Properties

Label 2304.3.h.a.2177.4
Level $2304$
Weight $3$
Character 2304.2177
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(2177,2304)]
 
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([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.2177");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.h (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: \(\Q(\zeta_{8})\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2177.4
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2304.2177
Dual form 2304.3.h.a.2177.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.07107 q^{5} -12.0000 q^{7} +O(q^{10})\) \(q+7.07107 q^{5} -12.0000 q^{7} -5.65685 q^{11} +8.00000i q^{13} -9.89949i q^{17} +16.0000i q^{19} -39.5980i q^{23} +25.0000 q^{25} +29.6985 q^{29} -4.00000 q^{31} -84.8528 q^{35} +30.0000i q^{37} -21.2132i q^{41} -8.00000i q^{43} +16.9706i q^{47} +95.0000 q^{49} +49.4975 q^{53} -40.0000 q^{55} +79.1960 q^{59} +14.0000i q^{61} +56.5685i q^{65} +88.0000i q^{67} -28.2843i q^{71} +80.0000 q^{73} +67.8823 q^{77} +100.000 q^{79} +130.108 q^{83} -70.0000i q^{85} +148.492i q^{89} -96.0000i q^{91} +113.137i q^{95} -112.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 48 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 48 q^{7} + 100 q^{25} - 16 q^{31} + 380 q^{49} - 160 q^{55} + 320 q^{73} + 400 q^{79} - 448 q^{97}+O(q^{100}) \) 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
\(5\) 7.07107 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(6\) 0 0
\(7\) −12.0000 −1.71429 −0.857143 0.515079i \(-0.827763\pi\)
−0.857143 + 0.515079i \(0.827763\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.65685 −0.514259 −0.257130 0.966377i \(-0.582777\pi\)
−0.257130 + 0.966377i \(0.582777\pi\)
\(12\) 0 0
\(13\) 8.00000i 0.615385i 0.951486 + 0.307692i \(0.0995567\pi\)
−0.951486 + 0.307692i \(0.900443\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 9.89949i − 0.582323i −0.956674 0.291162i \(-0.905958\pi\)
0.956674 0.291162i \(-0.0940417\pi\)
\(18\) 0 0
\(19\) 16.0000i 0.842105i 0.907036 + 0.421053i \(0.138339\pi\)
−0.907036 + 0.421053i \(0.861661\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 39.5980i − 1.72165i −0.508900 0.860826i \(-0.669948\pi\)
0.508900 0.860826i \(-0.330052\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 29.6985 1.02409 0.512043 0.858960i \(-0.328889\pi\)
0.512043 + 0.858960i \(0.328889\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.129032 −0.0645161 0.997917i \(-0.520550\pi\)
−0.0645161 + 0.997917i \(0.520550\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −84.8528 −2.42437
\(36\) 0 0
\(37\) 30.0000i 0.810811i 0.914137 + 0.405405i \(0.132870\pi\)
−0.914137 + 0.405405i \(0.867130\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 21.2132i − 0.517395i −0.965958 0.258698i \(-0.916707\pi\)
0.965958 0.258698i \(-0.0832933\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 0.186047i −0.995664 0.0930233i \(-0.970347\pi\)
0.995664 0.0930233i \(-0.0296531\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 16.9706i 0.361076i 0.983568 + 0.180538i \(0.0577838\pi\)
−0.983568 + 0.180538i \(0.942216\pi\)
\(48\) 0 0
\(49\) 95.0000 1.93878
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 49.4975 0.933915 0.466957 0.884280i \(-0.345350\pi\)
0.466957 + 0.884280i \(0.345350\pi\)
\(54\) 0 0
\(55\) −40.0000 −0.727273
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 79.1960 1.34230 0.671152 0.741320i \(-0.265800\pi\)
0.671152 + 0.741320i \(0.265800\pi\)
\(60\) 0 0
\(61\) 14.0000i 0.229508i 0.993394 + 0.114754i \(0.0366080\pi\)
−0.993394 + 0.114754i \(0.963392\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 56.5685i 0.870285i
\(66\) 0 0
\(67\) 88.0000i 1.31343i 0.754138 + 0.656716i \(0.228055\pi\)
−0.754138 + 0.656716i \(0.771945\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 28.2843i − 0.398370i −0.979962 0.199185i \(-0.936171\pi\)
0.979962 0.199185i \(-0.0638295\pi\)
\(72\) 0 0
\(73\) 80.0000 1.09589 0.547945 0.836514i \(-0.315410\pi\)
0.547945 + 0.836514i \(0.315410\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 67.8823 0.881588
\(78\) 0 0
\(79\) 100.000 1.26582 0.632911 0.774224i \(-0.281860\pi\)
0.632911 + 0.774224i \(0.281860\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 130.108 1.56756 0.783781 0.621037i \(-0.213288\pi\)
0.783781 + 0.621037i \(0.213288\pi\)
\(84\) 0 0
\(85\) − 70.0000i − 0.823529i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 148.492i 1.66845i 0.551421 + 0.834227i \(0.314086\pi\)
−0.551421 + 0.834227i \(0.685914\pi\)
\(90\) 0 0
\(91\) − 96.0000i − 1.05495i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 113.137i 1.19092i
\(96\) 0 0
\(97\) −112.000 −1.15464 −0.577320 0.816518i \(-0.695901\pi\)
−0.577320 + 0.816518i \(0.695901\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 63.6396 0.630095 0.315048 0.949076i \(-0.397980\pi\)
0.315048 + 0.949076i \(0.397980\pi\)
\(102\) 0 0
\(103\) 44.0000 0.427184 0.213592 0.976923i \(-0.431484\pi\)
0.213592 + 0.976923i \(0.431484\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) − 136.000i − 1.24771i −0.781542 0.623853i \(-0.785566\pi\)
0.781542 0.623853i \(-0.214434\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 21.2132i 0.187727i 0.995585 + 0.0938637i \(0.0299218\pi\)
−0.995585 + 0.0938637i \(0.970078\pi\)
\(114\) 0 0
\(115\) − 280.000i − 2.43478i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 118.794i 0.998268i
\(120\) 0 0
\(121\) −89.0000 −0.735537
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −20.0000 −0.157480 −0.0787402 0.996895i \(-0.525090\pi\)
−0.0787402 + 0.996895i \(0.525090\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −56.5685 −0.431821 −0.215910 0.976413i \(-0.569272\pi\)
−0.215910 + 0.976413i \(0.569272\pi\)
\(132\) 0 0
\(133\) − 192.000i − 1.44361i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 89.0955i 0.650332i 0.945657 + 0.325166i \(0.105420\pi\)
−0.945657 + 0.325166i \(0.894580\pi\)
\(138\) 0 0
\(139\) 120.000i 0.863309i 0.902039 + 0.431655i \(0.142070\pi\)
−0.902039 + 0.431655i \(0.857930\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 45.2548i − 0.316467i
\(144\) 0 0
\(145\) 210.000 1.44828
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 128.693 0.863714 0.431857 0.901942i \(-0.357858\pi\)
0.431857 + 0.901942i \(0.357858\pi\)
\(150\) 0 0
\(151\) 196.000 1.29801 0.649007 0.760783i \(-0.275185\pi\)
0.649007 + 0.760783i \(0.275185\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −28.2843 −0.182479
\(156\) 0 0
\(157\) − 238.000i − 1.51592i −0.652299 0.757962i \(-0.726195\pi\)
0.652299 0.757962i \(-0.273805\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 475.176i 2.95140i
\(162\) 0 0
\(163\) 168.000i 1.03067i 0.856987 + 0.515337i \(0.172333\pi\)
−0.856987 + 0.515337i \(0.827667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 169.706i 1.01620i 0.861298 + 0.508101i \(0.169652\pi\)
−0.861298 + 0.508101i \(0.830348\pi\)
\(168\) 0 0
\(169\) 105.000 0.621302
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 207.889 1.20167 0.600836 0.799372i \(-0.294834\pi\)
0.600836 + 0.799372i \(0.294834\pi\)
\(174\) 0 0
\(175\) −300.000 −1.71429
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 203.647 1.13769 0.568846 0.822444i \(-0.307390\pi\)
0.568846 + 0.822444i \(0.307390\pi\)
\(180\) 0 0
\(181\) − 216.000i − 1.19337i −0.802475 0.596685i \(-0.796484\pi\)
0.802475 0.596685i \(-0.203516\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 212.132i 1.14666i
\(186\) 0 0
\(187\) 56.0000i 0.299465i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 373.352i − 1.95472i −0.211573 0.977362i \(-0.567859\pi\)
0.211573 0.977362i \(-0.432141\pi\)
\(192\) 0 0
\(193\) −178.000 −0.922280 −0.461140 0.887327i \(-0.652559\pi\)
−0.461140 + 0.887327i \(0.652559\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −207.889 −1.05528 −0.527638 0.849469i \(-0.676922\pi\)
−0.527638 + 0.849469i \(0.676922\pi\)
\(198\) 0 0
\(199\) −196.000 −0.984925 −0.492462 0.870334i \(-0.663903\pi\)
−0.492462 + 0.870334i \(0.663903\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −356.382 −1.75558
\(204\) 0 0
\(205\) − 150.000i − 0.731707i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 90.5097i − 0.433061i
\(210\) 0 0
\(211\) 56.0000i 0.265403i 0.991156 + 0.132701i \(0.0423651\pi\)
−0.991156 + 0.132701i \(0.957635\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 56.5685i − 0.263109i
\(216\) 0 0
\(217\) 48.0000 0.221198
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 79.1960 0.358353
\(222\) 0 0
\(223\) 28.0000 0.125561 0.0627803 0.998027i \(-0.480003\pi\)
0.0627803 + 0.998027i \(0.480003\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −84.8528 −0.373801 −0.186900 0.982379i \(-0.559844\pi\)
−0.186900 + 0.982379i \(0.559844\pi\)
\(228\) 0 0
\(229\) 184.000i 0.803493i 0.915751 + 0.401747i \(0.131597\pi\)
−0.915751 + 0.401747i \(0.868403\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 304.056i 1.30496i 0.757806 + 0.652481i \(0.226272\pi\)
−0.757806 + 0.652481i \(0.773728\pi\)
\(234\) 0 0
\(235\) 120.000i 0.510638i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 226.274i − 0.946754i −0.880860 0.473377i \(-0.843035\pi\)
0.880860 0.473377i \(-0.156965\pi\)
\(240\) 0 0
\(241\) 384.000 1.59336 0.796680 0.604401i \(-0.206587\pi\)
0.796680 + 0.604401i \(0.206587\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 671.751 2.74184
\(246\) 0 0
\(247\) −128.000 −0.518219
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −197.990 −0.788804 −0.394402 0.918938i \(-0.629048\pi\)
−0.394402 + 0.918938i \(0.629048\pi\)
\(252\) 0 0
\(253\) 224.000i 0.885375i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 91.9239i 0.357680i 0.983878 + 0.178840i \(0.0572345\pi\)
−0.983878 + 0.178840i \(0.942765\pi\)
\(258\) 0 0
\(259\) − 360.000i − 1.38996i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 395.980i − 1.50563i −0.658234 0.752813i \(-0.728696\pi\)
0.658234 0.752813i \(-0.271304\pi\)
\(264\) 0 0
\(265\) 350.000 1.32075
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 83.4386 0.310181 0.155090 0.987900i \(-0.450433\pi\)
0.155090 + 0.987900i \(0.450433\pi\)
\(270\) 0 0
\(271\) 140.000 0.516605 0.258303 0.966064i \(-0.416837\pi\)
0.258303 + 0.966064i \(0.416837\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −141.421 −0.514259
\(276\) 0 0
\(277\) − 248.000i − 0.895307i −0.894207 0.447653i \(-0.852260\pi\)
0.894207 0.447653i \(-0.147740\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 168.291i − 0.598902i −0.954112 0.299451i \(-0.903197\pi\)
0.954112 0.299451i \(-0.0968035\pi\)
\(282\) 0 0
\(283\) 368.000i 1.30035i 0.759783 + 0.650177i \(0.225305\pi\)
−0.759783 + 0.650177i \(0.774695\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 254.558i 0.886963i
\(288\) 0 0
\(289\) 191.000 0.660900
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 207.889 0.709520 0.354760 0.934957i \(-0.384563\pi\)
0.354760 + 0.934957i \(0.384563\pi\)
\(294\) 0 0
\(295\) 560.000 1.89831
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 316.784 1.05948
\(300\) 0 0
\(301\) 96.0000i 0.318937i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 98.9949i 0.324574i
\(306\) 0 0
\(307\) 552.000i 1.79805i 0.437902 + 0.899023i \(0.355722\pi\)
−0.437902 + 0.899023i \(0.644278\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 33.9411i 0.109135i 0.998510 + 0.0545677i \(0.0173781\pi\)
−0.998510 + 0.0545677i \(0.982622\pi\)
\(312\) 0 0
\(313\) 222.000 0.709265 0.354633 0.935006i \(-0.384606\pi\)
0.354633 + 0.935006i \(0.384606\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 445.477 1.40529 0.702646 0.711540i \(-0.252002\pi\)
0.702646 + 0.711540i \(0.252002\pi\)
\(318\) 0 0
\(319\) −168.000 −0.526646
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 158.392 0.490377
\(324\) 0 0
\(325\) 200.000i 0.615385i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 203.647i − 0.618987i
\(330\) 0 0
\(331\) − 520.000i − 1.57100i −0.618864 0.785498i \(-0.712407\pi\)
0.618864 0.785498i \(-0.287593\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 622.254i 1.85747i
\(336\) 0 0
\(337\) 48.0000 0.142433 0.0712166 0.997461i \(-0.477312\pi\)
0.0712166 + 0.997461i \(0.477312\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 22.6274 0.0663561
\(342\) 0 0
\(343\) −552.000 −1.60933
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 356.382 1.02704 0.513518 0.858079i \(-0.328342\pi\)
0.513518 + 0.858079i \(0.328342\pi\)
\(348\) 0 0
\(349\) − 210.000i − 0.601719i −0.953668 0.300860i \(-0.902726\pi\)
0.953668 0.300860i \(-0.0972736\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 272.943i − 0.773210i −0.922245 0.386605i \(-0.873648\pi\)
0.922245 0.386605i \(-0.126352\pi\)
\(354\) 0 0
\(355\) − 200.000i − 0.563380i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 593.970i 1.65451i 0.561825 + 0.827256i \(0.310099\pi\)
−0.561825 + 0.827256i \(0.689901\pi\)
\(360\) 0 0
\(361\) 105.000 0.290859
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 565.685 1.54982
\(366\) 0 0
\(367\) 172.000 0.468665 0.234332 0.972157i \(-0.424710\pi\)
0.234332 + 0.972157i \(0.424710\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −593.970 −1.60100
\(372\) 0 0
\(373\) 2.00000i 0.00536193i 0.999996 + 0.00268097i \(0.000853379\pi\)
−0.999996 + 0.00268097i \(0.999147\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 237.588i 0.630207i
\(378\) 0 0
\(379\) − 736.000i − 1.94195i −0.239176 0.970976i \(-0.576877\pi\)
0.239176 0.970976i \(-0.423123\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 480.000 1.24675
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −459.619 −1.18154 −0.590770 0.806840i \(-0.701176\pi\)
−0.590770 + 0.806840i \(0.701176\pi\)
\(390\) 0 0
\(391\) −392.000 −1.00256
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 707.107 1.79014
\(396\) 0 0
\(397\) − 338.000i − 0.851385i −0.904868 0.425693i \(-0.860030\pi\)
0.904868 0.425693i \(-0.139970\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 148.492i 0.370305i 0.982710 + 0.185153i \(0.0592780\pi\)
−0.982710 + 0.185153i \(0.940722\pi\)
\(402\) 0 0
\(403\) − 32.0000i − 0.0794045i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 169.706i − 0.416967i
\(408\) 0 0
\(409\) −144.000 −0.352078 −0.176039 0.984383i \(-0.556329\pi\)
−0.176039 + 0.984383i \(0.556329\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −950.352 −2.30109
\(414\) 0 0
\(415\) 920.000 2.21687
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 763.675 1.82261 0.911307 0.411727i \(-0.135074\pi\)
0.911307 + 0.411727i \(0.135074\pi\)
\(420\) 0 0
\(421\) 680.000i 1.61520i 0.589729 + 0.807601i \(0.299234\pi\)
−0.589729 + 0.807601i \(0.700766\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 247.487i − 0.582323i
\(426\) 0 0
\(427\) − 168.000i − 0.393443i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 5.65685i − 0.0131250i −0.999978 0.00656248i \(-0.997911\pi\)
0.999978 0.00656248i \(-0.00208892\pi\)
\(432\) 0 0
\(433\) −482.000 −1.11316 −0.556582 0.830793i \(-0.687887\pi\)
−0.556582 + 0.830793i \(0.687887\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 633.568 1.44981
\(438\) 0 0
\(439\) 756.000 1.72210 0.861048 0.508524i \(-0.169809\pi\)
0.861048 + 0.508524i \(0.169809\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −197.990 −0.446930 −0.223465 0.974712i \(-0.571737\pi\)
−0.223465 + 0.974712i \(0.571737\pi\)
\(444\) 0 0
\(445\) 1050.00i 2.35955i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 77.7817i 0.173233i 0.996242 + 0.0866166i \(0.0276055\pi\)
−0.996242 + 0.0866166i \(0.972394\pi\)
\(450\) 0 0
\(451\) 120.000i 0.266075i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 678.823i − 1.49192i
\(456\) 0 0
\(457\) −464.000 −1.01532 −0.507659 0.861558i \(-0.669489\pi\)
−0.507659 + 0.861558i \(0.669489\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −49.4975 −0.107370 −0.0536849 0.998558i \(-0.517097\pi\)
−0.0536849 + 0.998558i \(0.517097\pi\)
\(462\) 0 0
\(463\) 20.0000 0.0431965 0.0215983 0.999767i \(-0.493125\pi\)
0.0215983 + 0.999767i \(0.493125\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −695.793 −1.48992 −0.744960 0.667109i \(-0.767532\pi\)
−0.744960 + 0.667109i \(0.767532\pi\)
\(468\) 0 0
\(469\) − 1056.00i − 2.25160i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 45.2548i 0.0956762i
\(474\) 0 0
\(475\) 400.000i 0.842105i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 673.166i 1.40536i 0.711508 + 0.702678i \(0.248013\pi\)
−0.711508 + 0.702678i \(0.751987\pi\)
\(480\) 0 0
\(481\) −240.000 −0.498960
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −791.960 −1.63291
\(486\) 0 0
\(487\) −388.000 −0.796715 −0.398357 0.917230i \(-0.630420\pi\)
−0.398357 + 0.917230i \(0.630420\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) − 294.000i − 0.596349i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 339.411i 0.682920i
\(498\) 0 0
\(499\) − 96.0000i − 0.192385i −0.995363 0.0961924i \(-0.969334\pi\)
0.995363 0.0961924i \(-0.0306664\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 424.264i − 0.843467i −0.906720 0.421734i \(-0.861422\pi\)
0.906720 0.421734i \(-0.138578\pi\)
\(504\) 0 0
\(505\) 450.000 0.891089
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −309.713 −0.608473 −0.304237 0.952597i \(-0.598401\pi\)
−0.304237 + 0.952597i \(0.598401\pi\)
\(510\) 0 0
\(511\) −960.000 −1.87867
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 311.127 0.604130
\(516\) 0 0
\(517\) − 96.0000i − 0.185687i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 722.663i 1.38707i 0.720423 + 0.693535i \(0.243948\pi\)
−0.720423 + 0.693535i \(0.756052\pi\)
\(522\) 0 0
\(523\) 352.000i 0.673040i 0.941676 + 0.336520i \(0.109250\pi\)
−0.941676 + 0.336520i \(0.890750\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 39.5980i 0.0751385i
\(528\) 0 0
\(529\) −1039.00 −1.96408
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 169.706 0.318397
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −537.401 −0.997034
\(540\) 0 0
\(541\) − 936.000i − 1.73013i −0.501660 0.865065i \(-0.667277\pi\)
0.501660 0.865065i \(-0.332723\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 961.665i − 1.76452i
\(546\) 0 0
\(547\) − 56.0000i − 0.102377i −0.998689 0.0511883i \(-0.983699\pi\)
0.998689 0.0511883i \(-0.0163009\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 475.176i 0.862388i
\(552\) 0 0
\(553\) −1200.00 −2.16998
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −321.026 −0.576349 −0.288175 0.957578i \(-0.593048\pi\)
−0.288175 + 0.957578i \(0.593048\pi\)
\(558\) 0 0
\(559\) 64.0000 0.114490
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −469.519 −0.833959 −0.416979 0.908916i \(-0.636911\pi\)
−0.416979 + 0.908916i \(0.636911\pi\)
\(564\) 0 0
\(565\) 150.000i 0.265487i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 408.708i − 0.718291i −0.933282 0.359146i \(-0.883068\pi\)
0.933282 0.359146i \(-0.116932\pi\)
\(570\) 0 0
\(571\) 816.000i 1.42907i 0.699599 + 0.714536i \(0.253362\pi\)
−0.699599 + 0.714536i \(0.746638\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 989.949i − 1.72165i
\(576\) 0 0
\(577\) 242.000 0.419411 0.209705 0.977765i \(-0.432750\pi\)
0.209705 + 0.977765i \(0.432750\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1561.29 −2.68725
\(582\) 0 0
\(583\) −280.000 −0.480274
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −395.980 −0.674582 −0.337291 0.941400i \(-0.609511\pi\)
−0.337291 + 0.941400i \(0.609511\pi\)
\(588\) 0 0
\(589\) − 64.0000i − 0.108659i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 205.061i 0.345803i 0.984939 + 0.172901i \(0.0553142\pi\)
−0.984939 + 0.172901i \(0.944686\pi\)
\(594\) 0 0
\(595\) 840.000i 1.41176i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 593.970i 0.991602i 0.868436 + 0.495801i \(0.165125\pi\)
−0.868436 + 0.495801i \(0.834875\pi\)
\(600\) 0 0
\(601\) −286.000 −0.475874 −0.237937 0.971281i \(-0.576471\pi\)
−0.237937 + 0.971281i \(0.576471\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −629.325 −1.04021
\(606\) 0 0
\(607\) −100.000 −0.164745 −0.0823723 0.996602i \(-0.526250\pi\)
−0.0823723 + 0.996602i \(0.526250\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −135.765 −0.222200
\(612\) 0 0
\(613\) 770.000i 1.25612i 0.778166 + 0.628059i \(0.216150\pi\)
−0.778166 + 0.628059i \(0.783850\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 869.741i − 1.40963i −0.709391 0.704815i \(-0.751030\pi\)
0.709391 0.704815i \(-0.248970\pi\)
\(618\) 0 0
\(619\) 256.000i 0.413570i 0.978386 + 0.206785i \(0.0663001\pi\)
−0.978386 + 0.206785i \(0.933700\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 1781.91i − 2.86021i
\(624\) 0 0
\(625\) −625.000 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 296.985 0.472154
\(630\) 0 0
\(631\) −196.000 −0.310618 −0.155309 0.987866i \(-0.549637\pi\)
−0.155309 + 0.987866i \(0.549637\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −141.421 −0.222711
\(636\) 0 0
\(637\) 760.000i 1.19309i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 643.467i − 1.00385i −0.864911 0.501924i \(-0.832626\pi\)
0.864911 0.501924i \(-0.167374\pi\)
\(642\) 0 0
\(643\) − 120.000i − 0.186625i −0.995637 0.0933126i \(-0.970254\pi\)
0.995637 0.0933126i \(-0.0297456\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 707.107i − 1.09290i −0.837491 0.546450i \(-0.815979\pi\)
0.837491 0.546450i \(-0.184021\pi\)
\(648\) 0 0
\(649\) −448.000 −0.690293
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −131.522 −0.201412 −0.100706 0.994916i \(-0.532110\pi\)
−0.100706 + 0.994916i \(0.532110\pi\)
\(654\) 0 0
\(655\) −400.000 −0.610687
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1187.94 1.80264 0.901320 0.433154i \(-0.142600\pi\)
0.901320 + 0.433154i \(0.142600\pi\)
\(660\) 0 0
\(661\) − 130.000i − 0.196672i −0.995153 0.0983359i \(-0.968648\pi\)
0.995153 0.0983359i \(-0.0313519\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 1357.65i − 2.04157i
\(666\) 0 0
\(667\) − 1176.00i − 1.76312i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 79.1960i − 0.118027i
\(672\) 0 0
\(673\) 610.000 0.906389 0.453195 0.891412i \(-0.350284\pi\)
0.453195 + 0.891412i \(0.350284\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −603.869 −0.891978 −0.445989 0.895038i \(-0.647148\pi\)
−0.445989 + 0.895038i \(0.647148\pi\)
\(678\) 0 0
\(679\) 1344.00 1.97938
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −989.949 −1.44941 −0.724707 0.689057i \(-0.758025\pi\)
−0.724707 + 0.689057i \(0.758025\pi\)
\(684\) 0 0
\(685\) 630.000i 0.919708i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 395.980i 0.574717i
\(690\) 0 0
\(691\) − 632.000i − 0.914616i −0.889308 0.457308i \(-0.848814\pi\)
0.889308 0.457308i \(-0.151186\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 848.528i 1.22090i
\(696\) 0 0
\(697\) −210.000 −0.301291
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 29.6985 0.0423659 0.0211829 0.999776i \(-0.493257\pi\)
0.0211829 + 0.999776i \(0.493257\pi\)
\(702\) 0 0
\(703\) −480.000 −0.682788
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −763.675 −1.08016
\(708\) 0 0
\(709\) 56.0000i 0.0789845i 0.999220 + 0.0394922i \(0.0125740\pi\)
−0.999220 + 0.0394922i \(0.987426\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 158.392i 0.222149i
\(714\) 0 0
\(715\) − 320.000i − 0.447552i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 616.597i 0.857576i 0.903405 + 0.428788i \(0.141059\pi\)
−0.903405 + 0.428788i \(0.858941\pi\)
\(720\) 0 0
\(721\) −528.000 −0.732316
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 742.462 1.02409
\(726\) 0 0
\(727\) −372.000 −0.511692 −0.255846 0.966718i \(-0.582354\pi\)
−0.255846 + 0.966718i \(0.582354\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −79.1960 −0.108339
\(732\) 0 0
\(733\) 520.000i 0.709413i 0.934978 + 0.354707i \(0.115419\pi\)
−0.934978 + 0.354707i \(0.884581\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 497.803i − 0.675445i
\(738\) 0 0
\(739\) − 1040.00i − 1.40731i −0.710543 0.703654i \(-0.751551\pi\)
0.710543 0.703654i \(-0.248449\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 622.254i 0.837489i 0.908104 + 0.418744i \(0.137530\pi\)
−0.908104 + 0.418744i \(0.862470\pi\)
\(744\) 0 0
\(745\) 910.000 1.22148
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −180.000 −0.239680 −0.119840 0.992793i \(-0.538238\pi\)
−0.119840 + 0.992793i \(0.538238\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1385.93 1.83567
\(756\) 0 0
\(757\) 152.000i 0.200793i 0.994948 + 0.100396i \(0.0320111\pi\)
−0.994948 + 0.100396i \(0.967989\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 544.472i 0.715469i 0.933823 + 0.357735i \(0.116451\pi\)
−0.933823 + 0.357735i \(0.883549\pi\)
\(762\) 0 0
\(763\) 1632.00i 2.13893i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 633.568i 0.826033i
\(768\) 0 0
\(769\) −526.000 −0.684005 −0.342003 0.939699i \(-0.611105\pi\)
−0.342003 + 0.939699i \(0.611105\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1421.28 −1.83866 −0.919330 0.393487i \(-0.871269\pi\)
−0.919330 + 0.393487i \(0.871269\pi\)
\(774\) 0 0
\(775\) −100.000 −0.129032
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 339.411 0.435701
\(780\) 0 0
\(781\) 160.000i 0.204866i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 1682.91i − 2.14384i
\(786\) 0 0
\(787\) − 1072.00i − 1.36213i −0.732221 0.681067i \(-0.761516\pi\)
0.732221 0.681067i \(-0.238484\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 254.558i − 0.321819i
\(792\) 0 0
\(793\) −112.000 −0.141236
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1534.42 −1.92525 −0.962623 0.270843i \(-0.912697\pi\)
−0.962623 + 0.270843i \(0.912697\pi\)
\(798\) 0 0
\(799\) 168.000 0.210263
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −452.548 −0.563572
\(804\) 0 0
\(805\) 3360.00i 4.17391i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 134.350i − 0.166070i −0.996547 0.0830348i \(-0.973539\pi\)
0.996547 0.0830348i \(-0.0264613\pi\)
\(810\) 0 0
\(811\) 544.000i 0.670777i 0.942080 + 0.335388i \(0.108868\pi\)
−0.942080 + 0.335388i \(0.891132\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1187.94i 1.45759i
\(816\) 0 0
\(817\) 128.000 0.156671
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 878.227 1.06970 0.534852 0.844946i \(-0.320367\pi\)
0.534852 + 0.844946i \(0.320367\pi\)
\(822\) 0 0
\(823\) 868.000 1.05468 0.527339 0.849655i \(-0.323190\pi\)
0.527339 + 0.849655i \(0.323190\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1414.21 1.71005 0.855026 0.518585i \(-0.173541\pi\)
0.855026 + 0.518585i \(0.173541\pi\)
\(828\) 0 0
\(829\) 1208.00i 1.45718i 0.684952 + 0.728589i \(0.259823\pi\)
−0.684952 + 0.728589i \(0.740177\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 940.452i − 1.12899i
\(834\) 0 0
\(835\) 1200.00i 1.43713i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1069.15i 1.27431i 0.770736 + 0.637155i \(0.219889\pi\)
−0.770736 + 0.637155i \(0.780111\pi\)
\(840\) 0 0
\(841\) 41.0000 0.0487515
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 742.462 0.878653
\(846\) 0 0
\(847\) 1068.00 1.26092
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1187.94 1.39593
\(852\) 0 0
\(853\) − 1282.00i − 1.50293i −0.659772 0.751465i \(-0.729347\pi\)
0.659772 0.751465i \(-0.270653\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1039.45i − 1.21289i −0.795125 0.606445i \(-0.792595\pi\)
0.795125 0.606445i \(-0.207405\pi\)
\(858\) 0 0
\(859\) − 1032.00i − 1.20140i −0.799476 0.600698i \(-0.794889\pi\)
0.799476 0.600698i \(-0.205111\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 56.5685i 0.0655487i 0.999463 + 0.0327744i \(0.0104343\pi\)
−0.999463 + 0.0327744i \(0.989566\pi\)
\(864\) 0 0
\(865\) 1470.00 1.69942
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −565.685 −0.650961
\(870\) 0 0
\(871\) −704.000 −0.808266
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 462.000i 0.526796i 0.964687 + 0.263398i \(0.0848432\pi\)
−0.964687 + 0.263398i \(0.915157\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 247.487i − 0.280916i −0.990087 0.140458i \(-0.955142\pi\)
0.990087 0.140458i \(-0.0448576\pi\)
\(882\) 0 0
\(883\) − 8.00000i − 0.00906002i −0.999990 0.00453001i \(-0.998558\pi\)
0.999990 0.00453001i \(-0.00144195\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 893.783i − 1.00765i −0.863807 0.503824i \(-0.831926\pi\)
0.863807 0.503824i \(-0.168074\pi\)
\(888\) 0 0
\(889\) 240.000 0.269966
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −271.529 −0.304064
\(894\) 0 0
\(895\) 1440.00 1.60894
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −118.794 −0.132140
\(900\) 0 0
\(901\) − 490.000i − 0.543840i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 1527.35i − 1.68768i
\(906\) 0 0
\(907\) − 200.000i − 0.220507i −0.993903 0.110254i \(-0.964834\pi\)
0.993903 0.110254i \(-0.0351663\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 79.1960i − 0.0869330i −0.999055 0.0434665i \(-0.986160\pi\)
0.999055 0.0434665i \(-0.0138402\pi\)
\(912\) 0 0
\(913\) −736.000 −0.806134
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 678.823 0.740264
\(918\) 0 0
\(919\) 884.000 0.961915 0.480958 0.876744i \(-0.340289\pi\)
0.480958 + 0.876744i \(0.340289\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 226.274 0.245151
\(924\) 0 0
\(925\) 750.000i 0.810811i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1076.22i 1.15847i 0.815161 + 0.579234i \(0.196648\pi\)
−0.815161 + 0.579234i \(0.803352\pi\)
\(930\) 0 0
\(931\) 1520.00i 1.63265i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 395.980i 0.423508i
\(936\) 0 0
\(937\) −78.0000 −0.0832444 −0.0416222 0.999133i \(-0.513253\pi\)
−0.0416222 + 0.999133i \(0.513253\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 267.286 0.284045 0.142023 0.989863i \(-0.454639\pi\)
0.142023 + 0.989863i \(0.454639\pi\)
\(942\) 0 0
\(943\) −840.000 −0.890774
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −576.999 −0.609292 −0.304646 0.952466i \(-0.598538\pi\)
−0.304646 + 0.952466i \(0.598538\pi\)
\(948\) 0 0
\(949\) 640.000i 0.674394i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 1336.43i − 1.40234i −0.712993 0.701171i \(-0.752661\pi\)
0.712993 0.701171i \(-0.247339\pi\)
\(954\) 0 0
\(955\) − 2640.00i − 2.76440i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 1069.15i − 1.11485i
\(960\) 0 0
\(961\) −945.000 −0.983351
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1258.65 −1.30430
\(966\) 0 0
\(967\) −20.0000 −0.0206825 −0.0103413 0.999947i \(-0.503292\pi\)
−0.0103413 + 0.999947i \(0.503292\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 118.794 0.122342 0.0611709 0.998127i \(-0.480517\pi\)
0.0611709 + 0.998127i \(0.480517\pi\)
\(972\) 0 0
\(973\) − 1440.00i − 1.47996i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 420.021i − 0.429909i −0.976624 0.214955i \(-0.931040\pi\)
0.976624 0.214955i \(-0.0689604\pi\)
\(978\) 0 0
\(979\) − 840.000i − 0.858018i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1187.94i − 1.20848i −0.796801 0.604242i \(-0.793476\pi\)
0.796801 0.604242i \(-0.206524\pi\)
\(984\) 0 0
\(985\) −1470.00 −1.49239
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −316.784 −0.320307
\(990\) 0 0
\(991\) −116.000 −0.117053 −0.0585267 0.998286i \(-0.518640\pi\)
−0.0585267 + 0.998286i \(0.518640\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1385.93 −1.39289
\(996\) 0 0
\(997\) 130.000i 0.130391i 0.997873 + 0.0651956i \(0.0207671\pi\)
−0.997873 + 0.0651956i \(0.979233\pi\)
\(998\) 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.h.a.2177.4 4
3.2 odd 2 inner 2304.3.h.a.2177.2 4
4.3 odd 2 2304.3.h.h.2177.4 4
8.3 odd 2 2304.3.h.h.2177.1 4
8.5 even 2 inner 2304.3.h.a.2177.1 4
12.11 even 2 2304.3.h.h.2177.2 4
16.3 odd 4 144.3.e.a.17.1 2
16.5 even 4 576.3.e.h.449.2 2
16.11 odd 4 576.3.e.a.449.2 2
16.13 even 4 72.3.e.a.17.1 2
24.5 odd 2 inner 2304.3.h.a.2177.3 4
24.11 even 2 2304.3.h.h.2177.3 4
48.5 odd 4 576.3.e.h.449.1 2
48.11 even 4 576.3.e.a.449.1 2
48.29 odd 4 72.3.e.a.17.2 yes 2
48.35 even 4 144.3.e.a.17.2 2
80.3 even 4 3600.3.c.c.449.3 4
80.13 odd 4 1800.3.c.a.449.2 4
80.19 odd 4 3600.3.l.l.1601.1 2
80.29 even 4 1800.3.l.a.1601.2 2
80.67 even 4 3600.3.c.c.449.1 4
80.77 odd 4 1800.3.c.a.449.4 4
112.13 odd 4 3528.3.d.a.1961.2 2
144.13 even 12 648.3.m.a.593.1 4
144.29 odd 12 648.3.m.a.377.1 4
144.61 even 12 648.3.m.a.377.2 4
144.67 odd 12 1296.3.q.k.593.1 4
144.77 odd 12 648.3.m.a.593.2 4
144.83 even 12 1296.3.q.k.1025.1 4
144.115 odd 12 1296.3.q.k.1025.2 4
144.131 even 12 1296.3.q.k.593.2 4
240.29 odd 4 1800.3.l.a.1601.1 2
240.77 even 4 1800.3.c.a.449.3 4
240.83 odd 4 3600.3.c.c.449.4 4
240.173 even 4 1800.3.c.a.449.1 4
240.179 even 4 3600.3.l.l.1601.2 2
240.227 odd 4 3600.3.c.c.449.2 4
336.125 even 4 3528.3.d.a.1961.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.3.e.a.17.1 2 16.13 even 4
72.3.e.a.17.2 yes 2 48.29 odd 4
144.3.e.a.17.1 2 16.3 odd 4
144.3.e.a.17.2 2 48.35 even 4
576.3.e.a.449.1 2 48.11 even 4
576.3.e.a.449.2 2 16.11 odd 4
576.3.e.h.449.1 2 48.5 odd 4
576.3.e.h.449.2 2 16.5 even 4
648.3.m.a.377.1 4 144.29 odd 12
648.3.m.a.377.2 4 144.61 even 12
648.3.m.a.593.1 4 144.13 even 12
648.3.m.a.593.2 4 144.77 odd 12
1296.3.q.k.593.1 4 144.67 odd 12
1296.3.q.k.593.2 4 144.131 even 12
1296.3.q.k.1025.1 4 144.83 even 12
1296.3.q.k.1025.2 4 144.115 odd 12
1800.3.c.a.449.1 4 240.173 even 4
1800.3.c.a.449.2 4 80.13 odd 4
1800.3.c.a.449.3 4 240.77 even 4
1800.3.c.a.449.4 4 80.77 odd 4
1800.3.l.a.1601.1 2 240.29 odd 4
1800.3.l.a.1601.2 2 80.29 even 4
2304.3.h.a.2177.1 4 8.5 even 2 inner
2304.3.h.a.2177.2 4 3.2 odd 2 inner
2304.3.h.a.2177.3 4 24.5 odd 2 inner
2304.3.h.a.2177.4 4 1.1 even 1 trivial
2304.3.h.h.2177.1 4 8.3 odd 2
2304.3.h.h.2177.2 4 12.11 even 2
2304.3.h.h.2177.3 4 24.11 even 2
2304.3.h.h.2177.4 4 4.3 odd 2
3528.3.d.a.1961.1 2 336.125 even 4
3528.3.d.a.1961.2 2 112.13 odd 4
3600.3.c.c.449.1 4 80.67 even 4
3600.3.c.c.449.2 4 240.227 odd 4
3600.3.c.c.449.3 4 80.3 even 4
3600.3.c.c.449.4 4 240.83 odd 4
3600.3.l.l.1601.1 2 80.19 odd 4
3600.3.l.l.1601.2 2 240.179 even 4