Properties

Label 2880.2.k.i.1441.1
Level $2880$
Weight $2$
Character 2880.1441
Analytic conductor $22.997$
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] = [2880,2,Mod(1441,2880)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2880.1441");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.k (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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1441.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2880.1441
Dual form 2880.2.k.i.1441.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{5} +O(q^{10})\) \(q-1.00000i q^{5} -3.46410i q^{11} +3.46410i q^{13} +3.46410 q^{17} +4.00000i q^{19} -1.00000 q^{25} +6.00000i q^{29} +3.46410 q^{31} -3.46410i q^{37} +6.92820 q^{41} -4.00000i q^{43} +12.0000 q^{47} -7.00000 q^{49} +6.00000i q^{53} -3.46410 q^{55} -10.3923i q^{59} +3.46410 q^{65} -4.00000i q^{67} +2.00000 q^{73} +3.46410 q^{79} -13.8564i q^{83} -3.46410i q^{85} +6.92820 q^{89} +4.00000 q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{25} + 48 q^{47} - 28 q^{49} + 8 q^{73} + 16 q^{95} + 40 q^{97}+O(q^{100}) \) 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
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.46410i − 1.04447i −0.852803 0.522233i \(-0.825099\pi\)
0.852803 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 3.46410 0.622171 0.311086 0.950382i \(-0.399307\pi\)
0.311086 + 0.950382i \(0.399307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 3.46410i − 0.569495i −0.958603 0.284747i \(-0.908090\pi\)
0.958603 0.284747i \(-0.0919097\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) −3.46410 −0.467099
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 10.3923i − 1.35296i −0.736460 0.676481i \(-0.763504\pi\)
0.736460 0.676481i \(-0.236496\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.46410 0.429669
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 3.46410 0.389742 0.194871 0.980829i \(-0.437571\pi\)
0.194871 + 0.980829i \(0.437571\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 13.8564i − 1.52094i −0.649374 0.760469i \(-0.724969\pi\)
0.649374 0.760469i \(-0.275031\pi\)
\(84\) 0 0
\(85\) − 3.46410i − 0.375735i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.92820 0.734388 0.367194 0.930144i \(-0.380318\pi\)
0.367194 + 0.930144i \(0.380318\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000i 0.597022i 0.954406 + 0.298511i \(0.0964900\pi\)
−0.954406 + 0.298511i \(0.903510\pi\)
\(102\) 0 0
\(103\) 6.92820 0.682656 0.341328 0.939944i \(-0.389123\pi\)
0.341328 + 0.939944i \(0.389123\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 6.92820i − 0.669775i −0.942258 0.334887i \(-0.891302\pi\)
0.942258 0.334887i \(-0.108698\pi\)
\(108\) 0 0
\(109\) 6.92820i 0.663602i 0.943349 + 0.331801i \(0.107656\pi\)
−0.943349 + 0.331801i \(0.892344\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 17.3205 1.62938 0.814688 0.579899i \(-0.196908\pi\)
0.814688 + 0.579899i \(0.196908\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) −20.7846 −1.84434 −0.922168 0.386790i \(-0.873584\pi\)
−0.922168 + 0.386790i \(0.873584\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 10.3923i − 0.907980i −0.891007 0.453990i \(-0.850000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.3205 1.47979 0.739895 0.672722i \(-0.234875\pi\)
0.739895 + 0.672722i \(0.234875\pi\)
\(138\) 0 0
\(139\) 4.00000i 0.339276i 0.985506 + 0.169638i \(0.0542598\pi\)
−0.985506 + 0.169638i \(0.945740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.00000i 0.491539i 0.969328 + 0.245770i \(0.0790407\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(150\) 0 0
\(151\) −10.3923 −0.845714 −0.422857 0.906196i \(-0.638973\pi\)
−0.422857 + 0.906196i \(0.638973\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 3.46410i − 0.278243i
\(156\) 0 0
\(157\) − 17.3205i − 1.38233i −0.722698 0.691164i \(-0.757098\pi\)
0.722698 0.691164i \(-0.242902\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 3.46410i − 0.258919i −0.991585 0.129460i \(-0.958676\pi\)
0.991585 0.129460i \(-0.0413242\pi\)
\(180\) 0 0
\(181\) 6.92820i 0.514969i 0.966282 + 0.257485i \(0.0828937\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.46410 −0.254686
\(186\) 0 0
\(187\) − 12.0000i − 0.877527i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) 24.2487 1.71895 0.859473 0.511182i \(-0.170792\pi\)
0.859473 + 0.511182i \(0.170792\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) − 6.92820i − 0.483887i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 13.8564 0.958468
\(210\) 0 0
\(211\) − 4.00000i − 0.275371i −0.990476 0.137686i \(-0.956034\pi\)
0.990476 0.137686i \(-0.0439664\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) 6.92820 0.463947 0.231973 0.972722i \(-0.425482\pi\)
0.231973 + 0.972722i \(0.425482\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 20.7846i − 1.37952i −0.724037 0.689761i \(-0.757715\pi\)
0.724037 0.689761i \(-0.242285\pi\)
\(228\) 0 0
\(229\) 27.7128i 1.83131i 0.401960 + 0.915657i \(0.368329\pi\)
−0.401960 + 0.915657i \(0.631671\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −10.3923 −0.680823 −0.340411 0.940277i \(-0.610566\pi\)
−0.340411 + 0.940277i \(0.610566\pi\)
\(234\) 0 0
\(235\) − 12.0000i − 0.782794i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 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
\(245\) 7.00000i 0.447214i
\(246\) 0 0
\(247\) −13.8564 −0.881662
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.46410i 0.218652i 0.994006 + 0.109326i \(0.0348693\pi\)
−0.994006 + 0.109326i \(0.965131\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.3923 −0.648254 −0.324127 0.946014i \(-0.605071\pi\)
−0.324127 + 0.946014i \(0.605071\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 18.0000i − 1.09748i −0.835993 0.548740i \(-0.815108\pi\)
0.835993 0.548740i \(-0.184892\pi\)
\(270\) 0 0
\(271\) 17.3205 1.05215 0.526073 0.850439i \(-0.323664\pi\)
0.526073 + 0.850439i \(0.323664\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.46410i 0.208893i
\(276\) 0 0
\(277\) − 24.2487i − 1.45696i −0.685065 0.728482i \(-0.740226\pi\)
0.685065 0.728482i \(-0.259774\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 20.7846 1.23991 0.619953 0.784639i \(-0.287152\pi\)
0.619953 + 0.784639i \(0.287152\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 18.0000i − 1.05157i −0.850617 0.525786i \(-0.823771\pi\)
0.850617 0.525786i \(-0.176229\pi\)
\(294\) 0 0
\(295\) −10.3923 −0.605063
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 16.0000i − 0.913168i −0.889680 0.456584i \(-0.849073\pi\)
0.889680 0.456584i \(-0.150927\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 20.7846 1.16371
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 13.8564i 0.770991i
\(324\) 0 0
\(325\) − 3.46410i − 0.192154i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 28.0000i 1.53902i 0.638635 + 0.769510i \(0.279499\pi\)
−0.638635 + 0.769510i \(0.720501\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 12.0000i − 0.649836i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.92820i 0.371925i 0.982557 + 0.185963i \(0.0595404\pi\)
−0.982557 + 0.185963i \(0.940460\pi\)
\(348\) 0 0
\(349\) 27.7128i 1.48343i 0.670714 + 0.741716i \(0.265988\pi\)
−0.670714 + 0.741716i \(0.734012\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.3923 0.553127 0.276563 0.960996i \(-0.410804\pi\)
0.276563 + 0.960996i \(0.410804\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 2.00000i − 0.104685i
\(366\) 0 0
\(367\) −27.7128 −1.44660 −0.723299 0.690535i \(-0.757375\pi\)
−0.723299 + 0.690535i \(0.757375\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 31.1769i 1.61428i 0.590360 + 0.807140i \(0.298986\pi\)
−0.590360 + 0.807140i \(0.701014\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −20.7846 −1.07046
\(378\) 0 0
\(379\) 20.0000i 1.02733i 0.857991 + 0.513665i \(0.171713\pi\)
−0.857991 + 0.513665i \(0.828287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 6.00000i − 0.304212i −0.988364 0.152106i \(-0.951394\pi\)
0.988364 0.152106i \(-0.0486055\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 3.46410i − 0.174298i
\(396\) 0 0
\(397\) − 24.2487i − 1.21701i −0.793551 0.608504i \(-0.791770\pi\)
0.793551 0.608504i \(-0.208230\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −27.7128 −1.38391 −0.691956 0.721940i \(-0.743251\pi\)
−0.691956 + 0.721940i \(0.743251\pi\)
\(402\) 0 0
\(403\) 12.0000i 0.597763i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −13.8564 −0.680184
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.2487i 1.18463i 0.805708 + 0.592314i \(0.201785\pi\)
−0.805708 + 0.592314i \(0.798215\pi\)
\(420\) 0 0
\(421\) − 13.8564i − 0.675320i −0.941268 0.337660i \(-0.890365\pi\)
0.941268 0.337660i \(-0.109635\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.46410 −0.168034
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 24.2487 1.15733 0.578664 0.815566i \(-0.303574\pi\)
0.578664 + 0.815566i \(0.303574\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 6.92820i − 0.329169i −0.986363 0.164584i \(-0.947372\pi\)
0.986363 0.164584i \(-0.0526283\pi\)
\(444\) 0 0
\(445\) − 6.92820i − 0.328428i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −27.7128 −1.30785 −0.653924 0.756560i \(-0.726879\pi\)
−0.653924 + 0.756560i \(0.726879\pi\)
\(450\) 0 0
\(451\) − 24.0000i − 1.13012i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 30.0000i − 1.39724i −0.715493 0.698620i \(-0.753798\pi\)
0.715493 0.698620i \(-0.246202\pi\)
\(462\) 0 0
\(463\) 13.8564 0.643962 0.321981 0.946746i \(-0.395651\pi\)
0.321981 + 0.946746i \(0.395651\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 41.5692i 1.92359i 0.273764 + 0.961797i \(0.411731\pi\)
−0.273764 + 0.961797i \(0.588269\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −13.8564 −0.637118
\(474\) 0 0
\(475\) − 4.00000i − 0.183533i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 10.0000i − 0.454077i
\(486\) 0 0
\(487\) −34.6410 −1.56973 −0.784867 0.619664i \(-0.787269\pi\)
−0.784867 + 0.619664i \(0.787269\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10.3923i 0.468998i 0.972116 + 0.234499i \(0.0753450\pi\)
−0.972116 + 0.234499i \(0.924655\pi\)
\(492\) 0 0
\(493\) 20.7846i 0.936092i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 20.0000i 0.895323i 0.894203 + 0.447661i \(0.147743\pi\)
−0.894203 + 0.447661i \(0.852257\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 18.0000i 0.797836i 0.916987 + 0.398918i \(0.130614\pi\)
−0.916987 + 0.398918i \(0.869386\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 6.92820i − 0.305293i
\(516\) 0 0
\(517\) − 41.5692i − 1.82821i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −27.7128 −1.21412 −0.607060 0.794656i \(-0.707651\pi\)
−0.607060 + 0.794656i \(0.707651\pi\)
\(522\) 0 0
\(523\) − 40.0000i − 1.74908i −0.484955 0.874539i \(-0.661164\pi\)
0.484955 0.874539i \(-0.338836\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 24.0000i 1.03956i
\(534\) 0 0
\(535\) −6.92820 −0.299532
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 24.2487i 1.04447i
\(540\) 0 0
\(541\) 20.7846i 0.893600i 0.894634 + 0.446800i \(0.147436\pi\)
−0.894634 + 0.446800i \(0.852564\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.92820 0.296772
\(546\) 0 0
\(547\) − 44.0000i − 1.88130i −0.339372 0.940652i \(-0.610215\pi\)
0.339372 0.940652i \(-0.389785\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 42.0000i − 1.77960i −0.456354 0.889799i \(-0.650845\pi\)
0.456354 0.889799i \(-0.349155\pi\)
\(558\) 0 0
\(559\) 13.8564 0.586064
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 20.7846i 0.875967i 0.898983 + 0.437983i \(0.144307\pi\)
−0.898983 + 0.437983i \(0.855693\pi\)
\(564\) 0 0
\(565\) − 17.3205i − 0.728679i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13.8564 −0.580891 −0.290445 0.956892i \(-0.593803\pi\)
−0.290445 + 0.956892i \(0.593803\pi\)
\(570\) 0 0
\(571\) 4.00000i 0.167395i 0.996491 + 0.0836974i \(0.0266729\pi\)
−0.996491 + 0.0836974i \(0.973327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 20.7846 0.860811
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.92820i 0.285958i 0.989726 + 0.142979i \(0.0456681\pi\)
−0.989726 + 0.142979i \(0.954332\pi\)
\(588\) 0 0
\(589\) 13.8564i 0.570943i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −38.1051 −1.56479 −0.782395 0.622783i \(-0.786002\pi\)
−0.782395 + 0.622783i \(0.786002\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −48.0000 −1.96123 −0.980613 0.195952i \(-0.937220\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.00000i 0.0406558i
\(606\) 0 0
\(607\) −27.7128 −1.12483 −0.562414 0.826856i \(-0.690127\pi\)
−0.562414 + 0.826856i \(0.690127\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 41.5692i 1.68171i
\(612\) 0 0
\(613\) − 17.3205i − 0.699569i −0.936830 0.349784i \(-0.886255\pi\)
0.936830 0.349784i \(-0.113745\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −10.3923 −0.418378 −0.209189 0.977875i \(-0.567082\pi\)
−0.209189 + 0.977875i \(0.567082\pi\)
\(618\) 0 0
\(619\) 28.0000i 1.12542i 0.826656 + 0.562708i \(0.190240\pi\)
−0.826656 + 0.562708i \(0.809760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 12.0000i − 0.478471i
\(630\) 0 0
\(631\) 24.2487 0.965326 0.482663 0.875806i \(-0.339670\pi\)
0.482663 + 0.875806i \(0.339670\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 20.7846i 0.824812i
\(636\) 0 0
\(637\) − 24.2487i − 0.960769i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20.7846 0.820943 0.410471 0.911873i \(-0.365364\pi\)
0.410471 + 0.911873i \(0.365364\pi\)
\(642\) 0 0
\(643\) 40.0000i 1.57745i 0.614749 + 0.788723i \(0.289257\pi\)
−0.614749 + 0.788723i \(0.710743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 42.0000i − 1.64359i −0.569785 0.821794i \(-0.692974\pi\)
0.569785 0.821794i \(-0.307026\pi\)
\(654\) 0 0
\(655\) −10.3923 −0.406061
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 38.1051i − 1.48436i −0.670198 0.742182i \(-0.733791\pi\)
0.670198 0.742182i \(-0.266209\pi\)
\(660\) 0 0
\(661\) 13.8564i 0.538952i 0.963007 + 0.269476i \(0.0868504\pi\)
−0.963007 + 0.269476i \(0.913150\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 30.0000i − 1.15299i −0.817099 0.576497i \(-0.804419\pi\)
0.817099 0.576497i \(-0.195581\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 27.7128i 1.06040i 0.847872 + 0.530201i \(0.177883\pi\)
−0.847872 + 0.530201i \(0.822117\pi\)
\(684\) 0 0
\(685\) − 17.3205i − 0.661783i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −20.7846 −0.791831
\(690\) 0 0
\(691\) − 20.0000i − 0.760836i −0.924815 0.380418i \(-0.875780\pi\)
0.924815 0.380418i \(-0.124220\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) 24.0000 0.909065
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 18.0000i 0.679851i 0.940452 + 0.339925i \(0.110402\pi\)
−0.940452 + 0.339925i \(0.889598\pi\)
\(702\) 0 0
\(703\) 13.8564 0.522604
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 41.5692i 1.56116i 0.625053 + 0.780582i \(0.285077\pi\)
−0.625053 + 0.780582i \(0.714923\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) − 12.0000i − 0.448775i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 6.00000i − 0.222834i
\(726\) 0 0
\(727\) −34.6410 −1.28476 −0.642382 0.766385i \(-0.722054\pi\)
−0.642382 + 0.766385i \(0.722054\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 13.8564i − 0.512498i
\(732\) 0 0
\(733\) 10.3923i 0.383849i 0.981410 + 0.191924i \(0.0614728\pi\)
−0.981410 + 0.191924i \(0.938527\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −13.8564 −0.510407
\(738\) 0 0
\(739\) 20.0000i 0.735712i 0.929883 + 0.367856i \(0.119908\pi\)
−0.929883 + 0.367856i \(0.880092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −38.1051 −1.39048 −0.695238 0.718780i \(-0.744701\pi\)
−0.695238 + 0.718780i \(0.744701\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 10.3923i 0.378215i
\(756\) 0 0
\(757\) − 10.3923i − 0.377715i −0.982005 0.188857i \(-0.939522\pi\)
0.982005 0.188857i \(-0.0604784\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −27.7128 −1.00459 −0.502294 0.864697i \(-0.667511\pi\)
−0.502294 + 0.864697i \(0.667511\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 36.0000 1.29988
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 18.0000i − 0.647415i −0.946157 0.323708i \(-0.895071\pi\)
0.946157 0.323708i \(-0.104929\pi\)
\(774\) 0 0
\(775\) −3.46410 −0.124434
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 27.7128i 0.992915i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −17.3205 −0.618195
\(786\) 0 0
\(787\) − 32.0000i − 1.14068i −0.821410 0.570338i \(-0.806812\pi\)
0.821410 0.570338i \(-0.193188\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6.00000i 0.212531i 0.994338 + 0.106265i \(0.0338893\pi\)
−0.994338 + 0.106265i \(0.966111\pi\)
\(798\) 0 0
\(799\) 41.5692 1.47061
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 6.92820i − 0.244491i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −20.7846 −0.730748 −0.365374 0.930861i \(-0.619059\pi\)
−0.365374 + 0.930861i \(0.619059\pi\)
\(810\) 0 0
\(811\) 44.0000i 1.54505i 0.634985 + 0.772524i \(0.281006\pi\)
−0.634985 + 0.772524i \(0.718994\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 18.0000i 0.628204i 0.949389 + 0.314102i \(0.101703\pi\)
−0.949389 + 0.314102i \(0.898297\pi\)
\(822\) 0 0
\(823\) 34.6410 1.20751 0.603755 0.797170i \(-0.293671\pi\)
0.603755 + 0.797170i \(0.293671\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.7128i 0.963669i 0.876262 + 0.481834i \(0.160029\pi\)
−0.876262 + 0.481834i \(0.839971\pi\)
\(828\) 0 0
\(829\) 34.6410i 1.20313i 0.798823 + 0.601566i \(0.205456\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −24.2487 −0.840168
\(834\) 0 0
\(835\) 12.0000i 0.415277i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 1.00000i − 0.0344010i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 24.2487i 0.830260i 0.909762 + 0.415130i \(0.136264\pi\)
−0.909762 + 0.415130i \(0.863736\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.2487 0.828320 0.414160 0.910204i \(-0.364075\pi\)
0.414160 + 0.910204i \(0.364075\pi\)
\(858\) 0 0
\(859\) − 4.00000i − 0.136478i −0.997669 0.0682391i \(-0.978262\pi\)
0.997669 0.0682391i \(-0.0217381\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 12.0000i − 0.407072i
\(870\) 0 0
\(871\) 13.8564 0.469506
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 31.1769i − 1.05277i −0.850246 0.526385i \(-0.823547\pi\)
0.850246 0.526385i \(-0.176453\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −20.7846 −0.700251 −0.350126 0.936703i \(-0.613861\pi\)
−0.350126 + 0.936703i \(0.613861\pi\)
\(882\) 0 0
\(883\) 40.0000i 1.34611i 0.739594 + 0.673054i \(0.235018\pi\)
−0.739594 + 0.673054i \(0.764982\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 48.0000i 1.60626i
\(894\) 0 0
\(895\) −3.46410 −0.115792
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 20.7846i 0.693206i
\(900\) 0 0
\(901\) 20.7846i 0.692436i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.92820 0.230301
\(906\) 0 0
\(907\) − 4.00000i − 0.132818i −0.997792 0.0664089i \(-0.978846\pi\)
0.997792 0.0664089i \(-0.0211542\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) −48.0000 −1.58857
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 24.2487 0.799891 0.399946 0.916539i \(-0.369029\pi\)
0.399946 + 0.916539i \(0.369029\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 3.46410i 0.113899i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 41.5692 1.36384 0.681921 0.731426i \(-0.261145\pi\)
0.681921 + 0.731426i \(0.261145\pi\)
\(930\) 0 0
\(931\) − 28.0000i − 0.917663i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12.0000 −0.392442
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 18.0000i − 0.586783i −0.955992 0.293392i \(-0.905216\pi\)
0.955992 0.293392i \(-0.0947840\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 6.92820i 0.224899i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 17.3205 0.561066 0.280533 0.959844i \(-0.409489\pi\)
0.280533 + 0.959844i \(0.409489\pi\)
\(954\) 0 0
\(955\) − 24.0000i − 0.776622i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 10.0000i 0.321911i
\(966\) 0 0
\(967\) −48.4974 −1.55957 −0.779786 0.626046i \(-0.784672\pi\)
−0.779786 + 0.626046i \(0.784672\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 17.3205i 0.555842i 0.960604 + 0.277921i \(0.0896453\pi\)
−0.960604 + 0.277921i \(0.910355\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 17.3205 0.554132 0.277066 0.960851i \(-0.410638\pi\)
0.277066 + 0.960851i \(0.410638\pi\)
\(978\) 0 0
\(979\) − 24.0000i − 0.767043i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 10.3923 0.330122 0.165061 0.986283i \(-0.447218\pi\)
0.165061 + 0.986283i \(0.447218\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 24.2487i − 0.768736i
\(996\) 0 0
\(997\) − 10.3923i − 0.329128i −0.986366 0.164564i \(-0.947378\pi\)
0.986366 0.164564i \(-0.0526216\pi\)
\(998\) 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.k.i.1441.1 yes 4
3.2 odd 2 2880.2.k.h.1441.4 yes 4
4.3 odd 2 2880.2.k.h.1441.2 yes 4
8.3 odd 2 2880.2.k.h.1441.3 yes 4
8.5 even 2 inner 2880.2.k.i.1441.4 yes 4
12.11 even 2 inner 2880.2.k.i.1441.3 yes 4
24.5 odd 2 2880.2.k.h.1441.1 4
24.11 even 2 inner 2880.2.k.i.1441.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2880.2.k.h.1441.1 4 24.5 odd 2
2880.2.k.h.1441.2 yes 4 4.3 odd 2
2880.2.k.h.1441.3 yes 4 8.3 odd 2
2880.2.k.h.1441.4 yes 4 3.2 odd 2
2880.2.k.i.1441.1 yes 4 1.1 even 1 trivial
2880.2.k.i.1441.2 yes 4 24.11 even 2 inner
2880.2.k.i.1441.3 yes 4 12.11 even 2 inner
2880.2.k.i.1441.4 yes 4 8.5 even 2 inner