Properties

Label 1568.2.i.r.1537.2
Level $1568$
Weight $2$
Character 1568.1537
Analytic conductor $12.521$
Analytic rank $0$
Dimension $4$
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] = [1568,2,Mod(961,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1568.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.i (of order \(3\), degree \(2\), 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: \(12.5205430369\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1537.2
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1568.1537
Dual form 1568.2.i.r.961.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 - 1.22474i) q^{3} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.707107 - 1.22474i) q^{3} +(0.500000 + 0.866025i) q^{9} +(1.00000 - 1.73205i) q^{11} +2.82843 q^{13} +(-2.12132 + 3.67423i) q^{17} +(-2.12132 - 3.67423i) q^{19} +(4.00000 + 6.92820i) q^{23} +(2.50000 - 4.33013i) q^{25} +5.65685 q^{27} +6.00000 q^{29} +(4.24264 - 7.34847i) q^{31} +(-1.41421 - 2.44949i) q^{33} +(1.00000 + 1.73205i) q^{37} +(2.00000 - 3.46410i) q^{39} -4.24264 q^{41} -6.00000 q^{43} +(-1.41421 - 2.44949i) q^{47} +(3.00000 + 5.19615i) q^{51} +(-3.00000 + 5.19615i) q^{53} -6.00000 q^{57} +(6.36396 - 11.0227i) q^{59} +(2.82843 + 4.89898i) q^{61} +(6.00000 - 10.3923i) q^{67} +11.3137 q^{69} +4.00000 q^{71} +(0.707107 - 1.22474i) q^{73} +(-3.53553 - 6.12372i) q^{75} +(6.00000 + 10.3923i) q^{79} +(2.50000 - 4.33013i) q^{81} +9.89949 q^{83} +(4.24264 - 7.34847i) q^{87} +(-2.12132 - 3.67423i) q^{89} +(-6.00000 - 10.3923i) q^{93} -18.3848 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} + 4 q^{11} + 16 q^{23} + 10 q^{25} + 24 q^{29} + 4 q^{37} + 8 q^{39} - 24 q^{43} + 12 q^{51} - 12 q^{53} - 24 q^{57} + 24 q^{67} + 16 q^{71} + 24 q^{79} + 10 q^{81} - 24 q^{93} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0.707107 1.22474i 0.408248 0.707107i −0.586445 0.809989i \(-0.699473\pi\)
0.994694 + 0.102882i \(0.0328064\pi\)
\(4\) 0 0
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) 1.00000 1.73205i 0.301511 0.522233i −0.674967 0.737848i \(-0.735842\pi\)
0.976478 + 0.215615i \(0.0691756\pi\)
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.12132 + 3.67423i −0.514496 + 0.891133i 0.485363 + 0.874313i \(0.338688\pi\)
−0.999859 + 0.0168199i \(0.994646\pi\)
\(18\) 0 0
\(19\) −2.12132 3.67423i −0.486664 0.842927i 0.513218 0.858258i \(-0.328453\pi\)
−0.999882 + 0.0153309i \(0.995120\pi\)
\(20\) 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
\(25\) 2.50000 4.33013i 0.500000 0.866025i
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.24264 7.34847i 0.762001 1.31982i −0.179817 0.983700i \(-0.557551\pi\)
0.941818 0.336124i \(-0.109116\pi\)
\(32\) 0 0
\(33\) −1.41421 2.44949i −0.246183 0.426401i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 + 1.73205i 0.164399 + 0.284747i 0.936442 0.350823i \(-0.114098\pi\)
−0.772043 + 0.635571i \(0.780765\pi\)
\(38\) 0 0
\(39\) 2.00000 3.46410i 0.320256 0.554700i
\(40\) 0 0
\(41\) −4.24264 −0.662589 −0.331295 0.943527i \(-0.607485\pi\)
−0.331295 + 0.943527i \(0.607485\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.41421 2.44949i −0.206284 0.357295i 0.744257 0.667893i \(-0.232804\pi\)
−0.950541 + 0.310599i \(0.899470\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 3.00000 + 5.19615i 0.420084 + 0.727607i
\(52\) 0 0
\(53\) −3.00000 + 5.19615i −0.412082 + 0.713746i −0.995117 0.0987002i \(-0.968532\pi\)
0.583036 + 0.812447i \(0.301865\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) 6.36396 11.0227i 0.828517 1.43503i −0.0706842 0.997499i \(-0.522518\pi\)
0.899201 0.437535i \(-0.144148\pi\)
\(60\) 0 0
\(61\) 2.82843 + 4.89898i 0.362143 + 0.627250i 0.988313 0.152436i \(-0.0487119\pi\)
−0.626170 + 0.779686i \(0.715379\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.00000 10.3923i 0.733017 1.26962i −0.222571 0.974916i \(-0.571445\pi\)
0.955588 0.294706i \(-0.0952216\pi\)
\(68\) 0 0
\(69\) 11.3137 1.36201
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 0.707107 1.22474i 0.0827606 0.143346i −0.821674 0.569958i \(-0.806960\pi\)
0.904435 + 0.426612i \(0.140293\pi\)
\(74\) 0 0
\(75\) −3.53553 6.12372i −0.408248 0.707107i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.00000 + 10.3923i 0.675053 + 1.16923i 0.976453 + 0.215728i \(0.0692125\pi\)
−0.301401 + 0.953498i \(0.597454\pi\)
\(80\) 0 0
\(81\) 2.50000 4.33013i 0.277778 0.481125i
\(82\) 0 0
\(83\) 9.89949 1.08661 0.543305 0.839535i \(-0.317173\pi\)
0.543305 + 0.839535i \(0.317173\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.24264 7.34847i 0.454859 0.787839i
\(88\) 0 0
\(89\) −2.12132 3.67423i −0.224860 0.389468i 0.731418 0.681930i \(-0.238859\pi\)
−0.956277 + 0.292462i \(0.905526\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −6.00000 10.3923i −0.622171 1.07763i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.3848 −1.86669 −0.933346 0.358979i \(-0.883125\pi\)
−0.933346 + 0.358979i \(0.883125\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 8.48528 14.6969i 0.844317 1.46240i −0.0418959 0.999122i \(-0.513340\pi\)
0.886213 0.463278i \(-0.153327\pi\)
\(102\) 0 0
\(103\) −4.24264 7.34847i −0.418040 0.724066i 0.577702 0.816247i \(-0.303949\pi\)
−0.995742 + 0.0921813i \(0.970616\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.00000 + 3.46410i 0.193347 + 0.334887i 0.946357 0.323122i \(-0.104732\pi\)
−0.753010 + 0.658009i \(0.771399\pi\)
\(108\) 0 0
\(109\) −9.00000 + 15.5885i −0.862044 + 1.49310i 0.00790932 + 0.999969i \(0.497482\pi\)
−0.869953 + 0.493135i \(0.835851\pi\)
\(110\) 0 0
\(111\) 2.82843 0.268462
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.41421 + 2.44949i 0.130744 + 0.226455i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.50000 + 6.06218i 0.318182 + 0.551107i
\(122\) 0 0
\(123\) −3.00000 + 5.19615i −0.270501 + 0.468521i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 0 0
\(129\) −4.24264 + 7.34847i −0.373544 + 0.646997i
\(130\) 0 0
\(131\) −0.707107 1.22474i −0.0617802 0.107006i 0.833481 0.552548i \(-0.186344\pi\)
−0.895261 + 0.445542i \(0.853011\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 10.3923i 0.512615 0.887875i −0.487278 0.873247i \(-0.662010\pi\)
0.999893 0.0146279i \(-0.00465636\pi\)
\(138\) 0 0
\(139\) −12.7279 −1.07957 −0.539784 0.841803i \(-0.681494\pi\)
−0.539784 + 0.841803i \(0.681494\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) 2.82843 4.89898i 0.236525 0.409673i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.00000 + 5.19615i 0.245770 + 0.425685i 0.962348 0.271821i \(-0.0876260\pi\)
−0.716578 + 0.697507i \(0.754293\pi\)
\(150\) 0 0
\(151\) 6.00000 10.3923i 0.488273 0.845714i −0.511636 0.859202i \(-0.670960\pi\)
0.999909 + 0.0134886i \(0.00429367\pi\)
\(152\) 0 0
\(153\) −4.24264 −0.342997
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.41421 2.44949i 0.112867 0.195491i −0.804058 0.594550i \(-0.797330\pi\)
0.916925 + 0.399060i \(0.130663\pi\)
\(158\) 0 0
\(159\) 4.24264 + 7.34847i 0.336463 + 0.582772i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.00000 + 5.19615i 0.234978 + 0.406994i 0.959266 0.282503i \(-0.0911648\pi\)
−0.724288 + 0.689497i \(0.757831\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.1421 1.09435 0.547176 0.837018i \(-0.315703\pi\)
0.547176 + 0.837018i \(0.315703\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 2.12132 3.67423i 0.162221 0.280976i
\(172\) 0 0
\(173\) 4.24264 + 7.34847i 0.322562 + 0.558694i 0.981016 0.193927i \(-0.0621226\pi\)
−0.658454 + 0.752621i \(0.728789\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −9.00000 15.5885i −0.676481 1.17170i
\(178\) 0 0
\(179\) −2.00000 + 3.46410i −0.149487 + 0.258919i −0.931038 0.364922i \(-0.881096\pi\)
0.781551 + 0.623841i \(0.214429\pi\)
\(180\) 0 0
\(181\) 14.1421 1.05118 0.525588 0.850739i \(-0.323845\pi\)
0.525588 + 0.850739i \(0.323845\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 4.24264 + 7.34847i 0.310253 + 0.537373i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.0000 17.3205i −0.723575 1.25327i −0.959558 0.281511i \(-0.909164\pi\)
0.235983 0.971757i \(-0.424169\pi\)
\(192\) 0 0
\(193\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −12.7279 + 22.0454i −0.902258 + 1.56276i −0.0777029 + 0.996977i \(0.524759\pi\)
−0.824556 + 0.565781i \(0.808575\pi\)
\(200\) 0 0
\(201\) −8.48528 14.6969i −0.598506 1.03664i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −4.00000 + 6.92820i −0.278019 + 0.481543i
\(208\) 0 0
\(209\) −8.48528 −0.586939
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 2.82843 4.89898i 0.193801 0.335673i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.00000 1.73205i −0.0675737 0.117041i
\(220\) 0 0
\(221\) −6.00000 + 10.3923i −0.403604 + 0.699062i
\(222\) 0 0
\(223\) −16.9706 −1.13643 −0.568216 0.822879i \(-0.692366\pi\)
−0.568216 + 0.822879i \(0.692366\pi\)
\(224\) 0 0
\(225\) 5.00000 0.333333
\(226\) 0 0
\(227\) −6.36396 + 11.0227i −0.422391 + 0.731603i −0.996173 0.0874056i \(-0.972142\pi\)
0.573782 + 0.819008i \(0.305476\pi\)
\(228\) 0 0
\(229\) −1.41421 2.44949i −0.0934539 0.161867i 0.815508 0.578745i \(-0.196457\pi\)
−0.908962 + 0.416878i \(0.863124\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.0000 + 20.7846i 0.786146 + 1.36165i 0.928312 + 0.371802i \(0.121260\pi\)
−0.142166 + 0.989843i \(0.545407\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 16.9706 1.10236
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −9.19239 + 15.9217i −0.592134 + 1.02561i 0.401811 + 0.915723i \(0.368381\pi\)
−0.993945 + 0.109883i \(0.964952\pi\)
\(242\) 0 0
\(243\) 4.94975 + 8.57321i 0.317526 + 0.549972i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.00000 10.3923i −0.381771 0.661247i
\(248\) 0 0
\(249\) 7.00000 12.1244i 0.443607 0.768350i
\(250\) 0 0
\(251\) −12.7279 −0.803379 −0.401690 0.915776i \(-0.631577\pi\)
−0.401690 + 0.915776i \(0.631577\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.6066 + 18.3712i 0.661622 + 1.14596i 0.980189 + 0.198062i \(0.0634648\pi\)
−0.318568 + 0.947900i \(0.603202\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 3.00000 + 5.19615i 0.185695 + 0.321634i
\(262\) 0 0
\(263\) −10.0000 + 17.3205i −0.616626 + 1.06803i 0.373470 + 0.927642i \(0.378168\pi\)
−0.990097 + 0.140386i \(0.955166\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) −12.7279 + 22.0454i −0.776035 + 1.34413i 0.158176 + 0.987411i \(0.449439\pi\)
−0.934211 + 0.356721i \(0.883895\pi\)
\(270\) 0 0
\(271\) −8.48528 14.6969i −0.515444 0.892775i −0.999839 0.0179261i \(-0.994294\pi\)
0.484395 0.874849i \(-0.339040\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.00000 8.66025i −0.301511 0.522233i
\(276\) 0 0
\(277\) 9.00000 15.5885i 0.540758 0.936620i −0.458103 0.888899i \(-0.651471\pi\)
0.998861 0.0477206i \(-0.0151957\pi\)
\(278\) 0 0
\(279\) 8.48528 0.508001
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −10.6066 + 18.3712i −0.630497 + 1.09205i 0.356953 + 0.934122i \(0.383816\pi\)
−0.987450 + 0.157931i \(0.949518\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.500000 0.866025i −0.0294118 0.0509427i
\(290\) 0 0
\(291\) −13.0000 + 22.5167i −0.762073 + 1.31995i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.65685 9.79796i 0.328244 0.568535i
\(298\) 0 0
\(299\) 11.3137 + 19.5959i 0.654289 + 1.13326i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −12.0000 20.7846i −0.689382 1.19404i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −21.2132 −1.21070 −0.605351 0.795959i \(-0.706967\pi\)
−0.605351 + 0.795959i \(0.706967\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) −5.65685 + 9.79796i −0.320771 + 0.555591i −0.980647 0.195783i \(-0.937275\pi\)
0.659877 + 0.751374i \(0.270609\pi\)
\(312\) 0 0
\(313\) −9.19239 15.9217i −0.519584 0.899947i −0.999741 0.0227638i \(-0.992753\pi\)
0.480156 0.877183i \(-0.340580\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.00000 15.5885i −0.505490 0.875535i −0.999980 0.00635137i \(-0.997978\pi\)
0.494489 0.869184i \(-0.335355\pi\)
\(318\) 0 0
\(319\) 6.00000 10.3923i 0.335936 0.581857i
\(320\) 0 0
\(321\) 5.65685 0.315735
\(322\) 0 0
\(323\) 18.0000 1.00155
\(324\) 0 0
\(325\) 7.07107 12.2474i 0.392232 0.679366i
\(326\) 0 0
\(327\) 12.7279 + 22.0454i 0.703856 + 1.21911i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −9.00000 15.5885i −0.494685 0.856819i 0.505296 0.862946i \(-0.331383\pi\)
−0.999981 + 0.00612670i \(0.998050\pi\)
\(332\) 0 0
\(333\) −1.00000 + 1.73205i −0.0547997 + 0.0949158i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) −8.48528 + 14.6969i −0.460857 + 0.798228i
\(340\) 0 0
\(341\) −8.48528 14.6969i −0.459504 0.795884i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5.00000 + 8.66025i −0.268414 + 0.464907i −0.968452 0.249198i \(-0.919833\pi\)
0.700038 + 0.714105i \(0.253166\pi\)
\(348\) 0 0
\(349\) −19.7990 −1.05982 −0.529908 0.848055i \(-0.677773\pi\)
−0.529908 + 0.848055i \(0.677773\pi\)
\(350\) 0 0
\(351\) 16.0000 0.854017
\(352\) 0 0
\(353\) −10.6066 + 18.3712i −0.564532 + 0.977799i 0.432561 + 0.901605i \(0.357610\pi\)
−0.997093 + 0.0761940i \(0.975723\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.00000 3.46410i −0.105556 0.182828i 0.808409 0.588621i \(-0.200329\pi\)
−0.913965 + 0.405793i \(0.866996\pi\)
\(360\) 0 0
\(361\) 0.500000 0.866025i 0.0263158 0.0455803i
\(362\) 0 0
\(363\) 9.89949 0.519589
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −16.9706 + 29.3939i −0.885856 + 1.53435i −0.0411270 + 0.999154i \(0.513095\pi\)
−0.844729 + 0.535194i \(0.820239\pi\)
\(368\) 0 0
\(369\) −2.12132 3.67423i −0.110432 0.191273i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 11.0000 + 19.0526i 0.569558 + 0.986504i 0.996610 + 0.0822766i \(0.0262191\pi\)
−0.427051 + 0.904227i \(0.640448\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.9706 0.874028
\(378\) 0 0
\(379\) −18.0000 −0.924598 −0.462299 0.886724i \(-0.652975\pi\)
−0.462299 + 0.886724i \(0.652975\pi\)
\(380\) 0 0
\(381\) −8.48528 + 14.6969i −0.434714 + 0.752947i
\(382\) 0 0
\(383\) 12.7279 + 22.0454i 0.650366 + 1.12647i 0.983034 + 0.183424i \(0.0587180\pi\)
−0.332668 + 0.943044i \(0.607949\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.00000 5.19615i −0.152499 0.264135i
\(388\) 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
\(391\) −33.9411 −1.71648
\(392\) 0 0
\(393\) −2.00000 −0.100887
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 18.3848 + 31.8434i 0.922705 + 1.59817i 0.795210 + 0.606334i \(0.207360\pi\)
0.127495 + 0.991839i \(0.459306\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.00000 5.19615i −0.149813 0.259483i 0.781345 0.624099i \(-0.214534\pi\)
−0.931158 + 0.364615i \(0.881200\pi\)
\(402\) 0 0
\(403\) 12.0000 20.7846i 0.597763 1.03536i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) 17.6777 30.6186i 0.874105 1.51399i 0.0163909 0.999866i \(-0.494782\pi\)
0.857714 0.514128i \(-0.171884\pi\)
\(410\) 0 0
\(411\) −8.48528 14.6969i −0.418548 0.724947i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −9.00000 + 15.5885i −0.440732 + 0.763370i
\(418\) 0 0
\(419\) 12.7279 0.621800 0.310900 0.950443i \(-0.399370\pi\)
0.310900 + 0.950443i \(0.399370\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) 0 0
\(423\) 1.41421 2.44949i 0.0687614 0.119098i
\(424\) 0 0
\(425\) 10.6066 + 18.3712i 0.514496 + 0.891133i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −4.00000 6.92820i −0.193122 0.334497i
\(430\) 0 0
\(431\) −20.0000 + 34.6410i −0.963366 + 1.66860i −0.249424 + 0.968394i \(0.580241\pi\)
−0.713942 + 0.700205i \(0.753092\pi\)
\(432\) 0 0
\(433\) −1.41421 −0.0679628 −0.0339814 0.999422i \(-0.510819\pi\)
−0.0339814 + 0.999422i \(0.510819\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 16.9706 29.3939i 0.811812 1.40610i
\(438\) 0 0
\(439\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.0000 17.3205i −0.475114 0.822922i 0.524479 0.851423i \(-0.324260\pi\)
−0.999594 + 0.0285009i \(0.990927\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 8.48528 0.401340
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −4.24264 + 7.34847i −0.199778 + 0.346026i
\(452\) 0 0
\(453\) −8.48528 14.6969i −0.398673 0.690522i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4.00000 6.92820i −0.187112 0.324088i 0.757174 0.653213i \(-0.226579\pi\)
−0.944286 + 0.329125i \(0.893246\pi\)
\(458\) 0 0
\(459\) −12.0000 + 20.7846i −0.560112 + 0.970143i
\(460\) 0 0
\(461\) −25.4558 −1.18560 −0.592798 0.805351i \(-0.701977\pi\)
−0.592798 + 0.805351i \(0.701977\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.36396 + 11.0227i 0.294489 + 0.510070i 0.974866 0.222792i \(-0.0715172\pi\)
−0.680377 + 0.732862i \(0.738184\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −2.00000 3.46410i −0.0921551 0.159617i
\(472\) 0 0
\(473\) −6.00000 + 10.3923i −0.275880 + 0.477839i
\(474\) 0 0
\(475\) −21.2132 −0.973329
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 1.41421 2.44949i 0.0646171 0.111920i −0.831907 0.554915i \(-0.812751\pi\)
0.896524 + 0.442995i \(0.146084\pi\)
\(480\) 0 0
\(481\) 2.82843 + 4.89898i 0.128965 + 0.223374i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(488\) 0 0
\(489\) 8.48528 0.383718
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) −12.7279 + 22.0454i −0.573237 + 0.992875i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −6.00000 10.3923i −0.268597 0.465223i 0.699903 0.714238i \(-0.253227\pi\)
−0.968500 + 0.249015i \(0.919893\pi\)
\(500\) 0 0
\(501\) 10.0000 17.3205i 0.446767 0.773823i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.53553 + 6.12372i −0.157019 + 0.271964i
\(508\) 0 0
\(509\) −21.2132 36.7423i −0.940259 1.62858i −0.764977 0.644058i \(-0.777250\pi\)
−0.175282 0.984518i \(-0.556084\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −12.0000 20.7846i −0.529813 0.917663i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −5.65685 −0.248788
\(518\) 0 0
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) 14.8492 25.7196i 0.650557 1.12680i −0.332431 0.943128i \(-0.607869\pi\)
0.982988 0.183670i \(-0.0587979\pi\)
\(522\) 0 0
\(523\) −6.36396 11.0227i −0.278277 0.481989i 0.692680 0.721245i \(-0.256430\pi\)
−0.970957 + 0.239256i \(0.923097\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 18.0000 + 31.1769i 0.784092 + 1.35809i
\(528\) 0 0
\(529\) −20.5000 + 35.5070i −0.891304 + 1.54378i
\(530\) 0 0
\(531\) 12.7279 0.552345
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 2.82843 + 4.89898i 0.122056 + 0.211407i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −9.00000 15.5885i −0.386940 0.670200i 0.605096 0.796152i \(-0.293135\pi\)
−0.992036 + 0.125952i \(0.959801\pi\)
\(542\) 0 0
\(543\) 10.0000 17.3205i 0.429141 0.743294i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −42.0000 −1.79579 −0.897895 0.440209i \(-0.854904\pi\)
−0.897895 + 0.440209i \(0.854904\pi\)
\(548\) 0 0
\(549\) −2.82843 + 4.89898i −0.120714 + 0.209083i
\(550\) 0 0
\(551\) −12.7279 22.0454i −0.542228 0.939166i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.0000 25.9808i 0.635570 1.10084i −0.350824 0.936442i \(-0.614098\pi\)
0.986394 0.164399i \(-0.0525683\pi\)
\(558\) 0 0
\(559\) −16.9706 −0.717778
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) 6.36396 11.0227i 0.268209 0.464552i −0.700190 0.713956i \(-0.746902\pi\)
0.968399 + 0.249405i \(0.0802349\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.0000 + 25.9808i 0.628833 + 1.08917i 0.987786 + 0.155815i \(0.0498003\pi\)
−0.358954 + 0.933355i \(0.616866\pi\)
\(570\) 0 0
\(571\) 21.0000 36.3731i 0.878823 1.52217i 0.0261885 0.999657i \(-0.491663\pi\)
0.852634 0.522508i \(-0.175004\pi\)
\(572\) 0 0
\(573\) −28.2843 −1.18159
\(574\) 0 0
\(575\) 40.0000 1.66812
\(576\) 0 0
\(577\) −0.707107 + 1.22474i −0.0294372 + 0.0509868i −0.880369 0.474290i \(-0.842705\pi\)
0.850931 + 0.525277i \(0.176038\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 6.00000 + 10.3923i 0.248495 + 0.430405i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 38.1838 1.57601 0.788006 0.615667i \(-0.211113\pi\)
0.788006 + 0.615667i \(0.211113\pi\)
\(588\) 0 0
\(589\) −36.0000 −1.48335
\(590\) 0 0
\(591\) 12.7279 22.0454i 0.523557 0.906827i
\(592\) 0 0
\(593\) 14.8492 + 25.7196i 0.609785 + 1.05618i 0.991275 + 0.131806i \(0.0420777\pi\)
−0.381490 + 0.924373i \(0.624589\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 18.0000 + 31.1769i 0.736691 + 1.27599i
\(598\) 0 0
\(599\) 4.00000 6.92820i 0.163436 0.283079i −0.772663 0.634816i \(-0.781076\pi\)
0.936099 + 0.351738i \(0.114409\pi\)
\(600\) 0 0
\(601\) 18.3848 0.749931 0.374965 0.927039i \(-0.377655\pi\)
0.374965 + 0.927039i \(0.377655\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 8.48528 + 14.6969i 0.344407 + 0.596530i 0.985246 0.171145i \(-0.0547467\pi\)
−0.640839 + 0.767675i \(0.721413\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.00000 6.92820i −0.161823 0.280285i
\(612\) 0 0
\(613\) 9.00000 15.5885i 0.363507 0.629612i −0.625029 0.780602i \(-0.714913\pi\)
0.988535 + 0.150990i \(0.0482461\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 2.12132 3.67423i 0.0852631 0.147680i −0.820240 0.572019i \(-0.806160\pi\)
0.905503 + 0.424339i \(0.139494\pi\)
\(620\) 0 0
\(621\) 22.6274 + 39.1918i 0.908007 + 1.57271i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.5000 21.6506i −0.500000 0.866025i
\(626\) 0 0
\(627\) −6.00000 + 10.3923i −0.239617 + 0.415029i
\(628\) 0 0
\(629\) −8.48528 −0.338330
\(630\) 0 0
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) 0 0
\(633\) 8.48528 14.6969i 0.337260 0.584151i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2.00000 + 3.46410i 0.0791188 + 0.137038i
\(640\) 0 0
\(641\) −9.00000 + 15.5885i −0.355479 + 0.615707i −0.987200 0.159489i \(-0.949015\pi\)
0.631721 + 0.775196i \(0.282349\pi\)
\(642\) 0 0
\(643\) −12.7279 −0.501940 −0.250970 0.967995i \(-0.580750\pi\)
−0.250970 + 0.967995i \(0.580750\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.89949 17.1464i 0.389189 0.674096i −0.603151 0.797627i \(-0.706089\pi\)
0.992341 + 0.123531i \(0.0394219\pi\)
\(648\) 0 0
\(649\) −12.7279 22.0454i −0.499615 0.865358i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9.00000 15.5885i −0.352197 0.610023i 0.634437 0.772975i \(-0.281232\pi\)
−0.986634 + 0.162951i \(0.947899\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.41421 0.0551737
\(658\) 0 0
\(659\) −34.0000 −1.32445 −0.662226 0.749304i \(-0.730388\pi\)
−0.662226 + 0.749304i \(0.730388\pi\)
\(660\) 0 0
\(661\) −11.3137 + 19.5959i −0.440052 + 0.762193i −0.997693 0.0678896i \(-0.978373\pi\)
0.557641 + 0.830083i \(0.311707\pi\)
\(662\) 0 0
\(663\) 8.48528 + 14.6969i 0.329541 + 0.570782i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 24.0000 + 41.5692i 0.929284 + 1.60957i
\(668\) 0 0
\(669\) −12.0000 + 20.7846i −0.463947 + 0.803579i
\(670\) 0 0
\(671\) 11.3137 0.436761
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 0 0
\(675\) 14.1421 24.4949i 0.544331 0.942809i
\(676\) 0 0
\(677\) −21.2132 36.7423i −0.815290 1.41212i −0.909120 0.416535i \(-0.863244\pi\)
0.0938301 0.995588i \(-0.470089\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 9.00000 + 15.5885i 0.344881 + 0.597351i
\(682\) 0 0
\(683\) 2.00000 3.46410i 0.0765279 0.132550i −0.825222 0.564809i \(-0.808950\pi\)
0.901750 + 0.432259i \(0.142283\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −4.00000 −0.152610
\(688\) 0 0
\(689\) −8.48528 + 14.6969i −0.323263 + 0.559909i
\(690\) 0 0
\(691\) −23.3345 40.4166i −0.887687 1.53752i −0.842602 0.538536i \(-0.818978\pi\)
−0.0450846 0.998983i \(-0.514356\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 9.00000 15.5885i 0.340899 0.590455i
\(698\) 0 0
\(699\) 33.9411 1.28377
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) 4.24264 7.34847i 0.160014 0.277153i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 9.00000 + 15.5885i 0.338002 + 0.585437i 0.984057 0.177854i \(-0.0569156\pi\)
−0.646055 + 0.763291i \(0.723582\pi\)
\(710\) 0 0
\(711\) −6.00000 + 10.3923i −0.225018 + 0.389742i
\(712\) 0 0
\(713\) 67.8823 2.54221
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −5.65685 + 9.79796i −0.211259 + 0.365911i
\(718\) 0 0
\(719\) −12.7279 22.0454i −0.474671 0.822155i 0.524908 0.851159i \(-0.324100\pi\)
−0.999579 + 0.0290041i \(0.990766\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 13.0000 + 22.5167i 0.483475 + 0.837404i
\(724\) 0 0
\(725\) 15.0000 25.9808i 0.557086 0.964901i
\(726\) 0 0
\(727\) −8.48528 −0.314702 −0.157351 0.987543i \(-0.550295\pi\)
−0.157351 + 0.987543i \(0.550295\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) 12.7279 22.0454i 0.470759 0.815379i
\(732\) 0 0
\(733\) −14.1421 24.4949i −0.522352 0.904740i −0.999662 0.0260048i \(-0.991721\pi\)
0.477310 0.878735i \(-0.341612\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12.0000 20.7846i −0.442026 0.765611i
\(738\) 0 0
\(739\) 15.0000 25.9808i 0.551784 0.955718i −0.446362 0.894852i \(-0.647281\pi\)
0.998146 0.0608653i \(-0.0193860\pi\)
\(740\) 0 0
\(741\) −16.9706 −0.623429
\(742\) 0 0
\(743\) −44.0000 −1.61420 −0.807102 0.590412i \(-0.798965\pi\)
−0.807102 + 0.590412i \(0.798965\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 4.94975 + 8.57321i 0.181102 + 0.313678i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 12.0000 + 20.7846i 0.437886 + 0.758441i 0.997526 0.0702946i \(-0.0223939\pi\)
−0.559640 + 0.828736i \(0.689061\pi\)
\(752\) 0 0
\(753\) −9.00000 + 15.5885i −0.327978 + 0.568075i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 0 0
\(759\) 11.3137 19.5959i 0.410662 0.711287i
\(760\) 0 0
\(761\) 6.36396 + 11.0227i 0.230693 + 0.399573i 0.958012 0.286727i \(-0.0925672\pi\)
−0.727319 + 0.686300i \(0.759234\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.0000 31.1769i 0.649942 1.12573i
\(768\) 0 0
\(769\) 35.3553 1.27495 0.637473 0.770473i \(-0.279980\pi\)
0.637473 + 0.770473i \(0.279980\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 0 0
\(773\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(774\) 0 0
\(775\) −21.2132 36.7423i −0.762001 1.31982i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.00000 + 15.5885i 0.322458 + 0.558514i
\(780\) 0 0
\(781\) 4.00000 6.92820i 0.143131 0.247911i
\(782\) 0 0
\(783\) 33.9411 1.21296
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 6.36396 11.0227i 0.226851 0.392917i −0.730022 0.683423i \(-0.760490\pi\)
0.956873 + 0.290506i \(0.0938237\pi\)
\(788\) 0 0
\(789\) 14.1421 + 24.4949i 0.503473 + 0.872041i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 8.00000 + 13.8564i 0.284088 + 0.492055i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −42.4264 −1.50282 −0.751410 0.659835i \(-0.770626\pi\)
−0.751410 + 0.659835i \(0.770626\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) 2.12132 3.67423i 0.0749532 0.129823i
\(802\) 0 0
\(803\) −1.41421 2.44949i −0.0499065 0.0864406i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 18.0000 + 31.1769i 0.633630 + 1.09748i
\(808\) 0 0
\(809\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(810\) 0 0
\(811\) −29.6985 −1.04285 −0.521427 0.853296i \(-0.674600\pi\)
−0.521427 + 0.853296i \(0.674600\pi\)
\(812\) 0 0
\(813\) −24.0000 −0.841717
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 12.7279 + 22.0454i 0.445294 + 0.771271i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.00000 5.19615i −0.104701 0.181347i 0.808915 0.587925i \(-0.200055\pi\)
−0.913616 + 0.406578i \(0.866722\pi\)
\(822\) 0 0
\(823\) 12.0000 20.7846i 0.418294 0.724506i −0.577474 0.816409i \(-0.695962\pi\)
0.995768 + 0.0919029i \(0.0292950\pi\)
\(824\) 0 0
\(825\) −14.1421 −0.492366
\(826\) 0 0
\(827\) 4.00000 0.139094 0.0695468 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) 0 0
\(829\) −19.7990 + 34.2929i −0.687647 + 1.19104i 0.284950 + 0.958542i \(0.408023\pi\)
−0.972597 + 0.232498i \(0.925310\pi\)
\(830\) 0 0
\(831\) −12.7279 22.0454i −0.441527 0.764747i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 24.0000 41.5692i 0.829561 1.43684i
\(838\) 0 0
\(839\) 48.0833 1.66002 0.830009 0.557750i \(-0.188335\pi\)
0.830009 + 0.557750i \(0.188335\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 15.0000 + 25.9808i 0.514799 + 0.891657i
\(850\) 0 0
\(851\) −8.00000 + 13.8564i −0.274236 + 0.474991i
\(852\) 0 0
\(853\) 19.7990 0.677905 0.338952 0.940804i \(-0.389927\pi\)
0.338952 + 0.940804i \(0.389927\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6.36396 + 11.0227i −0.217389 + 0.376528i −0.954009 0.299778i \(-0.903087\pi\)
0.736620 + 0.676307i \(0.236421\pi\)
\(858\) 0 0
\(859\) 2.12132 + 3.67423i 0.0723785 + 0.125363i 0.899943 0.436007i \(-0.143608\pi\)
−0.827565 + 0.561370i \(0.810274\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.00000 + 3.46410i 0.0680808 + 0.117919i 0.898056 0.439880i \(-0.144979\pi\)
−0.829976 + 0.557800i \(0.811646\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1.41421 −0.0480292
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 16.9706 29.3939i 0.575026 0.995974i
\(872\) 0 0
\(873\) −9.19239 15.9217i −0.311115 0.538867i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 9.00000 + 15.5885i 0.303908 + 0.526385i 0.977018 0.213158i \(-0.0683750\pi\)
−0.673109 + 0.739543i \(0.735042\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4.24264 −0.142938 −0.0714691 0.997443i \(-0.522769\pi\)
−0.0714691 + 0.997443i \(0.522769\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 18.3848 + 31.8434i 0.617300 + 1.06920i 0.989976 + 0.141234i \(0.0451070\pi\)
−0.372676 + 0.927962i \(0.621560\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −5.00000 8.66025i −0.167506 0.290129i
\(892\) 0 0
\(893\) −6.00000 + 10.3923i −0.200782 + 0.347765i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 32.0000 1.06845
\(898\) 0 0
\(899\) 25.4558 44.0908i 0.849000 1.47051i
\(900\) 0 0
\(901\) −12.7279 22.0454i −0.424029 0.734439i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 18.0000 31.1769i 0.597680 1.03521i −0.395482 0.918474i \(-0.629423\pi\)
0.993163 0.116739i \(-0.0372441\pi\)
\(908\) 0 0
\(909\) 16.9706 0.562878
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 0 0
\(913\) 9.89949 17.1464i 0.327625 0.567464i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(920\) 0 0
\(921\) −15.0000 + 25.9808i −0.494267 + 0.856095i
\(922\) 0 0
\(923\) 11.3137 0.372395
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) 0 0
\(927\) 4.24264 7.34847i 0.139347 0.241355i
\(928\) 0 0
\(929\) −2.12132 3.67423i −0.0695983 0.120548i 0.829126 0.559061i \(-0.188838\pi\)
−0.898725 + 0.438514i \(0.855505\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 8.00000 + 13.8564i 0.261908 + 0.453638i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −24.0416 −0.785406 −0.392703 0.919665i \(-0.628460\pi\)
−0.392703 + 0.919665i \(0.628460\pi\)
\(938\) 0 0
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) −16.9706 + 29.3939i −0.553225 + 0.958213i 0.444815 + 0.895623i \(0.353270\pi\)
−0.998039 + 0.0625904i \(0.980064\pi\)
\(942\) 0 0
\(943\) −16.9706 29.3939i −0.552638 0.957196i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 17.0000 + 29.4449i 0.552426 + 0.956830i 0.998099 + 0.0616337i \(0.0196311\pi\)
−0.445673 + 0.895196i \(0.647036\pi\)
\(948\) 0 0
\(949\) 2.00000 3.46410i 0.0649227 0.112449i
\(950\) 0 0
\(951\) −25.4558 −0.825462
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −8.48528 14.6969i −0.274290 0.475085i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −20.5000 35.5070i −0.661290 1.14539i
\(962\) 0 0
\(963\) −2.00000 + 3.46410i −0.0644491 + 0.111629i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) 12.7279 22.0454i 0.408880 0.708201i
\(970\) 0 0
\(971\) −21.9203 37.9671i −0.703456 1.21842i −0.967246 0.253842i \(-0.918306\pi\)
0.263790 0.964580i \(-0.415028\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −10.0000 17.3205i −0.320256 0.554700i
\(976\) 0 0
\(977\) 6.00000 10.3923i 0.191957 0.332479i −0.753942 0.656941i \(-0.771850\pi\)
0.945899 + 0.324462i \(0.105183\pi\)
\(978\) 0 0
\(979\) −8.48528 −0.271191
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 0 0
\(983\) 12.7279 22.0454i 0.405958 0.703139i −0.588475 0.808516i \(-0.700271\pi\)
0.994432 + 0.105376i \(0.0336047\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −24.0000 41.5692i −0.763156 1.32182i
\(990\) 0 0
\(991\) 24.0000 41.5692i 0.762385 1.32049i −0.179233 0.983807i \(-0.557362\pi\)
0.941618 0.336683i \(-0.109305\pi\)
\(992\) 0 0
\(993\) −25.4558 −0.807817
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 28.2843 48.9898i 0.895772 1.55152i 0.0629254 0.998018i \(-0.479957\pi\)
0.832846 0.553504i \(-0.186710\pi\)
\(998\) 0 0
\(999\) 5.65685 + 9.79796i 0.178975 + 0.309994i
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 1568.2.i.r.1537.2 4
4.3 odd 2 1568.2.i.q.1537.1 4
7.2 even 3 inner 1568.2.i.r.961.2 4
7.3 odd 6 1568.2.a.q.1.2 yes 2
7.4 even 3 1568.2.a.q.1.1 2
7.5 odd 6 inner 1568.2.i.r.961.1 4
7.6 odd 2 inner 1568.2.i.r.1537.1 4
28.3 even 6 1568.2.a.r.1.1 yes 2
28.11 odd 6 1568.2.a.r.1.2 yes 2
28.19 even 6 1568.2.i.q.961.2 4
28.23 odd 6 1568.2.i.q.961.1 4
28.27 even 2 1568.2.i.q.1537.2 4
56.3 even 6 3136.2.a.bl.1.2 2
56.11 odd 6 3136.2.a.bl.1.1 2
56.45 odd 6 3136.2.a.bo.1.1 2
56.53 even 6 3136.2.a.bo.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1568.2.a.q.1.1 2 7.4 even 3
1568.2.a.q.1.2 yes 2 7.3 odd 6
1568.2.a.r.1.1 yes 2 28.3 even 6
1568.2.a.r.1.2 yes 2 28.11 odd 6
1568.2.i.q.961.1 4 28.23 odd 6
1568.2.i.q.961.2 4 28.19 even 6
1568.2.i.q.1537.1 4 4.3 odd 2
1568.2.i.q.1537.2 4 28.27 even 2
1568.2.i.r.961.1 4 7.5 odd 6 inner
1568.2.i.r.961.2 4 7.2 even 3 inner
1568.2.i.r.1537.1 4 7.6 odd 2 inner
1568.2.i.r.1537.2 4 1.1 even 1 trivial
3136.2.a.bl.1.1 2 56.11 odd 6
3136.2.a.bl.1.2 2 56.3 even 6
3136.2.a.bo.1.1 2 56.45 odd 6
3136.2.a.bo.1.2 2 56.53 even 6