Properties

Label 2736.2.d.b.2015.9
Level $2736$
Weight $2$
Character 2736.2015
Analytic conductor $21.847$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,2,Mod(2015,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.2015");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.d (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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2015.9
Character \(\chi\) \(=\) 2736.2015
Dual form 2736.2.d.b.2015.16

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.28244i q^{5} -4.16899i q^{7} +O(q^{10})\) \(q-1.28244i q^{5} -4.16899i q^{7} +0.743817 q^{11} -4.42357 q^{13} -1.23027i q^{17} +1.00000i q^{19} +7.13146 q^{23} +3.35535 q^{25} -7.86993i q^{29} +0.885686i q^{31} -5.34648 q^{35} +0.287141 q^{37} -7.60097i q^{41} +5.02768i q^{43} -9.88028 q^{47} -10.3805 q^{49} -9.21504i q^{53} -0.953899i q^{55} -10.9619 q^{59} -11.2919 q^{61} +5.67295i q^{65} +7.62728i q^{67} -1.47154 q^{71} +10.9535 q^{73} -3.10097i q^{77} -3.62346i q^{79} -7.21360 q^{83} -1.57774 q^{85} +13.4121i q^{89} +18.4418i q^{91} +1.28244 q^{95} -7.73111 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 8 q^{25} - 32 q^{37} - 32 q^{49} + 8 q^{73} + 40 q^{85} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(5\) − 1.28244i − 0.573524i −0.958002 0.286762i \(-0.907421\pi\)
0.958002 0.286762i \(-0.0925788\pi\)
\(6\) 0 0
\(7\) − 4.16899i − 1.57573i −0.615847 0.787866i \(-0.711186\pi\)
0.615847 0.787866i \(-0.288814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.743817 0.224269 0.112135 0.993693i \(-0.464231\pi\)
0.112135 + 0.993693i \(0.464231\pi\)
\(12\) 0 0
\(13\) −4.42357 −1.22688 −0.613438 0.789743i \(-0.710214\pi\)
−0.613438 + 0.789743i \(0.710214\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.23027i − 0.298384i −0.988808 0.149192i \(-0.952333\pi\)
0.988808 0.149192i \(-0.0476673\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.13146 1.48701 0.743506 0.668729i \(-0.233161\pi\)
0.743506 + 0.668729i \(0.233161\pi\)
\(24\) 0 0
\(25\) 3.35535 0.671071
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 7.86993i − 1.46141i −0.682693 0.730705i \(-0.739191\pi\)
0.682693 0.730705i \(-0.260809\pi\)
\(30\) 0 0
\(31\) 0.885686i 0.159074i 0.996832 + 0.0795370i \(0.0253442\pi\)
−0.996832 + 0.0795370i \(0.974656\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.34648 −0.903719
\(36\) 0 0
\(37\) 0.287141 0.0472057 0.0236029 0.999721i \(-0.492486\pi\)
0.0236029 + 0.999721i \(0.492486\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 7.60097i − 1.18707i −0.804807 0.593536i \(-0.797731\pi\)
0.804807 0.593536i \(-0.202269\pi\)
\(42\) 0 0
\(43\) 5.02768i 0.766713i 0.923600 + 0.383357i \(0.125232\pi\)
−0.923600 + 0.383357i \(0.874768\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.88028 −1.44119 −0.720593 0.693358i \(-0.756131\pi\)
−0.720593 + 0.693358i \(0.756131\pi\)
\(48\) 0 0
\(49\) −10.3805 −1.48293
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 9.21504i − 1.26578i −0.774240 0.632892i \(-0.781868\pi\)
0.774240 0.632892i \(-0.218132\pi\)
\(54\) 0 0
\(55\) − 0.953899i − 0.128624i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.9619 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(60\) 0 0
\(61\) −11.2919 −1.44578 −0.722889 0.690964i \(-0.757186\pi\)
−0.722889 + 0.690964i \(0.757186\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.67295i 0.703643i
\(66\) 0 0
\(67\) 7.62728i 0.931821i 0.884832 + 0.465910i \(0.154273\pi\)
−0.884832 + 0.465910i \(0.845727\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.47154 −0.174640 −0.0873198 0.996180i \(-0.527830\pi\)
−0.0873198 + 0.996180i \(0.527830\pi\)
\(72\) 0 0
\(73\) 10.9535 1.28201 0.641004 0.767538i \(-0.278518\pi\)
0.641004 + 0.767538i \(0.278518\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.10097i − 0.353388i
\(78\) 0 0
\(79\) − 3.62346i − 0.407671i −0.979005 0.203835i \(-0.934659\pi\)
0.979005 0.203835i \(-0.0653407\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.21360 −0.791795 −0.395897 0.918295i \(-0.629566\pi\)
−0.395897 + 0.918295i \(0.629566\pi\)
\(84\) 0 0
\(85\) −1.57774 −0.171130
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.4121i 1.42168i 0.703352 + 0.710842i \(0.251686\pi\)
−0.703352 + 0.710842i \(0.748314\pi\)
\(90\) 0 0
\(91\) 18.4418i 1.93323i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.28244 0.131575
\(96\) 0 0
\(97\) −7.73111 −0.784976 −0.392488 0.919757i \(-0.628385\pi\)
−0.392488 + 0.919757i \(0.628385\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 7.71438i − 0.767609i −0.923414 0.383805i \(-0.874614\pi\)
0.923414 0.383805i \(-0.125386\pi\)
\(102\) 0 0
\(103\) 8.33416i 0.821189i 0.911818 + 0.410595i \(0.134679\pi\)
−0.911818 + 0.410595i \(0.865321\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.66786 −0.451259 −0.225629 0.974213i \(-0.572444\pi\)
−0.225629 + 0.974213i \(0.572444\pi\)
\(108\) 0 0
\(109\) 4.10834 0.393507 0.196754 0.980453i \(-0.436960\pi\)
0.196754 + 0.980453i \(0.436960\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 3.28491i − 0.309019i −0.987991 0.154509i \(-0.950620\pi\)
0.987991 0.154509i \(-0.0493796\pi\)
\(114\) 0 0
\(115\) − 9.14565i − 0.852836i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.12898 −0.470173
\(120\) 0 0
\(121\) −10.4467 −0.949703
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 10.7152i − 0.958398i
\(126\) 0 0
\(127\) 17.7389i 1.57407i 0.616906 + 0.787037i \(0.288386\pi\)
−0.616906 + 0.787037i \(0.711614\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 11.2751 0.985109 0.492555 0.870281i \(-0.336063\pi\)
0.492555 + 0.870281i \(0.336063\pi\)
\(132\) 0 0
\(133\) 4.16899 0.361498
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 0.233400i − 0.0199407i −0.999950 0.00997037i \(-0.996826\pi\)
0.999950 0.00997037i \(-0.00317372\pi\)
\(138\) 0 0
\(139\) − 4.11798i − 0.349282i −0.984632 0.174641i \(-0.944123\pi\)
0.984632 0.174641i \(-0.0558765\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.29032 −0.275151
\(144\) 0 0
\(145\) −10.0927 −0.838153
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16.0740i 1.31683i 0.752655 + 0.658415i \(0.228773\pi\)
−0.752655 + 0.658415i \(0.771227\pi\)
\(150\) 0 0
\(151\) − 17.4086i − 1.41669i −0.705866 0.708345i \(-0.749442\pi\)
0.705866 0.708345i \(-0.250558\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.13584 0.0912327
\(156\) 0 0
\(157\) −15.6960 −1.25268 −0.626340 0.779550i \(-0.715448\pi\)
−0.626340 + 0.779550i \(0.715448\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 29.7310i − 2.34313i
\(162\) 0 0
\(163\) 9.84447i 0.771078i 0.922691 + 0.385539i \(0.125985\pi\)
−0.922691 + 0.385539i \(0.874015\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.1322 −1.09358 −0.546790 0.837270i \(-0.684150\pi\)
−0.546790 + 0.837270i \(0.684150\pi\)
\(168\) 0 0
\(169\) 6.56793 0.505226
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 22.7579i − 1.73025i −0.501554 0.865126i \(-0.667238\pi\)
0.501554 0.865126i \(-0.332762\pi\)
\(174\) 0 0
\(175\) − 13.9884i − 1.05743i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.64721 0.347349 0.173674 0.984803i \(-0.444436\pi\)
0.173674 + 0.984803i \(0.444436\pi\)
\(180\) 0 0
\(181\) −21.4784 −1.59647 −0.798237 0.602344i \(-0.794234\pi\)
−0.798237 + 0.602344i \(0.794234\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 0.368241i − 0.0270736i
\(186\) 0 0
\(187\) − 0.915095i − 0.0669183i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.9024 1.36773 0.683864 0.729609i \(-0.260298\pi\)
0.683864 + 0.729609i \(0.260298\pi\)
\(192\) 0 0
\(193\) −15.3639 −1.10592 −0.552960 0.833208i \(-0.686502\pi\)
−0.552960 + 0.833208i \(0.686502\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 7.50966i − 0.535041i −0.963552 0.267521i \(-0.913796\pi\)
0.963552 0.267521i \(-0.0862044\pi\)
\(198\) 0 0
\(199\) 2.02502i 0.143550i 0.997421 + 0.0717748i \(0.0228663\pi\)
−0.997421 + 0.0717748i \(0.977134\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −32.8097 −2.30279
\(204\) 0 0
\(205\) −9.74777 −0.680814
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.743817i 0.0514509i
\(210\) 0 0
\(211\) 10.5645i 0.727290i 0.931538 + 0.363645i \(0.118468\pi\)
−0.931538 + 0.363645i \(0.881532\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.44768 0.439728
\(216\) 0 0
\(217\) 3.69242 0.250658
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.44218i 0.366080i
\(222\) 0 0
\(223\) − 9.26053i − 0.620131i −0.950715 0.310065i \(-0.899649\pi\)
0.950715 0.310065i \(-0.100351\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.8910 0.789232 0.394616 0.918846i \(-0.370878\pi\)
0.394616 + 0.918846i \(0.370878\pi\)
\(228\) 0 0
\(229\) 8.03944 0.531261 0.265631 0.964075i \(-0.414420\pi\)
0.265631 + 0.964075i \(0.414420\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 24.2635i − 1.58956i −0.606900 0.794778i \(-0.707587\pi\)
0.606900 0.794778i \(-0.292413\pi\)
\(234\) 0 0
\(235\) 12.6708i 0.826555i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 22.4847 1.45442 0.727208 0.686417i \(-0.240818\pi\)
0.727208 + 0.686417i \(0.240818\pi\)
\(240\) 0 0
\(241\) 19.5412 1.25876 0.629379 0.777099i \(-0.283309\pi\)
0.629379 + 0.777099i \(0.283309\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 13.3124i 0.850496i
\(246\) 0 0
\(247\) − 4.42357i − 0.281465i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.73713 0.109646 0.0548232 0.998496i \(-0.482540\pi\)
0.0548232 + 0.998496i \(0.482540\pi\)
\(252\) 0 0
\(253\) 5.30450 0.333491
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 28.8546i − 1.79990i −0.435994 0.899949i \(-0.643603\pi\)
0.435994 0.899949i \(-0.356397\pi\)
\(258\) 0 0
\(259\) − 1.19709i − 0.0743836i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.4759 0.830960 0.415480 0.909602i \(-0.363614\pi\)
0.415480 + 0.909602i \(0.363614\pi\)
\(264\) 0 0
\(265\) −11.8177 −0.725956
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 0.894031i − 0.0545100i −0.999629 0.0272550i \(-0.991323\pi\)
0.999629 0.0272550i \(-0.00867662\pi\)
\(270\) 0 0
\(271\) − 24.6977i − 1.50028i −0.661280 0.750139i \(-0.729986\pi\)
0.661280 0.750139i \(-0.270014\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.49577 0.150500
\(276\) 0 0
\(277\) 22.3676 1.34394 0.671971 0.740578i \(-0.265448\pi\)
0.671971 + 0.740578i \(0.265448\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 6.64344i − 0.396314i −0.980170 0.198157i \(-0.936504\pi\)
0.980170 0.198157i \(-0.0634957\pi\)
\(282\) 0 0
\(283\) 3.47351i 0.206479i 0.994657 + 0.103239i \(0.0329208\pi\)
−0.994657 + 0.103239i \(0.967079\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −31.6884 −1.87051
\(288\) 0 0
\(289\) 15.4864 0.910967
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 23.5716i 1.37707i 0.725204 + 0.688534i \(0.241745\pi\)
−0.725204 + 0.688534i \(0.758255\pi\)
\(294\) 0 0
\(295\) 14.0580i 0.818487i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −31.5465 −1.82438
\(300\) 0 0
\(301\) 20.9604 1.20813
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 14.4811i 0.829188i
\(306\) 0 0
\(307\) 12.2077i 0.696728i 0.937359 + 0.348364i \(0.113263\pi\)
−0.937359 + 0.348364i \(0.886737\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 17.4706 0.990665 0.495333 0.868703i \(-0.335046\pi\)
0.495333 + 0.868703i \(0.335046\pi\)
\(312\) 0 0
\(313\) −2.55101 −0.144191 −0.0720957 0.997398i \(-0.522969\pi\)
−0.0720957 + 0.997398i \(0.522969\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.4886i 0.757595i 0.925480 + 0.378798i \(0.123662\pi\)
−0.925480 + 0.378798i \(0.876338\pi\)
\(318\) 0 0
\(319\) − 5.85379i − 0.327749i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.23027 0.0684540
\(324\) 0 0
\(325\) −14.8426 −0.823321
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 41.1908i 2.27092i
\(330\) 0 0
\(331\) − 31.6269i − 1.73837i −0.494484 0.869187i \(-0.664643\pi\)
0.494484 0.869187i \(-0.335357\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 9.78151 0.534421
\(336\) 0 0
\(337\) −9.92289 −0.540534 −0.270267 0.962785i \(-0.587112\pi\)
−0.270267 + 0.962785i \(0.587112\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.658788i 0.0356754i
\(342\) 0 0
\(343\) 14.0933i 0.760969i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −13.9484 −0.748791 −0.374395 0.927269i \(-0.622150\pi\)
−0.374395 + 0.927269i \(0.622150\pi\)
\(348\) 0 0
\(349\) 10.9967 0.588638 0.294319 0.955707i \(-0.404907\pi\)
0.294319 + 0.955707i \(0.404907\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 23.7735i − 1.26534i −0.774423 0.632669i \(-0.781960\pi\)
0.774423 0.632669i \(-0.218040\pi\)
\(354\) 0 0
\(355\) 1.88716i 0.100160i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 13.3019 0.702050 0.351025 0.936366i \(-0.385833\pi\)
0.351025 + 0.936366i \(0.385833\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 14.0472i − 0.735262i
\(366\) 0 0
\(367\) 9.71599i 0.507170i 0.967313 + 0.253585i \(0.0816098\pi\)
−0.967313 + 0.253585i \(0.918390\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −38.4174 −1.99453
\(372\) 0 0
\(373\) 20.4428 1.05849 0.529245 0.848469i \(-0.322475\pi\)
0.529245 + 0.848469i \(0.322475\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 34.8132i 1.79297i
\(378\) 0 0
\(379\) − 12.3290i − 0.633297i −0.948543 0.316649i \(-0.897442\pi\)
0.948543 0.316649i \(-0.102558\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.32251 −0.169772 −0.0848861 0.996391i \(-0.527053\pi\)
−0.0848861 + 0.996391i \(0.527053\pi\)
\(384\) 0 0
\(385\) −3.97680 −0.202676
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.93846i 0.301092i 0.988603 + 0.150546i \(0.0481032\pi\)
−0.988603 + 0.150546i \(0.951897\pi\)
\(390\) 0 0
\(391\) − 8.77361i − 0.443701i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.64686 −0.233809
\(396\) 0 0
\(397\) −5.14026 −0.257982 −0.128991 0.991646i \(-0.541174\pi\)
−0.128991 + 0.991646i \(0.541174\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.4755i 0.523121i 0.965187 + 0.261561i \(0.0842371\pi\)
−0.965187 + 0.261561i \(0.915763\pi\)
\(402\) 0 0
\(403\) − 3.91789i − 0.195164i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.213580 0.0105868
\(408\) 0 0
\(409\) −0.282636 −0.0139755 −0.00698773 0.999976i \(-0.502224\pi\)
−0.00698773 + 0.999976i \(0.502224\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 45.7002i 2.24876i
\(414\) 0 0
\(415\) 9.25099i 0.454113i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 26.7445 1.30655 0.653277 0.757119i \(-0.273394\pi\)
0.653277 + 0.757119i \(0.273394\pi\)
\(420\) 0 0
\(421\) 12.6874 0.618346 0.309173 0.951006i \(-0.399948\pi\)
0.309173 + 0.951006i \(0.399948\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 4.12799i − 0.200237i
\(426\) 0 0
\(427\) 47.0758i 2.27816i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 33.8802 1.63195 0.815975 0.578087i \(-0.196201\pi\)
0.815975 + 0.578087i \(0.196201\pi\)
\(432\) 0 0
\(433\) −20.1806 −0.969818 −0.484909 0.874565i \(-0.661147\pi\)
−0.484909 + 0.874565i \(0.661147\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.13146i 0.341144i
\(438\) 0 0
\(439\) 21.3057i 1.01687i 0.861101 + 0.508434i \(0.169775\pi\)
−0.861101 + 0.508434i \(0.830225\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −32.6164 −1.54965 −0.774827 0.632174i \(-0.782163\pi\)
−0.774827 + 0.632174i \(0.782163\pi\)
\(444\) 0 0
\(445\) 17.2002 0.815369
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 11.4787i − 0.541713i −0.962620 0.270857i \(-0.912693\pi\)
0.962620 0.270857i \(-0.0873069\pi\)
\(450\) 0 0
\(451\) − 5.65373i − 0.266224i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 23.6505 1.10875
\(456\) 0 0
\(457\) 23.2828 1.08912 0.544561 0.838721i \(-0.316696\pi\)
0.544561 + 0.838721i \(0.316696\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 8.56760i − 0.399033i −0.979894 0.199516i \(-0.936063\pi\)
0.979894 0.199516i \(-0.0639371\pi\)
\(462\) 0 0
\(463\) − 37.6104i − 1.74790i −0.486012 0.873952i \(-0.661549\pi\)
0.486012 0.873952i \(-0.338451\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −32.1560 −1.48800 −0.744000 0.668180i \(-0.767074\pi\)
−0.744000 + 0.668180i \(0.767074\pi\)
\(468\) 0 0
\(469\) 31.7981 1.46830
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.73967i 0.171950i
\(474\) 0 0
\(475\) 3.35535i 0.153954i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 29.7819 1.36077 0.680385 0.732855i \(-0.261813\pi\)
0.680385 + 0.732855i \(0.261813\pi\)
\(480\) 0 0
\(481\) −1.27019 −0.0579156
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.91467i 0.450202i
\(486\) 0 0
\(487\) − 1.60249i − 0.0726158i −0.999341 0.0363079i \(-0.988440\pi\)
0.999341 0.0363079i \(-0.0115597\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.26773 −0.0572117 −0.0286058 0.999591i \(-0.509107\pi\)
−0.0286058 + 0.999591i \(0.509107\pi\)
\(492\) 0 0
\(493\) −9.68214 −0.436061
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.13484i 0.275185i
\(498\) 0 0
\(499\) − 0.186383i − 0.00834367i −0.999991 0.00417183i \(-0.998672\pi\)
0.999991 0.00417183i \(-0.00132794\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 25.6715 1.14464 0.572318 0.820031i \(-0.306044\pi\)
0.572318 + 0.820031i \(0.306044\pi\)
\(504\) 0 0
\(505\) −9.89321 −0.440242
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 37.1654i − 1.64733i −0.567078 0.823664i \(-0.691926\pi\)
0.567078 0.823664i \(-0.308074\pi\)
\(510\) 0 0
\(511\) − 45.6650i − 2.02010i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 10.6880 0.470972
\(516\) 0 0
\(517\) −7.34912 −0.323214
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 17.6596i − 0.773683i −0.922146 0.386841i \(-0.873566\pi\)
0.922146 0.386841i \(-0.126434\pi\)
\(522\) 0 0
\(523\) 5.89269i 0.257669i 0.991666 + 0.128835i \(0.0411236\pi\)
−0.991666 + 0.128835i \(0.958876\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.08963 0.0474651
\(528\) 0 0
\(529\) 27.8577 1.21120
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 33.6234i 1.45639i
\(534\) 0 0
\(535\) 5.98623i 0.258808i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −7.72120 −0.332576
\(540\) 0 0
\(541\) −40.7900 −1.75370 −0.876849 0.480767i \(-0.840358\pi\)
−0.876849 + 0.480767i \(0.840358\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 5.26869i − 0.225686i
\(546\) 0 0
\(547\) − 7.95762i − 0.340243i −0.985423 0.170122i \(-0.945584\pi\)
0.985423 0.170122i \(-0.0544161\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.86993 0.335270
\(552\) 0 0
\(553\) −15.1062 −0.642380
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.1166i 0.598138i 0.954232 + 0.299069i \(0.0966760\pi\)
−0.954232 + 0.299069i \(0.903324\pi\)
\(558\) 0 0
\(559\) − 22.2403i − 0.940663i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 25.8435 1.08917 0.544587 0.838704i \(-0.316686\pi\)
0.544587 + 0.838704i \(0.316686\pi\)
\(564\) 0 0
\(565\) −4.21270 −0.177229
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 34.8478i 1.46090i 0.682968 + 0.730449i \(0.260689\pi\)
−0.682968 + 0.730449i \(0.739311\pi\)
\(570\) 0 0
\(571\) 4.32290i 0.180908i 0.995901 + 0.0904538i \(0.0288317\pi\)
−0.995901 + 0.0904538i \(0.971168\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 23.9286 0.997890
\(576\) 0 0
\(577\) −9.93312 −0.413521 −0.206761 0.978392i \(-0.566292\pi\)
−0.206761 + 0.978392i \(0.566292\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 30.0734i 1.24766i
\(582\) 0 0
\(583\) − 6.85430i − 0.283876i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 43.4288 1.79250 0.896249 0.443551i \(-0.146281\pi\)
0.896249 + 0.443551i \(0.146281\pi\)
\(588\) 0 0
\(589\) −0.885686 −0.0364941
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 40.1735i 1.64973i 0.565331 + 0.824864i \(0.308748\pi\)
−0.565331 + 0.824864i \(0.691252\pi\)
\(594\) 0 0
\(595\) 6.57760i 0.269655i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −9.45838 −0.386459 −0.193229 0.981154i \(-0.561896\pi\)
−0.193229 + 0.981154i \(0.561896\pi\)
\(600\) 0 0
\(601\) 34.6402 1.41300 0.706501 0.707712i \(-0.250272\pi\)
0.706501 + 0.707712i \(0.250272\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 13.3973i 0.544677i
\(606\) 0 0
\(607\) − 41.8946i − 1.70045i −0.526419 0.850225i \(-0.676466\pi\)
0.526419 0.850225i \(-0.323534\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 43.7061 1.76816
\(612\) 0 0
\(613\) 1.25774 0.0507997 0.0253999 0.999677i \(-0.491914\pi\)
0.0253999 + 0.999677i \(0.491914\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 8.12508i − 0.327103i −0.986535 0.163552i \(-0.947705\pi\)
0.986535 0.163552i \(-0.0522951\pi\)
\(618\) 0 0
\(619\) 21.9388i 0.881794i 0.897558 + 0.440897i \(0.145340\pi\)
−0.897558 + 0.440897i \(0.854660\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 55.9151 2.24019
\(624\) 0 0
\(625\) 3.03516 0.121407
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 0.353261i − 0.0140854i
\(630\) 0 0
\(631\) − 18.9260i − 0.753434i −0.926328 0.376717i \(-0.877053\pi\)
0.926328 0.376717i \(-0.122947\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 22.7491 0.902769
\(636\) 0 0
\(637\) 45.9189 1.81937
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 23.6972i 0.935983i 0.883733 + 0.467991i \(0.155022\pi\)
−0.883733 + 0.467991i \(0.844978\pi\)
\(642\) 0 0
\(643\) 10.9864i 0.433261i 0.976254 + 0.216630i \(0.0695066\pi\)
−0.976254 + 0.216630i \(0.930493\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −29.8804 −1.17472 −0.587360 0.809326i \(-0.699833\pi\)
−0.587360 + 0.809326i \(0.699833\pi\)
\(648\) 0 0
\(649\) −8.15365 −0.320059
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.69439i 0.105440i 0.998609 + 0.0527199i \(0.0167890\pi\)
−0.998609 + 0.0527199i \(0.983211\pi\)
\(654\) 0 0
\(655\) − 14.4596i − 0.564983i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −35.6546 −1.38890 −0.694452 0.719539i \(-0.744353\pi\)
−0.694452 + 0.719539i \(0.744353\pi\)
\(660\) 0 0
\(661\) −15.6821 −0.609963 −0.304982 0.952358i \(-0.598650\pi\)
−0.304982 + 0.952358i \(0.598650\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 5.34648i − 0.207327i
\(666\) 0 0
\(667\) − 56.1241i − 2.17313i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −8.39909 −0.324243
\(672\) 0 0
\(673\) 22.0739 0.850886 0.425443 0.904985i \(-0.360118\pi\)
0.425443 + 0.904985i \(0.360118\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 27.1750i − 1.04442i −0.852817 0.522209i \(-0.825108\pi\)
0.852817 0.522209i \(-0.174892\pi\)
\(678\) 0 0
\(679\) 32.2310i 1.23691i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −11.4504 −0.438137 −0.219069 0.975709i \(-0.570302\pi\)
−0.219069 + 0.975709i \(0.570302\pi\)
\(684\) 0 0
\(685\) −0.299321 −0.0114365
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 40.7633i 1.55296i
\(690\) 0 0
\(691\) 1.60749i 0.0611518i 0.999532 + 0.0305759i \(0.00973412\pi\)
−0.999532 + 0.0305759i \(0.990266\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −5.28105 −0.200322
\(696\) 0 0
\(697\) −9.35124 −0.354204
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 20.4400i − 0.772008i −0.922497 0.386004i \(-0.873855\pi\)
0.922497 0.386004i \(-0.126145\pi\)
\(702\) 0 0
\(703\) 0.287141i 0.0108297i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −32.1612 −1.20955
\(708\) 0 0
\(709\) 36.6945 1.37809 0.689046 0.724718i \(-0.258030\pi\)
0.689046 + 0.724718i \(0.258030\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6.31624i 0.236545i
\(714\) 0 0
\(715\) 4.21963i 0.157805i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 15.7366 0.586878 0.293439 0.955978i \(-0.405200\pi\)
0.293439 + 0.955978i \(0.405200\pi\)
\(720\) 0 0
\(721\) 34.7451 1.29397
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 26.4064i − 0.980709i
\(726\) 0 0
\(727\) − 10.6478i − 0.394906i −0.980312 0.197453i \(-0.936733\pi\)
0.980312 0.197453i \(-0.0632670\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 6.18539 0.228775
\(732\) 0 0
\(733\) 23.7388 0.876811 0.438406 0.898777i \(-0.355543\pi\)
0.438406 + 0.898777i \(0.355543\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.67330i 0.208979i
\(738\) 0 0
\(739\) − 40.5762i − 1.49262i −0.665598 0.746310i \(-0.731824\pi\)
0.665598 0.746310i \(-0.268176\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −15.2145 −0.558164 −0.279082 0.960267i \(-0.590030\pi\)
−0.279082 + 0.960267i \(0.590030\pi\)
\(744\) 0 0
\(745\) 20.6139 0.755233
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 19.4603i 0.711063i
\(750\) 0 0
\(751\) 44.5365i 1.62516i 0.582851 + 0.812579i \(0.301937\pi\)
−0.582851 + 0.812579i \(0.698063\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −22.3254 −0.812505
\(756\) 0 0
\(757\) −35.7119 −1.29797 −0.648986 0.760800i \(-0.724807\pi\)
−0.648986 + 0.760800i \(0.724807\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 16.3436i − 0.592456i −0.955117 0.296228i \(-0.904271\pi\)
0.955117 0.296228i \(-0.0957288\pi\)
\(762\) 0 0
\(763\) − 17.1276i − 0.620062i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 48.4907 1.75090
\(768\) 0 0
\(769\) 13.4698 0.485734 0.242867 0.970060i \(-0.421912\pi\)
0.242867 + 0.970060i \(0.421912\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 29.9105i 1.07580i 0.843007 + 0.537902i \(0.180783\pi\)
−0.843007 + 0.537902i \(0.819217\pi\)
\(774\) 0 0
\(775\) 2.97179i 0.106750i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.60097 0.272333
\(780\) 0 0
\(781\) −1.09456 −0.0391663
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 20.1292i 0.718441i
\(786\) 0 0
\(787\) 13.3674i 0.476497i 0.971204 + 0.238248i \(0.0765732\pi\)
−0.971204 + 0.238248i \(0.923427\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −13.6948 −0.486930
\(792\) 0 0
\(793\) 49.9504 1.77379
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 21.9906i − 0.778946i −0.921038 0.389473i \(-0.872657\pi\)
0.921038 0.389473i \(-0.127343\pi\)
\(798\) 0 0
\(799\) 12.1554i 0.430027i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 8.14738 0.287515
\(804\) 0 0
\(805\) −38.1282 −1.34384
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.79728i 0.0983471i 0.998790 + 0.0491736i \(0.0156587\pi\)
−0.998790 + 0.0491736i \(0.984341\pi\)
\(810\) 0 0
\(811\) 20.9149i 0.734423i 0.930138 + 0.367211i \(0.119687\pi\)
−0.930138 + 0.367211i \(0.880313\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 12.6249 0.442232
\(816\) 0 0
\(817\) −5.02768 −0.175896
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 14.3274i − 0.500031i −0.968242 0.250015i \(-0.919564\pi\)
0.968242 0.250015i \(-0.0804357\pi\)
\(822\) 0 0
\(823\) 22.9377i 0.799557i 0.916612 + 0.399778i \(0.130913\pi\)
−0.916612 + 0.399778i \(0.869087\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −47.0384 −1.63568 −0.817842 0.575442i \(-0.804830\pi\)
−0.817842 + 0.575442i \(0.804830\pi\)
\(828\) 0 0
\(829\) −30.4073 −1.05609 −0.528044 0.849217i \(-0.677075\pi\)
−0.528044 + 0.849217i \(0.677075\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 12.7708i 0.442483i
\(834\) 0 0
\(835\) 18.1236i 0.627194i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −32.3644 −1.11734 −0.558672 0.829389i \(-0.688689\pi\)
−0.558672 + 0.829389i \(0.688689\pi\)
\(840\) 0 0
\(841\) −32.9359 −1.13572
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 8.42297i − 0.289759i
\(846\) 0 0
\(847\) 43.5524i 1.49648i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.04774 0.0701955
\(852\) 0 0
\(853\) 12.4556 0.426472 0.213236 0.977001i \(-0.431600\pi\)
0.213236 + 0.977001i \(0.431600\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.98270i 0.0677278i 0.999426 + 0.0338639i \(0.0107813\pi\)
−0.999426 + 0.0338639i \(0.989219\pi\)
\(858\) 0 0
\(859\) 45.4260i 1.54992i 0.632013 + 0.774958i \(0.282229\pi\)
−0.632013 + 0.774958i \(0.717771\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −12.2358 −0.416512 −0.208256 0.978074i \(-0.566779\pi\)
−0.208256 + 0.978074i \(0.566779\pi\)
\(864\) 0 0
\(865\) −29.1856 −0.992341
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 2.69519i − 0.0914280i
\(870\) 0 0
\(871\) − 33.7398i − 1.14323i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −44.6717 −1.51018
\(876\) 0 0
\(877\) 20.2691 0.684438 0.342219 0.939620i \(-0.388822\pi\)
0.342219 + 0.939620i \(0.388822\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 6.60011i 0.222363i 0.993800 + 0.111182i \(0.0354636\pi\)
−0.993800 + 0.111182i \(0.964536\pi\)
\(882\) 0 0
\(883\) − 58.0485i − 1.95349i −0.214410 0.976744i \(-0.568783\pi\)
0.214410 0.976744i \(-0.431217\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −45.3048 −1.52119 −0.760594 0.649228i \(-0.775092\pi\)
−0.760594 + 0.649228i \(0.775092\pi\)
\(888\) 0 0
\(889\) 73.9534 2.48032
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 9.88028i − 0.330631i
\(894\) 0 0
\(895\) − 5.95976i − 0.199213i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6.97029 0.232472
\(900\) 0 0
\(901\) −11.3370 −0.377690
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 27.5447i 0.915615i
\(906\) 0 0
\(907\) 28.5782i 0.948924i 0.880276 + 0.474462i \(0.157357\pi\)
−0.880276 + 0.474462i \(0.842643\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.45969 −0.0483618 −0.0241809 0.999708i \(-0.507698\pi\)
−0.0241809 + 0.999708i \(0.507698\pi\)
\(912\) 0 0
\(913\) −5.36559 −0.177575
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 47.0058i − 1.55227i
\(918\) 0 0
\(919\) − 57.9682i − 1.91219i −0.293049 0.956097i \(-0.594670\pi\)
0.293049 0.956097i \(-0.405330\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 6.50945 0.214261
\(924\) 0 0
\(925\) 0.963460 0.0316784
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 35.4982i − 1.16466i −0.812953 0.582329i \(-0.802141\pi\)
0.812953 0.582329i \(-0.197859\pi\)
\(930\) 0 0
\(931\) − 10.3805i − 0.340208i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.17355 −0.0383793
\(936\) 0 0
\(937\) 5.78900 0.189118 0.0945592 0.995519i \(-0.469856\pi\)
0.0945592 + 0.995519i \(0.469856\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 53.8831i 1.75654i 0.478165 + 0.878270i \(0.341302\pi\)
−0.478165 + 0.878270i \(0.658698\pi\)
\(942\) 0 0
\(943\) − 54.2060i − 1.76519i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −16.7495 −0.544287 −0.272143 0.962257i \(-0.587733\pi\)
−0.272143 + 0.962257i \(0.587733\pi\)
\(948\) 0 0
\(949\) −48.4534 −1.57286
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 8.25135i 0.267288i 0.991029 + 0.133644i \(0.0426678\pi\)
−0.991029 + 0.133644i \(0.957332\pi\)
\(954\) 0 0
\(955\) − 24.2411i − 0.784425i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −0.973045 −0.0314212
\(960\) 0 0
\(961\) 30.2156 0.974695
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 19.7033i 0.634271i
\(966\) 0 0
\(967\) 35.9341i 1.15556i 0.816192 + 0.577781i \(0.196081\pi\)
−0.816192 + 0.577781i \(0.803919\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 11.0795 0.355557 0.177779 0.984070i \(-0.443109\pi\)
0.177779 + 0.984070i \(0.443109\pi\)
\(972\) 0 0
\(973\) −17.1678 −0.550375
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 12.6243i − 0.403888i −0.979397 0.201944i \(-0.935274\pi\)
0.979397 0.201944i \(-0.0647259\pi\)
\(978\) 0 0
\(979\) 9.97617i 0.318840i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −16.3442 −0.521300 −0.260650 0.965433i \(-0.583937\pi\)
−0.260650 + 0.965433i \(0.583937\pi\)
\(984\) 0 0
\(985\) −9.63068 −0.306859
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 35.8547i 1.14011i
\(990\) 0 0
\(991\) 22.4374i 0.712748i 0.934343 + 0.356374i \(0.115987\pi\)
−0.934343 + 0.356374i \(0.884013\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.59696 0.0823291
\(996\) 0 0
\(997\) 58.8152 1.86270 0.931349 0.364129i \(-0.118633\pi\)
0.931349 + 0.364129i \(0.118633\pi\)
\(998\) 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 2736.2.d.b.2015.9 24
3.2 odd 2 inner 2736.2.d.b.2015.15 yes 24
4.3 odd 2 inner 2736.2.d.b.2015.10 yes 24
12.11 even 2 inner 2736.2.d.b.2015.16 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2736.2.d.b.2015.9 24 1.1 even 1 trivial
2736.2.d.b.2015.10 yes 24 4.3 odd 2 inner
2736.2.d.b.2015.15 yes 24 3.2 odd 2 inner
2736.2.d.b.2015.16 yes 24 12.11 even 2 inner