Properties

Label 4032.2.k.g.3905.15
Level $4032$
Weight $2$
Character 4032.3905
Analytic conductor $32.196$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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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: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{25} \)
Twist minimal: no (minimal twist has level 2016)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3905.15
Root \(-2.13875i\) of defining polynomial
Character \(\chi\) \(=\) 4032.3905
Dual form 4032.2.k.g.3905.16

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.61313 q^{5} +(1.25928 - 2.32685i) q^{7} +O(q^{10})\) \(q+2.61313 q^{5} +(1.25928 - 2.32685i) q^{7} -4.29945i q^{11} +1.53073i q^{13} +0.448342 q^{17} +1.92762i q^{19} +0.737669i q^{23} +1.82843 q^{25} +1.41421i q^{29} -6.58132i q^{31} +(3.29066 - 6.08034i) q^{35} +2.82843 q^{37} +6.94269 q^{41} +9.64212 q^{43} -2.72607 q^{47} +(-3.82843 - 5.86030i) q^{49} +2.58579i q^{53} -11.2350i q^{55} +9.30739 q^{59} -8.28772i q^{61} +4.00000i q^{65} -1.47534 q^{67} -11.4230i q^{71} +11.0866i q^{73} +(-10.0042 - 5.41421i) q^{77} -15.7225 q^{79} +15.8887 q^{83} +1.17157 q^{85} -17.3952 q^{89} +(3.56178 + 1.92762i) q^{91} +5.03712i q^{95} -6.75699i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{25} - 16 q^{49} + 64 q^{85}+O(q^{100}) \) 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
\(5\) 2.61313 1.16863 0.584313 0.811529i \(-0.301364\pi\)
0.584313 + 0.811529i \(0.301364\pi\)
\(6\) 0 0
\(7\) 1.25928 2.32685i 0.475963 0.879465i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.29945i 1.29633i −0.761499 0.648167i \(-0.775536\pi\)
0.761499 0.648167i \(-0.224464\pi\)
\(12\) 0 0
\(13\) 1.53073i 0.424549i 0.977210 + 0.212275i \(0.0680871\pi\)
−0.977210 + 0.212275i \(0.931913\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.448342 0.108739 0.0543694 0.998521i \(-0.482685\pi\)
0.0543694 + 0.998521i \(0.482685\pi\)
\(18\) 0 0
\(19\) 1.92762i 0.442227i 0.975248 + 0.221113i \(0.0709691\pi\)
−0.975248 + 0.221113i \(0.929031\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.737669i 0.153815i 0.997038 + 0.0769073i \(0.0245046\pi\)
−0.997038 + 0.0769073i \(0.975495\pi\)
\(24\) 0 0
\(25\) 1.82843 0.365685
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.41421i 0.262613i 0.991342 + 0.131306i \(0.0419172\pi\)
−0.991342 + 0.131306i \(0.958083\pi\)
\(30\) 0 0
\(31\) 6.58132i 1.18204i −0.806657 0.591020i \(-0.798726\pi\)
0.806657 0.591020i \(-0.201274\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.29066 6.08034i 0.556223 1.02777i
\(36\) 0 0
\(37\) 2.82843 0.464991 0.232495 0.972598i \(-0.425311\pi\)
0.232495 + 0.972598i \(0.425311\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.94269 1.08427 0.542133 0.840292i \(-0.317617\pi\)
0.542133 + 0.840292i \(0.317617\pi\)
\(42\) 0 0
\(43\) 9.64212 1.47041 0.735205 0.677845i \(-0.237086\pi\)
0.735205 + 0.677845i \(0.237086\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.72607 −0.397638 −0.198819 0.980036i \(-0.563711\pi\)
−0.198819 + 0.980036i \(0.563711\pi\)
\(48\) 0 0
\(49\) −3.82843 5.86030i −0.546918 0.837186i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.58579i 0.355185i 0.984104 + 0.177593i \(0.0568309\pi\)
−0.984104 + 0.177593i \(0.943169\pi\)
\(54\) 0 0
\(55\) 11.2350i 1.51493i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.30739 1.21172 0.605859 0.795572i \(-0.292829\pi\)
0.605859 + 0.795572i \(0.292829\pi\)
\(60\) 0 0
\(61\) 8.28772i 1.06113i −0.847643 0.530567i \(-0.821979\pi\)
0.847643 0.530567i \(-0.178021\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.00000i 0.496139i
\(66\) 0 0
\(67\) −1.47534 −0.180241 −0.0901206 0.995931i \(-0.528725\pi\)
−0.0901206 + 0.995931i \(0.528725\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.4230i 1.35566i −0.735218 0.677831i \(-0.762920\pi\)
0.735218 0.677831i \(-0.237080\pi\)
\(72\) 0 0
\(73\) 11.0866i 1.29758i 0.760966 + 0.648792i \(0.224725\pi\)
−0.760966 + 0.648792i \(0.775275\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −10.0042 5.41421i −1.14008 0.617007i
\(78\) 0 0
\(79\) −15.7225 −1.76892 −0.884458 0.466620i \(-0.845472\pi\)
−0.884458 + 0.466620i \(0.845472\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 15.8887 1.74401 0.872006 0.489496i \(-0.162819\pi\)
0.872006 + 0.489496i \(0.162819\pi\)
\(84\) 0 0
\(85\) 1.17157 0.127075
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −17.3952 −1.84389 −0.921944 0.387324i \(-0.873399\pi\)
−0.921944 + 0.387324i \(0.873399\pi\)
\(90\) 0 0
\(91\) 3.56178 + 1.92762i 0.373376 + 0.202070i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.03712i 0.516798i
\(96\) 0 0
\(97\) 6.75699i 0.686068i −0.939323 0.343034i \(-0.888545\pi\)
0.939323 0.343034i \(-0.111455\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.83938 0.780047 0.390024 0.920805i \(-0.372467\pi\)
0.390024 + 0.920805i \(0.372467\pi\)
\(102\) 0 0
\(103\) 15.8887i 1.56556i −0.622298 0.782780i \(-0.713801\pi\)
0.622298 0.782780i \(-0.286199\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.737669i 0.0713132i 0.999364 + 0.0356566i \(0.0113523\pi\)
−0.999364 + 0.0356566i \(0.988648\pi\)
\(108\) 0 0
\(109\) −15.3137 −1.46679 −0.733394 0.679804i \(-0.762065\pi\)
−0.733394 + 0.679804i \(0.762065\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.07107i 0.288902i 0.989512 + 0.144451i \(0.0461416\pi\)
−0.989512 + 0.144451i \(0.953858\pi\)
\(114\) 0 0
\(115\) 1.92762i 0.179752i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.564588 1.04322i 0.0517557 0.0956320i
\(120\) 0 0
\(121\) −7.48528 −0.680480
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −8.28772 −0.741276
\(126\) 0 0
\(127\) −8.59890 −0.763029 −0.381515 0.924363i \(-0.624597\pi\)
−0.381515 + 0.924363i \(0.624597\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 19.7439 1.72504 0.862518 0.506026i \(-0.168886\pi\)
0.862518 + 0.506026i \(0.168886\pi\)
\(132\) 0 0
\(133\) 4.48528 + 2.42742i 0.388923 + 0.210484i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.0711i 1.62935i −0.579917 0.814676i \(-0.696915\pi\)
0.579917 0.814676i \(-0.303085\pi\)
\(138\) 0 0
\(139\) 3.85525i 0.326998i 0.986544 + 0.163499i \(0.0522780\pi\)
−0.986544 + 0.163499i \(0.947722\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.58132 0.550357
\(144\) 0 0
\(145\) 3.69552i 0.306896i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.58579i 0.539529i −0.962926 0.269764i \(-0.913054\pi\)
0.962926 0.269764i \(-0.0869458\pi\)
\(150\) 0 0
\(151\) −14.6792 −1.19458 −0.597290 0.802025i \(-0.703756\pi\)
−0.597290 + 0.802025i \(0.703756\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 17.1978i 1.38136i
\(156\) 0 0
\(157\) 20.0083i 1.59684i 0.602102 + 0.798419i \(0.294330\pi\)
−0.602102 + 0.798419i \(0.705670\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.71644 + 0.928932i 0.135275 + 0.0732101i
\(162\) 0 0
\(163\) −18.6731 −1.46259 −0.731297 0.682059i \(-0.761085\pi\)
−0.731297 + 0.682059i \(0.761085\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.4366 0.807605 0.403803 0.914846i \(-0.367688\pi\)
0.403803 + 0.914846i \(0.367688\pi\)
\(168\) 0 0
\(169\) 10.6569 0.819758
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.10748 0.692429 0.346214 0.938155i \(-0.387467\pi\)
0.346214 + 0.938155i \(0.387467\pi\)
\(174\) 0 0
\(175\) 2.30250 4.25447i 0.174053 0.321608i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.82411i 0.211084i −0.994415 0.105542i \(-0.966342\pi\)
0.994415 0.105542i \(-0.0336578\pi\)
\(180\) 0 0
\(181\) 10.1899i 0.757407i 0.925518 + 0.378704i \(0.123630\pi\)
−0.925518 + 0.378704i \(0.876370\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.39104 0.543400
\(186\) 0 0
\(187\) 1.92762i 0.140962i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.33657i 0.675571i 0.941223 + 0.337785i \(0.109678\pi\)
−0.941223 + 0.337785i \(0.890322\pi\)
\(192\) 0 0
\(193\) 15.7990 1.13724 0.568618 0.822602i \(-0.307478\pi\)
0.568618 + 0.822602i \(0.307478\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.7279i 1.19182i 0.803053 + 0.595908i \(0.203208\pi\)
−0.803053 + 0.595908i \(0.796792\pi\)
\(198\) 0 0
\(199\) 13.9611i 0.989675i −0.868986 0.494837i \(-0.835228\pi\)
0.868986 0.494837i \(-0.164772\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.29066 + 1.78089i 0.230959 + 0.124994i
\(204\) 0 0
\(205\) 18.1421 1.26710
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.28772 0.573274
\(210\) 0 0
\(211\) −19.7164 −1.35733 −0.678665 0.734448i \(-0.737441\pi\)
−0.678665 + 0.734448i \(0.737441\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 25.1961 1.71836
\(216\) 0 0
\(217\) −15.3137 8.28772i −1.03956 0.562607i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.686292i 0.0461650i
\(222\) 0 0
\(223\) 4.65369i 0.311634i 0.987786 + 0.155817i \(0.0498011\pi\)
−0.987786 + 0.155817i \(0.950199\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.0179 1.12952 0.564758 0.825257i \(-0.308969\pi\)
0.564758 + 0.825257i \(0.308969\pi\)
\(228\) 0 0
\(229\) 28.0334i 1.85250i −0.376910 0.926250i \(-0.623013\pi\)
0.376910 0.926250i \(-0.376987\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.0711i 0.987338i 0.869650 + 0.493669i \(0.164345\pi\)
−0.869650 + 0.493669i \(0.835655\pi\)
\(234\) 0 0
\(235\) −7.12356 −0.464690
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.4601i 1.06472i 0.846519 + 0.532359i \(0.178694\pi\)
−0.846519 + 0.532359i \(0.821306\pi\)
\(240\) 0 0
\(241\) 17.2095i 1.10856i 0.832330 + 0.554280i \(0.187006\pi\)
−0.832330 + 0.554280i \(0.812994\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −10.0042 15.3137i −0.639142 0.978357i
\(246\) 0 0
\(247\) −2.95068 −0.187747
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0335 −0.759545 −0.379772 0.925080i \(-0.623998\pi\)
−0.379772 + 0.925080i \(0.623998\pi\)
\(252\) 0 0
\(253\) 3.17157 0.199395
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.61313 −0.163002 −0.0815012 0.996673i \(-0.525971\pi\)
−0.0815012 + 0.996673i \(0.525971\pi\)
\(258\) 0 0
\(259\) 3.56178 6.58132i 0.221318 0.408943i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.5466i 1.14363i −0.820382 0.571816i \(-0.806239\pi\)
0.820382 0.571816i \(-0.193761\pi\)
\(264\) 0 0
\(265\) 6.75699i 0.415078i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 17.0238 1.03796 0.518979 0.854787i \(-0.326312\pi\)
0.518979 + 0.854787i \(0.326312\pi\)
\(270\) 0 0
\(271\) 6.58132i 0.399786i 0.979818 + 0.199893i \(0.0640595\pi\)
−0.979818 + 0.199893i \(0.935940\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.86123i 0.474050i
\(276\) 0 0
\(277\) −27.4558 −1.64966 −0.824831 0.565380i \(-0.808730\pi\)
−0.824831 + 0.565380i \(0.808730\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.7279i 0.997904i 0.866630 + 0.498952i \(0.166282\pi\)
−0.866630 + 0.498952i \(0.833718\pi\)
\(282\) 0 0
\(283\) 11.2350i 0.667852i −0.942599 0.333926i \(-0.891626\pi\)
0.942599 0.333926i \(-0.108374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.74280 16.1546i 0.516071 0.953575i
\(288\) 0 0
\(289\) −16.7990 −0.988176
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.7975 0.689219 0.344609 0.938746i \(-0.388011\pi\)
0.344609 + 0.938746i \(0.388011\pi\)
\(294\) 0 0
\(295\) 24.3214 1.41604
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.12918 −0.0653019
\(300\) 0 0
\(301\) 12.1421 22.4357i 0.699861 1.29317i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 21.6569i 1.24007i
\(306\) 0 0
\(307\) 24.3976i 1.39245i −0.717825 0.696223i \(-0.754862\pi\)
0.717825 0.696223i \(-0.245138\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 17.0179 0.964995 0.482498 0.875897i \(-0.339730\pi\)
0.482498 + 0.875897i \(0.339730\pi\)
\(312\) 0 0
\(313\) 23.4412i 1.32498i −0.749073 0.662488i \(-0.769501\pi\)
0.749073 0.662488i \(-0.230499\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.10051i 0.342639i −0.985216 0.171319i \(-0.945197\pi\)
0.985216 0.171319i \(-0.0548030\pi\)
\(318\) 0 0
\(319\) 6.08034 0.340434
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.864233i 0.0480872i
\(324\) 0 0
\(325\) 2.79884i 0.155251i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.43289 + 6.34315i −0.189261 + 0.349709i
\(330\) 0 0
\(331\) 12.5928 0.692163 0.346081 0.938204i \(-0.387512\pi\)
0.346081 + 0.938204i \(0.387512\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.85525 −0.210635
\(336\) 0 0
\(337\) 21.6569 1.17972 0.589862 0.807504i \(-0.299182\pi\)
0.589862 + 0.807504i \(0.299182\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −28.2960 −1.53232
\(342\) 0 0
\(343\) −18.4571 + 1.52840i −0.996589 + 0.0825258i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 21.4973i 1.15403i 0.816732 + 0.577017i \(0.195783\pi\)
−0.816732 + 0.577017i \(0.804217\pi\)
\(348\) 0 0
\(349\) 3.43289i 0.183758i −0.995770 0.0918791i \(-0.970713\pi\)
0.995770 0.0918791i \(-0.0292873\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.83938 0.417248 0.208624 0.977996i \(-0.433102\pi\)
0.208624 + 0.977996i \(0.433102\pi\)
\(354\) 0 0
\(355\) 29.8498i 1.58426i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 30.0962i 1.58841i −0.607647 0.794207i \(-0.707886\pi\)
0.607647 0.794207i \(-0.292114\pi\)
\(360\) 0 0
\(361\) 15.2843 0.804435
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 28.9706i 1.51639i
\(366\) 0 0
\(367\) 30.9790i 1.61709i 0.588436 + 0.808544i \(0.299744\pi\)
−0.588436 + 0.808544i \(0.700256\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.01673 + 3.25623i 0.312373 + 0.169055i
\(372\) 0 0
\(373\) 14.9706 0.775146 0.387573 0.921839i \(-0.373313\pi\)
0.387573 + 0.921839i \(0.373313\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.16478 −0.111492
\(378\) 0 0
\(379\) 7.55568 0.388109 0.194055 0.980991i \(-0.437836\pi\)
0.194055 + 0.980991i \(0.437836\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −6.58132 −0.336289 −0.168145 0.985762i \(-0.553778\pi\)
−0.168145 + 0.985762i \(0.553778\pi\)
\(384\) 0 0
\(385\) −26.1421 14.1480i −1.33233 0.721050i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 20.9289i 1.06114i 0.847641 + 0.530569i \(0.178022\pi\)
−0.847641 + 0.530569i \(0.821978\pi\)
\(390\) 0 0
\(391\) 0.330728i 0.0167256i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −41.0848 −2.06720
\(396\) 0 0
\(397\) 24.8632i 1.24785i 0.781486 + 0.623923i \(0.214462\pi\)
−0.781486 + 0.623923i \(0.785538\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.5858i 0.528629i 0.964437 + 0.264314i \(0.0851457\pi\)
−0.964437 + 0.264314i \(0.914854\pi\)
\(402\) 0 0
\(403\) 10.0742 0.501834
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.1607i 0.602783i
\(408\) 0 0
\(409\) 15.9414i 0.788251i −0.919057 0.394125i \(-0.871048\pi\)
0.919057 0.394125i \(-0.128952\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.7206 21.6569i 0.576733 1.06566i
\(414\) 0 0
\(415\) 41.5192 2.03810
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 13.1626 0.643037 0.321518 0.946903i \(-0.395807\pi\)
0.321518 + 0.946903i \(0.395807\pi\)
\(420\) 0 0
\(421\) 9.51472 0.463719 0.231860 0.972749i \(-0.425519\pi\)
0.231860 + 0.972749i \(0.425519\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.819760 0.0397642
\(426\) 0 0
\(427\) −19.2842 10.4366i −0.933230 0.505061i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.8862i 1.00605i −0.864272 0.503025i \(-0.832220\pi\)
0.864272 0.503025i \(-0.167780\pi\)
\(432\) 0 0
\(433\) 11.7206i 0.563256i 0.959524 + 0.281628i \(0.0908745\pi\)
−0.959524 + 0.281628i \(0.909126\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.42195 −0.0680210
\(438\) 0 0
\(439\) 13.9611i 0.666326i 0.942869 + 0.333163i \(0.108116\pi\)
−0.942869 + 0.333163i \(0.891884\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.737669i 0.0350477i −0.999846 0.0175239i \(-0.994422\pi\)
0.999846 0.0175239i \(-0.00557830\pi\)
\(444\) 0 0
\(445\) −45.4558 −2.15481
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12.0416i 0.568280i 0.958783 + 0.284140i \(0.0917080\pi\)
−0.958783 + 0.284140i \(0.908292\pi\)
\(450\) 0 0
\(451\) 29.8498i 1.40557i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9.30739 + 5.03712i 0.436337 + 0.236144i
\(456\) 0 0
\(457\) −8.97056 −0.419625 −0.209813 0.977742i \(-0.567285\pi\)
−0.209813 + 0.977742i \(0.567285\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.50981 0.163468 0.0817341 0.996654i \(-0.473954\pi\)
0.0817341 + 0.996654i \(0.473954\pi\)
\(462\) 0 0
\(463\) −8.59890 −0.399625 −0.199812 0.979834i \(-0.564033\pi\)
−0.199812 + 0.979834i \(0.564033\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −30.6482 −1.41823 −0.709115 0.705092i \(-0.750905\pi\)
−0.709115 + 0.705092i \(0.750905\pi\)
\(468\) 0 0
\(469\) −1.85786 + 3.43289i −0.0857882 + 0.158516i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 41.4558i 1.90614i
\(474\) 0 0
\(475\) 3.52452i 0.161716i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −26.3253 −1.20283 −0.601416 0.798936i \(-0.705397\pi\)
−0.601416 + 0.798936i \(0.705397\pi\)
\(480\) 0 0
\(481\) 4.32957i 0.197411i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 17.6569i 0.801756i
\(486\) 0 0
\(487\) 11.7286 0.531472 0.265736 0.964046i \(-0.414385\pi\)
0.265736 + 0.964046i \(0.414385\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 22.1084i 0.997736i 0.866678 + 0.498868i \(0.166251\pi\)
−0.866678 + 0.498868i \(0.833749\pi\)
\(492\) 0 0
\(493\) 0.634051i 0.0285562i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −26.5796 14.3848i −1.19226 0.645245i
\(498\) 0 0
\(499\) 14.6792 0.657133 0.328567 0.944481i \(-0.393434\pi\)
0.328567 + 0.944481i \(0.393434\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 34.5035 1.53843 0.769217 0.638988i \(-0.220647\pi\)
0.769217 + 0.638988i \(0.220647\pi\)
\(504\) 0 0
\(505\) 20.4853 0.911583
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.04601 −0.267985 −0.133992 0.990982i \(-0.542780\pi\)
−0.133992 + 0.990982i \(0.542780\pi\)
\(510\) 0 0
\(511\) 25.7967 + 13.9611i 1.14118 + 0.617602i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 41.5192i 1.82955i
\(516\) 0 0
\(517\) 11.7206i 0.515472i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 20.4567 0.896223 0.448111 0.893978i \(-0.352097\pi\)
0.448111 + 0.893978i \(0.352097\pi\)
\(522\) 0 0
\(523\) 5.45214i 0.238405i 0.992870 + 0.119203i \(0.0380338\pi\)
−0.992870 + 0.119203i \(0.961966\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.95068i 0.128534i
\(528\) 0 0
\(529\) 22.4558 0.976341
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 10.6274i 0.460325i
\(534\) 0 0
\(535\) 1.92762i 0.0833384i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −25.1961 + 16.4601i −1.08527 + 0.708988i
\(540\) 0 0
\(541\) 33.7990 1.45313 0.726566 0.687097i \(-0.241115\pi\)
0.726566 + 0.687097i \(0.241115\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −40.0166 −1.71412
\(546\) 0 0
\(547\) −29.9696 −1.28141 −0.640704 0.767788i \(-0.721357\pi\)
−0.640704 + 0.767788i \(0.721357\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.72607 −0.116134
\(552\) 0 0
\(553\) −19.7990 + 36.5838i −0.841939 + 1.55570i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10.8701i 0.460579i −0.973122 0.230290i \(-0.926033\pi\)
0.973122 0.230290i \(-0.0739673\pi\)
\(558\) 0 0
\(559\) 14.7595i 0.624261i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.72607 0.114890 0.0574451 0.998349i \(-0.481705\pi\)
0.0574451 + 0.998349i \(0.481705\pi\)
\(564\) 0 0
\(565\) 8.02509i 0.337618i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 36.7279i 1.53971i 0.638216 + 0.769857i \(0.279673\pi\)
−0.638216 + 0.769857i \(0.720327\pi\)
\(570\) 0 0
\(571\) −8.59890 −0.359853 −0.179926 0.983680i \(-0.557586\pi\)
−0.179926 + 0.983680i \(0.557586\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.34877i 0.0562478i
\(576\) 0 0
\(577\) 28.2960i 1.17798i 0.808140 + 0.588990i \(0.200474\pi\)
−0.808140 + 0.588990i \(0.799526\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 20.0083 36.9706i 0.830085 1.53380i
\(582\) 0 0
\(583\) 11.1175 0.460438
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −23.5992 −0.974043 −0.487021 0.873390i \(-0.661917\pi\)
−0.487021 + 0.873390i \(0.661917\pi\)
\(588\) 0 0
\(589\) 12.6863 0.522730
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 34.3421 1.41026 0.705130 0.709078i \(-0.250889\pi\)
0.705130 + 0.709078i \(0.250889\pi\)
\(594\) 0 0
\(595\) 1.47534 2.72607i 0.0604830 0.111758i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 45.8186i 1.87210i 0.351869 + 0.936049i \(0.385546\pi\)
−0.351869 + 0.936049i \(0.614454\pi\)
\(600\) 0 0
\(601\) 28.2960i 1.15422i 0.816667 + 0.577110i \(0.195820\pi\)
−0.816667 + 0.577110i \(0.804180\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −19.5600 −0.795226
\(606\) 0 0
\(607\) 21.6716i 0.879622i 0.898090 + 0.439811i \(0.144955\pi\)
−0.898090 + 0.439811i \(0.855045\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.17289i 0.168817i
\(612\) 0 0
\(613\) −34.2843 −1.38473 −0.692364 0.721548i \(-0.743431\pi\)
−0.692364 + 0.721548i \(0.743431\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.04163i 0.162710i 0.996685 + 0.0813550i \(0.0259247\pi\)
−0.996685 + 0.0813550i \(0.974075\pi\)
\(618\) 0 0
\(619\) 46.5369i 1.87048i 0.354018 + 0.935238i \(0.384815\pi\)
−0.354018 + 0.935238i \(0.615185\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −21.9054 + 40.4760i −0.877622 + 1.62163i
\(624\) 0 0
\(625\) −30.7990 −1.23196
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.26810 0.0505625
\(630\) 0 0
\(631\) −35.8709 −1.42800 −0.714000 0.700146i \(-0.753118\pi\)
−0.714000 + 0.700146i \(0.753118\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −22.4700 −0.891695
\(636\) 0 0
\(637\) 8.97056 5.86030i 0.355427 0.232194i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 36.2426i 1.43150i −0.698357 0.715749i \(-0.746085\pi\)
0.698357 0.715749i \(-0.253915\pi\)
\(642\) 0 0
\(643\) 12.8319i 0.506041i −0.967461 0.253020i \(-0.918576\pi\)
0.967461 0.253020i \(-0.0814240\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 33.3743 1.31208 0.656039 0.754727i \(-0.272230\pi\)
0.656039 + 0.754727i \(0.272230\pi\)
\(648\) 0 0
\(649\) 40.0166i 1.57079i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8.72792i 0.341550i 0.985310 + 0.170775i \(0.0546271\pi\)
−0.985310 + 0.170775i \(0.945373\pi\)
\(654\) 0 0
\(655\) 51.5934 2.01592
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.6330i 0.803748i 0.915695 + 0.401874i \(0.131641\pi\)
−0.915695 + 0.401874i \(0.868359\pi\)
\(660\) 0 0
\(661\) 13.1426i 0.511186i 0.966784 + 0.255593i \(0.0822707\pi\)
−0.966784 + 0.255593i \(0.917729\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 11.7206 + 6.34315i 0.454506 + 0.245977i
\(666\) 0 0
\(667\) −1.04322 −0.0403937
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −35.6326 −1.37558
\(672\) 0 0
\(673\) 22.8284 0.879971 0.439986 0.898005i \(-0.354984\pi\)
0.439986 + 0.898005i \(0.354984\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −34.3421 −1.31987 −0.659936 0.751322i \(-0.729417\pi\)
−0.659936 + 0.751322i \(0.729417\pi\)
\(678\) 0 0
\(679\) −15.7225 8.50894i −0.603373 0.326543i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 39.3062i 1.50401i 0.659158 + 0.752004i \(0.270913\pi\)
−0.659158 + 0.752004i \(0.729087\pi\)
\(684\) 0 0
\(685\) 49.8351i 1.90410i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −3.95815 −0.150794
\(690\) 0 0
\(691\) 27.9222i 1.06221i 0.847306 + 0.531104i \(0.178223\pi\)
−0.847306 + 0.531104i \(0.821777\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.0742i 0.382138i
\(696\) 0 0
\(697\) 3.11270 0.117902
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 32.0416i 1.21020i 0.796151 + 0.605098i \(0.206866\pi\)
−0.796151 + 0.605098i \(0.793134\pi\)
\(702\) 0 0
\(703\) 5.45214i 0.205631i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.87197 18.2410i 0.371274 0.686024i
\(708\) 0 0
\(709\) 24.2843 0.912015 0.456007 0.889976i \(-0.349279\pi\)
0.456007 + 0.889976i \(0.349279\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.85483 0.181815
\(714\) 0 0
\(715\) 17.1978 0.643161
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20.8731 −0.778436 −0.389218 0.921146i \(-0.627255\pi\)
−0.389218 + 0.921146i \(0.627255\pi\)
\(720\) 0 0
\(721\) −36.9706 20.0083i −1.37686 0.745149i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.58579i 0.0960337i
\(726\) 0 0
\(727\) 26.7930i 0.993697i 0.867837 + 0.496848i \(0.165509\pi\)
−0.867837 + 0.496848i \(0.834491\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.32296 0.159891
\(732\) 0 0
\(733\) 23.1786i 0.856120i 0.903750 + 0.428060i \(0.140803\pi\)
−0.903750 + 0.428060i \(0.859197\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.34315i 0.233653i
\(738\) 0 0
\(739\) −4.42602 −0.162814 −0.0814068 0.996681i \(-0.525941\pi\)
−0.0814068 + 0.996681i \(0.525941\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.86123i 0.288401i −0.989549 0.144200i \(-0.953939\pi\)
0.989549 0.144200i \(-0.0460610\pi\)
\(744\) 0 0
\(745\) 17.2095i 0.630507i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.71644 + 0.928932i 0.0627175 + 0.0339424i
\(750\) 0 0
\(751\) −16.7657 −0.611789 −0.305894 0.952065i \(-0.598955\pi\)
−0.305894 + 0.952065i \(0.598955\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −38.3587 −1.39602
\(756\) 0 0
\(757\) 6.34315 0.230546 0.115273 0.993334i \(-0.463226\pi\)
0.115273 + 0.993334i \(0.463226\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 7.46796 0.270713 0.135357 0.990797i \(-0.456782\pi\)
0.135357 + 0.990797i \(0.456782\pi\)
\(762\) 0 0
\(763\) −19.2842 + 35.6326i −0.698137 + 1.28999i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14.2471i 0.514434i
\(768\) 0 0
\(769\) 35.1618i 1.26797i −0.773346 0.633984i \(-0.781419\pi\)
0.773346 0.633984i \(-0.218581\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 5.30318 0.190742 0.0953710 0.995442i \(-0.469596\pi\)
0.0953710 + 0.995442i \(0.469596\pi\)
\(774\) 0 0
\(775\) 12.0335i 0.432254i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 13.3829i 0.479492i
\(780\) 0 0
\(781\) −49.1127 −1.75739
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 52.2843i 1.86611i
\(786\) 0 0
\(787\) 7.71049i 0.274849i −0.990512 0.137425i \(-0.956118\pi\)
0.990512 0.137425i \(-0.0438825\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 7.14590 + 3.86733i 0.254079 + 0.137507i
\(792\) 0 0
\(793\) 12.6863 0.450503
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8.21080 0.290841 0.145421 0.989370i \(-0.453546\pi\)
0.145421 + 0.989370i \(0.453546\pi\)
\(798\) 0 0
\(799\) −1.22221 −0.0432387
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 47.6661 1.68210
\(804\) 0 0
\(805\) 4.48528 + 2.42742i 0.158085 + 0.0855552i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 43.3553i 1.52429i −0.647405 0.762146i \(-0.724146\pi\)
0.647405 0.762146i \(-0.275854\pi\)
\(810\) 0 0
\(811\) 37.2295i 1.30731i 0.756794 + 0.653653i \(0.226764\pi\)
−0.756794 + 0.653653i \(0.773236\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −48.7953 −1.70922
\(816\) 0 0
\(817\) 18.5864i 0.650255i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.5858i 0.369446i −0.982791 0.184723i \(-0.940861\pi\)
0.982791 0.184723i \(-0.0591389\pi\)
\(822\) 0 0
\(823\) −50.1181 −1.74701 −0.873503 0.486818i \(-0.838157\pi\)
−0.873503 + 0.486818i \(0.838157\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 38.6951i 1.34556i 0.739843 + 0.672780i \(0.234900\pi\)
−0.739843 + 0.672780i \(0.765100\pi\)
\(828\) 0 0
\(829\) 31.0949i 1.07997i −0.841675 0.539985i \(-0.818430\pi\)
0.841675 0.539985i \(-0.181570\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.71644 2.62742i −0.0594712 0.0910346i
\(834\) 0 0
\(835\) 27.2720 0.943788
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −49.2630 −1.70075 −0.850374 0.526179i \(-0.823624\pi\)
−0.850374 + 0.526179i \(0.823624\pi\)
\(840\) 0 0
\(841\) 27.0000 0.931034
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 27.8477 0.957990
\(846\) 0 0
\(847\) −9.42607 + 17.4171i −0.323883 + 0.598459i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.08644i 0.0715224i
\(852\) 0 0
\(853\) 8.28772i 0.283766i 0.989883 + 0.141883i \(0.0453157\pi\)
−0.989883 + 0.141883i \(0.954684\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −36.1354 −1.23436 −0.617181 0.786821i \(-0.711725\pi\)
−0.617181 + 0.786821i \(0.711725\pi\)
\(858\) 0 0
\(859\) 37.5603i 1.28154i 0.767733 + 0.640770i \(0.221385\pi\)
−0.767733 + 0.640770i \(0.778615\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.91056i 0.167157i 0.996501 + 0.0835786i \(0.0266350\pi\)
−0.996501 + 0.0835786i \(0.973365\pi\)
\(864\) 0 0
\(865\) 23.7990 0.809190
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 67.5980i 2.29310i
\(870\) 0 0
\(871\) 2.25835i 0.0765213i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −10.4366 + 19.2842i −0.352820 + 0.651927i
\(876\) 0 0
\(877\) 9.31371 0.314502 0.157251 0.987559i \(-0.449737\pi\)
0.157251 + 0.987559i \(0.449737\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −21.3533 −0.719413 −0.359706 0.933066i \(-0.617123\pi\)
−0.359706 + 0.933066i \(0.617123\pi\)
\(882\) 0 0
\(883\) 31.0128 1.04366 0.521832 0.853048i \(-0.325249\pi\)
0.521832 + 0.853048i \(0.325249\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −8.17821 −0.274597 −0.137299 0.990530i \(-0.543842\pi\)
−0.137299 + 0.990530i \(0.543842\pi\)
\(888\) 0 0
\(889\) −10.8284 + 20.0083i −0.363174 + 0.671058i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.25483i 0.175846i
\(894\) 0 0
\(895\) 7.37976i 0.246678i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9.30739 0.310419
\(900\) 0 0
\(901\) 1.15932i 0.0386224i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 26.6274i 0.885125i
\(906\) 0 0
\(907\) 22.6670 0.752647 0.376323 0.926488i \(-0.377188\pi\)
0.376323 + 0.926488i \(0.377188\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 37.2197i 1.23314i 0.787298 + 0.616572i \(0.211479\pi\)
−0.787298 + 0.616572i \(0.788521\pi\)
\(912\) 0 0
\(913\) 68.3127i 2.26082i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 24.8632 45.9411i 0.821054 1.51711i
\(918\) 0 0
\(919\) −16.7657 −0.553049 −0.276525 0.961007i \(-0.589183\pi\)
−0.276525 + 0.961007i \(0.589183\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 17.4856 0.575545
\(924\) 0 0
\(925\) 5.17157 0.170040
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 30.5378 1.00191 0.500956 0.865473i \(-0.332982\pi\)
0.500956 + 0.865473i \(0.332982\pi\)
\(930\) 0 0
\(931\) 11.2965 7.37976i 0.370226 0.241862i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.03712i 0.164731i
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −20.4567 −0.666868 −0.333434 0.942773i \(-0.608207\pi\)
−0.333434 + 0.942773i \(0.608207\pi\)
\(942\) 0 0
\(943\) 5.12141i 0.166776i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 17.0712i 0.554741i 0.960763 + 0.277370i \(0.0894629\pi\)
−0.960763 + 0.277370i \(0.910537\pi\)
\(948\) 0 0
\(949\) −16.9706 −0.550888
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 14.8701i 0.481688i 0.970564 + 0.240844i \(0.0774243\pi\)
−0.970564 + 0.240844i \(0.922576\pi\)
\(954\) 0 0
\(955\) 24.3976i 0.789489i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −44.3754 24.0158i −1.43296 0.775511i
\(960\) 0 0
\(961\) −12.3137 −0.397216
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 41.2848 1.32900
\(966\) 0 0
\(967\) 15.7225 0.505600 0.252800 0.967518i \(-0.418648\pi\)
0.252800 + 0.967518i \(0.418648\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −35.6326 −1.14351 −0.571753 0.820426i \(-0.693736\pi\)
−0.571753 + 0.820426i \(0.693736\pi\)
\(972\) 0 0
\(973\) 8.97056 + 4.85483i 0.287583 + 0.155639i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.58579i 0.0827266i −0.999144 0.0413633i \(-0.986830\pi\)
0.999144 0.0413633i \(-0.0131701\pi\)
\(978\) 0 0
\(979\) 74.7898i 2.39029i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −26.7930 −0.854563 −0.427282 0.904119i \(-0.640529\pi\)
−0.427282 + 0.904119i \(0.640529\pi\)
\(984\) 0 0
\(985\) 43.7122i 1.39279i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.11270i 0.226171i
\(990\) 0 0
\(991\) −25.9757 −0.825145 −0.412573 0.910925i \(-0.635370\pi\)
−0.412573 + 0.910925i \(0.635370\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 36.4821i 1.15656i
\(996\) 0 0
\(997\) 41.4386i 1.31237i −0.754599 0.656187i \(-0.772168\pi\)
0.754599 0.656187i \(-0.227832\pi\)
\(998\) 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.g.3905.15 16
3.2 odd 2 inner 4032.2.k.g.3905.3 16
4.3 odd 2 inner 4032.2.k.g.3905.14 16
7.6 odd 2 inner 4032.2.k.g.3905.4 16
8.3 odd 2 2016.2.k.a.1889.2 yes 16
8.5 even 2 2016.2.k.a.1889.3 yes 16
12.11 even 2 inner 4032.2.k.g.3905.2 16
21.20 even 2 inner 4032.2.k.g.3905.16 16
24.5 odd 2 2016.2.k.a.1889.15 yes 16
24.11 even 2 2016.2.k.a.1889.14 yes 16
28.27 even 2 inner 4032.2.k.g.3905.1 16
56.13 odd 2 2016.2.k.a.1889.16 yes 16
56.27 even 2 2016.2.k.a.1889.13 yes 16
84.83 odd 2 inner 4032.2.k.g.3905.13 16
168.83 odd 2 2016.2.k.a.1889.1 16
168.125 even 2 2016.2.k.a.1889.4 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2016.2.k.a.1889.1 16 168.83 odd 2
2016.2.k.a.1889.2 yes 16 8.3 odd 2
2016.2.k.a.1889.3 yes 16 8.5 even 2
2016.2.k.a.1889.4 yes 16 168.125 even 2
2016.2.k.a.1889.13 yes 16 56.27 even 2
2016.2.k.a.1889.14 yes 16 24.11 even 2
2016.2.k.a.1889.15 yes 16 24.5 odd 2
2016.2.k.a.1889.16 yes 16 56.13 odd 2
4032.2.k.g.3905.1 16 28.27 even 2 inner
4032.2.k.g.3905.2 16 12.11 even 2 inner
4032.2.k.g.3905.3 16 3.2 odd 2 inner
4032.2.k.g.3905.4 16 7.6 odd 2 inner
4032.2.k.g.3905.13 16 84.83 odd 2 inner
4032.2.k.g.3905.14 16 4.3 odd 2 inner
4032.2.k.g.3905.15 16 1.1 even 1 trivial
4032.2.k.g.3905.16 16 21.20 even 2 inner