Properties

Label 7056.2.k.e.881.5
Level $7056$
Weight $2$
Character 7056.881
Analytic conductor $56.342$
Analytic rank $0$
Dimension $8$
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] = [7056,2,Mod(881,7056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7056, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7056.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(56.3424436662\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1764)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.5
Root \(-0.382683 - 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 7056.881
Dual form 7056.2.k.e.881.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.765367 q^{5} +O(q^{10})\) \(q+0.765367 q^{5} -2.00000i q^{11} -0.317025i q^{13} +5.54328 q^{17} -3.69552i q^{19} -3.17157i q^{23} -4.41421 q^{25} -6.82843i q^{29} -6.75699i q^{31} -0.242641 q^{37} -2.74444 q^{41} -6.82843 q^{43} -11.9832 q^{47} +12.2426i q^{53} -1.53073i q^{55} -13.2513 q^{59} +3.56420i q^{61} -0.242641i q^{65} +4.48528 q^{67} +9.31371i q^{71} +11.8519i q^{73} +11.3137 q^{79} -4.32957 q^{83} +4.24264 q^{85} +1.66205 q^{89} -2.82843i q^{95} -11.8519i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{25} + 32 q^{37} - 32 q^{43} - 32 q^{67}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1765\) \(4609\) \(6175\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(5\) 0.765367 0.342282 0.171141 0.985247i \(-0.445255\pi\)
0.171141 + 0.985247i \(0.445255\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) − 0.317025i − 0.0879270i −0.999033 0.0439635i \(-0.986001\pi\)
0.999033 0.0439635i \(-0.0139985\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.54328 1.34444 0.672221 0.740350i \(-0.265340\pi\)
0.672221 + 0.740350i \(0.265340\pi\)
\(18\) 0 0
\(19\) − 3.69552i − 0.847810i −0.905707 0.423905i \(-0.860659\pi\)
0.905707 0.423905i \(-0.139341\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 3.17157i − 0.661319i −0.943750 0.330659i \(-0.892729\pi\)
0.943750 0.330659i \(-0.107271\pi\)
\(24\) 0 0
\(25\) −4.41421 −0.882843
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 6.82843i − 1.26801i −0.773330 0.634004i \(-0.781410\pi\)
0.773330 0.634004i \(-0.218590\pi\)
\(30\) 0 0
\(31\) − 6.75699i − 1.21359i −0.794858 0.606795i \(-0.792455\pi\)
0.794858 0.606795i \(-0.207545\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.242641 −0.0398899 −0.0199449 0.999801i \(-0.506349\pi\)
−0.0199449 + 0.999801i \(0.506349\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.74444 −0.428610 −0.214305 0.976767i \(-0.568749\pi\)
−0.214305 + 0.976767i \(0.568749\pi\)
\(42\) 0 0
\(43\) −6.82843 −1.04133 −0.520663 0.853762i \(-0.674315\pi\)
−0.520663 + 0.853762i \(0.674315\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.9832 −1.74793 −0.873967 0.485985i \(-0.838461\pi\)
−0.873967 + 0.485985i \(0.838461\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.2426i 1.68166i 0.541302 + 0.840828i \(0.317931\pi\)
−0.541302 + 0.840828i \(0.682069\pi\)
\(54\) 0 0
\(55\) − 1.53073i − 0.206404i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −13.2513 −1.72518 −0.862589 0.505906i \(-0.831158\pi\)
−0.862589 + 0.505906i \(0.831158\pi\)
\(60\) 0 0
\(61\) 3.56420i 0.456349i 0.973620 + 0.228175i \(0.0732757\pi\)
−0.973620 + 0.228175i \(0.926724\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 0.242641i − 0.0300959i
\(66\) 0 0
\(67\) 4.48528 0.547964 0.273982 0.961735i \(-0.411659\pi\)
0.273982 + 0.961735i \(0.411659\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.31371i 1.10533i 0.833402 + 0.552667i \(0.186390\pi\)
−0.833402 + 0.552667i \(0.813610\pi\)
\(72\) 0 0
\(73\) 11.8519i 1.38716i 0.720378 + 0.693581i \(0.243968\pi\)
−0.720378 + 0.693581i \(0.756032\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.32957 −0.475232 −0.237616 0.971359i \(-0.576366\pi\)
−0.237616 + 0.971359i \(0.576366\pi\)
\(84\) 0 0
\(85\) 4.24264 0.460179
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.66205 0.176177 0.0880885 0.996113i \(-0.471924\pi\)
0.0880885 + 0.996113i \(0.471924\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 2.82843i − 0.290191i
\(96\) 0 0
\(97\) − 11.8519i − 1.20338i −0.798730 0.601690i \(-0.794494\pi\)
0.798730 0.601690i \(-0.205506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 11.8519 1.17931 0.589655 0.807655i \(-0.299264\pi\)
0.589655 + 0.807655i \(0.299264\pi\)
\(102\) 0 0
\(103\) − 8.92177i − 0.879088i −0.898221 0.439544i \(-0.855140\pi\)
0.898221 0.439544i \(-0.144860\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7.17157i − 0.693302i −0.937994 0.346651i \(-0.887319\pi\)
0.937994 0.346651i \(-0.112681\pi\)
\(108\) 0 0
\(109\) −11.0711 −1.06042 −0.530208 0.847868i \(-0.677886\pi\)
−0.530208 + 0.847868i \(0.677886\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 10.5858i − 0.995827i −0.867227 0.497914i \(-0.834100\pi\)
0.867227 0.497914i \(-0.165900\pi\)
\(114\) 0 0
\(115\) − 2.42742i − 0.226358i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −7.20533 −0.644464
\(126\) 0 0
\(127\) −1.17157 −0.103960 −0.0519801 0.998648i \(-0.516553\pi\)
−0.0519801 + 0.998648i \(0.516553\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −16.9469 −1.48065 −0.740327 0.672247i \(-0.765329\pi\)
−0.740327 + 0.672247i \(0.765329\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.8284i 0.925135i 0.886584 + 0.462567i \(0.153072\pi\)
−0.886584 + 0.462567i \(0.846928\pi\)
\(138\) 0 0
\(139\) − 15.6788i − 1.32985i −0.746908 0.664927i \(-0.768462\pi\)
0.746908 0.664927i \(-0.231538\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.634051 −0.0530220
\(144\) 0 0
\(145\) − 5.22625i − 0.434017i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 7.07107i − 0.579284i −0.957135 0.289642i \(-0.906464\pi\)
0.957135 0.289642i \(-0.0935363\pi\)
\(150\) 0 0
\(151\) −10.1421 −0.825355 −0.412678 0.910877i \(-0.635406\pi\)
−0.412678 + 0.910877i \(0.635406\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 5.17157i − 0.415391i
\(156\) 0 0
\(157\) 6.17733i 0.493004i 0.969142 + 0.246502i \(0.0792813\pi\)
−0.969142 + 0.246502i \(0.920719\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2.82843 −0.221540 −0.110770 0.993846i \(-0.535332\pi\)
−0.110770 + 0.993846i \(0.535332\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.42742 −0.187839 −0.0939196 0.995580i \(-0.529940\pi\)
−0.0939196 + 0.995580i \(0.529940\pi\)
\(168\) 0 0
\(169\) 12.8995 0.992269
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −18.7946 −1.42893 −0.714464 0.699672i \(-0.753329\pi\)
−0.714464 + 0.699672i \(0.753329\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.31371i 0.696139i 0.937469 + 0.348070i \(0.113163\pi\)
−0.937469 + 0.348070i \(0.886837\pi\)
\(180\) 0 0
\(181\) 1.66205i 0.123539i 0.998090 + 0.0617696i \(0.0196744\pi\)
−0.998090 + 0.0617696i \(0.980326\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.185709 −0.0136536
\(186\) 0 0
\(187\) − 11.0866i − 0.810729i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 17.3137i − 1.25278i −0.779511 0.626388i \(-0.784533\pi\)
0.779511 0.626388i \(-0.215467\pi\)
\(192\) 0 0
\(193\) −9.65685 −0.695116 −0.347558 0.937659i \(-0.612989\pi\)
−0.347558 + 0.937659i \(0.612989\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.07107i 0.218805i 0.993998 + 0.109402i \(0.0348937\pi\)
−0.993998 + 0.109402i \(0.965106\pi\)
\(198\) 0 0
\(199\) 7.39104i 0.523937i 0.965076 + 0.261968i \(0.0843716\pi\)
−0.965076 + 0.261968i \(0.915628\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2.10051 −0.146706
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −7.39104 −0.511249
\(210\) 0 0
\(211\) 18.6274 1.28236 0.641182 0.767389i \(-0.278444\pi\)
0.641182 + 0.767389i \(0.278444\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.22625 −0.356427
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 1.75736i − 0.118213i
\(222\) 0 0
\(223\) 21.8017i 1.45995i 0.683474 + 0.729975i \(0.260468\pi\)
−0.683474 + 0.729975i \(0.739532\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.75699 0.448477 0.224238 0.974534i \(-0.428011\pi\)
0.224238 + 0.974534i \(0.428011\pi\)
\(228\) 0 0
\(229\) − 16.1815i − 1.06930i −0.845073 0.534651i \(-0.820443\pi\)
0.845073 0.534651i \(-0.179557\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 15.1716i − 0.993923i −0.867773 0.496961i \(-0.834449\pi\)
0.867773 0.496961i \(-0.165551\pi\)
\(234\) 0 0
\(235\) −9.17157 −0.598287
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 8.82843i − 0.571063i −0.958369 0.285532i \(-0.907830\pi\)
0.958369 0.285532i \(-0.0921702\pi\)
\(240\) 0 0
\(241\) − 25.5516i − 1.64592i −0.568097 0.822962i \(-0.692320\pi\)
0.568097 0.822962i \(-0.307680\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.17157 −0.0745454
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −25.4972 −1.60937 −0.804685 0.593702i \(-0.797666\pi\)
−0.804685 + 0.593702i \(0.797666\pi\)
\(252\) 0 0
\(253\) −6.34315 −0.398790
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −20.5880 −1.28424 −0.642122 0.766603i \(-0.721946\pi\)
−0.642122 + 0.766603i \(0.721946\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.6569i 1.21209i 0.795429 + 0.606047i \(0.207246\pi\)
−0.795429 + 0.606047i \(0.792754\pi\)
\(264\) 0 0
\(265\) 9.37011i 0.575601i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −17.8979 −1.09126 −0.545628 0.838027i \(-0.683709\pi\)
−0.545628 + 0.838027i \(0.683709\pi\)
\(270\) 0 0
\(271\) − 5.48888i − 0.333426i −0.986005 0.166713i \(-0.946685\pi\)
0.986005 0.166713i \(-0.0533153\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.82843i 0.532374i
\(276\) 0 0
\(277\) 12.9706 0.779326 0.389663 0.920958i \(-0.372592\pi\)
0.389663 + 0.920958i \(0.372592\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.4853i 0.625499i 0.949836 + 0.312750i \(0.101250\pi\)
−0.949836 + 0.312750i \(0.898750\pi\)
\(282\) 0 0
\(283\) − 4.06694i − 0.241754i −0.992667 0.120877i \(-0.961429\pi\)
0.992667 0.120877i \(-0.0385707\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 13.7279 0.807525
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −16.4441 −0.960676 −0.480338 0.877083i \(-0.659486\pi\)
−0.480338 + 0.877083i \(0.659486\pi\)
\(294\) 0 0
\(295\) −10.1421 −0.590498
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.00547 −0.0581478
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.72792i 0.156200i
\(306\) 0 0
\(307\) − 27.1367i − 1.54877i −0.632712 0.774387i \(-0.718058\pi\)
0.632712 0.774387i \(-0.281942\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 30.7235 1.74217 0.871084 0.491134i \(-0.163418\pi\)
0.871084 + 0.491134i \(0.163418\pi\)
\(312\) 0 0
\(313\) − 28.8757i − 1.63215i −0.577945 0.816076i \(-0.696145\pi\)
0.577945 0.816076i \(-0.303855\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 15.7574i − 0.885021i −0.896763 0.442511i \(-0.854088\pi\)
0.896763 0.442511i \(-0.145912\pi\)
\(318\) 0 0
\(319\) −13.6569 −0.764637
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 20.4853i − 1.13983i
\(324\) 0 0
\(325\) 1.39942i 0.0776257i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 27.3137 1.50130 0.750649 0.660702i \(-0.229741\pi\)
0.750649 + 0.660702i \(0.229741\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.43289 0.187559
\(336\) 0 0
\(337\) −27.0711 −1.47466 −0.737328 0.675535i \(-0.763913\pi\)
−0.737328 + 0.675535i \(0.763913\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −13.5140 −0.731823
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 29.7990i 1.59969i 0.600204 + 0.799847i \(0.295086\pi\)
−0.600204 + 0.799847i \(0.704914\pi\)
\(348\) 0 0
\(349\) − 31.9372i − 1.70956i −0.518993 0.854779i \(-0.673693\pi\)
0.518993 0.854779i \(-0.326307\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −21.1451 −1.12544 −0.562720 0.826647i \(-0.690245\pi\)
−0.562720 + 0.826647i \(0.690245\pi\)
\(354\) 0 0
\(355\) 7.12840i 0.378336i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 19.4558i − 1.02684i −0.858137 0.513420i \(-0.828378\pi\)
0.858137 0.513420i \(-0.171622\pi\)
\(360\) 0 0
\(361\) 5.34315 0.281218
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.07107i 0.474801i
\(366\) 0 0
\(367\) 36.0585i 1.88224i 0.338075 + 0.941119i \(0.390224\pi\)
−0.338075 + 0.941119i \(0.609776\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 30.6274 1.58583 0.792914 0.609334i \(-0.208563\pi\)
0.792914 + 0.609334i \(0.208563\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.16478 −0.111492
\(378\) 0 0
\(379\) 36.2843 1.86380 0.931899 0.362718i \(-0.118151\pi\)
0.931899 + 0.362718i \(0.118151\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 14.7821 0.755329 0.377664 0.925943i \(-0.376727\pi\)
0.377664 + 0.925943i \(0.376727\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 1.17157i − 0.0594011i −0.999559 0.0297006i \(-0.990545\pi\)
0.999559 0.0297006i \(-0.00945537\pi\)
\(390\) 0 0
\(391\) − 17.5809i − 0.889105i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.65914 0.435688
\(396\) 0 0
\(397\) − 14.0167i − 0.703478i −0.936098 0.351739i \(-0.885590\pi\)
0.936098 0.351739i \(-0.114410\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 5.51472i − 0.275392i −0.990475 0.137696i \(-0.956030\pi\)
0.990475 0.137696i \(-0.0439697\pi\)
\(402\) 0 0
\(403\) −2.14214 −0.106707
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.485281i 0.0240545i
\(408\) 0 0
\(409\) 1.13679i 0.0562104i 0.999605 + 0.0281052i \(0.00894734\pi\)
−0.999605 + 0.0281052i \(0.991053\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −3.31371 −0.162664
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −27.6620 −1.35138 −0.675688 0.737187i \(-0.736154\pi\)
−0.675688 + 0.737187i \(0.736154\pi\)
\(420\) 0 0
\(421\) 15.3137 0.746344 0.373172 0.927762i \(-0.378270\pi\)
0.373172 + 0.927762i \(0.378270\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −24.4692 −1.18693
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 33.3137i 1.60466i 0.596877 + 0.802332i \(0.296408\pi\)
−0.596877 + 0.802332i \(0.703592\pi\)
\(432\) 0 0
\(433\) − 0.502734i − 0.0241599i −0.999927 0.0120799i \(-0.996155\pi\)
0.999927 0.0120799i \(-0.00384526\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11.7206 −0.560673
\(438\) 0 0
\(439\) − 13.8854i − 0.662713i −0.943506 0.331357i \(-0.892494\pi\)
0.943506 0.331357i \(-0.107506\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 29.3137i 1.39274i 0.717685 + 0.696368i \(0.245202\pi\)
−0.717685 + 0.696368i \(0.754798\pi\)
\(444\) 0 0
\(445\) 1.27208 0.0603023
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 11.5563i 0.545378i 0.962102 + 0.272689i \(0.0879130\pi\)
−0.962102 + 0.272689i \(0.912087\pi\)
\(450\) 0 0
\(451\) 5.48888i 0.258461i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.17186 0.240877 0.120439 0.992721i \(-0.461570\pi\)
0.120439 + 0.992721i \(0.461570\pi\)
\(462\) 0 0
\(463\) 2.82843 0.131448 0.0657241 0.997838i \(-0.479064\pi\)
0.0657241 + 0.997838i \(0.479064\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.23172 −0.288370 −0.144185 0.989551i \(-0.546056\pi\)
−0.144185 + 0.989551i \(0.546056\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 13.6569i 0.627943i
\(474\) 0 0
\(475\) 16.3128i 0.748483i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 23.3324 1.06609 0.533043 0.846088i \(-0.321048\pi\)
0.533043 + 0.846088i \(0.321048\pi\)
\(480\) 0 0
\(481\) 0.0769232i 0.00350740i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 9.07107i − 0.411896i
\(486\) 0 0
\(487\) −19.7990 −0.897178 −0.448589 0.893738i \(-0.648073\pi\)
−0.448589 + 0.893738i \(0.648073\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 34.4853i − 1.55630i −0.628079 0.778149i \(-0.716159\pi\)
0.628079 0.778149i \(-0.283841\pi\)
\(492\) 0 0
\(493\) − 37.8519i − 1.70476i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −26.1421 −1.17028 −0.585141 0.810931i \(-0.698961\pi\)
−0.585141 + 0.810931i \(0.698961\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 29.5641 1.31820 0.659100 0.752055i \(-0.270937\pi\)
0.659100 + 0.752055i \(0.270937\pi\)
\(504\) 0 0
\(505\) 9.07107 0.403657
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −12.9343 −0.573303 −0.286652 0.958035i \(-0.592542\pi\)
−0.286652 + 0.958035i \(0.592542\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 6.82843i − 0.300896i
\(516\) 0 0
\(517\) 23.9665i 1.05404i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.03347 0.0890879 0.0445439 0.999007i \(-0.485817\pi\)
0.0445439 + 0.999007i \(0.485817\pi\)
\(522\) 0 0
\(523\) 18.2150i 0.796485i 0.917280 + 0.398242i \(0.130380\pi\)
−0.917280 + 0.398242i \(0.869620\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 37.4558i − 1.63160i
\(528\) 0 0
\(529\) 12.9411 0.562658
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.870058i 0.0376864i
\(534\) 0 0
\(535\) − 5.48888i − 0.237305i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.34315 0.358700 0.179350 0.983785i \(-0.442601\pi\)
0.179350 + 0.983785i \(0.442601\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8.47343 −0.362962
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −25.2346 −1.07503
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 16.7279i − 0.708785i −0.935097 0.354392i \(-0.884688\pi\)
0.935097 0.354392i \(-0.115312\pi\)
\(558\) 0 0
\(559\) 2.16478i 0.0915606i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −22.0643 −0.929900 −0.464950 0.885337i \(-0.653928\pi\)
−0.464950 + 0.885337i \(0.653928\pi\)
\(564\) 0 0
\(565\) − 8.10201i − 0.340854i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 0.485281i − 0.0203441i −0.999948 0.0101720i \(-0.996762\pi\)
0.999948 0.0101720i \(-0.00323791\pi\)
\(570\) 0 0
\(571\) 2.62742 0.109954 0.0549770 0.998488i \(-0.482491\pi\)
0.0549770 + 0.998488i \(0.482491\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14.0000i 0.583840i
\(576\) 0 0
\(577\) 19.8770i 0.827491i 0.910393 + 0.413745i \(0.135780\pi\)
−0.910393 + 0.413745i \(0.864220\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 24.4853 1.01408
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −28.0334 −1.15706 −0.578531 0.815660i \(-0.696374\pi\)
−0.578531 + 0.815660i \(0.696374\pi\)
\(588\) 0 0
\(589\) −24.9706 −1.02889
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −30.7779 −1.26389 −0.631947 0.775011i \(-0.717744\pi\)
−0.631947 + 0.775011i \(0.717744\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 9.31371i − 0.380548i −0.981731 0.190274i \(-0.939062\pi\)
0.981731 0.190274i \(-0.0609376\pi\)
\(600\) 0 0
\(601\) − 23.9121i − 0.975394i −0.873013 0.487697i \(-0.837837\pi\)
0.873013 0.487697i \(-0.162163\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.35757 0.217816
\(606\) 0 0
\(607\) − 12.6173i − 0.512120i −0.966661 0.256060i \(-0.917576\pi\)
0.966661 0.256060i \(-0.0824245\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.79899i 0.153691i
\(612\) 0 0
\(613\) −45.2132 −1.82614 −0.913072 0.407798i \(-0.866297\pi\)
−0.913072 + 0.407798i \(0.866297\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 17.4558i − 0.702746i −0.936236 0.351373i \(-0.885715\pi\)
0.936236 0.351373i \(-0.114285\pi\)
\(618\) 0 0
\(619\) − 1.79337i − 0.0720815i −0.999350 0.0360407i \(-0.988525\pi\)
0.999350 0.0360407i \(-0.0114746\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 16.5563 0.662254
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.34502 −0.0536296
\(630\) 0 0
\(631\) 12.4853 0.497031 0.248516 0.968628i \(-0.420057\pi\)
0.248516 + 0.968628i \(0.420057\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −0.896683 −0.0355838
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 23.4558i − 0.926450i −0.886241 0.463225i \(-0.846692\pi\)
0.886241 0.463225i \(-0.153308\pi\)
\(642\) 0 0
\(643\) − 32.5168i − 1.28234i −0.767400 0.641169i \(-0.778450\pi\)
0.767400 0.641169i \(-0.221550\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.38557 0.251043 0.125521 0.992091i \(-0.459940\pi\)
0.125521 + 0.992091i \(0.459940\pi\)
\(648\) 0 0
\(649\) 26.5027i 1.04032i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 5.85786i − 0.229236i −0.993410 0.114618i \(-0.963436\pi\)
0.993410 0.114618i \(-0.0365644\pi\)
\(654\) 0 0
\(655\) −12.9706 −0.506802
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.97056i 0.115717i 0.998325 + 0.0578583i \(0.0184272\pi\)
−0.998325 + 0.0578583i \(0.981573\pi\)
\(660\) 0 0
\(661\) 23.3868i 0.909642i 0.890583 + 0.454821i \(0.150297\pi\)
−0.890583 + 0.454821i \(0.849703\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −21.6569 −0.838557
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 7.12840 0.275189
\(672\) 0 0
\(673\) 20.0416 0.772548 0.386274 0.922384i \(-0.373762\pi\)
0.386274 + 0.922384i \(0.373762\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 27.9790 1.07532 0.537661 0.843161i \(-0.319308\pi\)
0.537661 + 0.843161i \(0.319308\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 46.9706i − 1.79728i −0.438688 0.898639i \(-0.644557\pi\)
0.438688 0.898639i \(-0.355443\pi\)
\(684\) 0 0
\(685\) 8.28772i 0.316657i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.88123 0.147863
\(690\) 0 0
\(691\) − 25.6060i − 0.974098i −0.873375 0.487049i \(-0.838073\pi\)
0.873375 0.487049i \(-0.161927\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 12.0000i − 0.455186i
\(696\) 0 0
\(697\) −15.2132 −0.576241
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 15.4558i − 0.583759i −0.956455 0.291880i \(-0.905719\pi\)
0.956455 0.291880i \(-0.0942807\pi\)
\(702\) 0 0
\(703\) 0.896683i 0.0338190i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −30.8701 −1.15935 −0.579675 0.814848i \(-0.696820\pi\)
−0.579675 + 0.814848i \(0.696820\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −21.4303 −0.802570
\(714\) 0 0
\(715\) −0.485281 −0.0181485
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19.6369 −0.732333 −0.366167 0.930549i \(-0.619330\pi\)
−0.366167 + 0.930549i \(0.619330\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 30.1421i 1.11945i
\(726\) 0 0
\(727\) 53.7933i 1.99508i 0.0700903 + 0.997541i \(0.477671\pi\)
−0.0700903 + 0.997541i \(0.522329\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −37.8519 −1.40000
\(732\) 0 0
\(733\) − 22.3044i − 0.823833i −0.911222 0.411916i \(-0.864860\pi\)
0.911222 0.411916i \(-0.135140\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 8.97056i − 0.330435i
\(738\) 0 0
\(739\) −2.54416 −0.0935883 −0.0467941 0.998905i \(-0.514900\pi\)
−0.0467941 + 0.998905i \(0.514900\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 43.6569i 1.60161i 0.598922 + 0.800807i \(0.295596\pi\)
−0.598922 + 0.800807i \(0.704404\pi\)
\(744\) 0 0
\(745\) − 5.41196i − 0.198279i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 36.2843 1.32403 0.662016 0.749490i \(-0.269701\pi\)
0.662016 + 0.749490i \(0.269701\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −7.76245 −0.282505
\(756\) 0 0
\(757\) 18.1005 0.657874 0.328937 0.944352i \(-0.393310\pi\)
0.328937 + 0.944352i \(0.393310\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 53.6619 1.94524 0.972622 0.232394i \(-0.0746557\pi\)
0.972622 + 0.232394i \(0.0746557\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.20101i 0.151690i
\(768\) 0 0
\(769\) 14.2793i 0.514926i 0.966288 + 0.257463i \(0.0828866\pi\)
−0.966288 + 0.257463i \(0.917113\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 33.5767 1.20767 0.603835 0.797110i \(-0.293639\pi\)
0.603835 + 0.797110i \(0.293639\pi\)
\(774\) 0 0
\(775\) 29.8268i 1.07141i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 10.1421i 0.363380i
\(780\) 0 0
\(781\) 18.6274 0.666541
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.72792i 0.168747i
\(786\) 0 0
\(787\) − 50.4692i − 1.79903i −0.436889 0.899515i \(-0.643920\pi\)
0.436889 0.899515i \(-0.356080\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.12994 0.0401254
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 26.3714 0.934122 0.467061 0.884225i \(-0.345313\pi\)
0.467061 + 0.884225i \(0.345313\pi\)
\(798\) 0 0
\(799\) −66.4264 −2.35000
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 23.7038 0.836490
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 21.2132i 0.745817i 0.927868 + 0.372908i \(0.121639\pi\)
−0.927868 + 0.372908i \(0.878361\pi\)
\(810\) 0 0
\(811\) − 42.1814i − 1.48119i −0.671951 0.740595i \(-0.734544\pi\)
0.671951 0.740595i \(-0.265456\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.16478 −0.0758291
\(816\) 0 0
\(817\) 25.2346i 0.882846i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 51.0711i 1.78239i 0.453618 + 0.891196i \(0.350133\pi\)
−0.453618 + 0.891196i \(0.649867\pi\)
\(822\) 0 0
\(823\) 40.2843 1.40422 0.702111 0.712068i \(-0.252241\pi\)
0.702111 + 0.712068i \(0.252241\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 18.4853i − 0.642796i −0.946944 0.321398i \(-0.895847\pi\)
0.946944 0.321398i \(-0.104153\pi\)
\(828\) 0 0
\(829\) 35.3701i 1.22845i 0.789130 + 0.614226i \(0.210532\pi\)
−0.789130 + 0.614226i \(0.789468\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1.85786 −0.0642940
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 32.8882 1.13543 0.567714 0.823226i \(-0.307828\pi\)
0.567714 + 0.823226i \(0.307828\pi\)
\(840\) 0 0
\(841\) −17.6274 −0.607842
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.87285 0.339636
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0.769553i 0.0263799i
\(852\) 0 0
\(853\) − 57.2805i − 1.96125i −0.195899 0.980624i \(-0.562763\pi\)
0.195899 0.980624i \(-0.437237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 39.6227 1.35349 0.676743 0.736219i \(-0.263391\pi\)
0.676743 + 0.736219i \(0.263391\pi\)
\(858\) 0 0
\(859\) − 13.7766i − 0.470052i −0.971989 0.235026i \(-0.924483\pi\)
0.971989 0.235026i \(-0.0755175\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 25.1127i 0.854846i 0.904052 + 0.427423i \(0.140579\pi\)
−0.904052 + 0.427423i \(0.859421\pi\)
\(864\) 0 0
\(865\) −14.3848 −0.489097
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 22.6274i − 0.767583i
\(870\) 0 0
\(871\) − 1.42195i − 0.0481809i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.928932 0.0313678 0.0156839 0.999877i \(-0.495007\pi\)
0.0156839 + 0.999877i \(0.495007\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 22.1187 0.745198 0.372599 0.927992i \(-0.378467\pi\)
0.372599 + 0.927992i \(0.378467\pi\)
\(882\) 0 0
\(883\) 21.8579 0.735576 0.367788 0.929910i \(-0.380115\pi\)
0.367788 + 0.929910i \(0.380115\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −19.8995 −0.668161 −0.334081 0.942545i \(-0.608426\pi\)
−0.334081 + 0.942545i \(0.608426\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 44.2843i 1.48192i
\(894\) 0 0
\(895\) 7.12840i 0.238276i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −46.1396 −1.53884
\(900\) 0 0
\(901\) 67.8644i 2.26089i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.27208i 0.0422853i
\(906\) 0 0
\(907\) 32.4853 1.07866 0.539328 0.842096i \(-0.318678\pi\)
0.539328 + 0.842096i \(0.318678\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 42.2843i 1.40094i 0.713682 + 0.700470i \(0.247026\pi\)
−0.713682 + 0.700470i \(0.752974\pi\)
\(912\) 0 0
\(913\) 8.65914i 0.286576i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 15.5147 0.511783 0.255892 0.966705i \(-0.417631\pi\)
0.255892 + 0.966705i \(0.417631\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.95268 0.0971887
\(924\) 0 0
\(925\) 1.07107 0.0352165
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 22.3044 0.731784 0.365892 0.930657i \(-0.380764\pi\)
0.365892 + 0.930657i \(0.380764\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 8.48528i − 0.277498i
\(936\) 0 0
\(937\) 16.5210i 0.539719i 0.962900 + 0.269860i \(0.0869773\pi\)
−0.962900 + 0.269860i \(0.913023\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −2.37302 −0.0773584 −0.0386792 0.999252i \(-0.512315\pi\)
−0.0386792 + 0.999252i \(0.512315\pi\)
\(942\) 0 0
\(943\) 8.70420i 0.283448i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 8.82843i − 0.286885i −0.989659 0.143443i \(-0.954183\pi\)
0.989659 0.143443i \(-0.0458173\pi\)
\(948\) 0 0
\(949\) 3.75736 0.121969
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 58.8701i − 1.90699i −0.301412 0.953494i \(-0.597458\pi\)
0.301412 0.953494i \(-0.402542\pi\)
\(954\) 0 0
\(955\) − 13.2513i − 0.428803i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −14.6569 −0.472802
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −7.39104 −0.237926
\(966\) 0 0
\(967\) −15.5147 −0.498920 −0.249460 0.968385i \(-0.580253\pi\)
−0.249460 + 0.968385i \(0.580253\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −25.2346 −0.809816 −0.404908 0.914357i \(-0.632696\pi\)
−0.404908 + 0.914357i \(0.632696\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 0.485281i − 0.0155255i −0.999970 0.00776276i \(-0.997529\pi\)
0.999970 0.00776276i \(-0.00247099\pi\)
\(978\) 0 0
\(979\) − 3.32410i − 0.106239i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −7.01962 −0.223891 −0.111946 0.993714i \(-0.535708\pi\)
−0.111946 + 0.993714i \(0.535708\pi\)
\(984\) 0 0
\(985\) 2.35049i 0.0748930i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 21.6569i 0.688648i
\(990\) 0 0
\(991\) 9.85786 0.313145 0.156573 0.987666i \(-0.449955\pi\)
0.156573 + 0.987666i \(0.449955\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 5.65685i 0.179334i
\(996\) 0 0
\(997\) − 15.4705i − 0.489956i −0.969529 0.244978i \(-0.921219\pi\)
0.969529 0.244978i \(-0.0787808\pi\)
\(998\) 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 7056.2.k.e.881.5 8
3.2 odd 2 inner 7056.2.k.e.881.4 8
4.3 odd 2 1764.2.f.b.881.6 yes 8
7.6 odd 2 inner 7056.2.k.e.881.3 8
12.11 even 2 1764.2.f.b.881.3 8
21.20 even 2 inner 7056.2.k.e.881.6 8
28.3 even 6 1764.2.t.c.1097.6 16
28.11 odd 6 1764.2.t.c.1097.4 16
28.19 even 6 1764.2.t.c.521.5 16
28.23 odd 6 1764.2.t.c.521.3 16
28.27 even 2 1764.2.f.b.881.4 yes 8
84.11 even 6 1764.2.t.c.1097.5 16
84.23 even 6 1764.2.t.c.521.6 16
84.47 odd 6 1764.2.t.c.521.4 16
84.59 odd 6 1764.2.t.c.1097.3 16
84.83 odd 2 1764.2.f.b.881.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1764.2.f.b.881.3 8 12.11 even 2
1764.2.f.b.881.4 yes 8 28.27 even 2
1764.2.f.b.881.5 yes 8 84.83 odd 2
1764.2.f.b.881.6 yes 8 4.3 odd 2
1764.2.t.c.521.3 16 28.23 odd 6
1764.2.t.c.521.4 16 84.47 odd 6
1764.2.t.c.521.5 16 28.19 even 6
1764.2.t.c.521.6 16 84.23 even 6
1764.2.t.c.1097.3 16 84.59 odd 6
1764.2.t.c.1097.4 16 28.11 odd 6
1764.2.t.c.1097.5 16 84.11 even 6
1764.2.t.c.1097.6 16 28.3 even 6
7056.2.k.e.881.3 8 7.6 odd 2 inner
7056.2.k.e.881.4 8 3.2 odd 2 inner
7056.2.k.e.881.5 8 1.1 even 1 trivial
7056.2.k.e.881.6 8 21.20 even 2 inner