Properties

Label 4032.2.k.h.3905.7
Level $4032$
Weight $2$
Character 4032.3905
Analytic conductor $32.196$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4032,2,Mod(3905,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.3905");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.k (of order \(2\), degree \(1\), 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: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.101415451701035401216.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 18x^{12} + 145x^{8} - 72x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 2016)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3905.7
Root \(0.137538 + 0.752908i\) of defining polynomial
Character \(\chi\) \(=\) 4032.3905
Dual form 4032.2.k.h.3905.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.841723 q^{5} +(1.16372 - 2.37608i) q^{7} +O(q^{10})\) \(q-0.841723 q^{5} +(1.16372 - 2.37608i) q^{7} -3.64575i q^{11} +5.53019i q^{13} +0.841723 q^{17} +3.06871i q^{19} -3.64575i q^{23} -4.29150 q^{25} +8.89753i q^{29} +7.82087i q^{31} +(-0.979531 + 2.00000i) q^{35} +3.29150 q^{37} +8.66259 q^{41} +2.32744 q^{43} +9.10132 q^{47} +(-4.29150 - 5.53019i) q^{49} -4.24264i q^{53} +3.06871i q^{55} +11.0604 q^{59} +9.10132i q^{61} -4.65489i q^{65} -8.48528 q^{67} +0.354249i q^{71} -14.6315i q^{73} +(-8.66259 - 4.24264i) q^{77} -8.48528 q^{79} +9.10132 q^{83} -0.708497 q^{85} +6.97915 q^{89} +(13.1402 + 6.43560i) q^{91} -2.58301i q^{95} -3.57113i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{25} - 32 q^{37} + 16 q^{49} - 96 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(5\) −0.841723 −0.376430 −0.188215 0.982128i \(-0.560270\pi\)
−0.188215 + 0.982128i \(0.560270\pi\)
\(6\) 0 0
\(7\) 1.16372 2.37608i 0.439846 0.898073i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.64575i 1.09924i −0.835416 0.549618i \(-0.814773\pi\)
0.835416 0.549618i \(-0.185227\pi\)
\(12\) 0 0
\(13\) 5.53019i 1.53380i 0.641767 + 0.766899i \(0.278201\pi\)
−0.641767 + 0.766899i \(0.721799\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.841723 0.204148 0.102074 0.994777i \(-0.467452\pi\)
0.102074 + 0.994777i \(0.467452\pi\)
\(18\) 0 0
\(19\) 3.06871i 0.704011i 0.935998 + 0.352005i \(0.114500\pi\)
−0.935998 + 0.352005i \(0.885500\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.64575i 0.760192i −0.924947 0.380096i \(-0.875891\pi\)
0.924947 0.380096i \(-0.124109\pi\)
\(24\) 0 0
\(25\) −4.29150 −0.858301
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.89753i 1.65223i 0.563502 + 0.826115i \(0.309454\pi\)
−0.563502 + 0.826115i \(0.690546\pi\)
\(30\) 0 0
\(31\) 7.82087i 1.40467i 0.711847 + 0.702335i \(0.247859\pi\)
−0.711847 + 0.702335i \(0.752141\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.979531 + 2.00000i −0.165571 + 0.338062i
\(36\) 0 0
\(37\) 3.29150 0.541120 0.270560 0.962703i \(-0.412791\pi\)
0.270560 + 0.962703i \(0.412791\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.66259 1.35287 0.676435 0.736502i \(-0.263524\pi\)
0.676435 + 0.736502i \(0.263524\pi\)
\(42\) 0 0
\(43\) 2.32744 0.354932 0.177466 0.984127i \(-0.443210\pi\)
0.177466 + 0.984127i \(0.443210\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.10132 1.32756 0.663782 0.747926i \(-0.268950\pi\)
0.663782 + 0.747926i \(0.268950\pi\)
\(48\) 0 0
\(49\) −4.29150 5.53019i −0.613072 0.790027i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.24264i 0.582772i −0.956606 0.291386i \(-0.905884\pi\)
0.956606 0.291386i \(-0.0941163\pi\)
\(54\) 0 0
\(55\) 3.06871i 0.413785i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.0604 1.43994 0.719969 0.694006i \(-0.244156\pi\)
0.719969 + 0.694006i \(0.244156\pi\)
\(60\) 0 0
\(61\) 9.10132i 1.16530i 0.812722 + 0.582652i \(0.197985\pi\)
−0.812722 + 0.582652i \(0.802015\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.65489i 0.577368i
\(66\) 0 0
\(67\) −8.48528 −1.03664 −0.518321 0.855186i \(-0.673443\pi\)
−0.518321 + 0.855186i \(0.673443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.354249i 0.0420416i 0.999779 + 0.0210208i \(0.00669162\pi\)
−0.999779 + 0.0210208i \(0.993308\pi\)
\(72\) 0 0
\(73\) 14.6315i 1.71249i −0.516571 0.856244i \(-0.672792\pi\)
0.516571 0.856244i \(-0.327208\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.66259 4.24264i −0.987194 0.483494i
\(78\) 0 0
\(79\) −8.48528 −0.954669 −0.477334 0.878722i \(-0.658397\pi\)
−0.477334 + 0.878722i \(0.658397\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.10132 0.998999 0.499500 0.866314i \(-0.333517\pi\)
0.499500 + 0.866314i \(0.333517\pi\)
\(84\) 0 0
\(85\) −0.708497 −0.0768473
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.97915 0.739788 0.369894 0.929074i \(-0.379394\pi\)
0.369894 + 0.929074i \(0.379394\pi\)
\(90\) 0 0
\(91\) 13.1402 + 6.43560i 1.37746 + 0.674635i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.58301i 0.265011i
\(96\) 0 0
\(97\) 3.57113i 0.362593i −0.983428 0.181297i \(-0.941971\pi\)
0.983428 0.181297i \(-0.0580294\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −14.8000 −1.47266 −0.736328 0.676624i \(-0.763442\pi\)
−0.736328 + 0.676624i \(0.763442\pi\)
\(102\) 0 0
\(103\) 5.05034i 0.497625i 0.968552 + 0.248812i \(0.0800402\pi\)
−0.968552 + 0.248812i \(0.919960\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.22876i 0.602157i −0.953599 0.301078i \(-0.902653\pi\)
0.953599 0.301078i \(-0.0973466\pi\)
\(108\) 0 0
\(109\) 10.5830 1.01367 0.506834 0.862044i \(-0.330816\pi\)
0.506834 + 0.862044i \(0.330816\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.412247i 0.0387809i −0.999812 0.0193905i \(-0.993827\pi\)
0.999812 0.0193905i \(-0.00617256\pi\)
\(114\) 0 0
\(115\) 3.06871i 0.286159i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.979531 2.00000i 0.0897935 0.183340i
\(120\) 0 0
\(121\) −2.29150 −0.208318
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 7.82087 0.699520
\(126\) 0 0
\(127\) 17.7951 1.57906 0.789528 0.613715i \(-0.210325\pi\)
0.789528 + 0.613715i \(0.210325\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.95906 0.171164 0.0855821 0.996331i \(-0.472725\pi\)
0.0855821 + 0.996331i \(0.472725\pi\)
\(132\) 0 0
\(133\) 7.29150 + 3.57113i 0.632253 + 0.309656i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.89753i 0.760167i −0.924952 0.380084i \(-0.875895\pi\)
0.924952 0.380084i \(-0.124105\pi\)
\(138\) 0 0
\(139\) 12.8712i 1.09172i 0.837876 + 0.545861i \(0.183797\pi\)
−0.837876 + 0.545861i \(0.816203\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 20.1617 1.68601
\(144\) 0 0
\(145\) 7.48925i 0.621949i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.89753i 0.728914i 0.931220 + 0.364457i \(0.118745\pi\)
−0.931220 + 0.364457i \(0.881255\pi\)
\(150\) 0 0
\(151\) 19.2980 1.57045 0.785225 0.619211i \(-0.212547\pi\)
0.785225 + 0.619211i \(0.212547\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.58301i 0.528760i
\(156\) 0 0
\(157\) 1.95906i 0.156350i 0.996940 + 0.0781751i \(0.0249093\pi\)
−0.996940 + 0.0781751i \(0.975091\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −8.66259 4.24264i −0.682708 0.334367i
\(162\) 0 0
\(163\) 8.48528 0.664619 0.332309 0.943170i \(-0.392172\pi\)
0.332309 + 0.943170i \(0.392172\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.0194 1.00747 0.503737 0.863857i \(-0.331958\pi\)
0.503737 + 0.863857i \(0.331958\pi\)
\(168\) 0 0
\(169\) −17.5830 −1.35254
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −12.0295 −0.914585 −0.457292 0.889316i \(-0.651181\pi\)
−0.457292 + 0.889316i \(0.651181\pi\)
\(174\) 0 0
\(175\) −4.99412 + 10.1969i −0.377520 + 0.770817i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.9373i 1.41544i −0.706495 0.707718i \(-0.749725\pi\)
0.706495 0.707718i \(-0.250275\pi\)
\(180\) 0 0
\(181\) 5.53019i 0.411056i 0.978651 + 0.205528i \(0.0658911\pi\)
−0.978651 + 0.205528i \(0.934109\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.77053 −0.203694
\(186\) 0 0
\(187\) 3.06871i 0.224406i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 25.5203i 1.84658i −0.384102 0.923291i \(-0.625489\pi\)
0.384102 0.923291i \(-0.374511\pi\)
\(192\) 0 0
\(193\) 15.2915 1.10071 0.550353 0.834932i \(-0.314493\pi\)
0.550353 + 0.834932i \(0.314493\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.06713i 0.361018i −0.983573 0.180509i \(-0.942225\pi\)
0.983573 0.180509i \(-0.0577746\pi\)
\(198\) 0 0
\(199\) 7.52269i 0.533269i −0.963798 0.266635i \(-0.914088\pi\)
0.963798 0.266635i \(-0.0859117\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 21.1412 + 10.3542i 1.48382 + 0.726726i
\(204\) 0 0
\(205\) −7.29150 −0.509261
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 11.1878 0.773874
\(210\) 0 0
\(211\) 6.98233 0.480684 0.240342 0.970688i \(-0.422740\pi\)
0.240342 + 0.970688i \(0.422740\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.95906 −0.133607
\(216\) 0 0
\(217\) 18.5830 + 9.10132i 1.26150 + 0.617838i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.65489i 0.313122i
\(222\) 0 0
\(223\) 4.75216i 0.318228i 0.987260 + 0.159114i \(0.0508637\pi\)
−0.987260 + 0.159114i \(0.949136\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.0604 −0.734103 −0.367052 0.930201i \(-0.619633\pi\)
−0.367052 + 0.930201i \(0.619633\pi\)
\(228\) 0 0
\(229\) 5.53019i 0.365445i 0.983165 + 0.182723i \(0.0584910\pi\)
−0.983165 + 0.182723i \(0.941509\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.5524i 0.887848i 0.896064 + 0.443924i \(0.146414\pi\)
−0.896064 + 0.443924i \(0.853586\pi\)
\(234\) 0 0
\(235\) −7.66079 −0.499735
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.06275i 0.327482i 0.986503 + 0.163741i \(0.0523561\pi\)
−0.986503 + 0.163741i \(0.947644\pi\)
\(240\) 0 0
\(241\) 7.48925i 0.482425i 0.970472 + 0.241213i \(0.0775451\pi\)
−0.970472 + 0.241213i \(0.922455\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.61226 + 4.65489i 0.230779 + 0.297390i
\(246\) 0 0
\(247\) −16.9706 −1.07981
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.1617 1.27260 0.636298 0.771444i \(-0.280465\pi\)
0.636298 + 0.771444i \(0.280465\pi\)
\(252\) 0 0
\(253\) −13.2915 −0.835630
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −12.0295 −0.750379 −0.375189 0.926948i \(-0.622422\pi\)
−0.375189 + 0.926948i \(0.622422\pi\)
\(258\) 0 0
\(259\) 3.83039 7.82087i 0.238009 0.485965i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 30.9373i 1.90767i 0.300327 + 0.953836i \(0.402904\pi\)
−0.300327 + 0.953836i \(0.597096\pi\)
\(264\) 0 0
\(265\) 3.57113i 0.219373i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 29.3547 1.78979 0.894893 0.446281i \(-0.147252\pi\)
0.894893 + 0.446281i \(0.147252\pi\)
\(270\) 0 0
\(271\) 23.4626i 1.42525i 0.701544 + 0.712626i \(0.252494\pi\)
−0.701544 + 0.712626i \(0.747506\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 15.6458i 0.943474i
\(276\) 0 0
\(277\) −3.87451 −0.232797 −0.116398 0.993203i \(-0.537135\pi\)
−0.116398 + 0.993203i \(0.537135\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 25.8681i 1.54316i 0.636132 + 0.771580i \(0.280533\pi\)
−0.636132 + 0.771580i \(0.719467\pi\)
\(282\) 0 0
\(283\) 25.4442i 1.51250i 0.654281 + 0.756251i \(0.272971\pi\)
−0.654281 + 0.756251i \(0.727029\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.0808 20.5830i 0.595054 1.21498i
\(288\) 0 0
\(289\) −16.2915 −0.958324
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6.97915 −0.407726 −0.203863 0.978999i \(-0.565350\pi\)
−0.203863 + 0.978999i \(0.565350\pi\)
\(294\) 0 0
\(295\) −9.30978 −0.542036
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 20.1617 1.16598
\(300\) 0 0
\(301\) 2.70850 5.53019i 0.156115 0.318755i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7.66079i 0.438655i
\(306\) 0 0
\(307\) 3.06871i 0.175141i 0.996158 + 0.0875703i \(0.0279103\pi\)
−0.996158 + 0.0875703i \(0.972090\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −11.0604 −0.627177 −0.313588 0.949559i \(-0.601531\pi\)
−0.313588 + 0.949559i \(0.601531\pi\)
\(312\) 0 0
\(313\) 22.1208i 1.25034i 0.780489 + 0.625170i \(0.214970\pi\)
−0.780489 + 0.625170i \(0.785030\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.89753i 0.499735i −0.968280 0.249867i \(-0.919613\pi\)
0.968280 0.249867i \(-0.0803871\pi\)
\(318\) 0 0
\(319\) 32.4382 1.81619
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.58301i 0.143722i
\(324\) 0 0
\(325\) 23.7328i 1.31646i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 10.5914 21.6255i 0.583923 1.19225i
\(330\) 0 0
\(331\) 19.2980 1.06071 0.530357 0.847774i \(-0.322058\pi\)
0.530357 + 0.847774i \(0.322058\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.14226 0.390223
\(336\) 0 0
\(337\) −9.41699 −0.512976 −0.256488 0.966547i \(-0.582565\pi\)
−0.256488 + 0.966547i \(0.582565\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 28.5129 1.54406
\(342\) 0 0
\(343\) −18.1343 + 3.76135i −0.979159 + 0.203094i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.9373i 1.01661i −0.861179 0.508303i \(-0.830273\pi\)
0.861179 0.508303i \(-0.169727\pi\)
\(348\) 0 0
\(349\) 20.1617i 1.07923i −0.841912 0.539615i \(-0.818570\pi\)
0.841912 0.539615i \(-0.181430\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 22.6209 1.20399 0.601994 0.798500i \(-0.294373\pi\)
0.601994 + 0.798500i \(0.294373\pi\)
\(354\) 0 0
\(355\) 0.298179i 0.0158257i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.77124i 0.0934827i −0.998907 0.0467413i \(-0.985116\pi\)
0.998907 0.0467413i \(-0.0148836\pi\)
\(360\) 0 0
\(361\) 9.58301 0.504369
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.3157i 0.644632i
\(366\) 0 0
\(367\) 4.75216i 0.248061i 0.992278 + 0.124030i \(0.0395820\pi\)
−0.992278 + 0.124030i \(0.960418\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −10.0808 4.93725i −0.523372 0.256329i
\(372\) 0 0
\(373\) −8.58301 −0.444411 −0.222206 0.975000i \(-0.571326\pi\)
−0.222206 + 0.975000i \(0.571326\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −49.2050 −2.53419
\(378\) 0 0
\(379\) −19.2980 −0.991272 −0.495636 0.868530i \(-0.665065\pi\)
−0.495636 + 0.868530i \(0.665065\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 24.0798 1.23042 0.615211 0.788363i \(-0.289071\pi\)
0.615211 + 0.788363i \(0.289071\pi\)
\(384\) 0 0
\(385\) 7.29150 + 3.57113i 0.371609 + 0.182002i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 21.2132i 1.07555i −0.843088 0.537776i \(-0.819265\pi\)
0.843088 0.537776i \(-0.180735\pi\)
\(390\) 0 0
\(391\) 3.06871i 0.155191i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.14226 0.359366
\(396\) 0 0
\(397\) 35.1402i 1.76364i −0.471590 0.881818i \(-0.656320\pi\)
0.471590 0.881818i \(-0.343680\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 17.3828i 0.868056i 0.900899 + 0.434028i \(0.142908\pi\)
−0.900899 + 0.434028i \(0.857092\pi\)
\(402\) 0 0
\(403\) −43.2509 −2.15448
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.0000i 0.594818i
\(408\) 0 0
\(409\) 7.48925i 0.370320i 0.982708 + 0.185160i \(0.0592803\pi\)
−0.982708 + 0.185160i \(0.940720\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 12.8712 26.2803i 0.633351 1.29317i
\(414\) 0 0
\(415\) −7.66079 −0.376053
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −40.3234 −1.96993 −0.984963 0.172763i \(-0.944731\pi\)
−0.984963 + 0.172763i \(0.944731\pi\)
\(420\) 0 0
\(421\) −9.29150 −0.452840 −0.226420 0.974030i \(-0.572702\pi\)
−0.226420 + 0.974030i \(0.572702\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.61226 −0.175220
\(426\) 0 0
\(427\) 21.6255 + 10.5914i 1.04653 + 0.512554i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.52026i 0.0732283i −0.999329 0.0366142i \(-0.988343\pi\)
0.999329 0.0366142i \(-0.0116573\pi\)
\(432\) 0 0
\(433\) 11.0604i 0.531528i 0.964038 + 0.265764i \(0.0856242\pi\)
−0.964038 + 0.265764i \(0.914376\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 11.1878 0.535183
\(438\) 0 0
\(439\) 30.4946i 1.45543i −0.685881 0.727713i \(-0.740583\pi\)
0.685881 0.727713i \(-0.259417\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9.06275i 0.430584i 0.976550 + 0.215292i \(0.0690703\pi\)
−0.976550 + 0.215292i \(0.930930\pi\)
\(444\) 0 0
\(445\) −5.87451 −0.278478
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 35.1779i 1.66015i 0.557655 + 0.830073i \(0.311701\pi\)
−0.557655 + 0.830073i \(0.688299\pi\)
\(450\) 0 0
\(451\) 31.5817i 1.48712i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −11.0604 5.41699i −0.518519 0.253953i
\(456\) 0 0
\(457\) 27.7490 1.29804 0.649022 0.760770i \(-0.275178\pi\)
0.649022 + 0.760770i \(0.275178\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −11.4331 −0.532494 −0.266247 0.963905i \(-0.585784\pi\)
−0.266247 + 0.963905i \(0.585784\pi\)
\(462\) 0 0
\(463\) 17.7951 0.827006 0.413503 0.910503i \(-0.364305\pi\)
0.413503 + 0.910503i \(0.364305\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.95906 −0.0906546 −0.0453273 0.998972i \(-0.514433\pi\)
−0.0453273 + 0.998972i \(0.514433\pi\)
\(468\) 0 0
\(469\) −9.87451 + 20.1617i −0.455962 + 0.930981i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.48528i 0.390154i
\(474\) 0 0
\(475\) 13.1694i 0.604253i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 18.2026i 0.829969i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.00590i 0.136491i
\(486\) 0 0
\(487\) 31.6137 1.43255 0.716276 0.697817i \(-0.245845\pi\)
0.716276 + 0.697817i \(0.245845\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 36.8118i 1.66129i −0.556801 0.830646i \(-0.687971\pi\)
0.556801 0.830646i \(-0.312029\pi\)
\(492\) 0 0
\(493\) 7.48925i 0.337299i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.841723 + 0.412247i 0.0377564 + 0.0184918i
\(498\) 0 0
\(499\) 40.9235 1.83199 0.915993 0.401195i \(-0.131405\pi\)
0.915993 + 0.401195i \(0.131405\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −35.1402 −1.56682 −0.783412 0.621503i \(-0.786523\pi\)
−0.783412 + 0.621503i \(0.786523\pi\)
\(504\) 0 0
\(505\) 12.4575 0.554352
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −19.8504 −0.879852 −0.439926 0.898034i \(-0.644995\pi\)
−0.439926 + 0.898034i \(0.644995\pi\)
\(510\) 0 0
\(511\) −34.7656 17.0270i −1.53794 0.753230i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.25098i 0.187321i
\(516\) 0 0
\(517\) 33.1811i 1.45930i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −32.1252 −1.40743 −0.703715 0.710482i \(-0.748477\pi\)
−0.703715 + 0.710482i \(0.748477\pi\)
\(522\) 0 0
\(523\) 41.3842i 1.80960i 0.425834 + 0.904801i \(0.359981\pi\)
−0.425834 + 0.904801i \(0.640019\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.58301i 0.286760i
\(528\) 0 0
\(529\) 9.70850 0.422109
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 47.9058i 2.07503i
\(534\) 0 0
\(535\) 5.24289i 0.226670i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −20.1617 + 15.6458i −0.868426 + 0.673910i
\(540\) 0 0
\(541\) 9.29150 0.399473 0.199736 0.979850i \(-0.435991\pi\)
0.199736 + 0.979850i \(0.435991\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8.90796 −0.381575
\(546\) 0 0
\(547\) −27.1048 −1.15892 −0.579459 0.815001i \(-0.696736\pi\)
−0.579459 + 0.815001i \(0.696736\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −27.3040 −1.16319
\(552\) 0 0
\(553\) −9.87451 + 20.1617i −0.419907 + 0.857363i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 36.0024i 1.52547i −0.646712 0.762734i \(-0.723856\pi\)
0.646712 0.762734i \(-0.276144\pi\)
\(558\) 0 0
\(559\) 12.8712i 0.544394i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 13.0194 0.548704 0.274352 0.961629i \(-0.411537\pi\)
0.274352 + 0.961629i \(0.411537\pi\)
\(564\) 0 0
\(565\) 0.346998i 0.0145983i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 31.3475i 1.31415i −0.753823 0.657077i \(-0.771793\pi\)
0.753823 0.657077i \(-0.228207\pi\)
\(570\) 0 0
\(571\) −16.1461 −0.675692 −0.337846 0.941201i \(-0.609698\pi\)
−0.337846 + 0.941201i \(0.609698\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 15.6458i 0.652473i
\(576\) 0 0
\(577\) 29.2630i 1.21824i 0.793080 + 0.609118i \(0.208476\pi\)
−0.793080 + 0.609118i \(0.791524\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 10.5914 21.6255i 0.439405 0.897175i
\(582\) 0 0
\(583\) −15.4676 −0.640603
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.0194 0.537370 0.268685 0.963228i \(-0.413411\pi\)
0.268685 + 0.963228i \(0.413411\pi\)
\(588\) 0 0
\(589\) −24.0000 −0.988903
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −40.5424 −1.66488 −0.832439 0.554117i \(-0.813056\pi\)
−0.832439 + 0.554117i \(0.813056\pi\)
\(594\) 0 0
\(595\) −0.824494 + 1.68345i −0.0338010 + 0.0690145i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 28.8118i 1.17722i 0.808418 + 0.588608i \(0.200324\pi\)
−0.808418 + 0.588608i \(0.799676\pi\)
\(600\) 0 0
\(601\) 14.9785i 0.610986i −0.952194 0.305493i \(-0.901179\pi\)
0.952194 0.305493i \(-0.0988213\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.92881 0.0784173
\(606\) 0 0
\(607\) 1.98162i 0.0804317i 0.999191 + 0.0402158i \(0.0128046\pi\)
−0.999191 + 0.0402158i \(0.987195\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 50.3320i 2.03622i
\(612\) 0 0
\(613\) 32.5830 1.31602 0.658008 0.753011i \(-0.271399\pi\)
0.658008 + 0.753011i \(0.271399\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 16.5583i 0.666613i 0.942819 + 0.333306i \(0.108164\pi\)
−0.942819 + 0.333306i \(0.891836\pi\)
\(618\) 0 0
\(619\) 28.5129i 1.14603i −0.819544 0.573016i \(-0.805773\pi\)
0.819544 0.573016i \(-0.194227\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.12179 16.5830i 0.325393 0.664384i
\(624\) 0 0
\(625\) 14.8745 0.594980
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.77053 0.110468
\(630\) 0 0
\(631\) −27.1048 −1.07903 −0.539513 0.841977i \(-0.681392\pi\)
−0.539513 + 0.841977i \(0.681392\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −14.9785 −0.594404
\(636\) 0 0
\(637\) 30.5830 23.7328i 1.21174 0.940329i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 39.0083i 1.54073i −0.637601 0.770367i \(-0.720073\pi\)
0.637601 0.770367i \(-0.279927\pi\)
\(642\) 0 0
\(643\) 34.3522i 1.35472i −0.735653 0.677359i \(-0.763124\pi\)
0.735653 0.677359i \(-0.236876\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −29.2630 −1.15045 −0.575224 0.817996i \(-0.695085\pi\)
−0.575224 + 0.817996i \(0.695085\pi\)
\(648\) 0 0
\(649\) 40.3234i 1.58283i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 13.5524i 0.530347i 0.964201 + 0.265174i \(0.0854292\pi\)
−0.964201 + 0.265174i \(0.914571\pi\)
\(654\) 0 0
\(655\) −1.64899 −0.0644313
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 28.8118i 1.12235i 0.827698 + 0.561173i \(0.189650\pi\)
−0.827698 + 0.561173i \(0.810350\pi\)
\(660\) 0 0
\(661\) 1.95906i 0.0761987i −0.999274 0.0380994i \(-0.987870\pi\)
0.999274 0.0380994i \(-0.0121303\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.13742 3.00590i −0.237999 0.116564i
\(666\) 0 0
\(667\) 32.4382 1.25601
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 33.1811 1.28094
\(672\) 0 0
\(673\) −15.2915 −0.589444 −0.294722 0.955583i \(-0.595227\pi\)
−0.294722 + 0.955583i \(0.595227\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.61226 −0.138830 −0.0694151 0.997588i \(-0.522113\pi\)
−0.0694151 + 0.997588i \(0.522113\pi\)
\(678\) 0 0
\(679\) −8.48528 4.15580i −0.325635 0.159485i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 30.2288i 1.15667i −0.815799 0.578336i \(-0.803702\pi\)
0.815799 0.578336i \(-0.196298\pi\)
\(684\) 0 0
\(685\) 7.48925i 0.286150i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 23.4626 0.893854
\(690\) 0 0
\(691\) 34.0540i 1.29548i −0.761863 0.647738i \(-0.775715\pi\)
0.761863 0.647738i \(-0.224285\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.8340i 0.410957i
\(696\) 0 0
\(697\) 7.29150 0.276185
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 33.5289i 1.26637i 0.774001 + 0.633184i \(0.218252\pi\)
−0.774001 + 0.633184i \(0.781748\pi\)
\(702\) 0 0
\(703\) 10.1007i 0.380954i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −17.2231 + 35.1660i −0.647742 + 1.32255i
\(708\) 0 0
\(709\) −28.0000 −1.05156 −0.525781 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 28.5129 1.06782
\(714\) 0 0
\(715\) −16.9706 −0.634663
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −18.2026 −0.678844 −0.339422 0.940634i \(-0.610231\pi\)
−0.339422 + 0.940634i \(0.610231\pi\)
\(720\) 0 0
\(721\) 12.0000 + 5.87719i 0.446903 + 0.218878i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 38.1838i 1.41811i
\(726\) 0 0
\(727\) 36.3338i 1.34755i −0.738938 0.673773i \(-0.764672\pi\)
0.738938 0.673773i \(-0.235328\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.95906 0.0724586
\(732\) 0 0
\(733\) 23.7328i 0.876592i 0.898831 + 0.438296i \(0.144418\pi\)
−0.898831 + 0.438296i \(0.855582\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 30.9352i 1.13951i
\(738\) 0 0
\(739\) −44.0754 −1.62134 −0.810670 0.585504i \(-0.800897\pi\)
−0.810670 + 0.585504i \(0.800897\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18.2288i 0.668748i 0.942440 + 0.334374i \(0.108525\pi\)
−0.942440 + 0.334374i \(0.891475\pi\)
\(744\) 0 0
\(745\) 7.48925i 0.274385i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −14.8000 7.24854i −0.540781 0.264856i
\(750\) 0 0
\(751\) 16.2921 0.594507 0.297254 0.954799i \(-0.403929\pi\)
0.297254 + 0.954799i \(0.403929\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −16.2436 −0.591164
\(756\) 0 0
\(757\) −41.1660 −1.49620 −0.748102 0.663584i \(-0.769035\pi\)
−0.748102 + 0.663584i \(0.769035\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 52.8173 1.91462 0.957312 0.289055i \(-0.0933411\pi\)
0.957312 + 0.289055i \(0.0933411\pi\)
\(762\) 0 0
\(763\) 12.3157 25.1461i 0.445857 0.910348i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 61.1660i 2.20858i
\(768\) 0 0
\(769\) 25.3449i 0.913960i −0.889477 0.456980i \(-0.848931\pi\)
0.889477 0.456980i \(-0.151069\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.20861 0.151373 0.0756867 0.997132i \(-0.475885\pi\)
0.0756867 + 0.997132i \(0.475885\pi\)
\(774\) 0 0
\(775\) 33.5633i 1.20563i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 26.5830i 0.952435i
\(780\) 0 0
\(781\) 1.29150 0.0462136
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.64899i 0.0588549i
\(786\) 0 0
\(787\) 25.7424i 0.917618i 0.888535 + 0.458809i \(0.151724\pi\)
−0.888535 + 0.458809i \(0.848276\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −0.979531 0.479741i −0.0348281 0.0170576i
\(792\) 0 0
\(793\) −50.3320 −1.78734
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0295 0.426106 0.213053 0.977041i \(-0.431659\pi\)
0.213053 + 0.977041i \(0.431659\pi\)
\(798\) 0 0
\(799\) 7.66079 0.271019
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −53.3428 −1.88243
\(804\) 0 0
\(805\) 7.29150 + 3.57113i 0.256992 + 0.125866i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 31.3475i 1.10212i 0.834466 + 0.551059i \(0.185776\pi\)
−0.834466 + 0.551059i \(0.814224\pi\)
\(810\) 0 0
\(811\) 31.2835i 1.09851i 0.835654 + 0.549256i \(0.185089\pi\)
−0.835654 + 0.549256i \(0.814911\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −7.14226 −0.250182
\(816\) 0 0
\(817\) 7.14226i 0.249876i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.5524i 0.472983i 0.971634 + 0.236491i \(0.0759975\pi\)
−0.971634 + 0.236491i \(0.924003\pi\)
\(822\) 0 0
\(823\) 25.4558 0.887335 0.443667 0.896191i \(-0.353677\pi\)
0.443667 + 0.896191i \(0.353677\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.3542i 0.429599i −0.976658 0.214800i \(-0.931090\pi\)
0.976658 0.214800i \(-0.0689099\pi\)
\(828\) 0 0
\(829\) 30.8751i 1.07234i 0.844111 + 0.536168i \(0.180129\pi\)
−0.844111 + 0.536168i \(0.819871\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.61226 4.65489i −0.125157 0.161282i
\(834\) 0 0
\(835\) −10.9588 −0.379244
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −38.3643 −1.32448 −0.662242 0.749290i \(-0.730395\pi\)
−0.662242 + 0.749290i \(0.730395\pi\)
\(840\) 0 0
\(841\) −50.1660 −1.72986
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 14.8000 0.509136
\(846\) 0 0
\(847\) −2.66667 + 5.44479i −0.0916279 + 0.187085i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 12.0000i 0.411355i
\(852\) 0 0
\(853\) 27.3040i 0.934870i 0.884028 + 0.467435i \(0.154822\pi\)
−0.884028 + 0.467435i \(0.845178\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11.4331 0.390548 0.195274 0.980749i \(-0.437440\pi\)
0.195274 + 0.980749i \(0.437440\pi\)
\(858\) 0 0
\(859\) 25.4442i 0.868146i −0.900878 0.434073i \(-0.857076\pi\)
0.900878 0.434073i \(-0.142924\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.64575i 0.124103i −0.998073 0.0620514i \(-0.980236\pi\)
0.998073 0.0620514i \(-0.0197643\pi\)
\(864\) 0 0
\(865\) 10.1255 0.344277
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 30.9352i 1.04941i
\(870\) 0 0
\(871\) 46.9252i 1.59000i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.10132 18.5830i 0.307681 0.628220i
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 4.69934 0.158325 0.0791624 0.996862i \(-0.474775\pi\)
0.0791624 + 0.996862i \(0.474775\pi\)
\(882\) 0 0
\(883\) 28.6078 0.962729 0.481364 0.876521i \(-0.340141\pi\)
0.481364 + 0.876521i \(0.340141\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −53.3428 −1.79108 −0.895539 0.444984i \(-0.853209\pi\)
−0.895539 + 0.444984i \(0.853209\pi\)
\(888\) 0 0
\(889\) 20.7085 42.2825i 0.694541 1.41811i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 27.9293i 0.934619i
\(894\) 0 0
\(895\) 15.9399i 0.532813i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −69.5864 −2.32084
\(900\) 0 0
\(901\) 3.57113i 0.118972i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.65489i 0.154734i
\(906\) 0 0
\(907\) −23.9529 −0.795343 −0.397671 0.917528i \(-0.630182\pi\)
−0.397671 + 0.917528i \(0.630182\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 23.6458i 0.783419i −0.920089 0.391709i \(-0.871884\pi\)
0.920089 0.391709i \(-0.128116\pi\)
\(912\) 0 0
\(913\) 33.1811i 1.09814i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.27980 4.65489i 0.0752858 0.153718i
\(918\) 0 0
\(919\) −54.8881 −1.81059 −0.905296 0.424781i \(-0.860351\pi\)
−0.905296 + 0.424781i \(0.860351\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.95906 −0.0644833
\(924\) 0 0
\(925\) −14.1255 −0.464443
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 44.9964 1.47628 0.738142 0.674645i \(-0.235703\pi\)
0.738142 + 0.674645i \(0.235703\pi\)
\(930\) 0 0
\(931\) 16.9706 13.1694i 0.556188 0.431609i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.58301i 0.0844733i
\(936\) 0 0
\(937\) 44.2415i 1.44531i 0.691210 + 0.722654i \(0.257078\pi\)
−0.691210 + 0.722654i \(0.742922\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −50.0467 −1.63148 −0.815739 0.578421i \(-0.803669\pi\)
−0.815739 + 0.578421i \(0.803669\pi\)
\(942\) 0 0
\(943\) 31.5817i 1.02844i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 10.2288i 0.332390i −0.986093 0.166195i \(-0.946852\pi\)
0.986093 0.166195i \(-0.0531481\pi\)
\(948\) 0 0
\(949\) 80.9150 2.62661
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 25.0436i 0.811242i 0.914041 + 0.405621i \(0.132945\pi\)
−0.914041 + 0.405621i \(0.867055\pi\)
\(954\) 0 0
\(955\) 21.4810i 0.695108i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −21.1412 10.3542i −0.682686 0.334356i
\(960\) 0 0
\(961\) −30.1660 −0.973097
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −12.8712 −0.414339
\(966\) 0 0
\(967\) 8.48528 0.272868 0.136434 0.990649i \(-0.456436\pi\)
0.136434 + 0.990649i \(0.456436\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 33.1811 1.06483 0.532417 0.846482i \(-0.321284\pi\)
0.532417 + 0.846482i \(0.321284\pi\)
\(972\) 0 0
\(973\) 30.5830 + 14.9785i 0.980446 + 0.480189i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 60.6337i 1.93984i −0.243415 0.969922i \(-0.578268\pi\)
0.243415 0.969922i \(-0.421732\pi\)
\(978\) 0 0
\(979\) 25.4442i 0.813201i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 13.0194 0.415256 0.207628 0.978208i \(-0.433426\pi\)
0.207628 + 0.978208i \(0.433426\pi\)
\(984\) 0 0
\(985\) 4.26512i 0.135898i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.48528i 0.269816i
\(990\) 0 0
\(991\) 54.8881 1.74358 0.871789 0.489881i \(-0.162960\pi\)
0.871789 + 0.489881i \(0.162960\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 6.33202i 0.200739i
\(996\) 0 0
\(997\) 9.10132i 0.288242i 0.989560 + 0.144121i \(0.0460354\pi\)
−0.989560 + 0.144121i \(0.953965\pi\)
\(998\) 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 4032.2.k.h.3905.7 16
3.2 odd 2 inner 4032.2.k.h.3905.11 16
4.3 odd 2 inner 4032.2.k.h.3905.6 16
7.6 odd 2 inner 4032.2.k.h.3905.12 16
8.3 odd 2 2016.2.k.b.1889.10 yes 16
8.5 even 2 2016.2.k.b.1889.11 yes 16
12.11 even 2 inner 4032.2.k.h.3905.10 16
21.20 even 2 inner 4032.2.k.h.3905.8 16
24.5 odd 2 2016.2.k.b.1889.7 yes 16
24.11 even 2 2016.2.k.b.1889.6 yes 16
28.27 even 2 inner 4032.2.k.h.3905.9 16
56.13 odd 2 2016.2.k.b.1889.8 yes 16
56.27 even 2 2016.2.k.b.1889.5 16
84.83 odd 2 inner 4032.2.k.h.3905.5 16
168.83 odd 2 2016.2.k.b.1889.9 yes 16
168.125 even 2 2016.2.k.b.1889.12 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2016.2.k.b.1889.5 16 56.27 even 2
2016.2.k.b.1889.6 yes 16 24.11 even 2
2016.2.k.b.1889.7 yes 16 24.5 odd 2
2016.2.k.b.1889.8 yes 16 56.13 odd 2
2016.2.k.b.1889.9 yes 16 168.83 odd 2
2016.2.k.b.1889.10 yes 16 8.3 odd 2
2016.2.k.b.1889.11 yes 16 8.5 even 2
2016.2.k.b.1889.12 yes 16 168.125 even 2
4032.2.k.h.3905.5 16 84.83 odd 2 inner
4032.2.k.h.3905.6 16 4.3 odd 2 inner
4032.2.k.h.3905.7 16 1.1 even 1 trivial
4032.2.k.h.3905.8 16 21.20 even 2 inner
4032.2.k.h.3905.9 16 28.27 even 2 inner
4032.2.k.h.3905.10 16 12.11 even 2 inner
4032.2.k.h.3905.11 16 3.2 odd 2 inner
4032.2.k.h.3905.12 16 7.6 odd 2 inner