Properties

Label 1568.2.t.c.177.1
Level $1568$
Weight $2$
Character 1568.177
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(177,1568)]
 
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, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.t (of order \(6\), 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 \(\Q(\zeta_{6})\)
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: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 177.1
Root \(-1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1568.177
Dual form 1568.2.t.c.753.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.22474 + 0.707107i) q^{3} +(1.22474 + 0.707107i) q^{5} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-1.22474 + 0.707107i) q^{3} +(1.22474 + 0.707107i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(-2.44949 + 1.41421i) q^{11} -4.24264i q^{13} -2.00000 q^{15} +(3.00000 + 5.19615i) q^{17} +(-3.67423 - 2.12132i) q^{19} +(-3.00000 + 5.19615i) q^{23} +(-1.50000 - 2.59808i) q^{25} -5.65685i q^{27} +2.82843i q^{29} +(-2.00000 - 3.46410i) q^{31} +(2.00000 - 3.46410i) q^{33} +(-7.34847 - 4.24264i) q^{37} +(3.00000 + 5.19615i) q^{39} +6.00000 q^{41} +8.48528i q^{43} +(-1.22474 + 0.707107i) q^{45} +(-7.34847 - 4.24264i) q^{51} +(-4.89898 + 2.82843i) q^{53} -4.00000 q^{55} +6.00000 q^{57} +(1.22474 - 0.707107i) q^{59} +(-11.0227 - 6.36396i) q^{61} +(3.00000 - 5.19615i) q^{65} -8.48528i q^{69} +(-1.00000 - 1.73205i) q^{73} +(3.67423 + 2.12132i) q^{75} +(4.00000 - 6.92820i) q^{79} +(2.50000 + 4.33013i) q^{81} -15.5563i q^{83} +8.48528i q^{85} +(-2.00000 - 3.46410i) q^{87} +(-3.00000 + 5.19615i) q^{89} +(4.89898 + 2.82843i) q^{93} +(-3.00000 - 5.19615i) q^{95} -10.0000 q^{97} -2.82843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{9} - 8 q^{15} + 12 q^{17} - 12 q^{23} - 6 q^{25} - 8 q^{31} + 8 q^{33} + 12 q^{39} + 24 q^{41} - 16 q^{55} + 24 q^{57} + 12 q^{65} - 4 q^{73} + 16 q^{79} + 10 q^{81} - 8 q^{87} - 12 q^{89} - 12 q^{95} - 40 q^{97}+O(q^{100}) \) 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
\(3\) −1.22474 + 0.707107i −0.707107 + 0.408248i −0.809989 0.586445i \(-0.800527\pi\)
0.102882 + 0.994694i \(0.467194\pi\)
\(4\) 0 0
\(5\) 1.22474 + 0.707107i 0.547723 + 0.316228i 0.748203 0.663470i \(-0.230917\pi\)
−0.200480 + 0.979698i \(0.564250\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −2.44949 + 1.41421i −0.738549 + 0.426401i −0.821541 0.570149i \(-0.806886\pi\)
0.0829925 + 0.996550i \(0.473552\pi\)
\(12\) 0 0
\(13\) 4.24264i 1.17670i −0.808608 0.588348i \(-0.799778\pi\)
0.808608 0.588348i \(-0.200222\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) 3.00000 + 5.19615i 0.727607 + 1.26025i 0.957892 + 0.287129i \(0.0927008\pi\)
−0.230285 + 0.973123i \(0.573966\pi\)
\(18\) 0 0
\(19\) −3.67423 2.12132i −0.842927 0.486664i 0.0153309 0.999882i \(-0.495120\pi\)
−0.858258 + 0.513218i \(0.828453\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 + 5.19615i −0.625543 + 1.08347i 0.362892 + 0.931831i \(0.381789\pi\)
−0.988436 + 0.151642i \(0.951544\pi\)
\(24\) 0 0
\(25\) −1.50000 2.59808i −0.300000 0.519615i
\(26\) 0 0
\(27\) 5.65685i 1.08866i
\(28\) 0 0
\(29\) 2.82843i 0.525226i 0.964901 + 0.262613i \(0.0845842\pi\)
−0.964901 + 0.262613i \(0.915416\pi\)
\(30\) 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
\(33\) 2.00000 3.46410i 0.348155 0.603023i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.34847 4.24264i −1.20808 0.697486i −0.245741 0.969335i \(-0.579031\pi\)
−0.962340 + 0.271850i \(0.912365\pi\)
\(38\) 0 0
\(39\) 3.00000 + 5.19615i 0.480384 + 0.832050i
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 8.48528i 1.29399i 0.762493 + 0.646997i \(0.223975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) −1.22474 + 0.707107i −0.182574 + 0.105409i
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −7.34847 4.24264i −1.02899 0.594089i
\(52\) 0 0
\(53\) −4.89898 + 2.82843i −0.672927 + 0.388514i −0.797185 0.603736i \(-0.793678\pi\)
0.124258 + 0.992250i \(0.460345\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 0 0
\(59\) 1.22474 0.707107i 0.159448 0.0920575i −0.418153 0.908377i \(-0.637322\pi\)
0.577601 + 0.816319i \(0.303989\pi\)
\(60\) 0 0
\(61\) −11.0227 6.36396i −1.41131 0.814822i −0.415800 0.909456i \(-0.636499\pi\)
−0.995512 + 0.0946341i \(0.969832\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 5.19615i 0.372104 0.644503i
\(66\) 0 0
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 0 0
\(69\) 8.48528i 1.02151i
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −1.00000 1.73205i −0.117041 0.202721i 0.801553 0.597924i \(-0.204008\pi\)
−0.918594 + 0.395203i \(0.870674\pi\)
\(74\) 0 0
\(75\) 3.67423 + 2.12132i 0.424264 + 0.244949i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 6.92820i 0.450035 0.779484i −0.548352 0.836247i \(-0.684745\pi\)
0.998388 + 0.0567635i \(0.0180781\pi\)
\(80\) 0 0
\(81\) 2.50000 + 4.33013i 0.277778 + 0.481125i
\(82\) 0 0
\(83\) 15.5563i 1.70753i −0.520658 0.853766i \(-0.674313\pi\)
0.520658 0.853766i \(-0.325687\pi\)
\(84\) 0 0
\(85\) 8.48528i 0.920358i
\(86\) 0 0
\(87\) −2.00000 3.46410i −0.214423 0.371391i
\(88\) 0 0
\(89\) −3.00000 + 5.19615i −0.317999 + 0.550791i −0.980071 0.198650i \(-0.936344\pi\)
0.662071 + 0.749441i \(0.269678\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.89898 + 2.82843i 0.508001 + 0.293294i
\(94\) 0 0
\(95\) −3.00000 5.19615i −0.307794 0.533114i
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 2.82843i 0.284268i
\(100\) 0 0
\(101\) −8.57321 + 4.94975i −0.853067 + 0.492518i −0.861684 0.507445i \(-0.830590\pi\)
0.00861771 + 0.999963i \(0.497257\pi\)
\(102\) 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
\(107\) −4.89898 2.82843i −0.473602 0.273434i 0.244144 0.969739i \(-0.421493\pi\)
−0.717746 + 0.696305i \(0.754826\pi\)
\(108\) 0 0
\(109\) −7.34847 + 4.24264i −0.703856 + 0.406371i −0.808782 0.588109i \(-0.799873\pi\)
0.104926 + 0.994480i \(0.466539\pi\)
\(110\) 0 0
\(111\) 12.0000 1.13899
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) −7.34847 + 4.24264i −0.685248 + 0.395628i
\(116\) 0 0
\(117\) 3.67423 + 2.12132i 0.339683 + 0.196116i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.50000 + 2.59808i −0.136364 + 0.236189i
\(122\) 0 0
\(123\) −7.34847 + 4.24264i −0.662589 + 0.382546i
\(124\) 0 0
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) −6.00000 10.3923i −0.528271 0.914991i
\(130\) 0 0
\(131\) −1.22474 0.707107i −0.107006 0.0617802i 0.445542 0.895261i \(-0.353011\pi\)
−0.552548 + 0.833481i \(0.686344\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 4.00000 6.92820i 0.344265 0.596285i
\(136\) 0 0
\(137\) −3.00000 5.19615i −0.256307 0.443937i 0.708942 0.705266i \(-0.249173\pi\)
−0.965250 + 0.261329i \(0.915839\pi\)
\(138\) 0 0
\(139\) 4.24264i 0.359856i 0.983680 + 0.179928i \(0.0575865\pi\)
−0.983680 + 0.179928i \(0.942414\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.00000 + 10.3923i 0.501745 + 0.869048i
\(144\) 0 0
\(145\) −2.00000 + 3.46410i −0.166091 + 0.287678i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.79796 5.65685i −0.802680 0.463428i 0.0417274 0.999129i \(-0.486714\pi\)
−0.844407 + 0.535701i \(0.820047\pi\)
\(150\) 0 0
\(151\) −5.00000 8.66025i −0.406894 0.704761i 0.587646 0.809118i \(-0.300055\pi\)
−0.994540 + 0.104357i \(0.966722\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 5.65685i 0.454369i
\(156\) 0 0
\(157\) 11.0227 6.36396i 0.879708 0.507899i 0.00914557 0.999958i \(-0.497089\pi\)
0.870562 + 0.492059i \(0.163756\pi\)
\(158\) 0 0
\(159\) 4.00000 6.92820i 0.317221 0.549442i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 7.34847 + 4.24264i 0.575577 + 0.332309i 0.759374 0.650655i \(-0.225506\pi\)
−0.183797 + 0.982964i \(0.558839\pi\)
\(164\) 0 0
\(165\) 4.89898 2.82843i 0.381385 0.220193i
\(166\) 0 0
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 3.67423 2.12132i 0.280976 0.162221i
\(172\) 0 0
\(173\) 8.57321 + 4.94975i 0.651809 + 0.376322i 0.789149 0.614202i \(-0.210522\pi\)
−0.137340 + 0.990524i \(0.543855\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.00000 + 1.73205i −0.0751646 + 0.130189i
\(178\) 0 0
\(179\) 4.89898 2.82843i 0.366167 0.211407i −0.305616 0.952155i \(-0.598862\pi\)
0.671783 + 0.740748i \(0.265529\pi\)
\(180\) 0 0
\(181\) 12.7279i 0.946059i 0.881047 + 0.473029i \(0.156840\pi\)
−0.881047 + 0.473029i \(0.843160\pi\)
\(182\) 0 0
\(183\) 18.0000 1.33060
\(184\) 0 0
\(185\) −6.00000 10.3923i −0.441129 0.764057i
\(186\) 0 0
\(187\) −14.6969 8.48528i −1.07475 0.620505i
\(188\) 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
\(193\) 2.00000 + 3.46410i 0.143963 + 0.249351i 0.928986 0.370116i \(-0.120682\pi\)
−0.785022 + 0.619467i \(0.787349\pi\)
\(194\) 0 0
\(195\) 8.48528i 0.607644i
\(196\) 0 0
\(197\) 5.65685i 0.403034i −0.979485 0.201517i \(-0.935413\pi\)
0.979485 0.201517i \(-0.0645872\pi\)
\(198\) 0 0
\(199\) 10.0000 + 17.3205i 0.708881 + 1.22782i 0.965272 + 0.261245i \(0.0841331\pi\)
−0.256391 + 0.966573i \(0.582534\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 7.34847 + 4.24264i 0.513239 + 0.296319i
\(206\) 0 0
\(207\) −3.00000 5.19615i −0.208514 0.361158i
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 16.9706i 1.16830i 0.811645 + 0.584151i \(0.198572\pi\)
−0.811645 + 0.584151i \(0.801428\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.00000 + 10.3923i −0.409197 + 0.708749i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2.44949 + 1.41421i 0.165521 + 0.0955637i
\(220\) 0 0
\(221\) 22.0454 12.7279i 1.48293 0.856173i
\(222\) 0 0
\(223\) 28.0000 1.87502 0.937509 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 0 0
\(227\) 8.57321 4.94975i 0.569024 0.328526i −0.187735 0.982220i \(-0.560115\pi\)
0.756760 + 0.653693i \(0.226781\pi\)
\(228\) 0 0
\(229\) −3.67423 2.12132i −0.242800 0.140181i 0.373663 0.927565i \(-0.378102\pi\)
−0.616463 + 0.787384i \(0.711435\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.00000 + 5.19615i −0.196537 + 0.340411i −0.947403 0.320043i \(-0.896303\pi\)
0.750867 + 0.660454i \(0.229636\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 11.3137i 0.734904i
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 5.00000 + 8.66025i 0.322078 + 0.557856i 0.980917 0.194429i \(-0.0622852\pi\)
−0.658838 + 0.752285i \(0.728952\pi\)
\(242\) 0 0
\(243\) 8.57321 + 4.94975i 0.549972 + 0.317526i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −9.00000 + 15.5885i −0.572656 + 0.991870i
\(248\) 0 0
\(249\) 11.0000 + 19.0526i 0.697097 + 1.20741i
\(250\) 0 0
\(251\) 18.3848i 1.16044i 0.814461 + 0.580218i \(0.197033\pi\)
−0.814461 + 0.580218i \(0.802967\pi\)
\(252\) 0 0
\(253\) 16.9706i 1.06693i
\(254\) 0 0
\(255\) −6.00000 10.3923i −0.375735 0.650791i
\(256\) 0 0
\(257\) 3.00000 5.19615i 0.187135 0.324127i −0.757159 0.653231i \(-0.773413\pi\)
0.944294 + 0.329104i \(0.106747\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.44949 1.41421i −0.151620 0.0875376i
\(262\) 0 0
\(263\) 12.0000 + 20.7846i 0.739952 + 1.28163i 0.952517 + 0.304487i \(0.0984850\pi\)
−0.212565 + 0.977147i \(0.568182\pi\)
\(264\) 0 0
\(265\) −8.00000 −0.491436
\(266\) 0 0
\(267\) 8.48528i 0.519291i
\(268\) 0 0
\(269\) 6.12372 3.53553i 0.373370 0.215565i −0.301560 0.953447i \(-0.597507\pi\)
0.674930 + 0.737882i \(0.264174\pi\)
\(270\) 0 0
\(271\) 10.0000 17.3205i 0.607457 1.05215i −0.384201 0.923249i \(-0.625523\pi\)
0.991658 0.128897i \(-0.0411435\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.34847 + 4.24264i 0.443129 + 0.255841i
\(276\) 0 0
\(277\) −14.6969 + 8.48528i −0.883053 + 0.509831i −0.871664 0.490104i \(-0.836959\pi\)
−0.0113895 + 0.999935i \(0.503625\pi\)
\(278\) 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
\(283\) −11.0227 + 6.36396i −0.655232 + 0.378298i −0.790458 0.612517i \(-0.790157\pi\)
0.135226 + 0.990815i \(0.456824\pi\)
\(284\) 0 0
\(285\) 7.34847 + 4.24264i 0.435286 + 0.251312i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −9.50000 + 16.4545i −0.558824 + 0.967911i
\(290\) 0 0
\(291\) 12.2474 7.07107i 0.717958 0.414513i
\(292\) 0 0
\(293\) 24.0416i 1.40453i 0.711917 + 0.702264i \(0.247827\pi\)
−0.711917 + 0.702264i \(0.752173\pi\)
\(294\) 0 0
\(295\) 2.00000 0.116445
\(296\) 0 0
\(297\) 8.00000 + 13.8564i 0.464207 + 0.804030i
\(298\) 0 0
\(299\) 22.0454 + 12.7279i 1.27492 + 0.736075i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 7.00000 12.1244i 0.402139 0.696526i
\(304\) 0 0
\(305\) −9.00000 15.5885i −0.515339 0.892592i
\(306\) 0 0
\(307\) 12.7279i 0.726421i 0.931707 + 0.363210i \(0.118319\pi\)
−0.931707 + 0.363210i \(0.881681\pi\)
\(308\) 0 0
\(309\) 5.65685i 0.321807i
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 5.00000 8.66025i 0.282617 0.489506i −0.689412 0.724370i \(-0.742131\pi\)
0.972028 + 0.234863i \(0.0754642\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 19.5959 + 11.3137i 1.10062 + 0.635441i 0.936383 0.350980i \(-0.114152\pi\)
0.164234 + 0.986421i \(0.447485\pi\)
\(318\) 0 0
\(319\) −4.00000 6.92820i −0.223957 0.387905i
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 25.4558i 1.41640i
\(324\) 0 0
\(325\) −11.0227 + 6.36396i −0.611430 + 0.353009i
\(326\) 0 0
\(327\) 6.00000 10.3923i 0.331801 0.574696i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 22.0454 + 12.7279i 1.21173 + 0.699590i 0.963135 0.269019i \(-0.0866994\pi\)
0.248590 + 0.968609i \(0.420033\pi\)
\(332\) 0 0
\(333\) 7.34847 4.24264i 0.402694 0.232495i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) 0 0
\(339\) 14.6969 8.48528i 0.798228 0.460857i
\(340\) 0 0
\(341\) 9.79796 + 5.65685i 0.530589 + 0.306336i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 6.00000 10.3923i 0.323029 0.559503i
\(346\) 0 0
\(347\) 12.2474 7.07107i 0.657477 0.379595i −0.133838 0.991003i \(-0.542730\pi\)
0.791315 + 0.611408i \(0.209397\pi\)
\(348\) 0 0
\(349\) 4.24264i 0.227103i 0.993532 + 0.113552i \(0.0362227\pi\)
−0.993532 + 0.113552i \(0.963777\pi\)
\(350\) 0 0
\(351\) −24.0000 −1.28103
\(352\) 0 0
\(353\) −3.00000 5.19615i −0.159674 0.276563i 0.775077 0.631867i \(-0.217711\pi\)
−0.934751 + 0.355303i \(0.884378\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.0000 + 25.9808i −0.791670 + 1.37121i 0.133263 + 0.991081i \(0.457455\pi\)
−0.924932 + 0.380131i \(0.875879\pi\)
\(360\) 0 0
\(361\) −0.500000 0.866025i −0.0263158 0.0455803i
\(362\) 0 0
\(363\) 4.24264i 0.222681i
\(364\) 0 0
\(365\) 2.82843i 0.148047i
\(366\) 0 0
\(367\) −14.0000 24.2487i −0.730794 1.26577i −0.956544 0.291587i \(-0.905817\pi\)
0.225750 0.974185i \(-0.427517\pi\)
\(368\) 0 0
\(369\) −3.00000 + 5.19615i −0.156174 + 0.270501i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 29.3939 + 16.9706i 1.52196 + 0.878702i 0.999664 + 0.0259367i \(0.00825685\pi\)
0.522294 + 0.852766i \(0.325076\pi\)
\(374\) 0 0
\(375\) 8.00000 + 13.8564i 0.413118 + 0.715542i
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 25.4558i 1.30758i −0.756677 0.653789i \(-0.773178\pi\)
0.756677 0.653789i \(-0.226822\pi\)
\(380\) 0 0
\(381\) 2.44949 1.41421i 0.125491 0.0724524i
\(382\) 0 0
\(383\) −12.0000 + 20.7846i −0.613171 + 1.06204i 0.377531 + 0.925997i \(0.376773\pi\)
−0.990702 + 0.136047i \(0.956560\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −7.34847 4.24264i −0.373544 0.215666i
\(388\) 0 0
\(389\) 2.44949 1.41421i 0.124194 0.0717035i −0.436616 0.899648i \(-0.643823\pi\)
0.560810 + 0.827945i \(0.310490\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 0 0
\(393\) 2.00000 0.100887
\(394\) 0 0
\(395\) 9.79796 5.65685i 0.492989 0.284627i
\(396\) 0 0
\(397\) −18.3712 10.6066i −0.922023 0.532330i −0.0377429 0.999287i \(-0.512017\pi\)
−0.884280 + 0.466957i \(0.845350\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 20.7846i 0.599251 1.03793i −0.393680 0.919247i \(-0.628798\pi\)
0.992932 0.118686i \(-0.0378683\pi\)
\(402\) 0 0
\(403\) −14.6969 + 8.48528i −0.732107 + 0.422682i
\(404\) 0 0
\(405\) 7.07107i 0.351364i
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) 11.0000 + 19.0526i 0.543915 + 0.942088i 0.998674 + 0.0514740i \(0.0163919\pi\)
−0.454759 + 0.890614i \(0.650275\pi\)
\(410\) 0 0
\(411\) 7.34847 + 4.24264i 0.362473 + 0.209274i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 11.0000 19.0526i 0.539969 0.935253i
\(416\) 0 0
\(417\) −3.00000 5.19615i −0.146911 0.254457i
\(418\) 0 0
\(419\) 26.8701i 1.31269i 0.754462 + 0.656344i \(0.227898\pi\)
−0.754462 + 0.656344i \(0.772102\pi\)
\(420\) 0 0
\(421\) 16.9706i 0.827095i 0.910483 + 0.413547i \(0.135710\pi\)
−0.910483 + 0.413547i \(0.864290\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 9.00000 15.5885i 0.436564 0.756151i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −14.6969 8.48528i −0.709575 0.409673i
\(430\) 0 0
\(431\) −3.00000 5.19615i −0.144505 0.250290i 0.784683 0.619897i \(-0.212826\pi\)
−0.929188 + 0.369607i \(0.879492\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 5.65685i 0.271225i
\(436\) 0 0
\(437\) 22.0454 12.7279i 1.05457 0.608859i
\(438\) 0 0
\(439\) −14.0000 + 24.2487i −0.668184 + 1.15733i 0.310228 + 0.950662i \(0.399595\pi\)
−0.978412 + 0.206666i \(0.933739\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −19.5959 11.3137i −0.931030 0.537531i −0.0438929 0.999036i \(-0.513976\pi\)
−0.887137 + 0.461506i \(0.847309\pi\)
\(444\) 0 0
\(445\) −7.34847 + 4.24264i −0.348351 + 0.201120i
\(446\) 0 0
\(447\) 16.0000 0.756774
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −14.6969 + 8.48528i −0.692052 + 0.399556i
\(452\) 0 0
\(453\) 12.2474 + 7.07107i 0.575435 + 0.332228i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 14.0000 24.2487i 0.654892 1.13431i −0.327028 0.945015i \(-0.606047\pi\)
0.981921 0.189292i \(-0.0606194\pi\)
\(458\) 0 0
\(459\) 29.3939 16.9706i 1.37199 0.792118i
\(460\) 0 0
\(461\) 15.5563i 0.724531i 0.932075 + 0.362266i \(0.117997\pi\)
−0.932075 + 0.362266i \(0.882003\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 0 0
\(465\) 4.00000 + 6.92820i 0.185496 + 0.321288i
\(466\) 0 0
\(467\) 6.12372 + 3.53553i 0.283372 + 0.163605i 0.634949 0.772554i \(-0.281021\pi\)
−0.351577 + 0.936159i \(0.614354\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −9.00000 + 15.5885i −0.414698 + 0.718278i
\(472\) 0 0
\(473\) −12.0000 20.7846i −0.551761 0.955677i
\(474\) 0 0
\(475\) 12.7279i 0.583997i
\(476\) 0 0
\(477\) 5.65685i 0.259010i
\(478\) 0 0
\(479\) −6.00000 10.3923i −0.274147 0.474837i 0.695773 0.718262i \(-0.255062\pi\)
−0.969920 + 0.243426i \(0.921729\pi\)
\(480\) 0 0
\(481\) −18.0000 + 31.1769i −0.820729 + 1.42154i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −12.2474 7.07107i −0.556128 0.321081i
\(486\) 0 0
\(487\) 1.00000 + 1.73205i 0.0453143 + 0.0784867i 0.887793 0.460243i \(-0.152238\pi\)
−0.842479 + 0.538730i \(0.818904\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 39.5980i 1.78703i 0.449032 + 0.893516i \(0.351769\pi\)
−0.449032 + 0.893516i \(0.648231\pi\)
\(492\) 0 0
\(493\) −14.6969 + 8.48528i −0.661917 + 0.382158i
\(494\) 0 0
\(495\) 2.00000 3.46410i 0.0898933 0.155700i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 14.6969 + 8.48528i 0.657925 + 0.379853i 0.791486 0.611187i \(-0.209308\pi\)
−0.133561 + 0.991041i \(0.542641\pi\)
\(500\) 0 0
\(501\) 29.3939 16.9706i 1.31322 0.758189i
\(502\) 0 0
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) −14.0000 −0.622992
\(506\) 0 0
\(507\) 6.12372 3.53553i 0.271964 0.157019i
\(508\) 0 0
\(509\) 1.22474 + 0.707107i 0.0542859 + 0.0313420i 0.526897 0.849929i \(-0.323355\pi\)
−0.472611 + 0.881271i \(0.656689\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −12.0000 + 20.7846i −0.529813 + 0.917663i
\(514\) 0 0
\(515\) −4.89898 + 2.82843i −0.215875 + 0.124635i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 3.00000 + 5.19615i 0.131432 + 0.227648i 0.924229 0.381839i \(-0.124709\pi\)
−0.792797 + 0.609486i \(0.791376\pi\)
\(522\) 0 0
\(523\) −25.7196 14.8492i −1.12464 0.649312i −0.182060 0.983287i \(-0.558276\pi\)
−0.942582 + 0.333975i \(0.891610\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 20.7846i 0.522728 0.905392i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 1.41421i 0.0613716i
\(532\) 0 0
\(533\) 25.4558i 1.10262i
\(534\) 0 0
\(535\) −4.00000 6.92820i −0.172935 0.299532i
\(536\) 0 0
\(537\) −4.00000 + 6.92820i −0.172613 + 0.298974i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 14.6969 + 8.48528i 0.631871 + 0.364811i 0.781476 0.623935i \(-0.214467\pi\)
−0.149605 + 0.988746i \(0.547800\pi\)
\(542\) 0 0
\(543\) −9.00000 15.5885i −0.386227 0.668965i
\(544\) 0 0
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) 8.48528i 0.362804i 0.983409 + 0.181402i \(0.0580636\pi\)
−0.983409 + 0.181402i \(0.941936\pi\)
\(548\) 0 0
\(549\) 11.0227 6.36396i 0.470438 0.271607i
\(550\) 0 0
\(551\) 6.00000 10.3923i 0.255609 0.442727i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 14.6969 + 8.48528i 0.623850 + 0.360180i
\(556\) 0 0
\(557\) −4.89898 + 2.82843i −0.207576 + 0.119844i −0.600185 0.799862i \(-0.704906\pi\)
0.392608 + 0.919706i \(0.371573\pi\)
\(558\) 0 0
\(559\) 36.0000 1.52264
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) 1.22474 0.707107i 0.0516168 0.0298010i −0.473970 0.880541i \(-0.657179\pi\)
0.525586 + 0.850740i \(0.323846\pi\)
\(564\) 0 0
\(565\) −14.6969 8.48528i −0.618305 0.356978i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) −22.0454 + 12.7279i −0.922572 + 0.532647i −0.884455 0.466626i \(-0.845469\pi\)
−0.0381170 + 0.999273i \(0.512136\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 18.0000 0.750652
\(576\) 0 0
\(577\) −19.0000 32.9090i −0.790980 1.37002i −0.925361 0.379088i \(-0.876238\pi\)
0.134380 0.990930i \(-0.457096\pi\)
\(578\) 0 0
\(579\) −4.89898 2.82843i −0.203595 0.117545i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 8.00000 13.8564i 0.331326 0.573874i
\(584\) 0 0
\(585\) 3.00000 + 5.19615i 0.124035 + 0.214834i
\(586\) 0 0
\(587\) 41.0122i 1.69275i −0.532584 0.846377i \(-0.678779\pi\)
0.532584 0.846377i \(-0.321221\pi\)
\(588\) 0 0
\(589\) 16.9706i 0.699260i
\(590\) 0 0
\(591\) 4.00000 + 6.92820i 0.164538 + 0.284988i
\(592\) 0 0
\(593\) 15.0000 25.9808i 0.615976 1.06690i −0.374236 0.927333i \(-0.622095\pi\)
0.990212 0.139569i \(-0.0445716\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −24.4949 14.1421i −1.00251 0.578799i
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.67423 + 2.12132i −0.149379 + 0.0862439i
\(606\) 0 0
\(607\) 16.0000 27.7128i 0.649420 1.12483i −0.333842 0.942629i \(-0.608345\pi\)
0.983262 0.182199i \(-0.0583216\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −7.34847 + 4.24264i −0.296802 + 0.171359i −0.641005 0.767536i \(-0.721482\pi\)
0.344203 + 0.938895i \(0.388149\pi\)
\(614\) 0 0
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) 24.0000 0.966204 0.483102 0.875564i \(-0.339510\pi\)
0.483102 + 0.875564i \(0.339510\pi\)
\(618\) 0 0
\(619\) 3.67423 2.12132i 0.147680 0.0852631i −0.424339 0.905503i \(-0.639494\pi\)
0.572019 + 0.820240i \(0.306160\pi\)
\(620\) 0 0
\(621\) 29.3939 + 16.9706i 1.17954 + 0.681005i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.500000 0.866025i 0.0200000 0.0346410i
\(626\) 0 0
\(627\) −14.6969 + 8.48528i −0.586939 + 0.338869i
\(628\) 0 0
\(629\) 50.9117i 2.02998i
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) −12.0000 20.7846i −0.476957 0.826114i
\(634\) 0 0
\(635\) −2.44949 1.41421i −0.0972050 0.0561214i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 24.0000 + 41.5692i 0.947943 + 1.64189i 0.749749 + 0.661723i \(0.230174\pi\)
0.198194 + 0.980163i \(0.436492\pi\)
\(642\) 0 0
\(643\) 21.2132i 0.836567i −0.908317 0.418284i \(-0.862632\pi\)
0.908317 0.418284i \(-0.137368\pi\)
\(644\) 0 0
\(645\) 16.9706i 0.668215i
\(646\) 0 0
\(647\) 6.00000 + 10.3923i 0.235884 + 0.408564i 0.959529 0.281609i \(-0.0908680\pi\)
−0.723645 + 0.690172i \(0.757535\pi\)
\(648\) 0 0
\(649\) −2.00000 + 3.46410i −0.0785069 + 0.135978i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −31.8434 18.3848i −1.24613 0.719452i −0.275792 0.961217i \(-0.588940\pi\)
−0.970335 + 0.241765i \(0.922274\pi\)
\(654\) 0 0
\(655\) −1.00000 1.73205i −0.0390732 0.0676768i
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 2.82843i 0.110180i −0.998481 0.0550899i \(-0.982455\pi\)
0.998481 0.0550899i \(-0.0175446\pi\)
\(660\) 0 0
\(661\) −33.0681 + 19.0919i −1.28620 + 0.742588i −0.977974 0.208725i \(-0.933069\pi\)
−0.308226 + 0.951313i \(0.599735\pi\)
\(662\) 0 0
\(663\) −18.0000 + 31.1769i −0.699062 + 1.21081i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −14.6969 8.48528i −0.569068 0.328551i
\(668\) 0 0
\(669\) −34.2929 + 19.7990i −1.32584 + 0.765473i
\(670\) 0 0
\(671\) 36.0000 1.38976
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 0 0
\(675\) −14.6969 + 8.48528i −0.565685 + 0.326599i
\(676\) 0 0
\(677\) 8.57321 + 4.94975i 0.329495 + 0.190234i 0.655617 0.755094i \(-0.272409\pi\)
−0.326122 + 0.945328i \(0.605742\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −7.00000 + 12.1244i −0.268241 + 0.464606i
\(682\) 0 0
\(683\) 4.89898 2.82843i 0.187454 0.108227i −0.403336 0.915052i \(-0.632149\pi\)
0.590790 + 0.806825i \(0.298816\pi\)
\(684\) 0 0
\(685\) 8.48528i 0.324206i
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) 0 0
\(689\) 12.0000 + 20.7846i 0.457164 + 0.791831i
\(690\) 0 0
\(691\) −11.0227 6.36396i −0.419323 0.242096i 0.275464 0.961311i \(-0.411168\pi\)
−0.694788 + 0.719215i \(0.744502\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.00000 + 5.19615i −0.113796 + 0.197101i
\(696\) 0 0
\(697\) 18.0000 + 31.1769i 0.681799 + 1.18091i
\(698\) 0 0
\(699\) 8.48528i 0.320943i
\(700\) 0 0
\(701\) 19.7990i 0.747798i 0.927470 + 0.373899i \(0.121979\pi\)
−0.927470 + 0.373899i \(0.878021\pi\)
\(702\) 0 0
\(703\) 18.0000 + 31.1769i 0.678883 + 1.17586i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 22.0454 + 12.7279i 0.827933 + 0.478007i 0.853144 0.521675i \(-0.174693\pi\)
−0.0252116 + 0.999682i \(0.508026\pi\)
\(710\) 0 0
\(711\) 4.00000 + 6.92820i 0.150012 + 0.259828i
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 16.9706i 0.634663i
\(716\) 0 0
\(717\) 7.34847 4.24264i 0.274434 0.158444i
\(718\) 0 0
\(719\) 12.0000 20.7846i 0.447524 0.775135i −0.550700 0.834703i \(-0.685639\pi\)
0.998224 + 0.0595683i \(0.0189724\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −12.2474 7.07107i −0.455488 0.262976i
\(724\) 0 0
\(725\) 7.34847 4.24264i 0.272915 0.157568i
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) −29.0000 −1.07407
\(730\) 0 0
\(731\) −44.0908 + 25.4558i −1.63076 + 0.941518i
\(732\) 0 0
\(733\) −25.7196 14.8492i −0.949977 0.548469i −0.0569030 0.998380i \(-0.518123\pi\)
−0.893074 + 0.449910i \(0.851456\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −36.7423 + 21.2132i −1.35159 + 0.780340i −0.988472 0.151403i \(-0.951621\pi\)
−0.363117 + 0.931744i \(0.618287\pi\)
\(740\) 0 0
\(741\) 25.4558i 0.935144i
\(742\) 0 0
\(743\) 30.0000 1.10059 0.550297 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) 0 0
\(745\) −8.00000 13.8564i −0.293097 0.507659i
\(746\) 0 0
\(747\) 13.4722 + 7.77817i 0.492922 + 0.284589i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −11.0000 + 19.0526i −0.401396 + 0.695238i −0.993895 0.110333i \(-0.964808\pi\)
0.592499 + 0.805571i \(0.298141\pi\)
\(752\) 0 0
\(753\) −13.0000 22.5167i −0.473746 0.820553i
\(754\) 0 0
\(755\) 14.1421i 0.514685i
\(756\) 0 0
\(757\) 25.4558i 0.925208i 0.886565 + 0.462604i \(0.153085\pi\)
−0.886565 + 0.462604i \(0.846915\pi\)
\(758\) 0 0
\(759\) 12.0000 + 20.7846i 0.435572 + 0.754434i
\(760\) 0 0
\(761\) −15.0000 + 25.9808i −0.543750 + 0.941802i 0.454935 + 0.890525i \(0.349663\pi\)
−0.998684 + 0.0512772i \(0.983671\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −7.34847 4.24264i −0.265684 0.153393i
\(766\) 0 0
\(767\) −3.00000 5.19615i −0.108324 0.187622i
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 8.48528i 0.305590i
\(772\) 0 0
\(773\) 28.1691 16.2635i 1.01317 0.584956i 0.101054 0.994881i \(-0.467778\pi\)
0.912119 + 0.409925i \(0.134445\pi\)
\(774\) 0 0
\(775\) −6.00000 + 10.3923i −0.215526 + 0.373303i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −22.0454 12.7279i −0.789859 0.456025i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 16.0000 0.571793
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) 33.0681 19.0919i 1.17875 0.680552i 0.223026 0.974813i \(-0.428407\pi\)
0.955725 + 0.294260i \(0.0950733\pi\)
\(788\) 0 0
\(789\) −29.3939 16.9706i −1.04645 0.604168i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −27.0000 + 46.7654i −0.958798 + 1.66069i
\(794\) 0 0
\(795\) 9.79796 5.65685i 0.347498 0.200628i
\(796\) 0 0
\(797\) 26.8701i 0.951786i −0.879503 0.475893i \(-0.842125\pi\)
0.879503 0.475893i \(-0.157875\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −3.00000 5.19615i −0.106000 0.183597i
\(802\) 0 0
\(803\) 4.89898 + 2.82843i 0.172881 + 0.0998130i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −5.00000 + 8.66025i −0.176008 + 0.304855i
\(808\) 0 0
\(809\) −6.00000 10.3923i −0.210949 0.365374i 0.741063 0.671436i \(-0.234322\pi\)
−0.952012 + 0.306062i \(0.900989\pi\)
\(810\) 0 0
\(811\) 12.7279i 0.446938i −0.974711 0.223469i \(-0.928262\pi\)
0.974711 0.223469i \(-0.0717381\pi\)
\(812\) 0 0
\(813\) 28.2843i 0.991973i
\(814\) 0 0
\(815\) 6.00000 + 10.3923i 0.210171 + 0.364027i
\(816\) 0 0
\(817\) 18.0000 31.1769i 0.629740 1.09074i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19.5959 + 11.3137i 0.683902 + 0.394851i 0.801324 0.598231i \(-0.204129\pi\)
−0.117421 + 0.993082i \(0.537463\pi\)
\(822\) 0 0
\(823\) 16.0000 + 27.7128i 0.557725 + 0.966008i 0.997686 + 0.0679910i \(0.0216589\pi\)
−0.439961 + 0.898017i \(0.645008\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) 45.2548i 1.57366i −0.617167 0.786832i \(-0.711720\pi\)
0.617167 0.786832i \(-0.288280\pi\)
\(828\) 0 0
\(829\) −18.3712 + 10.6066i −0.638057 + 0.368383i −0.783866 0.620930i \(-0.786755\pi\)
0.145809 + 0.989313i \(0.453422\pi\)
\(830\) 0 0
\(831\) 12.0000 20.7846i 0.416275 0.721010i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −29.3939 16.9706i −1.01722 0.587291i
\(836\) 0 0
\(837\) −19.5959 + 11.3137i −0.677334 + 0.391059i
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) 0 0
\(843\) 7.34847 4.24264i 0.253095 0.146124i
\(844\) 0 0
\(845\) −6.12372 3.53553i −0.210663 0.121626i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 9.00000 15.5885i 0.308879 0.534994i
\(850\) 0 0
\(851\) 44.0908 25.4558i 1.51141 0.872615i
\(852\) 0 0
\(853\) 4.24264i 0.145265i 0.997359 + 0.0726326i \(0.0231401\pi\)
−0.997359 + 0.0726326i \(0.976860\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 0 0
\(857\) −21.0000 36.3731i −0.717346 1.24248i −0.962048 0.272882i \(-0.912023\pi\)
0.244701 0.969599i \(-0.421310\pi\)
\(858\) 0 0
\(859\) −40.4166 23.3345i −1.37900 0.796164i −0.386957 0.922098i \(-0.626474\pi\)
−0.992039 + 0.125934i \(0.959807\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 12.0000 20.7846i 0.408485 0.707516i −0.586235 0.810141i \(-0.699391\pi\)
0.994720 + 0.102624i \(0.0327240\pi\)
\(864\) 0 0
\(865\) 7.00000 + 12.1244i 0.238007 + 0.412240i
\(866\) 0 0
\(867\) 26.8701i 0.912555i
\(868\) 0 0
\(869\) 22.6274i 0.767583i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 5.00000 8.66025i 0.169224 0.293105i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −36.7423 21.2132i −1.24070 0.716319i −0.271464 0.962449i \(-0.587508\pi\)
−0.969237 + 0.246130i \(0.920841\pi\)
\(878\) 0 0
\(879\) −17.0000 29.4449i −0.573396 0.993151i
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 50.9117i 1.71331i 0.515886 + 0.856657i \(0.327463\pi\)
−0.515886 + 0.856657i \(0.672537\pi\)
\(884\) 0 0
\(885\) −2.44949 + 1.41421i −0.0823387 + 0.0475383i
\(886\) 0 0
\(887\) 24.0000 41.5692i 0.805841 1.39576i −0.109881 0.993945i \(-0.535047\pi\)
0.915722 0.401813i \(-0.131620\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −12.2474 7.07107i −0.410305 0.236890i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 8.00000 0.267411
\(896\) 0 0
\(897\) −36.0000 −1.20201
\(898\) 0 0
\(899\) 9.79796 5.65685i 0.326780 0.188667i
\(900\) 0 0
\(901\) −29.3939 16.9706i −0.979252 0.565371i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9.00000 + 15.5885i −0.299170 + 0.518178i
\(906\) 0 0
\(907\) −29.3939 + 16.9706i −0.976008 + 0.563498i −0.901062 0.433689i \(-0.857212\pi\)
−0.0749452 + 0.997188i \(0.523878\pi\)
\(908\) 0 0
\(909\) 9.89949i 0.328346i
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 22.0000 + 38.1051i 0.728094 + 1.26110i
\(914\) 0 0
\(915\) 22.0454 + 12.7279i 0.728799 + 0.420772i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −8.00000 + 13.8564i −0.263896 + 0.457081i −0.967274 0.253735i \(-0.918341\pi\)
0.703378 + 0.710816i \(0.251674\pi\)
\(920\) 0 0
\(921\) −9.00000 15.5885i −0.296560 0.513657i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 25.4558i 0.836983i
\(926\) 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
\(933\) 0 0
\(934\) 0 0
\(935\) −12.0000 20.7846i −0.392442 0.679729i
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 14.1421i 0.461511i
\(940\) 0 0
\(941\) −45.3156 + 26.1630i −1.47725 + 0.852888i −0.999670 0.0257029i \(-0.991818\pi\)
−0.477575 + 0.878591i \(0.658484\pi\)
\(942\) 0 0
\(943\) −18.0000 + 31.1769i −0.586161 + 1.01526i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.44949 + 1.41421i 0.0795977 + 0.0459558i 0.539271 0.842133i \(-0.318700\pi\)
−0.459673 + 0.888088i \(0.652033\pi\)
\(948\) 0 0
\(949\) −7.34847 + 4.24264i −0.238541 + 0.137722i
\(950\) 0 0
\(951\) −32.0000 −1.03767
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 9.79796 + 5.65685i 0.316723 + 0.182860i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 7.50000 12.9904i 0.241935 0.419045i
\(962\) 0 0
\(963\) 4.89898 2.82843i 0.157867 0.0911448i
\(964\) 0 0
\(965\) 5.65685i 0.182101i
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) 18.0000 + 31.1769i 0.578243 + 1.00155i
\(970\) 0 0
\(971\) 28.1691 + 16.2635i 0.903990 + 0.521919i 0.878493 0.477756i \(-0.158550\pi\)
0.0254978 + 0.999675i \(0.491883\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 9.00000 15.5885i 0.288231 0.499230i
\(976\) 0 0
\(977\) 9.00000 + 15.5885i 0.287936 + 0.498719i 0.973317 0.229465i \(-0.0736978\pi\)
−0.685381 + 0.728184i \(0.740364\pi\)
\(978\) 0 0
\(979\) 16.9706i 0.542382i
\(980\) 0 0
\(981\) 8.48528i 0.270914i
\(982\) 0 0
\(983\) −6.00000 10.3923i −0.191370 0.331463i 0.754334 0.656490i \(-0.227960\pi\)
−0.945705 + 0.325027i \(0.894626\pi\)
\(984\) 0 0
\(985\) 4.00000 6.92820i 0.127451 0.220751i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −44.0908 25.4558i −1.40201 0.809449i
\(990\) 0 0
\(991\) −8.00000 13.8564i −0.254128 0.440163i 0.710530 0.703667i \(-0.248455\pi\)
−0.964658 + 0.263504i \(0.915122\pi\)
\(992\) 0 0
\(993\) −36.0000 −1.14243
\(994\) 0 0
\(995\) 28.2843i 0.896672i
\(996\) 0 0
\(997\) 18.3712 10.6066i 0.581821 0.335914i −0.180036 0.983660i \(-0.557621\pi\)
0.761857 + 0.647746i \(0.224288\pi\)
\(998\) 0 0
\(999\) −24.0000 + 41.5692i −0.759326 + 1.31519i
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.t.c.177.1 4
4.3 odd 2 392.2.p.a.373.1 4
7.2 even 3 224.2.b.a.113.1 2
7.3 odd 6 1568.2.t.b.753.1 4
7.4 even 3 inner 1568.2.t.c.753.2 4
7.5 odd 6 1568.2.b.a.785.2 2
7.6 odd 2 1568.2.t.b.177.2 4
8.3 odd 2 392.2.p.a.373.2 4
8.5 even 2 inner 1568.2.t.c.177.2 4
21.2 odd 6 2016.2.c.a.1009.2 2
28.3 even 6 392.2.p.b.165.2 4
28.11 odd 6 392.2.p.a.165.2 4
28.19 even 6 392.2.b.b.197.2 2
28.23 odd 6 56.2.b.a.29.2 yes 2
28.27 even 2 392.2.p.b.373.1 4
56.3 even 6 392.2.p.b.165.1 4
56.5 odd 6 1568.2.b.a.785.1 2
56.11 odd 6 392.2.p.a.165.1 4
56.13 odd 2 1568.2.t.b.177.1 4
56.19 even 6 392.2.b.b.197.1 2
56.27 even 2 392.2.p.b.373.2 4
56.37 even 6 224.2.b.a.113.2 2
56.45 odd 6 1568.2.t.b.753.2 4
56.51 odd 6 56.2.b.a.29.1 2
56.53 even 6 inner 1568.2.t.c.753.1 4
84.23 even 6 504.2.c.a.253.1 2
112.37 even 12 1792.2.a.p.1.1 2
112.51 odd 12 1792.2.a.n.1.1 2
112.93 even 12 1792.2.a.p.1.2 2
112.107 odd 12 1792.2.a.n.1.2 2
168.107 even 6 504.2.c.a.253.2 2
168.149 odd 6 2016.2.c.a.1009.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.b.a.29.1 2 56.51 odd 6
56.2.b.a.29.2 yes 2 28.23 odd 6
224.2.b.a.113.1 2 7.2 even 3
224.2.b.a.113.2 2 56.37 even 6
392.2.b.b.197.1 2 56.19 even 6
392.2.b.b.197.2 2 28.19 even 6
392.2.p.a.165.1 4 56.11 odd 6
392.2.p.a.165.2 4 28.11 odd 6
392.2.p.a.373.1 4 4.3 odd 2
392.2.p.a.373.2 4 8.3 odd 2
392.2.p.b.165.1 4 56.3 even 6
392.2.p.b.165.2 4 28.3 even 6
392.2.p.b.373.1 4 28.27 even 2
392.2.p.b.373.2 4 56.27 even 2
504.2.c.a.253.1 2 84.23 even 6
504.2.c.a.253.2 2 168.107 even 6
1568.2.b.a.785.1 2 56.5 odd 6
1568.2.b.a.785.2 2 7.5 odd 6
1568.2.t.b.177.1 4 56.13 odd 2
1568.2.t.b.177.2 4 7.6 odd 2
1568.2.t.b.753.1 4 7.3 odd 6
1568.2.t.b.753.2 4 56.45 odd 6
1568.2.t.c.177.1 4 1.1 even 1 trivial
1568.2.t.c.177.2 4 8.5 even 2 inner
1568.2.t.c.753.1 4 56.53 even 6 inner
1568.2.t.c.753.2 4 7.4 even 3 inner
1792.2.a.n.1.1 2 112.51 odd 12
1792.2.a.n.1.2 2 112.107 odd 12
1792.2.a.p.1.1 2 112.37 even 12
1792.2.a.p.1.2 2 112.93 even 12
2016.2.c.a.1009.1 2 168.149 odd 6
2016.2.c.a.1009.2 2 21.2 odd 6