Properties

Label 252.2.f.a.125.3
Level $252$
Weight $2$
Character 252.125
Analytic conductor $2.012$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,2,Mod(125,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("252.125");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.f (of order \(2\), degree \(1\), 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: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 125.3
Root \(0.517638i\) of defining polynomial
Character \(\chi\) \(=\) 252.125
Dual form 252.2.f.a.125.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.46410 q^{5} +(-1.00000 - 2.44949i) q^{7} +4.24264i q^{11} -4.89898i q^{13} +3.46410 q^{17} +4.89898i q^{19} -4.24264i q^{23} +7.00000 q^{25} +4.24264i q^{29} +(-3.46410 - 8.48528i) q^{35} -8.00000 q^{37} -3.46410 q^{41} -2.00000 q^{43} -6.92820 q^{47} +(-5.00000 + 4.89898i) q^{49} +12.7279i q^{53} +14.6969i q^{55} -13.8564 q^{59} -9.79796i q^{61} -16.9706i q^{65} +8.00000 q^{67} -4.24264i q^{71} +4.89898i q^{73} +(10.3923 - 4.24264i) q^{77} -4.00000 q^{79} +6.92820 q^{83} +12.0000 q^{85} -10.3923 q^{89} +(-12.0000 + 4.89898i) q^{91} +16.9706i q^{95} -4.89898i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7} + 28 q^{25} - 32 q^{37} - 8 q^{43} - 20 q^{49} + 32 q^{67} - 16 q^{79} + 48 q^{85} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) −1.00000 2.44949i −0.377964 0.925820i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.24264i 1.27920i 0.768706 + 0.639602i \(0.220901\pi\)
−0.768706 + 0.639602i \(0.779099\pi\)
\(12\) 0 0
\(13\) 4.89898i 1.35873i −0.733799 0.679366i \(-0.762255\pi\)
0.733799 0.679366i \(-0.237745\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 4.89898i 1.12390i 0.827170 + 0.561951i \(0.189949\pi\)
−0.827170 + 0.561951i \(0.810051\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.24264i 0.884652i −0.896854 0.442326i \(-0.854153\pi\)
0.896854 0.442326i \(-0.145847\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.24264i 0.787839i 0.919145 + 0.393919i \(0.128881\pi\)
−0.919145 + 0.393919i \(0.871119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.46410 8.48528i −0.585540 1.43427i
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) −5.00000 + 4.89898i −0.714286 + 0.699854i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.7279i 1.74831i 0.485643 + 0.874157i \(0.338586\pi\)
−0.485643 + 0.874157i \(0.661414\pi\)
\(54\) 0 0
\(55\) 14.6969i 1.98173i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −13.8564 −1.80395 −0.901975 0.431788i \(-0.857883\pi\)
−0.901975 + 0.431788i \(0.857883\pi\)
\(60\) 0 0
\(61\) 9.79796i 1.25450i −0.778818 0.627250i \(-0.784180\pi\)
0.778818 0.627250i \(-0.215820\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 16.9706i 2.10494i
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.24264i 0.503509i −0.967791 0.251754i \(-0.918992\pi\)
0.967791 0.251754i \(-0.0810075\pi\)
\(72\) 0 0
\(73\) 4.89898i 0.573382i 0.958023 + 0.286691i \(0.0925553\pi\)
−0.958023 + 0.286691i \(0.907445\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3923 4.24264i 1.18431 0.483494i
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.92820 0.760469 0.380235 0.924890i \(-0.375843\pi\)
0.380235 + 0.924890i \(0.375843\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.3923 −1.10158 −0.550791 0.834643i \(-0.685674\pi\)
−0.550791 + 0.834643i \(0.685674\pi\)
\(90\) 0 0
\(91\) −12.0000 + 4.89898i −1.25794 + 0.513553i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 16.9706i 1.74114i
\(96\) 0 0
\(97\) 4.89898i 0.497416i −0.968579 0.248708i \(-0.919994\pi\)
0.968579 0.248708i \(-0.0800060\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.3923 1.03407 0.517036 0.855963i \(-0.327035\pi\)
0.517036 + 0.855963i \(0.327035\pi\)
\(102\) 0 0
\(103\) 9.79796i 0.965422i −0.875780 0.482711i \(-0.839652\pi\)
0.875780 0.482711i \(-0.160348\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.24264i 0.410152i 0.978746 + 0.205076i \(0.0657441\pi\)
−0.978746 + 0.205076i \(0.934256\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.7279i 1.19734i 0.800995 + 0.598671i \(0.204304\pi\)
−0.800995 + 0.598671i \(0.795696\pi\)
\(114\) 0 0
\(115\) 14.6969i 1.37050i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.46410 8.48528i −0.317554 0.777844i
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 20.7846 1.81596 0.907980 0.419014i \(-0.137624\pi\)
0.907980 + 0.419014i \(0.137624\pi\)
\(132\) 0 0
\(133\) 12.0000 4.89898i 1.04053 0.424795i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.24264i 0.362473i 0.983440 + 0.181237i \(0.0580100\pi\)
−0.983440 + 0.181237i \(0.941990\pi\)
\(138\) 0 0
\(139\) 9.79796i 0.831052i −0.909581 0.415526i \(-0.863598\pi\)
0.909581 0.415526i \(-0.136402\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 20.7846 1.73810
\(144\) 0 0
\(145\) 14.6969i 1.22051i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.7279i 1.04271i −0.853339 0.521356i \(-0.825426\pi\)
0.853339 0.521356i \(-0.174574\pi\)
\(150\) 0 0
\(151\) 22.0000 1.79033 0.895167 0.445730i \(-0.147056\pi\)
0.895167 + 0.445730i \(0.147056\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −10.3923 + 4.24264i −0.819028 + 0.334367i
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.92820 −0.536120 −0.268060 0.963402i \(-0.586383\pi\)
−0.268060 + 0.963402i \(0.586383\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.3923 0.790112 0.395056 0.918657i \(-0.370725\pi\)
0.395056 + 0.918657i \(0.370725\pi\)
\(174\) 0 0
\(175\) −7.00000 17.1464i −0.529150 1.29615i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.24264i 0.317110i −0.987350 0.158555i \(-0.949317\pi\)
0.987350 0.158555i \(-0.0506835\pi\)
\(180\) 0 0
\(181\) 4.89898i 0.364138i −0.983286 0.182069i \(-0.941721\pi\)
0.983286 0.182069i \(-0.0582795\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −27.7128 −2.03749
\(186\) 0 0
\(187\) 14.6969i 1.07475i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.2132i 1.53493i 0.641089 + 0.767467i \(0.278483\pi\)
−0.641089 + 0.767467i \(0.721517\pi\)
\(192\) 0 0
\(193\) −20.0000 −1.43963 −0.719816 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.24264i 0.302276i 0.988513 + 0.151138i \(0.0482937\pi\)
−0.988513 + 0.151138i \(0.951706\pi\)
\(198\) 0 0
\(199\) 24.4949i 1.73640i 0.496217 + 0.868199i \(0.334722\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 10.3923 4.24264i 0.729397 0.297775i
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −20.7846 −1.43770
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.92820 −0.472500
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 16.9706i 1.14156i
\(222\) 0 0
\(223\) 14.6969i 0.984180i −0.870544 0.492090i \(-0.836233\pi\)
0.870544 0.492090i \(-0.163767\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 14.6969i 0.971201i 0.874181 + 0.485601i \(0.161399\pi\)
−0.874181 + 0.485601i \(0.838601\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.2132i 1.38972i −0.719144 0.694862i \(-0.755466\pi\)
0.719144 0.694862i \(-0.244534\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 29.6985i 1.92104i −0.278219 0.960518i \(-0.589744\pi\)
0.278219 0.960518i \(-0.410256\pi\)
\(240\) 0 0
\(241\) 4.89898i 0.315571i 0.987473 + 0.157786i \(0.0504355\pi\)
−0.987473 + 0.157786i \(0.949565\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −17.3205 + 16.9706i −1.10657 + 1.08421i
\(246\) 0 0
\(247\) 24.0000 1.52708
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.92820 −0.437304 −0.218652 0.975803i \(-0.570166\pi\)
−0.218652 + 0.975803i \(0.570166\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.3923 −0.648254 −0.324127 0.946014i \(-0.605071\pi\)
−0.324127 + 0.946014i \(0.605071\pi\)
\(258\) 0 0
\(259\) 8.00000 + 19.5959i 0.497096 + 1.21763i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.24264i 0.261612i −0.991408 0.130806i \(-0.958243\pi\)
0.991408 0.130806i \(-0.0417566\pi\)
\(264\) 0 0
\(265\) 44.0908i 2.70848i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.46410 −0.211210 −0.105605 0.994408i \(-0.533678\pi\)
−0.105605 + 0.994408i \(0.533678\pi\)
\(270\) 0 0
\(271\) 19.5959i 1.19037i −0.803590 0.595184i \(-0.797079\pi\)
0.803590 0.595184i \(-0.202921\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 29.6985i 1.79089i
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.24264i 0.253095i −0.991961 0.126547i \(-0.959610\pi\)
0.991961 0.126547i \(-0.0403896\pi\)
\(282\) 0 0
\(283\) 14.6969i 0.873642i −0.899548 0.436821i \(-0.856104\pi\)
0.899548 0.436821i \(-0.143896\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.46410 + 8.48528i 0.204479 + 0.500870i
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 17.3205 1.01187 0.505937 0.862570i \(-0.331147\pi\)
0.505937 + 0.862570i \(0.331147\pi\)
\(294\) 0 0
\(295\) −48.0000 −2.79467
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −20.7846 −1.20201
\(300\) 0 0
\(301\) 2.00000 + 4.89898i 0.115278 + 0.282372i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 33.9411i 1.94346i
\(306\) 0 0
\(307\) 14.6969i 0.838799i −0.907802 0.419399i \(-0.862241\pi\)
0.907802 0.419399i \(-0.137759\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.8564 0.785725 0.392862 0.919597i \(-0.371485\pi\)
0.392862 + 0.919597i \(0.371485\pi\)
\(312\) 0 0
\(313\) 19.5959i 1.10763i −0.832641 0.553813i \(-0.813172\pi\)
0.832641 0.553813i \(-0.186828\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.7279i 0.714871i 0.933938 + 0.357436i \(0.116349\pi\)
−0.933938 + 0.357436i \(0.883651\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 16.9706i 0.944267i
\(324\) 0 0
\(325\) 34.2929i 1.90223i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.92820 + 16.9706i 0.381964 + 0.935617i
\(330\) 0 0
\(331\) 22.0000 1.20923 0.604615 0.796518i \(-0.293327\pi\)
0.604615 + 0.796518i \(0.293327\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 27.7128 1.51411
\(336\) 0 0
\(337\) 16.0000 0.871576 0.435788 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 17.0000 + 7.34847i 0.917914 + 0.396780i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 21.2132i 1.13878i −0.822066 0.569392i \(-0.807179\pi\)
0.822066 0.569392i \(-0.192821\pi\)
\(348\) 0 0
\(349\) 19.5959i 1.04895i 0.851427 + 0.524473i \(0.175738\pi\)
−0.851427 + 0.524473i \(0.824262\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.46410 0.184376 0.0921878 0.995742i \(-0.470614\pi\)
0.0921878 + 0.995742i \(0.470614\pi\)
\(354\) 0 0
\(355\) 14.6969i 0.780033i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 21.2132i 1.11959i 0.828631 + 0.559795i \(0.189120\pi\)
−0.828631 + 0.559795i \(0.810880\pi\)
\(360\) 0 0
\(361\) −5.00000 −0.263158
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 16.9706i 0.888280i
\(366\) 0 0
\(367\) 24.4949i 1.27862i 0.768948 + 0.639312i \(0.220781\pi\)
−0.768948 + 0.639312i \(0.779219\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 31.1769 12.7279i 1.61862 0.660801i
\(372\) 0 0
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.7846 1.07046
\(378\) 0 0
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6.92820 0.354015 0.177007 0.984210i \(-0.443358\pi\)
0.177007 + 0.984210i \(0.443358\pi\)
\(384\) 0 0
\(385\) 36.0000 14.6969i 1.83473 0.749025i
\(386\) 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
\(391\) 14.6969i 0.743256i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −13.8564 −0.697191
\(396\) 0 0
\(397\) 29.3939i 1.47524i −0.675218 0.737618i \(-0.735950\pi\)
0.675218 0.737618i \(-0.264050\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 21.2132i 1.05934i 0.848205 + 0.529668i \(0.177684\pi\)
−0.848205 + 0.529668i \(0.822316\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 33.9411i 1.68240i
\(408\) 0 0
\(409\) 4.89898i 0.242239i 0.992638 + 0.121119i \(0.0386484\pi\)
−0.992638 + 0.121119i \(0.961352\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 13.8564 + 33.9411i 0.681829 + 1.67013i
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −27.7128 −1.35386 −0.676930 0.736048i \(-0.736690\pi\)
−0.676930 + 0.736048i \(0.736690\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 24.2487 1.17624
\(426\) 0 0
\(427\) −24.0000 + 9.79796i −1.16144 + 0.474156i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.24264i 0.204361i 0.994766 + 0.102180i \(0.0325819\pi\)
−0.994766 + 0.102180i \(0.967418\pi\)
\(432\) 0 0
\(433\) 29.3939i 1.41258i −0.707923 0.706290i \(-0.750368\pi\)
0.707923 0.706290i \(-0.249632\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.7846 0.994263
\(438\) 0 0
\(439\) 14.6969i 0.701447i 0.936479 + 0.350723i \(0.114064\pi\)
−0.936479 + 0.350723i \(0.885936\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.7279i 0.604722i 0.953194 + 0.302361i \(0.0977748\pi\)
−0.953194 + 0.302361i \(0.902225\pi\)
\(444\) 0 0
\(445\) −36.0000 −1.70656
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4.24264i 0.200223i −0.994976 0.100111i \(-0.968080\pi\)
0.994976 0.100111i \(-0.0319199\pi\)
\(450\) 0 0
\(451\) 14.6969i 0.692052i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −41.5692 + 16.9706i −1.94880 + 0.795592i
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.46410 −0.161339 −0.0806696 0.996741i \(-0.525706\pi\)
−0.0806696 + 0.996741i \(0.525706\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.7846 0.961797 0.480899 0.876776i \(-0.340311\pi\)
0.480899 + 0.876776i \(0.340311\pi\)
\(468\) 0 0
\(469\) −8.00000 19.5959i −0.369406 0.904855i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.48528i 0.390154i
\(474\) 0 0
\(475\) 34.2929i 1.57346i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −13.8564 −0.633115 −0.316558 0.948573i \(-0.602527\pi\)
−0.316558 + 0.948573i \(0.602527\pi\)
\(480\) 0 0
\(481\) 39.1918i 1.78699i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 16.9706i 0.770594i
\(486\) 0 0
\(487\) −14.0000 −0.634401 −0.317200 0.948359i \(-0.602743\pi\)
−0.317200 + 0.948359i \(0.602743\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.7279i 0.574403i −0.957870 0.287202i \(-0.907275\pi\)
0.957870 0.287202i \(-0.0927249\pi\)
\(492\) 0 0
\(493\) 14.6969i 0.661917i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10.3923 + 4.24264i −0.466159 + 0.190308i
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −34.6410 −1.54457 −0.772283 0.635278i \(-0.780885\pi\)
−0.772283 + 0.635278i \(0.780885\pi\)
\(504\) 0 0
\(505\) 36.0000 1.60198
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −24.2487 −1.07481 −0.537403 0.843326i \(-0.680594\pi\)
−0.537403 + 0.843326i \(0.680594\pi\)
\(510\) 0 0
\(511\) 12.0000 4.89898i 0.530849 0.216718i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 33.9411i 1.49562i
\(516\) 0 0
\(517\) 29.3939i 1.29274i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.46410 −0.151765 −0.0758825 0.997117i \(-0.524177\pi\)
−0.0758825 + 0.997117i \(0.524177\pi\)
\(522\) 0 0
\(523\) 39.1918i 1.71374i 0.515533 + 0.856870i \(0.327594\pi\)
−0.515533 + 0.856870i \(0.672406\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 5.00000 0.217391
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 16.9706i 0.735077i
\(534\) 0 0
\(535\) 14.6969i 0.635404i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −20.7846 21.2132i −0.895257 0.913717i
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 13.8564 0.593543
\(546\) 0 0
\(547\) −40.0000 −1.71028 −0.855138 0.518400i \(-0.826528\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −20.7846 −0.885454
\(552\) 0 0
\(553\) 4.00000 + 9.79796i 0.170097 + 0.416652i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.24264i 0.179766i 0.995952 + 0.0898832i \(0.0286494\pi\)
−0.995952 + 0.0898832i \(0.971351\pi\)
\(558\) 0 0
\(559\) 9.79796i 0.414410i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 20.7846 0.875967 0.437983 0.898983i \(-0.355693\pi\)
0.437983 + 0.898983i \(0.355693\pi\)
\(564\) 0 0
\(565\) 44.0908i 1.85491i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.24264i 0.177861i −0.996038 0.0889304i \(-0.971655\pi\)
0.996038 0.0889304i \(-0.0283449\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 29.6985i 1.23851i
\(576\) 0 0
\(577\) 9.79796i 0.407894i −0.978982 0.203947i \(-0.934623\pi\)
0.978982 0.203947i \(-0.0653771\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.92820 16.9706i −0.287430 0.704058i
\(582\) 0 0
\(583\) −54.0000 −2.23645
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20.7846 −0.857873 −0.428936 0.903335i \(-0.641112\pi\)
−0.428936 + 0.903335i \(0.641112\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 17.3205 0.711268 0.355634 0.934625i \(-0.384265\pi\)
0.355634 + 0.934625i \(0.384265\pi\)
\(594\) 0 0
\(595\) −12.0000 29.3939i −0.491952 1.20503i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.24264i 0.173350i 0.996237 + 0.0866748i \(0.0276241\pi\)
−0.996237 + 0.0866748i \(0.972376\pi\)
\(600\) 0 0
\(601\) 29.3939i 1.19900i −0.800374 0.599501i \(-0.795366\pi\)
0.800374 0.599501i \(-0.204634\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −24.2487 −0.985850
\(606\) 0 0
\(607\) 4.89898i 0.198843i −0.995045 0.0994217i \(-0.968301\pi\)
0.995045 0.0994217i \(-0.0316993\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 33.9411i 1.37311i
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.7279i 0.512407i 0.966623 + 0.256203i \(0.0824717\pi\)
−0.966623 + 0.256203i \(0.917528\pi\)
\(618\) 0 0
\(619\) 9.79796i 0.393813i 0.980422 + 0.196907i \(0.0630896\pi\)
−0.980422 + 0.196907i \(0.936910\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.3923 + 25.4558i 0.416359 + 1.01987i
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −27.7128 −1.10498
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −13.8564 −0.549875
\(636\) 0 0
\(637\) 24.0000 + 24.4949i 0.950915 + 0.970523i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 29.6985i 1.17302i 0.809942 + 0.586510i \(0.199498\pi\)
−0.809942 + 0.586510i \(0.800502\pi\)
\(642\) 0 0
\(643\) 14.6969i 0.579591i 0.957089 + 0.289795i \(0.0935872\pi\)
−0.957089 + 0.289795i \(0.906413\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −41.5692 −1.63425 −0.817127 0.576457i \(-0.804435\pi\)
−0.817127 + 0.576457i \(0.804435\pi\)
\(648\) 0 0
\(649\) 58.7878i 2.30762i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 46.6690i 1.82630i −0.407623 0.913150i \(-0.633642\pi\)
0.407623 0.913150i \(-0.366358\pi\)
\(654\) 0 0
\(655\) 72.0000 2.81327
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 46.6690i 1.81797i 0.416831 + 0.908984i \(0.363141\pi\)
−0.416831 + 0.908984i \(0.636859\pi\)
\(660\) 0 0
\(661\) 39.1918i 1.52439i 0.647350 + 0.762193i \(0.275877\pi\)
−0.647350 + 0.762193i \(0.724123\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 41.5692 16.9706i 1.61199 0.658090i
\(666\) 0 0
\(667\) 18.0000 0.696963
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 41.5692 1.60476
\(672\) 0 0
\(673\) −20.0000 −0.770943 −0.385472 0.922720i \(-0.625961\pi\)
−0.385472 + 0.922720i \(0.625961\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 31.1769 1.19823 0.599113 0.800664i \(-0.295520\pi\)
0.599113 + 0.800664i \(0.295520\pi\)
\(678\) 0 0
\(679\) −12.0000 + 4.89898i −0.460518 + 0.188006i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.7279i 0.487020i −0.969898 0.243510i \(-0.921701\pi\)
0.969898 0.243510i \(-0.0782989\pi\)
\(684\) 0 0
\(685\) 14.6969i 0.561541i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 62.3538 2.37549
\(690\) 0 0
\(691\) 29.3939i 1.11820i −0.829102 0.559098i \(-0.811148\pi\)
0.829102 0.559098i \(-0.188852\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 33.9411i 1.28746i
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 12.7279i 0.480727i −0.970683 0.240363i \(-0.922733\pi\)
0.970683 0.240363i \(-0.0772666\pi\)
\(702\) 0 0
\(703\) 39.1918i 1.47815i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10.3923 25.4558i −0.390843 0.957366i
\(708\) 0 0
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 72.0000 2.69265
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 27.7128 1.03351 0.516757 0.856132i \(-0.327139\pi\)
0.516757 + 0.856132i \(0.327139\pi\)
\(720\) 0 0
\(721\) −24.0000 + 9.79796i −0.893807 + 0.364895i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 29.6985i 1.10297i
\(726\) 0 0
\(727\) 29.3939i 1.09016i 0.838385 + 0.545079i \(0.183500\pi\)
−0.838385 + 0.545079i \(0.816500\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −6.92820 −0.256249
\(732\) 0 0
\(733\) 44.0908i 1.62853i −0.580492 0.814266i \(-0.697140\pi\)
0.580492 0.814266i \(-0.302860\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 33.9411i 1.25024i
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.24264i 0.155647i 0.996967 + 0.0778237i \(0.0247971\pi\)
−0.996967 + 0.0778237i \(0.975203\pi\)
\(744\) 0 0
\(745\) 44.0908i 1.61536i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 10.3923 4.24264i 0.379727 0.155023i
\(750\) 0 0
\(751\) 10.0000 0.364905 0.182453 0.983215i \(-0.441596\pi\)
0.182453 + 0.983215i \(0.441596\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 76.2102 2.77357
\(756\) 0 0
\(757\) 28.0000 1.01768 0.508839 0.860862i \(-0.330075\pi\)
0.508839 + 0.860862i \(0.330075\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −45.0333 −1.63246 −0.816228 0.577729i \(-0.803939\pi\)
−0.816228 + 0.577729i \(0.803939\pi\)
\(762\) 0 0
\(763\) −4.00000 9.79796i −0.144810 0.354710i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 67.8823i 2.45109i
\(768\) 0 0
\(769\) 9.79796i 0.353323i −0.984272 0.176662i \(-0.943470\pi\)
0.984272 0.176662i \(-0.0565299\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −17.3205 −0.622975 −0.311488 0.950250i \(-0.600827\pi\)
−0.311488 + 0.950250i \(0.600827\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 16.9706i 0.608034i
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 19.5959i 0.698519i −0.937026 0.349260i \(-0.886433\pi\)
0.937026 0.349260i \(-0.113567\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 31.1769 12.7279i 1.10852 0.452553i
\(792\) 0 0
\(793\) −48.0000 −1.70453
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −10.3923 −0.368114 −0.184057 0.982916i \(-0.558923\pi\)
−0.184057 + 0.982916i \(0.558923\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −20.7846 −0.733473
\(804\) 0 0
\(805\) −36.0000 + 14.6969i −1.26883 + 0.517999i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 38.1838i 1.34247i 0.741245 + 0.671235i \(0.234236\pi\)
−0.741245 + 0.671235i \(0.765764\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 27.7128 0.970737
\(816\) 0 0
\(817\) 9.79796i 0.342787i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.7279i 0.444208i −0.975023 0.222104i \(-0.928708\pi\)
0.975023 0.222104i \(-0.0712924\pi\)
\(822\) 0 0
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.7279i 0.442593i 0.975207 + 0.221297i \(0.0710289\pi\)
−0.975207 + 0.221297i \(0.928971\pi\)
\(828\) 0 0
\(829\) 24.4949i 0.850743i 0.905019 + 0.425371i \(0.139857\pi\)
−0.905019 + 0.425371i \(0.860143\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −17.3205 + 16.9706i −0.600120 + 0.587995i
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 34.6410 1.19594 0.597970 0.801518i \(-0.295974\pi\)
0.597970 + 0.801518i \(0.295974\pi\)
\(840\) 0 0
\(841\) 11.0000 0.379310
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −38.1051 −1.31086
\(846\) 0 0
\(847\) 7.00000 + 17.1464i 0.240523 + 0.589158i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 33.9411i 1.16349i
\(852\) 0 0
\(853\) 9.79796i 0.335476i 0.985832 + 0.167738i \(0.0536462\pi\)
−0.985832 + 0.167738i \(0.946354\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.2487 0.828320 0.414160 0.910204i \(-0.364075\pi\)
0.414160 + 0.910204i \(0.364075\pi\)
\(858\) 0 0
\(859\) 4.89898i 0.167151i −0.996501 0.0835755i \(-0.973366\pi\)
0.996501 0.0835755i \(-0.0266340\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 55.1543i 1.87748i −0.344633 0.938738i \(-0.611997\pi\)
0.344633 0.938738i \(-0.388003\pi\)
\(864\) 0 0
\(865\) 36.0000 1.22404
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 16.9706i 0.575687i
\(870\) 0 0
\(871\) 39.1918i 1.32796i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −6.92820 16.9706i −0.234216 0.573710i
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −31.1769 −1.05038 −0.525188 0.850986i \(-0.676005\pi\)
−0.525188 + 0.850986i \(0.676005\pi\)
\(882\) 0 0
\(883\) 46.0000 1.54802 0.774012 0.633171i \(-0.218247\pi\)
0.774012 + 0.633171i \(0.218247\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −6.92820 −0.232626 −0.116313 0.993213i \(-0.537108\pi\)
−0.116313 + 0.993213i \(0.537108\pi\)
\(888\) 0 0
\(889\) 4.00000 + 9.79796i 0.134156 + 0.328613i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 33.9411i 1.13580i
\(894\) 0 0
\(895\) 14.6969i 0.491264i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 44.0908i 1.46888i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16.9706i 0.564121i
\(906\) 0 0
\(907\) 22.0000 0.730498 0.365249 0.930910i \(-0.380984\pi\)
0.365249 + 0.930910i \(0.380984\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 21.2132i 0.702825i −0.936221 0.351412i \(-0.885702\pi\)
0.936221 0.351412i \(-0.114298\pi\)
\(912\) 0 0
\(913\) 29.3939i 0.972795i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −20.7846 50.9117i −0.686368 1.68125i
\(918\) 0 0
\(919\) −14.0000 −0.461817 −0.230909 0.972975i \(-0.574170\pi\)
−0.230909 + 0.972975i \(0.574170\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −20.7846 −0.684134
\(924\) 0 0
\(925\) −56.0000 −1.84127
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 45.0333 1.47750 0.738748 0.673982i \(-0.235418\pi\)
0.738748 + 0.673982i \(0.235418\pi\)
\(930\) 0 0
\(931\) −24.0000 24.4949i −0.786568 0.802788i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 50.9117i 1.66499i
\(936\) 0 0
\(937\) 19.5959i 0.640171i 0.947389 + 0.320085i \(0.103712\pi\)
−0.947389 + 0.320085i \(0.896288\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −45.0333 −1.46804 −0.734022 0.679126i \(-0.762359\pi\)
−0.734022 + 0.679126i \(0.762359\pi\)
\(942\) 0 0
\(943\) 14.6969i 0.478598i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 29.6985i 0.965071i −0.875876 0.482536i \(-0.839716\pi\)
0.875876 0.482536i \(-0.160284\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 38.1838i 1.23689i 0.785827 + 0.618447i \(0.212238\pi\)
−0.785827 + 0.618447i \(0.787762\pi\)
\(954\) 0 0
\(955\) 73.4847i 2.37791i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 10.3923 4.24264i 0.335585 0.137002i
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −69.2820 −2.23027
\(966\) 0 0
\(967\) −28.0000 −0.900419 −0.450210 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 27.7128 0.889346 0.444673 0.895693i \(-0.353320\pi\)
0.444673 + 0.895693i \(0.353320\pi\)
\(972\) 0 0
\(973\) −24.0000 + 9.79796i −0.769405 + 0.314108i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.24264i 0.135734i −0.997694 0.0678671i \(-0.978381\pi\)
0.997694 0.0678671i \(-0.0216194\pi\)
\(978\) 0 0
\(979\) 44.0908i 1.40915i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −34.6410 −1.10488 −0.552438 0.833554i \(-0.686303\pi\)
−0.552438 + 0.833554i \(0.686303\pi\)
\(984\) 0 0
\(985\) 14.6969i 0.468283i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.48528i 0.269816i
\(990\) 0 0
\(991\) −26.0000 −0.825917 −0.412959 0.910750i \(-0.635505\pi\)
−0.412959 + 0.910750i \(0.635505\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 84.8528i 2.69002i
\(996\) 0 0
\(997\) 29.3939i 0.930913i 0.885071 + 0.465457i \(0.154110\pi\)
−0.885071 + 0.465457i \(0.845890\pi\)
\(998\) 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 252.2.f.a.125.3 yes 4
3.2 odd 2 inner 252.2.f.a.125.1 4
4.3 odd 2 1008.2.k.b.881.4 4
5.2 odd 4 6300.2.f.b.3149.6 8
5.3 odd 4 6300.2.f.b.3149.4 8
5.4 even 2 6300.2.d.c.3401.4 4
7.2 even 3 1764.2.t.b.521.1 8
7.3 odd 6 1764.2.t.b.1097.4 8
7.4 even 3 1764.2.t.b.1097.2 8
7.5 odd 6 1764.2.t.b.521.3 8
7.6 odd 2 inner 252.2.f.a.125.2 yes 4
8.3 odd 2 4032.2.k.d.3905.2 4
8.5 even 2 4032.2.k.a.3905.1 4
9.2 odd 6 2268.2.x.i.377.3 8
9.4 even 3 2268.2.x.i.1889.2 8
9.5 odd 6 2268.2.x.i.1889.4 8
9.7 even 3 2268.2.x.i.377.1 8
12.11 even 2 1008.2.k.b.881.2 4
15.2 even 4 6300.2.f.b.3149.5 8
15.8 even 4 6300.2.f.b.3149.3 8
15.14 odd 2 6300.2.d.c.3401.3 4
21.2 odd 6 1764.2.t.b.521.4 8
21.5 even 6 1764.2.t.b.521.2 8
21.11 odd 6 1764.2.t.b.1097.3 8
21.17 even 6 1764.2.t.b.1097.1 8
21.20 even 2 inner 252.2.f.a.125.4 yes 4
24.5 odd 2 4032.2.k.a.3905.3 4
24.11 even 2 4032.2.k.d.3905.4 4
28.27 even 2 1008.2.k.b.881.1 4
35.13 even 4 6300.2.f.b.3149.8 8
35.27 even 4 6300.2.f.b.3149.2 8
35.34 odd 2 6300.2.d.c.3401.2 4
56.13 odd 2 4032.2.k.a.3905.4 4
56.27 even 2 4032.2.k.d.3905.3 4
63.13 odd 6 2268.2.x.i.1889.3 8
63.20 even 6 2268.2.x.i.377.2 8
63.34 odd 6 2268.2.x.i.377.4 8
63.41 even 6 2268.2.x.i.1889.1 8
84.83 odd 2 1008.2.k.b.881.3 4
105.62 odd 4 6300.2.f.b.3149.1 8
105.83 odd 4 6300.2.f.b.3149.7 8
105.104 even 2 6300.2.d.c.3401.1 4
168.83 odd 2 4032.2.k.d.3905.1 4
168.125 even 2 4032.2.k.a.3905.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.f.a.125.1 4 3.2 odd 2 inner
252.2.f.a.125.2 yes 4 7.6 odd 2 inner
252.2.f.a.125.3 yes 4 1.1 even 1 trivial
252.2.f.a.125.4 yes 4 21.20 even 2 inner
1008.2.k.b.881.1 4 28.27 even 2
1008.2.k.b.881.2 4 12.11 even 2
1008.2.k.b.881.3 4 84.83 odd 2
1008.2.k.b.881.4 4 4.3 odd 2
1764.2.t.b.521.1 8 7.2 even 3
1764.2.t.b.521.2 8 21.5 even 6
1764.2.t.b.521.3 8 7.5 odd 6
1764.2.t.b.521.4 8 21.2 odd 6
1764.2.t.b.1097.1 8 21.17 even 6
1764.2.t.b.1097.2 8 7.4 even 3
1764.2.t.b.1097.3 8 21.11 odd 6
1764.2.t.b.1097.4 8 7.3 odd 6
2268.2.x.i.377.1 8 9.7 even 3
2268.2.x.i.377.2 8 63.20 even 6
2268.2.x.i.377.3 8 9.2 odd 6
2268.2.x.i.377.4 8 63.34 odd 6
2268.2.x.i.1889.1 8 63.41 even 6
2268.2.x.i.1889.2 8 9.4 even 3
2268.2.x.i.1889.3 8 63.13 odd 6
2268.2.x.i.1889.4 8 9.5 odd 6
4032.2.k.a.3905.1 4 8.5 even 2
4032.2.k.a.3905.2 4 168.125 even 2
4032.2.k.a.3905.3 4 24.5 odd 2
4032.2.k.a.3905.4 4 56.13 odd 2
4032.2.k.d.3905.1 4 168.83 odd 2
4032.2.k.d.3905.2 4 8.3 odd 2
4032.2.k.d.3905.3 4 56.27 even 2
4032.2.k.d.3905.4 4 24.11 even 2
6300.2.d.c.3401.1 4 105.104 even 2
6300.2.d.c.3401.2 4 35.34 odd 2
6300.2.d.c.3401.3 4 15.14 odd 2
6300.2.d.c.3401.4 4 5.4 even 2
6300.2.f.b.3149.1 8 105.62 odd 4
6300.2.f.b.3149.2 8 35.27 even 4
6300.2.f.b.3149.3 8 15.8 even 4
6300.2.f.b.3149.4 8 5.3 odd 4
6300.2.f.b.3149.5 8 15.2 even 4
6300.2.f.b.3149.6 8 5.2 odd 4
6300.2.f.b.3149.7 8 105.83 odd 4
6300.2.f.b.3149.8 8 35.13 even 4