 + 0.0822443i
\(78\) 0 0
\(79\) 5.37359i 0.604576i 0.953217 + 0.302288i \(0.0977505\pi\)
−0.953217 + 0.302288i \(0.902250\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.43723 + 2.43723i 0.267520 + 0.267520i 0.828100 0.560580i \(-0.189422\pi\)
−0.560580 + 0.828100i \(0.689422\pi\)
\(84\) 0 0
\(85\) 4.72733 4.72733i 0.512751 0.512751i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.1796 1.18504 0.592518 0.805557i \(-0.298134\pi\)
0.592518 + 0.805557i \(0.298134\pi\)
\(90\) 0 0
\(91\) −1.47259 1.47259i −0.154370 0.154370i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.36850 −0.550797
\(96\) 0 0
\(97\) 2.81084 0.285398 0.142699 0.989766i \(-0.454422\pi\)
0.142699 + 0.989766i \(0.454422\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.95132 + 2.95132i 0.293667 + 0.293667i 0.838527 0.544860i \(-0.183417\pi\)
−0.544860 + 0.838527i \(0.683417\pi\)
\(102\) 0 0
\(103\) −3.36823 −0.331881 −0.165941 0.986136i \(-0.553066\pi\)
−0.165941 + 0.986136i \(0.553066\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.26471 2.26471i 0.218938 0.218938i −0.589113 0.808051i \(-0.700523\pi\)
0.808051 + 0.589113i \(0.200523\pi\)
\(108\) 0 0
\(109\) −1.05900 1.05900i −0.101433 0.101433i 0.654569 0.756002i \(-0.272850\pi\)
−0.756002 + 0.654569i \(0.772850\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 17.6370i 1.65915i −0.558395 0.829575i \(-0.688583\pi\)
0.558395 0.829575i \(-0.311417\pi\)
\(114\) 0 0
\(115\) −3.54240 + 3.54240i −0.330330 + 0.330330i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.52595i 0.323223i
\(120\) 0 0
\(121\) 7.25507i 0.659552i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −0.707107 + 0.707107i −0.0632456 + 0.0632456i
\(126\) 0 0
\(127\) 14.3319i 1.27175i −0.771791 0.635876i \(-0.780639\pi\)
0.771791 0.635876i \(-0.219361\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.42254 4.42254i −0.386399 0.386399i 0.487002 0.873401i \(-0.338091\pi\)
−0.873401 + 0.487002i \(0.838091\pi\)
\(132\) 0 0
\(133\) −2.00209 + 2.00209i −0.173603 + 0.173603i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.20075 0.102587 0.0512937 0.998684i \(-0.483666\pi\)
0.0512937 + 0.998684i \(0.483666\pi\)
\(138\) 0 0
\(139\) −7.61065 7.61065i −0.645527 0.645527i 0.306382 0.951909i \(-0.400882\pi\)
−0.951909 + 0.306382i \(0.900882\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 7.64145 0.639010
\(144\) 0 0
\(145\) 4.15962 0.345438
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.41054 + 5.41054i 0.443249 + 0.443249i 0.893102 0.449853i \(-0.148524\pi\)
−0.449853 + 0.893102i \(0.648524\pi\)
\(150\) 0 0
\(151\) −21.3534 −1.73772 −0.868858 0.495062i \(-0.835146\pi\)
−0.868858 + 0.495062i \(0.835146\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.39723 4.39723i 0.353194 0.353194i
\(156\) 0 0
\(157\) −16.1462 16.1462i −1.28861 1.28861i −0.935633 0.352975i \(-0.885170\pi\)
−0.352975 0.935633i \(-0.614830\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.64215i 0.208230i
\(162\) 0 0
\(163\) −14.6070 + 14.6070i −1.14411 + 1.14411i −0.156419 + 0.987691i \(0.549995\pi\)
−0.987691 + 0.156419i \(0.950005\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.8807i 0.996741i −0.866964 0.498370i \(-0.833932\pi\)
0.866964 0.498370i \(-0.166068\pi\)
\(168\) 0 0
\(169\) 2.59221i 0.199401i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.98756 + 1.98756i −0.151112 + 0.151112i −0.778614 0.627503i \(-0.784077\pi\)
0.627503 + 0.778614i \(0.284077\pi\)
\(174\) 0 0
\(175\) 0.527405i 0.0398681i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.6119 + 15.6119i 1.16689 + 1.16689i 0.982935 + 0.183954i \(0.0588899\pi\)
0.183954 + 0.982935i \(0.441110\pi\)
\(180\) 0 0
\(181\) −11.8656 + 11.8656i −0.881966 + 0.881966i −0.993734 0.111769i \(-0.964348\pi\)
0.111769 + 0.993734i \(0.464348\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −9.08542 −0.667974
\(186\) 0 0
\(187\) 9.14825 + 9.14825i 0.668986 + 0.668986i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −7.83695 −0.567062 −0.283531 0.958963i \(-0.591506\pi\)
−0.283531 + 0.958963i \(0.591506\pi\)
\(192\) 0 0
\(193\) 6.61910 0.476453 0.238226 0.971210i \(-0.423434\pi\)
0.238226 + 0.971210i \(0.423434\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.0068 11.0068i −0.784200 0.784200i 0.196336 0.980537i \(-0.437096\pi\)
−0.980537 + 0.196336i \(0.937096\pi\)
\(198\) 0 0
\(199\) 2.50842 0.177817 0.0889085 0.996040i \(-0.471662\pi\)
0.0889085 + 0.996040i \(0.471662\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.55125 1.55125i 0.108877 0.108877i
\(204\) 0 0
\(205\) 1.52741 + 1.52741i 0.106679 + 0.106679i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.3890i 0.718624i
\(210\) 0 0
\(211\) −15.6265 + 15.6265i −1.07577 + 1.07577i −0.0788863 + 0.996884i \(0.525136\pi\)
−0.996884 + 0.0788863i \(0.974864\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.66985i 0.454880i
\(216\) 0 0
\(217\) 3.27973i 0.222643i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −18.6668 + 18.6668i −1.25567 + 1.25567i
\(222\) 0 0
\(223\) 7.13280i 0.477647i −0.971063 0.238824i \(-0.923238\pi\)
0.971063 0.238824i \(-0.0767618\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.288353 + 0.288353i 0.0191386 + 0.0191386i 0.716611 0.697473i \(-0.245692\pi\)
−0.697473 + 0.716611i \(0.745692\pi\)
\(228\) 0 0
\(229\) −7.34086 + 7.34086i −0.485098 + 0.485098i −0.906755 0.421658i \(-0.861448\pi\)
0.421658 + 0.906755i \(0.361448\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.34352 0.481090 0.240545 0.970638i \(-0.422674\pi\)
0.240545 + 0.970638i \(0.422674\pi\)
\(234\) 0 0
\(235\) −8.02054 8.02054i −0.523202 0.523202i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −26.4178 −1.70882 −0.854411 0.519597i \(-0.826082\pi\)
−0.854411 + 0.519597i \(0.826082\pi\)
\(240\) 0 0
\(241\) 17.7860 1.14570 0.572849 0.819661i \(-0.305838\pi\)
0.572849 + 0.819661i \(0.305838\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.75306 4.75306i −0.303662 0.303662i
\(246\) 0 0
\(247\) 21.1986 1.34883
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.9075 16.9075i 1.06719 1.06719i 0.0696174 0.997574i \(-0.477822\pi\)
0.997574 0.0696174i \(-0.0221779\pi\)
\(252\) 0 0
\(253\) −6.85518 6.85518i −0.430982 0.430982i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 23.4607i 1.46344i −0.681605 0.731720i \(-0.738718\pi\)
0.681605 0.731720i \(-0.261282\pi\)
\(258\) 0 0
\(259\) −3.38824 + 3.38824i −0.210535 + 0.210535i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.85113i 0.545784i −0.962045 0.272892i \(-0.912020\pi\)
0.962045 0.272892i \(-0.0879801\pi\)
\(264\) 0 0
\(265\) 5.98789i 0.367833i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −19.2562 + 19.2562i −1.17407 + 1.17407i −0.192842 + 0.981230i \(0.561770\pi\)
−0.981230 + 0.192842i \(0.938230\pi\)
\(270\) 0 0
\(271\) 7.60120i 0.461740i −0.972985 0.230870i \(-0.925843\pi\)
0.972985 0.230870i \(-0.0741573\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.36838 1.36838i −0.0825164 0.0825164i
\(276\) 0 0
\(277\) 10.6425 10.6425i 0.639447 0.639447i −0.310972 0.950419i \(-0.600655\pi\)
0.950419 + 0.310972i \(0.100655\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 19.6112 1.16991 0.584954 0.811066i \(-0.301113\pi\)
0.584954 + 0.811066i \(0.301113\pi\)
\(282\) 0 0
\(283\) 17.3190 + 17.3190i 1.02951 + 1.02951i 0.999551 + 0.0299545i \(0.00953625\pi\)
0.0299545 + 0.999551i \(0.490464\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.13924 0.0672470
\(288\) 0 0
\(289\) −27.6953 −1.62914
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.26943 + 4.26943i 0.249423 + 0.249423i 0.820734 0.571311i \(-0.193565\pi\)
−0.571311 + 0.820734i \(0.693565\pi\)
\(294\) 0 0
\(295\) 0.456765 0.0265939
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 13.9879 13.9879i 0.808938 0.808938i
\(300\) 0 0
\(301\) −2.48740 2.48740i −0.143371 0.143371i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.25479i 0.0718489i
\(306\) 0 0
\(307\) 22.1738 22.1738i 1.26552 1.26552i 0.317147 0.948376i \(-0.397275\pi\)
0.948376 0.317147i \(-0.102725\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.5304i 0.710536i −0.934765 0.355268i \(-0.884390\pi\)
0.934765 0.355268i \(-0.115610\pi\)
\(312\) 0 0
\(313\) 16.9129i 0.955974i −0.878367 0.477987i \(-0.841367\pi\)
0.878367 0.477987i \(-0.158633\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.49210 + 6.49210i −0.364633 + 0.364633i −0.865515 0.500882i \(-0.833009\pi\)
0.500882 + 0.865515i \(0.333009\pi\)
\(318\) 0 0
\(319\) 8.04962i 0.450692i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 25.3787 + 25.3787i 1.41211 + 1.41211i
\(324\) 0 0
\(325\) 2.79215 2.79215i 0.154881 0.154881i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5.98223 −0.329811
\(330\) 0 0
\(331\) −0.294551 0.294551i −0.0161900 0.0161900i 0.698965 0.715155i \(-0.253644\pi\)
−0.715155 + 0.698965i \(0.753644\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.75886 −0.0960966
\(336\) 0 0
\(337\) −22.2233 −1.21058 −0.605291 0.796004i \(-0.706943\pi\)
−0.605291 + 0.796004i \(0.706943\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.50943 + 8.50943i 0.460812 + 0.460812i
\(342\) 0 0
\(343\) −7.23697 −0.390760
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −16.5776 + 16.5776i −0.889930 + 0.889930i −0.994516 0.104586i \(-0.966648\pi\)
0.104586 + 0.994516i \(0.466648\pi\)
\(348\) 0 0
\(349\) −8.60364 8.60364i −0.460543 0.460543i 0.438291 0.898833i \(-0.355584\pi\)
−0.898833 + 0.438291i \(0.855584\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 25.0941i 1.33563i 0.744329 + 0.667813i \(0.232769\pi\)
−0.744329 + 0.667813i \(0.767231\pi\)
\(354\) 0 0
\(355\) 10.1196 10.1196i 0.537093 0.537093i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 21.8516i 1.15328i 0.816998 + 0.576641i \(0.195637\pi\)
−0.816998 + 0.576641i \(0.804363\pi\)
\(360\) 0 0
\(361\) 9.82081i 0.516885i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.63827 + 5.63827i −0.295121 + 0.295121i
\(366\) 0 0
\(367\) 5.06483i 0.264382i 0.991224 + 0.132191i \(0.0422012\pi\)
−0.991224 + 0.132191i \(0.957799\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.23308 2.23308i −0.115935 0.115935i
\(372\) 0 0
\(373\) −21.6071 + 21.6071i −1.11878 + 1.11878i −0.126854 + 0.991921i \(0.540488\pi\)
−0.991921 + 0.126854i \(0.959512\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −16.4251 −0.845935
\(378\) 0 0
\(379\) 14.4241 + 14.4241i 0.740918 + 0.740918i 0.972755 0.231837i \(-0.0744735\pi\)
−0.231837 + 0.972755i \(0.574474\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.246311 0.0125859 0.00629296 0.999980i \(-0.497997\pi\)
0.00629296 + 0.999980i \(0.497997\pi\)
\(384\) 0 0
\(385\) −1.02063 −0.0520159
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 20.0404 + 20.0404i 1.01609 + 1.01609i 0.999868 + 0.0162190i \(0.00516289\pi\)
0.0162190 + 0.999868i \(0.494837\pi\)
\(390\) 0 0
\(391\) 33.4922 1.69377
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.79970 + 3.79970i −0.191184 + 0.191184i
\(396\) 0 0
\(397\) −5.46763 5.46763i −0.274413 0.274413i 0.556461 0.830874i \(-0.312159\pi\)
−0.830874 + 0.556461i \(0.812159\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 21.4643i 1.07187i 0.844258 + 0.535937i \(0.180041\pi\)
−0.844258 + 0.535937i \(0.819959\pi\)
\(402\) 0 0
\(403\) −17.3633 + 17.3633i −0.864928 + 0.864928i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 17.5819i 0.871505i
\(408\) 0 0
\(409\) 35.9009i 1.77518i 0.460631 + 0.887592i \(0.347623\pi\)
−0.460631 + 0.887592i \(0.652377\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.170342 0.170342i 0.00838200 0.00838200i
\(414\) 0 0
\(415\) 3.44676i 0.169195i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 25.1538 + 25.1538i 1.22884 + 1.22884i 0.964402 + 0.264440i \(0.0851872\pi\)
0.264440 + 0.964402i \(0.414813\pi\)
\(420\) 0 0
\(421\) 19.4663 19.4663i 0.948729 0.948729i −0.0500192 0.998748i \(-0.515928\pi\)
0.998748 + 0.0500192i \(0.0159283\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.68546 0.324292
\(426\) 0 0
\(427\) −0.467950 0.467950i −0.0226457 0.0226457i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.07105 0.340600 0.170300 0.985392i \(-0.445526\pi\)
0.170300 + 0.985392i \(0.445526\pi\)
\(432\) 0 0
\(433\) 33.9959 1.63374 0.816868 0.576824i \(-0.195708\pi\)
0.816868 + 0.576824i \(0.195708\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −19.0174 19.0174i −0.909724 0.909724i
\(438\) 0 0
\(439\) 14.9140 0.711804 0.355902 0.934523i \(-0.384174\pi\)
0.355902 + 0.934523i \(0.384174\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −19.3933 + 19.3933i −0.921401 + 0.921401i −0.997129 0.0757274i \(-0.975872\pi\)
0.0757274 + 0.997129i \(0.475872\pi\)
\(444\) 0 0
\(445\) 7.90518 + 7.90518i 0.374742 + 0.374742i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 34.7444i 1.63969i −0.572586 0.819844i \(-0.694060\pi\)
0.572586 0.819844i \(-0.305940\pi\)
\(450\) 0 0
\(451\) −2.95581 + 2.95581i −0.139184 + 0.139184i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.08256i 0.0976321i
\(456\) 0 0
\(457\) 4.46940i 0.209070i −0.994521 0.104535i \(-0.966665\pi\)
0.994521 0.104535i \(-0.0333354\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.21550 4.21550i 0.196335 0.196335i −0.602092 0.798427i \(-0.705666\pi\)
0.798427 + 0.602092i \(0.205666\pi\)
\(462\) 0 0
\(463\) 15.8356i 0.735943i −0.929837 0.367972i \(-0.880052\pi\)
0.929837 0.367972i \(-0.119948\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −19.1414 19.1414i −0.885760 0.885760i 0.108352 0.994113i \(-0.465442\pi\)
−0.994113 + 0.108352i \(0.965442\pi\)
\(468\) 0 0
\(469\) −0.655933 + 0.655933i −0.0302882 + 0.0302882i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12.9074 0.593481
\(474\) 0 0
\(475\) −3.79610 3.79610i −0.174177 0.174177i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 39.8319 1.81997 0.909983 0.414645i \(-0.136094\pi\)
0.909983 + 0.414645i \(0.136094\pi\)
\(480\) 0 0
\(481\) 35.8756 1.63579
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.98756 + 1.98756i 0.0902506 + 0.0902506i
\(486\) 0 0
\(487\) 18.6470 0.844978 0.422489 0.906368i \(-0.361156\pi\)
0.422489 + 0.906368i \(0.361156\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −2.48009 + 2.48009i −0.111925 + 0.111925i −0.760851 0.648926i \(-0.775218\pi\)
0.648926 + 0.760851i \(0.275218\pi\)
\(492\) 0 0
\(493\) −19.6639 19.6639i −0.885618 0.885618i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.54785i 0.338567i
\(498\) 0 0
\(499\) 3.41367 3.41367i 0.152817 0.152817i −0.626558 0.779375i \(-0.715537\pi\)
0.779375 + 0.626558i \(0.215537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.65191i 0.0736549i 0.999322 + 0.0368274i \(0.0117252\pi\)
−0.999322 + 0.0368274i \(0.988275\pi\)
\(504\) 0 0
\(505\) 4.17379i 0.185731i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4.87832 + 4.87832i −0.216228 + 0.216228i −0.806907 0.590679i \(-0.798860\pi\)
0.590679 + 0.806907i \(0.298860\pi\)
\(510\) 0 0
\(511\) 4.20538i 0.186035i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.38170 2.38170i −0.104950 0.104950i
\(516\) 0 0
\(517\) 15.5212 15.5212i 0.682622 0.682622i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −15.9371 −0.698218 −0.349109 0.937082i \(-0.613516\pi\)
−0.349109 + 0.937082i \(0.613516\pi\)
\(522\) 0 0
\(523\) 26.8688 + 26.8688i 1.17489 + 1.17489i 0.981029 + 0.193861i \(0.0621009\pi\)
0.193861 + 0.981029i \(0.437899\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −41.5743 −1.81100
\(528\) 0 0
\(529\) −2.09716 −0.0911807
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.03126 6.03126i −0.261243 0.261243i
\(534\) 0 0
\(535\) 3.20279 0.138469
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 9.19804 9.19804i 0.396188 0.396188i
\(540\) 0 0
\(541\) 4.22798 + 4.22798i 0.181775 + 0.181775i 0.792129 0.610354i \(-0.208973\pi\)
−0.610354 + 0.792129i \(0.708973\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.49765i 0.0641521i
\(546\) 0 0
\(547\) −0.243066 + 0.243066i −0.0103928 + 0.0103928i −0.712284 0.701891i \(-0.752339\pi\)
0.701891 + 0.712284i \(0.252339\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 22.3309i 0.951329i
\(552\) 0 0
\(553\) 2.83406i 0.120517i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19.5236 19.5236i 0.827242 0.827242i −0.159893 0.987134i \(-0.551115\pi\)
0.987134 + 0.159893i \(0.0511148\pi\)
\(558\) 0 0
\(559\) 26.3372i 1.11395i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −24.1772 24.1772i −1.01895 1.01895i −0.999817 0.0191314i \(-0.993910\pi\)
−0.0191314 0.999817i \(-0.506090\pi\)
\(564\) 0 0
\(565\) 12.4713 12.4713i 0.524670 0.524670i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 13.6423 0.571916 0.285958 0.958242i \(-0.407688\pi\)
0.285958 + 0.958242i \(0.407688\pi\)
\(570\) 0 0
\(571\) −17.4689 17.4689i −0.731052 0.731052i 0.239777 0.970828i \(-0.422926\pi\)
−0.970828 + 0.239777i \(0.922926\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.00971 −0.208919
\(576\) 0 0
\(577\) 46.6591 1.94244 0.971222 0.238176i \(-0.0765495\pi\)
0.971222 + 0.238176i \(0.0765495\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.28541 + 1.28541i 0.0533276 + 0.0533276i
\(582\) 0 0
\(583\) 11.5877 0.479912
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.11982 2.11982i 0.0874944 0.0874944i −0.662005 0.749499i \(-0.730294\pi\)
0.749499 + 0.662005i \(0.230294\pi\)
\(588\) 0 0
\(589\) 23.6065 + 23.6065i 0.972689 + 0.972689i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 29.5952i 1.21533i 0.794193 + 0.607665i \(0.207894\pi\)
−0.794193 + 0.607665i \(0.792106\pi\)
\(594\) 0 0
\(595\) 2.49322 2.49322i 0.102212 0.102212i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 23.8940i 0.976283i 0.872765 + 0.488141i \(0.162325\pi\)
−0.872765 + 0.488141i \(0.837675\pi\)
\(600\) 0 0
\(601\) 14.1676i 0.577909i 0.957343 + 0.288955i \(0.0933076\pi\)
−0.957343 + 0.288955i \(0.906692\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.13011 + 5.13011i −0.208569 + 0.208569i
\(606\) 0 0
\(607\) 34.3109i 1.39263i −0.717734 0.696317i \(-0.754821\pi\)
0.717734 0.696317i \(-0.245179\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 31.6707 + 31.6707i 1.28126 + 1.28126i
\(612\) 0 0
\(613\) −14.9012 + 14.9012i −0.601855 + 0.601855i −0.940805 0.338950i \(-0.889928\pi\)
0.338950 + 0.940805i \(0.389928\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −17.0403 −0.686015 −0.343008 0.939333i \(-0.611446\pi\)
−0.343008 + 0.939333i \(0.611446\pi\)
\(618\) 0 0
\(619\) −30.9842 30.9842i −1.24536 1.24536i −0.957747 0.287611i \(-0.907139\pi\)
−0.287611 0.957747i \(-0.592861\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.89619 0.236226
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 42.9498 + 42.9498i 1.71252 + 1.71252i
\(630\) 0 0
\(631\) −47.0036 −1.87118 −0.935591 0.353085i \(-0.885133\pi\)
−0.935591 + 0.353085i \(0.885133\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 10.1342 10.1342i 0.402164 0.402164i
\(636\) 0 0
\(637\) 18.7684 + 18.7684i 0.743631 + 0.743631i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.42235i 0.135175i 0.997713 + 0.0675873i \(0.0215301\pi\)
−0.997713 + 0.0675873i \(0.978470\pi\)
\(642\) 0 0
\(643\) 20.5347 20.5347i 0.809809 0.809809i −0.174796 0.984605i \(-0.555927\pi\)
0.984605 + 0.174796i \(0.0559266\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30.6843i 1.20633i 0.797618 + 0.603163i \(0.206093\pi\)
−0.797618 + 0.603163i \(0.793907\pi\)
\(648\) 0 0
\(649\) 0.883924i 0.0346971i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −5.93695 + 5.93695i −0.232331 + 0.232331i −0.813665 0.581334i \(-0.802531\pi\)
0.581334 + 0.813665i \(0.302531\pi\)
\(654\) 0 0
\(655\) 6.25442i 0.244380i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 21.5713 + 21.5713i 0.840297 + 0.840297i 0.988897 0.148601i \(-0.0474769\pi\)
−0.148601 + 0.988897i \(0.547477\pi\)
\(660\) 0 0
\(661\) 10.1017 10.1017i 0.392911 0.392911i −0.482813 0.875723i \(-0.660385\pi\)
0.875723 + 0.482813i \(0.160385\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.83138 −0.109796
\(666\) 0 0
\(667\) 14.7350 + 14.7350i 0.570543 + 0.570543i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.42824 0.0937412
\(672\) 0 0
\(673\) 5.35617 0.206465 0.103233 0.994657i \(-0.467081\pi\)
0.103233 + 0.994657i \(0.467081\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −11.3710 11.3710i −0.437022 0.437022i 0.453986 0.891009i \(-0.350002\pi\)
−0.891009 + 0.453986i \(0.850002\pi\)
\(678\) 0 0
\(679\) 1.48245 0.0568913
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.71058 6.71058i 0.256773 0.256773i −0.566967 0.823740i \(-0.691883\pi\)
0.823740 + 0.566967i \(0.191883\pi\)
\(684\) 0 0
\(685\) 0.849062 + 0.849062i 0.0324410 + 0.0324410i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 23.6444i 0.900779i
\(690\) 0 0
\(691\) −23.4167 + 23.4167i −0.890811 + 0.890811i −0.994599 0.103788i \(-0.966904\pi\)
0.103788 + 0.994599i \(0.466904\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.7631i 0.408267i
\(696\) 0 0
\(697\) 14.4411i 0.546996i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.68197 9.68197i 0.365683 0.365683i −0.500217 0.865900i \(-0.666746\pi\)
0.865900 + 0.500217i \(0.166746\pi\)
\(702\) 0 0
\(703\) 48.7751i 1.83959i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.55654 + 1.55654i 0.0585397 + 0.0585397i
\(708\) 0 0
\(709\) 30.3482 30.3482i 1.13975 1.13975i 0.151254 0.988495i \(-0.451669\pi\)
0.988495 0.151254i \(-0.0483313\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 31.1534 1.16671
\(714\) 0 0
\(715\) 5.40332 + 5.40332i 0.202073 + 0.202073i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −34.0246 −1.26890 −0.634451 0.772963i \(-0.718774\pi\)
−0.634451 + 0.772963i \(0.718774\pi\)
\(720\) 0 0
\(721\) −1.77642 −0.0661574
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.94130 + 2.94130i 0.109237 + 0.109237i
\(726\) 0 0
\(727\) 30.9373 1.14740 0.573700 0.819065i \(-0.305507\pi\)
0.573700 + 0.819065i \(0.305507\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −31.5306 + 31.5306i −1.16620 + 1.16620i
\(732\) 0 0
\(733\) 27.7826 + 27.7826i 1.02617 + 1.02617i 0.999648 + 0.0265245i \(0.00844400\pi\)
0.0265245 + 0.999648i \(0.491556\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.40371i 0.125377i
\(738\) 0 0
\(739\) 27.9617 27.9617i 1.02859 1.02859i 0.0290067 0.999579i \(-0.490766\pi\)
0.999579 0.0290067i \(-0.00923442\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0.366131i 0.0134321i −0.999977 0.00671603i \(-0.997862\pi\)
0.999977 0.00671603i \(-0.00213779\pi\)
\(744\) 0 0
\(745\) 7.65166i 0.280335i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.19442 1.19442i 0.0436432 0.0436432i
\(750\) 0 0
\(751\) 18.5853i 0.678186i −0.940753 0.339093i \(-0.889880\pi\)
0.940753 0.339093i \(-0.110120\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −15.0991 15.0991i −0.549514 0.549514i
\(756\) 0 0
\(757\) −7.11784 + 7.11784i −0.258702 + 0.258702i −0.824526 0.565824i \(-0.808558\pi\)
0.565824 + 0.824526i \(0.308558\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −21.2170 −0.769114 −0.384557 0.923101i \(-0.625646\pi\)
−0.384557 + 0.923101i \(0.625646\pi\)
\(762\) 0 0
\(763\) −0.558520 0.558520i −0.0202198 0.0202198i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.80363 −0.0651252
\(768\) 0 0
\(769\) −37.3964 −1.34855 −0.674273 0.738482i \(-0.735543\pi\)
−0.674273 + 0.738482i \(0.735543\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9.67007 + 9.67007i 0.347808 + 0.347808i 0.859293 0.511484i \(-0.170904\pi\)
−0.511484 + 0.859293i \(0.670904\pi\)
\(774\) 0 0
\(775\) 6.21862 0.223379
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.19988 + 8.19988i −0.293791 + 0.293791i
\(780\) 0 0
\(781\) 19.5833 + 19.5833i 0.700745 + 0.700745i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 22.8342i 0.814987i
\(786\) 0 0
\(787\) 2.50159 2.50159i 0.0891721 0.0891721i −0.661114 0.750286i \(-0.729916\pi\)
0.750286 + 0.661114i \(0.229916\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 9.30185i 0.330736i
\(792\) 0 0
\(793\) 4.95477i 0.175949i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −21.0921 + 21.0921i −0.747121 + 0.747121i −0.973937 0.226817i \(-0.927168\pi\)
0.226817 + 0.973937i \(0.427168\pi\)
\(798\) 0 0
\(799\) 75.8315i 2.68273i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −10.9111 10.9111i −0.385044 0.385044i
\(804\) 0 0
\(805\) −1.86828 + 1.86828i −0.0658482 + 0.0658482i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 53.9517 1.89684 0.948420 0.317016i \(-0.102681\pi\)
0.948420 + 0.317016i \(0.102681\pi\)
\(810\) 0 0
\(811\) 32.7497 + 32.7497i 1.15000 + 1.15000i 0.986552 + 0.163445i \(0.0522605\pi\)
0.163445 + 0.986552i \(0.447739\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −20.6574 −0.723598
\(816\) 0 0
\(817\) 35.8071 1.25273
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7.97077 7.97077i −0.278182 0.278182i 0.554201 0.832383i \(-0.313024\pi\)
−0.832383 + 0.554201i \(0.813024\pi\)
\(822\) 0 0
\(823\) −20.9488 −0.730230 −0.365115 0.930962i \(-0.618970\pi\)
−0.365115 + 0.930962i \(0.618970\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.12577 3.12577i 0.108694 0.108694i −0.650668 0.759362i \(-0.725511\pi\)
0.759362 + 0.650668i \(0.225511\pi\)
\(828\) 0 0
\(829\) 16.2577 + 16.2577i 0.564655 + 0.564655i 0.930626 0.365971i \(-0.119263\pi\)
−0.365971 + 0.930626i \(0.619263\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 44.9386i 1.55703i
\(834\) 0 0
\(835\) 9.10805 9.10805i 0.315197 0.315197i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 30.7937i 1.06312i 0.847021 + 0.531559i \(0.178394\pi\)
−0.847021 + 0.531559i \(0.821606\pi\)
\(840\) 0 0
\(841\) 11.6976i 0.403364i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.83297 + 1.83297i −0.0630560 + 0.0630560i
\(846\) 0 0
\(847\) 3.82636i 0.131475i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −32.1842 32.1842i −1.10326 1.10326i
\(852\) 0 0
\(853\) −0.973848 + 0.973848i −0.0333439 + 0.0333439i −0.723582 0.690238i \(-0.757506\pi\)
0.690238 + 0.723582i \(0.257506\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −18.1170 −0.618866 −0.309433 0.950921i \(-0.600139\pi\)
−0.309433 + 0.950921i \(0.600139\pi\)
\(858\) 0 0
\(859\) −31.9398 31.9398i −1.08977 1.08977i −0.995551 0.0942195i \(-0.969964\pi\)
−0.0942195 0.995551i \(-0.530036\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 49.5050 1.68517 0.842585 0.538563i \(-0.181033\pi\)
0.842585 + 0.538563i \(0.181033\pi\)
\(864\) 0 0
\(865\) −2.81084 −0.0955714
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −7.35312 7.35312i −0.249437 0.249437i
\(870\) 0 0
\(871\) 6.94519 0.235329
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.372932 + 0.372932i −0.0126074 + 0.0126074i
\(876\) 0 0
\(877\) −14.5440 14.5440i −0.491114 0.491114i 0.417543 0.908657i \(-0.362891\pi\)
−0.908657 + 0.417543i \(0.862891\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 4.39383i 0.148032i −0.997257 0.0740159i \(-0.976418\pi\)
0.997257 0.0740159i \(-0.0235816\pi\)
\(882\) 0 0
\(883\) 2.14779 2.14779i 0.0722788 0.0722788i −0.670043 0.742322i \(-0.733724\pi\)
0.742322 + 0.670043i \(0.233724\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.76736i 0.126496i 0.997998 + 0.0632478i \(0.0201458\pi\)
−0.997998 + 0.0632478i \(0.979854\pi\)
\(888\) 0 0
\(889\) 7.55873i 0.253512i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 43.0583 43.0583i 1.44089 1.44089i
\(894\) 0 0
\(895\) 22.0786i 0.738006i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −18.2908 18.2908i −0.610032 0.610032i
\(900\) 0 0
\(901\) −28.3068 + 28.3068i −0.943035 + 0.943035i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −16.7805 −0.557804
\(906\) 0 0
\(907\) −34.8859 34.8859i −1.15837 1.15837i −0.984827 0.173540i \(-0.944479\pi\)
−0.173540 0.984827i \(-0.555521\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 36.5990 1.21258 0.606290 0.795244i \(-0.292657\pi\)
0.606290 + 0.795244i \(0.292657\pi\)
\(912\) 0 0
\(913\) −6.67010 −0.220748
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.33247 2.33247i −0.0770250 0.0770250i
\(918\) 0 0
\(919\) −30.0080 −0.989874 −0.494937 0.868929i \(-0.664809\pi\)
−0.494937 + 0.868929i \(0.664809\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −39.9593 + 39.9593i −1.31528 + 1.31528i
\(924\) 0 0
\(925\) −6.42437 6.42437i −0.211232 0.211232i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 28.4741i 0.934206i 0.884203 + 0.467103i \(0.154702\pi\)
−0.884203 + 0.467103i \(0.845298\pi\)
\(930\) 0 0
\(931\) 25.5168 25.5168i 0.836280 0.836280i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 12.9376i 0.423104i
\(936\) 0 0
\(937\) 47.7787i 1.56086i −0.625242 0.780431i \(-0.715000\pi\)
0.625242 0.780431i \(-0.285000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 21.4584 21.4584i 0.699525 0.699525i −0.264783 0.964308i \(-0.585300\pi\)
0.964308 + 0.264783i \(0.0853002\pi\)
\(942\) 0 0
\(943\) 10.8214i 0.352392i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −13.5052 13.5052i −0.438859 0.438859i 0.452769 0.891628i \(-0.350436\pi\)
−0.891628 + 0.452769i \(0.850436\pi\)
\(948\) 0 0
\(949\) 22.2638 22.2638i 0.722714 0.722714i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −7.79285 −0.252435 −0.126218 0.992003i \(-0.540284\pi\)
−0.126218 + 0.992003i \(0.540284\pi\)
\(954\) 0 0
\(955\) −5.54156 5.54156i −0.179321 0.179321i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0.633284 0.0204498
\(960\) 0 0
\(961\) −7.67119 −0.247458
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.68041 + 4.68041i 0.150668 + 0.150668i
\(966\) 0 0
\(967\) −6.87264 −0.221009 −0.110505 0.993876i \(-0.535247\pi\)
−0.110505 + 0.993876i \(0.535247\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −10.6009 + 10.6009i −0.340200 + 0.340200i −0.856442 0.516243i \(-0.827330\pi\)
0.516243 + 0.856442i \(0.327330\pi\)
\(972\) 0 0
\(973\) −4.01390 4.01390i −0.128680 0.128680i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16.1936i 0.518080i −0.965867 0.259040i \(-0.916594\pi\)
0.965867 0.259040i \(-0.0834062\pi\)
\(978\) 0 0
\(979\) −15.2980 + 15.2980i −0.488925 + 0.488925i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 12.7286i 0.405978i 0.979181 + 0.202989i \(0.0650655\pi\)
−0.979181 + 0.202989i \(0.934934\pi\)
\(984\) 0 0
\(985\) 15.5659i 0.495972i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 23.6272 23.6272i 0.751303 0.751303i
\(990\) 0 0
\(991\) 21.4506i 0.681402i −0.940172 0.340701i \(-0.889336\pi\)
0.940172 0.340701i \(-0.110664\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.77372 + 1.77372i 0.0562306 + 0.0562306i
\(996\) 0 0
\(997\) 8.48964 8.48964i 0.268870 0.268870i −0.559775 0.828645i \(-0.689112\pi\)
0.828645 + 0.559775i \(0.189112\pi\)
\(998\) 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 2880.2.bl.b.431.9 24
3.2 odd 2 inner 2880.2.bl.b.431.3 24
4.3 odd 2 720.2.bl.b.611.9 yes 24
12.11 even 2 720.2.bl.b.611.4 yes 24
16.5 even 4 720.2.bl.b.251.4 24
16.11 odd 4 inner 2880.2.bl.b.1871.3 24
48.5 odd 4 720.2.bl.b.251.9 yes 24
48.11 even 4 inner 2880.2.bl.b.1871.9 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
720.2.bl.b.251.4 24 16.5 even 4
720.2.bl.b.251.9 yes 24 48.5 odd 4
720.2.bl.b.611.4 yes 24 12.11 even 2
720.2.bl.b.611.9 yes 24 4.3 odd 2
2880.2.bl.b.431.3 24 3.2 odd 2 inner
2880.2.bl.b.431.9 24 1.1 even 1 trivial
2880.2.bl.b.1871.3 24 16.11 odd 4 inner
2880.2.bl.b.1871.9 24 48.11 even 4 inner