Properties

Label 1568.2.i.c.961.1
Level $1568$
Weight $2$
Character 1568.961
Analytic conductor $12.521$
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] = [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: \(2\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1568.961
Dual form 1568.2.i.c.1537.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-1.00000 - 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{9} +(2.00000 + 3.46410i) q^{11} +4.00000 q^{13} +(-1.00000 - 1.73205i) q^{17} +(-3.00000 + 5.19615i) q^{19} +(-4.00000 + 6.92820i) q^{23} +(2.50000 + 4.33013i) q^{25} -4.00000 q^{27} +2.00000 q^{29} +(-2.00000 - 3.46410i) q^{31} +(4.00000 - 6.92820i) q^{33} +(-5.00000 + 8.66025i) q^{37} +(-4.00000 - 6.92820i) q^{39} +10.0000 q^{41} +4.00000 q^{43} +(2.00000 - 3.46410i) q^{47} +(-2.00000 + 3.46410i) q^{51} +(1.00000 + 1.73205i) q^{53} +12.0000 q^{57} +(5.00000 + 8.66025i) q^{59} +(-4.00000 + 6.92820i) q^{61} +(4.00000 + 6.92820i) q^{67} +16.0000 q^{69} +(-3.00000 - 5.19615i) q^{73} +(5.00000 - 8.66025i) q^{75} +(8.00000 - 13.8564i) q^{79} +(5.50000 + 9.52628i) q^{81} -2.00000 q^{83} +(-2.00000 - 3.46410i) q^{87} +(9.00000 - 15.5885i) q^{89} +(-4.00000 + 6.92820i) q^{93} +2.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - q^{9} + 4 q^{11} + 8 q^{13} - 2 q^{17} - 6 q^{19} - 8 q^{23} + 5 q^{25} - 8 q^{27} + 4 q^{29} - 4 q^{31} + 8 q^{33} - 10 q^{37} - 8 q^{39} + 20 q^{41} + 8 q^{43} + 4 q^{47} - 4 q^{51} + 2 q^{53} + 24 q^{57} + 10 q^{59} - 8 q^{61} + 8 q^{67} + 32 q^{69} - 6 q^{73} + 10 q^{75} + 16 q^{79} + 11 q^{81} - 4 q^{83} - 4 q^{87} + 18 q^{89} - 8 q^{93} + 4 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −1.00000 1.73205i −0.577350 1.00000i −0.995782 0.0917517i \(-0.970753\pi\)
0.418432 0.908248i \(-0.362580\pi\)
\(4\) 0 0
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 2.00000 + 3.46410i 0.603023 + 1.04447i 0.992361 + 0.123371i \(0.0393705\pi\)
−0.389338 + 0.921095i \(0.627296\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 1.73205i −0.242536 0.420084i 0.718900 0.695113i \(-0.244646\pi\)
−0.961436 + 0.275029i \(0.911312\pi\)
\(18\) 0 0
\(19\) −3.00000 + 5.19615i −0.688247 + 1.19208i 0.284157 + 0.958778i \(0.408286\pi\)
−0.972404 + 0.233301i \(0.925047\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 + 6.92820i −0.834058 + 1.44463i 0.0607377 + 0.998154i \(0.480655\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −2.00000 3.46410i −0.359211 0.622171i 0.628619 0.777714i \(-0.283621\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 4.00000 6.92820i 0.696311 1.20605i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 + 8.66025i −0.821995 + 1.42374i 0.0821995 + 0.996616i \(0.473806\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) −4.00000 6.92820i −0.640513 1.10940i
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.00000 3.46410i 0.291730 0.505291i −0.682489 0.730896i \(-0.739102\pi\)
0.974219 + 0.225605i \(0.0724358\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −2.00000 + 3.46410i −0.280056 + 0.485071i
\(52\) 0 0
\(53\) 1.00000 + 1.73205i 0.137361 + 0.237915i 0.926497 0.376303i \(-0.122805\pi\)
−0.789136 + 0.614218i \(0.789471\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 12.0000 1.58944
\(58\) 0 0
\(59\) 5.00000 + 8.66025i 0.650945 + 1.12747i 0.982894 + 0.184172i \(0.0589603\pi\)
−0.331949 + 0.943297i \(0.607706\pi\)
\(60\) 0 0
\(61\) −4.00000 + 6.92820i −0.512148 + 0.887066i 0.487753 + 0.872982i \(0.337817\pi\)
−0.999901 + 0.0140840i \(0.995517\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 + 6.92820i 0.488678 + 0.846415i 0.999915 0.0130248i \(-0.00414604\pi\)
−0.511237 + 0.859440i \(0.670813\pi\)
\(68\) 0 0
\(69\) 16.0000 1.92617
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −3.00000 5.19615i −0.351123 0.608164i 0.635323 0.772246i \(-0.280867\pi\)
−0.986447 + 0.164083i \(0.947534\pi\)
\(74\) 0 0
\(75\) 5.00000 8.66025i 0.577350 1.00000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 13.8564i 0.900070 1.55897i 0.0726692 0.997356i \(-0.476848\pi\)
0.827401 0.561611i \(-0.189818\pi\)
\(80\) 0 0
\(81\) 5.50000 + 9.52628i 0.611111 + 1.05848i
\(82\) 0 0
\(83\) −2.00000 −0.219529 −0.109764 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.00000 3.46410i −0.214423 0.371391i
\(88\) 0 0
\(89\) 9.00000 15.5885i 0.953998 1.65237i 0.217354 0.976093i \(-0.430258\pi\)
0.736644 0.676280i \(-0.236409\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.00000 + 6.92820i −0.414781 + 0.718421i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) −2.00000 + 3.46410i −0.197066 + 0.341328i −0.947576 0.319531i \(-0.896475\pi\)
0.750510 + 0.660859i \(0.229808\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000 13.8564i 0.773389 1.33955i −0.162306 0.986740i \(-0.551893\pi\)
0.935695 0.352809i \(-0.114773\pi\)
\(108\) 0 0
\(109\) −5.00000 8.66025i −0.478913 0.829502i 0.520794 0.853682i \(-0.325636\pi\)
−0.999708 + 0.0241802i \(0.992302\pi\)
\(110\) 0 0
\(111\) 20.0000 1.89832
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.00000 + 3.46410i −0.184900 + 0.320256i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.50000 + 4.33013i −0.227273 + 0.393648i
\(122\) 0 0
\(123\) −10.0000 17.3205i −0.901670 1.56174i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) −4.00000 6.92820i −0.352180 0.609994i
\(130\) 0 0
\(131\) −3.00000 + 5.19615i −0.262111 + 0.453990i −0.966803 0.255524i \(-0.917752\pi\)
0.704692 + 0.709514i \(0.251085\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.00000 + 12.1244i 0.598050 + 1.03585i 0.993109 + 0.117198i \(0.0373911\pi\)
−0.395058 + 0.918656i \(0.629276\pi\)
\(138\) 0 0
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 8.00000 + 13.8564i 0.668994 + 1.15873i
\(144\) 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 \(-0.912374\pi\)
0.716578 + 0.697507i \(0.245707\pi\)
\(150\) 0 0
\(151\) 8.00000 + 13.8564i 0.651031 + 1.12762i 0.982873 + 0.184284i \(0.0589965\pi\)
−0.331842 + 0.943335i \(0.607670\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.00000 + 10.3923i 0.478852 + 0.829396i 0.999706 0.0242497i \(-0.00771967\pi\)
−0.520854 + 0.853646i \(0.674386\pi\)
\(158\) 0 0
\(159\) 2.00000 3.46410i 0.158610 0.274721i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2.00000 + 3.46410i −0.156652 + 0.271329i −0.933659 0.358162i \(-0.883403\pi\)
0.777007 + 0.629492i \(0.216737\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −3.00000 5.19615i −0.229416 0.397360i
\(172\) 0 0
\(173\) −6.00000 + 10.3923i −0.456172 + 0.790112i −0.998755 0.0498898i \(-0.984113\pi\)
0.542583 + 0.840002i \(0.317446\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 10.0000 17.3205i 0.751646 1.30189i
\(178\) 0 0
\(179\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 0 0
\(183\) 16.0000 1.18275
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000 6.92820i 0.292509 0.506640i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(192\) 0 0
\(193\) −7.00000 12.1244i −0.503871 0.872730i −0.999990 0.00447566i \(-0.998575\pi\)
0.496119 0.868255i \(-0.334758\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 2.00000 + 3.46410i 0.141776 + 0.245564i 0.928166 0.372168i \(-0.121385\pi\)
−0.786389 + 0.617731i \(0.788052\pi\)
\(200\) 0 0
\(201\) 8.00000 13.8564i 0.564276 0.977356i
\(202\) 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
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −6.00000 + 10.3923i −0.405442 + 0.702247i
\(220\) 0 0
\(221\) −4.00000 6.92820i −0.269069 0.466041i
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) −11.0000 19.0526i −0.730096 1.26456i −0.956842 0.290609i \(-0.906142\pi\)
0.226746 0.973954i \(-0.427191\pi\)
\(228\) 0 0
\(229\) −10.0000 + 17.3205i −0.660819 + 1.14457i 0.319582 + 0.947559i \(0.396457\pi\)
−0.980401 + 0.197013i \(0.936876\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.00000 5.19615i 0.196537 0.340411i −0.750867 0.660454i \(-0.770364\pi\)
0.947403 + 0.320043i \(0.103697\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −32.0000 −2.07862
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 7.00000 + 12.1244i 0.450910 + 0.780998i 0.998443 0.0557856i \(-0.0177663\pi\)
−0.547533 + 0.836784i \(0.684433\pi\)
\(242\) 0 0
\(243\) 5.00000 8.66025i 0.320750 0.555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −12.0000 + 20.7846i −0.763542 + 1.32249i
\(248\) 0 0
\(249\) 2.00000 + 3.46410i 0.126745 + 0.219529i
\(250\) 0 0
\(251\) 10.0000 0.631194 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) 0 0
\(253\) −32.0000 −2.01182
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.00000 1.73205i 0.0623783 0.108042i −0.833150 0.553047i \(-0.813465\pi\)
0.895528 + 0.445005i \(0.146798\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.00000 + 1.73205i −0.0618984 + 0.107211i
\(262\) 0 0
\(263\) 8.00000 + 13.8564i 0.493301 + 0.854423i 0.999970 0.00771799i \(-0.00245674\pi\)
−0.506669 + 0.862141i \(0.669123\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −36.0000 −2.20316
\(268\) 0 0
\(269\) −6.00000 10.3923i −0.365826 0.633630i 0.623082 0.782157i \(-0.285880\pi\)
−0.988908 + 0.148527i \(0.952547\pi\)
\(270\) 0 0
\(271\) −8.00000 + 13.8564i −0.485965 + 0.841717i −0.999870 0.0161307i \(-0.994865\pi\)
0.513905 + 0.857847i \(0.328199\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −10.0000 + 17.3205i −0.603023 + 1.04447i
\(276\) 0 0
\(277\) −7.00000 12.1244i −0.420589 0.728482i 0.575408 0.817867i \(-0.304843\pi\)
−0.995997 + 0.0893846i \(0.971510\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 5.00000 + 8.66025i 0.297219 + 0.514799i 0.975499 0.220005i \(-0.0706075\pi\)
−0.678280 + 0.734804i \(0.737274\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.50000 11.2583i 0.382353 0.662255i
\(290\) 0 0
\(291\) −2.00000 3.46410i −0.117242 0.203069i
\(292\) 0 0
\(293\) −8.00000 −0.467365 −0.233682 0.972313i \(-0.575078\pi\)
−0.233682 + 0.972313i \(0.575078\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −8.00000 13.8564i −0.464207 0.804030i
\(298\) 0 0
\(299\) −16.0000 + 27.7128i −0.925304 + 1.60267i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −10.0000 −0.570730 −0.285365 0.958419i \(-0.592115\pi\)
−0.285365 + 0.958419i \(0.592115\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −12.0000 20.7846i −0.680458 1.17859i −0.974841 0.222900i \(-0.928448\pi\)
0.294384 0.955687i \(-0.404886\pi\)
\(312\) 0 0
\(313\) −13.0000 + 22.5167i −0.734803 + 1.27272i 0.220006 + 0.975499i \(0.429392\pi\)
−0.954810 + 0.297218i \(0.903941\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.0000 29.4449i 0.954815 1.65379i 0.220024 0.975494i \(-0.429386\pi\)
0.734791 0.678294i \(-0.237280\pi\)
\(318\) 0 0
\(319\) 4.00000 + 6.92820i 0.223957 + 0.387905i
\(320\) 0 0
\(321\) −32.0000 −1.78607
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) 10.0000 + 17.3205i 0.554700 + 0.960769i
\(326\) 0 0
\(327\) −10.0000 + 17.3205i −0.553001 + 0.957826i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6.00000 10.3923i 0.329790 0.571213i −0.652680 0.757634i \(-0.726355\pi\)
0.982470 + 0.186421i \(0.0596888\pi\)
\(332\) 0 0
\(333\) −5.00000 8.66025i −0.273998 0.474579i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) −6.00000 10.3923i −0.325875 0.564433i
\(340\) 0 0
\(341\) 8.00000 13.8564i 0.433224 0.750366i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.00000 + 10.3923i 0.322097 + 0.557888i 0.980921 0.194409i \(-0.0622790\pi\)
−0.658824 + 0.752297i \(0.728946\pi\)
\(348\) 0 0
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) 0 0
\(353\) 9.00000 + 15.5885i 0.479022 + 0.829690i 0.999711 0.0240566i \(-0.00765819\pi\)
−0.520689 + 0.853746i \(0.674325\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.00000 13.8564i 0.422224 0.731313i −0.573933 0.818902i \(-0.694583\pi\)
0.996157 + 0.0875892i \(0.0279163\pi\)
\(360\) 0 0
\(361\) −8.50000 14.7224i −0.447368 0.774865i
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.00000 + 6.92820i 0.208798 + 0.361649i 0.951336 0.308155i \(-0.0997115\pi\)
−0.742538 + 0.669804i \(0.766378\pi\)
\(368\) 0 0
\(369\) −5.00000 + 8.66025i −0.260290 + 0.450835i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −3.00000 + 5.19615i −0.155334 + 0.269047i −0.933181 0.359408i \(-0.882979\pi\)
0.777847 + 0.628454i \(0.216312\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 16.0000 + 27.7128i 0.819705 + 1.41977i
\(382\) 0 0
\(383\) 10.0000 17.3205i 0.510976 0.885037i −0.488943 0.872316i \(-0.662617\pi\)
0.999919 0.0127209i \(-0.00404928\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.00000 + 3.46410i −0.101666 + 0.176090i
\(388\) 0 0
\(389\) 3.00000 + 5.19615i 0.152106 + 0.263455i 0.932002 0.362454i \(-0.118061\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −14.0000 + 24.2487i −0.702640 + 1.21701i 0.264897 + 0.964277i \(0.414662\pi\)
−0.967537 + 0.252731i \(0.918671\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.00000 1.73205i 0.0499376 0.0864945i −0.839976 0.542623i \(-0.817431\pi\)
0.889914 + 0.456129i \(0.150764\pi\)
\(402\) 0 0
\(403\) −8.00000 13.8564i −0.398508 0.690237i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −40.0000 −1.98273
\(408\) 0 0
\(409\) −1.00000 1.73205i −0.0494468 0.0856444i 0.840243 0.542211i \(-0.182412\pi\)
−0.889689 + 0.456566i \(0.849079\pi\)
\(410\) 0 0
\(411\) 14.0000 24.2487i 0.690569 1.19610i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −10.0000 17.3205i −0.489702 0.848189i
\(418\) 0 0
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) 0 0
\(423\) 2.00000 + 3.46410i 0.0972433 + 0.168430i
\(424\) 0 0
\(425\) 5.00000 8.66025i 0.242536 0.420084i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 16.0000 27.7128i 0.772487 1.33799i
\(430\) 0 0
\(431\) −4.00000 6.92820i −0.192673 0.333720i 0.753462 0.657491i \(-0.228382\pi\)
−0.946135 + 0.323772i \(0.895049\pi\)
\(432\) 0 0
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −24.0000 41.5692i −1.14808 1.98853i
\(438\) 0 0
\(439\) 12.0000 20.7846i 0.572729 0.991995i −0.423556 0.905870i \(-0.639218\pi\)
0.996284 0.0861252i \(-0.0274485\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000 20.7846i 0.570137 0.987507i −0.426414 0.904528i \(-0.640223\pi\)
0.996551 0.0829786i \(-0.0264433\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 20.0000 + 34.6410i 0.941763 + 1.63118i
\(452\) 0 0
\(453\) 16.0000 27.7128i 0.751746 1.30206i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −11.0000 + 19.0526i −0.514558 + 0.891241i 0.485299 + 0.874348i \(0.338711\pi\)
−0.999857 + 0.0168929i \(0.994623\pi\)
\(458\) 0 0
\(459\) 4.00000 + 6.92820i 0.186704 + 0.323381i
\(460\) 0 0
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.00000 15.5885i 0.416470 0.721348i −0.579111 0.815249i \(-0.696600\pi\)
0.995582 + 0.0939008i \(0.0299336\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 12.0000 20.7846i 0.552931 0.957704i
\(472\) 0 0
\(473\) 8.00000 + 13.8564i 0.367840 + 0.637118i
\(474\) 0 0
\(475\) −30.0000 −1.37649
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) 6.00000 + 10.3923i 0.274147 + 0.474837i 0.969920 0.243426i \(-0.0782712\pi\)
−0.695773 + 0.718262i \(0.744938\pi\)
\(480\) 0 0
\(481\) −20.0000 + 34.6410i −0.911922 + 1.57949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −12.0000 20.7846i −0.543772 0.941841i −0.998683 0.0513038i \(-0.983662\pi\)
0.454911 0.890537i \(-0.349671\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) −2.00000 3.46410i −0.0900755 0.156015i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4.00000 6.92820i 0.179065 0.310149i −0.762496 0.646993i \(-0.776026\pi\)
0.941560 + 0.336844i \(0.109360\pi\)
\(500\) 0 0
\(501\) 20.0000 + 34.6410i 0.893534 + 1.54765i
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.00000 5.19615i −0.133235 0.230769i
\(508\) 0 0
\(509\) 10.0000 17.3205i 0.443242 0.767718i −0.554686 0.832060i \(-0.687161\pi\)
0.997928 + 0.0643419i \(0.0204948\pi\)
\(510\) 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
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) 19.0000 + 32.9090i 0.832405 + 1.44177i 0.896126 + 0.443800i \(0.146370\pi\)
−0.0637207 + 0.997968i \(0.520297\pi\)
\(522\) 0 0
\(523\) −7.00000 + 12.1244i −0.306089 + 0.530161i −0.977503 0.210921i \(-0.932354\pi\)
0.671414 + 0.741082i \(0.265687\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.00000 + 6.92820i −0.174243 + 0.301797i
\(528\) 0 0
\(529\) −20.5000 35.5070i −0.891304 1.54378i
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) 40.0000 1.73259
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −15.0000 + 25.9808i −0.644900 + 1.11700i 0.339424 + 0.940633i \(0.389768\pi\)
−0.984325 + 0.176367i \(0.943566\pi\)
\(542\) 0 0
\(543\) −12.0000 20.7846i −0.514969 0.891953i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) −4.00000 6.92820i −0.170716 0.295689i
\(550\) 0 0
\(551\) −6.00000 + 10.3923i −0.255609 + 0.442727i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −15.0000 25.9808i −0.635570 1.10084i −0.986394 0.164399i \(-0.947432\pi\)
0.350824 0.936442i \(-0.385902\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) 0 0
\(563\) 23.0000 + 39.8372i 0.969334 + 1.67894i 0.697489 + 0.716596i \(0.254301\pi\)
0.271846 + 0.962341i \(0.412366\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.00000 + 5.19615i −0.125767 + 0.217834i −0.922032 0.387113i \(-0.873472\pi\)
0.796266 + 0.604947i \(0.206806\pi\)
\(570\) 0 0
\(571\) 2.00000 + 3.46410i 0.0836974 + 0.144968i 0.904835 0.425762i \(-0.139994\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −40.0000 −1.66812
\(576\) 0 0
\(577\) −7.00000 12.1244i −0.291414 0.504744i 0.682730 0.730670i \(-0.260792\pi\)
−0.974144 + 0.225927i \(0.927459\pi\)
\(578\) 0 0
\(579\) −14.0000 + 24.2487i −0.581820 + 1.00774i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −4.00000 + 6.92820i −0.165663 + 0.286937i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) −6.00000 10.3923i −0.246807 0.427482i
\(592\) 0 0
\(593\) −3.00000 + 5.19615i −0.123195 + 0.213380i −0.921026 0.389501i \(-0.872647\pi\)
0.797831 + 0.602881i \(0.205981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.00000 6.92820i 0.163709 0.283552i
\(598\) 0 0
\(599\) −12.0000 20.7846i −0.490307 0.849236i 0.509631 0.860393i \(-0.329782\pi\)
−0.999938 + 0.0111569i \(0.996449\pi\)
\(600\) 0 0
\(601\) 46.0000 1.87638 0.938190 0.346122i \(-0.112502\pi\)
0.938190 + 0.346122i \(0.112502\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 24.0000 41.5692i 0.974130 1.68724i 0.291353 0.956616i \(-0.405895\pi\)
0.682777 0.730627i \(-0.260772\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 13.8564i 0.323645 0.560570i
\(612\) 0 0
\(613\) 11.0000 + 19.0526i 0.444286 + 0.769526i 0.998002 0.0631797i \(-0.0201241\pi\)
−0.553716 + 0.832705i \(0.686791\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) −11.0000 19.0526i −0.442127 0.765787i 0.555720 0.831370i \(-0.312443\pi\)
−0.997847 + 0.0655827i \(0.979109\pi\)
\(620\) 0 0
\(621\) 16.0000 27.7128i 0.642058 1.11208i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.5000 + 21.6506i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 24.0000 + 41.5692i 0.958468 + 1.66011i
\(628\) 0 0
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) −8.00000 13.8564i −0.317971 0.550743i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17.0000 + 29.4449i 0.671460 + 1.16300i 0.977490 + 0.210981i \(0.0676657\pi\)
−0.306031 + 0.952022i \(0.599001\pi\)
\(642\) 0 0
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.00000 10.3923i −0.235884 0.408564i 0.723645 0.690172i \(-0.242465\pi\)
−0.959529 + 0.281609i \(0.909132\pi\)
\(648\) 0 0
\(649\) −20.0000 + 34.6410i −0.785069 + 1.35978i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19.0000 32.9090i 0.743527 1.28783i −0.207352 0.978266i \(-0.566485\pi\)
0.950880 0.309561i \(-0.100182\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 12.0000 + 20.7846i 0.466746 + 0.808428i 0.999278 0.0379819i \(-0.0120929\pi\)
−0.532533 + 0.846410i \(0.678760\pi\)
\(662\) 0 0
\(663\) −8.00000 + 13.8564i −0.310694 + 0.538138i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.00000 + 13.8564i −0.309761 + 0.536522i
\(668\) 0 0
\(669\) 24.0000 + 41.5692i 0.927894 + 1.60716i
\(670\) 0 0
\(671\) −32.0000 −1.23535
\(672\) 0 0
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) 0 0
\(675\) −10.0000 17.3205i −0.384900 0.666667i
\(676\) 0 0
\(677\) 18.0000 31.1769i 0.691796 1.19823i −0.279453 0.960159i \(-0.590153\pi\)
0.971249 0.238067i \(-0.0765137\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −22.0000 + 38.1051i −0.843042 + 1.46019i
\(682\) 0 0
\(683\) 12.0000 + 20.7846i 0.459167 + 0.795301i 0.998917 0.0465244i \(-0.0148145\pi\)
−0.539750 + 0.841825i \(0.681481\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 40.0000 1.52610
\(688\) 0 0
\(689\) 4.00000 + 6.92820i 0.152388 + 0.263944i
\(690\) 0 0
\(691\) −11.0000 + 19.0526i −0.418460 + 0.724793i −0.995785 0.0917209i \(-0.970763\pi\)
0.577325 + 0.816514i \(0.304097\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −10.0000 17.3205i −0.378777 0.656061i
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) −30.0000 51.9615i −1.13147 1.95977i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −13.0000 + 22.5167i −0.488225 + 0.845631i −0.999908 0.0135434i \(-0.995689\pi\)
0.511683 + 0.859174i \(0.329022\pi\)
\(710\) 0 0
\(711\) 8.00000 + 13.8564i 0.300023 + 0.519656i
\(712\) 0 0
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −24.0000 41.5692i −0.896296 1.55243i
\(718\) 0 0
\(719\) −18.0000 + 31.1769i −0.671287 + 1.16270i 0.306253 + 0.951950i \(0.400925\pi\)
−0.977539 + 0.210752i \(0.932409\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 14.0000 24.2487i 0.520666 0.901819i
\(724\) 0 0
\(725\) 5.00000 + 8.66025i 0.185695 + 0.321634i
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −4.00000 6.92820i −0.147945 0.256249i
\(732\) 0 0
\(733\) 4.00000 6.92820i 0.147743 0.255899i −0.782650 0.622462i \(-0.786132\pi\)
0.930393 + 0.366563i \(0.119466\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −16.0000 + 27.7128i −0.589368 + 1.02081i
\(738\) 0 0
\(739\) −2.00000 3.46410i −0.0735712 0.127429i 0.826893 0.562360i \(-0.190106\pi\)
−0.900464 + 0.434930i \(0.856773\pi\)
\(740\) 0 0
\(741\) 48.0000 1.76332
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.00000 1.73205i 0.0365881 0.0633724i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −12.0000 + 20.7846i −0.437886 + 0.758441i −0.997526 0.0702946i \(-0.977606\pi\)
0.559640 + 0.828736i \(0.310939\pi\)
\(752\) 0 0
\(753\) −10.0000 17.3205i −0.364420 0.631194i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 0 0
\(759\) 32.0000 + 55.4256i 1.16153 + 2.01182i
\(760\) 0 0
\(761\) 15.0000 25.9808i 0.543750 0.941802i −0.454935 0.890525i \(-0.650337\pi\)
0.998684 0.0512772i \(-0.0163292\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.0000 + 34.6410i 0.722158 + 1.25081i
\(768\) 0 0
\(769\) −6.00000 −0.216366 −0.108183 0.994131i \(-0.534503\pi\)
−0.108183 + 0.994131i \(0.534503\pi\)
\(770\) 0 0
\(771\) −4.00000 −0.144056
\(772\) 0 0
\(773\) −12.0000 20.7846i −0.431610 0.747570i 0.565402 0.824815i \(-0.308721\pi\)
−0.997012 + 0.0772449i \(0.975388\pi\)
\(774\) 0 0
\(775\) 10.0000 17.3205i 0.359211 0.622171i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −30.0000 + 51.9615i −1.07486 + 1.86171i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −8.00000 −0.285897
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −23.0000 39.8372i −0.819861 1.42004i −0.905784 0.423740i \(-0.860717\pi\)
0.0859225 0.996302i \(-0.472616\pi\)
\(788\) 0 0
\(789\) 16.0000 27.7128i 0.569615 0.986602i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −16.0000 + 27.7128i −0.568177 + 0.984111i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −20.0000 −0.708436 −0.354218 0.935163i \(-0.615253\pi\)
−0.354218 + 0.935163i \(0.615253\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 9.00000 + 15.5885i 0.317999 + 0.550791i
\(802\) 0 0
\(803\) 12.0000 20.7846i 0.423471 0.733473i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −12.0000 + 20.7846i −0.422420 + 0.731653i
\(808\) 0 0
\(809\) 13.0000 + 22.5167i 0.457056 + 0.791644i 0.998804 0.0488972i \(-0.0155707\pi\)
−0.541748 + 0.840541i \(0.682237\pi\)
\(810\) 0 0
\(811\) −34.0000 −1.19390 −0.596951 0.802278i \(-0.703621\pi\)
−0.596951 + 0.802278i \(0.703621\pi\)
\(812\) 0 0
\(813\) 32.0000 1.12229
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −12.0000 + 20.7846i −0.419827 + 0.727161i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.00000 1.73205i 0.0349002 0.0604490i −0.848048 0.529920i \(-0.822222\pi\)
0.882948 + 0.469471i \(0.155555\pi\)
\(822\) 0 0
\(823\) −4.00000 6.92820i −0.139431 0.241502i 0.787850 0.615867i \(-0.211194\pi\)
−0.927281 + 0.374365i \(0.877861\pi\)
\(824\) 0 0
\(825\) 40.0000 1.39262
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 12.0000 + 20.7846i 0.416777 + 0.721879i 0.995613 0.0935647i \(-0.0298262\pi\)
−0.578836 + 0.815444i \(0.696493\pi\)
\(830\) 0 0
\(831\) −14.0000 + 24.2487i −0.485655 + 0.841178i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000 + 13.8564i 0.276520 + 0.478947i
\(838\) 0 0
\(839\) 52.0000 1.79524 0.897620 0.440771i \(-0.145295\pi\)
0.897620 + 0.440771i \(0.145295\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 6.00000 + 10.3923i 0.206651 + 0.357930i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 10.0000 17.3205i 0.343199 0.594438i
\(850\) 0 0
\(851\) −40.0000 69.2820i −1.37118 2.37496i
\(852\) 0 0
\(853\) 36.0000 1.23262 0.616308 0.787505i \(-0.288628\pi\)
0.616308 + 0.787505i \(0.288628\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −9.00000 15.5885i −0.307434 0.532492i 0.670366 0.742030i \(-0.266137\pi\)
−0.977800 + 0.209539i \(0.932804\pi\)
\(858\) 0 0
\(859\) −13.0000 + 22.5167i −0.443554 + 0.768259i −0.997950 0.0639945i \(-0.979616\pi\)
0.554396 + 0.832253i \(0.312949\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −24.0000 + 41.5692i −0.816970 + 1.41503i 0.0909355 + 0.995857i \(0.471014\pi\)
−0.907905 + 0.419176i \(0.862319\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −26.0000 −0.883006
\(868\) 0 0
\(869\) 64.0000 2.17105
\(870\) 0 0
\(871\) 16.0000 + 27.7128i 0.542139 + 0.939013i
\(872\) 0 0
\(873\) −1.00000 + 1.73205i −0.0338449 + 0.0586210i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.00000 12.1244i 0.236373 0.409410i −0.723298 0.690536i \(-0.757375\pi\)
0.959671 + 0.281126i \(0.0907079\pi\)
\(878\) 0 0
\(879\) 8.00000 + 13.8564i 0.269833 + 0.467365i
\(880\) 0 0
\(881\) 38.0000 1.28025 0.640126 0.768270i \(-0.278882\pi\)
0.640126 + 0.768270i \(0.278882\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 14.0000 24.2487i 0.470074 0.814192i −0.529340 0.848410i \(-0.677561\pi\)
0.999414 + 0.0342175i \(0.0108939\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −22.0000 + 38.1051i −0.737028 + 1.27657i
\(892\) 0 0
\(893\) 12.0000 + 20.7846i 0.401565 + 0.695530i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 64.0000 2.13690
\(898\) 0 0
\(899\) −4.00000 6.92820i −0.133407 0.231069i
\(900\) 0 0
\(901\) 2.00000 3.46410i 0.0666297 0.115406i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −16.0000 27.7128i −0.531271 0.920189i −0.999334 0.0364935i \(-0.988381\pi\)
0.468063 0.883695i \(-0.344952\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) −4.00000 6.92820i −0.132381 0.229290i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 20.0000 34.6410i 0.659739 1.14270i −0.320944 0.947098i \(-0.604000\pi\)
0.980683 0.195603i \(-0.0626666\pi\)
\(920\) 0 0
\(921\) 10.0000 + 17.3205i 0.329511 + 0.570730i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −50.0000 −1.64399
\(926\) 0 0
\(927\) −2.00000 3.46410i −0.0656886 0.113776i
\(928\) 0 0
\(929\) −9.00000 + 15.5885i −0.295280 + 0.511441i −0.975050 0.221985i \(-0.928746\pi\)
0.679770 + 0.733426i \(0.262080\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −24.0000 + 41.5692i −0.785725 + 1.36092i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) 52.0000 1.69696
\(940\) 0 0
\(941\) −20.0000 34.6410i −0.651981 1.12926i −0.982641 0.185515i \(-0.940605\pi\)
0.330660 0.943750i \(-0.392729\pi\)
\(942\) 0 0
\(943\) −40.0000 + 69.2820i −1.30258 + 2.25613i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.00000 3.46410i 0.0649913 0.112568i −0.831699 0.555227i \(-0.812631\pi\)
0.896690 + 0.442659i \(0.145965\pi\)
\(948\) 0 0
\(949\) −12.0000 20.7846i −0.389536 0.674697i
\(950\) 0 0
\(951\) −68.0000 −2.20505
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 8.00000 13.8564i 0.258603 0.447914i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 7.50000 12.9904i 0.241935 0.419045i
\(962\) 0 0
\(963\) 8.00000 + 13.8564i 0.257796 + 0.446516i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) −12.0000 20.7846i −0.385496 0.667698i
\(970\) 0 0
\(971\) 21.0000 36.3731i 0.673922 1.16727i −0.302861 0.953035i \(-0.597942\pi\)
0.976783 0.214232i \(-0.0687250\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 20.0000 34.6410i 0.640513 1.10940i
\(976\) 0 0
\(977\) −13.0000 22.5167i −0.415907 0.720372i 0.579616 0.814890i \(-0.303202\pi\)
−0.995523 + 0.0945177i \(0.969869\pi\)
\(978\) 0 0
\(979\) 72.0000 2.30113
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) −18.0000 31.1769i −0.574111 0.994389i −0.996138 0.0878058i \(-0.972015\pi\)
0.422027 0.906583i \(-0.361319\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −16.0000 + 27.7128i −0.508770 + 0.881216i
\(990\) 0 0
\(991\) 20.0000 + 34.6410i 0.635321 + 1.10041i 0.986447 + 0.164080i \(0.0524655\pi\)
−0.351126 + 0.936328i \(0.614201\pi\)
\(992\) 0 0
\(993\) −24.0000 −0.761617
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 4.00000 + 6.92820i 0.126681 + 0.219418i 0.922389 0.386263i \(-0.126234\pi\)
−0.795708 + 0.605681i \(0.792901\pi\)
\(998\) 0 0
\(999\) 20.0000 34.6410i 0.632772 1.09599i
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.c.961.1 2
4.3 odd 2 1568.2.i.j.961.1 2
7.2 even 3 1568.2.a.h.1.1 1
7.3 odd 6 1568.2.i.k.1537.1 2
7.4 even 3 inner 1568.2.i.c.1537.1 2
7.5 odd 6 224.2.a.a.1.1 1
7.6 odd 2 1568.2.i.k.961.1 2
21.5 even 6 2016.2.a.e.1.1 1
28.3 even 6 1568.2.i.b.1537.1 2
28.11 odd 6 1568.2.i.j.1537.1 2
28.19 even 6 224.2.a.b.1.1 yes 1
28.23 odd 6 1568.2.a.b.1.1 1
28.27 even 2 1568.2.i.b.961.1 2
35.19 odd 6 5600.2.a.t.1.1 1
56.5 odd 6 448.2.a.f.1.1 1
56.19 even 6 448.2.a.b.1.1 1
56.37 even 6 3136.2.a.f.1.1 1
56.51 odd 6 3136.2.a.y.1.1 1
84.47 odd 6 2016.2.a.g.1.1 1
112.5 odd 12 1792.2.b.f.897.2 2
112.19 even 12 1792.2.b.b.897.2 2
112.61 odd 12 1792.2.b.f.897.1 2
112.75 even 12 1792.2.b.b.897.1 2
140.19 even 6 5600.2.a.c.1.1 1
168.5 even 6 4032.2.a.p.1.1 1
168.131 odd 6 4032.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.a.a.1.1 1 7.5 odd 6
224.2.a.b.1.1 yes 1 28.19 even 6
448.2.a.b.1.1 1 56.19 even 6
448.2.a.f.1.1 1 56.5 odd 6
1568.2.a.b.1.1 1 28.23 odd 6
1568.2.a.h.1.1 1 7.2 even 3
1568.2.i.b.961.1 2 28.27 even 2
1568.2.i.b.1537.1 2 28.3 even 6
1568.2.i.c.961.1 2 1.1 even 1 trivial
1568.2.i.c.1537.1 2 7.4 even 3 inner
1568.2.i.j.961.1 2 4.3 odd 2
1568.2.i.j.1537.1 2 28.11 odd 6
1568.2.i.k.961.1 2 7.6 odd 2
1568.2.i.k.1537.1 2 7.3 odd 6
1792.2.b.b.897.1 2 112.75 even 12
1792.2.b.b.897.2 2 112.19 even 12
1792.2.b.f.897.1 2 112.61 odd 12
1792.2.b.f.897.2 2 112.5 odd 12
2016.2.a.e.1.1 1 21.5 even 6
2016.2.a.g.1.1 1 84.47 odd 6
3136.2.a.f.1.1 1 56.37 even 6
3136.2.a.y.1.1 1 56.51 odd 6
4032.2.a.p.1.1 1 168.5 even 6
4032.2.a.z.1.1 1 168.131 odd 6
5600.2.a.c.1.1 1 140.19 even 6
5600.2.a.t.1.1 1 35.19 odd 6