Properties

Label 4800.2.d.o.1249.1
Level $4800$
Weight $2$
Character 4800.1249
Analytic conductor $38.328$
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] = [4800,2,Mod(1249,4800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4800, 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("4800.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.3281929702\)
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: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 192)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 4800.1249
Dual form 4800.2.d.o.1249.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.46410i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.46410i q^{7} +1.00000 q^{9} -6.00000i q^{17} +4.00000i q^{19} -3.46410i q^{21} -6.92820i q^{23} +1.00000 q^{27} -3.46410i q^{29} -3.46410 q^{31} -6.92820 q^{37} -6.00000 q^{41} -4.00000 q^{43} +6.92820i q^{47} -5.00000 q^{49} -6.00000i q^{51} +3.46410 q^{53} +4.00000i q^{57} +12.0000i q^{59} -6.92820i q^{61} -3.46410i q^{63} +4.00000 q^{67} -6.92820i q^{69} +6.92820 q^{71} +2.00000i q^{73} -10.3923 q^{79} +1.00000 q^{81} -3.46410i q^{87} -6.00000 q^{89} -3.46410 q^{93} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} + 4 q^{27} - 24 q^{41} - 16 q^{43} - 20 q^{49} + 16 q^{67} + 4 q^{81} - 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(4351\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.46410i − 1.30931i −0.755929 0.654654i \(-0.772814\pi\)
0.755929 0.654654i \(-0.227186\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) − 3.46410i − 0.755929i
\(22\) 0 0
\(23\) − 6.92820i − 1.44463i −0.691564 0.722315i \(-0.743078\pi\)
0.691564 0.722315i \(-0.256922\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) − 3.46410i − 0.643268i −0.946864 0.321634i \(-0.895768\pi\)
0.946864 0.321634i \(-0.104232\pi\)
\(30\) 0 0
\(31\) −3.46410 −0.622171 −0.311086 0.950382i \(-0.600693\pi\)
−0.311086 + 0.950382i \(0.600693\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.92820 −1.13899 −0.569495 0.821995i \(-0.692861\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 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
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 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
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) − 6.00000i − 0.840168i
\(52\) 0 0
\(53\) 3.46410 0.475831 0.237915 0.971286i \(-0.423536\pi\)
0.237915 + 0.971286i \(0.423536\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) 12.0000i 1.56227i 0.624364 + 0.781133i \(0.285358\pi\)
−0.624364 + 0.781133i \(0.714642\pi\)
\(60\) 0 0
\(61\) − 6.92820i − 0.887066i −0.896258 0.443533i \(-0.853725\pi\)
0.896258 0.443533i \(-0.146275\pi\)
\(62\) 0 0
\(63\) − 3.46410i − 0.436436i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) − 6.92820i − 0.834058i
\(70\) 0 0
\(71\) 6.92820 0.822226 0.411113 0.911584i \(-0.365140\pi\)
0.411113 + 0.911584i \(0.365140\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.3923 −1.16923 −0.584613 0.811312i \(-0.698754\pi\)
−0.584613 + 0.811312i \(0.698754\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 3.46410i − 0.371391i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.46410 −0.359211
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 3.46410i − 0.344691i −0.985037 0.172345i \(-0.944865\pi\)
0.985037 0.172345i \(-0.0551346\pi\)
\(102\) 0 0
\(103\) − 17.3205i − 1.70664i −0.521387 0.853320i \(-0.674585\pi\)
0.521387 0.853320i \(-0.325415\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) − 13.8564i − 1.32720i −0.748086 0.663602i \(-0.769027\pi\)
0.748086 0.663602i \(-0.230973\pi\)
\(110\) 0 0
\(111\) −6.92820 −0.657596
\(112\) 0 0
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −20.7846 −1.90532
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 3.46410i − 0.307389i −0.988118 0.153695i \(-0.950883\pi\)
0.988118 0.153695i \(-0.0491172\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.0000i 1.04844i 0.851581 + 0.524222i \(0.175644\pi\)
−0.851581 + 0.524222i \(0.824356\pi\)
\(132\) 0 0
\(133\) 13.8564 1.20150
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) − 20.0000i − 1.69638i −0.529694 0.848189i \(-0.677693\pi\)
0.529694 0.848189i \(-0.322307\pi\)
\(140\) 0 0
\(141\) 6.92820i 0.583460i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −5.00000 −0.412393
\(148\) 0 0
\(149\) 10.3923i 0.851371i 0.904871 + 0.425685i \(0.139967\pi\)
−0.904871 + 0.425685i \(0.860033\pi\)
\(150\) 0 0
\(151\) 3.46410 0.281905 0.140952 0.990016i \(-0.454984\pi\)
0.140952 + 0.990016i \(0.454984\pi\)
\(152\) 0 0
\(153\) − 6.00000i − 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.92820 0.552931 0.276465 0.961024i \(-0.410837\pi\)
0.276465 + 0.961024i \(0.410837\pi\)
\(158\) 0 0
\(159\) 3.46410 0.274721
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.8564i 1.07224i 0.844141 + 0.536120i \(0.180111\pi\)
−0.844141 + 0.536120i \(0.819889\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 4.00000i 0.305888i
\(172\) 0 0
\(173\) 17.3205 1.31685 0.658427 0.752645i \(-0.271222\pi\)
0.658427 + 0.752645i \(0.271222\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000i 0.901975i
\(178\) 0 0
\(179\) 12.0000i 0.896922i 0.893802 + 0.448461i \(0.148028\pi\)
−0.893802 + 0.448461i \(0.851972\pi\)
\(180\) 0 0
\(181\) − 13.8564i − 1.02994i −0.857209 0.514969i \(-0.827803\pi\)
0.857209 0.514969i \(-0.172197\pi\)
\(182\) 0 0
\(183\) − 6.92820i − 0.512148i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) − 3.46410i − 0.251976i
\(190\) 0 0
\(191\) −13.8564 −1.00261 −0.501307 0.865269i \(-0.667147\pi\)
−0.501307 + 0.865269i \(0.667147\pi\)
\(192\) 0 0
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.3923 0.740421 0.370211 0.928948i \(-0.379286\pi\)
0.370211 + 0.928948i \(0.379286\pi\)
\(198\) 0 0
\(199\) −10.3923 −0.736691 −0.368345 0.929689i \(-0.620076\pi\)
−0.368345 + 0.929689i \(0.620076\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 6.92820i − 0.481543i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000i 0.275371i 0.990476 + 0.137686i \(0.0439664\pi\)
−0.990476 + 0.137686i \(0.956034\pi\)
\(212\) 0 0
\(213\) 6.92820 0.474713
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 12.0000i 0.814613i
\(218\) 0 0
\(219\) 2.00000i 0.135147i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 3.46410i 0.231973i 0.993251 + 0.115987i \(0.0370030\pi\)
−0.993251 + 0.115987i \(0.962997\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) − 27.7128i − 1.83131i −0.401960 0.915657i \(-0.631671\pi\)
0.401960 0.915657i \(-0.368329\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 18.0000i − 1.17922i −0.807688 0.589610i \(-0.799282\pi\)
0.807688 0.589610i \(-0.200718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −10.3923 −0.675053
\(238\) 0 0
\(239\) −27.7128 −1.79259 −0.896296 0.443455i \(-0.853752\pi\)
−0.896296 + 0.443455i \(0.853752\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) 24.0000i 1.49129i
\(260\) 0 0
\(261\) − 3.46410i − 0.214423i
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) − 31.1769i − 1.90089i −0.310893 0.950445i \(-0.600628\pi\)
0.310893 0.950445i \(-0.399372\pi\)
\(270\) 0 0
\(271\) 3.46410 0.210429 0.105215 0.994450i \(-0.466447\pi\)
0.105215 + 0.994450i \(0.466447\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −13.8564 −0.832551 −0.416275 0.909239i \(-0.636665\pi\)
−0.416275 + 0.909239i \(0.636665\pi\)
\(278\) 0 0
\(279\) −3.46410 −0.207390
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 20.7846i 1.22688i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 2.00000i 0.117242i
\(292\) 0 0
\(293\) 3.46410 0.202375 0.101187 0.994867i \(-0.467736\pi\)
0.101187 + 0.994867i \(0.467736\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 13.8564i 0.798670i
\(302\) 0 0
\(303\) − 3.46410i − 0.199007i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) − 17.3205i − 0.985329i
\(310\) 0 0
\(311\) −13.8564 −0.785725 −0.392862 0.919597i \(-0.628515\pi\)
−0.392862 + 0.919597i \(0.628515\pi\)
\(312\) 0 0
\(313\) − 26.0000i − 1.46961i −0.678280 0.734803i \(-0.737274\pi\)
0.678280 0.734803i \(-0.262726\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 31.1769 1.75107 0.875535 0.483155i \(-0.160509\pi\)
0.875535 + 0.483155i \(0.160509\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 13.8564i − 0.766261i
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 20.0000i 1.09930i 0.835395 + 0.549650i \(0.185239\pi\)
−0.835395 + 0.549650i \(0.814761\pi\)
\(332\) 0 0
\(333\) −6.92820 −0.379663
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 22.0000i − 1.19842i −0.800593 0.599208i \(-0.795482\pi\)
0.800593 0.599208i \(-0.204518\pi\)
\(338\) 0 0
\(339\) − 6.00000i − 0.325875i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 6.92820i − 0.374088i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) 6.92820i 0.370858i 0.982658 + 0.185429i \(0.0593675\pi\)
−0.982658 + 0.185429i \(0.940632\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −20.7846 −1.10004
\(358\) 0 0
\(359\) −6.92820 −0.365657 −0.182828 0.983145i \(-0.558525\pi\)
−0.182828 + 0.983145i \(0.558525\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 24.2487i − 1.26577i −0.774245 0.632886i \(-0.781870\pi\)
0.774245 0.632886i \(-0.218130\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) − 12.0000i − 0.623009i
\(372\) 0 0
\(373\) 20.7846 1.07619 0.538093 0.842885i \(-0.319145\pi\)
0.538093 + 0.842885i \(0.319145\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 20.0000i 1.02733i 0.857991 + 0.513665i \(0.171713\pi\)
−0.857991 + 0.513665i \(0.828287\pi\)
\(380\) 0 0
\(381\) − 3.46410i − 0.177471i
\(382\) 0 0
\(383\) − 13.8564i − 0.708029i −0.935240 0.354015i \(-0.884816\pi\)
0.935240 0.354015i \(-0.115184\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 24.2487i 1.22946i 0.788738 + 0.614729i \(0.210735\pi\)
−0.788738 + 0.614729i \(0.789265\pi\)
\(390\) 0 0
\(391\) −41.5692 −2.10225
\(392\) 0 0
\(393\) 12.0000i 0.605320i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −34.6410 −1.73858 −0.869291 0.494300i \(-0.835424\pi\)
−0.869291 + 0.494300i \(0.835424\pi\)
\(398\) 0 0
\(399\) 13.8564 0.693688
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) 6.00000i 0.295958i
\(412\) 0 0
\(413\) 41.5692 2.04549
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 20.0000i − 0.979404i
\(418\) 0 0
\(419\) 24.0000i 1.17248i 0.810139 + 0.586238i \(0.199392\pi\)
−0.810139 + 0.586238i \(0.800608\pi\)
\(420\) 0 0
\(421\) 13.8564i 0.675320i 0.941268 + 0.337660i \(0.109635\pi\)
−0.941268 + 0.337660i \(0.890365\pi\)
\(422\) 0 0
\(423\) 6.92820i 0.336861i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −24.0000 −1.16144
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.92820 0.333720 0.166860 0.985981i \(-0.446637\pi\)
0.166860 + 0.985981i \(0.446637\pi\)
\(432\) 0 0
\(433\) − 2.00000i − 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 27.7128 1.32568
\(438\) 0 0
\(439\) 31.1769 1.48799 0.743996 0.668184i \(-0.232928\pi\)
0.743996 + 0.668184i \(0.232928\pi\)
\(440\) 0 0
\(441\) −5.00000 −0.238095
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 10.3923i 0.491539i
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 3.46410 0.162758
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 22.0000i 1.02912i 0.857455 + 0.514558i \(0.172044\pi\)
−0.857455 + 0.514558i \(0.827956\pi\)
\(458\) 0 0
\(459\) − 6.00000i − 0.280056i
\(460\) 0 0
\(461\) − 31.1769i − 1.45205i −0.687666 0.726027i \(-0.741365\pi\)
0.687666 0.726027i \(-0.258635\pi\)
\(462\) 0 0
\(463\) − 10.3923i − 0.482971i −0.970404 0.241486i \(-0.922365\pi\)
0.970404 0.241486i \(-0.0776347\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 24.0000 1.11059 0.555294 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) 0 0
\(469\) − 13.8564i − 0.639829i
\(470\) 0 0
\(471\) 6.92820 0.319235
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.46410 0.158610
\(478\) 0 0
\(479\) −20.7846 −0.949673 −0.474837 0.880074i \(-0.657493\pi\)
−0.474837 + 0.880074i \(0.657493\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −24.0000 −1.09204
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 10.3923i 0.470920i 0.971884 + 0.235460i \(0.0756597\pi\)
−0.971884 + 0.235460i \(0.924340\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) − 36.0000i − 1.62466i −0.583200 0.812329i \(-0.698200\pi\)
0.583200 0.812329i \(-0.301800\pi\)
\(492\) 0 0
\(493\) −20.7846 −0.936092
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 24.0000i − 1.07655i
\(498\) 0 0
\(499\) − 28.0000i − 1.25345i −0.779240 0.626726i \(-0.784395\pi\)
0.779240 0.626726i \(-0.215605\pi\)
\(500\) 0 0
\(501\) 13.8564i 0.619059i
\(502\) 0 0
\(503\) 34.6410i 1.54457i 0.635278 + 0.772283i \(0.280885\pi\)
−0.635278 + 0.772283i \(0.719115\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −13.0000 −0.577350
\(508\) 0 0
\(509\) 24.2487i 1.07481i 0.843326 + 0.537403i \(0.180594\pi\)
−0.843326 + 0.537403i \(0.819406\pi\)
\(510\) 0 0
\(511\) 6.92820 0.306486
\(512\) 0 0
\(513\) 4.00000i 0.176604i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 17.3205 0.760286
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.7846i 0.905392i
\(528\) 0 0
\(529\) −25.0000 −1.08696
\(530\) 0 0
\(531\) 12.0000i 0.520756i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 12.0000i 0.517838i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 13.8564i 0.595733i 0.954607 + 0.297867i \(0.0962751\pi\)
−0.954607 + 0.297867i \(0.903725\pi\)
\(542\) 0 0
\(543\) − 13.8564i − 0.594635i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) − 6.92820i − 0.295689i
\(550\) 0 0
\(551\) 13.8564 0.590303
\(552\) 0 0
\(553\) 36.0000i 1.53088i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10.3923 0.440336 0.220168 0.975462i \(-0.429339\pi\)
0.220168 + 0.975462i \(0.429339\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 3.46410i − 0.145479i
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) − 4.00000i − 0.167395i −0.996491 0.0836974i \(-0.973327\pi\)
0.996491 0.0836974i \(-0.0266729\pi\)
\(572\) 0 0
\(573\) −13.8564 −0.578860
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) 0 0
\(579\) 2.00000i 0.0831172i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) − 13.8564i − 0.570943i
\(590\) 0 0
\(591\) 10.3923 0.427482
\(592\) 0 0
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −10.3923 −0.425329
\(598\) 0 0
\(599\) −34.6410 −1.41539 −0.707697 0.706516i \(-0.750266\pi\)
−0.707697 + 0.706516i \(0.750266\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 3.46410i − 0.140604i −0.997526 0.0703018i \(-0.977604\pi\)
0.997526 0.0703018i \(-0.0223962\pi\)
\(608\) 0 0
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −34.6410 −1.39914 −0.699569 0.714565i \(-0.746625\pi\)
−0.699569 + 0.714565i \(0.746625\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 0 0
\(619\) 4.00000i 0.160774i 0.996764 + 0.0803868i \(0.0256155\pi\)
−0.996764 + 0.0803868i \(0.974384\pi\)
\(620\) 0 0
\(621\) − 6.92820i − 0.278019i
\(622\) 0 0
\(623\) 20.7846i 0.832718i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 41.5692i 1.65747i
\(630\) 0 0
\(631\) 17.3205 0.689519 0.344759 0.938691i \(-0.387961\pi\)
0.344759 + 0.938691i \(0.387961\pi\)
\(632\) 0 0
\(633\) 4.00000i 0.158986i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.92820 0.274075
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 34.6410i 1.36188i 0.732340 + 0.680939i \(0.238428\pi\)
−0.732340 + 0.680939i \(0.761572\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 12.0000i 0.470317i
\(652\) 0 0
\(653\) −10.3923 −0.406682 −0.203341 0.979108i \(-0.565180\pi\)
−0.203341 + 0.979108i \(0.565180\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) − 12.0000i − 0.467454i −0.972302 0.233727i \(-0.924908\pi\)
0.972302 0.233727i \(-0.0750921\pi\)
\(660\) 0 0
\(661\) − 6.92820i − 0.269476i −0.990881 0.134738i \(-0.956981\pi\)
0.990881 0.134738i \(-0.0430193\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) 3.46410i 0.133930i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) − 2.00000i − 0.0770943i −0.999257 0.0385472i \(-0.987727\pi\)
0.999257 0.0385472i \(-0.0122730\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.46410 0.133136 0.0665681 0.997782i \(-0.478795\pi\)
0.0665681 + 0.997782i \(0.478795\pi\)
\(678\) 0 0
\(679\) 6.92820 0.265880
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 27.7128i − 1.05731i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 28.0000i − 1.06517i −0.846376 0.532585i \(-0.821221\pi\)
0.846376 0.532585i \(-0.178779\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000i 1.36360i
\(698\) 0 0
\(699\) − 18.0000i − 0.680823i
\(700\) 0 0
\(701\) − 10.3923i − 0.392512i −0.980553 0.196256i \(-0.937122\pi\)
0.980553 0.196256i \(-0.0628784\pi\)
\(702\) 0 0
\(703\) − 27.7128i − 1.04521i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) −10.3923 −0.389742
\(712\) 0 0
\(713\) 24.0000i 0.898807i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −27.7128 −1.03495
\(718\) 0 0
\(719\) 20.7846 0.775135 0.387568 0.921841i \(-0.373315\pi\)
0.387568 + 0.921841i \(0.373315\pi\)
\(720\) 0 0
\(721\) −60.0000 −2.23452
\(722\) 0 0
\(723\) −22.0000 −0.818189
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 10.3923i 0.385429i 0.981255 + 0.192715i \(0.0617292\pi\)
−0.981255 + 0.192715i \(0.938271\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000i 0.887672i
\(732\) 0 0
\(733\) 27.7128 1.02360 0.511798 0.859106i \(-0.328980\pi\)
0.511798 + 0.859106i \(0.328980\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 28.0000i − 1.03000i −0.857191 0.514998i \(-0.827793\pi\)
0.857191 0.514998i \(-0.172207\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27.7128i 1.01668i 0.861155 + 0.508342i \(0.169742\pi\)
−0.861155 + 0.508342i \(0.830258\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 41.5692i 1.51891i
\(750\) 0 0
\(751\) 24.2487 0.884848 0.442424 0.896806i \(-0.354119\pi\)
0.442424 + 0.896806i \(0.354119\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 27.7128 1.00724 0.503620 0.863925i \(-0.332001\pi\)
0.503620 + 0.863925i \(0.332001\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) −48.0000 −1.73772
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) − 18.0000i − 0.648254i
\(772\) 0 0
\(773\) −38.1051 −1.37055 −0.685273 0.728286i \(-0.740317\pi\)
−0.685273 + 0.728286i \(0.740317\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 24.0000i 0.860995i
\(778\) 0 0
\(779\) − 24.0000i − 0.859889i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 3.46410i − 0.123797i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −20.7846 −0.739016
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 24.2487 0.858933 0.429467 0.903083i \(-0.358702\pi\)
0.429467 + 0.903083i \(0.358702\pi\)
\(798\) 0 0
\(799\) 41.5692 1.47061
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 31.1769i − 1.09748i
\(808\) 0 0
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) 0 0
\(811\) − 20.0000i − 0.702295i −0.936320 0.351147i \(-0.885792\pi\)
0.936320 0.351147i \(-0.114208\pi\)
\(812\) 0 0
\(813\) 3.46410 0.121491
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 16.0000i − 0.559769i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 24.2487i 0.846286i 0.906063 + 0.423143i \(0.139073\pi\)
−0.906063 + 0.423143i \(0.860927\pi\)
\(822\) 0 0
\(823\) 31.1769i 1.08676i 0.839487 + 0.543379i \(0.182856\pi\)
−0.839487 + 0.543379i \(0.817144\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 41.5692i 1.44376i 0.692019 + 0.721879i \(0.256721\pi\)
−0.692019 + 0.721879i \(0.743279\pi\)
\(830\) 0 0
\(831\) −13.8564 −0.480673
\(832\) 0 0
\(833\) 30.0000i 1.03944i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −3.46410 −0.119737
\(838\) 0 0
\(839\) 6.92820 0.239188 0.119594 0.992823i \(-0.461841\pi\)
0.119594 + 0.992823i \(0.461841\pi\)
\(840\) 0 0
\(841\) 17.0000 0.586207
\(842\) 0 0
\(843\) 30.0000 1.03325
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 38.1051i − 1.30931i
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 48.0000i 1.64542i
\(852\) 0 0
\(853\) −48.4974 −1.66052 −0.830260 0.557376i \(-0.811808\pi\)
−0.830260 + 0.557376i \(0.811808\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 42.0000i − 1.43469i −0.696717 0.717346i \(-0.745357\pi\)
0.696717 0.717346i \(-0.254643\pi\)
\(858\) 0 0
\(859\) − 4.00000i − 0.136478i −0.997669 0.0682391i \(-0.978262\pi\)
0.997669 0.0682391i \(-0.0217381\pi\)
\(860\) 0 0
\(861\) 20.7846i 0.708338i
\(862\) 0 0
\(863\) − 13.8564i − 0.471678i −0.971792 0.235839i \(-0.924216\pi\)
0.971792 0.235839i \(-0.0757837\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.00000i 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6.92820 −0.233949 −0.116974 0.993135i \(-0.537320\pi\)
−0.116974 + 0.993135i \(0.537320\pi\)
\(878\) 0 0
\(879\) 3.46410 0.116841
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −27.7128 −0.927374
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 12.0000i 0.400222i
\(900\) 0 0
\(901\) − 20.7846i − 0.692436i
\(902\) 0 0
\(903\) 13.8564i 0.461112i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) 0 0
\(909\) − 3.46410i − 0.114897i
\(910\) 0 0
\(911\) 55.4256 1.83633 0.918166 0.396195i \(-0.129670\pi\)
0.918166 + 0.396195i \(0.129670\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 41.5692 1.37274
\(918\) 0 0
\(919\) 51.9615 1.71405 0.857026 0.515273i \(-0.172309\pi\)
0.857026 + 0.515273i \(0.172309\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 17.3205i − 0.568880i
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) − 20.0000i − 0.655474i
\(932\) 0 0
\(933\) −13.8564 −0.453638
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 26.0000i 0.849383i 0.905338 + 0.424691i \(0.139617\pi\)
−0.905338 + 0.424691i \(0.860383\pi\)
\(938\) 0 0
\(939\) − 26.0000i − 0.848478i
\(940\) 0 0
\(941\) 24.2487i 0.790485i 0.918577 + 0.395243i \(0.129340\pi\)
−0.918577 + 0.395243i \(0.870660\pi\)
\(942\) 0 0
\(943\) 41.5692i 1.35368i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 31.1769 1.01098
\(952\) 0 0
\(953\) − 30.0000i − 0.971795i −0.874016 0.485898i \(-0.838493\pi\)
0.874016 0.485898i \(-0.161507\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 20.7846 0.671170
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 38.1051i − 1.22538i −0.790324 0.612689i \(-0.790088\pi\)
0.790324 0.612689i \(-0.209912\pi\)
\(968\) 0 0
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) − 48.0000i − 1.54039i −0.637806 0.770197i \(-0.720158\pi\)
0.637806 0.770197i \(-0.279842\pi\)
\(972\) 0 0
\(973\) −69.2820 −2.22108
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 54.0000i − 1.72761i −0.503824 0.863807i \(-0.668074\pi\)
0.503824 0.863807i \(-0.331926\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) − 13.8564i − 0.442401i
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) 0 0
\(989\) 27.7128i 0.881216i
\(990\) 0 0
\(991\) 24.2487 0.770286 0.385143 0.922857i \(-0.374152\pi\)
0.385143 + 0.922857i \(0.374152\pi\)
\(992\) 0 0
\(993\) 20.0000i 0.634681i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −20.7846 −0.658255 −0.329128 0.944285i \(-0.606755\pi\)
−0.329128 + 0.944285i \(0.606755\pi\)
\(998\) 0 0
\(999\) −6.92820 −0.219199
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 4800.2.d.o.1249.1 4
4.3 odd 2 4800.2.d.j.1249.3 4
5.2 odd 4 192.2.d.a.97.2 yes 4
5.3 odd 4 4800.2.k.j.2401.3 4
5.4 even 2 4800.2.d.j.1249.4 4
8.3 odd 2 inner 4800.2.d.o.1249.3 4
8.5 even 2 4800.2.d.j.1249.1 4
15.2 even 4 576.2.d.b.289.2 4
20.3 even 4 4800.2.k.j.2401.2 4
20.7 even 4 192.2.d.a.97.4 yes 4
20.19 odd 2 inner 4800.2.d.o.1249.2 4
40.3 even 4 4800.2.k.j.2401.4 4
40.13 odd 4 4800.2.k.j.2401.1 4
40.19 odd 2 4800.2.d.j.1249.2 4
40.27 even 4 192.2.d.a.97.1 4
40.29 even 2 inner 4800.2.d.o.1249.4 4
40.37 odd 4 192.2.d.a.97.3 yes 4
60.47 odd 4 576.2.d.b.289.1 4
80.27 even 4 768.2.a.k.1.1 2
80.37 odd 4 768.2.a.j.1.1 2
80.67 even 4 768.2.a.j.1.2 2
80.77 odd 4 768.2.a.k.1.2 2
120.77 even 4 576.2.d.b.289.4 4
120.107 odd 4 576.2.d.b.289.3 4
240.77 even 4 2304.2.a.u.1.1 2
240.107 odd 4 2304.2.a.u.1.2 2
240.197 even 4 2304.2.a.s.1.2 2
240.227 odd 4 2304.2.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
192.2.d.a.97.1 4 40.27 even 4
192.2.d.a.97.2 yes 4 5.2 odd 4
192.2.d.a.97.3 yes 4 40.37 odd 4
192.2.d.a.97.4 yes 4 20.7 even 4
576.2.d.b.289.1 4 60.47 odd 4
576.2.d.b.289.2 4 15.2 even 4
576.2.d.b.289.3 4 120.107 odd 4
576.2.d.b.289.4 4 120.77 even 4
768.2.a.j.1.1 2 80.37 odd 4
768.2.a.j.1.2 2 80.67 even 4
768.2.a.k.1.1 2 80.27 even 4
768.2.a.k.1.2 2 80.77 odd 4
2304.2.a.s.1.1 2 240.227 odd 4
2304.2.a.s.1.2 2 240.197 even 4
2304.2.a.u.1.1 2 240.77 even 4
2304.2.a.u.1.2 2 240.107 odd 4
4800.2.d.j.1249.1 4 8.5 even 2
4800.2.d.j.1249.2 4 40.19 odd 2
4800.2.d.j.1249.3 4 4.3 odd 2
4800.2.d.j.1249.4 4 5.4 even 2
4800.2.d.o.1249.1 4 1.1 even 1 trivial
4800.2.d.o.1249.2 4 20.19 odd 2 inner
4800.2.d.o.1249.3 4 8.3 odd 2 inner
4800.2.d.o.1249.4 4 40.29 even 2 inner
4800.2.k.j.2401.1 4 40.13 odd 4
4800.2.k.j.2401.2 4 20.3 even 4
4800.2.k.j.2401.3 4 5.3 odd 4
4800.2.k.j.2401.4 4 40.3 even 4