Properties

Label 2880.2.d.i.289.8
Level $2880$
Weight $2$
Character 2880.289
Analytic conductor $22.997$
Analytic rank $0$
Dimension $8$
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] = [2880,2,Mod(289,2880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2880.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.d (of order \(2\), degree \(1\), 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: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 960)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 289.8
Root \(0.228425 + 1.39564i\) of defining polynomial
Character \(\chi\) \(=\) 2880.289
Dual form 2880.2.d.i.289.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.18890 + 0.456850i) q^{5} +0.913701i q^{7} +O(q^{10})\) \(q+(2.18890 + 0.456850i) q^{5} +0.913701i q^{7} +3.58258i q^{11} -0.913701 q^{13} +3.58258i q^{17} -4.00000i q^{19} +(4.58258 + 2.00000i) q^{25} +7.84190i q^{29} -5.29150 q^{31} +(-0.417424 + 2.00000i) q^{35} +7.84190 q^{37} -6.00000 q^{41} -7.16515 q^{43} +6.92820i q^{47} +6.16515 q^{49} +2.55040 q^{53} +(-1.63670 + 7.84190i) q^{55} +7.58258i q^{59} -10.5830i q^{61} +(-2.00000 - 0.417424i) q^{65} -15.1652 q^{67} +6.92820 q^{71} +12.0000i q^{73} -3.27340 q^{77} +5.29150 q^{79} -11.1652 q^{83} +(-1.63670 + 7.84190i) q^{85} +2.00000 q^{89} -0.834849i q^{91} +(1.82740 - 8.75560i) q^{95} +7.16515i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 40 q^{35} - 48 q^{41} + 16 q^{43} - 24 q^{49} - 16 q^{65} - 48 q^{67} - 16 q^{83} + 16 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(5\) 2.18890 + 0.456850i 0.978906 + 0.204310i
\(6\) 0 0
\(7\) 0.913701i 0.345346i 0.984979 + 0.172673i \(0.0552404\pi\)
−0.984979 + 0.172673i \(0.944760\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.58258i 1.08019i 0.841605 + 0.540094i \(0.181611\pi\)
−0.841605 + 0.540094i \(0.818389\pi\)
\(12\) 0 0
\(13\) −0.913701 −0.253415 −0.126707 0.991940i \(-0.540441\pi\)
−0.126707 + 0.991940i \(0.540441\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.58258i 0.868902i 0.900695 + 0.434451i \(0.143058\pi\)
−0.900695 + 0.434451i \(0.856942\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 4.58258 + 2.00000i 0.916515 + 0.400000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.84190i 1.45620i 0.685468 + 0.728102i \(0.259598\pi\)
−0.685468 + 0.728102i \(0.740402\pi\)
\(30\) 0 0
\(31\) −5.29150 −0.950382 −0.475191 0.879883i \(-0.657621\pi\)
−0.475191 + 0.879883i \(0.657621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.417424 + 2.00000i −0.0705576 + 0.338062i
\(36\) 0 0
\(37\) 7.84190 1.28920 0.644601 0.764520i \(-0.277024\pi\)
0.644601 + 0.764520i \(0.277024\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −7.16515 −1.09268 −0.546338 0.837565i \(-0.683978\pi\)
−0.546338 + 0.837565i \(0.683978\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.92820i 1.01058i 0.862949 + 0.505291i \(0.168615\pi\)
−0.862949 + 0.505291i \(0.831385\pi\)
\(48\) 0 0
\(49\) 6.16515 0.880736
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.55040 0.350325 0.175162 0.984540i \(-0.443955\pi\)
0.175162 + 0.984540i \(0.443955\pi\)
\(54\) 0 0
\(55\) −1.63670 + 7.84190i −0.220693 + 1.05740i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.58258i 0.987167i 0.869698 + 0.493584i \(0.164313\pi\)
−0.869698 + 0.493584i \(0.835687\pi\)
\(60\) 0 0
\(61\) 10.5830i 1.35501i −0.735516 0.677507i \(-0.763060\pi\)
0.735516 0.677507i \(-0.236940\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 0.417424i −0.248069 0.0517751i
\(66\) 0 0
\(67\) −15.1652 −1.85272 −0.926359 0.376642i \(-0.877079\pi\)
−0.926359 + 0.376642i \(0.877079\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.92820 0.822226 0.411113 0.911584i \(-0.365140\pi\)
0.411113 + 0.911584i \(0.365140\pi\)
\(72\) 0 0
\(73\) 12.0000i 1.40449i 0.711934 + 0.702247i \(0.247820\pi\)
−0.711934 + 0.702247i \(0.752180\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.27340 −0.373039
\(78\) 0 0
\(79\) 5.29150 0.595341 0.297670 0.954669i \(-0.403790\pi\)
0.297670 + 0.954669i \(0.403790\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −11.1652 −1.22553 −0.612767 0.790263i \(-0.709944\pi\)
−0.612767 + 0.790263i \(0.709944\pi\)
\(84\) 0 0
\(85\) −1.63670 + 7.84190i −0.177525 + 0.850574i
\(86\) 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
\(91\) 0.834849i 0.0875159i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.82740 8.75560i 0.187487 0.898306i
\(96\) 0 0
\(97\) 7.16515i 0.727511i 0.931494 + 0.363755i \(0.118506\pi\)
−0.931494 + 0.363755i \(0.881494\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.5975i 1.65151i −0.564026 0.825757i \(-0.690748\pi\)
0.564026 0.825757i \(-0.309252\pi\)
\(102\) 0 0
\(103\) 16.5975i 1.63540i 0.575644 + 0.817701i \(0.304751\pi\)
−0.575644 + 0.817701i \(0.695249\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 8.75560i 0.838635i −0.907840 0.419317i \(-0.862269\pi\)
0.907840 0.419317i \(-0.137731\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.41742i 0.415556i 0.978176 + 0.207778i \(0.0666232\pi\)
−0.978176 + 0.207778i \(0.933377\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.27340 −0.300072
\(120\) 0 0
\(121\) −1.83485 −0.166804
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.11710 + 6.47135i 0.815459 + 0.578815i
\(126\) 0 0
\(127\) 14.7701i 1.31064i −0.755354 0.655318i \(-0.772535\pi\)
0.755354 0.655318i \(-0.227465\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 15.5826i 1.36146i −0.732536 0.680728i \(-0.761664\pi\)
0.732536 0.680728i \(-0.238336\pi\)
\(132\) 0 0
\(133\) 3.65480 0.316912
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.5826i 1.67305i 0.547927 + 0.836526i \(0.315417\pi\)
−0.547927 + 0.836526i \(0.684583\pi\)
\(138\) 0 0
\(139\) 11.1652i 0.947016i 0.880790 + 0.473508i \(0.157012\pi\)
−0.880790 + 0.473508i \(0.842988\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.27340i 0.273736i
\(144\) 0 0
\(145\) −3.58258 + 17.1652i −0.297517 + 1.42549i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 11.4967i 0.941847i −0.882174 0.470923i \(-0.843921\pi\)
0.882174 0.470923i \(-0.156079\pi\)
\(150\) 0 0
\(151\) 15.8745 1.29185 0.645925 0.763401i \(-0.276472\pi\)
0.645925 + 0.763401i \(0.276472\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −11.5826 2.41742i −0.930335 0.194172i
\(156\) 0 0
\(157\) 21.6983 1.73171 0.865857 0.500292i \(-0.166774\pi\)
0.865857 + 0.500292i \(0.166774\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.6120i 1.74977i 0.484331 + 0.874885i \(0.339063\pi\)
−0.484331 + 0.874885i \(0.660937\pi\)
\(168\) 0 0
\(169\) −12.1652 −0.935781
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.3060 0.859580 0.429790 0.902929i \(-0.358588\pi\)
0.429790 + 0.902929i \(0.358588\pi\)
\(174\) 0 0
\(175\) −1.82740 + 4.18710i −0.138139 + 0.316515i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.58258i 0.566748i −0.959009 0.283374i \(-0.908546\pi\)
0.959009 0.283374i \(-0.0914538\pi\)
\(180\) 0 0
\(181\) 8.75560i 0.650799i 0.945577 + 0.325399i \(0.105499\pi\)
−0.945577 + 0.325399i \(0.894501\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 17.1652 + 3.58258i 1.26201 + 0.263396i
\(186\) 0 0
\(187\) −12.8348 −0.938577
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.8564 1.00261 0.501307 0.865269i \(-0.332853\pi\)
0.501307 + 0.865269i \(0.332853\pi\)
\(192\) 0 0
\(193\) 3.16515i 0.227833i 0.993490 + 0.113916i \(0.0363396\pi\)
−0.993490 + 0.113916i \(0.963660\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −25.5438 −1.81992 −0.909961 0.414695i \(-0.863888\pi\)
−0.909961 + 0.414695i \(0.863888\pi\)
\(198\) 0 0
\(199\) −1.63670 −0.116023 −0.0580113 0.998316i \(-0.518476\pi\)
−0.0580113 + 0.998316i \(0.518476\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −7.16515 −0.502895
\(204\) 0 0
\(205\) −13.1334 2.74110i −0.917277 0.191447i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 14.3303 0.991248
\(210\) 0 0
\(211\) 11.1652i 0.768641i 0.923200 + 0.384320i \(0.125564\pi\)
−0.923200 + 0.384320i \(0.874436\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −15.6838 3.27340i −1.06963 0.223244i
\(216\) 0 0
\(217\) 4.83485i 0.328211i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.27340i 0.220193i
\(222\) 0 0
\(223\) 0.913701i 0.0611859i 0.999532 + 0.0305930i \(0.00973956\pi\)
−0.999532 + 0.0305930i \(0.990260\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.16515 −0.210078 −0.105039 0.994468i \(-0.533497\pi\)
−0.105039 + 0.994468i \(0.533497\pi\)
\(228\) 0 0
\(229\) 22.6120i 1.49424i 0.664687 + 0.747122i \(0.268565\pi\)
−0.664687 + 0.747122i \(0.731435\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.5826i 1.28290i −0.767166 0.641449i \(-0.778334\pi\)
0.767166 0.641449i \(-0.221666\pi\)
\(234\) 0 0
\(235\) −3.16515 + 15.1652i −0.206472 + 0.989265i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −21.1660 −1.36912 −0.684558 0.728959i \(-0.740005\pi\)
−0.684558 + 0.728959i \(0.740005\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 13.4949 + 2.81655i 0.862158 + 0.179943i
\(246\) 0 0
\(247\) 3.65480i 0.232549i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 11.5826i 0.731086i −0.930795 0.365543i \(-0.880883\pi\)
0.930795 0.365543i \(-0.119117\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.74773i 0.171399i −0.996321 0.0856993i \(-0.972688\pi\)
0.996321 0.0856993i \(-0.0273124\pi\)
\(258\) 0 0
\(259\) 7.16515i 0.445221i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.75560i 0.539894i 0.962875 + 0.269947i \(0.0870061\pi\)
−0.962875 + 0.269947i \(0.912994\pi\)
\(264\) 0 0
\(265\) 5.58258 + 1.16515i 0.342935 + 0.0715747i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.66930i 0.589548i 0.955567 + 0.294774i \(0.0952444\pi\)
−0.955567 + 0.294774i \(0.904756\pi\)
\(270\) 0 0
\(271\) 26.0761 1.58401 0.792006 0.610514i \(-0.209037\pi\)
0.792006 + 0.610514i \(0.209037\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.16515 + 16.4174i −0.432075 + 0.990008i
\(276\) 0 0
\(277\) 14.7701 0.887450 0.443725 0.896163i \(-0.353657\pi\)
0.443725 + 0.896163i \(0.353657\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.33030 0.258324 0.129162 0.991623i \(-0.458771\pi\)
0.129162 + 0.991623i \(0.458771\pi\)
\(282\) 0 0
\(283\) 7.16515 0.425924 0.212962 0.977060i \(-0.431689\pi\)
0.212962 + 0.977060i \(0.431689\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.48220i 0.323604i
\(288\) 0 0
\(289\) 4.16515 0.245009
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.37780 0.255754 0.127877 0.991790i \(-0.459184\pi\)
0.127877 + 0.991790i \(0.459184\pi\)
\(294\) 0 0
\(295\) −3.46410 + 16.5975i −0.201688 + 0.966344i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 6.54680i 0.377351i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.83485 23.1652i 0.276843 1.32643i
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 17.5112 0.992970 0.496485 0.868045i \(-0.334624\pi\)
0.496485 + 0.868045i \(0.334624\pi\)
\(312\) 0 0
\(313\) 19.1652i 1.08328i 0.840611 + 0.541639i \(0.182196\pi\)
−0.840611 + 0.541639i \(0.817804\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.93180 0.164666 0.0823332 0.996605i \(-0.473763\pi\)
0.0823332 + 0.996605i \(0.473763\pi\)
\(318\) 0 0
\(319\) −28.0942 −1.57297
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 14.3303 0.797359
\(324\) 0 0
\(325\) −4.18710 1.82740i −0.232259 0.101366i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.33030 −0.349001
\(330\) 0 0
\(331\) 26.3303i 1.44724i −0.690196 0.723622i \(-0.742476\pi\)
0.690196 0.723622i \(-0.257524\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −33.1950 6.92820i −1.81364 0.378528i
\(336\) 0 0
\(337\) 10.3303i 0.562727i −0.959601 0.281364i \(-0.909213\pi\)
0.959601 0.281364i \(-0.0907867\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 18.9572i 1.02659i
\(342\) 0 0
\(343\) 12.0290i 0.649505i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.1652 1.02884 0.514420 0.857539i \(-0.328007\pi\)
0.514420 + 0.857539i \(0.328007\pi\)
\(348\) 0 0
\(349\) 3.27340i 0.175221i 0.996155 + 0.0876106i \(0.0279231\pi\)
−0.996155 + 0.0876106i \(0.972077\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 33.9129i 1.80500i 0.430689 + 0.902500i \(0.358270\pi\)
−0.430689 + 0.902500i \(0.641730\pi\)
\(354\) 0 0
\(355\) 15.1652 + 3.16515i 0.804883 + 0.167989i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.92820 0.365657 0.182828 0.983145i \(-0.441475\pi\)
0.182828 + 0.983145i \(0.441475\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.48220 + 26.2668i −0.286952 + 1.37487i
\(366\) 0 0
\(367\) 35.9361i 1.87585i −0.346838 0.937925i \(-0.612745\pi\)
0.346838 0.937925i \(-0.387255\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.33030i 0.120983i
\(372\) 0 0
\(373\) −6.01450 −0.311419 −0.155710 0.987803i \(-0.549766\pi\)
−0.155710 + 0.987803i \(0.549766\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.16515i 0.369024i
\(378\) 0 0
\(379\) 4.83485i 0.248349i 0.992260 + 0.124175i \(0.0396283\pi\)
−0.992260 + 0.124175i \(0.960372\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 22.9934i 1.17491i −0.809257 0.587454i \(-0.800130\pi\)
0.809257 0.587454i \(-0.199870\pi\)
\(384\) 0 0
\(385\) −7.16515 1.49545i −0.365170 0.0762154i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 18.4249i 0.934180i 0.884210 + 0.467090i \(0.154698\pi\)
−0.884210 + 0.467090i \(0.845302\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 11.5826 + 2.41742i 0.582783 + 0.121634i
\(396\) 0 0
\(397\) −7.84190 −0.393574 −0.196787 0.980446i \(-0.563051\pi\)
−0.196787 + 0.980446i \(0.563051\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) 4.83485 0.240841
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 28.0942i 1.39258i
\(408\) 0 0
\(409\) 18.8348 0.931323 0.465662 0.884963i \(-0.345816\pi\)
0.465662 + 0.884963i \(0.345816\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.92820 −0.340915
\(414\) 0 0
\(415\) −24.4394 5.10080i −1.19968 0.250389i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.7477i 0.525061i 0.964924 + 0.262530i \(0.0845570\pi\)
−0.964924 + 0.262530i \(0.915443\pi\)
\(420\) 0 0
\(421\) 18.9572i 0.923918i −0.886901 0.461959i \(-0.847147\pi\)
0.886901 0.461959i \(-0.152853\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −7.16515 + 16.4174i −0.347561 + 0.796362i
\(426\) 0 0
\(427\) 9.66970 0.467949
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 17.1298 0.825114 0.412557 0.910932i \(-0.364636\pi\)
0.412557 + 0.910932i \(0.364636\pi\)
\(432\) 0 0
\(433\) 38.3303i 1.84204i −0.389519 0.921018i \(-0.627359\pi\)
0.389519 0.921018i \(-0.372641\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 2.01810 0.0963187 0.0481594 0.998840i \(-0.484664\pi\)
0.0481594 + 0.998840i \(0.484664\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.8348 −0.609802 −0.304901 0.952384i \(-0.598623\pi\)
−0.304901 + 0.952384i \(0.598623\pi\)
\(444\) 0 0
\(445\) 4.37780 + 0.913701i 0.207528 + 0.0433136i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −8.33030 −0.393131 −0.196566 0.980491i \(-0.562979\pi\)
−0.196566 + 0.980491i \(0.562979\pi\)
\(450\) 0 0
\(451\) 21.4955i 1.01218i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.381401 1.82740i 0.0178803 0.0856699i
\(456\) 0 0
\(457\) 23.1652i 1.08362i −0.840501 0.541810i \(-0.817739\pi\)
0.840501 0.541810i \(-0.182261\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 25.3531i 1.18081i −0.807106 0.590406i \(-0.798968\pi\)
0.807106 0.590406i \(-0.201032\pi\)
\(462\) 0 0
\(463\) 39.5909i 1.83995i −0.391982 0.919973i \(-0.628210\pi\)
0.391982 0.919973i \(-0.371790\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 19.1652 0.886857 0.443429 0.896310i \(-0.353762\pi\)
0.443429 + 0.896310i \(0.353762\pi\)
\(468\) 0 0
\(469\) 13.8564i 0.639829i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 25.6697i 1.18029i
\(474\) 0 0
\(475\) 8.00000 18.3303i 0.367065 0.841052i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 31.7490 1.45065 0.725325 0.688407i \(-0.241690\pi\)
0.725325 + 0.688407i \(0.241690\pi\)
\(480\) 0 0
\(481\) −7.16515 −0.326703
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.27340 + 15.6838i −0.148637 + 0.712165i
\(486\) 0 0
\(487\) 20.2523i 0.917720i 0.888509 + 0.458860i \(0.151742\pi\)
−0.888509 + 0.458860i \(0.848258\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 14.7477i 0.665556i −0.943005 0.332778i \(-0.892014\pi\)
0.943005 0.332778i \(-0.107986\pi\)
\(492\) 0 0
\(493\) −28.0942 −1.26530
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.33030i 0.283953i
\(498\) 0 0
\(499\) 18.3303i 0.820577i −0.911956 0.410289i \(-0.865428\pi\)
0.911956 0.410289i \(-0.134572\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14.2378i 0.634832i −0.948286 0.317416i \(-0.897185\pi\)
0.948286 0.317416i \(-0.102815\pi\)
\(504\) 0 0
\(505\) 7.58258 36.3303i 0.337420 1.61668i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8.22330i 0.364492i −0.983253 0.182246i \(-0.941663\pi\)
0.983253 0.182246i \(-0.0583367\pi\)
\(510\) 0 0
\(511\) −10.9644 −0.485037
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.58258 + 36.3303i −0.334128 + 1.60090i
\(516\) 0 0
\(517\) −24.8208 −1.09162
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −38.6606 −1.69375 −0.846876 0.531791i \(-0.821519\pi\)
−0.846876 + 0.531791i \(0.821519\pi\)
\(522\) 0 0
\(523\) 18.3303 0.801528 0.400764 0.916181i \(-0.368745\pi\)
0.400764 + 0.916181i \(0.368745\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 18.9572i 0.825789i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5.48220 0.237461
\(534\) 0 0
\(535\) 8.75560 + 1.82740i 0.378538 + 0.0790054i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 22.0871i 0.951360i
\(540\) 0 0
\(541\) 16.0652i 0.690697i −0.938474 0.345349i \(-0.887761\pi\)
0.938474 0.345349i \(-0.112239\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.00000 19.1652i 0.171341 0.820945i
\(546\) 0 0
\(547\) 29.4955 1.26113 0.630567 0.776135i \(-0.282822\pi\)
0.630567 + 0.776135i \(0.282822\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 31.3676 1.33631
\(552\) 0 0
\(553\) 4.83485i 0.205599i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −20.0616 −0.850038 −0.425019 0.905185i \(-0.639733\pi\)
−0.425019 + 0.905185i \(0.639733\pi\)
\(558\) 0 0
\(559\) 6.54680 0.276900
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 25.4955 1.07451 0.537253 0.843421i \(-0.319462\pi\)
0.537253 + 0.843421i \(0.319462\pi\)
\(564\) 0 0
\(565\) −2.01810 + 9.66930i −0.0849022 + 0.406791i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.330303 0.0138470 0.00692351 0.999976i \(-0.497796\pi\)
0.00692351 + 0.999976i \(0.497796\pi\)
\(570\) 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
\(575\) 0 0
\(576\) 0 0
\(577\) 3.16515i 0.131767i −0.997827 0.0658835i \(-0.979013\pi\)
0.997827 0.0658835i \(-0.0209866\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 10.2016i 0.423234i
\(582\) 0 0
\(583\) 9.13701i 0.378416i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −42.3303 −1.74716 −0.873579 0.486682i \(-0.838207\pi\)
−0.873579 + 0.486682i \(0.838207\pi\)
\(588\) 0 0
\(589\) 21.1660i 0.872130i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9.91288i 0.407073i −0.979067 0.203537i \(-0.934756\pi\)
0.979067 0.203537i \(-0.0652436\pi\)
\(594\) 0 0
\(595\) −7.16515 1.49545i −0.293743 0.0613076i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −28.0942 −1.14790 −0.573949 0.818891i \(-0.694589\pi\)
−0.573949 + 0.818891i \(0.694589\pi\)
\(600\) 0 0
\(601\) −5.16515 −0.210691 −0.105345 0.994436i \(-0.533595\pi\)
−0.105345 + 0.994436i \(0.533595\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.01630 0.838251i −0.163286 0.0340798i
\(606\) 0 0
\(607\) 9.28790i 0.376984i −0.982075 0.188492i \(-0.939640\pi\)
0.982075 0.188492i \(-0.0603600\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.33030i 0.256097i
\(612\) 0 0
\(613\) −15.1515 −0.611964 −0.305982 0.952037i \(-0.598985\pi\)
−0.305982 + 0.952037i \(0.598985\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.41742i 0.177839i −0.996039 0.0889194i \(-0.971659\pi\)
0.996039 0.0889194i \(-0.0283414\pi\)
\(618\) 0 0
\(619\) 9.49545i 0.381655i −0.981624 0.190827i \(-0.938883\pi\)
0.981624 0.190827i \(-0.0611170\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.82740i 0.0732133i
\(624\) 0 0
\(625\) 17.0000 + 18.3303i 0.680000 + 0.733212i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 28.0942i 1.12019i
\(630\) 0 0
\(631\) 1.63670 0.0651560 0.0325780 0.999469i \(-0.489628\pi\)
0.0325780 + 0.999469i \(0.489628\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.74773 32.3303i 0.267775 1.28299i
\(636\) 0 0
\(637\) −5.63310 −0.223192
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −32.3303 −1.27697 −0.638485 0.769634i \(-0.720439\pi\)
−0.638485 + 0.769634i \(0.720439\pi\)
\(642\) 0 0
\(643\) 18.3303 0.722877 0.361438 0.932396i \(-0.382286\pi\)
0.361438 + 0.932396i \(0.382286\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 45.6054i 1.79293i −0.443110 0.896467i \(-0.646125\pi\)
0.443110 0.896467i \(-0.353875\pi\)
\(648\) 0 0
\(649\) −27.1652 −1.06633
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −25.5438 −0.999607 −0.499803 0.866139i \(-0.666594\pi\)
−0.499803 + 0.866139i \(0.666594\pi\)
\(654\) 0 0
\(655\) 7.11890 34.1087i 0.278159 1.33274i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 16.4174i 0.639532i −0.947497 0.319766i \(-0.896396\pi\)
0.947497 0.319766i \(-0.103604\pi\)
\(660\) 0 0
\(661\) 45.6054i 1.77385i −0.461918 0.886923i \(-0.652839\pi\)
0.461918 0.886923i \(-0.347161\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.00000 + 1.66970i 0.310227 + 0.0647481i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 37.9144 1.46367
\(672\) 0 0
\(673\) 40.0000i 1.54189i −0.636904 0.770943i \(-0.719785\pi\)
0.636904 0.770943i \(-0.280215\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 23.3350 0.896837 0.448419 0.893824i \(-0.351987\pi\)
0.448419 + 0.893824i \(0.351987\pi\)
\(678\) 0 0
\(679\) −6.54680 −0.251243
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 49.4955 1.89389 0.946945 0.321394i \(-0.104151\pi\)
0.946945 + 0.321394i \(0.104151\pi\)
\(684\) 0 0
\(685\) −8.94630 + 42.8643i −0.341821 + 1.63776i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.33030 −0.0887775
\(690\) 0 0
\(691\) 32.6606i 1.24247i 0.783625 + 0.621234i \(0.213368\pi\)
−0.783625 + 0.621234i \(0.786632\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −5.10080 + 24.4394i −0.193484 + 0.927040i
\(696\) 0 0
\(697\) 21.4955i 0.814198i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 40.6555i 1.53554i −0.640727 0.767769i \(-0.721367\pi\)
0.640727 0.767769i \(-0.278633\pi\)
\(702\) 0 0
\(703\) 31.3676i 1.18305i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.1652 0.570344
\(708\) 0 0
\(709\) 5.10080i 0.191565i −0.995402 0.0957823i \(-0.969465\pi\)
0.995402 0.0957823i \(-0.0305353\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 1.49545 7.16515i 0.0559268 0.267961i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −52.1522 −1.94495 −0.972475 0.233008i \(-0.925143\pi\)
−0.972475 + 0.233008i \(0.925143\pi\)
\(720\) 0 0
\(721\) −15.1652 −0.564780
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −15.6838 + 35.9361i −0.582482 + 1.33463i
\(726\) 0 0
\(727\) 23.1443i 0.858375i 0.903215 + 0.429187i \(0.141200\pi\)
−0.903215 + 0.429187i \(0.858800\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 25.6697i 0.949428i
\(732\) 0 0
\(733\) 39.5909 1.46232 0.731162 0.682204i \(-0.238978\pi\)
0.731162 + 0.682204i \(0.238978\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 54.3303i 2.00128i
\(738\) 0 0
\(739\) 20.0000i 0.735712i −0.929883 0.367856i \(-0.880092\pi\)
0.929883 0.367856i \(-0.119908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 26.6482i 0.977628i 0.872388 + 0.488814i \(0.162570\pi\)
−0.872388 + 0.488814i \(0.837430\pi\)
\(744\) 0 0
\(745\) 5.25227 25.1652i 0.192428 0.921980i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 3.65480i 0.133544i
\(750\) 0 0
\(751\) 22.4213 0.818165 0.409083 0.912497i \(-0.365849\pi\)
0.409083 + 0.912497i \(0.365849\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 34.7477 + 7.25227i 1.26460 + 0.263937i
\(756\) 0 0
\(757\) −12.9427 −0.470411 −0.235205 0.971946i \(-0.575576\pi\)
−0.235205 + 0.971946i \(0.575576\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.92820i 0.250163i
\(768\) 0 0
\(769\) −31.4955 −1.13576 −0.567878 0.823113i \(-0.692235\pi\)
−0.567878 + 0.823113i \(0.692235\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −14.5794 −0.524385 −0.262192 0.965016i \(-0.584446\pi\)
−0.262192 + 0.965016i \(0.584446\pi\)
\(774\) 0 0
\(775\) −24.2487 10.5830i −0.871039 0.380153i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.0000i 0.859889i
\(780\) 0 0
\(781\) 24.8208i 0.888159i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 47.4955 + 9.91288i 1.69519 + 0.353806i
\(786\) 0 0
\(787\) 36.0000 1.28326 0.641631 0.767014i \(-0.278258\pi\)
0.641631 + 0.767014i \(0.278258\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.03620 −0.143511
\(792\) 0 0
\(793\) 9.66970i 0.343381i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 26.6084 0.942518 0.471259 0.881995i \(-0.343800\pi\)
0.471259 + 0.881995i \(0.343800\pi\)
\(798\) 0 0
\(799\) −24.8208 −0.878097
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −42.9909 −1.51712
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 19.1652i 0.672979i 0.941687 + 0.336490i \(0.109240\pi\)
−0.941687 + 0.336490i \(0.890760\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −8.75560 1.82740i −0.306695 0.0640111i
\(816\) 0 0
\(817\) 28.6606i 1.00271i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9.66930i 0.337461i −0.985662 0.168731i \(-0.946033\pi\)
0.985662 0.168731i \(-0.0539668\pi\)
\(822\) 0 0
\(823\) 14.7701i 0.514854i 0.966298 + 0.257427i \(0.0828746\pi\)
−0.966298 + 0.257427i \(0.917125\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) 16.0652i 0.557968i 0.960296 + 0.278984i \(0.0899976\pi\)
−0.960296 + 0.278984i \(0.910002\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 22.0871i 0.765273i
\(834\) 0 0
\(835\) −10.3303 + 49.4955i −0.357495 + 1.71286i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −13.4750 −0.465209 −0.232604 0.972571i \(-0.574725\pi\)
−0.232604 + 0.972571i \(0.574725\pi\)
\(840\) 0 0
\(841\) −32.4955 −1.12053
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −26.6283 5.55765i −0.916042 0.191189i
\(846\) 0 0
\(847\) 1.67650i 0.0576053i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 23.5257 0.805505 0.402753 0.915309i \(-0.368054\pi\)
0.402753 + 0.915309i \(0.368054\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 20.4174i 0.697446i 0.937226 + 0.348723i \(0.113385\pi\)
−0.937226 + 0.348723i \(0.886615\pi\)
\(858\) 0 0
\(859\) 43.1652i 1.47278i 0.676559 + 0.736388i \(0.263470\pi\)
−0.676559 + 0.736388i \(0.736530\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.20880i 0.0751885i −0.999293 0.0375942i \(-0.988031\pi\)
0.999293 0.0375942i \(-0.0119694\pi\)
\(864\) 0 0
\(865\) 24.7477 + 5.16515i 0.841448 + 0.175620i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 18.9572i 0.643079i
\(870\) 0 0
\(871\) 13.8564 0.469506
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.91288 + 8.33030i −0.199892 + 0.281616i
\(876\) 0 0
\(877\) −54.8933 −1.85362 −0.926808 0.375536i \(-0.877459\pi\)
−0.926808 + 0.375536i \(0.877459\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0.330303 0.0111282 0.00556409 0.999985i \(-0.498229\pi\)
0.00556409 + 0.999985i \(0.498229\pi\)
\(882\) 0 0
\(883\) 2.33030 0.0784209 0.0392105 0.999231i \(-0.487516\pi\)
0.0392105 + 0.999231i \(0.487516\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 15.3024i 0.513805i −0.966437 0.256902i \(-0.917298\pi\)
0.966437 0.256902i \(-0.0827018\pi\)
\(888\) 0 0
\(889\) 13.4955 0.452623
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 27.7128 0.927374
\(894\) 0 0
\(895\) 3.46410 16.5975i 0.115792 0.554794i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 41.4955i 1.38395i
\(900\) 0 0
\(901\) 9.13701i 0.304398i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.00000 + 19.1652i −0.132964 + 0.637071i
\(906\) 0 0
\(907\) 37.4955 1.24502 0.622508 0.782613i \(-0.286114\pi\)
0.622508 + 0.782613i \(0.286114\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −35.0224 −1.16034 −0.580172 0.814494i \(-0.697015\pi\)
−0.580172 + 0.814494i \(0.697015\pi\)
\(912\) 0 0
\(913\) 40.0000i 1.32381i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 14.2378 0.470174
\(918\) 0 0
\(919\) 43.5873 1.43781 0.718907 0.695107i \(-0.244643\pi\)
0.718907 + 0.695107i \(0.244643\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −6.33030 −0.208364
\(924\) 0 0
\(925\) 35.9361 + 15.6838i 1.18157 + 0.515680i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 42.6606 1.39965 0.699825 0.714315i \(-0.253262\pi\)
0.699825 + 0.714315i \(0.253262\pi\)
\(930\) 0 0
\(931\) 24.6606i 0.808219i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −28.0942 5.86360i −0.918779 0.191760i
\(936\) 0 0
\(937\) 22.3303i 0.729499i 0.931106 + 0.364750i \(0.118845\pi\)
−0.931106 + 0.364750i \(0.881155\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 44.3103i 1.44448i −0.691645 0.722238i \(-0.743114\pi\)
0.691645 0.722238i \(-0.256886\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) 10.9644i 0.355920i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 56.2432i 1.82190i −0.412522 0.910948i \(-0.635352\pi\)
0.412522 0.910948i \(-0.364648\pi\)
\(954\) 0 0
\(955\) 30.3303 + 6.33030i 0.981466 + 0.204844i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −17.8926 −0.577782
\(960\) 0 0
\(961\) −3.00000 −0.0967742
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.44600 + 6.92820i −0.0465484 + 0.223027i
\(966\) 0 0
\(967\) 11.1153i 0.357444i 0.983900 + 0.178722i \(0.0571963\pi\)
−0.983900 + 0.178722i \(0.942804\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 49.9129i 1.60178i −0.598811 0.800890i \(-0.704360\pi\)
0.598811 0.800890i \(-0.295640\pi\)
\(972\) 0 0
\(973\) −10.2016 −0.327048
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 34.7477i 1.11168i −0.831290 0.555839i \(-0.812397\pi\)
0.831290 0.555839i \(-0.187603\pi\)
\(978\) 0 0
\(979\) 7.16515i 0.228999i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 50.7062i 1.61728i 0.588306 + 0.808639i \(0.299795\pi\)
−0.588306 + 0.808639i \(0.700205\pi\)
\(984\) 0 0
\(985\) −55.9129 11.6697i −1.78153 0.371827i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −43.5873 −1.38460 −0.692298 0.721611i \(-0.743402\pi\)
−0.692298 + 0.721611i \(0.743402\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3.58258 0.747727i −0.113575 0.0237045i
\(996\) 0 0
\(997\) −39.2095 −1.24178 −0.620889 0.783899i \(-0.713228\pi\)
−0.620889 + 0.783899i \(0.713228\pi\)
\(998\) 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 2880.2.d.i.289.8 8
3.2 odd 2 960.2.d.f.289.1 yes 8
4.3 odd 2 2880.2.d.j.289.8 8
5.4 even 2 2880.2.d.j.289.2 8
8.3 odd 2 inner 2880.2.d.i.289.1 8
8.5 even 2 2880.2.d.j.289.1 8
12.11 even 2 960.2.d.e.289.1 8
15.2 even 4 4800.2.k.q.2401.2 8
15.8 even 4 4800.2.k.r.2401.7 8
15.14 odd 2 960.2.d.e.289.7 yes 8
20.19 odd 2 inner 2880.2.d.i.289.2 8
24.5 odd 2 960.2.d.e.289.8 yes 8
24.11 even 2 960.2.d.f.289.8 yes 8
40.19 odd 2 2880.2.d.j.289.7 8
40.29 even 2 inner 2880.2.d.i.289.7 8
48.5 odd 4 3840.2.f.i.769.3 8
48.11 even 4 3840.2.f.k.769.7 8
48.29 odd 4 3840.2.f.k.769.6 8
48.35 even 4 3840.2.f.i.769.2 8
60.23 odd 4 4800.2.k.r.2401.2 8
60.47 odd 4 4800.2.k.q.2401.7 8
60.59 even 2 960.2.d.f.289.7 yes 8
120.29 odd 2 960.2.d.f.289.2 yes 8
120.53 even 4 4800.2.k.r.2401.3 8
120.59 even 2 960.2.d.e.289.2 yes 8
120.77 even 4 4800.2.k.q.2401.6 8
120.83 odd 4 4800.2.k.r.2401.6 8
120.107 odd 4 4800.2.k.q.2401.3 8
240.29 odd 4 3840.2.f.k.769.2 8
240.59 even 4 3840.2.f.k.769.3 8
240.149 odd 4 3840.2.f.i.769.7 8
240.179 even 4 3840.2.f.i.769.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
960.2.d.e.289.1 8 12.11 even 2
960.2.d.e.289.2 yes 8 120.59 even 2
960.2.d.e.289.7 yes 8 15.14 odd 2
960.2.d.e.289.8 yes 8 24.5 odd 2
960.2.d.f.289.1 yes 8 3.2 odd 2
960.2.d.f.289.2 yes 8 120.29 odd 2
960.2.d.f.289.7 yes 8 60.59 even 2
960.2.d.f.289.8 yes 8 24.11 even 2
2880.2.d.i.289.1 8 8.3 odd 2 inner
2880.2.d.i.289.2 8 20.19 odd 2 inner
2880.2.d.i.289.7 8 40.29 even 2 inner
2880.2.d.i.289.8 8 1.1 even 1 trivial
2880.2.d.j.289.1 8 8.5 even 2
2880.2.d.j.289.2 8 5.4 even 2
2880.2.d.j.289.7 8 40.19 odd 2
2880.2.d.j.289.8 8 4.3 odd 2
3840.2.f.i.769.2 8 48.35 even 4
3840.2.f.i.769.3 8 48.5 odd 4
3840.2.f.i.769.6 8 240.179 even 4
3840.2.f.i.769.7 8 240.149 odd 4
3840.2.f.k.769.2 8 240.29 odd 4
3840.2.f.k.769.3 8 240.59 even 4
3840.2.f.k.769.6 8 48.29 odd 4
3840.2.f.k.769.7 8 48.11 even 4
4800.2.k.q.2401.2 8 15.2 even 4
4800.2.k.q.2401.3 8 120.107 odd 4
4800.2.k.q.2401.6 8 120.77 even 4
4800.2.k.q.2401.7 8 60.47 odd 4
4800.2.k.r.2401.2 8 60.23 odd 4
4800.2.k.r.2401.3 8 120.53 even 4
4800.2.k.r.2401.6 8 120.83 odd 4
4800.2.k.r.2401.7 8 15.8 even 4