Properties

Label 1449.1.bq.b.1315.1
Level $1449$
Weight $1$
Character 1449.1315
Analytic conductor $0.723$
Analytic rank $0$
Dimension $20$
Projective image $D_{22}$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1449,1,Mod(55,1449)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1449, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 11, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1449.55");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1449.bq (of order \(22\), degree \(10\), 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: \(0.723145203305\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{44})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{22}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{22} - \cdots)\)

Embedding invariants

Embedding label 1315.1
Root \(-0.755750 - 0.654861i\) of defining polynomial
Character \(\chi\) \(=\) 1449.1315
Dual form 1449.1.bq.b.811.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.627899 - 1.37491i) q^{2} +(-0.841254 + 0.970858i) q^{4} +(0.142315 + 0.989821i) q^{7} +(0.412791 + 0.121206i) q^{8} +O(q^{10})\) \(q+(-0.627899 - 1.37491i) q^{2} +(-0.841254 + 0.970858i) q^{4} +(0.142315 + 0.989821i) q^{7} +(0.412791 + 0.121206i) q^{8} +(-0.234072 + 0.512546i) q^{11} +(1.27155 - 0.817178i) q^{14} +(0.0902783 + 0.627899i) q^{16} +0.851677 q^{22} +(0.540641 - 0.841254i) q^{23} +(0.415415 + 0.909632i) q^{25} +(-1.08070 - 0.694523i) q^{28} +(1.19136 + 1.37491i) q^{29} +(1.16854 - 0.750975i) q^{32} +(0.239446 - 0.153882i) q^{37} +(1.61435 - 0.474017i) q^{43} +(-0.300696 - 0.658432i) q^{44} +(-1.49611 - 0.215109i) q^{46} +(-0.959493 + 0.281733i) q^{49} +(0.989821 - 1.14231i) q^{50} +(0.258908 + 1.80075i) q^{53} +(-0.0612263 + 0.425839i) q^{56} +(1.14231 - 2.50132i) q^{58} +(-1.23259 - 0.792140i) q^{64} +(-0.544078 - 1.19136i) q^{67} +(-0.234072 - 0.512546i) q^{71} +(-0.361922 - 0.232593i) q^{74} +(-0.540641 - 0.158746i) q^{77} +(-0.118239 + 0.822373i) q^{79} +(-1.66538 - 1.92195i) q^{86} +(-0.158746 + 0.183203i) q^{88} +(0.361922 + 1.23259i) q^{92} +(0.989821 + 1.14231i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{4} + 2 q^{7} - 24 q^{16} - 2 q^{25} - 2 q^{28} + 4 q^{37} + 4 q^{43} - 2 q^{49} + 22 q^{58} + 2 q^{64} - 4 q^{67} - 4 q^{79} - 22 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1449\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(829\) \(1289\)
\(\chi(n)\) \(e\left(\frac{2}{11}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.627899 1.37491i −0.627899 1.37491i −0.909632 0.415415i \(-0.863636\pi\)
0.281733 0.959493i \(-0.409091\pi\)
\(3\) 0 0
\(4\) −0.841254 + 0.970858i −0.841254 + 0.970858i
\(5\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(6\) 0 0
\(7\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(8\) 0.412791 + 0.121206i 0.412791 + 0.121206i
\(9\) 0 0
\(10\) 0 0
\(11\) −0.234072 + 0.512546i −0.234072 + 0.512546i −0.989821 0.142315i \(-0.954545\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(12\) 0 0
\(13\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(14\) 1.27155 0.817178i 1.27155 0.817178i
\(15\) 0 0
\(16\) 0.0902783 + 0.627899i 0.0902783 + 0.627899i
\(17\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(18\) 0 0
\(19\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.851677 0.851677
\(23\) 0.540641 0.841254i 0.540641 0.841254i
\(24\) 0 0
\(25\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(26\) 0 0
\(27\) 0 0
\(28\) −1.08070 0.694523i −1.08070 0.694523i
\(29\) 1.19136 + 1.37491i 1.19136 + 1.37491i 0.909632 + 0.415415i \(0.136364\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(30\) 0 0
\(31\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(32\) 1.16854 0.750975i 1.16854 0.750975i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.239446 0.153882i 0.239446 0.153882i −0.415415 0.909632i \(-0.636364\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(42\) 0 0
\(43\) 1.61435 0.474017i 1.61435 0.474017i 0.654861 0.755750i \(-0.272727\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(44\) −0.300696 0.658432i −0.300696 0.658432i
\(45\) 0 0
\(46\) −1.49611 0.215109i −1.49611 0.215109i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(50\) 0.989821 1.14231i 0.989821 1.14231i
\(51\) 0 0
\(52\) 0 0
\(53\) 0.258908 + 1.80075i 0.258908 + 1.80075i 0.540641 + 0.841254i \(0.318182\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.0612263 + 0.425839i −0.0612263 + 0.425839i
\(57\) 0 0
\(58\) 1.14231 2.50132i 1.14231 2.50132i
\(59\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(60\) 0 0
\(61\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.23259 0.792140i −1.23259 0.792140i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.544078 1.19136i −0.544078 1.19136i −0.959493 0.281733i \(-0.909091\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.234072 0.512546i −0.234072 0.512546i 0.755750 0.654861i \(-0.227273\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(72\) 0 0
\(73\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(74\) −0.361922 0.232593i −0.361922 0.232593i
\(75\) 0 0
\(76\) 0 0
\(77\) −0.540641 0.158746i −0.540641 0.158746i
\(78\) 0 0
\(79\) −0.118239 + 0.822373i −0.118239 + 0.822373i 0.841254 + 0.540641i \(0.181818\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.66538 1.92195i −1.66538 1.92195i
\(87\) 0 0
\(88\) −0.158746 + 0.183203i −0.158746 + 0.183203i
\(89\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.361922 + 1.23259i 0.361922 + 1.23259i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(98\) 0.989821 + 1.14231i 0.989821 + 1.14231i
\(99\) 0 0
\(100\) −1.23259 0.361922i −1.23259 0.361922i
\(101\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(102\) 0 0
\(103\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 2.31329 1.48666i 2.31329 1.48666i
\(107\) −1.89945 0.557730i −1.89945 0.557730i −0.989821 0.142315i \(-0.954545\pi\)
−0.909632 0.415415i \(-0.863636\pi\)
\(108\) 0 0
\(109\) −0.857685 0.989821i −0.857685 0.989821i 0.142315 0.989821i \(-0.454545\pi\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.608660 + 0.178719i −0.608660 + 0.178719i
\(113\) 0.449181 + 0.983568i 0.449181 + 0.983568i 0.989821 + 0.142315i \(0.0454545\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.33708 −2.33708
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.446947 + 0.515804i 0.446947 + 0.515804i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0.698939 1.53046i 0.698939 1.53046i −0.142315 0.989821i \(-0.545455\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(128\) −0.117492 + 0.817178i −0.117492 + 0.817178i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.29639 + 1.49611i −1.29639 + 1.49611i
\(135\) 0 0
\(136\) 0 0
\(137\) −1.51150 −1.51150 −0.755750 0.654861i \(-0.772727\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.557730 + 0.643655i −0.557730 + 0.643655i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −0.0520365 + 0.361922i −0.0520365 + 0.361922i
\(149\) −0.449181 + 0.983568i −0.449181 + 0.983568i 0.540641 + 0.841254i \(0.318182\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(150\) 0 0
\(151\) 0.118239 0.822373i 0.118239 0.822373i −0.841254 0.540641i \(-0.818182\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0.121206 + 0.843008i 0.121206 + 0.843008i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(158\) 1.20493 0.353799i 1.20493 0.353799i
\(159\) 0 0
\(160\) 0 0
\(161\) 0.909632 + 0.415415i 0.909632 + 0.415415i
\(162\) 0 0
\(163\) 0.345139 + 0.755750i 0.345139 + 0.755750i 1.00000 \(0\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(168\) 0 0
\(169\) −0.959493 0.281733i −0.959493 0.281733i
\(170\) 0 0
\(171\) 0 0
\(172\) −0.897877 + 1.96608i −0.897877 + 1.96608i
\(173\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(174\) 0 0
\(175\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(176\) −0.342959 0.100702i −0.342959 0.100702i
\(177\) 0 0
\(178\) 0 0
\(179\) −0.909632 0.584585i −0.909632 0.584585i 1.00000i \(-0.5\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(180\) 0 0
\(181\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0.325137 0.281733i 0.325137 0.281733i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.281733 + 1.95949i 0.281733 + 1.95949i 0.281733 + 0.959493i \(0.409091\pi\)
1.00000i \(0.500000\pi\)
\(192\) 0 0
\(193\) 1.61435 1.03748i 1.61435 1.03748i 0.654861 0.755750i \(-0.272727\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.533654 1.16854i 0.533654 1.16854i
\(197\) 0.153882 1.07028i 0.153882 1.07028i −0.755750 0.654861i \(-0.772727\pi\)
0.909632 0.415415i \(-0.136364\pi\)
\(198\) 0 0
\(199\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(200\) 0.0612263 + 0.425839i 0.0612263 + 0.425839i
\(201\) 0 0
\(202\) 0 0
\(203\) −1.19136 + 1.37491i −1.19136 + 1.37491i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.25667 + 1.45027i −1.25667 + 1.45027i −0.415415 + 0.909632i \(0.636364\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(212\) −1.96608 1.26352i −1.96608 1.26352i
\(213\) 0 0
\(214\) 0.425839 + 2.96177i 0.425839 + 2.96177i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.822373 + 1.80075i −0.822373 + 1.80075i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(224\) 0.909632 + 1.04977i 0.909632 + 1.04977i
\(225\) 0 0
\(226\) 1.07028 1.23516i 1.07028 1.23516i
\(227\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.325137 + 0.711950i 0.325137 + 0.711950i
\(233\) 1.89945 0.557730i 1.89945 0.557730i 0.909632 0.415415i \(-0.136364\pi\)
0.989821 0.142315i \(-0.0454545\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.66538 1.07028i 1.66538 1.07028i 0.755750 0.654861i \(-0.227273\pi\)
0.909632 0.415415i \(-0.136364\pi\)
\(240\) 0 0
\(241\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(242\) 0.428546 0.938384i 0.428546 0.938384i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(252\) 0 0
\(253\) 0.304632 + 0.474017i 0.304632 + 0.474017i
\(254\) −2.54311 −2.54311
\(255\) 0 0
\(256\) −0.208518 + 0.0612263i −0.208518 + 0.0612263i
\(257\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(258\) 0 0
\(259\) 0.186393 + 0.215109i 0.186393 + 0.215109i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.215109 1.49611i 0.215109 1.49611i −0.540641 0.841254i \(-0.681818\pi\)
0.755750 0.654861i \(-0.227273\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 1.61435 + 0.474017i 1.61435 + 0.474017i
\(269\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(270\) 0 0
\(271\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0.949069 + 2.07817i 0.949069 + 2.07817i
\(275\) −0.563465 −0.563465
\(276\) 0 0
\(277\) −0.284630 −0.284630 −0.142315 0.989821i \(-0.545455\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.53046 + 0.983568i 1.53046 + 0.983568i 0.989821 + 0.142315i \(0.0454545\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(282\) 0 0
\(283\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(284\) 0.694523 + 0.203930i 0.694523 + 0.203930i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.117492 0.0344989i 0.117492 0.0344989i
\(297\) 0 0
\(298\) 1.63436 1.63436
\(299\) 0 0
\(300\) 0 0
\(301\) 0.698939 + 1.53046i 0.698939 + 1.53046i
\(302\) −1.20493 + 0.353799i −1.20493 + 0.353799i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(308\) 0.608936 0.391340i 0.608936 0.391340i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(312\) 0 0
\(313\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −0.698939 0.806618i −0.698939 0.806618i
\(317\) 1.66538 + 1.07028i 1.66538 + 1.07028i 0.909632 + 0.415415i \(0.136364\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(318\) 0 0
\(319\) −0.983568 + 0.288802i −0.983568 + 0.288802i
\(320\) 0 0
\(321\) 0 0
\(322\) 1.51150i 1.51150i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0.822373 0.949069i 0.822373 0.949069i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.41542 + 0.909632i −1.41542 + 0.909632i −0.415415 + 0.909632i \(0.636364\pi\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.84125 0.540641i −1.84125 0.540641i −0.841254 0.540641i \(-0.818182\pi\)
−1.00000 \(\pi\)
\(338\) 0.215109 + 1.49611i 0.215109 + 1.49611i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.415415 0.909632i −0.415415 0.909632i
\(344\) 0.723844 0.723844
\(345\) 0 0
\(346\) 0 0
\(347\) −0.822373 1.80075i −0.822373 1.80075i −0.540641 0.841254i \(-0.681818\pi\)
−0.281733 0.959493i \(-0.590909\pi\)
\(348\) 0 0
\(349\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(350\) 1.27155 + 0.817178i 1.27155 + 0.817178i
\(351\) 0 0
\(352\) 0.111387 + 0.774713i 0.111387 + 0.774713i
\(353\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −0.232593 + 1.61772i −0.232593 + 1.61772i
\(359\) −0.909632 + 0.584585i −0.909632 + 0.584585i −0.909632 0.415415i \(-0.863636\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −0.142315 0.989821i −0.142315 0.989821i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0.577031 + 0.263521i 0.577031 + 0.263521i
\(369\) 0 0
\(370\) 0 0
\(371\) −1.74557 + 0.512546i −1.74557 + 0.512546i
\(372\) 0 0
\(373\) −1.61435 1.03748i −1.61435 1.03748i −0.959493 0.281733i \(-0.909091\pi\)
−0.654861 0.755750i \(-0.727273\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.345139 + 0.755750i −0.345139 + 0.755750i 0.654861 + 0.755750i \(0.272727\pi\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 2.51722 1.61772i 2.51722 1.61772i
\(383\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.44009 1.56815i −2.44009 1.56815i
\(387\) 0 0
\(388\) 0 0
\(389\) 0.234072 + 0.512546i 0.234072 + 0.512546i 0.989821 0.142315i \(-0.0454545\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.430218 −0.430218
\(393\) 0 0
\(394\) −1.56815 + 0.460451i −1.56815 + 0.460451i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.533654 + 0.342959i −0.533654 + 0.342959i
\(401\) 0.281733 1.95949i 0.281733 1.95949i 1.00000i \(-0.5\pi\)
0.281733 0.959493i \(-0.409091\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 2.63843 + 0.774713i 2.63843 + 0.774713i
\(407\) 0.0228243 + 0.158746i 0.0228243 + 0.158746i
\(408\) 0 0
\(409\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(420\) 0 0
\(421\) −0.239446 1.66538i −0.239446 1.66538i −0.654861 0.755750i \(-0.727273\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(422\) 2.78305 + 0.817178i 2.78305 + 0.817178i
\(423\) 0 0
\(424\) −0.111387 + 0.774713i −0.111387 + 0.774713i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 2.13940 1.37491i 2.13940 1.37491i
\(429\) 0 0
\(430\) 0 0
\(431\) −0.708089 0.817178i −0.708089 0.817178i 0.281733 0.959493i \(-0.409091\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(432\) 0 0
\(433\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.68251 1.68251
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −0.368991 0.425839i −0.368991 0.425839i 0.540641 0.841254i \(-0.318182\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0.608660 1.33278i 0.608660 1.33278i
\(449\) 0.822373 1.80075i 0.822373 1.80075i 0.281733 0.959493i \(-0.409091\pi\)
0.540641 0.841254i \(-0.318182\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −1.33278 0.391340i −1.33278 0.391340i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.797176 0.234072i 0.797176 0.234072i 0.142315 0.989821i \(-0.454545\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 1.25667 0.368991i 1.25667 0.368991i 0.415415 0.909632i \(-0.363636\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(464\) −0.755750 + 0.872182i −0.755750 + 0.872182i
\(465\) 0 0
\(466\) −1.95949 2.26138i −1.95949 2.26138i
\(467\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(468\) 0 0
\(469\) 1.10181 0.708089i 1.10181 0.708089i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.134919 + 0.938384i −0.134919 + 0.938384i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −2.51722 1.61772i −2.51722 1.61772i
\(479\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −0.876769 −0.876769
\(485\) 0 0
\(486\) 0 0
\(487\) 1.10181 1.27155i 1.10181 1.27155i 0.142315 0.989821i \(-0.454545\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.45027 0.425839i −1.45027 0.425839i −0.540641 0.841254i \(-0.681818\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.474017 0.304632i 0.474017 0.304632i
\(498\) 0 0
\(499\) −0.186393 1.29639i −0.186393 1.29639i −0.841254 0.540641i \(-0.818182\pi\)
0.654861 0.755750i \(-0.272727\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0.460451 0.716476i 0.460451 0.716476i
\(507\) 0 0
\(508\) 0.897877 + 1.96608i 0.897877 + 1.96608i
\(509\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.755750 + 0.872182i 0.755750 + 0.872182i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0.178719 0.391340i 0.178719 0.391340i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(522\) 0 0
\(523\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −2.19209 + 0.643655i −2.19209 + 0.643655i
\(527\) 0 0
\(528\) 0 0
\(529\) −0.415415 0.909632i −0.415415 0.909632i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −0.0801894 0.557730i −0.0801894 0.557730i
\(537\) 0 0
\(538\) 0 0
\(539\) 0.0801894 0.557730i 0.0801894 0.557730i
\(540\) 0 0
\(541\) −0.544078 + 1.19136i −0.544078 + 1.19136i 0.415415 + 0.909632i \(0.363636\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.61435 1.03748i −1.61435 1.03748i −0.959493 0.281733i \(-0.909091\pi\)
−0.654861 0.755750i \(-0.727273\pi\)
\(548\) 1.27155 1.46745i 1.27155 1.46745i
\(549\) 0 0
\(550\) 0.353799 + 0.774713i 0.353799 + 0.774713i
\(551\) 0 0
\(552\) 0 0
\(553\) −0.830830 −0.830830
\(554\) 0.178719 + 0.391340i 0.178719 + 0.391340i
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0.391340 2.72183i 0.391340 2.72183i
\(563\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −0.0344989 0.239945i −0.0344989 0.239945i
\(569\) −1.29639 1.49611i −1.29639 1.49611i −0.755750 0.654861i \(-0.772727\pi\)
−0.540641 0.841254i \(-0.681818\pi\)
\(570\) 0 0
\(571\) 0.186393 0.215109i 0.186393 0.215109i −0.654861 0.755750i \(-0.727273\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.989821 + 0.142315i 0.989821 + 0.142315i
\(576\) 0 0
\(577\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(578\) 1.45027 0.425839i 1.45027 0.425839i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.983568 0.288802i −0.983568 0.288802i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0.118239 + 0.136456i 0.118239 + 0.136456i
\(593\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.577031 1.26352i −0.577031 1.26352i
\(597\) 0 0
\(598\) 0 0
\(599\) 1.51150 1.51150 0.755750 0.654861i \(-0.227273\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(600\) 0 0
\(601\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(602\) 1.66538 1.92195i 1.66538 1.92195i
\(603\) 0 0
\(604\) 0.698939 + 0.806618i 0.698939 + 0.806618i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.25667 + 0.368991i 1.25667 + 0.368991i 0.841254 0.540641i \(-0.181818\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −0.203930 0.131058i −0.203930 0.131058i
\(617\) −0.989821 + 1.14231i −0.989821 + 1.14231i 1.00000i \(0.5\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(618\) 0 0
\(619\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −0.239446 + 1.66538i −0.239446 + 1.66538i 0.415415 + 0.909632i \(0.363636\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(632\) −0.148485 + 0.325137i −0.148485 + 0.325137i
\(633\) 0 0
\(634\) 0.425839 2.96177i 0.425839 2.96177i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 1.01466 + 1.17098i 1.01466 + 1.17098i
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −1.16854 + 0.533654i −1.16854 + 0.533654i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −1.02408 0.300696i −1.02408 0.300696i
\(653\) 1.27155 0.817178i 1.27155 0.817178i 0.281733 0.959493i \(-0.409091\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.74557 + 0.512546i 1.74557 + 0.512546i 0.989821 0.142315i \(-0.0454545\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(660\) 0 0
\(661\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(662\) 2.13940 + 1.37491i 2.13940 + 1.37491i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.80075 0.258908i 1.80075 0.258908i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.857685 0.989821i −0.857685 0.989821i 0.142315 0.989821i \(-0.454545\pi\)
−1.00000 \(\pi\)
\(674\) 0.412791 + 2.87102i 0.412791 + 2.87102i
\(675\) 0 0
\(676\) 1.08070 0.694523i 1.08070 0.694523i
\(677\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −0.281733 1.95949i −0.281733 1.95949i −0.281733 0.959493i \(-0.590909\pi\)
1.00000i \(-0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −0.989821 + 1.14231i −0.989821 + 1.14231i
\(687\) 0 0
\(688\) 0.443376 + 0.970858i 0.443376 + 0.970858i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −1.95949 + 2.26138i −1.95949 + 2.26138i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.182822 1.27155i 0.182822 1.27155i
\(701\) 0.627899 1.37491i 0.627899 1.37491i −0.281733 0.959493i \(-0.590909\pi\)
0.909632 0.415415i \(-0.136364\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.694523 0.446343i 0.694523 0.446343i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.25667 + 1.45027i −1.25667 + 1.45027i −0.415415 + 0.909632i \(0.636364\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.33278 0.391340i 1.33278 0.391340i
\(717\) 0 0
\(718\) 1.37491 + 0.883600i 1.37491 + 0.883600i
\(719\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.27155 + 0.817178i −1.27155 + 0.817178i
\(723\) 0 0
\(724\) 0 0
\(725\) −0.755750 + 1.65486i −0.755750 + 1.65486i
\(726\) 0 0
\(727\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 1.38905i 1.38905i
\(737\) 0.737982 0.737982
\(738\) 0 0
\(739\) 1.25667 0.368991i 1.25667 0.368991i 0.415415 0.909632i \(-0.363636\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.80075 + 2.07817i 1.80075 + 2.07817i
\(743\) −0.215109 1.49611i −0.215109 1.49611i −0.755750 0.654861i \(-0.772727\pi\)
0.540641 0.841254i \(-0.318182\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −0.412791 + 2.87102i −0.412791 + 2.87102i
\(747\) 0 0
\(748\) 0 0
\(749\) 0.281733 1.95949i 0.281733 1.95949i
\(750\) 0 0
\(751\) 0.797176 + 0.234072i 0.797176 + 0.234072i 0.654861 0.755750i \(-0.272727\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.797176 + 1.74557i 0.797176 + 1.74557i 0.654861 + 0.755750i \(0.272727\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(758\) 1.25580 1.25580
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(762\) 0 0
\(763\) 0.857685 0.989821i 0.857685 0.989821i
\(764\) −2.13940 1.37491i −2.13940 1.37491i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.350833 + 2.44009i −0.350833 + 2.44009i
\(773\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0.557730 0.643655i 0.557730 0.643655i
\(779\) 0 0
\(780\) 0 0
\(781\) 0.317493 0.317493
\(782\) 0 0
\(783\) 0 0
\(784\) −0.263521 0.577031i −0.263521 0.577031i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(788\) 0.909632 + 1.04977i 0.909632 + 1.04977i
\(789\) 0 0
\(790\) 0 0
\(791\) −0.909632 + 0.584585i −0.909632 + 0.584585i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.16854 + 0.750975i 1.16854 + 0.750975i
\(801\) 0 0
\(802\) −2.87102 + 0.843008i −2.87102 + 0.843008i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.19136 + 1.37491i −1.19136 + 1.37491i −0.281733 + 0.959493i \(0.590909\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(810\) 0 0
\(811\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(812\) −0.332601 2.31329i −0.332601 2.31329i
\(813\) 0 0
\(814\) 0.203930 0.131058i 0.203930 0.131058i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0.0801894 + 0.557730i 0.0801894 + 0.557730i 0.989821 + 0.142315i \(0.0454545\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(822\) 0 0
\(823\) 0.698939 + 0.449181i 0.698939 + 0.449181i 0.841254 0.540641i \(-0.181818\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.97964 −1.97964 −0.989821 0.142315i \(-0.954545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(840\) 0 0
\(841\) −0.328708 + 2.28621i −0.328708 + 2.28621i
\(842\) −2.13940 + 1.37491i −2.13940 + 1.37491i
\(843\) 0 0
\(844\) −0.350833 2.44009i −0.350833 2.44009i
\(845\) 0 0
\(846\) 0 0
\(847\) −0.446947 + 0.515804i −0.446947 + 0.515804i
\(848\) −1.10731 + 0.325137i −1.10731 + 0.325137i
\(849\) 0 0
\(850\) 0 0
\(851\) 0.284630i 0.284630i
\(852\) 0 0
\(853\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −0.716476 0.460451i −0.716476 0.460451i
\(857\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(858\) 0 0
\(859\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −0.678936 + 1.48666i −0.678936 + 1.48666i
\(863\) 0.234072 0.512546i 0.234072 0.512546i −0.755750 0.654861i \(-0.772727\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −0.393828 0.253098i −0.393828 0.253098i
\(870\) 0 0
\(871\) 0 0
\(872\) −0.234072 0.512546i −0.234072 0.512546i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.84125 0.540641i 1.84125 0.540641i 0.841254 0.540641i \(-0.181818\pi\)
1.00000 \(0\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(882\) 0 0
\(883\) −0.698939 + 0.449181i −0.698939 + 0.449181i −0.841254 0.540641i \(-0.818182\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −0.353799 + 0.774713i −0.353799 + 0.774713i
\(887\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(888\) 0 0
\(889\) 1.61435 + 0.474017i 1.61435 + 0.474017i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −0.825581 −0.825581
\(897\) 0 0
\(898\) −2.99223 −2.99223
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0.0662029 + 0.460451i 0.0662029 + 0.460451i
\(905\) 0 0
\(906\) 0 0
\(907\) −0.186393 + 1.29639i −0.186393 + 1.29639i 0.654861 + 0.755750i \(0.272727\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.27155 0.817178i 1.27155 0.817178i 0.281733 0.959493i \(-0.409091\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.822373 0.949069i −0.822373 0.949069i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0.284630 0.284630 0.142315 0.989821i \(-0.454545\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0.239446 + 0.153882i 0.239446 + 0.153882i
\(926\) −1.29639 1.49611i −1.29639 1.49611i
\(927\) 0 0
\(928\) 2.42468 + 0.711950i 2.42468 + 0.711950i
\(929\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.05645 + 2.31329i −1.05645 + 2.31329i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(938\) −1.66538 1.07028i −1.66538 1.07028i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 1.37491 0.403709i 1.37491 0.403709i
\(947\) −1.29639 + 1.49611i −1.29639 + 1.49611i −0.540641 + 0.841254i \(0.681818\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −0.281733 + 1.95949i −0.281733 + 1.95949i 1.00000i \(0.5\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −0.361922 + 2.51722i −0.361922 + 2.51722i
\(957\) 0 0
\(958\) 0 0
\(959\) −0.215109 1.49611i −0.215109 1.49611i
\(960\) 0 0
\(961\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.91899 −1.91899 −0.959493 0.281733i \(-0.909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(968\) 0.121977 + 0.267092i 0.121977 + 0.267092i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −2.44009 0.716476i −2.44009 0.716476i
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0.325137 + 2.26138i 0.325137 + 2.26138i
\(983\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0.474017 1.61435i 0.474017 1.61435i
\(990\) 0 0
\(991\) 0.544078 + 1.19136i 0.544078 + 1.19136i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −0.716476 0.460451i −0.716476 0.460451i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(998\) −1.66538 + 1.07028i −1.66538 + 1.07028i
\(999\) 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 1449.1.bq.b.1315.1 yes 20
3.2 odd 2 inner 1449.1.bq.b.1315.2 yes 20
7.6 odd 2 CM 1449.1.bq.b.1315.1 yes 20
21.20 even 2 inner 1449.1.bq.b.1315.2 yes 20
23.6 even 11 inner 1449.1.bq.b.811.1 20
69.29 odd 22 inner 1449.1.bq.b.811.2 yes 20
161.6 odd 22 inner 1449.1.bq.b.811.1 20
483.167 even 22 inner 1449.1.bq.b.811.2 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1449.1.bq.b.811.1 20 23.6 even 11 inner
1449.1.bq.b.811.1 20 161.6 odd 22 inner
1449.1.bq.b.811.2 yes 20 69.29 odd 22 inner
1449.1.bq.b.811.2 yes 20 483.167 even 22 inner
1449.1.bq.b.1315.1 yes 20 1.1 even 1 trivial
1449.1.bq.b.1315.1 yes 20 7.6 odd 2 CM
1449.1.bq.b.1315.2 yes 20 3.2 odd 2 inner
1449.1.bq.b.1315.2 yes 20 21.20 even 2 inner