Properties

Label 2016.2.s.n.865.1
Level $2016$
Weight $2$
Character 2016.865
Analytic conductor $16.098$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2016,2,Mod(289,2016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.s (of order \(3\), degree \(2\), 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: \(16.0978410475\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 672)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2016.865
Dual form 2016.2.s.n.289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 + 2.59808i) q^{5} +(0.500000 + 2.59808i) q^{7} +O(q^{10})\) \(q+(1.50000 + 2.59808i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-0.500000 + 0.866025i) q^{11} -4.00000 q^{13} +(2.00000 - 3.46410i) q^{17} +(4.00000 + 6.92820i) q^{23} +(-2.00000 + 3.46410i) q^{25} +7.00000 q^{29} +(-5.50000 + 9.52628i) q^{31} +(-6.00000 + 5.19615i) q^{35} +(-2.00000 - 3.46410i) q^{37} +4.00000 q^{41} -2.00000 q^{43} +(-1.00000 - 1.73205i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(-5.50000 + 9.52628i) q^{53} -3.00000 q^{55} +(3.50000 - 6.06218i) q^{59} +(-5.00000 - 8.66025i) q^{61} +(-6.00000 - 10.3923i) q^{65} +(-5.00000 + 8.66025i) q^{67} -6.00000 q^{71} +(3.00000 - 5.19615i) q^{73} +(-2.50000 - 0.866025i) q^{77} +(-5.50000 - 9.52628i) q^{79} -11.0000 q^{83} +12.0000 q^{85} +(3.00000 + 5.19615i) q^{89} +(-2.00000 - 10.3923i) q^{91} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{5} + q^{7} - q^{11} - 8 q^{13} + 4 q^{17} + 8 q^{23} - 4 q^{25} + 14 q^{29} - 11 q^{31} - 12 q^{35} - 4 q^{37} + 8 q^{41} - 4 q^{43} - 2 q^{47} - 13 q^{49} - 11 q^{53} - 6 q^{55} + 7 q^{59} - 10 q^{61} - 12 q^{65} - 10 q^{67} - 12 q^{71} + 6 q^{73} - 5 q^{77} - 11 q^{79} - 22 q^{83} + 24 q^{85} + 6 q^{89} - 4 q^{91} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(5\) 1.50000 + 2.59808i 0.670820 + 1.16190i 0.977672 + 0.210138i \(0.0673912\pi\)
−0.306851 + 0.951757i \(0.599275\pi\)
\(6\) 0 0
\(7\) 0.500000 + 2.59808i 0.188982 + 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.500000 + 0.866025i −0.150756 + 0.261116i −0.931505 0.363727i \(-0.881504\pi\)
0.780750 + 0.624844i \(0.214837\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 3.46410i 0.485071 0.840168i −0.514782 0.857321i \(-0.672127\pi\)
0.999853 + 0.0171533i \(0.00546033\pi\)
\(18\) 0 0
\(19\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 + 6.92820i 0.834058 + 1.44463i 0.894795 + 0.446476i \(0.147321\pi\)
−0.0607377 + 0.998154i \(0.519345\pi\)
\(24\) 0 0
\(25\) −2.00000 + 3.46410i −0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.00000 1.29987 0.649934 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) 0 0
\(31\) −5.50000 + 9.52628i −0.987829 + 1.71097i −0.359211 + 0.933257i \(0.616954\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.00000 + 5.19615i −1.01419 + 0.878310i
\(36\) 0 0
\(37\) −2.00000 3.46410i −0.328798 0.569495i 0.653476 0.756948i \(-0.273310\pi\)
−0.982274 + 0.187453i \(0.939977\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 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
\(47\) −1.00000 1.73205i −0.145865 0.252646i 0.783830 0.620975i \(-0.213263\pi\)
−0.929695 + 0.368329i \(0.879930\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.50000 + 9.52628i −0.755483 + 1.30854i 0.189651 + 0.981852i \(0.439264\pi\)
−0.945134 + 0.326683i \(0.894069\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.50000 6.06218i 0.455661 0.789228i −0.543065 0.839691i \(-0.682736\pi\)
0.998726 + 0.0504625i \(0.0160695\pi\)
\(60\) 0 0
\(61\) −5.00000 8.66025i −0.640184 1.10883i −0.985391 0.170305i \(-0.945525\pi\)
0.345207 0.938527i \(-0.387809\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.00000 10.3923i −0.744208 1.28901i
\(66\) 0 0
\(67\) −5.00000 + 8.66025i −0.610847 + 1.05802i 0.380251 + 0.924883i \(0.375838\pi\)
−0.991098 + 0.133135i \(0.957496\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 3.00000 5.19615i 0.351123 0.608164i −0.635323 0.772246i \(-0.719133\pi\)
0.986447 + 0.164083i \(0.0524664\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.50000 0.866025i −0.284901 0.0986928i
\(78\) 0 0
\(79\) −5.50000 9.52628i −0.618798 1.07179i −0.989705 0.143120i \(-0.954286\pi\)
0.370907 0.928670i \(-0.379047\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −11.0000 −1.20741 −0.603703 0.797209i \(-0.706309\pi\)
−0.603703 + 0.797209i \(0.706309\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 + 5.19615i 0.317999 + 0.550791i 0.980071 0.198650i \(-0.0636557\pi\)
−0.662071 + 0.749441i \(0.730322\pi\)
\(90\) 0 0
\(91\) −2.00000 10.3923i −0.209657 1.08941i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.00000 5.19615i 0.298511 0.517036i −0.677284 0.735721i \(-0.736843\pi\)
0.975796 + 0.218685i \(0.0701767\pi\)
\(102\) 0 0
\(103\) 8.00000 + 13.8564i 0.788263 + 1.36531i 0.927030 + 0.374987i \(0.122353\pi\)
−0.138767 + 0.990325i \(0.544314\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.50000 6.06218i −0.338358 0.586053i 0.645766 0.763535i \(-0.276538\pi\)
−0.984124 + 0.177482i \(0.943205\pi\)
\(108\) 0 0
\(109\) −5.00000 + 8.66025i −0.478913 + 0.829502i −0.999708 0.0241802i \(-0.992302\pi\)
0.520794 + 0.853682i \(0.325636\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) −12.0000 + 20.7846i −1.11901 + 1.93817i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 10.0000 + 3.46410i 0.916698 + 0.317554i
\(120\) 0 0
\(121\) 5.00000 + 8.66025i 0.454545 + 0.787296i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 17.0000 1.50851 0.754253 0.656584i \(-0.227999\pi\)
0.754253 + 0.656584i \(0.227999\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.50000 + 2.59808i 0.131056 + 0.226995i 0.924084 0.382190i \(-0.124830\pi\)
−0.793028 + 0.609185i \(0.791497\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.00000 + 1.73205i −0.0854358 + 0.147979i −0.905577 0.424182i \(-0.860562\pi\)
0.820141 + 0.572161i \(0.193895\pi\)
\(138\) 0 0
\(139\) −22.0000 −1.86602 −0.933008 0.359856i \(-0.882826\pi\)
−0.933008 + 0.359856i \(0.882826\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00000 3.46410i 0.167248 0.289683i
\(144\) 0 0
\(145\) 10.5000 + 18.1865i 0.871978 + 1.51031i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.00000 5.19615i −0.245770 0.425685i 0.716578 0.697507i \(-0.245707\pi\)
−0.962348 + 0.271821i \(0.912374\pi\)
\(150\) 0 0
\(151\) −5.50000 + 9.52628i −0.447584 + 0.775238i −0.998228 0.0595022i \(-0.981049\pi\)
0.550645 + 0.834740i \(0.314382\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −33.0000 −2.65062
\(156\) 0 0
\(157\) 6.00000 10.3923i 0.478852 0.829396i −0.520854 0.853646i \(-0.674386\pi\)
0.999706 + 0.0242497i \(0.00771967\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −16.0000 + 13.8564i −1.26098 + 1.09204i
\(162\) 0 0
\(163\) 4.00000 + 6.92820i 0.313304 + 0.542659i 0.979076 0.203497i \(-0.0652307\pi\)
−0.665771 + 0.746156i \(0.731897\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.0000 1.70241 0.851206 0.524832i \(-0.175872\pi\)
0.851206 + 0.524832i \(0.175872\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.00000 5.19615i −0.228086 0.395056i 0.729155 0.684349i \(-0.239913\pi\)
−0.957241 + 0.289292i \(0.906580\pi\)
\(174\) 0 0
\(175\) −10.0000 3.46410i −0.755929 0.261861i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.00000 + 10.3923i −0.448461 + 0.776757i −0.998286 0.0585225i \(-0.981361\pi\)
0.549825 + 0.835280i \(0.314694\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.00000 10.3923i 0.441129 0.764057i
\(186\) 0 0
\(187\) 2.00000 + 3.46410i 0.146254 + 0.253320i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 + 20.7846i 0.868290 + 1.50392i 0.863743 + 0.503932i \(0.168114\pi\)
0.00454614 + 0.999990i \(0.498553\pi\)
\(192\) 0 0
\(193\) 9.50000 16.4545i 0.683825 1.18442i −0.289980 0.957033i \(-0.593649\pi\)
0.973805 0.227387i \(-0.0730182\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 10.0000 17.3205i 0.708881 1.22782i −0.256391 0.966573i \(-0.582534\pi\)
0.965272 0.261245i \(-0.0841331\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.50000 + 18.1865i 0.245652 + 1.27644i
\(204\) 0 0
\(205\) 6.00000 + 10.3923i 0.419058 + 0.725830i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.00000 5.19615i −0.204598 0.354375i
\(216\) 0 0
\(217\) −27.5000 9.52628i −1.86682 0.646686i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −8.00000 + 13.8564i −0.538138 + 0.932083i
\(222\) 0 0
\(223\) 9.00000 0.602685 0.301342 0.953516i \(-0.402565\pi\)
0.301342 + 0.953516i \(0.402565\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.50000 + 6.06218i −0.232303 + 0.402361i −0.958485 0.285141i \(-0.907959\pi\)
0.726182 + 0.687502i \(0.241293\pi\)
\(228\) 0 0
\(229\) 4.00000 + 6.92820i 0.264327 + 0.457829i 0.967387 0.253302i \(-0.0815167\pi\)
−0.703060 + 0.711131i \(0.748183\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 + 10.3923i 0.393073 + 0.680823i 0.992853 0.119342i \(-0.0380786\pi\)
−0.599780 + 0.800165i \(0.704745\pi\)
\(234\) 0 0
\(235\) 3.00000 5.19615i 0.195698 0.338960i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 12.5000 21.6506i 0.805196 1.39464i −0.110963 0.993825i \(-0.535394\pi\)
0.916159 0.400815i \(-0.131273\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −16.5000 12.9904i −1.05415 0.829925i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 25.0000 1.57799 0.788993 0.614402i \(-0.210603\pi\)
0.788993 + 0.614402i \(0.210603\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.00000 + 12.1244i 0.436648 + 0.756297i 0.997429 0.0716680i \(-0.0228322\pi\)
−0.560781 + 0.827964i \(0.689499\pi\)
\(258\) 0 0
\(259\) 8.00000 6.92820i 0.497096 0.430498i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.0000 22.5167i 0.801614 1.38844i −0.116939 0.993139i \(-0.537308\pi\)
0.918553 0.395298i \(-0.129359\pi\)
\(264\) 0 0
\(265\) −33.0000 −2.02717
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.5000 18.1865i 0.640196 1.10885i −0.345192 0.938532i \(-0.612186\pi\)
0.985389 0.170321i \(-0.0544803\pi\)
\(270\) 0 0
\(271\) 0.500000 + 0.866025i 0.0303728 + 0.0526073i 0.880812 0.473466i \(-0.156997\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.00000 3.46410i −0.120605 0.208893i
\(276\) 0 0
\(277\) −16.0000 + 27.7128i −0.961347 + 1.66510i −0.242222 + 0.970221i \(0.577876\pi\)
−0.719125 + 0.694881i \(0.755457\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 1.00000 1.73205i 0.0594438 0.102960i −0.834772 0.550596i \(-0.814401\pi\)
0.894216 + 0.447636i \(0.147734\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.00000 + 10.3923i 0.118056 + 0.613438i
\(288\) 0 0
\(289\) 0.500000 + 0.866025i 0.0294118 + 0.0509427i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.00000 0.408944 0.204472 0.978872i \(-0.434452\pi\)
0.204472 + 0.978872i \(0.434452\pi\)
\(294\) 0 0
\(295\) 21.0000 1.22267
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −16.0000 27.7128i −0.925304 1.60267i
\(300\) 0 0
\(301\) −1.00000 5.19615i −0.0576390 0.299501i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 15.0000 25.9808i 0.858898 1.48765i
\(306\) 0 0
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.00000 10.3923i 0.340229 0.589294i −0.644246 0.764818i \(-0.722829\pi\)
0.984475 + 0.175525i \(0.0561621\pi\)
\(312\) 0 0
\(313\) −0.500000 0.866025i −0.0282617 0.0489506i 0.851549 0.524276i \(-0.175664\pi\)
−0.879810 + 0.475325i \(0.842331\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.50000 + 14.7224i 0.477408 + 0.826894i 0.999665 0.0258939i \(-0.00824321\pi\)
−0.522257 + 0.852788i \(0.674910\pi\)
\(318\) 0 0
\(319\) −3.50000 + 6.06218i −0.195962 + 0.339417i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 8.00000 13.8564i 0.443760 0.768615i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.00000 3.46410i 0.220527 0.190982i
\(330\) 0 0
\(331\) −6.00000 10.3923i −0.329790 0.571213i 0.652680 0.757634i \(-0.273645\pi\)
−0.982470 + 0.186421i \(0.940311\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −30.0000 −1.63908
\(336\) 0 0
\(337\) −15.0000 −0.817102 −0.408551 0.912735i \(-0.633966\pi\)
−0.408551 + 0.912735i \(0.633966\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −5.50000 9.52628i −0.297842 0.515877i
\(342\) 0 0
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.00000 10.3923i 0.322097 0.557888i −0.658824 0.752297i \(-0.728946\pi\)
0.980921 + 0.194409i \(0.0622790\pi\)
\(348\) 0 0
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.0000 + 20.7846i −0.638696 + 1.10625i 0.347024 + 0.937856i \(0.387192\pi\)
−0.985719 + 0.168397i \(0.946141\pi\)
\(354\) 0 0
\(355\) −9.00000 15.5885i −0.477670 0.827349i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.00000 + 1.73205i 0.0527780 + 0.0914141i 0.891207 0.453596i \(-0.149859\pi\)
−0.838429 + 0.545010i \(0.816526\pi\)
\(360\) 0 0
\(361\) 9.50000 16.4545i 0.500000 0.866025i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 18.0000 0.942163
\(366\) 0 0
\(367\) −8.50000 + 14.7224i −0.443696 + 0.768505i −0.997960 0.0638362i \(-0.979666\pi\)
0.554264 + 0.832341i \(0.313000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −27.5000 9.52628i −1.42773 0.494580i
\(372\) 0 0
\(373\) 6.00000 + 10.3923i 0.310668 + 0.538093i 0.978507 0.206213i \(-0.0661139\pi\)
−0.667839 + 0.744306i \(0.732781\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −28.0000 −1.44207
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.00000 + 1.73205i 0.0510976 + 0.0885037i 0.890443 0.455095i \(-0.150395\pi\)
−0.839345 + 0.543599i \(0.817061\pi\)
\(384\) 0 0
\(385\) −1.50000 7.79423i −0.0764471 0.397231i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.00000 + 5.19615i −0.152106 + 0.263455i −0.932002 0.362454i \(-0.881939\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(390\) 0 0
\(391\) 32.0000 1.61831
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 16.5000 28.5788i 0.830205 1.43796i
\(396\) 0 0
\(397\) 2.00000 + 3.46410i 0.100377 + 0.173858i 0.911840 0.410546i \(-0.134662\pi\)
−0.811463 + 0.584404i \(0.801328\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8.00000 + 13.8564i 0.399501 + 0.691956i 0.993664 0.112388i \(-0.0358501\pi\)
−0.594163 + 0.804344i \(0.702517\pi\)
\(402\) 0 0
\(403\) 22.0000 38.1051i 1.09590 1.89815i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) −3.50000 + 6.06218i −0.173064 + 0.299755i −0.939490 0.342578i \(-0.888700\pi\)
0.766426 + 0.642333i \(0.222033\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 17.5000 + 6.06218i 0.861119 + 0.298300i
\(414\) 0 0
\(415\) −16.5000 28.5788i −0.809953 1.40288i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 8.00000 + 13.8564i 0.388057 + 0.672134i
\(426\) 0 0
\(427\) 20.0000 17.3205i 0.967868 0.838198i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.00000 6.92820i 0.192673 0.333720i −0.753462 0.657491i \(-0.771618\pi\)
0.946135 + 0.323772i \(0.104951\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −7.50000 12.9904i −0.357955 0.619997i 0.629664 0.776868i \(-0.283193\pi\)
−0.987619 + 0.156871i \(0.949859\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.5000 18.1865i −0.498870 0.864068i 0.501129 0.865373i \(-0.332918\pi\)
−0.999999 + 0.00130426i \(0.999585\pi\)
\(444\) 0 0
\(445\) −9.00000 + 15.5885i −0.426641 + 0.738964i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) −2.00000 + 3.46410i −0.0941763 + 0.163118i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 24.0000 20.7846i 1.12514 0.974398i
\(456\) 0 0
\(457\) 8.50000 + 14.7224i 0.397613 + 0.688686i 0.993431 0.114433i \(-0.0365053\pi\)
−0.595818 + 0.803120i \(0.703172\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.00000 + 10.3923i 0.277647 + 0.480899i 0.970799 0.239892i \(-0.0771121\pi\)
−0.693153 + 0.720791i \(0.743779\pi\)
\(468\) 0 0
\(469\) −25.0000 8.66025i −1.15439 0.399893i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.00000 1.73205i 0.0459800 0.0796398i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.00000 5.19615i 0.137073 0.237418i −0.789314 0.613990i \(-0.789564\pi\)
0.926388 + 0.376571i \(0.122897\pi\)
\(480\) 0 0
\(481\) 8.00000 + 13.8564i 0.364769 + 0.631798i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 10.5000 + 18.1865i 0.476780 + 0.825808i
\(486\) 0 0
\(487\) 1.50000 2.59808i 0.0679715 0.117730i −0.830037 0.557709i \(-0.811681\pi\)
0.898008 + 0.439979i \(0.145014\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 37.0000 1.66979 0.834893 0.550412i \(-0.185529\pi\)
0.834893 + 0.550412i \(0.185529\pi\)
\(492\) 0 0
\(493\) 14.0000 24.2487i 0.630528 1.09211i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.00000 15.5885i −0.134568 0.699238i
\(498\) 0 0
\(499\) −5.00000 8.66025i −0.223831 0.387686i 0.732137 0.681157i \(-0.238523\pi\)
−0.955968 + 0.293471i \(0.905190\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −32.0000 −1.42681 −0.713405 0.700752i \(-0.752848\pi\)
−0.713405 + 0.700752i \(0.752848\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −21.5000 37.2391i −0.952971 1.65059i −0.738945 0.673766i \(-0.764676\pi\)
−0.214026 0.976828i \(-0.568658\pi\)
\(510\) 0 0
\(511\) 15.0000 + 5.19615i 0.663561 + 0.229864i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −24.0000 + 41.5692i −1.05757 + 1.83176i
\(516\) 0 0
\(517\) 2.00000 0.0879599
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.00000 12.1244i 0.306676 0.531178i −0.670957 0.741496i \(-0.734117\pi\)
0.977633 + 0.210318i \(0.0674500\pi\)
\(522\) 0 0
\(523\) 10.0000 + 17.3205i 0.437269 + 0.757373i 0.997478 0.0709788i \(-0.0226123\pi\)
−0.560208 + 0.828352i \(0.689279\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.0000 + 38.1051i 0.958335 + 1.65989i
\(528\) 0 0
\(529\) −20.5000 + 35.5070i −0.891304 + 1.54378i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −16.0000 −0.693037
\(534\) 0 0
\(535\) 10.5000 18.1865i 0.453955 0.786272i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.00000 6.92820i 0.0430730 0.298419i
\(540\) 0 0
\(541\) 15.0000 + 25.9808i 0.644900 + 1.11700i 0.984325 + 0.176367i \(0.0564345\pi\)
−0.339424 + 0.940633i \(0.610232\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −30.0000 −1.28506
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 22.0000 19.0526i 0.935535 0.810197i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.50000 2.59808i 0.0635570 0.110084i −0.832496 0.554031i \(-0.813089\pi\)
0.896053 + 0.443947i \(0.146422\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.50000 16.4545i 0.400377 0.693474i −0.593394 0.804912i \(-0.702212\pi\)
0.993771 + 0.111438i \(0.0355457\pi\)
\(564\) 0 0
\(565\) −18.0000 31.1769i −0.757266 1.31162i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(570\) 0 0
\(571\) 11.0000 19.0526i 0.460336 0.797325i −0.538642 0.842535i \(-0.681062\pi\)
0.998978 + 0.0452101i \(0.0143957\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −32.0000 −1.33449
\(576\) 0 0
\(577\) 8.50000 14.7224i 0.353860 0.612903i −0.633062 0.774101i \(-0.718202\pi\)
0.986922 + 0.161198i \(0.0515357\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5.50000 28.5788i −0.228178 1.18565i
\(582\) 0 0
\(583\) −5.50000 9.52628i −0.227787 0.394538i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15.0000 0.619116 0.309558 0.950881i \(-0.399819\pi\)
0.309558 + 0.950881i \(0.399819\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 6.00000 + 31.1769i 0.245976 + 1.27813i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 9.00000 15.5885i 0.367730 0.636927i −0.621480 0.783430i \(-0.713468\pi\)
0.989210 + 0.146503i \(0.0468017\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −15.0000 + 25.9808i −0.609837 + 1.05627i
\(606\) 0 0
\(607\) 13.5000 + 23.3827i 0.547948 + 0.949074i 0.998415 + 0.0562808i \(0.0179242\pi\)
−0.450467 + 0.892793i \(0.648742\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.00000 + 6.92820i 0.161823 + 0.280285i
\(612\) 0 0
\(613\) 8.00000 13.8564i 0.323117 0.559655i −0.658012 0.753007i \(-0.728603\pi\)
0.981129 + 0.193352i \(0.0619359\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) 11.0000 19.0526i 0.442127 0.765787i −0.555720 0.831370i \(-0.687557\pi\)
0.997847 + 0.0655827i \(0.0208906\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.0000 + 10.3923i −0.480770 + 0.416359i
\(624\) 0 0
\(625\) 14.5000 + 25.1147i 0.580000 + 1.00459i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) −43.0000 −1.71180 −0.855901 0.517139i \(-0.826997\pi\)
−0.855901 + 0.517139i \(0.826997\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 25.5000 + 44.1673i 1.01194 + 1.75273i
\(636\) 0 0
\(637\) 26.0000 10.3923i 1.03016 0.411758i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −5.00000 + 8.66025i −0.197488 + 0.342059i −0.947713 0.319123i \(-0.896612\pi\)
0.750225 + 0.661182i \(0.229945\pi\)
\(642\) 0 0
\(643\) 2.00000 0.0788723 0.0394362 0.999222i \(-0.487444\pi\)
0.0394362 + 0.999222i \(0.487444\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.0000 + 25.9808i −0.589711 + 1.02141i 0.404559 + 0.914512i \(0.367425\pi\)
−0.994270 + 0.106897i \(0.965908\pi\)
\(648\) 0 0
\(649\) 3.50000 + 6.06218i 0.137387 + 0.237961i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 21.5000 + 37.2391i 0.841360 + 1.45728i 0.888745 + 0.458402i \(0.151578\pi\)
−0.0473852 + 0.998877i \(0.515089\pi\)
\(654\) 0 0
\(655\) −4.50000 + 7.79423i −0.175830 + 0.304546i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 21.0000 36.3731i 0.816805 1.41475i −0.0912190 0.995831i \(-0.529076\pi\)
0.908024 0.418917i \(-0.137590\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 28.0000 + 48.4974i 1.08416 + 1.87783i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) 45.0000 1.73462 0.867311 0.497766i \(-0.165846\pi\)
0.867311 + 0.497766i \(0.165846\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7.50000 + 12.9904i 0.288248 + 0.499261i 0.973392 0.229147i \(-0.0735938\pi\)
−0.685143 + 0.728408i \(0.740260\pi\)
\(678\) 0 0
\(679\) 3.50000 + 18.1865i 0.134318 + 0.697935i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10.5000 18.1865i 0.401771 0.695888i −0.592168 0.805814i \(-0.701728\pi\)
0.993940 + 0.109926i \(0.0350613\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 22.0000 38.1051i 0.838133 1.45169i
\(690\) 0 0
\(691\) −10.0000 17.3205i −0.380418 0.658903i 0.610704 0.791859i \(-0.290887\pi\)
−0.991122 + 0.132956i \(0.957553\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −33.0000 57.1577i −1.25176 2.16811i
\(696\) 0 0
\(697\) 8.00000 13.8564i 0.303022 0.524849i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −33.0000 −1.24639 −0.623196 0.782065i \(-0.714166\pi\)
−0.623196 + 0.782065i \(0.714166\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.0000 + 5.19615i 0.564133 + 0.195421i
\(708\) 0 0
\(709\) 17.0000 + 29.4449i 0.638448 + 1.10583i 0.985773 + 0.168080i \(0.0537568\pi\)
−0.347325 + 0.937745i \(0.612910\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −88.0000 −3.29563
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −9.00000 15.5885i −0.335643 0.581351i 0.647965 0.761670i \(-0.275620\pi\)
−0.983608 + 0.180319i \(0.942287\pi\)
\(720\) 0 0
\(721\) −32.0000 + 27.7128i −1.19174 + 1.03208i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −14.0000 + 24.2487i −0.519947 + 0.900575i
\(726\) 0 0
\(727\) 7.00000 0.259616 0.129808 0.991539i \(-0.458564\pi\)
0.129808 + 0.991539i \(0.458564\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −4.00000 + 6.92820i −0.147945 + 0.256249i
\(732\) 0 0
\(733\) −7.00000 12.1244i −0.258551 0.447823i 0.707303 0.706910i \(-0.249912\pi\)
−0.965854 + 0.259087i \(0.916578\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.00000 8.66025i −0.184177 0.319005i
\(738\) 0 0
\(739\) −17.0000 + 29.4449i −0.625355 + 1.08315i 0.363117 + 0.931744i \(0.381713\pi\)
−0.988472 + 0.151403i \(0.951621\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 50.0000 1.83432 0.917161 0.398517i \(-0.130475\pi\)
0.917161 + 0.398517i \(0.130475\pi\)
\(744\) 0 0
\(745\) 9.00000 15.5885i 0.329734 0.571117i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 14.0000 12.1244i 0.511549 0.443014i
\(750\) 0 0
\(751\) 1.50000 + 2.59808i 0.0547358 + 0.0948051i 0.892095 0.451848i \(-0.149235\pi\)
−0.837359 + 0.546653i \(0.815902\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −33.0000 −1.20099
\(756\) 0 0
\(757\) −14.0000 −0.508839 −0.254419 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 10.3923i −0.217500 0.376721i 0.736543 0.676391i \(-0.236457\pi\)
−0.954043 + 0.299670i \(0.903123\pi\)
\(762\) 0 0
\(763\) −25.0000 8.66025i −0.905061 0.313522i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −14.0000 + 24.2487i −0.505511 + 0.875570i
\(768\) 0 0
\(769\) −3.00000 −0.108183 −0.0540914 0.998536i \(-0.517226\pi\)
−0.0540914 + 0.998536i \(0.517226\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 15.0000 25.9808i 0.539513 0.934463i −0.459418 0.888220i \(-0.651942\pi\)
0.998930 0.0462427i \(-0.0147248\pi\)
\(774\) 0 0
\(775\) −22.0000 38.1051i −0.790263 1.36878i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 3.00000 5.19615i 0.107348 0.185933i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 36.0000 1.28490
\(786\) 0 0
\(787\) 17.0000 29.4449i 0.605985 1.04960i −0.385911 0.922536i \(-0.626113\pi\)
0.991895 0.127060i \(-0.0405540\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 31.1769i −0.213335 1.10852i
\(792\) 0 0
\(793\) 20.0000 + 34.6410i 0.710221 + 1.23014i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 25.0000 0.885545 0.442773 0.896634i \(-0.353995\pi\)
0.442773 + 0.896634i \(0.353995\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.00000 + 5.19615i 0.105868 + 0.183368i
\(804\) 0 0
\(805\) −60.0000 20.7846i −2.11472 0.732561i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 20.0000 34.6410i 0.703163 1.21791i −0.264188 0.964471i \(-0.585104\pi\)
0.967351 0.253442i \(-0.0815627\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −12.0000 + 20.7846i −0.420342 + 0.728053i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 18.5000 + 32.0429i 0.645654 + 1.11831i 0.984150 + 0.177338i \(0.0567487\pi\)
−0.338495 + 0.940968i \(0.609918\pi\)
\(822\) 0 0
\(823\) 8.00000 13.8564i 0.278862 0.483004i −0.692240 0.721668i \(-0.743376\pi\)
0.971102 + 0.238664i \(0.0767093\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 21.0000 0.730242 0.365121 0.930960i \(-0.381028\pi\)
0.365121 + 0.930960i \(0.381028\pi\)
\(828\) 0 0
\(829\) −12.0000 + 20.7846i −0.416777 + 0.721879i −0.995613 0.0935647i \(-0.970174\pi\)
0.578836 + 0.815444i \(0.303507\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.00000 + 27.7128i −0.138592 + 0.960192i
\(834\) 0 0
\(835\) 33.0000 + 57.1577i 1.14201 + 1.97802i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.50000 + 7.79423i 0.154805 + 0.268130i
\(846\) 0 0
\(847\) −20.0000 + 17.3205i −0.687208 + 0.595140i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 16.0000 27.7128i 0.548473 0.949983i
\(852\) 0 0
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21.0000 36.3731i 0.717346 1.24248i −0.244701 0.969599i \(-0.578690\pi\)
0.962048 0.272882i \(-0.0879768\pi\)
\(858\) 0 0
\(859\) −17.0000 29.4449i −0.580033 1.00465i −0.995475 0.0950262i \(-0.969707\pi\)
0.415442 0.909620i \(-0.363627\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −9.00000 15.5885i −0.306364 0.530637i 0.671200 0.741276i \(-0.265779\pi\)
−0.977564 + 0.210639i \(0.932446\pi\)
\(864\) 0 0
\(865\) 9.00000 15.5885i 0.306009 0.530023i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 11.0000 0.373149
\(870\) 0 0
\(871\) 20.0000 34.6410i 0.677674 1.17377i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.50000 + 7.79423i 0.0507093 + 0.263493i
\(876\) 0 0
\(877\) 16.0000 + 27.7128i 0.540282 + 0.935795i 0.998888 + 0.0471555i \(0.0150156\pi\)
−0.458606 + 0.888640i \(0.651651\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −4.00000 6.92820i −0.134307 0.232626i 0.791026 0.611783i \(-0.209547\pi\)
−0.925332 + 0.379157i \(0.876214\pi\)
\(888\) 0 0
\(889\) 8.50000 + 44.1673i 0.285081 + 1.48132i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −36.0000 −1.20335
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −38.5000 + 66.6840i −1.28405 + 2.22403i
\(900\) 0 0
\(901\) 22.0000 + 38.1051i 0.732926 + 1.26947i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18.0000 + 31.1769i 0.598340 + 1.03636i
\(906\) 0 0
\(907\) −16.0000 + 27.7128i −0.531271 + 0.920189i 0.468063 + 0.883695i \(0.344952\pi\)
−0.999334 + 0.0364935i \(0.988381\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −58.0000 −1.92163 −0.960813 0.277198i \(-0.910594\pi\)
−0.960813 + 0.277198i \(0.910594\pi\)
\(912\) 0 0
\(913\) 5.50000 9.52628i 0.182023 0.315274i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −6.00000 + 5.19615i −0.198137 + 0.171592i
\(918\) 0 0
\(919\) −4.00000 6.92820i −0.131948 0.228540i 0.792480 0.609898i \(-0.208790\pi\)
−0.924427 + 0.381358i \(0.875456\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 16.0000 0.526077
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 9.00000 + 15.5885i 0.295280 + 0.511441i 0.975050 0.221985i \(-0.0712536\pi\)
−0.679770 + 0.733426i \(0.737920\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6.00000 + 10.3923i −0.196221 + 0.339865i
\(936\) 0 0
\(937\) −13.0000 −0.424691 −0.212346 0.977195i \(-0.568110\pi\)
−0.212346 + 0.977195i \(0.568110\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −24.5000 + 42.4352i −0.798677 + 1.38335i 0.121801 + 0.992555i \(0.461133\pi\)
−0.920478 + 0.390795i \(0.872200\pi\)
\(942\) 0 0
\(943\) 16.0000 + 27.7128i 0.521032 + 0.902453i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −20.0000 34.6410i −0.649913 1.12568i −0.983143 0.182836i \(-0.941472\pi\)
0.333231 0.942845i \(-0.391861\pi\)
\(948\) 0 0
\(949\) −12.0000 + 20.7846i −0.389536 + 0.674697i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2.00000 −0.0647864 −0.0323932 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 0 0
\(955\) −36.0000 + 62.3538i −1.16493 + 2.01772i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −5.00000 1.73205i −0.161458 0.0559308i
\(960\) 0 0
\(961\) −45.0000 77.9423i −1.45161 2.51427i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 57.0000 1.83489
\(966\) 0 0
\(967\) −5.00000 −0.160789 −0.0803946 0.996763i \(-0.525618\pi\)
−0.0803946 + 0.996763i \(0.525618\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −13.5000 23.3827i −0.433236 0.750386i 0.563914 0.825833i \(-0.309295\pi\)
−0.997150 + 0.0754473i \(0.975962\pi\)
\(972\) 0 0
\(973\) −11.0000 57.1577i −0.352644 1.83239i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 13.0000 22.5167i 0.415907 0.720372i −0.579616 0.814890i \(-0.696798\pi\)
0.995523 + 0.0945177i \(0.0301309\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −24.0000 + 41.5692i −0.765481 + 1.32585i 0.174511 + 0.984655i \(0.444166\pi\)
−0.939992 + 0.341197i \(0.889168\pi\)
\(984\) 0 0
\(985\) −9.00000 15.5885i −0.286764 0.496690i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.00000 13.8564i −0.254385 0.440608i
\(990\) 0 0
\(991\) −11.5000 + 19.9186i −0.365310 + 0.632735i −0.988826 0.149076i \(-0.952370\pi\)
0.623516 + 0.781810i \(0.285704\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 60.0000 1.90213
\(996\) 0 0
\(997\) −1.00000 + 1.73205i −0.0316703 + 0.0548546i −0.881426 0.472322i \(-0.843416\pi\)
0.849756 + 0.527176i \(0.176749\pi\)
\(998\) 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 2016.2.s.n.865.1 2
3.2 odd 2 672.2.q.a.193.1 2
4.3 odd 2 2016.2.s.k.865.1 2
7.2 even 3 inner 2016.2.s.n.289.1 2
12.11 even 2 672.2.q.f.193.1 yes 2
21.2 odd 6 672.2.q.a.289.1 yes 2
21.11 odd 6 4704.2.a.bf.1.1 1
21.17 even 6 4704.2.a.b.1.1 1
24.5 odd 2 1344.2.q.u.193.1 2
24.11 even 2 1344.2.q.k.193.1 2
28.23 odd 6 2016.2.s.k.289.1 2
84.11 even 6 4704.2.a.o.1.1 1
84.23 even 6 672.2.q.f.289.1 yes 2
84.59 odd 6 4704.2.a.s.1.1 1
168.11 even 6 9408.2.a.bt.1.1 1
168.53 odd 6 9408.2.a.e.1.1 1
168.59 odd 6 9408.2.a.bl.1.1 1
168.101 even 6 9408.2.a.dc.1.1 1
168.107 even 6 1344.2.q.k.961.1 2
168.149 odd 6 1344.2.q.u.961.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.a.193.1 2 3.2 odd 2
672.2.q.a.289.1 yes 2 21.2 odd 6
672.2.q.f.193.1 yes 2 12.11 even 2
672.2.q.f.289.1 yes 2 84.23 even 6
1344.2.q.k.193.1 2 24.11 even 2
1344.2.q.k.961.1 2 168.107 even 6
1344.2.q.u.193.1 2 24.5 odd 2
1344.2.q.u.961.1 2 168.149 odd 6
2016.2.s.k.289.1 2 28.23 odd 6
2016.2.s.k.865.1 2 4.3 odd 2
2016.2.s.n.289.1 2 7.2 even 3 inner
2016.2.s.n.865.1 2 1.1 even 1 trivial
4704.2.a.b.1.1 1 21.17 even 6
4704.2.a.o.1.1 1 84.11 even 6
4704.2.a.s.1.1 1 84.59 odd 6
4704.2.a.bf.1.1 1 21.11 odd 6
9408.2.a.e.1.1 1 168.53 odd 6
9408.2.a.bl.1.1 1 168.59 odd 6
9408.2.a.bt.1.1 1 168.11 even 6
9408.2.a.dc.1.1 1 168.101 even 6