Properties

Label 576.2.l.a.431.2
Level $576$
Weight $2$
Character 576.431
Analytic conductor $4.599$
Analytic rank $0$
Dimension $16$
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] = [576,2,Mod(143,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.143");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.l (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: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 431.2
Root \(0.944649 + 1.05244i\) of defining polynomial
Character \(\chi\) \(=\) 576.431
Dual form 576.2.l.a.143.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.10489 - 2.10489i) q^{5} +4.40731 q^{7} +O(q^{10})\) \(q+(-2.10489 - 2.10489i) q^{5} +4.40731 q^{7} +(-0.215589 + 0.215589i) q^{11} +(-2.73544 - 2.73544i) q^{13} -2.36438i q^{17} +(-0.758681 + 0.758681i) q^{19} -1.75549i q^{23} +3.86110i q^{25} +(5.54221 - 5.54221i) q^{29} -9.01709i q^{31} +(-9.27690 - 9.27690i) q^{35} +(3.10242 - 3.10242i) q^{37} -10.1014 q^{41} +(3.54621 + 3.54621i) q^{43} +3.90136 q^{47} +12.4244 q^{49} +(2.71378 + 2.71378i) q^{53} +0.907583 q^{55} +(3.40445 - 3.40445i) q^{59} +(-1.75868 - 1.75868i) q^{61} +11.5156i q^{65} +(-9.11951 + 9.11951i) q^{67} +11.8897i q^{71} -0.482639i q^{73} +(-0.950169 + 0.950169i) q^{77} +6.88995i q^{79} +(4.79951 + 4.79951i) q^{83} +(-4.97676 + 4.97676i) q^{85} +7.00534 q^{89} +(-12.0559 - 12.0559i) q^{91} +3.19387 q^{95} -3.34374 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{19} + 32 q^{43} + 16 q^{49} + 64 q^{55} - 32 q^{61} + 16 q^{67} - 32 q^{85} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −2.10489 2.10489i −0.941334 0.941334i 0.0570377 0.998372i \(-0.481834\pi\)
−0.998372 + 0.0570377i \(0.981834\pi\)
\(6\) 0 0
\(7\) 4.40731 1.66581 0.832904 0.553418i \(-0.186677\pi\)
0.832904 + 0.553418i \(0.186677\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.215589 + 0.215589i −0.0650026 + 0.0650026i −0.738861 0.673858i \(-0.764636\pi\)
0.673858 + 0.738861i \(0.264636\pi\)
\(12\) 0 0
\(13\) −2.73544 2.73544i −0.758675 0.758675i 0.217406 0.976081i \(-0.430240\pi\)
−0.976081 + 0.217406i \(0.930240\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.36438i 0.573447i −0.958013 0.286724i \(-0.907434\pi\)
0.958013 0.286724i \(-0.0925661\pi\)
\(18\) 0 0
\(19\) −0.758681 + 0.758681i −0.174053 + 0.174053i −0.788758 0.614704i \(-0.789275\pi\)
0.614704 + 0.788758i \(0.289275\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.75549i 0.366045i −0.983109 0.183023i \(-0.941412\pi\)
0.983109 0.183023i \(-0.0585881\pi\)
\(24\) 0 0
\(25\) 3.86110i 0.772221i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.54221 5.54221i 1.02916 1.02916i 0.0296002 0.999562i \(-0.490577\pi\)
0.999562 0.0296002i \(-0.00942341\pi\)
\(30\) 0 0
\(31\) 9.01709i 1.61952i −0.586763 0.809759i \(-0.699598\pi\)
0.586763 0.809759i \(-0.300402\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.27690 9.27690i −1.56808 1.56808i
\(36\) 0 0
\(37\) 3.10242 3.10242i 0.510035 0.510035i −0.404502 0.914537i \(-0.632555\pi\)
0.914537 + 0.404502i \(0.132555\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.1014 −1.57757 −0.788785 0.614669i \(-0.789290\pi\)
−0.788785 + 0.614669i \(0.789290\pi\)
\(42\) 0 0
\(43\) 3.54621 + 3.54621i 0.540792 + 0.540792i 0.923761 0.382969i \(-0.125099\pi\)
−0.382969 + 0.923761i \(0.625099\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.90136 0.569072 0.284536 0.958665i \(-0.408160\pi\)
0.284536 + 0.958665i \(0.408160\pi\)
\(48\) 0 0
\(49\) 12.4244 1.77491
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.71378 + 2.71378i 0.372766 + 0.372766i 0.868484 0.495717i \(-0.165095\pi\)
−0.495717 + 0.868484i \(0.665095\pi\)
\(54\) 0 0
\(55\) 0.907583 0.122378
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.40445 3.40445i 0.443222 0.443222i −0.449871 0.893093i \(-0.648530\pi\)
0.893093 + 0.449871i \(0.148530\pi\)
\(60\) 0 0
\(61\) −1.75868 1.75868i −0.225176 0.225176i 0.585498 0.810674i \(-0.300899\pi\)
−0.810674 + 0.585498i \(0.800899\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 11.5156i 1.42833i
\(66\) 0 0
\(67\) −9.11951 + 9.11951i −1.11413 + 1.11413i −0.121539 + 0.992587i \(0.538783\pi\)
−0.992587 + 0.121539i \(0.961217\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.8897i 1.41105i 0.708684 + 0.705526i \(0.249289\pi\)
−0.708684 + 0.705526i \(0.750711\pi\)
\(72\) 0 0
\(73\) 0.482639i 0.0564886i −0.999601 0.0282443i \(-0.991008\pi\)
0.999601 0.0282443i \(-0.00899163\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.950169 + 0.950169i −0.108282 + 0.108282i
\(78\) 0 0
\(79\) 6.88995i 0.775180i 0.921832 + 0.387590i \(0.126692\pi\)
−0.921832 + 0.387590i \(0.873308\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.79951 + 4.79951i 0.526814 + 0.526814i 0.919621 0.392807i \(-0.128496\pi\)
−0.392807 + 0.919621i \(0.628496\pi\)
\(84\) 0 0
\(85\) −4.97676 + 4.97676i −0.539805 + 0.539805i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.00534 0.742564 0.371282 0.928520i \(-0.378918\pi\)
0.371282 + 0.928520i \(0.378918\pi\)
\(90\) 0 0
\(91\) −12.0559 12.0559i −1.26381 1.26381i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.19387 0.327685
\(96\) 0 0
\(97\) −3.34374 −0.339506 −0.169753 0.985487i \(-0.554297\pi\)
−0.169753 + 0.985487i \(0.554297\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.76361 1.76361i −0.175486 0.175486i 0.613899 0.789385i \(-0.289600\pi\)
−0.789385 + 0.613899i \(0.789600\pi\)
\(102\) 0 0
\(103\) 1.01709 0.100217 0.0501085 0.998744i \(-0.484043\pi\)
0.0501085 + 0.998744i \(0.484043\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.33152 + 2.33152i −0.225396 + 0.225396i −0.810766 0.585370i \(-0.800949\pi\)
0.585370 + 0.810766i \(0.300949\pi\)
\(108\) 0 0
\(109\) −8.07918 8.07918i −0.773845 0.773845i 0.204931 0.978776i \(-0.434303\pi\)
−0.978776 + 0.204931i \(0.934303\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.09677i 0.197247i 0.995125 + 0.0986237i \(0.0314440\pi\)
−0.995125 + 0.0986237i \(0.968556\pi\)
\(114\) 0 0
\(115\) −3.69511 + 3.69511i −0.344571 + 0.344571i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 10.4206i 0.955253i
\(120\) 0 0
\(121\) 10.9070i 0.991549i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.39725 + 2.39725i −0.214416 + 0.214416i
\(126\) 0 0
\(127\) 10.2802i 0.912218i −0.889924 0.456109i \(-0.849243\pi\)
0.889924 0.456109i \(-0.150757\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.80891 + 6.80891i 0.594897 + 0.594897i 0.938950 0.344053i \(-0.111800\pi\)
−0.344053 + 0.938950i \(0.611800\pi\)
\(132\) 0 0
\(133\) −3.34374 + 3.34374i −0.289939 + 0.289939i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.7922 1.17834 0.589172 0.808008i \(-0.299454\pi\)
0.589172 + 0.808008i \(0.299454\pi\)
\(138\) 0 0
\(139\) 13.1195 + 13.1195i 1.11278 + 1.11278i 0.992773 + 0.120010i \(0.0382927\pi\)
0.120010 + 0.992773i \(0.461707\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.17946 0.0986317
\(144\) 0 0
\(145\) −23.3314 −1.93757
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.76174 + 7.76174i 0.635867 + 0.635867i 0.949533 0.313667i \(-0.101557\pi\)
−0.313667 + 0.949533i \(0.601557\pi\)
\(150\) 0 0
\(151\) 0.202466 0.0164765 0.00823823 0.999966i \(-0.497378\pi\)
0.00823823 + 0.999966i \(0.497378\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −18.9800 + 18.9800i −1.52451 + 1.52451i
\(156\) 0 0
\(157\) 3.75868 + 3.75868i 0.299975 + 0.299975i 0.841004 0.541029i \(-0.181965\pi\)
−0.541029 + 0.841004i \(0.681965\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.73700i 0.609761i
\(162\) 0 0
\(163\) 13.2684 13.2684i 1.03926 1.03926i 0.0400655 0.999197i \(-0.487243\pi\)
0.999197 0.0400655i \(-0.0127567\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 20.1644i 1.56037i 0.625548 + 0.780186i \(0.284875\pi\)
−0.625548 + 0.780186i \(0.715125\pi\)
\(168\) 0 0
\(169\) 1.96528i 0.151175i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.98315 + 3.98315i −0.302833 + 0.302833i −0.842121 0.539288i \(-0.818693\pi\)
0.539288 + 0.842121i \(0.318693\pi\)
\(174\) 0 0
\(175\) 17.0171i 1.28637i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 14.7182 + 14.7182i 1.10009 + 1.10009i 0.994399 + 0.105688i \(0.0337044\pi\)
0.105688 + 0.994399i \(0.466296\pi\)
\(180\) 0 0
\(181\) 3.26456 3.26456i 0.242653 0.242653i −0.575294 0.817947i \(-0.695112\pi\)
0.817947 + 0.575294i \(0.195112\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −13.0605 −0.960227
\(186\) 0 0
\(187\) 0.509736 + 0.509736i 0.0372756 + 0.0372756i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −7.69868 −0.557057 −0.278528 0.960428i \(-0.589847\pi\)
−0.278528 + 0.960428i \(0.589847\pi\)
\(192\) 0 0
\(193\) −3.51736 −0.253185 −0.126593 0.991955i \(-0.540404\pi\)
−0.126593 + 0.991955i \(0.540404\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.40456 + 6.40456i 0.456306 + 0.456306i 0.897441 0.441135i \(-0.145424\pi\)
−0.441135 + 0.897441i \(0.645424\pi\)
\(198\) 0 0
\(199\) −23.3491 −1.65517 −0.827586 0.561339i \(-0.810286\pi\)
−0.827586 + 0.561339i \(0.810286\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 24.4262 24.4262i 1.71439 1.71439i
\(204\) 0 0
\(205\) 21.2623 + 21.2623i 1.48502 + 1.48502i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.327127i 0.0226278i
\(210\) 0 0
\(211\) 6.63688 6.63688i 0.456901 0.456901i −0.440736 0.897637i \(-0.645282\pi\)
0.897637 + 0.440736i \(0.145282\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 14.9287i 1.01813i
\(216\) 0 0
\(217\) 39.7411i 2.69780i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.46763 + 6.46763i −0.435060 + 0.435060i
\(222\) 0 0
\(223\) 13.2219i 0.885406i 0.896668 + 0.442703i \(0.145980\pi\)
−0.896668 + 0.442703i \(0.854020\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −16.4029 16.4029i −1.08870 1.08870i −0.995663 0.0930369i \(-0.970343\pi\)
−0.0930369 0.995663i \(-0.529657\pi\)
\(228\) 0 0
\(229\) −4.98677 + 4.98677i −0.329535 + 0.329535i −0.852410 0.522875i \(-0.824860\pi\)
0.522875 + 0.852410i \(0.324860\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.05879 −0.0693634 −0.0346817 0.999398i \(-0.511042\pi\)
−0.0346817 + 0.999398i \(0.511042\pi\)
\(234\) 0 0
\(235\) −8.21193 8.21193i −0.535687 0.535687i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.317107 0.0205119 0.0102560 0.999947i \(-0.496735\pi\)
0.0102560 + 0.999947i \(0.496735\pi\)
\(240\) 0 0
\(241\) 11.7334 0.755816 0.377908 0.925843i \(-0.376644\pi\)
0.377908 + 0.925843i \(0.376644\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −26.1520 26.1520i −1.67079 1.67079i
\(246\) 0 0
\(247\) 4.15065 0.264100
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7.02450 7.02450i 0.443382 0.443382i −0.449765 0.893147i \(-0.648492\pi\)
0.893147 + 0.449765i \(0.148492\pi\)
\(252\) 0 0
\(253\) 0.378465 + 0.378465i 0.0237939 + 0.0237939i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 23.5830i 1.47107i −0.677487 0.735535i \(-0.736931\pi\)
0.677487 0.735535i \(-0.263069\pi\)
\(258\) 0 0
\(259\) 13.6733 13.6733i 0.849621 0.849621i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 20.1370i 1.24170i −0.783928 0.620851i \(-0.786787\pi\)
0.783928 0.620851i \(-0.213213\pi\)
\(264\) 0 0
\(265\) 11.4244i 0.701796i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8.62201 8.62201i 0.525693 0.525693i −0.393592 0.919285i \(-0.628768\pi\)
0.919285 + 0.393592i \(0.128768\pi\)
\(270\) 0 0
\(271\) 2.18722i 0.132864i −0.997791 0.0664319i \(-0.978838\pi\)
0.997791 0.0664319i \(-0.0211615\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.832413 0.832413i −0.0501964 0.0501964i
\(276\) 0 0
\(277\) 8.64248 8.64248i 0.519277 0.519277i −0.398076 0.917352i \(-0.630322\pi\)
0.917352 + 0.398076i \(0.130322\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 14.8081 0.883375 0.441688 0.897169i \(-0.354380\pi\)
0.441688 + 0.897169i \(0.354380\pi\)
\(282\) 0 0
\(283\) −6.12714 6.12714i −0.364221 0.364221i 0.501144 0.865364i \(-0.332913\pi\)
−0.865364 + 0.501144i \(0.832913\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −44.5199 −2.62793
\(288\) 0 0
\(289\) 11.4097 0.671158
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.8979 + 11.8979i 0.695080 + 0.695080i 0.963345 0.268265i \(-0.0864503\pi\)
−0.268265 + 0.963345i \(0.586450\pi\)
\(294\) 0 0
\(295\) −14.3320 −0.834441
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.80204 + 4.80204i −0.277709 + 0.277709i
\(300\) 0 0
\(301\) 15.6293 + 15.6293i 0.900855 + 0.900855i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7.40365i 0.423932i
\(306\) 0 0
\(307\) −4.91467 + 4.91467i −0.280495 + 0.280495i −0.833306 0.552811i \(-0.813555\pi\)
0.552811 + 0.833306i \(0.313555\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.7712i 1.17783i −0.808195 0.588915i \(-0.799555\pi\)
0.808195 0.588915i \(-0.200445\pi\)
\(312\) 0 0
\(313\) 16.3897i 0.926400i 0.886254 + 0.463200i \(0.153299\pi\)
−0.886254 + 0.463200i \(0.846701\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.3592 + 22.3592i −1.25582 + 1.25582i −0.302749 + 0.953070i \(0.597904\pi\)
−0.953070 + 0.302749i \(0.902096\pi\)
\(318\) 0 0
\(319\) 2.38968i 0.133796i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.79381 + 1.79381i 0.0998103 + 0.0998103i
\(324\) 0 0
\(325\) 10.5618 10.5618i 0.585865 0.585865i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 17.1945 0.947965
\(330\) 0 0
\(331\) 7.08533 + 7.08533i 0.389445 + 0.389445i 0.874490 0.485044i \(-0.161197\pi\)
−0.485044 + 0.874490i \(0.661197\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 38.3911 2.09753
\(336\) 0 0
\(337\) −23.3314 −1.27094 −0.635472 0.772124i \(-0.719195\pi\)
−0.635472 + 0.772124i \(0.719195\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.94399 + 1.94399i 0.105273 + 0.105273i
\(342\) 0 0
\(343\) 23.9070 1.29086
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −21.2074 + 21.2074i −1.13847 + 1.13847i −0.149751 + 0.988724i \(0.547847\pi\)
−0.988724 + 0.149751i \(0.952153\pi\)
\(348\) 0 0
\(349\) 5.38022 + 5.38022i 0.287996 + 0.287996i 0.836288 0.548291i \(-0.184721\pi\)
−0.548291 + 0.836288i \(0.684721\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.5163i 0.559727i −0.960040 0.279864i \(-0.909711\pi\)
0.960040 0.279864i \(-0.0902892\pi\)
\(354\) 0 0
\(355\) 25.0266 25.0266i 1.32827 1.32827i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.7583i 0.620578i −0.950642 0.310289i \(-0.899574\pi\)
0.950642 0.310289i \(-0.100426\pi\)
\(360\) 0 0
\(361\) 17.8488i 0.939411i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.01590 + 1.01590i −0.0531747 + 0.0531747i
\(366\) 0 0
\(367\) 3.49973i 0.182684i −0.995820 0.0913422i \(-0.970884\pi\)
0.995820 0.0913422i \(-0.0291157\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.9605 + 11.9605i 0.620957 + 0.620957i
\(372\) 0 0
\(373\) 19.1831 19.1831i 0.993262 0.993262i −0.00671500 0.999977i \(-0.502137\pi\)
0.999977 + 0.00671500i \(0.00213747\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −30.3208 −1.56160
\(378\) 0 0
\(379\) 14.9977 + 14.9977i 0.770381 + 0.770381i 0.978173 0.207792i \(-0.0666278\pi\)
−0.207792 + 0.978173i \(0.566628\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −30.4027 −1.55351 −0.776754 0.629805i \(-0.783135\pi\)
−0.776754 + 0.629805i \(0.783135\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −19.4571 19.4571i −0.986516 0.986516i 0.0133943 0.999910i \(-0.495736\pi\)
−0.999910 + 0.0133943i \(0.995736\pi\)
\(390\) 0 0
\(391\) −4.15065 −0.209908
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 14.5026 14.5026i 0.729704 0.729704i
\(396\) 0 0
\(397\) −20.4339 20.4339i −1.02555 1.02555i −0.999665 0.0258815i \(-0.991761\pi\)
−0.0258815 0.999665i \(-0.508239\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.5671i 1.12695i −0.826133 0.563475i \(-0.809464\pi\)
0.826133 0.563475i \(-0.190536\pi\)
\(402\) 0 0
\(403\) −24.6657 + 24.6657i −1.22869 + 1.22869i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.33770i 0.0663073i
\(408\) 0 0
\(409\) 9.78286i 0.483731i −0.970310 0.241866i \(-0.922241\pi\)
0.970310 0.241866i \(-0.0777593\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 15.0045 15.0045i 0.738323 0.738323i
\(414\) 0 0
\(415\) 20.2048i 0.991817i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −9.09125 9.09125i −0.444137 0.444137i 0.449263 0.893400i \(-0.351687\pi\)
−0.893400 + 0.449263i \(0.851687\pi\)
\(420\) 0 0
\(421\) −17.9862 + 17.9862i −0.876595 + 0.876595i −0.993181 0.116586i \(-0.962805\pi\)
0.116586 + 0.993181i \(0.462805\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 9.12913 0.442828
\(426\) 0 0
\(427\) −7.75106 7.75106i −0.375100 0.375100i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 30.7707 1.48217 0.741085 0.671411i \(-0.234311\pi\)
0.741085 + 0.671411i \(0.234311\pi\)
\(432\) 0 0
\(433\) 3.49735 0.168072 0.0840360 0.996463i \(-0.473219\pi\)
0.0840360 + 0.996463i \(0.473219\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.33186 + 1.33186i 0.0637113 + 0.0637113i
\(438\) 0 0
\(439\) 21.1290 1.00843 0.504216 0.863578i \(-0.331782\pi\)
0.504216 + 0.863578i \(0.331782\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.6022 + 12.6022i −0.598750 + 0.598750i −0.939980 0.341230i \(-0.889157\pi\)
0.341230 + 0.939980i \(0.389157\pi\)
\(444\) 0 0
\(445\) −14.7454 14.7454i −0.699001 0.699001i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.98731i 0.471330i 0.971834 + 0.235665i \(0.0757268\pi\)
−0.971834 + 0.235665i \(0.924273\pi\)
\(450\) 0 0
\(451\) 2.17775 2.17775i 0.102546 0.102546i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 50.7528i 2.37933i
\(456\) 0 0
\(457\) 15.2508i 0.713402i −0.934219 0.356701i \(-0.883902\pi\)
0.934219 0.356701i \(-0.116098\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.3850 14.3850i 0.669976 0.669976i −0.287734 0.957710i \(-0.592902\pi\)
0.957710 + 0.287734i \(0.0929019\pi\)
\(462\) 0 0
\(463\) 22.1295i 1.02845i 0.857657 + 0.514223i \(0.171920\pi\)
−0.857657 + 0.514223i \(0.828080\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.79951 + 4.79951i 0.222095 + 0.222095i 0.809380 0.587285i \(-0.199803\pi\)
−0.587285 + 0.809380i \(0.699803\pi\)
\(468\) 0 0
\(469\) −40.1926 + 40.1926i −1.85592 + 1.85592i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.52905 −0.0703058
\(474\) 0 0
\(475\) −2.92934 2.92934i −0.134408 0.134408i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 23.2801 1.06369 0.531847 0.846840i \(-0.321498\pi\)
0.531847 + 0.846840i \(0.321498\pi\)
\(480\) 0 0
\(481\) −16.9730 −0.773902
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.03820 + 7.03820i 0.319588 + 0.319588i
\(486\) 0 0
\(487\) −21.0748 −0.954990 −0.477495 0.878635i \(-0.658455\pi\)
−0.477495 + 0.878635i \(0.658455\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.1459 26.1459i 1.17995 1.17995i 0.200194 0.979756i \(-0.435843\pi\)
0.979756 0.200194i \(-0.0641571\pi\)
\(492\) 0 0
\(493\) −13.1039 13.1039i −0.590170 0.590170i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 52.4018i 2.35054i
\(498\) 0 0
\(499\) −21.0195 + 21.0195i −0.940961 + 0.940961i −0.998352 0.0573910i \(-0.981722\pi\)
0.0573910 + 0.998352i \(0.481722\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 44.4925i 1.98382i 0.126929 + 0.991912i \(0.459488\pi\)
−0.126929 + 0.991912i \(0.540512\pi\)
\(504\) 0 0
\(505\) 7.42440i 0.330382i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 11.7664 11.7664i 0.521536 0.521536i −0.396499 0.918035i \(-0.629775\pi\)
0.918035 + 0.396499i \(0.129775\pi\)
\(510\) 0 0
\(511\) 2.12714i 0.0940991i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.14086 2.14086i −0.0943377 0.0943377i
\(516\) 0 0
\(517\) −0.841092 + 0.841092i −0.0369912 + 0.0369912i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −29.6216 −1.29775 −0.648873 0.760897i \(-0.724759\pi\)
−0.648873 + 0.760897i \(0.724759\pi\)
\(522\) 0 0
\(523\) −4.96353 4.96353i −0.217040 0.217040i 0.590210 0.807250i \(-0.299045\pi\)
−0.807250 + 0.590210i \(0.799045\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −21.3199 −0.928708
\(528\) 0 0
\(529\) 19.9183 0.866011
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 27.6317 + 27.6317i 1.19686 + 1.19686i
\(534\) 0 0
\(535\) 9.81517 0.424347
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.67857 + 2.67857i −0.115374 + 0.115374i
\(540\) 0 0
\(541\) 10.0792 + 10.0792i 0.433338 + 0.433338i 0.889762 0.456424i \(-0.150870\pi\)
−0.456424 + 0.889762i \(0.650870\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 34.0115i 1.45689i
\(546\) 0 0
\(547\) −2.48797 + 2.48797i −0.106378 + 0.106378i −0.758293 0.651914i \(-0.773966\pi\)
0.651914 + 0.758293i \(0.273966\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.40953i 0.358258i
\(552\) 0 0
\(553\) 30.3662i 1.29130i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −24.2575 + 24.2575i −1.02782 + 1.02782i −0.0282205 + 0.999602i \(0.508984\pi\)
−0.999602 + 0.0282205i \(0.991016\pi\)
\(558\) 0 0
\(559\) 19.4009i 0.820570i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −14.4061 14.4061i −0.607143 0.607143i 0.335055 0.942198i \(-0.391245\pi\)
−0.942198 + 0.335055i \(0.891245\pi\)
\(564\) 0 0
\(565\) 4.41346 4.41346i 0.185676 0.185676i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 33.5254 1.40546 0.702729 0.711457i \(-0.251964\pi\)
0.702729 + 0.711457i \(0.251964\pi\)
\(570\) 0 0
\(571\) −14.5368 14.5368i −0.608348 0.608348i 0.334167 0.942514i \(-0.391545\pi\)
−0.942514 + 0.334167i \(0.891545\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.77813 0.282668
\(576\) 0 0
\(577\) 20.3662 0.847855 0.423927 0.905696i \(-0.360651\pi\)
0.423927 + 0.905696i \(0.360651\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 21.1529 + 21.1529i 0.877571 + 0.877571i
\(582\) 0 0
\(583\) −1.17012 −0.0484616
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −17.4809 + 17.4809i −0.721512 + 0.721512i −0.968913 0.247401i \(-0.920424\pi\)
0.247401 + 0.968913i \(0.420424\pi\)
\(588\) 0 0
\(589\) 6.84109 + 6.84109i 0.281882 + 0.281882i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 25.6632i 1.05386i −0.849908 0.526930i \(-0.823343\pi\)
0.849908 0.526930i \(-0.176657\pi\)
\(594\) 0 0
\(595\) −21.9341 + 21.9341i −0.899212 + 0.899212i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 22.4652i 0.917902i −0.888461 0.458951i \(-0.848225\pi\)
0.888461 0.458951i \(-0.151775\pi\)
\(600\) 0 0
\(601\) 31.8106i 1.29758i 0.760967 + 0.648790i \(0.224725\pi\)
−0.760967 + 0.648790i \(0.775275\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 22.9581 22.9581i 0.933379 0.933379i
\(606\) 0 0
\(607\) 14.9829i 0.608138i −0.952650 0.304069i \(-0.901655\pi\)
0.952650 0.304069i \(-0.0983453\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −10.6720 10.6720i −0.431741 0.431741i
\(612\) 0 0
\(613\) 25.6734 25.6734i 1.03694 1.03694i 0.0376494 0.999291i \(-0.488013\pi\)
0.999291 0.0376494i \(-0.0119870\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −27.1598 −1.09341 −0.546705 0.837325i \(-0.684118\pi\)
−0.546705 + 0.837325i \(0.684118\pi\)
\(618\) 0 0
\(619\) 29.4244 + 29.4244i 1.18267 + 1.18267i 0.979051 + 0.203616i \(0.0652693\pi\)
0.203616 + 0.979051i \(0.434731\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 30.8747 1.23697
\(624\) 0 0
\(625\) 29.3974 1.17590
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −7.33532 7.33532i −0.292478 0.292478i
\(630\) 0 0
\(631\) 7.84697 0.312383 0.156191 0.987727i \(-0.450078\pi\)
0.156191 + 0.987727i \(0.450078\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −21.6386 + 21.6386i −0.858702 + 0.858702i
\(636\) 0 0
\(637\) −33.9862 33.9862i −1.34658 1.34658i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 37.7956i 1.49284i 0.665477 + 0.746418i \(0.268228\pi\)
−0.665477 + 0.746418i \(0.731772\pi\)
\(642\) 0 0
\(643\) −27.3026 + 27.3026i −1.07671 + 1.07671i −0.0799071 + 0.996802i \(0.525462\pi\)
−0.996802 + 0.0799071i \(0.974538\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.100686i 0.00395836i 0.999998 + 0.00197918i \(0.000629993\pi\)
−0.999998 + 0.00197918i \(0.999370\pi\)
\(648\) 0 0
\(649\) 1.46793i 0.0576212i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.10477 9.10477i 0.356297 0.356297i −0.506149 0.862446i \(-0.668931\pi\)
0.862446 + 0.506149i \(0.168931\pi\)
\(654\) 0 0
\(655\) 28.6640i 1.11999i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 14.6141 + 14.6141i 0.569285 + 0.569285i 0.931928 0.362643i \(-0.118126\pi\)
−0.362643 + 0.931928i \(0.618126\pi\)
\(660\) 0 0
\(661\) 5.87057 5.87057i 0.228339 0.228339i −0.583660 0.811998i \(-0.698380\pi\)
0.811998 + 0.583660i \(0.198380\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 14.0764 0.545860
\(666\) 0 0
\(667\) −9.72929 9.72929i −0.376720 0.376720i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.758305 0.0292741
\(672\) 0 0
\(673\) −27.3515 −1.05432 −0.527161 0.849766i \(-0.676743\pi\)
−0.527161 + 0.849766i \(0.676743\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −31.2462 31.2462i −1.20089 1.20089i −0.973897 0.226992i \(-0.927111\pi\)
−0.226992 0.973897i \(-0.572889\pi\)
\(678\) 0 0
\(679\) −14.7369 −0.565551
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15.0378 15.0378i 0.575406 0.575406i −0.358228 0.933634i \(-0.616619\pi\)
0.933634 + 0.358228i \(0.116619\pi\)
\(684\) 0 0
\(685\) −29.0310 29.0310i −1.10922 1.10922i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 14.8468i 0.565617i
\(690\) 0 0
\(691\) −4.24894 + 4.24894i −0.161637 + 0.161637i −0.783292 0.621654i \(-0.786461\pi\)
0.621654 + 0.783292i \(0.286461\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 55.2302i 2.09500i
\(696\) 0 0
\(697\) 23.8835i 0.904653i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −23.4629 + 23.4629i −0.886183 + 0.886183i −0.994154 0.107971i \(-0.965565\pi\)
0.107971 + 0.994154i \(0.465565\pi\)
\(702\) 0 0
\(703\) 4.70750i 0.177547i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7.77278 7.77278i −0.292326 0.292326i
\(708\) 0 0
\(709\) 2.79314 2.79314i 0.104898 0.104898i −0.652710 0.757608i \(-0.726368\pi\)
0.757608 + 0.652710i \(0.226368\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −15.8294 −0.592816
\(714\) 0 0
\(715\) −2.48264 2.48264i −0.0928454 0.0928454i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 33.8130 1.26101 0.630507 0.776184i \(-0.282847\pi\)
0.630507 + 0.776184i \(0.282847\pi\)
\(720\) 0 0
\(721\) 4.48264 0.166942
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 21.3990 + 21.3990i 0.794740 + 0.794740i
\(726\) 0 0
\(727\) 2.42732 0.0900245 0.0450122 0.998986i \(-0.485667\pi\)
0.0450122 + 0.998986i \(0.485667\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.38460 8.38460i 0.310115 0.310115i
\(732\) 0 0
\(733\) 16.8596 + 16.8596i 0.622725 + 0.622725i 0.946227 0.323503i \(-0.104860\pi\)
−0.323503 + 0.946227i \(0.604860\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.93214i 0.144842i
\(738\) 0 0
\(739\) −25.2243 + 25.2243i −0.927892 + 0.927892i −0.997570 0.0696780i \(-0.977803\pi\)
0.0696780 + 0.997570i \(0.477803\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 33.4926i 1.22872i 0.789025 + 0.614361i \(0.210586\pi\)
−0.789025 + 0.614361i \(0.789414\pi\)
\(744\) 0 0
\(745\) 32.6752i 1.19713i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −10.2757 + 10.2757i −0.375467 + 0.375467i
\(750\) 0 0
\(751\) 17.9247i 0.654081i 0.945010 + 0.327040i \(0.106051\pi\)
−0.945010 + 0.327040i \(0.893949\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −0.426168 0.426168i −0.0155099 0.0155099i
\(756\) 0 0
\(757\) 3.10619 3.10619i 0.112896 0.112896i −0.648402 0.761298i \(-0.724562\pi\)
0.761298 + 0.648402i \(0.224562\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.10917 0.0402073 0.0201037 0.999798i \(-0.493600\pi\)
0.0201037 + 0.999798i \(0.493600\pi\)
\(762\) 0 0
\(763\) −35.6075 35.6075i −1.28908 1.28908i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −18.6254 −0.672523
\(768\) 0 0
\(769\) 22.4591 0.809897 0.404948 0.914340i \(-0.367289\pi\)
0.404948 + 0.914340i \(0.367289\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −22.6327 22.6327i −0.814041 0.814041i 0.171196 0.985237i \(-0.445237\pi\)
−0.985237 + 0.171196i \(0.945237\pi\)
\(774\) 0 0
\(775\) 34.8159 1.25062
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.66372 7.66372i 0.274581 0.274581i
\(780\) 0 0
\(781\) −2.56330 2.56330i −0.0917221 0.0917221i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 15.8232i 0.564754i
\(786\) 0 0
\(787\) 38.6505 38.6505i 1.37774 1.37774i 0.529315 0.848425i \(-0.322449\pi\)
0.848425 0.529315i \(-0.177551\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 9.24111i 0.328576i
\(792\) 0 0
\(793\) 9.62153i 0.341671i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37.8969 37.8969i 1.34238 1.34238i 0.448686 0.893689i \(-0.351892\pi\)
0.893689 0.448686i \(-0.148108\pi\)
\(798\) 0 0
\(799\) 9.22432i 0.326333i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0.104052 + 0.104052i 0.00367191 + 0.00367191i
\(804\) 0 0
\(805\) −16.2855 + 16.2855i −0.573989 + 0.573989i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 27.0719 0.951799 0.475899 0.879500i \(-0.342123\pi\)
0.475899 + 0.879500i \(0.342123\pi\)
\(810\) 0 0
\(811\) 19.0554 + 19.0554i 0.669126 + 0.669126i 0.957514 0.288388i \(-0.0931193\pi\)
−0.288388 + 0.957514i \(0.593119\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −55.8571 −1.95659
\(816\) 0 0
\(817\) −5.38088 −0.188253
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −37.5393 37.5393i −1.31013 1.31013i −0.921315 0.388817i \(-0.872884\pi\)
−0.388817 0.921315i \(-0.627116\pi\)
\(822\) 0 0
\(823\) 27.9589 0.974584 0.487292 0.873239i \(-0.337985\pi\)
0.487292 + 0.873239i \(0.337985\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.04182 2.04182i 0.0710011 0.0710011i −0.670715 0.741716i \(-0.734012\pi\)
0.741716 + 0.670715i \(0.234012\pi\)
\(828\) 0 0
\(829\) 34.5808 + 34.5808i 1.20104 + 1.20104i 0.973851 + 0.227188i \(0.0729533\pi\)
0.227188 + 0.973851i \(0.427047\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 29.3760i 1.01782i
\(834\) 0 0
\(835\) 42.4439 42.4439i 1.46883 1.46883i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0.865723i 0.0298881i 0.999888 + 0.0149440i \(0.00475701\pi\)
−0.999888 + 0.0149440i \(0.995243\pi\)
\(840\) 0 0
\(841\) 32.4321i 1.11835i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.13669 4.13669i 0.142306 0.142306i
\(846\) 0 0
\(847\) 48.0707i 1.65173i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −5.44628 5.44628i −0.186696 0.186696i
\(852\) 0 0
\(853\) −14.4262 + 14.4262i −0.493942 + 0.493942i −0.909546 0.415604i \(-0.863570\pi\)
0.415604 + 0.909546i \(0.363570\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −37.6532 −1.28621 −0.643105 0.765778i \(-0.722354\pi\)
−0.643105 + 0.765778i \(0.722354\pi\)
\(858\) 0 0
\(859\) 38.7470 + 38.7470i 1.32203 + 1.32203i 0.912130 + 0.409901i \(0.134437\pi\)
0.409901 + 0.912130i \(0.365563\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −14.8672 −0.506085 −0.253043 0.967455i \(-0.581431\pi\)
−0.253043 + 0.967455i \(0.581431\pi\)
\(864\) 0 0
\(865\) 16.7681 0.570134
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1.48540 1.48540i −0.0503887 0.0503887i
\(870\) 0 0
\(871\) 49.8918 1.69052
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −10.5654 + 10.5654i −0.357176 + 0.357176i
\(876\) 0 0
\(877\) −13.7699 13.7699i −0.464976 0.464976i 0.435306 0.900283i \(-0.356640\pi\)
−0.900283 + 0.435306i \(0.856640\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 20.6232i 0.694812i −0.937715 0.347406i \(-0.887063\pi\)
0.937715 0.347406i \(-0.112937\pi\)
\(882\) 0 0
\(883\) 41.5416 41.5416i 1.39799 1.39799i 0.592185 0.805802i \(-0.298265\pi\)
0.805802 0.592185i \(-0.201735\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7.44312i 0.249916i −0.992162 0.124958i \(-0.960120\pi\)
0.992162 0.124958i \(-0.0398795\pi\)
\(888\) 0 0
\(889\) 45.3079i 1.51958i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.95989 + 2.95989i −0.0990489 + 0.0990489i
\(894\) 0 0
\(895\) 61.9602i 2.07110i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −49.9746 49.9746i −1.66675 1.66675i
\(900\) 0 0
\(901\) 6.41642 6.41642i 0.213762 0.213762i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −13.7431 −0.456835
\(906\) 0 0
\(907\) −3.75106 3.75106i −0.124552 0.124552i 0.642083 0.766635i \(-0.278070\pi\)
−0.766635 + 0.642083i \(0.778070\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −29.9816 −0.993334 −0.496667 0.867941i \(-0.665443\pi\)
−0.496667 + 0.867941i \(0.665443\pi\)
\(912\) 0 0
\(913\) −2.06944 −0.0684886
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 30.0090 + 30.0090i 0.990984 + 0.990984i
\(918\) 0 0
\(919\) −38.1402 −1.25813 −0.629064 0.777353i \(-0.716562\pi\)
−0.629064 + 0.777353i \(0.716562\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 32.5237 32.5237i 1.07053 1.07053i
\(924\) 0 0
\(925\) 11.9788 + 11.9788i 0.393860 + 0.393860i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 13.8738i 0.455183i −0.973757 0.227592i \(-0.926915\pi\)
0.973757 0.227592i \(-0.0730851\pi\)
\(930\) 0 0
\(931\) −9.42615 + 9.42615i −0.308930 + 0.308930i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.14587i 0.0701775i
\(936\) 0 0
\(937\) 26.4097i 0.862767i 0.902169 + 0.431384i \(0.141974\pi\)
−0.902169 + 0.431384i \(0.858026\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −4.91123 + 4.91123i −0.160102 + 0.160102i −0.782612 0.622510i \(-0.786113\pi\)
0.622510 + 0.782612i \(0.286113\pi\)
\(942\) 0 0
\(943\) 17.7329i 0.577462i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.84386 + 6.84386i 0.222396 + 0.222396i 0.809507 0.587111i \(-0.199735\pi\)
−0.587111 + 0.809507i \(0.699735\pi\)
\(948\) 0 0
\(949\) −1.32023 + 1.32023i −0.0428565 + 0.0428565i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 4.33045 0.140277 0.0701386 0.997537i \(-0.477656\pi\)
0.0701386 + 0.997537i \(0.477656\pi\)
\(954\) 0 0
\(955\) 16.2048 + 16.2048i 0.524377 + 0.524377i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 60.7864 1.96289
\(960\) 0 0
\(961\) −50.3079 −1.62284
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 7.40365 + 7.40365i 0.238332 + 0.238332i
\(966\) 0 0
\(967\) −40.6664 −1.30774 −0.653871 0.756606i \(-0.726856\pi\)
−0.653871 + 0.756606i \(0.726856\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −24.0742 + 24.0742i −0.772577 + 0.772577i −0.978556 0.205979i \(-0.933962\pi\)
0.205979 + 0.978556i \(0.433962\pi\)
\(972\) 0 0
\(973\) 57.8218 + 57.8218i 1.85368 + 1.85368i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 21.0381i 0.673068i 0.941671 + 0.336534i \(0.109255\pi\)
−0.941671 + 0.336534i \(0.890745\pi\)
\(978\) 0 0
\(979\) −1.51028 + 1.51028i −0.0482686 + 0.0482686i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 14.1206i 0.450376i −0.974315 0.225188i \(-0.927700\pi\)
0.974315 0.225188i \(-0.0722997\pi\)
\(984\) 0 0
\(985\) 26.9618i 0.859074i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6.22534 6.22534i 0.197954 0.197954i
\(990\) 0 0
\(991\) 33.3338i 1.05888i 0.848346 + 0.529442i \(0.177599\pi\)
−0.848346 + 0.529442i \(0.822401\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 49.1472 + 49.1472i 1.55807 + 1.55807i
\(996\) 0 0
\(997\) 24.8358 24.8358i 0.786559 0.786559i −0.194369 0.980928i \(-0.562266\pi\)
0.980928 + 0.194369i \(0.0622660\pi\)
\(998\) 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 576.2.l.a.431.2 16
3.2 odd 2 inner 576.2.l.a.431.7 16
4.3 odd 2 144.2.l.a.35.1 16
8.3 odd 2 1152.2.l.a.863.7 16
8.5 even 2 1152.2.l.b.863.7 16
12.11 even 2 144.2.l.a.35.8 yes 16
16.3 odd 4 1152.2.l.b.287.2 16
16.5 even 4 144.2.l.a.107.8 yes 16
16.11 odd 4 inner 576.2.l.a.143.7 16
16.13 even 4 1152.2.l.a.287.2 16
24.5 odd 2 1152.2.l.b.863.2 16
24.11 even 2 1152.2.l.a.863.2 16
48.5 odd 4 144.2.l.a.107.1 yes 16
48.11 even 4 inner 576.2.l.a.143.2 16
48.29 odd 4 1152.2.l.a.287.7 16
48.35 even 4 1152.2.l.b.287.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.l.a.35.1 16 4.3 odd 2
144.2.l.a.35.8 yes 16 12.11 even 2
144.2.l.a.107.1 yes 16 48.5 odd 4
144.2.l.a.107.8 yes 16 16.5 even 4
576.2.l.a.143.2 16 48.11 even 4 inner
576.2.l.a.143.7 16 16.11 odd 4 inner
576.2.l.a.431.2 16 1.1 even 1 trivial
576.2.l.a.431.7 16 3.2 odd 2 inner
1152.2.l.a.287.2 16 16.13 even 4
1152.2.l.a.287.7 16 48.29 odd 4
1152.2.l.a.863.2 16 24.11 even 2
1152.2.l.a.863.7 16 8.3 odd 2
1152.2.l.b.287.2 16 16.3 odd 4
1152.2.l.b.287.7 16 48.35 even 4
1152.2.l.b.863.2 16 24.5 odd 2
1152.2.l.b.863.7 16 8.5 even 2