Properties

Label 4608.2.d.o.2305.4
Level $4608$
Weight $2$
Character 4608.2305
Analytic conductor $36.795$
Analytic rank $0$
Dimension $4$
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] = [4608,2,Mod(2305,4608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4608, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4608.2305");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.d (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: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1536)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2305.4
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 4608.2305
Dual form 4608.2.d.o.2305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.41421i q^{5} +0.585786 q^{7} +O(q^{10})\) \(q+3.41421i q^{5} +0.585786 q^{7} -2.00000i q^{11} -2.82843i q^{13} +7.65685 q^{17} -5.65685i q^{19} -6.82843 q^{23} -6.65685 q^{25} -3.41421i q^{29} -7.41421 q^{31} +2.00000i q^{35} -1.65685i q^{37} -0.343146 q^{41} -9.65685i q^{43} -4.48528 q^{47} -6.65685 q^{49} -7.89949i q^{53} +6.82843 q^{55} +4.00000i q^{59} +1.65685i q^{61} +9.65685 q^{65} +8.00000i q^{67} -14.8284 q^{71} -9.65685 q^{73} -1.17157i q^{77} -14.2426 q^{79} -13.3137i q^{83} +26.1421i q^{85} +2.00000 q^{89} -1.65685i q^{91} +19.3137 q^{95} -9.31371 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{7} + 8 q^{17} - 16 q^{23} - 4 q^{25} - 24 q^{31} - 24 q^{41} + 16 q^{47} - 4 q^{49} + 16 q^{55} + 16 q^{65} - 48 q^{71} - 16 q^{73} - 40 q^{79} + 8 q^{89} + 32 q^{95} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(3583\) \(4097\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) 3.41421i 1.52688i 0.645877 + 0.763441i \(0.276492\pi\)
−0.645877 + 0.763441i \(0.723508\pi\)
\(6\) 0 0
\(7\) 0.585786 0.221406 0.110703 0.993854i \(-0.464690\pi\)
0.110703 + 0.993854i \(0.464690\pi\)
\(8\) 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
\(13\) − 2.82843i − 0.784465i −0.919866 0.392232i \(-0.871703\pi\)
0.919866 0.392232i \(-0.128297\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.65685 1.85706 0.928530 0.371257i \(-0.121073\pi\)
0.928530 + 0.371257i \(0.121073\pi\)
\(18\) 0 0
\(19\) − 5.65685i − 1.29777i −0.760886 0.648886i \(-0.775235\pi\)
0.760886 0.648886i \(-0.224765\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.82843 −1.42383 −0.711913 0.702268i \(-0.752171\pi\)
−0.711913 + 0.702268i \(0.752171\pi\)
\(24\) 0 0
\(25\) −6.65685 −1.33137
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 3.41421i − 0.634004i −0.948425 0.317002i \(-0.897324\pi\)
0.948425 0.317002i \(-0.102676\pi\)
\(30\) 0 0
\(31\) −7.41421 −1.33163 −0.665816 0.746116i \(-0.731916\pi\)
−0.665816 + 0.746116i \(0.731916\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000i 0.338062i
\(36\) 0 0
\(37\) − 1.65685i − 0.272385i −0.990682 0.136193i \(-0.956513\pi\)
0.990682 0.136193i \(-0.0434866\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.343146 −0.0535904 −0.0267952 0.999641i \(-0.508530\pi\)
−0.0267952 + 0.999641i \(0.508530\pi\)
\(42\) 0 0
\(43\) − 9.65685i − 1.47266i −0.676625 0.736328i \(-0.736558\pi\)
0.676625 0.736328i \(-0.263442\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.48528 −0.654246 −0.327123 0.944982i \(-0.606079\pi\)
−0.327123 + 0.944982i \(0.606079\pi\)
\(48\) 0 0
\(49\) −6.65685 −0.950979
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 7.89949i − 1.08508i −0.840030 0.542540i \(-0.817463\pi\)
0.840030 0.542540i \(-0.182537\pi\)
\(54\) 0 0
\(55\) 6.82843 0.920745
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) 1.65685i 0.212138i 0.994359 + 0.106069i \(0.0338265\pi\)
−0.994359 + 0.106069i \(0.966173\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.65685 1.19779
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.8284 −1.75981 −0.879905 0.475149i \(-0.842394\pi\)
−0.879905 + 0.475149i \(0.842394\pi\)
\(72\) 0 0
\(73\) −9.65685 −1.13025 −0.565125 0.825006i \(-0.691172\pi\)
−0.565125 + 0.825006i \(0.691172\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.17157i − 0.133513i
\(78\) 0 0
\(79\) −14.2426 −1.60242 −0.801211 0.598382i \(-0.795811\pi\)
−0.801211 + 0.598382i \(0.795811\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 13.3137i − 1.46137i −0.682715 0.730685i \(-0.739201\pi\)
0.682715 0.730685i \(-0.260799\pi\)
\(84\) 0 0
\(85\) 26.1421i 2.83551i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) − 1.65685i − 0.173686i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 19.3137 1.98154
\(96\) 0 0
\(97\) −9.31371 −0.945664 −0.472832 0.881153i \(-0.656768\pi\)
−0.472832 + 0.881153i \(0.656768\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.5563i 1.34891i 0.738317 + 0.674454i \(0.235621\pi\)
−0.738317 + 0.674454i \(0.764379\pi\)
\(102\) 0 0
\(103\) −5.07107 −0.499667 −0.249834 0.968289i \(-0.580376\pi\)
−0.249834 + 0.968289i \(0.580376\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7.31371i − 0.707043i −0.935426 0.353521i \(-0.884984\pi\)
0.935426 0.353521i \(-0.115016\pi\)
\(108\) 0 0
\(109\) − 18.8284i − 1.80344i −0.432324 0.901718i \(-0.642306\pi\)
0.432324 0.901718i \(-0.357694\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) − 23.3137i − 2.17401i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.48528 0.411165
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 5.65685i − 0.505964i
\(126\) 0 0
\(127\) 18.7279 1.66183 0.830917 0.556396i \(-0.187816\pi\)
0.830917 + 0.556396i \(0.187816\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 4.00000i − 0.349482i −0.984614 0.174741i \(-0.944091\pi\)
0.984614 0.174741i \(-0.0559088\pi\)
\(132\) 0 0
\(133\) − 3.31371i − 0.287335i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.6569 1.33766 0.668828 0.743417i \(-0.266796\pi\)
0.668828 + 0.743417i \(0.266796\pi\)
\(138\) 0 0
\(139\) − 3.31371i − 0.281065i −0.990076 0.140533i \(-0.955119\pi\)
0.990076 0.140533i \(-0.0448815\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.65685 −0.473050
\(144\) 0 0
\(145\) 11.6569 0.968049
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.75736i 0.471661i 0.971794 + 0.235831i \(0.0757811\pi\)
−0.971794 + 0.235831i \(0.924219\pi\)
\(150\) 0 0
\(151\) 7.41421 0.603360 0.301680 0.953409i \(-0.402453\pi\)
0.301680 + 0.953409i \(0.402453\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 25.3137i − 2.03325i
\(156\) 0 0
\(157\) − 20.0000i − 1.59617i −0.602542 0.798087i \(-0.705846\pi\)
0.602542 0.798087i \(-0.294154\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) − 17.6569i − 1.38299i −0.722380 0.691496i \(-0.756952\pi\)
0.722380 0.691496i \(-0.243048\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.3431 0.800377 0.400188 0.916433i \(-0.368945\pi\)
0.400188 + 0.916433i \(0.368945\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.72792i 0.511514i 0.966741 + 0.255757i \(0.0823248\pi\)
−0.966741 + 0.255757i \(0.917675\pi\)
\(174\) 0 0
\(175\) −3.89949 −0.294774
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 4.00000i − 0.298974i −0.988764 0.149487i \(-0.952238\pi\)
0.988764 0.149487i \(-0.0477622\pi\)
\(180\) 0 0
\(181\) 8.48528i 0.630706i 0.948974 + 0.315353i \(0.102123\pi\)
−0.948974 + 0.315353i \(0.897877\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.65685 0.415900
\(186\) 0 0
\(187\) − 15.3137i − 1.11985i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.9706 1.22795 0.613973 0.789327i \(-0.289570\pi\)
0.613973 + 0.789327i \(0.289570\pi\)
\(192\) 0 0
\(193\) −10.3431 −0.744516 −0.372258 0.928129i \(-0.621416\pi\)
−0.372258 + 0.928129i \(0.621416\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.0711i 1.21626i 0.793836 + 0.608132i \(0.208081\pi\)
−0.793836 + 0.608132i \(0.791919\pi\)
\(198\) 0 0
\(199\) 17.5563 1.24454 0.622268 0.782804i \(-0.286211\pi\)
0.622268 + 0.782804i \(0.286211\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 2.00000i − 0.140372i
\(204\) 0 0
\(205\) − 1.17157i − 0.0818262i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −11.3137 −0.782586
\(210\) 0 0
\(211\) − 3.31371i − 0.228125i −0.993474 0.114063i \(-0.963614\pi\)
0.993474 0.114063i \(-0.0363864\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 32.9706 2.24857
\(216\) 0 0
\(217\) −4.34315 −0.294832
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 21.6569i − 1.45680i
\(222\) 0 0
\(223\) 11.8995 0.796849 0.398425 0.917201i \(-0.369557\pi\)
0.398425 + 0.917201i \(0.369557\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.68629i 0.443785i 0.975071 + 0.221892i \(0.0712234\pi\)
−0.975071 + 0.221892i \(0.928777\pi\)
\(228\) 0 0
\(229\) − 0.485281i − 0.0320683i −0.999871 0.0160341i \(-0.994896\pi\)
0.999871 0.0160341i \(-0.00510405\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −13.3137 −0.872210 −0.436105 0.899896i \(-0.643642\pi\)
−0.436105 + 0.899896i \(0.643642\pi\)
\(234\) 0 0
\(235\) − 15.3137i − 0.998956i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −16.9706 −1.09773 −0.548867 0.835910i \(-0.684941\pi\)
−0.548867 + 0.835910i \(0.684941\pi\)
\(240\) 0 0
\(241\) −6.34315 −0.408598 −0.204299 0.978909i \(-0.565491\pi\)
−0.204299 + 0.978909i \(0.565491\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 22.7279i − 1.45203i
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 22.0000i 1.38863i 0.719672 + 0.694314i \(0.244292\pi\)
−0.719672 + 0.694314i \(0.755708\pi\)
\(252\) 0 0
\(253\) 13.6569i 0.858599i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −21.3137 −1.32951 −0.664756 0.747060i \(-0.731465\pi\)
−0.664756 + 0.747060i \(0.731465\pi\)
\(258\) 0 0
\(259\) − 0.970563i − 0.0603078i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.34315 0.144485 0.0722423 0.997387i \(-0.476985\pi\)
0.0722423 + 0.997387i \(0.476985\pi\)
\(264\) 0 0
\(265\) 26.9706 1.65679
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 2.24264i − 0.136736i −0.997660 0.0683681i \(-0.978221\pi\)
0.997660 0.0683681i \(-0.0217792\pi\)
\(270\) 0 0
\(271\) 16.5858 1.00751 0.503757 0.863845i \(-0.331951\pi\)
0.503757 + 0.863845i \(0.331951\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 13.3137i 0.802847i
\(276\) 0 0
\(277\) 24.4853i 1.47118i 0.677428 + 0.735589i \(0.263094\pi\)
−0.677428 + 0.735589i \(0.736906\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) − 7.31371i − 0.434755i −0.976088 0.217377i \(-0.930250\pi\)
0.976088 0.217377i \(-0.0697502\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.201010 −0.0118653
\(288\) 0 0
\(289\) 41.6274 2.44867
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.10051i 0.473237i 0.971603 + 0.236618i \(0.0760391\pi\)
−0.971603 + 0.236618i \(0.923961\pi\)
\(294\) 0 0
\(295\) −13.6569 −0.795133
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 19.3137i 1.11694i
\(300\) 0 0
\(301\) − 5.65685i − 0.326056i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5.65685 −0.323911
\(306\) 0 0
\(307\) 16.0000i 0.913168i 0.889680 + 0.456584i \(0.150927\pi\)
−0.889680 + 0.456584i \(0.849073\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 11.3137 0.641542 0.320771 0.947157i \(-0.396058\pi\)
0.320771 + 0.947157i \(0.396058\pi\)
\(312\) 0 0
\(313\) 16.9706 0.959233 0.479616 0.877478i \(-0.340776\pi\)
0.479616 + 0.877478i \(0.340776\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.07107i 0.509482i 0.967009 + 0.254741i \(0.0819902\pi\)
−0.967009 + 0.254741i \(0.918010\pi\)
\(318\) 0 0
\(319\) −6.82843 −0.382319
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 43.3137i − 2.41004i
\(324\) 0 0
\(325\) 18.8284i 1.04441i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.62742 −0.144854
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −27.3137 −1.49231
\(336\) 0 0
\(337\) −12.9706 −0.706552 −0.353276 0.935519i \(-0.614932\pi\)
−0.353276 + 0.935519i \(0.614932\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 14.8284i 0.803004i
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.68629i − 0.144208i −0.997397 0.0721038i \(-0.977029\pi\)
0.997397 0.0721038i \(-0.0229713\pi\)
\(348\) 0 0
\(349\) − 23.3137i − 1.24795i −0.781443 0.623977i \(-0.785516\pi\)
0.781443 0.623977i \(-0.214484\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.31371 −0.495719 −0.247859 0.968796i \(-0.579727\pi\)
−0.247859 + 0.968796i \(0.579727\pi\)
\(354\) 0 0
\(355\) − 50.6274i − 2.68702i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −23.7990 −1.25606 −0.628031 0.778188i \(-0.716139\pi\)
−0.628031 + 0.778188i \(0.716139\pi\)
\(360\) 0 0
\(361\) −13.0000 −0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 32.9706i − 1.72576i
\(366\) 0 0
\(367\) −2.72792 −0.142396 −0.0711982 0.997462i \(-0.522682\pi\)
−0.0711982 + 0.997462i \(0.522682\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 4.62742i − 0.240244i
\(372\) 0 0
\(373\) − 32.2843i − 1.67162i −0.549022 0.835808i \(-0.685000\pi\)
0.549022 0.835808i \(-0.315000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9.65685 −0.497353
\(378\) 0 0
\(379\) − 29.6569i − 1.52337i −0.647947 0.761685i \(-0.724372\pi\)
0.647947 0.761685i \(-0.275628\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.31371 0.169323 0.0846613 0.996410i \(-0.473019\pi\)
0.0846613 + 0.996410i \(0.473019\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 13.7574i − 0.697526i −0.937211 0.348763i \(-0.886602\pi\)
0.937211 0.348763i \(-0.113398\pi\)
\(390\) 0 0
\(391\) −52.2843 −2.64413
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 48.6274i − 2.44671i
\(396\) 0 0
\(397\) − 4.00000i − 0.200754i −0.994949 0.100377i \(-0.967995\pi\)
0.994949 0.100377i \(-0.0320049\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 20.3431 1.01589 0.507944 0.861390i \(-0.330406\pi\)
0.507944 + 0.861390i \(0.330406\pi\)
\(402\) 0 0
\(403\) 20.9706i 1.04462i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.31371 −0.164254
\(408\) 0 0
\(409\) 21.3137 1.05390 0.526948 0.849898i \(-0.323336\pi\)
0.526948 + 0.849898i \(0.323336\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.34315i 0.115299i
\(414\) 0 0
\(415\) 45.4558 2.23134
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 21.3137i 1.04124i 0.853788 + 0.520621i \(0.174300\pi\)
−0.853788 + 0.520621i \(0.825700\pi\)
\(420\) 0 0
\(421\) 6.14214i 0.299349i 0.988735 + 0.149675i \(0.0478226\pi\)
−0.988735 + 0.149675i \(0.952177\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −50.9706 −2.47244
\(426\) 0 0
\(427\) 0.970563i 0.0469688i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −27.1127 −1.30597 −0.652986 0.757370i \(-0.726484\pi\)
−0.652986 + 0.757370i \(0.726484\pi\)
\(432\) 0 0
\(433\) 28.6274 1.37575 0.687873 0.725831i \(-0.258545\pi\)
0.687873 + 0.725831i \(0.258545\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 38.6274i 1.84780i
\(438\) 0 0
\(439\) 3.89949 0.186113 0.0930564 0.995661i \(-0.470336\pi\)
0.0930564 + 0.995661i \(0.470336\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 39.9411i − 1.89766i −0.315787 0.948830i \(-0.602269\pi\)
0.315787 0.948830i \(-0.397731\pi\)
\(444\) 0 0
\(445\) 6.82843i 0.323698i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3.65685 −0.172578 −0.0862888 0.996270i \(-0.527501\pi\)
−0.0862888 + 0.996270i \(0.527501\pi\)
\(450\) 0 0
\(451\) 0.686292i 0.0323162i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.65685 0.265197
\(456\) 0 0
\(457\) −5.31371 −0.248565 −0.124282 0.992247i \(-0.539663\pi\)
−0.124282 + 0.992247i \(0.539663\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 25.0711i 1.16768i 0.811870 + 0.583838i \(0.198450\pi\)
−0.811870 + 0.583838i \(0.801550\pi\)
\(462\) 0 0
\(463\) −33.5563 −1.55950 −0.779748 0.626094i \(-0.784653\pi\)
−0.779748 + 0.626094i \(0.784653\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13.3137i 0.616085i 0.951373 + 0.308042i \(0.0996739\pi\)
−0.951373 + 0.308042i \(0.900326\pi\)
\(468\) 0 0
\(469\) 4.68629i 0.216393i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −19.3137 −0.888045
\(474\) 0 0
\(475\) 37.6569i 1.72781i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 12.4853 0.570467 0.285234 0.958458i \(-0.407929\pi\)
0.285234 + 0.958458i \(0.407929\pi\)
\(480\) 0 0
\(481\) −4.68629 −0.213676
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 31.7990i − 1.44392i
\(486\) 0 0
\(487\) 25.5563 1.15807 0.579034 0.815303i \(-0.303430\pi\)
0.579034 + 0.815303i \(0.303430\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 23.3137i 1.05213i 0.850443 + 0.526066i \(0.176334\pi\)
−0.850443 + 0.526066i \(0.823666\pi\)
\(492\) 0 0
\(493\) − 26.1421i − 1.17738i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8.68629 −0.389633
\(498\) 0 0
\(499\) − 0.686292i − 0.0307226i −0.999882 0.0153613i \(-0.995110\pi\)
0.999882 0.0153613i \(-0.00488985\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.4853 −0.556691 −0.278346 0.960481i \(-0.589786\pi\)
−0.278346 + 0.960481i \(0.589786\pi\)
\(504\) 0 0
\(505\) −46.2843 −2.05962
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 19.2132i 0.851610i 0.904815 + 0.425805i \(0.140009\pi\)
−0.904815 + 0.425805i \(0.859991\pi\)
\(510\) 0 0
\(511\) −5.65685 −0.250244
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 17.3137i − 0.762933i
\(516\) 0 0
\(517\) 8.97056i 0.394525i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 22.9706 1.00636 0.503179 0.864182i \(-0.332164\pi\)
0.503179 + 0.864182i \(0.332164\pi\)
\(522\) 0 0
\(523\) 37.6569i 1.64662i 0.567593 + 0.823310i \(0.307875\pi\)
−0.567593 + 0.823310i \(0.692125\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −56.7696 −2.47292
\(528\) 0 0
\(529\) 23.6274 1.02728
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.970563i 0.0420397i
\(534\) 0 0
\(535\) 24.9706 1.07957
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 13.3137i 0.573462i
\(540\) 0 0
\(541\) − 10.8284i − 0.465550i −0.972531 0.232775i \(-0.925219\pi\)
0.972531 0.232775i \(-0.0747806\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 64.2843 2.75364
\(546\) 0 0
\(547\) 4.97056i 0.212526i 0.994338 + 0.106263i \(0.0338885\pi\)
−0.994338 + 0.106263i \(0.966111\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −19.3137 −0.822792
\(552\) 0 0
\(553\) −8.34315 −0.354787
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 1.07107i − 0.0453826i −0.999743 0.0226913i \(-0.992777\pi\)
0.999743 0.0226913i \(-0.00722349\pi\)
\(558\) 0 0
\(559\) −27.3137 −1.15525
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 16.6274i 0.700762i 0.936607 + 0.350381i \(0.113948\pi\)
−0.936607 + 0.350381i \(0.886052\pi\)
\(564\) 0 0
\(565\) − 20.4853i − 0.861822i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −14.9706 −0.627599 −0.313799 0.949489i \(-0.601602\pi\)
−0.313799 + 0.949489i \(0.601602\pi\)
\(570\) 0 0
\(571\) − 20.0000i − 0.836974i −0.908223 0.418487i \(-0.862561\pi\)
0.908223 0.418487i \(-0.137439\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 45.4558 1.89564
\(576\) 0 0
\(577\) −28.2843 −1.17749 −0.588745 0.808319i \(-0.700378\pi\)
−0.588745 + 0.808319i \(0.700378\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 7.79899i − 0.323557i
\(582\) 0 0
\(583\) −15.7990 −0.654327
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 31.3137i 1.29246i 0.763145 + 0.646228i \(0.223654\pi\)
−0.763145 + 0.646228i \(0.776346\pi\)
\(588\) 0 0
\(589\) 41.9411i 1.72815i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.00000 0.0821302 0.0410651 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 0 0
\(595\) 15.3137i 0.627801i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −11.5147 −0.470479 −0.235239 0.971937i \(-0.575587\pi\)
−0.235239 + 0.971937i \(0.575587\pi\)
\(600\) 0 0
\(601\) 5.65685 0.230748 0.115374 0.993322i \(-0.463193\pi\)
0.115374 + 0.993322i \(0.463193\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 23.8995i 0.971653i
\(606\) 0 0
\(607\) −3.89949 −0.158276 −0.0791378 0.996864i \(-0.525217\pi\)
−0.0791378 + 0.996864i \(0.525217\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.6863i 0.513232i
\(612\) 0 0
\(613\) 10.6274i 0.429237i 0.976698 + 0.214619i \(0.0688509\pi\)
−0.976698 + 0.214619i \(0.931149\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23.9411 0.963833 0.481917 0.876217i \(-0.339941\pi\)
0.481917 + 0.876217i \(0.339941\pi\)
\(618\) 0 0
\(619\) 4.00000i 0.160774i 0.996764 + 0.0803868i \(0.0256155\pi\)
−0.996764 + 0.0803868i \(0.974384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.17157 0.0469381
\(624\) 0 0
\(625\) −13.9706 −0.558823
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 12.6863i − 0.505836i
\(630\) 0 0
\(631\) 18.9289 0.753549 0.376774 0.926305i \(-0.377033\pi\)
0.376774 + 0.926305i \(0.377033\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 63.9411i 2.53743i
\(636\) 0 0
\(637\) 18.8284i 0.746009i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9.02944 −0.356641 −0.178321 0.983972i \(-0.557066\pi\)
−0.178321 + 0.983972i \(0.557066\pi\)
\(642\) 0 0
\(643\) 32.2843i 1.27317i 0.771208 + 0.636584i \(0.219653\pi\)
−0.771208 + 0.636584i \(0.780347\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 22.8284 0.897478 0.448739 0.893663i \(-0.351873\pi\)
0.448739 + 0.893663i \(0.351873\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 20.5858i − 0.805584i −0.915292 0.402792i \(-0.868040\pi\)
0.915292 0.402792i \(-0.131960\pi\)
\(654\) 0 0
\(655\) 13.6569 0.533617
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 39.3137i − 1.53144i −0.643171 0.765722i \(-0.722382\pi\)
0.643171 0.765722i \(-0.277618\pi\)
\(660\) 0 0
\(661\) − 39.3137i − 1.52913i −0.644549 0.764563i \(-0.722955\pi\)
0.644549 0.764563i \(-0.277045\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 11.3137 0.438727
\(666\) 0 0
\(667\) 23.3137i 0.902710i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3.31371 0.127924
\(672\) 0 0
\(673\) −8.62742 −0.332562 −0.166281 0.986078i \(-0.553176\pi\)
−0.166281 + 0.986078i \(0.553176\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 35.2132i − 1.35335i −0.736280 0.676677i \(-0.763420\pi\)
0.736280 0.676677i \(-0.236580\pi\)
\(678\) 0 0
\(679\) −5.45584 −0.209376
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.68629i 0.102788i 0.998678 + 0.0513940i \(0.0163664\pi\)
−0.998678 + 0.0513940i \(0.983634\pi\)
\(684\) 0 0
\(685\) 53.4558i 2.04244i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −22.3431 −0.851206
\(690\) 0 0
\(691\) 6.34315i 0.241305i 0.992695 + 0.120652i \(0.0384986\pi\)
−0.992695 + 0.120652i \(0.961501\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 11.3137 0.429153
\(696\) 0 0
\(697\) −2.62742 −0.0995205
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 4.58579i − 0.173203i −0.996243 0.0866014i \(-0.972399\pi\)
0.996243 0.0866014i \(-0.0276006\pi\)
\(702\) 0 0
\(703\) −9.37258 −0.353494
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.94113i 0.298657i
\(708\) 0 0
\(709\) 31.1127i 1.16846i 0.811587 + 0.584231i \(0.198604\pi\)
−0.811587 + 0.584231i \(0.801396\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 50.6274 1.89601
\(714\) 0 0
\(715\) − 19.3137i − 0.722292i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 30.8284 1.14971 0.574853 0.818257i \(-0.305059\pi\)
0.574853 + 0.818257i \(0.305059\pi\)
\(720\) 0 0
\(721\) −2.97056 −0.110630
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 22.7279i 0.844094i
\(726\) 0 0
\(727\) −22.2426 −0.824934 −0.412467 0.910973i \(-0.635333\pi\)
−0.412467 + 0.910973i \(0.635333\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 73.9411i − 2.73481i
\(732\) 0 0
\(733\) 23.5147i 0.868536i 0.900784 + 0.434268i \(0.142993\pi\)
−0.900784 + 0.434268i \(0.857007\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 10.6274i 0.390936i 0.980710 + 0.195468i \(0.0626226\pi\)
−0.980710 + 0.195468i \(0.937377\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) −19.6569 −0.720171
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 4.28427i − 0.156544i
\(750\) 0 0
\(751\) −5.27208 −0.192381 −0.0961904 0.995363i \(-0.530666\pi\)
−0.0961904 + 0.995363i \(0.530666\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 25.3137i 0.921260i
\(756\) 0 0
\(757\) − 32.4853i − 1.18070i −0.807148 0.590349i \(-0.798990\pi\)
0.807148 0.590349i \(-0.201010\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 35.6569 1.29256 0.646280 0.763100i \(-0.276324\pi\)
0.646280 + 0.763100i \(0.276324\pi\)
\(762\) 0 0
\(763\) − 11.0294i − 0.399292i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11.3137 0.408514
\(768\) 0 0
\(769\) −2.34315 −0.0844960 −0.0422480 0.999107i \(-0.513452\pi\)
−0.0422480 + 0.999107i \(0.513452\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 6.72792i − 0.241987i −0.992653 0.120993i \(-0.961392\pi\)
0.992653 0.120993i \(-0.0386079\pi\)
\(774\) 0 0
\(775\) 49.3553 1.77290
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.94113i 0.0695480i
\(780\) 0 0
\(781\) 29.6569i 1.06121i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 68.2843 2.43717
\(786\) 0 0
\(787\) − 21.6569i − 0.771983i −0.922502 0.385992i \(-0.873859\pi\)
0.922502 0.385992i \(-0.126141\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.51472 −0.124969
\(792\) 0 0
\(793\) 4.68629 0.166415
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 35.4142i − 1.25444i −0.778844 0.627218i \(-0.784194\pi\)
0.778844 0.627218i \(-0.215806\pi\)
\(798\) 0 0
\(799\) −34.3431 −1.21497
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 19.3137i 0.681566i
\(804\) 0 0
\(805\) − 13.6569i − 0.481341i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.02944 −0.176826 −0.0884128 0.996084i \(-0.528179\pi\)
−0.0884128 + 0.996084i \(0.528179\pi\)
\(810\) 0 0
\(811\) − 18.3431i − 0.644115i −0.946720 0.322057i \(-0.895626\pi\)
0.946720 0.322057i \(-0.104374\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 60.2843 2.11167
\(816\) 0 0
\(817\) −54.6274 −1.91117
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.87006i 0.309567i 0.987948 + 0.154784i \(0.0494680\pi\)
−0.987948 + 0.154784i \(0.950532\pi\)
\(822\) 0 0
\(823\) 39.2132 1.36689 0.683443 0.730004i \(-0.260482\pi\)
0.683443 + 0.730004i \(0.260482\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 29.9411i 1.04115i 0.853814 + 0.520577i \(0.174283\pi\)
−0.853814 + 0.520577i \(0.825717\pi\)
\(828\) 0 0
\(829\) − 52.7696i − 1.83276i −0.400307 0.916381i \(-0.631096\pi\)
0.400307 0.916381i \(-0.368904\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −50.9706 −1.76603
\(834\) 0 0
\(835\) 35.3137i 1.22208i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −28.4853 −0.983421 −0.491711 0.870759i \(-0.663628\pi\)
−0.491711 + 0.870759i \(0.663628\pi\)
\(840\) 0 0
\(841\) 17.3431 0.598040
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 17.0711i 0.587263i
\(846\) 0 0
\(847\) 4.10051 0.140895
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11.3137i 0.387829i
\(852\) 0 0
\(853\) − 34.6274i − 1.18562i −0.805342 0.592810i \(-0.798018\pi\)
0.805342 0.592810i \(-0.201982\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 41.5980 1.42096 0.710480 0.703717i \(-0.248478\pi\)
0.710480 + 0.703717i \(0.248478\pi\)
\(858\) 0 0
\(859\) 44.9706i 1.53438i 0.641422 + 0.767188i \(0.278345\pi\)
−0.641422 + 0.767188i \(0.721655\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 15.0294 0.511608 0.255804 0.966729i \(-0.417660\pi\)
0.255804 + 0.966729i \(0.417660\pi\)
\(864\) 0 0
\(865\) −22.9706 −0.781023
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 28.4853i 0.966297i
\(870\) 0 0
\(871\) 22.6274 0.766701
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 3.31371i − 0.112024i
\(876\) 0 0
\(877\) − 23.3137i − 0.787248i −0.919272 0.393624i \(-0.871221\pi\)
0.919272 0.393624i \(-0.128779\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −45.3137 −1.52666 −0.763329 0.646010i \(-0.776436\pi\)
−0.763329 + 0.646010i \(0.776436\pi\)
\(882\) 0 0
\(883\) − 30.3431i − 1.02113i −0.859840 0.510564i \(-0.829437\pi\)
0.859840 0.510564i \(-0.170563\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 29.6569 0.995780 0.497890 0.867240i \(-0.334108\pi\)
0.497890 + 0.867240i \(0.334108\pi\)
\(888\) 0 0
\(889\) 10.9706 0.367941
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 25.3726i 0.849061i
\(894\) 0 0
\(895\) 13.6569 0.456498
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 25.3137i 0.844259i
\(900\) 0 0
\(901\) − 60.4853i − 2.01506i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −28.9706 −0.963014
\(906\) 0 0
\(907\) 51.5980i 1.71328i 0.515912 + 0.856641i \(0.327453\pi\)
−0.515912 + 0.856641i \(0.672547\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 28.6863 0.950419 0.475210 0.879873i \(-0.342372\pi\)
0.475210 + 0.879873i \(0.342372\pi\)
\(912\) 0 0
\(913\) −26.6274 −0.881239
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 2.34315i − 0.0773775i
\(918\) 0 0
\(919\) −17.3553 −0.572500 −0.286250 0.958155i \(-0.592409\pi\)
−0.286250 + 0.958155i \(0.592409\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 41.9411i 1.38051i
\(924\) 0 0
\(925\) 11.0294i 0.362646i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −29.5980 −0.971078 −0.485539 0.874215i \(-0.661377\pi\)
−0.485539 + 0.874215i \(0.661377\pi\)
\(930\) 0 0
\(931\) 37.6569i 1.23415i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 52.2843 1.70988
\(936\) 0 0
\(937\) −36.6274 −1.19657 −0.598283 0.801285i \(-0.704150\pi\)
−0.598283 + 0.801285i \(0.704150\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 20.3848i 0.664525i 0.943187 + 0.332262i \(0.107812\pi\)
−0.943187 + 0.332262i \(0.892188\pi\)
\(942\) 0 0
\(943\) 2.34315 0.0763033
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 34.6274i − 1.12524i −0.826716 0.562620i \(-0.809793\pi\)
0.826716 0.562620i \(-0.190207\pi\)
\(948\) 0 0
\(949\) 27.3137i 0.886640i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 30.9706 1.00323 0.501617 0.865090i \(-0.332739\pi\)
0.501617 + 0.865090i \(0.332739\pi\)
\(954\) 0 0
\(955\) 57.9411i 1.87493i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 9.17157 0.296166
\(960\) 0 0
\(961\) 23.9706 0.773244
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 35.3137i − 1.13679i
\(966\) 0 0
\(967\) 1.75736 0.0565129 0.0282564 0.999601i \(-0.491005\pi\)
0.0282564 + 0.999601i \(0.491005\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 29.3137i − 0.940722i −0.882474 0.470361i \(-0.844124\pi\)
0.882474 0.470361i \(-0.155876\pi\)
\(972\) 0 0
\(973\) − 1.94113i − 0.0622296i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30.2843 0.968880 0.484440 0.874825i \(-0.339023\pi\)
0.484440 + 0.874825i \(0.339023\pi\)
\(978\) 0 0
\(979\) − 4.00000i − 0.127841i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4.28427 0.136647 0.0683235 0.997663i \(-0.478235\pi\)
0.0683235 + 0.997663i \(0.478235\pi\)
\(984\) 0 0
\(985\) −58.2843 −1.85709
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 65.9411i 2.09681i
\(990\) 0 0
\(991\) −11.8995 −0.378000 −0.189000 0.981977i \(-0.560525\pi\)
−0.189000 + 0.981977i \(0.560525\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 59.9411i 1.90026i
\(996\) 0 0
\(997\) − 12.9706i − 0.410782i −0.978680 0.205391i \(-0.934153\pi\)
0.978680 0.205391i \(-0.0658466\pi\)
\(998\) 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 4608.2.d.o.2305.4 4
3.2 odd 2 1536.2.d.f.769.1 4
4.3 odd 2 4608.2.d.c.2305.4 4
8.3 odd 2 4608.2.d.c.2305.1 4
8.5 even 2 inner 4608.2.d.o.2305.1 4
12.11 even 2 1536.2.d.a.769.3 4
16.3 odd 4 4608.2.a.r.1.2 2
16.5 even 4 4608.2.a.a.1.1 2
16.11 odd 4 4608.2.a.e.1.1 2
16.13 even 4 4608.2.a.n.1.2 2
24.5 odd 2 1536.2.d.f.769.4 4
24.11 even 2 1536.2.d.a.769.2 4
48.5 odd 4 1536.2.a.e.1.2 yes 2
48.11 even 4 1536.2.a.l.1.2 yes 2
48.29 odd 4 1536.2.a.g.1.1 yes 2
48.35 even 4 1536.2.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.b.1.1 2 48.35 even 4
1536.2.a.e.1.2 yes 2 48.5 odd 4
1536.2.a.g.1.1 yes 2 48.29 odd 4
1536.2.a.l.1.2 yes 2 48.11 even 4
1536.2.d.a.769.2 4 24.11 even 2
1536.2.d.a.769.3 4 12.11 even 2
1536.2.d.f.769.1 4 3.2 odd 2
1536.2.d.f.769.4 4 24.5 odd 2
4608.2.a.a.1.1 2 16.5 even 4
4608.2.a.e.1.1 2 16.11 odd 4
4608.2.a.n.1.2 2 16.13 even 4
4608.2.a.r.1.2 2 16.3 odd 4
4608.2.d.c.2305.1 4 8.3 odd 2
4608.2.d.c.2305.4 4 4.3 odd 2
4608.2.d.o.2305.1 4 8.5 even 2 inner
4608.2.d.o.2305.4 4 1.1 even 1 trivial