Properties

Label 2304.3.g.q.1279.2
Level $2304$
Weight $3$
Character 2304.1279
Analytic conductor $62.779$
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] = [2304,3,Mod(1279,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.1279");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.g (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: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 192)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1279.2
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1279
Dual form 2304.3.g.q.1279.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.46410 q^{5} +10.0000i q^{7} +O(q^{10})\) \(q-3.46410 q^{5} +10.0000i q^{7} +13.8564i q^{11} -6.92820 q^{13} -30.0000 q^{17} +6.92820i q^{19} -12.0000i q^{23} -13.0000 q^{25} -51.9615 q^{29} -14.0000i q^{31} -34.6410i q^{35} +55.4256 q^{37} -6.00000 q^{41} +62.3538i q^{43} -84.0000i q^{47} -51.0000 q^{49} +17.3205 q^{53} -48.0000i q^{55} +62.3538i q^{59} +96.9948 q^{61} +24.0000 q^{65} -48.4974i q^{67} +60.0000i q^{71} +86.0000 q^{73} -138.564 q^{77} -38.0000i q^{79} +13.8564i q^{83} +103.923 q^{85} +78.0000 q^{89} -69.2820i q^{91} -24.0000i q^{95} +62.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 120 q^{17} - 52 q^{25} - 24 q^{41} - 204 q^{49} + 96 q^{65} + 344 q^{73} + 312 q^{89} + 248 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) −3.46410 −0.692820 −0.346410 0.938083i \(-0.612599\pi\)
−0.346410 + 0.938083i \(0.612599\pi\)
\(6\) 0 0
\(7\) 10.0000i 1.42857i 0.699854 + 0.714286i \(0.253248\pi\)
−0.699854 + 0.714286i \(0.746752\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 13.8564i 1.25967i 0.776728 + 0.629837i \(0.216878\pi\)
−0.776728 + 0.629837i \(0.783122\pi\)
\(12\) 0 0
\(13\) −6.92820 −0.532939 −0.266469 0.963843i \(-0.585857\pi\)
−0.266469 + 0.963843i \(0.585857\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −30.0000 −1.76471 −0.882353 0.470588i \(-0.844042\pi\)
−0.882353 + 0.470588i \(0.844042\pi\)
\(18\) 0 0
\(19\) 6.92820i 0.364642i 0.983239 + 0.182321i \(0.0583610\pi\)
−0.983239 + 0.182321i \(0.941639\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 12.0000i − 0.521739i −0.965374 0.260870i \(-0.915991\pi\)
0.965374 0.260870i \(-0.0840093\pi\)
\(24\) 0 0
\(25\) −13.0000 −0.520000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −51.9615 −1.79178 −0.895888 0.444279i \(-0.853460\pi\)
−0.895888 + 0.444279i \(0.853460\pi\)
\(30\) 0 0
\(31\) − 14.0000i − 0.451613i −0.974172 0.225806i \(-0.927498\pi\)
0.974172 0.225806i \(-0.0725017\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 34.6410i − 0.989743i
\(36\) 0 0
\(37\) 55.4256 1.49799 0.748995 0.662576i \(-0.230537\pi\)
0.748995 + 0.662576i \(0.230537\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.146341 −0.0731707 0.997319i \(-0.523312\pi\)
−0.0731707 + 0.997319i \(0.523312\pi\)
\(42\) 0 0
\(43\) 62.3538i 1.45009i 0.688702 + 0.725045i \(0.258181\pi\)
−0.688702 + 0.725045i \(0.741819\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 84.0000i − 1.78723i −0.448830 0.893617i \(-0.648159\pi\)
0.448830 0.893617i \(-0.351841\pi\)
\(48\) 0 0
\(49\) −51.0000 −1.04082
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 17.3205 0.326802 0.163401 0.986560i \(-0.447754\pi\)
0.163401 + 0.986560i \(0.447754\pi\)
\(54\) 0 0
\(55\) − 48.0000i − 0.872727i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 62.3538i 1.05684i 0.848982 + 0.528422i \(0.177216\pi\)
−0.848982 + 0.528422i \(0.822784\pi\)
\(60\) 0 0
\(61\) 96.9948 1.59008 0.795040 0.606557i \(-0.207450\pi\)
0.795040 + 0.606557i \(0.207450\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 24.0000 0.369231
\(66\) 0 0
\(67\) − 48.4974i − 0.723842i −0.932209 0.361921i \(-0.882121\pi\)
0.932209 0.361921i \(-0.117879\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 60.0000i 0.845070i 0.906347 + 0.422535i \(0.138860\pi\)
−0.906347 + 0.422535i \(0.861140\pi\)
\(72\) 0 0
\(73\) 86.0000 1.17808 0.589041 0.808103i \(-0.299506\pi\)
0.589041 + 0.808103i \(0.299506\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −138.564 −1.79953
\(78\) 0 0
\(79\) − 38.0000i − 0.481013i −0.970648 0.240506i \(-0.922687\pi\)
0.970648 0.240506i \(-0.0773135\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.8564i 0.166945i 0.996510 + 0.0834723i \(0.0266010\pi\)
−0.996510 + 0.0834723i \(0.973399\pi\)
\(84\) 0 0
\(85\) 103.923 1.22262
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 78.0000 0.876404 0.438202 0.898876i \(-0.355615\pi\)
0.438202 + 0.898876i \(0.355615\pi\)
\(90\) 0 0
\(91\) − 69.2820i − 0.761341i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 24.0000i − 0.252632i
\(96\) 0 0
\(97\) 62.0000 0.639175 0.319588 0.947557i \(-0.396456\pi\)
0.319588 + 0.947557i \(0.396456\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −65.8179 −0.651663 −0.325831 0.945428i \(-0.605644\pi\)
−0.325831 + 0.945428i \(0.605644\pi\)
\(102\) 0 0
\(103\) 14.0000i 0.135922i 0.997688 + 0.0679612i \(0.0216494\pi\)
−0.997688 + 0.0679612i \(0.978351\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 76.2102i − 0.712245i −0.934439 0.356123i \(-0.884099\pi\)
0.934439 0.356123i \(-0.115901\pi\)
\(108\) 0 0
\(109\) −20.7846 −0.190684 −0.0953422 0.995445i \(-0.530395\pi\)
−0.0953422 + 0.995445i \(0.530395\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.00000 0.0530973 0.0265487 0.999648i \(-0.491548\pi\)
0.0265487 + 0.999648i \(0.491548\pi\)
\(114\) 0 0
\(115\) 41.5692i 0.361471i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 300.000i − 2.52101i
\(120\) 0 0
\(121\) −71.0000 −0.586777
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 131.636 1.05309
\(126\) 0 0
\(127\) − 82.0000i − 0.645669i −0.946455 0.322835i \(-0.895364\pi\)
0.946455 0.322835i \(-0.104636\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 76.2102i − 0.581758i −0.956760 0.290879i \(-0.906052\pi\)
0.956760 0.290879i \(-0.0939476\pi\)
\(132\) 0 0
\(133\) −69.2820 −0.520918
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 42.0000 0.306569 0.153285 0.988182i \(-0.451015\pi\)
0.153285 + 0.988182i \(0.451015\pi\)
\(138\) 0 0
\(139\) 48.4974i 0.348902i 0.984666 + 0.174451i \(0.0558151\pi\)
−0.984666 + 0.174451i \(0.944185\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 96.0000i − 0.671329i
\(144\) 0 0
\(145\) 180.000 1.24138
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −183.597 −1.23220 −0.616099 0.787669i \(-0.711288\pi\)
−0.616099 + 0.787669i \(0.711288\pi\)
\(150\) 0 0
\(151\) − 134.000i − 0.887417i −0.896171 0.443709i \(-0.853663\pi\)
0.896171 0.443709i \(-0.146337\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 48.4974i 0.312887i
\(156\) 0 0
\(157\) −83.1384 −0.529544 −0.264772 0.964311i \(-0.585297\pi\)
−0.264772 + 0.964311i \(0.585297\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 120.000 0.745342
\(162\) 0 0
\(163\) 76.2102i 0.467547i 0.972291 + 0.233774i \(0.0751075\pi\)
−0.972291 + 0.233774i \(0.924893\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 216.000i − 1.29341i −0.762739 0.646707i \(-0.776146\pi\)
0.762739 0.646707i \(-0.223854\pi\)
\(168\) 0 0
\(169\) −121.000 −0.715976
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 162.813 0.941114 0.470557 0.882370i \(-0.344053\pi\)
0.470557 + 0.882370i \(0.344053\pi\)
\(174\) 0 0
\(175\) − 130.000i − 0.742857i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 242.487i 1.35468i 0.735672 + 0.677338i \(0.236867\pi\)
−0.735672 + 0.677338i \(0.763133\pi\)
\(180\) 0 0
\(181\) −297.913 −1.64593 −0.822963 0.568094i \(-0.807681\pi\)
−0.822963 + 0.568094i \(0.807681\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −192.000 −1.03784
\(186\) 0 0
\(187\) − 415.692i − 2.22295i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 216.000i − 1.13089i −0.824786 0.565445i \(-0.808704\pi\)
0.824786 0.565445i \(-0.191296\pi\)
\(192\) 0 0
\(193\) −286.000 −1.48187 −0.740933 0.671579i \(-0.765616\pi\)
−0.740933 + 0.671579i \(0.765616\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 169.741 0.861629 0.430815 0.902440i \(-0.358226\pi\)
0.430815 + 0.902440i \(0.358226\pi\)
\(198\) 0 0
\(199\) − 130.000i − 0.653266i −0.945151 0.326633i \(-0.894086\pi\)
0.945151 0.326633i \(-0.105914\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 519.615i − 2.55968i
\(204\) 0 0
\(205\) 20.7846 0.101388
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −96.0000 −0.459330
\(210\) 0 0
\(211\) − 325.626i − 1.54325i −0.636078 0.771625i \(-0.719444\pi\)
0.636078 0.771625i \(-0.280556\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 216.000i − 1.00465i
\(216\) 0 0
\(217\) 140.000 0.645161
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 207.846 0.940480
\(222\) 0 0
\(223\) 302.000i 1.35426i 0.735863 + 0.677130i \(0.236777\pi\)
−0.735863 + 0.677130i \(0.763223\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 166.277i 0.732497i 0.930517 + 0.366249i \(0.119358\pi\)
−0.930517 + 0.366249i \(0.880642\pi\)
\(228\) 0 0
\(229\) 20.7846 0.0907625 0.0453812 0.998970i \(-0.485550\pi\)
0.0453812 + 0.998970i \(0.485550\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −282.000 −1.21030 −0.605150 0.796111i \(-0.706887\pi\)
−0.605150 + 0.796111i \(0.706887\pi\)
\(234\) 0 0
\(235\) 290.985i 1.23823i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 192.000i − 0.803347i −0.915783 0.401674i \(-0.868429\pi\)
0.915783 0.401674i \(-0.131571\pi\)
\(240\) 0 0
\(241\) −182.000 −0.755187 −0.377593 0.925972i \(-0.623248\pi\)
−0.377593 + 0.925972i \(0.623248\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 176.669 0.721099
\(246\) 0 0
\(247\) − 48.0000i − 0.194332i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 290.985i 1.15930i 0.814865 + 0.579650i \(0.196811\pi\)
−0.814865 + 0.579650i \(0.803189\pi\)
\(252\) 0 0
\(253\) 166.277 0.657221
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 78.0000 0.303502 0.151751 0.988419i \(-0.451509\pi\)
0.151751 + 0.988419i \(0.451509\pi\)
\(258\) 0 0
\(259\) 554.256i 2.13999i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 48.0000i 0.182510i 0.995828 + 0.0912548i \(0.0290878\pi\)
−0.995828 + 0.0912548i \(0.970912\pi\)
\(264\) 0 0
\(265\) −60.0000 −0.226415
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 280.592 1.04309 0.521547 0.853223i \(-0.325355\pi\)
0.521547 + 0.853223i \(0.325355\pi\)
\(270\) 0 0
\(271\) − 130.000i − 0.479705i −0.970809 0.239852i \(-0.922901\pi\)
0.970809 0.239852i \(-0.0770990\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 180.133i − 0.655030i
\(276\) 0 0
\(277\) −270.200 −0.975451 −0.487725 0.872997i \(-0.662173\pi\)
−0.487725 + 0.872997i \(0.662173\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −234.000 −0.832740 −0.416370 0.909195i \(-0.636698\pi\)
−0.416370 + 0.909195i \(0.636698\pi\)
\(282\) 0 0
\(283\) 311.769i 1.10166i 0.834618 + 0.550829i \(0.185688\pi\)
−0.834618 + 0.550829i \(0.814312\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 60.0000i − 0.209059i
\(288\) 0 0
\(289\) 611.000 2.11419
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 155.885 0.532029 0.266015 0.963969i \(-0.414293\pi\)
0.266015 + 0.963969i \(0.414293\pi\)
\(294\) 0 0
\(295\) − 216.000i − 0.732203i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 83.1384i 0.278055i
\(300\) 0 0
\(301\) −623.538 −2.07156
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −336.000 −1.10164
\(306\) 0 0
\(307\) − 103.923i − 0.338512i −0.985572 0.169256i \(-0.945864\pi\)
0.985572 0.169256i \(-0.0541364\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.0000i 0.0771704i 0.999255 + 0.0385852i \(0.0122851\pi\)
−0.999255 + 0.0385852i \(0.987715\pi\)
\(312\) 0 0
\(313\) −190.000 −0.607029 −0.303514 0.952827i \(-0.598160\pi\)
−0.303514 + 0.952827i \(0.598160\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −107.387 −0.338761 −0.169380 0.985551i \(-0.554177\pi\)
−0.169380 + 0.985551i \(0.554177\pi\)
\(318\) 0 0
\(319\) − 720.000i − 2.25705i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 207.846i − 0.643486i
\(324\) 0 0
\(325\) 90.0666 0.277128
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 840.000 2.55319
\(330\) 0 0
\(331\) − 34.6410i − 0.104656i −0.998630 0.0523278i \(-0.983336\pi\)
0.998630 0.0523278i \(-0.0166641\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 168.000i 0.501493i
\(336\) 0 0
\(337\) −490.000 −1.45401 −0.727003 0.686634i \(-0.759087\pi\)
−0.727003 + 0.686634i \(0.759087\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 193.990 0.568885
\(342\) 0 0
\(343\) − 20.0000i − 0.0583090i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 443.405i 1.27782i 0.769280 + 0.638912i \(0.220615\pi\)
−0.769280 + 0.638912i \(0.779385\pi\)
\(348\) 0 0
\(349\) −96.9948 −0.277922 −0.138961 0.990298i \(-0.544376\pi\)
−0.138961 + 0.990298i \(0.544376\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 246.000 0.696884 0.348442 0.937330i \(-0.386711\pi\)
0.348442 + 0.937330i \(0.386711\pi\)
\(354\) 0 0
\(355\) − 207.846i − 0.585482i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 708.000i − 1.97214i −0.166316 0.986072i \(-0.553187\pi\)
0.166316 0.986072i \(-0.446813\pi\)
\(360\) 0 0
\(361\) 313.000 0.867036
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −297.913 −0.816199
\(366\) 0 0
\(367\) 82.0000i 0.223433i 0.993740 + 0.111717i \(0.0356349\pi\)
−0.993740 + 0.111717i \(0.964365\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 173.205i 0.466860i
\(372\) 0 0
\(373\) 512.687 1.37450 0.687248 0.726423i \(-0.258819\pi\)
0.687248 + 0.726423i \(0.258819\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 360.000 0.954907
\(378\) 0 0
\(379\) − 533.472i − 1.40758i −0.710410 0.703788i \(-0.751490\pi\)
0.710410 0.703788i \(-0.248510\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 408.000i − 1.06527i −0.846344 0.532637i \(-0.821201\pi\)
0.846344 0.532637i \(-0.178799\pi\)
\(384\) 0 0
\(385\) 480.000 1.24675
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −142.028 −0.365111 −0.182555 0.983196i \(-0.558437\pi\)
−0.182555 + 0.983196i \(0.558437\pi\)
\(390\) 0 0
\(391\) 360.000i 0.920716i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 131.636i 0.333255i
\(396\) 0 0
\(397\) 41.5692 0.104708 0.0523542 0.998629i \(-0.483328\pi\)
0.0523542 + 0.998629i \(0.483328\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −270.000 −0.673317 −0.336658 0.941627i \(-0.609297\pi\)
−0.336658 + 0.941627i \(0.609297\pi\)
\(402\) 0 0
\(403\) 96.9948i 0.240682i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 768.000i 1.88698i
\(408\) 0 0
\(409\) 110.000 0.268949 0.134474 0.990917i \(-0.457065\pi\)
0.134474 + 0.990917i \(0.457065\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −623.538 −1.50978
\(414\) 0 0
\(415\) − 48.0000i − 0.115663i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 415.692i − 0.992105i −0.868292 0.496053i \(-0.834782\pi\)
0.868292 0.496053i \(-0.165218\pi\)
\(420\) 0 0
\(421\) −575.041 −1.36589 −0.682946 0.730468i \(-0.739302\pi\)
−0.682946 + 0.730468i \(0.739302\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 390.000 0.917647
\(426\) 0 0
\(427\) 969.948i 2.27154i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 396.000i − 0.918794i −0.888231 0.459397i \(-0.848066\pi\)
0.888231 0.459397i \(-0.151934\pi\)
\(432\) 0 0
\(433\) −386.000 −0.891455 −0.445727 0.895169i \(-0.647055\pi\)
−0.445727 + 0.895169i \(0.647055\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 83.1384 0.190248
\(438\) 0 0
\(439\) 470.000i 1.07062i 0.844657 + 0.535308i \(0.179804\pi\)
−0.844657 + 0.535308i \(0.820196\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 332.554i 0.750686i 0.926886 + 0.375343i \(0.122475\pi\)
−0.926886 + 0.375343i \(0.877525\pi\)
\(444\) 0 0
\(445\) −270.200 −0.607191
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 786.000 1.75056 0.875278 0.483619i \(-0.160678\pi\)
0.875278 + 0.483619i \(0.160678\pi\)
\(450\) 0 0
\(451\) − 83.1384i − 0.184342i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 240.000i 0.527473i
\(456\) 0 0
\(457\) −34.0000 −0.0743982 −0.0371991 0.999308i \(-0.511844\pi\)
−0.0371991 + 0.999308i \(0.511844\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −349.874 −0.758946 −0.379473 0.925203i \(-0.623895\pi\)
−0.379473 + 0.925203i \(0.623895\pi\)
\(462\) 0 0
\(463\) 614.000i 1.32613i 0.748560 + 0.663067i \(0.230746\pi\)
−0.748560 + 0.663067i \(0.769254\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 443.405i − 0.949475i −0.880127 0.474738i \(-0.842543\pi\)
0.880127 0.474738i \(-0.157457\pi\)
\(468\) 0 0
\(469\) 484.974 1.03406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −864.000 −1.82664
\(474\) 0 0
\(475\) − 90.0666i − 0.189614i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 300.000i 0.626305i 0.949703 + 0.313152i \(0.101385\pi\)
−0.949703 + 0.313152i \(0.898615\pi\)
\(480\) 0 0
\(481\) −384.000 −0.798337
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −214.774 −0.442834
\(486\) 0 0
\(487\) − 638.000i − 1.31006i −0.755602 0.655031i \(-0.772656\pi\)
0.755602 0.655031i \(-0.227344\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 672.036i − 1.36871i −0.729150 0.684354i \(-0.760084\pi\)
0.729150 0.684354i \(-0.239916\pi\)
\(492\) 0 0
\(493\) 1558.85 3.16196
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −600.000 −1.20724
\(498\) 0 0
\(499\) − 561.184i − 1.12462i −0.826927 0.562309i \(-0.809913\pi\)
0.826927 0.562309i \(-0.190087\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 588.000i 1.16899i 0.811399 + 0.584493i \(0.198707\pi\)
−0.811399 + 0.584493i \(0.801293\pi\)
\(504\) 0 0
\(505\) 228.000 0.451485
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 169.741 0.333479 0.166740 0.986001i \(-0.446676\pi\)
0.166740 + 0.986001i \(0.446676\pi\)
\(510\) 0 0
\(511\) 860.000i 1.68297i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 48.4974i − 0.0941698i
\(516\) 0 0
\(517\) 1163.94 2.25133
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −366.000 −0.702495 −0.351248 0.936283i \(-0.614242\pi\)
−0.351248 + 0.936283i \(0.614242\pi\)
\(522\) 0 0
\(523\) − 367.195i − 0.702093i −0.936358 0.351047i \(-0.885826\pi\)
0.936358 0.351047i \(-0.114174\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 420.000i 0.796964i
\(528\) 0 0
\(529\) 385.000 0.727788
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 41.5692 0.0779910
\(534\) 0 0
\(535\) 264.000i 0.493458i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 706.677i − 1.31109i
\(540\) 0 0
\(541\) −48.4974 −0.0896440 −0.0448220 0.998995i \(-0.514272\pi\)
−0.0448220 + 0.998995i \(0.514272\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 72.0000 0.132110
\(546\) 0 0
\(547\) − 422.620i − 0.772615i −0.922370 0.386307i \(-0.873750\pi\)
0.922370 0.386307i \(-0.126250\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 360.000i − 0.653358i
\(552\) 0 0
\(553\) 380.000 0.687161
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −557.720 −1.00129 −0.500647 0.865652i \(-0.666904\pi\)
−0.500647 + 0.865652i \(0.666904\pi\)
\(558\) 0 0
\(559\) − 432.000i − 0.772809i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 775.959i 1.37826i 0.724639 + 0.689129i \(0.242006\pi\)
−0.724639 + 0.689129i \(0.757994\pi\)
\(564\) 0 0
\(565\) −20.7846 −0.0367869
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −54.0000 −0.0949033 −0.0474517 0.998874i \(-0.515110\pi\)
−0.0474517 + 0.998874i \(0.515110\pi\)
\(570\) 0 0
\(571\) − 214.774i − 0.376137i −0.982156 0.188069i \(-0.939777\pi\)
0.982156 0.188069i \(-0.0602227\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 156.000i 0.271304i
\(576\) 0 0
\(577\) 46.0000 0.0797227 0.0398614 0.999205i \(-0.487308\pi\)
0.0398614 + 0.999205i \(0.487308\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −138.564 −0.238492
\(582\) 0 0
\(583\) 240.000i 0.411664i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 630.466i 1.07405i 0.843567 + 0.537024i \(0.180452\pi\)
−0.843567 + 0.537024i \(0.819548\pi\)
\(588\) 0 0
\(589\) 96.9948 0.164677
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −858.000 −1.44688 −0.723440 0.690387i \(-0.757440\pi\)
−0.723440 + 0.690387i \(0.757440\pi\)
\(594\) 0 0
\(595\) 1039.23i 1.74661i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 228.000i − 0.380634i −0.981723 0.190317i \(-0.939048\pi\)
0.981723 0.190317i \(-0.0609516\pi\)
\(600\) 0 0
\(601\) −1010.00 −1.68053 −0.840266 0.542174i \(-0.817601\pi\)
−0.840266 + 0.542174i \(0.817601\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 245.951 0.406531
\(606\) 0 0
\(607\) − 850.000i − 1.40033i −0.713981 0.700165i \(-0.753110\pi\)
0.713981 0.700165i \(-0.246890\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 581.969i 0.952486i
\(612\) 0 0
\(613\) −429.549 −0.700732 −0.350366 0.936613i \(-0.613943\pi\)
−0.350366 + 0.936613i \(0.613943\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −234.000 −0.379254 −0.189627 0.981856i \(-0.560728\pi\)
−0.189627 + 0.981856i \(0.560728\pi\)
\(618\) 0 0
\(619\) − 1129.30i − 1.82439i −0.409757 0.912195i \(-0.634386\pi\)
0.409757 0.912195i \(-0.365614\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 780.000i 1.25201i
\(624\) 0 0
\(625\) −131.000 −0.209600
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1662.77 −2.64351
\(630\) 0 0
\(631\) − 446.000i − 0.706815i −0.935470 0.353407i \(-0.885023\pi\)
0.935470 0.353407i \(-0.114977\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 284.056i 0.447333i
\(636\) 0 0
\(637\) 353.338 0.554691
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 522.000 0.814353 0.407176 0.913350i \(-0.366513\pi\)
0.407176 + 0.913350i \(0.366513\pi\)
\(642\) 0 0
\(643\) − 145.492i − 0.226271i −0.993580 0.113136i \(-0.963911\pi\)
0.993580 0.113136i \(-0.0360894\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 996.000i 1.53941i 0.638398 + 0.769706i \(0.279597\pi\)
−0.638398 + 0.769706i \(0.720403\pi\)
\(648\) 0 0
\(649\) −864.000 −1.33128
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 994.197 1.52251 0.761254 0.648454i \(-0.224584\pi\)
0.761254 + 0.648454i \(0.224584\pi\)
\(654\) 0 0
\(655\) 264.000i 0.403053i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 48.4974i − 0.0735924i −0.999323 0.0367962i \(-0.988285\pi\)
0.999323 0.0367962i \(-0.0117152\pi\)
\(660\) 0 0
\(661\) −138.564 −0.209628 −0.104814 0.994492i \(-0.533425\pi\)
−0.104814 + 0.994492i \(0.533425\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 240.000 0.360902
\(666\) 0 0
\(667\) 623.538i 0.934840i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1344.00i 2.00298i
\(672\) 0 0
\(673\) 46.0000 0.0683507 0.0341753 0.999416i \(-0.489120\pi\)
0.0341753 + 0.999416i \(0.489120\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −696.284 −1.02849 −0.514243 0.857645i \(-0.671927\pi\)
−0.514243 + 0.857645i \(0.671927\pi\)
\(678\) 0 0
\(679\) 620.000i 0.913108i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 124.708i − 0.182588i −0.995824 0.0912940i \(-0.970900\pi\)
0.995824 0.0912940i \(-0.0291003\pi\)
\(684\) 0 0
\(685\) −145.492 −0.212397
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −120.000 −0.174165
\(690\) 0 0
\(691\) 602.754i 0.872292i 0.899876 + 0.436146i \(0.143657\pi\)
−0.899876 + 0.436146i \(0.856343\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 168.000i − 0.241727i
\(696\) 0 0
\(697\) 180.000 0.258250
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 128.172 0.182841 0.0914207 0.995812i \(-0.470859\pi\)
0.0914207 + 0.995812i \(0.470859\pi\)
\(702\) 0 0
\(703\) 384.000i 0.546230i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 658.179i − 0.930947i
\(708\) 0 0
\(709\) −533.472 −0.752428 −0.376214 0.926533i \(-0.622774\pi\)
−0.376214 + 0.926533i \(0.622774\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −168.000 −0.235624
\(714\) 0 0
\(715\) 332.554i 0.465110i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1332.00i 1.85257i 0.376820 + 0.926287i \(0.377018\pi\)
−0.376820 + 0.926287i \(0.622982\pi\)
\(720\) 0 0
\(721\) −140.000 −0.194175
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 675.500 0.931724
\(726\) 0 0
\(727\) 754.000i 1.03714i 0.855036 + 0.518569i \(0.173535\pi\)
−0.855036 + 0.518569i \(0.826465\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 1870.61i − 2.55898i
\(732\) 0 0
\(733\) −727.461 −0.992444 −0.496222 0.868196i \(-0.665280\pi\)
−0.496222 + 0.868196i \(0.665280\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 672.000 0.911805
\(738\) 0 0
\(739\) − 145.492i − 0.196877i −0.995143 0.0984386i \(-0.968615\pi\)
0.995143 0.0984386i \(-0.0313848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 816.000i 1.09825i 0.835740 + 0.549125i \(0.185039\pi\)
−0.835740 + 0.549125i \(0.814961\pi\)
\(744\) 0 0
\(745\) 636.000 0.853691
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 762.102 1.01749
\(750\) 0 0
\(751\) 466.000i 0.620506i 0.950654 + 0.310253i \(0.100414\pi\)
−0.950654 + 0.310253i \(0.899586\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 464.190i 0.614821i
\(756\) 0 0
\(757\) −450.333 −0.594892 −0.297446 0.954739i \(-0.596135\pi\)
−0.297446 + 0.954739i \(0.596135\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.0236531 0.0118265 0.999930i \(-0.496235\pi\)
0.0118265 + 0.999930i \(0.496235\pi\)
\(762\) 0 0
\(763\) − 207.846i − 0.272406i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 432.000i − 0.563233i
\(768\) 0 0
\(769\) 1298.00 1.68791 0.843953 0.536417i \(-0.180222\pi\)
0.843953 + 0.536417i \(0.180222\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 197.454 0.255438 0.127719 0.991810i \(-0.459234\pi\)
0.127719 + 0.991810i \(0.459234\pi\)
\(774\) 0 0
\(775\) 182.000i 0.234839i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 41.5692i − 0.0533623i
\(780\) 0 0
\(781\) −831.384 −1.06451
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 288.000 0.366879
\(786\) 0 0
\(787\) 256.344i 0.325722i 0.986649 + 0.162861i \(0.0520723\pi\)
−0.986649 + 0.162861i \(0.947928\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 60.0000i 0.0758534i
\(792\) 0 0
\(793\) −672.000 −0.847415
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −959.556 −1.20396 −0.601980 0.798511i \(-0.705621\pi\)
−0.601980 + 0.798511i \(0.705621\pi\)
\(798\) 0 0
\(799\) 2520.00i 3.15394i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1191.65i 1.48400i
\(804\) 0 0
\(805\) −415.692 −0.516388
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 378.000 0.467244 0.233622 0.972328i \(-0.424942\pi\)
0.233622 + 0.972328i \(0.424942\pi\)
\(810\) 0 0
\(811\) − 228.631i − 0.281912i −0.990016 0.140956i \(-0.954982\pi\)
0.990016 0.140956i \(-0.0450176\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 264.000i − 0.323926i
\(816\) 0 0
\(817\) −432.000 −0.528764
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1236.68 1.50631 0.753157 0.657840i \(-0.228530\pi\)
0.753157 + 0.657840i \(0.228530\pi\)
\(822\) 0 0
\(823\) − 758.000i − 0.921021i −0.887654 0.460510i \(-0.847666\pi\)
0.887654 0.460510i \(-0.152334\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 644.323i 0.779109i 0.921003 + 0.389554i \(0.127371\pi\)
−0.921003 + 0.389554i \(0.872629\pi\)
\(828\) 0 0
\(829\) 921.451 1.11152 0.555761 0.831342i \(-0.312427\pi\)
0.555761 + 0.831342i \(0.312427\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1530.00 1.83673
\(834\) 0 0
\(835\) 748.246i 0.896103i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 612.000i 0.729440i 0.931117 + 0.364720i \(0.118835\pi\)
−0.931117 + 0.364720i \(0.881165\pi\)
\(840\) 0 0
\(841\) 1859.00 2.21046
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 419.156 0.496043
\(846\) 0 0
\(847\) − 710.000i − 0.838253i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 665.108i − 0.781560i
\(852\) 0 0
\(853\) 55.4256 0.0649773 0.0324886 0.999472i \(-0.489657\pi\)
0.0324886 + 0.999472i \(0.489657\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −582.000 −0.679113 −0.339557 0.940586i \(-0.610277\pi\)
−0.339557 + 0.940586i \(0.610277\pi\)
\(858\) 0 0
\(859\) 1364.86i 1.58889i 0.607336 + 0.794445i \(0.292238\pi\)
−0.607336 + 0.794445i \(0.707762\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 456.000i 0.528389i 0.964469 + 0.264195i \(0.0851061\pi\)
−0.964469 + 0.264195i \(0.914894\pi\)
\(864\) 0 0
\(865\) −564.000 −0.652023
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 526.543 0.605919
\(870\) 0 0
\(871\) 336.000i 0.385763i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1316.36i 1.50441i
\(876\) 0 0
\(877\) −13.8564 −0.0157998 −0.00789989 0.999969i \(-0.502515\pi\)
−0.00789989 + 0.999969i \(0.502515\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1362.00 −1.54597 −0.772985 0.634424i \(-0.781237\pi\)
−0.772985 + 0.634424i \(0.781237\pi\)
\(882\) 0 0
\(883\) − 1586.56i − 1.79678i −0.439197 0.898391i \(-0.644737\pi\)
0.439197 0.898391i \(-0.355263\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 240.000i 0.270575i 0.990806 + 0.135287i \(0.0431958\pi\)
−0.990806 + 0.135287i \(0.956804\pi\)
\(888\) 0 0
\(889\) 820.000 0.922385
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 581.969 0.651701
\(894\) 0 0
\(895\) − 840.000i − 0.938547i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 727.461i 0.809189i
\(900\) 0 0
\(901\) −519.615 −0.576709
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1032.00 1.14033
\(906\) 0 0
\(907\) − 159.349i − 0.175688i −0.996134 0.0878438i \(-0.972002\pi\)
0.996134 0.0878438i \(-0.0279976\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 1392.00i − 1.52799i −0.645221 0.763996i \(-0.723235\pi\)
0.645221 0.763996i \(-0.276765\pi\)
\(912\) 0 0
\(913\) −192.000 −0.210296
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 762.102 0.831082
\(918\) 0 0
\(919\) − 422.000i − 0.459195i −0.973286 0.229597i \(-0.926259\pi\)
0.973286 0.229597i \(-0.0737409\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 415.692i − 0.450371i
\(924\) 0 0
\(925\) −720.533 −0.778955
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1026.00 1.10441 0.552207 0.833707i \(-0.313786\pi\)
0.552207 + 0.833707i \(0.313786\pi\)
\(930\) 0 0
\(931\) − 353.338i − 0.379526i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1440.00i 1.54011i
\(936\) 0 0
\(937\) −814.000 −0.868730 −0.434365 0.900737i \(-0.643027\pi\)
−0.434365 + 0.900737i \(0.643027\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 980.341 1.04181 0.520904 0.853615i \(-0.325595\pi\)
0.520904 + 0.853615i \(0.325595\pi\)
\(942\) 0 0
\(943\) 72.0000i 0.0763521i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1503.42i − 1.58756i −0.608204 0.793780i \(-0.708110\pi\)
0.608204 0.793780i \(-0.291890\pi\)
\(948\) 0 0
\(949\) −595.825 −0.627846
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −366.000 −0.384050 −0.192025 0.981390i \(-0.561506\pi\)
−0.192025 + 0.981390i \(0.561506\pi\)
\(954\) 0 0
\(955\) 748.246i 0.783504i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 420.000i 0.437956i
\(960\) 0 0
\(961\) 765.000 0.796046
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 990.733 1.02667
\(966\) 0 0
\(967\) 686.000i 0.709411i 0.934978 + 0.354705i \(0.115419\pi\)
−0.934978 + 0.354705i \(0.884581\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1344.07i 1.38421i 0.721795 + 0.692107i \(0.243317\pi\)
−0.721795 + 0.692107i \(0.756683\pi\)
\(972\) 0 0
\(973\) −484.974 −0.498432
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1470.00 −1.50461 −0.752303 0.658817i \(-0.771057\pi\)
−0.752303 + 0.658817i \(0.771057\pi\)
\(978\) 0 0
\(979\) 1080.80i 1.10398i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 720.000i − 0.732452i −0.930526 0.366226i \(-0.880650\pi\)
0.930526 0.366226i \(-0.119350\pi\)
\(984\) 0 0
\(985\) −588.000 −0.596954
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 748.246 0.756568
\(990\) 0 0
\(991\) − 1598.00i − 1.61251i −0.591566 0.806256i \(-0.701490\pi\)
0.591566 0.806256i \(-0.298510\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 450.333i 0.452596i
\(996\) 0 0
\(997\) 1454.92 1.45930 0.729650 0.683820i \(-0.239683\pi\)
0.729650 + 0.683820i \(0.239683\pi\)
\(998\) 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 2304.3.g.q.1279.2 4
3.2 odd 2 768.3.g.f.511.2 4
4.3 odd 2 inner 2304.3.g.q.1279.1 4
8.3 odd 2 inner 2304.3.g.q.1279.3 4
8.5 even 2 inner 2304.3.g.q.1279.4 4
12.11 even 2 768.3.g.f.511.4 4
16.3 odd 4 576.3.b.a.415.4 4
16.5 even 4 576.3.b.a.415.1 4
16.11 odd 4 576.3.b.a.415.2 4
16.13 even 4 576.3.b.a.415.3 4
24.5 odd 2 768.3.g.f.511.3 4
24.11 even 2 768.3.g.f.511.1 4
48.5 odd 4 192.3.b.b.31.2 yes 4
48.11 even 4 192.3.b.b.31.4 yes 4
48.29 odd 4 192.3.b.b.31.3 yes 4
48.35 even 4 192.3.b.b.31.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
192.3.b.b.31.1 4 48.35 even 4
192.3.b.b.31.2 yes 4 48.5 odd 4
192.3.b.b.31.3 yes 4 48.29 odd 4
192.3.b.b.31.4 yes 4 48.11 even 4
576.3.b.a.415.1 4 16.5 even 4
576.3.b.a.415.2 4 16.11 odd 4
576.3.b.a.415.3 4 16.13 even 4
576.3.b.a.415.4 4 16.3 odd 4
768.3.g.f.511.1 4 24.11 even 2
768.3.g.f.511.2 4 3.2 odd 2
768.3.g.f.511.3 4 24.5 odd 2
768.3.g.f.511.4 4 12.11 even 2
2304.3.g.q.1279.1 4 4.3 odd 2 inner
2304.3.g.q.1279.2 4 1.1 even 1 trivial
2304.3.g.q.1279.3 4 8.3 odd 2 inner
2304.3.g.q.1279.4 4 8.5 even 2 inner